Structural, Electronic, and Mechanical Properties of Zr₂SeB and Zr₂SeN from First-Principle Investigations

Abstract

MAX phases have exhibited diverse physical properties, inspiring their promising applications in several important research fields. The introduction of a chalcogen atom into a phase of MAX has further facilitated the modulation of their physical properties and the extension of MAX family diversity. The physical characteristics of the novel chalcogen-containing MAX 211 phase Zr₂SeB and Zr₂SeN have been systematically investigated. The present investigation is conducted from a multi-faceted perspective that encompasses the stability, electronic structure, and mechanical properties of the system, via the employment of the first-principles density functional theory methodology. By replacing C with B/N in the chalcogen-containing MAX phase, it has been shown that their corresponding mechanical properties are appropriately tuned, which may offer a way to design novel MAX phase materials with enriched properties. In order to assess the dynamical and mechanical stability of the systems under investigation, a thorough evaluation has been carried out based on the analysis of phonon dispersions and elastic constants conditions. The predicted results reveal a strong interaction between zirconium and boron or nitrogen within the structures of Zr₂SeB and Zr₂SeN. The calculated band structures and electronic density of states for Zr₂SeB and Zr₂SeN demonstrate their metallic nature and anisotropic conductivity. The theoretically estimated Pugh and Poisson ratios imply that these phases are characterized by brittleness.

1. Introduction

The family of materials denoted as the MAX phases is a subject of great interest within the scientific community, with a general formulation expressed as M_n+1AX_n, where n can take on values of 1, 2, or 3. In this expression, M denotes an early transition metal, A typically represents an A-group element, and X is predominantly characterized by C or N. The origins of this family of materials can be traced back to the pioneering research of Nowotny et al. in the 1960s [1,2,3,4]. Until the 1990s, the interest in MAX phases was reignited by Barsoum et al. [5,6], who revealed their exceptional properties. These materials exhibit metallic like characteristics such as high electrical and thermal conductivity, as well as machinability and mechanical strength. Additionally, they possess exceptional mechanical properties at high temperatures and exhibit highly sensible corrosion and reaction resistance, which are similar to those of ceramics [7]. All these unique properties can be attributed to their nano-layered structures, where elemental A in the style of a single atomic layer is situated between M_n+1X_n sheets. Furthermore, the bond of M-A plays a pivotal role in determining the chemical and physical characteristics. This distinctive set of properties has resulted in the identification of over 150 MAX phases that can be utilized in various applications [8,9].

Given the growing demand for MAX phase materials, researchers have been exploring ways to enhance their structural versatility and performance flexibility. This has led to the development of novel MAX phase compounds or the recombination of M, A, and/or X elements in existing structures [8]. Recently, boron (an element with an atomic number of five) was introduced as an additional X element, resulting in an expanded range of MAX phases. The remarkable physical and chemical properties of boron and corresponding compounds make them highly desirable for high-temperature applications, thereby creating a pressing need for boride MAX phases [10]. As such, the boride MAX phase, much like its conventional MAX phase counterparts, has garnered significant research attention [11,12,13,14,15,16,17]. Khazaei [11] systematically investigated the structure and properties of the Sc₂AlB, Ti₂AlB, Cr₂AlB, Zr₂AlB, and Nb₂AlB MAX phase borides. The MAX phase borides Ti₂AlB, Ti₂GaB, and Ti₂InB have also been investigated via theoretical approaches [13]. In addition, the chalcogen-containing MAX phases with more robust mechanical properties received greater attention than the corresponding aluminum-containing MAX phases [18]. Currently, the MAX phases with sulfur elements at A sites are limited to Ti₂SC, Zr₂SC, Hf₂SC, Nb₂SC, and M₂SB (M = Zr, Hf, Nb) [19]. The experimental realization of Se occupying the A site in a Zr₂SeC MAX phase has expanded the family of nano-laminated ternary carbides [20]. Recently, the DFT method was utilized to investigate the physical characteristics of novel chalcogen-containing MAX phases, Hf₂SeC and Zr₂SeC, for high-temperature applications [21,22]. This study aims to have an in-depth study of the effect of replacing carbon with boron or nitrogen as the X element on the crystal lattice constant, electronic structure, and several physical properties of Se-containing MAX phases, which would enrich the materials’ properties and extend their potential applications. In addition, to obtain complete image of chalcogen-containing MAX phases, the electronic properties of Zr₂SeC was calculated for comprehensive comparison. In this work, the predicted results of the B/N atom replacing the C atom in the ternary 211 MAX-phase nano-laminates essentially hold the potential to enhance the properties of the MAX phase materials and broaden their applications in various fields.

2. Computational Details

Throughout the whole work, density functional theory (DFT) [23] calculations were performed using the Cambridge Serial Total Energy Package (CASTEP) [24]. The electronic exchange–correlation interaction was described using the generalized gradient approximation [25] with the Perdew–Burke–Ernzerh [26] (GGA-PBE) functional. The elemental core and valance electrons were implemented by norm-conserving pseudopotentials and a plane-wave basis functional with kinetic energy cutoff of 520 eV. The main calculated electronic configurations were Zr: 4s²4p⁶4d²5s², Se: 4s²4p⁴, N: 2s²2p³, and B: 2s²2p¹. To optimize the geometric structure and cell structure, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization scheme was chosen to equilibrize the structure. Throughout the entire self-consistent field (SCF) calculation process, the difference of total energy [27], Hellman–Feynman forces on each atom, atomic displacements, and the stresses were less than 1.0 × 10⁻⁷ eV/atom, 0.002 eV/Å, 1 × 10⁻³ Å, and 0.05 GPa, respectively, to achieve the convergence threshold.

3. Results and Discussions

3.1. Analysis of Crystal Structure of Zr₂SeX (X = B, N)

The atomic models of Zr₂SeB and Zr₂SeN cells were first constructed for the optimization of geometric configuration before the investigation of the electronic structure and their respective physical properties. As depicted in Figure 1, Zr₂SeX (with B or N located at the X site) with a hexagonal crystal structure and space group of P6₃/mmc, has eight atoms (four Zr atoms, two X atoms, and two Se atoms) in a unit cell, which is identical to that of the reported Zr₂SeC.

Based on the input parameters utilized in this investigation, the optimized lattice constants of the Zr₂SeC cells exhibit strong agreement with similar structures (

Table 1. The lattice parameter data (a, c) for Zr₂SeX (X = C, B, N).

	Methods	a (Å)	c (Å)	c/a
Zr2SeC	GGA-PBE	3.487	12.631	3.622	This work
Zr2SeB	GGA-PBE	3.578	12.734	3.559	This work
Zr2SeN	GGA-PBE	3.478	12.371	3.557	This work
Zr2SeC	Experiment	3.491	12.556	3.615	[20]
Zr2SeC	GGA-PBE	3.465	12.540	3.618	[21]
Zr2SB	GGA-PBE	3.516	12.313	3.502	[18]
Zr2SC	GGA-PBE	3.420	12.205	3.568	[18]

), alongside corresponding experimental and theoretical values. Moreover, we can see that our results of Zr₂SeB and Zr₂SeN are in good agreement with other reported data [28]. After the identification and comparison of calculation results, the lattice consists of the a and c values of Zr 211 MAX compounds, and the Se, at the A site, is larger than the S, which can be ascribed to the atomic size. Specifically, the calculated values of a and c for Zr₂SeC show a 0.63% and 0.72% increase, respectively, relative to prior theoretical results.

The dynamic stabilities of both configurations of Zr₂SeB and Zr₂SeN were revealed by the theoretical calculations of the phonon dispersions [29], as presented in Figure 2. A unit cell of Zr₂SeB or Zr₂SeN has eight atoms and thus has twenty-four phonon branches (three acoustic and twenty-one optical branches). They are labeled according to their symmetries at the Γ point: TA and LA modes are the in-plane transverse and longitudinal acoustic modes, the vibration planes of these two phonons are along the ab plane direction. The vibration plane of the other transverse acoustic branch (ZA) phonon is perpendicular to the ab plane. The slopes indicate their group velocities, and the slope of LA is the largest, while the slope of ZA is smaller than that of LA and TA. In addition, the irreducible representation can be classified as Γoptical = 2E1g + 4E2g + A1g + 4E1u + 2A2u + 4E2u + 2B2g + 2B1u. The E1g, E2g, and A1g modes are the Raman active vibration modes, and the E1u and A2u modes are the IR active vibration modes. Notably, the phonon frequencies of the Zr₂SeB structure were observed to be considerably higher than those of the corresponding Zr₂SeN structure, which could be ascribed to a lighter mass of the boron element than that of nitrogen atom.

3.2. Electronic Properties

On the basis of symmetry of hexagonal crystal system, to optimize the calculation process of electronic properties of two models, a high symmetry path of G-A-L-K-H towards the Brillouin zone was adopted to investigate the electronic band structures of Zr₂SeB and Zr₂SeN (Figure 3). Due to their comparable structures, there is a certain degree of similarity in their band structures. The electronic energy bands of both configurations, as shown in Figure 3, overlap near the Fermi level (EF), indicating a metal-like conductivity that is similar to Zr₂SeC.

According to the band structure (Figure 3), it can be seen that electronic conduction is naturally anisotropic. In the K-L and H-K directions, the energy dispersion with a unit area is small along the c-direction. Conversely, conductivity within the basal plane is demonstrated by G-M, and the L-A direction is higher than that within the basal plane, indicated by K-L and H-K directions with a unit area in the c-direction. Thus, it follows that Zr₂SeB and Zr₂SeN exhibit higher basal plane conductivity than the c-directional conductivity, a characteristic analogous to most traditional MAX phases reported in the literature [18].

The density of states (DOS) for Zr₂SeB and Zr₂SeN is depicted in Figure 4. The high degree of similarity between the electronic bands of the two materials demonstrates that the DOS diagrams of then would be similar consequently. In agreement with the analysis results of energy bands, Zr₂SeB and Zr₂SeN belong to electronic conductors, according to the several orbit states (such as B-p and Zr-d) that occupy the Fermi level. The contribution from the different states of Zr, Se, N, and B to the total DOS is confirmed from the partial DOS (PDOS) diagrams. For instance, Zr’s 4d orbit state provides the dominant contribution around the Fermi level, corresponding to the electronic structure of the reported Zr₂SeC [20,21]. Neither N nor Se contributes to the DOS at the Fermi level, which is consistent with prior experimental and theoretical results [18,21]. Moreover, obvious hybridization could be observed from the PDOS diagrams in both Zr₂SeB and Zr₂SeN. In Figure 4a, as for Zr₂SeB, B’s 2p state splits, owing to strong hybridization with Zr’s 4d state. In Zr₂SeN, similarly, a strong hybridization between N’s 2p state and Zr’s 4d state is observed in the energy range from −7.5 to −4.5 eV. Furthermore, in both Zr₂SeN and Zr₂SeB, it is non-negligible that Se’s p state is hybridized with Zr’s d states.

3.3. Mechanical Properties

Dynamic characteristics are significant for evaluating a material’s performance. The elastic constants predict these traits and behaviors, which can represent important macroscopic properties. Herein, the stress–strain method was applied to calculate the various elastic constants of Zr₂SeB and Zr₂SeN, and corresponding values are presented in

Table 2. The calculated mechanical properties ($C_{ij}$, $B$, $G$, and $E$ in GPa, ${\theta }_D$ in K, $\sigma$) of Zr₂SeB and Zr₂SeN compared with other theoretical and experimental results.

MAX	${\mathit{C}}_{11}$	${\mathit{C}}_{12}$	${\mathit{C}}_{13}$	${\mathit{C}}_{33}$	${\mathit{C}}_{44}$	${\mathit{C}}_{66}$	$\mathit{B}$	$\mathit{G}$	$\mathit{E}$	$\mathsf{\sigma }$	${\mathsf{\theta }}_{\mathit{D}}$	$\mathit{B}/\mathit{G}$	Ref.
Zr2SeC	276	75	93	289	126	100	151	109	264	0.20	512	1.38	[20]
Zr2SeB	264	107	105	303	138	78	162	106	260	0.23	498	1.54	This work
Zr2SeN	274	75	100	279	121	99	153	105	256	0.22	499	1.45	This work

. As the MAX phases are hexagonal in crystal symmetry, they possess six elastic constants $C_{ij}\left(C_{11},C_{12},C_{13},C_{33},C_{44}-C_{55},C_{66}\right)$ Of these, $C_{66}$ is dependent $\left(C_{66}=\left(C_{11}-C_{12}\right)/2\right)$. The mechanical stability of a material under load is a critical factor in practical applications, and the stability conditions [30] for hexagonal systems dictate that $C_{11}>\left|C_{12}\right|$, $\left(C_{11}+C_{12}\right)C_{33}>2C_{13}^2$, $C_{44}>0$, and $C_{66}>0$. Consequently, the mechanical stability of Zr₂SeB and Zr₂SeN should be fulfilled by the four aforementioned conditions. From the obtained elastic constants, other important parameters can also be calculated, such as the bulk modulus, $B$; the shear modulus, $G$; Young’s modulus, $E$; Poisson’s ratio, $\sigma$; and Debye’s temperature, ${\theta }_D$, using relevant equations with the software [31,32,33].

$$B_R=\frac{C_{33}\left(C_{11}+C_{12}\right)-2C_{13}^2}{C_{11}+C_{12}+2C_{33}-4C_{13}}$$

(1)

$$B_V=\frac{1}{9}\left[2\left(C_{11}+C_{12}\right)+C_{33}+4C_{13}\right]$$

(2)

$$G_R=\frac{5}{2}\frac{C_{44}\left(C_{11}-C_{12}\right)\left[C_{33}\left(C_{11}+C_{12}\right)-2C_{13}^2\right]}{3B_V{C}_{44}\left(C_{11}-C_{12}\right)+\left\{2C_{44}+\left(C_{11}-C_{12}\right)\left[C_{33}\left(C_{11}+C_{12}\right)-2C_{13}^2\right]\right\}}$$

(3)

$$G_V=\frac{1}{30}\left[C_{11}+C_{12}+2C_{33}-4C_{13}+12C_{44}+6\left(C_{11}-C_{12}\right)\right]$$

(4)

$$B=\frac{1}{2}(B_R+B_V)$$

(5)

$$G=\frac{1}{2}(G_R+G_V)$$

(6)

$$E=\frac{9BG}{3B+G}$$

(7)

$$\sigma =\frac{3B-2G}{2\left(3B+G\right)}$$

(8)

$$v_l=\sqrt{\frac{3B+4G}{3\rho }}$$

(9)

$$v_t=\sqrt{\frac{G}{\rho }}$$

(10)

$$v_m={\left[\frac{1}{3}\left(\frac{2}{v_t^3}+\frac{1}{v_l^3}\right)\right]}^{-1/3}$$

(11)

$${\theta }_D=\frac{h}{k}{\left[\frac{3n}{4\pi }\left(\frac{N_A\rho }{M}\right)\right]}^{1/3}{v}_m$$

(12)

Here, $v_l$ and $v_t$ represent longitudinal and transverse sound velocities, respectively. $\rho$ is the density of the cell, $v_m$ is the averaged sound velocity, $h$ is Planck’s constant, and $k$ is Boltzmann’s constant.

Elastic constants provide crucial insights into bonding behaviors across different crystallographic planes. Specifically, the Zr₂SeX (X = C, B, N) compounds exhibit greater compression along the c-axis than the a-axis. This observation is supported by the lattice parameters of the MAX phases, which indicates a preferential compression along the c-axis, as opposed to that along the a-axis. This trend is a common feature of MAX phases and is reflected in their elastic anisotropic characteristics.

To analyze and estimate the brittleness or toughness of a material, Poisson’s ratio ($v$), as a critical parameter, is usually tested. Traditionally, the transition value of 0.26 is the threshold for evaluating whether a material is brittle or ductile. As shown in Table 2, the Zr₂SeN and Zr₂SeB MAX phases are relatively brittle when compared to Zr₂SeC and Zr₂SB. Furthermore, Pugh’s ratio is a valuable tool in predicting ductile or brittle failure modes by examining the ratio of bulk to shear moduli. A critical value of 1.75 is used to classify materials as either ductile or brittle, with a B/G value greater than 1.75, indicating a ductile character. The MAX phases of Zr₂SeN and Zr₂SeB are classified as brittle, similar to the previously reported Zr₂SeC.

It can be inferred that Zr₂SeB possesses higher $B$ and $G$ values when compared to Zr₂SeN (Table 2). This observation indicates that Zr₂SeB necessitates greater pressure than Zr₂SeN for bulk and plastic deformation. Furthermore, the $E$ values of Zr₂SeB surpass those of Zr₂SeN, suggesting that Zr₂SeB exhibits greater hardness than Zr₂SeN. Moreover, $C_{44}$, an important indicator of material hardness, exhibits a strong correlation with hardness in comparison to other elasticity moduli. Consequently, Zr₂SeB is expected to possess a higher $C_{44}$ than Zr₂SeN, thereby enhancing its hardness. In contrast, Zr₂SeN’s hardness is lower than Zr₂SeC and Zr₂SeB. These findings offer new clues for tuning the X-composition of the substituted MAX phase materials, potentially leading to improved performance in various applications.

Debye’s temperature, ${\theta }_D$, helps to predict the application of the material at high temperatures. Using the Anderson model, the ${\theta }_D$ of Zr₂SeN and Zr₂SeB are 499 K and 498 K, respectively, which are lower than that of Zr₂SeC (512 K, calculated using Anderson’s model; 679 K, calculated via the quasi-harmonic Debye model [21]). The ${\theta }_D$ of Zr₂SeB (498 K), calculated via Anderson’s model, is lower than that of Zr₂SB (540 K, obtained using the quasi-harmonic Debye model [18]). Comparable results are seen in other carbides MAX 211 phases, such as the reported Zr₂SC, Hf₂SB/C, and Nb₂SC/B [18,34]. In the high-temperature applications of Zr₂SeN and Zr₂SeB, such as thermal barrier coating (TBC), the Debye temperatures are required.

Mechanical anisotropy is one of the non-negligible factors that are closely related to the potential applications of functional materials. For example, in practical applications, the material itself may produce micro-cracks or undergo deformation in different directions, which is limited by their intrinsic mechanical properties. Thus, the mechanical anisotropies of Zr₂SeB and Zr₂SeN are investigated, and corresponding data were recorded in the forms of 2D and 3D. Visually, as shown in Figure 5 and Figure 6, Young’s modulus and shear modulus of Zr₂SeB and Zr₂SeN are direction-dependent. Usually, in terms of measurement of elastic moduli, the spherical shape of the curved surface in 3D and the circular shape of plots in 2D indicate the isotropic mechanical behavior of solids. However, deviations from spherical/circular symmetry or symmetry breaking indicate that the mechanical properties of the measured object are anisotropic. Meanwhile, the degree of anisotropy of the elastic moduli of a substance is measured through the amount of deviation from a perfect sphere/circle. Figure 5 shows the directional dependence of E for Zr₂SeB and Zr₂SeN. As can be seen in Figure 5, E is isotropic in the xy plane, and its plot shape is uniformly circular. In contrast, E is anisotropic in the xz and yz planes. It can be seen in Figure 6 that the G of Zr₂SeB and Zr₂SeN does not change direction on the xy plane, and the two-dimensional graph is uniformly circular but changes direction on both the xz and yz planes. These are essentially identical to the symmetry of hexagonal crystals and are consistent with the findings of M.A. Hadi et al [18].

By mechanical analysis, the Zr₂SeN and Zr₂SeB phases can be tentatively identified as elastically anisotropic. The three shear anisotropy coefficients that depend on the hexagonal crystal of $C_{ij}$ [35], which quantify the degree of elastic anisotropy, can be obtained as follows [36]:

$$A_1=\frac{\left(C_{11}+C_{12}+2C_{33}-4C_{13}\right)}{6C_{44}}$$

(13)

which is associated with the {100} shear planes in the 〈011〉 and 〈010〉 directions;

$$A_2=\frac{2C_{44}}{C_{11}-C_{12}}$$

(14)

which is related to the {010} shear planes in the 〈101〉 and 〈001〉 directions;

$$A_3=\frac{\left(C_{11}+C_{12}+2C_{33}-4C_{13}\right)}{3\left(C_{11}-C_{12}\right)}$$

(15)

which denotes the shear anisotropy that occurs in the 001 shear planes in the 〈110〉 and 〈010〉 directions. As is well known, in anisotropic crystals, $A_i\left(i=1,2,3\right)$ would have a value other than unity. In contrast, all factors of $A_i\left(i=1,2,3\right)$ would have a unit value in all isotropic systems [37]. Moreover, the deviation of $A_i$ from unity ($\Delta A_i$) could determine the degree of elastic anisotropy in the shear. Subsequently, Zr₂SeN and Zr₂SeB exhibit elastically anisotropic in the shear (

Table 3. Elastic anisotropy factors for Zr₂SeX (X = C, B, N) MAX phases.

MAX	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$	${\mathit{k}}_{\mathit{c}}/{\mathit{k}}_{\mathit{a}}$	${\mathit{A}}_{\mathit{B}}$	${\mathit{A}}_{\mathit{G}}$	Ref.
Zr2SeC	0.73	1.57	1.14	0.84	0.64	0.30	[21]
Zr2SB	0.85	1.35	1.14	0.91	0.04	0.90	[18]
Zr2SeB	0.67	1.76	1.18	0.81	0.63	0.30	This work
Zr2SeN	0.70	1.22	0.85	0.83	0.63	0.31	This work

). Moreover, regarding the specific evaluation of elastic anisotropy for the hexagonal crystal, there is another anisotropy factor that is obtained from $C_{ij}$, $k_c/k_a=\left(C_{11}+C_{12}-2C_{13}\right)/\left(C_{33}-C_{13}\right)$ [38]. In the formula, $k_a$ and $k_c$, respectively, represent the linear compressibility coefficients along the a and c axes. All the values of $k_c/k_a$ (Table 3), which are different from unity ($\Delta k_c/k_a$), demonstrating the degree of anisotropy of Zr₂SeN and Zr₂SeB under a linear compression in the $a$ and $c$ directions.

Hill’s theory proposed a proportional relationship between difference between $B_V$ and $B_R$, and the elastic anisotropy level of crystals. The same relationship is appropriate for the difference between $G_V$ and $G_R$ as well. Then, the percentage of anisotropy factors $A_B$ and $A_G$ can be calculated as follows:

$$A_B=\frac{B_V-B_R}{B_V+B_R}\times 100\%$$

(16)

$$A_G=\frac{G_V-G_R}{G_V+G_R}\times 100\%$$

(17)

Considering the compressibility and shear, these two coefficients are assigned zero values for fully isotropic crystals. We can conclude that the Zr₂SeB and Zr₂SeN phases are anisotropic, as confirmed by the results in Table 3. The current knowledge about the anisotropy of Zr₂SeN and Zr₂SeB contributes to their mechanical stability during specific physical processes, including the occurrence of plastic deformation and the generation of microscale cracks.

4. Conclusions

In summary, the electronic structure and several mechanical properties of two chalcogen-containing ternary MAX phases, Zr₂SeB and Zr₂SeN, were investigated systematically via DFT calculations. The lattice parameters of Zr₂SeB and Zr₂SeN are consistent with those of Zr₂SeC and Zr₂SB, and these MAX phases exhibit dynamical and mechanical stability. Through the analyses of band structure and the density of states, the electronic character of Zr₂SeB and Zr₂SeN is identified as a metal, which is consistent with that of the conventional MAX phases. Furthermore, we investigate the mechanical properties of Zr₂SeB and Zr₂SeN and compare them with those of Zr₂SB and Zr₂SeC, obtained from previous studies. The intrinsic anisotropy of Zr₂SeB and Zr₂SeN in bonding strength along the a- and c-axis is revealed via calculations, and the mechanical properties of Zr₂SeB and Zr₂SeN, such as elastic anisotropic characteristics, brittleness, hardness, and mechanical stability, are comparable with those of other prior-reported chalcogenide–MAX phases. From the inspiration of this work, it is worth noting that the mechanical properties and electronic structure of MAX phases can be obviously modulated by designing the X element rationally. Thereby, the construction of two MAX-phase materials in this work and corresponding calculations provide an effective strategy for selecting and optimizing MAX phases towards broader applications.