Now let us see the simulation results of spin transport in following four kinds of operation systems, and discuss some interesting phenomena revealed by them.
Here the operation conditions are similar to that of the previous chapter Spin Relaxation, that is we have F a lT l , where l is the phase relaxation length, lT is the transport mean-free-path, a is the size of the operation system,
F is the Fermi wavelength, so that we can treat the transport cases in terms of ballistic cavities viewpoint and then apply semiclassical approximation to solve them as the statement in Chapter 6 of Part I.
Figure 8.1. Four kinds of operation systems for spin transport simulation.
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In our simulation the main purpose is to …nd out some signi…cant implication of the simulation results (that is the simulation patterns) related to the main simulation parameters. They are the operation systems[73] (here we divide them into four kinds, Fig. 8.1, (a) and (b) are ring-like open systems, they belong to regular system, the main di¤erence between them is the ratio of the radius of inner circle and the radius of outer circle, for (a) we have ri : ro = 29=30 : 1, for (b), ri : ro = 9=10 : 1, (c) and (d) are chaotic systems, the main di¤erence between them is the ratio of the radius of inner circle and the half length of the outer square, for (c) we have ri : ro = 9=10 : 1, (d) ri : ro = 8=10 : 1), the operation cases (here as the conditions discussed in Chapter 8 Spin Relaxation, we divide the operation cases into nine (or ten) kinds, they are Rashba term case, Dresselhause term cases (they include linear and cubic terms) for di¤erent magnitude of wave vector, and the combination of Rashba and Dresselhause terms cases for di¤erent magnitude of wave vector, Fig. 7.1, and we also consider a special case of operation case, which is the combination of Rashba term and Dresselhause linear term, and the strength (or magnitude) of the e¤ective magnetic …eld for both cases are the same), the initial spinor (mainly we vary the initial spinor from expectation value in +x direction, expectation value in +y direction, expectation value in +z direction to the expectation value in arbitrary direction), and the scale of the operation cases (that is we divide this parameter into two situation, one is we treat the maximal magnitude of the Bef f for various operation cases correspond to the same magnitude of a new parameter Lso which it means the
degree of precession and we assume jBef fj / 1=Lso, the purpose of such treatment is to o¤er the information of the e¤ect of the con…guration of Bef f releted to the simulation patterns, another is we treat the various operation cases as a whole, we assume the magnitude of Bef f in operation case, R case, as a reference, that is we assume that it corresponds to a speci…c Lso value, and then depending on the real situation of Bef f for di¤erent operation cases, the di¤erent magnitude of Bef f
corresponding to a scaled magnitude of Lso, we execute such simulation is try to o¤er the information to see what happened about the e¤ect of real con…guration of Bef f for various operation case related to simulation patterns), and so forth.
In this chapter we mainly divide it into three parts, part one says the overview of the simulation patterns, it talks about the characters and implication of the simulation patterns for all operation parameters, etc., in part two we talk about the equivalence of Rashba term case (i.e. R case) and Dresselhause linear case (generally speaking, it is the D k1case), and in part three we represent a special case, that is the combination of Rashba term and Dresselhause linear term case which the strengths of the maximal Bef f for the two cases are the same, we …nd a never decay interesting phenomena in simulation pattern.
8.1. Overview of the Conductance Decay Patterns (Bef f Con…guration Normalized Case)
In principle the conductance pattern is governed by the following factors, sys-tem categories, syssys-tem size, Bef f con…guration and the initial input of spinor, and
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Figure 8.2. Spin conductance hgxxi v.s. 1=Lso under the action of normalized Bef f in more narrow circular ring system.
so on. Figures 8.2, 8.3 and 8.4 show the conductance versus 1=Lso in circular ring with inner radius Ri = 29=30, outer radius Ro = 1 and the width of both inlet and outlet w = 0:1 for initial spinor (or polarization) in x, y and z direction under the action of various Bef f con…guration. Here we have normalized the maximum of Bef f to be 1 for various kinds of operation cases. In Fig. 8.2 we …nd the similarity between R case and RD k2 case, they show the hgxxi decays very fast to the xdirection and then gradually comes back to someone minus hgxxi value, that is
Figure 8.3. Spin conductance hgyyi v.s. 1=Lso under the action of normalized Bef f in more narrow circular ring system.
very interesting that the hgxxi seems not to approach zero even in the very large strength of Bef f, i.e. in large value of 1=Lso. We also …nd the similarity between D k2 and D k4 cases, they show the oscillated decay to zero as 1=Lso goes by. And we also …nd the slightly decay pattern in D k3 and RD k4 cases, they don’t exhibit the suddenly deep decay behavior at the decay portion. We also
…nd the tendency of di¤erent oscillation decay period between RD k2, RD k3 and RD k4cases, and for the three cases they seem to exhibit the similar decay
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Figure 8.4. Spin conductance hgzzi v.s. 1=Lso under the action of normalized Bef f in more narrow circular ring system.
pattern. The most interesting decay pattern is D k1 case and RD k1 case, in D k1 case the decay pattern shows a very beautiful cone shape with almost the same period of oscillation, the theoretical prediction has been done by A.G.
Mal’shukov et al. [67], and the simulation has been shown by C.H. Chang et al. [72], the special regular behavior exhibits great applicability in future. For the RD k1 case, it shows a very special decay oscillation pattern. It seems to exhibit a tendency similar to the pattern of D k1 (or D k4) case, but with
an oscillation tendency attached in this decay pattern. In principle it shows a deep implication that if we vary the initial spinor input, we could obtain the very slowly decay pattern, or even creat a never decay pattern (we should discuss this situation in below sections). Well! anyhow these various kinds of decay pattern response the various Bef f con…guration of di¤erent operation cases, see Fig. 7.1.
I think that the reason is very obvious, but it implies that we could creat various conductance pattern as possibly as we could by create various Bef f con…guration for the possible application in future.
Now let’s focus on Fig. 8.3 and see what happened. At …rst glance we have a strong impression about the similarity and strong correlation between the hgyyi of R, D k1, RD k1 cases and hgxxi of R, D k1, RD k1 cases, Fig. 8.2.
The reason is due to the equivalence between pure Rashba term e¤ect and pure Dresselhause linear term e¤ect, we should give a detailed explanation for them in below section. We also …nd the similarity between R and RD k2 cases, the phenomena also be exhibited in Fig. 8.2, the reason is due to the similarity of Bef f
con…guration of R and RD k2 cases, Fig. 7.1. Besides we also …nd an obvious di¤erence between hgyyi of RD k1 case and hgxxi of RD k1 case, it implies that we can’t ignore the e¤ect of initial spinor input, and it also implies that someone deep correlation between operation system, operation case (i.e. the con…guration of Bef f), and initial spinor input, and so on. For other operation cases they show the similar decay pattern to that of Fig. 8.3.
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Next let’s see the pattern of Fig. 8.4. Well! it seems to exhibit a particular decay pattern of its own. At …rst we …nd the similarity between R and D k1 cases, it seems to tell us that something symmetry implication between the circular ring operation system and R and D k1 cases. And we …nd a very interesting phenomena that for R, D k1 s D k4 and RD k2 cases, they exhibit a residue decay value, the …nal decay value hgyyi seems not to approach to be zero. It seems to tell us that if the Bef f con…guration exhibits an equial distribution with respect to di¤erent k, the residue character exhibited. And if the Bef f con…guration owns an un-equal distrubution, like RD k1, RD k3 and RD k4 cases, the conductance should approach zero as 1=Lso goes by. Of course this suggestion is happened under the circular ring system and the initial spinor in z direction, we don’t see the tendency for the initial spinor input in x and y direction, see Figures 8.2 and 8.3.
Of course, for these all patterns shown in Figures 8.2, 8.3 and 8.4, the universal conductance oscillation phenomenon is very obvious, it can be realized easily by the point of view of the technique of the semiclassical approach analysis.
Now let us look at what happened as we vary the width of the circular ring.
The more details about the e¤ect of the size and the width of the circular ring related to the oscillation pattern should be described in Chapter 10. Figures 8.5, 8.6 and 8.7 show the conductance pattern in circular ring which with the same operation conditions except for Ri = 9=10 and Ro = 1. In Fig. 8.5 we …nd almost the same decay feature to the corresponding case of the wider circular ring
Figure 8.5. Spin conductance hgxxi v.s. 1=Lso under the action of normalized Bef f in less narrow circular ring system.
operation system, see Fig. 8.2, the main di¤erences are the conductance is larger and the conductance oscillation is faster and more radical, so they show indented oscillation feature. These features re‡ect that as we increase the width of the ring, more electrons can go through the outlet after they orbit several times of the ring, and since the width is increased, the oscillation behaviors of the trajectories of the electrons exhibit more radical aspects. Well! we also note a very interesting aspect
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Figure 8.6. Spin conductance hgyyi v.s. 1=Lso under the action of normalized Bef f in less narrow circular ring system.
in R case, the conductance hgxxi exhibits arc-like shape decrement as 1=Lso goes by.
Next we …nd that Fig. 8.6 exhibits the same tendency to Fig. 8.3. The same phenomenon also happens between Fig. 8.7 and Fig. 8.4. But wait a moment, from Figures 8.3, 8.4 and Figures 8.6, 8.7, and even from Figures 8.2, 8.5, we …nd the RD k2case seems to show a diminutive conduction pattern compared with R case. From the Bef f con…guration we could realize the phenomenon, since they
Figure 8.7. Spin conductance hgzzi v.s. 1=Lso under the action of normalized Bef f in less narrow circular ring system.
exhibit the similar Bef f con…guration, see Fig. 7.1, but I think it is interesting to theorize these phenomena.
Now let us turn our focus to another operation system, Sinai billiard, to see what happened about their conductance decay patterns. Here we explore two di¤erent sizes of Sinai billiard, in Figures 8.8, 8.9, 8.10, and 8.11, 8.12, 8.13, the Ri = 9=10 and Ro = 1, the widths of both inlet and outlet are 0:2, and in Figures 8.14, 8.15, 8.16 and 8.17, 8.18, 8.19, the Ri = 8=10 and Ro = 1, the widths of
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Figure 8.8. Spin conductance hgxxi v.s. 1=Lso under the action of normalized Bef f in more narrow Sinai billiard system.
both inlet and outlet are 0:2. Compare Fig. 8.8 with Fig. 8.2, Fig. 8.9 with Fig. 8.3, Fig. 8.10 with Fig. 8.4, Fig. 8.14 with Fig. 8.5, Fig. 8.15 with Fig.
8.6, Fig. 8.16 with Fig. 8.7 each other, we …nd the almost the same tendency between these paired patterns. The most striking aspect is in the D k2 case, we observe the appearance of most slow decay pattern. Reference to the Bef f
con…guration, see Fig. 7.1 we could understand the aspect, the reason is due to the Bef f disappeared in some region of k, so the spin in these region doesn’t relax.
Figure 8.9. Spin conductance hgyyi v.s. 1=Lso under the action of normalized Bef f in more narrow Sinai billiard system.
And we also …nd the particular decay aspects in R, D k1 and RD k1 cases, they exhibit the most chiseled decay pattern. For R and D k1cases, they also show the correlation implication between their own patterns, the reason is due to the equivalence between R and D k1 cases. For D k1 case, the special Bef f
con…guration causes the particular decay pattern. We will discuss that the special Bef f con…guration could create a never decay conductance aspect in the section below. Figures 8.11, 8.12, 8.13 and 8.17, 8.18, 8.19 show the sum up aspects of the
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Figure 8.10. Spin conductance hgzzi v.s. 1=Lso under the action of normalized Bef f in more narrow Sinai billiard system.
conductance patterns for each di¤erent initial spinor input cases of the di¤erent size of Sinai billiard. We could clearly see the special aspects for the D k2case (light green line), it exhibits the most slow decay aspects, and the most clean up decay pattern in R (black line), D k1(red line) and RD k1(purple line) cases. Finally we observe that in Sinai billiard system, as 1=Lso goes by, the conductance decaies to zero for all operation cases. It seems to imply that the particular aspects is due to the operation system is chaotic system, we could see that the conductance
Figure 8.11. Sum up of spin conductance hgxxi v.s. 1=Lso under the action of nine kinds of normalized Bef f in more narrow Sinai billiard system.
Figure 8.12. Sum up of spin conductance hgyyi v.s. 1=Lso under the action of nine kinds of normalized Bef f in more narrow Sinai billiard system.
decay doesn’t always approach zero for various operation cases in circular ring system, since it belongs to regular system. We also …nd the same appearance for spin relaxation simulation in last chapter.
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Figure 8.13. Sum up of spin conductance hgzzi v.s. 1=Lso under the action of nine kinds of normalized Bef f in more narrow Sinai billiard system.
8.2. Some Aspects of Conductance Decay Patterns (Bef f Con…guration in Real Material)
In last section we talk about the decay patterns under the action of normal-ized Bef f con…guration, that is we treat the maximum of Bef f as 1. The purpose of such treatment is to try to o¤er the information about the e¤ect of Bef f
con-…guration related to the decay pattern. Well! as the statement of last section we found out many interesting phenomena. Here we want to see what happened about the conductance decay pattern under the action of Bef f con…guration in real material (although it is also deduced from theoretical calculation). Fig. 7.1 shows the Bef f con…guration in real material, the larger and bolder arrow means the larger (stronger) Bef f, so under di¤erent standpoint of these Bef f (i.e. not to be normalized aspect) the decay patterns may be re‡ected something di¤erent and interesting aspects.
Figure 8.14. Spin conductance hgxxi v.s. 1=Lso under the action of normalized Bef f in less narrow Sinai billiard system.
Here we don’t intend to give an overview about the e¤ect of Bef f con…guration in real material, we just pick some representative operation cases to indicate the e¤ect of Bef f con…guration in real material. Fig. 8.20 indicates the obvious dif-ference between the normalized Bef f con…guration and Bef f con…guration in real material for D k2 case in circular ring system with Ri = 29=30 , Ro = 1 and widths of both inlet and outlet w = 0:1. As Fig. 7.1 shown, in D k2 case the maximum of Bef f is smaller than one of Bef f in R case (which the magnitude of
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Figure 8.15. Spin conductance hgyyi v.s. 1=Lso under the action of normalized Bef f in less narrow Sinai billiard system.
Bef f is setted to be 1), so we obtain a slow down decay pattern, i.e. we obtain a stretched-out and shifted toward right decay pattern (red line) as comparison with the decay pattern of normalized Bef f con…guration (black line). Fig. 8.21 shows a reversed trend, since we …nd the maximum of Bef f in RD k4case is much larger than the one in R case, that is meant that the e¤ect of Bef f is much stronger in Bef f con…guration in real material. So the decay pattern exhibits contracted and
Figure 8.16. Spin conductance hgzzi v.s. 1=Lso under the action of normalized Bef f in less narrow Sinai billiard system.
shifted toward left decay aspects (red line) compared with the decay pattern of normalized Bef f con…guration (black line).
The same trend happened in Figures 8.22 and 8.23, for them we proceed the simulation in Sinai billiard with Ri = 8=10 , Ro = 1 and widths of both inlet and outlet w = 0:2 under D k1 and RD k4 cases. For D k1 case we …nd out the smaller magnitude of Bef f than the one of Bef f in R case from Fig. 7.1, so we predict to obtain a slow down decay pattern as Fig. 8.22 shown. Note that for
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Figure 8.17. Sum up of spin conductance hgxxi v.s. 1=Lso under the action of nine kinds of normalized Bef f in less narrow Sinai billiard system.
Figure 8.18. Sum up of spin conductance hgyyi v.s. 1=Lso under the action of nine kinds of normalized Bef f in less narrow Sinai billiard system.
these two paired cases, the initial conductances don’t vary, the reason is obvious since the variation of the magnitude of Bef f doesn’t a¤ect the number of electrons which leave o¤ the outlet, so we get the same conductance initial values.
Figure 8.19. Sum up of spin conductance hgzzi v.s. 1=Lso under the action of nine kinds of normalized Bef f in less narrow Sinai billiard system.
8.3. Equivalence between R and D k1 Modi…ed Cases
Now let us talk about the very interesting topic, the equivalence between R case and D k1modi…ed case (i.e. we just consider the linear term of Dresselhause Hamiltonian).
See Fig. 8.24, we …nd out the result
(8.1) SR;x_y_zPn;R;x_y_z = SD;y_x_ zPn;D;y_x_ z = SD;x_y_zPn;D;x_y_z
where we set Pn;R;x_y_z as the initial spinor for Rashba term case (R case)
(8.2) Pn;R;x_y_z = cos2
sin2ei
!
;
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Figure 8.20. Comparison between the normalized Bef f con…guration and Bef f con…guration in real material for D k2 case in circular ring system.
and the corresponding Pn;D;x_y_z as the initial spinor for Dresselhause linear term case (D k1modi…ed case)
(8.3) Pn;D;x_y_z = cos12( )
sin12( ) ei(2 )
! :
SR;x_y_z is the equivalent rotation operator for someone speci…c electron travels half-tour of circular ring under the action of R case viewed in x_y_z coordinates
Figure 8.21. Comparison between the normalized Bef f con…guration and Bef f con…guration in real material for RD k4case in circular ring system.
system, and SD;x_y_z is the equivalent rotation operator for someone speci…c elec-tron travels half-tour of circular ring under the action of D k1 modi…ed case viewed in x_y_z coordinates system. And SD;y_x_ z is the equivalent rotation operator for someone speci…c electron travels half-tour of circular ring under the action of D k1 modi…ed case viewed in y_x_ z coordinates system, here we
…nd that SR;x_y_z is the same as SD;y_x_ z, reference Eq. 3.30. And Pn;D;y_x_ z
is the initial spinor viewed in y_x_ z coordinates system, it has the same form as Pn;D;x_y_z which viewed in x_y_z coordinates system.
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Figure 8.22. Comparison between the normalized Bef f con…guration and Bef f con…guration in real material for D k1 case in Sinai billiard.
Then let us apply the above result to give two examples for illustration. See Fig.
8.25, the …rst example we proceeded is the circular ring system with Ri = 29=30, Ro = 1 and widths of both inlet and outlet w = 0:1 in R case, the initial spinor
8.25, the …rst example we proceeded is the circular ring system with Ri = 29=30, Ro = 1 and widths of both inlet and outlet w = 0:1 in R case, the initial spinor