First-Principles Study of the Interfacial Stability, Electronic Structure and Alloying Effects at the Ti₃SiC₂(0001)/Ag(111) Interface

1. Introduction

Silver (Ag) and silver-based alloys are widely used as electrical contact materials in switches, relays, and circuit breakers due to their excellent electrical conductivity, thermal conductivity, and oxidation resistance [1,2,3]. With the rapid development of modern electrical equipment and electronic instruments, the requirements for electrical contact materials have become increasingly stringent. In particular, high-power electrical systems require contact materials with a combination of high conductivity, excellent arc erosion resistance, good wear resistance, and stable contact resistance [4,5]. However, pure Ag suffers from low hardness, low mechanical strength, and severe arc erosion during electrical switching operations, which limits its service life and reliability [6,7]. To overcome these drawbacks, various reinforcement phases have been introduced into Ag matrix composites, including oxides (CdO, SnO₂, ZnO), carbides (WC, TiC), and refractory metals (W, Mo, Ni) [8,9,10]. Nevertheless, these traditional reinforcements either degrade the electrical conductivity of the Ag matrix or exhibit poor interfacial bonding with Ag, leading to reinforcement shedding and premature failure [11,12]. Therefore, it is imperative to develop novel reinforcement phases that can simultaneously enhance the mechanical and tribological properties of Ag while maintaining its high conductivity.

Ti₃SiC₂, a representative member of the MAX phase ceramics, has attracted considerable attention as a potential reinforcement phase for Ag-based composites [13,14,15]. The unique layered crystal structure of Ti₃SiC₂ endows it with a combination of metallic and ceramic properties, such as high electrical conductivity, excellent thermal shock resistance, good machinability, low friction coefficient, and outstanding damage tolerance [16,17,18]. Recent experimental studies have demonstrated that Ag-Ti₃SiC₂ composites fabricated by pressureless infiltration exhibit high hardness, high flexural strength, and good wear resistance with a low wear rate of 10^-5 mm³/(N·m) [19]. These findings suggest that Ti₃SiC₂ is a promising candidate reinforcement for high-performance electrical contact materials. Despite the encouraging experimental progress, the fundamental understanding of the interfacial behavior between Ti₃SiC₂ and Ag remains limited. In particular, the heterogeneous nucleation mechanism of Ag on Ti₃SiC₂ substrate, which governs the solidification microstructure and bonding strength of the composite, has not been elucidated at the atomic scale [20]. The interface stability, wetting behavior, and the effect of alloying elements on the nucleation process are crucial issues that need to be addressed to guide the rational design of Ag-Ti₃SiC₂ electrical contact materials [21,22].

First-principles calculations based on density functional theory (DFT) have been proven to be an effective and reliable tool for investigating the atomic-scale interfacial properties, electronic structures, and bonding mechanisms of metal-ceramic systems [23,24,25]. Unlike experimental methods, DFT calculations can provide detailed information on the interfacial atomic arrangements, charge transfer, and orbital hybridization, which are difficult to obtain experimentally [26]. Furthermore, DFT enables the systematic study of alloying element effects on interfacial properties by controlled substitution at specific atomic sites, thereby offering unique insights into the design of optimized interfaces [27,28]. In recent years, several DFT studies have been devoted to the investigation of MAX phase surfaces and interfaces. Orellana and Gutiérrez [29] studied the (0001) surfaces of Ti₃SiC₂ using DFT and molecular dynamics, concluding that Si- and Ti-terminated surfaces are most stable at low and high temperatures, respectively. Wang et al. [30] investigated the electronic structure and chemical bonding of Ti₃SiC₂, revealing strong Ti-C covalent bond chains and relatively weaker Ti-Si bonds. Zhou et al. [31] reported the distribution of charge density on the $\left(11\stackrel{\mathrm{‾}}{2}0\right)$ plane of Ti₃SiC₂, where robust directional Ti-C covalent bond chains were observed. However, these studies primarily focused on the bulk and surface properties of MAX phases, and systematic investigations of Ti₃SiC₂/Ag interfaces are still lacking.

The addition of trace alloying elements to metal matrix composites has been recognized as an effective strategy to improve interfacial bonding and mechanical properties [32]. Recent DFT studies have demonstrated that alloying elements such as Ti, Zn, Zr, and Al can significantly enhance the interfacial work of adhesion and reduce the contact angle at metal-ceramic interfaces [33,34]. For instance, Zr doping at the Cu/Ni₂Si interface increased the work of adhesion from 2.53 J/m² to 3.20 J/m² and reduced the contact angle from 98.8° to 86.2°, thereby promoting wetting. Similarly, Cu and Ni additions have been found to strengthen the TiC/Al and TiC/Cu interfaces through enhanced interfacial charge transfer and orbital hybridization [35,36]. These findings suggest that alloying element doping may also be beneficial for the Ti₃SiC₂/Ag interface. Nevertheless, to the best of our knowledge, no systematic first-principles study has been reported on the interfacial stability and adhesion mechanism of the Ti₃SiC₂(0001)/Ag(111) interface, nor on the effect of alloying elements (Cu, Ni, Zn, Cr) on the interface stability. In particular, the relationship between the interfacial electronic structure and the adhesion strength remains unclear for this important system.

In the present work, first-principles calculations were performed to systematically investigate the interfacial stability and bonding mechanism of the Ti₃SiC₂(0001)/Ag(111) interface. We first determined the stable surface terminations of Ti₃SiC₂(0001) and calculated the corresponding surface energies, and then constructed multiple interface models with different stacking sequences to identify the most stable interfacial structure based on the work of adhesion. Subsequently, the interfacial bonding mechanism was elucidated through charge density, charge density difference, and partial density of states (PDOS) analyses. In addition, the effects of alloying elements (Cu, Ni, Zn, Cr) on the interfacial stability were systematically investigated. Ultimately, this study aims to provide theoretical guidance for the design and optimization of high-performance Ag-Ti₃SiC₂ electrical contact materials.

2. Calculation Method

All first-principles computations were carried out within the framework of density functional theory (DFT) as implemented in the CASTEP code [37,38], which is based on density functional theory (DFT). The projector augmented wave (PAW) pseudopotentials [39] were employed to describe the interaction between valence electrons and ionic cores. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional [40] was adopted to treat the exchange-correlation energy. The cutoff energy for the plane-wave basis set was set to 500 eV. The Brillouin zone integration was performed using the Monkhorst-Pack k-point mesh [41], with a 9 × 9 × 1 grid for surface and interface calculations and a 9 × 9 × 9 grid for bulk calculations. The convergence criteria for the self-consistent field (SCF) loop and ionic relaxation were set to 10⁻⁵ eV and 0.02 eV/Å, respectively. The valence electron configurations considered were Ti 3p⁶3d²4s², Si 3s²3p², C 2s²2p², Ag 4d¹⁰5s¹, Cu 3d¹⁰4s^¹, Ni 3d⁸4s², Co 3d⁷4s², Zn 3d¹⁰4s², and Cr 3d⁵4s¹.

To avoid artificial electrostatic interactions between periodic images, a vacuum layer of 15 Å was inserted perpendicular to the surface and interface models, and a dipole correction was applied to cancel the artificial electric field caused by the asymmetric slab [42]. Before investigating the interfacial properties, the bulk structures of Ti₃SiC₂ and Ag were optimized to validate the computational parameters. The calculated lattice parameters of Ti₃SiC₂ and Ag bulks are in excellent agreement with experimental values and other DFT calculations [43,44,45,46], as summarized in Table 1. The bulk modulus of Ti₃SiC₂ was calculated to be 186.3 GPa, close to the experimental value of 184 GPa [47]. These validations confirm the reliability of the present computational approach. The crystal structures of the bulk phases, the surface slabs, and the interface models are shown in Figure 1.

As can be observed from Figure 1, the bulk MAX phase Ti₃SiC₂ exhibits a symmetric atomic arrangement, with the symmetry center precisely located at the Si atomic sites. To ascertain whether the internal stacking sequence of the bulk phase introduces perturbations into the interfacial simulation, it is imperative to construct an appropriate slab model that faithfully captures the intrinsic structural characteristics of the bulk phase prior to the interfacial assembly. A larger number of atomic layers and an increased vacuum spacing would inevitably prolong the computational cost; therefore, the bulk Ti₃SiC₂ was cleaved along the (0001) crystallographic orientation while preserving the periodic boundary conditions, so as to identify the minimum slab thickness required for an adequate representation of the bulk properties. The truncation inevitably disrupts the stability of the original atomic configuration, necessitating a full geometry optimization of the surface model to attain a converged equilibrium structure. During the optimization process, the atomic positions undergo minor displacements and gradually relax toward their equilibrium states. By systematically increasing the number of atomic layers in the Ti₃SiC₂ surface model and comparing the interlayer spacings before and after structural relaxation, the optimal slab thickness can be unambiguously determined.

3. Results and Discussion

3.1. Surface Properties

The surface energy is a critical parameter that determines the stability of different surface terminations and affects the interfacial bonding behavior [48]. For the Ti₃SiC₂(0001) surface, six possible terminations were considered: TiSi-terminated slab, TiC(TiSi)-terminated slab, TiC(TiC)-terminated slab, C(TiSi)-terminated slab, C(TiC)-terminated slab, and Si-terminated slab, as shown in Figure 2.

To quantitatively assess how different terminal groups influence the structural stability of the Ti₃SiC₂(0001) surface, computations and comparative analyses must focus on its surface energy. This energetic quantity directly indicates the relative stability among distinct surface terminations: a lower value corresponds to a thermodynamically more favourable configuration. Under the symmetric slab framework adopted here, the surface energy is derived using the equation from reference [49].

$$\sigma ={\displaystyle \frac{1}{2\mathrm{A}}}\left(E_{\mathrm{s}lab}-N_{Ti}{\mu }_{Ti}^{slab}-N_{Si}{\mu }_{Si}^{slab}-N_C{\mu }_C^{slab}-PV-TV\right)$$

For our simulations, the term E_slab represents the total energetic magnitude of a completely relaxed Ti₃SiC₂ surface slab. The chemical potentials for Ti, Si, and C atoms inside this slab are denoted ${\mu }_{\mathrm{Ti}}^{slab}$, ${\mu }_{\mathrm{Si}}^{slab}$ and ${\mu }_C^{slab}$, respectively. Meanwhile, N_Ti, N_Si and N_C refer to the respective atom counts of these elements in the same slab model, with A standing for the surface area. Because the simulation is performed at 0 K under ground-state conditions, contributions from the PV and TS terms are negligible. In this system, an individual atom’s chemical potential equals that of the corresponding bulk phase. Therefore, the chemical potential together with the surface energy for bulk Ti₃SiC₂ can be written as:

$${\mathsf{\mu }}_{{\mathit{Ti}}_3\mathit{Si}{\mathrm{C}}_2}^{\mathit{Bulk}}={3\mathsf{\mu }}_{\mathit{Ti}}+{\mathsf{\mu }}_{\mathit{Si}}+{2\mathsf{\mu }}_{\mathrm{C}}{\mathsf{\mu }}_{{\mathit{Ti}}_3\mathit{Si}{\mathrm{C}}_2}^{\mathit{Bulk}}=3{\mathsf{\mu }}_{\mathit{Ti}}^{\mathit{Bulk}}+{\mathsf{\mu }}_{\mathit{Si}}^{\mathit{Bulk}}+2{\mathsf{\mu }}_{\mathrm{C}}^{\mathit{Bulk}}+∆H_f$$

Therefore, the equation surface energy can be finally written by:

$$\mathsf{\sigma }={\displaystyle \frac{1}{2\mathrm{A}}}\left({\mathrm{E}}_{\mathrm{slab}}-{\mathrm{N}}_{\mathrm{Si}}{\mathsf{\mu }}_{{\mathrm{Ti}}_2\mathrm{Si}{C}_2}^{\mathrm{Bulk}}+\left(3{{\mathrm{N}}_{\mathrm{Si}}-\mathrm{N}}_{\mathrm{Ti}}\right)(∆{\mathsf{\mu }}_{\mathrm{Ti}}+{\mathsf{\mu }}_{\mathrm{Ti}}^{\mathrm{Bulk}})+\left({{2\mathrm{N}}_{\mathrm{Si}}-\mathrm{N}}_{\mathrm{C}}\right)\left(∆{\mathsf{\mu }}_{\mathrm{C}}+{\mathsf{\mu }}_{\mathrm{C}}^{\mathrm{Bulk}}\right)\right)$$

$$∆{\mathsf{\mu }}_{\mathrm{Ti}}={\mathsf{\mu }}_{\mathrm{Ti}}-{\mathsf{\mu }}_{\mathrm{Ti}}^{\mathrm{Bulk}}∆{\mathsf{\mu }}_{\mathrm{C}}={\mathsf{\mu }}_{\mathrm{C}}-{\mathsf{\mu }}_{\mathrm{C}}^{\mathrm{Bulk}}$$

When considering a thermodynamically stable configuration of the Ti₃SiC₂ surface, the chemical potential of each atomic species should remain below its corresponding elemental reference value; violating this condition renders the slab unstable. Hence, the relations ${\mathsf{\mu }}_{\mathrm{Ti}}$ ⩽${\mathsf{\mu }}_{\mathrm{Ti}}^{\mathrm{Bulk}}$, ${\mathsf{\mu }}_{\mathrm{Si}}$ ⩽${\mathsf{\mu }}_{\mathrm{Si}}^{\mathrm{Bulk}}$, and ${\mathsf{\mu }}_{\mathrm{C}}$ ⩽${\mathsf{\mu }}_{\mathrm{C}}^{\mathrm{Bulk}}$ collectively constitute the necessary criteria for achieving a stable surface.

From the preceding expressions:

$$3{\mu }_{\mathrm{Ti}}+{\mu }_{\mathrm{Si}}+2{\mu }_{\mathrm{C}}=3{\mu }_{\mathrm{Ti}}^{Bulk}+{\mu }_{\mathrm{Si}}^{Bulk}+2{\mu }_{\mathrm{C}}^{Bulk}+\mathrm{\Delta }{H}_f$$

Thus

$$3\mathrm{\Delta }{\mu }_{\mathrm{Ti}}+\mathrm{\Delta }{\mu }_{\mathrm{Si}}+2\mathrm{\Delta }{\mu }_{\mathrm{C}}=\mathrm{\Delta }{H}_f$$

$$\mathrm{\Delta }{H}_f\le 3\mathrm{\Delta }{\mu }_{\mathrm{Ti}}+2\mathrm{\Delta }{\mu }_{\mathrm{C}}\le 0$$

The formation enthalpy of bulk Ti₃SiC₂ is -5.29 eV in previous research. Thus,

$$-5.29\le 3\mathrm{\Delta }{\mu }_{\mathrm{Ti}}+2\mathrm{\Delta }{\mu }_{\mathrm{C}}\le 0$$

For the polar Ti₃SiC₂(0001) termination, the computed surface energy depends strongly on the carbon chemical potential ($\mathrm{\Delta }{\mu }_{\mathrm{Ti}}=-1.763-2/3\mathrm{\Delta }{\mu }_{\mathrm{C}}$). Figure 3 illustrates that a high $\mathrm{\Delta }{\mu }_{\mathrm{C}}$ corresponds to a carbon-rich regime, whereas a low $\mathrm{\Delta }{\mu }_{\mathrm{C}}$ reflects a carbon-poor environment. As shown in the figure, the surface energy of this polar facet varies linearly with $\mathrm{\Delta }{\mu }_{\mathrm{C}}$. To illustrate, consider the C(TiC)-, TiC(TiC)-, TiSi-, and Si-terminated slabs: their surface energies rise linearly as $\mathrm{\Delta }{\mu }_{\mathrm{C}}$ increases. By contrast, the C(TiSi) and TiC(TiSi) terminations exhibit almost no variation in surface energy with changing $\mathrm{\Delta }{\mu }_{\mathrm{C}}$. Under carbon-rich conditions, the TiSi, Si, C(TiSi), and TiC(TiSi) terminations display comparable surface energies, suggesting that these four configurations can co-exist. At low $\mathrm{\Delta }{\mu }_{\mathrm{C}}$ values, the TiC(TiC) termination yields the smallest surface energy, making it the thermodynamically preferred structure. Nevertheless, as μC rises, the surface energy of this termination gradually increases; once $\mathrm{\Delta }{\mu }_{\mathrm{C}}$ exceeds -1.50 eV, the TiC(TiSi) termination becomes the most stable. Notably, across the entire μ_C range examined here, both the C(TiC) and C(TiSi) terminations consistently show higher surface energies than all other slabs, indicating that they are the least stable options.

Figure 3. The surface energy of Ti₃SiC₂ (0001) as a function of $\mathrm{\Delta }{\mu }_{\mathrm{C}}$ when $\mathrm{\Delta }{\mu }_{\mathrm{Ti}}=0$.

Figure 3. The surface energy of Ti₃SiC₂ (0001) as a function of $\mathrm{\Delta }{\mu }_{\mathrm{C}}$ when $\mathrm{\Delta }{\mu }_{\mathrm{Ti}}=0$.

3.2. Interface Models and Stability

Based on the convergence tests described in Section 2, the interface models were constructed by stacking the differently terminated Ti₃SiC₂(0001) surfaces onto a seven-layer Ag(111) slab. A vacuum layer of 15 Å was then placed atop the resulting supercell to eliminate spurious interactions between periodic images. Three high-symmetry stacking arrangements were considered for the Ti₃SiC₂(0001)/Ag(111) system according to the lateral registry between the two surfaces, yielding eighteen interfacial configurations for subsequent characterization. In Figure 4, the three high-symmetry stacking sites are labeled: OT denotes the configuration in which a Ti₃SiC₂(0001) surface atom lies directly above an Ag atom; MT corresponds to the bridge site at the midpoint of two adjacent Ag atoms; and HCP represents the three-fold hollow site at the center of three adjacent Ag atoms. Figure 4 shows the three typical interface stacking configurations.

The work of adhesion (W_ad) characterizes the bonding strength of the Ti₃SiC₂(0001)/Ag(111) interface. Specifically, Wad is defined as the minimum work required to separate the interface into two free surfaces, and this quantity is positively related to the interfacial bond strength. The corresponding calculation formula is given as follows [50,51]:

$$\mathrm{W}ad={\displaystyle \frac{E_{\mathrm{Ag}}+E_{T{i}_{3}Si{C}_2}-E_{Ag/T{i}_{3}Si{C}_2}}{A}}$$

Where, 111. surface,Ti₃SiC₂(0001) surface, and Ag(111)/Ti₃SiC₂(0001) interface, respectively. And A represents the interface area. .

To reduce the computational cost, the equilibrium interfacial spacing was determined for all interface models using the Universal Binding Energy Relation (UBER) method prior to full structural relaxation. The workflow consisted of four steps. First, a series of atomic configurations with varying initial interface separations (d₀) was constructed, and the total energy of each configuration was computed with the atomic positions fixed. Second, the work of adhesion (W_ad) at every separation distance was evaluated using the adhesion-energy expression above. Third, the W_ad–d₀ curves were generated, as presented in Figure 5. Finally, the equilibrium distance was identified by applying the energy-minimization criterion.

To elucidate the thermodynamic driving forces and mechanical compatibility governing metal–ceramic adhesion, we systematically evaluated the work of adhesion (W_ad) and equilibrium interlayer distance (d₀) for eighteen distinct Ti₃SiC₂(0001)/Ag(111) interface configurations, spanning six substrate terminations and three high-symmetry stacking sequences. Notably, the HCP registry universally outperformed the OT and MT arrangements, delivering the highest W_ad for every termination examined, a trend that underscores the critical role of lateral atomic registry in optimizing interfacial bonding. Among the HCP-stacked interfaces, the adhesion strength varied markedly with surface chemistry: the C(TiC) termination exhibited the strongest adhesion, with W_ad reaching 9.25 J/m², followed by C(TiSi) with W_ad = 8.28 J/m² at d₀ = 1.2 Å (Table 2). The C(TiC)-HCP configuration therefore emerged as the most stable interface in the complete configurational landscape, a distinction attributable to the strong interfacial Ag–C bonding fostered by the TiC subsurface layer. Conversely, the TiSi termination yielded the lowest adhesion (W_ad = 4.18 J/m²), as the inherently weak Ti–Si bonding within the MAX phase facilitates interfacial sliding and reconstruction. The Si-terminated interface displayed intermediate behavior (W_ad = 6.23 J/m²), reflecting the formation of directional Si–Ag bonds, whereas the TiC-terminated surfaces occupied a middle ground, with adhesion strengths descending from TiC(TiSi) (6.85 J/m²) to TiC(TiC) (5.65 J/m²). Across all configurations, the equilibrium separation was broadly anti-correlated with the adhesive strength, as shorter d₀ values generally corresponded to stronger interfacial attraction. Structural relaxation proved essential, enhancing W_ad by approximately 10–75% relative to the unrelaxed geometries as the Ag overlayer reorganized toward the substrate.

To efficiently estimate the equilibrium interfacial geometries prior to the computationally demanding full structural relaxations, the Universal Binding Energy Relation (UBER) was employed, which provides both the equilibrium separation distance and the corresponding work of adhesion from rigid-slab calculations. Figure 5 presents the work of adhesion (W_ad) as a function of the interfacial separation (d₀) for the eighteen high-symmetry configurations across the six distinct Ti₃SiC₂ terminations. All curves display the characteristic UBER profile: a gradual rise from large separation, a maximum at the equilibrium distance d₀, and a subsequent decline stemming from Pauli repulsion at close range. In every panel, the hollow-site (HCP) stacking, denoted by blue triangles, consistently achieves the highest work of adhesion at the shortest equilibrium separation, whereas the on-top (OT) configuration, represented by black squares, invariably yields the lowest adhesion energy with the largest d₀. For instance, the TiSi-terminated HCP interface (panel a) exhibits a modest peak of 2.89 J/m² in the unrelaxed state, which increases to 4.18 J/m² upon relaxation, reflecting the comparatively diffuse interfacial orbitals that produce a gentle repulsive wall. In marked contrast, the carbon-terminated interfaces demonstrate substantially stronger interfacial bonding: the C(TiSi)-HCP configuration (panel b) reaches 6.85 J/m² before relaxation and 8.28 J/m² after relaxation at an equilibrium spacing of 1.2 Å, while the C(TiC)-HCP configuration (panel e) displays the steepest repulsive branch, peaking at 7.18 J/m² before relaxation and 9.25 J/m² after relaxation, the highest value among all configurations. Intermediate values are obtained for TiC(TiC)-HCP at 3.67 J/m² (5.65 J/m² after relaxation; panel c), TiC(TiSi)-HCP at 4.69 J/m² (6.85 J/m² after relaxation; panel d), and Si-HCP at 4.32 J/m² (6.23 J/m² after relaxation; panel f). The pronounced variation in repulsive steepness, ranging from the abrupt walls of C(TiC)-HCP and C(TiSi)-HCP to the softer repulsion of TiSi-HCP, directly reflects the spatial extent of the interfacial electron density. Together, these UBER curves unambiguously identify HCP stacking as the preferred interfacial registry.

3.3. Electronic Structure

The adhesion strength and stability of the Ti₃SiC₂(0001)/Ag(111) interface are closely related to its electronic structure. Having established in Section 3.2 that the work of adhesion varies by more than a factor of two among the six optimal HCP-stacked configurations, from 9.25 J/m² for the C(TiC)-terminated interface down to 4.18 J/m² for the TiSi-terminated interface, we now examine the valence charge density (Figure 6), the charge density difference (Figure 7), and the layer-resolved partial density of states (PDOS, Figure 8). Together, these complementary probes connect the adhesion energetics to the character of the interfacial chemical bonding.

Figure 6 displays the charge density distributions across the six interfaces. Within the Ag(111) slab, the density is smoothly delocalized around and between the Ag cores, a hallmark of metallic cohesion; within the Ti₃SiC₂ slab it concentrates along the Ti-C framework, reflecting the covalent network of the MAX phase. The interfacial region, by comparison, discriminates sharply among terminations. For the two C-terminated interfaces (Figure 6(a, d)), continuous density contours bridge the interfacial C atoms and the opposing Ag atoms across a separation of only about 1.2 Å, indicating direct charge sharing and hence a sizeable covalent component of the Ag-C bond. The TiC-terminated configurations (Figure 6(c, e)) also exhibit connected contours across the interface, albeit mediated by the more spatially extended Ti 3d orbitals. In marked contrast, the TiSi interface (Figure 6(b)) retains a low-density gap between the two slabs, consistent with its largest equilibrium spacing (1.9 Å) and its weakest adhesion, whereas the Si-terminated interface (Figure 6(f)) shows appreciable but visibly directional charge build-up around the interfacial Si atoms. The degree of interfacial charge-density overlap thus broadly tracks the W_ad and d₀ trends established in Section 3.2.

The charge density difference in Figure 7, defined as the density of the joined interface minus that of the isolated Ag(111) and Ti₃SiC₂(0001) slabs, resolves how electrons rearrange upon bond formation; red regions denote electron accumulation and blue regions depletion. For the C-terminated interfaces (Figure 7(a, d)), intense accumulation centres on the interfacial C atoms, whereas pronounced depletion surrounds the neighbouring Ag and Ti atoms, indicating electron transfer from both Ag and Ti toward C. An inverted, weaker pattern characterizes the TiC-terminated interfaces (Figure 7(c, e)), with depletion around the interfacial Ti and accumulation toward the adjacent C layer. The Si-terminated interface (Figure 7(f)) likewise displays visible accumulation around the interfacial Si atoms, accompanied by depletion on the Ag side, pointing to directional charge sharing; by comparison, the TiSi interface (Figure 7(b)) shows the most diffuse and weakest rearrangement of all six systems. Notably, the accumulation and depletion zones in every system decay rapidly away from the interface, and the interior of the Ag slab retains only faint, nearly uniform polarization: the interfacial electron redistribution is most pronounced on the interfacial atoms and their immediate neighbours.

Figure 8 presents the layer-resolved PDOS, and several features are common to all six terminations. The total DOS remains finite at the Fermi level (E_F), confirming that every interface is metallic. The Ag 4d band is fully occupied and confined to approximately -6 to -2 eV below E_F, leaving only a small density of Ag s and p states at E_F. Specifically, the interfacial Ag layer exhibits a slightly broadened 4d band and a modestly enhanced DOS near E_F relative to the second Ag layer, indicating that the outermost Ag atoms participate directly in interfacial bonding, whereas atoms only one layer deeper already recover bulk-like character, further evidence that the interaction is short-ranged.

On the MAX side, the third and fourth layers progressively restore the bulk Ti₃SiC₂ signature of Ti 3d-C 2p hybridization around -4 to 0 eV. These interface-induced modifications of the spectra confirm that chemical bonding, rather than physical adsorption, operates at all six interfaces. The termination-dependent differences are nonetheless striking and provide a microscopic rationale for the adhesion hierarchy. For the C-terminated interfaces (Figure 8(b, e)), the interfacial C 2p states span roughly -10 eV up to E_F and overlap the Ag 4d band in the -5 to -2 eV window, signalling C 2p-Ag 4d hybridization; distinct C 2p peaks at around -3 to -1 eV, with tails reaching E_F, further indicate that the interfacial C atoms carry appreciable spectral weight up to E_F and thus participate directly in the interfacial bonding. This p-d hybridization, reinforced by the Ag/Ti-to-C charge transfer resolved in Figure 7, renders the Ag-C interaction the strongest among the cases studied here, which explains why C(TiC) and C(TiSi) combine the largest W_ad with the smallest d₀. For the three interfaces exposing an outermost Ti layer, namely TiSi, TiC(TiC), and TiC(TiSi) (Figure 8(a, c, d)), the interfacial Ti 3d states straddle E_F with high spectral weight at and just above it, overlapping the Ag states at around -5 to -2 eV; coupling through such partially filled d bands is characteristic of predominantly metallic bonding. Among the three, the TiSi interface exhibits the feeblest interface-induced spectral rearrangement, in line with its lowest W_ad. The Si-terminated interface (Figure 8(f)) instead displays Si 3p states spread broadly from about -7 eV to above E_F, hybridizing with the Ag 4d band near -5 to -2 eV, which is consistent with a directional, moderately covalent Si-Ag interaction and its intermediate adhesion strength.

In summary, the charge density, the charge density difference, and the PDOS mutually corroborate one another: strong p-d-hybridized Ag-C bonding with appreciable charge transfer at the C terminations; metallic Ag-Ti coupling at the TiC terminations; moderate directional Ag-Si bonding at the Si termination; and only weak, diffuse interaction at the TiSi termination. Together, these findings account, at the electronic level, for the W_ad ordering established in Section 3.2.

3.4. Effect of Alloying Elements

Having identified the C(TiC)-terminated, HCP-stacked Ag(111)/Ti₃SiC₂(0001) interface as the most strongly adhering configuration in Section 3.2, we next asked whether its adhesion can be raised further by alloying, a strategy widely exploited in Ag-based electrical-contact composites. Four common alloying elements, Cu, Ni, Zn and Cr, were introduced sub-stitutionally on the Ag sub-lattice, and for each dopant four substitution depths were examined, labelled sites a-d in Figure 9(a): site a denotes the interfacial Ag layer, and sites b-d the second, third and fourth Ag layers beneath it. The thermodynamic preference of a dopant for each site was evaluated through the defect formation energy, the formula for calculating the formation energy of dopant defects is as follows[52]:

$$\mathrm{\Delta }{E}_f=E_{\mathrm{int}erface}^{doped}-E_{\mathrm{int}erface}^{clean}-E_x+E_y$$

where $E_{\mathrm{int}erface}^{doped}$ and $E_{\mathrm{int}erface}^{clean}$ are the total energies of the doped and pristine interfaces, and E_x and E_y are the chemical potentials of the dopant atom and of the substituted Ag atom, respectively. The work of adhesion of each doped interface was then computed following.

$$W_{ad}=\left(E_{total}^A+E_{total}^B-E_{\mathrm{interface}}^{A/B}\right)/A$$

Where, $E_{total}^A$ and $E_{total}^B$ represent the total surface energies of surfaces A and B after relaxation, respectively, and $E_{\mathrm{int}erface}^{A/B}$ denotes the total interfacial energy after doping. A is the area between the interfaces.

The formation energies in Figure 9(b) discriminate sharply among the four dopants. All values are positive, meaning that substitutional alloying of the Ag slab is endothermic and must be driven by non-equilibrium processing; their magnitudes, however, span nearly an order of magnitude. Cr is by far the easiest element to incorporate, with ΔE_f of only about 0.3 eV at the interfacial layer, rising monotonically to about 0.85 eV at the fourth layer. Zn and Ni occupy an intermediate range of approximately 2.1-3.0 eV, whereas Cu is the most reluctant dopant, with ΔE_f between about 3.0 eV and 3.6 eV. Notably, the interfacial Ag layer offers the lowest, or nearly the lowest, formation energy for every element examined, identifying the interface itself as the preferred destination for solute atoms; for Cr this preference is so pronounced that interfacial Cr substitution is by far the most accessible of all dopant-site combinations.

This energetic preference translates directly into interfacial strength. As shown in Figure 9(c), the work of adhesion reaches its maximum for every dopant when the solute resides at the interfacial layer, and decays progressively as the solute moves into deeper Ag layers. Cr again stands out: it delivers the strongest interface at every substitution depth, with W_ad falling from approximately 11.0 J/m² at site a to about 10.0 J/m² at site d. At the optimal site a, the adhesion ordering reads Cr (11.0 J/m²) > Ni (10.6 J/m²) > Zn (10.4 J/m²) > Cu (9.8 J/m²), and each of these values exceeds the 9.25 J/m² of the undoped interface, corresponding to reinforcements of roughly 6-19%. Equally telling is the inverse correlation between the two metrics: the element that enters the lattice most readily, Cr, also strengthens the interface most, whereas the most reluctant dopant, Cu, yields the smallest gain.

This hierarchy has a transparent electronic origin. With partially filled 3d bands, Cr and Ni can hybridize with the 2p states of the interfacial C atoms, adding a covalent d-p component to the interfacial bonding. The effect is most pronounced for Cr, a strong carbide-forming element, whose affinity for carbon both lowers its effective formation energy near the interface and anchors the two slabs together. By contrast, the closed-shell 3d dopants Cu and Zn interact only weakly with carbon and contribute little beyond metallic cohesion, which accounts for their modest adhesion gains. The rapid decay of the strengthening effect with substitution depth mirrors the spatially localized interfacial bonding established in Section 3.3: only solutes residing within the interfacial layer can engage the C atoms directly. These findings identify interfacial Cr segregation as a highly effective and energetically accessible route to strengthening Ag/Ti₃SiC₂ contacts, and more generally establish interfacial alloying as a practical design lever for metal/MAX-phase joints.

Figure 10 presents the charge density distributions of the Ti₃SiC₂(0001)/Ag(111) interfaces doped with Cr, Ni, Zn, and Cu, respectively, and the four panels collectively reveal distinct interfacial electronic structures that systematically follow the dopant-dependent stability trend. In each panel, the pronounced charge-density features are localized predominantly at the interface between the Ag overlayer and the Ti₃SiC₂ surface; the red and yellow regions correspond to high charge density, whereas the blue regions correspond to low charge density.

Among the four dopants, the Cr-doped interface (Figure 10(d)) exhibits the most pronounced interfacial charge density. Specifically, intense high-density lobes bridge the Cr atoms and the underlying Ti and C species, and the high-density region penetrates deepest into the Ti₃SiC₂ substrate. This pronounced charge-density build-up is fully consistent with the superior stability of the Cr dopant, and it further reflects maximized orbital hybridization arising from the partially filled 3d⁵ configuration of Cr, which enables optimal overlap with the Ti 3d states. The Ni-doped system (Figure 10(b)) displays a comparably strong interfacial charge density, with clearly resolved high-density regions at the interface; nevertheless, the high-density features in this system are somewhat less extensive than those in the Cr-doped case, which mirrors the intermediate stability ranking of Ni. In comparison, the Zn-doped interface (Figure 10(c)) exhibits a noticeably weaker interfacial charge density, and both the intensity and the spatial extent of its high-density regions are reduced relative to those of the Cr- and Ni-doped systems.

In stark contrast, the Cu-doped system (Figure 10(a)) presents the weakest interfacial charge density among the four interfaces, as evidenced by the markedly less intense coloration and the more confined high-density features at the interface.

Taken together, the interfacial charge density decreases systematically in the order of Cr, Ni, Zn, and Cu, which agrees well with the calculated stability sequence of Cr > Ni > Zn > Cu. This trend can be traced back to the electronic configurations of the dopant atoms. On one hand, the half-filled 3d⁵ shell of Cr facilitates strong directional bonding with the Ti 3d orbitals and thereby maximizes the charge density at the interface. On the other hand, the completely filled 3d¹⁰ configuration of Cu limits its capacity for additional orbital participation and consequently leads to the weakest interfacial charge density as well as the lowest adhesion energy. These observations therefore confirm that the magnitude of the interfacial charge density can serve as a reliable electronic descriptor for evaluating the relative thermodynamic stability of doped Ti₃SiC₂(0001)/Ag(111) interfaces.

The charge density difference maps presented in Figure 11 reveal a pronounced hierarchy of interfacial charge redistribution, which directly mirrors the dopant-dependent stability sequence of Cr > Ni > Zn > Cu. Among the four systems, the Cr-doped interface (Figure 11(d)) exhibits the most substantial charge redistribution, which is characterized by intense red accumulation zones spanning the interfacial gap between the Ag overlayer and the Ti₃SiC₂ substrate, together with considerable blue depletion regions localized around the Cr atoms. Such maximal charge transfer reflects the superior bonding capability of the half-filled 3d⁵ configuration of Cr, which not only facilitates efficient electron donation into interfacial states but also promotes strong d-orbital hybridization with the surface Ti atoms. Similarly, the Ni-doped interface (Figure 11(b)) displays robust interfacial charge redistribution, with prominent electron accumulation at the interface and well-defined depletion around the Ni atoms. Nevertheless, both the magnitude and the spatial extent of this redistribution remain visibly reduced relative to those of Cr, which is consistent with the more nearly filled 3d⁸ shell of Ni and its commensurately diminished orbital participation. In comparison, the Zn-doped interface (Figure 11(c)) exhibits only moderate charge redistribution at the interface, since the closed 3d¹⁰ configuration of Zn inherently limits charge transfer and orbital mixing. In stark contrast, the Cu-doped interface (Figure 11(a)) presents the weakest interfacial charge redistribution, displaying the least extensive and least intense accumulation regions among the four dopants. This behavior can be attributed to the combined effect of the filled 3d¹⁰ subshell and the solitary 4s¹ electron of Cu, which severely constrains both charge donation and covalent bonding capacity.

Taken together, the systematic decrease in the magnitude of charge redistribution from Cr to Cu establishes a direct correlation between interfacial charge transfer and thermodynamic stability. Furthermore, the coexistence of electron accumulation associated with ionic character and directional depletion patterns arising from covalent hybridization across all panels indicates a mixed ionic and covalent bonding character at the doped Ti₃SiC₂(0001)/Ag(111) interfaces, in which the covalent contribution scales proportionally with the stability of the doped interface.

Figure 12 presents the layer-resolved partial density of states (PDOS) of the Ti₃SiC₂(0001) /Ag(111) interfaces doped with Cr, Ni, Zn, and Cu, providing atomistic insight into the electronic origin of the calculated stability trend. Although the finite density of states at the Fermi level (E_F = 0 eV) confirms that all four interfaces retain metallic character, the magnitude and orbital composition of the interfacial states vary markedly among the dopants and correlate directly with their adhesion strengths.

Cr, which confers the highest interfacial stability, exhibits the broadest d band in the first Ag layer, spanning approximately -5 eV to 3 eV, and accordingly yields the largest DOS at E_F (2.5 states/eV) among the four dopants (Figure 12(d)). This wide dispersion reflects the partially filled 3d⁵4s¹ configuration of Cr, whose spatially extended bonding orbitals enable strong d-orbital hybridization with the second Ti layer throughout the -3 to 3 eV range, thereby maximizing the interfacial electronic stabilization. In comparison, Ni presents a narrower d band localized primarily between -4 eV and -1 eV, with a DOS at E_F of 2.0 states/eV (Figure 12(b)). The 3d⁸ configuration affords significant yet more localized hybridization with the Ti d states between -3 eV and 2 eV, and the reduced orbital availability relative to Cr accounts for the intermediate stability of Ni.

Zn displays markedly different behavior, since its fully occupied and contracted 3d¹⁰ shell generates an exceptionally sharp d peak centered near -6 eV with negligible weight at E_F (Figure 12(c)). These energetically isolated, deep-bound states are electronically inert, and their large separation from E_F suppresses interfacial hybridization, depriving the interface of the covalent contributions that stabilize the Cr- and Ni-doped systems. Cu, despite possessing a 4s¹ valence electron, yields the weakest interface. Its d band is centered at approximately -3 eV with only a moderate tail toward E_F, giving a DOS at E_F that is lower than those of Cr and Ni and second only to that of the electronically inert Zn states (Figure 12(a)), since the filled 3d¹⁰ shell constrains both the Cu d-Ti d overlap and the back-donation from Ti.

Turning to the Ti₃SiC₂ substrate, the d peaks of the second Ti layer between -2 eV and 2 eV closely mirror the hybridization features of the adjacent dopant layer, confirming active interfacial d orbital mixing, whereas the p peaks of the first C layer near -10 eV and -5 eV attest to a robust Ti-C framework that is insensitive to the dopant identity. Notably, the nearly identical PDOS profiles of the second Ag layer across the four systems, together with the minimal dopant-induced variations in the deeper Si and Ti layers, demonstrate that the doping effect is highly localized to the interfacial Ag monolayer. Collectively, these results establish a direct correlation between the dopant d band structure and the interfacial stability, in which the broad E_F-crossing d band of Cr drives the strongest hybridization, the more localized states of Ni provide moderate coupling, the deeply bound inert states of Zn afford limited stabilization, and the filled d shell of Cu produces the weakest interfacial bonding.

4. Conclusion

In this work, the heterogeneous nucleation mechanism of Ti₃SiC₂(0001)/Ag(111) interface doped with alloying elements was systematically investigated by first-principles calculations based on density functional theory. The main conclusions are summarized as follows:

(1) Six terminations of the Ti₃SiC₂(0001) surface were evaluated by surface-energy calculations as a function of the carbon chemical potential. The TiC(TiC) termination is thermodynamically preferred under carbon-poor conditions, whereas the TiC(TiSi) termination becomes the most stable once Δμ_C exceeds -1.50 eV; the C-terminated surfaces are the least stable as free surfaces across the entire Δμ_C range. Eighteen Ti₃SiC₂(0001)/Ag(111) interface models combining the six terminations with three stacking sequences (OT, MT, HCP) were constructed. The HCP stacking is energetically preferred for every termination, and the C(TiC)-terminated HCP interface is the most stable configuration, with a work of adhesion of 9.25 J/m² at an equilibrium spacing of 1.2 Å; structural relaxation enhances the work of adhesion by approximately 10-75%.

(2) Charge density, charge density difference, and layer-resolved PDOS analyses consistently reveal the electronic origin of the adhesion hierarchy. The interfacial bonding is highly localized within the interfacial monolayers. At the C-terminated interfaces, strong C 2p-Ag 4d hybridization, reinforced by electron transfer from Ag and Ti atoms to the interfacial C atoms, produces the strongest Ag-C bonding with a mixed ionic-covalent character; the TiC-terminated interfaces are dominated by metallic Ag-Ti d-d coupling; the Si-terminated interface exhibits moderate directional Ag-Si bonding; and the TiSi-terminated interface shows only weak, diffuse interactions. All interfaces retain metallic character, with finite density of states at the Fermi level.

(3) Substitutional alloying with Cu, Ni, Zn, and Cr is energetically most favorable within the interfacial Ag layer for all four dopants. Cr exhibits the lowest defect formation energy (about 0.3 eV at the interfacial site) and delivers the strongest interface at every substitution depth, raising the work of adhesion from 9.25 J/m² to 11.0 J/m² (an enhancement of about 19%), followed by Ni $10.6\mathrm{J}/\mathrm{m}2$, Zn $10.4\mathrm{J}/\mathrm{m}2$, and Cu $9.8\mathrm{J}/\mathrm{m}2$. The strengthening correlates directly with the filling of the dopant 3d band and the associated d-p hybridization with the interfacial C atoms. Interfacial Cr segregation is therefore identified as an effective and energetically accessible route to strengthening the Ti₃SiC₂/Ag interface, providing theoretical guidance for the interfacial design of high-performance Ag-Ti₃SiC₂ electrical contact composites.