Magnetoconductance fluctuations in ballistic silver point-contacts

Superlattices and Microstructures,

MAGNETOCONDUCTANCE P.A.M.

Vol. 17, No. 2, 1992

229

FLUCTUATIONS IN BALLISTIC SILVER POINT-CONTACTS

Holweg(l), J.A. Kokkedee(*), J. Care(l), A.H. Verbruggencl), S. Radelaar(l), A.G.M. Jansen(2) and P. Wyder@)

(I) Delft Instituteof Microelectronics and Submicron Technology

Delft Universityof Technology, Lorentzweg I 2628 CJ De@, The Netherlands (2) Hochfeld-Magnetlabor, Max-Planck-InstitutfiirFestkiirperforschung 166X, F-38042 Grenoble CEDEX, France (Received 19 May 1991)

The magnetoconductance of three-dimensional ballistic silver point-contacts shows reproducible fluctuations. The zero bias rms amplitude 6G of the fluctuations for an 11 R Ag point-contact is 1.8x10-2 e2/h. The amplitude of the fluctuations is found to be independent of temperature from 0.1 K to 5 K and for bias voltages up to 10 mV. An interpretation of the measured magnetic and voltage cornelation lengths in terms of quantum interference of electrons shows that the length scale of the interference loops is the elastic electron mean free nath instead of the much smaller constriction size of the point-contact.

1. Introduction

2. Experimental Results

A point-contact between two metals is a “classic” system in the physics of nanostructuresl. The effective sample dimension of such a three-dimensional contact is the size of the contact diameter 2a (tvpicallv 5-20 nm), since an __ applied voltage drops almost completely across.the contact region. At low temperatures, both the elastic and inelastic electronic mean free path in pure point-contacts are much larger than 2~. In passing the contact ballistically, electrons gain an energy eV which can be as large as lOOmeV, enabling studies of the energy dependence of the transport (point-contact spectroscopy). So far, point-contact spectroscopy was mainly applied to study the (in principle sample independent) electron-phonon interaction. Recently, point-contacts gained renewed interest as a result of the development of a fabrication technique2 which produces devices of high stabilitv. enabling studies of reuroducible I sample depenhent (me&scopic) phenonena such as twolevel resistance fluctuations caused by the reversible motion of a single defect3. In this paper we report on the first observation of mesoscopic magnetoconductance fluctuations in ballistic metallic point-conatcts. Fabrication of our point-contacts starts by patterning a single hole (diameter 5-20 nm) in a 20 nm thick silicon nitride membrane using e-beam lithograuhv and drv etching. Then, by evaporating under Ul?V’conditions a 200 nm metal layer onto both sides of the membrane, the hole is filled with metal and a point-contact is formed between the two metal layers. In the experiments the differential conductance aI/& was measured usinz a low frequency ac bridge with an excitation voltage of 6.3 mV rms. The point-contact spectrum -c&/&2 was measured with a lock-in technique. The measurements were performed in a Kelvinox dilution refrigerator equipped with a 14 T superconducting solenoid.

Figure 1 shows magnetoconductance traces at different voltages for an 11 R Ag point-contact, measured at T=400 mK, with the magnetic field oriented parallel to the constriction axis. Reproducible conductance fluctuations, superimposed on a bias dependent background are clearly visible. As the bias voltage is changed in small steps the typical fluctuation pattern in a conductance trace (magnetofingerprint) changes gradually. When the change in bias exceeds the value of about 1 mV (as in Fig. 1) a completely new fingerprint results. With increasing bias voltage, the typical “period” of the fluctuations, characterized by the magnetic correlation length B,, is found to increase from B,=0.07 T to B,=l T between V=OmV and V=12 mV. At zero bias the rms amplitude of the conductance fluctuations is &=1.8x10-2 e2/h. We have found that for zero bias the fluctuations do not depend on temperature from 0.1 K to 5 K. Above 5 K the amplitude decreases. Figure 2 shows the rms amplitude of the fluctuations for a large number of biases, including those of Fig. 1, together with the point-contact spectrum, which is proportional to the point-contact variant a2Fp of the Eliashberg electron-phonon coupling function4. After an initial increase, the amplitude of the fluctuations is roughly constant, whereafter a clear decrease starts at 10 mV, the position of the transverse acoustic (TA) phonon peak.In our nanobridges the conductance is not symmetric around zero voltage (see Fig. 3). The rms amplitude of the fluctuations of the anti-symmetric component [G(V)-G(-V)]l2 increases with voltage up to 1 mV and then remains constant. This observation provides a clear example of the lack of inversion-center symmetry of the impurity configuration in mesoscopic sampless. The conductance fluctuations in our ballistic point-

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Superlattices

and Microstructures,

Vol. 7 1, No. 2, 1992

OmV 0 2.2 4.4 6.6 Voltage (mV) Figure 2. Rms amplitude 6G of the fluctuations as a function of applied voltage (left hand scale) and the pointcontact spectrum (tight hand scale), both of the same Ag point-contact. The dashed curve is calculated from the point-contact spectrum (see text).

14.3 o

1

2

3

4

5

6

7

1.1 mV 0.83 0.55 0.28 0 -0.28 -0.55 -0.83 -1.1

8

Magnetic field (T) Figure 1. Magnetoconductance fluctuations of an 11 R Ag point-contact at T=400 mK for different applied voltages across the contact. For clarity, the curves have been shifted with respect to each other. The trace at zero voltage is shown for magnetic field sweep up and down to demonstrate the reproducibility of the fluctuations.

012345 contacts are in appearance very similar to universal conductance fluctuations6 (UCF). This strongly suggests that also in this case quantum interference of electron waves is the underlying mechanism. The area A of the flux enclosing electron trajectories can be obtained from the magnetic correlation length using B,=(h/e)/A. From the experimental values of B, (0.07 T to l.OT) we find lengths dA ranging from 245 nm to 65 nm. This is rather surprising because at first sight one expects that only electron trajectories in the constriction region (a=5 nm, calculated from Eq. (1) below) will contribute to the interference. The values for dA compare well with the elastic mean free path f,=240nm (determined from the resistance ratio of one of the metal layers and of the pointcontact itself). We interpret this as strong evidence that impurity scattering outside the immediate constriction region plays a crucial role in determining the length-scale of the flux-enclosing paths.

By using a method introduced by Lee7, which focusses on the ensemble averaged transmission probability of an electron, an expression can be derived for the fluctuation amplitude. To this end we relate the interpolation formula of WexlerS for the resistance of a point-contact

7

8

Magnetic field (T) Figure 3. Magnetoconductance traces measured at closely spaced voltages around V=O showing that the traces gradually evolve to a completely new fingerprint. For clarity the curves have been shifted with respect to each other and the background has been subtracted.

Rp=R~+l-R~=R~[l+l-g

37r a &]

(where RS=4di/e2k$a2 is the Sharvin resistance, RM=plZa is the Maxwell resistance and Tis a slowly varying function of 1,/a, with r(O)=1 and r(=~)=9&128), to the generalized Landauer formula for the conductance G G = ya;$@2

3. Amplitude of the Fluctuations

6

.

(2)

where lre$ i. 1s the transmission probability from an incoming channel a to an outgoing channel Band N is the number of channels. The number of channels N=a2k$/4 follows from equating the conductance Gp=l/Rp of a purely ballistic point-contact (i.e. I,== and lro,5i2= @)to , the ensemble average of the conductance given by

Superlattices

and Microstructures,

Vol. 11, No. 2, 1992

Eq. (2). In case of elastic scattering &cm) the ensemble averaged transmission probability is given by ilt,~1*>=[1-~(3~/8)(a/l,)l/N. By transforming transmission to reflection probabilities to include correlations in the transmission coefficients, we find for the T=O rms amplitude of the fluctuations * .-- * (3) In contrast to the well known T=O UCF magnitude e*/h, the amplitude given by Eq. (3) depends on effective sample size and degree of disorder, as expressed by the factor all,. With a=5 nm and le=240 nm Eq. (3) gives &=3.3~10-~ e*/h, in fair agreement with the experimental zero bias result &=1.8x10-* e2/h. 4. Inelastic Processes The decrease of 6G starting at 10 mV (see Fig. 2) can be understood from the strong enhancement of the electronphonon coupling in the energy range of the TA nhonon peak, resulting l’n a reduced Gela& mean free path Ltn. This reduction of Ltn destroys phase coherence of the larger interference loops.. The energy dependent inelastic mean free path Ltn(eV n simply be calculated from4 cz*Ft,(e)de, where VF is the Fermi determmed from the point-contact for the inelastic orocesses. we multiplied Eq. (3) by the factor exp[-le?Lt,,(eV)], the orobabilitv that an electron of excess enerev eV is inelastically scattered due to phonon emission on Ke length scale 1,. The resultine function is nlotted in Fig. 2 fdashed curve); its magnitudk fitted to coincide with The roughly constant level of 6G between 3 and 9 mV. The calculated curve agrees well with the experimental data. 5. Voltage Correlation Length From the conductance traces shown in Fig. 3 we calculated the correlation function in bias voltage, from which we obtained V,=l mV as the voltage correlation length. Remarkably, however, 6G is roughly constant for voltages up to 10 mV (see- Fig. 2). This is in strong contrast with the observed behaviour in diffusive mesoscopic samples, where for voltages exceeding the voltage correlation length V, averaging over Vlv, coherent subsystems occurs, resulting in a decrease of the fluctuation amplitude with voltageg. However, in a ballistic point-contact the impurities are located at distance of order 1, from the constriction region and because the potential drop occurs entirely close to the constriction, the impurity potential involved in the quantum interference effects is not affected by the applied voltage. Bv measuring the conductance differe&lly, only t< energy bands at &cess energy 0 and eV and of width defined bv the modulation voltage, are probed. Increasing the bias voltage merely changes the excess energy of the electrons. Therefore, there is no averaging and the fluctuation amplitude remains constant even for V>>V,. In this sense, with a point-

231

contact one performs a spectroscopy on the quantum interference effects. From the correlation energy eV,=tiDIL*, with V,=l mV and D=O. 1 m*/s, we find L=270 nm, again remarkably close to the value for the elastic mean free path. 6. Conclusions Microfabricated metallic nanobridges operating in the ballistic regime show reproducible aperiodic conductance fluctuations. The fluctuations are well described by quantum interference on a characteristic length scale of the elastic mean free path, the trajectories of the interferine _ . electrons extending beyond the immediate constriction region. The amplitude of the fluctuations is well described by SG=1.6(e*/h)(all,), and is determined by both constriction size and degree of disorder. The spectroscopic measurement of the conductance fluctuations results in a constant 6G, even for voltages larger than the voltage correlation length. Inelastic processes introduced by the voltage controlled strength of the electron-phonon interaction destroy the quantum interference effects. Acknowledgement-We wish to thank H.M. Appelboom of the Solid State Physics Group of Delft University of Technology for assistance with the evaporation. This work is part of the research program of the Stichting Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Grganisatie voor Wetenschappelijk Onderzoek (NWO).

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[Soviet Journal of Law Temperature Physics 9, 343 A.G.M. Jansen, A.P. van Gelder and (1983)]; P. Wyder, Journal of Physics C 13, 6073 (1980). B.L. Al’tschuler and D.E. Khmel’nitskii, Pis’ma v Zhurnal Eksperimental’noi Teoreticheskoi Fiziki 42, 291 (1985) [JETPLetters 42, 359 (1985)]. P.A. Lee and A.D. Stone, Physical Review Letters 55, 1622 (1985). P.A. Lee, Physica A 140, 169 (1986). G. Wexler, Proceedings of the Physical Society 89, 927 (1966). R.A. Webb, S. Washburn and C.P. Umbach, Physical Review B 37 8455 (1988).