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(л/a) and v(−/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 ħ²k² 2m* Ek = Eo + where m* = mo[1 − (32m²aª/hªñª)v(π/a)²]¹ is the effective mass of the electron at k = 0.
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(л/a) and v(−/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 ħ²k² 2m* Ek = Eo + where m* = mo[1 − (32m²aª/hªñª)v(π/a)²]¹ is the effective mass of the electron at k = 0.
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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(n/a) and v(-π/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
where m* =
Ek
electron at k = 0.
= Eo +
= mo[1 − (32m²aª / hªлª)v(π/a)²]¯¹ is the effective mass of the
ħ²k²
2m*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e064dca-cd35-4766-b0c6-265b46df649a%2Fc762fd47-f3ce-472c-9ed9-eb194719d497%2Fp9wo2lr_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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(n/a) and v(-π/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
where m* =
Ek
electron at k = 0.
= Eo +
= mo[1 − (32m²aª / hªлª)v(π/a)²]¯¹ is the effective mass of the
ħ²k²
2m*
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