The document summarizes key information about the CdGa2S4 semiconductor. It belongs to the ordered vacancy compound family, which contains direct bandgap materials less than 4 eV that can be used in optoelectronic devices. CdGa2S4 has a defect chalcopyrite structure at ambient conditions and undergoes a phase transition to a disordered rocksalt structure above 18.8 GPa. First-principles density functional theory calculations were performed to investigate the structural stability and electronic properties of both phases under varying pressures. The results show that the defect chalcopyrite phase is more stable at lower pressures and becomes metallic above the transition pressure where the structures change.

1. 1. Introduction
The cadmium thiogallate 𝐶𝑑𝐺𝑎2𝑆4 is associated with the ordered vacancy compounds (OVCs)
family, which encloses the ternary semiconducting compounds 𝐴𝐼𝐼
𝐵2
𝐼𝐼𝐼
𝑋4
𝑉𝐼
(𝐴𝐼𝐼
=
𝑍𝑛, 𝐶𝑑, 𝐻𝑔; 𝐵𝐼𝐼𝐼
= 𝐴𝑙, 𝐺𝑎, 𝐼𝑛; 𝐶𝑉𝐼
= 𝑂, 𝑆, 𝑆𝑒, 𝑇𝑒). This family contains a wide range of direct
bandgap materials and the forbidden energy gap is less or equal to 4 𝑒𝑉. These compounds can
be used in photoluminescence and optoelectronic devices [1-10], i.e., laser diodes, optical fiber,
photodiodes, and solar cells. The photographic characteristic of these semiconductors is
described by electrophotographic layers and provides superior quality outputs [11]. The
𝐶𝑑𝐺𝑎2𝑆4 composite can be used in a wide range of light deflectors and temperature sensors
[12-14]. Therefore, a greater focus on the importance of this crystal in optics draws attention
to the handling of its bandgap by exerting pressure. These compounds act as optically
birefringent [2] with the occupation of strong anisotropy and the absence of cubic symmetry
and are best suited for phase matching applications [15]. The electronic, optical, structural,
mechanical, and elastic properties have been investigated by the first principle DFT study of
𝐶𝑑𝐺𝑎2𝑆4 crystal. The defect chalcopyrite (DC) semiconductor’s band structure is described by
Jiang and Lambrecht [16] to coordinate with normal chalcopyrite semiconductors. The
𝐶𝑑𝐺𝑎2𝑆4 semiconductor’s electronic and structural properties variation have been calculated
at different pressure and temperature [17]. The vibrational properties of 𝐶𝑑𝐺𝑎2𝑆4
semiconductors have been evaluated by Hecht et al. [18-19]. For 𝐶𝑑𝐺𝑎2𝑆4 and 𝐶𝑑𝐺𝑎2𝑆𝑒4
semiconductors, the PL, and absorption spectra have been observed in the range of 2.3 −
3.8 𝑒𝑉 and 370 − 600 𝑛𝑚, respectively by V.F. Zhitar [20] and N.V. Joshi [21]. The 𝐶𝑑𝐺𝑎2𝑆4
thin film’s dielectric properties and electrical conductivity have been measured by confining
the process of traditional thermal evaporation [22]. The polar phonon’s dispersion is directional
for DC semiconductors and is investigated by Razzetti et al. [23-24]. The iodine is used as a
transport agent in this process. Hamide Vaezi et al. [57] have been described the optical spectra
of 𝐶𝑑𝐺𝑎2𝑆4 semiconductor’s crystalline form.
The phase transition in the DC semiconductors under pressure has been tried. At high pressure,
the Raman spectroscopy, X-ray diffraction, and optical absorption process have been
accomplished on 𝐶𝑑𝐺𝑎2𝑆𝑒4 [25], 𝑍𝑛𝐺𝑎2𝑇𝑒4[26] and 𝐶𝑑𝐺𝑎2𝑆4 [19] semiconductors. The
Raman modes and phase transitions have been examined by Raman scattering spectroscopy
under different pressures [27-30]. Noticeably, the DC materials go through a structural phase
transition from tetragonal structure to rocksalt structure at room temperature and high pressure
[31-32]. Incidentally, the thiospinels have been demonstrated the partially controlled rocksalt

2. structure [33] under pressure. However, the 𝐶𝑑𝐺𝑎2𝑆4 the structure is least studied than other
defect chalcopyrite semiconductors according to the literature review. Thus, it was considered
interesting to investigate various properties of this semiconductor under pressure. In this paper,
we have utilized generalized gradient approximation (GGA) inside the system of first-
principles density functional theory (FP-DFT) to examine both defect chalcopyrite and rocksalt
structure’s stability, electronic, elastic, and optical properties at different pressures.
2. Computational Details
The density functional theory (DFT) system [34] has been applied to crystal structure with first
principal evaluations by using Vienna Ab initio Simulation Package (VASP) [35]. This process
follows the full-potential linearized augmented plane waves (FP-LAPW) method [3]. The
generalized gradient approximation (GGA) has been used with Perdew-Burke-Ernzerhof
(PBE) parameterization [36-37] to outline electrons exchange-correlation interaction. In
this paper, the wave functions consist of valence states of 𝐶𝑑(3𝑑, 4𝑠), 𝐺𝑎(3𝑑, 4𝑠, 4𝑝) and
𝑆(3𝑠, 3𝑝) by using pseudo-atomic OMX basis sets [38-39]. The atom’s semi-core states are
used by these basis sets for precise calculations. The Monkhorst-Pack grids [40], i.e., 6 × 6 × 6
and 8 × 8 × 8 have been used for defect chalcopyrite (DC) and disordered rocksalt (DR) phase,
respectively by Brillouin zone integration technique. The mesh cut-off for the DC and DR
phase has been set to 480 𝑒𝑉 and 500 𝑒𝑉, respectively. The observed structures have been
fully relaxed to their optimized configuration by the forces and stress tensor calculations in this
paper. The limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) optimization
technique [41] has been applied for achieving an optimized molecular geometry. The value of
force and stress tolerance are 0.0001 𝑒𝑉 Å
⁄ and 0.009 𝐺𝑃𝑎, respectively in the optimized
configurations. For self-consistent calculations, the total energy differences are less than
0.272 × 10−6
𝑒𝑉.
3. Results and Discussion
3.1 Structural Results and Properties
𝐶𝑑𝐺𝑎2𝑆4 includes body-centered cubic (BCC) geometry, tetragonal structure and space group
𝐼4 (#82). This crystal is a defect chalcopyrite (DC) semiconductor and contains seven atoms
per unit cell. The DC-phase of 𝐶𝑑𝐺𝑎2𝑆4 structure’s lattice parameters are represented in Fig.
1(a) and given as [57] 𝑎 = 5.642 Å and 𝑐 = 10.44 Å, where 𝑎 = 𝑏 ≠ 𝑐
The Wyckoff positions of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 structure is
𝐶𝑑 (0,0,0), 𝐺𝑎1 (0,0,0.5), 𝐺𝑎2 (0,0.5,0.25) and 𝑆 (0.273,0.265,0.138). The specified atoms
occupy these positions in the traditional unit cell. The disordered rocksalt (DR) phase of

3. 𝐶𝑑𝐺𝑎2𝑆4 structure [43] contains space group 𝐹𝑚 − 3𝑚 (#225), as illustrated in Fig. 1(b). At
diffusive pressure, the volume vs total energy plots has been evaluated for both (DC and DR)
phases and are represented in Fig. 2. In Table 1, the optimized lattice parameters are represented
with comparable experimental and theoretical values. For 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 and 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4
structures, the bulk modulus, and the energy band-gap have been calculated and are also shown
in Table 1.
Fig. 2 represents that the DC-phase is more poised than the DR-phase of 𝐶𝑑𝐺𝑎2𝑆4 structure at
absolute zero temperature and zero pressure. The Gibbs free energy has been calculated for
determining the phase transition of 𝐶𝑑𝐺𝑎2𝑆4 structure from DC to DR phase. The Gibbs free
energy is expressed as
𝐺 = 𝑈 + 𝑃𝑉 − 𝑇𝑆 (1)
In equation (1), the system’s internal energy, work done by structure volume, and vibrational
energy are denoted as 𝑈, 𝑃𝑉, and 𝑇𝑆, respectively. This equation can be rewritten in terms of
enthalpy, given as
𝐺 = 𝐻 − 𝑇𝑆 (2)
where enthalpy 𝐻 = 𝑈 + 𝑃𝑉.
In this paper, the enthalpies have been calculated for optimized 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 and 𝐷𝑅 −
𝐶𝑑𝐺𝑎2𝑆4 structures at 5, 10, 15, 18.8, 20, 25, and 30 𝐺𝑃𝑎 pressures. The 𝐶𝑑𝐺𝑎2𝑆4 structure’s
phase transition from DC to DR phase is achieved at 18.8 𝐺𝑃𝑎, as represented in Fig. 3. Two
enthalpy curves intersection point describes this phase transition. Moreover, the transition
pressure’s experimental value [57] is 17 𝐺𝑃𝑎 which is less than the calculated transition
pressure. The total difference between theoretical and experimental transition pressure is
1.8 𝐺𝑃𝑎 due to temperature variation and taken powder’s purity. The enthalpy change turns
negative at 18.8 𝐺𝑃𝑎 for DR structure. Consequently, the stability of the DR structure has
increased after transition pressure 18.8 𝐺𝑃𝑎 but the DC structure is stable until phase transition
occurred.
The normalized cell volume (𝑉 𝑉0
⁄ ) has been calculated through phase transition analysis from
DC to DR structure. The traditional unit cell’s volume at a certain pressure and the DC phase’s
equivalent volume at zero pressure is denoted as 𝑉 and 𝑉0, respectively. Fig. 4 shows the
pressure vs normalized cell volume plot. As it turns out to be volume collapse (Δ𝑉 𝑉0
⁄ ) at the
transition pressure, the particular ratio is 0.586 or 58.6 %.
Fig. 5(a) represents the band structure of 𝐶𝑑𝐺𝑎2𝑆4 semiconductor’s DC-phase. The DC-
phase’s band structure is evaluated at 0 𝐺𝑃𝑎 and the marked energy range is −14 𝑒𝑉 to 6 𝑒𝑉.

4. It is found that the 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 semiconductor’s energy bandgap is 2.09 𝑒𝑉 and is
represented in Table 1. The Fermi level is fixed to zero. The experimental value [58] of the
bandgap is quite high than the theoretical value due to temperature variation. Fig. 5(b)
illustrates the band structure of 𝐶𝑑𝐺𝑎2𝑆4 structure’s DC-phase at 18.8 𝐺𝑃𝑎. It is found that
the overlapping has occurred in valence and conduction bands due to its metallic nature.
The distinct electronic state's energy distribution can be reflected by the total density of states
(TDOS) and partial density of states (PDOS). It is well known that DOS calculations are used
for observing the structural electronic behavior of 𝐶𝑑𝐺𝑎2𝑆4 semiconductor’s both phases. The
total and partial DOS of both phases of 𝐶𝑑𝐺𝑎2𝑆4 the structure is illustrated in Fig. 6(a) and
6(b) at 0 𝐺𝑃𝑎 and 18.8 𝐺𝑃𝑎, respectively.
As reported by partial DOS in Fig. 6(a), the atomic states 𝑆 − 3𝑠 with the combination of 𝐺𝑎 −
3𝑑, 4𝑠, and 4𝑝 states result from the electronic states over −13 𝑒𝑉 to −11 𝑒𝑉. The dominant
peak in energy interval of −7.5 𝑒𝑉 to −7 𝑒𝑉 arises due to 𝐶𝑑 − 4𝑑 atomic states. The
combination of 𝐺𝑎 − 4𝑠, 4𝑝 and 𝑆 − 3𝑝 atomic states produce this peak. The valence and
conduction electronic states are the most significant parts of DOS. Below the fermi level, the
valence states can be splitted into three different regions. The valence band’s lower-level states
are primarily shaped by 𝐺𝑎 − 4𝑠 states with the 𝑆 − 3𝑠 atomic states. The union of 𝑆 −
3𝑝, 𝐺𝑎 − 4𝑝, and 𝐶𝑑 − 5𝑠 atomic states form the intermediate region. Moreover, the upper
level of states is placed just below the fermi level and is also called the valence band’s top
region. The chief contribution of 𝑆 − 3𝑝 atomic states with the least combination of 𝐶𝑑 − 4𝑑
and 𝐺𝑎 − 4𝑝 form upper states of the valence band. In the same way, the conduction band can
also divide into three regions. The hybridization of atomic states 𝐺𝑎 − 4𝑠 with 𝑆 − 3𝑠 and 3𝑝
states form the conduction band’s bottom region. In the intermediate region, the peak arises
after 3 𝑒𝑉 to 5 𝑒𝑉 and comes from atomic states 𝐺𝑎 − 4𝑝 with a strong contribution of 𝑆 − 3𝑝
and 𝐶𝑑 − 5𝑠 states. The combination of 𝐺𝑎 − 4𝑝 and 𝑆 − 3𝑝 atomic states form the upper
level of conduction states.
Similarly, the atomic states 𝑆 − 3𝑠 with the combination of 𝐺𝑎 − 3𝑑, 4𝑠, 4𝑝 and 𝐶𝑑 − 5𝑠 states
result from the electronic states over −13 𝑒𝑉 to −11 𝑒𝑉, as reported by PDOS in Fig. 6(b). the
leading peak in energy interval of −8 𝑒𝑉 to −7 𝑒𝑉 arises due to 𝐶𝑑 − 4𝑑 atomic states. The
combination of 𝐺𝑎 − 4𝑠, 4𝑝 and 𝑆 − 3𝑠 atomic states produce this peak. The valence and
conduction electronic states are the most important parts of DOS and can be divided into three
regions: lower, intermediate, and upper level. This will follow the same process as described
in the above (DC-phase) paragraph. It is important to acknowledge the fact that the results

5. presented in this paper are comparable with literature results [44-45]. It is observed that the
electronic states are more responsive to high pressure by the both (DC and DR) phases DOS
comparisons due to greater angular momentum.
Figure 1: The unit cell of (a) 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 and (b) 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4. In this figure, dark green
balls represent Cd atoms, light green balls represent Ga atoms and yellow balls represent S
atoms

6. Figure 2: Total energy vs volume curves of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 and 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4.
Figure 3: Variation of enthalpies as a function of pressure in 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 and 𝐷𝑅 −
𝐶𝑑𝐺𝑎2𝑆4.
Table 1
Lattice parameters 𝑎(Å) and 𝑐(Å), bulk modulus (𝐵) (in GPa) and energy gap (𝐸𝑔) (in eV) of
𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 and 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 phase semiconductor.
Parameters Proposed work Experimental Other theoretical
results
𝑫𝑪 − 𝑪𝒅𝑮𝒂𝟐𝑺𝟒
a 5.642 5.56 [58], 5.536 [59] 5.553 [57]
c 10.44 10.18 [58], 10.160
[60]
10.272 [57]

7. B 47.2a
64 [57-58] 40.8 [57]
𝐸𝑔 2.093 3.23 [58] 1.96 [57]
𝑫𝑹 − 𝑪𝒅𝑮𝒂𝟐𝑺𝟒
a 5.1892 5.4355 [59]
B 159.45
𝐸𝑔 0 0 0
a
2nd
order EOS
Figure 4: Variation of the normalized volume (𝑉 𝑉0
⁄ ) of 𝐶𝑑𝐺𝑎2𝑆4 as a function of pressure.
3.2 Elastic Properties
The material’s mechanical strength, hardness, brittleness, ductility, stiffness, and durability
have been described by elastic parameters. The crystallization of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4
structure occurs in 𝐼4 the space group and placed into tetragonal shaped material group TII. In
VASP, seven elastic constants 𝐶11, 𝐶12, 𝐶13, 𝐶33, 𝐶44, 𝐶66 and 𝐶16 can be calculated by using
MT elastic simulation for this group. Due to the occupation of shear elastic constant 𝐶16 in a
slanted direction, the elastic moduli can’t be derived by these elastic constants. TI crystal can
calculate elastic moduli from six independent elastic constants. This elastic co-relation is not
possible in the TII laue group of tetragonal structure.

8. The 𝐶16 must be zero for converting TII crystal’s seven elastic constants to TI crystal’s six
elastic constants. The rotation around the z-axis with the angle 𝜙𝜅,𝛾 can transform these elastic
constants are is expressed as
𝜙𝜅,𝛾 =
1
4
𝑡𝑎𝑛−1
(
4𝐶16
𝐶11−𝐶12−2𝐶66
) (3)
where 𝜙𝜅 ranges from 0 to 𝜋 2
⁄ and by employing the relation 𝜙𝛾 = 𝜙𝜅 + 𝜋 4
⁄ , the value of
𝜙𝛾 can be detected. From equation (3), the value of 𝜙𝜅 is −0.0010
at 0 𝐺𝑃𝑎 for DC-phase of
𝐶𝑑𝐺𝑎2𝑆4 structure. Using the relationship described in Ref. [46], the TII crystal’s six
independent elastic constants have been evaluated at 0 𝐺𝑃𝑎. In Table 2, the six new elastic
constants are recorded at 0, 5, 10, 15, and 18.8 𝐺𝑃𝑎. The calculation of elastic constants is very
useful at applied pressure for finding the mechanical properties of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 structure.
The Born-Huang stability criteria [47] describes the mechanical stability at zero pressure for
tetragonal structure. According to this criterion, the conditions for elastic constants are termed
as
1) The 𝐶11, 𝐶44 and 𝐶66 must be greater than zero.
2) The difference in 𝐶11 and 𝐶12 should be greater than zero.
3) Half of the sum of the product of 𝐶11, 𝐶33 and 𝐶12, 𝐶33 should be less than a square of
𝐶13.
There are no experimental and theoretical elastic parameter values exist for comparing with
computed results, as per our knowledge. The Born stability conditions have been altered due
to applied pressure at a tetragonal structure. The stability conditions [47-48] for elastic
constants are termed as
1) The difference in 𝐶11 and applied pressure (𝑃) must be greater than zero.
2) The difference in 𝐶44 and 𝑃 should be greater than zero.
3) The half of the difference in 𝐶11 and 𝐶44 should be less than pressure (P).
4) The half of the product of the difference in 𝐶33 and 𝑃 and summation of 𝐶11 and 𝐶12
must be less than the whole square of 𝐶13 and 𝑃 summation.

9. Figure 5: Electronic band structure of (a) DC-phase (at 0 𝐺𝑃𝑎) and (b) DR-phase (at
18.8 𝐺𝑃𝑎) of 𝐶𝑑𝐺𝑎2𝑆4.

10. The elastic constants 𝐶11, 𝐶12, 𝐶13, 𝐶33 and 𝐶44 rise linearly with increasing pressure up to
18.8 𝐺𝑃𝑎 except 𝐶66 for 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 structure. The 𝐶66 intially rise till 5 GPa, therefore
fall up to 10 𝐺𝑃𝑎 and then rise again with increasing pressure. The value of 𝐶66 decreases with
pressure in the range of 5 − 10 𝐺𝑃𝑎 due to the change in temperature and decrease in material’s
stiffness at subsequent pressure range. Table 2 lists the computed values of DC structure’s
elastic constants towards 20 𝐺𝑃𝑎 pressure. It is noted that the elastic constants values fall with
the increasing pressure ranges between 18.8 − 20 𝐺𝑃𝑎. By the way, the Born’s stability
criteria [47-48] is retracted by 𝐶44 value because of the phase transition appears at 18.8 𝐺𝑃𝑎.
Hence, the DC phase of 𝐶𝑑𝐺𝑎2𝑆4 semiconductor is stable until 18.8 𝐺𝑃𝑎 pressure and
becomes less stable after transition pressure 18.8 𝐺𝑃𝑎, as illustrated in prior literature [49].

11. Figure 6: (a) Density of states of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 at 0 GPa. (b) The density of states of 𝐷𝑅 −
𝐶𝑑𝐺𝑎2𝑆4 at 18.8 𝐺𝑃𝑎.
The 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 the structure’s calculated elastic constants are represented in Fig. 7(a)
with applied pressure. The values of 𝐶11 and 𝐶33 rises with increasing pressure which describes
the confinement resistance along with prominent a and c-axes. At different pressures, the DC
structure comprises lower resistance to shear than to compression because of the values of
𝐶12, 𝐶13 and 𝐶66 are the half of the 𝐶11 and 𝐶33 values. However, the value of 𝐶44 is 61% of
the 𝐶11 and 𝐶33 values. Using the relationship mentioned in Ref. [26], the elastic moduli
containing bulk modulus (𝐵), shear modulus (𝐺), Young’s modulus (𝐸) and Poisson’s
ratio (𝜎) have been calculated at 0, 5, 10, 15, and 18.8 𝐺𝑃𝑎. The calculation of elastic moduli
has been executed by using 𝐶𝑑𝐺𝑎2𝑆4 the structure’s six elastic constants and recorded in Table
2. The elastic moduli of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 the structure is represented in Fig. 7(b) and displays
the pressure dependence nature. It is noticeable that the 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 structure’s bulk
modulus (𝐵) rises linearly with increasing pressure till 18.8 𝐺𝑃𝑎. Moreover, the value of 𝐺
and 𝐸 rises linearly with increasing pressure up to 5 𝐺𝑃𝑎, therefore falls up to 10 𝐺𝑃𝑎 and then
rises again with increasing pressure for DC-structure. The values of 𝐵 and 𝐸 reduce with
pressure in the range of 5 − 10 𝐺𝑃𝑎 because of the reduction in material’s stiffness and
temperature variation at subsequent pressure range. It is noted that the calculated bulk modulus
by structure’s elastic constants is comparable with attained bulk modulus via 2nd
order energy
of states (EOS), which is recorded in Table 1. It is also discovered that the value of 𝐺 is way
less than 𝐵 value, which displays the lower resistance to shear deformation than compression
deformation. The material’s stiffness is approximated by Young’s modulus 𝐸.
The material’s brittleness and ductility are approximated by 𝐵 𝐺
⁄ ratio. This ratio is coined by
Pugh’s theory [49] to play a significant role in material characterization. The material is ductile
when 𝐵 𝐺
⁄ ratio overtakes 1.75 otherwise possess brittle behavior. This ratio is 1.79 for 𝐷𝐶 −
𝐶𝑑𝐺𝑎2𝑆4 at 0 𝐺𝑃𝑎, hence material possesses ductile behavior. The 𝐵 𝐺
⁄ the ratio rises with
increasing pressure and achieves the peak value of 3.52 at transient pressure (18.8 𝐺𝑃𝑎), as
represented in Fig. 9. Hence the material’s ductility has been enhanced due to the applied
pressure.

12. Table 2: Calculated elastic constants (𝐶𝑖𝑗 𝑖𝑛 𝐺𝑃𝑎), bulk modulus (B in GPa), shear modulus
(𝐺 in GPa), Young modulus (𝐸 in GPa), Poisson’s ratio (𝜎), and 𝐵 𝐺
⁄ ratio and Zener
anisotropy factor (𝐴) of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 and 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 the structure under different
pressures.
Phase 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4
Pressure
(GPa)
0 5 10 15 18.8 20 20 25 30
𝐶11 66.68 79.26 102.05 119.69 133.35 75.93 265.29 292.82 338.11
𝐶12 32.92 39.49 55.49 73.94 86.45 42.24 108.27 116.44 132.28
𝐶13 39.4 57.72 81.09 101.76 113.82 55.06
𝐶33 61.96 80.97 103.05 134.22 147.94 83.65
𝐶44 42.25 43.45 47.7 52.56 54.64 49.78 19.92 17.6 16.62
𝐶66 37.43 42.29 33.83 39.14 41.77 37.01
𝐵 46.52 60.15 80.5 99.55 112.63 159.46 180.53 200.92
𝐺 26.06 25.98 25 28.76 30.45 46.29 49.1 52.57
𝐸 65.69 67.85 67.6 78.48 83.59 126.24 134.59 144.33
𝜎 0.26 0.31 0.36 0.37 0.38 0.37 0.38 0.38
𝐵 𝐺
⁄ 1.79 2.31 3.22 3.46 3.7 3.45 3.68 3.82
𝐴 2.22 2.13 1.45 1.71 1.78 0.43 0.427 0.38
The material’s bonding nature is described by Poisson’s ratio 𝜎, which ranges from 0.26 to
0.38. The material’s bonding is ionic when the value of 𝜎 confines from 0.26 to 0.5, but if 𝜎
is less than 0.26, it contains covalent bonding. Fig. 9 represents the change in 𝜎 value with
applied pressure. The 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 structure’s bonding is ionic because the value of 𝜎 is
0.26 and this ionic bonding becomes strong with an increase in pressure. As well as, the
material’s compressibility [50] on the dominant axis of a lattice have been approximated by
axial compressibilities 𝜒𝑎 and 𝜒𝑐, expressed as
𝜒𝑎 =
𝐶33−𝐶13
𝐶33(𝐶11+𝐶12)−2𝐶13
2 (4)
𝜒𝑐 =
𝐶11+𝐶12−2𝐶13
𝐶33(𝐶11+𝐶12)−2𝐶13
2 (5)

13. where 𝜒𝑎 = 7.36 × 10−3
𝐺𝑃𝑎−1
and 𝜒𝑐 = 6.78 × 10−3
𝐺𝑃𝑎−1
for 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 at zero
pressure. The volume compressibility 𝜒 can be calculated by using axial compressibilities,
expressed as
𝜒 = −𝑉−1 𝜕𝑉
𝜕𝑃
= 2𝜒𝑎 + 𝜒𝑐 (6)
From equation (6), the value of 𝜒 is 0.0215 𝐺𝑃𝑎−1
. The 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 structure’s bulk
modulus (𝐵) is 46.51 𝐺𝑃𝑎 considering the inverse operation of 𝜒. This approach’s reliability
and results precision have been validated from the fact that the calculated value of 𝐵 is
comparable with 2𝑛𝑑
order Birch-Murnaghan energy of state’s bulk modulus value.
Figure 7: Calculated elastic constants and elastic moduli of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 as a function of
pressure.

14. The specific material’s elastic anisotropy strength is determined by Zener anisotropy factor 𝐴.
The material exhibits elastic isotropy when 𝐴 = 1 otherwise elastic anisotropy. A large
anisotropic strength may likely lead to microcracks induced in the material that disable the
devices. The relationship between 𝐴 and DC structure’s elastic constants [49] are represented
as
𝐴 =
2𝐶66
(𝐶11−𝐶12)
(7)
The anisotropic factor from the 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 the structure was calculated at different
pressures and is shown in Table 2. The DC-phase’s anisotropic factor falls with increasing
pressure up to 18.8 𝐺𝑃𝑎.
The conversion from DC to DR structure for 𝐶𝑑𝐺𝑎2𝑆4 semiconductor has occurred at
18.8 𝐺𝑃𝑎. The 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 the structure is illustrated by three independent elastic
constants (𝐶11, 𝐶12 and 𝐶44) and exhibits a cubic-shaped lattice. Fig. 8(a) represents the
calculated elastic constants at 20, 25, and 30 𝐺𝑃𝑎 pressures and are recorded in Table 2. The
mechanical stability of steady cubic structure must satisfy the Born stability criteria [48],
illustrated as
1) The difference in 𝐶11, 𝐶12 and 𝑃 should be greater than zero.
2) The half of the difference between 𝑃 and 𝐶11 should be less than 𝐶12.
3) The difference in 𝐶44 and 𝑃 should be greater than zero.

15. Figure 8: Calculated elastic constants and elastic moduli of 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 as a function of
pressure.
The 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 structure’s elastic constants 𝐶11 and 𝐶12 rise tediously with increasing
pressure but 𝐶44 falls tediously with an increase in pressure. It is noted that the 𝐶11 rises rapidly
with pressure than 𝐶12. Hence, the DR structure contains higher resistance to compression than
to shear. The atomic bonding’s angular character is illustrated by Cauchy pressure for the cubic
structure’s elastic constants. The Cauchy pressure [51] has been determined by the difference
in 𝐶12 and 𝐶44. The metallic and non-metallic nature of this structure is defined by the Cauchy
pressure’s positive and negative values, respectively. It is found that the 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 the
structure exhibits metallic behavior due to positive values of calculated Cauchy pressure at
20, 25, and 30 𝐺𝑃𝑎. The Cauchy pressure’s positive values increase with rising pressure and
indicate strong metallic property. The 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 is ductile in nature because of the
Cauchy pressure’s positive values. It also fits well with our computed electronic band structure
and DOS demonstrating its conductive character after phase transition occurs.
Using the relationship in Refs. [26], [52], the 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 structure’s bulk modulus (𝐵),
shear modulus (𝐺), Young’s modulus (𝐸) and Poisson’s ratio (𝜎) have been evaluated at
20, 25, and 30 𝐺𝑃𝑎. The 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 structure’s elastic moduli are dependent on applied
pressures are recorded in Table 2. The value of 𝐵 is more than a value of 𝐺 for DR structure,
as illustrated in Fig. 8(b). Hence, the 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 the structure exhibits higher resistance to
compression than to shear stress. Fig. 8(b) also represents the material’s stiffness increases with
rising pressure ranges from 20 − 30 𝐺𝑃𝑎.

16. Figure 9: Pressure dependence of 𝐵 𝐺
⁄ , 𝜎 and 𝐴 of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 and 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4.
For 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4, the 𝐵 𝐺
⁄ the ratio rises from 3.45 at 20 𝐺𝑃𝑎 to 3.82 at 30 𝐺𝑃𝑎, as
illustrated in Fig. 9. It is revealed that the DR structure exhibits ductile nature at applied
pressure 20 𝐺𝑃𝑎 and beyond.
The computed value of 𝜎 is 0.37 at 20 𝐺𝑃𝑎 for the DR structure and exposes the strong ionic
bonding. The value of 𝜎 rises linearly with increasing pressure up to 25 𝐺𝑃𝑎 and then displays
the constant state.
The relation between anisotropic factor 𝐴 and elastic constants of DR structure [53] is
represented as
𝐴 =
2𝐶44
(𝐶11−𝐶12)
(8)
For 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 structure, the values of 𝐴 are calculated at different pressures and are
recorded in Table 2. It is found that the value of 𝐴 decrease linearly with an increase in pressure
for 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4, as shown in Fig. 9. Hence, the DR structure of 𝐶𝑑𝐺𝑎2𝑆4 exhibits elastic
anisotropic nature at applied pressure ranges from 20 − 30 𝐺𝑃𝑎.
3.3 Optical Properties
The material’s optical properties can be evaluated by complex dielectric function 𝜖(𝜔). The
real and imaginary parts of the dielectric function (in the x-direction) are computed by the inter-
band transitions [56] between the valence band and the conduction bands. The complex
dielectric tensors are represented as
𝜖𝛼𝛽(𝜔) = 𝜖1 + 𝑖𝜖2 (9)

17. 𝐼𝑚 𝜖𝛼𝛽(𝜔) =
ℏ2𝑒2
𝜋𝑚2𝜔2
∑ ∫ 𝑑𝑘 < 𝑐𝑘|𝑃𝛼
|𝑣𝑘 >< 𝑣𝑘|𝑃𝛽
|𝑐𝑘 > 𝛿(𝜖𝑐𝑘 − 𝜖𝑣𝑘 − 𝜔)
𝑐,𝑣 (10)
𝑅𝑒 𝜖𝛼𝛽(𝜔) = 𝛿𝛼𝛽 +
2
𝜋
𝑃 ∫
𝜔1𝐼𝑚 𝜖𝛼𝛽(𝜔1)
𝜔12
−𝜔2
𝑑𝜔1
∞
0
(11)
where P is the integral’s principal value. From equation (10), the reduced Plank’s constant,
electron’s mass, angular frequency, speed of light, and initial charge of the electron are denoted
by ℏ, 𝑚, 𝜔, 𝑐 and 𝑒, respectively. The 𝛿 are represented as a partial change in parameters.
The dispersion and absorption of incident photons determine the dielectric function’s real part
𝜖1(𝜔) and imaginary part 𝜖2(𝜔), respectively. The elements of the momentum matrix are
bounded by the occupied and unoccupied wave functions to calculate the imaginary part 𝜖2(𝜔).
Using the Kramer-Kronig relationship [54], the real part 𝜖1(𝜔) is obtained from 𝜖2(𝜔).
The imaginary part of the dielectric function 𝜖2(𝜔) is represented in Fig. 10. The fundamental
absorption edge (at 0 𝐺𝑃𝑎) is detected at 2.08 𝑒𝑉, which reveals the inter-band transition
among the lowermost conduction band and the uppermost valence band. The dominant states
of the uppermost valence band are 𝑆 − 3𝑝 and 𝐶𝑑 − 5𝑠. Due to the transition in 𝑆 − 3𝑝 and
𝐺𝑎 − 4𝑝 states, the curve’s peak achieves at 5.94 𝑒𝑉 towards 0 𝐺𝑃𝑎 pressure. The 𝜖2(𝜔) peaks
appear at 5.17 and 5.43 𝑒𝑉 towards 10 𝐺𝑃𝑎 and 18.8 𝐺𝑃𝑎 pressures, respectively. Due to the
blue shift phenomena [55], the curve peaks tend towards greater energies with increasing
pressure ranges between 10 to 18.8 𝐺𝑃𝑎.
Figure 10: The calculated imaginary part of the dielectric function of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4.
The photon energy falls with increasing pressure ranges between 5 to 10 𝐺𝑃𝑎 because of the
instant variation in wavelength. This blue shift phenomena are dependent on the applied

18. pressure and also imitates other optical properties. Due to energy bandgap falls with increasing
pressure ranges from 0 𝐺𝑃𝑎 to 18.8 𝐺𝑃𝑎, the sharp falling edge tilt towards the higher energies.
Fig. 11 illustrates the dielectric function’s real part. The photon energy ranges from 0 𝑒𝑉 to
35 𝑒𝑉 in this curve. At zero pressure, the DC structure’s static dielectric constant is found to
be 6.42 and is recorded in Table 3 including theoretical and experimental results. It is crucial
to acknowledge the fact that the results presented in this section are comparable with the
literature results.
Optical Parameters
under pressure
𝜀1(0) 𝜂(0) 𝑅(0)
This
work
Literature
Values
This
work
Literature
Values
This
work
Literature
Values
0 GPa 6.42 5.86a
2.534 2.35a
, 2.67b
0.188 0.157a
5 GPa 6.83 2.613 0.199
10 GPa 7.24 2.69 0.21
15 GPa 7.63 2.762 0.219
18.8 GPa 7.88 2.807 0.198
a
Theoretical Ref. [58].
b
Experimental Ref. [60].
Table 3 also represents the calculated value of 𝜖1(0) at 10 𝐺𝑃𝑎 and 18.8 𝐺𝑃𝑎 pressures. It is
found that the 𝜖1(0), 𝜂(0) and 𝑅(0) values rise with increasing pressure, also described in Ref.
[29].
Fig. 12 represents the DC structure’s refractive index 𝜂(𝜔) and is calculated at 0, 10, and
18.8 𝐺𝑃𝑎. At 0 𝐺𝑃𝑎, the static refractive index is found to be 2.53 and is recorded in Table 3.
The refractive index 𝜂(𝜔) is analogous of 𝜖1(𝜔) curve. The calculated result is nearer to the
experimental value [58] than the theoretical literature value [60]. At 10 𝐺𝑃𝑎 and 18.8 𝐺𝑃𝑎
pressures, the 𝜂(0) values are 2.69 and 2.81, respectively.

19. Figure 11: The calculated real part of the dielectric function of the 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 under
pressure.
Fig. 12 represents the DC structure’s refractive index 𝜂(𝜔) and is calculated at 0, 10, and
18.8 𝐺𝑃𝑎. At 0 𝐺𝑃𝑎, the static refractive index is found to be 2.53 and is recorded in Table 3.
The refractive index 𝜂(𝜔) is analogous of 𝜖1(𝜔) curve. The calculated result is nearer to the
experimental value than the theoretical literature value. At 10 𝐺𝑃𝑎 and 18.8 𝐺𝑃𝑎 pressures,
the 𝜂(0) values are 2.69 and 2.81, respectively.
Due to a blue shift, the peaks tend to have higher energies. The 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 structure can
be used for photonic applications due to the high value of the refractive index.
The DC structure’s extinction coefficient 𝑘(𝜔) is illustrated in Fig. 13 and is calculated at
0, 10, and 18.8 𝐺𝑃𝑎. The 𝑘(𝜔) curve’s peak achieves at 7.27 𝑒𝑉 towards 0 𝐺𝑃𝑎 pressure. The
absorption is dominant at this point and describes the zero value of 𝜖1(𝜔). At 10 𝐺𝑃𝑎 and
18.8 𝐺𝑃𝑎 pressures, the peak values of 𝑘(𝜔) are achieved at 7.28 and 7.63 𝑒𝑉, respectively.

20. Figure 12: The calculated refractive index of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4.
Figure 13: The calculated extinction coefficient of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4.

21. Figure 14: The calculated absorption coefficient of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4.
The infiltration limit of light particles into material containing specific wavelength has been
described by absorption coefficient 𝛼(𝜔) until the material’s absorption process is initiated.
The 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 structure’s absorption coefficient 𝛼(𝜔) is illustrated in Fig. 14 and is
calculated at 0, 10, and 18.8 𝐺𝑃𝑎. The photon energy ranges from 0 to 35 𝑒𝑉 in this curve.
Due to the blue shift phenomena, the absorption curves tend towards greater energies with
increasing pressure ranges from 10 − 18.8 𝐺𝑃𝑎. The peak values of 𝛼(𝜔) are found at
9.05 𝑒𝑉, 8.08 𝑒𝑉 and 9.33 𝑒𝑉 towards 0, 10, and 18.8 𝐺𝑃𝑎 pressures, respectively.
Fig. 15 represents the DC structure’s reflectivity 𝑅(𝜔) in the effect of applied pressure. At
0, 10 and 18.8 𝐺𝑃𝑎 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑠, the peak values of 𝑅(𝜔) are found at 9.5 𝑒𝑉, 10.18 𝑒𝑉, and
9.83 𝑒𝑉, respectively. The reflectivity curve peaks shift towards higher energies with
increasing pressure ranges from 0 to 10 𝐺𝑃𝑎. But, this curve’s peak falls with a further increase
in pressure because of the instant variation in blue shift and material’s stiffness. If 𝜖1 tends to
zero, the energy spectrum belongs to the region of high reflectivity.

22. Figure 15: The calculated reflectivity spectrum 𝑅(𝜔) of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4.
As illustrated in Fig. 15, the 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 structure’s incident light radiation is found to be
18.8% at zero pressure. It is important to acknowledge the fact that calculated results are closer
to theoretical literature results and are recorded in Table 3. At 10 𝐺𝑃𝑎 and 18.8 𝐺𝑃𝑎 pressures,
the reflection rises to 21% and falls to 19.8%, respectively due to the varied stiffness of the
material. The 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 the structure can be used as a coating agent for high-frequency
UV radiation’s shielding process.
The energy loss function 𝐿(𝜔) depicts the energy forfeit by an electron moving rapidly through
the semiconductor. The DC structure’s loss function is represented in Fig. 16. After applying
0, 10, and 18.8 𝐺𝑃𝑎 pressures, the screening plasma frequency [61] is detected at
18.3 𝑒𝑉, 19.55 𝑒𝑉, and 20.18 𝑒𝑉, respectively. The zero-crossing of 𝜖1(0) and concurrent
decrement of 𝑅(𝜔) the curve has been defined by the loss function’s peak.

23. Figure 16: The calculated loss function 𝐿(𝜔) of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4.
Figure 17: The calculate bandgaps and refractive index of 𝐷𝐶 − 𝐶𝑑𝐺𝑎2𝑆4 with pressure.
Fig. 17 demonstrates the bandgap and refractive index variation with pressure for 𝐷𝐶 −
𝐶𝑑𝐺𝑎2𝑆4 structure. The bandgap falls monotonically with an increase in pressure whereas the
refractive index rises linearly with increasing pressure. The maximum value of bandgap and
refractive index are 2.09 and 2.81, which is achieved at 0 𝐺𝑃𝑎 and 18.8 𝐺𝑃𝑎 pressures,
respectively. Noticeably, the bandgap is in the visible region at zero pressure and tends towards
the ultraviolet (UV) region with increasing pressure.
4. Conclusion
In the paper presented, the 𝐶𝑑𝐺𝑎2𝑆4 semiconductor’s electronic, structural, elastic and optical
properties were studied using the full potential linearized augmented plane waves (FP-LAPW)

24. method as part of the GGA function. The 𝐶𝑑𝐺𝑎2𝑆4 semiconductor’s phase transition has been
detected from DC to DR structure at 18.8 𝐺𝑃𝑎 pressure. It is found that the stability of DC
structure is higher than the DR structure until 18.8 𝐺𝑃𝑎 pressure and then vica versa after
18.8 𝐺𝑃𝑎 pressure. At 0 𝐺𝑃𝑎 and 18.8 𝐺𝑃𝑎 pressures, the optimal lattice constants, bulk
moduli and energy band gaps have been determined for both DC and DR structures. Table 1
records the computed values, which are compared to the experimental and theoretical literature
values that are accessible. The computed values of lattice constants 𝑎 and 𝑐 are overestimated
by 2.5% of experimental values. Therefore, the density of states of 𝐶𝑑𝐺𝑎2𝑆4 semiconductor’s
DC and DR (both) phases have been described and are contributed in band structure
calculation. Table 2 shows the computed elastic stiffness constants (𝐶𝑖𝑗), 𝐵, 𝐺, 𝐸, 𝜎, 𝐴 and 𝐵 𝐺
⁄
ratio for DC and DR structures, respectively for pressures of 0, 5, 10, 15 and 18.8 𝐺𝑃𝑎 for DC
and 20, 25 and 30 𝐺𝑃𝑎 for DR. Moreover, no comparable data is available, thus our estimates
will be used to guide future research on this semiconductor. At zero pressure, the computed
values of 𝐵 𝐺
⁄ and 𝜎 are 1.79 and 0.26, respectively and demonstrates the ductile and ionic
nature for DC structure. Further, the computed value of the 𝐵 𝐺
⁄ and 𝜎 reveal the ductile nature
and ionic behavior of 𝐷𝑅 − 𝐶𝑑𝐺𝑎2𝑆4 at 20 𝐺𝑃𝑎 pressure. The optical spectra of 𝐷𝐶 −
𝐶𝑑𝐺𝑎2𝑆4 involves dielectric function 𝜖(𝜔), reflectivity 𝑅(𝜔), absorption coefficient 𝛼(𝜔),
refractive index 𝜂(𝜔), extinction coefficient 𝜅(𝜔) and energy loss function 𝐿(𝜔), which are
computed for pressures of 0, 10 and 18.8 𝐺𝑃𝑎. Table 3 lists the computed values of 𝜖1(0), 𝜂(0)
and 𝑅(0) for pressures of 0, 10 and 18.8 𝐺𝑃𝑎. The computed values of real dielectric function,
refractive index and reflectivity are compared with theoretical literature values and
experimental values. It is noted that the value of refractive index rises with increase in pressure
and decrease in energy band gap. The calculated values agree better with experimental values
than previous literature values, indicating the accuracy of the present computations while
awaiting experimental conformation.
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