# Superexchange dominates in magnetic topological insulators

Cezary Śliwa<EMAIL_ADDRESS>Institute of Physics, Polish Academy of
Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland  Carmine Autieri ID
International Research Centre MagTop, Institute of Physics, Polish Academy of
Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland  Consiglio Nazionale
delle Ricerche CNR-SPIN, UOS Salerno, I-84084 Fisciano (Salerno), Italy  Jacek
A. Majewski  Institute of Theoretical Physics, Faculty of Physics, University
of Warsaw, ul. Pasteura 5, PL-02093 Warsaw, Poland  Tomasz Dietl ID
<EMAIL_ADDRESS>International Research Centre MagTop, Institute of
Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw,
Poland  WPI-Advanced Institute for Materials Research, Tohoku University,
Sendai 980-8577, Japan

###### Abstract

It has been suggested that the enlarged spin susceptibility in topological
insulators, described by interband Van Vleck’s formalism, accounts for the
ferromagnetism of bismuth-antimony topological chalcogenides doped with
transition metal impurities. In contrast, earlier studies of HgTe and related
topological systems pointed out that the interband analog of the Ruderman-
Kittel-Kasuya-Yosida interaction (the Bloembergen-Rowland mechanism) leads to
antiferromagnetic coupling between pairs of localized spins. Here, we
critically revisit these two approaches, show their shortcomings, and
elucidate why the magnitude of the interband contribution is small even in
topological systems. From the proposed theoretical approach and our
computational studies of magnetism in Mn-doped HgTe and CdTe, we conclude that
in the absence of band carriers, the superexchange dominates, and its sign
depends on the coordination and charge state of magnetic impurities rather
than on the topological class of the host material.

In the traditional approach to localized magnetism in solids, one considers
pairwise exchange interactions between spins $J_{ij}$ comprising the Ruderman-
Kittel-Kasuya-Yosida (RKKY) coupling brought about by band carriers and the
Anderson-Goodenough-Kanamori superexchange mediated mainly by anion orbitals
White (2007). However, it has been demonstrated that in the case of p-type
dilute magnetic semiconductors (DMSs), the Zener model Zener (1951) is
remarkably versatile, in which the local magnetization
$\mathbf{M}(\mathbf{r})$ plays a role of a continuous order parameter. This
approach has allowed understanding the physics of bound magnetic polarons
Dietl and Spałek (1982); Dietl (2015) and ferromagnetism of p-type DMSs Dietl
_et al._ (2000), and subsequently describing quantitatively a wealth of
micromagnetic properties and spintronic functionalities of (Ga,Mn)As and
related ferromagnets Dietl and Ohno (2014); Jungwirth  _et al._ (2014).
Notably, the equivalence between the RKKY and Zener models was established
within the mean-field approximation (MFA) Dietl  _et al._ (1997).

Ferromagnetic topological insulators Ke  _et al._ (2018); Tokura  _et al._
(2019), such as (Bi,Sb,Cr)2Te3, have made possible the experimental
realization of the quantum anomalous Hall effect Chang  _et al._ (2013), the
axion insulator Xiao  _et al._ (2018), efficient magnetization reversal by
spin currents Fan  _et al._ (2014), and the much disputed chiral Majorana
fermions He  _et al._ (2017); Kayyalha  _et al._ (2020). Interestingly, the
appearance of ferromagnetism in these systems is also attributed to their
topological character, as the inverted band structure enhances the interband
spin susceptibility leading to carrier-independent spin-spin coupling Yu  _et
al._ (2010), referred to as the Van Vleck magnetism Ke  _et al._ (2018);
Tokura  _et al._ (2019); Yu  _et al._ (2010). That appears surprising,
however, as early studies of spin-spin coupling mediated by an interband
analog of the RKKY interaction (the Bloembergen-Rowland (BR) mechanism
Bloembergen and Rowland (1955)) found predominately antiferromagnetism in
topological Mn-doped zero-gap topological HgTe Bastard and Lewiner (1979);
Lewiner and Bastard (1980); Lee (1988).

In this Letter, we resolve this puzzle by demonstrating that the mean-field
Zener-Van Vleck model fails in the case of magnetism associated with interband
bulk excitations in insulators. Furthermore, by making use of the recent
progress in the theory of the indirect exchange interaction Śliwa and Dietl
(2018) and in the quantitative description of exchange splitting of bands in
the whole Brillouin zone (BZ) Autieri  _et al._ (2021), we determine various
contributions to spin-spin coupling in non-topological CdTe and topological
HgTe doped with Mn ions (see Supplemental Material at the manuscript end for
an introduction to the topic). We find that the superexchange dominates not
only in (Cd,Mn)Te Larson  _et al._ (1988); Savoyant  _et al._ (2014), but also
in topological (Hg,Mn)Te. We also show that the conclusion about the
predominant role of the superexchange substantiates the experimental results
on (Cd,Hg,Mn)Te Spałek  _et al._ (1986); Lewicki  _et al._ (1988) and explains
hitherto challenging chemical trends in the magnetic properties of V, Cr, Mn,
and Fe-doped tetradymite topological insulators observed experimentally Ke
_et al._ (2018); Tokura  _et al._ (2019); Satake  _et al._ (2020) and found in
ab initio studies Vergniory  _et al._ (2014); Kim  _et al._ (2017); Peixoto
_et al._ (2020).

RKKY-BR vs. Zener-Van Vleck models. For concreteness, we consider $xN$
randomly distributed Mn spins $S=5/2$ in zero-gap Hg1-xMnxTe in which both the
conduction and valence bands are of $\Gamma_{8}$ symmetry at the BZ center. In
the high-temperature expansion of the partition function for the pairwise
interactions Spałek  _et al._ (1986), the contribution of the RKKY-BR term to
the Curie-Weiss temperature (equal to spin ordering temperature $T_{\text{c}}$
within MFA) assumes the form Bastard and Lewiner (1979); Lee (1988); Dietl
(1994),

$\Theta_{\text{CW}}=\frac{xS(S+1)}{3N{\mathcal{V}}}\sum_{i\neq
j,\mathbf{q}}\exp[\mbox{i}\mathbf{q}\cdot(\mathbf{R}_{i}-\mathbf{R}_{j})]\sum_{\mathbf{k},n,n^{\prime},\sigma,\sigma^{\prime}}\frac{2|\langle
u_{n,\mathbf{k},\sigma}|\beta
s_{z}|u_{n^{\prime},\mathbf{k+q},\sigma^{\prime}}\rangle|^{2}}{{\mathcal{V}}(E_{n^{\prime},\mathbf{k+q}}-E_{n,\mathbf{k}})}f_{n,\mathbf{k}}(1-f_{n^{\prime},\mathbf{k+q}}),$
(1)

where the Boltzmann constant $k_{\text{B}}=1$; ${\cal{V}}$ is the crystal
volume, and $\mathbf{k},\mathbf{k+q}\in\text{BZ}$. We see that if the $p$-$d$
exchange integral $\beta$ were $\mathbf{k}$-independent, the summation over
$\mathbf{k}$ would provide the spin susceptibility $\tilde{\chi}(\mathbf{q})$
of the $\Gamma_{8}$ bands $n$ and $n^{\prime}$, as defined in Refs. Dietl  _et
al._ , 2001 and Ferrand  _et al._ , 2001. Furthermore, if the contribution of
the self-interaction energy were small compared to interaction energies for
$i\neq j$, the term $i=j$ could be included in Eq. 1, transferring the sum
over the cation positions $\mathbf{R}_{i}$ and $\mathbf{R}_{j}$ into the
structure factor that is non-zero for $\mathbf{q}=0$ only. This is the case of
the long-range RKKY coupling, for which the sum over $i,j$ can be approximated
by $N^{2}\beta^{2}\tilde{\chi}(0)$, as presumed within the Zener-Van Vleck
model Dietl  _et al._ (2001); Ferrand  _et al._ (2001); Yu  _et al._ (2010).
However, in the case of the BR mechanism, the decay of the interaction with
the inter-spin distance is faster Bloembergen and Rowland (1955); Bastard and
Lewiner (1979); Lewiner and Bastard (1980); Lee (1988) and, therefore,
$\tilde{\chi}(q)$ beyond $q=0$ determines the sign and magnitude of
$\Theta_{\text{CW}}$ and $T_{\text{C}}$. In conclusion, atomistic computations
of pair exchange energies $J_{ij}$ are necessary in order to meaningfully
evaluate the role of the interband contribution Ginter  _et al._ (1979).

Theoretical methodology. We consider exchange interactions between Anderson
magnetic impurities occupying cation substitutional positions in considered
semiconductor compounds, whose band structures are described within the
empirical tight-binding approximation taking into account spin-orbit
interactions. This approach Blinowski and Kacman (1995); Blinowski  _et al._
(1996); Simserides  _et al._ (2014) was successfully applied to elucidate the
nature of ferromagnetism in (Ga,Mn)N Sawicki  _et al._ (2012); Stefanowicz
_et al._ (2013) and has recently been generalized by us to simultaneously take
into account various contributions to the spin pair exchange energy Śliwa and
Dietl (2018), including the BR interband term. Assuming time-reversal symmetry
(no spontaneous magnetization) and within the fourth order perturbation theory
in the $p$-$d$ hybridization energy $V_{\text{hyb}}$ between band states
$E_{\mathbf{k},n}$ and Mn $d$ orbitals residing at $E_{d}=E(d^{5})-E(d^{4})$
and $E_{d}+U=E(d^{6})-E(d^{5})$, the spin Hamiltonian is,

$\hat{\cal{H}}_{\mathrm{eff}}^{(4)}=-\sum_{i\neq
j}J^{(4)}_{ij,\alpha\beta}\hat{S}_{i,\alpha}\hat{S}_{j,\beta},$ (2)

where the tensor of exchange integrals (parameters) for spins at sites $(i,j)$
can be written as a double integral over the BZ [$k\equiv(\mathbf{k},n)$]:

$J^{(4)}_{ij,\alpha\beta}=-\frac{1}{2(2S)^{2}}\sum_{m,m^{\prime}}\sum_{k,k^{\prime}}A^{(4)}_{kk^{\prime}}W_{i\alpha,k^{\prime}k,m}W_{j\beta,kk^{\prime},m^{\prime}},$
(3)

where $m$ labels the $d$ orbitals and

$\displaystyle W_{i\alpha,k^{\prime}k,m}=$ (4) $\displaystyle=$
$\displaystyle\sum_{a,b=\uparrow,\downarrow}\left<k^{\prime}\middle|V_{\mathrm{hyb}}^{\dagger}\middle|d_{i}ma\right>\left<a\middle|\sigma_{\alpha}\middle|b\right>\left<d_{i}mb\middle|V_{\mathrm{hyb}}\middle|k\right>,$

where $a$ and $b$ are spin directions and $\sigma_{\alpha}$ the Pauli
matrices. For an insulator in a zero-temperature approximation and using the
notation,

$\displaystyle w_{k}=\frac{1}{E_{d}+U-E_{k}};\quad
w^{\prime}_{k}=\frac{1}{E_{d}-E_{k}},$ (5) $\displaystyle
w_{k^{\prime}}=\frac{1}{E_{d}+U-E_{k^{\prime}}};\quad
w^{\prime}_{k^{\prime}}=\frac{1}{E_{d}-E_{k^{\prime}}},$ (6)

one can write $A^{(4)}_{kk^{\prime}}$ in terms of the Heaviside step function
$\Theta$ as,

$\displaystyle A^{(4)}_{kk^{\prime}}\approx$
$\displaystyle\Theta(E_{F}-E_{k})\Theta(E_{F}-E_{k^{\prime}})w_{k}w_{k^{\prime}}\left(w_{k}+w_{k^{\prime}}\right)+\Theta(E_{F}-E_{k})\Theta(E_{k^{\prime}}-E_{F})\frac{\left(w_{k}-w^{\prime}_{k^{\prime}}\right)^{2}}{E_{k}-E_{k^{\prime}}}$
(7)
$\displaystyle{}+\Theta(E_{k}-E_{F})\Theta(E_{F}-E_{k^{\prime}})\frac{\left(w_{k^{\prime}}-w^{\prime}_{k}\right)^{2}}{E_{k^{\prime}}-E_{k}}-\Theta(E_{k}-E_{F})\Theta(E_{k^{\prime}}-E_{F})w^{\prime}_{k}w^{\prime}_{k^{\prime}}\left(w^{\prime}_{k}+w^{\prime}_{k^{\prime}}\right)$
$\displaystyle+\frac{2}{U}\Biggl{[}\Theta(E_{F}-E_{k})w_{k}+\Theta(E_{k}-E_{F})w^{\prime}_{k}\Biggr{]}\Biggl{[}\Theta(E_{F}-E_{k^{\prime}})w_{k^{\prime}}+\Theta(E_{k^{\prime}}-E_{F})w^{\prime}_{k^{\prime}}\Biggr{]},$

where we assume that the Fermi energy $E_{F}$ lies in the range
$E_{d}<E_{F}<E_{d}+U$. Average values of the exchange integrals and
$\Theta_{0}=\Theta_{\text{CW}}/x$ are obtained by tracing the tensors,

$J^{(4)}_{ij}=\frac{1}{3}\sum_{\alpha}J^{(4)}_{ij,\alpha\alpha};\,\,\,\Theta_{0}=\frac{2}{3}S(S+1)\sum_{i\geq
1}z_{i}J^{(4)}_{0i},$ (8)

where $z_{i}$ is a number of cation sites in the consecutive coordination
spheres $i\geq 1$.

Figure 1: Convergence of the interband $he$ term for Mn pairs in HgTe with
respect to the number $m$ of $\vartheta$-points in each space direction in the
trapezoids quadrature: squares — the sum of exchange integrals including
$J_{0}$ calculated for the $4\times 4\times 4$ supercell as a single BZ
integral (abscissa provides $m/4$); triangles — the value of $J_{0}$ to be
subtracted in order to obtain $\Theta_{\text{CW}}$ (abscissa provides $m$).

We represent $A^{(4)}_{kk^{\prime}}$ as a sum of three contributions Larson
_et al._ (1988); Kacman (2001): the superexchange (or $hh$) term includes
contributions proportional to
$\Theta(E_{F}-E_{k})\Theta(E_{F}-E_{k^{\prime}})$, the two-electron (or $ee$)
term includes those proportional to
$\Theta(E_{k}-E_{F})\Theta(E_{k^{\prime}}-E_{F})$, and the electron-hole
($he$) includes those proportional to
$\Theta(E_{F}-E_{k})\Theta(E_{k^{\prime}}-E_{F})$ or
$\Theta(E_{k}-E_{F})\Theta(E_{F}-E_{k^{\prime}})$. Such a decomposition leads
to the analogous decomposition of $J^{(4)}_{ij}$ and
$\Theta_{0}=\Theta_{hh}+\Theta_{ee}+\Theta_{he}$.

We use a 16-orbital $sp^{3}$ tight-binding model of the band structure
together with the parameter values obtained recently by us employing a
modified GGA + U ab initio approach with $U_{\text{Mn}}=5$ eV. These
parameters are presented in Tables II and III of Ref. Autieri  _et al._ ,
2021. In particular, $E_{d}$ in our Eqs. 5-7 is the mean value of
$E_{eg\uparrow}$ and $E_{t2g\uparrow}$, whereas $E_{d}+U$ is the mean values
of $E_{eg\downarrow}$ and $E_{t2g\downarrow}$. Finally, matrix elements of
$V_{\text{hyb}}$ are spin-averaged values of $V_{sd}\sigma$, $V_{pd}\sigma$,
and $V_{pd}\pi$ Autieri  _et al._ (2021). Extensive magnetooptical data
collected for Cd1-xMnxTe and Hg1-xMnxTe at the $\Gamma$ and $L$ points of the
BZ served to benchmark the model Autieri  _et al._ (2021). Importantly, our
tight-binding model reconfirms for these compounds stronger hybridization of
$t_{2g}$ orbitals with the band states, compared to the $e_{g}$ case, which is
crucial for the sign and magnitude of the interaction between localized spins.

In topological materials, the most interesting is the $he$ term. It appears
whenever transitions between the fully occupied valence bands and the empty
conduction bands are symmetry-allowed (the BR mechanism), or when there is a
non-zero density of states at the Fermi level (the RKKY mechanism). This term
features an energetic denominator $E_{k}-E_{k^{\prime}}$. In insulators, the
latter is guaranteed to be non-zero by the Heaviside-$\Theta$ prefactors (it
is understood that each term vanishes whenever the zero Heaviside–$\Theta$
prefactor does, even despite singular denominators). However, in a semimetal,
the denominator is singular at the Fermi level (i.e., when either $E_{k}\to
E_{F}^{+}$ and $E_{k^{\prime}}\to E_{F}^{-}$, or vice versa). In undoped (or
isoelectronically doped) HgTe, this happens at the $\Gamma$ point of the BZ.
For this reason, it has been essential to elaborate a special integration
method. The Supplemental Material at the manuscript end presents issues
associated with $k$ and $k^{\prime}$ integrations in simple models Bastard and
Lewiner (1979); Lewiner and Bastard (1980); Lee (1988) and a comparison of the
second-order perturbation theory in the $p$-$d$ exchange integral compared to
the fourth-order perturbation theory in the hybridization matrix element
$V_{\text{hyb}}$ employed here.

Brillouin-zone integration. Although our goal is to find the exchange
integrals $J^{(4)}_{ij}$ in the limit of an infinitely large system, it is
typical in numerical calculations to replace the BZ integration by a summation
over a discrete set of $\mathbf{k}$-points in the BZ. As pointed out in Ref.
Froyen, 1989, such discrete $\mathbf{k}$-point mesh may be defined through the
introduction of the superlattice vectors $\\{\mathbf{g}_{i}\\}$, being the
linear combinations of primitive crystal translations $\\{\mathbf{a}_{i}\\}$,

$\mathbf{g}_{i}=\sum_{j=1}^{3}\mathbf{a}_{j}M_{ji}.$ (9)

In cubic systems the three $\mathbf{g}_{i}$ can be taken as vectors along the
three Cartesian axes with length $La$. This defines the equidistant
$\mathbf{k}$-point mesh
$\mathbf{\kappa}_{\\{m\\}}=\frac{2\pi}{La}(m_{x},m_{y},m_{z})$ of $L^{3}$ grid
points. In addition, the grid points can be shifted.

However, the expressions for the $J_{ij}$ tensor components involve double
integration over the BZ. Here, we present an efficient method to deal with the
double BZ integration facilitated by the specifics of the integrand. In
particular, this method allows for accurate treatment in the $he$ contribution
for $J_{ij}$ in the case of zero-gap systems. Indeed, the product
$W_{i\alpha,k^{\prime}k,m}W_{j\beta,kk^{\prime},m^{\prime}}$ includes a phase
factor $\exp[\mbox{i}(\kappa-\kappa^{\prime})(R_{i}-R_{j})]$, and the
summation over the images of $R_{i}$,
$R_{i}+La_{\text{lat}}(n_{x},n_{y},n_{z})$ yields (by the principle of the
Poisson summation) a set of Dirac deltas at
$\kappa-\kappa^{\prime}=\frac{2\pi}{La_{\text{lat}}}(m_{x},m_{y},m_{z})$,

$\displaystyle\sum_{n_{x},n_{y},n_{z}}\exp\left\\{\mbox{i}(\kappa-\kappa^{\prime})[R_{i}-R_{j}+La_{\text{lat}}(n_{x},n_{y},n_{z})]\right\\}=$
(10) $\displaystyle=$
$\displaystyle\left(\frac{2\pi}{La_{\text{lat}}}\right)^{3}\exp\left[\mbox{i}(\kappa-\kappa^{\prime})(R_{i}-R_{j})\right]\sum_{m_{x},m_{y},m_{z}}\delta\left(\kappa-\kappa^{\prime}+\frac{2\pi}{La_{\text{lat}}}(m_{x},m_{y},m_{z})\right),\quad$

where the delta suppresses only one integration. To handle this issue, we
first sum over a shifted grid of equidistantly spaced $k$-points,

$\kappa=\frac{2\pi}{La_{\text{lat}}}(m_{x}+\frac{\vartheta_{x}}{2\pi},m_{y}+\frac{\vartheta_{y}}{2\pi},m_{z}+\frac{\vartheta_{z}}{2\pi});\,\,\,\,\kappa^{\prime}=\frac{2\pi}{La_{\text{lat}}}(m_{x}^{\prime}+\frac{\vartheta_{x}}{2\pi},m_{y}^{\prime}+\frac{\vartheta_{y}}{2\pi},m_{z}^{\prime}+\frac{\vartheta_{z}}{2\pi}),$
(11)

then integrate over the common shift
$(\vartheta_{x},\vartheta_{y},\vartheta_{z})\in[0,2\pi)^{3}\equiv T^{3}$
($T^{3}$ stands for the three-dimensional torus).

In order to calculate $J_{0}+\sum_{i\geq 1}z_{i}J_{i}$ we rewrite (10) with
$L=1$ as

$\sum_{j}\exp\left[\mbox{i}(\kappa-\kappa^{\prime})(R_{i}-R_{j})\right]=\left(\frac{2\pi}{a_{\text{lat}}}\right)^{3}\sum_{m_{x},m_{y},m_{z}}\delta\left(\kappa-\kappa^{\prime}+\frac{2\pi}{a_{\text{lat}}}(m_{x},m_{y},m_{z})\right).$
(12)

Since $2\pi/a_{\text{lat}}$ is the lattice constant of the reciprocal lattice,
only the term with $m_{x}=m_{y}=m_{z}=0$ remains in the last sum, and the
Dirac delta suppresses one integration. Therefore, $J_{0}+\sum_{i\geq
1}z_{i}J_{i}$ and, thus, $\tilde{\chi}(0)$ can be computed as a single
integral over the BZ.

In insulators, the integration over $\vartheta$’s is approximated by a sum
over an equally spaced grid (the trapezoids method). The number of required
$\vartheta$-points in each Cartesian direction gets smaller as the larger
supercells are considered. Ultimately, just one $\vartheta$-point is
sufficient; it can be chosen as, e.g.,
$\vartheta_{x}=\vartheta_{y}=\vartheta_{z}=0$ or
$\vartheta_{x}=\vartheta_{y}=\vartheta_{z}=\pi$, in correspondence with the
boundary conditions imposed on the electronic wavefunctions. In contrast, in
semimetals, if the boundary conditions dictate $\vartheta=0$, finite summation
is not appropriate because the denominator $E_{k}-E_{k^{\prime}}$ in Eq. 7
renders the quantity undefined. Therefore, a special set of $\vartheta$-points
has been chosen here, which is equivalent to the transformation
$\vartheta_{i}^{\prime}\mapsto\vartheta_{i}=\vartheta_{i}^{\prime}-\sin\vartheta_{i}^{\prime}$
under the BZ integral. This transformation, besides possessing analytic
properties, preserves periodicity while
$d\vartheta_{i}/d\vartheta_{i}^{\prime}=0$ at $\vartheta_{i}^{\prime}=0$. As a
result, the divergence of the integral at the origin cancels with the zero of
the Jacobian of the transformation and the integral can be computed with the
trapezoids method. Indeed, the singularity of the integrand at $\vartheta=0$
is integrable, as long as the dimensionality is sufficient and the band
structure is well-behaved ($k$-linear terms, $k^{3}$-terms, and the anisotropy
may play a role here). The RKKY (i.e., intraband) term is omitted in this
discussion, as the density of states vanishes if the Fermi energy $E_{F}\to
0$.

The computations have been performed with an efficient algorithm based on the
fast Fourier transform on a $16\times 16\times 16$ supercell (16384 cation
lattice sites) that also determines the grid density of $k$ and $k^{\prime}$
points employing periodic boundary conditions. For Cd1-xMnxTe and Hg1-xMnxTe,
the grids with up to two and eight different $\vartheta$ values have been
employed, respectively. Figure 1 demonstrates that the magnitude of
$\Theta_{he}$ in Hg1-xMnxTe converges with the number of employed $\vartheta$
values. This means that in contrast to the static dielectric function Liu and
Tosatti (1969); Broerman (1970), the spin susceptibility, though enhanced, is
not singular at $q\to 0$ in the symmetry-induced zero-gap semiconductors.
Aitken’s delta-squared process served to accelerate the integration
convergence. We have checked that $\Theta_{0}$ calculated by the single
integral, as outlined in a preceding paragraph, and after subtracting $J_{0}$,
is in excellent numerical agreement (better than 1 K) with the value obtained
by summing $z_{i}J_{i}$, $i\geq 1$, obtained by the double BZ integration.
Actually, the coupling to the nearest neighbors contributes over 50% to the
value of $\Theta_{0}$.

over 50

Table 1: Consecutive nearest-neighbor exchange energies
$-(J_{i}=J_{i,hh}+J_{i,ee}+J_{i,he})$, $i=1,2,3,4$ (from the fourth-order
perturbation theory) and Curie-Weiss parameter $-\Theta_{0}$ in Kelvins
compared to experimental results. Contributions from the superexchange ($hh$),
electron-electron ($ee$), and interband ($he$) terms to $\Theta_{0}$ are also
shown.

| Cd1-xMnxTe | Hg1-xMnxTe
---|---|---
| theory | expl | theory | expl
$-J_{1}$ | $9.77$ |  $6.3\pm 0.3$ Larson  _et al._ (1986) | $6.46$ |  $5.1\pm 0.5$ Galazka  _et al._ (1988)
|  |  $6.15\pm 0.05$ Cherbunin  _et al._ (2020) |  |  $4.3\pm 0.5$ Lascaray  _et al._ (1989)
$-J_{2}$ | $0.810$ |  $1.9\pm 1.1$ Larson  _et al._ (1986) | $0.842$ |
|  |  $1.80\pm 0.05$ Cherbunin  _et al._ (2020) |  |
$-J_{3}$ | $0.352$ |  $0.4\pm 0.3$ Larson  _et al._ (1986) | $0.394$ |
|  |  $1.39\pm 0.05$ Cherbunin  _et al._ (2020) |  |
$-J_{4}$ | $0.255$ |  $0.81\pm 0.05$ Cherbunin  _et al._ (2020) | $0.467$ |
$-\Theta_{0}$ | $801$ |  $470\pm 34$ Spałek  _et al._ (1986) | $666$ |  $500\pm 10$Spałek  _et al._ (1986)
|  |  |  |  $660\pm 88$Lewicki  _et al._ (1988)

theory |  |
---|---|---
$-J_{1,he}$ | $-0.396$ | $-\Theta_{hh}$ | $772$ | $-1.583$ | $651$ |
$-J_{2,he}$ | $0.421$ | $-\Theta_{ee}$ | $8.9$ | $0.492$ | $19.0$ |
$-J_{3,he}$ | $0.131$ | $-\Theta_{he}$ | $20.0$ | $0.112$ | $-3.9$ |
$-J_{4,he}$ | $0.061$ |  |  | $0.154$ |  |

Figure 2: Computed total exchange energies $J_{i}$ (a,b) and the interband BR
contribution $J_{i,he}$ (c,d) for Mn pairs (including the self-interaction
values $J_{0}$) vs. Mn-Mn distances $d_{i}$ in the unit of the lattice
parameter $a_{\text{lat}}$ for CdTe (a,c) and HgTe (b,d). Dashed lines
indicate $J_{i}\sim d_{i}^{-n}$ with $n=6.8$ and $4.8$, as found
experimentally for Cd1-xMnxTe Twardowski  _et al._ (1987) and Hg1-xMnxTe
Gała̧zka (1995), respectively.

Discussion of theoretical results vis-à-vis experimental data. As expected
theoretically for the random distribution of magnetic ions, experimental
values of $\Theta_{\text{CW}}$ show linear dependence on $x$ in II-VI DMSs
Spałek  _et al._ (1986); Sawicki  _et al._ (2013). As shown in Table I, our
theory and together with the employed tight-binding parametrization explains
the interaction sign but overestimate by 60% the absolute values of $J_{1}$
and $\Theta_{0}$ in Cd1-xMnxTe and by 30% in the case of Hg1-xMnxTe.

Extensive experimental studies of spin-glass freezing temperature $T_{f}$ in
wide-gap Mn- and Co-based DMSs, including Cd1-xMnxTe, indicate that
$\Theta_{\text{CW}}\gg T_{f}\sim x^{\alpha}$, where $\alpha=2.25\pm 0.1$
Galazka  _et al._ (1980); Twardowski  _et al._ (1987); Gała̧zka (1995);
Stefanowicz  _et al._ (2013). A scaling argument Twardowski  _et al._ (1987);
Smith (1975) then implies $J_{i}\sim d_{i}^{-n}$, where $d_{i}$ is the
distance between spin pairs and $n=3\alpha=6.8\pm 0.3$ Twardowski  _et al._
(1987). Figure 2(a) demonstrates that the dependence of $J_{i}$ on $d_{i}$
obtained here for Cd1-xMnxTe is in agreement with the experimentally
determined power law. However, an exponential decay would describe the
computed data over a wider range of $d_{i}$.

Comparing $J_{i,he}$ values displayed in Figs. 2(c,d) to $J_{l}$ data in Figs.
2(a,b), we find that the BR mechanism dominates at large $d$. It decays
exponentially with $d$ in the wide-gap Cd1-xMnxTe but for topological zero-gap
Hg1-xMnxTe, $J_{i,he}(d)$ shows a power-law dependence, also at large $d$.
This behavior accounts for a relatively weak decay of $T_{f}$ with decreasing
$x$ in Hg1-xMnxTe Mycielski  _et al._ (1984), leading to $n=4.8$ Gała̧zka
(1995). As seen in Fig. 2(b), this value of $n$ is consistent with our
theoretical results, though we have to note that a considerable shift of bands
with $x$ is expected in topological materials, while our computations have
been performed for Mn pairs in HgTe. As shown in Table 1, the relevant
ferromagnetic and antiferromagnetic contributions $J_{i,he},\,i\geq 1$
actually cancel each other in $\Theta_{he}$, making the contribution of the BR
mechanism to $\Theta_{0}$ negligible in both compounds. This explains why no
effect of gap opening on $\Theta_{\text{CW}}$ was found in Hg1-x-yCdyMnxTe
Lewicki  _et al._ (1988). At the same time, it is clear from Fig. 1 that the
inclusion of the self-interaction term $J_{0,he}$, would drastically increase
the magnitude of $\Theta_{he}$.

Conclusions and outlook. Our results demonstrate that the interband BR term
changes the sign from ferromagnetic to antiferromagnetic as a function of Mn
pair distance, with the behavior contradicting the Van Vleck-like approach
that predicts only the ferromagnetic coupling Yu  _et al._ (2010). Such an
alternating sign, reflecting the presence of both ferromagnetic and
antiferromagnetic excitations in $\tilde{\chi}(q)$ Bednik (2020),
significantly reduces the role of the interband contribution making the
superexchange to determine whether a spin-glass or a ferromagnet becomes the
magnetic ground state, the case of Mn2+ in II-VI compounds and Mn3+ in GaN,
respectively.

There are persisting uncertainties concerning the distribution (random vs.
clustering Chang  _et al._ (2014)) and the location of transition metal (TM)
impurities in the tetradymite lattice (substitutional vs. interstitial
positions in the van der Waals gap Růžička  _et al._ (2015)). Nevertheless, a
series of arguments allows extending the conclusion about the dominance of the
superexchange to topological tetradymite chalcogenides doped by substitutional
V, Cr, or Fe ions, whose magnetism has so far been merely attributed to the
Van Vleck mechanism Ke  _et al._ (2018); Tokura  _et al._ (2019). (i) These
impurities appear isoelectronic Ke  _et al._ (2018); Tokura  _et al._ (2019);
Satake  _et al._ (2020), which means that $d$ orbitals remain fully occupied
or empty. Moreover, as in other DMSs, correlations, together with the Jahn-
Teller effect and dilution, enhance the $d$ orbital localization further on.
Accordingly, the double-exchange scenario, put forward when interpreting ab
initio results Vergniory  _et al._ (2014); Peixoto  _et al._ (2020), is not
valid. (ii) Another ab initio study reveals the insensitivity of the spin-spin
coupling energy of the band inversion Kim  _et al._ (2017), the finding
contradicting the Van Vleck model. (iii) Recent studies of x-ray magnetic
circular dichroism and resonant photoelectron spectroscopy demonstrate
similarities of $p$-$d$ hybridization effects in V- or Cr-doped (BixSb1-x)2Te3
Peixoto  _et al._ (2020) and II-VI DMSs Kacman (2001), in particular, stronger
hybridization of $t_{2g}$ TM levels compared to the $e_{g}$ case, which
implies a similar physics of spin-spin coupling as found in tetrahedrally
coordinated DMSs. (iv) As superexchange prevails over the interband Van Vleck
mechanism in the zero-gap case, it should dominate even more strongly in the
gapped topological systems. (v) From the direct computations for tetrahedral
systems carried out here and previously Blinowski  _et al._ (1996); Simserides
_et al._ (2014); Blinowski  _et al._ (1995), supported by experimental data
Stefanowicz  _et al._ (2013); Twardowski  _et al._ (1987); Gała̧zka (1995);
Twardowski (1990); Watanabe  _et al._ (2019), we know that the superexchange
is ferromagnetic for $d^{3}$ and $d^{4}$, whereas it is antiferromagnetic for
$d^{5}$ and $d^{6}$ cases. According to experimental results Ke  _et al._
(2018); Tokura  _et al._ (2019); Satake  _et al._ (2020) and ab initio studies
Vergniory  _et al._ (2014), the same sequence occurs in tetradymite
topological insulators, except for the Mn case, as Mn acts as an acceptor Hor
_et al._ (2010), so that the RKKY interaction accompanies the
antiferromagnetic superexchange, such as (Ga,Mn)As Dietl and Ohno (2014); Sato
_et al._ (2010). Altogether, these arguments indicate that the TM charge state
and coordination, more than a topological class, govern the magnetic
properties of DMSs.

Acknowledgments. The work is supported by the Foundation for Polish Science
through the International Research Agendas program co-financed by the European
Union within the Smart Growth Operational Programme. One of us (JAM)
acknowledges support of the National Science Centre (Poland) under the grant
UMO-218/31/BST3/03758. We acknowledge the access to the computing facilities
of the Interdisciplinary Center of Modeling at the University of Warsaw, Grant
No. G68-12.

Supplemental Material

##  I Overview of various exchange mechanisms

Since the exchange part of the Coulomb energy decays exponentially with the
distance between magnetic ions, the spin-spin interaction in transition-metal
compounds is dominated by indirect coupling via non-magnetic states residing
in-between localized spins White (2007); Kacman (2001). Quite generally, the
presence of $d$ orbitals brought about by transition metals leads to
additional delocalization of valence electrons, i.e., to the lowering of their
quantum mechanical kinetic energy, as illustrated in Fig. S1 Bonanni and Dietl
(2010). The effect is spin dependent as, according to the Hund’s rule and due
to the on-site Hubbard energy $U$, $d$-orbitals are spin-polarized in
transition metals.

Figure S1: Schematic illustration of open $d^{5}$ orbitals at the transition
metal cations and entirely occupied $p$ orbitals at the anion without (a) and
with (b) $p$-$d$ hybridization. A spin-dependent shift of the orbitals
(described by the fourth order perturbation theory), occurring for an
antiferromagnetic arrangement of Mn spins (red arrows) shown in (b), leads to
lowering of the quantum kinetic energy associated with delocalization of
electrons over the $p$ and $d$ orbitals. Note that virtual $p$-$d$ transitions
occurs only for antiparallel orientation of Mn (red arrows) and anion spins
(blue arrows).

Recently, we proposed a formal theory of the indirect spin-dependent
interaction resulting from hybridization between $d$-orbitals of magnetic ions
and $sp$ band states Śliwa and Dietl (2018). That theory, within the second
order in the hybridization energy $V_{\text{hyb}}$ provides exchange integrals
describing Kondo-like coupling between localized spins and band states
$J_{sp-d}$ Kacman (2001); Śliwa and Dietl (2018); Autieri  _et al._ (2021),
whereas the fourth order in $V_{\text{hyb}}$ allows to obtain the exchange
tensor of the interaction between pairs of the localized spins Kacman (2001);
Śliwa and Dietl (2018); Larson  _et al._ (1988). This procedure encompasses on
equal footing various spin-spin exchange mechanisms, including superexchange,
Ruderman-Kittel-Kasuya-Yosida (RKKY) and Bloembergen-Rowland-Van Vleck (BR-VV)
interactions, the latter two usually described within the second order
perturbation theory in $J_{sp-d}$, as discussed in the next section. We recall
that these two mechanisms of coupling between localized spins can be viewed as
mediated by spin-polarization of band, carriers and/or occupied bands, in an
analogy to intraband and interband dielectric polarizations. However, in the
absence of band carriers, the dominant interaction is the superexchange
described in the fourth order in $V_{\text{hyb}}$ by a diagram depicted in
Fig. S2(a) Kacman (2001). As shown in the main body of our paper, if the self-
interaction term is treated correctly, i.e., omitted when considering pairwise
interactions, the superexchange [Fig. S2(a)] dominates over the interband BR-
VV contribution [Fig. S2(b)], even in the case of zero-gap Hg1-xMnxTe.

Figure S2: Examples of four virtual transitions allowed by hybridization
between band states and $d$ orbitals of two Mn ions (labeled 1 and 2) in zero-
gap HgTe. These processes account for (a) superexchange (the $hh$
contribution) and (b) Bloembergen-Rowland-Van Vleck coupling (the $eh$
contribution).

##  II Fourth-order vs. second-order perturbation theory

Typically, as mentioned in the previous Section, the RKKY interaction is
assumed to originate from a Kondo-like contact interaction
$\hat{\cal{H}}_{pd}$,

$\hat{\cal{H}}_{pd}=-\frac{1}{\mathcal{V}}\,\left(\hat{J}_{pd}(\mathbf{r})\hat{\mathbf{s}}\right)\cdot\hat{\mathbf{S}}_{i},$
(S1)

where $\hat{\mathbf{s}}$ and $\hat{\mathbf{S}}_{i}$ are the band electron’s
and impurity’s spin operators, respectively, whereas the usual exchange
integral $J_{pd}$ has been replaced with a Hermitian operator
$\hat{J}_{pd}(\hat{\mathbf{r}})$. For an Anderson impurity,
$\hat{\cal{H}}_{pd}$ is considered a second-order effective Hamiltonian that
can be obtained via the Schrieffer-Wolff transformation Schrieffer and Wolff
(1966). We imagine that this operator is a product,
$\hat{J}_{pd}(\hat{\mathbf{r}})\equiv J_{pd}\hat{P}(\hat{\mathbf{r}})$, of a
constant $J_{pd}$ and a Hermitian operator $\hat{P}(\hat{\mathbf{r}})$, the
latter restricting the interaction to the tight-binding orbitals of
appropriate symmetry [namely, the $p$ orbitals of the anion; in the effective-
mass approximation $\hat{P}$ includes a Dirac delta at the impurity position
$\mathbf{R}_{i}$, $\delta(\hat{\mathbf{r}}-\mathbf{R}_{i})$]. The Hamiltonian
for two impurities $(i,j)$ reads

$\hat{\cal{H}}_{\text{eff}(2)}=-\frac{1}{\mathcal{V}}\,\hat{J}_{pd}(\hat{\mathbf{r}})\hat{\mathbf{s}}\cdot\left(\hat{\mathbf{S}}_{i}+\hat{\mathbf{S}}_{j}\right).$
(S2)

By applying a second order thermodynamic perturbation theory we obtain the
Landau free energy $\Omega$ for the gas of band electrons,

$\Omega^{(2\times
2)}=\sum_{k,k^{\prime}}\frac{f(E_{k})}{E_{k}-E_{k^{\prime}}}\left<k\middle|\hat{H}_{\text{eff}(2)}\middle|k^{\prime}\right>\left<k^{\prime}\middle|\hat{H}_{\text{eff}(2)}\middle|k\right>$
(S3)

($2\times 2$ stands for the second order perturbation theory on the second
order effective Hamiltonian, yielding effectively the fourth order). It
includes an interaction term,

$\Omega^{(2\times
2)}_{i,j}=\sum_{\alpha\beta}\frac{2S_{i\alpha}S_{j\beta}}{\mathcal{V}^{2}}\sum_{k,k^{\prime}}\frac{f(E_{k})}{E_{k}-E_{k^{\prime}}}\left<k^{\prime}\middle|\hat{J}_{pd}(\mathbf{r})\hat{s}_{\alpha}\middle|k\right>\left<k\middle|\hat{J}_{pd}(\hat{\mathbf{r}})\hat{s}_{\beta}\middle|k^{\prime}\right>,$
(S4)

where the impurities are assumed to be in spin-coherent states,
$\left<\hat{S}_{i\alpha}\right>=S_{i\alpha}$. Therefore the low-temperature
interaction constants, $J^{(2\times 2)}_{ij,\alpha\beta}$, are given by

$J^{(2\times
2)}_{ij,\alpha\beta}=-\frac{J_{pd}^{2}}{\mathcal{V}^{2}}\sum_{k,k^{\prime}}\frac{f(E_{k})}{E_{k}-E_{k^{\prime}}}\left<k^{\prime}\middle|\hat{P}\hat{s}_{\alpha}\middle|k\right>\left<k\middle|\hat{P}\hat{s}_{\beta}\middle|k^{\prime}\right>.$
(S5)

Since (according to Schrieffer and Wolff Schrieffer and Wolff (1966))

$J_{pd}=\frac{1}{S}\frac{U}{(E_{d}-E_{F})(E_{d}+U-E_{F})}\left|\tilde{V}_{k_{F}}\right|^{2},$
(S6)

we obtain

$J^{(2\times
2)}_{ij,\alpha\beta}=-\frac{1}{2(2S)^{2}}\frac{U^{2}}{(E_{d}-E_{F})^{2}(E_{d}+U-E_{F})^{2}}\sum_{k,k^{\prime}}\frac{f(E_{k})-f(E_{k^{\prime}})}{E_{k}-E_{k^{\prime}}}\left<k^{\prime}\middle|\hat{Q}\hat{\sigma}_{\alpha}\middle|k\right>\left<k\middle|\hat{Q}\hat{\sigma}_{\beta}\middle|k^{\prime}\right>,\qquad$
(S7)

with $\hat{Q}=\frac{1}{\mathcal{V}}\left|\tilde{V}_{k_{F}}\right|^{2}\hat{P}$.
Furthermore, by writing

$\hat{Q}=\hat{V}_{\mathrm{hyb}}^{\dagger}\left(\sum_{m;a=\uparrow,\downarrow}\left|d_{i}ma\right>\left<d_{i}ma\right|\right)\hat{V}_{\mathrm{hyb}}$
(S8)

and assuming that the hybridization is spin-independent:

$\hat{Q}\hat{\sigma}_{\alpha}=\hat{V}_{\mathrm{hyb}}^{\dagger}\left(\sum_{m;a,b=\uparrow,\downarrow}\left|d_{i}ma\right>\left<a\middle|\sigma_{\alpha}\middle|b\right>\left<d_{i}mb\right|\right)\hat{V}_{\mathrm{hyb}},$
(S9)

we reproduce some of the $he$ terms (those with the singular denominator),
neglecting the dependence on the band energies $(E_{k},E_{k^{\prime}})$.

The remaining $he$ terms (those proportional to $2/U$) can be derived from the
second-order quasi-degenerate perturbation theory Winkler (2003) as follows.
Consider the six-dimensional Hilbert space for two Anderson impurities
occupied by two electrons. The ground state corresponds to a single occupation
of each impurity and is four-fold spin-degenerate, the remaining two states
are higher in energy by $U$. The $2/U$ terms appear on restricting the
Hamiltonian to the four-dimensional ground state via application of the
Winkler’s perturbation theory (in the second order).

##  III Bloembergen-Rowland mechanism: integration convergence

In this Section, the BR-VV interaction in a zero-gap semiconductor ($E_{g}=0$)
with spherical, parabolic bands is considered in the case of accidental
degeneracy, $E_{c(v)}=+(-)\hbar^{2}k^{2}/2m_{c(v)}$. The double reciprocal-
space integral that arises is not absolutely convergent. It may be truncated
at a finite momentum Bloembergen and Rowland (1955); Lee (1988) or understood
in the improper sense Bastard and Lewiner (1979); Lewiner and Bastard (1980),
i.e., as the limit
$\lim_{k_{\text{max}}\to{+\infty}}\int_{k=0}^{k=k_{\text{max}}}$. It turns out
that the order of taking the limit in the two integrals (if the integral is
calculated as an iterated one) may affect the result of the double-integration
procedure, as evidenced by the violation of the expected symmetry between the
valence and conduction bands ($m_{c}\leftrightarrow m_{v}$) Bastard and
Lewiner (1979). In the extreme case, even the condition $J_{0}+\sum_{i\geq
1}z_{i}J_{i}>0$, may be violated. This difficulties appear as unavoidable and
make estimation of the interaction within a simple model rather complex, as
elaborated previously. With the improper sense in place and the Fermi level in
the conduction band ($\hbar k_{F}=\sqrt{2m_{c}E_{F}}$), the Hamiltonian for a
system of two localized spins assumes the form:

$\displaystyle{\cal{H}}(R_{ij})=-\frac{2J_{cv}^{2}}{(2\pi)^{3}\hbar^{2}}\frac{(m_{c}m_{v})^{3/2}}{(m_{c}+m_{v})^{2}}\frac{\exp(-\sqrt{m_{v}/m_{c}}k_{F}R_{ij})}{R_{ij}^{4}}\times{}$
(S10)
$\displaystyle{}\times\Biggl{[}\left(2+\frac{m_{c}+m_{v}}{\sqrt{m_{c}m_{v}}}k_{F}R_{ij}\right)\cos(k_{F}R_{ij})+\left(\frac{m_{v}-m_{c}}{\sqrt{m_{c}m_{v}}}+\frac{m_{c}+m_{v}}{m_{c}}k_{F}R_{ij}\right)\sin(k_{F}R_{ij})\Biggr{]}\mathbf{S}_{i}\cdot\mathbf{S}_{j},$

where $J_{cv}$ is an interband $sp$-$d$ exchange integral. In contrast to Ref.
Bloembergen and Rowland, 1955, where a hard momentum cut-off $k_{t}$ was
introduced, the expression is not integrable at $R_{ij}=0$, and as such, is
significantly different from Eq. 32 in Ref. Bloembergen and Rowland, 1955.
Assuming a real-space cut-off at $R_{\text{min}}$,

$\displaystyle\Theta_{he}=N_{0}\left(\frac{J_{cv}}{\pi\hbar}\right)^{2}\frac{m_{c}m_{v}}{(m_{c}+m_{v})^{2}}\frac{\exp(-\sqrt{m_{v}/m_{c}}k_{F}R_{\text{min}})}{R_{\text{min}}}\times{}$
(S11)
$\displaystyle{}\times\left[2\sqrt{m_{c}m_{v}}\cos(k_{F}R_{\text{min}})-(m_{c}-m_{v})\sin(k_{F}R_{\text{min}})\right]\frac{S(S+1)}{3}.$

The oscillations as a function of $k_{F}R_{\text{min}}$ are damped
exponentially with $\alpha=\sqrt{m_{v}/m_{c}}$ and shifted in phase by
$\phi=\arctan[(m_{c}-m_{v})/2\sqrt{m_{c}m_{v}}]$. Taking, for instance,
$R_{\text{min}}=(4\pi N_{0})^{-1/3}$, and parameters for intrinsic HgTe,
$k_{F}=0$, $a_{\text{lat}}=0.646$ nm, $N_{0}J_{cv}=-0.6$ eV; $m_{c}=0.3m_{0}$,
and $m_{v}=0.45m_{0}$, we obtain ferromagnetic $\Theta_{he}=85$ K.

An alternative procedure is proposed here which amounts to regularizing the
momentum integrals by introducing under the integral an additional weight,
$dk\to\exp(-ak)\,dk$, $a\to 0^{+}$. One proceeds with integration over one of
the momenta ($k$) and takes the limit $a\to 0^{+}$. Then, an additional factor
appears which exponentially divergent and must be suppressed by
$dk^{\prime}\to\exp(-a^{\prime}k^{\prime})\,dk^{\prime}$ with sufficiently
large $a^{\prime}$. However, the result of the second integration is analytic
with respect to $a^{\prime}$ and can be continued to $a^{\prime}=0^{+}$, where
its value vanishes as $a^{\prime}\to 0^{+}$. Furthermore, although the
exchange constant diverges as $1/a^{\prime}$ if integrated over the real space
($R_{ij}$), the real space integral vanishes again if the vicinity of
$R_{ij}=0$ is omitted from the integration. One concludes that this simple
model predicts $\Theta_{he}=0$, and the interaction vanishes except for
$R_{ij}\to 0$. In practice, values of $a$ and $a^{\prime}$ of the order of the
lattice constant are appropriate, and result in a real-space-distance
dependence of the exchange constant similar to that presented in Fig. 2 in the
main text.

## References

  * White (2007) Robert M. White,  _Quantum Theory of Magnetism_ (Springer-Verlag, Berlin, 2007) pp. 1–213.
  * Zener (1951) C. Zener, “Interaction between the d shells in the transition metals,” Phys. Rev. 81, 440 (1951).
  * Dietl and Spałek (1982) T. Dietl and J. Spałek, “Effect of fluctuations of magnetization on the bound magnetic polaron: Comparison with experiment,” Phys. Rev. Lett. 48, 355–358 (1982).
  * Dietl (2015) T. Dietl, “Spin dynamics of a confined electron interacting with magnetic or nuclear spins: A semiclassical approach,” Phys. Rev. B 91, 125204 (2015).
  * Dietl  _et al._ (2000) T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Ferrand, “Zener model description of ferromagnetism in zinc-blende magnetic semiconductors,” Science 287, 1019–1022 (2000).
  * Dietl and Ohno (2014) T. Dietl and H. Ohno, “Dilute ferromagnetic semiconductors: Physics and spintronic structures,” Rev. Mod. Phys. 86, 187–251 (2014).
  * Jungwirth  _et al._ (2014) T. Jungwirth, J. Wunderlich, V. Novák, K. Olejník, B. L. Gallagher, R. P. Campion, K. W. Edmonds, A. W. Rushforth, A. J. Ferguson, and P. Němec, “Spin-dependent phenomena and device concepts explored in (Ga,Mn)As,” Rev. Mod. Phys. 86, 855–896 (2014).
  * Dietl  _et al._ (1997) T. Dietl, A. Haury, and Y. Merle d’Aubigne, “Free carrier-induced ferromagnetism in structures of diluted magnetic semiconductors,” Phys. Rev. B 55, R3347 (1997).
  * Ke  _et al._ (2018) He Ke, Yayu Wang, and Qi-Kun Xue, “Topological materials: quantum anomalous Hall system,” Annu. Rev. Cond. Mat. Phys. 9, 329–344 (2018).
  * Tokura  _et al._ (2019) Y. Tokura, K. Yasuda, and A. Tsukazaki, “Magnetic topological insulators,” Nat. Rev. Phys. 1, 126–143 (2019).
  * Chang  _et al._ (2013) Cui-Zu Chang, Jinsong Zhang, Xiao Feng, Jie Shen, Zuocheng Zhang, Minghua Guo, Kang Li, Yunbo Ou, Pang Wei, Li-Li Wang, Zhong-Qing Ji, Yang Feng, Shuaihua Ji, Xi Chen, Jinfeng Jia, Xi Dai, Zhong Fang, Shou-Cheng Zhang, Ke He, Yayu Wang, Li Lu, Xu-Cun Ma, and Qi-Kun Xue, “Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator,” Science 340, 167–170 (2013).
  * Xiao  _et al._ (2018) Di Xiao, Jue Jiang, Jae-Ho Shin, Wenbo Wang, Fei Wang, Yi-Fan Zhao, Chaoxing Liu, Weida Wu, Moses H. W. Chan, N. Samarth, and Cui-Zu Chang, “Realization of the axion insulator state in quantum anomalous Hall sandwich heterostructures,” Phys. Rev. Lett. 120, 056801 (2018).
  * Fan  _et al._ (2014) Yabin Fan, P. Upadhyaya, Xufeng Kou, Murong Lang, So Takei, Zhenxing Wang, Jianshi Tang, Liang He, Li-Te Chang, M. Montazeri, Guoqiang Yu, Wanjun Jiang, Tianxiao Nie, R. N. Schwartz, Y Tserkovnyak, and Kang L. Wang, “Magnetization switching through giant spin-orbit torque in a magnetically doped topological insulator heterostructure,” Nat. Mater. 13, 699–704 (2014).
  * He  _et al._ (2017) Qing Lin He, Lei Pan, A. L. Stern, E. C. Burks, Xiaoyu Che, Gen Yin, Jing Wang, Biao Lian, Quan Zhou, Eun Sang Choi, K. Murata, Xufeng Kou, Zhijie Chen, Tianxiao Nie, Qiming Shao, Yabin Fan, Shou-Cheng Zhang, Kai Liu, Jing Xia, and Kang L. Wang, “Chiral Majorana fermion modes in a quantum anomalous Hall insulator–superconductor structure,” Science 357, 294–299 (2017).
  * Kayyalha  _et al._ (2020) M. Kayyalha, Di Xiao, Ruoxi Zhang, Jaeho Shin, Jue Jiang, Fei Wang, Yi-Fan Zhao, Run Xiao, Ling Zhang, K. M. Fijalkowski, Pankaj Mandal, M. Winnerlein, C. Gould, Qi Li, L. W. Molenkamp, Moses H. W. Chan, N. Samarth, and Cui-Zu Chang, “Absence of evidence for chiral Majorana modes in quantum anomalous Hall-superconductor devices,” Science 367, 64–67 (2020).
  * Yu  _et al._ (2010) Rui Yu, Wei Zhang, Hai-Jun Zhang, Shou-Cheng Zhang, Xi Dai, and Zhong Fang, “Quantized anomalous Hall effect in magnetic topological insulators,” Science 329, 61–64 (2010), see Eq. (S1) in Supporting Online Material.
  * Bloembergen and Rowland (1955) N. Bloembergen and T. J. Rowland, “Nuclear spin exchange in solids: ${\mathrm{Tl}}^{203}$ and ${\mathrm{Tl}}^{205}$ magnetic resonance in thallium and thallic oxide,” Phys. Rev. 97, 1679–1698 (1955).
  * Bastard and Lewiner (1979) G. Bastard and C. Lewiner, “Indirect-exchange interactions in zero-gap semiconductors,” Phys. Rev. B 20, 4256–4267 (1979).
  * Lewiner and Bastard (1980) C. Lewiner and G. Bastard, “Indirect exchange interaction in extremely non-parabolic zero-gap semiconductors,” J. Phys. C: Solid State Physics 13, 2347–2360 (1980).
  * Lee (1988) Ven-Chung Lee, “Indirect exchange interaction in the zero-gap semiconductor ${\mathrm{Hg}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Mn}}_{\mathrm{x}}$Te,” Phys. Rev. B 37, 8849–8854 (1988).
  * Śliwa and Dietl (2018) C. Śliwa and T. Dietl, “Thermodynamic perturbation theory for noninteracting quantum particles with application to spin-spin interactions in solids,” Phys. Rev. B 98, 035105 (2018).
  * Autieri  _et al._ (2021) C. Autieri, C. Śliwa, R. Islam, G. Cuono, and T. Dietl, “Momentum-resolved spin splitting in Mn-doped trivial CdTe and topological HgTe semiconductors,” Phys. Rev. B 103, 115209 (2021).
  * Larson  _et al._ (1988) B. E. Larson, K. C. Hass, H. Ehrenreich, and A. E. Carlsson, “Theory of exchange interactions and chemical trends in diluted magnetic semiconductors,” Phys. Rev. B 37, 4137–4154 (1988).
  * Savoyant  _et al._ (2014) A. Savoyant, S. D’Ambrosio, R. O. Kuzian, A. M. Daré, and A. Stepanov, “Exchange integrals in Mn- and Co-doped II-VI semiconductors,” Phys. Rev. B 90, 075205 (2014).
  * Spałek  _et al._ (1986) J. Spałek, A. Lewicki, Z. Tarnawski, J. K. Furdyna, R. R. Galazka, and Z. Obuszko, “Magnetic susceptibility of semimagnetic semiconductors: The high-temperature regime and the role of superexchange,” Phys. Rev. B 33, 3407–3418 (1986).
  * Lewicki  _et al._ (1988) A. Lewicki, J. Spałek, J. K. Furdyna, and R. R. Gałazka, “Magnetic susceptibility of diluted magnetic (semimagnetic) semiconductors: Further evidence for superexchange,” Phys. Rev. B 37, 1860–1863 (1988).
  * Satake  _et al._ (2020) Y. Satake, J. Shiogai, G. P. Mazur, S. Kimura, S. Awaji, K. Fujiwara, T. Nojima, K. Nomura, S. Souma, T. Sato, T. Dietl, and A. Tsukazaki, “Magnetic-field-induced topological phase transition in Fe-doped ${(\mathrm{Bi},\mathrm{Sb})}_{2}\mathrm{S}{\mathrm{e}}_{3}$ heterostructures,” Phys. Rev. Materials 4, 044202 (2020).
  * Vergniory  _et al._ (2014) M. G. Vergniory, M. M. Otrokov, D. Thonig, M. Hoffmann, I. V. Maznichenko, M. Geilhufe, X. Zubizarreta, S. Ostanin, A. Marmodoro, J. Henk, W. Hergert, I. Mertig, E. V. Chulkov, and A. Ernst, “Exchange interaction and its tuning in magnetic binary chalcogenides,” Phys. Rev. B 89, 165202 (2014).
  * Kim  _et al._ (2017) Jeongwoo Kim, Seung-Hoon Jhi, A. H. MacDonald, and Ruqian Wu, “Ordering mechanism and quantum anomalous Hall effect of magnetically doped topological insulators,” Phys. Rev. B 96, 140410(R) (2017).
  * Peixoto  _et al._ (2020) T. R. F. Peixoto, H. Bentmann, P. Rüßmann, A.-V. Tcakaev, M. Winnerlein, S. Schreyeck, S. Schatz, R. C. Vidal, F. Stier, V. Zabolotnyy, R. J. Green, Chul Hee Min, C. I. Fornari, Maaß H., H. B. Vasili, P. Gargiani, M. Valvidares, A. Barla, J. Buck, M. Hoesch, F. Diekmann, S. Rohlf, M. Kalläne, K. Rossnagel, Ch. Gould, K. Brunner, S. Blügel, V. Hinkov, L. W. Molenkamp, and F. Reinert, “Non-local effect of impurity states on the exchange coupling mechanism in magnetic topological insulators,” npj Quantum Materials 5, 87 (2020).
  * Dietl (1994) T. Dietl, “(Diluted) Magnetic Semiconductors,” in  _Handbook of Semiconductors_ , Vol. 3B, edited by S. Mahajan (North Holland, Amsterdam, 1994) p. 1270.
  * Dietl  _et al._ (2001) T. Dietl, H. Ohno, and F. Matsukura, “Hole-mediated ferromagnetism in tetrahedrally coordinated semiconductors,” Phys. Rev. B 63, 195205 (2001).
  * Ferrand  _et al._ (2001) D. Ferrand, J. Cibert, A. Wasiela, C. Bourgognon, S. Tatarenko, G. Fishman, T. Andrearczyk, J. Jaroszynski, S. Kolesnik, T. Dietl, B. Barbara, and D. Dufeu, “Carrier-induced ferromagnetism in p-Zn1-xMnxTe,” Phys. Rev. B 63, 085201 (2001).
  * Ginter  _et al._ (1979) J. Ginter, J. Kossut, and L. Świerkowski, “Indirect exchange interaction via electrons in spin-orbit coupled bands in semiconductors,” phys. stat. solidi (b) 96, 735–744 (1979).
  * Blinowski and Kacman (1995) J. Blinowski and P. Kacman, “Superexchange in $A^{\mathrm{II}}B^{\mathrm{VI}}$ diluted magnetic semiconductors with Mn, Fe and Co ions,” Acta Phys. Pol. A 87, 539–542 (1995).
  * Blinowski  _et al._ (1996) J. Blinowski, P. Kacman, and J. A. Majewski, “Ferromagnetic superexchange in Cr-based diluted magnetic semiconductors,” Phys. Rev. B 53, 9524 (1996).
  * Simserides  _et al._ (2014)
  * Simserides  _et al._ (2014) C. Simserides, J. A. Majewski, K. N. Trohidou, and T. Dietl, “Theory of ferromagnetism driven by superexchange in dilute magnetic semi-conductors,” EPJ Web of Conferences 75, 01003 (2014).
  * Sawicki  _et al._ (2012) M. Sawicki, T. Devillers, S. Gałȩski, C. Simserides, S. Dobkowska, B. Faina, A. Grois, A. Navarro-Quezada, K. N. Trohidou, J. A. Majewski, T. Dietl, and A. Bonanni, “Origin of low-temperature magnetic ordering in Ga1-xMnxN,” Phys. Rev. B 85, 205204 (2012).
  * Stefanowicz  _et al._ (2013) S. Stefanowicz, G. Kunert, C. Simserides, J. A. Majewski, W. Stefanowicz, C. Kruse, S. Figge, Tian Li, R. Jakieła, K. N. Trohidou, A. Bonanni, D. Hommel, M. Sawicki, and T. Dietl, “Phase diagram and critical behavior of a random ferromagnet Ga1-xMnxN,” Phys. Rev. B 88, 081201(R) (2013).
  * Kacman (2001) P. Kacman, “Spin interactions in diluted magnetic semiconductors and magnetic semiconductor structures,” Semicon. Sci. Technol. 16, R25–R39 (2001).
  * Froyen (1989) S. Froyen, “Brillouin-zone integration by Fourier quadrature: Special points for superlattice and supercell calculations,” Phys. Rev. B 39, 3168–3172 (1989).
  * Liu and Tosatti (1969) L. Liu and E. Tosatti, “Dielectric constant and mobility of a doped zero-gap semiconductor,” Phys. Rev. Lett. 23, 772–774 (1969).
  * Broerman (1970) J. G. Broerman, “Anomalous mobility and dielectric singularity of $\alpha$-Sn,” Phys. Rev. Lett. 24, 450–451 (1970).
  * Larson  _et al._ (1986) B. E. Larson, K. C. Hass, and R. L. Aggarwal, “Effects of internal exchange fields on magnetization steps in diluted magnetic semiconductors,” Phys. Rev. B 33, 1789–1796 (1986).
  * Galazka  _et al._ (1988) R. R. Galazka, W. Dobrowolski, J. P. Lascaray, M. Nawrocki, A. Bruno, J. M. Broto, and J. C. Ousset, “Nearest neighbor exchange constants in Hg1-xMnxTe, Hg1-xMnxSe and other semimagnetic semiconductors,” J. Magn. Magn. Mater. 72, 174–180 (1988).
  * Cherbunin  _et al._ (2020) R. V. Cherbunin, V. M. Litviak, I. I. Ryzhov, A. V. Koudinov, S. Elsässer, A. Knapp, T. Kiessling, J. Geurts, S. Chusnutdinow, T. Wojtowicz, and G. Karczewski, “High-resolution resonance spin-flip Raman spectroscopy of pairs of manganese ions in a CdTe quantum well,” Phys. Rev. B 101, 241301(R) (2020).
  * Lascaray  _et al._ (1989) J. P. Lascaray, A. Bruno, J. C. Ousset, H. Rakoto, J. M. Broto, and S. Askenazy, “Quasi-static pulsed high magnetic field used to study magnetic properties of semimagnetic semiconductors,” Physica B 155, 353–356 (1989).
  * Twardowski  _et al._ (1987) A. Twardowski, H. J. M. Swagten, W. J. M. de Jonge, and M. Demianiuk, “Magnetic behavior of the diluted magnetic semiconductor ${\mathrm{Zn}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Mn}}_{\mathrm{x}}$Se,” Phys. Rev. B 36, 7013–7023 (1987).
  * Gała̧zka (1995) R. R. Gała̧zka, “Influence of electron subsystem on magnetic properties of semimagnetic semiconductors,” J. Magn. Magn. Mater. 140-144, 13–16 (1995).
  * Sawicki  _et al._ (2013) M. Sawicki, E. Guziewicz, M. I. Łukasiewicz, O. Proselkov, I. A. Kowalik, W. Lisowski, P. Dluzewski, A. Wittlin, M. Jaworski, A. Wolska, W. Paszkowicz, R. Jakiela, B. S. Witkowski, L. Wachnicki, M. T. Klepka, F. J. Luque, D. Arvanitis, J. W. Sobczak, M. Krawczyk, A. Jablonski, W. Stefanowicz, D. Sztenkiel, M. Godlewski, and T. Dietl, “Homogeneous and heterogeneous magnetism in (Zn,Co)O: From a random antiferromagnet to a dipolar superferromagnet by changing the growth temperature,” Phys. Rev. B 88, 085204 (2013).
  * Galazka  _et al._ (1980) R. R. Galazka, S. Nagata, and P. H. Keesom, “Paramagnetic–spin-glass–antiferromagnetic phase transitions in Cd1-xMnxTe from specific heat and magnetic susceptibility measurements,” Phys. Rev. B 22, 3344 (1980).
  * Smith (1975) D. A. Smith, “The zero-frequency susceptibility of spin glasses and micromagnets,” J. Phys. F 5, 2148–2167 (1975).
  * Mycielski  _et al._ (1984) A. Mycielski, C. Rigaux, M. Menant, T. Dietl, and M. Otto, “Spin glass phase transition in Hg1-kMnkTe semimagnetic semiconductors,” Solid State Commun. 50, 257 – 260 (1984).
  * Bednik (2020) G. Bednik, “Antiferromagnetism and spin density waves in three-dimensional Dirac metals,” Phys. Rev. B 102, 125119 (2020).
  * Chang  _et al._ (2014) Cui-Zu Chang, Peizhe Tang, Yi-Lin Wang, Xiao Feng, Kang Li, Zuocheng Zhang, Yayu Wang, Li-Li Wang, Xi Chen, Chaoxing Liu, Wenhui Duan, Ke He, Xu-Cun Ma, and Qi-Kun Xue, “Chemical-potential-dependent gap opening at the dirac surface states of ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ induced by aggregated substitutional Cr atoms,” Phys. Rev. Lett. 112, 056801 (2014).
  * Růžička  _et al._ (2015) J. Růžička, O. Caha, V. Holý, H. Steiner, V. Volobuiev, A. Ney, G. Bauer, T. Duchoň, K. Veltruská, I. Khalakhan, V. Matolín, E. F. Schwier, .H Iwasawa, K. Shimada, and G. Springholz, “Structural and electronic properties of manganese-doped Bi2Te3 epitaxial layers,” New J. Phys. 17, 013028 (2015).
  * Blinowski  _et al._ (1995) J. Blinowski, P. Kacman, and J. A. Majewski, “Superexchange in diluted magnetic semiconductors,” Mater. Sci. Forum 182-184, 779–782 (1995).
  * Twardowski (1990) A. Twardowski, “Magnetic properties of Fe-based diluted magnetic semiconductors,” J. Appl. Phys. 67, 5108–5113 (1990).
  * Watanabe  _et al._ (2019) R. Watanabe, R. Yoshimi, M. Kawamura, M. Mogi, A. Tsukazaki, X. Z. Yu, K. Nakajima, K. S. Takahashi, M. Kawasaki, and Y. Tokura, “Quantum anomalous Hall effect driven by magnetic proximity coupling in all-telluride based heterostructure,” Appl. Phys. Lett. 115, 102403 (2019).
  * Hor  _et al._ (2010) Y. S. Hor, P. Roushan, H. Beidenkopf, J. Seo, D. Qu, J. G. Checkelsky, L. A. Wray, D. Hsieh, Y. Xia, S.-Y. Xu, D. Qian, M. Z. Hasan, N. P. Ong, A. Yazdani, and R. J. Cava, “Development of ferromagnetism in the doped topological insulator ${\text{Bi}}_{2-x}{\text{Mn}}_{x}{\text{Te}}_{3}$,” Phys. Rev. B 81, 195203 (2010).
  * Sato  _et al._ (2010) K. Sato, L. Bergqvist, J. Kudrnovský, P. H. Dederichs, O. Eriksson, I. Turek, B. Sanyal, G. Bouzerar, H. Katayama-Yoshida, V. A. Dinh, T. Fukushima, H. Kizaki, and R. Zeller, “First-principles theory of dilute magnetic semiconductors,” Rev. Mod. Phys. 82, 1633–1690 (2010).
  * Bonanni and Dietl (2010) A. Bonanni and T. Dietl, “A story of high-temperature ferromagnetism in semiconductors,” Chem. Soc. Rev. 39, 528–539 (2010).
  * Schrieffer and Wolff (1966) J. R. Schrieffer and P. A. Wolff, “Relation between the Anderson and Kondo hamiltonians,” Phys. Rev. 149, 491 (1966).
  * Winkler (2003) R. Winkler,  _Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems_, Springer Tracts in Modern Physics, Vol. 191 (Springer, Berlin, Heidelberg, 2003).