# What is Fermi Factor

## Fermi energy

With the **Fermi energy** can you the **highest energy** one **Particle** in one **system** of **Fermions** at a**temperature** of **zero****Kelvin** () to calculate. In this article we will explain what the **Fermi energy** exactly **describes **and how you like them **to calculate** and **derive** can. You will also learn how to use her materials, in particular **Metals, semiconductors** and **Isolators, characterize** can and what at **Temperatures higher** as **0 Kelvin** happens. At the end of the article we will also look at a specific one **example** view by using the **Fermi energy**explicit for** copper** to calculate.

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### Fermi energy explained in simple terms

The **Fermi energy** gives the energy of **highest occupied energy levels** in the **Basic state** of a system **Fermions** at.

**Fermion,**such as a

**electron**in the

**Basic state,**may have.

Are you looking at the **absolute zero** (0 Kelvin) then all energy levels are up to **Fermi energy** fully occupied and the **higher energy levels** are vacant. A system is in the basic state when it is in the state** lowest possible energy** is located.

### Fermi energy formula

The **Fermi energy** lets out for a gas **non-interacting fermions** calculate with the formula

It represents the **reduced Planck quantum of action,** the **Mass of the fermion** and the **Fermi wave vector**

The **Particle density** is the quotient from the **Particle number** and the **volume**. Using the **Fermi energy** you can also do the **Fermi speed** just calculate with

The **Fermi speed** is the speed of a fermion at which the **kinetic energy** equal to the **Fermi energy** corresponds to

With the **Fermi wave vector** can also be called the **Fermi wavelength** express

**Fermions,** such as electrons, both propagate as **wave** as well as **Particle** out. The **Fermi wavelength** is the wavelength of the electrons, which they at the **Fermi energy** have. In addition, the so-called **Fermi temperature** to calculate

in which the **Boltzmann constant** designated.

### Fermi energy derivation

In this section we will use the formula for that**Fermi energy** derive. We assume a square **Potential box** with the volume out. In addition, the **Fermions** as **non-interacting particles** approached and the system is in the ground state at a temperature of **0 Kelvin**. Because one of a **infinite, periodic potential** goes out, applies to the **Wave function**

By solving the **stationary Schrödinger equation** with the **Bloch function** is then obtained as a condition for the **Wavenumber vector**

(1) With .

From particle physics, there is also the following relationship between the **frequency** and the **energy** known

In addition, you also know that the **Wavenumber** proportional to **pulse** is

If one uses these relationships, one obtains for them **kinetic energy** the following **formula**

If you put in this formula for the **Wave vector** the **Fermi wave vector** one, then the result is the **Fermi energy**

(2)

All energy levels are then in the basic state up to this **Fermi energy** with each **two spins occupied.** These occupied states are in the **Fermi sphere** with the **volume**

The **volume**, which contains exactly one state, has in the **reciprocal space** a volume of

If you divide the volume of the **Fermi sphere** by volume , then you get the **number** the **Electrons** in the **Potential box**

The factor 2 comes from there **each state has two electrons** can accommodate. With this, the **Electron density** determine

If you model this equation around

and plug this into formula (2), then you can use the **Fermi energy** also with the **Particle density** express

### Fermi energy semiconductor, insulator, metal

With the **Fermi energy** you can now explain why some materials are conductive and some are not. Is the **Fermi energy** in the **Band gap** between the **fully occupied valence band** and the **empty conduction band** so, for example, (sufficiently strong) **thermal excitation,** Electrons in that **Conduction band** reach. This becomes the material **conductive** and in this case one speaks of one **Semiconductor.** Is the **Band gap** however too big, so that the electrons get this by excitation **not bridge** can, then the material **not conductive** and one speaks of one **Insulator.** A metal is a **conductive material,** since the **Fermi energy** in the** Conduction band** is located and thus the **Line band partially occupied** is. And partially occupied bands are precisely the prerequisite for a material to be conductive.

### Fermi edge at higher temperatures

The states are at a temperature of **0 Kelvin** just until **Fermi level **occupied. In this context one speaks of a **Fermi edge,** because the **Fermi distribution** For a **Edge** having. The **Fermi distribution** gives the **probability** with which a state with the **energy** at a **certain temperature ** is busy

Here is the **chemical potential** and for applies . To **energy** the **Fermi edge** are so **all states** occupied and the states** higher energy** are all vacant. If the temperature is starting from 0 Kelvin **elevated,** so it gives way **Fermi edge** more and more up and so will **conditions** above that **Fermi level** occupied. This can be seen in the following illustration.

It must, however, apply that the **thermal energy** is much smaller than that **Fermi energy**

### Calculate Fermi energy example

With the above formula, the **Fermi energy** calculate various elements, such as the **Fermi energy sodium **or the** Fermi energy copper.** Here we want the example **Fermi energy copper** to calculate. The **free electrons** in copper have at a **density** of , the **Mass of electrons** is given by and the **reduced Planck quantum of action** With . So you get the **Fermi energy copper**

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