
IntroductionWe measured the temperature dependence of a niobium (Nb) superconductor’s energy gap in a Josephson Junction. The Josephson Junction was placed in a liquid helium (He) cryostat where we took voltage measurements at a controlled, constant temperature between 2 and 10K. The energy gap of the superconductor can be inferred from the behavior of the voltage as we vary the current across the junction. Josephson Junctions can be used in superconducting quantum interference devices (SQUIDs) that measure extremely weak (< 5x10^{18} T) magnetic fields. Understanding the superconductor’s temperature dependence will allow us to understand the use of SQUIDs at these temperatures. A Josephson Junction consists of a thin insulator sandwiched between two superconductors. In this experiment we used a Nb superconductor with an aluminum oxide insulator to measure the energy gap dependence on temperature. In a superconductor, the positive ion lattice energetically favors electrons in a Cooper pair configuration. As the temperature decreases, the binding energy of the electrons in Cooper pairs decreases below the energy of electrons in the normal state, creating an energy gap. These cooper pairs tunnel through the insulator in the so called Josephson Effect. BCS theory gives a relation for the energy gap of a superconductor. We compared this theory to the energy gap inferred from our experimental current versus voltage (IV) plot. On the plot, we expect to see a voltage around 3 mV ^{7,5} where the current suddenly appears. The voltage where this current appears represents the energy gap of the Cooper pair. However, since there are two layers of the superconductor, this value is actually twice the energy gap for one pair of electrons.^{7} TheoryNiobium (Nb) is one of several elementary metals that acts as a superconductor at low temperatures. Superconductors’ resistance suddenly drops to zero below a specific critical temperature, a property useful in many electronics. Nb’s critical temperature is 9.2K. Below the critical temperature electrons in a superconductor pair together into a Cooper pair configuration. Since Cooper pairs are energetically favored by the positive ion lattice they are able to flow along the material with no resistance.^{4} Above the critical temperature, Cooper pairs begin to break apart into normal state electrons which have resistance when traveling in a material. As the temperature of the superconductor decreases further below the critical temperature, the difference in binding energy increases between Cooper paired electrons and electrons in the nonsuperconducting state. This difference in energy is called the energy gap of the superconductor and is equivalent to the amount of energy needed to excite the electrons back to a normal resistive state.^{4} In a moment we will characterize this energy gap mathematically. In our experiment we exploit a property of superconductors called the DC Josephson Effect in which Cooper pairs tunnel through the thin insulator in a Josephson Junction creating a current. This phenomenon is analogous to particles tunneling through thin potential barriers in quantum mechanics. Provided the insulator in our Josephson Junction is thin enough, and the current applied to the junction is below the critical current, all of the Cooper pairs will tunnel through the insulator without resistance.^{4} The resulting current that flows across this junction can be seen in Equ. 1 where δ represents the phase difference between the superconductors.^{5} (1) The critical current represents the maximum amount of current that can travel through the superconductor before Cooper pairs begin to break apart and create resistance within the insulator. For currents below the critical current there is no voltage difference between the superconductors. Above the critical current, however, the density of Cooper pairs traveling through the insulator causes some of them to break apart creating a resistance and thus a potential difference. In this experiment we will be inducing a current in the junction and measuring the resulting voltage to create plots of voltage versus current (IV). Fig. 1 shows our expected results.
Figure 1. Expected Results of Current vs. Voltage for a Josephson Junction. Once the voltage goes above Vc, there is a finite resistance. The current corresponding with this voltage is the critical current. Vc represents twice the energy gap of the superconductor. The critical voltage, V_{c}, corresponding to the critical current represents the potential difference between the Cooper pairs and electrons in the normal configuration. Since we have two superconductors in our junction this potential difference is equivalent to twice the energy gap, E_{g}, as shown in Equ. 2. (2) Plotting IV for several temperatures will allow us to determine the energy gap’s dependence on temperature. BCS theory predicts the dependence of the energy gap on temperature to be given by a hyperbolic tangent function.^{7} (3) Near the critical temperature, T_{c}, the relationship can be simplified to a power law where the energy gap is proportional to the square root of the temperature T^{6} (4) where k_{B} is Boltzmann’s constant. Townsend and Sutton investigated the temperature dependence of the energy gap in 1960 and their results are shown in Fig. 2.^{7}
Figure 2.: Reduced values for the energy gaps for lead, tin, tantalum and niobium as a function of reduced temperature, and compared to the theoretical curve (solid line) of BCS theory. (P. Townsend, J. Sutton, “Investigation by Electron Tunneling of the Superconducting Energy Gaps in Nb, Ta, Sn and Pb,” Phys. Rev. 128 (2), 591595 (1962).) References ^{1} M. Bryan, J. Mattsen, “Josephson Junctions,” Methods of Experimental Physics, Spring 2009, University of Minnesota, April 5 2011. <http://mxp.physics.umn.edu/s09/Projects/S09_ShapiroSteps/> ^{ 2} Ermolov, Marchenko, Chizhov, “Magneticfield penetration into superconducting niobium,” Pis’ma Zh. Eksp. Teor. Fiz. 43 (2), 8285 (1986). ^{ 3} A. Gray, J. Pham, “Josephson Junctions: Magnetic Interference Effects and Shapiro Steps,” Methods of Experimental Physics, Spring 2003, University of Minnesota, Feb 19 2011. <http://mxp.physics.umn.edu/s03/Projects/S03Josephson/>. ^{ 4} C. Kittel. Introduction to Solid State Physics, 4 ed. New York: John Wiley & Sons, 1971. ^{ 5} R. Maunu, S. Vig, “Josephson Effect: Determining the magnetic flux quantum using niobiumaluminum oxide Josephson junctions,” Methods of Experimental Physics, Spring 2010, University of Minnesota, Feb 19 2011. <http://mxp.physics.umn.edu/s10/Projects/S10_JosephsonEffect/>. ^{ 6} D. J. Thouless, “StrongCoupling Limit in the Theory of Superconductivity,” Phys. Rev. 117 (5), 12561260 (1960). ^{ 7} P. Townsend, J. Sutton, “Investigation by Electron Tunneling of the Superconducting Energy Gaps in Nb, Ta, Sn and Pb,” Phys. Rev. 128 (2), 591595 (1962).
Last Updated: 05/13/2011 