Solid State Physics - Intermediate Modern Physics - Lecture Notes | PHY 3101, Study notes of Physics

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Solid State Physics^ Solid State Physics

Section 10-4,

Topics^ Topics

z^ Heat Capacity of Electron Gas z^ Band Theory of Solids z^ Conductors, Insulators and Semiconductors z^ Summary

Special Extra Credit^ Special Extra Credit^2

2 (^1) U M^ 2

r^ kT ω =^

(^2) =

2

0

0

0

/^

/^

R^

R^

r^

r

ρ^

ρ

=^

=^2

2

U^

M^

r^

E

ω

=^

1 2 (^

)

E^ n

n

ε

=^

Derive the temperaturedependence of R/R

by 0

computing the average potential energy of a lattice ion

rather than

En

= n

ε^ as

assuming that the energylevel of the nthvibrational state isEinstein had assumed^ Due:before classes end

Heat Capacity of Electron Gas^ Heat Capacity of Electron Gas

By definition, the heat capacity (at constant volume) of the electron gas is given by

V

dU C^

dT

where U is the total energy of the gas. For a gas of N electrons, each with average energy , the total energy is given by

U^

N^

E

Heat Capacity of Electron Gas^ Heat Capacity of Electron Gas

At

T= 0

, the total energy of the electron gas is

3 5

F

U^

N E

N^

E ⎛^

⎞

=^

=^

⎜^

⎟ ⎝^

⎠

For

0 < T << T

, only a small fraction F

kT/E

F

of the electrons can be excited to higher energy

states

Moreover, the energy ofeach is increased byroughly

kT

Heat Capacity of Electron Gas^ Heat Capacity of Electron Gas

Therefore, the total energy can be written as

3 5

F

kT F

U^

NE

NkT E ⎛ α

⎞

=^

+^

⎜^

⎟ ⎝^

⎠

where

α^

=^ π

2 /4, as first shown by Sommerfeld

(^22)

V

F

dU

T

C^

Nk

dT

T

=^

=

The heat capacity of theelectron gas is predicted tobe

Band Theory of Solids^ Band Theory of Solids

So far we have neglected the lattice of^ positively charged ions Moreover, we have ignored the Coulomb^ repulsion between the electrons and the^ attraction between the lattice and the^ electrons The

band theory of solids

takes into account

the interaction between the electrons and thelattice ions

Band Theory of Solids^ Band Theory of Solids

Consider the potential energy of a1-dimensional solidwhich we approximate by the

Kronig-Penney Model

13

Band Theory of Solids^ Band Theory of Solids

For

periodic potentials

, Felix Bloch showed that

the solution of the Schrödinger equation must be of the form

ikx k

x^

u^

x e

ψ^

and the wavefunction mustreflect the periodicity ofthe lattice:

(^

)

(^

(^

b nik^ a

a x

b

x

e

n

ψ

ψ^

Band Theory of Solids^ Band Theory of Solids

By requiring the wavefunction and its derivativeto be continuous everywhere, one finds energylevels that are grouped into

bands

separated by

energy gaps

. The gaps occur at

ka

n^ π

=^ ±

The energy gapsare basically energylevels that cannotoccur in the solid

Band Theory of Solids^ Band Theory of Solids

When,

the wavefunctions become

standing waves

. One wave peaks at the lattice

sites, and another peaks between them.

Ψ^2

, has

lower energy

ka^

n^ π

than

Ψ^1

. Moreover, there is a

jump in energy

between these states, hence the energy gap

Band Theory of Solids^ Band Theory of Solids

The allowed ranges of the wave vector

k^

are

called

Brillouin zones

zone 1:

- π

/a < k <

π /a

; zone 2:

π /a < -

π /a

zone 3:

π /a < k < 2

π /a

etc. The theory can explain why somesubstances are

conductors

, some

insulators

and others semiconductors

Conductors, Insulators,^ Conductors, Insulators,

Semiconductors^ Semiconductors

NaCl is an insulator, with a band gap of

2 eV

which is much larger than the thermal energy atT=300K Therefore, only a tiny fraction of electrons arein the conduction band

Conductors, Insulators,^ Conductors, Insulators,

Semiconductors^ Semiconductors

Silicon and germanium have band gaps of 1 eV and0.7 eV, respectively. At room temperature, a smallfraction of the electrons are in the conductionband. Si and Ge are

intrinsic semiconductors