The efficiency of photovoltaic solar cells at low temperatures Honours Thesis Mathew Guenette August 4th 2006

Abstract This study examines the operation of photovoltaic solar cells at low tempertaures(T < 300K). The efficiency at which a GaAs solar cell converts light into electrical power was measured as a function of temperature. The efficiency was found to increase as temperature decreased and then begin to plateau at T ∼ 100K. These results can be explained by a decrease in energy loss from emission of radiation(radiative recombination) and most importantly a decrease in the activity of electronic defects in the solar cell (Shockley-Read-Hall recombination). Computer simulations showed that the relative contribution of radiative and Shockley-Read-Hall recombination increases towards the ideal radiative regime at low temperatures providing an environment which can be considered to be free of parasitic losses due to Shockley-Read-Hall recombination however experimentally this was found not to be the case due to the complex behaviour of Shockley-Read-Hall capture cross section with temperature.

1

Acknowledgements Type acknowledgements here.

Statement of Student Contribution All Matlab code written by myself and all computer simulations run by myself except for computer simulation of radiative efficiency in section 3.3 done with N.J. Ekins-Daukes. Building of experimental setup and all experimental measurements as outlined in sections 4 and 5 were done by myself.

I certify that this report contains work carried out by myself except where otherwise acknowledged. Signed ......................................... Date:

Contents 1

Introduction

2

Survey of the literature 2.1 World energy market . . . . . . . . . . . . . . . 2.2 Solar energy . . . . . . . . . . . . . . . . . . . . 2.3 The Landsberg efficiency . . . . . . . . . . . . . 2.4 Semiconductor solar cells . . . . . . . . . . . . . 2.5 Quasi-Fermi levels . . . . . . . . . . . . . . . . 2.6 The p-n junction solar cell . . . . . . . . . . . . 2.7 Electron-hole generation . . . . . . . . . . . . . 2.8 Electron-hole radiative recombination . . . . . . 2.9 Current-voltage characteristics . . . . . . . . . . 2.10 Shockley-Queisser efficiency limit . . . . . . . . 2.11 Shockley-Read-Hall non-radiative recombination 2.12 Advanced design concepts . . . . . . . . . . . .

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Theoretical model and computer simulation 3.1 Model overview . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Temperature dependant Shockley-Queisser calculation . . . 3.3 Temperature dependance of radiative efficiency . . . . . . . 3.4 Shockley-Queisser efficiency including SRH recombination . Experimental setup 4.1 Overview . . . . . . . . . . . . 4.2 In the vacuum chamber . . . . . 4.2.1 Temperature sensor . . . 4.2.2 Heater . . . . . . . . . . 4.2.3 Sample solar cell . . . . 4.3 Outside the vacuum chamber . . 4.3.1 Cryogenic system . . . . 4.3.2 Temperature Controller . 4.3.3 Variable light source . . 4.3.4 I-V source measure unit 4.4 Photogenerated current . . . . .

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Contents 5

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Results 5.1 Current-Voltage and power measurements . . . . . . . . . . 5.2 Variation of capture cross section with temperature . . . . . 5.3 Temperature vs efficiency with varying capture cross section 5.4 Uncertainties with power conversion efficiency . . . . . . . 5.4.1 Theoretical uncertainty . . . . . . . . . . . . . . . . 5.4.2 Experimental uncertainty . . . . . . . . . . . . . . . 5.5 Dark I-V curves . . . . . . . . . . . . . . . . . . . . . . . .

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Conclusion

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A Matlab Code

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B Solar Cell Mount Designs

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C Capture Cross Section Data and I-V curves C.1 Capture cross section data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Experimental uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Light I-V curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 42 42 43

References

46

Chapter 1 Introduction In the past extensive studies have been performed to test how the power conversion efficiency of photovoltaic solar cells changes with increased temperature however none have considered operation at low temperature. This is primarily due to the energy required for cooling far outweighing any benefits from increase of efficiency. This regime is interesting as it is hypothesised electronic defects will effectively become ’frozen out’ and the only efficiency losses will come from emission of radiation, which is thermodynamically unavoidable. This provides an environment in which solar cells can operate in the ideal radiative limit and a testing ground for new solar cell device concepts such as the intermediate band solar cell[1] which currently do not operate efficiently at room temperature. This thesis is organised into 6 chapters. Chapter 2 is a survey of the literature on the subject of photovoltaic solar cells and provides most of the background and theory necessary for this thesis. Chapter 3 discusses the theory and computer model used to predict the behaviour of solar cells and contains the results from several computer simulations. Chapter 4 outlines the experimental setup and equipment used to measure efficiency. Chapter 5 gives the results of the experiment and discusses how the results correlate with theory. Chapter 6 concludes the thesis and discusses possible associated future work.

4

Chapter 2 Survey of the literature 2.1 World energy market Most sources of energy on Earth originate from the Sun, the most dominant in today’s world energy market being the burning of fossil fuels. At the current rate of consumption it is estimated world supplies of oil will only last approximately another 30 years and coal supplies lasting another 250 years[2]. The urgency to find an efficient alternative energy source is compounded by the effect of global warming, predicted to result in catastrophic consequences including rising sea levels and increased frequency of extreme weather events.

2.2

Solar energy

The power density from the Sun just outside the Earth’s atmosphere is given by the solar constant 1353 ± 21W/m2 [3], dropping to approximately 1000W/m2 on average at the Earth’s surface[4] attenuated by absorption in the atmosphere. Figure 2.1 compares the extraterrestrial solar spectrum, Air Mass 0 (AM0) to the standard terrestrial AM1.5 spectrum. The standard for solar cell efficiency measurements is the AM1.5 spectrum with an energy density of 1000W/m2 . The extraterrestrial solar spectrum can be closely modeled as a 6000K blackbody spectrum with the generalised Planck equation[5]. n(E, T, µ) = ²(E)

2π E2 c2 h3 e E−µ kT − 1

(2.1)

where n is the photon flux as a function of energy E, ² is the emissivity,µ is the photon chemical potential and T is the temperature. For a blackbody, ² = 1 for all energies and µ = 0. The total energy density from the sun can be obtained by multiplying the photon flux by the photon energy, E, and integrating over all energies to obtain the Stefan-Boltzmann law. Z ∞ En(E, T )dE = σT 4 (2.2) 0

When T = Tsun = 6000K equation 2.1 describes the photon flux at the surface of the sun and equation 2.2 describes the energy density at the surface of the sun. On Earth we receive sunlight from 5

Chapter 2. Survey of the literature

6

Figure 2.1 Extraterrestrial AM0 or 6000K blackbody spectrum (solid line) compared with terrestrial AM1.5 spectrum(Dashed line).

the solar disc subtending only a fraction of the hemisphere visible to the solar cell thus a dilution factor of fω = 2.16 × 10−5 [6] must be included to calculate the photon flux and energy density on Earth.

2.3

The Landsberg efficiency

At it’s most basic level a solar cell is an electronic heat engine that is driven by the temperature difference between the Sun and the cell. The Landsberg efficiency[7] is the ultimate thermodynamic efficiency limit at which solar energy can be harnessed. It can be calculated by considering some general absorber(Figure 2.2) which perfectly absorbs the energy flux Eabs from the Sun. The Landsberg efficiency unrealistically assumes the absorber can absorb photons over all angles via the use of an infinite lens with no reflection and no entropy is generated upon absorption[8]. For an absorber with unit area, Eabs is given by the Stefan-Boltzmann law, σTS 4 , where TS is the temperature of the Sun. This energy flux also has an associated entropy flux related via maxwells relations, Sabs = 34 ETabs , which can be calculated to be Sabs = 34 σTS 3 [9]. Any body which absorbs radiation from the sun, must also emit radiation back to the sun according to its temperature. For an absorber with temperature TA , Eemit = σTA 4 and Semit = 43 σTA 3 . Work is defined to be entropy free energy, so the excess entropy (and associated energy) not radiated back to the sun must be dumped to some reservoir with temperature TR in order to extract useful work from the system, with TR = TA for maximum efficiency. The efficiency for such a perfect absorber can be calculated to be 4 TA 1 TA 4 Ework ) ≡ ηL = 1 − ( − Eabs 3 TS 3 TS 4

(2.3)

This is only dependant on the temperature difference between the Sun and the absorber with ηL = 93.3% for TA = 300K and approaching 100% as TA approaches 0K.

Chapter 2. Survey of the literature

7

Figure 2.2 A general absorber showing energy and entropy absorbed from the sun, radiated back to the sun and dumped to a reservoir, resulting in useful work.

The Landsberg efficiency is the efficiency limit for a general absorber and does not consider the realistic unavoidable entropy generation upon photon absorption[10]. When this is considered the efficiency at 300K drops to 86%.

2.4

Semiconductor solar cells

Conventional solar cells are made from semiconductors and incur additional efficiency losses over the general absorber considered in the Landsberg efficiency. Solar cells are based on the idea of the photovoltaic effect, where an electron can be excited across the band gap by absorbing a single photon of sufficient energy. No electronic states exist in a semiconductor’s band-gap and consequently cannot be described as a blackbody. Figure 2.3 (a) outlines the generation processes in a semiconductor solar cell with band-gap energy Eg . (1) Any incoming photon with an energy E < Eg will not have enough energy to excite an electron across the band-gap, the solar cell is transparent to these photons and their energy is wasted. (2) A photon with energy E = Eg will have just enough energy to excite an electron across the bandgap and into the conduction band creating an electron-hole pair. (3) A photon with E > Eg will be excited high into the conduction band and will quickly relax back to the band-gap edge via many phonon interactions. The difference in energy between the band-gap energy and the photon energy is lost in the form of heat. Figure 2.3 (b) outlines additional losses associated with the recombination of electrons and holes in a semiconductor solar cell. (4) Radiative recombination occurs as an electron in the conduction band drops down across the band-gap to the valence band and recombines with a hole, losing energy in the form of an emitted a photon. (5) Shockley-Read-Hall (SRH) recombination occurs when an electron recombines with a hole via defect impurity states in the semiconductor[11, 12]. Generation and recombination processes will be discussed in more detail in later sections.

Chapter 2. Survey of the literature

8

(a) Generation

(b) Recombination

Figure 2.3 (a) Radiative generation in a semiconductor solar cell. (1) Transparent for incoming photon with E < Eg .(2) Incoming photon with E = Eg will have enough energy to excite an electron across the band-gap.(3) Incoming photon with energy E > Eg will excite an electron high into the conduction band and quickly relax to the band gap edge(b)Radiative and non-radiative recombination in a semiconductor solar cell.(4) Electron drops to across the band-gap and emits a photon. (5) SRH recombination, electrons and holes recombine via defects

2.5

Quasi-Fermi levels

A Fermi level describes the electron energy at which 50% occupancy is attained. For an intrinsic semiconductor in equilibrium the Fermi energy is given by Ec + Ev 3kT mh + ln (2.4) 2 4 me where Ec and Ev are the energy levels of the conduction and valence bands respectively, k is Boltzmann’s constant, T is the temperature and mh and me are the effective masses of the holes and electrons respectively[13]. At room temperatures and lower the second term is small hence the Fermi energy lies very close to the centre of the band gap. When an impurity is introduced to an intrinsic semiconductor the Fermi energy must shift to ensure charge neutrality. When a donor state is introduced with an energy level close to the conduction band (n-type), the Fermi energy will shift towards the conduction band. When a acceptor state is introduced with an energy close to the valence band(p-type), the Fermi energy will shift down towards the valence band. A single Fermi level is sufficient to describe a solar cell in equilibrium however when the cell is illuminated and/or a voltage bias is applied it is no longer in equilibrium. Quasi-Fermi levels Ef =

Chapter 2. Survey of the literature

9

Figure 2.4 Energy levels for a p-n junction. (a) In equilibrium (b)With an applied bias resulting in a separation of quasi-Fermi levels µ

describe the occupancy of the conduction and valence band individually when the electron and hole populations are in respective equilibrium, but not in equilibrium with each other. The dashed lines on figure 2.3 indicate the quasi-Fermi levels of the conduction and valence band with an applied bias V. The separation of the quasi-Fermi levels is µ which is equal to the applied bias V at the terminals when E is measured in electron-volts[5].

2.6

The p-n junction solar cell

Solar cells require some form of built in asymmetry that will allow useful power to be extracted before electrons and holes recombine. The majority of solar cells consist of a p-n or p-i-n junction to allow high carrier mobility and current to only flow in one direction[14, 13]. A typical energy level diagram for a p-n junction is illustrated in figure 2.4. It should be noted the separation of quasi-Fermi levels stays constant throughout the device assuming infinite mobility, in reality this is a reasonable approximation for good quality solar cells[15]. When there is an abrupt transition from p-type doping to n-type doping the electrons and holes will diffuse to form a region of lower electron and hole concentration known as the depletion region[13]. The depletion region width is given by s NA ND 1 1 2²s kTcell ln ( ( + )) (2.5) W = 2 2 e ni NA ND where ²s is the permittivity of the material, k is Boltzmann’s constant, Tcell is the temperature of the cell, e is the charge of an electron, NA and ND are the acceptor and donor densities respectively (ie the p-type and n-type doping densities) and ni is the intrinsic density of states which is given by ni = (

p

−Eg

Nc Nv )e 2kTcell

(2.6) ∗

3

cell )2 , where Eg is the band gap, Nc is the density of states in the conduction band Nc = 2( 2πmehkT 2

Chapter 2. Survey of the literature

10

Nv is the density of states in the valence band Nv = 2( electron and hole masses respectively.

2.7

2πm∗h kTcell 3 )2 h2

and m∗e and m∗h are the effective

Electron-hole generation

The current produced in a solar cell can be calculated by considering the generalised Planck equation. Recall that solar cells cannot be trated as a blackbody thus the generalised Planck equation needs to be modified by including a term describing the absorptivity of the solar cell, α(E) in order to model a ”grey body”. Z ∞ α(E)n(E, T, µ)dE (2.7) 0

As discussed in section 2.4, any photon with energy E < Eg will not have enough energy to be absorbed by the solar cell and hence α(E) = 0 for E < Eg . Assuming the device has a quantum efficiency of 1 for energies E > Eg (i.e. 100% of photons with E > Eg will be absorbed by the solar cell), α(E) = 1 for E > Eg . The chemical potential of a blackbody is µ = 0. The total number of photons from the sun absorbed by a solar cell of unit area per second will be Z ∞ fω n(E, TSun , 0)dE (2.8) Eg

In an ideal solar cell one photon produces one electron-hole pair, thus equation 2.8 can be multiplied by the charge of an electron, e to give the current density generated in a solar cell. efω 2π Jabs (E) = 2 3 ch

Z

∞

E2

dE (2.9) E e kTSun − 1 where E is measured in electron-volts. This integral has no easy analytical solution and must be evaluated numerically[16]. Eg

2.8 Electron-hole radiative recombination Radiative recombination rates can be calculated using a similar ’grey body’ approach. Only photons with energy E > Eg will be emitted from the solar cell as no electronic states exist within the bandgap. The emissivity ²(E) = 0 for E < Eg and ²(E) = 1 for E > Eg . A solar cell will emit radiation over all angles and the intensity is dependant on it’s temperature Tcell . Recall from section 2.5 that µ =V, the applied bias voltage. The photon flux emitted from a solar cell of unit area is given by Z ∞ n(E, Tcell , V )dE (2.10) Eg

The current density loss due to electron-hole radiative recombination can be calculated by multiplying equation 2.10 by the charge of an electron. e2π Jrad (E, V ) = 2 3 ch

Z

∞

E2 E−V

Eg

e kTcell − 1

dE

(2.11)

Chapter 2. Survey of the literature

11

2.9 Current-voltage characteristics

Figure 2.5 Idealised I-V curves for a solar cell in the dark(Green) and in the light(Red) indicating Jsc , Voc , the voltage and current for maximum power Vop and Jop and the associated rectangle of maximum power.

A typical idealised current-voltage (I-V) curve for a p-n junction solar cell in the dark and light is shown in figure 2.5. The dark current curve (green) shows only the recombination effects in a solar cell as no radiation is incident on the cell to produce generation effects. The light current curve (red) is the same shape as the dark current curve but offset by a factor of Jsc . Jsc is the short circuit current, defined to be the current when the applied bias is zero, it can be interpreted as the photogenerated current resulting from illumination of the cell. Voc is the open circuit voltage, the voltage at which there is zero current. Voc can be interpreted as the voltage at which the photogenerated current matches and cancels out the recombination current. The maximum power output of a solar cell occurs at some optimum applied bias Vop and a corresponding current Jop . The fill factor is defined to be Vop Jop Voc Jsc The efficiency of a solar cell under incident power density Pin is defined as FF =

η=

Vop Jop Pin

(2.12)

(2.13)

2.10 Shockley-Queisser efficiency limit Recalling the solar spectrum flux distribution in figure 2.1 it is evident that the size of the band-gap plays an important role in determining the limiting efficiency of a solar cell. A cell with a large band-

Chapter 2. Survey of the literature

12

gap will be transparent to all but the highest energy photons. A cell with a relatively low band-gap will have more electrons excited across the band gap but they will have less energy. Using the principle of detailed balance Shockley and Queisser calculated the efficiency limits for different band-gaps[17]. The Shockley-Queisser calculation ignores all parasitic losses such as Shockley-Read-Hall recombination, leaving only radiative recombination, as required by the second law of thermodynamics[18]. The net power output for such a cell will be the difference between the current generated from the sun and the current lost from radiative recombination multiplied by the applied voltage P (Eg , V ) = V × (Jabs − Jrad ) efω 2π P (Eg , V ) = V × ( 2 3 ch

Z

∞

E2

e2π dE − 2 3 ch −1

(2.14) Z

∞

E2

dE) (2.15) E−V E−V Eg e kTcell − 1 e kTSun Finding Vop to maximise power output is done numerically for different band-gaps, the most efficient band-gap being approximately 1.3eV with an efficiency of η ≈ 30%. The efficiency is calculated by dividing the maximum power output by the power input from the sun given by the StefanBoltzmann law multiplied by the dilution factor fω = 2.16e − 5. Eg

η=

P (Eg , Vop 4 fω σTsun

(2.16)

The results from Shockley and Queisser are summarised in figure 2.6 (a). The Shockley-Queisser limit for a terrestrial AM1.5 spectrum is shown in figure 2.6 (b) with the best experimental results[19] for single junction devices shown as red dots. The optimum band gap for the AM1.5 spectrum is approximately 1.4eV which corresponds closely to the band-gap of 1.42eV for GaAs.

2.11 Shockley-Read-Hall non-radiative recombination SRH recombination is the dominant efficiency loss in most real solar cells1 and occurs through defects or traps in the semiconductor. These impurities have energy levels deep in the band gap and should be differentiated from doping impurities which have energy levels close to the band gap edge. SRH traps are localised and can act as recombination centres for electrons and holes[11, 12]. Delocalised electrons or holes may fall into these traps. An electron may then be thermally excited back to the conduction band or a hole thermally excited back to the valence band however if both an electron and hole exist in the same trap simultaneously, recombination occurs and this results in losses in current and efficiency. In real solar cells many different trap levels may exist and it is no easy task to discover their density and energy level. A good approximation for SRH recombination is to assume there is one trap level which dominates and that the majority of recombination takes place in the depletion region[20]. The SRH recombination rate for a single trap level in the depletion region is given by[13] JSRH (V ) = 1

qV eW σvth Nt ni e 2kTcell 2

(2.17)

For example good GaAs cells are not SRH dominant at the operating voltage, the main recombination mechanism is radiative recombination. This will be explored in the next chapter

Chapter 2. Survey of the literature

(a) AM0

13

(b) AM1.5

Figure 2.6 (a)The Shockley-Queisser efficiency limit for a single-junction cell published in the original Shockley and Queisser paper in 1961[17] for an AM0 incident spectrum (6000K blackbody), including the best experimental efficiency for Silicon cells in 1961 and a semi-empirical curve thought to be the efficiency limit before the Shockley-Queisser paper was published(b)The Shockley-Queisser limit with an AM1.5 incident spectrum, best experimental results for single junction devices are shown as red dots.Image (b) from N.J. Ekins-Daukes

where e is the charge of an electron, W is the depletion layer width given by equation 2.5,q σ is the cell capture cross sectional area, vth is the thermal RMS velocity of an electron given by vth = 3kT m∗e , Nt is the trap density, ni is the intrinsic density of states given by equation 2.6 and V is the applied bias. The capture cross section is the probability of an electron or holer being captured in a defect state expressed as an area. The only terms which cannot be calculated easily are the capture cross section and the trap density which generally must be measured using capacitance spectroscopy[21, 22] or deep-level transient spectroscopy[23].

2.12 Advanced design concepts Several design concepts have been proposed in order to attempt to break the Shockley-Queisser efficiency limit[1]. The intermediate band cell is one such design which contains two band gaps allowing photons with energies less than the band gap to be absorbed[24]. The operation of such devices is complex and not essential to understanding this thesis so will be omitted however it should be noted that to work efficiently these designs need to operate int he radiative limit, where SRH recombination is negligible and the only recombination is radiative which is thermodynamically unavoidable.

Chapter 3 Theoretical model and computer simulation 3.1 Model overview The full matlab code of the model can be found in Appendix A. The net current in a solar cell can be modeled as Net Current = Photogenerated Current - Recombination current where the recombination current has two components Recombination current = Radiative Recombination Current + SRH recombination current The photogenerated current is given by equation 2.9 and is calculated in the model by numerically integrating from the band gap, Eg to 10eV 1 using the trapezium rule method with Tsun = 6000K. The recombination current calculation consists of two parts dependant on the bias voltage, radiative and non-radiative SRH recombination. Radiative recombination is calculated by numerically integrating equation 2.11 from Eg to 10eV for a bias voltage range of 0 to 0.99Eg . SRH recombination is calculated by simply evaluating equation 2.17 over the same bias voltage range. The net current can then be plotted as a function of voltage. The power of the solar cell as a function of voltage can be calculated by multiplying the current curve by the voltage. The maximum power can easily be found from this curve and divided by the power input to find the efficiency.

3.2

Temperature dependant Shockley-Queisser calculation

The Shockley-Queisser efficiency is discussed in Chapter 1 and is illustrated in figure 2.6. This curve was reproduced with the Matlab model for the case of a 300K cell and 6000K sun confirming the accuracy of the photogenerated current and radiative recombination current calculation. The same Shockley-Queisser type calculations were run for different temperatures 300K, 200K, 100K and 50K. These results are illustrated in figure 3.2 10eV can be used instead of ∞ as the intensity of the solar spectrum is low at such high photon energies and increasing the limits of integration to higher numbers such as 20eV and 100eV were found to produce the same results 1

14

Chapter 3. Theoretical model and computer simulation

15

The results are interesting to conceptually show a trend for increasing efficiencies with decreasing temperatures however could not be considered to be a theoretical prediction for real solar cells as SRH recombination has not been included. Increased efficiency is not particularly surprising as the photogenerated current will not change with cell temperature for a fixed band gap however radiative recombination is strongly dependant on temperature with decreased recombination occurring at lower temperatures. What is surprising though is the fact that the band gap at which the peak efficiency occurs shifts with temperature, however only slightly from approximately 1.3eV at 300K towards 1.2eV at 50K.

(a) I-V Curve

(b) Power Curve

Figure 3.1 Shockley-Queisser calculation at 300K and 50K for band gap Eg = 1.42eV (a) Current-Voltage curve. Note this curve is of the same form as the one shown in figure 2.5, current density is simply measured in the opposite direction. (b) Power curve

Chapter 3. Theoretical model and computer simulation

16

Figure 3.2 Temperature dependant Shockley-Queisser type calculation, shows a general trend for increasing efficiencies with decreasing temperatures. 300K - Blue (Lowest), 200K - Green (2nd lowest), 100K - Red (2nd highest) and 50K - Light Blue (Highest)

3.3

Temperature dependance of radiative efficiency

The relative rate of radiative and non-radiative SRH recombination is strongly dependant on bias and temperature. Whilst both radiative and SRH recombination will decrease with decreasing temperatures simulations suggest at low temperatures SRH recombination becomes negligible compared to radiative recombination due to low carrier thermal velocity. Radiative efficiency is defined as Jrad (3.1) Jrad + JSRH and provides a measurement of the relative contributions of radiative and SRH recombination. High radiative efficiency is desirable as it means there is negligible parasitic current losses and recombination is predominantly radiative. The dark current of a good 24.4% efficient GaAs cell[25] is shown in figure 3.4 along with calculated radiative and SRH recombination components.

Figure 3.3 Photon recycling, emitted photons internally reflected off the front surface then absorbed into the substrate

Chapter 3. Theoretical model and computer simulation

17

The radiative recombination component is calculated using the same method as in the Shockley-Queisser calculation except now photon recycling is taken into account. The rate of radiative recombination increases by a factor of 2n2 where n is the refractive index[26] due to total internal reflection of emitted photons off the front of the cell and photon absorption in the substrate. The refractive index used for GaAs was n = 3.62. e2π Jrad (E, V ) = 2n 2 3 ch 2

Z

∞

E2

dE (3.2) e −1 The SRH recombination component was calculated using equation 2.17 as in the previous section. For simplicity the capture cross section, trap density, effective mass and band-gap were taken to be independent of temperature, the validity of these assumptions will be examined in future sections. A trap density of Nt = 4.5 × 1014 cm−3 was required to fit the non-radiative component with a capture cross section of σ = 1 × 10−16 cm2 and a cell temperature of 307K was required to fit the radiative component. Under the illumination of one sun this cell has Jsc = 27.6 mA/m2 as indicated by the dashed line in figure 3.4. The radiative efficiency is calculated at the corresponding Voc point giving this cell a high radiative efficiency of 88%. The radiative efficiency of this cell is plotted in figure 3.5(a) as a function of temperature along with calculations for a GaAs cell with a much larger trap density Nt = 1 × 1016 cm−3 . Jsc was taken to be the same for both cells, at all temperatures. It is clear that both cells display an increasing radiative efficiency with decreasing temperature, the cell with the larger trap density showing a more dramatic increase. The power conversion efficiency of the two cells are shown in figure 3.5(b). The larger trap density is a considerable problem at room temperature but becomes insignificant at low temperatures converging with the low trap density cell. These results indicate that the low temperature regime is effectively a environment where defects in the cell are ’frozen out’ and the cell can operate in the ideal radiative limit. Eg

E−V kTcell

3.4 Shockley-Queisser efficiency including SRH recombination The model was used to calculate how efficiency changed with band gap with the inclusion of SRH recombination. In order to find reasonable values for the capture cross section and trap density in the SRH equation the data for Voc and Jsc given in the Solar Cell Efficiency Tables(Version 27)[19] was used. The capture cross section was fixed at σ = 1 × 10−16 cm2 and the trap density, Nt used as a fitting parameter. GaAs is one of the best materials to make solar cells out of and has the highest efficiency record to date for single junction devices. The world record GaAs cell has Jsc = 28.2 mA/cm−2 and Voc = 1.022 V. Jsc was fixed to 28.2 mA/cm−2 and and the trap density which predicted the same Voc was found to be Nt = (1.0 ± 0.1) × 1016 cm−3 . The model predicted an efficiency of 23 ± 2% whereas the record is 25.1 ± 0.8%. CdTe is an inexpensive material to make solar cells from, however the purity of the material is low leading to a large SRH recombination rate. The wordl record CdTe cell has Jsc = 25.9 mA/cm−2 and

Chapter 3. Theoretical model and computer simulation

18

Voc = 0.845 V. Once again Jsc was fixed to 25.9 mA/cm−2 and and the trap density which predicted the same Voc was found to be Nt = 4 ± 0.1 × 1018 cm−3 . The model predicted an efficiency of 17.1 ± 0.4% whereas the record is 16.5 ± 0.5%. Although within errors the inconsistency between the measured efficiency and the efficiency predicted by the model is due to the model incorrectly predicting the fill factor of the cell. This comes from not correctly predicting the rate of change between the voltage and the recombination current. The motivation for this is to find a reasonable value for Nt for a good quality solar cell (GaAs) and a poor quality solar cell (CdTe). Band-Gap vs efficiency for a good cell, Nt = 4 × 1018 cm−3 and a poor cell, Nt = 1 × 1016 cm−3 are illustrated in figure 3.6 along with a mediocre quality cell, Nt = 2 × 1017 cm−3 and a cell with a extremely low trap density, Nt = 4.5 × 1014 cm−3 . The interesting and somewhat surprising result from this simulation is that the band gap where peak efficiency occurs is dependant on the trap density. The very low trap density of Nt = 4.5 × 1014 cm−3 approaches the Shockley-Queisser efficiency curve as expected, with the optimum band gap being approximately 1.3eV . The highest trap density of Nt = 4.5 × 1018 cm−3 corresponds to an optimum band gap of approximately 1.7eV . These results suggest that if the intention is to purposely make a cheap solar cell suffering from high rates of SRH recombination it might be better focusing on materials with higher band gaps around the 1.7eV range rather than the band gaps traditionally thought to produce the highest efficiencies predicted by the Shockley-Queisser efficiency curve. In the SRH calculation the data for effective electron and hole mass used was that of GaAs, ∗ me = 0.45m0 and m∗e = 0.067m0 where m0 is the mass of the electron. The permittivity of GaAs was also used ²s = 13.1²0 .

Chapter 3. Theoretical model and computer simulation

19

Figure 3.4 Dark current data of a good GaAs cell[25]. Dashed line refers to Jsc under AM1.5 illumination. Image from N.J. Ekins-Daukes

(a) Radiative Efficiency

(b) Power Conversion Efficiency

Figure 3.5 (a) Radiative efficiency and (b) power conversion efficiency as a function of temperature for two GaAs cells, Nt = 4.5 × 1014 cm−3 and Nt = 1 × 1016 cm−3

Chapter 3. Theoretical model and computer simulation

20

Figure 3.6 Shockley-Queisser type calculation including SRH recombination, shows a general trend for increasing optimum band gap with increaasing trap density, capture cross section kept constant at σ = 1 × 10−16 cm−2 . Nt = 4.5 × 1014 cm−3 - Blue (Highest), Nt = 1 × 1016 cm−3 - Green (2nd highest), Nt = 2 × 1017 cm−3 - Red (2nd Lowest) and Nt = 4.5 × 1018 cm−3 - Light Blue (Lowest)

Chapter 4 Experimental setup 4.1 Overview

(a) Schematic

(b) Photograph

Figure 4.1 (a) Schematic of experimental setup (b) Photograph of experimental setup

The experimental setup is shown in figure 4.1. The sample solar cell is held by a cryostat in a specifically designed mount inside the vacuum chamber. The vacuum is necessary to insulate the sample and reach very low temperatures efficiently. It was discovered no diffusion pump was required, only a rotary pump was necessary to reach temperatures as low as ∼ 10K using the cryogenic system. A nichrome wire heater and diode temperature sensor were attached to the mount in the vacuum chamber which were controlled by the temperature controller. The I-V source / measure unit was 21

Chapter 4. Experimental setup

22

used via the laptop to measure the current-voltage characteristics of the sample. The sample was illuminated by the variable light source via an optic fibre with a collimator through diffusing glass. These components will be discussed in more detail in the next sections.

4.2 In the vacuum chamber

(a) Schematic

(b) Photograph

Figure 4.2 (a) Schematic of inside the chamber (b) Photograph of inside the chamber

The vacuum chamber is shown in figure 4.2. Substantial work was done in building the experiment. A new temperature sensor with a 4-point measurement system was installed. The heater failed several times throughout the course of the experiment and needed to be rewired after cooling the system one or two times. Plugs required to connect to the sample solar cell were constructed and connected via a 4-point measurement to the I-V source / measure unit. A specific size mount was designed to hold the sample solar cell and fit the cryostat, the design of the mount can be found in Appendix B.

4.2.1

Temperature sensor

The diode temperature sensor used was a Lakeshore DT-670-CO-1.4H which has a extremely small maximum uncertainty of ±30mK. This error was negligible compared to the error associated with being unable to stabilise the temperature. A 4 point measurement system was installed with the temperature sensor, measuring voltage and current independently to eliminate any errors due to lead

Chapter 4. Experimental setup

23

resistance(∼ 5Ω). The temperature sensor was clamped down to the base of the mount with a spring loaded screw, the spring is required to compensate for any thermal contraction that occurs during cooling to avoid crushing the diode.

4.2.2

Heater

The heater consisted of ∼ 1.5m Nichrome 32 AWG wire wrapped around the base of the mount with a total resistance ∼ 50Ω which gives maximum heating capacity from the temperature controller. The heater is required to stabilise the temperature by counteracting the cryogenic system which provides cooling at a constant rate.

4.2.3 Sample solar cell

(a) Schematic

(b) Photograph

Figure 4.3 (a) Schematic of sample cell, cell is 1mm in diameter, the black region indicating the shading of the cell from the metal cap (b) Photograph of 6 sample cells

The sample solar cell used in this experiment was QT1405R, designed by the Department of Physics at Imperial College, London. It is a GaAs pn junction grown by metal organic chemical vapour deposition (MOCVD) with parameters as specified on the growth profile - Dielectric Constant ²s = 13.18²0 - Band Gap Eg = 1.424 eV - Refractive index n = 3.660 - P-type doping NA = 2 × 1018 cm−3 - N-type doping ND = 2 × 1017 cm−3

Chapter 4. Experimental setup

24

- P-type region thickness 0.3000µm - N-type region thickness 3.000µm As can be seen in the photograph in figure 4.3 the sample actually consists of 6 small cells, each with significant shading. Each cell is circular with a diameter of 1.000 ± 0.002mm, resulting in a total area of (7.85 ± 0.03) × 10−7 m2 . Each cell is covered by by a metal cap with a circular opening with diameter of 0.65mm which significantly shades approximately 60% of the cell from illumination. The precise measure of shading is required for this experiment as we are only interested in the relative change in efficiency.

4.3

Outside the vacuum chamber

4.3.1

Cryogenic system

The cryogenic system consists of three parts, CTI cryogenics 8001 controller, CTI cryogenics 8300 compressor and a CTI cryogenics model 22 refrigerator. The system works on the Gifford-McMahon refrigeration cycle[27, 28] using compressed Helium. The system used in this experiment was able to reach temperatures as low as ∼ 10K however stabilising the temperature this low was impossible. Theoretically the Gifford-McMahon cycle can reach 4.2K which is achievable with more sophisticated systems but such low temperatures were not required in this experiment.

4.3.2

Temperature Controller

The temperature controller unit used was a Lakeshore 331 temperature controller. This unit came from the manufacturer already calibrated for the diode temperature sensor purchased at the same time. It features a PID(Proportional, Integral, Differential) heater control system common with most temperature control systems. It calculates the heater output using three terms Z de Heater Output = P (e + I e dt + D ) (4.1) dt where P, I and D are input parameters and e is the error, e = Setpoint - Feedback Reading. For this experiment the auto tune function was used which automatically calculates the optimum P, I and D settings as cooling occurs. This was found to be the quickest method of stabilising the temperature.

4.3.3

Variable light source

The light source was a Oriel DC regulated broad band quartz halogen source model 77501. It features a 100W quartz halogen lamp of which the intensity can be controlled. The lamp was focussed by a series of lenses to propagate through an optic fibre which is held in place via several optical stands seen in figure 4.1. The end of the fibre was connected to a collimator to focus the light on the glass window of the vacuum chamber. The glass window acted as a diffusor in order to obtain minimal variation of intensity on the surface of the sample solar cell.

Chapter 4. Experimental setup

25

The spectrum of this light source is different from the spectrum of the sun and hence this experiment is not capable of accurately measuring the efficiency under one sun illumination. The light source is a tungsten halogen light bulb producing a 3200K blackbody spectrum whereas the sun produces a ∼ 6000K blackbody spectrum. However this experiment is primarily concerned with measuring the relative change in efficiency at with temperature and it is not a requirement we know what the efficiency under one sun illumination as long as we have the same illumination source for all temperatures.

4.3.4

I-V source measure unit

A Keithley 2440 5A source meter was used to measure the IV curves of the sample solar cell. A 4 point measurement system was used to eliminate errors due to lead resistance(∼ 5ω). In this experiment voltage was used as the source and the current was measured. The system has a rated accuracy for source voltage of ±(0.02%+600µV) in the 2V range. Over the range of which currents were measured the measured current has a maximum error of ±(0.027%60nA) in the 1mA range. These errors whilst small were worth considering. The error from line noise was negligible as the measurement was 1 integrated over 10 line cycles (ie measured current was averaged over 10 × 50Hz = 0.2seconds) This was confirmed by measuring Jsc of a solar cell under constant illumination 10 times in a row and getting the same result to 4 significant figures. The unit was connected to a laptop via a GPIB-USB interface and run through labview software provided by Keithley which allowed collection of large amounts of data relatively quickly.

4.4 Photogenerated current The band gap of a semiconductor as a function of temperature can be calculated from the empirical equation Eg (T ) = Eg (0) −

αT 2 T +β

(4.2)

where Eg (0) = 1.52eV, α = 5.4×10−4 and β = 204 for GaAS[13]. Band gap decreases with with increased temperature is due to thermal expansion of the lattice in the semiconductor. An increase in interatomic spacing decreases the potential of the electron and consequently the band gap. Under constant illumination the photogenerated current in a cell is going to change with temperature due to the temperature dependance of the band gap. To further complicate the situation in this experiment the location of the solar cell is going to change due to the thermal contraction of the mount at low temperatures making it very difficult to experimentally keep the same level of illumination on the cell. It was decided for simplicity to keep Jsc constant by adjusting the position and intensity of the light source for each measurement. It is known that under illumination of 1000 W m− 2 at 300K the QT1405R cell has a Jsc of 63 ± 0.5µA[29]. It was found that the lamp used in the experimental setup could not produce enough power at low temperatures to create such a Jsc however it could easily produce Jsc of 21µA for all cell temperatures, equivalent to illumination under 1000 W m− 2 or 2.6 × 10−4 W over the front of the 3

Chapter 4. Experimental setup

26

cell(including shading). Jsc varied slightly for measurements at different temperatures between the range 21 ± 1µA, this was taken into account when the power conversion efficiency was calculated.

Chapter 5 Results 5.1 Current-Voltage and power measurements The I-V and power curve of QT1405R at 300K is shown in figure 5.1. The 300K theoretical curve was fitted to the data by fixing Jsc at the experimental value and varying Nt and σ to match the experimental Voc . These were found to be Nt = 1.75×1016 cm−3 and σ = (1.0±0.1)×10−16 cm2 . This gave very good agreement between theoretical and experimental efficiency, ηtheory = 5.5% ± 0.5% and ηexp = 5.5% ± 0.05%. The error associated with the current and voltage measurements discussed in the previous chapter was calculated and is too small to be visible on the graphs.

(a) I-V Curve

(b) Power Curve

Figure 5.1 (a) Theoretical and experimental I-V curve of Q1405R at 300K (b) Theoretical and experimental power curve of Q1405R at 300 ± 1K. There is an error of ±1K for the experimental curve.

27

Chapter 5. Results

28

However at lower temperatures the model underestimated the recombination current leading to an overestimate of the volatge at a particular current as evident in figure 5.2. At 200K the discrepancy is moderate yet beyond experimental error and at 50K the discrepancy is large. The magnitude of the error is revealed in figure 5.3 where the experimental and theoretical efficiency as a function of temperature is shown. Experimental efficiency as a function of temperature shows a very strong linear relationship at temperatures greater than ∼ 100K and then begins to plateau at lower temperatures. Good agreement is obtained at 300K where the fitting parameters were established but a significant deviation from the theoretical curve towards lower efficiencies at low temperatures is observed experimentally. These results indicate that the fitting parameters have some temperature dependance.

(a) 200K I-V Curve

(b) 50K I-V Curve

Figure 5.2 (a) Theoretical and experimental I-V curve of Q1405R at 200 ± 1K and (b)50 ± 5K. Theory keeps σ constant with temperature.

5.2

Variation of capture cross section with temperature

The trap density, Nt is not expected to change with temperature as it is a only a measure of the concentration of impurity atoms in the semiconductor. Recall that the capture cross section is the probability if a defect state capturing an electron or hole and it is not surprising that it may change with temperature. The assumption that capture cross section does not change significantly with temperature may be reasonable over small temperature ranges however this experiment was performed over a large range of temperatures and this simplification was not acceptable. Capture cross section varies with temperature in a complex way which is not well understood. Figure 5.4(a) shows how capture cross section varies with inverse temperature for a variety of deep

Chapter 5. Results

29

Figure 5.3 Experimental and Theoretical efficiency Vs temperature. Theory keeps σ constant with temperature

level traps in GaAs and GaP[30]. No information is known about the defect levels in QT1405R and thus attempting to theoretically model how σ changes with temperature was not possible. Nt = 1.75 × 1016 cm−3 was kept constant and σ was used as a fitting parameter for each curve such that σ was varied until the experimental Voc matched the theoretical Voc . A plot of fitted capture cross section vs inverse temperature is shown in figure 5.4 (b). The shape of the capture cross section bears resemblance to the shape of the deep level defect ’A’ in figure 5.4 lending credibility to this method of approximating σ(T ) and suggesting that the unknown deep level defect ’A’ may be the dominant defect present in QT1405R. Measuring the capture cross section of defects in solar cell can be performed using capacitance spectroscopy and deep level transient spectroscopy techniques[23] though this was not done due to complexity and time restraints.

5.3

Temperature vs efficiency with varying capture cross section

Theoretical predictions of how efficiency varies with temperature was recalculated taking into account the variation of capture cross section with temperature. The temperature vs efficiency graph in fig-

Chapter 5. Results

30

Figure 5.4 (a)Capture cross section vs Inverse temperature for a number of defects in GaAs and GaP. n denotes electron capture whilst p denotes hole capture. A and B are common but unidentified deep level defects Image from C.H. Henry and D.V. Lang[30]. (b) Inverse temperature vs fitted capture cross section for QT1405R.

ure 5.5 shows much better agreement between theory and experiment in the range 150K . T < 300K but still displays significant variation at the lowest measured temperatures (T . 150K). The most likely explanation behind this deviation between experiment and theory is due to a reduction in electron density at low temperatures where free electrons and holes return to their respective donor and acceptor impurity states.[13]. Electron density can be considered to be constant in the range between approximately 150K and 500K and equal to the donor impurity concentration, ND . However when the thermal energy of the lattice is not high enough to ionize the donor impurities, the electron density decreases dramatically with temperature and is less than the donor impurity concentration, this is known as the’freeze out region’. Incorporating this theoretically into calculations of I-V curves would require a model and computer package with a higher degree of sophistication and is beyond the scope of this thesis. For completeness the full set of experimental I-V curves and their corresponding theoretical curves are in appendix C.

5.4 Uncertainties with power conversion efficiency 5.4.1

Theoretical uncertainty

The dominant error in the theoretical efficiency predictions is the uncertainty of fitting the capture cross section, σ. As explained in the previous sections the capture cross section was used as a fitting parameter for each curve by matching the experimental and theoretical Voc . Generally the experimental curves did not have data points at the Voc point, J=0. σ was varied until the theoretical curve crossed the V axis in between the two data points on either side of J=0. The error is sigma was the

Chapter 5. Results

31

Figure 5.5 Experimental and Theoretical efficiency Vs temperature. Theory varies σ with temperature.

half the difference in the σ value needed to match the points on either side of J=0. The values of σ and associated error is summarised in appendix C.

5.4.2 Experimental uncertainty There are two things to consider when calculating the experimental efficiency error, the errors due to measurement and errors due to uncertainty of the location of the maximum power point. The errors associated with experimental efficiency measurements are the source voltage error(0.02%), measured current error(0.027%) and the³ error´due to only knowing the Jsc of the cell under one sun illumination to 2 significant figures 0.5µA . Combining these errors gives a total experimental 63µA = 0.8%. efficiency fractional error associated with the accuracy of a measured point of ∆η η The second component to consider is the fact that the actual maximum power point could lie anywhere between the points immediately before or after the measured maximum power point. The error associated with this was found to vary between 0.1% and 0.8% for different I-V curves and thus was included on a case by case basis. Overall the dominant error associated with experimental efficiency is limitation of knowing Jsc of the cell under one sun illumination to 2 significant figures.

Chapter 5. Results

32

The error associated with the temperature of the cell varies for different temperatures and is determined by the inability to keep the temperature stable during measurement. Experimental uncertainty data is summarised in figure C.2 in Appendix C.

5.5 Dark I-V curves Recall that the I-V curve of a solar cell in the dark only describes the recombination occurring in the cell. As discussed in section 3.3 there are two recombination regimes a solar cell can operate in, radiatively dominated or non-radiatively dominated. A clear change in gradient is observed on a semi-log dark I-V plot when this occurs. The dark I-V curves of QT1405R for temperatures 300K, 175K and 100K are shown in figure 5.6. As also evident in figure 5.5 the theoretical model is unable to accurately predict the rate of non-radiative recombination at the lowest temperatures. Data at very low currents (. 10−7 A)are due to the I-V source/measure unit being unable to resolve such small currents in the solar cell. At all temperatures QT1405 did not enter the radiative regime at the operating current Jsc = 63µA. The current had to be limited to a maximum of 10−3 A as it is known the bond wires will melt at higher currents thus the cell was never able to become radiative at high voltages. Radiative recombination is not dependant on any properties particular to the sample, only the band gap and refractive index of GaAs. The model was able to predict the correct radiative recombination for a different GaAs cell(figure 3.4) so it is safely assumed it would correctly predict radiative recombination for this cell. The radiative efficiency of QT1405R as a function of temperature is shown in figure 5.7 displaying a slightly increasing radiative efficiency until T=100K then sharply dropping by 3 orders of magnitude. This sudden decrease in radiative efficiency is attributed to the sharp increase in the capture cross section at temperatures less than 100K, as illustrated in the plot of capture cross section as a function of temperature on the secondary axis in figure 5.7. QT1405R does not demonstrate increasing radiative efficiency as expected from section 3.3 due to the complex behaviour of the capture cross section with temperature. Despite this result the power conversion efficiency of the cell still increases consistently with decreasing temperature as in figure 5.5. The suggestion in section 3.3 that a low temperature environment could be used as a potential testing ground for new solar cell design concepts discussed in section 2.12 appears to be flawed as the radiative efficiency did not increase with temperature. Considering figure 5.4 (a) the capture cross section for some defect levels increase with temperature such as in QT1405R whilst others decrease dramatically with temperature which would consequently decrease SRH recombination and increase radiative efficiency at low temperatures. The low temperature environment may act as a radiatively dominated regime for cells only with defects where the capture cross section decreases with temperature however it must be concluded that the radiative regime can not generally be reliably achieved by simply cooling solar cells to low temperatures.

Chapter 5. Results

33

(a) 300K dark I-V curve

(b) 175K dark I-V curve

(c) 100K dark I-V curve

Figure 5.6 Dark I-V curves for QT1405R at 300K, 175k and 100K. The dashed line represents the cell’s Jsc

Chapter 5. Results

34

Figure 5.7 Radiative efficiency of QTQ1405R as a function of temperature, a sharp decrease is observed at T < 100K due to the sharp increase in capture cross section shown on the secondary axis

Chapter 6 Conclusion The power conversion efficiency of a photovoltaic solar cell was predicted to increase as the temperature of the cell was decreased due to a reduction in radiative and non-radiative recombination. A simple matlab model for calculatin I-V curves and efficiency theoretically predicted how the efficiency varied with temperature. Experimental results revealed that the efficiency of our sample GaAs solar cell increased consistently with decreasing temperature however discrepancies between theory and experiment were apparent. This was due to the complex behaviour of the capture cross section for non-radiative recombination with temperature. Capture cross section was adjusted to vary with temperature showing significantly improved agreement between theory and experimental yet still large discrepancies at the lowest temperatures. The most likely reason for this is the thermal energy of the lattice not providing enough energy to ionise donor and acceptor impurity atoms decreasing the electron density of the semiconductor. The experimental radiative efficiency of the cell was found to be approximately constant above 100K and dropping dramatically at lower temperatures to almost zero. These results indicate that the sample cell did not become radiatively efficient at low temperatures with parasitic SRH recombination still the dominant recombination process in the cell. The decrease in experimental radiative efficiency contradicted computer simulations again due to the complex behaviour of the capture cross section with temperature. It was concluded that the low temperature environment is generally not a reliable testing ground for new solar cell design concepts such as the intermediate band cell unless the capture cross section for the defects present in a particular cell are known to decrease with temperature. Future work associated with this project would be to employ a more sophisticated computer model for increased accuracy in predicting I-V curves and understand how the electron density varies at low temperatures. The low current measurements on the dark I-V curves could be improved by better wiring and shielding of wires from interference in the cryostat. By improving the bonding of wires to the solar cell increased currents could be used in the cell without the risk of melting the bond wires thus being able to push the cell into the radiative regime at high voltages. In this work the capture cross section and trap density are used as fitting parameters having only values relative to each other. Capacitance spectroscopy and deep level transient spectroscopy techniques would allow accurate measurement of both parameters as a function of temperature.

35

Appendix A Matlab Code %Solar Cell Recombination Model % %Mathew Guenette, October 2006 %Everything in electronvolts T=300 %temp of cell sigma=1e-16/10000;

%Cross Sectional Area mˆ2%

q=1.60217646e-19; %charge on electron fw=2.16e-5; %dilution factor from sun c=299792458; %speed of light h=6.626068e-34/q; %Plancks constant in eV k=1.3806503e-23/q; %Boltzmann’s constant in eV Tsun=6000; %temp of sun eg=1.519 - ((5.405e-4*Tˆ2)/(T+204)); % band gap of GaAs A=7.854e-7; %Area Of Cell Qt1450R nt=1.75e16*1000000; %Number Of Traps mˆ-3% %nc=4.7e17*1e6; Density of states (electrons) (unused) %nv=7e18*1e6; Density of states (holes) (unused) kb=1.3806503e-23; %boltzmanns constant in joules% me=9.10938188e-31; %Mass of Electron% mestar=0.067.*me; %effective elctron mass mhstar=0.45.*me; %effective hole mass hj=6.626068e-34; %Planck’s Constant in Joules% eps0=8.85418782e-12; %epsilon0 epsGaAs=13.1*eps0; %dielectric constant for GaAs

36

Appendix A. Matlab Code

37

Na=2e18*1000000; %Doping contrations for QT1405R Nd=2e17*1000000; %Define More Constants dependant on other constants% nc=2*((2*pi*mestar*kb*T)/(hjˆ2))ˆ(3/2); %Electron Density of states nv=2*((2*pi*mhstar*kb*T)/(hjˆ2))ˆ(3/2); %Hole Density of states vth=((3.*kb.*T)./(mestar))ˆ0.5; %Thermal Velocity ni=((nc.*nv)ˆ0.5)*exp(-eg/(2*k*T)); %Intrinsic Density Of States w=((2*epsGaAs*kb*T)/(qˆ2)*reallog((Na*Nd)/(niˆ2))*((1/Na)+(1/Nd)))ˆ0.5 %ˆdepletion region width %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Shockley-Read-Hall current expression% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V=linspace(0,0.99*eg,1000); const=((q*w*sigma*vth*nt*ni)/2); JSRH=const*exp((V)/(2*k*T)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Calculate photgenerated current from the Sun %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e=linspace(eg,20,1000); n=1:1:1000; square=e.ˆ2; cube=e.ˆ3; energy(n)=e; J(n)=(fw*q*2*pi)/(cˆ2*hˆ3)*(square)./(exp((e)./(k*Tsun))-1); JSC=trapz(e,J); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %calculate radiative recombination% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for j=1:1000; v= j*0.99*(eg)/1000; m = 1:1:1000; en=linspace(eg,20,1000); energy(m)=en;

Appendix A. Matlab Code

38

planck(m)=((26.2*q*2*pi)/(cˆ2*hˆ3)*en.ˆ2)./(exp((en-v)/(k*T))-1); bias(j)=v; int(j)=trapz(energy, planck); %Radiative Current Loss% end

%ExpV and ExpI is experimental data% yv=expV; yi=-expI; yp=yv.*yi; yierror=0.00027*yi+60e-9 %Calculate experimental Voc dooexp=(yi).ˆ2; [cooexp,lexp]=min(dooexp); Vocexp=yv(lexp) %Calculating the experimental efficiency% yep=expV.ˆ2; [a,b]=min(yep); JSC=yi(b) f=6.3e-5/JSC; [maxpowers,maxplace]=max(yp); etaEXPERIMNTAL=100*maxpowers*f/7.854e-4 %experimental efficiency %experimental error from finding position of maximum power etaEXPerror=100*f*0.5/7.854e-4*abs(maxpowers- (yp(maxplace - 1)* + yp(maxplace + 1))) current=JSC-(JSRH+int)*A; %theoretical light I-V curve power=current.*bias; %theoretical power curve [maxpower,maxt]=max(power); stef=fw*5.67e-8*Tsunˆ4; doo=(current).ˆ2; [coo,l]=min(doo); Voc=bias(l)

Appendix A. Matlab Code

39

Eff=100*maxpower*f/(A*1000)

%plotting stuff %Light curves figure(3) plot(bias,current,yv,yi,’*’) title([’Light I-V curve at ’,int2str(T),’\pm1K’]) xlabel(’Applied Bias (V)’); ylabel(’Current (A)’); legend(’Theoretical’,’Experimental’) ylim([0 0.3e-4]) xlim([0 1.5]) %dark curves figure(9) semilogy(bias,JSRH2,bias,int2,bias,dark,expV2,expI2,’*’)%yv,yi,’*’) title([’Dark I-V curve at ’,int2str(T),’\pm1K’]) xlabel(’Applied Bias (V)’); ylabel(’Current (A)’); legend(’Theoretical SRH’,’Theoretical radiative’, ’Theoretical SRH + radiative’,’Experimental’)%,’dark shifted by JSC’) %power curves figure(4) plot(bias,power,yv,yp,’*’)%,expV2,pow3) title([’Power curve at ’,int2str(T),’\pm1K’]) xlabel(’Applied Bias (V)’); ylabel(’Power (W)’); legend(’Theoretical’,’Experimental’)%,’dark shifted by JSC’) ylim([0 0.4e-4]) xlim([0 1.5]) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Appendix B Solar Cell Mount Designs Designs of mount and cover plate for sample solar cell, material is copper.

40

Appendix B. Solar Cell Mount Designs

41

Appendix C Capture Cross Section Data and I-V curves C.1

Capture cross section data

This section contains a table of the full set of data for capture cross section used to match the experimental and theoretical Voc as in chapter 5.

Temperature (K) 300 275 250 225 200 175 150 125 100 75 50

C.2

Capture Cross Section (cm2 ) (1.0 ± 0.1) × 10−16 (1.2 ± 0.1) × 10−16 (1.5 ± 0.1) × 10−16 (1.6 ± 0.1) × 10−16 (2.3 ± 0.1) × 10−16 (3.3 ± 0.2) × 10−16 (4.2 ± 0.3) × 10−16 (1.0 ± 0.1) × 10−15 (3.6 ± 0.2) × 10−15 (9.4 ± 0.5) × 10−14 (3.6 ± 0.1) × 10−11

Experimental uncertainties

42

Appendix C. Capture Cross Section Data and I-V curves Temperature (K) Position Error (%) 300 ± 1 ±0.6 275 ± 1 ±0.7 250 ± 1 ±0.8 225 ± 1 ±0.1 200 ± 1 ±0.4 175 ± 2 ±0.5 150 ± 5 ±0.3 125 ± 5 ±0.1 100 ± 5 ±0.5 75 ± 5 ±0.1 50 ± 5 ±0.2

Measurement Error (%) ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8

43 Total Error (%) ±1 ±1 ±1 ±0.8 ±0.9 ±0.9 ±0.9 ±0.8 ±0.9 ±0.8 ±0.8

Figure C.1 Table showing the experimental error associated with Temperature, the position of the maximum power point, measurement of the maximum power point and total combined error for power conversion efficiency

C.3

Light I-V curves

The full set of experimental light I-V curves and corresponding theoretical curve with varying capture cross section is contained in this section.

Appendix C. Capture Cross Section Data and I-V curves

44

(a) 300K

(b) 275K

(c) 250K

(d) 225K

(e) 200K

(f) 175K

Figure C.2 I-V curves for temperatures between 175K and 300K

Appendix C. Capture Cross Section Data and I-V curves

45

(a) 150K

(b) 125K

(c) 100K

(d) 75K

(e) 50K

Figure C.3 I-V curves for temperatures between 50K and 175K

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