Tight-binding theory of spin-spin interactions, Curie temperatures, and quantum Hall effects in topological (Hg,Cr)Te in comparison to non-topological (Zn,Cr)Te, and (Ga,Mn)N

TL;DR

This paper extends theoretical models of magnetic interactions in topological and non-topological semiconductors, analyzing their effects on Curie temperatures and quantum Hall phenomena, with implications for spintronic applications.

Contribution

It introduces extended exchange interaction calculations including beyond superexchange terms and applies them to topological HgTe and other semiconductors, linking microscopic interactions to macroscopic magnetic and quantum Hall effects.

Findings

Calculated exchange integrals and Curie temperatures match experimental data.

Identified conditions for spin-glass phase in HgCrTe.

Predicted stabilization of quantum spin Hall effect with magnetic competition.

Abstract

Earlier theoretical results on $p$-$d$ and $d$-$d$ exchange interactions for zinc-blende semiconductors with $\mathrm{Cr}^{2{+}}$ and $\mathrm{Mn}^{3{+}}$ ions are revisited and extended by including contributions beyond the dominating ferromagnetic (FM) superexchange term [i.e., the interband Bloembergen-Rowland-Van Vleck contribution and antiferromagnetic (AFM) two-electron term], and applied to topological Cr-doped HgTe and non-topological (Zn,Cr)Te and (Ga,Mn)N in zinc-blende and wurtzite crystallographic structures. From the obtained values of the $d$-$d$ exchange integrals $J_{ij}$, and by combining the Monte-Carlo simulations with the percolation theory for randomly distributed magnetic ions, we determine magnitudes of Curie temperatures $T_{\text{C}}(x)$ for $\mathrm{Zn}_{1-x}\mathrm{Cr}_x\mathrm{Te}$ and $\mathrm{Ga}_{1-x}\mathrm{Mn}_x\mathrm{N}$ and compare to available experimental data. Furthermore, we find that competition between FM and AFM $d$-$d$ interactions can lead to a spin-glass phase in the case of $\mathrm{Hg}_{1-x}\mathrm{Cr}_x\mathrm{Te}$. This competition, along with a relatively large magnitude of the AF $p$-$d$ exchange energy $N_0\beta$ can stabilize the quantum spin Hall effect, but may require the application of tilted magnetic field to observe the quantum anomalous Hall effect in HgTe quantum wells doped with Cr, as confirmed by the Chern number determination.