Diluted magnetic Dirac-Weyl materials: Susceptibility and ferromagnetism in three-dimensional chiral gapless semimetals

We theoretically investigate the temperature-dependent static susceptibility and long-range magnetic coupling of three-dimensional (3D) chiral gapless electron-hole systems (semimetals) with arbitrary band dispersion [i.e., $\varepsilon \left(k\right)\sim k^N$, where $k$ is the wave vector and $N$ is a positive integer]. We study the magnetic properties of these systems in the presence of dilute random magnetic impurities. Assuming carrier-mediated Ruderman-Kittel-Kasuya-Yosida indirect exchange interaction, we find that the magnetic ordering of intrinsic 3D chiral semimetals in the presence of dilute magnetic impurities is ferromagnetic for all values of $N$. Using finite-temperature self-consistent field approximation, we calculate the ferromagnetic transition temperature ($T_{\mathrm{c}}$). We find that $T_{\mathrm{c}}$ increases with increasing $N$ due to the enhanced density of states, and the calculated $T_{\mathrm{c}}$ is experimentally accessible, assuming reasonable coupling between the magnetic impurities and itinerant carriers.

Figure 1

The calculated finite-temperature static susceptibility $\chi (\mathit{q},T)$ as a function of wave vector for various temperatures $T=0,0.02,0.04,0.06,0.08,$ and $0.1T_0$, and for different values of $N$ (a) $N=1$, (b) $N=2$, (c) $N=3$, and (d) $N=4$. Here, $T_0={\varepsilon }_0/k_{\mathrm{B}}, D_1\left(a^{-1}\right)=\frac{g{k}_0}{2{\pi }^2{\varepsilon }_0{a}^2}$, and $a=0.343$ nm (lattice constant of TaAs). For $N=1$, the finite momentum cutoff $a^{-1}$ is used for the convergence of the integral.

Figure 2

The range function $\chi (\mathit{r},T)$ as a function of distance for different values of $N$ (a) $N=1$, (b) $N=2$, (c) $N=3$, and (d) $N=4$. In each figure, the curves with different colors represent different temperatures $T=0$, 0.1, 0.2, 0.3, 0.4, and 0.5 $T_0$. Here, $D_0=D_1\left(a^{-1}\right)/a^3$. In this calculation, the ultraviolet momentum cutoff $q_{\mathrm{c}}=a^{-1}$ is used.

Figure 3

The calculated effective coupling (solid lines) as a function of temperature for different values of $N=1,2,3,4$. In this calculation, the ultraviolet cutoff $q_{\mathrm{c}}=a^{-1}$ and exponential cutoff $R=100a$ are used. Here, the normalization factor $J_{\mathrm{eff}}^{\left(0\right)}=4\pi {\left[J_{\mathrm{ex}}{a}^3\right]}^2{D}_1\left(a^{-1}\right)/4{\mathrm{\Omega }}_{\mathrm{unit}}$ is independent of $N$ and temperature $T$. The dashed line represents $3k_{\mathrm{B}}T/\left[S(S+1)x\right]$, and the intersections with $J_{\mathrm{eff}}\left(T\right)$ indicate the transition temperatures solved self-consistently. Here, $J_{\mathrm{ex}}=0.1$ eV, $x=0.05$ and $S=5/2$.

Figure 4

The calculated transition temperature $T_{\mathrm{c}}$ as a function of (a) the exchange coupling $J_{\mathrm{ex}}$, (b) the magnetic impurity concentration $x=n_{\mathrm{imp}}{a}^3$, and (c) the degeneracy factor $g$ for different values of $N=1,2,3,4$. Here, for fixed parameters, we used $J_{\mathrm{ex}}=0.1$ eV, $x=0.05$ and $g=4$.