# What is Fermi energy: Definition and 104 Discussions

The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi gas, the lowest occupied state is taken to have zero kinetic energy, whereas in a metal, the lowest occupied state is typically taken to mean the bottom of the conduction band.
The term "Fermi energy" is often used to refer to a different yet closely related concept, the Fermi level (also called electrochemical potential).
There are a few key differences between the Fermi level and Fermi energy, at least as they are used in this article:

The Fermi energy is only defined at absolute zero, while the Fermi level is defined for any temperature.
The Fermi energy is an energy difference (usually corresponding to a kinetic energy), whereas the Fermi level is a total energy level including kinetic energy and potential energy.
The Fermi energy can only be defined for non-interacting fermions (where the potential energy or band edge is a static, well defined quantity), whereas the Fermi level remains well defined even in complex interacting systems, at thermodynamic equilibrium.Since the Fermi level in a metal at absolute zero is the energy of the highest occupied single particle state,
then the Fermi energy in a metal is the energy difference between the Fermi level and lowest occupied single-particle state, at zero-temperature.

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1. ### I Fermi energy for a Fermion gas with a multiplicity function ##g_n##

I ran across the following problem : Statement: Consider a gas of ## N ## fermions and suppose that each energy level ## \varepsilon_n## has a multiplicity of ## g_n = (n+1)^2 ##. What is the Fermi energy and the average energy of this gas when ## N \rightarrow \infty## ? My attempt: The...
2. ### Condensed Matter Physics - Fermi velocity, etc.

I have made solutions a-d, but my fermi velocity seems too big.
3. ### I Determining the Number of Points in the n-Space

Electron gas is a collection of non - interacting electrons. If these electrons are confined to certain volume (for example, cube of metal), their behavior can be described by the wavefunction which is a solution to the particle in a box problem in quantum mechanics. Allowed energy states for...
4. ### Statistical physics, using the ideas of Fermi Energies, etc. for a star

a) V=(4/3)pi(r^3) N=M/m_n (M=mass of neutron star, m_n=mass of neutron) Subbed into E_f = (hbar^2 / 2m) (3(pi^2)N / V)^(2/3). T_F = E_F / k_B b) dU = (dU/dS)_s dS + (dU/dV)_s dV p = -(dU/dV)_s dV V=(4/3)pi(r^3) -> r = cubedroot(3V/4pi) subbed into U_g = -(3/5)(G M^2 / r) take (dU/dV) plug into...
5. ### I Fermi energy definition and Fermi-Dirac distribution

1)In my book , there is a definition of fermi energy as topmost filled level in the ground state of an N electron system. This definition holds only for absolute zero,right? If it is not absolute zero,fermi energy is the energy at which the probability of a state being occupied is 50 percent...
6. ### I The meaning of the Fermi energy

The Fermi energy Ef is defined as the energy of the topmost filled level in the ground state of the N electron system. Ground state is n=1 level. And in the ground state there can be only one orbital right? One orbital can have only up to 2 electrons. Does this mean that fermy energy is the...
7. ### Fermi energy for potassium

I have completed part a, from which I got the expression: Cv = 3KTn/(T_f) For part b, the first term is the electron contribution and the second term is the phonon contribution. I'm stuck on how to estimate the fermi energy for the potassium metal. I'm thinking I only need to consider the...

9. ### Fermi energy in a quantum well

Problem Statement: See below Relevant Equations: 2D density of states ##g\left(E \right)=\frac{m^{*}}{\pi\hbar^2}## Fermi energy in a quantum cascade laser ##E_{F}=E_{i}+k_{B}Tln\left[exp\left(\frac{\pi\hbar^2n^{2D}}{k_{B}Tm^{*}} \right)-1\right]## I've been stuck on this problem for a few...

Attempt:
11. ### I Some questions about electrons and the Fermi energy

Hello ,evreyone.I have two questions about fermi energy. 1,Can I claim that 'fermi energy ' play the role of chemical potential? 2,I have learned from thermal physics that only electrons near fermi level can conduct in metals.How can electrons behave like this? I can't figure out why only...
12. ### I Fermi Energy Calculations About Non Parabolic Dispersions

Greetings! It is easy to understand that for a free electron, we can easily define the energy state density, and by doing the integration of the State density* Fermi-Dirac distribution we will be able to figure out the chemical potential at zero kelvin, which is the Fermi-Energy. Hence, we can...
13. ### Calculate velocity of the fastest neutron inside a 96Mo nucl

Homework Statement Calculate the velocity of the fastest neutron in a 96Mo nucleus and, based on this, explain whether or not we are safe to consider such nucleons in a non-relativistic way. Hint: first calculate the Fermi energy. Homework Equations Fermi energy from Fermi gas model...
14. ### I Valence band and Fermi level difference?

I was wondering what the difference was between the valence band and fermi level? How do we distinguish between the two? Thanks in advance.
15. ### Conduction band, valence band and Fermi energy

Homework Statement Homework Equations The Attempt at a Solution The probability of getting a state with energy ## E_v## is ## \frac { N_v } { N_v +N_c } = \frac1{ e^{-(E_v – E_f)/k_BT} +1} ## ………….(1) Since, ## E_v < E_f, e^{-(E_v – E_f)/k_BT}>>1 ## as ## E_f – E_v>> k_BT ##……….(2)...
16. ### A Zero Energy Wavefunctions in 1D superconductor

\bf{Setup} Hi! I am trying to derive the wavefunctions of the zero energy solutions of the Schrodinger equation in a 1D p-wave superconductor (Kitaev model). I am starting with the Hamiltonian  \begin{equation} H = \left[\begin{array}{cc} \epsilon_k & \Delta^{\ast}_k\\ \Delta_k & -\epsilon_k...
17. ### B Fermi Energy simplification

Here is a simplification of a problem given in my book. I don't get the simplification part. Could you please help at this ? It seems book has done the calculation wrongly.
18. ### Fermi energy for two spin states equal in equilibrium?

Homework Statement Let's consider conduction electrons (at T=0K) that are put in a magnetic field. The electrons can have spin that is parallel or antiparallel to the magnetic field. Below is the density of occupied states for such a system (horizontally) as a function of energy (vertically)...
19. ### [PoM] Electrons Fermi level in a crystal

Homework Statement The conduction band of a hypothetical crystal of one-dimensional Cesium reticular with step a=300 pm (1 atom per cell) is characterized by the ε dispersion law ##\epsilon (k) = V_0 + \frac{\hbar^2}{m_e}(\frac{1}{2}k^2 - \frac{a}{3\pi}|k|^3## where ##V_0 = -4 eV##, is set so...
20. ### Fermi Energy and Conduction Electrons

The problem: A simple cubic metal has an electron density such that the Fermi energy just touches the edge of the first Brillouin zone. Calculate the number of conduction electrons per atom for this condition to be fulfilled. The attempt at a solution: I know that the electron density for a...
21. ### I Negative Magnetoresistance?

A free electron gas would have zero magnetoresistance; it takes two carrier types to get ordinary magnetoresistance, which is always positive in sign. Beal-Monod and Weiner explain the negative magnetoresistance found in very dilute magnetic alloys, in terms of the spin-flip scattering of...
22. ### Always calculating Fermi energy incorrectly

Homework Statement This isn't really a homework question but I didn't know where to put it. Homework Equations ##E_f = \frac{h^2 k_f^2}{2m}## ##k_f = (3 \pi^2n)^{\frac{1}{3}}## The Attempt at a Solution I'm going through a lot of examples and every time I punch the numbers in I get an...
23. ### I Differential number of particles in Fermi gas model

I'm practicing for the Physics GRE, and came across a question that has me stumped. "In elementary nuclear physics, we learn about the Fermi gas model of the nucleus. The Fermi energy for normal nuclear density (ρ0) is 38.4 MeV. Suppose that the nucleus is compressed, for example in a heavy ion...
24. ### I Quasi Fermi level and intrinsic Fermi energy

Please explain me. I don't understand what is quasi fermi level f(E) and fermi energy Ef. For example (GaAs) at the room temperature (T=300K) Eg = 1.42 eV; (energy band gap) mc = 0.067 me; (effective mass of electron in conduction band) mv = 0.45 me; (effective mass of hole in valance band) kT =...
25. ### I Position of Fermi Energy for Semi-Conductors

what i mean exactly what is the position of Fermi energy for semi conductor materials 1- at the highest of valance band 2- at the mid-way in energy gap like Fermi level at 0 K
26. ### A Fermi energy Ef changes with applied electric field?

Hi people, I don't understand why when we apply the electric field to the metal Ef remains the same. Ef as translation energy of electrons remains the same but we accelerate the electrons with applied electric field so the translation energy increases too? In other hand according the formula...
27. ### Relation between electronic band structure and Fermi energy

I have some qualitative questions about the relation between band structure, density of states, and Fermi energy (or Fermi level). 1) Say you have a given electronic band structure (energy as a function of k) obtained by any method. How do you relate this to the Fermi energy (or Fermi level) ...
28. ### What is Fermi energy? given the number of electrons and (L)

hi every body question is : Consider that 5.5E22 free electrons are confined in a potential barrier of length 3.2A . find its Fermi energy ? the main point is that i was confident about the answer of question .. but the doctor said it is wrong .. he said i should use the relation between...
29. ### Fermi energy question help

Homework Statement Find the densities of states 0.08 eV above the conduction band edge and 0.08 eV below the valence band edge for germanium. Find the volume density of states (i.e. number of states per unit volume) with energies between the conduction band edge and 0.4 eV above the conduction...
30. ### Fermi energy question

Homework Statement Part 1) Use the fermi dirac probability function for t=150k, t=300k, and t=600k to fill in the table below. Part 2) Also show a sample calculation for (e-ef)=0.06eV and T=300k. Part 3)(Same as part 2?) Calculate the probabilities of a state at E -EF =0.06 eV being empty for...
31. ### Fermi energy of multiple electrons, infinite potential well

Homework Statement [/B] Five free electrons exist in a three-dimensional infinite potential well with all three widths equal to a 12 angstroms. Determine the Fermi energy level at T 0 K. Homework Equations E = [(h_bar*pi)2/(2*m*a2)]*(nx2 + ny2 + nz2) The Attempt at a Solution Tried using EF...
32. ### Energy gap between brillouin zones?

Homework Statement Consider a monovalent 2D crystal with a rectangular lattice constants ##a## and ##b##. Find expressions for the fermi energy and fermi wavevector in 2D. Show that the fermi surface extends beyond first zone if ## 2a > b\pi##. If the crystal is now divalent, estimate the...
33. ### Fermi Energy's plausible, but why define Fermi Temperature?

I get that in a single particle of a metal, Fermi energy is defined at T = 0 as the maximum energy that electrons can reach. I get that, but my book defines this concept called Fermi Temperature. Is Fermi Temperature the temperature where electrons can reach the next empty energy band in...
34. ### Solid State Chemistry Question Regarding Fermi Energy

Hi there, I am new to electron theory, and have a question regarding fermi energy. The book I am reading plots the Fermi energy distribuiton function vs Energy for T=0 ( upper right graph in pcture) and for T not equal to zero. The book says that, when T does not equal zero, the decrease in the...
35. ### Fermi energy in metals approximately doubling

Between Cs and Na, the fermi energy in metal approximately doubles. why doesn't the carrier concentration also double?
36. ### Calculating The Fermi Energy - Condensed Matter Physics

Homework Statement Calculate the Fermi energy, EF at 0K for potassium (atomic weight = 39, density = 860 kgm3). Homework Equations KF3 = 3π2n Fermi Momentum ρ = h(bar)KF The Attempt at a Solution :[/B] For the first part: Using: E = ρ2/ 2m Can substitute Fermi momentum into that to get: EF...
37. ### Free electron concentration range between semiconductors and metals

A structure with free electron density around 10^26 m^-3 is considered as a highly doped semiconductor or a metal? Or in other words, what is the lowest possible free electron concentration for a metal and what is the highest possible free electron concentration for a doped semiconductor?
38. ### Fermi Energy in Intrinsic Si, Ge, and GaAs

Homework Statement Using the values of the density of states effective masses me* and mh* in table 5.1, find the position of the Fermi energy in intrinsic Si, Ge, and GaAs with respect to the middle of the bandgap (Eg/2). Table 5.1 shows the following density of states effective masses me*/me...
39. ### Fermi energy condensed matter exam problem

Homework Statement Hello, I am preparing a condensed matter exam and I was wondering if I could get some help on the following question from a past exam paper: Show that for the free electron gas at zero temperature the Fermi energy is given by: ε_{F}=\frac{\hbar^{2}}{2m}(3π^{2}N)^{2/3} and...
40. ### Fermi energy and Fermi momentum from first principles

Hi everyone! I run the software ABINIT. It allows one to model solids as a plane-wave pseudopotential. Since it is based on first principles DFT, we are able to obtain the energies over many k-points. This would give us a band structure. Besides this, the calculation would print out a...
41. ### Is is possible to find unoccupied states below fermi energy?

Is is possible to find unoccupied states below fermi energy?? Or all states below fermi energy are always occupied?
42. ### Fermi energy of simple 3D gas

Homework Statement Consider a 3D gas of N non-interacting fermions in a volume V at temperature T << Ef / k. Suppose that the particles in the energy range [0.25 Ef, 0.5 Ef] are suddenly removed. Calculate the Fermi energy of the remaining particles after the system reaches its new thermal...
43. ### What is the Fermi energy of (undoped) graphene?

All of the sources I have found for this online have been wildly unclear. Many use the phrase "Fermi energy" to refer to the "Fermi level" (which is emphatically not what I'm looking for; I want the Fermi energy as defined in this Wikipedia article: http://en.wikipedia.org/wiki/Fermi_energy )...
44. ### Fermi energy in semiconductors

From thermodynamics we have dU=Tds-Pdv+\mu dN. So the chemical potential is the energy change due to adding an extra particle when S and V are constant. Now consider an intrinsic semiconductor at T=0 in which the valence band is all-occupied and conduction band is empty. If we add an extra...
45. ### Fermi Energy Carbon Nanotubes

What would be a good Internet link that would properly explain Fermi Energy or (Fermi Level)? How does the Fermi Energy of a material vary with external influence, such as doping of the material, and applied electric and magnetic fields for instance? What other factors can effect Fermi Energy...
46. ### Semiconductor and fermi energy

hey guys i just wanted to confirm something; so, for systems of continuous energy states (or small separations of discrete energy states), we can plot a graph like this and call the fermi energy the middle point where Probability=1/2. like this where, if T=0K, the transition from...
47. ### Finding the fermi energy in doped quantum well

Homework Statement Regular Doping: Let's say that we want to introduce some electrons into a quantum well, by adding dopant atoms (donors) to the semiconductor. The "regular" way to do this would be to deposit our semiconductor films, first a "barrier" layer with large band gap, then the...
48. ### Fermi Energy: Basics & Definition

what is fermi energy?
49. ### What is band bending and how does it relate to Fermi Energy?

Homework Statement Consider a pn junction in Si at 300K (other parameters given), with doping NA = 1021/m3 and ND = 1023/m3. Assume all impurities are ionized. On this basis find the Fermi level on each side. From this find the band bending VB and make a sketch of the pn junction. Homework...
50. ### How do I calculate the Fermi Energy of a compound?

Homework Statement 1) This question has to do with pure InAs, with a bandgap 0.33 eV, electron mass 0.02, hole mass 0.41. (a) Evaluate the number of electrons/m3 int he conduction band at 300K. For this purpose you can assume the Fermi Energy is exactly at the center of the energy gap...