Electrons & Phonons
• Ohm’s & Fourier’s Laws
• Mobility & Thermal Conductivity
• Heat Capacity
• Wiedemann-Franz Relationship
• Size Effects and Breakdown of Classical Laws
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
1
Thermal Resistance, Electrical Resistance
P = I2 × R
∆T = P × RTH
∆V=I×R
R = f(∆T)
Fourier’s Law (1822)
© 2010 Eric Pop, UIUC
Ohm’s Law (1827)
ECE 598EP: Hot Chips
2
Poisson and Fourier’s Equations
·(òV )    0
J E  V  qDn
drift
·(k T )  p  0
diffusion
JT  kT
diffusion only
Note: check units!
Some other differences:
• Charge can be fixed or mobile; Fermions vs. bosons…
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
3
Mosquitoes on a Windy Day
• Some are slow
• Some are fast
• Some go against the wind
n
n(v0)dv
wind
dv
Also characterized by some
spatial distribution and average
n(x,y,z)
© 2010 Eric Pop, UIUC
v
Can be characterized by
some velocity histogram
(distribution and average)
ECE 598EP: Hot Chips
4
Calculating Mosquito Current Density
Area A
dr
• Current
=
# Mosquitoes through A per second
• NA = n(r,v)*Vol = n(r,v)*A*dr
• Per area per second: JA = dNA/dt = n(dr/dt) = nv
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
5
Charge and Energy Current Density
Area A
dr
• What if mosquitoes carry charge (q) or energy (E) each?
• Charge current: Jq = qnv
 Units?
• Energy current: JE = Env
 Units?
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
6
Particles or Waves?
• Recall, particles are also “matter waves” (de Broglie)
• Momentum can be written in either picture
pmv k 
*
2

• So can energy
E 
2 2
m*v 2
p2
k
E


2
2m* 2m*
1 E
v
k
• Acknowledging this, we usually write n(k) or n(E)
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
7
Charge and Energy Flux (Current)
• Total number of particles in the distribution:
N   n( k )   g ( k ) f ( k )
k
k
• Charge & energy current density (flux):
J q  q  n ( k )v ( k )  q   g ( k ) f ( k )  v ( k )
k
k
J E   E (k )n(k )v(k )   E (k )  g (k ) f (k )  v(k )
k
k
• Of course, these are usually integrals (Slide 11)
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
8
What is the Density of States g(k)?
• Number of parking spaces in a parking lot
• g(k) = number of quantum states in device per unit volume
• How “big” is one state and how many particles in it?
L  n  n
L
L
dk vol ,d
2
k
1  2 
 

s L 
d
“d” dimensions
“s” spins or polarizations
L
Ld  L, A, or V
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
9
Constant Energy Surface in 1-, 2-, 3-D
 2 


 L 
0
2
ky
kz
k
kx
ky
kx
kx
kz
Si
• For isotropic m*, the constant
energy surface is a sphere in 3-D kspace, circle in 2-D k-space, etc.
ml ≈ 0.91m0
mt ≈ 0.19m0
ky
kx
• Ellipsoids for Si conduction band
• Odd shapes for most metals
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
10
Counting States in 1-, 2-, or 3-D
ky
kz
 2 


 L 
 2 


 L 
kx
0
3
k
ky
kx
kx
• System has N particles (n=N/V), total energy U (u=U/V)
• This is obtained by counting states, e.g. in 3-D:

N   g (k ) f (k )dk
0

U 1
u    E (k ) g (k ) f (k )dk
V V 0
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
11
What is the Probability Distribution?
• Probability (0 ≤ f(k) ≤ 1) that a parking spot is occupied
• Probability must be properly normalized
1   f (k )
k
• Just like the number of particles must add up
N   n( k )   g ( k ) f ( k )
k
k
• But people generally prefer to work with energy
distributions, so we convert everything to E instead of k,
and work with integrals rather than sums
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
12
Fermi-Dirac vs. Bose-Einstein Statistics
1
T=0
fBE
fFD
T=300
0.6
0.4
T=0
1.5
0.8
T=1000
1
T=300
T=1000
0.5
0.2
0
-0.2
∞
2
1.2
0
-0.4
-0.2
0
0.2
0.4
0.6
0
E-EF (eV)
f FD ( E ) 
1
 E  EF
exp 
 k BT
f BE ( E ) 

 1

Fermions = half-integer spin
(electrons, protons, neutrons)
0.2
E (eV)
0.4
1
 E
exp 
 k BT
0.6

 1

Bosons = integer spin
(phonons, photons, 12C nuclei)
• In the limit of high energy, both reduce to the classical
Boltzmann statistics:
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
13
Temperature and (Non-)Equilibrium
• Statistical distributions (FD or BE) establish a link
between temperature, quantum state, and energy level
• Particles in thermal equilibrium obey FD or BE statistics
• Hence temperature is a measure of the internal energy
of a system in thermal equilibrium
1
0.8
• Result: “hot electrons” with effective
temperature, e.g., at T=1000 K
• Non-equilibrium distribution
© 2010 Eric Pop, UIUC
T=300 K
0.6
fFD
• What if (through an external
process) I greatly boost occupation
of electrons at E=0.1 eV?
ECE 598EP: Hot Chips
T=1000 K
0.4
0.2
0
-0.1
0
0.1
0.2
0.3
0.4
E-EF (eV)
14
Counting Free Electrons in a 3-D Metal


N 1
1
4 k 2 dk
n    g (k ) f (k )dk  3  2
f (k )
3
V V 0
L 0  2 / L 
E
spin
T>0
EF
# states in k-space
spherical shell
 2 


 L 
3
kz
dk
k
g  E   E1/2
ky
kx
• Convert integral over k to integral over E since we know
energy distribution (for electrons, Fermi-Dirac)
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
15
Counting Free Electrons Over E
• At T = 0, all states up to EF are filled, and above are empty
• This also serves as a definition of EF
1,   EF
f    
0,   EF

n   g   f   d 
0
EF

0


1  2m 
g   d   2  2 
3 

2
h2  3 
2 2/3
EF 
3 n 
 n
2m
8m   
 6.6 10 
34
EF 
8  9 10
31
2/3
3/ 2
1  2m 
g 3D    2  2   1/ 2
2   
3/2
 EF 
3/2
E
EF
2

11029

2/3
J  6 eV
TF  EF / k B  few 104 K
So EF >> kBT for most temperatures
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
g  E   E1/2
16
Density of States in 1-, 2-, 3-D
(with the nearly-free electron model and quadratic energy bands!)
1  2m 
E  2  2 
2 

*
g
3D
3/2
E
1/2
g
g 2D  E  
© 2010 Eric Pop, UIUC
m*

2
ECE 598EP: Hot Chips
1D
E 
m*3/2 21/2

E 1/2
van Hove
singularities
17
Electronic Properties of Real Metals
Metal
n (1028 m-3)
m*/m0
vF (106 m/s)
Cu
8.45
1.38
1.57
Ag
5.85
1.00
1.39
Au
5.90
1.14
1.39
Al
18.06
1.48
2.02
Pb
13.20
1.97
1.82
Fermi equi-energy surface
usually not a sphere
source: C. Kittel (1996)
• In practice, EF and m* must be determined experimentally
• m* ≠ m0 due to electron-electron and electron-ion
interactions (electrons are not entirely “free”)
• Fermi energy: EF = m*vF2/2
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
18
Energy Density and Heat Capacity


U 1
1
4 k 2 dk
u    E (k ) g (k ) f (k )dk  3  2 E (k )
f (k )
3
V V 0
L 0
 2 / L 
similar to before,
but don’t forget E(k)

u
f ( E )
CV 
E
g ( E )dE
T 0
T
2
k BT
CV 
nk B
2
EF
• For nearly-free electron gas, heat capacity is linear in T
and << 3/2nkB we’d get from equipartition
• Why so small? And what are the units for CV?
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
19
Heat Capacity of Non-Interacting Gas
• Simplest example, e.g. low-pressure monatomic gas
• Back to the mosquito cloud
• From classical equipartition each molecule has energy
1/2kBT per degree of motion (here, translational)
• Hence the heat capacity is simply CV 
• Why is CV of electrons in metal so
much lower if they are nearly free?
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
3
nk B
2
EF
k BT
20
Current Density and Energy Flux
• Current density:
J q  q n(k ) v(k )  q  g (k ) f (k ) v(k )dk
k
k
1 E
v
k
• What if f(k) is a symmetric distribution?
• Energy flux:
J E   n(k ) v(k ) E (k )   g (k ) f (k ) v(k ) E (k )dk
k
k
• Check units?
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
21
Conduction in Metals
• Balance equation for forces on electrons (q < 0)
dv
v
m
 m  q (ε  v  B)
dt

• Balance equation for energy of electrons
dE
E

 IV
dt

• Current (only electrons near EF contribute!)
J  q( n)vF
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
22
Boltzmann Transport Equation (BTE)
• The particle distribution (mosquitoes or electrons) evolves
in a seven-dimensional phase space f(x,y,z,kx,ky,kz,t)
• The BTE is just a way of “bookkeeping” particles which:
– move in geometric space (dx = vdt)
– accelerate in momentum space (dv = adt)
– scatter
• Consider a small control volume drdk
Rate of change
of particles both
in dr and dk
Net spatial
= inflow due to
velocity v(k)
+
Net inflow due
to acceleration by +
external forces
f
F
f
  vr f  k f 
t
t
© 2010 Eric Pop, UIUC
Net inflow due
to scattering
scat
ECE 598EP: Hot Chips
23
Relaxation Time Approximation (RTA)
f
F
f
  vr f  k f 
t
t
• Where recall v =
scat
and F =
• In the “relaxation time approximation” (RTA) the system is
not driven too far from equilibrium
f
t

scat
f (k )  f 0 ( E )
 scat

f '( E )
 scat
• Where f 0(E) is the equilibrium distribution (e.g. Fermi-Dirac)
• And f’(E) is the distribution departure from equilibrium
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
24
BTE in Steady-State (∂f/∂t = 0)
vr f 
F
k f  
f'

• Still in RTA, approximate f on left-side by f 0
0
F
F

f
0
• Furthermore by chain rule:
k f  k E

E
• So we can obtain the non-equilibrium distribution
f 0
f '   vr f   F  v
E
0
• And now we can directly calculate the current & flux
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
25
Current Density and Mobility
0



f
0
J q  q  g (k ) f '(k ) v(k )dk  q  g (k )  vr f  F  v
 vdk
E 

k
k
• This may be converted to an integral over energy
• Assume F is along z: vz2  v 2 / d (dimension d=1,2,3)
J q,z
q  
q   
 
 n  *  Fz   k BT  *  n 
z 
 m 
 m  
(Lundstrom, 2000)
(Ferry, 1997)
• Assumption: position-independent relaxation time (τ)
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
26
Energy Bands & Fermi Surface of Cu
• Fermi surface of Cu is just a slightly distorted sphere 
nearly-free electron model is a good approximation
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
27
Why the Energy Banding in Solids?
 k ( x)  uk ( x)eikx
uk ( x)  uk ( x  a)
• Key result of wave mechanics (Felix Bloch, 1928)
– Plane wave in a periodic potential
– Wave momentum only unique up to 2π/a
– Electron waves with allowed (k,E) can propagate (theoretically)
unimpeded in perfectly periodic lattice
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
28
What Do Bloch Waves Look Like?
uk ( x)  uk ( x  a)
• Periodic potential has a very small effect on the plane-wave character
of a free electron wavefunction
• Explains why the free electron model works well in most simple metals
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
29
Semiconductors Are Not Metals
empty states
empty
conduction
band
E
EF
0
electron states
in isolated atom
filled
valence
band
electron states
in metal
• Filled bands cannot conduct current
electron states
in semiconductor
ne 2 e pe 2 h


me
mh
• What about at T > 0 K?
• Where is the EF of a semiconductor?
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
30
Semiconductor Energy Bands
Si
GaAs
Equi-energy surfaces at
bottom of conduction band
Si
GaAs
• Silicon: six equivalent ellipsoidal pockets (ml*, mt*)
• GaAs: spherical conduction band minimum (m*)
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
31