Appendix A. Reciprocal Lattice Vector and Discretized Wavevector K. Ohtaka and K. Inoue 1 Reciprocal Lattice Vectors and First Brillouin Zone Reciprocal lattice vectors of a lattice are defined to be the wavevectors h that satisfy exp(ih · R) = 1, (1) for any lattice translation vector R given by (2) Here Pl, P2, P3 are three arbitrary integers and a1, a2, a3 are three primitive translation vectors that define the lattice. For the three special cases of R = a1, a2 and a3, (1) leads to h · a1 = 2nn1, h · a2 = 2nn2, h · a3 = 2nn3, (3) repectively, using three integers. They are linear coupled equations for h = (hx, hy, hz). Resolving the vectors into the cartesian components, we obtain the solution h from Cramer's rule of linear algebra: (4) where (5) The solution ( 4) obtained as a necessary condition is obviously a sufficient condition for (1) to hold for an arbitrary lattice translation R. The three vectors b1 , b2 and b3 define the primitive translation vectors of the reciprocal 304 K. Ohtaka lattice. The lattice points spanned by b1, b2 and b3 are the reciprocal lattice points. Reciprocal lattice and real lattice have important relations. Using three integers l, m, n, we can define the reciprocal lattice points h(l, m, n) by h(l, m, n) = lb1 + mbz + nb3. It is shown that the direction of the vector h(l, m, n) is perpendicular to the (real) lattice plane of Miller indices (l, m, n). Also, the length lh(l, m, n)l is inverse of the spacing d(l, m, n) of the (real) lattice planes (l, m, n), i.e., lh(l,m,n)l = 2njd(l,m,n). (6) Miller indices are defined using three coprime numbers. If l, m, n are not coprime numbers like h(2, 4, 6), then lh(2,4,6)1 = 2(2n/d(1,2,3)). The first BZ is the Wigner-Seitz cell around h = 0, the origin of the recipro­ cal space, which is defined to be the region of reciprocal space that is closer to the point h = 0 than any other lattice point. The Wigner-Seitz cell contains one lattice point in it and fills all the space when translated through all recip­ rocal lattice vectors. In other words, the first BZ is the territorial region that belongs to the point h = 0 in the reciprocal space and is constructed as the smallest volume entirely closed by a set of planes that are the perpendicular bisectors of various reciprocal lattice vectors drawn from the origin. Notice that such planes are particularly important in the theory of wave propagation in crystals, because a wave with a wavevector drawn from the origin termi­ nating on any of these planes should satisfy without fail the conditions for diffraction. The volume of the Wigner-Seitz cell of the real lattice, i.e., the territorial region of one lattice point, is equal to the unit cell volume Vc, the volume of the parallelepiped formed by a 1 , a 2 , a 3 . The volume of the first BZ, i.e., the volume of the Wigner-Seitz cell in the reciprocal lattice space is equal to the volume of the parallelepiped formed by b1, bz, b3, because obviously both are the region occupied by one reciprocal lattice point. From (5), therefore, the volume of the first BZ is calculated to be (2n) 3 fvc. Examples of the first BZ and the names of the special points inside it are given in Fig.l. Let us consider, as an example, how to obtain the first BZ of a 2D trian­ gular lattice, depicted in the upper right of Fig. 1. Letting a be the lattice constant, we have a 1 = a(1, 0), and a 2 = a(1/2, v'3/2) in the xy plane, and we take a 3 such that a 3 = cz parallel to the z axis, with an arbitrary con­ stant c. Then, since two vectors b1 and b2 are in the (at, a 2 ) plane, we can construct the 2D reciprocal lattice by using b1 and b2 . From (5), we obtain 2n b1 = y'3a (y'3, -1), 4n bz = ---;-(0, 1), y3a Appendix A. Reciprocal Lattice Vector and Discrete Wavevector 305 M M K X X r v/j. z z M M T z .;:.... ··--+-~ky .··r·· .. ky I· M 21lla u Fig. 1. First Brillouin zone of various lattices: 2D square lattice (upper left), 2D triangular lattice (upper right), simple cubic lattice (lower left) and fcc lattice (lower right). The names of symmetry points are shown and Ib1l = Ib2l = 4n I (.J3a). They form a triangular lattice of lattice constant 4n I (.J3a), the direction of the lattice being rotated by 1r I 6 with respect to the real-space lattice. Bisecting b1 and b2 and their equivalents, we obtain the first BZ. In the first BZ, there are three high-symmetry points, marked as r, M and Kin the figure, positioned at (0, 0), (n la)(l, 11 .J3), and (n la)(4l3, 0), respectively, and their equivalents. A 2D triangular lattice is an important example of PCs, which will be frequently treated in this book. In this con­ nection, it is remarked that the symbols X and J are used often instead of M and K, respectively. Any 2D or 3D BZ is obained similarly. 2 Density of States The number of states is calculated only if the values taken by k, and hence those taken by the band frequency wn(k), are discrete (countable). By im­ posing the periodic boundary condition using a large integer N 1 , given by 306 K. Ohtaka Ek(r + N1a1) =eik·N 1 a 1 Ek(r) (7) = Ek(r) (the first line is the Bloch theorem) and similarly for a 2 and a 3 with integer N 2 and N 3 , respectively, we can discretize the values of the wavevector k of band states. Expressing in the first BZ, we find k = E.:!:_ b1 + p2 b2 + p3 b3 (8) N1 N2 N3 with three arbitrary integers Pl, P2, P3· The spacings between two allowed values of k in the b1, b2 and b3 directions are given by etc. Therefore the region of the first BZ of volume (2n)3 fvc is divided into N 1N 2N 3 small cells, each containing one allowed k point given by (8). Here we call this unit of k space a k cell. The volume of the k cell is thus (2n)3 _ (2n)3 --- "total being the volume of the system on which the periodic boundary con­ dition was imposed. Usually the volume of the system itself is used as "total· The total number of discretized k points in a given volume i1k = i1kxi1kyi1kz is equal to the number of k cells in it and is given by dk _ vtotal dk (2n)3 /vtotal - (2n)3 · From this rule, for an arbitrary function F(kx, ky, kz) it follows that 2 3 ( n) LF(k) --4 JdkF(k) (9) vtotal k in the limit "total --4 oo, because the sum of F(k) over the discrete k points, each multiplied by the k cell volume, becomes an integral over the k space. In the lD system of lattice constant a and system length L, used for the quantization of k, the allowed values of k are k = 2np = ]!__ 2n L N a for p = 0, ±1, ±2, ... , where N is the number of unit cells in the length L. The total number of allowed points in the first BZ, which is the region Appendix A. Reciprocal Lattice Vector and Discrete Wavevector 307 -~ ::::; k < ~' is precisely equal to N. Note that the equality sign to define the first BZ may be added either to - ~ or ~ but not both, for these two points are equivalent. In a 2D square lattice spanned by a 1 =ax and a 2 = aiJ, we obtain k - 21rPx k _ 2npy x- Na' Y- Na (- ~ ::::; Px, Py < ~) · The number of allowed points of k of the first BZ is N 2 , which is equal to the total number of lattice points of the system. In this way we can show for any lattice that the number of allowed values of k within the first BZ is exactly equal to the total number of lattice points. This is why we can obtain information of any system by considering only the states within the first BZ. The density of states p( w) is defined to be the number of states of the system per unit frequency at the frequency w. By "per unit frequency", we mean that an infinitesimally small interval [w, w + Llw] has the number of states p(w)Llw. To calculate p(w), we have only to know the volume Llk in the first BZ, where all the discrete k points therein give the band energies within the frequency frequency interval. This volume divided by the k cell volume is p(w)Llw. If we know the k dependence of the band frequency wn(k), we can calculate it for each band. In Fig. 2, which depicts a modellD-dispersion curve, we show that p(w)Llw is equal to the number of discrete points of kin the corresponding interval Llk of k space. We find (see also the part following (9) of AppendixB) L 1 (10) p(w) = 2n dw(k)/dk Fig. 2. Density of discretized states in a lD system. The spacing between neigh­ boring k values is equidistant, while that of the w values is not.