1

Lecture 3 1

Lecture 3

3. The Single-Electron Model

References:1. Marder, Chapters 6

2. Kittel, Chapter 6, pp.144-1563. Ashcroft and Mermin, Chapter 8

4. Ziman, Chapter 3, pp.77-915. Kaxiras, Chapter 36. Phillips, Chapter 1

7. Ibach, Chapter 6

… or Free Electron Fermi Gas

Free electrons subject to the Pauli principle

nnn

n dx

d

mH ψεψψ =−=

2

22

2

hIn 1D:

∑∑≠ −

+=' '

'2

ˆ2

1

2ˆ

ll ll

ll

l l

l

RR

qq

m

pH )

)

Lecture 3 2

Approximations

∑∑≠ −

+=' '

'2

ˆ2

1

2ˆ

ll ll

ll

l l

l

RR

qq

m

pH )

)

Born-Oppenheimer approximation :nuclei treated as classical potentials; many electrons combined with nuclei in closed shells to form ions; ions treated as static potential

All electronic degrees of freedom eliminated; effective potentials for ions. Phonon dynamics, cohesive energy

Single electron approximation: Coulomb interaction incorporated into lattice potential

Jellium: interacting electrons move in uniform positive potential

Ions arranged in a lattice, forming periodic potential for single electrons. Weak ionic potentials justified by pseudopotentials

Free Fermi gas: Ionic potential eliminated, surprisingly effective for alkali metals

2

Lecture 3 3

Metals

• For an isolated atom the valence electrons are in the potential well due to the nucleus and core electrons

• As the atoms approach each other to build a crystal, the overlap of the atomic potentials causes the valence electrons to be in an effective potential that is lower than in the isolated atoms

• For metals can consider a “smooth” potential well W• Drude’s Model (1900): an application of the kinetic theory of electron gas in

a solid; uses average energy and Maxwell-Bolzmann distribution of the electron velocities

• Sommerfeld theory : quantum mechanics approach to find energy and velocities; uses Pauli exclusion principle to fill up the quantized states

Lecture 3 4

Important Terms

� Free electron gas� Occupation number� Fermi energy� Fermi surface� Density of states� Fermi-Dirac Statistics and Temperature Effects� Sommerfeld expansion� Heat Capacity of the Electron Gas and Sommerfeld parameter

3

Lecture 3 5

Basic Hamiltonian

Single-electron model:

)...()...()(2 11

1

22

NN

N

ll

l rrrrrUm

Hrrrrrh Ψ=Ψ

+∇−=Ψ ∑

=

ε

N conduction electrons, interacting with external potential U but does not interacting with the other conduction electrons

Find eigenfunctions for single electrons obeying:)( ll rrψ

)()()(2

22

rrrUm lll

rrrhψεψ =

+

∇−

⇒ eigenfunctions describing many particles are products of one-particle functions

)...()...(2 11

1

22

NN

N

ll rrrr

m

rrrrh Ψ=Ψ∇−∑

=

ε

Free electron gas: (no external potential U)

Lecture 3 6

Energy Levels in 1D

The boundary conditions are imposed by the infinite potential energy barriers:

nnn

n dx

d

mH ψεψψ =−=

2

22

2

hIn 1D:

0)(;0)0( == Lnn ψψThey are satisfied by sinelike wavefunction:

LnxA nn

n =

= λ

λπψ

2

1 ;

2sin

−=

= xL

n

L

nA

dx

dx

L

n

L

nA

dx

d nn ππψππψsin ;cos

2

2

2

The energy is given by:22

2

=L

n

mn

πε h

Accommodate N electrons, recall Pauli exclusion principle (n, mS=±1/2)

The Fermi energy εεεεF is defined as the energy of the topmost filled level in the ground state of the N electron system:

2222

22

=

=L

N

mL

n

mF

F

ππε hh

4

Lecture 3 7

Free Electron Gas in 3D

Apply periodic boundary conditions (period L):

)()(2 2

2

2

2

2

22

rrzyxm kkk

rrh ψεψ =

∂∂+

∂∂+

∂∂−In 3D:

),,(),,( );,,(),,( );,,(),,( LzyxzyxzLyxzyxzyLxzyx +=+=+= ψψψψψψ

Wavefunctions satisfying Schrödinger equation and boundary conditions are:

),exp()( rkirk

rrrr ⋅=ψ

where the components of the wavevector are:kr

;...4

;2

;0LL

kx

ππ ±±=

We can confirm that the values of kx satisfy periodic boundary condition:

The eigenvalue corresponding to the wavefunction is::

( )2222

22

22 zyxkkkk

mk

m++== hrh

rε

Lecture 3 8

k – or reciprocal space

VLV

Lk

kk

33

3)2(

or ,2

is stateper volumespace

,L

2 of distancesby separated points gneighborinwith

space, reciprocalor in lattice cubic aoccupy aboveequation by described states

ππ

π

⇒=

rr

5

Lecture 3 9

Occupation number

The ground state of N electrons is build from…

…by putting two electrons into the lowest state |k|=0;

next into all states with and so on

Occupation number fk of a state indexed by

is 1 if this one-electron state is part of the ground state, and 0 otherwise

,2

Lk

π=r

kr

In the ground state of a system of N free electrons the occupied orbitals are points inside a sphere in k-space

- the energy at the surface of the sphere is the Fermi energy:2

2

2 FF km

h=ε

The total number of orbitals in the sphere of volume is 3

4 3Fkπ

Lecture 3 10

Fermi Energy

The total number of orbitals in the sphere is 323

33

32

)2(3

4F

F kVLk

Nππ

π =×=

Radius kF can be defined as (only N dependent): 3/123

=

V

NkF

π

3/2222

2 3

22

==

V

N

mk

m FF

πε hh

Fermi energy is related to the electron concentrati on N/V!!!

6

Lecture 3 11

Definition of Density of states, D

Density of electronic states or density of levels

Defined so that

3)2(

12

π=

kDr

kdFV

kdFFL

F

kkk

k

kk

rrrr

r

r

r

r

∫∫∑

∑

⇒=

3

3

)2(

2

,

ππ

kdFDVFkk

kk

rrr

r

r ∫∑ =

Several separate functions D are all referred to as density of states

⇒ distinguish by their argument

Energy density of states D(E)

Lecture 3 12

Energy Density of States

To find D(E), note that

∫∑ = dEEFEDVEFk

k)()()(

σr

r

The number of orbitals per unit energy range, D(E) – density states

3

2/32/3

23

2

)2(

33 h

me

Vk

VN FF ππ

==

Note dimensionality effect:

• for 3D: D(E) ∝ E1/2

• for 2D: D(E) ∝ constant

• for 1D: D(E) ∝ E-1/2

7

Lecture 3 13

Results for Free Electrons

The Fermi wave vector kf is related tot the density of electrons (n=N/V) by

31212/13

2/3

210812.6

)2(

2)( −−×=

= cmeV

eV

EE

mVED

hπ

Radius parameter , rs

Fermi energy , EF or Fermi level

Fermi surface , electrons with energy EF

Lecture 3 14

Experimental confirmation

Soft X-ray emission spectra :If Al metal is bombarded by electrons with enough energy to knock a 2p-

electron from the Al core then X-rays are emitted

Al 1s22s22p63s23p1

see Kittel

8

Lecture 3 15

Fermi-Dirac Statistics and Temperature Effects

Recall : in the Drude theory we used a single average energy, av. velocity, etcNeed a proper statistical approach to characterize thermal occupancy of the

allowed quantum states. Particles with a spin of ½ obeys Fermi-Diracstatistics

For an ideal electron gas in thermal equilibrium with a heat bath at a temperature T, the probability that an allowed state, with energy E, will be occupied is

1exp

1)(

+

−=

Tk

EEEf

B

F

At low T (→ 0 K)

Note: when E = EF, f(E) =1/2

f(E) Fermi function or the occupation probability

Lecture 3 16

Temperature Effects

At higher T : the kinetic energy of the electron gas increases as the temperature is increased: some energy levels are occupied which were vacant at absolute zero

when E-µ >> kBT; the exponent term is dominant

⇒ ?

1exp

1)(

+

−=

Tk

EEf

B

µ

Fermi temperature:B

FF k

ET =

9

Lecture 3 17

Elements as free electron gases

Lecture 3 18

Sommerfeld Expansion

Paradox that density of states too small solved by)( Fv TDc ε∝

Derivatives of Fermi function for various kBT

f is only nonzero over an energy range around the Fermi energy

10

Lecture 3 19

Sommerfeld Expansion

∫ ∑

∫

∞−

∞

=

∞

∞−

+=

=

µ

1

...)(

)()(

n

dEEHH

dEEfEHH

Follow Marder, p. 148, or Ashcroft, p.760-761

[ ] [ ] ...)('''360

7)('

6)( 4

42

2

+++= ∫∞−

µπµπµ

HTkHTkdEEHH BB

for quantum-mechanical thermal averages at low T

Chapter 5 20

Heat Capacity of the Electron Gas

Classical statistical mechanics :

free particle heat capacity 3/2 kB (for system of N atoms – 3/2 NkB) ⇒ experimental value 0.01 ×3/2 N kB

When sample is heated, only those electrons in orbitals within an energy range kBT of the Fermi level are excited thermally ⇒ only a fraction of the order T/TF can be excited thermally at temperature T

At low temperature (kBT<<EF) we can derive a quantitative expression for the electronic heat capacity

TkT

NTU B

F

≈

∫∫ −=∆∞ FE

dEEEDdEEfEEDU00

)()()(

11

Lecture 3 21

Specific Heat

For non-interacting electrons at low T

NVV T

E

Vc |

1

∂∂=

=V

E

TV

V

NV NT

N

T |

||

µ

µ µ

∂∂∂∂

−=∂∂

∫= 'dEVN

=∂∂

NVT|

µ

Lecture 3 22

Specialize to Free Fermi Gas

As predicted, cV ∝ D (EF) TLinear coefficient, Sommerfeld parameter

)(

)(')(

62

2

F

FBF ED

EDTkE

πµ −=

T

cV=γ

12

Lecture 3 23

Experimental Heat Capacity of Metals

At T below both the Debye temperature and the Fermi temperature, the heat capacity has electron and phonon contributions:

C = γ T + A T3,where γ (Sommerfeld parameter) and A are constants characteristic of the

material

C /T = γ + A T2

Lecture 3 24

Comparison of free-electron estimate of Sommerfeld parameter with experiment

Marder, Table 6.2