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[转载]石墨烯(graphene)的色散关系

2010-12-15 23:28阅读：

http://blog.sina.cn/dpool/blog/u/1721153897

石墨烯(graphene)的色散关系
石墨烯(graphene)作为近年来的研究热点，有很多优异的物理性质。利用紧束缚方法(Tight-binding approach)计算出的能量色散关系(energy dispersion relation)可以发现很多有趣的东西，同时对碳纳米管的研究也有一定的指导意义。

The energy dispersion relation formula.

Honeycomb lattice and its Brillouin zone.(以上公式和图见Rev. Mod. Phys., Vol. 81,No. 1,109-162,2009.)

The values of Brillouin zone
A number of interesting and peculiar features emerge from this figure. First it is clear that the valence and the conduction bands touch each other at a number of finite momentum values. The momentum values at which the two bands touch are termed Dirac points (there are two in the Brillouin zone) and are represented by the momentum vectors K and K’. As a consequence, graphene’s spectrum does not have an energy gap. On the other hand, since the bands only touch at two momentum points the density of states is zero at the corresponding energy. Therefore, grapheme is sometimes termed a zero-gap semiconductor with vanishing density of states at the Fermi energy.(J. Phys.: Condens. Matter 21 (2009) 323201)

Dirac points附近的数值。

附Matlab程序
%单层石墨（graphene）的色散关系
%Copyright 2009 Zheng-Wei Zuo
clc;clear all;close all
%清除之前的内存空间变量、图形等
syms x y %d定义自变量，x代表Kx，以此类推;
t=2.7;%the nearest-neighbor hopping energy (hopping between different sublattices).
m=-0.2*t;%the next nearest-neighbor hopping energy (hopping in the same sublattices).
a=1.42;%the carbon-carbon distance.
E1=t*sqrt(3+2*cos(sqrt(3)*y*a)+4*cos(sqrt(3)/2*y*a)*cos(3/2*x*a))-m*(2*cos(sqrt(3)*y*a)+4*cos(sqrt(3)/2*y*a)*cos(3/2*x*a));%the upper band
E2=-t*sqrt(3+2*cos(sqrt(3)*y*a)+4*cos(sqrt(3)/2*y*a)*cos(3/2*x*a))-m*(2*cos(sqrt(3)*y*a)+4*cos(sqrt(3)/2*y*a)*cos(3/2*x*a));%the lower band
ezsurf(E1,[-4*sqrt(3)*pi/(9*a),4*sqrt(3)*pi/(9*a),-4*sqrt(3)*pi/(9*a),4*sqrt(3)*pi/(9*a)])%plot the upper
hold on %继续在当前图形上画图
ezsurf(E2,[-4*sqrt(3)*pi/(9*a),4*sqrt(3)*pi/(9*a),-4*sqrt(3)*pi/(9*a),4*sqrt(3)*pi/(9*a)])%plot the lower
shading interp; % 每个等高面用不同的颜色进行区分
box on%加外框
axis tight
figure; %另外画一图形
ezsurf(E2,[-sqrt(3)*pi/(18*a),sqrt(3)*pi/(18*a),7*sqrt(3)*pi/(18*a),sqrt(3)*pi/(2*a)])%plot the upper
hold on %继续在当前图形上画图
ezsurf(E1,[-sqrt(3)*pi/(18*a),sqrt(3)*pi/(18*a),7*sqrt(3)*pi/(18*a),sqrt(3)*pi/(2*a)])%plot the lower
shading interp; % 每个等高面用不同的颜色进行区分
box on%加外框
axis tight
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