Spin filter using a semiconductor quantum ring side-coupled to a quantum wire

Transcription

1 Spin ilter using a semiconductor quantum ring side-coupled to a quantum wire Minchul Lee, 2 and C. Bruder Department o Physics and Astronomy, University o Basel, CH-456 Basel, Switzerland 2 Department o Physics, Korea University, Seoul 36-7, Korea We introduce a new spin ilter based on spin-resolved Fano resonances due to spin-split levels in a quantum ring (QR) side-coupled to a quantum wire (QW). Spin-orbit coupling inside the QR, together with external magnetic ields, induces spin splitting, and the Fano resonances due to the spin-split levels result in perect or considerable suppression o the transport o either spin direction. Using the numerical renormalization group method, we ind that the Coulomb interaction in the QR enhances the spin ilter operation by widening the separation between dips in conductances or dierent spins and by allowing perect blocking or one spin direction and perect transmission or the other. The spin-ilter eect persists as long as the temperature is less than the broadening o QR levels due to the QW-QR coupling. We discuss realistic conditions or the QR-based spin ilter and its advantages to other similar devices. PACS numbers: d, 7.7.Ej, Hk, 5..Cc Introduction. Spintronics [] that utilizes the electron s spin degree o reedom rather than its charge or inormation processing and storage has been a subject o intense interest in recent decades. The practical realization o spin-based electronic circuits requires the development o eicient means to generate spin-polarized currents, and to manipulate and detect spins. Spin ilters that block the transport o one spin direction have been proposed as a device to generate and detect spin currents [2]. The basic scheme o a spin ilter exploits spin-dependent transport through systems lacking timereversal symmetry or having nontrivial geometric structures with spin-dependent interactions; such systems include erromagnetic junctions [2], and nanostructures like quantum dots [3] (QD s) and rings [4, 5]. A simple but eective spin-ilter implementation without coupling to magnetic materials has been suggested to exploit the spin-dependent resonance through a QD with Zeeman splitting that is embedded [3] in, or sidecoupled [6, 7] to a quantum wire (QW). While in both cases the spin-dependent transport is based on scattering rom spin-split QD levels and can be tuned by varying the gate voltage or the external magnetic ield, the sidecoupled coniguration is more eective because the Fano resonance in this case can lead to perect blocking o one spin direction and almost total transmission o the other. Side-coupled QD systems show two dips corresponding to the total suppression in the conductance o spin up and down [6, 7]. Since such spin iltering deteriorates rapidly with increasing temperature T, however, an ideal operation o the device requires large magnetic ields B or high g-actors such that gµ B B k B T, k B T K where µ B is the Bohr magneton and T K is the Kondo temperature [7]. Recently, quantum rings (QR s) with Rashba spinorbit coupling have been proposed or spin injection devices [5]. Spin precession due to the momentumdependent eective magnetic ield and the ollowing spin intererence o two quantum states propagating in opposite directions can not only modulate the charge conductance [8] but also induce spin currents through leads attached to the ring [5]. When an unpolarized charge current is injected through one o leads, quantum intererence can produce pure spin currents through one o other leads. However, this intererence-based spin-ilter operation requires more than two leads to be linked to dierent positions o the QR. In this Letter we propose another kind o spin ilter that consists o a QR side-coupled to a QW; see Fig. (a). Spin-resolved Fano resonances due to spin-split levels ormed in the QR in the presence o Rashba spin-orbit coupling and external magnetic ields [9] lead to a complete suppression o transport o either spin component at a set o gate voltage values, resulting in a series o valleys in the spin-resolved conductance. The separations between valleys are observed to be o the order o the Coulomb interaction energy. This QR-based spin ilter has three advantages: () It does not require strong magnetic ields and high g-actors like the QD-based system. In the presence o spin-orbit coupling, a weak magnetic ield applied to a small QR [] can induce a large energy splitting between spin levels because it is the magnetic lux that causes the level splitting in the ring geometry. (2) Only a single contact o the QR to the external circuit such as leads or wires is necessary. (3) Finally, on-board tuning o polarization direction o spin currents is possible via the control o spin-orbit coupling strength. Model. First, we examine the energy level structure o a non-interacting QR with Rashba spin-orbit coupling in the presence o an external magnetic ield. In the ideal one-dimensional limit where the radial width is much smaller than the radius R, only the lowest radial subband is occupied [], and the eective Hamiltonian projected to the lowest radial mode can be written in polar

2 2 (a) (b) En mev B T FIG. : (a) Schematic view o a quantum ring side-coupled to a quantum wire. Both systems are described by tight-binding models. The number o sites in the quantum ring is N R; in our study the limit N R is taken. (b) Energy spectrum o a non-interacting QR as a unction o the magnetic ield B or the corresponding lux threading the QR with a radius R = 2nm. The dotted and solid lines correspond to spin index µ = + and levels, respectively. Here, the material parameters or GaAs are taken such that m =.67m e and α =.53 evm in which case 2/ cos θ = 3. coordinates [9] H RN = 2 2mR 2 ( i ) 2 φ + α {σ ˆr, i + }, () 2R φ where m is the eective electron mass, σ the Pauli matrices, ˆr a unit vector in radial direction, and {A, B} = AB+BA. The energy o the radial mode is omitted. The spin-orbit coupling strength α deines the spin-lip length l so π 2 /mα, and the external magnetic ield induces the normalized magnetic lux πbr 2 /Φ threading the ring; Φ = hc/e is the lux quantum. The Zeeman splitting can be ignored compared to the kinetic energy E 2 /2mR 2 as long as gµ B B/E = g(m/m e ) is well satisied; this holds in usual semiconducting materials with. Following the standard procedure [2], one can set up the tight-binding version o the Hamiltonian in terms o spin-µ electron operators a nµ, a nµ deined at site n o the QR, and the tight-binding Hamiltonian can be diagonalized through the Fourier transormation such that H RN = N R m= µ=± ɛ mµ d mµ d mµ. (2) The operator d mµ creates an electron in orbital mode m and with spin index µ having space-dependent polarization ˆn = ẑ cos θ + ˆr sin θ that does not depend on the orbital index m due to the assumption that the Zeeman splitting is negligible [9]. In the limit N R, the eigenenergy ɛ mµ is given by [ ( ɛ mµ = E m+ 2 + µ ) 2 + ( ) ] 2 cos θ 4 cos 2 (3) θ with the polarization angle θ = arctan [ N so ], where N so 2πR/l so is the number o spin lips around the ring. The resulting energy spectrum is periodic not only in but also in /2 cos θ (excluding the overall shit due to the last term in Eq. (3)). Moreover, the energy gaps between neighboring spin-split levels reach their maxima whenever /2 cos θ = (2l + )/4, whereas the spinsplitting disappears at /2 cos θ = l/2, or integer l >. Throughout our Letter 2/ cos θ is assumed to be an odd integer to maximize the spin-splitting. Figure (b) shows the energy spectrum or realistic material parameters or GaAs. The ring size is taken to be R = 2nm, which is easible using current abrication technology []. The spectrum shows that a small magnetic ield < 5mT is enough to induce a spin-splitting energy gap comparable to to 3K. This large splitting that exists even in the absence o a strong external magnetic ield deinitely makes the QR a good candidate or ideal spin ilter operation. To take into account the electron-electron Coulomb interaction in the small QR, we adopt a simple capacitive model where the Coulomb interaction depends only on the total number o electrons: H RI = (U/2) [ N 2 2N g N ] with N mµ d mµd mµ. Here U e 2 /(C + C g ) and N g C g V/ e denote the interaction strength and the gate charge (in units o e ), respectively, in terms o sel and gate capacitances, C and C g. The total Hamiltonian or a QR side-coupled to a QW can then be written as H = H RN + H RI + H W + H WR (4) with H W = t w nµ (c n+µ c nµ + h.c.) and H WR = t wr µ (c µ a N Rµ + h.c.), where the operator c nµ (c nµ ) destroys (creates) an electron with spin index µ at site n o the wire. H W models the QW as an ininite tightbinding chain with a hopping energy t w between neighboring sites, and H WR a spin-independent tunneling with strength t wr between site o the wire and site N R o the ring. Note that the spin quantization axis or the QW has been rotated to align with the spin axis at site N R o the QR, ˆn at ˆr = ˆx. Spin ilter. We have calculated the zero-bias conductance G µ or spin µ at the Fermi level ɛ F = under the assumption that two electron reservoirs with nearly the same chemical potentials are attached at both ends o the QW [3]. The non-equilibrium scattering ormalism [4] enables us to express the conductance in terms o the Green s unction Gµ R (ɛ) or a spin-µ electron at site o the QW: G µ = e2 h dɛ (ɛ) ɛ Im Γ(ɛ)Gµ R (ɛ). (5) Here, (ɛ) is the Fermi distribution unction with ɛ F = and the symmetric coupling Γ(ɛ) o site to the let and right sides o the QW is given by Γ(ɛ) = (2t w / ) sin χ w (ɛ) with χ w (ɛ) arccos[ ɛ/2t w ]. In the non-interacting case

3 3 c UN g mev a b a b 2 c.2 G. FIG. 2: (Color online) Net spin conductance G as a unction o gate voltage UN g and magnetic lux in the noninteracting case at zero temperature. The plus and minus signs indicate the sign o G and are assumed to be repeated periodically along the UN g axis. Here we have used the same QR parameter values as in Fig. (b) and set t w = 5meV, t wr =.4meV. The right and bottom igures show the spin-resolved conductances G + (dotted) and G (solid) taken along the dashed lines in the main igure. (C C g and U ), by solving the Dyson equation or Gµ R, we obtain the spin-dependent transmission probability T µ (ɛ) = Im Γ(ɛ)Gµ R (ɛ) = /( + [Q µ (ɛ)] 2 ) with Q µ (ɛ) Γ(ɛ) (t wr / ) 2 m gr mµ(ɛ) = (ɛ) m /(ɛ ɛ mµ ), where gmµ R is the Green s unction or the uncoupled QR and (ɛ) t 2 wr / Γ(ɛ) is the level broadening due to the QW-QR coupling. Since Q µ diverges at ɛ = ɛ mµ, the transmission probability T µ vanishes whenever a resonant state with spin µ is ormed in the QR, giving rise to perect suppression o the transport o spin-µ electrons. Note that this blocking condition is independent o any characteristics o the wire. Figure 2 shows the ormation o a series o spin-split dips in the zero-bias conductances G µ as unctions o the gate voltage at zero temperature. The width o the valleys is restricted by the minimum o the energy splitting between neighboring levels and the level broadening (ɛ F ). The spindependent conductance can also be controlled by varying the lux. The condition or total transmission (T µ = ), sin 2π (ɛ F UN g )/E =, does not depend on spin, thus the peak positions in G µ are the same or both spins. This spin-dependent transmission generates a net spin low through the wire: G G + G = (e 2 /h)([q ] 2 [Q + ] 2 )/[(+[Q + ] 2 )(+[Q ] 2 )] at zero temperature. The net spin conductance G has local maxima or minima whenever one o the Q µ diverges; see Fig. 2. The peak height in G reaches almost the maximum value e 2 /h i the spin splitting δɛ is larger than the broadening (ɛ F ), in which case the unblocked states with opposite spin are transmitted almost completely. The ideal operation o the spin ilter, thereore, requires δɛ (ɛ F ). In addi- G N g.2. FIG. 3: (Color online) Net spin conductance G as a unction o gate charge N g and magnetic lux in the interacting case with U =.5meV. Lower igures show spin-resolved conductances G ± or = and /4, respectively. The same plot styles and parameter values as in Fig. 2 are used. For = /4, the conductances (blue lines) with a larger coupling t wr =.8meV are also shown. tion, to avoid temperature-induced broadening through Eq. (5), both the spin splitting and the broadening should be larger than the temperature T as well. Interestingly, at = /4, G reaches e 2 /h at its peaks regardless o UN g, implying perect blocking or one spin direction and perect transmission or the other. Also, the peak widths are maximal at = /4. This is related to the appearance o degenerate levels with the same spin at = /4 [see Fig. (b)], which merges two peaks separated at /4 into one peak and strengthens the Fano resonances with broader width. This observation indicates that the best perormance o the spin ilter can be achieved at = /4. It should be noted that the polarization direction ˆn = ẑ cos θ + ˆx sin θ o the spin current can be rotated by tuning the strength o the spin-orbit coupling, which should be still adjusted to satisy the odd-integer condition o 2/ cos θ to achieve the maximal separation between spin-split levels. I the spin current is measured along a direction ˆn other than ˆn, the net spin current decreases via G ˆn = G ˆn cos ζ, where ζ is the relative angle between ˆn and ˆn. Coulomb interaction. We now turn on the selcharging interaction in the QR with moderate values o U and investigate its eect on the transport at inite temperatures. The numerical renormalization group method, proven to be an excellent numerical tool or Andersontype impurity systems, was applied to calculate the spinresolved local density o states ρ µ (ɛ) on site N R o the ring. The transmission amplitude can then be calculated using the Dyson equation: T µ (ɛ) = π (ɛ)ρ µ (ɛ). Figure 3 shows the dependence o the zero-temperature conductances G µ and G on magnetic lux and gate voltage which has been tuned to such large values that high

4 4.5 a T T 3 T T 2 T T T b N g FIG. 4: (Color online) Finite-temperature conductance G or = /4. The temperatures are given by k BT =.3 (ɛ F ), k BT 2 = (ɛ F ), and k BT 3 = 8. (ɛ F ). (a) t wr =.4meV and (ɛ F ) =.6meV. (b) t wr =.8meV and (ɛ F ) =.64meV. Other parameters as in Fig. 3. QR levels with δɛ (ɛ F ) contribute to the transport: the dips in conductance correspond to the QR levels with m = 3 and 4. At = the correlation between spindegenerate QR levels and QW conduction electrons induces the Kondo eect whenever the QR contains an odd number o electrons. As a consequence, the Fano resonance due to the resultant eective resonant level at the Fermi level suppresses the charge transport regardless o the spin direction [5]. Each broad valley in G µ at =, however, splits into two spin-dependent sharp dips as the spin-splitting δɛ due to inite magnetic ield ( ) becomes larger than the Kondo temperature T K. The Coulomb repulsion widens the separations between dips in G µ or G, which is now o the order o U [6, 7], or O() in terms o N g even at = /4, while the width o valleys, still o the order o (ɛ F ), is not aected. As long as U > (ɛ F ), the broadened separation due to the Coulomb interaction contributes toward perect spin iltering at /4. As in the non-interacting case, the dip width increases as goes to /4. For large QW-QR coupling, but still δɛ > (ɛ F ), the valleys can overlap, opening wide gate-voltage windows or inducing a inite spin current; see the uppermost igure in Fig. 3. Larger coupling with δɛ (ɛ F ), however, leads to concurrent suppression o both spins and smaller net spin current. Thermal luctuations diminish the resonance-based suppression as soon as k B T (ɛ F ); see Fig. 4. First, thermal broadening in the QW, via the smoothened peak in / ɛ, obscures the resonance as the temperature becomes comparable to the resonance width in ρ µ (ɛ), or the broadening (ɛ F ). Second, thermal luctuations invoke transitions between QR levels that also weaken the resonance and consequently diminish the peak height in ρ µ (ɛ). The rapid degradation o the spin ilter eect at k B T (ɛ F ) can be attributed to these thermal luctuations. Consequently, our system will show ideal spinilter operation i k B T (ɛ F ) δɛ. Since k B T K δɛ in most cases with., the Kondo temperature is irrelevant in spin iltering. Discussion. I the ring width is not narrow compared to the radius, higher radial modes will contribute to the transport. While the large energy gap between radial modes prohibits the direct excitation to higher modes at low temperatures, the spin can experience dephasing or relaxation due to the spin-orbit interaction that couples dierent radial modes with opposite spins [9]. Also, nonmagnetic impurity scattering in such a thick ring with spin-orbit coupling can smear out the spin iltering. Conclusions. Our spin ilter takes advantage o two ingredients: () the relatively large spin-splitting in a small QR with Rashba spin-orbit coupling and (2) the Fano resonances due to the spin-split levels in the QR that is side-coupled to a QW with one conduction channel. We predict perect or considerable suppression o the transport o either o spin direction under real experimental conditions that are accessible using current technology. We would like to thank V. Golovach and M.-S. Choi or helpul discussions. This work was inancially supported by the SKORE-A program, the Swiss NSF, and the NCCR Nanoscience. [] G.A. Prinz, Science 282, 66 (998); S.A. Wol, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molnár, M.L. Roukes, A.Y. Chtchelkanova, and D.M. Treger, Science 294, 488 (2). [2] See, e.g., I. Zutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (24), and reerences therein. [3] P. Recher, E.V. Sukhorukov, and D. Loss, Phys. Rev. Lett 85, 962 (2); T.A. Costi, Phys. Rev. B 64, 243(R) (2). [4] D. Frustaglia, M. Hentschel, and K. Richter, Phys. Rev. Lett. 87, (2); M. Popp, D. Frustaglia, and K. Richter, Nanotechnology 4, 347 (23). [5] A.A. Kiselev and K.W. Kim, J. Appl. Phys. 94, 4 (23); S. Souma and B. Nikolić, Phys. Rev. Lett. 94, 662 (25). [6] M.E. Torio, K. Hallberg, S. Flach, A.E. Miroshnichenko, and M. Titov, Eur. Phys. J. B 37, 399 (24). [7] A.A. Aligia and L.A. Salguero, Phys. Rev. B 7, 7537 (24). [8] J. Nitta, F.E. Meijer, and H. Takayanagi, Appl. Phys. Lett. 75, 695 (999); D. Frustaglia and K. Richter, Phys. Rev. B 69, 2353 (24). [9] F.E. Meijer, A.F. Morpurgo, and T.M. Klapwijk, Phys. Rev. B 66, 337 (22); J. Splettstoesser, M. Governale, and U. Zülicke, Phys. Rev. B 68, 6534 (23). [] U.F. Keyser, C. Fühner, S. Borck, R.J. Haug, M. Bichler, G. Abstreiter, and W. Wegscheider, Phys. Rev. Lett. 9, 966 (23); A. Fuhrer, T. Ihn, K. Ensslin, W.Wegscheider, and M. Bichler, Phys. Rev. Lett. 93, 7683 (24). [] A. Fuhrer, S. Lüscher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, Nature (London) 43, 822 (2); U.F. Keyser, S. Borck, R.J. Haug, M. Bichler, G.

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