Inorganic and organic halide perovskites of the form ABX₃ (where A is either organic or inorganic
monovalent entity, B = Sn, Pb or Ge and X = Cl, Br or I) have got significant attention because of
their easy processing methods as well as stupendous opto-electronic properties[1]. Halide perovskites
are generally flexible to external stimuli, therefore, the band gap can be tuned in the range of
visible spectrum so that opto-electronic properties of pure phase can be enhanced by tuning band
topology[2]. From the structural point view, halide perovskites crystallizes in cubic, tetragonal and
orthorhomobic phases at different temperature ranges[3]. Here, lower symmetry phases are distorted
from cubic phases and are characterized by neighboring B-X-B octahedral rotations in a − b as well
as a − c planes. The less investigated lower symmetry phases draw the major attention as most of
them crystallize at room temperature or slightly above. The characteristic feature of these phases i.e,
octahedral rotation, plays a vital role in band gap engineering. Despite of vast number of works, we
found following things need to be addressed with much greater detail: (I) The unification of band
gap engineering process in halide perovskite family under strain case to tune the opto-electronic
properties. (II) Analysis of combined effect of octahedral rotation angle and linear strain
on the band gap as these parameters plays vital role in chemical substitution and other external
stimuli in lower symmetry phases. In the present study, through density functional calculations
and parametric tight-binding methods, we explore the trivial and non-trivial insulating phases as
a function of biaxial strain, on different poly-morphs of prototype CsSnI₃ . At high temperature
(> 425 K), the compound CsSnI₃ crystallizes in the high symmetric cubic (α) phase (Pm-3m).
With lowering in the temperature, the distortion occurs to stabilize the compound in the tetragonal
(β) phase (P4/mbm) (351-425 K) and on further reduction (< 351 K) in the temperature, the
compound adopts the lesser symmetric orthorhombic (γ) phase (Pnam)[3–5]. Considering all the
three structure also helps in developing a comprehensive understanding of the perovskites family in
general as most of the halide perovskite family members exhibit these three polymorphs.The salient
features of the present work include, non-trivial phase in this prototype compound in all the three
phases under bi-axial strain. The Wannier function based surface band structure confirms the bulk
TI state. Exact diagonalization of Slater-Koster TB Hamiltonian reveals five effective characteristic
TB parameters that govern the band gap engineering. In the tetragonal phase, which has only
in-plane rotation, our study reveals a phase spanned by shear strain and linear strain. The phase
diagram reveal, non-trivial and trivial phases. The variation of characteristic TB parameters as a
function of octahedral rotation angle as well as strain reveals the universality picture of the halide
perovskite family members. As mentioned earlier, the lower symmetry structures are distorted from
cubic phases, we map the low symmetry structures on cubic phase, hence we explain the properties
of lower structures on mapped cubic phase. Finally, we formulate the surface TB band Hamiltonian
to obtain the surface band structure