Band Structure Engineering of MXenes for Low‐Loss Visible Epsilon‐Near‐Zero Properties by First‐Principles Calculation

Epsilon‐near‐zero (ENZ) photonics, which exhibit extraordinary capabilities in terms of controlling light–matter interactions, have been attracting increased interest in recent years. However, several challenges still lie ahead, such as large optical loss and the rarity of ENZ candidates, especially in the visible range. Here, by first‐principles calculations, this work proposes dozens of MXenes with promising plasmonic properties that could be potential ENZ candidates with very low loss in the visible range by band structure engineering. Because of the special electronic structures of these MXenes, they all possess quite large screened plasmonic frequencies: ωp/ε∞\[{{\bm{\omega }}_{\bf p}}{\rm{/}}\sqrt {{{\bm{\varepsilon }}_{\bm{\infty }}}} \] , but less than the interband transition onsets, which cause the ENZ properties in these MXenes to appear in the visible range with very low loss (ε2 less than 0.2) by eliminating the interband transition loss at the ENZ frequencies. Furthermore, it is also demonstrated that the ENZ properties in MXenes, including the ENZ frequency and loss, can be effectively tuned upon surface modifications, such as −Cl and −OH. These outstanding ENZ properties, including tunablity and low‐loss, make these MXenes great for potential applications in ENZ photonics, especially in the visible‐light range.


Introduction
Based on electromagnetic wave theory, tailoring epsilon (ε), the relative permittivity of materials or artificial metamaterials, is a vital method to manipulate the light-matter interactions and www.advelectronicmat.de some of them are demonstrated by experiments. [26] Among the many 2D metals, MXenes are the most attractive ones because of their highly desirable applications in many fields, [27,28] such as energy storage, [29,30] electromagnetic interference shielding, [31] and superconductivity. [32] Recently, MXenes were demonstrated to act as broadband absorbers [33] and photodetectors [34] due to their promising plasmonic properties. The Gogotsi group also reported that the plasmonic properties of MXenes can be tunable by compositions [35] or surface modifications. [36] As we know, the potential applications of MXenes in many fields are just part of reason for their great attention. More importantly, MXenes offer a powerful platform for predesigning various physicochemical properties, functions and thus applications, based on their highly tunable compositions by changing M (standing for the early transition metals, such as Sc, Ti, and V), X (standing for carbon or nitrogen), and even surface terminations. [37,38] It inspires us to investigate MXenes as a potential platform to design low-loss ENZ properties for visible light.
Here, to find low-loss visible ENZ candidates based on MXenes, we design lots of potential MXenes by changing the compositions to modify the electronic structures of Ti 3 C 2 F 2 and Ti 2 CF 2 . As a result, dozens of MXenes, including nitride MXenes and double transition metal (DTM) carbide MXenes, are selected as ENZ candidates for their promising ENZ properties: very low loss at the ENZ frequencies in the range from 1.0 to 2.0 eV. Furthermore, their ENZ properties are also proved to be tunable upon the surface modifications. These findings indicate that MXenes can serve as promising candidates for low-loss ENZ photonics for visible light.

Strategy for Low-Loss ENZ based on MXenes
As we know, the optical properties in the visible range are usually strongly related to the electronic properties of materials. For metals and doped semiconductors, they can be described by: [22] where ε ∞ is the static dielectric constant, the second term is due to the intraband transition described by a Drude model, while the last term describes the interband transitions. According to Equation 1, there could be lots of possibilities to realize the ENZ properties: ε 1 = 0 (ε 1 : the real part of dielectric function). But the optical loss, a critical parameter for many applications of ENZ devices, should be as low as possible, which is determined by the imaginary part of dielectric function: ε 2 . Comparing to intraband transitions, the interband transitions usually result in much higher loss. Thus, to get the loss as low as possible, a general strategy is to have the onset of the interband transition (ω onset ) higher than / p ω ε ∞ . In this case, the interband transitions do not contribute to the optical response at the ENZ frequency. ε 1 will reach a zero value around / p ω ε ∞ with a relatively small ε 2 , which is only impacted by intraband transitions. Meanwhile, a large ω p is also essential for the ENZ properties in the visible range.
Generally, the surface plasmon is a collective behavior of free electrons around the Fermi level. ω p is given by ω p 2 = ne 2 /ε 0 m, in which n is the carrier density and m is the effective mass of the carrier. As a result, the candidate should contain a well dispersed single band at the Fermi level for large plasma frequency, while multibands at Fermi level would introduce the interband loss. Meanwhile, to satisfy the condition of ω onset > / p ω ε ∞ , the conduction bands (CBs) and the valence bands (VBs) should be well separated from the band crossing Fermi level. Thus, a schematic electronic structure, which could meet the two requirements for low-loss visible-light ENZ properties, is displayed in Figure 1a.
However, few materials exhibit this type electronic structure. It was reported that the electronic properties of MXenes are highly tunable upon their compositions, [37,38] they can be semiconductors, metals, topological insulators, and even superconductors. Thus, we think MXenes might also be a platform to design low-loss visible ENZ properties. Here, we firstly check the electronic structures of two typical MXenes, Ti 2 CF 2 and Ti 3 C 2 F 2 , as displayed in Figure 1b,c. Neither of them can meet the electronic structure proposed in Figure 1a. Both of them have several bands crossing the Fermi levels, suggesting the high optical loss due to the interband transitions, labeled by green arrows. But interestingly, they all possess free-electron-like band structures below the Fermi levels: well dispersed single-bands around the M points. If the Fermi levels can be moved down, as illustrated by the dashed red lines in Figure 1c,e, they would have well dispersed single-bands crossing the Fermi levels. Meanwhile, the subbands at the Fermi levels disappear, and the interband transitions will also be blue-shifted as shown by the blue arrows in Figure 1c,e. As a result, they could exhibit the proposed electronic structure for low-loss ENZ properties.

Band Structure Engineering for MXenes
To adjust the Fermi levels, we can use chemical doping or electric field gating. Usually, electric field gating is not an effective method to modify the Fermi levels of metals. Here, we try to change the compositions to adjust the Fermi levels of Ti 3 C 2 F 2 and Ti 2 CF 2 , because the properties of MXenes have been demonstrated to be highly tunable by changing transition metals, main group elements, and even surface modifications. Moving down the Fermi level could be regarded as reducing the valence electrons in MXenes. The main group elements X only can be C or N. N even has one more electron than C. Thus, to reduce the valence electrons, only Y, Sc, Ti, Zr, and Hf elements could be selected for early transition metals, and for the surface termination, O can be used to replace F, because O has one electron less than F. So, 21 potential MXenes with fewer valence electrons than those of Ti 3 C 2 F 2 and Ti 2 CF 2 are illustrated in Figure 2. They are classified by valence electron numbers comparing to Ti 3 C 2 F 2 and Ti 2 CF 2 . Firstly, we prescreen them by their electronic structures. We find that for the MXenes with one valence electron fewer than those of Ti 3 C 2 F 2 and Ti 2 CF 2 , they all possess the similar electronic structures we proposed, including Y 3 N 2 F 2 , Sc 3 N 2 F 2 , Y 2 NF 2 , Sc 2 NF 2 , Ti 2 NO 2 , Zr 2 NO 2 , and Hf 2 NO 2 . When there are two electrons fewer than www.advelectronicmat.de Ti 3 C 2 F 2 and Ti 2 CF 2 , the Fermi levels would be further moved down. As a result, they become semiconductors, [27,37] such as Ti 3 C 2 O 2 , Ti 2 CO 2 , Sc 2 CF 2 , and Y 2 CF 2 . With fewer electrons, Fermi levels will keep on moving down. They will be metals again, but not the similar electronic structures we proposed. So, we focus our attention on the seven nitride MXenes with one valence electron less than Ti 3 C 2 F 2 and Ti 2 CF 2 .
Before investigating the electronic and optical properties of the selected candidates, we examine the dynamic stabilities of their structures. The structure parameters of these MXenes are supplied in the Supporting Information. Phonon spectrums indicate that Y 3 N 2 F 2 , Y 2 NF 2 , Sc 3 N 2 F 2 , and Sc 2 NF 2 can be dynamically stable (see Figure S1, Supporting Information), while Ti 2 NO 2 , Zr 2 NO 2 , and Hf 2 NO 2 are unstable, due to their imaginary frequencies ( Figure S2, Supporting Informa-tion). Thus, we systematically investigate the electronic and optical properties of the remaining four candidates: Y 3 N 2 F 2 , Y 2 NF 2 , Sc 3 N 2 F 2 , and Sc 2 NF 2 . Both Y 3 N 2 F 2 and Sc 3 N 2 F 2 are oneelectron less than Ti 3 C 2 F 2 , while the other two are one-electron less than Ti 2 CF 2 . They can be regarded as one hole doped for Ti 3 C 2 F 2 and Ti 2 CF 2 , respectively. Thus, the Fermi levels would be shifted down in principle, luckily they are still metals. The PBE exchange-correlation functional usually underestimates both band gaps of semiconductors and interband transition energies in metals. After optimizing their crystal structures, we applied a hybrid functional (HSE06) to compute the electronic structures of MXenes, which generally yield more reliable band energies. [50] The electronic band structures of Y 3 N 2 F 2 , Y 2 NF 2 , Sc 3 N 2 F 2 , and Sc 2 NF 2 obtained with the HSE06 functional are displayed in Figure 3a-d, respectively. We find that they have similar features in band structures: each of them has a highly dispersed single-band crossing the Fermi level. These dispersed bands are mainly composed of the d orbitals of Y or Sc. For the free-electron-like band structures, ω p 2 is proportion to n/m. All of four candidates could be assumed to have one electron per unit cell. They have the similar charge carrier density: ≈1.0 × 10 15 cm −2 , very close to that in TaSe 2 , [23,39] which exhibits plasmons in the near infrared range. For TaSe 2 , the bandwidth of the band crossing the Fermi level is less than 1 eV. But for Sc 2 NF 2 , the band crossing Fermi level distributes from −1.0 to 1.3 eV, much more delocalized than that in TaSe 2 , indicating that the effective masses of charge carrier in MXenes are much smaller than that in TaSe 2 . Similar dispersions are found in the other three candidates. Based on our calculations, the plasma frequencies of these candidates are listed in Table 1. The plasma frequencies (ω p ) can reach about 2.95, 2.84, 2.38, and 2.66 eV for Y 2 NF 2 , Sc 2 NF 2 , Y 3 N 2 F 2 , and Sc 3 N 2 F 2 , respectively,  www.advelectronicmat.de much higher than that of TaSe 2 . Thus, we can anticipate that these four candidates may extend the low-loss ENZ properties into the visible range.

ENZ Properties of MXenes
As a metal, the dielectric properties are contributed by two parts: the intraband transitions and interband transitions. The intraband transitions can be descripted by a Drude model, while the interband transitions can be obtained from first-principles calculations. The dielectric functions of MXenes were calculated by using the random phase approximation (RPA) 52 with the PBE functional, which usually underestimates interband transition energies in metals. Here, the difference between the onsets of interband transitions by PBE and HSE06 was used as a scissor operator to correct the dielectric functions. As a result, the dielectric properties (ε 1 and ε 2 ) of these four candidates, including the contributions from both intraband transitions and interband transitions, are shown in Figure 4a-d, respectively. Several properties of these candidates are summarized in Table 1, including ε ∞ , ω p , ω onset , ENZ frequency (ω ENZ ), and ε 2 at ω ENZ . The onset of direct band transitions of these four candidates occur between the bands indicated by the green arrows in Figure 3a-d, respectively. The ω onset for these four candidates Y 2 NF 2 , Sc 2 NF 2 , Y 3 N 2 F 2 , and Sc 3 N 2 F 2 , are 2.04, 1.94, 2.06, and 2.06 eV, respectively. As we discussed above, if we have the relation: ω onset > / p ω ε ∞ , the low-loss ENZ property will occur around / p ω ε ∞ . As summarized in Table 1, all of the four candidates can satisfy this requirement. As displayed in Figure 4a-d, the ENZ behaviors of Sc 2 NF 2 and Y 2 NF 2 appear in the visible range: around 1.91 and 1.95 eV, respectively, while www.advelectronicmat.de they appear in the near infrared range for Y 3 N 2 F 2 and Sc 3 N 2 F 2 , 1.31 and 1.44 eV, respectively. Thus, the losses at the ENZ frequency deserve further consideration for these four candidates.
The losses in metals originate from the electron transitions between unoccupied and occupied states of the band structure, which also can be divided into interband transitions (direct and indirect) and intraband transitions. Usually, the indirect interband transitions are much weaker than direct band transitions, indicating loss from interband transitions are dominated by the direct interband transitions. So, for these candidates, the losses at ENZ frequencies are dominated by the intraband transitions, which are proportional to γ/ω p (γ is the constant relaxation rate of the free carriers). We adopt classical models to estimate γ for the candidates: by the relation between dc conductivity σ and plasma frequency ω p . ε 2 of the four MXenes are summarized in Table 1, quite low: around 0.2 at their ENZ frequencies, implying very low loss for ENZ properties. For Y 3 N 2 F 2 , ε 2 is only about 0.19 at 1.31 eV, while 0.17 at 1.91 eV for Sc 2 NF 2 . We also compare the ENZ properties of these MXenes, including ENZ frequency and ε 2 , with reported ENZ materials, such as polar semiconductors, doped semiconductors, and metals in Figure 5. As we see, the ε 2 of these MXenes are even lower than those in some doped semiconductors, [6,11,17] such as ITO, and AZO. They are even comparable with the ε 2 in SiC, SiO 2 , and AlN, which are due to phonon polaritons. [11,15] More importantly, these MXenes can have ENZ properties in the visible range, but for doped semiconductors and polar semiconductors, they are located in the infrared region. Metals could show the ENZ properties in the visible range, but with much higher  www.advelectronicmat.de losses due to interband transitions. For example, the ε 2 of Ag is about 0.5 at the wavelength of about 360 nm and it is about 3 for Au at 520 nm. [11,16] Many metal alloys and metal nitrides have been developed for ENZ properties, [11,18,19] but it is hard to have the ε 2 less than 1. ENZ photonics has been emerging as an important research field in last 10 years, for its extraordinary capabilities in manipulating the light-matter interactions. However, as shown in Figure 5, in the energy region >1 eV, almost no candidate can show ε 2 less than 1 at their ENZ frequency, [11] which greatly limits their fundamental research and industrial applications. Luckily, these MXenes with quite low loss in visible range can fill this gap, which endows them with great potential applications in visible ENZ photonics.

Tunable ENZ Properties by Surface Terminations
Doped semiconductors attracted great attention for the applications in ENZ photonics, not only due to their low losses in the near infrared range, but also their tunable ENZ properties upon applied external fields. [40] Figure S3, Supporting Information), but still remain in the visible range. Luckily, the value of / p ω ε ∞ in Sc 2 NCl 2 is still lower than the onset of interband transition, which can keep the low ε 2 (0.25) at the ENZ frequency. We also find that replacing F by Cl for Y 3 N 2 F 2 : Y 3 N 2 Cl 2 , could also tune the ENZ properties to visible range from near infrared range ( Figure S4, Supporting Information). Different with Cl, replacing F with OH changes the band structure significantly, especially around the Fermi level, though it does not change the valence electron number. There is still a well-dispersed single band at the M point at the Fermi level, as shown in Figure S5 (Supporting Information). But around the Γ point, another two bands contributed by the OH groups appear near the Fermi level. As a result, the interband transitions will occur between these two bands, which significantly increases the ε 2 to 0.48 at the ENZ frequency of 1.77 eV (Table 1). But it is still lower than those of typical metals. As we see, the surface terminations can effectively modulate the ENZ properties of MXenes, including ENZ frequency and the losses. Therefore, finding methods to modulate the surface terminations experimentally plays an important role in modifying the ENZ properties in MXenes. Recently, Maleski et al. [36] modified the surface terminations by changing the solution of etchant, and observed tunable optical properties in Ti 3 C 2 T x and Ti 2 CT x due to different surface terminations. They ascribed them to the plasma properties upon the surface terminations. It suggests that the tunable ENZ properties obtained by modifying the surface terminations in MXenes should be approachable in experiments.
The most popular method to obtain MXenes is etching Al from MAXs by LiF-HCl solution. [26,27] In this case, MXenes have a random mixture of surface terminations (OH, Cl, and F), which can significantly influence their ENZ properties. Thus, solution etching method may be not a suitable way to obtain the MXenes for the applications in ENZ photonics. How to acquire MXenes with controllable surface-termination becomes vital issue for their applications. Recently, Huang et al. [41] developed a Lewis acidic etching method to prepare MXenes in nonaqueous electrolyte, which show the ability to control the surface terminations of MXenes. Besides, in 1986, Hwu et al. [42] have successfully synthesized Sc 2 NCl 2 (one of the MXene candidates we proposed) through a high-temperature solid-state reaction with metal nitrides (ScN) as the N source. Just recently, Druffel et al. [43] also reported the Y 2 CF 2 which were also synthesized by a solid-state reaction with Y 2 C, YC, and YF 3 . These works imply that the ENZ properties in MXenes with controllable surface terminations can be possibly obtained in experiments.

ENZ Candidates based on Double Transition Metal Carbide MXenes
To synthesize these proposed nitride MXenes through a solid-state reaction, it is crucial to find a precursor for the N source. Comparing with the limitation of N source for nitride MXenes, carbon can be directly used as source for carbide MXenes. Thus, it is better to find carbide MXenes for ENZ candidates, for it is more convenient to synthesize. But as shown in Figure 2, it is hard to find suitable ENZ candidates   [44,45] Introducing one more transition metal in MXenes would give more flexibility to design ENZ materials. Several DTM carbide MXenes: MM' 2 C 2 F 2 (M = Sc, Y; M' = Ti, Zr, Hf), with one valence electron less than Ti 3 C 2 F 2 are also proposed, and their ENZ properties are also evaluated, as shown in Figure 6. The optimized crystal structures are supplied in the Supporting Information. Phonon spectrums indicate that these DTM carbide MXenes can be dynamically stable ( Figure S6, Supporting Information). They also show similar electronic structures and ENZ properties with the nitride MXenes we proposed. We take YHf 2 C 2 F 2 as an example to illustrate their ENZ properties. Others are supplied in Figures S7-S11 (Supporting Information). As shown in Figure 6, ENZ properties of YHf 2 C 2 F 2 appear at 1.43 eV, with the ε 2 of only 0.23. Their properties, including the other four DTM carbide MXenes, are also summarized in Table 1 and Figure 5. YHf 2 C 2 F 2 and ScHf 2 C 2 F 2 can show ENZ properties for red light, while the other four DTM carbide MXenes are in near infrared range. All of them have very low optical losses, because all of them can satisfy the relation of ω onset > / p ω ε ∞ . According to Figure 5, the ENZ properties of DTM carbide MXenes, including ENZ frequencies and loss, are competitive with traditional ENZ materials and also the nitride MXenes we proposed. More importantly, due to more choices for carbon sources, it would be much easier to synthesize carbide MXenes than nitride MXenes by the solid-state reaction. Thus, we can expect that the low-loss visible-light ENZ properties in these MXenes can be realized in the near future.

Conclusion
By modifying the electronic structures of Ti 3 C 2 F 2 and Ti 2 CF 2 , we have successfully designed dozens of low-loss visible ENZ candidates based on MXenes. They could exhibit very low optical loss (with ε 2 less than 0.3) at their ENZ frequencies, ranging from 1.0 to 2.0 eV. Moreover, their ENZ properties are suggested to be tunable upon the surface terminations of MXenes, including ENZ frequency and loss. These Mxenesbased ENZ candidates, including nitride MXenes and DTM carbide MXenes, not only fill the gap of low loss ENZ materials in the visible range, but also are beneficial to the discovery of unprecedented phenomena and applications based on their excellent ENZ properties. Furthermore, our work also indicates much broader material choices beyond traditional plasmonic materials, not only for ENZ photonics, since there are plenty of 2D metals to be explored.

Experimental Section
Electronic Structure Calculations: Density functional theory (DFT) computations were performed with the Vienna ab initio simulation package (VASP) code. [46,47] The exchange-correlation interactions were treated by the generalized gradient approximation (GGA) in the form of Perdew-Burke-Ernzerhof (PBE). [48,49] More than 20 Å vacuum space perpendicular to the surface is added to avoid interactions between periodic slabs. The cutoff energy for plane-wave basis sets is 500 eV. The convergence thresholds for force and total energy are 10 −2 eV Å −1 and 10 −5 eV, respectively. The band structures of MXenes were calculated with a k-point density of 10 Å −1 by hybrid functional (HSE06). [50] Their linear response optical properties were calculated by using RPA for the density response function. [51] A Monkhorst-Pack k-point sampling of 40 × 40 × 1 was employed to calculate the optical response. This work applied a scissor operator at the onset of the interband transitions of MXenes. The scissor operator was determined by the difference between the onsets of interband transitions by PBE and HSE06, respectively.
Dielectric Properties Calculations: This work applied the independent particle approximation to describe of the dielectric function from electronic band structure calculations. The dielectric function ε(ω) = ε 1 (ω) + iε 2 (ω) (where ε 1 is the real part, and ε 2 is the imaginary part) of metals in the low-frequency region was contributed by two important processes, interband (ε inter (ω)) and intraband (ε intra (ω)) transitions.
The imaginary part of the dielectric function contributed by interband transition is calculated by: [51] where e is the electron charge, Ω is the primitive cell volume, w k is the weight of the k-points. The indices c and v refer to the CB and VB states, respectively. The parameter E jk is the single-electron energy state of band j at wave vector k, u jk is the periodic part of the Bloch wave www.advelectronicmat.de function corresponding to the eigenvalue E ik (i = c, v), and δ is the delta function, which depends upon the method used for calculation. The real part of the dielectric function contributed by interband transition is obtained from the Kramers-Kronig relation. The dielectric function for intraband transitions, ε intra (ω), is known as the dielectric function of a free electron gas. The contribution by intraband transitions, ε intra (ω), are obtained by Drude expression for a given plasma frequency ω p and damping constant (γ): [52] while ω p can be calculated from the electronic band structure as follows: [53] 8 ,( where v nkα is the quasiparticle group velocity at k for the nth band, ε n k is the energy eigenvalue and f n k is the Fermi-Dirac distribution function. The total dielectric function is the sum of the dielectric functions for inter-and intraband transitions: ε(ω) = ε inter (ω) + ε intra (ω).

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.