E-k diagram

FIGURE 13.9. Behavior of the energy bands of 13.41 on distortion to the structure of 13.42. (a) Block diagram, (b) e k) diagrams. The three bands corresponding to the levels of the undistorted geometry split apart into pairs (labeled + and ) on distortion. In (b) the middle panel shows how the bands... 

FIGURE 3.18 E-k diagram (left) and its projection of contour (right) in first Brillouin zone, with the indication of electronic occupancy until Fermi level, for an insulating crystal after (Further Readings on Quantum Solid 1936-1967 Putz, 2006). 

Waves interact with each other to raise or lower their combined energies. Graphically, when curves on the E(k) diagram intersect they may interfere resulting in local changes in their energies. Note that... 

The energy perturbation given in Equation 2.10 technically applies only at the boundary, ki=k2. As one moves away from the boundary, the magnitude of this interaction decreases quadratically with k. This is the same behavior as at the bottom of the free-electron curve. Another way to view this behavior is that near any extremum (maximum or minimum) of an arbitrary function, a power-law expansion of that function is always quadratic. Thus, near enough to any local maximum or minimum of an E(k) diagram, the behavior of an electron will always appear free-electron like. This provides a partial, if circular, justification of the approximations made above. A rigorous justification is provided by quantum mechanical perturbation theory and may be found in most quantum mechanics texts. 

Figure 2.9 A schematic of the transitions produced by a phonon (horizontal) and a photon (vertical) on an E(k) diagram. Indirect transitions require both and thus are less likely.

In other words, the effective mass of an electron Bloch wave is inversely related to the curvature of the E(k) diagram. Strong curvatures mean small effective masses and vice versa. The effective mass is generally expressed in units of electron rest mass in a vacuum, i.e. m /mo. 

Compounds involving increasingly ionic bonds have less dispersion in their bands across momentum space (the E(k) diagrams). 

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