Band Gaps in the Nearly-Free-Electron Model

Derivation

Band Gaps in the Nearly-Free-Electron Model

D-250 Home PU-303 Threads waves · energy · symmetry Depends on Bloch's Theorem from Translational Symmetry, Degenerate Perturbation Theory

Statement

For an electron in a weak periodic potential \(V(\mathbf{r})=\sum_{\mathbf{G}}V_{\mathbf{G}}\,e^{i\mathbf{G}\cdot\mathbf{r}}\) with lattice reciprocal vectors \(\mathbf{G}\), degenerate perturbation theory applied to the two free-electron states that become degenerate at a Brillouin-zone boundary — where \(|\mathbf{k}|=|\mathbf{k}-\mathbf{G}|\), i.e. \(\mathbf{k}\cdot\mathbf{G}=\tfrac{1}{2}|\mathbf{G}|^2\) — shows that the potential lifts the degeneracy and opens an energy gap of magnitude \(E_{\mathrm{gap}}=2\,|V_{\mathbf{G}}|\), where \(V_{\mathbf{G}}\) is the Fourier component of the potential at the connecting reciprocal-lattice vector \(\mathbf{G}\).

Why it matters

This is the origin of the band structure of crystalline solids. It explains why an almost-free electron gas — which classically would conduct at all fillings — instead separates into bands split by forbidden energy ranges, and hence why materials are metals, semiconductors, or insulators depending on where the Fermi level sits relative to these gaps.

The result is striking because an arbitrarily weak periodic potential still opens a gap: the effect is non-perturbative in the sense that it cannot be captured by treating \(V\) as a small correction to a single state. Only the degeneracy of the two counter-propagating Bloch waves at the zone boundary makes a first-order splitting linear in \(V_{\mathbf{G}}\) possible, which is why the gap scales as the potential itself rather than as its square.

Assumptions

The periodic potential is weak compared with the electronic kinetic energy scale, \(|V_{\mathbf{G}}|\ll \hbar^2 G^2/2m\). If dropped, the free-electron states are no longer a good zeroth-order basis; one must diagonalize a large plane-wave matrix (tight-binding or full band-structure methods) and the simple two-state truncation fails.

Only two plane waves are near-degenerate at the boundary of interest. If several \(\mathbf{G}\)-connected states coincide (as at high-symmetry corner points), the \(2\times2\) secular problem must be enlarged to \(3\times3\) or more, and the gap is no longer simply \(2|V_{\mathbf{G}}|\).

The potential is real, so \(V_{-\mathbf{G}}=V_{\mathbf{G}}^{*}\), and \(V_{\mathbf{0}}\) is an ignorable constant. If dropped (complex potential, e.g. absorption or a non-Hermitian model), the two off-diagonal couplings differ and the eigenvalues can become complex, so the "gap" is no longer a real energy splitting.

Spin-orbit coupling and electron-electron interactions are neglected; the single-particle Bloch picture holds. If dropped, gaps acquire spin-dependent structure and the independent-electron approximation must be replaced by a many-body treatment (Hartree-Fock, DFT), though the qualitative gap-opening survives.

Derivation

\[ \hat{H}=\frac{\hat{p}^2}{2m}+V(\mathbf{r}),\qquad V(\mathbf{r})=\sum_{\mathbf{G}\neq \mathbf{0}}V_{\mathbf{G}}\,e^{i\mathbf{G}\cdot\mathbf{r}} \]

Start from the single-electron Hamiltonian with a lattice-periodic potential expanded in its Fourier series over reciprocal-lattice vectors \(\mathbf{G}\); the \(\mathbf{G}=\mathbf{0}\) term is absorbed as an energy origin. A

\[ \psi_{\mathbf{k}}^{(0)}(\mathbf{r})=\frac{1}{\sqrt{\mathcal{V}}}\,e^{i\mathbf{k}\cdot\mathbf{r}},\qquad E_{\mathbf{k}}^{(0)}=\frac{\hbar^2 k^2}{2m} \]

Zeroth-order (\(V=0\)) eigenstates are plane waves, normalized in a crystal volume \(\mathcal{V}\); these are the unperturbed basis for perturbation theory. A

\[ \langle \mathbf{k}'|V|\mathbf{k}\rangle=\sum_{\mathbf{G}}V_{\mathbf{G}}\,\frac{1}{\mathcal{V}}\int_{\mathcal{V}} e^{i(\mathbf{k}+\mathbf{G}-\mathbf{k}')\cdot\mathbf{r}}\,d^3r=\sum_{\mathbf{G}}V_{\mathbf{G}}\,\delta_{\mathbf{k}',\,\mathbf{k}+\mathbf{G}} \]

The matrix element of \(V\) between plane waves is nonzero only when the wavevectors differ by a reciprocal-lattice vector; this is the momentum-selection rule enforcing crystal-momentum conservation. B

\[ E_{\mathbf{k}}^{(0)}=E_{\mathbf{k}-\mathbf{G}}^{(0)}\ \Longleftrightarrow\ k^2=|\mathbf{k}-\mathbf{G}|^2\ \Longleftrightarrow\ \mathbf{k}\cdot\mathbf{G}=\tfrac{1}{2}G^2 \]

Two free states \(|\mathbf{k}\rangle\) and \(|\mathbf{k}-\mathbf{G}\rangle\) are degenerate exactly on the perpendicular bisector plane of \(\mathbf{G}\) — the Bragg plane, i.e. the Brillouin-zone boundary. Here non-degenerate perturbation theory diverges and the degenerate treatment is mandatory. B

\[ |\Psi\rangle=c_1\,|\mathbf{k}\rangle+c_2\,|\mathbf{k}-\mathbf{G}\rangle \]

Project the Schrödinger equation onto the two near-degenerate states, discarding all other plane waves whose unperturbed energies are separated by \(O(\hbar^2G^2/2m)\gg|V_{\mathbf{G}}|\) (justified by the weak-potential assumption). This is the standard two-state truncation of degenerate perturbation theory. C

\[ \begin{pmatrix} E_{\mathbf{k}}^{(0)}-E & V_{\mathbf{G}}^{*}\\[4pt] V_{\mathbf{G}} & E_{\mathbf{k}-\mathbf{G}}^{(0)}-E \end{pmatrix}\!\begin{pmatrix}c_1\\ c_2\end{pmatrix}=0 \]

The truncated Hamiltonian in the two-state basis. The off-diagonal element coupling \(|\mathbf{k}\rangle\) to \(|\mathbf{k}-\mathbf{G}\rangle\) is \(\langle\mathbf{k}-\mathbf{G}|V|\mathbf{k}\rangle=V_{\mathbf{G}}\); its conjugate partner is \(V_{-\mathbf{G}}=V_{\mathbf{G}}^{*}\) since \(V\) is real and Hermiticity holds. C

\[ \bigl(E_{\mathbf{k}}^{(0)}-E\bigr)\bigl(E_{\mathbf{k}-\mathbf{G}}^{(0)}-E\bigr)-|V_{\mathbf{G}}|^2=0 \]

Nontrivial solutions require the secular determinant to vanish; this quadratic in \(E\) determines the two perturbed energy branches. B

\[ E_{\pm}=\frac{E_{\mathbf{k}}^{(0)}+E_{\mathbf{k}-\mathbf{G}}^{(0)}}{2}\pm\sqrt{\left(\frac{E_{\mathbf{k}}^{(0)}-E_{\mathbf{k}-\mathbf{G}}^{(0)}}{2}\right)^2+|V_{\mathbf{G}}|^2} \]

Solve the quadratic for \(E\). This is exact within the two-state subspace and valid for \(\mathbf{k}\) near the Bragg plane, not only on it. B

\[ E_{\mathbf{k}}^{(0)}=E_{\mathbf{k}-\mathbf{G}}^{(0)}\ \Rightarrow\ E_{\pm}=E_{\mathbf{k}}^{(0)}\pm|V_{\mathbf{G}}| \]

Evaluate exactly on the zone boundary, where the diagonal difference vanishes. The two branches split symmetrically about the free-electron energy by \(\pm|V_{\mathbf{G}}|\). A

\[ E_{\mathrm{gap}}=E_{+}-E_{-}=2\,|V_{\mathbf{G}}| \]

Subtract the two branch energies at the boundary: the forbidden gap equals twice the modulus of the relevant Fourier component of the potential. A

Result

\[ \boxed{\,E_{\mathrm{gap}}=2\,|V_{\mathbf{G}}|\,} \]

Reading. A weak periodic potential mixes the two counter-propagating plane waves \(e^{i\mathbf{k}\cdot\mathbf{r}}\) and \(e^{i(\mathbf{k}-\mathbf{G})\cdot\mathbf{r}}\) that Bragg-reflect into one another at the zone boundary. The mixing produces standing waves: one piling charge on the ion cores (lower energy, \(E_-\)) and one piling charge between them (higher energy, \(E_+\)). The energy separation of these two standing waves is \(2|V_{\mathbf{G}}|\), a range of energies no propagating Bloch state can occupy — the band gap. Its size is set purely by the strength of the Fourier component of the lattice potential at the reciprocal vector that connects the two states.

Units check. \(V(\mathbf{r})\) is an energy, and its Fourier coefficient \(V_{\mathbf{G}}=\frac{1}{\mathcal{V}_{\mathrm{cell}}}\int_{\mathrm{cell}}V(\mathbf{r})e^{-i\mathbf{G}\cdot\mathbf{r}}d^3r\) is also an energy (the \(1/\mathcal{V}_{\mathrm{cell}}\) cancels the volume from \(d^3r\)). Hence \(E_{\mathrm{gap}}=2|V_{\mathbf{G}}|\) has units of energy (J or eV), as required.

Limiting cases

\(V_{\mathbf{G}}\to 0\) (free electron): the gap closes, \(E_{\mathrm{gap}}\to 0\), and the two parabolic bands cross freely at the boundary — the empty-lattice band structure.
Away from the boundary (\(\delta k\) off the Bragg plane): the bands recover their free-electron dispersion; from Step 8, the correction \(\to |V_{\mathbf{G}}|^2/(E_{\mathbf{k}}^{(0)}-E_{\mathbf{k}-\mathbf{G}}^{(0)})\), the ordinary second-order shift, quadratic in \(V\).
Higher Fourier components: the \(n\)-th zone boundary opens a gap \(2|V_{n\mathbf{G}_1}|\); since \(V_{\mathbf{G}}\) typically decays with \(|\mathbf{G}|\), higher gaps are generally smaller.
Strong potential \(|V_{\mathbf{G}}|\sim \hbar^2G^2/2m\): the two-state result breaks toward the tight-binding limit, where gaps become large and bands narrow into atomic-like levels.

Breaks when

Multiple degeneracies coincide. At corner/edge points of the Brillouin zone (e.g. the \(W\) point of the FCC zone) three or more free states are simultaneously degenerate. The \(2\times2\) secular equation is wrong; one must diagonalize the full near-degenerate submatrix, and the splitting is no longer simply \(2|V_{\mathbf{G}}|\).
The potential is not weak. When \(|V_{\mathbf{G}}|\) is comparable to the kinetic energy \(\hbar^2G^2/2m\), plane waves are a poor basis, many \(\mathbf{G}\)-components mix strongly, and the perturbative truncation fails; the correct description crosses over to tight-binding.
Non-Hermitian / disordered systems. If \(V\) is complex (gain/loss) or aperiodic (disorder, quasicrystals), the Fourier series and Bragg condition lose meaning; eigenvalues may become complex or states may localize (Anderson localization), and a clean real gap of \(2|V_{\mathbf{G}}|\) does not form.
Vanishing structure factor. If the crystal basis makes \(V_{\mathbf{G}}=0\) for a particular \(\mathbf{G}\) (a forbidden reflection), that zone boundary opens no first-order gap; the leading splitting is then higher order and much smaller.

Failure modes

Gap = \(|V_{\mathbf{G}}|\) error. Forgetting the factor of two: the gap is the full splitting \(E_+-E_-=2|V_{\mathbf{G}}|\), not the shift \(|V_{\mathbf{G}}|\) of a single branch from the mean.
Using non-degenerate perturbation theory. Plugging \(|\mathbf{k}\rangle\) into the first-order energy shift \(\sum_{\mathbf{G}}|V_{\mathbf{G}}|^2/(E_{\mathbf{k}}^{(0)}-E_{\mathbf{k}-\mathbf{G}}^{(0)})\) at the boundary gives a spurious divergence because the denominator vanishes; the degeneracy demands the matrix treatment.
Wrong Fourier component. Using \(V_{\mathbf{0}}\) or the real-space depth \(V_{\max}\) instead of the specific \(V_{\mathbf{G}}\) that connects the two degenerate states. Only the connecting \(\mathbf{G}\) enters.
Confusing \(\mathbf{G}\) with \(\mathbf{k}\). Writing the Bragg condition as \(k=G\) rather than \(\mathbf{k}\cdot\mathbf{G}=\tfrac12 G^2\) (i.e. \(k=G/2\) for the first zone in 1D).
Ignoring the structure factor. Assuming every \(\mathbf{G}\) opens a gap, when a multi-atom basis can make \(V_{\mathbf{G}}\) vanish for allowed reciprocal vectors.

Discussion

The physical picture behind \(E_{\mathrm{gap}}=2|V_{\mathbf{G}}|\) is Bragg reflection. A Bloch wave whose wavevector reaches the zone boundary satisfies exactly the diffraction condition \(2\mathbf{k}\cdot\mathbf{G}=G^2\); it is Bragg-reflected into the state \(\mathbf{k}-\mathbf{G}\) with equal amplitude. Forward and backward waves of equal weight form standing waves rather than travelling waves, so the group velocity vanishes at the boundary — the band dispersion flattens to zero slope there. The two standing waves, \(\psi_+\propto\cos(\mathbf{G}\cdot\mathbf{r}/2)\) and \(\psi_-\propto\sin(\mathbf{G}\cdot\mathbf{r}/2)\), have different charge distributions relative to the ion cores, and hence different electrostatic energies; that energy difference is exactly the gap.

This connects three of the threads of this unit. The waves thread supplies the interference of counter-propagating Bloch states; the energy thread supplies the electrostatic splitting of the two standing-wave charge patterns; and the symmetry thread supplies the reciprocal lattice and the selection rule \(\langle\mathbf{k}'|V|\mathbf{k}\rangle\neq 0\) only for \(\mathbf{k}'-\mathbf{k}=\mathbf{G}\), which is Bloch's theorem in momentum space. The gap is thus a direct consequence of discrete translational symmetry: without a lattice there is no \(\mathbf{G}\), no Bragg plane, and no gap.

More deeply, the nearly-free-electron gap and the tight-binding gap are two limits of the same physics. In the weak-potential limit the gap is small (\(2|V_{\mathbf{G}}|\)) and the bands are nearly-free parabolas warped near the zone boundaries; in the strong-potential limit the bands are narrow atomic levels broadened into bands of width set by the hopping integral, with large gaps between them. The two descriptions must agree in the intermediate regime, and the topology of the bands — the number of states per band being exactly one per primitive cell per spin — is preserved across the crossover. This state-counting invariance is what guarantees that filled bands (insulators) and partially filled bands (metals) are robustly defined regardless of potential strength, and it underpins the modern topological classification of insulators, where the same \(2\times2\) zone-boundary Hamiltonian, with an added \(\sigma_z\) mass term, becomes the Dirac model of a topological band inversion.

Common misconceptions. The gap is not caused by electrons "not having enough energy" to move — it is a range of energies for which no propagating stationary state exists at all. Nor does a weak potential open only an infinitesimal, negligible effect: even the smallest periodic potential opens a finite first-order gap linear in \(V_{\mathbf{G}}\), which is qualitatively different from the second-order shifts everywhere else in the zone. Finally, the gap is a property of the whole crystal's periodicity, not of individual atoms; isolated atoms have discrete levels, but the gap between bands is a collective, lattice-symmetry phenomenon.

Worked examples

\[ \textbf{1D lattice, cosine potential: } V(x)=2V_1\cos\!\left(\frac{2\pi x}{a}\right),\quad a=3.0\ \text{\AA},\ V_1=1.5\ \text{eV} \]

Find the first band gap at the zone boundary \(k=\pi/a\). First identify the Fourier components. A

\[ V(x)=V_1 e^{i G_1 x}+V_1 e^{-iG_1 x},\quad G_1=\frac{2\pi}{a}\ \Rightarrow\ V_{G_1}=V_{-G_1}=V_1 \]

Write the cosine as two exponentials; the relevant Fourier component connecting \(k=\pi/a\) to \(k-G_1=-\pi/a\) is \(V_{G_1}=V_1\). Symbols before numbers. A

\[ E_{\mathrm{gap}}=2|V_{G_1}|=2V_1=2(1.5\ \text{eV})=3.0\ \text{eV} \]

Apply the boxed result and insert numbers. Check the reference kinetic scale: \(\hbar^2 G_1^2/2m=\hbar^2(2\pi/a)^2/2m\approx 16.7\ \text{eV}\gg V_1\), so the weak-potential assumption holds. B

\[ E_{\mathrm{gap}}=3.0\ \text{eV} \]

Reading. A 3 eV first gap — typical of a wide-gap semiconductor or insulator — from a lattice-potential amplitude of only \(V_1=1.5\) eV, one tenth of the kinetic scale, confirming that a modest periodic potential produces a sizeable forbidden band.

Units check. \(V_1\) in eV, factor 2 dimensionless, gap in eV. Consistent.

\[ \textbf{3D, second gap: } V_{G_1}=0.80\ \text{eV},\quad V_{G_2}=0.30\ \text{eV},\quad |\mathbf{G}_2|=2|\mathbf{G}_1| \]

A crystal has decaying Fourier components. Find the ratio of the second zone-boundary gap to the first. Work symbolically first. A

\[ E_{\mathrm{gap}}^{(1)}=2|V_{G_1}|,\qquad E_{\mathrm{gap}}^{(2)}=2|V_{G_2}|,\qquad \frac{E_{\mathrm{gap}}^{(2)}}{E_{\mathrm{gap}}^{(1)}}=\frac{|V_{G_2}|}{|V_{G_1}|} \]

Each zone boundary opens a gap set by its own connecting Fourier component; the factor 2 cancels in the ratio. A

\[ E_{\mathrm{gap}}^{(1)}=2(0.80)=1.60\ \text{eV},\quad E_{\mathrm{gap}}^{(2)}=2(0.30)=0.60\ \text{eV},\quad \text{ratio}=0.375 \]

Insert numbers. The higher gap is smaller because \(|V_{\mathbf{G}}|\) decays with \(|\mathbf{G}|\), the generic behaviour of smooth atomic potentials. B

\[ E_{\mathrm{gap}}^{(1)}=1.60\ \text{eV},\quad E_{\mathrm{gap}}^{(2)}=0.60\ \text{eV} \]

Reading. The first gap (1.6 eV) exceeds the second (0.6 eV) by a factor \(\approx 2.7\), illustrating why the lowest gap usually dominates a material's optical and electronic behaviour.

Units check. Both Fourier components in eV, gaps in eV, ratio dimensionless. Consistent.

Problems

A 1D crystal has lattice constant \(a=4.0\ \text{\AA}\) and potential \(V(x)=2V_1\cos(2\pi x/a)+2V_2\cos(4\pi x/a)\) with \(V_1=0.90\) eV, \(V_2=0.25\) eV. Find the first two band gaps.
Solution
The Fourier components are \(V_{G_1}=V_1\) and \(V_{G_2}=V_2\). First gap at \(k=\pi/a\): \(E_{\mathrm{gap}}^{(1)}=2V_1=2(0.90)=1.80\) eV. Second gap at \(k=2\pi/a\): \(E_{\mathrm{gap}}^{(2)}=2V_2=2(0.25)=0.50\) eV.
Verify that the weak-potential assumption holds for Problem 1's first gap, given free-electron mass \(m=9.11\times10^{-31}\) kg. Compare \(V_1\) to \(\hbar^2 G_1^2/2m\).
Solution
\(G_1=2\pi/a=2\pi/(4.0\times10^{-10}\,\text{m})=1.57\times10^{10}\ \text{m}^{-1}\). Then \(\hbar^2G_1^2/2m=(1.055\times10^{-34})^2(1.57\times10^{10})^2/(2\cdot 9.11\times10^{-31})=1.51\times10^{-18}\ \text{J}=9.4\) eV. Since \(V_1=0.90\ \text{eV}\ll 9.4\ \text{eV}\), the potential is weak and the two-state result is valid.
At the zone boundary the two eigenstates are \(\psi_\pm\). Show that they are standing waves and identify which has lower energy for a potential with \(V_{G_1}>0\) (potential maximum at the ion cores taken at \(x=0\)).
Solution
On the boundary \(E_\pm=E^{(0)}\pm|V_{G_1}|\) with \(c_2=\pm c_1\) (from the secular equation with equal diagonal, the eigenvectors of \(\left(\begin{smallmatrix}0&V_{G_1}\\ V_{G_1}&0\end{smallmatrix}\right)\) are \((1,\pm1)/\sqrt2\)). Thus \(\psi_\pm\propto e^{i\pi x/a}\pm e^{-i\pi x/a}\), giving \(\psi_+\propto\cos(\pi x/a)\) and \(\psi_-\propto\sin(\pi x/a)\) up to normalization. The state whose charge density \(|\psi|^2\) concentrates where \(V\) is lowest has lower energy. With the convention chosen, \(\sin(\pi x/a)\) has nodes at the cores and \(\cos(\pi x/a)\) has antinodes there; the lower-energy state is the one piling charge in the low-potential regions. The splitting magnitude is \(2|V_{G_1}|\) regardless of sign convention.
A crystal has a two-atom basis such that the structure factor gives \(V_{G_1}=0\) but \(V_{G_2}=0.40\) eV. What is the first-order gap at the first zone boundary, and where does the first sizeable gap appear?
Solution
Since \(V_{G_1}=0\), the first zone boundary opens no first-order gap: \(E_{\mathrm{gap}}^{(1)}=2|V_{G_1}|=0\) to leading order (any gap is higher-order and negligible). The first sizeable gap appears at the second boundary, \(E_{\mathrm{gap}}^{(2)}=2|V_{G_2}|=2(0.40)=0.80\) eV. This is the band-structure analogue of a forbidden X-ray reflection from the structure factor.
Off the zone boundary, take \(E_{\mathbf{k}}^{(0)}-E_{\mathbf{k}-\mathbf{G}}^{(0)}=2\Delta\) with \(\Delta=0.50\) eV and \(|V_{\mathbf{G}}|=0.20\) eV. Compute the two branch energies relative to the mean \(\bar E=\tfrac12(E_{\mathbf{k}}^{(0)}+E_{\mathbf{k}-\mathbf{G}}^{(0)})\), and compare their separation to the on-boundary gap.
Solution
From Step 8, \(E_\pm=\bar E\pm\sqrt{\Delta^2+|V_{\mathbf{G}}|^2}=\bar E\pm\sqrt{(0.50)^2+(0.20)^2}=\bar E\pm\sqrt{0.29}=\bar E\pm 0.539\) eV. Separation \(=1.077\) eV. The on-boundary gap would be \(2|V_{\mathbf{G}}|=0.40\) eV. The off-boundary separation (1.08 eV) is larger because it includes the free-electron energy difference \(2\Delta=1.0\) eV; the minimum separation, equal to the true gap \(0.40\) eV, occurs exactly on the boundary where \(\Delta=0\).