NEARLY FREE ELECTRON MODEL IN 1D: Consider an electron in a weak periodic potent

Question

NEARLY FREE ELECTRON MODEL IN 1D:

Consider an electron in a weak periodic potential in one dimension V (x) = V (x+a). where the sum is over the reciprocal lattice G = 2n/a, where G is a reciprocal lattice vector such that |k| is close to |k + G|. For an electron of mass m with wavevector (k) exactly at a BRILLOUIN zone boundry , the eigenenergies at this wave vector are given by: E =(h^2 k^2)/2m+ V0 ± |VG|

V(G) = 1/L _0^Ldx V(x)e^(-i G x) L=Na

where G is chosen so |k| = |k + G|.

compute V_(2/a)

compute V 4/a _

and compute V_(6/a)

for the following the potential:

) V(x)= 2 ( U_1 cos2x/a + U_2 cos4x/a

Explanation / Answer

Here,   V(x)= 2 ( U_1 cos2x/a + U_2 cos4x/a

=> V_(2/a)   =   2U_1 + 2U_2

V_(4/a)    =   2U_1 + U_2

V_(6/a)    =   U_1 + 2U_2

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