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一维链式晶格能带和态密度

2023-08-26 12:42 作者:syr56  | 我要投稿

一维链式晶格

图1 一维链式晶格(基元包含一个原子)

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2个最近邻原子)格点的跃迁,

设最近邻格点间距为a_0%0A%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B%5Ctextbf%7Ba%7D_0%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%20a_0%5Chat%7Bx%7D%0A

%3C%5Ctextbf%7Bij%7D%3E'表示仅考虑电子与次近邻(NNN,每个原子有2个次近邻原子)格点的跃迁,

次近邻格点间距为2a_0%0A%5Ctextbf%7Bl%7D_%5Ctextbf%7Bi%7D%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%2B2%5Ctextbf%7Ba%7D_0%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%202a_0%5Chat%7Bx%7D

%3C%5Ctextbf%7Bij%7D%3E''%0A表示仅考虑电子与第三近邻(TNN,每个原子有2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%2B2%5Ctextbf%7Ba%7D_0%3D%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%5Cpm%203a_0%5Chat%7Bx%7D

通过傅里叶变换可以得到动量空间中的哈密顿量为:

H(k)%3D%5Csum_%7Bk%7D%5B-2t_1%5Ccos(ka_0)-2t_2%5Ccos(3ka_0)-2t_3%5Ccos(3ka_0)%5Dc_k%5E%7B%5Cdagger%7Dc_k%0A%5Ctag%7B2%7D%0A

因此能带函数为

%5Cbegin%7Baligned%7D%20%5Cmathrm%7BNN%3A%7D%20%20%26%5Cquad%20E(k)%3D-2t_1%5Ccos(ka_0)%20%5C%5C%20%5Cmathrm%7BNNN%3A%7D%20%26%5Cquad%20E(k)%3D-2t_1%5Ccos(ka_0)-2t_2%5Ccos(3ka_0)%20%20%5C%5C%20%5Cmathrm%7BTNN%3A%7D%20%26%5Cquad%20E(k)%3D-2t_1%5Ccos(ka_0)-2t_2%5Ccos(3ka_0)-2t_3%5Ccos(3ka_0)%20%20%5C%5C%20%5Cend%7Baligned%7D%0A%5Ctag%7B3%7D%0A

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

a_0%3D1%0A时,第一布里渊区(简约布里渊区)的范围为:-%5Cpi%3C%5Cpi%2C%5CDelta%20k%3D2%5Cpi%0A;能带和态密度图像如下(t_1%3Dt_2%3Dt_3%3D1)

图2 能带(左)和态密度(右)

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

以最近邻跃迁项为例

%5Cbegin%7Baligned%7D%0A-t_1%5Csum_%7B%3C%5Ctextbf%7Bi%7D%5Ctextbf%7Bj%7D%3E%7Dc_%5Ctextbf%7Bi%7D%5E%7B%5Cdagger%7Dc_%5Ctextbf%7Bj%7D%26%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_0%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_0)%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%7D%2C%5Ctextbf%7Bkk%7D'%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7D%5Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D%2B%20a_0%5Chat%7Bx%7D)%7De%5E%7Bi%5Ctextbf%7Bk%7D'%5Ccdot%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%7D%2Be%5E%7B-i%5Ctextbf%7Bk%7D%5Ccdot(%5Ctextbf%7Bl%7D_%7B%5Ctextbf%7Bj%7D%7D-a_0%5Chat%7Bx%7D)%7De%5E%7Bi%5Ctextbf%7Bk%7D'%5Ccdot%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%7D%5D%20%5C%5C%0A%26%3D-t_1%5Csum_%7B%5Ctextbf%7Bkk%7D'%7D%5Cfrac%7B1%7D%7BN%7D%5Csum_%7B%5Ctextbf%7Bj%7D%7De%5E%7B-i(%5Ctextbf%7Bk%7D-%5Ctextbf%7Bk%7D')%5Ccdot%20%5Ctextbf%7Bl%7D_%5Ctextbf%7Bj%7D%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7D(e%5E%7B-ika_0%7D%2Be%5E%7Bik'a_0%7D)%20%5C%5C%0A%26%3D-t_1%5Csum_%7B%5Ctextbf%7Bkk%7D'%7Dc_%7B%5Ctextbf%7Bk%7D%7D%5E%7B%5Cdagger%7Dc_%7B%5Ctextbf%7Bk%7D'%7D(e%5E%7B-ika_0%7D%2Be%5E%7Bik'a_0%7D)%5Cdelta_%7B%5Ctextbf%7Bkk%7D'%7D%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(e%5E%7B-ika_0%7D%2Be%5E%7Bika_0%7D)%3D%5Csum_%7Bk%7D-2t_1%5Ccos(ka_0)c_%7Bk%7D%5E%7B%5Cdagger%7Dc_%7Bk%7D%0A%5Cend%7Baligned%7D%0A%5Ctag%7B6%7D

【代码】

绘制能带曲线


绘制态密度曲线


【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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