Abstract
MXenes are emerging layered materials that are promising for electrochemical energy storage and (opto)electronic applications. A fundamental understanding of charge transport in MXenes is essential for such applications, but has remained under debate. While theoretical studies pointed to efficient band transport, device measurements have revealed thermally activated, hoppingtype transport. Here we present a unifying picture of charge transport in two model MXenes by combining ultrafast terahertz and static electrical transport measurements to distinguish the short and longrange transport characteristics. We find that bandlike transport dominates shortrange, intraflake charge conduction in MXenes, whereas longrange, interflake transport occurs through thermally activated hopping, and limits charge percolation across the MXene flakes. Our analysis of the intraflake charge carrier scattering rate shows that it is dominated by scattering from longitudinal optical phonons with a small coupling constant (α ≈ 1), for both semiconducting and metallic MXenes. This indicates the formation of large polarons in MXenes. Our work therefore provides insight into the polaronic nature of free charges in MXenes, and unveils intra and interflake transport mechanisms in the MXene materials, which are relevant for both fundamental studies and applications.
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Main
Twodimensional (2D) transition metal carbides and/or nitrides, known as MXenes, have attracted considerable attention as a consequence of their outstanding electronic, optical, mechanical and thermal properties^{1,2,3}. The general formula of MXenes is M_{n + 1}X_{n}T_{x} (n = 1–4), where M represents the transition metal, X carbon and/or nitrogen and T_{x} the surface termination. A unique combination of tunable electronic properties (through the M_{n + 1}X_{n} backbone) and versatile surface chemistry (from T_{x} functionalization) results in many intriguing properties such as the metaltoinsulator transition, topological insulators and superconductivity^{4}. As such, MXenes hold great promise for a variety of electronic and optoelectronic applications^{1,2,3}.
In many applications, the electrical transport properties of MXenes are quintessential. Understanding the mechanism governing the electrical conductivity of MXenes is therefore important. Accordingly, many previous studies have investigated the conductivity of charge carriers, specifically as a function of temperature (T). However, controversial results, mainly between experiments and theoretical calculations, have been reported. Experimentally^{5,6,7,8}, based on static electrical transport studies using fourprobe methods, it has been shown that the conductivity σ increases at elevated temperature T (that is, dσ/dT > 0). This dependence was attributed to a thermally activated hopping transport. On the other hand, theoretical studies have predicted bandlike transport^{9,10,11}. For band transport, the charge carrier mobility typically increases with decreasing T (that is, dσ/dT < 0) as a result of the decreased carrier–phonon scattering probability at lower T. This contradiction between the theoretical and experimental results raises questions regarding the nature of the charge carriers and charge transport mechanism in MXenes. Owing to the polar nature of MXene crystals^{3}, charge carriers can induce lattice distortions via Coulomb interactions. A quasiparticle, a socalled polaron, is normally introduced to describe the charge carriers dressed with lattice distortions. The carrier–phonon coupling constant α characterizes the strength of the coupling between charge carriers and phonons^{12}. For sufficiently strong carrier–phonon coupling (α ≫ 6), selftrapped, localized polaron states, or socalled small polarons (with the lattice distortion limited to ~1 lattice unit), occur. Transport of small polarons requires thermal activation, which may explain the observed hoppinglike transport behaviour in MXenes. For weaker coupling (α < 6), large polarons are formed; these are characterized by bandlike transport of charge carriers with a renormalized effective mass. Large polaron mobility typically increases with decreasing temperature^{12,13}. We note that other charge localization effects, for example, due to the presence of a high density of defects (depending on the quality of the samples), could lead to hoppinglike transport as well^{14}. Given the increasing interest in MXenes for electrical and electrochemical applications, it is crucial to understand the nature of charge species and their transport mechanism across materials and interfaces.
In this Article, we present a unifying picture of charge transport in liquidexfoliated MXenes, using semiconductorlike Nb_{4}C_{3}T_{x} and metallic Ti_{3}C_{2}T_{x} as model systems. By comprehensively employing ultrafast terahertz and static electrical transport measurements, we distinguish the short and longrange transport characteristics of MXene films. We reveal that while bandlike transport dominates intraflake charge conduction in MXenes, interflake transport occurs via thermally activated hopping processes and becomes the limiting step for charge percolation across the MXene flakes (see the illustration in Fig. 1a). More importantly, by analysing contributions from impurities, acoustic and longitudinal optical (LO) phonons to the charge scattering mechanisms, we further show that carrier–LO phonon scattering dominates the carrier transport at room T, with a small carrier–LO phonon coupling constant (α ≈ 1). The small α indicates the formation of large polarons in the materials (see the illustration in Fig. 1b). These results suggest that, for both semiconducting and metallic MXenes, large polaron formation is a generic property that fundamentally affects the intrinsic charge transport and carrier lifetime. Our results reconcile the debate between previous theoretical and experimental studies and shed light on the charge species and their transport mechanisms in MXenes.
Results and discussion
We present detailed structural characterizations for Nb_{4}C_{3}T_{x} MXene in Fig. 1, and show those for the wellestablished Ti_{3}C_{2}T_{x} MXene in Supplementary Fig. 1. Figure 1c presents the Xray diffraction (XRD) pattern of the Nb_{4}AlC_{3} MAX precursor and the resultant Nb_{4}C_{3}T_{x} MXene following the selective etching of aluminium element by hydrogen fluoride. After etching, the diffraction peaks corresponding to the MAX phase either disappear or greatly diminish in intensity, along with the appearance of a new peak at a 2θ of ~5°. This new peak can be assigned to the (002) facet for Nb_{4}C_{3}T_{x} MXene^{6}. Furthermore, Raman spectroscopy displays two pronounced peaks at ~240 cm^{−1} and 675 cm^{−1}, corresponding to Nb–O and Nb–C vibrational modes, respectively (Fig. 1d). The emergence of the Nb–O peak at ~240 cm^{−1} points to (partial) surface oxidation, which is unavoidable during the synthesis process. Such surface oxidation might partially contribute to the semiconducting nature of Nb_{4}C_{3}T_{x} MXene in our study. In other MXenes (for example, Ti_{4}N_{3}T_{x}), surface oxidation has been reported to lead to a bandgap opening, thus provoking a metaltosemiconductor transition^{15}. In addition, as shown in Xray photoelectron spectroscopy (XPS) results, the assynthesized Nb_{4}C_{3}T_{x} has mixed termination (Supplementary Fig. 2 and Supplementary Discussion). Highangle annular darkfield scanning transmission electron microscopy (HAADFSTEM) and energydispersive X‐ray analysis (EDX) display uniform distributions of Nb, C, F and O in Nb_{4}C_{3}T_{x} MXene (Supplementary Fig. 3). A selectedarea electron diffraction (SAED) pattern of a Nb_{4}C_{3}T_{x} flake demonstrates the high crystallinity and hexagonal symmetry (P6_{3}/mmc), in line with typical MXene structures^{3,6}. We further conducted highresolution atomic force microscope (AFM) measurements, which are described in Supplementary Fig. 4. Based on the analysis, over ~60% of the MXene flakes are ~5 nm in thickness, corresponding to a bilayer structure for Nb_{4}C_{3}T_{x} MXene^{6}. The homogeneity and good quality of our samples are further confirmed by transmission electron microscopy (TEM) and scanning electron microscopy characterizations in Fig. 1e and Supplementary Fig. 5, showing that the lateral dimension is on the order of several micrometres.
Niobiumbased MXene has been predicted to have a narrow bandgap (on the order of ~0.1 eV)^{16}. In line with this prediction, the bandgap of Nb_{4}C_{3}T_{x} in our study was determined to be ~0.19 eV from the Tauc plot method (Supplementary Fig. 7). Static terahertz measurements further support the narrowgap nature of the Nb_{4}C_{3}T_{x} MXene. We find negligible conductivity (or, equivalently, absorption) by free charge carriers in our 0–2 THz window (corresponding to 0–8 meV; Fig. 2a). In contrast, the metallic Ti_{3}C_{2}T_{x} exhibits substantial conductivity (absorption) in this terahertz region (Fig. 2a).
The different—semiconducting and metallic—natures of Nb_{4}C_{3}T_{x} and Ti_{3}C_{2}T_{x} MXenes, respectively, give rise to distinct photoresponses, as shown in Fig. 2b,c. Photoexcitation of a semiconductor like Nb_{4}C_{3}T_{x} promotes electrons to the conduction band, increasing the material’s conductivity from ~0 (Fig. 2a) to a finite value (that is, giving rise to a positive photoconductivity)^{17}. In contrast, for metallic materials such as Ti_{3}C_{2}T_{x}, optical excitation of conducting electrons results in heating up of the electron gas, which gives rise to a transiently reduced conductivity (that is, negative photoconductivity)^{18}. The samples’ photoconductivity is determined by opticalpump terahertzprobe (OPTP) spectroscopy. OPTP spectroscopy provides insight into the electrical transport properties of charge carriers in a contactfree and noninvasive fashion^{17,19,20,21}. Thanks to the transient nature of terahertz probe pulses (with duration of ~1 ps), terahertz spectroscopy characterizes shortrange, intraflake transport (over approximately tens of nanometres) in MXenes^{17,22}. By monitoring the photoinduced terahertz absorption (ΔE) at various pump–probe delays, the photoconductivity (Δσ_{opt}) dynamics can be quantified using the thinfilm approximation as follows^{17}:
where Z_{0} = 377 Ω is the impedance of free space, n_{sub} = 1.95 is the refractive index of the fusedsilica substrate in the terahertz range and L is the excitation thickness. E_{pump} and E_{0} represent the transmitted terahertz electric field with and without photoexcitation, respectively (see Methods for details).
The photoconductivity of Nb_{4}C_{3}T_{x} MXene (Fig. 2b) decays swiftly within several picoseconds, followed by a slow decay (>1 ns, limited by our temporal probe window). We attribute the fast photoconductivity decay in the Nb_{4}C_{3}T_{x} MXene to the trapping of free charge carriers at defects. This claim is supported by the fluencedependent OPTP dynamics analysis, in which trap filling is observed (Supplementary Fig. 8 and Supplementary Discussion)^{23}. In addition, by comparing different sample batches, we observe that, for samples with a slightly higher degree of oxidation (characterized by Raman spectroscopy), the relative weight of the fast decay is higher (Supplementary Fig. 9). As such, surface oxidation is a potential candidate for defect formation and charge carrier trapping^{15}. The negative photoconductivity in Ti_{3}C_{2}T_{x} MXene confirms the metallic nature of Ti_{3}C_{2}T_{x}. The transient photoconductivity decrease in metallic Ti_{3}C_{2}T_{x} MXene is consistent with previous reports^{24}. Therefore, the ultrafast terahertz photoconductivity traces shown in Fig. 2 confirm the semiconductorlike behaviour in Nb_{4}C_{3}T_{x} MXene and a metallic response in Ti_{3}C_{2}T_{x} MXene.
To investigate the charge transport mechanism in Nb_{4}C_{3}T_{x} MXenes, we complementarily employed temperaturedependent ultrafast OPTP and static electrical measurements. To provide a fair comparison between terahertz and electrical studies, we followed the same film deposition recipe to make thin films of Nb_{4}C_{3}T_{x}. As shown in Fig. 3a, the photoconductivity increases at reduced temperatures by a factor of 3–4, that is, dΔσ_{opt}/dT < 0. Meanwhile, a substantial, yet opposite trend (dσ_{ele}/dT > 0) is observed in electrically measured conductivity. The spectroscopic result suggests that bandlike transport dominates the charge transport mechanism in Nb_{4}C_{3}T_{x} MXene, with thermally populated phonons limiting the charge mobility through carrier–phonon scattering. Notably, a temperature dependence T^{−β} of the conductivity has been predicted in molybdenum and titaniumbased MXenes, with 1.5 < β < 2.1 (refs. ^{9,10}). Our results reveal β ≈ 1 for the averaged longlived photoconductivity (as shown in Fig. 3b). This experimentally observed weaker temperature dependence of the conductivity suggests that, besides the dominant role of phonons, impurity scattering may also contribute to the limitation of the charge transport in Nb_{4}C_{3}T_{x} MXene.
We note that the dΔσ_{opt}/dT < 0 in our spectroscopic result is in line with previous theoretical studies^{9,10,11}, but in stark contrast to the dσ_{ele}/dT > 0 in device results shown here and also reported previously for other MXene materials measured using the fourprobe method^{5,6,7}. This apparent contradiction can be explained as follows. In the terahertz photoconductivity measurements, the transient, picosecondduration terahertz field drives the charge carrier over approximately tens of nanometres. Therefore, terahertz spectroscopy provides the local, intraflake charge transport information (given the micrometre size of the flakes). Meanwhile, static electrical transport studies provide longrange charge carrier conductance over macroscopic distances between the electrodes under d.c. bias^{14,25}. As such, we can rationalize the results by postulating bandlike transport dominating intraflake charge conduction, while the interflake hopping is the ratelimiting step for charge percolation through devices consisting of many MXene flakes. Importantly, such a scenario reconciles the debate between previous theoretical and experimental studies on the charge transport mechanism in MXenes. Moreover, these findings are relevant for the optimization of MXenebased devices: for example, interface molecular engineering to enhance the interflake transport or increasing the geometrical size is therefore recommended^{26}.
To further confirm and investigate the bandlike, intraflake charge transport in MXenes, we disentangled the contribution of charge mobility (μ) from carrier density (ΔN) to the photoconductivity Δσ_{opt} (= ΔNeμ, where e is the elementary charge) by measuring the temperaturedependent photoconductivity spectra employing terahertz timedomain spectroscopies (THzTDS; for details of the measurement and data analysis, see Methods). We recorded the conductivity spectra at a delay of ~200 ps after photoexcitation to ensure a quasisteadystate charge carrier situation. The frequencyresolved complex photoconductivity spectra are shown in Fig. 4a. We analysed the spectra using the Drude–Smith (DS) model and found that the model describes the data well. The DS model describes the transport behaviour of free carriers in a medium, where charges experience a preferentially backscattering effect due to nanoscale confinements (the presence of grain boundaries, for example)^{17,27,28}. The DS model equation reads
where τ, ω_{p}, ε_{0} and m* are the effective carrier momentum scattering time, plasma frequency, vacuum permittivity and charge effective mass, respectively. The parameter c characterizes the probability that the charges backscatter between the scattering events, with the values ranging from 0 (isotropic scattering) to −1 (completely backscattering). The best fit to the DS model yields the parameters of τ, ω_{p} and c, which are plotted as a function of temperature (Fig. 4b and Supplementary Fig. 10). Interestingly, we find that ω_{p} and c depend very weakly on temperature, whereas τ increases drastically with lowering temperatures, from 38 ± 6 fs at 404 K to 95 ± 3 fs at 78 K. This substantial change demonstrates the dominant role of scattering time τ and thus effective charge mobility µ \((={\frac{{e \, \tau }}{{m^\ast }}}{\left( {1 + c} \right)})\) (refs. ^{17,27}) in the d.c. limit in governing the photoconductivity amplitude as shown in Fig. 2. The threefold increase in scattering time is quantitatively consistent with the approximately three to fourfold increase in photoconductivity amplitude. Based on these parameters, the intraflake charge carrier mobility in Nb_{4}C_{3}T_{x} can readily reach ~10^{3} cm^{2} V^{−1} s^{−1} at room temperature, using a reduced effective carrier mass of 0.04m_{0} (ref. ^{16}).
To quantitatively analyse the Tdependent scattering mechanism, we consider charge scattering contributions from both phonons and impurities following Matthiessen’s Law (for an extended discussion see Methods)^{25}:
where γ is the total scattering rate, and γ_{ac}, γ_{LO} and γ_{imp} are the scattering rates due to acoustic phonons, LO phonons and impurities, respectively.
Under the effective mass approximation, the acoustic phonon scattering rate reads^{29}
where k_{B}, c_{ii} and E_{def} represent the Boltzmann constant, the elastic constant and the electron deformation potential, respectively (see Methods for more details).
The term γ_{LO}(T) describes the scattering from LO phonons following^{30}
where ω_{LO}, m_{p} and α are the angular frequency of the LO phonon, the polaron effective mass and the carrier–LO phonon coupling constant, respectively. This expression is valid in the limit of weak electron–LO phonon coupling (α < 6, justified below). f(α) is a dimensionless function, slowly varying from 1.0 to 1.2 for sufficiently small coupling constant (0 < α < 4)^{30}. Here we use ω_{LO} = 19.5 THz, estimated from other MXene systems^{9,31,32}. This ~20THz LO phonon mode is associated with the inplane vibration in the carbonbased MXene structures, which is theoretically predicted to dominate the carrier–LO phonon interaction over all other phonon modes^{9,33}. The calculated phonon linewidth (which is positively correlated with the electron–phonon coupling strength) for ω_{LO} (= 19.5 THz) is several times higher than those for other optical phonon branches.
In contrast to the phononscattering mechanism, defect scattering, for example, from ionic impurities, increases as the temperature decreases. The typical temperature dependence for ionic impurity scattering rate scales as^{25}
where A is a prefactor characterizing the strength of impurity scattering.
Following the above model (equations (3)–(6)), we reproduce the temperature dependence of the effective scattering rate (defined by \({\gamma} = {\frac{1}{\tau }}\)) using only two adjustable parameters (A and α) (for details see Methods). As shown in Fig. 4c, the LOphonon scattering dominates the charge transport over the entire temperature range, while impurity scattering becomes notable only at lower temperatures (T < 150 K). Acoustic phonon scattering is not required to describe the data. This is in line with both density functional theory (DFT) calculations, which show γ_{ac}(T) < 1 THz (see below and Supplementary Fig. 15), and previous theoretical predictions^{9}. The analysis yields the value of α to be 0.77 ± 0.10. The small value of α (<6) indicates the formation of large polarons in Nb_{4}C_{3}T_{x} thin films.
The large polaron limit is further confirmed by the more general Feynman polaron theory. This theory is nonperturbative and applicable for arbitrary coupling strength^{34,35,36}. By variationally solving the Feynman polaron model^{36}, the data analysis provides the carrier–LO phonon coupling constant in the range of 0.5 < α < 4 for Nb_{4}C_{3}T_{x} (Supplementary Fig. 14a and Supplementary Discussion).
An independent estimate of the electron–phonon coupling strength can be obtained from the Landau–Pekar model (equation (7)), which is widely employed to estimate the charge–LO phonon coupling strengths^{12,13,36}. Strictly speaking, the model possesses an intrinsic limitation, as it is designed to treat continuum Fröhlich polarons, but formally justified results can only be obtained in the strong electron–phonon coupling regime^{13,34,37,38}:
where ε_{opt} and ε_{s} are, respectively, the optical and static dielectric constants, and ħ is the reduced Planck constant. The dielectric function for Nb_{4}C_{3}T_{x} MXene versus the photon energy in the infrared to ultraviolet (IR–UV) range (0.4–3.1 eV) is shown in Supplementary Fig. 11. In the measured range, our spectra are very similar to the theoretical results of that in Ti_{3}C_{2}T_{x} MXene^{39}. Therefore, considering equation (7) as well as the dielectric properties (ε_{opt} ≈ 4–6 from our measurement, ε_{s} = 22 based on the theoretical study), we infer 0.74 < α < 1.29. This quantitative agreement between α obtained from the experiment and that obtained from the Fröhlich polaron theory supports the occurrence of large polaron transport in Nb_{4}C_{3}T_{x} MXene materials.
Additional evidence for the dominating role of optical phonons in limiting the electrical transport properties of MXenes is provided by firstprinciples DFT calculations (for details see Methods). In short, we have performed numerical simulations of the full Boltzmann transport equation on Nb_{4}C_{3}, Nb_{4}C_{3}T_{x} (with T_{x} = O_{2} and (OH)_{2}), Ti_{3}C_{2} and Ti_{3}C_{2}T_{x} (with T_{x} = (OH)_{2}), using a constant relaxation time for acoustic phonons that is calculated ab initio in the deformation potential approximation. The calculated scattering rates vary with both the nature of the metal ion (being systematically smaller for Ti than for Nb) and of the functional groups (with a reduction when oxygenated species are present), but are consistently below 1 THz for all cases in the T range considered (Supplementary Fig. 15). This supports our claim of the minor role of acoustic phonon scattering in determining the electrical conductivity in MXenes. Furthermore, if we apply the same formalism but using the measured scattering rates (thus accounting for optical phonons), the charge mobility values at room temperature decrease from ~10^{4} cm^{2} V^{−1} s^{−1} in the acoustic limit down to ~10^{3} cm^{2} V^{−1} s^{−1}, in good agreement with the experimental data (Supplementary Fig. 16).
Given the very similar dielectric constants for different types of MXene structure^{39}, we expect that the formation of large polarons in MXenes is universal. To validate such a statement, we further studied the charge transport mechanism in metallic Ti_{3}C_{2}T_{x}—the model MXene widely used for electrochemical energy storage applications^{1,2,3}. The metallic nature of Ti_{3}C_{2}T_{x} MXene allows us to evaluate its complex conductivity by performing static THzTDS measurements without involving optical excitations. In static THzTDS measurements, we compared the amplitude and phase of the terahertz pulses transmitted through the substrate alone and the sample on the substrate in the frequency domain^{17,22}. We conducted Tdependent static THzTDS measurement from 78 to 397 K (Fig. 4d). Analysing the complex terahertz conductivity with the DS model yields scattering time τ, plasma frequency ω_{p} and backscattering rate c, at various T. As shown in Fig. 4e, at elevated temperatures, scattering by phonons limits the scattering time, revealing a bandlike charge transport nature. Interestingly, with rising T, the \({\omega _{\rm{p}}^2}\) (∝ carrier density) increases, probably due to the detrapping of charge carriers by thermal activation (Fig. 4e, inset). The excess carrier density in the highT regime (>288 K) could impose the additional carrier–carrier interaction channel onto the scattering mechanism. This is also supported by the Tdependent backscattering rate c, where c remains constant from 78 to 269 K, but increases in absolute magnitude when T exceeds 288 K (Supplementary Fig. 12). As such, to prevent the contribution from carrier–carrier scattering, we limited our analysis of the scattering rate in Ti_{3}C_{2}T_{x} MXene to T ranging from 78 to 269 K. Following Matthiessen’s rule and the model discussed for Nb_{4}C_{3}T_{x}, we find that the Tdependent scattering rate can be described properly by considering the scattering from phonons and impurity (for details see the Methods). As shown in Fig. 4f, the carrier–LO phonon scattering dominates the overall scattering rate in the highT regime (>100 K), whereas the impurity scattering governs in the lowT range. Notably, the analysis yields carrier–LO phonon coupling α of 0.50 ± 0.05, indicating the formation of large polarons also in Ti_{3}C_{2}T_{x} MXene. Independently, we further calculated α following the Landau–Pekar polaron model (equation (7)) to be between 0.44 and 1.04, and, by solving the Feynman polaron model (Supplementary Fig. 14b), we obtained the carrier–LO phonon coupling constant in the range of 0.5 < α < 3 for Ti_{3}C_{2}T_{x}. Interestingly, the inferred carrierimpurity scattering rates in both Ti_{3}C_{2}T_{x} and Nb_{4}C_{3}T_{x} MXenes are very similar (Supplementary Fig. 13). We observed variations in trapping due to varying defect densities in different samples, yet the charge scattering time in Nb_{4}C_{3}T_{x} is independent of the defect level (Supplementary Fig. 9). These results suggest that charge carriers might be protected as large polarons, leading to the reduced charge scattering, despite the unavoidable high defect density in MXenes^{40,41}. We note that angleresolved photoemission spectroscopy (ARPES) is instructive for identifying the presence of polaronic features^{42,43}. A recent ARPES study has provided the evidence of polaron states in single crystals of the parent phase of MXene, MAX compounds, consistent with our claim of polaron formation in MXenes^{44}.
Considering Fröhlich polaron theory and the similar dielectric constants for different types of MXene structure, weak carrier–LO phonon coupling is probably a generic property of MXenes, leading to large polaron formation. Experimentally, we indicate the presence of large polarons in two popular MXenes, including semiconducting Nb_{4}C_{3}T_{x} and metallic Ti_{3}C_{2}T_{x} MXenes. Large polaron formation may determine not only the intrinsic charge transport, but also carrier lifetimes in MXenes. For example, in organic–inorganic perovskites, the formation of large polarons has been proposed to effectively screen the defect potential, resulting in mobile charge carriers with high defect tolerance^{40}. The indication of large polaron formation in MXenes may partially rationalize the reported extremely high carrier mobility in this class of 2D materials, despite structural and surface defects.
Conclusions
In this Article, we reconcile the debate between previous theoretical and experimental studies on the charge transport mechanism and propose a unifying picture of charge transport in MXenes. This is achieved by combining terahertz and static electrical transport measurements to distinguish the short and longrange transport characteristics. We reveal that the bandlike charge transport governs intraflake charge transport in MXenes, whereas interflake transport occurs by hopping and becomes the limiting step for charge percolation through the network of MXene flakes. Furthermore, by analysing the charge carrier scattering following Matthiessen’s law, we find that carrier–LO phonon scattering dominates the intraflake carrier transport for both semiconducting and metallic MXenes. The resultant small carrier–LO coupling constant (α ≈ 1) indicates large polaron formation in MXenes. Our work sheds light on the polaronic nature of free charges in MXenes, and unveils intra and interflake transport mechanisms in the MXene network, relevant for both fundamental studies and applications.
Methods
Material preparation
Nb_{4}C_{3}T_{x} MXene was synthesized according to the previously reported hydrogen fluoride etching method^{45}. Briefly, 0.8 g of Nb_{4}AlC_{3} MAX was etched by 20 ml of 49 wt% hydrogen fluoride aqueous solution at room temperature for 96 h. After washing by deionized water until the pH was above 6.0, the obtained multilayer Nb_{4}C_{3}T_{x} was further delaminated by tetramethylammonium hydroxide (TMAOH; 25% in H_{2}O) with magnetic stirring for 5 h. The Nb_{4}C_{3}T_{x} MXene nanosheets were obtained by repeatedly washing and sonication in deionized water for 1 h. The sediment without delamination was removed by centrifugation at 2,000 r.p.m. for 20 min and the delaminated MXene flakes in supernatant were collected for subsequent measurements.
Material characterizations
The morphology of each sample was characterized via scanning electron microscopy (Zeiss Gemini S4 500) and TEM (FEI Talos F200X, 200 kV). Chemical compositions of the samples were analysed by XRD using a PW1820 Xray diffractometer with Cu Kα radiation. Raman spectra were measured on a Renishaw inVia reflex spectrometer with a laser excitation of 532 nm. XPS spectra were recorded with a a Kratos Axis Ultra^{DLD} spectrometer using the monochromatic Al Kα source (1,486.6 eV). The static electrical conductivities were measured using the van der Pauw fourprobe method, based on a commercial Lakeshore Hall system (9700A).
Terahertz spectroscopy setup
The terahertz spectroscopy was powered by a femtosecond laser source providing pulses with a duration of ~50 fs and a repeating frequency of 1 kHz, at a central wavelength of 800 nm. The terahertz radiation was generated by the optical rectification effect in a ZnTe nonlinear crystal (along 〈110〉 orientation), and the terahertz probe field was mapped out through electrooptical sampling by femtosecond laser pulses in a second ZnTe detection crystal. For opticalpump terahertz probe measurements, the terahertz absorption induced by optical excitations (here, 800 nm) was monitored by fixing the sampling beam to the peak of the terahertz field. In this configuration, we measured timedependent pumpinduced terahertz absorption by tuning the time delay between the pump and terahertz probe. The entire terahertz setup was kept under nitrogen purging to avoid terahertz absorption by vapours. The samples were either purged by dry N_{2} or placed under vacuum conditions (<2 × 10^{−4} mbar) during measurements.
Modelling the charge scattering rate in MXenes
The charge scattering contributions were analysed following Matthiessen’s law, by considering the carrier interaction with acoustic phonons, LO phonons and impurities (equation (3)). We considered the carrier–acoustic phonon scattering under the effective mass approximation (equation (4))^{29}, and we took the elastic constant c_{ii} = 3.86 × 10^{12} dyn cm^{−2} from Nb_{4}C_{3}T_{x} MXene in the literature^{6}. The electron deformation potential, E_{def}, defined as the linear displacement coefficient of the single electron band energy, was used to characterize the electron–acoustic phonon coupling strength. Neither m* nor E_{def} has been reported previously in Nb_{4}C_{3}T_{x} MXene materials. Considering the similarity of the chemical structures of Nb_{4}C_{3}T_{x} and Nb_{2}CT_{x} MXenes, we estimated the values of m* and E_{def} from those in Nb_{2}CT_{x} MXene^{16,46}. The modification of the lower oxidation state in the inner transition metal atoms to m* and E_{def} is expected to be small^{47,48,49}. For carrier–LO phonon scattering rate γ_{LO}(T) (equation (5)), in the moderate coupling regime (α < 6) we took into account the polaron mass m_{p} and the effective charge mass m* by \({\frac{{m_{\rm{p}}}}{{m^\ast }}} = {\left( {{1} + {\frac{\alpha }{6}} + {\frac{{\alpha ^2}}{{40}}} + {\cdots }}\right)}\) (ref. ^{30}). As for carrierimpurity scattering (equation (6)), we primarily considered the contribution from ionic impurity scattering, where the scattering increases as the temperature decreases. For modelling the scattering rate in Ti_{3}C_{2}T_{x} MXene, we applied the same method and took the values of c_{ii}, m* and E_{def} from the literature^{46,50,51}.
Computational details
DFT calculations were performed with the projectoraugmented wave (PAW) basis set, as implemented in the VASP code^{52,53}, with the exchange and correlation effects treated at the Perdew−Burke−Ernzerhof (PBE) level^{54}. Dispersion forces by Grimme correction (PBE+D2)^{55} and dipole moment correction along the c axis (z direction and perpendicular to the MXene surface) were incorporated with a kinetic energy cutoff of 500 eV and using a Monkhorst–Pack sampling of 3 × 3 × 1 for the Brillouin zone integration for all geometry optimizations. Vacuum spacing was set to be 30 Å to avoid the interaction with periodic images. Subsequent selfconsistent field calculations were performed using a dense Kpoint grid of 100 × 100 × 1 for the Brillouin zone integration. Electron conductivities and mobilities were estimated using the semiclassical Boltzmann transport theory within both the deformation potential approach and the relaxation time approximations^{56,57,58}.
Data availability
All data generated or analysed in this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.
Code availability
The code used to analyse the data and perform numerical simulations is available from the corresponding authors upon reasonable request.
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Acknowledgements
Financial support by the Max Planck Society is acknowledged. This work was financially supported by the European Union’s Horizon 2020 research and innovation programme (GrapheneCore3 881603) and Deutsche Forschungsgemeinschaft (MXOSMOPED project and CRC 1415 (grant no. 417590517)). The work in Mons is financially supported by FLAGERA JTC 2017 project MXOSMOPED, the Belgian National Fund for Scientific Research (FRSFNRS), the Consortium des Équipements de Calcul Intensif (CÉCI) under grant no. 2.5020.11 and by the Walloon Region (ZENOBE Tier1 supercomputer, grant no. 1117545). D.B. is FNRS Research Director. We thank K. Krewer, M. Grechko, M. Ballabio, X. Jia and A. Tries for fruitful discussions. We acknowledge P. Kumar, H. Kim and R. Ulbricht for constructive comments on the manuscript. We are grateful to H. Burg and R. Berger for conducting AFM image measurements. S.F. acknowledges fellowship support from the Chinese Scholarship Council (CSC). L.D.V. acknowledges support from the EU Horizon 2020 Framework Programme (grant no. 811284).
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H.I.W. and M.B. conceived and designed the project. W.Z. conducted the terahertz spectroscopy experiments, analysed the data and performed the modelling. Z.L. performed van der Pauw conductivity measurements. B.S. and D.L. synthesized the sample and provided basic structural characterizations, under the supervision of M.Y. and X.F. S.M.G. and D.B. performed the DFT calculations. W.Z., H.I.W. and M.B. drafted the manuscript with input from all other authors.
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Zheng, W., Sun, B., Li, D. et al. Band transport by large Fröhlich polarons in MXenes. Nat. Phys. 18, 544–550 (2022). https://doi.org/10.1038/s4156702201541y
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DOI: https://doi.org/10.1038/s4156702201541y
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