Wave Equation of Electrons in Periodic Potentials & Energy-Band Formation
The potential experienced by an electron in a 1-D crystal lattice, due to the array of positively charged ions, exhibits translational invariance with a lattice spacing a. This means the potential at any point x is identical to the potential at x+a:
Periodic Potential and Its Fourier Representation
U(x)=U(x+a)
Any function possessing this exact periodicity can be precisely represented as a Fourier series. The terms in this series are indexed by reciprocal-lattice vectors G, which are defined as integral multiples of 2\pi/a:
G=\dfrac{2\pi n}{a},\;n\in\mathbb Z
The Fourier series expansion for the periodic potential is given by:
U(x)=\sumG UG e^{iGx} \qquad (22)
Physical constraints are placed upon the Fourier coefficients U_G:
For the potential U(x) to be real, the coefficients must satisfy the condition U{-G}=UG^{* }. If the lattice possesses inversion symmetry (centro-symmetric), the potential is symmetric, allowing U_G to be real (meaning its phase can be set to zero).
The magnitude of these coefficients, |U_G|, typically decays rapidly as the magnitude of the reciprocal vector |G| increases. For instance, for a bare Coulomb potential, the decay follows a 1/G^2 proportionality.
One-Electron Schrödinger Equation
Within the one-electron (or independent-electron) approximation, each electron is considered to move independently in the average periodic potential created by the ions and all other electrons. The Hamiltonian for a single electron is:
H=\frac{p^2}{2m}+U(x)
The fundamental eigenproblem to be solved for the electron's states and energies is:
H\,\psi(x)=\varepsilon\,\psi(x) \qquad (24)
The solutions \psi(x) to this equation are referred to as eigenfunctions, orbitals, or, more specifically in a periodic potential, Bloch functions.
Fourier Expansion of the Wavefunction
To properly describe the system, we impose periodic boundary conditions over a macroscopic length L=N a, which effectively represents a ring of N unit cells. This condition determines the allowed values of the crystal momentum k. The allowed crystal momenta are discrete and given by:
k=\dfrac{2\pi n}{L},\;n\in\mathbb Z
The electron's orbital \psi(x) can be expanded as a superposition of plane waves, reflecting the delocalized nature of electrons in a crystal:
\psi(x)=\sum_{k} C(k) e^{ikx} \qquad (25)
A critical property, known as the Bloch property (which is proven subsequently), dictates that if a plane-wave component with momentum k is present in the expansion, then all components k+G (where G is any reciprocal lattice vector) must also be present.
This concept is visually represented in Fig. 7: the lower row depicts all possible 2\pi n/L values, while the upper row illustrates the specific subset k+G linked to a fixed "label" k chosen within the first Brillouin zone.
Restatement & Proof of Bloch Theorem
The Fourier expansion of the wavefunction can be ingeniously rearranged to explicitly show the Bloch form:
\psi(x)=e^{ikx}\,u_k(x)
where uk(x)=\sumG C(k-G) e^{iGx}. This function u_k(x) is often called the cell-periodic part of the Bloch function.
Because the sum defining uk(x) consists of terms e^{iGx} (where G are reciprocal lattice vectors), a lattice translation by T=ma (an integer multiple of the lattice spacing) will multiply each term by e^{iG(ma)}=e^{i(2\pi n/a)(ma)}=e^{i2\pi nm}=1. Consequently, uk(x+T)=uk(x), meaning uk(x) is periodic in real space with the same periodicity as the lattice potential.
This periodicity directly leads to the Bloch theorem for the overall wavefunction:
\psik(x+T)=e^{ikT}\,\psik(x).
This theorem is fundamental: it states that electron wavefunctions in a periodic potential can be written as a plane wave e^{ikx} multiplied by a function u_k(x) that has the same periodicity as the lattice itself. The phase factor e^{ikT} is crucial.
Central (Algebraic) Equation
To find the allowed solutions and energies, we substitute the plane-wave series (25) for \psi(x) and the Fourier series (22) for U(x) into the Schrödinger Equation (24). By equating coefficients of each Fourier component e^{ik'x} (where k' belongs to the set of allowed momenta), we obtain an infinite set of linear algebraic equations. For every G, a general component is:
\Big[\frac{\hbar^2}{2m}(k-G)^2-\varepsilon\Big]C(k-G)+\sum{G'}U{G-G'}\,C(k-G')=0
In the simplified but illustrative scenario where the potential contains only a single dominant Fourier amplitude, say U\equiv U_G (and where |G| represents the shortest reciprocal vector, often 2\pi/a), the infinite set of coupled equations truncates significantly. For electrons near a zone boundary, where k and k-G components are strongly coupled, the relevant equations collapse to:
\Big(\frac{\hbar^2}{2m}k^2-\varepsilon\Big)C(k)+U\,C(k-G)=0
\Big(\frac{\hbar^2}{2m}(k-G)^2-\varepsilon\Big)C(k-G)+U\,C(k)=0
And similarly for other harmonics if included.
This infinite set of linear equations can be written in matrix form. Non-trivial solutions for the coefficients C(k) (and thus for \psi(x)) exist only if the determinant of this matrix (the secular condition) is zero. In practical calculations, this infinite matrix is always truncated (e.g., to a 2×2, 4×4, or larger block matrix) because the higher Fourier components (U_G for large |G|) decay rapidly, making their contribution negligible.
Crystal Momentum & Selection Rules
Under a lattice translation T, the Bloch wavefunction transforms as:
\psik(r+T)=e^{ik\cdot T}\,\psik(r)
The phase factor e^{ikT} signifies that the quantity \hbar k behaves as a momentum-like quantity in the crystal, hence it is termed crystal momentum. Unlike canonical momentum, crystal momentum is only conserved modulo a reciprocal lattice vector.
In scattering processes involving electrons within the crystal, the conservation law for crystal momentum is modified to:
k{initial}+q{scattered}=k_{final}+G
For example, if an electron in state k absorbs a phonon with momentum q, it transitions to a new state k'. This process is only allowed if the sum of the initial crystal momentum and the phonon momentum equals the final crystal momentum plus (or minus) some reciprocal lattice vector G. This rule dictates selection rules for various phenomena like optical absorption, phonon scattering, and X-ray diffraction in crystals.
Kronig–Penney Model in Reciprocal Space
The Kronig–Penney model offers an exactly solvable and insightful illustration of band theory. It employs a highly simplified, yet physically relevant, periodic potential: an array of Dirac \delta -functions, each of strength A, placed at every lattice point x=s a, where s is an integer:
U(x)=A a \sum_s \delta(x-s a)
This potential can be expressed as a Fourier series: 2\sumG U\cos(Gx), where all Fourier components UG are equal to A a (i.e., U_G = A a for all G \ne 0).
When this specific potential is substituted into the central equation and simplified, it yields:
(\varepsilon-\varepsilonk)C(k)+A\sumn C(k-2\pi n/a)=0 \quad (35)
An auxiliary sum is defined as f(k)=\sum_n C(k-2\pi n/a). By applying periodic shifts to this sum and manipulating the equations, one can derive a relationship for C(k):
C(k)=\dfrac{A}{\varepsilon_k-\varepsilon}f(k)
Following a standard mathematical procedure involving cotangent summation (which arises from the specific form of the \delta -function potential in Fourier space), the celebrated dispersion relation for the Kronig-Penney model emerges:
\boxed{\Big(\frac{m a}{\hbar^2 K a}\Big)\sin(K a)+\cos(K a)=\cos(k a)} \qquad (43)
Here, K=\sqrt{2m\varepsilon}/\hbar. This equation is identical to the result obtained using real-space methods for the Kronig–Penney model, with the parameter P=m A a/\hbar^2 representing the barrier strength.
Figs. 5 and 6 (as referenced in the original context) illustrate key features:
Fig. 5 shows the left-hand side of Eq. (43), the function P\sin Ka/(Ka)+\cos Ka, plotted against Ka. The allowed energy bands correspond to the segments where this function lies between \pm1. Regions where it falls outside this range represent forbidden energy gaps.
Fig. 6 displays the resulting energy \varepsilon versus crystal momentum k, strikingly revealing the presence of forbidden energy gaps (band gaps) at ka=\pi,\,2\pi,3\pi,\dots, which correspond to the Brillouin zone boundaries.
Empty Lattice Approximation ("Reduced-Zone" Picture)
In the case of very weak potentials, the electronic band structure closely resembles that of free electrons. The empty lattice approximation serves as a crucial starting point for understanding and calculating more complex band structures. It involves "folding" the parabolic free-electron dispersion curve \varepsilon=\hbar^2 k^2/2m back into the first Brillouin zone.
For an arbitrary wave vector k outside the first Brillouin zone, a reciprocal lattice vector G is chosen such that k' = k - G lies within the first Brillouin zone.
The energy is then written as \varepsilon(k)=\dfrac{\hbar^2}{2m}(k+G)^2 but is plotted versus this reduced wave vector k'. This creates multiple "folded" parabolic branches within the first Brillouin zone.
For example, in a simple cubic lattice with \hat k along the [100] direction (and taking \hbar^2/2m=1 for simplicity), the first few folded bands correspond to contributions from different plane-wave indices like (000), (100), (010), (001), (110), etc. Fig. 8 (bold curves within the zone) visually depicts this reduced-zone scheme, showing how the free-electron parabola effectively wraps back into the central zone.
Two-Wave ((2\times2)) Approximation Near a Zone Boundary
This approximation is particularly insightful near a Brillouin zone boundary, specifically where k=G/2. At this point, two free-electron plane waves, e^{ikx} and e^{i(k-G)x}, have degenerate energies (\varepsilonk=\varepsilon{k-G}). This degeneracy is lifted by the periodic potential, leading to the formation of an energy gap.
In this 2\times2 approximation, we retain only the coefficients C(k) and C(k-G) (where here we let C(k) denote C(+G/2) and C(k-G) denote C(-G/2)). The central equations simplify to a pair of coupled linear equations:
\big(\varepsilon_k-\varepsilon\big)C(k)+U C(k-G)=0 \qquad (44)
\big(\varepsilon_{k-G}-\varepsilon\big)C(k-G)+U C(k)=0 \qquad (45)
For non-trivial solutions (i.e., not all coefficients being zero), the determinant of the coefficient matrix must vanish:
\big(\varepsilonk-\varepsilon\big)\big(\varepsilon{k-G}-\varepsilon\big)-U^2=0 \qquad (46)
Exactly at the zone boundary where k=G/2, we have \varepsilonk=\varepsilon{k-G}=\varepsilon_0=\dfrac{\hbar^2 G^2}{8m}. Substituting this into Eq. (46) yields a quadratic equation for \varepsilon, providing two distinct eigenvalues:
\varepsilon{\pm}=\varepsilon0\,\pm U \qquad (47)
This result demonstrates the opening of an energy gap of width 2|U| precisely at the zone edge, due to the interaction of the degenerate plane waves caused by the periodic potential.
The ratio of the coefficients for the two resulting eigenstates can be found from either Eq. (44) or (45):
\dfrac{C(k-G)}{C(k)}=-\dfrac{\varepsilon_k-\varepsilon}{U}=\mp1 \qquad (48)
This implies that the eigenfunctions at the zone boundary are combinations of the two degenerate plane waves that form even and odd standing waves, respectively:
For the lower energy state (bonding-like): \psi_{\text{lower}}\propto\cos\big(G x/2\big) (where electron density is concentrated between ions, attractive).
For the upper energy state (antibonding-like): \psi_{\text{upper}}\propto\sin\big(G x/2\big) (where electron density is concentrated at ion positions, repulsive).
For k slightly perturbed away from the zone boundary, let's define a small deviation \tilde K=k-G/2. The two-band dispersion relation, obtained by expanding Eq. (46) to leading order in \tilde K, becomes:
\varepsilon{\pm}(k)=\varepsilon0+\frac{\hbar^2 G}{m}\tilde K\pm\sqrt{U^2+\Big(\frac{\hbar^2 G}{2m}\tilde K\Big)^2} \qquad (51)
This can be symmetrically expressed as:
\varepsilon{\pm}(k)=\frac{\varepsilon{+}+\varepsilon{-}}{2}\,\pm\,\sqrt{U^2+\big(\hbar vF \tilde K\big)^2} \qquad (52)
Here, v_F=\hbar G/2m represents a velocity equivalent to the Fermi velocity at the crossing point of the free-electron bands before the potential is turned on.
Fig. 9 (as referenced) clearly illustrates how the interacting (split) bands diverge from the free-electron parabola near the zone boundary, showing the energy gap. Fig. 10 (as referenced) plots the probability weights |C(k)|^2 versus k, vividly demonstrating the gradual transfer of weight between the two interacting plane-wave components as k moves across the zone boundary.
Physical & Conceptual Take-Aways
The fundamental Bloch’s theorem, enabled by lattice periodicity, transforms the complex differential Schrödinger equation into a more manageable algebraic (matrix) problem in reciprocal space.
Electron states in a periodic potential are organized into bands of allowed energies, which are separated by gaps (forbidden energy regions). These gaps intrinsically arise when a Fourier component of the periodic potential causes degenerate plane waves to mix and split their energy levels.
Crucially, crystal momentum \hbar k is conserved in scattering and optical processes, but only up to an additive (or subtractive) reciprocal lattice vector \hbar G. This unique conservation law dictates the selection rules that govern all electron-related quantum transitions in crystals.
The Kronig–Penney model serves as an invaluable, exactly solvable example that beautifully links underlying concepts: Fourier analysis, the implications of boundary conditions, and the profound emergence of forbidden energy regions, providing a clear microscopic picture of band formation.
The empty-lattice approximation provides the foundational starting point for practical band-structure calculations. It folds the unperturbed free-electron parabola into the first Brillouin zone. When a weak periodic potential is subsequently "turned on," it primarily "pushes apart" these nearly degenerate folded branches at the Brillouin zone edges, leading to band gaps.
In computational practice, only a limited number of Fourier components are usually retained due to their rapid decay. The 2\times2 approximation (focusing on two interacting plane waves) provides remarkably transparent analytic expressions for the dispersion relation and the character of wavefunctions in the vicinity of these fundamental energy gaps.
Connections & Broader Relevance
The central-equation machinery and the concepts of dispersion relations and gaps are not exclusive to electron physics in crystals. They reappear in analogous forms across diverse fields:
X-ray diffraction: Described by Maxwell's equations for electromagnetic waves propagating through a periodic dielectric medium, leading to Bragg's law and photonic band gaps.
Phonon dispersion: Governed by the classical equations of motion for vibrations in a periodic mass-spring lattice, leading to acoustic and optical phonon branches and phonon band gaps.
Photonic/phononic crystals: Engineered materials where periodic structures are designed to create specific band gaps (for photons or phonons) that are exploited in advanced optical and acoustic technologies.
From an ethical and practical standpoint, a deep understanding of band gaps is the cornerstone of semiconductor design. This knowledge directly underpins the functionality of all modern electronic devices, from microprocessors to transistors, and is also fundamental to the development of efficient photovoltaic cells and other energy technologies.