Andreev Reflections in Micrometer-Scale Normal-Insulator-Superconductor Tunnel Junctions
Peter J. Lowell, Galen C. O'Neil, Jason M. Underwood, Joel N. Ullom
JJournal of Low Temperature Physics manuscript No. (will be inserted by the editor)
Peter J. Lowell · Galen C. O’Neil · Jason M.Underwood · Joel N. Ullom
Andreev Reflections in Micrometer-ScaleNormal-Insulator-Superconductor TunnelJunctions (cid:63)
21 October 2011
Keywords
Andreev Reflection · Microrefrigerators · Subgap Conductance · Superconducting Tunnel Junctions
Abstract
Understanding the subgap behavior of Normal-Insulator-Superconductor(NIS) tunnel junctions is important in order to be able to accurately model thethermal properties of the junctions. Hekking and Nazarov developed a theory inwhich NIS subgap current in thin-film structures can be modeled by multiple An-dreev reflections. In their theory, the current due to Andreev reflections dependson the junction area and the junction resistance area product. We have measuredthe current due to Andreev reflections in NIS tunnel junctions for various junctionsizes and junction resistance area products and found that the multiple reflectiontheory is in agreement with our data. PACS numbers : 74.45.+c 74.50.+r
Accurate modeling of the current-voltage (IV) characteristics of NIS junctions isrequired in order to use the junctions in applications such as primary thermometersor solid-state refrigerators. The Bardeen-Cooper-Schrieffer (BCS) theory of super-conductivity accurately predicts the IV characteristics of NIS junctions above thesuperconducting energy gap ∆ , while predicting almost no current when the junc-tion is biased below the gap . When NIS junctions are measured, subgap currentsgreater than BCS predictions are often measured. This excess current can be ex-plained by Andreev reflections , where an electron (hole) in the normal metal isreflected from the NS interface as a hole (electon), which allows a Cooper pair to (cid:63) Contribution of the US government; not subject to copyright in the United States.National Institute of Standards and Technology
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E-mail: [email protected] a r X i v : . [ c ond - m a t . s up r- c on ] O c t enter (leave) the superconductor. In interfaces where the electrons and quasiparti-cles are in the ballistic regime and can be represented as a plane wave, the Andreevreflection is described by the Blonder-Tinkham-Klapwijk (BTK) theory . This ex-cess power load in NIS junctions was modeled by Bardas and Averin . In realisticinterfaces, the electrons and quasiparticles are no longer in the ballistic regimebut behave diffusively, because they can reflect off the barrier and surroundingsurfaces many times before tunneling, which will cause a higher current than pre-dicted by the BTK theory. Hekking and Nazarov developed a model to account forthe extra current due to multiple Andreev reflections. Rajauria et al. have mademeasurements of SINIS IV curves, and their subgap data agreed with Hekkingand Nazarov’s theory when they multiplied their data by a scaling factor of 1.37 .However, Rajauria et al. preformed measurements on only a single junction sizeand a single oxidation thickness. In order to understand NIS junctions, it must beunderstood how junction area and interface resistance affect the Andreev current.In this paper, we provide a more robust test of Hekking and Nazarov’s theory bycomparing it with measurements of multiple NIS devices with different junctionareas and resistance area products. BCS theory predicts that current will flow through an NIS junction at voltage biasgreater than the superconducting energy gap. When the bias voltage is less thanthe superconducting gap, little current should flow because electron tunneling islimited by the density of states in the superconductor. However, a current belowthe gap is possible from mechanisms such as Andreev reflections. Hekking andNazarov predict an additional current below the subgap caused by Andreev reflec-tions: I Andreev = ¯ he R n S ν N d N tanh ( eV / T ) + ¯ he R n S ν S d S eV π∆ (cid:112) − eV / ∆ (1)where e is the electron charge, R n is the normal state resistance of the junction, S is the junction area, ν N , S is the density of states of the normal metal (superconduc-tor), d N , S is the thickness of the normal metal (superconductor), V is the voltagebias and T is the temperature. By rearranging Eq. 1, we find that the dimensionlessquantity eIR n / ∆ scales inversely with the resistance area product.Equation 1 is valid as long as the junction dimensions are larger than thecoherence length in each metal. The coherence length of the superconductor isgiven by ξ S = (cid:112) ¯ hD S / ∆ , and the coherence length of the normal metal is givenby ξ N = (cid:112) ¯ hD N / k b T , where D S , N is the diffusion constant for the superconductorand the normal metal, respectively. We measured the resistivity of our supercon-ductor, Al, to be ρ Al = 0.005 Ω × µ m, and measured the resistivity of our normalmetal, AlMn , to be ρ AlMn = 0.10 Ω × µ m. Using these values, we calculatedthe coherence lengths to be ξ S = 444 nm and ξ N = 571 nm. Our smallest junctiondimension is 2 µ m, which is greater than both coherence lengths. Therefore, it isvalid to use Eq. 1 to model our junctions. Fig. 1 (Color online) Optical image of a device used in our measurements. The junction is madeby overlapping normal metal and superconducting wires. The junction area is defined by a viain a SiO layer between the two metal layers. Figure 1 shows a typical device used in this experiment. The junctions were cre-ated by intersecting normal metal and superconducting wires, and the junctionarea is defined by a via in a layer of SiO separating the two metal layers. Thejunctions were fabricated on a Si wafer by first sputter depositing 30 nm of Aldoped with Mn to 4000 ppm by atomic percent. The AlMn was patterned usingstandard photolithographic techniques and was etched in an acid bath. A 90 nmthick layer of SiO was deposited by use of plasma-enhanced chemical vapor de-position (PECVD) and vias were created by use of a plasma etch to define the junc-tion area. The devices were ion milled to remove any native oxide from the normalmetal, and then exposed to oxygen to form the insulating layer. A resistance areaproduct, R SP ≡ R n × S , of 30 Ω µ m was created by exposing the devices to 0.1torr-s of oxygen and a R SP of 200 Ω µ m was created by exposing the devices to42 torr-s of oxygen. Finally, the superconducting Al counter electrode was sputterdeposited and then wet etched by use of standard photolithographic techniques.The 30 Ω µ m devices were fabricated on top of a 150 nm layer of SiO depositedby PECVD to increase the quality of the junctions, but no observable differencewas measured between these junctions and junctions that were fabricated on justthe thermal oxide.The junctions were then screened to determine their quality before measure-ments were performed. To determine the quality of our junctions, we use the qual-ity factor Q , where Q ≡ R leak / R n . The leakage resistance, R leak , is defined as thehighest resistance of the junction in the subgap. Both wafers produced deviceswith a Q ≈ µ m by 3 µ m and 4 µ m by 4 µ m from the waferwith a R SP = 200 Ω µ m , and devices 2 µ m by 2 µ m and 3 µ m by 3 µ m from thewafer with a R SP = 30 Ω µ m . Devices from the same wafer were chosen from the Device Area ( µ m ) R n ( Ω ) R SP ( Ω µ m ) d N (nm) d S (nm) ∆ ( µ eV)1 16 11.2 179 30 525 1852 9 21.5 193 30 525 1853 9 2.98 27 31 230 1854 4 8.25 33 31 230 185 Table 1
Measured values of device parameters used in our experiment. − − − − − − − − − C u rr en t ( e I R n / Δ ) Voltage (eV/ Δ ) Device 1 dataDevice 1 theoryDevice 2 dataDevice 2 theoryDevice 3 dataDevice 3 theoryDevice 4 dataDevice 4 theory Fig. 2 (Color online) Current vs voltage for the four devices that we measured. The data arerepresented by points and the theory as dashed lines. The error bars for the uncertainty in themeasurement are smaller than the data points. The theory was calculated using the device pa-rameters shown in Table 1. The data are in excellent agreement with the theory at lower voltagebiases. At higher biases, a detailed thermal model is needed to match the theory with data. same chip to make sure that the junction properties, such as the metal thicknessesand R SP , were as similar as possible.IV measurements of the devices were made by four wire measurements in anadiabatic demagnetization refrigerator at 100 mK. Current biasing was accom-plished with a low-noise voltage source in series with a 10 M Ω resistor. Table 1shows the properties of the devices that were measured in this experiment. Thenormal state resistance was measured from the differential resistance of the de-vices. The normal metal and superconductor thicknesses were measured for eachdevice by use of a profilometer. The results of our experiment are shown in Fig. 2. The uncertainty in the measure-ment is ±
50 pA and ± µ V, which makes the error bars smaller than the markersin the plot. The current was scaled by eR n / ∆ , and the voltage was scaled by e / ∆ in order to make the axes unitless. The theory lines were calculated by using Eq.1 with the measured device parameters shown in Table 1. The divergence of thedata with the theory at higher voltage biases, around 0.7 ∆ / e and above, is due tothe non-isothermal behavior of the NIS junctions. In this work, we modeled the − − − − − − − − − C u rr en t ( e I R n / Δ ) Voltage (eV/ Δ ) Device 1Device 2Dynes Parameter = 2e − − − − − Fig. 3 (Color online) Current vs voltage for the first two devices plotted against NIS curvesgenerated using various Dynes parameters. The Dynes theory does not fit the data in the subgap.The Dynes theory predicts a steadily rising current in the subgap, which we do not see in ourdata. Instead, we see a flat plateau below the subgap, which is consistent with Andreev reflection. devices as being isothermal. However, NIS junctions are known for their abilityto cool electrons in the normal metal at bias voltages near the superconductinggap . Therefore, in order to calculate the true behavior of these junctions, the IVcurves need to be modeled by solving a complex power balance equation. Sincewe are interested only in current due to Andreev reflections, which occurs belowthe gap, the junctions behave isothermally in the region of interest and no powerbalance modeling is required.As Fig. 2 shows, our data are in excellent agreement with the theory belowthe superconducting gap. Devices with the same R SP have the same dimensionlesscurrent, and the current due to Andreev reflections is inversely proportional to R SP ,as theory predicts. Dividing the subgap data by the theory, Devices 1 and 2 agreewith theory within 7 %, Device 3 agrees within 18 % and Device 4 agrees within15 %.For comparison, we also fit our data with NIS IV curves based on the phe-nomenological Dyne’s parameter . The Dynes parameter is added to the BCSdensity of states in an attempt to account for the broadening of the gap edge and/orthe presence of subgap states, as shown in Eq. 2. I = eR N (cid:90) ∞ [ f N ( E − eV ) − f N ( E + eV )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re (cid:34) E − i Γ (cid:112) ( E − i Γ ) − ∆ (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d E (2)where f N is the Fermi function of the normal metal and Γ is the Dynes parameter.As Fig. 3 shows, the Dynes theory does not provide a good fit to our data. TheDynes theory predicts a steadily rising current below the gap, while in our data,the current plateaus, which is consistant with Hekking and Nazarov’s theory. Thissupports that we are measuring Andreev reflections and not subgap conductancedue to the presence of subgap states. In this paper, we have measured the current due to multiple Andreev reflections inNIS junctions and found that it is in excellent agreement with the theory presentedby Hekking and Nazarov. Their theory is valid only for voltage biases below thesuperconducting gap, and we are working to incorporate their theory with ourpower balance equations in order to make more accurate thermal models of NISjunctions. These models will allow us to better predict the cooling properties ofjunctions with a small R SP , for which the Andreev current is significant. Acknowledgements
This work is supported by the NASA APRA program.
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