臺灣博碩士論文加值系統

在晶格電子只能做最近鄰與次近鄰的跳躍下, 我們討論外加一均勻磁場對
系統能量與霍爾電導的影響. 當每單位晶格的磁通量等於一有理數 p/q
(p,q為互質的兩個整數), 則一初始能帶將會分裂成 q 個次能帶. 而電子
填滿一個次能帶時, 由庫柏公式 (kubo formula) 來計算電子對霍爾電導
的貢獻, 其結果為一整數. 這和 Laughlin 的理論相符合. 另一方面, 當
次近鄰的跳躍強度在某一特定值時, 兩個相鄰的次能帶可能重疊在一起.
而當跳躍強度再偏離此一特定值時, 電子對霍爾電導的貢獻將會改變
e^2/h 的整數值 (0,1,2,...). 我們發現霍爾電導的改變量, 與在簡併附
近次能帶的結構有密切的關係; 其改變可由一整數, 即貝利指數 (Berry
index) 來描述.

We examine the energy and the Hall conductance of the tight-
binding electronson a square lattice with the nearest-neighbor
(NN) hopping as well as thenext-nearest-neighbor (NNN) hopping.
In a uniform magnetic field perpendicularto the square lattice,
the original band will split into q subbands when the flux per
unit cell equal to p/q (p,q are relatively prime). From Kubo
formula,we can determine the Hall conductance of a filled
subband to be an integer,which is consistent with the Laughlin
argument. Two subbands may touch atcertain values of NNN hopping
strength and the Hall conducatnce changes by aninteger multiple
of e^2/h after the degeneracy is lifted. We give an
explanationof the change by the Berry index which is closely
related to the local structurearound the degenerate point where
subbands touch.