6.1. Chaotic Scattering
The study of a physical system from the viewpoint of Quantum Chaos usually starts with its classical dynamics. In open systems, e.g., the quantum transport, we must consider a classical scattering problem. Since the trajectory exit the scattering region after a …nite amount of time in open systems, the concepts of chaos which developed for closed systems, and related to the long-time properties of the trajectories must be re-examined. Here, we don’t attempt to review the
…eld of Chaotic Scattering [56], but just roughly present the information needed to understand the quantum properties of ballistic cavities, and o¤er an example [57] to illustrate.
For a scattering problem, the transient chaos is characterized by the in…nite set of trajectories that stay in the scattering region forever. The periodic unstable orbits of the scattering region (the strange repeller ) and their stable manifold (the trajectories that converge to the previous ones in the in…nite-time limit) form the set. Chaotic scattering is gained when the dynamics in the neighborhood of the repeller is chaotic in the usual sense, and this set has a fractal dimension in the space of classical trajectories. When an incoming particle enters the scattering
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region it approaches the strange repeller, bounces around close to this set for a while and it is eventually ejected from the scattering region (if the initial conditions to be trapped is lost). If we scan a set of scattering trajectories (say, we …x the initial position y at the left entrance of the cavity of Fig. 0.1 (a) and we vary the initial injection angle ) studing the time that the particle spends in the interaction region, then we could get a fractal curve for ( ). We …nd an interesting aspect, the in…nitely trapped trajectories give the divergences of ( ) and determine its self-similar structure. To determine if our scattering is chaotic, the study of ( ) is a quick way. The rate at which particles escape from the scattering region ( ) results from a balance between the rate in which nearly trajectories diverge away from the repeller (it could be characterized by its largest Lyapunov exponent ) and the rate at which the chaotic escaping trajectories are folded back into the scattering region (depending on the density of the repeller, that is measured by its fractal dimension d). More precise speaking, if we start (or inject) particles into the scattering region, the survival probability at time will be p ( ) = e , with = (1 d) [58]. We may interpret the escape rate as the inverse of the typical time spend by the particle in the scattering region. Let us see the examples [57] shown in Fig. 6.1, the length distribution (which in billiards is equivalent to the length distribution) for a cavity with the shape of stadium, and verify the exponential law (solid line), p (L) = e clL (with cl = &=v and v the constant velocity of the scattering particles). We notice that the exponential law sets in very fast, just after a length corresponding to a few bounces.
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Figure 6.1. Classicsl distribution of length for stadium (solid line) and rectangular (dash line) billiards. In the stadium, the distribution is close to exponential after a short transient region and are very di¤erent from the distributions for the rectangle, which show the power-law behavior characteristic of non-chaotic systems.
We don’t surprise the appearance of a single scale, since in chaotic scattering the particle moves ergodically over the whole energy surface while in the scattering region. We also can estimate the value of the escape rate from general arguments of ergodicity in the case of chaotic cavities with small openings, where the typical trajectory bounces aroung many times before it escapes [59]. Assuming that the instantaneous distribution of trajectories is uniform on the energy surface, the escape rate is simply given by = F=A, where F is the ‡ux through the holes (equal to the size of the holes time v= , the factor of comes from integration over the departing angles), A is the area of the two-dimensional scattering domain. We
…nd that in the case of small holes this simple estimate reproduces remarkably well the escape rates obtained from the numerical determination of the survival probability using classical trajectories.
The reason we talk about the escape rate is due to the fact that, the energy scale of the conductance ‡uctuation is given by this classical quantity. In addition, we also …nd that the conductance ‡uctuations as a function of magnetic …eld are governed by the area distribution.
6.2. Scattering Approach to the Electric Conductance
According to the Landauer-Büttiker approach, in the phase-coherent regiom the resistance is not an intensve resistivity of the type de…ned in standard con-densed matter books [60], e.g., electron-phonon interaction, but arises from the elastic scattering that electrons su¤er which traversing mesoscopic sample between the measuring devices. We treat the measuring devices as macroscopic electron reservoirs. They are characterized by an electrochemical potential which does not vary while giving and accepting electrons. The role of the reservoirs is impor-tant as they render the total system in…nite, and the spectrum continuous. It is only in the reservoirs that the randomization of electron phases is asumed to take place.
Figure 0.1 shows the simplest experimental set up with two-probe measure-ment, where the sample is attached between two reservoirs whose electrochemical potentials di¤er by the value of the applied voltage V , which is supposed to be very small ( 1 2 = eV 1).
The scattering description necessitates a set of asymptotic states. In this case such a set is provided by the propagating channels of the leads connecting the
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sample with the reservoirs. The three key elements of ballistic transport are sample, reservoirs and leads.
Assuming the leads to be disorder-free, with hard walls (of width W ) in the y-direction and in…nite in the x-y-direction, their eigenstates with energy " are products particle-in a box wave -function
(6.1) a(y) =
r 2
W sin ay W
(a is an integer) in the transverse direction and plane-waves propagating in the longitudinal direction, with wave-vectors kasuch that " = ~2= (2m) (a =W )2+ ka2 . The N transverse momenta which satisfy this relationship with k2a > 0 de…ne the 2N propagating channels of the leads with energy ". Then we …nd that the incom-ing lead-state are
(6.2) ( )1(2);";a(r) = 1 va1=2
e ikax a(y) ; r = (x; y) ; a = 1; 2; :::; N
The normalization factor va1=2 = (m=~ka)1=2 is chosen in order to have a unit of incoming ‡ux in each channel. The subindex 1 (2) corresponding to channels propagating from the left (right) reservoir with longitudinal momenta ka ( ka), and ka explicitly positive. The outgoing lead-state (+)1(2);";a(r) are de…ned as in Eq. 6:2, but with the of the exponent inverted. The time order of outgoing
and incoming lead-states is obtained by giving an in…nitesimal positive (negative) imaginary part to ka.
Then we get the scattering states corresponding to an electron incoming from lead 1 (2) with energy ", in the mode a are given, in the asymptotic regions, by
(6.3) (+)1;";a(r) =f
The 2N 2N scattering matrix S, relating incoming ‡ux and outgoing ‡ux, can be written in terms of the N N re‡ection and transmission matrices r and t (r0 and t0) from the left (right) as
From the current conservation, we …nd that the incoming ‡ux should be equal to the outgoing ‡ux, and therefore S is unitary (SSy = I):In terms of the total transmission (T =P
a;bjtbaj2) and the re‡ection (R =P
a;bjtbaj2) coe¢ cients, the unitarity condition is expressed as T +R = N . Also, unitarity dictates that T = T0
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and R = R0. In the absence of magnetic …eld, time reversal invariance furthermore dictates that S is symmetric S = ST . For the case of cavities with geometrical symmetries (up-down or right-left) are described by scattering matrices with a block structure [61].
The set { (+)1(2);";a} constitutes an orthogonal (but not orthonormal) basis [62][63],
(6.6) R
dr (+)l;";a (r) (+)
l0;"0;a0(r) = 2
va aa0 (ka ka0) u0
Using the spectral decomposition of the retarded Green function in this basis and taking into account the analytical properties of the transmission amplitudes in the complex k-plane, we can relate the Green function to the scattering ampli-tudes. Or alternatively, the formal theory of scattering (Lippmann-Schinger) can
be adapted to wave-guides and obtain [64].
(6.7)
where the integration take place at the transverse cross sections Sx on the left lead and Sx0 on the right (left) lead for the transmission (re‡ection) amplitudes.
We then have that the physical observables are obtained from the transmission and re‡ection coe¢ cients (T = jtbaj2 and R = jtbaj2) between modes, which, by current conservation, do not depend on the choice of the transverse cross sections.
And we will use this freedom to take Sx and Sx0 at the entrance and exit of the cavity (or both at the entrance for Eq. 6.8), and we will omit the x and x0 dependences henceforth.The above equations give us an intuitive interpretation as a particle arriving at the cavity in mode a, propagating inside (through the Green function), and exsiting in mode b is quite straightforward. Expressing the scattering amplitudes in terms of Green function is extremely useful for analytical and numerical computations. The diagrammatic perturbation theory, as well as semiclassical expansions, are built on Green functions.
We have presented the scattering theory for samples connected to wave-guides so far. Now, we reproduce the standard counting argument to relate conductance with scattering [65][1]. At the beginning of this section, we assume that the left reservoir has an electrochemical potential 1 slightly than the one of the right reservoir ( 1 2 = eV). In the energy interval eV between 2 and 1 electrons are injected into right-going states emerging from reservoir 1, but none are injected into left-going states emerging from reservoir 2. Therefore, there is a net right-going current proportional to the number of states in the interval 1 2, given by
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N is the number of propagating channels at the energy 1, the factor gspin = 2 takes into account spin degeneracy, P
b=1~N Tba is the probability for an electron coming in the mode a to traverse the system, dna=d" quasi-one-dimensional den-sity of states (which for noninteracting particles satis…es that dna=d" = 1=hva).
Then we …nd that the two-probe conductance is just proportional to the total transmission coe¢ cient of the microstructure.
(6.10) g = I
Note that the magnetic …elds that we consider will always be very weak, and therefore the zero…eld formulation of the conductance that we presented is su¢ -cient for our purposes.
6.3. Semiclassical Transmission Amplitudes
The scattering formalism presented in the last section is the base for the semi-classical theory of ballistic transport that we develop here. Our goal is to calculate the conductance through a cavity (like one in Fig. 0.1) by using Eq. 6.10 and last part within a semiclassical approach. The Green function is the Laplace transform of the propagator. The Van Vleck expression, together with a stationary-phase integration on the time variable, leads to the semiclassical approximation for the Green function [13] The sum is over classical trajectories S, with energy E, going between the initial and …nal points r = (x; y) and r0 = x0; y0 . Ss = R
Csp dq is the action integral along the path Cs. In the case of billiards without magnetic …eld Ss=~ = kLs, where Ls is the trajectory length. The factor Ds describing the evolution of the classical probability can be expressed as a determinant of second derivatives of the action [13]. As the geometry shown in Fig. 0.1 (a), if we denote by and 0 the incoming and outgoing angles of the trajectory with the x-axis, Ds =
v cos 0 =m
1
@ =@y0 y . Here we include in the phase s the Maslov index counting the number of constant-energy conjugate points and the phase acquired
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at the bounces with the walls when those are giving by an in…nite potential (hard wall). We will always take the spatial dimensionality d = 2 in our calculation.
In the case of hard-wall leads, the transverse wave-functions have the sinusoidal form of Eq. 6.1. Using the semiclassical expression, Eq. 6.12, of the Green function appeared in last section, we see that, for lrge integers a, integral over y will be dominated by the stationary-phase contribution occurring for trajectories starting at points y0 de…ned by
(6.13) @s
@y y0
= py = a~
W ; a = a:
The dominant trajectories are those entering the cavity with the angles asuch that sin a = a =kW. Thus, the initial transverse momentum of the trajectories equals the momentum of the transverse wave-function. As always in this type of reasoning, we have assumed that we could interchange the order of the integration and the sum over trajectories. Integrating the gaussian ‡uctuations we have
(6.14)
(6.15) S y0; a; E = S y0; y0 a; y0 ; E + ~ a
W y0 a; y0 :
The prefactor is now given by D = (v cos 0) 1 @y=@y0 , and the new index v(that we still call Maslov index) is increased by one if @ =@y0 y0 is positive. At this intermediate stage we have a mixed representation, with trajectories starting with …xed angle ( a) and …nishing at points y0. A new stationary-phase over y0 calls for. Here we will assume that trajectories are isolated and we can perform the y0 integration by stationary phase. The …nal points y00 are selected according to
(6.16) @s
@y0 a = @s
@y0 y = py0 = b~
W ; b = b
implying that the trajectories have an outgoing angle b such that sin b = b =kW. Therefore the semiclassical expression for the transmission amplitude can then be casted as [66]
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(6.18) S b; a; E = S y~ 00; y0; E + ~ a
W y0 ~ b
W y00:
For billiards it can be written as ~S = ~k ~L, with ~L = L + ky0sin a ky00sin b: The prefactor is now given by
(6.19) D~s= 1
where H is the Heaviside step function.
The similar arguments can be used to write the semiclassical re‡ection ampli-tude in terms of trajectories leaving and returning to the cross section at the left entrance with appropriate quantized angles. We note that there are two kinds of trajectories contributing to G y0; y; E in the case of re‡ected paths: those which penetrate into the cavity and those which go directly from y to y0 staying on the cross section of the lead. It is only trajectories of the …rst kind which contribute to the semiclassical re‡ection amplitude, as trajectories of the second kind merely cancel the ba of Eq. 6.8.
Here we notice that from the quantum point of view, since the Gutzwiller trace formula must reproduce a deltd-function spectrum, it can be conditionally conver-gent at most, while the quantum transmission amplitude is a smooth function of the Fermi energy and so the semiclassical sum can be absolutely convergent . And we also …nd that the simple prescription for the Maslov indices makes possible the numerical evolution of the semiclassical transmission amplitude.
6.4. Spin Conductance
Following the point of view of scattering approach to the charge conductance to study the spin dependent conductance [67], we de…ne
(6.21) g = e2
h P
a;b
tbatba
where tba is the transmission amplitude of an electron at Fermi energy EF
propagating from the channel (or mode) a and the spin state in the injector to the channel (or mode) b and the spin state in the collector, the same symbol-ization to tba. And tba is the element of the matrix tba which operates on spin states. Then we have the usual spin independent electric conductance is simply g = e2=hP
a;b; ; tba
2
.
If from the injector a spin oriented in the x axis is injected, and its orientation turns to the y axis when the spin is collected at the collector, let gyx represent this spin current passing through the loop. Then the matrix elements can then be written as
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where i are Pauli matrices with i = x; y; z. By measuring the polarization of the emitted light [68][69][70], the spin orientation can be detected. In such an experiment, the polarization matrix of the emitted photons can be derived if we know gji. By applying the saddle point approximation to the path integral representation of the transmission amplitude, the quasiclassical expression of g can then be obtained as
where t0(p)is the spin independent transmission amplitude for the pth classical trajectory, Sp is the matrix element of the operator of evolution of spin state along the pth trajectory, the same symbolization to Sq . In Fig. 5.2, one such trajectory is schematically plotted as the zigzap line. The explicit expression [66][57][71] of t0(p) is not needed for the present work, because we only need to know its general statistical properties determined by the particle chaotic motions. The dependence of the transmission on spin degrees of freedom is represented by the spin evolution operator Sp along the p trajectory. From Eq. 5.1 Sp could be expressed as
(6.24) Sp = Tsexp i
~ R
pHsodt
The symbol Tsmeans a Ts order such that the operators in the integrand in Eq.
6.24 are ordered along the path with the operator corresponding to the later part of the path length operating …rst. We also …nd for di¤erent spin-orbit interaction (SOI), the Sp can also be expressed as Eqs. 5.20 and 5.21.
In Equation 6.23 each quasiclassical amplitude t0(s) contains a phase factor exp(2 ils= ). Since the path lengths lp and lq of the trajectories in Eq. 6.23 are much longer than the electron wavelength , in the sum the terms with p 6= q oscillate rapidlly even for a small variation of the particle energy, as well as for a slight change of the loop shape and/or the con…guration of charged impurities.
On the other hand, the terms with p 6= q do not oscillate. If one is not interested in mesoscopic ‡uctuations of the spin conductance, only the terms with p = q need to be retained. Accordingly, from Eq. 6.23 we obtain the so averaged spin conductance hg i as
(6.25) hg i = e2
h P
p jt0(p)j2Dp ;
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where
(6.26) Dp = x Spy S Sp x ;
and where x means the unit initial spinor with the spin state , S is the spin operator for measure the spin state .
For means i, i = x; y; z, according to Eq. 6.22, we get the so averaged spin conductance hgjii as
(6.27) hgjii = e2
h P
p jt0(p)j2Djip; where
(6.28) Djip = T r iSp jSpy :
The so averaged electric conductance is simply 2e2=hP
p jt0(p)j2, and is spin independent.
In the semiclassical approximation, the spin independent transmission rate jt0(p)j2 in Eqs. 6.25 and 6.27 is approached by the transmission ratio of the classical trajectory ensemble, in which jt0(p)j2 is described by [57][66]
(6.29) jt0(p)j2 = fp(y; p) cos ( p) =N;
where N is the total number of the injected trajectories and fp(y; p) = 1, if the trajectory with initial conditions (y; p) is transmitted and fp(y; p) = 0 otherwise.
6.5. Spin Evolution
Now, let us give the calculation basis of the simulation for spin relaxation case.
The procedure of the deduction of the calculation formulae is almost similar to the one of spin conductance as listed above. The calculation basis for the simulation in spin relaxation case is deduced from equation
(6.30) Pci(t) = Pj(0) 2
RRij(r; r0; t)j (r0 R)j2d2r0
here it is the expression of the semiconductor spin polarization. And
(6.31) Rij(r; r0; t) = T r iU (t; r; r0) jUy(t; r; r0)
where U (t; r; r0)could be represented as
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(6.32) U (t t0; r; r0) = T exp i
~ R
pHsodt ,
it is the unitary matrix represents the spin dependence part of the Green’s function, and Hso is the Hamiltonian of spin-orbit interaction under the e¤ect of Rashba and Dresselhause term case, see Eq. 5.1, and T is time ordering operator, reference Eq. 6.24. Equation 6.30 describes the spin evolution of a particle initially distributed around the point R with the probability density j (r0 R)j2: The particle starts its classical motion from the point r0 with the momentum ~k at time zero and arrives in the position r at time t.
We do not intend to deduce detailedly the expression 6.30, the detail deduction procedure had been done by C.H. Chang et al [72], the basic idea about the deduction is given the two component spinor eigenfunction 'n corresponding to the nth quantized energy level En, and then by applying it we could get the time-dependent wave packet
(6.33) (r; t) =P
n
Cn'n(r) e iEnt=~,
where
(6.34) Cn=R
'yn(r) (r) d2r
and then in terms of (r; t) the time dependent spin polarization could be expressed as
(6.35) P(t) =P R
(r; t) (r; t) d2r.
Then by introducing the retarded and advanced Green’s function and their cor-responding semiclassical approximation and using the saddle point approximation in them, we get the desired Green’s function
(6.36) Gr(t t0; r; r0) = 1 2
P
p
pJ (r; r0)e~iS0(t t0;r;r0)U (t t0; r; r0)
which it is a sum over all classical trajectories p, and J (r; r0) is the spin inde-pendent monodromy and S0(t t0; r; r0) is the spin independent classical action, U (t t0; r; r0)is the spin dependent part of the Green’s function shown in Eq. 6.32.
And then by applying the Green’s function into Eq. 6.35 we obtain a semiclassical expression for the spin polarization. Finally we proceed some simpli…ed procedure in it, we get the …nal form, Eq. 6.30, which o¤ers the ingredient to deduce the
And then by applying the Green’s function into Eq. 6.35 we obtain a semiclassical expression for the spin polarization. Finally we proceed some simpli…ed procedure in it, we get the …nal form, Eq. 6.30, which o¤ers the ingredient to deduce the