# #2 This is a solid state physics problem. ( I use this book : kittel introduction...

#2 This is a solid state physics problem. ( I use this book : kittel introduction to soild state physocs ) thank you !

5. [Graphene] Consider graphene, a hexagonal lattice of carbon atoms as shown in the figure. The distance between neighboring carbon atoms is a0.143 nm (a) B A (a) Write down the unit lattice vectors ā, and a and unit reciprocal lattice vectors b, and b,. [5] a2 (b) The two sites A and B are not equivalent Therefore, one can write the wave function of an electron in this lattice as (F) a(F)by(F), (7-F,) and (F)= N/2^%n(F-) are the where (F) N2 wave functions for electrons located at A and B sites, respectively. p(f) is the orbital of a carbon atom located at F=0. Use tight-binding approximations to find the energy eigenvalue E() such that Hy(F)= E(k)(F), where H is the Hamiltonian. [15 Consider only the nearest-neighbor interaction and assume that the overlap of wave functions between A and B sites is negligible. For graphene, the hopping energy y(F-H |F- F,)= 2.8 eV 27 27 (c) Consider the K point of the Brillouin zone K= If one writes 3a'3a k K, one can Taylor expand E(k) near the K point in terms of . Show that E(R) « K. [10]

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