Josephson junctions due to domain
configuration of mesoscopic
Sr
2
RuO
4
THESIS
submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE in PHYSICS Author : W.G. Stam, MSc. Student ID : s0801542 Supervisors : R. Fermin, MSc. Prof. Dr. J. Aarts
2ndcorrector : Prof. Dr. J. van Ruitenbeek
Josephson junctions due to domain
configuration of mesoscopic
Sr
2
RuO
4
W.G. Stam, MSc.
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
July 21, 2020
Abstract
The pairing symmetry of the superconducting material Strontium Ruthenate (Sr2RuO4), despite much research, has not been established.
The experimental evidence until now points to a pairing symmetry compatible with the existence of superconducting domains. The boundaries of these domains are expected to act as Josephson junctions.
Although bulk samples contain a random and non-predictable domain structure, mesoscopic samples feature a controllable configuration of the
domains. The behavior of singly or doubly presupposed connected domain boundary is investigated by the electronic transport properties of
mesoscopic samples of Sr2RuO4, in which the domain walls are pinned to
the geometry. The characteristic critical current oscillations and Shapiro steps of the Josephson junction are established, as well as current
switchable states due to in-plane (IP) fields.
Keywords: Sr2RuO4, unconventional superconductivity, Josephson junctions,
Contents
Acknowledgements iii List of Abbreviations iv List of Figures vi List of Tables vi 1 Introduction 11.1 Unconventional superconductivity and pairing symmetry 1
1.2 The unconventionality of Sr2RuO4 3
1.2.1 Triplet superconductivity 4
1.2.2 TRS breaking 5
1.2.3 3K-phase and possible solutions 6
1.3 Goal 7
2 Josephson Junctions 9
2.1 Josephson Junctions 10
2.2 RCSJ model 11
2.3 Shapiro steps 15
2.4 Interference patterns in Josephson Junctions 17
3 Experimental Method & Setup 19
3.1 Sample Preparation 19
3.2 Experimental setting 21
3.3 Detailed Sample Preparation 22
3.3.1 Exfoliation of Sr2RuO4flakes 22
3.3.3 Shaping into mesoscopic device with focused
(Gal-lium) ion beam (FIB) 23
3.3.4 Pitfalls 24
4 Experimental Results 25
4.1 Sample C15F5: Disk, ring and arm 27
4.1.1 Out-of-plane magnetic fields 29
4.1.2 Shapiro steps 32
4.1.3 In-plane magnetic fields 35
4.2 Sample A2: Two-step Bridge 38
4.3 Sample D4F1: Ring 41
4.4 Sample D8F1: Double bridge 43
5 Discussion 45
5.1 Out-of-plane fields and interference patterns 45
5.2 Shapiro steps 46
5.3 In-plane fields 46
6 Conclusion and Outlook 49
6.1 Conclusion 49
6.2 Outlook 49
A Additional Experimental Results 53
A.1 Sample C7F2: Bulky bridge 53
A.2 Sample C15F5: In-plane field 7 mT anomaly 55
Acknowledgement
This report is written by a single author, but I was not alone in my endeav-ors. First and foremost I want to thank my supervisor Remko Fermin, with whom I worked together closely during this project. Secondly, I want to thank Dr. Kaveh Lahabi for fruitful discussions and help in the lab. I am also indebted to Douwe Scholma and Thomas Mechielsen for equipment training and help when technical things went wrong. Last but not least I want to thank Prof. Dr. Jan Aarts for his guidance and for basically giving me a free rein to use the laboratory equipment and the help of the people in his group to conduct this research.
About halfway during this research, Corona struck the country, and students were sent away from the University. However, due to the diligent sample preparations of Remko Fermin, for which I want to thank him yet again, we were able to continue without much delay. During these corona months through remote desktop applications, we were able to continue the work, although technical problems loomed and steadily halted more of the progress. Technical issues with the insert and RF cable were discovered and addressed far too late. Therefore, although we have been able to measure the Shapiro steps, we have not been able to manipulate them or discover when and why they appear.
FC field cooled
FIB focused (Gallium) ion beam
IP in-plane
IV current-voltage
IVT current-voltage-temperature
µ-SR muon spin relaxation
OOP out-of-plane
RF radio frequency
SEM scanning electron microscopy
SQUID Superconducting QUantum Interference Device
Sr2RuO4 Strontium Ruthenate
SrTiO3 Strontium Titanate
List of Figures
1.1 Pairing Symmetries 3 2.1 Josephson Junction 10 2.2 RSCJ schematic 11 2.3 Washboard Potential 12 2.4 Normal retrapping 142.5 Exotic Josephson Junctions 15
2.6 Shapiro steps 16
2.7 Single and Double Josephson Junction Interference Pattern 18
3.1 Sample production process 20
4.1 Sample C15F5: SEM 27
4.2 Sample C15F5: IVT 28
4.3 Sample C15F5: OOP 30
4.4 Sample C15F5: Critical Field 32
4.5 Sample C15F5: Shapiro steps 33
4.6 Sample C15F5: Shapiro steps CD2 and CD3 34
4.7 Sample C15F5: IP CD4 IV 35
4.8 Sample C15F5: IP CD3 36
4.9 Sample C15F5: IP CD4 37
4.10 Sample A2: SEM image 38
4.11 Sample A2: IVT 39
4.12 Sample A2: OOP and Telegraph Noise 39
4.13 Sample A2: FC Shapiro test 40
4.14 Sample D4F1: SEM image 41
4.15 Sample D4F1: OOP and Shapiro step 42
4.16 Sample D8F1: SEM image 43
A.3 Sample C15F5: IP 7mT anomaly 55
A.4 Cobalt Sample G1: Shapiro steps 56
List of Tables
Chapter
1
Introduction
The pairing symmetry of the unconventional superconductivity of the
peculiar superconductor Sr2RuO4 has confounded researchers since its
discovery in 1994. [1] Most evidence, until very recently, pointed in the direction of Sr2RuO4being a chiral p-wave superconductor. In very recent
literature published in 2020, key evidence for the chiral p-wave picture was refuted. [2, 3] This reopened the problem of the superconducting pairing
symmetry of Sr2RuO4for renewed interpretation of experimental results
and theoretical investigation. In this thesis, we will present experimental results that may advance our understanding of the pairing symmetry of Sr2RuO4. In particular, our results will show the existence of
superconduct-ing domains in Sr2RuO4.
1.1
Unconventional superconductivity and
pair-ing symmetry
Since the discovery of superconductivity in 1911 by Heike Kamerlingh Onnes in Leiden, [4, 5] it has become clear that superconductivity occurs when electrons through some interaction condense into Cooper pairs. In-dividual electrons are Fermions, but when they pair up, the aggregate Cooper pair is a boson and therefore the Cooper pairs can condense into a Bose-Einstein condensate in which all Cooper pairs are in the ground state.
Because the set of Cooper pairs are all in the same state, they can collec-tively be described by a single macroscopic wavefunction, also called the order parameter. For a conventional superconductor, this can be written as
ψ(r) = pns(r)exp(iφ(r)), where ns is the density of the superconducting
Cooper pairs and φ is a position-dependent phase.
For conventional superconductors, first described in what is now known as BCS theory, [6] the attractive interaction between the electrons responsible for condensing them into pairs is electron-phonon coupling. This is not the only way there can be a net attractive interaction. In a many-body electron system, there is a myriad of ways to attain an attractive electron-electron coupling. Given the right parameters, this attractive interaction can lead to superconductivity. It is, however, often a very difficult theoretical problem to establish the exact mechanism by which superconductivity occurs in these complex many-electron systems. One key ingredient in our understanding of unconventional superconductivity is establishing the pairing symmetry of the electrons; the symmetry of the wavefunction constituting the order parameter, which determines much of the behavior of the superconductor.
The order parameter can be categorized by the pairing symmetry of the individual electrons. Electrons are still fermions and therefore obey the Pauli exclusion principle, so the total combined wavefunction must be antisymmetrical. This can be done by antisymmetrizing the spin, orbital, or frequency degrees of freedom of the wavefunction of the electrons. Combining these three degrees of freedom of the wavefunction yields various ways of creating antisymmetric wavefunctions. The spin part can be either even (triplet) or odd (singlet). The orbital part of the wavefunction can be even parity: s-, d-wave and so on, or odd parity: p-, f -wave, where the letters indicate the orbital angular momentum states as customary and defined as in atomic physics. See Figure 1.1 for an overview of the possibilities. In conventional superconductors, the electrons are paired with opposite spins into a spin-singlet state, and the momentum part is s-wave. Another possibility is for instance a spin-triplet state (even) with p-wave momentum symmetry (odd). Determining the pairing symmetry is a key element required for understanding superconductivity beyond conventional superconductors.
Unconventional superconductors are those that cannot be described by Bardeen-Cooper-Schrieffer theory or Bogolyubov’s theory, which only
de-1.2 The unconventionality of Sr2RuO4 3
Figure 1.1:Possible pairing symmetries according to the three degrees of freedom of the electron wavefunctions. The drawings on the right correspond to the allowed momentum or orbital part of the wavefunction. Image taken from Lahabi. [7, p. 12]
scribe s-wave spin-singlet superconductors. Therefore, neglecting the fre-quency freedom at the moment, unconventional superconductors are char-acterized by a non-uniform phase in momentum space, because they have a higher order orbital part of the wavefunction (p-wave, d-wave or higher).
1.2
The unconventionality of Sr
2RuO
4From its discovery in 1994, Sr2RuO4has been the subject of much research,
indicated by the sheer amount of review papers. [8–15] Already in the initial paper, [1] the authors theorize that due to the strong electron-electron
interactions at the Fermi level, Sr2RuO4 could have an unconventional
pairing symmetry. A year later, it is proposed that Sr2RuO4 is a chiral
p-wave superconductor, [16] the electrical analogue of the superfluid3He. [17, 18] This picture would remain the dominant picture until very recently. It quickly became clear from the strong suppression of the critical
tem-perature due to impurities that Sr2RuO4 was indeed an unconventional
superconductor with an anisotropic superconducting gap function. [19–22] If the gap function is isotropic, which is only true for s-wave supercon-ductors, the scattering events induced by the impurities do not break the
Cooper pairs. In case the gap function is not isotropic, for all higher-order orbital superconductors, the scattering events lead to a loss of supercon-ductivity. This initially posed an experimental difficulty to investigate the material properly, because samples with little impurities were hard to grow and the obtained critical temperatures were relatively low. Most samples are nowadays grown with a technique called the floating zone method, which yields very pure samples with a critical temperature of TC =1.50 K.
This method has been developed for Sr2RuO4by Mao et al. in 2000, [23]
but is still being refined until now. [24]
1.2.1
Triplet superconductivity
In establishing the pairing symmetry of Sr2RuO4, one major component is
establishing the spin symmetry of the wavefunction. Early on, the NMR Knight-shift experiment by Ishida et al., [25–27] established consensus that the spin pairing of Sr2RuO4is triplet. In this experiment Ishida et al.
de-duce from the NMR Knight-shift of individual atoms in the crystal that the spin susceptibility of the Cooper pairs does not change passing through the critical temperature TC, indicating triplet superconductivity. This
exper-iment also held up in follow-up experexper-iments by the same group. [28–30] However, other experiments that investigate the phase transition at high magnetic fields reported a first-order phase transition, which contradicts the no Zeeman energy gain from Knight shift measurements. [31–33] And indeed, in 2019 the original NMR Knight-shift experiment was refuted. [2] The original authors of the original experiment collaborated their find-ings and figured out their original results were unreliable due to transient heating. [34]
Another early experiment with polarized neutron measurements by Duffy et al. [35] reported no change in spin susceptibility through TCwhich the
authors found to be compatible only with spin-triplet pairing. However, also most recently, this experiment was repeated at fields below the critical field, in which they did find a reduction of the spin susceptibility. [3] The authors of this paper exclude chiral p-wave, but do not exclude triplet superconductivity per se.
The discussion on triplet versus singlet superconductivity of Sr2RuO4is not
1.2 The unconventionality of Sr2RuO4 5
superconductivity: Andreev reflection, [36] Half quantum vortices [37,
38] and direct penetration of superconductivity in Sr2RuO4-ferromagnet
junctions. [39] However, what used to be the strongest evidence in support of spin-triplet superconductivity – the NMR Knight shift – has now become the strongest evidence against spin-triplet superconductivity. [40] The
general tendency within the community is that Sr2RuO4must be a singlet
superconductor with a higher-order orbital part of the wavefunction.
1.2.2
TRS breaking
The second major experimental result for establishing the pairing
sym-metry of Sr2RuO4 show that time reversal symmetry (TRS) is broken in
Sr2RuO4. Sr2RuO4exhibits spontaneous magnetism locally, established by
two important experiments: a muon spin relaxation (µ-SR) experiment [41, 42] and a non-zero Kerr rotation experiment. [43]
These experiments were indicative of chiral p-wave at first, as one of the main predictions of chiral p-wave is that the material will split up in so-called chiral domains. Combining the pairing symmetry with the electronic
structure further pins down the form of the order parameter. Sr2RuO4
has a cylindrical symmetrical Fermi surface. In the case Sr2RuO4 is a
spin-triplet p-wave superconductor, the total wavefunction is composed of three independent gap functions describing the spin symmetry. This yields seven different unitary states with p-wave symmetry. See Lahabi [7] and references therein for a more thorough exposition of the different possible unitary states. [16, 44, 45] Most importantly, one of these unitary states is the so-called chiral state. The spin projection of this state is zero (i.e. lies in the ab-plane of the crystal) but the orbital angular momentum of this state can be either plus or minus 1. The orbital angular momentum indicates the chirality of the state and the two options are degenerate in energy. A bulk crystal of Sr2RuO4 is therefore thought to segregate into
domains of different chirality. An important consequence is that within each domain, all electron pairs have the same rotation direction and orbital angular momentum, which yields a net magnetization and therefore breaks TRS. It must be noted that it is not necessary to have p-wave symmetry to possibly obtain chiral degenerate states. This should be possible with any higher-order orbital part of the wavefunction. It is not the purpose of this thesis to derive these options theoretically.
These chiral domains are supposed to give rise to edge currents and show a net magnetization due to the orbital angular momentum adding up for all the Cooper pairs within the domain. The net magnetization is detected by the µ-SR and Kerr rotation experiments, but edge currents have not been observed by scanning SQUID microscopy [46] and scanning Hall bar microscopy. [47, 48] The main difference between these type of experiments is that the µ-SR and Kerr rotation experiments probe magnetism locally, while the latter establish the phenomena for the bulk of the material. Theo-rists have argued that the edge currents need not have been measurable. [49–54] In light of the doubts about whether Sr2RuO4 has triplet pairing
symmetry, it may be better to shift the focus away from the p-wave picture to the consequences of TRS breaking in Sr2RuO4. The main consequence of
which is a degenerate ground state and segregation into domains.
There have been several experiments indicating the existence of these chiral domains. Kidwiringa et al., following a proposal for a Josephson ring topology [55, 56], found telegraph noise and Josephson interference in a Sr2RuO4-Pb hybrid ring structure indicative of the chiral domains. [57,
58] Anwar et al. reports anomalous switching in the IV characteristic in
Nb-Ru-Sr2RuO4 junctions, according to the authors caused by domain
wall motion. They also report similar telegraph noise in the timescale of minutes, meaning that the domains remain stationary below this timescale.
[59, 60] In varying the width of Nb-Sr2RuO4Josephson junctions, Saitoh
et al. estimated the size of a single domain to the order of several microns.
[61] Last but not least, our group produced microrings of pure Sr2RuO4
and found critical current oscillations indicative of Josephson junctions due to a chiral domain wall. [62] At the domain wall, there must a local decrease in superconductivity. When there is such a short interruption of superconductivity, the device can still superconduct due to the proximity effect. This type of junction is called a Josephson junction. Therefore, if the domain walls are there, they will act as Josephson junctions. In this research, we will investigate this type of samples further.
1.2.3
3K-phase and possible solutions
What has muddled a lot of research is the fact that under uniaxial strain, SRO exhibits a second, different superconducting state with a higher critical temperature and no TRS breaking state. The first indications of this phase
1.3 Goal 7
were experiments on Sr2RuO4with metallic microdomains in the form of
ruthenium inclusions. [63] It was later discovered by Hicks et al. that these inclusions caused a strain on the material, and this strain was responsible for the so-called 3K-phase. [64] This phase is characterized by a strong increase in TCunder uniaxial stress, but no cusp in TCat zero strain that was
expected for a TRS breaking order parameter. [65, 66] The maximum critical temperature of TC =3.4 K occurs at (or near) a Lifshitz transition when the
Fermi level passes a Van Hove singularity. [67, 68] Interestingly, recent µ-SR experiments were done including strain and show a strain-induced splitting of the critical temperature TCand onset temperature of TRS breaking TTRSB,
consistent with expectations for chiral superconductivity. [69]
Theoreticians have attempted to reconcile the results of these new measure-ments by proposing more exotic forms of chiral superconductivity. Notably, Kivelson et al. propose that a near accidental degeneracy between the dx2−y2 and gxy(x2−y2) electrons, can lead to a d+ig-wave pairing symmetry,
which breaks into a single component pairing symmetry under strain. [40] Moreover, in a very recent paper, Wu et al. have shown that the assump-tion that the electronic structure of Sr2RuO4shares the four-fold rotation
symmetry with the crystal lattice is not correct, but that it rather has a two-fold symmetry. This means that the pairing symmetries compatible with Sr2RuO4proposed by theoreticians need to be reconsidered. [70]
It is clear that the question of the pairing symmetry of SRO is still alive and active after 26 years of research. It has been more or less established that the material does segregate into domains, but visualizing these domains still requires further investigation.
1.3
Goal
Most experiments until now have worked with bulk Sr2RuO4 samples
or hybrid junctions between Sr2RuO4 and another superconductor. In
hybrid structures, the behavior is usually complex and it is hard to isolate the phenomena. In bulk experiments, it is very difficult to establish the existence of domains because there will be many of them in the sample and randomly distributed, blurring or averaging out the phenomena associated with the domains or their walls.
In this research, single or double domain wall configurations are inves-tigated. This can be done by creating a mesoscopic device out of single
crystals of Sr2RuO4. The geometry of the device will then pin the
su-perconducting domains, similarly to how magnetic domains are pinned to geometrical features in ferromagnets. The process by which we have created these samples is described in Chapter 3.
The idea of a mesoscopic device is that it is much larger than the dimensions of the fundamental effect, here, the coherence length of the Cooper pairs (ξ(0) = 66 nm in the ab-plane for Sr2RuO4[8]), but small enough to isolate
certain phenomena otherwise only accessible by their statistical collective behavior. Thus, the dimensions of our devices are chosen in the regime much larger than the coherence length to exclude point-contact (SsS) junc-tion effects, but of the size of the presupposed superconducting domains, such that we obtain a Josephson junction due to the domain walls, which are in turn pinned down by the geometry. A more detailed description of Josephson junctions and their behavior can be found in Chapter 2.
In this group, for a ring geometry of Sr2RuO4 with two alleged domain
walls, some aspects of Josephson junctions have been established. Such a structure is called a Superconducting QUantum Interference Device (SQUID) and exhibits so-called SQUID oscillations of the critical current. The SQUID oscillations have been reported by this group in previous research. [62] Another hallmark effect of a Josephson junction is the occur-rence of Shapiro steps. These occur when the sample is bathed in radio frequency (RF) radiation. See for details Section 2.3.
The goal of this thesis is to investigate and establish the existence of do-mains in Sr2RuO4made visible by the behavior of the Josephson junctions
that they form in singly and doubly connected devices. In particular, we will look for Shapiro steps in our produced mesoscopic devices and investi-gate their behavior.
This thesis is further comprised of five chapters. In Chapter 2 the theory of Josephson Junctions and Shapiro steps is discussed. Chapter 3 dis-cusses sample production and the utilized experimental method. Chapter 4 presents the experimental results per sample, Chapter 5 discusses these results per topic and Chapter 6 concludes this thesis.
Chapter
2
Josephson Junctions
In this chapter, the fundamentals of Josephson Junctions and Shapiro steps will be discussed. The content of this chapter has been composed by consulting various textbooks. [71–73]
The basic superconducting electronic element that needs to be considered for this thesis is a Josephson junction. A Josephson junction consists of two weakly coupled superconducting electrodes separated by an obstruction in which there can be no superconductivity by itself. This can be a small gap between the superconductors, a thin piece of isolating material (SIS junction), or a normal metal (SNS junction) between the superconductors or even a junction made out of a point contact of the superconductor (SsS junction), where the contacting region is smaller than the coherence length of the Cooper pairs. A common example is a micron-sized ring made out of aluminum. Because of the large coherence length of aluminum, such a device also exhibits the typical interference pattern of a squid device that will be discussed later in this section. [74] Another way to create a Josephson junction is with superconducting domains. At the domain wall, the superconducting state is locally diminished and this too can act as a Josephson junction. [62]
Due to the proximity effect, the two superconducting wavefunctions over-lap and a supercurrent can flow across the barrier. This is the Josephson effect named after its discoverer. [75]
Ψ
l
Ψ
r
Figure 2.1:A Josephson junction depicted the amplitudes of the superconducting wavefunctions of the left and right electrode (gray parts). The yellow part is the barrier in which the superconducting wavefunctions are exponentially decaying. Image taken from Lahabi. [7]
2.1
Josephson Junctions
The supercurrent across the junction of an ideal Josephson junction is governed by the two Josephson relations. The d.c.-Josephson relation:
Is = Icsin(φ), (2.1)
where Ic is the critical current across the junction and φ ≡ φ2−φ1 is the
phase drop across the junction. This current flows even at zero voltage. The a.c.-Josephson equation relates the change of phase difference with a voltage:
˙φ= 2e
¯h V = 2π
Φ0V , (2.2)
where Φ0 = h/2e = 2.068×1015V/Hz is the magnetic flux quantum.
This last equation is responsible for the a.c.-Josephson effect. Applying
a constant voltage V 6= 0, we obtain by integration of Equation 2.2 and
substituting in Equation 2.1: Is(t) = ICsin 2π Φ0 Vt , (2.3)
which is an alternating current with Josephson frequency νJ = V/Φ0.
Experimentally, we can access this phenomenon by, for example, applying microwave radiation to obtain current steps in the d.c. IV characteristic at constant voltages. These are called Shapiro steps and occur at integer steps
of the Josephson frequency nνJ =V/Φ0. [76] We will discuss these steps
2.2 RCSJ model 11
2.2
RCSJ model
For a real Josephson junction, in addition to the two Josephson relations above, a displacement current and a quasi-particle current need to be considered. These currents are often approximated by an Ohmic shunt resistor and a capacitor connected parallel to the Josephson current. This is called the ’resistively-’ and ’capacitively-’ shunted junction model or RCSJ model.
R C
I
sI
bias
Figure 2.2:Electronic schematic of the RCSJ model. The Josephson junction with supercurrent Isis connected in parallel with a resistor and a capacitor. The junction
is driven by a bias current Ibias.
For our purposes now, we will neglect the capacitor and consider a parallel ohmic resistor only. Adding both currents together, we find for the total current:
I(φ) = Is+IR =Icsin φ+
V
R . (2.4)
Substituting the a.c.-Josephson relation, we find a first order differential equation in φ
I(φ) = Icsin φ+ Φ0
2πR ˙φ (2.5)
, which can be solved for the time-averaged voltage:
V = ( RqI2−I2 C for I >IC 0 for I <IC, (2.6)
oscillating with the Josephson frequency νJ.1 The average voltage behavior
is similar to a normal superconductor. Increasing the current, there is no
1The derivation of this is too lengthy to include here, so we will leave this as an exercise
voltage until a critical current IC and in the high current regime the voltage
approaches V = IR according to Ohm’s law in which R is the normal state
resistance of the junction.
I/I
CUJ
φ
Figure 2.3:The washboard potential including a bias current I, tilting the cosine potential profile. The phase remains stationary in a potential well unless the bias current is ramped up high enough such that the minima disappear.
If we also include the parallel capacitance (add I =C ˙V) we obtain a second order differential equation:
I(φ) −Icsin φ = Φ0 2π ˙φ R +C ¨φ . (2.7)
An intuitive way to grasp the dynamics implied by this equation is by transforming it into a potential equation, the so called tilted washboard potential. It has been noted that the equation can be understood with a mechanical analogy of the motion of a particle of mass m and friction coefficient ξ moving along the x coordinate through a series of potential wells V driven by an external force F:
m¨x+ξ ˙x =F−∂V
∂x = −
∂(V−Fx)
∂x . (2.8)
In term of the Josephson junction parameters we have bias current I acting as the external force; the potential V is given by IC(1−cos φ), the mass by
2πC/Φ0and the friction term by 2π/Φ0R. In other words we can write
C ¨φ+ 1 R ˙φ = 2π Φ0 I− ∂IC(1−cos(φ)) ∂φ
2.2 RCSJ model 13 = −2π Φ0 ∂(IC(1−cos(φ)) −Iφ) ∂φ ≡ − 2π Φ0 ∂UJ ∂φ , (2.9)
where the potential wells are given by UJ = Φ0 2π (IC(1−cos φ) −Iφ) = EJ 1−cos φ− I IC φ , (2.10)
with EJ ≡ Φ2π0IC is the Josephson coupling energy. The bias current I
cor-responding to the external force tilts the cosine-shaped potential. In the case where the bias current is smaller than the critical current I < IC, the
particle will remain in one of the potential minima where it oscillates with the plasma frequency
ωP = s 2π Φ0C IC2 −I21/4 . (2.11)
In this model, the time-averaged change in φ signifies the voltage drop over the junction. therefore V =0 if I <IC and the phase difference φ remains
constant. As the bias current exceeds the critical current I >IC, the minima
disappear and the particle starts to roll down the potential ’curve’ with a fixed velocity. This gives a nonzero ˙φ which, by the Josephson relation, corresponding to a d.c. voltage across the junction.
The dynamics implied by the washboard potential can yield some inter-esting phenomena. In an upward sweep of the bias current I. The phase
difference φ will remain stationary until I = IC from where it starts to
roll down the potential curve. However, if we ’tilt the potential back’, in other words, decrease the bias current I again, due to the finite velocity of the phase difference it is not immediately trapped inside a potential well when crossing I = IC, but will be retrapped only when it has lost its phase
velocity. This yields a hysteresis in the IV: When sweeping the current upward a finite voltage starts te appear at a higher current (equal to IC)
than when sweeping the current downward. There the voltage becomes
zero at the retrapping current IR ≈4IC/πQ, with Q =ωPRC the quality
factor of the junction. Due to thermal fluctuations, retrapping can occur earlier than the retrapping current. Important to note is that for ordinary Josephson junctions the reptrapping current is always smaller than the critical current.
For an arbitrary Josephson junction, it is by no means necessary that the washboard potential is of the shape of a perfect cosine. Even if it is of
Figure 2.4:Normal retrapping or IV hysteresis in the washboard potential model. Image taken from Tinkham. [71, p. 208]
the form of the cosine, the energy minimum need not be at zero phase
difference. If the energy minimum is at φ = 0, the Josephson junction is
called a 0-junction. In addition, it is also possible to have a π-junction,
where the minimum is at φ=π. One can also think of an arbitrary phase
difference φ0as the minimum: a φ0-junction. [77] Lastly, the minima need
not be 2π separated from each other. This is called a φ-junction, where 2φ
is the distance between the minima. [78] While minima at φ = 0, π still
yield zero current at the ground state of the junction, the more exotic φ and
φ0junctions have a non-zero (and non-π) minima leading to the peculiar
effect of a non-zero current across the junction even without any applied bias current.
Assuming chiral domains in Sr2RuO4 acting as the Josephson junctions,
depending on the respective orientations of the chiral domains, the phase difference across the junction can take any value between 0 and π,
corre-sponding to a φ0-junction. Moreover, the distance between the minima
need not be 2π, so the chiral domain wall may well be a combination between a φ0and φ junction. [62, 79]
It is possible that in a certain geometrical configuration of the chiral do-mains, that the orientation of the chiral domain walls relax into an orienta-tion yielding one type of Josephson juncorienta-tion. It is then an open experimental question whether it is possible to change these orientations and placement of the chiral domains by applying in-plane magnetic fields. The possi-bility of a generalized Josephson washboard potential has consequences for retrapping phenomena. Depending on the energy landscape, the re-trapping current can higher than the critical current on one side of the IV
2.3 Shapiro steps 15 d kx-iky kx+iky a 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 b 1.0 0.5 0.0 0.5 1.0 c α/π α/π α/π Chir al Domain w all ener gy U J( θ ,α ) θ=0 θ=-π/8 θ=-π/4 θ=0 θ=-π/8 θ=-π/4 φ+ φ -φ' φ0 0
Figure 2.5:More exotic Josephson junctions that can be made with chiral domain walls. In a-c the energy profiles are seen of the chiral domains at various orienta-tions θ relative to each other at the junction. α is the phase difference between the two sides of the junction. Three different type of Josephson junctionscan be made. The first type (a), for which the relative orientations are matched is a 0-junction, which is an ordinary Josephson junction. The second type (b) has a slight mismatch in orientations, resulting in a φ0-junction with two minima. The right φ0minimum
is a stable minimum, but the left φ0minumum is only metastable. As θ approaches
π/4, the two minima become degenerate (c) and this is a φ-junction. In (d) it is
depicted how the phase can be matched for two opposite chiral states. Image taken from Lahabi, [7] who in turn adopted it from Sigrist & Achterberg. [79]
characteristic. [7, pp. 122-123] This will be discussed in more detail in Chapter 4.
2.3
Shapiro steps
The first experimental evidence of the a.c. Josephson effect was given by Shapiro in 1963, just one year after Josephson’s seminal paper. [76] Shapiro showed that by applying external RF radiation to a Josephson junction,
at exact multiples of the Josephson frequency νJ = V/Φ0, constant d.c.
voltage steps appear in the IV characteristic. This happens because of the synchronization or phase locking of the motion of the particle with νJ and
washboard is rocked with frequency νRF. If it is rocked hard enough, within
one period of oscillation the particle can drop along n minima and become trapped again when the RF radiation switches sign. The average velocity of the particle is now dependent on the frequency of the applied radiation and not on the d.c. bias current I which tilts the washboard statically. If the washboard potential is rocked harder, i.e. with higher RF power, the particles can drop down n minima and reach a higher average velocity, corresponding to a higher voltage step. The average velocity translates into constant d.c. voltage steps in the IV characteristic appearing at:
V =nνΦ0. (2.12)
for the n-th step. From the tilted washboard potential, it is also evident that the Shapiro steps start to appear near IC, as the bias current makes it easier
for the particles to roll down the potential.
0 0.5 1 1.5 2 2.5 d.c. Current (I C) 0 1 2 3 4 Voltage ( 0 RF )
Figure 2.6:Shapiro steps due to RF radiation (red) on top of the normal Josephson IV curve (blue) according to Equation 2.6. The Shapiro steps appear at integer multiples ofΦ0νRF.
The width of the n-th step depends strongly on the amplitude and fre-quency of the RF radiation and shows oscillatory behavior with increasing
power. In the limit of small frequency νRF/νC 1 (with νC = ICR/Φ0
the characteristic frequency of the Josephson junction) and large quality parameter Q of the junction, the n-th step width can be approximated as:
∆I = IC Jn RIRF νRFΦ0 , (2.13)
2.4 Interference patterns in Josephson Junctions 17
with Jn the n-th Bessel function. From this relation, as Jn(x) varies as xn
for small values of x, it is clear that the lowest step appears first as the IRF
drive is increased from zero.
From our discussion of the more exotic Josephson junctions such as φ-junctions, it is clear that the minima of the washboard potential can be unevenly spaced. Translating this to Shapiro steps it seems that it is also possible to obtain non-integer Shapiro steps, according to the distance of the minima.
2.4
Interference patterns in Josephson Junctions
In applying an out-of-plane magnetic field to a Josephson junction, inter-esting wave-like phenomena will occur. Because of the Meissner effect, the superconductor will try to neutralize the penetrating magnetic flux by inducing screening currents, which decrease the maximum supercurrent across the junction, as the total supercurrent available must be preserved. Because the phase difference across the junction must be single-valued, magnetic flux can enter the junction, but it must be quantized according to multiples of 2π in the so-called Josephson vortex. For a single Josephson junction, this will result in a single slit Fraunhofer pattern, in complete analogy with light passing through a single-slit:
I(H) = Ic sinc πΦsingle Φ0 , (2.14)
whereΦsingleis the total flux going through the junction itself. Because the
size of the Josephson junction is often very small, this interference pattern is not always visible as it is possible that the first zero of the envelope is at a magnetic field higher than the critical magnetic field of the superconductor. Most interestingly, in a ring topology with two Josephson junctions, a double-slit interference pattern appears:
I(H) = 2Ic sinc πΦsingle Φ0 · |cos(πΦdouble/Φ0)| , (2.15)
where Φdouble is the total flux entering the entire ring geometry. Because
flux changes much faster with respect to the magnetic field, and generally we will see many oscillations within the single slit Fraunhofer envelope. This structure is therefore very sensitive to a change in flux, and this is the basis for the much-used SQUID magnetometer, one of the most sensitive measurement devices for magnetic flux.
-3 -2 -1 0 1 2 3 / 0 0 0.2 0.4 0.6 0.8 1 I(H)/2I C
(a)Single Josephson Junction
-20 -10 0 10 20 / 0 0 0.2 0.4 0.6 0.8 1 I(H)/2I C
(b)Double Josephson Junction
Chapter
3
Experimental Method & Setup
This chapter is divided into two parts. In the first part, comprised of two sections, a general overview is given on the sample preparation and experimental setup. Followed by a more detailed description of sample fabrication.
3.1
Sample Preparation
Three main steps are used in the production of a mesoscopic Sr2RuO4
device from a bulk crystal. The bulk crystals were supplied by the research group of Maeno from Japan and have been produced using the floating zone method. [23] The bulk crystals have a critical temperature of 1.5 K, indicating that the crystals have very few impurities. [19–22]
The first step in the production is using mechanical exfoliation to procure the Sr2RuO4flakes with a thickness of the order of 0.3−1 µm and
deposit-ing them on an Strontium Titanate (SrTiO3) substrate, which has the same
crystal structure as Sr2RuO4and matched thermal properties. [62]
In the second step, gold contacts are grown on top of the Sr2RuO4flake,
shaped by means of e-beam lithography and high vacuum Argon plasma sputtering. A thin layer of chrome is sputtered as an intermediate sticking
(a)Step 1, 100x, Sr2RuO4flake on substrate (b) Step 2, 100x, after e-beam, before gold deposition
(c)Step 2, 5x, after liftoff
(d)Step 2, SEM, before milling
(e)Step 3, SEM, overview during milling (f) Step 3, SEM, final device
Figure 3.1:Sample production process of sample C15F5. In panel (a) the Sr2RuO4
is deposited on the SrTiO3 substrate. Panel (b) shows the result of the e-beam
lithography. Panels (c-d) the result after liftoff. Panels (e-f) show the two steps in the milling process. Panels (a-c) are taken with an optical microscope and panels (d-e) with the FIB equipped with a scanning electron microscopy (SEM).
3.2 Experimental setting 21
layer between the Sr2RuO4/SrTiO3and the gold.
Finally, in the third step, the mesoscopic device is milled in the flake by making use of FIB lithography. Before milling, the rest of the sample is protected from the Ga+ions by a sputtered layer of silicon dioxide (SiO2).
The result is a micron-sized ring or bridge attached to four contacts made out of Sr2RuO4only. A detailed recipe can be found in Section 3.3.
3.2
Experimental setting
The sample is loaded in an Oxford Heliox4He pulse-tube cryostat equipped
with an3He-insert which cools the sample further by sorption pumping.
This yields stable working temperatures down to 300 mK with a stable temperature for up to 24 hours. The cryostat is also equipped with a Nb3Sn
vector magnet. The superconducting Nb3Sn vector magnet can set very
stable and homogeneous magnetic fields with steps of 0.1 mT up to fields of the order of 1 T in any direction. In our setup, it is also possible to bathe the sample with RF radiation with a microwave source (Rohde und Schwarz SMIQ 04b) using a semi-rigid co-ax cable with an open-ended antenna inside the insert in very close proximity to the sample. [80]
In the experiments, the electronic transport properties of the device are investigated using the four contacts of the device as a 4-point-probe. The current and voltage are measured in DC or AC with a Keithley nanovolt-meter or a Syntek MCL1-540 lock-in amplifier respectively. These devices set a d.c. current and measure the voltage. In a typical measurement, we sweep the current and measure the response of the voltage.
All in all, we investigate the electronic transport of the mesoscopic Sr2RuO4
device controlling three ambient properties: temperature, magnetic field, and RF radiation.
3.3
Detailed Sample Preparation
3.3.1
Exfoliation of Sr
2RuO
4flakes
Before the exfoliation of the Sr2RuO4flakes, the surface of the SrTiO3
sub-strate is cleaned by submerging it in acetone and isopropanol consecutively while ultrasonicating the sample for 10 minutes each submersion. After taking the substrate out of the isopropanol, the surface is dried by blowing on it with gaseous nitrogen. Lastly, the substrate is cleaned and prepared
for bonding with the Sr2RuO4 flakes by etching with an oxygen plasma
(Oxford PlasmaLab 90+) for three minutes.
The Sr2RuO4flakes are exfoliated by placing the bulk Sr2RuO4crystal on
scotch tape and ripping an undetermined amount of layers off the crystal. This process is repeated several times to decrease the thickness of the flakes and increasing the number of flakes on the sticky side of the scotch tape. Subsequently, where the concentration of flakes is thought to be high, the
SrTiO3substrate is placed on top. The scotch tape and substrate are now
placed on top of a glass microscope slide and baked on 100◦C for 2 minutes to harden the scotch tape glue and let the flake adhere to the substrate. The scotch tape is now removed under a 45-degree angle from the glass slide
and substrate and hopefully, some Sr2RuO4flakes will have remained on
the surface. The substrate is now gently cleaned with acetone, isopropanol, and dried with N2to remove the glue of the scotch tape.
At every step of depositing the Sr2RuO4flakes onto the SrTiO3substrate,
progress can be checked with optical microscopy. Indeed at the end of this process, the substrate and exact location of the flake is mapped using an optical microscope. An example of such an optical image is shown in Figure 3.1 (a). It must be noted that this process does not give much control over the thickness of the samples, but this is not a problem because suitable samples can be post-selected.
3.3.2
E-beam lithography
To prepare the sample for e-beam lithography, two layers of positive resist (950K and 600K PPMA A4) are spin-coated on the substrate with a
stan-3.3 Detailed Sample Preparation 23
dard recipe of 4000 rpm for 1 minute. Each layer is finalized by baking
it at 180◦C for 1 minute on a hot plate. Because the substrate does not
conduct well, two layers of Electra-92 (same spin-coat recipe, no baking) are added on top of the sample. The samples are now loaded into the e-beam lithography machine (EBPG Raith-100) and four contact pads are written in the resist with an average exposure of 350 µC cm−1, after which the resist is developed for 45 s in the PPMA developer, which is finally cleaned off with isopropanol and dried with nitrogen. The progress thus far is shown in Figure 3.1 (b).
In the next step, the contacts are sputtered onto the substrate with the Leybold Z-400 diode sputtering system and Argon as plasma. Two layers are sputtered: A 5 nm thick layer of chrome as a sticking layer and an approximately 300 nm thick layer of gold. Finally, the rest of the resist is lifted off the sample by leaving the sample overnight in acetone and removing the leftover gold with pincers and very cautiously ultrasonicated on the 50% setting of the device. In Figure 3.1 (c-d), we observe that this succeeds reasonably well, but at times, there is some leftover gold on the sample. For this sample, the gold liftoff process failed, but this could still be fixed by milling through the gold with the FIB in the third step. However, it could not always be helped that there was an unconnected layer of gold on top of the Sr2RuO4flake.
To make sure the Sr2RuO4 flake makes good electrical contact with the
gold, the contacts can be strengthened by using an e-beam ion deposition technique (EBID). However, in the samples most relevant for this thesis, this was not necessary.
3.3.3
Shaping into mesoscopic device with FIB
In the third step, the required constriction on the Sr2RuO4flake is milled
from the flake by making use of focused (Gallium) ion beam (FIB) lithog-raphy. Before milling, the rest of the sample is protected by a 200 nm sputtered layer of silicon dioxide. The larger features are milled using a 10 pC beam current, and the smaller features are milled using 1.5 pC. In this step, the thickness of the Sr2RuO4flakes is also measured. See Figure 3.1
Lastly, the sample is placed on a puck where the contacts are wire-bonded with aluminum wires (K&S-4700 Convertible dual wire bonder) to the macroscopic electronics.
3.3.4
Pitfalls
Sample degradation: It is known that Sr2RuO4flakes degrade in air. [81]
Therefore, we have taken heed to keep the samples in a vacuum as much as possible. We have noted that the critical temperature of the samples is usually lower than the ideal critical temperature of 1.5 K (, but still above 1 K). Part of this can be explained by the slight degradation of the samples, or that one of the preparation steps introduce impurities into the flake. It is most likely that our cleaving method cleaves the crystal at areas where the concentration of impurities is highest, as these areas have the lowest mechanical integrity.
3K-phase and strain: In the samples used for this research, we have not
observed the 3K-phase that can occur due to strain in the sample. [64–66] This strain can be induced by the FIB process by Gallium implantation. In previous research by this group, milled samples that were not protected by a SiO2layer did show the 3K-phase occasionally. [62]
Electrostatic discharge:From the point where the sample is wire-bonded
until it was safely loaded and cooled in the cryostat, we believe electrostatic discharge plays a major role in destroying samples. We have had the majority of the samples ’blowing up’ during loading. This happened especially often for the smaller/thinner samples. We believe a major cause of this was faults in the electronics of the cryostat insert. A slight millivolt offset could already destroy such a sample.
Chapter
4
Experimental Results
Due to the nature of working with mesoscopic samples, – ’no sample is the same’ – in this chapter we will present each sample separately and discuss all interesting results pertaining that particular sample. Many samples broke during loading and some did not show a superconducting transition, so we will present here a select few samples that we were able to measure extensively on.
Another reason for choosing this type of presentation is that each sample is unique in its geometry and tells its own part of the story. Three main types of geometry have been used: A (single) bridge (sample A2, C7F2), a ring (samples C15F5, D4F1) and a double (rectangular) bridge (sample D8F1). All samples have been produced according to the method described in Chapter 3. The most extensively measured sample, which survived four cooldowns, is sample C15F5, which tells most of the story and we will present first. Lastly, as a control experiment of the microwave radiation setup, we cooled down an on-purpose designed Josephson junction made from a Nb/Co bilayer trenched disk to measure its Shapiro steps, see
Section A.3. An overview of the used Sr2RuO4samples can be found in
Table 4.1.
As discussed in Chapter 3, during the experiments we will measure current-voltage (IV) characteristics, setting a d.c. bias current while controlling three ambient parameters: temperature, magnetic field and RF radiation. This means we have four types of experiment. The first is the
current-voltage-temperature (IVT) measurement, which determines the general behavior of the superconductor, its critical temperature and the tempera-ture dependence of the critical current. Each IVT measurement consists of sweeping the temperature, and at every temperature increment record-ing an IV characteristic, which consists of settrecord-ing the bias current and measuring the voltage across the junction, yielding voltage data for each bias current and temperature. Often, it will be more insightful to plot the resistance however, so we usually plot the dV/dI, the derivative of the measured voltage with respect to the set bias current. The second is the out-of-plane (OOP) field measurement, which records a full IV sweeping an out-of-plane magnetic field while keeping the temperature stable. This measurement shows possible interference patterns due to the Josephson junctions in our devices. The third are in-plane field measurements where we investigate the phenomenon of ‘current switchable states’ and the fourth are Shapiro measurements, bathing the sample in RF radiation, to look for Shapiro steps.
4.1 Sample C15F5: Disk, ring and arm 27
4.1
Sample C15F5: Disk, ring and arm
Sample C15F5 is the sample on which we have performed most measure-ments and we have cooled it down four separate times, each cooldown with a different geometry. In Figure 4.1 a SEM image shows the four different geometries of the four respective cooldowns. The crystal is 450 nm thick and is shorted by a layer of gold on top due to a failed lift-off, which as far as we could tell did not influence the measurements as gold is not a superconductor.
Figure 4.1:Sample C15F5: SEM images. From left to right and top to bottom are images of the respective cooldowns, indicated as CD1-4.
In the first cooldown, the sample has the shape of a disk with a diameter of 1 µm and neck width of 0.43 µm. In the IVT measurement, seen in Figure 4.2 top left, we see a clean(TC−T)1/2transition at TC =1.35 K. This geometry
was chosen to contrast with the results of the subsequent geometries. In the second cooldown, the central part of the disc was milled away to obtain a ring structure with arms of 0.32 µm wide. This type of structure should yield a SQUID device, as the two presumed domain walls in the
1 . 2 0 1 . 2 5 1 . 3 0 1 . 3 5 1 . 4 0 1 . 4 5 1 . 5 0 1 . 5 5 0 2 0 4 0 6 0 8 0 1 0 0 d .c . C u rr e n t (µ A ) T e m p e r a t u r e ( K ) C D 2 C D 3 C D 1 C D 4 1 . 0 1 . 1 1 . 2 1 . 3 1 . 4 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 d .c . C u rr e n t (µ A ) T e m p e r a t u r e ( K ) 1 . 0 1 . 1 1 . 2 1 . 3 1 . 4 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 d .c . C u rr e n t (µ A ) T e m p e r a t u r e ( K ) 1 . 0 1 . 1 1 . 2 1 . 3 1 . 4 1 . 5 0 1 0 2 0 3 0 4 0 5 0 6 0 d .c . C u rr e n t (µ A ) T e m p e r a t u r e ( K ) 0 . 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 1 . 3 1 . 5 1 . 8 2 . 0 d V / d I ( R N)
Figure 4.2: Sample C15F5: IVT measurements. In this figure four IVT measure-ments are shown of the four respective cooldowns. The bias current has been swept in steps of 1 µA from low to high and the temperature in 30, 2, 5 and 5 mK steps for the respective cooldowns, always from low to high. The colors depict the dV/dI, the derivative of the measured voltage with respect to the set bias current normalized to the normal state resistance RN.
middle of the arms will act as two parallel Josephson junctions. In the middle we have left an small island to investigate whether such an island may have influence on the interference patterns, as it would host a single superconducting domain. In the IVT measurement, visible in Figure 4.2 top right, the original transition of the disk is still visible as the blue band tracing the critical current transition of the first cooldown, which leads us to believe that this is the superconducting transition of the neck. We also see that the superconducting transition is much slower in increasing the critical current, though appears at the same critical temperature of
TC = 1.35 K. This is a common feature that we find for superconducting
Sr2RuO4samples that show signs of Josephson junctions.
In the third cooldown, we removed the small island and thinned the arms of the ring to 0.27 µm. As we can see in Figure 4.2 bottom left, this procedure has had little influence on the shape of the critical current curve visible in the IVT, but the critical current at 1.0 K decreased according to the thinner
4.1 Sample C15F5: Disk, ring and arm 29
arms of the ring.
In the fourth cooldown, we milled a gap in one of the arms of the ring. Here, we see that the normal state resistance has doubled, as it should, but the critical current at 1.0 K only decreases to about 3/4th, as we can see
comparing the two IVT measurements in Figure 4.2.1 With a single arm, the
SQUID interference pattern should vanish during this cooldown, but there is still one domain wall in the arm of the ring acting as a single Josephson junction, so Shapiro steps should still appear.
4.1.1
Out-of-plane magnetic fields
In each of the cooldowns, interesting phenomena emerge sweeping the OOP magnetic field. We will adress the interesting features again per cooldown.
In the first cooldown of the disk, we expected to find no indications of interference. For the higher temperature, the critical current vs field pattern at 1.2 K, seen in Figure 4.3 in the top-right figure, seems Gaussian. Instead, for the lower temperatures, we find an onset of an interference pattern, perhaps indicating a domain wall configuration in the neck of the disk. Moving on to the second cooldown of the ring with an island. Immediately clear from the second row of Figure 4.3 is the SQUID interference pattern indicative of Josephson junctions in both arms of the ring. The periodicity
of 3.6±0.2 mT matches the dimensions of the ring with effective area of
Ae f f =0.6±0.1 µm2according toΦ0 =∆Φ=µ0∆H·Ae f f.
On the right, we also see that the periodicity does not depend on tempera-ture, ruling out other types of interference patterns than SQUID patterns. [7] Together with the previous results of this group, [62] we are confident that such a pattern will always appear for any doubly connected
supercon-1The IVT of the fourth cooldown looks a bit funny because the measurement has been
done with a.c. current-voltage settings. There was a slight problem with the voltage offset measured by the lockin, therefore the data has been patched together in two different ways.
One (T>1.2 K) using the IV slope with the lock-in provides directly and one (T<1.2 K)
with the derivative of the voltage and current that the lock-in calculates indirectly. The first method yields a much less noisy signal, but was not possible for part of the measurement with the faulty voltage offset.
- 2 0 - 1 0 0 1 0 2 0 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 d .c . C u rr e n t (µ A ) M a g n e t i c F i e l d ( m T ) 0 . 0 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8 0 . 0 d V / d I ( m Ω) T = 1 0 0 0 m K z e r o f i e l d c o o l e d C D 1 - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 2 0 0 m K 1 0 0 0 m K 8 0 0 m K C ri ti c a l C u rr e n t (µ A ) M a g n e t i c F i e l d ( m T ) C D 1 - 5 0 5 1 0 1 5 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 d .c . C u rr e n t (µ A ) M a g n e t i c F i e l d ( m T ) 0 . 0 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8 0 . 0 d V / d I ( m Ω) T = 1 2 0 0 m K z e r o f i e l d c o o l e d C D 2 - 1 0 - 5 0 5 1 0 - 1 4 0 - 1 2 0 - 1 0 0 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 C ri ti c a l C u rr e n t (µ A ) M a g n e t i c F i e l d ( m T ) 4 0 0 m K 1 0 0 0 m K 1 2 0 0 m K 1 2 0 0 m K C D 2 - 2 0 - 1 0 0 1 0 2 0 3 0 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 d .c . C u rr e n t (µ A ) M a g n e t i c F i e l d ( m T ) 0 . 0 0 1 2 . 5 2 5 . 0 3 7 . 5 5 0 . 0 6 2 . 5 7 5 . 0 8 7 . 5 1 0 0 d V / d I ( m Ω) T = 1 0 0 0 m K z e r o f i e l d c o o l e d C D 3 - 2 0 - 1 5 - 1 0 - 5 0 5 1 0 1 5 2 0 - 1 4 0 - 1 2 0 - 1 0 0 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 C ri ti c a l C u rr e n t (µ A ) M a g n e t i c F i e l d ( m T ) 4 0 0 m K 8 0 0 m K 1 0 0 0 m K 1 2 0 0 m K C D 3 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 - 1 0 0 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 d .c . C u rr e n t (µ A ) M a g n e t i c F i e l d ( m T ) 0 . 0 0 2 1 . 9 4 3 . 8 6 5 . 6 8 7 . 5 1 0 9 1 3 1 1 5 3 1 7 5 d V / d I ( m Ω) T = 8 0 0 m K z e r o f i e l d c o o l e d C D 4 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 4 0 0 m K 8 0 0 m K 1 0 0 0 m K 1 2 0 0 m K C ri ti c a l C u rr e n t (µ A ) M a g n e t i c F i e l d ( m T ) C D 4
Figure 4.3: Sample C15F5: OOP magnetic field measurements. Each row corre-sponds to one of the four cooldowns in order from top to bottom. On the left is one of the OOP measurements at 1, 1.2, 1 and 0.8 K respectively. On the right are the critical currents of the first transitions plotted versus OOP magnetic field for various temperatures. The current steps are typically 0.5-1 µA and the field steps are 1 mT for the first and fourth cooldown, and 0.2-0.25 mT for the second and third cooldowns.
4.1 Sample C15F5: Disk, ring and arm 31
ducting structure of Sr2RuO4of similar dimensions.
If the island hosts a single chiral domain, it may produce a magnetic field measurable by the SQUID device. However, it seems that the island did not have any perceivable influence on the measurements. Perhaps because its size is too small compared to the coherence length of the Cooper pairs in Sr2RuO4to host a superconducting domain.2
In the third cooldown, where we took the most extensive field measure-ments, the qualitative phenomenon is the same, and the periodicity of 3.3±0.2 mT was also roughly the same. At higher currents or fields around the SQUID pattern, a second transition is visible, probably indicating the superconducting transition of the neck of the ring.
This second transition is most visible in the fourth cooldown, depicted in the fourth row of Figure 4.3. Here, we cut through one of the arms of the ring and as was to be expected, the SQUID pattern has vanished, but there also seems no indication of a single-slit interference pattern corresponding to a single Josephson junction. A quick estimate of the junction area, assuming the coherence length ξ(0) =66 nm as the length of the Josephson junction and the arm width as its width, yields a periodicity of 111 mT, which is higher than the critical field. Therefore, it is logical that there no interference pattern is visible.
Indeed, we measured the critical field for the full operational range of the cryostat down to 300 mK, as can be seen in Figure 4.4, and it appears that for this geometry and assumption of domain configuration, it is impossible to measure any interference pattern due to a single Josephson junction. In Figure 4.4, the critical field of the arm of the ring (fourth cooldown) fol-lows a(TC−T)1/2type pattern, though the critical temperature TC seems
somewhat lower than the IVT measurement, due to the high bias current used to do this measurement. At each temperature the field is swept at a constant bias current of 10 µA. As soon as a high enough voltage is measured, the critical field is found. Note that the critical field at 300 mK is higher than the bulk critical field of 75 mT reported by Mackenzie and Maeno. [8] This yields a slightly lower higher coherence length of ap-proximately ξ(0) = pΦ0/2πHc2(0) = 57 nm. This phenomenon has been
reported by Uchida et al. for extremely thin films with thickness t =50 nm.
[82] 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 1 . 1 1 . 2 1 . 3 1 . 4 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 C ri ti c a l F ie ld ( m T ) T e m p e r a t u r e ( K )
Figure 4.4: Sample C15F5: Critical Field measurement of the arm of the ring (fourth cooldown). Here, the critical field, measured at 10 µA is depicted versus temperature (in 30 mK steps).
4.1.2
Shapiro steps
In the fourth cooldown we have performed two sets of measurements that clearly show Shapiro steps at given frequencies. For these measurements we measured the IV characteristic while bathing the sample in RF radiation with the maximum power of our RF source. The results of these type of measurements are shown in Figure 4.5 for two temperatures. On the left of these graphs the resistance is plotted as the color scale, with the frequency on the horizontal axis and the normalized voltage on the vertical axis, with normalization constantΦ0νRF. The blue spots at integer normalized voltage
are dips in the IV and represent integer Shapiro steps. We see that for most frequencies, only the first step is visible. As one can see on the right, the Shapiro steps are not as pronounced that they appear as flat regions in the IV curve, except for the 640 MHz step, but rather regions where the slope is less steep. For 640 MHz a second integer step is also visible. These measurements give clear evidence that there is a Josephson junction in the single arm of the ring. Specifically, cutting the ring did not remove the Josephson junction.
4.1 Sample C15F5: Disk, ring and arm 33 5 5 0 6 0 0 6 5 0 7 0 0 7 5 0 8 0 0 8 5 0 9 0 0 9 5 0 1 0 0 0 - 3 - 2 - 1 0 1 2 3 V o lt a g e ( Φ0 νRF ) F r e q u e n c y ( M H z ) 0 . 0 0 0 . 0 3 1 3 0 . 0 6 2 5 0 . 0 9 3 8 0 . 1 2 5 0 . 1 5 6 0 . 1 8 8 0 . 2 1 9 0 . 2 5 0 d V / d I (Ω) T = 1 0 0 0 m K - 6 0 - 5 5 - 5 0 - 4 5 - 4 0 - 3 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 - 3 - 2 - 1 0 1 2 3 6 4 0 M H z 7 2 0 M H z 8 8 0 M H z 1 0 0 0 M H z V o lt a g e ( Φ0 νRF ) d . c . C u r r e n t ( µ A ) T = 1 0 0 0 m K 3 1 0 3 2 0 5 5 0 5 6 0 6 2 0 6 3 0 6 4 0 6 5 0 7 1 0 7 2 0 8 7 0 8 8 0 9 7 0 9 8 0 9 9 0 1 0 0 0 1 0 9 0 1 1 0 0 1 1 1 0 1 1 2 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 V o lt a g e ( Φ0 νRF ) F r e q u e n c y ( M H z ) 0 . 0 0 5 0 . 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 d V / d I ( m Ω) T = 8 0 0 m K 5 5 6 0 6 5 7 0 7 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 3 2 0 M H z 5 5 5 M H z 6 4 0 M H z 7 2 0 M H z 1 1 1 0 M H z V o lt a g e ( Φ0 νRF ) d . c . C u r r e n t ( µ A ) T = 8 0 0 m K
Figure 4.5: Sample C15F5: Shapiro steps. On the left is a detailed frequency sweep with a step size of 10 MHz for the 1 K measurement and 1 MHz for the 0.8 K measurement. The frequency is plotted against the normalized voltage. The Shapiro steps are visible as blue spots in the resistance. On the right are a few selected IV characteristics of the same measurement.
An interesting aspect, seen in the 0.8 K measurement, is that for the exact halves of the frequencies of the most pronounced steps at 640 MHz and 1110 MHz we find that the steps at 320 MHz and 555 MHz skip the first step and only show the second integer step. However, as for Shapiro steps the first step should always be the most pronounced, we believe that these are harmonic resonances and that the sample is actually excited by the 640 MHz and 1110 MHz frequencies, whilst radiating them with 320 MHz and 555 MHz.
The reason for the Shapiro steps not appearing at every frequency has to do with the power transfer of the RF radiation to the sample. At the source, we apply a constant power, but the transfer function to the sample is highly frequency dependent and depends highly on the geometry of the inner vacuum chamber and the distance between the open-ended coax cable and the sample.
We must also mention that at some point after these measurements there was a leak in the inner vacuum chamber of the cryostat and we had to repump the system at 10 K. This probably rearranged the RF cable in such a way that the power dissipation was not optimal anymore. After this pumping, only the steps at 640 MHz and 1110 MHz remained visible, but barely. With a new RF cable in a fifth cooldown of this sample, only small Shapiro steps have been seen, indicating that the power transfer to the sample is still not optimal.
In hindsight, we have reanalyzed the frequency sweeps that we mea-sured in the second and third cooldown and found Shapiro steps in both cooldowns, seen in Figure 4.6. However, these steps were also barely vis-ible. It is unknown whether the geometry of the sample made it more difficult for Shapiro steps to appear or that the power transfer was not optimal during these cooldowns.
- 1 0 0 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 6 4 0 M H z 1 1 1 0 M H z V o lt a g e ( Φ0 νRF ) d . c . C u r r e n t ( µ A ) T = 1 0 0 0 m K
(a)Second CD (ring with island).
8 0 8 5 9 0 9 5 1 0 0 1 0 5 1 1 0 1 1 5 1 2 0 1 2 5 1 3 0 1 3 5 1 4 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 1 0 1 0 M H z 1 1 4 0 M H z V o lt a g e ( Φ0 νRF ) d . c . C u r r e n t ( µ A ) T = 4 0 0 m K
(b)Third CD (ring without island).
Figure 4.6: Sample C15F5: Shapiro steps in the second and third cooldown. For both cooldowns, IV characteristics are plotted that show a faint Shapiro step.
4.1 Sample C15F5: Disk, ring and arm 35
4.1.3
In-plane magnetic fields
Probing the sample with IP fields yields an interesting phenomenon that we will try to describe here. Setting an IP field seems to make the sample susceptible to ‘current switchable states’ in the IV curve. In Figure 4.7, we observe a higher critical current state corresponding to the blue curve, and a lower critical current state corresponding to the red curve. These IV curves are measured with an IP field of 15 mT at an angle of 55° (where 90° corresponds to the current direction), roughly corresponding to the angle of the neck into the arm of the ring.
- 1 2 0 - 1 0 0 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 - 8 - 6 - 4 - 2 0 2 4 6 8 d o w n u p n o f i e l d V o lt a g e ( µ V ) d . c . C u r r e n t ( µ A ) T = 6 0 0 m K , µ 0H I P = 1 5 m T , 5 5 ° 0 - - > P - - > N - - > 0 - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 - 3 - 2 - 1 0 1 2 3 d o w n u p V o lt a g e ( µ V ) d . c . C u r r e n t ( µ A ) 0 - - > P - - > N - - > 0 T = 1 0 0 0 m K , µ 0H I P = 1 5 m T , 5 5 °
Figure 4.7: Sample C15F5: IV measurement with IP field at 55° in the fourth cooldown (arm of the ring). In this figure, two IV characteristics are shown for 0.6 K on the left and 1.0 K on the right. On the left, an IV curve without any field is added for comparison in black. Both are taken sweeping the current from 0 to positive to negative and back to 0. The red curve corresponds to the upward sweep, the blue curve to the downward sweep.
To reproduce such a hysteretic curve, a certain order of sweeping the current is required. There are two ’switching’ points in the IV curve at bias currents of around±60 µA. To obtain the lower critical current state, one must first sweep the bias current past the negative switching point. To obtain the higher critical current state, corresponding to the blue curve, one must first sweep the bias current past the positive switching point. If the bias current is too low to reach the switching point, the material will remain in the previous state. We are led to believe there are two relatively stable states, accessible by sweeping the bias current in one direction or the other. Furthermore, the anomalous IV pattern switches polarity by reversing the field direction.
