We report conductance measurements of ballistic one-dimensional (1D) wires defined in GaAs/AlGaAs heterostructures in an in-plane magnetic field, B. When the Zeeman energy is equal to the 1D subband energy spacing, the spin-split subband Nup arrow intersects (N+1)down arrow, where N is the index of the spin-degenerate 1D subband. At the crossing of N=1up arrow and N=2down arrow subbands, there is a spontaneous splitting giving rise to an additional conductance structure evolving from the 1.5(2e(2)/h) plateau. With further increase in B, the structure develops into a plateau and lowers to 2e(2)/h. With increasing temperature and magnetic field the structure shows characteristics of the 0.7 structure. Our results suggest that at low densities a spontaneous spin splitting occurs whenever two 1D subbands of opposite spins cross

Cooper, NR

Graham, AC

Pepper, M

Ritchie, DA

Simmons, MY

Thomas, KJ

arXiv

UCL Discovery

P H Y S I C A L R E V I E W L E T T E R S week ending26 SEPTEMBER 2003VOLUME 91, NUMBER 13Interaction Effects at Crossings of Spin-Polarized One-Dimensional SubbandsA. C. Graham, K. J. Thomas, M. Pepper, N. R. Cooper, M.Y. Simmons,* and D. A. RitchieCavendish Laboratory, Madingley Road, Cambridge, CB3 OHE, United Kingdom(Received 7 May 2003; published 25 September 2003)136404-1We report conductance measurements of ballistic one-dimensional (1D) wires defined in GaAs/AlGaAs heterostructures in an in-plane magnetic field, B. When the Zeeman energy is equal to the 1Dsubband energy spacing, the spin-split subband N " intersects N  1 # , where N is the index of thespin-degenerate 1D subband. At the crossing of N  1 " and N  2 # subbands, there is a spontaneoussplitting giving rise to an additional conductance structure evolving from the 1:52e2=h plateau. Withfurther increase in B, the structure develops into a plateau and lowers to 2e2=h. With increasingtemperature and magnetic field the structure shows characteristics of the 0.7 structure. Our resultssuggest that at low densities a spontaneous spin splitting occurs whenever two 1D subbands of oppositespins cross.DOI: 10.1103/PhysRevLett.91.136404 PACS numbers: 71.70.–d, 72.25.Dc, 73.21.Hb, 73.23.AdIn this Letter, we show that at the crossing of Zeeman-split 1D subbands of opposite spins and different spatialthe split levels, for example, N " and N  1 # convergeand pass through a crossing point (see Q in the inset).Studies of ballistic transport in one dimension (1D)have shown that a spontaneous spin splitting may occur atzero magnetic field, as indicated by a conductance struc-ture at 0:72e2=h, which drops to 0:52e2=h in an in-plane magnetic field [1]. This so-called 0.7 structure iswidely reported in various types of ballistic 1D wiresdefined in GaAs [1–4] and Si heterostructures [5]. Insome cases, at very low electron densities a structurehas been observed at 0:52e2=h at zero magnetic fieldwhich strengthened with in-plane magnetic field [6], in-dicating a complete spin polarization [7,8].By studying Zeeman splitting of 1D subbands, it wasshown that the 0.7 structure is accompanied by an en-hancement of the Lande g value as the 1D subbands aredepopulated, and the energy difference between the spin-split 1D subbands tends to a finite value at zero magneticfield [1]. The conductance plateau at 0:52e2=h, eitherobserved in zero magnetic field or induced by Zeemaneffect, rises to 0:62e2=h with increasing temperature[9]. None of these characteristics can be explained withina single particle model. As the situation is dynamic, forshort ballistic 1D wires, zero-field spin splitting may notbe in conflict with the theorem of Lieb and Mattis [10].The discovery of the 0.7 structure in ballistic 1D wireshas stimulated much theoretical work in one dimension,some of which focused on zero-field spin polarization[11–13], spin density wave formation [14], pairing ofelectrons [15], singlet-triplet formation [16], Kondo-likeinteractions [4,17,18], and electron-phonon effects [19].Experimental studies [1–3] of the 0.7 structure have, ingeneral, indicated that due to a spin splitting at zeromagnetic field, a complex many-body state may exist ina ballistic 1D constriction. In order to further study therole of spin, a strong magnetic field is applied parallel to aquantum wire to produce large Zeeman splitting andinduce crossings between spin-split 1D subbands [20].0031-9007=03=91(13)=136404(4)$20.00wave functions, a spontaneous splitting sets in, givingrise to new conductance structures exhibiting character-istics of the 0.7 structure; we call this new structure a 0.7analog. We have observed such 0.7 analogs in eightsamples, with magnetic field applied in both in-planedirections.Split-gate devices were defined by electron beam li-thography on a Hall bar etched from a high mobilityGaAs=AlxGa1xAs heterostructure. Samples A and Bused in this work have a length 0:4 	m and widths 0.6and 0:5 	m. The two-dimensional electron gas (2DEG)formed 292 nm below the surface has a mobility of 1:1106 cm2=Vs and a carrier density of 1:15 1011 cm2.Conductance measurements were performed in a dilutionrefrigerator using an excitation voltage of 10 	V at 77 Hz.The samples were mounted with the magnetic field, Bk,parallel to the current direction. By monitoring the Hallvoltage, the out-of-plane misalignment was measured tobe less than 0:5 .Figure 1 shows differential conductance G  dIdV traces,measured as a function of split-gate voltage Vg at fixedmagnetic fields, Bk. The inset shows a schematic illustra-tion of linear Zeeman splitting of 1D energy subbands fora parabolic potential confinement. The left trace of themain figure shows conductance plateaus quantized atN2e2=h and the 0.7 structure at Bk  0. As Bk is in-cremented to 15.6 T (right trace), the overall conductancecharacteristics undergo three major changes. These cor-respond to P, Q and R in Fig. 1 inset. First, each spin-degenerate 1D subband N splits into two, N " and N # (seeP in the inset), and new conductance plateaus appear athalf-integer values of 2e2=h. Second, with further in-crease of Bk, the half-integer plateaus strengthen andinteger plateaus weaken. When the Zeeman energy,g	BBk, is equal to the subband energy spacing,EN;N1, integer plateaus disappear. This happens when 2003 The American Physical Society 136404-1FIG. 2. (a) Gray-scale plot of transconductance, dG=dVg, as afunction of Vg and Bk for sample B. (b) GVg traces (offsethorizontally) for Bk  10 T to 13 T incremented by 0:2 T(sample B, different cooldown). (c) gray-scale plot as in (a),for sample A. (d) GVg traces as in (b), for sample A.FIG. 1. Differential conductance, GVg, traces at Bk incre-mented in steps of 0.6 T. For clarity, successive traces are offsethorizontally. Inset: Schematic energy diagram for a linearZeeman splitting of 1D subbands and subsequent crossings.P H Y S I C A L R E V I E W L E T T E R S week ending26 SEPTEMBER 2003VOLUME 91, NUMBER 13Finally, with further increase of Bk, half-integer plateausweaken and integer plateaus reappear as the N " and N 1 # diverge again after the crossing (see R in the inset).For example, the plateau at 1:52e2=h weakens and the2e2=h plateau reappears for Bk > 8 T. However, this isaccompanied by the evolution of a weak structure fromthe edge of the 1:52e2=h plateau, which gradually low-ers to 2e2=h and develops into a plateau. This resemblesthe evolution of the 0.7 structure to 0:52e2=h withincreasing Bk. For this reason, we call the structure atthe crossing a 0.7 analog. It may be noted that the reap-pearing 2e2=h plateau now carries the opposite spin tothat before crossing. The lowest subband 1 # , however,does not encounter a crossing; therefore the plateau at0:52e2=h is intact, and does not change its spin.The evolution of conductance characteristics with thesplitting of 1D subbands in Bk can be clearly representedin a gray-scale plot of the transconductance dG=dVg,obtained by numerical differentiation of GVg character-istics. Figure 2(a) shows dG=dVg plots of sample B as afunction of Bk and Vg. White regions represent low trans-conductance [plateaus in GVg] and the dark thick linescorrespond to high transconductance (transitions betweenplateaus). Each dark line splits into two as Bk increases.This can be interpreted as the splitting of each 1D sub-band into two subbands of opposite spins as shown by P inthe Fig. 1, inset. On the left of Fig. 2(a), for N  1, thereare two distinct dark lines at Bk  0. The white regionbetween these two dark lines represents the 0.7 structure,marked by 0. As the gap between the N  1 # and N 1 " widens with Bk, the 0.7 structure evolves into a plateau136404-2at 0:52e2=h and the white region (0) in Fig. 2(a) broad-ens. At Bk  11 T, dark lines corresponding to N  1 "and N  2 # subbands cross. After the crossing, the lineN  1 " shows a discontinuous shift of Vg  23 mVfrom the crossing point, marked by1. This discontinuityin the dark line N  1 " corresponds to the appearance ofthe 0.7 analog. Figure 2(b) shows GVg traces in thevicinity of 1 from a different cooldown of the sample,highlighting the evolution of the 0.7 analog from the edgeof 1:52e2=h plateau to 2e2=h with increasing Bk. It isobserved that the 1:52e2=h plateau, though weakening,remains visible when the 0.7 analog evolves.The discontinuous evolution of the right-moving darklines ( " -spin subbands) can also be observed at the cross-ing ofN  2 " with N  3 # lines marked by , and at thesecond crossing of N  1 " with N  3 # line, marked by2. Figures 2(c) and 2(d) show results of sample A in adifferent cooldown from that of Fig. 1. In addition to thesplittings at the crossing of peaks as observed in sampleB, in this case there is also a splitting just before thecrossing of N  1 " and N  2 # peaks, marked by anasterisk in Fig. 2(c). In one of the cooldowns, sample Balso showed a weak splitting before the crossing. It is notclear whether this splitting marked by asterisk is relatedto 1, the 0.7 analog. It may be observed that this corre-sponds to a slower rate of suppression of the 2e2=h plateau(1 " -subband) in sample Awith Bk as shown in Fig. 2(d),compared to sample B in Fig. 2(b).136404-2P H Y S I C A L R E V I E W L E T T E R S week ending26 SEPTEMBER 2003VOLUME 91, NUMBER 13In the following, we suggest that the main features ofour observations arise from strong electron-electron in-teractions. To clarify this, we first describe the expectedbehavior in the case of noninteracting electrons. In par-ticular, we show that the diamagnetic shifts of the sub-band energies in an in-plane magnetic field have only avery limited influence on the most important features ofour observations.Figure 3(a) shows a gray-scale plot of sample B withmany occupied 1D subbands, part of which is shown inFig. 2(a). Figure 3(b) shows the positions of the calculatedtransconductance peaks as a function of electron densityand Bk for a model of noninteracting electrons in aninfinite 1D wire. This model includes the diamagneticeffects with Bk, assuming parabolic confinement in thetransverse and vertical (quantum well) directions withsubband spacings of 1.85 and 15 meV [21], respectively. Ag value of 1.9 is used in this model in accordance with thevalue measured at low Bk in our samples.If one assumes that the electron density in the wire islinearly related to Vg, then Figs. 3(a) and 3(b) can becompared directly. Clearly the model of noninteractingelectrons accounts well for the general trends in theFIG. 3. (a) Gray-scale plot of dG=dVg as a function of Vg andBk for sample B, showing the splitting of five 1D subbands.(b) Calculated Zeeman splitting with diamagnetic shift of the1D subbands. The solid lines represent parallel spin and thedotted lines represent antiparallel spin.136404-3evolution of the transconductance peaks with Vg andBk. However, the model cannot capture the appearanceof discontinuities in the positions of the transconducancepeaks at the crossings,1, 2, and  in Fig. 2. As we havedescribed above, these are the regions where the conduc-tance displays the 0.7 analogs.Figure 4 shows the temperature dependence of the 0.7analog at the first crossing of 1 " and 2 # subbands. Adefining characteristic of the 0.7 structure is its unusualtemperature dependence. In addition, at low tempera-tures, the 0.7 structure becomes well defined only at lowelectron densities [6]. For a higher 1D density, a highertemperature (typically, T  1 K in GaAs split-gate de-vices [1]) is required to induce the 0.7 structure [2]. Athigh Bk, it is well known that the 0.7 structure developsinto a plateau at 0:52e2=h; however, with an increase inT, the plateau rises to 0:62e2=h [9]. These trends in thetemperature dependence of the 0.7 structure and0:52e2=h plateau are also observed in the new 0.7 analogstructure at Bk  10 T and 2e2=h plateau at Bk  13 T.Figure 4 shows that, at Bk  10 T, the 0.7 analog presentjust below 1:52e2=h (shown by a down arrow) drops as Trises; but at Bk  13 T, the plateau at 2e2=h (shown by anup arrow) rises with increasing T. In the inset of Fig. 4,conductance of the 0.7 analog is plotted as a function ofBk for a range of temperatures. These data compare wellto the temperature dependence of the 0.7 structure [9],showing the crossover of G as a function of T at acharacteristic Bk.Nonquantized conductance structures can be due to achange in the transmission probability caused by scatter-ing or a many-body effect in the 1D channel. We haveobserved 0.7 analogs in eight samples, and they areFIG. 4. Temperature dependence of GVg for sample B [samecooldown as Fig. 2(b)] at 10 and 13 T. Inset: The height of the0.7 analog as a function of Bk for various temperatures.136404-3P H Y S I C A L R E V I E W L E T T E R S week ending26 SEPTEMBER 2003VOLUME 91, NUMBER 13independent of cooldowns, and occur only at the crossingof spin-split subbands of opposite spins. Because of thehigh reproducibility of the 0.7 analogs, a disorder-inducedscattering effect can be discounted.When two energy levels are brought together, an anti-crossing may occur. However, this depends on the sym-metry of the two wave functions. In our case, the two 1Dlevels that cross in Bk have different spins and subbandindices; therefore such anticrossings should be very weak.Experimentally, we do not observe anticrossings of 1Dsubbands, rather a gap forms abruptly after the crossing.As in the case of the 0.7 structure, we believe that the new0.7 analog is a consequence of strong exchange interac-tions. In the former case, there is a lifting of the zero-fieldspin degeneracy, whereas in the latter case, the degeneracyat the crossing point is lifted. We can quantify thestrength of the exchange interactions by measuring thegate voltage splitting at 1. From dc source-drain biascalibration [22], 1 is measured to be 0.5 meV, which is athird of the subband spacing (E1;2  1:6 meV) at zeromagnetic field.Given the strong similarity between the 0.7 analog andthe 0.7 structure, we consider whether theories for the 0.7structure could apply to our results. A recently proposedelectron-phonon scattering mechanism for the 0.7 struc-ture [19] cannot account for the observed 0.7 analog.Consider Fig. 2(b): if the strong 0.7 analog in these traceswere the result of a conductance suppression caused byelectron-phonon scattering, then one should expect atleast as strong a suppression at the spin-polarized0:52e2=h plateau; there is no such indication.The behavior of the 0.7 analog does not seem to beconsistent with the most simple extension of a proposed‘‘Kondo model’’ for the 0.7 structure [17] to the presentsituation—in which a Kondo impurity forms from quasi-bound states of the 1 " and 2 # levels which becomedegenerate at some nonzero Bk. In this model, one wouldexpect the 0.7analog feature to fall onto the 2e2=h plateauas Bk is increased or decreased away from the point ofdegeneracy in either direction. In contrast, the 0.7 analogevolves asymmetrically about the midpoint of the cross-ing. A detailed analysis within the Kondo model will beconsidered in a later publication.To conclude, we have observed the crossings of spin-split 1D subbands of different spins and spatial wavefunctions in a 1D electron gas. At crossings, there is aspontaneous splitting giving rise to new conductancestructures. There are no indications of anticrossings,but an energy splitting may occur due to exchangeinteractions whenever two 1D subbands are degenerate.The magnetic field and temperature dependences showthat the new structures strongly resemble the zero-field 0.7 structure. We believe that these 0.7 analog struc-tures may provide the key to a fuller understanding of136404-4the role of electron-electron interactions in ballistic1D wires.We thank J. T. Nicholls, D. Khmelnitskii, V. Tripathi,and C. J. B. Ford for useful discussions. This work wassupported by EPSRC, U.K. K. J.T. acknowledges supportfrom the Royal Society.*Current address: University of New South Wales, Schoolof Physics, Sydney, NSW 2052, Australia.[1] K. J. Thomas, J. T. Nicholls, M.Y. Simmons, M. Pepper,D. R. Mace, and D. A. Ritchie, Phys. Rev. Lett. 77, 135(1996).[2] A. Kristensen et al., Physica B (Amsterdam) 251, 180(1998).[3] D. J. Reilly et al., Phys. Rev. B 63, 121311 (2001).[4] S. M. Cronenwett, H. J. Lynch, D. Goldhaber-Gordon,L. P. Kouwenhoven, C. M. Marcus, K. Hirose, N. S.Wingreen, and V. Umansky, Phys. Rev. Lett. 88,226805 (2002).[5] N.T. Bagraev, A. D. Buravlev, L. E. Klyachkin, A. M.Malyarenko, W. Gehlhoff, V. K. Ivanov, and I. A.Shelykh, Semiconductors 36, 439 (2002).[6] K. J. Thomas, J. T. Nicholls, M. Pepper, W. R. Tribe, M.Y.Simmons, and D. A. Ritchie, Phys. Rev. B 61, 13 365(2000).[7] C.-K. Wang and K.-F. Berggren, Phys. Rev. B 54, 14 257(1996).[8] A. Gold and L. Calmels, Philos. Mag. Lett. 74, 33 (1996).[9] K. J. Thomas, J. T. Nicholls, M. Pepper, M.Y. Simmons,D. R. Mace, and D. A. Ritchie, Physica E (Amsterdam)12, 708 (2002).[10] E. Lieb and D. Mattis, Phys. Rev. 125, 164 (1962).[11] B. Spivak and F. Zhou, Phys. Rev. B 61, 16 730 (2000).[12] H. Bruus, V.V. Cheianov, and K. Flensberg, Physica E(Amsterdam) 10, 97 (2001).[13] A. A. Starikov, I. I. Yakimenko, and K.-F. Berggren, Phys.Rev. B 67, 235319 (2003).[14] S. M. Reimann, M. Koskinen, and M. Manninen, Phys.Rev. B 59, 1613 (1999).[15] V.V. Flambaum and M.Y. Kuchiev, Phys. Rev. B 61, 7869(2000).[16] T. Rejec, A. Ramsak, and J. H. Jefferson, Phys. Rev. B 62,12 985 (2000).[17] Y. Meir, K. Hirose, and N. S. Wingreen, Phys. Rev. Lett.89, 196802 (2002).[18] P. E. Lindelof, Proc. SPIE Int. Soc. Opt. Eng. 4415, 12(2001).[19] G. Seelig and K. A. Matveev, Phys. Rev. Lett. 90, 176804(2003).[20] A. J. Daneshvar, C. J. B. Ford, A. R. Hamilton, M.Y.Simmons, M. Pepper, and D. A. Ritchie, Phys. Rev. B55, 13 409 (1997).[21] T. Ando, J. Phys. Soc. Jpn. 51, 3893 (1982).[22] N. K. Patel, J.T. Nicholls, L. 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Interaction effects at crossings of spin-polarized one-dimensional subbands

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Interaction effects at crossings of spin-polarized one-dimensional subbands

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