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76 CHAPTER 5. BANDSTRUCTURE OF REAL MATERIALS • You will also often see particular bands labelled either along lines or at points by greek or latin capital letters with a subscript. These notations label the group representation of the state (symmetry) and we won’t discuss them further here. Density of states We have dealt earlier with the density of states of a free electron band in 2. The maxima E max and minima E min of all bands must have a locally quadratic dispersion with respect to momenta measured from the minima or maxima. Hence the density of states (in 3D) near the minima will be the same g(E > ∼ E min ) = V ( m ∗ 2m ∗ (E − E min ) π 2 h¯ 2 h¯ 2 ) 1 2 . (5.10) as before, with now however the replacement of the bare mass by an effective mass m ∗ = (m ∗ xm ∗ ym ∗ z) 1/3 averaging the curvature of the bands in the three directions 3 . A similar form must apply near the band maxima, but with now g(E) ∝ (E max − E) 1 2 . Notice that the flatter the band, the larger the effective mass, and the larger the density of states 4 . Since every band is a surface it will have saddle points (in two dimensions or greater) which are points where the bands are flat but the curvature is of opposite signs in different directions. Examples of the generic behaviour of the density of states in one, two and three dimensions are shown in Fig. 5.1. The saddle points give rise to cusps in the density of states in 3D, and a logarithmic singularity in 2D. For any form of E(k), the density of states is g(E) = ∑ n g n (E) = ∑ n ∫ dk 4π 3 δ(E − E n(k)) , (5.11) Because of the δ-function in (5.11), the momentum integral is actually over a surface in k-space S n which depends on the energy E; S n (E F ) is the Fermi surface. We can separate the integral in k into a two-dimensional surface integral along a contour of constant energy, and an integral perpendicular to this surface dk ⊥ (see Fig. 5.2). Thus g n (E) = = ∫ ∫ S n(E) S n(E) ∫ dS 4π 3 dk ⊥ (k) δ(E − E n (k)) where ∇ ⊥ E n (k) is the derivative of the energy in the normal direction. 5 dS 4π 3 1 |∇ ⊥ E n (k)| , (5.12) Notice the appearance of the gradient term in the denominator of (5.12), which must vanish at the edges of the band, and also at saddle points, which exist generically in two and three dimensional bands. Maxima, minima, and saddle points are all generically described by dispersion (measured relative to the 3 Since the energy E(k) is a quadratic form about the minimum, the effective masses are defined by h¯ 2 m ∗ α ∣ along the principal axes α of the ellipsoid of energy. ∂ 2 E(k) ∂k ∣ α 2 kmin 4 The functional forms are different in one and two dimensions. 5 We are making use of the standard relation δ(f(x) − f(x 0 )) = δ(x − x 0 )/|f ′ (x 0 )| =
5.1. BANDS AND BRILLOUIN ZONES 77 1D 2D 3D Figure 5.1: Density of states in one (top curve), two (middle curve) and three (lower curve) dimensions Figure 5.2: Surface of constant energy critical point) of E(k) = E 0 ± h¯2 k 2 h¯2 x ± k 2 h¯2 y ± kz 2 (5.13) 2m x 2m y 2m z If all the signs in (5.13) are positive, this is a band minimum; if all negative, this is a band maximum; when the signs are mixed there is a saddle point. In the vicinity of each of these critical points, also called van Hove singularities, the density of states (or its derivative) is singular. In two dimensions, a saddle point gives rise to a logarithmically singular density of states, whereas in three dimensions there is a discontinuity in the derivative.
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