Pressure-induced band-gap evolution governed by competing orbital coupling and lattice anharmonicity in Cu/Ag-stuffed zinc-blende X₂YZ compounds

Abstract

Lattice contraction has attracted extensive attention in high-pressure research because of its dominant influence on phonon and electron transport properties. However, how hydrostatic pressure regulates chemical bonding and orbital coupling strengths remains under debate. We systematically investigate the pressure-dependent band-gap evolution of stuffed zinc-blende X₂YZ compounds (X = Li, Na; Y = Cu, Ag; Z = As, Sb). We show that compression has little effect on the semimetallic character of Na₂AgAs. The band gap either increases or remains nearly unchanged under pressure, governed by the competition between interband s−p coupling and intraband p−d coupling within the nonstandard tetrahedral coordination. Beyond gap modulation, pressure also reconstructs the electronic dispersion near the Fermi level. In Li₂CuAs, compression induces convergence of multiple conduction-band valleys, thereby enhancing the power factor. Notably, Li₂CuAs exhibits an unusually low lattice thermal conductivity of 0.74 W·m⁻¹·K⁻¹ due to enhanced four-phonon scattering, contrary to the conventional expectation that lightweight compounds possess high thermal conductivity. Consequently, n-type Li₂CuAs reaches an optimal ZT of 3.0 at 700 K under pressure. These findings establish a unified microscopic framework for pressure-driven band-gap modulation and provide guidance for the design of strain-tunable semiconductors based on stuffed zinc-blende frameworks.

Graphical abstract

Keywords

pressure / band gap / power factor

1 Introduction

The electronic band gap (E_g) is a key parameter that governs the performance of transistors, photodetectors, sensors, and thermoelectric (TE) materials [1−6]. In thermoelectric energy conversion, E_g and the dispersion of electronic states near the Fermi level strongly affect electrical conductivity and thermopower, thereby determining the overall conversion efficiency. This efficiency is commonly described by the dimensionless figure of merit ZT = S²σT/(κ_e+κ_L) [7], where S, σ, and T are the Seebeck coefficient, electrical conductivity, and absolute temperature, respectively, while κ_e and κ_L are the electronic and lattice contributions to thermal conductivity. A suitably small E_g is favorable for optimizing the thermoelectric power factor [8, 9]. Previous studies have shown that semimetals, despite the absence of an E_g, often exhibit higher intrinsic carrier concentrations and mobilities than heavily doped semiconductors used in thermoelectric applications [10, 11]. In addition, lattice distortions and electronic correlations cooperatively modulate the E_g and reshape the electronic structure [12−17]. Therefore, rational control of the E_g and a fundamental understanding of its coupling to lattice distortion are crucial for designing high-performance thermoelectric materials.

The E_g can be tuned by external perturbations, among which thermal expansion is one of the most direct approaches [18, 19]. In general, E_g decreases with increasing temperature according to the empirical Varshni relation [18]. However, in chalcopyrite-type ZnSnAs₂ [20], E_g first increases and then decreases with temperature, whereas in CuCl [21], it increases monotonically. These unusual trends arise because temperature simultaneously modifies electron-phonon interactions and lattice expansion [22, 23]. The thermal response is typically slow and inefficient. This limits dynamic, selective, and precise control of the electronic structure. Although empirical data on the temperature dependence of E_g have been reported [18], establishing clear correlations remains challenging. In particular, the relationships between thermal effects and fundamental factors such as elemental composition and crystal symmetry are still not well understood.

Hydrostatic pressure provides an effective route for precise tuning of the E_g in many systems. As a thermodynamic variable, pressure can reversibly modify both electronic and phononic properties [24, 25]. In diamond- and zinc-blende-type semiconductors, moderate pressure modifies valence-orbital energies and interorbital coupling. This can induce lattice distortions and shift the valence-band maximum and conduction-band minimum [26−28]. In situ high-pressure ultrafast pump-probe spectroscopy has been used to probe pressure-driven changes in the electronic structure of SnSe [3, 29]. These studies show that pressure not only adjusts the band-gap magnitude, but also reshapes the band dispersion near the Fermi level through lattice deformation [30−32]. Moreover, pressure-induced shifts of the Fermi level strongly influence electrical conductivity and the Seebeck coefficient, providing an effective pathway to optimize thermoelectric performance. However, a systematic and in-depth understanding of the microscopic coupling between the pressure-dependent E_g and lattice distortion remains limited.

In this work, we investigate the evolution of the E_g in X₂YZ compounds (X = Li, Na; Y = Cu, Ag; Z = As, Sb) under lattice contraction. The nonmonotonic E_g behavior originates from the competition between interband s−p and intraband p−d orbital coupling in non-ideal tetrahedral Cu/Ag-stuffed zinc-blende structures. Pressure also renormalizes the conduction-band valleys near the Fermi level. In Li₂CuAs, multiple conduction valleys become energetically aligned under compression, which mitigates the reduction in electrical conductivity during band-gap widening. In contrast, Li₂CuSb is dominated by a single conduction valley, resulting in inferior electron transport performance. Although Li₂CuAs exhibits stronger chemical bonding and higher phonon group velocities than Li₂CuSb, its higher phonon frequencies widen the phonon gap and enhance four-phonon scattering, leading to a lower lattice thermal conductivity of 0.74 W·m⁻¹·K⁻¹ at 300 K. As a result of the favorable balance between electronic and lattice transport properties, Li₂CuAs achieves a ZT of approximately 3.0 at 700 K.

2 Computational methodology

Our first-principles calculations were performed within density functional theory (DFT) [33, 34], as implemented in the Vienna ab initio Simulation Package (VASP) [35, 36] using projector augmented-wave pseudopotentials. For Brillouin zone sampling, a plane-wave energy cutoff of 550 eV and Γ-centered k-point meshes with a resolution of 2π × 0.05 Å⁻¹ were adopted, ensuring convergence of 10⁻⁷ eV in the self-consistent calculations. To account for electron−electron correlations involving d orbitals in copper- and silver-based compounds, structure optimizations and electronic band structures were computed using the Heyd−Scuseria−Ernzerhof (HSE) [37, 38] hybrid functional, with the fraction of nonlocal Fock exchange set to 0.25.

The phonon dispersions, second-order interatomic force constants (IFCs), and third-order IFCs were calculated using the HIPHIVE Python package [39], which generates results through a Monte Carlo-based random displacement method to fit a cluster expansion model to the training data. We used a simple Monte Carlo procedure to generate 100 configurations, which imposed a lower limit on the interatomic distances in the final structure. Subsequently, 90 configurations were selected to fit both second-order and third-order IFCs. The lattice thermal conductivity, based on three-phonon (3ph) scattering, is predicted by solving the phonon Boltzmann transport equation using the SHENGBTE code [40]. Moreover, the effect of four-phonon scattering (4ph) on thermal transport is verified using the FOURPHONON package [41]. Detailed convergence tests of lattice thermal conductivity with respect to sample size and Ngrid density for the three-phonon (3ph) and combined three- and four-phonon (3+4ph) processes at 300 K are presented in Fig. S1 of the Supplemental Material.

The electronic transport coefficients are calculated by solving the electronic Boltzmann transport equation using the AMSET software package [42]. The scattering mechanism-dependent transport properties were considered via three different processes, including acoustic deformation potential (ADP), ionized impurity (IMP), and polar optical phonon (POP) scattering. Accordingly, ADP and IMP are treated as elastic scattering processes, while POP is treated as an inelastic scattering process. The elastic scattering rate from state |n\mathit{k}⟩ to state |m\mathit{k}+\mathit{q}⟩ is calculated using the momentum relaxation time approximation [43]

(1)\begin{array}{rl}{\tilde{\tau }}_{nk}^{-1}=& \frac{2\pi }{\hslash }\sum _m\int \frac{{\mathrm{d}}^3\mathit{q}}{\mathrm{\Omega }}\left(1-\frac{{\mathit{v}}_{n\mathit{k}}\cdot {\mathit{v}}_{m\mathit{k}+\mathit{q}}}{{|{\mathit{v}}_{n\mathit{k}}|}^2}\right)\\ & \times {\left|g_{nm}\left(\mathit{k},\mathit{q}\right)\right|}^2\delta \left({\epsilon }_{n\mathit{k}}-{\epsilon }_{m\mathit{k}+\mathit{q}}\right),\end{array}

where \mathrm{\Omega } denotes the volume of the Brillouin zone, {\mathit{v}}_{n\mathit{k}} and {\epsilon }_{n\mathit{k}} denote the group velocity and energy of the state |n\mathit{k}⟩, respectively, and g_{nm}\left(\mathit{k},\mathit{q}\right) is the electron−phonon coupling matrix associated with different scattering mechanisms. For inelastic scattering, the self-energy relaxation time approximation is employed [44]:

(2)\begin{array}{rl}{\tilde{\tau }}_{nk}^{-1}=& \frac{2\pi }{\hslash }\sum _m\int \frac{{\mathrm{d}}^3\mathit{q}}{\mathrm{\Omega }}{\left|g_{nm}\left(\mathit{k},\mathit{q}\right)\right|}^2\\ & \times (n_{\mathit{q}}+1-f_{m\mathit{k}+\mathit{q}})\delta \left({\epsilon }_{n\mathit{k}}-{\epsilon }_{m\mathit{k}+\mathit{q}}-\hslash {\omega }_{po}\right)\\ & \left(n_{\mathit{q}}+f_{m\mathit{k}+\mathit{q}}\right)\delta \left({\epsilon }_{n\mathit{k}}-{\epsilon }_{m\mathit{k}+\mathit{q}}+\hslash {\omega }_{po}\right),\end{array}

where n_{\mathit{q}} is the Bose−Einstein distribution function, f is the Fermi−Dirac distribution function, {\omega }_{po} is an effective phonon frequency, and \pm \hslash {\omega }_{po} corresponds to scattering via phonon absorption and emission, respectively. The total scattering rate is calculated using Matthiessen’s rule [45]:

(3){\displaystyle \frac{1}{{\tau }_e}}={\displaystyle \frac{1}{{\tau }_e^{\mathrm{A}\mathrm{D}\mathrm{P}}}}+{\displaystyle \frac{1}{{\tau }_e^{\mathrm{I}\mathrm{M}\mathrm{P}}}}+{\displaystyle \frac{1}{{\tau }_e^{\mathrm{P}\mathrm{O}\mathrm{P}}}}.

After the scattering rates were determined, the BoltzTrap2 code [46] was used to interpolate electron eigenvalues to compute transport parameters.

3 Results and discussion

The X₂YZ compounds (X = Li, Na; Y = Cu, Ag; Z = As, Sb) crystallize in cubic symmetry with the F\overline{4}3m space group under ambient conditions, as shown in Fig. 1(a). Y and Z atoms alternate to form a zinc-blende-like sublattice with a four-connected network, while the remaining two equivalent sites are partially occupied by X atoms, forming stuffed zinc-blende structures derived from conventional half-Heusler phases [47−49]. The chemical bonding in the stuffed zinc-blende structure is characterized by a combination of strong ionic and metallic interactions. The electronic charge density is mainly localized around the As or Sb atoms. The low electronegativity of Li or Na leads to pronounced charge delocalization, whereas Cu or Ag atoms exhibit mixed localization and delocalization behavior, with Cu showing slightly stronger localization than Ag (Fig. S2). The bond heterogeneity introduces pronounced lattice anharmonicity, which suppresses phonon transport and simultaneously affects the electronic properties [50]. In the Y-centered (Y = Cu, Ag) tetrahedral clusters, the local tetragonal distortion in the X₂YZ compounds is primarily governed by the competition among Y−Z bond lengths, bond strengths, and their spatial orientations. The Li−Li bond is the weakest among all atomic interactions (Fig. S3).

As depicted in Fig. 1(b) and Fig. S4, the Cu/Ag-s and As/Sb-s orbitals primarily contribute to the unoccupied states, whereas the Cu/Ag-d and As/Sb-p orbitals dominate the occupied states near the band edge. In the ideal T_d point group, the occupied d orbitals split into doubly degenerate e and triply degenerate t₂ states. According to group theory analysis, only p−d coupling is symmetry-allowed in the occupied states. However, pressure-induced volume contraction and lattice distortion generate dynamic off-centering of the Y-centered (Y = Cu, Ag) tetrahedral clusters, thereby lowering the local symmetry, as shown in Fig. 1(c). For instance, when a Cu atom deviates from the tetrahedral center, the T_d symmetry is reduced to C_3v, leading to the splitting of the triply degenerate t₂ states into e and a₁ components [51−55]. Because s−p and p−d states belong to compatible irreducible representations, coupling can occur both between the valence and conduction bands and within the valence band.

The s−p interaction lowers the energy of occupied states, whereas p−d coupling within the valence band raises the energy of the valence states. As pressure increases, lattice contraction shortens bond lengths and enhances orbital hybridization, while stronger s−d coupling further reduces the energy separation of states associated with anharmonicity [56, 57]. The competition between s−p and p−d orbital couplings leads to distinct variations in the band E_g, which either increases or remains nearly unchanged under stress. The following section examines how s−p and p−d interactions regulate the energies of the valence band maximum (VBM) and conduction band minimum (CBM), and consequently, the evolution of E_g.

The relative variation of pressure-dependent E_g in the X₂YZ compounds is shown in Fig. 2, where the materials can be categorized into two types: those exhibiting a monotonic increase in E_g and those showing nonmonotonic or minor variations. In particular, with increasing structural compression, the E_g of Na₂AgAs remains nearly invariant until the compressive strain reaches approximately 5% under the HSE06 functional and 11% under the PBE functional. Beyond these thresholds, the E_g begins to deviate significantly from zero. To elucidate the physical origin of the distinct pressure-dependent E_g responses among different structures, three representative compounds were selected for detailed analysis, each exemplifying a characteristic trend. Na₂CuAs and Na₂CuSb display a pronounced increase in E_g under compression, whereas Na₂AgAs exhibits semimetallic behavior with the most resistant E_g opening.

First, for Na₂CuAs and Na₂CuSb, with the Na cation unchanged, replacing Sb with As shifts the anion p-orbital energy upward, while the Cu d-orbital energy remains nearly constant. This upward shift weakens the p−d coupling within the valence band but strengthens the s−p coupling between the valence and conduction bands. Both effects lower the energy of the occupied states, as illustrated in Fig. 3, thereby suppressing the pressure-induced reduction in E_g. Consequently, the increase in E_g with pressure is more pronounced in Na₂CuAs than in Na₂CuSb under both PBE and HSE06 functionals. For Na₂CuAs and Na₂AgAs, replacing Cu with Ag causes a downward shift of the d-orbital energy, which weakens the p−d coupling within the valence band. Meanwhile, the As s-orbital energy remains nearly unchanged, so the s−p coupling is unaffected. The Ag 4d orbitals are broader and more spatially extended than the Cu 3d orbitals, contributing more significantly to the VBM, as confirmed by the projected band structures (Fig. S5). The enhanced d-orbital contribution increases the hybridization strength, making E_g widening more difficult. Even under pressure, the Ag d-states remain more energetically stable and less responsive to compression than the Cu d-states, leading to a weaker pressure-induced increase in E_g.

Next, we analyze the effect of volume contraction on the band structure near the Fermi level and track the energy evolution along the selected Brillouin zone path Γ−X−U−K−Γ−L−W−X for the X₂YZ cubic structure, as shown in Fig. 4(a). The VBM of X₂YZ is located at Γ and exhibits threefold band degeneracy (Γ_1, Γ₂, Γ₃). With increasing volume contraction, the conduction band Γ valley shifts upwards while the valence band at Γ shifts downwards in energy, leading to a gradual widening of E_g. For the conduction states, in addition to the upward shift at Γ, the energies at non-Γ k-points also evolve with volume contraction. This results in an “alignment” of conduction-band energies across different k-points at a specific contraction level. In Li₂CuAs, for instance, when the volume is reduced to 97% of its original value, the conduction-band states at all k-points converge to the same energy [Figs. 4(b) and (c)]. Both PBE and HSE06 calculations reveal multi-valley alignment in the conduction band of Li₂CuAs as the volume contracts (Fig. 5). In contrast, pressure causes the multi-valley states near the Fermi level in Li₂CuSb to separate, and the upward shift of the L-point conduction band reduces carrier transport anisotropy. Meanwhile, the threefold band degeneracy at the VBM remains intact for both Li₂CuAs and Li₂CuSb. This combination of VBM band convergence and conduction-band multi-valley alignment, as illustrated in Fig. 4(c), is expected to facilitate enhanced carrier transport, further enhancing the TE efficiency [58−60].

Based on our calculations, we analyze the TE properties of Li₂CuAs for both p-type (hole) and n-type (electron) carriers, as shown in Fig. 6. For comparison, Li₂CuSb is also considered, in which conduction-band alignment is less pronounced. By applying a 3% lattice contraction, we examine the chemical bonding strength and lattice anharmonicity of both compounds at the intrinsic lattice parameter (a₀) and the contracted lattice parameter (a₀−3%a₀). We further investigate their effects on phonon and electron transport. The valence- and conduction-band dispersions near the band edges differ significantly between the two compounds, resulting in distinct carrier effective masses (Table S1). This relates to the S and \sigma according to the equations S=2k_B^2{m}_d^{\ast }T{[\pi /(3n)]}^{2/3}/(3e{\hslash }^2), \sigma =n{e}^2\tau /m_c^{\ast }, where m^{\ast } is the carrier effective mass, n is the carrier concentration, \tau is the relaxation time, and m_d^{\ast } is the density-of-states (DOS) effective mass. The DOS effective mass is given by m_d^{\ast }=N_V^{2/3}{m}^{\ast }, which is proportional to the Fermi surface complexity factor N_VK* in semiconductors with band degeneracy and nonspherical Fermi surfaces, where K* is the anisotropy parameter [61].

S and σ exhibit opposite dependencies on m^{\ast } and n. As shown in Figs. 6(a) and (b), the calculated |S| values at 300 K with a carrier concentration of 10¹⁹ cm⁻³ are 68.2 μV·K⁻¹ (n-type) and 298.7 μV·K⁻¹ (p-type) for Li₂CuAs, and 319.7 μV·K⁻¹ (n-type) and 264.5 μV·K⁻¹ (p-type) for Li₂CuSb. The flatter and more asymmetric bands of Li₂CuSb near the Fermi level along the L−Γ and L−W directions yield larger values of the anisotropy factor K* and m*, thereby increasing the DOS effective mass m_d^{\ast } and enhancing its n-type S. In Li₂CuAs, by contrast, the degenerate valence bands contribute to a higher p-type S than the n-type case. Furthermore, Figs. 6(c) and (d) show that \sigma of both compounds increases with n and is inversely proportional to m^{\ast }, consistent with m_c^{\ast }={[\frac{1}{3}{\sum }_{\alpha \beta }1/m^{\ast }]}^{-1}, where m_c^{\ast } is the conductivity effective mass. As illustrated in Fig. 5, the small m^{\ast } along the Γ₁−L and Γ₁−K directions at the VBM indicates light p-type carriers in both compounds. Consequently, the p-type \sigma exceeds the n-type \sigma in both Li₂CuAs and Li₂CuSb. At 300 K under hole doping, the low-carrier-concentration regime is dominated by ionized-impurity scattering, whereas acoustic-deformation-potential scattering governs carrier mobility at higher concentrations. With increasing temperature, both S and σ decrease due to enhanced phonon scattering, reduced carrier mobility, and thermally induced carrier excitation. Moreover, the electronic thermal conductivity κ_e follows a trend similar to that of \sigma (Fig. S6), consistent with the Wiedemann−Franz relation, κ_e = L\sigma \mathrm{T}, where L represents the Lorenz number and is conventionally treated as a constant.

The effect of pressure on electronic transport properties is further examined. As shown in Fig. 5, pressure has little influence on the band degeneracy near the Fermi level in either compound. However, in Li₂CuAs, pressure enhances conduction-band anisotropy near the Fermi level, thereby improving S under n-type doping (Fig. 6). In contrast, for Li₂CuSb, pressure drives the conduction band toward a more parabolic dispersion, resulting in only minor changes in its n-type S. The pressure-induced widening of E_g suppresses carrier transport, leading to a decrease in σ. However, the n-type σ of Li₂CuAs decreases more gradually than that of Li₂CuSb and remains higher under lattice compression. This behavior arises from the pressure-induced alignment of multiple conduction-band minima at comparable energy levels in Li₂CuAs, which alleviates the reduction in σ. Moreover, as shown in Figs. 6(e) and (f), weaker electron scattering by polar optical phonons in Li₂CuAs further enhances carrier mobility compared with Li₂CuSb. Finally, the calculated power factor (PF = S²\sigma) initially increases and then decreases with carrier concentration (Fig. S7). The p-type transport shows better performance than n-type transport because of its higher \sigma. For Li₂CuSb, pressure exerts opposite effects on the PF of p-type and n-type carriers: the p-type PF increases, whereas the n-type PF decreases due to the pronounced pressure-induced reduction in \sigma. In contrast, for Li₂CuAs, pressure enhances the PF under both electron and hole doping. This enhancement originates from the unique electronic structure of Li₂CuAs, where pressure strengthens conduction-band anisotropy and promotes multi-valley alignment near the Fermi level.

Understanding thermal transport behavior is essential for enhancing TE performance. Figure 7(a) compares the intrinsic {\kappa }_L of the two compounds at 300 K with that under a 3% lattice contraction (a₀−3%a₀) induced by pressure. Li₂CuAs exhibits a lower intrinsic {\kappa }_L, and the inclusion of four-phonon (3+4ph) scattering has a more pronounced effect on Li₂CuAs, reducing its κ_3+4ph to 0.74 W·m⁻¹·K⁻¹, corresponding to a ~ 67% reduction relative to κ_3ph. Upon applying pressure, {\kappa }_L increases for both compounds, with a more pronounced increase in the three-phonon component κ_3ph of Li₂CuAs. This behavior is attributed to the increased stiffness of chemical bonds under compressive stress. As shown in Table 1, Li₂CuAs exhibits a higher Young’s modulus, indicating that greater pressure is required to achieve a 3% lattice contraction (a₀−3%a₀). This increased bonding rigidity leads to higher phonon group velocities (Fig. S8). The decrease in {\kappa }_L with increasing temperature becomes more pronounced for both Li₂CuAs and Li₂CuSb, as shown in Fig. 7(b). Heavy Cu and Sb atoms primarily contribute to low-frequency vibrational modes below 5 THz, as indicated by the phonon density of states in Fig. S8. According to the thermal conductivity spectrum {\kappa }_L(\omega ) and the cumulative {\kappa }_L at 300 K from the 3+4ph model [Fig. 7(c)], the dominant contribution to {\kappa }_L arises from low-frequency phonons. Furthermore, phonon transport is dominated by out-of-plane transverse acoustic (ZA) and longitudinal acoustic (LA) modes at 300 K (Fig. S9). This is because high-frequency phonons require higher excitation energies, particularly in materials with a gap between acoustic and optical branches, as described by the Bose−Einstein distribution.

To quantify the contributions of different phonon modes, the {\kappa }_L can be expressed within kinetic theory as {\kappa }_L=\sum C_v{\overline{v}}^2\overline{\tau }, where C_v is the specific heat, and \overline{v} and \overline{\tau } are the average group velocity and relaxation time, respectively. A high scattering rate (1/τ) and low group velocity collectively lead to a reduced {\kappa }_L. As shown in Fig. S8, a significant decrease in group velocity is observed when going from Li₂CuAs to Li₂CuSb, consistent with the increased atomic mass from As to Sb. Meanwhile, lattice contraction leads to phonon-frequency hardening, accompanied by a decrease in the absolute value of the Grüneisen γ. γ is defined as the rate of phonon frequency with respect to volume and reflects the degree of lattice anharmonicity. A larger absolute value of γ indicates stronger anharmonicity. Under applied pressure, both Li₂CuAs and Li₂CuSb exhibit reduced anharmonicity, suggesting that compression enhances the harmonic character of their lattices. Meanwhile, the heavier Sb atom in Li₂CuSb gives rise to softer optical and acoustic phonon modes with lower frequency due to weaker chemical bonding, resulting in flatter low-frequency phonon dispersions (Fig. S8).

Figures 7(d) and (e) show the phonon scattering rates at 300 K for three-phonon and four-phonon processes. The four-phonon scattering is less constrained by selection rules and is generally more favorable for energy conservation than three-phonon scattering. In the phonon dispersion relations, each branch has distinct upper and lower energy bounds that determine possible transitions. For a four-phonon combination process (\lambda +{\lambda }^{{}^{\prime }}+{\lambda }^{{}^{″}}\to {\lambda }^{{}^{‴}}), assuming the absorbed phonons originate from different branches, the corresponding maximum and minimum summed frequencies are denoted as {\omega }_{\mathrm{m}\mathrm{a}\mathrm{x}} and {\omega }_{\mathrm{m}\mathrm{i}\mathrm{n}}. The branch emitting the phonon must lie within this range, providing a general energy criterion for any branch. Notably, the redistribution process (\lambda +{\lambda }^{{}^{\prime }}\to {\lambda }^{{}^{″}}+{\lambda }^{{}^{‴}}) more readily satisfies energy conservation and therefore dominates the scattering rate across the full frequency spectrum in both Li₂CuAs and Li₂CuSb. The increase in atomic mass from As to Sb reduces the acoustic-optical phonon gap, allowing optical branches to appear at lower frequencies and enhancing acoustic-optical coupling. In addition, the smaller atomic mass of As makes the Li₂CuAs lattice lighter, shifting the phonon spectrum upward. The wider phonon energy gap is a critical factor enhancing four-phonon scattering [62, 63]. Consequently, the contribution of the redistribution process decreases from Li₂CuAs to Li₂CuSb. A similar trend is observed in Na₂AgSb: the heavier atomic mass leads to lower phonon frequencies (Fig. S8) and reduced phonon group velocities, resulting in a lower {\kappa }_L (Fig. S10). Moreover, a smaller acoustic-optical phonon gap weakens four-phonon scattering, and the reduced {\kappa }_L is primarily governed by the low phonon group velocity.

Upon a 3% pressure-induced lattice contraction, three-phonon scattering is markedly suppressed, primarily due to the reduced lattice constant, enhanced bond strength, and pronounced hardening of the low-frequency phonon branches. These effects weaken the interactions between low-frequency optical and acoustic modes. In both Li₂CuAs and Li₂CuSb, the concurrent hardening of low- and high-frequency phonons results in an enlarged acoustic-optical gap, while the four-phonon scattering intensity remains nearly unaffected above 5 THz.

Figure 8 demonstrates that Li₂CuAs and Li₂CuSb exhibit promising TE performance in their intrinsic configurations. At 300 K, a hole concentration of 10¹⁹ cm⁻³ yields ZT values of approximately 1.0 for both compounds. At 700 K with the same doping level, the ZT values rise to about 4.5 for Li₂CuAs and 4.0 for Li₂CuSb. Under pressure-induced lattice contraction to 0.97a₀, the ZT values under hole doping decrease notably in both systems, whereas electron doping at 3.0 × 10¹⁹ cm⁻³ enhances the ZT of Li₂CuAs at both 300 and 700 K.

4 Conclusion

In conclusion, we investigated the electronic band structure of Cu/Ag-stuffed zinc-blende X₂YZ compounds (X = Li, Na; Y = Cu, Ag; Z = As, Sb) under pressure. The pressure-dependent E_g is governed by the competition between interband s−p coupling and intraband p−d coupling. Compared with Cu 3d orbitals, the lower-lying Ag 4d orbitals contribute more strongly to the VBM, thereby suppressing the pressure-induced widening of E_g, whereas the higher As-p orbital energy weakens p−d coupling and leads to a larger gap increase. Li₂CuAs exhibits pressure-induced multi-valley alignment in the conduction band, which mitigates the reduction in electrical conductivity. In phonon transport, the lighter As atom shifts the phonon spectrum toward higher frequencies, enhancing four-phonon scattering and thus reducing the lattice thermal conductivity of Li₂CuAs. The combination of favorable electronic transport and low lattice conductivity yields remarkable TE performance: under pressure, Li₂CuAs achieves an optimal ZT value of 3.0 at 700 K with a hole concentration of 10¹⁹ cm⁻³. This work provides fundamental insights into orbital coupling and phonon scattering mechanisms, offering guidance for optimizing thermoelectric materials through pressure or compositional design.

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