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Experiments in Superconductivity: the Meissner and Josephson Effects

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Experiments in Superconductivity: the Meissner and Josephson Effects Experiments in Superconductivity: The Meissner and Josephson Effects Edwin Ng∗ MIT Department of Physics (Dated: April 9, 2012) We perform a series of basic experiments in superconductivity on vanadium, lead and niobium bulk samples using liquid helium cryogenics. We measure Tc for each sample via the Meissner effect and demonstrate for vanadium a dependence of Tc on an external B field. We observe the presence of persistent currents by measuring the trapped flux in a superconducting lead cylinder. Finally, we observe the DC Josephson effect through a Nb-Al2O3-Nb junction, and we measure the critical current as a function of an external B field to determine the fundamental flux quantum, which we −7 2 find to be (1:93 ± 0:01stat. ± 0:1syst.) × 10 G cm . I. INTRODUCTION AND THEORY[1] I.1. The Meissner Effect Superconductivity (SC) is a phenomenon which occurs The Meissner effect sets superconductors apart from in various materials at low temperatures, including thirty perfect classical conductors. Generally speaking, the elements and thousands of compounds. One aspect of Meissner effect is the empirical observation, made by superconductivity is zero resistance; in this sense, a su- F.W. Meissner in 1933, that superconductors exclude perconductor is similar to a perfect classical conductor. magnetic fields from their interior. The London equa- However, superconductors also exhibit other distinctive tions, proposed by Fritz and Heinz London in 1935, give properties. In this lab, we study the Meissner effect, the a phenomenological account for this effect. hallmark of SC, as well as the Josephson effect, which has More specifically, the London equations show that a −z/λL important applications to high-precision measurements. superconductor obeys Bsc = Be , where B is the ex- Superconductivity occurs only under certain condi- ternal field, z is the depth from the surface of the SC, and tions. In the absence of an external magnetic B field, λL is the material-dependent London penetration depth. Thus, magnetic fields can exist only at the surface of the there is a critical temperature Tc below which the ma- terial becomes superconducting. When B 6= 0, however, SC; for depths z larger than λL, Bsc is effectively zero. the transition temperature is lowered; empirically, the We can therefore also view supercondutors as exhibiting condition for SC is perfect diamagnetism. Physically, this effect is achieved through the presence " # T 2 of surface currents. These surface currents act to pre- B ≤ B0 1 − ; (1) cisely cancel out the field in the interior of the SC, thus Tc bringing about the Meissner effect. In this lab, we take where B0 is the field above which SC cannot occur, even advantage of this flux exclusion to detect the transition at T ! 0. Thus, we can think of the transition to SC as a into SC, which allows us to determine Tc. phase transition in B and T , with the SC phase delimited Moreover, since superconductors also have zero re- according to Equation1. sistence, these surface currents can potentially be persis- This condition and other properties of superconduc- tent. That is, once these surface currents are set up, they tors, such as heat capacities, can be derived with BCS can be made to flow indefinitely, as long as the sample theory, formulated by Bardeen, Cooper, and Schrieffer remains below Tc and the B = 0 condition is respected. in 1957. In BCS theory, electrons at low temperatures couple to form Cooper pairs via vibrations of the lattice. I.2. The Josephson Effect These Cooper pairs then condense into a ground state, where they flow freely without resistance. The Josephson effect involves the tunneling of the There is also a distinction between the so-called Type Cooper pairs across a narrow insulating gap, called a I and Type II superconductors. Type I superconductors Josephson junction, and is named after B.D. Josephson include most of the elemental superconductors and are for his discovery of it in 1962. characterized by a sharp transition at Tc. Type II su- We consider the application of a DC voltage V0 across perconductors, on the other hand, develop non-SC vor- two superconductors separated by a narrow insulating tices near Tc, resulting in a mixture of SC and non-SC gap. Because the electrons are bound into Cooper pairs, properties during the transition; they are therefore char- there are no free electrons for single-particle tunneling acterized by wider transitions and more persistent SC across the junction, and so there should be no resulting behavior around Tc.[2] Of the samples we use, lead is a current until we exceed the binding energy of the Cooper Type I SC while vanadium and niobium are Type II. pairs, upon which we get a nonlinear return to an Ohmic response as the Cooper pairs break up. Nevertheless, Josephson discovered that when the gap ∗ [email protected] is narrow enough, the two superconductors can still cou- 2 ple together and result in the tunneling of the Cooper of a standard 30 L liquid helium dewar and kept in place pairs themselves. This Josephson current density is[3] by a lock collar. Circuitry extends out of the tube to con- necting ports on the top. Detailed pictures can be found 1 J(t) = J0 sin δ0 + V0t ; in [1], and we discuss the essential components and logic Φ0 of each probe in the following subsections. Temperature control for probes I and II is achieved by where δ0 is a constant phase, J0 is the critical current pumping helium out through the probe neck|we control density, and Φ0 = h=2e is the flux quantum. the temperature by adjusting the airflow speed and the Thus, when V0 = 0, we get a finite and constant depth of the probe head. We typically pump with dewar Josephson current, proportional to J0. On the other pressures down to −500 mbar. Probe III does not use hand, when V0 6= 0, the Josephson current oscillates with pumping, and temperature is controlled by lowering or high frequency (dictated by 1=Φ0); this current averages raising the probe. Temperature readouts vary by probes; out to zero, until V0 surpasses the binding energy of the Cooper pairs and we return to Ohmic response. we use a calibrated silicon diode for probe I, a carbon This DC Josephson effect is of particular interest to us resistor for probe II, and a digital readout for probe III. because the critical current J0 is itself related to Φ0. In particular, if we apply a static magnetic field B = Bz^ II.1. Probe I: Measurements of T perpendicular to the gap, then the coupling of the super- c conductors across the junction can be described by[4] The head of probe I consists of an outer driving 1 Z solenoid with 2200 turns, length 31:0 mm, inner diam- g(r) = g exp 2πi · A · dr ; 0 Φ eter 14:0 mm, and outer diameter 16:9 mm. Inside this 0 C solenoid is a test-coil solenoid consisting of 810 turns, for g0 a constant and B = r × A under some gauge length 12:0 mm, inner diameter 7:1 mm, and outer diam- choice. The integral is taken over the path C travelled by eter 10:5 mm. The sample is inserted into the inner coil the Cooper pairs as they cross the gap and is analogous and kept in place with a brass spacer fastened with a to an Aharonov-Bohm effect. Finally, we can relate this threaded loop of wire. coupling to the measured J0 by the relation[4] We then drive an AC voltage across the outer solenoid ZZ using an Agilent function generator, set to 200 Hz at ap- 2 J0 = g(r) d r ; proximately 500 mVrms. This causes the inner coil to S pick up an induced EMF VC . When the sample transi- where S is the x-y cross-section of the junction. tions from non-SC to SC, the excluded field due to the As described in Section II.3, the setup of our Josephson Meissner effect causes the flux within the inner coil to junction is approximately a cylindrical superconducting drop dramatically. Observing the variation of VC against wire of radius R split by a cylindrical gap of width D. temperature gives us essentially the SC transition curve. 1 Hence, we take C to be along the axial direction x^, S to We measure the RMS value of VC using a 6- ⁄2 digit Ag- be the circular x-y cross-section, and A = −By x^ along ilent multimeter. Typical RMS values for VC are about the gap and zero elsewhere. If we define Φ = (2RD)B to 3 mV, although this depends on the function generator be the flux perpendicular to the junction, then voltage. However, upon transitioning to the SC phase, this value usually drops by 1 mV for the type II (V and 2 2g0R πΦ Nb) samples, and by about 0:2 mV for the type I Pb sam- J0 = J1 ; (2) Φ=Φ0 Φ0 ple (following a gradual decline of about 0:5 mV due to the conductivity of Pb at low but non-SC temperatures). where Jn denotes the nth Bessel function of the first kind. This is the basic model we will use in determining Temperature readout is made by a silicon diode sit- the fundamental flux quantum. uated 1 cm above the sample, calibrated in [1] with a 10 µA, 9 V battery source. The calibrated diode response ranges from about 0:5 V at 300 K to about 1:7 V at 1:4 K. 1 II.
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