Higher-order topological phase diagram revealed by anomalous Nernst effect in a Janus ScClI monolayer

Higher-order topological properties of two-dimensional (2D) magnetic materials have recently been proposed. In 2D ferromagnetic Janus materials, we find that ScClI is a second-order topological insulator. Using the tight-binding approximation, we develop a multiorbital model that adequately describes the high-order topological states of ScClI. Further, we give the complete high-order topological phase diagram of ScClI, based on the external field modulation of the magnetovalley coupling and energy levels. The 2D ScClI has a pronounced valley polarization. This type of magnetovalleytronics results in different insulating phases to exhibit completely different anomalous Nernst conductance. Through valleytronics, we establish a link between the topological insulator and the valley Nernst effect, thus constructing an anomalous valley Nernst conductance map that corresponds to the topological phase diagram. We utilize the characteristics of valley electronics to link higher-order topological materials with the anomalous Nernst effect, which has potential implications for high-order topological insulators and valley electronics.

Figure 1

(a) Top and side views of ScClI. (b) In the out-of-plane ferromagnetic state, a fat band of projected orbitals is present. (c) The projected density of states (PDOS) of the system. (d) Spin-polarized energy bands near the Fermi level. Red and blue lines indicate spin-up and spin-down states.

Figure 2

(a) The complete phase diagram as a function of the magnetic moment direction ${\theta }_z$ and the energy level difference ${\mathrm{\Delta }}_e$. (b) Energy spectrum of triangular armchair nanosheets, and the total wave-function distribution in real space for the corner states within the band gap of the energy spectrum. (c) The quantum dot energy spectrum of QAVHI is displayed, with blue dots marking the boundary states near the Fermi energy. The inset shows the wave-function distribution. (d) The energy spectrum of NI, illustrating the wave-function distribution of bulk states.

Figure 3

(a) Projected energy band evolution in the $1.5\%\text{–}3\%$ strain range. WCC and surface band of QAVHI at $2.2\%$ strain are shown in (b) and (c), respectively. (d) Plot of quantum anomalous Hall conductance vs energy.

Figure 4

(a)–(c) present the Berry curvature, ANC, and ANV of three insulation correlations: SOTI, QAVHI, and NI, respectively, concerning the $K/K^{\prime }$ valley. In (d), ANV is plotted as a function of energy and gap. Meanwhile, (e) illustrates the relationship between ANV, energy, and the direction of the magnetic moment ${\theta }_z$. Phase boundaries are marked by gray dashed lines.