NNSE 508 EM Lecture #6

1

Lecture contents

• Metals: Drude model

– Conductivity – frequency dependence

• Plasma waves

• Difficulties of classical free electron model

Paul Karl Ludwig Drude (German: [ˈdʀuːdə]; July 12, 1863 – July 5, 1906)

NNSE 508 EM Lecture #6

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Phenomenology of electron transport: relaxation time

• In conductors, valence electrons are treated as free

electrons: “free particle swarm” (Drude model)

– Electron motion in the field:

• Electrons experience collisions similar to gas

molecules in the kinetic theory of gasses

– Extra average velocity due to electric field:

– Equivalent to a friction force on a “free”

electron :

– Thermal velocity is much higher than drift

velocity”

– Motion in real space = thermal motion + drift

+ scattering

dvm qE

dt

d

qv E

m

Relaxation time

1 2

7310B

th d

k T cmv v

m s

d ddv vm qE m

dt

NNSE 508 EM Lecture #6

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Phenomenology of electron transport: mobility

• Current density is proportional to drift velocity of

carriers

• Concentration n is taken as a density of valence

electrons

• In the steady state drift velocity is proportional to the

field (m –drift mobility):

- with mobility m [cm2/V-s] introduced in metals and

semiconductors

• And current density (s –conductivity) gives Ohm’s

law:

q

m

m

dJ qnv

dJ qnv qn E Em s

2qqn n

m

s m

d

qv E E

m

m

NNSE 508 EM Lecture #6

4 Resistivity of metals

Room temperature resistivity of single

crystalline metals • Drude theory was successful to

describe basic tendencies of

metal conductivity:

Dependence on

• Crystalline quality

• Temperature

• Alloying

• Frequency

NNSE 508 EM Lecture #6

5 Frequency dependence of conductivity of metals

• Let’s use simple microscopic picture:

(relaxation time approximation) and find how

the conductivity depends on frequency of EM

field

Assuming the electric field along x-direction:

• We get the displacement

• The response of the material can be described

with polarization (not at DC)

• And dielectric function:

where s0– static conductivity)

x

m x q E x B

Drift velocity which can

change in time and space

Smaller than electric

0

i txE E e

2

1qx E

mi

0P qnx E

2

00 0

2 0

1 11

q n

m ii

s

2

0we used q

nm

s

NNSE 508 EM Lecture #6

6 Plasma frequency

• Dielectric function:

• At low frequency, we return to

static conductivity

• At high frequency

With introduced plasma frequency

• Let’s estimate plasma frequency in metals:

1

1

1 22

15

0

12.2 10

2 2

p

p

q nf Hz

m

136 nm

Deep UV

2

00 0

2 0

1 11

q n

m ii

s

0 00

0 0

1 is s

Compare with general

phenomenological dispersion relation:

1 is

22 2

0 0 02 2 2

0

11 1

pq n q n

m m

becomes real – no attenuation !

1 2 1 220

0 0p

q n

m

s

does not depend on !

NNSE 508 EM Lecture #6

7 Drude optical properties of metals

• Dielectric function:

With plasma frequency and damping frequency

• We can write optical constants (refraction and extinction indexes)

2

00 0 2

20

11 1

p

di i

s

1 22

0p

q n

m

1d

2

2 21

2 20

1p

d

n

2

2

2 20

2pd

d

n

From Hummel, 2001

p

1n

1

At low frequency

NNSE 508 EM Lecture #6

8 Drude optical properties of metals

• Plasma frequency

• To improve accuracy of Drude model, effective number of free electron is usually introduced

• Damping frequency

• Damping frequency (scattering time) generally correlates with conductivity but not accurately

1 22

10

2p

q n

m

2

1

1

(observed)

(calculated)effN

From Hummel, 2001

2

12d

NNSE 508 EM Lecture #6

9 Optical properties of metals and dielectrics

• Reflectivity

• In metals, the major feature is plasma edge, also some interband transitions appear at higher (UV) frequencies

• Above plasma frequency there is no difference between metals and dielectrics

• In dielectrics, there are vibration-related features in IR and band features in UV as in metals

22 2

2

11

1 1

nnR

n n

NNSE 508 EM Lecture #6

10 Plasma waves

• Plasma frequency can be considered as maximum frequency of plasma response. It corresponds to internal electrostatic oscillations of plasma

• The electric field pulls the electrons back towards equilibrium, where they exactly neutralize the ion charge, but the kinetic energy gained in this process causes the electrons to overshoot to a new displacement on the other side.

• Let’s consider the simplest mode of plasma oscillations – 1D oscillations, resulting in B=0

• The 2nd Maxwell equation is

• Assuming that oscillations are faster than scattering time , the motion equation of carriers is

• Current density is as usual

• We get equation for oscillator with frequency p :

• In this simplest case the wavevector does not depend on frequency

• More sophisticated analysis considering thermal motion gives weak dependence on wavevector

22p

q n

m

0D

H Jt

0B

Et

0E

Jt

vm qE

t

J qnv

substitute

1

2 2

20

v nqv

t m

2 2 23p thkv

NNSE 508 EM Lecture #6

11 Difficulties of classical free electron model: electric

properties • Mass of electron obtained for cyclotron resonance may differ

significantly from free electron mass

• Hall effect may show positive sign of carriers transporting current

• Does not explain temperature dependence of conductivity in metals

– Experimentally

– Since kinetic energy of electron and scattering time

– We can expect

– And conclude that -ad hoc assumption

of the model

• Crystal structure effects are ignored

– Periodicity of crystal is ignored

– Anisotropy of conductivity in some non-cubic metals

• Predicts two orders higher paramagnetic susceptibility

than measured in experiment

1Ts

21 1 2

th

qn v T

m

s

23

2 2

thmvE kT

f

th

l

v

1 2

fl T

Effect of temperature on resistivity of

metals

NNSE 508 EM Lecture #6

12 Difficulties of classical free electron mode: thermal

properties • Problem even with the best triumph of the free-electron

model: Wiedemann-Franz law – relationship between thermal conductivity K and electron conductivity s :

WF

KC T

s

3

2E kT

3

2v

dEC n nk

dT

– Exact statistical analysis according to kinetic theory of gases shows that

– Wiedemann-Franz constant is about 2 times smaller than experimental value

• Problem with the specific heat:

– Average energy of electron is

– Specific heat = change of average energy per unit volume with temperature

– The lattice with atom density N will contribute at room temperature

– But typically specific heat density of metals is not higher than that of dielectrics

2

8

2

31.22 10

2

K dE k Wn T T

dT e SKs

, 3v lattC Nk

NNSE 508 EM Lecture #6

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Quantum theory of solids

• Quantum mechanical treatment of carriers: wave functions, bands

• Periodic potential – Bloch formalism: symmetry points

• Fermi statistics

• Note: Classical free electron model is extremely useful in semiconductors