Free electron Fermi gas (Sommerfeld, 1928 - phy.ntnu.edu.twchangmc/Teach/SS/SS_note/chap06.pdf ·...

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.

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

…

L. Hoddeson et al, Out of the crystal maze, p.104

• 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

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

= = =

⇒ = ∈

= ∈

= ∈


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.


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


• 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


• 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=

∆∫ ∫


• Free electron DOS (per unit volume) in 1D, 2D, and 3D

• Multiple bound states in 2D

z

⊥


Examples of low-dimensional electron system

Poly-

acetylene

Graphene

Fe chains on

Pb surface

Quantum well







• Hall effect



• 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 ≠


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)


2

2e B

F

TC Nk

T

π=

A better

result,(see Kittel p.142)

eC

Tγ =

• In general C = Ce + Cp

= γT + aT 3

Ce is important only at very low T.







• Hall effect



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


Drift velocity

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:


dmv k= ∆

ℏDrift velocity

• 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=


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


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)


optionalA surprising discovery (von Klitzing, 1980)

RH deviates from (h/e2)/n by less than 3 ppm on the very first report.


25 years of QHE, von Klitzing

classical

quantum

Quantum Hall effect

• h/e2=25812.80745 Ω exact (1990)

• This result is independent of

the shape/size of sample.

optional

1985


An accurate and stable resistance standard (1990)

optional


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).

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


Thermal conductivity

• Wiedemann-Franz law (1853)

for metal, a good electrical conductor

is also a good thermal conductor.

273K


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

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


wrong by roughly a factor of 2

21

3

T

V

J K T

K c v τ

= − ∇

=

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 .


Free electron Fermi gas (Sommerfeld, 1928 - phy.ntnu.edu.twchangmc/Teach/SS/SS_note/chap06.pdf ·...

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