Introduction to solid state physics

L3 Licence Physique et Applications

Anuradha JagannathanLaboratoire de physique des solides

Bât.510Université Paris-Sud

Orsay

Periodic table of the elements

3 dimensional lattices

Ch I. Electronic properties

Electrical properties

Solids can show widely differing resistances to passage of electrical current

Current,voltage and resistance

Ohm's law : V = IR ↔ j = σEwhere

V=voltage (potential difference) = EL (E:electric field, L : length)

I = jS (j:current density, S : crosssection)

R = ρL/S (ρ = 1/σ : electrical resistivity)

R depends on the shape of the solid whereas ρ is an intrinsic property

Some measured resistivities

The birth of a conduction electron

In the solid state, atoms may lose their outermost (or valence) electrons

Example: sodium (1 valence electron) ionizes to Na+ and the valence electron becomes delocalized.

Classical free electron gas

In this first chapter We assume that the ions create a fixed periodic positive charge We assume that the electron-electron Coulomb interactions can be

neglected

If we assume, in addition, that the positive charge is uniformly distributed, we

have a gas of free electrons

A solid (left)modelled by a gasof conduction electrons (right)

Classical theory of gases

The kinetic theory of gases for particles of mass m in a volume V at a temperature T gives

according to this formula, at ambiant temperature thetypical speed of an electronIs approximately 105 m/s

electrical current in a conductor

For a fluid of identical charged particles subjected to an electric field, one can For a fluid of identical charged particles subjected to an electric field, one can show thatshow that

j = nqvj = nqvwhere

n :carrier concentration

v : velocity

q : charge per carrier

In a conductor,

q = -e (charge of the electron)

we can calculate n (next slide)

then, we need to calculate v(E) 

Drude model

Drude's observation : In the solid, the electrons are scattered and the direction of the velocities is randomly changed after a collision

Drude introduced a « relaxation time » or, average time between collisions : τ

In the absence of a field,the mean velocity is zero

In the presence of a field, the mean velocity is parallel to E

vd = Eeτ/m σ = ne2τ/m

Other predictions of the classical model

Calculation of thermal conductivity κ

Calculation of Hall resistivity, RH

Optical conductivity, plasma frequency (color of metals)

Calculation of the specific heat per mole due to the conduction electrons

C = dU/dT = 3/2 NAkB (≈12.3 J/mol K )

But this value is not observed in experiments !

Quantum theory of free electron gas

Electrons are fermions obeying the Pauli exclusion principle

Spin=1/2

Solution to the Schroedinger equation for a free particle

The stationary solutions of

where V=0, assuming periodic BC, are plane waves

Plane wave in 1d,

One dimensional gas Electron of mass m in a box of length L

3d quantum gas The energy levels depend on three quantum numbers

plot of energy levels for positive k-values(those in the first quadrant) with eachpoint in k-space colored according to the Energy (ε0 = h2/2mL2)

Fermi circle/sphere

Two dimensional case

Specific heat

The specific heat is much much smaller than the value expected from the classical theory of electron gas. Agrees with experiments.

But : we know (from observations) that the lattice must play an important rôle for electronic properties.

How can we take the crystal structure into account ?

Electrons in crystals

Electrons in a periodic potential

We begin with a one-dimensional crystal and the Schrődinger equation with V(x+a)=V(x)

Fourier series for V(x)

V(x) = Σ Vn e ingx

where g =2π/a

Perturbation theory

This term blows up when

the denominator→0 but numerator ≠ 0

Condition for this relation to break down :

E(K) = E(K') ….....that is, when K' = K+ng

Energy bands and gaps