二維方晶格能帶和態(tài)密度

2023-08-28 15:27 作者:syr56 0人讀過 | 我要投稿

二維方晶格

緊束縛近似下哈密頓量為

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

$%3C%5Ctextbf%7Bij%7D%3E$ 表示僅考慮電子與最近鄰(NN，每個原子有4個最近鄰原子)格點的躍遷，

最近鄰格點間距為 $a_0$ ， $%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%5Ctextbf%7Ba%7D_%7B%5Cmathrm%7BNN%7D%7D%3D%5Cleft%5C%7B%5Carray%7B%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%20a_0%5Chat%7Bx%7D%20%5C%5C%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%20a_0%5Chat%7By%7D%7D%20%5Cright.$

$%3C%5Ctextbf%7Bij%7D%3E'$ 表示僅考慮電子與次近鄰(NNN，每個原子有4個次近鄰原子)格點的躍遷，

次近鄰格點間距為 $%5Csqrt%7B2%7Da_0$ ， $%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%5Ctextbf%7Ba%7D_%7B%5Cmathrm%7BNNN%7D%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%20(%5Cpm%20a_0%5Chat%7Bx%7D%20%5Cpm%20a_0%5Chat%7By%7D)$

$%3C%5Ctextbf%7Bij%7D%3E''$ 表示僅考慮電子與第三近鄰(TNN，每個原子有4個第三近鄰原子)格點的躍遷，第三近鄰格點間距為 $2a_0$ ， $%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B2%5Ctextbf%7Ba%7D_%7B%5Cmathrm%7BTNN%7D%7D%3D%5Cleft%5C%7B%5Carray%7B%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%202a_0%5Chat%7Bx%7D%20%5C%5C%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%202a_0%5Chat%7By%7D%7D%20%5Cright.$

通過傅里葉變換可以得到動量空間中的哈密頓量為：

$%5Cbegin%7Baligned%7D%20H(%5Ctextbf%7Bk%7D)%3D%5Csum_%7B%5Ctextbf%7Bk%7D%7D%5C%7B%26-2t_1%5B(%5Ccos(k_xa_0)%2B%5Ccos(k_ya_0)%5D%5C%5C%20%26-2t_2%5B%5Ccos(k_xa_0%2Bk_ya_0)%2B%5Ccos(k_xa_0-k_ya_0)%5D%20%20%5C%5C%20%26-2t_3%5B%5Ccos(2k_xa_0)%2B%5Ccos(2k_ya_0)%5D%5C%7D%20c_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%20%5Cend%7Baligned%7D%0A%5Ctag%7B2%7D$

能帶函數(shù)為：

$%5Cbegin%7Baligned%7D%20%5Cmathrm%7BNN%3A%7D%20%20%26%5Cquad%20E(%5Ctextbf%7Bk%7D)%3D-2t_1%5B%5Ccos(k_xa_0)%2B%5Ccos%7Bk_ya_0%7D%5D%20%5C%5C%20%5Cmathrm%7BNNN%3A%7D%20%26%5Cquad%20E(%5Ctextbf%7Bk%7D)%3D-2t_1%5B%5Ccos(k_xa_0)%2B%5Ccos%7Bk_ya_0%7D%5D-2t_2%5B%5Ccos(k_xa_0%2Bk_ya_0)%2B%5Ccos(k_xa_0-k_ya_0)%5D%20%20%5C%5C%20%5Cmathrm%7BTNN%3A%7D%20%26%5Cquad%20E(%5Ctextbf%7Bk%7D)%3D-2t_1%5B%5Ccos(k_xa_0)%2B%5Ccos%7Bk_ya_0%7D%5D-2t_2%5B%5Ccos(k_xa_0%2Bk_ya_0)%2B%5Ccos(k_xa_0-k_ya_0)%5D%20-2t_3%5B%5Ccos(2k_xa_0)%2B%5Ccos(2k_ya_0)%5D%20%20%5C%5C%20%5Cend%7Baligned%7D%0A%5Ctag%7B3%7D$

態(tài)密度為：

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

取 $a_0%3D1$ ，第一布里淵區(qū)的范圍為：

$-%5Cpi%3Ck_x%3C%5Cpi%2C-%5Cpi%3Ck_y%3C%5Cpi%3B%5CDelta%20k_x%3D2%5Cpi%2C%5CDelta%20k_y%3D2%5Cpi%EF%BC%9B$

得到三維能帶和能帶投影如下( $t_1%3Dt_2%3Dt_3%3D1$ )

第一布里淵和費米面( $E(%5Ctextbf%7Bk%7D)%3D0$ )圖像如下

沿高對稱路徑( $%5CGamma-M-K-%5CGamma$ )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

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

最近鄰項的傅里葉變換

$%5Cbegin%7Baligned%7D%0A-t_1%26%5Csum_%7B%3C%5Ctextbf%7Bi%7D%5Ctextbf%7Bj%7D%3E%7Dc_%5Ctextbf%7Bi%7D%5E%7B%5Cdagger%7Dc_%5Ctextbf%7Bj%7D%3D-%5Cfrac%7Bt_1%7D%7BN%7D%5Csum_%7B%3C%5Ctextbf%7Bij%7D%3E%2C%5Ctextbf%7Bkk%7D'%7Dc_%5Ctextbf%7Bk%7D%5E%7B%5Cdagger%7De%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%7Dc_%7B%5Ctextbf%7Bk%7D'%7De%5E%7Bi%5Ctextbf%7Bk%7D'%5Ccdot%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%7D%3D-%5Cfrac%7Bt_1%7D%7BN%7D%5Csum_%7B%5Ctextbf%7Bj%7D%2C%5Ctextbf%7Bkk%7D'%2C%5Ctextbf%7Ba%7D_%7B%5Cmathrm%7BNN%7D%7D%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7De%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%2B%5Ctextbf%7Ba%7D_%5Cmathrm%7BNN%7D)%7De%5E%7Bi%5Ctextbf%7Bk%7D'%5Ccdot%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%7D%20%20%5C%5C%0A%26%3D-%5Cfrac%7Bt_1%7D%7BN%7D%5Csum_%7B%5Ctextbf%7Bj%2Ckk'%7D%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7De%5E%7Bi%5Ctextbf%7Bk%7D'%5Ccdot%5Ctextbf%7Bl%7D_j%7D%5Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot%20(%5Ctextbf%7Bl%7D_j%2Ba_0)%5Chat%7Bx%7D%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot%20(%5Ctextbf%7Bl%7D_j-a_0)%5Chat%7Bx%7D%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot%20(%5Ctextbf%7Bl%7D_j%2Ba_0)%5Chat%7By%7D%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot%20(%5Ctextbf%7Bl%7D_j-a_0)%5Chat%7By%7D%7D%5D%20%20%5C%5C%0A%26%3D-t_1%5Csum_%7B%5Ctextbf%7Bk%7D%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5Be%5E%7Bik_xa_0%7D%2Be%5E%7B-ik_xa_0%7D%2Be%5E%7Bik_ya_0%7D%2Be%5E%7B-ik_ya_0%7D%5D%20%20%5C%5C%0A%26%3D%5Csum_%7B%5Ctextbf%7Bk%7D%7D-2t_1%5B%5Ccos(k_xa_0)%2Bcos(k_ya_0)%5Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%0A%5Cend%7Baligned%7D%0A%5Ctag%7B6%7D$

類似地，第三近鄰項的傅里葉變換為

$-t_3%5Csum_%7B%3C%5Ctextbf%7Bij%7D%3E''%7Dc_%7B%5Ctextbf%7Bi%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bj%7D%7D%3D%5Csum_%7B%5Ctextbf%7Bk%7D%7D-2t_3%5B%5Ccos(2k_xa_0)%2Bcos(2k_ya_0)%5Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%0A%5Ctag%7B7%7D$

次近鄰項的傅里葉變換為

$%5Cbegin%7Baligned%7D%0A%26-t_2%5Csum_%7B%3C%5Ctextbf%7Bij%7D%3E'%7Dc_%7B%5Ctextbf%7Bi%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bj%7D%7D%3D-%5Cfrac%7Bt_2%7D%7BN%7D%5Csum_%7B%3C%5Ctextbf%7Bij%7D%3E'%5Ctextbf%7Bkk%7D'%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7De%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%7De%5E%7Bi%5Ctextbf%7Bk%7D%5Ccdot%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%7D%0A%3D-%5Cfrac%7Bt_2%7D%7BN%7D%5Csum_%7B%5Ctextbf%7Bj%7D%2C%5Ctextbf%7Bkk%7D'%2C%5Ctextbf%7Ba%7D_%7B%5Ctextbf%7BNNN%7D%7D%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7De%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_j%2B%5Ctextbf%7Ba%7D_%7B%5Ctextbf%7BNNN%7D%7D)%7De%5E%7Bi%5Ctextbf%7Bk%7D%5Ccdot%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%7D%20%20%20%5C%5C%5C%0A%26%3D-%5Cfrac%7Bt_2%7D%7BN%7D%5Csum_%7B%5Ctextbf%7Bj%7D%2C%5Ctextbf%7Bkk%7D'%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7D%20e%5E%7B-i%5Ctextbf%7Bk%7D'%5Ccdot%5Ctextbf%7Bl%7D_%7Bj%7D%7D%5Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7Bj%7D%2Ba_0%5Chat%7Bx%7D%2Ba_0%5Chat%7By%7D)%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7Bj%7D-a_0%5Chat%7Bx%7D%2Ba_0%5Chat%7By%7D)%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7Bj%7D%2Ba_0%5Chat%7Bx%7D-a_0%5Chat%7By%7D)%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7Bj%7D-a_0%5Chat%7Bx%7D-a_0%5Chat%7By%7D)%7D%5D%20%5C%5C%0A%26%3D-t_2%5Csum_%7B%5Ctextbf%7Bk%7D%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5Be%5E%7B-i(k_xa_0%2Bk_ya_0)%7D%2Be%5E%7Bi(k_xa_0%2Bk_ya_0)%7D%2Be%5E%7B-i(k_xa_0-k_ya_0)%7D%2Be%5E%7Bi(k_xa_0-k_ya_0)%7D%5D%20%20%5C%5C%0A%26%3D%5Csum_%7B%5Ctextbf%7Bk%7D%7D-2t_2%5B%5Ccos(k_xa_0%2Bk_ya_0)%2B%5Ccos(k_xa_0-k_xa_0)%5Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D%7D%0A%5Cend%7Baligned%7D%0A%5Ctag%7B8%7D$

【代碼】

能帶、費米面圖像繪制

態(tài)密度圖像繪制

【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：

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