二維三角晶格能帶和態(tài)密度

2023-09-11 11:05 作者:syr56 0人讀過 | 我要投稿

二維三角晶格

緊束縛近似下的哈密頓量為

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

$%3C%5Ctextbf%7Bij%7D%3E$ 表示僅考慮電子與最近鄰(NN，每個(gè)原子有6個(gè)最近鄰原子)格點(diǎn)的躍遷，

最近鄰格點(diǎn)間距為 $a_0$ ， $%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%5Ctextbf%7Ba%7D_%7B%5Ctextbf%7BNN%7D%7D%3D%5Cleft%5C%7B%20%5Carray%7B%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%20a_0%5Chat%7Bx%7D%20%5C%5C%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B(%5Cpm%20%5Cfrac%7B1%7D%7B2%7Da_0%5Chat%7Bx%7D%5Cpm%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0%5Chat%7By%7D)%7D%20%5Cright.$ ；

<\textbf{ij}>' $AAA$ 表示僅考慮電子與次近鄰(NNN，每個(gè)原子有6個(gè)次近鄰原子)格點(diǎn)的躍遷，

次近鄰格點(diǎn)間距為 $2a_0$ ， $%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%5Ctextbf%7Ba%7D_%7B%5Ctextbf%7BNNN%7D%7D%3D%5Cleft%5C%7B%20%5Carray%7B%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%20%5Csqrt%7B3%7Da_0%5Chat%7By%7D%20%5C%5C%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B(%5Cpm%20%5Cfrac%7B3%7D%7B2%7Da_0%5Chat%7Bx%7D%5Cpm%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0%5Chat%7By%7D)%7D%20%5Cright.$ ；

<\textbf{ij}>'' $AAA$ 表示僅考慮電子與第三近鄰(TNN，每個(gè)原子有6個(gè)第三近鄰原子)格點(diǎn)的躍遷，

第三近鄰格點(diǎn)間距為 $3a_0$ ， $%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%5Ctextbf%7Ba%7D_%7B%5Ctextbf%7BTNN%7D%7D%3D%5Cleft%5C%7B%20%5Carray%7B%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%202a_0%5Chat%7Bx%7D%20%5C%5C%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B(%5Cpm%20a_0%5Chat%7Bx%7D%5Cpm%20%5Csqrt%7B3%7Da_0%5Chat%7By%7D)%7D%20%5Cright.$ 。

通過傅里葉變換可以得到動(dòng)量空間中的哈密頓量為：

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

能帶函數(shù)為：

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

態(tài)密度為：

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

三角晶格的實(shí)空間基矢可取為 $%5Cleft%5C%7B%5Cbegin%7Baligned%7D%5Ctextbf%7Ba%7D_1%26%3Da_0%5Chat%7Bx%7D%20%5C%5C%5Ctextbf%7Ba%7D_2%26%3D%5Cfrac%7B1%7D%7B2%7Da_0%5Chat%7Bx%7D%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Da_0%5Chat%7By%7D%5Cend%7Baligned%7D%20%5Cright.$ ，由公式 $%5Ctext%7Ba%7D_%7Bi%7D%5Ccdot%20%5Ctextbf%7Bb%7D_j%3D2%5Cpi%5Cdelta_%7Bij%7D$ ，可以求得動(dòng)量空間的基矢為 $%5Cleft%5C%7B%5Cbegin%7Baligned%7D%5Ctextbf%7Bb%7D_1%26%3D%5Cfrac%7B2%5Cpi%7D%7Ba_0%7D%5Chat%7Bk%7D_x-%5Cfrac%7B2%5Cpi%7D%7B%5Csqrt%7B3%7Da_0%7D%5Chat%7Bk%7D_y%20%5C%5C%5Ctextbf%7Bb%7D_2%26%3D%5Cfrac%7B4%5Cpi%7D%7B%5Csqrt%7B3%7Da_0%7D%5Chat%7Bk%7D_y%5Cend%7Baligned%7D%20%5Cright.$ ，取 $a_0%3D1$ ，

則有 $%5Cleft%5C%7B%5Cbegin%7Baligned%7D%5Ctextbf%7Bb%7D_1%26%3D2%5Cpi%5Chat%7Bk%7D_x-%5Cfrac%7B2%5Cpi%7D%7B%5Csqrt%7B3%7D%7D%5Chat%7Bk%7D_y%20%5C%5C%5Ctextbf%7Bb%7D_2%26%3D%5Cfrac%7B4%5Cpi%7D%7B%5Csqrt%7B3%7D%7D%5Chat%7Bk%7D_y%5Cend%7Baligned%7D%20%5Cright.$ ，因此可以得到三角晶格的第一布里淵區(qū)為一個(gè)正六邊形，同時(shí)和費(fèi)米面一起繪制出來，其中 $t_1%3Dt_2%3Dt_3%3D1$ ，如下圖

圖2 第一布里淵區(qū)和費(fèi)米面：NN(左)，NNN(中)，TNN(右)

三維能帶和能帶投影如下

沿高對稱路徑( $%5CGamma-K-M-%5CGamma$ )的能帶和態(tài)密度圖像如下

附：

【哈密頓量的傅里葉變換過程】

傅里葉變換公式，參見《固體理論》--李正中

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

最近鄰項(xiàng)的傅里葉變換

$%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，可得到次鄰項(xiàng)和第三近鄰項(xiàng)為

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

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

【代碼】

能帶、費(fèi)米面圖像繪制

態(tài)密度繪制

【英文縮寫】

? ?NN：Nearest Neighbor，最近鄰

NNN：Next?Nearest Neighbor，次近鄰

TNN：Third?Nearest Neighbor，第三近鄰

標(biāo)簽：

最美情侣中文字幕电影,在线麻豆精品传媒,在线网站高清黄,久久黄色视频

二維三角晶格能帶和態(tài)密度

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