PROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR

Transcription

1 PROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR The aim of this project is to present the student with a perspective on the notion of electronic energy band structures and energy band gaps in solids. Energy band gaps are also discussed in the lectures and further considered in one of the mandatory hand-in exercises. To complete this project successfully, it is sufficient that student performs correctly the required tasks. The students participating to the project are encouraged to contact Claudio Verdozzi at any time for further clarifications. 1. Presentation & motivation In many cases, a solid piece of matter can be seen as a regular array of atoms (crystalline solids). The periodicity of such array (or lattice) of atoms has a profound and characteristic impact on the energy spectrum of a solid. The energy spectrum of a solid consists of continua, called energy bands, that may or may not overlap; there are also energy regions, called energy gaps, which do not belong to any band and, as a result, correspond to zero density of states. The aim of this project is to present the student with a specific perspectives on the notion of electronic energy band structures and electronic energy band gaps. This will require the use of quantum mechanics as introduced in the course. The task will be to produce results for simplified band structures, obtained from a 3D, large cluster of atoms ( 1000), which are arranged in a tetrahedral structure. Each atom is described by 4 orbitals (s, p x, p y, p z ). The model system provides a simplified description at the atomic level of homogeneous bulk semiconductors, such as, e.g., silicon. The notions and concepts required in the project (in particular, the exact diagonalization approach) are essentially those made available during the lectures and the laboratory, and a good knowledge of MATLAB is necessary. However, a new concept will be required to carry out the project, namely the Tight Binding Method for electronic structure (this is a quite important conceptual -and practical - approach to describe the electronic structure of solids). The fundamentals notions of i) exact numerical diagonalization and ii) the tight binding method will be presented below in Section 3, to enable the student to carry out the project. Some further details necessary to implement the Hamiltonian in MATLAB will also be provided. 2. Description of the project: a model 3D tetrahedrally bonded semiconductor Consider the following Tight-Binding Hamiltonian: H = V 1 φ j i φj i + H = V 2 φ j i φj i i j j j i i 1

2 2 PROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR Here φ j i is the Dirac ket representing the electron state in the j-th orbital at site i. The φ orbitals can be thought as coming from hybridization of s and p x, p y, p z orbitals in atoms, to produce the 4 tetrahedrally arranged sp 3 hybrid orbitals (those enter, for example, the structure of the CH 4 methane molecule; see the last section of this document). The Hamiltonian matrix elements V 1 are essentially responsible for the broadening of the levels, while V 2 is responsible for the formation of the split of the levels (V 1 and V 2 are usually referred to as banding and bonding terms. This model exhibits a simple analytic structure of the allowed energy levels, once one makes use of the so-called Bloch s theorem. However, here another approach is followed, which is largely based on numerical methods. Figure 1. System 3: Diamond structure in which hybridized orbitals are shown in repeated unit structure. Bonds 1-4 and 5- are those belonging to the unit cell. So In the Hamiltonian, contact is made between the different unit cells via matrix elements of the black orbitals within a cell, and the others in the neighboring cells. Assume that we can describe the infinite system by approximating it with a large but finite cluster. More precisely let us consider a cubic array of N N N cells. Each of this cell contains eight orbitals, as shown in Figure 2. Thus, in this system there are in total N 3 orbitals. By labeling the generic cell in the cluster by the cartesian triplet (i,j,k) ( 1 i N and cyclic), compute and store with MATLAB the N 3 N 3 Hamiltonian matrix Hint 1 The (ijk)-th cell can be seen as a subunit of the Hamiltonian matrix. Hint 2 In each of these subunits (which correspond to an submatrix) order the 1- orbitals as in Figure 2

3 PROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR 3 Hint 3 Some elements of such submatrix are shown here: 0 V 1 V 1 V 1?? V 1 0 V 1 V 1 0?? 0 0 V 1 V 1 0 V 1 0 0?? 0 V 1 V 1 V ???? 0 0 0???? 0?? 0 0???? 0 0?? 0???? 0 0 0????? where? and?? stands for entries to be filled. The entries?? are quite straightforward to fill (and involve V 2 only), whilst those marked by? involve V 1 only and require taking in consideration the neighboring units to the one under consideration (see again in Fig.2 to see how the black orbitals at the corners of the cell connect to the other hybrid orbitals in neighboring cells. Use periodic boundary conditions, i.e. close the system on itself. Example: if the cell (ijn) is being connected by V 1 to the cell (ijn+1), replace the cell (ijn+1) with (ij1). Or, if the cell (1ik) is being connected by V 1 to the cell (0ik), replace the latter with( Nik). Diagonalize numerically with MATLAB the N 3 N 3 Hamiltonian matrix. This should be done initially for very small N, to be progressively increased up to N=10 when the computer code is working correctly. Discard all the eigenvalues λ = V 1 + V 2 and V 1 V 2 (they give rise to artificial structures that disappear when a more refined Hamiltionian is introduced). Determine the minimum λ min and maximum λ Max of the remaining eigenvalues, and plot the density of states D(E) = 1 δ(e λ). M λ in the interval (2λ min, 2λ Max ), and where M is a normalization factor such that D(E)dE = 1. Since the system in question is discrete, the density of states is made of delta function spikes. To conveniently represent D(E), convolve it with a normalized Lorentzian L(E) = Γ 1, i.e. plot D Γ (E) = π E 2 +Γ 2 de D(E )L Γ (E E ). Plot the results for V 2 = 1 and V 1 = 0.25, 0.5, 1.0 and Γ = Discuss the results as a function of the parameters, for example how the energy gap changes for different values of V 1. Suppose that V 1 = 0 (or V 2 = 0). Can you solve the model analytically? And what kind of solutions does one obtain?

4 4 PROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR 3. Support material 3.1. Exact numerical diagonalization in a nutshell. In quantum mechanics, the change in time for a system is described by the time-dependent Schrödinger equation (1) i ψ t = H(t) ψ, where H(t) is the is the Hamiltonian operator the system. A special class of wave functions are those stationary in time, and for them H is independent of time, and the Schrödinger equation specializes to: (2) H ψ = E ψ. Here we use Dirac s bracket notation in which the wave function is given by a ket, and is denoted by ψ. Such ψ :s satisfying Eq. (2) are called eigenfunctions, and E are the relative eigenvalues. To solve the time-independent Schrödinger Equation, one expresses eigenfunctions in terms of pre-assigned basis sets of the associated Hilbert Space, thereby transforming equation (2) into a matrix equation. The goal is then to calculate the eigenvalues and eigenvectors of this matrix. Very few quantum-mechanical systems admit an exact analytic solution of the associated eigen-problem. In most cases, this can be done only numerically, and the method is commonly referred to as exact (or numerical) diagonalization. In the last few decades, due to the availability of increasingly powerful computer resources, the method of numerical diagonalization has gained favor as a quite effective tool, complementary to other numerical methods. The practicality of the method is essentially limited by the size of the problem (i.e. of the matrix to be diagonalized). For small systems it is a simple task to set up the respective matrices and directly plug them into our favorite computer program/software. However, for large systems, the diagonalization procedure can be computationally very demanding, if not practically impossible. In fact, it should also be noted that what can and cannot be done with numerical diagonalization methods improves steadily with increasing computer power and with the introduction of new algorithms. For a more comprehensive discussion of the method, we defer to the lectures in the course; in the following, we provide the basic element necessary to carry out the project. Say that we have an Hamiltonian H and are interested in its ground state g, to compute some averages of physical quantities g Ôg. We choose a basis { b }, and expand the generic eigenstates of H in that basis: (3) H λ = E λ λ, λ = b c λ b b. Acting with H on the expansion, and multiplying from the left with basis vector b, (4) c λ b b Hb = E λ c λ b δ bb, b b

5 PROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR 5 from which we get (5) c λ b [ b Hb E λ δ bb ] = 0 b This is a homogeneous linear system, and to have a solution beside the trivial one, we must impose that det(h E λ I) = 0, where I is the identity matrix, and the matrix elements are computed in the b basis. Numerically, this translates in writing the matrix H b b = b Hb and using some numerical procedure, to find the eigevector(s) and eigenvalue(s) of interest The Tight Binding Method: a sketchy introduction. One can adopt two perspectives to obtain the the electronic eigenvalues and eigenvectors of crystalline materials. One approach starts from the free electron picture and analyzes how this is modified when a weak periodic perturbation is introduced. This is the nearly free electron model, that we do not consider any further in this project. The other approach views the solids as being made up of atoms brought together from an infinite relative distance. It is then natural (as for molecules) to use linear combinations of atomic orbitals (LCAO) to build up the eigenfunctions. In general one can have more than one atom per primitive cell, and there are obviously several atomic orbital (in principle an infinite number) of atomic orbitals per atom. It can also been shown that, in general, while the atomic orbitals at a given atom are mutually orthogonal to each other (zero overlap), atomic orbitals from different atoms have in general a nonzero overlap. This requires specific modifications of the formulation, since one is working with a basis set which is not orthonormal anymore. However, to gain a qualitative insight in the problem (as we intend to do in this project), we can imagine that we have a finite (somewhat minimal) number of orbitals per atom The simplest case of all is when we consider only one atom per primitive crystal cell, only one atomic orbital per atom, nearest-neighbor coupling only, and we assume orthonormality between atomic orbitals from different atoms. This oversimplified version of the LCAO is often refereed to as the single-band tight-binding model (TBM) and he atomic orbital associated with the atom located at site will be symbolized by w(r l) = r l. We wish again to emphasize that for more realistic calculations one needs to take into account several complicating factors: Usually one needs several orbitals per atom, e.g., tetrahedral solids (C, Si, Ge, etc.) require at least four orbitals per site (one s-like and three p-like), while transition metals require in addition five d-like orbitals. Furthermore, one may need to employ hybrid atomic orbitals, such as the the tetrahedrally oriented hybridized sp 3 orbitals (see part 3.3 below). One may have more than one atom per primitive crystalline cell. The matrix elements between orbitals at different sites may not decay fast enough so that more than nearest-neighbor matrix elements may be needed. The atomiclike orbitals at different sites may not be orthogonal to each other. Indeed, true atomic orbitals are not orthogonal. The direct calculation of the matrix elements in this atomiclike orbital basis is in general a very difficult task which we do not discuss further here. In fact we assume that the value of such matrix element has been made available from more sophisticated calculations, so to us such matrix elements are input parameters. As mentioned before, in the simplest formulation of TBM (one atom per cell, one orbital per atom, with orthonormality between

6 PROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR atomic orbitals from different atoms), the basic set of functions for a 3D solid consists of orthonormal, identical, atomiclike orbitals, each one centered at the lattice sites m = m 1 a 1 +m 2 a 2 +m 3 a 3. Here, a 1, a 2, a 3 are the three primitive vectors of the 3D lattice, and m 1, m 2, m 3 take all integer values. Thus r m = w(r m), the orbital centered at m. The matrix element of the Hamiltonian within this subspace are m H n = V mn. The periodicity of the Hamiltonian implies that one can take V mn V m n and, for m = n, V mm = 0. It should be stressed that the Hamiltonian, which describes a real periodic solid, has matrix elements outside the subspace spanned by the m vectors and that this subspace is coupled with the rest of the Hilbert space. Nevertheless, we restrict ourselves to this subspace for the sake of simplicity. The price for this approximation can be considered reasonable, since many important qualitative features are retained in spite of this drastic simplification. Furthermore, bands arising from atomic orbitals weakly overlapping with their neighbors (i.e., tightly bound to their atoms) can be described rather accurately by working within the above-defined subspace. For this reason, the Hamiltonian V mn, which is confined within the subspace spanned by m,where m runs over all lattice sites, is called the tight-binding Hamiltonian (TBH) or the tight-binding model (TBM). Such Hamiltonian in the bra/ket notation, can be equivalently written as () H = mn V mn m n Growing a little in complication, one can also consider a more general case where each primitive cell has more than one atom, and each atom has more than one orbital. In this case, the Hamiltonian can be written H = (7) V mµ,nν mµ nν, mn µν where mµ denotes the µ-th orbital at the site m. For a tetrahedral lattice, by taking i) V mµ,nν = V 1 when µ ν and m = n; ii) V mµ,nν = V 2 when µ = ν and (m, n) are nearest neighbor sites on the tetrahedral lattice; and iii) V mµ,nν = 0 otherwise, we obtain the Hamiltonian used in this project, if the label µ spans the four sp 3 hybridized orbitals. In this case, to easily manipulate the Hamiltonian, it is convenient to think of a primitive cubic cell with two atoms (i.e. eight orbitals, four per atom) per cell. Such cell is then periodically repeated to form a 3D cubic array. This is the way in which the Hamiltonian is represented in the Section of the project devoted to system Brief discussion of the sp 3 orbitals. To quickly illustrate the concept of sp 3 orbitals, we consider the case of methane (CH 4 ). In methane, experiment shows that it is energetically equivalent to break any of the four C H bonds, which suggests symmetrical bonds with equal energies and equal strenght, the hybridised sp 3 bonds. In more detail, in methane the carbon atom orbitals s and p undergo sp 3 hybridisation and forms four sp 3 orbitals. Each of the hybrid sp 3 orbitals of carbon overlaps axially with the s orbital of the hydrogen atom, forming four bonds called sigma (σ) bonds. The four hybridised orbitals are directed towards the corner of a regular tetrahedron, making an angle of 109 degrees (see Figure 2). This kind of hybridisation occurs also in semiconductors like silicon, where the outer s and p electrons are the active valence orbitals to be considered.

7 PROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR 7 Figure 2. The methane molecule 3.4. Graphical aid to the computational setup of the atom coordinates. Figure 3 shows the relation between the points in the unit cell and the points in the nearest units cells, in terms of the atoms coordinates. This information may be of help in visualizing the connection between the coordinates of two bonds in different units cell. (-1,-1,0) 5 (0,0,0) (-1,0,-1) (0,-1,-1) 7 (-1,0,1) (0,0,0) 5 (0,1,1) (-1,1,0) 7 k j (0,-1,1) (1,-1,0) 5 (1,0,1) 7 (0,0,0) (0,0,0) (1,0,-1) 5 (1,1,0) 7 (0,1,-1) i Figure 3. Geometrical aspects of the setup procedure for the atom coordinated within the supercell