We have experimentally studied shot noise of chaotic cavities defined by two
quantum point contacts in series. The cavity noise is determined as 1/4*2e|I|
in agreement with theory and can be well distinguished from other contributions
to noise generated at the contacts. Subsequently, we have found that cavity
noise decreases if one of the contacts is further opened and reaches nearly
zero for a highly asymmetric cavity.Comment: 4 pages, 4 figures, REVTe

A. G. Huibers

A. H. Steinbach

A. Kumar

B. J. van Wees

C. Schönenberger

C. Strunk

C. W. J. Beenakker

D. A. Wharam

E. V. Sukhorukov

G. B. Lesovik

G. F. Giuliani

K. E. Nagaev

L. P. Kouwenhoven

M. Büttiker

M. Henny

M. Holland

M. J. M. de Jong

O. Agam

R. A. Jalabert

R. J. Schoelkopf

R. Landauer

S. Oberholzer

T. Heinzel

V. I. Kozub

Ya. M. Blanter

Yu. V. Nazarov

English

arXiv

Crossref

Physical Review Letters

Shot Noise by Quantum Scattering in Chaotic Cavities

We have experimentally studied shot noise of chaotic cavities defined by two quantum point contacts in series. The cavity noise is determined as (1/4)2e[1] in agreement with theory and can be well distinguished from other contributions to noise generated at the contacts. Subsequently, we have found that cavity noise decreases if one of the contacts is further opened and reaches neatly zero for a highly asymmetric cavity. Heating inside the cavity due to electron-electron interaction can slightly enhance the noise of large cavities and is also discussed quantitatively

Oberholzer, Stefan

Sukhorukov, Eugene V.

Strunk, Christoph

Schönenberger, Christian

Heinzel, T.

Holland, M.

University of Regensburg Publication Server

VOLUME 86, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 5 MARCH 20012114Shot Noise by Quantum Scattering in Chaotic CavitiesS. Oberholzer, E. V. Sukhorukov, C. Strunk,* and C. SchönenbergerInstitut für Physik, Universität Basel, Klingelbergstrasse 82, CH-4056 Basel, SwitzerlandT. HeinzelSolid State Physics Laboratory, ETH-Zürich, CH-8093 Zürich, SwitzerlandM. HollandDepartment of Electronics, University of Glasgow, Glasgow G12 8QQ, United Kingdom(Received 6 September 2000)We have experimentally studied shot noise of chaotic cavities defined by two quantum point contactsin series. The cavity noise is determined as 142ejIj in agreement with theory and can be well dis-tinguished from other contributions to noise generated at the contacts. Subsequently, we have foundthat cavity noise decreases if one of the contacts is further opened and reaches nearly zero for a highlyasymmetric cavity. Heating inside the cavity due to electron-electron interaction can slightly enhancethe noise of large cavities and is also discussed quantitatively.DOI: 10.1103/PhysRevLett.86.2114 PACS numbers: 73.23.Ad, 72.70.+m, 73.50.TdThe nonequilibrium time dependent fluctuations of theelectrical current, known as shot noise, are caused bythe randomness of charge transfer in units of e [1]. Ifthe electron transfer can be described by a Poissonianprocess, the spectral density S of the current fluctuationsis SPoisson  2ejIj. Correlations imposed by fermionicstatistics as well as Coulomb interaction may change shotnoise from SPoisson. This is expressed by the Fano factorF defined as F  SSPoisson. A quantum wire with anintermediate barrier with energy-independent transmissionprobability G, G  12, for example, has a Fano factor of1 2 G  12 [2,3]. This suppression is due to binominalinstead of Poissonian statistics. Here, we explore whathappens if the barrier is replaced by a chaotic cavity[Fig. 1, inset (a)]. For a symmetric and open cavity,which is a cavity connected to ideal reservoirs via twoidentical noiseless (barrier-free) quantum wires, the meantransmission probability is 12, too. But surprisingly,the Fano factor is predicted to be only 14 [4,5]. The14 Fano factor, valid for open and symmetric chaoticcavities, was first derived by random matrix theory, whichis based on phase-coherent quantum mechanical transport[4]. Recently, a semiclassical analysis using the “principleof minimal correlations” arrived at the same result [5].Similar to metallic diffusive wires, where F  13[6–8], the Fano factor 14 for chaotic cavity is universalin the sense that it is insensitive to microscopic properties[4,5,9,10]. Nevertheless, there is an important differencebetween these two systems concerning the origin ofresistance and noise. In a diffusive conductor resistanceand shot noise are both generated locally at scatteringcenters, which are homogeneously distributed along thewire. In an open chaotic cavity resistance and shot noiseare generated differently. The resistance is due to thefundamental quantum resistance of the contacts. Althoughthe source of resistance, the open contacts do not con-0031-90070186(10)2114(4)$15.00tribute to noise because electrons are transmitted withunit probability. Shot noise arises inside the cavity due toquantum mechanical diffraction which splits the electronwave packet into two partial waves leaving the two exits.In the semiclassical approach cavity noise is determinedby the average fluctuations of the state occupancy insidethe cavity given, at T  0, by [5]S  2GZdE fC1 2 fC . (1)Here fCE denotes the distribution function inside thecavity, which is homogeneous and isotropic. The total con-ductance G  G0NLNRNL 1 NR with G0  2e2his equal to the series conductance of the left and rightFIG. 1. Scanning electron microscope picture of a Hall barwith three QPCs in series used to define chaotic cavities of dif-ferent size. (a) The ratio of the number of modes h  NLNR GLGR can be adjusted by varying the openings of the left andright contacts, independently. (b) QPC conductance vs gate volt-age of one of the contacts.© 2001 The American Physical SocietyVOLUME 86, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 5 MARCH 2001contacts with NL (NR) open channels (i.e., G1,...,NL,R  1,G.NL,R  0). For noninteracting electrons the distributionfunction in the cavity fC just equals the weighted aver-age of the distribution functions fL and fR in the leftand right reservoirs. In the symmetric case NL  NR ,i.e., fC 12  fL 1 fR, Eq. (1) yields a Fano factor of14. For very asymmetric contacts (NL ¿ NR) shot noiseapproaches zero, since the system can then be regardedas a single contact with NR open and therefore noiselesschannels. The general Fano factor F  S2ejIj for cavitynoise isFh NLNRNL 1 NR2h1 1 h2, (2)where we introduce the parameter h  NLNR measuringthe symmetry of the cavity.Experimentally, we have realized chaotic cavities bytwo quantum point contacts (QPC) in series. These areelectrostatically defined in a two dimensional electron gas(2DEG) by metallic split gates on top (see Fig. 1) [11].The opening of the contacts can be individually tunedby varying the applied gate voltages independently. The2DEG forms 80 nm below the surface at the interface of astandard GaAsAl0.3Ga0.7As heterojunction. Magnetore-sistance measurements yield a carrier density of 2.7 31015 m22, corresponding to a Fermi energy of106 K anda mobility of 83 V sm22 resulting in a mean free path of7 mm comparable to the size of the cavity. Three QPCsin series as shown in Fig. 1 enable one to define two cavi-ties of different size: either the outer gates A and C with themiddle gate B kept completely open can be used to define arelatively large cavity of 11 3 8 mm or two of the innergates (A,B or B,C) create a smaller cavity of 5 3 8 mm.The conductance of the QPCs is quantized according to theLandauer formulaG  G0Pn Gn [12] [inset (b) of Fig. 1].An open cavity is defined when both QPCs are adjusted toa conductance plateau, where N modes are fully transmit-ted (G  1) and the others are totally reflected (G  0).Since the gates do not influence each other, the transmis-sion of both contacts can be determined independently bycompletely opening one of them while the conductance ofthe other is measured. The two-terminal conductance Gis experimentally found to correspond to the series con-ductance of the two contacts GLGRGL 1 GR with anaccuracy of less than 1% [4,5]. Therefore, direct transmis-sion of electrons from the left to the right contact can beexcluded, as well as quantum corrections [13,14].Two independent low-noise amplifiers (EG&G 5184)operating at room temperature are used to detect the volt-age fluctuations across the cavity. A spectrum analyzer(HP 89410A) calculates the cross-correlation spectrum ofthe two amplified signals. This technique allows one to re-duce uncorrelated noise contributions which do not origi-nate from the sample itself. Experimental details can befound in [8,15]. Furthermore, the whole setup is filteredagainst RF interference at low temperatures by a shieldedsample box and lossy microcoaxes to minimize heatingby radiation. Voltage noise is typically measured at fre-quencies around 6 kHz where the noise is frequency inde-pendent (white) up to the maximum bias current #50 nAused in the experiment. The sensitivity for voltage noisemeasurements is of the order 5 3 10221 V2 s. The mea-sured noise is calibrated against equilibrium Nyquist noiseat different bath temperatures. From the Nyquist relationSV  4kBRT the voltage gain as well as the offset in thevoltage noise SoffI R2 caused by the finite current noiseSoffI of the amplifiers can be determined with high accu-racy. Although shot noise is a nonequilibrium phenome-non observed in its purest form in the limit eV ¿ kBT ,in this experiment bias voltages are limited to 8kBTe,only. This is to avoid nonlinearities of the current-voltagecharacteristics of the QPCs [16] and 1f-noise contribu-tions occurring at larger currents [15]. Within this limit,the differential resistance, recorded for all noise measure-ments, changes by &2.5%. The current noise is finallyobtained from the measured voltage fluctuations by SI SVdVdI2 2 SoffI .Figure 2 shows shot noise measurements of a cavitydefined by gates A and B with a size of 5 3 8 mm fordifferent symmetry parameters h  GLGR . The solidcurves describe the crossover from thermal to shot noisefor the measured value of h given by [5]S  SeqΩ1 1 Fh∑eV2kBTcothµeV2kBT∂2 1∏æ. (3)0 10 20 30 4034567890 5 10 15 2012T = 270 mKS I  (10-27 A2s)current I  (nA) S I  (10-27 A2s)current I  (nA) η = 1for  Σ Γ  = 1.16n n1/4 η = 1.0 η = 2.7 η = 6.2 η = 41FIG. 2. Shot noise of a chaotic cavity with ideal con-tacts (GL,RG0  integer) for different conductance ratiosh  GLGR . The data for h  1.0, 2.7, and 6.2 are offset forclarity by 20, 15, and 5 3 10228 A2 s, respectively. Inset: Shotnoise is larger than 142ejIj if there is additional partitioningdue to nonideal contacts (GL,RG0 fi integer). The curves arenumerical calculations assuming no mode mixing (dotted line)and for slight mode mixing of 10% (solid line).2115VOLUME 86, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 5 MARCH 2001Seq  4kBTG denotes the equilibrium noise and Fh theFano factor [Eq. (2)]. In the symmetric case (h  1)with NL  NR  5 we obtain a very good agreementbetween the experimental data and the theoretical pre-diction of 142ejIj. When the right contact is furtheropened (GR . GL), h increases from 1 (symmetric) to41 (asymmetric). Thereby, shot noise gradually disap-pears for larger values of h as expected from Eq. (2). Forpartial transmission in the contacts shot noise is larger than142ejIj because additional noise is generated at the con-tacts. This is shown in the inset of Fig. 2 where the con-ductance of each point contact equals 1.16G0. The dottedcurve is a numerical calculation for no mode mixing. Thedata are approximated the best assuming slight mode mix-ing of 10% (solid line) with G1  0.90 and G2  0.26.Up to now we have assumed that inelastic electron scat-tering inside the cavity can be neglected. In general, heat-ing caused by electron-electron interaction enhances shotnoise [1]. The Fano factor of a diffusive wire, for example,changes from 13 for noninteracting (cold) electrons top34 for interacting (hot) electrons [17]. Heating also af-fects the shot noise of a chaotic cavity. The Fano factor ismodified to [18]Fh p3NLNRpNL 1 NRp3hp1 1 h, (4)and the crossover from thermal to shot noise is describedbyS Seq2(1 1s1 1 Fh2µeVkBT∂2). (5)For a symmetric cavity Fh  1  0.276 for hot elec-trons, which is only slightly larger than Fh  1  0.25for cold electrons. The inset of Fig. 3 comparesSeVkBT  in the hot and cold electron regime for adiffusive wire and a cavity. As is evident, the differencesare very small, in particular, in case of a cavity whereeven a crossing at eVkBT  15 occurs. In Fig. 3 themeasured noise for h  1 of Fig. 2 is replotted andcompared to the prediction for cold (solid line) and forhot electrons (dashed line). Although the data points lieclearly closer to the prediction for cold electrons, thisalone is not sufficient to decide which regime is realizedin the cavity, because of the finite experimental accuracy.An additional criterion is needed.In order to decide whether the cold or hot electron theoryis appropriate for the comparison with the measurements,the electron-electron scattering time tee is compared withthe dwell time for electrons inside the cavity. We argue thatthermalization is present if tD ¿ tee. The average dwelltime is the product of the ballistic flight time across the cav-ity tF  LyF with the number of scattering events insidethe cavity given by the ratio of the cavity size L and thewidth of the contacts W  WL 1 WR lF2 NL 1 NR:21160 10 20 30 403450 5 10 15 20 25510T = 270 mKS I  (10-27 A2s)current I  (nA)   cold   hot   η = 1 3 / 2π 3 / 4S I  (10-27 A2s)eV / kBT1 / 41 / 3FIG. 3. Shot noise of a symmetric cavity and theoretical pre-dictions for cold (solid line) and hot electrons (dashed line).Inset: comparison of the noise of a chaotic cavity (14 andp32p) with a diffusive wire (13 and p34) for cold and hotelectrons.tD 2p h¯EFµLlF∂2 1NL 1 NR. (6)The electron-electron scattering rate t21ee in a two dimen-sional electron system is given by [19]t21ee EF2p h¯µkBTeEF∂2 ∑lnµEFkBTe∂1 lnµ2qkF∂1 1∏(7)with the Thomas-Fermi screening wave vectorq  2me2ere0h¯2. Because the system is out of equilib-rium the temperature Te in Eq. (7) has to be replaced bythe effective electron temperature Teff given by Teff 1kBRdefC1 2 fC [20]. The ratio tDtee is plottedin the inset of Fig. 4 as a function of h  GLGR for thetwo different types of cavities taking tee from Eq. (7) forTeff corresponding to the largest applied voltage V in theexperiment. The upper curve belongs to the large cavity(11 3 8 mm), where the right contact is nearly closed(GR fixed to G0). In this case, tD ¿ tee. The lowercurve corresponds to the smaller cavity (5 3 8 mm)with a 5 times larger opening of the right contact. For thistype of cavity we find tD , tee.According to this argument we use Eq. (5) valid for hotelectrons to fit the noise data obtained for chaotic cavitieswith tDtee . 1. The Fano factor F is the only fitting pa-rameter. On the other hand, we use Eq. (3) valid for coldelectrons if tDtee , 1. The Fano factors F  S2ejIjobtained according to this procedure are plotted as a func-tion of the measured h for the two different cavities de-scribed above. For the solid squares, which belong tothe large cavity with nearly closed contacts (large dwelltime), we find good agreement with the theoretical FanoVOLUME 86, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 5 MARCH 2001 GR = G0    largeGR = 5*G0  small1 10 1000.000.050.100.150.200.250.300.351 10 1000123456F = S I / 2eI τ D / τe-eη = GL / GR hot electrons cold electronsη = GL / GR GR = G0    large GR = 5*G 0  smallFIG. 4. Fano factor F  S2ejIj vs the symmetry parameterh for (open circles) a small cavity with widely opened contacts(tD , tee) and for (solid squares) a large cavity with nearlyclosed contacts (tD ¿ tee). Predictions for cold electrons (solidline) and hot electrons (dashed line). Inset: tDtee vs h for thetwo different types of cavity.factor for hot electrons given by Eq. (4) (dashed line). Theopen circles are results for the small cavity with wideropened contacts (small dwell time) which are consistentwith the prediction for noninteracting electrons describedby Eq. (2). If we use the formula for cold electrons in-stead of the one for hot electrons to fit the data obtainedfor the larger cavity, the solid squares move only slightlydownwards by0.02 0.03. They still lie clearly above theopen circles, demonstrating that heating is indeed impor-tant for the larger cavity. Good agreement between theoryand experiment is found for both regimes with the excep-tion of very asymmetric contacts, i.e., h ¿ 1. Here weattribute the deviations to slight mode mixing within theQPCs, which is difficult to avoid [15]. Let us assume,as an example for the data point at h  180, that twomodes instead of one participate in the left contact trans-mitting, respectively, with G1  0.97 and G2  0.03 in-stead of G1  1.00 and G2  0. This yields a Fano factorof 0.06 in agreement to what is experimentally observed.In conclusion, we have experimentally studied shotnoise of open chaotic cavities defined by two QPCs inseries. In the regime of noninteracting electrons a Fanofactor F  S2ejIj of 14 has been measured as theoreti-cally predicted for symmetric cavities. The origin of thisshot noise is partitioning of the electron wave functionby quantum mechanical diffraction inside the cavity. Thecontacts themselves, which actually define the resistanceof the system, do not contribute to noise. In addition,we have also investigated heating effects due to inelasticelectron-electron scattering by changing the opening ofthe contacts as well as the size of the cavity. Similar toother mesoscopic systems heating increases shot noise inagreement with theory. Shot noise in chaotic cavities is apurely quantum phenomenon. It would be interesting tostudy the crossover from “quantum chaos” to “classicalchaos,” where shot noise is predicted to be absent [21].The authors thank Ya. M. Blanter for valuable com-ments. This work was supported by the Swiss NationalScience Foundation.*Present address: Institute for Experimental and AppliedPhysics, University of Regensburg, D-93040 Regensburg,Germany.[1] For a recent review, see Ya. M. Blanter and M. Büttiker,Phys. Rep. 336, 1 (2000).[2] G. B. Lesovik, JETP Lett. 49, 592 (1989).[3] M. Büttiker, Phys. Rev. Lett. 65, 2901 (1990).[4] R. A. 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Lett. 76, 2778 (1996).[16] L. P. Kouwenhoven et al., Phys. Rev. B 39, 8040 (1989).[17] K. E. Nagaev, Phys. Rev. B 52, 4740 (1995); V. I. Kozuband A. M. Rudin, Phys. Rev. B 52, 7853 (1995); A. H.Steinbach, J. M. Martinis, and M. H. Devoret, Phys. Rev.Lett. 76, 3806 (1996).[18] M. J. M. de Jong and C. W. J. Beenakker, in MesoscopicElectron Transport, edited by L. P. Kouwenhoven,G. Schön, and L. L. Sohn, NATO ASI, Series E, Vol. 345(Kluwer Academic, Dordrecht, 1996).[19] G. F. Giuliani and J. J. Quinn, Phys. Rev. B 26, 4421(1982).[20] Thermalization has been assumed for fC which couldslightly overestimate t21ee .[21] O. Agam, I. Aleiner, and A. Larkin, Phys. Rev. Lett. 85,3153 (2000).2117

Shot noise by quantum scattering in chaotic cavities

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Shot Noise by Quantum Scattering in Chaotic Cavities

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