Vortex creep at very low temperatures in single crystals of the extreme type-II superconductor Rh$_9$In$_4$S$_4$

TL;DR

This study visualizes vortex creep at very low temperatures in Rh$_9$In$_4$S$_4$ using STM, revealing thermal fluctuations as the primary cause and showing the vortex lattice remains hexagonal during creep.

Contribution

First direct imaging of vortex creep at millikelvin temperatures in a superconductor, analyzing thermal and quantum barriers and vortex lattice dynamics.

Findings

Vortex creep occurs at temperatures as low as 150 mK.

Vortex lattice remains hexagonal during creep.

Creep velocity depends on local pinning arrangements.

Abstract

We image vortex creep at very low temperatures using Scanning Tunneling Microscopy (STM) in the superconductor Rh$_9$In$_4$S$_4$ ($T_c$=2.25 K). We measure the superconducting gap of Rh$_9$In$_4$S$_4$, finding $\Delta\approx 0.33$meV and image a hexagonal vortex lattice up to close to H$_{c2}$, observing slow vortex creep at temperatures as low as 150 mK. We estimate thermal and quantum barriers for vortex motion and show that thermal fluctuations likely cause vortex creep, in spite of being at temperatures $T/T_c<0.1$. We study creeping vortex lattices by making images during long times and show that the vortex lattice remains hexagonal during creep with vortices moving along one of the high symmetry axis of the vortex lattice. Furthermore, the creep velocity changes with the scanning window suggesting that creep depends on the local arrangements of pinning centers. Vortices fluctuate on small scale erratic paths, indicating that the vortex lattice makes jumps trying different arrangements during its travel along the main direction for creep. The images provide a visual account of how vortex lattice motion maintains hexagonal order, while showing dynamic properties characteristic of a glass.

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Full text

I Introduction

In type-II superconductors, vortex-vortex repulsion favors an ordered vortex lattice. This competes with vortex pinning and thermal fluctuations that favor disordered vortex arrangementsBlatter et al. [1994], Brandt [1995]. Under the action of a current, vortices move and the superconductor leaves the zero resistance state. Vortex motion producing a residual dissipation in absence of an applied current has been observed in many materials and is termed vortex creep. Vortex creep has awakened interest of experiment and theory alike for a long time, because it ultimately limits achieving a true dissipationless state in a superconductor.

Vortex creep can occur in any practical situation in a superconducting specimen. For example, when changing the magnetic field below the superconducting critical temperature $T_{c}$ in a type-II superconductor, vortices enter from the edges of the sample jumping over pinning barriers and filling the interiorBlatter et al. [1994], Brandt [1995], Larkin and Ovchinnikov [1979], Altshuler and Johansen [2004]. The magnetization usually acquires a near equilibrium situation after some time, which is mostly quite short. However, often the magnetization continues to vary over much longer times due to vortex creep. Vortex creep is also observed when field cooling from above $T_{c}$, because strongly pinned metastable vortex lattices are created when crossing the peak effect regionHenderson et al. [1996], Paltiel et al. [2000]. Vortex creep is driven by a current density $j$, which results from the difference between the actual magnetization from the metastable vortex configurations and its equilibrium value. $j$ is well below the critical current density $j_{c}$Feigel’man et al. [1989].

A large Ginzburg-Levanyuk number $G_{L}$ Blatter et al. [1994], Brandt [1995], Feigel’man et al. [1989], Kes et al. [1989], Yeshurun and Malozemoff [1988], Blatter and Geshkenbein [1993], Kierfeld et al. [2000], Konczykowski et al. [2012], Eley et al. [2016], Levanyuk [1959], Ginzburg [1960] favors vortex creep. $G_{L}$ is the ratio between the critical temperature and the superconducting condensation energy ($\propto H_{c}^{2}\xi^{3}$, with $H_{c}$ the thermodynamic critical field) in a volume of size of the coherence length $\xi$ and is given by $G_{L}=\frac{1}{2}\left(\frac{\mu_{o}k_{B}T_{c}}{4\pi H_{c}^{2}\xi^{3}}\right)^{2}$. $G_{L}$ quantifies the relevance of thermal fluctuations in a superconducting material and ranges from $\approx 10^{-1}$ in cuprate superconductors to $\approx 10^{-10}$ in conventional low $T_{c}$ superconductors.

Vortex creep has been studied thoroughly using macroscopic techniques, such as magnetization or resistivity (see e.g. Refs.Nideröst et al. [1998], Dumont and Mota [2002], Prozorov et al. [2008], Konczykowski et al. [2012], Fruchter et al. [1991], Klein et al. [2014], Eley et al. [2016]). However, imaging experiments at low temperatures are, to our knowledge, scarce.

Previous imaging studies mostly address vortex flow driven by a metastable magnetic field configuration within the peak effect regime or at high temperaturesPardo et al. [1998], Troyanovski et al. [1999, 2002], Guillamon et al. [2011], Lee et al. [2011]. Moving vortex lattices have been imaged in real space at very low magnetic fields using magnetic decoration and at high magnetic fields using STMPardo et al. [1998], Troyanovski et al. [1999, 2002], Guillamon et al. [2011], Lee et al. [2011]. They have shown different structure factors changing from hexagonal to smectic-like Olson et al. [1998]. Smectic structures forming channels where vortex positions are uncorrelated along the flow direction but correlated perpendicular to it appear more often at low magnetic field where intervortex interaction is weak Moon et al. [1996]. In contrast, crystalline-like hexagonal moving lattices are predicted for stiff dense lattices under high driving currents suggesting that motion can induce order in the vortex lattice. The latter, known as moving Bragg glass, are free from topological defects and show long range positional and orientational correlations Giamarchi and Le Doussal [1994, 1995]. For instance, the smectic driven phase has been observed in 2H-NbSe2 at extremely low fields and high driving currents, with crystalline order appearing at higher fields Pardo et al. [1998]. In our experiment, contrary to previous cases, the flow velocity is low and we measure at high fields in the dense lattice regime and at very low temperatures where the stiffness is maximum, so that motion is expected to be correlated.

Furthermore, in previous experiments, vortex motion occurs due to currents of order of $j_{c}$. Here we directly observe slow vortex creep at millikelvin temperatures for a situation where $j<<j_{c}$. We find that in this regime vortices move by jumps with large directional changes and show strong fluctuations on top of the overall motion in the direction for creep.

We have used single crystals of the recently discovered superconducting material Rh9In4S4 ($T_{c}$=2.25 K) Kaluarachchi et al. [2016]. The low temperature coherence length is of $\xi\approx 9.4$ nm. The penetration depth is very large, of nearly 600 nm and the mean free path is rather low, of approximately 5 nm (see Appendix). The Ginzburg-Landau parameter is of $\kappa\approx 61$, showing that this material is an extreme type-II superconductor. The Ginzburg-Levanyuk parameter is of $G_{L}\approx 1.78\times 10^{-5}$. This value is above the one found usually in low $T_{c}$ superconductors and shows that thermal effects are important for the vortex lattice of Rh9In4S4. The peak effect is observed in a magnetic field range similar to that found in many other strongly disordered superconducting single crystalline materials Kaluarachchi et al. [2016]. As we show below, vortex creep is particularly apparent in this compound, probably due to the combination of a large peak effect region and a relatively large Ginzburg-Levanyuk parameter.

First we show basic characterization of this material from tunneling density of states and vortex imaging. We find a superconducting tunneling conductance at 150 mK that is spatially homogeneous at zero magnetic field with superconducting gap $\Delta=0.33$ meV. We also find that the vortex lattice is hexagonal at low temperatures. Then we image creeping vortex lattices and determine the main parameters of vortex motion (direction and velocity) directly from our images by following each vortex during experiments lasting for many hours.

II Experimental Methods

Single crystals of Rh9In4S4 were grown using a solution growth technique Canfield and Fisk [1992], Canfield and Fisher [2001] from high-purity Rh, In and S elements. Details of the synthesis and sample characterization are described in Ref. Kaluarachchi et al. [2016]. We used crystals with lateral sizes between 1-3 mm and transversal areas less than 1 mm2 perpendicular to [100]. Figure 1 shows a picture of the samples. To prepare the surface, we tried to cleave the samples in both ambient and cryogenic conditions using the pulling system described in Ref.Suderow et al. [2011]. We combine this with either a ceramic blade or a piece of brass glued on top of the crystals so as to break the crystals by pushing on the slab at low temperatures. Most often crystals did not break and the few ones that broke revealed irregular surfaces which were not shiny and where we did not achieve stable tunneling conditions as we find usually in metals with a cleaving plane. We obtained however excellent results by using the pristine as-grown samples. Scanning conditions at low temperatures were reproducible and independent of the tunneling conductance.

To obtain the tunneling conductance, we take the numerical derivatie of the I-V curve, as in previous workSuderow et al. [2011], Crespo et al. [2006], Herrera et al. [2015]. The magnetic field is applied parallel to the $c$-axis. To obtain the vortex lattice images shown in this work, we map the zero bias conductance normalized to its value at high bias. Neither filtering nor image treatment is made to the images we present here.

III Tunneling conductance and vortex lattice of Rh9In4S4

Fig.1a shows the crystal structure of Rh9In4S4. In the topographic STM image shown there (Fig.1b), we can find square symmetric features in the image, separated by steps (Fig.1c, in the appendix we provide further topographic images). The step height is an integer multiple of the $c$-axis unit cell, pointing out that the observed intertwined squares and rectangles are due to the growth procedure that can be expected in a single crystalline material with tetragonal symmetry.

The tunneling conductance as a function of bias voltage for different temperatures is shown in Fig.2a (left panel). At the lowest temperatures we observe a smeared tunneling conductance that deviates strongly from the high quasiparticle peaks and zero bias conductance expected in a standard BCS s-wave superconductor (see e.g.Herrera et al. [2015]). The tunneling conductance remains mostly unaltered as a function of the position over the surface. In order to find the superconducting gap, we deconvolute the density of states from the tunneling conductance, see Fig.2a (right panel) (using the derivative of the Fermi function at each temperature, see e.g. Ref.Crespo et al. [2006]). We observe that the density of states becomes featureless at a temperature of 2.3 K, which coincides with the bulk transition temperature $T_{c}$Kaluarachchi et al. [2016]. The quasiparticle peaks in the density of states are located at an energy of 0.33 meV, which is the value expected for the superconducting gap within simple s-wave BCS theory ($\Delta$=$1.76k_{B}T_{c}$). The temperature dependence of the position of the quasiparticle peaks also follows expectations from BCS theory (Fig.2c).

The size of the superconducting gap found with our measurements is somewhat below the one extracted from macroscopic specific heat measurements. These lead to a slightly enhanced at the superconducting transition observed in specific heat, pointing out possible strong coupling effectsKaluarachchi et al. [2016]. Our measurements show rather smeared density of states, which points out that there can be a distribution of values of the superconducting gap with larger values than the one obtained from the above analysis. More striking is however the presence of a finite density of states at the Fermi level. The bulk density of states is clearly zero within the gap, as shown by specific heat measurementsKaluarachchi et al. [2016]. Other intermetallic compounds with surfaces prepared in similar conditions show zero tunneling conductance within the gapRubio-Bollinger et al. [2001], Herrera et al. [2015]. Some sort of surface contamination could eventually lead to a normal contribution to the tunneling conductance by, for instance, a normal surface layer or pair breaking. This being said, it is quite remarkable that we find the bulk $T_{c}$ and $H_{c2}$, as well as the value of $\Delta$ expected within s-wave BCS theory from our tunneling experiments.

Results under magnetic fields are shown in Fig.3. We find a hexagonal vortex lattice between 0.2 T up to 2.0 T (H${}_{c2}\approx$ $2.5$ T Kaluarachchi et al. [2016]), with the intervortex distance decreasing with field as expected. We do not observe signatures of Caroli-de Gennes-Matricon Andreev core statesCaroli et al. [1964], Hess et al. [1990], Guillamon et al. [2008]. Rather, the normalized tunneling conductance at the vortex center reaches one (Fig.3c). This is not surprising, taking into account that Rh9In4S4 is strongly in the dirty limit and that the discrete core states are expected to turn into a continuum in the dirty limitRenner et al. [1991].

The vortex lattice is mostly oriented along the directions defined by the steps observed at the sample surface (see Fig. 1 and Fig.9 at the appendix). Fig.3(b) shows the position of the six lowest Bragg peaks of the Fourier transform of the vortex lattice images obtained at different magnetic fields vs the direction of the surface steps. The vortex lattice orients along one of the two crystalline directions of the square atomic lattice at the basal plane. Likely this shows the influcence of large scale defects, such as grain boundaries or large size dislocations, that follow the symmetry of the crystalline structure and pin the vortex lattice.

We have also analized the vortex core radius $\cal C$, which we define as in Refs.Fente et al. [2016], Ichioka et al. [2016], namely the inverse of the slope of the radial variation of the superconducting order parameter from the vortex center. $\cal C$ can be obtained from the STM images as described in Refs.Fente et al. [2016], Ichioka et al. [2016]. To this end, we make an angular average of the tunneling conductance $\sigma$ (normalized in such a way as to provide unity at the vortex center and zero in between vortices) around the center of a vortex and fit the result to the radial dependence of the density of states discussed in Fente et al. [2016], Ichioka et al. [2016]. The experimental result is shown as points and the fits as lines in Fig.3c. We use only images made at two magnetic fields, where we obtained sufficient contrast to perform the analysis. This is admitedly too little to provide a serious magnetic field dependence. Still, we find values for $\cal C$ (inset of Fig.3c) that are consistent with the magnetic field dependence proposed in Ref. Fente et al. [2016] $\cal C$ $\propto 1/\sqrt{H}$. We can adjust the $\cal C$ vs H dependence in such a way as to cross the two measured points and obtain at the same time an extrapolation to Hc2 that provides the same value as the superconducting coherence length obtained from Hc2, $\xi\approx$ 9.4 nm.

Finally, we have analyzed the intervortex density of states by taking tunneling conductance curves at the lowest temperatures in the middle between vortices. The result is shown in Fig.3d. The intervortex density of states at the Fermi level increases linearly with the magnetic field.

IV Vortex creep

We find that vortices are slowly moving during imaging. Images in Fig.3 were made within a few minutes to obtain static views of the vortex lattice. In Fig.4 we show experiments made in different locations and magnetic fields. Motion remains unaltered (with similar velocity and direction of motion) at temperatures as low as 150 mK and after more than one day. We acquired consecutively images taken in a few minutes each during a period of several hours. In Fig.4 we show the average over a set of consecutive images (left panels of Fig.4) and the vortex trajectories during the whole experiment (right panels of Fig.4) (see Appendix for a larger set of vortex images at each magnetic field). We include in this figure zero field cooled (upper and lower panels) and field cooled experiments (middle panel). We find that the vortex motion strongly depends on location and magnetic field-temperature history. Generally, we observed either net vortex motion along a well defined direction (upper panel), net motion involving changes in the directions (middle panel) and fluctuations around the vortex positions (lower panel). The average of vortex velocities estimated by considering positions of vortices in subsequent frames is of $100$  nm/h in the upper panel, of $40$ nm/h in the middle panel and close to zero in the lower panel.

In the location where we took images of the upper panel, we find that vortices move along a main axis of the vortex lattice and that vortex motion is smooth and correlated, without any signature of jumps or steps. In the location of the middle panel, vortex motion is also along one main axis of the vortex lattice, but vortices move along another axis after a certain time. In the location of the lower panel, we only observe wiggling, without a net motion along a given direction.

To show the correlated nature of net vortex motion we have acquired an image (Fig.5) during 16 hours at the same scanning window where we did the experiment shown in the upper panel of Fig.4. Scanning was very slow, so that vortices were moving below the tip. The fast scanning axis is along a direction nearly perpendicular to vortex motion. Therefore, vortices appear compressed along the slow scanning direction (see (Fig.5b). By stretching the image with the intervortex distance at this magnetic field (0.4 T), we reproduce a hexagonal vortex lattice (Fig.5d). To understand this, we must consider the relation between vortex and tip motion. Vortices have to move with a constant velocity over the time frame given by the time a vortex remains below the tip, which is of approximately 12 minutes. The vortex lattice moves in the opposite direction to the slow scanning motion. The tip velocity along this direction is of $v_{tip}=18$ nm/h. Using this velocity and the ratio between the intervortex distance expected for this field, $d$, and the value found in the compressed vortex lattice, $d^{\prime}$, we obtain for the vortex velocity $v=(d/d^{\prime}-1)v_{tip}=$ 100 nm/h. This value coincides with the value measured by making fast consecutive images discussed above in the same field of view (upper panels of Fig.4).

Creep can be thermal and/or quantum activated. The relevant scales are given, respectively, by the collective pinning energy $U_{c}=H_{c}^{2}\xi^{3}(j_{c}/j_{0})^{1/2}$ and the Euclidean action $S_{E}=(\hbar/Q_{u})(j_{0}/j_{c})^{1/2}$ with $H_{c}$ the thermodynamic critical field, $\xi$ the coherence length, $j_{c}$ and $j_{0}$ the critical and depairing current densities and $Q_{u}$ the quantum resistance Blatter et al. [1994]. The crossover temperature $T_{0}=\hbar U_{c}/S_{E}$ determines the temperature below which quantum effects dominate. In Rh9In4S4, we find $U_{c}=3.5K$ using $\mu_{0}H_{c}=25mT$, $\xi=9.4$ nm Kaluarachchi et al. [2016], $j_{c}=3\cdot 10^{8}A/m^{2}$ Dew-Hughes [2001] and $j_{0}=3\cdot 10^{10}A/m^{2}$. On the other hand, superconductors with a large normal state resistivity and small coherence lengths, i.e. quantum resistance $Q_{u}=(e^{2}/\hbar)(\rho_{n}/\xi)$ of order of a k$\Omega$ are candidates for quantum creepBlatter and Geshkenbein [1993]. In Rh9In4S4, $Q_{u}$ is of $4\cdot 10^{-2}$ in between the values reported for conventional superconductors ($Q_{u}\sim 10^{-3}$) where quantum effects are negligible and cuprate high $T_{c}$ superconductors ($Q_{u}\sim 10^{-1}$) where quantum fluctuations are largeBlatter et al. [1994], Ephron et al. [1996], Eley et al. [2016]. When comparing thermal and quantum contributions, we find that the crossover temperature $T_{0}$ is of about $15mK$ indicating that, even at 150 mK, motion is likely thermally activated.

V Discussion and conclusions

In summary, we have imaged the hexagonal superconducting vortex lattice in the extreme type-II s-wave superconductor Rh9In4S4. We determine the superconducting coherence length and find values comparable to those obtained using macroscopic upper critical field measurements. We observe directly vortex creep at very low temperatures, $T/T_{c}<0.1$ in nearly equilibrium conditions. From our direct observation, we conclude that the creep velocity strongly dependens on each experiment. During creep, vortices move by jumping over the pinning barriers due to thermal excitation. The vortex lattice remains hexagonal over length scales well above the image size, comprising here several hexagons. For these small sized vortex bundles and the small driving currents present in our experiment, pinning of isolated vortices plays a minor role in the motion, which rather reflects collective activated motion of the hexagonal lattice over the pinning landscape.

On the other hand, we should consider the intrinsic disorder of the vortex lattice, characterized by absence of positional ordering at large length scales Larkin and et al [1970], Nattermann [1990], Giamarchi and Le Doussal [1994, 1995]. The consequence of disorder is expected to manifest in the dynamic properties of the vortex lattice by the fact that the height of the pinning barriers $U$ depends on $j$ for $j<<j_{c}$. In particular, $U$ was shown to diverge as $U\propto j^{-\alpha}$ with decreasing $j$Feigel’man et al. [1989]. This actually leads to a truly superconducting (dissipationless) phase also in systems with large vortex creepBlatter et al. [1994], Brandt [1995]. In our experiments we observe very different dynamic behavior for configurations in which the current producing vortex motion $j<<j_{c}$, which depends on the particular pinning landscape at each location, has been certainly very different too. The fact that we observe moving as well as nearly static vortex lattices shows that the barrier height $U$ for vortex motion strongly depends on location and magnetic field-temperature history.

Interestingly, we observe that the velocity fluctuates significantly around the average values provided above (see Fig.6). If we follow the velocities of all vortices as a function of time, we observe that all motion is accompanied by wiggling vortices, resulting in a roughly Gaussian distribution of velocities with a width between 16 nm/h and 23 nm/h. The resulting fluctuations in the vortex positions are somewhat smaller but of order of the intervortex distance (as can be readily seen in the Fig.4). Instead of moving smoothly, the vortex lattice makes jumps trying adjacent metastable configurations, showing that the pinning potential has many small minima that are superimposed to a large variation producing creepBlatter et al. [1994]. As the barrier for motion is large for small currents, the wiggles just provide intermediate and unstable arrangements.

So our images of creeping vortex lattices visually show the capability of the vortex lattice to deform and adapt to the pinning landscape, giving glassy dynamic properties while maintaining hexagonal order.

VI Acknowlegments

We acknowledge R. Willa and V.G. Kogan for helpful and critical comments and discussions with S. Vieira. We wish to acknowledge the support of Departamento Administrativo de Ciencia, Tecnología e Innovación, COLCIENCIAS (Colombia) Programa Doctorados en el Exterior Convocatoria 568-2012. We further acknowledge support by the Spanish Ministry of Economy and Competitiveness (FIS2014-54498-R, MDM-2014-0377), by the Comunidad de Madrid through program Nanofrontmag-CM (S2013/MIT-2850), by EU (IG, European Research Council PNICTEYES grant agreement 679080, FP7-PEOPLE-2013-CIG 618321, Cost MP-1201) and by Axa Research Fund. SEGAINVEX-UAM and Banco Santander are also acknowledged. Work done in Ames Lab was supported by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358.

VII Appendix

Figure 7 shows the Hall response of the single crystals of Rh9In4S4. Calculated carrier density using single band model at 5 K is $2.8\times 10^{21}$ cm*-3*. Using this value for the carrier density, the Drude model and the resistivity Kaluarachchi et al. [2016] we estimate a mean free path of $\ell\approx 5$ nm. In Figure 8 we show a set of the frames from the experiment shown in Fig.4. In Fig.9 we show topographic images obtained in several places of the sample.