Aula 5-6 Phoenix Solid State Physics PDF

# Aula 3 Solid State Physics ## Chapter 8: Electron levels in a periodic potential: General properties ### Contents: 1. Introduction 2. The periodic potential and Bloch's theorem 3. The Born-Von Karman Boundary condition 4. Second proof of Bloch's theorem 5. Band structure of the solid 6. The Ferm...

# Aula 3 Solid State Physics ## Chapter 8: Electron levels in a periodic potential: General properties ### Contents: 1. Introduction 2. The periodic potential and Bloch's theorem 3. The Born-Von Karman Boundary condition 4. Second proof of Bloch's theorem 5. Band structure of the solid 6. The Fermi surface 7. Density of states ## 1. Introduction Because the ions in a perfect crystal are arranged in a regular periodic array, we are led to consider the problem of an electron in a potential $U(r)$ with the periodicity of the underlying Bravais lattice; i.e., $U(r + R) = U(r)$ (8.1) for all Bravais lattice vectors R. The periodicity of the potential is a consequence of the lattice periodicity! In this chapter, we shall discuss those properties of the electronic levels that depend only on the periodicity of the potential, without regard to its particular form. The discussion will be continued in Chapters 9 and 10 in two limiting cases of great physical interest that provide more concrete illustrations of the general results of this chapter. ### Warnings!!! We emphasize at the outset that perfect periodicity is an idealization. Real solids are never absolutely pure, and in the neighborhood of the impurity atoms the solid is not the same as elsewhere in the crystal. Furthermore, there is always a slight temperature-dependent probability of finding missing or misplaced ions (Chapter 30) that destroy the perfect translational symmetry of even an absolutely pure crystal. Finally, the ions are not in fact stationary, but continually undergo thermal vibrations about their equilibrium positions. ## 2. The periodic potential and Bloch's theorem The problem of electrons in a solid is in principle a many-electron problem, for the full Hamiltonian of the solid contains not only the one-electron potentials describing the interactions of the electrons with the massive atomic nuclei, but also pair potentials describing the electron-electron interactions. $A = T_k + T_e + V_{ke} + V_{ee} + V_{nn}$ Non-relativistic Hamiltonian In the independent electron approximation, interactions can be represented by an effective one-electron potential, whose Schrodinger equation can be given by: $H\psi = (-ħ^2/2m)∇^2 + U(r) ψ = εψ$, (8.2) Only electron-nucleus Independent electrons, each of which obeys a one-electron Schrödinger equation with a periodic potential, are known as Bloch electrons (in contrast to "free electrons," to which Bloch electrons reduce when the periodic potential is identically zero). The stationary states of Bloch electrons have the following very important property as a general consequence of the periodicity of the potential U: ### Theorem: The eigenstates $ψ$ of the one-electron Hamiltonian $H= -ħ^2/2m + U(r)$, where $U(r + R) = U(r)$ for all R in a Bravais lattice, can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais lattice: $ψ_{nk}(r) = e^{ikr}u_{nk}(r)$, (8.3) where $u_{nk}(r + R) = u_{nk}(r)$ (8.4) where: * $ψ_{nk}(r)$ is the Bloch wavefunction. * $e^{ikr}$ is a plane wave with wavevector k. * $u_{nk}(r)$ is a function with the same periodicity as the crystal lattice. * n is the band index, and k is the wavevector within the first Brillouin zone. Bloch's theorem is a fundamental result in solid-state physics that describes the behavior of electrons in a periodic potential, such as the periodic potential created by atoms in a crystalline solid. The theorem states that the wavefunctions of electrons (also known as Bloch functions) in a periodic potential can be expressed as a plane wave modulated by a function with the same periodicity as the lattice. Note that Eqs. (8.3) and (8.4) imply that $ψ_{nk}(r + R) = e^{ikR}ψ_{nk}(r)$. (8.5) Bloch's theorem is sometimes stated in this alternative form: the eigenstates of H can be chosen so that associated with each $ψ$ is a wave vector k such that $ψ(r + R)=e^{ikR}ψ(r)$, (8.6) for every R in the Bravais lattice. ## 3. The Born-Von Karman Boundary condition * The definition of a crystalline lattice is an infinite set of points, vectors, or unit cells, but all real crystals are finite in size. The best quality single crystals consist of grains that contain 1019-1021 unit cells, so the lattice is, at best, a quasi-infinite set. * Such dimensions, however, make computations of electronic structures and other properties intractable, so an approximation is needed to lower the order of the lattice set while keeping important characteristics of the crystal. * The Born-von Karman boundary condition is a mathematical concept used in solid-state physics to model the periodicity of a crystal lattice. It simplifies the problem of dealing with the infinite extent of a crystal by imposing a periodic boundary condition, which means that the crystal is treated as if it is repeating itself indefinitely in all directions. * Periodic (Born-von Karman) boundary conditions are designed to make physical problems for crystalline structures reasonably solvable and involve selecting a large, finite subset of lattice points for any crystalline lattice. By imposing an appropriate boundary condition on the wave functions, we can demonstrate that the wave vector k must be real, and arrive at a condition restricting the allowed values of k. The condition generally chosen is the natural generalization of the condition (2.5) used in the Sommerfeld theory of free electrons in a cubical box. $\psi(x,y,z + L) = \psi(x,y,z)$, $\psi(x, y + L, z) = \psi(x, y, z)$, $\psi(x + L, y, z) = \psi(x,y, z)$. (2.5) Unless the Bravais lattice is cubic and L is an integral multiple of "a", it is not convenient to continue to work in a cubic volume of side L. Instead, it is more convenient to work in a primitive unit cell. Then the periodic boundary condition (2.5) now becomes: $\psi(r + Na_i) = \psi(r)$, $i = 1, 2, 3$, (8.22) Applying Bloch's theorem (8.6) to the boundary condition (8.22) we find that $(\psi{r + Na_i} = e^{ikNa_i}ψ(r)$, $i = 1, 2, 3$, (8.23) which requires that $e^{ikNa_1} = 1$, i = 1, 2, 3. (8.24) When k has the form (8.20), Eq. (8.24) requires that $e^{2πin_i} = 1$, (8.25) and consequently we must have $x_i = {m_i}/{N_i}$, $m_i$ integral. (8.26) Therefore the general form for allowed Bloch wave vectors is $k = Σ_{i=1}^{3} {m_i}/{N_i}b_i$, $m_i$ integral. (8.27) It follows from (8.27) that the volume $Δk$ of k-space per allowed value of k is just the volume of the little parallelogram with edges b₁/N: $Δk = {b_1}/{N_1}·{b_2}/{N_2}·{b_3}/{N_3} = {1}/{N}(b_1 × b_2 × b_3)$. (8.28) Since $b_1 · (b_2 × b_3)$ is the volume of a reciprocal lattice primitive cell, Eq. (8.28) asserts that the number of allowed wave vectors in a primitive cell of the reciprocal lattice is equal to the number of sites in the crystal. The volume of a reciprocal lattice primitive cell is $(2π)^3/v$, where $v = V/N$ is the volume of a direct lattice primitive cell, so Eq. (8.28) can be written in the alternative $Δk = {(2π)^3}/{V}$. (8.29) ## 4. Second proof of Bloch's theorem By performing this second proof, now accounting for the BVK boundary condition, we arrive at the following equation: $(ħ^2/(2m)(k - K)^2 - ε ) c_{k - K} + Σ_K' U_{k - K', k - k} c_{k-k'} = 0.$ (8.41) * This is a Schrodinger equation in momentum space, simplified by the periodicity of the potential. * For a fixed k in the 1st BZ, the set of equations (8.41) for all reciprocal lattice vectors K couples only those coefficients whose wave vectors differ from k by a reciprocal lattice vector. * Thus, the original problem has been separated into N independent problems: one for each allowed value of k in 1st BZ. * Each such problem has a solution that is a superposition of plane waves containing only the wave vector k and wave vectors differing from k by a reciprocal lattice vector. The problem instead to solved in the real space (direct space) $H\psi = (-ħ^2/(2m)∇^2 + U(r)) ψ = εψ$, (8.2) Become a problem in reciprocal space $(ħ^2/(2m)(k - K)^2 - ε ) c_{k - K} + Σ_K' U_{k - K', k - k} c_{k-k'} = 0.$ (8.41) ## 5. Band structure of the solid * Based on the quantum model of free electrons in a metal, the electrical and thermal conductivity of a material can be clarified. However, this theory was not sufficient to describe the behavior of semiconductor and insulating materials. * The explanation of the microscopic mechanisms of insulating and semiconducting materials was only possible with the development of the theory of solids and, in particular, the theory of electronic bands. * Knowledge of the band structure of a material allows us, for example, to classify whether it is a metal, semiconductor or electrical insulator. This is possible because in band structure models arise the concept of allowed and forbidden energy ranges. * When electrons move in a periodic potential, energy gaps arise in their dispersion relation at BZ boundaries, as will be seen from a mathematical point of view. It is interesting to note that the "n" band index for a given k appears in the Bloch theorem, i.e., $ψ_{nk}(r) = e^{ikr}u_{nk}(r)$, (8.3) For a given n, the eigenstates and eigenvalues are periodic functions of k in reciprocal space, i.e. $ψ_{nk+K}(r) = ψ_{nk}(r)$, $ε_{nk+K} = ε_{nk}$. (8.50) This fact leads to a description of the energy levels of an electron in a periodic potential in terms of a family of a continuous function $ε_{nk}$ (or $ε_n(k)$). The information contained in these functions is referred to as the band structure of the solid. ## 6. The Fermi Surface The ground state of N free electrons is constructed by occupying all one-electron levels k with energies $ε(k) = ħ^2k^2/2m$ less than $ε_F$, where $ε_F$ is determined by requiring the total number of one-electron levels with energies less than $ε_F$ to be equal to the total number of electrons (Chapter 2). The ground state of N Bloch electrons is similarly constructed, except that the one-electron levels are now labeled by the quantum numbers n and k, $ε(k)$ does not have the simple explicit free electron form, and k must be confined to a single primitive cell of the reciprocal lattice if each level is to be counted only once. When the lowest of these levels are filled by a specified number of electrons, two quite distinct types of configuration can result: * A certain number of bands may be completely filled, all others remaining empty. The difference in energy between the highest occupied level and the lowest unoccupied level (i.e., between the "top" of the highest occupied band and the "bottom" of the lowest empty band) is known as the band gap. We shall find that solids with a band gap greatly in excess of kBT (T near room temperature) are insulators (Chapter 12). If the band gap is comparable to kBT, the solid is known as an intrinsic semiconductor (Chapter 28). * A number of bands may be partially filled. When this occurs, the energy of the highest occupied level, the Fermi energy $ε_F$, lies within the energy range of one or more bands. For each partially filled band there will be a surface in k-space separating the occupied from the unoccupied levels. The set of all such surfaces is known as the Fermi surface, and is the generalization to Bloch electrons of the free electron Fermi sphere. The Fermi surface is a geometrical representation of all the wave vectors (k-points) that correspond to electrons having the Fermi energy. For example, in a free electron model (where electrons move in a uniform background without any crystal lattice potential), the Fermi surface is a sphere in 3D or a circle in 2D. Some alkali metals [Li (1s22s1), Na ([Ne] 3s1, K [Ar]4s1 and Rb [Kr]5s1) behave in this form, as shown below. The Figures are from calculations of the BCC crystal structure of these compounds. The Fermi surface (FS) can become more complex, leading to various shapes that reflect the anisotropy and band structure of the material. * Cesium is an alkali metal with a single valence electron per atom ([Xe]6s1). In its solid form, the electrons in cesium behave similarly to nearly free electrons, which means the band structure can be relatively simple compared to more complex materials. Its FS is mostly spherical, but there can be slight distortions due to effects like spin-orbit coupling, especially given cesium's heavy atomic mass. * Scandium is characterized by a complex Fermi surface (multiple sheets with different topologies) due to the contributions from the 3d3 and 4s1 orbitals, and the anisotropy introduced by its hexagonal close-packed crystal structure. * Aluminum is characterized as nearly spherical, consistent with its classification as a nearly free-electron metal. However, due to the FCC structure and interactions between electrons and the crystal lattice, distortions occur, particularly near the Brillouin zone boundaries. These distortions introduce features like necks or pockets in the Fermi surface. The Fermi surface provides critical information about several physical properties of a metal, including: * **Electrical Conductivity:** The shape and topology of the Fermi surface directly influence how electrons move through the material under an applied electric field. Metals with a simple, spherical Fermi surface (like alkali metals) typically have high electrical conductivity because electrons can move easily in all directions. Complex Fermi surfaces can lead to anisotropic conductivity, meaning the electrical conductivity varies depending on the direction of the crystal. * **Magnetic properties:** For instance, in materials with open or nested Fermi surfaces, magnetic instabilities like spin-density waves can occur. ## 7. Density of states (DOS) It is defined as the number of available electronic states per unit energy interval at a given energy. The DOS is expressed in units of states per energy per volume, such as states/eV/m3. In a solid, it can be mathematically represented by: $Ω = 2 Σ_k Ω_n(k)$, (8.53) where for each n the sum is over all allowed k lying in a single primitive cell and the factor 2 is because each level specified by n and k can accommodate two electrons of opposite spin. Thus, we assume that ... does not depend on electron spin. If it does, the factor 2 must be replaced by a sum on spin. In the limit of a large crystal, the allowed values (8.27) of k get very close together, and the sum may be replaced with an integral. Since the volume of k-space per allowed k (Eq. (8.29)) has the same value as in the free electron case, the prescription derived in the free electron case (Eq. (2.29)) remains valid, and we find that $Ω = lim_{V→∞} {1}/{V} = 2 Σ_n ∫ {dk}/{(2π)^3} Ω_n(k)$, (8.54) where the integral is over a primitive cell. If, as is often the case, Ω_n(k) depends on n and k only through the energy $ε_n(k)$, then in further analogy to the free electron case one can define a density of levels per unit volume (or "density of levels" for short) g(ε) so that q has the form (cf. (2.60)): $q = ∫ dε g(ε)Ω(ε)$. (8.55) Comparing (8.54) and (8.55) we find that $g(ε) = Σ_n g_n(ε)$, (8.56) where $g_n(ε)$, the density of levels in the nth band, is given by $g_n(ε) = ∫ {dk}/{(4π)^3} δ(ε - ε_n(k))$, (8.57) where the integral is over any primitive cell. The DOS helps identify energy ranges where electronic states are available. Peaks in the DOS correspond to energy bands where many states are available, while gaps indicate energy ranges with few or no states, corresponding to band gaps in semiconductors and insulators. The position of the Fermi level (the highest occupied energy level at absolute zero) within the DOS diagram helps determine the material's conductivity. In metals, the Fermi level lies within a band of states, whereas in insulators and semiconductors, it lies within a band gap. A high DOS at the Fermi level suggests that the material has many available states for electrons to occupy, contributing to higher electrical conductivity (as in metals). The DOS influences the specific heat and thermal conductivity of a material. At low temperatures, the electronic contribution to specific heat is proportional to the DOS at the Fermi level.