Find the general effective mass tensor

Question

Problem 1 - Can you work the preceding problem without making the simplifying assumption suggested above, and shown that

e^ε_f/kT = -½ + √((½)² + e^ε_e/kT(e^ε_v/kT - 1)) ?

You should now be able to show that this answer reduces to the one found previously under the right circumstances, and that in the limit of large temperatures, ε_f = ε_v - kT.

Problem 2 - In Bloch wave functions of the form (8.2 - 21), both ψ(x) and u(x) are in general comples quantities. By writing the wave function in polar form, ψ(x) = R(x)expi[kx + φ(x)], where R, k, and φ are all real quantities, show that the function R and φ satisfy differential equations of the form

d²R/dx² - (k + (dφ/dx))² R + (2m/h²)[ε - V(x)]R(x) = 0

and

R d²φ/dx² + 2(k + (dφ/dx)) · dR/dx = 0.

Hint: Substitute a solution of the above form into Schrodinger's equation and equate real and imaginary parts.

Problem 3 - By separating variables in the second of the above equations and integrating, show that these solutions can be written as

vR² = v₀R₀² = const., where v(x) = k + dφ/dx, and v₀, R₀ are initial values,

and

d²R/dx² - (v₀²R₀⁴/R³) + (2m/h)[ε - V(x)]R(x) = 0.

Problem 4 - Show from the second of the equations given in Question 2 that the function u(x) in the Bloch wave function cannot be a purely real quantity, except in the trivial instance when V(x) is zero and the problem reduces to that of a free electron in vacuum.

Problem 5 - In an anisotropic metallic crystal, the dispersion relation has the approximate form

ε - εb = (h²k_x²/2m_x) + (h²k_y²/2m_y) + (h²k_z²/2m_z).

Write the inverse effective mass tensor for this substance.

Problem 6 - Find the general effective mass tensor for the three-dimensional Kronig-Penney tight binding model described by the dispersion relation (8.6 - 5). Explain why this tensor for the electron distribution as a whole is isotropic, even though the tensor that applies to a single particle isn't of the form of a constant times the unit matrix.

Problem 7 - Find the volumes of the Brillouin zones associated with the b.c.c. and f.c.c. structures, expressed in terms of the edge of the cube that represents the simple cubic zone.

Problem 8 - Find the fraction of the zone volume occupied by a sphere of the largest possible radius inscribed in (a) the simple cubic  Brillouin zone, (b) the b.c.c. zone, and (c) the f.c.c. zone.

Problem 9 - Cu, Ag, and Au are all monovalent metals whose structure in the pure state is f.c.c. When they are alloyed with divalent metals like zinc or cadmium, random solid solutions are formed in which divalent atoms occupy f.c.c. sites at random. One effect of the alloying process is to increase the electron concentration to values increasingly larger than one electron per atom as more and more of the alloying is added. It is observed that as the electron-per-atom ratio reaches about 1.36, a phase change occurs in which the structure of the alloy changes from f.c.c. to b.c.c. As the electron-per-atom ratio is further increased, at a ratio of roughly 1.5 another phase change to a much more complex structure occurs. Can you explain these figures in terms of a model-obviously oversimplified-that involves a constantly expanding, roughly spherical Fermi surface in the Brillouin zone?

Problem 10 - A certain one-dimensional periodic potential gives rise to a dispersion relation of the form

ε - εb = (h²/2m)(k_x² - Ck_x⁴) (-π/a < kx < π/a).

Evaluate the constant C in terms of the other parameters in this expression. Find the effective mass at the bottom and top if the band.

Problem 11 - Show that the effect of including the next term in the series in (8.5 - 32) is to give a dispersion relation of the form

ε - ε_b = (D₁/C₁) - √((D₁²/C₁²) - 2D₁(1- coska)), where 1/D₁ = (d²f/dε²)b.

Show that this reduces to ε - ε_b = C₁(1-coska) when 1/D₁ approaches zero.