Solid State PhysicsUNIST, Jungwoo Yoo
1. What holds atoms together - interatomic forces (Ch. 1.6)2. Arrangement of atoms in solid - crystal structure (Ch. 1.1-4) - Elementary crystallography - Typical crystal structures - X-ray Crystallography3. Atomic vibration in solid - lattice vibration (Ch. 2) - Sound waves - Lattice vibrations - Heat capacity from lattice vibration - Thermal conductivity4. Free electron gas - an early look at metals (Ch. 3) - The free electron model, Transport properties of the conduction electrons---------------------------------------------------------------------------------------------------------(Midterm I) 5. Free electron in crystal - the effect of periodic potential (Ch. 4) - Nearly free electron theory - Block's theorem (Ch. 11.3) - The tight binding approach - Insulator, semiconductor, or metal - Band structure and optical properties6. Waves in crystal (Ch. 11) - Elastic scattering of waves by a crystal - Wavelike normal modes - Block's theorem - Normal modes, reciprocal lattice, brillouin zone7. Semiconductors (Ch. 5) - Electrons and holes - Methods of providing electrons and holes - Transport properties - Non-equilibrium carrier densities8. Semiconductor devices (Ch. 6) - The p-n junction - Other devices based on p-n junction - Metal-oxide-semiconductor field-effect transistor (MOSFET)---------------------------------------------------------------------------------------------------------------(Final)
All about atoms
backstage
All about electrons
Main character
Main applications
Solid State PhysicsUNIST, Jungwoo Yoo
Free Electrons in Metals
1. The electron as a wave
2. Quantum mechanical description
3. Introduction
4. The free electron model
5. Transport properties of the conduction electrons
A number of electrical properties of metals can be well described with the model of free electrons, in which we ignored the attractive interaction between electrons and ions. In the free electron model, the valence electron move freely in the specimen of size L solid.
Solid State PhysicsUNIST, Jungwoo Yoo
The electron as a wave
Electron also have wavelike properties too!
A good example is the interference of electron waves in the experiment of Davisson and Germer in 1927
A
B
Screen with holes
Target Screen
Gun
A
B
Source of waves
Source of particles
nx
2
1 nx
Interference:Constructive interference
Destructive interference
Solid State PhysicsUNIST, Jungwoo Yoo
The electron as a wave
De Broglie’s matter wave:According to de Broglie’s hypothesis, particles of matter (such as electrons) have wave properties.
mv
h
Davisson and Germer’s experiment
Electron gunIncident beam Reflected beam
Detector
Path difference a sin2d
The reflected beam displayed an interference pattern a The wavelike nature of the electron
was conclusively demonstrated
Condition for constructive inteference
n (Bragg condition)
Solid State PhysicsUNIST, Jungwoo Yoo
The electron as a wave
sin2d n (Bragg condition)
The difference in angle between two successive maxima is of the order of
d/~
For good resolution, d~ (lattice spacing)
16111031
34
ms1025.7mJskg10101.9
106.6
m
hv
Then, the accelerating voltage in e-gun to produce electron with sufficient energy is
eVmv 2
2
1
V150Cskgm106.12
)1025.7(101.9
2122
19
26312
e
mvVa
Electron ? Proton ? Neutron ?
How about bullet ? ,kg10 3m ,ms10 13 v a m10 34Extremely difficult to observe !
Solid State PhysicsUNIST, Jungwoo Yoo
The electron microscope
The usage of wave nature of electron
The resolving power of an optical microscope is fundamentally limited by the wave-length of the light. For greater resolution, we need a shorter wavelength.
Electron can have very short wavelength and can be easily focused by electric and magnetic field
Solid State PhysicsUNIST, Jungwoo Yoo
Some properties of waves
A wave of frequency, w, and wave number, k, can be described as )()( tkziaezu
Then, the phase velocity can be defined as
fkt
zvp
constant
For a single frequency wave this is fairly obvious.
But, what happens when several waves are superimposed ?
The resultant wave is given by )()( tzki
nn
nneazu For the continuum case,
dkekazu tkkzi ))(()()(
Here, a(k) and w(k) are functions of k
Solid State PhysicsUNIST, Jungwoo Yoo
Some properties of waves
Let’s consider for the time being, t = 0, than
dkekazu ikz)()(
Now, let’s investigate the relationship between a(k) and u(z)
Consider the simplest possible case of
1)( ka
0)( ka
for22 0
kkk
k
k
)(ka
1
k
2/
2/
0
0
)(kk
kk
ikzdkezu a
2/
2/sin)( 0
kz
kzkezu zik
z
z
)(zu
Wave packetWidth of packet is determined by the point where the amplitude drops to 0.63 of its maximum value
22
kz
a 2 zk
Solid State PhysicsUNIST, Jungwoo Yoo
Some properties of waves
Let’s consider for the time-varying case, and a(k) is zero beyond Dk.
2/
2/
)(0
0
)()(kk
kk
tkzi dkekazu
Rewrite the formula as the following form
)( 00),()( tzkietzAzu
Then,
2/
2/
))()((0
0
00)(),(kk
kk
tzkki dkekatzA
Now, we can define two velocity,
I. Phase velocity: the velocity with which the central components propagate
00 / kvp
II. Group velocity: the velocity with which the envelop of the wave packet propagate
constant.)()( 00 zkkt
a ,0
0
kkt
zvg
a0kk
g kv
Solid State PhysicsUNIST, Jungwoo Yoo
Applications to electrons
Now, let’s try to apply the properties of waves to the particular case of the electrons
I) Position of electron as a wave,
If wave ripples are uniformly distributed in space, as is the case for a single fre-quency wave, the electron can be everywhere
If wave ripples are concentrated in space in the form of a wave packet, the posi-tion of an electron is defined with certain uncertainty. We also can identify the velocity of the wave packet with the electron velocity.
I) Energy of the electron
A photon of frequency w has an energy
hfEAnalogously, the energy of an electron in a wave packet cnetred at the fre-quency w is given by the same formula.
hfETaking the potential energy as a zero,
2
2
1mvE a
k
vmv
kg
g
ak
vm g
a gmvk
agmv
h
Solid State PhysicsUNIST, Jungwoo Yoo
Applications to electrons
Uncertainty principle
The electron of a wave packet have a width of position Dz, an uncer-tainty about the position of the electron
From the previous relationship in the wave packet
2 zk and gmvk
We get
hzp
Ex) if we know the position of the electron with an accuracy of 10-9m then the uncertainty in momentum is
125 kgms106.6 p a 15 ms107 vThe uncertainty in velocity is quite appre-ciable
For Dx ~ 10-9m for the uncertainty in position, and a bullet with a mass of 10-3kg, the uncertainty in velocity decreases to
122 ms106.6 v
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
Important consequences from Quantum mechanical description
I) Discreet energy level
II) Tunneling
III) Spin & Magnetism
III) Theory of Conventional Superconductor (BCS theory)
Ex) particle in 1D box, potential well, harmonic oscillation, hydrogen atom, etcand new particles with fundamental unit of energy, photon, phonon, magnon, etc
II) Band theory
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
The electron as a wave
2
22
2ˆ
zmH
H
In a free space, V=0, the Schröndinger eqn. is
The general solution for the time dependent Schröndinger eqn. is
),(ˆ),( trHtrt
i
The general solution for the Schröndinger eqn. is plane wave propagating left and right
)exp()exp( ikzBikzA m
kE
2
22
)exp()exp()exp( ikzBikzAti E
The momentum of the electron is equal tok
Then, how about the position of the electron ?
For simplicity, take B=0 (the case for the forward traveling wave)
a The probability of finding the electron at any particular point is unity
.const)(2 z
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
The electron as a particle
02 2
22
Ezm
The Schröndinger eqn.
is the linear differential equation, hence the sum of the solutions is still a solution.
Therefore, a wave packet (a sum of many waves) is also solution of Schröndinger eqn.
A wave packet represents an electron as a particle because is appre-ciably different from zero only within the packet.
2)(z
With the choice in the interval , the probability of finding the elec-tron is given by
22
)2/(
)2/sin(K)(
kz
kzz
1)( ka k
Solid State PhysicsUNIST, Jungwoo Yoo
Particle in 1D box
An electron of mass m is confined to a length L by infinite barriers
From the schrödinger equation H 2
22
2ˆ
xmH
From the fixed boundary condition, 0)()0( L
integral number of half wavelength should fit in size L
22
2
L
n
mn
Ln )2/(L
nk
a
Wavefunction can be given by the form kxAx sin)(
Brief Review of Quantum Mechanics
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
Harmonic Oscillator )
2
1( nn
,2
ˆ
2
ˆˆ22 x
Km
pH
m
K
,2
ˆ
2
ˆ 22
2 xm
m
p
)2
1ˆˆ( aa
N
m
pix
ma
2
ˆˆ
2ˆ
m
pix
ma
2
ˆˆ
2ˆ
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
The electron meeting a potential barrier
For finite potential barrier
z
V
Region I Region II
,ˆ H )(2
ˆ2
2
xVxm
H
VI=0 VII=V
,11 xikxikI BeAe E
m
k
2
21
2For region I,
,2xikII Ce VE
m
k
2
22
2For region II,
From boundary condition (or matching cond.), At z=0, both and should be continuous
z
CBA
,)( 21 CikBAik
a ,21
21
kk
kk
A
B
.2
21
1
kk
k
A
C
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
The electron meeting a potential barrier
For finite potential barrier
z
V
Region I Region II
For E > V
,ˆ H )(2
ˆ2
2
xVxm
H
VI=0 VII=V
Oscillatory solution in region II, and finite amount of re-flection
022 k a k is real
For E < V
022 k a k is imaginary
The amplitude of wave de-cline exponentially in region II (there is finite, though de-clining, probability of finding the electron at z > 0)
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
Tunneling
For finite potential barrier
x
V
Region I Region II Region III
For E > V
,11 xikxikI BeAe E
m
k
2
21
2
,22 xikxikII DeCe VE
m
k
2
22
2
xikIII Fe 1 E
m
k
2
21
2
-a a
)2(sin)(4
11
1
22
2
akVEE
VT
,ˆ H )(2
ˆ2
2
xVxm
H
,2
A
FT
2
A
BR 1RT
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
From boundary condition at x = a and x = -a
aikaikaikaik eA
De
A
Ce
A
Be 2211
aikaikaikaik e
A
De
A
Cke
A
Bek 2211
21
aikaikaik eA
Fe
A
De
A
C122
aikaikaik eA
Fke
A
De
A
Ck 122
12
1
221
22
21
22 )2sin(
2)2cos(2
akkk
kkiake
A
F aik
)2sin(2 221
21
22 ak
kk
kk
A
Fi
A
B
122
A
B
A
FRT
)2(sin4
11
12
2
21
22
21
2
akkk
kk
F
A
T
a )2(sin
)(4
11
12
22
akVEE
V
T
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
For finite potential barrier
x
V
Region I Region II Region III
For E < V
,11 xikxikI BeAe E
m
k
2
21
2
,xxII DeCe 0
2
22
EVm
xikIII Fe 1 E
m
k
2
21
2
-a a
)2(sinh)(4
11
1
22
aEVE
VT
,ˆ H )(2
ˆ2
2
xVxm
H
,2
A
FT
2
A
BR 1RT
Tunneling
2ik
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
Tunneling
For finite potential barrier
x
V
Region I
Region II Region III
-a a
For E < V
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
The Ramsauer effect
x
V
Region I Region II Region III
,11 xikxikI BeAe E
m
k
2
21
2
,22 xikxikII DeCe VEEV
m
k
2
22
2
xikIII Fe 1 E
m
k
2
21
2
-a a
)2(sin)(4
11
1
22
2
akVEE
VT
,ˆ H )(2
ˆ2
2
xVxm
H
,2
A
FT
2
A
BR 1RT
For E > 0
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
Potential well
,xI Ae 0
2
22
Em
,ikxikxII CeBe 0
2
22
EVm
k
,xIII De 0
2
22
Em
,ˆ H )(2
ˆ2
2
xVxm
H
x
V
Region I Region II Region III
-a a
For E < 0
From boundary condition (or matching cond.), At x=a,-a both and should be continuous
z
1E2E
3E
Solid State PhysicsUNIST, Jungwoo Yoo
Brief Review of Quantum Mechanics
The uncertainty principle
V
V
dV
dVAA 2
ˆ
The average value of a physically measurable quantity is
The rms value of position and momentum are
2/12xxx 2/12
ppp
The uncertainty relationship is
hpx
htE
Solid State PhysicsUNIST, Jungwoo Yoo
Introduction
Electrical resistivity of materials
Materials fall into 3 main classes:
Metals: resistivities between 10-8 to 10-5 Wm
Semiconductors: resistivities between 10-5 to 10 Wm
Insulators: resistivities above 10 Wm
Resistivity increases by addition of small amount impurities
Resistivity decreases by addition of small amount impurities
Solid State PhysicsUNIST, Jungwoo Yoo
Introduction
For most metals: T For most semiconductors:TkBe /
tend to become insulator at low T
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Nucleus and many electrons
Most electrons are boundedElectrons at outer shell mainly address properties of atomsElectrons at Fermi energy mainly ad-dress physical properties of materi-als electrical, magnetic, optical
Ions and electrons
Solid
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Solid
Drude model: an early look at metal (proposed by Paul Drude in 1900)
It is based on the elemental kinetic theory of gases
eliminate all the electron ion interactions and replace them by a single pa-rameter t.
t is collision time
Ch 3.3 transport properties of conduction electrons is based on Drude model.
How is solid so transparent to conduction electrons ? i) The electron matter wave can propagate freely in a periodic structure ii) The conduction electron is scattered very rarely
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Solid
Sommerfeld model: impose Fermi-Dirac distribution on Drude model
treat electrons as Fermions
a Free electron model
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
• The valance electrons are responsible for the conduction of electricity, and for this reason these electrons are termed conduction electrons.
• Na11 → 1s2 2s2 2p6 3s1
• This valance electron, which occupies the third atomic shell, is the electron which is responsible chemical properties of Na.
• When Na atoms come closer to form a Na metal, Solid state of Na atoms overlap slightly. Then, a valance electron is no longer attached to a particular ion, but belongs to the whole crystal, since it can move readily from one ion to its neighbour, and then the neighbour’s neighbour, and so on.
• This mobile electron becomes a conduction electron in a solid.
Free electron model
Valance electron (loosely bound)
Core electrons
+
+ + +
+ + The removal of the valance elec-trons leaves a positively charged ion.
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Consider 1D case
One-dimentional periodic potential associated with a chain of identical atoms
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
In free electron model: the positive ion cores is spread uniformly throughout the metal so that the electrons move in a constant electrostatic potential. No details of crystal structure No el-ions interaction, No el-el interaction
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Particle in 1D box
An electron of mass m is confined to a length L by infinite barriers
From the schrödinger equation H 2
22
2ˆ
xmH
From the fixed boundary condition, 0)()0( L
integral number of half wavelength should fit in size L
Ln )2/(L
nk
22
2
L
n
mn
a
Wavefunction can be given by the form kxAx sin)(
Solid State PhysicsUNIST, Jungwoo Yoo
For 3D
The Free Electron Model
H 22
2ˆ
mH
From the periodic boundary condition,
),,(),,( zyxLzLyLx
Solution of the schrödinger equation
)(
2/12/1
11),,( zkykxkirki zyxe
Ve
Vzyx
pNa pNa
k2
a
2xk p
L
2yk q
L
2zk r
L
m
k
2
22 )(
2222
2
zyx kkkm
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
kdkdkkg R
3)()( dkk
Vkd
L 23
33
482
2
the number of allowed k values inside a spherical shell of k-space of radius k
,2
2
dkVk
For spin up and down
dkkgdg )()(
d
dkkgg )()(
m
k
2
22
m
k
dk
d 2
2
2
m
k
2/12/33222222
2
)2(2
2
22)(2)(
m
VmVm
k
mVk
d
dkkgg
2/3320
2/12/3320
)2(3
)2(2
)( FmV
dmV
dgNFF
Solid State PhysicsUNIST, Jungwoo Yoo
When all states are filled up to a certain energy, this upper limit energy is being called Fermi energy EF. The Fermi energy is obtained by integrating density of states between 0 and EF, should equal N. Hence
The Free Electron Model
2/12/332
)2(2
)(
mV
Eg
m
k
V
N
mF
F 2
3
2
223/222
3/222 3
V
NkF
kz
ky
kx
Fermi surfaceE=EF
kF
2/3320
2/12/3320
)2(3
)2(2
)( FmV
dmV
dgNFF
)(3
2FF g
2/12/332
)2(2
)( FF mV
g
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron ModelEx) Monovalent potassium metal as an example; the atomic density (the same as valence electron density) N/V is 1.402x1028 m-3
Only at a temperature of the order of TF that the particles in a classical gas can attain (gain) kinetic energies as high as EF
At T > TF , the free electron gas behave like a classical gasAt T < TF , behaviour of the free electron gas is dominated by Pauli exclusion principle
Fermi energy:
Fermi wavenumber:
Fermi temperature:
Fermi momentum:
Fermi velocity:
eV12.2J1040.33
219
3/222
V
N
mF
746.03
3/12
V
NkF
Å-1
16 ms1086.0 e
F
e
FF m
k
m
Pv
FF kP
Kk
TB
FF
41046.2
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Free electron gas at finite T
At a temperature T , the probability of occupation of an electron state of energy E is given by the Fermi distribution function
1
1),(
/)( TkBe
Tf
EFE<EF E>EF
0.5
fFD(E,T)
E
At absolute zero Temp. T=0K
11
1)( /)(
TkBe
f For
01
1)( /)(
TkBe
f For
At T = 0K, m is eF
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Free electron gas at finite T
The number of electrons per unit energy range in thermal equilibrium is given by
The # of electrons per unit energy =
Density of state g(e) ⅹ probability of occupation f(e, T)
),()(),( TfgTn
T > 0
n(e,T)
e
g(e)
eF
T = 0
TkB~The effect of finite temperature
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Heat capacity of the free electron gas
T > 0
n(e,T)
e
g(e)
eF
T = 0
TkB~
The amount of energy absorbed by electrons at finite T = shaded area ⅹ kBT
½ ⅹ height (½ g(eF)) ⅹ base (2kBT)
Thermal energy =
2))((2
1~)0()( TkgETE BF
TkgT
EC BFv
2)(
)(
3
2FF gN
FBFF Tk
NNg
2
3
2
3)(
FBv T
TNkC
2
3
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Heat capacity of the free electron gas
FBv T
TNkC
2
3
For exact calculation,
0
),( dTnN
0
),()( dTnTE
FBBFv T
TNkTkgC
2)(
3
22
2
The lattice heat capa-city,
at room T, 3NkB and falls off T3 below Debye temperature
The total heat capacity at low temperat-ure,
ElectronicHeat capacity
Lattice HeatCapacity
3TTC
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron Model
Heat capacity of the free electron gas
Intercept gives g
slope gives bFor metal:
For insulator:
Heat capacity of KCl
Heat capacity of K
21KmolmJ08.2 212
KmolmJ67.12
F
B
T
Nk
KTF41046.2
No contribution from conduction electrons
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron ModelHeat capacity of the free electron gas
The discrepancy between predicted values and experimental values can often be corrected by introducing effective mass of electron m*, which dif-fers from their bare mass
Effective mass correction
For change by a factor,F ,FT ,Fvm
m
For change by a factor),( Fg ,vCm
m
For a theoretical calculation m*, electron-phonon interaction and electron-electron interaction need to be considered
The effective mass m*, associated with any physical property provides a use-ful way of quantifying departures of that property from the free electron prediction
For Potassium, 25.1
m
m
Solid State PhysicsUNIST, Jungwoo Yoo
The Free Electron ModelMetallic binding
22
2
L
n
mn
2
22
1 2ma
a
5a
2
22
1 50ma
2
22
5 2ma
Consider 4 electrons in each 5 atoms
Delocalization of electrons reduction of kinetic energy metallic bonding
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electronsThe equation of motion of the electrons
In the presence of electric field (E) and magnetic field (B), the force on an electron of charge e and mass me
BveEedt
vdmF e
In free electron model, the electron is being described as a plane wave states, which extend through the crystal.
To describe with classical equation of motion for a plane wave states of elec-tronsNeed to introduce wave packet with well defined position and momentum to give a particle like entity for a electrons.
The velocity of wave packet is the group velocity of the waves
ee m
p
m
k
kd
d
kd
dv
1
The size of wave packet should be much larger than atomic spacing with well defined positionto approximate electron-ion interaction
to use equation of motion~ 10 a
For 10a, k ~ kF/10 P ~ PF/10
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electronsThe equation of motion of the electrons
In the presence of electric field (E) and magnetic field (B), the force on an electron of charge e and mass me
BveEedt
vdmF e
In the absence of magnetic field, the applied E results a constant acceleration but this will not cause a continuous increase in current. Since electrons suffer collisions withI) phonons II)electrons
a impose collision by adding v
me
BveEev
dt
vdme
In the absence of electric and magnetic field, v decay exponentially to zero, a v is drift velocity, is departure from the thermal equilibrium state given by Fermi distribution
Drude model
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electronsThe equation of motion of the electrons
In the presence of dc electric field (E) only
The steady state solution (no acceleration) is
Em
ev
e
eMobility:
The Ohm’s law for the electric current density J
EEm
nevnqj
e
2
where n=N/V
ee
nem
ne 2
Electrical Resistivity and Resistance
A
LR
,
1
Collistions: 1) electron-phonon a as T g 0, tph(T) infinity 2) electron with impurity atoms gives a finite scattering time t0 even at T g 0
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
At finite temperature, the electron scattering rate
0
1
)(
11
Tph
term for perfect crystal(phonon scattering)
term for scattering with impurity
Valid if two scattering mechanism is independent, for low impurity material
The electrical resistivity (Mattheisen’s rule):
00
222)(
1
)(
111
Tne
m
Tne
m
ne
mI
e
ph
ee
Ideal resistivity Residual resistivity
Residual resistivity ratio: 0T
RT
can be as high as 106
Solid State PhysicsUNIST, Jungwoo Yoo
Temperature (K)
pureimpure
Rel
ativ
e re
sist
ance
104
R/R
29
0K
Transport properties of the conduction electrons
The resistivity vs T graphs for different spec-imens of the same mate-rial differ only by a dis-placement, This dis-placement is associated with the variation in r0 due to different imper-fection densities
The r vs T curves for sodium specimens of differing purity
For sodium: 117 m100.2 RT 114 m103.5 residual
Taking mmn e ,m107.2 328s106.2~ 14
2
ne
ms100.7~ 11
at room temperature
at T = 0
Fermi velocity: 16
3/22
ms101.13
V
N
mm
kv
ee
FF
a electron mean free path: 29 nm at RT 77 mm at T g 0
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electronsThe thermal conductivity
Electrons coming from a hotter region of the metal carry more thermal energy than those from a cooler region, resulting in a net flow of heat. The thermal conductivity
Typically, due to the contribution of conduction electrons metalnonmetal KK
From elementary kinetic theorylvCK FV3
1
Mean free path of conduction electrons: Fvl
2
3
1FV vCK a
FBV T
Tk
V
NC
2
2 2
2
1FF mv
ae
B
m
TnkK
3
22
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
Wiedemann-Franz law
e
B
m
TnkK
3
22 e
e
nem
ne 2
The ratio of the electrical and thermal conductivities
a independent of the electron gas parameters;
2822
KW1045.23
e
k
T
K B
Lorentz number
28 KW1023.2 T
KL
For copper at 0 C
The collision time limiting the flow of electric and heat currents are the same
Electrical and Thermal Conductivity
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
The group velocity of the electronic waves
ee m
p
m
k
kd
d
kd
dv
1 BveEe
v
dt
vdme
The drift velocity associ-ated with electric current
Em
ev
e
The change in the wavevector of each elec-tron
Ee
vm
k e
The electric current carrying state corresponds to a shift by of the whole Fermi sphere
k
For 2D
E
For a current density of 107 Am-2
Fv
ne
jv
8
1
11928
7
10~
mms10~
ms1010
10
Electrical and Thermal Conductivity
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
E
k
At finite T, T≠0
xk
yk
0 Tx HotCold
Electrical and Thermal Conductivity
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
At finite T, T≠0
xk
yk
0 Tx HotCold
Electrical and Thermal Conductivity
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
Scattering associated with electrical and thermal conductivity by electrons
fk
ik
q
fi kqk
Phonon absorption
fk
ik
q
qkk fi
Phonon emission
Electron-phonon scattering
Need to satisfy conservation of momentum and energy
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
E
k
xk
yk
HotCold
Scattering associated with electrical and thermal conductivity by electrons
A typical relaxation processes associated with electric-current carrying state
Momentum of phonon is importnat
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
At finite T, T≠0
xk
yk
0 Tx HotCold
Electrical and Thermal Conductivity
A typical relaxation processes associated with heat-current carrying state
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
xk
yk
At very high T , maximum phonon energy a to satisfy conservation of energy only states near the Fermi energy involve scattering with phonon
But, maximum phonon momentum
a both large and small momentum transfer can occur by phonon scatter-
ing
)( DT
Scattering associated with electrical and thermal conductivity by electrons
q
Fk
~
DBk FBTk
Since wavelength for is scale of atomic spacingand Fermi wavelength is alsoscale of atomic spacing
DBk
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
xk
yk
At low T , typical phonon energy
a phonon energy has momentum of order
a allow scattering between the electron states close to each other
in the vicinity of Fermi energy
)( DT
Scattering associated with electrical and thermal conductivity by electrons
TkB
FD pT )/( TkB
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
Temperature Dependence of Electrical and Thermal Conductivity
At high T ( ), the typical phonon energy is . and electron mean free path lph is
inversely proportional to the phonon number.
Lattice vibration energy is and each phonon energy ( ) is nearly const.
a phonon # a the electron scattering time T 1Tph
,)(22
Tlne
vm
ne
mT
ph
Fe
ph
eI
T
TNkB3
Bk
Bk
T
D
TI
T
Temperature Dependence of Electrical Conductivity
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
Temperature Dependence of Electrical and Thermal Conductivity
At high T ( ), the typical phonon energy is . and electron mean free path lph is
inversely proportional to the phonon number.
Lattice vibration energy is and each phonon energy ( ) is nearly const.
a phonon # a the electron scattering time T 1TphTNkB3
Bk
Bk
T
K
D T
Temperature Dependence of Thermal Conductivity
0TK
022
3T
m
TnkK
e
B
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
Temperature Dependence of Electrical and Thermal Conductivity
At very low T ( ), scattering dominated by impurity
D
0
T
Temperature Dependence of Electrical Conductivity
0T const.ph
.)(2
constne
mT
ph
eI
a
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
Temperature Dependence of Electrical and Thermal Conductivity
K
D T
Temperature Dependence of Thermal Conductivity
a Tm
TnkK
e
B 3
22
TK
At very low T ( ), scattering dominated by impurity0T const.ph
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
Temperature Dependence of Electrical and Thermal Conductivity
At low T ( ), the average phonon energy is .
Lattice vibration energy is and
phonon energy only allow low momentum transfer scattering.
a mean free path and scattering time are inversly proportional to the phonon #
K
D T
,3th
T
TkB
TkB
Temperature Dependence of Thermal Conductivity
T4T
a 222
3 T
m
TnkK
e
B
2TK
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
Temperature Dependence of Electrical and Thermal Conductivity
At low T ( ), the average phonon energy is .
phonon energy has momentum of order
TkB
TkBT
FD pT )/(
D
5TI
T
Temperature Dependence of Electrical Conductivity
Too small to introduce Large momentum change
Fk
Fk q
k
22
2T
k
q
F
For small theta
5
21
~1
TT
thDel
Therefore,
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
Temperature Dependence of Electrical and Thermal Conductivity
D
0 5TI
TI
T
K
D T
2TK0TK
Temperature Dependence of Electrical Conductivity
Temperature Dependence of Thermal Conductivity
TK
Impurity scattering dominant
Phonon scattering dominant
Impurity scattering dominant
Phonon scattering dominant
Wiedemann-Franz law fails
Wiedemann-Franz law22
3
e
k
T
K B
Solid State PhysicsUNIST, Jungwoo Yoo
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electrons
The Hall Effect
The origin of hall ef-fect
a Lorentz force
Bve
The Lorentz force tends to deflect the electrons downwards and this results in the rapid build up of a negative charge den-sity in –y side of the bar
In steady state, the Lorentz force on the electrons is just balanced by
the force due to the Hall field
Bve
HEe
HE
jBRE HH
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electronsThe Hall Effect
In steady state, the Lorentz force on the electrons is just balanced by
the force due to the Hall field
Bve
HEe
HE
BveEev
me
z
x
B
v
kji
e
00
00
xxe eEvm /
)(0 BvEe xy
a
neRH /1 HRand
BjRneBjBvE xHxxy )/(
Solid State PhysicsUNIST, Jungwoo Yoo
Transport properties of the conduction electronsThe Hall Effect
Metal Group )/(1 NeRH
Na I +0.9Ka +1.1
Cu IB +1.3Au +1.5
Be II -0.2Mg +1.5
Cd IIB -2.2
Al III +3.5
Hall coefficient of various met-als
Predicted as the number of conduction electrons per atom
HR is positive a charge carriers are holes
Solid State PhysicsUNIST, Jungwoo Yoo
Summary
The Free Electron Model: By ignoring all interaction, we describe the electrons as a free particles confined in size L of specimen.
How we describe these electrons ? Using quantum mechanism, electron has wave and particle duality
)(
2/12/1
11),,( zkykxkirki zyxe
Ve
Vzyx
)(22
222222
zyx kkkmm
k
kdkdkkg s
3)()( dkk
Vkd
L 23
33
482
,2 2
2
dkVk
2/12/33222222
2
)2(2
2
22)(2)(
m
VmVm
k
mVk
d
dkkgg
m
k
V
N
mF
F 2
3
2
223/222
2/3320
2/12/3320
)2(3
)2(2
)( FmV
dmV
dgNFF
3/222 3
V
NkF
Solid State PhysicsUNIST, Jungwoo Yoo
Fermi energy:
Fermi wavenumber:
Fermi temperature:
Fermi momentum:
Fermi velocity:
eV12.2J1040.33
219
3/222
V
N
mF
746.03
3/22
V
NkF
Å-1
16 ms1086.0 e
F
e
FF m
k
m
Pv
FF kP
Kk
TB
FF
41046.2
SummaryThe Free Electron Model: For potassium metal
of N/V = 1.402x1028 m-3
Atomic scale
Much higher than Debye T
At finite T the probability of the occupation of an electron state of energy E
1
1),( /)( TkBe
Tf
EFE<EF E>EF
0.5
fFD(E,T)
E
Solid State PhysicsUNIST, Jungwoo Yoo
Summary
The Free Electron Model:
The # of electrons per unit energy = ),()(),( TfgTn
T > 0
n(e,T)
e
g(e)
eF
T = 0
TkB~
The amount of energy absorbed by electrons at finite T = shaded area ⅹ kBT
Thermal energy =
,))((2
1~)0()( 2TkgETE BF )(
3
2FF gN
FBv T
TNkC
2
3
ElectronicHeat capacity
Lattice HeatCapacity
3TTC
Solid State PhysicsUNIST, Jungwoo Yoo
SummaryTransport properties of the conduction electrons
BveEedt
vdmF e
a impose collision by adding v
me
BveEev
dt
vdme
Force applied to an electron in the presence of E&M fields
Em
ev
e
a
Steady state solution when B=0
The Ohm’s law for the electric current density J
EEm
nevnqj
e
2
ee
nem
ne 2
a
0
1
)(
11
Tph
term for perfect crystal(phonon scattering)
term for scattering with impurity
00
222)(
1
)(
111
Tne
m
Tne
m
ne
mI
e
ph
eea
Ideal resistivity Residual resistivity
From elementary kinetic theorylvCK FV3
1 a
e
B
m
TnkK
3
22
Solid State PhysicsUNIST, Jungwoo Yoo
SummaryTransport properties of the conduction electrons
Wiedemann-Franz law
e
B
m
TnkK
3
22 e
e
nem
ne 2
2822
KW1045.23
e
k
T
K B
Lorentz number
Scattering associated with electrical and thermal conductivity by electrons:
a Absorption and emission of phonon fk
ik
q
fk
ik
q
xk
yk
0 Tx
HotCold
A typical relaxation processes associated with heat-current carrying state
E
xk
yk
k
A typical relaxation processes associated with electric-current carrying state
Solid State PhysicsUNIST, Jungwoo Yoo
D
0 5TI
TI
T
K
D T
2TK0TK
Temperature Dependence of Electrical Conductivity
Temperature Dependence of Thermal Conductivity
TK
Impurity scattering dominant
Phonon scattering dominant
Impurity scattering dominant
Phonon scattering dominant
Wiedemann-Franz law fails
SummaryTransport properties of the conduction electrons
At very low T ( ), scattering dominated by impurity0T const.phAt low T ( ), the average phonon energy is and momentum of T TkB
a mean free path and scattering time are inversely proportional to the phonon
#
3th
TFD pT )/(
At high T ( ), the typical phonon energy is .T Bk
a mean free path and scattering time are inversely proportional to the phonon
#
1Tph
e
B
m
TnkK
3
22 e
e
nem
ne 2
Solid State PhysicsUNIST, Jungwoo Yoo
The Hall Effect
In steady state, the Lorentz force on the electrons is just balanced by the force due to the Hall field
Bve
HEe
HE
jBRE HH
BveEev
me
z
x
B
v
kji
e
00
00
xxe eEvm /
)(0 BvEe xy
a
neRH /1 HRand
BjRneBjBvE xHxxy )/(
Summary