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Ferroelectricity and ferromagnetism in EuTiO3 nanowires

We predicted the ferroelectric-ferromagnetic multiferroic properties of EuTiO3 nanowires and generated the phase diagrams in coordinates of temperature and wire radii. The calculations were performed within the Landau-Ginzburg-Devonshire theory with phenomenological parameters extracted from tabulated experimental data and first principles calculations. Since bulk EuTiO3 is antiferromagnetic at temperatures lower than 5.5 K and paraelectric at all temperatures, our goal was to investigate the possibility of inducing the ferroelectric and ferromagnetic properties of EuTiO3 by reducing the bulk to nanosystems. Our results indicate that ferroelectric spontaneous polarization of ~0.1-0.5C/m2 is induced in EuTiO3 nanowires due to the intrinsic surface stress, which is inversely proportional to the nanowire radius. The spontaneous polarization exists at temperatures lower than 300 K, for the wire radius less than 1 nm and typical surface stress coefficients ~ 15 N/m. Due to the strong biquadratic magnetoelectric coupling, the spontaneous polarization in turn induces the ferromagnetic phase at temperatures lower than 30 K for 2 nm nanowire, and at temperatures lower than 10 K for 4 nm nanowire in EuTiO3. Thus we predicted that the EuTiO3 nanowires can be the new ferroelectric-ferromagnetic multiferroic.


Brief history of the question. Bulk quantum paraelectric EuTiO 3 is antiferromagnetic at
temperatures lower than 5.5 K and paraelectric at all other temperatures [1,2]. Oxide materials with large magnetoelectric (ME) coupling are very important for magnetoelectric applications based on the magnetic field control of the material dielectric permittivity [1,2]. Although multiferroicity is a rare event, the search for new multiferroic is very important because of various applications.
Using ab initio calculations, Fennie and Rabe [3] predicted theoretically that (001) EuTiO 3 thin films subjected to the compressive epitaxial strain become ferromagnetic and ferroelectric simultaneously for strains exceeding 1.2%. Recently Lee et al. [4]] demonstrated experimentally that an epitaxial strain indeed turns EuTiO 3 into multiferroic. In particular, they demonstrated that EuTiO 3 thin film epitaxially grown on DyScO 3 substrate (corresponding to a tensile misfit strain of more that 1%) becomes ferromagnetic at temperatures lower than 4.24 K and ferroelectric at temperatures lower than 250 K. Lee et al. explained the appearance of the ferromagnetism in EuTiO 3 thin film by the strong spin-lattice biquadratic ME coupling.

Motivation of our study.
While the idea to induce the new multiferroic properties by elastic strain seems very attractive, its realization could meet some difficulties in epitaxial films, since relatively high misfit strains (~1-3 %) between the film and substrate relaxes because of e.g. the appearance of misfit dislocations [5,6]. It is extremely difficult to synthesize a strongly strained epitaxial film without rather special, complex and thus high-cost deposition processes. It has been demonstrated that strain relaxation to the values lower than 1% eliminates ferroelectric-ferromagnetic phase appearance in EuTiO 3 thin film [3].
However, in nanowires and nanorods (Fig.1a), the intrinsic surface stress exists spontaneously due to the surface curvature and typically does not relax in nanowires and nanorods. Surface stress is inversely proportional to the wire radius and directly proportional to the surface stress tensor (similar to Laplace surface tension). The intrinsic surface stress should depend both on the growth conditions and the surface termination morphology [7,8]. Although surface tension appears even for the case of non-reconstructed geometrical surfaces due to the surface curvature [8], surface reconstruction should affect the surface tension value or even be responsible for the appearance of surface stresses [9,10].
It is shown that the intrinsic surface stress can induce ferroelectricity, ferromagnetism and increase corresponding phase transition temperatures in conventional and quantum paraelectric nanorods and nanowires [11,12,13,14,15]. These facts motivated us to explore the possibility of inducing simultaneous ferroelectricity and ferromagnetism in EuTiO 3

Background of the used method.
The application of the continuum media LGD theory to the description of nanowire polar and magnetic properties requires justification due to the small size of the object of study. The continuum media theory was successfully used for the analysis of elastic properties of metallic, semiconductor, dielectric or polymeric nanowires and nanotubes [16,17,18,19] and the piezoelectric response [20]. For nanosized ferroics, the applicability of the continuous media phenomenological theory is corroborated by the fact that the critical sizes (~2-10 lattice constants) of the appearance of long-range order calculated from atomistic [21] and phenomenological theories [13,14,15] are in good agreement with each other [22,23] as well as with experimental results [24]. The long-range order appears for sizes larger than critical ones. Once the long-range order is established, it is possible to apply the mean field LGD theory [25]. Thus the agreement between the magnitudes of the critical sizes calculated from LGD and atomistic theories are extremely important

II. Basic Formalism
LGD free energy F depends on the polarization vector P, magnetic sublattices (a,b) magnetization vector and antimagnetization vector where is polarization-dependent, is magnetization-dependent and is the biquadratic magnetoelectric (ME) coupling energy. The ME coupling coefficients 0 < η FM for ferromagnetics and η for antiferromagnetics [4]. Following Lee et al. [4] we can regard that for numerical calculations, as anticipated for equivalent magnetic Eu ions with antiparallel spin ordering in a bulk EuTiO 3 (see Fig.1b).
Considering very long cylindrical EuTiO 3 nanowires with polarization and external electric field (if any) directed along the cylinder axes z one could neglect the depolarization field if the wire length h is much higher than its radius R, namely 3 10 >> R h [11]. Thus we considered very high aspect ratio wires, for which the polarization vector ( ) ( ) ρ = 3 , 0 , 0 P P appears in the z-direction along the wire axis, and depends only on the polar coordinate due to the radial symmetry of the problem (as shown in Fig.1a). The single-component polarization P 3 and elastic strains u ij dependent part of the free energy is is the Curie temperature (negative for EuTiO 3 [1]) as it has to be for quantum paraelectric, T q is the quantum vibration temperature, which is positive. c ijkl is the stiffness tensor and is the electrostriction tensor component.
For correct phenomenological description of the spatially confined system the surface energy should be considered, the contribution of which increases with the decrease in the wire radii. The surface energy coefficient α is isotropic and weakly temperature dependent. Thus, higher terms can be neglected in the surface energy expansion. µ is the intrinsic surface stress tensor that exists under the curved surface of solid body and determines the excess pressure on the surface [8]. We will use the isotropic approximation , where the scalar coefficient µ varies in the typical range of 3−30 N/m [26,27,28,29]. This surface stress coefficient can be manipulated in some range by the choice of the ambient of the nanowire (template material, composite glassy matrix, glue, gel or gas), and depend on surface reconstruction. is the tensor of the surface piezoelectric effect originated from the disappearance of inversion symmetry in the vicinity of the surface [30,31].
The magnetization-dependent part of the free energy is  LGD-gradient coefficient g fitting results of [3,4] DFT LGD-coefficients γ P m 9 /(C 4 F) <<10 11 Not enough data to determine exactly Electrostriction coefficients Q ijkl (Voigt notation)

DFT [this paper]
LGD-coefficient   [37] DFT data. # Elastic compliance matrix is calculated from the stiffness matrix as inverse matrix Note, that the surface piezoeffects coupled with the intrinsic surface stress can favor the longrange order appearance in the nanosized system [32]. Since no reliable data is known about the surface piezoelectric and piezomagnetic coefficients and surface energy coefficients α and , we did not include these into the numerical calculations. It was shown [11][12][13]38], that for the case , corresponding polarization, magnetization and strain spatial distributions become quasihomogeneous inside the wire cross-section, i.e. almost independent of the polar coordinate ρ, but their average values are dependent on the wire radius R. Thus we have considered the case α , which allowed us to list and analyze only the average values of the properties and do not concentrate on their spatial distributions.
The nonzero components of the strain tensor inside a cylindrical wire of radius R subjected to the intrinsic surface stress, electrostriction and magnetostriction effects have the following form: Here s ij 's are the elastic compliances inherent to the bulk material. It is important to mention that the elastic compliances for nanowire and thin films may be different from those of the bulk material [39,40,41], but LGD deals with bulk coefficients and obtain their renormalization self-consistently.
Elastic strains calculated from Eqs.(2) are shown in Fig.2. For wire radii R >2.5 nm the absolute value of compressive strains u becomes less than 1.2% and should lead to the disappearance of ferroelectricity, which is in agreement with Ref. [3]. 22 , 11 The difference between the axial tensile u and compressive u strains radial dependencies are rather small, but visible (compare top and bottom curves in Fig.2). Namely, the small impact of the electrostriction coupling is seen for the tensile strain only (breaks on blue and red curves appeared when the spontaneous polarization appears at the critical radius), while the peculiarities are almost invisible for u due to the fact that the electrostriction coefficient 33 may be used for estimations (see also Refs. [11,42]). The effect u 33 u ≠ ρρ , represented by Eqs. (2), leads to the unit cell tetragonality, namely to its compression in the radial direction (since u ) and elongation along the wire axes (since ) [42]. The strain radius dependence is well fitted by the law proving that the strains are primarily originated from the intrinsic surface stress. Wire radius R (nm) as directed inside the wire cross-section (see Fig.1a and Refs. [11][12][13]25]).
Using the expressions for strains and direct variational method [11][12][13], the spontaneous polarization , magnetization and antiferromagnetic order parameter averaged over the wire radius R was derived as: and α are given by It is seen from the Using the condition was derived as: The form of Eq.(5) follows from the Barrett law; the positive term is the contribution of intrinsic surface stress R µ − = σ ρρ . The second term in Eq.(5) increases T , since (see Table 1). and antiferromagnetic (AFM) phases were derived as:
Following Lee et al. [4], we assigned , so that the expression for the difference in transition temperatures is given as: The last term in Eq. (7) becomes positive because 0 > η AFM [4] and N α is always positive. So the condition ( ) ( ) can be reached for EuTiO 3 nanowires, as observed experimentally for EuTiO 3 thin strained films [4].
Similarly, the boundary between the FM+FE phase and FM+PE phase is determined from the . The boundary between the AFM+FE phase and AFM+PE phase is determined from the condition 0 ) , ( Following the expressions in Eqs. (3)(4)(5)(6) it is obvious that by changing the wire radius one can predicted to be ferroelectric at temperatures lower than 300 K. The wire of radius ~4 lattice constants is predicted to be ferroelectric at temperatures lower than 100 K (Fig. 3(b)). For µ=10 N/m and wire radius of 2-4 lattice constants (~0.8 -1.6 nm), the spontaneous polarization of the nanowire reaches the values of ~0.5-0.1 C/m 2 at temperatures lower than 100 K. Note, that LGD phenomenology predicts higher enhancement of ( R T P , 3 ) values for nanowires with radius R = 1 lattice constant, but LGD continuum approach is not quantitatively correct for such small size.
The cusp or peculiarity on polarization temperature dependence (marked by filled circles in  nanowires of radius less than 3 lattice constants at temperatures lower than 10 K. For µ=30 N/m the surface stress and ME coupling induce the radius-dependent FE+FM phase in EuTiO 3 nanowires of radius less than 3 lattice constants at temperatures lower than 30 K. For µ=30 N/m and radius less than 10 lattice constants (~4 nm), FE+FM phase appears at temperatures lower than 10 K. Thus, the higher is the coefficient µ, the wider is the region of the multiferroic FE+FM phase (compare plots c, d). The region of the multiferroic phase increases with the decrease in wire radius.
In Figs. 4b,c the boundary between PM+PE and PM+FE phase appeared to be vertical at temperatures lower than 30 K. This indicates that the boundary becomes virtually independent on temperature in this condition. This is because at temperatures essentially lower than T (that is about 12 becomes independent of temperature. The temperature independence of ( ) at T in turn originated from the temperature "plateau" at T q T << for the coefficient which is in accordance with the Barrett law. It is evident from our theoretical analysis that changing EuTiO 3 from bulk to nanowire it is possible to induce multiferroic properties. Manipulating the wire radii and surface stresses, the multiferroic nature of these nanowires can be tuned. Quantitatively, we demonstrated how the spontaneous polarization (induced by the intrinsic surface stress) in turn induces the ferromagnetic phase in EuTiO 3 nanowires with radii less than 1-10 lattice constants (~0.4-4 nm) due to the biquadratic ME coupling.

Summary
Using Landau-Ginzburg-Devonshire free energy with material parameters extracted from the first principle calculations and experimental data, we calculated the radii and temperature dependence of the paraelectric, ferroelectric, paramagnetic, ferromagnetic, antiferromagnetic and multiferroic phases in EuTiO 3 nanowires. Due to the strong biquadratic magnetoelectric coupling, the spontaneous polarization in turn induces the ferromagnetic phase in EuTiO 3 nanowires at temperatures lower than 30 K. Thus we predicted that the EuTiO 3 nanowires becomes ferroelectric-ferromagnetic, i.e. multiferroic, which can be important for their potential applications.
Regarding the applications, multiferroic ferromagnetic-ferroelectric EuTiO 3 nanowires potentially can win the competition with strained EuTiO 3 thin films, which becomes ferromagnetic at temperatures lower than 4.24 K, since we predicted ferromagnetism at temperatures lower than 30 K in the nanowires. In general, nanosized EuTiO 3 wires as new multiferroic is favourable in comparison with a bulk material, since bulk EuTiO 3 is antiferromagnetic at temperatures less than 5.5 K and paraelectric at all other temperatures, i.e. it has neither polarization nor magnetization.
We hope that our prediction can stimulate both synthesis of the EuTiO 3 nanowires (e.g. in the form of nanopowders or arrays embedded into the different matrices) and experimental studies of their ferroic properties as well as the first-principles calculations of the spontaneous dipole moment and magnetization induced by the intrinsic stress under the curved surface.

Supplementary materials to "Ferroelectricity and ferromagnetism in EuTiO 3 nanowires"
In the main article, we have reported the multiferroic nature of EuTiO 3 nanowires based on wire radii, temperature, and surface characteristics (surface stress and surface stress coefficient) using Landau-Ginzburg-Devonshire (LGD) theory. All the values estimated from the LGD continuum approach are based on the bulk EuTiO 3 values mentioned in Table 1. In this supplement we have extended the information in the text and addressed some of the key assumptions.  Figure S1(a) illustrates the bulk EuTiO 3 . The dotted circle is an example of a nanowire that can be defined from the bulk systems.

Appendix A. Effect of termination, rumpling and stoichiometry
It depends on polarization P, magnetization M and "antimagentization" L (order parameter of antiferroelectric phase). This free energy is stable at high values of order parameters only under the conditions: Possible phases are listed below

Para phase
with and free energy 0 . This phase is stable at , and

FE-phase ("P") with
This phase exists and is stable under the This phase exists and is stable under the This phase exists and is stable under the This phase exists and is stable under the conditions Katsufuji and Takagi [1] reported about temperature and magnetic field dependence of dielectric permittivity ε and magnetic susceptibility of EuTiO 3 (ETO) in the range T = 2-100 K and magnetic fields from 0 to 5 Tesla. From these data they deduced temperature dependence of P α (Barret equation) and α corresponding to the bulk para-phase.   We extracted the Bulk modulus B from Ranjan et al [37] as (172.6±10) GPa.
For the sake of simplicity we suppose that we could approximate a solid as elastically isotropic, i.e.