本工作就石墨系統之具有ABC 堆疊構形者，以緊束縛模型研究其磁電子性質。該模型係基於磁場下的平移對稱性，不做微擾近似。在此對稱性下，哈密頓矩陣的行列維度是取決於磁場下的一個擴大晶胞。所得到的巨大矩陣則以共對角程序來處理，以便計算其本徵值和本徵向量，而給出藍道能階能譜和波函數。該等藍道能階波函數俱由六角形晶胞所含六個次晶格的六個磁緊束縛布洛赫函數所構成。根據波函數的振盪模可定義其有效量子數，用以指標藍道能階。藍道能階能譜反映零磁場下的次能帶特點。ABC 堆疊三層石墨有三群藍道能階，對應於三對零磁場次能帶。其中，較高的二群強烈混合。較低的一群包括一道賦有局域化量子態的藍道能階。這些量子態是局域化在外層中無垂直鄰層原子的次晶格上。在無限多層模型化的菱方石墨中，藍道能階能譜係類似單層石墨中所已知者，但有某些非類似的特點。本研究工作的緊束縛模型呈現了藍道能階的完整特徵，此乃模型能於哈密頓算符中納入全部的最近和次近
層間電子跳躍作用所致。同樣的哈密頓算符，在某些層間電子跳躍作用的影響下，其在有效質量近似下所表出的矩陣會是無限大的，從而難以精確計算分析。本研究的結果顯示，藍道能階能譜以費米能為準一般是不對稱的；在ABC 堆疊三層石墨中，則有三重藍道能階處於費米能附近，係由某些層間電子跳躍作用造成的細微分裂；在菱方石墨中，層間電子跳躍作用導致三維藍道次能帶，表徵了該系統與單層石墨的非類似點。

The low-energy magnetoelectronic properties of ABC-stacked trilayer graphenes are investigated by the tight-binding model, based on the magnetic translation symmetry without perturbative approximation. The Hamiltonian matrix under the symmetry has the dimension of a magnetic enlarged cell.
Co-diagonal procedures are developed to solve the eigenvalues and eigenvectors of such a huge matrix,
thus giving the energy spectra and the wave functions of the Landau levels. The Landau-level wave functions are each composed of six magnetic tight-binding Bloch functions for the six sublattices with respect to the hexagonal unit cell. Effective quantum numbers are defined according to the oscillation modes of the wave functions, being used to index the Landau levels. The Landau-level energy spectrum reflects the features of the zero-field subbands. ABC-stacked trilayer graphene possesses three groups of Landau levels corresponding to the three pairs of the zero-field subbands. The two higher-lying groups are strongly mixing. Of the lower-lying group, there exists a level having localized states. They are localized, in the outer layers, at the sublattices
with no vertical neighbors in contiguous layers. In rhombohedral graphite modelled with infinite ABC-stacked layers, the Landau level spectrum is similar to what has been realized in monolayer graphene but has certain dissimilar features. The tight-binding model displays the full characteristics of the Landau levels, as a consequence of including all the nearest and next-nearest interlayer hoppings into the Hamiltonian. In general, the energy spectrum about the Fermi energy is asymmetric. In ABC-stacked trilayer graphene, there are triplet levels in the vicinity of the Fermi energy, split finely by certain interlayer hoppings. In rhombohedral graphite,
three-dimensional Landau subbands are identified, characterizing the dissimilarities to monolayer graphene. The same Hamiltonian would be intractable within the effective-mass approximation because the Hamiltonian matrix becomes infinite due to certain interlayer hoppings.

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Ch.5
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