Lecture 5: Populated Bands and Electron Kinematics

A band structure describes available one-particle energies, not which states a material has prepared or how those occupied states contribute to motion. This lecture adds population and phase-space geometry to the periodic eigenproblem, turning bands into a Fermi surface, particle count, density of states, velocity, and curvature. Those same structures later control screening, one-particle spectra, pairing, Hall state counting, and reservoir transport, but with different preparations and observables. A spectrum is a space of possibilities; physical kinematics begins only after a state or statistical preparation assigns weight within that space.

An alkali metal makes the missing information visible. Its low-energy conduction states are often close enough to a single broad, nearly free band that a sphere in momentum space is a useful first model. Even there, the radius of the sphere is not fixed by the dispersion alone: it follows from how many electrons occupy the band. To test the spherical model’s limits, we will also use an explicitly declared ellipsoidal band. Its directional masses distinguish particle count from velocity and curvature. The isotropic gas supplies the accounting; the ellipsoid tests which conclusions depend on rotational symmetry.

Three experiments read different pieces of this populated geometry. ARPES maps occupied removal-energy features, quantum oscillations select extremal cross-sectional areas of the Fermi surface, and the low-temperature electronic heat capacity weights the density of states near the Fermi energy. A correct model must connect all three to the same band and occupation convention, but they do not generally return one interchangeable “effective mass.”

We will populate the isotropic anchor from normalized probability weights, count its states, derive its density of states and velocities, and then test which conclusions survive the anisotropy of the declared ellipsoidal model.

Temperature, chemical potential, and equilibrium occupation are derived locally from an explicit comparison of probabilities. The same band then returns as a density of states, a Fermi surface, and a source of wave-packet velocity and curvature.

The organizing question for this lecture is therefore

Given a Bloch band and a preparation specified by temperature and chemical potential, which states are occupied, how many particles result, and what motion follows from the dispersion alone?

The answer has practical consequences. Low-temperature electronic transport and the electronic heat capacity are controlled mainly by a narrow region about a Fermi surface even though the solid contains electrons at much lower energies, and the location of that region is set by the preparation rather than by the dispersion. The three probes named above therefore have to be read against one occupation convention before any of them can be compared with a band calculation.

Use Lecture 4’s hybridization plot, panel (a): the upper solid branch at ${g/t=0.4}$ , with horizontal coordinate ${ka/\pi\in[0,1]}$ and vertical coordinate ${E/t}$ , where ${t>0}$ . Treat the endpoints as points of the smooth periodic band, not hard-wall boundaries. Perform four actions before using any thermal formula.

Mark a minimum, a maximum, and a point with nonzero slope.
Predict the sign of the group velocity at the sloping point.
State whether positive or negative curvature occurs at each extremum.
Explain what additional information is needed to decide which states are occupied.

The first three are band-geometry questions. The fourth exposes this lecture’s need: a Hamiltonian supplies possible energies, not a preparation.

Preparation populates a band

This section separates the three descriptions that the opening ran together. The Hamiltonian supplies the available bands; an equilibrium preparation supplies their occupations; and, once the occupied band has been counted, a wave packet built from one band supplies collisionless motion. The first two descriptions are developed in this section and the next, and each carries its validity statement at the point where it is used.

Reservoir weights and the Fermi function

Four objects must remain separate as the calculation develops. The band is a one-particle state family and dispersion; the preparation assigns occupations; the Fermi surface is a level set of that occupied-state problem; and a wavepacket is a coherent superposition narrow enough in crystal momentum for semiclassical motion. Effective mass will enter as a local Hessian controlling acceleration, not as a literal replacement for the electron’s bare mass. Transport still requires a nonequilibrium occupation and a relaxation model, so no current will be inferred from the dispersion alone. The group-velocity law derived here is the dispersion-only limit; geometric corrections that can appear for a band with nontrivial state-space geometry are identified as a boundary here. Their ingredients are supplied later by local-basis connection language and projector Berry curvature, without importing those later calculations into the present kinematic model.

Let the spatial dimension be ${d\in\{1,2,3\}}$ , let ${\mathrm{BZ}=\mathbb R^d/\Lambda^*}$ be the Brillouin torus, and let

$\varepsilon_n:\mathrm{BZ}\longrightarrow\mathbb R$

be the energy function of band ${n}$ . The variable ${\mathbf k\in\mathrm{BZ}}$ is a bold-upright classical wave vector with units of inverse length. We retain the exact Lecture 4 definitions of Bloch Hamiltonian, signed indirect separation, and indirect insulating gap. We initially neglect interactions; state counting and the equilibrium occupation derived below are then exact within this declared model. In every general state-counting formula, the band index ${n}$ is resolved: it includes spin and every other retained internal label. When a spin-degenerate orbital band is written only once, its degeneracy is instead inserted explicitly as ${g_{\mathrm s}=2}$ ; these two conventions must never be used simultaneously.

Suppose a large environment can exchange energy and electrons with the solid. Two possible preparations ${r}$ and ${s}$ have energies ${E_r,E_s}$ , electron numbers ${N_r,N_s}$ , and probabilities ${P_r,P_s}$ . Here ${P_r}$ is a dimensionless probability assigned to a discrete alternative, not a probability density and not a wave function.

Why should equilibrium assign the relative weight used below? Let the large environment have multiplicity ${\Omega_{\mathrm{env}}(E,N)}$ and entropy ${S_{\mathrm{env}}(E,N)=k_{\mathrm B}\ln\Omega_{\mathrm{env}}(E,N)}$ . Here ${N}$ is treated through the smooth large-system interpolation of the integer particle number, and reservoir volume and other external controls are held fixed.

Let ${D\subset\mathbb R^2}$ be an open region of reservoir energy–particle- number states on which ${S_{\mathrm{env}}:D\to\mathbb R}$ is differentiable and ${{(\partial S_{\mathrm{env}}/\partial E)}_N>0}$ . At a reference state ${(E,N)\in D}$ , the reservoir temperature ${T(E,N)>0}$ and chemical potential ${\mu(E,N)\in\mathbb R}$ are defined locally by $\frac1T :=\left(\frac{\partial S_{\mathrm{env}}}{\partial E}\right)_N, \qquad \mu :=-T\left(\frac{\partial S_{\mathrm{env}}}{\partial N}\right)_E, \qquad \beta:=\frac1{k_{\mathrm B}T}.$ Thus ${T}$ , ${\mu}$ , and ${\beta}$ are scalar functions of the declared reservoir state and fixed external controls; they are not additional one-particle band energies.

Fix total energy ${E_{\mathrm{tot}}}$ and particle number ${N_{\mathrm{tot}}}$ . Assume weak subsystem–environment coupling, equiprobability of compatible combined microstates, and a twice- differentiable reservoir entropy near the reference state. The combined microstates compatible with subsystem alternative ${r}$ are counted by ${\Omega_{\mathrm{env}}(E_{\mathrm{tot}}-E_r, N_{\mathrm{tot}}-N_r)}$ . Equiprobability of those combined microstates gives

$\ln\frac{P_s}{P_r} =\frac{S_{\mathrm{env}}(E_{\mathrm{tot}}-E_s, N_{\mathrm{tot}}-N_s) -S_{\mathrm{env}}(E_{\mathrm{tot}}-E_r, N_{\mathrm{tot}}-N_r)}{k_{\mathrm B}}.$

Choose the environment compatible with alternative ${r}$ as reference: ${\mathbf X_r=(E_{\mathrm{tot}}-E_r,N_{\mathrm{tot}}-N_r)}$ , and write ${\Delta\mathbf X=(E_s-E_r,N_s-N_r)}$ . The environment for ${s}$ is ${\mathbf X_r-\Delta\mathbf X}$ . Its first-order entropy change is $-\nabla S_{\mathrm{env}}(\mathbf X_r)\mathbin{\cdot}\Delta\mathbf X =-\frac{E_s-E_r}{T}+\frac{\mu(N_s-N_r)}{T}.$ The exact second-order integral remainder along this segment is $\mathcal R_{sr} =\int_0^1(1-z)\, \Delta\mathbf X^{\mathsf T} H_S(\mathbf X_r-z\Delta\mathbf X)\Delta\mathbf X\,\mathrm dz,$ where ${H_S}$ is the reservoir entropy Hessian and the segment must remain inside its differentiable thermodynamic region. Dividing by ${k_{\mathrm B}}$ gives $\ln\frac{P_s}{P_r} =-\beta\left[(E_s-E_r)-\mu(N_s-N_r)\right] +\frac{\mathcal R_{sr}}{k_{\mathrm B}},$ where ${\mathcal R_{sr}}$ is second order in the exchanges. Assume an extensive reservoir entropy, ${S_{\mathrm{env}}(E,N,V)=V s(E/V,N/V)}$ , with bounded second derivatives of ${s}$ at the fixed densities under consideration. Then ${H_S=O(V^{-1})}$ ; for bounded exchanges, ${\mathcal R_{sr}=O(V^{-1})}$ vanishes as the reservoir grows. This estimate requires a regular thermodynamic region and cannot simply be carried through a singular susceptibility. A large finite reservoir therefore gives the first-order approximation $\frac{P_s}{P_r} \simeq \exp\!\left[-\beta\left((E_s-E_r)-\mu(N_s-N_r)\right)\right].$ In the ideal thermodynamic-reservoir limit adopted for the remainder of this lecture, the neglected remainder vanishes and the operational rule is

(5.1)

$\frac{P_s}{P_r} =\exp\!\left[-\beta\left((E_s-E_r)-\mu(N_s-N_r)\right)\right], \qquad \beta=\frac{1}{k_{\mathrm B}T}.$

Temperature ${T>0}$ is measured in kelvin, Boltzmann’s constant ${k_{\mathrm B}}$ converts it to an energy scale, and ${\beta}$ has units of inverse energy. The chemical potential ${\mu}$ has units of energy and records the exchange cost of one electron. This lecture uses the rule operationally: it compares stated alternatives and then normalizes their probabilities. Canonical and grand-canonical ensembles, density operators, traces, and partition functions are constructed in Lectures 9–10, which show why the same ratio follows from them.

The sign in the exponential has an immediate physical test. At fixed particle number, if ${E_s>E_r}$ , cooling increases ${\beta}$ and suppresses ${P_s/P_r}$ . If the formula predicted the reverse, it would populate higher energies more strongly as the system cooled.

Filling one resolved fermion level

Determine the empty and occupied probabilities for one resolved fermion state of energy ${\varepsilon}$ . The allowed occupations are ${n=0,1}$ , with probabilities ${p_0,p_1}$ . At ${T>0}$ , the reservoir supplies ${\beta=1/(k_{\mathrm B}T)}$ , chemical potential ${\mu}$ , and the rules

$\frac{p_1}{p_0}=\mathrm e^{-\beta(\varepsilon-\mu)}, \qquad p_0+p_1=1.$ Find the mean occupation as well.

Pauli exclusion fixes the two alternatives, but the reservoir rule and normalization fix their probabilities. Solve those two equations together.

Set ${w:=\mathrm e^{-\beta(\varepsilon-\mu)}>0}$ . Then $p_1=wp_0, \qquad (1+w)p_0=1, \qquad p_0=\frac1{1+w}, \qquad p_1=\frac{w}{1+w} =\frac1{\mathrm e^{\beta(\varepsilon-\mu)}+1}.$ The ${+1}$ is therefore the normalization of the empty and occupied alternatives, not an unexplained correction to a Boltzmann factor. Solving these equations gives the Fermi occupation.

Both probabilities lie between zero and one and their sum is one. When ${\varepsilon>\mu}$ , ${0<w<1}$ , so ${p_1<p_0}$ : raising the energy relative to the exchange cost suppresses occupancy. At ${\varepsilon=\mu}$ , both probabilities are ${1/2}$ for every ${T>0}$ . At fixed nonzero ${\varepsilon-\mu}$ , taking ${T\to0^+}$ gives occupation zero above ${\mu}$ and one below it, an independent boundary check. The mean is ${\langle n\rangle=0p_0+1p_1=p_1}$ . This construction assigns a probability to one available mode; it does not yet count how many modes a band contains. Its first boundary is the model: the two alternatives must describe one resolved fermionic level with the stated reservoir ratio.

Fix ${T>0}$ and ${\mu\in\mathbb R}$ , and set ${\beta={(k_{\mathrm B}T)}^{-1}}$ . The Fermi occupation function is the ordinary function $f_{T,\mu}:\mathbb R\longrightarrow(0,1), \qquad f_{T,\mu}(\varepsilon) =\frac{1}{\mathrm e^{\beta(\varepsilon-\mu)}+1},$ whose value is the mean occupation of one resolved fermionic mode of energy ${\varepsilon}$ . Its zero-temperature extension is ${f_{0,\mu}(\varepsilon)=1}$ for ${\varepsilon<\mu}$ and ${f_{0,\mu}(\varepsilon)=0}$ for ${\varepsilon>\mu}$ . The value assigned at ${\varepsilon=\mu}$ does not affect state-count integrals only when states at exactly that energy carry zero counting weight. A degenerate sector with nonzero weight, including a flat band at ${\mu}$ , requires its preparation or limiting prescription to be stated. The object ${f_{T,\mu}}$ is a function of mode energy, not a probability density over energy.

Historical lineage

[Fer26, pp. 902–912] — Fermi’s 1926 paper gives the earlier published exclusion-principle route to ideal-gas quantum statistics.
[Dir26, pp. 667–673] — Dirac’s later paper develops a distinct antisymmetric many-particle route; “distinct” here does not assert no prior awareness.
[PI22, pp. 14–15] — Pérez and Ibáñez compare the chronology and surviving evidence, including the reason to treat the two routes as historically related.

When ${T}$ and ${\mu}$ are fixed and no ambiguity can result, we abbreviate ${f_{T,\mu}}$ to ${f}$ .

The function ${f}$ is the mean occupation of a resolved state. Because the only possible occupations are zero and one, ${\langle n\rangle=0p_0+1p_1=p_1}$ ; in this special two-alternative case the mean equals the probability of occupancy. It is not a distribution of the positions or energies of labeled electrons. That distinction will matter for bosons, whose mean occupation can exceed one.

Before introducing the state count, write a response to this proposed normalization: ${\int f(\varepsilon)\,\mathrm d\varepsilon=1}$ . Which object was actually normalized in Example 5.1? Use units to test the proposal and name the additional information needed to turn an occupation function into a particle density. Compare only after recording your explanation.

Three limits give more information than the memorized curve.

At ${\varepsilon=\mu}$ , empty and occupied alternatives have equal weight, so ${f=1/2}$ at every positive temperature.
If ${\varepsilon-\mu\gg k_{\mathrm B}T}$ , occupation is exponentially suppressed: ${f\simeq\mathrm e^{-\beta(\varepsilon-\mu)}}$ .
As ${T\to0}$ , ${f}$ approaches the Heaviside function ${\Theta(\mu-\varepsilon)}$ away from the jump.

For the last limit, the exponential tends to infinity above ${\mu}$ and to zero below it. The equality point remains ${1/2}$ at every positive temperature. Its value is immaterial for regular dispersive pockets whose states at exactly ${\mu}$ have zero counting weight, but not for a flat band there. In the measure notation introduced below, the condition is ${\mathcal N_{\mathrm{DOS}}(\{\mu\})=0}$ . Fixed- ${\mu}$ thermal continuation selects ${1/2}$ even in a sector of nonzero weight; a different preparation or a different limiting path ${\mu(T)}$ must be declared.

Cover the one-level calculation. An occupied level is three times as likely as its empty alternative. Complete ${p_1=3p_0}$ , ${p_0+p_1=1}$ , and ${\langle n\rangle=0p_0+1p_1}$ . State the sign of ${\varepsilon-\mu}$ consistent with this preparation at ${T>0}$ , and check the normalization before comparing your response.

At zero temperature, the chemical potential of a normal metal is denoted ${E_{\mathrm F}}$ . Before taking its boundary, define how much of a declared band subspace is populated.

From occupation to filling and the Fermi surface

Consider a periodic sample with ${N_{\mathrm c}}$ primitive cells and the discrete set ${K_N}$ of its ${N_{\mathrm c}}$ allowed crystal momenta, as declared in Lecture 4 for the finite ring. Let ${\mathcal I}$ be a finite set of resolved band labels, including every retained spin or internal copy. Its capacity is ${N_{\mathrm c}|\mathcal I|}$ one-particle states. The expected particle count per primitive cell and the normalized band filling are $\nu_{\mathcal I}(T,\mu) =\frac{1}{N_{\mathrm c}} \sum_{n\in\mathcal I}\sum_{\mathbf k\in K_N} f_{T,\mu}[\varepsilon_n(\mathbf k)], \qquad \phi_{\mathcal I}(T,\mu) =\frac{\nu_{\mathcal I}(T,\mu)}{|\mathcal I|}\in[0,1].$ Thus ${\phi_{\mathcal I}=0}$ and ${1}$ mean empty and full relative to the declared capacity. At ${T=0}$ , replace ${f_{T,\mu}}$ by its step-function extension. If filling is instead normalized per volume, area, formula unit, or orbital, that alternative denominator must be stated explicitly.

Applying the zero-temperature step to a band defines an occupied region in momentum space. At a regular Fermi energy, its boundary is the following level set. At a critical energy, a level-set point need not bound the strict sublevel region: a local minimum at that energy can be isolated from the occupied region even when other momenta in the band are occupied.

For differentiable resolved band functions ${\varepsilon_n:\mathrm{BZ}\to\mathbb R}$ at zero temperature and chemical potential ${E_{\mathrm F}}$ , the Fermi surface is $\mathcal I_{\mathrm{pf}} =\left\{n: \min_{\mathbf k\in\mathrm{BZ}}\varepsilon_n(\mathbf k) <E_{\mathrm F}< \max_{\mathbf k\in\mathrm{BZ}}\varepsilon_n(\mathbf k) \right\}, \qquad \mathcal S_{\mathrm F} =\bigcup_{n\in\mathcal I_{\mathrm{pf}}} \left\{\mathbf k\in\mathrm{BZ}: \varepsilon_n(\mathbf k)=E_{\mathrm F}\right\},$ where the union is restricted to the declared one-particle model. An empty or completely filled isolated band contributes no component. Equality of ${E_{\mathrm F}}$ with a band extremum is a critical onset or disappearance, not a regular Fermi surface under this convention, and must be named separately. At a regular point the set is a codimension-one surface; at a band touching, saddle, or other critical point it may be nonsmooth or change topology.

Let ${\varepsilon_n}$ be continuously differentiable on the Brillouin torus. If ${E_{\mathrm F}}$ is a regular value—equivalently, ${\nabla_{\mathbf k}\varepsilon_n(\mathbf k)\neq\mathbf0}$ at every point of ${\varepsilon_n^{-1}(E_{\mathrm F})}$ —then the regular-level-set theorem makes that component locally a continuously differentiable codimension-one submanifold. If the gradient vanishes on the level set, this conclusion fails: the boundary can be nonsmooth, components can merge, or a Lifshitz change of topology can occur. For the three-dimensional parabolic anchor with ${E_{\mathrm F}>0}$ , the gradient is nonzero on ${|\mathbf k|=k_{\mathrm F}}$ , so the Fermi surface is a smooth sphere.

Fermi surfaces and quantum oscillations

[Som28, pp. 1–5] — Sommerfeld applies Fermi statistics to the electron gas in metals.
[Yu17, PDF p. 20] — A Euclidean regular-level-set statement supports the local smooth-surface step; the Brillouin-torus formulation above is the course’s chartwise adaptation.
[Ons52, pp. 1007–1008] — Onsager relates magnetic oscillations to closed orbits and extremal Fermi-surface cross sections.
[LK56, pp. 640–644] — Lifshitz and Kosevich develop the extremal-area amplitude and temperature refinement.
[MS99, pp. 2147, 2149] — Mikitik and Sharlai give a later, model-dependent Berry-phase correction for orbits linked with a band-contact line; it is not a universal replacement for Onsager’s relation.

A completely filled isolated band contains electrons but has no Fermi surface. Conversely, a band plot with a curve crossing a chosen horizontal line has not established a physical Fermi surface until the particle number or chemical potential fixes that line.

For the nearly spherical alkali-metal anchor, this level set becomes a sphere of radius ${k_{\mathrm F}}$ . Quantum oscillations do not photograph that sphere. As the magnetic field selects cyclotron orbits, an oscillation frequency constrains an extremal cross-sectional area perpendicular to the field. Rotating the field therefore tests shape. An ellipsoidal distortion can produce direction-dependent areas even when the enclosed volume—and hence the electron count in a one-band description—remains fixed. This is why “the Fermi wave vector” is safe for the isotropic model and potentially misleading for a real multiorbital metal.

Counting how many states live near an energy

The occupation answers how much one resolved state is populated. A particle density also needs the number of available states.

States per momentum volume

For a macroscopic ${d}$ -dimensional periodic solid, let every ${n}$ denote one resolved band, including spin and other internal degeneracies. The density-of-states measure per physical ${d}$ -volume is the Borel measure ${\mathcal N_{\mathrm{DOS}}}$ on energy defined for each Borel set ${A\subset\mathbb R}$ by the pushforward $\mathcal N_{\mathrm{DOS}}(A) =\sum_n\int_{\mathrm{BZ}} \frac{\mathrm d^d k}{{(2\pi)}^d} \mathbf1_A[\varepsilon_n(\mathbf k)].$ When ${\mathcal N_{\mathrm{DOS}}}$ is absolutely continuous with respect to the energy Lebesgue measure, its Radon–Nikodym density ${g=\mathrm d\mathcal N_{\mathrm{DOS}}/\mathrm d\varepsilon}$ is the density of states, written distributionally as $g(\varepsilon) =\sum_n\int_{\mathrm{BZ}} \frac{\mathrm d^d k}{{(2\pi)}^d} \delta\!\left(\varepsilon-\varepsilon_n(\mathbf k)\right).$ It has units of states per energy per physical ${d}$ -volume. A per-cell, per-area, or per-spin DOS is a different normalization and must be labeled. For a finite isolated sample the exact measure is discrete and is represented by a sum of Dirac distributions; a broadened plot is an ordinary function obtained only after a stated smoothing prescription.

The Dirac ${\delta}$ is a distribution, not an ordinary function. For a smooth compactly supported test function ${\varphi:\mathbb R\to\mathbb R}$ ,

$\left\langle\delta(\varepsilon-\varepsilon_0),\varphi\right\rangle =\varphi(\varepsilon_0).$

In the momentum integral it selects the constant-energy surface and weights it by the inverse band slope. The final ${g}$ is ordinarily a nonnegative, locally integrable function after the continuum limit or physical broadening has been taken; for a finite isolated crystal, the exact spectrum is instead a sum of delta distributions. These objects must not be conflated merely because both are written against an energy axis.

A finite-bin DOS estimate. To separate momentum sampling from energy binning, consider the dimensionless toy dispersion ${x=q^2}$ on the single branch ${0\leq q<2}$ . This example counts the measure ${\mathrm dq}$ , not the spin-inclusive three-dimensional state density. Sample ${q_m=(m+\tfrac12)\Delta q}$ , with ${m=0,\ldots,M-1}$ and ${\Delta q=2/M}$ . For energy bins ${[x_j,x_{j+1})}$ of equal width ${\Delta x}$ , let ${C_j}$ count the sampled energies in that bin. The piecewise-constant estimate is $\rho_j=\frac{C_j\Delta q}{\Delta x},\qquad \sum_j\rho_j\Delta x=2,$ where the bins cover the full interval ${[0,4]}$ . The height is a density; the area, not the height, is the counted weight. For a bin within that interval, the continuum comparison is its exact bin average $\overline\rho_j =\frac{\sqrt{x_{j+1}}-\sqrt{x_j}}{\Delta x}.$ The first bin has finite weight even though the pointwise continuum density ${1/(2\sqrt{x})}$ diverges at the band bottom. Such a nonanalytic point of a density of states is a Van Hove singularity, named for Van Hove’s 1953 analysis of nonanalytic structure in the vibrational frequency distribution of a periodic harmonic crystal [Hov53, publisher abstract and printed pp. 1189–1193]; the electronic usage transfers the same critical-point geometry to a band-energy density of states, and the name does not imply that every peak in a plotted DOS is singular. Narrowing bins at fixed ${M}$ exposes sparse discrete levels; it does not by itself produce a smooth continuum DOS. Increasing ${M}$ at a fixed bin width instead tests convergence to the displayed bin average. The angular and spin factors of the physical three-dimensional count are supplied separately below.

Figure 5.1. Finite-bin state counting for the toy dispersion. Uniform midpoint samples in q map to energy bins; bar area counts momentum weight. The continuum comparison gives bin averages, not pointwise values at the singular band bottom. In HTML, change sample count and bin count separately and explain why narrowing bins at fixed sample count need not improve smoothness.

The occupied number and energy densities are

(5.2)

$n_{\mathrm e} =\int_{-\infty}^{\infty}\mathrm d\varepsilon\, g(\varepsilon)f(\varepsilon), \qquad u_{\mathrm e} =\int_{-\infty}^{\infty}\mathrm d\varepsilon\, \varepsilon g(\varepsilon)f(\varepsilon).$

The units expose a frequent error. With ${g}$ measured in states per energy per volume, ${g\,\mathrm d\varepsilon}$ is a number density and ${\varepsilon g\,\mathrm d\varepsilon}$ is an energy density. If a spin factor has already been included in ${g}$ , a second factor of two double counts the states.

The shell Jacobian

For a parabolic pocket, a thin shell makes the two counting operations explicit: count momenta in its volume, then convert its thickness from wave vector to energy. The occupation step will fill the resulting state measure.

Find the density of available states per energy and the Fermi energy at a specified number density ${n_{\mathrm e}}$ at ${T=0}$ . Take one isotropic three-dimensional band

$\varepsilon(\mathbf k)=\frac{\hbar^2{|\mathbf k|}^2}{2m^*}, \qquad m^*>0,$

with two spin states. The function is defined on a region where the parabolic approximation is valid; it is not asserted across an entire real Brillouin zone. Use a periodic box of side lengths ${L_i}$ and volume ${V}$ , and assume the occupied pocket remains within that regime. Work in the large-volume continuum limit: a momentum-volume count approximates the finite box’s discrete staircase.

Count the allowed momenta before changing variables to energy. In the periodic box, allowed momenta are separated by ${2\pi/L_i}$ . Thus each state occupies momentum volume ${{(2\pi)}^3/V}$ , and the continuum count inside a sphere of radius ${k}$ is

$\frac{N(k)}{V} =2\frac{1}{{(2\pi)}^3}\frac{4\pi k^3}{3} =\frac{k^3}{3\pi^2}.$

Figure 5.2. Counting cells in a three-dimensional momentum sphere. The illustrated radius is the Fermi radius when the sphere is filled. One octant is cut away only for visibility; the count uses the full sphere. Orange denotes one allowed-momentum cell, with spin counted separately. Cubic grid cells illustrate equal box lengths; the volume formula applies also to unequal lengths.

Because ${k(\varepsilon)=\sqrt{2m^*\varepsilon}/\hbar}$ , differentiation must transform both the spherical shell and its radial coordinate. Equivalently,

(5.3)

$g_{\mathrm{3D}}(\varepsilon) =2\frac{4\pi k^2}{{(2\pi)}^3}\frac{\mathrm dk}{\mathrm d\varepsilon}, \qquad \frac{\mathrm dk}{\mathrm d\varepsilon} =\frac{m^*}{\hbar^2k}.$

Substituting ${k=\sqrt{2m^*\varepsilon}/\hbar}$ gives

$g_{\mathrm{3D}}(\varepsilon) =\frac{1}{2\pi^2} {\left(\frac{2m^*}{\hbar^2}\right)}^{3/2} \sqrt{\varepsilon}\,\Theta(\varepsilon).$

Before reading Figure 5.3, use Equation (5.3) to identify which factor comes from the number of states in a shell and which comes from changing the shell coordinate from ${k}$ to energy. The two factors must be transformed together.

Figure 5.3. A momentum shell contributes a surface-area factor times its radial thickness in three dimensions; changing variables through the parabolic dispersion turns that shell count into the square-root density of states.

The annulus in panel (a) is a central section of a three-dimensional spherical shell, not a two-dimensional state count. Panel (b) records the same states after the change of variables: ${g(\varepsilon)\,\mathrm d\varepsilon=\mathrm dN/V}$ . This picture illustrates the two state-counting factors above; it is not an independent derivation.

The dimensions check before integration: ${m^*/\hbar^2}$ has units ${{(\text{energy}\times\text{length}^2)}^{-1}}$ , so the equivalent form ${g_{\mathrm{3D}}=m^*k/(\pi^2\hbar^2)}$ has units ${{(\text{energy}\times\text{volume})}^{-1}}$ . At ${T=0}$ , use ${\int_0^{E_{\mathrm F}}\sqrt{\varepsilon}\,\mathrm d\varepsilon ={[2\varepsilon^{3/2}/3]}_0^{E_{\mathrm F}}}$ to obtain

$n_{\mathrm e}=\int_0^{E_{\mathrm F}}g_{\mathrm{3D}}(\varepsilon) \,\mathrm d\varepsilon =\frac{1}{3\pi^2} {\left(\frac{2m^*E_{\mathrm F}}{\hbar^2}\right)}^{3/2} =\frac{k_{\mathrm F}^3}{3\pi^2}, \qquad k_{\mathrm F}={(3\pi^2n_{\mathrm e})}^{1/3}, \qquad E_{\mathrm F}=\frac{\hbar^2k_{\mathrm F}^2}{2m^*}.$

The density integral therefore agrees with counting the occupied sphere directly. The shell factor measures available modes, while Definition 5.2 selects which are occupied. Both results require the occupied pocket to remain inside the parabolic regime; extrapolating this unbounded model through an entire lattice band would invent extra states.

Close the derivation for seven minutes and reconstruct the three-dimensional density of states from scratch: the sphere-counting proportionality ${g_{\mathrm{3D}}(\varepsilon)\propto\sqrt{\varepsilon}}$ , the spin-degeneracy and momentum-cell factors that fix the prefactor, and the units that check it; then recover ${n_{\mathrm e}\propto k_{\mathrm F}^3}$ by integrating the reconstruction up to ${E_{\mathrm F}}$ . A wrong power of energy is repaired at the Jacobian; a wrong prefactor is repaired at the degeneracy or the ${2\pi}$ counting — locate the step before rereading it.

Cover Example 5.2. For one spin species the spherical count is ${N(k)/V=k^3/(6\pi^2)}$ . Starting from ${\varepsilon=\hbar^2k^2/(2m^*)}$ , complete the radial Jacobian and ${g(\varepsilon)=[k^2/(2\pi^2)]\,\mathrm dk/\mathrm d\varepsilon}$ . Check units and state which factor changes when the second spin species is restored. Record the intermediate expression before comparing.

The same counting operation changes in two dimensions. With spin included, a disk gives $\frac{N(k)}{A}=2\frac{\pi k^2}{{(2\pi)}^2}=\frac{k^2}{2\pi}, \qquad g_{\mathrm{2D}}(\varepsilon) =2\frac{2\pi k}{{(2\pi)}^2}\frac{m^*}{\hbar^2k}.$ The circumference factor cancels the inverse radial slope, giving

$g_{\mathrm{2D}}(\varepsilon) =\frac{m^*}{\pi\hbar^2}\Theta(\varepsilon)$

including spin. The power of energy is a geometric consequence of the dimension, not a universal signature of a parabolic band.

An ellipsoid changes the counting volume

The spherical calculation separates geometry from occupation. To test which part depended on isotropy, keep the same two spin states and equilibrium preparation but stretch the pocket in different directions.

Determine the carrier density at a specified Fermi energy after the spherical pocket becomes anisotropic. Retain two spin states, ${T=0}$ , and ${E_{\mathrm F}>\varepsilon_{\mathrm c}}$ . Obtain the density of states as an intermediate result. In principal-axis coordinates take $\varepsilon(\mathbf k)=\varepsilon_{\mathrm c} +\frac{\hbar^2}{2}\left(\frac{k_x^2}{m_x} +\frac{k_y^2}{m_y}+\frac{k_z^2}{m_z}\right), \qquad m_x,m_y,m_z>0.$ The origin is the pocket minimum and the quadratic approximation is used only over its valid energy range, including the occupied pocket. Retain the large-volume continuum limit of Example 5.2.

Transform the ellipsoid to a sphere while retaining the change in momentum volume. If the positive quadratic form is initially nondiagonal, an orthogonal principal-axis rotation preserves momentum volume; the unequal eigenvalues then give the displayed masses. We reuse the momentum-cell normalization of Example 5.2, not its spherical shape in the original coordinates.

Choose a reference mass ${m_0>0}$ and set $q_i=\sqrt{\frac{m_0}{m_i}}\,k_i, \qquad \varepsilon-\varepsilon_{\mathrm c} =\frac{\hbar^2|\mathbf q|^2}{2m_0}.$ Both ${q_i}$ and ${k_i}$ have inverse-length units. Before evaluating the measure, predict whether increasing ${m_x}$ enlarges or reduces the original momentum volume below a fixed energy. The inverse transformation and its Jacobian give $k_i=\sqrt{\frac{m_i}{m_0}}\,q_i, \qquad \left|\det\frac{\partial\mathbf k}{\partial\mathbf q}\right| =\sqrt{\frac{m_xm_ym_z}{m_0^3}}, \qquad \mathrm d^3k=\sqrt{\frac{m_xm_ym_z}{m_0^3}}\,\mathrm d^3q.$ For ${\xi=\varepsilon-\varepsilon_{\mathrm c}>0}$ , the radius in the transformed coordinates is ${Q=\sqrt{2m_0\xi}/\hbar}$ . The count below that energy is therefore $\frac{N(\xi)}{V} =\frac{2}{{(2\pi)}^3}\sqrt{\frac{m_xm_ym_z}{m_0^3}} \frac{4\pi Q^3}{3} =\frac{\sqrt{m_xm_ym_z}}{3\pi^2} {\left(\frac{2\xi}{\hbar^2}\right)}^{3/2}.$ The arbitrary ${m_0}$ cancels. Differentiation now yields $g_{\mathrm{ell}}(\varepsilon) =\frac{\sqrt{m_xm_ym_z}}{2\pi^2} {\left(\frac{2}{\hbar^2}\right)}^{3/2} \sqrt{\varepsilon-\varepsilon_{\mathrm c}}\, \Theta(\varepsilon-\varepsilon_{\mathrm c}).$ At ${T=0}$ , for ${\xi_{\mathrm F}=E_{\mathrm F}-\varepsilon_{\mathrm c}>0}$ , the occupation step fills precisely this ellipsoid: $n_{\mathrm e}=\int_{\varepsilon_{\mathrm c}}^{E_{\mathrm F}} g_{\mathrm{ell}}(\varepsilon)\,\mathrm d\varepsilon =\frac{\sqrt{m_xm_ym_z}}{3\pi^2} {\left(\frac{2\xi_{\mathrm F}}{\hbar^2}\right)}^{3/2}.$

There are three useful checks. Setting all masses to ${m^*}$ recovers the spherical formulas at the shifted energy ${\xi}$ . Independently, the ellipsoid semiaxes are ${k_{{\mathrm F}i}=\sqrt{2m_i\xi_{\mathrm F}}/\hbar}$ ; its volume ${4\pi k_{{\mathrm F}x}k_{{\mathrm F}y}k_{{\mathrm F}z}/3}$ times ${2/{(2\pi)}^3}$ gives the same density. Finally, ${\sqrt{m_xm_ym_z}/\hbar^3}$ has units ${\text{energy}^{-3/2}\text{length}^{-3}}$ , so ${g_{\mathrm{ell}}}$ has the required inverse-energy, inverse-volume units.

One may summarize the count by ${m_{\mathrm{DOS}}={(m_xm_ym_z)}^{1/3}}$ , because ${m_{\mathrm{DOS}}^{3/2}=\sqrt{m_xm_ym_z}}$ . This single number retains the volume dependence but loses the three semiaxes. It therefore cannot determine directional motion. The construction also fails for a saddle: a negative principal curvature does not enclose the finite ellipsoid used in the count, and the surrounding band and its cutoff become essential.

Cover the worked ellipsoid calculation. Let ${(m_x,m_y,m_z)=(4m_0,m_0/4,m_0)}$ . Complete the three inverse coordinate rescalings and their determinant. At the same positive ${E_{\mathrm F}-\varepsilon_{\mathrm c}}$ , compare the particle count and the three semiaxes with the isotropic pocket of mass ${m_0}$ . Use the semiaxis product as a second check; record the comparison before consulting the response. As a check on the direction of the Jacobian, change only ${m_x}$ to ${8m_0}$ and repeat the determinant.

The next figure returns to the isotropic model so that temperature can change while its geometry is held fixed. The ellipsoid calculation has established which part of the count changes when geometry is changed instead. Directional motion will require additional information from the band derivatives.

Heating a populated band under two constraints

Before viewing the default fixed-chemical-potential state of Figure 5.4, predict the energy at which all Fermi curves cross, the width of the energy interval in which they differ appreciably, and the shape of the occupied zero-temperature region for the isotropic band.

Figure 5.4. Fermi occupation under two preparation constraints. The adjustable green occupation curve uses the left axis; gray available DOS and dark occupied DOS use the right state-density axis. Occupied DOS is available DOS multiplied by occupation. The default holds chemical potential fixed; HTML controls can instead hold particle density fixed. Linked horizontal guides select energies μ ± c kBT (default c = 1). At the default they intersect the green curve at energies μ − kBT (blue) and μ + kBT (orange), with occupations approximately 0.731 and 0.269. Matching contours in panel (b) represent the same energies. The blue surface is ε = μ; it coincides with the zero-temperature reference Fermi sphere in the default free-electron state, where μ = EF0. The dotted reference contour stays at EF0. One octant is cut away for visibility, not physically emptied. The guide contours delimit a conventional thermal-response window, not a sharp finite-temperature occupation boundary. Switching the panel heading reveals the same available and occupied state densities in a separate panel.

The occupation equals one half at ${\varepsilon=\mu}$ , while occupation changes are appreciable mainly within an energy window of order ${k_{\mathrm B}T}$ . The figure does not show spectral broadening: the band states retain their energies while their mean occupations change. The right axis plots the gray available DOS ${gE_{\mathrm F0}/n_0}$ and its dark occupied part ${gfE_{\mathrm F0}/n_0}$ ; the left axis plots occupation ${f}$ . The area under the dark curve over the full energy range is ${n_{\mathrm e}/n_0}$ ; the finite plotted window omits the reported high-energy tail. The alternate density panel shows these same state densities. The momentum view shows the blue constant-energy surface ${\varepsilon=\mu}$ , with a visual octant cutaway, while the dotted contour marks the fixed zero-temperature reference energy ${E_{\mathrm F0}}$ . The blue surface is not a sharp finite-temperature occupation boundary. At fixed density, each temperature has its own ${\mu(T)}$ : the two marked energies move with it. All axes retain the fixed reference scales ${E_{\mathrm F0}}$ and ${k_{\mathrm F0}}$ .

Use that distinction to make a figure decision. Raise ${\mu}$ slightly while holding the plotted DOS fixed, shade the newly occupied spectral weight, and predict the sign of ${\delta n_{\mathrm e}}$ . Then raise ${T}$ while holding ${\mu}$ and the DOS fixed, as the default comparison does, and identify redistribution about ${\mu}$ , not a broadening of the Hamiltonian’s energy levels. This fixed- ${\mu}$ comparison need not conserve particle density. If instead ${n_{\mathrm e}=n_0}$ is held fixed, the chemical potential becomes the implicitly defined function ${\mu(T)}$ satisfying $n_0=\int_{-\infty}^{\infty}\mathrm d\varepsilon\, g(\varepsilon)f_{T,\mu(T)}(\varepsilon).$ The distinction changes the global population constraint but not the local conclusion that only an energy window of order ${k_{\mathrm B}T}$ is substantially rearranged in a degenerate metal.

For a numerical comparison keep a fixed reference density ${n_0}$ and its zero-temperature Fermi energy ${E_{\mathrm F0}>0}$ . In the same spin-inclusive three-dimensional parabolic band, set ${z=\varepsilon/E_{\mathrm F0}}$ , ${\theta=k_{\mathrm B}T/E_{\mathrm F0}}$ , and ${\alpha=\mu/E_{\mathrm F0}}$ . The density integral becomes

(5.4)

$f(z)=\frac{1}{1+\mathrm e^{(z-\alpha)/\theta}},\qquad \frac{n_{\mathrm e}}{n_0}=\frac32\int_0^\infty \sqrt z\,f(z)\,\mathrm dz.$

At fixed chemical potential hold the selected ${\alpha}$ constant; the reference comparison sets ${\alpha=1}$ . At fixed density choose ${\alpha}$ so the integral equals one. The energy scale stays fixed in both preparations. The free-electron comparison below uses ${0.02\leq\theta\leq0.60}$ and an unbounded parabolic model, not a fitted full crystal band. Its plotted window stops at ${z=2.5}$ , but the density readout includes the tail beyond it.

For this comparison, use the figure above.

Predict whether heating at fixed chemical potential preserves the occupied state count. Raise the temperature in that preparation, then switch to fixed density and compare the chemical potential and occupied spectral weight. Panel (a) shows occupation probabilities. In fixed-chemical-potential mode, drag its blue handle horizontally to select ${\mu}$ ; in fixed-density mode, ${\mu(T)}$ is instead determined by the density constraint. Switch panel (b) from the constant-energy surface to available and occupied state densities. For the free-electron model their normalization is given by Equation (5.4); lattice models use their own DOS and reference density. Drag the temperature control or edit its green legend value; drag either horizontal guide to vary the linked window width ${c}$ (default 1). The guide levels ${1/(1+e^{-c})}$ and ${1/(1+e^c)}$ add to one, and their crossings track ${\mu\mp c k_{\mathrm B}T}$ , respectively, corresponding to the inner and outer contours in panel (b). This changes the chosen window, not the prepared distribution. The display range ${0.1\le c\le2}$ is a viewing choice. If the lower energy falls below the band edge, there is no inner crossing or contour. Occupation probability and state density are different quantities.

The model menu also offers a simple-cubic lattice and weakly coupled square layers. Their small real-space diagrams identify nearest-neighbor hopping strengths: ${t_x=t_y=t_z=t}$ , or ${t_x=t_y=t,\ t_z=0.2t}$ , respectively. Both use one spinless orbital per site and ${\epsilon(\mathbf k)=2\sum_i t_i[1-\cos(k_i a)]}$ , measured from the band bottom. The displayed reference energies are ${E_{\mathrm F0}=3.6t}$ and ${3t}$ ; each model has its own zero-temperature reference density ${n_0}$ . These are contrasts with, not assumptions of, the isotropic calculation above. The gray horizontal line ${f=1/2}$ crosses at the current ${\mu}$ , and the blue surface follows the same energy ${\varepsilon=\mu}$ . The dotted reference contour stays at ${E_{\mathrm F0}}$ . A selected energy outside the band’s range has no corresponding constant-energy surface. The tight-binding surface is a numerical mesh; the DOS uses precomputed samples and interpolation, with chemical-potential seeds for the density solve.

For the free-electron model, at fixed chemical potential the particle density increases for this increasing parabolic DOS; at fixed density the chemical potential falls to compensate. The band energies and DOS have not broadened. The density beyond the plotted window is reported separately and included in the total. The static form shows the default fixed-chemical-potential case; the integral defines the alternative.

The thermal window follows from

$-\frac{\partial f}{\partial\varepsilon} =\frac{1}{4k_{\mathrm B}T} \operatorname{sech}^2\!\left( \frac{\varepsilon-\mu}{2k_{\mathrm B}T}\right).$

This is an ordinary positive normalized function for ${T>0}$ . As ${T\to0}$ , it converges to ${\delta(\varepsilon-\mu)}$ as a distribution. Thus a smooth observable weighted by ${-\partial f/\partial\varepsilon}$ samples the Fermi surface. The statement does not mean that every electron has energy ${k_{\mathrm B}T}$ ; most electrons remain deep in the filled sea and are blocked from small changes.

This thermal-window statement connects the density of states to heat capacity. Raising the temperature rearranges occupations only within an energy range of order ${k_{\mathrm B}T}$ about ${E_{\mathrm F}}$ , so the leading electronic heat capacity is controlled by ${g(E_{\mathrm F})}$ , not by the total depth of the filled sea. The numerical coefficient requires a low-temperature expansion that is not derived here; the present conclusion is the physically prior one. ARPES, by contrast, resolves occupied dispersion and spectral weight over momentum and energy, while an oscillation frequency measures an orbit area. Agreement among these probes is therefore a consistency test across spectral weight, geometry, and thermodynamic phase space—not a demand that each yield the same fitted mass.

A calculation for a three-dimensional spin-degenerate parabolic pocket writes $g_{\mathrm{bad}}(\varepsilon) =2\frac{2\pi k}{{(2\pi)}^3}\frac{m^*}{\hbar^2k}.$ Locate the first invalid geometrical step and repair the calculation. Check both the energy power and physical units. A second calculation uses the correct three-dimensional shell but omits the leading spin factor. Explain why units alone miss that error and give a counting check that catches it. Write both repairs before comparing with the instructor response.

The occupied band begins to move

Exact crystal-momentum flow on an infinite lattice

There is an exact statement behind the electric-field part of the packet equations. Take an infinite one-dimensional crystal, no magnetic field or scattering, and a constant force ${F=-eE}$ . On ${L^2(\mathbb R)}$ , let ${\hat H=\hat H_0-F\hat x}$ , where ${\hat H_0=\hat p^2/(2m_{\mathrm e})+V(\hat x)}$ and ${V(x+a)=V(x)}$ . Assume a regular real periodic potential and use a common translation-invariant operator core for the following commutators. This infinite-line problem is not a finite ring with a single-valued linear potential.

Derive position in the Bloch basis first. Write ${\phi_{mk}(x)=e^{ikx}u_{mk}(x)}$ and expand ${\Psi(x)=\sum_m\int_{\mathrm{BZ}}c_m(k)\phi_{mk}(x)\,dk}$ . Choose delta-normalized Bloch states and the corresponding normalized cell inner product, so ${\sum_m\int|c_m|^2dk=1}$ . Differentiating the basis gives $x\phi_{mk}=-i\partial_k\phi_{mk}+i e^{ikx}\partial_k u_{mk}.$ First take smooth coefficients supported within a smooth momentum patch. Integration by parts moves the first derivative onto the coefficient, with no endpoint term. Expanding the second term in the complete cell basis gives ${(\hat Xc)}_n=i\partial_k c_n+\sum_m\mathcal A_{nm}(k)c_m, \qquad \mathcal A_{nm}=i{\langle u_{nk}|\partial_k u_{mk}\rangle}_{\mathrm{cell}}.$ Thus position is a derivative plus the change of the Bloch basis, not just a derivative of the coefficient. The matrix Berry connection ${\mathcal A}$ has length units. For a full-zone state, use consistently sewn Bloch gauges and coefficient functions at the zone boundary; their boundary terms cancel. We restrict to states in the position-operator domain, so the following position mean is defined. The periodic momentum coordinate belongs to the infinite lattice; it does not impose a finite real-space ring.

Recall lattice translation in the convention ${(\hat S_a\psi)(x)=\psi(x+a)}$ . It is unitary and has Bloch eigenvalue ${e^{ika}}$ . An eigenoperator of the commutator map is a nonzero operator ${\hat O}$ satisfying ${[\hat H,\hat O]=\lambda\hat O}$ , on the stated common domain. Here ${\lambda}$ has energy units. Translation, not a globally single-valued crystal-momentum coordinate, is such an operator: $[\hat H_0,\hat S_a]=0,\qquad [\hat x,\hat S_a]=-a\hat S_a,\qquad [\hat H,\hat S_a]=Fa\hat S_a.$ Indeed, ${x\psi(x+a)-(x+a)\psi(x+a)=-a\psi(x+a)}$ . Therefore its Heisenberg evolution is $\hat S_a(t)=e^{i\hat Ht/\hbar}\hat S_a e^{-i\hat Ht/\hbar} =e^{iFat/\hbar}\hat S_a.$ An initial generalized Bloch state in sector ${k_0}$ thus evolves within the moving translation sector $e^{iak(t)}=e^{ia(k_0+Ft/\hbar)},\qquad k(t)=k_0+\frac{Ft}{\hbar}\pmod{2\pi/a}.$ The generalized state is not a normalizable localized particle. For a normalizable superposition, the entire translation-sector probability distribution is shifted around the Brillouin circle. This result is exact for the stated Hamiltonian, even if the field mixes bands within each moving sector. It does not make a Bloch state an energy eigenstate of the tilted Hamiltonian. Nor is ${\hat k}$ itself an eigenoperator: a chosen reduced-zone coordinate has a cut. The derivative ${\hbar\dot k=F}$ describes a continuous lift of the moving phase, not a jump of a fixed label at that cut.

Where the single-band approximation enters. Substituting the derived position operator into ${\hat H=\hat H_0-F\hat X}$ gives the exact all-band coefficient equation $i\hbar\left(\partial_t+\frac{F}{\hbar}\partial_k\right)c_n =\varepsilon_n(k)c_n-F\sum_m\mathcal A_{nm}(k)c_m.$ The derivative on the left follows the moving sector. The off-diagonal terms on the right transfer amplitude between bands; they do not change its translation eigenvalue. Along an isolated band, neglect these terms only when interband transfer is negligible over the time of interest, including passage through the smallest gaps. Then the remaining real diagonal term changes phase, and $\left(\partial_t+\frac{F}{\hbar}\partial_k\right)|c_n|^2=0, \qquad |c_n(k,t)|^2=|c_n(k-Ft/\hbar,0)|^2.$ These equations are exact within the projected one-band model; using that model for a multiband crystal is an approximation. They need no narrow momentum envelope. Local Bloch gauges must be patched consistently at the zone boundary; the probability density and translation phase are unambiguous.

Derive the center velocity. Project onto the isolated band, writing ${\hat X_n=i\partial_k+\mathcal A_{nn}}$ and ${\hat H_n=\varepsilon_n(k)-F\hat X_n}$ . The product rule gives $[\varepsilon_n,\hat X_n]c =\varepsilon_n i\partial_kc-i\partial_k(\varepsilon_n c) =-i\varepsilon_n'c, \qquad \hat v_n\equiv\frac{i}{\hbar}[\hat H_n,\hat X_n] =\frac{\varepsilon_n'(k)}{\hbar}.$ The connection commutes with the scalar band energy, and the force term commutes with position. This one-dimensional projected identity needs no narrow-packet assumption; it is not an all-band velocity identity.

From flow to Bloch oscillation. Write ${w_0(k)=|c_n(k,0)|^2}$ , with unit integral over one Brillouin zone. For a state with a well-defined position mean, the projected one-dimensional velocity and its time integral give, for ${F\ne0}$ , $\langle v\rangle(t)=\int_{\mathrm{BZ}}w_0(k) \frac{\varepsilon_n'(k+Ft/\hbar)}{\hbar}\,dk,$ $\langle x\rangle(t)-\langle x\rangle(0) =\frac1F\int_{\mathrm{BZ}}w_0(k) \left[\varepsilon_n(k+Ft/\hbar)-\varepsilon_n(k)\right]dk.$ Periodicity ${\varepsilon_n(k+2\pi/a)=\varepsilon_n(k)}$ returns the mean position after ${T_{\mathrm B}=2\pi\hbar/(|F|a)}$ . A narrow distribution reduces the velocity integral to ${\varepsilon_n'(k(t))/\hbar}$ , recovering the packet-center picture. For the cosine band below, the changing band slope reverses the velocity even while ${k(t)}$ keeps flowing. A uniform distribution instead has zero mean velocity: exact momentum flow does not guarantee an oscillating center for every preparation. Interband transitions or scattering remove the guarantee of this isolated-band recurrence, and no dc conductivity follows. The general-field extension below requires additional semiclassical conditions; the exact argument here concerns uniform electric forcing in one dimension only.

Figure 5.5. Exact translation-sector flow on an infinite lattice. Write q = ka and let p(q,t) be the probability density with respect to dq, summed over bands. For E > 0 its initial peak at q = −π/2 moves clockwise on the phase circle. The edge peaks at one quarter period are one continuous distribution across the identified zone boundary. No single-band approximation is needed for this sector flow.

Figure 5.6. Different preparations under the same momentum flow. In the isolated cosine band, normalized initial densities proportional to exp[κ cos(q − π/2)] have κ = 20 (narrow), κ = 1 (broad), or κ = 0 (uniform). The plotted mean velocity and displacement are probability-weighted band integrals, not trajectories assigned to a single momentum. Broad weight reduces the mean oscillation; uniform weight gives a stationary mean despite sector flow. Interband transitions and scattering are excluded.

Find crystal wave number, velocity, displacement and the period-averaged velocity of a packet initially at ${(x_0,k_0)}$ . Take one spatial dimension, ${B=0}$ , a nonzero constant field ${E}$ , and

$\varepsilon(k)=\varepsilon_0-2J\cos(ka), \qquad J>0.$

Here ${J}$ is a positive hopping magnitude: Lecture 4’s signed hopping in ${\varepsilon_0+2t_{\mathrm{L4}}\cos ka}$ is ${t_{\mathrm{L4}}=-J}$ , not ${+J}$ . The symbol ${t}$ below denotes time only. Use the collisionless one-band model with electron charge ${-e}$ , ${e>0}$ , and exclude interband tunneling. Here ${x(t),k(t)}$ are packet-center coordinates in the bulk approximation. For a finite sample, assume that the packet’s width and its entire excursion fit inside the sample without appreciable overlap with an edge. A Brillouin-zone boundary is a momentum-coordinate cut, not a physical edge.

Integrate the crystal wave number first, evaluate the band velocity on that trajectory, and then integrate position. The acceleration theorem integrates exactly within the declared collisionless one-band model:

$\hbar\dot k=-eE \quad\Longrightarrow\quad k(t)=k_0-\frac{eE}{\hbar}t.$

The band velocity is periodic rather than unbounded,

$v(t)=\frac{1}{\hbar}\frac{\mathrm d\varepsilon}{\mathrm dk} =\frac{2Ja}{\hbar}\sin\!\left(k_0a-\omega_{\mathrm B}t\right), \qquad \omega_{\mathrm B}=\frac{eEa}{\hbar}.$

The sign of the next integral matters. With ${s}$ a time integration variable, $\int_0^t\sin(k_0a-\omega_{\mathrm B}s)\,\mathrm ds ={\left[\frac{\cos(k_0a-\omega_{\mathrm B}s)}{\omega_{\mathrm B}}\right]}_0^t,$ since differentiating the cosine supplies two minus signs. Multiplying by ${2Ja/\hbar}$ and imposing ${x(0)=x_0}$ gives

$x(t)-x_0 =\frac{2J}{eE} \left[ \cos\!\left(k_0a-\omega_{\mathrm B}t\right)-\cos(k_0a) \right].$

The trajectory returns after

$T_{\mathrm B}=\frac{2\pi}{|\omega_{\mathrm B}|} =\frac{2\pi\hbar}{e|E|a}.$

Consequently the full-period mean velocity is $\overline v=\frac1{T_{\mathrm B}}\int_0^{T_{\mathrm B}}v(t)\,\mathrm dt =\frac{x(T_{\mathrm B})-x_0}{T_{\mathrm B}}=0.$ The same conclusion follows from the velocity alone: a sine integrates to zero over any complete cycle, whatever its initial phase, so the vanishing mean does not depend on the displacement calculation.

Before viewing Figure 5.7, take ${E>0}$ and ${k_0a=\pi/2}$ . Predict how the linearly decreasing unwrapped value of ${k(t)a/\pi}$ appears after it is represented in ${[-1,1)}$ , and predict whether ${x(t)-x_0}$ has a secular part after one period.

Figure 5.7. For a collisionless electron in a one-dimensional cosine band, the unwrapped crystal momentum decreases linearly while its reduced-zone representative wraps at zone boundaries; the corresponding real-space displacement is bounded and returns after one Bloch period.

At each time, the unwrapped wave number and its first-Brillouin-zone representative describe the same crystal-momentum class, but the field changes that class continuously with time. The right panel uses the same time coordinate and shows no secular displacement: the packet returns after ${T_{\mathrm B}}$ . These statements follow from the collisionless periodic band. They do not imply that a driven sample has zero measured current once scattering, reservoirs, and interband processes are supplied.

Dimensions provide an early checkpoint: ${2J/(eE)}$ is a length and ${eEa/\hbar}$ is a frequency. Reversing ${E}$ reverses traversal in ${k}$ but not the period. In the parabolic approximation the same integration would predict indefinite acceleration; the return occurs because the full band is periodic. This is collisionless Bloch motion, not a prediction of dc current: scattering, reservoirs, and interband tunneling have all been excluded.

There is also a regular zero-field limit at fixed observation time. Expanding the cosine difference to first order in ${\omega_{\mathrm B}t}$ gives ${x(t)-x_0\to(2Ja/\hbar)\sin(k_0a)t=v(k_0)t}$ . At ${E=0}$ itself the packet drifts with its initial group velocity and there is no finite Bloch period. The apparently divergent prefactor is canceled by the vanishing cosine difference.

Cover the trajectory above. A packet has ${v(\tau)=v_a\sin(\phi-\omega\tau)}$ , where ${v_a}$ is a constant velocity, ${\phi}$ is dimensionless, and the signed frequency ${\omega\ne0}$ may be negative. Complete ${x(\tau)-x(0)=\int_0^\tau v(s)\,\mathrm ds}$ . Differentiate your result, check its units, and test return after ${2\pi/|\omega|}$ . Record the signed antiderivative before comparing.

After completing that antiderivative, compare two field directions on the same positive period scale. For ${\tau=t/T_{\mathrm B}}$ , ${\phi=k_0a/\pi}$ , ${\sigma=\operatorname{sgn}E}$ , and ${v_{\max}=2Ja/\hbar}$ , the derived trajectory is

(5.5)

$q=\frac{k(t)a}{\pi}=\phi-2\sigma\tau,\qquad \frac{v}{v_{\max}}=\sin(\pi q),\qquad \frac{e|E|(x-x_0)}{2J}=\sigma[\cos(\pi q)-\cos(\pi\phi)].$

Choose an initial wave number, move the clock through two periods, and reverse the electric field. Predict the direction of crystal-momentum traversal before each reversal. Velocity and displacement are continuous; only the reduced-zone representative jumps when it is returned to ${[-1,1)}$ . The readouts distinguish that representative from the unwrapped wave number. Changing the field sign compares two constant-field trajectories from the same initial state; it does not reverse a running field partway through an orbit. Continuity refers to motion along each of those trajectories. Try the band minimum ${\phi=0}$ as well as the default ${\phi=1/2}$ : at the latter special initial phase, field reversal leaves these scaled velocity and displacement curves unchanged even though momentum traversal reverses. Every complete period returns to the initial position. This is the same collisionless, single-band, nonzero-field model as the worked example, not a dc-current prediction. Zero field has no finite Bloch period and is excluded from this clock; its fixed-time limit remains the one derived above.

Large finite samples. The infinite-lattice result describes bulk motion when the evolving packet’s full spatial extent and its oscillation excursion remain well separated from the ends over the observation time. A large number of cells alone is not enough: edge reflection invalidates this bulk description. A finite ring instead requires the time-dependent-vector-potential description discussed in the following qualifications, not a single-valued linear scalar potential.

General packet motion and finite-size qualifications

From allowed labels to a packet center. The discrete momenta used in state counting label basis states. Here the dynamical variables are the position center ${\mathbf r_{\mathrm c}(t)}$ and the crystal-wave-vector center ${\mathbf k_{\mathrm c}(t)}$ of a wave packet. A center need not be one of the allowed labels. To see the distinction in one dimension, expand a normalized state in a fixed field-free Bloch basis with discrete labels ${k_j}$ . Within a momentum-coordinate patch containing the narrow packet, $|\Psi(t)\rangle=\sum_j c_j(t)|n,k_j\rangle,\qquad \sum_j|c_j(t)|^2=1,\qquad k_{\mathrm c}(t)=\sum_j k_j|c_j(t)|^2.$ For differentiable coefficients, the weights, and hence their mean, can vary smoothly even though the labels remain fixed: $\dot k_{\mathrm c}(t)=\sum_j k_j\frac{d|c_j(t)|^2}{dt}.$ This identity explains differentiability; it does not derive a force law. Nor does it assert that a driven state remains in one band exactly. Because crystal momentum is defined modulo reciprocal-lattice vectors, use a locally continuous representative and continue it across zone boundaries; an arithmetic mean across the cut of a fixed reduced zone can be misleading.

The smooth-envelope approximation connects this description to a bulk band. For a one-dimensional periodic-box representation of length ${L=Na}$ , a useful hierarchy is $\frac{2\pi}{L}\ll\Delta k_{\mathrm{packet}}\ll\frac1a.$ The packet then contains many allowed momenta while sampling a narrow region of the band. This continuum approximation to the envelope is stronger than the mere existence of a differentiable center. In a finite physical sample, the packet must also remain well inside the sample throughout the time interval considered. The smooth band is the bulk Bloch family, not an assertion that a finite sample has a continuous energy spectrum.

State counting identifies which momenta matter, but not how a localized disturbance propagates. Construct a wave packet from one isolated, nondegenerate band, narrow about ${\mathbf k_{\mathrm c}}$ and broad compared with a unit cell. For a nearby component ${\mathbf k_{\mathrm c}+\mathbf q}$ , expand

$\varepsilon_n(\mathbf k_{\mathrm c}+\mathbf q) =\varepsilon_n(\mathbf k_{\mathrm c}) +\mathbf q\mathbin{\cdot}\nabla_{\mathbf k} \varepsilon_n(\mathbf k_{\mathrm c})+O({|\mathbf q|}^2).$

The constant term changes the overall phase. The linear term translates the packet envelope, giving the ordinary vector field

$\mathbf v_n:\mathrm{BZ}\to\mathbb R^d, \qquad \mathbf v_n(\mathbf k) =\frac1\hbar\nabla_{\mathbf k}\varepsilon_n(\mathbf k).$

This is group velocity. It generally differs from ${\hbar\mathbf k/m_{\mathrm e}}$ : crystal momentum labels a translation sector rather than the mechanical momentum. For Lecture 3’s local scalar Hamiltonian ${\hat H=\hat{\mathbf p}^{\,2}/(2m_{\mathrm e})+V(\hat{\mathbf r})}$ , the bulk wavepacket velocity is ${\hat{\mathbf v}=[\hat{\mathbf r},\hat H]/(i\hbar) =\hat{\mathbf p}/m_{\mathrm e}}$ on a common domain where this commutator is defined. Multiplication by position does not preserve the finite torus’s periodic operator domain, so this is not a commutator identity on every finite-periodic Bloch state.

For that setting use the cell Hamiltonian ${\hat H(\mathbf k)={(\hat{\mathbf p}+\hbar\mathbf k)}^2/(2m_{\mathrm e})+V}$ on its common periodic domain. Differentiate a normalized, differentiable isolated eigenpair ${\hat H(\mathbf k)u_n=\varepsilon_n u_n}$ and multiply by ${\langle u_n|}$ . Self-adjointness cancels the two terms containing ${\partial_{k_j}u_n}$ , leaving $\frac1\hbar\partial_{k_j}\varepsilon_n =\frac1\hbar\langle u_n|\partial_{k_j}\hat H(\mathbf k)|u_n\rangle =\left\langle u_n\left|\frac{\hat p_j+\hbar k_j}{m_{\mathrm e}}\right|u_n\right\rangle.$ This is the Bloch state’s canonical-momentum expectation divided by ${m_{\mathrm e}}$ , not generally ${\hbar k_j/m_{\mathrm e}}$ . At a degeneracy the differentiable isolated-branch argument needs a separate subspace treatment. More general effective Hamiltonians require their own velocity operator; periodicity alone does not break the equality above.

Write the magnetic force law in three spatial dimensions, where the cross product is defined. For an electron of charge ${-e}$ , with ${e>0}$ , slowly varying fields ${\mathbf E,\mathbf B:\mathbb R^3\times\mathbb R\to\mathbb R^3}$ change the packet center according to the following equations. The lower- dimensional applications here take ${\mathbf B=\mathbf0}$ and use the corresponding electric-field components. Why can we still use crystal momentum? In the scalar-potential description of a uniform electric field, the electron potential energy is ${e\mathbf E\cdot\mathbf r}$ , which is not lattice-periodic along the field. The full Hamiltonian therefore does not conserve the field-free Bloch label in that direction. The Bloch states of the periodic Hamiltonian nevertheless remain a basis: in the isolated-band wavepacket approximation, ${\mathbf k(t)}$ denotes the center of the packet’s distribution in that basis, not an exact stationary-state label of the tilted crystal. In these equations we abbreviate ${\mathbf r_{\mathrm c},\mathbf k_{\mathrm c}}$ as ${\mathbf r,\mathbf k}$ . The dot differentiates the packet-center coordinate in time, not an individual fixed Bloch label. We invoke the semiclassical acceleration theorem under the conditions below; it does not follow from the weighted-mean identity alone. For ${\mathbf B=\mathbf0}$ , it evolves according to ${\hbar\dot{\mathbf k}=-e\mathbf E}$ ; the periodic dispersion belongs to the underlying field-free crystal. Negligible interband transitions and a packet narrow in momentum are essential to this packet-center approximation.

(5.6)

$\dot{\mathbf r} =\frac1\hbar\nabla_{\mathbf k}\varepsilon_n(\mathbf k), \qquad \hbar\dot{\mathbf k} =-e\left(\mathbf E+\dot{\mathbf r}\times\mathbf B\right),$

${(m^{*-1})}_{ij}(\mathbf k) =\frac1{\hbar^2} \frac{\partial^2\varepsilon_n(\mathbf k)} {\partial k_i\partial k_j}.$

The object ${m^{*-1}}$ is a real symmetric rank-two tensor field on the band region, not a scalar unless its principal values are equal, as symmetry can enforce. With ${\mathbf B=\mathbf0}$ , differentiating the band velocity gives

$\dot v_i =\frac1\hbar \frac{\partial^2\varepsilon_n}{\partial k_i\partial k_j} \dot k_j ={(m^{*-1})}_{ij}(-eE_j),$

where repeated Cartesian indices are summed. Positive curvature near a band minimum gives electron-like acceleration. Negative curvature near a maximum gives the opposite response for the same charge. A saddle can have opposite signs along two principal axes, making one scalar mass actively misleading.

A controlled test of mixed curvature. Compare two local quadratic bands $\varepsilon_\eta(\mathbf k)=\frac{\hbar^2}{2m_0} \bigl(k_x^2+k_y^2+2\eta k_xk_y\bigr),\qquad m_0>0,\qquad \eta=0\ \text{or}\ \tfrac12.$ Their inverse-mass matrices are $m^{*-1}=\frac1{m_0} \begin{pmatrix}1&\eta\\\eta&1\end{pmatrix}.$ Both bands have a minimum at the origin, since their principal curvatures are proportional to ${1\pm\eta>0}$ . Start both packets at rest there and apply the same constant force ${\mathbf F=(F,0)}$ , with ${F>0}$ and ${\mathbf B=\mathbf0}$ . Then $\hbar\dot{\mathbf k}=\mathbf F,\qquad \dot{\mathbf v}=\frac{F}{m_0}(1,\eta).$ The momentum step is horizontal in both cases. The longitudinal acceleration is also identical, but only the mixed-curvature band has a transverse acceleration, ${F/(2m_0)}$ . For an electron the electric field points opposite ${\mathbf F}$ . The energy surface is drawn over momentum space, not real space: its gradient gives velocity, and the change of that gradient along the forced momentum step gives acceleration.

Figure 5.8. Controlled comparison of band curvature. Both packets start at rest and receive the same horizontal force. Without mixed curvature the acceleration is parallel to that force; with eta equal to one half, the same longitudinal response is accompanied by transverse acceleration. The perspective surfaces and top-view contours describe energy over momentum, not real-space trajectories.

The ellipsoid in Example 5.3 now supplies a direct comparison: its inverse-mass tensor is ${\operatorname{diag}(1/m_x,1/m_y,1/m_z)}$ . Equal products ${m_xm_ym_z}$ give equal DOS at the same energy above the minimum, but they need not give equal acceleration along a specified field. The static count and directional dynamics consume different parts of the same dispersion.

Return to the same upper solid branch in Lecture 4’s hybridization plot, panel (a). A zero slope means zero group velocity at that point; it does not determine the effective mass. The curvature, not the slope, determines the inverse mass. State both before performing a derivative.

For the parabolic band, consider ${\mathbf k}$ and ${-\mathbf k}$ at the same nonzero energy in equilibrium. Explain separately how their occupations, their contributions to the available-state count, and their velocities compare. Which distinction would be lost by reading a DOS peak as a large directed velocity? State what further state or relaxation information a current claim would require. Record your explanation before comparing.

Let a wave packet be built from one isolated, sufficiently smooth band, remain narrow in ${\mathbf k}$ and broad relative to a unit cell, and experience external fields that vary slowly on both lattice and packet scales. Require fields weak enough to retain the field-free band description, and neglect Berry-curvature and orbital-moment corrections. During an interval in which interband tunneling and collisions are negligible and the packet remains away from physical sample boundaries, its centers obey, to the retained semiclassical order, $\dot{\mathbf r}=\hbar^{-1}\nabla_{\mathbf k}\varepsilon_n(\mathbf k), \qquad \hbar\dot{\mathbf k} =-e\bigl(\mathbf E+\dot{\mathbf r}\times\mathbf B\bigr).$ These equations supply collisionless one-band kinematics only. They cease to be sufficient near unresolved degeneracies, under appreciable interband tunneling, when the packet or fields violate the scale separation, or when scattering or geometric velocity corrections cannot be neglected. They do not determine a lifetime or a dc transport coefficient.

Semiclassical dynamics

[Hou40, pp. 184–186] — Houston develops crystal-wave-vector acceleration in a uniform field and marks interband transitions as a boundary of one-band motion.
[XCN10, secs. V, VII] — Xiao, Chang, and Niu give the modern electron-sign equations, separate ordinary group velocity from Berry-curvature corrections, and discuss the one-dimensional Bloch-oscillation specialization.

In particular, a dispersion can give velocity but cannot give a lifetime. If a separate scattering model supplies ${\tau}$ , a mean free path may be estimated as ${\ell=|\mathbf v|\tau}$ ; two samples with the same band can still have very different ${\ell}$ and conductivity.

Boundary conditions and electromagnetic gauge. The scalar-potential argument applies in the bulk of a sample with boundaries; a linear potential is not single-valued on a periodic ring. A uniform field on a ring can instead be represented by a time-dependent vector potential ${A(t)}$ , with ${E=-\dot A}$ , physically corresponding to changing flux through the ring. The canonical labels ${k_j}$ stay discrete and fixed, while the band argument for an electron shifts as $\kappa_j(t)=k_j+\frac{eA(t)}{\hbar},\qquad \hbar\dot\kappa_j=-eE.$ This is a different representation, not motion of the canonical labels. In the general electromagnetic equations above, ${\mathbf k}$ denotes the local gauge-invariant crystal wave vector of the packet, rather than a gauge-dependent canonical Fourier label. It is not generally the mechanical momentum divided by ${\hbar}$ . [XCN10, secs. III.A, V.A–V.B]

Simulating an optical test of the Bloch timescale

The calculated trajectory in Figure 5.7 is not an experimental trace. What could a real detector measure, and which part of the prediction could that signal test? Feldmann and collaborators used a semiconductor superlattice: 91 repeats of a ${9.5\,\mathrm{nm}}$ GaAs well and a ${1.5\,\mathrm{nm}}$ alloy barrier. The repeat distance ${d=11\,\mathrm{nm}}$ , rather than an atomic lattice spacing, sets the relevant Bloch period. Figure 5.9 supplies the physical structure and the measurement principle before the optical-response calculation. Tunneling between neighboring wells forms a miniband along the growth direction, so this is the periodic band to which the one-band argument applies. At fixed field a longer period shortens the Bloch period. Resolving a recurrence also requires coherence to survive on that timescale; a longer period alone does not ensure observability. We now change preparation: the optical pulses create coherent interband excitations, not the equilibrium occupation of the preceding counting problem. [Fel+92, sample and optical arrangement, pp. 7252–7253]

Figure 5.9. Physical system and optical measurement. (a) Four representative periods of the 91-period intrinsic stack; well-to-barrier thickness ratio is preserved, lateral extent is schematic, and contacts are omitted. (b) Two pulses generate light in the selected diffracted-light direction shown in the panel; a slow detector records the time-integrated signal at each pulse delay. This is a functional signal diagram, not an optical-bench layout. [Fel+92, pp. 7252–7253]

The first pulse excites an optical polarization; the second arrives after delay ${\tau}$ . Repeating the experiment while scanning that delay produces ${S(\tau)}$ , the integrated diffracted light. It is neither the electron’s position nor its velocity or a dc current. For the experiment modeled below, the source uses approximately ${5\,\mathrm K}$ and ${110\,\mathrm{fs}}$ pulses. An optical polarization is a coherent sum of induced interband dipoles. Their relative phases evolve between the pulses. In the optical-response description, the second pulse can rephase these dipoles, producing a photon echo. The field changes the crystal wave vector and hence the interband transition frequencies during the interval. This modulates rephasing: delays corresponding to a complete Bloch cycle restore rephasing at the echo time and enhance the integrated response. Thus scanning the pulse separation can reveal the recurrence even though the detector does not resolve the emission in time. This is a qualitative optical-response argument, not a derivation of intensity from the packet’s position. [Fel+92, pp. 7253–7254] Before reading the calculated late features, estimate the expected recurrence time from the reported energy drop per period, ${e|E|d=6.2\,\mathrm{meV}}$ : $T_{\mathrm B}=\frac{2\pi\hbar}{e|E|d} \simeq0.667\,\mathrm{ps}.$

Figure 5.10. Calculated optical response, not measured data or a fit. (a) Adding static broadening and carrier escape suppresses late revivals; both scans use the baseline maximum for normalization. Dotted guides mark predicted Bloch periods, not fitted peaks. (b) Revised-model early emission at one Bloch-period delay, normalized separately; integration includes the later tail. Both models use 110 fs pulses and a 6.2 meV energy drop. The revised model assumes a 1 ps escape lifetime and Gaussian transition broadening of 4 meV FWHM; these are not measured parameters for this trace. [Fel+92, fig. 2 and discussion, pp. 7253–7255]

The paper reports experimental revivals near ${0}$ , ${0.7}$ , and ${1.4\,\mathrm{ps}}$ , compatible with this timescale; those data are not plotted here. Our independent-transition, two-miniband calculation evolves the optical coherence along accelerated momenta, extracts the third-order polarization in the selected emission direction, then integrates its squared coherent momentum average over emission time. The model uses the paper’s 21 meV electron and 2 meV heavy-hole miniband widths. Additional assumptions are transform-limited Gaussian pulses with 110 fs intensity FWHM, constant optical dipole, a uniform field, and excitation centered halfway between the minus-one and minus-two ladder transitions relative to the mean-gap reference. The baseline uses a two-picosecond coherence time, borrowed from the paper’s calculation at a different field and pulse duration rather than measured for this trace. The revised model retains that rate as background pure dephasing and adds irreversible excited-state escape into optically inactive states, with an assumed one-picosecond lifetime. Escape contributes half its population-loss rate to coherence decay, giving a one-picosecond homogeneous coherence time here; ground-state depletion is retained rather than treating escape as relaxation back to the ground state. This mechanism is motivated by the paper’s attribution of enhanced dephasing to carrier escape, but its rate is not fitted or measured here. The paper also reports about 4 meV of inhomogeneous transition broadening and models it through in-plane dispersion. We approximate this by static Gaussian detunings with 4 meV FWHM: the Gaussian shape and FWHM interpretation are additional assumptions. Polarizations are averaged over these detunings before squaring; static phase spreading can rephase into an echo and is not an extra irreversible decay rate. The model does not reproduce the reported 20 meV pulse spectrum independently, and temperature is not an explicit input. Coulomb effects, microscopic scattering, screening, finite-stack boundaries and optical propagation are omitted. The curves explain a possible optical recurrence, not quantitative agreement with the experiment or the ideal packet’s displacement. The original paper supplies the experimental evidence and a more specific optical-response comparison. [Fel+92, fig. 2 and discussion, pp. 7253–7255; note 20]

Bloch motion and its boundary

[Hou40, pp. 184–186] — The historical acceleration theory supports the one-band lineage, not the exact cosine-band calculation above.
[Zen34, pp. 523–529] — Zener analyzes field-driven excitation from a filled to an unfilled band, the interband process excluded from the worked example.
[Fel+92, pp. 7252–7255] — Feldmann and collaborators report periodic optical modulations in a semiconductor superlattice with the expected Bloch period.

The familiar one-band Hall coefficient illustrates the same boundary. For a single isotropic parabolic band, one carrier type, a steady Drude state, and the axis convention ${R_{\mathrm H}=E_y/(j_xB_z)}$ , transverse force balance gives

$R_{\mathrm H}=-\frac1{n_{\mathrm e}e}.$

This sign is not a universal carrier counter. Multiple bands, different mobilities, anisotropy, and Berry curvature can change the interpretation. The result is retained here only to show exactly where extra assumptions enter after band kinematics.

Estimate the Fermi wave number, energy and velocity, the thermal energy ratio, and the fractional density change produced by a small chemical-potential shift. Use the spin-degenerate isotropic parabolic model with ${m^*=m_{\mathrm e}}$ and ${n_{\mathrm e}=8.0\times10^{28}\,\mathrm m^{-3}}$ . Evaluate the thermal ratio at ${T=300\,\mathrm K}$ and the zero-temperature density change for ${\delta\mu=1\,\mathrm{meV}}$ .

Use the occupied momentum volume to fix the Fermi boundary, then evaluate the energy, velocity and density derivative at that boundary. Reuse the continuum state count of Example 5.2 with the stated mass and density:

$k_{\mathrm F}={(3\pi^2n_{\mathrm e})}^{1/3} \simeq1.33\times10^{10}\,\mathrm m^{-1},$

$E_{\mathrm F} =\frac{\hbar^2k_{\mathrm F}^2}{2m_{\mathrm e}} \simeq6.8\,\mathrm{eV}, \qquad v_{\mathrm F} =\frac{\hbar k_{\mathrm F}}{m_{\mathrm e}} \simeq1.5\times10^6\,\mathrm{m\,s^{-1}}.$

At ${T=300\,\mathrm K}$ , ${k_{\mathrm B}T/E_{\mathrm F}\simeq3.8\times10^{-3}}$ . The small ratio does not say the metal contains little energy. Thermal changes in occupation are concentrated within an energy range of order ${k_{\mathrm B}T}$ around the Fermi energy.

Differentiate the count ${n_{\mathrm e}(E)=C E^{3/2}}$ , with its mass-dependent constant ${C}$ fixed. Then ${g(E)=(3/2)C\sqrt E}$ , so

$g(E_{\mathrm F})=\frac{3n_{\mathrm e}}{2E_{\mathrm F}}.$

At zero temperature, a small shift ${\delta\mu}$ therefore changes the density by

$\frac{\delta n_{\mathrm e}}{n_{\mathrm e}} \simeq\frac32\frac{\delta\mu}{E_{\mathrm F}}.$

For ${\delta\mu=1\,\mathrm{meV}}$ , the fractional change is about ${2.2\times10^{-4}}$ . This small change shows why low-energy density response is controlled by states at the Fermi boundary rather than by all electrons equally.

Two consistency checks connect the outputs. Substitution gives ${k_{\mathrm F}^3/(3\pi^2)=n_{\mathrm e}}$ , recovering the supplied density, and ${m_{\mathrm e}v_{\mathrm F}^2/2=E_{\mathrm F}}$ , recovering the same energy from the velocity. The DOS formula has units of density per energy; ${\delta n_{\mathrm e}\to0}$ as ${\delta\mu\to0}$ . These estimates assume that the filled pocket lies within the parabolic regime and that ${|\delta\mu|\ll E_{\mathrm F}}$ ; they supply no scattering lifetime.

Changed dispersion. Replace the parabola by ${\varepsilon(k)=\varepsilon_0-2J\cos(ka)}$ , with ${J>0}$ , corresponding to Lecture 4’s signed hopping ${t_{\mathrm{L4}}=-J}$ . Near ${k=0}$ the curvature is positive; near ${k=\pi/a}$ it is negative. The electric force has the same sign in both regions, yet the acceleration changes sign. No positive scalar substituted for ${m_{\mathrm e}}$ can represent both limits.

A model DOS is constant and equal to ${g_0}$ in a window about ${\mu}$ . At ${T=0}$ , predict the first-order density change when ${\mu}$ rises by ${\delta\mu}$ . At nonzero temperature, identify the function that replaces the sharp boundary and state its width.

Choose the required calculation before writing a formula. Work without the worked examples, then compare your route and checks with the instructor response.

A reservoir specifies ${T>0}$ , ${\mu}$ , and the energy of one resolved fermion mode. To predict its mean occupation, which preparation assumptions and normalization are needed? Would a momentum-space volume count answer this question?
In the quadratic ellipsoid, hold the particle density and band minimum fixed at ${T=0}$ , but multiply ${m_x}$ by ${\lambda>0}$ while leaving ${m_y,m_z}$ fixed. Determine how ${E_{\mathrm F}-\varepsilon_{\mathrm c}}$ changes. Choose the state-count or evolution route explicitly and check ${\lambda=1}$ ; assume both pockets stay within the quadratic regime.
For a packet in the cosine band under a nonzero constant electric field, which equations determine the displacement over one reciprocal-space cycle? Does knowing the equilibrium density alone settle that displacement or a dc conductivity? Name the extra assumptions behind each claim.

Lecture 5 has now turned a set of possible energies into a populated and moving reference state:

$\varepsilon_n(\mathbf k) \longrightarrow f(\varepsilon) \longrightarrow g(\varepsilon)f(\varepsilon) \longrightarrow\mathcal S_{\mathrm F} \longrightarrow\mathbf v_n,m^{*-1}.$

Reconstruct the chain without looking at the displayed formulas. For one supplied band, state (i) the two alternatives whose relative weight produces the fermionic occupation, (ii) the measure and degeneracy convention used in the density integral, (iii) the level set that becomes the zero-temperature Fermi surface, and (iv) the two derivatives of the band that control velocity and acceleration. Then identify one quantity that none of these data determines: a lifetime, a conductivity, or a probe intensity. A complete answer must distinguish a property of the Hamiltonian from a property of the preparation and from a property of the measurement protocol.

The chain still cannot name what an experiment creates. Adding an electron, removing one, or exciting a lattice mode are different changes relative to the populated reference. Lecture 6 develops that distinction.

The larger course role of this lecture is now explicit. Bloch theory supplied possible one-electron states; equilibrium filling selected a many-electron reference state; the Fermi surface identified its low-energy boundary; and band derivatives supplied kinematics. Screening, pairing, transport, and many-body decay will all use this same boundary as phase space. The spherical alkali-metal model is valuable because every factor can be reconstructed. The ellipsoidal model marks one boundary: anisotropy already requires directional curvatures. Multiple sheets, interactions, and lifetimes require still more information rather than a better choice of one scalar mass.

A complete kinematic claim now has four declarations: the band-state family, the preparation that occupies it, the wavepacket and field scale that licenses semiclassical motion, and the probe or scattering model used for observation. Changing a reciprocal representative changes none of these physical outputs, whereas changing occupation or relaxation can change transport without changing the dispersion. The effective-mass tensor is local curvature data inside this contract, not a new fundamental mass.

Problems

1. Occupation window and density

A spin-inclusive density of states is constant, ${g(\varepsilon)=g_0}$ , on ${[0,W]}$ and zero outside.

At ${T=0}$ and ${0<\mu<W}$ , calculate the particle density.
Find the first-order change produced by ${\mu\to\mu+\delta\mu}$ .
At low nonzero temperature, use ${-\partial f/\partial\varepsilon}$ to explain which energy window responds.
State how every answer changes if ${g_0}$ is instead quoted per spin.
If this density of states represents one isolated band with capacity ${g_0W}$ per unit volume, state the zero-temperature filling fraction and whether a Fermi surface exists for ${0<\mu<W}$ .

2. Band kinematics and scattering lifetimes

For the anisotropic band

$\varepsilon(\mathbf k) =\frac{\hbar^2k_x^2}{2m_x} +\frac{\hbar^2k_y^2}{2m_y} +2t_z[1-\cos(k_za)],$

with ${m_x,m_y,t_z,a>0}$ ,

derive ${\mathbf v(\mathbf k)}$ and the inverse-mass tensor. Under ${\mathbf B=\mathbf0}$ , determine ${\dot{\mathbf k}}$ and ${\dot{\mathbf v}}$ for a uniform electric field. Explain why the same data do not determine conductivity, and state two conditions under which the one-band semiclassical equations fail.

3. Bloch oscillation as a kinematic boundary

Consider an electron in the one-dimensional band with nearest- and next-nearest-neighbor hopping,

$\varepsilon(k)=-2J\cos(ka)-2J'\cos(2ka), \qquad J>0,\quad J'\geq0,$

under a nonzero constant electric field ${E\mathbf e_x}$ , with no magnetic field and no scattering. The worked cosine-band cycle is the case ${J'=0}$ ; reuse its integration, but work every step that the second harmonic changes.

Starting from the semiclassical equations, find ${k(t)}$ and ${v(t)}$ for an initial ${k_0}$ . Identify the two harmonics of ${v(t)}$ and their relative amplitude.
Derive the real-space displacement ${x(t)-x(0)}$ and show that the Bloch period is the same as for ${J'=0}$ , although ${v(t)}$ now contains a component with half that period.
Show that the velocity averaged over one full period vanishes even though the field is nonzero. Then determine the condition on ${J'/J}$ under which the band acquires additional stationary points, so that the packet reverses direction more than twice per period, and locate those extra turning points in ${k}$ .
Explain why this collisionless result is neither an ordinary constant acceleration nor a prediction of a finite dc conductivity. State one physical effect that must be added before a transport coefficient can be calculated.

(advanced): When a Scalar Effective Mass Fails

This optional appendix is not a provider for Lecture 7. It retains the core wave-packet operation but changes the band geometry enough that one scalar mass cannot describe the result.

Effective-mass evidence and terminology

[DKK55, pp. 368, 375] — Dresselhaus, Kip, and Kittel measure anisotropic cyclotron-resonance masses in silicon and germanium.
[HAK12, pp. 1–2] — Hilton, Arikawa, and Kono distinguish curvature, density-of-states, cyclotron, and spectroscopic effective masses.

Let a two-dimensional band near ${\mathbf k_0}$ have

$\varepsilon(\mathbf k_0+\mathbf q) =\varepsilon_0+\hbar\mathbf v_0\mathbin{\cdot}\mathbf q +\frac{\hbar^2}{2}q_iA_{ij}q_j+O({|\mathbf q|}^3),$

where ${A=A^{\mathsf T}}$ is a real inverse-mass matrix. Under an orthogonal rotation ${O}$ , ${O^{\mathsf T}AO=\operatorname{diag}(a_1,a_2)}$ . The principal accelerations in zero magnetic field are

$\dot v'_i=-e a_i E'_i.$

At a nondegenerate stationary point ( ${\mathbf v_0=\mathbf0}$ and both ${a_i\neq0}$ ), a minimum has ${a_1,a_2>0}$ , a maximum has both negative, and a saddle has opposite signs. If a Hessian eigenvalue vanishes, higher-order terms may be needed to classify the stationary point; the quadratic sign test alone is inconclusive. Replacing the tensor by a scalar average can predict the wrong acceleration sign along one direction.

Consider the explicit periodic model

(5.7)

$\varepsilon(k_x,k_y) =-2t_x\cos(k_xa)-2t_y\cos(k_ya), \qquad t_x,t_y>0.$

At ${(0,0)}$ both curvatures are positive. At ${(\pi/a,0)}$ , the ${x}$ -curvature is negative and the ${y}$ -curvature positive: this is a saddle point. Its density of states is enhanced because the energy changes slowly along competing directions, but the local motion cannot be summarized by “heavy” or “light” without specifying direction.

Writing ${k_x=\pi/a+q_x}$ and ${k_y=q_y}$ , the quadratic expansion is

$\varepsilon-\varepsilon_{\mathrm s} \simeq -t_xa^2q_x^2+t_ya^2q_y^2, \qquad m^{*-1}=\frac{2a^2}{\hbar^2} \begin{pmatrix}-t_x&0\\0&t_y\end{pmatrix}.$

Consequently ${E_x>0}$ produces ${\dot v_x=+2ea^2t_xE_x/\hbar^2}$ , opposite to the acceleration near the band minimum, while an equal-sign ${E_y}$ still gives negative electron acceleration. The opposite eigenvalue signs are the quantitative content of “saddle”; any scalar effective mass would discard one of them.

The expansion also has a wave-packet condition. If the packet has momentum widths ${\Delta k_i}$ , the neglected cubic and higher terms must remain small across those widths, and the packet’s driven trajectory must stay away from band degeneracies. A nearly flat direction reduces a principal curvature but also makes higher-order terms important sooner; an apparently infinite scalar mass may signal that the quadratic approximation has lost its resolving power.

Berry-curvature velocity is a separate geometric correction developed in Lectures 21–24.

For the periodic model in Equation (5.7), compute the Hessians at ${(0,0)}$ , ${(\pi/a,\pi/a)}$ , and ${(\pi/a,0)}$ . Classify each as a minimum, maximum, or saddle and predict its two principal accelerations under ${\mathbf B=0}$ and a uniform electric field ${(E_x,E_y)}$ , for charge ${-e}$ with ${e>0}$ . Use ${m^{*-1}=\hbar^{-2}\operatorname{Hess}\varepsilon}$ and compare with the worked saddle before consulting the instructor response.

References

[Dir26] Dirac, P. A. M. (1926). “On the theory of quantum mechanics.” Proceedings of the Royal Society A 112 (762), 661–677. https://doi.org/10.1098/rspa.1926.0133.

[DKK55] Dresselhaus, G., Kip, A. F., and Kittel, C. (1955). “Cyclotron resonance of electrons and holes in silicon and germanium crystals.” Physical Review 98, 368–384. https://doi.org/10.1103/PhysRev.98.368.

[Fel+92] Feldmann, J., Leo, K., Shah, J., Miller, D. A. B., Cunningham, J. E., Meier, T., Plessen, G. von, Schulze, A., Thomas, P., and Schmitt-Rink, S. (1992). “Optical investigation of bloch oscillations in a semiconductor superlattice.” Physical Review B 46 (11), 7252–7255. https://doi.org/10.1103/PhysRevB.46.7252.

[Fer26] Fermi, E. (1926). “Zur quantelung des idealen einatomigen gases.” Zeitschrift für Physik 36, 902–912. https://doi.org/10.1007/BF01400221.

[HAK12] Hilton, D. J., Arikawa, T., and Kono, J. (2012). “Cyclotron resonance.” in Characterization of materials, Wiley, pp. 1–15. https://doi.org/10.1002/0471266965.com068.pub2.

[Hou40] Houston, W. V. (1940). “Acceleration of electrons in a crystal lattice.” Physical Review 57, 184–186. https://doi.org/10.1103/PhysRev.57.184.

[LK56] Lifshitz, I. M., and Kosevich, A. M. (1956). “Theory of magnetic susceptibility in metals at low temperature.” Soviet Physics JETP 2 (4), 636–645. https://jetp.ras.ru/cgi-bin/e/index/e/2/4/p636?a=list.

[MS99] Mikitik, G. P., and Sharlai, Yu. V. (1999). “Manifestation of berry’s phase in metal physics.” Physical Review Letters 82, 2147–2150. https://doi.org/10.1103/PhysRevLett.82.2147.

[Ons52] Onsager, L. (1952). “Interpretation of the de haas–van alphen effect.” Philosophical Magazine 43, 1006–1008. https://doi.org/10.1080/14786440908521019.

[PI22] Pérez, E., and Ibáñez, J. (2022). “Indistinguishable elements in the origins of quantum statistics: The case of fermi–dirac statistics.” European Physical Journal H 47, 1–25. https://doi.org/10.1140/epjh/s13129-021-00033-6.

[Som28] Sommerfeld, A. (1928). “Zur elektronentheorie der metalle auf grund der fermischen statistik.” Zeitschrift für Physik 47, 1–32. https://doi.org/10.1007/BF01391052.

[Hov53] Van Hove, L. (1953). “The occurrence of singularities in the elastic frequency distribution of a crystal.” Physical Review 89 (6), 1189–1193. https://doi.org/10.1103/PhysRev.89.1189.

[XCN10] Xiao, D., Chang, M.-C., and Niu, Q. (2010). “Berry phase effects on electronic properties.” Reviews of Modern Physics 82 (3), 1959–2007. https://doi.org/10.1103/RevModPhys.82.1959.

[Yu17] Yu, T. (2017). Lecture 4: Higher dimensional regular surfaces. Drexel University. https://www.math.drexel.edu/~tyu/Math538/Lecture4.pdf.

[Zen34] Zener, C. (1934). “A theory of the electrical breakdown of solid dielectrics.” Proceedings of the Royal Society A 145, 523–529. https://doi.org/10.1098/rspa.1934.0116.

Lecture 5: Populated Bands and Electron Kinematics

5.1 Preparation populates a band

5.1.1 Reservoir weights and the Fermi function

5.2 Filling one resolved fermion level

Historical lineage

5.2.1 From occupation to filling and the Fermi surface

Fermi surfaces and quantum oscillations

5.3 Counting how many states live near an energy

5.3.1 States per momentum volume

5.3.2 The shell Jacobian

5.3.3 An ellipsoid changes the counting volume

5.3.4 Heating a populated band under two constraints

5.4 The occupied band begins to move

5.4.1 Exact crystal-momentum flow on an infinite lattice

5.4.2 General packet motion and finite-size qualifications