4.1. Using the nearly free-electron approximation for a one-dimensional (1-D) crystal lattice and assuming that the only nonvanishing Fourier coefficients of the crystal potential are v(7/a) and v(-n/a) in (4.73), show that near the band edge at k = 0, the dependence of electron energy on the wave vector k is given by Ek = Eo + 2m* where m* = mo[l– (32mza*/h*n*)v(7/a)²]¬l is the effective mass of the electron at k = 0. - 4.2. The E-k relation of a simple cubic lattice given by (4.79) is derived from the tight-binding approximation. Show that near k 0 this relation can be expressed by
4.1. Using the nearly free-electron approximation for a one-dimensional (1-D) crystal lattice and assuming that the only nonvanishing Fourier coefficients of the crystal potential are v(7/a) and v(-n/a) in (4.73), show that near the band edge at k = 0, the dependence of electron energy on the wave vector k is given by Ek = Eo + 2m* where m* = mo[l– (32mza*/h*n*)v(7/a)²]¬l is the effective mass of the electron at k = 0. - 4.2. The E-k relation of a simple cubic lattice given by (4.79) is derived from the tight-binding approximation. Show that near k 0 this relation can be expressed by
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4.1. Using the nearly free-electron approximation for a one-dimensional (1-D)
crystal lattice and assuming that the only nonvanishing Fourier coefficients
of the crystal potential are v(7/a) and v(-n/a) in (4.73), show that near the
band edge at k = 0, the dependence of electron energy on the wave vector
k is given by
h?k?
Eo +
2m*
Ek =
where m* = mo[l – (32mza*/h*n*)v(T/a)²]¬l is the effective mass of the
electron at k = 0.
4.2. The E-k relation of a simple cubic lattice given by (4.79) is derived from
the tight-binding approximation. Show that near k 0 this relation can be
expressed by
E = Eno +
2m*
where m* = h? /2B,a?.
And for k T /a, show that the E-k relation is given by
Ek = Eno +
2m*
where m* = -h? /2B,a?.
43. If the conductivity and the density-of-states effective masses of electrons are](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F36cea953-d702-47c0-924e-5b4d7b70d5ed%2F71f59e46-f8c3-4420-84b0-a0dedc4fddee%2F73nt4se_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problems
4.1. Using the nearly free-electron approximation for a one-dimensional (1-D)
crystal lattice and assuming that the only nonvanishing Fourier coefficients
of the crystal potential are v(7/a) and v(-n/a) in (4.73), show that near the
band edge at k = 0, the dependence of electron energy on the wave vector
k is given by
h?k?
Eo +
2m*
Ek =
where m* = mo[l – (32mza*/h*n*)v(T/a)²]¬l is the effective mass of the
electron at k = 0.
4.2. The E-k relation of a simple cubic lattice given by (4.79) is derived from
the tight-binding approximation. Show that near k 0 this relation can be
expressed by
E = Eno +
2m*
where m* = h? /2B,a?.
And for k T /a, show that the E-k relation is given by
Ek = Eno +
2m*
where m* = -h? /2B,a?.
43. If the conductivity and the density-of-states effective masses of electrons are
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