Anomalous Enhancement of Upper Critical Field in Sr2RuO4 Thin Films

TL;DR

This study reports a significant enhancement of the upper critical magnetic field in Sr2RuO4 thin films, indicating altered superconducting properties possibly linked to a triplet d-vector rotation, advancing understanding of exotic superconductivity.

Contribution

It reveals a dramatic enhancement of Hc2 in Sr2RuO4 thin films due to dimensional crossover, differing from bulk behavior and suggesting new superconducting phenomena.

Findings

Hc2 is dramatically enhanced in thin films compared to bulk

The enhancement follows a distinct relation from bulk Hc2 and transition temperature

Possible rotation of the triplet d-vector associated with the enhancement

Abstract

We report large enhancement of upper critical field Hc2 observed in superconducting Sr2RuO4 thin films. Through dimensional crossover approaching two dimensions, Hc2 except the in-plane field direction is dramatically enhanced compared to bulks, following a definite relation distinct from bulk one between Hc2 and the transition temperature. The anomalous enhancement of Hc2 is highly suggestive of important changes of the superconducting properties, possibly accompanied with rotation of the triplet d-vector. Our findings will become a crucial step to further explore exotic properties by employing Sr2RuO4 thin films.

Equations

$${\left(\frac{H_{c2}(\theta )\sin \theta }{H_{c2||c}}\right)}^2+{\left(\frac{H_{c2}(\theta )\cos \theta }{H_{c2||a}}\right)}^2=1$$

$$\left|\frac{H_{c2}(\theta )\sin \theta }{H_{c2||c}}\right|+{\left(\frac{H_{c2}(\theta )\cos \theta }{H_{c2||a}}\right)}^2=1$$

$$\alpha \left|\frac{H_{c2}(\theta )\sin \theta }{H_{c2||c}}\right|+(1-\alpha ){\left(\frac{H_{c2}(\theta )\sin \theta }{H_{c2||c}}\right)}^2$$

$$+{\left(\frac{H_{c2}(\theta )\cos \theta }{H_{c2||a}}\right)}^2=1$$

$$\frac{H_{\mathrm{c}2\parallel c}}{{T_{\mathrm{c}}}^2}=\frac{\pi {\Phi }_0}{8}{\left(\frac{a{k}_{\mathrm{B}}}{\hslash v_{\mathrm{F},ab}}\right)}^2$$

Full text

Superconductors with a multicomponent order parameter, represented by spin-triplet superconductors, have attracted great interest as a ground of rich physics originating in the internal degrees of freedom. Among them, a layered-perovskite superconductor Sr2RuO4 has been a leading candidate possibly having chiral $p$-wave symmetry SRO ; SROreview1 ; SROreview2 , which is one of topological superconducting states supporting Majorana modes at edges and vortices SROreview3desiringfilm ; SROreview4desiringfilm . For further investigation and possible applications of the unique properties, the use of Sr2RuO4 thin films has been increasingly demanded in recent years SROreview3desiringfilm ; SROreview4desiringfilm .

In general, bulk superconducting state or pairing symmetry can be altered in the thin film form, affected by dimensionality change, inversion symmetry breaking, and/or epitaxial strain book . Spin-triplet superconducting states are characterized by $d$-vector, which represents the pair amplitude for the spin component perpendicular to the corresponding basis. Particularly in the case of Sr2RuO4, it has been theoretically suggested that the $d$-vector can flip from perpendicular (chiral $p$-wave) to parallel to the RuO2 $ab$ plane in the reduced dimensions, while the system still can host the Majorana modes thinSROdvector . Also, in helium-3 superfluid phases, changes of the $p$-wave order parameter have been experimentally demonstrated by mesoscopically confining it in a two-dimensional (2D) cavity thin3Hedvector . In this context, it is indispensable to examine fundamental superconducting properties of Sr2RuO4 thin films. While growth of the superconducting films had been extremely challenging over the past decades since the discovery of Sr2RuO4 YoshiharuPLD ; RobinsonPLD , the reproducible and controllable growth has been recently achieved by refining molecular beam epitaxy techniques UchidaMBE ; HariMBE .

Upper critical field $H_{\mathrm{c}2}$ is one of the fundamental superconducting parameters related to superconducting symmetry, and thus has been intensively investigated in the study of Sr2RuO4 bulks Hc2kittaka_graph ; Hc2akimacohelength_graph ; Hc2mackenzie_graph ; Hc2yoshida_graph ; Hc2deguchigap2 ; Hc2deguchigap1phase2 ; Hc2Pauli1 ; Hc2Pauli2 ; Hc2yonezawafirst1 ; Hc2eutecticRu_3K1 ; Hc2eutecticRu_3K4 ; Hc2inplane ; Hc2calc2inplane ; Hc2strain ; WHHforSRO1 . While its behavior is generally consistent, some features have been interpreted as incompatible with the simple $p_{x}\pm ip_{y}$ model SROreview4desiringfilm . In particular, $H_{\mathrm{c}2}$ observed for the in-plane field direction is much more suppressed than expected at low temperatures, also accompanied with the first-order superconducting transition Hc2kittaka_graph ; Hc2yonezawafirst1 . This suppression implies that $H_{\mathrm{c}2}$ for $H\parallel a$ might be affected by the paramagnetic pair breaking induced by the Zeeman splitting, called the Pauli limit SROreview4desiringfilm .

Here we report detailed dependences of $H_{\mathrm{c}2}$ in Sr2RuO4 thin films, by measuring low-temperature magnetotransport systematically changing the field angle. The superconducting films are grown on a lattice-matched cubic substrate, yielding extremely limited defects in the films UchidaMBE . In addition to dimensional crossover confirmed in the field angle and temperature dependences, $H_{\mathrm{c}2}$ in the films is largely enhanced over a wide range of field angles except the in-plane direction, up to about four times the bulk value. This anomalous enhancement indicates that the triplet $d$-vector in thin films may be aligned on the $ab$ plane, consistent with the recent theoretical prediction thinSROdvector .

Superconducting single crystalline Sr2RuO4 films as displayed in Figs. 1(a) and (b) were epitaxially grown on cubic (LaAlO3)0.3(SrAl0.5Ta0.5O3)0.7 (LSAT) (001) substrates by oxide molecular beam epitaxy, following the same procedures detailed in Ref. UchidaMBE . Sr and Ru elemental fluxes were simultaneously supplied from a conventional Knudsen cell and an electron beam evaporator, respectively. The deposition was performed flowing distilled 100% ozone with a pressure of $1\times 10^{-6}$ Torr and heating the substrate at 900 *∘*C. The film thickness is typically 50 nm along the $c$ axis and the channel area of each sample is approximately 500 $\mu$m $\times$ 200 $\mu$m in the $ab$ plane. Four-point measurements of the longitudinal resistivity were performed using low-frequency lock-in techniques with an excitation current of 3 $\mu$A along the $a$ axis. Two samples were cooled down to 60 mK in a 3He-4He dilution refrigerator equipped with a superconducting magnet. As shown in Figs. 1(c) and (d), a superconducting transition with $T_{\mathrm{c}}\sim 1.3$ K (onset) is confirmed for a typical sample. While the present films do not yet reach the high standard quality of Sr2RuO4 bulk single crystals highestbulk , the transition temperature and its sharpness are now qualitatively comparable to the first reported bulk single crystal SRO . For field rotation in the $ac$ plane, the samples were set on a single-axis rotating stage mounted on the mixing chamber.

Figures 1(e) and (f) show field dependence of the in-plane resistivity in a 50 nm thick Sr2RuO4 film, taken for $H\parallel a$ and $H\parallel c$ geometries at various temperatures down to 60 mK. Unlike the Ru eutectic phase Hc2eutecticRu_3K1 ; Hc2eutecticRu_3K4 or uniaxially strained phase Hc2strain , hysteresis between the upward and downward sweeps is not detected in the resistivity. With increasing field, the resistivity changes from zero to a normal-state value due to the suppression of superconductivity through $H_{\mathrm{c}2}$. Reflecting anisotropic superconductivity of this compound, the superconducting state is maintained up to higher fields for $H\parallel a$ than for $H\parallel c$.

Detailed field angle dependence of $H_{\mathrm{c}2}$ approaches 2D behavior with reducing the system thickness. Figure 2 compares field angle dependence between a Sr2RuO4 bulk Hc2kittaka_graph and the film, where the out-of-plane field angle $\theta$ is measured from the $a$ axis. In the bulk, the angle dependence except for a very low angle region is well described by the following anisotropic 3D Ginzburg-Landau (GL) model Hc2kittaka_graph ; Hc2yonezawafirst1 .

[TABLE]

$H_{\mathrm{c}2}$ is assumed to be dominated by the diamagnetic pair breaking process originating from the screening currents, known as the orbital limit. The coherence length along the $c$ axis $\xi_{c}$, calculated from the GL expression $\xi_{c}=\sqrt{\Phi_{0}H_{\mathrm{c2}||c}/2\pi{H_{\mathrm{c2}||a}}^{2}}$, is 3.2 nm, which is much larger than the lattice spacing of the RuO2 layers. In this regard, superconductivity in the Sr2RuO4 bulk is not classified into ideal 2D systems Hc2kittaka_graph . On the other hand, the angle dependence in the 2D limit is explained by the Tinkham model.

[TABLE]

As shown in Figs. 2(c) and (d), angle dependence observed in the Sr2RuO4 film is fitted better by the 2D model. Assuming that both $H_{\mathrm{c}2\parallel a}$ and $H_{\mathrm{c}2\parallel c}$ are determined by the orbital limit as described by the GL equations, the effective superconducting thickness $d$ is estimated at 23 nm from $d=\sqrt{6\Phi_{0}H_{\mathrm{c2}||c}/\pi{H_{\mathrm{c2}||a}}^{2}}$. Considering that the film thickness is 50 nm, the film can be understood to be located in a dimensional crossover region.

In a very low angle region, $H_{\mathrm{c2}}$ seems suppressed compared to the 2D model. One possible origin of the deviation is the 2D-3D crossover. In such an intermediate superconducting state, the following empirical model interpolating Eqs. 1 and 2 has been proposed to explain the transitional angle dependence quasi2D1 .

[TABLE]

The curve fitting is improved by adopting this model with $\alpha$ ranging about from 0.8 to 0.9 (for details see Supplemental Material supplemental ), also suggesting that the system is located in the crossover region. $H_{\mathrm{c}2}$ around $H\parallel a$ may be affected also by the presence of the Pauli limit, as discussed later.

Fig. 3(a) summarizes the $H$-$T$ phase diagram obtained for the Sr2RuO4 film. Surprisingly, $H_{\mathrm{c}2}$ for $H\parallel c$ shows linear temperature dependence down to the lowest temperature without suppression as in the Werthamer-Helfand-Hohenberg (WHH) theory WHH1 , as also clearly confirmed in the raw data in Fig. 1(f). The linear dependence without any suppression at low temperatures may be related to $d$-vector flipping from perpendicular to parallel to the $ab$ plane in thin films, where the Pauli limit is no longer effective for the out-of-plane direction. For $H\parallel a$, $H_{\mathrm{c}2}$ follows the WHH-like curve but is rather weakly suppressed at low temperatures, in comparison to the bulk, as clearly seen in Fig. 3(b). Such a deviation from clear square-root temperature dependence expected in the 2D GL model has been also confirmed in other crossover systems showing the transitional field angle dependence with $0<\alpha<1$ quasi2D2 . $h^{\ast}$, $H_{\mathrm{c}2}$ normalized by the initial slope at $T_{\mathrm{c}}$, is saturated at about 0.64 for $H\parallel a$, which is even higher than the value of 0.42 measured for the bulk Hc2kittaka_graph .

While the superconducting state approaches 2D like in the Sr2RuO4 thin film, the anisotropy ratio $\Gamma=H_{\mathrm{c}2\parallel a}/H_{\mathrm{c}2\parallel c}$ itself is reduced to 10 near $T_{\mathrm{c}}$ and 6 at the lowest temperature. This primarily results from increase in $H_{\mathrm{c}2\parallel c}$, about four times over the bulk. As confirmed in Fig. 2(a), $H_{\mathrm{c}2}$ is anomalously enhanced over a wide range of field angles centered at $H\parallel c$. Figure 4(a) plots the correlation between $H_{\mathrm{c}2\parallel c}$ and $T_{\mathrm{c}}$ for Sr2RuO4 bulks and films including previously reported other superconducting samples YoshiharuPLD ; UchidaMBE . Almost independent of the sample quality, the bulk and film $H_{\mathrm{c}2\parallel c}$ follow each universal curve, which is roughly proportional to $T_{\mathrm{c}}^{2}$ as expected for the orbital-limiting $H_{\mathrm{c}2}$. In the case of dirty samples, $\xi$ decreases with decrease of the mean free path $l$. This results in the extrinsic enhancement of $H_{\mathrm{c}2}$, and this trend can be confirmed for MgB2 and YBa2Cu3O7 as positive correlation in the $l$-$\xi$ plot in Fig. 4(b). In the case of clean samples, on the other hand, $\xi$ increases with decrease of $l$, accompanied by the decrease of $T_{\mathrm{c}}$ or superconducting gap $\Delta_{0}$. This trend appears as negative correlation in the $l$-$\xi$ plot. The Sr2RuO4 films and also bulks independently show the clean-limit trend, excluding the extrinsic effects as a possible origin of increase of $H_{\mathrm{c}2\parallel c}$.

By assuming that the GL in-plane coherence length $\xi_{ab}=\sqrt{\Phi_{0}/2\pi H_{\mathrm{c2}||c}}$ is equal to the Pippard one $\xi_{ab,0}=\hbar v_{\mathrm{F},ab}/\pi\Delta_{0}$ at the lowest temperature and using the superconducting gap relation $2\Delta_{0}=ak_{\mathrm{B}}T_{\mathrm{c}}$, the following relation can be derived.

[TABLE]

In the right hand side, material dependent parameters are only the coupling ratio $a$ and the in-plane Fermi velocity $v_{\mathrm{F},ab}$. For example, if we assume the BCS limit $a=3.5$ and take an experimental value of $v_{\mathrm{F},ab}=9.3\times 10^{4}$ m/s averaged on the active $\gamma$ band FSparameterization ; BROstrain , the dashed curve in Fig. 4(a) is obtained in rough agreement with but somewhat below the bulk trend, although a detailed analysis is surely dependent on the momentum-dependent gap structure Hc2deguchigap2 as well as the multi-band effect Hc2deguchigap1phase2 . In the case of thin films, on the other hand, other intrinsic origins should cause the further enhancement of $H_{\mathrm{c}2\parallel c}$ from the bulk trend. In terms of the epitaxial strain effect, a change in the in-plane lattice parameter compared to bulks is as small as $-0.07$% at room temperature UchidaMBE , which can be further reduced to $+0.03$% at low temperatures SROparameter ; LSATparameter . In addition, angle-resolved photoemission spectroscopy on strained Sr2RuO4 films grown on various substrates has demonstrated that the in-plane effective mass shows weak monotonic dependence on the strain value (less than 5% for the 1% in-plane lattice change) for all the three bands BROstrain , indicating that $v_{\mathrm{F},ab}$ is not a principal factor determining the enhancement. The uniaxial strain effect on bulks and films Hc2strain ; HariMBE is also excluded, as the present tiny strain is biaxial. Instead, an increase in the coupling ratio $a$ (almost double) is one plausible origin. The enhancement of $H_{\mathrm{c}2\parallel c}$ and the rather two-dimensional-like field-angle dependence are commonly observed in the films, regardless of the definition of $T_{\mathrm{c}}$ nor the film quality, as shown in Supplemental Material supplemental . Because all the other possible origins, such as film quality, epitaxial strain, and quantum confinement, are carefully excluded, the enhancement of $H_{\mathrm{c}2\parallel c}$ or $a$ is most likely related to the observed dimensional crossover. Electrons may couple more strongly in the real space through the dimensional crossover, resulting in the shorter $\xi_{ab}$. Its microscopic mechanism will need to be further elucidated from theoretical aspects, while it may be also consistent with the recent theoretical prediction on two-dimensional Sr2RuO4 films, as discussed below.

While $H_{\mathrm{c}2}$ is largely enhanced centered at $H\parallel c$, it remains relatively low for $H\parallel a$. One origin of this difference is a change in the out-of-plane electronic structure by quantum confinement in films. However, an increase in the out-of-plane Fermi velocity $v_{\mathrm{F},c}$, which may account for the elongation of $\xi_{c}$, is less likely in terms of the mass enhancement due to the confinement. Another possible origin of this relative suppression is the Pauli limit. The presence of the Pauli limit for $H\parallel a$ is not generally consistent with the $d$-vector direction ($d\parallel c$) in the 2D $p_{x}\pm ip_{y}$ state SROreview4desiringfilm . Therefore, the suppression of $H_{\mathrm{c}2\parallel a}$ suggests a change of the pairing symmetry, possibly accompanied with the $d$-vector flipping ($d\parallel ab$) suggested for thin films thinSROdvector . This is also consistent with the disappearance of the suppression newly observed in the temperature dependence of $H_{\mathrm{c}2\parallel c}$ (Fig. 3(b)), indicating the absence of the Pauli limit for $H\parallel c$ in thin films.

In summary, we have revealed changes of the Sr2RuO4 superconducting state induced by confining it into thin films. Through the dimensional crossover, $H_{\mathrm{c}2}$ is intrinsically enhanced centered at $H\parallel c$ compared to bulks, while it remains suppressed for $H\parallel a$. The anomalous enhancement of $H_{\mathrm{c}2}$ suggests important changes of the spin-triplet superconducting state in the reduced dimensions. Taken together, these findings are compatible with the triplet state with the $d$-vector flipped parallel to the RuO2 plane, which still could support the Majorana modes at edges and vortices thinSROdvector . Our study will provide the significant basis for further investigating superconducting properties of Sr2RuO4 thin films and applying its exotic states to junction devices.

We acknowledge T. Nojima, S. Kittaka, D. Manske, T. Shibauchi, Y. Mizukami, Y. Iwasa, Y. Saito, and Y. Nakagawa for fruitful discussions. This work was supported by Grant-in-Aids for Scientific Research (B) No. JP18H01866 and Scientific Research on Innovative Areas “Topological Materials Science” No. JP16H00980 from MEXT, Japan and JST PRESTO No. JPMJPR18L2 and CREST Grant No. JPMJCR16F1, Japan.