Tight-Binding Band Dispersion

Derivation

Tight-Binding Band Dispersion

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Statement

For a crystal with one localized atomic orbital \(\phi\) per site, the single-band energy is the lattice Fourier transform of the Hamiltonian matrix elements between orbitals. Expanding the Bloch eigenstate as a Bloch sum of orbitals, \(|\psi_{\mathbf{k}}\rangle=\frac{1}{\sqrt{N}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}|\phi_{\mathbf{R}}\rangle\), and assuming the orbitals are orthonormal, the band dispersion is \(E(\mathbf{k})=\sum_{\mathbf{S}}e^{i\mathbf{k}\cdot\mathbf{S}}\,h(\mathbf{S})\) with \(h(\mathbf{S})=\langle\phi_{\mathbf{0}}|\hat{H}|\phi_{\mathbf{S}}\rangle\), which for nearest-neighbour hopping on a one-dimensional chain of spacing \(a\) reduces to \(E(k)=\epsilon-2t\cos(ka)\).

Why it matters

The tight-binding model is the minimal microscopic route from atoms to bands. It shows that a discrete band of Bloch states emerges from a single atomic level once neighbouring wavefunctions overlap, and it makes the bandwidth, the effective mass, and the shape of the Fermi surface explicit functions of a handful of hopping integrals. It is the language of graphene, of Hubbard-model condensed matter, of Wannier-based DFT "downfolding", and of essentially every lattice model in solid-state physics.

Because it starts from localized orbitals rather than plane waves, it is complementary to the nearly-free-electron picture: it is accurate precisely where the free-electron expansion fails, namely for narrow bands built from tightly bound \(d\)- and \(f\)-electrons, and it exposes how chemistry (which orbitals, which neighbours) controls the electronic structure.

Assumptions

One orbital per site, energetically isolated band.If a second orbital lies within a hopping energy, the two Bloch sums hybridize and a single-band scalar dispersion is replaced by a matrix eigenvalue problem; the formula gives the wrong band and misses avoided crossings. Localized orbitals that are orthonormal, \(\langle\phi_{\mathbf{R}}|\phi_{\mathbf{R'}}\rangle=\delta_{\mathbf{R}\mathbf{R'}}\).If orbitals on different sites overlap, the normalization \(\langle\psi_{\mathbf{k}}|\psi_{\mathbf{k}}\rangle=S(\mathbf{k})\neq1\) must be divided out and the problem becomes generalized (Löwdin/Wannier orthogonalization needed); dropping this over-counts the bandwidth and breaks electron-hole symmetry. Translational invariance of the lattice and of \(\hat{H}\).Without it \(\langle\phi_{\mathbf{R'}}|\hat{H}|\phi_{\mathbf{R}}\rangle\) is not a function of \(\mathbf{R}-\mathbf{R'}\) alone, the double sum does not collapse, and \(\mathbf{k}\) ceases to be a good quantum number (Bloch's theorem fails). Hopping truncated to a finite shell (here nearest neighbours).Long-range hopping adds higher harmonics \(\cos(2ka),\dots\) to the dispersion; truncating too early distorts the band shape and can misplace van Hove singularities. Static lattice, single particle.Neglecting electron-phonon coupling and electron-electron interaction; if the on-site repulsion \(U\) is comparable to the bandwidth the band picture itself collapses (Mott physics) and the dispersion no longer describes the excitations.

Derivation

\[ |\psi_{n\mathbf{k}}\rangle=\frac{1}{\sqrt{N}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}\,|\phi_{n}(\mathbf{r}-\mathbf{R})\rangle \equiv \frac{1}{\sqrt{N}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}\,|\phi_{\mathbf{R}}\rangle \]

Ansatz: build the trial state as a phase-weighted Bloch sum of one orbital per lattice site \(\mathbf{R}\). This automatically satisfies Bloch's theorem \(\psi_{\mathbf{k}}(\mathbf{r}+\mathbf{R}_0)=e^{i\mathbf{k}\cdot\mathbf{R}_0}\psi_{\mathbf{k}}(\mathbf{r})\), so it is the correct symmetry-adapted basis. B

\[ E(\mathbf{k})=\frac{\langle\psi_{\mathbf{k}}|\hat{H}|\psi_{\mathbf{k}}\rangle}{\langle\psi_{\mathbf{k}}|\psi_{\mathbf{k}}\rangle} \]

The band energy is the expectation value in the Bloch state. For a single isolated band the Bloch sum is the exact eigenstate (up to the truncation), so the Rayleigh quotient returns the eigenvalue. B

\[ \langle\psi_{\mathbf{k}}|\hat{H}|\psi_{\mathbf{k}}\rangle=\frac{1}{N}\sum_{\mathbf{R}}\sum_{\mathbf{R'}}e^{i\mathbf{k}\cdot(\mathbf{R}-\mathbf{R'})}\,\langle\phi_{\mathbf{R'}}|\hat{H}|\phi_{\mathbf{R}}\rangle \]

Substitute the sum, using \(\langle\psi_{\mathbf{k}}|=N^{-1/2}\sum_{\mathbf{R'}}e^{-i\mathbf{k}\cdot\mathbf{R'}}\langle\phi_{\mathbf{R'}}|\). Linearity of \(\hat{H}\) and of the inner product lets the two lattice sums come outside. A

\[ \langle\phi_{\mathbf{R'}}|\hat{H}|\phi_{\mathbf{R}}\rangle=h(\mathbf{R}-\mathbf{R'}),\qquad h(\mathbf{S})\equiv\langle\phi_{\mathbf{0}}|\hat{H}|\phi_{\mathbf{S}}\rangle \]

Translational invariance: shifting both orbitals by any lattice vector leaves \(\hat{H}\) and the integral unchanged, so the matrix element depends only on the separation \(\mathbf{S}=\mathbf{R}-\mathbf{R'}\). This is the step that converts a double sum into a lattice Fourier series. C

\[ \langle\psi_{\mathbf{k}}|\hat{H}|\psi_{\mathbf{k}}\rangle=\frac{1}{N}\sum_{\mathbf{R'}}\sum_{\mathbf{S}}e^{i\mathbf{k}\cdot\mathbf{S}}\,h(\mathbf{S})=\sum_{\mathbf{S}}e^{i\mathbf{k}\cdot\mathbf{S}}\,h(\mathbf{S}) \]

Change variable \(\mathbf{R}\to\mathbf{S}=\mathbf{R}-\mathbf{R'}\) at fixed \(\mathbf{R'}\). The summand no longer depends on \(\mathbf{R'}\), so \(\sum_{\mathbf{R'}}1=N\) cancels the prefactor \(1/N\). This telescoping is exactly the diagonalization Bloch's theorem promises. C

\[ \langle\psi_{\mathbf{k}}|\psi_{\mathbf{k}}\rangle=\sum_{\mathbf{S}}e^{i\mathbf{k}\cdot\mathbf{S}}\,\langle\phi_{\mathbf{0}}|\phi_{\mathbf{S}}\rangle=\sum_{\mathbf{S}}e^{i\mathbf{k}\cdot\mathbf{S}}\,\delta_{\mathbf{S},\mathbf{0}}=1 \]

The overlap denominator collapses by the same manipulation; with the orthonormality assumption only \(\mathbf{S}=\mathbf{0}\) survives, giving unity. This is where the "orthonormal orbitals" assumption enters. B

\[ \boxed{\,E(\mathbf{k})=\sum_{\mathbf{S}}e^{i\mathbf{k}\cdot\mathbf{S}}\,h(\mathbf{S})\,}\qquad h(\mathbf{S})=\langle\phi_{\mathbf{0}}|\hat{H}|\phi_{\mathbf{S}}\rangle \]

Divide numerator by the unit denominator. The band energy is the lattice Fourier transform of the hopping integrals. This is the general single-band tight-binding result. A

\[ E(\mathbf{k})=\underbrace{h(\mathbf{0})}_{\epsilon}+\sum_{\mathbf{S}\neq\mathbf{0}}e^{i\mathbf{k}\cdot\mathbf{S}}\,h(\mathbf{S}) =\epsilon-\sum_{\mathbf{S}\neq\mathbf{0}}t(\mathbf{S})\,e^{i\mathbf{k}\cdot\mathbf{S}} \]

Separate the on-site term \(\epsilon=h(\mathbf{0})\) (a rigid, \(\mathbf{k}\)-independent shift) and define the transfer/hopping integral \(t(\mathbf{S})=-h(\mathbf{S})=-\langle\phi_{\mathbf{0}}|\hat{H}|\phi_{\mathbf{S}}\rangle\) with the conventional minus sign so \(t>0\) for \(s\)-orbitals. B

\[ E(\mathbf{k})=\epsilon-t\sum_{\boldsymbol{\delta}}e^{i\mathbf{k}\cdot\boldsymbol{\delta}} \]

Nearest-neighbour truncation: keep only the vectors \(\boldsymbol{\delta}\) to the \(z\) nearest neighbours, where by point symmetry all \(t(\boldsymbol{\delta})\equiv t\) are equal for an \(s\)-orbital. Farther shells are exponentially smaller because orbital overlap decays with distance. B

\[ E(k)=\epsilon-t\left(e^{ika}+e^{-ika}\right)=\epsilon-2t\cos(ka) \]

Specialize to the 1D chain: the two neighbours sit at \(\boldsymbol{\delta}=\pm a\hat{x}\). Euler's identity \(e^{ika}+e^{-ika}=2\cos(ka)\) gives the closed-form dispersion. The generalization to simple cubic follows immediately: \(E(\mathbf{k})=\epsilon-2t\big[\cos(k_xa)+\cos(k_ya)+\cos(k_za)\big]\). A

Result

\[ E(\mathbf{k})=\sum_{\mathbf{S}}e^{i\mathbf{k}\cdot\mathbf{S}}\,h(\mathbf{S}) \;\;\xrightarrow[\text{1D chain}]{\text{NN}}\;\; E(k)=\epsilon-2t\cos(ka) \]

Reading. A single atomic level of energy \(\epsilon\) broadens into a continuous band as orbitals on neighbouring sites overlap and let the electron hop with amplitude \(t\). The band is a cosine of total width \(W=2zt\) (\(=4t\) for the chain, \(z=2\); \(=12t\) for simple cubic, \(z=6\)): larger overlap means wider bands and lighter carriers. Near the band bottom the cosine is parabolic, \(E\simeq(\epsilon-2t)+t a^2 k^2\), so the electron behaves as a free particle with effective mass \(m^{*}=\hbar^{2}/(2ta^{2})\) — the lattice has replaced the bare mass with one set by the hopping.

Units check. \(h(\mathbf{S})\) and \(t\) are Hamiltonian matrix elements, hence energies (J or eV); \(e^{i\mathbf{k}\cdot\mathbf{S}}\) is dimensionless, so \(E(\mathbf{k})\) is an energy. In \(m^{*}=\hbar^{2}/(2ta^{2})\): \(\dfrac{(\mathrm{J\,s})^{2}}{\mathrm{J}\cdot\mathrm{m}^{2}}=\dfrac{\mathrm{J\,s}^{2}}{\mathrm{m}^{2}}=\mathrm{kg}\). Correct.

Limiting cases

Atomic limit \(t\to0\): \(E(\mathbf{k})\to\epsilon\), a flat, dispersionless band — the electron is trapped on its site and \(m^{*}\to\infty\).
Band bottom, 1D: \(ka\to0\) gives \(E\simeq\epsilon-2t+ta^{2}k^{2}\), a free-electron parabola with \(m^{*}=\hbar^{2}/(2ta^{2})>0\).
Band top, 1D: \(ka\to\pi\) gives \(E\simeq\epsilon+2t-ta^{2}(k-\pi/a)^{2}\), an inverted parabola with negative curvature, \(m^{*}=-\hbar^{2}/(2ta^{2})\) (hole-like).
Half-filling on a bipartite lattice: the \(-2t\cos\) band is symmetric about \(\epsilon\); the Fermi level sits at the band centre and the density of states shows the characteristic van Hove peaks (1D edge divergences, 2D log saddle).
Wide-band / free-electron limit: as \(t\) grows so \(W\gg\epsilon\)-spacing, bands overlap and the single-orbital picture crosses over to nearly-free-electron behaviour.

Breaks when

Strong correlation (Mott regime). When the on-site Coulomb repulsion \(U\gtrsim W=2zt\), double occupancy is forbidden and the electrons localize into a Mott insulator; the non-interacting band \(E(\mathbf{k})\) no longer describes the low-energy excitations even though the lattice is metallic by band counting.
Overlapping / non-orthogonal orbitals. If \(\langle\phi_{\mathbf{0}}|\phi_{\mathbf{S}}\rangle=s(\mathbf{S})\neq0\), the denominator \(S(\mathbf{k})=1+\sum_{\mathbf{S}\neq0}s(\mathbf{S})e^{i\mathbf{k}\cdot\mathbf{S}}\) cannot be set to one; the true dispersion is \(E(\mathbf{k})=\big[\epsilon+\sum h(\mathbf{S})e^{i\mathbf{k}\cdot\mathbf{S}}\big]/S(\mathbf{k})\), which is asymmetric between band top and bottom.
Band mixing. If another orbital lies within \(\sim t\), the scalar formula is invalid and one must diagonalize a multi-band \(H_{mn}(\mathbf{k})\); at points where bands would cross, hybridization opens gaps the single-band cosine cannot capture.
Broken translational symmetry. Disorder, defects, a surface, or an applied field make \(\mathbf{k}\) not conserved; Bloch's theorem fails, the double sum does not collapse, and eigenstates may localize (Anderson).

Failure modes

Dropping the overlap denominator. Setting \(\langle\psi_{\mathbf{k}}|\psi_{\mathbf{k}}\rangle=1\) when the orbitals are non-orthogonal; this silently rescales the band and destroys top-bottom asymmetry.
Sign of \(t\). Forgetting the conventional minus sign in \(t=-h(\mathbf{S})\), which flips band bottom and top and gives the wrong sign of \(m^{*}\).
Treating \(\epsilon\) as physical curvature. The on-site term is a rigid shift; students sometimes let it "bend" the band. It never enters \(m^{*}\) or the bandwidth.
Using \(|\mathbf{k}|\) instead of the vector \(\mathbf{k}\). The phase is \(\mathbf{k}\cdot\mathbf{S}\); replacing it with \(ka\) in 2D/3D loses the anisotropy and mislocates van Hove points.
Double-counting neighbours. Summing over ordered pairs, or counting \(+\boldsymbol{\delta}\) and \(-\boldsymbol{\delta}\) as one, which doubles or halves the bandwidth.
Reading \(m^{*}\) at the wrong band edge. Taking the curvature at the band top for electrons at the bottom, getting a spurious negative mass.
Confusing bandwidth \(2zt\) with hopping \(t\). Quoting \(W=t\) or \(W=zt\); the correct 1D width is \(4t\), not \(2t\).

Discussion

The derivation is, at heart, a diagonalization by symmetry. Bloch's theorem tells us that \(\mathbf{k}\) labels the irreducible representations of the translation group, so the Hamiltonian is already block-diagonal in the Bloch-sum basis. All the double sum does is confirm this: the \(N\times N\) hopping matrix reduces, block by block, to a single number \(E(\mathbf{k})\) which is the discrete Fourier transform of the real-space hoppings. The band and the hopping integrals form a Fourier-transform pair — measure the band by ARPES and you can, in principle, invert to read off \(t(\mathbf{S})\).

Physically the model quantifies the trade-off at the heart of chemistry: localization versus delocalization. A tightly bound electron has low kinetic energy but, once orbitals touch, gains energy \(\sim zt\) by spreading over the crystal. The bandwidth \(W=2zt\) is therefore set jointly by how many neighbours there are (\(z\), the coordination) and how strongly adjacent orbitals overlap (\(t\)). Narrow \(d\)- and \(f\)-bands, broad \(s\)-\(p\) bands, and the pressure-driven metallization of insulators are all this one competition in different guises.

The orthonormality assumption deserves scrutiny. Atomic orbitals on different sites are not orthogonal, so the honest object is the Wannier function — the maximally localized, mutually orthogonal orbital obtained by Löwdin-symmetrizing the atomic set, \(|w_{\mathbf{R}}\rangle=\sum_{\mathbf{R'}}(S^{-1/2})_{\mathbf{R}\mathbf{R'}}|\phi_{\mathbf{R'}}\rangle\). In the Wannier basis the derivation is exact for an isolated band: \(t(\mathbf{S})=\langle w_{\mathbf{0}}|\hat{H}|w_{\mathbf{S}}\rangle\) are the true Fourier coefficients of the ab-initio band, and modern DFT codes "downfold" onto exactly this tight-binding form. The naive non-orthogonal calculation instead yields a generalized eigenvalue problem \(H(\mathbf{k})c=E(\mathbf{k})S(\mathbf{k})c\), whose asymmetry between band top and bottom is a real, measurable fingerprint of overlap.

The single-band cosine is also the parent of richer models. Add a second site to the basis and you get graphene's honeycomb Dirac cones; add spin-orbit coupling to the hopping and you get topological insulators; add on-site repulsion \(U\) and you get the Hubbard model and Mott physics. In every case the kinetic backbone is the lattice Fourier sum derived here. Common misconceptions: the band does not exist "inside" a single atom — it is a collective, crystal-wide phenomenon that vanishes if any one link \(t\) is cut across the whole lattice; \(t\) is not a hopping rate but an energy amplitude (the rate involves \(t/\hbar\)); and a filled tight-binding band still carries no net current, because \(\sum_{\mathbf{k}}\nabla_{\mathbf{k}}E=0\) over the full Brillouin zone regardless of how large \(t\) is.

Worked examples

Effective mass of a 1D chain. Given \(t=1.0\ \mathrm{eV}\), \(a=3.0\ \mathrm{\AA}\). Find the bandwidth and the band-bottom effective mass \(m^{*}/m_e\).

Symbols first, numbers last. A

\[ W=2zt=2(2)t=4t \]

1D chain has \(z=2\) nearest neighbours, so bandwidth \(W=4t\). A

\[ E(k)\simeq(\epsilon-2t)+ta^{2}k^{2}\ \Rightarrow\ \frac{\hbar^{2}}{2m^{*}}=ta^{2}\ \Rightarrow\ m^{*}=\frac{\hbar^{2}}{2ta^{2}} \]

Match the small-\(k\) parabola to \(\hbar^2k^2/2m^*\). B

\[ W=4(1.0\ \mathrm{eV})=4.0\ \mathrm{eV} \]

\[ m^{*}=\frac{(1.055\times10^{-34})^{2}}{2(1.602\times10^{-19})(3.0\times10^{-10})^{2}}=\frac{1.113\times10^{-68}}{2.884\times10^{-38}}=3.86\times10^{-31}\ \mathrm{kg} \]

\[ \frac{m^{*}}{m_e}=\frac{3.86\times10^{-31}}{9.11\times10^{-31}}=0.42 \]

Insert numbers with \(1\ \mathrm{eV}=1.602\times10^{-19}\ \mathrm{J}\), \(\hbar=1.055\times10^{-34}\ \mathrm{J\,s}\). A

\[ W=4.0\ \mathrm{eV},\qquad m^{*}=3.9\times10^{-31}\ \mathrm{kg}\approx0.42\,m_e \]

Reading. A wide 1D band gives a light carrier; halving \(t\) would double \(m^{*}\) and halve the width. Units: eV for \(W\), kg for \(m^{*}\).

Group velocity on a simple-cubic band. Given \(t=0.50\ \mathrm{eV}\), \(a=4.0\ \mathrm{\AA}\). Find the bandwidth and the \(x\)-component of the group velocity at \(\mathbf{k}=(\pi/2a,0,0)\).

Symbols first. A

\[ E(\mathbf{k})=\epsilon-2t\big[\cos(k_xa)+\cos(k_ya)+\cos(k_za)\big],\qquad W=2zt=12t \]

Simple cubic has \(z=6\); each cosine ranges over \([-1,1]\), so \(E\in[\epsilon-6t,\ \epsilon+6t]\), width \(12t\). A

\[ v_x=\frac{1}{\hbar}\frac{\partial E}{\partial k_x}=\frac{2ta}{\hbar}\sin(k_xa) \]

Group velocity \(\mathbf{v}=\hbar^{-1}\nabla_{\mathbf{k}}E\). B

\[ k_xa=\frac{\pi}{2}\ \Rightarrow\ \sin(k_xa)=1 \]

\[ W=12(0.50\ \mathrm{eV})=6.0\ \mathrm{eV} \]

\[ v_x=\frac{2(0.50\times1.602\times10^{-19})(4.0\times10^{-10})}{1.055\times10^{-34}}(1)=\frac{6.41\times10^{-29}}{1.055\times10^{-34}}=6.1\times10^{5}\ \mathrm{m\,s^{-1}} \]

Numbers last, with \(t\) converted to joules. A

\[ W=6.0\ \mathrm{eV},\qquad v_x=6.1\times10^{5}\ \mathrm{m\,s^{-1}} \]

Reading. The velocity is maximal at the zone-face midpoint (\(k_xa=\pi/2\)) and vanishes at the zone centre and boundary, where the band is flat. Units: eV for \(W\), \(\mathrm{m\,s^{-1}}\) for \(v_x\).

Problems

(Band edges, easy.) A 1D chain has \(\epsilon=-2.0\ \mathrm{eV}\) and \(t=0.80\ \mathrm{eV}\). Find the minimum and maximum band energies and the bandwidth.
Solution
\(E(k)=\epsilon-2t\cos(ka)\). Minimum at \(ka=0\): \(E_{\min}=\epsilon-2t=-2.0-1.6=-3.6\ \mathrm{eV}\). Maximum at \(ka=\pi\): \(E_{\max}=\epsilon+2t=-2.0+1.6=-0.4\ \mathrm{eV}\). Bandwidth \(W=E_{\max}-E_{\min}=4t=3.2\ \mathrm{eV}\).
(Hole mass at band top.) For the same chain (\(t=0.80\ \mathrm{eV}\), \(a=2.5\ \mathrm{\AA}\)), find the effective mass at the top of the band and state its sign.
Solution
Near \(ka=\pi\) write \(q=k-\pi/a\): \(\cos(ka)\simeq-1+\tfrac12(qa)^2\), so \(E\simeq\epsilon+2t-ta^2q^2\), giving \(\partial^2E/\partial k^2=-2ta^2<0\) and \(m^*=\hbar^2/(\partial^2E/\partial k^2)=-\hbar^2/(2ta^2)\). Magnitude: \(|m^*|=(1.055\times10^{-34})^2/[2(0.80\times1.602\times10^{-19})(2.5\times10^{-10})^2]=1.113\times10^{-68}/(1.602\times10^{-38})=6.9\times10^{-31}\ \mathrm{kg}\approx0.76\,m_e\). The mass is negative (hole-like); carriers near the top respond to a field as positive charges.
(2D square lattice, van Hove.) For \(E(\mathbf{k})=\epsilon-2t(\cos k_xa+\cos k_ya)\), find the energies at \(\Gamma=(0,0)\), \(X=(\pi/a,0)\), and \(M=(\pi/a,\pi/a)\), the bandwidth, and identify which point is a saddle.
Solution
\(\Gamma\): \(E=\epsilon-2t(1+1)=\epsilon-4t\) (band minimum). \(X\): \(E=\epsilon-2t(-1+1)=\epsilon\) (band centre). \(M\): \(E=\epsilon-2t(-1-1)=\epsilon+4t\) (band maximum). Bandwidth \(W=8t=2zt\) with \(z=4\). At \(X\) the curvature is positive along \(k_y\) but negative along \(k_x\) (\(\partial^2E/\partial k_x^2=2ta^2\cos k_xa=-2ta^2\), \(\partial^2E/\partial k_y^2=+2ta^2\)), so \(X\) is a saddle point — the source of the logarithmic van Hove singularity in the 2D density of states at \(E=\epsilon\).
(Where the current vanishes.) For the 1D chain show where the group velocity is zero and evaluate its maximum magnitude for \(t=1.2\ \mathrm{eV}\), \(a=3.0\ \mathrm{\AA}\).
Solution
\(v(k)=\hbar^{-1}dE/dk=(2ta/\hbar)\sin(ka)\). It vanishes where \(\sin(ka)=0\), i.e. at the zone centre \(k=0\) and zone boundary \(k=\pi/a\) — both band edges are flat. Maximum at \(ka=\pi/2\): \(v_{\max}=2ta/\hbar=2(1.2\times1.602\times10^{-19})(3.0\times10^{-10})/(1.055\times10^{-34})=1.153\times10^{-28}/1.055\times10^{-34}=1.1\times10^{6}\ \mathrm{m\,s^{-1}}\).
(Non-orthogonal correction, hard.) Including nearest-neighbour overlap \(s\), the 1D dispersion is \(E(k)=\dfrac{\epsilon+2\beta\cos(ka)}{1+2s\cos(ka)}\), where \(\beta=h(a)=-t\). Take \(\epsilon=0\), \(t=1.0\ \mathrm{eV}\), \(s=0.10\). Find the band bottom (\(ka=0\)) and top (\(ka=\pi\)) energies, show the band is asymmetric, and confirm it reduces to \(-2t\cos(ka)\) as \(s\to0\).
Solution
With \(\epsilon=0\), \(\beta=-t\): \(E(k)=-2t\cos(ka)/[1+2s\cos(ka)]\). Bottom, \(ka=0\): \(E_{\min}=-2t/(1+2s)=-2.0/1.2=-1.67\ \mathrm{eV}\). Top, \(ka=\pi\): \(E_{\max}=+2t/(1-2s)=2.0/0.8=+2.50\ \mathrm{eV}\). The magnitudes differ (\(1.67\) vs \(2.50\ \mathrm{eV}\)): overlap pushes the antibonding top up more than it pushes the bonding bottom down, so the band is asymmetric about \(\epsilon=0\) — a direct signature of non-orthogonality. As \(s\to0\) the denominator \(\to1\) and \(E(k)\to-2t\cos(ka)\), recovering the orthonormal result with symmetric edges \(\mp2t\).