CND-07

Fermi energy

E_F = (ħ² / 2m) (3π² n)^{2/3}. Fermi energy of a 3-D free-electron gas.

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Electrons in solidsFermi energy

Governing equation

E_F=\dfrac{\hbar^2}{2m}(3\pi^2 n)^{2/3}

$n \in \left[0.1, 20\right]$

where

n: Density (10²⁸/m³)
E_F: Fermi energy (eV)
T_F: Fermi temperature (K)

The figure, part by part

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①

Governing law — red

E_F=\dfrac{\hbar^2}{2m}(3\pi^2 n)^{2/3}

$n \in \left[0.1, 20\right]$

$n = 0.1$ $E_F = 0.365$

Along the curve

②

Domain of the variable — blue

n_{\min} = 0.1,\quad n_{\max} = 20

$n \in \left[0.1, 20\right]$

$n = 0.1$ $n = 0.1$

Along the curve

③

Response curve — green

E_F=\dfrac{\hbar^2}{2m}(3\pi^2 n)^{2/3}

$n \in \left[0.1, 20\right]$

$n = 0.1$ $E_F = 0.365$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E_F}{\mathrm{d}n}

$n \in \left[0.1, 20\right]$

$n = 0.225$ $E_F^{\prime} = 1.903$

Along the curve

Lecture brief

Historical brief

Drude electrons, Bloch waves, BCS pairing (1957) and Wiedemann–Franz heat are the first solids-and-metals laws. The lab is conductivity, gap and phonon heat in closed form. This sheet (CND-07 — Fermi energy) is the form associated with Fermi energy. Working symbols: $n$ $\rightarrow$ $E_F$ , $T_F$ . Copper sits near 7 eV. Only a slice k_B T around E_F is thermally active.

Purpose

Purpose: compute

E_F

T_F

from

n

in Condensed matter via

E_F=\dfrac{\hbar^2}{2m}(3\pi^2 n)^{2/3}

E_F = (ħ² / 2m) (3π² n)^{2/3}. Fermi energy of a 3-D free-electron gas. Use it when a real condensed matter question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n = 8.500\,\mathrm{10^{2}⁸/m^{3}}

, the governing relation

E_F=\dfrac{\hbar^2}{2m}(3\pi^2 n)^{2/3}

yields

E_F = 7.049\,\mathrm{eV}

T_F = 81804\,\mathrm{K}

. A filled Fermi sea, a surface. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Density n8.5 10²⁸/m³

Outputs

Fermi energy E_F7.049 eV
Fermi temperature T_F81804 K

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Narration of this film

A filled Fermi sea, a surface.

Copper sits near 7 eV. Only a slice k_B T around E_F is thermally active.

Reading speed

Watch on YouTube