Introduction to solid state physics

L3 Licence Physique et Applications

Anuradha JagannathanLaboratoire de physique des solides

Bât.510Université Paris-Sud

Orsay

references

C.Kittel Physique de l'état solide

H.Alloul Physique des électrons dans les solides

N.W. Ashcroft and N.D.Mermin Physique des solides

Types of solids

The solid state macroscopic characteristics: rigidity, resistance to deformation

microscopic viewpoint: each atom has a well-defined mean position

Two types of solid matter Crystalline: the distribution of atoms is periodic in space

Amorphous: atoms are randomly situated

Periodic table of the elements

3 dimensional lattices

2 dimensional lattices

Example of a 2D lattice  the « 123 » high-Tc parent compound

(left) one unit cell of the YBa2Cu3O7 compound (right) one plane of Cu and O atoms

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.

the 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 an electron gas in zero field

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

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

Charged particles in a field

Problem :

A free electron in an electric field is subject to a constant force, and accelerates without limit, that is, its velocity (and the electrical current) → ∞ !

Solution :

The Drude model was introduced to explain the fact that the mean velocity remains finite, in a conductor, and that

j = σ E where σ is the electrical conductivity, which depends on the material, the

temperature, and eventually other factors (pressure, doping,defects,...)

We will next derive the Drude formula, which expresses σ as a function of the quantities n, e and τ

Microscopic model of electrical transport

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

C C j = nqv j = 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)