Fermi Surfaces, Effective Mass, and Metals vs Insulators

The band structure is a set of energy sheets $E_{n} (k)$ over the Brillouin zone. It becomes physics only when the electrons are poured into it. The Pauli principle fills the lowest available Bloch states up to the Fermi energy, and the shape of that filling — which bands are full, which are partly occupied, where the highest occupied states sit — decides every zeroth-order property of the solid: whether it is a metal, an insulator, or a semiconductor, and how it responds to fields. This lesson establishes the rules of band filling and the semiclassical dynamics of the electrons that occupy the partly filled bands.

A filled band carries no current

The electrical current from a band is the charge times the average velocity of its occupied states. The velocity of a Bloch electron is the group velocity of its wavepacket, set by the slope of the band,

v_{n} (k) = \frac{1}{ℏ} \nabla_{k} E_{n} (k) .

Summing $- e v$ over the occupied states, with the density of $k$ -points $V / (2 π)^{3}$ , the current density from band $n$ is

j_{n} = - \frac{e}{( 2 π ) ^{3}} occ \int v_{n} (k) d^{3} k = - \frac{e}{( 2 π ) ^{3} ℏ} occ \int \nabla_{k} E_{n} (k) d^{3} k .

If the band is completely filled the integral runs over the entire Brillouin zone. Time-reversal symmetry makes each band even in $k$ , $E_{n} (- k) = E_{n} (k)$ , so the velocity is odd, $v_{n} (- k) = - v_{n} (k)$ , and the contributions from $k$ and $- k$ cancel in pairs. Equivalently, the integral of a gradient over the periodic zone (a region without boundary) is zero. Either way,

j_{n} = 0 for a filled band.

A filled band is inert: it holds exactly $2 N$ electrons whose velocities cancel in every direction, and no electric field can produce a net current from it because there are no empty states nearby to scatter into. Only a partly filled band conducts. A field shifts the occupied region of $k$ -space slightly, unbalancing the velocity sum, and a net current flows.

In a filled band every occupied k has a partner at minus k with opposite velocity, so the velocities cancel and no current flows. In a partly filled band an applied field displaces the occupied region, leaving a net velocity and a current.

Metals, insulators, and semimetals

Each band holds $2 N$ electrons and the crystal has some integer number of valence electrons per primitive cell. The parity of that number is decisive.

Odd electrons per cell. The valence electrons cannot fill an integer number of bands; the highest occupied band is half-filled and the Fermi level lies inside it. The crystal is a metal. The alkali metals (one $s$ -electron per cell) are the cleanest examples.
Even electrons per cell, no band overlap. The electrons exactly fill an integer number of bands, and a gap separates the highest filled band from the lowest empty one. No partly filled band exists; the crystal is an insulator (or, if the gap is small enough for thermal excitation, a semiconductor).
Even electrons per cell, with band overlap. If a filled band's maximum rises above an empty band's minimum, electrons spill from the first into the second, leaving both partly filled. The crystal conducts as a semimetal; the divalent metals (calcium, the group-II elements) work this way, and it is why an even electron count does not guarantee an insulator.

The count of $2 N$ states per band, combined with the presence or absence of a gap, is the entire zeroth-order theory of why some solids conduct.

Band filling decides the character. A half-filled band (metal) has states at the Fermi level. A filled band below a gap with an empty band above (insulator) has none. Overlapping bands (semimetal) leave two bands partly filled. Shaded regions are occupied.

The Fermi surface

In a metal the boundary in $k$ -space between occupied and empty states is the Fermi surface, the constant-energy surface $E_{n} (k) = E_{F}$ . For the free-electron gas it is a sphere of radius $k_{F}$ ; the periodic potential distorts it, and where it crosses a Bragg plane the gap makes it meet the zone boundary at right angles. Every low-temperature property of a metal — its conductivity, its heat capacity, its magnetic response — is governed by the electrons on this surface, since only they have empty neighbouring states to move into.

Harrison's construction builds the free-electron Fermi surface in the reduced zone quickly. Draw the free-electron sphere of radius $k_{F}$ centred on every reciprocal-lattice point. A point of the first zone lies in the first-band Fermi sea if it is inside one sphere, in the second band if inside two overlapping spheres, and so on. Translating each region back into the first zone by its reciprocal-lattice vector assembles the electron pockets of the higher bands and the hole pockets of the lower ones. The construction is only a starting point — the real potential rounds the sharp corners where spheres intersect — but it predicts the topology of the Fermi surface of the simple metals correctly.

Harrison's construction. Free-electron spheres of radius k_F are drawn about each reciprocal-lattice point; their overlaps, folded back into the first zone, give the Fermi-surface pockets of successive bands. Doubly covered regions (darker) belong to the second band.

Holes

A band that is nearly full is more economically described by its few empty states. The current of a band with a handful of empty states $k_{e}$ is

j = - e occ \sum v (k) = - e (all \sum v - empty \sum v) = + e empty \sum v (k_{e}),

using the filled-band theorem $\sum_{all} v = 0$ . The current of the nearly-full band is identical to that of positively charged particles occupying the empty states. These fictitious positive carriers are holes. A hole has charge $+ e$ , wavevector $- k_{e}$ , and — because the empty states sit near the top of the band where $E (k)$ curves downward — a positive effective mass. In a semiconductor conduction proceeds through electrons in the nearly empty conduction band and holes in the nearly full valence band, and the hole picture is what makes the positive Hall coefficient of some metals intelligible.

Effective mass

Near a band extremum the dispersion is parabolic, and expanding about the minimum $k_{0}$ gives

E (k) \approx E (k_{0}) + \frac{ℏ ^{2}}{2} i, j \sum (k - k_{0})_{i} (M^{- 1})_{ij} (k - k_{0})_{j}, (M^{- 1})_{ij} = \frac{1}{ℏ ^{2}} \frac{\partial ^{2} E}{\partial k _{i} \partial k _{j}} .

The effective-mass tensor $M$ is the inverse curvature of the band. A Bloch electron responds to forces as if it were free but with mass $M$ in place of the bare $m$ , absorbing the whole effect of the periodic potential into this one tensor. Sharp curvature (a narrow band) means a heavy effective mass and sluggish carriers; gentle curvature (a wide band) means a light effective mass. At a band maximum the curvature is negative, so the electron effective mass is negative — another way of seeing that the excitations there are better described as holes with positive mass.

The effective mass is the inverse curvature of the band. At the band minimum the curvature is positive (light or heavy electron mass by how sharp); at the maximum it is negative, so the natural carriers there are holes of positive mass.

Semiclassical dynamics and Bloch oscillations

Between collisions a Bloch electron moves as a wavepacket obeying two semiclassical equations. The position advances at the group velocity, and the crystal momentum responds to the external forces,

\dot{r} = v_{n} (k) = \frac{1}{ℏ} \nabla_{k} E_{n} (k), ℏ \dot{k} = - e (E + v_{n} \times B) .

The force law involves only the external fields, not the far larger periodic force of the lattice, which is already folded into $E_{n} (k)$ through the effective mass. Crystal momentum, not true momentum, is the quantity the external force drives.

A striking consequence follows from a constant electric field alone. Then $ℏ \dot{k} = - e E$ is constant, so $k$ advances uniformly through the Brillouin zone,

k (t) = k (0) - \frac{e E}{ℏ} t,

and on reaching the zone boundary it re-enters at the opposite face (the two are the same state). The velocity $v_{n} = ℏ^{- 1} \nabla_{k} E_{n}$ therefore oscillates as $k$ sweeps the periodic band, and the electron executes a periodic real-space motion — a Bloch oscillation — of period, for a one-dimensional band of lattice constant $a$ ,

T_{Bloch} = \frac{h}{e E a} .

In an ordinary crystal the electron scatters off phonons or impurities in about $1 0^{- 14} s$ , far shorter than $T_{Bloch}$ at attainable fields, so it never completes a period and the motion averages to a steady drift — Ohm's law. Bloch oscillations are observed only in engineered superlattices, whose large period $a$ shortens $T_{Bloch}$ below the scattering time.

Under a constant field the crystal momentum sweeps uniformly across the zone and wraps around at the boundary; the group velocity (slope of the band) oscillates in sign, so the electron oscillates in real space rather than accelerating away.

Band theory now stands complete in outline: Bloch states organized into bands, bands filled to a Fermi surface, and the electrons on that surface moving by the semiclassical equations. The semiconductor module applies this machinery to the technologically central case of a small gap, where a modest thermal population of electrons and holes, tuned by doping, carries the current of every solid-state device.