1

Lecture contents

• Coupled oscillators
• Dispersion relationship
• Acoustical and optical lattice vibrations
• Acoustical and optical phonons
• Phonon statistics
• Acoustical phonon scattering

NNSE 618 Lecture #11

 Few concepts from Solid State Physics 2

1. Adiabatic approximation
When valence and core ectrons are separated, general Schrödinger equation for a
condensed medium without spin

= H L + He

H( R, r )  E( R, r )
• Mass of ions >1000 (for most
( R, r )   (r, R0 )( R)
semiconductors >102 times greater than
mass of electrons
H L ( R)  EL ( R)
• Ion velocities >100 times slower
H e (r , R)  Ee (r , R)
• Electrons adjust ‘instantaneously” to the
positions of atoms

• Separate ion and electron motion

(accuracy ~m/M)

NNSE 618 Lecture #11

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Few concepts from Solid State Physics
2. Phonons
Hamiltonian for lattice motion (harmonic oscillations) :
Phonon dispersion

 
pl2 relation in GaAs
 U 0 Rl0  Rm0   Cl ,m ul  um   U anhar
1
HL  
2

l 2M l l ,m l ,m 2

Displacements show up as plane waves with weak

interaction via anharmonicity:

uk ,  u0eikr it
Energy in a mode:

 1
E k ,     nk ,    
 2
Equilibrium distribution (Bose Einstein):

1
n( ) 
  
exp   1
 kT 
NNSE 618 Lecture #11
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Lattice vibrations

U R 
pl2
Lattice Hamiltonian: HL  
l
2M l

l ,m
0
l  Rm0

Binding energy vs. interatomic distance in a crystal

Expanding binding energy around the

equilibrium position R0 :

0
Linear term is zero at minimum

Neglecting anharmonic terms:

U ( R)  U ( R0 )  C R 2
1
2
with a force constant C

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Diatomic chain

Let’s consider diatomic chain to

demonstrate acoustical and optical
dispersion branches

Masses are connected by springs with

equal spring constants, C, for
simplicity
Force = - C .R

d 2u s
With u and v, the displacements of M1  C vs  vs 1  2u s 
dt 2
respective atoms, we can write down
d 2 vs
classical equation of motion (second M2 2
 C u s 1  u s  2vs 
Newton law) dt

The solution for displacements in the

u s  ueiksait
vs  ve iksait
chain will be searched as traveling
waves: From Singh, 2003
NNSE 618 Lecture #11
Solution for diatomic chain 6

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Dispersion relations for diatomic chain

Solutions for small k :

Solutions for the edge of Brillouin

zone k=p/a :

From Singh, 2003

NNSE 618 Lecture #11
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Acoustical and optical waves
For acoustical branch in long
wavelength limit (at small k): uv or u s  vs
d a C
Sound velocity: vs  
dk 2 M av

For optical branch at k=0 :

(Two atoms vibrate against their M2
center of masses) u v
M1

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Dispersion curves in semiconductor crystals
• For each wavevector there are 1 longitudinal mode Example: shell model
and 2 transverse modes
• The frequencies are determined by force constants
• Usually longitudinal mode (LA) is stiffer
• Energy scales (for similar crystals) as M-1/2
• Atomic vibrations are in THz range

Si GaAs InAs

From Singh, 2003

NNSE 618 Lecture #11
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Anisotropy of phonon dispersion curves

Experimental (points) and calculated phonon dispersion curves for Si

From Yu, Cordona, 2002

NNSE 618 Lecture #11

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Quantum harmonic oscillator
C
2
p 1
Quantum harmonic oscillator: Hamiltonian H  Cx 2
2M 2 M M
Solution gives resonance frequency (as in C E
classical mechanics) 2 
M
And quantum oscillation spectrum:  1 
(n may be considered as number of En   n  
 2
“quasiparticles”)
x

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Quantization of lattice vibrations: phonons
For a single oscillator the frequency is fixed, but
when many oscillators interact we have a
number of modes (normal modes)
k

 1
Each mode is occupied by nk phonons Ek   nk   k
 2

2pn N
For a 1D chain states are determined as: k ; for n  0,1,...
Na 2 Bose-Einstein distribution
function
Occupancy of modes is given by Bose-statistics:
1
n( ) 
  
exp   1
 kT 

NNSE 618 Lecture #11

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Optical phonons: Raman scattering
• Inelastic light scattering = Raman scattering gives
information on optically active vibrations in a
material
• Wavevector of photons is SMALL
• Stokes (creation of vibration) and anti-Stokes
(emission of vibration) GaAs
• Symmetry and selection rules: Raman scattering
intensity depends on geometry and polarization

From Yu and Cordona, 2003

NNSE 618 Lecture #11
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Lattice scattering rate calculation
Goal: calculation of the scattering integral or relaxation time:

f f  f0 d 3k '
t coll


 f k' 1  f k W k ' , k   f k 1  f k ' W k , k '
2p 3
Step 1. Determine scattering potential H  eiqrit

Step 2. Calculate matrix elements from k’ to k


H k 'k   k*' H k d 3 r
V

2p
H k 'k  E (k )  E (k ' )   
Step 3. Calculate transition rate from k’ to k using 2
W (k ' , k ) 
“golden Fermi rule” 

f f f0
Step 4. Calculate state relaxation time
t coll

 (k )
 f k' 1  f k W (k ' , k )  f k 1  f k ' W (k , k ' )

Step 5. Average relaxation time  (k )

NNSE 618 Lecture #11