DFT-driven insights into the electronic, magnetic, and transport properties of a 2D Nb₃C₂ MXene for high-performance Li-ion batteries

Abstract

‘MXenes’, two-dimensional materials, have attained significant attention for their outstanding characteristics inherent to their nanostructures. However, to date, the Nb₃C₂ MXene, particularly in Cr-doped form, has not been theoretically explored for Li-ion battery applications. In this work, first-principles calculations were performed to explore the structural, electronic, magnetic, and transport properties of newly designed pristine Nb₃C₂ and Cr-doped Nb₃C₂, along with their energy storage potential, using the FP-LAPW approach. Both structures are dynamically and thermally stable, as confirmed from phonon dispersion and AIMD simulations. Electronic properties, including the band structure and density of states, indicate metallic behavior in both structures with an indirect band gap, fulfilling a key requirement for electrode materials in energy storage systems. Pristine Nb₃C₂ exhibits an essentially non-magnetic ground state, while Cr-doped Nb₃C₂ exhibits ferromagnetic behavior. Theoretical capacities of 169 mAh g⁻¹ for pristine Nb₃C₂ and 280 mAh g⁻¹ for Cr-doped Nb₃C₂ were obtained, indicating a substantial enhancement upon Cr doping and exceeding that of pristine Nb₂C (170 mAh g⁻¹) reported in the literature. The predicted electrochemical properties unveil that both pristine and Cr-doped Nb₃C₂ possess favorable open-circuit voltages within the desirable range for anode materials, along with high electronic conductivity and improved gravimetric capacity. Furthermore, transport property analysis based on semi-classical Boltzmann theory highlights their promising thermoelectric behavior, complementing their electrochemical performance and providing a comprehensive evaluation of MXenes in energy storage devices.

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1. Introduction

The successful synthesis and research on graphene have compelled scientists to explore more graphene-like 2D materials, both experimentally and theoretically, made of elements beyond carbon, such as phosphorene, hexagonal boron nitride (h-BN), and transition metal dichalcogenides. These materials possess unique properties, such as a high surface area-to-volume ratio and excellent electrical and thermal conductivity, making them promising candidates for use in supercapacitors and batteries.¹

However, the performance of current batteries is insufficient to adequately satisfy the escalating demand for extensive energy storage. In order to address and comprehend this challenge, an immediate imperative arises to actively pursue novel materials that exhibit enhanced performance attributes within energy storage systems.² Recently, a new class of 2D materials known as MXenes have been drawing much interest because of their exceptional geometries and electronic topologies, i.e., excellent conductivity, fast ion diffusion, and hydrophilic nature, which make them competent materials for storage systems.³ MXenes are graphene-like layered 2D early transition metal carbides, nitrides, or carbonitrides that have been prepared by selectively etching the A layers from 3D MAX phases (M_n+1AX_n, where n ranges from 1 to 3), for example, M₂AX, M₃AX₂, and M₄AX₃, where M denotes a transition metal (e.g., Ti, Nb, Mo, Ta, Sc, V, Zr, etc.); A is an element from group III-A or IV-A (Al, Si, or Ga); and X corresponds to carbides, nitrides, or carbonitrides.⁴ Because the M–X bond is stronger than the M–A bond, the A atom can be selectively etched to yield 2D MXenes.⁵

A number of studies have been carried out on Nb₂C and Nb₄C₃ MXenes because of their potential use in energy storage devices, catalysis, and biomedical applications. Ghidiu et al. reported the synthesis and characterization of the Nb₄C₃ MXene obtained by selectively etching Nb₄AlC₃.⁶ Shen et al. demonstrated the synthesis of the Nb₂C MXene and explored its electrochemical performance for supercapacitors.⁷ Nishat et al. described the electrochemical performance of Nb₂C and Nb₂CO₂ for Na- and Li-ion batteries using an ab initio approach.⁸ Although several studies have been conducted on Nb-based MXenes, such as Nb₂C and Nb₄C₃, studies on Nb₃C₂ remain limited. Furthermore, Cr doping in Nb₂C and Nb₄C₃ MXenes has not yet been explored.^6,7 In particular, a systematic investigation of their structural stability and electronic, magnetic, and electrochemical properties upon transition metal doping, especially with Cr, has not been explored. Hence, a detailed first-principles study is required to evaluate the potential of Cr-doped Nb₃C₂ MXenes for high-performance Li-ion battery applications.

Furthermore, niobium-based MXenes show superior performance as anode materials in Li-ion batteries compared to Ti-based MXenes, as reported in the literature.⁹ Nb₃C₂ holds a different M/X ratio compared to other members of the MXene family, making it unique and influencing its layer thickness, electron density, surface chemistry, and mechanical strength, thereby positioning it as a promising and versatile candidate for future energy devices.¹⁰ Due to its layered configuration, the Nb₃C₂ MXene possesses a relatively small interlayer spacing, which hinders the intercalation and deintercalation of Li⁺ ions during battery operation.¹¹ To overcome this limitation, various dopant atoms have been introduced to expand the interlayer distance and investigate their influence on the electrocatalytic behavior of MXenes.¹² Doping not only increases the lattice parameters but also tunes the band gap and enhances the electrical conductivity, thereby improving the overall electrochemical performance of the material.¹³

In recent years, several studies have demonstrated that doping can effectively modify the interlayer spacing and surface chemistry of MXenes, thereby improving their electrochemical performance.¹⁴ Among potential dopants, Cr is notable for its good electrical conductivity and ionic radius (0.62 Å), closely matching that of niobium (0.64 Å), making it a promising candidate for Nb₃C₂ MXene doping.¹⁵

Herein, a systematic first-principles investigation of pristine Nb₃C₂ and Cr-doped Nb₃C₂ MXenes is performed to explore their structural stability, electronic structure, magnetic behavior, transport properties, and electrochemical performance as potential anode materials for Li-ion batteries. Furthermore, the effect of Cr doping on the structural, electronic, and electrochemical properties of Nb₃C₂ is comprehensively examined.

The findings of this work provide valuable theoretical insights into the properties of the Nb₃C₂ MXene and highlight the potential of Cr doping to improve its electrochemical performance, thereby supporting future experimental investigations and the development of advanced MXene-based energy storage devices.

2. Computational method

All computational calculations were executed within the density functional theory (DFT) framework, employing the full-potential augmented plane waves plus local orbital (FP-LAPW+lo) method, as implemented through the WIEN2k code.¹⁶ The exchange–correlation potential energies of the electrons were calculated in the generalized gradient approximation (GGA) via the Perdew–Burke–Ernzerhof (PBE) functional.¹⁷

The two-dimensional Nb₃C₂ MXene structure was constructed by selectively etching the A-group element from its corresponding MAX phase precursor to obtain a Nb₃C₂ monolayer. A vacuum layer of 15–20 Å was introduced along the c-direction to avoid the interactions between periodic layers. To evaluate the effect of substitutional Cr doping, one Nb atom in the supercell was substituted with a Cr atom. All structures were fully optimized prior to the electronic, magnetic and transport property calculations.^2,6

Firstly, the unit cell and supercell structure of Nb₃C₂ and Cr-doped Nb₃C₂ were generated and optimized at 1000 (19 × 19 × 2) and 200 k-points within a 2 × 12 × 7 k-mesh in the irreducible Brillouin zone (IBZ), respectively, using GGA-PBE. The expansion of valence wave functions within the Nb spheres was conducted up to l_max = 10. Additionally, the Fourier expansion of the charge density was pursued up to G_max = 14. For structural and electronic calculations, the potential was expanded within the Nb spheres, employing a combination of spherical harmonic functions and the Brillouin zone. The cut-off energy for the separation of valence and core electrons was chosen as −6 Ry, which corresponds to RMT K_max = 7, where RMT is the smallest of all atomic sphere radii and K_max determines the truncation of the reciprocal lattice expansion of the wave functions in the interstitial region.^18,19 Convergence of the total energy was set to 10⁻⁵ eV between successive iterations, with a Gaussian smearing factor of 0.05 eV. Atomic positions were fully relaxed until the maximum force on each atom was below 10⁻³ eV Å⁻¹. The plane-wave basis set was taken with an energy cut-off of 600 eV. For the electronic structure analysis, the Fermi level (E_F), defined as the highest occupied electronic state at 0 K, was set to 0 eV as the reference energy.^20,21

Phonon dispersion calculations were performed using the PHONOPY package interfaced with WIEN2k to evaluate the dynamical stability of pristine and Cr-doped Nb₃C₂ structures. In addition, ab initio molecular dynamics (AIMD) simulations were carried out to examine thermal stability. The AIMD simulations were conducted in the canonical ensemble (NVT) using a Nosé–Hoover thermostat at 500 K. A time step of 1 fs was employed, and the total simulation time was 100 ps, corresponding to 10 000 MD steps.²²

Spin-polarization calculations were carried out to determine the magnetic ordering of the 2D MXenes. Pristine Nb₃C₂ exhibited negligible magnetism (very small magnetic moments within numerical error), while Cr-doping stabilized a ferromagnetic state. The magnetic behavior was further evaluated using both GGA-PBE and GGA-PBE+U, where U represents the Hubbard potential. The U values of Nb (0.22 Ry = 3 eV)²⁰ and Cr (U = 0.15 Ry = 2 eV) were adopted from the literature, where similar parameters successfully captured the localized d-electron correlations.²³

Although applying the U slightly increased the bandgap, the valence and conduction band levels near the Fermi level remained consistent, ensuring reliable results for electronic and optical properties.²⁴ The simplicity of this approach, along with the availability of all required inputs from first principles, makes it highly suitable for high-throughput computations, where it has gained wide acceptance.²⁵ The electronic transport properties were calculated using the BoltzTraP2 code interfaced with WIEN2k under the constant relaxation time approximation (RTA), yielding the thermal and electrical conductivities.²⁶ To evaluate the Li storage capability of pristine and Cr-doped Nb₃C₂ for Li-ion batteries, we computed the adsorption energy,²⁷ the open circuit voltage (OCV),²⁸ and the theoretical capacity (Q).²⁹

The adsorption energy (E_ads) was determined using:

The open-circuit voltage (OCV) was calculated using:

V = [E_total(Nb₃C₂) + xE_total(Li) − E_total(Li_xNb₃C₂)]/x (2)

The theoretical capacity (Q) was determined from:Here, E_total(Nb₃C₂) and E_total(Li_xNb₃C₂) denote the total energies per formula unit before and after Li intercalation, while E_total(Li) is the energy per Li atom in bulk lithium; multiplying by x accounts for the total energy of x intercalated Li atoms during the process. In the theoretical capacity formula, ‘n’ is the number of electrons transferred per formula unit, F is the Faraday constant, and M_f is the mass of the formula unit.

3. Results and discussion

3.1. Structure prediction of novel MXenes

According to results obtained via first-principles DFT calculations, the newly designed Nb₃C₂ MXene structure was found to belong to the hexagonal geometry with the space group number 194 (P6₃/mmc) and lattice parameters a = b = 3.131 Å, c = 19.24 Å with α = β = 90° and γ = 120°. Two inequivalent niobium atoms were identified as Nb(1) and Nb(2), along with one carbon atom, having fractional coordinates Nb(1) (2/3, 1/3, 0.875), Nb(2) (0, 0, 0.5) and C: (1/3, 2/3, 0.924). The obtained lattice constant values are very close to the reported literature value of 3.3008 Å.³⁰

To elucidate the effect of Cr doping in the Nb₃C₂ structure, a 2 × 1 × 1 supercell was generated. This supercell contains six Nb atoms and two C atoms. A 25% Cr-doped Nb₃C₂ structure was then optimized using 1000 k-points within a 2 × 12 × 7 k-mesh in the irreducible Brillouin zone (IBZ). The schematic views of pristine and Cr-doped Nb₃C₂ are shown in Fig. 1.

Fig. 1 (a) Crystal structure of pristine Nb₃C₂. (b) Schematic view of the 2 × 1 × 1 supercell of Cr-doped Nb₃C₂.

3.2. Structural and thermal stability

The phonon dispersion curves for both pristine and Cr-doped Nb₃C₂ are presented in Fig. 2. It is evident that no imaginary (negative) frequencies are present throughout the Brillouin zone, confirming the dynamical stability of both structures. A slight shift in phonon band separation is observed in the frequency range of approximately 10–15 THz between pristine and doped Nb₃C₂ (Fig. 2a and b), indicating modifications in interatomic interactions and bonding strength due to doping. Both systems exhibit three acoustic and several optical branches, as typically observed in two-dimensional materials.

Fig. 2 Phonon dispersion curves of (a) pristine Nb₃C₂ and (b) Cr-doped Nb₃C₂ MXenes.

The slope of the acoustic branches is slightly higher in pristine Nb₃C₂ as shown in Fig. 2a, compared to the doped structure in Fig. 2(b), suggesting a relatively higher phonon group velocity. Furthermore, the absence of a distinct gap between acoustic and optical modes may enhance acoustic–optical phonon scattering, which can influence the thermal conductivity of both pristine and doped Nb₃C₂ MXene layers.³¹

The energy evolution of pristine and Cr-doped Nb₃C₂ MXene systems during the AIMD simulations is presented in Fig. 3, with pristine Nb₃C₂ shown in Fig. 3a and Cr-doped Nb₃C₂ in Fig. 3b. The total energy of both structures remains nearly constant throughout the simulation, indicating good numerical stability. The potential, kinetic, and non-bonding energies exhibited moderate fluctuations around their average values, which are expected due to the thermal motion of atoms. For pristine Nb₃C₂ (Fig. 3a), energy variations remain confined within a narrow range over the 10 ps simulation, suggesting that the structural framework remains intact without significant distortion. A similar trend is observed for the Cr-doped Nb₃C₂ system (Fig. 3b), where the energy components fluctuate around equilibrium values without any noticeable drift in total energy. These stable energy oscillations confirm that Cr incorporation does not compromise the structural integrity of the MXene lattice, demonstrating that both systems maintain thermal stability under the simulated conditions.²²

Fig. 3 AIMD simulation of (a) pristine Nb₃C₂ and (b) Cr-doped Nb₃C₂ at 500 K, showing energy evolution with atoms in their most stable position.

3.3. Electronic properties of pristine and Cr-doped Nb₃C₂

Exploring the electronic properties of a potential electrode requires determining whether the material exhibits metallic, semiconducting, or insulating behavior.³² In this study, Cr atoms were employed as dopants to investigate the electronic characteristics, including charge density distribution, band structure, and density of states (DOS). The energy gaps between valence and conduction bands were identified from the calculated density of states (DOS) and band structure.³³

Charge density plots provide direct insight into the bonding characteristics of the system.³⁴ Fig. 4(a) and (b) shows the electron density of pristine and Cr-doped Nb₃C₂ along the (100) and (110) planes. It is revealed that Cr-doping modifies the bonding nature between Nb and C atoms. As shown in Fig. 4(a), the charge density contour of pristine Nb₃C₂ indicates a covalent bond between C and Nb atoms, while in Fig. 4(b), Cr-doped Nb₃C₂ exhibits a covalent bond between Cr and C atoms due to the overlap of their valence electrons. In addition, the spherical charge distribution around Nb atoms in Nb₃C₂ indicates ionic bonding between the electronegative C atoms and electropositive Nb atoms. In the Cr-doped Nb₃C₂, the weak covalent interaction between Nb and C atoms suggests a weaker hybridization tendency of Nb towards C, as compared with pristine Nb₃C₂.³⁵ These features confirm that Cr doping significantly alters the bonding environment and electronic interactions in Nb₃C₂.

Fig. 4 Charge density distribution of (a) pristine Nb₃C₂ and (b) Cr doped Nb₃C₂ along the (100) plane and (110) crystallographic planes. The plots illustrate charge localization and bonding characteristics between Nb–C and Nb–Cr atoms, highlighting the electronic modification of electronic interactions upon Cr doping.

The interpretation of the band structure provides detailed insights into the electronic behaviour of materials, serving as a fundamental indicator of their conductivity and their responsiveness to doping-induced modifications. The band structure of pristine Nb₃C₂ (Fig. 5a) indicated metallic behavior, as the band gap was found to be zero.³⁶

Fig. 5 Band structures of (a) pristine Nb₃C₂ calculated using sp-GGA and (b) Cr-doped Nb₃C₂.

In contrast, Cr-doping (Fig. 3b) increased the density of electronic states around the Fermi level; however, the band gap for the Cr doped Nb₃C₂ structure also remained zero. This confirms that Nb₃C₂ retains its metallic character even after Cr doping. Moreover, Cr doping increases the electronic states above the Fermi level due to the additional energy levels introduced by Cr atoms within the crystal lattice, as evident from Fig. 5b.³⁷

This expansion is shown to be due to the n-type doping effect of Cr atoms, bringing about distinct modifications in the pristine Nb₃C₂ band structure, which in turn influences the electrical conductivity of the material.³⁸ Band gap analysis therefore suggests that both pristine and doped Nb₃C₂ exhibit effective electrical conductivity, as this enables efficient electron transport during electrochemical processes.³⁹

To estimate the number of states at different energy levels occupied by electrons, the partial and total density of states (PDOS and TDOS) for pristine and Cr-doped Nb₃C₂ were calculated, as shown in Fig. 6 and 7, respectively. For the TDOS, Fig. 6(a) shows a maximum peak in the density of states around −6.5 eV, indicating a high concentration of electrons at this energy level.⁴⁰

Fig. 6 (a) Total density of states (TDOS) of pristine Nb₃C₂ and (b)–(d) partial density of states (PDOS) of C, Nb₁, and Nb₂ atoms, respectively.

Fig. 7 Partial density of states (PDOS) of Cr-doped Nb₃C₂ in a 2 × 1 × 1 supercell: (a) Cr₁, (b) Nb₁, (c) Cr₂, (d) Nb₂, (e) Nb₃, (f) Nb₄, and (g) C atoms.

For the PDOS of pristine Nb₃C₂ (Fig. 6a–d), the maximum electronic states at the Fermi level (E_F) mainly originate from the Nb₁-4d orbital states.⁴¹ The valence band is dominated by C-2p states, with an admixture of Nb d-orbital contributions, confirming covalent Nb–C bonding.⁴² Above the Fermi level, the conduction band is primarily composed of Nb₁-4d orbitals, with additional contributions from Nb₂-4d states, suggesting anisotropic electronic transport.⁴³ The dominance of Nb-4d orbitals near the Fermi level confirms its metallic conductivity, which is beneficial for electrochemical performance when used as an electrode material.⁴⁴

Cr doping significantly increases the density of states per electron volt. At the Fermi level, the dominant contributions arise from Cr₁-3d and Cr₂-3d orbitals, with smaller contributions from Nb₁-4d and Nb₂-4d states. The upper part of the conduction band consists of states from Cr₁, Cr₂, Nb₁, Nb₂, Nb₃, and Nb₄ d orbitals. In the energy range of 2–4 eV above E_F, the conduction band is mainly dominated by Nb₃-4d and Nb₄-4d orbitals, with minor contributions from Cr₁-3d, Cr₂-3d, and Nb₁-4d and Nb₂-4d states. Meanwhile, C-2p orbitals contribute negligibly to the conduction band but play a major role in the valence band through hybridization with Nb₃-4d and Nb₄-4d orbitals.

As evident from Fig. 7, the density of states in both the valence and conduction bands cross the Fermi level without opening a gap, thereby confirming the metallic character of Cr-doped Nb₃C₂.⁴⁵ The PDOS plots show the orbital contributions of Cr, Nb, and C atoms in Cr-doped Nb₃C₂. Cr atoms (a) and (c) mainly contribute through d-states near the Fermi level, indicating strong hybridization. Nb atoms (b)–(f) also show dominant d-orbital contributions, crucial for electronic conductivity. C atoms (g) contribute primarily through p-states, supporting the overall electronic interaction in the heterostructure.

3.4. Magnetic properties

Magnetism in MXenes has not been extensively explored for the M₃C₂ phase, either theoretically or experimentally. For comparison, Ti₃C₂ magnetism has been briefly studied by Shien et al., who reported ferromagnetic behavior.^46,47

Herein, pristine Nb₃C₂ exhibited an essentially non-magnetic ground state, as the calculated magnetic moments for Nb₂ were −0.00002µ_B (GGA) and −0.00007µ_B (GGA+U), which were negligibly small and within the margin of computational error.⁴⁸ Although the PDOS of Nb₂ shows a slight asymmetry between spin-up and spin-down states (Fig. 8a), the imbalance is too weak to generate a measurable magnetic response in the whole compound. Consequently, the net magnetic moment is effectively zero, confirming the non-magnetic nature of pristine Nb₃C₂.

Fig. 8 Density of states (DOS) of pristine Nb₃C₂ calculated using spin-polarized GGA (sp-GGA): (a) total DOS (TDOS), (b) DOS and PDOS of C-2p states, (c) TDOS and PDOS of Nb₁-4d states, and (d) TDOS and PDOS of Nb₂-4d states.

Fig. 8(b)–(d) further illustrates the DOS and PDOS of C-2p, Nb₁-4d, and Nb₂-4d states. By applying the Hubbard potential (GGA+U), the magnetic moment of the carbon atom became aligned with that of Nb₂ and anti-parallel to Nb₁, which enhanced the overall magnetic moment. This increase arises because the GGA+U correction shifts the partial density of states of Nb₁ and Nb₂, accounting for stronger electron–electron correlation effects⁴⁹ as shown in Fig. 9. Cr doping in Nb₃C₂ enhanced the magnetism and stabilized a ferromagnetic ground state.

Fig. 9 Density of states (DOS) of pristine Nb₃C₂ calculated using spin-polarized GGA with Hubbard U correction (sp-GGA+U): (a) total DOS (TDOS), (b) TDOS and PDOS of C-2p states, (c) TDOS and PDOS of Nb₁-4d states, and (d) TDOS and PDOS of Nb₂-4d states.

Table 1 shows that pristine Nb₃C₂ is nearly non-magnetic with negligible total magnetic moment using both PBE-GGA and PBE-GGA+U calculations. Upon Cr doping, the system exhibits a substantial increase in magnetism (the total moment rising from ∼7.5 to ∼12.0µ_B), primarily originating from the strong spin polarization of Cr-3d states and significant interstitial contributions, while Nb and C atoms provide smaller site-dependent moments. Fig. 8 further supports this, as the spin-polarized PDOS confirms the ferromagnetic nature through asymmetry of electronic states below and above the Fermi level for majority and minority spin channels.⁵⁰ The Cr-d orbitals (Fig. 10c and d) dominate the magnetism with strong spin polarization, whereas C-p orbitals (Fig. 10b) remain nearly symmetric. Nb-d states (Fig. 10e–h) display moderate asymmetry, indicating induced contributions through hybridization with Cr-d states. Overall, the net magnetic moment mainly arises from unpaired Cr-d electrons, with Nb atoms providing secondary polarization, thereby stabilizing robust ferromagnetism in Cr-doped Nb₃C₂, which is beneficial for charge storage and transport in lithium-ion batteries.

Table 1 Magnetic moments of pristine Nb₃C₂ and Cr-doped Nb₃C₂

Compound	Atoms	Magnetic moment (PBE-GGA)	Magnetic moment (PBE-GGA+U)
Pristine Nb3C2	Nb1 (µB)	0.00003	0.00026
	Nb2 (µB)	−0.00002	−0.00007
	C (µB)	0.00001	−0.000001
	Interstitial (µB)	0.00013	0.00059
	Total (µB)	0.00024	0.00146
Cr-doped Nb3C2	Cr1 (µB)	1.76157	2.51618
	Cr2 (µB)	1.76010	2.55740
	Nb1–Nb4 (µB)	−0.03548	0.11141
		−0.03698	0.16468
		−0.02989	−0.07683
		−0.02989	−0.07441
	C (µB)	−0.04485	−0.07782
		−0.04473	−0.08433
		−0.01336	−0.02645
		−0.01323	−0.03146
	Interstitial (µB)	0.94320	2.08746
	Total (µB)	7.48972	12.04418

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Fig. 10 Density of states (DOS) for Cr-doped Nb₃C₂ calculated using sp-GGA: (a) total DOS, (b) TDOS and PDOS of C-2p states, (c) and (d) TDOS and PDOS for Cr₁ and Cr₂ (total and 3d states, respectively), and (e)–(h) TDOS and PDOS for Nb₁, Nb₂, Nb₃ and Nb₄ (total and 3d states, respectively).

The calculated magnetic moment values for pristine Nb₃C₂ and Cr-doped Nb₃C₂ are summarized in Table 1.

3.5. Transport properties

The transport properties, including electrical conductivity, Seebeck coefficient, and thermal conductivity, were evaluated as a function of chemical potential at temperatures of 300 K, 500 K, and 700 K. These parameters provide insight into the charge-carrier behavior and electron transport efficiency in Nb₃C₂, which is essential for assessing its potential as an electrode material for energy storage applications.⁵¹ Transport properties were also estimated for the lithiated systems of both pristine and Cr-doped Nb₃C₂, since lithiation and delithiation during Li-ion battery charge/discharge significantly influence charge-carrier mobility and the density of states near the Fermi level.⁵²

The Seebeck coefficient is a key parameter for understanding charge-carrier behavior and transport efficiency.⁵³ Fig. 11 depicts the Seebeck coefficient as a function of chemical potential for various Nb₃C₂-based compounds. For pristine Nb₃C₂, Fig. 11(a) shows four distinct peaks at chemical potentials of −5.9 eV, −5.2 eV, −3.2 eV, and −3.0 eV. Outside this range, the Seebeck coefficient rapidly approaches zero, which indicates good transport performance. In contrast, lithiated Nb₃C₂, Cr-doped Nb₃C₂, and lithiated Cr-doped Nb₃C₂ (Fig. 11(b)–(d)) show smaller Seebeck values over the entire chemical potential range, consistent with strong metallic behavior. In Li-ion battery electrodes, metallic conductivity is essential for efficient charge transport.⁵⁴ These results suggest that the lithiated phases, particularly Li₂Nb₃C₂ and Li₄Cr-doped Nb₃C₂, exhibit high metallic conductivity and may provide more efficient electronic transport during Li-ion battery operation compared to pristine Nb₃C₂.

Fig. 11 Seebeck coefficient as a function of chemical potential at three different temperatures for the compounds: (a) pristine Nb₃C₂, (b) Li₂Nb₃C₂, (c) Cr-doped Nb₃C₂, and (d) Li₄–Cr-doped Nb₃C₂.

The maximum values of the Seebeck coefficient for the different compounds at various temperatures are summarized in Table 2. The results show that, with increasing temperature, the carrier concentration increases, leading to a reduction in the Seebeck coefficient, as expected for metallic systems. This behavior is consistent with the metallic nature of all compounds.

Table 2 Seebeck coefficient at different temperatures for lithiated and de-lithiated phases of pristine and Cr-doped Nb₃C₂

Temperature (K)	Nb3C2	Li2Nb3C2	Cr-doped Nb3C2	Li4CrNb3C2
300	4.2 × 10^−4	1.5 × 10^−4	2.9 × 10^−4	8.0 × 10^−5
500	2.0 × 10^−4	1.3 × 10^−4	1.8 × 10^−4	6.5 × 10^−5
700	1.0 × 10^−4	1.2 × 10^−4	1.2 × 10^−4	6.0 × 10^−5

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Electrical conductivity reflects the ease of electron flow in a material, and materials with efficient transport typically exhibit high conductivity. In metallic systems, conductivity generally decreases with increasing temperature but increases with chemical potential, due to higher carrier concentration and improved mobility.⁵⁵ As shown in Fig. 12a and c, the electrical conductivity approaches zero near −5.5 eV at 300 K for both delithiated Nb₃C₂ and delithiated Cr-doped Nb₃C₂. In contrast, the lithiated phases shown in Fig. 12(b) and (d) exhibited significantly higher conductivity than their delithiated counterparts, indicating enhanced electron conduction during Li-ion battery operation. The maximum values of electrical conductivity are summarized in Table 3.

Fig. 12 Electronic conductivity as a function of chemical potential at three different temperatures for: (a) pristine Nb₃C₂, (b) Li₂Nb₃C₂, (c) Cr-doped Nb₃C₂, and (d) Li₄–Cr-doped Nb₃C₂.

Table 3 Electronic conductivity of lithiated and delithiated phases of pristine and Cr-doped Nb₃C₂ at room temperature

Compound	Electrical conductivity (S m^−1)
Li2Nb3C2	4.7 × 10^20
Nb3C2	4.4 × 10^20
LiCr–Nb3C2	1.9 × 10^20
Cr–Nb3C2	1.7 × 10^20

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Fig. 13 depicts the thermal conductivity of Nb₃C₂-based systems as a function of chemical potential at three different temperatures. For all compounds, κ increases with temperature, consistent with their metallic nature.⁵⁶ Pristine Nb₃C₂ shows a smoother variation in thermal conductivity with chemical potential, whereas the other systems exhibit stronger variations, maintaining higher conductivity over a broader chemical potential (μ) range. Combined with the Seebeck coefficient and electrical conductivity results, these trends indicate that lithiation significantly enhances both charge and heat transport.⁵⁷ In particular, Li₂Nb₃C₂ and Li₄–Cr-doped Nb₃C₂ demonstrate excellent transport properties, making them promising candidates for efficient Li-ion battery electrodes.

Fig. 13 Thermal conductivity as a function of chemical potential at three different temperatures for the compounds: (a) pristine Nb₃C₂, (b) Li₂Nb₃C₂, (c) Cr-doped Nb₃C₂, and (d) Li₄–Cr-doped Nb₃C₂.

3.6. Electrochemical performance of MXenes as electrode materials

Metallic behavior is a desirable characteristic for electrode materials.⁵⁸ The electronic properties confirm that both pristine Nb₃C₂ and Cr-doped Nb₃C₂ are metallic, thereby fulfilling this requirement. In electrochemical applications, intercalating species play a crucial role, with Li⁺ ions being the most widely used in Li-ion batteries.⁵⁹ In this study, Li⁺ intercalation into pristine Nb₃C₂ and Cr-doped Nb₃C₂ MXenes was investigated to evaluate their storage capability as electrode materials. The electrochemical suitability of these MXenes was examined through adsorption energy analysis.

3.6.1. Adsorption energy. Eqn (1) was employed to calculate the adsorption energies of lithium ions adsorbed on pristine and Cr-doped Nb₃C₂. The adsorption energy was further used to estimate the average intercalation potential of the materials.⁶⁰ In this study, the calculated adsorption energies of a Li atom on pristine Nb₃C₂ and Cr-doped Nb₃C₂ were −1.20 eV and −1.48 eV, respectively. A negative value of E_ad indicates that the corresponding atom favors adsorption on the surface rather than forming clusters.^61,62 Cr-doped Nb₃C₂ exhibits a more negative adsorption energy, indicating that Li adsorption is energetically more favorable compared to that on pristine Nb₃C₂. Similar trends have been reported in other doped MXenes; for instance, V-doped Ti₂Co₂ shows more negative energies in the range of −1.9 eV to −2.8 eV compared to its undoped system.⁶³ This behavior is attributed to the additional electronic states introduced by the dopant atoms, which enhance charge transfer and orbital hybridization with Li, thereby strengthening the binding interaction.⁶⁴ In the present case, the introduction of Cr atoms modifies the electronic density of states near the adsorption site, increasing the interaction with Li atoms and thereby improving adsorption favorability. According to our calculations, a unit cell and a 2 × 1 × 1 supercell can accommodate only 2 and 4 Li atoms per formula unit, respectively. Beyond the adsorption of two Li ions per formula unit, the system reaches a saturation point. The saturation point in a structure varies depending on coulombic repulsion, structural distortion, and the lack of favorable adsorption sites.⁶⁵

3.6.2. Open circuit voltage (OCV). The open-circuit voltage (OCV) is an important parameter for evaluating the electrochemical performance of electrode materials in lithium-ion batteries. It provides insights into the average lithiation potential of the material, which directly influences both energy density and cycling stability.⁶⁶ In this work, the OCV was determined using eqn (2) for pristine and Cr-doped Nb₃C₂ systems with the adsorption of 2 and 4 Li atoms per formula unit, respectively.

V = [E_total(Nb₃C₂) + xE_total(Li) − E_total(Li₂Nb₃C₂)]/x (4)

V = [E_total(CrNb₃C₂) + xE_total(Li) − E_total(Li₄CrNb₃C₂)]/x (5)

Here, E_total(Nb₃C₂) and E_total(Li₂Nb₃C₂) represent the total energies per formula unit of pristine Nb₃C₂ without and with Li intercalation, respectively. Similarly, E_total(Li₄CrNb₃C₂) and E_total(CrNb₃C₂) correspond to the total energies of the Cr doped system before and after Li intercalation, respectively. E_total(Li) denotes the total energy of an isolated Li atom.⁶⁷

The corresponding intercalation reactions can be expressed as:

Nb₃C₂ + 2Li → Li₂Nb₃C₂ (6)

CrNb₃C₂ + 4Li → Li₄CrNb₃C₂ (7)

The calculated OCV values for Li⁺ extraction are 1.20 V for Li₂Nb₃C₂ and 1.48 V for Li₄CrNb₃C₂. The positive voltages indicate energetically favorable Li intercalation. Notably, Cr-doped Nb₃C₂ exhibits a higher OCV compared to the pristine system. This enhancement can be attributed to the additional electronic states introduced by Cr atoms, which strengthen the interaction with Li and stabilize the intercalated structure. These values are consistent with previously reported OCV values for other MXene materials; for example, Ti₂C exhibits an OCV of approximately 1.22 V.⁶⁸

3.6.3. Theoretical capacity. The theoretical capacity is a key property of energy storage materials. Here, the theoretical capacity (Q) was estimated using eqn (3) for both pristine and Cr-doped Nb₃C₂ systems.

The theoretical capacity of pristine Nb₃C₂ for Li-ion batteries was found to be 169 mAh g⁻¹, which is comparable to the reported value of Nb₂C (170 mAh g⁻¹).⁶⁹ In contrast, Cr-doped Nb₃C₂ exhibits a significantly higher storage capacity of 280 mAh g⁻¹, indicating that Cr doping substantially enhances the Li-ion storage performance of the MXene.

4. Conclusions

Using first-principles calculations, we systematically investigated the structural, electronic, magnetic, and transport properties of pristine Nb₃C₂ and Cr-doped Nb₃C₂, as well as their potential as electrode materials for Li-ion batteries. Structural analysis confirmed that Nb₃C₂ and Cr-doped Nb₃C₂ adopt hexagonal (P6₃/mmc) and orthorhombic (Pmm2) crystal structures, respectively. Phonon and AIMD results confirm that both pristine Nb₃C₂ and Cr-doped Nb₃C₂ are dynamically and thermally stable, with no significant structural distortions observed. These findings validate the reliability of the subsequent electronic and magnetic property calculations.

Charge density contours revealed that Cr doping substantially modifies the Nb–C bonding nature, while the band structures and density of states (DOS) confirm metallic behavior (indirect zero-band gap) in both systems, primarily originating from Nb 4d states. Moreover, substitutional Cr doping transforms non-magnetic pristine Nb₃C₂ into a ferromagnetic material with a finite magnetic moment, making it a promising candidate for 2D spintronic applications.

Transport property evaluations based on Boltzmann transport theory demonstrate that lithiation significantly enhances charge and heat transport. In particular, Li₂Nb₃C₂ and Li₄Cr-doped Nb₃C₂ exhibit high electrical and thermal conductivities with low Seebeck coefficients, confirming their metallic character and suitability as Li-ion battery electrodes. Furthermore, Cr-doped Nb₃C₂ outperforms pristine Nb₃C₂, suggesting superior transport performance for high-efficiency energy storage applications.

Electrochemical analysis further supports Cr-doped Nb₃C₂ as a promising electrode material, owing to its higher electronic conductivity, higher open-circuit voltage, and improved theoretical capacity compared to pristine Nb₃C₂. These improved electrochemical and transport properties make Cr-doped Nb₃C₂ a potential candidate for high performance lithium-ion battery electrodes, particularly in applications requiring fast charge transport and high energy density, such as portable electronic devices and electric vehicles. Overall, these findings demonstrate that Cr doping is an effective strategy for tailoring the properties of Nb₃C₂ MXenes, underscoring their strong potential for next-generation spintronic and energy storage applications.

Conflicts of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Data availability

The computational data that support the findings of this study are available upon request. All the related data are presented in the form of figures and tables in the manuscript.

Acknowledgements

The authors gratefully acknowledge the School of Interdisciplinary Engineering & Science (SINES) Islamabad, National University of Science and Technology, for providing computational resources and facilities to support this work.

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