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EXCITON BEHAVIOR IN CARBON NANOTUBES: DIELECTRIC SCREENING AND DECAY DYNAMICS ANDREW GERALD WALSH Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy &

%

BOSTON UNIVERSITY

BOSTON UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES

Dissertation

EXCITON BEHAVIOR IN CARBON NANOTUBES: DIELECTRIC SCREENING AND DECAY DYNAMICS

by

ANDREW GERALD WALSH B. S., Cornell University, 1992

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2009

Approved by

First Reader Anna K. Swan, Ph.D. Associate Professor of Electrical and Computer Engineering

Second Reader Bennett B. Goldberg, Ph.D. Professor of Physics

ACKNOWLEDGEMENTS

My time at Boston University has been both challenging and rewarding as my wife and I struggled to balance our careers with the raising of three small, beautiful girls (they take after their mother). Therefore, I must first and foremost thank my wife, Kate, the love of my life, for her constant support as I pursued my degree. Thank you, Kate. And thank you to you, Meghan, Anna, and Clara, for bringing joy to my life on a daily basis. I am blessed beyond what words can describe. I must also thank my de facto advisor, Prof. Anna Swan, for her unwaivering support as well. Anna, I could not have had a more understanding and flexible mentor.

I have also been incredibly fortunate to have been surrounded over the last five and half years by truly talented professors and students who shared not only my love of science but also of life (e.g. beer, the Red Sox, an occasional off-color YouTube clip...). Bennett, Selim, and Anna, you should be proud of the positive environment that you have engendered within your lab. You don’t have to talk to too many other graduate students to know that not all labs are this way.

On a professional level, thank you to Anna, Bennett, and Selim for your mentorship and support. To say that I have learned a lot (after cramming my way through Cornell and then taking a ten year hiatus from science) would be an understatement of enormous proportions. Not having been one to refrain from asking questions, I most certainly have single-handedly disproved the whole “there is no such thing as a stupid question” paradigm many times over - thank you for your patience. iii

And thank you to Nick and Yan for your help over the years also. I am well aware that I aimed more than a few stupid questions your direction as well - thanks for refraining from outright ridicule. Also, thank you for helping to make the lab an enjoyable place to work.

To Mark, Craig, and Michele, thank you for all your support on the nanotube project. I hope my efforts to instill some of my nanotube wisdom didn’t screw you up too much.

Thank you to Jude Schneck and Prof. Ziegler for all your help. It has been a struggle for me to reconcile your chemist’s view with my solid state my way of looking at things but, as is often the case, that struggle has paid huge dividends in my understanding of spectroscopy of carbon nanotubes.

To Marc, Hakan, Mehmet, Ayca, Constanze, Sebastian, Emre, and Alex, thanks for all your support as well and for making the lab fun.

Thank you to Mirtha for all your administrative help and for sharing stories of parental struggles.

I should also thank Steve C., Wolfgang, Ernie, Maurice, and Jared for your support during your time in the lab. We have been truly fortunate to have had you.

Finally, thank you to Anlee Krupp for your support and for introducing me to MTPV Corp!

iv

EXCITON BEHAVIOR IN CARBON NANOTUBES: DIELECTRIC SCREENING AND DECAY DYNAMICS (Order No.

)

ANDREW GERALD WALSH Boston University Graduate School of Arts and Sciences, 2009 Major Professor: Anna K. Swan, Associate Professor of Electrical and Computer Engineering

ABSTRACT Despite more than a decade of intense research, electronic relaxation in carbon nanotubes (CNTs) is still not well understood. For instance, the optical properties of the electronic energy levels change with environment, and the intrinsic lifetimes and oscillator strengths of the excited states remain unknown. The dominant non-radiative relaxation mechanisms are still a matter of debate. These fundamental electronic parameters place theoretical limits on quantities such as optical quantum efficiency and thus are of great importance to a wide range of potential carbon nanotube applications. In this thesis, we present our efforts to understand the impact of environment on CNT exciton energy levels and relaxation dynamics using a variety of continuous wave and timeresolved spectroscopies. First, we use resonant Raman spectroscopy to probe changes in the optical transition energies of individual nanotubes with external dielectric screening. The small optical shifts are shown to derive from a competition between large changes in the underlying exciton binding and band gap renormalization energies. The large shift in the renormalized free particle band gap has major consequences for nanotube transport. We develop a scaling relationship between the exciton binding energy and the actual external dielectric value. The measured exciton binding and band gap renormalization energies

v

approach one electron volt but decrease by half upon screening by water. In order to probe population relaxation dynamics, we use transient absorption (TA) spectroscopy on a highly homogeneous suspension of (6,5) CNTs. The TA signal exhibits stretched exponential behavior consistent with one dimensional random walk dynamics which suggests that the ultimate optical quantum yield may be limited by CNT length. The power dependence of the TA signal is used to measure both the dipole length, ∼0.1 nm, and the homogeneous dephasing line width. Comparison with the photoluminescence line width indicates a significant heterogeneous contribution. We also discuss how forward scattered Rayleigh and photon echo spectroscopies are used to remove incoherent and heterogeneous contributions from the electronic line width. Finally, we discuss density of states-modified black body emission from carbon nanotube devices.

vi

Contents 1 Introduction

1

2 Carbon Nanotube Basics

6

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.3

Tight Binding Electronic Structure . . . . . . . . . . . . . . . . . . . . . . .

7

2.4

Many-Body Corrections to CNT Electronic Structure . . . . . . . . . . . . .

12

2.5

Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3 Dielectric Screening of Particle Interaction Energies

18

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3.2

Screening Theory and Scaling Relationships . . . . . . . . . . . . . . . . . .

19

3.3

Resonant Raman Spectroscopy Theory . . . . . . . . . . . . . . . . . . . . .

26

3.4

Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.5

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.6

Conclusion

38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 (6,5) Carbon Nanotube Exciton Dynamics

39

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.2

Transient Absorption Theory . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.3

Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.4

Three Level Stretched Exponential Model . . . . . . . . . . . . . . . . . . .

45

4.5

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.6

Conclusion

55

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

5 Photon Echo Spectroscopy

56

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

5.2

Photon Echo Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

5.3

Conclusion

63

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Forward Scattered Rayleigh Experiment

64

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

6.2

Forward Scattered Rayleigh Theory

. . . . . . . . . . . . . . . . . . . . . .

65

6.3

The Forward Scattered Rayleigh Microscope . . . . . . . . . . . . . . . . . .

71

6.4

Conclusion

77

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Spectroscopy of Individual Nano-scale Emitters

78

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

7.2

Psuedo-Spectral Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

7.3

Measuring the Focal Spot Profile . . . . . . . . . . . . . . . . . . . . . . . .

85

7.4

Slit Width Effects

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

7.5

Extended Emitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

7.6

Conclusion

94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A Denisty Matrix Derivation of the Raman Excitation Profile

95

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

A.2 The Perturbative Density Matrix Formalism . . . . . . . . . . . . . . . . . .

95

A.3 Density Matrix Derivation of the Raman Excitation Profile . . . . . . . . .

101

A.4 Comparison of REP Models . . . . . . . . . . . . . . . . . . . . . . . . . . .

106

A.5 Conclusion

110

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B Modified Black Body Emission

111

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

B.2 Black Body Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

B.3 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

B.4 Data Fitting and Results

114

. . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

B.5 Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

References

119

Curriculum Vitae

126

ix

List of Tables 3.1

Tabulated results of red shifts for 2 carbon nanotubes. . . . . . . . . . . . .

36

3.2

Particle interaction energies. . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

7.1

Renishaw 1000B μRaman spectrometer parameters. . . . . . . . . . . . . .

85

x

List of Figures 2·1

CNT tight binding electronic structure. . . . . . . . . . . . . . . . . . . . .

8

2·2

CNT many-body electronic structure. . . . . . . . . . . . . . . . . . . . . .

14

2·3

Raman spectra from a single, suspended carbon nanotube. . . . . . . . . . .

16

3·1

Cartoon depicting the electron-electron and electron-hole Coulomb interactions. 19

3·2

Energy diagrams of the effect of the band gap renormalization and exciton binding energies on the optical transition energy. . . . . . . . . . . . . . . .

21

3·3

Coulomb potential and the cutoff parameter. . . . . . . . . . . . . . . . . .

22

3·4

Scaling of particle interaction energies with external screening. . . . . . . .

25

3·5

Cartoon of the dependence of the Raman excitation profile on external screening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3·6

Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3·7

REP Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4·1

Transient absorption experiment. . . . . . . . . . . . . . . . . . . . . . . . .

40

4·2

Sample A and B absorption data. . . . . . . . . . . . . . . . . . . . . . . . .

42

4·3

Low and high fluence TA data with stretched exponential fits. . . . . . . . .

43

4·4

Power dependence of the transient absorption signal. . . . . . . . . . . . . .

52

5·1

Photon echo phase unwinding. . . . . . . . . . . . . . . . . . . . . . . . . .

59

5·2

Photon echo spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

6·1

Forward scattered Rayleigh. . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

6·2

Forward scattered Rayleigh experimental setup. . . . . . . . . . . . . . . . .

71

6·3

Kataura plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

xi

7·1

Nano-scale spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

7·2

Effects of off-optical axis emission from a nano-emitter.

. . . . . . . . . . .

83

7·3

Slit width effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

7·4

Extended emitter effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

7·5

FWHM as a function of slit width. . . . . . . . . . . . . . . . . . . . . . . .

93

A·1 Raman WMEL diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100

A·2 REPs predicted by different models. . . . . . . . . . . . . . . . . . . . . . .

108

B·1 Standard black body theory. . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

B·2 Kataura plot of the optical transition energies. . . . . . . . . . . . . . . . .

115

B·3 Three different functional fits to the same electroluminescence data. . . . .

117

B·4 Fits to the electroluminescence data at different applied voltages using six Lorentzians and a black body envelope. . . . . . . . . . . . . . . . . . . . .

xii

118

List of Abbreviations

1D

..............

1 Dimension(al)

2D

..............

2 Dimension(al)

3D

..............

3 Dimension(al)

BGR

..............

Band Gap Renormalization

CCD

..............

Charge Coupled Device

CNT

..............

Carbon Nanotube

CW

..............

Continuous Wave

DOS

..............

Density of States

dt

..............

Nanotube Diameter

DT

..............

Differential Transmission

FID

..............

Free Induction Decay

FWHM

..............

Full Width at Half Maximum

MRI

..............

Magnetic Resonance Imaging

NA

..............

Numerical Aperture

OBE

..............

Optical Bloch Equations

OPA

..............

Optical Parametric Amplifier

PE

..............

Photon Echo

PL

..............

Photoluminescence

PSF

..............

Point Spread Function

QD

..............

Quantum Dot

QE

..............

Quantum Efficiency

xiii

RBM

..............

Radial Breathing Mode

REP

..............

Raman Excitation Profile

RIE

..............

Reactive Ion Etch

RWA

..............

Rotating Wave Approximation

RRS

..............

Resonant Raman Spectroscopy

SEM

..............

Scanning Electron Microscope

SWNT

..............

Single Wall Carbon Nanotube

TA

..............

Transient Absorption

TB

..............

Tight Binding

vHs

..............

van Hove singularity

WMEL

..............

Wave Mixing Energy Level

xiv

“A witty saying proves nothing.” – Voltaire, 1767

Fly Navy!

xv

1

Chapter 1

Introduction Carbon nanotubes, or CNTs, have been the focus of intense research since their discovery in the early 1990’s. Specifically, the carbon-carbon sp2 hybridized bond imparts to the nanotube several very desirable characteristics including extremely high tensile strength, and superior heat and electrical conduction, while their one-dimensional structure leads to striking optical properties which are the primary focus of this work. These properties of CNTs promise to lead to new advances in everything from electronic devices, optoelectronic sensors, flexible conducting membranes, and strong lightweight materials to perhaps even a so-called “space elevator.” Therefore, understanding the basic electronic and vibrational properties of CNTs is critical. However, despite intense research on the part of many groups around the world, the basic electronic decay mechanisms are not yet well understood. The intrinsic radiative lifetime has yet to be measured, the oscillator strengths of the fundamental optical transitions are not well know, and a unified picture of exciton dynamics has yet to be demonstrated that adequately explains all the observed relaxation data in a coherent way. Much of the difficulty arises from the inherent sensitivity of carbon nanotubes to their environment. In this thesis, we preset our efforts to understand how different environmental factors affect nanotube electronic structure and relaxation dynamics by means of an array of spectroscopies, each probing different contributions to the line width and, together, providing a more complete understanding. In the Optical Characterization and Nanophotonics laboratory at Boston University, we use lasers, both single line and broadband, continuous wave and pulsed, to probe the basic physics of CNTs. Specifically, we use Raman, Rayleigh, photoluminescence, transient absorption, and photon echo spectroscopies to probe the effect of the environment on the

2 electronic structure and optical properties of CNTs. The one dimensional nature of CNTs and the attendant reduction in phase space, in conjunction with a lack of screening found in bulk materials, favor the formation of strongly bound excitons, electron-hole pairs bound by Coulomb forces, with binding energies that are large fractions of an electron volt. In contrast, exciton binding energies in three dimensional materials are typically between one and two orders of magnitude smaller. In fact, the binding energy in CNTs can be nearly the size of the single particle band gap, i.e. the band gap calculated by ignoring particle interactions. And in addition to the binding energy due to the Coulomb attraction between electron and hole, there is also Coulomb repulsion between electrons. This repulsive interaction leads to a large increase of the single particle band gap. This band gap renormalization, or BGR, energy is usually slightly larger than the exciton binding energy and, since the two terms enter into the Hamiltonian with opposite signs, they largely cancel each other. Measuring these energies is important for optical applications and is absolutely critical to CNT device performance since the free particle band gap, which is the relevant energy level for conduction processes, lies energetically at the renormalized band gap, i.e. at the onset of exciton ionization, and not at the optical transition energy. Furthermore, these interaction energies are extremely sensitive to environmental screening effects. Unlike three dimensional materials where the majority of the constituent atoms reside inside the bulk of the solid, in CNTs, every atom resides at the surface of the nanotube. Thus, changing the CNT environment from air to, say, water, which is a highly polar molecule, can be expected to greatly increase screening. That is, the presence of an external dielectric can be expected to significantly reduce the relevant interaction energies, changing the CNT spectrum. In this dissertation, after an introduction to CNT “basics” in Chapter 2, we discuss several experiments designed to probe environmental effects on the electronic structure, and the exciton relaxation dynamics in CNTs. First, in Chapter 3, we use Raman spectroscopy to determine the underlying electronic transition energies as a function of external dielectric. In order for the reader to understand the experiment, we first define Raman excitation profiles, or REPs, in which the height of a given Raman peak is plotted against the laser

3 excitation energy as that laser is tuned through a resonance of the CNT. We compare the REP line shape derived using third-order perturbation theory within a solid state framework with the line shape derived by third order polarization density matrix expansion. The line shape from each approach is then compared to the data. We use Raman spectroscopy to measure the transition energy of a single CNT which has been grown across a trench in order to remove substrate screening. We measure this transition energy, for the same CNT, in air, in high humidity, and in water. This is done by tuning the incident laser through the resonance of the CNT, which, in one dimension, is a sharp van Hove singularity. The Raman peak intensity (for any given Raman active mode) will be maximized when the incident laser is resonant with the electronic transition. In order to extract the underlying particle interaction energies from the data, we derive a scaling relationship between both BGR and exciton binding energies and the external dielectric value. The data shows that these energies decrease by hundreds of meV upon screening by water when compared to air, a decrease of about fifty percent. Raman spectroscopy has several advantages over photoluminescence spectroscopy. First, Raman spectroscopy can probe any electronic transition, not just the transition associated with the band gap of the material and, in one dimensional materials, there are a series of sharp, higher transition energies, very much like a molecular system. Second, Raman can probe metallic CNTs which do not photoluminesce. Third, even semiconducting CNTs will not appreciably photoluminesce when immersed in water whereas appreciable Raman signals persist. In a second set of experiments, we use femtosecond transient absorption (TA) and photon echo spectroscopies to directly probe the electronic relaxation dynamics of a highly mono-dispersed solution of (6,5) CNTs. In Chapter 4, we show how the TA data is well fit at both high and low fluence and in both the short and long time regimes by a three level stretched exponential model, whereas previously published work has resorted to segregating the data and fitting it with separate functional forms. We show that the data is consistent with exciton diffusion without the need to invoke exciton-exciton annihilation. We then

4 show how, in the limit of ultrafast pulses shorter than the thermalization timescale, carbon nanotubes behave as two level systems. The power dependence of the zero time delay TA signal is fit using the optical Bloch equations in the short time scale limit with negligible population loss. We measure a dipole length of ∼0.1 nm, slightly less than predicted, and show that the data implies a large heterogeneous contribution to the line width. The absorption cross section is shown to dominate the total cross section on resonance. In Chapter 5, we explain how photon echo spectroscopy can be used to differentiate between the homogeneous and inhomogeneous contributions to the total electronic line width. In Chapter 6, we describe our efforts to measure the dipole moment and intrinsic line width of a single CNT suspended over a fully etched gap by tuning a diode laser through the electronic resonance. As the laser approaches resonance, a detector positioned directly beneath the CNT will register a dip in laser light transmission due to absorption and/or scattering by the CNT. Because the absolute signal is small, lock-in techniques are used. Specifically, the polarization of the laser is modulated. Because of a strong antenna effect, the scattering by the CNT is thus modulated at the same frequency. By finely tuning the laser through resonance, we can directly measure the line width of the electronic level where the resolution of the measurement is now limited by the laser wavelength step size and not by, for example, spectrometer resolution. The product of the modulation depth and width is shown to be a direct measure of the oscillator strength of the resonant transition. Finally, in Chapter 7, we discuss the additional difficulties attendant to the spectroscopy of nano-scale objects not usually considered when dealing with bulk or even micron-scale objects. Specifically, we discuss how failure to precisely position the object on optical axis results in a pseudo spectral shift and how the width of the slit at the entrance to the spectrometer influences the measured spectral width of any feature being measured. Also, we describe a novel way to use nano-scale objects to directly measure the diffraction limited spot of a laser in the object plane. We model the results using a full vector treatment of a linearly polarized Gaussian laser beam incident on the back aperture of a high numerical

5 aperture objective. In Appendix A, we derive the lineshape of the Raman excitation profile (REP) using perturbative density matrix formalism where a four level system is used to represent the electronic and vibrational ground and excited states of a single carbon nanotube. In Appendix B, we fit the electroluminescence from carbon nanotube-based electronic devices using a density of states-modified black body emission model.

6

Chapter 2

Carbon Nanotube Basics 2.1

Introduction

In this chapter, we will review the basics of carbon nanotube electronic and vibrational structure, covering the salient concepts that pertain to this thesis. We will not attempt to rederive electronic or vibrational structure here. For a more in depth discussion of carbon nanotube basics, the reader is directed to the excellent book by Reich, et al. (Reich et al., 2004) Section 2.2 briefly reviews the fundamental properties that make CNTs so promising for a wide range of potential applications. Section 2.3 covers the basic carbon nanotube electronic structure as derived by a simple tight binding model. In section 2.4, we will discuss the many-body corrections to the basic electronic structure discussed in section 2.3. Section 2.5 will cover the relevant phonon modes that appear in Raman spectroscopy and play major rolls in electronic relaxation.

2.2

Overview

Since their discovery in 1991, carbon nanotubes have been the focus of intense research. The carbon bond that forms the basis of graphite, graphene (i.e. single layer graphite), and carbon nanotubes imparts to CNTs a host of desirable physical properties. These include tensile strength much larger than steel, and heat and electrical conductiviity far superior to silicon or copper. In addition, the electronic band gap is direct in semiconducting carbon nanotubes unlike in silicon. Paired with a large density of states at the band edges due to their one dimensional nature, CNTs have enormous promise for use in various optical applications, including optical sensors and electro-optical devices. And this all comes at a

7 truly nanometer scale with single wall carbon nanotube diameters typically ranging from less than 1 nm up to 5 nm. However, despite the intense study over the last decade and a half, many of the fundamental parameters of interest are still not well known, such as intrinsic electronic lifetimes and oscillator strenghs. Much of this ambiguity is direcly related to the one-dimensional nature of carbon nanotubes. In addition to a phase space restriction that leads to strong many-body effects, in CNTs every carbon atom lies at the surface of the nanotube and thus is exceedingly sensitive to environment. Environmental perturbations can screen many-body Coulomb interactions, dope the nanotube, lead to local strains, etc. The research covered in this thesis presents our efforts to, by use of various spectroscopies that probe different dynamics, parse the electronic linewidth into its different contributions and, in addition, study how the relevant electronic transition rates, energies, and strengths change with environment.

2.3

Tight Binding Electronic Structure

Carbon has a number of allotropes, depending on formation temperatures, pressures, etc. In diamond, the outer 2s, 2px , 2py and 2pz orbitals hybridize in a maximally symmetric way forming sp3 orbitals with a tetrahedral arrangement. At standard pressures and temperatures, graphite is energetically favored, where the 2s and two of the 2p orbitals hybridize into sp2 orbitals which form a planar, hexagonal lattice. The double bonds thus formed between neighboring carbon atoms are known as σ bonds and, being strongly bonded, are ignored in simple tight binding calculations of the electronic structure of graphite or graphene (single layer graphite). The 2pz orbital (where now z is defined perpendicular to the hexagonal lattice plane) extends out of plane and weakly bonds to neighboring hexagonal layer. Since the interlayer interactions in graphite are small and since carbon nanotubes can be thought of, as a first order approximation, as rolled up graphene, the electronic structure of graphene forms the starting point for carbon nanotube electronic structure and so is discussed here. In simple tight binding, or TB, calculations, the 2pz orbital serves as the basis set from which the electronic wavefunction of the lattice will be formed. That is, the electron is

8

(a) TBGraphene ElectronicBand Structure

(b)

+

QuantizedK

ŏ

Metallic

=

(c)

(d)

(e)

Bands

Projection

Density ofstates

E

Conduction Band DOS

E Valence Band DOS

Semiconducting Figure 2·1: CNT tight binding electronic structure. (a) Graphene electronic structure calculated using simple tight binding. (b) Cutting lines, shown in green, due to quantization in the circumferential direction. Top and bottom rows show metallic and semiconducting CNTs, respectively. (c) 3D depiction of the Dirac cone after quantization. (d) Projection of the Dirac cone slices onto a plane, translating linear to orbital momentum. (e) 1D density of states with van Hove singularities.

9 assumed to be tightly bound to its ion core and thus the resulting wavefunction of the electron on the lattice is presumed to look something like the 2pz orbital. The electron is then assumed to tunnel, or hop, from lattice site to lattice site, from one 2pz orbital to another. This is in contrast to the free electron gas approximation where the electron is assumed to be largely free of its ion core and thus plane waves are used as the basis set. The tight binding model ignores electron-electron and electron-hole interactions and yet, perhaps somewhat surprisingly, often does a reasonable job modeling electronic structure in 3D semiconductors. However, as we shall see in Section 2.4, many-body corrections in 1D are large, of order the size of the fundamental bang gap, and can not be ignored. Still, the tight binding model serves as the starting point when writing down the Hamiltonian onto which are added particle interaction terms. The simple tight-binding electronic structure of graphite was solved in 1947 (Wallace, 1947). The graphene electronic structure is shown graphically in Fig. 2·1(a) with the high symmetry points labelled, i.e. the Γ, M, and K points. At optical energies, the relevant region is at the K-point which, to first approximation, may be modelled as a cone in kspace, known as a Dirac cone, so called because the linear dispersion results in zero electron effective mass, a case to which the Dirac equation applies. That is, in graphene, the valence and conduction bands come together at a point and thus graphene is metallic, albeit with zero density of states at the Dirac point. However, when the graphene sheet is “rolled up,” the electronic wavefunction becomes quantized in the circumferential direction in addition to the delta function localization in the direction orthogonal to the sheet present already in graphene. The nanotube electronic wavefunction is free in only one dimension - along the nanotube axis. The nanotube is not literally formed by rolling up a sheet of graphene but rather grown typically at high temperatures in a CVD furnace with ferrite catalyst particles. We use different sample types depending on the spectroscopy and thus sample growth and preparation will be discussed in later sections. The quantization of the wavefunction in the circumferential direction results in lines of allowed k values in the graphene k space, shown in green in Fig. 2·1(b). k is not restricted in the direction parallel to the nanotube axis,

10 i.e. along the green lines, but can only take certain values in the circumferential direction. These “cutting lines” in k space define the graphene valence and conduction bands, shown in Fig. 2·1(c). One can imagine rolling up the graphene sheet into cylinders in numerous ways, each of which results in a nanotube of some diameter and chirality, refered to as the nanotube’s “species.” The diameter of each species can be plotted against its electronic transition energies resulting in a so-called “Kataura Plot” which we show in Fig. B·4 in Appendix B. The diameter and chirality of the nanotube will determine the spacing between and angle of the cutting lines with respect to the nanotube axis, respectively. If one of the cutting lines happens to include the K point, i.e. the Dirac point, then its valence and conduction bands will touch as they do in graphene and, ignoring higher order corrections, the nanotube will be metallic. This will happen 1/3 of the time and is the case depicted in the top row of graphs in Fig. 2·1. The other 2/3 of the time, the cutting line misses the K point and a gap is opened, resulting in a semiconducting nanotube, as is the case depicted in the bottom row of graphs in Fig. 2·1. The smaller the diameter of the nanotube, the further apart the cutting lines in k space will be and thus the further away from the K point the closest cutting line will be, resulting in a larger band gap. Note that the band gap depends also on the chiral angle of the nanotube since it will determine the angle of the cutting line in k space and thus affect how close the cutting line comes to the K point. The dispersion relation associated with each cutting line is projected onto a common E versus k plot, as shown in Fig. 2·1(d). An interesting subtlety here is that upon roll-up one component of linear momenta (i.e. the vector from the Γ point to a given allowed point in the graphene k space) changes to representing an angular momentum around the nanotube circumference. So upon projection, each cutting line and its associated valence and conduction band have an angular momentum quantum number associated with it. Conservation of linear momentum in the graphene plane maps to angular momentum conservation selection rules in CNTs. Also notice that the lowest valence and conduction bands, i.e. the fundamental band gap which results from the cutting line nearest the K point (and far from the zero momentum Γ point), thus represent a high angular momentum electronic state.

11 To keep things simpler however, a naming convention has been adopted where the cutting line closest to the K point is labelled “1”. Thus the lowest conduction and highest valence band are both labelled with a “1” as well, since they both derive from that same cutting line, and thus the fundamental band gap transition energy is designated “E11 ”. Similarly, the transition between the second valence and second conduction band (formed from the second closest cutting line to the K point) is designated “E22 ”. There can also be transverse transitions between, say, first valence and second conduction band, i.e. E12 , but transitions of the form Eii are more common due to selection rules and a strong antenna effect. Fig. 2·1(e) plots the density of states (DOS) of the metallic and semiconducting nanotubes versus energy. Since CNTs are one dimensional, i.e. there is only k along the tube and thus no integration over kx or ky , the DOS, for a given band, is simply related to the inverse of its slope. The DOS at the extrema of the conduction and valence bands diverge in what are referred to as van Hove singularities (vHs). Hence, there is expected to be very strong absorption or scattering of photons which match a particular CNT transition energy and not much scattering when the photon is not resonant with an interband transition since the DOS falls off very quickly. In this respect, CNTs behave like molecular systems or quantum dots. An important fact for our later discussion exciton dynamics is that the graphene lattice is formed from a two atom basis which leads to inequivalent K and K’ points in the graphene Brillioun zone. This graphene K ↔ K’ degeneracy maps to, in nanotubes, a symmetry of the electronic wavefunction rotating either clockwise or counterclockwise around the nanotube axis. Excitons are formed from combinations of electronic states (from the conduction bands) and hole states (from the valence bands) from both K and K’ points are therefore 16 fold degenerate when including spin. Two more nanotube properties should be mentioned before moving on to a discussion of many-body effects in CNTs. First, the nanotube geometry leads to a strong antenna, or depolarization, effect. That is, the nanotube is able to respond strongly to photons polarized parallel to the nanotube axis by separating charge along its length but can respond only

12 weakly to photons polarized in the transverse direction. Thus, in the most common experimental arrangement, the laser is focussed with a k vector perpendicular to the nanotube axis and polarized parallel to it. Second, angular momentum selection rules dictate to which interband transition an incident photon can couple. Specifically, single photon excitations with linear polarization parallel to the nanotube axis induce transistions between valence and conducution bands arising from the same cutting line, i.e. transitions of the form Eii . Single photon excitations with linear polarization perpendicular to the nanotube axis induce transistions between valence and conducution bands arising from the neighboring cutting lines, i.e. transitions of the form Ei,i±1 . In other words, parallel polarized photons couple valence and conduction bands directly above each other in the graphene k space whereas perpendicular polarized photons couple valence and conduction bands slightly offset from each other in the graphene k space. Finally, since optical wavelengths are much larger than the lattice spacing (as is usually the case in solid state physics), the photon momentum is very small on the scale of electronic wavevectors. Conservation of momentum dictates that optical transitions in nanotubes will appear vertical on nanotube dispersion diagrams, such as the ones in Fig. 2·1(d). In fact, in the typical geometry where the photon k vector is perpendicular to the nanotube axis, the transition is exactly vertical.

2.4

Many-Body Corrections to CNT Electronic Structure

In the previous section, electron-electron and electron-hole interactions were ignored. There is not, of course, just a single electron hopping around on an otherwise empty lattice and thus the Hamiltonian should include terms representing the Coulomb energy associated with electrons interacting with other electrons and also of excited state electrons interacting with the holes they have left behind. In bulk 3D materials, however, ignoring these terms leads, somewhat surprisingly, to accurate predictions. Whereas band gaps are often of order 1 eV, exciton (bound electron-hole pairs) binding energies are only of order a few meV. Thus, at room temperature, the bound electron-hole pair is expected to be ionized by thermalization.

13 There are two reasons for the small exciton binding and electron self-energies. First, 3D phase space favors larger separations; there are more ways to be further away than close. Second, in a bulk material, the other electrons will move around in such a way as to minimize the total Coulomb interaction energy. Given two electrons, the other electrons will move in such a way as to screen the Coulomb interaction between them, thus minimizing the interaction energy. Conversely, in 1D, the electrons are confined to the nanotube axis; phase space no longer favors larger separation and the electrons are not free to move in three dimensions in such a way as to screen the Coulomb interaction. In other words, there is no other material bulk present outside of the nanotube to screen most of the field lines, which extend into or live in all three dimensions, between two electrons or an electron-hole pair. Thus, the resulting interaction energies are large and the fundamental electronic excitations of the system become quasiparticles known as excitons, not free electrons. In fact, where the single particle band gap of order 1 eV, the electron-electron interaction and excition binding energies have been calculated to be of order 1 eV and clearly can not be ignored. There are a number of good theory papers (Ando, 1997; Kane and Mele, 2004; Vamivakas et al., 2006a; Spataru et al., 2004) to which the reader is directed for the details of the calculation. In this section, we will cover the salient results relevant to our experimental work. There are a number of complexities involved when transitioning from the single, or free, electron and hole picture to the excitonic picture. First, the free electron band gap is blue shifted by an amount equal to the electron self-energy in a process known as band gap renormalization, BGR. The electron self-energy term in the Hamiltonion is thus referred to as EBGR . The electron-hole Coulomb interaction leads to a bound electron-hole pairs, known as excitons, where the lowest excitonic state appears in the band gap below the renormalized band edge by an amount known as the exciton binding energy, or EBind , as shown in Figs. 2·2(b) and 2·2(c). In fact, there exists a hydrogenic-like series of excitonic states associated with each interband transition. This situation is depicted in Fig. 2·2(c). When solving for this excitonic spectrum, the reference frame is shifted to the center of

14

Conduction Band

E Exciton

….

Continuum

EBGR EBind

ESP KŒ

Valence Band

E11

2s, 2p

EOptical ESP

E22

Continuum 3s, 3p, 3d

d d b d

~ 7 meV

1s

GS

KCM

KCM

GS

Single Particle Picture

Excitonic Picture

Excitonic Series at Each Band

Bright and Dark Excitons

(a)

(b)

(c)

(d)

Figure 2·2: CNT many-body electronic structure. (a) Single particle electronic band gap with optical transition energy equal to ESP . (b) Excitonic band structure showing EBGR and EBind corrections to ESP and the resulting optical transition energy EOptical . (c) Hydrogenic series associated with each band. (d) Four-fold degeneracy of the spin singlet excitons broken into bright (b) and dark (d) states. mass frame of the exciton, as is also done when solving for the spectrum of the hydrogen atom. The point here is that the k axis becomes, in excitonic energy band diagrams, the center of mass momentum of the exciton, kCM , and there are also no longer valence and conduction bands but simply the excitonic states which are formed from superpositions of free electron and hole states. More specifically, the verical transition allowed in the single electron band picture becomes constrained to a vertical transistion precisely at kCM =0 by virtue of conservation of momentum. Also, unlike free electrons or holes, the exciton is localized in real space and, therefore, delocalized in k space. That is, the exciton is formed from a superposition of electron and hole states. (This process is typically done using the Bethe-Salpeter equation. (Dresselhaus et al., 2007)) Recall that, when an electron is promoted from the valence to the conduction band, the hole that it leaves behind, though moving in the same direction as the electron, has momentum opposite to it by virtue of its negative mass. Therefore, excitons created from electron and hole states both originating

15 near the K (or both near the K’) point have small center of mass momenta. Antisymmetric linear combinations of these excitons are termed A2 excitons and are dipole allowed, i.e. bright, whereas excitons formed from symmetric linear combinations are termed A1 excitons and are dipole forbidden, or dark. Conversely, excitons created from electron and hole states from opposite sides of the Brillouin zone, termed E and E* excitons, have large center of mass momenta are dark by momentum conservation. In addition to K ↔ K’ degeneracy, electron and hole spins lead to singlet (L=0) and triplet (L=1) spin states. Only the singlet state is accessible by photon absorption. All told then, each excitonic state is sixteen-fold degenerate yet only one of these states is optically accessible (the spin singlet A2 exciton). In addition, the vast majority of the spectral weight is shifted from the free particle band edge to the lowest exciton and thus the optical response of carbon nanotubes is dominated by the lowest spin singlet A2 exciton at each band. (Ando, 1997; Kane and Mele, 2004; Vamivakas et al., 2006a) The reader is directed to the excellent work of Lee et al. who are able to probe both the bound excitonic states as well as the continuum band edge using photocurrent spectroscopy on split-gate nanotube devices. (Lee et al., 2007) Fig. 2·2(d) depicts the splitting of the four-fold degeneracy of the lowest spin singlet exciton into three dark and one bright state. It is of major consequence to the optical response of carbon nanotubes that the lowest lying spin singlet exciton is in fact dark (A1 ), about 10 meV below the bright (A2 ) exciton. This, along with thermalization described in Chapter 4, directly leads to much lower quantum efficiency, QE = number photons out / number photons in, than might otherwise be the case and must be considered when discussing dynamics of the nanotube electronic system as we will in this thesis.

2.5

Phonons

In this section, we will discuss the important Raman active vibrational modes relevant to the work presented in this thesis. For a more in-depth discussion, the reader is directed to our group’s work on the derivation of phonon modes in carbon nanotubes of arbitrary

16 chiral angle using a mass and spring model. (Vamivakas et al., 2006b; Mu et al., 2006) A knowledge of group theory is necessary to determine which modes are Raman and/or IR active and is not discussed here.

RBM

Figure 2·3: Raman spectra from a single, suspended carbon nanotube with Lorentzian fits showing the G+ , G− , and 1734 cm−1 modes. The optical modes from which the G+ and G− peaks derive are shown schematically. Inset: The radial breathing mode. Fig. 2·3 shows a typical Raman spectra taken from a single carbon nanotube suspended across a gap in order to remove substrate effects. The high energy in-plane graphitic, or G, Raman active mode found in graphite and graphene splits into G+ and G− modes when the CNT is rolled up due to symmetry breaking. The spectrum shows the raw data in black with each peak fit by a Lorentzian, shown in various colors. The mode at 1734 cm−1 is ascribed to a combination mode of two K point phonons (Yin et al., 2008) while the G+ and G− modes arise from single Γ point LO and TO phonons, respectively. The G+ phonon is the primary source of electron scattering in CNT field effect transistors and limits ballistic transport. (Perebeinos et al., 2004) Measuring the electron-phonon coupling strength is therefore of great import and we are the first group to do so experimentally by use of resonant Raman spectroscopy, RRS, of the G+ phonon mode. (Yin et al., 2007; Yin

17 et al., 2006; Yin et al., 2008) Our RRS experimental setup and procedure will be discussed in Chapter 3 where we will describe how we use RRS to measure the effect of screening on electron-electron and electron-hole interaction energies. The low energy radial breathing mode, or RBM, is shown in the inset of Fig. 2·3. It is of particular interest not just because it is often easily seen in the spectrum but also because its energy is inversely related to its diameter. (Rao et al., 1997b) Thus, the RBM, if it is visible in the spectrum, can be used to quickly narrow down to (typically) three to five the possible nanotube species when attempting to identify which nanotube species is being investigated. Identifying the particular nanotube species, i.e. the diameter and chirality, is somewhat of an art form and is often imprecise. There are a range of clues that can be derived from the various Raman modes and from photoluminesence (PL) which can be used to determine the species but the precise “(n,m)” value is not as important to this work as knowing the diameter and thus many of the details are not discussed here. Rather the reader is again directed to the book by Reich et al. (Reich et al., 2004) For the purposes of this thesis, the RBM is used to determine diameter and, along with the electronic transistion energy derived from RRS, make a rough species assignment.

18

Chapter 3

Dielectric Screening of Particle Interaction Energies 3.1

Introduction

It has been predicted for some time that the optical response of carbon nanotubes should be dominated by many-body Coulomb interactions and that those interaction energies should depend strongly upon local environment. (Ando, 1997; Kane and Mele, 2004; Kane and Mele, 2003; Spataru et al., 2004; Perebeinos et al., 2004) Recent two-photon experiments have demonstrated that the optical resonances are, in fact, excitonic in nature (F. Wang et al., 2005; Maultzsch et al., 2005; Dukovic et al., 2005) but were performed in heavily screened environments. The unscreened exciton binding and band gap renormalization energies were still unknown but predicted to be much larger, of order 1 eV, compared to the measured values in the 400 meV range. In addition, no one had yet developed a model that, given a particular dielectric environment, could predict those particle interaction energies since all published models relied upon a single dielectric value used as a fit parameter and representing an average value over the nanotube and its environment rather than using two separate values for nanotube and environment. In this chapter, we develop such and model and describe our measurement of the dependence of the electron-electron and electron-hole interaction energies (E BGR and E Bind respectively) with external dielectric screening. (Walsh et al., 2007; Walsh et al., 2008) We begin with a general discussion of how screening affects these particle interaction energies individually and the net effect on the optical transition energy E Opt . We derive scaling relationships of both E Bind and E BGR on external dielectric. We then discuss how resonant

19 Raman spectroscopy, or RRS, is used to meaasure the electronic transition energy with which the laser is resonant. Following the discussion of RRS theory, we will describe our sample preparation, experimental setup, and many of the nuances involved in obtaining high quality Raman excitation profiles, or REPs. Finally, we will show experimental data demonstrating a red shift of the optical transition energy with increased dielectric screening and, using our scaling relationships, extract the underlying screened and unscreened particle interaction energies.

3.2

Screening Theory and Scaling Relationships

ee-

eȯ1

p+

ȯ2

EBGR

EBind

Figure 3·1: Cartoon depicting the electron-electron and electron-hole Coulomb interactions. The dielectric function of the carbon nanotube and its enviroment are given by 1 and 2 , respectively. In this section, we will discuss how screening by the nanotube enviroment affects the particle interaction energies and the resulting optical transition energy. We will then derive scaling relationships between the particle interaction energies and the actual external dielectric value. We will discuss how our model improves upon previously published work on dielectric scaling. The optical and electronic properties of single wall carbon nanotubes (SWCNTs) are calculated to be dominated by strong Coulomb interactions between electrons and electrons and between electrons and holes. (Ando, 1997; Kane and Mele, 2004; Kane and Mele, 2003; Spataru et al., 2004; Perebeinos et al., 2004) Two-photon experiments have measured

20 exciton binding energies of several hundred meV. (F. Wang et al., 2005; Maultzsch et al., 2005; Dukovic et al., 2005) However, these measurements were performed on SWCNTs in screened environments; the intrinsic, chirality dependent, unscreened exciton binding energies are predicted to be significantly larger. (Capaz et al., 2006) Understanding how these particle interaction energies change with screening by the nanotube environment is critical when designing opto-electronic devices, carbon nanotube field effect transistors, etc. Previous theoretical models of particle interaction energies in carbon nanotubes (CNTs) typically include a single variable for the dielectric function and treat it as a fit parameter. (Perebeinos et al., 2004; Capaz et al., 2006) Thus, in these models, the dielectric function represents some average value of the heterogeneous dielectric environment and can not be used as an input parameter even when the value of the dieletric function external to the CNT is known. In this chapter, we derive a scaling relationship that uses the actual external dielectric function. We use this model later in this chapter to fit resonance Raman data taken from single CNTs suspended across trenches as the dielectric environment is altered. The results show that the particle interaction energies are about two times larger in air than when screened in water. However the measured energy shift of the optical transition energy is small since the band gap renormalization and exciton binding energies, defined in Section 2.4, have opposite signs. (Ando, 1997; Kane and Mele, 2004) The changes in these underlying interaction energies may be separately quite large but their difference relatively small, in agreement with reported solvatochromic shifts. (Choi and Strano, 2007; Lefebvre et al., 2004b; Moore et al., 2003) As discussed in Chapter 2, the one-dimensional nature of carbon nanotubes leads to smaller Coulomb screening and larger particle interactions compared to two- and threedimensional materials. Thus, Coulomb interaction energies can not be ignored when attempting to understand the electronic structure and optical properties in one-dimensional systems. The one-dimensional nature of carbon nanotubes also makes their electronic structure very sensitive to their environment and changes therein. Fig. 3·1 depicts schematically the screening of (or lack thereof) the electric field lines in carbon nanotubes of dielectric

21 a)

UNSCREENED

b)

SCREENED

Continuum Continuum Exciton

EBGR EBind

EBind

ESP

EOptical

ǻEOPT

EOptical ESP GS

Exciton

EBGR

KCM

GS

KCM

Figure 3·2: Energy diagrams of the effect of the band gap renormalization and exciton binding energies on the optical transition energy in (a) unscreened and (b) screened environments. value 1 in an external dielectric environment 2 . Electronic interactions lead to large blue shifts of the free particle band gap, a process known as band gap renormalization, while electron-hole interactions lead to a series of bound excitonic states well inside the band gap. (Ando, 1997) We label these interaction energies E BGR and E Bind (refering to the lowest optically active exciton which dominates the optical response (Vamivakas et al., 2006a)), respectively. Almost by definition, the single particle energy given by the tight binding approximation will not depend on external dielectric environment. In theory, the external dielectric could screen the pz orbitals of the carbon atoms upon which the tight binding approximation is based leading to reshaping of the valence and conduction bands and of the electron and hole effective masses. We ignore these higher order corrections here since they are expected to be small. Screening is then modeled as a decrease in both E BGR and E Bind with increasing 2 , as shown in Fig. 3·2, but leaving single particle energies unchanged. In fact, these particle interaction energies in an unscreened environment are calculated to be on the order of one electron volt for one nanometer diameter single wall carbon nanotubes, about the size of the single particle band gap. However, since they enter into the Hamil-

22 tonian with opposite signs,(Kane and Mele, 2004) they largely cancel each other and the resulting optical transition energy, E Opt, is only slightly higher than the transition energy predicted by single particle models, E SP . That is, E Opt = E SP + E BGR - E Bind . We now derive an expression for the scaling dependence of the exciton binding energy, E Bind , on the external dielectric, 2 .

b)

0.0

0.3 -0.2 -0.4

H2 = 4

-0.6

H2 = 2

-0.8 -1.0

H1 = 4

H2 = 1.3

-2

-1

0.2

0.1

H2 = 1

LX , LY = 1.1 nm -3

Z 0 / a b*

Coulomb Potential / Rh*

a)

0

1

Z / ab*

2

3

0.0

1

2

3

4

H1 /H

f Figure 3·3: Coulomb potential and the cutoff parameter. (a) VEf 1D as a function of electron hole separation, z, for four values of the external dielectric 2 . Truncated Coulomb potential fits to the numerically calculated data are shown as solid lines. a∗b =1 2 /μe2 is the bulk exciton Bohr radius. (b) The best fit cut off parameter z0 from (a) as a function of external dielectric 2 . The dashed red line highlights the linear dependence of z0 on 1/2 . f We use a one-dimensional effective potential, VEf 1D (z), for a quantum wire of dielectric

1 in an environment 2 , integrating over the lateral x, y dimensions which yields a function of z, the electron-hole separation. (Ogawa and Takagahara, 1991b; Banyai et al., 1987) Numerical results from Ogawa and Takagahara (Ogawa and Takagahara, 1991b) are shown in Fig. 3·3a for four different values of the ratio of 1 /2 . 1 is taken as 4 for graphite. (Pedersen, 2003a; Taft and Philipp, 1965) 2 , the external dielectric, spans values from 1, i.e. unscreened, to 4. Here, the binding energy of the exciton dictates that we use the optical value of the dielectric function, which, for water, is about 1.78 (Hayashi et al., 2000). The resulting curves are fit with a truncated Coulomb potential of the form 1 / ( |z|+z0 ). The fit

23 parameter z0 is known as the cutoff parameter. By incorporating z0 , the divergence as the electron-hole separation z→ 0 is removed. This also reflects the geometry of the problem, i.e. the carbon nanotube is not truly one-dimensional but has a finite diameter implying a minimum electron-hole separation. The fits are clearly excellent. The best fit value of z0 for each ratio 1 /2 is plotted in Fig. 3·3b. For the range of external dielectric values of interest here, the dependence is almost perfectly linear. That is, we can say z0 scales with −1 2 . The same scaling relationship is found using the expression for the one-dimensional effective potential of Banyai et al. (Banyai et al., 1987) as well. This result allows us to derive a scaling relationship between the exciton binding energy and the external dielectric function using Loudon (Loundon, 1959) which analytically solved the binding energy for the one dimensional hydrogen atom using the truncated Coulomb potential. Specifically, Loudon (Loundon, 1959) found that the binding energy E Bind =R∗ /λ2 where R∗ is an effective Rydberg equal to μe4 /22 2 , μ is the exciton effective mass, and  is the dielectric constant, a poorly defined quantity when applied to a heterogeneous environment. The quantum number, λ, is not, in general, integer and is a complicated function of the cutoff parameter, z0 . However, Combescot and Guillet (Combescot and Guillet, 2003) −1 show that λ scales as ∼ z0.4 0 which, in turn, we have shown scales as 2 . Thus, E Bind

scales as R∗ x 2∗0.4 . Further, the depth of the effective potential, which is proportional to e2 /, on Fig. 3·3a is found to scale with 1/2 over this external dielectric range. Thus, the effective Rydberg, R∗ , which is proportional to e4 /2 , is presumed to scale as 1/22 . The overall scaling of the exciton binding energy is then 2∗0.4 /2 = 1/1.2 , the central result of this work. We emphasize that this scaling relationship is based on the actual external dielectric value and is thus of practical use. The value of the scaling exponent, α, is very close to the value α = 1.4 derived in Perebeinos et al. (Perebeinos et al., 2004) where the model contained a single  and thus represented a sort of averaging over the heterogeneous dielectric environment. Also, since R∗ is independent of the radius, r , and z0 scales with r, (Ogawa and Takagahara, 1991b; Ogawa and Takagahara, 1991a) the binding energy scales with 1/r0.8 , close to the 1/r0.6 dependence found by Ref. (Pedersen, 2003b) using a

24 variational method. We now combine this scaling result with the scaling behavior of the band gap renormalization energy, E BGR , in order to address how the eletronic structure, which depends on both electron-electron and electron-hole interactions, scales with the external dielectric environment. Specifically, Kane and Mele (Kane and Mele, 2004) found E BGR scales approximately as 1/. Having found the dependence of R∗ on 2 was the same as the dependence on  for this dielectric range, we assume E BGR scales with 1/2 . Then the expression E Opt = E SP + E BGR - E Bind becomes

2 =1 2 =1 /2 - E Bind /1.2 E Opt(2 ) = E SP + E BGR 2 ,

ignoring the possible small dependence of the single particle term on external dielectric through the exciton effective mass, μ, which is a function of valence and conduction band curvature. Thus, upon changing the environmental screening,

2 =1 −1.2 −1 −1.2 2 =1 (−1 ΔE Opt = ΔE BGR - ΔE Bind = E BGR 2,F inal - 2,Initial ) - E Bind (2,F inal - 2,Initial ).

Fig. 3·4a depicts the scaling behavior of the constituent particle interaction energies. 2 =1 2 =1 is taken as 730 meV and E Bind as 580 meV (Walsh et al., 2007), where these are E BGR

the interaction energies associated with the second valence band to second conduction band transition, E22 . The values 2 = 1 and 1.78 (water) are highlighted with vertical dashed lines. It is important to note the relatively weak dependence of E Opt with screening due to the opposite sings of E BGR and E Bind , in accordance with the picture described by Ando and Kane and Mele (Ando, 1997; Kane and Mele, 2004) and reported in the literature. (Lefebvre et al., 2004b; Moore et al., 2003) First order, single particle models, such as the nearest-neighbor tight-binding model, predict a constant ratio of E 22 /E 11 equal to 2. Experiments found that ratio, on average, to be closer to a value of 1.7 (Bachilo et al., 2002) and was dubbed the “ratio problem”. As explained by Kane and Mele (Kane and

25

400

+

200 EOPT = ESP + EBGR + EBind 0

-200 -400 -600 -800

580

1

ESP

-1.2 EBind DH2

ESP = 0 meV EBGRH2=1) = 730 meV EBindH2=1) = 580 meV

200 150

D = 1.4

100 50

D = 1.2 Water

+

Water

600

b)

Unscreened

D = 1.2

EOpt - ESP ( meV )

-1 EBGR DH2

+

Unscreened

ǻ Energy ( meV )

a) + 800 730

D = 1.0

0

2

External Dielectric H2

3

1

2

External Dielectric H2

3

Figure 3·4: Scaling of particle interaction energies with external screening, referenced to the single particle energy, E SP , which is taken to be zero. 2 =1 (a) E BGR is shown as a blue dashed line and scales with −1 2 . E BGR is assumed to be 730 meV. E Bind is shown as a red dashed line and scales with 2 =1 is assumed to be -580 meV. The resulting behavior of E Opt 2−1.2 . E Bind with screening is shown as a black line. (b) The behavior of E Opt (minus the constant E SP ) as a function of α. Notice larger values of α lead to blue shifts of E Opt with small screening.

26 Mele, 2003), particle interaction energies resolve this discrepancy since E BGR and E Bind do not exactly cancel but lead to an overall blue shift of E Opt compared to E SP , for each subband E ii . Both the numerator and the denominator in the ratio E 22 /E 11 are slightly blue shifted and thus the ratio is decreased (ignoring chirality effects.) Note, however, that the free particle band gap is significantly altered compared to the single particle value and can change dramatically upon perturbation of the environment. This is, of course, a critical consideration with regard to CNT electronic device design and operation. Fig. 3·4b shows 2 =1 the effect of the scaling exponent, α, on the behavior of E Opt . For the same E BGR and 2 =1 , as in Fig. 3·4a, notice how larger values of α can actually lead to blue shifts in E Opt E Bind

in the small screening limit.

3.3

Resonant Raman Spectroscopy Theory

Resonant Raman spectroscopy, or RRS, was used to determine the optical transition energies, E 22 , for two single CNTs suspended across trenches (in order to remove substrate effects) as the dielectric environment was altered from dry N2 , to high humidity N2 , to water. In this section, I will describe how we use RRS to measure the optical transition energy of a single carbon nanotube. The screening theory discussed above is then applied to RRS measurements in the following sections. RRS has two significant advantages over photoluminescence (PL) spectroscopy in relation to this experiment. First, PL spectroscopy is necessarily restricted to the lowest sub-band E 11 whereas RRS can be used to probe the effect of screening on the exciton associated with any sub-band. Further, most dielectric environments will tend to quench the PL signal almost entirely making the type of measurements made in this work impossible using PL. Recall from Chapter 2 that the one-dimensional nature of carbon nanotubes leads to van Hove singularities at every band edge or, in the excitonic picture, at the bottom of each exciton band at kCM =0. The density of states (DOS) for the bright exciton is large at one specific optical transition energy and small elsewhere. Furthermore, for typical nanotube lengths, there is often only one (or at most a few) exitonic state in each band of sufficiently

Raman Peak Height

27

Outgoing Resonances

Increasing Screening

Incoming Resonances

1.4

1.5

1.6

1.7

Excitation ( eV ) Figure 3·5: Cartoon of the dependence of the Raman excitation profile on external screening. The peaks associated with incoming and outgoing resonances are highlighted by dashed circles. small momemtum to directly emit or absorb a photon. This point is critical to the results of Chapter 4 and is thus discussed in more detail there. Thus, the nanotube optical response is similar to that of an atomic or molecular system in that the electronic system may be modeled, as a first approximation, as simply a series of energy levels representing the exciton at each E ii . This model holds well for absorption and Raman spectroscopies where the dark exciton 10 meV below the bright exciton need not be considered. For other spectroscopies such as PL and transient absorption, the lower dark exciton is critical and must be included in the model. Consider a simple two level system where the upper level represents, in our case, the bright exciton at E 22 . As a tunable laser is tuned through this resonance, the Raman scattering will increase dramatically by virtue of the increased electron-photon scattering. When the incident photon is resonant, the Raman peak will be at a maximum and this is referred to as an “incoming resonance.” Similarly, if the laser is tuned further (towards higher energy), the Raman scattered photon (at lower energy than the incident photon for the Stokes process) can also be resonant. Thus, in RRS, plotting the Raman peak height for a given Raman active mode against laser excitation energy

28 yields, in general, a curve with two peaks, known as the Raman excitation profile, or REP. Fig. 3·5 shows a cartoon of the REP. The peak at lower (higher) energy corresponds to the situation where the incident (scattered) photon is resonant with the optical transition energy. Fig. 3·5 depicts the REP in the case where the phonon energy is large enough that the two peaks are spectrally separated, as is the case with the Raman active G+ vibrational mode.1 In contrast, the REP associated with the RBM phonon mode will appear single peaked by virtue of the much smaller phonon energy. In both cases, the effect of screening is to red-shift the underlying optical transition energy and thus the REP red-shifts as well. The REP can be fit using a one-phonon, exciton-mediated lineshape (Vamivakas et al., 2006a) for each dielectric condition and the shift in the optical transition energy determined. Notice how Raman spectroscopy is being used to extract the energy of the underlying electronic resonance. By fitting a one phonon exciton-mediated REP line shape (Vamivakas et al., 2006a), we can determine the optical transition energy, E ii , with which the photons are resonant. One can also measure the anti-Stokes REP (phonon absorption rather than emission) which would also be double peaked and should yield the same E ii . In fact, the REP associated with each phonon mode should yield the same E ii since each REP is based on the same electronic resonance. There are important subtleties here which should be pointed out. The language used in the previous paragraph inherently applies to a solid state approach. The electronic and vibrational systems are treated under a full Born-Oppenheimer approximation, i.e. each electronic and vibrational degrees of freedom are assumed separable. (Vamivakas et al., 2006a) Thus the two level system discussed above represents two electronic levels and the vibrational modes are assumed unaffected by the electronic state of the system. In a crystalline system with 1023 atoms, where the momentum k is a good quantum number and the electronic wavefunction is delocalized in real space, this is usually a good approximation. This is not usually so in atoms and molecules. Indeed, the system is usually modeled by molecular energy eigenstates and the coupling strength between electron and phonon 1

See Appendix A for a discussion of sources of REP asymmetry as shown in the figure.

29 depends on the electronic state of the system. There is a distinction made between ground and excited state phonons (of the same mode) and the strongest single photon excitation of a molecular system often involves the generation of one or more phonons. This obviously has consequences for the REP of an atom or molecule and, specifically, leads to an additional degree of freedom in the REP lineshape where the “incoming” and “outgoing” resonances can have dramatically different heights. Carbon nanotubes fall somewhere in between a 3D crystal and a large molecule with, typically, 104 to 106 atoms. A perturbative density matrix approach to deriving the CNT REP line shape is used in Appendix A where the CNT is modeled by four molecular levels representing the ground electronic state with and without a phonon and the excited electronic state with and without a phonon. Notice how the language and the idea of “resonance” changes. In the two level solid state approach, at “outgoing resonance,” only the outgoing photon is considered resonant (hence the name) and the incident photon is not, i.e. it is viewed as coupling the ground state to a virtual excited (electronic) state. In contrast, in the molecular approach, this incident photon couples two real states, the ground electronic state with no phonon and the excited electronic state with one phonon. The outgoing photon then couples this state to the ground electronic state with one phonon, i.e. the nanotube is left in a vibrationally excited molecular state. The reader is directed to Appendix A for more details. The solid state and denisty matrix derivations, however, yield different line shapes and one model, in theory, should fit the experimental data better than the other. In addition to the added degree of freedom in the ground vs. excited electron-phonon coupling mentioned previously, they also predict different amounts of spectral weight between incoming and outgoing resonance. Unfortunately, the RBM REP, being single peaked, is equally well fit by either model and obtaining a good REP for the G+ phonon, including both peaks, is challenging since it involves tuning over a 100 nm range (in the spectral range of our experimental setup), four different laser line rejection filters, and changing system response. However, given the right nanotube with the right resonant energy and by using (perhaps) the Si 520 cm−1 Raman line as a standard candle, it may well be possible to distinguish between the two models.

30

3.4

Experimental Details

In this section, we briefly cover some of the experimental details that are important to obtaining good REP data. We discuss sample preparation first and then the experimental apparatus. For the data shown in this chapter, individual single-wall carbon nanotubes were grown over trenches 1 to 2 μm wide in order to remove substrate effects, including screening effects which we wish to quantify. The trenches were etched by reactive ion etching (RIE) in a CF4 plasma. A chromium layer patterned by electron beam lithography and wet chemical etching was used to mask the quartz during the RIE process. SWNTs were then grown over the trenches by chemical vapor deposition (CVD) in methane gas at 900o C using a 1 nm thick film of Fe as the growth catalyst. The samples for the REP studies were grown at Harvard University by our collaborator, Stephen Cronin. The measured nanotubes are presumed to be individual based on several observations: (i) the low CNT density as determined by SEM, (ii) the measured values of E22 which are very close to published values for individual CNTs, and (iii) no indications of resonant modes from other CNTs over the entire tunable range. Data was collected using a customized Renishaw 1000B spectrometer, as shown in Fig. 3·6. A Ti:Sapphire laser, tunable from 705 to 855 nm, was used as the excitation source. A narrow band-pass filter was used to “clean up” the laser in order to remove laser spectral weight in the region 100 to 300 cm−1 from the laser where the RBMs lie. The band-pass filter was “twist-tuned” away from normal incidence, blue-shifting its spectral pass band, in order to follow the tunable laser line. This twist-tuning offsets the beam path spatially, however, and must be compensated by use of a microscope slide or other glass blank twist-tuned in opposition. This compensation was calibrated by observing the movement of the focused laser spot relative to the optical crosshairs of the microscope and adjusting the compensating slide as needed to return the laser spot to its original position at the center of the crosshairs. The laser was then collimated using a pair of lenses. Laser

31

Spectrometer Edgeornotchfilter 90/10beamsplitter Broadbandhalfwaveplate

600g/mmgrating InGaAs(NIR)

Microscope Ͳ XYstagecontrol Ͳ Gratingcontrol Ͳ CCDcontrol

100x0.90NAor 50xLWD.5NA Objective

CCD(Visible) Flipmirror

Cryostat

TunableTi:Sapphire

Compensatingslide

Bandpassfilter

Figure 3·6: Experimental setup. The excitation source is a tunable Ti:Sapphire laser whose emission is cleaned up by a twist-tuned narrow band pass filter. Any resulting beam displacement is compensated by use of a counter-rotated glass slide. After collimation, the beam is sent via a 92/8 pellicle beam splitter to the microscope, high NA objective, and onto the suspended CNT. The beam path is shown in red. The Raman spectra are collected in a reflection geometry with the same objective and sent back through the 92/8 beam splitter. The laser is filtered with either a narrow notch filter or a sharp long pass filter. The scattered light is then dispersed by a 600 g/mm NIR grating and sent to either, by use of a flip mirror, a CCD (visible) or InGaAs (NIR) detector.

32 power was kept constant by directly measuring the beam at each wavelength and rotating a continuous neutral density filter wheel as needed. The typical trick of splitting off 4 percent of the laser power with a glass slide and sending it to a fixed detector was not used since we found fringing effects in the slide were wavelength dependent and effected the splitting ratio, enough so that we decided to manually insert and remove the detector head, before taking data at each wavelength, from the beam path at the point just before it enters the microscope. Inside the Renishaw, the beam was steered and split using a nominally 92/8 broadband pellicle beam splitter. A pellicle beam splitter, by virute of its 1 to 2 μm thickness, removes spatial fringes (or ghosting) in the object plane created by normal beam splitters. The spatial position of any such fringes is wavelength dependent and thus would change the intensity felt locally by the nanotube as the laser is tuned. This would significantly deteriorate the REP line shape. We did find the pellicle spectral response to be far from flat, but it could be calibrated and the spectra corrected if needed. The response, however, was usually slowly varying over the scanned wavelength range. The laser was then focused by use of a high NA objective (typically 100X/0.9 NA) onto the suspended nanotube, located spatially by optimizing either the G+ or RBM peak height. The pellicle itself and a second mirror were used to align the beam precisely to the center of the crosshairs of the microscope which were themselves previously calibrated to lie at the exact optical axis of the microscope. Poor spectrometer alignment or sample positioning has a number of spectral ramifications discussed at length in Chapter 7. The diffraction limited spot size, being smaller than the trench width, probed the portion of the nanotube suspended over the trench and not in contact with the substrate. The Raman scattered signal was collected in a reflection geometry using the same objective and sent to the spectrometer. After the beam splitter, a sharp long pass filter was used to reject the laser line while passing the low energy RBM signal that lies spectrally close to the laser. Like the band-pass filter, this filter was twist-tuned to follow the laser. No compensating glass was necessary,

33 however, as the attendant spatial offset was in the so-called “non-spectral” direction. That is, the offset of the signal beam led to an offset parallel to the grooves of the grating rather than transverse to them and thus led to an offset of the signal at the detector array in the direction perpendicular to the spectral direction as defined by the grooves of the grating. The signal at the detector was integrated over a sufficient number of pixels in this nonspectral direction so as to avoid clipping of the signal with laser tuning. The reader is referred to Chapter 7 for more details The slit at the entrance of the spectrometer was kept constant, typically 40 μm yielding a good signal to noise ratio. Typical integration times were 30 seconds. Spectra were taken at each wavelength in both the RBM and G+ region of the spectrum. In order to demonstrate reproducibility of the REP data, the laser wavelength was typically stepped in 4 nm increments through the nanotube resonance and then stepped again in 4 nm increments through the resonance but sampling the alternate/intermediate wavelengths, resulting in what were referred to as “Run 1” and “Run 2”.

3.5

Results

We measured REPs of individually resonant CNTs in dry N2 in an enclosed chamber, and then again after adding water vapor to the nitrogen. The humidity was measured with a hygrometer. Finally, the sample is directly immersed in water and the experiment repeated a third time, all on the same singly resonant CNT. The set of REPs yield E ii for each phonon mode and dielectric environment, and thus measure the shift in the electronic level with increasing . The shifts measured by each phonon mode are all identical for the same CNT, as they should be. The results of increasing external dielectric environment on the E ii for two different Raman modes from the same CNT are shown in Fig. 3·7. Contour plots are shown on the left with the resulting REPs shown on the right (verticle line cuts of the contour plots). The data clearly shows the outgoing peak of the REP red shifting with increasing external dielectric for both the G+ and 1734 cm−1 Raman modes. Though the origin of the 1734

34

82% RH N2

In Water

b) 1

1.67 1.66 1.65 -

+

G

Intensity(Arb. (Arb. Units) Units) Intensity

1.68 -

Dry N2

Normalized Intensity

Excitation Energy (eV)

a)

Dry N2

82% RH N2

In Water

1.64 0

1.64

1560 1600 1640 1560 1600 1640 1560 1600 1640 Raman Shift (cm-1)

Dry N2 82% RH N2 In Water

d) 1

1.68 1.67 1.66 1.65 1.64 -

0

1720 1740 1760 1720 1740 1760 1720 1740 1760 Raman Shift (cm-1)

Intensity (Arb. Units)

1.69 -

Normalized Intensity

Excitation Energy (eV)

c)

1.66 1.68 Excitation (eV) Excitation (eV)

1.70

1734 cm-1 Dry N2

82% RH N2

In Water

1.64

1.66 1.68 Excitation (eV)

1.70

Figure 3·7: REP Data. (a) Contour plot showing normalized intensity of the G+ outgoing resonance as a function of both excitation energy and Raman shift measured in dry N2 , 82% RH N2 , and immersed in water. (b) REPs of the data shown in (a). Plots are offset in the vertical direction for clarity. Data measured in dry N2 is shown in blue squares with the accompanying REP fit shown with a solid blue line. Data measured in 82% RH N2 and in water are shown similarly in green and red, respectively. (c) and (d) Same as in (a) and (b), respectively, but for the 1734 cm−1 Raman mode.

35 cm−1 mode is still disputed in the literature, the exact line shape used to fit the data is not critical since we are looking for shifts in E 22 rather than the absolute value. (Brar et al., 2002; Yin et al., 2008) The measurements in water are noisier due to lower signal level as a result of perturbation of the wave fronts by the water-coverslip and coverslip-air interfaces. Despite this, shifts in the resonance peak energies are clearly visible. High relative humidity N2 introduces a ∼10 meV red shift in E 22 . Liquid water red shifts E 22 by ∼30 meV, similar to the ensemble average differences reported by Lefebvre et al. (Lefebvre et al., 2004b) Table 3.1 shows tabulated results for two different nanotubes. Note the consistency in the observed shifts between each dielectric environment. In order to use our results to quantify the effect of screening on the particle interaction energies in CNTs, we first must discount other possible environmental influences on the electronic transition energies including temperature (Cronin et al., 2006; Capaz et al., 2005; Lefebvre et al., 2004a), mechanical strain (Cronin et al., 2006; Cronin et al., 2004; Cronin et al., 2005), and charge transfer (Shim et al., 2001; Zahab et al., 2000; Collins et al., 2000; Lee et al., 1997; Kong et al., 2000; Shim et al., 2006; Rao et al., 1997a; Claye et al., 2001). All measurements were taken at room temperature and laser power kept sufficiently low to avoid heating of the CNT. (Cronin et al., 2006) Mechanical strain changes the C-C bond lengths which shifts the energy levels of the nanotube (Cronin et al., 2006; Capaz et al., 2004) and can lead to E ii shifts depending on the type of strain (uniaxial, isotropic, radial), (n-m)mod3 value, sub-band index, and chiral angle. But strain causes a change in the observed phonon energies (Cronin et al., 2006; Cronin et al., 2004; Cronin et al., 2005) and we observe no changes in any phonon mode energies (to within ∼ 1 cm−1 ). A number of studies have investigated charge transfer and its effect on transport(Shim et al., 2001; Zahab et al., 2000; Collins et al., 2000; Lee et al., 1997; Kong et al., 2000) but it is difficult to separate the effect of charging from that of screening in such experiments. However, charge transfer has been shown to be associated with a change in the tangential phonon energy(Shim et al., 2006; Rao et al., 1997a; Claye et al., 2001) and since, again, we observe no changes in any phonon mode energies, we believe that charge transfer is negligible, in

36 agreement the diameter dependent activation model proposed by Shim et al. (Shim et al., 2006). Thus the primary mechanism for the observed shifting of the electronic transitions is screening of the particle interactions. CNT 1 RBM = 208 cm-1

E22 Dry E22 “Wet N2” ( eV ) ( eV )

“Wet N2” Shift (meV)

E22 in H20 ( eV )

H20 Shift (meV)

Incoming G+

1.472

1.465

-7

Outgoing G+

1.476

1.467

-9

1.449

- 27

Mode 1734

1.474

1.467

-7

1.441

- 33

Outgoing G+

1.437

1.427

- 10

1.411

- 26

Mode 1734

1.434

1.423

- 11

1.405

- 29

CNT 2 RBM = 212 cm-1

Table 3.1: Tabulated results of red shifts for 2 carbon nanotubes. E 22 , as determined by REP fit, is shown for each phonon mode and resonance condition. We can now use the simple scaling relationships derived in Section 3.2 to extract the effect of screening on the particle interaction energies. As previously discussed, the tightbinding term in the Hamiltonian is assumed to be independent of external dielectric and 2 =1 /2 and E Bind = thus we may write ΔE Opt = ΔE BGR - ΔE Bind . Further E BGR = E BGR 2 =1 /1.2 E Bind 2 . In our experiment, 1 ∼4 for graphite (Taft and Philipp, 1965; Pedersen, 2003a),

initial 2 =1 (dry N2 ), and final 2 =1.332 =1.78 (in water). Having directly measured ΔE Opt and by using an unscreened exciton binding energy of 580 meV for nanotube (12,4) (Capaz et al., 2006) we are able to extract values for the screened exciton binding energy and for the screened and unscreened BGR energies. Note that this calculated E 11 exciton binding energy is expected to be a slightly smaller (Jiang et al., 2007) than that at E 22 , so the derived values are conservative. Specifically, this analysis yields an exciton binding energy of ∼290 meV after immersion in water, an unscreened BGR energy of ∼730 meV, and a screened BGR energy of ∼410 meV. Thus large reductions in the exciton binding energy and BGR energy of ∼290 meV and ∼320 meV, respectively, lead to the small 30 meV red shift measured in the optical transition energy, depicted schematically in Fig. 3·2. Limitations

37 of our model include the assumption of solid wires rather than cylindrical shells (in both quantum wire models) and the value of 2 is not known precisely. We can also compare the exciton binding and BGR energies at 2 =1 for these nanotubes. Theory predicts that the BGR energy should be larger than the exciton binding energy which is used to explain the so-called “ratio problem” (Kane and Mele, 2004; Kane and 2=1 Mele, 2003; Ando, 1997). Indeed, we find E 2=1 BGR - E Bind ∼ +150 meV, a BGR energy larger

than the exciton binding energy at 2 =1 by about 25 percent for this particular nanotube in an unscreened environment. Qualitatively, this result is fairly insensitive to the choice of α, going as low as +70 meV at α=1 or as high as +220 meV at α=1.4 at F2 inal =1.78.

Į Ł 1.0

Į Ł 1.2

Į Ł 1.4 ǻE Opt {

EBGR

EBind

EBGR

EBind

EBGR

EBind

671

580

648

580

625

580

ȯ2 = 1

377

326

364

326

351

326

ȯ2 = 1.78

294

254

284

254

274

254

752

580

730

580

707

580

ȯ2 = 1

422

290

410

290

397

290

ȯ2 = 1.78

330

290

320

290

310

290

824

580

801

580

778

580

ȯ2 = 1

463

259

450

259

437

259

ȯ2 = 1.78

361

321

351

321

341

321

-40

-30

ǻE

ǻE

ǻE

-20

( All energies in meV )

Table 3.2: Particle interaction energies as a function of the exciton scaling exponent, α, the external dielectric, 2 , and the change in the optical transition energy, ΔE Opt. An unscreened exciton binding energy of 580 meV (Capaz et al., 2006) is used as an input parameter. The values below the line in each box show the changes of the BGR and exciton binding energies with screening. Their difference is ΔE Opt. The dashed circle highlights the numbers quoted in the text. 2=1 Although E 2=1 BGR is greater than E Bind , the change in E BGR is not necessarily larger than

the change in E Bind with small screening by virtue of their different scaling exponents. In fact, our model predicts that values of α greater than 1.4 lead to negligible or even blue shifts with small screening; the initial red shift measured in high humidity N2 indicates a

38 monotonic decrease in the electronic exciton energy level with increasing screening and thus supports a value of α less than 1.4. Tab. 3.2 shows the underlying variation of the particle interaction energies predicted by the model as a function of the external dielectric value, the scaling exponent, and optical transition energy shifts. CNT 2 in Tab. 3.1 is assigned as a (13,2) nanotube. It belongs to the same branch (2n+m) and family as the (12,4) nanotube; thus the particle interaction energies determined by the model are very similar.

3.6

Conclusion

In summary, we have derived the scaling relationship between exciton binding energy and the actual value of the external dieletric which is of practical importance when designing a variety of CNT based technologies. The scaling exponent, α, is found to have a value of 1.2. We experimentally show a monotonic decrease in the optical transition energy with increasing dielectric environment. Our model explains the small observed shifts despite large changes in the underlying particle interaction energies. The relevant particle interaction energies are shown to decrease on order of 50% upon screening by water. Further, we demonstrate that the band gap renormalization energy is significantly larger than the exciton binding energy at =1 and that the optical transition energy scales much more weakly with increasing external dielectric.

39

Chapter 4

(6,5) Carbon Nanotube Exciton Dynamics 4.1

Introduction

The details of exciton relaxation dynamics in carbon nanotubes (CNTS) are still not well understood and yet these dynamics will play a critical role in a variety of potential CNT applications. In addition to a complicated electronic structure as described previously, nanotube emission characteristics vary widely in the literature due to the inherent sensitivity of nanotubes to their environment, different defect levels, different surfactants and dielectric environments, lengths, etc. Thus different groups report, for example, very different lifetimes or different best fit functional forms to their data despite using the same or similar spectroscopic techniques. Much of this variation arises from the difficulty of deconvolving the contributions to the signal of several nanotube species present in the sample. In addition, at moderate power levels there arises the possibility of exciton-exciton interactions. In this chapter, we address both of these issues by using a sample of virtually only one nanotube species and by using ultra-short pulses shorter than the thermalization time scale. We demonstrate three level relaxation dynamics in an ultra-pure sample of (6,5) carbon nanotubes (Arnold et al., 2006) using transient absorption (TA) spectroscopy. Whereas in prior work the TA data is often segmented into short and long time scale data or low and high fluence data and subsequently fit by separate functional forms, we show that the TA dynamics are exceptionally well fit by a single three level stretched exponential model in both time and fluence regimes. Multi-exciton formation within a given nanotube is precluded by the use of ultrafast pulses shorter than the thermalization time scale in nanotubes short enough such that only one optically accessible state exists per nanotube. Thus, the

40 observed stretched exponential behavior is ascribed to diffusion limited contact quenching and/or sample inhomogeneities rather than exciton-exciton interactions as is observed is previous studies. (Wang et al., 2004; Ma et al., 2005; Valkunas et al., 2006) The power dependence of the zero time delay TA signal is measured over four orders of magnitude and shown to behave as a two level system. The optical Bloch equations (OBE) are solved in the short time limit of zero population loss. This model is shown to fit the data exceptionally well. A dipole length of ∼0.1 nm is measured as is the homogeneous line width.

Transient Absorption Theory Splitter

(a) Dump

(b)

Delay ǻIJ

Sample

Population Relaxation

t = ǻIJ

t=0 Excited

Initial

(c)

Probe

Pump Detector

Incident Pulse

Transmission

4.2

Delay ǻIJ

Excited

Relaxing

(d)

Probe

Pump

E22

2 Ȗ21 Ȗ20

E11

1 Ȗ1 0

Figure 4·1: Transient absorption experiment. (a) Experimental setup. (b) Cartoon of two level TA dynamics. (c) Cartoon of TA signal. (d) Schematic of the three level model used to fit the data. Transient absorption spectroscopy is fairly simple to understand and is depicted schematically in Fig. 4·1. TA uses a pump-probe configuration (shown in Fig. 4·1a and discussed in more detail in the next section) where the pump pulse is used to set up an initial excited

41 state population at t=0, as shown in Fig. 4·1b. The absorption of the subsequent probe pulse at t=Δτ is then reduced since the system is still partially excited already. The delay Δτ can be varied from t=0 to as long as signal to noise or experimental limitations will allow. The transmitted power of the probe beam is measured at each delay time. The resulting curve will typically look something like the curve shown in Fig. 4·1c. In the ideal two-level system with a single decay constant, the functional form of the curve will be a simple exponential decay which may be fit in order to measure the excited state lifetime. Things can get more complicated. For instance, the lower state need not be the ground state in which case the transmission of the probe beam will also depend on the population dynamics of the lower state. Indeed, we use a three-level model, described below and depicted in Fig. 4·1d, where the signal primarily depends on the recovery of the ground state, and the relaxation dynamics are best described by a distribution of decay rates.

4.3

Experimental Details

In these experiments we use nanotubes micelle encapsulated and suspended in solution (O’Connell et al., 2002) inside a cuvette. TA experiments benefit greatly by using highly homogenous samples, as do most time resolved experiments, since it quickly becomes difficult to analyze data from a heterogenous sample of emitters with a range of decay rates all on the same time scale, especially when the population distribution of the emitters is not precisely known. For this reason, we use samples from the Hersam group at Northwestern which are known to be very homogeneous, as demonstrated by the absorption data shown in Fig. 4·2. This is not a simple process (Arnold et al., 2006) and involves both micelle encapsulation using sodium chlorate, in order to isolate the nanotubes from each other, and density gradient ultracentrifugation, in order to remove bundles and segregate the nanotubes by species. The resulting sample consists predominantly of (6,5) nanotubes. The sample is placed in a 2 mm cuvette with a resulting optical density of ∼0.3. The laser beam is only weakly focused (details below) and thus the measured signal at the detector results from the combined absorption of thousands of nanotubes of virtually all one species.

Normalized Absorbance

42

Sample A Sample B Previously Published

(6,5) E22

600

(6,5) E11

Other species

800

1000

1200

Wavelength (nm) Figure 4·2: Sample A and B absorption data showing the (6,5) nanotube E 11 and E 22 transitions. Minor peaks are visible corresponding to the presence of small amounts of other nanotube species. A 100 kHz train of pulses is generated by pumping an optical parametric amplifier (OPA) with a titanium sapphire regenerative amplifier. The OPA may be tuned to resonance with either the E 11 or E 22 transition. The OPA output pulses are compressed to ∼38 fs using a prism pair and the beam split with a beam splitter. The experimental configuration is shown in Fig. 4·1a. The probe pulses are delayed using a standard delay line and the pump and probe beams are both focused to a beam waist of ∼ 19 μm using the same 12 inch focal length lens. Since the beams are slightly laterally offset from each other at the lens, they focus onto the sample at slightly different angles which allows for spatial separation of the probe beam from the pulse beam at the probe detector. The transmitted power of the probe pulses are measured using a silicon photodiode and lock-in amplifier. The probe pulses can also be spectrally dispersed by use of a monochrometer prior to the detector which can then be used to record the TA signal at each wavelength. Data is taken at multiple fluences in order to look for non-linear behavior indicative of many-body effects such as exciton-exciton annihilation via Auger-like processes where one

43

-3

a)

4.0 x10 3.5

Low Fluence

8

High Fluence

3.0 2.5 'T/T

6

'T/T

-1

b)

x10 10

4

2.0 1.5 1.0

2

0.5

0

0.0

Resid

5 0 -5

20 40 60 80 100 120 140 160

x10

0

-2

10

20 40 60 80 100 120 140 160

0

20 40 60 80 100 120 140 160

0

20 40 60 80 100 120 140 160

-3

Resid

0

-4

2 0 -2

x10

-1

'T/T

'T/T

10

-2

10 -3

10

x10

1

10

-3

100

Resid

Resid

-4

5 0 -5

1

10 Delay W (ps)

100

2 0 -2

x10

1

10

100

1

10 Delay W (ps)

100

Figure 4·3: Low and high fluence TA data with stretched exponential fits. The data are shown as black squares on both linear and logarithmic scales, with residuals. The three level stretched exponential fits are shown in red. (a) Low fluence (2.2x1014 photons/pulse/cm2 ). (b) High fluence (5.4x1016 photons/pulse/cm2 ).

44 exciton decays non-radiatively by transfer of energy to a second exciton which is excited into a higher energy state. Low and high fluence data with the laser at E 22 resonance are shown in Fig. 4·1 in both linear and logarithmic form with residuals shown below. The data is best fit by a three level model using stretched exponentials, as described in the next section. Using published values for the scattering cross section of ∼1x10−13 cm2 (Carlson et al., 2007; Islam et al., 2004), we estimate that even at the lowest powers used, we may still be creating several excitons per 100 nm of nanotube length. Specifically, the published crosssection of 1x10−13 cm2 per CNT for (6,5) CNTs of average length of 380 nm (resonant at E 11 ) (Carlson et al., 2007) can be converted to a cross-section of 2.9x10−18 cm2 per C atom by knowing the (6,5) CNT has 90 C atoms per nm of length. At 571 nm (E 22 ), 0.1 mW power, and a 100 kHz rep rate, there are 2.9x109 photons per incident pulse. The focusing lens creates a 19 μm FWHM beam spot (measured using the standard razor edge and error function fitting method) which translates to about 2.5x1014 photons per pulse per cm2 , albeit at higher fluences near the Gaussian beam center than at the edges of the beam spot. Multiplying by the cross-section and, for 100 nm of CNT length (i.e. 9000 C atoms), we expect about 6 to 7 excitons at this fluence. Again, this calculation uses the higher of the two published cross-sections by a factor of about 2. In addition, it is actually the value given (to only one significant digit) for the E 11 transition. Both the literature (Islam et al., 2004) and the absorption data shown in Fig. 4·2 indicate a lower E 22 cross-section by a factor of about 2. Thus, the calculated number of excitons is a upper limit estimate. The applicability of the published cross sections used in this calculation, however, are suspect in two important ways. First, the published cross sections are total cross sections and do not differentiate between contributions from absorption and from scattering. Only the former will contribute to the TA signal. Second, they derive from experiments using CW excitation which allows for multiple excitations per nanotube due to thermalization. In our experiment, because the pulse duration is shorter than the thermalization time scale, we believe there is only one exciton created per CNT regardless of the fluence. This is

45 discussed in more depth below.

4.4

Three Level Stretched Exponential Model

The data shown in Fig. 4·3 is taken with the laser resonant with the E 22 optical transition. We therefore model the data using a three level system, as shown in Fig. 4·1d, where the excited states correspond to the E 11 and E 22 exciton manifolds. The 38 fs pump pulse (1/e full width) is assumed to instantaneously create an excited state population at E 22 which then relaxes to the ground state. We allow for two distinct decay rates γ20 and γ21 from E 22 to the ground state and to E 11 , respectively. The E 11 state decays to the ground state at a rate γ1 . We expect from the literature that the upper excited state population relaxes extremely quickly. However this does not mean the TA signal recovers on the same time scale since γ21 feeds the lower excited state E 11 which is expected to be longer lived. TA using a probe pulse resonant with E 22 is sensitive not just to loss of upper state population but also to recovery of ground state population. In this work, it is the latter that leads to the long time scale behavior out to hundreds of picoseconds as shown in Fig. 4·2. This model ignores the excitonic structure at both the E 11 and E 22 manifolds. However, TA spectroscopy probes optically allowed transitions only, of which there is only one at each manifold. However, in previously published work by Zhu et al., (Zhu et al., 2007), using spectrally resolved TA, they find an initial rise in the 7.5 meV blue shifted TA signal on the 6 ps time scale which they ascribe to a transition from the A1 dark excition to the A1 +A2 two exciton manifold. We see similar spectral signatures in our data at the E 22 manifold. This interpretation implies that the dark exciton state, which is initially empty and therefore does not attentuate the probe, becomes populated on the 6 ps time scale by thermalization. This transient population then quickly decays by non-radiative relaxation. The higher lying bright exciton has also decayed by this time by primarily non-radiative means and thus does not continue to feed the lower dark state. Zhu et al. (Zhu et al., 2007) fit the long time scale behavior using a power-law model. They find a best fit exponent of 0.45 and ascribe this result to sub-diffusive trapping of

46 dark excitons, where it is sub-diffusive due to disorder. We find, however, the long time behavior can be marginally well fit by both power laws and simple exponentials but both of these begin to show systematic divergence from the data above the ∼150 ps time scale. We therefore extended our measurements to longer times in order to investigate this behavior. We find the data is best fit by using stretched exponentials. Rather than the simple linear dependence of the rate on the population as is the case in simple exponential decay models, the decay coefficient has time dependence governed by the stretch exponential, β. Specifically, the rate equations in the mono-exponential three level case look like:

n˙ 2 = −γ20 n2 − γ21 n2 = −γ2 n2

( γ2 ≡ γ20 + γ21 )

(4.1a)

n˙ 1 = +γ21 n2 − γ1 n1

(4.1b)

n˙ 0 = +γ20 n2 + γ1 n1

(4.1c)

where it is the ground state population/recovery, i.e. 4.1c, that dominates the TA signal in the long time scale limit. These rate equations are readily solved:

n2 (t) = ne−γ2 t γ21 (e−γ1 t − e−γ2 t ) γ2 − γ1 γ21 γ20 − γ1 −γ2 t e−γ1 t − n e n0 (t) = n − n γ2 − γ1 γ2 − γ1

n1 (t) = n

(4.2a) (4.2b) (4.2c)

where n is defined as the initial number of excitons generated in the upper excited state by the pump pulse. 4.2c, however, does not fit the entire data well, nor does a power law. We instead try stretched exponentials which have the form

47

β

n(t) = n(0)e−(γt)

(4.3)

which implies rate equations of the form

n˙ = −βγ β tβ−1 n.

(4.4)

4.1 can be adjusted accordingly:

β20 β20 −1 β21 β21 −1 t n2 − β21 γ21 t n2 n˙ 2 = −β20 γ20

(4.5a)

β21 β21 −1 t n2 − β1 γ1β1 tβ1 −1 n1 n˙ 1 = +β21 γ21

(4.5b)

β20 β20 −1 t n2 + β1 γ1β1 tβ1 −1 n1 n˙ 0 = +β20 γ20

(4.5c)

This system of coupled rate equations can be solved analytically when β1 = β20 = β21 ≡ β. The solutions are

β

β

β

n2 (t) = ne−(γ20 +γ21 )t n1 (t) = n

β γ21 β γ20

n0 (t) = n −

β γ21

(4.6a) β β

β

β

β

(e−γ1 t − e−(γ20 +γ21 )t )

+ − γ1β β γ21 n β β γ20 + γ21 −

−γ1β tβ

e β

γ1

−n

(4.6b)

β γ20 − γ1β β γ20

+

β γ21

β

β

β

e−(γ20 +γ21 )t β

− γ1

(4.6c)

Notice the parallel form of the solution to 4.2. The TA signal is dominated by the recovery of the ground state, 4.6c, after the first ∼1 ps due to the rapid relaxation of the upper excited state. The data is fit extremely well using this stretched exponential model, as shown in Fig. 4·3. Notice the fit to the long time scale data with no systematic divergence. We now discuss the physical interpretation of using stretched exponentials. Any monotonically decaying signal may be decomposed into a distribution of exponential decay rates by Laplace transform. Given a signal S(t), there is a weighted distribution of decay rates

48 given by ∞ S(t) =

H(γ)e−γt

(4.7)

0

where the distribution function H(γ) can be found by inverse Laplace transform. A subset β

of all possible S(t) are of the form e−(γt) , i.e. a stretched exponential. In this case, the associated distribution function H(γ) depends on β and does not have an analytic solution though it may be expressed as a convergent power series. (Berberan-Santos et al., 2005) The particular value of β can have physical meaning or can simply reflect heterogeneity in the sample.

4.5

Results

We find that the TA data is fit extremely well by our three level stretched exponential model at all fluences. We note that previously published works have failed to fit the low and high fluence regimes with one consistent model or have even had to, at a given fluence, divide the TA curve into short and long time data sets and fit these two regimes separately with different functional forms. Our model fits both the short and long time scale TA curve in both the low and high fluence regimes. We find β ∼0.5 which is consistent with diffusion-controlled contact quenching for nuetral carriers (Rice, 1985). We first discuss the implications for the long time scale data and then for the short time scale data. The initial, rapid ∼ 1 ps decay of the TA signal implies that the nanotubes quickly reach a state where there is at most one exciton remaining per CNT. Exactly how that condition is reached is still open to debate and is discussed below. The stretched exponential fit to the long term dynamics may reflect simple heterogeneity within the sample due to a distribution of nanotube lengths, changes in local environment, or defects. It could also indicate other diffusion limited processes such as defect trapping of dark excitons as postulated by Zhu et al. (Zhu et al., 2007) We find the long time scale experimental data is fit better using the stretched exponential model (and we emphasize that our sample is expected to be

49 at least as homogeneous as any used in previously published works) than the power law dependence used by other groups. The stretched exponential behavior is consistent with either heterogeneity or diffusional quenching. Indeed, the dynamics are likely to contain contributions from both. Photon echo spectroscopy, discussed in the next chapter, will help differentiate between these two contributions. In previously published work, the short time scale data shows an increasing contribution to the signal with increasing incident flux of a ∼1 ps rapid initial decay component. The picture described in the literature is that of multiple excitons on one nanotube diffusing along the length of the tube in a random walk type fashion until coming into contact with another exciton. These excitons quickly annihilate each other until only one exciton per CNT remains. Many previous published works, however, used the standard three dimensional form for the Auger process to fit that initial rapid decay, i.e. that the rate of population loss from the upper state is simply proportional to the square of the upper state population via a time independent constant. Valkunas et al. (Valkunas et al., 2006) discuss the proper functional form for one dimensional Auger processes, which differs for the 3D functional form, and show that the initial fast decay is not well fit by 1D Auger. They instead develop a coherent exciton model but, ostensibily due to a complicated, non-analytical form, that model is not used to directly fit the data but is rather just shown to have a similar shape. The point here is that a stretched exponential form is able to fit the data very well and is still consistent with excitonic diffusion whereas the 1D Auger functional form does not fit the data well. It is an area of future work to determine whether the stretched exponential form might also be consistent with the coherent multi-exciton excited states of Valkunas et al. The multi-excitonic states of Valkunas et al. imply long range interactions, rather than contact interactions, between excitons. Further evidence for long range excitation comes ogele et al. in the form of photon anti-bunching (g(2) ) from single carbon nanotubes by H¨ (Hogele et al., 2008) These results are indeed quite intriguing and seem to contradict the Auger-like picture of independent excitons diffusing in a random walk type fashion along the length of the nanotube until coming into contact with another exciton, defect, or nanotube

50 endcap. However, calculating the number of allowed k states that fall within the light cone, i.e. satisfy momentum conservation upon photon emission or absorption, for our average CNT length of 600 nm, we find there is only one state per exciton band, i.e there is only one bright exciton per nanotube that can be excited at one time. That is not to say that only one exciton may exist at a time on a given CNT since the initial exciton can thermalize up the band thereby vacating the zero momentum k state and allowing another subsequent photon absorption. The thermalization effect is clearly important in CW experiments or in time-resolved spectroscopies where the pulse width exceeds the thermalization time scale, estimated at 70 fs by Cognet et al. (Cognet et al., 2007) Specifically, they measured discreet jumps or steps, both positive and negative, in the photoluminescence decay from single CNTs as they became functionalized by various reactants in solution. These results support/imply the exciton random walk behavior referred to above. The diffusional range was measured to be 90 nm with a hopping period of 70 fs (identified with the dephasing timescale) and step size of 2 nm, corresponding nicely with the theoretical exciton size. These results imply that, for the pulse width less than 70 fs, as is this case in our work, only one optically excited photon can be created per CNT. This is critical in interpreting our data as it rules out exciton-exciton annihilation, at least within the same nanotube. This also might very naturally explain, of course, the photon anti-bunching results of H¨ ogele et al. (Hogele et al., 2008) though their pulse width was 130 fs. If indeed we are below the thermalization time scale, then our short time scale dynamics must be explained within a single exciton context. As with the long time scale data, the stretched exponential still implies a distribution of decay rates due to physical processes, perhaps diffusional in nature, that could be either intrinsic to all nanotubes within the sample or due simply to sample heterogeneities. However, a β value of 1/2 indicates a t1/2 time dependent rate coefficient and implies the presence of diffusion limited contact quenching. (Rice, 1985; Redner, 2001) That is, a β value of 1/2 is consistent with exciton random walk dynamics (Redner, 2001) where the exciton is diffusing along the nanotube length until contacting defects, protona-

51 tion sites, (O’Connell et al., 2002; Strano, 2003) or perhaps the end caps of the nanotube.1 This picture is in agreement with the stepwise quenching of photoluminescence observed by Cognet et al. (Cognet et al., 2007) Note that sample hetergeneities, e.g. variation in CNT lengths, defect densities, and surfactant wrappings, i.e. a distribution of linear trap densities, ρ, would result in a distribution of decay rates, γij , but would not alter the value of the stretch parameter as long as you remain in the short time limit. (Redner, 2001) We now look at the power dependence of the zero probe delay TA signal and show that the nanotubes act as a two level systems in the ultrafast limit, i.e. below the thermalization time scale. If our pulses are indeed short enough and there is only one allowed resonant state, then our power dependence data may be modeled as an ensemble of two level systems, one per CNT, where the pump is treated as a square pulse turned on at t=0 and off 38 fs (FWHM) later. The pump acts as a quasi-CW field which starts to build an initial upper state population. This population is then read out by the probe at a delay time significantly less than the lifetime, i.e. before the loss of population from the ecited state. The results, for a nominally zero delay time, are shown in Fig. 4·4. The data is initially modeled using a standard Einstein coefficient rate equation approach for a two level system with appropriate input values (Loudon, 2000) and turns out to be a sensitive measure of the exciton dipole length since it enters into the argument of the exponential term squared. In fact, all parameters in the model, other than the dipole moment, are known fairly well or can be calculated. The modeling, however, is complicated by the high density of CNTs in the sample (higher than the photon density at low fluences) and possibly due to coherence effects since the pulse width is less than the dephasing time scale (by design). The spontaneous decay rate given by Loudon (Loudon, 2000) should be modified in one dimension; the form is given by Capaz et al. (Capaz et al., 2006) The derivation, however, assumes a continuous density of states within the light cone which is not valid for nanotubes of 600 nm average length. Therefore, the CNTs are more accurately modeled by the standard two level 1

Technically, in the short time limit defined by t < 1/(Dρ2 ) where D is the diffusion coefficient and ρ is the linear trap density, (Redner, 2001) approximately given by t < 670 ps using D∼0.4cm2 /s(Cognet et al., 2007) and ρ is at most 1 trap per 10 nm.

52 approach since there is only one optically accessible resonant state. In fact, the spontaneous emission rate predicted by the 1D model of Capaz et al. (Capaz et al., 2006) is two orders of magnitude larger than that for a simple two level system and completely fails to fit the power dependence data. Dipole moments using the two level model and consistent with the data fall in the 0.04 nm rage, and are thus much smaller than previously published values of 0.25 to 0.5 nm, though these are for the E 11 transition. (Zhao and Mazumdar, 2004; Spataru et al., 2005) Further, the Einstein model is a monotonically increasing function and thus can not fit the turn over in the power dependence data.

0.4 ( ǻT / T )max

10-1 10-2 10-3 10-4

0 0

2x1016

4x1016

1012 1013 1014 1015 1016 1017

Fluence ( Photons / pulse / cm2 ) Figure 4·4: Power dependence (over four orders of magnitude) of the τ =0 transient absorption signal in linear and log scales. The optical Bloch equations fit is shown in red. We therefore turn to the optical Bloch equations (OBEs) which include coherent effects between the upper and lower states. Loudon(Loudon, 2000) specifies two limits in which the previous rate equation approach should be correct: (1) when the excitation band width exceeds the atomic transition line width, or (2) when the elastic dephasing greatly exceeds the radiative line width. Given that the electronic decay in CNTs is dominated by nonradiative relaxation and given the measured absorption line width of ∼90 meV, typical in CNTs, greatly exceeds the spontaneous line width, it would seem, at first, that limit (2) would apply and the use of OBEs unnecessary. However, our pulse width is so short that

53 we are near the dephasing time scale implied by the absorption line width and so neither (1) or (2) may apply. In addition, the above comparison with line widths assumes that the room temperature absorption line width is largely homogeneous. It has been shown that the room temperature linewidth is inhomogeneously broadened to a large degree(H. Htoon and Klimov, 2004) and that the homogeneous line width, i.e. that of the individual two level systems, is much less making the dephasing time scale even longer relative to the pulse width. We find below that these two time scales are comparable and thus a posteriori justify the use of OBEs. Further, the Rabi period, i.e. the time scale on which the population is driven into the upper state, is longer than the pulse width until only the highest fluences are reached at which point these two time scales become comparable. And again, if we are to assume the turn over in the TA data at high fluence is related to two level behavior and not associated with, for instance, heating of the solvent (which is expected to be small), then we must use an OBE approach since rate equations could never be expected to fit the turn over in the high fluence behavior. We solve the OBEs in the limit of large dephasing and negligible population loss (on the 10’s of fs time scale). We find (using Mathematica ) 1

e− 2 γt 1 1 1 (γ sinh( λt) + λ cosh( λt))) (4.8) n2 (t) = (1 − 2 λ 2 2  where γ is the dephasing, λ= γ 2 − 4ν 2 , and ν is the bare Rabi frequency. The dipole moment appears via the Rabi frequency and is one fit parameter in the model. γ is taken as a fraction of the measured absorption line width and thus is a indicator of the amount of the total line width that is due to inhomogeneity. Finally, an overall scaling factor is included which is related to the absorption cross section since the TA signal is sensitive to population and not scattering. The detuning from resonance, Δ, can also be included in the model by √ use of the modified Rabi frequency Ω= Δ2 + ν 2 in place of the bare Rabi frequency but this correction is found to be negligible since the pump and probe are on resonance and the pulse width nearly matches the transition line width. The dot product between the dipole moment and the applied electric field that appears in the Rabi frequency is critically important since

54 the nanotubes within the sample are randomly oriented. The antenna effect(Reich et al., 2004) in nanotubes is also important since CNTs aligned with the probe polarization will contribute to the signal most. These nanotubes, being orthogonal to the pump, have the smallest Rabi frequency for a given dipole moment and applied field strength and thus result in the slowest build up of population into the upper state. Nanotubes oriented parallel to the pump will have the fastest population rise times but contribute less by virtue of the antenna effect. We therefore average Eq. 4.8 over all orientations within the sample. γ is also allowed to take on a range of values in order to account for inhomogeneity from one CNT to another, as discussed above. The best fit results of Eq. 4.8 are shown in Fig. 4·4. Eq. 4.8 clearly does an excellent job of modeling the turn over in the high fluence regime representing the weighted contributions from Rabi oscillations of different frequencies. Specifically, we find the dipole moment to be just over 0.1 nm at the E 22 transition. Given the factor of about two smaller cross section at E 22 than at E 11 , (Islam et al., 2004) this is in excellent agreement with the 0.25 nm dipole length predicted by Spataru et al.(Spataru et al., 2004) for the comparably sized (10,0) CNT at E 11 . We find a best fit γ value of about 40% of the measured absorption line width indicating the presence of a substantial amount of inhomogeneous broadening. This value a posteriori supports the use of OBEs since the homogeneous line width is about equal to the excitation band width, i.e. γ defines a dephasing time scale of 56 fs, slightly longer than the pump pulse width. Note that this value is also in excellent agreement with the 70 fs dephasing time scale measured by Cognet et al.(Cognet et al., 2007) The inhomogeneity in the linewidth likely arises from local variations in dielectric environment which alters the optical transition energies. (Walsh et al., 2007; Lefebvre et al., 2004b) Finally, we can estimate both the total and absorption cross sections. Given a CNT number density of 1.3 to 1.9x1014 per mL and a measured optical density of 0.3 in the linear regime for a 2 mm path length and 19 μm focussed beam spot size, we estimate the total cross section at 2.2x10−14 cm2 for a 600 nm long (6,5) CNT. This is about a factor of three to four smaller than previously reported(Carlson et al., 2007) even after correcting

55 for average CNT length and the factor of 2 for E 22 rather than E 11 . Since the TA signal is sensitive to excited state population, it is a measure of the absorption cross section rather than the total cross section. From the overall scaling prefactor of the τ =0 TA signal, which scales with twice the ratio of the absorption to the total cross section, we find the total cross section on resonance is dominated by absorption with relatively little scattering.

4.6

Conclusion

In summary, we have demonstrated stretched exponential relaxation dynamics in a highly homogeneous sample of (6,5) CNTs, well fit by a single model in both the high and low fluence regimes. We find that power law, Auger, and simple exponential models may appear to fit the TA data fairly well in various separate regimes but systematically deviate from the long time scale data or fail to fit the whole data set in a coherent way. The stretched exponential behavior with β ∼0.5 indicates the presence of random walk dynamics. Quantum confinement in the axial direction leads to a single resonant optically accessible exciton per CNT which dictates that, given a pulse width of order or less than the thermalization time scale, only one exciton can be excited per nanotube. The power dependence of the τ ∼ 0 TA signal is well modeled by an OBE approach for a two level system in the large dephasing but short time scale limit providing further evidence of two level behavior in CNTs. The dipole length for the E 22 transition is found to be ∼0.1 nm.

56

Chapter 5

Photon Echo Spectroscopy 5.1

Introduction

Since carbon nanotubes are so sensitive to their environment, it is likely that the measured line width from an ensemble of many CNTs would have a large inhomogenous contribution. In addition to hetergeneities in the local environment, sample preparation techniques invariably lead to a distribution in nanotube lengths and numbers of defects. One way to deal with this problem would be to perform measurements on individual carbon nanotubes. However, in addition to this being a more difficult experiment, single nanotube experiments do not yield any information about how environment, length distribution, etc. change the measured quantity of interest unless one were to repeat the single CNT experiment on the same nanotube in many different environments which is often not practical and is certainly a non-trivial task. Thus, the degree of inhomogeneity that is present in the electronic line width is another outstanding question within the nanotube community. In this chapter, we discuss future work using photon echo (PE) spectroscopy which has the ability differentiate between homogenous and inhomogenous contributions to the CNT emission line width from an ensemble. Specifically, PE can be used to measure the homogenous linewidth free from heterogenous contributions by “rewinding” the optical coherence that leads to the photon echo. In this way, photon echo is the electric dipole analogue to magnetic resonance imaging, MRI. We hope to use PE in order to discern whether the stretched exponential behavior measured using TA spectroscopy and discussed in the previous section indicates simple heterogeneity within the sample or reflects an intrinsic spread in decay rates due to some relaxation physics/dynamics such as diffusional contact quenching.

57

5.2

Photon Echo Theory

An excellent discussion of free induction decay, FID, and photon echo spectroscopy can be found in Allen and Eberly (Allen and Eberly, 1987). For more in depth discussion, there are a number of good references in the literature. (Ohta et al., 2001; Kirkwood and Albrecht, 2000; Cho et al., 1996; Jimenez et al., 1997; Joo et al., 1996) The best way to understand photon echos is to start from the beginning with an outline of damped driven harmonic oscillator theory, in the non-relativistic limit, that serves as the starting point for the classical treatment of light-matter interactions. The harmonic oscillator with natural frequency ω0 is assumed to be driven by an external electric field of frequency ω. The damping term in the governing differential equation arises from the radiative energy emitted by the oscillating dipole and can be characterized (in the limit of slow radiative loss rate relative to the dipole oscillation frequency) by a simple exponential decay rate, 1/T. This exponential decay in time results in a Lorentzian response, or line shape, in frequency with width 1/T. All dipoles in a given system decay exponentially on a time scale characterized by T, the so-called homogeneous linewidth. In the literature, the lifetime associated with spontaneous emission due to coupling of the oscillator to the electromagnetic vacuum modes is often referred to as T1 , or the longitudinal lifetime in MRI literature. This line width is phenomenolically broadened to include those processes which cause all emitters to dephase equally, resulting in the homogeneous line width referred to as T above. Naming conventions are not entirely uniform in the literature. In Allen and Eberly, the total homogeneous line 



width is referred to as 1/T2 and is given by 1/T2 = 1/T1 + 1/T2 where T2 is the pure dephasing rate. Those processes which cause one emitter to dephase at a rate differently 

from another leads to the inhomogeneous line width 1/T∗2 . T2 and T∗2 are referred to as transverse lifetimes in MRI literature. The total line width is then given by 1/T2 + 1/T∗2 . The primary contribution to the inhomogeneous line width 1/T∗2 arises from a distribution of slightly different emitter frequencies, as shown in Fig. 5·1a, resulting from, for example, locally different strains on the dipole. Clearly, a distribution of frequencies will cause the

58 individual dipoles to dephase with respect to one another. Indeed, the measured line width can be dominated by this inhomogeneous dephasing. The macroscopic polarization, P(t), which results from the interference of the emission of all dipoles in the sample, can decay on a time scale much faster than that of the individual dipole moments, d(t). This phenomenon, where the macroscopic emission, governed by P(t), decays much faster than the intrinsic or homogeneous rate, governed by d(t), is known as free induction decay. This also means that there exists the potential to “unwind” the relative phases of the emitters before they have 

each decayed individually on the time scale given by T2 . This is because the accumulation of phases of the individual dipoles relative to each other is deterministic. The relative dephasing on the T∗2 time scale arises from the slight frequency differences between emitters so that the phase ΔΦ accumumated over a time t between two emitters differing in emission frequency by an amount Δω is simply ΔΦ = Δω t. Obviously, these phases will not rephase on their own but there is, as it turns out, a way to reverse the dephasing process by suitable application of external fields. The emitters can then “unwind” or re-phase, resulting in a signal at some later time, although weaker in strength as governed by the homogeneous decay of of the individual dipole moments, d(t). The longer the time delay before rewinding the phases, the smaller the echo. This is the basic concept behind photon echos. This phase unwinding concept is shown schematically in Fig. 5·1. Figs. 5·1b and c show the decay of 

P(t) and d(t) on the T∗2 and T2 time scales, respectively, emphasizing how the individual dipoles may continue to “ring” long after the macroscopic signal has decayed away. A more detailed understanding of photon echos requires a brief review of the optical Bloch equations. The interaction of light with a single two level system is characterized by the optical Bloch equations which are a set of three coupled differential equations. The deriviation of these equations is very similar to the derivation for the damped, driven harmonic oscillator and the details can be found in various texts. (Loudon, 2000; Allen and Eberly, 1987) The three variables are w, u, and v where w represents the population in the upper (w = +1) and lower (w = -1) states and u and v represent the in phase and in quadrature components of the dipole moment with the driving field. The time evolution

59

a)

d) # of dipoles

ʌ pulse Heterogeneous advancement of dipole phases

Frequency b) P(t)

u

t

T2*

v

d(t)

c)

T2'

t

Dipoles initially in ground state

ʌ/2 pulse w

Figure 5·1: Photon echo phase unwinding. (a) Distribution of dipole frequencies. (b) Decay of the macroscopic polarization P(t) on the characteristic time scale T∗2 . (c) Decay of the individual dipole moments d(t) on the  characteristic time scale T2 . (d) All dipoles are initially in the ground state. A π/2 pulse is applied and the individual dipoles begin to deterministically dephase with respect to each other. At some later time, a π pulse is applied, rotating all phase vectors around the u axis and the individual dipoles begin to rephase. A photon echo is observed when the individual dipoles rephase along the u axis. The echo strength is given by the extent to which the individual dipoles have decayed during this time, governed by d(t).

60 of the state of a two level system can then be fully described by a vector of unit length on the so-called Bloch sphere, depicted in Fig. 5·1d, with axes w, u, and v. A sample with many resonant dipoles is then represented by many vectors on the sphere, one per dipole. Without heterogeneity, all vectors would evolve in the same manner. However, in the presence of heterogeneity, the vectors will diverge. Specifically, Fig. 5·1d shows all dipoles initially in the ground state. Following excitation by an appropriately tailored π/2 pulse, the individual dipole vectors begin to dephase relative to each other on the time scale given by T∗2 in a deterministic fashion as explained above. By application of a π pulse, which rotates all vectors by 180 degrees around the u axis, the vectors will begin to come back together and ultimately rephase at the u axis. This results in the PE signal. The time scale over which you can wait to apply the π pulse and still hope to get an appreciable PE signal back is determined by the homogeneous decay of the individual dipoles. Photon echos are often measured in a three beam configuration that is slightly more complicated than the process we have just described but the basic concept is the same. The WMEL diagrams for PE and FID are shown in Figs. 5·2a and b (complex conjugate diagrams not shown). The positive or negative Fourier components of the incident fields dictated by the rotating wave approximation (RWA) are also shown. These WMEL diagrams look similar to that for absorption followed by re-emission but the vector nature of the photon fields is important. A full derivation of the PE signal using the third order perturbative density matrix expansion is not given here but follows closely that given for Raman spectroscopy in Appendix A. In Fig. 5·2c, the three incident beams (not to scale), each of which provides one of the fields depicted in the WMEL diagram, are shown with the detector placed in the “phase matched” direction. Notice that the detector is not placed directly in any of the three beam paths meaning that, although PE signal strengths may be low, signal to noise ratios can be quite good as long as the scattering from the sample is not too great. The relevant phase term that arises in the derivation of the PE signal is shown in Fig. 5·2d which, by appropriate selection of delay times τ (the coherence time) and T (the population time), can become equal to one at a time later than t=0. That is, it allows

61

Photon Echo

a)

Free Induction Decay

b)

e

IJ

IJ

T

+ȍ -ȍ -k1 +k2

-ȍ +k3

ELO

Detectorin phasematched direction

c)

e

-ȍ +k1

T

K1

+ȍ -ȍ -k2 +k3

ELO

K2 K3

i t1

t2

t3

t

t1

t2

t3

t

i Ȧei [ ( t1 - t2 ) – ( t3 – t ) ]

phasecangotozero Phase“unwinding”alaMRI

PESignal

Signal Į e

True/static inhomogeneity

f)

e)

d)

PeakShift(x2)

CoherenceTimeIJ (fs) T=0

PeakShift

i

Sample

PopulationTimeͲ T(fs)

Figure 5·2: Photon echo spectroscopy. (a) WMEL diagram of PE spectroscopy. The time delays τ (coherence time) and T (population time) are shown as well as the positive or negative fourier components dictated by the rotating wave approximation. (b) WMEL diagram of free induction decay (FID) spectroscopy. (c) Schematic of three beam PE spectroscopy with the detector placed in the phase matched direction. (d) Phase term appearing in PE spectroscopy. (e) Cartoon of photon echos as a function of coherence time τ at T=0 population time. (f) Cartoon of the peak shift as a function of population time.

62 for the possibility of rephasing of the dipoles in the phase matched direction resulting in an echo in the manner explained above. Note that the corresponding phase term in FID spectroscopy can not similarly be set to zero. In general, PE is a three beam experiment where each beam provides one of the interacting fields shown in the WMEL diagrams and together act in a manner similar to the π/2 and π pulses described above. Specifically, the first beam acts to set up a coherence during which time (the coherence time τ ) the dipoles begin to dephase, i.e. the off diagonal elements of the density matrix evolve in time. If T = 0, however, then a population is created for a time T, the evolution of which is characterized by T1 (which is usually assumed to be long compared to T∗2 ) and the dipoles do not dephase homogeneously. The third beam then, in essence, completes the π pulse and the dipoles subsequently start to rephase. If the second and third fields were to act simultaneously, i.e. T = 0, then these two beams together act as a π pulse and the dipoles begin to rephase as described above. This situation corresponds to a two beam form of PE where the same laser provides both the second and third field interactions. In the two beam version of PE, the population time T can not be varied from zero but is otherwise identical to three beam PE. In all versions of PE, the central idea remains the same, that the time dependence of the PE amplitude depends only on the homogeneous decay of the individual dipoles and, thus, PE can be used to differentiate between homogeneous and inhomogeneous contributions the line width. In our time-resolved work with carbon nanotubes, we hope to use PE to differentiate between the two possible interpretations of the data implied by the stretched exponential model as discussed previously. In practice, rather than looking at the time dependence of the PE amplitude to quantify the homogeneous life time, a parameter known as the “peak shift” is more often used to not only quantify the amount of inhomogeneity but also to further differentiate between dynamic and static contributions to the inhomogeneity. The distribution of transition frequencies shown in Fig. 5·1a could arise from a distribution of CNT lengths, local strains due to different surfactant wrappings, or defect concentrations. The resultant spread in transition

63 frequencies associated with these causes would be expected to be static. On the other hand, doppler shifts, for example, could result in a distribution of frequencies that could vary on the time scale of these measurements. Since the PE signal depends on the individual dipoles retaining a memory of their own transition frequency in order to rephase properly, dynamic contributions to the heterogeneity will prevent the extent to which the dipoles can rephase as time progresses. The ablilty of the system to rephase is represented by the so-called peak shift. First notice that the roles that the beams k1 and k2 play in the WMEL diagram can be reversed. This is equivalent to a negative τ and leads to an echo in a separate phase matched direction. Theoretically, the two signals are identical and only one need be measured. In practice, the point τ =0 is not easy to define unless both PE signals are measured, as shown in Fig. 5·2e. The separation between the PE peaks is referred to as the peak shift (x2). The peak shift may be plotted as a function of population time, as shown in Fig. 5·2f, and shows how the system’s ability to rephase decreases in time due to dynamic contributions to the frequency distribution. In the long population time limit, the peak shift in many systems goes to zero indicating that there is no static distribution of transition frequencies. In our case, a non-zero, long population time peak shift would indicate the presence of a static distribution of frequencies and naturally explain the stretched exponential form of the TA data. On the other hand, a zero peak shift would imply that the stretched exponential behavior is intrinsic. Notice that the initial (T=0) peak shift value is also a measure of the electron-phonon coupling strength since a strong coupling would rapidly dephase the individual dipoles in a non-deterministic manner and thus reduce its ability to produce an echo, i.e. produce a small initial peak shift.

5.3

Conclusion

In this section we reviewed how photon echo spectroscopy can be used to measure the inhomogeneous contribution to the line width. Specifically, PE can be used to determine to what extent the measured spread in relaxation rates in carbon nanotubes samples arise from sample heterogeneity and to what extent this distribution is intrinsic.

64

Chapter 6

Forward Scattered Rayleigh Experiment 6.1

Introduction

In this chapter, we present our efforts to measure the intrinsic nanotube electronic line width and oscillator strength for the optically accessible E 11 transition using forward scattered Rayleigh spectroscopy. These very fundamental quantities, which characterizethe intrinsic lifetime and strength of a given optical transition, respectively, are still not unknown. This is due in large part to heterogeneity in the sample and in the local environment, as discussed in the previous chapter. In addition, the intrinsic radiative lifetime is difficult to measure due to rapid dephasing which dominates the linewidth. Forward scattered Rayleigh spectroscopy, also known as differential transmission, or DT, relies upon interference between the forwarded scattered light and the incident laser and is thus inherently insensitive to incoherent contributions to the line width. Thus, DT is an ideal spectroscopy for such a measurement. It is complicated, however, by the requirement for very low excitation powers, discussed below, and thus low signal to noise levels. By cooling to liquid helium temperatures and maintaining sufficiently low powers, we remove thermalization and power broadening contribtions to the line width. By observing the foward scattered signal as a diode laser is stepped through resonance, our spectral resolution of that line width is no longer limited by typical spectrometer resolutions of ∼1 cm−1 but rather only by the step size of the tunable laser. We begin by reviewing foward scattered Rayleigh theory and then discuss the experimental details. As of this writing, our efforts continue to grow long, suspended nanotubes with optical transitions which are resonant with our tunable diode lasers.

65

6.2

Forward Scattered Rayleigh Theory

In this section, we review the derivation of the forward scattered Rayleigh signal in the weak scattering limit. It is shown that the product of the modulation width and the modulation depth of the forward transmitted signal is a direct measure of the oscillator strength, or dipole moment, of the transition with which the laser is resonant. In the sufficiently low power limit and at low temperature, the measured line width (the resolution of which is limited only by the laser wavelength step size) is free from thermalization effects, radiative broadening, and, since the experiment is on a single nanotube, inhomogeneous broadening, as well. Thus it is possible to measure the intrinsic spontaneous lifetime of the optical transition. We now briefly review the salient points of the semi-classical derivation given by Karrai and Warburton. (Karrai and Warburton, 2003) The critical concept comes from the optical theorem which relates the forward scattered amplitude to the total cross section. By measuring the elastic signal in the forward direction, we have essentially measured the total cross section of the scatterer. Conceptually this is actually not hard to understand: the scattering event removes photons at the incident wavelength from the forward direction and therefore the forward scattered Rayleigh signal is a measure of the total cross section. Recall that the stimulated emission portion of the signal in the forward direction is indistinguishable from the incident field itself and so the modulation of the signal in the forward direction will relate to the spontaneous lifetime of the excited state (we are assuming the lower state is the ground state.) The first step in calculating the electric field at a detector in the forward direction is to

i , plus the scattered

T , is given by the sum of the incident filed, E assume the total field, E

s: field, E

i + E

s.

T = E E

(6.1)

i | and that we are in

s |  E Here we assume the perturbative treatment is valid with |E

66 the linear regime. The nano-scale emitter is also assumed to sit at the center of the focused

Tunable Polarized CW Laser

Lock-In Amplifier

ES

Ge Detector

(d)

(a)

CNT

0

T=1-

Į0

Į0 Ȗ2 į 2 + Ȗ2 Wavelength

(e) kBT

Light Cone

2Ȗ

0.99

ʄȦ

ǻ 20 ȝm

1.00

Normalized Transmission

Ei

(c)

Polarization Modulation

Normalized Transmission

(b)

Ȝ

|Es|2

Ei Es

KCM Wavelength

Ȝ

Figure 6·1: Forward scattered Rayleigh. (a) Single carbon nanotube suspended across a trench and illuminated by super-continuum light. (b) Cartoon of the experiment. (c) Schematic of the forward Rayleigh scattered signal. (d) Cartoon of the thermal broadening effect. (e) Contribution of

s |2 and interference terms to the forward Rayleigh signal the |E

i may be treated as a laser beam spot, of order the wavelength of the laser, and thus E quasi-plane wave. The situation is depicted in Figs. 6·1a and b. The scattered field in the forward direction is then given by

s = 1 e r˙ (t − z/c) E A 20 c

(6.2)

where A is the area of the flat phase front over the focussed spot, c is the speed of light in vacuum, e is the free electron charge, 0 is the free space permittivity, r is the dipole oscillation amplitude, and z is the distance from the emitter on axis in the forward direction. The two level system response is modeled as a single damped, driven harmonic oscillator yielding the usual Lorentzian frequency dependence of the dipole oscillation amplitude

67

r = −

i f eE 2 m0 ω0 − ω 2 − iωΓ

(6.3)

where m0 is the free electron mass, f is the oscillator strength of the transition, ω0 is the (angular) transition frequency, ω is the (angular) frequency of the incident field, and Γ is

i and thus the time the total dephasing rate. The time dependence comes from the eiωt in E derivative in 6.3 is trivial. 6.2 then becomes, in the (realistic) limit of a sharp resonance where Γ  ω0 ,

i

s = α0 −iγ E E 2 δ + iγ

(6.4)

where γ ≡ Γ/2, δ ≡ ω − ω0 and is referred to as the detuning, and

α0 ≡

1 e2 f . A 0 cm0 Γ

(6.5)

The transmission coefficient, T, is given by

T =|

Ei + Es 2 | Ei

(6.6)

which, using 6.4, becomes

T = (1 −

α0 γδ α0 γ 2 )2 + ( )2 . 2 δ2 + γ 2 2 δ2 + γ 2

(6.7)

In the weak scattering limit, i.e. α0 1, we see this expression simplifies to

T = 1 − α0

γ2 δ2 + γ 2

(6.8)

where the second term arises from the beating of the scattered field with the incident field and α0 is now seen in Fig. 6·1c to be the contrast in the transmission curve at zero detuning, i.e. the maximum contrast. Notice that, from 6.5, that the product of the modulation depth and the width of the transmission curve is

68

α0 Γ =

1 e2 f , A 0 cm0

(6.9)

a direct measure of the oscillator strength f since all other parameters are known quantities! Furthermore, since

f=

2m0 ω0 μie 2 ( ) ,  e

(6.10)

the product α0 Γ is also a measure of the dipole moment μie between the ground state and the excited state exciton. It is the primary goal of this experiment to directly measure the product α0 Γ and thus quantify f and μie . We now calculate the expected maximum contrast given at minimum broadening, i.e. at Γ=Γsp , the spontaneous emission rate. From Loudon (Loudon, 2000),

Γsp =

ω03 μ2ie . 3π0 c3

(6.11)

Using 6.9 and 6.10, the maximum contrast is then given by

αsp =

2πc 3 λ2 ;λ ≡ 2π A ω0

(6.12)

This is an interesting result; the maximum contrast in the absence of other dephasing processes depends only on λ and how tightly the incident laser is focused. Since 0.6λ 2 ) , NA

(6.13)

3 NA 2 ( ) = 1.33(N A)2 . 2π 0.6

(6.14)

A∼( we find

αsp ∼

For a 0.4 NA objective, αsp ∼21%, which should be experimentally accessible, especially using the lock-in techniques discussed in the following chapter. Clearly higher NA improves contrast; indeed, if one were able to reduce the excitation spot to the size of the cross section

69 of the scatterer, the transmission in theory would go to zero. Before proceeding further, however, we now need to re-evaluate one of our orginal assumptions, namely, that the scattered field strength would be small enough such that we could ignore the contribution to the signal from the beating of the scattered field with itself,

s |2 = Is . Having just calculated the magnitude of interference of the scattered field i.e. |E

i | and

s | ∼ 0.2|E with the incident field and found a modulation of ∼20%, this implies |E thus Is ∼ 0.04Ii or 1/5th the magnitude of interference term. This contribution to the signal will be positive and thus lessen the observed contrast, as depicted schematically in Fig. 6·1e, and should not be ignored. The real challenge, however, lies in saturation. Incident power beyond the linear regime, in addition to increasing the noise background, leads to power broadening of the transition. Γ increases from its minumum value of 2γsp to 1 eEi μie 2 + ν 2 )1/2 ; ν ≡ Γ = 2(γsp 2 

(6.15)

where ν is the Rabi frequency. (Loudon, 2000) The attendent transmission line shape is then given by (Karrai and Warburton, 2003; Ayache, 2006)

α(δ) = αsp

2 γsp 2 (1 + δ2 + γsp

ν2 2 ) 2γsp

(6.16)

Notice how increasing incident power, via the Rabi frequency ν, leads to both power broadening and decreased contrast. We now recalculate the product α0 Γ in the presence of power broadening. The transmission contrast at zero detuning is now given by

α0 =

αsp 1+

ν2 2 2γsp

(6.17)

and the FWHM by

Γ = Γsp (1 +

ν 2 1/2 ) 2 2γsp

(6.18)

70 and therefore the product

αΓ = αsp Γsp

1 (1 +

ν 2 1/2 2 ) 2γsp

(6.19)

no longer directly gives a measure of the oscillator strength f in αsp but must be corrected by the power broadening factor which itself contains αsp . One has a few options: (1) work in power regime where power broadening is known to be negligible, (2) estimate the correction factor, or (3) measure α(δ) at a series of powers and fit the α(0) curve to extract f . Coupling to a bath or intrinsic scattering sources can also lead to pure dephasing on the characteristic time scale T∗2 which, by definition, will remove phase coherence between the scattered and incident field and thus broaden the transition and reduce the contrast. However, since (1) we hope to probe the CNT at or near mid-trench away from any potential bath, (2) measurements will be taken in vacuum and at cryogenic temperatures, (3) nanotubes are known to be largely free from defects, and (4) our nanotubes are not micelle enscapsulated, any such T∗2 contributions are expected to be small. We should also briefly discuss the validity of modeling our complicated nanotube electronic structure, discussed in Chapter 2, as a two level system. The salient point here is that the dip in the forward scattered Rayleigh signal on resonance is necessarily a coherent effect requiring a dipole allowed transition between the two levels. Not only does the density of states diverge sharply defining a narrow energy range for each exciton, but there is also only one bright exciton at zone center that is optically accessible. The next higher exciton within the same manifold, i.e. associated with the same E ii , is at least a few hundred meV higher in energy and has a much smaller oscillator strength. (Ando, 1997; Vamivakas et al., 2006a) After initial optical excitation, population up the excitonic band due to thermalization, as shown in Fig. 6·1d, removes excitons from within the light cone; those states are dark due to momentum conservation and thus this contribution to the electronic line width, present in PL or Raman spectroscopy, should be absent in this experiment.

71

6.3

The Forward Scattered Rayleigh Microscope

As stated in previous section, we intend to measure the product α0 Γ of the forward scattered Rayleigh differential transmission curve. We will use lock-in techniques to extract the small signal from a large background. In this section, we discuss the experimental details and difficulties. The experimental setup is depicted in Fig. 6·2. PolaMight Polarization Maintaining Fiber PolaRite

PolaRite Half-Wave Plate Camera

Tunable Diode Laser

Fiber Bundle Function Generator

Lock-In Amplifier

Objective

Reference

Cryostat Current Amplifier

Signal

Linear Polarizer ( Removable )

Sample Detector

Figure 6·2: Forward scattered Rayleigh experimental setup. In order to measure the forward scattered Rayleigh signal, CNTs are grown across trenches 20 to 30 μm wide that have been etched completely through in a Si wafer using RIE. A thermally grown layer of SiO2 is used to prevent catalyst migration and clumping at high growth temperatures. Small (1-3 nm) Fe particles are deposited directly on the surface by use of ferritin cages in solution. The ferritin cages are removed in situ upon ramp up to growth temperatures in the 850 to 910 degrees C range. Methane is flown as the growth gas for, typically, 10 to 30 minutes. The resulting CNT diameters tended to be in the 2 to 2.5 nm range, as determined primarily by G+ - G− splitting in the Raman spectrum. The lowest optical transition energies for this diamter range are below the tunable range of the diode laser centered at 1300 nm that we hope to use in this experiment. Efforts therefore continue to adjust growth parameters that result in smaller CNT diameters with the appropriate optical transition energy. Having said that, it might be possible to perform

72 the experiment at the E 22 transition of these larger CNTs, though the resonance is expected to be broader since it lies within the continuum of the lower lying E 11 transition. Nanotubes with E 11 transitions potentially within the tunable range of our diode laser are initially identified by Raman spectroscopy on a separate spectrometer (described previously in Chapter 3). Verification can made simultaneously by PL spectroscopy using a custom added InGaAs camera which unabmiguously identifies the E 11 transition energy. However, the transition energy can blue shift on the order of ∼10 eV upon cooling from room temperature to 4 K (Capaz et al., 2005; Cronin et al., 2006), depending on chirality, an effect which should be considered. The nanotube is also illuminated from the side of the sample, via a second long working distance objective, with super-continuum light generated using a photonic crystal fiber and a femtosecond pulsed laser operating at 800 nm, as shown in Fig. 7·1c. This capability is available on both the Raman spectrometer and on the forward scattered Rayleigh microscope described in this section. The details of the supercontinuum generation can be found elsewhere1 as well as details on using super-continuum Rayleigh scattering from CNTs to identify their optical transitions (Wang et al., 2006); however, the central idea is that by having incident powers on the order of 1 mW across a spectral range from ∼400 nm to 1 μm, the illuminated nanotube will have an optical transition somewhere in that range. Single line, CW illumination may be off-resonance and takes significant time and effort to tune to each wavelength to find suspended nanotubes, super-continuum illumination can be used to quickly find suspended nanotubes since the Rayleigh scattered light, collected by the objective of the microscope at normal incidence to the sample, can be seen using only the inspection camera on the microscope. In addition to simply finding nanotubes, the nanotube can be literally imaged (albeit diffraction limited) using the Rayleigh scattered light and thus the growth direction with respect to the trench directly observed. Forking of the “roots” at the side walls of the trenches is also clearly visible indicating the presence of a small rope of CNTs. These images can be saved which greatly simplifies CNTs location later when the CNT is not under super-continuum 1

See Newport AppNote at http://www.newport.com/file store/Optics and Mechanics/AppsNote28.pdf

73 illumination. We should clarify here that the super-continuum Rayleigh scattering used here to locate suspended CNTs is distinct from the single line Rayleigh scattering of the main experiment discussed in this chapter. Also, the use of the term “Rayleigh scattering” is often used to refer to elastic scattering though the proper use refers to scattering from an object much smaller than the wavelength of the incident light. The sample is placed directly onto the detector in a helium cryostat capable of reaching 4 K. The laser is focused onto the CNT through a coverslip using a cover slip corrected, 0.42 NA long working distance objective. A function generator is used to drive the polarization modulation of the incident laser exploiting the strong polarization dependence of the nanotube as described in Chapter 2. The excitation, when oriented parallel to the nanotube, will be scattered strongly due to the antenna effect since it is resonant with the E 11 (or possibly E 22 ) transition. Conversely, when the excitation is polarized perpendicular to the nanotube, the excitation is weakly scattered. The scattered field at the detector beneath the suspended nanotube will be modulated at the frequency of the polarization modulation and thus the beating term at the detector will also be modulated at the frequency of the polarization modulation. The output from the detector and the output of the function generatorare sent to the lock-in amplifier where they are beat together resulting in a DC signal. The DC signal strength is recorded at each incident wavelength as the diode laser is tuned through resonance, tracing out α(ω). Details of the polarization modulation theory and setup are provided elsewhere. (Ayache, 2006) We emphasize that the spectral resolution of α(ω) is limited only by the minimum laser step size, ∼3 pm, which translates to a resolution of ∼2 μm eV or ∼0.02 cm−1 which is significantly better than the resolution of a typical spectrometer. There are a number of experimental difficulties and considerations which we now discuss. First, though higher NA improves contrast, working distance is an issue since these experiments must be performed at cryogenic temperatures. The sample is therefore placed close to the bottom of the entrance window of the cryostat and a long working distance objective used for excitation. Sample drift and stability are also considerations though,

74 given sufficient time for temperature stabilization, initial indications are that these are not significant. We did replace the usual incandescent illumination source used for imaging with a fiber bundle light source since heating of the microscope frame and objective were evident using the former. As mentioned in the previous section, saturation of the transition is a major concern in this experiment. Low powers in the 1 nW range are required and the detector signal must be amplified. (Ayache, 2006) However, calculations indicate peak to noise should exceed 100:1 at resonance. An in depth analysis of amplifier noise and signal strength is provided elsewhere. (Ayache, 2006) As mentioned earlier, the contrast is expected to be significant and so lock-in techniques may not be necessary. Without using lock-in techniques, we must ensure the incident polarization is parallel to the nanotube axis. Determination of the polarization of the incident beam has turned out not to be trivial. The first attempt used a detector and polarizer/analyzer mounted in a side arm of the microscope with part of the incident beam split off with a beam splitter. The issue arises from the very different reflectivities of the S and P polarizations of the beam which is incident on the beam splitter at a ∼45 degree angle. In theory, one could rotate the incident polarization such that it has equal parts S and P with respect to the splitter but this defines a single polarization not only at the splitter but also at the nanotube which grows across the trench within a wide range of angles. We could, in theory rotate the sample each time to align it to the beam polarization but this is not easily done as there is no rotation mount on the cryostat’s cold finger. Therefore, we wish to design the microscope such that the incident beam polarization may be rotated to arbitrary angle (to match the nanotube orientation) by use of a halfwave plate just beneath the fiber collimating lens. We therefore now directly insert a linear polarizer just above the objective which is rotated to directly measure the two othogonal polarization directions and then removed before taking data. Maximizing the signal on the detector beneath the CNT by rotating the polarizer yields one polarization direction. The polarizer is then rotated to find the second polarization axis created by the PolaRite/PolaMight and collimation optics.

75 Ideally, the two axes are 90 degrees to eachother with one axis aligned to the CNT. If not, the PolaRite and PolaMight are adjusted. This is not the total solution, however, since we can not be sure that the modulated polarization coming out of the fiber is necessarily of equal intensity for both orthogonal polarizations. As of this writing, this appears not to be the case. The intensity at the detector of one polarization, as measured using the analyzer as described in the previous paragraph, is not the same as the intensity of the orthogonal polarization. This remains the case even with all beam spitters removed and when focusing off sample and directly onto the detector so that any possible effect of birefringence from the trenches is removed. It is possible that the fiber collimating lens or objective is birefringent but, more likely, the PolaMight/PolaRite combination (Ayache, 2006) is generating orthogonal polarization states at the drive frequency but not of equal intensity. Experimental efforts to generate equal intensity othogonal polarizations continues. In addition, the edges of the trenches can scatter one polarization more strongly than the other. Ideally, the nanotube would be grown at a 45 degree angle to the trench side walls and thus, when the polarization is modulated between parallel and perpendicular with respect to the nanotube axis, these two polarization states are symmetric with respect to the side walls. We do not have, of course, this precision in growth orientation. However, if we are successful in growing nanotubes of the right diameter range across these large gaps, then perhaps selecting nanotubes in this orientation might be preferable. On the other hand, manipulation of the PolaRite and PolaMight combination should be able, in theory, to compensate for differential side wall scattering or any other birefringence in the incident beam optics. Having discussed at some length our efforts to generate equal intensity orthogonal polarizations, in practice, this may not be critical. Since it is really the frequency dependence of α(ω) that we are after, unequal intensities in the two incident orthogonal polarization states should appear in the data as a simple offset at the lock-in which can be background substracted. This assumes, of course, that the incident polarization state is either frequency

76 independent or a sufficiently slowly varying function of frequency such that it may be ignored. The same must be true of any other birefringence of the system. This may not be true of the PolaMight/PolaRite system, i.e. the polarization states may change with laser wavelength. Readjusting and re-measuring the polarization states at each wavelength is not practical. However, it should be possible to measure the frequency dependence, if any, first off-nanotube and then the experiment repeated on-nanotube and the data appropriately corrected.

Optical Transition Energy Eii (nm)

1500

1400

(12, 4) 1326 nm 210 cm-1 (9, 7) 1306 nm 218 cm-1 (13, 2) 1291 nm 215 cm-1 (14, 0) 1279 nm 216 cm-1

1300

E11

1200

E11

Ȟ = +1 Ȟ = -1

1100 5

6

7

8

9

10

11

12

13

14

15

Nanotube Diameter (Å)

Figure 6·3: Kataura plot with four CNT species resonant with the diode laser. (n,m) species, optical transition wavelengths, and RBM energies are shown. The tunable range of about 1270 nm to 1330 nm of the Sacher Lasertechnik TEC 5001310-05 diode laser defines a handful resonant species of CNTs. These species, along with their predicted room temperature E 11 transition energies and RBM energies, are shown in Fig. 6·3 Finally, we make a few comparisons between our expected results and measurements from single quantum dots. Specifically, InAs QD line widths, as measured by differential transmission (Vamivakas et al., 2007), have been shown to be as narrow as ∼ 1.5 μeV,

77 corresponding to a lifetime of about 1.5 ns. These width were obtained at powers in the tens of pW. Power broadening was significant at 1 nW, leading to line widths about 10X as broad with very little contrast. We expect to work in a similar power regime for this reason, as indicated by preliminary calculations. (Ayache, 2006) The contrast, which depends primarily on NA as discussed previously, should be comparable. Our intrinsic line widths, however, are expected to be somewhat broader. In fact, the intrinsic bright exciton lifetime is predicted to be on the order of 10 ps for an infinitely long CNT. (Spataru et al., 2005) At 4 K, the effective lifetime due to thermalization of excitons out of the light cone is still considerably longer, perhaps 10X longer, but differential transmission spectroscopy is insensitive to these incoherent effects. Thus, our intrinsic spontaneous linewidth could be as much as 100X broader than that of a single QD. However, as demonstrated and discussed in some detail in Chapter 4, finite length effects can lead to two level behavior and call into question the 1D assumption used to derive a 10 ps intrinsic lifetime. In this experiment our suspended CNTs will be 20 to 30 μm in length, significantly longer than those in Chapter 4, but could very likely still result in longer lifetimes than those predicted for infinitely long CNTs.

6.4

Conclusion

We showed in this section how the forward Rayleigh scattered signal can be used to directly measure the oscillator strength of an allowed excitonic transition in a single suspended carbon nanotube. We discussed some of the experimental difficulties involved and how these challenges will be met. As of this writing, we are “ready to go” as soon as we can successfully suspend nanotubes of sufficiently small diameter such that their E 11 transitions are resonant with our tunable diode laser.

78

Chapter 7

Spectroscopy of Individual Nano-scale Emitters 7.1

Introduction

Spectroscopy of individual nano-scale emitters presents an array of unique challenges not associated with extended samples. These challenges go well beyond reduced signal strength and are often overlooked. With the advent of the nanotechnology era, there are a rapidly growing number of nano-sized emitters under investigation, each of which require careful, informed interpretation of the spectroscopic data. In this chapter, we illustrate a number of potential pitfalls attendant to the interpretion of spectra taken from single nano-scale emitters. Spectra of individual nano-scale emitters must be carefully interpreted in order to avoid drawing erroneous conclusions. We show how lack of due diligence can lead to mistaken conclusions about spectral widths, positions, and strengths. Specifically, we discuss (1) the relationship between spatial position of the nano-scale emitter in the object plane and its spectral position at the detector, and (2) the effect of spectrometer configuration and (mis)alignment on the spectra of those emitters. In Section 2 of this chapter, we use a single, suspended carbon nanotube to demonstrate how off-optical axis emission leads to erroneous spectral position, which we refer to as a pseudo-spectal shift. This spatial-spectral correlation is especially relevant as spectroscopy of individual nano-scale emitters becomes commonplace. We believe this effect is often overlooked and can lead to variation in any number of values derived from spectra of individual nano-scale emitters which are reported in the literature. In Section 3, we demonstrate the use of a sub-diffraction-limited emitter to directly measure the intensity profile of the exciting laser focused by the objective. This ability to

79 directly measure the beam profile is of potentially great use to the nano-optics community. In Section 4, we demonstrate how aperture stop effects, chromatic aberration, emitter position, and misalignment of the optical axis with respect to the slit at the entrance to the spectrometer alter spectral intensity, position, and resolution. We provide a series of “telltale” indicators, sometime subtle, that indicate the presence of chromatic aberration and/or spectrometer misalignment. Precise alignment of the spectrometer and of the emitter to the optical axis are shown to be critical in obtaining accurate results. Furthermore, certain effects are both unavoidable and significant, such as chromatic aberration, especially when the spectroscopy of choice involves inelastic processes. Using a model with inputs such as slit and sample displacement in the longitudinal and lateral directions, we explain how we are able to extract from the spectroscopic data information about our own spectrometer alignment and resolution. Many of these results are specific to individual sub-diffraction-limited emitters; others can be seen in extended emitters, as well, but are often not considered, and so we demonstrate the effect of certain spectrometers parameters on the spectra of extended emitters as well. Finally, in Section 5, we perform similar measurements on extended emitters for comparison. Fig. 7·1a shows a schematic of our experimental configuration which is fairly generic for the purposes of this discussion. The spectroscopic effects we will discuss in this chapter are applicable to virtually any spectrometer since the effects we describe require only the presence of an intermediate image plane at (or near) a slit postitioned at the entrance to the spectrometer which is then “imaged” onto the detector after the grating. Central to a fundamental understanding of spectroscopy is the concept of mapping. In simple imaging, ray tracing provides a point by point mapping of the object plane onto the image plane. Similarly, in spectroscopy, each point in the object plane maps to a point in the intermediate image plane and then, for a given wavelength and grating order, to a pixel on the detector (assumed to be a 2D array.) Furthermore, the orientation of the grooves on the diffraction grating will define a “spectral direction” at the detector, as well as an orthogonal “non-

80

Optical Axis

a)

Pixel Array

CCD

Spectral Direction

Obj

b) Grating

Slit

Emitter Pre-Slit Lens

Non-Spectral Direction

c) 0

d)

30

Optical Crosshairs

60

270

90

Focal Spot

240 120 210 180 150

Spectral Direction

330 300

ʄ2

20 ȝm

Trench

Supercontinuum

Sample Translation CNT

Slit

ʄ1

820 nm Super-continuum

Figure 7·1: Nano-scale spectroscopy. (a) Schematic of a typical (simplified) spectrometer setup. (b) CCD pixel array, showing the spectral and nonspectral directions defined by the grating orientation. (c) Image of a single suspended CNT using supercontinuum illumination. The incident supercontinuum direction is shown with a white arrow and is inclined approximately 30 degrees from the sample plane. The black arrow shows the direction of sample movement through the red laser focal spot which is aligned to the white crosshairs. The spectral direction, mapped from the CCD back into the object plane, is also indicated. Upper Left Inset: Polar plot of G+ Raman mode intensity as a function of incident polarization. Lower Right Inset: Schematic of single line (820 nm) illumination in reflection mode and side illumination by the supercontinuum. (d) Blow up of the intermediate image plane and slit at the entrance to the spectrometer, showing the effect of chromatic aberration.

81 spectral direction,” as depicted in Fig. 7·1b. These spectral and non-spectral directions at the detector map back to spectral and non-spectral directions in the object plane, as shown by the blue arrow in Fig. 7·1c. In this chapter we focus primarily on effects arising from (1) small (sub-diffraction limit) displacements (in the spectral direction) of the emitter from the optical axis, which lead to false, measurable spectral shifts, and (2) aperture stop effects by the slit as a results of chromatic aberration and/or misalignment, which lead to artificial spectral broadening, reduction in spectral intensity, and possibly pseudo-shifts as well. The connection between spatial displacement in the object plane and spectral signature is intrinsic to all spectrometers though is often overlooked. The aperture stop and chromatic aberration effects, on the other hand, can often be greatly reduced by use of doublets, acromats, etc. though they can never be removed entirely, especially in inelastic spectroscopies, and spectrometer alignment is never perfect no matter how much care is taken.

7.2

Psuedo-Spectral Shifts

Our nano-emitter is a single, resonant carbon nanotube suspended across a gap of approximately 20μm. The sample is shown in Fig. 7·1c with the CNT visible (along with some scattering from the edges of the trench), upon super-continuum illumination (Wang et al., 2006) 1 , to the inspection camera on the microscope for the reasons discussed in Section 6.3. Results are discussed below and shown in Fig. 7·2. The diameter of the CNT is approximately 1.5 nm, well below the diffraction limit of the spectrometer, and thus can be treated as a point source emitter in the direction orthogonal to the nanotube length. Furthermore, there are no other emitters in the diffraction limited excitation focal spot, shown as a red circle (not to scale) in Fig. 7·1c. The lower right inset of Fig. 7·1c shows the two excitation paths used in the experiment, with the continuous wave (CW), single line, 820 nm light focused by the objective aligned vertically. The incident super-continuum light is also focused onto the CNT from the side with a second objective and is used, for the purposes of 1

See Newport AppNote at http://www.newport.com/file store/Optics and Mechanics/AppsNote28.pdf

82 this work, to locate the CNT and align the sample. In order to demonstrate the effect of off optical axis emission, we aligned the CNT to the non-spectral direction and walked the CNT, taking spectra at each step, through the focal spot of the CW laser, in the direction shown. The emission of choice for this work is the LO tangential, or G+ , Raman mode (1591 cm−1 ) of the nanotube (Reich et al., 2004) and is collected in a reflection geometry. The upper left inset of Fig. 7·1c shows the intensity of the Raman signal as a function of polarization angle of the incident laser, illustrating the strong antenna effect of the CNT and providing an independent means of verifying the orientation of the CNT or, knowing the CNT orientation a priori, as we do, of verifying the incident polarization direction. Fig. 7·1d shows a blow up of the slit which, ideally, lies at the intermediate image plane, i.e. the beam waist of a point emitter in the object plane. However, due either to chromatic aberration and/or misalignment, that is not exactly so usually. When this is the case, the slit can restrict the acceptance angles of the spectrometer, i.e. act as the aperture stop for the system. The attendant effects are discussed below and illustrated in Fig. 7·3. At this point, it is worth clarifying the meaning of the term “optical axis.” A spectrometer is aligned by use of a source, in our case a neon lamp positioned beneath a pinhole, with known spectral lines which are passed to the spectrometer software. The source is placed beneath the objective and the grating scanned as the software matches up the measured spectral lines to the known input spectrum. In addition, the grating is sent to zero order and a sample of some sort imaged onto the detector array. The slit, in the intermediate image plane, is then closed down in order to ensure that whatever lies at the crosshairs (of an inspection camera and/or microscope) in the object plane is also centered precisely on the slit, which is, in turn, imaged onto the center pixel of the detector array. This step is critical, as we shall see below. This ray that connects the point in the object plane, defined by the crosshairs, to the center of the slit to (typically) the center pixel of the detector (in the spectral direction) is referred to as the optical axis. When the grating is scanned to calibrate the system, it is critical that the calibration source lies on the optical axis, i.e. at the crosshairs. In this way, placing any future sample at the crosshairs ensures the spectral

83 content maps properly onto the detector array without asymmetric clipping by the slit. MOS

a)

MSD

Optical Axis

Slit

Grating

On Axis Emitter Obj Pre-Slit Lens 6 5

CNT 100X

CNT 50X

0 -1 -2 -3

Si

-4 -5 -2

-1

Step Direction

Optical Axis

d)

4 3 2 1

Crosshairs

Normalized G+ Peak Height

Raman Pseudo-Shift (cm-1)

c)

Laser Spot

CCD

Spectral Direction

Off Axis Emitter

CNT

b)

0

Position (ȝm)

1

2

1

50X 100X 0 -2

-1

0

1

2

Position (ȝm)

Figure 7·2: Effects of off-optical axis emission from a nano-emitter. (a) Ray traces for emission from on (solid red line) and off (dashed red line) optical axis. (b) Experimental arrangement for the data shown in (c) and (d). The CNT is stepped through the laser focal spot which is aligned to the optical axis defined by the crosshairs. (c) Pseudo-Raman shifts using both the 100X (red) and 50X (blue) objectives. The measured and calculated slopes for the 50X objective were -1.36 and -1.50 cm−1 /μm, respectively, and -3.12 and 3.00 cm−1 /μm for the 100X. For the extended emitter (Si, shown in green), there is no effect. (d) Profiles of the laser spot focussed by the 100X and 50X objectives, shown in red and blue, respectively, as measured by the CNT. Calculated ASR model results for a 1.46 mm Gaussian beam incident on the objective back aperture are shown with solid lines. The measured FWHM for the 50X and 100X are 0.95 and 0.54 μm, respectively. Fig. 3·2 illustrates the effects of off optical axis emission. Fig. 3·2a shows ray traces in solid and dashed red lines for an emitter positioned on and off optical axis, respectively, and how this maps to a displacement at the detector. If the emitter is displaced in the spectral direction then the detector will register a pseudo spectral shift. We demonstrate this effect by stepping the sample, and thus the nanotube, through the focal spot of the CW laser (which has been carefully aligned to the optical axis) and taking spectra at each position, shown schematically in Fig. 7·2b. The spectral position of the G+ Raman mode

84 is plotted against spatial position of the nanotube in Fig. 7·2c clearly showing the pseudo shift effect. The dispersion for the 100X objective, shown in red, is twice that of the 50X objective, shown in blue, by virtue of the factor of two greater magnification. Given the grating lines/mm, diffraction order, and center wavelength, and the detector lens focal length, laser wavelength, Raman shift, and pixel size, the dispersion in wavenumbers per pixel can be calculated. Specifically, a displacement in the object plane Δxobj is imaged to a displacement at the slit, Δxslit = Δxobj MOS . The second path both magnifies and disperses the light. The physical displacement on the CCD is recorded as a change in wavelength. Hence we can calculate the shift in spectral position Δσ due to a displacement Δxobj of the nano-emitter from the optical axis along the spectral direction2

Δσ = Δxobj MOS MSD dσ/dx

(7.1)

where Δσ is the shift in cm−1 due to displacement Δxobj and the linear dispersion dσ/dx = λ−2 cos(β)/knLB is calculated from the wavelength λ, grating order k, groove density n, exit focal length LB and exit angle β. The dispersion is calculated using Eq. 7.1 and shows good agreement with the measurements. The parameter values for our spectrometer are listed in Tab. 7.1. Since the center position, and hence the true wavelength, can be determined with the help of the intensity profile discussed below, it is possible to use the spectral information to locate the nano-emitter relative to the optical axis to better than the diffraction limit. We consider the 100X objective, which collects more light and hence gives a better signal to noise ratio than the 50X objective. From peak fitting, the spectral position is determined, conservatively, to a precision of better than a quarter of a wavenumber. Using the measured linear dispersion of 3.1 cm−1 /μm, the position of the nanotube can be determined to 80 nm precision, or ∼ λ/12. We note that with emitters of known and nonoverlapping spectral signatures, the distance between emitters in the spectral direction can then also be determined with similar sub-wavelength precision. Higher magnifications MOS 2

http://www.jobinyvon.com/SiteResources/Data/Templates/1divisional.asp?DocID=566&v1ID=&lang

85

k=1, Ȝ0 = 943 nm

Spectrometer Parameters (50X Objective) MOS

MSD

50 mm 12.5

1.6

LPreSlit

LB

n

Pixel Size

Į

22 ȝm

31.9

250 mm 600 g/mm

ȕ o

2.1

dı/dx o

0.075 cm-1/ȝm

Table 7.1: Renishaw 1000B μRaman spectrometer parameters.

and MSD increase the sensitivity to position. The linear dispersion is inversely proportional to the number of grooves, so a more dispersive grating will decrease the linear dispersion and hence decrease the sensitivity. However, it will also provide more data points of the spectral feature, and can therefore provide a more precise determination of the measured spectral position. We also show in green in Fig. 7·2c the absence of any pseudo shift for an extended emitter, in this case the 520 cm−1 Raman line of silicon.

7.3

Measuring the Focal Spot Profile

We now demonstrate the use of a nano-emitter to directly measure the excitation spot size of the focusing objective. We are careful here to differentiate between the excitation spot size of the objective from its point spread function (PSF) (or that of the whole spectrometer) which refers to the extent to which the objective or system can image a point emitter. Here, our laser beam incident on the back aperture of the objective has a Gaussian profile with a full width at half maximum (FWHM) of 1.46 mm (as determined by recording the power as a razor is moved through the beam by micrometer screw and the resutling curve fit with the error function.) The intensity of the G+ Raman mode is plotted as a function of sample position for both a 100X/0.9 NA and a 50X/0.5 NA (long working distance) objective in Fig. 7·2d. The result is the beam profile since the nanotube samples the focal spot intensity point by point in the scan direction while the detector integrates the signal in the orthogonal direction which, for the Gaussian function (which is a good approximation to our focal spot intensity profile), is separable. This presumes, of course, a

86 linear response of the emitter with excitation power, which we have verified experimentally. For reasons discussed below, the slit must be kept open far enough to prevent clipping of off optical axis emission. Data are compared to modeling using a full vector treatment in the angular spectrum representation (ASR),(Novotny and Hecht, 2006) shown as solid lines in Fig. 7·2d. Typically in microscopy or spectroscopy, the diffraction limited spot size is approximated by the Rayleigh criterion, 0.61λ/NA,(Hecht, 2002) but this assumes scalar plane wave illumination of the back aperture in the paraxial limit for low NA; the full vector treatment for a Guassian profile laser beam, including possible underfilling of the back aperture and higher numerical apertures yields much richer results but is more complicated. Here, due to the antenna effect, the nanotube probes the same polarization component as the incident linear polarization. The two other polarization components are present in the focal plane but are significantly smaller. However, these polarization components could be important if, for example, you are attempting to couple to a dipole oriented in the z direction, as is common in quantum dots. We also note that the focal spot will be slightly asymmetric in the focal plane for a linearly polarized laser; here, the CNT is measuring the beam profile in the slightly more tightly focused direction (orthogonal direction to the incident polarization). The measured data matches the ASR results to within 5 percent and yields FWHMs of 0.96 and 0.54 μm for the 50X and 100X objectives, respectively.

7.4

Slit Width Effects

Poor choice of slit width can also lead to misinterpretation of the spectral data, as demonstrated in Fig. 7·3. Fig. 7·3a shows schematically the slit acting as an aperture stop. The solid red line in the expanded view shows the Gaussian beam waist in the absence of the slit. When the slit is not located at the beam waist, however, it can begin to restrict the acceptance angles of the system, as shown by the dashed red line. This reduces the effective back aperture of the collection objective, reducing its effective numerical aperture, and results in a larger point spread function at the slit, as shown in the expanded view of Fig. 7·3a. This non-ideal slit position is not (necessarily) the result of poor alignment. In fact, this effect

b)

a) Slit

Emitter

Obj Pre-Slit Lens

0

G+ Peak FWHM (cm-1)

1590

1589

1588

0

50

100 150 200 250 300

Slit Width (ȝm)

50

100 150 200 250 300

Slit Width (ȝm)

d)

c)

G+ Raman Shift (cm-1)

G+ Integrated Intensity

87

12 10 8 6 4 2 0 0

50

100 150 200 250 300

Slit Width (ȝm)

Figure 7·3: Slit width effects. (a) Slit offset from plane of the beam waist leads to a reduction in angles accepted by the optical system, i.e. acts as an aperture stop. (b), (c) and (d) G+ Raman mode integrated intensity, spectral position, and spectral width, respectively, as a function of slit width. The simultaneous best fits from the model are shown as solid red lines. is unavoidable, to a greater or lesser degree, due to chromatic aberration. We calculate, using Abbe numbers(Hecht, 2002), that for a typical, simple, 0.05 NA, pre-slit lens, the beam waist for G+ Raman line, 1591 cm−1 , of the nanotube will fall approximately two Rayleigh ranges beyond the plane of the beam waist of the excitation at 820 nm and will thus begin to act as an aperture stop at small slit widths. Short of realigning the system for each wavelength of interest, chromatic aberration can potentially play a significant role in understanding spectral results, especially at small slit widths. We demonstrate these effects below. We should note that (1) the use of a doublet for a pre-slit lens can greatly reduce the chromatic aberration and the attendant aperture stop effects at the slit, and (2) the use of spherical mirrors rather than lenses in the spectrometer itself can also reduce effects related to chromatic aberration. For a nano-scale emitter, the image at the intermediate image plane at (or near) the slit is determined by the PSF of the objective (assuming the pre-slit lens has a large enough pupil

88 such that it does not act as an aperture stop) multiplied by the magnification, MOS (given by the ratio of focal lengths of the objective and pre-slit lens), assuming the spectrometer is designed in such a way as to avoid any intermediate field stop effects. This intermediate image is then magnified and mapped onto the detector (for a given wavelength and diffration order) and thus the spatial extent of the image, convolved with the true spectral width, determines the measured spectral width of a given feature. For our system, MOS , for the 50X/0.5 NA objective and a 50 mm focal length preslit lens, is 12.5 and the PSF at 820 nm is a little less than 1 μm. Thus, we expect the intermediate image size at the slit to be, ignoring the aperture stop effects disccussed below, about 12 μm FWHM. At minimum slit width of 10 μm we can expect the slit to clip the image, i.e. act as a field stop, and thus reduce the integrated area of our spectral features at the detector. This also implies a smaller spatial extent of the image at the detector and thus a smaller line width. However, as we shall see below, aperture stop effects can dominate leading to increased line width at small slit width. Furthermore, the intermediate image may be laterally offset with respect to the slit. This effect is also discussed below. Fig. 7·3b shows the effect of slit width on integrated intensity at the detector. If the slit is closed far enough, the image will begin to act as a field stop, the intermediate image will be clipped, and thus the integrated area of the spectral features at the detector reduced. The shape of this curve is complicated by slit width effects, however, as described above. The image that is being clipped by the slit in the intermediate image plane is also changing size as the slit begins to act as an aperture stop, decreasing effective numerical aperture of the objective and increasing the PSF of the system. The integrated area of the G+ Raman feature is plotted as a function of slit width as black squares in Fig. 7·3b while the model results are plotted with a red line. Our model uses as its fit parameters the longitudinal displacement of the slit from the plane of the beam waist, the lateral displacement of the intermediate image from the exact optical axis (again, defined as a ray passing through the center of the slit), and the true spectral width of the feature. Figs. 7·3c and d show the effect of slit width on the measured Raman shift and spectral width and are derived from

89 the same data set as in Fig. 7·3b. We therefore required a simultaneous best fit by our model to the data in Figs. 7·3b through d, shown by red fit curves. The fits are quite good and indicate a longitudinal slit displacement of two to three Rayleigh ranges, in agreement with the calculation of the effect of chromatic aberration using Abbe numbers. The fit to the spectral shift shown in Fig. 7·3c indicates either that the emitter is laterally offset from the optical axis in the object plane by about 500 nm and/or the intermediate image is not exactly centered on the slit. As the slit is closed, the image is clipped asymmetrically and leads to another form of pseudo shift distinct from that described in Fig. 7·2 (though the latter effect manifests itself in the data as an overall shift.) Note that the sign of the shift indicates to which side the emitter or intermediate image is offset. A 500 nm offset of the emitter from the optical axis in the object plane is unlikely since we are able to position the emitter at the crosshairs with better than 200 nm accuracy (stage limited), as confirmed by Rayleigh scattering of super-continuum light described above and shown in Fig. 7·1c. More likely, in addition to some small displacement of the emitter in the object plane, the optical axis (i.e. the crosshairs) is slightly offset, with respect to the center of the slit. This offset, however, is less than the slit width at its smallest (10 μm) and translates to a fraction of one pixel at the detector making any further alignment difficult and very time consuming. Thus, this effect is, in practice, one that must also be understood to properly interpret the spectral data. Fig 7·3d shows the effect of slit width on measured spectral width. As an input, we have used 7 cm−1 as the true spectral feature width to fit the data at large slit width. At small slit width, the slit clearly begins to act as an aperture stop, artificially broadening the spectral line width. Again, ideally the slit would be positioned precisely at the beam waist and no such broadening would occur. In fact, field stop effects leading to a smaller image at the detector should tend to decrease the line width, which is what most spectroscopists would probably expect to see. Here, however, aperture stop effects clearly predominate. Note that proper positioning of the emitter in the focal plane of the objective is also critical. It will influence the position of the intermediate image plane with respect to the slit and thus

90 the results we have shown. By having measured the focused excitation beam profile, as discussed above and shown in Fig 7·2d, and obtained a FWHM value within a few percent of the value predicted by the ASR model, we know that our emitter lies in the focal plane of the collection objective for the excitation wavelength (820 nm.) At this point we emphasize that we endeavored to align our spectrometer well (optimized for the near IR) and that these effects we which have demonstrated are not the result of intentional misalignment or carelessness. More precisely, certain of these effects, such as slit width/aperture stop spectral broadening in the presence of chromatic aberration, are not the result of misalignment at all but are intrinsic to any spectrometer and thus need to be understood by anyone wishing to properly interpret the spectra of nano-scale emitters. In addition, other effects, such as the pseudo spectral shifting with slit width demonstrated in Fig. 7·3c, could, in theory, be eliminated with much effort, but, in practice, this is often not possible.

7.5

Extended Emitters

In Fig. 7·4, we show pseudo shift and slit width effects using extended emitters. In Figs. 7·3a and b, the focused beam spot is intentionally offset from the optical axis in the spectral direction, effectively creating an diffraction limited sized, off axis emitter in a homogenous piece of silicon. Fig. 7·3a shows the raw spectra of the 520

−1

Raman line of Si for three

positions of the focused spot. Fig 7·3b plots the peak spectral positions from Fig. 7·3a as a function of focal spot position and demonstrates the pseudo shift effect for an extended emitter due not to small sample displacement, as in the case of nano-scale emitters in Fig 7·2c, but due rather to misalignment of the excitation beam to the optical axis. In Figs. 7·3c and d, the extended emitter is either a 3 μm or 100 μm pinhole illuminated with a neon lamp from below. The spectra are collected with the pinhole positioned on optical axis in the focal plane of either a 5X/0.12 NA or a 50X/0.5 NA objective. In Fig 7·4c, we see the effect of slit width on the spectral width of two different neon lines using the 3 μm pinhole and 50X objective. For both wavelengths, the spectral width decreases with

a)

b)

Si 520 cm-1 Raman Intensity (Arb .Units)

Optical Axis Focal Spot

Si 520 cm-1 Raman Pseudo-Shift (cm-1)

91

20 10 0 -10 -20 - 10

11500

11600

11700

11800

7

10354 cm-1

6 14220 cm-1

5 4 3 2 1 0 0

50

100 150 200 250 300

Slit Width (ȝm)

d)

Integrated Intensity

Peak FWHM (cm-1)

Wavenumber c)

-5

0

5

10

Focal Spot Offset from Optical Axis (ȝm) 5X 3 ȝm 50X 3 ȝm 5X 100 ȝm 50X 100 ȝm

0

50

100 150 200 250 300

Slit Width (ȝm)

Figure 7·4: Extended emitter effects. (a) Spectra of the Si 520 cm−1 Raman peak for different focal spot positions (red dots), offset from the optical axis (black cross-hairs) in the spectral direction. (b) Peak positions from (a) as a function of off-axis displacement of the focal spot. Measured slope of the best fit line, shown in black, is -1.6 cm−1 /μm compared to the calculated value -1.8 cm−1 /μm. (c) Peak spectral width vs. slit width for two different emission lines of a neon lamp beneath a 3 μm diameter pinhole. The blue circle highlights aperture stop effects resulting from chromatic aberration of the longer wavelength. (d) Integrated peak area of the neon 14220 cm−1 line vs. slit width for different pinhole sizes and objectives showing the effect of emitter image size at the slit.

92 slit width as the slit begins to act as a field stop. The slit clips the intermediate image which leads to smaller spatial extent when mapped onto the detector and thus results in a smaller line width. However, for the longer wavelength, the chromatic aberration also leads to an aperture stop effect at very small slit widths which, in turn, begins to artificially broaden the neon line. In contrast, for the shorter wavelength, which lies spectrally closer to the wavelength where the spectrometer was optimized, the slit falls closer to or at the intermediate image plane and thus does not act as an aperture stop and thus no artificial broadening is visible. In Fig 7·4d, we see the effect of slit width on the integrated area of the 14220 cm−1 neon line, for both objectives and pinholes, due to field stop effects. For the 3 μm pinhole, the intermediate image is smaller than that of the 100 μm pinhole and thus the neon line reaches its maximum intensity at smaller slit width. Similarly, the intermediate image is smaller for the 5X than for the 50X objective. Note that the intermediate image is so large for the 100 μm pinhole and 50X objective that field stop effects are still in play even at 300 μm slit width. The measured spectral width of a given feature from a macroscopic object depends on a convolution of the intrinsic line-width, Δσint , with the instrument broadening which is given by the contribution from the widths of entrance slit and CCD pixel size, Δσslit and Δσpixel .3 Assuming Gaussian line profiles for a simple expression of the convolution4 we have

 ΔσF W HM,macro =

2 + Δσ 2 + Δσ 2 Δσint slit pixel

(macroscopic source)

where Δσpixel gives the lowest instrument broadening if the slit is infinitely narrow. For a nano-scale emitter, the image size at the intermediate image plane slit is determined by the 3 In most systems, the contribution from the grating resolution can be ignored. Here, Δσg contributes < 2% of the instrumental broadening. 4 Here we use the Gaussian form to simply illuminate the different contributions to the line width even though the slit and the pixel have a rectangular form. The result in Figure 7.5 is obtained from a proper convolution.

93

FWHM (cm-1)

a)

b)

60

Extended Emitter

50 40

Į

30 20

Wg’

CNT 10

Wg’’ ɴ

0 0

50

100

150

200

250

300

Slit Width (ȝm) Figure 7·5: (a) FWHM as a function of slit width. CNT G+ Raman mode widths are shown in blue circles. Linewidths of a spatially extended emitter (703.2 nm Ne line) are shown in green triangles. The system responses for macroscopic and point sources are shown with green and blue lines, as governed by Eqs. 7.2 and 7.3, respectively. (b) Close up of the grating and associated beam angles and widths.

point spread function of the microscope objective, ΔxP SF , multiplied by the magnification MOS . The intermediate image is magnified and mapped onto the detector and gives the contribution ΔσP SF to the spectral width 



ΔσP SF = ΔxP SF MOS MSD (Wg /Wg )dσ/dx 

(7.2)



where Wg / Wg =cos(α) / cos(β) (Fig. 7·5b), so that

 ΔσF W HM,nano =

2 + Δσ 2 2 Δσint P SF + Δσpixel

(point source)

Note that it is the diffraction limited point spread function that enters into the line width and that the line width is independent of the slit width.5 Hence a fully open slit does not contribute to a broadening of the line shape for a nano-scale emitter, in contrast 5

When the slit starts to clip the diffraction limited spot size, the line width contribution from the instrument response decreases. However, the system parameters are often chosen such that the pixel size is approximately commensurate with the diffraction limited spot size at the slit making this effect moot.

94 to the case for spectroscopy of a macroscopic object, where a slit width larger than the corresponding pixel width always contributes to instrument broadening. The measured line width for the CNT emitter is shown in Fig. 7·5a with blue circles using the 50X/0.5NA objective. The solid blue line shows the contribution from the instrument resolution using Eq. 7.3 where ΔσP SF is approximated by the Rayleigh criterion. By removing this system contribution to the measured line width of 7.5 cm−1 , the intrinsic CNT Raman line width for the G band of a suspended nanotube is calculated to be 6.9 cm−1 . For contrast, Fig. 7·5a also shows, in green triangles, the measured line width from an atomically sharp transition (the Ne 703.2 nm line) where the line width is dominated by the system response due to the large physical extent of the source in the object plane. Specifically, the objective is focused on a 100 μm pinhole which is illuminated from below by a Ne lamp. The calculated line width contribution from the system for this case, as given by Eq. 7.2, is shown by the solid green line. Of course, in many cases, the emitter size is equal to the diffraction limited laser spot size used to excite the sample and so the measured spectral line width dependence on slit width will fall somewhere between Eqs. 7.2 and 7.3.

7.6

Conclusion

In summary, we demonstrated some of the challenges inherent to spectroscopy of individual nano-scale emitters. Though some effects, such as pseudo spectral shifts, can, with effort, be largely, but often not completely, eliminated, others, such as those due to chromatic aberration, are intrinsic and thus must be understood by spectroscopists of nano-emitters.

Appendix A

Denisty Matrix Derivation of the Raman Excitation Profile A.1

Introduction

In this Appendix, we derive the Raman excitation profile, REP, using a third order, perturbative density matrix approach applied to a four level system representing the ground and excitated electronic state, each with zero or one phonon. The four level model is justified given the van Hove singularities in the DOS in 1D and the momentum conservation requirement for photon and phonon absorption and emission, which effectively restricts optical activity to the zone center. We discuss the assumptions made in such an approach and contrast it with traditional solid state approaches. Finally, we comare the line shapes obtained from this approach, the Raman transform method (Shreve et al., 2007) assuming Lorentzian absorption line shapes, the free electron hole solid state approach (Cardona and G¨ untherodt, 1982), and the solid state approach incorporating excitonic effects. (Vamivakas et al., 2006a)

A.2

The Perturbative Density Matrix Formalism

We start by reviewing the density matrix formalism (following CH653 class notes (Zeigler, 2007)). We first expand the pure state Ψ in the basis un using the Schrodinger representation:

Ψ(t) =



C(t)n |un > .

n

The expectation value of observable Aˆ is then given by 95

(A.1)

96

ˆ >= < A(t)



ˆ p >= Cn∗ (t) Cp (t) < un |A|u

n,p



Cn∗ (t) Cp (t) Anp .

(A.2)

n,p

Using the orthonormality of the basis set un , we find from A.1 that

Cp (t) =< up |Ψ(t) >; Cn∗ (t) =< Ψ(t)|un > .

(A.3)

Inserting this result into A.2, we find

ˆ >= < A(t)



< up |Ψ(t) >< Ψ(t)|un > Anp =

n,p



< up |ˆ ρ(t)|un > Anp

(A.4)

n,p

where we have defined

ρˆ(t) ≡ |Ψ(t) >< Ψ(t)|

(A.5)

known as the density operator. The on- and off-diagonal components are referred to as populations and coherences, respectively, for reasons that will become apparent. The components of the matrix representation of the density operator are basis set dependent and are given by, for the basis set un ,

ρ(t)|un >=< up |Ψ(t) >< Ψ(t)|un >= Cp (t) Cn∗ (t). ρˆpn (t) =< up |ˆ

(A.6)

Note that the populations are real, Tr[ρ(t)]=1 in any basis, and the off-diagonal elements ρpn =ρ∗np . Using A.4 and A.6 and by removing a complete set of states, we find

ˆ >= T r[ˆ ˆ < A(t) ρ(t)A].

(A.7)

Specifically, we will calculate the induced polarization density, P, using the expectation value of the dipole moment, μ ˆ, and the density matrix:

P = T r[ˆ ρ(t)ˆ μ].

(A.8)

97 More precisely, we will derive below P for the resonant Raman process using the density matrix calculated to third order in perturbation theory in the dipole approximation. The dependence of P yields the resonant Raman excitation profile, REP, which we are after. Note that if the basis set un are the eigenstates of the full Hamiltonian of the system including the electron-photon interaction and, perhaps, electron-phonon interactions, then Ψ will equal some un and thus the off-diagonal ρpn =< p|Ψ >< Ψ|n >=0. That is, the basis states are decoupled. Conversely, non-zero coherences relate to coupling between basis states of, e.g., the electronic eigenstates due to some interaction, commonly the dipole coupling of the electromagnetic field to the electronic system. In this work, the basis states un represent molecular states, i.e. eigenstates of the unperturbed Hamiltonian including the electron-phonon interaction with the particular phonon mode represented by the eigenstate. The perturbation will be the interaction of the molecule with the radiation field in the dipole and Condon approximations. We now derive the time dependence of the density operator:

d d d dˆ ρ(t) = (|Ψ(t) >< Ψ(t)|) = (|Ψ(t) >) < Ψ(t) + |Ψ(t) > (< Ψ(t)|) dt dt dt dt

(A.9)

and, using the Schr¨ odinger equation, we obtain the Quantum Louisville equation:

i ˆ i i ˆ dˆ ρ(t) ˆ = − H(t)|Ψ(t) >< Ψ(t) + |Ψ(t) >< Ψ(t)|H(t) = − [H(t), ρˆ(t)]. dt   

(A.10)

If we apply basis states un of the unperturbed time-independent Hamiltonian H0 from left and right, and define ωnm =En - Em , we find dˆ ρnm (t) = −iωnm ρˆnm (t) dt

(A.11)

the solution to which is, of course, a simple exponential for the coherences oscillating at ωnm . As expected, the populations do not change in time (ωnn =0) since the basis states un

98 were taken as the eigenstates of H0 . We now phenomenologically add damping γnm to A.10 to obtain the damped Louisville equation: i ˆ dˆ ρnm (t) = − [H ˆ(t)]nm − γnm ρˆnm (t) = −(iωnm + γnm )ˆ ρnm (t) 0 (t), ρ dt 

(A.12)

with solutions of the form

ρˆnm (t) = ρˆnm (t0 )e−(iωnm +γnm )(t−t0 ) .

(A.13)

where γnm (=γmn ) is referred to as the total dephasing rate. It is commonly expressed as

γnm ≡

1 1 quasi = (Γn + Γm ) + γnm T2 2

quasi = where the Γi = T11i represent loss of coherence due to population decay and γnm

(A.14) 1  T2

rep-

resents loss of coherence due to quasi-elastic interactions with the environment. Often the lower state um is the ground state of the system and thus Γm may be taken as zero and is referred to simply as

1 T1 .

1 T1n

In this case, 1 1 1 = +  T2 T1 T2

(A.15)

For the diagonal elements of ρ, γnn simply equals Γn , ωnn =0, and so

ρˆnn (t) = ρˆnn (t0 )e−Γn (t−t0 )

(A.16)

representing simple exponential population decay. We now include coupling to the radiation field, invoking the dipole approximation, i.e.

ˆ  (t); ˆ ˆ0 + H H(t) =H

ˆ  (t) = −ˆ ˆ H μ · E(t)

(A.17)

ˆ ˆ  (t) is assumed to be where E(t) is the applied external electric field operator. As usual, H ˆ 0 with H ˆ in A.12, we find ˆ 0 . Replacing H H

99

dˆ ρnm (t) i ˆ i = − [H(t), ρˆ(t)]nm − γnm ρˆnm (t) = −(iωnm + γnm )ˆ ρnm (t) − [Hˆ  (t), ρˆ(t)]nm dt   (A.18) which is solved by power series expanding the density operator and grouping terms of equal order in the perturbative expansion. We find that the kth order solution ρˆ(k) (t) depends on the k-1th order term in the following way: i dˆ ρnm (t) (k) = −(iωnm + γnm )ˆ ρnm (t) − [Hˆ  (t), ρˆ(k−1) (t)]nm dt 

(A.19)

and thus the solution at order k is found by iteratively solving lower order terms. The zeroth ˆ  (t) and the ρˆ(0) are given by a thermal order solution thus contains no application of H ii distribution with no coherences. Higher order terms are found to be given by

(k) (t) ρˆnm

i =− 

t

 dt [Hˆ  (t), ρˆ(k−1) (t)]nm e(iωnm +γnm )(t −t) .

(A.20)

−∞

ˆ  (t) and by opening the commutator and Finally, by using the dipole approximation for H using a summation to represent the matrix multiplication, we find

i (k) (t) = ρˆnm  −

i 

t

ˆ ) dt E(t

−∞ t

ˆ ) dt E(t

−∞



(k−1)

μ ˆnl ρˆlm



(t )e(iωnm +γnm )(t −t)

l



(k−1)

ρˆnl



(t )ˆ μlm e(iωnm +γnm )(t −t) .

(A.21)

l

This is the central result of the perturbative density matrix approach in the electric dipole (k−1)

approximation. Note that the first term takes ρˆlm

(k)

(t ) → ρˆnm (t), i.e., |l >< m| → |n >
is taken to have initial population (0)

ρˆii (0) with higher lying levels assumed to have negligible intitial thermal population by

100 virtue of the Boltzmann factor. Since all initial coherences are also assumed to be zero, the number of contributing elements in the sum over l at each order k is greatly reduced, simplifying the calculation. Also notice that in the dipole approximation the density matrix (and any observable derived from it) at kth order will be given by k applications of the electric field. “Incoming Resonance”

“Outgoing Resonance”

e’

e’

e

e

-ȍ1 +ȍ2 +ȍ1

-ȍ1’ +ȍ2’ +ȍ1’

f

f

i

i t1

i

C. C.

i e

i

t2

t3

f

i

t1 f

f

e

f

i

i e’

i

t2

t3

f

i

f f

f

e’

Figure A·1: Raman WMEL diagrams corresponding to so-called “incoming” and “outgoing” resonances in the REP. WMEL diagrams with ket side evolution first are shown. The corresponding bra side first WMEL diagrams are given by the complex conjugate. The evolution of the density operator is shown below the diagrams. We now introduce wave-mixing energy level, or WMEL, diagrams used to describe the time evolution of a sytem at the amplitude level, as shown in Fig. A·1. Unlike the more common energy level diagrams where upward or downward going arrows represent gain or loss of population, in WMEL diagrams, each arrow represents one application of the electric field, developing either coherences or populations. Thus, WMEL diagrams represent evolution of the system at the amplitude level. Solid arrows represent ket side evolution of the density matrix or, equivalently, the state of the system, and dashed arrows represent bra side evolution. Electric fields act at times t1 , t2 , ... tn with time increasing along the

101 horizontal direction and tn > ... > t2 > t1 . In the dipole approximation, the perturbation order will match the number of applications of the electric field or arrows. The last arrow represents the field at the detector and must form a population in the final state for the spectroscopy of interest, e.g. in the vibrationally excited state |f >< f | for (Stokes) Raman spectroscopy. The appropriate coherence or population is calculated using A.21 at each (0)

order k following calculation at order k-1, usually starting from ρˆii (0)=0, as discussed above. The contributions to the density matrix from all possible WMEL diagrams for the spectroscopy of interest are added together. In this work, A.8 is then used to find Pˆ (3) , i.e. the third order polarization density amplitude, which is then beat against the local oscillator field, ELO , to find the signal at the detector. The signal at third order is given by

S(3) =

−ΩLO ∗ ˆLO Im[E · Pˆ (3) ]. 2

(A.22)

where Ω is the local oscillator frequency and Pˆ (3) is defined by

P(3) (t) = Pˆ (3) eiΩt + Pˆ ∗(3) e−iΩt .

(A.23)

. That is, the polarization response of the system at frequency Ω is beat against the field at the detector, either from a stimulating beam or the vacuum, at frequency ΩLO equal to Ω.

A.3

Density Matrix Derivation of the Raman Excitation Profile

We now use the WMEL diagrams in Fig. A·1, which describe the (Stokes) Raman process, and repeated application of A.21 to guide our derivation of the Raman excitation profile. One can see by inspection of the WMEL diagrams that the Raman process requires three applications of the electric field and is thus referred to as a third order scattering process and involves calculation of the density matrix to third order. One could also write down WMEL diagrams where the order of arrows at t2 and t3 are reversed. The line shapes associated with these processes, however, form excited state populations and thus can be

102 shown to involve the broader electronic line width and, although they are valid WMEL diagrams, contribute to flourescence processes and not the Raman process. The application of the electric field at t1 in the first WMEL diagram in Fig. A·1 takes (0)

(1)

ρii (0) to ρei (t1 ), calculated using the first term in A.21 for ket side evolution:

(1) ρei (t1 )

i = 

t1

ˆ1 (t ) dt E



(0)



μ ˆel ρˆli (t )e(iωei +γei )(t −t1 ) .

(A.24)

l

−∞

The sum over l is has only one non-zero term at l=i since only ρii is initially non-zero (thermally populated) at t=0. Depending on phonon energy, level f might also have some small (0)

initial thermal population as well, i.e. ρf f (0)=0; however, this would only be important if we were to calculate the anti-Stokes REP. The incident electric field is assumed to be a linearly polarized, monochromatic plane wave, i.e. of the form ˆ r) ˆ ˆ1 (t ) = E1 ei(Ω1 t−k·ˆ eˆ + E1∗ e−i(Ω1 t−k·ˆr) eˆ∗ . E

(A.25)

where E1 is the amplitude (constant and, from here on, assumed real) and Ω1 the incident frequency. The subscript differentiates fields of different frequencies and amplitudes. Formally, this derivation first calculates the REP for stimulated Raman involving a second ˆ2 at Ω2 , perhaps in a second direction or perhaps supplied by the same broadapplied field E ˆ2 |2 that appears is replaced band laser encompassing both frequencies. Later, the term |E by a term representing spontanteous emission into the vacuum. This “fix up,” discussed in more detail below, is necessary since the semi-classical approach used here can not properly treat spontaneous emission. In either case, Ω2 represents the scattered frequency. We also mention here that the dot products between, for example, the incident field and the dipole moment, are ignored since we are after the frequency dependence of the Raman signal and not an absolute cross-section for the Raman process. The electric field is assumed parallel to the induced dipole and thus the dot product a maximum; other geometries could lead to an overall signal reduction but, given a fixed direction to the detector, will not affect the frequency dependence of the signal at that detector. Obviously, there may be dipole

103 non-allowed directions where we would want to avoid placing our detector. The integral in A.24 is solved by using the rotating wave approximation, or RWA. We note that only one term in A.25 will significantly contribute to the integral in A.24 while for the other term the integrand oscillates quickly over the range of the limits of integration and thus its contribution time averages to zero. The frequency component that is retained in the RWA is shown in Fig. A·2. Using a simple change of variables and defining

Δ1 ≡ ωei − Ω1

(A.26)

we now can directly integrate A.24 to find

(1)

ρei (t1 ) =

i 1 (0) E1 μ ˆei ρˆii e−iΩ1 t1 .  iΔ1 + γei

(A.27)

Δ1 ≡ ωe i − Ω1 ,

(A.28)

Defining

the second WMEL diagram in Fig. A·1 similarly yields

(1)

ρe i (t1 ) =

i 1 (0) E1 μ ˆe i ρˆii e−iΩ1 t1 .  iΔ1 + γe i

(A.29) (1)

The second electric field interaction is calculated by applying A.21 again which takes ρei (t1 ) (2)

to ρf i (t2 ):

(2) ρf i (t2 )

i = 

t2 ˆ2 (t1 ) dt1 E −∞



(1)

μ ˆf l ρˆli (t1 )e(iωf i +γf i )(t1 −t2 ) .

(A.30)

l (1)

(1)

The sum over l now has two terms since both ρei (t1 ) and ρe i (t1 ) are non-zero. We again invoke the RWA to find

μ ˆf e μ μ ˆ f e μ ˆei ˆ e i i 1 (2) (0) + )ei(Ω2 −Ω1 )t2 . ρf i (t2 ) = ( )2 E1 E2 ρˆii (  iΔ1 + γei iΔ1 + γe i iΔv + γf i

(A.31)

104 where we have defined

Δv ≡ ω f i + Ω2 − Ω 1

(A.32)

and made the assumption that Ω2 − Ω1 ∼ Ω2 − Ω1 , i.e. the energy of the phonon is negligibly affected by the electronic state of the nanotube. (1)

(1)

ρf e (t3 ) and ρf e (t3 ) are now similarly calculated using bra side evolution:

μ ˆ f e μ ˆ e i μ ˆf e μ ˆei i 1 1 (3) (0) + )ˆ μie eiΩ2 t3 . ρf e (t3 ) = −( )3 |E1 |2 E2 ρˆii (    iΔ1 + γei iΔ1 + γe i iΔv + γf i −iΔ2 + γf e (A.33)

μ ˆf e μ μ ˆ f e μ ˆei ˆ e i i 1 1 (3) (0) + )ˆ μie eiΩ2 t3 . ρf e (t3 ) = −( )3 |E1 |2 E2 ρˆii (  iΔ1 + γei iΔ1 + γe i iΔv + γf i −iΔ2 + γf e (A.34) where we have defined

Δ2 ≡ ωef − Ω2 ; Δ2 ≡ ωe f − Ω2

(A.35)

We are now ready to calculate the third order induced polarization density P(3) using A.8. Specifically,

(3)

(3)

ρˆ(3) = ρˆf e + ρˆf e + complex conjugate terms.

(A.36)

Using A.23 (which removes the exponential terms), we find that the induced polarization density amplitude is then given by

μ ˆf e μ μ ˆ f e μ μ ˆef μ μ ˆ e f μ ˆei ˆ e i ˆie ˆie i 1 (0) + )( + ) . Pˆ (3) = − 3 |E1 |2 E2 ρˆii (  iΔ1 + γei iΔ1 + γe i −iΔ2 + γf e −iΔ2 + γf e iΔv + γf i (A.37) We now make two assumptions: (1) that Δ1 ∼ Δ2 ≡ Δ and Δ1 ∼ Δ2 ≡ Δ , i.e. that

105 the emitted photon energy is ∼ ELaser − EPhonon , which is a very good assumption for the narrow Raman line widths here, and (2) that γie ∼ γie ∼ γf e ∼ γf e ≡ γ, i.e. that all the rates involving electronic transitions are dominated by the electronic decay rate and are thus approximately equal. Note that γf i , which contains only the vibrational decay rate, is not included in the approximation and will determine the Raman line width. Using ˆ∗ji and A.22, we find μ ˆij = μ S(3) =

μ ˆ f e μ ˆf e μ ˆei ˆe i 2 γf i Ω2 (0) μ + | |E1 |2 |E2 |2 ρˆii | . 3  2 iΔ + γ iΔ + γ Δ2v + γf2i

(A.38)

Note that the signal is linear in both I1 = |E1 |2 and I2 = |E2 |2 . This is, in fact, the stimulated Raman signal in the linear regime (i.e. where the assumption of small perturbation is valid). To obtain the spontaneous Raman signal, we replace |E2 |2 with Ω32 4Π2 c3 .

This factor comes from counting vacuum modes and this “fix up” can be shown to

yield the correct result using a fully quantum mechanical treatment rather than the semiclassical approach used here. However, this is beyond the scope of this thesis. The result for spontaneous Raman is

S(3) =

μ ˆ f e μ ˆe i 2 γf i ˆf e μ ˆei Ω42 2 (0) μ + | |E | ρ ˆ | . 1 ii 82 π 2 c3 iΔ + γ iΔ + γ Δ2v + γf2i

(A.39)

Notice the Lorenztian lineshape centered at Δv =0 with width γf i for a given Ω1 , i.e., for a given E Laser . The Raman excitation profile is given by sitting at Δv =0 and varying E Laser . This line shape is discussed in detail in the next section and contrasted with REP line shapes from three other models. Note that the line shape will peak when either Δ or Δ go to zero, reflecting the “incoming” and “outgoing” resonances of the solid state approach discussed in Chapter 3. Also note that the linewidths of these peaks are given by γ, i.e., the REP line width reflects the dominant electronic decay line width with which the laser is resonant. Finally, note the non-resonant background proportional to the fourth power of the scattered frequency, Ω2 .

106

A.4

Comparison of REP Models

In this section we compare the REP line shape derived in the previous section with the REP line shape from three other models, namely, (1) the free electron hole solid state approach, (2) the solid state approach including excitonic corrections, and (3) Raman transform theory. We will not derive each of these line shapes here, of course, but simply state the results, ignoring prefactors since we are interested here only in the shape of the curves and not any absolute cross section. We first write down explicitly the REP line shape from the previous section, taking Δv =0 and ignoring prefactors:

S∝|

μ ˆ f e μ ˆei ˆ e i 2 μ ˆf e μ + | .  iΔ + γ iΔ + γ

(A.40)

The typical solid state approach assuming free electrons and holes yields the following line shape (Cardona and G¨ untherodt, 1982):

S ∝ |√

1 1 |2 . −√  iΔ + γ iΔ + γ

(A.41)

Including excitonic interactions and ignoring contributions from all but the lowest exciton (which dominates the optical response of the system) yields the following line shape (Vamivakas et al., 2006a):

S∝|

1 1 |2  iΔ + γ iΔ + γ

(A.42)

Finally, the Raman transform method which uses a Kramers-Heisenberg approach to find the REP from the absorption spectrum. Here we go into a bit more depth though the complete derivation is lengthy and somewhat complicated. (Stallard et al., 1983) Specifically, the adiabatic approximation is invoked with respect to the vibrational degree of freedom of interest. That is, the total wavefunction of the molecule, nanotube, etc. is assumed to be separable into a direct product of a part that is a function of the vibrational coordinate

107 and a part that is not, i.e. depends on everything else. In the Condon approximation, the electronic excitation is assumed to be virtually instantaneous on the time scale of the nuclear recoil by virute of the very different masses of the excited electron and the nucleus. The matrix element is assumed independent of nuclear coordinate and thus the total cross section for the process in question will include an overlap integral between the initial and final vibrational states. This overlap integral is refered to as the Franck-Condon factor and is not necessarily largest for initial and final states with equal numbers of vibrational quanta. Molecular transitions, which are proporational to the square of the overlap integral, often favor the creation of one or more vibrational quanta upon electronic excitation. These are known as “vibronic” transitions. It is possible to relax the assumption that the electronic transition matrix element does not depend on the nuclear coordinate and rather to do an expansion of the matrix element about its equilibrium value. (Stallard et al., 1983) This process introduces so-called “non-Condon” factors, C, which reflect the degree to which the Condon approximation is (not) valid. The resulting REP line shape is given by (Shreve et al., 2007; Albrecht et al., 1994):

S ∝ |(1 + C)Φ(Δ) − (1 − C)Φ(Δ )|2

(A.43)

where Φ is calculated using a Kramers-Heisenberg transform of the absorption spectrum A(ω): 1 Φ(ω) = P π

∞ ∞

dω 

iA(ω) A(ω) + − ω) ω

ω  (ω 

(A.44)

where P denotes taking the principle part. In order to compare this model to the others, we assume a Lorentzian line shape centered at ωei with width γ for the absorption spectrum A(ω) which we then transform (by use of Mathematica ) to find Φ and thus the line shape. In Fig. A·2 we plot the normalized line shapes from each model using the same relevant energies, as shown, representing typical values for the G+ phonon for a CNT with dt ∼1 nm. In Fig. A·2a the non-Condon parameter is set to zero for the Raman transform model and

108

Raman Transform

Exciton Model

a)

Free e- hole

4 Level Molecular

b) 1

1

0

Eii

C = -0.3 ȝie’ = ȝfe ȝie = 0.5 ȝfe’

Normalized Intensity

Normalized Intensity

C=0 ȝie’ = ȝfe ȝie = ȝfe’

Eii + Ephonon Eii = 1.5 eV

0

Eph = 0.2 eV

Eii

Eii + Ephonon

ī = 15 meV

Figure A·2: REPs predicted by different models. (a) Non-Condon factor set to zero in Raman transform model; all dipole moments equal in four level molecular model. The Raman transform and exciton model curves are identical in this case. (b) Non-Condon factor set to -0.3; μie = 0.5 μf e .

109 all dipole moments are assumed equal in the four level molecular model. Notice the much greater spectral weight between peaks predicted by the free electron hole model, shown in green, and the near zero lack of spectral weight predicted by the 4 level molecular model, shown in black. Also note that, for C=0, the line shapes are identical for the excitonic and Raman transform methods. In Fig. A·2b, with C=-0.3 and μie = 0.5 μf e , the Raman transform line shape, shown in red, has split off from the excitonic line shape, shown in blue. These parameters have the effect of increasing the outgoing resonance relative to the incoming resonance, as supported by both the experimental work of Shreve et al. (Shreve et al., 2007), who find best fits to RBM REPs using C=0, and by our own preliminary data which shows a larger outgoing resonance peak. Note that this degree of freedom is not present in the two solid state approaches; these results imply that the traditional solid state approach fails to capture the coupling between the electronic and vibrational degrees of freedom. Clearly, given a full G+ REP, with its widely separated peaks and with sufficient signal to noise, the data could be fit by each model in order to differentiate them. This, however, is experimentally difficult to do since it requires tuning the laser over a large range, would require at least four different laser line rejection filters, and spans a wide specrtal range over which the system response varies widely. Having said that, given a good standard candle such as, perhaps, the 520 cm−1 Si Raman peak to which the data might be normalized, this should be possible and constitutes future work. The full RBM REP, though easier to obtain experimentally, would not easily differentiate between the various models since the incoming and outgoing peaks overlap. Either the Raman transform or the 4 level molecular REP can be used to extract information about how good the adiabatic approximation is in carbon nanotubes, at least with respect to the G+ phonon degree of freedom. Further, the 4 level molecular REP could perhaps be used to quantify the relative dipole moment strengths of the various molecular transitions.

110

A.5

Conclusion

In this section we reviewed the density matrix formalism which we used to perturbatively derive the REP for carbon nanotubes. The CNT is modeled by a four level molecular system. The resulting REP line shape is compared with three other REP models and possible future work described which could differentiate between these various models.

Appendix B

Modified Black Body Emission B.1

Introduction

In this Appendix, we present our work modeling the electroluminescence from carbon nanotube devices under applied voltage. We start by briefly reviewing black body theory. We then describe the samples and provide experimental details before fitting the data. We show that a density of state modified black body model fits the data well.

B.2

Black Body Theory

We begin by briefly outlining standard black body theory. The reader is directed to Loudon’s The Quantum Theory of Light (Loudon, 2000) (or other suitable reference) for the details of the derivation. In Fig.B·1 we illustrate mode counting in a box of volume V. The mode density is found to scale with ω 2 and the classical treatment stops here. Obviously, there is a problem at high frequency, referred to as the ultraviolet catastrophe. The resolution to the problem lies in quantizing the electromagnetic field and the resulting Bose-Einstein distribution. The short story is that, though more modes are available at higher frequency, those modes are exponentially less likely to be populated. This is typically where an undergraduate treatment of black body radiation ends but a further review is needed here before proceeding. A review of the limited literature relating to black body emission from carbon nanotubes reveals a misunderstanding of the subject. Often, CNT emission is treated at a sum of emission relating to electronic transitions, specifically photoluminescence, and black body radiation treated as a background. In fact, all emission must consider the modes of the

111

112

Quantization Mode Density

x

L

e

L

n n n 2 y

!Z

E

2

2 z

4L O2

!Z k bT

1

2000 K

=

Energy Density

V

2 x

Black Body

Bose Einstein Distribution

1750 K

1250 K

Wavelength (nm)

Figure B·1: Standard black body theory. electromagnetic vacuum into which the sample emits. For instance, the Purcell effect arises when the volume of the “box,” e.g. a microcavity, in which the emitter sits is made small, thereby modifying the mode density and minimum mode energy of the surroundings into which the sample emits. The energy spectrum of both the emitter and the environment must be considered, even when the environment is the lab. The proper treatment is given by

S(ω, T ) = (ω, T ) ρ(ω) < E(ω, T ) >

(B.1)

where the spectrum S depends on the sample emissivity , the electromagnetic mode density ρ of the “box” into which the sample is emitting, the average energy per mode < E >, and the temperature T. The spectral emission of any material in thermal equilibrium includes an emissivity prefactor that modifies the blackbody response. While often taken as a constant or as a slowly varying function of wavelength, the emissivity should properly reflect the dielectric function of the material and is therefore a complex function of the electronic system, phonons, plasmons, etc. Formally, the emissivity is defined as the ratio of the energy radiated by the material to that emitted by an equally size perfect black body at the same temperature and is ≤1 for all frequencies. When it is assumed be frequency

113 independent, i.e. a constant, the material is termed a “grey-body.” In the visible and near infrared region of the spectrum, the emissivity is dominated by the electronic structure of the carbon nanotubes and will be modeled accordingly below. In a 1D material such as carbon nanotubes, with sharp peaks in the density of states in the optical region of the spectrum, the absorptivity, α, is obviously far from being a slowly varying function of the frequency and, in thermal equilibrium, the emissivity of a material equals its absorptivity. (Butcher, 1951) This equivalance, known as Kirchhoff’s Law, arises from simple energy conservation and the principle of detailed balance. (Burkhard et al., 1972) Therefore the emissivity  may be replaced by the absorptivity α in B.1. The validity of the thermal equilibrium assumption in our experiment is discussed in the next section.

B.3

Experimental Details

The samples used in this expertiment were fabricated and by Nantero, Inc. of Woburn, MA. Mats of single wall carbon nanotubes were grown between contacts by the CVD process and include all different chiralities with diameters between approximately 0.8 to 1.5 nm. Further specifics of the device are not presented due to trade secret considerations. However, the concept is simple: a DC voltage is applied between contacts resulting in light emission, following either electron-phonon scattering (Perebeinos et al., 2005) or carrier impact ionization (Chen et al., 2005), from the lowest bright exciton at E 11 and possibly E 22 from each of the various nanotube species present in the mat. The device is operated at room temperature in nitrogen gas in a custom-made flow cell in order to prevent burn out of the nanotubes resulting from reactivity with oxygen. The electroluminescence is collected using a long working distance, high numerical aperture objective (50X / 0.5 NA) through a quartz coverslip inspection window in the flow cell and sent to the spectrometer and a silicon CCD detector. The raw spectra are corrected for grating and CCD response before fitting. The primary difficulty lay in the transient nature of the emission signal. Even in the flow cell, the current appeared to follow a winding and time varying path from source to

114 drain through the CNT mat as nanotubes burnt out and the current sought a new path of least resistance. The emission was typically bright enough at spots to be visible to the inspection camera on the microscope but could jump around spatially at higher applied biases. We were therefore careful to operate at a voltage low enough such that the emission appeared stable over the time scale that the spectra shown here were collected. Having said that, it would seem that perhaps the thermal equilibrium assumption is in doubt. However, the requirement of thermal equilibrium does not imply the CNT mat need be at the same temperature as the substrate or even the radiation field (Burkhard et al., 1972). Further, thermalization in carbon nanotubes is known to occur on the sub-picosecond time scale. Finally, the excellent fit to the data provides a posteriori confirmation of the applicability of Kirchhoff’s law and the use of the black body model.

B.4

Data Fitting and Results

In this section, we fit the electroluminescence data using a density of states modified black body curve. The emissivity is modeled with six Lorentzian curves. The choice of six Lorentzian components is not an arbitrary one. In order to explain this choice we must first revisit how nanotubes may be thought of as being formed from a single layer of sp2 bonded carbon atoms known as graphene. As discussed briefly in Chapter 2, there are a variety of nanotube species of various diameters and chiralities. The graphene wrapping vectors, shown in the inset of Fig. B·2, that specify these various nanotube species are characterized by the multiples, n and m, of the graphene translation vectors, a and b, respectively, used to form them. For example, the (n,m)=(5,1) nanotube is formed using a wrapping vector = 5 a+1 b (which defines a circumference of the nanotube.) When the electronic transitions E 11 and E 22 of the semiconducting nanotubes (i.e. those that will electroluminesce) are plotted against their diameter, as shown in Fig. B·2, so-called “family” patterns emerge. (Weisman and Bachilo, 2003) Families are grouped, for example, by having a common 2n+m value. Further, they are split into tubes with either ν ≡(n-m)Mod3=+1 or -1. The specifics of how these patterns emerge is not critical to our discussion of black body emission; however,

115

1200 a 1100

Optical Transition Energy Eii (nm)

CCD Detection Range

b 988 +/- 6 nm

1000

879 +/- 3 nm 900

842 +/- 1 nm 796 79 7 6 +/- 5 nm

800

740 +/- 4 nm 700

600

E11(Mod Ȟ = +1 1 -) E11 E Ȟ = -1 2 +) E1111(Mod E22 (Mod Ȟ = +1 1 +) E22 E Ȟ = -1 2 -) E22 22 (Mod

602 +/- 2 nm nm2

500 5

6

7

8

9

10

11

12

13

14

15

Nanotube Diameter (Å) Approximate CNT Diameter Distribution

Figure B·2: Kataura plot of the E 11 and E 22 transition energies. Detector sensitivity and nanotube diameter ranges are highlighted. The six Lorentzian centers from the fits in Fig. B·3 (range of fit values given after the ±) are shown correlated to each ν=-1 CNT family. Inset: Graphene translation vectors and the (5,1) nanotube wrapping vector.

116 it turns out the (n-m)Mod3=-1 nanotubes have, on average, a stronger electron-phonon coupling (Goupalov et al., 2006; Yin et al., 2007). Since the ballistic transport in nanotubes is limited by LO phonon scattering (Perebeinos et al., 2005) we expect electroluminescence from the (n-m)Mod3=-1 family to be stronger since only approximately zero center of mass momentum excitons may emit a photon. The exciton must scatter in order to give up its mometum before it can radiatively decay. Refering to the Kataura plot in Fig. B·2, we see that our detector sensitivity range and nanotube diameter distribution define six (n-m)Mod3=-1 families from which we expect to see electrolumiscence. These families fall within fairly narrow spectral ranges which match very well the best fit values, shown in Fig. B·2, of the the six Lorentzian peaks used to fit the data shown below in Fig. B·4. The ± values shown give the range of best fit values found from fitting the data, shown below, at the different voltages. In essence, we have measured the emissivity of the mat which is dominated by the electronic system in this spectral range. In theory, we could calculate the emissivity using published values for the electron-phonon and electron-photon matrix elements for each species and then compare to the measured emissivity; however, the relative numbers of the different nanotube species present in the mat is unknown and can not be assumed constant. Therefore, no so such calculation is attempted here. We show in Fig. B·3 fits to the same (corrected) data using three different functional forms. In Fig. B·3a, we fit the data using four Lorentzians, each allowed to vary in center wavelength, width, and height. These Lorentzians are meant to model the optical response of the system, i.e. the emissivity prefactor in B.1. What electronic transitions these Lorentzians might represent is discussed below. In Fig. B·3b, we fit the same data using four Lorentzians with a black body envelope which does a much better job fitting the high energy tail but is clearly missing a component in the peak at ∼ 850 nm. Notice that treating the emission as the addition of black body radiation and optical response can not reduce the contribution of the Lorentzian tails to the spectral weight at high energy and thus will not fit the data either, as has been suggested in the literature. In the proper treatment used here, the high energy behavior of the black body curve exponentially suppresses the high

117 Corrected Data

600

700

800

900

1000

500

1000 500 600600 700 700 800 800 900 900 1000 Wavelength (nm) Wavelength (nm)

500 600 Wavelength 700 800 900 1000 (nm) Wavelength (nm)

Can’t fit shoulder Poor peak and at ~ 600 nm due shoulder fit to tail of Lorentzian

4 Lorentzians x Black Body Envelope

(c)

6 Lorentzians x Black Body Envelope

Intensity (Arb. Units)

Intensity (Arb. Units)

500

Intensity

(b)

4 Lorentzian Fit

Intensity

Intensity (Arb. Units)

(a)

Fit

Intensity

Raw Data

500

Fits ~ 600 nm peak and shoulder but flat peak implies another Lorentzian component is needed

600

700

800

900

1000

500 600 Wavelength 700 800 900 1000 (nm) Wavelength (nm) Nearly perfect fit

Data taken at 15.5 V / 18.1 ȝA

Figure B·3: Three different functional fits to the same electroluminescence data. The raw data, shown in black, is corrected for CCD and grating efficiency with the result shown in green. Fit curves are shown in red. The corrected data is fit using (a) four Lorentzians, (b) four Lorentzians with a black body envelope, and (c) six Lorentzians with a black body envelope. Deficiencies in the fits in (a) and (b) are highlighted. energy Lorentzian tails. In Fig. B·3b we add two more Lorenztian components resulting in an excellent fit to the data. In Fig. B·4 the six Lorentzian functional form is fit to five data sets taken at different applied voltages. The best fit temperatures range from 1235 to 1300 K. The best fit peak center values varied at most ±4 nm and best fit FWHM values ±5% across the applied voltage range.

B.5

Conclusion

In this Appendix, we discussed how treating nanotube emission as the sum of photoluminescence and black body radiation is erroneous and demonstrated how electroluminescence data is fit well using an emissivity modified black body curve. The best fit optical response of the CNT mat is shown to correlate extremely well with nanotube family structure.

118

700 800 500 600600 Wavelength 700 (nm) 800 900900 1000 1000 Wavelength (nm)

Intensity

Intensity (Arb. Units)

Intensity

Intensity (Arb. Units)

500 600600 700700 800800 900900 1000 1000 Wavelength (nm) Wavelength (nm)

500

15.0 V 17.9 ȝA

14.0 V 17.0 ȝA

Intensity

Intensity (Arb. Units)

Intensity 500

500 600600 700700 800800 900900 1000 1000 Wavelength (nm) Wavelength (nm)

500

14.5 V 17.6 ȝA

13.5 V 16.3 ȝA

Intensity (Arb. Units)

y(

Intensity

)

13.0 V 13.6 ȝA

500 600600 700700 800800 90090010001000 Wavelength (nm) Wavelength (nm)

500

Raw Data Corrected Data Fit

500

700 500 600600 Wavelength 700 800 800 900900 1000 1000 (nm) Wavelength (nm)

Temperatures 1235 K to 1300 K

Figure B·4: Fits to the electroluminescence data at different applied voltages using six Lorentzians and a black body envelope. Note the persistent excellent fit.

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Curriculum Vitae Andrew Gerald Walsh Contact information: Department of Physics Boston University 590 Commonwealth Avenue Boston, MA 02215, USA

Email: [email protected] Phone: (+1) 617 353-1712 Fax: (+1) 617 353-9947

Date & Place of birth: March 11th 1970, Chelsea, MA USA.

Education: Ph.D., Physics, Boston University (2003-2009). B.S., Applied and Engineering Physics, Cornell University (1988-1992).

Licensure: Physics 8-12, Preliminary, Massachusetts Department of Education. Mathematics 8-12, Preliminary, Massachusetts Department of Education. Mathematics 5-8, Preliminary, Massachusetts Department of Education.

Fellowships: 2007–2008 Photonics Center Fellow, Boston University 2004–2007 Research Assistant, Department of Physics, Boston University 2003–2004 NSF GK-12 Fellow, Department of Physics, Boston University

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Research papers: 9. Spectroscopic properties unique to nano-emitters A. G. Walsh, W. Bacsa, A. N. Vamivakas, and A. K. Swan Submitted 8. Scaling of exciton binding energy with external dielectric function in carbon nanotubes ¨ u, B. B. Goldberg, A. G. Walsh, A. N. Vamivakas, Y. Yin, S. B. Cronin, M. S. Unl¨ and A. K. Swan Physica E., 7 2375 (2008). 7. Screening of Excitons in Suspended, Single Wall Carbon Nanotubes ¨ u, B. B. Goldberg, A. G. Walsh, A. N. Vamivakas, Y. Yin, S. B. Cronin, M. S. Unl¨ and A. K. Swan Nano Letters, 7 1485 (2007). 6. Optical determination of electron-phonon coupling in carbon nanotubes ¨ u, B. B. Goldberg, and A. K. Swan Y. Yin, A. N. Vamivakas, A. G. Walsh, M. S. Unl¨ Physical Review Letters, 98 037404 (2007). 5. Exciton-mediated one-phonon resonant Raman scattering from one-dimensional systems ¨ u, B. B. Goldberg, and A. K. Swan A. N. Vamivakas, A. G. Walsh, Y. Yin, M. S. Unl¨ Physical Review B, 74 205405 (2006). 4. Tunable resonant Raman scattering from singly resonant single wall carbon nanotubes Y. Yin, A. G. Walsh, A. N. Vamivakas, S. B. Cronin, A. Stolyarov, M. Tinkham, W. ¨ u, B. B. Goldberg, and A. K. Swan Bacsa, M. S. Unl¨ IEEE Journal of Selected Topics in Quantum Electronics, 12 1083 (2006). 3. Temperature dependence of the electronic transition energies in carbon nanotubes: The role of electron-phonon coupling and thermal expansion S. B. Cronin, Y. Yin, A. G. Walsh, R. B. Capaz, A. Stolyarov, P. Tangney, M. L. ¨ u, B. B. Goldberg, and M. Tinkham Cohen, S. G. Louie, A. K. Swan, M. S. Unl¨ Physical Review Letters, 96 127403 (2006). 2. Chirality dependence of the radial breathing phonon mode density in single wall carbon nanotubes ¨ u, B. B. Goldberg, and A. K. Swan A. N. Vamivakas, Y. Yin, A. G. Walsh, M. S. Unl¨ arXiv:cond-mat/0609197. 1. Gray-scale response of multiple-quantum-well spatial light modulators W. S. Rabinovich, S. R. Bowman, R. Mahon, A. G. Walsh, G. Beadie, C. L. Adler, D. S. Katzer, K. Ikossi-Anastasiou Journal of the Optical Society of America B, 13 2235 (1996).

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Conference Papers & Presentations: 14. Time-resolved studies of carbon nanotubes Future of Light Symposium 2008 Boston, MA, USA, June 2008. 13. Combining optical spectroscopy and miccroscopy to image objects smaller than the size of the focal point Proceedings of the Nanotechnology Conference and Trade Show 2008 Boston, MA, USA, June 2008. 12. Time-resolved studies of carbon nanotubes Boston University Science and Engineering Day 2008 Boston, MA, USA, April 2008. 11. Pitfalls of nano-spectroscopy Bulletin of APS Meeting 2008 New Orleans, LA, USA, March 2008. 10. Screening of excitons in single suspended carbon nanotubes Boston University Science and Engineering Day 2007 Boston, MA, USA, April 2007. 9. Controlled screening of excitons in single, suspended carbon nanotubes Bulletin of APS Meeting 2007 Denver, CO, USA, March 2007. 8. One and two-phonon resonant Raman scattering from single wall carbon nanotubes MRS Fall Meeting 2006 Boston, MA, USA, November 2006. 7. The effect of excitons on Raman excitation profiles in one-dimensional systems Proceedings of IEEE Lasers and Electro-Optics Society Annual Meeting 2006 Montreal, Canada, October 2006. 6. Screening of excitons in single, suspended carbon nanotubes OSA Frontiers in Optics 2006 / Laser Science XXII Rochester, NY, USA, October 2006. 5. One and two-phonon resonant Raman scattering from single wall carbon nanotubes Near Field Optics 9 2006 Lausanne, Switzerland, September 2006. 4. Environmental manipulation of the electronic structure of suspended carbon nanotubes Bulletin of APS Meeting 2006 Baltimore, MD, USA, March 2006. 3. Manipulation of the Electronic Structure of Suspended Carbon Nanotubes MRS Fall Meeting 2005 Boston, MA, USA, November 2005.

129 2. Inelastic light scattering and light emission from single and double wall carbon nanotubes Proceedings of the Nanotechnology Conference and Trade Show 2005 Anaheim, CA, USA, May 2005. 1. Signal suppression in resonance Raman spectra from suspended carbon nanotubes Bulletin of APS Meeting 2005 Los Angeles, CA, USA, March 2005.