Ferromagnetic (Ga,Mn)As Layers and ... - OPUS Würzburg

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58 4. Finite Element Simulations of Strain Relaxation 1 .0 x 1 0 -3 8 .0 x 1 0 -4 6 .0 x 1 0 -4 4 .0 x 1 0 -4 S tra in 2 .0 x 1 0 -4 0 .0 -2 .0 x 1 0 -4 -4 .0 x 1 0 -4 e (1 1 0 ) e z e x y e x = e y e x (1 0 0 s trip e ) e z (1 0 0 s trip e ) -6 .0 x 1 0 -4 -8 .0 x 1 0 -4 -1 .0 x 1 0 -3 2 4 6 8 1 0 R e la tiv e W id th Figure 4.8: Simulation of the average strain in a (Ga,Mn)As stripe aligned along the [ 110 ] crystal direction. The solid lines represent the strain in the cubic directions (blue: e x = e y , red: e z ), the shear strain (green: e xy ), and the calculated relaxation perpendicular to the stripe axis (black: e [110] ). The dotted lines are the simulation results for an identical stripe aligned along [100], taken from Fig. 4.4. By comparing Eqn. (4.19) and Eqn. (4.20), we can now identify: e [110] = 1 2 (e x − 2e xy + e y ) (4.21) e [110] = 1 2 (e x + 2e xy + e y ) (4.22) For the given material parameters of the stipe, the strain in [1¯10] is equal to −1.5 · 10 −3 , as no strain relaxation takes place in this direction. The strain e [110] can be directly compared to e x for a stripe along [100] to determine the difference in perpendicular relaxation for both geometries. We note that in the rotated matrix E ′ , the shear strain vanishes, as we would expect for a stripe aligned with the coordinate axes. The strain simulation shown in Fig. 4.8 was compiled by varying the relative width of a (Ga,Mn)As stripe with identical parameters as discussed in Section 4.3.1. By comparing the relaxation perpendicular to the stripe, e [110] , with the the corresponding e x (dotted blue line) of the nonrotated stripe, we can immediately see, that we achieve a lesser degree of relaxation in this geometry for otherwise identical structures. This fact is also evident in the larger value of e z in the rotated stripe. The
4.5. Stripes Along [1¯10] 59 in-plane strain relaxation is split equally between e x and e y , and therefore significantly different from the [100] case, where e x represents the full in-plane relaxation and e y remains at the pseudomorphic value. The shear strain e xy is a new factor whose influence on the magnetic anisotropy will be investigated in Chapter 5. For large values of rw, the [1¯10] stripes also approach the limit of an unpatterned pseudomorphic layer, with e xy → 0.
Page 1 and 2:
Ferromagnetic (Ga,Mn)As Layers and
Page 3 and 4:
Contents Zusammenfassung 5 Summary
Page 5 and 6:
Zusammenfassung Ferromagnetische Ha
Page 7 and 8: Anisotropiekontrolle in (Ga,Mn)As i
Page 9 and 10: Summary The great promise of ferrom
Page 11 and 12: Chapter 1 Introduction The event co
Page 13 and 14: 13 investigated with SQUID and magn
Page 15 and 16: Chapter 2 The (Ga,Mn)As Material Sy
Page 17 and 18: 2.1. Ferromagnetism in (Ga,Mn)As 17
Page 19 and 20: 2.2. Band Structure Calculations in
Page 21 and 22: 2.3. Magnetic Anisotropy 21 growth
Page 23 and 24: 2.3. Magnetic Anisotropy 23 E n e r
Page 25 and 26: 2.3. Magnetic Anisotropy 25 Figure
Page 27 and 28: Chapter 3 MBE Growth of Ferromagnet
Page 29 and 30: 3.2. Epitaxial Growth of (Ga,Mn)As
Page 31 and 32: 3.3. Crystal Defects 31 lead to a d
Page 33 and 34: 3.4. RHEED 33 Figure 3.5: (a) Ewald
Page 35 and 36: 3.5. X-Ray Diffraction 35 1 0 7 1 0
Page 37 and 38: 3.5. X-Ray Diffraction 37 between t
Page 39 and 40: 3.5. X-Ray Diffraction 39 lattice d
Page 41 and 42: 3.5. X-Ray Diffraction 41 substrate
Page 43 and 44: Chapter 4 Finite Element Simulation
Page 45 and 46: 4.1. Derivation of the Equation Sys
Page 47 and 48: 4.2. The FlexPDE Software 47 4.2 Th
Page 49 and 50: 4.3. Simulation Results - Physical
Page 55 and 56: 4.4. Simulation Results - (In,Ga)As
Page 57: 4.5. Stripes Along [1¯10] 57 Figur
Page 61 and 62: Chapter 5 Local Anisotropy Control
Page 63 and 64: 5.1. Patterning and Structural Char
Page 73 and 74: 5.2. Magnetic Characterization 73 p
Page 75 and 76: 5.2. Magnetic Characterization 75 1
Page 77 and 78: 5.2. Magnetic Characterization 77 8
Page 79 and 80: 5.3. Shape Anisotropy 79 ∼ 0.2 k
Page 81 and 82: 5.4. A Model for Anisotropy Orienta
Page 83 and 84: 5.4. A Model for Anisotropy Orienta
Page 85 and 86: Chapter 6 Device Application As sta
Page 87 and 88: 6.2. The Role of the Constriction 8
Page 93 and 94: Chapter 7 Conclusion and Outlook Th
Page 95 and 96: Appendix A Band Structure Hamiltoni
Page 97 and 98: Appendix B Sample FlexPDE Input Fil
Page 99 and 100: 99 BOUNDARIES s u r f a c e ’ s u
Page 101 and 102: Bibliography [Abol 01] M. Abolfath,
Page 103 and 104: 103 [McGu 75] T. R. McGuire and R.
Page 105 and 106: Publications • J. Wenisch, C. Gou
Page 107: Acknowledgements There is a number
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Ferromagnetic (Ga,Mn)As Layers and ... - OPUS Würzburg

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