Holes

A hole is the absence of an electron relative to a specified nearly full band, described as an excitation with charge $+e$. It is not an additional microscopic particle. Hole language is useful because a few missing occupations are easier to track than almost all the electrons in the band, and because the resulting excitation has a consistent positive-charge current description.

For the physical-momentum convention used throughout this page,

$$h_{\bar a\mathbf k_h}^{\dagger} := c_{a,-\mathbf k_h}, \qquad \varepsilon_{h,\bar a}(\mathbf k_h) := E_{v,a}-\varepsilon_{v,a}(-\mathbf k_h),$$

so that, modulo a reciprocal-lattice vector,

$$\mathbf k_h \equiv -\mathbf k_{\mathrm{removed}}, \qquad q_h=+e, \qquad \mathbf v_{h,\bar a}(\mathbf k_h) = \mathbf v_{e,a}(-\mathbf k_h).$$

The band-edge energy above is nonnegative when $E_{v,a}$ is the maximum of that band. A removal cost relative to a reservoir uses a different energy zero, explained below. The momentum reversal and unchanged group velocity are linked consequences of the definition, not independent sign conventions.

Required background. Band Theory Overview supplies band labels, Brillouin-zone periodicity, filling, and the distinction among model, Kohn–Sham, and quasiparticle bands.

Helpful background. Effective Mass supplies the curvature tensor; Fermi Surface supplies pocket geometry; and Metals, Insulators, and Semiconductors supplies equilibrium carrier statistics and material classification.

A Missing Electron in a Nearly Full Band

Consider one finite crystal with $N_c$ primitive cells and periodic boundary conditions. A nondegenerate band contains $N_c$ allowed Bloch states. If an unresolved multiplicity $g$ remains outside the band label—for example, a genuine twofold spin degeneracy—the capacity is

$$N_{\mathrm{states}} = gN_c.$$

If $N_e$ of those states are occupied, the number of holes is the exact complement

$$N_h = N_{\mathrm{states}}-N_e.$$

No approximation enters this count. The approximation enters when the missing occupations are treated as independent or weakly interacting quasiparticles with a well-defined dispersion.

The hole representation is economical when $N_h\ll N_{\mathrm{states}}$. Near an otherwise full valence band, a many-electron occupation pattern can then be specified by listing only the empty modes. Near an otherwise empty conduction band, the electron representation is more economical. Near half filling neither description has a privileged small carrier number, even though algebraic particle–hole transformations may still be useful in a model.

The Filled-Band Reference and Hole Operators

Let $\lambda$ run over every one-electron state in the retained valence band, including momentum and any internal label. Choose a fixed ordering and define the filled reference

$$\lvert F_v\rangle := \prod_{\lambda\in v} c_{\lambda}^{\dagger} \lvert0\rangle.$$

The ordering changes an overall fermionic phase convention but not physical matrix elements. Define a hole creation operator by removing the corresponding electron,

$$h_{\lambda}^{\dagger} := c_{\lambda}, \qquad h_{\lambda} := c_{\lambda}^{\dagger}.$$

The electron anticommutation relations immediately give

$$\left\{ h_{\lambda},h_{\lambda'}^{\dagger} \right\} = \delta_{\lambda\lambda'}, \qquad h_{\lambda}\lvert F_v\rangle=0.$$

Thus the filled electronic band is the hole vacuum. The occupation identity is

$$c_{\lambda}^{\dagger}c_{\lambda} = 1-h_{\lambda}^{\dagger}h_{\lambda}.$$

For a diagonal one-body band Hamiltonian,

$$H_v = \sum_{\lambda\in v} \varepsilon_{\lambda} c_{\lambda}^{\dagger}c_{\lambda},$$

normal ordering about the filled reference gives

$$H_v = E_{F_v} - \sum_{\lambda\in v} \varepsilon_{\lambda} h_{\lambda}^{\dagger}h_{\lambda}, \qquad E_{F_v} := \sum_{\lambda\in v} \varepsilon_{\lambda}.$$

The minus sign does not describe a negative-energy particle. Absolute one-electron energies can be shifted by a constant, and removing an electron changes particle number. A positive hole energy appears only after the relevant reference generator and energy zero are declared.

The transformation is exact for the chosen finite set of fermionic modes. It does not by itself make interactions disappear. A two-body electron Hamiltonian becomes a hole Hamiltonian with a reference constant, one-body terms produced by contractions with the filled band, and residual hole–hole interactions. Calling holes “noninteracting” is an additional physical approximation.

Hole Energy, Charge, and State Count

Three useful energy conventions

A band maximum defines the hole’s kinetic-energy zero, a reservoir fixes the cost of changing particle number, and an electron–hole promotion preserves the number. These are distinct energies:

Quantity	Definition for removing an electron from $\lambda$	Use
band-edge hole energy	$\varepsilon_h^{(v)}=E_v-\varepsilon_{v\lambda}$	dispersion and effective-mass expansion near a valence maximum
grand-canonical removal cost	$E_h^{(\mu)}=\mu-\varepsilon_{v\lambda}$	exchange with a reservoir governed by $H-\mu N$
electron–hole pair energy	$E_{eh}=\varepsilon_{c\lambda'}-\varepsilon_{v\lambda}$	number-conserving promotion across bands

For a full valence band below the chemical potential,

$$E_h^{(\mu)} = \mu-E_v + \varepsilon_h^{(v)}.$$

If the hole is exchanged with that reservoir, the first term is the cost of creating it at the valence edge; the second is its kinetic energy measured downward from that edge. In a number-conserving electron–hole promotion, the chemical potential cancels. In an interacting system, the exact removal energies are differences between $N$- and $(N-1)$-electron eigenenergies; a sharp hole band exists only when the removal spectrum contains a sufficiently coherent pole.

Positive charge relative to the filled reference

With $e>0$, the electronic charge operator in the retained band is

$$Q_v = -e \sum_{\lambda\in v} c_{\lambda}^{\dagger}c_{\lambda}.$$

Relative to the filled-band charge $Q_{F_v}=-eN_{\mathrm{states}}$,

$$Q_v-Q_{F_v} = +e \sum_{\lambda\in v} h_{\lambda}^{\dagger}h_{\lambda}.$$

An electronic hole therefore carries charge $+e$ relative to the reference. This is an additive-charge statement, not a claim that the vacuum contains a new elementary positively charged particle.

State density

When the band and internal multiplicities are explicit, the hole density is

$$p = \sum_a \int_{\mathrm{BZ}} \frac{d^dk}{(2\pi)^d} \left[ 1-f_{v,a}(\mathbf k) \right].$$

Here $a$ includes every retained internal state. An external degeneracy factor should be inserted only for labels omitted from $a$. The formula counts missing states per real-space $d$-volume; multiplying by the primitive-cell volume gives holes per cell. The equilibrium form of $f$, intrinsic and doped carrier densities, and charge-neutrality equations remain with Metals, Insulators, and Semiconductors.

As a finite check, a spin-degenerate band in a crystal of eight cells has sixteen states. Fourteen electrons leave exactly two holes, not two holes per spin and not eighteen total carriers. If spin is already part of the displayed band index, multiplying by two again is an error.

Momentum Labels, Velocity, and Current

Two legitimate hole labels

Let an electron be removed from $(a,\mathbf k_e)$. The many-body crystal momentum changes by

$$\Delta\mathbf K = -\hbar\mathbf k_e \pmod{\hbar\mathbf G}.$$

Two conventions are common.

Convention	Creation operator	Meaning of label	Physical crystal momentum change
removed-state label	$h_{a\mathbf k_e}^{\dagger}=c_{a\mathbf k_e}$	names the missing electron state	$-\hbar\mathbf k_e$ modulo $\hbar\mathbf G$
physical-momentum label	$\widetilde h_{\bar a\mathbf k_h}^{\dagger}=c_{a,-\mathbf k_h}$	label equals the hole wave vector	$+\hbar\mathbf k_h$, with $\mathbf k_h\equiv-\mathbf k_e\pmod{\mathbf G}$

The bar on $\bar a$ denotes the conjugate hole label associated with electron state $a$; it does not by itself assume time-reversal symmetry. In spinful bands, authors often choose a time-reversed basis. Any phase or unitary sewing matrix used to convert to that basis must then accompany the operator definition rather than being inferred from the word hole.

This page uses the physical-momentum label below and drops the tilde. Its band-edge energy is

$$\varepsilon_{h,\bar a}(\mathbf k_h) = E_{v,a} - \varepsilon_{v,a}(-\mathbf k_h).$$

The hole group velocity is therefore

$$\begin{aligned} \mathbf v_{h,\bar a}(\mathbf k_h) &= \frac{1}{\hbar} \nabla_{\mathbf k_h} \varepsilon_{h,\bar a}(\mathbf k_h) \\ &= \frac{1}{\hbar} \nabla_{\mathbf k_e} \varepsilon_{v,a}(\mathbf k_e) \bigg|_{\mathbf k_e=-\mathbf k_h} \\ &= \mathbf v_{e,a}(-\mathbf k_h). \end{aligned}$$

With the removed-state label instead, differentiating $E_v-\varepsilon_v(\mathbf k_e)$ gives $-\mathbf v_e(\mathbf k_e)$. That derivative is not a contradiction; the label $\mathbf k_e$ is then the negative of the hole’s physical momentum coordinate.

The current sign

For ordinary group-velocity current in one band and sample volume $V$,

$$\mathbf J_e = -\frac{e}{V} \sum_{a,\mathbf k} \mathbf v_e(\mathbf k) n_e(a,\mathbf k).$$

In the bulk limit, the group current of a filled band vanishes because the Brillouin-zone integral of the gradient of a periodic energy is zero. The reference-subtracted identity below is exact even in a finite calculation where one keeps the explicit reference current. Using

$$n_e(a,\mathbf k) = 1-n_h(\bar a,-\mathbf k),$$

and subtracting the filled reference gives

$$\mathbf J_e-\mathbf J_{F_v} = \frac{e}{V} \sum_{\bar a,\mathbf k_h} \mathbf v_{h,\bar a}(\mathbf k_h) n_h(\bar a,\mathbf k_h).$$

The missing electron current is exactly the current of a carrier with charge $+e$ and velocity $\mathbf v_h$. This is the clean reason the signs must be fixed together.

Removing a state at $k_e=k_v+q$ creates a hole at $k_h=-k_v-q$. The valence maximum at $k_v$ becomes a hole minimum at $-k_v$. At the marked states both slopes, and therefore both group velocities, point left. Removing the electron’s current gives the charge-$+e$ hole current. The drawing uses a local parabola; inversion or time-reversal symmetry is not needed for the transformation.

In a uniform electric field, the relabeling turns $\hbar\dot{\mathbf k}_e=-e\mathbf E$ into $\hbar\dot{\mathbf k}_h=+e\mathbf E$. Semiclassical Dynamics of Bloch Electrons derives the more general field equations. The group-current calculation above does not remove polarization, orbital magnetization, interband response or quantized Hall response from a filled band; those require their corresponding response terms.

Hole curvature near a valence maximum

Let a smooth nondegenerate valence band have a maximum at $\mathbf k_v$. Write the removed electron wave vector as

$$\mathbf k_e = \mathbf k_v+\mathbf q_e,$$

and the physical hole wave vector relative to its corresponding extremum as

$$\mathbf q_h := \mathbf k_h+\mathbf k_v = -\mathbf q_e \pmod{\mathbf G}.$$

If the valence band has the local expansion

$$\varepsilon_v(\mathbf k_v+\mathbf q_e) \approx E_v - \frac{\hbar^2}{2} \mathbf q_e^{\mathsf T} \mathsf M_h^{-1} \mathbf q_e,$$

then the hole excitation has

$$\varepsilon_h(\mathbf q_h) \approx \frac{\hbar^2}{2} \mathbf q_h^{\mathsf T} \mathsf M_h^{-1} \mathbf q_h.$$

The inverse hole-mass tensor is the negative of the valence electron inverse-mass tensor at the maximum. This conversion does not give the electron a negative rest mass, nor does it make every response mass equal. Effective Mass derives the tensor and distinguishes the masses extracted from different responses.

The expansion is local. Farther from the extremum, nonparabolicity, warping, nearby avoided crossings, and band mixing can make one constant tensor inadequate. At an exact degeneracy, an arbitrarily chosen vector in the degenerate subspace need not remain an energy eigenvector away from that momentum. Treat the coupled subspace before identifying smooth branches and their curvatures.

Worked Reorganization of a Cosine Band

Consider a one-dimensional valence band with lattice spacing $a$,

$$\varepsilon_v(k) = E_v - 2t \left[ 1-\cos(ka) \right], \qquad t>0.$$

The maximum is at $k_v=0$. Remove an electron at $k_e=-k_h$. The band-edge hole energy is

$$\varepsilon_h(k_h) = E_v-\varepsilon_v(-k_h) = 2t \left[ 1-\cos(k_ha) \right].$$

The velocities are

$$v_e(k) = -\frac{2ta}{\hbar} \sin(ka),$$

and

$$v_h(k_h) = \frac{2ta}{\hbar} \sin(k_ha) = v_e(-k_h).$$

Near the band edge,

$$\varepsilon_h(k_h) \approx t a^2k_h^2 = \frac{\hbar^2k_h^2}{2m_h}, \qquad m_h = \frac{\hbar^2}{2ta^2}.$$

Suppose the band has $N_c$ momentum states and only the electron state at $k_e$ is missing. The electron current of that occupied state would have been $-ev_e(k_e)/V$. Removing it changes the current by

$$\Delta J = +\frac{e}{V}v_e(k_e) = +\frac{e}{V}v_h(k_h).$$

Energy, velocity, momentum, charge, and current therefore agree under one convention. If one instead labels the hole by $k_e$, the physical momentum remains $-\hbar k_e$; changing only the symbol without changing the energy argument is the usual source of a sign error.

Hole Pockets Versus Hole Quasiparticles

A hole excitation is a missing occupation relative to a filled reference. A hole pocket is a Fermi-surface geometry: a compact sheet enclosing a small unoccupied region near a band maximum. They often occur together, but they are not synonyms.

Fermi Surface explains pocket orientation, compensation and orbit geometry. In particular, a measured Hall sign is not a model-independent pocket label. In a multiband material it depends on carrier densities, mobilities, anisotropic scattering, field regime, and sometimes Berry or interband effects. Hall Effect derives these response weights.

The word quasiparticle adds another requirement. In an interacting crystal, applying $c_{a\mathbf k}$ to the ground state produces a vector in the $N-1$ sector, if it is nonzero. Its squared norm is the occupation $\langle c_{a\mathbf k}^\dagger c_{a\mathbf k}\rangle$, not generally one; normalize it before treating it as a state. It need not overlap mainly with one long-lived eigenstate. The electron-removal spectral function can contain

• a narrow quasihole pole with residue $0<Z\le 1$;
• incoherent satellites or continua;
• several mixed orbital branches;
• broad features whose linewidth is comparable to their excitation energy.

A sharp hole dispersion is justified only for those branches and energy–momentum windows containing a sufficiently narrow pole with $0<Z\le 1$; incoherent weight may coexist with that pole. Spectral Functions derives the pole and continuum contributions. Angle-Resolved Photoemission Spectroscopy explains how occupation, matrix elements, surfaces and resolution weight the intensity; a bright ARPES ridge is evidence for a removal feature, not automatically a complete bulk hole Hamiltonian.

An optical electron–hole pair is neutral as a whole, and an exciton is a correlated bound neutral excitation. Neither is an isolated charge-$+e$ hole. Band Gaps distinguishes charge, quasiparticle, neutral and optical thresholds.

Degenerate Valence Bands and Multiband Limits

The scalar picture is controlled only when one smooth band is isolated from the others over the wave packet’s support. Many technologically important valence edges fail this test.

In common cubic semiconductors, atomic spin–orbit coupling and crystal symmetry organize the valence states near the zone center into coupled angular-momentum subspaces. For the usual ordering of the p-derived valence states, the effective $J=3/2$ manifold supports heavy-hole and light-hole branches, while a $J=1/2$ split-off manifold lies below it. These are near-zone-center angular-momentum labels for cubic symmetry representations, not a claim of full rotational symmetry throughout the crystal. In an ideal bulk crystal the heavy- and light-hole branches meet at the zone center; strain, confinement, fields, and lower symmetry can lift or reorganize them.

For a retained electron matrix $\mathsf H_v(\mathbf k)$, transform the generator $H_v-E_{\mathrm{ref}}N$ about the filled reference. The exact one-body hole matrix is

$$\mathsf H_h(\mathbf k_h) =E_{\mathrm{ref}}\mathsf I-\mathsf H_v^{\mathsf T}(-\mathbf k_h).$$

The transpose follows from reversing the order of fermionic operators. A change to a time-reversed hole basis introduces the corresponding unitary transformation as well. One cannot generally replace the matrix by $-\mathsf H_v$; complex off-diagonal entries distinguish the two operations.

Use one common $E_{\mathrm{ref}}$ when comparing or mixing hole branches. Separate band-edge zeros $E_{v,a}$ are useful for each branch’s kinetic energy, but erase the physical offsets if used without restoring those offsets in a multiband Hamiltonian or occupation calculation.

The Luttinger parameters describe the leading quadratic invariants in the cubic $J=3/2$ subspace. Off-diagonal terms mix the states, and “heavy” and “light” refer to curvatures along specified directions. They are not immutable particle species. Strong mixing, field-driven interband transitions and geometric corrections require the corresponding multiband wave-packet description; broad incoherent removal spectra require spectral functions rather than a single hole band.

The internal quantum numbers of a hole also transform contragrediently to those of the removed electron. For spin-1/2, a time-reversed labeling can make the hole basis look familiar. For orbital multiplets, simply copying the electron’s magnetic quantum number can give the wrong sign for an additive generator. State the basis and determine observables from the many-body change, not from a mnemonic.

A hole is also distinct from a positron, an exciton and a Bogoliubov quasiparticle. A positron is an elementary antiparticle; an exciton is a neutral correlated electron–hole excitation; a Bogoliubov quasiparticle mixes particle creation and annihilation. The ordinary band-hole transformation simply rewrites the occupations relative to a filled reference.

Exercises

1. Finite-band transformation

A nondegenerate band has four Bloch modes labeled $1,\ldots,4$, with

$$H = \sum_{i=1}^{4} \varepsilon_i c_i^{\dagger}c_i.$$

Define the filled reference and hole operators. Show that the two-hole state $h_2^{\dagger}h_4^{\dagger}\lvert F\rangle$ has electron occupations $n_2=n_4=0$ and $n_1=n_3=1$. Rewrite $H$ in hole-normal-ordered form.

Solution

Choose a fixed product order,

$$\lvert F\rangle = c_1^{\dagger}c_2^{\dagger} c_3^{\dagger}c_4^{\dagger} \lvert0\rangle,$$

and set $h_i^{\dagger}=c_i$, $h_i=c_i^{\dagger}$. Since

$$n_i^{e} = c_i^{\dagger}c_i = 1-h_i^{\dagger}h_i,$$

one hole in modes 2 and 4 gives $n_2^e=n_4^e=0$; the other modes retain $n_1^e=n_3^e=1$. Fermionic reordering may change the state’s overall sign but not these occupations. With $E_F=\sum_i\varepsilon_i$,

$$H = E_F - \sum_{i=1}^{4} \varepsilon_i h_i^{\dagger}h_i.$$

2. Do not mix energy zeros

A valence state has $\varepsilon_v=0.20\,\mathrm{eV}$, the valence maximum is $E_v=0.50\,\mathrm{eV}$, the chemical potential is $\mu=0.80\,\mathrm{eV}$, and a conduction state has $\varepsilon_c=1.40\,\mathrm{eV}$. Find the band-edge hole energy, grand-canonical removal cost, and number-conserving electron–hole pair energy.

Solution

The three quantities are

$$\varepsilon_h^{(v)} = E_v-\varepsilon_v = 0.30\,\mathrm{eV},$$ $$E_h^{(\mu)} = \mu-\varepsilon_v = 0.60\,\mathrm{eV},$$

and

$$E_{eh} = \varepsilon_c-\varepsilon_v = 1.20\,\mathrm{eV}.$$

They differ because the first measures kinetic energy down from the valence edge, the second changes particle number relative to $H-\mu N$, and the third preserves particle number.

3. Current without inversion symmetry

Consider the shifted band

$$\varepsilon_v(k)=E_v-2t[1-\cos(ka-\phi)], \qquad \phi=\pi/4.$$

Time-reversal symmetry is not assumed. For a hole at $k_h=\pi/(12a)$, find its band-edge energy, velocity, removed-electron momentum and contribution to current density. Why would replacing $\varepsilon_v(-k_h)$ by $\varepsilon_v(k_h)$ give the wrong answer?

Solution

The removed electron is at $k_e=-\pi/(12a)$. Since $k_ha+\phi=\pi/3$,

$$\varepsilon_h(k_h)=2t[1-\cos(k_ha+\phi)]=t,$$ $$v_h(k_h)=\frac{2ta}{\hbar}\sin(k_ha+\phi) =\frac{\sqrt3ta}{\hbar}=v_e(k_e).$$

Thus $\Delta J=e v_h/V=\sqrt3eta/(\hbar V)$. The electron dispersion is not even in $k$ for this fixed $\phi$. Omitting the momentum reversal would give $2t(1-\sqrt3/2)$ instead of $t$ and a different velocity. The correct transformation requires no symmetry relating $k$ and $-k$.

4. Complex matrix elements in the hole Hamiltonian

For two modes, let $H_e=\sum_{ij}c_i^\dagger A_{ij}c_j$ with

$$A=\begin{pmatrix}-2&i\lambda\\-i\lambda&-1\end{pmatrix}, \qquad \lambda\in\mathbb R.$$

Using $h_i^\dagger=c_i$ and a common reference energy zero, derive the hole matrix and reference constant. Does multiplying $A$ by $-1$ give the right matrix in this basis?

Solution

Fermion anticommutation gives

$$\sum_{ij}c_i^\dagger A_{ij}c_j =\operatorname{Tr}A-\sum_{ij}h_i^\dagger A_{ji}h_j.$$

Therefore

$$E_F=-3,\qquad A_h=-A^{\mathsf T}= \begin{pmatrix}2&i\lambda\\-i\lambda&1\end{pmatrix}.$$

The off-diagonal entries differ from those in $-A$. Although the two matrices have the same eigenvalues, they do not encode the same matrix elements in the declared hole basis. For a physical Bloch Hamiltonian the transpose is accompanied by $\mathbf k\mapsto-\mathbf k_h$.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston (1976), Chapters 12–13 and 28.
• E. O. Kane, “Band Structure of Indium Antimonide,” Journal of Physics and Chemistry of Solids 1, 249–261 (1957), doi:10.1016/0022-3697(57)90013-6.
• C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley (2004), Chapters 6 and 8.
• J. M. Luttinger, “Quantum Theory of Cyclotron Resonance in Semiconductors: General Theory,” Physical Review 102, 1030–1041 (1956), doi:10.1103/PhysRev.102.1030.
• J. M. Luttinger and W. Kohn, “Motion of Electrons and Holes in Perturbed Periodic Fields,” Physical Review 97, 869–883 (1955), doi:10.1103/PhysRev.97.869.
• M. P. Marder, Condensed Matter Physics, 2nd ed., Wiley (2010), Chapters 7–9.
• R. M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press (2004), doi:10.1017/CBO9780511805769.
• S. H. Simon, The Oxford Solid State Basics, Oxford University Press (2013), Chapters 6–8.
• G. Sundaram and Q. Niu, “Wave-Packet Dynamics in Slowly Perturbed Crystals: Gradient Corrections and Berry-Phase Effects,” Physical Review B 59, 14915–14925 (1999), doi:10.1103/PhysRevB.59.14915.
• R. Winkler, Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, Springer Tracts in Modern Physics 191, Springer (2003), doi:10.1007/b13586.
• P. Y. Yu and M. Cardona, Fundamentals of Semiconductors: Physics and Materials Properties, 4th ed., Springer (2010), doi:10.1007/978-3-642-00710-1.