A Spin Cell for Spin Current

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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
      Center for the Physics of Materials and Department of Physics, McGill University, Montreal, PQ, Canada H3A 2T8
                            Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
                   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 ˆ 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# † ˆ 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 ˆ …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 ˆ 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  ˆ 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  .

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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=2†gB , 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 
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† ˆ  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† ˆ P                        ^             ^
                                                                                  expfi t dt0  W …t0 †D g, where D 
                                                                P y                    y
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 ˆ ‰ ÿ gB =2Šdy 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:Š
QD flows out to the right electrode after absorption of a                  k
photon, indicated by A ‡ . This way, driven by the po-               ‡        ‰tC ei    0
                                                                                            dt0  cos!t0 y
                                                                                                        dL dR   ‡ H:c:Š;                 (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 …† ˆ
                                                                                                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 ˆ         n is the time-averaged intradot electron occupation
ÿIs;R ˆ 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 ˆ ‰ ‡ W …t† ÿ …1=2†gB Šdy d    
                                                                from the fluctuation-dissipation theorem: g< …† ˆ     
       ‡ U dy d" dy d# ‡         ‰k ‡ W …t†Šay ak       ÿf ‰gr …† ÿ ga …†Š. With these preparations, the
                  "    #                          k
          X                    k X                           Green’s function Gr and G< of the total Hamiltonian H
       ‡     ‰tk ay d ‡ H:c:Š ‡ ‰tC dy dR ‡ H:c:Š;          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 …† ˆ gr …† ‡ Gr …†r …†gr …†;
                                                                     ;mn          ;mn             ;mk     ;kn    ;nn
tively. The left and right QDs include a single energy level                                           k

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where       Gr …†  Gr
              ;mn          ;nÿm … ‡ m!†,    and       the                              charge current I e             spin current I
quantity Gn …† is the Fourier expansion of G…t; t0 † [8].                                                A−
The retarded self-energy r …† is the Fourier trans-                           1.5
form of r …t1 ; t2R where r …t1 ; t2 † ˆ r …t1 ; t2 † ˆ

                                                                  frequency ω
                                LR          RL                               1.4
…t1 ÿ t2 †tC exp‰i t1 dt0  cos!t0 Š, and r ˆ r ˆ
                     0                      LL      RR
0. We obtain r                   r
                 LR;mn …† ˆ RL;nm …† ˆ tC Jnÿm …=!†.                      1.3             A
The Green’s function gr …† is gr;mn          ;mn …† ˆ
 mn gr … ‡ m!†. Then Gr …† can be solved                                  1.2
from the above Dyson equation [10]:                                             1.1
       Gr                    r
        ;mn …† ˆ mn =‰…g;mm †   ÿ A;mm Š;
                                                                               1.0                       A+

                                                                                      0   0.2   0.4   0.6      0.8   0   0.2   0.4   0.6    0.8   1
       Gr ;mn …† ˆ 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 …t†i 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 ˆ R ˆ 0, ÿL ˆ ÿR ˆ kB T ˆ 0:1, tC ˆ
                                                                 0:02, UL ˆ 1, UR ˆ 0:9, gBL ˆ 0:2, gBR ˆ ÿ0:4, VgL ˆ
  I  hI …t†i                                                0:5, Vg ˆ 0, and =! ˆ 1:0.
      ˆ ÿIm …d=2†ÿ …†‰G<
                           ;00 …†

                                 ‡ 2f …†Gr
                                           ;00 …†Š; (4)
                                                                    In the following we focus on the spin-cell operation by
                                                                 fixing gate voltage VgR ˆ 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 ˆ ÿi …d=2†G<         ;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! ˆ 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! ˆ 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! ˆ R" ‡ UR ÿ
                                                                large values for a wide range of Vg : this range is in the
L" ˆ 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 ˆ 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 ˆ 0 identically
example, in the case of Fig. 1(c), an open circuit will lead     when BL ˆ BR if UL ˆ UR , or Is  0 if UL Þ UR .
to L" < L# and R" > R# .                                     However, Fig. 3(b) demonstrates that we need only a
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                                        (b)                               (c)
                                                                                knowledge Dr. Baigeng Wang for many discussions and
                                                                                inputs on the physics of spin current.
                    3                                                 A
 I and I (X10 3 )


                                    − 0.4                                        [1] S. A. Wolf et al., Science 294, 1488 (2001).

                                                 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

                                                                                     by a rotating magnetic field in a unipole device. A uni-
                                                                                     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. ! ˆ                              5528 (1994).
1:25 in (a); 2! ˆ UL ‡ UR ‡ g…BL ÿ BR † in (b). VgR ˆ                           [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 =! ˆ 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 ˆ n hhak j dy iir ,
                                                                                     hhay  d d j dy iir ˆ hhak dy  d j dy iir ˆ 0. In
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 ˆ !=2 ˆ
                                                                                     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! ˆ
pling ÿ to be 20 times less than U as in typical QD                                 L # ‡ UL ‡ R " ‡ UR ÿ L" ÿ R# ˆ UL ‡ UR ‡
experiments, i.e., kB T ˆ 100 mK and ÿ ˆ 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.

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