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Ab-initio Methods for Quantum Calculations on Surface Energies
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本研究係基於量子物理探討表面能之計算方法，包括塊材能量近似法、能量增益近似法、能量適調近似法以及真空近似法等四種不同方法，並以Ge(001)薄膜表面能之收斂性及GaN(0001)/(000-1)薄膜劈裂能之收斂性進行分析比較。在Ge(001)薄膜表面能研究中，首先，我們由能量適調近似法計算Ge(001)薄膜表面能得到0.086 eV/Å2，而塊材能量近似法對Ge(001)薄膜表面能做估計時，Ge(001)薄膜表面能值會隨著薄膜層數增加逐漸趨於發散，能量增益近似法對Ge(001)薄膜表面能做估計時，藉由收斂的能量增益值所估計的表面能在層數上升時即達到與收斂值相一致的結果，真空近似法估計Ge(001)薄膜表面能時，當真空厚度大於5.77Å[約為Ge(001)薄膜厚度的0.25倍]時，所計算出的Ge(001)薄膜表面能已能達到與收斂值相同之結果。經比較能量增益近似法及真空近似法表面能計算結果，我們發現能量增益近似法及真空近似法所計算之Ge(001)薄膜表面能值均與能量適調近似法所得0.086 eV/Å2相一致。在GaN(0001)/(000-1)薄膜劈裂能研究中，我們由能量適調近似法計算GaN(0001)/(000-1)薄膜之劈裂能可得到0.363 eV/Å2，而塊材能量近似法對GaN(0001)/(000-1)薄膜劈裂能做估計時，GaN(0001)/(000-1)薄膜劈裂能會隨著薄膜層數增加趨近於與收斂值相等，藉由收斂的能量增益值所估計的GaN(0001)/(000-1)薄膜劈裂能會隨著薄膜層數增加趨近於與收斂值相等，利用真空近似法估計GaN(0001)/(000-1)的薄膜劈裂能時，當真空厚度為5.24Å[約為GaN(0001)/(000-1)薄膜厚度的0.5倍]以上時，所計算出的GaN(0001)/(000-1)薄膜劈裂能會與收斂值趨於相等。經比較塊材能量近似法、能量增益近似法及真空近似法之劈裂能計算結果，我們發現塊材能量近似法、能量增益近似法及真空近似法所估計的GaN(0001)/(000-1)薄膜劈裂能分別為0.370 eV/Å2、0.368 eV/Å2與0.369 eV/Å2。綜上所述，能量適調近似法與真空近似法皆適合用來估算Ge(001)薄膜表面能與GaN(0001)/(000-1)劈裂能，而真空近似法的優勢在估算Ge(001)薄膜表面能及GaN(0001)/(000-1)薄膜劈裂能時，僅需藉由少數模型計算即可達到與收斂值吻合的結果。
We conduct ab-initio calculations to estimate the convergence of the surface energies for Ge(001) thin films and the cleavage energies for GaN(0001)/(000-1) thin films by using the bulk-energy approach, incremental energy approach, energy fit approach and vacuum approach. The surface energy of Ge(001) thin films is 0.086 eV/Å2 calculated from the energy fit approach. The surface energies of Ge(001) thin films calculated from the bulk-energy approach diverge with increasing the slab thickness. The surface energies of Ge(001) thin films calculated from the incremental energy approach converge with increasing the slab thickness. The surface energies of Ge(001) thin films calculated from the vacuum approach are consistent with the convergent value of 0.086 eV/Å2, which the vacuum thickness is 5.77Å(0.25 times larger than slab thickness). We find that both the surface energies of Ge(001) thin films calculated from incremental energy approach and from vacuum approach are identical to the value of 0.086 eV/Å2, which is estimated from energy fit approach. The cleavage energy of GaN(0001)/(000-1) thin films is 0.363 eV/Å2 calculated from the energy fit approach. The cleavage energies of GaN(0001)/(000-1) thin films from the bulk-energy approach converge with increasing slab thickness. The cleavage energies of GaN(0001)/(000-1) thin films from the incremental energy approach converge with increasing slab thickness. The cleavage energies of GaN(0001)/(000-1) thin films calculated from vacuum approach are close to the convergent value, which the vacuum thickness is 5.24Å(0.5 times larger than slab thickness). The converged cleavage energies of GaN(0001)/(000-1) thin films calculated from the bulk-energy approach, incremental energy approach and vacuum approach are 0.370 eV/Å2, 0.368 eV/Å2 and 0.369 eV/Å2, respectively. Both the energy fit approach and the vacuum approach are suitable for estimating the surface energy of Ge(001) thin films and the cleavage energies of GaN(0001)/(000-1) thin films. The advantage of vacuum approach is few computational cost of the accuracy surface/cleavage energies for Ge(001) thin films and GaN(0001)/(000-1) thin films.
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