The Quantized Free Electron Theory

+ + + + + + + +

Ene

rgy

E

Spatial coordinate x

Nucleus withlocalized coreelectrons

Jellium model:

electrons shield potential to a large extent

Electron “sees” effective smeared potential

Electron in a box

In one dimension:

In three dimensions:

)r(E)r()r(V)r(m

2

2

where

otherwise

Lz,y,xfor.constV

)z,y,x(V

00

222222

22 zyx kkkmm

kE

where zzyyxx nL

k,nL

k,nL

k

2222

2

8zyx nnn

mL

hE

and ,...,,n,n,n zyx 321

zksinyksinxksinL

)r( zyx

/ 232

Fixed boundary conditions:

+ + + + + + + + x

0 L

)Lx()x( 00

+

+

+ +

+ +

+ +

Periodic boundary conditions:

)z,y,x()Lz,Ly,Lx(

rki/

eL

)r(23

1

zzyyxx nL

k,nL

k,nL

k

222

and ,...,,,n,n,n zyx 3210

kx

m

kx

2

22

Ldkx

2

“free electron parabola”

density of states

Remember the concept of

dE

# of statesin ]dEE,E[

41111

)k(E 1 )k(E 2 E

))k(EE( 1

1. approach use the technique already applied for phonon density of states

k

))k(EE()E(D~

E

EE

E

dE)E(D~1

1

k

EE

E

dE))k(EE(1

1

where )E(D~

V:)E(D1

Density of states per unit volume

3

3k

Vd k

2

1/ Volume occupied by a state in k-space

k

))k(EE()E(D~

kx

ky

kz

L

2

L

2

L

2

Volume( )

VL

3322

Free electron gas: m

k

m

kE

22

2222

Independent from and

dkkkd 23 4

Independent from and

mEk 21

dE

E

mdk

2

1

dEE

mmE))k(EE()E(D

~V

)E(D2

124

2

1123

Em

)E(D//

3

2321

2

2

2

1

2

Each k-state can be occupied with 2 electrons of spin up/down

Em

)E(D/ 23

22

2

2

1

k2 dk

2. approach calculate the volume in k-space enclosed by the spheres

.constm

k)k(E

2

22and .constdE)k(E

kx

ky

L

2

32

2

4

L/

dkkdk)k(D

~

# of states between spheres with k and k+dk :

dEE

mdk

2

1

22 2

mE

k

with )E(D~

V)E(D

12

2 spin states

Em

)E(D/ 23

22

2

2

1

E

D(E)

E’ E’+dE

D(E)dE =# of states in dE / Volume

Statistics of the electrons (fermions)

Fermions are indistinguishable particles which obey the Pauli exclusion principle

T=0

En=1

En=2

En=3

En=4

En=5

Let us distribute4 electrons spin

4 electrons spin

En=6 Occupationnumber 0for state

4 nEE

Occupation number 1 for states

Fn E:EE 4of a given spin

Ef(

E,T

=0)

Probability that a qm state is occupied

EF

1

x

Fermi Dirac distribution function at T>0

With accuracy sufficient for many estimations: f(E,T) linearized at EF

1

1

TBk

E

e

)T,E(f here chemical potential

FE

Fermi energy

More detailed approach to Fermi statistics

The grand canonical ensemble

Heat Reservoir RT=const.

Particle reservoir

System

n nn

U E

n nn

N N

1 nn

Average energy

Average particle #

Normalized probabilities

Now we consider independent particles i ii

E n

Total energy of N fermion system

occupation # ni=0,1 of single particle state i with energy i

i ii i

N n n where1 2

1 2( , ,...)

( , ,...)j jn n

n n n n

i ii

U E n

average occupation of state j is given by

( )

{ }

( )

{ }

i i

i

i i

i

n

jn

j n

n

n e

ne

Chemical potentialP,TV,TV,S N

G

N

F

N

U

For details see & additional info see

1 2( , ,...)n n

where the summation ...

jn means

1 2( , ,...)

...n n

( )

{ }

( )

{ }

i i

i

i i

i

n

jn

j n

n

n e

ne

1 1 2

1 2 3

1 1 2

1 2 3

, ,...

, ,...

i ii

i ii

nn

jn n n

nn

n n n

e n e

e e

1

0...

j j

j

j j

j

n

jn

n

n

n e

e

Repeat this step

( )

1( , )

1ij jn f Te

The Fermi gas at T=0

E

f(E

,T=

0)

EF

1

E

D(E)

EF0

0

dE)T,E(f)E(Dn

Electron density

#of states in [E,E+dE]/volume

Fermi energydepends on T

Probability that state is occupied

0

0

FE

dE)E(D dEEm FE/

0

0

23

22

2

2

1

3222

0 32

/

F nm

E T=0

00 5

3FEnU

0

00

FE

dE)E(DEUEnergy of the electron gas @ T=0: dEEEm FE/

0

0

23

22

2

2

1

25023

22 5

22

2

1 /

F

/

Em

2300

23

22 5

121 /

FF

/

EEm

3222

0 32

/

F nm

E

there is an average energy of 0

5

3FE per electron without thermal stimulation

with electron density 322 1

10cm

n we obtain KT@eVTkeVE BF 30040

11240

Click for a table of Fermi energies,

Fermi temperatures and Fermi velocities

Specific Heat of a Degenerate Electron Gas

here: strong deviation from classical value

only a few electrons in the vicinity of EF can be scattered by thermal energy into free states

Specific heat much smaller than expected from classical consideration

D(E)

Den

sity

of

occu

pied

sta

tes

EEF

energy of electron state

0

dE)T,E(f)E(DEU

#states in [E,E+dE]

probability of occupation,average occupation #

2kBT

Before we calculate U let us estimate:

These Tk)E(D

BF 222

1 # of electrons

increase energy from TkE BF to TkE BF TknE

TkTk)E(DU B

F

BBF 2

2Tk)E(DU BF Tk)E(DC BFel2 π2

3

subsequent more precise calculation

Calculation of Cel from

0

dE)T,E(f)E(DEU

0

dET

f)E(DE

T

UC

Vel

22

1

TBkFEE

TBkFEE

B

F

e

e

Tk

EE

T

f

0

dET

f)E(DEE F

0

0 dET

f)E(DE

T

nE FFTrick:

Significant contributions only in the vicinity of EF

)E(DTkC FBel2

2

3

3

2

0

dET

f)E(DEEC Fel

with Tk

EE:x

B

F and dxTkdE B

E

D(E

)

EF

)E(D)E(D F

0

dET

fEE)E(DC FFel

21

x

x

e

e

T

x

T

f

TBk/FEx

x

FBel dxe

ex)E(DTkC

2

22

1

decreases rapidly to zero for x

dx

e

ex)E(DTkC

x

x

FBel 2

22

1

)E(DTkC FBel2

2

3

F

/

F Em

)E(D23

22

2

2

1

with 322

20 3

2

/

F nm

E

and

F

BBel E

TkknC

2

2 in comparison with

Bclassical

el knC2

3

1 for relevant temperatures

W.H. Lien and N.E. Phillips, Phys. Rev. 133, A1370 (1964)

Heat capacity of a metal:

3ATTC

electronic contribution lattice contribution@ T<<ӨD

Selected phenomena which don’t require detailed knowledge of the band structure

Temperature dependence of the electrical resistance

T

residual

5T

T

Scattering of electrons: deviations from a perfect periodic potential

Impurities: temperature independent imperfection scattering phonon scattering

)T()T(phonresidual Matthiessen’s rule:

D

TBkD

d

eM

03

1

1

2

13

Simple approach to understand Tphon for T>>ӨD

Remember Drude expression:m

ne

2

11scattering rate

Fv

V

N1

#of scattering centers/volume

scattering cross section

scattering cross section

2u

tcosuu 0

D

d)(DuN

u0

22

3

1

02

2

1 uuFermi velocity of electrons: m/Ev FF 2

23

9

D

N

compare lecture notes:Thermal Properties of Crystal Lattices

D

D

dN

uN

0

23

20 9

2

1

3

1

202

2

1 uM)(E

1

1

2

1

TBke

Tu

211

D

TBkD

d

eM

u0

32

1

1

2

13

with

Tkx

B

and dx

Tkd B

T/D

xB

D

dxe

xTk

Mu

0

2

32

1

1

2

13

Let us consider the high temperature limit: T>>ӨD

T/D

xB

D

dxe

xTk

Mu

0

2

32

1

1

2

13

11

1

2

10

xxex

2

23

DB

T

kM

Note: T5 –low temperature dependence not described by this simple approach

Lindemann melting temperature TM: 2

22 3

D

M

BMT

T

kMu

22sm

MTrxu where

emperical value 10.average atomicspacing

Thermionic Emission

Finite barrier height of the potential

E

x

EF

Fvac EE: work function

Evac

Fx E

m

k

2

22

xx vnqj

Current density for homogeneous velocity

generalized k

xx )k(vV

qj

Current density for k-dependent velocity

3

3k

Vd k

2

Again:

kd)k(v

qj xx

332

Fx E

m

k

2

22Occupied and

minxk

xxzy ))T,k(E(fkdkdkdkm

q 32

2

Spin degeneracy

Since TkB

Fermi distribution approximated by Maxwell Boltzmann distribution

1

1

TBkFEE

e

)T,E(f approximatedTBkEFE

e)T,E(f

minxk

TBkFE

TBkmxk

xxTBkmzk

zTBkmyk

yx eekdkedkedkm

qj 2

22

2

22

2

22

32

2

TBkm

)zkykxk(

TBkFE

ee 2

2222

minxk

TBkFE

TBkmxk

xx eekdk 2

22

Let us investigate the integral

22

2

22

2

2

1

/)FE(m

TBkFE

TBkmxk

x ee)k(d TBkB eTmk

2

Fx E

m

k

2

22

TBkmzk

zedk 2

22

Remember integrals of the type:

dxeTkm xB 22

dxTkm

dk Bz

2

Tkm

kx

B

z

2

Tkm B

2

TBkBB

x eTkmTmk

m

qj

223

2

2

2

TBkBx e)Tk(

h

emj

23

4

Richardson-Dushman

Richardson Constant

TBkBx eT)r(k

h

emj

223

14

Typical value of Tungsten: eV.54

Nobel prize  in 1928

"for his work on the thermionic phenomenon and

especially for the discovery of the law named after him".

Owen Willans Richardson

Universal constant: A=1.2 X 106 A/m2K

Reflection at the potential step

A (1-r)=0.72 X 106 A/m2 K

Vacuum tube

Field-Aided Emission

EF

Evac

E

x

Image potential

Electric field in x-direction

xEx)x(V

@ very high electric fields of m/V810 tunneling through thin barrier cold emission