---
title: Fermi Surfaces, Effective Mass, and Metals vs Insulators
module: Band Theory
moduleNumber: 6
lessonNumber: 4
order: 604
summary: >
  Filling the bands settles which crystals conduct. A filled band carries no
  current, so a crystal with filled bands and a gap is an insulator, while a
  partly filled band makes a metal. This lesson derives the no-current theorem
  for a filled band, defines the Fermi surface and Harrison's construction,
  introduces holes and the effective mass from band curvature, and states the
  semiclassical equations of motion that lead to Bloch oscillations.
topics: [Band Theory]
draft: false
sources:
  - book: Ashcroft & Mermin
    ref: "Ch. 12 — The Semiclassical Model of Electron Dynamics; Ch. 8–9"
  - book: Kittel
    ref: "Ch. 9 — Fermi Surfaces and Metals"
---

The [band structure](/condensed-matter/band-theory/blochs-theorem-and-energy-bands)
is a set of energy sheets $E_n(\vec k)$ 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,

$$
\vec v_n(\vec k) = \frac{1}{\hbar}\nabla_{\vec k} E_n(\vec k).
$$

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

$$
\vec j_n = -\frac{e}{(2\pi)^3}\int_{\text{occ}} \vec v_n(\vec k)\, \d^3 k
= -\frac{e}{(2\pi)^3 \hbar}\int_{\text{occ}} \nabla_{\vec k} E_n(\vec k)\, \d^3 k.
$$

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

$$
\vec j_n = 0 \qquad \text{for a filled band.}
$$

A filled band is inert: it holds exactly $2N$ 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 $\vec k$-space
slightly, unbalancing the velocity sum, and a net current flows.

$$
% caption: In a filled band every occupied k has a partner at minus k with
% opposite velocity, so the velocities cancel and no current flows. In a partly
% filled band an applied field displaces the occupied region, leaving a net
% velocity and a current.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % filled band (left)
  \begin{scope}
    \draw[black, ->] (-2.0,0) -- (2.2,0) node[below, font=\scriptsize] {$k$};
    \draw[black, ->] (0,-0.3) -- (0,2.2) node[left, font=\scriptsize] {$E$};
    \draw[acc, very thick, domain=-1.8:1.8, samples=80, variable=\x] plot ({\x},{0.9*\x*\x/3.24});
    \foreach \x in {-1.6,-1.2,-0.8,-0.4,0,0.4,0.8,1.2,1.6}{ \fill[acc] (\x,{0.9*\x*\x/3.24}) circle (1.5pt); }
    \draw[black, ->] (-1.0,{0.9*1.0/3.24}) -- (-1.5,{0.9*1.0/3.24});
    \draw[black, ->] (1.0,{0.9*1.0/3.24}) -- (1.5,{0.9*1.0/3.24});
    \node[black, anchor=south, font=\scriptsize] at (0,1.7) {full band: $v$ cancels};
  \end{scope}
  % partly filled (right), displaced occupancy
  \begin{scope}[xshift=6.4cm]
    \draw[black, ->] (-2.0,0) -- (2.2,0) node[below, font=\scriptsize] {$k$};
    \draw[black, ->] (0,-0.3) -- (0,2.2) node[left, font=\scriptsize] {$E$};
    \draw[acc, very thick, domain=-1.8:1.8, samples=80, variable=\x] plot ({\x},{0.9*\x*\x/3.24});
    \foreach \x in {-0.8,-0.4,0,0.4,0.8,1.2}{ \fill[acc] (\x,{0.9*\x*\x/3.24}) circle (1.5pt); }
    \node[black, anchor=south, font=\scriptsize] at (0,1.7) {shifted: net $v$};
    \draw[black, ->] (-1.9,1.2) -- (-1.1,1.2) node[right, font=\scriptsize] {force};
  \end{scope}
\end{tikzpicture}
$$

## Metals, insulators, and semimetals

Each band holds $2N$ 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](/condensed-matter/semiconductors/intrinsic-and-extrinsic-semiconductors)).
- **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 $2N$ states per band, combined with the presence or absence of a
gap, is the entire zeroth-order theory of why some solids conduct.

$$
% caption: Band filling decides the character. A half-filled band (metal) has
% states at the Fermi level. A filled band below a gap with an empty band above
% (insulator) has none. Overlapping bands (semimetal) leave two bands partly
% filled. Shaded regions are occupied.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % metal
  \begin{scope}
    \node[black, anchor=south, font=\scriptsize] at (0.6,3.0) {metal};
    \draw[black] (0,0) rectangle (1.2,1.4);
    \fill[acc!22] (0,0) rectangle (1.2,0.7);
    \draw[acc, thick] (0,0.7) -- (1.2,0.7);
    \node[black, anchor=west, font=\scriptsize] at (1.35,0.7) {$E_F$};
  \end{scope}
  % insulator
  \begin{scope}[xshift=4.2cm]
    \node[black, anchor=south, font=\scriptsize] at (0.6,3.0) {insulator};
    \draw[black] (0,0) rectangle (1.2,1.0);
    \fill[acc!22] (0,0) rectangle (1.2,1.0);
    \draw[black] (0,2.2) rectangle (1.2,3.0);
    \node[black, anchor=west, font=\scriptsize] at (1.35,1.6) {gap};
    \draw[black, <->] (0.6,1.0) -- (0.6,2.2);
  \end{scope}
  % semimetal
  \begin{scope}[xshift=8.4cm]
    \node[black, anchor=south, font=\scriptsize] at (0.6,3.0) {semimetal};
    \draw[black] (0,0) rectangle (1.2,1.6);
    \fill[acc!22] (0,0) rectangle (1.2,1.35);
    \draw[black] (0,1.1) rectangle (1.2,2.7);
    \fill[acc!22] (0,1.1) rectangle (1.2,1.35);
    \node[black, anchor=west, font=\scriptsize] at (1.35,1.2) {overlap};
  \end{scope}
\end{tikzpicture}
$$

## The Fermi surface

In a metal the boundary in $\vec k$-space between occupied and empty states is
the **Fermi surface**, the constant-energy surface $E_n(\vec k) = E_F$. 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.

> **Definition (Fermi surface).** The **Fermi surface** is the surface in
> reciprocal space on which the band energy equals the Fermi energy,
> $E_n(\vec k) = E_F$, separating occupied from unoccupied Bloch states at zero
> temperature. A metal has a Fermi surface; an insulator (all bands filled or
> empty) does not.

**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.

$$
% caption: Harrison's construction. Free-electron spheres of radius k_F are drawn
% about each reciprocal-lattice point; their overlaps, folded back into the first
% zone, give the Fermi-surface pockets of successive bands. Doubly covered
% regions (darker) belong to the second band.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % reciprocal lattice points
  \foreach \i in {-1,0,1}{ \foreach \j in {-1,0,1}{ \fill[black] (2*\i,2*\j) circle (1.6pt); }}
  % zone boundary (square around origin)
  \draw[black, thick] (-1,-1) rectangle (1,1);
  % free-electron circles radius > half-diagonal so they overlap
  \foreach \i in {-1,0,1}{ \foreach \j in {-1,0,1}{
    \draw[acc, thick] (2*\i,2*\j) circle (1.28);
  }}
  \node[black, anchor=south, font=\scriptsize] at (0,2.35) {reciprocal lattice};
  \node[acc, anchor=west, font=\scriptsize] at (1.4,-2.0) {Fermi spheres, radius $k_F$};
\end{tikzpicture}
$$

## 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 $\vec k_e$ is

$$
\vec j = -e\sum_{\text{occ}} \vec v(\vec k) = -e\left(\sum_{\text{all}} \vec v - \sum_{\text{empty}} \vec v\right) = +e\sum_{\text{empty}} \vec v(\vec k_e),
$$

using the filled-band theorem $\sum_{\text{all}}\vec v = 0$. 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 $-\vec k_e$, and — because the empty states sit near the
top of the band where $E(\vec k)$ curves downward — a positive effective mass. In
a [semiconductor](/condensed-matter/semiconductors/carrier-transport-and-recombination)
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
$\vec k_0$ gives

$$
E(\vec k) \approx E(\vec k_0) + \frac{\hbar^2}{2}\sum_{i,j}(k - k_0)_i\, (M^{-1})_{ij}\, (k - k_0)_j,
\qquad (M^{-1})_{ij} = \frac{1}{\hbar^2}\frac{\partial^2 E}{\partial k_i\,\partial k_j}.
$$

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.

$$
% caption: The effective mass is the inverse curvature of the band. At the band
% minimum the curvature is positive (light or heavy electron mass by how sharp);
% at the maximum it is negative, so the natural carriers there are holes of
% positive mass.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (-3.6,0) -- (3.8,0) node[below] {$k$};
  \draw[black, ->] (0,-0.3) -- (0,3.4) node[left] {$E$};
  % cosine band: min at centre (curvature > 0), max at zone edge (curvature < 0)
  \draw[acc, very thick, domain=-3.0:3.0, samples=140, variable=\x]
    plot ({\x},{1.7-1.1*cos(180*\x/3.0)});
  % osculating parabola at the band minimum (opens up)
  \draw[black, dashed, domain=-1.4:1.4, samples=40, variable=\x] plot ({\x},{0.6+0.40*\x*\x});
  % osculating parabola at the band maximum, near the zone edge (opens down)
  \draw[black, dashed, domain=1.7:3.0, samples=40, variable=\x] plot ({\x},{2.8-0.40*(\x-3)*(\x-3)});
  \fill[acc] (0,0.6) circle (1.8pt);
  \fill[acc] (3.0,2.8) circle (1.8pt);
  \node[acc, anchor=north, font=\scriptsize] at (0,0.5) {curvature $>0$: electron};
  \node[acc, anchor=south east, font=\scriptsize] at (3.0,2.9) {curvature $<0$: hole};
\end{tikzpicture}
$$

## 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,

$$
\dot{\vec r} = \vec v_n(\vec k) = \frac{1}{\hbar}\nabla_{\vec k} E_n(\vec k),
\qquad
\hbar\dot{\vec k} = -e\left(\vec E + \vec v_n\times\vec B\right).
$$

The force law involves only the external fields, not the far larger periodic
force of the lattice, which is already folded into $E_n(\vec k)$ 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
$\hbar\dot{\vec k} = -e\vec E$ is constant, so $\vec k$ advances uniformly through
the Brillouin zone,

$$
\vec k(t) = \vec k(0) - \frac{e\vec E}{\hbar}t,
$$

and on reaching the zone boundary it re-enters at the opposite face (the two are
the same state). The velocity $\vec v_n = \hbar^{-1}\nabla_{\vec k}E_n$ therefore
oscillates as $\vec k$ 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 $a$,

$$
T_{\text{Bloch}} = \frac{h}{eEa}.
$$

In an ordinary crystal the electron scatters off phonons or impurities in about
$10^{-14}\ \text{s}$, far shorter than $T_{\text{Bloch}}$ 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](/condensed-matter/nanostructures/quantum-wells-wires-and-dots),
whose large period $a$ shortens $T_{\text{Bloch}}$ below the scattering time.

$$
% caption: Under a constant field the crystal momentum sweeps uniformly across
% the zone and wraps around at the boundary; the group velocity (slope of the
% band) oscillates in sign, so the electron oscillates in real space rather than
% accelerating away.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % velocity vs k over one period: v ~ sin(ka)
  \draw[black, ->] (-3.4,0) -- (3.6,0) node[below] {$k$};
  \draw[black, ->] (0,-1.8) -- (0,1.9) node[left] {$v$};
  \draw[black, dashed] (3.0,-1.6) -- (3.0,1.6) node[above, black, font=\scriptsize] {zone edge};
  \draw[black, dashed] (-3.0,-1.6) -- (-3.0,1.6);
  \draw[acc, very thick, domain=-3.0:3.0, samples=120, variable=\x]
    plot ({\x},{1.5*sin(180*\x/3.0)});
  \draw[black, ->] (-2.6,0.3) -- (-1.6,0.3) node[right, font=\scriptsize] {$k$ drifts};
\end{tikzpicture}
$$

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](/condensed-matter/semiconductors/semiconductor-bands-and-junctions)
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.

[^am]: Ashcroft & Mermin, Ch. 12; Ch. 8–9.
[^kittel]: Kittel, Ch. 9.