Magnetic Interactions and Cluster Formation: Boosting Surface Thermopower in Topological Insulators

Abstract

This study presents a theoretical investigation of the thermoelectric properties of three-dimensional magnetic topological insulators (TIs), with a focus on the role of exchange interactions between magnetic dopants. The presence of these magnetic atoms on the TI surface modulates the local magnetic order, which in turn alters the electronic band structure and surface transport phenomena. Magnetic correlations, such as those arising from ferromagnetic or antiferromagnetic exchange, promote cluster formation, magnetic domain structures, and spin fluctuations, all of which critically influence thermoelectric responses. Using extensive Monte Carlo simulations based on Ising and Heisenberg models of these surface exchange interactions, we analyze how magnetic clustering, particularly near the surface critical temperature, affects relaxation dynamics, electrical and thermal resistivity, the Seebeck coefficient, and the thermoelectric figure of merit. Our results demonstrate that exchange-driven magnetic clustering enhances the scattering of Dirac surface states, thereby increasing the thermoelectric power factor. Specifically, optimized interlayer and intralayer exchange interactions can elevate the surface thermopower beyond levels observed in conventional spin-based thermoelectric materials. These findings highlight the significant potential of magnetic TIs for thermoelectric applications and provide a foundation for future experimental and theoretical studies of magnetic correlations in topologically nontrivial systems.

Figures (15)

• Figure 1: Left: A triangular lattice of ${\rm Mn}$ atoms, with exchange constants $J_1$ to $J_4$ representing interactions between Mn atoms with spin $\tilde{s} = 5/2$ from NN to fourth NNs. In $\rm MnBi_2Te_4$, they are characterized by the parameters $\tilde{s} J_1\approx 0.30$, $\tilde{s} J_2\approx -0.083$, $\tilde{s} J_3 \approx 0$, and $\tilde{s} J_4 \approx 0.023$ millielectron-Volt (${\rm meV}$), as reported in Ref. PRL_MagneticInteractions. Middle: the lattice structure of the antiferromagnetic $\rm MnBi_2Te_4$, which consists of two septuple layers separated by a van der Waals gap, with $J_c$ representing the interlayer interaction between magnetic atoms. Right: the lattice structure of the ferromagnetic TI $\rm MnBi_6Te_{10}$, in which two $\rm Bi_2Te_3$ layers are placed between two $\rm MnBi_2Te_4$ layers.
• Figure 2: The MC results (Ising Hamiltonian) for the number of magnetic clusters per unit area, $\bar{n}_c$ (left), and the size of magnetic clusters, $\xi/a$ (right), as functions of temperature $\widetilde{T}=k_{\rm B}T/(J_1\tilde{s}^2)$. Here, $k_{\rm B}$ denotes the Boltzmann constant, $\tilde{s}$ represents the spin of the magnetic atoms, $J_1$ is the NN exchange interaction, and $a$ is the triangular lattice constant. The interaction of magnetic atoms on the surface of the magnetic TI is described by the Ising Hamiltonian. The number and size of clusters are independent of the nature of the interlayer interaction, whether antiferromagnetic ($J_c<0$) or ferromagnetic ($J_c>0$). In our analysis, the interlayer coupling $|J_c|$ is scaled to $J_1$ and set to 0.24, as referenced in PRL_MagneticInteractions. The critical temperature is approximately $\widetilde{T}_c \simeq 4.2$ for both the antiferromagnetic and ferromagnetic cases.
• Figure 3: Clustering of magnetic atoms located on the triangular lattice sites on the TI surface, driven by the Heisenberg interaction (see Eq. (\ref{['eq:H1']})). The different clusters are distinguished by the ratio $s_z/s$. Here, $\widetilde{T}=k_{\rm B}T/(J_1\tilde{s}^2)$, $J_1$ is the NN exchange coupling, and the exchange couplings $J_2$ and $J_4$, the uniaxial single-ion anisotropy $D$, and the interlayer exchange $J_c$ are scaled to $J_1$ with values set to $-0.28$, $0.08$, $0.4$, and $-0.18$, respectively. Different colors denote various cluster types. In the ordered phase, below the critical temperature $\widetilde{T}_c=1.42$, all clusters exhibit a negative spin value $s_z<0$ (surface magnetization is along the $+z$-direction). As the system crosses the critical point, clusters with $s_z>0$ emerge on the surface. Above the critical point, increasing temperature results in smaller cluster sizes.
• Figure 4: The MC results for the mean size and number of clusters of different types at temperatures, below ($\widetilde{T} = 1.2$) and above ($\widetilde{T} = 2.8$) the critical temperature ($\widetilde{T}_c = 1.42$). Here, $\widetilde{T} = k_{\rm B} T / (J_1 \tilde{s}^2)$, with $J_1$ being the NN exchange constant. When the interactions between magentic atoms are described by the Heisenberg Hamiltonian, various types of clusters form, characterized by their $s_z$ component or, equivalently, their polar angles $\theta_s$ relative to the $z$-axis (see also the MC results presented in Fig. \ref{['fig: xiH']}). For each $\theta_s$, both the size and the number of clusters are independent of the azimuthal angle $\phi_s$ (above four plots). In the bottom panel, we schematically depict two types of clusters, categorized by their sizes and spin angles: $\xi_1$ with angle $\theta_{s_1}$ (dark blue), and $\xi_2$ with angle $\theta_{s_2}$ (light blue). The spins within clusters of the same type are oriented such that their in-plane components cancel each other out on average.
• Figure 5: The MC results for the mean size and number of clusters of different types, labeled by $s_z/s=\cos\theta_s$, as functions of $\widetilde{T} = k_{\rm B} T / (J_1 \tilde{s}^2)$. The interaction of magnetic atoms on the surface of the magnetic TI is described by the Heisenberg Hamiltonian. The interlayer interactions are antiferromagnetic ($J_c<0$), with $|J_c|/J_1=0.18$. At temperatures below the critical temperature, only clusters with negative $s_z$, depicted in red, appear on the surface of the TI, while above the critical temperature, all types of clusters emerge.