# Effects of charge dopants in quantum spin Hall materials

Tomasz Dietl<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, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

###### Abstract

Semiconductors’ sensitivity to electrostatic gating and doping accounts for
their widespread use in information communication and new energy technologies.
It is demonstrated quantitatively and with no adjustable parameters that the
presence of paramagnetic acceptor dopants elucidates a variety of hitherto
puzzling properties of two-dimensional topological semiconductors at the
topological phase transition and in the regime of the quantum spin Hall
effect. The concepts of resonant states, charge correlation, Coulomb gap,
exchange interaction between conducting electrons and holes localized on
acceptors, strong coupling limit of the Kondo effect, and bound magnetic
polaron explain a short topological protection length, high hole mobilities
compared with electron mobilities, and different temperature dependence of the
spin Hall resistance in HgTe and (Hg,Mn)Te quantum wells.

Introduction–Quantized Hall resistance is a hallmark of two-dimensional (2D)
topological electronic systems von Klitzing  _et al._ (2020). The integer
quantum Hall effect’s high-precision quantization is behind a new definition
of units Rigosi and Elmquist (2019), whereas other quantum Hall phenomena lead
to many far-reaching developments von Klitzing  _et al._ (2020). Surprisingly,
however, although the quantum spin Hall effect (QSHE) has been known for more
than a decade Kane and Mele (2005); Bernevig  _et al._ (2006); König  _et al._
(2007), experimental resistance magnitudes attain the expected value
$h/2e^{2}$ only in mesoscopic samples, such as micron-size HgTe-based quantum
wells (QWs) Bendias  _et al._ (2018); Shamim  _et al._ (2021) and sub-100-nm
atomically thin 1T’-WTe2 2D monolayers Fei  _et al._ (2017); Wu  _et al._
(2018). Moreover, although several theoretical models have been proposed Hsu
_et al._ (2021); Yevtushenko and Yudson (2022), a short experimentally found
protection length has usually been assigned Bendias  _et al._ (2018); Shamim
_et al._ (2021); Fei  _et al._ (2017); Wu  _et al._ (2018) to unidentified
charge puddles that trap edge carriers and within which spin-flip, allowing
for scattering between helical edges, occurs Väyrynen  _et al._ (2014).

We claim here that the challenging properties of QSHE semiconductors result
from the presence of native acceptors in these materials. Quantitative
agreement between experimental and theoretical values of the topological
protection lengths supports this claim. The starting point for this work is a
quantitative theory of acceptor states in HgTe QWs, which provides positions
of acceptor levels with respect to bands and topological edge states as a
function of the QW thickness. With this information, we contend that the
acceptor density is determined by the gate voltage range in which edge states
carry the electric current König  _et al._ (2007); Grabecki  _et al._ (2013);
Bendias  _et al._ (2018); Lunczer  _et al._ (2019); Shamim  _et al._ (2021);
Fei  _et al._ (2017); Wu  _et al._ (2018). Furthermore, considering charge
correlation and Coulomb-gap effects Wilamowski  _et al._ (1990), the acceptor
scenario explains why at the 2D topological phase transition, the mobility of
holes is significantly greater than that of electrons Shamim  _et al._ (2020);
Yahniuk  _et al._ , as well as elucidates the origin of high-frequency
conductivity Dartiailh  _et al._ (2020) and gating hystereses Lunczer  _et
al._ (2019). As a next step, a theory of exchange coupling between electrons
and acceptor holes Śliwa and Dietl (2008) is employed to demonstrate that, in
topological materials, the interaction between edge electrons with acceptor
holes reaches the strong coupling limit of the Kondo effect, where the spin
dephasing rate assumes, up to a material-specific logarithmic correction, a
universal behavior discussed in the context of magnetic impurities Maciejko
_et al._ (2009); Tanaka  _et al._ (2011); Micklitz  _et al._ (2006). The
central result of this work is that, in this limit, the topological protection
length $L_{\text{p}}$ in the Ohmic conductivity regime is given by a product
of the inverse of one-dimensional (1D) acceptor hole density in the edge
region and the anisotropy of exchange coupling to hole spins. This finding
elucidates the magnitude of $L_{\text{p}}$ in HgTe QWs and WTe2 2D monolayers.
Finally, we demonstrate that the formation of acceptor bound magnetic polarons
explains a difference in carrier mobilities and the temperature dependence of
the edge resistivity of topological HgTe and Hg1-xMnxTe QWs Shamim  _et al._
(2021). The result presented here are supported and extended in the companion
paper Dietl (2023).

Acceptor levels–Electrically active point centers, together with planar and
linear defects, such as dislocations, account for differences between devices
fabricated to be similar. However, steady and impressive progress in the
quality of MBE-grown modulation-doped III-V Umansky  _et al._ (2009); Chung
_et al._ (2021) and II-VI Tsukazaki  _et al._ (2010); Piot  _et al._ (2010);
Betthausen  _et al._ (2014) heterostructures has been achieved by increasing
pumping efficiency and improving chemical purity of constituting elements,
which points to the dominant role of background ionized donor and acceptor
impurities in epitaxial structures of those compound semiconductors.
Similarly, native acceptors in bulk compound semiconductors have been
frequently assigned to metal vacancies giving double acceptors ($Z=-2$) in II-
VI materials, but the case of ZnTe and CdTe indicates that residual charged
impurities, such as Cu ($Z=-1$), are involved Pautrat  _et al._ (1985).

To describe charge dopant states in topological QWs, the Kohn-Luttinger
effective mass theory developed for acceptors in GaAs and HgTe QWs taking into
account four $\Gamma_{8}$ valence bands (including spin) Fraizzoli and
Pasquarello (1991); Kozlov  _et al._ (2019) is extended in this Letter to the
case, in which also $\Gamma_{6}$ and $\Gamma_{7}$ bands are relevant Novik
_et al._ (2005). In the companion paper Dietl (2023), we present an explicit
form of the wave functions that diagonalize the eight bands’ QW Hamiltonians
without and with a charged impurity, as well as examine the validity range of
the axial approximation, employed routinely for acceptors in zinc-blende QWs
Fraizzoli and Pasquarello (1991); Kozlov  _et al._ (2019). Within that
approximation, the eigenfunctions are found to be labeled by eigenvalues of
$F_{z}=j_{z}+l_{z}$, where $j_{z}$ and $l_{z}$ denote the components
perpendicular to the QW plane of the angular momenta corresponding to the
Kohn-Luttinger amplitudes and the associated envelope functions, respectively,
confirming that $F_{z}$ commutes with axial eight bands’ Hamiltonians Sercel
and Vahala (1990). The resultant wave functions are mainly composed of either
$p_{\pm 1/2}$ and $s_{\pm 1/2}$ ($j_{z}=\pm 1/2$) or $p_{\pm 3/2}$ ($j_{z}=\pm
3/2$) Kohn-Luttinger amplitudes, respectively, where $s_{sz}$ and $p_{jz}$
transform under the point group operations like $s$ and $p$ atomic orbitals.
The corresponding binding energies of the ground-state Kramers doublets are
denoted $E_{1/2}$ or $E_{3/2}$, and are usually referred to as light and heavy
hole acceptors, respectively.

Figure 1: Band structure and positions of acceptor levels in HgTe QWs of
different thicknesses computed with band structure parameters given in Ref.
Novik  _et al._ , 2005. (a, b) Band energies $E$ vs. wavevector $k$ for
unstrained QW thickness of 6 and 8 nm sandwiched between Hg0.3Cd0.7Te
barriers. Red rectangles depict the band region displayed in (c). (c) Band
edges and acceptor levels (symbols connected by dashed lines vs. QW thickness
$d_{\text{QW}}$. Except for the orange circles computed for the doubly ionized
acceptor ($Z=-2$), other symbols represent the single acceptor ($Z=-1$). The
orange symbols ($E_{3/2}$) correspond to acceptors associated with the valence
band around $k=0$; the blue symbols ($E_{1/2}$) with valence band side maxima
visible in (a) and (b). Full symbols represent the acceptors residing in the
QW center; the open symbols represent acceptors at the distances
$d_{\text{QW}}/4$, $d_{\text{QW}}/2$, and $3d_{\text{QW}}/2$ of the QW center.
Colors represent the participation of the $p_{\pm 3/2}$ Kohn–Luttinger
amplitude in the wave functions. The discontinuity in the orbital content
occurring at $k=0$ and $E_{\text{g}}\rightarrow 0$ [see (a)], is blurred in
(c) by contributions with $k\neq 0$.

Figure 1 depicts energies of relevant QW bands and acceptor ground-state
levels for a range of the HgTe QW widths $d_{\text{QW}}$ with colors
representing a fraction of the $p_{\pm 3/2}$ amplitude in the carrier wave
function. Three distinct areas are observed in Fig. 1(c): (i) normal band
ordering (cation $s$ states above anion $p$ states) at small $d_{\text{QW}}$
values; (ii) the range of the topological phase transition centered around the
bandgap $E_{\text{g}}=0$ and $d_{\text{c}}\approx 5.8$ nm; (iii) the
topological region $d_{\text{w}}>d_{\text{c}}$, where the band ordering is
inverted, resulting in 1D topological gapless edge states Kane and Mele
(2005); König  _et al._ (2007) to be discussed later. Such a band diagram is
generic for this class of 2D topological systems, however the value of
$d_{\text{c}}$ depends on strain (set to zero here) and Cd or Mn content in
the barriers and well Shamim  _et al._ (2020); Dietl (2023).

We note that the binding energies of the doubly ionized acceptors $E^{(2-/-)}$
are irrelevant for the low-energy physics. In contrast, $E^{(-/0)}$ levels,
residing near band edges or in the gap, are essential. They originate from
either single acceptors ($Z=-1$) or singly ionized double acceptors that, in
the mean-field approach, have the same binding energy as single acceptors. As
seen in Fig. 1(c), in the regions of interest here ($d_{\text{QW}}\approx
d_{\text{c}}$ and $d_{\text{QW}}>d_{\text{c}}$), the ground state corresponds
to the level $E_{1/2}$ associated with the side maximum of the valence band
visible in Figs. 1(a) and 1(b). Notably, the acceptor levels form a band, as
the hole binding energy depends on the location of the parent acceptor
impurity with respect to the QW center, as shown in Fig. 1(c).

Within this model, the range of gate voltage corresponding to sweeping over
the bandgap $E_{\text{g}}$ at $d_{\text{QW}}>d_{\text{c}}$ directly provides
the 2D areal density of relevant acceptors $N_{\text{a}}$, with the
experimental data implying $N_{\text{a}}\approx 10^{11}$ cm-2 for HgTe QWs
Bendias  _et al._ (2018); Yahniuk  _et al._ , the value consistent with the
areal hole concentration in undoped QWs König  _et al._ (2013), and
$N_{\text{a}}\approx 10^{13}$ cm-2 for WTe2 Fei  _et al._ (2017). The
$N_{\text{a}}$ for HgTe QWs corresponds to the three dimensional (3D)
concentration of the order of $N_{\text{A}}=3\cdot 10^{16}$ cm-3, a typical
magnitude for bulk HgTe Szlenk (1979) and Hg1-xMnxTe Sawicki  _et al._ (1983).
For such a concentration, the holes are localized, as for the evaluated Bohr
radius of 5 nm, the Mott critical concentration is $1.4\cdot 10^{17}$ cm-3.
Next, we demonstrate that the presence of acceptors explains several hitherto
puzzling properties of 2D topological insulators.

Figure 2:  Schematic picture of carrier and acceptor bands at the topological
phase transition ($E_{\text{g}}=0$). (a) Bulk 3D case with the Fermi energy
pinned in the conduction band (c. b.) by acceptors negatively charged below
the Fermi level. Coulomb gap at $E_{\text{F}}$ is also shown. (b,c) The same
for the 2D case and two positions of the Fermi level. The acceptor band is
wide as the binding energy depends on the acceptor location with respect to
the QW center.

Region of topological phase transition–One of the rather surprising facts is
that low-temperature electron mobility $\mu_{e}$ in modulation donor-doped
HgTe QWs $d_{\text{QW}}$ barely reaches $0.4\cdot 10^{6}$ cm2/Vs Bendias  _et
al._ (2018), whereas $\mu_{e}$ in bulk HgTe as well as in Hg1-xCdxTe and
Hg1-xMnxTe near the 3D topological transition approaches or exceeds $1\cdot
10^{6}$ cm2/Vs Szlenk (1979); Dubowski  _et al._ (1981); Sawicki  _et al._
(1983) with the onset of the Shubnikov de Haas oscillations at 10 mT Sawicki
_et al._ (1983). Even more surprisingly, in the vicinity of the topological
phase transition in 2D QWs, the hole mobility $\mu_{h}$ is larger than
$\mu_{e}$ Shamim  _et al._ (2020); Yahniuk  _et al._ , reaching
$\mu_{h}=0.9\cdot 10^{6}$ cm2/Vs, for which the integer quantum Hall effect
(QHE) plateau is resolved in 50 mT in Hg0.976Mn0.024Te Shamim  _et al._
(2020), which is relevant for the QHE metrology Yahniuk  _et al._ . In
addition, the QW hole concentration evaluated from the Hall effect, is
significantly smaller than the charge density generated by the gate voltage
Shamim  _et al._ (2020); Yahniuk  _et al._ .

Figure 2 elucidates those findings using information obtained from Fig. 1. In
the 3D bulk case (Fig. 2(a)), as previously discussed in detail Wilamowski
_et al._ (1990), the acceptor band resides in the conduction band. In
addition, due to a small electron mass value, we are on the metallic side of
the Anderson–Mott transition so that donors do not bind electrons at any
position of the Fermi energy $E_{\text{F}}$. Now, if the donor concentration
$N_{\text{D}}\ll N_{\text{A}}$, most of the acceptors are neutral.
Furthermore, under these conditions, to reduce the Coulomb energy, only
acceptors in close vicinity to donors are ionized. The resulting dipole
formation substantially reduces the electron scattering rate. Furthermore, the
presence of the Efros–Shklovskii Coulomb gap precludes resonant scattering. By
fine hydrostatic pressure tuning of the band structure toward the 3D
topological transition at $E_{\text{g}}=0$, $\mu_{e}=20\cdot 10^{6}$ cm2/Vs
was registered in Hg0.94Mn0.06Te at 2 K Sawicki  _et al._ (1983).

The situation is entirely different at the topological phase transition in the
2D case. As shown in Fig. 2(b), for the Fermi level in the conduction band,
obtained through modulation donor doping, all acceptors are ionized,
explaining the low electron mobility. In contrast, in the hole transport
regime [Fig. 2(c)], achieved by gating-induced discharging of acceptors, the
aforementioned charge correlation occurs, which along with the small effective
mass of holes in the Dirac cone and the formation of the Coulomb gap
$E_{\text{C}}$, results in high hole mobilities at
$k_{\text{B}}T<E_{\text{C}}\approx 0.5$ meV Dietl (2023). However, with a
growth of hole density, the hole effective mass increases [see, Fig. 1(a)] and
the hole mobility tends to diminish Shamim  _et al._ (2020); Yahniuk  _et al._
. Interestingly, higher carrier mobilities were observed in Mn-containing
samples Shamim  _et al._ (2020); Sawicki  _et al._ (1983) compared to the HgTe
case. We note that $E_{\text{C}}$ is enlarged by the acceptor bound magnetic
polaron (BMP) energy $E_{\text{p}}$, where for $x_{\text{Mn}}=0.02$,
$E_{\text{p}}>0.3$ meV at $T<2$ K Dietl (2023). Further systematic
experimental investigations would help verify the resonant BMP model proposed
here. The presence of the Coulomb gap explains also a large thermal stability
of the QSHE in WTe2 Wu  _et al._ (2018); Dietl (2023).

Edge transport range–Having elucidated the role of acceptors in the region of
the topological phase transition we focus on the region
$d_{\text{QW}}>d_{\text{c}}$ (Fig. 2(c)). Here, the Coulomb gap diminishes d.
c. hoping conductivity. However, since there is no Coulomb gap for
electron–hole excitations, the presence of the acceptor band explains the
origin of puzzling gap states detected by high-frequency conductivity
Dartiailh  _et al._ (2020). Moreover, under these conditions, one can
anticipate the appearance of the exchange interaction ${\cal{H}}_{eh}$ between
spins of electrons in the topological edge states, ${\vec{s}}$, and
paramagnetic acceptor holes, ${\vec{j}}$.

To reveal the striking consequences of this suggestion, it worth recalling
that a long-range component of this coupling originates from the third order
perturbation theory (second in _kp_ and first in the Coulomb interaction), for
which the exchange energy ${\cal{J}}_{eh}\propto 1/E_{eh}^{2}$, where $E_{eh}$
represents the electron–hole energy distance Bir (1974). According to the
theory Śliwa and Dietl (2008), which is quantitatively verified for the
interaction between photoelectrons at the bottom of the conduction and holes
on Mn acceptors in GaAs Myers  _et al._ (2005), ${\cal{H}}_{eh}$ assumes a
scalar (Heisenberg) form,
${\cal{H}}_{eh}=-{\cal{J}}_{eh}{\vec{s}}\cdot{\vec{j}}$, where $j=3/2$ and
${\cal{J}}_{eh}=-0.23$ eV Śliwa and Dietl (2008). When $E_{eh}=1.4$ eV in
GaAs:Mn, the lower bound of $E_{eh}$ is as small as $E_{\text{C}}\approx 0.3$
meV for the topological edge electrons and acceptor holes. Hence, the
antiferromagnetic ${\cal{J}}_{eh}$ is the largest relevant energy, and despite
a small DOS magnitude at $E_{\text{F}}$ in the 1D channels, drives the system
to a strong coupling limit of the Kondo effect Dietl (2023), specified in QWs
by a wide distribution of Kondo temperatures $T_{\text{K}}$. For the parameter
values specifying HgTe QWs, i.e., the Fermi velocity $v_{\text{F}}=4\cdot
10^{5}$ m/s and the penetration length of the edge electron wave function into
the QW, $b=5$ nm, a broad distribution of $T_{\text{K}}$ values up to 100 K is
expected Dietl (2023). Importantly, for areal hole density $N_{h}=0.5\cdot
10^{11}$ cm-2, the number of edge electrons per unit length for $E_{\text{F}}$
in the gap center $n_{e}=E_{\text{g}}/2\pi\hbar v_{\text{F}}=12/\mu$m is
greater than the number of acceptor holes in the edge region,
$n_{h}=N_{h}b=3/\mu$m. In the case of double acceptors, if the Hund’s rule is
obeyed, gating changes the value of holes’ spin rather than $n_{h}$.

Thus, we can quantitatively verify numerous theoretical studies on the Kondo
effect in QSHE materials Maciejko  _et al._ (2009); Tanaka  _et al._ (2011);
Altshuler  _et al._ (2013); Eriksson (2013) and on the role of exchange
anisotropy that allows for net backscattering of edge electrons Tanaka  _et
al._ (2011); Altshuler  _et al._ (2013); Kimme  _et al._ (2016). It worth
noting in this context that for transition metal impurities such as Mn,
$T_{\text{K}}\ll 1$ mK in HgTe QWs and the exchange anisotropy vanishes if the
transition metal is an orbital singlet state Dietl (2023).

Figure 3: Destructive role of charge dopants in the quantum spin Hall effect.
If axial symmetry is maintained
(${\cal{J}}_{x}={\cal{J}}_{y}),{\cal{J}}_{yz}=0,D_{x}=0$) only spin-flop
($\Uparrow\downarrow\,\leftrightarrows\,\Downarrow\uparrow$) transitions occur
(case 1 in the figure), so that edge current is conserved in the spin–momentum
locking situation. Note that arrows up and down refer to time reversal
partners rather than to spin up and down. However, if exchange interaction is
anisotropic, $\Uparrow\uparrow\,\leftrightarrows\,\Downarrow\downarrow$
transitions that violate pseudospin conservation are also allowed (case 2),
leading to net backscattering after a chain of spin-dependent interactions of
electrons with an acceptor. For realistic concentrations of charge dopants,
backscattering is efficient in the strong coupling limit of the Kondo effect.

In general, the exchange Hamiltonian between pseudospins of edge electrons
$s=1/2$ and acceptor holes $j=1/2$ assumes a form,
${\cal{H}}_{eh}=-\sum_{\alpha,\beta}s_{\alpha}{\cal{J}}^{(\alpha,\beta)}j_{\beta}$,
where ${\cal{J}}^{(\alpha,\beta)}$ is a real tensor, whereas $\alpha$ and
$\beta$ refer to the vector components $x,y,z$. It is convenient to introduce
the notation ${\cal{J}}_{\alpha}={\cal{J}}^{(\alpha,\alpha)}$ for
$\alpha=\beta$ and if $\alpha\neq\beta$,
${\cal{J}}_{\alpha,\beta}^{(s)}=({\cal{J}}^{(\alpha,\beta)}+{\cal{J}}^{(\beta,\alpha)})/2$
and
$D_{\alpha}=\sum_{\beta,\gamma}\epsilon_{\alpha\beta\gamma}{\cal{J}}^{(\beta,\gamma)}/2$,
where $D_{\alpha}$ are vector components of the Dzyaloshinskii-Moriya
contribution and $\epsilon_{\alpha\beta\gamma}$ is the antisymmetric Levi-
Civita tensor. If axial symmetry is maintained, the group theory implies
${\cal{J}}_{x}={\cal{J}}_{y},{\cal{J}}_{yz}=0,D_{x}=0$. In such a situation,
only spin-flop ($\Uparrow\downarrow\,\leftrightarrows\,\Downarrow\uparrow$)
are allowed, which precludes net backscattering in the spin-momentum locking
case, as sketched in Fig. 3 Tanaka  _et al._ (2011); Altshuler  _et al._
(2013); Kimme  _et al._ (2016). Since, however, the edge breaks the axial
symmetry and a random distribution of holes breaks the inversion symmetry,
there appear anisotropic contributions ${\cal{J}}_{\text{an}}$ of the form
${\cal{J}}_{x}-{\cal{J}}_{y}$, ${\cal{J}}_{yz}$, and $D_{x}$. The presence of
such spin non-conserving terms ensures the leak of electron angular momentum
to crystal orbital momentum in a chain of scattering events and, thus, leads
to net backscattering of edge electrons Tanaka  _et al._ (2011); Altshuler
_et al._ (2013); Kimme  _et al._ (2016). The resulting backscattering rate,
compared to the conventional spin dephasing rate $\gamma_{\text{s}}$, is
reduced by a factor $r$, so that $\gamma_{\text{b}}=r\gamma_{s}$, where
$r=[2{\cal{J}}_{\text{an}}/(({\cal{J}}_{x}+{\cal{J}}_{y})]^{2}$ Tanaka  _et
al._ (2011); Kimme  _et al._ (2016).

Using $\gamma_{\text{s}}$ determined by Wilson’s numerical renormalization
group approach for 1D systems in the Kondo regime Micklitz  _et al._ (2006),
and noting that the topological protection length
$L_{\text{p}}=v_{\text{F}}/\gamma_{b}$, we arrive to the main result of this
Letter,

$L_{\text{p}}^{-1}=\sum_{i}r^{(i)}f(T/T_{\text{K}}^{(i)})/L_{x},$ (1)

where the summation is over all QW holes bound to acceptors for a given gate
voltage $V_{\text{g}}$. The function $F(x)=1$ for $x=1$, it decays to zero for
$x\rightarrow 0$ and slowly decreases with $x$ for $x>1$ (for $x=0.2$ and 10,
$F(x)=0.5$ and 0.6, respectively) Micklitz  _et al._ (2006). The $r$ value is
not universal, but varies with the hole position in respect to the edge and QW
center. To estimate $L_{\text{p}}$ we adopt Dietl (2023): $N_{h}=0.5\cdot
10^{11}$ cm-2, an average value of $r^{(i)}$ as $r_{Dx}=0.13$ , the cut-off
length beyond which strong coupling of holes and electrons tends to vanish
$y_{\text{c}}=2b=10$ nm, and an average value of $f(T/T_{\text{K}})=0.4$.
These numbers lead to $L_{\text{p}}=4$ $\mu$m, the order of magnitude
consistent with experimental findings König  _et al._ (2007); Lunczer  _et
al._ (2019); Majewicz (2019). A more elaborated approach Dietl (2023) provides
conductance values and temperature dependence $G(T)$ that agree with
experimental observations, if the influence of Luttinger correlation effects
upon $r$ Väyrynen  _et al._ (2016) is taken into account.

Equation 1 implies that $L_{\text{p}}^{-1}$ scales linearly with $N_{h}$. This
fact explains a two orders of magnitude longer $L_{\text{p}}$ in HgTe QWs
König  _et al._ (2007); Lunczer  _et al._ (2019); Majewicz (2019) compared to
1T’-WTe2 2D monolayers Fei  _et al._ (2017); Wu  _et al._ (2018), as gating
experiments point to correspondingly different acceptor concentrations in
these two systems $N_{\text{a}}=10^{11}$ and $10^{13}$ cm-2, respectively
Bendias  _et al._ (2018); Yahniuk  _et al._ ; Fei  _et al._ (2017).
Furthermore, a small number of relevant acceptor holes leads to reproducible
resistance fluctuations König  _et al._ (2007); Shamim  _et al._ (2021); Fei
_et al._ (2017); Wu  _et al._ (2018). At the same time, a decrease of
conductance seen in scanning gate microscopy experiments König  _et al._
(2013) results from a local increase in the number of acceptor holes in the
edge region. Filamentary charging and discharging of barrier acceptors under a
strong gate electric field may account for hystereses and irreversibilities in
low-temperature transport properties when cycling the gate voltage Lunczer
_et al._ (2019); Shamim  _et al._ (2021).

An unexpected appearance of quantized resistance below 0.3 K in a
Hg0.988Mn0.012Te QW Shamim  _et al._ (2021) can be elucidated using the
acceptor model by spin splitting $\Delta$ of hole states originating from the
BMP effect, as for $x=0.012$, $\Delta>k_{\text{B}}T$ at $T<3.5$ K Dietl
(2023). Interestingly, the existing theories on the disappearance of the Kondo
effect in a magnetic field assume the same $\Delta$ for the impurity and band
states Filippone  _et al._ (2018), which is not the case in the presence of
BMPs.

Conclusions and outlook–The proposed impurity band model can elucidate the
critical properties of QSHE materials. In addition to controlling carriers’
densities and mobilities, the charge dopants enlarge the spin Hall plateau
width, but diminish the quantization precision. However, the resistance
quantization accuracy can be recovered by doping topological QWs with
isoelectronic magnetic impurities, as the formation of the bound magnetic
polarons weakens the Kondo effect. Similarly, impurities with a negative value
of Hubbard’s $U$ can pin the Fermi level in the gap, but will not contribute
to backscattering, provided two trapped carriers form a spin singlet. Even if
such DX- or AX+ centers are unstable under ambient conditions Chatratin  _et
al._ (2023), fast gate sweeping, light, or hydrostatic pressure might serve
for their activation Suski  _et al._ (1990).

In this Letter, the model’s quantitative predictions have been compared to
experimental data on HgTe and Hg1-xMnxTe QWs as well as on 1T’-WTe2 2D
monolayers, however, it would be interesting to verify the model in the case
of other QSHE candidate materials, such as $\alpha$-Sn and Bi films, other 2D
monolayers, and Heusler compounds with an inverted band structure. More
generally, while electrostatic gating is widely used to reveal the unique
properties of quantum materials, the results presented here demonstrate that
charge dopants play an important and unanticipated role in the physics and
applications of topological semiconductors. Finally, we mention that our
theory have been limited to the Ohmic range. A pallet of new phenomena is
expected beyond the linear response regime Väyrynen  _et al._ (2016); Lunde
and Platero (2012); Del Maestro  _et al._ (2013).

Acknowledgments–This project was 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 (MAB/2017/1).

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