Figure 1.
Superconductors can be described, phenomenologically, by the two fluid model, in which the metal contains a mixture of normal and superconducting electrons. Below a transition temperature Tc, normal electrons begin condensing into superconducting pairs, this creates a gap 2D in the normal electron (or quasiparticle) energy spectrum much as the band gap in a semiconductor. As the temperature decreases and a greater fraction of the normal electrons condense into pairs, the energy gap widens. A typical graph of the temperature dependence of the energy gap is shown in Fig. 1.
Figure 1.
This gap can be observed in the I-V curves of tunnel junctions with superconducting electrodes, since the quasiparticle current is blocked by the gap for voltages V < 2 D. You need a strong background in solid state physics to understand the microscopic theory leading to this gap will likely be beyond you unless . The results of the theory along with a phenomenological discussion of quasi-particle tunneling are presented e.g. in Van Duzer [1] sec. 2.11-2.16.
In the experiment you will measure the I-V curves of niobium tunnel junctions for temperatures ranging from 4.2 K to over 9 K enabling you to determine the temperature dependence of D along with that of both the pair and quasiparticle tunnel currents. For "weak coupling" superconductors the value of the gap at T = 0 is related to the superconducting transition temperature Tc by
D(0) = 1.76 kB Tc (1)
For Nb, some corrections to weak coupling results are required. This is discussed e.g. in Ref. [3]. The exact properties of the Nb depend on just how the films were made, but Ref. [3] should serve as a guide to what sort of corrections to expect. The ratio d= D(T) / D(0) of the gap at finite and zero temperatures is a universal function of t = T / Tc. This function is given implicitly by an integral equation (Van Duzer 2.11, Eq. 3). The numerical solutions to this equation have been published [2]. Near Tc there is an analytic form for this,
. (2)
This result is valid only very near to Tc.
An empirical formula, which fits the gap function rather well
thought the whole temperature range is given in Ref. [4] as
; (3)
in fact this is how the curve on Fig. 1. was produced. Another, very good, approximate formula for the gap is given in ref. [6].
In addition to the quasiparticle tunnel current, the junction also carries a supercurrent of paired electrons which flows even when V = 0 between the electrodes of the junction. Junctions which carry supercurrents are called Josephson junctions and are discussed in Van Duzer, chapter 4. The maximum value of the this zero voltage current, Ic is related to the energy gap by,
(4)
where Rn is the junction resistance for voltages well above the gap voltage.
A second striking feature of superconductors, in addition to zero resistance, is the Meissner effect, that is, the expulsion of magnetic flux from bulk superconductors. A magnetic field will only penetrate into a superconductor for a depth of the order of 100nm (for niobium) called the penetration depth, l. This depth depends on the density of superconducting pairs and is therefore temperature dependent. A phenomenological discussion of this effect is presented in Chapter 3 of Van Duzer. l can be measured from the magnetic field dependence of Ic of Josephson junctions. As discussed in Van Duzer 4.05, a magnetic field applied in the plane of the junction will modulate Ic in the form of a Fraunhofer diffraction pattern. The minima of this pattern are separated by the fields needed to create one quantum of magnetic flux F0 = 2.07 x 10-15Wb through the junction. The flux through the junction is F = w teff B, where w is the junction width, B is the magnetic field and teff is the effective thickness of the barrier in the junction. teff = 2l + d where d is the oxide thickness of about 5nm. Thus, the periodicity of Ic vs. B enables one to measure l.
To do a reasonable job on this experiment, one must understand something of the theory of superconductors and Josephson junctions as referenced above. Since this material is usually not covered in courses, you should plan to spend a fair amount of time reading the reference material if you want to do a good job at this experiment.
Some of the cables used to connect the circuit elements have TWO leads in the shielding. These (low noise) cables and connectors look similar to the regular BNC cables and connectors, except for the blue color. When making connections with these cables the two inner leads of the connector have to matched carefully.
The main switch of the amplifier rack is on the left back side.
The Josephson junctions are located on a vacuum isolated platform on a probe which can be inserted into a liquid He storage Dewar. There is a weak thermal coupling between the platform and the He bath by a copper wire. The temperature of the platform is measured by a Ge resistance thermometer. You can regulate the platform temperature above the He bath temperature (4.2K) by supplying current through a 1kW heater on the platform. There is a small superconducting magnet surrounding the vacuum can. According to our calibration, 1mA current corresponds to 0.25 G ± 5% . The magnet will supply fields up to about 125 G, parallel to the plane of the junctions. Do not leave the magnetic field turned up, unless you need it, since significatn power is dissipated in the magnet, boiling off the helium unnecessarily. One of the staff will assist you in cooling down the probe and inserting it into the Dewar. Do not attempt do this yourself.
Figure 3.
The circuit for the temperature measurement consists of a variable voltage source, which drives a current through a decade resistor and the thermometer resistor in the probe. VT can be measured, without changing any leads, by switching the DMM to the DC voltage mode. You can change VT either by changing the voltage source or the decade resistor. The DMM, when set in the ratio mode, measures the ratio of the voltages across the thermometer and the (known) decade resistor. This allows to easily calculate the thermometer resistance - easily, assuming you pick a sensible value for the decade resistor. You should consult the DMM manual to determine the accuracy of the ratio measurement as well as the best conditions for achieving high accuracy. You can test your understanding of the measurement system by using it to measure some of the known resistors in the "ersatz probe" box located on the amplifier rack. See Figure 5 for the circuit diagram for this box, which contains a number of circuits that mimic those in the actual probe. This allows you to check out both the electronics and your understanding of the measurement circuits before you have helium.
The principle of the measurement is illustrated in the Figure 3. For the simple I-V measurements the lock-in amplifier is not needed, and the X-Y mode of the oscilloscope can be used to observe the junction characteristics. The lock in allows to detect the first derivative of the I-V curve, and therefore it is useful at higher temperatures, when the gap feature is weak.
The "ersatz probe" box , located on the rack, also contains a "pseudo-sample". The pseudo-sample has an I-V curve a bit like that of the junctions so you can get some practice before you use liquid helium.
The junctions, which are located on a silicon chip
in thermal contact with the temperature controlled platform, were
fabricated at Stony Brook using a process similar to that described
in Ref. [5]. It is important for you to convince yourself that
the sample and the platform are at the same temperature during
your measurements. One possible cause for a temperature difference
is the presence of helium gas in the vacuum can. This gas is
put in the can in order to provide thermal contact while cooling
the probe. Its presence provides a direct thermal path from the
sample to the He bath-bypassing the temperature regulated platform.
It is necessary to pump on the vacuum can for at least 1/2 hour
with the diffusion pump in order to adequately remove this gas.
Figure 5.
Four junctions, with different sizes and orientations, are connected. The wiring diagram for the junctions in the probe along with a micrograph of the junctions is available in the lab. A particular junction is selected through the 4 position switch on the "PROBE" box located on the amplifier relay rack. This switch simultaneously switches the current and voltage leads. You must always place the shorting switch (just below the 4 throw switch) in the shorted (ST) position before switching junctions.
In the actual realization of the measurement, differential amplifiers are used as shown in Figure 5. A slowly varying current, usually a triangle wave with an amplitude of about 1 mA, is passed through a current monitoring (and limiting) resistor connected to the sample current leads. The voltage across this resistor is amplified by a differential amplifier whose output is connected to the Y channel of a digital scope (see Figure 4 and Figure 5). The junction voltage is amplified and connected to the X channel of the scope. The scope display can be dumped to a printer. Note also that you can use the scope for signal averaging in the X, Y vs. time mode in order to increase the signal to noise ratio. It is also possible to couple a small ac current (about mA at 1 kHz) through the sample and monitor the ac voltage with a lockin amplifier. The lockin then gives a low frequency output which is proportional to the differential resistance of the junction. This is useful for locating more subtle features in the I-V curves near Tc.
Figure
6.
References
