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Band Theory

Field: Solid State Physics

A theoretical model for the electronic structure of solids.

Sequence of Expressions

Definition

Energy Bands

The allowed energy spectrum EE for an electron in a crystal lattice is defined by the function E(k)E(\mathbf{k}), where k\mathbf{k} is the crystal momentum restricted to the first Brillouin zone (FBZ). The existence of these continuous bands is a direct consequence of the periodicity of the potential V(r)V(\mathbf{r}) and the Bloch theorem. Mathematically, the allowed energies EE are the eigenvalues of the Hamiltonian HH for which the secular equation has a solution for k∈FBZ\mathbf{k} \in \text{FBZ}. The bands are characterized by the dispersion relation En(k)E_n(\mathbf{k}), where nn is the band index.
Definition

Band Gap

A band gap is a range of forbidden energies ΔE=(Egap,min,Egap,max)\Delta E = (E_{gap, min}, E_{gap, max}) such that the time-independent Schrödinger equation, (−ℏ22m∇2+V(r)−E)ψ=0\left(-\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) - E\right) \psi = 0, possesses no solution ψ\psi satisfying the boundary conditions of the crystal lattice for any crystal momentum k\mathbf{k}. Formally, the gap exists when the determinant condition derived from the periodic potential yields no real solution for the wave vector β\beta for a given energy EE. The gap width is ΔE=Egap,max−Egap,min\Delta E = E_{gap, max} - E_{gap, min}.
Definition

Fermi Level

Let μ\mu be the chemical potential (Fermi level). The probability P(E)P(E) that an energy state EE is occupied by an electron at temperature TT is given by the Fermi-Dirac distribution function:\nP(E)=f(E)=1e(E−μ)/kBT+1P(E) = f(E) = \frac{1}{e^{(E - \mu)/k_B T} + 1}
For a system of fermions, the occupation number nin_i of any single quantum state ii must satisfy the constraint:\nni∈{0,1}n_i \in \{0, 1\}
Let V(r)V(\mathbf{r}) be a periodic potential with lattice constant R\mathbf{R}, such that V(r)=V(r+R)V(\mathbf{r}) = V(\mathbf{r} + \mathbf{R}). For the time-independent Schrödinger equation, (−ℏ22m∇2+V(r))ψ(r)=Eψ(r)\left(-\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r})\right) \psi(\mathbf{r}) = E \psi(\mathbf{r}), the solutions ψk(r)\psi_k(\mathbf{r}) must take the form:\nψk(r)=eik⋅ruk(r)\psi_k(\mathbf{r}) = e^{i \mathbf{k} \cdot \mathbf{r}} u_k(\mathbf{r})\nwhere k\mathbf{k} is the crystal momentum (or wave vector), and uk(r)u_k(\mathbf{r}) is a function that has the same periodicity as the lattice: uk(r)=uk(r+R)u_k(\mathbf{r}) = u_k(\mathbf{r} + \mathbf{R}). This implies that the allowed energies EE are functions of k\mathbf{k}, E(k)E(\mathbf{k}).
Consider the Hamiltonian H=H0+V(r)H = H_0 + V(\mathbf{r}), where H0=p22mH_0 = \frac{\mathbf{p}^2}{2m} is the free electron Hamiltonian and V(r)V(\mathbf{r}) is the periodic potential. Using the plane wave basis set, ψk(r)=∑GcGeiG⋅r\psi_k(\mathbf{r}) = \sum_{\mathbf{G}} c_{\mathbf{G}} e^{i \mathbf{G} \cdot \mathbf{r}}, the energy eigenvalues EE and coefficients cGc_{\mathbf{G}} are determined by solving the secular equation derived from the matrix representation of HH: \ndet⁡((E−ℏ22mk⋅k)δG,G′−VG−G′)=0\det \left( \left(E - \frac{\hbar^2}{2m} \mathbf{k} \cdot \mathbf{k} \right) \delta_{\mathbf{G}, \mathbf{G}'} - V_{\mathbf{G}-\mathbf{G}'} \right) = 0 \nwhere G\mathbf{G} and G′\mathbf{G}' are reciprocal lattice vectors, and VG−G′V_{\mathbf{G}-\mathbf{G}'} are the Fourier components of the potential V(r)V(\mathbf{r}).
Define the periodic potential V(x)V(x) as a sequence of rectangular barriers of width aa and height V0V_0, separated by wells of width bb, such that the period is L=a+bL = a+b. The transmission coefficient TT for an electron with energy EE across one period is given by the transfer matrix method. The condition for wave propagation (i.e., T=1T=1) leads to the transcendental equation:\ncos⁡(βL)=cos⁡(βb)cosh⁡(κa)−12γ(γsinh⁡(κa)sin⁡(βb))\cos(\beta L) = \cos(\beta b) \cosh(\kappa a) - \frac{1}{2 \gamma} \left( \gamma \sinh(\kappa a) \sin(\beta b) \right) \nwhere β=2mE/ℏ\beta = \sqrt{2mE}/\hbar, γ=V0/E−1\gamma = V_0/E - 1, and κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0 - E)}/\hbar. Allowed energies EE correspond to real values of βL\beta L, while forbidden energies correspond to complex βL\beta L.
The density of states g(E)g(E) for a system with periodic boundary conditions in a volume VV is defined by the summation over the Brillouin zone ΩB\Omega_B in k\mathbf{k}-space:\ng(E)=V(2π)3∑kδ(E−Ek)g(E) = \frac{V}{(2\pi)^3} \sum_{\mathbf{k}} \delta(E - E_{\mathbf{k}})
The effective mass tensor mij∗m^*_{ij} for an electron band E(k)E(\mathbf{k}) is defined by the inverse of the curvature of the dispersion relation with respect to the wave vector k\mathbf{k}:\n(1m∗)ij=1ℏ2∂2E(k)∂ki∂kj∣k=k0\left( \frac{1}{m^*} \right)_{ij} = \frac{1}{\hbar^2} \frac{\partial^2 E(\mathbf{k})}{\partial k_i \partial k_j} \bigg|_{\mathbf{k}=\mathbf{k}_0}
For a band En(k)E_n(\mathbf{k}) in a crystal with periodic potential V(r)V(\mathbf{r}), the energy spectrum must satisfy the symmetry condition relating the energy at k\mathbf{k} and k+G\mathbf{k} + \mathbf{G}, where G\mathbf{G} is a reciprocal lattice vector:\nEn(k)+En(k+G)=En(k)+En(k+G)(This is a general symmetry property, often simplified to En(k)=En(−k) for specific cases.)E_n(\mathbf{k}) + E_n(\mathbf{k} + \mathbf{G}) = E_n(\mathbf{k}) + E_n(\mathbf{k} + \mathbf{G}) \quad \text{(This is a general symmetry property, often simplified to } E_n(\mathbf{k}) = E_n(-\mathbf{k}) \text{ for specific cases.)}