Ann. Physik 2 (1993) 591-601 Annalen der Physik 0 Johann Ambrosius Barth 1993 Discrete resistance fluctuations in pressure-type point contacts 3. A. Kokkedee, C Thier, and A. G. M. Jansen Max-Planck-lnstitut fifr FestkOrperforschung, HochfeId-Magnetlabor, B.P. 166, F-38042 Grenoble Cedex 9, France Received 26 April 1993, revised version 14 July 1993, accepted 17 July 1993 Abstract. In high ohmic pressure-type metallic point contacts (resistance range 50 Q to 3 kQ) the point-contact resistance is observed to switch randomly between two or more discrete levels. This effect can be explained by the motion or reorientation of single defects, thereby changing their cross section for electron scattering. From the temperature- and voltage-dependence of the characteristic times of the fluctuations, electromigrationparameters for a defect in silver are extracted. Keywords: Metallic point-contacts; l/f noise; Electromigration. 1 Introduction Electrical noise in solid state systems can have different origins due to random processes in electron transport. There is Johnson noise, which is voltage noise induced by thermal fluctuations of the velocity of the electrons. The open-circuit Johnson noise generated by a-resistor R at temperature T is given by Vnoise= ( ~ I ? ~ T R B(with ) ” ~ B the experimental bandwidth), obeys a Gaussian amplitude distribution and has a flat power spectrum (“white noise”). Another type of noise in a solid state system is the so-called shot noise, which is current noise and finds its origin in the discrete nature of an electrical current. The finiteness of the charge quantum e results in statistical fluctuations of a dc current Idc, given by I2Agt = (2eIdcB)”2.This type of noise, like the Johnson noise, is Gaussian and has a white spectrum. A third and most puzzling type of noise in a small conductor is the flicker or I/$ noise. Nearly all kind of resistors (metals, semi-metals, semiconductors, tunneljunctions) exhibit voltage fluctuations with a power spectrum inversely proportional with the frequency$ There is evidence that these fluctuations AVare a result of fluctuations AR in the resistance, since voltage fluctuations are not observed in the absence of a driving current I,, through a sample and the spectral density of the observed voltage fluctuations increases with the square of the current (i.e. AV = I A R ) . Although there is still a great deal of controversy around the fundamental origins of l/fnoise, substantial new insight in the l/fnoise problem has been obtained through the study of noise properties of small area devices. The resistance of different types of extremely small devices (i.e. Si-inversion layers, MOS structures, point contacts) at low temperatures is found to switch between two or more discrete levels (random telegraph signal). The power spectrum of these resistance fluctuations has a Lorentzian shape. At high temperatures or in the large scale variant of the same device, noise with a l/fspectrum is observed. Ann. Physik 2 (1993) 592 These observations lead to the idea that the l/fnoise in small solid state structures originates from a superposition of Lorentzians, where each Lorentzian corresponds to an elementary noise source, which is observable only in very small structures as a random telegraph signal (McWorther [l] and Dutta et al. [2]). For a complete review on l/f noise we refer to Weissman [3]. Resistance fluctuations in the form of a.random telegraph signal were observed in different semiconductor devices (Si-inversionlayers (Ralls et al. [4]),MOS-structures (Uren et al. [5] and Farmer et al. [6]),quantum point contacts (Dekker et al. [7])) and are caused by the trapping of carriers into states in the oxide-semiconductor interface leading to small changes in the electron density. In normal metal tunnel junctions (NbNb20S-PbBi tunnel junctions, Rogers et al. [S]) the discrete resistance switches can be attributed to the charging and the discharging of single localized electronic trap states in the barrier material. Recently, two level fluctuators (TLF’s) were observed in ballistic metallic point contacts (Ralls et al. [9] and Holweg et al. [lo]). This random switching between discrete resistance levels is caused by a defect jumping back and forth in a double well potential. The point contacts used in these experiments were made by micro fabrication technique and were extremely stable This fabrication technique of pointcontact devices involves the evaporation of metals on an amorphous membrane. The resistance of these so-called nanobridges is often quite low, which hinders the observation of TLF’s. In this paper we discuss an attempt to observe TLF’s in pressure-type point contacts. Pressure-type point contacts are in general not as stable as the point contacts made by micro fabrication techniques, but can be made very high ohmic and of a large number of different materials (metals, semi-metals, inter-metallic compounds, etc.). After a discussion of the basic ideas around two level fluctuators as a result of defect motion, we describe the experimental setup for measuring these TLF’s in pressure-type point contacts and present the experimental results. 2 Two-level resistance fluctuations The electrical conductivity of disordered metals is reduced, because of the scattering of the conduction electrons with the impurities or defects in the material. In the simple classical Drude model the multiple scattering leads to a total resistance R which is constant in time, and varies monotonically as a function of an applied voltage t: magnetic field B or temperature 1: In a quantum theory of interfering electron waves diffusing through a medium with fixed scatterers, fluctuations with an universal amplitude 6G = e 2 / h were predicted [ll], and observed in the conductance G(B)of a small sample as a function of the magnetic field B [12]. However these fluctuations are independent of time and are completely reversible in the magnetic field B. When the defects are not fixed at their positions, but vibrate between meta-stable positions or are completely mobile they can modulate the resistance as a function of time. The dynamics of .the defects at low temperatures can be described by jumping back and forth in a double-well potential leading to a thermally activated behaviour of the mean time ri spent in one of the wells: Ti= To,i exp ($) ( i = 1,2) . J. A. Kokkedee et al., Resistance fluctuations in point contacts 593 Here ro,iis the attempt time and ei the thermal activation energy needed to make a transition. The motion in the double-well potential is associated with a spatial displacement or a reorientation of the defect between two positions or orientations. Equivalently, the scattering cross section r~ of the defect jumps in between two discrete values with a transition rate '5; (Eq. (I)), leading to discrete jumps in the resistance. According to Machlup [I31 the autocorrelation function of a signal that switches randomly between two levels, is exponentialy leading to a Lorentzian power spectrum given by: S(w)= SrJ T l+02r2' with S0/4 the integrated power and r-' the transition rate. In the case of a resistance change AR between two levels, these quantities are: So=4AR2- r and T-'=T;*+T;' , (3) T1 +r2 where 71 and r2 are the mean times spent in the two levels. In a large conductor, the discrete resistance changes corresponding with the cross section changes of the individual defects lead to a total noise signal. In this case it is possible to build a l/fpower spectrum out of many Lorentzians with different characteristic times T . Such a model to explain l/f noise in metals as a superposition of Lorentzians was proposed by McWorther [I] and later generalized for random activated processes (like defect motion) PI. When a conductor gets sufficiently small it is possible to resolve the resistance fluctuations due to the motion of individual defects. A suitable system for these kind of experiments is a point contact, e.g. a small constriction with radius a dividing two metallic electrodes. An applied voltage V across a point contact drops nearly completely over a distance of the order 2a, and hence defines the length scale of the system (Ymson [I41 and Jansen [15]). The resistance RP of a point contact can be written as an interpolation between the Maxwell resistance RM= p / 2 a and the Sharvin resistance Rs = 4p 1/3 IC a2: (4) where 1 is the electronic mean free path, p is the resistivity and T ( a / l ) a slow varying function between 0.694 ( l e a ) and 1 (Is-a).When l > a the first resistivity-independent term (since p oc 1 / l ) in Eq. (4) dominates the resistance Rp.In this regime, an applied voltage Vacross the constriction accelerates the electrons to a well defined excess energy e r/: and at low temperatures the derivative dR/dV of the resistance with respect to the voltage is directly proportional with a point-contact variant of the Eliashberg function a 2F(eV) for the electron-phonon interaction, an experimental technique known as point-contact spectroscopy [1 51. A simple straightforward way of realising a point contact is by moving a sharply etched metallic needle in a flat surface or by pressing together two wedge-shaped pieces of metal. In this way the electron-phonon interaction was studied in nearly all metals [I61 594 Ann. Physik 2 (1993) and in a large number of intermetallic compounds [17]. The advantage of the pressuretype method is the possibility to use a large number of materials and the option to control the resistance of the contact, by changing the pressure between the electrodes. The diameter d in real experimental situations can be made as small as d = 50 A.The disadvantage is the noise inherent to the mechanical instability which limits the experimental sensitivity to A R / R = In addition, the lifetime of pressure-type point contacts is in general small. A more sophisticated method to produce a point contact was applied by Ralls [18] and Holweg [19] in a study of noise in ballistic constrictions. They used electron-beam lithography to pattern a small hole in a suspended membrane, and evaporated metal onto both sides of the membrane. In this way very stable devices operating in the ballistic limit were obtained. The micro-fabricated point-contacts devices were small and stable enough to observe two-level fluctuators in the resistance in a temperature range from T = 4.2- 300 K. For low applied bias voltage across the constriction no bias dependence was found in the TLF's and the average times spent in the high and low resistance states were well described by Eq. (1) in the temperature range 20< T < 150 K [9]. The attempt times ro and activation energies E for different TLF's in different devices were found to vary from so lo-" - lo-'* s and E 30-400meV [9, 101. With Eq. (4) the change in scattering-cross~ resistance change AR/R section ha of a defect can be approximated from t h relative of the observed TLF yielding in most cases A a 10 A2, supporting the idea of individual atomic-sized defect motion as the origin for the noise At high temperatures (T> 150 K) many defect fluctuators are active simultaneously resulting in a l/f shape in the power spectrum of the noise signal. After thermal cycling to room temperature, in general completely different TLF's with different activation energies and attempt times are found, indicating that the defects are rearranged in the 'defect potential after thermal diffusion. Keeping the temperature fixed but raising the bias, has a similar effect as increasing the temperature [lo,201. At a certain bias a TLF appears, which increases rapidly its transition rate 7 - * with increasing bias, and eventually the fluctuator leaves the experimental bandwidth, whereafter a new TLF (with different attempt times and activation energies E , , ~ ) appears. The high current density in a point contact (up to 10'' A/m2) results in an electron-wind force on the defect, which is one of the driving forces for electromigration. When the scattering of the ballistic electrons with the defect is inelastic, the defect heats up to a temperature TD above the lattice temperature. To account for these effects, the switching rates r:J1,2 as a function of applied bias for a specific TLF are described by a slightly modified version of Eq. (1) [20]: - - ( i = 1,2) . (5) Here Ciis the electromigration parameter and TD the temperature of the defect, which depends on the amount of inelastic electron-defect scattering and hence on the applied voltage across the constriction. By fitting the observed mean times 7; as a function of the voltage V to Eq. (5), electromigration parameters for Cu, A1 and Pd could be extracted [20]. Similar parameters were obtained in Au nanobridges [lo]. 3 Experiment So far, the dynamics of two-level fluctuators in metals were mainly studied in ballistic nano-bridges, which are very stable but so-far restricted to evaporable metals. Here, we J.A. Kokkedee et al., Resistance fluctuations in point contacts 595 report on the observation of two-level fluctuators in pressure-type point contacts, made by pressing a very sharply etched metallic needle in a thin evaporated metallic film. The rather simple realization of pressure-type point contacts opens the investigation of noise phenomena in small systems to more complicated systems (alloys, compounds). As a first test we used Ag and Cu point contacts. To obtain a needle we electrochemically etched Ag wires in a HN0,-solution and Cu wires in a H3P04-solution to a sharp tip. Thin evaporated Ag and Cu films (thickness 3000 A) served as counter electrodes. The samples were placed in the vacuum space of an insert with a differentialscrew approaching system, which was placed in a 4He cryostat. By filling the vacuum around the insert with He-gas at low pressure, thermal contact with the He-bath was achieved. With a heater we could raise the temperature up to T = 20K. To measure the point-contact resistance RP as a function of time t we used a low noise dc circuit. A dc current, coming from a battery with a series resistor R,$R, was passed through the point contact. The voltage across the contact was amplified by a low noise LT 1028 amplifier, which was necessary to obtain a proper impedance matching with next stage PAR113 differential amplifier. The amplified signal was sent to a AD 3525 FFT-analyzer, which measured at a maximum frequency of 200 kHz. The experimental bandwidth ranged form 1 Hz to 10 kHz. If possible, we tried to measure the spectrum d 2 Vd12(V)of the point contacts. For this we applied standard phase-sensitive lock-in techniques. - 4 Experimental results With the above described pressure-type technique we were able to produce point contacts in the resistance range 50 !2 <R < 3 kQ, high enough (with small enough diameter d ) to observe the two-level fluctuators. In Fig. 1 we plotted characteristic resistance-versus-time traces of point contacts with different applied voltages. The relative amplitude For low bias, the low frequency of the observed TLF’s was A R / R - 5 x lod4behaviour of the point-contact resistance is dominated by fluctuations between two distinct levels (Fig. 1(a)). This is the simplest type of noise we have observed. For higher bias often more than two levels are visible, interpreted as more than one TLF active simultaneously. Fig. 1@) shows a resistance-time trace in which three discrete levels are visible. The trace can be divided into two random telegraph signals with small and large characteristic time ‘c. Apparently two TLF’s corresponding to two defects with different activation energies E and attempt times ‘co are active. The observation of three resistance levels instead of four leads to the conclusion that the change in scattering cross section ha of the two defects is alike. In Fig. 1 (c) four distinct levels (more often observed than three levels) are visible pointing to two TLF’s as a result of the positional change or reorientation of two independent defects with a different ha. Interactions between two defects, where the motion of one defect in a double-well potential modifies the double-well potential of another defect, can also result in three or more levels [91. Finally, we plotted in Fig. 1 (d) one of the rare observations of more than four levels in a resistance-time trace. No less than six different resistance levels can be resolved, indicating that at least three defects are changing their cross section. In Fig. 2 we plotted resistance-time traces of a point contact for three increasing applied voltages. The mean resistance of this point contact was R = 860 SZ, and remained more or less constant in the applied bias range. The transition rates rl;i between the two levels increase with increasing bias, and finally the TLF is no more observable Ann. Physik 2 (1993) 596 .. (b) . - V=78mV Time Fig. 1 Resistance versus time for AgAg point contacts for different applied biases at T = 4.2 K. The relative amplitude AR/R lo-' The starting resistance of this point contact was R = 100 a at low bias. For voltages smaller than V = 40 mV only two-level fluctuators (a) are visible. With increasing bias more TLF's become active simultaneously. In trace (d) no less than 6 discrete resistance levels can be observed. The time scale of the traces can be anywhere between and 10-l~; this mean time of a high and low resistance state depends strongly on the bias. The mean resistance of the point contact was not constant, but changed considerably with increasing bias as a result of the instability of the contact. - 4.0 vr 0.0. -4.0; - I I V=43mV I 0 . . 10 . . 20 30 40 50 60 70 80 t [ms1 Fig. 2 Resistance versus time for a Ag point contact with R = 860 voltages across the contact. at T = 4.2 K, for three increasing 597 J.A. Kokkedee et al., Resistance fluctuations in point contacts -’ (T > 10 kHz). Note that the duty cycle, defined as T,/(T,+ rz),is not a constant but changes from about 30% at V = 33 mV to about 95% at Y = 43 mV. A further increase in bias results in the observation of a new TLF with different characteristic times 51,2 and amplitude A R / R . The change in effective cross section can be approximated from the relative resistance change A R / R between the two levels using Eq. (4). Assuming that only one single defect is placed in a clean constriction, the backscattering of electrons on this impurity results in small changes in the Sharvin resistance of a clean contact. A small change Aha in scattering cross section 0 of a single impurity is similar to a change AS in contact area S, leading to: This expression serves as a lower limit on Aha for two reasons. In the first place, the defect is assumed to be in the center of the constriction. A defect further away from the center of the constriction is less effective on the resistance and hence appears to have a smaller cross section. In the second place, the area of the point contact is determined by the Sharvin formula, which is only correct in the absolute ballistic limit. When the transport is in the intermediate Wexler regime, the contact area S is larger than 4p1/3Rp. In somewhat more elaborate calculations a similar result is found [lo,211. The relative resistance change A R / R of a number of point contacts as a function of the contact resistance R is plotted in Fig. 3. With Eq. (6) we find for these contacts cross section changes in the range 0.5 A a < 5 A’,in agreement with observations in Au nanobridges [lo] but smaller than observed in Cu, A1 and Pd [20]. A’< o.10 0.010 f / I Fig. 3 The relative resistance change A R / R as a function of the contact resistance R for different two- level fluctuators. The straight line corresponds to a conductance change AG = e2/h. 100 1000 R [a1 Although we were able to produce very high ohmic point contacts (up to 3000 a),the noise experiments were strongly disturbed by sudden changes of the mean resistance of the point contact. The rather low ohmic point contacts turned out to be the most stable but corresponded to small resistance changes, while the high ohmic point contacts (which were often very unstable) showed the largest TLF’s (Fig. 3). In the same figure, the straight line corresponds to the change in resistance A R = -R 2AG for a change in 598 Ann. Physik 2 (1993) conductance AG = e2/h. The value e 2 / h equals the conductance of one channel for the electron transport in the framework of the transmission approach for ballistic transport [l I]. One can argue that the unit of conductance e 2 / h determines the upper limit of the change in resistance due to the introduction of an impurity in the center of a ballistic contact [21]. The observed changes in resistance of a TLF were always found to be smaller. In the very high-ohmic point contacts (R> 500 SZ) it is practically impossible to measure a proper point-contact spectrum for the electron-phonon interaction, and hence we cannot claim with certainty that the transport in these point contacts is ballistic In Fig. 4 we plotted a d 2Vd12 spectrum of a R = 190 SZ point contact, together with a R = 28 SZ spectrum. Both spectra show clearly structure at voltages corresponding to the transverse (at 12 mV) and longitudinal (at 20 mV) phonon frequencies of bulk Ag, as usually observed in point-contact spectroscopy and interpreted in a direct measurement of the Eliasbergh function for the electron-phonon interaction [15, 161. The point-contact spectra of high-ohmic contacts are mostly more broadened. The 190 SZ contact in Fig. 4 shows an anomalous, irreproducible structure around the longitudinal phonon energies. The appearance of phonon-related structures in the two spectra leads to the conclusion that the ballistic transport in these high-ohmic point contacts is a reasonable assumption. Because of the high mechanical instability of the point contacts it turned out to be practically impossible to study the voltage and temperature dynamics of the observed TLF’s. Only one point contact appeared to be stable enough to measure the average times of the two resistance states for four different temperatures and two applied voltages. These characteristic times were obtained by fitting the power spectrum to the Lorentzian, given by Eq. (2) and (3). The results’areplotted in Fig. 5, and are reasonable described by Eq. (9,if we use for the defect temperature To = T+5e I V1/16kB [lo]. For the quantitative description of the characteristic times given in Fig. 5 we used the electromigration parameters T~~~~ = s, eUp= 210 meV, Cup = -5.0 meV/mV and q-,down = s, &down = 200 mev, = -3.7 meV/mV. Although these vaiues are of the same order of magnitude as observed in Cu, A1 and Pd [20], and in Au point contacts [lo](except for the larger values of [), more data extending over a larger temperature and voltage interval are necessary for a more reliable determination of the electromigration parameters. cdown 0 10 20 Voltage [mv] Fig. 4 The d 2VdZz(V) spectra for a high (R= 190 Q) and a low (R= 28 n) ohmic Ag point contact showing the structure of the Eliashberg function a2F of the electronphonon interaction. J.A. Kokkedee et al., Resistance fluctuations in point contacts 599 Fig. 5 The temperature dependence of the ~ a TLF for two characteristic times T , ,of different applied voltages. The circles correspond to down states, the squares to up states. Open symbols correspond to V = 46 mV, closed symbols to V = 58 mV. The dashed curves give a description of the data according to Eq. (5). The mean resistance of this Ag point contact was R=8051. (b) V = 3 6 m V .r( ul aJ p: (d) V = 6 5 m V Time - Fig. 6 Resistance-time traces for two Ag point contacts (R 100 51) with applied voltage. Each shot is about 100 ms long. The noise characteristics at such high bias are constantly changing in time. Transitions to different resistance levels (from four to three levels (a)), or the sudden appearance or disappearance of a TLF (b) are observed. The average resistance remains constant, but starts to change for voltages larger than V = 150 mV. For very high applied voltages V = 80 mV (this varies from contact to contact) across the constriction the noise characteristics are in general changing in time, indicating that interactions between the defects start to govern the noise properties. Two examples are given in Fig. 6, where we plotted resistance-time traces of two different point contacts 600 Ann. Physik 2 (1993) with high applied bias. The average resistance of these point contacts remained more or less constant, but the noise characteristics changed continuously in time. Apparently the motion or reorientation of one defect influences the dynamics of an other nearby defect. Similar indications for interactions between defects were previously reported in point contacts at T = 300 K [lo, 201. They suggest, that this long range interaction occurs through the defect strain field, and they argue that the interaction is strong enough to dominate the defect dynamics of a few defects in a small volume (na2= 30 A2), resulting in a l/fpower spectrum at high temperatures or voltages. The range of the interaction is at least of the same order as the distance between the interacting objects. The mean distance between defects can be estimated from the elastic mean free path I,= 1000 A and the scattering cross section 0 - 5 A2, via Limp-imp = (1e0)”3 - 20 A. Higher voltages (V> 100 mV) across the constriction result in general in a change (increase or decrease) of the average point-contact resistance, indicating that the defects start to move irreversibly to or away from the constriction. The lifetime of the pressure-type point contacts at such high voltages was in general too small to study these electromigration effects. 5 Conclusions We have studied noise properties of metallic Ag-Ag point contacts, fabricated by the pressure-type technique. The resistance of high ohmic point contacts, corresponding to point contacts with a very small diameter, switches randomly between two or more discrete resistance levels, as a result of the motion or reorientation of single defects. The change in resistance is smaller than the corresponding e2/h-change in conductance, the contribution to the conductance of one channel in the transmission picture of ballistic transport of electrons. A higher bias voltage results in general in reversible motion of a few interacting defects, as manifested by the observation of more than two resistance levels and in changing noise characteristics emerging in the well-known l/fpower spectrum. References [l] A. L. McWorther, in: Semiconductor Surface Physics, R. H. Kingston (ed.), University of Pennsylvania, Philadelphia 1957 [2] P. Dutta, P. Dimon, P.M. Horn, Phys. Rev. Lett. 43 (1979)646 131 M.B. Weissman, Rev. Mod. Phys. 60 (1988) 537 [4] K. S. Ralls, W. J. Skocpol, L. D. Jackel, R.E. Howard, L. A. Fetter, R. W. Epworth, D. M. Tennant, Phys. Rev. Lett. 52 (1984)228 [5] M. J. Uren, D. J. Day, M. J. Kirton, Appl. Phys. Lett. 47 (1987) 1195 [6] K.R. Farmer, C.T. Rogers, R.A. Buhrman, Phys. Rev. Lett. 58 (1987)2255 [7] C. Dekker, A. J. Scholten, F. Liefrink, R. Eppenga, H. van Houten, C.T. Foxon, Phys. Rev. Lett. 66 (1991)2148 [8] C.T. Rogers, R.A. Buhrman, Phys. Rev. Lett. 53 (1984)1272 [9] K. S. Ralls, R. A. Buhrman, Phys. Rev. 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