CND-04

Bloch wavevector

k = n π / a in 1-D, E = ħ² k² / (2 m*). A free-electron parabolic snapshot.

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Electrons in solidsBloch

Governing equation

E=\dfrac{\hbar^2 k^2}{2m^*},\quad k=\dfrac{n\pi}{a}

$n \in \left[1, 20\right]$

where

n: Index n (—)
a: Lattice constant (Å)
m^*/m: Effective mass (—)
k: Wavevector (1/Å)
E: Energy (eV)

The figure, part by part

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①

Governing law — red

E=\dfrac{\hbar^2 k^2}{2m^*},\quad k=\dfrac{n\pi}{a}

$n \in \left[1, 20\right]$

$n = 1$ $k = 2.901$

Along the curve

②

Domain of the variable — blue

n_{\min} = 1,\quad n_{\max} = 20

$n \in \left[1, 20\right]$

$n = 1$ $n = 1$

Along the curve

③

Response curve — green

E=\dfrac{\hbar^2 k^2}{2m^*},\quad k=\dfrac{n\pi}{a}

$n \in \left[1, 20\right]$

$n = 1$ $k = 2.901$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}k}{\mathrm{d}n}

$n \in \left[1, 20\right]$

$n = 1.119$ $k^{\prime} = 6.496$

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-04 — Bloch wavevector) is the form associated with Bloch. Working symbols: $n$ , $a$ , $m^*/m$ $\rightarrow$ $k$ , $E$ . Bloch's theorem: ψ = u_k e^{i k·r} with u periodic. The zone edge is π/a.

Purpose

Purpose: compute

k

E

from

n

a

m^*/m

in Condensed matter via

E=\dfrac{\hbar^2 k^2}{2m^*},\quad k=\dfrac{n\pi}{a}

k = n π / a in 1-D, E = ħ² k² / (2 m*). A free-electron parabolic snapshot. 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 = 1.000\,\mathrm{—}

a = 3.600\,\mathrm{Å}

m^*/m = 1.000\,\mathrm{—}

, the governing relation

E=\dfrac{\hbar^2 k^2}{2m^*},\quad k=\dfrac{n\pi}{a}

yields

k = 0.873\,\mathrm{1/Å}

E = 2.901\,\mathrm{eV}

. A lattice, a k-point, a parabola. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Index n n1 —Lattice constant a3.6 ÅEffective mass m^*/m1 —

Outputs

Wavevector k0.873 1/Å
Energy E2.901 eV

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Narration of this film

A lattice, a k-point, a parabola.

Bloch's theorem: ψ = u_k e^{i k·r} with u periodic. The zone edge is π/a.

Reading speed

Watch on YouTube