Abstract
Multiferroic materials, in which ferroelectric and magnetic ordering coexist, are of fundamental interest for the development of multistate memory devices that allow for electrical writing and nondestructive magnetic readout operation. The great challenge is to create multiferroic materials that operate at room temperature and have a large ferroelectric polarization P. Cupric oxide, CuO, is promising because it exhibits a significant polarization, that is, P~0.1 μC cm^{−2}, for a spinspiral multiferroic. Unfortunately, CuO is only ferroelectric in a temperature range of 20 K, from 210 to 230 K. Here, by using a combination of density functional theory and Monte Carlo calculations, we establish that pressuredriven phase competition induces a giant stabilization of the multiferroic phase of CuO, which at 20–40 GPa becomes stable in a domain larger than 300 K, from 0 to T>300 K. Thus, under high pressure, CuO is predicted to be a roomtemperature multiferroic with large polarization.
Introduction
Since the first observation of multiferroicity^{1,2} in cupric oxide (CuO) by Kimura et al.^{3}, it has been established that CuO is a typeII multiferroic, so that ferroelectricity occurs as a result of magnetic ordering^{3,4} and, therefore, the multiferroic ordering temperature equals the magnetic ordering temperature T_{N}=230 K. Moreover, CuO is a quasionedimensional (1D) magnetic system with a large magnetic coupling J_{z}~80 meV (refs 5, 6, 7), which explains the high ordering temperature T_{N}. In addition, upon cooling, a polar incommensurate antiferromagnetic (AF) spinspiral ordering, referred to as AF_{2}, appears below T_{N}=230 K and a nonpolar commensurate AF spin structure, AF_{1}, below the lockin temperature T_{L}=213 K. Finally, the Dzyaloshinskii–Moriya (DM) ‘cycloidal’ interactions have been shown to have a major role in the emergence of the electric polarization in CuO^{8,9,10}. Different aspects of the interplay between the magnetic, orbital and electronic degrees of freedom in CuO have been studied intensely^{8,9,10,11,12,13,14}. Recently, we have shown that by applying a pressure of 8.8 GPa to CuO^{14}, the magnetic exchange interactions can increase by 46%. This holds the promise that under pressure T_{N} will increase, perhaps even to room temperature (RT). Indeed, the monoclinic phase of CuO is known to be stable up to at least 70 GPa^{15}, even if detailed structural refinements are only available at pressures lower than 10 GPa^{16}. Establishing the stability of the multiferroic phase under pressure, this topic not only requires a calculation of the magnetic exchange interactions by density functional theory (DFT) but also a determination of T_{L}, T_{N} and the temperature dependence of the polarization P by complementary methodologies.
Here we establish that pressuredriven phase competition renders CuO multiferroic at RT with a large P. For this, we employ both a semiempirical ansatz as well as unbiased classical Monte Carlo (MC) simulations. More specifically, using a combination of DFT and MC calculations, we demonstrate that upon applying pressure the effective magnetic dimensionality initially decreases, passes through a minimum and subsequently increases, the magnetocrystalline anisotropy (MAE) is reduced and the stability range of the multiferroic state strongly increases by lowering T_{L} and increasing T_{N}.
Results
Pressure effect on the magnetic exchange parameters
CuO consists of corner and edgesharing squareplanar CuO_{4} units, which form (–Cu–O–)_{∞} zigzag chains running along the [10–1] and [101] directions of the unit cell^{17}. The lowT magnetic structure, AF_{1}, consists of Cu moments arranged antiferromagnetically along [10–1] and ferromagnetically along [101], with the [010] direction as the easy axis^{18}. The exchange interactions are captured by the Heisenberg Hamiltonian H_{H}=∑_{ij}J_{ij} S_{i}· S_{j}, which, to properly describe the magnetic properties of CuO, requires at least five magnetic exchange coupling parameters, that is, four superexchange interactions (J_{a}, J_{b}, J_{x} and J_{z}) and one supersuperexchange interaction (J_{2a})^{7,14,19,20} (see Fig. 1a). The pressure dependence of the unit cell volume and of the Jvalues is shown in Fig. 1b, c, respectively.
The predictive power of our DFT geometry optimization is confirmed by its capacity to reproduce the volume decrease with pressure as reported up to 17 GPa for nanocrystalline CuO samples^{21}. The pressure dependence of the Jvalues, determined for the optimized atomic structures, compare very well with previous calculations^{14}, that use the available experimental structures up to 8.8 GPa (ref. 16). Most importantly, J_{z} strongly increases, whereas J_{2a} is nearly constant for pressures up to ~20 GPa, after which they increase in a similar manner as J_{z}. Of the three smaller Jvalues, J_{a} is most affected by pressure and becomes ferromagnetic beyond about 2 GPa. The magnetic frustration, that is, the competition between J_{a} and J_{2a} (ref. 14), is therefore strongly enhanced by pressure. The change in ratio between the two largest Jvalues, that is, J_{z}/J_{2a}, evidences that the effective magnetic dimensionality is also affected by pressure. As shown in Fig. 2a, the quasi1D character of the magnetic structure is enhanced for pressures up to 20 GPa but then is reduced.
Pressure effect on the Neel temperature
Having determined the pressure dependence of the magnetic exchange coupling constants, we calculate the multiferroic ordering temperature T_{N} up to 200 GPa. We first evaluate it using the semiempirical random phase approximation (RPA) expression for the quasi1D AF Heisenberg model on a cubic lattice with intrachain and interchain couplings J and J′, respectively^{22}. The resulting T_{N} is shown in Fig. 2b. Choosing the parameterization such that it reproduces the transition temperature T_{N}=230 K at ambient pressure, we observe a monotonic and substantial increase of T_{N} with pressure. This result coincides with the experimental pressure dependence of T_{N}, as was measured up to 1.8 GPa (ref. 23) and reaches RT at ~20 GPa.
Pressure effect on the magnetic anisotropy
To substantiate this prediction, however, one needs to go beyond the semiempirical approach. For this purpose, we employed a classical MC technique to explore the competition between the different magnetic states as a function of both pressure and temperature, with the Hamiltonian H=H_{H}+H_{UA}+H_{DM}+H_{MA}, where H_{H} is the Heisenberg exchange, H_{UA} is uniaxial anisotropy (UA), H_{DM} the Dzyaloshinskii Moriya (DM) term and H_{MA} is the multiaxial (MA) anisotropy term. All these terms are relevant, but in particular the anisotropy terms are shown by our DFT calculations to be crucial for describing the effects of pressure. The relevant magnetocrystalline anisotropy energies (MAEs) of CuO have been calculated for the groundstate (GS) AF_{1} magnetic structure with MAE=E[uvw]–E[010], where E[uvw] is the energy deduced from spin–orbit (SO) calculations with the magnetization along the [uvw] crystallographic direction. Figure 3a shows the anisotropy energy surface^{24} for CuO in the AF_{1} magnetic order at a pressure of 0 GPa. Two minima are observed along the [010] direction and equivalently [0–10] direction. Thus, the SO DFT calculations properly predict that the b axis is the easy axis of magnetization of CuO for the lowT magnetic phase AF_{1}. A similar result is obtained for the entire pressure domain, that is, from 0 to 200 GPa. However, the MAE values are rapidly decreasing with pressure as evidenced in Fig. 3b, in which the MAE in the (a,c) plane is plotted as a function of the angle ϕ, such that ϕ=0° corresponds to the [101] direction. It also turns out that the hardest axis of magnetization (largest MAE value) is close to the [10–1] direction, that is, the antiferromagnetic direction. The pressure dependence of the MAE approximately follows an exponential decay, as is illustrated in Fig. 3c for the [−101] direction.
Pressure effect on the electronic polarization
Before discussing the MC data, we can estimate the order of magnitude of the ferroelectric polarization, P, by using the empirical formula proposed by Katsura et al.^{25}: P=(V/Δ)^{3}, where V is the Cu–O electronic overlap integral and Δ is the p–d splitting. The superexchange parameter is approximately given by, J=V^{4}/Δ^{3}. The relevant superexchange interactions for the ferroelectric nature of CuO are J_{a} and J_{b}. Therefore, taking J=5 meV and Δ=1.4 eV (refs 7, 26), we find P=0.15 μC cm^{−2}, which is close to the experimental value^{27} of about 0.1 μC cm^{−2}. Alternatively, we can estimate the ferroelectric polarization directly from our DFT calculations using the Berry phase method^{28}. As previously demonstrated theoretically^{8}, the lattice contribution to the polarization (P_{l}) is small compared with the electronic one (P_{e}) in CuO, that is, P_{l}~0.050 μC cm^{−2} and P_{e}~0.200 μC cm^{−2}. Here, taking into account the SO coupling, P_{e} was calculated to be 0.286 μC m^{−2} at 0 GPa. It should be noted that we find an electronic polarization along the b direction (P_{e}~0 along a and c directions), in good agreement with the experiments.
To determine the pressure effect on the value of P_{e}, we have considered two other pressure values, that is, 20 and 40 GPa. Our calculations show that P_{e}=0.286, 0.379 and 0.455 μC cm^{−2} at 0, 20 and 40 GPa, respectively, and at zero temperature. These polarization values are along the b direction, and under pressure the polarization along the a and c directions remains zero. This clearly confirms that applying pressure on CuO leads to an increase of the electric polarization, which is predicted to be along the b direction only.
Discussion
The finite temperature MC simulations use the Jvalues obtained from the DFT calculations and, in particular, incorporate the H_{MA} term that is rapidly decreasing with pressure. Figure 4a shows the resulting spin current as a function of pressure, which is a quantity that is directly proportional to P. We observe that at ambient pressure close to the paramagnetic to AF1 transition, a spontaneous polarization is induced. This polarization is found to be nonzero between T_{N}=200 and T_{L}=150 K, which compares well with the experimentally observed stability domain of the incommensurate AF_{2} magnetic order, between T_{N}=230 and T_{L}=213 K.
The fact that the calculated values are somewhat lower than the experimental ones is due to the model approximations involved and indicates that the MC results are conservative in the sense that they rather tend to underestimate the stability of the multiferroic phase. When the pressure is increased the polarization grows, in agreement with our DFT result for P_{e} dependence with pressure, and extends to a larger temperature range. For instance, at 30 GPa an increase of about 20% is observed with respect to the polarization at 0 GPa, and the temperature range is larger and in between 245 and 115 K. At 200 GPa, the multiferroic phase (AF_{2}) extends down to zero temperature and the ferroelectric polarization is more than doubled. The MC results confirm the increase of T_{N} with pressure, in accordance with the experimental observations for pressures up to 2 GPa and the results from the semiempirical RPA expressions. We find good quantitative agreement for the values of the differential pressure increase of T_{N} from experiment, 2.7 (0.2) K GPa^{−1} (ref. 23), and from the RPA and MC results, 3.5 (0.3) K GPa^{−1} and 3.0 (0.3) K GPa^{−1}, respectively. The calculated temperature–pressure phase diagram of CuO (see Fig. 4b) shows, in addition, a monotonic decrease of T_{L} with pressure. As a consequence, the nonpolar AF_{1} phase disappears from the phase diagram with increasing pressure, at the benefit of the multiferroic AF_{2} phase. The MC simulations indicate that T_{N} reaches RT at ~40 GPa, which is higher than the ~20 GPa, obtained from the semiempirical RPA expressions, underlining that the MC pressure of ~40 GPa is a conservative estimate for the critical pressure value.
Finally, the present temperature–pressure phase diagram of CuO evidences under high pressure a large increase of the stability range of the incommensurate multiferroic AF_{2} phase, which is stable in a domain of only 20 K (from 210 to 230 K) at 0 GPa, and in a domain larger than 300 K (from 0 to T>300 K) at 20–40 GPa. Such a giant stabilization of a multiferroic phase by pressure has never been observed or proposed. Indeed, except for CuO, all the reported pressure–temperature phase diagrams of multiferroic materials (Ni_{3}V_{2}O_{8}, MnWO_{4}, TbMnO_{3} and RMn_{2}O_{5} with R=Tb, Dy, Ho…)^{29,30,31,32} lead to the same conclusion, namely that the stability range of the incommensurate magnetic phase is reduced by pressure. The fact that our theoretical and predictive approach correctly reproduces the experimental lowpressure results gives considerable credit to our predictions. In contrast to the multiferroic compounds mentioned above, the geometrical modifications (bond distances and angles) under pressure in CuO induce an increase of the magnetic frustration, as previously demonstrated up to 8.8 GPa (ref. 14) and confirmed here for a larger pressure domain from 0 to 200 GPa. Further theoretical efforts on the dependence of the magnetic frustration versus pressure (chemical or physical) in CuO and the related compounds is clearly needed and will be of direct interest in the quest of typeII multiferroics with tunable ferroelectric and magnetic properties.
The first RT binary multiferroic material is thus within reach: CuO at pressures of 20–40 GPa. To be practical, for technical applications, the highpressure form of CuO must be made stable at ambient conditions. To achieve this, there are at least two strategies. Very special for CuO is the possibility to stabilize its high pressure form at a nanoscale level by applying highenergy ion irradiation at high pressures^{33}. In such experiments, the quenched highpressure structure remains even after releasing pressure. Another promising strategy can be a coreshell synthesis^{34} according to which CuO nanoparticles are embedded in a shell material that has a negative thermal expansion coefficient, which then acts as an effective pressure medium for the CuO core.
Methods
DFT calculations
The DFT calculations have been carried out by using two different codes: Vienna Ab initio Simulation Package (VASP)^{35} for the geometry optimization at the different pressure values and WIEN2k programme package^{36} for the calculation of the magnetic exchange, J_{ij}, and magnetocrystalline anisotropy energy, MAE, values.
For the geometry optimizations, a 16 formula units cell has been used, that is, 2a × 2b × 2c, with a, b and c being the crystallographic cell parameters. The GS magnetic order (AF_{1}) has been considered for the geometry optimization. The parameters used in the VASP calculations are the following. We have used the GGA+U approach with U_{eff}=6.5 eV for the Cu(3d) states, as in our previous investigation^{8,20}. It allows having a proper description of the structural properties of CuO. The wave functions are expanded in a plane wave basis set with kinetic energy below 500 eV. The VASP package is used with the projector augmented wave method of Blöchl^{37}. The integration in the Brillouin Zone is done by the Methfessel–Paxton method^{38} on a 3 × 3 × 3 set of kpoints determined by the Monkhorst–Pack scheme^{39}. All atoms were then allowed to relax by following a conjugate gradient minimization of the total energy scheme (3 × 10^{−2} eV Å^{−1}).
The magnetic exchange parameters (J_{ij} values) were estimated based on the optimized atomic structures and for each pressure (from 0 to 200 GPa), and using the WIEN2k programme package with the onsite PBE0 (ref. 40) hybrid functional for 8 and 32 f.u. cells. The J_{ij} values have been deduced from a leastsquares fit procedure and the quality of the fits is shown in Fig. 5a,b. It should be noticed that the choice of the PBE0 onsite hybrid functional in WIEN2k was motivated by its ability to properly reproduce the magnetic exchange coupling in a series of copper oxide compounds and its dependence with the Cu–O–Cu bond angle^{14}.
The MAE value has been estimated for the AF_{1} GS magnetic structure, using the code WIEN2k with the PBE0 hybrid functional and including the SO coupling. MAE corresponds to an energy difference between two directions of the magnetization density. Here we use the [010] direction, that is, the easy axis, as the reference:
E[uvw] is the energy deduced from SO calculations with magnetization along the [uvw] crystallographic direction. It should be noticed that MAE is very sensitive to the kmesh. The quality of the kmesh has been carefully chosen, leading to the use of a 5 × 12 × 6 set of kpoints for the 8 f.u. cell.
The electronic contribution (P_{e}) to the polarization P was evaluated using the Berry phase approach^{28}. As previously shown^{8}, the lattice contribution (P_{l}) is small compared with the electronic one (P_{e}) in CuO: P_{e}~0.200 μC cm^{−2} and P_{l}~0.050 μC cm^{−2}. Here we have redone such calculations using the VASP code with the GGA+U formalism and a U_{eff} value of 7.5 eV, to match the J_{ij} values obtained using the PBE0 hybrid functional in our WIEN2k calculations. The noncollinear magnetic structure, AF_{2}, previously discussed in ref. 8 has been used. To have adequate electric polarization values, it was crucial to turn on the SO coupling during the structural relaxation. Indeed, at 0 GPa, P_{e}=0.053 and 0.286 μC cm^{−2}, respectively, for the atomic structure relaxed without and with SO.
Estimation of T_{N} based on the RPA formula
The above equation has been developed for the estimation of T_{N} of a quasi1D AF Heisenberg model on a cubic lattice with J and J′, the intrachain and interchain couplings, respectively^{22}. The related GS magnetic order leads to the following energy expression:
Although CuO is a quasi1D magnetic system, it exhibits a more complex magnetic order due to the low symmetry of its atomic structure (monoclinic space group: C2/c). As a consequence, its GS magnetic order (AF_{1}) leads to the following energy expression:
with J_{2a} being the predominant supersuperexchange interaction. For more details, see (refs 7 and 20). Considering E(GS)=E(AF_{1}) and J_{z} as the intrachain coupling, that is, J_{z}=J, we can define J′ as:
Our detailed results are illustrated in Fig. 6.
Classical MC simulations
The model Hamiltonian used in MC simulations is given by,
where, H_{H} is the Heisenberg part, H_{UA} is UA term, H_{DM} is DM term and H_{MA} is the MA anisotropy. The Heisenberg term and the UA term can be written together as,
where refer to exchange parameters at pressure P. The various exchange parameters are J_{z}, J_{x}, J_{2a}, J_{a} and J_{b} (see Fig. 1a). The UA term has been included in the Heisenberg part by making the following replacement:
This allows a lower energy for a collinear AF_{1} state in which all the spins are aligned along the y axis. The DM term is given by
As we are mainly interested in the competition between two magnetic states, the collinear AF_{1} and the noncollinear AF_{2}, we restrict the spins to reside in y–z plane only. Therefore, we only introduce a DM vector pointing along x direction. The experimental observation that the AF_{2} state is stable in a narrow temperature window between 230 and 213 K is reproduced in an effective manner by introducing a MA anisotropy term, in the Hamiltonian of the form^{24},
The Heisenberg term, H_{H}, alone leads to a degeneracy of GSs. The GS manifold consists of perfectly ordered, interpenetrating sublattices with vector order parameters whose relative orientation is left completely undetermined by the Heisenberg term alone. The UA term prefers a collinear magnetic state, where the two sublattice order parameters align along the y axis. This is precisely the experimentally observed AF_{1} state of CuO. The DM coupling favours a noncollinear (but coplanar) state and, therefore, it competes with the H_{UA}. There is a critical strength of the DM coupling, D_{c}, above which the AF_{1} is not the GS. Given that the AF_{1} is the GS at ambient pressure, we conclude that D<D_{c}.
We use the Metropolis algorithm to perform Markov Chain MC simulations on classical spins. The simulations are started with a completely random spin configuration at high temperature. Because of the presence of many competing interactions and nearly degenerate GSs, the simulations require a large number of equilibration and averaging steps. We use ~10^{6} MC steps for equilibration and a similar number of steps for averaging at each temperature. The temperature is then reduced in small steps (~5 K) and the system is allowed to anneal towards the GS spin configuration. The main quantity of interest is the spin current, which is defined as ‹e_{ij} × (S_{i} × S_{j})›,where e_{ij} is a vector connecting spins S_{i} and S_{j}, and the angular brackets denote thermal as well as spatial average. The ferroelectric polarization is proportional to the spin current with a prefactor estimated to be 0.150 μC cm^{−2} following Katsura et al.^{25}
The simulations are carried out on lattices with N=12^{3} sites. We have checked the stability of our results for larger sizes (up to N=32^{3}) for selected values of pressure (Supplementary Fig. S1). The procedure used to estimate T_{L} and T_{N} for 2 pressures (0 and 30 GPa) is shown in Fig. 7.
The UA parameter λ=0.02 is kept constant. The MA anisotropy B decreases exponentially with increasing pressure; we use B=500 e^{−P/10} (B=500 for P=0 and B=24.9 for P=30). The exponential decrease is motivated by the DFT results presented in Fig. 3. The large value of B at P=0 is required to obtain the narrow range of stability of AF_{2} state at high temperatures. Although UA term is also decreasing with pressure, this does not lead to any crucial changes in the phase diagram shown in Fig. 4. We also keep the DM coupling fixed to D=0.8D_{c}. The choice of the parameter D is not very crucial, as long as D is smaller than D_{c}. To illustrate this point, we show the results for spin current for various values of D at P=0 GPa and P=30 GPa in Supplementary Fig. S2.
Additional information
How to cite this article: Rocquefelte, X. et al. Roomtemperature spinspiral multiferroicity in highpressure cupric oxide. Nat. Commun. 4:2511 doi: 10.1038/ncomms3511 (2013).
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Acknowledgements
X.R. thanks the IMN (Nantes) and the CCIPL (Centre de Calcul Intensif des Pays de la Loire) for computing facilities. S.K. acknowledges the use of High Performance Computing Facility at IISER Mohali, and support from DST, India. P.B. was supported by the Austrian Science Foundation FWF (SFB F41 ‘Vicom’).
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X.R. designed and performed the DFT calculations. X.R. wrote the paper with contributions from all coauthors. S.K. performed the MC simulations. All coauthors contributed to analysing and discussing the results.
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Rocquefelte, X., Schwarz, K., Blaha, P. et al. Roomtemperature spinspiral multiferroicity in highpressure cupric oxide. Nat Commun 4, 2511 (2013). https://doi.org/10.1038/ncomms3511
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DOI: https://doi.org/10.1038/ncomms3511
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