HW6-Energy-Bands - Physics 463 W12 Name(First Last VI Energy Bands in Solids(due Mar 29 Reading Kittels Chap 7 8(1st half 9 p161-204 p221-235 242-255

HW6-Energy-Bands - Physics 463 W12 Name(First Last VI...

This preview shows page 1 - 2 out of 2 pages.

Physics 463, W12 Name (First, Last): _____________________ VI. Energy Bands in Solids (due Mar. 29) Reading: Kittel’s Chap 7, 8 (1 st half) & 9: p161-204, p221-235, 242-255. Ashcroft & Mermin Chap 8-10 & 12. 1. Periodic functions. (15 pts) A function 𝑉 ( 𝒓 �⃗ ) has the same periodicity of a Bravais lattice. Prove that 𝑉 ( 𝒓 �⃗ ) can be expanded in the form: 𝑉 ( 𝒓 �⃗ ) = ∑ 𝑉 𝑮 ��⃗ 𝑒 𝑖𝑮 ��⃗ ⋅𝒓 �⃗ 𝑮 ��⃗ , where 𝑮 ��⃗ are reciprocal lattice vectors. 2. 1D electron band for a weak potential. (30 pts) Electrons of mass 𝑚 are confined to one dimension. A weak periodic potential of period 𝑎 , 𝑉 ( 𝑥 ) = 𝑉 0 𝑐𝑜𝑠 (2 𝜋𝑥 / 𝑎 ) , is applied. a. Under what conditions will the nearly free-electron approximation work? Assuming that the condition is satisfied, sketch the three lowest energy bands in the first Brillouin zone. Number the energy bands (starting from one at the lowest band). b. Calculate (to first-order in 𝑉 0 ) the energy gap between the first and second band, and at 𝑘 = 𝜋 / 𝑎
Image of page 1
Image of page 2

You've reached the end of your free preview.

Want to read both pages?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture