Advanced Laboratory, Physics 407

This experiment consists of three parts.

The details are described in the following text which contains a section for each of the three parts. The Appendix contains the squid electrical and mechanical specifications.

1. Chapter 21, pp. 21-1 to 21-18 in Feynman Lectures on Physics Vol.3, by

Feynman, Leighton, and Sands.

The directions that follow assume that you have some familiarity with the basics of superconductivity including zero resistance, flux quantization, and the Josephson effect. If not, you may wish to read some of Section V ("An Introduction to Superconductivity and SQUIDs") before going any further.

The Mr. SQUID chip contains a dc SQUID made with thin film YBCO superconductor. The dc SQUID is a simple circuit that can be schematically represented like this:

The dc SQUID is simply a ring of superconductor containing two Josephson junctions, which are marked by the X's labelled J₁ and J₂ in the circuit diagram above. The Josephson junctions can be thought of as weakened areas of the superconductor, which nevertheless still allow a certain amount of resistanceless current to flow. The simplest experiment to perform upon this circuit is to pass current "I" from left to right across the ring and to measure the voltage "V" that appears across the ring. If the two junctions in the SQUID are identical, in the absence of any magnetic field, the current will divide evenly and half of it will pass through each junction before recombining at the right side.

The Mr. SQUID control box allows you to perform this experiment quite easily. If you have set up the system according to the instructions in the previous section, you are ready to try this voltage-current experiment.

Once again, we assume that you are familiar with the operation of your oscilloscope or x-y recorder. For initial settings, try an x sensitivity of 0.2 volts per division (or whatever value is closest to this on your instrument) and a y sensitivity of around 50 mV per division. You may want to adjust these values as needed later. Now refer to the diagram of the Mr. SQUID control box (Figure 3.2) below.

STEP 1:

If you are using an oscilloscope, set the oscillator frequency toggle switch (7) to the osc

(high-speed) position. If you are using an x-y recorder, use the x-y (low-speed) position.

STEP 2:

Before turning on the power, make sure the amplitude control (4) is at its minimum (fully counter-clockwise) position and set the flux bias control (2) and the current bias control (3) to their 12 -o^’clock (straight up) positions.

STEP 3:

Make sure the function switch (1) is in the V/I position. In this position, the BNC cable connected from the X output on the front panel (5) reads the current being fed through the SQUID. The cable connected to the Y output (6) reads the voltage across the SQUID. Set the power switch (9) to the ON position. On an oscilloscope, there should be a small bright spot on the center of the screen. On an x-y recorder, the pen should be stationary in the middle of the page. You may have to adjust the offset controls on your display device to achieve this in either case.

The current bias control (3) varies the current being sent through the SQUID. Slowly turn this knob in either direction. The spot on the oscilloscope screen or the plotter pen will move in response to the changing current. As you turn the knob back and forth, you will trace out a curve representing the relationship between the current fed through the SQUID and the voltage across the SQUID. This curve is called the V-I curve for the SQUID.

Return the current bias control to the 12 o'clock position and now slowly turn the amplitude control (4) in the clockwise direction. This function sweeps the current through the SQUID back and forth between preset values, thus automating the procedure you performed by hand using the current bias control. A solid curve should now appear on the oscilloscope screen; alternatively, the pen on your plotter will repeatedly trace out the curve on the page.

If you now turn the current bias control, the center point of the curve being traced on the screen or page will move. The current bias control sets a single value of current being passed through the SQUID and the amplitude control sweeps the current back and forth about that set value.

Your output device acts like a voltmeter. The sensitivity settings on it determine how much voltage corresponds to a division on the screen or on the page of graph paper. The current output on the Mr. SQUID box (5) actually represents the voltage across a 10,000W resistor in the electronics box. According to Ohm's Law (I = V/R), the current flowing through the resistor is therefore equal to the voltage across it divided by 10,000 W .

The typical voltage levels from the SQUID are small enough that we have provided amplification in the Mr. SQUID control box. Thus, to calculate the actual voltages across the SQUID, the measured value on the oscilloscope or x-y plotter should also be divided by 10,000.

If the settings on the Mr. SQUID box and your output device are correct, and if the SQUID is behaving properly, you will see a curve that looks more or less like this on the screen or page.

It is important that there is a flat region in the center of the curve as shown above, although its width may vary from device to device. If what you see looks like a straight line, as in Figure 3.4 below, then

either you don't have enough liquid nitrogen in the dewar, you have not let the probe get cold, or there may be trapped magnetic flux in the SQUID. The latter is a very common occurrence because the SQUID is very sensitive to magnetic fields in its external environment. Refer to the section on trapped flux in Section VI ("Troubleshooting and Getting Help"), if this appears to be the problem. Assuming you see the proper curve, how can we understand its shape?

What you are looking at is the V-I characteristic of two Josephson junctions connected in parallel with one another. Assuming they are identical junctions (in practice, they are at least very similar), the V-I characteristic you see is the same as for a single junction. The most important feature of the curve is the flat spot in the middle, called the critical current. In this region, there is current flowing with no voltage - it is a supercurrent. This is the dc Josephson effect: a resistanceless current that flows through a superconductor Josephson junction.

A Josephson junction consists of two regions of superconductor that are weakly coupled together. The meaning of this statement is that the junction behaves like a superconductor but can only carry a small amount of resistanceless current before it becomes resistive. Any superconductor has the same property: it ceases to be resistanceless as soon as the current it is carrying exceeds a maximum value called the critical current. A Josephson junction is a weak link between two regions of superconductor, and this weak link carries far less resistanceless current than the superconductor on either side. The maximum supercurrent that can flow through a Josephson junction is called the critical current of the junction.

At this point, you might want to adjust the flux bias control (2) on the control box. This control feeds current into a small gold coil placed above the SQUID and this current applies a magnetic field upon the loop the SQUID. As explained in Section V, a magnetic field will modulate the critical current in the SQUID in a very specific manner. By turning the flux bias control knob, the critical current in the junction will change visibly on the oscilloscope screen or on your plotter page. At this point, try to adjust the flux bias current such that the flat region of the V-I curve is widest. The response to these changes may be quite sensitive; it may take some practice to tune the critical current to its maximum value. This procedure may be especially useful if some small amount of magnetic field was already present in the SQUID loop. Zero applied field will yield the largest critical current in the junction.

You can determine the critical current of the junctions in Mr. SQUID by measuring the width (in volts) of the flat part of the V-I curve and dividing that number by 10,000 W to convert your answer into amperes of current. If the flat region is 1 volt wide, for example, then the corresponding current is 100 microamps. However, you are looking at the current through both of the junctions in the SQUID, not just one. Therefore, assuming the junctions are identical, the current through one junction is half the value you are measuring (in our example, 50 microamps). Is this the critical current of the junction? Not quite. The curve you are looking at drives the current symmetrically about zero (marked in the drawing). The current you have measured is actually composed of a contribution in the positive

direction and a contribution in the negative direction. If the curve is symmetric, they are equal. As a result, the real critical current of the junction is half the value you infer from the measurement. Thus, in our example, the critical current is 25 microamps, 1/4 of the original measurement. The flat region on the curve corresponds to 4 times the junction critical current.

Apart from the critical current of the junction (usually written as ), there is also a parameter of the junction known as the normal-state resistance or . You can determine it by measuring the slope of the V-I curve out at the ends where it is essentially a straight line. To obtain a resistance, you must convert the x-axis value into amperes from volts and take into account the amplification of the y-axis signal. This is all very simple since the conversion factor for each axis is the same value of 10,000. Remember that the dc SQUID contains two junctions in parallel, so that the measured resistance corresponds to half the resistance of a single junction (assuming they are identical.) Thus, simply taking the slope numbers in volts off your oscilloscope or x-y recorder will give you one-half the normal-state resistance in ohms. The product of the critical current and the normal state resistance is a voltage that is an important parameter for the operation of a SQUID. Make a note of it now for use later. Note that for one of the junctions has the same value for the SQUID, which has two junctions in parallel. For the junctions in Mr. SQUID operating in liquid nitrogen, you will probably obtain a value between 1 and 10 microvolts. This value sets the maximum voltage change in the SQUID by an individual magnetic flux quantum, and is discussed later in this section.

As you observe the properties of the Josephson junctions in Mr. SQUID, realize that such an experiment required liquid helium and several pieces of electronic instrumentation just a few years ago. In this era of high-temperature superconductivity, Josephson electronics is becoming a far more accessible technology.

Up to this point, we have been looking at the properties of Josephson junctions. Now we will turn our attention to the properties of the dc SQUID itself.

The dc SQUID has the remarkable property that there is a periodic relationship between the output voltage of the SQUID and applied magnetic flux. This relationship comes from the flux quantization property of superconducting rings that is discussed in detail in

Section V. The Mr. SQUID control box will allow you to observe this periodic relationship in the form of V-F characteristics on your oscilloscope screen or x-y plotter page.

You have already observed the effects of a magnetic field on the V-I characteristics of the SQUID by adjusting the flux bias control current. The V-F characteristics are basically an automatically plotted version of this behavior. The physics underlying the V-F curve is

discussed in Section V.

As we saw before, in the V-I operating mode one can apply a magnetic field to the SQUID using the flux bias control. This dial controls a current that is applied to the modulation coil, simply a one-turn gold coil that creates a magnetic field inside the loop of the SQUID, as shown in Figure 3.6. The second coil shown in the photograph is the external modulation coil that allows you to couple other sources of current to the SQUID.

If you slowly turn the flux bias control knob, you can see the change in the critical current and the overall V-I curve that arises from magnetic flux threading the loop of the SQUID. Another way to see the sensitivity of the SQUID to external fields is to rotate a

small horseshoe magnet slowly in the vicinity of the dewar. if you experiment carefully with the flux bias control, you will see that the critical current of the SQUID oscillates between a maximum value (the flat region of the V-I curve is at its widest) and a minimum value at which point the V-I curve may be more-or-less linear.

The Mr. SQUID control box allows you to view the periodic behavior of the SQUID in a convenient, automated way To obtain the V-F plot, you will set the bias current so that the SQUID voltage is most sensitive to changes in applied magnetic field. This occurs at the knee of the V-I curve, the area highlighted in Figure 3.7 below.

To find this point, first adjust the flux bias control (2) so that the critical current is at its largest value. Then, turn down the amplitude control (4) all the way so that only a point is visible on the oscilloscope screen or so that the pen is stationary on the plotter page. In this mode, you can then sweep the point up and down the V-I curve by adjusting the current bias control (3), as you did initially. Set the current bias level so that the point rests at either knee in the V-I curve. Now turn the flux bias control (2), which controls the amount of magnetic flux penetrating through the hole in the SQUID loop. You should see the point move back and forth vertically on the screen or page. This periodic motion arises because screening currents in the SQUID body are sensitive to differences between the magnetic flux applied and integral multiples of the magnetic flux quantum () This phenomenon is a manifestation of the macroscopic quantum nature of superconductivity.

To automate this procedure, switch the function switch (1) from V-I to V-F . In this new mode, all of the controls on the Mr. SQUID control box work the same way as before, except for the amplitude control (4). Instead of controlling the sweep of the drive current through the SQUID, it now controls the amplitude of the current through the gold modulation coil, which is linearly related to the magnetic field applied to the SQUID. To view as many V- F periods as possible, turn the amplitude control (4) in the clockwise direction. Depending on the particular SQUID and coil in your probe, you should be able to see at least four or five periods of oscillation, as shown in Figure 3.8.

The voltage change that occurs due to the influence of the magnetic field appears now on the vertical axis of your display device. The voltage amplification provided by the Mr. SQUID box is unchanged from before. To determine the actual voltage across the SQUID, you must divide by 10,000. However, since the SQUID modulation signals are smaller than the junction voltages you examined earlier, you will need to increase the sensitivity of the vertical scale on your display device. The peak-to-peak voltage of the SQUID modulation in Mr. SQUID (D V) is typically between 0.5 and 5 m V (or 5 to 50 mV output from the front panel of the Mr. SQUID control box). Depending on the setting of the amplitude control, you may wish to change the x sensitivity to more conveniently view the oscillations of the SQUID.

In addition, it may be necessary to use the position knob on your output device to center the curve. If your oscilloscope can be operated in an AC-coupled mode on the vertical channel,

you can use this mode for more convenient viewing of the V-F curve (never use the AC mode to look at the V-I curve, or it will be completely distorted on the oscilloscope screen.) You should see a curve that resembles the one shown in Figure 3.8. Depending on the particular device and on the settings on the control box, you may see many more periods than shown here. At this point, you can try to maximize the signal by fine-tuning the current bias control (3) to the most sensitive part of the V-I curve (just keep your eye on the V-F curve as you tweak the current bias control until you get the maximum modulation). There will be some setting of the current bias control that gives the largest modulation amplitude.

Notice the effect that the flux bias control (2) has on the V-F curve. It allows you to set the central value of applied flux about which the amplitude control sweep varies. In other words, this control allows you to apply a static magnetic field to the SQUID on top of the oscillating field applied with the amplitude control. Turning the flux bias control will therefore allow you to move left and right along the V-F curve and thereby explore more of it than the amplitude sweep permits. You can accomplish the same thing in a less controlled manner by waving a small permanent magnet near the probe. Another way to see the effect of an additional applied field is to rotate the entire dewar. This rotates the SQUID with respect to the earth's magnetic field and causes a shift. Actually, most of the earth's field is screened out by the magnetic shield on the probe, but the amount of field that gets by the shield is quite enough to move the V-F pattern noticeably on the screen.

Later on, you may want to warm up the probe to room temperature and carefully remove the magnetic shield from the bottom of the probe, by loosening the small set-screw that holds the shield. Set both the shield and the screw aside in a safe place so that they can be reinstalled later. If you now cool down the probe and set up the V-F measurement, you will find that the SQUID is tremendously more sensitive to its environment. In fact, you may have a great deal of difficulty in getting a good V-I curve without flux trapping (see the discussion in Section VI). Assuming you succeed in observing the V-F curve without the magnetic shield (which may be impossible in many environments), you will find that almost any magnetic disturbance anywhere near the dewar will be visible on an oscilloscope screen.

The most sensitive technique for SQUID operation consists of operating the SQUID in a "flux-locked loop" which essentially uses a feedback mechanism to counteract any change in external magnetic field. One actually measures the amount of feedback current, a technique that permits the detection of fields corresponding to a tiny fraction of a flux quantum.

Now that you can measure a variety of properties of the SQUID in the Mr. SQUID probe, you can determine a key parameter of the device, namely the parameter. We define it by:

(3.1)

where L is the inductance of the SQUID loop.

Earlier we mentioned that the modulation voltage depth D V as measured from the V-F curve is related to the