Boundary-mediated electron-electron interactions in quantum point contacts
Vincent Thomas Francois Renard, O. A. Tkachenko, V.A. Tkachenko, T. Ota, N. Kumada, J.-C. Portal, Y. Hirayama
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Boundary-mediated electron-electron interactions in quantum point contacts
V. T. Renard , , O. A. Tkachenko , , V. A. Tkachenko , T. Ota , N. Kumada , J-C Portal , , Y. Hirayama , , . NTT Basic Research Laboratories, NTT Corporation 3-1 Morinosato Wakamiya, Atsugi 243-0198, Japan GHMFL-CNRS, BP 166, F-38042, Grenoble Cedex 9, France, Institute of Semiconductor Physics, 630090 Novosibirsk, Russia, INSA and Institut Universitaire de France, F-31077, Toulouse Cedex 4, France. SORST-JST 4-1-8 Honmachi, Kawaguchi, Saitama 331-0012, Japan and Department of Physics, Tohoku University, Sendai 980-8578, Japan (Dated: March 28, 2008)An unusual increase of the conductance with temperature is observed in clean quantum pointcontacts for conductances larger than 2 e /h . At the same time a positive magnetoresistance arisesat high temperatures. A model accounting for electron-electron interactions mediated by bound-aries (scattering on Friedel oscillations) qualitatively describes the observation. It is supported bynumerical simulation at zero magnetic field. Quantum point contacts (QPC) are usually formedwhen two wide 2D conducting regions are connected bya small constriction. They exhibit a number of intrigu-ing phenomena among which conductance quantization[1, 2] is the most emblematic. Nowadays QPCs are verycommon tools for condensed matter physicists. Examplesof recent applications include nuclear spin manipulation,solid state electron optics or precise electron counting [3].Recently, the puzzling low conductance “0.7 anomaly”[4]has attracted most of the attention since it is relatedto electron-electron (e-e) interactions. But apart “0.7anomaly” it is commonly believed that the physics ofQPCs is well understood using a single-particle picture(See Ref.5 and references therein). On the contrary, it iswell known that the properties of two-dimensional elec-tron gases (2DEGs) dramatically depend on interactions[6, 7, 8, 9, 10]. The related corrections to the conduc-tance and tunnelling density of states have their origin inFriedel oscillations (FO) of electron density around im-purities. Friedel oscillations are also known to appearat boundaries [11, 12, 13] and could therefore affect theproperties of nano-scale devices. Interestingly, bound-ary mediated FOs were recently shown to be possiblyinvolved in ”0.7 anomaly” physics [14] but this subject isstill highly debated [15].In this context we show that e-e interactions can havea significant influence on transport properties of cleanquantum point contacts even in the large conductanceregime (
G >> e /h ). We used a combination of rela-tively low electron densities, high mobility and low seriesresistance to clearly uncover the effects of e-e interac-tions. It increases interactions due to reduced screeningand ensures that impurity scattering can be disregarded.Starting from the second conductance step, the averageslope of the conductance versus gate voltage linearly in-creases with increasing temperature T . At the same time,the low field magneto-resistance (MR) is non-monotonicat high T and strongly temperature dependent. Someof our findings are present, although not as evident, inprevious works [16, 17, 18, 19]. They are consistent with a model of e-e interaction mediated by boundaries whichis supported by numerical simulation at B = 0 T.The quantum point contacts were defined employing asplit-gates lateral depletion technique [20] on high mobil-ity GaAs quantum wells [21]. Needle and square shapedsamples of various sizes were used to check the influenceof the geometry (see Table I). All samples a have backgate to tune the electron density and some samples havea 0 . µ m-wide center gate to obtain better defined con-ductance quantization steps when grounded [22]. Four-terminal resistance measurements were carried out be-tween 350 mK and 10 K with a standard low-frequencytechnique at small excitation voltage < µ V to avoidheating effects. Any obvious temperature dependance(activated parallel conduction, anomalous temperaturedependance of the series 2DEG or leakage of the gates)was excluded [23]. More than ten samples showed a qual-itatively similar behavior.
TABLE I: Summary of the samples (for the wafer and Hallbar see [21]).Sample Wafer Hall bar point contact center gateSquare1 1 1 W=0.6 µ m; L=0.4 µ m yesSquare2 1 1 W=0.6 µ m; L=1 µ m yesNeedle1 2 2 W=0.8 µ m noNeedle2 2 2 W=0.6 µ m no Figure 1a shows the conductance G of the sampleSquare1 as the split-gate voltage V sp is varied for dif-ferent temperatures. The small ( ∼ T the conductanceis quantized to exact integer values of 2 × e /h . Notethat the “0.7 anomaly” is observed. Increasing temper-ature not only shrinks the plateaus but also increasesthe overall slope of G ( V sp ) resulting in the increase ofconductance with T . Such a change in the slope of theconductance is not expected from a simple energy averag-ing, which produces fix temperature independent points -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.901234567 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.90123456 V sp (V) G ( / h ) G ( / h ) a V*sp (V) b T (K)
FIG. 1: a) Conductance of the sample Square1 as a function of V sp (From top to bottom T =4 K, 3 K, 2 K, 1 K and 350 mK).The inset shows SEM images illustrating possible shapes ofthe point contacts. b) Conductance as a function of the scaledsplit-gate voltage V ∗ sp for the same temperatures. The insetshows the scaling parameter α . in the conductance located at integers of e /h [24]. Re-gions with alternatively positive or negative temperaturedependence should be separated by these fixed points.This is not observed in our experiment. It is howeverpossible to restore this behavior by a phenomenologicalapproach. For each temperature a new effective split-gatevoltage can be calculated V ∗ sp = V p + α ( T ) × ( V sp − V p ) tosuperpose the curve to the low temperature one (See Fig.1b). Here, V p is the pinch off voltage and α is the scalingparameter. The strong linear temperature dependence of α points at quantum effects, possibly e-e interactions.In the Hartee-Fock approximation scattering on thepotential created by all the other electrons can be con-sidered as the origin of e-e interactions effects [6, 7, 8, 9,12, 13]. This potential is connected to the formation ofFriedel oscillations of the electron density close to scatter-ers (defects or boundaries) due to interferences betweenincoming and reflected electron waves. Spatial variationsof their density results in a varying effective potentialseen by conducting electrons. In our case the QPC isclean and FOs are created at the boundaries of the con-striction. They affect its conductance in two differentways. At low T when the thermal length l T exceeds W the QPC width, FOs located in the 2DEG can reducethe conductance at a plateau via scattering of emittedelectrons back to the QPC [11]. As T is increased theseoscillations are damped and conductance plateaus reachideal integer value of 2 × e /h [25]. When l T ≈ W thereremain only FOs located inside the QPC. This corre-sponds to our experimental conditions ( l T ≈ µ m at T = 300 mK). As T is further increased the transverseoscillations inside the QPC are damped thus effectivelywidening the constriction (or equivalently shifting 1D en-ergy levels). One has to compensate by applying a lower V sp to obtain a comparable value of conductance (thecurves shift to the left). The scaling can be qualitativelyunderstood as follows. Electrostatically defined QPCshave a parabolic cross section potential U = m ∗ ω x / m ∗ is the effective mass. The width of the channel can bedefined as W = (2 /ω ) p E F /m ∗ , where E F is the Fermilevel in 2DEG reservoirs. One-dimensional energy levelsare written E n = ( n + 1 / hω and in a constant capaci-tance model W is proportional to V sp − V p . It follows that∆ E n /E n ∝ ∆ ω/ω ∝ ∆ W/W ∝ ∆( V sp − V p ) / ( V sp − V p ).In analogy to the 2D case where the conductivity σ isrenormalized as ∆ σ/σ ∼ β × T /E F in the ballistic regimeof interaction [8] we expect that ∆ E n /E n ∼ β × T /E F when E n ∼ E F (the interaction-induced shift ∆ E n of theclosest to Fermi level subband only is considered). Here, β is the interaction parameter. This leads to the rescal-ing V sp → V ∗ sp = V p + α ( T )( V sp − V p ) with α ( T ) ∝ T asmeasured.This result is consistant with a numerical non-self-consistent Hartree-Fock simulation. The principle of thecomputation is first to calculate the local electron density n ( x, y, T, µ ) solving the one-particle Shr¨odinger equationin a “soft wall” electrostatic potential for different chem-ical potential µ and temperature T . Indeed, instead ofcalculating the conductance as a function of V sp for avarying potential U ( x, y, V sp ) and fixed µ , we chose tofix the electrostatic potential and vary the chemical po-tential. This greatly simplifies the calculation becausethe wave-functions are calculated only once. It is validfor small interval of split gate voltage (between two con-ductance steps). Numerical technique and parametersare described in Ref. 26, 27. In analogy to the 2D case,the interaction correction to the potential is computedas follows δU ( x, y, T, µ ) = βδn ( x, y, T, T ∗ , µ ) /D , where β is the interaction parameter, D is the 2D density ofstates, δn ( x, y, T, µ ) = n ( x, y, T, µ ) − n ( x, y, T ∗ , µ ) and T ∗ a temperature high enough to suppress FOs. The ob-tained correction which includes only T -dependent termsis then used to solve the transmission problem in the to-tal potential which is now a function of T and µ [26].Contrary to 2D systems containing impurities [8, 9] it isnot possible to relate the interaction parameter β to theinteraction constant F in a simple way. Close to theboundaries the electron density drops to zero. This dropis not completely abrupt and the first Friedel oscillation FIG. 2: Calculated correction to theelectron density due to Friedel oscil-lations at G ≈ × e /h for T=3.5K (a) and T=350 mK (b). Thescale is m − . This clearly illustratethe rise of the interaction potentialas T decreases. c) Conductance ofthe sample Needle1 for different tem-perature. The vertical arrow corre-sponds to the voltage at which Fig.3a was recorded. d) Calculated con-ductance at a similar value of con-ductance. The inset shows the calcu-lated conductance neglecting e-e in-teractions. develops in rarefied weakly-screening electron gas wheree-e interactions become particularly strong. This is takeninto account in our simulation by the large value of theparameter β = −
6. Figures 2a and b show the obtainedcorrection to the electrons density δn ( x, y ) in the vicinityof the constriction at G ≈ × e /h for T =3.5 K (a) and T =350 mK (b). This clearly illustrates the formation ofFOs as T is decreased. Note that pronounced featuresare demonstrated despite the use of a “soft wall” model.Figures 2c and d compare the experimental conductanceof the sample Needle1 to the conductance calculated bythe method described above. In both cases the fix tem-perature independent points are missing and the conduc-tance is on average increasing with temperature. A testof the model consists in neglecting the interaction term ofthe scattering potential (i.e. setting the interaction pa-rameter β to 0) which restores the dependence for energyaveraging as expected (Inset to Fig. 2d). An overall goodqualitative agreement is obtained between computer andreal experiments which further confirms the qualitativeunderstanding.In general, the magnetoresistance reveals important in-formations about scattering, coherent processes and e-einteractions [9]. Figure 3a shows that beside the unusualtemperature dependence at B = 0 T the samples presentan unexpected field dependence. The data displayed weremeasured for sample Needle1 at G ≈ × e /h (see ar-row in Fig. 2c). At high temperature, the magneto-resistance presents a maximum around B = 10 mT andbecomes negative at higher fields. At low T the resis-tance decreases with B at all magnetic fields. The highfield slope of the MR is T -dependent. Note that the max-imum moves to lower fields in QPCs with larger width(not shown). A linear negative MR with a slope inversely propor-tional to the width of the channel is known to appear inQPCs [28]. The increased slope at low temperature thatwe observe is consistent with the observation at B = 0 Twhich lead to the conclusion that e-e interactions couldin principle reduce the effective width of the channel atlow T . As for the positive MR, it cannot be attributedto diffuse boundary scattering which has very differentcharacteristics [29]. In particular it is T -independent andabsent in short constrictions. Similarly, the interplay ofboundary scattering and electron collisions [30] can bediscarded. Finally, commensurability of electron trajec-tories with the voltage probes can be ruled out since verydifferent Hall bar geometries were tested (Table. I).The absence of positive MR at low T confirms thatthe observed quantum effect is due to e-e interaction andis not an interference effect (i.e. Weak anti-localization)which should increase at low T . Similarly to 2D systems[31], it can most likely be explained by a variation of theparameter β at small magnetic fields which is only visibleat high T . Note that simulations with B -independent β did not produce the positive MR. Figure 3b and c demon-strates that the magneto-resistance depends on the elec-tron density and detailed geometry of the point contacts.Carefully adjusting the back- and split-gate voltages, theconductance G ( T = 4K , B = 0T) of the sample Needle2was tuned to the same value ( ≈ × e /h ) for two dif-ferent electron densities. Although qualitatively similar,the effect is found to be more pronounced at the lowerdensity. This is consistent with the Friedel oscillation pic-ture which should have in principle smaller effect at highdensities when the system resembles a non-interactingFermi liquid. Figure 3c compares the results of the sam-ple Needle2 to the effect obtained in the sample Square2 -30 0 30 60 900.80.91.01.11.2 -30 0 30 60 900.80.91.01.11.2 -40 0 40 80 120 160 200 2401.21.41.61.82.0 n s =10 cm n s =1.5*10 cm -2 b R ( k ) B (mT)
T= 1.5 K
T= 2 K T= 3 K T= 4 K
Needle shape
Square shape R ( k ) T= 1.5 K T= 2 K T= 3 K T= 4 K c B (mT) T =350 mK 700 mK 900 mK 1.3 K 1.7 K R ( k ) B (mT)a
FIG. 3: a) Resistance of the sample Needle1 at the gate volt-age indicated by an arrow in Fig. 2c. b) Resistance of thesample Needle2 for the electron density n s = 1 × cm − (dash) and n s = 1 . × cm − (solid) for G ( T = 4K) ≈ × e /h . c) Resistance of Needle2 (solid) and Square2 (dot)for n s = 1 . × cm − and G ( T = 4K) ≈ × e /h . at the same density and conductance. The temperaturedependence at B = 0 T is comparable but the field de-pendence is much more pronounced for the sample Nee-dle2 which has smoother entrances. The geometry de-pendence therefore appears to be a very interesting toolto study the effect of magnetic field on the measured ef-fect. The positive MR raises interesting questions aboutthe influence of low magnetic field on Friedel oscillationsand requires additional theoretical work.We believe that the presented deviations have not beenobserved in regular point contacts (for example in Ref.[24]) due to the lower mobility and higher density in theseexperiments. Indeed, Friedel oscillations are known todepend exponentially on disorder and interaction. How-ever, recent measurements on similar samples show sim-ilar phenomenology (see Ref. 16, 17, 18 for the conduc-tance quantization and Ref. 19 for the MR) indicatingthat our observation is a general effect. It could have par-ticular importance in nano-scale electronics since bound-aries dominate transport properties in nano-devices (elec-trostatically defined quantum dots, rings, Y-junctionsetc.).We are very grateful to I. Gornyi, A. Dmitriev, K.Takashina, Y. Tokura, K. Pyshkin for helpful discus-sions and K. Muraki for providing some samples. O.A.T.acknowledges UJF for the invitation as “Maˆıtre deConf´erences”, the Supercomputing Siberian center and CNRS/IDRIS (project 61778). Part of this work was sup-ported by CNRS/RAS agreement between ISP Novosi-birsk and GHMFL Grenoble. [1] B. J. Van Wees et al . Phys. Rev. Lett. , 848 (1988).[2] D. A. Wharam et al . J. Phys. C. , L209 (1988).[3] G. Yusa et al . Nature , 1001 (2005); B. J. Le Roy etal . Phys. Rev. Lett. , 126801 (2005); T. Fujisawa et alScience. , 1634 (2006).[4] K. J. Thomas et al . Phys. Rev. Lett. , 135 (1996).[5] C. W. J. Beenakker & H. Van Houten Solid State Physics. , 1 (1991).[6] B. L. Altshuler & A. G. Aronov Electron-electron inter-action in disordered systems. (A. L. Efros, M. Pollak,Amsterdam, 1985).[7] A. Rudin, I. Aleiner & L. Glazman Phys. Rev. B ,9322 (1997).[8] G. Zala, B. N. Narozhny, & I. Aleiner Phys. Rev. B ,214204 (2001).[9] I. V. Gornyi & A. D.Mirlin Phys. Rev. lett. , 076801(2003); Phys. Rev. B , 045313 (2004).[10] V. T. Renard et al . Phys. Rev. B , 075313 (2005).[11] A. Alekseev & V. Cheianov Phys. Rev. B , 6834 (1998).[12] I. Aleiner & L. Glazman Phys. Rev. B , 9608 (1998).[13] V. A. Sablikov JETP Lett. , 404 (2006); B. S.Shchamkhalova & V. A. Sablikov J. Phys.: Condens.Matter, v. 19, 156221 (2007).[14] T. Rejec & Y. Meir
Nature , 900 (2006).[15] S. Ihnatsenka & I. Zozoulenko
Phys.Rev. B , 045338(2007).[16] A. E. Hansen, A. Kristensen, H. Bruus, Proceedings ofNANO-7/ECOSS-21. Malmo, Sweden. 24-28 June 2002 .[17] K. Pyshkin (Phd Thesis, University of Cambridge, 2000).[18] S. Cronenwett et al . Phys. Rev. lett. , 226805 (2002).[19] Y. Feng et al . J. Vac. Sci. Technol. A , 730 (1999).[20] T. J. Thornton et al Phys. Rev. lett. , 1198 (1986).[21] Two wafers were used with the mobility µ ≈ × cm /Vs and µ ≈ × cm /Vs for a density n s =1 . × and n s = 1 × . They were processed into10 µ m wide (type 1) and 100 µ m wide (type 2) Hall bars.[22] H. M. Lee, K. Muraki, E. Y. Chang, & Y. Hirayama, J.App. Phys. , 043701 (2006).[23] Hall measurements show that the electron density is T -independent. The resistance of the 2DEG decreases onlyby a few Ohms between high and low T which makes thissource of T -dependence incompatible with the observa-tion. Also, the split gate’s leakage current was monitoredto be smaller than 20 pA at all voltages and T.[24] B. J. Van Wees et al . Phys. Rev. B , 12431 (1991).[25] A. Yacoby et al . Phys. Rev. Lett. , 4612 (1996)[26] The computation region was 3 µ m long and 0.7 µ m wide.The constriction is 250 nm wide in the model. The calcu-lation step along x and y was 5 nm, the energy step was10 − meV. Usual hopping contant for GaAs were used.[27] T. Usuki et al Phys. Rev. B , 8244 (1995).[28] H. Van Houten et al Phys. Rev. B , 8534 (1988).[29] T. J. Thornton et al Phys. Rev. lett. , 2128 (1989); F.Rahman et al Semicond. Sci. Technol. , 478 (1999).[30] R. N. Gurzhi, A. N. Kalinenko, & A. I. Kopeliovich Phys.Rev. lett. , 3872 (1995). [31] G. Zala, B. N. Narozhny, & I. Aleiner Phys. Rev. B65