Phys 446: Solid State Physics / Optical Properties Lattice vibrations: Thermal, acoustic, and optical properties Fall 2015 Lecture 4

Andrei Sirenko, NJIT

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Solid State Physics Last weeks:

Lecture 4 (Ch. 3)

• Diffraction from crystals • Scattering factors and selection rules for diffraction

Today: • Lattice vibrations: Thermal, acoustic, and optical properties This Week: • Start with crystal lattice vibrations. • Elastic constants. Elastic waves. • Simple model of lattice vibrations – linear atomic chain

• HW1 and HW2 discussion

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Material to be included in the 1st QZ

• Crystalline structures. Diamond structure. Packing ratio 7 crystal systems and 14 Bravais lattices

• Crystallographic directions and Miller indices

d hkl 

• Definition of reciprocal lattice vectors: • What is Brillouin zone • Bragg formula: 2d·sinθ = mλ

;

n 12

 h2 k 2 l 2   2  2  2  b c  a

k = G

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• Factors affecting the diffraction amplitude: Atomic scattering factor (form factor): reflects distribution of electronic cloud. In case of spherical distribution

f a   n(r )e ik rl d 3 r r0

f a   4r 2 n(r ) 0

• Structure factor

F   f aj e

sin Δk  r  dr Δk  r

2i ( hu j  kv j  lw j )

j

where the summation is over all atoms in unit cell

• Be able to obtain scattering wave vector or frequency from

geometry and data for incident beam (x-rays, neutrons or light) 4

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Material to be included in the 2nd QZ TBD

Elastic stiffness and compliance. Strain and stress: definitions and relation between them in a linear regime (Hooke's law):

 ij   Sijkl kl

 ij   Cijkl  kl

kl

kl

• Elastic wave equation:

 u Ceff  u x  t 2  x 2 2

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Ceff

sound velocity v 



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• Lattice vibrations: acoustic and optical branches

In three-dimensional lattice with s atoms per unit cell there are 3s phonon branches: 3 acoustic, 3s - 3 optical

• Phonon - the quantum of lattice vibration. Energy ħω; momentum ħq

• Concept of the phonon density of states • Einstein and Debye models for lattice heat capacity. Debye temperature Low and high temperatures limits of Debye and Einstein models

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Formula for thermal conductivity

1 K  Cvl 3

• Be able to obtain scattering wave vector or frequency from

geometry and data for incident beam (x-rays, neutrons or light) 6

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Elastic properties Elastic properties are determined by forces acting on atoms when they are displaced from the equilibrium positions

Taylor series expansion of the energy near the minimum (equilibrium position):

U ( R)  U 0 

U R

( R  R0 )  R0

1  2U 2 R 2

( R  R0 )  ... R0

For small displacements, neglect higher terms. At equilibrium, So,

ku 2 U ( R)  U 0  2

where

k

 2U R 2

U R

0 R0

R0

u = R - R0 - displacement of an atom from equilibrium position

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F 

force F acting on an atom:

U  ku R

k - interatomic force constant. This is Hooke's law in simplest form. Valid only for small displacements. Characterizes a linear region in which the restoring force is linear with respect to the displacement of atoms. Elastic properties are described by considering a crystal as a homogeneous continuum medium rather than a periodic array of atoms In a general case the problem is formulated as follows:

• Applied forces are described in terms of stress , • Displacements of atoms are described in terms of strain . • Elastic constants C relate stress  and strain , so that  = C. In a general case of a 3D crystal the stress and the strain are tensors 9

Stress has the meaning of local applied “pressure”. Applied force F(Fx, Fy, Fz) 

Stress components ij (i,j = 1, 2, 3)

General case for stress: i.e ij

Fj 

 Fi dV  

V

V

 ij x j

x  1, y  2, z  3

 ij x j

dV    ij dS j S

Shear forces must come in pairs: ij = ji (no angular acceleration)  stress tensor is diagonal, generally has 6 components 10

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Stress has the meaning of local applied “pressure”. Applied force F(Fx, Fy, Fz) 

Stress components ij (i,j = 1, 2, 3)

General case for stress: i.e ij

Fj 

 ij

x  1, y  2, z  3

x j

Hydrostatic pressure – stress tensor is equivalent to a scalar: i.e xx =yy =zz

 p 0 0  ˆ  [ ij ]  0  p 0    0 0  p  Stress tensor is a “field tensor” that can have any symmetry not related to the crystal symmetry. Stress tensor can change the crystal symmetry 11

Stress has the meaning of local applied “pressure”. Applied force F(Fx, Fy, Fz) 

Stress components ij (i,j = 1, 2, 3)

Compression stress: i = j, i.e xx , yy , zz

 xx 

x  1, y  2, z  3

Fx Ax

Shear stress: i ≠ j, i.e xy , yx , xz zx , yz , zy

 yx 

Fy Ax

Shear forces must come in pairs: ij = ji (no angular acceleration)  stress tensor is diagonal, generally has 6 components

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Strain tensor 3x3

 ij 

ui x j

In 3D case, introduce the displacement vector as

u = uxx + uyy + uzz

Strain tensor components are defined as

 xx 0 0   ˆ  [ ij ]  0  yy 0  0 0   zz   V ' dV   xx   yy   zz  Tr (ˆ ) dV

ˆ

u x x u  xy  x y

 xx 

Share deformations:

 xx   yy   zz  Tr (ˆ )  0

Can be diagonalized in x-y-z coordinates at a certain point of space In other points the tensor is not necessarily diagonal

Strain tensor components are defined as

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 ij 

ui x j

 xx 

u x x

 xy 

u x y

Since ij and ji always applied together, we can define shear strains symmetrically:

1  ui u j    2  x j xi 

 ij   ji   

So, the strain tensor is also diagonal and has 6 components 14

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Elastic stiffness (C) and compliance (S) constants relate the strain and the stress in a linear fashion: This is a general form of the Hooke’s law. 6 components ij, 6 ij  36 elastic constants

 ij   Cijkl  kl kl

 ij   Sijkl kl kl

Notations: Cmn where 1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = zx, 6 = xy For example, C11 Cxxxx , C12 Cxxyy , C44 Cyzyz Therefore, the general form of the Hooke’s law is given by

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Elastic constants in cubic crystals Due to the symmetry (x, y, and z axes are equivalent) C11 = C22 = C33 ;

C12 = C21 = C13 = C31 = C23 = C32 ; C44 = C55 = C66 Also, the off diagonal shear components are zero:

C45 = C54 = C46 = C64 = C56 = C65 and mixed compression/shear coupling does not occur:

C45 = C54 = C46 = C64 = C56 = C65  the cubic elastic stiffness tensor has the form:

only 3 independent constants 16

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Elastic constants in cubic crystals Longitudinal compression (Young’s modulus):

C11 

 xx F A   xx u L L

Transverse expansion:

Shear modulus:

C12 

C44 

 xx  yy

 xy F A   xy  17

Uniaxial pressure setup for optical characterization of correlated oxides Pressure control

•Variables: Uniaxial pressure Temperature External magnetic field Measured sample properties: Far-IR Transmission / Reflection Raman scattering

Optical cryostat

sample Low T 18

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Elastic waves Considering lattice vibrations three major approximations are made: • atomic displacements are small: u > a, one may disregard the atomic nature – solid is treated as a continuous medium. Such vibrations are referred to as elastic (or acoustic) waves. 19

Elastic waves First, consider a longitudinal wave of compression/expansion mass density  segment of width dx at the point x; elastic displacement u  



 2u 1 F  xx   t 2 A x x

where F/A = xx

Assuming that the wave propagates along the  xx  C11 xx [100] direction, can write the Hooke’s law in the form Since

u  xx  x x

 2 u C11  2 u x  get wave equation: t 2  x 2

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A solution of the wave equation - longitudinal plane wave

u ( x, t )  Aei ( qx t ) vL 

C11



where q - wave vector; frequency ω = vLq - longitudinal sound velocity

Now consider a transverse wave which is controlled by shear stress and strain:

In this case



 2u  xy  t 2 x

where  xy  C44 xy

 wave equation is  2u

C44  2u x  t 2  x 2

 xy 

and

vT 

u x

- transverse sound velocity

C44



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Two independent transverse modes: displacements along y and z For q in the [100] direction in cubic crystals, by symmetry the velocities of these modes are the same - modes are degenerate Normally C11 > C44  vL > vT We considered wave along [100]. In other directions, the sound velocity depends on combinations of elastic constants:

v

Ceff



Ceff - an effective elastic constant. For cubic crystals:

Relation between ω and q - dispersion relation. For sound ω = vq 22

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Model of lattice vibrations one-dimensional lattice: linear chain of atoms harmonic approximation: force acting on the nth atom is

equation of motion (nearest neighbors interaction only):

 2u M 2  Fn  C (un 1  un )  C (un 1  un )  C (2un  un 1  un 1 ) t M is the atomic mass, C – force constant Now look for a solution of the form

u ( x, t )  Aei ( qxn t )

where xn is the equilibrium position of the n-th atom  xn = na obtain

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

 the dispersion relation is Note: we change q  q + 2/a the atomic displacements and frequency ω do not change  these solutions are physically identical  can consider only

i.e. q within the first Brillouin zone The maximum frequency is 2 C M

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At the boundaries of the Brillouin zone q = /a  un  A(1) n e  it standing wave Phase and group velocity phase velocity is defined as v p  group velocity

vg 

 q

d dq

vg  a

C qa cos M 2

vg = 0 at the boundaries of the Brillouin zone (q = /a)  no energy transfer – standing wave

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Long wavelength limit:  >> a ; q = 2/ >θD, x