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Temperature dependence of electron tunneling Neufeld, Philip David 1966

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TEMPERATURE DEPENDENCE OF ELECTRON TUNNELING by PHILIP DAVID NEUFELD B.Sc, The University of Waterloo, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE'DEGREE OF -MASTER OF SCIENCE in the Department of Physics We accept this thesis as conforming to the required standard ' THE UNIVERSITY OF BRITISH COLUMBIA August, 1966 In presenting this thesis in par t i a l fulfi lment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shal l make i t f reely aval].able for reference and study, I further agree that permission., for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for f inancia l gain shal l not be allowed without my written permission. Department The University of B r i t i sh Columbia Vancouver 8, Canada - i i -Abstract The variation with temperature of electron tunneling through thin-insulating films of A^O^ between aluminum film electrodes was studied at liquid helium temperatures (1.18°K to 4.2°K) and liquid oxygen temperatures (65°K to 90°K). Samples were prepared in a vacuum evaporating system using the method developed by Fisher and Giaever (1961). The aluminum oxide films were grown in air at room temperature. Resistance was measured as a function of voltage by means of a d.c. Wheatstone bridge for sample currents in the region of 10 ^ amps. Conduction due to electron tunneling was indicated by the non-ohmic vol-tage dependence of resistance and by the fact that the resistance increased with a decrease in temperature. The voltage dependence of the tunneling resistance was found to be in good agreement with the theory as derived by Simmons (1963) , l i h d ^ -was parabolic for low voltages. Reliable, and reproducible data for the temperature dependence of resistance were difficult to obtain because of instabilities in the samples and a general increase in resistance due to aging. Comparison of the observed temperature dependence with the theory of Simmons (1964) showed the variation to be approximately twice as great as predicted. - i i i -Table of Contents Chapter Page I Introduction 1 II Theory Notation 4 Tunneling Current at Absolute Zero of Temperature 4 The Low Voltage Limit 7 The Intermediate Voltage Range 7 The Image Force 8 The Tunneling Current.at Finite Temperatures 10 Discussion 12 III Practical Tunneling Junctions Thin Aluminum Oxide Films 13 Growth of: Aluminum Oxide Films 13 Experimental Behaviour of Tunneling Junctions 15 Conclusions 18 IV Experiment Sample Preparation 19 Substrates and Cleaning 21 Evaporating Sources 21 Measurement of Evaporated Film Thickness 22 Oxidation 22 Filamentary Bridges 23 Electrical Leads 24 Measuring Technique 24 The Cryostat 27 V Results:and Discussion First Trial 28 Second Trial 31 Third Trial 33 Analysis of Data 35 VI Conclusions 39 VII Bibliography 41 - iv List of Figures Figure Page 1 Diagram of Potential Barrier 5 2 The Function Y(E ) 5 x 3 Voltage Dependence of Tunneling Resistance 9 4 Preparation of a Tunneling Junction 20 5 Measuring Circuit 25 6 Experimental Results for Sample 1 30 7 Experimental Results for Sample 2 32 8 Experimental Results for Sample 3 34 9 Resistance vs. Square of Voltage for Sample 1 36 - v -Acknowledgments I would like to thank my supervisor, Dr. P. W. Matthews for his advice and guidance throughout this project, and Dr. P. R. Critchlow for reading the thesis and making,valuable suggestions. I also wish to thank: R. Weissbach and G. Brooks for technical assistance in building the apparatus; J. Lees for the glass work; D. L. Johnson and J. D. Jones for helpful discussions; Dr. L. Young and N. Owen of the Electrical Engineering Department for the use of the Veeco 400 evaporator and much invaluable advice and instruction in preparing the samples. Financial assistance in the form of a National Research Council Bursary is gratefully acknowledged. INTRODUCTION When two metal electrodes are separated by an insulator, clas-sical electrons can cross the junction only i f they have enough energy to surmount the potential energy barrier of the insulator and flow in its con-duction band. For a good insulator and normal applied voltages, the con-duction band energy is so much greater than the average energy of the elec-trons in the metal that the current is essentially zero. According to classical physics, an electron of energy E approaching a potential barrier of height greater than E cannot possibly cross the barrier; i t has no choice but to be reflected, However, quantum mechanics predicts that this is not strictly true; the electron has a finite probability of crossing the barrier even i f i t does not have the energy to "climb over the top"ff The electron impinges on one side of the barrier and simply reappears on the other side. This remarkable phenomenon, known appropriately as "tunneling", arises from the wave nature of the electron. Theoretically, the finite probability of tunneling applies not just to electrons but to any object approaching any barrier- There must obviously be some quantitative restriction which ensures that for example, an automobile approaching a brick wall at fifty miles per hour has a truly negligible chance of emerging undamaged on the other side, in agreement with empirical fact. This restriction is imposed by the quantum mechanical transmission coefficient which determines the probability of tunneling and depends exponentially on the height and width of the barrier. If the height of the barrier is much greater than the energy of the electron and the width is more than a few tens of angstroms the tunneling probability - 2 -rapidly becomes infinitesimal. It was this stringent requirement on the width of the insulating barrier that, until fairly recently, made experimental metal - insulator -metal tunneling junctions impossible. The general theory, however, was first worked out in 1933 by Sommerfeld and Bethe, who considered the cases of very low and very high voltages applied across two metal electrodes separated by a thin insulating film. Holm (1951) developed the theory for intermediate voltages. In 1960, Fisher and Giaever (1961) developed a technique for pro-ducing experimental tunneling junctions by evaporating an aluminum film onto a glass substrate, allowing the film to oxidize and then evaporating another layer of aluminum over i t . The aluminum oxide film, being about 50 A thick, was a thin enough potential barrier to allow a measurable tun-neling current for applied; potentials of about 1 volt. This development sparked a great interest in tunneling junction^, both theoretically and experimentally. Simmons (1963) extensively refined the theory, developing an accurate treatment of the shape-of the barrier potential under the influence of image forces, generalizing the results for a barrier of arbitrary shape," and correcting several discrepancies in Holm's theory. He also expressed the tunneling current explicitly in terms of the measurable parameters such as applied voltage, barrier height and oxide film thickness. Experimentally, Morris and Pollack (1964) improved on the sample preparation and current measuring techniques of Fisher and Giaever and verified Simmons' theory,for the current-voltage characteristics of asym-metric junctions (those in which the electrodes are of different metals) - 3 -over a current range of, nine orders of magnitude. Their measurements, taken at 77° K, were in good agreement with Simmons1 theory for 0° K. Soon afterwards, Simmons (1964) extended his:theory to- the-case of finite temperatures and derived expressions for the. temperature,depen-dence of the tunneling current. This work was the basis for the experiment described here: the study of the temperature dependence of the tunneling current for symmetric junctions at low temperatures (1° K to 90° K). - 4 -THEORY The work of Simmons ( 1 9 6 3 , 1964) is the most complete and accu-rate study of the theory of tunneling junctions to date. Like the analyses of Sommerfeld and Bethe ( 1933) and Holm ( 1 9 5 1 ) , i t is based on the WBK approximation of the transmission coefficient of an electron through a potential barrier. The analysis is first carried out for an assumed tem-perature of 0 ° K and is then extended to the case of non-zero temperatures. Notation m = mass of electron r) = Fermi level energy e = charge of electron f(E) = Fermi function h = Planck's constant V = work function of metal electrodes s = insulating film thickness % = height of rectangular barrier s^, = limits of barrier at ^ = mean barrier height Fermi level " € = permittivity of insulating film A s = s^-s^ K = dielectric constant of insulating J = tunneling current density film V = applied voltage R = tunneling resistance Tunneling Current at Absolute Zero of Temperature The probability that an electron whose x component of energy is Ex = % mv will tunnel through the barrier from left to right is given by the transmission coefficient ~ ~hj^ fern) L V(x) ~ E x J CLX j ( 1 ) The number of electrons tunneling from left to right is Hx = J Vx TI (VX) D ( E * ) c t v x o Since Ex = Va. Tf\ Vx* and d.E* = mVxdv* , N = 4 f B r i C v x ) DC Ex) d £ x 1 o -5 -F f R M I " LEVEL FERMI I el NEGATIVE. E L E C T R O D E POSIT/VE E L E C T / R O D E FIG. 1 The potential barrier due to an insulator between two metal electrodes. tone . ^ E ) /( •Ex) .= ^ r ^ - f [ f ( E ) - f ( E + eVJoLE FIG. 2 The function Y(E ) is the integral of the difference between two Fermi functions. - 6 -Em is the maximum energy of the electrons in the electrodes, v is the cor-m 3 responding velocity, and n(vx) is the number of electrons per cm with velocities between v and v + dv , and is given by x x x' ° J 7i(v x) = fem3/h3) f(E) cLvxdvydv* .\ n (Vx) = (2ynYk%) f f f ( E ) JLvydvi eo -co 2 If we substitute v = v + v and Er = r r y z Similarly, the number of electrons tunneling from right to left is = (V/rn/k 3) / D(Ex)JLB» f f(E-heV) dLEr o o If the right-hand electrode is positive^ the Fermi function is written f(E + ev) . The transmission coefficient is the same in both directions. The net number of electrons'crossing from left to right is then J = eN = e(/Vt-N^) - / V E X U E * {(^fmeAl)J[m-HE^V)UEr} (2) Let If E x ) = (Htrme / k B ) J t H E ) - f ( E + e V ) ] c L E r ^irme/h1) j^WE)-f ( t ^eV)J clE -Then J = jer*D(E*)Y(Ex)££x (3) Y(Ejf) is the integral of the difference between two Fermi functions, the shaded area in the.diagram of Fig. 2. l^n-me/h})eV f o r 0<E*O?-eV W E * ) = ^  (T/rweA3)(T]-Ex) f o r 7|-eV< E x * 77 0 f o r . E x > 77 J (4) From Fig. 1 we see that V(x) = n + ^ (x). Then the transmission coefficient becomes -S^ (, £(E*) = exp{-(Wh)f2m)'/;i J L \ + T V X ) - E J } - 7 -This can be written approximately as D f E x ) » expf-zACr) + <?- E x ) / a l (5) where. A = \ h / v^ W"/ and p is a correction factor 1; (See appendix in Simmons (1963)). Then the expression for the current density becomes ^ C,(^^p\rk{yi + ^ ^ , / z ] IBx ( 6 ) (7) If we make some suitable approximations, Eq. (6) integrates to . J = Srrk/jUs)* -{ ?expf-/»^ A)-fP+eV)exp[-/»f? +eV) / j tJ } The Low Voltage Limit (VSS 0) In this limit we simplify Eq. (7) making the approximation that eV « T°, (^ is usually 1 or 2 electron volts) , and also making use of to Va. the fact that A r >^ I. (For typical values of barrier thickness, the value of A *P lies between 10 and 50) . We then obtain T - — - (8) The important point to be-noted here is thatJ is linear in V, i.e. the tunneling junction is ohmic for very low voltages. Eq. (8) can be further simplified by noting that for very small Voltages,: the barrier shown in Fig. 1 is almost rectangular. Then As = s. - s, Pi s and the mean barrier height f can be considered equal to the rectangular barrier height ^. Then (8) becomes ,/ The Intermediate Voltage Range (V — In this range L\S — S - 8 -Also, for voltages below 0.75 the proportionality factor ^ can be set equal to unity with negligible error. Making these substitutions we obtain ] Expanding the exponentials to third order terms we obtain . c = and x = . 2 The junction resistance in ohm . cm is then R = V/J . R = 4rrlrsV;rp (c) f I + i£t * c - f r c * ) V * ( c C a ) I & (c -2) j If c » 2 , then A^e^ 2 s This expression is valid only i f V <. 1.- Typically, this condition is satisfied i f V is less than about 0.3 volts. Then from Eq. (12) i t can be seen that the resistance drops off parabolically from a low voltage limiting value. This behaviour is illustrated in Fig. 3. The Image Force So far we have considered the tunneling barrier to be rectangu-lar. This is assumed in the approximation leading to Eq.:(9) and is implied in the expression for the transmission coeficient (1). Actually, the barrier is slightly rounded by the image potential as indicated in Fig. 1. The image potential is a hyperbolic function which, when inc-luded in the integral of (1), makes i t soluble only by numerical methods. One alternative is to approximate the image potential by a symmetric - 9 -— 1 T 1 % T~ V (VOLTS) FIG. 3 Voltage dependence of tunneling resistance for various barrier thicknesses and a barrier height of 1 eV. From Simmons (1963). - 10 -parabola as has been done by Sommerfeld and Bethe and by Holm. However, this approximation is rather crude and works well only for low voltages, since an applied voltage makes the barrier asymmetric. Simmons has overcome this difficulty by approximating the true image potential by a simpler hyperbolic function which is quite accurate and can also be readily solved. Taking the image potential into account results in a corrected value for the barrier height j° , which is now a function of barrier thickness and the dielectric properties, K and £. , of the insulator. The limits of the barrier at the Fermi level, s^ and s^ are also functions of 6 and ^ o . In the low voltage case, the corrected value of barrier height reduces numerically to This value: of r' would then be used in Eq. (8) to obtain the low voltage current density. Similarly, a corrected expression for applicable to the intermediate voltage range would be used in Eq. (7). Simmons (1964) notes that in practice, especially for low or moderate voltages, the actual barrier is practically rectangular i f K >4. (For A I ^ O ^ J K varies from 8 to 10). Consequently, the image force will be neglected in our analysis. The Tunneling Current: at Finite Temperatures In the foregoing discussion we assumed a temperature of 0°K. The tunneling theory for finite temperatures is formally identical with that set out in the previous section, except that the Fermi functions in Eq. (2) are no longer degenerate. - 11 -In general, the integral of the Fermi function is / f ( E ) c L E = / 1 + e x p ( E - i ? ) / k T = - k T J ^ C l + e x p f 7 ? - E ) / k T j Then /1(E)oiE . = k T U C l + exp(>j-F«)/kT] Then the function Y(E ) of (3) becomes Y(E,) - ( 4/rme / h 3 ) ^ L 7 ( E ) - ^ E + e|/) i f Li + e^PCh-Ex-eVy/cTjJ Using the expression for D(E ) given in (5), we have X e x p f - . / \ ^ + dLE, . ' ( i 3 ) With the help of some simplifying approximations, (13) reduces to JUT) = ^ [ / ^ I k T J '"Xl+°*P(6eV)J <u) where B = k/Zf'/x When T = 0, (14) becomes ^ V ' Q ) = ^n-KfftAs)* * ? e xP ^xp [^ r?+e«f j ] ( i 5 ) ' Comparing (15) with (7), we see that they are identical except that (7) has a factor of ( r 5 + eV) in the last term where (15) has a factor of f 5 . This.discrepancy arises because of.an approximation used in deriving (15) Eq. (7) is the correct form because i t was obtained without resorting.to this approximation, but since eV^ <^ P, the difference is not significant. The temperature dependence of the tunneling current.at a given voltage is found from Eq. (14) : z • r„ r-» (AST) n 7 J(V, T) « J(V, o) |1 + L 3xi0 ip J J (16) 2 Here J is expressed in amps/cm , S in angstroms,. T in degrees Kelvin and T* in. volts. .Eq. (16) is the general result for any barrier shape or voltage - 12 -range. To apply i t to a particular voltage range with or without cor-rection for the image potential, the appropriate expression for J(V, 0) is used. Discussion We have seen that the tunneling resistance at a given tempera-ture is ohmic for very low voltages, and for v <^  *Po/e drops off roughly parabolically with increasing voltage. Similarly, at a fixed voltage, the temperature.dependence of the resistance is also parabolic as can be seen from.Eq. (16). However, the dependence is rather weak: for typical values of barrier thickness As and barrier height T° , the tunneling cur-rent is expected to change by only about 67. between the temperatures of 0°K and 77°K. The resistance is exponentially dependent on barrier thickness and height and furthermore,is strongly affected by the dielectric constant of the insulator, since this influences the image force and hence the shape and effective thickness of the barrier. (See Fig.'s 7 and 9 of Simmons (1963)) . Thus the tunneling current depends not only on the pro-perties of the electrodes but also on those of the insulator. - 13 -PRACTICAL TUNNELING JUNCTIONS Thin Aluminum Oxide Films The electrical properties of thin oxide films are the subject of intensive investigation and are s t i l l not well understood. Quite often samples .are irreproducible in their electrical behaviour and investigators will differ in their observations. The reason for this appears to be that the properties of an oxide film are critically dependent on many factors influencing i t during its preparation. As was mentioned earlier, Fisher and Giaever (1961). pioneered the technique of producing tunneling junctions by allowing a thin film of evaporated aluminum to oxidize and then evaporating another film of alu-minum or other metal, over i t . Obviously such considerations as pressure, temperature and cleanliness of the vacuum system are important but other apparently minor factors, such as the rate of deposition of both films, also affect the properties of the junction. Furthermore, the junctions exhibit a marked aging effect which also varies with the method of preparation of the sample and the condi-tions under which i t is subsequently stored. Growth of Aluminum Oxide Films Hunter and Fowle (1956) have made a study of the growth of Al^O^ films under various conditions. Film thickness was measured by making the sample (a piece of oxidized aluminum) the anode in an electro-lytic bath and noting the potential necessary to produce a normal leak-age current. The applied potential is a direct measure of the oxide thickness. However, this method measures only that thickness of oxide which is electrically insulating and i t is known that in general these - 14 films consist of an insulating "compact" or "barrier" layer covered by a porous, electrically permeable layer. The explanation of this is as follows: If a piece of aluminum is oxidized in dry air or oxygen, the oxide very quickly builds up to a limiting thickness which varies as the temperature,, confirming that the depth of penetration of the oxidation is determined by thermal motion of the metal atoms. This oxide layer is a l l of the compact type. If the sample is now exposed to moist air, the water vapour attacks the surface of the oxide and breaks i t down into a porous layer. As the.barrier is broken down, more of the metal oxidizes until finally, after months or even years, an equilibrium :barrier thickness is reached. This thickness is determined by the temperature.of the environment independently of the temperature at which the-original barrier layer was formed. If the sample is - oxidized in moist air, the oxidizing and hydrating effects go on simul-taneously. The final barrier thickness is the same.-as that formed in dry air at the same temperature but i t is now covered with a porous layer. It is thus: apparent that the maximum possible, barrier thickness is a function of temperature only. When a junction is made by evaporating a second layer of metal several hundred A thick over the newly formed oxide, the deterioration of the barrier oxide by water vapour should be greatly inhibited. This.is confirmed by the fact that the resistance of a metal - oxide - metal sand-wich almost invariably increases with age, whereas one would expect i t to decrease i f the insulating.oxide layer were being decomposed by water vapour. Furthermore, this increase has been observed even when the • sample is stored in an inert atmosphere or a vacuum. The.aging effect is much - 15 -smaller and the characteristics of the sample.are more stable and repro-o o ducible i f i t has been oxidized at temperatures;of 200 C to 350 C. Experimental Behaviour of Tunneling Junctions The original tunneling samples of Fisher and Giaever were made by evaporating a strip of aluminum about 3/64" wide and several hundred angstroms - thick along a glass microscope s l i d e , allowing i t to oxidize, and then evaporating several strips of aluminum of varying widths at right angles-across i t . Electrical contacts were. made, to the ends of the strips and d.c. J-V characteristics were measured with a volt-meter and ammeter at room temperature and at liquid nitrogen temperature. Voltages up to 1.4 volts were applied and currents up to about .10 milliamps were observed. It was found that the junctions were ohmic at very low voltages, the current increasing exponentially at higher potentials and being roughly proportional to the junction area. Samples oxidized at high temperatures (400°G) were found to be slightly rectifying and the preferred direction reversed i f the sample was cooled to 77°K. These samples were also photo-voltaic, devel oping Q. iticixXTnuni open ciir.cui.t. v o l to. ge of 35 mv. Neither of these effects was observed i n samples oxidized at room temperature. The thickness jof • the- oxide film was calculated from the e l e c t r i c a l capacitance -of the junction. Results agreed qualitatively with the theory of Holm but the currents were consistently higher than those predicted, usually cor-responding to the theoretical resistivity of an oxide, film 1/3 of the measured thickness. In the. course of verifying experimentally the improved tunneling theory developed by Simmons (1963), Morris and Pollack (1964) made several important discoveries concerning the preparation of tunneling junctions - 1 6 -and the measurement of their characteristics. Firstly they found that for samples oxidized at 23°C, a signi-ficant portion of the current came from edge emission at the places where the counter-electrode passed over the. edges of the oxidized electrode. This had theeffect of placing in parallel with the. oxide resistance a short-circuiting resistor of about 4 ohms permm.of edge, and was de-tected by the fact that the resistance of the samples-depended on edge length and not on junction area. In order to overcome this difficulty, a film of SiO about o 1000 A thick was evaporated over the edges of the oxidized electrode, leaving a central strip bare for contact with the counter-electrode. These shields also made i t possible to apply higher voltages to the sample without edge break-down. The second discovery was that at room temperature an applied potential of a few volts across the. sample created a field that was more than enough to cause aluminum ion migration through the oxide. This obviously changes the sample and means that the current can no longer be attributed to electron tunneling. However, this ion current required a finite length of time to set in, the time varying inversely with tempera-ture. If the measurements were taken quickly enough, reproducible tun-neling characteristics could be obtained. For currents below 10 ^ amps.at 77°K, the change in current with time was so small that d.c. measurements could be made. For measurements at higher currents, the•sample was placed across one arm of a Wheatstone bridge whose output voltage was directly proportional to the sample current. A triangular input voltage lasting from 2 to 60 seconds was applied and the output was automatically recorded - 1 7 -on.an.X-Y plotter. If two consecutive identical input pulses produced the same output trace i t was-assumed that the conduction was due predominantly to tunneling and not ion migration. At higher currents and temperatures the time required for ion migration to begin was so short that electronic pulse'techniques were-used. The ion migration at room temperature was found to produce hysteresis loops on a J-V graph and this anomalous behaviour could be quenched in by cooling the sample to liquid nitrogen temperature within 20 minutes of taking the measurement at room temperature. Consequently, although the sample is not permanently damaged by inducing;ion migration at room temperature, care should be taken not to apply a d.c. voltage to i t immediately before cooling. The energy-barrier model proposed by Morris and Pollack inc-ludes an n-type semiconducting transition region between the oxide-and the electrode on which i t was grown. This region has a small step-like barrier of about 0.1 eV but since i t is small compared to the oxide barrier height (about .2 eV), its main effect is to reduce the thickness of the-insulating barrier, and most of the voltage applied to the sample appears:across the oxide barrier. The counter-electrode•is known to•pene-o trate the oxide layer to a depth of about 5 A, but the electronic barrier interface is considered to be quite sharp. It was found that even for symmetric junctions in which both electrodes were aluminum, the barrier was trapezoidal rather than rectan-gular with a. difference in height of 0.92•eV between the two metal-oxide interfaces. Morris and Pollack found, as had Fisher and Giaever, that the - 18 -value of oxide thickness - determined by capacitive-measurement was greater by a factor of 1.5 to 2 than the value calculated from the tunneling characteristics. In view of the b u i l t - i n barrier asymmetry of 0.92 eV, the thick-ness measurements of Hunter and Fowle (1956), which were,determined from the applied potential required to produce a certain leakage current, would have to be corrected. This would mean that for a given oxidation tempera-ture, the barrier oxide thickness would be greater than-originally calcu-lated. This corrected value-agread f a i r l y well with the thickness deter-mined capacitively and the large.discrepancy between these values and the thickness calculated from tunneling is s t i l l unexplained. Conclusions From the foregoing discussion we can summarize several important considerations to be kept in mind during the preparation and measurement of tunneling junctions: 1. In.order to produce a compact oxide layer, the sample should be-oxidized in dry air and stored in a dry atmosphere or vacuum. 2. Shields of evaporated SiO should -be-used - to-cover the edges of the oxidized electrode to prevent edge•emission. 3. If.d.c. measuring techniques are to be used, currents must not exceed 10 ^ amps and temperatures should not be-much more than that o of liquid nitrogen (77 K), otherwise aluminum ion migration begins to set i n . 4. Samples can be-tested at room temperature without harming .them•as long as measuring currents are kept low and the sample is l e f t unbiased for a least 20 minutes before cooling to low temperatures. - 19 -EXPERIMENT Sample Preparation Samples were prepared in a Veeco 400 evaporator belonging to the Electrical Engineering department. This versatile system contained six evaporating chambers. The five masks used during the production of the samples were etched photographically from thin shim-stock copper sheets and each one was positioned over a separate evaporating chamber about 8" above the filament. The glass substrate was placed in a holder which could be rotated from outside the vacuum system so as to position the substrate over each of the masks in turn. The clearance between the substrate and the masks was about 1 mm. The steps in the production of samples are illustrated in Fig. 4. a First, nine strips of aluminum about 800 A thick are evaporated onto a 2 x 2" piece of optically flat substrate glass (a). Next, a gold film o o from 400 A to 800 A thick is deposited over the "lands" at the ends of the strips in.order to prevent these areas from oxidizing later, and to pro-vide a good surface for soldering electrical leads (b). Silicon oxide. o shields about 1500 A thick are evaporated over the edges of the strips (c), and the system is opened to the atmosphere to oxidize the aluminum (d). , After a typical oxidation time of 2 hours, the system is evacuated again and the nine counterelectrodes are deposited and their ends covered with a gold film, (e) and ( f ) . The electrodes are of varying 2 widths so as to produce junctions with areas of 1,2, 4 and 8 mm . This method produces nine samples under identical conditions. The samples are then cut.apart with a glass-cutter, and leads are attached. - 20 -FIG. 4 Steps in the preparation of a tunneling junction. - 21 -Substrates and Cleaning The substrates used were special Corning Pyrex 7059 borosilicate •substrate glass 2" square . These plates are optically flat to within o 60 A. The manufacturer recommends only very simple cleaning i f any, since the substrates are supposed to be exceptionally clean even i f taken straight from the package. Small lint particles can be removed with a brush and the glass may be washed in de-ionized water. During the course of the experiment, several cleaning methods were tried. Sometimes the substrate, was simply brushed free, of lint be-fore using, sometimes i t was further cleaned by means of a glow discharge in the vacuum system just prior to evaporation, and on one occasion i t was cleaned by means of ultra-sonic vibration in distilled, water. The dif-ference in cleaning techniques did not seem to have any. marked effect on the properties of the samples. Evaporating Sources The gold and aluminum used (both 99.999?c pure) were, in the form of wire and were wound on straight filaments of stranded tungsten. In a l l cases, the substrate was shielded from the source while the metal was melted, so as to wet the filament and remove any surface contaminants, like grease or aluminum oxide. The substrate was then exposed and the metal evaporated by passing 50 to 70 amps through the filament for 3 min. Usually the metal was completely evaporated within 2. min., signifying an O D evaporation rate of 5 to 7 A per sec. for aluminum and 4 or 5 A per sec. for gold. The silicon oxide was in the form of pellets which were crushed and placed in a tantalum fo i l box with a tungsten filament inside i t . A - 22 -current of 30 amps was passed through the filament for 5 to 10 min., o o evaporating about 1500 A of oxide at a rate of 2.5 to 5 A per sec. -6 -5 The pressure in the system varied from 5 x 10 to 2 x 10 torr during the evaporations, as measured by an ionization gauge situated a few inches, above the substrate. Measurement of Evaporated Film Thickness The thickness of the evaporated films was monitored by means of a quartz crystal with a resonant frequency of about 5 megacycles per sec. The crystal was driven by a transistorized oscillator and was situated just below and to one side of the substrate so as to be exposed to the evaporating source. As the metal film built up on the crystal, its mass increased and its resonant frequency correspondingly decreased. The change in frequency was registered on a Beckman digital display pulse counter. The crystals had been calibrated so that, by means of a simple conversion factor, the change in frequency during evaporation could be translated into film thickness in angstroms. Oxidation According to the study made by Hunter and Fowle (1956) , an oxide-layer formed on high purity aluminum in dry air-at room temperature o reaches an ultimate, thickness of about 10 A in two hours. If allowance is made for the barrier asymmetry as mentioned earlier, this value inc-o reases to about 20 A. In the present experiment, samples were usually oxidized in ordinary air at room temperature, for a period of 1 to 2 hrs., although times as short as 15 min. and as long as 60 hrs. were tried. In the for-mer case the resistance of the junctions was essentially zero, and in the - 23 -latter case i t was many megohms and could not be. measured with our equip-ment. It became apparent that the optimum time for producing resistances 4 5 of 10 to 10 ohms was about 2 hrs. There was, however, an extremely high failure rate. In a batch of nine samples i t was not unusual to find six of them short circuited. At first i t was thought that this was due to dust particles on the subst-rate, which left pin-holes in the oxide film and short circuited the junction. Accordingly, the various cleaning techniques mentioned above were tried, none of which appeared to have any effect on the failure rate. Even among samples that were not shorted, there was often an order of magnitude difference in the resistances of junctions from the same batch.and there did not.seem to be any correlation between resis-tance and junction area. Results were the same for samples oxidized in ordinary air and in dry air. A satisfactory explanation, of these effects is s t i l l lacking. Filamentary Bridges One of the possible reasons for the failure of a sample is that one or more filaments.of aluminum have penetrated the oxide layer through pin-holes or other faults in the structure., forming a conducting bridge, between the electrodes. The presence of these filaments can be ruled out if the samples are strongly non-ohmic, i f the. resistance Increases with decreasing temperature, or i f the resistance is above-a certain minimum. The last criterion is the easiest to check quickly with one room temperature resistance measurement and is fairly reliable. For i f we assume the filament has the resistivity of bulk aluminum, a thickness of o only 10 A, and a length equal to the thickness of the oxide layer, say - 24 50 A, then its resistance is only about 300 ohms. If the filament is o thicker than 10 A, or i f there is more, than one, the resistance will be even smaller. Furthermore, Fisher and Giaever point out that if conduction were due to such small filaments, i t would create unreasonably high local current densities. Consequently, we can assume that a sample is free of filaments i f its resistance is greater than about 300 ohms. Electrical Leads Electrical leads in the form of #37 bare copper wire were •attached to the gold films at the ends of the aluminum electrodes using indium as a solder. If the surface of the gold film is clean, the indium will wet i t quite readily without the use of any kind of flux. Indium also wets clean glass and usually diffuses through the metal films and sticks to the substrate, forming a strong, low resistance joint which is reliable at low temperatures. Since the samples are in the form of a cross with provision for making electrical contact at both ends of each electrode, i t was a simple matter to measure the resistance. •due to the aluminum film, the soldered joint, and the wire leads. This resistance was always about 2 ohms. Measuring Technique The electrical circuit used to measure the sample resistance is shown in Fig. 5. It consists of a Wheatstone bridge powered by an ad-justable d.c. current supply. The voltage, applied to the circuit is measured by means of a Leeds and Northrup potentiometer. Since this .instrument does not measure potentials greater than 75 mV, a precision voltage divider of 990 ohms and 10 ohms is used. The applied voltage is - 25 -FIG. 5 C i r c u i t used for measuring sample resistance. - 26 -then exactly 100 times the potentiometer reading. The resistor of 10^  ohms in series with the bridge limits the bridge current to a maximum of 3 x 10 ^ amps. The output of the bridge is amplified by a Guildline type 9460 photocell galvanometer amplifier, and a Hewlett-Packard model 425A micro-ammeter is used as a null detector. Since the balance of a Wheatstone bridge is most sensitive when all.four bridge resistors are equal, resistors and R^.are adjustable by decades from 10 ohms to 10"* ohms by means of a low resistance double-pole, five-way switch. Thus, although R^  is always equal to R^ , their, value can be selected, after a rough balance is obtained, to be within half an order of magnitude of the sample resistance for values up to 0.5 megohms. The resistance of the leads between the bridge and the sample is cancelled out by means of the arrangement shown, whereby the sample leads are placed in different arms of the bridge so that their resistances balance. One of the problems encountered'in using a d.c. measuring cir-cuit is that thermal emf's affect the bridge balance. The effect of these stray voltages is overcome by use of the current reversal switch. The bridge is adjusted so that reversing the current produces a minimum deflection of the null detector. However, this method of balancing the bridge also masks any asymmetries in the sample i f they should exist; the resistance so deter-mined will be the average of the forward and reverse resistances of the junction. The sample current and voltage can be calculated from a know-ledge of the input voltage as determined by the potentiometer, and the - 27 -values of the bridge and current limiting resistors. The Cryostat Samples were maintained at low temperatures in a conventional 4 He cryostat by direct contact with a bath of liquid He or liquid 0^. The temperature of the bath could be reduced below the normal boiling point of the liquid by pumping on the vapour above the bath with a Stokes pump in the case of liquid He, or a Cenco Megavac pump in the case of liquid 0^ . In this way, liquid He temperatures from 4.2°K to 1.18°K, and liquid temperatures from 90°K to 62°K could be attained. When pumping pure oxygen.with a rotary vacuum pump, there is always the danger that the oxygen and pump oil vapour will form an ex-plosive mixture which can ignite under compression in the pump. There-fore, the ordinary pump oil was drained from the Cenco Megavac pump and replaced with pure tricresyl phosphate. (TCP), which is non-explosive. The vapour pressure above the bath was measured by a mercury or oil manometer and the temperature was determined from vapour pressure vs. temperature tables for liquid helium and oxygen. - 28 -RESULTS AND DISCUSSION As was mentioned earlier, our method of sample preparation re-sulted in a discouragingly high proportion of failures. In a l l , 108 samples were made, of which perhaps half a dozen had resistances in the 3 6 range of 10 to 10 ohms. The high failure rate and non-uniformity of results from one evaporation to the next could have been caused by con-tamination of the vacuum system. This was possible because the evaporator was being used almost continuously by several different experimenters on various projects. However, these same circumstances made i t very dif-ficult to determine or eliminate the sources of contamination if they existed. The samples exhibited an aging effect characterized by a mono-tonic but sometimes erratic increase in resistance, which would often double within two days after formation of the sample. This was not un-expected in view of, the findings of previous workers. However, there were often large and very sudden changes in resistance which could not be easily explained. These and other effects will be described in more detail as we describe three experimental trials in chronological order. First Trial The first sample tested, hereafter referred to.as sample 1, was made by oxidizing the aluminum for 45 min. in air at room temperature. o. o The aluminum films were 650 A and 850 A thick, and the silicon oxide 0 2 shields were 1800 A thick. The junction area was 1 mm . The sample was stored in air and three days later.had a room temperature resistance of 74K ohms for a current of about 10 '''amps. Upon o cooling to liquid nitrogen temperature (77 K), the resistance increased - 29 -to 248K ohms. The sample was then cooled with liquid helium and two pre-liminary measurements were taken at 4.2°K and 2°K for a potential of 0.033 volts. At both temperatures, the resistance was 331K ohms. These measurements are represented as point A in Fig. 6. The voltage dependence of the resistance was not investigated at this time. The next day, the sample was immersed in liquid oxygen and the voltage dependent characteristics were taken at 81°K and 65°K. The sample was kept in liquid oxygen for three days, after which the oxygen was allowed to evaporate and liquid helium was again transferred into the dewar vessel. The value of the resistance at 4.2°K agreed fairly well with the value obtained four days previously at the same temperature and vol-tage (point A), but the 1,18°K measurements, taken about half an hour afterwards, were much higher than expected. According to Eq. (16), the current density, and hence the resistance at a given voltage should Change by only one part in 10^  between 4.2°K and 1.18°K. In view of the fact that no change in resistance was noted between these temperatures four days previously, it. appears that this large increase was due to some anomalous change in the barrier parameters of the sample and is not related to the temperature dependence of the tunneling current. Such large and unexplained increases in the resistance of samples were not uncommon; the resistance of a second sample being tested at the same time as sample 1 increased quite suddenly from .1.58 megohms to many megohms and could no longer be measured with our equipment. Since the superconducting transition temperature for bulk alu-- 30 -' o l d o!o2 o!o3 olo+ o!oS o!o6 1/ (VOLTS) FIG. 6 Experimental resistance vs. voltage characteristics for sample 1 at various temperatures. - 31 -minum is 1.2 K, and for thin films is generally higher, the question arose as to whether the large change in resistance between 4.2°K and 1.18°K was caused by some superconducting phenomenon, this seemed unlikely because one would expect the tunneling resistance to decrease.as the electrodes became superconducting. Furthermore, subsequent trials on other samples failed to show any variation in resistance between 4.2°K and 1.18°K. The anomalous result therefore remains unexplained. It is apparent from Fig. 6 however, that at a l l temperatures the voltage dependence of the resistance is parabolic, as predicted by Eq. (12). It can be seen that the low voltage data are on the threshold of the ohmic region. This makes the results of this trial particularly suited to analysis by Eq. (12) which is a low-voltage approximation. The detailed analysis of.these data will be described in a later section. Second Trial Sample 2 was oxidized at room temperature in atmospheric air for 3 hours and had a room temperature resistance of 182K ohms for a cur-rent of 10 ^  amps one day after formation. This value had increased to 330K ohms two days later. At 77°K, the resistance was 45OK ohms. The resistance vs. voltage characteristics were taken at 4.2°K, 2i4°K and 1.18°K. The curves for these three temperatures coincided and were reproducible over the voltage range, as shown in Fig.. 7. These re-sults strengthened our suspicion that the increase in resistance of sample 1 between 4.2°K and 1.18°K was not a temperature effect. They also indicated that the resistance was not pressure dependent, since the o o pressure varied from 760 mm Hg at 4.2 K to 0.5mm Hg at 1.18 K. Unfor-tunately sample 2 broke down before i t could be tested at higher - 32 -S A M P L E 2. - 33 -temperatures. Third Trial In preparing samples for this t r i a l , two new techniques were used. The substrate was cleaned by ultra-sonic vibration in distilled water for 5 min., and the sample was oxidized for 2 hrs. in dry air. Of the nine samples, three were short circuited, three had resistances too high to measure, and three were in the range of 18K to 39K ohms. The resistance of sample 3 was 21K ohms one day after formation, increased to 160K ohms at 77°K and remained fairly steady at this tempera-ture for 12 hrs. before increasing quite suddenly to 6.35 megohms. This large jump in resistance was almost certainly due to the operation of a Tesla spark coil near the apparatus. It was observed on several other occasions that samples were quite sensitive to such external electric fields. Because our measuring technique was to pass certain pre-selected values of current through the sample, the voltage developed across the junction was determined by its resistance. Consequently, the voltage range in which sample 3 was investigated was roughly an order of magnitude larger than for samples 1 and 2, and was well out of the ohmic region. Characteristic curves were taken,at 1.18 , 2.2 , 4.2 , 62 , 70 , 80°.and 90°K; The voltage dependence at a given temperature was usually reproducible within the experimental error. Fig. 8 shows an effect which was observed at 1.18°, 62°, 80° and 90°K: as the sample current is increased beyond a certain value, the resistance drops so sharply that the resultant voltage actually decreases. In a l l four cases, this'decrease occurred as the sample voltage reached - 34 SAMPLE 3 1 .28 °K 1 0.1 0\Z 0.3 0.4 V .(VOLTS) FIG. 8 Experimental resistance vs. voltage characteristics for sample 3 at 1.18°K. - 35 -0.4 volts. This effect is similar to what one would expect i f the Fermi level of the positively biased electrode were, depressed below the level of the conduction band of the negative electrode. (See Fig. 3(c) of Simmons, 1963). This occurs when eV >•( V+'Vj ) , typically for an applied potential of 1 or 2 volts* The observed value of 0.4 volts would seem to indicate a very low value for f . The temperature dependence of resistance for sample 3 was incon-clusive. As in the case of sample 2, no appreciable change in character-istics was observed in the-liquid helium temperature range. Although there was a variation with temperature in the liquid oxygen temperature range, the characteristics for 1.18°K were almost coincident with those for 90°K. Since the high temperature data were taken about 24 hours after the low, i t appears that the temperature induced decrease in resistance was offset by an increase due to aging. Analysis of Data A quantitative analysis was carried out for the data taken from sample 1. The parabolic voltage.dependence of the resistance, which is indicated in Fig. 6, is shown more strikingly in Fig. 9, in which the resistance is plotted against the square of the voltage. The data points for each temperature fa l l along a straight line. Since, for reasons discussed above, the characteristic curve taken at 1.18°K was rejected as anomalous, the 4.2°K data were taken as an,approximation to the behaviour of the sample at 0°K and were analyzed according to the low voltage, low temperature approximation, Eq. (12), repeated here for convenience: T H E O R E T I C A L ON 5X10 10X 10" FIG.9 1 5 X 1 0 ' ( V 0 L T s f RE S I S T A N C E VS. S Q U A R E O F V O L T A G E A T T H R E E T E M P E R A T U R E S , TOGETHER W I T H T H E O R E T I C A L HI6H T E M P E R A T U R E CHARACTERISTICS CALCULATED F R O M 0E H A W O O R A T * . 2 6 K . - 37 -R ( V' 0 ) " eMlmft)* t A *6 ¥>e Y j ( 1 2 ) Setting V = 0 in this equation gives an expression for R(0, 0) to be equated with the extrapolated zero voltage value of resistance from the 4.2°K data in Fig. 9. If we assume *P = 2eV, which is the average barrier height for a symmetric aluminum junction as determined by Morris and Pollack (1964), the only unknown in the theoretical expression for R(0, 0) is A, which is a function of the barrier thickness s: A - 4 g * ( 2 m ) & Thus, by equating the theoretical expression for R(0, 0) with the extrapolated value, we can obtain a value for s which will be quite precise because of the exponential dependence of R(0, 0) on A. This calculation yields s = 20.5 A. 2 Eq. (12) also shows that the graph of R(V, 0) vs.V is a e is given by • U r n { 96 <P, J straight line whose slope is given by hZs e#f Equating this expression for the slope with the value obtained graphically, (again taking *P•= 2eV), we can again solve for s. This yields o s = 22.2 A. These two approximations for s are in fairly good agreement and are reasonable values, according to the corrected results of Hunter and Fowle (1956), for the thickness of an oxide film grown in air at room temperature. The results from sample 1 are also quite consistent in that the lines in Fig. 9 a l l have the same slope, indicating that the voltage -38 -dependence is the same at a l l temperatures,, as predicted by theory. The 4.2°K line was used to predict the two theoretical lines in Fig. 9 for the behaviour at 65°K and 81°K. Writing Eq. (12) as r A z e z u*? R(V, 0) = R(0, 0) [ 1 - -§5^ r-Y J 2 and the slope of the line as dR/d(V ), we obtain j£ • = _ ±K V % v 1 V. <L(t) A ~e* R(o,0) • 2 Using the graphically obtained values for dR/d(V ) and R(0, 0) 2 2 we derive a value for A / Po and hence a value for s / f«. Since for small voltages, A s « s a n d ' P o , this value of s^/r'o can be used in the tem-perature dependent Eq. (16) for various values of V to obtain the theore-tical lines for 65°K and 81°K as shown in Fig. 9. The theoretical change in resistance between 4.2°K and 65°K was 4.6% as compared with an observed change of 8.67.. Between 4,2°K and 81°K, the theoretical and observed changes were 6.6% and 127. respectively. The variation in tunneling resistance with temperature was thus roughly twice as great as predicted. This was also true for the variation in resistance between 65°K and 81°K. 39 -CONCLUSIONS The purpose of t h i s experiment was to test quantitatively the theory of the temperature dependence of the tunneling resistance i n thin insulating films as derived by Simmons (1963, 1964). One of the problems encountered was the high f a i l u r e rate i n sample production. This did not appear to be due to dust on the substrates or moisture i n the a i r used for oxidation. I t i s known that cleanliness of the vacuum system i s a very c r i t i c a l factor i n the production of metallic t h i n films with uniform e l e c t r i c a l properties, and contamination may have been responsible for the poor r e s u l t s . C l e a r l y , i n order to ensure success, the conditions i n the vacuum system during evaporation must be very s t r i c t l y controlled. The samples were found to be e a s i l y changed or damaged by such influences as external e l e c t r i d f i e l d s , and besides the general increase i n resistance due to aging, often showed large and sudden jumps i n r e s i s -tance which could not be e a s i l y explained. These effects may have been due to unstable i o n i c configurations at the bar r i e r interfaces (see Morris and Pollack, 1964) . I n order to obtain reproducible and r e l i a b l e temperature dependence data, i t i s essential that the samples be stable and the aging effect as small as possible. This might be achieved by oxidizing the samples for extended periods of time at temperatures of several hundred degrees Centigrade. Although d.c. techniques are the most suitable for the i n v e s t i -gation of tunneling c h a r a c t e r i s t i c s , provision should be made for measuring the resistance of the junction i n each d i r e c t i o n i n order to obtain some idea of the b a r r i e r asymmetry even for all-aluminum samples. The voltage dependence of the tunneling resistance was found to - 40 -agree quite well with theory and was parabolic for low voltages as pre-dicted. Changes i n resistance of the sample due to aging or anomalous jumps did not destroy the voltage dependence, although they did make i t d i f f i c u l t to investigate the temperature dependence, since the temperature range could not be scanned as quickly and eas i l y as the voltage range. Analysis of:the temperature.dependence of resistance for sample 1 was carried out by taking the 4.2°K curve as an approximation of the behaviour at 0°K. By graphical analysis, an experimental value for was.derived and used i n the equation for the temperature dependence to predict lines for 65°K and 81°K. The theoretical change i n resistance between 4.2°K and 65°K was 4.6% as compared with an observed changeof 8.6%. Between 4.2°K and 81°K the observed temperature dependence was also about twice as great as expected - 12% as compared with a predicted value of 6.6%.. - 41 -BIBLIOGRAPHY Fisher,.J.C. and I. Giaever, J. Appl. Phys. 32, 172 (1961). Holm, R., J. Appl. Phys.. 22, 569 (1951). Hunter, M.S. and P. Fowle, J. Electrochem. Soc. 103_, 482 (1956). Pollack, S. R. and C. E. Morris, J. Appl. Phys. 35, 5 (1964). Simmons,. J . G., J. Appl. Phys. 34, 6 (1963). Simmons, J. G., J. Appl. Phys. 35, 9 (1964). Sommerfeld, A. and H. Bethe, Handbuch der Physik von Geiger und Scheel (Julius Springer Verlag, Berlin, 1933), Vol. 24/2, p. 450. 

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