Discovery of Intrinsic Quantum Anomalous Hall Effect in Organic Mn-DCA Lattice
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Supplementary information:
The stable magnetic configurations and Curie temperature in Mn-DCA lattice
We perform calculations on the electron spin polarization of the 2D Mn-DCA lattice. The degeneracy of the two spin channels is splitting and thus, the local magnetic moments mainly from half-filled Mn electrons. To visualize the spatial distribution of spins in Mn-DCA lattice, we plot the spin-polarized electron density, ∆ρ, which was calculated from the difference between the electron density of two spin channels, ∆ρ =ρ↑ − ρ↓, in Fig. S1(a). In this case, the origination of 4.0 μB local magnetic moments can also be understood in terms of the occupations of 3d electrons of Mn3+ ions in Mn-DCA lattice.
Fig. S1 Plot of the spin-polarized electron density isosurfaces of a Mn-DCA lattice.
To further estimate the Curie temperature, we employ an Ising model to calculate the nearest neighbor exchange parameter J0 of the local magnetic moment. The Hamiltonian of the Ising model can be expressed as
where the and are the local magnetic moment at i and j sites and the J0 is positive and denotes the strength of the FM interaction. Herein, for a triangular lattice, the value of J0 can be evaluated from the exchange energy Eex using the equation:
whereand factor of 1/3 is adopted because there are altogether three magnetic coupling interactions in the Mn-DCA lattice. So, the J0 is estimated to be 8.2 meV. Furthermore, we apply Monte Carlo (MC) simulations within the Ising Hamiltonian to investigate the variation of magnetic moment and magnetic susceptibility with the variation of temperature. The magnetic susceptibility can be deduced from the equation:
where L is two dimensional triangular lattice of size and m is the magnetic moment at certain temperature (T). It can be observed that the magnetic moment begins to drop dramatically at 220 K, and the paramagnetic state is achieved at a temperature of about 253 K, as shown in Fig. S2. To better understand the FM-Paramagnetic (PM) transition, we further calculate the heat capacity (CV) using the equation
where E is the corresponding energy of each magnetic moment. The calculated CV as a function of temperature is shown in the inset of Fig. S2, in which the FM-PM with a second-order phase transition at 253 K can be observed, demonstrating that it will be stabilized near the room-temperature.
Fig. S2 Monte Carlo simulations of the average magnetic moment and magnetic susceptibility in per unit cell. The insert shows the calculated heat capacity (CV) of the 2D Mn-DCA lattice.