Hubbard U library and high throughput exploration of spin Hamiltonian parameters for the rational design of metal trihalides MX$_3$ (M={Ti,V,Cr,Fe}, X={Cl,Br,I}) with high Curie temperature

Metal trihalides (MX3) are the most important family of 2D magnetic materials, being the chromium trihalides the most studied 2D magnets. The discovery of CrI3, the recent obtention of CrCl3 in-plane magnetic monolayer and the role of CrBr3 inducing topological superconductivity in NbSe2, may serve to showcase the active research being done in this family nowadays. Despite the high impact of these materials, most of the members in the MX3 family are still unexplored and constitute an untapped source of interesting physical properties. Stimulated by the most recent advances in straintronics and aware of the crucial role of the dielectric screening, we present here a high throughput methodology to automatize the exploration of 2D materials. Employing this methodology, we studied the MX3 family (M= Cr, Fe, V, Ti; X= Cl, Br, I) with the goal of advancing towards the solution of the most problematic issue in these materials, namely the Curie temperature. We use a particular case to show how this methodology allows us to obtain a complete description of the magnetic interaction picture (Jiso, Jxx, Jyy, Jzz) up to third neighbors condensed in a single effective equation per parameter, describing magnetic interaction in terms of the strain and the Hubbard U parameter. Additionally, and because of the important role of the Hubbard U in MX3 materials, we provide a library of self-consistent calculated Hubbard U for the principal pseudopotential families. The work presented herein advances in the description of the still unexplored MX3 materials, opening the door to a rational design of 2D magnetic materials. 2D magnetic materials provide an excellent platform to realize the hottest topics in condensed matter science, such as spintronics and quantum computation [1,2] They also provide a plethora of interesting phenomena such as topological properties and Moiré magnets . Also nascent fields as magnonics and straintronics find in 2D magnetic materials, a perfect environment to explore new properties of matter . Looking more in depth in the field of 2D materials, metal trihalides (MX3) are the most important family, being chromium trihalides the most studied 2D magnets . The discovery of CrI3 , the recent obtention of CrCl3 in-plane magnetic monolayer [14] and the role of CrBr3 inducing topological superconductivity in NbSe2 , may serve to showcase the active research being done in this family nowadays. Despite the high impact of these materials, most of the members in the MX3 family are still unexplored and constitute an untapped source of interesting physical properties. In opposition to the numerous advantages of these materials, the temperature necessary to maintain the magnetic order (Curie temperature) has shown to be specially low, being the most problematic issue of the family. Different approaches have been studied to improve this limitation . Among the strategies to obtain materials with improved properties, recent studies have shown the key role of strain and dielectric screening manipulation. Biaxial strain has raised as an important method to improve the properties of 2D materials [18, . On the other hand, to tune the properties of different systems modifying the dielectric screening using the Hubbard U parameter could provide important results as shown in recent works . An important advantage of these techniques is to provide an easy implementation using different substrates to apply certain dielectric screening and biaxial strain conditions. To obtain a detailed exploration of MX3 materials under the effect of these techniques using ab initio calculations is still limited by four important problems. First, the Hubbard model has a crucial role in the description of these systems, and is often not well considered. The absence of self-consistent Hubbard U parameters has produced in the community a tendency to use U values not properly calculated, which belong to calculations with a different pseudopotential, structure or even a different material. On the other hand, the obtention of the magnetic interaction parameters is not trivial, these parameters are often calculated using the total energy map analysis which requires the use of big supercells to take account of high order interaction. This ends in a high cost description in terms of simple Hamiltonians. Another important problem is related to Curie temperature, which calculation is often performed using classical methods as Monte Carlo simulations, or even with very simple analytical laws. Despite all these methods are able to provide correct results in most of the cases, a different approach is needed to perform a high throughput study of these properties, looking for a less expensive, easier to automatize and more precise method. Herein, we perform a high throughput exploration of the MX3 family (M= Cr, Fe, V, Ti; X= Cl, Br, I) to propose a different strategy to address these issues. We provide a self-consistent Hubbard U library for different pseudopotentials in the case of each material in order to encourage the use of a correct Hubbard U. We use Wannier90 [21] to connect our QuantumEspresso [22] DFT+U calculations with the Green function method, recently implemented in TB2J software , to obtain an inexpensive mapping into a Heisenberg Hamiltonian. With the magnetic interaction data acquired, we use a recently developed code [24] to obtain Tc as a solution of the quantum Heisenberg model in applied field conditions taking account up to third neighbors. Finally, to show that the accuracy and reliable cost of this method offers a solution to the high computational cost required to perform a full study of these materials, we perform a deeper study in the case of CrCl3 under strain and Hubbard U modifications. This methodology allows us to obtain a very complete description of the CrCl3 magnetic scenario and allows us to map the magnetic exchange and anisotropies up to third neighbors into effective equations providing an easy and instantaneous way to obtain an accurate approximation of the magnetic parameters under -5% to 5% strain conditions and in a 2-6 eV Hubbard U range. With this complete description of the system we are able to provide an infinite resolution heat map for Curie temperature, probing this approach as a valuable method for the rational design of 2D materials. RESULTS AND DISCUSSION The Hubbard model is a key element to describe the MottHubbard physics present in magnetic systems and thus a correct estimate of the Hubbard U becomes crucial. Despite of this, a complete self-consistent calculation of the Hubbard U parameters has not yet been performed for MX3 materials. As a consequence, the Hubbard U is usually extracted from a given calculation and then transferred to a calculation that relies on different codes or pseudopotentials or even is applied to different materials. To address this problem, we performed DFT calculations over the MX3 monolayer slab structures in order to calculate and converge the self-consistent Hubbard U for some of the most used families of pseudopotentials (see details in the Methods section). The Projected Augmented Wave (PAW) and “ultrasoft” (US) families of pseudopotentials from the QuantumEspresso [22] database are specially interesting because they provide a relativistic version of the pseudopotential, something needed to properly perform spin-orbit calculations using a Hubbard U that is consistent with the selected pseudopotential. Additionally, the Materials Cloud [25] database pseudopotentials were analyzed (in this particular case, GBRV [26] and PAW pseudopotentials were recommended) to calculate the Hubbard U. The results of this study are presented in Table 1, which could be used as a self-consistent calculated Hubbard U library. We take advantage of these results to point out the strongly dependence of the Hubbard U with the pseudopotential. We can see the differences are not always critical if we compare PAW and US pseudopotentials but the difference could be very severe if we compare the Hubbard U obtained with other pseudopotentials as the ones provided by the GBRV [26] library (shown in this paper), or with norm-conserving pseudopotentials. Due to the importance of spin orbit coupling in this work, from this point onwards we employ exclusively PAW and US pseudopotentials. Although the Hubbard U depends mostly on the metallic core, it is also affected by the exchange-correlation functional and the chemical environment of the localized orbitals. In our results we observe a tendency in the self-consistent U depening on the halide (figure 1). This tendency is independent from the pseudopotentials chosen and characteristic for each metal ion, increasing with rising atomic radius of the halide in the case of Cr and Fe halides, with halides of Ti and V presenting the opposite behavior. In all cases, the U ranges between 3.7 4.9 eV depending on the material. The evolution of the Hubbard U with respect to the pseudo-potential, metal ion and halide is well behaved, so that if any particular calculation does not converge it can be estimated with reasonable accuracy by knowing the results of neigh-bouring calculations. This fact allows to predict the Hubbard U for the different structures as we did in the case of non-converging calculations (TiI3, TiBr3, VI3) with the US pseudopotentials.