= Electron hole
{wiki}
= Hole
{synonym}
An <electron hole> is an unoccupied <electron> <quantum state> in an otherwise nearly full <energy band>, often the <valence band>. “Filled” refers to occupation of the allowed <quantum states>, not to filling all space with a continuous fluid of <electrons>. Each state, including its <spin> label, can hold at most one <electron> by the <Pauli exclusion principle>. The almost full <energy band> and its few missing occupants are therefore simpler to describe as a background plus a few <electron holes>. The <electron hole> is a <quasiparticle>, not a <Positron> inserted into the crystal.
Removing an <electron> of <electric charge> $-e$ changes the background's <electric charge> by $+e$. Thus an <electron hole> carries positive <electric charge> relative to that background. Promoting an <electron> into the <conduction band> creates an <electron hole> without changing the whole crystal's <electric charge>: $(-e)+(+e)=0$. An unbound pair still consists of two separate <charge carriers>; only a <bound state> of their relative motion is an <exciton>.
\Image[/electron-hole-band-occupation.png]
{title=<Energy band> occupation and the motion and filling of an <electron hole>}
{description=Original occupation schematic. Blue filled circles are occupied spin-resolved states; open circles are empty states. A promotion conserves the displayed <electron> count while changing occupation of the <valence band> and <conduction band>. The lower, localized cartoon shows an <electron> filling a neighboring vacancy so that the vacancy moves oppositely, and a <conduction band> <electron> filling a <valence band> <electron hole>. The <energy> axis is qualitative: the row diagrams are neither a <dispersion relation> nor single-electron levels for an <exciton>.}
{source=electron-hole-band-occupation.py}
The local cartoon explains why a vacancy can move: when a neighboring <electron> fills it, another vacancy appears where that <electron> was. The vacancy has moved opposite to that particular transfer of an <electron>. In an extended <Bloch state>, use <crystal momentum> instead of imagining a permanently empty atomic site. For a completely filled <energy band> the summed <electric current> vanishes. Omitting one <electron> with <group velocity> $\mathbf v_v$ removes its contribution $-e\mathbf v_v$, leaving a contribution $+e\mathbf v_v$. This fixes both the positive <electric charge> and the velocity convention of the <electron hole>.
More precisely, near a parabolic <valence band> maximum, set its <energy> $E_v(0)$ as reference and define the <electron hole> <wave vector> by $\mathbf k_h=-\mathbf k_v$. Then
$$
E_h(\mathbf k_h)=E_v(0)-E_v(-\mathbf k_h),\qquad
\mathbf v_h=\frac{1}{\hbar}\nabla_{\mathbf k_h}E_h=\mathbf v_v(-\mathbf k_h),\qquad
E_h(\mathbf k_h)\simeq\frac{\hbar^2k_h^2}{2m_h^*}.
$$
The occupied <electron> branch has negative curvature there, while the <electron hole> branch has positive <effective mass> $m_h^*$. The velocity equality compares the <electron hole> with the missing <Bloch state>'s <group velocity>; it does not contradict the opposite vacancy motion in a local transfer cartoon. The sign conventions are listed in <Phys 460, Fall 2006/Phys 460 Lecture 17>.
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