Holes

A hole is the absence of an electron relative to a specified nearly full band, reorganized as a positive-energy 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 reference band, energy zero, internal-state convention, and regime of validity must accompany these formulas. The generic particle–hole treatment owns pair continua and response kinematics. This page owns the crystalline nearly-full-band specialization: exact state counting per cell, the two common momentum labels, the group-current sign, and the point at which a scalar hole band must be replaced by a multiband or spectral description.

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.

Three declarations are needed before the word hole is quantitative:

1. Reference. State which band or retained subspace is regarded as full.
2. Band object. State whether its energies are model eigenvalues, Kohn–Sham eigenvalues, coherent quasiparticle poles, or another controlled object.
3. Labels. State whether the hole carries the label of the removed electron or a relabeled physical crystal momentum.

The first declaration fixes what is missing. The second fixes what, if anything, may be called a hole dispersion. The third fixes the signs of momentum and velocity.

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

Different questions attach different constants to the same missing occupation. They must not be mixed.

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,

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

The first term is the cost of bringing a hole to 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 $V$,

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

A completely filled ordinary band has zero net group-velocity current because the Brillouin-zone sum of the gradient of a periodic energy vanishes. 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.

The hole description is a change of reference and labels. Removing the electron at $\mathbf k_e$ creates one hole at $\mathbf k_h\equiv-\mathbf k_e\pmod{\mathbf G}$; the physical-momentum convention makes $q_h=+e$, $\mathbf v_h=\mathbf v_e(\mathbf k_e)$, and the current weight $q_h\mathbf v_h=+e\mathbf v_h$ mutually consistent. Dividing that weight by the sample volume gives one carrier’s contribution to the current density. An isolated band supports a scalar or tensor dispersion, whereas a degenerate or strongly mixed valence subspace requires a matrix Hamiltonian.

In a uniform electric field and in the simplest isolated-band approximation, the same relabeling turns $\hbar\dot{\mathbf k}_e=-e\mathbf E$ into $\hbar\dot{\mathbf k}_h=+e\mathbf E$. Semiclassical Dynamics of Bloch Electrons owns the complete signed field equations and their validity tests; this page retains the electron-to-hole charge, momentum, velocity, and current transformation. Collision terms, Berry-curvature corrections, and observable conductivities require the dedicated transport and geometric owners. The group-current derivation above also does not erase polarization, orbital magnetization, interband response, or quantized transverse response from filled bands.

Curvature and the Effective-Mass Handoff

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 positive hole-mass tensor is the negative of the valence electron curvature 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 owns the curvature derivation and the distinctions among curvature, density-of-states, conductivity, cyclotron, optical, and quasiparticle masses.

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, assigning a Hessian to one arbitrarily chosen eigenvector is generally not gauge invariant; the retained degenerate subspace must be treated together.

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 owns the orientation of electron and hole pockets, compensation, Fermi-volume counting, extremal orbits, and topology changes. 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 owns that inference.

The word quasiparticle adds another requirement. In an interacting crystal, annihilating an electron produces an exact state in the $N-1$ sector, but 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 lifetime 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 owns poles, residues, self-energies, and incoherent weight. Angle-Resolved Photoemission Spectroscopy owns the occupation-, matrix-element-, surface-, and resolution-weighted 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 owns the distinction among 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. The top $J=3/2$ manifold supports heavy-hole and light-hole branches, while a $J=1/2$ split-off manifold lies at a material-dependent separation. 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.

The correct low-energy object is then a matrix Hamiltonian acting on the retained valence subspace,

$$H_h(\mathbf k) \longrightarrow \mathsf H_h(\mathbf k),$$

whose off-diagonal terms encode symmetry-allowed mixing. The Luttinger parameters summarize the leading quadratic invariants for a cubic $J=3/2$ manifold. “Heavy” and “light” refer to curvatures along specified directions and in specified regimes; they are not immutable particle species throughout the Brillouin zone.

Use the following escalation test.

Situation	Minimum adequate description
smooth isolated nondegenerate valence band	scalar hole dispersion, generally with a mass tensor
two or more nearly degenerate valence bands	coupled multiband Hamiltonian and subspace projectors
strong spin–orbit coupling or warping	momentum-dependent spinor states and anisotropic response
wave packet sampling an avoided crossing	multiband dynamics; check interband transitions
coherent pole with strong interaction renormalization	quasihole dispersion, residue, and lifetime
broad or fractionalized removal spectrum	spectral description; no single-hole band claim
important Berry curvature or orbital moment	geometric wave-packet treatment in the retained subspace

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.

Semiconductor and Probe Handoffs

Hole bookkeeping is an input to several larger calculations. It does not replace them.

Question	Canonical owner	What the hole ledger alone does not determine
How many carriers are present at temperature $T$?	Metals, Insulators, and Semiconductors	chemical potential, dopant ionization, charge neutrality, degeneracy regime
Which mass enters a measured response?	Effective Mass	density-of-states, conductivity, cyclotron, optical, or quasiparticle definition
Is a Fermi-surface sheet electron-like or hole-like?	Fermi Surface	pocket geometry, compensation, orbit orientation
What is the conductivity or mobility?	Boltzmann Transport	collision operator, relaxation times, distribution, vertex corrections
What fixes the Hall sign?	Hall Effect	multicarrier weighting, field regime, anomalous terms
Is a removal feature a coherent quasihole?	Spectral Functions and ARPES	residue, lifetime, matrix elements, surface sensitivity, resolution
What is the optical threshold?	Band Gaps	directness, selection rules, exciton binding, phonon assistance
What is a generic neutral particle–hole continuum?	Particle–Hole Excitations	joint phase space, response vertices, collective poles

Likewise, acceptor chemistry, junction electrostatics, confinement, and devices require their own material and boundary-condition models. The statement “acceptors create holes” is a carrier-count summary, not a microscopic account of the impurity wave function, ionization energy, screening, or compensation.

Validity and Reporting Checklist

Before using a hole Hamiltonian, record:

1. Reference occupancy. Which band or subspace is nearly full, and how small is $N_h/N_{\mathrm{states}}$?
2. Energy object. Are the inputs model, Kohn–Sham, quasiparticle, or measured removal energies?
3. Energy zero. Is the quoted value referenced to $E_v$, to $\mu$, or to a number-conserving electron–hole transition?
4. Momentum convention. Does the label name the removed electron state or the physical hole momentum?
5. Internal labels. Are spin, valley, orbital, Kramers, and degeneracy indices explicit, and is any multiplicity counted exactly once?
6. Isolation test. Is one band separated from nearby bands across the wave-packet and field-induced momentum range?
7. Coherence test. Does the removal spectrum contain a pole narrow enough for the intended time and energy scale?
8. Response handoff. Which scattering, vertex, geometric, electrostatic, or probe model converts the hole inputs into the requested observable?
9. Failure test. Would degeneracy, nonparabolicity, strong fields, disorder, interactions, or topology invalidate the scalar description?

A defensible result has the form: “relative to this filled reference, using this energy and momentum convention, these missing states behave as charge-$+e$ holes over this specified window.” It is not a universal claim about every valence-band feature or every experimental sign.

Common Mistakes

Calling a hole a microscopic positive particle. A hole is a missing electronic occupation relative to a reference. Its positive charge is the change in electronic charge when an electron is removed.

Giving a hole the removed electron’s momentum without qualification. If the hole is labeled by $\mathbf k_e$, its physical crystal momentum change is $-\hbar\mathbf k_e$. If its label is to equal physical momentum, relabel $\mathbf k_h\equiv-\mathbf k_e\pmod{\mathbf G}$.

Mixing the energy references. $E_v-\varepsilon_v$, $\mu-\varepsilon_v$, and $\varepsilon_c-\varepsilon_v$ answer different questions. A constant is harmless only while the hole number is fixed.

Using charge $+e$ with the wrong velocity. Current consistency requires the momentum relabeling and the energy argument to be transformed together. Copying one sign from each convention produces a spurious minus sign.

Multiplying degeneracy twice. A band index that already includes spin or Kramers partners needs no extra factor of two. Count the actual one-particle labels in the retained subspace.

Treating every valence edge as one parabolic band. Degenerate, warped, or strongly spin–orbit-coupled valence manifolds require a matrix Hamiltonian. Heavy-hole and light-hole labels are regime- and direction-dependent.

Inferring carrier type from one probe sign. Hall, thermopower, optical, and ARPES signals contain different weights and assumptions. Route each inference through its response owner.

Confusing a hole with a positron, an exciton, or a Bogoliubov quasiparticle. A positron is an antiparticle, an exciton is a neutral correlated electron–hole excitation, and a Bogoliubov quasiparticle coherently mixes particle creation and annihilation. Ordinary band-hole bookkeeping does none of those things.

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. Momentum conventions on a Brillouin torus

In a one-dimensional lattice with Brillouin zone $[-\pi/a,\pi/a)$, remove an electron at $k_e=0.7\pi/a$. Give (a) the removed-state hole label and its physical momentum, and (b) a physical-momentum hole label in the same zone. Explain why adding $2\pi/a$ does not create a different crystal-momentum sector.

Solution

In the removed-state convention the hole is labeled $k_e=0.7\pi/a$, but the many-body momentum change is

$$\Delta K = -0.7\pi\hbar/a \pmod{2\pi\hbar/a}.$$

In the physical-momentum convention choose

$$k_h = -0.7\pi/a.$$

The alternative representative $1.3\pi/a$ differs by the reciprocal vector $2\pi/a$ and therefore labels the same crystal momentum on the Brillouin torus.

4. Current consistency in the cosine band

For the worked cosine band, take a hole at $k_h=\pi/(3a)$. Compute $\varepsilon_h$, $v_h$, the removed electron wave vector, and the current carried by one hole in volume $V$.

Solution

The removed electron was at $k_e=-\pi/(3a)$. Therefore

$$\varepsilon_h = 2t \left[ 1-\cos(\pi/3) \right] = t,$$

and

$$v_h = \frac{2ta}{\hbar} \sin(\pi/3) = \frac{\sqrt{3}ta}{\hbar}.$$

The electron velocity at $k_e$ has the same value. Removing its charge-$-e$ current changes the total current by

$$\Delta J = \frac{e}{V}v_h = \frac{\sqrt{3}eta}{\hbar V}.$$

5. Degeneracy-aware state counting

A crystal has $10^6$ primitive cells. A valence manifold contains two explicitly listed spin–orbit-split bands, each with one state per $\mathbf k$, and has $1.9998\times10^6$ electrons. How many holes are present? What error results from adding an extra spin factor of two?

Solution

The two explicit bands contain

$$N_{\mathrm{states}} = 2\times10^6$$

states, so

$$N_h = 2\times10^6 - 1.9998\times10^6 = 200.$$

The spin–orbit-split bands already carry their internal labels. Multiplying their capacity by two again would predict $4\times10^6$ states and the nonsensical count $2{,}000{,}200$ holes.

6. Decide whether one hole mass is enough

At a zone-center valence edge, two branches are exactly degenerate, split linearly under strain, and mix strongly away from a high-symmetry axis. A fit along that axis reports one “heavy-hole mass.” List the minimum additional information needed before using that number in a three-dimensional conductivity model.

Solution

At minimum one needs:

1. the retained degenerate subspace and basis;
2. the matrix Hamiltonian or symmetry-allowed parameters away from the fitted axis;
3. the strain tensor and its splitting of the branches;
4. the carrier-density and momentum window sampled;
5. direction-dependent velocities or curvature tensors after mixing;
6. scattering and interband-coherence assumptions for conductivity.

One axial curvature is not a global species label or a complete transport mass. The problem requires a multiband hole Hamiltonian before the transport calculation.

7. Route four experimental statements

Assign each statement to the owner needed beyond hole bookkeeping:

1. “The ARPES ridge has a linewidth of $25\,\mathrm{meV}$.”
2. “The Hall coefficient is positive.”
3. “The optical onset is $1.7\,\mathrm{eV}$.”
4. “The hole density at $300\,\mathrm K$ is $4\times10^{16}\,\mathrm{cm}^{-3}$.”

State one reason each number cannot be interpreted from charge $+e$ alone.

Solution

1. Route the linewidth through Spectral Functions and ARPES; it depends on self-energy, matrix elements, surface sensitivity, and resolution.
2. Route the sign through Hall Effect; multiple carriers, mobilities, anisotropy, and field regime enter.
3. Route the onset through Band Gaps and the optical-probe model; directness, selection rules, excitons, and phonons can shift or hide thresholds.
4. Route the density through Metals, Insulators, and Semiconductors; the chemical potential, density of states, dopant ionization, and charge neutrality determine it.

The hole ledger fixes the relative charge and state complement, not these response-specific forward models.

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., Cambridge University Press (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.