Magnetically driven anisotropic structural changes in the atomic laminate $\mathrm{M}{\mathrm{n}}_2\mathrm{GaC}$

Inherently layered magnetic materials, such as magnetic $M_{n+1}A{X}_n$ (MAX) phases, offer an intriguing perspective for use in spintronics applications and as ideal model systems for fundamental studies of complex magnetic phenomena. The MAX phase composition ${M_n}_{+1}A{X}_n$ consists of $M_{n+1}{X}_n$ blocks separated by atomically thin $A$-layers where $M$ is a transition metal, $A$ an A-group element, $X$ refers to carbon and/or nitrogen, and $n$ is typically 1, 2, or 3. Here, we show that the recently discovered magnetic $\mathrm{M}{\mathrm{n}}_2\mathrm{GaC}$ MAX phase displays structural changes linked to the magnetic anisotropy, and a rich magnetic phase diagram which can be manipulated through temperature and magnetic field. Using first-principles calculations and Monte Carlo simulations, an essentially one-dimensional (1D) interlayer plethora of two-dimensioanl (2D) Mn-C-Mn trilayers with robust intralayer ferromagnetic spin coupling was revealed. The complex transitions between them were observed to induce magnetically driven anisotropic structural changes. The magnetic behavior as well as structural changes dependent on the temperature and applied magnetic field are explained by the large number of low energy, i.e., close to degenerate, collinear and noncollinear spin configurations that become accessible to the system with a change in volume. These results indicate that the magnetic state can be directly controlled by an applied pressure or through the introduction of stress and show promise for the use of $\mathrm{M}{\mathrm{n}}_2\mathrm{GaC}$ MAX phases in future magnetoelectric and magnetocaloric applications.

Figure 1

Schematic illustration of considered collinear magnetic spin configurations for (a) $\mathrm{FM}$, (b) $\mathrm{AFM}{\left[0001\right]}_1$, (c) $\mathrm{AFM}{\left[0001\right]}_2^A$, (d) $\mathrm{AFM}{\left[0001\right]}_2^X$, (e) in-$\mathrm{AFM}1$, (f) in-$\mathrm{AFM}2$, (g) $\mathrm{AFM}{\left[0001\right]}_4^A$, (h) $\mathrm{AFM}{\left[0001\right]}_6^A$, (i) $\mathrm{AFM}{\left[0001\right]}_8^A$, (j) $\mathrm{AFM}{\left[0001\right]}_4^X$, (k) $\mathrm{AFM}{\left[0001\right]}_6^X$, and (l) $\mathrm{AFM}{\left[0001\right]}_8^X$ seen along the $\left[11\overline{2}0\right]$ direction. Spin directions of Mn atoms are indicated by up and down arrows.

Figure 2

The in-plane magnetization measured with VSM for selected temperatures in the range (a) 3 to 50 K and (b) 50 to 300 K. Below 230 K a sharp FM transition is seen, see inset in (a) at low fields, followed by an increasing magnetization with field. Above 250 K there is no FM response and a new field-driven transition is observed. The inset in panel (b) shows the magnetic moment at 5 T, $m_{5\mathrm{T}}$, as a function of temperature. Note that symbols used for the different temperatures in (a) and (b) are only shown for every 50 to 100 data points whereas the lines represent every data point.

Figure 3

Stability and structure from first-principles calculations. (a) Energy as a function of volume for different collinear spin configurations of $\mathrm{M}{\mathrm{n}}_2\mathrm{GaC}$. Energies are given relative to the minimum energy of the nonmagnetic state $E_0^{\mathrm{NM}}$. The measured volume at room temperature is shown by a vertical line [37]. (b) Magnetic unit cell schematics of selected low-energy spin configurations showing three different interlayer distances within the lattice. In (c) the Mn-Ga-Mn interlayer distance is displayed as a function of volume for the three configurations displayed in (b) showing two distinct distances, $d_A^{+/+}$ and $d_A^{+/-}$, for the $\mathrm{AFM}{\left[0001\right]}_4^A$. Panel (d) shows the $c$ and $a$ lattice parameters as a function of energy difference for relaxed FM, $\mathrm{AFM}{\left[0001\right]}_{\alpha }^A$, and $\mathrm{AFM}{\left[0001\right]}_{\alpha }^X$ spin configurations. All data presented here are based on first-principles calculations employing the generalized gradient approximation (GGA), see Sec. 2 for more details.

Figure 4

(a) $2\theta -\omega$ scans of $\mathrm{M}{\mathrm{n}}_2\mathrm{GaC}$ at room temperature (RT) and 150 K. A shift in the 0006 peak position is accompanied by a new peak appearing at ∼35° for the 150 K scan, compared to RT. The theoretical peak positions and their relative intensities for three different spin configurations are also presented. The inset shows a close-up of the 0006 peak shift. (b) Remanent moment $m_r$ measured with MOKE (in arbitrary units) as a function of temperature $T$, and the relative change in $c$-lattice parameter with respect to measurement at RT.

Figure 5

Heisenberg Monte Carlo simulations with (a) net moment of lowest-energy configurations as a function of volume. Results from different chain lengths for GGA () and LDA () based MEI, with corresponding average in solid red and dashed green lines, respectively. (b) Selected lowest-energy spin spirals at different volumes, as indicated in (a), where each arrow represents the spin direction of a Mn-C-Mn supermoment.

Figure 6

Schematic of a possible transition between magnetic states and explanation for the observed FM response. (a) Schematic of a canting transition from $\mathrm{AFM}{\left[0001\right]}_4^A$ to FM using the coarse-grained model with Mn moments in a Mn-C-Mn trilayer represented by a supermoment. The crystal configuration of $\mathrm{AFM}{\left[0001\right]}_4^A$ and FM is also seen as a reference. (b) Change in energy of the canted $\mathrm{AFM}{\left[0001\right]}_4^A$ with canting angle (filled circles) along with the corresponding net moment (open circles). Note that ${\theta }_{\mathrm{c}}=90$ corresponds to an FM state. The gray box indicates the region where the angle is degenerate to within 0.2 meV/atom. Also shown are the energies for selected low-energy spin spirals IV (), V (), and VI () as well as for $\mathrm{AFM}{\left[0001\right]}_4^A$ (), relative to that of $\mathrm{AFM}{\left[0001\right]}_4^A$.