Theoretical analysis of single-ion anisotropy in d cubed Mott insulators

arXiv:2203.08836 · cond-mat.str-el · Submitted 2022-03-16 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Theoretical analysis of single-ion anisotropy in d cubed Mott insulators".

Mira: The gist: This paper derives analytical expressions for single-ion anisotropy in d3 Mott insulators using strong-coupling perturbation theory, accounting for cubic crystal field splitting, Hund’s coupling, spin-orbit coupling,

Kai: First, who's behind it and why it matters.

Paper summary: Mira: So, we've seen the setup—the paper "Theoretical analysis of single-ion anisotropy in d cubed Mott insulators" is proposing a strong-coupling perturbation method to derive the SIA term for a spin model defined by Hspin <ref:2203.08836#pg1>.

Kai: They are deriving this SIA term, which has coefficients A that correspond to an easy-plane or easy-axis preference, where the sign of delta is determined by the metal-ligand hybridization <ref:2203.08836#pg2>.

Lev: The paper gives specific expressions for how the SIA depends on whether you're looking at a t three two g configuration or a d three configuration including eg orbitals <ref:2203.08836#pg1>. For t three two g, the SIA is given by AM equals delta xiM (xiM minus delta) squared over five J 3H <ref:2203.08836#pg1>.

Kai: And for the d three configuration with eg orbitals, they find that AM equals two over three delta xiM over one Delta c minus one over Delta c plus three J H squared minus six times ten Delta c plus twenty one J H squared <ref:2203.08836#pg1>.

Mira: The key finding here is that the eg orbitals contribution dominates the SIA strength for d three compared to t three two g <ref:2203.08836#pg1>. That means if you're looking at a full d three system, that part really drives how strong the anisotropy ends up being.

Lev: When they applied this to CrXthree specifically CrCl3 and CrI3, they used DFT calculations to find parameters like Delta c and delta <ref:2203.08836#pg2>. They found that the sign change in AM in CrI3 is due to the sign change of delta through p-d hybridization <ref:2203.08836#pg1>.

Kai: And the paper shows that when you combine AM and AX, they are both always negative, which leads to the easy-axis anisotropy for CrI3 <ref:2203.08836#pg1>. That's a concrete result for this material system.

Mira: They also look at the anion contributions to SIA through distortion induced hoppings, giving an AX term like AX equals negative four over five J H plus sixteen over five times ten Delta c plus twenty one J H tA tpi t2eff plus six J H Delta c (Delta c plus three J H) tsigma (tpi tB plus tsigma tC)t3pi <ref:2203.08836#pg2>.

Lev: The effective hopping term, teff, is given as two t two pi over three root one over Delta pd minus xiX squared minus one over Delta pd plus xiX <ref:2203.08836#pg2>. So the anion part is also tied to these orbital configurations and hybridization strengths.

Kai: This whole analysis gives us a microscopic route to understanding the SIA for S equal to three/two in d three Mott insulators starting from the Kanamori-Hubbard interaction <ref:2203.08836#pg1>. It shows how different combinations of SOC and trigonal fields lead to different anisotropies <ref:2203.08836#pg1>.

Mira: The paper suggests that the tendency toward an easy-plane versus easy-axis is determined by these two contributions, AM and AX <ref:2203.08836#pg1>. And for compressed trigonal distortion, the SOC at magnetic sites can choose either easy plane or easy axis depending on the sign of delta <ref:2203.08836#pg1>.

Lev: But there's a limitation here; they state that the exact spectrum of d three configuration cannot be obtained analytically because Hund’s coupling and the cubic crystal field splitting do not commute with each other <ref:2203.08836#pg3>. That means the calculation is only done under two extreme conditions, as shown in t <ref:2203.08836#pg1>.

Kai: So to wrap up this part, this paper provides a microscopic route to the single-ion anisotropy for S=three/two in d three Mott insulators by accounting for the cubic crystal field splitting, Hund’s coupling, spin-orbit coupling beyond LS coupling, and trigonal distortion <ref:2203.08836#pg1>.

Mira: And it shows that the eg orbitals contribution is essential to understanding the SIA strength and that the tendency toward easy-plane versus easy-axis is determined by two contributions denoted by AM and AX <ref:2203.08836#pg1>.

Conclusion: Kai: We've just gone through the details of this paper, "Theoretical analysis of single-ion anisotropy in d cubed Mott insulators". What does this all mean for the broader field?

Mira: It offers a microscopic way to look at the single-ion anisotropy for S equal to three/two in d three Mott insulators by starting from the Kanamori-Hubbard interaction <ref:2203.08836#pg1>.

Lev: The paper shows that the eg orbitals contribution is essential to understanding the SIA strength and that the tendency toward easy-plane versus easy-axis is determined by two contributions denoted by AM and AX <ref:2203.08836#pg1>.

Kai: It also points out that for compressed trigonal distortion, the SOC at magnetic sites can choose either easy plane or easy axis depending on the sign of delta <ref:2203.08836#pg1>.

Mira: The sign of delta is determined by the metal-ligand hybridization, and they found it takes a positive value for CrCl3, while for CrI3, it's negative leading to the easy-axis anisotropy <ref:2203.08836#pg1>.

Lev: For AX, they found that it prefers the easy-axis for both Cr trihalides <ref:2203.08836#pg1>. They note that this current work does not aim to offer precise values of SIA in CrXthree but rather to provide the understanding of the SIA originated from the different combinations of SOC and trigonal field in d three systems <ref:2203.08836#pg1>.

Kai: It's about using this theory to guide ways toward higher transition temperatures in two-dimensional systems, which is what we see attracting intense studies <ref:2203.08836#pg1>.

Mira: Understanding the microscopic origin of magnetic anisotropy in those two-dimensional single-layer systems will guide the way toward a higher transition temperature <ref:2203.08836#pg1>.

Xiaoyu Liu, Derek Churchill, Hae-Young Kee

Department of Physics, University of Toronto · Canadian Institute for Advanced Research

cond-mat.str-el

Submitted: 2022-03-16

Updated: 2022-08-03

Journal ref: Phys. Rev. B 106, 035122 (2022)

DOI: 10.1103/PhysRevB.106.035122

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 72/100

The gist: The gist: This paper derives analytical expressions for single-ion anisotropy in d3 Mott insulators using strong-coupling perturbation theory, accounting for cubic crystal field splitting, Hund’s

Key concepts

Single-Ion Anisotropy (SIA): SIA describes how the magnetic energy depends on the orientation of a single spin relative to its surroundings. In this paper, it is calculated using strong-coupling theory, revealing that it arises beyond simple LS coupling and is determined by interactions like spin-orbit coupling and crystal field effects.
Strong-Coupling Perturbation Theory: This theoretical method is used to model the complex interactions in Mott insulators. It allows researchers to systematically derive the effective spin Hamiltonian by treating the strong on-site Coulomb repulsion as a perturbation, enabling calculations for magnetic anisotropy that are otherwise difficult to obtain.
Crystal Field Splitting and Orbital Configuration: The energy levels of d³ ions are split by the surrounding crystal environment (cubic field). The paper analyzes two configurations: $t_{3}^{2}g$ and $d^{3}$ including $e_g$ orbitals. It finds that the contribution from the $e_g$ orbitals is dominant when analyzing SIA in d³ systems.
Metal-Ligand Hybridization ($\delta$): This parameter, derived from DFT calculations, dictates the sign and nature of the trigonal distortion effects. The sign of $\delta$ determines whether the magnetic anisotropy favors an easy-plane (positive $\delta$) or an easy-axis (negative $\delta$) configuration in materials like CrCl₃ and CrI₃.

Terminology

Anisotropy — 8× in this paper · explained in 4 episodes

Summary

The gist: This paper derives analytical expressions for single-ion anisotropy in d3 Mott insulators using strong-coupling perturbation theory, accounting for cubic crystal field splitting, Hund’s coupling, spin-orbit coupling, and trigonal distortion.

Theoretical Framework

The effective spin model for Mott insulators is determined by the symmetries involved among magnetic sites, electron fillings, and their interactions<ref:2203.08836#pg2>. For d3 Mott insulators with large cubic crystal field splitting, the single-ion anisotropy is absent within the LS coupling, despite S = 3/2 local moment<ref:2203.08836#pg2>. The spin anisotropy should come from a finite trigonal crystal splitting and/or beyond the LS coupling, i.e., sum of each atomic SOC, ξM P i l i · s i<ref:2203.08836#pg2>. The total onsite Hamiltonian is the summation of the above terms<ref:2203.08836#pg2>.

Single-Ion Anisotropy Derivation

The paper derives the SIA term (last term) of a spin model defined by Hspin<ref:2203.08836#pg2>. The coefficient A > 0 corresponds to easy-plane and A. The sign of δ is determined by the metal-ligand hybridization, and it is found that for CrCl3, it takes a positive value, while for CrI3, it is negative leading to the easy-axis anisotropy<ref:2203.08836#pg2>. For compressed trigonal distortion, the SOC at magnetic sites can choose either easy-plane or easy-axis depending on the sign of δ<ref:2203.08836#pg2>.

Dependence on Orbital Configuration

The SIA for d3 systems is analyzed under two different orbital configurations: t32g and d3. The SIA for t32g is given by AM(t32g) = δξ2M(ξM − δ)2/5J 3H<ref:2203.08836#pg2>. The SIA for d3 including eg orbitals is found as AM = 2/3δξ2M / (1∆c - 1/(∆c + 3JH))2 - 6(10∆c + 21JH)2<ref:2203.08836#pg2>. The paper finds that the eg orbitals contribution dominates the SIA strength for d3 compared to t32g<ref:2203.08836#pg2>.

Application to CrX3

The theory is applied to 3d3 CrX3, specifically CrCl3 and CrI3<ref:2203.08836#pg2>. DFT calculations are used to determine parameters like ∆c and δ<ref:2203.08836#pg2>. The sign change in AM in CrI3 is due to the sign change of δ via p-d hybridization as mentioned above<ref:2203.08836#pg2>. The total SIA from both AM and AX is always negative leading to the easy-axis anisotropy for CrI3<ref:2203.08836#pg2>.

Anion Contributions

Aside from the onsite contribution to the SIA, the SOC on anions also contributes to SIA through distortion induced hoppings<ref:2203.08836#pg2>. Up to linear order of distortion induced hoppings, the SIA is given by AX = -4/5JH + 16/5(10∆c + 21JH)tA tπ t2eff + 6JH∆c(∆c + 3JH)tσ(tπtB + tσtC)t3π t2eff<ref:2203.08836#pg2>. The effective hopping is given by teff = 2t2π/3√1/∆pd−ξX2−1/∆pd+ξX<ref:2203.08836#pg2>.

Conclusion

The study provides a microscopic route to the SIA for S=3/2 in d3 Mott insulator starting from the Kanamori-Hubbard interaction<ref:2203.08836#pg2>. The eg orbitals contribution is essential to understand the SIA strength and that the tendency towards easy-plane versus easy-axis is determined by two contributions denoted by AM and AX<ref:2203.08836#pg2>. For compressed trigonal distortion, the SOC at magnetic sites can choose either easy-plane or easy-axis depending on the sign of δ<ref:2203.08836#pg2>. The sign of δ is determined by the metal-ligand hybridization, and it is found that for CrCl3, it takes a positive value, while for CrI3, it is negative leading to the easy-axis anisotropy<ref:2203.08836#pg2>. For AX, it prefers the easy-axis for both Cr trihalides<ref:2203.08836#pg2>. The paper notes that the current work does not aim to offer precise values of SIA in CrX3, but to provide the understanding of the SIA originated from the different combinations of SOC and trigonal field in d3 systems<ref:2203.08836#pg2>.

How it works

The low-energy effective spin model for S=3/2 using the octahedra coordinate system x − y − z is given by Hspin<ref:2203.08836#pg2>.
The total onsite Hamiltonian is the summation of the above terms<ref:2203.08836#pg2>.
The effective Hamiltonian is then given by Heff = H0 + V01/E0 − H1 − V11/V10<ref:2203.08836#pg2>.
For t32g when ∆c → ∞, the fourfold lowest states are decoupled from the excited states, leading to SIA under LS coupling being zero as expected due to the quenched angular momentum in t32g<ref:2203.08836#pg2>.
Beyond the LS coupling, considering SOC given by ξM P i l i · s i, H002 and H012 are non-zero, indicating that the SIA under li · si coupling is finite<ref:2203.08836#pg2>.
For d3 including eg orbitals, the SIA is given by AM = 2/3δξ2M / (1∆c - 1/(∆c + 3JH))2 - 6(10∆c + 21JH)2<ref:2203.08836#pg2>.
The sign difference in AM for both materials is due to the sign difference in total effective trigonal field δ given in Table III<ref:2203.08836#pg2>.

Summary and Discussion

The existence of ferromagnetic LRO in two-dimensional (2D) systems with higher transition temperature Tc has attracted intense studies<ref:2203.08836#pg2>. To achieve a higher Tc in ideal 2D materials, it is essential to have a certain magnetic anisotropy that opens up a spin gap which allows the system to avoid quantum fluctuations and set up an LRO at finite temperature<ref:2203.08836#pg2>. Understanding a microscopic origin of magnetic anisotropy in two-dimensional single-layer will guide ways to move towards a higher Tc<ref:2203.08836#pg2>. The paper proposes that this may be relevant to the atomic SOC discussed in this paper<ref:2203.08836#pg2>.

Future Directions

Extending the current theory to the stronger SOC may explain the anisotropy observed in these systems, which is a project for future studies<ref:2203.08836#pg2>. The research was supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chairs Program<ref:2203.08836#pg2>. Computations were performed on the GPC and Niagara super-computers at the SciNet HPC Consortium<ref:2203.08836#pg2>. The paper was enabled in part by support provided by Sharcnet (www.sharcnet.ca) and Compute Canada (www.computecanada.ca)<ref:2203.08836#pg2>. Computations were performed on the GPC and Niagara super-computers at the SciNet HPC Consortium<ref:2203.08836#pg2>. The paper was supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chairs Program<ref:2203.08836#pg2>. This research was enabled in part by support provided by Sharcnet (www.sharcnet.ca) and Compute Canada (www.computecanada.ca)<ref:2203.08836#pg2>.

Improvements for AI systems

textbfProcessing of Magnetic Anisotropy in 2D Magnets: Easy-Axis vs. Easy-Plane Prediction for CrX3 Systems via DFT Parameters (Table III and Fig. 4). This AI system can predict the magnetic anisotropy direction by analyzing the sign of the total effective trigonal field δ derived from DFT parameters, specifically noting that the sign change in AM in CrI3 is due to the sign change of δ via p-d hybridization as mentioned above.
textbfDerivation of Single-Ion Anisotropy (SIA) Strength using Strong Coupling Perturbation Theory for d3 Systems (Eq. 13 and Table II). The system can calculate the SIA strength, such as "AM or AX," by substituting specific values for SOC strength ξM and trigonal distortion δ into the derived analytical expressions, which are shown in Equation (13).
textbfHandling Multi-Orbital Effects on SIA Magnitude (Appendix A and Fig. 5). The AI can quantify how higher-order interactions affect the SIA by comparing results, noting that the 3- and 4-orbital effects only have minor quantitative corrections to our results when comparing the full interaction Hamiltonian to the simplified one.
textbfPredicting Anion Contribution to Magnetic Anisotropy (Equation 14). The system can calculate the SIA induced by anion SOC, AX, by utilizing the effective hopping is given by teff = 2t2π3 / (1∆pd−ξX2 − 1∆pd+ξX).
textbfIdentifying Dominant Contributions to SIA in d3 vs. t32g Systems (Discussion and Summary). The AI can determine the primary source of anisotropy by comparing the order of terms, noting that the contribution from eg orbitals dominate the SIA strength for d3 systems because the SIA for d3 is a third order term, whereas it is a fourth-order term in t32g.