# Quantum spin Hall phase in GeSn heterostructures on silicon

B. M. Ferrari  Dipartimento di Scienza dei Materiali, Università di Milano-
Bicocca, LNESS and BiQuTe, via Cozzi 55, 20125 Milano, Italy  F. Marcantonio
Dipartimento di Scienza dei Materiali, Università di Milano-Bicocca, LNESS and
BiQuTe, via Cozzi 55, 20125 Milano, Italy  F. Murphy-Armando  Tyndall National
Institute, Cork T12R5CP, Ireland  M. Virgilio  Dipartimento di Fisica,
Università di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy  F. Pezzoli
<EMAIL_ADDRESS>Dipartimento di Scienza dei Materiali, Università di
Milano-Bicocca, LNESS and BiQuTe, via Cozzi 55, 20125 Milano, Italy

###### Abstract

Quantum phases of solid-state electron systems look poised to sustain exotic
phenomena and a very rich spin physics. We propose a practical silicon-based
architecture that spontaneously sustains topological properties, while being
fully compatible with the high-volume manufacturing capabilities of modern
microelectronic foundries. Here we show how $\mathrm{Ge_{1-x}Sn_{x}}$ alloys,
an emerging group IV semiconductor, can be engineered into junctions that
demonstrate a broken gap alignment. We predict such basic building block
undergo a quantum phase transition that can elegantly accommodate the
existence of gate-controlled chiral edge states directly on Si. This will
enable tantalizing prospects for designing integrated circuits hosting quantum
spin hall insulators and advanced topological functionalities.

The hybridization between valence and conduction states at surfaces and
interfaces has been recognized as a key-enabling solution for the realization
of topological protected states [1, 2, 3, 4]. In heterostructures possessing a
broken-gap (BG) alignment of the band edges, electron and hole subbands can be
localized at opposite sides of the junctions, while simultaneously retaining
the inverted ordering regime [3]. This leads to the emergence of helical
boundary states, which are characteristic fingerprints of the so-called
quantum spin Hall (QSH) phase [1, 2, 5, 6, 7, 8, 9, 10]. Challenges, however,
remain before these concepts can be harnessed for their practical use in
everyday applications. Were QSH capabilities realised in group IV materials,
this would bring exceptional topological properties into mass-produced
devices. Yet, the crucial BG band line-up is the missing ingredient that
presently hampers the achievement of genuine quantum phases in such systems.

In this Letter we introduce group IV architectures that will enable quantum
phase transitions in ubiquitous Si-based devices. More precisely, we discover
the coveted BG in heterojunctions based on $\mathrm{Ge_{1-x}Sn_{x}}$ and
single out, by the model-solid theory [11], unique designs capable of
sustaining the QSH phase. We leverage multiband $k\cdot p$ calculations to
fully capture the topological phase transitions in specific
$\mathrm{Ge_{1-x}Sn_{x}/Ge_{1-y}Sn_{y}}$ superlattices (SL). We show that
strain and quantum confinement offer practical degrees of freedom to engineer
the band inversion. Finally, we extend our numerical investigation to finite
size SLs connected to gate leads using self-consistent modelling. In so doing,
we explore realistic device geometries and validate the advantageous addition
of the electrical tunability owned by these unique topological architectures.

Lately, alloying Ge with Sn has increasingly attracted attention because it
provides a means to carefully induce a transition of the fundamental energy
gap [12, 13, 14]. Present-day epitaxy of $\mathrm{Ge_{1-x}Sn_{x}}$ on Si is
well established and industrial techniques have already succeeded in
synthesizing materials with Sn contents above 30 at.% [15, 16]: well beyond
the equilibrium solid solubility of the constituent elements, i.e., less than
1%. Here we envision an heteroepitaxial stack grown on the so-called virtual
substrate (VS), which accommodates the mismatch between the lattice parameters
($\mathrm{a_{Sn}>a_{Ge}>a_{Si}}$), and consists of a $\mathrm{Ge_{1-s}Sn_{s}}$
buffer deposited on a common Si(001) wafer. The topmost VS film is a relaxed
alloy, which can then be used as a template for the repeated deposition of
$\mathrm{Ge_{1-x}Sn_{x}/Ge_{1-y}Sn_{y}}$ bilayers in a state of biaxial
tension or compression, owning to the lattice mismatch with the selected VS.
The most feasible method to fabricate VSs would be a standard compositionally
graded film [17, 18, 19], but other strategies have been also shown viable
[20].

Figure 1: Band-edge profiles of Ge/$\mathrm{Ge_{1-y}Sn_{y}}$/Ge (a) and
$\mathrm{Ge_{0.88}Sn_{0.12}/Ge_{1-y}Sn_{y}/Ge_{0.88}Sn_{0.12}}$ (b)
heterojunctions as a function of the Sn concentration of the relaxed
substrate. The well/barrier stack is coherent with respect to the lattice
spacing pertaining to the $\mathrm{Ge_{1-s}Sn_{s}}$ substrate. The schematics
demonstrate a type I (yellow), type II (green), and broken gap (BG red, light
and dark blue) alignments. The region where the band inversion occurs within
the same layer as in bulk topological materials is also shown (INV, grey). In
(b) the barriers feature a direct gap regardless of the Sn content of the
substrate. Minimum energy difference ($\Delta E$) between conduction and
valence bands across Ge/$\mathrm{Ge_{1-y}Sn_{y}}$/Ge (c) and
$\mathrm{Ge_{0.88}Sn_{0.12}/Ge_{1-y}Sn_{y}/Ge_{0.88}Sn_{0.12}}$ (d).
Schematics of the heterostructures (not to scale) are shown as insets. Figure
2: Band structure of a $\mathrm{Ge_{0.78}Sn_{0.22}}$(10
nm)/$\mathrm{Ge_{0.88}Sn_{0.12}}$(25 nm) superlattice (SL) coherently grown on
relaxed buffer layers, namely $\mathrm{Ge_{0.75}Sn_{0.25}}$ (a) and
$\mathrm{Ge_{0.71}Sn_{0.29}}$ (b). (c) Energy levels of a 10 nm thick
$\mathrm{Ge_{0.78}Sn_{0.22}}$ QW embedded in $\mathrm{Ge_{0.88}Sn_{0.12}}$
barriers of 25 nm as a function of the Sn molar fraction of the
$\mathrm{Ge_{1-s}Sn_{s}}$ template. The lowest subband energies for the
conduction and valence bands are indicated as red and blue lines. (d) Energy
of the confined states of a $\mathrm{Ge_{0.78}Sn_{0.22}}$ QW embedded in
$\mathrm{Ge_{0.88}Sn_{0.12}}$ barriers as a function of the well thickness.
The substrate is a relaxed $\mathrm{Ge_{0.7}Sn_{0.3}}$.

To disclose a rational design of QSH insulators on Si we begin by considering
the band alignment established at the $\mathrm{Ge_{1-x}Sn_{x}/Ge_{1-y}Sn_{y}}$
heterointerface. We assume the films to be in full registry (coherent) with
the lattice of a fully relaxed $\mathrm{Ge_{1-s}Sn_{s}}$ buffer. To minimize
the computational burden, we rely on the flexible approach offered by the
well-established model-solid theory that enable us to compute interface line-
ups by including strain effects [11]. The parameters and validation of the
model are provided in the Supplemental Material [21]. Another important point
is that we will focus on strained films that are predicted to have a direct
band gap. This can provide a viable solution to avoid the competitive presence
of absolute minima at the zone edges, like the characteristic L-valleys in
relaxed Ge, that could impede the observation of the QSH phase in future
experiments.

Figure 3: (a) Schematics of a $\mathrm{Ge_{0.78}Sn_{0.22}}$(25
nm)/$\mathrm{Ge_{0.88}Sn_{0.12}}$(25 nm) SL, which is fully strained with
respect to a $\mathrm{Ge_{0.7}Sn_{0.3}}$ buffered substrate. The finite-size
Hall bar has a width of 300 nm and extends along the j direction (not to
scale). (b) Band structure of the two-dimensional ribbon. The red lines
highlight the topological edge states. (c) Colour-coded maps showing the
probability density $\left|\Psi^{2}\right|$ associated to the surface states
(P and $\mathrm{P^{\prime}}$).

For consistency with the literature, we consider a symmetric
$\mathrm{Ge_{1-x}Sn_{x}/Ge_{1-y}Sn_{y}/Ge_{1-x}Sn_{x}}$ heterostructure.
Without loss of generality, the two external identical
$\mathrm{Ge_{1-x}Sn_{x}}$ films will be dubbed barriers, whereas the
intermediate $\mathrm{Ge_{1-y}Sn_{y}}$ layer will be termed well. The line-ups
at the heterointerface have been obtained by varying the strain in the
epitaxial stack through the modification of the Sn molar fraction of the
$\mathrm{Ge_{1-s}Sn_{s}}$ buffer. Specifically, we utilized elemental Ge and a
$\mathrm{Ge_{0.70}Sn_{0.30}}$ alloy. The latter has been chosen as un upper
limit to guarantee experimental feasibility using state-of-the-art deposition
techniques. For the same reason, we restricted the barriers to a Sn-diluted
regime, namely $x<12$%. Even though in this work we focus on the
experimentally accessible region of the parameter space, the
$\mathrm{Ge_{1-y}Sn_{y}}$ well was allowed to freely sample the whole
composition range ($0<y<1$) to give a better overview of all the domains
potentially characterized by the sought-after BG. Fig. 1 summarizes the result
that were obtained when opposite ends of the y range are considered [21]. As a
guideline, Figure 1a and 1b also report a schematic of the offsets at the
conduction band (black lines) and at the top of the valence band (red lines).
The strain dependence is shown in Fig. S3 [21].

It is interesting to notice the rich atlas of line-ups unleashed in the
$\mathrm{Ge_{1-x}Sn_{x}}$ system. Insights can be obtained by the gross
features that exquisitely appear in Fig. 1a-b. When the Sn molar fraction of
the substrate is higher than $\sim 0.26$, a Sn bulk-like band structure can be
achieved within the films (INV, grey in Fig. 1a and b). However, when
$s<0.26$, an inverted regime can be prominently observed when the Sn content
of the $\mathrm{Ge_{1-y}Sn_{y}}$ wells exceeds $\sim 50$%. Specifically, a
region exhibiting an inverted type-I feature, as in HgTe/CdTe, opens below
60%, while a distinct InAs/GaSb-like BG develops well above 60% (Fig. 1a). At
a first glance, $\mathrm{Ge_{1-y}Sn_{y}}$ wells with a Sn molar fraction
larger than 50% can be regarded as strong candidates for hosting topologically
nontrivial phases. It should be noticed, however, that although very
promising, alloys of high Sn content are hard to synthesise. In the low Sn
content scenario, namely when $\mathrm{Ge_{1-y}Sn_{y}}$ wells with $y<0.5$ are
considered, the strain imprinted by the substrate appears to restore the
normal band sorting, featuring either a type-I (yellow area) or a staggered
type-II (green area) alignment (Fig. 1a). In the latter case, charges are set
apart at opposite sides of the heterointerface, where a well-defined
insulating gap is retained. Surprisingly, in this diluted alloy regime there
exists a region in which our calculations predict the emergence of a unique
BG. This occurs around $0.23<s<0.3$ and $0<y<0.4$ that is marked in red in
Fig. 1a-b. Such a band has the advantage of having a direct gap in both the
well and substrate and of being manifested in heterostructures that are easier
to grow through epitaxial techniques. Another central finding regarding the
interface alignments is the energy difference $\Delta E$ calculated between
the bottommost conduction band edges and the topmost valence band, which is
established across the junction. Fig. 1c-d demonstrate negative (positive)
$\Delta E$ when the inverted (normal) band ordering sets in. Interestingly,
$\Delta E$ can be extraordinarily large as these gaps possess remarkable
amplitudes in the range of several hundreds of meV. This offers indeed
plausible prospects for topological applications well above cryogenic
temperatures.

Figure 4: Energy spectrum of a ten-fold $\mathrm{Ge_{0.88}Sn_{0.12}}$(25
nm)/$\mathrm{Ge_{0.78}Sn_{0.22}}$(25nm) SL grown on a
$\mathrm{Ge_{0.75}Sn_{0.25}}$ buffer. The SL is unbiased (a) or under an
electric field of 0.95 (b) or 4.75 kV/cm (c). The red branches indicate the
topological states (see also [21]). Band edges and squared wave functions for
the lowest energy subband states at the centre of the Brillouin zone refers to
an unbiased device (d) or when electric fields of 0.95 (e) or 4.75 kV/cm (f)
are applied.

In the following we turn our attention to a
$\mathrm{Ge_{0.88}Sn_{0.12}/Ge_{0.78}Sn_{0.22}}$ multilayer stack and analyse
the evolution of the electronic states as a function of the thickness of the
layers as well as strain relative to the substrate. In Fig. 1 such
heterointerface lays within the parameter space previously disclosed by the
model-solid theory as the source of a BG.

We now leverage a multiband $k\cdot p$ model [22] to unveil the richness in
terms of quantum phases and to fully capture the essential physics of this new
and intriguing $\mathrm{Ge_{1-x}Sn_{x}}$ platform. At first, we investigate an
infinite periodic $\mathrm{Ge_{0.88}Sn_{0.12}/Ge_{0.78}Sn_{0.22}}$ SL. We set
the thickness of the $\mathrm{Ge_{0.78}Sn_{0.22}}$ well at 10 nm, while the
width of the $\mathrm{Ge_{0.88}Sn_{0.12}}$ barrier is maintained relatively
large, e.g., 25 nm, to ensure a reduced penetration of the wavefunctions
throughout the whole SL period. The energy spectra calculated by the exact
diagonalization of the $k\cdot p$ model along the well plane when the virtual
substrate terminates with $\mathrm{Ge_{0.75}Sn_{0.25}}$ (Fig. 2a), and
$\mathrm{Ge_{0.71}Sn_{0.29}}$ (Fig. 2b) suggest strain to be a pivotal driving
force for the closure of the band gap and the likely cause of the
hybridization between electron and hole states. Fig. 2c summarizes the
calculated shift of the energy levels of the SL as a function of the Sn
content of the buffer. A striking transition from a trivial to a topological
insulator can be clearly seen to occur at the critical value $s_{c}=0.27$. The
strain introduced by the substrate provokes indeed an opposite behaviour for
the electron and hole subbands, yielding a crossover that drives the SL
structure into an inverted scheme when $s>s_{c}$ (Fig. 2c). Notice that in the
topological regime, that is, above $s_{c}$, the opening of the bulk insulating
gap can be tuned in a very drastic way, as it remarkably approaches a
magnitude of more than 1 eV in the limiting, although unpractical, case of
growth on almost pure Sn buffers. We can find reassurance on these results by
the agreement with the $\Delta E$ band offsets disclosed by the model-solid
theory. Remarkably, the amplitude of the gapped bulk of our
$\mathrm{Ge_{1-x}Sn_{x}}$ proposal is exceedingly large compared to
conventional 2D QSH insulators, whose value typically lays below 10 meV [1, 3,
7, 8], and compares well with atomically thin materials [10].

Next, we consider a $\mathrm{Ge_{0.7}Sn_{0.3}}$ buffer whose Sn content is
above $s_{c}$. Fig. 2d shows no crossing points in the investigated range of
the well thickness, thereby indicating that width fluctuations cannot destroy
the attained normal and nontrivial insulator regimes. In the following, we
focus on the study of the edge state spectrum to explicitly demonstrate the
emergence of the gapless helical states. To achieve this purpose, we set an
equal thickness of 25 nm for both the well and barrier layers and break the
translational symmetry of the SL by opening free surfaces along the
crystallographic direction j lying in the QW plane, while keeping periodic
boundary conditions along k as shown in Fig. 3a. The lateral dimension of the
nanoribbon along j is set to 300 nm to minimize interactions between states
laying at opposite facets of the Hall bar (Fig. 3a). In figure 3b we report
the electronic band structure in the neighbour of the $\Gamma$ point along the
i direction chosen parallel to the $[001]$ crystallographic axis. Fig. 3b
shows two linearly dispersing energy states (red) appearing inside of the
gapped bulk (black) and crossing each other at the $\Gamma$ point. Fig. 3c
reports the square modulus of the wavefunction associated to these interesting
states, which are confined along the growth axis k and behave as plane wave in
the i direction, which is in the SL plane. Above all, their vanishing square
amplitude in the bulk of the SL, that is, along j, openly unveils that these
are Kramers′s pair tightly localized at the opposite sides of the nanoribbon.
This eventually provides a compelling proof that the QSH phase indeed exists
in group IV heterostructures. Calculations suggest that degenerate bulk states
can possibly occur in these heterostructures, thus indicating that, as for
bulk HgTe and $\alpha$-Sn, special care will be needed to single out chiral
edge states in future experimental works [23].

The unique coexistence and concomitant spatial separation of 2D electron and
hole gases in the $\mathrm{Ge_{0.88}Sn_{0.12}/Ge_{0.78}Sn_{0.22}}$ QSH
insulator spontaneously establishes a dipole layer across the interface. This
raises the attractive prospect of controlling the hybridization through
electric fields. We discuss this possibility by focusing on the model
$\mathrm{Ge_{0.88}Sn_{0.12}}$(25 nm)/$\mathrm{Ge_{0.78}Sn_{0.22}}$(25nm) SL in
the form of a finite size stack comprising 10 periods, which we have studied
for different Sn concentrations covering the range were a phase transition can
occur. This allows us to identify, via self-consistent $k\cdot p$
calculations, the impact on the QSH phase of an external electrical bias, as
we applied ohmic contacts at the two closing ends of the SL right along the k
growth axis.

As we scan the Sn molar fraction of the substrate at zero bias, we recover the
opening of a minigap within the valence band with the distinct formation of
two complementary localized conductive edge states that become more pronounced
as the Sn content, i.e., strain, increases (see also Fig S4 and Ref. [24]).
The resulting density distribution further supports our conclusions as we find
these states to be confined at the top and bottom surfaces when $s>0.2$. It is
worth noting that in this calculation the topological states emerge solely
because of the finite dimension of the SL along the growth direction. That is
distinct from the results obtained for the previously discussed infinite SL in
the nanoribbon geometry (Fig. 3), thereby reinforcing the case for
$\mathrm{Ge_{1-x}Sn_{x}}$-based heterostructures as robust hosts of the
nontrivial QSH phase. We therefore study the influence of an external bias on
the topological structure focussing on a SL deposited on a
$\mathrm{Ge_{0.75}Sn_{0.25}}$ buffer layer. The results are summarized in Fig.
4. Charge confinement manifests itself at 0 bias as a bending in a
neighbourhood of the final layers of the SL, while the profile at the centre
remains flat.

By applying a gentle positive bias, the edge states are more tightly confined
and even more peaked at the free surfaces. At the same time, the external
electric field drives the accumulation of charge density at each well/barrier
interface as demonstrated by the characteristic bending profile, which is
seemingly different from the flat band condition. Notably, the further
increase of the polarization causes the edge states to fade away (see panels c
and f), inducing a phase transition of the SL from a QSH to a trivial
insulator. A finding that provides a novel route towards the future
implementation of topological field effect devices.

In conclusion, we unveiled quantum phases in silicon-compatible materials
beyond conventional systems. Our platform naturally permits strain and
electric field control of the quantum phases on Si. This concept is highly
flexible and can have a longstanding impact, stimulating new research
directions towards Si topology.

###### Acknowledgements.

We acknowledge C. Colombo and A. Filippi for technical assistance with model-
solid theory and $k\cdot p$ calculations, respectively. F.M.-A. would like to
acknowledge Science Foundation Ireland grant SFI/19/FFP/6953.

## References

  * Bernevig  _et al._ [2006] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells, Science 314, 1757 (2006).
  * König  _et al._ [2007] M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, a. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Quantum Spin Hall Insulator State in HgTe Quantum Wells, Science 318, 766 (2007).
  * Liu  _et al._ [2008] C. Liu, T. L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang, Quantum Spin Hall Effect in Inverted Type-II Semiconductors, Phys. Rev. Lett. 100, 236601 (2008).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Reviews of Modern Physics 82, 3045 (2010).
  * Knez  _et al._ [2011] I. Knez, R.-R. Du, and G. Sullivan, Evidence for Helical Edge Modes in Inverted InAs/GaSb Quantum Wells, Phys. Rev. Lett. 107, 136603 (2011).
  * Du  _et al._ [2015] L. Du, I. Knez, G. Sullivan, and R.-R. Du, Robust Helical Edge Transport in Gated InAs/GaSb Bilayers, Phys. Rev. Lett. 114, 096802 (2015).
  * Miao  _et al._ [2012] M. S. Miao, Q. Yan, C. G. Van de Walle, W. K. Lou, L. L. Li, and K. Chang, Polarization-Driven Topological Insulator Transition in a GaN/InN/GaN Quantum Well, Phys. Rev. Lett. 109, 186803 (2012).
  * Zhang  _et al._ [2013] D. Zhang, W. Lou, M. Miao, S.-c. Zhang, and K. Chang, Interface-Induced Topological Insulator Transition in GaAs/Ge/GaAs Quantum Wells, Phys. Rev. Lett. 111, 156402 (2013).
  * Xu  _et al._ [2013] Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S.-C. Zhang, Large-gap quantum spin hall insulators in tin films, Phys. Rev. Lett. 111, 136804 (2013).
  * Lodge  _et al._ [2021] M. S. Lodge, S. A. Yang, S. Mukherjee, and B. Weber, Atomically Thin Quantum Spin Hall Insulators, Adv. Mater. 33, 2008029 (2021).
  * Van de Walle [1989] C. G. Van de Walle, Band lineups and deformation potentials in the model-solid theory, Phys. Rev. B 39, 1871 (1989).
  * Soref and Friedman [1993] R. A. Soref and L. Friedman, Direct-gap Ge/GeSn/Si and GeSn/Ge/Si heterostructures, Superlattices Microst. 14, 189 (1993).
  * Chang  _et al._ [2010] G.-E. Chang, S.-W. Chang, and S. L. Chuang, Strain-balanced ${\rm{ge}}_{z}{\rm sn}_{1-z}\hbox{--}{\rm si}_{x}{\rm ge}_{y}{\rm sn}_{1-x-y}$ multiple-quantum-well lasers, IEEE J. Quantum Elect. 46, 1813 (2010).
  * Wirths  _et al._ [2016] S. Wirths, D. Buca, and S. Mantl, Si–Ge–Sn alloys: From growth to applications, Prog. Cryst. Growth Ch. 62, 1 (2016).
  * Xu  _et al._ [2019] C. Xu, D. Ringwala, D. Wang, L. Liu, C. D. Poweleit, S. L. Y. Chang, H. L. Zhuang, J. Menéndez, and J. Kouvetakis, Synthesis and Fundamental Studies of Si-Compatible (Si)GeSn and GeSn Mid-IR Systems with Ultrahigh Sn Contents, Chem. Mater. 31, 9831 (2019).
  * Zheng  _et al._ [2018] J. Zheng, Z. Liu, Y. Zhang, Y. Zuo, C. Li, C. Xue, B. Cheng, and Q. Wang, Growth of high-Sn content (28%) GeSn alloy films by sputtering epitaxy, J. Cryst. Growth 492, 29 (2018).
  * Dou  _et al._ [2018a] W. Dou, Y. Zhou, J. Margetis, S. A. Ghetmiri, S. Al-Kabi, W. Du, J. Liu, G. Sun, R. A. Soref, J. Tolle, B. Li, M. Mortazavi, and S.-Q. Yu, Optically pumped lasing at 3 $\mu$m from compositionally graded GeSn with tin up to 22.3%, Opt. Lett. 43, 4558 (2018a).
  * Dou  _et al._ [2018b] W. Dou, M. Benamara, A. Mosleh, J. Margetis, P. Grant, Y. Zhou, S. Al-Kabi, W. Du, J. Tolle, B. Li, M. Mortazavi, and S.-Q. Yu, Investigation of GeSn Strain Relaxation and Spontaneous Composition Gradient for Low-Defect and High-Sn Alloy Growth, Sci. Rep. 8, 5640 (2018b).
  * Liu  _et al._ [2021] X. Liu, J. Zheng, M. Li, F. Wan, C. Niu, Z. Liu, Y. Zuo, C. Xue, and B. Cheng, Growth of relaxed GeSn film with high Sn content via Sn component-grade buffer layer structure, J. Phys. D Appl. Phys. 54, 435101 (2021).
  * Assali  _et al._ [2019] S. Assali, J. Nicolas, and O. Moutanabbir, Enhanced Sn incorporation in GeSn epitaxial semiconductors via strain relaxation, J. Appl. Phys. 125, 025304 (2019).
  * [21] See Supplemental Material.
  * Birner  _et al._ [2007] S. Birner, T. Zibold, T. Andlauer, T. Kubis, M. Sabathil, A. Trellakis, and P. Vogl, nextnano: General Purpose 3-D Simulations, IEEE T. Electron Dev. 54, 2137 (2007).
  * Khaetskii  _et al._ [2022] A. Khaetskii, V. Golovach, and A. Kiefer, Revisiting the physical origin and nature of surface states in inverted-band semiconductors, Phys. Rev. B 105, 035305 (2022).
  * Belopolski  _et al._ [2017] I. Belopolski, S.-Y. Xu, N. Koirala, C. Liu, G. Bian, V. N. Strocov, G. Chang, M. Neupane, N. Alidoust, D. Sanchez, H. Zheng, M. Brahlek, V. Rogalev, T. Kim, N. C. Plumb, C. Chen, F. Bertran, P. L. Fèvre, A. Taleb-Ibrahimi, M.-C. Asensio, M. Shi, H. Lin, M. Hoesch, S. Oh, and M. Z. Hasan, A novel artificial condensed matter lattice and a new platform for one-dimensional topological phases, Sci. Adv. 3, e1501692 (2017).