Download Lecture 9

PHYS 342/555
Introduction to solid state physics
Instructor: Dr. Pengcheng Dai
Associate Professor of Physics
The University of Tennessee
(Room 407A, Nielsen, 974-1509)
(Office hours: TR 1:10AM-2:00 PM)
Lecture 9, room 512 Nielsen
Chapter 6: Free electron Fermi gas
Lecture in pdf format will be available at:
http://www.phys.utk.edu
1
Chapter 6: Free electron Fermi gas
Under quantum theory and the Pauli exclusion principle, we consider
N noninteracting electrons confined to a volume V ( L3 ). If the wave function
G
of single electron is ψ (r ), then
=2 2 G
=2 ⎛ ∂ 2
∂2
∂2 ⎞ G
G
ψ
(
r
)
ψ
(
r
)
εψ
(
r
).
−
+
+
=
−
∇
=
⎜ 2
2
2 ⎟
2m ⎝ ∂x ∂y ∂z ⎠
2m
Applying boundary condition
ψ ( x, y, z + L) = ψ ( x, y, z );ψ ( x, y + L, z ) = ψ ( x, y, z );
ψ ( x + L, y, z + L) = ψ ( x, y, z ). The solutions are
1 ikG⋅rG
=2k 2
G
. Note the probability of
ψ K (r ) =
e , ε (k ) =
2m
V
G G 2
finding the electron somewhere in the volume is 1 = ∫ dr ψ (r ) .
Dai/PHYS 342/555 Spring 2006
Chapter 6-2
G
Note that ψ K (r ) is an eigenstate of the momentum operator,
G ikG⋅rG
G = ∂ = = ∂ ikG⋅rG
p=
G = ∇,
G e = =ke
i ∂r i
i ∂r
G
G
G
an electron in the level ψ K (r ) has a momentum p = =k and a
G
G G
velocity v = p / m = =k / m, where λ = 2π / k .
Periodic boundary condition requires
2π n y
2π nx
2π nz
ik y L
ik x L
ik z L
, ky =
, kz =
e = e = e = 1 or k x =
L
L
L
Thus in a 3-D k -space, the allowed wavevectors are those along
2π
the three axes given integer mutiples of
.
L
Dai/PHYS 342/555 Spring 2006
Chapter 6-3
To calculate the allowed states in a region of k -space volume Ω,
Ω
ΩV
=
or the number of allowed k -values per unit
3
3
(2π / L)
(2π )
volume of k -space (known as the k -space density of levels) is
V
. Because the electrons are noninteracting we can built up
3
(2π )
the N -electron ground state by placing electrons into the allowed
one-electron levels. Pauli exclusion principle allows each wavevector
to have 2 electronic levels with spins up and down.
Dai/PHYS 342/555 Spring 2006
Chapter 6-4
Since the energy of a one-electron level is directly proportional
to k 2 , when N is enormous the occupied region will be indistinguishable
from a sphere. The radius of this sphere is called k F (F for Fermi),
G
3
and its volume Ω is 4π k F / 3. The # of allowed k within the sphere is:
⎛ 4π k F3
⎜
⎝ 3
⎞ ⎛ V ⎞ k F3
⎟ ⎜ 3 ⎟ = 2 V . Since each allowed k -value leads to two
⎠ ⎝ 8π ⎠ 6π
k F3
k F3
one-electron levels, we must have N = 2 2 V = 2 V .
6π
3π
If electron density is n = N / V , then we have n = k F3 / 3π 2 .
Dai/PHYS 342/555 Spring 2006
Chapter 6-5
The sphere of radius k F containing the occupied one eletron levels
is called the Fermi sphere.
The Surface of the Fermi sphere, which separate the occupied form
the unoccupied levels is called the Fermi surface.
1/3
⎛ 3π N ⎞
The momentum pF = =k F == ⎜
⎟ of the occupied
⎝ V ⎠
one-electron levels of highest energy is the Fermi momentum.
2
ε F = = 2 k F2 / 2m =
vF = p F / m = (
= ⎛ 3π N ⎞
⎜
⎟
2m ⎝ V ⎠
2
2
is the Fermi energy;
1/3
= ⎛ 3π N ⎞
)⎜
⎟
m ⎝ V ⎠
2
2/3
is the Fermi velocity.
Dai/PHYS 342/555 Spring 2006
Chapter 6-6
Dai/PHYS 342/555 Spring 2006
Chapter 6-7
To find an expression for the
number of allowed electron states
per unit energy, D(ε ), we use
V ⎛ 2mε ⎞
N= 2⎜ 2 ⎟
3π ⎝ = ⎠
3/ 2
, and
3/ 2
dN
V ⎛ 2m ⎞
= 2 ⎜ 2 ⎟ ε 1/ 2 .
d ε 2π ⎝ = ⎠
3
ln N = ln ε + const ;
2
dN 3 d ε
dN 3 N
, where D(ε ) ≡
.
=
=
N 2 ε
d ε 2ε
D(ε ) ≡
Dai/PHYS 342/555 Spring 2006
Chapter 6-8
Effect of temperature on the Fermi-Dirac distribution
1
. μ is chemical potential,
f (ε ) =
exp[(ε − μ ) / k BT ] + 1
f (ε ) = 1/ 2 when ε = μ .
Dai/PHYS 342/555 Spring 2006
Chapter 6-9
Heat capacity of the electron gas
In a gas of free and independent electrons, the one electron levels
G
G
are specified by wavevector k and spin quantum # s with
G
ε ( k ) = = 2 k 2 / 2m
G
G
As T → 0, we have lim f = 1, for ε (k ) < μ and zero for ε (k ) > μ .
T →0
U
⎛ ∂u ⎞
,
.
u
=
cV = ⎜
⎟
V
⎝ ∂T ⎠V
G
G
ε (k ) f (ε (k )). As V → ∞, we have
U = 2∑
G
k
G
G
G
G
G
1
dk
ε (k ) f (ε (k )) = 2∫ 3 ε (k ) f (ε (k ))
u = lim 2∑
G
V →∞ V
8π
k
G
G
dk
If the electron density n = N / V , n = ∫ 3 f (ε (k )).
4π
Dai/PHYS 342/555 Spring 2006
Chapter 6-10
Dai/PHYS 342/555 Spring 2006
Chapter 6-11
At low temperature k BT ε F , the increase in energy when heated
∞
∞
0
0
to a temperature T is ΔU = ∫ d ε D(ε )ε f (ε ) − ∫ d ε D (ε )ε .
(∫
εF
0
+∫
∞
εF
) dε D(ε )ε
εF
F
f (ε ) = ∫ d ε D(ε )ε F .
0
∞
εF
εF
0
ΔU = ∫ d ε D(ε )(ε − ε F ) f (ε ) + ∫ d ε D (ε )(ε F − ε )[1 − f (ε )].
∞
∞
dU
df
df
= ∫ d ε D(ε )(ε − ε F )
≈ D(ε F ) ∫ d ε (ε − ε F )
Cel =
0
0
dT
dT
dT
exp[(ε − ε F ) / τ ]
df ε − ε F
=
⋅
2
τ
{exp[(ε − ε F ) / τ ] + 1}2
dT
Define x ≡ (ε − ε F ) / τ , we have
2 x
1 2
x
e
2
≈
π
ε
(
)
Cel = k B2TD(ε F ) ∫
dx x
D
k
F
BT
2
−ε F /τ
(e + 1)
3
∞
Dai/PHYS 342/555 Spring 2006
Chapter 6-12
Experimental heat capacity of metals
At sufficient low temperatures, CV = γ T + AT 3 . Where γ is the
Sommerfeld parameter. The ratio of the observed to the free
electron values of the electronic heat capacity is related to
thermal effective mass as:
mth γ (observed)
≡
γ (free)
m
Dai/PHYS 342/555 Spring 2006
Chapter 6-13
Electrical conductivity and Ohm’s law
Considering Newton's second law, we have
G
G
G
G 1G G
dv
dk
==
= − e( E + v × B )
F =m
dt
dt
c
G
G
The displacement of the Fermi sphere, δ k = −eEt / =.
G
G
If collision time is τ , the incremental velocity is v = −eEτ / m.
G
In a constant electric field E and n electrons per volume, the
G
G
G
G
2
electric current density is j = nqv = ne Eτ / m = σ E.
The electrical conductivity σ = ne 2τ / m.
Dai/PHYS 342/555 Spring 2006
Chapter 6-14
Dai/PHYS 342/555 Spring 2006
Chapter 6-15
Experimental electrical resistivity of metals and
Umklapp scattering
Dai/PHYS 342/555 Spring 2006
Chapter 6-16
Motion in magnetic fields
The Lorentz force on an electron is
G
G
G
G 1G G
dv
dk
==
= − e( E + v × B )
F =m
dt
dt
c
G
G
For a static magnetic field B lie along the z axis, we have
d 1
B
m ( + ) v x = − e( E x + v y )
dt τ
c
d 1
B
m ( + ) v y = − e( E y − v x )
dt τ
c
d 1
m( + )vz = −eEz
dt τ
In the steady state in a static electric field, the drift velocity is
eτ
eτ
eτ
vx = − Ex − ωcτ v y ; v y = − E y + ωcτ vx ; vz = − Ez
m
m
m
ωc ≡ eB / mc is the cyclotron frequency.
Dai/PHYS 342/555 Spring 2006
Chapter 6-17
Hall effect
E y = −ω cτ Ex = −
RH =
Dai/PHYS 342/555 Spring 2006
Ey
jx B
eBτ
Ex
mc
is called Hall coefficient
Chapter 6-18
1
and is a fermion. The density of liquid He3 is
2
0.081 gcm -3 near absolute zero. Calculate the Fermi energy
ε F and the Fermi temperature TF .
The atom He3 has spin-
Dai/PHYS 342/555 Spring 2006
Chapter 6-19
Thermal conductivity of metals
Wiedemann-Franz law
Thermal conductivity for a Fermi gas
π 2 nk B2Tτ
1
π 2 nk B2T
⋅
⋅ vF ⋅ l =
K el = Cvl =
2
3
3 mvF
3m
The Wiedemann-Franz law states that for metals at not
too low temperatures the ratio of the thermal conductivity
to the electrical conductivity is directly proportional to the
temperature, independent of the particular metal.
K el
σ
=
π nk Tτ / 3m π ⎛ k B ⎞
=
⎜ ⎟ T ≡ LT .
ne τ / m
3 ⎝ e ⎠
2
2
B
2
2
2
Lorenz number L = 2.45 × 10−8 watt-ohm/deg 2
Dai/PHYS 342/555 Spring 2006
Chapter 6-20
Dai/PHYS 342/555 Spring 2006
Chapter 6-21
Dai/PHYS 342/555 Spring 2006
Chapter 6-22