We demonstrate the time reversal Aharonov-Casher (AC) effect in small arrays
of mesoscopic semiconductor rings. By using an electrostatic gate we can
control the spin precession rate and follow the AC phase over several
interference periods. We show that we control the precession rate in two
different gate voltage ranges; in the lower range the gate voltage dependence
is strong and linear and in the higher range the dependence in almost an order
of magnitude weaker. We also see the second harmonic of the AC interference,
oscillating with half the period. We finally map the AC phase to the spin-orbit
interaction parameter alpha and find it is consistent with Shubnikov-de Haas
analysis.Comment: 4 pages, 6 figure

Bergsten, Tobias

Kobayashi, Toshiyuki

Nitta, Junsaku

Sekine, Yoshiaki

Physical Review Letters

English

arXiv

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Experimental Demonstration of the Time Reversal Aharonov-Casher Effect

Tohoku University Repository (TOUR) / 東北大学機関リポジトリ

Experimental Demonstration of the TimeReversal Aharonov-Casher Effect著者 Bergsten  Tobias, Kobayashi  Toshiyuki, Sekine Yoshiaki, Nitta  Junsakujournal orpublication titlePhysical Review Lettersvolume 97number 19page range 196803year 2006URL http://hdl.handle.net/10097/52963doi: 10.1103/PhysRevLett.97.196803Experimental Demonstration of the Time Reversal Aharonov-Casher EffectTobias Bergsten,1,3 Toshiyuki Kobayashi,1 Yoshiaki Sekine,1 and Junsaku Nitta1,2,31NTT Basic Research Labs, 3-1 Morinosato-Wakamiya, Atsugi-shi, Kanagawa 243-0198, Japan2Graduate School of Engineering, Tohoku University, 6-6-02 Aramaki-Aza Aoba, Aoba-ku, Sendai 980-8579, Japan3CREST-Japan Science and Technology Agency, Kawaguchi Center Building,4-1-8, Hon-cho, Kawaguchi-shi, Saitama 332-0012, Japan(Received 12 December 2005; published 8 November 2006)We demonstrate the time reversal Aharonov-Casher (AC) effect in small arrays of mesoscopicsemiconductor rings. By using an electrostatic gate we can control the spin precession rate and followthe AC phase over several interference periods. We show that we control the precession rate in twodifferent gate voltage ranges; in the lower range the gate voltage dependence is strong and linear and in thehigher range the dependence in almost an order of magnitude weaker. We also see the second harmonic ofthe AC interference, oscillating with half the period. We finally map the AC phase to the spin-orbitinteraction parameter  and find it is consistent with Shubnikov–de Haas analysis.DOI: 10.1103/PhysRevLett.97.196803 PACS numbers: 85.35.Ds, 71.70.Ej, 73.23.bSpintronics is the art of generating, manipulating, anddetecting the spin of electrons in solid state electronicdevices. While this has traditionally involved ferromag-netic materials and external magnetic fields, we can alsomanipulate spins with purely electric fields via the spin-orbit interaction (SOI) between a moving spin particle andan electric field. In particular, we can design a semicon-ductor heterostructure with a two-dimensional electron gas(2DEG) which has an internal electric field perpendicularto the 2DEG due to an asymmetric quantum well. We willthen have SOI even without external electric fields. This iscalled the Rashba effect [1,2].The SOI is a relativistic effect on a particle with spinwhich is moving through an electric field. In the particle’sframe of reference there will be a magnetic field perpen-dicular to the electric field and the direction of movement.The spin direction will precess around the axis parallel tothis magnetic field and the precession rate depends on thespin-orbit interaction strength  (  2kF,  is thespin splitting energy), and the value of  can be controlledby a gate voltage [3]. This allows us to control the spinprecession rate with an electrostatic gate on top of theheterostructure.In this Letter we present evidence of quantum interfer-ence effects due to spin precession in small arrays ofmesoscopic 2DEG rings. This interference is anAharonov-Casher (AC) effect [4] of time reversal symmet-ric paths and is the electromagnetic dual [5] to theAl’tshuler-Aronov-Spivak (AAS) effect [6]. Previouslythe AC effect has been detected in a neutron interferometryexperiment [7]. As the AAS effect and the relatedAharonov-Bohm (AB) effect have proven to be importanttools in research, we can expect that the AC effect will be apowerful tool for understanding quantum interactions andmaterial properties. We further show that we can controlthe spin precession rate with an electrostatic gate andmodulate the interference pattern over several periods.Earlier experiments on square loop arrays yielded similarresults, but only for up to one interference period [8]. Wealso see the second order AC effect where the oscillationperiod is half the period of the first order effect and theoscillations correspond to up to 26 spin precession angle.Second and third order harmonics have also recently beenobserved in square loop arrays [9]. We map the interfer-ence pattern to changes in  and combine the results withmeasurements of  using Shubnikov–de Haas (SdH) beat-ing patterns, which are consistent with the spin interferencedata. Other related experiments include single rings whichgave inconclusive results [10], and Aharonov-Bohm typeAC effect in a single ring with a complex gate voltagedependence [11].The ring arrays were etched out in a dry-etching processfrom an InAlAs=InGaAs based 2DEG (Fig. 1), similar tosample 1 in Ref. [12]. The electron mobility was7–11 m2=Vs depending on the carrier density and theeffective electron mass m was 0:050me as determinedfrom the temperature dependence of SdH oscillations. Byusing arrays rather than single rings we get a strongersignal and we average out some of the universal conduc-FIG. 1. An SEM image of an array of 0:5 m radius rings. Thearray is covered by a 50 nm SiO2 insulator layer and an 80 nmAu gate. The width of the rings is 100 nm after dry etching.PRL 97, 196803 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 NOVEMBER 20060031-9007=06=97(19)=196803(4) 196803-1 © 2006 The American Physical Societytance fluctuations (UCF) and AB oscillations [13]. Thearrays consisted of between 3 3 and 6 6 rings. The(average) radius of the rings was between 0.5 and 1:1 mand the width was 20% of the radius. The rings werecovered with a 50 nm thick SiO2 insulator layer, depositedby ECR sputtering, and on top of that was an Au gate, usedto control the carrier density and the SOI strength . Theadvantage of using a small number of rings rather than alarge array is that the gate leakage is much smaller and wecan use relatively high gate voltage. This makes it possibleto see several oscillations of AC interference.Close to the arrays and in the same current path andunder the same gate was a Hall bar, 5 m wide and 20 mlong, used to measure the carrier density. For the SdHmeasurements we used a larger Hall bar, 20 m by120 m, covered by an identical gate structure.The resistance of a mesoscopic ring is affected by sev-eral quantum interference effects. The well-known ABeffect results in a resistance oscillation with a magneticflux period of h=e. Another is the AAS effect, which worksin parallel with the AB effect and occurs when the twowave function parts travel a full turn around the ring inopposite directions and interfere at the entry point.Contrary to the AB effect, the two parts follow the exactsame path but in different directions, or time reversalsymmetric paths. This means that it is independent of theFermi momentum (and consequently the carrier density)and the interference is always constructive in the absenceof magnetic flux. However, the AAS effect is sensitive tothe spin phase (the AC phase) as we show in this Letter.When the flux is increased the resistance oscillates with theperiod h=2e, but the amplitude decays after a few periodsbecause of averaging between different paths in the ring,with different areas.In Fig. 2 we display the resistance versus magnetic fieldfor an array of rings. In the top graph we can see bothinterference effects described above. The fast oscillationsnear the center are the h=2e oscillations and the slower arethe h=e oscillations which do not decay significantly athigher magnetic fields [14]. The background peak is mostlikely due to weak localization. In our experiments we areinterested in the h=2e oscillations and in order to see themmore clearly we feed the raw data through a digital band-pass filter [15] (bandwidth approximately 40% of the peakfrequency), as shown in the lower graph.If there is SOI in the ring, the electron spin will startprecessing around the effective magnetic field. The pre-cession axes for the two parts of the wave function areopposite and therefore the relative precession angle istwice the angle of each part. Depending on this angle theinterference is constructive or destructive, and in the caseof destructive interference the phase of the h=2e oscilla-tions will shift by , which we interpret as negativeamplitude.In the following, precession angle (rate) refers to theprecession of a single path.The precession angle of an electron moving along astraight narrow channel is 0  2mL=@2, where L isthe distance travelled [16]. In a ring the precession is a bitmore complicated because of the precession axis con-stantly changing direction. The h=2e oscillation amplitudecan be written as [17–19] RR0 cos212m@2 r2s ; (1)with R and R0 being the h=2e oscillation amplitudewith and without SOI, respectively, and r the radius of thering. In the limit of strong SOI or large rings the argumentof the cosine reduces to 0 because the distance travelledaround the ring is 2r.The experiment was carried out in a 3He cryostat at thebase temperature which varied between 220 and 270 mK.The sample was in the core of a superconducting magnetwith the field B perpendicular to the 2DEG plane. Wemeasured the resistance R of the ring array simultaneouslywith the Hall resistance RH of the Hall bar close to therings, while stepping the magnetic field and the gate volt-age VG. Figure 3 shows the result after digital band-passfiltering of the h=2e oscillations which are visible asvertical bands in the figure. We can clearly see the oscil-lations switching phase as we increase the gate voltage.In order to reduce noise and UCF effects we averagedten resistance versus magnetic field (R versus B) curveswith slightly different gate voltages. This averaging pre-serves the h=2e oscillations but the averaging of M curvesreduces the h=e oscillation amplitude roughly as M1=2[10,20].We calculated the h=2e oscillation amplitude by inte-gration of the h=2e peak of the FFT spectra of R versus Bcurves. We then plotted the amplitude against the gatevoltage. Figure 4 shows the results from three different6.386.366.346.32Resistance (kΩ)-2.00.02.0Resistance (Ω)-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0Magnetic field (mT)(a)(b)FIG. 2 (color online). The resistance of an array of rings vs themagnetic field. The array consisted of 4 4 rings with 1:1 mradius. (a) Raw data, averaged from eight sweeps with slightlydifferent gate voltages (1.9–2.0 V). (b) The same data aftergoing through a digital band-pass filter, showing the h=2eoscillations clearly with the amplitude falling off with higherfield.PRL 97, 196803 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 NOVEMBER 2006196803-2ring arrays. We also calculated the carrier density ne fromthe slope of the RH versusB (n1e  edRH=dB) and plottedit in the same diagrams. It was important to measure nesimultaneously with R rather than measuring the ne versusVG dependence separately because the dependence shiftedconsiderably between different gate voltage sweeps. Thisis obvious from Fig. 4(a) and 4(b) which are measured onarrays with the same ring radius, 1:0 m. All the featuresof graph (b) are faithfully reproduced in graph (a), except itis shifted by about 1 V. If we plot the amplitude against thecarrier density instead [inset of Fig. 4(c)] we see that thetwo curves agree very well.As we see in Fig. 4 the h=2e amplitude oscillates as wechange the carrier concentration using the top gate. As thegate voltage is changed, the SOI strength  changes with itand as we expect from Eq. (1) the h=2e amplitude crosseszero, inverting the h=2e oscillations. Each period repre-sents one extra 2 spin precession of an electron movingaround a ring. However, the top two graphs have an un-expected ‘‘half oscillation‘‘ (a negative peak which doesnot cross zero) at a carrier concentration of 1:71016 m2. There is no theoretical explanation for thisresult. We also notice from the top and bottom graphsthat the carrier density saturates around 2:2 1016 m2,and that the h=2e amplitude is still changing, but muchmore slowly.Since we are measuring the electron density in a Hall barrather than in the rings directly, there may be some differ-ence due to depletion of the 2DEG near the edges.However, the fact that the density in the Hall bar saturatesat about the same voltage as where the oscillations in therings slow down indicates that the difference is small.In the FFT spectra there is also a small peak at h=4e.This is due to the wave function parts going twice aroundthe ring before interfering. We may also have interferencefrom paths going around two neighboring rings, with thesame period, but we do not know the amplitude of thiscontribution. We expect, however, that the period is 8 timeslonger with respect to VG [21], and thus longer than ourmeasurement range. If we do the same analysis on thispeak we get an oscillating amplitude with half the periodcompared to the h=2e amplitude (Fig. 5). This is expectedbecause the distance is twice and therefore the precessionangle is also twice. The first peak we can distinguish fromthe noise, at 2.3 V, corresponds to a precession angle of26. In the h=4e curve the negative peaks correspond tothe zero crossings of the first harmonic, while the positivepeaks correspond to the peaks, positive and negative. Theamplitudes of the h=4e oscillations are between 40 and 1002.01.51.00.510.09.08.07.06.05.04.03.02.01.00.0-1.02.00.0-2.02.01.51.00.56.05.04.03.02.01.00.04.02.00.0-2.010.5π11.5π12.5π13.5πθ=14.5π2.22.01.81.61.41.28.07.06.05.04.03.02.01.00.0Gate voltage (V)-2.00.02.04.5πθ=5.5π-4.00.04.02.01.51.00.5Carrier density (1016m-2)AAS amplitude (Ω)(a)(b)(c)×1016m-2FIG. 4 (color online). The time reversal AC effect. h=2e(AAS) oscillation amplitude and carrier density plotted againstthe gate voltage for three different ring arrays: (a) 3 3 array,1:0 m radius; (b) 5 5 array, 1:0 m; (c) 6 6 array,0:5 m. Inset of (c): the amplitudes of (a) [solid lines] and (b)[dotted lines] plotted against the carrier density. The precessionangle  corresponds to the argument of the cosine in Eq. (1).-150-100-50050100150Amplitude (m)6.05.04.03.02.01.00.0Gate voltage (V)-6.0-4.0-2.00.02.04.06.0Amplitude ()FIG. 5 (color online). The h=4e amplitude () oscillates withhalf the period compared to the h=2e amplitude (). These dataare from the same measurement as Fig. 4(b).5.04.03.02.01.0Gate voltage (V)-5.0 0.0 5.0Magnetic field (mT)FIG. 3 (color online). The resistance vs gate voltage andmagnetic field after digital filtering [see Fig. 2(b)]. Color scaleis black-red-yellow-white-blue. We clearly see the h=2e oscil-lations switching phase as the gate voltage is increased. Thesample is a 5 5 array of 1:0 m radius rings.PRL 97, 196803 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 NOVEMBER 2006196803-3times smaller than the h=2e oscillations, depending on thegate voltage. Since the amplitude decreases exponentiallywith the electron path length [13] we can write the ratio ofthe amplitudes as Ah=2e=Ah=4e  exp	4r 2r=L,which gives a coherence length L between 1.4 and1:7 m. We note one interesting point: the negative peakat 5.2 V in the h=4e curve corresponds to the ‘‘half oscil-lation‘‘ in the first harmonic rather than a zero crossing.We can use the oscillations to map  against ne, like inRef. [8]. According to Eq. (1) the h=2e amplitude is zerowhen    @28mr	2N  12  16q; (2)with integer N > 1. The zero crossings in the graphscorrespond to consecutive values of N. However, thisdoes not tell us the absolute value of , so we measuredbeatings in SdH oscillations in order to anchor the string ofzero crossings to an absolute value [22].We measured SdH oscillations at different carrier den-sities in a separate Hall bar and calculated  from thebeating patterns (black dots in Fig. 6). We know fromsimilar heterostructures that the value of  is positive[12]. Now we can map the spin interference graphs toabsolute  values by choosing suitable values of N, seeFig. 6. The gate voltage sensitivity =VG is0:46–0:57 peVm=V in the range below carrier densitysaturation. In the saturation region the sensitivity is muchsmaller, 0:05–0:11 peVm=V.To conclude, we have shown that the spin precession ratecan be controlled in a precise and predictable way with anelectrostatic gate. We have experimentally demonstratedthe time reversal AC effect, the electromagnetic dual to theAAS effect, in small arrays of rings, including the secondharmonic of this effect. Our spin interference data agreewith SdH measurements of . The precise spin precessioncontrol is important in order to realize semiconductorspintronics devices based on SOI, e.g., the Datta-Das tran-sistor [16].[1] E. I. Rashba, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960)[Sov. Phys. Solid State 2, 1109 (1960 )].[2] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039(1984).[3] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.Rev. Lett. 78, 1335 (1997).[4] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319(1984).[5] H. Mathur and A. D. Stone, Phys. Rev. Lett. 68, 2964(1992).[6] B. L. Al’tshuler, A. G. Aronov, and B. Z. Spivak, Pis’maZh. Eksp. Teor. Fiz. 33, 101 (1981) [JETP Lett. 33, 94(1981)].[7] A. Cimmino, G. I. Opat, A. G. Klein, H. Kaiser, S. A.Werner, M. Arif, and R. Clothier, Phys. Rev. Lett. 63,380 (1989).[8] T. Koga, Y. Sekine, and J. Nitta, Phys. Rev. B 74, 041302(2006).[9] Y. Sekine, T. Koga, and J. Nitta (to be published).[10] J. Nitta, T. Koga, and H. Takayanagi, Physica(Amsterdam) 12E, 753 (2002).[11] M. Ko¨nig, A. Tschetschetkin, E. M. Hankiewicz,J. Sinova, V. Hock, V. Daumer, M. Scha¨fer, C. R.Becker, H. Buhmann, and L. W. Molenkamp, Phys. Rev.Lett. 96, 076804 (2006).[12] T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi, Phys.Rev. Lett. 89, 046801 (2002).[13] C. P. Umbach, C. Van Haesendonck, R. B. Laibowitz,S. Washburn, and R. A. Webb, Phys. Rev. Lett. 56, 386(1986).[14] A. D. Stone and Y. Imry, Phys. Rev. Lett. 56, 189 (1986).[15] The digital filter used was the Digital FIR Filter.vi, in-cluded in LabVIEW from National Instruments. It imple-ments the Remez/Parks-McClellan algorithm; for details,see, e.g., T. W. Parks and J. J. McClellan, IEEE Trans.Circuits Syst. 19, 189 (1972).[16] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).[17] J. Nitta, F. E. Meijer, and H. Takayanagi, Appl. Phys. Lett.75, 695 (1999).[18] D. Frustaglia and K. Richter, Phys. Rev. B 69, 235310(2004).[19] M. J. van Veenhuizen, T. Koga, and J. Nitta, Phys. Rev. B73, 235315 (2006).[20] A. F. Morpurgo, J. P. Heida, T. M. Klapwijk, B. J. vanWees, and G. Borghs, Phys. Rev. Lett. 80, 1050 (1998).[21] M. J. van Veenhuizen (private communication).[22] Th. Scha¨pers, G. Engels, J. Lange, Th. Klocke, M. Holl-felder, and H. Lu¨th, J. Appl. Phys. 83, 4324 (1998).7.06.05.04.03.0α (peV m)2.01.81.61.41.21.00.80.60.4Carrier density (1016m-2)9.08.07.06.05.04.0Precession rate (rad/µm) Shubnikov-de Haas 0.5 µm rings, 6x6 1.0 µm rings, 3x3 1.0 µm rings, 5x5FIG. 6 (color online). By combining  values derived fromShubnikov–de Haas measurements () and from the zero cross-ings of the spin interference graphs (, , and 5) we can map values over a wide range of carrier densities. The right axisshows the corresponding precession rates 0=2r. The smallsymbols are the actual precession rates in the rings [the argumentof the cosine in Eq. (1)] which are slightly higher, particularlyfor the 0:5 m rings.PRL 97, 196803 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 NOVEMBER 2006196803-4

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Experimental demonstration of the time reversal Aharonov-Casher effect

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