Transcript PowerPoint 프레젠테이션 - Micro Thermal System

THERMAL PROPERTIES OF
SOLIDS AMD THE SIZE EFFECT
• Specific Heat of Solids
• Quantum Size Effect on the Specific Heat
• Electrical and Thermal Conductivities of Solids
• Thermoelectricity
• Classical Size Effect on Conductivities and Quantum
Conductance
Specific Heat of Solids
Specific heat of bulk solids:
lattice vibration, free electron
macroscopic behavior from microscopic point of view
 Lattice Vibration in Solids: Phonon Gas
 Atoms in solids
Inter-atomic forces keep them
in position
only move around by vibrations
near their equilibrium positions
 Lattice
Periodical array of atoms
Lattice vibration contribute to thermal energy
storage and heat conduction
 Einstein Specific Heat Model
 Dulong-Petit law
cv  3 R
3 vibrational degree of freedom
high-temperature limit of the specific heat of
elementary solids
good prediction for specific heat of solid at high
temperature
 Einstein Specific Heat Model
simple harmonic oscillator model
each atom : independent oscillator
All atoms vibrate at the same frequency.
1

quantized energy level  i   h , i  1, 2,...
2

specific heat as a function of temperature
Quantized energy of atoms
1

 i   i   h , i  1, 2,...
2

Vibrational partition function
 i  
 1
ZV   gi exp  
   gi exp[  i   h / kBT ]
 2
i 0
 kBT  i 0


1

  E / 2T
 iE / T
 E  h / kB
  exp[  i    E / T ]  e
e

2


i 0
i 0
Einstein temperature


 e E / 2T 1  e E / T  e 2E / T    

 e E / 2T 1  e E / T  (e E / T )2     
e E / 2T

1  e E / T
Internal energy
  ln Z 
U  NkT 
,

 T V
2
E
ln Z  
 ln 1  e E / T
2T


1
1

U  NkB  E   E / T

1
2 e
Specific heat
vibration along single axis
E 2
e E / T
 U 
cv (T )  
 NkB 2

T e E / T  1
 T v

vibration along three axes
E2
e E / T
cv (T )  3 R 2
T e E / T  1


2

2
E 2
e E / T
R 2
T e E / T  1


2
Limitation of Einstein model
2E
eE / T
cv (T )  3 R 2
T eE / T  1


2
cv  0 as T  0, cv  3R as T   E
Einstein specific heat is
significantly lower than the
experimental data in the
intermediate range.
Force-spring interaction
must be considered.
 Debye Specific Heat Model
 Assumption
Velocity of sound: same in all crystalline directions and for
all frequencies
a high-frequency cutoff and no vibration beyond this
frequency (shortest wavelength of lattice wave should be
on the order of lattice constants)
Upper bound m, determined by the total number of
vibration modes 3N (N : number of atoms)
Vibrations inside the whole crystal just like standing waves
 Equilibrium distribution
Phonon : Bose-Einstein statistics
h
  h , p 
vp
The total number of phonons is not conserved since it depends
on temperature.
 BE statistics
Ni
1
   i
gi e e  1
total number of phonons is not conserved
  0,   1/ k BT
Ni
1
  i / kBT
gi e
1
The energy levels are so closely spaced that it can be
regarded as a continuous function : BE distribution
function
dN
1
fBE ( ) 
 h / kBT
dg e
1
 Degeneracy of phonons
The number of quantum states per unit volume in the
phase space
Given volume V and within a spherical shell in the
momentum space (from p to p + dp)
h
4 Vp2dp 4 V 2d
p
dg 

3
3
vp
h
v
p
dg g( )d
4 2

 D( )d  3 d
V
V
vp
D() : density of states of phonons
the number of quantum states per unit volume per unit
frequency or energy (h) interval
2


4 2
1
2
12

D( )  3  4 2  3  3  
3
vp
v
v
v
t 
a
 l
va: weighted average
speed
Relation between f() and fBE()
dN
1
fBE ( ) 
 h / kBT
dg e
1
dN  f BE ( )dg  f BE ( ) g ( )d
f BE ( ) g( )d
dN
dn 

V
V
dg g( )d

 D( )d
V
V
dn  f BE ( ) g ( )d  f ( )d


0
0
n   f BE ( ) D( )d   f ( )d
Total number of quantum states must be equal to 3N
dg g( )d

 D( )d
V
V
2
 dg

 m 12
3N
0 V  V  0 D( )d  0 va3 d
3N 4 m3
 3
V
va
upper limit of the frequency (due to lattice constant)
 3na 
m  

4



1/ 3
va
na = N/V : number density of atoms
Debye temperature
h m
h  3na 
D 

kB
kB  4 
1/ 3
va
Vibration contribution to the internal energy
Distribution function for phonons
1 dN
f ( ) 
 D( ) f BE ( )
V d
12 2
1
 3na 
D( ) 
, f BE ( )  h / kBT
, m  

3
va
4

e
1


f ( ) 
12 2
v
3
a
e
h / kBT
1


9na 2

3
m
e
h / kBT
1


m
0
0
U  U 0   f ( )Vh d  

9 N 3 h
m
0

3
m
e
h / kBT
V
3
m
e
1

f ( )Vh d
d
va
9 N 2
vibration contribution to internal energy

1/ 3
h / kBT
1

,  m
Let
h
x
k BT
kBTx
kBTdx

 d 
h
h
3
 kBT  3
9N 
x h

3
m
x
kBT
9 N h
h 
D

0  3 e h / kBT  1 d  0  3 e x  1 h dx
m
m



 kBT 
U  U 0  9 NkBT 

 h m 
 T 
 9 NkBT 


 D
3

xD
0
3

xD
0
x3
dx
x
e 1
x3
dx
x
e 1

xD 
h m  D

k BT
T
Molar specific heat
3
 T  xD x 3
U  U 0  9 NkBT 
dx
 0 x
e 1
 D 
 T 
u  u0  9 RT 


 D
3

xD
0
R  N A kB
x3
dx
x
e 1
 T 
 u 
cv (T )  
 36 R 


 T V
 D 
3

xD
0
x3
dx
x
e 1
3
 T    xD x 3

9 RT 
dx 

 0 x
e 1 
  D  T 

 xD
  xD x 3
  xD x 3
dx  
dx 
 0 x
 0 x
T 
e  1  xD 
e  1  T
h m  D
xD 

k BT
T
 h m    xD x 3

xD xD3
1 xD4
 
dx   

 0 x
xD
xD
2 
k
T

x
e

1
T
T
e

1
e
1

 B  D

xD
0
 x

4 x


x
1 x
xe
D 1


dx



dx


2
x

0 4
x


ex 1
4
e

1

0
e

1


3
xD
4


1 xD x 4 e x
1 xD4
 
dx 
2
4 0 ex 1
4 e xD  1

 T 
cv (T )  36 R 


 D

 x

4
4 x
x
1
x
e
1
D
D


dx

2
x

4 0 ex 1
4 e D  1


3


3
 T  xD4
 9R 
 xD

 D  e 1
 T 
cv (T )  9 R 


 D
3

xD
0
e
x 4e x
x
1

2
dx
T   D
When
xD   D / T  0,

xD

xD
0
0

x 4e x
e 1
x

2
ex 1  x
dx  4
xD
0
xD4
x3
dx  xD
x
e 1
e 1
xD4
xD3
x3
3

x
dx

,
D,
xD
x
e 1
e 1
3
 T 
cv (T )  9 R 


 D
3

xD
0
3
e
x 4e x
x
1

2
 T  xD3
 9R 
 3R

 D  3
dx

xD
0

xD3
dx 
2
x
3
e 1
x 4e x

When T   D
Relative difference between calculated value and
experimental data : about 5%
 Riemann zeta function




0
0

x ne x
e 1
x

2
dx  n 

0
x n1
dx
x
e 1
x n1
dx   n  1 ! ( n)
x
e 1
1
1
1
 ( n)  1  n  n  n    
2
3
4
n
1
 ( n)

2
2
6
3
1.202 ...
4
4
90
5
1.037 ...
6
6
945
7
1.008 ...
8
8
9450
When T   D

xD

xD



0
0
0

x 4e x
e 1
x
x 4e x
e 1
x


2
dx  4
xD
2
dx  4
xD
0
0
h m  D
xD 

k BT
T
xD4
x3
dx  xD
x
e 1
e 1
x3
 4 4 4
dx  4  3! (4)  4  6 

x
e 1
90 15
x n1
dx   n  1 ! ( n)
x
e 1
 T 
cv (T )  9 R 


 D
T /  D  0.1
3

xD
0
3
 T 
xe
12
3
dx

R

T


(e x  1)2
5

 D
4
x
4
agrees with experiments within a few
percent
Debye Model vs. Einstein Model
 Free Electron Gas in Metals
Free electron
Translational motion of free electrons within the solid: largely
responsible to the electrical and thermal conductivities of
metals
 Fermi-Dirac distribution
dN
1
f FD ( ) 
 (   )/ kBT
dg e
1
 F : Fermi energy
 at 0 K
Degeneracy for electrons
dg  2  4 V ( me / h)3 v 2dv
due to positive and negative spins
Distribution function
in terms of the electron speed v
dN
dn 
 f (v )dv
V
3
1 dN 1 dg dN
v2
 me 
f (v ) 

 8 
 e (   ) / kBT  1
V dv V dv dg
h


in terms of the kinetic energy of the electron using
f (v )dv  f ( )d
me v 2

2
me v 2

, v
2
2
, dv 
me
2 1 1

me 2 
d
2me
dN
dn 
 f ( )d 
V
3
1 dN
v2
 me 
f (v ) 
 8 
 e (   )/ kBT  1
V dv
h


1 dN 1 dN dv 1 dN
f ( ) 


V d  V dv d  V dv
1
2 me
2 / me
1
 me 
 8 
 e (   ) / kBT  1
h
2 me


3
 2me 
 4  2 
 h 

3/ 2
e
(    ) / k BT
1
 f FD ( ) D( )
dN
f FD (v ) 
dg
dN  f FD (v )dg  f FD (v ) g (v )dv
f FD (v ) g(v )dv
dN
dn 

V
V
dg g(v )d

 D(v )dv
V
V
dn  f FD (v ) D(v )dv  f (v )dv


0
0
n   f FD (v ) D(v )dv   f (v )dv
 2me 
f ( )  4  2 
 h 

3/ 2
e
(    )/ kBT
1
 f FD ( ) D( )
 2m 
 Density of states for free electrons D( )  4  2 e 
 h 
 Fermi Level,  F
3/ 2

The number density of electrons as T  0


Ne
 2me 
ne 
 lim  f ( )d   lim  4  2 
T 0 0
T 0 0
V
 h 

3/ 2
e
(    )/ kBT
Since  –  < 0,

F
0
 2me 
4  2 
 h 
 3ne 
 F  

 8 
3/ 2
2/ 3
 2me 
 d   4  2 
 h 
h
h  3ne 

2me 8me   
2
2
3/ 2
2/ 3
2 3/ 2
F
3
1
d
 Sommerfeld expansion
Apply FD statistics to study free electrons in metals
Resolve the difficulty in the classical theory for electron
specific heat

ne   D( ) f FD ( , T )d 
0
F

0
D( )d      F  D( F ) 

2
 kBT 
6
2
D( F )
Since the difference between (T) and F is small,
ne  
F
0
D( )d 
Number of electrons does not depends on temperature
    F  D(  F ) 

2
 kBT 
6
2
D( F )  0
    F  D(  F ) 

 kBT 
2
6
 2me 
D( )  4  2 
 h 
D( F )  0
3/ 2
,
 2 me 
   F  4  h2 


   F  F 
2
3/ 2
 ( kBT )
2
12
F 
2
 2me 
D( )  4  2 
 h 
 ( kBT )
2
2
6
3/ 2
 2 me 
4  2 
 h 
1
2 
3/ 2
1
2 F
0
 2 ( kBT )2
  F  
12F
 0,
2

  ( kBT ) 
 ( kBT )
1   kBT  
  F 
 F  1 
  F 1  
 
2
12F
12F 
 3  2F  

2
2
2
2
Internal energy


0
0
U  V   f ( )d   V   f FD ( ) D( )d 
F
U
   D( )d 
0
V
 F    F  D( F )  F
F
   D( )d  
0
 2me 
D( )  4  2 
 h 
3/ 2
 2 ( kBT )2
 2 ( kBT )2
6
6
D( F ) 
6
D( F )
D( F )
h2  3ne 
 , F 
8me   
2
2

3
5  kBT  
U  N F 1 

 
5
12  F  


 2 ( kBT )2
2/ 3
3ne
 D( F ) 
2 F
Specific heat of free electrons
cv ,e
 5 2  kBT  kB   2 kBT
3
 u 

 N A F 
R

 

 T V 5
 6  F  F  2 F
Electronic contribution to the specific heat of solids is negligible
except at very low temperatures (a few kelvins or less)
cv (T )   sT  BT 3
= Electronic contribution + Lattice contribution
Quantum Size Effect on the Specific Heat
For nanoscale structures such as 2-D thin film and
supperlattices, 1-D nanowires and nanotubes, or 0-D
quantum dots and nanocrystals, substitution of
summation by integration is no longer appropriate.
2-D thin film: confined in one dimension
1-D nanowires: in two dimensions
0-D quantum dots: in three dimensions
1-D chain of N + 1 atoms in a solid with dimension L
with end nodes being fixed in position
min
min  2L0
max  2L
max L

N
min L0
L0
0
L
N number of vibrational
modes
eigenfunctions
x
x
 2 x 
 3 x 
 N x 
sin 
, sin 
, sin 
,  , sin 



  sin 

L
L
L
L








L
 0 
 Periodic Boundary Conditions
Born-von Kármán lattice model
medium: an infinite extension with periodic
boundary conditions
standing wave solutions for a solid with
dimensions of Lx, Ly, Lz (see Appendix B.7)

u( r , t )  A exp ik  r  i t

k : lattice wavevector
k
2

k  k x xˆ  k y yˆ  k z zˆ
 k x2  k y2  k z2
total number of modes = total number of atoms
along the 1-D chain
General Expressions of Lattice Specific Heat
lattice vibrational energy
1
1

u(T )  u0       / k T
 
1 2
P K
e
B
u0 : static energy at T = 0 K
f BE 
1
e
h / kBT
1

1
e
 / kBT
1
1
 : zero-point energy
2
K : wavevector index, dispersion relation    ( k )
P : polarization index
specific heat
u(T )  u0   
P
K


e
1
 / kBT
1
 
1 2
f BE
 u 
cv (T )  
  

T
 T  v P K
in terms of density of states
density of states : the number of states (or modes)
per unit volume and per unit energy interval
cv (T )   
P
f BE

D( )d
T

0
 kB  
P

0
  


k
T
 B 
2
e
e
 / kBT
 / kB T
1

2
D( )d
 Dimensionality: density of states
3-D reciprocal lattice space or k-space
k z volume of one
number of quantum
states per unit volume :
quantum state
k
 4 / 3  k 3
N
 2 / Lx   2 / Ly   2 / Lz 
L L L k


x
ky
kx
3-D view
y
z
6 2
3
Vk 3

6 2
1 dN
d  k3 
D( ) 



V d d  6 2 
3k 2 dk
k 2 dk


2
6 d 2 2 d
For a linear dispersion relation   va k
2


1
2
12

D( )  4 2  3  3  
3
v
v
v
t 
a
 l
2
k dk
1   1
2
D( ) 



2
2 
2 d 2  va  va 2 2va3
2
For a single polarization
4 2
D( )  3
va
Eq.(5-7)
D( )d  D( )d
4
4    d
2
D( )d  3 d  3 

d

2 3
va
va  2  2 2 va
2
2
2-D reciprocal lattice space or k-space
thin film, superlattice
volume of one
unit cell
2 / Lx
ky
number of quantum
states per unit area :
k
kx
2
Ly
N
 k2
 2 / Lx   2 / Ly 
Ak 2

4
1 dN
d  k2 
D( ) 



A d d  4 
2-D projection

When   va k , D( ) 
2 va2
2k dk
k dk


4 d 2 d
1-D reciprocal lattice space or k-space
nanowires, nanotubes
2 / Lx
kx
k
number of quantum states per unit length :
2k
Lk
N

2 / Lx

1 dN
d  k  1 dk
D( ) 




L d d     d
When   va k , D( ) 
1
 va
independent of frequency
Thin Films Including Quantum Wells
Ex 5-4: specific heat of thin film
thin film made of a monatomic solid
film thickness L with q monatomic layers, L = qL0
average acoustic speed va independent of temperature
kz
L0
L
.
.
.
ky
kx
molar specific heat
L0
specific heat per unit volume
f BE

T
cv (T )  
P
.
.
.
K
VN A
VN A kB
N
VR
Vcv 
cv  cv 
cv 
cv 
cv
NA
N
NkB
NkB
f BE
3VR
cv (T ) 


NkB k k ,k
T
x,
y
z
total number of modes

 q  1 
2
4
, 
,  , 
 0, 
L
L
L
kz  
 0,  2 ,  ,   q  2   ,  q

L
L
L
2
k z 
L
for q = 1, 3, 5, …
for q = 2, 4, 6, …
L
f BE
3VR

cv (T ) 

T
NkB k k ,k
x,
y
z
The lattice is infinitely extended in the x and y directions.
 2  k D2  k z2
L0
L
.
.
.
kz
kz
kD
ky
 kky
x
d

kx
ky
kx
 k
cv (T ) 

 0
3
 2  NkB k 
3VR
2
D
 k z2
z
f BE


2 d  k z
T

 2  k x2  k y2 , kz  2 / L
kD : cutoff value determined by setting the total number
of modes equal to the number of atoms per unit area
total number of modes
2
Ak
N

4

A k x2  k y2
4
  A
4
2


A k D2  k z2
k D2  kz2
N

A k
4
z
N
q
 2
total number of atoms
A L0

4
L0
.
.
.
L
kz
ky
kx
k D2  kz2
N
q

 2
A k
4
L0
z
k D2  kz2
1
1
q 2 1
q
2
2
2

kD 
kz 
kD 
kz  2




4 k  q  14
4 k
L0…
4 24 k 4
k
for q = 1, 3, 5,
, 
,  , 
 0, 
1/ 2
L 2 L
L
2
 4
kz 
k z 2 4
kz
k D  2  
2 , k D qL22   qq
for q = 2, 4, 6, …
L0  k ,q  ,   0 ,k  
0,

L
L
L
z
z
z
z
z
z
For a single layer
 4
k
kD   2  
L
 0 k q
2
z
z
 4

  2  0
 L0

1/ 2



1/ 2
 4

2
  2   kz 
L

k
0


2  3.54


L0
L0
z
1/ 2
3.982
As q  , k D 
L0
In 3-D case, k D 
3
6 2 3.90

L0
L0
 k
cv (T ) 

 0
3
 2  NkB k 
3VR
2
D
 k z2

z
f BE

2 d  k z
T

dispersion relation   va k  va  2  k z2
transformation  2   2  k z2
3 RA  kBT 
cv (T ) 


2 N  va 
2

kz
xD
xz
e
x 3e x
x
1

2
dx

va k 
x

k
T
B


Electrical and Thermal Conductivities of Solids
 Electrical Conductivity
R  re
re 
L
L

Ac  Ac
A
L
e-
V
z
1

V
R
I
L
V
I
V



 Ac
I
Ac
L
J E
re: resistivity, : conductivity
Ac: cross-sectional area, J: current density
E: electric filed
J E
Neweton’s 2nd law
ud
dv
F   eE  me
 me
dt

-eE
ud 
me
Current density J
3
 eE ne e 2
J   ene ud   ene 

E
me
me
ne e 2
J E 
E
me
ne e 2


me
ud : drift velocity
: relaxation time
ne: electron number
 Thermal Conductivity of Metals
 thermal conductivity (kinetic theory)
1
   cv ,e vF  e
3
 cv , e
ne 2 k B2 T

,
2F
  ne me ,
cv ,e
2 2  F
vF  e  vF   vF     v  

me
me
me R  k B
 2 k BT
 u 


R

2 F
 T V
 cv ,e
ne 2 k B2 T
 2 k BT
 ne me 
R
2 F
2F
2
F
1
  me v F2
2
2
2
vF 
me
1
1 ne 2 k B2 T 2F ne 2 k B2 T
   cv ,e vF  e  



3
3
2F
me
3me
 Wiedemann-Franz law

 T  Lz

ne e 2
ne 2 k B2 T

,  

me
3 me
Lz: Lorentz number
ne 2 k B2 T

3 me

1   2 k B2
Lz 

  2
2
ne e
T
3 e
T
me


2
 1   kB 
 

3
e



2
1   8.617  10 eV/K 
  2.44  108 V 2 / K 2  W   / K 2 
= 

3
e


5
 Derivation of Conductivities from the BTE
 Distribution function
local equilibrium, relaxation time approximation
f (r , v , t )  f ( r , p, t ), p  me v
h
2
h
kh
 f ( r , k , t ), p  , k 
, h=
, p 
k


2
 2
2 2
p2
k
 f ( , T ),  =

, T  T (r , t )
2me 2me
dk
f ( r , k , t )dk  f1 ( ,T ) d  f1 ( ,T )D( )d 
d
D() : density of states
f
f
f  f 
v
a
 
t
r
v  t  coll
steady state, relaxation time approximation
electric filed E along with temperature gradient in z
direction
f 0  f1
f 1
f 1
vz
 az

z
v z
 ( )
dv z
 eE
Fz   eE z  me
 me a z  a z 
dt
me
f 0  f1
f1 eE f1
vz


 z me v z
 ( )
f 0 ( , T )  f1 ( , T )
f1  eE f1 T
vz



  z m e T v z
 ( )
 eE f1 
 f 1 T 
f1 ( , T )  f 0 ( , T )   ( ) 
 vz

m



v

T

z
z
 e

f 0  f FD

1
1
2
  me v  me v x2  v 2y  v z2
2
2


 me v z
v z
Assume under local equilibrium
f 1 f 0

,
 
f 1 f 0

T T
In the case of no Temperature gradient
T
0
z
 eE f1 
 f 1 T 
f1 ( , T )  f 0 ( , T )   ( ) 
 vz

m



v

T

z
z
 e

f FD
eE
f1 ( , T )  f FD ( , T )   ( )
me v z
me

f FD
 f1 ( , T )  f FD ( , T )   ( )v z eE


ne   f FD D( )d : electron number density
0

J e   e  v z  f FD D( )d    eJ N
0
J e  e 

0


0
f FD

v z  f FD ( , T )   ( )v z eE



 D( )d

v z f FD ( , T ) D( )d  0
f FD
J e   e  v  ( )eE
D( )d 
0

1 2
2
2
Je   E    Je / E
vz  v 
3
3me

2
z
f FD
   e   ( )v
D( )d 
0

2e 2  f FD

 ( ) D( )d 

0
3me

2

2
z
f FD
Note that
  (   )

  (   ) : Dirac delta fucrtion



f ( x ) ( x  a )dx  f (a )
 (T )  F , (T )   F
Assume that
2e 2
 
3me
2e 2

3me


0


0
f FD
 ( ) D( )d 

 (   ) ( ) D( )d 
2e 2

 F  F D(  F )
3me
3/ 2
 2me 
D( )  4  2 
 h 
2/ 3
2
h  3ne 
F 
8me   
 (5.18)
(5.20)
 2me 
 D( F )  4  2 
 h 
3/ 2
2me
1  3ne 
 2 
h
4F   
 1  3ne 
F  4 
 4F   

3ne ne e 2
2e 2
2e 2

 F  F D(  F ) 
 F F

F
3me
3me
2 F
me
Drude-Lorentz expression
ne e 2


me
2/ 3




2/ 3
3/ 2

1/2
F
3ne

2 F
 Thermal Conductivity
In the case of no electric field, E = 0
 eE f1 
 f 1 T 
f1 ( , T )  f 0 ( , T )   ( ) 
 vz

m



v

T

z
z
 e

for an open system of fixed volume,
dU   Q  dN
qz  J E   J N

J E    v z f FD D( )d
0

J N   v z f FD D( )d

0
energy flux
particle flux


0
0
qz    v z f FD D( )d    v z f FD D( )d   v z (   ) f FD D( )d
0
f FD
f 1 T 

 f FD ( , T )   ( )  v z 


T

z



qz   v z (   ) f FD D( )d
0


f FD T  

  v z (   )  f FD ( , T )   ( )  v z 
D( )d


T z  


0
qz  
dT
dz


f FD T  
dz
dz

   qz
   v z (   )   ( )  v z 
D( )d


dT
T z  
dT


0

f FD
2
2
v
where z  2 3me
 v z  (   ) ( )
D( )d 
0
T
f FD
f FD
2 
2 

(   ) ( )
D( )d  
(   ) ( )
D( )d 


0
0
3 me
T
3 me
T
f FD
f FD     

Eq.B.82 (Appendix B.8)


T
  T 
2

3 me


0
2

3 me
f FD
D( )d 
(   ) ( )
T

0
2

3 meT


0

f FD     
(   ) ( )
D( )d 


  T 


0
f FD
 ( ) (   )
D( )d 

2
2
2

f

(
k
T
)


B
G ( )(   ) 2  FD  d  
G (  ) Appendix B.8
3
  
 2 ( k BT ) 2
2

 (  F )  F D(  F )
3meT
3
3ne  2 ( k BT )2 ne 2 k B2 T
2

 F F

F
3meT
2F
3
3 me
same result as simple kinetic theory
 Thermal Conductivity of Insulators
1
1
kinetic theory    cv v g  ph   cv v g ( )v g ( ) ( )
3
3

cv   
f BE ( , T )
T
P K
2
  
e  k BT
 kB   
D( )d (5.31)

 k BT
2
0
 1)
P
 k BT  ( e
under local equilibrium

2
 k BT



1

e
  k B    ( )v g2 ( ) 
D( )d

 k BT
2
0
3
 1)
P
 k BT  ( e
for isotropic distribution in k-space
1 dk
1 4 k 2 dk
1
k2
2
D( ) 



3
3
2
2 2
d

d

2

d

/
dk
2

v pv g
 2 
 2 
d

dk = 4 k dk , v g (k) 
, v p (k) 
dk
k
2
2
 k BT
2



1

e

  k B    ( )v g2 ( ) 
d


k
T
2
2
2
B
0
3
 1) 2 v p v g
P
 k BT  ( e
x

kBT
 
2
kBT
 kBT  2
x   
dx
 x  d 


kBT
2
for a large system with isotropic dispersion
x
xm
1
e
1
2
2

k
 ( x )v g ( x ) x x
2 B  0
2
2
3  2
(
e

1)
v
P
pv g
k B  k BT 


6 2 

3

P
xm
0
2
 k B T  2 k BT
dx

 x


x 4e x v g ( x )
 ( x)  x
dx
2
2
(e  1) v p ( x )
xm: corresponds to maximum frequency of
each phonon polarization