Computational Physics

6.1 Introduction: definitions 123 x y z a 1 a 1 a 1 a 2 a (^2) a 2 a 3 a 3 a 3 Figure 6.1. Lattice structure of the simple cubic ...

124 Solving the Schrödinger equation in periodic solids 6.1.2 Reciprocal lattice A function which is periodic on a Bravais latti ...

6.2 Band structures and Bloch’s theorem 125 three instead of two atoms: then the atomic levels split into three different ones, ...

126 Solving the Schrödinger equation in periodic solids The eigenstates of the Hamiltonian can thus be written in the form of a ...

6.3 Approximations 127 Energy k Figure 6.2. Nearly free electron spectrum for a periodic potential in one dimension. consisting ...

128 Solving the Schrödinger equation in periodic solids k Γ XW L Γ K Energy (atomic units) 0.0 0.1 0.2 0.3 0.4 –0.1 0.5 Figure 6 ...

6.3 Approximations 129 neighbours. The numerical solution of the generalised eigenvalue problemHC= ESCis treated inChapter 3. Of ...

130 Solving the Schrödinger equation in periodic solids Figure 6.4. Hexagonal lattice of the graphene sheet with basis vectorsa ...

6.3 Approximations 131 It follows that the energiesE(k)are given as E±(k)= −(− 2 E 0 +E 1 )± √ (− 2 E 0 +E 1 )^2 − 4 E 2 E 3 2 E ...

132 Solving the Schrödinger equation in periodic solids M Γ K (^) Γ 10 –8 –6 –4 –2 0 2 4 6 8 10 K M Figure 6.5. Tight-binding ...

6.4 Band structure methods and basis functions 133 Γ X Γ X –6 –4 –2 0 2 4 6 –6 –4 –2 0 2 4 6 Figure 6.6. Energy bands for a(12, ...

134 Solving the Schrödinger equation in periodic solids V(r) Ψ(r) Figure 6.7. Valence state and Coulomb potential in a crystal. ...

6.5 Augmented plane wave methods 135 Figure 6.8. The muffin tin approximation. In the next sections, we consider the augmented p ...

136 Solving the Schrödinger equation in periodic solids where the functionsRl(r)are the solutions of the radial Schrödinger equa ...

6.5 Augmented plane wave methods 137 Summarising the results so far, we can say that in the APW method the wave function is appr ...

138 Solving the Schrödinger equation in periodic solids given by Hij=〈k+Ki|H|k+Kj〉=−EAij+Bij+ l∑max l= 0 Cijl R′l(R) Rl(R) . (6. ...

6.5 Augmented plane wave methods 139 kz ky kx X W K L Γ Figure 6.9. Brillouin zone of the fcc lattice. For a given vectorkin the ...

140 Solving the Schrödinger equation in periodic solids Γ XW L Γ K k E 0.0 0.1 0.2 0.3 Figure 6.10. Band structure of fcc copper ...

6.6 The linearised APW (LAPW) method 141 programming exercise Write a program for calculating the determinant|H−E|. CheckCheck t ...

142 Solving the Schrödinger equation in periodic solids In comparison with the APW method, we have twice as many radial function ...