# What factors affect the Fermi level

## Ideal Fermigas

As Fermigas (after Enrico Fermi, who first presented it in 1926) In quantum physics, one describes a system of identical particles of the Fermion type, which are present in such large numbers that one has to limit oneself to statistical statements to describe the system. In contrast to gas in classical physics, the quantum-theoretical exclusion principle applies here.

The ideal Fermigas is a model for this, in which the mutual interaction of the particles is completely neglected, analogous to the ideal gas in classical physics. This represents a great simplification, but simplifies the formulas in such a way that physically correct predictions can be made in many practically important cases, e.g. B. for

### Ground state (vanishing temperature)

Since only a few particles can occupy the (single-particle) level with the lowest possible energy (set as \$ E = 0 \$) due to the principle of exclusion, most of the particles must occupy higher levels in the lowest possible energy state of the entire gas. The energy of the highest occupied level is called the Fermi energy \$ E_ \ mathrm {F} \$. It depends on the particle density \$ \ rho \$ (number per volume):

\$ E_ \ mathrm {F} = \ frac {\ hbar ^ 2} {2m} \; (3 \ pi ^ 2 \ rho) ^ \ frac {2} {3} \ approx \ frac {\ rho ^ \ frac { 2} {3}} {m} h ^ 2 \ cdot 0 {,} 121215345 \$

In it is

The formula applies to particles with spin \$ \ tfrac {1} {2} \$ such as B. electrons and is justified in quantum statistics.

With a spatial density of 1022 Particles per cm3 (like conduction electrons in metal) the Fermi energy results in a few electron volts. This is in the same order of magnitude as the energy of atomic excitations and has a clear effect on the macroscopic behavior of the gas. One then speaks of one degenerate Fermigas. The Fermi energy forms its salient characteristic, which has far-reaching consequences for the physical properties of (condensed) matter.

The Fermi energy can only be neglected in extremely dilute Fermi gas. It then behaves “not degenerate”, i.e. H. like a normal (classic) diluted gas.

### Simplified derivation

If a gas consisting of \$ \, N \$ particles in a spatial volume \$ \, V \$ (with potential energy zero) adopts the ground state, then from the bottom on so many states with different kinetic energies \$ \, E_ \ mathrm {kin} \ mathord = \ tfrac {p ^ 2} {2m} \ ge 0 \$ occupied until all particles are accommodated. The highest energy achieved in this way is \$ \, E_ \ mathrm {F} \ mathord = \ tfrac {{p_ \ mathrm {F}} ^ 2} {2m} \$, where \$ \, p_ \ mathrm {F} \$ as Fermi impulse referred to as. In three-dimensional Momentum space Then all particle impulses occur between \$ \, p \ mathord = 0 \$ and \$ \, p \ mathord = p_ \ mathrm {F} \$, in all directions. They form a sphere (Fermi sphere) with radius \$ \, p_ \ mathrm {F} \$ and volume \$ V_p \ mathord = \ tfrac {4 \ pi} {3} {p_ \ mathrm {F}} ^ 3 \$. If the particles were mass points, they would fill the volume \$ \, \ Omega = V \ cdot V_p \$ in their 6-dimensional phase space. For particles with spin \$ \, s \$, the spin multiplicity \$ \ mathord (2s + 1) \$ has to be multiplied. Since every (linearly independent) state in phase space occupies a cell of size \$ (2 \ pi \, \ hbar) ^ 3 \$, we get \$ \, (2s + 1) \ Omega / (2 \ pi \ hbar) ^ 3 \$ different states, each of which can accommodate one of the \$ \, N \$ particles (Occupation number 1):

\$ N = \ frac {(2s + 1) \ Omega} {(2 \ pi \ hbar) ^ 3} = \ frac {(2s + 1) V \ cdot V_p} {(2 \ pi \ hbar) ^ 3} = \ frac {(2s + 1) V} {(2 \ pi \ hbar) ^ 3} \ frac {4 \ pi} {3} {p_ \ mathrm {F}} ^ 3 \$

By converting to \$ \, E_ \ mathrm {F} \ mathord = \ tfrac {{p_ \ mathrm {F}} ^ 2} {2m} \$ and inserting \$ \, s = \ tfrac {1} {2} \$ the above formula follows.

### Excited state (finite temperature)

It becomes an ideal Fermigas at the hypothetical temperature that cannot be achieved in reality T = 0 K (→ Third law of thermodynamics) Supplied with energy, particles must move from levels below the Fermi energy to levels above. In thermal equilibrium, the occupation numbers develop for the levels, which continuously decreases from one to zero. This course, which is of great importance in various physical fields, is called the Fermi distribution or Fermi-Dirac distribution. The medium occupation\$ \ lang n_i \ rang \$ of a state \$ | i \ rang \$ with the energy \$ E_i \$ is:

\$ \ lang n_i \ rang = \ frac {1} {e ^ {\ frac {E_i - \ mu} {k_ \ mathrm {B} T}} + 1} \$

Here is

• \$ \ mu \$ the Fermi level or chemical potential
• \$ T \$ the temperature and
• \$ k_ \ mathrm {B} \$ the Boltzmann constant.

The Fermi distribution can be derived within the framework of statistical physics with the help of the grand canonical ensemble.

### Simplified derivation

A simple derivation using classic Boltzmann statistics, the principle of detailed equilibrium and the principle of exclusion, follows here:

Let us consider the equilibrium state of a Fermi gas at temperature T in thermal contact with a classic gas. A fermion with energy \$ E_1 \$ can then absorb energy from a particle of the classical system and change into a state with energy \$ E_2 \$. Because of the conservation of energy, the classical particle changes its state in the opposite sense from \$ E_2 '\$ to \$ E_1' \$, where \$ E_2'-E_1 '= E_2-E_1 \$. The occupation numbers are \$ n_1 \$ and \$ n_2 \$ for the two states of the fermion, \$ n_1 '\$ and \$ n_2' \$ for the two states of the classical particle. So that these processes do not change the equilibrium distribution, they must occur forwards and backwards with the same frequency overall. The frequency (or total transition rate) is determined from the product of the transition probability \$ W \$, as it applies to individual particles if there were no other particles, with statistical factors that take into account the presence of the other particles:

\$ n_1 \ cdot n_2 '\ cdot (1-n_2) \ cdot W_ {1 \ rightarrow 2} = n_2 \ cdot n_1' \ cdot (1-n_1) \ cdot W_ {2 \ rightarrow 1} \$

In words: The total number of transitions of a fermion from \$ E_1 \$ to \$ E_2 \$ (left side of the equation) is proportional to the number of fermions in state 1, to the number of reaction partner particles in state 2 ', and - so that the principle of exclusion is taken into account - on the number of free places for the fermion in state 2. Analogous for the reverse reaction (right side of the equation). Since, according to the principle of detailed equilibrium, \$ W \$ has the same value for jumping back and forth (\$ W_ {2 \ rightarrow 1} \ mathord = W_ {1 \ rightarrow 2} \$), the statistical factors are also the same. Now the Boltzmann factor applies to the classical particles

\$ \ frac {n_2 '} {n_1'} = e ^ {- \ tfrac {E_2'-E_1 '} {k_ \ mathrm {B} T}}. \$

Substituting this relationship and using the above equation \$ E_2'-E_1 '= E_2-E_1 \$ it follows:

\$ \ frac {n_1} {1-n_1} \; e ^ {\ tfrac {E_1} {k_ \ mathrm {B} T}} = \ frac {n_2} {1-n_2} \; e ^ {\ tfrac {E_2} {k_ \ mathrm {B} T}}. \$

This quantity has the same value for both states of the fermion. Since the choice of these states was free, this equality holds for all possible states, so it represents a constant for all single-particle states in the whole of Fermigas, which we can use with \$ e ^ {\ tfrac {\ mu} {k_ \ mathrm {B} T }} \$ parameterize:

\$ \ frac {n} {1-n} \; e ^ {\ tfrac {E} {k_ \ mathrm {B} T}} = e ^ {\ tfrac {\ mu} {k_ \ mathrm {B} T}}. \$

Resolved after n follows:

\$ n = \ frac {1} {e ^ {\ tfrac {E- \ mu} {k_ \ mathrm {B} T}} + 1}. \$

The parameter \$ \ mu \$ of this derivation thus turns out to be the Fermi level.