我們利用轉換矩陣方法嚴格求解自由電子模型近似下的三維水丁格方程式，必據以計算單位壘和雙位壘磁性穿隧元件的電流自旋極化和磁阻效應。在單位壘元件的計算中，我們發現磁阻效應將隨偏壓增大而變小，在某些偏壓範圍甚至呈現負直，並隨偏壓變化有一震盪的關係。在雙位壘元件的研究中，我們發現共振穿隧能級扮演一個很重要的角色，適當的設計共振能級的位置可以達到最佳磁阻和自旋電流極化源的效果。

Current spin polarization and tunneling magnetoresistance (MR) of
single and double magnetic tunneling junctions have been calculated within the free electron model by solving rigorously the 3D Schrodinger equation for different bias voltages and barrier widths. For single barrier junction, it is found that in general, the MR would decrease with voltage and become negative at some large voltages, a trend in agreement with both earlier and recent experiments. Also, it is revealed that for certain barrier widths, the MR would oscillate with bias voltage which is a prediction to be confirmed by experiments. We also describe a principle for designing double barrier magnetic tunneling structure that could produce spin current with controllable spin direction or have optimal spin valve effect. The principle is based on the finding that by tuning the energy position of spin dependent resonant tunneling level of a double barrier junction, fully spin-polarized current and optimal MR effect could be
readily achieved.

Contents
1 Introduction
2 Model and Method
2.0 The model
2.1 Transfer Matrix Method
2.2 Interpretation of M matrix
2.3 Current integration
3 Single Barrier TMR
3.1 Transmission Coefficient - Dispersion Relation
3.2 Current Plot and TMR Analysis
4 Double Barrier TMR
4.1 Configuration of Magnetization Axes
4.2 Spin Valve
4.3 Spin current source
4.3.1 Type I: NM/Is/Ic/FM/Ic/Is/NM structure
4.3.2 Type II: S/I/FM/I/S structure
5. Summary and Disscussion
References

References
1. J.S. Moodera, Lisa R. Kinder, Terrilyn M. Wong, and R. Meservey, Phys. Rev. Lett., 74, 3273, (1995).
2. J. S. Moodera and Lisa R. Kinder, J. Appl. Phy., 79, 4724 (1996).
3. M. Julliere, Phys. Lett., 54A, 225 (1975).
4. J. C. Slonczewski, Phys. Rev., B 39, 6995 (1989).
5. T Miyazaki, et. al., J. Phys. D: Apply. Phys. 31, 630 (1998).
6. Manish Sharma, et. al., Phys. Rev. Lett., 82, 616 (1999).
7. A. Voskoboynikov, et. al., Phys. Rev. B, (1999).
8. V. V. Paranjape, Phys. Rev. B 52, 10740 (1995)..
9. Xiangdong Zhang, et al., Phys. Rev. B 56, 5484 (1997).
10. S. T. Chui, Phys. Rev. B 55, 5600 (1997)
11. A. M. Bratkovsky. Phys. Rev. B 56, 2344 (1997)..
12. J. S. Moodera, et al. Phys. Rev. Lett. 70, 853 (1993).
13. Mary Beth STEARNS, J. Mag. Mag. Material 5 167 (1977)
14. Principles of Electron Tunneling Spectroscopy, E.L. Wolf (Oxford, Oxfordshire, 1985)
15. Handbook of Mathematical Functions, edited by M. Abramowittz and I. A. Stegun (Dover, New York, 1965).
16. X. Hao, et al., Phys. Rev. B 42, 8235 (1990).
17. J. L. Simonds, Physics Today, Apiral, 26 (1995)