Читать книгу Introduction to Solid State Physics for Materials Engineers - Emil Zolotoyabko - Страница 15

With no doubts, leading crystal symmetry is translational symmetry, which is of great importance to the foundations of solid state physics. In particular, it allows us to deeply understand the essential features of wave propagation in periodic media, which influence a majority of physical phenomena in crystals. We start now with the symmetry-based analysis of wave propagation following the ideas of Leon Brillouin.

Let us consider, first, the propagation of the plane electron wave, Y = Y₀ exp[i(kr − ωt)], in a homogeneous medium. Here, Y₀ is the wave amplitude, k is the wavevector, and ω is the wave angular frequency, whereas r and t are the spatial and temporal coordinates. The phase of plane wave is ϕ = (kr − ωt), i.e. Y = Y₀ exp(iϕ). According to the Emmy Noether theorem, the homogeneity of space leads to the momentum conservation law. It means that an electron wave having wavevector, k_i, at a certain point in its trajectory, will continue to propagate with the same wavevector since the wavevector, k, is linearly related to the momentum, P, via the reduced Planck constant ℏ, i.e. P = ℏk. The latter relationship follows from the de Broglie definition of the particle wavelength (de Broglie wavelength) via its momentum: .

The situation drastically changes for a non-homogeneous medium, in which the momentum conservation law, generally, is not valid because of the breaking of the aforementioned symmetry (homogeneity of space). Consequently, in such a medium, one can find wavevectors, k_f, differing from the initial wavevector, k_i.

Our focus here is on a non-homogeneous medium with translational symmetry, which comprises scattering centers in specific points, r_s, given by Eq. (1.1). Based on the translational symmetry only, we can say that in an infinite medium with no absorption, the magnitude of plane wave, Y, should be the same near each lattice node. It means that the amplitude, Y₀, is the same at all points, r_s, whereas the phase, ϕ = kr − ωt, can differ by an integer number m of 2π.Let us suppose that the plane wave has wavevector, k_i, at starting point r₀ = 0 and time moment, t₀ = 0, and hence ϕ(0) = 0. If so, at point r_s, the phase, ϕ(r_s), should be equal:

Figure 1.13 Illustration of the wave scattering in a periodic medium.

(1.23)

Note that the change of the wavevector from k_i to k_f physically means that the wave experiences scattering in point, r_s (Figure 1.13). For elastic scattering (with no energy change):

(1.24)

where λ is the electron wavelength. Furthermore, the time interval, t, for wave propagation between points, r₀ = 0 and r_s, equals

(1.25)

where

(1.26)

is the phase wave velocity. Substituting Eqs. (1.24—1.26) into Eq. (1.23) yields:

(1.27)

Introducing a new vector, G, which is called vector of reciprocal lattice,

(1.28)

and combining Eqs. (1.27, 1.28), we find

(1.29)

According to Eqs. (1.28, 1.29), different values of k_f = k_i + 2πG are permitted in a periodic medium, but only those that provide scalar products of certain vectors, G, with all possible vectors, r_s, to be integer numbers, m. By substituting Eq. (1.1) into Eq. (1.29), we finally obtain:

(1.30)

In order to find the set of allowed vectors, G, satisfying Eq. (1.30), the reciprocal space is built, which is based on three non-coplanar vectors b₁, b₂, and b₃. Real (direct) space and reciprocal space are interrelated by the orthogonality (reciprocity) conditions:

(1.31)

where δ_ij is the Kronecker symbol, equal to 1 for i = j or 0 for i ≠ j (i, j = 1, 2, 3). To produce the reciprocal space from real space, we use the following mathematical procedure:

(1.32)

where V_c stands for the volume of the parallelepiped (unit cell) built in real space on vectors a₁, a₂, a₃:

(1.33)

By using Eqs. (1.32, 1.33), it is easy to directly check that the procedure (1.32) provides proper orthogonality conditions (1.31). For example, a₁ · b₁ = a₁ · [a₂ × a₃]/V_c = V_c/V_c = 1, whereas a₂ · b₁ = a₂ · [a₂ × a₃]/ V_c = 0. Certainly, the volume of the unit cell, V_rec, in reciprocal space

(1.34)

is inverse to V_c. To prove this statement, we use the relationship well-known in vector algebra:

(1.35)

In the reciprocal space, the allowed vectors, G, are linear combinations of the basis vectors, b₁, b₂, b₃:

(1.36)

with integer projections (hkl), known as Miller indices. The ends of vectors, G, being constructed from the common origin (000), produce the nodes of a reciprocal lattice. For all vectors, G, Eq. (1.30) is automatically valid due to the orthogonality conditions (1.31). We repeat that in the medium with translational symmetry, only those wavevectors, k_f, may exist, which are in rigid interrelation with the initial wavevector k_i, satisfying Eq. (1.28). Sometimes Eq. (1.28) is called as quasi-momentum (or quasi-wavevector) conservation law in the medium with translational symmetry, which should be used instead of the momentum conservation law in a homogeneous medium. We remind that the latter law means 2πG = k_f − k_i = 0, i.e. k_f = k_i.

Note that each vector of reciprocal lattice, G = hb₁ + kb₂ + lb₃, is perpendicular to a specific crystallographic plane in real space. This statement directly follows from Eq. (1.29), which defines the geometric plane for the ends of certain vectors, r_s, the plane being perpendicular to the vector G (Figure 1.14). Bearing in mind possible wave diffraction when propagating through a periodic medium, it is worth to introduce a set of parallel planes of this type (i.e. those given by Eq. (1.29)), which are separated by the d-spacing

(1.37)

Figure 1.14 Sketch of a crystal plane, normal to the vector of reciprocal lattice, G, which contains the ends of vectors, r_s, satisfying Eq. (1.29).

Figure 1.15 Graphical interrelation between wavevectors of the incident (k_i) and scattered (k_f) waves and the vector of reciprocal lattice, G.

In fact, using graphical representation of Eq. (1.28) (Figure 1.15) and solving the wavevector triangle, we find (with the aid of Eq. (1.24)) that

(1.38)

Substituting Eq. (1.37) into Eq. (1.38), we finally obtain the so-called Bragg law:

(1.39)

which provides the relationship between the possible directions for the diffracted wave propagation (via Bragg angles, Θ_B) and inter-planar spacings (d-spacings), d, in crystals. We stress that if λ > 2d, Bragg diffraction is not possible.

Note that for quasicrystals, the diffraction conditions (like Eq. (1.28)) can be deduced from the quasi-momentum (quasi-wavevector) conservation law in the n-dimensional space (hyperspace, n > 3), in which the vectors of reciprocal lattice, G^qc, are:

(1.40)

In case of icosahedral symmetry, n = 6, and the set of basis vectors has the following form:

(1.41)

Figure 1.16 The traces of isoenergetic surfaces (red curves) in reciprocal space for the incident (k_i) and diffracted (k_f) waves. The point of degeneracy of quantum states is marked by the letter D.

where G₀ is some constant and are unit vectors expressed via the golden ratio (Eq. (1.6)), as:

(1.42)

Using this terminology, regular crystals constitute the largest class, for which n = 3 (see Eq. (1.36)).

Considering the latter, we stress that in an infinite periodic medium, the waves having wavevectors k_i and k_f = k_i + 2πG are identical from quantum-mechanical point of view that leads to the degeneracy of the corresponding quantum states. The degeneracy point (marked by letter D in Figure 1.16) is located at the intersection of the iso-energetic surfaces (red lines in Figure 1.16) for the incident (k_i) and diffracted (k_f) waves. Being projected onto vector G in reciprocal space, this point called as the Brillouin zone boundary is in the middle between the 0 and G-nodes of the reciprocal lattice (Figure 1.16). Construction based on lines and surfaces normally cutting corresponding vectors of reciprocal lattices in their middles is used to build Brillouin zones in two and three dimensions (see Chapter 2). As we also will see in Chapter 2, the degeneracy of states at the Brillouin zone boundary is removed by an interaction of electron waves with periodic lattice potential that results in the formation of the forbidden energy zones (gaps). Within these gaps, the electron states do not exist. Therefore, we can say that energy gaps in crystals are formed due to diffraction of free (or almost free) valence electrons, having wavevectors comparable with vectors of reciprocal lattice.

Figure 1.17 Illustration of the restrictions imposed by translational symmetry on permitted types of rotation axes in crystals.