Pressure-induced modulation of structural, mechanical, electronic, optical, and thermodynamic properties of KZn₄As₃ and CsCd₄As₃ zintl compounds: implications for thermal and optoelectronic applications

Abstract

In this work, using DFT, we systematically investigate the structural, dynamic, elastic, thermodynamic, electronic, and optical properties of ternary arsenide Zintl phases KZn₄As₃ and CsCd₄As₃ under hydrostatic pressures of 0–15 GPa. The optimized lattice parameters at ambient pressure agree closely with the available experimental data, validating the computational approach. Both compounds undergo pronounced anisotropic lattice compression, with the c-axis contracting more than the a-axis, and reduction in unit-cell volumes by approximately 22–24% over the studied pressure range. Phonon dispersion calculations confirm that both phases remain dynamically stable, with no imaginary frequency modes across the Brillouin zone at any pressure. The compounds satisfy the Born mechanical stability criteria under ambient and applied pressures, confirming elastic stability throughout the studied range. Poisson's ratio increases from 0.29 to 0.36 for KZn₄As₃ and from 0.286 to 0.36 for CsCd₄As₃, confirming that both compounds remain ductile across the entire pressure range. Thermodynamic parameters such as average sound velocity, Debye temperature, and melting temperature, all derived from the elastic constants, increase with pressure, with KZn₄As₃ consistently exhibiting stronger interatomic bonding and higher thermal robustness than CsCd₄As₃. GGA-PBE predicts KZn₄As₃ to be metallic across all pressures, while HSE06 reveals a narrowing semiconducting gap of 0.921–0.557 eV, reflecting the known underestimation of GGA-PBE. For CsCd₄As₃, both functionals show pressure-induced gap reduction, with HSE06 consistently predicting larger gaps. Strong optical absorption, conductivity, and energy-loss features in the near-UV to UV region (4–6 eV), combined with low static reflectivity, highlight both compounds as mechanically robust, pressure-tunable Zintl phases suitable for infrared and UV optoelectronic devices, high-pressure structural applications, and thermoelectric energy conversion.

In this work, using DFT, we systematically investigate the structural, dynamic, elastic, thermodynamic, electronic, and optical properties of ternary arsenide Zintl phases KZn₄As₃ and CsCd₄As₃ under hydrostatic pressures of 0–15 GPa.

1. Introduction

Zintl compounds represent a fascinating and technologically significant class of intermetallic materials, named in recognition of Eduard Zintl's pioneering work in the 1930s on electropositive metals combined with main-group metalloids.^1,2 The defining characteristic of Zintl phases is the formal electron transfer from electropositive elements (typically alkali or alkaline earth metals from groups 1 or 2) to more electronegative main-group elements (from groups 13–15).^1,2 This electron transfer results in the formation of anionic clusters or extended networks of the main-group elements, which then forms covalent bonds amongst themselves. The Zintl-Klemm concept formalizes this electron bookkeeping, postulating that the anionic sublattice achieves a valence-satisfied state, often resembling an isoelectronic neutral element.² For instance, in NaTl, thallium (Tl), a group 13 element, accepts one electron from sodium to become Tl⁻, which is isoelectronic with group 14 elements, like silicon, and forms a tetrahedral network. Similarly, in CaSi, silicon (Si) accepts two electrons from calcium to become Si²⁻, which is isoelectronic with group 16 elements, like sulfur, leading to zigzag chains of silicon atoms. The unique bonding characteristics of Zintl compounds impart a phonon-glass, electron-crystal behavior, making them particularly promising for thermoelectric applications.³ This property arises from the combination of rigid, covalently bonded frameworks, which can efficiently conduct electrons, engage in soft, ionic interactions with cations, and scatter phonons effectively, leading to low lattice thermal conductivity. This decoupling of electrical and thermal transport properties is crucial for achieving high thermoelectric performance.

Beyond thermoelectrics, Zintl compounds are garnering significant interest across various scientific domains due to their diverse structures and tunable electronic properties. A multitude of Zintl thermoelectrics have been explored, including 1–2–2, 3–1–3, 5–2–6, 9–4–9, and 10–3–10 phases.³ Specific examples include Eu₂ZnSb₂, which exhibits a remarkable combination of high mobility and low thermal conductivity due to intrinsic nanostructures and partial ordering.^4–6 Other promising thermoelectric Zintl phases under investigation include AB₂Sb (e.g., A = Ca, Eu, and Sr), CaAl₂Si₂-type compounds,⁵ and SmMg₂Sb₂-based materials.⁴ First-principles studies are actively identifying new candidates, such as BaSr₂X₂ (X = As, P, and Sb)⁷ and BaMg₂X₂ (X = P, As, and Sb) alloys⁸ for their excellent thermoelectric response. The low lattice thermal conductivity of Zintl phase compounds is a key area of research, often studied using techniques like neutron scattering⁹ and machine learning-assisted discovery.¹⁰

Many Zintl compounds are direct or narrow bandgap semiconductors, making them attractive for optoelectronic devices and solar energy conversion.⁷ Examples include Ag₂Se₂ (A = Sr and Ba) compounds with direct bandgaps of 1.76 eV and 1.35 eV, respectively,¹¹ BaAg₂Te₂ with a direct bandgap of 1.445 eV,¹² and BaCd₂X₂ (X = P, As, and Sb) compounds.¹³ First-principles investigations are also exploring the optoelectronic properties of novel nitride cadmium Zintl compounds AB₂C₂ (A = Mg, Ca, Sr, and Ba; B = Cd and C = N),¹⁴ XIn₂Z₂ (X = Ca and Sr and Z = As and Sb),¹⁵ and SrCd₂Pn₂ (Pn = P/As) compounds.¹⁶ A₃InAs₃ (A = Sr and Eu) Zintl phases are also identified as n-type narrow band gap semiconductors with direct bandgaps ranging from 0.50 to 0.74 eV.¹⁷

Zintl compounds with unique electronic structures are being investigated because of their superconductivity and topological insulating properties. Layered SnAs-based Zintl compounds like SrSn₂As₂ have shown pressure-induced superconductivity, with a dome-like evolution of T_c at approximately 20.8 GPa.¹⁸ Topological electronic structures can emerge from vacancy ordering in materials like Eu₂ZnSb₂.⁴ Furthermore, monolayer Zintl A₂MX₂ compounds (A = Ca, Sr, or Yb; M = Zn or Cd; X = P, As, Sb, or Bi) are being studied for large-gap topological insulating phases and anisotropic Rashba and chiral spin textures.¹⁹ Computational searches have also predicted topological Dirac semimetal phases in CaM₂X₂ Zintl layered compounds.²⁰ Some Zintl phases exhibit interesting magnetic behaviors. The layered Zintl-phase compound Eu₇Ga₆Sb₈ shows colossal magnetoresistance and a structural phase transition with accompanying antiferromagnetic ordering.^21,22 EuIn₂As₂, another Zintl phase, is an antiferromagnetic metal whose spin dynamics are being studied using techniques like time-resolved magneto-optical Kerr effect.²¹ The Zintl-Klemm concept is being extended to understand and predict bonding in the amorphous state, which is particularly relevant for materials like Na–P battery anodes.²³ Hua He, Chauntae Tyson, and Svilen Bobev reported two new series of Zintl phases, ACd₄Pn₃ and AZn₄Pn₃ (A = Na, K, Rb, and Cs; Pn = As and P), whose structures feature complex atomic arrangements based on four- and eight-coordinated arsenic and phosphorus. A total of twelve compounds were synthesized from the corresponding elements via high-temperature reactions and structurally characterized by single-crystal X-ray diffraction. This study established the experimental crystal structures of KZn₄As₃ and CsCd₄As₃, the two compounds investigated in the present first-principles work, and therefore provides the essential structural foundation and motivation for the present computational study of their pressure-dependent properties.²⁴ Moreover, Ono and co-workers investigated the thermoelectric performance of RbZn_4−xCu_xAs₃, which crystallizes into the same KCu₄S₃-type (tetragonal, P4/mmm) structure as KZn₄As₃ and CsCd₄As₃. Hole carriers were doped by substituting Zn with Cu (up to x = 0.02), increasing the power factor from 0.049 to 0.52 mW mK⁻² at 797 K. The lattice thermal conductivity was found to be substantially lower (1.61 W mK⁻¹ at 312 K), attributed to large vibrational amplitudes of the Rb atoms, as confirmed by neutron diffraction. A maximum dimensionless figure of merit, ZT, of 0.53 at 797 K was reported, the highest value found so far for this class of “143”-Zintl compounds, indicating that this compound family constitutes a promising new class of high-performance thermoelectric materials.²⁵

This research is motivated by the growing demand for advanced materials with tunable properties for energy conversion and technological devices.^26–28 In this study, we investigated the pressure-induced structural, mechanical, optical, and thermal properties of AB₄P₃ Zintl compounds to assess their suitability for thermal and optoelectronic applications.

2. Computational details

The DFT-based CASTEP computer software with the generalized gradient approximation GGA + PBE, GGA + PBEsol and HSE06 was used for first-principles calculations.^29,30 For structural optimization and optical property calculations, the GGA-PBE functional was used due to its balanced accuracy and widespread applicability in predicting electronic and optical behavior in semiconductors and insulators.³¹ For elastic constant and thermodynamic property evaluations, the GGA-PBEsol functional was adopted, as it offers improved accuracy in predicting equilibrium structural parameters and mechanical properties of densely packed solids.³² For band structure calculations, a combination of GGA-PBE, GGA-PBEsol, and the hybrid HSE06 functional was utilized to ensure a comprehensive and accurate description of electronic states. The HSE06 functional, in particular, is known for its superior performance in predicting band gaps by incorporating a portion of exact exchange.³³

The electromagnetic wave function extension in a plane-wave basis set used an energy cutoff of 500 eV, and the Brillouin zone (BZ) was sampled using 6 × 6 × 2 k-points for all calculations. A convergence test was performed for the plane-wave cutoff energy and k-point sampling, increasing the cutoff beyond 500 eV and the k-point sampling beyond 6 × 6 × 2. Hydrostatic pressure was applied using the variable-cell optimization method implemented in CASTEP, where the target external pressure (0, 5, 10, and 15 GPa) was imposed as a constraint during geometry optimization. At each pressure point, the unit cell parameters and atomic positions were fully relaxed until the enthalpy was minimized, with the stress tensor components converging to within 0.10 GPa of the target pressure. This approach directly yields the equilibrium structure at each specified pressure, eliminating the need for separate equation-of-state fitting to determine the pressure. The geometric structures were optimized using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization technique.³⁴ Vanderbilt-type ultrasoft pseudopotentials (UPP) were used to describe the ion–electron interactions, considering the valence electron configurations of K (3s²3p⁶4s¹), Cs (5s²5p⁶6s¹), Zn (3d¹⁰4s²), Cd (4d¹⁰5s²), and As (4s²4p³). This approach achieves a substantially softer pseudo-wave function, requiring significantly fewer plane waves for calculations while achieving the same precision.³⁵ The elastic constants were obtained using first-principles methods, where a series of small, predefined homogeneous strains (up to ±0.5%) were applied to the optimized structures. The phonon dispersion relations were calculated using the finite displacement method with a cutoff radius of 5 Å, ensuring accurate representation of interatomic force constants and dynamical stability of the compounds. Additionally, for accurate evaluation of optical properties, the Brillouin zone was sampled using a Monkhorst–Pack k-point grid of 10 × 10 × 4, ensuring a dense and uniform distribution.

3. Results and discussions

3.1. Structural properties

The investigated compounds KZn₄As₃ and CsCd₄As₃ both crystallize in the tetragonal structure with space group P4/mmm (123), as shown in Fig. 1.³⁶ The detailed atomic positions are as follows: the K/Cs atoms occupy the 1b Wyckoff position (0, 0, 1/2), the Zn/Cd atoms occupy the 4i position (1/2, 0, z), and As atoms occupy two distinct sites: As1 at the 1a position (0, 0, 0) and As2 at the 2h position (1/2, 1/2, z). The coordination environment around each atom type reflects the mixed bonding character: As atoms are tetrahedrally coordinated by Zn/Cd atoms, forming the covalent backbone, while K/Cs atoms are surrounded by As atoms in a cage-like arrangement.

Fig. 1. Crystal structures of KZn₄As₃ and CsCd₄As₃.

For CsCd₄As₃, the experimental lattice parameters are a = 4.4990 Å and c = 10.7870 Å,³⁷ obtained from prior single-crystal X-ray diffraction studies, which are in good agreement with our GGA-PBEsol values a = 4.47 Å, c = 10.71 Å and GGA-PBE values a = 4.57 Å, c = 11.05 Å. For KZn₄As₃, single-crystal data could not be obtained from the literature; instead, powder X-ray diffraction reported in the same literature source confirmed isomorphism with the tetragonal structure of CsZn₄As₃ and RbZn₄As₃,³⁷ with refined unit cell parameters of a = 4.154 Å and c = 10.159 Å. The structural properties of KZn₄As₃ and CsCd₄As₃ were examined at hydrostatic pressures of 0, 5, 10, and 15 GPa to understand the pressure response of their crystal frameworks. The optimized lattice parameters, unit-cell volumes, total energies and bulk moduli are listed in Table 1.

Calculated lattice parameters (a, c), unit cell volume (V), total energy (E), bulk modulus (B) and structural description of KZn₄As₃ and CsCd₄As₃ under hydrostatic pressure.

		KZn4As3				CsCd4As3
		0 GPa	5 GPa	10 GPa	15 GPa	0 GPa	5 GPa	10 GPa	15 GPa
GGA-PBE	a	4.17	4.05	3.97	3.90	4.57	4.43	4.34	4.27
	c	10.34	9.73	9.43	9.23	11.05	10.22	9.82	9.58
	c/a	2.479616	2.402469	2.375315	2.366667	2.417943	2.306998	2.262673	2.24356
	V	180.32	160.21	148.87	140.86	231.63	201	185.45	175.02
	E	−17831.43	−17831.17	−17830.66	−17830.04	−15537.69	−15537.29	−15536.59	−15535.78
	B	22.34	49.99	90.64	99.45	40.11	64.40	78.57	80.86
GGA-PBEsol	a	4.09	3.98	3.91	3.85	4.47	4.35	4.27	4.21
	c	10.10	9.56	9.29	9.11	10.71	10	9.66	9.46
	c/a	2.46	2.40	2.37	2.36	2.39	2.29	2.26	2.24
	V	169.25	152.31	142.44	135.27	214.36	190.04	176.98	167.93
	E	−17810.16	−17809.94	−17809.49	−17808.94	−15522.74	−15522.42	−15521.83	−15521.13
	B	34.55	63.34	109.92	107.97	20.57	41.05	76.06	87.79
		Other works37
RbZn4As3	a	4.17	Exp	CsCd4As3	a	4.4990	Exp
	c	10.35			c	10.78
RbZn4P3	a	4.05	Exp	CsZn4As3	a	4.17	Exp
	c	10.06			c	10.57

Atom	Wyckoff site	x	y	z	Atom	Wyckoff site	x	y	z
K	1b	0	0	1/2	Cd	1b	0	0	1/2
Zn	4i	0	1/2	0.156	Zn	4i	0	1/2	0.163
As1	2h	1/2	1/2	0.284	As1	2h	1/2	1/2	0.298
As2	1a	0	0	0	As2	1a	0	0	0

For KZn₄As₃ at ambient pressure, the GGA-PBE functional yields lattice parameters of a = 4.17 Å and c = 10.34 Å, while GGA-PBEsol gives slightly smaller values of a = 4.09 Å and c = 10.10 Å. The PBEsol results are generally considered more reliable for equilibrium structural parameters of solids due to its optimization for condensed matter systems.³⁸ For CsCd₄As₃, the lattice constants are larger (a = 4.57 Å and c = 11.05 Å) with GGA-PBE at 0 GPa, consistent with the larger ionic radii of Cs⁺ compared to K⁺ and Cd²⁺ compared to Zn²⁺. For KZn₄As₃ GGA-PBE, the lattice parameter a decreases from 4.17 Å at 0 GPa to 3.90 Å at 15 GPa, while c decreases from 10.34 Å to 9.43 Å, as shown in Fig. 2a and b. The variation of the c/a ratio in Fig. 2c with pressure reveals anisotropic compression of the tetragonal lattice. Specifically, the c-axis contracts more strongly than the a-axis, leading to a systematic decrease in c/a with increasing pressure. For CsCd₄As₃, the experimental lattice parameters are a = 4.4990 Å and c = 10.7870 Å,³⁹ which agree closely with our GGA-PBEsol values (a = 4.47 Å, c = 10.71 Å) and GGA-PBE values (a = 4.57 Å and c = 11.05 Å). The unit cell volume for KZn₄As₃ decreases by approximately 22% over the range 0 to 15 GPa. A similar trend is observed for CsCd₄As₃, where the volume reduction from 0 to 15 GPa is about 24% with GGA-PBE. The bulk modulus, which reflects a material's resistance to compression, also increases with pressure for both compounds. This indicates that the material becomes progressively harder and less compressible under compression.

Fig. 2. Pressure dependence of the lattice constants a and c, and the c/a ratio for KZn₄As₃ and CsCd₄As₃ calculated using GGA-PBE and GGA-PBESol.

3.2. Dynamical stability

Dynamical stability is a fundamental prerequisite for a crystalline material to exist as a stable phase. A structure is dynamically stable if all phonon frequencies across the entire Brillouin zone are positive (real), meaning no imaginary frequency modes are present. The presence of imaginary (negative) frequencies indicates a structural instability, implying that the crystal would spontaneously distort to a lower-energy configuration.⁴⁰ Phonon dispersion curves were computed along the high-symmetry path G-F-Q-Z-G of the tetragonal Brillouin zone at hydrostatic pressures of 0, 5, 10, and 15 GPa. As clearly shown in Fig. 3, both KZn₄As₃ and CsCd₄As₃ exhibit entirely positive phonon frequencies throughout the Brillouin zone at all investigated pressures, confirming their dynamical stability under these conditions. The absence of imaginary modes at any high-symmetry points or along any reciprocal path provides strong evidence that these compounds are mechanically and dynamically robust across the studied pressure range.⁴¹

Fig. 3. Pressure-dependent phonon dispersion curves of KZn₄As₃ and CsCd₄As₃ at selected pressures.

In Fig. 3, for CsCd₄As₃, the low-frequency acoustic branches highlighted in color for clarity span up to approximately 2.5–2.8 THz near the zone boundary, and the optical branches extend to a maximum of ∼8 THz. For KZn₄As₃, the phonon spectrum is slightly lower in frequency compared to that of CsCd₄As₃, reaching ∼7 THz in the optical region. The acoustic branches display a multi-branch character near the Z point, showing subtle avoided crossings that indicate anharmonic coupling between acoustic and lower optical modes. The observed pressure-induced trends in KZn₄As₃ and CsCd₄As₃, where acoustic phonon branches progressively shift and reveal signatures of lattice stiffening or softening depending on mode character, resonate with the behavior reported for ZrAs₂. In ZrAs₂, Raman and DFT studies demonstrated a crossover from phonon hardening to softening of the lowest A₁g mode above 11 GPa, linked to strong electron-phonon coupling and potentially foreshadowing superconductivity. Similarly, the downward movement of acoustic modes in our compounds suggests pressure-driven phonon softening, which may act as a precursor to dynamical instability or an emergent electronic phase.⁴²

The constant-volume heat capacity (C_V) of CsCd₄As₃ and KZn₄As₃ was calculated from the phonon density of states within the quasi-harmonic approximation (QHA), as a function of temperature (0–1000 K) at pressures of 0, 5, 10, and 15 GPa. The heat capacity provides direct information about the thermal energy storage capacity of a material and is intimately connected to the phonon population and vibrational entropy.⁴³ As shown in Fig. 4, both compounds exhibit the characteristic Debye-like behavior: C_V rises steeply at low temperatures following the T³ law, reflecting the dominant contribution of low-energy acoustic phonons at low temperatures. This T³ dependence is a hallmark of the Debye model and has been reported across a broad class of crystalline solids.^44,45 Above approximately 200–300 K, the heat capacity saturates progressively toward the classical Dulong-Petit limit of 3nR ≈ 48 Cal/cell K (where n is the number of atoms per unit cell, and R is the gas constant), which corresponds to the fully classical equipartition regime in which all phonon modes are thermally populated.

Fig. 4. Heat capacity C_v of KZn₄As₃ and CsCd₄As₃ as a function of temperature at different hydrostatic pressures.

For CsCd₄As₃, the curves at different pressures (0–15 GPa) remain nearly indistinguishable at intermediate temperatures but show a minor divergence above 600 K, with the 15 GPa curve slightly exceeding the 0 GPa curve at elevated temperatures. For KZn₄As₃, the heat capacity curves at all four pressures are essentially superimposed throughout the entire temperature range, indicating that pressure has a negligible effect on C_V in this compound. Only at the highest temperatures (>800 K) does the 15 GPa curve (purple) marginally deviate above the rest. The saturation value for both compounds approaches ∼48 Cal/cell K, consistent with the Dulong-Petit prediction for a compound with 8 atoms per formula unit (48 Cal/mol K). The thermal behavior of these compounds is characteristic of phonon-mediated heat storage, with minimal anharmonic renormalization under pressure. The near pressure-independence of C_V indicates robust thermal stability for both CsCd₄As₃ and KZn₄As₃ across the studied pressure range, making them promising candidates for thermoelectric energy conversion and high-pressure electronic device applications where consistent thermal response under variable operating conditions is essential.^46,47

3.3. Elastic properties

Elastic constants are fundamental for evaluating a material's mechanical stability, stiffness, and resistance to deformation, providing a direct link between microscopic bonding and macroscopic mechanical response since they describe a material's mechanical features and elastic limit under externally applied stress, while also revealing stability, elasticity, toughness, ductility, and anisotropic behavior. At high pressures, where direct experimental measurement becomes difficult, DFT-based elastic constant calculations are of vital importance both for exploring materials' properties and for guiding new materials design, including checking structural stability against Born elastic stability criteria.^48–50 The elastic properties of KZn₄As₃ and CsCd₄As₃ were investigated under hydrostatic pressures of 0, 5, 10, and 15 GPa in order to assess their mechanical stability, stiffness, and potential for high-pressure applications. The second-order elastic constants C_ij were obtained from stress–strain calculations within the optimized crystal structures at each pressure point.⁵¹ In tetragonal crystals, six independent elastic constants C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, and C₆₆ must be determined to describe the material's response to external stresses.⁵² In this work, we employed Stadler's energy method within the framework of density functional theory (DFT) to calculate these constants.⁵³ The approach evaluates the ground-state total energy of a slightly deformed system at fixed volume.⁵⁴ Within Hooke's law, the deformation energy can be expressed as a Taylor series expansion in terms of the deformation parameters (δ_k), the system volume (V), and the elastic constants (C_ij):

1

In the absence of deformation, the elastic constants are determined by

2

3.3.1. The stiffness constant: mechanical stability

For the present tetragonal systems, the independent elastic constants and their pressure evolution are summarized in Table 2. To check whether these compounds are mechanically stable, the calculated elastic constants were tested against the Born stability criteria for tetragonal crystals,⁵⁵ which require C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, (C₁₁ − C₁₂) > 0, (C₁₁ + C₃₃ − 2C₁₃)>0, and (2C₁₁ + C₃₃ + 2C₁₂ + 4C₁₃) > 0. For all considered pressures, these inequalities are satisfied for both KZn₄As₃ and CsCd₄As₃, confirming that the two compounds remain mechanically stable at ambient pressure. At ambient pressure, KZn₄As₃ exhibits C₁₁ = 103.34 GPa and C₃₃ = 50.27 GPa, while CsCd₄As₃ shows C₁₁ = 79.25 GPa and C₃₃ = 40.63 GPa. The larger C₁₁ values compared to C₃₃ indicate stronger atomic bonding within the basal plane than along the stacking direction. The shear-related constants C₄₄ and C₆₆ also increase with pressure, reflecting an enhanced resistance to shear distortions in both materials.

Table 2. Pressure-dependent generalized Born stability criteria for KZn₄As₃ and CsCd₄As₃.

Compound	P (GPa)	C̃ 11 (GPa)	C̃ 11 − C̃12 (GPa)	(C̃11 + C̃12) C̃33 − 2C̃132 (GPa2)	C̃ 44 (GPa)	C̃ 66 (GPa)	Stability
KZn4As3	0	103.34	90.71	5200.0	15.18	14.56	Stable
	5	131.25	96.86	10 325.3	17.57	16.70	Stable
	10	150.37	91.32	15 114.5	16.60	17.70	Stable
	15	168.35	85.54	18 672.7	15.28	18.49	Stable
CsCd4As3	0	79.25	61.91	3167.8	16.45	17.33	Stable
	5	104.96	58.98	5343.1	22.18	23.04	Stable
	10	124.38	51.13	7054.7	23.89	27.17	Stable
	15	137.67	37.23	7256.5	23.74	30.43	Stable

Using the modified elastic constants C̃₁₁ = C₁₁ − P, C̃₃₃ = C₃₃ − P, C̃₄₄ = C₄₄ − P, C̃₆₆ = C₆₆ − P, C̃₁₂ = C₁₂ + P, and C̃₁₃ = C₁₃ + P, we verified the five conditions of C̃₁₁ > 0, C̃₁₁ – C̃₁₂ > 0, (C̃₁₁ + C̃₁₂) C̃₃₃ – 2C̃₁₃² > 0, C̃₄₄ > 0, and C̃₆₆ > 0 at each investigated pressure (0, 5, 10, and 15 GPa). As shown in Table 2, all five criteria are satisfied at every pressure for both KZn₄As₃ and CsCd₄As₃, confirming that both compounds remain mechanically stable throughout the pressure range studied.^56–58

3.3.2. Elastic moduli and mechanical parameters

The polycrystalline bulk (B) and shear (G) moduli were estimated using the Voigt–Reuss–Hill (VRH) approximation.⁵⁹ In the Voigt approximation, uniform strain is assumed throughout the polycrystal, while the Reuss approximation assumes uniform stress; the Hill average provides realistic estimates for bulk polycrystalline properties by averaging the two bounds. For a tetragonal crystal, the Voigt bulk and shear moduli are given by^59,60

3

The Voigt shear modulus is given by

4

The Reuss bulk modulus is given by⁶¹

5

The Reuss shear modulus is given by

6

where

C² = (C₁₁ + C₁²)C₃₃ − 2C²₁₃ 7

and

M = C₁₁ + C₁₂ + 2C₃₃ − 4C₁₃ 8

The following relationships define Poisson's ratio (ν) and Young's modulus (E):

9

10

The effective bulk and shear moduli are found according to the Voigt–Reuss–Hill (VRH) approximation as follows:

11

12

All calculated elastic parameters for KZn₄As₃ and CsCd₄As₃ at different pressures are listed in Tables 3 and4, including the single-crystal elastic constants, polycrystalline moduli, Poisson's ratio, Pugh's ratio, and elastic anisotropy factor, to provide a comprehensive picture of their pressure-dependent mechanical properties.

Table 3. Elastic constants of KZn₄As₃ and CsCd₄As₃ under applied pressures (0–15 GPa).

	KZn4As3				CsCd4As3
	0 GPa	5 GPa	10 GPa	15 GPa	0 GPa	5 GPa	10 GPa	15 GPa
C 11	103.34	136.25	160.37	183.35	79.25	109.96	134.38	152.67
C 12	12.63	29.39	49.05	67.81	17.34	40.98	63.25	85.44
C 13	17.76	39.68	62.34	79.78	19.45	45.23	68.69	88.82
C 22	104.64	136.01	159.33	182.93	77.99	109.59	134.33	153.37
C 33	50.27	91.44	132.15	160.88	40.63	73.83	108.36	136.01
C 44	15.18	22.57	26.60	30.28	16.45	27.18	33.89	38.74
C 55	16.46	22.11	26.65	30	16.46	27.11	33.88	38.74
C 66	14.56	21.70	27.70	33.49	17.33	28.04	37.17	45.43
C 2	5198.97	11 997.12	19 900.98	27 676.92	3168.21	7052.39	11 977.47	16 607.36
M	145.47	189.8	224.35	253.8	100.08	117.68	139.58	154.85

Table 4. Calculated bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (ν), B/G ratio, elastic anisotropy factors (A₁, A₂, and A₃), and anisotropy indices (A_B, A_G, and A^U) of KZn₄As₃ and CsCd₄As₃ under applied pressures (0–15 GPa).

	P (GPa)	B V	B R	B H	G V	G R	G H	E	ν	B/G	A 1 = A2	A 3	A B (%)	A G (%)	A U
KZn4As3	0	39.25	35.73	37.49	22.90	13.08	17.99	46.53	0.29	2.08	0.51	0.31	4.68	27.29	3.85
	5	64.60	63.20	63.90	30.38	19.04	24.71	65.67	0.32	2.58	0.60	0.40	1.09	22.93	2.99
	10	88.92	88.70	88.81	34.79	23.12	28.95	78.35	0.35	3.06	0.63	0.49	0.12	20.14	2.52
	15	109.14	109.05	109.09	38.82	26.78	32.80	89.45	0.36	3.32	0.65	0.58	0.04	18.34	2.24
CsCd4As3	0	34.61	31.65	33.13	19.57	13.39	16.48	42.40	0.286	2.01	0.81	0.56	4.46	18.74	2.40
	5	61.84	59.92	60.88	27.30	19.97	23.63	62.78	0.32	2.57	1.16	0.81	1.57	15.49	1.86
	10	86.48	85.81	86.14	32.75	24.72	28.73	77.58	0.34	2.99	1.28	1.04	0.39	13.96	1.63
	15	107.50	107.24	107.37	36.46	27.94	32.20	87.83	0.36	3.33	1.39	1.34	0.11	13.22	1.52

As pressure rises, the elastic constants also increase, which indicates that the compounds are less likely to undergo shear deformation.⁶² The bulk modulus measures the resistance to volume compression, while the shear and Young's moduli characterize the resistance to shape change and the overall stiffness, respectively.⁶³ The bulk moduli of 37.49 GPa for KZn₄As₃ and 33.13 GPa for CsCd₄As₃ obtained here fall within the range reported for metal iodates, the high-pressure behavior of which is comprehensively reviewed by Liang et al.⁶⁴ In contrast, KZn₄As₃ and CsCd₄As₃ derive their compressibility from covalent Zn–As/Cd–As tetrahedral bonding combined with a more ionic alkali-metal interlayer, with no lone-pair-driven mechanism, explaining why their bulk moduli increase smoothly with pressure rather than showing the discontinuous behavior typical of iodates. As seen from Table 4 and Fig. 5b and c, all three moduli increase with pressure for both KZn₄As₃ and CsCd₄As₃, indicating that their lattices become less compressible and more rigid at higher pressures. The bulk modulus shows the most significant increase, suggesting that compression becomes increasingly difficult as pressure rises. Between the two compounds, KZn₄As₃ has higher values of B, G, and Young's modulus at a given pressure, implying it is mechanically stiffer and less compressible. This pressure-enhanced stiffening trend mirrors that reported for other pressure-stable compounds, where increasing elastic moduli under compression have been linked to exceptional suitability for applications under extreme conditions such as high-pressure sensors, protective coatings, and structural components in deep-earth or aerospace environments.^65,66

Fig. 5. (a–d) Pressure-dependent elastic moduli and Poisson's ratio of KZn₄As₃ and CsCd₄As₃.

3.3.3. Ductility and brittleness

Poisson's ratio serves as a fundamental parameter for describing how a material responds laterally when subjected to uniaxial stress. This coefficient indicates the tendency of a material to expand in directions perpendicular to the applied force. Based on the established criterion, materials with Poisson's ratio exceeding 0.26 are classified as ductile, whereas those with values below this threshold are considered brittle.⁶⁷ For KZn₄As₃, the Poisson's ratio increases steadily with pressure, rising from 0.29 at ambient conditions to 0.36 at 15 GPa. CsCd₄As₃ exhibits an identical trend, with values progressing from 0.28 at 0 GPa to 0.36 at 15 GPa. This upward shift, shown in Fig. 5d, demonstrates that both compounds become progressively more ductile under compression. As pressure intensifies, the materials gain enhanced plasticity, enabling them to absorb greater mechanical energy before fracture occurs. Poisson's ratio serves as a useful indicator of the dominant bonding character in a solid, since covalently bonded materials typically exhibit Poisson's values of 0.1–0.2, ionic compounds fall in an intermediate range around 0.25 and metallically bonded materials approach v = 0.33 or higher.⁶⁷ For our investigated compounds, the v values lie above the ionic threshold and increase steadily with pressure, indicating predominantly mixed ionic-metallic bonding character at ambient pressure, which becomes progressively more metallic-like under compression.

Pugh's ratio, defined as the ratio of bulk modulus (B) to shear modulus (G), provides an additional metric for assessing ductile versus brittle behavior. According to the widely adopted criterion, a B/G value greater than 1.75 signifies ductile behavior, while values below this threshold indicate brittleness.⁶⁸ For KZn₄As₃, Pugh's ratio starts at 2.08 at 0 GPa, which exceeds the ductility threshold, and as pressure increases to 15 GPa, the ratio rises significantly to 3.32, placing the compound firmly within the ductile regime. For CsCd₄As₃ at ambient pressure, the ratio is 2.01, already slightly above the ductility threshold, and rises to 3.33 at 15 GPa. This progression indicates a clear strengthening of ductile character with increasing pressure.

3.3.4. Elastic anisotropy factors

Elastic anisotropy describes the directional dependence of a material's mechanical response to applied stress. In isotropic solids, elastic properties such as bulk modulus, shear modulus, and Young's modulus are identical in all directions. However, in crystals with lower symmetry (such as tetragonal systems), these properties vary depending on the crystallographic orientation;^69,70 deviations from unity imply different stiffness along different crystallographic directions. The calculated anisotropy factors at 0–15 GPa shown in Fig. 6a–d reveal how pressure tunes the directional nature of bonding and may have implications for crack propagation and mechanical reliability.

Fig. 6. (a–d) Pressure dependence of the elastic anisotropy of KZn₄As₃ and CsCd₄As₃: (a) universal anisotropy index (A^U), (b) shear anisotropy factors (A₁ = A₂ and A₃), (c) percentage anisotropy in compressibility (A_B), and (d) percentage anisotropy in shear (A_G).

Three shear anisotropy factors can be defined for a tetragonal crystal. For the {100} shear plane:^71,72

13

for the {010} shear plane:

14

and for the {001} shear plane:

15

A ₁ = A₂ for the {100}/{010} shear planes and A₃ for the {001} plane, with both deviating from the isotropic value of 1 at every pressure, confirming that both compounds are elastically anisotropic. For KZn₄As₃, A₁ and A₃ increase toward unity with pressure from 0.51 to 0.65 and from 0.31 to 0.58, respectively, indicating progressively reduced shear anisotropy under compression. For CsCd₄As₃, A₁ rises above unity between 5 and 10 GPa from 0.81 to 1.39, and A₃ similarly crosses unity, indicating a change in the character of the shear-plane anisotropy partway through the pressure range. Elastic anisotropy is directly relevant to the propensity for microcrack formation and anisotropic thermal expansion in device applications, so this pressure dependence is important for assessing mechanical reliability under operating conditions involving pressure.⁷³

Solid-elastic anisotropy can be quantified using indices such as the percentage anisotropy in bulk and shear moduli (A_B and A_G). These values are calculated from general expressions valid for crystals of any symmetry, as given in eqn (16) and (17), which define compressibility anisotropy (A_B) and shear anisotropy (A_G),⁷⁴ respectively.

16

17

A _B compressibility anisotropy falls sharply with pressure in both compounds, from 4.68% to 0.04% for KZn₄As₃ and from 4.46% to 0.11% for CsCd₄As₃, showing that the volumetric response becomes nearly isotropic under compression. A_G shear anisotropy also decreases substantially with pressure but remains much larger than A_B throughout 27.29% to 18.34% for KZn₄As₃ and 18.74% to 13.22% for CsCd₄As₃, showing that the residual elastic anisotropy at all pressures is predominantly of shear character. This is directly relevant to anisotropic thermal transport, a property of particular interest for the thermoelectric applications already reported for this Zintl-phase family.⁷⁵

Another useful measure of elastic anisotropy is the universal anisotropy index (A^U) proposed by Ranganathan and Ostoja-Starzewski.⁷⁶ This index provides a general estimate of anisotropic elasticity for crystals of any symmetry. For an isotropic solid, A^U = 0, while deviations from zero indicate the degree of anisotropy.

18

A ^U combines both the bulk and shear anisotropy into a single index, with A^U = 0 corresponding to perfect isotropy.⁷⁷A^U decreases substantially with pressure for both compounds from 3.85 to 2.24 for KZn₄As₃ and from 2.40 to 1.52 for CsCd₄As₃, confirming an overall reduction in elastic anisotropy under compression, consistent with the A_B and A_G trends above. Because a lower A^U corresponds to a more isotropic mechanical response, this suggests that the mechanical reliability of both compounds in terms of uniform load-bearing behavior in the polycrystalline form improves under pressure, which may be favorable for pressure-cell or high-pressure device applications where isotropic mechanical response reduces the risk of localized failure.

3.4. Thermodynamic properties

The thermodynamic behavior of KZn₄As₃ and CsCd₄As₃ is characterized through key parameters such as sound velocity, Debye temperature, Grüneisen (γ) parameter and melting temperature, all of which are derived from the elastic constants obtained from first-principles calculations.^78,79 The longitudinal (v₁), transverse (v_t), and average (v_m) sound velocities are significant acoustic properties that substantially influence the thermal conductivity and mechanical behavior of solids. In polycrystalline materials, these velocities can be estimated from the shear modulus (G) and bulk modulus (B) using Navier's relations:⁸⁰

19

The average sound velocity v_m is then obtained from the longitudinal and transverse velocities as follows:

20

The Debye temperature θ_D is a fundamental parameter that links the elastic properties to the thermodynamic behavior of a material. It represents the temperature above which nearly all vibrational modes are excited and is directly related to the strength of interatomic bonds, lattice stiffness, and thermal conductivity. A higher Debye temperature indicates stronger atomic bonding and higher thermal conductivity. The Debye temperature can be determined from the average sound velocity using the Anderson model:⁸¹

21

where h is Planck's constant, k_B is Boltzmann's constant, N_A is Avogadro's number, n is the number of atoms per formula unit, ρ is the density, and M is the molecular weight.

The melting temperature T_m is a critical parameter for assessing the thermal stability and high-temperature applicability of materials. It represents the temperature at which the solid and liquid phases coexist in equilibrium under standard atmospheric pressure. Understanding the melting temperature is essential for determining the operational limits of materials in high-temperature environments. For tetragonal compounds, the melting temperature can be estimated using an empirical relationship that links it to the elastic constants, specifically C₁₁ and C₃₃, which represent the stiffness along the principal axes:^61,82,83

22

The calculated sound velocities, Debye temperature, and melting temperature for KZn₄As₃ and CsCd₄As₃ at different pressures (0, 5, 10, and 15 GPa) are presented in Table 5. For KZn₄As₃, the average sound velocity increases steadily from 2186.91 m s⁻¹ at ambient pressure to 2695.53 m s⁻¹ at 15 GPa. Correspondingly, the Debye temperature rises from 235.41 K to 312.66 K over the same pressure range, while the melting temperature increases from 739.43 K to 1145.37 K. For CsCd₄As₃, the average sound velocity increases from 1903.24 m s⁻¹ at 0 GPa to 2378.59 m s⁻¹ at 15 GPa. For CsCd₄As₃, the Debye temperature increases from 189.36 K to 256.71 K, and the melting temperature rises from 652.70 K to 1016.03 K over the same pressure range. The lower Debye temperature of CsCd₄As₃ compared to KZn₄As₃ at all equivalent pressures indicates that CsCd₄As₃ has weaker interatomic bonding and is expected to exhibit lower lattice thermal conductivity. This trend is consistent with the elastic moduli discussed in Section 3.2, where CsCd₄As₃₃ showed softer mechanical behavior with lower bulk and shear moduli. The higher melting temperature of KZn₄As₃ compared to CsCd₄As₃ at equivalent pressures (739.43 K vs. 652.70 K at 0 GPa and 1145.37 K vs. 1016.03 K at 15 GPa) suggests that KZn₄As₃ possesses stronger interatomic bonding and can withstand higher operating temperatures, making it more suitable for high-temperature applications. However, the melting temperatures were estimated using an empirical relationship based on the elastic constants, and no experimental melting data are currently available for these compounds. We now emphasize that these predictions should be regarded as guidance rather than definitive values, and that experimental verification would require challenging high-pressure/high-temperature measurements, such as laser-heating studies.⁸⁴ We have also noted that our computational estimates can provide useful motivation for future experimental investigations aimed at validating the thermal stability of these materials.

Table 5. Calculated average sound velocity (v_m in m s⁻¹), Debye temperature (θᴅ in K), Grüneisen parameter (γ) and melting temperature (T_m in K) of KZn₄As₃ and CsCd₄As₃ at different pressures.

	KZn4As3				CsCd4As3
	0 GPa	5 GPa	10 GPa	15 GPa	0 GPa	10 GPa	5 GPa	15 GPa
Sound velocity (m s−1)	2186.91	2456.12	2596.15	2695.53	1903.24	2310.04	2168.24	2378.59
Debye temperature (K)	235.41	273.85	296	312.66	189.36	244.99	224.56	256.71
Melting temperature	739.43	899.91	1033.34	1145.37	652.7	794.63	919.68	1016.03
Grüneisen parameter (γ)	1.73	1.96	2.15	2.24	1.69	1.96	2.13	2.25

The pressure-induced increase in sound velocity, Debye temperature, and melting temperature shown in Fig. 7a–c for both compounds reflects the progressive stiffening of the lattice under compression. As external pressure is applied, interatomic distances decrease, leading to stronger force constants between neighboring atoms. This results in faster propagation of acoustic waves, higher sound velocity, a higher Debye temperature, and an elevated melting point, indicating that the materials become thermally more robust at elevated pressures.

Fig. 7. (a–d): Pressure dependence of sound velocity, Debye temperature, melting temperature and Grüneisen (γ) parameter, respectively, for KZn₄As₃ and CsCd₄As₃.

The Grüneisen parameter (γ) is a key thermodynamic quantity that varies with both volume and temperature, serving as a direct measure of phonon–phonon interaction strength within a crystal. It governs several critical material properties, including thermal conductivity, thermal expansion, the temperature dependence of elastic moduli, acoustic attenuation, and bond anharmonicity.⁸⁵ Larger γ values signify stronger anharmonic effects, typically associated with reduced thermal conductivity and heightened sensitivity of thermal behavior to changes in temperature and pressure. This parameter can be evaluated from Poisson's ratio using the following relation.⁸⁶

23

For the studied compounds, the calculated γ values listed in Table 5 increase steadily with pressure, from 1.73 to 2.24 for KZn₄As₃ and from 1.69 to 2.25 for CsCd₄As₃ across the 0–15 GPa range. From Fig. 7d, the results suggest that both compounds exhibit significant and progressively increasing anharmonicity under compression, implying a corresponding reduction in lattice thermal conductivity with increasing pressure. At ambient pressure, CsCd₄As₃ shows a marginally lower γ (1.69) than KZn₄As₃ (1.73), indicating slightly weaker phonon–phonon scattering and thus somewhat higher thermal transport efficiency at low pressure; however, the two compounds converge closely by 15 GPa to 2.24 and 2.25, respectively, indicating that their anharmonic behavior becomes nearly indistinguishable under strong compression. This pressure-induced increase in γ, and the corresponding suppression of lattice thermal conductivity, has direct relevance for the practical application of these compounds.⁸⁷ Reduced lattice thermal conductivity is a generally desirable property for thermoelectric materials, since the thermoelectric figure of merit ZT is inversely proportional to the total thermal conductivity. The present results therefore suggest that hydrostatic pressure could serve as a means of further suppressing lattice thermal conductivity and thereby enhancing the thermoelectric performance of this class of Zintl-phase compounds.^88,89

3.5. Electronic properties

The electronic band structure of KZn₄As₃ and CsCd₄As₃ provides crucial information about their electronic properties, including their electrical conductivity and response to external forces. Finding out the band structure of a material helps determine whether it is metallic, semi-metallic, or insulating.⁹⁰ The electronic band gap is one of the most important parameters governing the technological applicability of a semiconducting material, directly determining its optical absorption edge, its suitability for photovoltaic or photodetector applications, and its electrical transport behavior. In the present work, the electronic band gaps of KZn₄As₃ and CsCd₄As₃ were calculated at 0, 5, 10, and 15 GPa using both the standard GGA-PBE exchange-correlation functional and the screened hybrid HSE06 functional⁹¹ (Table 6). These band gap values enable the evaluation of the functional dependence of the predicted electronic character and its evolution under hydrostatic pressure. The band structures were computed along the high-symmetry directions G-F-Q-Z-G of the tetragonal Brillouin zone, and the results are presented in Fig. 8 and 9.

Table 6. Calculated band gap using the GGA-PBE and HSE06 functionals for KZn₄As₃ and CsCd₄As₃ at different pressures.

	KZn4As3				CsCd4As3
	0 GPa	5 GPa	10 GPa	15 GPa	0 GPa	10 GPa	5 GPa	15 GPa	Other work
GGA-PBE	—	—	—	—	0.258	0.007	—	—	WC-GGA
									1.112 (ref. 96)
HSE06	0.921	0.796	0.677	0.557	0.741	0.574	0.434	0.295	0.6697

Fig. 8. (a–h) Pressure-dependent electronic band structures of (a–d) KZn₄As₃ and (e–h) CsCd₄As₃ using the GGA-PBE functional.

Fig. 9. (a–h) Pressure-dependent electronic band structures of (a–d) KZn₄As₃ and (e–h) CsCd₄As₃ using the HSE06 functional.

It is well established in the electronic-structure literature that semi-local functionals such as GGA-PBE systematically underestimate the fundamental band gap of semiconductors and insulators, a consequence of the self-interaction error and the resulting derivative discontinuity error inherent to these functionals. This underestimation is not merely quantitative: for materials with intrinsically narrow gaps, GGA-PBE frequently predicts a qualitatively incorrect metallic ground state where the true material is in fact a semiconductor. Hybrid functionals, such as HSE06, which incorporate a fraction of exact Hartree–Fock exchange over a screened short-range interaction while retaining GGA-level exchange-correlation at long range, substantially correct this underestimation and typically yield band gaps in much closer quantitative agreement with experiment across a wide range of semiconductor families.^92–95 This well-documented functional dependence is precisely the behavior observed for the present compounds, and provides an important cautionary and instructive example for the interpretation of the electronic structure of this material class.

At the GGA-PBE level, KZn₄As₃ is predicted to be metallic at every pressure investigated across the full 0–15 GPa range, which would, taken at face value, imply that this compound has no semiconducting character whatsoever under any of the conditions studied. However, the HSE06 calculation reveals a substantial gap of 0.921 eV at ambient pressure, a value that places the material within the optimal range for infrared detection and low-bandgap thermoelectric energy conversion.^97,98 Also, the band gap of 0.921 eV for KZn₄As₃ at ambient pressure is ideal for infrared photodetectors operating in the short-wave infrared (SWIR) region, which is crucial for telecommunications and night-vision technologies.⁹² This gap narrows steadily and monotonically with increasing pressure, decreasing to 0.557 eV at 15 GPa. Importantly, the tunability down to 0.557 eV at 15 GPa extends detection into the mid-wave infrared (MWIR) region, underscoring the potential of these materials for pressure-controlled infrared sensing applications.⁹⁷ The complete disappearance of the gap at the GGA-PBE level, despite a clear and substantial HSE06 gap of well over 0.5 eV persisting even at the highest pressure studied, is a striking and instructive illustration of the well-documented tendency of GGA functionals to qualitatively misinterpret narrow and moderate gap semiconductors as metals. This consistent reduction suggests that pressure induces a gradual closure of the band gap, potentially driving the material toward a semi-metallic or topological phase transition at higher pressures, a phenomenon observed in related Zintl compounds such as AEIn₂As₂bib99.⁹⁹

Similarly, for CsCd₄As₃, the GGA-PBE functional initially predicts a small but clearly non-zero gap of 0.258 eV at ambient pressure, already substantially smaller than the corresponding HSE06 value, consistent with the general tendency of GGA to underestimate rather than fully eliminate the gap in wider-gap systems. However, this already-underestimated gap collapses rapidly under pressure: it falls to a near-vanishing 0.007 eV by 5 GPa, and is predicted to be fully closed metallic by both 10 and 15 GPa. The HSE06 functional, by contrast, predicts a substantially larger ambient-pressure gap of 0.741 eV, nearly three times the GGA-PBE value, which narrows smoothly and monotonically with pressure to 0.574 eV at 5 GPa and to 0.295 eV at 15 GPa. The significant reduction of the gap to 0.295 eV at 15 GPa suggests that, under extreme conditions, this material could exhibit semi-metallic behavior with high carrier mobility, beneficial for specific high-speed electronic applications, though likely detrimental to thermoelectric efficiency due to increased bipolar conduction.⁹⁷ Even at the highest pressure studied, the HSE06-predicted gap remains, which is a clear qualitative contrast to the band gap predicted by GGA-PBE beyond 5 GPa. This discrepancy underscores the importance of employing a hybrid or otherwise corrected functional when the electronic semiconducting versus metallic nature of a compound is itself the property of interest, rather than relying on GGA-PBE alone.^100,101

The total density of states (TDOS) reveals how electronic states are distributed across different energy levels and serves as a fundamental descriptor of a material's electronic nature.¹⁰² The behavior of states around the Fermi level (E_F) directly influences electrical conductivity, magnetic response, and potential superconducting characteristics. The TDOS of KZn₄As₃ and CsCd₄As₃ under hydrostatic pressures of 0, 5, 10, and 15 GPa are illustrated in Fig. 10a and b. When pressure is applied, the TDOS undergoes systematic modifications. The most notable change is observed at the Fermi level: N(E_F) decreases progressively with increasing pressure for both KZn₄As₃ and CsCd₄As₃. This reduction in the density of available electronic states at E_F indicates that compression gradually suppresses the electronic states near the chemical potential.

Fig. 10. Projected density of states of (a) KZn₄As₃ and (b) CsCd₄As₃ under hydrostatic pressure.

3.6. Optical properties

The calculation of optical features of a material is necessary to assess its optoelectronic and photovoltaic applications.^103,104 The optical parameters such as dielectric function (with both real and imaginary parts), conductivity, absorption coefficient, reflectivity and loss function of KZn₄As₃ and CsCd₄As₃ were computed for pressures between 0 GPa and 15 GPa. All the parameters are presented in Fig. 11 as a function of energy. Fig. 11 presents numerous physical properties of the materials. Fig. 11a presents the real part of the dielectric function ε₁(ω) for both compounds. The value at zero frequency, ε₁(0), is called the static dielectric function (SDF). The SDF for KZn₄As₃ is found to be about 11.95 at 0 GPa and about 11.83 at 15 GPa, while for CsCd₄As₃ it is about 23.75 at 0 GPa and about 13.78 at 15 GPa. Under hydrostatic pressure, the static dielectric function decreases slightly for both compounds. The larger value of SDF for CsCd₄As₃ compared to KZn₄As₃ also signifies its stronger polarization.¹⁰⁵ Another study on Zintl phases XIn₂Z₂ (X = Ca and Sr; Z = As and Sb) reported SDF values of 10.1 and 12.61 for CaIn₂As₂ and SrIn₂As₂, and 12.73, 14.24, 12.76, and 14.27 for CaIn₂Sb₂ and SrIn₂Sb₂, respectively.¹⁰⁶

Fig. 11. (a–f) Calculated optical properties of KZn₄As₃ and CsCd₄As₃ at 0, 5, 10, and 15 GPa: (a) real part of the dielectric function, (b) imaginary part of the dielectric function, (c) optical conductivity, (d) absorption coefficient, (e) reflectivity, and (f) loss function.

After reaching their static values, the spectra of ε₁(ω) grow and reach maximum peaks at specific photon energies. For KZn₄As₃ at 0 GPa, a maximum peak is observed around 2.06 eV, while for CsCd₄As₃ the peak appears near 0.01 eV. Beyond these peaks, the real dielectric function decreases and eventually becomes negative at higher energies. When ε₁(ω) becomes negative, the material begins to exhibit metallic behavior, completely reflecting incident electromagnetic waves in those energy regions.¹⁰⁷ The threshold energy for negative ε₁(ω) shifts slightly with pressure, indicating a modification of the electronic structure under compression.

Fig. 11b represents the imaginary part of the dielectric function, which determines the optical absorption of the material. The maximum peak is found around 3.56 eV at 0 GPa for KZn₄As₃, and for CsCd₄As₃ the peak energy is observed at 2.47 eV at 15 GPa. For both compounds, the imaginary part ε₂(ω) shows a pronounced absorption peak in the near-ultraviolet region (4–6 eV), while the lower-energy region below about 3 eV does not exhibit any clear peak. Beyond this range, ε₂(ω) generally decreases but remains extended over higher photon energies, indicating that the optical absorption is stronger in the ultraviolet and vacuum ultraviolet regions for both compounds. Previously reported Zintl phases show varying absorption characteristics. Our KZn₄As₃ exhibits a slightly lower peak position (3.66 eV at 0 GPa) that increases to 4.23 eV at 15 GPa, demonstrating pressure-induced tunability not reported for CsCd₄As₃. For XIn₂Z₂ compounds (X = Ca and Sr; Z = As and Sb), the ε₂(ω) peaks were observed in the range of 2.86–4.00 eV,¹⁰⁶ which aligns well with our findings at lower pressures. For K₃Ag₃As₂, the imaginary dielectric function exhibited a large absorption peak between 0.6 and 3.78 eV, covering the visible to near-infrared region.¹⁰⁸ Our compounds show absorption at relatively high energies (4–6 eV), making them more suitable for ultraviolet optoelectronic applications. For BaMg₂P₂, the ε₂(ω) peaks were observed at 1.7–2.4 eV and 5.1–5.6 eV, depending on the functional used.¹⁰⁹ The higher-energy peaks in BaMg₂P₂ (∼5.6 eV with mBJ) align well with our observed UV absorption peaks.

Fig. 11c shows the optical conductivity spectra of KZn₄As₃ and CsCd₄As₃ as a function of photon energy. In both compounds, the conductivity is very small in the low-energy region and then increases sharply with rising energy, reaching pronounced maxima in the ultraviolet region, as shown in Table 7. For KZn₄As₃, the conductivity peak appears at a lower energy than in CsCd₄As₃, and its magnitude is also slightly more sensitive to pressure. By contrast, CsCd₄As₃ shows a broader and stronger high-energy response, indicating delayed but more extended photon-driven carrier excitation. This behavior signifies that those optical transitions become more probable at higher energies, confirming stronger photon absorption and improved interband excitation in the ultraviolet range. The optical conductivity of BaMg₂P₂ was reported to follow a pattern similar to its dielectric function, with maximum values observed at higher photon energies.¹⁰⁹ Our compounds exhibit a comparable trend, with conductivity increasing sharply above the band edge. For K₃Ag₃As₂, the optical response was strongest in the visible region (0.6–3.78 eV), making it suitable for photovoltaics.¹¹⁰ In contrast, our KZn₄As₃ and CsCd₄As₃ show maximum conductivity in the UV region, making them better suited for UV photodetectors and high-energy optoelectronic devices. The pressure sensitivity of our compounds, particularly KZn₄As₃, offers a tunability advantage not reported for K₃Ag₃As₂ or BaMg₂P₂. For CsCd₄As₃, the conductivity showed minimal pressure dependence,¹¹¹ whereas our KZn₄As₃ exhibits a more pronounced pressure response.

Table 7. Static dielectric, static reflectivity, maximum dielectric function, and conductivity of Zintl compounds KZn₄As₃ and CsCd₄As₃.

	KZn4As3				CsCd4As3
	0 GPa	5 GPa	10 GPa	15 GPa	0 GPa	5 GPa	10 GPa	15 GPa
ε 1(0)	11.95	11.69	11.47	11.83	23.75	13.56	13.45	13.78
ε 1(Max)	16.65	17.51	17.58	17.50	23.14	16.21	16.79	17.14
Max conductivity	8.43 (3.66 eV)	10.21 (4.14 eV)	11.17 (4.55 eV)	11.84 (4.52 eV)	6.21 (3.02 eV)	7.42 (3.56 eV)	8.70 (3.92 eV)	9.09 (4.31)
Static reflectivity R(0)	0.30	0.29	0.30	0.30	0.43	0.32	0.32	0.33

Fig. 11d presents the absorption coefficient spectra of the two compounds. The absorption coefficient remains weak in the low-energy region, then rises rapidly with photon energy and reaches strong peaks in the ultraviolet region. For both materials, the absorption edge shifts with pressure, showing that compression modifies the electronic structure and lowers or changes the energy required for optical excitation. KZn₄As₃ generally exhibits an earlier onset of absorption, whereas CsCd₄As₃ shows a stronger absorption response at higher photon energies. This comparison suggests that both materials are efficient ultraviolet absorbers, but KZn₄As₃ is more favorable for lower-energy optical activity, while CsCd₄As₃ is better suited for higher-energy ultraviolet harvesting. The absorption coefficient of K₃Ag₃As₂ showed strong absorption from 0.6 eV to 20.0 eV, with dispersion shoulders at 1.21 eV and 1.67 eV, covering a broad spectral range from visible to UV.¹¹⁰ Our compounds show more concentrated absorption in the UV region (4–6 eV), which is advantageous for applications requiring selective UV absorption without visible-light interference. For BaMg₂P₂, the absorption increased gradually with energy, with maximum absorption observed in the UV region.¹⁰⁹ For CsZn₄P₃ and CsZn₄P₃, the maximum absorption coefficients were reported as 184 × 10⁴ cm⁻¹ and 162 × 10⁴ cm⁻¹, respectively.¹¹¹ Our compounds exhibit comparable absorption coefficients, with the advantage of pressure-induced tunability. For XIn₂Z₂ compounds, absorption was primarily active in the infrared region (3–6 eV),¹⁰⁶ whereas our compounds show a broader absorption profile.

Fig. 11e illustrates the reflectivity spectra. In the low-energy region, both compounds show relatively modest reflectivity, followed by a rise or fluctuation with increasing photon energy, particularly in the ultraviolet range. KZn₄As₃ and CsZn₄P₃ both exhibit pressure-dependent changes in reflectivity, but CsZn₄P₃ tends to maintain a stronger reflectance over a wider energy interval. The reflectivity behavior also indicates that the materials do not strongly reflect low-energy light, while their optical response becomes more complex at higher energies. This signifies that both compounds may have useful surface optical behavior, with CsCd₄As₃ offering stronger interaction with incident radiation. For K₃Ag₃As₂, strong reflectivity characteristics were observed in the lower energy region (0–10 eV) compared to the higher energy region.¹¹⁰ Our compounds show modest reflectivity across the spectrum, which is advantageous for antireflective coating applications. BaMg₂P₂ exhibited reflectivity patterns closely following its dielectric function, with maximum values in the UV region.¹⁰⁹ For CsZn₄P₃ and CsZn₄As₃, the static reflectivity R(0) was reported as 0.22 and 0.28, respectively.¹¹¹ For XIn₂Z₂ compounds, static reflectivity was reported in the range of 50–60% for arsenides and 50–55% for antimonides.¹⁰⁶ Our compounds exhibit lower static reflectivity, which is advantageous for minimizing reflection losses in optical devices.

Fig. 11f shows the energy loss function, which describes the energy dissipation of fast electrons traveling through the materials. In both KZn₄As₃ and CsCd₄As₃, the loss function exhibits peaks at higher photon energies, while its value remains small in the low-energy region. The peak positions shift slightly with pressure, reflecting changes in the plasma frequency and electronic screening. Compared with KZn₄As₃, CsCd₄As₃ generally displays a stronger or broader loss feature at high energy, suggesting a more pronounced plasmonic response. This means that the plasma oscillation and collective electron excitation are more significant in the high-energy region, which is important for understanding the electron dynamics of these compounds. The energy loss function for K₃Ag₃As₂ showed strong energy loss between 5.0 eV and 17.0 eV, with a peak in the UV region.¹¹⁰ Our compounds exhibit loss peaks in a similar energy range, indicating comparable plasmonic behavior. For BaMg₂P₂, the maximum energy loss peaks were observed at 12.15–12.97 eV, depending on the functional.¹⁰⁹ For CsZn₄P₃ and CsZn₄As₃, the energy loss peaks were observed at approximately 10.27 eV and 4.82 eV, respectively.¹¹¹ Our compounds show loss peaks in a comparable range. The pressure-induced shift in loss peak position observed in our compounds is a novel feature not reported in previous Zintl studies.

4. Conclusion

This study presents a first-principles investigation of the pressure-dependent structural, dynamical, elastic, thermodynamic, electronic, and optical properties of the tetragonal Zintl-phase arsenides KZn₄As₃ and CsCd₄As₃ across the 0–15 GPa range, using the CASTEP plane-wave DFT code with the GGA-PBE, GGA-PBEsol, and HSE06 functionals. The optimized ambient-pressure lattice parameters agree closely with available experimental and isostructural analogue data, lending confidence to the predicted trends. Both compounds undergo substantial, anisotropic lattice contraction under compression, with unit-cell volumes decreasing by roughly 22–24% between 0 and 15 GPa. Phonon dispersion and Born stability analysis confirm that both compounds remain dynamically and mechanically stable throughout the studied pressure range, with the bulk, shear, and Young's moduli increasing steadily with pressure; a pressure-induced stiffening that, together with the reduction in elastic anisotropy, supports the suitability of both compounds for structural components in high-pressure sensors and protective coatings operating under variable mechanical load. The pressure-driven increase in Poisson's and Pugh's ratios indicates a transition toward more ductile, fracture-resistant behavior, which is favorable for the mechanical reliability of pressure-cell or high-pressure device applications. The steady rise in Debye temperature and melting point, combined with the near pressure-independence of the heat capacity, points to robust thermal stability suited to high-temperature and high-pressure electronic device environments, while the pressure-induced increase in the Grüneisen parameter and the corresponding suppression of lattice thermal conductivity suggest that hydrostatic pressure could be used as a tuning parameter to enhance the thermoelectric figure of merit (ZT) of this Zintl-phase family. The electronic band structure determined using HSE06 reveals a sizeable, pressure-tunable band gap for both compounds, with KZn₄As₃ (0.921–0.557 eV) and CsCd₄As₃ (0.741–0.295 eV) spanning the short-wave to mid-wave infrared range; consequently, they are deemed suitable candidates for pressure-tunable infrared photodetectors relevant to telecommunications and night-vision applications. The pronounced optical absorption, conductivity, and energy-loss features concentrated in the near-UV to UV region (4–6 eV), together with comparatively low static reflectivity, further identify both compounds as promising candidates for selective ultraviolet photodetectors and UV-harvesting optoelectronic devices, rather than visible-light photovoltaics. Taken together, these results establish KZn₄As₃ and CsCd₄As₃ as mechanically robust, thermally stable, and electronically and optically tunable multifunctional Zintl phases with clear potential for high-pressure structural applications, infrared and ultraviolet optoelectronic devices, and pressure-enhanced thermoelectric energy conversion.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The data supporting the findings of this study are available from the corresponding author upon reasonable request. No restrictions apply to data sharing.