The Central Equation


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1 The Central Equation Mervyn Roy May 6, Derivation of the central equation The single particle Schrödinger equation is, ( H E nk ψ nk = 0, ) ( + v s(r) E nk ψ nk = 0. (1) We can solve Eq. (1) at given wavevector, k, and band index, n, to find the energy, E nk, and the single particle wavefunction, ψ nk. In a crystal, the eigenstates, ψ nk, obey Bloch s theorem, and ψ nk (r) = e ik r u nk (r), () where u nk has the periodicity of the lattice. So u nk (r + T) = u nk (r) if the translation vector, T = n 1 a 1 + n a + n 3 a 3 is a sum over unit cell vectors, a 1, a, a 3, of the crystal. As the u nk have the periodicity of the lattice we can expand them as a Fourier series, u nk = c (k)e i r, so that ψ nk = c (k)e i(k+) r, (3) where are reciprocal lattice vectors of the crystal, and the c (k) are the expansion coefficients of the single particle wavefunction at each k. The potential, v s (r), must also have the periodicity of the lattice, so v s (r) = V e i r. (4) 1
2 Before substituting into the Schrödinger equation (Eq. (1)) we note that, and ψ nk = v s (r)ψ nk = k + c e i(k+) r, (5) V c Q (k)e i( +Q+k) r. (6) Q Then, in Eq. (1), we have ( ) k + E nk e i(k+) r + V c Q (k)e i( +Q+k) r = 0. (7) It is now advantageous to rearrange the double sum involving V so that the plane wave part in the double sum becomes e i(k+) r. To do this, we write Q in terms of another reciprocal lattice vector, = + Q and replace Q with. Because the sums are over an infinite set of reciprocal lattice vectors, for each fixed, we can replace the infinite sum over Q with an infinite sum over the equivalent set of vectors,. Then, ( ) k + E nk c (k)e i(k+) r + V c (k)e i(+k) r = 0, (8) or, { ( k ) + e i(k+) r E nk c (k) + } V c (k) = 0. (9) The left hand side of this equation can only be zero if, for each, the term inside the curly brackets {...} is zero. Hence, ( ) k + E nk c (k) + V c (k) = 0. (10) Equation (10) is called the central equation. The form above is similar to that given by Tanner (Introduction to the physics of electrons in solids). We can also rearrange the central equation in a number of different ways. To obtain the form used by Martin (Electronic Structure) we must rewrite the infinite sum over reciprocal lattice vectors in Eq. (10). We choose a new reciprocal lattice vector, =, and replace the infinite sum over with an equivalent infinite sum over. Then, ( ) k + E nk c (k) + V c (k) = 0. (11) Q
3 Because both terms on the left now contain a coefficient that looks like c Q we can place both terms within the sum over and factorise out the c. [( k + ) ] E nk δ( ) + V c (k) = 0, (1) where the delta function δ( ) ensures we only get the required = terms in the first term in the sum over. Solving the central equation From Eq. (1) it should be clear that the central equation is an infinite set of coupled simultaneous equations  with one equation for each possible reciprocal lattice vector. We can solve the central equation numerically or, in some very simple cases, analytically. In whatever scheme we use we must truncate the infinite sum over reciprocal lattice vectors at some point. When solving the central equation it is common to take V 0 = 0. The V 0 Fourier component is the average value of the potential and we can always set the average of the potential to zero by adding or subtracting a constant offset to the energies..1 Truncating the series: example with 3 terms To explicitly see how Eq. (1) becomes a set of coupled simultaneous equations let us imagine that the infinite sum over reciprocal lattice vectors is restricted to just three possibilities, g, 0, and g. In this case in Eq. (1) has three possible values ( g, 0, g) and, in the sum over reciprocal lattice vectors, must sum over the three possible values g, 0, g. The central equation will therefore give us three simultaneous equations each containing three terms. For example, if we select = g we can write the first of the three simultaneous equations as, [( k + ) ] E nk δ( g ) + V g c (k) = 0. (13) = g,0,g In the first term in the sum, = g and, because we have picked = g, we get a contribution from the part of the equation containing the delta 3
4 function. In this case, when g, we obviously only retain the terms containing the Fourier components of the potential V. Remembering that V 0 = 0 we can therefore write Eq. (13) as, ( ) k g E nk c g (k) + V g c 0 (k) + V g c g (k) = 0. (14) Similarly, we can write down another equation for the case when = 0. In this case we obviously do not get a contribution from the delta function term if = g or = g, and ( ) k V g c g (k) + E nk c 0 (k) + V g c g (k) = 0. (15) Finally, we get the third equation in the set by choosing = g, ( ) k + g V g c g (k) + V g c 0 (k) + E c g (k) = 0. (16) We can write this set of simultaneous equations as a matrix equation, ( 1 k ) g E nk ( V g V g V 1 ) c g (k) g k E nk ( V g V g V 1 g k + ) c 0 (k) = 0,(17) g E nk c g(k) and this only has nontrivial solutions if the determinant, ( 1 k ) g E nk V g V g ( V 1 ) g k E nk V g ( V g V 1 g k + ) = 0. (18) g E nk. eneral form In shorthand form we can write our matrix equation (Eq. (17)) as, [H EI] c = 0, (19) where I is the unit matrix and E are the eigenvalues E nk. From our previous example (with the sum restricted to 3 reciprocal lattice vectors) H is the hamiltonian matrix, ( 1 k g ) V g V g H = ( V 1 g k ) ( V g V g V 1 g k + g ) 4, (0)
5 and c is the eigenvector that contains the expansion coefficients, c g (k) c = c 0 (k). (1) c g (k) The notation in Eq. (19) is completely general. So, to find the single particle energies, we must always find the values of E so that the determinant, H EI = 0. () In our previous example H was a 3 3 matrix, but we could have restricted the infinite series in the central equation to any number of terms. For example, if we had included N reciprocal lattice vectors we would have an N N hamiltonian matrix..3 The origin of band gaps at the Brillouin zone boundary.3.1 The empty lattice If we set v s (r) = 0 then all of the offdiagonal terms in the hamiltonian matrix are zero and we get a simple set of uncoupled equations that immediately give us the allowed energies, E nk = 1 k + n. (3) These are essentially the familiar free electron solutions, E = p /m, but, in a periodic crystal wavevectors k and k + are equivalent. We can use this equivalence to simplify the way in which we represent the states: any wavevector k that is larger than a reciprocal lattice vector,, can be represented by a shorter wavevector inside the first Brillouin zone, k = k, because of the periodicity of the reciprocal lattice. In the reduced zone scheme we represent all of the possible E nk in terms of wavevectors within the first Brillouin zone. This is illustrated schematically in Fig Weak periodic potential: solution at k = g/ First we assume that the potential is weak and periodic such that v s (r) = V cos(g r), only the V g Fourier components of the potential are nonzero, and V g = V g. Then, as V is small, we might imagine that the wavefunctions ψ nk = c e i(k+) r will be similar to the free electron solutions, ψ nk = 5
6 3.5 g Energy (ev) g 0 g g g/ Γ g/ g g Figure 1: Illustration of the reduced zone scheme. The parts of the Ek curve outside the first Brillouin zone are folded back into the first Brillouin zone by translating by an integer number of reciprocal lattice vectors. The dotted vertical lines denote the edges of the first Brillouin zone at g/ and g/. k e ik r and it should therefore be possible to restrict the infinite sum in Eq. (1) to just a few terms. As we are interested in solutions near the Brillouin zone boundary, where k = g/ we can restrict this sum to cover just two reciprocal lattice vectors. Near k = g/ we assume that the = 0 and = g components are the most important so that, ψ nk = c 0 e ik r + c g e i(k g) r. (4) The hamiltonian matrix is then just a matrix and, by retaining only the = g and = 0 terms in Eq. (17) we can find the allowed energies from, ( 1 k ) g E nk V g ( V 1 ) g k E nk = 0. (5) If we use the shorthand λ k = 1 k then the allowed energies are obtained by solving the resultant quadratic equation (λ k g E)(λ k E) Vg = 0. The solutions for the allowed energies near k = g/ are then, E = 1 (λ k + λ k g ) ± 1 [ ] 1/ (λ k λ k g ) + 4Vg. (6) At the zone boundary, k = g/ and λ k = λ k g = g /8. So, at the zone boundary there are two possible values of the energy, E = g /8 ± V g. (7) 6
7 A periodic potential, no matter how small, therefore opens up a gap at the Brillouin zone boundary of V g. This is illustrated in figure. Energy 3g /8 g /8 V g Figure : E(k) relation in the presence of a weak periodic potential (solid red line). The faint dotted line shows the empty lattice solutions. The weak periodic potential introduces a gap of V g at the Brillouin zone boundary where k = g/. Γ g/ k.4 Numerical solution In principle, from Eq. (1), the hamiltonian matrix is infinite in extent, but to solve the central equation we must always truncate the sum over reciprocal lattice vectors at some point. By doing this we restrict the reciprocal lattice vectors in our Fourier expansion of the singleparticle wavefunction (Eq. 3) to be less than some maximum length max, ψ nk = c (k)e i(k+) r. (8) < max As with any Fourier series, we can include more and more terms in the expansion until the wavefunction is as accurate as we need it to be. How do we know how many terms to include? As always, we can use the variational theorem to help us decide. The variational theorem tells us that each E nk will decrease as max increases and our approximate form for the wavefunction improves. We can therefore examine the convergence of E nk with max to determine how many reciprocal lattice vectors we must include to obtain the singleparticle energies to a given accuracy (see, for example, figure 3). Many computer codes have been developed to solve systems of equations, like the central equation, numerically. To solve Eq. (1) we must essentially give our computer code just two pieces of information. First we need to specify how many reciprocal lattice vectors we would like to include in our 7
8 expansion of the single particle wavefunction (this sets the size of the N N hamiltonian in Eq. (19)). Second we need to supply the Fourier components of the potential, V, that set each element of the hamiltonian matrix in Eq. (19). There exist many sophisticated numerical routines which will then diagonalise this matrix  essentially, calculate and report the values of E that satisfy H EI = Link to density functional theory In a density functional theory (DFT) calculation the single particle Schrödinger equations (Eq. (1)) are the equations for the fictitious KohnSham orbitals. These orbitals give the real electron density of the system by n(r) = nk ψ nk, where the sum is over the occupied single particle energy levels. This density can then be used to calculate the total energy, E = T s [n] + v s [n] = T s [n] + v(r)n(r)dr + E H [n] + E XC [n], (9) where T s is the kinetic energy functional, E H is the Hartree energy, and E XC is the exchange and correlation energy. As the variational theorem tells us that the total energy, E, must decrease as we increase max, we can examine the convergence of E with max to determine how many Fourier components we need to retain in the expansion Eq. (3). In a DFT calculation it is common to use a plane wave cutoff energy, E cut, to specify the maximum number of reciprocal lattice vectors to be included in the calculation at each k point. The cutoff energy, E cut sets max according to, E cut = 1 k + max. (30) 8
9 8.58 Total Energy, E (H) E(E cut )E(4) E cut (H) E cut (H) Figure 3: Example calculation showing total energy, E in Hartrees (H) against the plane wave cutoff energy, E cut, which determines the number of plane waves in the expansion, Eq. (3). The faint horizontal line denotes the converged energy of E = H at E cut = 4. The inset shows the difference between the total energy calculated at each E cut and the converged energy calculated at E cut = 4 H. 9
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