Mott Insulator Ca2RuO4 under External Electric Field

We have investigated the structural, electronic and magnetic properties of the Mott insulator Ca 2 RuO 4 under the application of a static external electric field in two regimes: bulk systems at small fields and thin films at large electric fields. Ca 2 RuO 4 presents S- and L-Pbca phases with short and long c lattice constants and with large and small band gaps, respectively. Using density functional perturbation theory, we have calculated the Born effective charges as response functions. Once we break the inversion symmetry by off-centering the Ru atoms, we calculate the piezoelectric properties of the system that suggest an elongation of the system under an electric field. Finally, we investigated a four-unit cell slab in larger electric fields, and we found insulator–metal transitions induced by the electric field. By looking at the local density of states, we have found that the gap gets closed on surface layers while the rest of the sample is insulating. Correlated to the electric-field-driven gap closure, there is an increase in the lattice constant c. Regarding the magnetic properties, we have identified two phase transitions in the magnetic moments with one surface that gets completely demagnetized at the largest field investigated. In all cases, the static electric field increases the lattice constant c and reduces the band gap of Ca 2 RuO 4 , playing a role in the competition between the L-phase and the S-phase.

In this paper, we study the structural, electronic and magnetic properties of Ca 2 RuO 4 under the application of an external electric field with a theoretical, computational analysis by using the ab initio density functional theory (DFT) method. We analyze both the regions of small and large applied electric fields. In the first region, we focus on the bulk, and we calculate the response functions, while in the second one, we build a four-unit cell slab, and we investigate the local density of the states, equilibrium c-axis, band gap and magnetic moments of the Ru atoms. The paper is organized as follows: in the next section, we report the computational details, while in the third section, we present our results. Finally, in the last section, we draw our conclusions.

The Mott transition happens in the paramagnetic phase of the material at T M I . However, it was found that insulator–metal transitions in CRO can be obtained under electric fields [ 26 ] or currents [ 27 , 28 , 29 ] at lower temperatures. Recently, a way to induce a pattern formation by means of an applied electric field in CRO has also been shown [ 30 ]. The electric field in this compound can modify the lattice constant c, producing a competition between structural phases with different c lattice constant. In this paper, we want to focus on the effect of the electric field on the experimental insulating S-Pbca phase and on the hypothetical insulating L-Pbca phase as benchmarks. The latter cannot be observed experimentally without an electric field, but the insulating L-Pbca could be stabilized by the electric field in future experiments.

Ca 2 RuO 4 (CRO) is a system that lends itself to analysis in the electrical field for its many phases and states [ 15 , 16 , 17 , 18 ] and where the scale associated with the Mott transition at T M I = 357 K is much greater than that associated with antiferromagnetism at T N = 110 K. The magnetic and electronic properties of CRO are sensitive to the coupling of spin, charge and the orbital degrees of freedom [ 19 , 20 ]. It presents a Mott metal-insulator phase transition at T M I = 357 K from a low-temperature Mott insulating phase to a high-temperature metallic phase [ 21 ]. This transition is accompanied by a structural transition, namely, the compound has a small c-axis in the low-temperature phase, called S-Pbca, and a longer c lattice constant in the high-temperature phase, named L-Pbca configuration; both S- and L-Pbca configurations are orthorombic. We know from the literature that the L-Pbca phase is metallic and the S-Pbca phase is insulating, and the lower energy orbital xy is full [ 22 , 23 ]. The unit cell contains four formula units with the RuO 6 octahedra settled in corner-shared planes alternated by CaO layers, as shown in . It was shown that the Mott transition occurs because of the structural transition [ 22 ], and the occupation of the orbitals changes from a configuration with xz and yz occupied to one with xy occupied, bringing the system to an orbitally ordered state at low temperatures [ 24 , 25 ].

Furthermore, an electric field can also break the inversion symmetry. As a consequence of the breaking of the inversion symmetry, in theoretical models, the hamiltonian is more anisotropic [ 6 ] and other terms in the hamiltonian are allowed as the spin-orbit Rashba [ 7 , 8 ] and the orbital Rahsba [ 9 ]. We have to mention that the breaking of the inversion symmetry can be introduced in many different ways, not only with an external electric field but also in the presence of surface, interfaces or inclusions [ 10 , 11 ]. In particular, the interface between ferroelectric and magnetic materials has been widely investigated in the last decade [ 12 , 13 , 14 ].

The study of compounds under the application of an external electric field has recently aroused great interest. One of the most important phenomena induced by the electric field is the control of the electronic properties of the systems. Among the several cases, great attention has recently been devoted to the manipulation of the insulator–metal Mott transition [ 1 ] via an external electric field. The control of the Mott transition can be useful for electronic devices, for example, as resistance RAM [ 2 , 3 ]. The application of the electric field is complementary to the application of pressure, with the important difference that pressure influences the electronic states by modifying the structural parameters, while the electric field directly controls the electronic states, with many technological advantages. A large electric field can control the carrier density in a region of an insulator; this is called electrostatic carrier doping [ 4 , 5 ].

Our DFT simulations have been performed by employing the Vienna ab initio simulation package (VASP) [ 31 , 32 , 33 ]. The projector augmented wave (PAW) [ 34 ] technique has been used for the core and the valence electrons, with a cutoff of 480 eV for the plane-wave basis. The calculations have been performed with an 11 × 11 × 4 k-points grid for the bulk and a 14 × 14 × 1 k-points grid for the slab, all centered in Γ . The Local Density Approximation is enough to describe the metallic phases of ruthenates. For the treatment of exchange correlation, the Perdew–Burke–Ernzerhof (PBE) [ 35 ] generalized gradient approximation (GGA) has been used, and we have also considered the correlations for the Ru-4d states by using a Coulomb repulsion U = 3 eV on the Ru atoms in the antiferromagnetic insulating state [ 36 ]. For the Hund coupling, we have used a value in agreement with the literature for the 4 d / 5 d electrons [ 37 ], namely, J H = 0.15 U has been employed. The experimental lattice constants are a s h o r t = 5.3945 Å, b s h o r t = 5.5999 Å, c s h o r t = 11.7653 Å in the S-Pbca phase and a l o n g = 5.3606 Å, b l o n g = 5.3507 Å, c l o n g = 12.2637 Å in the L-Pbca phase [ 38 ]. For the calculation of the Born effective charges Z * and the piezoelectric tensor in the bulk, we have used the modern theory of polarization [ 39 , 40 ] and the self-consistent response to finite electric fields [ 41 , 42 ] for bulk systems as implemented in VASP. For the thin film, we have constructed four-unit cells along the (001) direction, and we have added the electric field in the direction (001) perpendicular to thin film surfaces performing the same strategy used for several 2D materials in the electric field [ 43 , 44 ]. In the case of the slab, we have considered the dipole corrections to the potential as implemented in VASP in order to avoid interactions between the periodically repeated images [ 45 ]. Density functional theory does not fully reproduce the properties of the Mott insulator at high temperatures. Therefore, deviations from the DFT results are expected once we would include many-body and dynamical effects in the self-energy, which is a relevant property of many-body systems, especially in the non-magnetic phase and close to the Mott transition at T M I .

3. Results

We divide our results into three subsections. In the first subsection, we report the investigation of the bulk with inversion symmetry. In the second one, we calculate the piezoelectric tensor after breaking the inversion symmetry by shifting the positions of the Ru atoms along the z-axis. The direction of the displacements of the Ru is shown in a. In the third subsection, we analyze the properties of a four-unit cell slab, shown in b, without and with the application of an external electric field.

3.1. Properties of the Bulk with Inversion Symmetry

In this subsection, we analyze the Ca2RuO4 bulk without and with the application of an external electric field. First, we have compressed and elongated the system along the c-axis both in the S- and L-Pbca phases and we have investigated how the energy of the compound varies as a function of the lattice constant c. We have taken into consideration both the non-magnetic and the magnetic cases. In , we report the energy of the S- and L-Pbca phases as a function of the c-axis in panel (a) in the non-magnetic phase while in panel (b) in the magnetic phase. In the latter case, the Ru atoms of the primitive cell are in the checkerboard antiferromagnetic configuration. In both cases, the L-Pbca phase has a theoretical value of the c lattice constant larger than the S-phase. The difference between the theoretical and experimental c lattice constants is of the order of 1% for the non-magnetic phases and Of the order of 2% for the magnetic phases. The non-magnetic phases are metallic, while the antiferromagnetic phases are insulating. Even without considering THE dynamical effects, we obtained that in the metallic phase, the L-Pbca phase is the ground state, while in the insulating phase, the S-phase is the ground state in agreement with the experimental results. In the bulk, the band gaps of the antiferromagnetic phases are 0.88 and 0.72 eV for the S- and L-Pbca phases, respectively.

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Then, we applied an electric field along the z-direction for the S-Pbca phase, with qE = 5 × 10−4 eV/Å = 5 × 104 eV/cm, where E is the electric field, and q is the elementary charge, and we calculated the response functions of the system, such as the Born effective charges Z*. For larger electric fields, the numerical simulations do not converge because we are close to the onset of Zener tunneling [42]. Under electric fields of this order of magnitude, the system is an insulator, and the gap is 0.60 eV for the short crystal structure. Therefore, even a relatively small electric field tends to close the band gap and could favor the formation of the L-phase that has a smaller gap. Within the density functional perturbation theory, Z* are defined as [46]:

Zij*A=1q∂FiA∂Ej,

(1)

with i,j=x,y,z, and where Ej is the electric field applied along the j-direction while FiA is the force that acts on the ion in the i-direction on atom A.

The Z* are reported in . As expected, diagonal Z* are positive for the cation Ca and Ru, while they are negative for the oxygen anions. We have obtained large Z*, especially for the cation Ca. Similarly, large Z* were also observed in other transition-metal oxides [47]. The largest Z* diagonal element for the Ru atoms is given by Zzz* = 1.193 |e|, which indicates that under an electric field along the z-direction, a component of the force acting on the Ru atoms is along the z-axis. The diagonal terms are the largest for both Ca and O atoms but not for the Ru. Indeed, we have found large off-diagonal terms for the Ru atoms up to Zyz* = 3.121 |e|, which is much larger than Zzz*. Therefore, under an electric field in the z-direction, there is also a component of the force on the Ru atoms acting along the y-axis. Large Z* off-diagonal terms were already observed for highly distorted transition metals perovskites [48], even if we did not find cases where off-diagonal Z* terms are larger than diagonal Z* terms as they appear for the Ru in the literature. Due to the symmetry of the system, the Zij* tensor is not diagonalizable, as we can see for the Ru case with spin up: Zij*Ru↑=0.045−0.464−0.1190.4120.0373.1210.4090.3771.193. A diagonal Zij* tensor in perovskites can be obtained in the absence of octahedral tilts and distortions, however, the undistorted Ca2RuO4 would be metallic as for the Sr2RuO4.

Table 1

Ion
Zxx*

Zxy*

Zxz*

Zyx*

Zyy*

Zyz*

Zzx*

Zzy*

Zzz*
Ca9.331−0.0890.087−0.1389.288−2.5940.1720.0888.875O1−4.794−1.1010.128−1.003−4.656−1.4710.1290.054−3.558O2−4.558−0.0170.383−0.299−4.651−1.3550.2270.126−5.915Ru↑0.045−0.464−0.1190.4120.0373.121−0.4090.3771.193Ru↓0.0450.4640.119−0.4120.0373.1210.4090.3771.193Open in a separate window

3.2. Properties of the Bulk without Inversion Symmetry

We also calculate the piezoelectric tensor, which is zero for centrosymmetric crystal structures [49]. Therefore, in order to obtain a piezoelectric tensor different from zero in systems such as Ca2RuO4 we have to break the inversion symmetry [50,51,52]. We have shifted the Ru atoms along the z-axis to break the inversion symmetry, as shown in a, and we have calculated the components of the tensor. The Ru atoms have been moved by 0.023, 0.046 and 0.069 Å along the positive direction of the z-axis, and we studied how the piezoelectric components vary at different values of the positions of the Ru atoms for both clamped and relaxed contributions. The piezoelectric tensor is defined as [53]:

ϵij0=−∂σi∂Ej,

(2)

where i=xx,yy,zz,xy,yz,zx and j=x,y,z. We can also use the notation i = 1, 2, 3, 4, 5, 6 and j = 1, 2, 3.

The results are reported in . We report both the ion-clamped (panels a and b) and the relaxed contributions (panels c and d), where the latter ones are the contributions that include ionic relaxation. With diagonal and off-diagonal we refer to the components of the stress tensor, namely, the diagonal components are i=xx,yy,zz (i = 1, 2, 3), while off-diagonal are those with i=xy,yz,zx (i = 4, 5, 6). We can see from the figure that, both in the cases of ion-clamped and relaxed contributions, the diagonal terms are one order of magnitude greater than the off-diagonal elements. The diagonal terms are zero in the case of centrosymmetric crystal structure, while they become different from zero when we break the inversion symmetry, and they are almost constant at different values of the displacement of the Ru atoms, while the off-diagonal terms show a linear behavior in the case of ion-clamped contributions while a more complex trend in the case of the relaxed contributions. From these results, we can state that in the case of breaking of the inversion symmetry due, for example, to the presence of interfaces or electric fields, the compound can show piezoelectric features. The positive value of the diagonal components means that the system would increase its volume under an external static electric field.

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