The transmission of conductance modes in atom-size gold contacts is
investigated by simultaneously measuring conductance and shot noise. The
results give unambiguous evidence that the current in the smallest gold
contacts is mostly carried by nearly fully transmitted modes. In particular,
for a single-atom contact the contribution of additional modes is only a few
percent. In contrast, the trivalent metal aluminum does not show this property.Comment: Fig. 2 replaced, small errors correcte

Brom, H. E. van den

van Ruitenbeek, J. M.

English

arXiv

H. E. van den Brom

J. M. van Ruitenbeek

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Quantum Suppression of Shot Noise in Atom-Size Metallic Contacts

Brom, H.E. van den

Ruitenbeek, J.M. van

Leiden University Scholary Publications

VOLUME 82, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 15 FEBRUARY 1999ndsouslyin thegle-etal1526Quantum Suppression of Shot Noise in Atom-Size Metallic ContactsH. E. van den Brom and J. M. van RuitenbeekKamerlingh Onnes Laboratorium, Leiden University, Postbus 9504, 2300 RA Leiden, The Netherla(Received 22 June 1998)The transmission of conductance modes in atom-size gold contacts is investigated by simultanemeasuring conductance and shot noise. The results give unambiguous evidence that the currentsmallest gold contacts is mostly carried by nearly fully transmitted modes. In particular, for a sinatom contact the contribution of additional modes is only a few percent. In contrast, the trivalent maluminum does not show this property. [S0031-9007(99)08411-2]PACS numbers: 72.70.+m, 72.15.Eb, 73.23.Ad, 73.40.Jnothe-,s ofeinttr8]:hotw,er-see.atondisesreIn 1918, Schottky mentioned shot noise as a fundamtal shortcoming of vacuum diodes. He realized that tdiscreteness of electron chargee causes the current to bea Poisson process, and calculated the corresponding msquare current fluctuations to be equal to the producte and the average currentI divided by the total time ofaveraging [1]. This type of noise is present in all kindof devices, including microscopic conductors. In the ladecade, it has become clear that it can actually be uto obtain information on the electron transport mechanis[2–12]. For example, in a ballistic quantum point contact (QPC) in a two-dimensional electron gas (2DEG), tconductanceG as a function of contact diameter showsstepwise increase by integer multiples of the conductanquantum,G0 ; 2e2yh [13]. Recent 2DEG experimentsshowed that shot noise was strongly suppressed at qutized conductance values [2,3], in accordance with theretical predictions [4–8]. In this Letter, measuremenof shot noise are performed for the first time to analythe electronic transport properties of atom-size metacontacts.For a metal, the size of an atom is comparable to hthe Fermi wavelengthlF of the conduction electrons.Therefore, the equivalence of electronic propertiesQPCs in a 2DEG and in a metal is far from trivial. Inparticular, it inhibits a direct observation of the effecof the formation of discrete electron modes in a metalQPC, i.e., quantization of the conductance. In fact, usia combined scanning tunneling microscopy and atomforce microscopy setup [14] it has been shown that stein the conductance, observed when stretching the contare the result of atomic rearrangements (see also [1Primarily, evidence for quantization of the conductanin metals is derived from histograms of the conductanvalues, which, for gold [16] and sodium [17], show peakclose to integer multiples ofG0. However, this evidenceis not unambiguous, as demonstrated for aluminum, whshows clear peaks near quantized conductance valuethe histograms [18], while up to three modes contributethe conductance nearG ­ G0 [19].In a ballistic QPC with perfect transmission of electrons there are no fluctuations in the occupation numb0031-9007y99y82(7)y1526(4)$15.00en-heeanofsstsedm-heacean-o-tszellicalfoftlicngicpsact,5]).cecesichs into-ersof left and right moving electrons, suppressing all shnoise [4–8]. For a contact of size comparable to tFermi wavelengthlF , electron transport is described using the Landauer-Büttiker formalism. In this formalismthe shot noise spectral density can be expressed in termthe transmission probabilitiesTn of the conducting chan-nels [5],PI ­ 2eV2e2hXnTns1 2 Tnd , (1)where the sum is over all modes which fulfill thquantization condition set by the boundaries of the pocontact. The labelI indicates that we consider currenfluctuations kDI2l. From Eq. (1) we see that, for amode with a transmission probabilityTn close to 1, theshot noise is indeed suppressed. If allTn are small, theclassical full shot noise formula,PI ­ 2eI, is recovered.Including the thermal excitations of quasiparticles fofinite temperatures leads to the following expression [7,PI ­ 2eV coth√eV2kT!2e2hXnTns1 2 Tnd1 4kT2e2hXnT2n . (2)In equilibrium (i.e., V ­ 0) or in the high temperaturelimit, the Johnson-Nyquist thermal noise,4kTG, is recov-ered. For zero temperature, Eq. (2) reduces to the snoise formula (1). In the experiments described belowe are interested in the amount of noise above the thmal noise, which we will call excess noise,PexcsId ;PsI , T d 2 Ps0, T d.In order to obtain a stable atomic scale contact, we uthe mechanically controllable break-junction techniquA notched, 99.998% pure, gold wire is glued on top ofphosphor bronze substrate, which is insulated with kapfoil. This is mounted into a vacuum can and cooledown to 4.2 K. By bending the substrate, the wirebroken, after which contact between the fracture surfacis controlled using a piezoelectric element. For a mocomplete description of the technique, see Ref. [20].© 1999 The American Physical SocietyVOLUME 82, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 15 FEBRUARY 1999ticx-s,here-henereri-x-nce.onall-p.es.reichessthertoattheviort:heherredtedalThe noise level we are expecting is on the ordernVypHz. In order to measure such small signals, wuse the experimental setup schematically drawn in Fig.The signal to be measured is first amplified105 timesby two stages of low noise wide band preamplifie(dc–1 MHz and 0.5 Hz–1 MHz, respectively). In ordeto suppress the noise of the preamplifiers, we use tsets of preamplifiers in parallel and feed the signainto a spectrum analyzer, which calculates the croscorrelation Fourier spectrum. The conductance of tsample is measured using the dc voltage after the fiamplifier; the bias current is sent using a battery (09 V) with a large series resistance, the latter being4.2 K close to the sample. Since we are interestedcontacts with resistance around13 kV, and our wiringhas a capacitance of around 250 pF, this introduces apass filter with a cutoff frequency of around 50 kHz. Tcorrect the measured spectra,PV sId, for the electronictransfer function of the system, we send white noise,PsrcV ,from a calibration source through a1 V series resistanceand measure the frequency response of the setupPcalV .The excess noise then equalsPexcV sId ­PV sId 2 PV s0dPcalV 2 PV s0d3 PsrcV . (3)The labelV indicates that here we consider the spectrdensity of voltage fluctuationskDV 2l [21].Obtaining one single spectrum in the interval 250 Hz100 kHz and averaging 10 000 times takes about 1 mFor each contact setting, we have to take several specone for calibration, one for thermal noise, and severwith different dc current biases. During this procedurthe contact is verified to be stable by measuring tconductance between each two spectra. Measuringconductance twice with different polarity of the biacurrent eliminates dc offset of the preamplifiers.For frequencies smaller than the inverse dephasing tiand obeyinghf ,, eV , kT the spectrum of both thermaland shot noise is white, i.e., frequency independent. Tis not the case for so-called1yf noise, which is generallybelieved to be a consequence of defect motion a~calibrationnoisesourceDC voltagesource1 OhmSAMPLEspectrumanalyzerlow noisepre-amps10 MOhmX100 X1000VDC voltmeter4.2 KX100 X1000FIG. 1. Wiring diagram of the experimental setup for shonoise measurements on atom-size point contacts. The aenclosed by the dashed lines indicates the part at 4.2 K.ofe1.rsrwolss-herst–atinlowoal–in.tra:ale,hethesmehisndtreadecreases at higher frequencies asf21. External vibra-tions (e.g., sound) and pickup of external electromagnesignals can be recognized in the Fourier spectrum. Eternal vibrations are mostly visible at low frequenciein the same range where1yf noise dominates, which iswhy we concentrate on the higher frequency part of tspectrum. Electromagnetic pickup, present despite caful shielding, is seen as sharp peaks and is removed wwe take the difference (3). In addition, the preamplifinoise left over after taking the cross correlation is furthremoved this way. Furthermore, the effect of external vbrations as well as1yf noise is quadratic in the suppliedcurrent, while shot noise is linear. We verify that the ecess noise we measure has the proper current dependeA typical example of the raw data,PV sId, we obtaincan be seen in Fig. 2a, where we show measurementsa contact of8.4 kV, or 1.53G0. The zero bias noise atlow frequencies is in agreement with the expected thermnoise,4kTR. At higher frequencies, we observe a roloff, which is due to the frequency response of the setuWhen applying a bias current the noise level increasApart from an increase in noise level over the entispectrum, we see a rapid rise at low frequencies, whwe attribute to1yf noise and external vibrations. Theexcess noise,PexcV sId, corrected for the transfer functionaccording to Eq. (3), is plotted in Fig. 2b and approacha white spectrum for higher frequencies. At the highefrequencies, a small decrease is observed due to higorder stray capacitance corrections, which we take inaccount in the error bars of Figs. 3 and 4. To show thwe are indeed measuring shot noise, we investigatedcurrent dependence and verified the expected behagiven by Eq. (2), which is nearly linear for higher currenIn Fig. 3, we compare the data for a contact withG ­1.02G0 to full shot noise, 2eI, and to Eq. (2) withFIG. 2. (a) Raw data for a gold contact of1.53G0 at 4.2 K,without correction for the electronic transfer of the setup. Tlowest curve shows the thermal noise, a few percent higthan the theoretical value2.0 3 10218 V2, which is due to thepreamplifier noise. The upper curves are the total measunoise for increasing current,I ­ 0.1, 0.2, 0.3, . . . , 0.9 mA.(b) Excess noise, calculated from the data in (a) and correcaccording to Eq. (3). Note that the effect of mechanicvibrations is visible at lower frequencies.1527VOLUME 82, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 15 FEBRUARY 1999rre-uc-woeisis-ntedingc-edredsbetheofovearer.3es.lylsteereasan-m-nsm-ld0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.5 Full shot noise T1=0.51, T2=0.51 T1=0.99, T2=0.03 T1=1.00, T2=0.02Excess noise (10 -26 A2 /Hz)Bias current ( µA)FIG. 3. Measured excess noise values for a contact wG ­ 1.02G0, as a function of bias current. For comparison wplot full shot noise,2eI, and Eq. (2) for the case of one singlepartially transmitted wave (i.e.,T1 ­ 1 andT2 ­ 0.02) and forthe case of two equally transmitted waves (T1 ­ T2 ­ 0.51).A good description of the data is obtained withT1 ­ 0.99 andT2 ­ 0.03.different combinations of transmission values. As can bseen in the figure, the data are well described by oalmost fully transmitted mode together with one modwith a very small transmission probability.A number of contacts with conductances in the rangs0.7 4.1dG0 have been similarly analyzed. In Fig. 4, weplot the excess noise values corresponding to the msured voltage noise at maximum bias current (0.9 mA)as a function of conductance for 27 different contactThe figure shows that all values are small, comparto 2eI. A smallest value ofs0.02 6 0.005d2eI is ob-served for a contact with conductance very nearG0. Forcomparison, we show the expected behavior, basedEq. (2), when the conductance is due to only fully tranFIG. 4. Measured excess noise values for 27 gold contacts4.2 K with a bias current of0.9 mA. Comparison is made withcalculations in the case of one single partially transmitted mo(full curve) and for various amounts of contributions of othemodes according to the model described in the inset (dashcurves). In the limit of zero conductance, these curvesconverge to full shot noise, i.e.,2.9 3 10225 A2yHz. Inset:transmission of modes in the case ofx ­ 10% contributionfrom neighboring modes.1528itheeneeeea-s.edons-atderedallmitted modes (Tn ­ 1) plus a single partially transmit-ted mode (full curve). All measured points are at oabove this curve, as is expected. From our measuments we cannot determine the complete sethTnj, sincewe have only two measured parameters (i.e., condtance and shot noise), and we can obtain at most tindependent parametersTn. To visualize the effect ofcontributions of different modes to the conductance, wuse the model described in the inset of Fig. 4. In thmodel the conductance betweens 2 1dG0 and nG0 isbuilt up as G ­ sn 2 2dG0 1 sTn21 1 Tn 1 Tn11dG0,where the three partially open channels have transmsions which increase linearly, and the sum ofs1 2 Tn21dand Tn11 is a constant fractionx. This model has nophysical basis but serves merely to illustrate the exteto which additional, partially open channels are requirto describe the measured shot noise. The correspondbehavior of the excess noise as a function of condutance, calculated from Eq. (2), is shown as the dashcurves in Fig. 4 forx ­ 5%, 10%, and 20%. We seethat for G , G0 the data are very close to thex ­ 0%curve, while forG0 , G , 2G0 the data are closer to thex ­ 10% curve. In particular, the minimum atG ­ G0is very sharp, while also a minimum nearG ­ 2G0 isvisible. ForG . 2G0, the contribution of other partiallyopen channels continues to grow. Each point is measufor a different contact, and the contribution of modeto the conductance of each contact can, in principle,widely different. For example, just above2G0 we find apoint with x , 5%, while the next point hasx . 20%.Hence, the scatter in the data compared to any ofcurves is not due to statistical errors but is a resultthe intrinsic variation in the properties of the contact.It should be stressed that the results described abare obtained in a gold QPC. We performed similexperiments on aluminum, which showed much weaksuppression of shot noise. For Al contacts between0.8G0and 2.5G0 the obtained shot noise values vary from 0to 0.6s2eId. These values fit the theory only if oneassumes contributions from a much wider set of modIn particular, the conductance atG ­ G0 is not carried byone fully transmitted mode but by at least two partialtransmitted modes instead.In model simulations for gold, Brandbygeet al. [22]find nearly full transmission of the channels forG near 1and3G0. However, they report two half-opened channearoundG ­ 2G0, which is at variance with the presenexperimental results.For atomic-size contacts of superconductors, Schet al. [19,23] performed experiments on current-voltagcharacteristics in the superconducting state. This wused to demonstrate that the number of conducting chnels in a single atom contact is determined by the nuber of valence orbitals [23,24]. However, reservatiowere made about the results on gold, since the proxiity effect was used to induce superconductivity in a goVOLUME 82, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 15 FEBRUARY 1999..,ys.QPC, causing a modified quasiparticle density of statOur results forG , G0 show unambiguously that, in themonovalent metal gold, the current through a single atocontact is indeed almost exclusively carried by one singconductance channel. Our results for a single atom cotact of aluminum (which has a valency of three) confirmthat several modes are transmitted [19,23,24]. In adtion, for gold we find that the conductance for larger cotacts is well-described by a set of conductance channwhere all are fully open, except one, which carries thfraction of conductance above the integer value. Thproperty was also inferred from measurements of condutance fluctuations [25] and thermopower [26] by Ludopet al., and was referred to as “saturation of channel tranmission.” From Fig. 4, we can even quantify the accurato which this rule is obeyed. Where the interpretationRefs. [25,26] depends on ensemble averaged propertthe present results show that saturation of channel tramission is observed forindividual contacts, and is inde-pendent of any adjustable parameters.From conductance histograms, the quantum condtance properties cannot be identified as unambiguouslymetals [18] as they are in 2DEG contacts. In gold QPCthe conductance does not show the pronounced preferefor integer multiples ofG0 [16,27] as would be expectedfor truly quantized conductance. In contrast, the quantusuppression of shot noise observed here is even more pnounced than in 2DEG systems [2,3]. By this method tquantum nature of the conductance in atom-size metapoint contacts is now straightforwardly revealed.This work is part of the research program of the Stiching voor Fundamenteel Onderzoek der Materie (FOMwhich is financially supported by NWO. We thank D. CGlattli, J. I. Dijkhuis, and M. G. Peters for helpful discussions and L. J. de Jongh for his stimulating interest in thwork.[1] W. Schottky, Ann. Phys. (Leipzig)57, 541 (1918).es.mlen-di-n-els,eisc-hs-cyofies,ns-uc-ins,ncemro-hellict-),.-is[2] M. Reznikovet al., Phys. Rev. Lett.75, 3340 (1995).[3] A. Kumar et al., Phys. Rev. Lett.76, 2778 (1996).[4] G. B. Lesovik, Sov. Phys. JETP Lett.49, 592 (1989).[5] M. Büttiker, Phys. Rev. Lett.65, 2901 (1990).[6] C. W. J. Beenakker and H. Van Houten, Phys. Rev. B43,12 066 (1991).[7] Th. Martin and R. Landauer, Phys. Rev. B45, 1742(1992).[8] M. Büttiker, Phys. Rev. B46, 12 485 (1992).[9] C. W. J. Beenakker and M. Büttiker, Phys. Rev. B46, 1889(1992); M. J. M. de Jong and C. W. J. Beenakker, PhysRev. B 46, 13 400 (1992); K. E. Nagaev, Phys. Lett. A169, 103 (1992).[10] F. Liefrink et al., Phys. Rev. B49, 14 066 (1994).[11] A. H. Steinbach, J. M. Martinis, and M. H. Devoret, Phys.Rev. Lett.76, 3806 (1996).[12] L. Saminadayaret al., Phys. Rev. Lett.79, 2526 (1997).[13] B. J. van Weeset al., Phys. Rev. Lett.60, 848 (1988);D. A. Wharamet al., J. Phys. C21, L209 (1988).[14] G. Rubio, N. Agraı¨t, and S. Vieira, Phys. Rev. Lett.76,2302 (1996).[15] H. E. van den Brom, A. I. Yanson, and J. M. van Ruiten-beek, Physica (Amsterdam)252B, 69 (1998).[16] M. Brandbygeet al., Phys. Rev. B52, 8499 (1995).[17] J. M. Kranset al., Nature (London)375, 767 (1995).[18] A. I. Yanson and J. M. van Ruitenbeek, Phys. Rev. Lett79, 2157 (1997).[19] E. Scheeret al., Phys. Rev. Lett.78, 3535 (1997).[20] C. J. Muller, J. M. van Ruitenbeek, and L. J. de JonghPhysica (Amsterdam)191C, 485 (1992); Phys. Rev. Lett.69, 140 (1992).[21] The relation between current and voltage noise is simplPV ­ PI yG2.[22] M. Brandbyge, M. R. Sørensen, and K. W. Jacobsen, PhyRev. B56, 14 956 (1997).[23] E. Scheeret al., Nature (London)394, 154 (1998).[24] J. C. Cuevas, A. Levy Yeyati, and A. Martı´n-Rodero, Phys.Rev. Lett.80, 1066 (1998).[25] B. Ludoph, D. Esteve, C. Urbina, M. H. Devoret, andJ. M. van Ruitenbeek, following Letter, Phys. Rev. Lett.82, 1530 (1999).[26] B. Ludoph and J. M. van Ruitenbeek (to be published).[27] Z. Gai et al., Phys. Rev. B53, 1042 (1996).1529

Quantum suppression of shot noise in atom-size metallic contacts

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Quantum suppression of shot noise in atom-size metallic contacts

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