Monolayer ${\mathrm{CrCl}}_3$ as an Ideal Test Bed for the Universality Classes of 2D Magnetism

The monolayer halides ${\mathrm{CrX}}_3$ ($X=\mathrm{Cl}$, Br, I) attract significant attention for realizing 2D magnets with genuine long-range order (LRO), challenging the Mermin-Wagner theorem. Here, we show that monolayer ${\mathrm{CrCl}}_3$ has the unique benefit of exhibiting tunable magnetic anisotropy upon applying a compressive strain. This opens the possibility to use ${\mathrm{CrCl}}_3$ for producing and studying both ferromagnetic and antiferromagnetic 2D Ising-type LRO as well as the Berezinskii-Kosterlitz-Thouless (BKT) regime of 2D magnetism with quasi-LRO. Using state-of-the-art density functional theory, we explain how realistic compressive strain could be used to tune the monolayer’s magnetic properties so that it could exhibit any of these phases. Building on large-scale quantum Monte Carlo simulations, we compute the phase diagram of strained ${\mathrm{CrCl}}_3$, as well as the magnon spectrum with spin-wave theory. Our results highlight the eminent suitability of monolayer ${\mathrm{CrCl}}_3$ to achieve very high BKT transition temperatures, around 50 K, due to their singular dependence on the weak easy-plane anisotropy of the material.

Figure 1

(a) Crystal structure of monolayer ${\mathrm{CrCl}}_3$. The Cr and Cl atoms are represented in blue and orange, respectively. The $\mathcal{A}$ and $\mathcal{B}$ sublattices of Cr is indicated. Dashed lines denote the unit cell with basic vectors ${\mathit{\delta }}_1=a_{ϵ}(0,1)$, ${\mathit{\delta }}_2=a_{ϵ}(\sqrt{3}/2,-1/2)$, ${\mathit{\delta }}_3=a_{ϵ}(-\sqrt{3}/2,-1/2)$, with strain-dependent lattice constant $a_{ϵ}$. (b)–(c) Magnetic nearest-neighbor superexchange $J$ and anisotropy $K$ of Hamiltonian (1), respectively, computed via DFT as a function of monolayer strain $ϵ$. (d) Finite-temperature phase diagram of the monolayer ${\mathrm{CrCl}}_3$ versus strain $ϵ$, obtained by QMC simulations for the $S=3/2$ model of Eq. (1) on the 2D honeycomb lattice. Strain drives the monolayer into three different finite-temperature magnetic phases: BKT quasi-LRO phase for $ϵ\lesssim -2.4\%$, AFM Ising for $-2.4\%\lesssim ϵ\lesssim -1.2\%$, and FM Ising for $ϵ\gtrsim -1.2\%$. At zero temperature, the BKT quasi-LRO turns into genuine $XY$ LRO, separated from the AFM Ising order by an isotropic Heisenberg point displaying Néel order (where $K_{ϵ}$ vanishes). The AFM and FM Ising phases are separated by a trivial paramagnetic point (where $J_{ϵ}$ vanishes). The colored lines are fits to the form of Eq. (2).

Figure 2

(a)–(b) Obtaining $T_c$ (dashed vertical lines) for onset of magnetic LRO from scaling analysis of QMC values of the order parameter in the EA regime, using 2D Ising critical exponents $\beta =1/8$ and $\nu =1$. (a) Magnetization density vs temperature for different $L$ at $ϵ=0\%$ (FM Ising), with $T_{\mathrm{c}}=14.84(1)\mathrm{K}$. (b) Staggered magnetization density vs temperature for different $L$ at $ϵ=-2\%$ (AFM Ising), with $T_{\mathrm{c}}=12.6(1)\mathrm{K}$. (c) Finite-size scaling analysis of QMC-computed spin stiffness ${\rho }_{\mathrm{s}}(L)$ at different $L$ in EA regime, at $ϵ=-4\%$ (BKT Quasi-LRO). Dashed line shows $2T/\pi$. (d) $T^{\star }(L)$ extracted from (c) vs $(\ln L{)}^{-2}$ for $ϵ=-4\%$. $T_{\mathrm{BKT}}=48(2)\mathrm{K}$ is extracted from fitting with $T_{\mathrm{BKT}}+C/(\ln L{)}^2$ (dashed line).

Figure 3

Critical temperature $T_{\mathrm{c}}$ (AFM Ising, orange) and $T_{\mathrm{BKT}}$ (BKT Quasi-LRO, green) vs $|K_{ϵ}|/J_{ϵ}$. Points with explicit $ϵ$ values correspond to $K_{ϵ}/J_{ϵ}$ for monolayer ${\mathrm{CrCl}}_3$ at such strain, see Fig. 1. Dashed lines show fit to $A/(\ln |J_{ϵ}/K_{ϵ}|+B)$ in the small anisotropy limit $|K_{ϵ}|/J_{ϵ}\le 0.1$, with $A$ and $B$ fitting parameters. For BKT, $A=14.6(3)$, $B=9.2(3)$. For AFM Ising, $A=14.1(5)$, $B=6.5(4)$. Both $A$ values are compatible with the analytical prediction $A=4\pi {\rho }_{\mathrm{s}}^{(0)}=16(1)$ in Eq. (2).

Figure 4

Band structure of magnons in (a) easy-axis FM and (b) easy-plane AFM. The blue and magenta curves represent $\alpha$ and $\beta$ bands, respectively. Inset in panel (a) shows gap opening at the bottom of spectrum close to $\mathrm{\Gamma }$ due to anisotropy while the one in panel (b) displays breaking of band degeneracy close to $\mathrm{\Gamma }$. The Brillouin zone of the honeycomb lattice is displayed in (b), along with the high symmetry points $\mathit{q}=a_{ϵ}^{-1}(q_x,q_y)$: $\mathrm{\Gamma }=a_{ϵ}^{-1}(0,0)$, $\mathrm{M}=a_{ϵ}^{-1}(4\pi /3,0)$, and $\mathrm{K}=a_{ϵ}^{-1}(\pi ,\pi /\sqrt{3})$ with strain-dependent lattice constant $a_{ϵ}$.