6Free Electron Fermi Gas

6.1. Electrons in a metal

6.1.1. Electrons in one atom

One electron in an atom (a hydrogen-like atom): the nucleon has charge +Z e, where Z is the atomic

number, and there is one electron moving around this nucleon

Four quantum number: n, l and lz, sz.

Energy levels En with n = 1, 2, 3 ...

(6.1)En = -Z

2 Μ e4

32 Π2 Ε02

Ñ2

1

n2

where Μ = me mN � Hme + mN L » me where me is the mass of an electron and mN is the mass of the nucleon.

(6.2)En = -H13.6 eV L Z

2

n2

For each n, the angular momentum quantum number [L2 Ψ = lHl + 1L Ψ] can take the values of l = 0, 1, 2, 3, 4 … , n - 1. These states are known

as the s, p, d, f , g,… states

For each l, the quantum number for Lz can be any integer between -l and +l Hlz = -l, -l + 1, … , l - 1, lL.For fixed n, l and lz, the spin quantum number sz can be +1/2 or -1/2 (up or down).

At each n, there are 2 n2 quantum states.

In a real atom

In a hydrogen-like atom, for the same n, all the 2 n2 quantum number has the same energy. In a real atom, the energy level splits according to the

total angular momentum quantum number l.

Typically, the state with lower l has lower energy (not always true). Due to the rotational symmetry, states with the same l has the same energy (if

we ignore the spin-orbital coupling, etc.).

… … … …

n = 4 l = 1 4 s 2

n = 3 l = 2 3 d 10

n = 3 l = 1 3 p 6

n = 3 l = 0 3 s 2

n = 2 l = 1 2 p 6

n = 2 l = 0 2 s 2

n = 1 l = 0 1 s 2

Phys463.nb 33

Many electrons at T = 0

Electrons are fermions. The Pauli exclusive principle requires that we can have at most 1 electron per quantum state.

At T = 0, to minimize the total energy, the electrons want to state at the lowest Ne quantum state.

Ne Element No. of electrons on each shell

1 H 1 s1

2 He 1 s2

3 Li 1 s2

2 s1

4 Be 1 s2

2 s2

5 B 1 s2

2 s2

2 p1

6 C 1 s2

2 s2

2 p2

… … …

10 Ne 1 s2

2 s2

2 p6

11 Na 1 s2

2 s2

2 p2

3 s1

12 Mg 1 s2

2 s2

2 p2

3 s2

... … ...

Valence electrons

Electrons in the low energy states (inner layers) are bonded tightly to the nucleon. (It costs about 1-10 eV to remove one of these electrons from an

atom. 1eV is 104

K. It is a very high energy cost in solid state physics).

Electrons in high energy states (outer layers) are loosely bonded to the nucleon (easy to remove). These electrons are called the valence electrons

and these energy states are called the valence shells.

Valence electrons are the electrons in a atom which can participate in the formation of chemical bonds. They are typically electrons in the

outermost (or second-outer-most) shell.

6.1.2. Electrons in a metal

In a metal, an atom may lose some or all of its valence electrons and thus turns into an ion (nucleon+inner electrons). These ions form a crystal and

their motions are phonons. The valence electrons are no longer bonded to nucleons. They can move freely in a crystal.

(6.3)A crystal = A lattice of ions + valence electrons

(6.4)Motions in a crystal = phonons + valence electrons

At low temperature, the interactions between phonons are typically very weak. So we can consider treat them as a quantum gas (a Bose gas).

In a metal, because valence electrons can move around, we can treat them as a quantum fluid (a fermion fluid). Typically, we call this fluid a Fermi

liquid. It is called a Fermi liquid, instead of a Fermi gas, because the interactions between electrons (Coulomb interactions) are typically pretty

strong (comparable to the kinetic energy in most metals).

A solid can be considered as the mixture of two type of fluids:

(6.5)A Bose gas of phonons + A Fermi liquid of electrons

6.1.3. Free electrons in 1D at zero temperature T = 0

Here, we start from the simplest situation: a free Fermi gas

The word “free” here means two things.

1. We ignore interactions between electrons

2. We ignore interactions between electrons and ions (nucleons)

Within this approximation, electrons are free particles. So their Hamiltonian is (for a 1D system)

(6.6)H =p

2

2 m

=1

2 m

-ä Ñ

â

â x

2

= -Ñ

2

2 m

â2

â x2

One electron in 1D

The Schrodinger equation is

34 Phys463.nb