Spin exchange, electronic correlation, and thermoelectric transport in Rb₂GeMBr₆ (M = V, Mn, Ni) halide double perovskites from first principles calculations

Abstract

Lead-free halide magnetic perovskites have emerged as environmentally benign materials with broad multifunctional potential spanning spintronic, optoelectronic, and thermoelectric domains. Here, we present a comprehensive first-principles investigation of transition-metal-based double perovskites Rb₂GeMBr₆ (M = V, Mn, Ni) to elucidate their structural, electronic, magnetic, and thermoelectric properties relevant to next-generation multifunctional devices. Structural robustness is confirmed through favorable Goldschmidt tolerance factor values (t_dp ≈ 0.95–0.96) and octahedral factor values (µ ≈ 0.36–0.40), while dynamical stability is verified by phonon spectra free of imaginary frequencies. Thermodynamic stability is established through negative formation energies, positive cohesive energies, and favorable positions on the convex hull. Finite-temperature ab initio molecular dynamics simulations at 500 K further verify their structural resilience under thermal perturbation. The electronic structures, computed using both GGA and GGA+mBJ schemes, reveal intrinsic ferromagnetism driven by crystal-field-stabilized high-spin 3d states of the transition-metal ions, yielding magnetic moments of 3µ_B (V), 5µ_B (Mn), and 2µ_B (Ni), respectively. Mean-field estimates predict Curie temperatures in the range of 520–680 K, confirming robust magnetic ordering well above room temperature. Spin-polarized electronic band structures exhibit indirect ferromagnetic semiconducting gaps of 0.8–3.3 eV, with distinct exchange-induced splitting evident at both valence and conduction band edges. Calculated carrier effective masses indicate favorable charge transport, with electrons showing lower effective masses than holes . Thermoelectric analysis reveals large Seebeck coefficients (up to 3000 µV K⁻¹ within optimal chemical potential ranges), ultralow lattice thermal conductivities (<0.7 W m⁻¹ K⁻¹), and high Grüneisen parameters (γ ≈ 2.0–2.5), reflecting strong lattice anharmonicity. These synergistic features yield figures of merit approaching unity across the series. Collectively, Rb₂GeMBr₆ compounds represent a new class of lead-free, intrinsically ferromagnetic semiconductors with low thermal conductivity and spin-dependent transport characteristics, offering promising prospects for spin-caloritronic, semiconductor spintronic, and thermoelectric energy-conversion technologies.

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1. Introduction

The accelerating advancement of information technologies and the global transition toward sustainable energy solutions underscore the need for multifunctional materials capable of optimizing both information-processing and energy-conversion efficiencies. Spintronics, which harnesses the electron's spin degree of freedom in addition to its charge, represents a transformative paradigm beyond conventional charge-based electronics, as it enables non-volatility, ultrafast switching, and significantly reduced energy dissipation.^1–3 These attributes make spin-based devices particularly promising for magnetoresistive random-access memory (MRAM), spin valves, and next-generation high-sensitivity magnetic sensors.^4,5 A key prerequisite for such technologies is the realization of magnetic semiconductors that integrate semiconducting charge transport with robust intrinsic ferromagnetism.⁶ Classical magnetic semiconductors including europium and manganese chalcogenides,^7,8 spinel oxides,^9,10 and manganese-doped II–VI and III–V semiconductors^11–14 provide foundational insight into spin-dependent transport, but their practical use is limited by their low Curie temperature, incomplete spin polarization, and structural instability. These limitations redirect attention to perovskite families that afford structural tunability, strong electronic correlation, and favorable charge transport.¹⁵ While oxide perovskites constitute a mature platform for magnetic functionality, halide perovskites, originally developed for optoelectronics, increasingly attract interest for spin-based applications.¹⁶ An early demonstration by Náfrádi et al.¹⁷ reports weak magnetism in manganese doped CH₃NH₃PbI₃ and thereby highlights the potential for spin interactions within halide frameworks. Nevertheless, many halide perovskite systems still suffer from Curie temperatures below 15 K, small spin splitting of less than 0.2 eV, and phase instability, which together impede device-level implementation. Intrinsic magnetic semiconductors that exhibit spontaneous magnetism without extrinsic doping therefore present a promising route to overcome these shortcomings. Double perovskite oxides are particularly attractive owing to their chemical flexibility and robust thermodynamic stability¹⁸; however, their halide counterparts remain comparatively less explored despite exhibiting significant potential for multifunctional applications. Recently, halide double perovskites have gained attention as lead-free alternatives with tunable electronic and magnetic properties. For instance, Cai et al.¹⁹ reported intrinsic ferromagnetism in lead-free Cs₂GeBX₆ halides, demonstrating that halide double perovskites can sustain spontaneous magnetic ordering while circumventing toxic elements. Consequently, ongoing theoretical and experimental efforts are directed toward harnessing their structural tunability to achieve materials with high Curie temperatures and strong spin polarization.

Concurrently, global efforts to harvest waste heat elevate the importance of thermoelectric materials that convert thermal gradients to electrical power.²⁰ Benchmark thermoelectrics such as Bi₂Te₃ and PbTe achieve high conversion efficiency, but face limitations associated with toxicity, scarcity, and complex processing.^21,22 Organic thermoelectrics and quantum dot composites mitigate some issues, yet remain constrained by limited stability and scalability.^23,24 In contrast, halide perovskites display ultralow lattice thermal conductivity, large Seebeck coefficient, and adequate carrier mobility, and thus align with the phonon glass electron crystal paradigm that is beneficial for thermoelectric performance.²⁵ The first observations of ultralow thermal conductivity in CH₃NH₃PbI₃ catalyse increasing interest in exploiting halide frameworks for energy conversion.²⁶

Recent studies on lead-free double perovskites broaden the multifunctional landscape by coupling optoelectronic, magnetic, and thermoelectric properties. Mustafa et al.²⁷ investigated Tl₂Os(Cl Br)₆ double perovskites and reported coupled optoelectronic and thermoelectric characteristics with relevance to renewable energy harvesting. Khan et al.²⁸ computationally assessed A₂YBiO₆ (A = Mg, Ca, Ba) double perovskites and identified favorable thermodynamic stability, strong optical absorption, and promising thermoelectric response, thereby positioning these oxide-based double perovskites as green energy alternatives. Nazir et al.²⁹ performed first principles calculations on K₂InBiX₆ (X = Cl, Br, I) halide double perovskites and emphasized their solar cell applicability, which illustrates the sensitivity of electronic structure to A site substitution. Yasir et al.³⁰ demonstrated that computational strain engineering combined with Hubbard U correction in Li₂CuWX₆ (X = Cl, Br) effectively tunes magnetic ordering and electronic structure, offering an explicit pathway to engineer spin-based functionalities in double perovskites. Importantly, Safdar et al.³¹ reported an ordered perovskite CaCu₃Mn₂Ir₂O₁₂ that exhibits a Curie temperature above room temperature, complete spin polarization, and a high thermoelectric response, thereby providing a compelling example where magnetic ordering and thermoelectric performance coexist at practically relevant temperatures.

Despite these encouraging advances, the coupled interplay among intrinsic magnetism, spin resolved electronic structure, and thermoelectric transport in halide based double perovskites is not yet fully elucidated. Filling this gap requires a systematic and comparative study that connects chemical composition and electronic correlation to magnetic ordering and transport coefficients. In this work, we present a comprehensive first principles investigation of Rb₂GeMBr₆ (M = V, Mn, Ni) double perovskites. Using density functional theory and post-DFT approaches, we examine structural stability, magnetic ground states, spin resolved electronic structure, and thermoelectric transport properties. Our calculations reveal intrinsic ferromagnetic semiconducting states with pronounced spin splitting, robust Curie temperature estimates, and favorable thermoelectric efficiency, thereby positioning Rb₂GeMBr₆ as a lead-free class of multifunctional materials for future spin based and energy harvesting devices.

2. Computational details

First-principles calculations are performed using the WIEN2k computational package in conjunction with its integrated auxiliary modules.³² The electronic structure is obtained by solving the Kohn–Sham equations within the framework of the generalized gradient approximation (GGA) as formulated by Perdew, Burke, and Ernzerhof (PBE).³³ To overcome the limitations of conventional GGA in describing localized d-states and in accurately predicting band gaps, both GGA+U and GGA+mBJ are considered.^34,35 However, the results are presented using the Tran–Blaha modified Becke–Johnson (mBJ) exchange potential, as it provides an optimal balance between computational efficiency and accuracy. The mBJ potential significantly reduces the systematic underestimation of band gaps inherent in GGA and yields results that are in close agreement with experimental measurements and hybrid functional (HSE06) calculations.^36,37 Extensive benchmark studies reveal that the modified Becke–Johnson (mBJ) potential accurately reproduces the band gaps of semiconductors and halide perovskites, with deviations typically ranging from 0.1 to 0.3 eV relative to the HSE06 hybrid functional, yet with substantially lower computational expense than hybrid or many body methods.³⁷ For the present halide double perovskite systems, the GGA+mBJ method provides quantitatively reliable band structures and band gaps that closely reflect experimental trends, offering a good foundation for subsequent analysis of their optical, electronic, and thermoelectric behavior. The calculations are carried out using the full-potential linearized augmented plane wave (FP-LAPW) method, where the unit cell is partitioned into non-overlapping muffin-tin (MT) spheres, centered at the atomic sites occupying the Wyckoff positions (4a, 4b, 8c, and 24e), and an interstitial region that fills the remaining space. Inside each MT sphere, wavefunctions are expanded in spherical harmonics up to an angular momentum quantum number l_max = 10, while the interstitial region is treated using plane waves. The basis set size is optimized using the criterion R_MT × K_max = 7, which ensures an appropriate balance between computational accuracy and efficiency. Self-consistency is considered achieved when the total energy difference between successive iterations is below 0.0001 Ry and the charge density variation is less than 0.0001 electrons. A dense k-point grid containing 1000 points in the irreducible Brillouin zone is used for the accurate evaluation of electronic, magnetic, and elastic properties. The elastic properties are evaluated using the Thomas Charpin cubic elastic code within the GGA–PBE framework.³⁸ Thermodynamic parameters are derived using the Gibbs2 code,³⁹ which is interfaced with WIEN2k to compute temperature dependent quantities such as vibrational free energy, vibrational entropy, vibrational internal energy, and specific heat capacity. Transport coefficients, including the Seebeck coefficient (S), electrical conductivity (σ), and the electronic contribution to thermal conductivity, are determined using semiclassical Boltzmann transport theory, as implemented in the BoltzTraP code.⁴⁰ Carrier dynamics are modelled within the rigid band approximation and the constant relaxation time approximation (CRTA). These thermoelectric coefficients are systematically evaluated as functions of the chemical potential (µ) for the halide perovskite systems under investigation, ensuring accurate prediction of their electronic transport behavior.Here, Ξ(ε)is the transport distribution function specified by where v^α_k represents the αth component of the group velocity with wave vector k. A denser k-point mesh comprising 100 000 points is employed to improve the precision of thermoelectric property evaluations and ensure accurate transport predictions. In addition, the lattice thermal conductivity is computed by solving the phonon Boltzmann transport equation within the relaxation time approximation as implemented in the ShengBTE code.⁴¹ This approach employs second-order force constants obtained from density functional perturbation theory and third-order force constants generated through finite displacement supercell calculations. These inputs enable ShengBTE to determine phonon lifetimes and scattering rates, providing reliable estimates of the lattice contribution to thermal conductivity.

3. Results and discussion

The physical properties of the novel halide perovskites Rb₂GeMBr₆ (M = V, Mn, Ni) are systematically examined within the framework of density functional theory using the GGA–PBE and GGA+mBJ exchange correlation potentials.

3.1. Structural and magnetic phase stability

The structural feasibility of Rb₂GeMBr₆ (M = V, Mn, Ni) halide perovskites is evaluated using two key geometric descriptors, the Goldschmidt tolerance factor (t_dp) and the octahedral factor (µ), which predict the formation and stability of cubic perovskite structures. The tolerance factor is defined as⁴², where r_Rb and r_Br represent the ionic radii of rubidium and bromine atoms, respectively, and r_avg denotes the average radius of the B site cations, that is, germanium and the transition metal M (V, Mn, Ni). The octahedral factor is expressed as⁴². The calculated tolerance factor values range from 0.81 to 1.11, consistent with the formation of a cubic perovskite phase (Table 1). Octahedral factor values between 0.29 and 0.55 further support structural stability through favorable BX₆ coordination (Table 1). The optimized structures adopt a three-dimensional cubic unit cell with Fm3̄m symmetry (Fig. 1), where rubidium occupies the twelvefold-coordinated A sites, while Ge and M occupy the B and B′ sites. Each B and B′ cation is octahedrally coordinated by six bromine anions, forming corner-sharing [GeBr₆] and [MBr₆] octahedra. The tolerance and octahedral factor analysis strongly supports the persistence of the cubic phase. This inference aligns with experimental reports on structurally analogous halide double perovskites, including Cs₂AgBiBr₆, Cs₂AgInBr₆, and Rb₂AgBiX₆ (X = Cl, Br), which retain the cubic framework across wide temperature ranges (up to approximately 500–600 K) and under moderate pressure.^43–45 X-ray diffraction, Raman spectroscopy, and thermogravimetric analysis collectively confirm their thermodynamic, dynamic, and mechanical stability. Considering the close structural and chemical correspondence between these systems and the Rb₂GeMBr₆ series, the cubic phase of the present compounds is expected to remain stable under realistic thermal and mechanical conditions. Next, the stability of competing magnetic phases is evaluated through total energy calculations to identify the ground-state configuration. Equilibrium structural parameters are subsequently determined by fitting the Birch–Murnaghan equation of state:⁴⁶Here, E(V) represents the energy at volume V, E₀ is the minimum energy, B₀ is the bulk modulus, and is its pressure derivative. The energy–volume curves for the different spin configurations (Fig. 2(a)) show that the ferromagnetic state is energetically most favorable. The corresponding total energies for the FM, antiferromagnetic (AFM), and nonmagnetic (NM) phases are listed in Table 1, showing that the FM configuration consistently exhibits the lowest energy for all three compounds, thereby confirming the ferromagnetic ground state. Moreover, the positive energy differences E_AFM/E_NM − E_FM further substantiate the stability of the FM phase. The equilibrium lattice constants obtained from the energy minima are 10.9 Å for Rb₂GeVBr₆, 10.99 Å for Rb₂GeMnBr₆, and 10.76 Å for Rb₂GeNiBr₆. Their close agreement with prior theoretical predictions¹⁹ confirms the reliability of the present computational results.

Table 1 Calculated values of the optimized lattice constant (a₀ in Å), bulk modulus (B in GPa), pressure derivative of the bulk modulus (, unitless), enthalpy of formation (ΔH in eV/atom), cohesive energy (E_C in eV), tolerance factor (t_dp, unitless), octahedral factor (μ, unitless), and total energies for ferromagnetic (E_FM), antiferromagnetic (E_AFM), and nonmagnetic (E_NM) configurations (in eV) for Rb₂GeMBr₆ (M = V, Mn, Ni). Values from previous works are included for comparison

Parameter	Rb2GeVBr6	Rb2GeMnBr6	Rb2GeNiBr6
a 0	10.95	10.99	10.76
a 0 (previous studies)	11.08^19	11.10^19	10.90^19
B	25.64	24.77	24.49
	4.10	4.81	4.04
ΔH	−1.37	−1.35	−1.22
E C	29.12	24.52	23.19
t dp	0.96	0.95	0.96
µ	0.39	0.40	0.36
E FM	−67 0524.90	−67 6219.06	−686 068.57
E AFM	−670 524.75	−676 218.88	−686 068.70
E NM	−670 523.09	−676 215.80	−686 068.32

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Fig. 1 Crystal structure of Rb₂GeMBr₆ (M = V, Mn, Ni) double halide perovskites.

Fig. 2 Geometric properties for Rb₂GeMBr₆ (M = V, Mn, Ni): (a) variation of total energy with unit cell volume obtained using the GGA–PBE functional and (b) simulated XRD patterns derived from density functional theory calculations.

Next, X-ray diffraction simulations are performed for the ideal cubic phase of these compounds. The predicted diffraction patterns reproduce the expected peak positions and intensities, thereby offering reference data for future experimental validation of phase purity and structural confirmation (Fig. 2(b)).

In addition to structural stability, the thermodynamic stability of these compounds is further assessed by calculating the enthalpy of formation (ΔH) using the relation:⁴⁷

where is the total DFT energy of the compound, and E_Rb, E_Ge, E_M, and E_Br correspond to the energies of the stable elemental phases. All computed values of ΔH are negative (Table 1), signifying thermodynamic stability.

Cohesive energy (E_Coh) is also evaluated to quantify the binding strength of the constituent atoms:⁴⁷

where E_Rb, E_Ge, E_m and E_Br correspond to the total energies of isolated Rb, Ge, M (V/Mn/Ni), and Br atoms, respectively, and represents the optimized ground-state energy of the fully relaxed compound. The positive E_Coh values obtained for all three compositions (Table 1) confirm strong interatomic bonding and intrinsic structural stability.

To further assess synthetic feasibility, phase stability is evaluated using energy-above-hull data from the Quantum Open Database.⁴⁸ All three compounds are found to lie on or very close to the convex hull, with small offsets of 0.020 eV (V), 0.012 eV (Mn), and 0.023 eV (Ni), indicating a high likelihood of experimental realization. The corresponding convex hull diagrams are provided in the SI (Section I). While the hull energies indicate thermodynamic accessibility at zero Kelvin, it is also essential to examine the behaviour of these compounds under finite-temperature conditions. The thermal stability of Rb₂GeMBr₆ halide perovskites is assessed using ab initio molecular dynamics within the Born–Oppenheimer molecular dynamics framework in the Quantum ESPRESSO code.^49–51 The simulations are performed in the canonical (NVT) ensemble using the Nose–Hoover thermostat to maintain a temperature of 500 K. A 5 × 5 × 2 supercell is employed to minimise periodic boundary effects.^52,53 Each simulation runs for 8000 steps with a time step of 1 fs. The total energy versus time profiles as presented in Fig. 3 show that the energies of Rb₂GeVBr₆, Rb₂GeMnBr₆ and Rb₂GeNiBr₆ fluctuate around stable average values without any discernible drift, indicating that thermal equilibrium is achieved. No abrupt fluctuations are observed, and the atomic configurations remain intact throughout the trajectory. The absence of bond breaking or structural distortion confirms that all three compounds maintain their structural integrity at 500 K, demonstrating their thermal robustness under finite temperature conditions.

Fig. 3 AIMD simulations at 500 K for 8000 steps confirm the thermodynamic stability of Rb₂GeMBr₆, as evidenced by stable total energy fluctuations and preserved structural integrity.

3.2. Vibrational thermodynamic properties

Vibrational thermodynamics, characterized through descriptors such as internal energy (E), Helmholtz free energy (A), vibrational entropy (S_v), and constant-volume heat capacity (C_v), is crucial for elucidating temperature-dependent variations in material properties. The quasi-harmonic Debye model,^54,55 which accounts for the dependence of phonon frequencies on both volume and temperature, offers a rigorous theoretical framework for evaluating these quantities. Within this formalism, thermal functions are derived from well-established thermodynamic relations, providing a reliable foundation for assessing structural stability and functional performance over a broad temperature range. The vibrational Helmholtz free energy is expressed aswhere n is the number of atoms per formula unit, k_B is the Boltzmann constant, θ_D is the Debye temperature, and D denotes the Debye integral.

The Debye temperature is given by

where h is Planck's constant, V is the unit cell volume, N is the number of atoms, f(v) is a function of the Poisson ratio, B_s is the adiabatic bulk modulus, and M is the molecular mass of the unit cell.

The vibrational entropy is obtained as follows:

Vibrational entropy (S_v):

and the constant volume specific heat capacity is expressed as

These relations enable systematic evaluation of thermodynamic stability and heat transport characteristics over a broad temperature window.

The vibrational thermodynamic analysis of Rb₂GeMBr₆ compounds elucidates the underlying lattice dynamics governing their thermal behavior. Tracking the evolution of vibrational internal energy (E), Helmholtz free energy (A), vibrational entropy (S_v), and specific heat capacity (C_v) as a function of temperature enables a rigorous understanding of their high temperature behavior. As shown in Fig. 4(a), the vibrational internal energy (E) increases nearly linearly with temperature, reflecting the progressive accumulation of thermal energy within the lattice. This trend originates from the enhancement of atomic vibrations, which collectively elevate both the kinetic and potential components of the lattice energy. The linear dependence underscores the enthalpic predictability of these compounds, confirming their capacity to store energy in a systematic manner under thermal excitation. In contrast, the Helmholtz free energy (A), shown in Fig. 4(b), decreases monotonically with increasing temperature, consistent with the reduction in the energy available to perform useful work as thermal disorder intensifies. This thermodynamic behavior is consistent across all Rb₂GeMBr₆ compositions, indicating that substitution of the transition metal at the M site introduces only a minor perturbation to the overall vibrational landscape. Since the bromide anion framework remains invariant throughout the series, the dominant phonon modes governing lattice dynamics are maintained, thereby yielding a relatively stable free energy profile irrespective of cationic substitution. Fig. 4(c) illustrates the temperature driven evolution of vibrational entropy (S_v), which increases steadily with rising temperature because of enhanced phonon population and configurational disorder. The close similarity in entropy trends across all Rb₂GeMBr₆ compositions indicates that the bromide sublattice imposes a dominant vibrational environment. Although transition metal substitution may introduce subtle variations in local bond stiffness and vibrational frequencies, these effects exert only a minor influence on the phonon density of states, resulting in a nearly uniform entropy response throughout the series.

Fig. 4 Thermodynamic vibrational functions for Rb₂GeMBr₆ (M = V, Mn, Ni) halide perovskites against a selected range of temperature: (a) vibrational internal energy (E), (b) vibrational Helmholtz free energy (A), (c) vibrational entropy (S_v) and (d) specific heat at a constant volume (C_v).

The temperature dependence of the specific heat at a constant volume (C_v), shown in Fig. 4(d), is in excellent agreement with theoretical expectations from the Debye formalism. In the low temperature regime (T ≪ θ_D), C_v follows the characteristic T³ dependence, reflecting the thermal activation of only long wavelength acoustic phonon modes. As temperature increases beyond the Debye threshold (T ≫ θ_D), C_v saturates at the classical Dulong–Petit limit, C_v ≈ 3nR, where R is the universal gas constant (8.314 J mol⁻¹ K⁻¹).⁵⁶ This asymptotic behavior arises from the complete excitation of phonon modes, which ensures equipartition of vibrational energy across all available degrees of freedom.

Further quantitative thermodynamic parameters, including the Debye temperature (θ_D), zero-point vibrational energy (E₀), and the Grüneisen parameter (γ), are computed and presented in Table 2. The calculated Debye temperatures for Rb₂GeMBr₆ exhibit only marginal variation across the series, reflecting the preservation of the Fm3̄m symmetry and the nearly identical bonding environment enforced by the bromide sublattice. Although substitution of the transition-metal cation alters the atomic mass, the effect on the vibrational density of states remains minimal, resulting in comparable θ_D values. To evaluate lattice anharmonicity, we determine the Grüneisen parameter using , where ε denotes Poisson's ratio.⁵⁷ The similar γ values obtained for all compositions suggest that their anharmonic lattice dynamics and thermal expansion responses are essentially conserved. These observations highlight the dominant role of the halide framework in governing the vibrational and thermodynamic behavior of bromide-based double perovskites.

Table 2 Computed values for the Debye temperature (θ_D) measured in Kelvin, zero point energy (E₀) expressed in kilojoules per mole, and the Gruneisen parameter (γ) (dimensionless)

Parameter	E 0	θ D	γ
Rb2GeVBr6	27	283	2.49
Rb2GeMnBr6	27	281	2.05
Rb2GeNiBr6	26	273	1.97

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3.3. Phonon stability

The vibrational stability and phonon-mediated thermal transport properties of crystalline solids are intrinsically governed by their phonon dispersion characteristics. In this study, the phonon spectra of Rb₂GeMBr₆ perovskite compounds are computed using density functional perturbation theory (DFPT) as implemented in the Quantum ESPRESSO package.⁵⁰ The dynamical stability of a material is confirmed when all phonon frequencies across the Brillouin zone are real and positive. Conversely, the emergence of imaginary (negative) frequencies indicates structural instability. As shown in Fig. 5, the calculated phonon dispersion relations for all compounds exhibit no imaginary modes, confirming their dynamical stability. Given that the primitive unit cell comprises ten atoms, the phonon spectrum consists of three acoustic and twenty-seven optical branches. At the Γ-point, the frequencies of the longitudinal acoustic (LA) and transverse acoustic (TA) modes naturally approach zero, consistent with the behavior expected for mechanically stable crystal lattices. The optical phonon modes, which encompass Raman-active, infrared-active, and silent modes, exhibit finite non-zero frequencies. While they contribute to various spectroscopic and vibrational phenomena, their role in heat conduction is limited due to their relatively low group velocities. The phonon group velocity v_g = dω/dk, which is the derivative of the phonon frequency (ω) with respect to wavevector (k), reflects how rapidly vibrational energy propagates through the crystal. A higher slope in the dispersion curve indicates greater group velocity, thus enhancing phonon-driven thermal conductivity. Among the phonon branches, the acoustic modes, especially the LA branch, demonstrate nearly linear dispersion relations with steep slopes. This makes them the primary carriers of thermal energy, as they facilitate efficient phonon transport. In contrast, the optical modes are characterized by flatter dispersion, resulting in lower group velocities and diminished influence on thermal conduction. Overall, thermal transport in these halide double perovskites is predominantly governed by acoustic phonon behavior, highlighting their potential for applications in thermal management and thermoelectric energy conversion.

Fig. 5 Phonon band structures of Rb₂GeMBr₆ (M = V, Mn, Ni) halide perovskites.

3.4. Mechanical stability

The mechanical robustness of crystalline materials originates from their atomic arrangement, which dictates their response to external mechanical perturbations. For Rb₂GeMBr₆ halide perovskites, the elastic constants are evaluated using the energy–strain formalism, accounting for both tetrahedral and rhombohedral distortions within the cubic symmetry framework.³⁸ As listed in Table 3, the computed elastic constants fulfill the Born–Huang mechanical stability criteria⁵⁸ (C₁₁ > 0, C₁₂ > 0, C₄₄ > 0, and C₁₁ + 2C₁₂ > 0), thereby confirming the intrinsic mechanical stability of these compounds. Based on these constants, the fundamental mechanical moduli such as bulk, shear and Young moduli are determined using Voigt–Reuss–Hill approximations.^59–61 The bulk modulus (B) characterizes the resistance to isotropic compression and is inversely proportional to the volume change. The shear modulus (G) reflects resistance to shape deformation, whereas Young's modulus (Y) represents tensile stiffness. The consistent trend Y > B > G observed across all compositions indicates a greater resistance to shear distortion compared with volumetric compression, highlighting their structural rigidity. Furthermore, the Kleinman parameter (ζ), obtained from the elastic tensor, provides insight into the internal bonding characteristics. Lower ζ values indicate that bond stretching predominates over bond-angle bending, signifying strong lattice cohesion under applied strain.⁶²

Table 3 Calculated second-order elastic constants (C₁₁, C₁₂, and C₄₄ in GPa); bulk modulus (B, GPa); shear modulus (G, GPa); Young's modulus (Y, GPa); Poisson's ratio (ν); B/G ratio; Cauchy's pressure (C₁₂ − C₄₄); Kleinman's parameter (ξ); Zener anisotropy constant (A_Z); and universal anisotropy index (A_U) for Rb₂GeMBr₆ double halide perovskites

Parameter	Rb2GeVBr6	Rb2GeMnBr6	Rb2GeNiBr6
C 11	52.90	47.95	48.85
C 12	10.41	12.10	13.15
C 44	11.97	13.08	14.43
B	24.57	24.05	25.05
G	15.09	14.84	15.72
Y	37.50	36.90	38.91
ν	0.245	0.244	0.240
B/G	1.63	1.62	1.59
C 12–C44	–1.56	–0.98	–1.28
ξ	0.41	0.42	0.43
A Z	0.57	0.50	0.51
A U	0.34	0.27	0.26

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The classification of mechanical behaviour into ductile or brittle regimes is determined using several empirical indicators, including Pugh's ratio, Cauchy's pressure, and Poisson's ratio. A positive Cauchy's pressure, a Pugh's ratio exceeding 1.75, and a Poisson's ratio greater than 0.26 are indicative of ductile behavior, whereas negative Cauchy's pressure, lower B/G values (<1.75), and ν < 0.26 signify brittle characteristics.^62–64 As summarized in Table 3, all three Rb₂GeMBr₆ compounds exhibit negative or relatively low values of these indices, indicating that they are predominantly brittle, with limited ductility, which has implications for their mechanical reliability and potential applications in flexible or damage-resistant devices.

Mechanical anisotropy, which describes the directional dependence of elastic behaviour, is assessed using the Zener anisotropy ratio and the universal anisotropic index .^65,66 The computed values of A_Z and A_U, as presented in Table 3, deviate from unity, thereby confirming the presence of anisotropic elastic responses and reinforcing the direction-dependent mechanical characteristics of these compounds.

To further elucidate anisotropy, three-dimensional (3D) elastic property surfaces are generated using the Elate software for Young's modulus (Y), shear modulus (G), Poisson's ratio (ν), and linear compressibility (β).⁶⁷ The contour plots presented in Fig. 6 exhibit clear deviations from spherical symmetry, most notably for Y and G, thereby confirming anisotropic elasticity across the crystallographic planes. The directional maxima and minima of the elastic parameters, listed in Table S1 of the SI, reveal pronounced anisotropy, particularly along the [100] and [110] crystallographic orientations. In contrast, the bulk modulus surfaces exhibit nearly spherical contours, indicating an essentially isotropic compressibility. This isotropy is further corroborated by the close agreement among the Voigt (B_V), Reuss (B_R), and Hill (B_H) bulk moduli, as summarized in Table S1 of the SI (Section II). The anisotropy observed in the Young's modulus (Y) and shear modulus (G) arises from variations in atomic bonding strength and elastic wave propagation along different crystallographic directions, reflecting the intrinsic symmetry of the lattice. This directional dependence governs the material's response to external stress and plays a critical role in determining the mechanisms of crack initiation and propagation. The isotropic nature of the bulk modulus, however, indicates uniform resistance to volumetric strain, ensuring mechanical stability under hydrostatic compression. Together, these findings highlight a delicate balance between anisotropic shear responses and isotropic compressibility, which is crucial for predicting the mechanical resilience and potential applicability of these compounds in advanced device environments.

Fig. 6 Three-dimensional representations of (a) Young's modulus, (b) linear compressibility, (c) shear modulus, and (d) Poisson's ratio for Rb₂GeVBr₆ (i), Rb₂GeMnBr₆ (ii), and Rb₂GeNiBr₆ (iii), illustrating the anisotropic elastic behavior of the compounds.

Furthermore, insights into the average sound velocity (V_av), derived from the transverse and longitudinal components, provide valuable information on dynamic elastic behaviour and phonon propagation.⁶⁸ The average sound velocity calculated using the relation⁶⁸ shows a gradual decline from the V to Ni based perovskites (Table 4), consistent with the increasing trend in density. This inverse relationship between sound velocity and density is particularly relevant for thermoelectric materials, as phonon propagation directly influences lattice thermal conductivity.

Table 4 Computed values of transverse (V_t in m s⁻¹), longitudinal (V_l in m s⁻¹) and average sound velocity (V_av in m s⁻¹), Frantsevich's ratio (G/B), machinability factor (µ_m) (unitless), Vickers hardness (H_V) (unitless), and melting temperature (T_m in K) for Rb₂GeMBr₆ perovskites

Parameter	Rb2GeVBr6	Rb2GeMnBr6	Rb2GeNiBr6
V t	1962	1956	1942
V l	3376	3361	3322
V av	2177	2170	2153
G/B	0.614	0.617	0.628
µ m	2.05	1.84	1.74
(Hv)Teter	2.279	2.241	2.374
(Hv)Tian	3.611	3.587	3.809
T m ± 300	866 ± 300	836 ± 300	842 ± 300

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Additional mechanical descriptors further strengthen the assessment of practical applicability. The Vickers hardness (H_V), estimated using both Teter's model ((H_V)_Teter = 0.151 G)⁶⁹ and Tian's model (H_V)_Tian = 0.92(G/B)^1.137G^0.708,⁷⁰ indicates considerable resistance to plastic deformation. The machinability index ⁶² suggests favourable workability, while the melting temperature (T_m (K) = [553 (K) + (5.911)C₁₁]GPa ± 300)⁷¹ confirms excellent thermal stability. A moderate G/B ratio reflects a balanced combination of stiffness and ductility, which is desirable for maintaining structural integrity under mechanical stress. All the calculated values of these parameters are summarized in Table 4. Collectively, the high hardness, brittle response, thermal robustness, and anisotropic elastic behaviour underscore the potential of Rb₂GeMBr₆ halide perovskites for advanced applications, particularly in thermoelectric energy conversion and high temperature structural devices.

3.5. Electronic structure

The electronic structure of Rb₂GeMBr₆ halide double perovskites is investigated through spin-polarized band structure and density of states calculations within the generalized gradient approximation (GGA) and the Tran–Blaha modified Becke–Johnson (mBJ) exchange potential. The mBJ functional substantially improves the description of the electronic band gap by correcting the well-known underestimation of GGA, thereby providing a more accurate representation of low-energy electronic excitations.³⁶ All three compounds exhibit semiconducting behavior, characterized by finite band gaps in both spin channels under both approximations, as shown in Fig. 7. In each case, the spin-up states approach the valence band maximum, while the spin-down states lie above the Fermi level, confirming a ferromagnetic semiconducting ground state. The mBJ corrected band structures reveal indirect band gaps accompanied by pronounced spin dependent asymmetry, reflecting strong exchange splitting between the spin up and spin down subbands. This exchange originates from Hund's coupling between the localized 3d orbitals of the transition metal cations and the itinerant Ge–Br framework. Distinct spin asymmetry appears near the Fermi level, where the conduction band minimum (CBM) and valence band maximum (VBM) exhibit substantial exchange induced splitting, denoted as ΔE_CBM and ΔE_VBM, respectively. The computed band gaps and the corresponding spin splitting values obtained from both GGA and mBJ approximations are presented in Table 5. As expected, GGA tends to underestimate the band gaps, whereas the mBJ potential provides a more accurate description of the electronic structure and yield values in closer agreement with experimental and advanced theoretical results. For the present family of Rb₂GeMBr₆ halide double perovskites, the GGA+mBJ computed band gaps fall within the range of approximately 0.8 to 3.3 eV. This range corresponds closely to experimentally measured band gaps of related lead-free halide double perovskites. For instance, Cs₂AgBiBr₆ exhibits a band gap of about 2.12 eV in thin film and single crystal forms,⁷² while Filip et al. demonstrated strong consistency between many body perturbation theory and experimental results for Cs₂BiAgCl₆ and Cs₂BiAgBr₆, confirming that advanced theoretical frameworks reliably reproduce the observed optical gaps for this class of materials.⁷³ In addition to this experimental consistency, the present Rb₂GeMBr₆ compounds exhibit band gaps that closely match those of analogous Cs₂GeMX₆ systems,¹⁹ further validating that the mBJ approach accurately captures the intrinsic electronic structure of lead-free halide double perovskites. Quantitative analysis indicates that ΔE_CBM reaches 0.74 eV for Rb₂GeNiBr₆, while ΔE_VBM attains 0.78 eV for Rb₂GeMnBr₆, both of which exceed the values reported for Cs-based analogues.¹⁹ The smaller ionic radius of Rb⁺ induces lattice contraction that enhances orbital hybridization and crystal field interactions. As a result, the Rb-based compounds display wider spin resolved band gaps and stronger exchange splitting compared to their C_s based counterparts, where weaker exchange coupling and narrower gaps are observed even within the GGA+U framework.

Fig. 7 Electronic band structures of Rb₂GeMBr₆ halide double perovskites calculated using GGA and GGA+mBJ exchange–correlation approximations. The upward arrow represents the spin-up channel, while the downward arrow signifies the spin-down channel.

Table 5 Calculated spin-resolved electronic properties of Rb₂GeMBr₆ (M = V, Mn, Ni) using GGA and mBJ approaches. The parameters include spin-up and spin-down band gaps, spin splitting gaps (ΔE_CBM and ΔE_VBM), and carrier effective masses . Literature-reported values for Cs₂GeMBr₆ (GGA+U) are included for comparative analysis

Compound/method	Band gap (↑)	Band gap (↓)	ΔECBM	ΔEVBM
Rb2GeVBr6 (GGA)	0.98	2.85	−0.83	1.05	—	—
Rb2GeVBr6 (GGA+mBJ)	2.20	3.30	−0.32	0.86	0.28/0.55	0.91/0.42
Rb2GeMnBr6 (GGA)	1.04	2.35	−0.06	1.20	—	—
Rb2GeMnBr6 (GGA+mBJ)	1.92	2.95	−0.03	0.75	0.41/0.88	0.75/0.52
Rb2GeNiBr6 (GGA)	1.25	0.95	0.89	0.51	—	—
Rb2GeNiBr6 (GGA+mBJ)	1.85	1.54	1.00	0.68	0.51/0.95	0.62/0.42
Cs2GeVBr6 (GGA+U)^19	2.04	—	−0.30	0.71	—	—
Cs2GeMnBr6 (GGA+U)^19	1.59	—	−0.39	0.78	—	—
Cs2GeNiBr6 (GGA+U)^19	1.37	—	0.32	0.48	—	—

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The curvature of the band dispersion along high symmetry directions provides direct insight into carrier effective masses, defined as⁷⁴. Steep and dispersive electronic bands correspond to small carrier effective masses, which enhance group velocities and consequently increase carrier mobility, as mobility scales inversely with mass within the semiclassical framework. Analysis of the GGA+mBJ band structures (Fig. 7) reveals that the conduction bands in the spin-up channel are highly dispersive, resulting in relatively small electron effective masses ranging from 0.25 to 0.29m₀ (Table 5). In contrast, the flatter valence bands in the spin-down channel yield heavier hole masses, reaching up to 0.91m₀. This pronounced asymmetry underscores the spin-dependent nature of charge transport, wherein electrons are expected to propagate more efficiently than holes owing to their lighter effective masses. The anisotropy in spin dependent band dispersion governs the thermoelectric response. Flat valence bands generate heavy hole states near the valence band maximum, which enhance the Seebeck coefficient through a steeper density of states gradient. Conversely, light and dispersive conduction bands promote efficient charge transport due to reduced effective mass, thereby improving electrical conductivity. The combination of entropy-driven voltage generation and high electrical conductivity enhances the thermoelectric power factor, thereby improving the overall figure of merit. In the context of spintronics, the pronounced separation between spin-up and spin-down states suppresses spin–flip scattering, supporting extended spin coherence and efficient spin transport. Exchange induced splitting, together with asymmetric effective masses, enables spin selective transport pathways that are critical for functional spintronic operation. The coexistence of dispersive, mobile spin polarized electrons and localized exchange stabilized states establishes Rb₂GeMBr₆ halide double perovskites as promising candidates for advanced thermoelectric and spintronic architectures.

3.5.1. Density of states. The electronic structure of Rb₂GeMBr₆ perovskites is further elucidated through a detailed analysis of their density of states (DOS) and partial density of states (PDOS), computed using the GGA+mBJ approach (Fig. 8). These results confirm the ferromagnetic semiconducting nature of all three compounds, with excellent agreement between the DOS profiles and the corresponding spin-resolved band structures. The composition of the valence and conduction bands directly reflects the electronic environment of the transition metal centre, highlighting its pivotal role in governing the spin-dependent electronic properties. In Rb₂GeMBr₆, the transition metal ion (V²⁺, Mn²⁺, or Ni²⁺) is octahedrally coordinated by six bromide ligands. This octahedral crystal field lifts the degeneracy of the five d-orbitals via electrostatic interactions, splitting them into two sets: the lower-energy, triply degenerate t_2g orbitals (d_xy, d_yz, d_zx) and the higher-energy, doubly degenerate e_g orbitals (d_x²−y², d_z²). The energy difference between these two sets of orbitals, known as the crystal field splitting Δ_o, is small because bromine is a weak-field ligand. Consequently, electrons adhere to Hund's rule, favoring maximal spin multiplicity and yielding high-spin electronic configurations. The detailed d-orbital occupancies, unpaired electron counts, spin states, and corresponding crystal field stabilization energies (CFSEs) for the transition metal ions in Rb₂GeMBr₆, summarized in Table 6, are crucial for interpreting the spin-polarized DOS features and the underlying mechanisms governing the semiconducting behavior. In Rb₂GeVBr₆, the vanadium ion adopts a d³ high-spin configuration. All three t_2g orbitals are singly occupied in the spin-up channel (↑), with e_g and all spin-down states remaining unfilled (Fig. 9). This configuration results in localized, spin-polarized states in the valence band originating from V-t_2g↑ orbitals, while the conduction band is governed by empty V-e_g↑ and V-t_2g↓ states. The CFSE here is quantified as –1/2Δ, reflecting the energy stabilization from fully occupying the t_2g subset (Table 6). For Rb₂GeMnBr₆, Mn²⁺ presents a d⁵ high-spin arrangement. The t_2g and e_g orbitals are filled with five electrons, all in the spin-up channel, leaving the spin-down manifold entirely vacant. As a result, the valence band comprises Mn-t_2g↑ and Mn-e_g↑ contributions, while the conduction band stems from unoccupied Mn-t_2g↓ and Mn-e_g↓ states (Fig. 9). This distribution again reinforces a semiconducting gap due to spin asymmetry and exchange splitting. With no net CFSE (0), the energy balance arises purely from exchange energy favouring the high-spin arrangement. In Rb₂GeNiBr₆, the Ni²⁺ ion with its d⁸ electron count displays a high-spin configuration as well. The t_2g orbitals are fully occupied in both spin channels (↑ ↓), and the e_g orbitals contain one electron each in the spin-up channel (Fig. 9). The spin-down e_g orbitals remain vacant, forming the lower edge of the conduction band. This partial occupancy pattern, along with a CFSE of –1/2Δ, contributes to a stable yet spin-polarized electronic ground state consistent with ferromagnetic semiconducting behavior. Across all compounds, the DOS and PDOS profiles underscore the dominant role of transition metal d-orbital physics in dictating the semiconducting gap. The precise nature of these gaps is inherently spin-dependent, arising from the symmetry and occupancy of crystal-field-split d-states within an octahedral ligand environment. The alignment of electronic states across spin channels both confirms ferromagnetic ordering and influences the fundamental bandgap, reinforcing their relevance in spintronic and optoelectronic applications. These materials exhibit controlled carrier dynamics and spin-resolved excitations, making them strong candidates for advanced photonic devices such as light-emitting diodes and semiconductor lasers, where spin-dependent recombination plays a pivotal role.

Fig. 8 Graphical representation of total density of states (TDOS) and projected density of states (PDOS) for Rb₂GeMBr₆ (M = V, Mn, Ni) double perovskites determined via the GGA+mBJ approach.

Table 6 Crystal field analysis and spin states of 3d transition metal ions in octahedral fields

Ion	d-electron count	t2g occupancy	eg occupancy	Unpaired electrons	Spin state	Total spin (S)	CFSE
V^2+	d^3	↑ ↑ ↑	– –	3	3/2	3/2	−1.2
Mn^2+	d^5	↑ ↑ ↑	↑ ↑	5	5/2	5/2	0.0
Ni^2+	d^8	↑↓ ↑↓ ↑↓	↑ ↑ –	2	1	1	−1.2

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Fig. 9 Crystal field splitting and spin configuration of 3d orbitals (t_2g and e_g) for V²⁺, Mn²⁺, and Ni²⁺ in Rb₂GeMBr₆ (M = V, Mn, Ni) double perovskites, illustrating the energy separation (Δ_o), electronic occupancy, and magnetic spin alignment.

3.5.2. Magnetism and Curie temperature. The magnetic behavior of the Rb₂GeMBr₆ perovskites is thoroughly examined using various exchange–correlation approximations, with special emphasis on the Tran–Blaha modified Becke–Johnson (TB-mBJ) potential. This method is particularly well-suited for capturing the complex exchange interactions governing the magnetism in transition metal-based systems. The computed total and atomic-resolved spin magnetic moments using the GGA+mBJ formalism are summarized in Table 7. For Rb₂GeVBr₆, Rb₂GeMnBr₆, and Rb₂GeNiBr₆, the total magnetic moments are found to be approximately 3µ_B, 5µ_B, and 2µ_B, respectively. These values directly correspond to the number of unpaired electrons contributed by the respective transition metal cations, consistent with their expected magnetic moments. The observed integer magnetic moments are rooted in the crystal field-induced splitting of the 3d orbitals under the influence of the octahedral Br⁻ ligand environment. In these systems, the transition metals adopt the following high-spin configurations: V²⁺ (3d³) fills the t_2g orbitals with three unpaired electrons (↑↑↑), Mn²⁺ (3d⁵) distributes five unpaired electrons across both t_2g and e_g orbitals (↑↑↑, ↑↑), while Ni²⁺ (3d⁸) places two unpaired electrons in the t_2g orbitals (↑↑), in agreement with Hund's rule of maximum multiplicity. The resultant spin-only moments from these unpaired electrons yield magnetic moments of 3µ_B, 5µ_B, and 2µ_B, respectively, aligning precisely with our first-principles results. Moreover, the high magnetic moments and strong spin polarization near the Fermi level underscore the capability of these compounds to support spin-polarized transport. This characteristic is particularly desirable for spintronic applications, where efficient spin injection and manipulation are crucial. The robust magnetic ordering and full spin polarization exhibited by Rb₂GeMBr₆ alloys position them as promising candidates for next-generation spin-based electronic devices.

Table 7 Calculated values of spin magnetic moments of mixed charge density for Rb₂GeMBr₆ (M = V, Mn, Ni) compounds using TB-mBJ approximations

Material	Method	Rb (µB)	Ge (µB)	M (µB)	Br (µB)	Interstitial (µB)	Total (µB)
Rb2GeVBr6	GGA+mBJ	0.00	−0.00	2.71	0.00	0.29	3.00
Rb2GeMnBr6	GGA+mBJ	0.00	0.05	4.64	0.02	0.17	5.00
Rb2GeNiBr6	GGA+mBJ	0.00	0.04	1.74	0.03	0.02	2.00

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The Curie temperature (T_C), which defines the thermal threshold above which a ferromagnetic system loses long-range magnetic ordering and becomes paramagnetic, is evaluated for the Rb₂GeMBr₆ perovskite compounds within the framework of mean-field theory (MFT) derived from the Heisenberg spin model. T_C is calculated using the relation^19,75 where ΔE is the total energy difference between the ferromagnetic (FM) and antiferromagnetic (AFM) ground states, k_B is the Boltzmann constant, and x represents the fraction of magnetic atoms per formula unit, which is unity in this case as there is one transition metal ion (M) per unit cell. ΔE is obtained from density functional theory (DFT) calculations by comparing the total energies of FM and AFM (or non-magnetic, NM) configurations across volume-optimized energy landscapes: ΔE = E_AFM/NM − E_FM. The calculated Curie temperatures are approximately 599.6 K, 680.2 K, and 520.5 K for Rb₂GeVBr₆, Rb₂GeMnBr₆, and Rb₂GeNiBr₆, respectively. These elevated T_C values exceed those commonly reported for oxide-based double perovskites, indicating stronger ferromagnetic exchange interactions in the halide framework. The robust magnetic coupling inferred from these results suggests that Rb₂GeMBr₆ systems maintain long-range magnetic ordering well above room temperature, a crucial requirement for practical spintronic applications. Their combination of thermal stability and strong ferromagnetism makes them promising candidates for devices that demand consistent magnetic performance under high-temperature operating conditions.

To elucidate the origin of the notably high Curie temperatures (T_C) in Rb₂GeMBr₆ halide double perovskites, a comparative analysis is performed with well-known oxide-based analogues such as Bi₂CrOsO₆. The study indicates that the enhanced ferromagnetic stability in the halide compounds is predominantly driven by indirect exchange interactions mediated by the Ge–Br₆ octahedral framework. These interactions facilitate magnetic coupling between neighboring transition metal ions via the bridging Ge–Br units. As shown in Fig. 8, a significant hybridization occurs between the 3d orbitals of the transition metals (V, Mn, Ni) and the 4p orbitals of Br. This strong orbital overlap arises from a quasi-90-degree superexchange mechanism involving M-3d and Ge–Br₆σs–p* antibonding orbitals. In contrast to the conventional 180-degree superexchange pathway observed in oxide double perovskites, such as the Cr(d)–O(p)–Os(d) chain in Bi₂CrOsO₆, the 90-degree interaction in the halide framework provides a more effective channel for stabilizing ferromagnetic alignment, consistent with the Pauli exclusion principle. This distinctive exchange geometry is a principal factor contributing to the elevated T_C values in the halide series.

The stability of the ferromagnetic ground state, previously inferred from total energy versus volume curves (Fig. 2(a)), is further corroborated by temperature-dependent magnetic susceptibility calculations. Magnetic susceptibility (χ) is evaluated using the Curie–Weiss model, expressed as ,⁷⁶ where C denotes the Curie constant associated with the effective magnetic moment, T is the absolute temperature, and Θ represents the Weiss temperature that reflects the prevailing magnetic exchange interactions. A positive Θ corresponds to ferromagnetic coupling, whereas a negative value indicates antiferromagnetic behavior. As illustrated in Fig. 10, χ and its reciprocal (χ⁻¹) are plotted as functions of temperature for Rb₂GeMBr₆ perovskites. The magnetic susceptibility decreases monotonically with increasing temperature, consistent with Curie–Weiss behavior. The reciprocal plots (χ⁻¹vs. T) exhibit linear trends, from which positive Weiss temperatures of approximately 140 K (Rb₂GeVBr₆), 170 K (Rb₂GeMnBr₆), and 100 K (Rb₂GeNiBr₆) are obtained. These positive Θ values confirm dominant ferromagnetic interactions in all three compositions. The observed trends are in good agreement with the Curie–Weiss law,⁶³ indicating a robust intrinsic ferromagnetic ground state. The combination of strong magnetic ordering and high Curie temperatures indicates that Rb₂GeMBr₆ halide perovskites are possible candidates for spintronic applications operating at elevated temperatures.

Fig. 10 Plot of (a) magnetic susceptibility (χ) and (b) its reciprocal (χ⁻¹) against temperature for Rb₂GeMBr₆ (M = V, Mn, Ni) halide perovskites.

3.6. Electron charge density distribution

The investigation of electronic charge density provides critical insights into the bonding environment and chemical stability of a material. Fig. 11 illustrates the spin-resolved electronic charge density distribution of Rb₂GeVBr₆ projected along the (111) crystallographic plane. Since the charge density characteristics of the three compounds follow a similar trend, this representative case is presented for detailed analysis. The distribution reveals pronounced charge accumulation around atomic centres, reflecting the intrinsic bonding characteristics within the lattice. The Rb and Br sites display nearly spherical charge density contours, consistent with ionic interactions where electron transfer occurs from electropositive Rb to electronegative Br atoms. In contrast, the Ge–Br and V–Br linkages exhibit anisotropic, lobe-like charge distributions directed along the bond axis, indicative of orbital overlap and shared electron density. These features signify the presence of covalent contributions superimposed on the predominantly ionic framework. The coexistence of ionic alkali–halide interactions with covalent transition-metal–halide coordination establishes a polar covalent bonding scheme. Such a mixed bonding nature is crucial, as it simultaneously enhances lattice cohesion, mechanical resilience, and chemical robustness in Rb₂GeMBr₆ perovskites.

Fig. 11 Spin resolved valence charge density of Rb₂GeVBr₆ projected along the (111) plane.

3.7. Thermoelectric coefficients

The thermoelectric effect, which facilitates the direct conversion between heat and electrical energy, is fundamentally evaluated using the dimensionless figure of merit, zT. This parameter is mathematically defined as where S is the Seebeck coefficient, σ denotes the electrical conductivity, κ is the total thermal conductivity, and T is the absolute temperature. Optimizing thermoelectric efficiency thus involves enhancing the power factor (S²σ) while minimizing thermal conductivity (κ). In this study, transport coefficients are computed using the semi-classical Boltzmann transport formalism within the constant relaxation time approximation and the rigid band model, as implemented in the BoltzTraP code.⁴⁰ The relaxation time (τ), which describes the average interval between electron scattering events, is assumed to be nearly constant across the investigated temperature range. This approach provides a reliable framework for calculating conductivity (σ/τ), the electronic contribution to thermal conductivity (κ_e/τ), and the figure of merit (zT), thereby ensuring consistency in the evaluation of thermoelectric performance across the Rb₂GeMBr₆ compounds. Building upon this framework, a rigorous evaluation of the thermoelectric response is performed by systematically analyzing both temperature- and chemical potential–dependent transport characteristics. The temperature variation of the Seebeck coefficient, electrical conductivity, and electronic thermal conductivity in the range of 200–800 K, presented in Fig. 12(a–b, d), elucidates the intrinsic evolution of carrier dynamics under thermal excitation. Complementarily, the dependence of σ/τ, S, and the dimensionless figure of merit (zT) on chemical potential between −2 eV and +2 eV at representative temperatures, shown in Fig. 13(a–c), reveals the impact of carrier concentration tuning on transport asymmetry and overall thermoelectric efficiency. This dual perspective provides a comprehensive framework for correlating the electronic structure, carrier scattering processes, and energy conversion performance in these halide double perovskites. Given the intrinsic ferromagnetic nature of the investigated compounds, a spin-resolved treatment of transport properties becomes indispensable, as spin asymmetry exerts a pronounced influence on both electrical and thermal conduction mechanisms. To accurately account for these effects, the well-established two-current model is employed, wherein the total transport coefficients are obtained by independently evaluating the spin-up (↑) and spin-down (↓) electronic channels. Within this framework, total electrical conductivity and electronic thermal conductivity are expressed as σ = σ↑ + σ ↓, κ_e = κ_e↑ + κ_e ↓. The effective Seebeck coefficient, which accounts for the spin-dependent transport, is derived using the weighted average: ,⁶² where S↑, S↓ and σ↑, σ↓ denote the Seebeck coefficients and electrical conductivities for spin-up and spin-down electrons, respectively.

Fig. 12 Temperature dependence of (a) Seebeck coefficient (S), (b) electrical conductivity (σ/τ), (c) lattice thermal conductivity (κ_l), and (d) electronic thermal conductivity (κ_e/τ) for Rb₂GeMBr₆ (M = V, Mn, Ni) halide perovskites.

Fig. 13 Thermoelectric plots of (a) Seebeck coefficient (S), (b) electrical conductivity (σ/t), and (c) figure of merit (zT) against chemical potential at different temperatures (300 K, 600 K, 800 K) for Rb₂GeMBr₆ (M = V, Mn, Ni) halide perovskites.

3.7.1. Seebeck coefficient. The Seebeck coefficient (S), which characterizes a material's ability to convert a temperature gradient into electrical voltage, is a critical parameter for evaluating thermoelectric efficiency. This phenomenon, governed by the relation ΔV = S × ΔT, is fundamentally influenced by the electronic structure, effective mass of charge carriers, and the density of states (DOS) near the Fermi level. The temperature-dependent Seebeck behavior of Rb₂GeVBr₆, Rb₂GeMnBr₆, and Rb₂GeNiBr₆ is presented in Fig. 12(a). At low temperatures (∼100 K), these compounds exhibit high thermopower, with S values reaching ∼600 µV K⁻¹, 550 µV K⁻¹, and 450 µV K⁻¹, respectively. With increasing temperature up to 800 K, S decreases gradually to about 275 µV K⁻¹ for the V-based, 260 µV K⁻¹ for the Mn-based, and 250 µV K⁻¹ for the Ni-based compounds. This decline reflects the expected semiconductor response, wherein thermally excited carriers of both polarities begin to dominate transport, lowering the entropy per carrier and weakening the energy filtering effect, which in turn reduces S. This trend is consistent with the inverse dependence of S on carrier density (n), as described by the approximate relation:⁴⁷ where m* is the carrier effective mass and n is the carrier concentration. Based on spin-resolved GGA+mBJ calculations (Table 5), the effective mass anisotropy among these systems governs their Seebeck trends. Rb₂GeVBr₆ exhibits a low electron effective mass and a relatively heavy hole mass in the spin-up channel. This combination supports high carrier mobility for electrons and an enhanced DOS slope at the valence band edge, thereby promoting large Seebeck coefficients. Conversely, Rb₂GeNiBr₆ displays heavier electrons and lighter holes in the spin-down channel, leading to reduced DOS asymmetry near the Fermi level and consequently reduced thermopower. Rb₂GeMnBr₆ exhibits an intermediate range of effective masses , producing a balanced compromise between carrier transport and Seebeck magnitude.

The influence of chemical potential (µ) on S is illustrated in Fig. 13(a). Positive S values correspond to hole (p-type) conduction, while negative values indicate electron (n-type) transport. All three compounds exhibit sharp Seebeck peaks under low carrier concentration regimes (µ ≈ 0–1.5 eV), where DOS asymmetry strongly enhances the energy selectivity of carrier transport. At room temperature, the maximum S values are approximately ±3000 µV K⁻¹ for Rb₂GeVBr₆, ±2000 µV K⁻¹ for Rb₂GeMnBr₆, and ±700 µV K⁻¹ for Rb₂GeNiBr₆ in the scanned chemical-potential window (µ = 0 to +1.0 eV). These trends correlate with the GGA+mBJ-derived band gaps, which decrease in the order Rb₂GeVBr₆ (3.3 eV, spin-down) > Rb₂GeMnBr₆ (2.95 eV) > Rb₂GeNiBr₆ (1.85 eV). The larger gaps in the V- and Mn-based systems delay intrinsic excitation, suppress bipolar conduction at elevated temperatures, and thus sustain higher thermopower values. In contrast, the narrower gap and higher intrinsic carrier density of Rb₂GeNiBr₆ lead to faster thermal degradation of S.

To gain further insight into the origin of the high Seebeck values, Fig. 14 correlates the Seebeck coefficient (S) with the density of states (DOS) and carrier concentration (n) as functions of chemical potential (µ) for Rb₂GeVBr₆, while plots for other compounds are omitted for brevity. The spin-up channel exhibits pronounced S peaks within µ ≈ −0.5 to 0.5 eV, whereas the spin-down channel shows broader maxima extending over µ ≈ −1.0 to 0.0 eV. These peaks arise from regions of suppressed DOS, where reduced carrier availability enhances the energy dependence of electronic transport, thereby amplifying the Seebeck coefficient. The observed asymmetry in DOS and evident spin-resolved band-edge splitting around the Fermi level facilitate strong energy-filtering effects and selective carrier transport, which collectively account for the exceptionally large thermopower observed in Rb₂GeVBr₆.

Fig. 14 Comparison of the Seebeck coefficient (S), electrical conductivity (σ/τ), figure of merit (zT), carrier concentration (n), and volumetric density of states (DOS) as a function of chemical potential at 300 K for the Rb₂GeVBr₆ halide perovskite. The upward and downward arrows represent the spin-up and spin-down directions, respectively.

Spin splitting in the conduction and valence bands (ΔE_CBM and ΔE_VBM) further contributes to spin-dependent asymmetry in S. Particularly in Rb₂GeVBr₆ and Rb₂GeMnBr₆, significant splitting near the valence band maximum (∼0.75–0.86 eV) enhances spin polarization, suppresses bipolar diffusion, and stabilizes the Seebeck coefficient even at elevated temperatures. At ∼800 K, S values decline across all systems due to enhanced bipolar conduction, reducing peak magnitudes to about ±1050 µV K⁻¹ (V), ±850 µV K⁻¹ (Mn), and ±600 µV K⁻¹ (Ni). This reduction follows the Mott relation:⁷⁷ where the derivative of the logarithmic conductivity near µ dictates the magnitude of S. As temperature rises, both electrons and holes are thermally activated, and their opposing contributions partially cancel each other. Nevertheless, the relatively large band gaps (>1.5 eV) of all compounds prevent complete compensation, maintaining finite thermopower even at high temperatures.

In essence, the combined effects of effective mass asymmetry, strong DOS features near the Fermi level, and pronounced spin-resolved band splitting underpin the remarkable Seebeck behavior of Rb₂GeMBr₆ perovskites. The persistence of sizable S values, coupled with their spin-dependent transport stability, positions these halide perovskites as promising lead-free materials for solid-state thermoelectric generators, coolers, thermocouples, and precision temperature sensors.

3.7.2. Electrical conductivity. The normalized electrical conductivity (σ/τ) serves as a fundamental descriptor of carrier transport, governed by temperature, electronic structure, and spin-resolved dynamics. For the Rb₂GeMBr₆ halide perovskites, σ/τ increases markedly between 200 and 800 K (Fig. 12(b)), reflecting intrinsic semiconducting behavior. At low temperatures, limited thermal excitation across the band gap restricts carrier generation, resulting in weak conductivity. With increasing temperature, enhanced thermal activation promotes carriers across the band edges, increasing carrier concentration (n) and, consequently, σ, following the relation σ = neµ, where e is the elementary charge and µ is the carrier mobility. Among the three compositions, Rb₂GeNiBr₆ exhibits the highest σ/τ at 800 K (∼2.65 × 10¹⁹ Ω⁻¹ m⁻¹ s⁻¹), facilitated by its narrow band gap (1.54–1.85 eV), which enables efficient thermal activation of charge carriers. Despite its relatively large electron effective mass , the high carrier density compensates to yield strong conductivity. Rb₂GeMnBr₆, with a band gap of 2.35–2.95 eV, reaches σ/τ values of around 2.2 × 10¹⁹ Ω⁻¹ m⁻¹ s⁻¹. Rb₂GeVBr₆ shows the lowest σ/τ (∼1.75 × 10¹⁹ Ω⁻¹ m⁻¹ s⁻¹) at 800 K, constrained by its wider band gap (2.85–3.3 eV), although this is partially offset by its low effective mass .

The dependence of σ/τ on chemical potential (µ), shown in Fig. 13(b), reflects the underlying distribution of electronic states and band alignment. Conductivity maxima are observed between −2 eV and 0 eV, corresponding to regions of high DOS where abundant electronic states actively contribute to charge transport. In Rb₂GeVBr₆, these peaks appear farther from the Fermi level, consistent with its relatively wider band gap and delayed onset of conduction, whereas Rb₂GeNiBr₆ displays broader peaks centered near µ ≈ 0 eV, indicative of higher intrinsic carrier availability. Spin-resolved transport further modulates this behavior, and in Rb₂GeVBr₆ and Rb₂GeMnBr₆, the spin-up channel predominantly contributes to σ/τ within defined energy ranges owing to pronounced spin splitting at the band edges (ΔE_CBM and ΔE_VBM up to ∼1 eV). This splitting originates from exchange interactions between the magnetic cations and conduction states, which enhance spin-dependent scattering and result in distinct transport anisotropies across the spin channels. These asymmetries give rise to spin-polarized conduction pathways, reinforcing their classification as ferromagnetic semiconductors with intrinsically anisotropic transport characteristics.

To gain deeper insight, the interdependence of σ/τ, carrier concentration (n), and the density of states (DOS) with respect to chemical potential (µ) is analyzed for Rb₂GeVBr₆ (other compounds omitted for brevity), as shown in Fig. 14. Peaks for σ/τ coincide with DOS maxima, indicating regions where a high electronic state density enables efficient carrier transport through enhanced scattering relaxation times. In contrast, near the pseudo-gap region (µ ≈ −0.5 to 0.3 eV), the carrier concentration and σ/τ exhibit noticeable suppression, whereas S displays sharp enhancement. This opposite trend between S and σ/τ reflects the intrinsic transport trade-off characteristic of thermoelectric materials, where the suppression of low-energy carriers and the dominance of high-energy ones contribute to larger thermopower via the Mott-type energy-filtering mechanism. Thus, the coexistence of localized DOS depressions and moderate n values enhances the slope of the transport distribution function, improving S without severely compromising electrical conductivity.

3.7.3. Thermal conductivity and its components. In thermoelectric materials, efficient energy conversion critically depends on their ability to suppress heat transport while maintaining high electrical performance. The total thermal conductivity (κ) represents the principal parameter governing this balance and consists of two distinct contributions: the electronic component (κ_e), arising from charge-carrier motion, and the lattice component (κ_l), originating from phonon-mediated heat propagation. Fig. 12(c and d) present the temperature-dependent thermal conductivity of Rb₂GeMBr₆ compounds, providing insight into their intrinsic heat transport characteristics. The lattice thermal conductivity (κ_l) is obtained by solving the phonon Boltzmann transport equation within the relaxation time approximation using ShengBTE,⁴¹ while the electronic thermal conductivity (κ_e) is calculated from BoltzTraP based on the electronic band structure within the constant relaxation time approximation.⁴⁰ The lattice component exhibits a pronounced temperature dependence, as shown in Fig. 12(c). κ_l decreases rapidly from approximately 2.50 W m⁻¹ K⁻¹ for Rb₂GeVBr₆, 1.90 W m⁻¹ K⁻¹ for Rb₂GeMnBr₆, and 1.50 W m⁻¹ K⁻¹ for Rb₂GeNiBr₆ at 100 K to values in the range of 0.15–0.20 W m⁻¹ K⁻¹ at 800 K. At room temperature, κ_l is 0.67 W m⁻¹ K⁻¹ for Rb₂GeVBr₆, 0.50 W m⁻¹ K⁻¹ for Rb₂GeMnBr₆, and 0.34 W m⁻¹ K⁻¹ for Rb₂GeNiBr₆, underscoring the intrinsically weak phonon-mediated heat transport in these materials. This strong suppression originates from dominant anharmonic phonon–phonon (Umklapp) scattering, which significantly limits phonon lifetimes and reduces their ability to carry heat. Furthermore, the structural complexity of halide double perovskites, characterized by multi-atom unit cells and heavy constituent elements, promotes additional scattering through mass disorder and lattice anharmonicity. Collectively, these effects ensure that phonon transport is governed by short-lived, strongly scattered modes with limited mean free paths, yielding intrinsically low κ_l values. Such a combination of strong anharmonicity, complex bonding environments, and enhanced scattering makes these compounds promising candidates for thermoelectric applications.

In contrast, the electronic contribution increases with temperature as carriers are thermally activated across the band gap, as shown in Fig. 12(d). BoltzTraP provides κ_e in the form κ_e/τ, which allows explicit scaling with the carrier relaxation time τ. In the reported units, κ_e/τ increases monotonically with temperature, with Rb₂GeNiBr₆ exhibiting the largest values, followed by Rb₂GeMnBr₆ and Rb₂GeVBr₆ (Fig. 12(d)). Because κ_e is presented as κ_e/τ, the absolute magnitude of κ_e depends on the choice of τ. Within realistic τ values used in the literature, κ_e emerges as a non-negligible fraction of the total thermal conductivity above ∼600 K. The interplay between the strongly suppressed lattice contribution and the steadily rising electronic term leads to an overall low total thermal conductivity (κ_t = κ_l + κ_e) across the studied temperature range. This balance is advantageous for thermoelectric performance, as it preserves favorable electronic transport while minimizing parasitic heat conduction through the lattice.

3.7.4. Thermoelectric figure of merit (zT). The dimensionless figure of merit (zT) encapsulates the collective influence of the Seebeck coefficient (S), electrical conductivity (σ), absolute temperature (T), and total thermal conductivity (κ), serving as a comprehensive measure of thermoelectric efficiency. The calculated zT maps as functions of chemical potential and temperature for Rb₂GeMBr₆, shown in Fig. 13(c), reveal peak values approaching unity, highlighting their promising thermoelectric potential. At 300 K, within the scanned chemical-potential window (µ = –1.0 to +2.0 eV), the maximum figures of merit are zT_max ≈ 1.00 for Rb₂GeVBr₆, zT_max ≈ 0.99 for Rb₂GeMnBr₆, and zT_max ≈ 0.98 for Rb₂GeNiBr₆. These enhanced values arise from the synergistic balance between large Seebeck coefficients, moderate-to-high electrical conductivity, and intrinsically low lattice thermal conductivity, the latter being governed by soft phonon modes and the structural complexity of the double perovskite framework. It should be emphasized, however, that these results correspond to peak zT values within the simulated chemical-potential range, representing optimal performance under specific Fermi-level shifts (i.e., effective doping conditions). They do not represent absolute bulk zT and thus require explicit carrier-concentration-dependent calculations or experimental validation for quantitative confirmation.

The superior performance of Rb₂GeVBr₆ is primarily attributed to its relatively wide band gap (2.85–3.3 eV), which promotes energy filtering and suppresses bipolar diffusion. This effect sustains high thermopower at elevated temperatures while maintaining a low intrinsic carrier concentration. The temperature dependence of zT exhibits a non-monotonic profile: at intermediate temperatures, thermally activated carriers enhance σ, compensating for the gradual decline in S and thereby sustaining high zT values. Beyond a critical temperature, however, the increase in κ and the reduction in S due to carrier overpopulation collectively lead to a moderate decrease in performance. Despite this decline, all three compositions retain zT values above 0.75 at 800 K, confirming their thermoelectric stability and suitability for high-temperature applications.

The variation of zT, DOS, and n with µ further highlights the spin-dependent transport asymmetry intrinsic to Rb₂GeVBr₆ (others omitted for brevity). The spin-resolved plots at 300 K (Fig. 14) reveal distinct zT maxima in the spin-up channel across µ ≈ 0.5 to 1.5 eV and in the spin-down channel across µ ≈ −0.8 to 0.5 eV. These zT peaks coincide with sharp DOS fluctuations and localized minima for n, where the increased energy dependence of carrier mobility amplifies thermoelectric efficiency. Such spin-specific modulation of DOS and n near the band edges indicates that transport in Rb₂GeVBr₆ is strongly governed by asymmetric spin channels and orbital hybridization effects. The combination of spin-resolved band asymmetry, optimized effective mass, and moderate carrier densities gives rise to enhanced thermopower and a high zT response. Overall, these results confirm that Rb₂GeVBr₆ and related halide double perovskites possess an intrinsic capability for energy-selective carrier transport, making them promising materials for high-temperature thermoelectric and spin-caloritronic applications.

Further enhancement of zT can be envisioned through rational band-engineering strategies, including isovalent or aliovalent doping, epitaxial strain tuning, and nanoscale structuring.⁷⁸ These approaches can modify band curvature, optimize Fermi-level alignment, and intensify phonon scattering to further suppress κ_l without degrading σ. Given that Rb₂GeMBr₆ compounds already exhibit optimized electronic transport and intrinsically low lattice thermal conductivity owing to heavy constituent atoms and pronounced bonding anharmonicity, they provide ideal platforms for such targeted design strategies.

Overall, the thermoelectric metrics summarized in Table 8, together with comparisons to reported halide double perovskites, confirm that Rb₂GeMBr₆ (M = V, Mn, Ni) outperform many previously studied analogues in both Seebeck coefficient and figure of merit. Their high zT values originate from the favorable interplay of large power factors, suppressed lattice thermal conductivity, and spin-dependent electronic dispersion. These features establish Rb₂GeMBr₆ compounds as promising candidates for next-generation thermoelectric technologies, including waste-heat recovery, micro-power generation, and precision thermal sensing, owing to their outstanding energy-conversion efficiency and structural tunability.

Table 8 Calculated thermoelectric and optical parameters for Rb₂GeMBr₆ (M = V, Mn, Ni) halide perovskites in comparison with other theoretical results

Material	Thermoelectric parameters (300 K)		Optical parameters (0–5 eV)
	Seebeck coefficient (µV K^−1)	Figure of merit (zTmax)	Real part ε1(ω)	Absorption coefficient α(ω)
The asterisk (*) denotes the maximum value obtained within the scanned chemical potential range under the constant relaxation time approximation (CRTA). These values correspond to low carrier concentrations and represent peak theoretical limits rather than absolute values, as they are evaluated within the CRTA framework.
Rb2GeVBr6	3000*	1.00*	8.0	63
Rb2GeMnBr6	2000*	0.99*	8.4	52
Rb2GeNiBr6	700*	0.98*	8.6	43
Mg2YBiO6^29	2.04 × 10^−4	0.75	6.8	55
Ca2YBiO6^29	2.52 × 10^−4	0.80	6.5	52
Ba2YBiO6^29	2.66 × 10^−4	0.81	7.1	50
Cs2AgBiCl6^25	236	0.74	4.1^30	49^30
Cs2AgBiBr6^25	234	0.73	6.2^30	58^30

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3.8. Optical dielectric characteristics

The investigation of optical and dielectric properties is essential for evaluating the suitability of halide perovskites in optoelectronic applications, since these parameters govern light–matter interactions and device performance. The complex dielectric function, ε(ω) = ε₁(ω) + iε₂(ω), provides a direct connection between the electronic structure and optical response by describing both photon–electron interactions and energy dissipation processes.⁷⁴ In this work, particular attention is given to the real component ε₁(ω), which reflects polarization and screening effects, and to the absorption coefficient α(ω), derived from the imaginary part ε₂(ω), as key descriptors of optical performance. As shown in Fig. 15(a), ε₁(ω) exhibits distinct energy-dependent behavior across the three compositions, with Rb₂GeNiBr₆ displaying the highest static dielectric constant, indicative of stronger electronic polarizability. This behavior correlates with its relatively narrow band gap, which facilitates improved electronic polarization and dielectric screening. An elevated static dielectric constant reflects efficient response to external electric fields, especially important for reducing charge recombination in optoelectronic devices. The maximum values of ε₁(ω) are observed at different photon energy ranges for each composition: Rb₂GeNiBr₆ peaks at around 9.2 near 1.0 eV, Rb₂GeMnBr₆ reaches approximately 8.0 at 2.0 eV, and Rb₂GeVBr₆ attains 8.4 at around 2.5 eV. This trend implies that as the band gap widens (from Ni to V), the optical response shifts toward higher energy regions, consistent with their respective electronic band structures.

Fig. 15 Plot of the dielectric constant (a) real part (ε₁(ω)) and (b) absorption coefficient (α(ω)) against photon energy for Rb₂GeMBr₆ halide perovskites.

In Fig. 15(b), the absorption coefficient α(ω) is presented to elucidate the photon absorption efficiency of Rb₂GeMBr₆ halide perovskites. Among the three compositions, Rb₂GeNiBr₆ shows the earliest absorption onset, beginning just above 1.0 eV, followed by a steep rise that reaches ∼40–45 at 2.0 eV and continues to increase across the visible spectrum. This pronounced low-energy response originates from its narrower band gap, enabling strong interaction with visible photons and highlighting its suitability for photovoltaic energy harvesting. Rb₂GeMnBr₆ exhibits a delayed absorption onset above 2.0 eV, with a pronounced peak near 3.0 eV where α(ω) reaches ∼50–55. These features indicate dominant interband transitions from Br-4p orbitals to Mn-3d and Ge-p conduction states, emphasizing its optical activity in the near-visible to ultraviolet region. For Rb₂GeVBr₆, the absorption edge appears close to 2.5 eV, with a gradual rise and broad maxima between 3.5 and 4.0 eV, where α(ω) reaches ∼60–65. These higher-energy absorptions are attributed to deeper-lying valence-to-conduction transitions influenced by the V-d states, which impart a distinctive contribution to the band dispersion. Taken together, these results identify Rb₂GeNiBr₆ as the most efficient absorber in the visible region, while Rb₂GeMnBr₆ and Rb₂GeVBr₆ exhibit stronger optical responses at higher photon energies. The combination of a high dielectric response and a broad absorption range highlights the strong light-harvesting capability of these materials. A detailed summary of the calculated dielectric constants and absorption characteristics, along with comparisons to structurally related halide perovskites, is provided in Table 8. Notably, the dielectric performance reported here surpasses that of earlier studied systems,^28,30 reinforcing the potential of lead-free Rb₂GeMBr₆ perovskites as promising candidates for optoelectronic applications, particularly in photovoltaics and photodetectors.

4. Conclusions

To summarize, this study presents a comprehensive first-principles investigation of the structural, electronic, magnetic, and thermoelectric properties of Rb₂GeMBr₆ (M = V, Mn, Ni) halide double perovskites. The calculated tolerance factors, negative formation enthalpies, and positive cohesive energies confirm their intrinsic structural and thermodynamic stability. Finite-temperature ab initio molecular dynamics simulations at 500 K further validate their robustness against thermal perturbations, while the fulfillment of the Born mechanical stability criteria supports their mechanical reliability. Mechanical analysis reveals an optimal balance between bulk and shear moduli, indicating strong resistance to external deformation. Magnetic calculations confirm stable ferromagnetic ordering arising from the spin-polarized 3d orbitals of the transition-metal cations, with Curie temperatures estimated in the range of 520–680 K. Spin-resolved electronic structure analysis reveals crystal-field-induced splitting of the 3d orbitals, resulting in spin-polarized band gaps and pronounced spin splitting at both conduction and valence band edges (ΔE_CBM and ΔE_VBM), which enhances their spintronic potential. The projected density of states shows that the 3d states of transition metals dominate the electronic spectrum near the Fermi level, governing exchange interactions and charge transport. The energy separation between t_2g and e_g orbitals, dictated by the octahedral crystal field, plays a decisive role in determining the magnetic and electronic response. Thermoelectric evaluation highlights the multifunctional nature of these compounds. Rb₂GeVBr₆ attains the highest Seebeck coefficient of approximately 3000 µV K⁻¹ within a specific chemical potential window, attributed to its wide band gap and large hole effective mass (0.91m₀). In contrast, Rb₂GeNiBr₆ exhibits superior electrical conductivity due to its smaller hole effective mass (0.42m₀), while Rb₂GeMnBr₆ displays balanced transport characteristics. The dimensionless figures of merit (zT) for all three-systems approach unity, confirming their efficiency for thermoelectric applications. Overall, the Rb₂GeMBr₆ family integrates semiconducting, ferromagnetic, and thermoelectric functionalities within a single structural framework, establishing them as promising candidates for next-generation spintronic and energy conversion technologies. Future work may focus on performance enhancement through controlled doping, strain modulation, or nanostructuring to further optimize their multifunctional properties.

Conflicts of interest

The authors declare no competing interest.

Data availability

The datasets generated and thereafter analysed would be available from the corresponding author upon reasonable request.

The Supplementary Information accompanying this article contains additional computational and analytical details, including: Phase stability diagrams of Rb₂GeMBr₆ (M = V, Mn, Ni), Elastic property data obtained using Voigt, Reuss, and Hill averaging schemes (Table S1), Supporting figures and tables relevant to the thermodynamic and mechanical stability analyses. Supplementary information (SI) is available. See DOI: https://doi.org/10.1039/d5tc01842k.

Acknowledgements

Mudasir Younis Sofi, a recipient of the Prime Minister Research Fellowship (PMRF) award (Letter No. 3302522), wishes to express sincere gratitude to the PMRF agency (Ministry of Education (MoE), Govt. of India) for the financial support.

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