30 Jun 2025
Energy Bands and the Kronig-Penney Model
Bloch bands, Brillouin-zone gaps, the Kronig-Penney dispersion, effective mass, and metal, semiconductor, and insulator band structures.
Isolated atoms have discrete levels. In a crystal, overlap between equivalent atomic states and the periodic ionic potential broaden those levels into allowed bands separated by forbidden energy gaps.
Bloch form and bands
For a one-dimensional lattice of period $a$,
\[V(x+a)=V(x).\]The translation operator by $a$ commutes with the Hamiltonian, so an energy eigenfunction may also be an eigenfunction of translation:
\[\psi_k(x+a)=e^{ika}\psi_k(x).\]Writing $u_k(x)=e^{-ikx}\psi_k(x)$ gives
\[\boxed{\psi_k(x)=e^{ikx}u_k(x)}, \qquad u_k(x+a)=u_k(x).\]This is Bloch’s form. Since $e^{i(k+G)na}=e^{ikna}$ for $G=2\pi m/a$, wave vectors differing by $G$ are equivalent; one may take $-\pi/a<k\leq\pi/a$.
For each $k$, the periodic boundary-value problem has a sequence $E_n(k)$. As $k$ varies continuously through the first Brillouin zone, each sequence forms an energy band.
Origin of a gap at a Brillouin-zone boundary
For a weak periodic potential
\[V(x)=\sum_GU_Ge^{iGx},\]the Fourier component $U_G$ couples free-electron waves $\lvert k\rangle$ and $\lvert k-G\rangle$. At the zone boundary $k=G/2$ their free energies are equal:
\[E_k^0=\frac{\hbar^2k^2}{2m} =\frac{\hbar^2(k-G)^2}{2m}=E_{k-G}^0.\]Within this degenerate pair the Schrödinger equation becomes
\[\begin{pmatrix} E_0-E&U_G\\ U_G^*&E_0-E \end{pmatrix} \begin{pmatrix}c_k\\c_{k-G}\end{pmatrix}=0.\]The determinant condition is
\[(E_0-E)^2-\lvert U_G\rvert^2=0,\]so
\[E_{\pm}=E_0\pm\lvert U_G\rvert, \qquad \boxed{E_g=E_+-E_-=2\lvert U_G\rvert}.\]The two combinations form standing waves with different probability densities relative to the ions and therefore different potential energies. Bragg reflection at $k=G/2$ is the reciprocal-space origin of the band gap.
Delta-barrier Kronig-Penney model
An exactly soluble periodic model uses barriers of strength $H$:
\[V(x)=H\sum_{n=-\infty}^{\infty}\delta(x-na),\]where $H$ has unit $\mathrm{J\,m}$. Between barriers, $V=0$ and
\[\frac{d^2\psi}{dx^2}+q^2\psi=0, \qquad q=\frac{\sqrt{2mE}}{\hbar}.\]Across a barrier at $x=0$, $\psi$ is continuous. Integrating the Schrödinger equation from $-\epsilon$ to $+\epsilon$ gives
\[-\frac{\hbar^2}{2m}\left[\psi'(0^+)-\psi'(0^-)\right] +H\psi(0)=0,\]so
\[\boxed{\psi'(0^+)-\psi'(0^-)=\frac{2mH}{\hbar^2}\psi(0)}.\]Represent the state by the column $(\psi,\psi^{\prime})^T$. Free propagation through distance $a$ gives
\[P_a= \begin{pmatrix} \cos qa&\sin qa/q\\ -q\sin qa&\cos qa \end{pmatrix},\]while crossing a delta barrier gives
\[D= \begin{pmatrix} 1&0\\ 2mH/\hbar^2&1 \end{pmatrix}.\]One cell has transfer matrix $M=DP_a$. Bloch’s condition requires its eigenvalues to be $e^{\pm ika}$. Since $\det M=1$, their sum equals the trace:
\[2\cos ka=\operatorname{Tr}M =2\cos qa+\frac{2mH}{\hbar^2q}\sin qa.\]Defining the dimensionless barrier strength
\[P=\frac{mHa}{\hbar^2}\]gives the Kronig-Penney relation
\[\boxed{\cos ka=\cos qa+P\frac{\sin qa}{qa}}.\]The left side must lie between $-1$ and $1$. Therefore an energy $E=\hbar^2q^2/(2m)$ is allowed only when
\[\boxed{\left\lvert\cos qa+P\frac{\sin qa}{qa}\right\rvert\leq1}.\]Intervals violating this condition are forbidden gaps. Increasing $P$ strengthens the periodic potential, narrows the allowed bands, and widens the gaps.
Effective mass
The group velocity of a wave packet in band $n$ is
\[v_n(k)=\frac1\hbar\frac{dE_n}{dk}.\]Under a slowly varying external force $F$, crystal momentum obeys
\[\hbar\frac{dk}{dt}=F.\]Therefore
\[\frac{dv}{dt} =\frac1\hbar\frac{d^2E}{dk^2}\frac{dk}{dt} =\frac{F}{\hbar^2}\frac{d^2E}{dk^2}.\]Writing this as $F=m^*dv/dt$ defines
\[\boxed{\frac1{m^*}=\frac1{\hbar^2}\frac{d^2E}{dk^2}}.\]Near an extremum $k_0$,
\[E(k)\simeq E(k_0)+\frac{\hbar^2(k-k_0)^2}{2m^*}.\]A band minimum has positive curvature and $m^*>0$; a band maximum has negative electron effective mass. Missing electrons near a nearly full band are more conveniently described as positively charged holes with positive effective mass.
Metals, semiconductors, and insulators
At zero temperature, electrons fill states up to the Fermi level.
- A metal has a partially filled band or overlapping bands. Empty states arbitrarily close in energy allow an electric field to change occupations and produce current.
- A semiconductor has a filled valence band and empty conduction band separated by a relatively small gap. Thermal or optical excitation produces mobile conduction electrons and valence-band holes.
- An insulator has the same filled-band arrangement but a sufficiently large gap that ordinary thermal excitation produces negligible carriers.
The classification depends on band filling and the gap, not simply on whether isolated atoms possess bound electrons.
Direct and indirect band gaps
Let $E_c(k)$ be the conduction band and $E_v(k)$ the valence band. If the conduction minimum and valence maximum occur at the same wave vector $k_0$, then
\[\boxed{E_g^{\rm dir}=E_c(k_0)-E_v(k_0)}.\]A photon carries negligible crystal momentum compared with a Brillouin-zone dimension, so a direct optical transition is approximately vertical in an $E$-$k$ diagram: $\Delta k\simeq0$.
For an indirect semiconductor, the extrema occur at $k_c\ne k_v$:
\[\boxed{E_g^{\rm ind}=E_c(k_c)-E_v(k_v)}.\]Momentum conservation then requires a phonon as well as a photon:
\[\mathbf k_c=\mathbf k_v\pm\mathbf q_{\rm ph}+\mathbf G.\]The reciprocal vector $\mathbf G$ allows crystal momentum to be conserved modulo the reciprocal lattice. This additional phonon requirement makes near-edge optical absorption and emission weaker than in a direct-gap material.
Discussion