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PHYSICA L R EVIEW LET T ERS week ending VOLUME 90, N UMBER 25 27 JUNE 2003 A Spin Cell for Spin Current Qing-feng Sun,1,2 Hong Guo,1,2 and Jian Wang3 1 Center for the Physics of Materials and Department of Physics, McGill University, Montreal, PQ, Canada H3A 2T8 2 Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China 3 Department of Physics, The University of Hong Kong, Pokfulam Rood, Hong Kong, China (Received 12 December 2002; published 26 June 2003) We propose and investigate a spin-cell device which provides the necessary spin-motive force to drive a spin current for future spintronic circuits. Our spin cell has four basic characteristics: (i) it has two poles so that a spin current ﬂows in from one pole and out from the other pole, and in this way a complete spin circuit can be established; (ii) it has a source of energy to drive the spin current; (iii) it maintains spin coherence so that a sizable spin current can be delivered; (iv) it drives a spin current without a charge current. The proposed spin cell for spin current should be realizable using technologies presently available. DOI: 10.1103/PhysRevLett.90.258301 PACS numbers: 85.35.–p, 73.23.–b, 72.25.Pn, 73.40.Gk Traditional electronics is based on the ﬂow of charge: plane. An extreme case of nonuniformity is BR ÿBL , the spin of the electron is ignored. The emerging tech- i.e., equal in value but opposite in direction. This particu- nology of spintronics will make the leap such that the lar magnetic ﬁeld distribution is not necessary at all for ﬂow of spin, in addition to charge, will be used for the operation of our spin cell, but it helps us to discuss its electronic applications [1,2]. A spin current is produced physics. Finally, the energy source of our spin cell is by the motion of spin-polarized electrons; therefore spin provided by shining a microwave radiation with strengths current is typically associated with the spin-polarized L =R for the left/right QDs. Because, typically, the charge current [1]. Nevertheless, if one can generate an microwave frequency is far less than the plasma fre- ideal situation, as shown in Fig. 1(a), where spin-up quency of the material covering the QDs, the effect of electrons move to the right while an equal number of the microwave ﬁeld is to induce a high frequency poten- spin-down electrons move to the left, then there will be no tial variation L=R cos!t in the left/right QD and their net charge current because Ie e I" I# 0, where leads [4]. When L Þ R , a time-dependent potential eI" ; eI# are charge currents due to spin-up and spin- difference, cos!t L ÿ R cos!t, exists between down electrons, respectively. There will be, however, a the two QDs. An ac electric ﬁeld E t in the middle ﬁnite spin current: Is h I" ÿ I# where h is the reduced 2 barrier is therefore established due to microwave radia- Planck constant. Considering the interesting and impor- tion [see Fig. 1(c)]. Then, electrons can absorb photons tant future perspective of spin-current circuits, it is when they pass the middle barrier of the device. The crucial to have a spin cell that satisﬁes the four character- establishment of E t across the two QDs is necessary istics discussed in the abstract and it produces the ﬂow pattern of Fig. 1(a) [3]. In this paper we theoretically propose and analyze such a spin cell. Our spin cell is schematically shown in Fig. 1(b). It consists of a double quantum dot (QD) fabricated in two- dimensional electron gas (2DEG) with split gate technol- ogy, and each QD is contacted by an electrode. Note that no magnetic material is involved. The two QDs and their associated contacts to the electrodes serve as the ‘‘posi- tive’’ or ‘‘negative’’ poles of the spin cell. The two elec- trodes maintain the same electrochemical potential L R (i.e., no bias voltage is applied on them). The size of the spin-cell structure is assumed to be within the spin-coherence length which can be as long as many microns for 2DEG. We control the QD energy levels by gate voltages Vg where L; R indicates the left/right QD. Both QD levels are controlled by an overall gate FIG. 1. (a) Schematic diagram for a conductor which has a voltage Vg ; see gate arrangements in Fig. 1(b). In order spin current with zero charge current; (b) schematic diagram to distinguish spin-up electrons from spin-down elec- for the double quantum dot spin cell; (c) schematic plot for the trons, a spatially nonuniform external magnetic ﬁeld B spin-cell operation via photon assisted tunneling processes is applied to the two QDs—perpendicular to the QD indicated by A . 258301-1 0031-9007=03=90(25)=258301(4)$20.00 2003 The American Physical Society 258301-1 PHYSICA L R EVIEW LET T ERS week ending VOLUME 90, N UMBER 25 27 JUNE 2003 for our spin cell to work; here, we use a nonuniform that has spin index and intradot Coulomb interaction microwave radiation to achieve this effect as has already U . To account for the magnetic ﬁeld B, the left/right been carried out experimentally [5], but other possibil- QD’s single particle energy has a term ÿ 1=2gB , in ities also exist. which we have required a different magnetic ﬁeld Before we present theoretical and numerical results of strength for the two QDs, i.e., BL Þ BR . tC and ÿ P the device in Fig. 1(b), we ﬁrst discuss why it works as a 2 k jtk j2 ÿ k describe the coupling strength be- spin cell. The physics is summarized in Fig. 1(c). To be tween the two QDs, and between electrode and its speciﬁc, let BR point to the ÿz direction and BL to the z corresponding QD, respectively. The microwave irradia- direction. Because of the Zeeman effect, a spin-degener- tion is given by W t cos!t [4,6] and it produces ate level R on the right QD is now split into spin-down/ an adiabatic change for the single particle energy. Here spin-up levels R# < R" . On the left QD, it is L" < L# . we permit the microwave ﬁeld to irradiate the entire Electrons in the electrodes can now tunnel into the QD: device including the electrodes, and we require a differ- on the right a spin-down electron is easier to tunnel ence in the radiation strength L Þ R . because level R# is lower, while a spin-up electron is Our theoretical analysis of the spin cell is based on easier to tunnel into the left QD. Once levels R# ; L" are standard Keldysh nonequilibrium Green’s function theory occupied, the charging energies UR ; UL of the two QDs [4,6] which we brieﬂy outline here. First, we perform a push the other two levels R" ; L# to higher energies R" of P unitary transformationR the Hamiltonian with a unitary UR ; L# UL , and the energy level positions indicated by operator U t P ^ ^ expfi t dt0 W t0 D g, where D P y y 0 the solid horizontal lines of Fig. 1(c) are established. k ak ak d d . The Hamiltonian H is trans- Next, the spin-down electron on the right QD can absorb formed to the following form: a photon and make a transition to the level at L# UL on X X the left QD: afterwards it easily ﬂows out to the left H ÿ gB =2dy d U dy d" dy d# " # electrode because L# UL > L . This process is indi- X X cated as A ÿ . Similarly the spin-up electron on the left k ay ak k y tk ak d H:c: QD ﬂows out to the right electrode after absorption of a k Rt k X photon, indicated by A . This way, driven by the po- tC ei 0 dt0 cos!t0 y dL dR H:c:; (2) tential variations of the QD induced by the microwave ﬁeld, a spin-down electron ﬂows to the left while a spin- up electron ﬂows to the right of the spin cell, and the where L ÿ R . In (2), we take the last term which continuation of the Aÿ; A processes generates a dc spin explicitly depends on time t as the interacting part HI and current that ﬂows from the left electrode, through the spin the remaining part as H0 H ÿ HI . The Green’s func- cell, and out to the right electrode. Clearly, if the two tion of H0 , gr , can be easily obtained with a decoupling processes are absolutely equivalent, there will be no approximation at the Hartree level [7]: charge current and only a spin current. Finally, since ÿ U n the spin-motive force is provided by a time-dependent gr ÿ i ÿ ÿ ; (3) ÿ 2 U n change of the electronic potential landscape of the QD, there is no spin-ﬂip mechanism and the spin current where ÿ ÿ ÿ U , ÿ gB =2, and ﬂowing through the spin cell is conserved, i.e., Is;L n is the time-averaged intradot electron occupation ÿIs;R Is . Our device then satisﬁes the four character- number at the state in the QD which we solve self- istics of a spin cell discussed in the abstract. consistently. It is worth mentioning that gr in Eq. (3) The last paragraph discusses the operation principle of has two resonances: one is at , while its associated the spin cell for spin current, but there are other interest- state at is empty; the other resonance is at U , ing device details which can be obtained only by detailed while its associated state is occupied. Notice, in H0 theoretical and numerical analysis for which we now turn. the left part of the spin cell (i.e., the left lead and the left The spin cell of Fig. 1(b) is described by the following QD) is not coupled with the right part of the spin cell, Hamiltonian [4,6]: therefore they are in equilibrium respectively. Hence the X Keldysh Green’s function g< for H0 can be solved H W t ÿ 1=2gB dy d X from the ﬂuctuation-dissipation theorem: g< X U dy d" dy d# k W tay ak ÿf gr ÿ ga . With these preparations, the " # k X k X Green’s function Gr and G< of the total Hamiltonian H tk ay d H:c: tC dy dR H:c:; can be solved. In particular, we calculate Gr t; t0 k L k ÿi t ÿ t0 hfd t; dy t0 gi by iterating the Dyson equa- (1) tion. In Fourier space, the Dyson equation can be reduced to [8,9] where ay k (ak ) anddy (d ) are creation (annihila- X tion) operators in the electrode and the dot , respec- Gr gr Gr r gr ; ;mn ;mn ;mk ;kn ;nn tively. The left and right QDs include a single energy level k 258301-2 258301-2 PHYSICA L R EVIEW LET T ERS week ending VOLUME 90, N UMBER 25 27 JUNE 2003 where Gr Gr ;mn ;nÿm m!, and the charge current I e spin current I s quantity Gn is the Fourier expansion of G t; t0 [8]. A− The retarded self-energy r is the Fourier trans- 1.5 ;kn form of r t1 ; t2R where r t1 ; t2 r t1 ; t2 frequency ω LR RL 1.4 t1 ÿ t2 tC expi t1 dt0 cos!t0 , and r r 0 LL RR 0. We obtain r r LR;mn RL;nm tC Jnÿm =!. 1.3 A The Green’s function gr is gr;mn ;mn mn gr m!. Then Gr can be solved 1.2 ;mn from the above Dyson equation [10]: 1.1 ÿ1 Gr r ;mn mn = g;mm ÿ A;mm ; 1.0 A+ 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 Gr ;mn Gr r r ;mm ;mn g ;nn ; vgR vgR P 2 where A;mm k jtC j2 Jkÿm =!gr ;kk . After- FIG. 2. The charge current Ie and spin current Is versus gate wards, the total Keldysh Green’s function G< t; t0 voltage VgR for different frequencies !. Different curves have ihdy t0 d ti is easily obtained from the Keldysh equa- been offset such that the vertical axis gives the frequency. Two tion. Finally, we obtain the time-averaged current in lead dotted oblique lines A indicate the position of the peaks. The from parameters are L R 0, ÿL ÿR kB T 0:1, tC 0:02, UL 1, UR 0:9, gBL 0:2, gBR ÿ0:4, VgL I hI ti 0:5, Vg 0, and =! 1:0. Z ÿIm d=2ÿ G< ;00 2f Gr ;00 ; (4) In the following we focus on the spin-cell operation by ﬁxing gate voltage VgR 0:45 which is its value at point and the self-consistent equation for the intradot occupa- R A of Fig. 2. We investigate Ie ; Is as functions of the overall tion number n : n ÿi d=2G< ;00 . gate potential Vg [Fig. 3(a)], magnetic ﬁeld gBL Figure 2 shows the calculated charge current Ie (in [Fig. 3(b)], and frequency ! [Fig. 3(c)]. The different units of e) and the spin current Is (in units of h=2) versus curves in Fig. 3 correspond to different microwave the gate voltage VgR at different microwave frequency !. strength L ÿ R . In all situations Ie 0, and we Ie shows a positive peak due to the A process and a do not discuss it anymore. Figure 3(c) shows that Is has negative peak by the Aÿ process [see Fig. 1(c)], but Is has several peaks and dips when we vary !: the large peak two positive peaks. As we tune the gate voltage VgR , the indicated by A is the spin-cell operation discussed above, right QD level is shifted so that when h! L# UL ÿ but peaks at C and D correspond to double- and triple- R# , the Aÿ process occurs with high probability leading photon processes which connect the A transitions of to a positive peak in Is and a negative peak in Ie . On the Fig. 1(c). The dip at B originates from less probable other hand we get positive peaks in both Ie and Is when transitions connecting levels indicated by the dashed h! R" UR ÿ L" , for the A process. lines of Fig. 1(c), while the dip at E is its two-photon The peak positions in Ie ; Is due to the A processes process. Now, ﬁxing ! at ! , i.e., at the spin-cell opera- shift linearly with the microwave frequency !, as shown tion point A, the value of Is can be tuned by the overall by the dotted lines in Fig. 2. Eventually, at a special gate voltage Vg as shown in Fig. 3(a). However, Is keeps frequency indicated by A, i.e., when h! R" UR ÿ large values for a wide range of Vg : this range is in the L" L# UL ÿ R# , the two peaks overlap so that the Coulomb interaction scale U=e. This is important, be- net charge current Ie cancels exactly due to the cancella- cause in an experimental situation any background charge tion of the A processes, at the same time the spin current or environmental effect near the spin cell may alter the Is doubles its value. At this special frequency, the full overall potential, and Fig. 3(a) shows that the spin-cell operation of the spin cell occurs so that a spin current is operation is not critically altered by this effect. When Vg driven across the spin cell, from the left electrode to the becomes very large so that L# UL and R" UR are right electrode, without a charge current. If we connect below the chemical potential , or L" and R# are above the spin cell to complete an external circuit, a spin current , Is diminishes because the A processes can no longer will be driven and will continue to ﬂow across the spin occur [see Fig. 1(c)]. Finally, a very important result is cell into the circuit [11]. On the other hand, if we let the shown in Fig. 3(b), where we ﬁxed gBR ÿ0:4 while two poles of the spin cell open, although Is must be zero, a varying gBL at the spin-cell operation point A [12]. spin-motive force in the two poles of the spin cell will Figure 3(b) shows clearly that Is increases with an still be induced so that chemical potential " Þ # . For increasing difference of BL ÿ BR : Is 0 identically example, in the case of Fig. 1(c), an open circuit will lead when BL BR if UL UR , or Is 0 if UL Þ UR . to L" < L# and R" > R# . However, Fig. 3(b) demonstrates that we need only a 258301-3 258301-3 PHYSICA L R EVIEW LET T ERS week ending VOLUME 90, N UMBER 25 27 JUNE 2003 (a) 3 (b) (c) knowledge Dr. Baigeng Wang for many discussions and 2 inputs on the physics of spin current. 3 A I and I (X10 3 ) 1 − 2 0 − 0.4 [1] S. A. Wolf et al., Science 294, 1488 (2001). e 0 0.4 1 gµBL [2] G. A. Prinz, Science 282, 1660 (1998). C [3] There has been some work on the spin-current generation D s by a rotating magnetic ﬁeld in a unipole device. A uni- 0 pole system, however, cannot function as a spin cell E B because it cannot complete a spin circuit. See A. −1 0 1 0.5 1 1.5 Brataas, Y. Tserkovnyak, G. E.W. Bauer, and B. I. vg frequency ω Halperin, Phys. Rev. B 66, 060404 (2002); B. Wang, J. Wang, and H. Guo, ibid. 67, 092408 (2003). FIG. 3. (a) –(c) are Ie and Is versus gate voltage Vg , the [4] A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B 50, magnetic ﬁeld gBL , and frequency !, respectively. ! 5528 (1994). 1:25 in (a); 2! UL UR g BL ÿ BR in (b). VgR [5] T. H. Oosterkamp et al., Nature (London) 395, 873 (1998). 0:45, and other parameters are the same as Fig. 2. The solid, [6] N. S. Wingreen, A.-P. Jauho, and Y. Meir, Phys. Rev. B 48, dotted, and dashed lines correspond to =! 2:0, 1.0, and 0.5, 8487 (1993). respectively. Notice that the three curves of charge current [7] In deriving the (nonperturbed) retarded Green’s overlap and they are all essentially zero. function of H0 , we have taken a decoupling approxi- mation as hhak dy d j dy iir n hhak j dy iir , hhay d d j dy iir hhak dy d j dy iir 0. In k slight difference in BL and BR , at a scale of the coupling this approximation the level renormalization has been constant ÿ , to generate a substantial Is . The most im- neglected. Because our system is in the Coulomb block- portant fact is that BL and BR do not have to point to ade regime, the level renormalization is very small and opposite directions which is experimentally difﬁcult to this approximation is reasonable. If the level renormal- do. In fact, if the two QDs are fabricated with different ization is included, it does not affect the working prin- ciple of the spin cell. materials so that the g factors are different, one can [8] Q.-f. Sun, J. Wang, and T.-h. Lin, Phys. Rev. B 59, 13126 actually use a uniform magnetic ﬁeld throughout. (1999). The proposed spin cell for spin current should be [9] Because H0 has the interaction U , this Dyson equation experimentally feasible using present technologies. is not exact, but is a good approximation. First, the double-QD structures can and have been fab- [10] Here we took the same approximation as that of Ref. [8] ricated by several laboratories. Second, microwave as- which is justiﬁable when h! max ÿ ; tC . sisted quantum transport measurements have recently [11] If resistances of external circuits for spin-up and spin- been reported [5,13,14]. In particular, the asymmetrical down channels are slightly different, the spin cell will microwave radiation on the double-QD device (i.e., L Þ drive a spin current but perhaps with a small charge R ) has already been carried out experimentally [5]. current. However, by regulating the gate voltage Vg Third, the asymmetric magnetic ﬁeld should be feasible which makes the spin-motive force slightly different for spin-up and spin-down electrons, we can still obtain as we have discussed above. If one takes f !=2 a spin current with zero charge current. 50 GHz, arranges the corresponding U h! [12] The frequency ! of the spin-cell operation point A [see 0:2 meV, and ﬁxes the temperature scale KB T and cou- Fig. 1(c)] actually depends on the value of BL : 2h! pling ÿ to be 20 times less than U as in typical QD L # UL R " UR ÿ L" ÿ R# UL UR experiments, i.e., kB T 100 mK and ÿ 10 eV, the g BL ÿ BR . To plot Fig. 3(b) we varied ! for each corresponding magnetic ﬁeld difference is [g BL ÿ value of BL accordingly. BR ÿ] jBL ÿ BR j 0:16=g tesla. These QD parameters [13] T. H. Oosterkamp et al., Phys. Rev. Lett. 78, 1536 (1997). have already been realized by present technology. Finally, [14] L. P. Kouwenhoven et al., Phys. Rev. Lett. 73, 3443 it is not difﬁcult to show that by adjusting the gate (1994). voltages one can easily calibrate the spin-cell operating [15] In order to calibrate experimental conditions at the spin- point [15]. cell operating point, one needs a method to detect spin current outside the spin cell. Recently, Hirsch has ad- We gratefully acknowledge ﬁnancial support from vanced a theoretical idea for this purpose which works NSERC of Canada, FCAR of Quebec (Q. S., H. G), the even in the absence of a charge current: J. E. Hirsch, National Science Foundation of China, the Chinese Phys. Rev. Lett. 83, 1834 (1999). Moreover, the detection Academy of Sciences (Q. S.), and a RGC grant from the can be made easier if we allow and then detect a small SAR Government of Hong Kong under Grant No. HKU charge current that ﬂows through the spin cell, using the 7091/01P (J.W.). H. G. thanks Dr. Junren Shi for a dis- two panels of Fig. 2 as a ‘‘map’’ between the charge and cussion on photon assisted tunneling. We gratefully ac- spin currents. 258301-4 258301-4

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