# Strain-driven topological quantum phase transition in the family of halide
perovskites

Ankita Phutela<EMAIL_ADDRESS>Sajjan Shoeran  Saswata
Bhattacharya<EMAIL_ADDRESS>[

###### Abstract

The centrosymmetric halide perovskites undergo a continuous phase transition
from a normal insulator to a topological insulator at the critical value of
strain. Contrarily, in noncentrosymmetric halide perovskites, this phase
transition is discontinuous. The noncentrosymmetry does not stabilize the
gapless state, causing a discontinuity in the bandgap. We have employed the
density functional theory and Slater-Koster formalism-based tight-binding
Hamiltonian studies to understand the evolution of band topology under the
compressive strain in the halide perovskites. Our study shows that both cubic
and pseudocubic FAPbI3 undergo a Pb s-p band inversion at $\gamma$ (V/V0) =
0.76 and 0.73, respectively. The cubic perovskite shows the surface state at
$\overline{M}$, whereas, the pseudocubic structure shows two conducting states
in the neighbourhood of $\overline{M}$, unlike the conventional topological
insulator. The Pb-Pb second nearest neighbor interactions determine this
topological phase transition. Alongside, we have modeled mixed cation halide
perovskites CsxMA1-xPbI3 (x = 0.25, 0.5 and 0.75) to study their topological
properties. Cs0.5MA0.5PbI3 shows non-trivial topology at $\gamma$ = 0.74. In
addition, we have checked the structural stability of different strained
configurations using ab initio molecular dynamics at operational temperature.
Their structural stability under compression strengthens the experimental
relevance.

###### keywords:

Density functional theory, Spin-orbit coupling, Surface states, Strain,
Topological insulator, Tight-binding model

+91-11-2659 1359 Indian Institute of Technology Delhi] Department of Physics,
Indian Institute of Technology Delhi, New Delhi, India

##  1 Introduction

Topological phases of matter such as topological insulators1, Dirac
semimetals2, 3, Weyl semimetals4, 5, topological metals6, and topological
crystalline insulators7 have occupied ample space in the research areas in the
last two decades. Among these phases, topological insulators are one of the
most promising materials for application in spintronics due to their
conducting surface states within the bulk band gap8, 9, 10. Numerous materials
like Bi-based selenides and tellurides11, 12, skutterudites13, Heusler
compounds14, 15, and Bi-based oxide perovskites (ABiO3)16, 17, 18 have been
extensively explored for non-trivial topology. More recently, halide
perovskites have also attracted attention for research in the area of band
topology19. These are the compounds with the formula ABX3, where A is an
inorganic or organic cation, B is a heavy element, and X is a halogen. These
materials are well known for their high performance in the field of
optoelectronics, photovoltaics, photocatalysts, and thermoelectrics20, 21, 22,
23. Their endeavors can be attributed to the compositional and structural
diversity that enables a wide array of functional properties. They possess a
suitable orbital symmetry and are structurally flexible to external stimuli
such as pressure, temperature, and chemical doping, which make them useful in
the field of topology24, 18. Furthermore, the molecular orbital picture of
topologically non-trivial cubic ABiO3, matches that of cubic halide
perovskites (CsSnX3)25. However, unlike oxide perovskites, the s-p inversion
on inclusion of spin-orbit coupling (SOC) is not shown by the halide
perovskites. Therefore, the family of halide perovskites is not known to show
topologically non-trivial phases at ambient conditions. Nevertheless, if we
tune certain aforementioned external parameters, a topological phase can be
realized in halide perovskites26. It is consistent with the recent theoretical
calculations that compressive strain can drive these materials into the
topological insulator region25. At a critical value of strain, a normal
insulator to topological insulator (NI-TI) phase transition takes place. In
materials preserving inversion symmetry, a continuous topological phase
transition (TPT) take place27. However, another type of phase transition,
which is discontinuous in nature, has been observed in MAPbI3 via. application
of pressure. It has been shown that in pseudocubic perovskites like MAPbI3,
the inversion symmetry breaking has a profound effect on the nature of phase
transition19. The discontinuous phase transition has also been observed in
Pb1-xSnxSe and TlBiS1-xSex28, 29, 30, 31. This is of key importance for device
applications based on tuning the topological properties of the system by
external means. Therefore, the realization of non-trivial phases under the
effect of strain in pseudocubic perovskite will bring several families of
compounds, beyond the intermetallic alloys, into the domain of research on
band topology. This could be important in providing various heterostructure
interfaces with multifunctional properties.
Motivated by this idea, we have looked for the NI-TI phase transition in
FAPbI3 (FA = formamidinium) under the effect of compressive strain. It has
been found that the nature of TPT is continuous for cubic FAPbI3. But, when
the centrosymmetry is broken in FAPbI3, an inverted band gap opens without
closing to zero. The absence of inversion symmetry brings a first-order
quantum phase transition on application of strain. We have applied volumetric
strain to trigger this non-trivial topology, by inducing a band inversion in
the bulk band structure. After that, the surface band structure is obtained to
see the origin of surface states with increasing strain. To explore the
underlying quantum mechanical process, which occurs due to structural
deformation caused by the application of strain, Slater-Koster tight-binding
(SK-TB) model have been employed. We have also designed the thermodynamically
stable mixed cation halide perovskites with the formula CsxMA1-xPbI3 (x =
0.25, 0.5, and 0.75). The bulk and surface band structures are plotted at
different values of strain to observe the TPT. Afterward, the structural
stability of all the unstrained and strained configurations at an operational
temperature of 300K has been confirmed to signify their practical usage.

Figure 1: Crystal structure of (a) cubic and (b) pseudocubic FAPbI3. (c) 3D BZ
along with the projected 2D BZ. (d) The molecular orbital picture for FAPbI3.
Bulk band structure of FAPbI3 in the (e) absence of SOC and (f) presence of
SOC.

FAPbI3 exists in cubic phase with Pm$\bar{3}$m space group at 298 K with
planar [H2N-CH-NH2]+ cation surrounded by [PbI3]- octahedra as shown in Figure
1(a). The lattice constant is taken as 6.32 Å , which is in close agreement
with the previous report32. The structural relaxation of FAPbI3 carried out
using DFT methods leads to a distortion in the octahedral cage, causing an
asymmetry in the structure33. The crystal structure becomes pseudocubic with
Pb-I-Pb angle $171.6^{\circ}$ as shown in Figure 1(b), which is similar to
that of MAPbI319. The bulk Brillouin zone (BZ) showing the high symmetry
k-points and the projected surface BZ is shown in Figure 1(c). The conduction
bands near the Fermi surface are mainly contributed by the Pb-6p orbitals,
whereas the valence bands are mainly contributed by Pb-6s and I-5p orbitals34.
The energy states of other elements in FA are mainly distributed in the deep
energy levels, and are not involved in bonding. The Pb-{s,p}–X-p covalent
interaction in the octahedron gives rise to four bonding and antibonding bands
along with five nonbonding bands35. FA contributes one electron, Pb
contributes four and I contribute five electrons per formula unit, hence, the
valence electron count becomes 20. Therefore, the states up to
$\sigma^{*}_{s-p}$ are filled and the Fermi level exists in between the lower-
lying Pb-s and upper-lying Pb-p antibonding bands (see Figure 1(d)). The SOC
arising due to the presence of heavy element affects the band structure near
the Fermi level. Figure 1(e) shows the electronic band structure of FAPbI3 in
the absence of SOC. On inclusion of SOC, the L.S coupling lifts the degeneracy
between the states J = $\frac{1}{2}$ and J = $\frac{3}{2}$ as shown in Figure
1(f). The strength of atomic SOC is the deterministic factor towards tailoring
the non-trivial topological phases in ABX3. However, the breaking down of the
inversion symmetry causes both the atomic SOC and Rashba SOC to become
deterministic in tailoring the electronic behavior of these noncentrosymmetric
quantum materials36.

Figure 2: Orbital resolved band structure in presence of SOC for cubic and
pseudocubic FAPbI3 (a), (d) without compression, and (b), (e) at critical
compression $\gamma$ = 0.76, and $\gamma$ = 0.73, respectively. Band gap at
different values of strain in (c) cubic FAPbI3 showing its continuous nature,
and (f) pseudocubic FAPbI3 showing a discontinuity at the critical value of
strain.

The experimentally reported band gap of FAPbI3 is 1.48 eV37. The closest band
gap of 1.56 eV is reproduced by G0W0@HSE06+SOC with $\alpha$=53%, which is in
agreement with the previous report38. We have observed that except for the
band gaps, other band features are similar irrespective of the choice of
functional (see section I of supporting information (SI)). Therefore, we have
used Perdew-Burke-Ernzerhof (PBE) functional for further calculations in view
of its low computational cost.

The compressional strain is a natural external stimulus to vary the bond
length while maintaining the symmetry of the lattice. Therefore, we have
investigated the effect of strain on the evolution of band structure. In the
case of cubic FAPbI3, at a critical compression $\gamma$ (V/V0) = 0.76, the
inversion between the Pb-s and Pb-p orbital takes place. This s-p inversion
takes place at R point and gives the initial indication of the existence of
non-trivial topology under the effect of strain. The orbital projected band
structure with the orbital contribution proportional to the thickness of the
symbols is shown in Figures 2(a) and (b). The continuous TPT takes place in
three steps, i) shrinking ii) closing iii) reemergence of the inverted band
gap, and is shown in the Figure 2(c). However, if the inversion symmetry is
not preserved in a material then the nature of phase transition changes. In
the case of noncentrosymmetric FAPbI3, as we keep on increasing the strain the
band gap decreases initially. But, after reaching a certain minimum value, a
new negative band gap emerges with VBM and CBM inverting their orbital
character. In this case, the value of this critical strain $\gamma$ is 0.73
(see Figures 2(d) and (e)). This flip in the character of band structure at
the critical value indicates the change in the topological nature of the
material. This type of phase transition in which the gapless state is not
stabilized, resulting in discontinuity in the band gap as shown in Figure
2(f), is termed a discontinuous TPT. It is also called a first-order phase
transition. The distortion caused in the octahedral cage of PbI due to
structural optimization is responsible for the discontinuous phase transition.
Subsequently, the structural stability of cubic and pseudocubic FAPbI3 along
with the strained configurations is confirmed using ab initio molecular
dynamics (AIMD). The radial distribution function (g(r)) plots at T = 0 K and
T = 300 K are shown in section II of SI.

SK-TB Hamiltonian studies are useful in providing information regarding the
underlying quantum mechanical process responsible for phase transition due to
structural deformation39. The TB model Hamiltonian in the second quantized
notation is written as40, 35

$H=\sum\limits_{i,\alpha}\epsilon_{i\alpha}c_{i\alpha}^{\dagger}c_{i\alpha}+\sum\limits_{ij:\alpha\beta}t_{i\alpha
j\beta}(c_{i\alpha}^{\dagger}c_{j\beta}+h.c.)+\lambda\textbf{L.S}+H_{R}.$ (1)

Here, i,j and $\alpha$, $\beta$ represents site and orbital indices,
respectively. $\epsilon_{i\alpha}$ is the onsite energy term and $t_{i\alpha
j\beta}$ gives the hopping term. In addition to the nearest neighbor Pb-I
interactions, the second neighbor Pb-Pb interactions are also included in the
Hamiltonian. The third term includes SOC, where $\lambda$ gives the SOC
strength. HR is included when the noncentrosymmetry is present in the crystal
structure and is called Rashba SOC. HR =
–$\frac{\mu_{B}}{2mc}$$\vec{\sigma}$.($\vec{E}$$\times$$\vec{p}$), where
$\vec{\textbf{E}}$ is the electric field experienced by the electron and
$\vec{\sigma}$ is the pauli matrix vector and $\vec{\textbf{p}}$ represents
the momenta41, 42. The basis set for the TB Hamiltonian contains one Pb-s,
three Pb-p and nine I-p orbitals. The rest orbitals are neglected as they do
not participate in the formation of the topological invariant states. The spin
independent Hamiltonian with the basis set in order $\bra{s^{Pb}}$,
$\bra{p_{x}^{Pb}}$, $\bra{p_{y}^{Pb}}$, $\bra{p_{z}^{Pb}}$,
$\bra{p_{x}^{I1}}$, $\bra{p_{y}^{I1}}$, $\bra{p_{z}^{I1}}$,$\bra{p_{x}^{I2}}$,
$\bra{p_{y}^{I2}}$, $\bra{p_{z}^{I2}}$, $\bra{p_{x}^{I3}}$,
$\bra{p_{y}^{I3}}$, $\bra{p_{z}^{I3}}$, takes the form

$H=\begin{pmatrix}M_{4\times 4}^{Pb-Pb}&M_{4\times 9}^{Pb-I}\\\ (M_{4\times
9}^{Pb-I})^{\dagger}&M_{9\times 9}^{I-I}\\\ \end{pmatrix}.$

The individual blocks of this matrix are as follows

$M_{4\times 4}^{Pb-Pb}=\begin{pmatrix}\epsilon_{s}+g_{1}&2it_{sp\sigma}^{Pb-
Pb}Sin(k_{x}a)&2it_{sp\sigma}^{Pb-Pb}Sin(k_{y}a)&2it_{sp\sigma}^{Pb-
Pb}Sin(k_{z}a)\\\ -2it_{sp\sigma}^{Pb-
Pb}Sin(k_{y}a)&\epsilon_{p1}+g_{2}&0&0\\\ -2it_{sp\sigma}^{Pb-
Pb}Sin(k_{x}a)&0&\epsilon_{p1}+g_{3}&0\\\ -2it_{sp\sigma}^{Pb-
Pb}Sin(k_{z}a)&0&0&\epsilon_{p1}+g_{4}\\\ \end{pmatrix}$

$M_{4\times
9}^{Pb-I}=\begin{pmatrix}t_{sp\sigma}^{Pb-I}S_{x}&0&0&0&t_{sp\sigma}^{Pb-I}S_{y}&0&0&0&t_{sp\sigma}^{Pb-I}S_{z}\\\
t_{pp\sigma}^{Pb-I}C_{x}&0&0&t_{pp\pi}^{Pb-I}C_{y}&0&0&t_{pp\pi}^{Pb-I}C_{z}&0&0\\\
0&t_{pp\pi}^{Pb-I}C_{x}&0&0&t_{pp\pi}^{Pb-I}C_{y}&0&0&t_{pp\pi}^{Pb-I}C_{z}&0\\\
0&0&t_{pp\pi}^{Pb-I}C_{x}&0&0&t_{pp\pi}^{Pb-I}C_{y}&0&0&t_{pp\sigma}^{Pb-I}C_{z}\\\
\end{pmatrix}$

$M_{9\times 9}^{I-I}=\begin{pmatrix}\epsilon_{p2}&0&0&0&0&0&0&0&0\\\
0&\epsilon_{p2}&0&0&0&0&0&0&0\\\ 0&0&\epsilon_{p2}&0&0&0&0&0&0\\\
0&0&0&\epsilon_{p2}&0&0&0&0&0\\\ 0&0&0&0&\epsilon_{p2}&0&0&0&0\\\
0&0&0&0&0&\epsilon_{p2}&0&0&0\\\ 0&0&0&0&0&0&\epsilon_{p2}&0&0\\\
0&0&0&0&0&0&0&\epsilon_{p2}&0\\\ 0&0&0&0&0&0&0&0&\epsilon_{p2}\\\
\end{pmatrix}.$

Here $\epsilon_{s}$ ,$\epsilon_{p1}$ , and $\epsilon_{p2}$ are the onsite
energies of Pb-s, Pb-p and I-p orbitals, respectively. The terms $C_{z}$ and
$S_{z}$ are the representations for 2$\cos(k_{z}a/2)$ and 2i$\sin(k_{z}a/2)$,
and $g_{i}$ (i = 1,2,3,4) are the k-dependent hopping integrals arising from
the Pb-Pb second neighbor interactions given by

$\displaystyle g_{1}=2t_{ss}^{Pb-Pb}[\cos(k_{x}a)+\cos(k_{y}a)+\cos(k_{z}a)],$
$\displaystyle g_{2}=2t_{pp\sigma}^{Pb-Pb}\cos(k_{x}a)+2t_{pp\pi}^{Pb-
Pb}[\cos(k_{y}a)+\cos(k_{z}a)],$ $\displaystyle g_{3}=2t_{pp\sigma}^{Pb-
Pb}\cos(k_{y}a)+2t_{pp\pi}^{Pb-Pb}[\cos(k_{x}a)+\cos(k_{z}a)],$ $\displaystyle
g_{4}=2t_{pp\sigma}^{Pb-Pb}\cos(k_{z}a)+2t_{pp\pi}^{Pb-
Pb}[\cos(k_{x}a)+\cos(k_{y}a)].$

The SOC component of the Hamiltonian is obtained by applying $\lambda$L.S on
the Pb-p states, as Pb being heavy is responsible for the SOC. We have
obtained the matrix elements for the SOC Hamiltonian with the basis set in the
order: $\ket{p_{x}^{Pb}\uparrow}$, $\ket{p_{y}^{Pb}\uparrow}$,
$\ket{p_{z}^{Pb}\uparrow}$, $\ket{p_{x}^{Pb}\downarrow}$,
$\ket{p_{y}^{Pb}\downarrow}$, $\ket{p_{z}^{Pb}\downarrow}$

$H_{SOC}=\lambda\begin{pmatrix}0&-i&0&0&0&1\\\ i&0&0&0&0&-i\\\ 0&0&0&1&-i&0\\\
0&0&1&0&i&0\\\ 0&0&i&-i&0&0\\\ 1&i&0&0&0&0\\\ \end{pmatrix}.$

Table 1: Interaction parameters and SOC strength for FAPbI3 from thirteen band
hamiltonian at zero and critical compression strength in units of eV.

Type $E_{Pb-s}$ $E_{Pb-p}$ $E_{I-p}$ ${t_{sp\sigma}^{Pb-I}}$
${t_{pp\sigma}^{Pb-I}}$ ${t_{pp\pi}^{Pb-I}}$ ${t_{ss\sigma}^{Pb-Pb}}$
${t_{sp\sigma}^{Pb-Pb}}$ ${t_{pp\sigma}^{Pb-Pb}}$ ${t_{pp\pi}^{Pb-Pb}}$
$\lambda$ Cubic ($\gamma$ = 1) -1 2.5 -2 0.15 0.4 0.3 -0.1 0.3 0.4 0.1 0.5
Cubic ($\gamma$ = 0.76) -1 2.5 -2 0.1 0.7 0.5 -0.1 0.3 0.76 0.17 0.5
Pseudocubic ($\gamma$ = 1) -0.8 3 -1 0.15 0.4 0.3 -0.07 0.45 0.5 0.4 0.4
Pseudocubic ($\gamma$ = 0.73) -1 3 -2 0.1 0.5 0.5 -0.11 0.5 0.95 0.07 0.4

Fitting this thirteen band model with the DFT bands, we get the effective
onsite and hopping terms. The parameters obtained at no strain and critical
strain are provided in Table 1. In the case of FAPbI3, the bands forming the
VBM and CBM are primerly composed of Pb-{s,p} orbitals. Also, the band
inversion is taking place between Pb-s, and Pb-p orbitals. Therefore, a
minimal basis Hamiltonian containing only essential interactions have been
developed. The second neighbor Pb-Pb interactions has been considered in
formulating the four band model. The SOC incorporated four band Hamiltonian is
written as

$\textbf{H}=\begin{pmatrix}H_{\uparrow\uparrow}&H_{\uparrow\downarrow}\\\
H^{\dagger}_{\downarrow\uparrow}&H_{\downarrow\downarrow}\\\ \end{pmatrix}$

where

$H_{\uparrow\downarrow}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&\lambda\\\
0&0&0&-i\lambda\\\ 0&0&\lambda&-i\lambda\\\ \end{pmatrix}$

$H_{\uparrow\uparrow}=\begin{pmatrix}\epsilon_{s}+g_{1}&2it_{sp\sigma}^{Pb-
Pb}Sin(k_{x}a)&2it_{sp\sigma}^{Pb-Pb}Sin(k_{y}a)&2it_{sp\sigma}^{Pb-
Pb}Sin(k_{z}a)\\\ -2it_{sp\sigma}^{Pb-
Pb}Sin(k_{y}a)&\epsilon_{p1}+g_{2}&0&0\\\ -2it_{sp\sigma}^{Pb-
Pb}Sin(k_{x}a)&0&\epsilon_{p1}+g_{3}&0\\\ -2it_{sp\sigma}^{Pb-
Pb}Sin(k_{z}a)&0&0&\epsilon_{p1}+g_{4}\\\ \end{pmatrix}$

Table 2: Interaction parameters and SOC strength for FAPbI3 from four band hamiltonian at zero and critical compression strength in units of eV. Type | $E_{Pb-s}$ | $E_{Pb-p}$ | ${t_{ss\sigma}^{Pb-Pb}}$ | ${t_{sp\sigma}^{Pb-Pb}}$ | ${t_{pp\sigma}^{Pb-Pb}}$ | ${t_{pp\pi}^{Pb-Pb}}$ | $\lambda$
---|---|---|---|---|---|---|---
Cubic ($\gamma$ = 1) | -0.8 | 2.5 | -0.09 | 0.3 | 0.4 | 0.1 | 0.5
Cubic ($\gamma$ = 0.76) | -1.2 | 2.5 | -0.19 | 0.44 | 0.75 | 0.16 | 0.5
Pseudocubic ($\gamma$ = 1) | -0.8 | 2.5 | -0.1 | 0.25 | 0.2 | 0.06 | 0.5
Pseudocubic ($\gamma$ = 0.73) | -1 | 3 | -0.12 | 0.4 | 0.8 | 0.1 | 0.5

The interaction parameters for both cubic and pseudocubic FAPbI3, from the
four band model fitting are provided in Table 2. Magnitude of all the
parameters increase with rise in strain. With the increase in compression, the
bond length decreases and hence the strength of interactions increases. In
comparison to Pb-Pb ss$\sigma$, sp$\sigma$ and pp$\pi$, the pp$\sigma$
interactions are most sensitive to the strain. For cubic structure the
pp$\sigma$ interaction rises from 0.4 to 0.75, while for pseudocubic, it rises
from 0.2 to 0.7, upon the application of respective critical strain. We have
also observed that $\lambda$ does not change with compression, therefore, SOC
is insensitive to the strain. From the electronic band structure calculations,
it is evident that the universality of the band structure arises from the
nearest-neighbor Pb-{s,p}–X-p covalent interactions. However, the TB model
elucidates that the second neighbor Pb-{s,p}–Pb-{s,p} interactions act as a
driving force for band gap engineering and non-trivial phases. This type of TB
model is universal, as it is independent of the choice of functional
adopted35.

Figure 3: Surface band structure in presence of SOC for cubic FAPbI3 (a)
without compression, (b) at critical compression $\gamma$ = 0.76, and
pseudocubic FAPbI3 (c) without compression, (d) at critical compression
$\gamma$ = 0.73.

The study of bulk electronic structure has enabled us to identify the
possibility of non-trivial topology and phase transition in halide
perovskites. However, the practical observation of these states comes from the
calculation of surface states43. The DFT based calculations on the slab
structure require a huge amount of computational time and memory to analyze
these properties. So, we have used the maximally localized wannier function
(MLWF) based TB approach to construct a semi-infinite slab along (001)
direction, minimizing the total spread of MLWF. As expected, there is an
absence of the surface states in FAPbI3 at ambient conditions (see Figures
3(a) and (c)). However, as the critical value of strain i.e., $\gamma$ = 0.76
is applied to cubic FAPbI3, the conducting states are formed at $\overline{M}$
point, similar to that of a conventional topological insulator. The invariant
surface states, exhibited by cubic FAPbI3 yields two interpenetrating Dirac
states. In this case, the Kramer pair i.e., doubly degenerate dirac cones,
each one arising from the VB and CB coincides at $\overline{M}$ point (see
Figure 3(b)). But when the inversion symmetry is broken, this degeneracy is
lifted and as a result one cone shifts towards the VB and the other towards
the CB19. Therefore, we see two Dirac states at $\overline{M}$ $\pm$ $\delta$,
along the path X–$\overline{M}$–X, as shown in Figure 3(d). The resulting
surface states illustrate the presence of discontinuous nature of NI-TI TPT.

Figure 4: Orbital resolved band structure in presence of SOC for
Cs0.5MA0.5PbI3 (a) without compression, (b) at critical compression $\gamma$ =
0.74. Surface band structure for Cs0.5MA0.5PbI3 (a) without compression, (b)
at critical compression $\gamma$ = 0.74.

We have studied the TPT in both inorganic (CsPbI3) and organic (MAPbI3,
FAPbI3) halide perovskites. CsPbI3 show continuous TPT at $\gamma$ = 0.76
because of its centrosymmetric nature, whereas MAPbI3 and FAPbI3 show similar
behaviour (see section III and IV of SI for CsPbI3 and MAPbI3, respectively).
It has been found that by partially replacing MA+ with Cs+, the stability can
be enhanced in ambient environment44. These mixed cation/anion perovskites
have attracted considerable attention due to their higher stability45. Thus,
we have designed the mixed cation halide perovskites to study the nature of
TPT in them. For this, three configurations i.e., Cs0.75MA0.5PbI3,
Cs0.5MA0.5PbI3, Cs0.25MA0.75PbI3, have been obtained by replacing MA with Cs
in the lattice of MAPbI3 (see section V of SI for crystal structures).
Firstly, we have calculated the thermodynamic stability of the mixed cation
halide perovkites i.e., Cs0.75MA0.5PbI3, Cs0.5MA0.5PbI3, Cs0.25MA0.75PbI3. The
precursors considered for the calculation of formation energy (Ef) of these
perovskites are MAPbI3, CsI and PbI2. The associated expression is

$E_{f}(x,y)=E_{sys}-E(MA_{4}Pb_{4}I_{12})-xE(CsI)-yE(PbI_{2})$ (2)

Here, coefficients x and y of each term on the right hand side are chosen such
that they stoichiometrically balance the equation. 1$\times$2$\times$2
supercell of MAPbI3 has been used. Ef(x,y) for Cs0.75MA0.25PbI3,
Cs0.5MA0.5PbI3, Cs0.25MA0.75PbI3 is -0.78 eV, -0.69 eV and -0.65 eV,
respectively. The negative formation energy shows that all the configurations
are thermodynamically stable. We have also checked the structural stability of
these configurations using AIMD simulations (see section V of SI). After this,
we have obtained the band structure for all the configurations with and
without SOC. When no strain is applied in Cs0.5MA0.5PbI3, the VBM is composed
of I-p orbitals, and CBM of Pb-p orbitals (see Figure 4(a)). The inclusion of
SOC does not depict any traits of non-trivial topological nature.
Subsequently, we have applied the compressional volumetric strain along with
the SOC. At the critical strain value of $\gamma$ = 0.74, the appearance of a
negative band gap is observed with the inverted orbital character as seen in
Figure 4(b). This suggests the possibility of TPT in Cs0.5MA0.5PbI3. Whereas,
in Cs0.75MA0.25PbI3 and Cs0.25MA0.75PbI3, no such parity inversion is observed
under the effect of compression. This may be attributed to the difference in
the ionic radii of Cs and MA cations, which is 1.88 Å and 2.117 Å ,
respectively. As the band inversion involves p orbitals of both I and Pb,
therefore, a four band TB model will not suffice in predicting the nature of
TPT. Thus, using the thirteen band model containing Pb-{s,p} and X-p orbitals,
we have calculated the onsite energies and hopping integrals (see section VI
of SI). All the interaction parameters increase on application of strain, but
the Pb-I pp$\pi$ interations rising from 1 to 1.5 are most affected by the
same. After this, we have plotted the surface state band structure for
Cs0.5MA0.5PbI3 at $\gamma$ = 0 and $\gamma$ = 0.74. As seen in the Figures
4(b) and (c), the origin of surface states with strain confirms the TPT at
$\gamma$ = 0.74.

The strain free lead halide perovskites are trivial band insulators, however,
a TPT can take place via. strain engineering. The presence of inversion
symmetry plays a vital role in the electronic properties of the halide
perovskites. The first-principles calculations suggest that FAPbI3 becomes
noncentrosymmetric on structural optimization. The centrosymmetric FAPbI3
shows a continuous phase transition on the application of strain. The
inversion symmetry breaking changes the nature of this quantum phase
transition. At a critical value of $\gamma$ = 0.73, pseudocubic FAPbI3
undergoes a first-order TPT. The discontinuity is evident from the band gaps
calculated at various values of strain, and the nature of surface states. This
study also shows that although thirteen orbitals participate in the chemical
bonding, a four band TB model is sufficient to capture energy dispersion near
the Fermi level. The Pb-Pb second neighbor interactions determine the topology
of this class of materials. The Pb-Pb pp$\sigma$ interactions are the most
affected by the application of strain. Hence, these interactions are the
driving forces in determining the phase transition. Furthermore, the mixed
cation perovskites Cs0.75MA0.25PbI3, Cs0.5MA0.5PbI3 and Cs0.25MA0.75PbI3 are
thermodynamically and structurally stable. Cs0.5MA0.5PbI3 shows TPT at a
compression of $\gamma$ = 0.74, which is shown by band inversion and the
nature of surface states. The structural stability of these perovskites under
strain strengthens that these phase transitions can be realized
experimentally.

##  2 Compulational Methods

The calculations are performed using DFT46, 47 with the Projected Augmented
Wave (PAW)48, 49 method implemented in Vienna Ab initio Simulation Package
(VASP)50 code. The self-consistent relativistic calculations are performed
using the Generalized Gradient Approximation (GGA) of PBE51. The structural
parameters are fully optimized until the Hellmann-Feynman forces are smaller
than 1 meV/Å. The cutoff energy of 520 eV is used for the plane wave basis set
such that the total energy calculations are converged within 10-5 eV. A
$\Gamma$-centered 4$\times$4$\times$4 k-grid is used to sample the irreducible
BZ of cubic phase with the Pm$\bar{3}$m space group. Throughout the
calculations, we have included SOC in order to consider the relativistic
effects due to the presence of heavy element in the systems. We have used
many-body perturbation method G0W0@HSE06+SOC for the better estimation of the
band gap52. For this, the number of bands is set to six times the number of
occupied bands. After this, to investigate the topological phases, the first-
principles calculations were first performed using fully relativistic norm-
conserving pseudopotentials as implemented in the QUANTUM ESPRESSO code53. The
results of these DFT calculations are then fed as input to WANNIER9054 for
constructing a TB model based on Maximally Localized Wannier Functions (MLWFs)
with p orbitals of I and sp orbitals Pb. The surface states are then
calculated using the Green’s function through an interactive method as
implemented in the Wannier-TOOLS package55. A TB model using Slater and Koster
approach is constructed in the two-center approximation using the package
Tight Binding Studio56. A nonlinear fitting algorithm is used to find the best
fit entries for both Hamiltonian and overlap matrices to reproduce the first-
principles data. Finally, the structural stability of different configurations
is checked at higher temperature using AIMD by running a 5 ps long simulation
with NVT ensemble (Nose–Hoover thermostat).

A.P. acknowledges IIT Delhi for the senior research fellowship. S.S.
acknowledges CSIR, India, for the senior research fellowship [grant no.
09/086(1432)/2019-EMR-I]. S.B. acknowledges financial support from SERB under
a core research grant (grant no. CRG/2019/000647) to set up his High
Performance Computing (HPC) facility “Veena” at IIT Delhi for computational
resources. Band profile of FAPbI3 using PBE+SOC and G0W0@HSE06+SOC
$\epsilon_{xc}$ functionals; Structural stability of cubic and pseudocubic
FAPbI3; Continous TPT in cubic CsPbI3; TPT in cubic and pseudocubic MAPbI3;
Crystal structures and structural stability of Cs0.5MA0.5PbI3,
Cs0.75MA0.25PbI3 and Cs0.25MA0.75PbI3; TB parameters for Cs0.5MA0.5PbI3.

## References

  * Bansil et al. 2016 Bansil, A.; Lin, H.; Das, T. Colloquium: Topological band theory. _Rev. Mod. Phys._ 2016, _88_ , 021004
  * Wang et al. 2012 Wang, Z.; Sun, Y.; Chen, X.-Q.; Franchini, C.; Xu, G.; Weng, H.; Dai, X.; Fang, Z. Dirac semimetal and topological phase transitions in A3Bi (A= Na, K, Rb). _Phys. Rev. B_ 2012, _85_ , 195320
  * Wang et al. 2013 Wang, Z.; Weng, H.; Wu, Q.; Dai, X.; Fang, Z. Three-dimensional Dirac semimetal and quantum transport in Cd3As2. _Phys. Rev. B_ 2013, _88_ , 125427
  * Wan et al. 2011 Wan, X.; Turner, A. M.; Vishwanath, A.; Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. _Phys. Rev. B_ 2011, _83_ , 205101
  * Xu et al. 2015 Xu, S.-Y.; Belopolski, I.; Alidoust, N.; Neupane, M.; Bian, G.; Zhang, C.; Sankar, R.; Chang, G.; Yuan, Z.; Lee, C.-C. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. _Science_ 2015, _349_ , 613–617
  * Burkov 2016 Burkov, A. Topological semimetals. _Nat. Mater._ 2016, _15_ , 1145–1148
  * Fu 2011 Fu, L. Topological crystalline insulators. _Phys. Rev. Lett._ 2011, _106_ , 106802
  * Hor et al. 2009 Hor, Y.; Richardella, A.; Roushan, P.; Xia, Y.; Checkelsky, J.; Yazdani, A.; Hasan, M.; Ong, N.; Cava, R. p-type Bi2Se3 for topological insulator and low-temperature thermoelectric applications. _Phys. Rev. B_ 2009, _79_ , 195208
  * Xu et al. 2014 Xu, Y.; Gan, Z.; Zhang, S.-C. Enhanced thermoelectric performance and anomalous Seebeck effects in topological insulators. _Phys. Rev. Lett._ 2014, _112_ , 226801
  * Pal et al. 2015 Pal, K.; Anand, S.; Waghmare, U. V. Thermoelectric properties of materials with nontrivial electronic topology. _J Mat. Chem. C_ 2015, _3_ , 12130–12139
  * Zhang et al. 2009 Zhang, H.; Liu, C.-X.; Qi, X.-L.; Dai, X.; Fang, Z.; Zhang, S.-C. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. _Nat. Phys._ 2009, _5_ , 438–442
  * Park et al. 2010 Park, K.; Heremans, J.; Scarola, V.; Minic, D. Robustness of topologically protected surface states in layering of Bi2Te3 thin films. _Phys. Rev. Lett._ 2010, _105_ , 186801
  * Yang and Liu 2014 Yang, M.; Liu, W.-M. The dp band-inversion topological insulator in bismuth-based skutterudites. _Sci. Rep._ 2014, _4_ , 1–7
  * Xiao et al. 2010 Xiao, D.; Yao, Y.; Feng, W.; Wen, J.; Zhu, W.; Chen, X.-Q.; Stocks, G. M.; Zhang, Z. Half-Heusler compounds as a new class of three-dimensional topological insulators. _Phys. Rev. Lett._ 2010, _105_ , 096404
  * Feng et al. 2010 Feng, W.; Xiao, D.; Zhang, Y.; Yao, Y. Half-Heusler topological insulators: A first-principles study with the Tran-Blaha modified Becke-Johnson density functional. _Phys. Rev. B_ 2010, _82_ , 235121
  * Yan et al. 2013 Yan, B.; Jansen, M.; Felser, C. A large-energy-gap oxide topological insulator based on the superconductor BaBiO3. _Nat. Phys._ 2013, _9_ , 709–711
  * Li et al. 2015 Li, G.; Yan, B.; Thomale, R.; Hanke, W. Topological nature and the multiple Dirac cones hidden in Bismuth high-Tc superconductors. _Sci. Rep._ 2015, _5_ , 1–8
  * Khamari et al. 2018 Khamari, B.; Kashikar, R.; Nanda, B. Topologically invariant double Dirac states in bismuth-based perovskites: Consequence of ambivalent charge states and covalent bonding. _Phys. Rev. B_ 2018, _97_ , 045149
  * Kore et al. 2020 Kore, A.; Kashikar, R.; Gupta, M.; Singh, P.; Nanda, B. Pressure and inversion symmetry breaking field-driven first-order phase transition and formation of Dirac circle in perovskites. _Phys. Rev. B_ 2020, _102_ , 035116
  * Jain et al. 2021 Jain, M.; Gill, D.; Bhumla, P.; Basera, P.; Bhattacharya, S. Theoretical insights to excitonic effect in lead bromide perovskites. _Appl. Phys, Lett._ 2021, _118_ , 192103
  * Jain et al. 2020 Jain, M.; Singh, A.; Basera, P.; Kumar, M.; Bhattacharya, S. Understanding the role of Sn substitution and Pb-$\boxtimes_{y}$ in enhancing the optical properties and solar cell efficiency of CH(NH2)2Pb1-x-ySnx$\boxtimes_{y}$Br3. _J. Mater. Chem. C_ 2020, _8_ , 10362–10368
  * Jain et al. 2023 Jain, M.; Gill, D.; Monga, S.; Bhattacharya, S. Theoretical evaluation of oxynitride, oxyfluoride and nitrofluoride perovskites with promising photon absorption properties for solar water splitting. _arXiv preprint arXiv:2301.04335_ 2023,
  * Bhumla et al. 2022 Bhumla, P.; Jain, M.; Sheoran, S.; Bhattacharya, S. Vacancy-Ordered Double Perovskites Cs2BI6 (B= Pt, Pd, Te, Sn): An Emerging Class of Thermoelectric Materials. _J. Phys. Chem. Lett._ 2022, _13_ , 11655–11662
  * Yang et al. 2012 Yang, K.; Setyawan, W.; Wang, S.; Buongiorno Nardelli, M.; Curtarolo, S. A search model for topological insulators with high-throughput robustness descriptors. _Nat. Mater._ 2012, _11_ , 614–619
  * Kashikar et al. 2018 Kashikar, R.; Khamari, B.; Nanda, B. Second-neighbor electron hopping and pressure induced topological quantum phase transition in insulating cubic perovskites. _Phys. Rev. Mater._ 2018, _2_ , 124204
  * Kashikar et al. 2020 Kashikar, R.; Gupta, M.; Nanda, B. Defining the topological influencers and predictive principles to engineer the band structure of halide perovskites. _Phys. Rev. B_ 2020, _101_ , 155102
  * Jin et al. 2012 Jin, H.; Im, J.; Freeman, A. J. Topological insulator phase in halide perovskite structures. _Phys. Rev. B_ 2012, _86_ , 121102
  * Krizman et al. 2018 Krizman, G.; Assaf, B.; Phuphachong, T.; Bauer, G.; Springholz, G.; de Vaulchier, L.; Guldner, Y. Dirac parameters and topological phase diagram of Pb1-xSnxSe from magnetospectroscopy. _Phys. Rev. B_ 2018, _98_ , 245202
  * Wojek et al. 2014 Wojek, B. M.; Dziawa, P.; Kowalski, B.; Szczerbakow, A.; Black-Schaffer, A. M.; Berntsen, M.; Balasubramanian, T.; Story, T.; Tjernberg, O. Band inversion and the topological phase transition in (Pb,Sn)Se. _Phys. Rev. B_ 2014, _90_ , 161202
  * Juričić et al. 2017 Juričić, V.; Abergel, D. S.; Balatsky, A. V. First-order quantum phase transition in three-dimensional topological band insulators. _Phys. Rev. B_ 2017, _95_ , 161403
  * Zeljkovic et al. 2015 Zeljkovic, I.; Okada, Y.; Serbyn, M.; Sankar, R.; Walkup, D.; Zhou, W.; Liu, J.; Chang, G.; Wang, Y. J.; Hasan, M. Z. et al. Dirac mass generation from crystal symmetry breaking on the surfaces of topological crystalline insulators. _Nat. Mater._ 2015, _14_ , 318–324
  * Weller et al. 2015 Weller, M. T.; Weber, O. J.; Frost, J. M.; Walsh, A. Cubic perovskite structure of black formamidinium lead iodide, $\alpha$-[HC(NH2)2]PbI3, at 298 K. _J. Phys. Chem. Lett._ 2015, _6_ , 3209–3212
  * Banerjee et al. 2019 Banerjee, A.; Chakraborty, S.; Ahuja, R. Rashba Triggered Electronic and Optical Properties Tuning in Mixed Cation–Mixed Halide Hybrid Perovskites. _ACS Appl. Energy Mater._ 2019, _2_ , 6990–6997
  * Wang et al. 2020 Wang, S.; Xiao, W.-b.; Wang, F. Structural, electronic, and optical properties of cubic formamidinium lead iodide perovskite: A first-principles investigation. _RSC Adv._ 2020, _10_ , 32364–32369
  * Kashikar et al. 2021 Kashikar, R.; Gupta, M.; Nanda, B. A generic Slater–Koster description of the electronic structure of centrosymmetric halide perovskites. _J Chem. Phys._ 2021, _154_ , 104706
  * Frohna et al. 2018 Frohna, K.; Deshpande, T.; Harter, J.; Peng, W.; Barker, B. A.; Neaton, J. B.; Louie, S. G.; Bakr, O. M.; Hsieh, D.; Bernardi, M. Inversion symmetry and bulk Rashba effect in methylammonium lead iodide perovskite single crystals. _Nat. Commun._ 2018, _9_ , 1–9
  * Aharon et al. 2015 Aharon, S.; Dymshits, A.; Rotem, A.; Etgar, L. Temperature dependence of hole conductor free formamidinium lead iodide perovskite based solar cells. _J. Mater. Chem. A_ 2015, _3_ , 9171–9178
  * Basera et al. 2021 Basera, P.; Singh, A.; Gill, D.; Bhattacharya, S. Capturing excitonic and polaronic effects in lead iodide perovskites using many-body perturbation theory. _J Mater. Chem. C_ 2021, _9_ , 17113–17123
  * Boyer-Richard et al. 2016 Boyer-Richard, S.; Katan, C.; Traore, B.; Scholz, R.; Jancu, J.-M.; Even, J. Symmetry-based tight binding modeling of halide perovskite semiconductors. _J. Phys. Chem. Lett._ 2016, _7_ , 3833–3840
  * Do et al. 2022 Do, T.-N.; Shih, P.-H.; Lin, H.; Huang, D.; Gumbs, G.; Chang, T.-R. Generalized Peierls substitution for the tight-binding model of twisted graphene systems in a magnetic field. _Phys. Rev. B_ 2022, _105_ , 235418
  * Sheoran et al. 2022 Sheoran, S.; Kumar, M.; Bhumla, P.; Bhattacharya, S. Rashba spin splitting and anomalous spin textures in the bulk ferroelectric oxide perovskite KIO3. _Mater. Adv._ 2022,
  * Sheoran et al. 2022 Sheoran, S.; Bhumla, P.; Bhattacharya, S. Emergence of cubic ordered full-plane persistent spin textures in lead-free materials. _Phys. Rev. Mater._ 2022, _6_ , 094602
  * Phutela et al. 2022 Phutela, A.; Bhumla, P.; Jain, M.; Bhattacharya, S. Exploring strong and weak topological states on isostructural substitutions in TlBiSe2. _arXiv preprint arXiv:2205.08950_ 2022,
  * Ma and Li 2018 Ma, X.-X.; Li, Z.-S. Substituting Cs for MA on the surface of MAPbI3 perovskite: A first-principles study. _Comput. Mater. Sci._ 2018, _150_ , 411–417
  * Gao et al. 2018 Gao, W.; Ran, C.; Li, J.; Dong, H.; Jiao, B.; Zhang, L.; Lan, X.; Hou, X.; Wu, Z. Robust stability of efficient lead-free formamidinium tin iodide perovskite solar cells realized by structural regulation. _J. Phys. Chem. Lett._ 2018, _9_ , 6999–7006
  * Hohenberg and Kohn 1964 Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. _Phys. Rev._ 1964, _136_ , B864
  * Kohn and Sham 1965 Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. _Phys. Rev._ 1965, _140_ , A1133
  * Kresse and Joubert 1999 Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. _Phys. Rev. B_ 1999, _59_ , 1758
  * Blöchl 1994 Blöchl, P. E. Projector augmented-wave method. _Phys. Rev. B_ 1994, _50_ , 17953
  * Kresse and Furthmüller 1996 Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. _Phys. Rev. B_ 1996, _54_ , 11169
  * Perdew et al. 1996 Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. _Phys. Rev. Lett._ 1996, _77_ , 3865
  * Hedin 1965 Hedin, L. New method for calculating the one-particle Green’s function with application to the electron-gas problem. _Phys. Rev._ 1965, _139_ , A796
  * Giannozzi et al. 2009 Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. _J. Phys.: Condens. Matter_ 2009, _21_ , 395502
  * Mostofi et al. 2008 Mostofi, A. A.; Yates, J. R.; Lee, Y.-S.; Souza, I.; Vanderbilt, D.; Marzari, N. wannier90: A tool for obtaining maximally-localised Wannier functions. _Comput. Phys. Commun._ 2008, _178_ , 685–699
  * Wu et al. 2018 Wu, Q.; Zhang, S.; Song, H.-F.; Troyer, M.; Soluyanov, A. A. WannierTools: An open-source software package for novel topological materials. _Comput. Phys. Commun._ 2018, _224_ , 405–416
  * Nakhaee et al. 2020 Nakhaee, M.; Ketabi, S. A.; Peeters, F. M. Tight-binding studio: A technical software package to find the parameters of tight-binding Hamiltonian. _Comput. Phys. Commun_ 2020, _254_ , 107379