Towards sub-mK effective electron
temperatures through inductive
measurements
THESIS
submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
in PHYSICS
Author : Matthijs van Weeren
Student ID : 1305425
Supervisor : Tjerk Oosterkamp
2ndcorrector : Jan van Ruitenbeek
Towards sub-mK effective electron
temperatures through inductive
measurements
Matthijs van Weeren
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
December 9, 2019
Abstract
In this research, we aim to achieve sub-mK effective electron measurements to better analyze effects that occur at these very low temperatures. We do this by using a Faraday cage, through which we send a signal using sets of inductors. In order to test this, we define an
effective frequency range for our signal by analyzing the theoretical electrical side effects that occur in our system. We perform multiple room temperature tests on our system within the defined frequency range, and
work towards testing the effectiveness of our system at millikelvin temperatures as well.
Chapter
1
Introduction
Achieving lower temperatures for experimental measurements has long been an important goal in many fields of physics. At lower temperatures, new interesting phenomena appear, which are fascinating to study. Nowa-days, achieving lower temperatures is still an important goal in various re-search. Reaching a temperature of 4K was groundbreaking in 1908, when Heike Kamerlingh Onnes managed to liquefy Helium at that temperature and discovering superconductivity. However, nowadays temperatures in the magnitude of millikelvin can be reached, with further research striv-ing to go even beyond that. Since Kamerlstriv-ingh Onnes achieved his feat in 1908 at the Leiden University, this area of research still has a particular importance in the university and as such a lot of research is carried out in this field. At the end of the last century much research was carried out at temperatures below 1 mK, in particular to study mixtures of He3 and He4. This research will focus on creating and evaluating a new setup to be used to do conductivity experiments on samples at temperatures down to 1 mK. Such experiments are particularly difficult because at such low tempera-tures, any outside noise coming in through e.g. electrical wiring can sig-nificantly affect the results’ accuracy and usually increases the effective electron temperature. To counter this, we propose a setup consisting of a closed copper cylinder which serves as a closed Faraday cage. To still cou-ple in a signal we place transformers at both ends, which enables signals to enter and leave the cylinder through their flux, which means no direct wire is coupled between the sample inside the cylinder and the outside world. Unlike ordinary transformers the two coils are on opposite sides of the wall of the Faraday cage. This setup may allow us to reach effec-tively lower temperatures, which give rise to a variety of effects, such as
the Shubnikov-de Haas effect, which will be explained in detail further. In this thesis we present a theoretical analysis of the concept, as well as some preliminary measurements.
Chapter
2
Theory
2.1
Shubnikov De Haas Effect
To have a sample to evaluate the effectiveness of the Faraday cage that is the topic of this research, we chose the Shubnikov-de Haas effect. This is an effect that occurs in quantum matter, most commonly a two-dimensional electron gas (2DEG). An electron gas is a gas in which the valence electrons of the atoms can be considered as free particles. The two-dimensional as-pect means that the gas is free to move in two dimensions, but is confined in the third, which gives rise to discrete energy levels, called Landau lev-els, separated by E = hωc, with ωc the cyclotron frequency of the
elec-trons. The Shubnikov-de Haas states that under sufficiently low temper-atures and in a sufficiently high enough magnetic field, the resistivity of a 2D electron gas sample will oscillate with the magnetic field strength B. Due to the applied magnetic field, the electrons in the sample will rotate in circular motion with the cyclotron frequency.
Because the cyclotron frequency is linearly proportional to the magnetic field strength B, the spacing between energy states will increase with higher B. As the energy levels themselves also rise with increasing B, they start to pass through the Fermi energy EF. As this happens, the material will start
to exhibit oscillations in conductivity (which is the inverse of resistivity) with B.
The reason why the Shubnikov-de Haas specifically is interesting to study is because of its low resistance, which allows for relatively high currents which will be challenging to measure through the walls of the Faraday cage, as well as the fact that it is noticeable even at low external magnetic
field strength. This is especially true for very low temperatures, which make the effect even more noticeable. These factors make the Shubnikov-de Haas effect a good candidate to study with our setup and the lower effective temperatures it should achieve.
In Figure 2.1 we show a plot of the Shubnikov-de Haas effect for different temperatures down to 23mK as presented by the group of Zumbuhl at the University of Basel.
Figure 2.1: This figure shows the Shubnikov-de Haas effect, dependent on the external magnetic field strength. It also shows the temperature dependency. [1]
The Shubnikov-de Haas effect has a variety of uses, which make it valu-able to be valu-able to achieve this effect with our setup. A primary use is the characterization of semiconductor devices.
In order to understand our setup and electron cooling in general, it is im-portant to take into consideration a number of side effects that have an effect on the possible bandwidth of our signal to pass through the cylin-der. Each effect has its own cut-off frequency and acts as either a low- or a high-pass filter, effectively creating a band-pass filter.
2.2 Experimental setup, Faraday cage 9
2.2
Experimental setup, Faraday cage
The goal of our setup is to separate samples from exterior effects, for which we needed to make use of materials and effects that would make this pos-sible. The basic principle of our setup is simple; we used a hollow copper cylinder closed off at both ends by copper plates that hold two pairs of inductors, one pair on each side. The inductors on the outer sides of the cylinder (see Figure 2.2) are connected to our signal source normally, but because of the separation between the inner and outer coils by metal foil, exterior harmful effects are filtered over a broad frequency range (up to THz). For a small frequency range, the signal we send in will still be able to enter the inner circuit through the flux generated by the outer coils, which induces a current in the inner coils.
Figure 2.2:Schematic drawing of the principle of our setup, the crossed blocks in-dicate a set of coils, the maroon component inin-dicates the sample. A small hole has been left in the exterior of the cylinder, to allow gas to escape from the cylinder in vacuum.
The equivalent circuit of the Faraday cage with the transformers coupling signals in and out of the Faraday cage wall is given in Figure 2.3.
Sample
SQUID Faraday cage
R
L1 L2 L3 L4 LSQ
Figure 2.3: Electric circuit diagram featuring the principle of our setup. Three of the coils making up the transformers have a resistance, because the coils are wound from copper wire. The last two coils are a superconducting coil wound from superconducting wire and a superconducting input coil on the SQUID chip.
2.3
Frequency range constraints
2.3.1
RL circuit
The first effect that comes to mind as having a big impact on the possible bandwidth of the signal frequency are the characteristics of an RL circuit. Because our setup consists of inductors and resistors, we are dealing with an RL circuit. RL circuits typically act as high-pass filters by definition, which in itself affects the bandwidth significantly. Finding the cut-off fre-quency of an RL circuit is simple using the formula
fRL = R
2πL (2.1)
with fRL the cut-off frequency of the filter, R the resistance in the circuit
and L the inductance of the coil in the circuit. Since we had already identi-fied a low-pass filter effect in our setup, we must make sure the high-pass filter effects do not completely overlap with these, or we would risk not
2.3 Frequency range constraints 11
having a good enough bandwidth for our measurements. If we take a typ-ical value of 50 Ohm for the resistance in the circuit, combined with our chosen inductors of 100µH we come to a cut-off frequency in the order of hundreds of kHz, with higher cut-off frequencies for higher resistance.
fRL = R
2πL =
50Ω
2π∗100µH ≈8∗10
4Hz (2.2)
This means the effect of the RL circuit would be noticeable throughout our measurements, unless we choose a very low resistance.
2.3.2
Eddy currents
Another significant impacting factor worth considering are Eddy currents. Eddy currents are an effect generated by magnetic fields positioned near conductive materials. If there is a physical motion of this magnetic field, or if the magnetic field changes in magnitude over time, such as when caused by AC currents, Eddy currents are created.
The basis of this is the fact that in conductors a current is induced when subjected to a magnetic field, as follows from Faraday’s law. Adding Lenz’s law to this, which states that a current induced in a conductor will always be directed in a way that the resulting magnetic field from this cur-rent will be opposed to the initial magnetic field that created the curcur-rent. This creates an undesirable repelling field that will, at least to some degree, affect in which frequency range we will effectively be able to operate.
Figure 2.4: Depiction of Eddy currents created in a conductive material with re-sulting magnetic field.[2]
The power loss due to Eddy currents is proportional to the squared fre-quency, as follows from the formula
P= π
2B Pd2f2
6kρD (2.3)
For low AC frequencies, the effect of the Eddy currents is negligible, but for higher frequencies, its effect becomes significant. Therefore the effect of the Eddy currents will function as a low-pass filter effect on our set-up, with higher frequencies being increasingly filtered out. The frequency range of this effect also varies with temperature, which is an important aspect for our measurements when making the switch from room temper-ature to measurements in liquid helium (4K).
The Eddy currents within our set-up will also give rise to other effects that will be significantly impactful on the frequency range in which we will be able to operate.
2.3.3
Skin effect
One of the resulting effects from the Eddy currents in our system is the skin effect. [3] The skin effect is a phenomenon that occurs in conducting metals. In essence, it means that the current density of alternating currents is not distributed along the metal evenly, but instead concentrates more around the surface, leaving the center with a lower current (and there-fore electron) density. The skin effect can be summarized in the following formula[4]
J = JSe−d/δ (2.4)
where JS is the current density at the surface, d is the depth from the
sur-face and δ is the so-called skin depth, a unique parameter based on the metal resistivity and permeability and on the frequency. This skin depth can be calculated by the formula
δ= s 2ρ ωµ = r ρ π f µ (2.5)
with ρ the resistivity of the metal, ω the angular frequency and µ the mag-netic permeability of the metal.
2.3 Frequency range constraints 13
Figure 2.5: Graphic impression showing the principle of the skin effect, shown here as a cross-section of a metal wire. It shows the current density difference between the surface of the wire and the center.[5]
The configuration of our setup is different from what the image shows. Instead of the skin effect in a metal wire, we deal with the skin effect in a flat metal plate, but the principle remains the same. The current density in the metal plates through which between our transformers is higher at the surfaces of these plates, with lower current density at the center.
The inhomogeneity caused by the skin effect has various consequences for measurements related to metal conductors. The primary result of the skin effect is that the resistance of the metal conductor will increase for higher input frequencies, as can be seen in the previous formulas. For higher frequencies, the skin depth δ will be lower, which results in the term e−d/δbeing lower and thus the current density will be lower as well. This means the resistance of the metal will be higher for higher frequencies. This means the skin effect functions as a low-pass filter on the transfer. The cut-off frequency of a filter is defined as the frequency at which the signal has decayed by a factor 1/e, i.e. when δ = d. We use a 100µm copper foil between our inductors, which means we can calculate the
cut-off frequency from the formula fcuto f f = ρ πµδ2 = ρ πµd2 = 1.68∗10−8Ωm π∗1.26∗10−6Ωs/m∗10−8m2 ≈400kHz (2.6) This result shows the cut-off frequency, taking into account the resistivity of copper at room temperature. If we take into account the restivity of copper at liquid helium temperatures ρ=0.62∗10−9, we get a cut-off fre-quency fcuto f f ≈ 16kHz, which is much lower than the cut-off frequency
at room temperature.
In conclusion we pose that the skin effect makes sure that signals, interfer-ence and noise above 400 kHz will be exponentially damped, if the wall of our Faraday cage is 100 micrometer at its thinnest parts. This cut-off will be at even lower frequencies considering that the resistivity of the copper is lower at cryogenic temperatutes. If the residual resistivity ratio RRR=100 (which is a conservative estimate), the cut off frequency will be 40 kHz. If these frequencies become impractically low, a different metal, such as brass can be used. It has a higher resistivity and a smaller temper-ature dependence.
2.3.4
Self-resonance of the transformer coils
The close proximity of the windings in the coils we use make for another potential impacting factor in our system. When the windings of an induc-tor are close together, as they mostly are, they will interact with each other, functioning as minuscule capacitors. This capacitance, together with the inductance, functions as an LC resonator, which one would like the avoid.
2.3.5
Cross-talk
Cross-talk is a phenomenon that creates undesired effects for electrical sys-tems. It occurs when one (part of) a circuit electrically connects with an-other in a way that is not intentional. This is because the electromagnetic fields generated by e.g. any electrical wire, can influence the signal in any other wire, unless it is properly protected against it. In order to shield wires from the effects caused by cross-talk, it would essentially be neces-sary to place some form of grounded material, usually a grounded wire, between the two wires that affect each other. By using a system based on inductance, we are especially vulnerable to these effects, as the magnetic flux of one inductor can induce a current in another inductor that it is not intended to be coupled with.
2.3 Frequency range constraints 15
To avoid coupling of a transformer on one side of the cylinder to the other side of the transformer, the transformers can be made gradiometric and if necessary they may be shielded.
2.3.6
Expected transfer
As we have now listed the possible effects that can occur and influence the transfer through the cylinder, we can try predicting what the results will look like. We know now that the signal will have the shape of a band-pass filter, as there are effects that act as a low-pass filter and effects that act as a high-pass filter. Combining the cut-off frequencies of these effects, we expect transfer to peak at around 100 kHz, sharply declining outside of this peak. However, this frequency range can change significantly with temperature.
To further quantify our expectations for the transfer we would get out of our cylinder, we decided to simulate our setup with a program called PSpice. This program allowed us to easily put together a basic electrical circuit and let us you simulate multiple parameters, such as voltage and noise. Below you can see what our setup looked like in PSpice.
Figure 2.6:Our cylinder circuit constructed in PSpice.
The figure shows the basic components of our cylinder setup, with two coupled pairs of inductors on both sides, in series with 1kΩ resistors. In order to attempt simulating the properties of the copper foil between the inductor pairs, we placed a single inductor-resistor (LR) circuit between
the inductor pairs to simulate the inductance and resistance of the copper foil. We came to the values of L and R by calculating these for a copper wire with a 100 µm diameter in a loop of the same diameter as our cylinder. For this, we use 3 separate coupling factors on both sides of the cylinder, two for the coupling of the copper foil to either inductor pair and one for the coupling between the two inductor pairs. This way, the LR proper-ties of our copper foil are approximately simulated, however other effects such as skin effect cannot be simulated properly. As such, this simulation is only shows a proper expectation for low-frequency transfer, whereas at higher frequencies the transfer goes down rapidly due to aforementioned effects. The K functions shown at the top of the circuit are used to couple the inductors with a coupling constant. We chose a coupling constant of k = 0.8 for the coupling between the copper foil and the inductor pairs, and k=0.7 for the coupling between the inductor pairs.
Note that in order for the PSpice program to run, every component must be connected to a ground, in this case we did this by connecting every separate ’block’ to the ground through a 1GΩ resistor.
1kHz 10 kHz 100 kHz 1 MHz 10 MHz 10 nV 1 mV 100 mV Frequency Signal
Figure 2.7: Result of our simulation, it shows a clear high-pass filter behaviour, however due to other effects, the actual transfer will quickly decline after about 100kHz, resulting in a much lower transfer.
Chapter
3
Experimental setup
3.1
Coils
As the inductor coils were a key component of our setup, it was impor-tant to choose their characteristics and their configuration carefully. We used three pairs of coils made of copper wire with an inductance of 100 µH, to serve as primary means of passing inductive signals through the cylinder. It was decided to use pairs instead of singular coils, to be able to reduce cross-talk to the other side of the cylinder. To this end, the coils we used were positioned in a way that each coil in a pair is wound in the opposite direction of the other coil. This serves to cancel out the effects of background magnetic fields, but leaves the coils the ability to pick up excitations from nearby fields, i.e. the incoming flux from the opposite coil pair.
Figure 3.1: Schematic drawing of the set of coils on top of a separating copper foil. As shown, the current flows through one coil in opposite direction to the other coil.
In order to pick up the weak signals from the current flowing in the circuit, we would use a SQUID (Superconducting Quantum Interference Device). In order to be able to use a SQUID, it is important that our fourth pair of coils be made of superconducting material.
It was important to decide which wire material to use for our supercon-ducting coils, as there are a few possible options. We opted to use niobium wire for our coils, as this is one of the most common superconducting ma-terials and is in ample supply. Of the available types of niobium wire, we used a single core niobium titanium (NbTi) alloy wire in a copper shield-ing, with a total wire diameter of 100µm. To decide on the number of windings we wanted our coils to have, we made a few rough calculations. The basis for this was that we would want the current generated within the SQUID to be in a desirable range. This is because in superconducting coils the flux cannot change, which means the coils will produce an exact opposite flux to the external flux that is applied. As a result, the current that is generated in the SQUID follows the simple formula
NΦ =LtotI (3.1)
where N is the number of turns of our superconducting coils,Φ is the ex-ternal flux, Ltotis the total inductance given by Ltot =Lcoil+Lwire+Linput.
3.2 Lock-in Amplifier 19
for different N with an online program.
For an optimal signal-to-noise ratio of our signal, we want the inductance of our superconducting coils to be of similar value to the inductance of the input of the SQUID. This inductance has a value of roughly 1 µH, as previ-ously determined by previous research. To get an inductance of 1 µH in a coil of similar dimensions, that is with the same core diameter and length, we need N ≈ 20, which was also a manageable amount to manually cre-ate. However, we decided that it would be easier to manufacture single row coils, for which we needed N =14. This is a relatively low N, as one would think more windings means more flux, creating a stronger signal, but since the inductance scales with N2, this would actually decrease the signal strength in the SQUID.
3.2
Lock-in Amplifier
To measure our signal we used the SR7280 lock-in amplifier by Signal Re-covery. As one might know, a lock-in amplifier is a machine that is used to measure small signals at specific frequencies, as it is able to filter out other frequency signals and amplifies the signal at a desired frequency. This model had the advantage of giving us access to specially made LabVIEW sub-VIs (functional building blocks for LabVIEW programs), allowing us to create an easy-to-use program for our frequency sweeps and other mea-surements specifically tailored to the functions of this machine. However, the downside of this is that we were not able to easily transfer to a different setup and use the same program there.
3.3
Cooled measurements
As the goal of our research is to devise a system that will show effects on samples at low temperatures, it is important to properly test our system at low temperatures. The tests we did involve measurements at a tempera-ture of 4K, using a vessel of helium to suspend our system into. In order to do this, we use a so-called ’dipstick’. A dipstick is a long metal rod, con-taining cables that lead from one end to the other. On one end the cylinder is connected, while the other end has connectors to be used to connect to our lock-in amplifier in order to read out the signal.
3.3.1
SQUID
As mentioned before, in order to measure weak signals in our cooled sys-tem, we will use a SQUID. A SQUID is a small electrical circuit that uses the principles of a Josephson junction and superconductivity to measure magnetic fields with high sensitivity [6].
There are two types of SQUIDs, DC (direct current) and RF (radio fre-quency) SQUIDs. Invented in 1964, the DC SQUID is a circuit with the primary component being two parallel coupled equal Josephson junctions, used to split the current flow. Logically, the current will be split between these two Josephson junctions equally if there are no external effects in-volved.
However, when an external magnetic field is applied, it will generate a current, called a screening current, in the loop of the Josephson junctions in such a way that it cancels out the external magnetic field flux, as stated in Lenz’s law. The resulting current in the loop will result in one side of the loop having a higher current I = I0
2 +Is, while the other side has a
lower current I = I0
2 −Is, with Is the screening current. Once the current
in either one of the sides exceeds a specific critical current, there will be a voltage across that junction when electrons tunnel through the junction, which can then be measured.
3.3 Cooled measurements 21
Figure 3.2: Schematic of our two-stage SQUID as shown in the MAGNICON manual, it shows the two separate stages and the wires connecting them.
In our setup, the SQUID is shielded, as to prevent outside disturbances and is mounted in an open cylindrical housing, to fit our copper cylinder. Furthermore, our SQUID is a two-stage device, which means that on top of a regular SQUID that acts as a sensor to magnetic fields, it has a second SQUID that is used to amplify the signal.
In order to be able measure a transfer signal with a SQUID, we had to manufacture a special metal shielding for the superconducting end of our set-up as well, in order to prevent exterior interference with the supercon-ducting signal.
Figure 3.3:Photograph of the shielding on the superconducting end of the cylin-der, the wires are made of a superconducting metal, which would connect to the SQUID.
With this shielding, we would be able to create an entirely shielded system on the superconducting end of our cylinder, safeguarding the signal.
3.3.2
Dilution refrigerator
In order to cool our setup to the millikelvin temperatures required to ex-hibit the Shubnikov-de Haas effect, we have to use a very powerful cool-ing machine, for which we use a dilution refrigerator setup, which the research group has developed and improved over time.
The concept of a dilution refrigerator is based on using a mixture of 3He and 4He and the unique properties of this mixture at very low temper-atures. Below approximately 0.8K, a phase separation will occur in the mixture, resulting in two separate liquid phases, a3He rich phase, consist-ing nearly entirely of3He, and a4He rich phase, consisting mostly of4He, but containing at least 6%3He. By pumping out3He, the equilibrium be-tween the two phases is disturbed, and in order to be restored3He moves from the3He rich state to the4He rich state. As this is an endothermic pro-cess, therefore it draws on energy from the surrounding liquid, effectively
3.3 Cooled measurements 23
cooling it. By connecting the helium mixing bath thermally to an exper-imental set-up, it is possible to perform experiments at (sub-)millikelvin temperatures.
Figure 3.4: Schematic representation of a generic dilution refrigerator, showing pump systems, the mixing bath and the connection to the experiment.[7]
Over the course of other experimental runs with the 4K stage of large di-lution refrigerator in the laboratory, it became apparent that our large re-frigerator was not cooling down as much as desired when the high current cables of the nuclear demagnetization stage (required to reach 1mK tem-peraturs) were connected. When looking for possible ways to improve the cooling down process, we decided to try changing the brass buckles used at the top of the refrigerator setup. In our setup, we used brass buckles with the dimensions of l =12.5mm, d=3.0mm, b=9.0mm.
The choice for brass as a material was made based on the electrical conduc-tivity and thermal conducconduc-tivity properties of this metal. It is imperative that the buckles have a low thermal conductivity to prevent passing too much heat into the system, however the electrical conductivity must still be high enough to properly send a signal into the system as well. In us-ing brass, these two properties come to a good middle ground, where the
thermal conductivity is low enough to not cause heat issues, whereas the electrical conductivity is still high enough. In order to remedy the cooling issues, we decided to try changing the current buckles for new buckles of the same material with a smaller cross-section.
We used the formula λA =
λ·A
t with λA the absolute thermal conduc-tivity, λ the specific thermal conductivity of brass, A the cross-section area of the buckle and t the thickness, in our case the length of the buckle. Following from this formula, it shows that the thermal conductivity is pro-portional to the cross-section area, with the other factors constant, mean-ing a smaller cross-section of the buckles will directly lead to a lower ther-mal conductivity, thus improving the heat isolation of our system.
We decided to use buckles of dimensions l = 12.5mm, d = 1.5mm, b =
3.0mm, which would result in the cross-section A 6 times smaller than that of the previously used buckles, resulting effectively in a thermal conduc-tivity through the buckles that is 6 times lower.
Figure 3.5: Photograph showing the old buckles after removing them from the refrigerator set-up compared to the smaller new buckles we installed.
3.3 Cooled measurements 25
Figure 3.6: Photographs showing the old buckles mounted at the top of our re-frigerator set-up and the new buckles after installing them.
After performing several cooling runs on our dilution refrigerator after mounting the thinner buckles into the system, they proved to be success-ful, being able to handle the 20A current that is sent through the buckles. This was a concern given the smaller buckle sizes, so it was good to know they were properly functional. The 4K plate of the refrigerator did heat up slightly more than without the buckles installed, however condensation in the mixing bath was still successful.
Chapter
4
Results
4.1
Single transformer measurements
In order to test the initial principle of passing an inductive current through a metal sheet with our inductor set-up, we started by measuring the trans-fer of a single coil pair through a single sheet of metal. We used similar coils to those we had made to use on our cylinder set-up, as well as similar sheet thickness. In doing this, we would be able to get a good idea of how functional our entire cylinder system would be.
We performed a number of measurements with three distinct situations at room temperature, with either no foil in-between the inductors, or either a 100µm copper or brass foil.
Figure 4.1:Photographs of our single transformer set-up, showing the simplistic set-up with which we tested the principle of our inductive measurement idea.
To measure the transfer through our single inductor set-up, we applied an input signal of 500mV, with a sensitivity on the lock-in amplifier of 100mV.
4.2 Cylinder measurements 29
Figure 4.2:Measurements on transfer of a single transformer plotted on a log-log scale, results with no foil separation shown in black, copper in red, brass in blue.
These results are representative of what we were already expecting to see, in the sense that having no foil between the inductors shows clear high-pass filter behavior, with a linear log-log increase up until the cut-off fre-quency around 100kHz, when it flattens out to a constant transfer.
From the results with both metal foils, we can see that this high-pass filter behavior is identical, however at higher frequencies, the aforementioned effects such as the skin effect start to become increasingly apparent, low-ering the transfer that we measure.
With these results, we are able to make a rough prediction on what the transfer for the whole cylinder might look like by simply squaring the results for the single transformer with copper foil. This can only be taken as an accurate prediction if the resistance within the cylinder is of similar value to the total resistance on the input side.
4.2
Cylinder measurements
After having made a start with seeing the inductive behavior through sheets of metal, we moved on to measure the transfer the transfer over
our entire cylinder set-up at room temperature. When initially measur-ing the transfer through the cylinder, we noticed big spikes in the results due to resonance, to solve this we added 1kΩ resistances on all incoming and outgoing wires, to suppress this effect. We performed these measure-ments a multitude of times, with varying input signal strengths, as well as varying lock-in amplifier sensitivity settings.
Figure 4.3: Results of single transformer compared to expectation and actual results over the whole cylinder. Single transformer results shown in black, quadratic form shown in red, cylinder results shown in blue.
In the results for the transfer over the whole cylinder, we can see a broad peak in the order of microvolts, was expected. We see that the transfer range corresponds well with the quadratic transfer, but the shape of the peak in the graph is less pronounced.
As our set-up functions based on inductive currents, we deduced before that there might be an issue with the inductive flux from one set of coils interacting with the other set of inductive coils at the other end of the cylin-der. This cross-talk between the inner two sets of coils, as well as the outer coils to a lesser degree, creates an undesirable noise effect that we wanted to suppress as much as possible. In order to measure the cross-talk, we performed several measurements with the wires within the cylinder
dis-4.2 Cylinder measurements 31
connected and found the cross-talk effect to be significant, to the point where compensating our cylinder transfer data with the cross-talk data, our transfer data looked significantly different.
To combat this cross-talk, we attempted a possible solution of wrapping the cylinder in aluminum foil to diminish the cross-talk between the outer coils, however this proved to be ineffective to reducing the cross-talk to negligible levels.
Figure 4.4: Cylinder transfer after we subtract the crosstalk data. Final transfer result shown in green.
After manufacturing the shielding required to be able to measure with our SQUID, we again measured the transfer through the cylinder, to see how it matched the data from our PSpice simulation. In the figure below we plot the two together and it shows that our transfer matches relatively well with the simulation at lower frequencies, but falls off rapidly in the 100kHz magnitude range.
Figure 4.5:Cylinder transfer data plotted against PSpice simulation data.
4.3
SQUID tests
In order to test if our SQUID actually functioned as intended, it was first necessary to test it properly in liquid Helium. However, there are a few connections that can be tested at room temperature. Using a multimeter, it is possible to measure the connections between+v/−V and the ground (GND), as well as the connections from I and GND, ΦX to GND and between+Φ and−Φ. These preliminary tests can already indicate if there
is any serious damage to the SQUID. To test the functionality of the sensor SQUID, cooling is necessary.
To protect it and the dipstick from damage, we used a stainless steel cone to envelop our SQUID. We used a simple vacuum pump to create a vac-uum inside the cone and make a strong seal. Due to poor thermal conduc-tance, it was necessary to use evaporated Helium from the dewer as ex-change gas, which we lead back into the dipstick. This cooled the SQUID enough in order for superconductivity in the SQUID to be possible. To test the SQUID, we simply followed the manual provided by the manufac-turer. Initial tests showed that the amplifier SQUID functioned properly, but the sensor SQUID showed no signal. To find out what the issue was, we had to open the SQUID and check all the connections. After mapping
4.3 SQUID tests 33
the structure of the SQUID we could not find anything wrong with it and after testing it without vacuum, the SQUID worked as intended.
Chapter
5
Discussion
On the whole, we were unable to conduct our experiment to the scope that we had initially envisioned. Unfortunately, due to external circum-stances, such as issues with the refrigerator set-up or delays in parts being made, we ran out of time to fully test our cylinder set-up within the dilu-tion refrigerator. We would have liked to be able to see at least on a basic level how our system functions at (sub-)millikelvin temperatures, since this would have given us enough of a basis to conduct further experimen-tation on with future projects.
Creating a fitting predictive simulation of what our data should look like was a challenge as well, as multiple effects came into play with the way our set-up was made, that were difficult to completely simulate for. The PSpice simulation we attempted serves as a very general idea of what our transfer might look like, however it did not line up with our actual results very well, primarily the low-pass behaviors were not taken into account as much. This is evident by how the simulation at first glance looks like a simple high-pass filter, whereas our transfer has a clear band-pass filter behavior, with a sharp drop-off at higher frequencies.
We were able to establish our cylinder set-up as a functioning principle, and our initial results have shown that it should be possible to achieve our initial goal of performing measurements at (sub-)millikelvin tempera-tures. We can see that the functioning frequency range of our signal lines up with our theoretical prediction. It is unfortunate we were unable to test this frequency range in our cooled set-up, and it remains challenging to determine how precisely and over which frequencies we would be able to perform our measurements.
Over the course of our measurements, a multitude of issues arose due to the novelty of our up and the challenging nature of creating a new set-up. These issues demanded a large amount of creative thinking, as well as a significant time investment, which further attributed to running out of time as mentioned before.
A significant problem was dealing with the cross-talk in our system. The key features of our set-up being the inductor coils, it was challenging to overcome an issue arising from the coils themselves, and we were unable to find a clear solution to this.
The accuracy of our measurements was a consistent problem throughout our project, and we had to continuously think of ways to improve this. The size of the cylinder was also a small problem, as it had to be attached to a dipstick in combination with a SQUID and its shielding. Therefore we trimmed the edges of the cylinder and opted not to use a protective cone, since we would need a new one of bigger size, which was impractical. Not using a protective cone did impose more risk on our own setup, as well as risk to damage the dipstick itself, preventing it from being able to be used in other experiments, therefore it was important to be cautious.
Because of the fact that our lock-in amplifier could only be operated by LabVIEW functions that were made specifically for this device, it was chal-lenging to create a program to use for our measurements. In order to speed this up, we were able to contact the company that manufactured the lock-in amplifier, which helped us to write a functionlock-ing program.
The SQUID that we used was left over from a previous experiment con-ducted a few years prior, which meant we had no clear idea on how the SQUID looked schematically. When we ran into an issue cooling the SQUID with a test in liquid helium, we had to open up the SQUID and map all the connections and wire-bonds to find where the issue was. For future experiments a new SQUID has been bought.
The accuracy of our measurements could be improved in the future, as we often noticed defects in our data that were not in line with the theoretical predictions, caused by things such as wire capacitance.
Chapter
6
Conclusion
In summary we have attempted to create a new experimental set-up which potentially can provide lower effective electron temperatures using a closed copper cylinder as shield with inductive flux through sets of coils to pass a signal through. This posed the challenge of how to pass through a sig-nal strong enough to effectively use for measurements. We attempted to define a suitable frequency range for our signal through analysing the var-ious effects that come into play when trying to send an a flux through a metal sheet.
The results we were able to obtain showed the potential of our set-up and confirmed certain assumptions and predictions we made at the be-ginning. Further research is still necessary to be able to draw definitive conclusions about its success, particularly at low temperatures, and de-termine whether the side effects we noticed will be able to be suppressed sufficiently for effective measurements.
The SQUID measurements will be crucial in achieving the sensitivity re-quired.
Bibliography
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[3] I. Vagner, B. Lembrikov, and P. Wyder, Electrodynamics of Magnetoactive Media, Springer Series in Solid-State Sciences, Springer Berlin Heidel-berg, 2003.
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[5] Skin effect graphic, https://en.wikipedia.org/wiki/File:Skin_ depth.svg, Accessed: 2019-22-07.
[6] J. Clarke and A. Braginski, The SQUID Handbook: Fundamentals and Technology of SQUIDs and SQUID Systems, Volume I, Wiley, 2006. [7] Sketch of helium dilution refrigerator, https://en.wikipedia.org/wiki/
