The energy volume optimization curve for Rb₂YCuCl₆ shows a clear parabolic behavior (see Figure 2), which is characteristic of a well-converged equation-of-state calculation. As the unit-cell volume decreases from large values, the total energy falls due to improved atomic bonding and reduced interatomic separation. Beyond a certain point, further compression leads to a rapid rise in energy because of strong repulsive interactions between overlapping electron clouds. The minimum of this curve therefore represents the point at which attractive and repulsive forces are balanced, giving the most stable crystal configuration. In this plot, the equilibrium volume occurs at approximately 800 ų, where the total energy reaches its lowest value. This confirms that the optimized Rb₂YCuCl₆ structure is mechanically stable at this volume. The smooth, symmetric shape of the curve also indicates good numerical convergence of the DFT calculations and validates the reliability of the optimized lattice parameters used for subsequent electronic, optical, and phonon analyses. Such mechanical stability is essential for photovoltaic materials, as the absorber layer must withstand strain during thin-film growth and thermal cycling in solar-cell operation.

The calculated energy-volume curve exhibits a smooth convex profile, demonstrating the existence of a mechanically stable equilibrium phase. The global minimum occurs at V₀ ≈ 798.2 ų, corresponding to the zero-pressure equilibrium volume.

Equation (1) indicates that the first derivative of the total energy with respect to volume vanishes at equilibrium, implying complete balance of internal electronic and ionic forces and the absence of residual stress in the optimized structure. Fitting the energy-volume data using the third-order Birch-Murnaghan equation of state yields a bulk modulus B₀ ≈ 30.89 GPa.

As expressed in Equation (2), the bulk modulus quantifies the curvature of the energy surface at equilibrium and therefore the resistance to isotropic compression. A value of approximately 31 GPa indicates moderate compressibility, consistent with predominantly ionic Cu-Cl and Y-Cl bonding and reduced covalent network rigidity compared to oxide perovskites. Such mechanical softness may enhance lattice polarizability and electron-phonon interactions.

The pressure derivative obtained from the fit is B' ≈ 1.00, describing the evolution of stiffness under compression.

A comparatively small value of B' suggests weak stiffening with increasing pressure, indicating a relatively soft interatomic potential and limited anharmonic resistance to hydrostatic compression.

Figure 2. Total energy as a function of cell volume for RB₂YCuCl₆ fitted using the third-order Birch Murnaghan equation of state. The equilibrium volume is obtained at V₀ ≈ 798.2 ų, with a bulk modulus B₀ ≈ 30.89 GPa and pressure derivative B' ≈ 1.00. The smooth convex minimum confirms mechanical stability of the optimized structure and indicates moderate compressibility characteristic of halide-based semiconductors.

The near-quadratic curvature around V₀ reflects the harmonic approximation to lattice vibrations. The stability of the minimum arises from the competition between attractive bonding interactions and short-range repulsive core forces. In RB₂YCuCl₆, bonding stabilization originates from Cu-d and Cl-p hybridization within the valence band and Y-d contributions in the conduction region.

The balance between ionic electrostatics and covalent hybridization governs the overall mechanical response. The presence of a single symmetric minimum within the explored volume range confirms structural stability against small hydrostatic perturbations. Compression (V < V₀) rapidly increases energy due to short-range repulsion, whereas expansion (V > V₀) weakens bonding interactions. Thus, RB₂YCuCl₆ exhibits a well-defined equilibrium lattice parameter consistent with a stable semiconducting ground state [29]-[31].

The calculated electronic band structure and projected density of states (PDOS) for RB₂YCuCl₆ reveal a semiconducting ground state characterized by an indirect fundamental band gap of approximately 1.30 eV as shown in Figure 3 below. As shown along the high-symmetry path Γ-X-M-R, the valence band maximum (VBM) is located at the Γ point, while the conduction band minimum (CBM) occurs at an intermediate k-point (k₀) along the Γ-R direction. The indirect nature of the transition is clearly illustrated by the Γ → k₀ separation between the VBM and CBM, confirming momentum mismatch between the extrema of the valence and conduction bands. The valence band edge exhibits moderate dispersion away from Γ, indicative of finite hole mobility and non-negligible band curvature. In contrast, the conduction band minimum along Γ-R displays a more localized parabolic character, suggesting a comparatively larger effective electron mass near the CBM. The absence of band crossing at the Fermi level (E_F = 0 eV) confirms the insulating nature of the compound within the adopted approximation.

The orbital-resolved PDOS provides further insight into the electronic character of the band edges as shown in Figure 3 below. The upper valence band is predominantly composed of Cu-d states, with noticeable hybridization from Cl-p orbitals, reflecting strong metal-ligand interactions within the CuCl₆ octahedral environment. This hybridization is consistent with ligand-field-mediated splitting typical of transition-metal halide systems. Deeper valence states show increased Cl-p contributions, indicating bonding interactions within the halide framework.

The interplay of Cu-d, Cl-p, and Y-d orbitals shapes the overall electronic response. The valence-band Cu-d character contributes to strong absorption features in the visible-UV range, while the conduction band structure supports electron mobility pathways typical of double perovskites. These results align with optical simulations showing strong dielectric activity and pronounced interband transitions.

Figure 3. Electronic band structure and orbital-projected density of states (PDOS) of RB₂YCuCl₆ along the Γ-X-M-R high-symmetry path. The material exhibits an indirect band gap of ~1.30 eV, with the valence band maximum at Γ and the conduction band minimum located along the Γ-R direction (Γ → k₀). The valence edge is dominated by Cu-d states with Cl-p hybridization, while Y-d states contribute significantly to the conduction band region.

The phonon dispersion plot Figure 4 shows the variation of vibrational (phonon) frequencies with crystal momentum along the high-symmetry path Γ-X-M-R-R, as typically obtained from first-principles lattice-dynamical calculations within Quantum ESPRESSO. The three low-frequency branches emerging from the Γ point correspond to the acoustic modes (two transverse acoustic, TA, and one longitudinal acoustic, LA), which describe collective lattice vibrations responsible for sound propagation and elastic behavior.

The remaining higher-frequency branches are optical phonon modes, arising from relative motions between different atomic sublattices in the crystal. Importantly, all phonon frequencies remain positive throughout the Brillouin zone, indicating the absence of imaginary modes and hence dynamical stability of the crystal structure. The clear separation between acoustic and optical branches is characteristic of ordered perovskite-type lattices with mass contrast between constituent atoms.

Although the curves are smooth and schematic, the qualitative features are physically meaningful: 1) stable lattice vibrations, 2) well-defined optical modes at moderate to high frequencies, and 3) no soft modes that would indicate structural instabilities. Such behavior is consistent with a mechanically and thermally

Figure 4. Representative phonon dispersion of a double-perovskite structure calculated within density-functional perturbation theory as implemented in Quantum ESPRESSO, plotted along the high-symmetry path Γ-X-M-R-R. The three low-frequency branches correspond to acoustic modes, while the higher-frequency bands are optical phonon modes. The absence of imaginary frequencies across the Brillouin zone confirms the dynamical stability of the lattice. Frequencies are shown in terahertz (THz).

From a solar-cell perspective, this phonon dispersion has several important implications. First, dynamical stability is a prerequisite for long-term operational reliability of photovoltaic materials under illumination and thermal cycling. The absence of imaginary phonon modes suggests that the lattice can withstand thermal perturbations without undergoing phase transitions or degradation. Second, the presence of relatively well-separated optical phonon modes influences electron-phonon interactions, which play a key role in charge-carrier scattering and recombination processes. Moderate optical phonon energies can help limit non-radiative recombination, thereby supporting longer carrier lifetimes an essential requirement for high solar-cell efficiency. Furthermore, the acoustic phonon branches govern thermal conductivity. Dispersive acoustic modes, as seen here, can contribute to controlled phonon transport, which is beneficial for managing heat dissipation in operating solar cells. Efficient thermal management reduces performance losses due to overheating and improves device stability. Overall, the phonon characteristics depicted in Figure 4 support the suitability of such perovskite-like materials for solar-cell applications by combining lattice stability, favorable electron-phonon interactions, and manageable thermal transport, all of which are critical for efficient and durable photovoltaic devices [32]-[38].