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On nonuniversal conductance quantization in high-quality quantum wires Anton Yu. Alekseev∗†, Vadim V. Cheianov∗† ∗ Institute of Theoretical Physics, Uppsala University, Box 803, S-75108, Uppsala, Sweden † Steklov Mathematical Institute, Fontanka 27, 191011, St.Petersburg, Russia (December1996) 8 9 We present a theoretical analysis of recent experimental results of Yacoby et al. on transport 9 properties of high quality quantum wires. We suggest an explanation of observed deviations of 1 2 the conductance from the universal value 2e /h per channel in the wire. We argue that at low n temperaturesandbiasesthedeviationcanbeaconsequenceofanomalouslyenhancedbackscattering a ofelectrons enteringthe2DEGfrom thewireandisnotconnectedtointrinsicpropertiesof1DEG. J PACS numbers: 73.20.Dx, 73.23.Ad 3 2 ] ll Recently Yacoby et al. [1] presented measurements of The second effect manifests itself in the differential a conductance and differential conductance of high qual- conductance on a plateau in a strongly biased quantum h ity quantum wires prepared using cleaved edge over- wireexceedingtheuniversalvalue2e2/h. Wedonotcon- - s growth technique. Despite the recent predictions that sider this effect in the paper. e m conductanceofapure adiabaticquantumwireshouldbe In the absence of impurity backscattering the conduc- quantized in units of 2e2/h independently of interaction tanceofanoninteracting1DEGisgivenbytheLandauer- t. strength in 1DEG, Yacoby et al. observed conspicuous Bu¨ttiker formula [2] a deviations from the predicted universality. Their experi- m mental results can be summarized as follows: e2 - (1)AtlowtemperatureastheFermienergyinthewire G=2 . (2) d h is variedthe dc conductanceshowsa number ofplateaus n o separated by the steps of approximately the same value c Thefactorof2accountsfortwospinorientationsofelec- [ e2 trons. It is believed that at low temperatures and low G=2g , (1) h biases the same universal result (2) holds true in the in- 2 teracting 1DEG [3]. In the derivation of the universal v where g < 1 is a temperature dependent factor which 7 varies from sample to sample and which may be as low conductance formula it is assumed that electrons leav- 7 ing the wire be irreversibly absorbed by the leads. In as 0.75. 1 ouropinion,thisconditionisnotsatisfiedinexperiments (2)Astemperatureisincreased(from0.3K upto25K) 2 performed by Yacoby et al. higher plateaus disappear while the factor g for lower 1 6 plateaus monotonically increases and approaches 1. Thequantumwireof[1]isformedbyedgestatesatthe 9 (3) The differential conductance of the wire mono- boundary of the 2DEG. At some segment of the bound- / tonically increases with applied bias and for sufficiently ary the 2DEG is depleted by applying the gate voltage t a strong biases (about 7 mV) exceeds the universal value and thereby is decoupled from the edge states. It is this m 2e2/h. segment of the boundary that models a quantum wire. - (4)Independence oftheconductanceonthe plateauof However, the edge states extend along the whole length d Fermi energy, rigid rise of plateaus when temperature is oftheboundary. Inanidealsystemtheedgestateswould n varied as well as the measurements of the conductance be orthogonal to the 2DEG states in the bulk. Thus, if o c dependence on the length of the wire suggest that there we had applied a bias voltageto the 2DEGleads of such : is no disorder induced backscattering in the wire which a system, there would have been no current at all. Of v might be responsible for the reduced value of conductiv- course, in the experimental device the 1DEG formed by i X ity. edge states in the leads (outside the true quantum wire r The purposeofthis paperis tosuggestanexplanation region)isstronglycoupledtothe2DEGinthebulk. This a oftheseexperimentaldata. Webelievethattwodifferent coupling is modeled by inserting randomly distributed effects are encountered which should be considered sep- scattering centers at the boundary. arately. The first effect is responsible for reduced values The model suggested in [1] introduces a possibility of the dc conductance at low temperatures and of the of backscattering in the edge channels and, presumably, differential conductance at low biases. In our opinion, it scatteringfromoneedgechanneltoanother. Theauthors is not related to the properties of 1DEG but rather is a of[1]introduce twocoefficient Γ andΓ characteriz- 2D BS consequence of anomalously enhanced backscattering of ing the rate of scattering to the 2DEG and of backscat- electrons in 2DEG. Only this effect will be consideredin tering to the edge states. Then by solving the Boltzman the present paper. equation they show that the conductance is reduced 1 2e2 2Γ −1/2 N N G= 1+ BS . (3) Pα = Pα = Aα 2. (6) h (cid:18) Γ2D (cid:19) Xi=1 i Xi=1| i| In our model we assume that the backscattering to the We assume that all modes of edge states are strongly edge states can be neglected Γ 0. The reason is BS ≈ coupled to the 2DEG. This means that sooner or later as follows. The region of the true quantum wire where an electron traveling along the edge is scattered into the the 1DEG is decoupled from the 2DEG is quite small. 2DEG. Then for each edge mode the total probability It was shown experimentally that the quantum wire is Pα 1. We shall see that this implies the universal be- shorter than the mean free path of an electron. So, we ≈ haviorofplateauswhentemperatureandbiasvoltageare can neglect the impurity scattering in this region. In varied. the region where the edge states are strongly coupled to We believe that at low temperatures and biases there the 2DEG the electron-electron interaction is irrelevant is strong backscattering of electrons entering the 2DEG [3]. If there had been some substantial backscattering from the edge states which leads to a non-zero probabil- intothe edgestatesinthisregiontheconductancewould ity that an electron coherently re-enters the wire after havebeen reducedby a temperature independent factor. having left it. The experiment [1] shows a strong temperature depen- Assumethatthemechanismofbackscatteringdoesnot dence of the conductance. Therefore, we conclude that mix different edge channels. Then the correction to the the backscatteringinto the edge states is notresponsible contributionintoconductanceoftheedgemodeαisgiven for the conductance reduction. bymultiplyingtheuniversalfactorof2e2/hbythetrans- Now choose a particular mode α in edge states. It mission coefficient [2]: is coupled to two 2DEG leads. As we neglect electron- electron interaction in 1DEG, we can consider the wave e2 function of a particular electron leaving the quantum G=2 tα. (7) h wire. It propagates along the edge and is scattered to the 2DEG by the scattering centers. The ith scatter- Infact,thescatteringateachcentermixesdifferentchan- ing center produces a contribution to the electrons wave nels. Hence, the matrix describing return amplitude for function in the 2DEG region an electron leaving some channel in the wire is not diagonal. InBornapproximationthe matrixofreturnampli- ψ =Aαψin(ρ,θ). (4) i p tudes is anti-Hermitian and, therefore, can be diagonal- ized by a unitary transformation. After diagonalization Here we denote by ρ = x2+y2 the distance from the the problem for N-channel edge looks as the problem of scattering center,the x-apxis is orthogonaland the y-axis N independent channels. is parallel to the edge of the sample, the origin of the In our model one can simplify the computation of coordinate system is placed at the scattering center, θ transmission coefficients tα. Let’s introduce the return is the angle between the direction of the radius vector ρ probability rα = 1 tα. As we neglect interference be- andthex-axis. Thewavefunctionψin(ρ,θ)isnormalized − p tweenthe incoming wavescreatedby differentscattering so as to correspond to the unit incident flux and can be centers one can compute rα as approximated by its asymptotic expansion N ψpin(ρ,θ)= √π2 ρe1ip/ρ2 cosθ. (5) rα =Xi=1r|Aαi|2, (8) Later we will argue why only this long distance asymp- where r is the return probability for an electroninjected toticsisphysicallyrelevant. Wehavechosenthesimplest into the 2DEG at some point at the boundary. It is formofthe wavefunction(5) proportionalto cosθ as we reasonable to assume that r does not depend on the assume that the backscattering is reasonably isotropic transversalstructureofthewavefunctioninthequantum andthis mode will give the leading contributioninto the well and, hence, is independent of α. By using equation scattering probability. (6) one easily sees that rα is also α-independent. At The probability for an electron to be scattered to the low biases the value of the return probability coefficient 2DEGbytheithscatteringcenterisequaltoPα = Aα 2. r is completely determined by scattering of electrons in i | i| As the mean free path in the wire is quite large (experi- 2DEGattheFermisurfaceof2DEG.Hence,itshouldde- mentallyitisoftheorderof102k−1)weconcludethatin pend neither on the position of the Fermi level in 1DEG F averagethe scatteringcentersaresituatedfarawayfrom nor on the number of filled channels. This explains in- eachotherandonecanneglectinterferencebetweenscat- dependence of the correction factor g of the number of tering wavescreated by different centers. Then the total filled channels as well as of their filling. probability for an electronto leave the edge state and to Let us mention that the value of r can strongly vary be scatteredtothe 2DEGatone ofthe impurities is just when the bias is increased as electrons enter 2DEG at the sum of probabilities P : higher energies. A similar phenomenon is observed in i 2 measurements of tunneling conductivity, see e.g. [4] and is the 2-dimensional Fourier transform of the Coulomb is generally known as zero-bias anomaly. At present a potential, ǫ 10 is the dielectric constant of AlGaAs. ≈ number ofdifferentmechanismsresponsibleforzero-bias The RPA polarization operator Π(q) of 2DEG at q = 0 anomalies is known. In our opinion, the relevant mecha- is equal to nismistheanomalousenhancementofthebackscattering ofelectrons entering2DEG with a sharpedge due to the m∗ Π(0)= , (14) formation of Friedel oscillations of the electron density 2π¯h2 near the edge of 2DEG [5]. The variation of the density of 2DEG near its bound- where m∗ is the effective mass of an electron [7]. Π(q) ary is approximately given by summing contributions to continuously decreases with growing q . A simple es- | | theelectrondensitybythesingle-electronstatesweighed timate shows that in the experiment of Yacoby et al. with the Fermi-Dirac distribution function Π(2kF)U0(2kF) Π(0)U0(2kF) = e2kF/(4ǫεF) 0.3. ≤ ≈ Hence, the exchange contribution is dominating. n(x)=4Z d2kπxd2kπy (cid:18)sin2(kxx)− 12(cid:19)f(εk−µ,T). (9) abNiliotwy tlehtautsaensteilmecattreonthceoohredreerntolfymrea-gennitteurdsethofetwheirperdoube- to backscattering on Friedel oscillations at zero temper- Hereεk isthekineticenergyofanelectronwiththewave ature. In the Bornapproximationthe return probability vector k, f(ε µ,T) is the Fermi-Dirac distribution at is given by − the chemical potential µ and the temperature T. At low temperatures the density variation (9) is a 1 t=r = 1 outV in 2, (15) slowly decaying oscillating function of x with the oscil- − (h¯v)2 |h | | i| lation period π/k where k is the Fermi wave vector F F [5]: where the wave function ψout is obtained by time rever- sal (complex conjugation) from the wave function ψ of in √2k2 3π an electron entering the 2DEG from the wire, v is the n(x) F sin 2k x+ . (10) ≈ π3/2(2k x)3/2 (cid:18) F 4 (cid:19) velocity of the incoming electron. As we expect that the F effect comes from the distances of the order of several This is an asymptotic formula valid for large kFx. Us- 1/kF, we can replace the exact wave function ψin by its ing formula (10) one can approximate the effective RPA asymptotic form (5). scattering potential by the following local form Astheeffectisexpectedtobemaximalforelectronsat Fermi energy of 2DEG, we compute the probability for v(x)=(U(0) U(2k ))n(x), (11) p=k . Takingintoaccountformulas(5),(10),(11)and F F − (15) we obtain the following expression for the probabil- whereU(q)istheFouriertransformoftheRPAscreened ity r: interaction potential. The term U(0) in (11) is responsible for the exchange interaction whereas U(2kf) stands n0 2 r =κ (U(0) U(2k )) , (16) forthescreenedCoulombpotential. Theoscillatingchar- 0 F (cid:18)πε − (cid:19) F acter of the scattering potential (11) leads to anomalously enhanced backscattering of the electrons entering where κ is a coefficient close to 1. Let us remark that the 2DEG at energies close to the Fermi energy [5]. instead of the linear dependence of the reflection coeffi- As the temperature is increased the oscillating tail of cient on the interaction potential derived in the work [5] the density distribution becomes suppressed at the dis- formula (16) predicts quadratic dependence. The differ- tances higher than enceisduetothefactthatinourmodelthetransmission coefficient is equal to 1 to the zeroth order in scattering 1 ε F potential whereas in the context of [5] the transmission l , T ∼ kFr T is weak. Substituting expressions (12), (13) and (14) into for- whereε istheFermienergy. Consequently,theanomaly F mula (16) one gets in the backscattering amplitude diminishes at higher temperatures. 2 4κ Π(0)U (2k ) 0 F The RPA expression for U(q) is r = 1 . (17) 0 π2 (cid:18) − 1+Π(2k )U (2k )(cid:19) F 0 F 1 U(q)=U0(q)1+Π(q)U (q), (12) Let us remark that in the regime when Π(0)U0(2kF) is 0 small the value of r is not very sensitive to parameters 0 where of the problem. If one neglects Π(0)U0(2kF) one gets a universal result r = 4κ/π2 0.4 . Note, however, that 0 2πe2 for so high a return probabil≈ity the Born approximation U (q)= (13) 0 ǫq would not be reliable anymore. | | 3 For the values of the electron density n = (1 2) As temperature in experiments [1] is always very small 0 1011cm−2 and of the Fermi energy ε = 10meV÷of th×e in comparisonto the Fermi energy (T (1 20)K,ε F F ∼ ÷ ∼ 2DEG in experiments [1] one gets r 0.2. This is in 10mV) one can approximate the dependence of t on the 0 ≈ a good agreement with experimental values of r = biasV by severalfirstterms ofthe Taylorexpansion. As exp (10 25) 10−2. Let us remark that our estimate is r(V,T) has a maximum at V =0 and behaves smoothly ÷ × basedontheassumptionthatthescreeningofthepoten- at finite T, the linear term in the Taylor expansion van- tial is purely 2-dimensional. This assumption is true if ishes. The higher order terms resultin the contributions the wave length of the Friedel oscillations π/k is much to G(T) proportional to the powers of T higher than 1 F greater than the width of the quantum well. In exper- andcanthereforebeneglectedinviewoftheexponential iments of Yacoby et al. π/k (25 40)nm whereas suppression of r(T) (19): F ≈ ÷ thethewidthofthewellvariesfrom14nmto40nm. The largestdeviationinconductanceisobservedforthinwells e2 r = (20 25) 10−2 and it decreases when the well G(T) 2 1 r e−αT/εF . (21) exp 0 width grow÷s to r× 10 10−2. We think that this ≈ h (cid:16) − (cid:17) exp ≈ × mayhappendueto cross-overfrompurely2-dimensional to 3-dimensional behavior. This dependence of G(T) is qualitatively similar to the Analysis of the dependence ofthe reflectioncoefficient one obtained in experiments [1]. r on the bias shows that for strong biases V ε the F ∼ The authors are grateful to J.Fröhlich, D.Maslov, asymptotic of r is given by T.M.Rice,A.RodinaandA.Yacobyforusefuldiscussions r(V) (εF +V)−3/2. (18) andcomments. One ofthe authors( V.C.) thanks ETH, ∼ Zu¨rich for hospitality. Numericalanalysisshowsthatthisasymptoticexpression workswellforV >0.3ε . Belowthis valuethe reflection F coefficientgrowssharplyasthe biasapproacheszeroand reaches the finite value r at V =0. 0 Therearetwodifferentfactorswhichmaketheconduc- tivity temperature dependent. First, the reflection coefficient decreases when temperature is increased because [1] A.Yacoby et al. Phys. Rev.Lett. 77, 4612 (1996). the Friedel oscillations are suppressed at high tempera- [2] R.Landauer, Philos. Mag. 21 863 (1970); tures. This dependence can be approximated by expo- M.Bu¨ttiker, Phys. Rev.Lett. 57, 1761 (1986). nential: [3] D.L.MaslovandM.Stone,Phys.Rev.B52,R5539(1995); r(T) e−αT/εF, (19) V.V.Ponomarenko, Phys. Rev.B 52, R8666 (1995); ∼ I.Safi and H.J.Schulz, Phys.Rev.B 52, R17040 (1995); where α is a coefficient numerically found to be of order A.Kawabata, J. Phys. Soc. Japan 65, 30 (1996). 1. Weexpectαtobesensitivetotheelectrondensitydis- [4] R.C.Ashoori et al. Phys. Rev.B 48, 4616 (1993). tribution near the edge. Second, at finite temperatures [5] L.Shekhtman and L.I.Glazman, Phys. Rev. B 52, R2297 the conductivityisrelatedtothe transmissioncoefficient (1995). t=1 r(V,T) via [6] C.L.Kane and M.P.A.Fisher, Phys. Rev. B 46 15233 − (1992). e2 ∂ [7] B.I.Halperin, P.A.Leeand N.Read,Phys.Rev.B 47 7312 G=2 dε t(ε µ,T) f(ε µ,T). (20) h Z − ∂µ − (1993). 4