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72 CHAPTER 4. ELECTRONIC STRUCTURE are plane wave like, but in the vicinity of the core they oscillate rapidly so as to be orthogonal to the core levels. We can now use the OPW’s as basis states for the diagonalisation in the same way that we used plane waves in the NFE, viz |ψ k >= ∑ α k−G |χ k−G > . (4.34) G This turns out to converge very rapidly, with very few coefficients, and only a few reciprocal lattice vectors are included in the sum. The following discussion explains why. Suppose we have solved our problem exactly and determined the coefficients α. Now consider the sum of plane waves familiar from the plane-wave expansion, but using the same coefficients, i.e. |φ k >= ∑ G α k−G |k − G > , (4.35) and then 4 it is easily shown that |ψ >= |φ > − ∑ n < f n |φ > |f n > . (4.36) Then substitute into the Schrodinger equation H|ψ >= E|ψ >, which gives us H|φ > + ∑ n (E − E n ) < f n |φ > |f n >= E|φ > (4.37) We may look upon this as a new Schrödinger equation with a pseudopotential defined by the operator V s |φ >= U|φ > + ∑ n (E − E n ) < f n |φ > |f n > (4.38) which may be written as a non-local operator in space ∫ (V s − U)φ(r) = V R (r, r ′ )φ(r ′ ) dr ′ , where V R (r, r ′ ) = ∑ n (E − E n )f n (r)f ∗ n(r ′ ) . (4.39) The pseudopotential acts on the smooth pseudo-wavefunctions |φ >, whereas the bare Hamiltonian acts on the highly oscillating wavefunctions |ψ >. One can see in (4.38) that there is cancellation between the two terms. The bare potential is large and attractive, especially near the atomic core at r ≈ 0; the second term V R is positive, and this cancellation reduces the total value of V s especially near the core. Away from the core, the pseudopotential approaches the bare potential. 4 Saving more notation by dropping the index k
Chapter 5 Bandstructure of real materials 5.1 Bands and Brillouin zones In the last chapter, we noticed that we get band gaps within nearly free electron theory by interference of degenerate forward- and backward going plane waves, which then mix to make standing waves. Brillouin zones. What is the condition that we get a gap in a three-dimensional band structure A gap will arise from the splitting of a degeneracy due to scattering from some Fourier component of the lattice potential, i.e. that E 0 (k) = E 0 (k − G) (5.1) which means (for a given G) to find the value of k such that |k| 2 = |k − G| 2 . Equivalently, this is k · G ∣ ∣∣∣ 2 2 = G 2 ∣ (5.2) which is satisfied by any vector lying in a plane perpendicular to, and bisecting G. This is, by definition, the boundary of a Brillouin zone; it is also the Bragg scattering condition, not at all coincidentally. 1 Electronic bands. We found that the energy eigenstates formed discrete bands E n (k), which are continuous functions of the momentum k and are additionally labelled by a band index n. The bandstructure is periodic in the reciprocal lattice E n (k + G) = E n (k) for any reciprocal lattice vector G. It is sometimes useful to plot the bands in repeated zones, but remember that these states are just being relabelled and are not physically different. 1 Notice that the Bragg condition applies to both the incoming and outgoing waves in the original discussion in Chapter 4, just with a relabelling of G → −G 73
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Quantum Condensed Matter Physics Pa
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