量子計算被視為是下一世代的科技發展，用於解決傳統電腦無法解決的問題。其中量子位元是其基本單位對應到傳統電腦中的位元。Si/SiO_2金屬氧化半導體和SiGe/Si/SiGe異質結構是兩個實現自旋量子位元的潛在候選者。然而，Si 導帶中的最低能階的六個簡併態受到Si/SiO_2或SiGe/Si介面的的拉伸應變破壞，導致基態提升，變成2-2-fold和4-fold的簡併能態。介面中的電場進一步導致2-fold能階谷分裂，對於量子位元的傳輸和讀取造成雜訊。因此，在矽基異質結構中，了解谷分裂對於雜訊控制問題至關重要。 我們利用有效質量理論和密度函數理論對非理想情況進行數值分析，例如 Ge 擴散到 Si 中、Ge 在勢壘中的變化濃度和界面粗糙度。 我們單獨計算非理想情況，我們發現 Ge 的擴散和變化的濃度會給谷分裂帶來大約 10% 的變化。 界面粗糙度也通過修改哈密頓量和包絡函數來計算。 通過修改，與理想情況相比，變化將達到 90% 左右。 我們還建立了一個90nm∙90nm∙40nm維度的真實模型來模擬包含所有非理想情況的真實情況。 結果表明，高達 94% 的數據會被抑制並接近實驗數據，解決了有效質量研究會高估能谷分裂值的問題。

Quantum computing has been considered as one the technologies of next generation and ben used to solve the problems that classical computers couldn’t solve. Quantum bits(Qubits) are the basic unit of quantum computer, compared to the bits in classical computer. SiGe/Si/SiGe heterostructures is the potential candidate to achieve the spin qubits. However, the degeneracy in the Si conduction band minimum is broken by the tensile strain and the ground state is further lifted, leading to the valley splitting by the electric field noise in the SiGe/Si interface. Therefore, understanding of valley splitting has been crucial to the noise controlling issue. We utilize the multivalley effective mass theory along with the density function theory to acquire numerical analysis of the non-ideal situations, such as diffusion of Ge into Si, variational concentration of Ge in the barrier and the interface roughness. We calculate non-ideal situations individually and we find the diffusion of Ge and variational concentration would give around 10% change to the valley splitting. Interface roughness is also calculated by modifying the Hamiltonian and the envelop function. With the modification, the change would be around 90% compared to the ideal case. We also build up a realistic model in the dimension of 90nm∙90nm∙40nm to simulate the real case with all the non-ideal situations included. The result shows that up to 94% would be suppress and is close to the experimental data and solve the problem that EMTs would over-estimated the value of valley splitting.

1. DiVincenzo, D. P. (2000). The physical implementation of quantum computation. Fortschritte der Physik: Progress of Physics, 48(9‐11), 771-783.
2. Kjaergaard, M., Schwartz, M. E., Braumüller, J., Krantz, P., Wang, J. I. J., Gustavsson, S., & Oliver, W. D. (2020). Superconducting qubits: Current state of play. Annual Review of Condensed Matter Physics, 11, 369-395.
3. Clarke, J., & Wilhelm, F. K. (2008). Superconducting quantum bits. Nature, 453(7198), 1031-1042.
4. Häffner, H., Roos, C. F., & Blatt, R. (2008). Quantum computing with trapped ions. Physics reports, 469(4), 155-203
5. Blatt, R., & Wineland, D. (2008). Entangled states of trapped atomic ions. Nature, 453(7198), 1008-1015.
6. Deutsch, I. H., Brennen, G. K., & Jessen, P. S. (2000). Quantum computing with neutral atoms in an optical lattice. Fortschritte der Physik: Progress of Physics, 48(9‐11), 925-943.
7. Monroe, C. (2002). Quantum information processing with atoms and photons. Nature, 416(6877), 238-246.
8. Jelezko, F., Gaebel, T., Popa, I., Domhan, M., Gruber, A., & Wrachtrup, J. (2004). Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate. Physical Review Letters, 93(13), 130501.
9. Nayak, C., Simon, S. H., Stern, A., Freedman, M., & Sarma, S. D. (2008). Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics, 80(3), 1083.
10. Stern, A. (2010). Non-Abelian states of matter. Nature, 464(7286), 187-193.
11. Loss, D., & DiVincenzo, D. P. (1998). Quantum computation with quantum dots. Physical Review A, 57(1), 120.
12. English, J. H., Gossard, A. C., Störmer, H. L., & Baldwin, K. W. (1987). GaAs structures with electron mobility of 5× 106 cm2/V s. Applied physics letters, 50(25), 1826-1828.
13. Zwanenburg, F. A., Dzurak, A. S., Morello, A., Simmons, M. Y., Hollenberg, L. C., Klimeck, G., ... & Eriksson, M. A. (2013). Silicon quantum electronics. Reviews of modern physics, 85(3), 961.
14. Watson, T. F., Philips, S. G. J., Kawakami, E., Ward, D. R., Scarlino, P., Veldhorst, M., ... & Vandersypen, L. M. K. (2018). A programmable two-qubit quantum processor in silicon. nature, 555(7698), 633-637.
15. Culcer, D., Cywiński, Ł., Li, Q., Hu, X., & Sarma, S. D. (2009). Realizing singlet-triplet qubits in multivalley Si quantum dots. Physical Review B, 80(20), 205302.
16. Kawakami, E., Scarlino, P., Ward, D. R., Braakman, F. R., Savage, D. E., Lagally, M. G., ... & Vandersypen, L. M. K. (2014). Electrical control of a long-lived spin qubit in a Si/SiGe quantum dot. Nature nanotechnology, 9(9), 666-670.
17. Mohiyaddin, F. A., Simion, G., Stuyck, N. D., Li, R., Ciubotaru, F., Eneman, G., ... & Raduimec, I. P. (2019, December). Multiphysics simulation & design of silicon quantum dot qubit devices. In 2019 IEEE International Electron Devices Meeting (IEDM) (pp. 39-5). IEEE.
18. Shehata, M. M. E. K., Simion, G., Li, R., Mohiyaddin, F. A., Wan, D., Mongillo, M., ... & Van Dorpe, P. (2022). Modelling semiconductor spin qubits and their charge noise environment for quantum gate fidelity estimation. arXiv preprint arXiv:2210.04539.
19. Machlup, S. (1954). Noise in semiconductors: spectrum of a two‐parameter random signal. Journal of Applied Physics, 25(3), 341-343.
20. Dutta, P., & Horn, P. M. (1981). Low-frequency fluctuations in solids: 1 f noise. Reviews of Modern physics, 53(3), 497.
21. Fleetwood, D. M. (2015). 1/f noise and defects in microelectronic materials and devices. IEEE Transactions on Nuclear Science, 62(4), 1462-1486.
22. Saraiva, A. L., Calderón, M. J., Hu, X., Sarma, S. D., & Koiller, B. (2009). Physical mechanisms of interface-mediated intervalley coupling in Si. Physical Review B, 80(8), 081305.
23. Saraiva, A. L., Calderón, M. J., Capaz, R. B., Hu, X., Sarma, S. D., & Koiller, B. (2011). Intervalley coupling for interface-bound electrons in silicon: an effective mass study. Physical Review B, 84(15), 155320.
24. Gamble, J. K., Harvey-Collard, P., Jacobson, N. T., Baczewski, A. D., Nielsen, E., Maurer, L., ... & Muller, R. P. (2016). Valley splitting of single-electron Si MOS quantum dots. Applied Physics Letters, 109(25), 253101.
25. Boykin, T. B., Klimeck, G., Friesen, M., Coppersmith, S. N., von Allmen, P., Oyafuso, F., & Lee, S. (2004). Valley splitting in low-density quantum-confined heterostructures studied using tight-binding models. Physical Review B, 70(16), 165325.
26. Zhang, L., Luo, J. W., Saraiva, A., Koiller, B., & Zunger, A. (2013). Genetic design of enhanced valley splitting towards a spin qubit in silicon. Nature communications, 4(1), 1-7.
27. Jiang, Z., Kharche, N., Boykin, T., & Klimeck, G. (2012). Effects of interface disorder on valley splitting in SiGe/Si/SiGe quantum wells. Applied Physics Letters, 100(10), 103502.
28. Ando, T., Fowler, A. B., & Stern, F. (1982). Electronic properties of two-dimensional systems. Reviews of Modern Physics, 54(2), 437.
29. Goswami, S., Slinker, K. A., Friesen, M., McGuire, L. M., Truitt, J. L., Tahan, C., ... & Eriksson, M. A. (2007). Controllable valley splitting in silicon quantum devices. Nature Physics, 3(1), 41-45.
30. Davies, J. H. (1998). The physics of low-dimensional semiconductors: an introduction. Cambridge university press.
31. Kittel, C., McEuen, P., & McEuen, P. (1996). Introduction to solid state physics (Vol. 8, pp. 105-130). New York: Wiley.
32. Shindo, K., & Nara, H. (1976). The effective mass equation for the multi-valley semiconductors. Journal of the Physical Society of Japan, 40(6), 1640-1644.
33. Sholl, D. S., & Steckel, J. A. (2022). Density functional theory: a practical introduction. John Wiley & Sons.
34. Hohenberg, P., & Kohn, W. (1964). Inhomogeneous electron gas. Physical review, 136(3B), B864.
35. Kresse, G., & Furthmüller, J. (1996). Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Computational materials science, 6(1), 15-50.
36. Kresse, G., & Furthmüller, J. (1996). Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical review B, 54(16), 11169.
37. Kohn, W., & Sham, L. J. (1965). Self-consistent equations including exchange and correlation effects. Physical review, 140(4A), A1133.
38. Becke, A. D. (1988). Density-functional exchange-energy approximation with correct asymptotic behavior. Physical review A, 38(6), 3098.
39. Ceperley, D. M., & Alder, B. J. (1980). Ground state of the electron gas by a stochastic method. Physical review letters, 45(7), 566.
40. Perdew, J. P., & Yue, W. (1986). Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation. Physical review B, 33(12), 8800.
41. Heyd, J., Scuseria, G. E., & Ernzerhof, M. (2003). Hybrid functionals based on a screened Coulomb potential. The Journal of chemical physics, 118(18), 8207-8215.
42. Kresse, G., & Joubert, D. (1999). From ultrasoft pseudopotentials to the projector augmented-wave method. Physical review b, 59(3), 1758.
43. Blöchl, P. E. (1994). Projector augmented-wave method. Physical review B, 50(24), 17953.
44. https://www.andrew.cmu.edu/user/feenstra/wavetrans/
45. Möttönen, M., Vartiainen, J. J., Bergholm, V., & Salomaa, M. M. (2004). Quantum circuits for general multiqubit gates. Physical review letters, 93(13), 130502.
46. Chiribella, G., D’Ariano, G. M., & Perinotti, P. (2008). Quantum circuit architecture. Physical review letters, 101(6), 060401.
47. Shor, P. W. (1994, November). Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science (pp. 124-134). Ieee.
48. Grover, L. K. (1996, July). A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing (pp. 212-219).
49. Peruzzo, A., McClean, J., Shadbolt, P., Yung, M. H., Zhou, X. Q., Love, P. J., ... & O’brien, J. L. (2014). A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5(1), 4213.
50. Dorner, U., Demkowicz-Dobrzanski, R., Smith, B. J., Lundeen, J. S., Wasilewski, W., Banaszek, K., & Walmsley, I. A. (2009). Optimal quantum phase estimation. Physical review letters, 102(4), 040403.
51. Jordan, P., & Wigner, E. P. (1993). Über das paulische äquivalenzverbot (pp. 109-129). Springer Berlin Heidelberg.
52. Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J. M., & Gambetta, J. M. (2017). Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature, 549(7671), 242-246.
53. Qiskit @misc{Qiskit,
author = {{Qiskit contributors}},
title = {Qiskit: An Open-source Framework for Quantum Computing},
year = {2023},
doi = {10.5281/zenodo.2573505} }