"3. Don't forget to save the data in between measurements! Even better, safe the data frequently while measuring and keep track of the file names in your logbook!\n",
"\n",
"4. Repeat warm up and cooldown five times for good statistics. Answer questions A to C.\n",
"4. Repeat warm up and cooldown at least five times for good statistics. Answer questions A to C.\n",
"\n",
"Measuring the critical current at different temperatures. Difficult part!\n",
"Measuring the critical current at different temperatures. Difficult part! You are welcome to try this at the end of your measurements sessions! \n",
"\n",
"5. Put the rod back on top of the dewar and rise the probe as high as possible. Lower the probe in equilly spaced steps into the liquid nitrogen (say 5-10 cm steps). Here, the nitrogen gas has a certain temperature gradient. Meaning the temperature will change the more you go down. \n",
"\n",
...
...
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# Mr. SQUID Manual
## The Josephson junction: Quantum tunnelling and interference in an electrical circuit
### by Jasper Franse and Gary A. Steele (December 3, 2020)
Be award that an old version of the Mr. SQUID Manual can be found in the TN2953-P QN Squid Practicum Teams -> General tab. The theory is the same as here, but the questions are out dated. Please ignore this file.
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# Contents
### 1. Introduction
### 2. Theory
2.1. Superconductivity
2.2. Superconductivity
2.3. Josephson Junctions
2.3.1 DC and AC Josephson Effects
2.3.2 DC SQUID (Direct Current Superconducting QUantum Interference Device)
### 3. Experiments
3.1 Equipment
3.2 Mr. SQUID probe: A closer look
3.3 Experiments:
3.3.1 A: Observing Superconductivity
3.3.2 B: Measuring Mr. SQUID's Critical Temperature
3.3.3 C: Observing the Flux Quantization
3.3.4 D: Proving Josephson Tunnelling
### 4. Practicum Requirements
4.1 General requirements
4.2 Report expectations
### 5. Literature
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## Chapter 1
# Introduction
Superconductivity was first discovered in 1911 in a sample of mercury metal that lost its electrical resistance just four degrees above absolute zero. The phenomenon of superconductivity has been the subject of both scientific research and application development ever since. The ability to perform experiments at temperatures close to absolute zero was rare
in the first half of the 20th century and superconductivity research proceeded in relatively few laboratories. The first experiments only revealed the zero resistance property of superconductors, and more than twenty years passed before the ability of superconductors to expel magnetic flux (the Meissner effect) was first observed. Magnetic flux quantization
\- the key to the operation of superconducting quantum interference devices (SQUIDs) \- was predicted theoretically only in 1950 and was finally observed in 1961. The Josephson effects were predicted (nobel prize for Brian Josephson in 1973) and experimentally verified a few years after that. SQUIDs, essentially two Josephson tunnel junctions incorporated into a superconducting loop, were first studied in the mid-1960s, soon after the first Josephson junctions were made. Practical superconducting wire for use in moving machines and magnets also became available in the 1960s. For the next twenty years, the field of superconductivity slowly progressed towards practical applications and to a more profound understanding of the underlying phenomena. A great revolution in superconductivity came in 1986 when the era of high-temperature superconductivity began (nobel prize 1987 for Bednorz and Mueller). The existence of superconductivity at liquid nitrogen temperatures ($T \approx$ 77 K) has opened the door to applications that are simpler and more convenient than what has ever been possible before. Nevertheless, the product you have in your hands today was made possible by many aspects of the more than 100 years of discovery that preceded it. Today, there is a wide variety of fields where SQUIDs are applied in, as for example for electrical measurements, thermometry, magnetic field sensors, geophysical measurements, biomagnetism and more recently for quantum science and quantum information technology.
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## Chapter 2
# Theory
### 2.1 Superconductivity
There are certain materials -- actually, many thousands of them by now -- that exhibit a remarkable transition in their ability to pass electrical currents: when they are cooled down to a sufficiently low temperature, the value of which depends on the material, their electrical resistance completely vanishes.
How this behaviour comes about was a mystery that occupied the minds of theoretical physicists for nearly half a century after it was first observed.
The answer turned out to be tied to the quantum-mechanical nature of solids, in particular, to the tendency of electrons to become paired.
All of these so-called "Cooper pairs" behave cooperatively and form a single quantum-mechanical state.
When the temperature is extremely low, the highest energy level which is occupied with free electrons in a normal metal is known as the Fermi level.
As the temperature starts to increase, these electrons will undergo excitations due to this increment over an energy range of $k_\mathrm{B}T$, leading to thermal occupation of energy levels above the Fermi level ($k_\mathrm{B}$ in the given expression is the Boltzmann constant and $T$ the temperature).
This zone is called conduction band and its electrons are responsible for transporting electrical charges.
When there is a flow of a current in the metal, or, in other words, when there is movement of electrical charges, the electrons will interact with the ions of the crystal lattice, causing displacements of their lattice positions by Coulomb interaction and hereby generating vibrational waves (phonons).
These processes are mainly responsible for the resistive behaviour of metals and the decrease of energy of their conducting electrons.
When it comes to superconducting materials, the situation is different and their behaviour can be explained based on the BCS theory (named after the theoretical physicists Bardeen, Cooper and Schrieffer, nobel prize in 1972).
All superconductors have a transition temperature $T_c$, at which their behaviour undergoes a big switch-over.
Below this temperature ($T < T_c$) the electrons in the conduction band fomr pairs known as Cooper-pairs or BCS pairs.
In this case all the pairs are described by a single quantum-mechanical wave-function and there is no interaction with free electrons.
For this reason there is no energy loss while interacting with the lattice (in the particle picture each scattering event is compensated by an opposite event of the Cooper partner) and there is no sign of an Ohmic behaviour when there is flow of a current (this is then called a supercurrent).
In addition, it is important to mention that the un-paired electrons, also known as quasi-particles, do not participate in the charge transfer unless the maximally possible Cooper-pair current is exceeded.
In this case ($I > I_c$), the material is no longer superconducting and an increase of the current will generate an increase in voltage.
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### 2.2 Superconducting Rings
A closed superconducting ring is a particularly convenient system to study for understanding the properties of superconductors.
It is also the basis of the SQUID.
Consider the following experiment.
We cool a ring of superconductor in a small magnetic field that corresponds to one flux quantum ($\Phi_0 = h/2e \approx 2.067 \cdot 10^{-15}\,$Wb) threading the ring.
(The magnetic flux through a loop area $A_l = \pi r^2$ in a homogeneous perpendicular magnetic field $B$ is given by $\Phi = B\cdot A_l$).
We now have a superconducting ring threaded by a single flux quantum.
Suppose we now turn off the applied field.
According to Faraday's law of induction, the moment that we change the flux threading the ring, a current flows around it.
This current tries to oppose the change in magnetic field by generating a field to replace the flux we removed.
In an ordinary metal, that current would rapidly decay due to Ohmic dissipation.
In a superconductor, something entirely different happens.
If the induced current decreased just a little bit in the ring, then the flux threading the ring would be a little less than a flux quantum.
This is not allowed.
The next allowable value of flux would be zero flux.
Therefore, the current would have to abruptly cease rather than decay away.
Because the superconducting state is composed of an enormous number of electrons that are paired up and occupying the same quantum state, a current reduction of the sort needed would require all the electrons to jump into another state simultaneously.
This is an extraordinarily unlikely event.
Practically speaking, it will never happen.
As a result, the current induced in a superconducting ring will flow infinitely long.
People have actually tried this experiment for years on end.
As long as the ring is kept cold, the current flows without resistance.
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### 2.3 Josephson Junctions
The Josephson effect is yet another manifestation of what we call the long-range quantum coherence of superconductors.
The simple picture of this is as follows: Two regions of superconducting material are placed very close to one another as shown in Fig. 2.1.
**Figure 2.1: Schematic diagram of two superconducting regions separated by a thin gap**
In both regions, the superconductiong charge carriers are described by a quantum wave function $\Psi_i = |\Psi_i|e^{i\phi_i}$ with an amplitude $|\Psi_i|$ and a quantum phase $\phi_i$ ($i = 1, 2$).
The phase on the left side of the gap is $\phi_1$ and on the right side it is $\phi_2$.
In a single piece of superconductor, the phases at two different positions have a specific relation to one another (macroscopic phase coherence).
This arrangement assures a lower energy ground state that results in superconductivity.
In the picture of Fig. 2.1, what will the phases of the two superconductors do?
As quantum mechanics predicts that the wave functions do not abruptly decay to zero at the boundary of the superconductors, but slightly extend into the outside region with an exponential decay.
Thus, the wave functions of the two pieces will overlap in the gap region and the surprising answer to the above question is that if the gap region is small enough, $\phi_1$ and $\phi_2$ will be related.
Practically speaking, this means that Cooper pairs from one superconductor can tunnel through the gap into the second and vice versa.
In other words, the two superconductors will essentially act like a single one, but with a small region where the ability to carry a supercurrent is reduced.
Still, electrical (tunnel) currents can flow between the two regions with zero electrical resistance.
Such tunnel currents are called Josephson currents and physical systems composed of two regions of superconductor connected via such a weak link are called Josephson junctions.
Strictly speaking, the resistance-less currents that flow in a Josephson junction are a manifestation of the DC (direct current) Josephson effect; a second property of Josephson junctions by which the Cooper pair current oscillates with high frequencies through the gap is called the AC (alternating current) Josephson effect.
Josephson junctions are the essential active devices of superconducting electronics, much as the transistor is the essential active device of semiconductor electronics.
Junctions can be used in a variety of electronic circuits as switching devices, sensors, variable inductors, oscillators (because of the AC Josephson effect), and other applications.
People have built Josephson electronic circuits that contain up to a million junctions.
At the opposite extreme, one of the most useful circuits made from Josephson junctions is the DC SQUID, which contains only two junctions.
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### 2.3.1 DC and AC Josephson Effects
With the phase difference $\delta = \phi_2 - \phi_1$, the two Josephson relations describing the supercurrent through and the voltage across a Josephson junction are given by
\begin{eqnarray}
I & = & I_c\sin{\delta} \tag{2.1}\\
V & = & \frac{\hbar}{2e}\frac{\partial \delta}{\partial t} \tag{2.2}.
\end{eqnarray}
An approach to study the phase state of a Josephson junction is using the equivalence of the equations of motion of a Josephson junction to the motion of a particle in a sinusoidal potential (a so-called washboard potential).
**Figure 2.2: Potential of a Josephson junction with no current (a) and a current (b) flowing**
(For the full equivalence between the two systems, the junction capacitance and an electrical resistor as damping term is included as well.)
For a junction, which is not biased with a current or a voltage, the potential energy landscape can be represented as a cosine potential as shown in Fig. 2.2 (a) and the particle just rests at $\delta = 0$.
The particle position in this model corresponds to the phase difference across the junction $\delta$.
Accordingly, the particle velocity correponds to the voltage across the junction.
When we apply a DC current to the Josephson junction, the potential in this model experiences a tilt as shown in Fig. 2.2 (b).
Even though a current is flowing through the junction now, however, the particle is still trapped in the potential minimum at a constant phase difference given by the first Josephson relation and its velocity is zero.
Thus, there is no voltage drop across the junction, which corresponds to a dissipationless current flow.
The tilt of the potential is given by the magnitude of the current, i.e., when increasing the current, the tilt gets larger.
So if we keep increasing the current through the junction, the potential will tilt further and further up to a point when the well trapping the particle vanishes and the particle starts to roll down the hill.
This critical tilting point is called the critical current $I_c$ of the Josephson junction and above this value, a voltage will drop across the junction.
A typical current-voltage characteristic of a Josephson junction obtained by sweeping the current is shown in Fig. 2.3.
**Figure 2.3: Characteristic function (V - I)**
Next, we assume that we first apply a DC current which is below the critical current.
As discussed above, the particle is then trapped and in rest.
But in addition, we send an AC current with an angular frequency $\omega$ through the junction now, which has the effect of shaking the potential around its equilibrium position as indicated in Fig. 2.4.
In an intuitive picture, the particle can then hop from one potential to the next during one shaking period, when the total current is above the critical current of the junction during the oscillation.
Depending on the parameters, i.e., on the amplitude of the two currents, the particle can also hop two wells per oscillations or three or four.
These hopping processes, when the particle moves a fixed number of wells, i.e., a fixed distance in phase, per oscillation period, lead to a step-like structure in the current-voltage characteristics and the occuring voltage steps, cf.
**Figure 2.4: Characteristic IV with visible (continuos line) and not visible (dashed line) shapiro steps**
We can easily calculate the height of these voltage steps.
The voltage is given by the second Josephson relation and the time derivative of the phase can be easily calculated using the washboard potential model.
When the particle moves a fixed number $k$ of wells per oscillation, its average velocity is $2\pi k/T$, where T is the time for one oscillation.
As $T=2\pi/\omega$, the average velocity is given by $\partial\delta/\partial t = k\omega$ and the $k$th voltage step has the height
\begin{equation}
V_k = k\frac{\hbar\omega}{2e}. \tag{2.3}
\end{equation}
This relation, which translates a given frequency to a voltage just by physical constants, is nowadays used to define a voltage standard.
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### 2.3.2 DC SQUID (Direct Current Superconducting QUantum Interference Device)
The DC SQUID is actually a rather simple devcice.
The device operation is essentially the same regardless of whether the SQUID is constructed using low-temperature superconductor (LTS) or a high-temperature superconductor (HTS) materials.
It consists of two Josephson junctions connected in parallel in a superconducting loop.
**Figure 2.5: A schematic representation of a DC SQUID with a magnetic flux crossing it**
As we have said above, a fundamental property of superconducting rings is that they enclose magnetic flux only in multiples of a universal constant called the flux quantum.
Because the flux quantum is very small, this physical effect can be exploited to produce an extraordinarily sensitive detector for changes in magnetic field (or anything that can be converted into a change of magnetic field) known as the Superconducting QUantum Interference Device, or SQUID.
In the case of no magnetic field applied to the SQUID, the maximum DC supercurrent which can be carried by the SQUID is just twice the critical current of a single Josephson junction (for the case of identical junctions).
Assuming now that a magnetic field $B_a$ is applied to this system (see Fig. 2.5) and remembering the flux quantization insuperconducting loops, it is easy to understand that a screening current will circulate around the SQUID loop, which is also going through the Josephson junctions.
The relationship between the phase differences across the two junctions, $\delta_1$ and $\delta_2$, can be written as
Thus, the critical current of the SQUID periodically modulates between $2I_c$ and $0$ with a periodicity of one flux quantum.
As conclusion, it is allowed to say that the applied magnetic field has lowered the critical current of the SQUID, i.e., has reduced the maximum amount of bias current that can pass through the ring without generating a voltage.
Additionally, when one of the junctions goes to the resistive state, all the current would have to pass through the other one, making it go resistive as well.
At this point both of the junctions are resistive and a voltage can be detected across the SQUID terminals.
Another situation worth being described here is what happens when the magnetic flux is increased from zero to illustrate the periodic behaviour.
In this case the screening current increases as well keeping the total flux at $0\Phi_0$, until it is energetically favorable for the SQUID to let one flux quantum into the loop and switch the direction of the screening current to fulfill $\Phi = 1\Phi_0$.
When the magnetic flux is then further increased towards one external flux quantum, the screening current decreases until one full flux quantum is applied and the screening current is zero.
This pattern repeats then with the periodicity of $\Phi_0$, the resulting current is plotted in Fig. 2.6.
**Figure 2.6: Variation of the screening current of a SQUID with the magnetic flux**
Remembering that the critical current depends on the screening current, and knowing that this last on has a periodic dependence on external magnetic flux, we conclude once again that the critical current shows a periodic modulation with magnetic flux as well.
As illustrated in Fig. 2.7, this modulation of the critical current can be used to detect small changes of magnetic flux by biasing the SQUID above the critical current and measure the voltage across it.
**Figure 2.7: Modulation of the voltage across a SQUID with the magnetic flux**
When the critical current is changing, also the voltage at a fixed bias point slightly above the critical current changes due to the nonlinear I-V characteristics in this regime.
Thus, also the voltage in the resistive state will show a periodic modulation with external flux.
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## Chapter 3
# Experiments
The experiment will be divided into parts and for each of them a few questions should be answered. The students should have answered the theory questions before starting the experiments.
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### 3.1 Equipment and schematic
* Mr. SQUID probe
<imgsrc="figs/probe.jpg"width=500px;height=700px>
**Figure 3.1: Mr.SQUID probe**
Mr. SQUID is a DC Superconducting QUantum Interference Device (SQUID) magnetometer incorporating a high-temperature superconductor (HTS) thin-film SQUID chip, two feedback coils to modulate the SQUID and to couple an external signal to the SQUID, a cryogenic probe with a removable magnetic shield, an electronic control box containing all the circuits needed to operate the SQUID, and a cable to connect the probe to the electronics box.
The probe is designed to be immersed in a liquid nitrogen bath that should be provided by the supervisors.
* Red Pitaya
**Figure 3.2: Red Pitaya**
The Red Pitaya is an open-source hardware project intended to be alternative for many expensive laboratory measurements and control instruments. Here, the Red Pitaya is programmed with Python to sent over signals to the SMA (SubMiniature version A) output ports and receive the data from the Mr. SQUID probe back at the SMA input port.
Extra information can be found on their website: https://www.redpitaya.com/
* Arduino Due
**Figure 3.3: Arduino Due**
The Arduino is an open source microcontroller. This platform is aimed at hobbyists, performers, artists and anyone interested in creating and designing smart and creative objects. With Arduino it is possible to create devices and objects that respond to their environment by means of digital and analog input signals. In our setup, we connect the Arduino (Due) to the PT1000 sensor (see below) to measure the temperature of the Mr. SQUID probe.
Extra information on Arduinos can be found here: https://www.arduino.cc/ .
The temperature sensor is a thermistor that changes its resistance with temperature and it will be used in experiment C to measure Mr.SQUID's critical temperature.
The conversion factor is $0.385\,\Omega/^{\circ}{\rm C}$ with $100\,\Omega$ at $0^{\circ}{\rm C}$.
**Figure 3.5: Measurement setup where the Red Pitaya and Arduino are both controlled by the measurement PC and sent (receive) data to (from) the Mr. SQUID probe or, respectively, PT1000 temperature sensor.**
Here, both the Arduino Due and Red Pitaya are controlled by the measurement PC. The Arduino is connected to a twisted pair cable that goes down the dewar, which is filled with liquid nitrogen. Here the twisted pair cable is connected to a PT1000, which is placed at the same height as the SQUID probe. Note that a voltage divider (with a 1.1 k$\Omega$ resistor) is placed in the circuit to read out the unknown resistance of the PT1000 sensor.
The input and output port of the Red pitaya are connected to the X and Y port of the Mr. SQUID control box. In turn the Mr. SQUID box sents/receives a signal, via the cable, to the Mr. SQUID probe placed inside the dewar.
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### 3.2 Mr. SQUID probe: A closer look (Old BOX! Explanation and description still relevant)
This section is meant to give a quick explanation of the procedures for operating the Mr. SQUID probe and the features of the Mr. SQUID control box. These are also explained in the videos inside of the general folder of Microsoft Teams folder.
The Mr. SQUID electronics box is designed to provide all the electronics necessary to observe the basic functions of a DC SQUID.
Included is a low-noise amplifier section that amplifies the output voltage of the SQUID (with switch-configurable gains of 100, 1,000 and 10,000; the latter is the factory-configured default setting for Mr. SQUID and already taken into account in the measuring software Gary built), current driver circuits to bias the SQUID and drive the feedback coil, and the switching required for the various functions.
An triangle wave test signal (generated by the Red Pitaya) is used to display the SQUID V-I and V-$\Phi$ characteristics in the measuring software.
The default frequency is 20 Hz but may be adjusted. For the once who are interested are welcome to look around in the code of the measuring software to see what frequencies are possible.
The front panel controls and connectors on the previous version of the Mr. SQUID electronics control box are shown in the figure below:
**Figure 3.6: Front panel of the Mr.SQUID electronics control box (previouse version, picture needs to be updated!)**
POWER Switch: A Three-position toggle switch that selects power ON to the system (up), power OFF to the system (middle) or test battery (down).
OUTPUT DISPLAY: A two-position toggle switch that selects between a 2-channel oscilloscope display (up) or X-Y display (down).
MODE Switch: A two-position knob that selects between the V-I (left) and V-$\Phi$ (right) modes.
* In V-I mode, the triangle wave test signal is applied across the SQUID bias terminals.
* In V-$\Phi$ mode, the triangle wave test signal is applied to a feedback coil that is inductively coupled to the SQUID.
VOLTAGE Output (X port <-> port 1 of Red Pitaya): A BNC connector providing the amplified voltage across the SQUID.
The Mr. SQUID electronics box includes an amplifier circuit that can be configured for a gain of 100, 1,000 or 10,000 using an internal switch.
For use with Mr. SQUID, the default factory-configured gain setting is 10,000.
Thus, the actual voltage across the SQUID is the voltage at the VOLTAGE output divided by 10,000.
(i.e., 1 V at the VOLTAGE output corresponds to 100 $\mu$V across the SQUID).
CURRENT Output (Y port <-> port 2 of Red Pitaya): The CURRENT output signal is the voltage output of an operational amplifier inside the Mr. SQUID electronics box that measures the voltage drop across a 10$\Omega$ resistor through which the current flows.
According to Ohm's Law (I = V/R), the current flowing through this resistor is equal to the voltage across it divided by the resistance, 10$\Omega$.
The operational amplifier is configured with a gain of 1,000 so a voltage of 1 V at the CURRENT monitor output corresponds to a current of (1/1,000)/10 = 1/10,000 A or 100 $\mu$A.
* In the V-I mode, the CURRENT output represents the current through the SQUID (the sum of the triangle wave plus the DC bias offset current set by the CURRENT OFFSET control).
* In the V-$\Phi$ mode, the CURRENT output represents the current through the feedback coil (the sum of the triangle wave plus the DC flux offset current set by the Flux Bias control).
AMPLITUDE: Sets the amplitude of the triangle wave test signal in either the V-I or V-$\Phi$ mode. In either mode, use the AMPLITUDE knob to set the width of the test signal sweep. In V-I mode, the triangle wave is applied to the bias (I) terminals of the SQUID; in V-$\Phi$ mode, the triangle wave is applied to the internal feedback coil.
CURRENT BIAS: Applies a positive or negative DC bias current to the SQUID. In the 12 o'clock position, this current is approximately zero. This control is used to apply a DC bias offset current to the SQUID.
FLUX BIAS: Applies a positive or negative DC current to the internal feedback coil, which produces a DC magnetic field that is coupled to the SQUID. In the 12 o'clock position, this current is approximately zero. This control is used to modulate the critical current of the SQUID manually by the application of an external magnetic flux produced by the current in the feedback coil.
The rear panel of the Mr. SQUID control box is shown in the figure below:
**Figure 3.7: Back panel of the Mr.SQUID electronics control box (previouse version, picture needs to be updated!)**
POWER: A five-pin DIN socket for the external $\pm$12 VDC power supply (or optional battery pack, which is used in this setup!!).
PROBE: Nine-pin DB-9 socket for connections to the Mr. SQUID probe.
EXT INPUT: BNC connector to couple an external voltage signal to the external feedback coil on the Mr. SQUID chip. A MODE switch inside the Mr. SQUID electronics box (accessible by removing the top cover) selects whether this signal is routed directly through a 100 mA fuse at location F1 inside the box to the external feedback coil on the Mr. SQUID chip (switch position DIR) or is converted to true differential using a buffer amplifier and then routed to the feedback coil (switch position BUF). The current output from the buffer amplifier (i.e., the current applied to the external feedback coil) is 100$\mu$A/V. The buffered configuration is the default set at the factory. (NOT NEEDED FOR THE EXPERIMENTS!)
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### 3.3 Experiments
For a better understanding of the experimental procedures and handeling of the software, please watch the videos in the general folder on Teams. Go to Microsoft Teams(web app) -> TN2953-P QN Squid Practicum -> General -> Files -> Instructional Videos. To save time, watch the videos before your first measurements session!
### 3.3.1 A: Observing Superconductivity
**Goal**:The goal of this experiment is too cool down the MR.SQUID probe and to observe the flow of a supercurrent, that is the zero resistance flow of cooper pairs.
### Experimental procedure
1. Open the Mr. SQUID software as shown in the videos in Teams
2. Confirm that the Mr. SQUID control box is turned on, set to V-I and Oscillations (not X-Y). Before you start adjusting and find the sweet spot for your IV curves, place the knobs flux bias, current bias and amplitude in the 12 'o clock position.
3. Confirm that the Red Pitaya is connected to the PC and to the MR. SQUID control box at the X and Y port.
4. Double check if the Arduino is seen by the PC. Go to the Arduino software on the Desktop. Go to tools/port and double check if it is set to "COM7(Arduino Due (programming port))". If not double check the USB cable in the Arduino.
5. Check if the Nine-pin DB-9 cable is plugged in and goes from the Arduino to the rod with the probe.
6. Check if the Mr. SQUID probe is covert by a magnetic shield (black metalic tube) and aluminum foil (for electric shielding). When the aluminum foil is already covering the probe DO NOT remove it! If you want to have a better look at the probe, you can find a spare probe in the room. In addition, we posted micrographs of the probe in the General channel of Teams.
7. Before lowering the probe into the liquid nitrogen, start up the SQUID measurements software as shown in the videos. Take an V-I trace (IV live display cell) and answer questions A and B below.
8. Now it's time to cool down Mr.SQUID: At this point you should already have received the liquid nitrogen. Put on the goggles and cryogenic gloves. Carefully (as shown in the videos) move the dewar to the designated place near the measurement equipment. After this the probe can be slowly dipped inside of the dewar.(Fig. 3.8)
<imgsrc="figs/dewar.jpg"width=300px;height=500px>
**Figure 3.8: Photo of the measurement setup**
9. The V-I curve should appear on your IV live display in the Jupiter Notebook. The CURRENT BIAS can be used to symmetrize the trace, if necessary, and the FLUX BIAS can be adjusted to maximize the critical current. Answer all other questions following question B. A typical Mr. SQUID V-I curve is shown in the figure below:
<imgsrc="figs/iv.png"width=400px;height=500px>
**Figure 3.9: A typical Mr. SQUID V-I characteristic (50 $\mu$A/$\Phi_0$} horizontal, 200 $\mu$V/div vertical) (previouse version, picture needs to be updated!)**
### Questions
A. Plot V versus I. What do you see? Explain why we see this behaviour for a superconducting material at room temperature.
B. What is the normal state resistance of the SQUID?
C. Since the voltage and current values needed to operate the SQUID are very small, the Mr.SQUID electronics box includes amplifiers. This means the values of voltage and current that you observe on the oscilloscope are higher compared to the ones measured on the SQUID. By consulting the Mr.SQUID user guide, write down the current and voltage conversions. Fortunately, these conversions are already included in the software.
D. When the cable of the temperature sensor (going from the Arduino) is attached to the rod and the probe is cooled down, plot the V-I curve. Is this what you would expect from a superconducting material? Explain why.
E. Let's detach the temperature sensor cable for now and assume the probe is at liquid nitrogen temperature (~77 K). Plot the V-I curve once more. It seems that the curve has changed around the origin. Try to explain what the influence of the cable is on your measurement.
F. Until a certain current is reached, there is no increase in voltage. What explains this behaviour? And how low is this resistance?
G. What was the measured critical current? And what is the definition of this current?
P.S. Don't forget to save all your plotted curves!
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### 3.3.2 B: Measuring Mr.SQUID's Critical Temperature
**Goal**:The goal of the experiment is to measure the critical current of the Mr.SQUID probe for different temperatures and estimate its critical temperature. (The temperature where the superconductivity is destroyed.)
### Experimental procedure
1. After having measured a IV curve, reconnect the temperature sensor cable and slowly start to remove the Mr. SQUID probe from the dewar (as shown in the videos). After you made sure there's no liquid Nitrogen leaking out of the aluminum foil socket, place the probe + rod on the table. Start the temperature trace recorder software and put the measurement delay (s) to 0. For more information look at the video in Teams! Make sure you start the trace on time so the temperature will rise from a equilibrium low (liquid nitrogen) to equilibrium high ($T_{Room}$) temperature point.
2. Wait for the probe to reach room temperature (RT) while taking data with the temperature trace recording software. When the RT is reached, save your data, refresh the temperature trace recording software, start the measurement and place the probe back in to the liquid Nitrogen dewar. Wait till the temperature reaches equilibrium (temperature of liquid Nitrogen).
3. Don't forget to save the data in between measurements! Even better, safe the data frequently while measuring and keep track of the file names in your logbook!
4. Repeat warm up and cooldown five times for good statistics. Answer questions A to C.
4. Repeat warm up and cooldown at least five times for good statistics. Answer questions A to C.
Measuring the critical current at different temperatures. Difficult part!
Measuring the critical current at different temperatures. Difficult part! You are welcome to try this at the end of your measurements sessions!
5. Put the rod back on top of the dewar and rise the probe as high as possible. Lower the probe in equilly spaced steps into the liquid nitrogen (say 5-10 cm steps). Here, the nitrogen gas has a certain temperature gradient. Meaning the temperature will change the more you go down.
6. Lower the probe in equilly sized steps down into the liquid nitrogen. For each step wait for the temperature to reach equilibrium (take a look via the temperature trace software). Then for each step, i.e. different temperture, record the V-I curve using the IV live display and save the plot. Don't forget to write in your notebook which plot is for which temperature!
7. Do this for multiple temperatures. Lower and rise the probe once! Answer the remaining questions.
### Questions
A. Plot R vs T for several cool downs and warm-ups in one graph (so one graph for one cooldown + one warm-up). At what temperature becomes the SQUID superconducting (or is the superconductivity destroyed when warming up? When the temperatures don't agree which each other, try to explain what is going on.
B. Plot the average R vs T for both the warm-up and cooldown in one graph.
C. What happens to the critical current of the SQUID during the loading/unloading procedure?
D. Determine the critical current for multiple temperatures and make a table.
E. Plot the critical current of the SQUID vs temperature (in Kelvin) separately for the loading and unloading experiment.
F. What is the critical temperature of Mr.SQUID and what happens to the probe when this temperature is reached?
G. Compare the obtained critical temperature for the loading and unloading procedure and the theoretical expected value.
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### 3.3.3 C: Observing the Flux Quantization
**Goal**: The goal of the experiment is to observe a typical voltage modulation with magnetic flux and verify that the periodicity of such modulation is one flux quantum. Afterwards we will see how this effect can be used for high precision current detection.
### Experimental procedure
1. Turn down the AMPLITUDE and use the CURRENT BIAS knob to bias the SQUID just above the critical current (at the knee of the V-I curve). This will be the most sensitive point in the curve.
2. You can now manually modulate the SQUID with the FLUX BIAS. The point on the oscilloscope screen will move up and down in response to changing the FLUX BIAS.
3. Move the MODE switch to the V-$\Phi$ (right) position. Increasing the AMPLITUDE sweeps the current (different from the current in step 1) through the feedback coil, which produces a magnetic field that couples to the SQUID.
4. As the AMPLITUDE is increased, a V-$\Phi$ curve similar to Fig. 3.10 should appear on your display (for more information look at the videos in Teams).
5. The modulation depth is smaller than the voltages you were measuring before, so increase the vertical sensitivity of your oscilloscope, and, if your oscilloscope can be AC coupled, use this mode for the vertical axis. You may also want to tweak the CURRENT BIAS to maximize the modulation depth. Adjusting the FLUX BIAS will allow you to select a region of the V-$\Phi$ curve for observation.
6. A typical Mr. SQUID voltage-flux characteristic appears in the following figure: Depending upon the particulars of the individual SQUID and coil in your probe, you should be able to see at least four or five oscillations on your output device.
<imgsrc="figs/flux.png"width=400px;height=500px>
**Figure 3.10: A typical Mr. SQUID V-$\Phi$ characteristic (50 $\mu$A/$\Phi_0$ horizontal, 10 $\mu$V/div vertical).)**
### Questions
A. Show how the voltage across the SQUID modulates with magnetic flux. Plot V vs flux bias current!
B. Present a diagram where you explain what is happening in the SQUID probe during the experiment.
C. What happens if you bias the SQUID to a point far below or far above the critical current? Explain why you choose a point close to the critical current to observe a V-$\Phi$ curve?
D. How smooth is the curve you just measured? How much is the noise compared to the measurement? (In this point show how you fit your data in order to get this value)
E. Remeasure the V-$\Phi$ curve for different values of CURRENT BIAS. What happens?
Make a plot of $\Delta$V vs CURRENT BIAS.
F. What is the sensitivity of the device (in A/$\sqrt{\mathrm{Hz}}$)? (What is the minimum change in $\Delta$V generated by sweeping the AMPLITUDE)
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## Chapter 4
# Practicum Requirements
For the Mr.SQUID practicum, the supervising team has a set of requeriments that each group should fulfill.
### 4.1 General requirements
1. Before the measurements are started the students will need to contact the supervisors to schedule an online introduction meeting. It is expected that the students have at least read through the introduction and theory section of the Mr. SQUID manual. Any further questions of the students (on the theory) can be discussed during the online introduction meeting.
2. During the meeting we will schedule:
* 3 half days for the experiments
* an intermediate meeting to discuss progress and plan for the rest of the experiments.
* when to hand in the concept report
* when to hand in the final version of the report
3. The group should work at least 3 half days on the experiments. The schedule for the working hours should be arranged together with the supervising team ahead of time. The group is allowed to work longer on the experiments after the half day is over. However, supervision isn't guaranteed to be quick in response outside of the scheduled half days.
4. The group should always follow the provided safety rule sheet when handeling liquid nitrogen. For better instructions look at the videos in the general Teams folder.
5. The group should keep a logbook where the results of the different experiments are documented.
6. The group should always explain to the supervisors the concept of the experiment and the experimental working plan before starting each experiment.
7. The contact with the supervising team should preferabily be done via teams, email or whatsapp. The phone number is provided below:
- Jasper Franse: 06-57997843
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### 4.2 Report expectations
1. An abstract, introduction and theory chapters.
2. An experimental chapter, where, for each experiment you:
- Formulate a research goal
- Describe the set-up and measurement scheme for each of the experiments with corresponding photos
- Show the obtained data and describe your observations
- Answer the questions at the end of each experiment chapter
- Compare your results with the expected theory values
3. A conclusion where you present a summary of your results and if the goal of the experiments was achieved.
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## Chapter 5
# Literature
1. J. Bardren, L. N. Cooper and J. R. Schrieffer, "Theory of Superconductivity", Physical Review, Vol. 108, N.5, pp.1175-1204, December 1, 1957
2. Barone and G. Paterno, "Physics and Applications of the Josephon Effect", John Wiley and Sons, 1982
3. J. Clarke and A. I. Braginski, "The SQUID Handbook", John Wiley and Sons, Vol I, 2005.
4. R. W. Simon, M. J. Burns, M. S. Colclough, G. Zaharchuk and R. Cantor, "Mr.Squid User's Guide", STAR Cryoelectronics, LLC 25 Bisbee Court, Suite A Santa F, 2004
5. Aviv, "SQUIDS - Superconducting Quantum Interference Devices", University of Negev, 2008
