Solid State Physics. Philip Hofmann

Solid State Physics

a 1 dot left-parenthesis bold a 2 times bold a 3 right-parenthesis EndFraction comma bold b 2 equals 2 normal pi StartFraction bold a 3 times bold a 1 Over bold a 1 dot left-parenthesis bold a 2 times bold a 3 right-parenthesis EndFraction comma bold b 3 equals 2 normal pi StartFraction bold a 1 times bold a 2 Over bold a 1 dot left-parenthesis bold a 2 times bold a 3 right-parenthesis EndFraction period"/>

From this, one can derive the simple but useful property²

(1.17) bold a Subscript i Baseline dot bold b Subscript j Baseline equals 2 normal pi normal delta Subscript i j Baseline comma

which can easily be verified. Equation (1.17) can then be used to verify that the reciprocal lattice vectors defined by Eqs. (1.15) and (1.16) do indeed fulfill the fundamental property of Eq. (1.13) defining the reciprocal lattice (see also Problem 6).

Another way to view the vectors of the reciprocal lattice is as wave vectors that yield plane waves with the periodicity of the Bravais lattice, because

(1.18) normal e Superscript normal i bold upper G dot bold r Baseline equals normal e Superscript normal i bold upper G dot bold r Baseline normal e Superscript normal i bold upper G dot bold upper R Baseline equals normal e Superscript normal i bold upper G dot left-parenthesis bold r plus bold upper R right-parenthesis Baseline period

Using the reciprocal lattice, we can finally define the Miller indices in a much simpler way: The Miller indices left-parenthesis i comma j comma k right-parenthesis define a plane that is perpendicular to the reciprocal lattice vector i bold b bold 1 plus j bold b bold 2 plus k bold b bold 3 (see Problem 9).

1.3.1.5 The Meaning of the Reciprocal Lattice

We have now defined the reciprocal lattice in a proper way, and we will give some simple examples of its usefulness. The most important feature of the reciprocal lattice is that it facilitates the description of functions with the same periodicity as that of the lattice. To see this, consider a one‐dimensional lattice, a chain of points with a lattice constant (Fig. 1.11). We are interested in a function with the periodicity of the lattice, such as the electron concentration rho left-parenthesis x right-parenthesis along the chain, . We can write this as a Fourier series of the form

(1.19) rho left-parenthesis x right-parenthesis equals upper C plus sigma-summation Underscript n equals 1 Overscript infinity Endscripts left-brace upper C Subscript n Baseline cosine left-parenthesis x Baseline 2 normal pi n slash a right-parenthesis plus upper S Subscript n Baseline sine left-parenthesis x Baseline 2 normal pi n slash a right-parenthesis right-brace

with real coefficients upper C Subscript n and upper S Subscript n . The sum starts at n equals 1 , the constant upper C is therefore outside the summation. Using complex coefficients rho Subscript n , we can also write this in the more compact form

(1.20) rho left-parenthesis x right-parenthesis equals sigma-summation Underscript n equals negative infinity Overscript infinity Endscripts rho Subscript n Baseline normal e Superscript normal i x n Baseline 2 normal pi slash a Baseline period

To ensure that rho left-parenthesis x right-parenthesis is still a real function, we require that

(1.21) rho Subscript negative n Superscript asterisk Baseline equals rho Subscript n Baseline comma

that is, that the coefficient rho Subscript negative n must be the complex conjugate of the coefficient . This description is more elegant than the one with the sine and cosine functions. How is it related to the reciprocal lattice? In one dimension, the reciprocal lattice of a chain of points with lattice constant is also a chain of points, now with spacing 2 normal pi slash a [see Eq. (1.17)]. This means that we can write a general reciprocal lattice “vector” as

(1.22) g equals n StartFraction 2 normal pi Over a EndFraction comma

where is an integer. Exactly these reciprocal lattice “vectors” appear in Eq. (1.20). In fact, Eq. (1.20) is a sum of functions with a periodicity corresponding to the lattice vector, weighted by the coefficients rho Subscript n . Figure 1.11 illustrates these ideas by showing the lattice and reciprocal lattice for such a chain as well as two lattice‐periodic functions, both in real space and as Fourier coefficients on the reciprocal lattice points. The advantage of describing these functions by the coefficients rho Subscript n is immediately obvious: Instead of giving rho left-parenthesis x right-parenthesis for every point in a range of 0 less-than-or-slanted-equals x less-than a , the Fourier description consists of just three numbers for the upper function and five numbers for the lower function. Actually, these even reduce to two and three numbers, respectively, because of Eq. (1.21).

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