E in units of hC/2m 0.27- 0.26- 0.25 0.24- 0.23 Energy gap 0 0.01 0.005 Imaginary...

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E in units of h²C²/2m 0.27- 0.26- 0.25 0.24- 0.23 Energy gap 0 0.01 0.005 Imaginary part of k/G 0.48 0.49 0.51 0.52 0.50 Real part of k/G Figure 12 In the energy gap there exist solutions of the wave equation for complex values of the wavevector. At the boundary of the first zone the real part of the wavevector is G. The imaginary part of k in the gap is plotted in the approximation of two plane waves, for U = 0.01 h'G/2m. In an infinite unbounded crystal the wavevector must be real, or else the amplitude will increase with- out limit. But on a surface or at a junction there can exist solutions with complex wavevector. 5. Complex wavevectors in the energy gap. Find an expression for the imaginary part of the wavevector in the energy gap at the boundary of the first Brillouin zone, in the approximation that led to Eq. (46). Give the result for the Im(k) at the center of the energy gap. The result for small Im(k) is (ħ²/2m) [Im(k)]2. ≈ 2mU²/h²G². The form as plotted in Fig. 12 is of importance in the theory of Zener tunneling from one band to another in the presence of a strong electric field. E in units of h²C²/2m 0.27- 0.26- 0.25 0.24- 0.23 Energy gap 0 0.01 0.005 Imaginary part of k/G 0.48 0.49 0.51 0.52 0.50 Real part of k/G Figure 12 In the energy gap there exist solutions of the wave equation for complex values of the wavevector. At the boundary of the first zone the real part of the wavevector is G. The imaginary part of k in the gap is plotted in the approximation of two plane waves, for U = 0.01 h'G/2m. In an infinite unbounded crystal the wavevector must be real, or else the amplitude will increase with- out limit. But on a surface or at a junction there can exist solutions with complex wavevector. 5. Complex wavevectors in the energy gap. Find an expression for the imaginary part of the wavevector in the energy gap at the boundary of the first Brillouin zone, in the approximation that led to Eq. (46). Give the result for the Im(k) at the center of the energy gap. The result for small Im(k) is (ħ²/2m) [Im(k)]2. ≈ 2mU²/h²G². The form as plotted in Fig. 12 is of importance in the theory of Zener tunneling from one band to another in the presence of a strong electric field.

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Solution 027 026 025 Energy gap 024 0005 ... View the full answer