Electronic Properties of Solids - ecourse2.ccu.edu.tw

Electronic properties of solids:

Free Electron Fermi Gas

2

Outline

• Overview - role of electrons in solids

⚫ Drude model

• Free electron model (Sommerfeld theory)

– Pauli Exclusion Principle, Fermi Statistics

– Energy levels in 1 and 3 dimensions

• Similarities, differences from vibration waves

• Density of states, heat capacity

• Fermi surface

3

Role of Electrons in Solids

• Electrons are responsible for binding of crystals: they are the

‘glue’ that hold the nuclei together

Van der Waals - electronic polarizability

Ionic - electron transfer

Covalent - electron bonds

Metallic

• Electrons are responsible for important properties:

– Electrical conductivity in metals

(But why are some solids insulators?)

– Magnetism

– Optical properties

– . . . .

4

Starting Point for Understanding

Electrons in Solids

• Nature of a metal:

Electrons can become “free of the nuclei” and move between

nuclei since we observe electrical conductivity.

• Electron Gas

Simplest possible model for a metal - electrons are completely

“free of the nuclei”

-- nuclei are replaced by a smooth background

-- “Electrons in a box”

5

Electron Gas - History

• Electron Gas model predates quantum mechanics

• Electrons Discovered in 1897 – J. J. Thomson

• Drude-Lorentz Model: Electrons - classical particles

free to move in a box

6

Drude Model (1900)

Three assumptions:

• Electrons have a scattering time τ. The probability of

scattering within a time interval dt is dt/τ.

• Once a scattering event occurs, we assume the electron returns

to momentum p = 0.

• In between scattering events, the electrons, which are charge

−e particles, respond to the externally applied electric field E

and magnetic field B.

• In 1900 Paul Drude realized that he could apply Boltzmann’s

kinetic theory of gases to understanding electron motion

within metals.

7

Drude-Lorentz Model (1900-1905)

( ) (1 )( ( ) ) 0dt dt

p t dt p t Fdt

+ = − + +

( )( ) (1 )( ( ) )

d p t dtp t dt p t Fdt

dt + = − +

( ) ( )d p t p tF

dt = −

keeping linear terms (dt)

under an electric field( ) ( )d p t p t

eEdt

= − −

for a steady state 0mv

eE

+ =

8


2e n

j env E Em

= − = =

e Ev

m

= −

under electric and magnetic fields

( ) ( )( )

d p t p te E v B

dt = − + −

for a steady state

0 ( )j B m

eE jn en

= − − +

E j=

9


2

2

2

0

0

0 0

m B

ne ne

B mE j

ne ne

m

ne

= −

When the magnetic field is along the z direction,

Hall resistivity

for a steady state E j=0 ( )j B m

eE jn en

= − − +

10

Edwin Hall experiment (1879)

11


2

2

2

0

0

0 0

m B

ne ne

B mE j

ne ne

m

ne

= −

Hall resistivity

Hall coefficient 1yx

HR

B ne

= = − from Drude model

When the magnetic field is along the z direction,

We could measure the density of electrons.

On the other hand, if the density of electrons is known, we could

measure the magnetic field.

Hall sensor

12


The Oxford Solid State Physics by Simon (2017)

Hall coefficient1yx

HR

B ne

= = − from Drude model

13


2e n

j env E Em

= − = =

Heat currentdT

J Kdx

= −

1 1

3 3v

K Cvl nc v v= =3

2v B

c k= 2 3B

k Tv

m=

2

831.11 10

2

BkK

T e

− =

WΩ/K2

the ratio of thermal conductivity to electric conductivity:

14

Drude-Lorentz Model (1900-1905)2

831.11 10

2

BkK

T e

− =

WΩ/K2


15

Wiedemann–Franz law (1853)

The ratio of the electronic contribution of the thermal

conductivity (K) to the electrical conductivity (σ) of a metal is

constant at the same temperature.

Ludivg Lorenz (1872): The ratio is proportional to T.

8, 2 10

KcT c

−=

The simple model seems work!

16

Peltier effect

Peltier effect: Running electrical current through a material

also transports heat.

qj j=

31( )

3 2

B

q

k Tj nv= j env= −

2

Bk T

e = −

44.3 10

2

Bk

ST e

−= = − = − V/K

Seebeck coefficient

17

Seebeck coefficient S

44.3 10S

−= − V/KThe Oxford Solid State Physics by Simon (2017)

100 times smaller…

The actual specific heat per particle is much lower.

18

Quantum Mechanics

• 1911: Bohr Model for H atom

• 1923: Wave Nature of Particles, Louie de Broglie

• 1924-26: Development of Quantum Mechanics -

Schrödinger equation

• 1924: Bose-Einstein Statistics for identical particles

(phonons, ...)

• 1925-26: Pauli Exclusion Principle, Fermi-Dirac

statistics (electrons, ...)

• 1925: Spin of the electron (spin = 1/2), G. E. Uhlenbeck

and S. Goudsmit

19

Schrödinger Equation : 1-d line

• Suppose particles can move freely on a line with position x,

0 < x < L

• Schrödinger Eq. in 1-d with V = 00 L

2 2

2 ( ) ( )

2

dx x

m dx − =

( )

1

2

2 2 2* 2

0

2sin( ), , 1, 2, 3, ...

( ) ( ) 1, ( )2 2

L

n

x kx k n nL L

k nx x dx

m m L

= = =

= = =

Normalization

Boundary

condition• Solution with (x) = 0 at x = 0, L

20

Electrons on a line

• For electrons in a box, the energy is just the kinetic energy which is quantized because the waves must fit into the box

2 2

, , 1, 2, 3, ...2

kk n n

m L

= = =

Approaches continuum

as L becomes large

k

E

21

Electrons in 3-d

• Schrödinger Eq. in 3D with V = 0

• Solution:

2 2 2 22 2

2 2 2( ) ( ),

2

d d dr r

m dx dy dz − = = + +

2 2

( ) ,

2= ,

2

i k r i k rr e ike

k mk

m

= =

=

ˆ( ) 2, 1 (Same for , )

2Thus = ( , , )

xi k Lik r Lx i k r

x x y z

x y z

e e e k n k kL

k n n nL

+ = = =

22

Electrons in 3-d• From previous slide, where we have imposed periodic

boundary conditions, the energy can be written as

2 2 22 2 2

( )2 2

x y z

kk k k

m m = = + +

23

Density of States (DOS)

Solid-State Physics by Ibachand Lueth (2009)

24

DOS in 3-d

• Key point - exactly the same as for vibration waves - the values of

kx, ky, kz are equally spaced : ∆kx= 2/L, etc.

• Thus the volume in k space per state is (2/L)3 and the number of

states N with |k| < k0 is

• Density of states per unit energy is ( )dN dN dk

D EdE dk dE

= =

3 3

30

02

4( )

2 3 6

kL VN N k

= =

2 2 2

2

k dE kE

m dk m= → =

2

3/2

2 2 2 2 22( ) ( )

2 2 2

Vk m Vmk V mD E E

k = = =

25

DOS in 3-d

• Consider electrons with spin ½ , there are two allowed values of

the spin quantum number for each allowed value of k. Therefore

the density of states is

3 / 2

2 2

2( ) ( )

2

V mD E E E

=

D(E)

E

Filled

EF

Empty

At 0 K

26

What is special about electrons?

• Fermions - obey Pauli exclusion principle

• Electrons have spin s = 1/2 - two electrons (spin up and spin down)

can occupy each state

• Kinetic energy = p2/2m = (ħ2/2m) k2

• If we know the number of electrons per unit volume Nelec/V, the

lowest allowed energy state is for the lowest Nelec/2 states to be

filled with 2 electrons each, and all the (infinite) number of other

states to be empty.

27

What is special about electrons?

• All states are filled up to the Fermi momentum kF and Fermi

energy EF = (ħ2/2m) kF2 given by

3

3

4

(2 ) 3 2F

V Nk

=

21/33

( )F

Nk

V

=

2 22/33

( )2

F

NE

m V

=

28

Fermi-Dirac distribution

• At finite temperatures, electrons are not all in the lowest energy

states

• Fermi-Dirac distribution gives the probability that a state at

energy will be occupied in an ideal electron gas in thermal

equilibrium

where is the chemical potential. At T = 0 K, = εF.

B

1( )

exp[( ) / ] 1f

k T

=

− +

f ()

E

1

½

f ()

E

1

½

T = 0 K T ≠ 0 K

29Introduction to Solid State Physics by Kittel (2005)



( )dN

D E EdE

=

3 3/2

3

2 4

(2 ) 3F

VN k E

=

3D

( )dN

D EdE

=

2

2

2

(2 )F

VN k E

=

2D

independent of E

1( )

dND E

dE E=

1/22

(2 )F

LN k E

=

1D

32

Typical values for electrons

• Conclusion: For typical metals the Fermi energy (or the

Fermi temperature) is much greater than ordinary

temperatures.



34

Heat Capacity for Electrons

• Just as for phonons the definition of heat capacity is C = dU/dT

where U = total internal energy

~ kBT

f ()

E

1

½

• For T << TF = EF / kB it is easy to see that roughly

U ~ U0 + Nelec (T/ TF) kBT so that

C = dU/dT ~ Nelec kB (T/TF)

35

• Quantitative evaluation: F

0

0 0

( ) ( ) ( ) ( ) ( )e e

U T U T U d D f d D

= − = −

0

( ) 1 ( ) ( ) ( ) ( )F

F

f D d d D f

= − − +

0 0

= ( ) 1 ( ) ( ) 1 ( ) ( )

+ ( ) ( ) ( ) ( ) ( )

F F

F F

F F

F F

f D d f D d

D f d f D d

− − − −

− +

0

0 0

= ( ) ( ) 1 ( ) + ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

F

F

F F

F

F F

F F F

d D f d D f

D d f D d f D d

− − −

− + +

36

• Quantitative evaluation: F

0

0 0

( ) ( ) ( ) ( ) ( )e e

U T U T U d D f d D

= − = −

0

0 0

= ( ) ( ) 1 ( ) + ( ) ( ) ( )

( ) ( ( ) ( ) ( ) ( ) )

F

F

F F

F

F F

F F

d D f d D f

D d f D d f D d

− − −

− + +

Energy cost of

empty state below F

Energy cost of occupied

state above F

N N

0

= ( ) ( ) 1 ( ) + ( ) ( ) ( ) F

F

F Fd D f d D f

− − −

37

F

FF F

0( ) ( ) ( ) ( ) ( ) ( )[1 ( )]

eU T d D f d D f

= − + − −

( )0

( , )( )

el e F

d df TC U d D

dT dT

= = −

Introduction to Solid State Physics by Kittel (2005)

38

( )0

( , )( )

el F F

df TC D d

dT

−

2 2

exp[( ) / ]( , )

(exp[( ) / ] 1)

F F

F

df

d

− −=

− +

Fx

−=

2 2

2( )

( 1)F

x

el B F x

eC k TD x dx

e

−

+ 100F

2 2

2( )

( 1)

x

el B F x

eC k TD x dx

e

−

+

F

FF F

0( ) ( ) ( ) ( ) ( ) ( )[1 ( )]

eU T d D f d D f

= − + − −

( )0

( , )( )

el e F

d df TC U d D

dT dT

= = −

39

2 2

2( )

( 1)

x

el B F x

eC k TD x dx

e

−

+

2

2 2

0

0

12 ( ) 4 ( )

1 1B F B Fx x

xk TD k TD x dx

e e

= − ++ +

2 2 1( )

( 1)( 1)B F x x

k TD x dxe e

−

−

=+ + even function

2 2

0

1[ ]

12 ( ) x

B F

dek TD x dx

dx

+= −

1

1 1

x

x x

e

e e

−

−=

+ + 1

( )1 ( )

x

x m

xm

ee

e

− −

−=

−= − = − −

− −

40

2

10

4 ( ) ( ) x m

el B F

m

C k TD x e dx

−

=

= − −

2

21 0

( 1)4 ( )

m

y

B F

m

k TD ye dym

−

=

−= −

2

21

( 1)4 ( )

m

B F

m

k TDm

=

−= −

41

2 2 2 2 2 21

( 1) 1 1 1 1 1 1( ...)1 2 3 4 5 6

m

m m

=

−= − − + − + − +

2 2 2 2 2 21

1 1 1 1 1 1 1... (2)

1 2 3 4 5 6m m

=

= + + + + + + =

2 2 2 2 2 2 21

1 1 1 1 1 1 1 (2)...

2 2 4 6 8 10 4m m

=

= + + + + + =

2 2 2 2 2 2

1 1 1 1 1 1 2... (2) (2)

1 2 3 4 5 6 2 − + − + − + = −

21

( 1) (2)

2

m

m m

=

−= −

42

2 FB F

BF

1,

2

el

TC N k T

kT

= =

2

21

( 1)4 ( )

m

el B F

m

C k TDm

=

−= −

3( )

2F

F

ND

=

2 214 ( )( (2)) 2 ( ) (2)

2B F B F

k TD k TD = − − =

22 ( ) (2)

B Fk TD =

2

22 ( )

6B F

k TD

=

43

2 FB F

BF

1,

2

el

TC N k T

kT

= =

Free electron model:

Drude model:

B

3

2el

C N k=

2 2 2 21 1

( ) ( )3 4 3 4

q B B

F F

T Tj Nk v k j

T e T

= = −

qj ST j=

21

( )3 4

B

F

TS k

e T

= −

2

Bk

e−

1

3q el

j U v=

j env= −

44

2e n

j env E Em

= − = =

21 1

3 3 2B

F

TK Cvl n k v v

T

= =

2 2B F

k Tv

m=

2 FB F

BF

1,

2

el

TC N k T

kT

= =

23

2

Bk Tn

m=

2 3B

k Tv

m=

22

3

Bk Tn

m

=

Drude model:

1 1 3

3 3 2B

K Cvl n k v v= =

45

Heat capacity

• Comparison of electrons in a metal with phonons

– Electrons dominate at low T in a metal

– Phonons dominate at high T because of reduction factor T/TF

Phonons approach

classical limit

C ~ 3 Natom kB

C

T

T 3 Electrons have

C ~ Nelec kB (T/TF)

T

46

Heat Transport

• Both electron and phonon can carry thermal energy

(Electrons dominate in metals).

• Similar to electric conduction, only the electrons near the

Fermi energy can contribute thermal current.

• Effects of electrons and phonons are comparable in poor

metals like alloys.

• Phonons dominate in non-metals.

47

Heat capacity

• Experimental results for metals

C/T = + A T 2 + ….

theo = (2/2) (Nelec/EF) kB2 is the free electron gas result.

Solid-State Physics by Ibach and Lueth (2009)

48

Effective mass of electrons


49

DOS of TM ions

Solid-State Physics by Ibach and Lueth (2009)

50

Heat capacity

• Experimental results for metals

C/T = + A T 2 + ….

It is most informative to find the ratio exp / theo where theo =

(2/2) (Nelec/EF) kB2 is the free electron gas result. Equivalently

since EF 1/m, we can consider the ratio exp / theo = mth / m,

where mth is a thermal effective mass for electrons in the metal.

51

Thermal effect mass

Causes of (free):

• Electron-ion interaction

• Electron-phonon interaction

• Electron-electron interaction

• Heavy fermion

• 4f or 5f electron wave-functions overlap with

neighboring ions

•

• CeAl3, CeCu2Si2, UPt3, UBe13

2 3th 10 -10 m

m

52

Interpretation of Ohm’s law:

Electrons act like a gas

• An electron is a particle - like a molecule.

• Electrons come to equilibrium by scattering like molecules

(electron scattering is due to defects, phonons, and electron-

electron scattering).

• Electrical conductivity occurs because the electrons are

charged and they move and equilibrate.

• What is different from usual molecules?

Electrons obey the Pauli exclusion principle. This limits the

allowed scattering which means that electrons act like a

weakly interacting gas.

53

The origin of resistivity

• Impurities - wrong atoms, missing atoms, extra atoms, …

– Proportional to concentration

(Dominated at low T)


• Lattice vibrations - atoms out of their ideal places

– Proportional to mean square displacement

(Dominated at room temperature)

54


lattice impurity = +

T impurityn Matthiessen’s rule


Impurity scattering

dominates at low T

55


( ) (0)lattice impurity

T = −

The resistivity ratio of a

specimen is usually defined as

the ratio of its resistivity at

room temperature to its

residual resistivity.


A copper specimen with a

resistivity ratio of 1000

corresponding to an impurity

concentration of about 20 ppm.

56


( ) (0)lattice impurity

T = −

For copper single crystal,

8 9(1.57 10 cm/s)(2 10 s)

Fl v −= =

In exceptionally pure specimens the resistivity ratio may be as

high as 106, whereas in some alloys (e.g., manganin) it is as low

as 1.1.

0.3cm=


58


Fermi surface

• The filling of the states is described by the Fermi surface –

the surface in k-space that separates filled from empty states

• For the electron gas this is a sphere of radius kF.

59

Why Drude model works


• The shift of Fermi sea has finite averaged velocity, drift

velocity.

• The electron with higher kinetic energy would scatter back.

60

Summary

• Role of electrons in solids:

Determine binding

“electronic” properties (conductivity, …)

• The starting point for understanding electrons in solids is

completely different from that for nuclei

• Simplest model - Electron Gas

– Failure of classical mechanics (Drude model)

– Success of quantum mechanics (Sommerfeld theory)

– Pauli Exclusion Principle, Fermi Statistics

• Similarities, differences from vibration waves

• Density of States, Heat Capacity

61

Shortcomings of the free electron model

• The mean free path of electrons in metal is around 10 nm at

RT and even 1 mm at low temperatures. Why the electrons

would not be scattered by the charged atoms?

• How about insulators where there are no free electrons?

• There are many electrons in atoms. Why do core electrons

not contribute to anything?

• Why the Hall coefficients of some materials are with wrong

sign?

• In optical spectra of metal there are usually many features

and thus metal have their characteristic colors. How to

explain it via the free electron model?

Documents

Electronic Properties of Solids - ecourse2.ccu.edu.tw