Understanding Antiferromagnetic Coupling in Lead-Free Halide Double Perovskite Semiconductors

Solution-processable semiconductors with antiferromagnetic (AFM) order are attractive for future spintronics and information storage technology. Halide perovskites containing magnetic ions have emerged as multifunctional materials, demonstrating a cross-link between structural, optical, electrical, and magnetic properties. However, stable optoelectronic halide perovskites that are antiferromagnetic remain sparse, and the critical design rules to optimize magnetic coupling still must be developed. Here, we combine the complementary magnetometry and electron-spin-resonance experiments, together with first-principles calculations to study the antiferromagnetic coupling in stable Cs2(Ag:Na)FeCl6 bulk semiconductor alloys grown by the hydrothermal method. We show the importance of nonmagnetic monovalence ions at the BI site (Na/Ag) in facilitating the superexchange interaction via orbital hybridization, offering the tunability of the Curie–Weiss parameters between −27 and −210 K, with a potential to promote magnetic frustration via alloying the nonmagnetic BI site (Ag:Na ratio). Combining our experimental evidence with first-principles calculations, we draw a cohesive picture of the material design for B-site-ordered antiferromagnetic halide double perovskites.

In general, magnetic Hamiltonian can be assumed as Here, owing to weak relativistic effects, the non-linear Dzyaloshinskii -Moriya interactions ( ) and   Kitaev interactions ( ) are expected not to contribute significantly compared to the dominant   Heisenberg exchange interactions term and anisotropy constants (K x , K y , K z ). and represent the       magnitudes of spins of atoms i and j.
Therefore, the magnetic Hamiltonian reduces to a simple Heisenberg Hamiltonian We then extracted the exchange interactions between any two sites i and j given by Here, refers to the energy of the system when the spins at sites i and j are aligned antiparallel to  ↑↓ each other w.r.t. a reference configuration, which is assumed to be AFM-1 for our calculation.For simplicity, the exchange interaction between an Fe atom and its i th nearest neighbour can be represented by .The anisotropy constants were calculated using the difference in calculated energy   between configurations with spins aligned along different crystal directions within the same magnetic structure.Since the crystal is symmetric along x and y axes, change in direction of the magnetization vector within the x -y plane does not change the total energy, while that in y -z or x -z plane does.
Finally, the magnetic transition temperatures were obtained by performing a temperature sweep starting from a temperature of 600 K and going down to 2 K in order to observe the magnetic phase transition.
At each temperature step, the system was equilibrated through 10 5 steps before sampling the configuration space (C.S.).Since we deal with low temperatures, the Heat Bath Monte Carlo algorithm was used to effectively sample the C.S.
The pathway of the exchange interaction of the spin are shown in Fig. S2.In this work, the atomic spins are set to unit magnitude and should be scaled appropriately while making a comparison to works where this is not the case.
The 0K DFT calculations also capture the tetragonal distortion in case of AFM -1 magnetic configuration of Cs 2 AgFeCl 6 .The same is not observed for the ferromagnetic configuration on the Fe sublattice, indicating that the distortion is likely owed to presence of spin -lattice coupling, which leads to the stabilization of low temperature AFM-1 phase.The fact that this distortion and resulting anisotropy is nearly negligible in case of the stable AFM phase for Cs 2 NaFeCl 6 , also highlights the difference in the strength of magnetic exchange interactions owing to the choice of mediating monovalent cation.Details on chemical bonding were obtained from a crystal orbital Hamiltonian population (COHP) 12 analysis as implemented in the LOBSTER package [13][14][15] .

The antiferromagnetic resonance and the magnetic anisotropy
We used an antiferromagnetic resonance (AFMR) technique to examine antiferromagnetic properties (e.g.magnon modes and magnetic anisotropy).According to the measured Curie-Weiss parameters and the DFT predictions, we expect the AFMR frequency of all samples to be beyond a standard X-to W-band (10-100 GHz) EPR spectrometer.Therefore, we resorted to high-frequency/highmagnetic field, far-infrared absorption experiments.Here, 360-750 GHz far-infrared radiation was applied to the powder samples of all crystals.The transmission was detected via a He-cooled barometer with an applied magnetic field in the range of 10-30 Tesla (the range where the free electron spin resonance condition is expected).The sample temperature was kept at 1.7K, where all three samples were in the AFM state.
The AFMR results are shown in Figure S3A-C.Under these conditions, the AFMR frequency in all three samples scales linearly with the magnetic field strength (see Figure S3D).This implies that, within this field range, the Zeeman interaction dominates over the AFM exchange interaction.The linear field dependence yields a g-factor of 2.020.01 in all samples, in agreement with our EPR results of the Fe 3+ on the B III site.The observed dominance of the Zeeman interaction suggests that magnon frequencies in all samples are below our instrumental limit (<360 GHz).While it is only possible to accurately determine the magnetic anisotropy by resolving the magnon modes and the critical spin-flop field, the AFMR linewidth at low resonant frequencies could be used to examine the magnetic anisotropy present in the samples.At a low AFM frequency, the contribution of the antiferromagnetic spinexchange term emerges.In a powder system, low-frequency AFMR modes at fixed resonant fields spread over a broad range of frequency due to the AFM anisotropy, resulting in linewidth broadening.
The low-frequency linewidth broadening is seen in Cs 2 AgFeCl 6, indicating that the anisotropy term plays a role in the field range of 14-16T.This strengthens our DFT prediction for magnetic anisotropy in Cs 2 AgFeCl 6.For both Cs 2 NaFeCl 6 and Cs 2 Ag 0.6 Na 0.4 FeCl 6, the DFT calculation predicts much less magnetic anisotropy and, therefore, no observable effect of magnetic anisotropy over the range of AFMR frequency of 400-750 GHz, in agreement with our experimental observation.

Figure S2.
The super-exchange interaction pathway of J 1 and J 2 inside a Cs 2 (Ag:Na)FeCl 6 crystal.

Figure S4 .
Figure S4.The density of states for the Cs 2 (Ag 0.6 Na 0.4 )FeCl 6 alloy modelled via a special quasi-random structure.A supercell with a random distribution of Ag and Na atoms on the B I site is considered.Owing to a lower concentration, lesser number of Ag(d) states near the valence band maxima with respect to Cl(p) states are observed in comparison to pristine Cs 2 AgFeCl 6 double perovskite.