摘 要
鍵結軌道模型是一個功能非常強的方法,運用在半導體及其異質結構之理論計算。此方法可視為緊密束縛法(tight-binding method) 與k‧p法之連結，且同時具有此兩種方法之優點。鍵結軌道模型包含取多緊密束縛法之優點，且無冗長累人之求參數過程。因其能帶數目與k‧p法相同，故計算時間與k‧p法接近。而其處理邊界問題很直接，不像波包函數(k‧p)法一樣複雜。其另一種優點為能計算半導體元件複雜的幾何結構。
(001)成長方向之鍵結軌道法已刊登在其它文獻由張亞中教授著述。為了得到(hkl)任意成長方向之漢米爾頓(Hamiltonian)，我們使用正交轉換矩陣旋轉上述(001)方向漢米爾頓而得到。此論文已計算完成(11N)方向有關量子井與超晶格之能帶結構(bandstructure) 。為了得到精確之能帶結構中高對稱點(除了 點，也包含X點)，必須把第二近鄰之互相作用也包含在鍵結軌道模型之架構內。
事實上，鍵結軌道模型是在緊密束縛法匡架下，以類似s與p之函數當基底。由於此基底是在晶胞大小尺度，所以微觀之界面微擾效應(晶胞內部效應)被鍵結軌道模型忽略。為了改進此缺點，我們把此基底展開成四角體鍵基底表示與位能運算子取代純量位能，如此便能把微觀之現象粹取出來，此稱為修正式的鍵結軌道模型。對非共同原子(no-common-atom)之量子井與超晶格，例如：InAs/GaSb和(InGa)As/InP，異種鍵存在於界面，將造成對稱性降低。根據我們的修正式方法，得到不尋常之位能矩陣在界面，而非純量位能。此修正法能直接探討異質結構(heterostructure)界面之化學鍵對稱性。
過去，張亞中教授曾計算(10,10) InAs/GaSb超晶格，用鍵結軌道模型與緊密束縛兩方法。他指出此兩種方法算得之能帶結構有巨大的差異。此差異源自界面之微擾效應，而被鍵結軌道模型忽略掉，但可被其修正鍵結軌道模型解決。因此修正法可視為上述兩方法之連結。對非共同原子量子井(InGa)As/InP之(001)、 (111)、 (110)方向樣本，其界面反非對稱性效應已在此本論文有詳細分析，並產生零電磁場自旋分裂(zero-field spin splitting)與界面方向之非等向性(in-plane anisotropy) 。用修正式鍵結軌道模型計算(11N)方向之InAs/GaSb超晶格，我們發現新的自旋分裂現象。此分裂發生在成長方向，但(001)與(111)方向之超晶格則無此分裂。此分裂與Dresselhaus效應類似，故稱其為Dresselhaus-like自旋分裂。此論文同時也探討(001)、 (111)、 與(110)方向InAs/GaSb超晶格之半金屬現象。此半金屬現象(負能隙)源自於三種貢獻：能帶之非等向性、自旋分裂、與多重能帶耦合，其顯示與成長方向有強烈關係。而導電帶與價電帶之子能帶之交叉(crossing)行為產生零能隙也在此論文內討論。
此論文之另一個主題為：若用泰勒展開鍵結軌道模型之漢米爾頓至動量(k)之第二階近似，我們可以得到相同過去微擾法算得之k‧p漢米爾頓。過去k‧p法之矩陣元內的動量相關項目均合併成一項，而我們推得之k‧p矩陣元內之動量相關項卻是分離的。在k‧p有限差分法(finite difference method)之計算上，我們的分離型式之k‧p能算得較精確結果。取最適當之微分距離在有限差分法過程，令人驚訝地，我們發現：在成長方向，k‧p法與鍵結軌道法有完全相同能帶。在鍵結軌道模型匡架上，包括多點能帶或多點作用鄰近晶胞，以上述方法作用在此模型，如此得到之k‧p法更精確，不僅 點，其它高對稱點均可同時精確。
再者，我們也計算量子井在帶間之光矩陣。除了(001) 與(111)方向之量子井外，光之非等向性(偏極化)皆可看到，此現象基於 點的混成效應(mixing effect) 。除了(001) 與(111)方向， 點重電洞與輕電洞之狀態皆有混成效應。此混成效應與光之非等向性皆源自於低對稱性。

Abstract
The bond orbital model (BOM) is a very powerful method for theoretical calculating on the semiconductors and their heterostructures. This method is a link of the tight-binding and k‧p method, while it combines the merit above two methods. The BOM contains many virtues of the empirical tight-binding method, while avoiding the tedious fitting procedure. The computational effort required to the BOM is comparable to the k‧p method involving the same number of bands. Unlike the envelope-function (k‧p) method, the boundary conditions of the BOM are straight-forward. The another advantage of the BOM is its flexibility to accommodate otherwise awkward geometries of the device structure.
The (001)-oriented BOM had published in detail by Chang. To obtain the (hkl)-oriented BOM Hamiltonian, we rotate the above-mentioned (001) Hamiltonian by the orthogonal transform matrix. The (11N)-oriented bandstructres of bulks quantum wells, and superlattices have been finished in this dissertation. To fit the high-symmetry points, not only the -point and the X-point, the second-neighbor interaction is included into the BOM framework.
The BOM is a tight-binding-like form with s- and p-like basis functions. Due to the unit-cell-scale basis set, the microscopic interface perturbation (intracell effect) is neglected by the BOM. To improve this problem, we expand the BOM basis in terms of the tetrahedral (anti)bonding orbitals and use the potential operator instead of the scalar potential for the extraction of the microscopic information, this is a so-called modified bond orbital model (MBOM). For the no-common-atom quantum wells or superlattices, such as InAs/GaSb and (InGa)As/InP, the heterobonds exist at the interfaces, which result in the symmetry reduction. According to our MBOM calculation, the unusual potential matrix emerges at the interfaces, but not a scalar potential. The MBOM provides the direct insinght into the microscopic symmetry of the crystal chemical bonds in the vicinity of the heterostructure interfaces.
In the past, Chang has calculated the (10,10) InAs/GaSb superlattice with the BOM and tight-binding method. He pointed out the sizable differences of the bandstructure between these two methods. These differences originate from the interface perturbation, which is neglected by the BOM but can solved by the MBOM. Therefore, the MBOM can regarded as a link between the BOM and tight-binding method. For the no-common-atom (InGa)As/InP quantum wells grown on the (001)-, (111)-, (110)-substrates, the interface inversion asymmetry effects are analyzed in detail on this dissertation, which result in the zero-field spin splitting and in-plane anisotropy. Based on the MBOM calculation on the (11N)-oriented InAs/GaSb superlattices, the new phenomenon of zero-field spin splitting is discovered. This spin splitting happens on the growth direction of the (hkl)-oriented no-common-atom superlattices other than the (001) and (111). Moreover, this spin splitting is based on a similar phenomenon of the Dresselhaus effect, so we term it as Dresselhaus-like spin splitting. In the semimetal regime, the bandstructures of (001)-, (111)-, and (110)-oriented InAs/GaSb superlattices are studied by the MBOM. The semimetal phenomenon (a negative band gap) may originates from three possible contributions: the band anisotropy, the spin splitting, and the multiband coupling, which is strongly growth-direction dependent. The crossing behavior (a zero gap) between the conduction and valence subbands is also discussed.
In the second topic of this dissertation, we take the Taylor-expansion on the BOM Hamiltonian up to the second order in k and then obtain the same k‧p Hamiltonian as usual. The usual k‧p Hamiltonian makes the k-dependent terms for each matrix elements lumped together (all-in-one). However, the k-dependent terms of our k‧p matrix are discrete with the different interaction neighbors, which are easily used by the k‧p finite difference method to obtain the most accurate result. With the optimum step length in the differential calculation, it is so surprisingly found that the k‧p bandstructure in the longitudinal direction can yield as same as that of the BOM. Taking the same process depicted above on the BOM with more bands and more interaction neighbors, we can obtain a more accurate k‧p formalism which can fit not only the -point but also other high-symmetry points.
In addition, the optical matrix elements of interband transistion in quantum wells are estimated. Apart from the (001) and (111) quantum wells, the optical anisotropy can be seen, which is in accordance with the zone-center mixing effect. The (11N) Hamiltonian other than the (001) and (111) has the zone-center mixing effect between the heavy-hole and light-hole states. Due to the low symmetry, this mixing effect and in-plane anisotropy exist on the semiconductors and their heterostructures.

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