二维三角晶格能带和态密度

2023-09-11 11:05 作者:syr56 0人读过 | 我要投稿

二维三角晶格

紧束缚近似下的哈密顿量为

$H%3D-t_1%5Csum_%7B%3C%5Ctextbf%7Bi%7D%5Ctextbf%7Bj%7D%3E%7D%20c_%5Ctextbf%7Bi%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bj%7D%7D-t_2%5Csum_%7B%3C%5Ctextbf%7Bi%7D%5Ctextbf%7Bj%7D%3E'%7D%20c_%5Ctextbf%7Bi%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bj%7D%7D-t_3%5Csum_%7B%3C%5Ctextbf%7Bi%7D%5Ctextbf%7Bj%7D%3E''%7D%20c_%5Ctextbf%7Bi%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bj%7D%7D%0A%5Ctag%7B1%7D$

$%3C%5Ctextbf%7Bij%7D%3E$ 表示仅考虑电子与最近邻(NN，每个原子有6个最近邻原子)格点的跃迁，

最近邻格点间距为 $a_0$ ， $%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%5Ctextbf%7Ba%7D_%7B%5Ctextbf%7BNN%7D%7D%3D%5Cleft%5C%7B%20%5Carray%7B%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%20a_0%5Chat%7Bx%7D%20%5C%5C%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B(%5Cpm%20%5Cfrac%7B1%7D%7B2%7Da_0%5Chat%7Bx%7D%5Cpm%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0%5Chat%7By%7D)%7D%20%5Cright.$ ；

<\textbf{ij}>' $AAA$ 表示仅考虑电子与次近邻(NNN，每个原子有6个次近邻原子)格点的跃迁，

次近邻格点间距为 $2a_0$ ， $%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%5Ctextbf%7Ba%7D_%7B%5Ctextbf%7BNNN%7D%7D%3D%5Cleft%5C%7B%20%5Carray%7B%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%20%5Csqrt%7B3%7Da_0%5Chat%7By%7D%20%5C%5C%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B(%5Cpm%20%5Cfrac%7B3%7D%7B2%7Da_0%5Chat%7Bx%7D%5Cpm%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0%5Chat%7By%7D)%7D%20%5Cright.$ ；

<\textbf{ij}>'' $AAA$ 表示仅考虑电子与第三近邻(TNN，每个原子有6个第三近邻原子)格点的跃迁，

第三近邻格点间距为 $3a_0$ ， $%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%5Ctextbf%7Ba%7D_%7B%5Ctextbf%7BTNN%7D%7D%3D%5Cleft%5C%7B%20%5Carray%7B%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%202a_0%5Chat%7Bx%7D%20%5C%5C%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B(%5Cpm%20a_0%5Chat%7Bx%7D%5Cpm%20%5Csqrt%7B3%7Da_0%5Chat%7By%7D)%7D%20%5Cright.$ 。

通过傅里叶变换可以得到动量空间中的哈密顿量为：

$%5Cbegin%7Baligned%7D%20H(%5Ctextbf%7Bk%7D)%3D%5Csum_%7B%5Ctextbf%7Bk%7D%7D%5C%7B%26-2t_1%5B%5Ccos(a_0k_x)%2B%5Ccos(%5Cfrac%7B1%7D%7B2%7Da_0k_x%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%2B%5Ccos(%5Cfrac%7B1%7D%7B2%7Da_0k_x-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%5D%20%20%20%5C%5C%20%26-2t_2%5B%5Ccos(a_0k_y)%2B%5Ccos(%5Cfrac%7B3%7D%7B2%7Da_0k_x%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%2B%5Ccos(%5Cfrac%7B3%7D%7B2%7Da_0k_x-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%5D%20%20%5C%5C%20%26%5Cquad%20-t_3%5B%5Ccos(2a_0k_x)%2B%5Ccos(a_0k_x%2B%5Csqrt%7B3%7Da_0k_y)%2B%5Ccos(a_0k_x-%5Csqrt%7B3%7Da_0k_y)%5D%5C%7D%20%5Cend%7Baligned%7D%0A%5Ctag%7B2%7D$

能带函数为：

$%5Cbegin%7Baligned%7D%20E_1(%5Ctextbf%7Bk%7D)%26%3D-2t_1%5B%5Ccos(a_0k_x)%2B%5Ccos(%5Cfrac%7B1%7D%7B2%7Da_0k_x%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%2B%5Ccos(%5Cfrac%7B1%7D%7B2%7Da_0k_x-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%5D%20%20%5C%5C%20E_2(%5Ctextbf%7Bk%7D)%26%3D-2t_2%5B%5Ccos(%5Csqrt%7B3%7Da_0k_y)%2B%5Ccos(%5Cfrac%7B3%7D%7B2%7Da_0k_x%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%2B%5Ccos(%5Cfrac%7B3%7D%7B2%7Da_0k_x-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%5D%20%20%5C%5C%20E_3(%5Ctextbf%7Bk%7D)%26%3D-2t_3%5B%5Ccos(2a_0k_x)%2B%5Ccos(a_0k_x%2B%5Csqrt%7B3%7Da_0k_y)%2B%5Ccos(a_0k_x-%5Csqrt%7B3%7Da_0k_y)%5D%20%20%5C%5C%20%5C%5C%20%5Cmathrm%7BNN%7D%3A%26%5Cquad%20E(%5Ctextbf%7Bk%7D)%3DE_1(%5Ctextbf%7Bk%7D)%20%5C%5C%20%5Cmathrm%7BNNN%7D%3A%26%5Cquad%20E(%5Ctextbf%7Bk%7D)%3DE_1(%5Ctextbf%7Bk%7D)%2BE_2(%5Ctextbf%7Bk%7D)%20%5C%5C%20%5Cmathrm%7BTNN%7D%3A%26%20%5Cquad%20E(%5Ctextbf%7Bk%7D)%3DE_1(%5Ctextbf%7Bk%7D)%2BE_2(%5Ctextbf%7Bk%7D)%2BE_3(%5Ctextbf%7Bk%7D)%20%5Cend%7Baligned%7D%0A%5Ctag%7B3%7D$

态密度为：

$%5Crho(%5Comega)%3D-%5Cfrac%7B1%7D%7BN%5Cpi%7D%5Cmathrm%7BIm%7D%20%5Csum_%7Bn%2C%5Ctextbf%7Bk%7D%7D%5Cfrac%7B1%7D%7B%5Comega-E_n(%5Ctextbf%7Bk%7D)%2Bi%5CGamma%7D%3D-%5Cfrac%7B1%7D%7BN%5Cpi%7D%5Csum_%7Bn%2C%5Ctextbf%7Bk%7D%7D%5Cfrac%7B%5CGamma%7D%7B%5B%5Comega-E_n(%5Ctextbf%7Bk%7D)%5D%5E2%2B%5CGamma%5E2%7D%20%20%0A%5Ctag%7B4%7D%0A$

三角晶格的实空间基矢可取为 $%5Cleft%5C%7B%5Cbegin%7Baligned%7D%5Ctextbf%7Ba%7D_1%26%3Da_0%5Chat%7Bx%7D%20%5C%5C%5Ctextbf%7Ba%7D_2%26%3D%5Cfrac%7B1%7D%7B2%7Da_0%5Chat%7Bx%7D%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0%5Chat%7By%7D%5Cend%7Baligned%7D%20%5Cright.$ ，由公式 $%5Ctext%7Ba%7D_%7Bi%7D%5Ccdot%20%5Ctextbf%7Bb%7D_j%3D2%5Cpi%5Cdelta_%7Bij%7D$ ，可以求得动量空间的基矢为 $%5Cleft%5C%7B%5Cbegin%7Baligned%7D%5Ctextbf%7Bb%7D_1%26%3D%5Cfrac%7B2%5Cpi%7D%7Ba_0%7D%5Chat%7Bk%7D_x-%5Cfrac%7B2%5Cpi%7D%7B%5Csqrt%7B3%7Da_0%7D%5Chat%7Bk%7D_y%20%5C%5C%5Ctextbf%7Bb%7D_2%26%3D%5Cfrac%7B4%5Cpi%7D%7B%5Csqrt%7B3%7Da_0%7D%5Chat%7Bk%7D_y%5Cend%7Baligned%7D%20%5Cright.$ ，取 $a_0%3D1$ ，

则有 $%5Cleft%5C%7B%5Cbegin%7Baligned%7D%5Ctextbf%7Bb%7D_1%26%3D2%5Cpi%5Chat%7Bk%7D_x-%5Cfrac%7B2%5Cpi%7D%7B%5Csqrt%7B3%7D%7D%5Chat%7Bk%7D_y%20%5C%5C%5Ctextbf%7Bb%7D_2%26%3D%5Cfrac%7B4%5Cpi%7D%7B%5Csqrt%7B3%7D%7D%5Chat%7Bk%7D_y%5Cend%7Baligned%7D%20%5Cright.$ ，因此可以得到三角晶格的第一布里渊区为一个正六边形，同时和费米面一起绘制出来，其中 $t_1%3Dt_2%3Dt_3%3D1$ ，如下图

三维能带和能带投影如下

沿高对称路径( $%5CGamma-K-M-%5CGamma$ )的能带和态密度图像如下

附：

【哈密顿量的傅里叶变换过程】

傅里叶变换公式，参见《固体理论》--李正中

$%5Cbegin%7Baligned%7D%0A%20%20%20%20%09c_%7Bnl%7D%26%3D%5Cfrac%7B1%7D%7B%5Csqrt%7BN%7D%7D%5Csum_%7Bk%5Cin%20BZ%7Dc_%7Bnk%7De%5E%7Bi%5Cvec%7Bk%7D%5Ccdot%5Cvec%7Bl%7D%7D%20%5Cqquad%20c_%7Bnk%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%7BN%7D%7D%5Csum_%7Bl%7Dc_%7Bnl%7De%5E%7B-i%5Cvec%7Bk%7D%5Ccdot%20%5Cvec%7Bl%7D%7D%20%20%5C%5C%0A%20%20%20%20%09c_%7Bnl%7D%5E%7B%5Cdagger%7D%26%3D%5Cfrac%7B1%7D%7B%5Csqrt%7BN%7D%7D%5Csum_%7Bk%5Cin%20BZ%7Dc_%7Bnk%7D%5E%7B%5Cdagger%7De%5E%7B-i%5Cvec%7Bk%7D%5Ccdot%5Cvec%7Bl%7D%7D%20%5Cqquad%20c_%7Bnk%7D%5E%7B%5Cdagger%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%7BN%7D%7D%5Csum_%7Bl%7Dc_%7Bnl%7D%5E%7B%5Cdagger%7De%5E%7Bi%5Cvec%7Bk%7D%5Ccdot%20%5Cvec%7Bl%7D%7D%20%5C%5C%0A%20%20%20%20%09~%5C%5C%0A%20%20%20%20%09%5Cfrac%7B1%7D%7BN%7D%26%5Csum_%7Bl%7De%5E%7B%5Cpm%20i(%5Cvec%7Bk%7D-%5Cvec%7Bk%7D')%5Ccdot%20%5Cvec%7Bl%7D%7D%20%3D%20%5Cdelta_%7Bkk'%7D%3B%20%5Cqquad%20%5Cfrac%7B1%7D%7BN%7D%5Csum_%7Bk%5Cin%20BZ%7De%5E%7B%5Cpm%20i%5Cvec%7Bk%7D%5Ccdot%20(%5Cvec%7Bl%7D-%5Cvec%7Bl%7D')%7D%3D%5Cdelta_%7Bll'%7D%0A%20%20%20%20%09%5Cend%7Baligned%7D%0A%5Ctag%7B5%7D$

最近邻项的傅里叶变换

$%5Cbegin%7Baligned%7D%0A%26-t_1%5Csum_%7B%3C%5Ctextbf%7Bij%7D%3E%7Dc_%7B%5Ctextbf%7Bi%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bj%7D%7D%3D-%5Cfrac%7Bt_2%7D%7BN%7D%5Csum_%7B%3C%5Ctextbf%7Bij%7D%3E%2C%5Ctextbf%7Bkk%7D'%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7De%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bi%7D%7D%7D%20e%5E%7Bi%5Ctextbf%7Bk%7D'%5Ccdot%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%7D%0A%3D-%5Cfrac%7Bt_1%7D%7BN%7D%5Csum_%7B%5Ctextbf%7Bj%7D%2C%5Ctextbf%7Bkk%7D'%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7D%20e%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%2B%5Ctextbf%7Ba%7D_%7B%5Ctextbf%7BNN%7D%7D)%7D%20e%5E%7Bi%5Ctextbf%7Bk%7D'%5Ccdot%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%7D%20%20%5C%5C%0A%26%3D-%5Cfrac%7Bt_1%7D%7BN%7D%5Csum_%7B%5Ctextbf%7Bj%7D%2C%5Ctextbf%7Bkk%7D'%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7D%20e%5E%7Bi%5Ctextbf%7Bk%7D'%5Ccdot%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%7D%5Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%2Ba_0%5Chat%7Bx%7D)%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D-a_0%5Chat%7Bx%7D)%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%2B%5Cfrac%7B1%7D%7B2%7Da_0%5Chat%7Bx%7D%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Chat%7By%7D)%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D-%5Cfrac%7B1%7D%7B2%7Da_0%5Chat%7Bx%7D-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Chat%7By%7D)%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D-%5Cfrac%7B1%7D%7B2%7Da_0%5Chat%7Bx%7D%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Chat%7By%7D)%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%2B%5Cfrac%7B1%7D%7B2%7Da_0%5Chat%7Bx%7D-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Chat%7By%7D)%7D%5D%20%20%20%5C%5C%0A%26%3D-t_1%5Csum_%7B%5Ctextbf%7Bk%7D%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5Be%5E%7B-ia_0k_x%7D%2Be%5E%7Bia_0k_x%7D%2Be%5E%7B-i(%5Cfrac%7B1%7D%7B2%7Da_0k_x%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%7D%2Be%5E%7Bi(%5Cfrac%7B1%7D%7B2%7Da_0k_x%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Dk_y)%7D%2Be%5E%7Bi(%5Cfrac%7B1%7D%7B2%7Da_0k_x-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%7D%2Be%5E%7B-i(%5Cfrac%7B1%7D%7B2%7Da_0k_x-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%7D%5D%20%20%5C%5C%0A%26%3D%5Csum_%7B%5Ctextbf%7Bk%7D%7D-2t_1%5B%5Ccos(a_0k_x)%2B%5Ccos(%5Cfrac%7B1%7D%7B2%7Da_0k_x%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%2B%5Ccos(%5Cfrac%7B1%7D%7B2%7Da_0k_x-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%5Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%0A%5Cend%7Baligned%7D%0A%5Ctag%7B6%7D$

同理，可得到次邻项和第三近邻项为

$-t_2%5Csum_%7B%3C%5Ctextbf%7Bi%7D%5Ctextbf%7Bj%7D%3E'%7D%20c_%5Ctextbf%7Bi%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bj%7D%7D%0A%3D%5Csum_%7B%5Ctextbf%7Bk%7D%7D-2t_2%5B%5Ccos(a_0k_y)%2B%5Ccos(%5Cfrac%7B3%7D%7B2%7Da_0k_x%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%2B%5Ccos(%5Cfrac%7B3%7D%7B2%7Da_0k_x-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0k_y)%5Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%0A%5Ctag%7B7%7D$

$-t_3%5Csum_%7B%3C%5Ctextbf%7Bi%7D%5Ctextbf%7Bj%7D%3E''%7D%3D%5Csum_%7B%5Ctextbf%7Bk%7D%7D-2t_3%5B%5Ccos(2a_0k_x)%2B%5Ccos(a_0k_x%2B%5Csqrt%7B3%7Da_0k_y)%2B%5Ccos(a_0k_x-%5Csqrt%7B3%7Da_0k_y)%5Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%0A%5Ctag%7B8%7D$

【代码】

能带、费米面图像绘制

态密度绘制

【英文缩写】

NN：Nearest Neighbor，最近邻

NNN：Next Nearest Neighbor，次近邻

TNN：Third Nearest Neighbor，第三近邻

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二维三角晶格能带和态密度

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