Electromagnetic Pulse Driven Spin-dependent Currents in Semiconductor Quantum Rings
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Electromagnetic Pulse Driven Spin-dependent Currents inSemiconductor Quantum Rings
Zhen-Gang Zhu, and Jamal Berakdar
Institut f¨ur Physik, Martin-Luther Universit¨at Halle-Wittenberg,Heinrich-Damerow-Str.4 06120 Halle, Germany
Abstract
We investigate the non-equilibrium charge and spin-dependent currents in a quantum ring witha Rashba spin orbit interaction (SOI) driven by two asymmetric picosecond electromagnetic pulses.The equilibrium persistent charge and persistent spin-dependent currents are investigated as well.It is shown that the dynamical charge and the dynamical spin-dependent currents vary smoothlywith a static external magnetic flux and the SOI provides a SU(2) effective flux that changesthe phases of the dynamic charge and the dynamic spin-dependent currents. The period of theoscillation of the total charge current with the delay time between the pulses is larger in a quantumring with a larger radius. The parameters of the pulse fields control to a certain extent thetotal charge and the total spin-dependent currents. The calculations are applicable to nano-meterrings fabricated in heterojuctions of III-V and II-VI semiconductors containing several hundredselectrons.
PACS numbers: 78.67.-n, 71.70.Ej, 42.65.Re, 72.25.Fe . INTRODUCTION Study of the spin-orbit interaction (SOI) in semiconductor low dimensional structuresand its application for spintronics devices have attracted much attention recently [1]. Thereare two important kinds of SOI in conventional semiconductors: one is the DresselhausSOI induced by bulk inversion asymmetry [2], and the other is the Rashba SOI caused bystructure inversion asymmetry [3]. As pointed out in [4], the Rashba SOI is dominant in anarrow gap semiconductor system and the strength of the Rashba SOI can be tuned by anexternal gate voltage in HgTe [5], InAs [6], In x Ga − x As [7], and GaAs [3, 8] quantum wells.Recent research is focused on the electrically-induced generation of a spin-dependent current(SC) mediated by SOI-type mechanism, e.g. as in the intrinsic spin Hall effect in a 3D p-doped semiconductor [9] and in a 2D electron gas with Rashba SOI [10]. Here we study a highquality spin-interacting quantum rings (QRs) with a radius on the nanometer scale [11, 12].These systems show Aharonov-Bohm-type (AB) spin-interferences [13, 14]. In particularwe investigate the dynamics triggered by time-dependent electric fields as provided by time-asymmetric pulses [15] or tailored laser pulse sequences [16]. The quantity under study is thespin-resolved pulse-driven current, in analogy to the spin-independent case [17, 18, 19]. In aprevious work [20], we investigated the dynamical response of the charge polarization to thepulse application. No net charge or spin-dependent current is generated because the clock-wise and anti-clock-wise symmetry of the carrier is not broken by one pulse or a series ofpulses having the same linear polarization axis. This symmetry is lifted if two time-delayedpulses with non-collinear polarization axes are applied [17]. However, to our knowledge allprevious studies on light-induced currents in quantum rings did not consider the coupling ofthe spin to the orbital motion (and hence to the light field), which is addressed in this work.As detailed below, having done that, it is possible to control dynamically the spin-dependentcurrent in a 1D quantum ring with Rashba SOI by using two time-delayed linearly polarizedelectromagnetic pulses. For transparent interpretation of the results only the Rashba SOIis considered in this work. The presence of the Dresselhaus SOI may change qualitativelythe results presented here for the spin-dependent non-equilibrium dynamic of the carriers,which can be anticipated from the findings on for the equilibrium case [21].2
I. THEORETICAL MODEL
We study the response of charges and spins confined in a one-dimensional (1D) ballisticQR with SOI to the application of two short time-delayed linearly polarized asymmetricelectromagnetic pulses [17, 22]. The effective single particle Hamiltonian reads ˆ H ′ = ˆ H SOI +ˆ H ( t ) [20], withˆ H SOI = p m ∗ + V ( r ) + α R ~ (ˆ σ × p ) z , ˆ H ( t ) = − e r · E ( t ) + µ B B ( t ) · ˆ σ. (1) E ( t ) and B ( t ) are the electric and the magnetic fields of the pulse. Integrating out the r dependence ˆ H SOI reads in cylindrical coordinates [23, 24, 25, 26, 27]ˆ H SOI = ~ ω i∂ ϕ + φφ − ω R ω σ r ) − ( ω R ω ) + ω B ω σ z ] . (2) ∂ ϕ = ∂∂ϕ , φ = h/e is the flux unit, φ = Bπa is the magnetic flux threading the ring, a is theradius of the ring, ~ ω = ~ / ( m ∗ a ) = 2 E , ~ ω R = 2 α R /a , ~ ω B = 2 µ B B and B are due to apossible external static magnetic field B = B ˆ e z . The single-particle eigenstates of ˆ H SOI arerepresented as Ψ Sn ( ϕ ) = e i ( n +1 / ϕ ν S ( γ, ϕ ) where ν S ( γ, ϕ ) = ( a S e − iϕ/ , b S e iϕ/ ) T are spinorsin the angle dependent local frame, and a ↑ = cos( γ/ , b ↑ = sin( γ/ , a ↓ = − sin( γ/ , b ↓ =cos( γ/ , ( T means transposed) where tan γ = − Q R = − ω R /ω (if we ignore the Zeemansplitting caused by the static magnetic field [27, 28]). γ describes the direction of the spinquantization axis, as illustrated in Fig. (1a). The energy spectrum of the QR with the SOIreads [20, 24, 25, 26, 27, 28] E Sn = ~ ω (cid:20) ( n − φ/φ + 1 − Sw − Q R (cid:21) , (3) w = q Q R = 1 / cos γ, where S = +1 ( S = −
1) stands for spin up (spin down) in the local frame.
III. PULSE-DRIVEN SINGLE-PARTICLE DYNAMICS
We apply two time-asymmetric pulses to the system (see Fig. (1b)) . The first one (at t = 0) propagates in the z direction and has a duration τ d . Its E-field is along the x direction. τ d is chosen much shorter than the ballistic time of the carriers in which case the QR states3 t t F F E(t) t (c)(b) S n E n e r gy (a) up down x y z t =t + FIG. 1: (Color online) (a) Schematic graph of the geometry, spin configuration and the appliedpulses is shown. (b) Time-delayed asymmetric pulses are schematically drawn. (c) Energy spectrumfor a ring with spin orbit interaction. ∆ S defines the distance between the spectrum symmetryaxis and the smallest nearest integer. develop as [17, 22, 29]Ψ Sn ( ϕ, t >
0) = Ψ Sn ( ϕ, t < e iα cos ϕ , α = eap/ ~ , p = − Z τ d E ( t ) dt, (4)where E ( t ) = F f ( t ), F and f ( t ) describe the amplitude and the time dependence of theelectric field of the pulse respectively. In the following, we use F and F to characterize thefirst and the second pulses. The pulse effect is encapsulated entirely in the action parameter α . With the initial conditions n ( t <
0) = n and S ( t <
0) = S and using Eq. (4) one findsΨ S n ( ϕ, t ) = 1 √ π X ns C Sn ( n , S , t ) e i ( n +1 / ϕ e − iE Sn t/ ~ | ν S i , (5)with C Sn = δ SS δ nn for t ≤ ,δ SS i n − n J n − n ( α ) for t > , (6)where J n is the n-th order Bessel function. For the time-dependent energy we find E S n ( t >
0) = E S n ( t <
0) + ~ ω α , (7)4ith E S n ( t <
0) is given by Eq. (3). Applying a second pulse at t = τ with the same duration τ d but the electric field being along the y axis (see Fig. (1b)) , the wave functions developas Ψ S n ( ϕ, t > τ ) = Ψ S n ( ϕ, t < τ ) e iα sin ϕ , where α is the action parameter associated withthe second pulse. Ψ S n ( ϕ, t = τ − ) follows from Eq. (5). For t > τ the expansion coefficientsbehave as C S ′ n ′ ( n , S , t > τ ) = P n δ S ′ S [ i n − n J n − n ( α ) J n ′ − n ( α )] e i ( E S ′ n ′ − E S n ) τ/ ~ . IV. NONEQUILIBRIUM SPIN AND CHARGE CURRENTS
A single pulse does not generate in QR any net charge current because of the degeneracyof the orbital states. However, the charge will be polarized [20, 22] and corresponding dipolemoments oscillate in the x direction with an associated optical emission. Applying a secondpulse as described above leads to a non-equilibrium net current, in addition to the persistentcharge current caused by the static flux and the SOI which causes a SU(2) vector potentialand manifests itself in an induced spin-dependent persistent charge current [27, 28, 30, 31].Consequently, a non-equilibrium spin-dependent current is induced.The line velocity operator is [32]ˆ v ϕ = ˆ e ϕ (cid:26) − i ~ m ∗ a ∂ ϕ − ~ m ∗ a φφ + α R ~ σ r (cid:27) which is associated to the operator of the angular velocity ˆ v ϕ /a [33]. Contributions to thepersistent charge current from each QR level read [34] I n ,S = 12 π Z π dϕ Z r r dr j ϕn ,S ( r ′ , t > τ ) , where j ϕn ,S = e ℜ [Ψ S , † n ( r ′ , t )ˆ v ϕ Ψ S n ( r ′ , t )] . Upon algebraic manipulations we find I n ,S ( t > τ ) = I (0) n ,S ( t > τ ) + I (1) n ,S ( t > τ ) . (8)The index “(0)” stands for the static persistent charge current (PCC) which exists in theabsence of pulse field, whereas the index “(1)” indicates the pulse-induced dynamic chargecurrent (DCC). The PCC is caused by a magnetic U(1) flux and has been studied extensively[35] without [34, 36, 37] or with the spin interactions [33]. It has been experimentallyobserved both in gold rings of radius with 1.2 and 2.0 µ m [38] and in a GaAs-AlGaAs ring5f radius about 1 µ m [39]. The SOI scattering effects were also studied [40]. The PCCcarried by the states characterized by n and S reads (please note the current in this workis defined as flow of positive charges, which is opposite to the direction of flow of electrons) I (0) n ,S ( t > τ ) = ˆ e ϕ I (cid:18) n − φφ + 1 − S w (cid:19) , (9)where I = 2 E a/φ is the unit of CC, the second term on the right hand side of Eq. (9)stems from the static magnetic field; the third term is a consequence of the SU(2) flux ofthe SOI [27]. The DCC part is I (1) n ,S ( t > τ ) = ˆ e ϕ I (cid:8) α h cos ϕ i S n ( τ ) (cid:9) , (10)where h cos ϕ i S n ( τ ) = α h (Ω τ ) sin b τ cos[2( n − φφ + 1 − S w b τ ] ,b τ = ω τ / , Ω τ = α p − cos(2 b τ )) ,h (Ω τ ) = J (Ω τ ) + J (Ω τ ) . To obtain the total persistent charge current and the dynamic current we have to considerthe spin-resolved occupations of the single particle states. For simplicity we operate at zerotemperatures and ignore the relaxation caused by phonons or other mechanisms, i.e. weconfine ourself to times shorter than the relaxation time. The general case can be developedalong the line of Ref. [41].At first we introduce an effective flux as φ S = φ − φ − Sw . (11)As evident from Eq. (3) the spectrum is symmetric with respect to x S = φ S /φ . Further wedefine the shift ∆ S = x S − l ( l ′ ), where S = ↑ or ↓ . Here l ( l ′ ) = [ x ↑ ( ↓ ) ] where [ x ] means thenearest integer which is less than x . ∆ S is shown in Fig. 1. When ∆ S = 1, it is equivalentto ∆ S = 0. Furthermore, ¯∆ S = | / − ∆ S | is the distance between the x S and the nearesthalf integer. 6 . Spinless Particles For N spinless particles we distinguish two cases: N is an even or an odd integer. Case (1) : If N is an even integer then I (0) even(∆) = sgn(∆) N ( 12 − ∆) ,I (1) even(∆) = α α h (Ω τ ) sin( N b τ ) cos(1 − b τ , (12)where sgn( x ) equals +1, for x >
0; 0 for x = 0, and -1 for x < Case (2) : If N is an oddinteger then I (0) odd(∆) = − sgn(∆)sgn( 12 − ∆) N ( 12 − ¯∆) ,I (1) odd(∆) = α α h (Ω τ ) sin( N b τ ) cos(1 − b τ . (13) B. Particles with 1/2 spin
For spin 1/2 particles we consider four cases.
Case (0) : For an even number of particles’ pairs, i.e. N = 4 m , where m is an integer wefind I (0) S (∆ S ) = I (0) even(∆ S ) , I (1) S (∆ S ) = I (1) even(∆ S ) , (14) Case (1) : For an odd number of particles’ pairs, i.e. N = 4 m + 2 we obtain I (0) S (∆ S ) = I (0) odd(∆ S ) , I (1) S (∆ S ) = I (1) odd(∆ S ) . (15) Case (2) : For an even number of pairs plus one extra particle, i.e. N = 4 m + 1 (there isone particle whose spin is unpaired as compared with case (0)) we find I (0) ext ,S (∆ S ) = − sgn(∆ S )sgn( 12 − ∆ S )( N −
14 + 12 − ¯∆ S ) ,I (1) ext ,S (∆ S ) = α α h (Ω τ ) sin( b τ ) cos( N −
12 + 1 − S ) b τ . (16)To determine which spin state is occupied by the extra particle one compares the distanceof the symmetric axis to the nearest half integral axis, i.e. ¯∆ S . The one with the largerdistance will be occupied. Case (3) : For an odd number of pairs plus one extra particle, i.e. N = 4 m + 3. Here weuse case (1) and determine the contribution to the current from the extra particle I (0) ext,S(∆ S ) = sgn(∆ S )sgn( 12 − ∆ S )( N −
34 + 12 + ¯∆ S ) , (1) ext,S(∆ S ) = α α h (Ω τ ) sin( b τ ) cos( N −
32 + 1 + 2 ¯∆ S ) b τ . (17)Which spin state is occupied by the extra particle is governed by ¯∆ S . The level with thesmaller ¯∆ S is populated. V. SPIN-DEPENDENT CURRENT (SC)
In presence of a static magnetic field and the SOI but in the absence of the pulse fieldthe PCC is accompanied with a persistent SC (PSC). Switching on the pulse field generatesa spin-dependent charge currents due to the SOI, and also a dynamic SC (DSC) that canbe controlled by the parameters of the pulse field. The SC density is j sn ,S ( r ′ , t ) = ℜ{ Ψ S , † n ( r ′ , t )ˆ v ′ ˆ s Ψ S n ( r ′ , t ) } , where ˆ s = ( ~ /
2) ˆ σ z δ ( r ′ − r )is the local spin density. The SC associated with level n , S is I sn ,S ( t > τ ) = 12 π Z π dϕ Z r r dr ′ j sn ,S ( r ′ , t ) (18)and can be evaluated as I s z n ,S = I s ℜ X n | C S n ( n , S , t ) | D S n , (19)where I s = ˆ e ϕ E a/ (2 π )sets the unit SC and D S n = [( a S ) − ( b S ) ]( n − φφ ) − ( b S ) . (20)Here ( a S ) − ( b S ) = S cos γ, and S = ±
1. The SC after applying two pulses to the ring is a sum of two parts I s z n ,S ( t > τ ) = I s z , (0) n ,S ( t > τ ) + I s z , (1) n ,S ( t > τ ) , (21)8here I s z , (0) n ,S ( t > τ ) = I s [ S cos γ ][( n − φφ ) + 12 − S γ ] , = I s [ S cos γ ] I (0) n ,S ( t > τ ) I , (22)is the static PSC [27] and the DSC part is I s z , (1) n ,S ( t > τ ) = I s [ S cos γ ][ α h cos ϕ i n ,S ( τ )] , = I s [ S cos γ ] I (1) n ,S ( t > τ ) I . (23)Summing over all occupied energy levels we find I s z S ( t > τ ) = I s z , (0) S ( t > τ ) + I s z , (1) S ( t > τ ) , (24)where ( I (0) , (1) S ( t > τ ) are PCC and DCC) I s z , (0) , (1) S ( t > τ ) = I s [ S cos γ ] I (0) , (1) S ( t > τ ) I . (25) VI. NUMERICAL RESULTS AND DISCUSSIONS
We performed calculations for pulse-driven ballistic quantum rings fabricated by an ap-propriate confinement in a quantum well of In x Ga − x As/InP [42]. Our results are also validfor other III-V or II-VI semiconductor quantum rings with spin orbit, e.g. GaAs-AlGaAsquantum well, or HgTe/HgCdTe quantum ring [43]. We shall present the total charge current(TCC) which is a sum of PCC and DCC over all the occupied states. Total spin-dependentcurrent (TSC) is obtained in the same way [28].Fig. 2 shows how the flux and the SOI affect the PCC, DCC and TCC. Without theSOI, the jump of the PCC occurs at integer flux for even pair occupation, shown in Fig.2. The jumps are different in other occupations (see [28]), here we only focus on the evenpair occupation case for clarity. The periodic sawtooth dependence of the PCC on theflux exhibits has been studied before, e.g. [27, 28]. At finite SOI the jumps in PCC areshifted to φ/φ = l + (1 ∓ w ) /
2; the two jumps are the consequences of a superpositionof the contributions from the two spin channels. When the SOI strength is such that γ = − arccos(1 / n ), ( n = 1 , , · · · ) the two jumps become at the half integer which is just9 D CC / I T CC / I / P CC / I FIG. 2: (Color online) Persistent charge current (PCC), dynamic charge current (DCC) and totalcharge current (TCC) are shown in (a), (b) and (c) respectively. The spin orbit angle are γ =0 ◦ , − ◦ , − ◦ and − ◦ for the solid lines, the dash lines, the dot lines and the dash-dot linesrespectively in all graphs. The other parameters are N = 100, a = 100 nm, τ = 26 . F = F = 1 kV/cm. the case of 4 n + 2 occupation in absence of the SOI [28]. The slope ratio between the twojumps is the same. As can be inferred from the analytic expressions DCC (cf. Fig. 2)depends smoothly on the flux. SOI results in a phase shift moving or even exchanging thepositions of the minima and maxima, as is for γ = − ◦ . The origin of the shape of TCCis deduced from those of PCC and DCC. Here the magnitudes of the two contributions iscrucial: The PCC magnitude is related to the numbers of charge carriers, while the DCCmagnitude is determined primarily by the product of the α and α (that can be externallyvaried by changing the pulse intensities), the delay time τ , and the ring radius.Fig. 3 shows the TCC dependence on the ring radius (in the absence of the SOI). Asexpected, a larger α enhances the DCC. On the other hand, α enters the Bessel functionargument whose increase suppresses the magnitude of DCC. It can be shown that the periodof the oscillation with τ increases with increasing the radius. The magnitudes of the maximaand minima are larger with larger radius.Now we discuss the spin-dependent current projected onto the z direction [27], i.e. I S z .The spin-dependent current projected onto the γ direction (e.g. the quantization axis of the10 b) -80-243280-0.50 -0.25 0.00 0.25 0.50081624 [ p s ] (c)/ -1008116200 [ p s ] -0.50 -0.25 0.00 0.25 0.50 (d)/ -250-17217450081624-60-3003060 (a) FIG. 3: (Color online) Contour plots of TCC on φ and τ are shown for different radius of thering. a = 100 nm, 200 nm, 300 nm and 400 nm in (a), (b), (c) and (d) respectively. The otherparameters are N = 100, γ = 0 ◦ , F = F = 500 V/cm. local spin frame) is I S z = I γ cos γ [28]. -30030-40040-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-80-40040 PCC PSC(a)(b) C h a r g e c u rr e n t a nd s p i n c u rr e n t DCC DSC(c) / TCC TSC
FIG. 4: (Color online) PCC and PSC, DCC and DSC, and TCC and TSC are shown in (a), (b)and (c) respectively. The solid lines are the charge current, and the dash lines are the spin current. γ = − ◦ , the other parameters are the same to Fig. 2. PSC posses steps at PCC jumps (Fig. 4(a)) as a function of φ . This can be understoodfrom the ratio of PCC for different spins; the SOI only introduces a relative effective flux11hift (see Eq. (11)) leading to a constant spin-dependent current between the jumps. ForDSC vs. φ (Fig. 4(b)) the effective flux leads to a shift of the DSC along the flux axis. Thephysics behind this shift is that SOI provides a SU(2) flux, meaning that the pulse-driven(local frame) spin-up electrons experience a different flux than those with down spin, leadingto a substantial spin-dependent current. In contrast, a static magnetic flux does not inducea spin-dependent current in the absence of the SOI. This shift and the jumps in the stepfunction of the PSC explain the behaviour of TCC in Fig. 4(c). -1.6 -0.8 0.0 0.8 1.6051015202530 F =F [kV/cm] -300.0600.01500 -1.6 -0.8 0.0 0.8 1.6 051015202530 [ p s ] [ p s ] (b) F =F [kV/cm] -6.00-5.82-5.64-5.50 (a) FIG. 5: (Color online) TCC (a) and TSC (b) vs. pulses strengths F = F and delay time τ . γ = − ◦ , φ = 0 .
2. Other parameters are N = 100, a = 400 nm. The control of TCC and TSC by tuning the pulse-field parameters is demonstrated in Fig.5. Because the two pulses transfer a net angular momentum setting the electrons in motionbut they do not couple directly to the spins the TCC and TSC show the same pattern with τ and F . From an experimental point of view it is essential to note that we are dealing withnon-equilibrium quantities which opens the way for their detection via their emission. E.g.,the TCC and TSC can be detected by measuring the current-induced magnetization of thering and the generated electrostatic potential [44]. VII. GENERAL AND CONCLUDING REMARKS
In presence of spin-orbit coupling, two time-delayed appropriately shaped electromag-netic pulses generate spin-dependent charge currents. As shown previously for the spin-12ndependent case [17], the sign of the current and its magnitude are controllable via thedelay time and the strengths of the pulses. From a symmetry viewpoint, similar phenomenamay be expected to occur for other geometries (wires, squares, etc). However, as shown forunbiased superlattices [45] (without SOI) details of the generated currents may differ quali-tatively. Application of an appropriate train of pulses open the possibility of controlling oreven stopping the current [41]. For increasing the magnitude of the current more intensepulses should be applied.For generating currents in quantum rings one may also apply circular polarized laserpulses [18, 19]. In this context we note the following: From an electrodynamics pointof view, generating currents by our pulses is a completely classical effect, i.e. currentsare generated in a completely classical system, even though in our case the subsequentexcited carrier evolution is quantum mechanical. For this reason our current is robustto disorder and geometry modifications. In addition, the tunable time delay between thepulses allows an ultra-fast control the current properties. Using circular polarized laser pulsesgenerates currents for quantized systems (in which case the rotating-wave approximation canbe applied). For systems with level broadening on the order of level spacing no appreciablecurrent is generated. Our disadvantage however is that our pulses are much more demandingto realize experimentally, whereas laser pulses are readily available, in particular at high lightintensities allowing thus for a strong current generation.The DSC is proportional to the DCC which can be comparable to the PCC for small ormoderate occupation number case as seen in Fig. (4b). The DCC depends on the strengthof the field and the delayed time between the two pulses sensitively and dramatically. Weprovide now an explicit calculation for the typical values of the CC and SC. For In x Ga − x As/InP quantum well [42] we have m ∗ = 0 . m . For a ring with radius of 100 nm, the linevelocity is then about 5000 m/s , and the current unit is I /a ∼ nA which corresponds tothe angular velocity current for one particle. If we convert it into the unit of an inducedmagnetization it is a radius-independent quantity M ≈ M ≈ πa ( I /a ) valid for rings considered here [17]).The work is support by the cluster of excellence ”Nanostructured Materials” of the state13axony-Anhalt. [1] For a review on spintronics, see, e.g., S. A. Wolf, et al., Science , 1488 (2001).[2] G. Dresselhaus, Phys. Rev. , 580 (1955).[3] E. I. Rashba, Sov. Phys. Solid State , 1109 (1960); Y. A. Bychkov, and E. I. Rashba, J. Phys.C , 6039 (1984).[4] G. Lommer, et al., Phys. Rev. Lett. , 728 (1988).[5] M. Schultz, et al., Semicond. Sci. Technol. , 1168 (1996); X. C. Zhang, et al., Phys. Rev. B , 245305 (2001); Y. S. Gui, et al., ibid. , 115328 (2004).[6] J. Luo, et al., Phys. Rev. B , 7685 (1990).[7] J. Nitta, et al., Phys. Rev. Lett. , 1335 (1997); C. -M. Hu, et al., ibid. , 728 (1988); G.Engels, et al., Phys. Rev. B , R1958 (1997); Th. Sch¨apers, et al., J. Appl. Phys. , 4324(1998).[8] F. Malcher et al., Superlatt. Microstruc. , 267 (1986).[9] S. Murakami, N. Nagaosa, and S. C. Zhang, Science , 1348 (2003).[10] J. Sinova, et al., Phys. Rev. Lett. , 126603 (2004).[11] A. Fuhrer, et al., Nature , 822 (2001); Microelectronic Engineering , 47 (2002); Phys.Rev. Lett. , 206802 (2003); ibid. , 176803 (2004).[12] R. J. Warburton, et al., Nature , 926 (2000); A. Lorke, et al., Phys. Rev. Lett. , 2223(2000); M. Bayer, et al., ibid. , 186801 (2003); U. F. Keyser, et al., ibid. , 196601 (2003);B. Al´en, et al., Phys. Rev. B , 45319 (2007).[13] J. Nitta, F. E. Meijer, and H. Takayanagi, Appl. Phys. Lett. , 695 (1999).[14] J. Nitta, and T. Koga, J. Supercon. , 689 (2003).[15] D. You, R. R. Jones, P. H. Bucksbaum and D. R. Dykaar, Opt. Lett. , 290 (1993); T. J.Bensky, G. Haeffler, and R. R. Jones, Phys. Rev. Lett. , 2018 (1997).[16] C. H. Bennett, and D. P. DiVincenzo, Nature , 247 (2000).[17] A. Matos-Abiague, and J. Berakdar, Phys. Rev. Lett. , 166801 (2005).[18] Y. V. Pershin, and C. Piermarocchi, Phys. Rev. B , 245331 (2005).[19] E. R¨as¨anen, et al., Phys. Rev. Lett. , 157404 (2007).[20] Z.-G. Zhu, J. Berakdar, Phys. Rev. B , 235438 (2008).
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