ARTICLE IN PRESS

Physica E 42 (2010) 1571–1578

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physe

Single spin-qubit rotators based on nanojunctions: A semiclassical pathintegral approach

S. Bellucci a,�, P. Onorato a,b

a INFN, Laboratori Nazionali di Frascati, P.O. Box 13, 00044 Frascati, Italyb Department of Physics ‘‘A. Volta’’, University of Pavia, Via Bassi 6, I-27100 Pavia, Italy

a r t i c l e i n f o

Article history:

Received 31 August 2009

Accepted 23 December 2009Available online 4 January 2010

Keywords:

Quantum computers

Qubit rotators

Quantum nanojunctions

Path integral

Rashba spin–orbit interaction

77/$ - see front matter & 2010 Elsevier B.V. A

016/j.physe.2009.12.047

esponding author.

ail address: bellucci@lnf.infn.it (S. Bellucci).

a b s t r a c t

We employ a path integral semiclassical approach to compute the properties for electrons transported

across quantum nanojunctions and with Rashba spin–orbit interaction (SOI). We use a piecewise

semiclassical approximation for the particle orbital motion and solve the spin dynamics exactly by

formulating a semiclassical theory for low dimensional systems with SOI. We obtain analytical results

that can be compared with the calculations made by using the quantum waveguide approach.

The proposed devices, with P and T geometry, can be analyzed within the scope both of spin

filtering and spin rotation. In fact some devices action results in a transformation of the qubit state

carried by the spin, i.e. act as one-qubit spintronic quantum gates whose properties can be varied by

tuning the strength of the SOI or by changing the geometrical and physical parameters in the

experimentally feasible range.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Quantum computers represent a new paradigm for computingdevices: computers whose components are individual quantumsystems, and which exploit the properties of quantum mechanicsto make possible new algorithms and new ways of processinginformation. Recently the interest in spin-based informationprocessing is largely increased and the field of semiconductorspintronics [1,2] was born. Spintronics designates the range oftechnologies that use the spin degrees of freedom of the electronsas the carriers [1,3] of qubits, the fundamental units in quantuminformation processing. A physical system with two distinguish-able states (as the electron spin) is called a quantum bit (q-bit orqubit). The electron spin in motion can be assumed as ‘‘flyingq-bits’’ while the ones confined in a quantum dot can be assumedas ‘‘static’’ ones.

Thus many basic building blocks are today investigatedtheoretically and experimentally in order to realize a fully spinbased circuitry. Among them, a particular relevance is covered by:(i) pure spin current generation and spin filtering [4,5], (ii) voltagecontrol of the spin polarization of a current and (iii) the electricdetection of this polarization. Moreover in order to implementquantum operations on electron spins of a flying q-bit appropriateelementary logical operations, gates, are necessary [6–8].

ll rights reserved.

It follows that several fundamental quantum phenomenawhich involve electron spin, have been investigated in order togenerate, manipulate and measure spin polarized currents.Among these studies many are focused on the role of spin–orbitinteractions (SOIs) in condensed matter systems.

The SOI, which plays a central role in the spintronic devices,can be described by the Hamiltonian [9]

HSO ¼�l2

0

‘m0eEðrÞ � ½r � p�: ð1Þ

Here EðrÞ is the electric field, r¼ ðsx;sy;szÞ are the Pauli matricesdefining the spin operators s ¼ 1

2‘r, p is the canonical momen-tum operator r is a 3D position vector, l2

0 ¼ ‘ 2=ð2m0cÞ2 and m0 theelectron mass in vacuum while in materials m0 and l0 aresubstituted by their effective values m� and l. The SOI term can beseen as the interaction of the electron spin with the magneticfield, Beff , appearing in the rest frame of the electron.

Recently, semiconductor technology allows to grow samples(for instance made by InAs or InGaAs) with sizeable SOI. At theinterface of two different materials the electrons are entrapped ina potential well and a two-dimensional electron gas (2DEG) isformed. Due to the lack of inversion symmetry along the growthdirection z of the heterostructure the electrons in the 2DEG aresubject to the so-called Rashba SOI [10] because they are movingin an effective electric field [11], Ez, along the z direction. TheHamiltonian in Eq. (1) will take the form [12]

HaSO ¼

a‘½rxpy�rypx�: ð2Þ

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Fig. 1. The P junctions (top) and the T junctions (bottom) can be assumed as

crossing junctions between three or two Q1D wires, respectively. The ballistic one-

dimensional wire is a nanometric solid-state device in which the transverse

motion is quantized into discrete modes. The width, W, of the wires, ranges

between � 25 and 100 nm, and the longitudinal motion is free.

S. Bellucci, P. Onorato / Physica E 42 (2010) 1571–15781572

a, which in vacuum is given by l20Eze, in semiconductor

heterostructures takes values typically [13–15] within the range� 10�11

210�12 eV m while its highest value is close to 10�10 eV mas reported in Refs. [16,17]. Moreover for some applications it isessential that the strength of the Rashba effect and thus the spinsplitting can be controlled by means of a gate electrode [18,19].

As we discussed in Ref. [20] in a quantum nanojunction with aP geometry and two external leads, the spin properties of anincoming electron are modified by the SOI, resulting in atransformation of the qubit state carried by the spin due toquantum interference. The properties of the quantum gates can bevaried by tuning the strength of the SOI, as well as by changingthe relative position of the junctions.

In this paper we focus on the aspects of spin-interference inballistic quasi-one-dimensional (Q1D) nanojunction by employ-ing a path integral semiclassical approach. In Section 2 weintroduce the path integral approach to the spin problem and thesemiclassical theory for the ballistic transport. In Section 3 wediscuss in detail the behavior of a T-shaped nanojunction withtwo external leads and the one of a P- shaped nanojunction withfour external leads shown in Fig. 1. Thus, by taking the point ofview of Ref. [21], we focus explicitly on the spin transformationproperties of the devices proposed, and show that those can beappropriately controlled by varying its geometrical and physicalparameters in the experimentally feasible range.

2. Semiclassical approaches

2.1. Semiclassical theories with spin

We introduce a general theory for electron systems startingfrom the discussion reported in Ref. [22] with spin s¼ 1=2 with a(mean-field) Hamiltonian linear in the spin operators

H ¼ H0ðr; pÞþHso ð3Þ

with

H0ðr; pÞ ¼p

2

2mþVðrÞ; Hso ¼ ‘kr � Cðr; pÞ: ð4Þ

The second term is a general spin–orbit interaction, where Cðr; pÞis an arbitrary vector function of coordinate and k is a couplingstrength. In the non-relativistic reduction of the Dirac equationwith an external electrostatic potential VðrÞ in vacuum, oneobtains

Cðr; pÞ ¼ ½=VðrÞ � p�; k¼ l20=‘

2: ð5Þ

The usual semiclassical approaches for the spin problems arethe following:

(i)

SCL: Littlejohn and Flynn [23] developed a semiclassicaltheory, treating the spin matrices quantum mechanicallywhile Wigner transforming the matrix operator to theclassical phase space ðr;pÞ, keeping the leading terms in an‘ expansion. Diagonalisation leads to a pair of Hamiltonians

H7 ðr;pÞ ¼H0ðr;pÞ7‘kjCðr;pÞj ðs¼ 1=2Þ; ð6Þ

where H0ðr;pÞ and Cðr;pÞ are the Wigner transforms (tolowest order in ‘ ) of the corresponding quantum operators.H7 can be considered as two classical adiabatic Hamiltonianswith opposite spin polarizations. This approach is oftenreferred to as the ‘‘strong coupling limit’’ (SCL), since itbecomes valid in the formal limit k-1 and ‘-0 with ‘kkept finite [23,24].

(ii)

WCL: Bolte and Keppeler [24] have derived a semiclassicaltheory from the Dirac equation. In the ‘‘weak coupling limit’’(WCL) the orbits are given by the dynamics of theunperturbed Hamiltonian H0 and the effect of spin precessionaround the local ‘‘magnetic field’’ kCðr;pÞ appears through asimple modulation factor. This approach neglects terms ofhigher than first order in ‘k and therefore is valid in the limitof weak SOI.An interesting application of the WLC approach was given fora 2DEG with a Rashba type [25] SOI C ¼ ð�py; px;0Þ in anexternal homogeneous magnetic field where p ¼ p�eA=c. Forthis system the exact quantum spectrum is explicitly known[25], and analytical trace formulae have been given for boththe exact quantum-mechanical level density and the semi-classical WC and SC limits [26], from which the limitations ofthese two approaches become evident.

(iii)

Spin coherent states. A semiclassical approach in which thespin degrees of freedom were introduced through spincoherent states [27] has been presented in Refs. [28,29].Tanks to the introduction of the spin coherent states, jz; sS,one can define classical spin components n¼ ðnx;ny;nzÞ ¼

/z; sjsjz; sS=‘ s and to enlarge the classical phase space byonly one pair of canonical variables, independently of thevalue of the spin s. Starting from the path integral in the SU(2)spin coherent state representation [30] and making the usualstationary-phase approximation in its evaluation, the semi-classical dynamics of the system in the extended phase space

ðr;p; v;uÞ is then determined [28] by the Hamiltonian

Hðr;p; v;uÞ ¼H0ðr;pÞþ‘k2snðv;uÞ � Cðr;pÞ: ð7Þ

2.2. Spin Hamiltonian in 1D wire

In this paper we discuss the cases of strictly 1D wires wherethe semiclassical approach usually plays very well. In order tohave a comparison between the semiclassical and the quantumapproaches in one- and two-dimensional electron system withRSOI also in the presence of magnetic field we can refer to Ref. [31].

The ballistic one-dimensional wire is a nanometric solid-statedevice in which the transverse motion (e.g. along x for the probe 1in Fig. 1) is quantized into discrete modes, and the longitudinalmotion (y direction for the probe 1 in Fig. 1) is free. If we can

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Fig. 2. First, second and third order paths of the transmission amplitude from

leads 1 to 4 including backscattering at the nodes.

S. Bellucci, P. Onorato / Physica E 42 (2010) 1571–1578 1573

assume D¼ ‘ 2=ð2m�W2Þ very large (typically ranges from tens toone hundred meV) transverse motion in both the x and z directionis quantum mechanically frozen out (i.e. with a mean value /pxSwhich corresponds to assume the px ¼ 0 in the semiclassicalHamiltonian). In this case, electrons propagate freely along to aclean narrow pipe and electronic transport with no scattering canoccur. The total Hamiltonian for one electron moving in the QW is

Hðy;py;rÞ ¼p2

y

2m�þa‘

pysx ¼p2

y

2m�7

‘ 2kRpy

m�; ð8Þ

where kR �m�a=‘ 2¼ pR=‘ , which has a natural value [32,33] for

a� 10�12210�10 eV m and kR � 10�5

210�2 nm�1. The r.h.s.equation is obtained applying the SCL approach, where Cðr;pÞ �ðpy;0;0Þ.

It follows a spin splitting in the energies:

eðpy; sxÞ ¼1

2m�ððpy7pRÞ

2�p2

RÞ: ð9Þ

The 7 sign corresponds to the spin polarization along the x axis.Hence we can conclude that four-split channels are present for afixed Fermi energy, eF , corresponding to 7py and sx ¼ 71,

py � sxpR7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2

Rþ2m�eF

q� sxpR7pe:

2.3. Semiclassical theory for ballistic transport

In describing ballistic transport semiclassical methods haveattracted much interest, since they establish a direct link betweenquantum transport properties and features of the correspondingclassical dynamics, e.g. chaotic, integrable, or mixed behavior [34, 35].

In order to study the transport properties of the system subjectto a constant, low bias voltage (linear regime) we calculate thezero temperature conductance G based on the Landauer formula[36–39].

G¼ gse2

hT ¼ gs

e2

h

XMn0 ;n ¼ 0

Xs0 ;sjts

0sn0n j

2; ð10Þ

where ts0s

n0n ðeÞ denote the transmission amplitudes betweenincoming channels n and outgoing channels n0 in the leads. Thelatter amplitudes can be written in terms of the projections of theGreen function on the transverse modes fnðxÞ in the leads (whichin our case of strictly 1D wire can be assumed as Dirac deltafunctions),

ts0s

n0n ðeÞ ¼�i‘ffiffiffiffiffiffiffiffiffiffiffivnvmp

Zdx0Z

dxf�nðxÞfnðx0ÞGðx; s;x0; s0; eÞ;

where the vn denote longitudinal velocities. The integrals aretaken over the cross sections of the leads at the entrance and theexit. Following the theory reported in Ref. [35] we can write thesemiclassical representation of transmission amplitudes

tnmC�

ffiffiffiffiffiffiffiffiffiffiffiffip‘

2ww0

r Xgðn ;mÞ

Fgexp½ði=‘ ÞSg�: ð11Þ

The sum runs over all lead-connecting trajectories g. In Eq. (11), Sgis the classical action, and Fg is a phase factor where mg containsthe Morse index. In our case n¼m¼ 0 and vn ¼ vm.

The propagation amplitude for an electron entering nanojunc-tion with spin polarization s to exit with spin polarization s0, atenergy e can be also written, analogously to Ref. [40] as

Aðmf ;m0jeÞ ¼Z 1

0

dtf

t0eietf =‘/~r f ; s; tf j~r0; s

0; t0S; ð12Þ

where /~rf ; s; tf j~r0; s0; t0S is the amplitude for a particle entering

the junction at the point ~r0 and at the time t0 with spin

polarization s0 to exit at the point ~rf at the time tf with spinpolarization s.

3. Results

For a realistic device, at each lead one has to take into accountthree possible scattering processes, consistently with the con-servation of the total current. This is described in terms of aunitary S-matrix or transmission reflection coefficients [40]. Thedetails about the calculation of the scattering matrix elements arereported in Appendix A.

3.1. T-junction and spin filtering

Now we discuss the case of a two terminal T-junction in thepresence of Rashba SOI. In the case of 1D wires we are able tocalculate all the possible paths in this kind of interferometer.These paths are shown in Fig. 2 where they are classified.

According to the calculation reported in Appendix B we obtainthat the spin dependent transmission amplitudes can be writtenas

T¼ Te�idU; ð13Þ

where U is a unitary, unimodular matrix in the form

U¼1ffiffiffi2p

eiðdxþdyÞpR=‘ eiðdx�dyÞpR=‘

e�iðdx�dyÞpR=‘ e�iðdx þdyÞpR=‘

!: ð14Þ

It is this unitary part that is independent of the Fermi energy eF

and performs a spin transformation in the qubit space as wediscuss below. Moreover in our case we have

T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 sin2

ðpebÞ

3sin2ðpebÞþ1

vuut ;

d¼peðdxþdyÞ

‘þarctan

cosðpebÞ

2sinðpebÞ

� �:

In the equations above T is a non-negative quantity which can beconsidered as the efficiency of the spintronic gate or rotator thatwe want to realize [21]. This efficiency can be optimized by actingon the Fermi energy or on the length of the left arm. In Fig. 3 weshow the efficiency as a function of the length of the left arm, d

and of the Fermi energy, eF .The first result that we obtain is that the transmission

amplitudes derived by the semiclassical approach are in perfect(analytical) agreement with the ones calculated by using thequantum mechanical waveguide approach [41].

Spin filtering and spin precession: In our semiclassical theory thespin components si can be assumed as classical dynamicalvariables. Thus the spin oscillations analyzed in the past [42] in

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kR (nm-1)d (nm)

d (n

m)

�F

� F� F

Tjunction

0 50 100 150 200 250 0.1 0.2 0.3 0.4 0.50

0.20.40.60.8

1

T

d=100nm - Tjunction Π − junction

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

0 50 100 150 200 250 0 5 10 15 200

100

200

300

400

500Π junction

Fig. 3. (Top left) Density plot of the efficiency jTj for the T nanojunction as a

function of the left arm length, d and the kinetic energy eF from 0 to the 0:5D (we

can suppose D� 50 meV corresponding to W � 20 nm). (Bottom Left) Density plot

of the efficiency for the P nanojunction as a function of the distance, d, between

the two crossing points and the kinetic energy. (Top right) A comparison between

TT and TP at a fixed value of the interferometric distance. While is possible to

obtain a lossless operator by using a T junction for the P junction discussed here

T2 is always below 1=4. (Bottom right) The cosine of the rotation angle, g, around

the y axis as a function of the experimental parameters kR and d (black

corresponds to cosðgÞ ¼�1 white to cosðgÞ ¼ 1).

S. Bellucci, P. Onorato / Physica E 42 (2010) 1571–15781574

terms of mean values /SS can now be written in terms ofclassical variables.

The value of sy in the exiting lead vanishes everywhere whilethe two orthogonal component sx and sz oscillates. We now focuson the out of plane szðx; tÞ component of the spin obtained in theright probe by using the semiclassical equation of motion for thespin components (see Appendix B). By replacing the time weobtain

/SzðxÞS¼ syðxÞ ¼‘2

cosð2pRxÞ: ð15Þ

From Eq. (15) we obtain the known spin oscillations, upon whichthe Datta–Das device for the spin filtering is also based [3,31]while these oscillations can disappear in a device where the SOIvanishes out of the crossing zone.

This device can be used for the spin filtering, i.e. for theproduction of a spin polarized current (or a spin accumulation). Infact the discussed oscillations can disappear in a device like theone proposed in Ref. [43], where the SOI vanishes in an externalpart of the lead 4 or if we introduce a collector as in Ref. [42]. Inthe latter case, fixing the collector in lead 4 at a distance L� p=2kR

from the node of the junction we could measure also a significantspin polarization along the out of plane direction. This behavior isunchanged if we substitute the collectors by ideal leads withvanishing SOI.

3.2. P�junction and spintronic applications

Now we discuss the case of a four terminal P�junction in thepresence of Rashba SOI.

In the case of 1D wires we are able to calculate all the possiblepaths in this kind of interferometer. These paths are shown in

Fig. 5 where they are classified and the calculation is discussedwith more details in Appendix C. We can write the resultsaccording to Eq. (13) where

T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4

16þ9 sin2ðqdÞ

s

and

U¼cosðkRdÞ isinðkRdÞ

isinðkRdÞ cosðkRdÞ

!: ð16Þ

Also in this case we obtain that the transmission amplitudesderived by the semiclassical approach are in perfect (analytical)agreement with the ones calculated by using the quantummechanical waveguide approach.

In order to discuss in detail the spin rotation we can startfrom the equation of motion of the classical spin componentsthat can be solved in the interferometric region between thetwo crossings of the P junction. At the first node the spinis injected with initial orientation corresponding to s¼ ðcosðWÞ;sinðWÞcosðjÞ; sinðWÞsinðjÞÞ. It follows, according the equationobtained in Appendix B that the classical spin in the node 2 isgiven by

sx ¼ cosðgÞcosðWÞ�sinðgÞsinðWÞsinðjÞ;

sy ¼ sinðWÞcosðjÞ;

sz ¼ sinðgÞcosðWÞþcosðgÞsinðWÞsinðjÞ;

i.e. the P junction rotates the spin along axis around the y axis byan angle

g� 2kRd: ð17Þ

Thus also in this case the spin precession can be discussed interms of classical dynamics.

Spin rotation—Now we can discuss some details about theapplication of the proposed device to the manipulation of a spinquantum bit. As we show in Fig. 3 top left, by changing thestrength of the SOI (kR) or the distance d between the junctions,according to the equation above the values of g can be varied from0 up to 2p. Notice that the unitary part U of the transformationgiven by Eq. (16) is independent of the Fermi energy eF .

In the language of quantum informatics [44], the transforma-tion discussed above represents a rather general single-qubit gate.This can be further extended by coupling two or more suchdevices in series. A transformation of the form U with g¼ p=2 isessentially a so-called quantum NOT gate, which plays adistinguished role in quantum algorithms. Thus the p- nanojunc-tion proposed here allows a significantly class of spin transforma-tions as the ones proposed in Refs. [20,21].

If we want use the natural Rashba SOI we can design the gateby fixing d in order to obtain the required rotation angle g (if weassume a InGaAs device with the SOI strength near to the naturalvalue we obtain d� Lso=4 with Lso which rages between some tensand some hundreds nanometers). In Fig. 3 (bottom right) we showthe density plot of cosðgÞ as a function of d and kR. Thus in order tooptimize the efficiency we have to act on the Fermi energy (eF)which has to be varied between 0 and some tens of meV. Fig. 3(bottom left) shows the gate efficiency T2 as a function of d.

One sees that, however, the transformation is never strictlyunitary (lossless gate), with T ¼ 1. Thus we can improve theefficiency by a suitable choice of the working conditions but withan intrinsic limit on To1=2. We note that in principle a numberof other gates can be constructed by coupling several of thoseP�junctions. For instance, two P�junctions one with g1 ¼�p=4and one other with g2 ¼ p=2 yields a NOT gate with the correctphase (U�p=4Up=2 ¼UNOT ). Thus if one couples such devices in

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S. Bellucci, P. Onorato / Physica E 42 (2010) 1571–1578 1575

series, then obviously the resulting transformation will be theproduct of the corresponding rotations, and with a furtherreduced efficiency.

4. Discussion

In this paper we developed a semiclassical approach to discusstwo realistic or theoretical devices, capable of acting as spintronicgates based on the Rashba SOI.

The feasibility of the proposed device has to be analyzed bydiscussing mainly three points: the reliability of a 1D channel, thecoherent ballistic transport, the SOI strength.

(i)

We analyzed an idealized model system in which transport isballistic and one-dimensional, i.e., the finite width, W, of thewire was not included in the calculation. It corresponds toassume one degree of freedom frozen because the energy gapD is larger both than the energy eF and than the thermalenergy kBT . The results we have shown were obtained usingvalues of well width within those given by presentlyavailable 2DEGs and nanolithography techniques in fact thelithographical width of a wire defined in a 2DEG can be assmall as 20 nm [45]. The choice of these narrow wires alsojustifies the assumption of single-mode propagation.

(ii)

In recent years, high mobility devices have become availablesuch that at cryogenic temperatures transport is found to beballistic over tens of microns. This is the regime in whichscattering with impurities can be neglected and the dimen-sions of the sample are reduced below the mean free path ofthe electrons. Here we proposed conductors smaller than thedephasing length for low temperatures. In fact phasecoherence and spin coherence lengths [46] have been foundup to 100mm while, recently, it was found that the finitewidth of the wires has a small effect on the loss of coherenceof the spin state. A possible non-ideal coupling to the leadscan be described through effective tunnel barriers. But, inmost of the current experimental systems the leads areconnected in a rather adiabatic way, which makes thecoupling very close to ideal.

(iii)

The SOI strengths, which was shortly discussed above, havebeen theoretically evaluated for some semiconductors com-pounds [47]. In a QW patterned in InGaAs/InP heterostruc-tures [32,33], the natural values read a\10�10 eV m whilefor GaAs–AsGaAl interface, one typically observes [13] valuesfor a10�11 eV m, whereas for HgTe based heterostructures acan be more than three times larger [48]. Thus values of kR upto � 10�2 nm�1 can be assumed due to the natural SOI, i.e. tothe structure inversion asymmetry of the heterostructurequantum well. On the contrary values larger have to beobtained by controlling the transverse electric field, e.g. bytuning the voltage on the gate electrode [18,19].

Fig. 4. Schematic representation of the transmission and reflection at a crossing

point between two wires for a spinfull particle.

5. Conclusions

In this paper, we have shown that a quantum nanojunctions withRashba-type SOI can serve as spin filters or as a one-qubit quantumgate for electron spins. The spin transformation properties of thegates can be extended by coupling such nanojunctions in series alsoif the problems due to the efficiency have to be taken in account.Different types of gates can be realized by tuning the electric fieldstrength and changing the geometry of the device. The consideredparameters are within the experimentally feasible range.

Moreover we have shown how the semiclassical path integralapproach is able to describe the behavior of one-dimensional

nanodevices also in the presence of SOI. In detail we havediscussed how the results obtained by using this approach areanalytically comparable with the ones obtained by using thequantum waveguide technique. Moreover we have shown how thespin accumulation (or the spin oscillations) can clearly discussedstarting from a classical model for the spin which is suitable inorder to study the physics of quantum wires in the presence of SOI.

Appendix A. Scattering matrix

Rules for the calculation: Next we approach just the cases ofstrictly 1D wires connected at some points (intersections ornodes). For a realistic device, at each lead one has to take intoaccount three possible scattering processes, consistently with theconservation of the total current. This is described in terms of aunitary S-matrix or transmission reflection coefficients.

Next we suppose that at the node the crossing wires areorthogonal and that at each node we have three legs. Theseconditions could be also modified and can be generalized to ageneral angle.

(i)

In each wire the quantum object can travel in the oppositedirection while the spin can be polarized orthogonally to thewire direction with opposite polarizations.

(ii)

At each node between the leg i (injection) and legs j (ejection)the quantum object can be reflected or transmitted with anamplitude ti;i ðti;jÞ.

(iii)

We can classify the three legs T-nodes according the twodifferent geometries shown in Fig. 4.It follows that in general for the first one Fig. 4 we have todefine ts;s0

j with j� fR; L; Ig while for the second geometry wehave to define ~t

s;s0

j with j� fR; T; Ig.

(iv) In each reflection at nodes there is no spin flip

(ts;�sI ¼ ~t

s;�sI ¼ 0), and analogously the spin which goes

straight through the node is undeflected (~ts;�sT ¼ 0).

T and ‘ junctions: Here we discuss the case of preservedmomentum, i.e. jpj is the same in any legs.

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Fig. 5. Basic bricks for calculating all order paths included in the calculation of the transmission amplitude across the P junction.

S. Bellucci, P. Onorato / Physica E 42 (2010) 1571–15781576

We now define the spin polarization as u (d) in the horizontalwires while r (l) for the vertical ones. We start with the T junction(top panel)

tl;lI ¼ tr;r

I ¼�13; tl;r

I ¼ tr;lI ¼ 0;

tl;uR ¼ tr;u

R ¼ tr;dR ¼

ffiffi2p

3 ; tl;dR ¼�

ffiffi2p

3 ;

tl;uL ¼ tr;u

L ¼ tr;dL ¼

ffiffi2p

3 ; tl;dL ¼�

ffiffi2p

3 ; ðA:1Þ

while for the ‘ junction we obtain

~tl;lI ¼

~tr;rI ¼�

13;

~tl;rI ¼

~tr;lI ¼ 0;

~tl;uR ¼

~tr;uR ¼

~tr;dR ¼

ffiffi2p

3 ; ~tl;dR ¼�

ffiffi2p

3 ;

~tl;lT ¼

~tr;rT ¼

23;

~tl;rT ¼

~tr;lT ¼ 0: ðA:2Þ

Appendix B. Transmission amplitude for T junction

B.1. Spinless case

We start from the case of a spinless particle injected in the lead 1and revealed in lead 4. We fix the starting point at qi ¼ � ð0;�dyÞ

and the final position at qf ¼ � ðdx;0Þ thus di from the nearest node.Next we can fix the energy eF and calculate the action SðeF ; qi; qf Þ bytaking in account the additivity properties of the action.

The basic bricks of our calculation are:

(a)

the action of the electron traveling from qi to qf along the Iorder path (see Fig. 5),

SI � SðeF ;qi;qf Þ ¼ 2ðdxþdyÞp;

where p¼ffiffiffiffiffiffiffiffiffiffi2mep

;

(b) the internal path action, related to one particle reflected in the

left arm

SO � SðeF ;O;OÞ ¼ 2bp;

where O� ð0;0Þ corresponds to the center of the junction;

(c)

the scattering matrix elements are determined according therules discussed above.

Now we can evaluate the actions for all the orders of paths,

SII ¼ SIþSO; SIII ¼ SIþ2SO; . . . ; Sn ¼ SIþnSO:

Thus we evaluate the transmission amplitude a sum over paths

t0 ¼ eiSI=‘ tRþtL~tT ð�1ÞeiSO=‘ þtL

~tT ð�1Þ2ð~t IÞei2SO=‘ þ � � �

� �

¼ eiSI=‘ tR�tL~tT

Xn

ð~t IÞneinSO=‘

!

¼ eiSI=‘2sin

SO

‘

� �

2sinSO

‘

� ��icos

SO

‘

� � : ðB:1Þ

The latter results agree with the one that can be obtained byapplying the quantum waveguide theory for 1D cross junctions.

B.2. Spinfull case

If we include the spin in the calculation:

(i)

The actions for one electron in the presence of Rashba SOI,SI7 ;7 ¼ ðdxþdyÞpþð7dx7dyÞpR replace the spinless ones

while SO is spin independent.

(ii) Because at any reflections an transmission along the same

wire the spin is unchanged the spin dependent transmissionamplitudes are easily obtained by replacing tR and tL with

tss0

R tss0

L .

Thus we obtain the transmission amplitudes as a matrix

t¼t0ffiffiffi

2p

eiðdxþdyÞpR=‘ eiðdx�dyÞpR=‘

e�iðdx�dyÞpR=‘ e�iðdxþdyÞpR=‘

!: ðB:2Þ

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B.3. Semiclassical spin precession

The right arm (2) is along the x axis thus is described by theHamiltonian

H¼p2

2m�þ

pRp

m�sy:

Thus we have

xðtÞ ¼pem�

t;

where we fix time t¼ 0 when the electron arrive from theinjecting probe at the crossing point. We assume that in theinjecting probe the spin is polarized along the x directioni.e. Sð0Þ ¼ ð�1;0;0Þ‘ =2. The Hamilton equations of motion forthe classical spin degrees of freedom,

trr14 ¼ eiðSr

1þSr

4Þ=‘ tr;u

R~t

u;rR eiSu

N=‘Xn ¼ 0

ð~tr;rI Þ

2neinS0=‘ þtr;dR~t

d;rR eiSd

N=‘Xn ¼ 0

ð~tl;lI Þ

2neinS0=‘

!����������2

¼4

9

eiped2cosðpRd=‘ Þ

1þ1

9ei2ped=‘

��������������2

¼4cos2ðpRd=‘ Þ

16þ9sin2ðped=‘ Þ

¼ T2cos2ðpRd=‘ Þ:

ðC:4Þ

_sy ¼ 0; _sz ¼�osx; _sx ¼osz

have a simple solution, with two oscillating components of thespin vector with frequency

o¼ pRpem�‘

:

Once we have fixed the boundary conditions we obtain

sy ¼ 0; sz ¼‘2

sinðotÞ; sx ¼�‘2

cosðotÞ:

Next we can write the values of the classical spin components as afunction of the position along the right probe and we obtain

sy ¼ 0; sz ¼‘2

sinpRx

‘

� �; sx ¼�

‘2

cospRx

‘

� �:

It can be related with the out of plane /SzðrÞS component of thespin accumulation obtained by using the usual quantum wave-guide theory as a simple function of the expansion coefficients.

Appendix C. Transmission amplitude for P junction

We fix the leads in each probe, i, at a distance di from thenearest node. Next we can fix the energy eF and calculate theaction SðeF ; s; qi; qf Þ by taking in account the additivity propertiesof the action. The basic bricks of our calculation are the actions ofthe electron traveling from the lead 1 to the node, i.e. the injectedelectron,

Ssi � SðeF ; s;�d1;0Þ ¼ psd1;

where s¼ þ1 (�1) for the right (left) polarization, i.e., Sri ¼

d1ðpeþpRÞ and Sli ¼ d1ðpe�pRÞ. For the electrons traveling from the

node to the leads the actions are easily computed as

Sr1 ¼ d1ðpe�pRÞ; Sl

1 ¼ d1ðpeþpRÞ;

Sr4 ¼ d4ðpe�pRÞ; Sl

4 ¼ d4ðpeþpRÞ;

Su2 ¼ d2ðpeþpRÞ; Sd

2 ¼ d2ðpe�pRÞ;

Su4 ¼ d4ðpe�pRÞ; Sd

4 ¼ d4ðpeþpRÞ; ðC:1Þ

while for the electrons traveling from one node to the other one(the distance between the nodes is d)

SuN ¼ dðpe�pRÞ; Sd

N ¼ dðpeþpRÞ: ðC:2Þ

The last contribution comes from the internal path, (Cu and Cd)where we obtain

S0 ¼ 2dpe ðC:3Þ

which does not depend on the spin polarization.Now we can calculate the probability of finding a particle in

the point of the lead 4 obtained from the sum over paths . Westart from a spin r in lead 1 and we want find a spin r in lead 4 sothat we have two families of paths, the first one is the sum ofCruþnCu (see figure) the second one is CrdþnCd where n is anarbitrary integer number. The corresponding actions areSr

i þSr4þSu

NþnS0 and Sri þSr

4þSdNþnS0 thus

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