Impact of thermal fluctuations on transport in antiferromagnetic semimetals

Recent demonstrations on manipulating antiferromagnetic (AF) order have triggered a growing interest in antiferromagnetic metal, and potential high-density spintronic applications demand further improvements in the anisotropic magnetoresistance (AMR). The antiferromagnetic semimetals (AFS) are newly discovered materials that possess massless Dirac fermions that are protected by the crystalline symmetries. In this material, a reorientation of the AF order may break the underlying symmetries and induce a finite energy gap. As such, the possible phase transition from the semimetallic to insulating phase gives us a choice for a wide range of resistance, ensuring a large AMR. To further understand the robustness of the phase transition, we study thermal fluctuations of the AF order in AFS at a finite temperature. For macroscopic samples, we find that the thermal fluctuations effectively decrease the magnitude of the AF order by renormalizing the effective Hamiltonian. Our finding suggests that the insulating phase exhibits a gap narrowing at elevated temperatures, which leads to a substantial decrease in AMR. We also examine spatially correlated thermal fluctuations for microscopic samples by solving the microscopic Landau-Lifshitz-Gilbert equation, finding a quantitative difference in the gap narrowing effect from that of the macroscopic sample. For both cases, the semimetallic phase shows a minimal change in its transmission spectrum, illustrating the robustness of the symmetry-protected states in AFS. Our finding may serve as a guideline for estimating and maximizing AMR of the AFS samples at elevated temperatures.

Figure 1

(a) A lattice structure consists of two sublattice atoms. Two sublattice atoms are indicated as $A$ and $B$, having an opposite spin configuration along the Néel vector $\widehat{\mathit{n}}$ (AF order) enforced by the exchange coupling. (b) The lattice structure has nonsymmorphic symmetry ${\mathcal{G}}_x$ for $\widehat{\mathit{n}}\left|\right|\left[100\right]$ and ${\mathcal{G}}_y$ for $\widehat{\mathit{n}}\left|\right|\left[010\right]$.

Figure 2

(a) A system comprised of a set of subsystems (patches) with an averaged Néel vector indicated as an arrow. Note that each patch contains enough unit cells so that the region is well described by the momentum space Hamiltonian in Eq. (5). (b) In the presence of a random thermal fluctuation, we assume that the Néel vector orientation deviates from the original configuration by an azimuthal angle ${\phi }_v$ and polar angle ${\theta }_v$. The original Néel vector orientation is aligned with $\widehat{z}$ direction as a gray solid arrow, and a possible Néel vector deviation is described by a black solid arrow.

Figure 3

(a) The numerically computed coefficients of the impurity potential self-energy when the averaged Néel vector is $\widehat{\mathit{n}}\left|\right|\left[100\right]$ (gapless). The mass term introduced by ${\mathrm{\Sigma }}_1$ may gap out the Hamiltonian, whereas ${\mathrm{\Sigma }}_3$ merely shifts the Dirac points in momentum space. The dotted line shows ${\mathrm{\Sigma }}_1^{\left(2\right)}=(A_{1+}-A_{1-}){\mathrm{\Delta }}^2$, which is the lowest correction term for ${\mathrm{\Sigma }}_1$. The dashed line and solid line show ${\mathrm{\Sigma }}_3^{\left(2\right)}=(A_{3+}-A_{3-}){\mathrm{\Delta }}^2$ and ${\mathrm{\Sigma }}_3^{\left(1\right)}=\mathrm{\Delta }$, which correspond to the second-order and first-order correction terms in ${\mathrm{\Sigma }}_3$, respectively. (b) The same calculation, but the averaged Néel vector is $\widehat{\mathit{n}}\left|\right|\left[001\right]$ (gapped). The dotted line shows ${\mathrm{\Sigma }}_1^{\left(2\right)}=(A_{1+}-A_{1-}){\mathrm{\Delta }}^2$, which is the lowest correction term for ${\mathrm{\Sigma }}_1$. The dashed line and solid line show ${\mathrm{\Sigma }}_5^{\left(2\right)}=(A_{5+}-A_{5-}){\mathrm{\Delta }}^2$ and ${\mathrm{\Sigma }}_5^{\left(1\right)}=\mathrm{\Delta }$, which corresponds to the second-order and first-order correction term in ${\mathrm{\Sigma }}_5$, respectively. The parameters used for simulation are $t=1,t_{xy}=t,t_z=0.5t,t_{xy}^{\prime }=0.05t,t_z^{\prime }=0.05t,\lambda =0.8t,{\lambda }_z=0.3t$, and $\mathrm{\Delta }=0.3t$, which is identical with that used in Wang [22]. We plot the results within the range $0<\mathrm{\Delta }/\lambda <0.9$, where the system is guaranteed to have at least two Dirac nodal lines [22].

Figure 4

(a) A plot of transmission as a function of energy with the maximum fluctuation angle of ${\theta }_{\max }=\pi /8$. 20 different random configurations are averaged and the resultant transmission is presented with red and blue lines for the gapped ($\widehat{n}\left|\right|\left[001\right]$) and gapless ($\widehat{n}\left|\right|\left[100\right]$) phase, respectively. The corresponding transmission without Néel vector fluctuation is plotted with a dotted line for comparison. (b) The same plot with (a), but with the maximum angle of ${\theta }_{\max }=5\pi /8$. 20 different random configurations are plotted in gray lines. (c) A plot of transmission as a function of maximum Néel vector fluctuation angle ${\theta }_{\max }$ at $E=-40$ meV. The gray lines indicate the results from 20 different random configurations, and the blue circular and red rectangular symbols are obtained by averaging those different configurations for the gapless and gapped phase, respectively. The dotted line is a fitting result by using $T=f_B{e}^{-f_A{cos}^2({\theta }_{\max }/2)}$, where $f_A$ and $f_B$ are fitting parameters. The parameters used for the 2D Hamiltonian in Eq. (28) are $t=1,t_{xy}=t,t_{xy}^{\prime }=0.05t,\lambda =0.8t$, and $\mathrm{\Delta }=0.3t$. The device dimension is fixed as $N_x\times N_y=50\times 100$, where $N_x$ is width and $N_y$ is length of the device.

Figure 5

(a) 2D lattice structure under consideration. The $A$ and $B$ sublattice structure exhibit staggered magnetic order, together forming an antiferromagnetic order. To maintain an antiferromagnetic order, we introduce an exchange coefficient $-A_{\mathrm{ex}}$ between $A$ and $B$ sublattices. Meanwhile, we use $A_{\mathrm{ex}}$ between the same sublattices in order to maintain ferromagnetic order. (b) We form a Néel vector $\widehat{\mathit{n}}$ by postprocessing micromagnetics simulation results, and the resultant unit-cell structure is shown in a green circle. The configuration is imported as an input for NEGF calculation.

Figure 6

The two-terminal current of our AFS device with a spatially correlated random thermal fluctuation model. We directly import the Néel vector configurations from micromagnetics simulations, and each ensemble average current is obtained for 20 different snapshots uniformly sampled from 2 to 6 ns spin evolution time. The same procedures are repeated over ten different random seeds, and the resultant averaged current is presented with a standard error. The red circular symbol represents the current from the gapped phase ($\widehat{\mathit{n}}\left|\right|\left[001\right]$), and the blue rectangular symbol shows the current of the gapless phase ($\widehat{\mathit{n}}\left|\left[100\right]\right|$). The thermal barrier $U_{\mathrm{th}}=K_{u}V/k_{B}T$ is adjusted in order to effectively tune the maximum fluctuation angle in micromagnetics simulation. We use the following micromagnetics simulation parameters: $\mathrm{\Delta }x=1$ nm, $m_s=800$ kA/m, $A_{\mathrm{ex}}=10$ pJ/m, $\alpha =0.01$, and $T=300$ K. We use the same NEGF parameters used in Fig. 4, and the device for both (a) and (b) is $N_x\times N_y=50\times 100$.

Figure 7

(a) The transmission plot as a function of energy for an $L_x\times L_y=50\times 50{\mathrm{nm}}^2$ system at one specific random seed. The gray line shows 20 different transmission plots, which corresponds to the different Néel vector configurations uniformly sampled from 2- to 6-ns spin evolutions. The averaged transmission values for the gapped and gapless phase are indicated as red and blue lines, respectively. The plot emphasizes that the averaged transmission for the gapped phase is dominantly determined by the one random configuration which shows an abnormally high transmission plot indicated by a circular gray symbol. To examine more details of this particular configuration, we plot spatially the resolved local density of states (LDOS) in (b). (b) The plot of LDOS that corresponds to the Néel vector configuration which produces the circular gray symbol in (a). The LDOS is plotted for $E=0$ eV, and its magnitude is plotted in log scale. (c) The corresponding Néel vector configuration. Here, we present only the $\widehat{z}$- and $\widehat{y}$-directional Néel vector components in the left and right sides, respectively. The majority lattice site has a Néel vector aligned with the $\widehat{z}$ direction, as the system is initially in the gapped phase. Meanwhile, we observe puddles in the leftmost side of the device, whose net Néel vector is oriented in a negative $\widehat{y}$ direction (highlighted by the white dashed line). The puddles locally close the gap and provide available states even at $E=0$ eV, as shown in the LDOS in (b).