Homework6_Solutions - Homework#6 Solutions Spring 2014 1.a For free electrons in 2 dimensions the density of states in k-space is(accounting for 2 A

Homework6_Solutions - Homework#6 Solutions Spring 2014 1.a...

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Homework #6 Solutions Spring 2014 2
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1 st Zone: 2 2 2 π a 2 nd Zone: k f = 2 π a 3 rd Zone: k f = 5 2 2 π a 4 th Zone: k f = 5 2 6 2 π a 5 th Zone: k f = 10 2 2 π a 6 th Zone: k f = 10 2 2 π a 7 th Zone: k f = 10 2 2 π a Based on this, our table for the first 13 electrons looks like: Brillouin Zone m k f / 2 π a 1 2 3 4 5 6 7 1 .399 P E E E E E E 2 .564 P P E E E E E 3 .691 P P E E E E E 4 .798 F P P P E E E 5 .892 F P P P E E E 6 .977 F P P P E E E 7 1.055 F F P P P E E
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8 1.128 F F F P P P E 9 1.197 F F F F P P E 10 1.262 F F F F P P P 11 1.323 F F F F P P P 12 1.382 F F F F P P P 13 1.438 F F F F P P P At m=12 electrons, the Fermi radius in units of 2 π a is 1.382. The shortest distance to the edge of the 7 th BZ (i.e. the point at which the eighth BZ would begin to be filled) occurs along the diagonal of the reciprocal lattice (see Figure 9.15). At that point, the radius of the Fermi circle would be: k f = 2 π a 2 = 1.404 2 π a This Fermi wavevector is generated between m=12 electrons and m=13 electrons. Thus for m=12 electrons only zones less than or equal to 7 are occupied while for m=13 electrons we begin to fill the eight and higher zones. 2. a.
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