CND-24

Fermi velocity

v_F = ħ k_F / m, k_F = (3π² n)^{1/3}. Speed at the Fermi surface.

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Electrons in solidsFermi velocity

Governing equation

v_F=\hbar k_F/m,\quad k_F=(3\pi^2 n)^{1/3}

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

where

n: Density (10²⁸/m³)
m^*/m: Effective mass (—)
k_F: Fermi wavevector (1/Å)
v_F: Fermi speed (Mm/s)

The figure, part by part

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①

Governing law — red

v_F=\hbar k_F/m,\quad k_F=(3\pi^2 n)^{1/3}

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

$n = 0.1$ $k_F = 0.358$

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

v_F=\hbar k_F/m,\quad k_F=(3\pi^2 n)^{1/3}

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

$n = 0.1$ $k_F = 0.358$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}k_F}{\mathrm{d}n}

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

$n = 0.225$ $k_F^{\prime} = 0.742$

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-24 — Fermi velocity) is the form associated with Fermi velocity. Working symbols: $n$ , $m^*/m$ $\rightarrow$ $k_F$ , $v_F$ . Typical metals: v_F ~ 10⁶ m/s, a percent of c. Graphene is linear, not parabolic.

Purpose

Purpose: compute

k_F

v_F

from

n

m^*/m

in Condensed matter via

v_F=\hbar k_F/m,\quad k_F=(3\pi^2 n)^{1/3}

v_F = ħ k_F / m, k_F = (3π² n)^{1/3}. Speed at the Fermi surface. 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}}

m^*/m = 1.000\,\mathrm{—}

, the governing relation

v_F=\hbar k_F/m,\quad k_F=(3\pi^2 n)^{1/3}

yields

k_F = 1.360\,\mathrm{1/Å}

v_F = 1.575\,\mathrm{Mm/s}

. A Fermi sphere, a radial speed. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Density n8.5 10²⁸/m³Effective mass m^*/m1 —

Outputs

Fermi wavevector k_F1.360 1/Å
Fermi speed v_F1.575 Mm/s

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Narration of this film

A Fermi sphere, a radial speed.

Typical metals: v_F ~ 10⁶ m/s, a percent of c. Graphene is linear, not parabolic.

Reading speed

Watch on YouTube