As we mention in Part I, the key indicator in this thesis is that we treat the electrons transport in the mesoscopic system (e.g., regular or chaotic quantum systems, say circular quantum dot, quantum ring, etc.) as rigid balls travel in the cavity it may collide with the impurities and the wall of the cavity, that is we have a ballistic cavity since F a lT l .
We note that for the spin relaxation simulation the main parameters are the categories of operation systems (e.g., regular or chaotic systems), operation cases (e.g., Rashba term case or Dresselhause term case or the combination of Rashba and Dresselhause terms cases, with various wave vectors, here we denote them as
Figure 7.1. Con…guration of Bef f for various k in real material.
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Figure 7.2. Five operation systems.
k1, k2, k3 and k4, shown in Fig. 7.1), Lso (which means the strength of e¤ective magnetic …eld Bef f), the initial spinor (i.e., the direction of initial polarization), mean free path, lm, the size of operation system, and the number of acting particles, and so on. In this chapter, we shall mainly talk about the following topics[73], (1) the overview of spin relaxation in the regular systems (include (A) circular dot with smooth boundary, Fig. 7.2, (B) triangular dot with smooth boundary, Fig. 7.2), and in the chaotic systems (include (C) circular dot with rough boundary, Fig. 7.2, (D) Sinai quantum dot, we create a triangle-like quantum dot but with a section cut in one of its acute angles, Fig. 7.2, and (F) two-dimensional bulk-like cavity, see Fig. 7.2, for this system we minik there are impurities existing in the cavity), (2) the equivalence between Rashba term e¤ect and Dresselhause linear term e¤ect in spin relaxation, (3) the equivalence between RM S of Bef f and Bef f con…guration,
that is for di¤erent traveling segment-like trajectories, their corresponding Bef f
which with various magnitude and direction, could be treated as they own the same magnitude, i.e., the root-mean-square (RM S) value of the whole Bef f, and their corresponding direction is the same as the original individual Bef f, (4) cases of slow down spin relaxation and never relaxation of spin conductance, and so on.
7.1. Overview of Spin Relaxation Pattern (Bef f Con…guration Normalized Case)
At …rst let us see the overview of spin relaxation patterns for …ve di¤erent systems (we roughly divide them into regular systems, Sys: = 1, the circular dot with smooth boundary, Sys: = 2, the triangular dot, and chaotic systems, Sys: = 3, the circular dot with rough boundary, Sys: = 4, the Sinai billiard, and the more chaotic system, Sys: = 5, two-dimensional bulk-like system, we must note that in fact the system is a stochastic open system, but not closed system, the elastic collision length l distributed according to the Poisson law P rob(l) = e l=lm=lm, where lm is the mean free path. This is just the system where the conventional D’yakonov-perel’spin relaxation has to be observed), and under the action of nine (or ten) di¤erent kinds of Bef f con…gurations (we denote them as R case, Rashba term case, D k1 s D k4cases, Dresselhause linear and cubic terms cases, as k(wave vector) varies from small to large, and D k1modi…ed case, it is just the Dresselhause linear term case, and RD k1 s RD k4 cases, the combination of Rashba term and Dresselhause linear and cubic terms cases, as k varies from
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small to large). In gererally, here we assume Lso= 2 (indicator of strength of Bef f, we normalize the whole Bef f con…gurations for nine operation cases, that is the maximum of Bef f denoted by Lso = 2), lm (mean free path) = 1, R (radius of circular dot) = 1, and adjust the size of triangular dot and Sinai billiard in order to get the all …ve systems with the same area, and we also assume about 1000 particles (electrons) to be operated initially, …nally we treat the polarization of initial spinor orientated in z direction and the spin detection also orientated in z direction, the situation is symboled by Pcz.
Well! at …rst glance, what do we see in Fig. 7.3? We …nd the resembles between the nine cases except for RD k1 case, this phenomena seem to mean that the dominated factor which causes the relaxation pattern is the operation system, not operation cases. In general speaking, the regular system (black and red lines), after quick drop of the magnitude of initial spinor, we set Pcz(t = 0) = 1, they remain as constant as time goes by. And the drop magnitudes for the two regular systems, circular dot and triangular dot, depend on the operation cases. it’s a so magic thing that we observe that in D k2, RD k1cases, the spin conductance of circular dot system drops more large than that of triangular dot system. Does it mean the Bef f
con…guration has something deep implication with the shape of operation system?
For chaotic systems (green and blue lines), they exhibit relaxation aspects as time goes by. But we note that the more sensitive to operation cases for Sys: = 3 (green line), since it behaves like more chaotic system (Sys: = 5) under the more
‡uctuated operation cases (e.g. D k2 s D k4, RD k2 s RD k4). This is a
Figure 7.3. Spin relaxation rate under the viewpoint of e¤ect of di-rection of Bef f in the …ve operation systems.
so interesting phenomenon worth studying advancedly. And we also note that the almost same aspecte between R and D k1 cases. Yes! it is true, R and D k1 cases are equivalent, we should mention them in Sec. Equivalence between R and D k1 Modi…ed Cases below. We also …nd the most special case, RD k1case, it seems to own the ability to retain the spin polarization regardless the operation systems what they belong. Of course this is due to the special Bef f con…guration for this case, we also should discuss it more detailedly in Sections Relaxation Rate
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Figure 7.4. Sum up of nine operation cases into one under the view-point of e¤ect of direction of Bef f in the …ve operation systems.
Slow Down Case and Relaxation Rate Never Decay Case below. Finally we see the very fast relaxation system, Sys: = 5, the more chaotic system consistutes a particular category of systems. Since in basically it belongs to open system, it owns its special relaxation pattern, we should talk about it latter.
Fig. 7.4 shows the sum up of nine di¤erent operation cases into one for di¤erent systems. From the …gure we could obviously distinguish the e¤ect of systems. For Sys: = 1, it dominates the relaxation pattern regardless the operation cases, but for Sys: = 2, intrinsically it dominates the relaxation pattern, but the operation cases could a¤ect the spin polarization drop signi…cantly. And for Sys: = 3, we
…nd the signi…cant role of operation cases, di¤erent Bef f con…guration could causes various relaxation pattern, we see that the more regular Bef f con…guration, the
Figure 7.5. The e¤ect of Lso in relaxation rate.
more slow down relaxation aspects appeared. Sys: = 4 shows the typical character of chaotic system, it is less a¤ected by operation cases. Sys: = 5 also show the dominated role of the system, we should talk about it more detailedly in Sec.
Equivalence between RM S of Bef f and Bef f Con…guration below.
Next let’s see what happened as we vary the Lso (that is we vary the corre-spondent magnitude of Lso with the magnitude of the normalized maximal Bef f).
It is obvious that the more larger Lso, the more less relaxation trend appeared, Fig. 7.5.
We also get that if we vary the direction of polarization of initial spinor and the direction of polarization of detection, the much di¤erent relaxation patterns appeared. Fig. 7.6 shows the so much informations about the e¤ect of the direction
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Figure 7.6. The initial spinor input in +x and +y directions for R, D k1, RD k1 and RD k4 cases in all …ve operation systems.
of initial spinor input. We maybe retain the advanced research about it in other thesis in future, so here we just indicate few astonishing aspects come from it. We
…nd that the equivalence between R and D k1 cases in all …ve systems both for the initial spinor input in +x and +y directions, it is not so surprised. The surprised thing is that the much di¤erent relaxation patterns between di¤erent kinds of categories of regular and chaotic systems, say circular-dot-like systems (e.g. Sys: = 1 and Sys: = 3) and triangular-dot-like systems (e.g. Sys: = 2 and Sys: = 4), we …nd that the aspect of faster relaxation rate in circular-dot-like systems than that of triangular-dot-like systems in R, D k1, RD k1 and RD k4 cases and the aspect of reversed relaxation rate for the regular and chaotic cases
of circular-dot-like systems compared with that of triangular-dot-like systems in R, D k1 and RD k1 cases. And the so astonishing relaxation patterns in R, D k1 and RD k1 cases in Sys: = 1, they show so quick drop and oscillated aspects about the relaxation patterns. I think that the reason is due to since cases of R, D k1, RD k1 exhibit more regular con…guration, they in‡uence the intrinsic character of regular and chaotic systems. And for R, D k1, RD k1 and RD k4 cases in Sys: = 5 we also …nd a so much interesting phenomenon which seems to relate to spin relaxation time T1 (often called longitudinal or spin-lattice time, it seems to be the case of intial spinor input in +z direction) and spin dephasing time T2(also called transverse or decoherence time, it seems to be the case of intial spinor input in +x and +y directions), look at these cases and compare them with Fig. 7.9 in Sys: = 5; if we add the e¤ect of B0 (in our simulation B0 = 0, the existence of B0 should slow down the relaxation rate in the case of intial spinor input in +z direction in Sys: = 5) as described in subsection Spin Relaxation Time and Spin Dephasing Time of chapter 1 we seem to able to get the consistency about the statement T2 6 2T1. Well! we stop to discuss it more, but we must note that it is not a simple thing to indicate the correlation between the relaxation pattern and the various operation factors.
Before we end this section, we indicate an interesting thing, we …nd a similar motional narrowing aspect for spin relaxation, we observe that as we reduce the size of the system, the less relaxation aspect happened. The same phenomenon
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Figure 7.7. The e¤ects of lm (length of mean free path) and size of system in relaxation rate.
also be observed as we change the length of the mean free path lm in Sys: = 5, the more longer the lm, the less relaxation trend exhibited, see Fig. 7.7.
Fig. 7.7 also shows the e¤ect of number of particles (electrons) for simulation, the much larger amount of particles, the relaxation curve exhibits less ‡uctuant aspect.
7.2. Some Aspects of Spin Relaxation Patterns (Bef f Con…guration in Real Material)
Before we discuss the other interesting aspects about the relaxation patterns, now let’s represent the case of spin relaxation in real material.
Fig. 7.1 shows the nine di¤erent operation cases in real material, here we assume the strength of Bef f of R case as the reference, that is it corresponds to
Figure 7.8. Spin relaxation rate under the viewpoint of e¤ect of real (calculated) aspect of Bef f in the …ve operation systems.
Lso= 2, for the stronger Bef f, the corresponding Lso is smaller, and reversely the weaker Bef f corresponds to larger Lso.
At …rst glance these relaxation patterns shown in Fig. 7.8 seem to exhibit less correspondence. Compare with Fig. 7.3 and look at Fig. 7.1, for systems 1~4, we
…nd that since the Bef f con…gurations are weaker in D k1and D k2cases, so their spin relaxation patterns exhibit slow down aspect, but for D k4, RD k1
~RD k4cases, the more stronger Bef f con…guration, the more quick relaxation
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Figure 7.9. Sum up of nine operation cases into one under the view-point of e¤ect real (calculated) aspect of Bef f in the …ve operation systems.
patterns appeared. And for Sys: = 5, the variation of strength of Bef f also plays the same role as statement above. We should see the details latter.
Fig. 7.9 shows the sum up of nine di¤erent operation cases into one for di¤erent systems. Compare with Fig. 7.4, we observe the similar trend between them. The most di¤erence is the variation of Bef f con…guration reduces the intrinsic chacter of systems, especially for Sys: = 1 and Sys: = 4. these aspects imply that the strenth of Bef f con…guration owns the more powerful ability to a¤ect the relax-ation pattern than the various Bef f con…guration itself for some typical operation systems. Another important thing should be noted is that in our simulation we use unitless time parameter. In order to indicate the reasonableness about our
simulation, we should convert the unitless time parameter into real unit. We …nd that for chaotic system the spin relaxation time (up to zero) is about 20(ns), for more chaotic system the relaxation time (up to zero) is estimated about 100(ps).
Well! they lie in the reasonable spin relaxation time which spans from several pico-seconds to several micro-seconds [21].
7.3. Equivalence between R and D k1 Modi…ed Cases
See Fig. 7.3 we …nd the spin relaxation patterns in Rashba term case (R case) and Dresselhause linear and cubic terms case (D k1case) (here D k1modi…ed case (Dresselhause linear term case) is almost the same as D k1case) are almost the same. The reason is due to the con…guration of Bef f for R case and D k1case are almost the same except for the rotation trend, for R case if we view the wave vector variation in counter-clock-wise (C.C.W.), the corresponding Bef f rotates in C.C.W. and they all have the same magnitude, and for D k1 case if we view the wave vector variation in C.C.W., we …nd the corresponding Bef f rotates clock-wise (C.W.) and they also own the same magnitude Fig. 8.24. We then …nd that for the ensemble electrons which own di¤erent propagating direction individually, the C.C.W. or C.W. rotation of Bef f doesn’t matter at all about the ensemble results of spin evolution, so we get the interesting result. We should discuss it more detailedly in Chapter 8.
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Figure 7.10. Equivalence between RM S of Bef f and Bef f con…g-uration in the relaxation under various operation cases in two-dimensional bulk-like system.
7.4. Equivalence between RM S of Bef f and Bef f Con…guration Reference Fig. 7.10 we …nd that in the two-dimensional bulk-like system the spin relaxation patterns under the action of each operation cases are almost the same as that under the action of their corresponding root-mean-square (RMS) magnitude of the e¤ective magnetic …eld Bef f. The reason is due to, for exam-ple the RD k3 case, in someone moment each particles in the two-dimensional
bulk-like system own various propagating direction, and regardless of the corre-sponding direction of Bef f for someone speci…c propagating direction (i.e. the con…guration of Bef f), each particles su¤ered di¤erent degree of precession, we assume the equivalence between the RMS of Bef f and the Bef f con…guration for someone operation case is meant that we assume the e¤ect of the average of the more ‡uctuated degree of precession for each particles in RD k3 case is almost the same as that of the average of the less ‡uctuated degree of precession for each particles in the RMS of Bef f of RD k3 case. Then we obtain the funny result and the mathematical representation is shown below
PN n=1
Pn;someone specif ic operation casez (t) (7.1) N
PN n=1
Pn;RM S ofz B
ef f of someone specif ic operation case(t) N
where N is the number of particles used in simulation, N 1000 in our simu-lation.
Look at the R case of this …gure, we also …nd the excellent …t between simu-lation result and theoretical prediction of longitudinal DP relaxation, that is the dash lines indicate the results of well-known expression for the longitudinal DP relaxation PDP(t) = exp( 4tlm=L2so), the relevant discussion could be found in the thesis of C.H. Chang et al. [72]
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Figure 7.11. Relaxation rate slow down case under speci…c operation cases with speci…c initial spinors input in three di¤erent systems.
7.5. Relaxation Rate Slow Down Case
From the Bef f con…guration, Fig. 7.1, we get a hint that if we assume someone special initial polarization direction, maybe we could get something special results, we should continue this talk in Chapter 8. Fig. 7.11 shows three systems, Sys: = 1; 3 and 5, each with two di¤erent initial spinors input, spinor pn :
= (cos( =2=2);
sin( =2=2) exp(i =4))and spinor pm :
= (cos( =2=2); sin( =2=2) exp(i3 =4)), acting on RD k1and RD k4cases individually. From this …gure we observe the trend that if the adequate match between initial spinor and operation case existence, we could get the relaxation slow down aspects. The aspects is more astonishing in systems 3 and 5 under the operation of RD k1 case. In fact, we should …nd in
Figure 7.12. Relaxation rate never decay case under speci…c opera-tion cases with speci…c initial spinors input in three di¤erent sys-tems.
next section and Chapter 8, the key role which causes such aspect is the operation case, that is if we could create someone special spinor evolution behavior which owns something special correlation with operation case, theoretically we could get many various relaxation aspects (patterns) as we desire.
7.6. Relaxation Rate Never Decay Case
O.K. let us end this chapter by a never decay case in terms of practical ap-plication. We will give the mathematical description in Chapter 8. Here we just represent the simulation results. Fig. 7.12 shows a never relaxation (decay) case as we create a special Bef f con…guration, that is the combination of the same strength of R case and D k1 modi…ed case, the direction of Bef f of such special operation case always direct in the same direction (or in opposite direction). Then if we input a corresponding initial spinor which parallels (or anti-parallels) to this
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direction, we get a never relaxation polarization output as the spinor evolution. In fact it is the ideal goal which the engineers desire to create in spintronics in future.