free electron fermi gas (sommerfeld, 1928 - phy.ntnu.edu.twchangmc/teach/ss/ss_note/chap06.pdf ·...
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Free electron Fermi gas (Sommerfeld, 1928)
M.C. Chang
Dept of Phys
• counting of states
• Fermi energy, Fermi surface
• thermal property: heat capacity
• transport property
• electrical conductivity
• Hall effect
• thermal conductivity
• In the free electron model, there is no lattice, and no electron-electron
interaction, but it gives nice result on electron heat capacity, electric/
thermal conductivity… etc.
• Free electron model is most accurate for alkali metals.
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Early history of solid state physics (Ref: Chap 4 of 半導體的故事, by 李雅明)
Electron
1897 – Thomson discovered electron
1899 – Drude theory of classical electron gas,
explained Wiedemann-Franz law
1924 – Bose-Einstein statistics
1926 – Fermi-Dirac statistics
(1925,6 – Heisenberg/Schrodinger theory)
1927 – Electron diffraction by crystal (Davisson
and Germer)
1927 – Sommerfeld’s quantum theory of metal
Atomic lattice
1878 – Crookes, Cathode ray
1895 – Rontgen, x-ray
1912 – von Laue, x-ray
diffraction by crystal
1908 – Einstein model of
specific heat
1910 – Debye model, T3 law
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…
L. Hoddeson et al, Out of the crystal maze, p.104
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• Plane wave solution
“Box” BC
k = π/L, 2π/L, 3π/L…
2 2( ) , ( ) / 2ikx ikxx Ae Be k k mψ ε−= + = ℏ
Periodic BC (PBC)
k = ±2π/L, ±4π/L, ±6π/L…
Quantization of k in a 1-dim box
Advantage:
allows travelling
waves
Counting of states Heat capacity Electron transport Thermal transport
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Free electron in a 3-dim box
22
22
2
22
2
22
2
2 2 2
2
0 ( ) ( )2
separable, assume
( ) ( ) ( ) ( )
then
( ) ( ) 0,
( ) ( ) 0,
( ) ( ) 0,
1( )
2
x y z
x x x
ik
y y y
z z z
y z
r
x
r rm
r x y z
dx k x
dx
dy k y
dy
dz k z
d
r eV
z
mk k k
ψ εψ
ψ φ φ φ
φ φ
φ φ
φ
ε
φ
φ
⋅
− ∇ + =
=
+ =
+ =
+ =
+ + =
=⇒
ℏ
ℏ
x
y
z
Lx
Ly
Lz
:
( ) (0); ( ) (0);
Periodic Boundary C
( ) (0)
2,
2,
2
ondition
,
x x x y y y z z z
x x x
x
y y y
y
z z z
z
L L L
k n n ZL
k n n ZL
k n n ZL
φ φ φ φ φ φ
π
π
π
= = =
⇒ = ∈
= ∈
= ∈
Counting of states Heat capacity Electron transport Thermal transport
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Quantization of k in a 3-dim box
box BC periodic BC
• Different BCs give the same Fermi wave vector and Fermi energy
box BC periodic BC
• Each point can have 2 electrons (because of spin). After filling in
N electrons, the result is a spherical sea of electrons called the
Fermi sphere. Its radius is called the Fermi wave vector, and the
energy of the outermost electron is called the Fermi energy.
Counting of states Heat capacity Electron transport Thermal transport
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Connection between electron density and Fermi energy
( )
( )
3
1/32
33
2
22/3
2
2
4
323
32
3
FF
F
F
L
kk
N V
m
k n
n
π
π
π
π
ε π
= =
⇒
⇒ =
=
ℏ
• For K, the electron density n=1.4×1028 m-3, therefore
193.40 10 2.12 eVF Jε −= × =10.746Fk A−=
• εF is of the order of the atomic energy levels.
• kF is of the order of a-1.
Fig from Baggioli’s paper
Counting of states Heat capacity Electron transport Thermal transport
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• The Fermi temperature is of the order of 104 K
2
2
also,
F B F F
F F
mk T v
k mv
ε = =
=ℏ
Fermi temperature and Fermi velocity
Counting of states Heat capacity Electron transport Thermal transport
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• For a 3D Fermi sphere,
3 2 2
3
3/22
3 2 2
( ) 2 ( )(2 / ) 2
4 2 2 ( )
(2 / ) 2
shell
d k kD d k
L m
k dk V mD
L
ε ε επ
πε
π πε
= =
= ⇒ =
∫ ℏ
ℏ
Density of states D(ε) (DOS, 態密度)
3 3
3
3
2( ,
2) shell
d kD d k
k L
πε ε
= ∆ =
∆
∫
• D(ε)dε is the number of states within the
energy surfaces of ε and ε+dε
important
3
32 ( ) ( ) ( )
k
d kh d D h
kε ε ε ε=
∆∫ ∫
Counting of states Heat capacity Electron transport Thermal transport
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• Free electron DOS (per unit volume) in 1D, 2D, and 3D
• Multiple bound states in 2D
z
⊥
Counting of states Heat capacity Electron transport Thermal transport
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Examples of low-dimensional electron system
Poly-
acetylene
Graphene
Fe chains on
Pb surface
Quantum well
Counting of states Heat capacity Electron transport Thermal transport
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• counting of states
• Fermi energy, Fermi surface
• thermal property: heat capacity
• transport property
• electrical conductivity
• Hall effect
• thermal conductivity
Counting of states Heat capacity Electron transport Thermal transport
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• Thermal distribution of electrons (fermions)
• Combine DOS D(E) and thermal dist f(E,T)
33
0
3Recall that 2 ( ) 2 ( ) ( ) ( )k
k kk
d kf E f E dED E f E
k
∆ →→ =∆
∑ ∫ ∫
D(ε) D(ε)
( )
( )/
: chemical pote
1where ( )
1
( ) 1/ 2
( 0 )
( 0 ) in 3-di
tial
m
n
E kT
F
F
f Ee
f
T K E
T K E
µ
µ
µ
µ
µ
−=
+
=
= =
> <
important
tourists
3
32
(( , )
2)
/(
)dED EN
LE T
df
k
π= →∫ ∫
3
3( ) 2 ( ) ( ) ( , )
(2 / )
d kU T E k dED E f E T
LE
π= →∫ ∫
Hotel rooms
money
0T ≠
0T ≠
Counting of states Heat capacity Electron transport Thermal transport
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Electronic heat capacity, a heuristic argument
• Energy absorbed by the electrons,
U(T)−U(0) ~ ∆N (kBT )
=N (kBT)2/EF
• Heat capacity Ce = dU /dT
(per mole) ~ 2R kBT /EF
= 2R T /TF
A factor of T/TF smaller than classical result 3R/2.
• T /TF ~ 0.01: Electron heat capacity is often negligible compared to phonon’s.
• Only the electrons near Fermi surface are excited by thermal energy kT.
Number of excited electrons is roughly
∆N = N (kBT / EF)
Counting of states Heat capacity Electron transport Thermal transport
2
2e B
F
TC Nk
T
π=
A better
result,(see Kittel p.142)
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eC
Tγ =
• In general C = Ce + Cp
= γT + aT 3
Ce is important only at very low T.
Counting of states Heat capacity Electron transport Thermal transport
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• counting of states
• Fermi energy, Fermi surface
• thermal property: heat capacity
• transport property
• electrical conductivity
• Hall effect
• thermal conductivity
Counting of states Heat capacity Electron transport Thermal transport
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Electrical transport (q=−e<0)
Classical view
• If these two types of scatterings are not related, then
scattering rates can be added up:
( )Matthiessenns rule1 1 1
+ ( )i ph Tτ τ τ
=
at steady state
e e
e
d v vm eE m
dt
ev E
m
τ
τ
= − −
⇒ = −
• Current density (n is electron density)
( )2
2
( ) Ohmn's law
nej e n v
e
m
E Em
n
τσ
τσ =
= − = =
Electric
conductivity
• Electric resistance comes from electron scattering with defects and phonons.
Relaxation
(scattering)
time
Counting of states Heat capacity Electron transport Thermal transport
Drift velocity
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Semi-classical view
• Center of Fermi sphere is shifted by ∆k
• One can show that when ∆k<<kF, V沒重疊
/V重疊
~3/2(∆k/kF).
• Therefore, the number of electrons away from equilibrium is about
(∆k/kF)Ne, or (vd/vF)Ne
• Semiclassical view vs classical view:
( ) dF d
F
vj e n v env
v
= − = −
2ne
j Em
τ∴ =
/
( / )
dv k m
e m Eτ
= ∆
= −
ℏ
( ) (0)
dkeE
dt
ek k k Eτ τ
= −
⇒ ∆ = − = −
ℏ
ℏ
• vF vs vd (differ by 109 !)
• (vd /vF) Ne vs Ne
The results are the same,
but the microscopic pictures are very different:
Counting of states Heat capacity Electron transport Thermal transport
dmv k= ∆
ℏDrift velocity
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• Cu at room temp . Electron density
→ τ = m/ρne2 = 2.5×10-14 s
81.7 10 mρ −= × Ω328105.8 −×= mn
Calculate the scattering time τ from measured resistivity ρ
• Fermi velocity of copper: ∴ mean free path ℓ = vFτ = 40 nm.16106.1 −× ms
• For very pure Cu crystal at 4K, the resistivity reduces by a factor of 105,
which means that ℓ increases by the same amount (ℓ = 0.4 cm!).
This cannot be explained using classical theory.
Residual resistance
at T=0
dirty
clean
K
• For a crystal without any defect, the only resistance
comes from phonon. Therefore, at very low T, the
electron mean free path theoretically is infinite.
2 2
1 1+
( )i ph
m m
ne ne Tρ
τ τ=
Counting of states Heat capacity Electron transport Thermal transport
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xy
Hall effect (1879)
Classical view:
(in 2-dimension)
mdv
dteE ev B m
v
B Bz dv dt
e e
= − − × −
= =
τ
ɵ; / 0 at steady state
/
/
x x
y y
v Em eBe
v EeB m
τ
τ
= −
−
2
2
,x x
y y
m BE jne ne
jE jB m
ne ne
τρ
τ
= = −
ɶ
2,
0
,
Lx L x
y
y H Hx
m
neE j
j
neE
Bj
ρτ
ρ
ρ
ρ
=
= ⇒ =
=
= −
|ρH|
B
1
2 2
or
1 L H
H LL H
σ ρ
ρ ρ
ρ ρρ ρ
−=
=
−+
ɶ ɶ
j env= −
site
s.g
oo
gle
.co
m/s
ite/josh
shalle
ffectc
ache
Counting of states Heat capacity Electron transport Thermal transport
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One can determine electron density
n by measuring the Hall coefficient.
Positive Hall coefficient?
Can’t be explained by free electron theory.
Band theory (next chap) is required.
Hall coefficient
1y
H
x
ER
j B ne≡ = −
(ρH)
Counting of states Heat capacity Electron transport Thermal transport
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optionalA surprising discovery (von Klitzing, 1980)
RH deviates from (h/e2)/n by less than 3 ppm on the very first report.
Counting of states Heat capacity Electron transport Thermal transport
25 years of QHE, von Klitzing
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classical
quantum
Quantum Hall effect
• h/e2=25812.80745 Ω exact (1990)
• This result is independent of
the shape/size of sample.
optional
1985
Counting of states Heat capacity Electron transport Thermal transport
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An accurate and stable resistance standard (1990)
optional
Counting of states Heat capacity Electron transport Thermal transport
Offers one of the most accurate way to determine the Planck constant h.
Update: the values of c,h,e are now defined (therefore exact).
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Thermal conduction in metal
( )2 2
22
1
3
1
3 2
3
v
BF F
F
B
K c v
nk Tv v
nk T
m
πτ
ε
πτ
=
=
=
ℓ
• Both electron and phonon can carry thermal energy
(Electrons dominate in metal).
• Similar to electrical conduction, only the electrons near
the Fermi energy can contribute to the thermal current.
2
2
BV B
F
k Tc nk
π
ε=
Heat capacity per unit volume
Counting of states Heat capacity Electron transport Thermal transport
Thermal conductivity
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• Wiedemann-Franz law (1853)
for metal, a good electrical conductor
is also a good thermal conductor.
273K
Counting of states Heat capacity Electron transport Thermal transport
KLT
σ=
K/σ~T is discovered
by L. Lorenz (1872)
22
22
2
3
3
B
B
nk TkK m T
ne e
m
πτ
π
τσ
= =
= 2.45×10-8 W-ohm/K2
Lorenz number
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Thermal conduction in metal
• Both electron and phonon can carry thermal energy.
In metal, electrons are dominant
2
2 2
2 22
22
2
3 1 3 3 3,
2 2 2 2 2
1,
2 2 3
3
BV B B B
V
F
BB F B
kn Kc nk m v k T K k T
m T e
T kK
T
nc nk m v K k
eT
T m
τ
σ
π τ πε
π
σ
= = ⇒ = ⇒ =
=
= ⇒ = ⇒
=
Classical:
Semi-classical:
• Thermal current density
Thermal
conductivity
Counting of states Heat capacity Electron transport Thermal transport
wrong by roughly a factor of 2
21
3
T
V
J K T
K c v τ
= − ∇
=
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Free-electron model can explain
• Electron heat capacity
• Electrical conductivity
• Thermal conductivity
• Magnetic susceptibility
• Hall effect
However, it fails to explain
• Why there is insulator
• Positive Hall coefficient
• …
Next step, take lattice into account.
Note: Is electron-electron interaction important?
Not really for many materials. But when it’s important,
the physics behind is interesting .
Counting of states Heat capacity Electron transport Thermal transport