Thermoelectric properties of X₃N₂O₂ (X = Hf, Zr) MXene monolayers: a first-principles study

Abstract

MXene monolayers have received increasing attention due to their unique properties, particularly their high conductivity, which shows great potential in thermoelectric materials. In this paper, we present a theoretical study of the thermoelectric properties of X₃N₂O₂ (X = Hf, Zr) MXene monolayers, taking electron–phonon coupling into consideration. Owing to their similar geometrical structures, electronic band structures, and phonon dispersions, X₃N₂O₂ MXene monolayers exhibit homogeneous electron and phonon transport properties. The conduction band shows multi-valley characteristics which leads to better n-type electron transport properties than p-type ones. The maximum values of the n-type power factor can reach 32 μW cm⁻¹ K⁻² for the Hf₃N₂O₂ monolayer and 23 μW cm⁻¹ K⁻² for the Zr₃N₂O₂ monolayer. In terms of phonon transport, the lattice thermal conductivity for the Zr₃N₂O₂ monolayer is higher than that for the Hf₃N₂O₂ monolayer, due to larger phonon group velocity. Our results show that the Hf₃N₂O₂ monolayer is more suitable for thermoelectric materials than the Zr₃N₂O₂ monolayer, with optimal n-type thermoelectric figure of merit (ZT) values of 0.36 and 0.15 at 700 K, respectively. These findings may be useful for the development of wearable thermoelectric devices and sensor applications based on X₃N₂O₂ MXene monolayers.

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1 Introduction

Two-dimensional (2D) materials are extensively researched due to their unique properties, such as high surface area,¹ high conductivity,² and tunable band gaps,³ which differ from their bulk counterparts. These properties make them promising materials for a wide range of applications, including electronics,⁴ energy storage,⁵ catalysis,⁶ and sensors.⁷ Furthermore, 2D materials can be easily integrated with other materials, making them a versatile platform for developing new technologies. Researchers continue to explore the potential of 2D materials and their applications, leading to exciting new discoveries and advancements in various fields. MXenes are a family of 2D transition metal carbides, nitrides, and carbonitrides. They were first discovered in 2011.⁸ The general formula for MXenes is M_n+1X_nT_x, where M is a transition metal, X is carbon or nitrogen, and T is a surface termination such as O, OH, F, and/or Cl.⁹ The structure of an MXene monolayer consists of layers of transition metal atoms stacked on top of each other, with the X and T atoms occupying the interstitial sites between them. The transition metal atoms are connected by covalent bonds, while the X and T atoms are connected to the transition metal atoms by ionic bonds. The MXene monolayer can be viewed as a sandwich structure, where the transition metal atoms form the bread and the X and T atoms form the filling. The sandwich structure can be peeled off layer by layer to produce MXene flakes, which have a thickness of a few nanometers.¹⁰ MXene monolayers usually possess high electrical conductivity,¹¹ efficient absorption of electromagnetic waves,¹² and tunable mechanical properties.¹³ Modifying the functionalized surfaces can be an effective way to modulate the properties of MXene monolayers.¹⁴ These unique properties make MXenes promising materials for a variety of applications.^15–17

Thermoelectric materials are a special type of material that can convert temperature differences into electrical energy and vice versa.¹⁸ The efficiency of a thermoelectric material is measured by its thermoelectric figure of merit, ZT.¹⁹ The higher the ZT value of a material, the more efficient it is in converting temperature differences into electrical energy. The thermoelectric figure of merit, ZT, is defined by the equation: ZT = S²σT/(κ_l + κ_e), where S is the Seebeck coefficient, σ is the electrical conductivity, T is the absolute temperature, and κ_l and κ_e are the lattice and electronic thermal conductivity, respectively. As mentioned previously, the high conductivity of MXenes makes them promising applications for thermoelectric materials. There have already been many studies on the thermoelectric properties of MXenes. For example, Kumar et al.²⁰ investigated the electronic and thermal properties of different functionalizations in the Sc₂C MXenes. They found that Sc₂C(OH)₂ possesses high electrical conductivity and relatively low lattice thermal conductivity, indicating it is a candidate for intermediate-temperature thermoelectric. Meanwhile, the role of functionalized surfaces on thermoelectric properties in M₂C and M₂N MXenes was also studied.²¹ The results showed that Mo₂C can gain a large Seebeck coefficient. Flexible film electrodes have been designed using a ternary composite of Ti₃C₂T_x MXene.²² Recently, Rana et al. investigated the narrow bandgap Janus monolayer MoWCO₂.²³ The thermoelectric figure of merit was 0.33 (p-type) and 0.08 (n-type) at 700 K. Experimental work was also carried out, and the power factor (PF) of Mo₂TiC₂T_x film can reach 309 μW cm⁻¹ K⁻² at 800 K.²⁴

However, the thermoelectric properties of the X₃N₂O₂ MXene monolayer have not been studied yet. In this paper, we calculate these properties for X₃N₂O₂ (X = Hf, Zr) using first principles and Boltzmann transport theory, including electron–phonon coupling. Our work provides a full understanding of the electron–phonon coupling on MXene monolayers with high accuracy, and can be expanded to the study of more thermoelectric materials.

2 Computational methods

All first-principles calculations were performed using density functional theory (DFT), as implemented in the Quantum ESPRESSO (QE) code.^25,26 The exchange-correlated interactions were described using the generalized gradient approximation of the Perdew–Burke–Ernzerhof (GGA-PBE).²⁷ The corresponding pseudopotential files were obtained from the Standard solid-state pseudopotentials (SSSP) libraries.²⁸ The wave function was expanded using a plane-wave cutoff energy of 80 Ry. The Brillouin zone was sampled using an 11 × 11 × 1 k-point mesh according to the Monkhorst–Pack scheme.²⁹ A vacuum slab of 15 Å was applied along the c-direction. The energy and force convergence thresholds were set to 10⁻¹⁰ Ry and 10⁻⁴ Ry, respectively, in order to fully relax the structure.

Electrical transport coefficients were calculated using Boltzmann transport theory as implemented in the BoltzTraP code.³⁰ Under the electron–phonon averaged (EPA) approximation,³¹ the carrier relaxation time (τ_e) was evaluated using the following equation:

Here, ε is the energy of the carriers, μ is the chemical potential, Ω is the volume of the primitive unit cell, ℏ is the reduced Planck's constant, g_s is the spin degeneracy, ν is the phonon mode index, g_ν² is the averaged electron–phonon matrix, [small omega, Greek, macron] × ν is the averaged phonon mode energy, n([small omega, Greek, macron] × ν, T) is the Bose–Einstein distribution function, f(ε + [small omega, Greek, macron]ν, μ, T) is the Fermi–Dirac distribution function, and ρ is the density of states per unit energy and unit volume. Details about this method can be found in the ref. 31.

The value of κ_l is obtained by solving phonon Boltzmann transport theory, as implemented in the ShengBTE code.³² The second-order harmonic interatomic force constants (IFCs) were calculated by density functional perturbation theory (DFPT) using a supercell of 5 × 5 × 1. The third-order anharmonic IFCs were calculated using the finite displacement method with the same supercell. To obtain a convergent κ_l, a cutoff distance of up to the 9th nearest atomic neighbors was used. The q-point grid was set to 60 × 60 × 1, and the smearing parameter was set to 1.0.

3 Results and discussion

The geometrical structures of Hf₃N₂O₂ and Zr₃N₂O₂ monolayers are illustrated in Fig. 1(a) and (b), respectively. These monolayers possess a hexagonal crystal structure with a space group of P6/mmm. This space group has a six-fold rotational symmetry axis and mirror planes that intersect at the c-axis of the crystal. The optimized lattice constants of Hf₃N₂O₂ and Zr₃N₂O₂ monolayers are 3.210 Å and 3.245 Å, respectively. Their lattice constants are close, indicating that their bond strengths are similar.

Fig. 1 The crystal structure of Hf₃N₂O₂ and Zr₃N₂O₂ monolayer on the (a) top view and (b) side view. (c) The band structure of Hf₃N₂O₂ and Zr₃N₂O₂ monolayer.

Fig. 1(c) shows a comparison of the band structures of Hf₃N₂O₂ and Zr₃N₂O₂ monolayers. The bands are similar, with the conduction band minimum (CBM) at the M point and the valence band maximum (VBM) at the K point. The band gap of Hf₃N₂O₂ and Zr₃N₂O₂ monolayers is 0.328 eV and 0.394 eV, respectively. The band gap of Hf₃N₂O₂ is smaller than that of Zr₃N₂O₂. Moreover, the conduction band has multiple valleys, while the valence band is relatively simple. The multiple valleys in the band result in a high density of states (DOS) and a high S value.

To gain a better understanding of the electronic properties of monolayers Hf₃N₂O₂ and Zr₃N₂O₂, we show their projected density of states (pDOS) in Fig. 2(a) and (b), respectively. This paper ignores the projected DOS for other orbitals due to their ultralow values. The pDOS of Hf₃N₂O₂ is similar to that of Zr₃N₂O₂, with the former mainly contributed by the Hf 5d electrons and the latter mainly contributed by the Zr 4d electrons. Furthermore, the DOS of Zr₃N₂O₂ is higher than that of Hf₃N₂O₂. The multiple valleys in the conduction band can be attributed to the Hf 5d or Zr 4d electrons.

Fig. 2 The projected density of states (DOS) for (a) Hf₃N₂O₂ and (b) Zr₃N₂O₂ monolayer.

The phonon dispersion curves of Hf₃N₂O₂ and Zr₃N₂O₂ monolayers are shown in Fig. 3(a) and (b), respectively. It can be seen that the phonon dispersion curves of both monolayers are generally similar. Both Hf₃N₂O₂ and Zr₃N₂O₂ monolayers are dynamically stable due to the absence of imaginary modes. There is a clear band gap around 300 cm⁻¹, with Hf₃N₂O₂ monolayer exhibiting a wider gap than the Zr₃N₂O₂ monolayer. The low frequency part (< 300 cm⁻¹) of Hf₃N₂O₂ monolayer is lower than that of Zr₃N₂O₂ monolayer. The slopes of the acoustic modes (marked by another color) for Zr₃N₂O₂ monolayer are larger than those of Hf₃N₂O₂ monolayer. This suggests that Zr₃N₂O₂ monolayer would exhibit larger phonon group velocity (v_g), which will be further confirmed.

Fig. 3 The harmonic phonon dispersion for (a) Hf₃N₂O₂ and (b) Zr₃N₂O₂ monolayer.

To analyze the thermoelectric transport properties, we first illustrate the absolute values of Seebeck coefficient (|S|) in Fig. 4(a) and (b). S can be attribute to the steep slope of the DOS,³³ i.e., ∂n(ε)/∂ε. Owing the DOS peak in conduction caused by Hf 5d or Zr 4d electrons, the absolute values of n-type S are much higher than p-type ones. We fit the steep slope of the DOS between 0.2–0.7 eV, which corresponds to a moderate n-type carrier concentration. The fitted steep slopes of the DOS are 6.55 and 10.47 for Hf₃N₂O₂ and Zr₃N₂O₂, respectively. These values are consistent with the n-type |S| (62 and 138 μV K⁻¹ for Hf₃N₂O₂ and Zr₃N₂O₂, respectively) at 300 K. At a moderate concentration (10⁻¹⁴ cm⁻²), n-type |S| can reach 200 μV K⁻¹ at 700 K, while p-type |S| is only 30 μV K⁻¹. The high n-type S implies that Zr₃N₂O₂ monolayers are potential n-type thermoelectric materials. Additionally, the absolute values of p-type S are higher than those of n-type S.

Fig. 4 The (a), (b) |S| and (c), (d) σ as a function of carrier concentration for Hf₃N₂O₂ and Zr₃N₂O₂ monolayer.

Since σ and κ_e are dependent on electronic relaxation time (τ_e), we need to analyze τ_e first. Fig. 5(a) and (b) show τ_e as a function of energy at various temperatures. Unlike the constant relaxation time approximation method and deformation potential theory, τ_e calculated by EPA depends firmly on the carrier density, which is more accurate. As energy approaches the Fermi level, τ_e increases. All X₃N₂O₂ monolayers have much higher τ_e for holes than for electrons. We attribute this phenomenon to the more dispersive valence band than the conduction band. There are many conduction band valleys near the Fermi level, which scatter charge carriers more easily and lead to a lower τ_e for electrons. As temperature increases, charge carrier scattering becomes stronger and a lower τ_e is observed at higher temperatures.

Fig. 5 The electronic relaxation time (τ_e) of (a) Hf₃N₂O₂ and (b) Zr₃N₂O₂ monolayers as a function of the energy.

Fig. 4(c) and (d) plot σ as a function of energy at different temperatures, based on τ_e. Contrary to S, the σ of n-type doping are much lower than that of their p-type doping. For a moderate concentration (10⁻¹⁴ cm⁻²), p-type σ for Hf₃N₂O₂ monolayer can exceed 7 × 10⁵ Ω⁻¹ m⁻¹ at room temperature. As expected, σ increases for increasing higher carrier concentration. As temperature increases, σ decreases due to stronger carrier scattering, which is the same as τ. According to the Wiedemann–Franz law, κ_e is proportional to σ: κ_e = LσT, where L is the Lorenz number.³⁴ Therefore, there is no need to analyze κ_e separately.

By combining the results of S and σ, we can determine the power factor (PF) of monolayers of X₃N₂O₂, as shown in Fig. 6(a) and (b). The PF of the Hf₃N₂O₂ monolayer is higher than that of the Zr₃N₂O₂ monolayer. Additionally, the p-type PF is much lower than the n-type PF, indicating that X₃N₂O₂ monolayers are more suited for n-type thermoelectric materials. Interestingly, the PF of the Zr₃N₂O₂ monolayer reaches its maximum value at 300 K (23 μW cm⁻¹ K⁻²) and decreases with temperature, while the PF of the Hf₃N₂O₂ monolayer is weakly temperature-dependent. The n-type maximum PF values for Hf₃N₂O₂ can reach 32 μW cm⁻¹ K⁻² at 500 K. For both materials, the maximum p-type PF values are half that of n-type.

Fig. 6 The (a) n-type nad (b) p-type power factor (PF) values as a function of carrier concentration for Hf₃N₂O₂ and Zr₃N₂O₂ monolayer.

The phononic transport properties of X₃N₂O₂ monolayers, including κ_l, Gruneisen parameter, phonon group velocity, and phonon relaxation time, are shown in Fig. 7(a)–(d). At room temperature, κ_l for Hf₃N₂O₂ and Zr₃N₂O₂ monolayer is 14.5 and 16 W mK⁻¹, respectively. As with other crystalline materials, κ_l decreases as temperature increases, following a 1/T dependence. Higher temperature causes intrinsic enhancement in phonon–phonon scattering. With temperature increasing to 700 K, κ_l for Hf₃N₂O₂ and Zr₃N₂O₂ monolayer decreases to 5.55 and 6.18 W mK⁻¹ which is relatively acceptable for thermoelectric applications.

Fig. 7 (a) κ_l, (b) phonon group velocity, (c) phonon lifetime, and (d) Gruneisen parameter for Hf₃N₂O₂ and Zr₃N₂O₂ monolayer.

It can be noticed that κ_l for Hf₃N₂O₂ monolayer is always lower than that of Zr₃N₂O₂ monolayer. κ_l is determined mainly by phonon group velocity (v_g) and phonon relaxation time (τ_ph), as κ_l ∝ v_gτ_ph. The v_g for Zr₃N₂O₂ monolayer is higher than that of Hf₃N₂O₂ monolayer for low frequency part (<300 cm⁻¹) as shown in Fig. 7(b). The v_g at Γ point can reach 8 km s⁻¹ for Zr₃N₂O₂ monolayer, while the corresponding v_g of Hf₃N₂O₂ monolayer is 6 km s⁻¹. For high frequency part (>300 cm⁻¹), v_g for X₃N₂O₂ monolayers are similar. This can be confirmed by the phonon dispersion curves in Fig. 3, as v_g = ∂ω_λ,q/∂q. Different from v_g, τ_ph for X₃N₂O₂ monolayers are close, as plotted in Fig. 7(c). Especially for low frequency (<50 cm⁻¹), τ_ph for X₃N₂O₂ monolayers are nearly the same. We further analyze the Gruneisen parameter (γ). γ reflects the anharmonic scattering strength³⁵ which is given by γ = −(V/ω_λ,q)∂ω_λ,q/∂V. Larger |γ| means stronger anharmonicity and phonon–phonon scattering. Obviously, γ for X₃N₂O₂ monolayers are basically the same, as shown in Fig. 7(d). The difference in v_g caused by phonon dispersions is the main reason for lower κ_l of Hf₃N₂O₂ monolayer than Zr₃N₂O₂ monolayer, while τ_ph and anharmonic scattering strength for X₃N₂O₂ monolayers are similar.

Combining with calculated PF and κ_l, we calculated the ZT values as a function of n as plotted in Fig. 8(a) and (b). X₃N₂O₂ monolayers exhibit higher n-type ZT values than p-type ones. Additionally, the ZT values of Hf₃N₂O₂ monolayer are higher than those of Zr₃N₂O₂ monolayer. For n-type Hf₃N₂O₂ monolayer, ZT values present an behavior of monotonically increasing with temperature and optimal n gradually move to the low concentration. More specifically, maximum ZT values can reach 0.36 at 700 K. For p-type Hf₃N₂O₂ monolayer, ZT values are relatively low and the optimal n is around 10²⁰ cm⁻³. The maximum ZT value for n-type Zr₃N₂O₂ monolayer is 0.15 at 700 K. Although all X₃N₂O₂ monolayers exhibit high power factors (PF), their final ZT values are relatively low due to their high κ_l. Future work should focus on reducing κ_l to improve the ZT values. For example, alloying with other elements to reduce the phonon group velocity and/or increasing the phonon scattering strength by introducing defects or nanostructures.

Fig. 8 The (a) n-type and (b) p-type ZT values as a function of carrier concentration for Hf₃N₂O₂ and Zr₃N₂O₂ monolayer.

4 Conclusions

In this paper, we studied the thermoelectric properties of the X₃N₂O₂ (X = Hf, Zr) MXene monolayers using first-principles calculations and Boltzmann transport theory. To obtain accurate electrical transport properties, we took electron–phonon coupling into consideration. We found that Hf₃N₂O₂ and Zr₃N₂O₂ monolayers have similar structures, band structures, and phonon dispersions. The narrow band gap (0.3 eV) ensures high electrical conductivity. The Hf₃N₂O₂ monolayer exhibits a better power factor and lower lattice thermal conductivity than the Zr₃N₂O₂ monolayer. The maximum ZT value achieved for n-type Hf₃N₂O₂ monolayer at 700 K is 0.36.

Author contributions

X. Y. and W. C. conceived the study. W. C. performed the calculations and analyzed the data. X. Y. and H. L. wrote, reviewed, and edited the manuscript. All authors have read and agreed to the published version of the manuscript.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This research was funded by the Fundamental Research Funds for the Central Universities: 2042022kf1078 and the Knowledge Innovation Program of Wuhan-Basic Research (No. 2022020801010473).

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