**Preface**

Three authors of this paper, namely Rustem Khasanov, Alexander Shengelaya, and Hugo Keller, had the opportunity to work with Alex Müller during his stay at the University of Zürich. By being specialized in measurements of the magnetic penetration depth (*λ*) by means of muon-spin rotation/relaxation and magnetization techniques, we try to test the prediction of Alex Müller that the symmetry of the superconducting order parameter in cuprate high-temperature superconductors (HTSs) changes "from purely *d* at the surface to more *s* inside" (Müller, K.A. *Phil. Mag. Lett.* **2002**, *82*, 279–288). As a result of our studies, complex order parameters were detected in *λ*(*T*) measurements for hole-doped HTSs such as La1.83Sr0.17CuO4, YBa2Cu3O7−*δ*, and YBa2Cu4O8. Here, we present evidence that the mixed order parameter symmetry is realized in Sr0.9La0.1CuO2, i.e., in a superconductor belonging to the family of electron-doped cuprate HTSs.

## **1. Introduction**

The order parameter in cuprate high-temperature superconductors (HTSs) is generally considered to be of pure *d*-wave symmetry. Pertinent evidence for this assumption for both the electron- and the hole-doped classes of HTSs stems from experiments where mainly surface phenomena are probed (e.g., angular resolved photoemission [1–4] or tricrystal experiments [5–8]). On the other hand, experimental data obtained by using techniques that probe the bulk of the material, such as nuclear magnetic resonance [9], Raman scattering [10,11], neutron crystal-field spectroscopy [12–14], and muon-spin rotation/relaxation (*μ*SR) [15–17] provide strong evidence for the presence of a substantial *s*-wave component in the order parameter. Based on these experimental findings and on an earlier idea that in HTSs, two superconducting condensates with different order parameter symmetries (*<sup>s</sup>*- and *d*-wave) coexist [18–20], Müller [21–24] proposed a scenario in which the order parameter symmetry in HTSs changes from primarily *d*-wave at the surface to more *d* + *s*-wave in the bulk. At first glance, this scenario seems to contradict the accepted possible symmetries of the order parameter in HTSs [7]. However, by applying an interacting boson-model used in nuclear physics theory to the *D*4*h* symmetry of HTSs, Iachello [25,26] demonstrated that a crossover from a *d*-wave order parameter symmetry at the surface to a *d* + *s*-wave symmetry in the bulk is indeed possible from a group theoretical point of view.

This scenario [21,22] can be directly tested by investigating the temperature dependence of the magnetic penetration depth *λ* near the surface and in the bulk of an HTS, since the behavior of *<sup>λ</sup>*−<sup>2</sup>(*T*) for a *d*-wave and a *s*-wave superconductor differs considerably. An isotropic *s*-wave pairing state leads to an almost constant value of the superfluid density *ρs* ∝ *λ*−<sup>2</sup> for *T* ≤ 0.3 *Tc* [27–30], while the presence of nodes in the gap gives rise to a continuum of low lying excitations, resulting in a linear temperature dependence of *<sup>λ</sup>*−<sup>2</sup>(*T*) at low temperatures [30–33].

Here, we report on a magnetization study of the magnetic penetration depth near the surface and in the bulk of the electron-doped HTS Sr0.9La0.1CuO2. The in-plane (*λab*) and the out-of-plane (*λc*) components of the magnetic penetration depth were extracted from measurements of the AC susceptibility on a *c*-axis oriented powder sample in the Meissner state and analyzed within a two-gap *s* + *d*-wave scenario. The present results are compared with previous muon-spin rotation (*μ*SR) experiments on Sr0.9La0.1CuO2 [34], which probe the magnetic penetration depth in the bulk of the sample.

## **2. Experimental Details**

Details on the sample preparation for Sr0.9La0.1CuO2 can be found elsewhere [35]. The Sr0.9La0.1CuO2 sample used in the present study was the same as measured by means of *μ*SR in [34].

The sintered Sr0.9La0.1CuO2 sample was grounded for about 20 min in order to obtain very small grains needed for the determination of the magnetic penetration depth from magnetic susceptibility measurements. In order to perform measurements of the in-plane and the out-of plane components of the magnetic penetration depth, samples were *c*-axis oriented in a static 9T magnetic field and cured at elevated temperatures in epoxy resin. In total, four *c*-axis oriented samples were prepared and measured.

The grain size distribution of the powder was determined by analyzing scanning electron microscope photographs (see Figure 1a). The measured particle size distribution *N*(*R*) (*R* is the grain radius) is shown in Figure 1b.

The AC magnetization ( *MAC*) experiments were performed by using a commercial PPMS Quantum Design magnetometer (*μ*0*HAC* = 0.3 mT, *ν* = 333 Hz) at temperatures between 1.75 K and 50 K. In two sets of experiments, the AC field was applied parallel and perpendicular to the *c*-axis. The absence of weak links between grains was confirmed for both field orientations by the linear magnetic field dependence of *MAC* at *T* = 10 K for AC fields ranging from 0.1 to 1.0 mT and frequencies between 49 and 599 Hz.

**Figure 1.** (**a**) An example of the scanning electron microscope photograph of the powdered Sr0.9La0.1CuO2 sample. (**b**) The grain size distribution *N*(*R*) determined from scanning electron microscope photographs. The thin vertical lines represent the statistical errors (±*N*(*R*)).

#### **3. Results and Discussion**

The temperature dependence of the magnetic penetration depth was extracted from the measured *MAC* by using the Shoenberg formula [36], modified for the known grain size distribution *N*(*R*) [37]:

$$\chi(T)\_{\parallel,\perp} = -\frac{3}{2} \int\_0^\infty \left( 1 - \frac{3\lambda\_{\parallel,\perp}^\*(T)}{R} \coth \frac{R}{\lambda\_{\parallel,\perp}^\*(T)} + \frac{3[\lambda\_{\parallel,\perp}^\*(T)]^2}{R^2} \right) N(R)R^3 dR / \int\_0^\infty N(R)R^3 dR. \tag{1}$$

Here, *χ*,<sup>⊥</sup> = *MAC* · *ρX*/*<sup>m</sup>* (*m* is the sample mass, and *ρX* is the X-ray density of Sr0.9La0.1CuO2) is the volume susceptibility, and *<sup>λ</sup>*∗,<sup>⊥</sup> is the effective magnetic penetration depth for the magnetic field applied parallel () and perpendicular (⊥) to the *c*-axis. In order to determine *λab* and *λc* from the measured *<sup>λ</sup>*∗,⊥, we followed the procedure of Porch et al. [37]:


Bearing this in mind, the temperature dependencies of the in-plane and the out-of plane components of the magnetic penetration depth are determined as:

$$
\lambda\_{ab}(T) = \lambda\_{\parallel}^\* \quad \text{and} \quad \lambda\_{\mathcal{E}} = 1.43 \lambda\_{\perp'}^\*
$$

respectively.

> The resulting temperature dependencies of *λ*−<sup>2</sup> *ab*and *λ*−<sup>2</sup> *c*are shown in Figure 2.

The zero-temperature values of *λab* and *λc* were obtained by a linear extrapolation of *λ*−<sup>2</sup> *ab*,*<sup>c</sup>*(*T*) for *T* < 5 K to *T* = 0, yielding *<sup>λ</sup>ab*(0) 157 nm and *<sup>λ</sup>c*(0) 1140 nm. An uncertainty in the absolute values of *<sup>λ</sup>ab*,*<sup>c</sup>* was considered by taking into account the statistical nature of the grain size distribution (*N*(*R*) ± *<sup>N</sup>*(*R*); see Figure 1b), which resulted in a relative error of about ∼7% for both *λab* and *λ<sup>c</sup>*.

An additional source of uncertainty stemmed from the deviation of the grain shapes from the spherical one. Assuming a small deviation of the demagnetization factor (1/3 ± 10%), the relative error of *<sup>λ</sup>ab*,*<sup>c</sup>*(0) was of the order of 3%. Taking both sources of errors into account yielded: *<sup>λ</sup>ab*(0) = 157(15) nm and *<sup>λ</sup>c*(0) = 1140(100) nm. The value of *<sup>λ</sup>ab*(0) obtained here was in a good agreemen<sup>t</sup> with the *<sup>λ</sup>ab*(0) = 147(7) nm reported by Kim et al. [38] based on the analysis of reversible magnetization data.

**Figure 2.** Temperature dependencies of *λ*−<sup>2</sup> *ab* (**a**) and *λ*−<sup>2</sup> *c* (**b**) for Sr0.9La0.1CuO2 extracted from the measured *MAC*(*T*) by using Equation (1). Solid lines represent fits with the two-gap *s* + *d*-wave model. *λ*−<sup>2</sup> *ab* (*T*) and *λ*−<sup>2</sup> *c* (*T*) were analyzed simultaneously by means of Equation (2) with *<sup>ω</sup>ab*, *ωc*, *<sup>λ</sup>ab*(0), and *<sup>λ</sup>c*(0) as the individual fitting parameters and common *s*-wave and anisotropic *d*-wave gap functions as described by Equations (4)–(6). See the text for details.

In order to test the predictions of [18–22] and in analogy to our previous results on cuprate HTSs [15–17,34], the experimental data presented in Figure 2 were analyzed by decomposing *λ*−<sup>2</sup> *ab* (*T*) and *λ*−<sup>2</sup> *c* (*T*) into two contributions with *s*-wave and *d*-wave symmetry [39]:

$$\frac{\lambda\_{\rm ab,\varepsilon}^{-2}(T)}{\lambda\_{\rm ab,\varepsilon}^{-2}(0)} = \omega\_{\rm ab,\varepsilon} \cdot \frac{\lambda\_{\rm ab,\varepsilon}^{-2}(T,\Delta^{s})}{\lambda\_{\rm ab,\varepsilon}^{-2}(0,\Delta^{s})} + (1-\omega\_{\rm ab,\varepsilon}) \cdot \frac{\lambda\_{\rm ab,\varepsilon}^{-2}(T,\Delta^{d})}{\lambda\_{\rm ab,\varepsilon}^{-2}(0,\Delta^{d})}.\tag{2}$$

Here, Δ*s* and Δ*<sup>d</sup>* denote the *s*-wave and the *d*-wave gap, respectively, and *<sup>ω</sup>ab*,*<sup>c</sup>* is the weighting factor (0 ≤ *<sup>ω</sup>ab*,*<sup>c</sup>* ≤ 1), representing the relative contribution of the *s*-wave gap to *λ*−<sup>2</sup> *ab*,*c*. Both (*<sup>s</sup>*- and *d*-wave) components can be expressed by [16]:

$$\frac{\lambda\_{ab,\varepsilon}^{-2}(T,\Lambda^{s,d})}{\lambda\_{ab,\varepsilon}^{-2}(0,\Lambda^{s,d})} = 1 + \frac{1}{\pi} \int\_0^{2\pi} \int\_{\Lambda^{s,d}(T,\rho)}^{\infty} \left(\frac{\partial f}{\partial E}\right) \frac{E}{\sqrt{E^2 - \Lambda^{s,d}(T,\rho)^2}} \, dEd\rho \,. \tag{3}$$

Here, *f* = [1 + exp(*E*/*kBT*)]−<sup>1</sup> is the Fermi function, *ϕ* is the angle along the Fermi surface (*ϕ* = *π*/4 corresponds to a zone diagonal), and:

$$
\Delta^{s,d}(T,\varphi) = \Delta\_0^{s,d} \, \delta(T/T\_c) \, \lg^{s,d}(\varphi). \tag{4}
$$

Here, <sup>Δ</sup>*<sup>s</sup>*,*<sup>d</sup>* 0 is the maximum value of the gap at *T* = 0. The temperature dependence of the gap is approximated by *δ*(*T*/*Tc*) = tanh{1.82[1.018(*Tc*/*T* − 1)]0.51} [40,41]. The function *<sup>g</sup><sup>s</sup>*,*<sup>d</sup>*(*ϕ*) describes the angular dependence of the gap and is given by:

$$\mathcal{g}^s(\mathcal{q}) = 1 \tag{5}$$

for the *s*-wave gap and:

$$g^{d\_{An}}(\varphi) = \frac{3\sqrt{3a}}{2} \frac{\cos 2\varphi}{(1 + a\cos^2 2\varphi)^{3/2}} \tag{6}$$

for the anisotropic *d*-wave gap [42] (*a* is a constant). We want to stress that series of experimental [3,43] and theoretical works [42] sugges<sup>t</sup> that the angular dependence of the gap in the electron-doped HTSs differs significantly from the simple functional form Δ0 cos 2*ϕ*, observed for various hole-doped HTSs, and has the so-called anisotropic *d*-wave symmetry (with the gap maximum in between the nodal and the antinodal points on the Fermi surface).

The measured *λ*−<sup>2</sup> *ab* (*T*) and *λ*−<sup>2</sup> *c* (*T*) displayed in Figure 2 were analyzed simultaneously by means of Equation (2) with *ωab*, *ωc*, *<sup>λ</sup>ab*(0), and *<sup>λ</sup>c*(0) as the individual fitting parameters and common *s*-wave and anisotropic *d*-wave gap functions, as described by Equations (4)–(6). The results of the analysis are summarized in Figure 2 and Table 1.

Panels (a) and (b) of Figure 3 represent the angular dependencies of the *s* and the *dAn* superconducting energy gaps at *T* = 0. The solid blue and the red lines in Figure 3c correspond to the individual *s*-wave and *d*-wave contributions, respectively.

**Figure 3.** (**a**) The angular dependence of the *s*-wave gap at *T* = 0 (Δ*<sup>s</sup>*0 · *g<sup>s</sup>*(*ϕ*); see Equation (5) and Table 1). (**b**) The angular dependence of the anisotropic *d*-wave gap (Δ*d*0 · *g<sup>d</sup>*(*ϕ*); Equation (6) and Table 1). (**c**) Contributions of the *s*-wave (blue line) and the anisotropic *d*-wave (red line) gaps to the superfluid density (*ρs* ∝ *<sup>λ</sup>*−2) obtained by means of Equation (3). The dashed-dotted line represents the *T*<sup>2</sup> behavior, which is generally observed in various electron-doped HTSs (see [30] and the references therein).

From the results presented in Figures 2 and 3 and Table 1, the following important points emerge:


assume that its temperature dependence is mainly determined by surface properties and therefore follows the one expected for a *d*-wave superconductor. In contrast, *<sup>λ</sup>c*(0) is almost a factor 10 larger than *<sup>λ</sup>ab*(0), and thus, *λ*−<sup>2</sup> *c* (*T*) contains contributions from both the surface and the bulk (mixed *s* + *d*-wave order parameter).

**Table 1.** Summary of the analysis of *λ*−<sup>2</sup> *ab* (*T*) and *λ*−<sup>2</sup> *c* (*T*) for Sr0.9La0.1CuO2 by means of Equation (2). The absolute errors of *<sup>λ</sup>ab*,*<sup>c</sup>*(0) account for the uncertainties in the grain size distribution *N*(*R*) ±*N*(*R*) and that of the demagnetization factor 1/3 ± 10%; see the text for details. TF-*μ*SR, denotes the transverse-field muon-spin rotation/relaxation (TF-*μ*SR) experiments.


Additional arguments pointing to the validity of this scenario [21,22] come from the comparison of the in-plane magnetic penetration depth *<sup>λ</sup>ab*(*T*) measured near the surface in this work with that determined in *μ*SR experiments on a similar Sr0.9La0.1CuO2 sample [34] (see Figure 4). Note that *μ*SR is a powerful technique to probe the magnetic penetration depth in the bulk of a superconductor in the vortex state [31,47]. It is evident that *λ*−<sup>2</sup> *ab* (*T*) measured near the surface decreases more strongly with increasing temperature than that obtained in the bulk. On the other hand, *λ*−<sup>2</sup> *ab* (*T*)/*λ*−<sup>2</sup> *ab* (0) (*μ*SR) is relatively close to *λ*−<sup>2</sup> *c* (*T*)/*λ*−<sup>2</sup> *c* (0) determined by AC magnetization experiments, thus indicating that *λ*−<sup>2</sup> *c* (*T*) is mainly governed by bulk properties. Another interesting issue stems from the comparison of the absolute *λab* values obtained in the bulk and the surface sensitive experiments. The analysis of *λ*−<sup>2</sup> *ab* (*T*) (*μ*SR) within the above described scheme, with Δ*s*0 and Δ*dAn* 0 fixed to the values determined from the AC susceptibility experiments, results in *<sup>λ</sup>ab*(0) = 93(2) nm (see Table 1), which is more than 50% shorter than *<sup>λ</sup>ab*(0) = 157(15) nm obtained near the surface. We may assume, therefore, that the difference in the absolute values and the temperature dependencies of the bulk and the surface *λab* can be explained within the scenario proposed in [21,22] and is caused by a substantial reduction of the *s*-wave contribution near the surface (*<sup>ω</sup>ab* 4%) in comparison with that in the bulk (*<sup>ω</sup>ab* 72%; see Table 1).

To the best of our knowledge, the above presented results give a first example of different order parameter symmetries near the surface and in the bulk in electron-doped high-temperature cuprate superconductors. As for the hole-doped representatives of HTSs, the comprehensive analysis was made in a series of works of K.A. Müller [18–24]. By comparing the results of the "surface" and the "bulk" sensitive experiments, it was concluded that the superconducting order parameter in the hole-doped cuprate HTSs changes from purely *d* at the surface to the mixture of *s* and *d* in the bulk. The theory explanation was given by Iachello [25,26], based on purely symmetry considerations and in analogy with atomic nuclei. The model consists of *s* and *d* pairs (approximated as bosons) in a two-dimensional Fermi system with a surface. The transition takes place between the phase where only one type of boson condensate to the phase consisting of a mixture of two type of bosons.

**Figure 4.** The normalized superfluid density *λ*−<sup>2</sup> *ab* (*T*)/*λ*−<sup>2</sup> *ab* (0) (open circles and squares) and *λ*−<sup>2</sup> *c* (*T*)/*λ*−<sup>2</sup> *c* (0) (closed circles) obtained in the present study (closed and open circles) and by the transverse-field *μ*SR experiments (open squares) in [34]. The solid lines correspond to fits by means of Equation (2) with the parameters summarized in Table 1.

## **4. Conclusions**

In conclusion, the temperature dependence of the in-plane (*λab*) and the out-of-plane (*λc*) components of the magnetic penetration depth of Sr0.9La0.1CuO2 were determined in the Meissner state from AC susceptibility measurements. The temperature dependence of *λ*−<sup>2</sup> *ab* is well described by assuming that the superconducting order parameter is mainly of *d*-wave symmetry ( 96%) with *<sup>λ</sup>ab*(0) = 157(15) nm. The out-of-plane component was found to be much longer, *<sup>λ</sup>c*(0) 1140(100) nm. The temperature dependence of *λ*−<sup>2</sup> *c* is in accordance with a mixed *s* + *d*-wave order parameter with a substantial *s*-wave component of more than 50%. A comparison of *λ*−<sup>2</sup> *ab* (*T*) reported in this work with that obtained in the bulk by *μ*SR [34] reveals that the *s*-wave component of the order parameter is strongly suppressed near the surface of the superconductor, associated with a substantial reduction of the superfluid density by more than a factor of two. The results presented here are consistent with the scenario of a complex mixed *s* + *d*-wave symmetry order parameter proposed by Müller [18–24]. In particular, the prediction of a strongly suppressed *s*-wave component near the surface was confirmed experimentally. This study clearly demonstrates that special care must be taken when experimental results obtained by surface sensitive and bulk sensitive techniques are compared, since they do not necessarily probe the same properties of high-temperature superconductors.

**Author Contributions:** R.K., A.S., and H.K. specified the topic of the studies. R.K. performed the experiment, analyzed the data, and wrote the manuscript. R.B. performed the electron-microscopy experiments. A.S. and H.K. contributed to finalizing the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Swiss National Science Foundation, by the K. Alex Müller Foundation, and in part by SCOPES Grant No. IB7420-110784.

**Acknowledgments:** The authors are grateful to A. Bussmann-Holder and K.A. Müller for stimulating discussions. D.J. Jang and S.-I. Lee are acknowledged for providing the samples used for the experiments.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References and Note**




© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Review* **SrTiO 3—Glimpses of an Inexhaustible Source of Novel Solid State Phenomena**

**Wolfgang Kleemann 1,\*, Jan Dec 2, Alexander Tkach 3 and Paula M. Vilarinho 3**


Received: 6 August 2020; Accepted: 30 September 2020; Published: 4 October 2020

**Abstract:** The purpose of this selective review is primarily to demonstrate the large versatility of the insulating quantum paraelectric perovskite SrTiO3 explained in "Introduction" part, and "Routes of SrTiO3 toward ferroelectricity and other collective states" part. Apart from ferroelectricity under various boundary conditions, it exhibits regular electronic and superconductivity via doping or external fields and is capable of displaying diverse coupled states. "Magnetoelectric multiglass (Sr,Mn)TiO3" part, deals with mesoscopic physics of the solid solution SrTiO3:Mn<sup>2</sup>+. It is at the origin of both polar and spin cluster glass forming and is altogether a novel multiferroic system. Independent transitions at different glass temperatures, power law dynamic criticality, divergent third-order susceptibilities, and higher order magneto-electric interactions are convincing fingerprints.

**Keywords:** strontium titanate; quantum paraelectricity; quantum fluctuations; ferroelectricity; isotope exchange; external stress; polar metal; superconductivity; phase coexistence; magnetoelectric multiglass

## **1. Introduction**

In this review, we focus onto two research lines of strontium titanate, SrTiO3 (STO):

(1) the low-temperature phases around the quantum critical point of pure STO, and (2) the disordered electric and magnetic dipolar glassy phases in the solid solution STO: Mn. It is not intended to describe the full extent of all phenomena observed and to detail all of their properties from abundantly issued publications. We merely try to give an impression of some actual fields, which are partially related to our earlier cooperation with K. A. Müller and J. G. Bednorz.

STO is probably the most versatile perovskite-type oxide and one of the richest materials in terms of functionalities. In 1979, Müller and Burkard [1] reported that the planar permittivity of STO strongly increased upon cooling from ≈ 300 at room temperature and saturated with ε 110 ≈ 2.5·10<sup>4</sup> as *T* → 0 (comparable to ε 100 vs. *T*, cf. Figure 1, curve 1 [2]). They conjectured a "quantum paraelectric" ground state in close proximity to a ferroelectric (FE) one, where the centrosymmetric tetragonal lattice structure of STO becomes stabilized by quantum fluctuations of the nearly softening in-plane *F*1*u* lattice mode.

Quantum corrections to the temperature were proposed to describe the critical behavior of STO from the beginning [1]. In order to account for the obvious deviations from the mean-field Curie–Weiss behavior, some of us proposed a generalized modified "quantum Curie–Weiss law" [3].

$$\varepsilon' = \mathbb{C} / \left( T^{\mathbb{Q}} - T\_0^{\mathbb{Q}} \right)^{\mathbb{V}} \tag{1}$$

**Figure 1.** Temperature dependence of the dielectric permittivity ε [100] of Sr1-*x*Ca*x*TiO3 crystals with 0 ≤ *x* ≤ 0.12 [2].

where *C* stands for the Curie constant, γ for the critical exponent, and *TQ*0 = *TS*cot <sup>h</sup>(*TS*/*T*0) for the quantum critical temperature with classic critical temperature *T*0 and saturation temperature *TS* being related to the ground state energy of the quantum oscillator, *E*0 = *kBTS*. Crucial novel ingredients are the free parameter γ and the quantum temperature scale, *TQ* = *TS*cot <sup>h</sup>(*TS*/*T*), which replaces *T.* The best fit to the STO data [3] within 2 ≤ *T* ≤ 110 K yields *C* = (3 ± 2) 106, *TS* = (17 ± 1) K, *TQ*0 ≈ 0, and the highly non-classic exponent γ = 1.7 ± 0.2. Interestingly, a very similar value, γ ≈ 2, was recently obtained on pure STO from a conventional power-law fit for 4 ≤ *T* ≤ 50 K [4], where saturation effects deviated below ≈ 4 K. While our approach matches with this extreme quantum regime without any extra conditions, the Φ<sup>4</sup> model of [4] requires corrections due to long-range dipolar interactions and coupling of the electric polarization field to acoustic phonons.

#### **2. Routes of SrTiO3 toward Ferroelectricity and Other Collective States**

Overcoming quantum paraelectricity and reaching stable long-range ordered states by proper treatments has remained a major challenge for ongoing research on STO.


A combined method of substitution and reduction was utilized in the case of Cr-doped STO [11]. A significant spatial correlation between oxygen vacancies and Cr3+ ions in bulk was established in thermally reduced Cr-doped STO. In the presence of electron donors, the Cr atoms change their valence from 4+ to 3+. Consequently, this reduction drives a symmetry change of the crystal field experienced by the Cr ions from cubic to axial [11], which may be controlled by means of thermal annealing and/or doping with electron donors. This capability of controlling oxygen vacancies or transition-metal dopants has been essential for the development of semiconducting electronic devices [13].

(v) Theoretical predictions of superconductivity in degenerate semiconductors motivated research on reduced *n*-type STO, which revealed the critical temperature *Tc* ≈ 0.28 K as early as 1964 [14]. However, 32 years later, perovskite-like cuprates were to open the door to modern high-*Tc* superconductivity with *Tc* ≈ 30 K [15] and to the physics Nobel Prize [16]. On the other hand, gated *n*-type STO has reached at most only *Tc* ≈ 0.6 K [17].

Meanwhile numerous other processes have made STO a nearly inexhaustible source of activating novel solid state phenomena that sugges<sup>t</sup> future applications. This has remained an attractive research goal even more than 60 years after the pioneering experiments. Extending the initial idea of breaking the local symmetry by stress [5], Bednorz and Müller [2] introduced an *A*-site doping route by random replacement of Sr2+ ions with smaller Ca2+ ions in single crystals of Sr1-*x*Ca*x*TiO3 (SCT). The local decrease of volume creates random strain ("negative stress field"), which has an enormous effect on the dielectric response for doping levels 0.002 ≤ *x* ≤ 0.12, as shown in Figure 1 (curves 2–11). Sharp peaks occur at finite temperatures, 10 < *Tm* < 40 K, which clearly hint at polar phase transitions (PTs). Their easy axes are actually lying along [110] and [110] within the basal *xy*-plane and yield, e.g., ε110 *max* = 1.1·10<sup>5</sup> for *x* = 0.0107 [2]. Discussion within a random-field concept of PTs reveals *xy-*type quantum ferroelectricity above *x*c = 0.0018 along the *a* axes of the paraelectric parent phase and a PT into a random phase above *x*r ≈ 0.016 (Figure 1). Quantum corrections to the temperature (see Equation (1)) are essential to describe the critical behavior.

In order to understand more details, some of us measured the optical linear birefringence (LB), Δ*nac* = *nc* − *na*, where *nc* and *na* are the principal refractive indices at light wavelength λ = 589.3 nm, being linearly polarized along the *c* and *a* axes of the SCT crystal, respectively, as functions of temperature, *T* [18]. It is well-known that the LB is sensitive to both the axial rotation of the TiO6 octahedra, <sup>&</sup>lt;ΔΦ2>, below the antiferrodistortive phase transition temperature *Ta* = 105 K (for *x* = 0), and to the FE short-range order parameter, <sup>&</sup>lt;*P*2*x*>, where *x* <110>*c* (Figure 1). Indeed, non-zero LB arises in pure STO at the transition temperature *T*a = 105 K and at 115, 140, and 255 K (arrows) for *x* = 0.002, 0.0107, and 0.058, respectively, as shown in Figure 2. Additional FE anomalies, δ(Δ*nac*), are superposed at low *T*. Being non-morphic, they start smoothly with fluctuation tails and bend over into steeply rising long-range order parts below inflection points *T*1 ≈ 15, 28, and 50 K, respectively (arrows). These temperatures systematically exceed the ε vs. *T* peak temperatures, *Tm* = 14, 26, and 35 K, respectively (Figure 1), where discontinuities of d(Δ*nac*)/d*T* would be expected in case of PTs into long-range order. Absence of anomalies of this type and increasing differences, *T*1–*Tm*, at increasing *x* hint at continuously growing smearing of the PTs. Simultaneously, as *T* → 0, the polarization was calculated by use of the ordinary refractive index *n*o = 2.41 and the electro-optic coefficient difference *g*11-*g*31 = 0.14 m<sup>4</sup>/C<sup>2</sup> as < *P*2*x* >1/2 = {2δ(Δnac)/[no3(g11-g31)]} = 9.8, 29.4, and 42.6 mC/m<sup>2</sup> for *x* = 0.002, 0.0107, and 0.058, respectively [18]. Since < *P*2*x* >1/2 varies less than proportionally with *x*, comparatively incomplete FE order is observed. Further, the low-*T* polarization saturates, albeit slowly, at increasing electric field, *E.* This strongly hints at random-field induced nanodomains, whose average size increases with an applied ordering field. The increase of the average order parameter gives credit for disappearing domain walls as known from the domain-state FE K0.974Li0.026TaO3 [19].

**Figure 2.** Linear birefringence Δ*nac* vs. *T* measured at λ = 589.3 nm on crystallographic single domains of Srl-*x*Ca*x*TiO3 with *x* = 0.002 (1), 0.0107 (2), and 0.058 (3), respectively [18].

Further insight into FE SCT is gained from its relaxational behavior. Figure 3 shows the temperature dependence of the real and imaginary parts of the dielectric permittivity of SCT (*x* = 0.002), ε , and ε vs. *T*, at frequencies 10<sup>3</sup> ≤ *f* ≤ 10<sup>4</sup> Hz [20]. In view of the rounded peaks of ε (*T*), a polydomain state of this FE is conjectured. This has first been interpreted within the concept of "dynamical heterogeneity" [21], which assumes a manifold of mesoscopic "dynamically correlated domains", relaxing exponentially with uniform single relaxation times. Their superposition defines the observed polydispersivity of the sample. It represents aggregates of polar clusters surrounding the quenched off-center Ca2+ dopant dipoles.

**Figure 3.** Real and imaginary parts of the permittivity, ε and ε" vs. *T*, of Sr0.998Ca0.002TiO3 measured within 1.5 ≤ *T* ≤ 15 K at frequencies 10−<sup>3</sup> ≤ *f* ≤ 10<sup>4</sup> Hz [20]. *Tg* ≈ 3.8 K is indicated by an arrow.

It is noticed that ε vs. *T* peaks at a "glass temperature", *Tg* ≈ 3.8 K (Figure 3, arrow), in the quasi-static limit, *f* = 1 mHz, although at first glance, no glassy criticality as in spin glass is expected. However, in view of recently ascertained magnetic superspin glasses (SSG) of dipolarly coupled magnetic nanoparticles at low concentration [22], a related electric superdipolar glass (SDG) has become envisaged. It should behave like a relaxor ferroelectric [23] in terms of a superglassy critical power law behavior of the ε (*f*) vs. *T* peak position *Tm*.

$$f(T\_m) \propto \left(T\_m - T\_{\mathcal{S}}^c\right)^{z\upsilon}.\tag{2}$$

Evaluation over the whole range of frequencies, 10−<sup>3</sup> ≤ *f* ≤ 10<sup>4</sup> Hz, yields the expected dynamic critical exponent *z*ν ≈ 10 at *f* > 1Hz, while systematic deviations occur at lower *f* due to the well-known additional tunneling dynamics. Tests on the expected non-ergodicity of the SDG phase at *T* < *Tg* upon zero-field- and field-cooled temperature cycles, respectively (cf. Section 3) are in preparation.

At higher concentration of Ca2+, the polar nanoregions (PNRs) percolate into an FE ground state, as proven by first-order Raman scattering at the softening *F*1*u* phonon mode in SCT (*x* = 0.007) at *T* < *T*0 = 18 K [24]. SCT thus succeeds in demonstrating stable ferroelectricity. However, systematic research at increasing Ca content showed that *T*0 is limited to ≈35 K, where the dielectric anomaly becomes increasingly smeared [25]. Better success was achieved by the classic method of stress-induced ferroelectricity in pure STO [5]. To this end Haeni et al. [26] utilized 50 nm thick films of STO, which were epitaxially grown with approximately +1.5% biaxial tensile strain on a (110) DyScO3 substrate, while −0.9% uniform compression due to a (LaAlO3)0.29(SrAl0.5Ta0.5O3)0.71 (LSAT) substrate was barely active in this respect (Figure 4). The high permittivity in the films on DyScO3, ε up to 7000 at 10 GHz and room temperature, as well as its sharp dependence on an electric field is promising for device applications [4,26]. The observation of stress-induced ferroelectricity in STO films has confirmed theoretical predictions of Pertsev et al. [27]. While substrate induced tensile strain in epitaxial STO films via lattice parameter mismatch favors in-plane FE, compressive strain provides out-of-plane directed ferroelectricity [27]. This was observed by Fuchs et al. [28] at an STO film epitaxially grown on an STO substrate coated by compressive YBa2Cu3O7.

**Figure 4.** In-plane permittivity ε vs. *T* of a strained 50 nm epitaxial STO/(110)DyScO3 film at *f* = 10 GHz as compared to a compressed STO/LSAT film. The inset shows a Curie–Weiss fit to (<sup>ε</sup>*r*)−<sup>1</sup> with *T*0 ≈ 260 K [26].

Another realization of room temperature ferroelectricity in STO confirms theoretical predictions of proximity effects at interfaces of metals to oxides containing PNRs such as, for example, STO [29]. Lee et al. [30] reported emergence of room temperature ferroelectricity at reduced dimensions, thus refuting a long-standing contradicting notion. Piezoelectric force microscopy (PFM) was able to evidence room-temperature ferroelectricity in strain-free epitaxial films with 24 unit-cell-thickness of otherwise non-ferroelectric STO (Figure 5). Following arguments from defect engineering in SCT, the authors claimed that electrically induced alignments of PNRs at Sr deficiency related defects are responsible for the appearance of a stable net of ferroelectric polarization in these films. This insight might be useful for the development of low-*D* materials of emerging nanoelectronic devices.

To systematically control ferroelectricity in thin films of STO at room temperature, Kang et al. [31] selectively engineered elemental vacancies by pulsed laser epitaxy (PLE). Sr2+ vacancies play an essential role in inducing the cubic-to-tetragonal transition, since they break the inversion symmetry, which is necessary for switchable electric polarization. The tetragonality turns out to increase with increasing vacancy density, thus strengthening the ferroelectricity, as shown in Figure 6a. This research has

optimized tetragonality-induced ferroelectricity in STO with reliable growth control of the behavior. PFM yields stable hysteresis loops at room temperature, as shown in Figure 6b, where low and high laser fluences during PLE clearly demonstrate their key role in creating FE polarization. Similar propositions were made by the Barthélémy–Bibes group, which invoked both an electric field-switchable two-dimensional electron gas emerging in ferroelectric SCT films [32] and the non-volatile electric control of spin–charge conversion in an STO Rashba system [33].

**Figure 5.** Polarization hysteresis of 24 and 120 unit-cell-thick STO films at room temperature, measured by using the double-wave PUND technique with a triangular *ac* electric field of 10 kHz (see schematic inset). The hysteresis component is obtained by subtracting the non-hysteretic (up (U) and down (D)) from the total (positive (P) and negative (N)) polarization runs [30].

**Figure 6.** (**a**) Sr/Ti elemental concentration ratio (blue circles) and tetragonality measured at room temperature (red circles) plotted as functions of the laser fluence during pulsed laser epitaxy (PLE). (**b**) Ferroelectric hysteresis loops recorded by piezoelectric force microscopy (PFM) at 5-nm-thick SrTiO3 films grown with low and high laser fluences (LF and HF, respectively; see (**a**)) on different bottom electrodes (STO:Nb and SrRuO3/STO) [33].

Only recently has another insight into the ferroelectric state of compressively strained STO become available from high-angle annular dark-field imaging in scanning transmission electron microscopy. Salmani–Rezaie et al. [34] observed local polar regions in the room-temperature paraelectric phase of (001)-strained STO films, which were grown on (001) faces of LSAT and underwent an FE transition at low T. This unexpected feature was explained by a locally dipolar-ordered, but globally random phase of displaced Ti4<sup>+</sup> columns, which underwent a disorder–order transition on cooling.

This Section started with different methods to establish long-range order, such as ferroelectricity (FE), metallicity (MT), or superconductivity (SC) in suitably modified STO [2,9,14]. Lately, more demanding procedures have become successful to stabilize the co-existence of apparently contradictory properties, e.g., FE-SC and FE-MT, which appear self-excluding at first glance. In this context, obscure terms such as, for example, "polar metal" and "metallic ferroelectric" or "ferroelectric metal" have been used interchangeably by the research community. Only recently have subtle distinctions of these variants with respect to their electric field switchability been clarified [12], although this topic still remains under debate.

Rischau et al. [35] showed that SC can coexist with an FE-like instability in oxygen-reduced ("*n-*doped") Sr1-*x*Ca*x*TiO3-<sup>δ</sup> (0.002 < *x* < 0.009, 0 < δ < 0.001), where both long-range orders are intimately linked. The FE transition of insulating SCT was found to survive in this reduced modification. Owing to its metallic conductivity, the latter does not show a bulk reversible electric polarization and hence cannot be a true ferroelectric. However, it shows anomalies in various physical properties at the Curie temperature of the insulator, e.g., Raman scattering evidences that the hardening of the FE soft mode in the dilute metal is identical with what is seen in the insulator. The anomaly in resistivity was found to terminate at a threshold carrier density (*n*\*), near to which the SC transition temperature is enhanced [35]. This evidences the link between SC pairing and FE dipolar ordering, a subject of current attention [36].

Moreover, it is widely accepted that the low-*T* phase of STO lies in the vicinity of a quantum critical point, where di fferent phases (i.e., paraelectric, antiferrodistortive, FE, MT, and SC) with similar energies compete, while weak residual interactions may stabilize one or several of these states [37–39]. The coexistence of MT and FE states in STO has been addressed under the keyword charge transport in a polar metal by Wang et al. [40], who studied the low-*T* electrical resistivity in several Sr1-*x*Ca*x*TiO3-<sup>δ</sup> single-crystals at δ > 0 within 0.002 < *x* < 0.01 (Figure 7). Since both MT and FE are dilute, the distance between mobile MT electrons and fixed FE dipoles can be separately tuned but kept much longer than the interatomic distance. This opens the chance of activating a Ruderman–Kittel–Kasuya–Yosida-like interaction [41] of carriers with local electric moments, which was originally proposed by Glinchuk and Kondakova [42]. They introduced this indirect interaction of FE o ff-center ions with conduction electrons in order to explain high FE transition temperatures in certain narrow-gap semiconductors with high conductivity, such as Pb1-*x*Ge*x*Te. In agreemen<sup>t</sup> with this theory, it is expected that the threshold concentration of carriers, *n*\*, is proportional to *x*, which indicates that it occurs at a fixed ratio between inter-carrier and inter-dipole distances.

**Figure 7.** Ferroelectric (FE) phase transition temperatures *T*c in insulating Sr1-*x*Ca*x*TiO3 as functions of *x* (black balls [2]) and in metallic Sr1-*x*Ca*x*TiO3-<sup>δ</sup> (δ > 0) as functions of charge carrier density *n* and *x* = 0.0022, 0.0045 and 0.009 (green, blue, and red balls, respectively [40]).

Tomioka et al. [17] finally demonstrated the simultaneous occurrence of three states, FE, MT, and SC, by independently controlling two concentrations of electron-doped Sr1−*<sup>x</sup>*La*x*Ti(<sup>16</sup> O1−*<sup>z</sup>*<sup>18</sup> O*z*)3 single crystals. They precisely controlled the "dome-like" SC characteristic by *n* doping via the La3<sup>+</sup> content, while independently enhancing *T*c by substitution of 18 O2- ions for 16 O2<sup>−</sup>. At an electron concentration of *n* ≈ 5 × 10<sup>19</sup> cm<sup>−</sup>3, they found the apex of the SC dome at *Tc* ≈ 0.44 K, where they subsequently shifted its height to a record-high *Tc* ≈ 0.6 K by adjusting *z*(18O).

Being arbitrarily close to the quantum critical point of non-centrosymmetric SC, experiments have thus come into reach to probe mixed-parity pairing mechanisms with topological aspects to their SC states, such as extremely large and highly anisotropic upper critical fields and topologically protected spin currents. A decisive step toward this aim was done by Schumann et al. [43] using La3<sup>+</sup> or Sm3+ *n*-doped STO films on (001)-strained LSAT substrates. Being in their polar phase, they reveal enhanced superconducting *Tc*, while some of them show signatures of an unusual SC state, where the in-plane critical field is higher than both the paramagnetic and orbital pair breaking limits. Moreover, nonreciprocal transport is observed, which reflects the ratio of odd versus even pairing interactions. A similar highlight was observed in a gate-induced 2D SC of interfacial STO [44]. Due to its Rashba-type spin orbit interaction, it reveals nonreciprocal transport, where the inequivalent rightward and leftward currents reflect simultaneous spatial inversion and time-reversal symmetry breaking—an exciting prospect of forthcoming research on STO.

#### **3. The Magnetoelectric Multiglass (Sr,Mn)TiO3**

The nature of glassy states in disordered materials has long been controversially discussed. In the magnetic community, generic spin glasses have long been accepted to undergo phase transitions at a static glass temperature *Tg*, where they exhibit criticality and originate well-defined order parameters [45]. In addition, disordered polar systems are expected to transit into generic "dipolar" or "orientational glass" states [46], which fulfil similar criteria as spin glasses. Hence, it appears quite natural to introduce the term "multiglass" for a new kind of multiferroic material revealing both polar and spin glass properties, which were discovered by some of us in the ceramic solid solution Sr0.98Mn0.02TiO3 [47]. By various experimental methods [48–50] it has been ascertained that the Mn2<sup>+</sup> ions are randomly substituting Sr2+ ions on *A*-sites in quantum paraelectric STO (Figure 8a), where they become o ff-centered due to their small ionic size and undergo covalent bonding with one of the twelve nearest neighboring O2− ions. These elementary dipoles readily form polar nanoclusters with frustrated dipolar interactions, as illustrated in Figure 8b. It depicts the local cluster formation of Mn2<sup>+</sup> ions with antiparallel electric dipole moments and antiferromagnetically correlated spins.

**Figure 8.** (**a**) *A* site substituted Mn2<sup>+</sup> ion in its cage of 12 nearest neighboring oxygen ions in the ABO3 lattice of STO going o ff-center along <100> [46]. (**b**) Schematic structure of SrTiO3: Mn2<sup>+</sup> highlighting a percolating multiglass path of randomly distributed Mn2<sup>+</sup> ions (red–blue broken line) carrying dipole moments σ*j* (blue lines) and spins *Sj* (red arrows) with electric dipolar and antiferromagnetic correlations, respectively, within polar STO clusters (red "clouds") [51].

The dipolar glass formation can easily be judged from the asymptotic shift of the dynamic dielectric susceptibility peak, *Tm*(*f*), at frequencies within the range 10−<sup>1</sup> ≤ *f* ≤ 10<sup>6</sup> Hz in Figure 9a. It obeys glassy critical behavior according to Equation (2), where *z*ν = 8.5 is the dynamic critical exponent and *Te g* ≈ 38 K the electric glass temperature [51]. On the other hand, frustrated and random Mn2<sup>+</sup>–O2−–Mn2<sup>+</sup> superexchange is at the origin of spin glass formation below the magnetic glass temperature *Tm g* ≈ 34 K. This temperature marks the confluence of three characteristic magnetization curves recorded in μ*0H* = 10 mT after zero-field cooling (ZFC) to *T* = 5 K upon field heating ( *mZFC*), upon subsequent field cooling ( *mFC*), and thereafter the thermoremanence ( *mTRM*) upon zero-field heating (ZFH) as shown in Figure 9c. It should be noticed that both glassy states have unanimously been confirmed by clear-cut individual aging, rejuvenation, and memory e ffects in their respective *dc* susceptibilities [51]. "Holes" burnt into the electric and magnetic susceptibilities by waiting in zero external field for 10.5 h at 32.8 K and for 2.8 h at 33 K, respectively, and subsequent heating with weak electric or magnetic probing fields are shown in Figure 9b,d, respectively. They corroborate the glassy ground states of both polar and magnetic subsystems and their compatibility with spin glass theory [45]. Observation of the biquadratic ME interaction in the free energy [47],

$$F(E,H) = F\_0 - (\delta/2)E\_i E\_j H\_k H\_l(i,j,k,l = 1,2,3),\tag{3}$$

is compatible with the low symmetry of the compound and is thought to crucially reinforce the spin glass ordering, as schematically depicted in Figure 8b [51]. Similarly to the dielectric anomaly [52], the magnetic anomaly has been found to depend not only on the frequency, but crucially also on the Mn content, confirming its intrinsic origin [53]. Furthermore, apart from ceramics, both glassy states have also been detected in equivalent thin films [54].

**Figure 9.** (**a**) Dielectric susceptibility ε'(*T*) of Sr0.98Mn0.02TiO3 ceramics recorded at frequencies 10−<sup>1</sup> ≤ *f* ≤ 10<sup>6</sup> Hz and (**c**) magnetization measured in *B* = 10 mT on field heating after ZFC (mZFC), on FC (m*FC*), and on ZFH after FC (m*TRM*). Holes Δε(*T*) and Δ*m*(*T*) burnt in zero fields at *T*wait = 32.5 K for 10.5 h (**b**) and *T*wait = 33 K for 2.8 h (**d**) confirm memory and rejuvenation of both electric and magnetic glassy subsystems [47].

Starting from a mean-field ansatz within the framework of a transverse Ising model [51], the complete theory of the ME multiglass is still under debate. In particular, the final steps for establishing the spin glass are missing. It is thought to emerge from multipolar interaction of spin clusters (Figure 8b) and probably comes close to the formation of a superspin glass as in systems of magnetic nanoparticles [22]. Since these probably consist of antiferromagnetic MnTiO3 and carry merely surface magnetization [55], special care has to be taken.

In search of other ME multiglasses, we successfully examined also Mn2<sup>+</sup> doped KTaO3, which in the undoped case is a quantum paraelectric like STO, but nevertheless has slightly different properties on doping [56]. Other research groups have made similar experiments, and all of them reported considerable complexity [57–60]. Moreover, various other ME multiglasses have also been observed in disordered solid solutions such as CuFe0.5V0.5O2 [61], La2NiMnO6 [62], Fe2TiO5 [63], and (Ba3NbFe3Si2O14):Sr [64].

## **4. Conclusions**

STO still enjoys vivid interest in research and technological development. Having overcome the low-T bottleneck by advanced nanotechnologies, STO belongs to the most promising nanoelectronic materials. Unusual properties around the quantum critical point such as the co-existence of regular and superconductivity with ferroelectricity are still the focus of attention. On the other hand, the novel disordered phases of a superglass in Sr0.998Ca0.002TiO3 and a multiglass in Sr0.98Mn0.02TiO3 also still require dedicated activity.

**Author Contributions:** Conceptualization, W.K.; writing—original draft preparation, W.K.; writing—review and editing, W.K., J.D., A.T. and P.M.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** We are grateful to K.A. Müller and J.G. Bednorz for providing their outstanding single crystal samples of SCT and cooperating within common publications. In addition, we acknowledge valuable cooperation with A. Albertini, S. Bedanta, U. Bianchi, P. Borisov, A. Hochstrat, S. Miga, F.J. Schäfer, and V.V. Shvartsman.

**Conflicts of Interest:** The authors declare no conflict of interest.
