Experimental Verification of a Spin-Interference Device Action
Yu Iwasaki, Yoshiaki Hashimoto, Taketomo Nakamura, Shingo Katsumoto
EExperimental Verification of a Spin-InterferenceDevice Action
Y Iwasaki, Y Hashimoto, T Nakamura and S Katsumoto
Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,Chiba 277-8581, JapanE-mail: [email protected]
July 2016
Abstract.
We report the detection of spin interference signal in an Aharonov-Bohmtype interferometer with quantum dots on the conduction paths. We have found thatresonators like quantum dots can work as efficient spin rotators. The interferencesignal appears only when spin-polarized electrons are injected into the device. Theinterference pattern in the gate voltage-magnetic field plane is checker board like,ensuring the modulation of spin wavefunction’s phase as well as the orbital phase.
1. Introduction
In spite of quantum mechanical nature of the spin freedom in electrons, it has beenmostly regarded as a classical variable in traditional spintronics. That is, a spin istreated as a single (classical) bit, which carries information up or down [1]. Actuallya spin can work as a quantum bit (qubit) or even as a “flying qubit”, which isexpressed as a linear combination of the two (up and down) eigenstates. This factenables us to compose quantum spintronics devices, which would gain novel functions forinformation processing and greatly widen the field of spintronics. The quantum natureof spins prominently appears in the interference [2, 3]. A device which utilizes suchspin interference of electrons traversing through it was proposed by Aharony et al. [4].They showed that a diamond like simple interference device can serve as an efficientand precise spin rotator and an analyzer. This device requires spin-orbit interaction(SOI) to rotate the spin of electrons traversing over the diamond structure and theAharonov-Bohm (AB) phase gained from magnetic flux.Here we report experimental verification of such device action in an ABinterferometer with quantum dots on the two transmission paths. Since the spin rotationwas achieved by quantum dot resonators in our device, its magnitude depends on initialspin polarization. This effect enables us to confirm the operation of the spin rotatorsby injecting electrons from a quantum point contact (QPC), because the interferencesignal appears only when spin-polarized electrons were injected and QPC can tune itspolarization. a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l xperimental Verification of a Spin-Interference Device Action
2. Methods
Figure 1(a) shows the cross-sectional view of layered structure grown with ordinarymolecular beam epitaxy onto a (001) GaAs substrate. A pseudo-morphic In . Ga . Asquantum well is placed next to an Al . Ga . As spacer. It is well known that structuralasymmetry and narrowness in the energy gap bring about strong SOI [5, 6] and thepresent structure is reported to have a comparatively strong Rashba-type SOI [7].The electron mobility µ = 65000 cm /Vs and the sheet carrier concentration n =1 . × / cm were obtained from the Hall and the Shubnikov-de Haas measurementat 4.2 K. An advantage of the present structure over InAs quantum well with strongerSOI’s is availability of the conventional metallic split-gate technique for defining finestructures such as quantum point contacts (QPCs). Figure 1(b) is a scanning electronmicrograph of the sample, which consists of a QPC (electron emitter) and an AB-typeinterferometer. Each arm of the AB interferometer has a short side gate for the controlof the conductance indicated as V L and V R in the figure. It will be shown later thatthese gates can pinch the paths and form quantum dots on them. The electrodes andthe gates are numbered as superposed on the figure for convenience of indication.The specimen was cooled down to 0.15 K in a dilution fridge under a bias-coolingcondition of the gate electrodes for leakage free application of gate voltages [8]. Externalmagnetic field was applied perpendicular to the growth plane with a superconductingsolenoid. Two-wire and four-wire resistances were measured with conventional lock-intechnique with frequencies lower than 80 Hz.First we need to check action of each “part” in the device. Figure 2(a) shows thetwo-terminal conductance G (1 and 4 indicate the terminal numbers) as a functionof the magnetic field for open-arm condition ( V L = V R = 0). A clear Aharonov-Bohm A ~ V ~ ~ μ m1 23 45QPC side gate V R V L V g V g V b GaAs 2 nmAl Ga As 5 nmAl Ga As 7.5 nmIn Ga As 20 nmGaAs 700 nmGaAs substrateSuperlattice buffer(GaAs, AlGaAs 2.5 nm) x 10 n + -Al Ga As 25.5 nm (a) (b)
Figure 1. (a) Cross sectional view of the layered structure. (b) Scanning electronmicrograph of the sample with the terminal numbers and schematic non-local resistancemeasurement wiring. xperimental Verification of a Spin-Interference Device Action -0.2-0.2-0.8-0.8 V L ( V ) V R (V) G (2e /h) (d) -0.9-1.32.01.00.0 G Q P C ( / h ) V g (V) (e) -20-402.602.652.702.75 0 20 40 B (mT) G ( / h ) (a) -0.0300.03-0.8 -0.2-0.4-0.6 V R (V) Δ G ( / h ) (b) V b ( m V ) V R (V) -0.10.1 ΔG (2e /h) (c) Figure 2. (a) AB oscillation in two-terminal (local) conductance G . (b)Conductance oscillation ∆ G versus gate voltage V R where V L is kept to be − .
79 V.A linear background was subtracted. (c) A gray scale plot of ∆ G on V R . Coulombdiamond structures are indicated by yellow lines. (d) Gray scale plot of the deviceconductance as a function of V R and V L . Crossing white lines are the Coulomb peaks.(e) QPC conductance G QPC as a function of the gate voltage V g . The bias voltage V b was − . G q . Thetemperature for the measurements is 150 mK. (AB) oscillation with the period for magnetic flux φ ≡ h/e (flux quantum) in the ringarea is observed to be symmetric to the zero-field due to the Onsager reciprocity [9].The result certifies that the ring is working as a quantum interferometer.Next we apply negative voltages to the two gates on arms of the ring. Theconductance G decreases with the negative gate voltages and before pinch-offthresholds, aperiodic oscillations versus the gate voltages appear as shown in Fig.2(b)for V R . The oscillation can be viewed as an overlap of two Coulomb oscillations: onewith narrow peaks and short periods coming from localized states weakly coupled to theelectrodes; the other with broad peaks and wide periods from states strongly coupled tothe electrodes (referred as strongly-coupled states ) [10]. Actually as exhibited in Fig.2(c),the current-voltage characteristics form several sizes of Coulomb diamonds overlappingeach other. Hence we interpret the oscillation as formation of quantum dots around theends of gate electrodes, which phenomenon often happens, e.g. , around the pinch-off ofQPC [11,12]. The irregularity of the oscillations indicates that the quantum confinementto the dots also contributes to the conductance spectra and each peak corresponds to asingle electron level. The gate capacitance is estimated from the smallest peak intervalas about 20 aF, which is reasonable from the geometric dimensions. Figure 2(d) shows xperimental Verification of a Spin-Interference Device Action G q ≡ e /h ) in addition to the one around a full G q . This is the signthat the SOI is strong enough to realize a spin filter on the half G q plateau and a spinrotator on the full G q plateau [13, 14].We then proceed to look for the spin interference effect. The simple crossings ( i.e. ,no avoided-crossing) in Fig.2(d) suggest that the coherent portion in the conductance issmall and to extract such transport, we need to change the terminal configuration. It iswell known that in so called non-local configuration, not only the coherent portionis emphasized in the total resistance, but also the coherence itself is enhanced byblocking of the external voltage fluctuation [15, 16] naturally built in the non-localconfigurations. Figure 1(b) schematically displays the terminal configuration for theresistance measurement. The electric current flows between terminals 1 and 3 throughthe QPC. This causes a local nonequilibrium around the current path,which propagatescoherently from the current path to the detector.The four-terminal transport characteristics expected for a spin-interference devicecan be calculated in the Landauer-B¨uttiker formalism [17]. Let T ij be the transmissioncoefficient between terminals i and j , and R ij,kl be the resistance for current flow between i and j , voltage between k and l . From an S-matrix analysis [18], the transmissioncoefficients across the ring can be generally written as [19–21] T ij = a ij + b ij cos( ω/
2) cos( φ + δ ij )1 + c ij f ( φ, δ ij , ω ) , (1)where φ , δ ij , ω are the phase shifts due to the AB effect, the electrostatic potential, andthe spin rotation respectively. In the numerator, coefficient a ij represents the imbalancebetween the two interference paths. The second term can be understood within thesimplest AB approximation. That is, the probability amplitude at the interference nodewith perfect balance is (cid:34)(cid:32) cos θ θ + ω (cid:33) (cid:104) α | + (cid:32) sin θ θ + ω (cid:33) (cid:104) β | (cid:35) (cid:104) ψ | (1 + e − iφ ) × (c . c . ) ∝ cos ω φ, where | α (cid:105) and | β (cid:105) are spin up and down eigenstates. With addition of channel dependentkinetic phase gained from electrostatic potential δ ij , the second term of the numeratorin eq.(1) is reproduced. f ( φ, δ ij , ω ) in the denominator also consists of trigonometricfunctions representing the effect of multiple circulation on the ring. This is smaller than1 and can be negligible in the discussion of interference pattern. As can be seen inFig.1(b), the ring part is common for T ij ’s in (1) and it is natural to assume the ratio a ij /b ij ≡ a is common in (1). On the other hand, “bendings” of the paths outside thering are connected in series and provide difference in the amplitudes of the transmission.We can thus write T ij ∝ [ a + cos ω cos( φ + δ )] for transmissions across the ring. xperimental Verification of a Spin-Interference Device Action π π -2 π -2 π δϕ π π -2 π -2 π δ=ωϕ π π -2 π -2 π ωϕ (b) (c)(a) Figure 3. (a) Interference pattern of four terminal non-local resistance R , calculated from eq.(3). The form of RHS is plotted against the plane of the ABphase φ and the electro-statically shifted kinetic phase δ . The phase shift in spin-space ω is kept constant (=0). (b) The same quantity as that in (a) but ω is proportional(in this plot equal) to δ . (c) δ is kept to 0 and the same is plotted versus ω − φ plane. The general formula for the four-terminal resistance at absolute zero on thisnotation is given as eq.(8) in Ref.17. In the present non-local measurement, it reads R , = 1 G q T T − T T D , (2)where D ≡ G − S ( α α − α α ), S ≡ T + T + T + T . And α ij are sums of T kl and T kl T mn products, of which we do not show the tedious explicit forms. As canbe guessed from the configuration shown in Fig.1(b), T and T are more than anorder of magnitude larger than other coefficients and get much weaker effects from theinterference on the ring. This leads to an approximation α α − α α ∼ T T .Because S consists of transmission coefficients across the ring, the effect of interferenceon the denominator D can be summarized as D ∝ [ a + cos ω cos( φ + δ )]. On theother hand, terms in the numerator has a common factor [ a + cos ω cos( φ + δ )] and weeventually get R , ∝ (cid:20) a + cos ω φ + δ ) (cid:21) . (3)Figure 3 shows the interference patterns for (a) ω = 0, (b) ω = δ , (c) δ = 0. It iswell known that δ can be tuned with Schottky gate voltages [15] and a pattern similarto Fig.3(a) was actually reported in refs.15, 22. Then, if the gate voltage influences thephase ω simultaneously, a cross-hatching pattern like Fig.3(b) should appear and withincreasing the sensitivity in ω , the pattern should change to a checker board patternin Fig.3(c). The above analysis can then be summarized that simple linear “flow” ofAB oscillation with a gate voltage on one arm is the sign of electrostatic modulationof kinetic phase and with overlapping of spin-interference, the angle of cross-hatchingincreases from 0 to 90 ◦ . Hence what we should look for is the behavior in Fig.3(b) orultimately in Fig.3(c). xperimental Verification of a Spin-Interference Device Action
3. Results
We measured 4-terminal non-local resistance ( R , ) as a function of the side gatevoltage V R with different AB phase φ and QPC conductance G QPC . AB phase φ isdefined as φ = ( B − B ) /φ where B = −
30 mT and φ is AB oscillation periodcalculated from the ring area (Fig.1(a)). In order to obtain sufficient signal-to-noiseratio in the non-local measurement, a comparatively high bias voltage of 2 mV for theemitter QPC is required. This is reasonable considering low transmission coefficientsover the AB ring.In the case of G QPC = 1 . G q , Fig.4 (b, d) exhibit some peaks, though they werealmost stable against φ . On the other hand, by adjusting G QPC to 1 . G q peaks startedto oscillate with a period of AB oscillation φ (Fig.4(a, c)). Actually we observeda non-local AB oscillation when G QPC = 1 . G q (Fig.4(e)), which is asymmetric tothe zero-magnetic field ensuring 4-terminal measurement is realized. There appearedin Fig.4(a) 3 peaks around V R = − .
48 V (peak A), V R = − .
38 V (peak B) and V R = − .
60 V (peak C). The height of peak A and B can be fitted by sinusoidal curvewith phase difference ∼ π as illustrated in Fig.4(f). This phase shift matches with thecross-hatching pattern in Fig.3(c). In fact Fig.5 shows such plots, where in the case of1 . G q we clearly observe oscillations depending on V R and φ . The patterns were stable bychanging from electron to hole injection, which agrees well with the Landauer-B¨uttikerformalism [17] and thus guaranteed reproducibility of the measurement. Comparing tothe case of G QPC = 1 . G q , Fig.5 for G QPC = 1 . G q did not exhibit any cross-hatchingpattern. However, for G QPC ∼ . G q , actually some weak trace of cross hatching patternis observed. These results can be explained as following. As noted before, high spinpolarization at the emitter can be obtained when the QPC conductance is on the plateauof 0.5 G q and on the plateau of 1.0 G q . However as in Ref.14, the spin polarization onthe 0.5 G q plateau is strongly reduced with increasing the bias voltage and around 2 mVthe spin polarization is very small. On the other hand, the spin polarization is almostconstant or even increases with the bias voltage on 1.0 G q plateau. We can thus interpretthe result in Fig.5 that the cross-hatching spin interference pattern appears when largepure spin current is injected into the device, reflecting the spin polarization of injectedelectrons.
4. Discussion
The results shown in Fig.5 were not predicted in the theory in Ref.4, in which the ABoscillation amplitude does not directly reflect the initial degree of spin polarization. Thiscomes from the special symmetric device setup in Ref.4. There is also a quantitativedifficulty in the interpretation of the above result. Assuming Rashba SOI as the originof spin phase shift, we can estimate the oscillation period in the gate voltage as [23]∆ V R = 2 π/ω = ¯ h / m ∗ αL ∼
50 V , (4) xperimental Verification of a Spin-Interference Device Action -4042-2-20 20100-10 B (mT) Δ R , ( Ω ) (e) (f)(a) (b) (c) (d) Δ R , ( Ω ) -0.3-0.7 V R (V) ϕ = 0 ϕ = ϕ Δ R , ( Ω ) -0.3-0.7 V R (V) ϕ = 0 ϕ = ϕ Δ R , ( Ω ) -0.25-0.65 V R (V) ϕ = 0 ϕ = ϕ Δ R , ( Ω ) -0.25-0.65 V R (V) ϕ = 0 ϕ = ϕ ϕ / ϕ Δ R , ( Ω ) peak Apeak B A BC
Figure 4. (a-d) ∆ R , as a function of V R with φ = 0 (bottom) to φ = φ (top).(a) and (c) corresponds to G QPC = 1 . G QPC = 1 . V b was − . . R , . A background with periodlarger than 20 mT was subtracted. (f) Markers are plots of peak height vs AB phase φ/φ for A and B peak illustrated in (a). Dashed lines are fitting sinusoidal curvescalculated from least square method. which is much larger than the observed result ∼
100 mV. Rather the observed valuematches well with the Coulomb peak period of strongly-coupled states. This urges usto look for a different origin of spin rotation around the quantum dot.Here we point out an efficient spin rotation mechanism in a quantum resonator likea quantum dot. A quantum dot resonator can be modeled with two potential barriersas in Fig.6(a). We assume it has a classically ellipsoidal orbit as illustrated in the figure.It is well known that confinement potentials U for semiconductor quantum dots arewell approximated as harmonic potentials and thus have large potential gradients ∇ U .Through the SOI H SOI ∼ σ · ( p × ∇ U ) , (5) xperimental Verification of a Spin-Interference Device Action V b = - . m V G QPC = 0.5 G q G QPC = 1.0 G q G QPC = 1.8 G q V b = + . m V R ( Ω ) B ( m T ) V R ( V ) R ( Ω ) B ( m T ) V R ( V ) R ( Ω ) B ( m T ) V R ( V ) R ( Ω ) B ( m T ) V R ( V ) R ( Ω ) B ( m T ) V R ( V ) R ( Ω ) B ( m T ) V R ( V ) Figure 5.
Color plots of R , on the plane of V R and magnetic field B . The columnsfrom left to right are for G QPC = 0 . G q , 1 . G q , 1 . G q respectively. The upper row is forthe QPC bias voltage V b = − V b = +2 mV(hole injection). this gives an effective magnetic field perpendicular to the conduction plane andconsequent spin phase rotation χ . With repetition of reflection, spin standing waveemerge with accumulated spin rotation ω . The scatterings at the barriers canmathematically be expressed with S-matrices as S = S = (cid:32) r e + iχσ z t t r e + iχσ z (cid:33) , S = (cid:32) (cid:33) , (6)where σ z is the z component of Pauli matrix. After calculating the composition ofS-matrices, we get the following expression for the total transmission, t total = (cid:32) t (cid:46) (1 − r e +2 iχ ) 00 t (cid:46) (1 − r e − iχ ) (cid:33) ≡ (cid:32) t + t − (cid:33) . (7)Figure 6(b) are the transmission coefficient T = | t + | + | t − | and the accumulatedspin rotation ω = arctan( t − /t + ) respectively, which are calculated from eq.(7) and themodel circuit shown in Fig.6(a). As expected the spin rotates by 2 π within a peak-to-peak interval. Note that the resonances appearing in Fig.6(b) are pure “spin resonance”though in the actual system, spin and orbital are entangled through SOI and a π rotationbetween adjacent Coulomb peaks is expected. This perfectly agrees with the presentobservation because the phase shift through a quantum dot is dominated by the strongly-coupled states [10] and the cross-hatch period should be the one for strongly-coupledstates observed in Fig.2(b). xperimental Verification of a Spin-Interference Device Action Figure 6. (a) Simple resonator model of a quantum dot with a ellipsoidal orbit inside.The potential barrier gradient ∇ U and the transverse velocity v produce a effectivemagnetic field B eff rotating the spin. These are hence further abstracted as the S-matrix circuit shown in the bottom. (b) Transmission coefficient T and accumulatedspin rotation ω/π as a function of spin phase χ at the potential wall in the model in(a). The parameter is r = 0 . We should comment on why high spin current is essential for a clear cross hatchedpattern. The total spin phase depends on the initial spin state as calculated in eq.(7).Thus spin phase of unpolarized electrons, that is electrons with random polarization,has random distribution resulting in disappearance of the interference pattern. Thoughthis effect had made it difficult for us to observe the spin interference, now we can makeuse of it conversely to check the operation of the quantum dot spin rotators by virtueof the QPC spin injector. Fig.5 clearly exhibits the disappearance of the interferencepattern when unpolarized electrons were injected ( G QPC ∼ . G q ). It guarantees that V R did modulate the spin phase via the quantum dot spin rotators.
5. Conclusion
In summary, we have fabricated a spin interference device with a spin injector, whichutilizes a QPC and Rashba SOI. A clear spin interference signal is obtained only whenhighly spin-polarized electrons or holes are injected. From the observation we havepresented a reasonable model of a resonator with a SOI and shown that such a resonatorcan work as an efficient spin rotator controllable with gate voltage. xperimental Verification of a Spin-Interference Device Action Acknowledgments
We thank A. Aharony and O. Entin-Wohlman for valuable discussion. This work wassupported by Grant-in-Aid for Scientific Research on Innovative Area, “Nano SpinConversion Science” (Grant No.26103003), also by Grant No.25247051 and by SpecialCoordination Funds for Promoting Science and Technology. Iwasaki was also supportedby Japan Society for the Promotion of Science through Program for Leading GraduateSchools (MERIT).
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