6. Free Electron Fermi Gas

• Energy Levels in One Dimension

• Effect of Temperature on the Fermi-Dirac Distribution

• Free Electron Gas in Three Dimensions

• Heat Capacity of the Electron Gas

• Electrical Conductivity and Ohm’s Law

• Motion in Magnetic Fields

• Thermal Conductivity of Metals

• Nanostructures

Introduction

Free electron model:Works best for alkali metals (Group I: Li, Na, K, Cs, Rb)Na: ionic radius ~ .98A, n.n. dist ~ 1.83A.

Successes of classical model:Ohm’s law.σ / κ

Failures of classical model:Heat capacity.Magnetic susceptibility.Mean free path.

Quantum model ~ Drude model

Energy Levels in One Dimension

2 2

22n n n

dH

m dx

Orbital: solution of a 1-e Schrodinger equation

Boundary conditions: 0 0n n L

sinn

nA x

L

2n L

n

Particle in a box

2sin

n

A x

1,2,n

22

2n

n

m L

Pauli-exclusion principle: No two electrons can occupy the same quantum state.

Quantum numbers for free electrons: (n, ms ) ,sm

Degeneracy: number of orbitals having the same energy.

Fermi energy εF = energy of topmost filled orbital when system is in ground state.

N free electrons:22

2F

F

n

m L

2F

Nn

Effect of Temperature on the Fermi-Dirac Distribution

Fermi-Dirac distribution : 1

1f

e

1

Bk T

Chemical potential μ = μ(T) is determined by N d g f g = density of states

At T = 0: 1for

0f

→ 0 F

1

2f For all

T :

For ε >> μ : f e

(Boltzmann distribution)

3D e-gas

Free Electron Gas in Three Dimensions

2 2 2 2

2 2 22

d d dH

m dx dy dz

r r

Particle in a box (fixed) boundary conditions:

0, , , , ,0, , , , ,0 , , 0y z L y z x z x L z x y x y L n n n n n n

sin sin sinyx znn n

A x y zL L L

n

Periodic boundary conditions:

Standing waves

, , , , , , , ,x y z x L y z x y L z x y z L k k k k

→

→iA e k r

k2 i

i

nk

L

0, 1, 2,in

2 2

2

k

m k

Traveling waves

1, 2,in

i k kp

kk → ψk is a momentum eigenstate with eigenvalue k.

p km

k

v

N free electrons: 33

42

8 3 F

VN k

1/323F

Nk

V

2 2

2F

F

k

m

2/32 23

2

N

m V

FF

kv

m

1/323 N

m V

Density of states:

32

8

dSVD

k

k

k k

2

3 2

4

4 /

V k

k m

2 2

V mk

3/2

2 2

2

2

V m

323 F

VN k

3/2

2 2

2

3FmV

→

3

2FF

ND

3

2 F F

ND

Heat Capacity of the Electron Gas

(Classical) partition theorem: kinetic energy per particle = (3/2) kBT.

N free electrons:3

2e BC N k ( 2 orders of magnitude too large at room temp)

Pauli exclusion principle → ~e BF

TC N k

T TF ~ 104 K for metal

U d D f

1

1f

e

1

Bk T 3

2 F F

ND

free electronsUsing the Sommerfeld expansion formula

2 1

22 1

2 11

2 2 2n

nnB n

n

d Hd H f d H n k T

d

2 42 B

dDU d D k T D O T

d

2 42 B

dDN d D k T O T

d

2 42 B

dDU d D k T D O T

d

2 42F

F

F F F B F F

dDd D D k T D O T

d

2 42 B

dDN d D k T O T

d

2 42F

F

F F B

dDd D D k T O T

d

→ 22 0

F

F F B

dDD k T

d

22

F

F B

dDk T

D d

2 42F

B Fd D k T D O T

F

N d D

2

2

3V F B

N

UC D k T

T

2

26

2

2B

V BF

k TC N k

3-D e-gas

1

2

d D

D d

for 3-D e-gas

22

F

F B

dDk T

D d

Bk T

1

2

d D

D d

for 3-D e-gas

1

2

d D

D d

for 1-D e-gas

Experimental Heat Capacity of Metals

For T << and T << TF : 3C T A T el + ph

2CA T

T

thobsm

m e gas

2

2

3 F BC D k T 3

2FF

ND

2 2

3 2

2 F

N m

k

1/323F

Nk

V

Deviation from e-gas value is described by mth :

thobsm

m e gas

Possible causes: e-ph interaction e-e interaction

Heavy fermion: mth ~ 1000 m

UBe3 , CeAl3, CeCu2Si2.

Electrical Conductivity and Ohm’s Law

d

dt

pFLorentz force on free electron:

1e

c E v B d

dt

k

No collision: 0e t

t E

k k t k

Collision time :

nqj v n e

m

k 2ne

m

E

Ohm’s law

2 1ne

m

Heisenberg picture: ,d

i Hdt

p

p , q p E r q i E d

qdt

p

EFree particle in constant E field

Experimental Electrical Resistivity of Metals

Dominant mechanismshigh T: e-ph collision.low T: e-impurity collision.

phonon impurity1 1 1

ph imp

ph imp

Matthiessen’s rule:

0 imp Sample dependent

ph impT T Sample independent

Residual resistivity:

Resistivity ratio: room

imp

T

imp ~ 1 ohm-cm per atomic percent of impurity

K

imp indep of T

(collision freq

additive)

Consider Cu with resistivity ratio of 1000:

32951.7 10 ohm-cm

resistivity ratioimp

K

3 21.7 10 10ic Impurity concentration: = 17 ppm

Very pure Cu sample: 54 10 300K K

94 2 10K s 4 4 0.3Fl K v K cm 8 11.57 10Fv cm s

For T > : T See App.J

From Table 3, we have 295 1.7 ohm-cmL K

imp ~ 1 ohm-cm per atomic percent of impurity

Umklapp Scattering

Normal: k k q

Umklapp:

k k q G

Large scattering angle ( ~ ) possible

Number of phonon available for U-process exp(U /T )

For Fermi sphere completely inside BZ, U-processes are possible only for q > q0

q0 = 0.267 kF for 1e /atom Fermi sphere inside a bcc BZ.

For K, U = 23K, = 91K U-process negligible for T < 2K

Motion in Magnetic Fields

1d

dt

k FEquation of motion with relaxation time : 1q

c E v B

1 1dm q

dt c

v E v B

/ / / /

1dm q

dt

v E

1 1 2

1 1dm q B

dt c

v E v

be a right-handed orthogonal basis 1 2 / /ˆ, , e e e BLet

2 2 1

1 1dm q B

dt c

v E v

Steady state:

1 1 2c

q q

m q

v E v

2 2 1c

q q

m q

v E v

/ / / /

q

m

v E

c

q B

m c = cyclotron frequency

q = –e for electrons

Hall Effect

0yj →

0y c x

q qE v

m q

x x

qv E

m

0z z

qv E

m

y c x

qE E

q x

qBE

mc

Hall coefficient:

yH

x

ER

j B 2

x

x

qBE

mcnq

E Bm

1

nqc

electrons

Thermal Conductivity of Metals

From Chap 5:1

3K C v l

Fermi gas:21

3 2B

el B F FF

k TK N k v v

2

3B

B

k TN k

m

In pure metal, Kel >> Kph for all T.

Wiedemann-Franz Law:2

23

BB

k TN kK m

nqm

22

3Bk

Tq

T

Lorenz number:K

LT

22

3Bk

q

8 22.45 10 watt-ohm/deg

for free electrons

8 22.45 10 watt-ohm/degL for free electrons