# Quantitative theory of backscattering in topological HgTe and (Hg,Mn)Te
quantum wells:
acceptor states, Kondo effect, precessional dephasing, and bound magnetic
polaron

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

We present the theory and numerical evaluations of the backscattering rate
determined by acceptor holes or Mn spins in HgTe and (Hg,Mn)Te quantum wells
in the quantum spin Hall regime. The role of anisotropic $s$–$p$ and $sp$–$d$
exchange interactions, Kondo coupling, Luttinger liquid effects, precessional
dephasing, and bound magnetic polarons is quantified. The determined magnitude
and temperature dependence of conductance are in accord with experimental
results for HgTe and (Hg,Mn)Te quantum wells.

##  I Introduction

This paper, supporting and extending the companion report Dietl , presents the
theory and provides numerical evaluations of conductance in the regime of the
quantum spin Hall effect (QSHE) in HgTe König  _et al._ (2007) and (Hg,Mn)Te
quantum wells Shamim  _et al._ (2020). We have quantitatively determined
acceptor energies, the Coulomb gap, exchange coupling between edge electrons
and acceptor holes, Kondo temperatures for both acceptor holes and Mn spins,
spin-flip and backscattering rates in strong and weak coupling regimes, and
effects of bound magnetic polarons. In addition to the Kondo effect discussed
already in the context of backscattering Maciejko  _et al._ (2009); Altshuler
_et al._ (2013), we incorporate into our approach spin-nonconserving
anisotropic components of the exchange coupling, allowing for backscattering
even in the spin-momentum locking case Tanaka  _et al._ (2011); Altshuler  _et
al._ (2013); Lunde and Platero (2013); Kimme  _et al._ (2016).

Our theoretical and numerical results substantiate the main conclusion of the
companion paper Dietl  that the value of the exchange interaction between edge
electrons and acceptor holes is large enough to drive the system to the strong
coupling region of the Kondo effect, where the spin-flip scattering rate
attains the unitary limit. Together with the demonstrated sizable exchange
anisotropy, the unitary scattering rate explains the observed topological
protection length in HgTe QWs in the Ohmic conductivity regime König  _et al._
(2007); Bendias  _et al._ (2018); Lunczer  _et al._ (2019) and the
corresponding temperature dependence of conductance, if the influence of
Luttinger liquid effects upon the anisotropic exchange are taken into account
Väyrynen  _et al._ (2016).

In contrast to occupied acceptor dopants, our results indicate that no Kondo
effect is expected for impurities with open $d$ or $f$ shells in dilute
magnetic semiconductors (DMSs), at least for typical values of
antiferromagnetic $sp$–$df$ exchange integrals. Furthermore, for magnetic ions
with orbital momentum $L=0$, such as Mn2+, anisotropic components of the
exchange coupling to edge states vanish, suggesting a minor role of Mn spins
in backscattering. Nevertheless, we suggest that backscattering by a dense
bath of interacting Mn spins can operate, as for the motionally-narrowed
precessional spin-dephasing process, the constraint of local spin momentum
conservation is relaxed. At the same time, we show that the formation of bound
magnetic polarons by acceptor holes reduces the role of the Kondo effect, the
observation elucidating the origin of the recovery of quantized conductance at
low temperatures in (Hg,Mn)Te QWs Shamim  _et al._ (2021).

##  II Theoretical results

###  II.1 Quantum-well band states

A starting point of our theory is the eight-band Kohn–Luttinger Hamiltonian
with boundary conditions previously used to obtain the subband structure in
HgTe QWs Novik  _et al._ (2005) and the four–band model developed for
determining, in the axial approximation, acceptor levels in GaAs QWs Fraizzoli
and Pasquarello (1991). Within that approximation, discussed in Sec. II.1, and
in the absence of the impurity potential, the eight relevant electron wave
functions $\psi_{j}$ can be taken in the form,

$\psi_{j}^{(m)}(\rho,\varphi,z)=h_{j}^{(m)}(\rho,\varphi)f_{j}(z)u_{j},$ (1)

where $m$ is the integer orbital quantum number; $\rho$, $\varphi$, and $z$
denote the electron position in the cylindrical coordinates with $z$ along the
growth direction; $u_{j}$ is the set of the Kohn–Luttinger amplitudes:
$s_{1/2,1/2}$, $s_{1/2,-1/2}$, $p_{3/2,3/2}$, $p_{3/2,1/2}$, $p_{3/2,-1/2}$,
$p_{3/2,-3/2}$, $p_{1/2,1/2}$, and $p_{1/2,-1/2}$ Novik  _et al._ (2005),
whereas the envelope functions $h_{j}^{(m)}(\rho,\phi)$ and $f_{j}(z)$ are
given by,

$h_{j}^{(m)}(\rho,\varphi)=N_{\text{o}}i^{|m_{j}|}\exp(im_{j}\varphi)J_{|m_{j}|}(k\rho),$
(2)

where $N_{\text{o}}$ is a normalization factor; $m_{j}=m$ for $j=1$, 4, and 7;
$m_{j}=m+1$ for $j=2$, 5, and 8; $m_{j}=m-1$ for $j=3$, and $m_{j}=m+2$ for
$j=6$; $J_{|m_{j}|}(k\rho)$ is the Bessel function; $k$ is a module of the in-
plane wavevector, and

$f_{j}(z)=(1/\sqrt{L_{z}})\sum_{n=-n_{\text{max}}}^{n_{\text{max}}}c^{(j)}_{n}\exp(2\pi
inz/L_{z}),$ (3)

where $L_{z}$ is the total thickness of the structure, including the two
Cd0.7Hg0.3Te barriers and HgTe QW, assumed here as $L_{z}=60$ nm +
$d_{\text{QW}}$; the expansion coefficients $c^{(j)}_{n}$ are to be determined
by a diagonalization procedure; $n_{\text{max}}=50$ ensures an appropriate
numerical convergence, as discussed in Sec. II.4.

We adopt the identical values of ${\vec{k}}\cdot{\vec{p}}$ parameters, as in
Ref. Novik  _et al._ (2005), except for the Luttinger $\kappa$ that we assume
to be the same in the barriers and QW. In this way, no Rashba-like splitting
occurs in symmetric QWs. Computations have been performed assuming the absence
of strain. Its role is discussed in Sec. II.3.

Figures 1(a,b) in the companion paper Dietl  present QW subband dispersions
$E(k)$ for $d_{\text{QW}}=6$ and 8 nm. The magnitudes of $f_{j}(z)$ for
$d_{\text{QW}}=8$ nm and $k=0$ are shown in Fig. 1. A character of their
inversion symmetry ($z\rightarrow-z$) plays an essential role in the
anisotropy of the electron-hole exchange interaction, as discussed in Sec.
II.6.

Figure 1: Envelope functions $f_{1}(z)$, $f_{3}(z)$, $f_{4}(z)$, and
$f_{7}(z)$ at $k=0$ for E1 and H1 subbands in HgTe quatum well of 8 nm-width.
The envelopes $f_{3}$ and $f_{6}=f_{3}$ correspond to the Kramer’s pair for
the H1 subband, whereas $f_{1}$, $f_{4}$, $f_{7}$ and $f_{2}=-f_{1}$,
$f_{5}=-f_{4}$, $f_{8}=f_{7}$ for the E1 subband with a relative weight 53.6,
45.9, and 0.5%, respectively. As seen, the values of $f_{j}$ are either real
and symmetric or imaginary and antisymmetric in respect to the QW center at
$z=0$.

###  II.2 Axial approximation

Figure 2 shows subband dispersions $E({\vec{k}})$ computed without and with
the axial approximation of a HgTe QW. As the energy differences between
${\vec{k}}\|\langle 10\rangle$ and ${\vec{k}}\|\langle 11\rangle$ near valence
band top are significantly smaller than the acceptor binding energies in Fig.
1(c) of the companion paper Dietl , the axial approximation holds.

Figure 2:  Verification of axial approximation. Subband dispersions for a HgTe
QW of the thickness $d_{\text{QW}}=6$ nm computed for ${\vec{k}}\|\langle
11\rangle$ (dotted lines), ${\vec{k}}\|\langle 11\rangle$ (dashed line) and
within the axial approximation (solid line).

###  II.3 Strain effects

We use a conventional sign definition of biaxial epitaxial strain Bir (1974);
O’Reilly (1989); Dietl and Ohno (2014), i.e., a positive value of
$\epsilon_{xx}=\epsilon_{yy}$ corresponds to QW under the tensile strain. In
particular, in the case of HgTe QW, $\epsilon_{xx}=0.31$% corresponds to
strain for a CdTe substrate, whereas a Cd1-xZnxTe substrate generates a
compressive strain Leubner  _et al._ (2016), $\epsilon_{xx}<0$. The evolution
of the QW band structure with biaxial strain is shown in Fig. 3 for HgTe QW of
the thickness $d_{\text{QW}}=6$ nm and $\epsilon_{xx}=0.31$%, 0, and $-0.31$%.

Figure 3: Effects of biaxial strain on band dispersion in HgTe QWs. (a)
$\epsilon_{xx}=0.31$% (CdTe substrate); (b) $\epsilon_{xx}=0$; (c)
$\epsilon_{xx}=-0.31$%. Colors describe the participation of the $p_{3/2,\pm
3/2}$ Kohn-Luttinger amplitude in the electronic wave function; HgTe QW
thickness $d_{\text{QW}}=6$ nm.

The strain, and also temperature, shift the QW thickness corresponding to the
topological phase transition ($E_{\text{g}}=0$), but the overall band picture
remains intact, in agreement with previous conclusions Novik  _et al._ (2005);
Leubner  _et al._ (2016); Kadykov  _et al._ (2018).

###  II.4 Determination of acceptor level energies

To determine energies of levels brought about by charge dopants, we supplement
the Hamiltonian by the Coulomb potential,

$V(r)=-Ze^{2}/4\pi\epsilon_{0}\epsilon r,$ (4)

and by the potential of image charges in the barriers Fraizzoli and
Pasquarello (1991), for which the dielectric constant is
$\epsilon_{\text{b}}=(1-x_{\text{b}})\epsilon+x_{\text{b}}\epsilon_{\text{e}}$,
where in our case $\epsilon=19.5$, $\epsilon_{\text{e}}=10.5$, and
$x_{\text{b}}=0.7$. We neglect central cell corrections and the image charge
in the gate metal, which is typically more than 100 nm apart.

Furthermore, we replace the Bessel function $J_{|m_{j}|}(k\rho)$ in Eq. 2 by

$J_{|m_{j}|}(k\rho)\rightarrow\rho^{|m_{j}|}\sum_{l=1}^{l_{\text{max}}}d^{(j)}_{l}\exp(-\rho/a_{l}),$
(5)

where the coefficients $d^{(j)}_{l}$ are to be determined by the
diagonalization procedure for a given set of ${a_{l}}$ values. We take
${a_{l}}$ as a geometrical series with a common ratio of $r=1.5$ and, for a
typical number of exponential functions $l_{\text{max}}=15$, the starting
value of $a_{1}=0.5$ nm. Since, the exponential functions with real exponents
are not orthogonal, a generalized eigenvalue solver has been employed to
obtain electronic energies. At the same time, the participation number serves
to evaluate an effective in-plane localization radius $a^{*}$.

Figure 4: Example of the identification of acceptor levels $E_{3/2}$ $(m=1)$
degenerate with the QW band states. (a) Energy levels $E$ as a function of the
center charge $Z$. (b) $E$ as a function of the number of exponential
functions $l_{\text{max}}$ in Eq. 5 and for $n_{\text{max}}=25$ (Eq. 3). (c)
$E$ as a function of the distance of acceptor to QW center $z_{0}$ for
$l_{\text{max}}=15$ and $n_{\text{max}}=50$. Color scale depicts the magnitude
of an effective in-plane Bohr radius $a^{*}$ in the logarithmic scale. Red
solid lines show the ground state $E_{3/2}$ acceptor level; dotted lines
represent examples of acceptor excited states. HgTe QW thickness
$d_{\text{QW}}=6$ nm; no strain.

The presence of the impurity potential breaks the degeneracy of the states
with respect to the quantum number $m$. However, due to time–reversal
symmetry, the impurity levels remain at least doubly degenerate. The use of
exponential functions (Eq. 5) is suitable for determining the localized levels
but not for oscillating extended states. Accordingly, the values of band
energies do not converge with increasing $l_{\text{max}}$.

In our case, identifying acceptor levels that overlap with the continuum of
band states appears difficult Buczko and Bassani (1992); Zholudev  _et al._
(2020). Figure 4 depicts level energies determined for $m=1$ in a HgTe QW with
$d_{\text{QW}}=6$ nm. As shown, the magnitude of $a^{*}$, and the evolution of
the level energies with $Z$ [Fig. (a)], $l_{\text{max}}$ [Fig. (b)], and with
the acceptor location of the QW center [Fig. (c)] unambiguously tell the band
and resonant impurity states. According to the result presented in the
companion paper Dietl , the ground state corresponds to $m=0$ or $m=-1$, and
is predominately built of the $p_{3/2,\pm 1/2}$ and $s_{1/2,\pm 1/2}$ Kohn-
Luttinger amplitudes. Its energy is referred to as $E_{1/2}$.

###  II.5 Coulomb gap

The Efros-Shklovskii Coulomb gap Shklovskii and Efros (1984) of the width
$E_{\text{C}}$ reaching 10 meV was observed in a single-layer of 1T’-WTe2 with
different coverage of surface by potassium Song  _et al._ (2018). Furthermore,
numerical simulations revealed the presence of the Coulomb gap for resonant
donor states in HgSe:Fe Wilamowski  _et al._ (1990).

In terms of 2D DOS of the acceptor band taken as
$\nu_{2}=N_{\text{a}}/E_{\text{b}}$, where $E_{\text{b}}$ is the acceptor
bandwidth Shklovskii and Efros (1984),

$E_{\text{C}}=2\nu_{2}e^{4}/(4\pi\epsilon_{0}\epsilon)^{2}.$ (6)

For HgTe QW, where $N_{\text{a}}=10^{11}$ cm-2; $E_{\text{b}}=30$ meV, and the
QW dielectric constant $\epsilon=19.5$, the magnitude of $E_{\text{C}}=0.34$
meV. This value sets the temperature scale above which the hole mobility
$\mu_{h}$ decreases and conductance quantization deteriorates. Thus, the
Coulomb gap model explains the stability of the QSHE up to 100 K in WTe2
monolayers Wu  _et al._ (2018), where – as mentioned above – $E_{\text{C}}$
attains 10 meV Song  _et al._ (2018).

###  II.6 Electron-hole spin exchange

Taking into account previous insight that flip-flop transitions conserving
total spin of edge electrons and, thus, due to spin-momentum locking edge
current Tanaka  _et al._ (2011), we are interested in determining the degree
$r$ of axial symmetry breaking by the QW edge for the exchange interaction
between topological edge electrons and acceptor holes in HgTe QWs. One of
possible mechanisms could be kinetic exchange discussed in semiconductors in
the context of $p$–$d$ coupling Kacman (2001) and the Kondo effect in quantum
dots Pustilnik and Glazman (2004). However, we demonstrate here that the
Bir–Pikus theory, originally developed for excitons Bir (1974), and later
extended to the case of exchange coupling between band electrons and acceptor
holes in bulk semiconductors Śliwa and Dietl (2008), satisfactorily explains
the experimental results for HgTe QWs.

According to the Bernevig-Hughes-Zhang (BHZ) theory, the wave function of
helical states at the edge along the $x$-direction assumes the form,

$\displaystyle\psi_{e}^{(+)}(\vec{r})$ $\displaystyle=$
$\displaystyle\frac{\exp(ikx)}{\sqrt{L_{x}}}[f_{1}(\vec{\mathtt{r}})u_{1}+f_{4}(\vec{\mathtt{r}})u_{4}+f_{3}(\vec{\mathtt{r}})u_{3}];$
(7) $\displaystyle\psi_{e}^{(-)}(\vec{r})$ $\displaystyle=$
$\displaystyle{\mathcal{T}}\psi_{e}^{(+)}(\vec{r}),$ (8)

where $L_{x}$ is the edge length; $f_{i}$ are electron envelope functions that
depend on $\vec{\mathtt{r}}=(y,z)$; $u_{i}$ are the Luttinger-Kohn amplitudes,
and ${\mathcal{T}}=-i\sigma_{y}K$ is a time reversal operator. Similarly, for
the acceptor ground state,

$\displaystyle\psi_{h}^{(+)}(\vec{r})$ $\displaystyle=$
$\displaystyle{\mathcal{C}}[F_{1h}(\vec{r})u_{1}+F_{4h}(\vec{r})u_{4}];$ (9)
$\displaystyle\psi_{h}^{(-)}(\vec{r})$ $\displaystyle=$
$\displaystyle{\mathcal{T}}\psi_{h}^{(+)}(\vec{r}),$ (10)

where ${\mathcal{C}}=\sigma_{y}K$ is a charge conjugation operator
transforming the acceptor wave function from the electron representation,
employed in previous sections for the determination of electronic states, to
the hole representation relevant here. In contrast, topological protection of
the helical edge states, and the resulting continuity of the carrier group
velocity, make the electron representation valid both above and below the
Dirac point.

With these wave functions we consider a long-range contribution to the
electron-hole exchange Bir (1974); Śliwa and Dietl (2008). This interaction is
nonlocal and represented, in the case under consideration, by the matrix
elements,

$\displaystyle{\cal{H}}_{eh}^{\sigma,\sigma\prime;\mu,\mu\prime}=-\frac{\hbar^{2}}{m_{0}^{2}}\sum_{\alpha,\beta}\int
d\vec{r}_{1}d\vec{r}_{2}\partial^{2}_{r_{1}^{(\alpha)}r_{2}^{(\beta)}}V(r_{12})$
$\displaystyle\cdot\langle\psi_{e}^{\sigma}(\vec{r}_{1})|p_{\alpha}|\psi_{h}^{\mu}(\vec{(}r_{1})\rangle_{\Omega}\langle\psi_{e}^{\sigma\prime}(\vec{r}_{2})|p_{\beta}|\psi_{h}^{\mu\prime}(\vec{(}r_{2})\rangle_{\Omega},$
(11)

where $\sigma$ and $\mu=\pm$; $\alpha,\beta=x,y,z$;
${\vec{r}}_{12}=\vec{r}_{1}-\vec{r}_{2}$. Furthermore, in accord with the
effective mass theory, internal matrix elements containing momentum operators
$p_{\alpha}$ and $p_{\beta}$ are over the elementary cell volume $\Omega$ and
involve the amplitudes $u_{i}$, whereas the envelope functions are assumed
constant within this volume. As seen, non-vanishing matrix elements correspond
to coupling of $s$-type and $p$-type edge states with $p$-type and $s$-type
acceptor states, respectively. Neglecting electrostatic image charges, the
Coulomb energy has a standard form,

$V(r_{12})=\frac{e^{2}}{4\pi\epsilon_{0}\epsilon r_{12}}.$ (12)

It is convenient to present a spatial function in the Coulomb term as a sum of
the local monopole and non-local dipole components,

$\displaystyle\partial^{2}_{r_{1}^{(\alpha)}r_{1}^{(\beta)}}r_{12}^{-1}=-\frac{4\pi}{3}\delta_{\alpha\beta}\delta(\vec{r}_{12})$
$\displaystyle-\frac{(r_{12})^{2}\delta_{\alpha\beta}-3\vec{r}_{12}^{(\alpha)}\vec{r}_{12}^{(\beta)}}{r_{12}^{5}}.$
(13)

In the case of electrons at the bottom of the conduction band and holes
localized on acceptor impurities in bulk zinc-blende semiconductors, the
exchange coupling has a scalar (Heisenberg) form,
${\cal{H}}_{eh}=-{\cal{J}}\vec{s}\cdot\vec{J}$, where $J=3/2$ and
${\cal{J}}<0$ Śliwa and Dietl (2008). Since the QW and the edge break
rotational symmetry, we expect, in the presence of intratomic spin-orbit
coupling, a nonscalar form of the exchange interaction,

${\cal{H}}_{eh}=-\sum_{\alpha,\beta}s_{\alpha}{\cal{J}}^{(\alpha,\beta)}j_{\beta},$
(14)

where ${\cal{J}}^{(\alpha,\beta)}$ is a real tensor and, in a standard
notation, ${\cal{J}}_{\alpha}={\cal{J}}^{(\alpha,\alpha)}$, and if
$\alpha\neq\beta$,
${\cal{J}}_{\alpha,\beta}^{(s)}=({\cal{J}}^{(\alpha,\beta)}+{\cal{J}}^{(\beta,\alpha)}/2$;
$D_{\alpha}=\sum_{\beta,\gamma}\epsilon_{\alpha\beta\gamma}{\cal{J}}^{(\beta,\gamma)}/2$,
where $D_{\alpha}$ are vector components of the Dzyaloshinskii-Moriya (DM)
contribution and $\epsilon_{\alpha\beta\gamma}$ is the antisymmetric Levi-
Civita tensor.

By comparing matrix elements of Hamiltonians given in Eqs. 11 and 14,
considering both monopole and dipole contributions of the long-range electron-
hole exchange interaction (Eq. 13), and taking into account that
${\cal{H}}_{eh}=-{\cal{H}}_{ee}$, we arrive to final forms of non-zero
exchange tensor components and a spin-independent part of the Fock energy
$\Delta_{eh}$,

$\displaystyle{\cal{J}}_{x,y}=A\left(\frac{16\pi}{9}{\cal{M}}_{21}+\frac{4}{3}{\cal{M}}_{31}-3{\cal{M}}_{52}\pm{\cal{M}}_{51}\right);$
(15)
$\displaystyle{\cal{J}}_{z}=A\left[\frac{8\pi}{9}({\cal{M}}_{11}-3{\cal{M}}_{12})+\frac{5}{3}{\cal{M}}_{41}+{\cal{M}}_{42}+2\sqrt{3}i{\cal{M}}_{63}\right];$
(16)
$\displaystyle{\cal{J}}_{yz}^{(s)}=iA\left(\frac{8\sqrt{3}\pi}{9}{\cal{M}}_{13}-\frac{\sqrt{3}}{3}{\cal{M}}_{43}-\sqrt{3}{\cal{M}}_{53}\right);$
(17)
$\displaystyle{\cal{D}}_{x}=-iA\left(\frac{8\sqrt{3}\pi}{9}{\cal{M}}_{13}+\frac{\sqrt{3}}{3}{\cal{M}}_{43}+\sqrt{3}{\cal{M}}_{53}\right);$
(18)
$\displaystyle\Delta_{eh}=A\left[\frac{2\pi}{3}({-\cal{M}}_{11}+{\cal{M}}_{12})-\frac{1}{4}{\cal{M}}_{41}-\frac{1}{4}{\cal{M}}_{42}-\frac{\sqrt{3}i}{2}{\cal{M}}_{63}\right].$
(19)

In the above formulae the prefactor $A$ contains information about the
strength of the Coulomb interaction and $\vec{k}\cdot\vec{p}$ coupling,

$A=\frac{e^{2}}{4\pi\epsilon_{0}\epsilon}\frac{P^{2}}{E_{eh}^{2}L_{x}},$ (20)

where $P$ is Kane’s $s$-$p$ momentum matrix element and $E_{he}$ is the energy
distance between hole and electron states in question.

Particular matrix elements ${\cal{M}}_{ij}$ are given by

${\cal{M}}_{ij}=\int
d\vec{r}_{1}d\vec{r}_{2}g_{i}(\vec{r}_{1},\vec{r}_{2})O_{j}(\vec{r}_{1},\vec{r}_{2}),$
(21)

where

$\displaystyle g_{1}({\vec{r}_{1}},{\vec{r}_{2}})$ $\displaystyle=$
$\displaystyle\delta(\vec{r}_{12});$ (22) $\displaystyle
g_{2}({\vec{r}_{1}},{\vec{r}_{2}})$ $\displaystyle=$
$\displaystyle\cos[k(x_{1}+x_{2})]\delta(\vec{r}_{12});$ (23) $\displaystyle
g_{3}({\vec{r}_{1}},{\vec{r}_{2}})$ $\displaystyle=$
$\displaystyle\cos[k(x_{1}+x_{2})](r_{12}^{2}-3z_{12}^{2})/r_{12}^{5};$ (24)
$\displaystyle g_{4}({\vec{r}_{1}},{\vec{r}_{2}})$ $\displaystyle=$
$\displaystyle\cos[k(x_{1}-x_{2})](r_{12}^{2}-3z_{12}^{2})/r_{12}^{5};$ (25)
$\displaystyle g_{5}({\vec{r}_{1}},{\vec{r}_{2}})$ $\displaystyle=$
$\displaystyle\cos[k(x_{1}+x_{2})](x_{12}^{2}-y_{12}^{2})/r_{12}^{5};$ (26)
$\displaystyle g_{6}({\vec{r}_{1}},{\vec{r}_{2}})$ $\displaystyle=$
$\displaystyle\sin[k(x_{1}-x_{2})]x_{12}z_{12}/r_{12}^{5},$ (27)

Overlap functions between edge-electron and acceptor-hole envelopes read

$\displaystyle
O_{1}(\vec{r}_{1},\vec{r}_{2})=f_{4}(\vec{\mathtt{r}}_{1})F_{1h}(\vec{r}_{1})-f_{1}(\vec{\mathtt{r}}_{1})F_{4h}(\vec{r}_{1})$
$\displaystyle\cdot(f_{4}(\vec{\mathtt{r}}_{2})F_{1h}(\vec{r}_{2})-f_{1}(\vec{\mathtt{r}}_{2})F_{4h}(\vec{r}_{2});$
(28) $\displaystyle
O_{2}(\vec{r}_{1},\vec{r}_{2})=f_{3}(\vec{\mathtt{r}}_{1})F_{1h}(\vec{r}_{1})f_{3}(\vec{\mathtt{r}}_{2})F_{1h}(\vec{r}_{2});$
(29) $\displaystyle
O_{3}(\vec{r}_{1},\vec{r}_{2})=f_{3}(\vec{\mathtt{r}}_{1})F_{1h}(\vec{r}_{1})$
$\displaystyle\cdot[f_{4}(\vec{\mathtt{r}}_{2})F_{1h}(\vec{r}_{2})-f_{1}(\vec{\mathtt{r}}_{2})F_{4h}(\vec{r}_{2})],$
(30)

where for the adopted phase convention, in accord with the results presented
in Fig. 1, $f_{1}(\vec{\mathtt{r}}),f_{3}(\vec{\mathtt{r}})$, and
$F_{1h}(\vec{r})$ are real, $f_{3}(\vec{\mathtt{r}})$ is negative, whereas
$f_{4}(\vec{\mathtt{r}})$ and $F_{4h}(\vec{r})$ are imaginary and change sign
as a function of $z$.

Inspection of the above equations show that exchange integrals
${\cal{J}}_{\alpha}$ are negative, implying antiferromagnetic coupling between
edge electrons and localized holes, which allows for the Kondo coupling. At
the same time, the presence of terms breaking the axial symmetry leads to spin
non-conserving transitions ($r\neq 0$) and, hence to net backscattering of
edge electrons. The off-diagonal exchange tensor components
${\cal{J}}_{yz}^{(s)}$ and ${\cal{D}}_{x}$ are non-zero for both monopole and
dipole coupling but only if the inversion symmetry is broken, i.e., the
acceptor resides away of the QW center, so that the hole envelope functions
$F_{1h}$ and $F_{4h}(\vec{r})$ cease to be symmetric and antisymmetric with
respect to $z=0$, respectively. The presence of such terms was noted for
dipole interactions of edge electrons with nuclear spins Lunde and Platero
(2013) and for the Heisenberg interaction with magnetic impurities Kimme  _et
al._ (2016). However, in our case, even if inversion symmetry is maintained
and, moreover, even if the $s$-type component in the hole wave function is
negligible ($F_{1h}=0$), spin non-conserving transitions are still allowed by
the edge-induced breaking of the axial symmetry, leading to
${\cal{J}}_{x}\neq{\cal{J}}_{y}$ in the dipole contribution.

For numerical evaluations, guided by theoretical results obtained for the edge
states Lunde and Platero (2013); Papaj  _et al._ (2016); Krishtopenko and
Teppe (2018) and our data presented in Secs. II.1 and II.4, we assume the
electron and hole envelope functions in an approximate form,

$\displaystyle f_{1}(y,z)=2e_{s}\exp(-y/b)\Theta(y)\cos(\pi
z/d_{\text{QW}})\Theta(d_{\text{QW}}/2-|z|)/(bd_{\text{QW}})^{1/2};$ (31)
$\displaystyle f_{4}(y,z)=2ie_{p}\exp(-y/b)\Theta(y)\sin(2\pi
z/d_{\text{QW}})\Theta(d_{\text{QW}}/2-|z|)/(bd_{\text{QW}})^{1/2};$ (32)
$\displaystyle f_{3}(y,z)=2e_{H}\exp(-y/b)\Theta(y)\cos(\pi
z/d_{\text{QW}})\Theta(d_{\text{QW}}/2-|z|)/(bd_{\text{QW}})^{1/2};$ (33)
$\displaystyle
F_{1h}({\vec{r}})=h_{s}\exp[-(x^{2}+(y-y_{\text{m}})^{2})^{1/2}/a^{*}]\Theta(y)\cos[\pi(z-z_{\text{m}})/d_{\text{QW}}]\Theta(d_{\text{QW}}/2-|z|)/N_{\text{o}};$
(34) $\displaystyle
F_{4h}({\vec{r}})=ih_{p}\exp[-(x^{2}+(y-y_{\text{m}})^{2})^{1/2}/a^{*}]\Theta(y)\sin[2\pi(z-z_{\text{m}})/d_{\text{QW}}]\Theta(d_{\text{QW}}/2-|z|)/N_{\text{o}}.$
(35)

Here $b$ describes the penetration length of the edge electron wave function
into the QW; $\Theta(x)$ is the Heaviside step function, $y_{\text{m}}$ and
$z_{\text{m}}$ are distances of a maximum of the hole wave function from the
sample edge located at $y=0$ and the QW center residing at $z=0$,
respectively, and $N_{\text{o}}$ is an in-plane normalization factor. We note
that one expects $|z_{\text{m}}|<d_{\text{QW}}/2$ even for acceptors localized
ouside the QW, $|z_{0}|>d{\text{QW}}/2$. Making use of previous results
concerning the BHZ model Lunde and Platero (2013) as well as of our data
presented in Secs. II.1 and II.4, we take $e_{s}=0.26$; $e_{p}=0.24$;
$e_{H}=-0.93$; $h_{s}=0.31$; and $h_{p}=0.92$; as the factors determining the
participation of particular orbital components in the total electron and hole
wave functions. Furthermore, the computations have been performed for
$d_{\text{QW}}=8$ nm and $a^{*}=5$ nm. A Monte Carlo method has been used to
evaluate six dimensional integrals. Such a method minimizes systematic errors
but necessarily leads to a statistical scatter of the results. The chosen
number of evaluation points insures that the accuracy of the obtained data is
better than one sigma.

Figure 5: Computed values (points) of exchange integrals
${\cal{J}}_{\alpha}/A$, the Dzyaloshinskii-Moriya energy $D_{x}/A$, and the
spin-independent term $\Delta_{eh}$ for coupling between edge electrons and
acceptor holes in topological HgTe QW, plotted as a function of the distance
$y_{\text{m}}$ between the acceptor wave function maximum and the edge; (a)
the electron wavevector $k=0.05$ nm-1 and the penetration of helical state
into the QW, $b=7$ nm; (b) $k=0.1$ nm-1 $b=5$ nm. The material constant $A$ is
defined in Eq. 20. Lines are linear fits to the computed points.

Figure 5 presents exchange tensor components as a function of $y_{\text{m}}$
for two values of $k=0.05$ and 0.1 nm-1 and the corresponding magnitudes of
$b=7$ and 5 nm, respectively Lunde and Platero (2013); Papaj  _et al._ (2016);
Krishtopenko and Teppe (2018). As could be expected for the exchange
interaction, ${\cal{J}}_{\alpha}/A$ exponentially decay with $y_{\text{m}}$
but, not surprisingly, this decay is weaker for
$r_{xy}=[(2({\cal{J}}_{x}-{\cal{J}}_{y})/({\cal{J}}_{x}+{\cal{J}}_{y})]^{2}$,
as shown in Fig. 6. The magnitude of $r_{xy}$ is independent of $A$ and
unaffected by a shift of the acceptor away from the QW center. However, for
$z_{\text{m}}\neq 0$, other axial symmetry breaking terms appear,
${\cal{J}}_{yz}$ and $D_{x}$, where $|{\cal{J}}_{yz}|\ll|D_{x}|$, and, for
$z_{\text{m}}=d_{\text{QW}}/4$,
$r_{Dx}=[(3D_{x}/({\cal{J}}_{x}+{\cal{J}}_{y}+{\cal{J}}_{z})]^{2}\gg r_{xy}$,
as depicted in Fig. 6. In the subsequent two sections, the obtained values of
$J{{}_{\alpha}}/A$ and $r_{i}$ serve to estimate the magnitude of Kondo
temperature and of the backscattering rate.

Figure 6: Computed magnitudes of exchange anisotropy ratio
$r_{xy}=[(2({\cal{J}}_{x}-{\cal{J}}_{y})/({\cal{{\cal{J}}}}_{x}+{\cal{J}}_{y})]^{2}$
and $r_{Dx}=[(3D_{x}/({\cal{J}}_{x}+{\cal{J}}_{y}+{\cal{J}}_{z})]^{2}$ for
edge electrons in topological HgTe QWs as a function of the distance
$y_{\text{m}}$ between the acceptor wave function maximum and the edge,
obtained for ${\cal{J}}_{\alpha}$ and $D_{x}$ values display in Fig. 5; the
results are for two electron wavevectors $k=0.05$ and $0.1$ nm-1, for which
the penetration of helical state into the QW $b=7$ and $5$ nm, respectively.
In the case of $r_{Dx}$, the distance between the QW center and the wave
function maximum $z_{\text{m}}=2$ nm (full points) or varies between 0.5 and
2.5 nm with 0.5 nm step (empty triangles); $k=0.1$ nm-1;
$r_{Dx}(z_{\text{m}}=0)=0)$. Lines are linear fits to the computed points.

Finally, we comment on the spin-independent part of the Fock energy
$\Delta_{eh}$, which together with Hartree terms originating from acceptor,
edge, and gate charges, contribute to a self-consistent potential in the edge
region, whose determination is beyond scope of the present work. Nevertheless,
to have an idea about the energy scale involved, we evaluate a contribution to
one-electron energy resulting from the Fock term in a self-consistent way
making use of Eqs. 19 and 20, and summing up over all holes with the areal
density $N_{a}$,

$E_{eh}^{(\text{F})}=\left[\frac{e^{2}N_{a}P^{2}}{4\pi\epsilon_{0}\epsilon}\int
dy|f(y)|^{2}\Delta_{eh}(y)/A\right]^{1/3}.$ (36)

where the dependence $\Delta_{eh}(y)/A$ is displayed in Fig. 5 for two values
of $k$ and $b$. Assuming $f(y)=\exp(-b/y)/\sqrt{b/2}$; $N_{a}=0.5\cdot
10^{11}$ cm-2; $P=8.28\cdot 10^{-8}$ eVcm, and $\epsilon=19.5$ we obtain
$E_{eh}^{(\text{F})}=4.8$ and $4.4$ meV for $b=5$ and 7 nm, respectively.

###  II.7 Kondo temperatures for acceptor holes and Mn spins

####  II.7.1 Acceptor holes

A Kondo collective state results from antiferromagnetic exchange coupling of
Fermi liquid with a single spin localized at ${\vec{R}}_{i}$. An
antiferromagnetic electron-hole interaction leads also to the Kondo effect, as
in the electron picture it corresponds to antiferromagnetic coupling of a
surplus localized electron with the electron Fermi liquid. To estimate an
order of magnitude of Kondo temperatures $T_{\text{K}}$, we use a time-honored
expression Daybell and Steyert (1968) costed to the form,

$k_{\text{B}}T_{\text{K}}\simeq W\exp[-1/(L^{d}\nu_{d}|{\mathcal{J}}_{d}|)],$
(37)

where $W$ is a energy width of carrier-containing states; $\nu_{d}$ is the
carrier DOS per spin, and ${\cal{J}}_{d}$ represents the exchange energy of
the antiferromagnetic interaction between spins of one carrier and a single
localized paramagnetic center. Both $\nu_{d}$ and ${\mathcal{J}}_{d}$
correspond to values at the Fermi level, and depend on the dimensionality $d$
of the Fermi liquid residing in a structure of the volume $L^{d}$.

For the case under consideration, $\nu_{1}=1/2\pi\hbar v_{\text{F}}$, where
$v_{\text{F}}=4\cdot 10^{5}$ m/s Lunde and Platero (2013); Papaj  _et al._
(2016); Krishtopenko and Teppe (2018), and
${\cal{J}}_{1}=({\cal{J}}_{x}+{\cal{J}}_{y})/2$ at given $y_{\text{m}}$, $k$,
$b$, and the energy of the acceptor hole with respect to the Fermi level,
$E_{eh}$. Furthermore, together with $P$ and $\epsilon$ values quoted in
previous sections, we take $W$ as a half of the bandgap, i.e., 15 meV for
$T_{\text{K}}$ estimations.

Figure 7: Calculated magnitudes (points) of Kondo temperatures, obtained for
${\cal{J}}_{x}$ and ${\cal{J}}_{y}$ values display in Fig. 5, for edge
electrons in HgTe QWs containing holes bound to acceptors as a function of the
distance $y_{\text{m}}$ between the acceptor wave function maximum and the
edge calculated for various values of the electron-hole energy interval
$E_{eh}$; full points: electron wavevector $k=0.05$ nm-1 and the penetration
of helical state into the QW, $b=7$ nm; open points: $k=0.1$ nm-1 $b=5$ nm.
Lines connect calculated points.

Several important conclusions emerge from $T_{\text{K}}$ magnitudes presented
in Fig. 7 as a function of $y_{\text{m}}$. First, for acceptor holes localized
close to the edge, $y_{\text{m}}<b$, $T_{\text{K}}$ reaches 100 K, indicating
that the Kondo physics is relevant for the system of edge electrons and
acceptor holes, making competing effects, such as exchange interactions
between hole spins, irrelevant. Second, due to the exponential dependencies of
$T_{\text{K}}$ on ${\cal{J}}_{\alpha}$ and ${\cal{J}}_{\alpha}$ on
$y_{\text{m}}$ there is a sharp cut-off $y{\text{c}}$, beyond which strong
coupling of holes and electrons vanishes. The magnitude of $y{\text{c}}$
depends on $b$ and $E_{eh}$, that is, also on the position of the Fermi level
in the acceptor band. Third, a broad distribution of $T_{\text{K}}$ values
means that at any experimentally relevant temperature, coupling of edge
electrons and holes corresponds to a superposition of weak and strong coupling
limits, including the crossover between them occurring at $T\simeq
T_{\text{K}}$.

####  II.7.2 Kondo effect for Mn spins

For comparison, we evaluate also the magnitude of Kondo temperature for a
transition metal impurity, such Mn, residing at $\vec{R}_{i}$ in a HgTe bulk
sample or in a topological edge channel. A $p$-type part of the Bloch function
is relevant, as the $p$–$d$ exchange integral $\beta$ is antiferromagnetic,
whereas the $s$–$d$ integral $\alpha$ is ferromagnetic Dietl (1994).
Accordingly, ${\cal{J}}_{d}$ appearing in Eq. 37 for $T_{\text{K}}$, can be
written in a form,

${\cal{J}}_{d}=\beta[|F_{3/2}(\vec{R}_{i})|^{2}+|F_{1/2}(\vec{R}_{i})|^{2}/3],$
(38)

where $F_{jz}$ are envelope functions accompanying the Kohn-Luttinger
amplitudes $p_{3/2,j_{z}}$. We see that an upper limit for the bulk case is
${\cal{J}}_{3}=\beta/{\cal{V}}$, where ${\cal{V}}$ is the sample volume.
Similarly, for a Mn ion localized at a topological channel, where
$\vec{R}_{i}=(x,0,0)$ and, thus, ${\cal{J}}_{1}=4\beta/(L_{x}bd_{\text{QW}})$.

Now, knowing that $N_{0}\beta=-0.7$ eV in HgTe Autieri  _et al._ (2021), where
$N_{0}=1.49\cdot 10^{22}$ cm-3 is the cation concentration, we are in position
to evaluate a lower limit of the exponent $w$ in Eq. 37 for $T_{\text{K}}$.
For the bulk case, where $\nu_{3}=k_{\text{F}}m^{*}/2\pi^{2}\hbar^{2}$ and
$m^{*}=0.45m_{0}$ Jȩdrzejczak and Dietl (1976), $w=-1.1\cdot 10^{3}$ for the
hole concentration $p=10^{16}$ cm-3 and $w=-49$ for $p=10^{20}$ cm-3, implying
$T_{\text{K}}\sim 10^{-20}$ K. For topological edge channels in HgTe QWs,
taking $b=5$ nm, $d_{\text{QW}}=8$ nm, and $v_{\text{F}}=4\cdot 10^{5}$ m/s,
we obtain $w=-350$.

Hence, it appears that for standard values of the $sp$-$df$ exchange
integrals, no presence of the Kondo effect is expected for spins tightly
localized on $d$ or $f$ shells in semiconductors with magnetic ions. The same
conclusion holds for nuclear spins for which the hyperfine coupling constant
is at least four orders of magnitude smaller compared to $N_{0}\beta$.

##  III Comparison to experimental results

###  III.1 Topological protection length: acceptor holes and Mn spins

####  III.1.1 Acceptor holes

Our approach to charge transport by topological edge channels in quantum spin
Hall (QSH) materials is built on several pillars put previously forward by
others, discussed in the companion paper Dietl , or elaborated in previous
sections of the present paper. First, charge dopants determine pertinent
properties of 2D topological systems, including the dependence of carrier
density on the gate voltage $V_{\text{g}}$ and the magnitudes of electron and
hole mobilities. Second, owing to a dependence of the dopant binding energy on
the position with respect to the QW center, the acceptors form a band
extending over the whole bandgap. The associated Coulomb gap in the acceptor
hole spectrum controls a contribution of QW states to charge transport in the
QSH regime Dietl . Third, due to a close energetic proximity, exchange
interactions between edge channel electrons and acceptor holes are strong
enough to bring the system to the Kondo regime (Sec. II.7). As known, spin-
dephasing rate reaches a unitary limit in that regime Maciejko  _et al._
(2009); Micklitz  _et al._ (2006). Fourth, since the magnitude of Kondo
temperature $T_{\text{K}}$ exponentially depends on the distance of the
acceptor hole to the edge and on the energy interval to the Fermi level,
$T_{\text{K}}$ values show a broad distribution covering the whole
experimentally relevant temperature range (Sec. II.7). This effect determines
also a spacial region from which acceptor holes contribute to spin-dephasing.
Fifth, it has been emphasized by many authors Tanaka  _et al._ (2011);
Altshuler  _et al._ (2013); Lunde and Platero (2013); Eriksson (2013); Kimme
_et al._ (2016) that because of spin-momentum locking, spin-conserving
transitions imply momentum conservation, i.e., no net backscattering between
helical channels in 2D topological insulators. However, spin-orbit
interactions can result in spin-nonconserving processes described by nonscalar
terms in the exchange coupling between edge electrons and paramagnetic centers
Tanaka  _et al._ (2011); Altshuler  _et al._ (2013); Lunde and Platero (2013);
Eriksson (2013); Kimme  _et al._ (2016). The magnitude of such terms has been
determined in Sec. II.6 for the case of QW holes bound to acceptor impurities.

We are interested in the topological protection length
$L_{\text{p}}=v_{\text{F}}/\gamma_{\text{b}}$, where $\gamma_{\text{b}}$ is a
backscattering rate for helical states. To see its relation to the two
terminal conductance $G=1/R$, we note that $R$ is non-zero due to the quantum
contact resistance $R_{\text{q}}=h/2e^{2}$ and the backscattering term which,
according to the Einstein relation, is given by
$R_{\text{b}}=L^{2-d}\gamma_{\text{b}}(2e^{2}\nu_{d}v_{\text{F}}^{2}/d)^{-1}$.
Hence, in our case, where $d=1$,

$G=(2e^{2}/h)(1+L_{x}/L_{\text{p}})^{-1}.$ (39)

To evaluate $\gamma_{\text{b}}$ we recall a form of the carrier spin-dephasing
rate in Kondo systems,

$\gamma_{s}=Cn_{S}f(T/T_{\text{K}})/2\pi\hbar\nu_{d},$ (40)

where $C$ is a numerical coefficient close to one; $n_{S}$ in the
concentration of magnetic impurities and $f(x)$ is a function obtained by the
numerical renormalization group approach Micklitz  _et al._ (2006); Costi  _et
al._ (2009) that is more accurate than the original Nagaoka-Suhl resummation
result, $f(x)=a/(a+\ln^{2}(T/T_{\text{K}})$, where $a=3\pi^{2}/4$. In either
case, $f(0)=0$; $f(1)\simeq 1$, and $f(10)\simeq 0.6$.

It has been previously noted that compared to $\gamma_{s}$, the backscattering
rate for helical channels is reduced by the exchange anisotropy ratio $r$,
$\gamma_{b}=r\gamma_{s}$ Tanaka  _et al._ (2011); Lunde and Platero (2013);
Kimme  _et al._ (2016). Thus, inverse $L_{\text{p}}$ can be written in an
appealingly simple form Dietl ,

$L_{\text{p}}^{-1}(T)=\sum_{i}r^{(i)}f(T/T_{\text{K}}^{(i)})/L_{x},$ (41)

where the summation is over all QW holes bound to acceptors for a given
$V_{\text{g}}$. As shown in Fig. 6, the $r$ value is not universal, but varies
with the hole position in respect to the edge and QW center, $y_{\text{m}}$
and $z_{\text{m}}$, respectively, and to a lesser degree with $k$ and $b$.
Similarly, according to results presented in Fig. 7, the $T_{\text{K}}$
magnitude strongly depends on $y_{\text{m}}$ and also on $E_{eh}$ that is
controlled by the Fermi level position and $z_{0}$, the distance of the
acceptor to the QW center, as shown in Fig. 1 of Ref. Dietl, . However, just
to see whether we are on the right track, we take the areal hole density as
$N_{\text{a}}=0.5\cdot 10^{11}$ cm-2, an average value of $r^{(i)}$ as
$r_{Dx}(z_{\text{m}}=2$ nm) = 0.13 (see, Fig. 6), the cut-off length beyond
which strong coupling of holes and electrons tends to vanish
$y_{\text{c}}=2b=10$ nm (see, Fig. 7), and an average value of
$f(T/T_{\text{K}})=0.4$. These numbers lead to the linear density
$N_{h}=5/\mu$m of holes participating in backscattering and $L_{\text{p}}=4$
$\mu$m, the order of magnitude consistent with experimental findings König
_et al._ (2007); Lunczer  _et al._ (2019); Majewicz (2019). A small number of
scattering centers has several important consequences, as discussed below.

Figure 8: Configurationally averaged two-terminal conductance $G$ in the units
$2e^{2}/h$ vs. temperature for a HgTe QW of the thickness $d_{\text{QW}}=8$ nm
and length $L_{x}=10$ $\mu$m computed for various energy distance of the Fermi
level to the acceptor state $E_{eh}$; $k=0.1$ nm-1 $b=5$ nm; full points: no
Luttinger liquid effects ($K=1$); empty points: Luttinger liquid effects taken
into account ($K=0.88$). Lines connect calculated points.

Figure 8 shows the temperature dependence of two-terminal conductance $G$ as
defined in Eq. 39, and obtained by integrating $L{\text{p}}^{-1}$ given in Eq.
41 in the $xy$ plane with the areal density of acceptor holes
$N_{\text{a}}=0.5\cdot 10^{11}$ cm-2. The latter changes with the gate voltage
and its overall magnitude vary from sample to sample. However, typically, the
gate voltage width corresponding to the gap region, $\Delta V_{\text{g}}$,
implies the total acceptor density of the order of $10^{11}$ cm-2. The
theoretical results are presented for three values of $E_{eh}$ and $L_{x}$. As
$r(y)$, we have used the values fitted to $r_{Dx}(z_{\text{m}}=2$ nm) in Fig.
6. Similarly, $T_{\text{K}}(y)$ is determined from fitted values of
${\cal{J}}_{x}$ and ${\cal{J}}_{y}$ shown by dashed lines in Fig. 5. Finally,
the function $f(T/T_{\text{K}})$ has been obtained from the data in Fig. 3 for
$d=1$ of Ref. Micklitz  _et al._ , 2006 and from the proposed there high-
temperature extrapolation.

A number of worthwhile conclusions can be drawn from data in Fig. 8. In
particular, lowering of temperature eliminates from backscattering spin
centers with $T_{\text{K}}>T$, so that $L_{\text{p}}$ and $G$ are seen to
steadily increase on cooling. However, the predicted recovery of $G=2e^{2}/h$
at $T\rightarrow 0$ Maciejko  _et al._ (2009); Väyrynen  _et al._ (2016) is
not found, as at any temperature there are centers far away from the channel
for which $T_{\text{K}}\gtrsim T$. Nevertheless our results are not in accord
with experimental findings in a sense that no systematic increase of $G$ on
decreasing $T$ has been observed. This disagreement can point to the onset of
localization, predicted for disordered Kondo systems Altshuler  _et al._
(2013) or to the presence of Tomonaga-Luttinger effects Maciejko  _et al._
(2009); Väyrynen  _et al._ (2016). In has been shown Väyrynen  _et al._ (2016)
that renormalization group equations combining Kondo and carrier correlation
phenomena imply a growth of the exchange anisotropy ratio $r$ with lowering
temperature down to $T^{*}$ according to,

$r(T)=r(W)(W/k_{\text{B}}T)^{2}(1-K),$ (42)

with

$k_{\text{B}}T^{*}=W\left[\frac{L_{x}\nu_{1}|{\cal{J}}_{1}|}{\sqrt{2}(1-K)}\right]^{1/(1-K)},$
(43)

where the Luttinger parameter $K<1$ for interacting 1D systems. As shown in
Fig. 8, incorporation of $r(T)$ into our theory makes $G$ virtually
independent of $T$ for a rather moderate interaction magnitude $K=0.88$. It
should be, however, noted that at any temperature only a part of acceptor
holes resides sufficiently far from the edge to satisfy the condition
$T>T^{*}$.

Up to now, we have discussed configuration-averaged behavior of an acceptors’
containing system. However, a small number of acceptor holes involved in
micron-size samples means that $G(V_{\text{g}}$ will show strong mesoscopic-
like conductance fluctuations, as observed König  _et al._ (2007); Grabecki
_et al._ (2013); Shamim  _et al._ (2021); Bubis  _et al._ (2021). Furthermore,
if at given $V_{\text{g}}$, acceptor holes close to the edge dominate for
which $T_{\text{K}}>T$, $f(T/T_{\text{K}})$ will increase and, hence, $G(T)$
decrease with temperature. By contrast, for distance acceptors
$T_{\text{K}}<T$, so that a falling down of $f(T/T_{\text{K}})$ with $T$ works
together with the Tomonaga-Luttinger effects to result in an increase of
$G(T)$ on heating. Such changes in sign of $dG(T)/dT$ with $V_{\text{g}}$ have
been observed experimentally, though regions with $dG(T)/dT>0$ appear to
prevail Bubis  _et al._ (2021).

####  III.1.2 Backscattering by Mn spins and precessional spin dephasing

As there are typically $10^{4}$ more Mn spins with $S=5/2$ in Hg0.99Mn0.01Te
compared with the number of acceptor holes, a question arises about the role
of Mn-induced backscattering of edge electrons. As the number of Mn ions in
the edge region is also much larger than the number of edge electrons, we
consider Mn spins in a continuous and classical approximation. Such an
approximation is not only employed in the description of static and dynamic
spintronics functionalities of ferromagnetic metals but has also been found
versatile and quantitatively accurate in the case of bound magnetic polarons
and single quantum dot electrons immersed in a nuclear spin bath in
semiconductors Dietl (2015) as well as when considering carrier-mediated
ferromagnetism in DMSs Dietl and Ohno (2014). Within this approach, it is
convenient to expand local magnetization $\vec{M}(\vec{r})$ into a Fourier
series,

$\vec{M}(\vec{r})=\sum_{\vec{q}}\vec{\eta}_{\vec{q}}\exp(i\vec{q}\cdot\vec{r})/\sqrt{{\cal{V}}}.$
(44)

For a uniform distribution of Mn ions over the volume ${\cal{V}}$ encompassing
the QW and under thermal equilibrium conditions, the fluctuation-dissipation
theorem implies,

$\langle\eta^{(\alpha)}_{\vec{q}}\eta^{(\beta)}_{\vec{q\prime}}\rangle=k_{\text{B}}T\chi_{\alpha,\beta}(\vec{q})\delta_{\vec{q},\vec{-q}\prime},$
(45)

where in cubic systems and in the absence of a magnetic field, the magnetic
susceptibility tensor
$\chi_{\alpha,\beta}(\vec{q})=\delta_{\alpha,\beta}\chi_{\text{Mn}}(q,T)$. We
then apply a standard weak-coupling procedure (Fermi’s golden rule) for spin-
flip transitions between 1D helical states,
$\psi_{e}^{(+)}(\vec{r})\rightarrow\psi_{e}^{(+)}(\vec{r})$, and adopt the
form of $\psi_{e}^{(\pm)}(\vec{r})$ given in Eq. 8. The scattering rate
becomes,

$\gamma_{\text{Mn}}=\frac{k_{\text{B}}T\chi_{\text{Mn}}(T)}{2\hbar^{2}g^{2}\mu_{\text{B}}^{2}v_{\text{F}}}\int
d\vec{\mathtt{r}}[\alpha^{2}|f_{1}(\vec{\mathtt{r}})|^{4}+\beta^{2}(|f_{3}(\vec{\mathtt{r}})|^{4}+|f_{4}(\vec{\mathtt{r}})|^{4})],$
(46)

where we have assumed that inverse $k$ and characteristic carrier confinement
lengths are much longer than an average distance between magnetic ions, so
that the $q$-dependence of magnetic susceptibility can be neglected, and
$g=2.0$ is the Landé factor of Mn spins. For a paramagnetic case and using the
form of the envelope functions given in Eq. 35,

$\gamma_{\text{Mn}}=\frac{S(S+1)xN_{0}}{4\hbar^{2}v_{\text{F}}bd_{\text{QW}}}[\alpha^{2}|e_{s}|^{4}+\beta^{2}(|e_{H}|^{4}+|e_{p}|^{4})].$
(47)

For the $s$–$d$ exchange energy $N_{0}\alpha=0.3$ eV Autieri  _et al._ (2021),
$x=0.01$ and other parameter values quoted in the previous paragraphs, the
topological protection length becomes, $L_{\text{p}}=7.2$
$\mu$m/$r_{\text{Mn}}$, where $r_{\text{Mn}}<1$ is a relevant exchange
anisotropy ratio. In principle, $r_{\text{Mn}}\ll 1$ for magnetic ions with
the orbital momentum $L=0$. There are however mechanisms that can enlarge
$r_{\text{Mn}}$. In particular, Mn ions residing out of the QW center locally
break the inversion symmetry, which leads to the appearance of anisotropic
exchange Kimme  _et al._ (2016), similarly to the case of spin scattering by
acceptor holes (Sec. II.6). This mechanism should work at $x\rightarrow 0$.

However, beyond the limit $x\rightarrow 0$, interactions between magnetic ions
is relevant. While the exchange mediated by edge electrons is weak Dietl and
Ohno (2014), antiferromagnetic superexchange is significant. This interaction
makes $\gamma_{\text{Mn}}$ to decrease with lowering temperature even stronger
than $T\chi(T)$ implies because spin scattering becomes inelastic. In the
spin-glass phase $T\leq T_{\text{g}}(x)$, Mn spins cease to contribute to
quantum decoherence and, accordingly, a recovery of universal conductance
fluctuations was found in nanostructures of Hg0.93Mn0.07Te at $T<T_{\text{g}}$
Jaroszyński  _et al._ (1998). All that might mean that Mn spins play a minor
role in backscattering.

Actually, it was noted that the presence of random Rashba fields effectively
enhances the $r$ value Kimme  _et al._ (2016). We argue that there are two
other effects. First, as discussed in the next section, magnetic poloron
formation around acceptor holes considerably weakness the Kondo effect and
associated backscattering in Hg1-xMnxTe compared to HgTe. Secondly, we suggest
that dephasing of carrier spins by a dense bath of interacting magnetic
moments originates in semiconductors from a chain of spin precession events
generated by local magnetization vectors rather than from flip-flop processes
each involving a single magnetic ion.

In order to evaluate the precession-induced dephasing rate, we follow a time-
honored Dykonov-Perel motional-narrowing approach to relaxation rates by spin-
orbit fields,
$\gamma_{\text{so}}\approx\tau_{\text{m}}\langle(g^{*}\mu_{\text{B}}B_{\text{so}}/2\hbar)^{2}/\rangle$,
where, in the case under consideration, the motion time of the carrier wave
packet extending over $\ell_{x}$, $\tau_{\text{m}}=\ell_{x}/v_{\text{F}}$ with
$k^{-1}\gg\ell_{x}\gg(N_{0}x)^{-1/3}$. This approach assumes that dynamics of
Mn spins is slow, $\tau_{\text{m}}(\ell_{x})<\tau_{\text{c}}(\ell_{x})$, where
$\tau_{\text{c}}$ is the Mn correlation time. Qualitatively, the magnitude of
local magnetization seen by a moving edge carrier,

$\langle
M_{\alpha}M_{\beta}\rangle=\delta_{\alpha\beta}k_{\text{B}}T\chi_{\text{Mn}}(T)\int
d\vec{r}|\psi(\vec{r})|^{4}.$ (48)

For the wavepacket,

$\displaystyle\psi(\vec{r})$ $\displaystyle=$
$\displaystyle\frac{2\exp(ikx-x^{2}/4\ell_{x}^{2}-y/b)\Theta(y)}{(2\pi)^{1/4}(\ell
bd_{\text{QW}})^{1/2}}$ (49) $\displaystyle\cdot\cos(\pi
z/d_{\text{QW}}\Theta(|z|-d_{\text{QW}}/2),$

and noting that spin splitting $\Delta={\cal{J}}_{sp-d}M/g\mu_{\text{B}}$ we
arrive to,

$\gamma_{\text{Mn}}=\frac{3S(S+1)xN_{0}{\cal{J}}_{sp-d}^{2}}{8\sqrt{\pi}\hbar^{2}v_{\text{F}}bd_{\text{QW}}}.$
(50)

We see that within a numerical factor of the order of one, the precession
approach leads to the same expression, as given in Eqs. 46 and 47. However,
within such a model, similarly to the case of spin-transfer torque and
electron precession around nuclear spins in a quantum dot, a change of
carriers’ spin momentum associated with backscattering is absorbed by the
ensemble of Mn spins, so that $r$ approaches 1. We conclude that
backscattering by Mn spins may not be negligible in Hg1-xMnxTe QWs, even in
the absence of Rashba fields.

Finally, we return to the role of spin dynamics. The magnitude of
$\tau_{\text{c}}$ is controlled by Mn spin diffusion and relaxation,
determined by scalar and non-scalar terms in the Mn-Mn interaction
Hamiltonian, respectively Dietl  _et al._ (1995). We note that the mechanisms
accounting for finite $\tau_{\text{c}}$ weakens building up of Mn
magnetization by electric current. Furthermore, it was suggested that spin
dynamics could promote depining of edge carriers localized by Kondo impurities
Altshuler  _et al._ (2013).

###  III.2 Magnetic polaron gap and zero-field spin-splitting

We consider again isoelectronic magnetic impurities, such as Mn in II-VI
compounds, in the paramagnetic phase. The presence of $sp$-$d$ exchange
interactions affect, by the bound magnetic polaron (BMP) effect, donor
electrons of acceptor holes even in the absence of macroscopic magnetization.
Optical studies provided the evidence for the presence of the acceptor BMP in
Hg1-xMnxTe Choi  _et al._ (1990); Zhu  _et al._ (2015). A contribution of the
BMP to thermally activated band conductivity Jaroszyński  _et al._ (1983) and
to the Coulomb gap in the hoping region Terry  _et al._ (1992) was found in
Cd1-xMnxTe.

According to the analytical solution of the central spin problem, the polaron
energy $\epsilon_{\text{p}}$ determines BMP energetics and thermodynamics in
the absence of an external magnetic field Dietl and Spałek (1982),

$\epsilon_{\text{p}}(T)={\cal{J}}_{sp-d}^{2}\chi_{\text{Mn}}(T)/[32(g\mu_{\text{B}}N_{0})^{2}\pi
a_{\text{B}}^{3}],$ (51)

where the magnitude of the exchange energy ${\cal{J}}_{sp-d}$ is given here by
the values of the $s$–$d$ and $p$–$d$ exchange integrals $N_{0}\alpha$ and
$N_{0}j_{z}\beta/3$, respectively, weighted by the corresponding orbital
content of the acceptor wave function; $\chi_{\text{Mn}}(T)$ is the Mn
susceptibility in the absence of acceptors; $a_{\text{B}}$ is the acceptor
localization radius determined from the participation number. In the case of
the doubly-occupied acceptor with $J=1$, $\epsilon_{\text{p}}(T)$ would be
four times greater.

We identify the polaron gap between occupied and non-occupied acceptor
centers, $E_{\text{P}}$, as the twice Fermi energy shift associated with the
polaronic effect. For the doubly-degenerate acceptor state $E_{\text{P}}$
assumes the form Jaroszyński  _et al._ (1983),

$E_{\text{P}}=\epsilon_{\text{p}}+2k_{\text{B}}T\ln(1+\epsilon_{\text{p}}/k_{\text{B}}T).$
(52)

For the $E_{3/2}$ state, ${\cal{J}}_{sp-d}=-0.7$ eV implying, neglecting
antiferromagnetic interactions between Mn spins, $E_{\text{P}}=1.7$ meV at
$T=2$ K, $x=0.024$, and $a_{\text{B}}=5$ nm. Similarly, for the more relevant
$E_{1/2}$ case, ${\cal{J}}_{sp-d}=-0.24$ eV, for which $E_{\text{P}}=0.31$
meV. For the $x$ value in question there are about 500 Mn ions within the
volume visited by the acceptor hole. We conclude that the formation of BMPs in
Hg1-xMnxTe may substantially enhance the magnitude of the gap at the Fermi
level in the acceptor band compared to the case of HgTe.

We are also interested in the magnitude of acceptor hole spin-splitting
$\Delta$. If larger than $k_{\text{B}}T_{\text{K}}$, BMPs reduce spin
dephasing in the Kondo regime and, thus, backscattering of edge electrons,
which improves the precision of resistance quantization in the quantum spin
Hall effect regime. The most probable magnitude of zero-field splitting
$\bar{\Delta}$ is given by an implicit equation Dietl and Spałek (1982),

$\displaystyle\bar{\Delta}^{2}-2\Delta\epsilon_{\text{p}}(T)\tanh(\bar{\Delta}/2k_{\text{B}}T)-8\epsilon_{\text{p}}(T)k_{\text{B}}T=0.$
(53)

For the parameters quoted above we obtain $\bar{\Delta}=2.7$ meV and 0.54 meV
at 2 K for the $E_{3/2}$ and $E_{1/2}$ level, respectively.

In the particular case of the sample with $x=0.012$ studied in Ref. Shamim
_et al._ (2021) and for the level $E_{1/2}$, $\bar{\Delta}\geq k_{B}T$ at
$T\leq 3.5$ K. It is, therefore, clear that the presence of Mn, $via$ the
formation of BMPs and the associated diminishing of the role played by the
Kondo effect, can significantly reduce backscattering of edge electrons,
leading to the recovery of quantized resistance at low temperature in
magnetically doped QWs, as observed in a Hg0.988Mn0.012Te QW at 0.2 K Shamim
_et al._ (2021).

##  IV Conclusions

The quantitative results presented here support the view that acceptor states
play a crucial role in the physics of quantum spin Hall effect in HgTe quantum
wells and related systems. On the one hand, discharging of acceptors accounts
for a non-zero width of the quantized plateaus and, on the other, the strong
Kondo coupling of edge electrons and acceptor holes leads to the unitary limit
of spin-flip scattering. A non-zero orbital momentum specific to $p$-type
Kohn-Luttinger amplitudes together with breaking of axial and inversion
symmetry by the edge and off center hole location allows for flow of edge
electron spin momentum to crystal orbital momentum and, thus, for efficient
backscattering, despite spin-momentum locking. Interestingly, the formation of
bound magnetic polarons in magnetically doped samples weakness this process,
as observed Shamim  _et al._ (2021).

As discussed in the companion paper Dietl , the acceptor band model
qualitatively elucidates several surprising properties in the vicinity of the
topological phase transition, where $E_{\text{g}}\simeq 0$ and, thus, acceptor
form resonant states. In particular, gate-induced discharging of the acceptor
states explains an unexpectedly slow rise of the itinerant hole concentration
with increasing negative gate voltage Shamim  _et al._ (2020); Yahniuk  _et
al._ and unusually wide integer quantum Hall plateaus in the same gate region
König  _et al._ (2007); Yahniuk  _et al._ (2019); Shamim  _et al._ (2020).
Furthermore, the BMP-induced reduction of Kondo carrier binding by acceptor
holes and the hardening of the Coulomb gap account for the hole mobility as
large as $0.9\cdot 10^{6}$ cm2/Vs at hole density as low as $p=3.5\cdot
10^{9}$ cm-2 in a Hg0.98Mn0.02Te QW at 20 mK Shamim  _et al._ (2020). In the
same way, the model qualitatively explains $\mu=20\cdot 10^{6}$ cm2/Vs at the
electron concentration $n=2.5\cdot 10^{13}$ cm-3 at 2 K in a bulk
Hg0.94Mn0.06Te under hydrostatic pressure, driving the system to the
topological phase transition Sawicki  _et al._ (1983).

We have also examined backscattering of electrons in helical states by
localized spins in DMSs. We have argued that precessional spin dephasing by
magnetization spacial fluctuations leads to sizable backscattering even in the
spin-momentum locking case, as the electron spin momentum is transferred to
the magnetic subsystem rather than to an individual magnetic ion.

## Acknowledgments

This work 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.

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|  Bond breaking and making in mixed clusters of fullerene and coronene
molecules following keV-ion impact
---|---
|  Naemi Florin,∗a Alicja Domaracka,b Patrick Rousseau,b Michael Gatchell,a
and Henning Zettergrena
|  We have performed classical molecular dynamics simulations of 3 keV Ar +
$(\mathrm{C}_{24}\mathrm{H}_{12})_{n}(\mathrm{C}_{60})_{m}$ collisions where
$(n,m)=(3,2),(1,4),(9,4)$ and $(2,11)$. The simulated mass spectra of
covalently bound reaction products reproduce the main features of the
corresponding experimental results reported by Domaracka et al., PCCP, 2018,
20, 15052. The present results support their conclusion that molecular growth
is mainly driven by knockout where individual atoms are promptly removed in
Rutherford type scattering processes. The so formed highly reactive fragments
may then bind with neighboring molecules in the clusters producing a rich
variety of growth products extending up to sizes containing several hundreds
of atoms, and here we show examples of such structures. In addition, knocked
out atoms may be absorbed such that e.g. hydrogenated coronene and fullerene
molecules are formed.

##

††footnotetext: a Department of Physics, Stockholm University, 106 91
Stockholm, Sweden. E-mail<EMAIL_ADDRESS>b Normandie
Univ, ENSICAEN, UNICAEN, CEA, CNRS, CIMAP, 14000 Caen, France. ††footnotetext:
† Electronic Supplementary Information (ESI) available: Additional figures for
small clusters containing five molecules. See DOI: 10.1039/cXCP00000x/

##  1 Introduction

The formation and breakage of bonds are at the heart of chemical reactions and
govern the evolution of molecules in terrestrial and extraterrestrial
environments.1 In recent years, there have been numerous studies reporting
molecular growth processes in gas-phase molecular clusters ignited by
different ionization and excitation agents (photon, electron, and ion impact).
Examples include peptide bond formation in clusters of amino acids,2, 3
formations of aromatic rings in clusters of small carbon chains,4 and growth
of complex carbon nanostructures in pure and mixed clusters of Polycyclic
Aromatic Hydrocarbons (PAHs) and fullerenes.5, 6, 7, 8, 9, 10 The cluster
environment is key here as it allows the excitation energy to be shared among
its molecular constituents such that the reaction products are formed
sufficiently cold to survive on extended timescales (microseconds and beyond).
The experiments reported so far have used various mass spectrometry techniques
to provide information on the mass-to-charge ratios of the reaction products,
which sometimes, as in the case of fullerenes, reveal the preferred cluster
packing types.11, 12 In general, however, they do not provide any structural
information of the cluster structures before they are exposed to radiation in
different forms, nor for any new species formed following such interactions.
Complementary theoretical studies aiming to determine inherent cluster
properties such as their stabilities and structures are thus needed to
characterize the experimental targets. Pioneering theoretical studies have for
instance shown that the most stable PAH clusters contain single or multiples
stacks depending on the cluster size13, while cluster of fullerenes prefer
icosahedral packing. This is consistent with mass spectrometry observations of
so-called magic numbers corresponding to particularly stable cluster sizes.14,
15 These structures may then be used as input in studies of intra-cluster bond
breaking and formation processes by means of, e.g., classical molecular
dynamics simulations to reveal detailed information about the reaction
pathways and final reaction products (structures and internal energies).

Fig. 1: The most stable (C24H12)9(C60)4 and (C24H12)2(C60)11 cluster
structures according to the calculations reported in Ref. 16.

In this work, we have performed such simulations of collision induced chemical
reactions inside of mixed clusters of PAH (C24H12) and fullerene (C60)
molecules. Here, we use the optimized structures from Ref. 16, see Fig. 1 for
two examples, and bombard them with 3 keV Ar-atoms mimicking the conditions in
the experimental study by Domaracka _et al_.10 There, they showed that a large
variety of new covalently bound reaction products are formed in the
collisions, and that their yields depend on the mixing of C24H12 and C60 in
the cluster targets. For clusters containing low amounts of C60, the molecular
growth products are mainly coronene-based and there are no signs of reactions
products containing C60-molecules alone, while growth products containing both
species become important as the C60 content in the clusters is increased. The
newly formed products were attributed to so-called knockout-driven reactions
as have been observed to drive the ion-induced chemistry in, e.g., pure
clusters of PAHs and fullerenes.5, 6, 7, 8 Single or multiple atoms are then
promptly removed in Rutherford-type scattering processes forming highly
reactive fragments inside the clusters that readily form new bonds with
neighboring molecules.17 These types of molecular growth processes are
particularly important in the so-called nuclear stopping regime, i.e. for
heavy keV-ion impact (velocities _below_ a few hundred km/s). In contrast,
lighter and faster keV projectiles (velocities _above_ a few hundred km/s)
mainly deposit energy to the molecular electron clouds (electron stopping)
that typically lead to strong cluster heating followed by evaporation of
intact molecules, albeit with a small probability for knockout driven
reactions.18, 5, 9, 7 In case the molecular building blocks are not as rigid
as fullerenes and PAHs, as in the case of, e.g., small carbon chains and amino
acids, molecular growth processes are more likely to also occur following
distant electron transfer processes. Ionization is often the main driving
force under such circumstances and the results are then similar to those
observed in electron impact induced molecular growth.

Here, we show that the results by Domaracka _et al._10 can be rationalized by
knockout-driven reactions, which we demonstrate through the close agreement
between their experimental and our simulated mass spectra for cluster targets
with different PAH-fullerene mixing ratios and cluster sizes. The present
results provide information on the structures of the covalently bound reaction
products, which range in complexity from hydrogenated coronene and fullerene
molecules to complex hydrocarbon and pure carbon structures containing
hundreds of atoms.

##  2 Methods

The MD simulations were carried out using the open source MD software LAMMPS
(Large-scale Atomic/Molecular Massively Parallel Simulator).19, 20 Our
approach follows that previously used to successfully model collisions between
keV atoms and pure PAH or fullerene clusters7, 8 and is briefly described
here. The AIREBO (Adaptive Intermolecular Reactive Empirical Bond Order)
potential21, 22 was used to describe bonding interactions between atoms within
each PAH and fullerene molecules, as well as to describe the long-range
interactions between molecules in the clusters. This type of reactive force
field allows bonds to be broken and formed dynamically in a realistic manner
during a simulation run, which is essential for this type of study. The
interactions between the projectile and the target were governed by the ZBL
(Ziegler-Biersack-Littmark) potential.23 The ZBL potential describes nuclear
scattering processes by treating two colliding atoms as point charges screened
by their respective electron clouds.

As targets, we used optimized geometries identified in our earlier study on
mixed coronene-C60 clusters.16 From those results, we selected the lowest
energy structures for four different cluster compositions,
$(\mathrm{C}_{24}\mathrm{H}_{12})_{n}(\mathrm{C}_{60})_{m}$ where
($n,m$)=(3,2), (1,4), (9,4), and (2,11) as targets for our simulations. These
were selected as they represent two distinct cluster sizes—5 and 13 molecules
in total, respectively—and four different mixing ratios that are close to
those reported by Domaracka _et al._10

In each simulation, a whole cluster as a unit was centered and randomly
orientated at the origin of a cubic box with 200 Å sides. We used 3 keV argon
atoms as projectiles, which were fired from a starting point at a fixed $z=15$
Å distance from the target center and parallel to the $z$ axis. The initial
$x$ and $y$ coordinates of the projectile was randomized in each run within a
square that was slightly larger than the projected size of the target cluster.
This setup ensured that all orientations and impact parameters within the
chosen square cross section were sampled with correct probabilities. To
improve computational efficiency, runs where the projectile would pass outside
the cluster and transfer only a negligible amount of energy (below about 1 eV)
were discarded before they were carried out. The simulations that were run
used a time step of $5\times 10^{-17}$ s for $2\times 10^{5}$ steps, giving a
total simulation time of 10 ps ($10^{-11}$ s).

Due to the force fields used, the simulations are limited to only describing
nuclear scattering processes between neutral projectiles and targets. For the
projectile used here, 3 keV Ar, electronic stopping will be significantly
weaker than nuclear stopping23 and its influence on the reactivity of
fragments within the clusters can safely be neglected.7 As for the charge,
coronene and C60 molecules are large, stable molecules with many delocalized
electrons. This gives them properties (e.g., dissociation and binding
energies) that change very little between neutral systems and ions with low
charge states. Our results are thus directly comparable to the experimental
results for cationic species. This has also been verified by past comparisons
between experiment and theory.7, 8, 24

Fig. 2: Upper panels: Mass spectra from molecular dynamics simulations of 3
keV Ar-projectiles colliding with
$(\mathrm{C}_{24}\mathrm{H}_{12})_{9}(\mathrm{C}_{60})_{4}$ (left) and
$(\mathrm{C}_{24}\mathrm{H}_{12})_{2}(\mathrm{C}_{60})_{11}$ (right). Lower
panels: Experimental mass-to-charge spectra for 3 keV Ar+ colliding with broad
distributions of mixed clusters of coronene and fullerene molecules 10. The
coronene/fullerene mixing ratios are R=2.8 and R=0.15 in the left and right
panel, respectively.

##  3 Results and discussion

We find that collisions with the larger cluster targets ($N=13$ molecules)
give reaction products extending up to maximum sizes that are slightly larger
than in collisions with the smaller clusters ($N=5$), which is expected since
more hydrocarbon material is available for bond forming reactions in larger
clusters. Apart from that, similar type of reaction products are produced in
collisions with all four cluster targets, suggesting that the growth
mechanisms are the same. The mixing ratio and cluster size instead merely
affects the branching ratios for the final distributions of reaction products.
In the following we will primarily focus on the growth mechanisms and the
structures of the products and we therefore only show the results for clusters
containing 13 molecules. The corresponding results for the smaller clusters
are given in the supporting information.

###  3.1 Simulated and experimental mass spectra

Figure 2 shows the simulated mass spectra for the cluster targets,
$(\mathrm{C}_{24}\mathrm{H}_{12})_{9}(\mathrm{C}_{60})_{4}$ (upper left
panel), $(\mathrm{C}_{24}\mathrm{H}_{12})_{2}(\mathrm{C}_{60})_{11}$ (upper
right panel), together with experimental results from Domaracka _et al._10 for
collisions with mixed clusters containing mostly coronene (lower left panel)
and fullerene molecules (lower right panel). In the simulated spectra, only
the covalently bound reaction products are shown, while in the experiments it
is not possible to distinguish those from any weakly bound cluster precursors
and remnants or larger clusters having the same mass. For this reason the
relative intensities of peaks at masses corresponding to intact clusters are
not directly comparable between experiment and theory. The four panels display
similar features with a wide range of reaction products spanning in size from
the intact coronene monomer ($m=300$ u) to slightly above that of the intact
fullerene trimer ($m=2160$ u). The most feature-rich spectrum from the
simulations is for
$(\mathrm{C}_{24}\mathrm{H}_{12})_{9}(\mathrm{C}_{60})_{4}$, which is
consistent with the experimental results showing a propensity for intracluster
reactions in clusters containing a higher fraction of coronene molecules
(lower left panel).

###  3.2 Covalently bound intact and defected mixed dimers

Fig. 3: Comparisons of simulated (blue) and experimental10 (black) mass
spectra in the region of the mixed dimer. The upper and lower panels show the
comparisons for clusters with high and low coronene contents, respectively.
Examples of reaction products from the simulations are shown in Fig. 4. Fig.
4: Examples of covalently bound reaction products with masses m=1008, 1020,
and 1032.

In Fig. 3 we show the simulated and measured mass spectra in the region of the
mixed dimer, $(\mathrm{C}_{24}\mathrm{H}_{12})(\mathrm{C}_{60})$, for which
$m=1020$. The upper two panels show the theoretical and experimental results
for targets with a high coronene content, while the lower two panels show the
equivalent data for clusters with a low coronene content. The simulated mass
spectra have been convoluted with a gaussian of width $\text{FWHM}=0.5$ u such
that the peak widths are similar to those in the corresponding experimental
mass spectra in order to aid the comparisons. The effective widths of the
peaks in Fig. 3 are the result of the underlying distributions of the number
of H atoms for fragments of similar masses. The simulations reproduce the main
features of the experimental results remarkably well. For collisions with
clusters containing a dominant fraction of coronene molecules the simulated
and experimental mass spectra contain clear peaks corresponding to reaction
products ranging in size from five carbon atoms less than the intact dimer
($m=960$ u) up to three additional carbon atoms ($m=1056$ u). These peaks are
separated by one carbon mass, which is a clear fingerprint for knockout driven
reactions with PAHs and fullerenes.5, 7, 17 Here, highly reactive fragments
from such processes form bonds with intact or defective neighbors leading to
reaction products with $m<1020$ u. Alternatively, knocked-out carbon atoms are
absorbed by molecules which then react with neighbors such that the reaction
products are heavier than the intact dimer ($m>1020$ u). In addition,
covalently bound reaction products with $m=1020$ u are formed in, e.g., fusion
of two intact molecules or in reactions with damaged molecules and molecules
that have absorbed one or several knocked out atoms. Typical examples of these
three types of reaction products are shown in Fig. 4. The left column shows
two species with mass $m=1008$ u. The upper structure is a coronene fragment
after single carbon knockout forming a single bond with an intact fullerene,
while the lower is a coronene that has lost one carbon atom and one hydrogen
atom and is forming a covalent bond with a fullerene that has absorbed a
knocked out hydrogen atom. The middle column shows structures corresponding to
the mass of an intact mixed dimer. In most cases such species are formed in
reactions between damaged but non-fragmented coronenes and intact fullerenes
as illustrated in the upper figure, but there are also examples where a
knockout fragment reacts with a molecule that has absorbed atom(s) with the
same mass as the knocked out atom(s) from the other molecule (not shown).
Furthermore, species consisting purely of carbon atoms (C85) are formed in
collisions that have lead to severe damage of a fullerene that binds to a
neighboring fullerene (see the lower middle panel of Fig. 4). Finally, the
right column of Fig. 4 shows structures with $m=1032$ u corresponding to the
addition of one carbon atom to an intact mixed dimer. An example of the most
typical binding situation is shown in the upper panel where a displaced carbon
is acting as a link between two intact molecules in a similar fashion as
dumbbell fullerene dimers (C121) observed when atomic carbon reacts with
fullerenes in helium nanodroplets.25 Another, more exotic, example is shown
below this where the fullerene and coronene molecules are bridged by two new
bonds, one where a H atom has been stripped from the PAH and one where an
additional CH fragment has formed a bond between the pair.

Fig. 5: Relative intensities of reaction products in the vicinity of the mixed
dimer (m=1020 u). Left: MD simulations. Right: Experimental results 10.

Fig. 5 shows a closer comparison of the simulated and experimental mass
distribution near 1020 u stemming from collisions with clusters containing a
high fraction of coronene molecules. The experimental distribution has been
extracted from the higher resolution spectrum reported in Ref. 10 where
individual peaks separated by one hydrogen mass are resolved. Peaks
corresponding to one or two hydrogen atoms fewer than for an intact mixed
dimer are clearly visible in the simulations and experiments. In the latter
case there is also a peak at $m=1021$ u corresponding to one additional
hydrogen, which is not seen in the present simulations. We attribute this to
the small cluster size in comparison with the typical ones in the experiments,
i.e. that there is a significantly lower probability that knocked out hydrogen
atoms are absorbed by a molecule in small cluster and hence secondary bond
formation processes involving such species and their neighbors are extremely
rare. In the simulations, there is a significantly higher fraction of $m=1020$
u in relation to the smaller masses compared to the experiments. This suggests
that the internal energies of such reaction products often are too high for
them to survive on the experimental timescales. The lower-mass products in
this range are expected to be more stable as hydrogen atoms have been lost
from coronene, which produces reactive sites where strong bonds to neighboring
molecules can be formed. The secondary loss of hydrogen after a bond-forming
reaction has taken place could also assist in stabilizing reaction products on
experimental timescales, an aspect not fully captured by the short timescales
of our simulations.

###  3.3 Dehydrogenated and hydrogenated coronene molecules

Fig. 6: Relative intensities of covalently bound reaction products in the
vicinity of the intact coronene molecule (m=300 u). Left: MD simulations.
Right: Experimental results 10. The inset shows the most common structure of
singly hydrogenated coronene (m=301 u) formed in the simulations.

Fig. 7: Examples of large covalently bound reaction products (see text).

The conclusion that hydrogen atoms are more likely to be absorbed by their
neighbors in collision with large clusters is further supported when comparing
the simulated and experimental mass spectra in the region of the intact
coronene molecule (see Fig. 6). The simulated mass spectrum shows that up to
four hydrogen atoms are knocked out ($m=296\ldots 299$ u) and in some cases
one hydrogen atom is picked up by another coronene molecule in the cluster
($m=301$ u). These reaction products are also seen in the experimental mass
spectrum, as are products from the loss of five and six hydrogen atoms
($m=295$ u and 296 u) and the absorption of 2–4 additional hydrogen atoms
($m=302\ldots 304$ u) in relation to the pristine coronene molecule. The
present simulations show that the hydrogen atom is predominantly attached to
the outer rim of the coronene molecule such that a CH2-group is formed (see
the inset in Fig. 6). In the experiments, more than one additional hydrogen is
common and it is likely that they are also predominantly located at different
positions at the outer rim. Such hydrogenated species have been suggested to
contribute to formation of molecular hydrogen in the interstellar medium and
may thus play key roles in star formation.26 In the simulations we also see
that hydrogenated fullerenes are formed, which are believed to exist in
circumstellar and interstellar environments.27

###  3.4 Exotic reaction products

In some cases the collisions ignite molecular growth processes where exotic
structures are formed through multiple bond breaking and forming processes.
These typically involve fusion of molecules that have been severely damaged by
the projectile. A few examples are highlighted in Fig. 7. The upper panel
shows two structures stemming from reactions involving one damaged coronene
and a fullerene that to a larger extent retains its original structure. These
are connected through two bonds forming large ring structures from the break-
ups of two and three rings in coronene, respectively. The left structure in
the second row stems from two damaged coronene molecules that are attached to
a fullerene. On the right hand side there is a pure carbon cluster consisting
of 116 atoms from a reaction of an intact fullerene with a large and open
fullerene fragment. The bottom two panels in Fig. 7 show the largest
covalently bound reaction products formed in collisions with clusters with a
high fraction of coronene and fullerene molecules, respectively. In the former
case, C289H23 is produced and consists of a significant fraction (63 %) of the
carbon atoms in the
$(\mathrm{C}_{24}\mathrm{H}_{12})_{9}(\mathrm{C}_{60})_{4}$ precursor cluster.
The fraction is even larger in collisions with
$(\mathrm{C}_{24}\mathrm{H}_{12})_{2}(\mathrm{C}_{60})_{11}$ where the largest
reaction product, C595 as shown in Fig. 7, contains 84 % of the carbon atoms
in a complex that largely preserves the structure of the fullerene component
of the precursor cluster.

##  4 Summary and conclusions

We have studied molecular growth processes in weakly bound mixed clusters of
coronene and fullerene molecules by means of classical molecular dynamic
simulations. These are ignited by 3 keV Ar impact and we find that:

  * •

The simulated mass spectra reproduce the experimental ones reported by
Domaracka _et al._10 Minor differences can be attributed to i) the different
timescales and hence survival probabilities probed in the simulations (up to
10 picoseconds) and in the experiments (up to microseconds), and ii) that
there is a broad distribution of cluster sizes in the experiments, while there
is a well-defined but rather small cluster size in the simulations.

  * •

The actual reaction products do not depend strongly on the fullerene/coronene
content in the clusters investigated in this study, only the branching ratios
for the final states.

  * •

There is a rich distribution of growth products extending in size from one
mass above the intact molecular cluster building blocks to pure carbon or
hydrocarbon structures containing hundreds of atoms. A plethora of new species
are thus formed with different and unique spectroscopic signatures (IR-
emission and absorption features).

  * •

Hydrogenated coronene and fullerene molecules are readily formed. The former
is believed to serve as an important reservoir for H2 formation - a key
ingredient for star formation.

  * •

The growth products are due to knockout-driven reactions which may be
important in certain astrophysical environments. Here, the keV Ar impact is
analogous to the processing of material by shock waves in the vicinity of old
stars28, 29 in a similar fashion as in the study by Bernal _et al._ where
fullerenes were observed to be produced following high energy Xe-atom
bombardment of a silicon carbide surface.30

## Author Contributions

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

M.G. and H.Z. acknowledge support from the Swedish Research Council (contracts
2020-03104 and 2020-03437, respectively). This work is a part of the project
“Probing charge- and mass-transfer reactions on the atomic level”, supported
by the Knut and Alice Wallenberg Foundation (Grant no. 2018.0028). This
article is based upon work from COST Actions CA18212 – Molecular Dynamics in
the GAS phase (MD-GAS) and CA21101 – Confined Molecular Systems: From a New
Generation of Materials to the Stars (COSY), supported by COST (European
Cooperation in Science and Technology).

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