Introduction

Utilized for the detection of specific gases, including toxic gases1, volatile organic compounds (VOCs)2, flammable and explosive gases3, chemiresistive gas sensors serve as electronic devices, paramount in safeguarding human health and safety4,5. In particular, two-dimensional (2D) materials, such as graphene6, transition metal dichalcogenides (TMDs)7, phosphorene8, and MXenes9, have risen to prominence as potential candidates for chemiresistive gas sensors due to their high surface-to-volume ratio, tunable surface functionalities, room temperature (RT) operation condition and low cost for sensor fabrication10,11,12. Within these sensors (Fig. 1a), the detection of target gases hinges on the monitoring of shifts in their electrical resistance13. It is well accepted that the resistance variations stem from the impact of gas adsorption on the sensing material’s carrier concentration and mobility6,14,15,16. As illustrated in Fig. 1b, the generation of response in 2D chemiresistive gas sensors involves a multitude of processes, including gas adsorption, modulation of material’s carrier concentration induced by the charge transfer between gas and sensing material, and carrier mobility influenced by factors like electron-phonon and ionized impurity scattering. Nevertheless, current studies fall short of comprehensively capturing these intertwined processes or quantifying the contribution of carrier concentration and mobility to the total gas response of sensing materials. The gap in understanding the intrinsic gas sensing mechanism underscores the crucial role of accurate simulations in advancing the field of 2D chemiresistive gas sensors.

Fig. 1: Schematics for calculating the response of 2D gas sensing materials.
figure 1

a Illustration of a chemiresistive gas sensor based on 2D MoS2 for NH3 detection. b Diagrammatic representation to explain the effect of gas molecules (NH3) adsorption on the carrier concentration (left inset) and mobility (right inset) of sensing material (2D MoS2). c First-principles workflow for calculating the response of 2D gas sensing materials.

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The depletion layer model was first proposed to qualitatively explain the chemiresistive mechanism for metal oxide semiconductors sensor14. It emphasizes that the resistance alterations of sensing materials are linked to the formation of carrier depletion or accumulation layers that triggered by the adsorption of target gases10,17. Building on insights from the depletion layer model, several theoretical studies employed density functional theory (DFT) to investigate the sensing behavior of chemiresistive gas sensors18,19,20,21,22. For instance, Dravid et al. calculated the charge transfer between MoS2 and specific gases, and discovered that NH3 and NO2 act as electron donors and acceptors to alter the carrier concentration of MoS2, clarifying the opposite resistance shifts of MoS2 with different gases adsorption in experiments7. Based on the calculated band structures by DFT, Lei et al. found that the adsorption of NH3 on the oxygen vacancy of MoO3 can increase the dispersion of gap state and narrow band gap, leading to a more pronounced enhancement in its conductivity, explaining its remarkable sensitivity to NH323. Luo et al. employed the charge transfer and adsorption energy to shed light on the excellent selectivity of CdS quantum dot (QD) gel to NO2 and the optimal combination of strong response and fast recovery achieved by PbCdSe QD gels24,25. Furthermore, Chen et al. introduced a charge-transfer-based approach to quantitatively access the carrier concentration of phosphorene exposed to NO2 and successfully explained the thickness dependence8.

However, current approaches, whether semi-quantitative or quantitative, solely based on charge transfer, leading to the following two major impediments: (1) the disregard for the impact of carrier mobility, and (2) the potential overestimation of carrier concentration changes. Moreover, the lack of a comprehensive simulation method for the full adsorption-response gas sensing process hinders the quantitative prediction of key metrics of gas sensing materials, such as response and limits of detection(LOD).

In this work, we introduce a first-principles method to calculate the response of 2D gas sensing materials, consisting of two modules: (1) a carrier concentration module, which provides more accurate carrier concentrations by considering the electronic structure around Fermi level, and (2) a carrier mobility module, which employs density functional perturbation theory (DFPT) and Wannier interpolation to compute carrier mobility that taking into account electron-phonon and ionized impurity scattering. Motivated by the abundance of experimental results of 2D MoS2 gas sensors7,12,26,27, we chose it as an illustrative prototype to demonstrate our computational method and its predictions. Our method can predict a reliable response that is accord with the experimental results, and give an accurate LOD. In contrast, we discover that the previous charge-transfer-based method tends to overestimate 2D MoS2’s response towards NH3, resulting in a predicted limit of detection (LOD) that is significantly more sensitive than the experimental values by nearly 2 orders of magnitude. The analysis that decouples carrier concentration and mobility from the conductivity of MoS2 offers a quantitative understanding of their contributions to the overall response of 2D MoS2 to NH3, demonstrating that its gas sensing mechanism is primarily dominated by carrier concentration, and the overestimated response or underestimated LOD in the charge-transfer-based method primarily stem from overestimating variations in carrier concentrations. Our findings not only demonstrate that the previous charge-transfer-based method offers a reasonable qualitative assessment but also reveal its limitations in providing quantitative predictions for the response of carrier concentration-dominated materials like MoS2. Moreover, our study opens exciting opportunities for exploring materials dominated by carrier mobility like SnO2 and alloyed MoS2-xSex15,28.

Results

First-principles framework for calculating response of 2D materials

As depicted in Fig. 1c, we start by constructing a sensing material surface from the relaxed 2D unit cell. Utilizing a statistical thermodynamics model8,29,30, we calculate the adsorption densities ρgas of the target gas on this surface across varying gas concentrations (details in Supplementary Note 1). We then introduce a linear interpolation method to determine the corresponding carrier concentrations n of the sensing material (refer to Supplementary Note 2). Based on the computed carrier concentrations n, the carrier relaxation time τ is ascertained through Fermi’s golden rule. Subsequently, the carrier mobility μ is derived from the Boltzmann transport equation (BTE), and the material’s conductivity σ is obtained as σ = e ⋅ n ⋅ μ. Ultimately, this enables us to gauge the response S of sensing material to various target gas concentrations (more details in Supplementary Note 3).

Carrier concentration module

To obtain the carrier concentration of 2D MoS2 at different NH3 concentrations, the corresponding NH3 adsorption density on MoS2 must first be determined. Figure 2a presents the Morse potential, represented by the black line, which is fitted using DFT results (shown as blue points). It describes the interaction between the NH3 molecule and MoS2 surface as a function of distance. With the fitted Morse potential, the adsorption densities of NH3 on the bilayer MoS2 surface upon different NH3 concentrations are deduced through a statistical thermodynamics model (Fig. 2b, see Supplementary Note 1 for calculation details), which has been employed to calculate the adsorption density of gases on graphene29,30 and phosphorene8 surfaces. Subsequently, to precisely ascertain the carrier concentration of MoS2 at arbitrary NH3 adsorption density, we introduce a linear interpolation method (details provided in Supplementary Note 2). As illustrated in Fig. 2c, given that NH3 acts as an electron donor, its adsorption results in an upward shift in the Fermi level of the MoS2 surface, leading to an increased carrier concentration. It should be pointed out that the band structure and band gap of MoS2 surface remain largely unchanged after NH3 adsorption, as evidenced by Supplementary Fig. 7. Moreover, the fat-band analysis (Supplementary Fig. 8) confirms that the energy bands of all atomic orbitals of NH3 lie outside the band gap and are distant from the conduction band minimum (CBM) and valence band maximum (VBM). Consequently, the carrier concentration of the MoS2 surface upon NH3 adsorption can be determined based on the corresponding shift of Fermi level. However, at low NH3 concentrations like ppm level, the adsorption density or coverage of NH3 on the MoS2 surface is notably low. This necessitates creating an exceptionally large MoS2 surface, which poses a prohibitive computational burden. As indicated in Fig. 2d, the linear interpolation method offers a credible estimation for the carrier concentration of MoS2 surface at low NH3 concentrations. By referencing the NH3 adsorption densities on MoS2 surface at various NH3 concentrations (Fig. 2b), we can determine the corresponding carrier concentrations (blue points in Fig. 2e, see detail in Supplementary Note 2). Moreover, it should be pointed out that the carrier concentrations of MoS2 calculated by charge-transfer-based method (orange points in Fig. 2e, see detail in Supplementary Note 7) are always higher than those obtained by our method, especially noticeable at elevated NH3 concentrations.

Fig. 2: NH3 adsorption density on bilayer MoS2 and resultant carrier concentrations of MoS2 under varying NH3 concentrations.
figure 2

a The Morse potential type of interaction strength between the single NH3 molecule and 4 × 4 bilayer MoS2 surface with respect to the distance which defined as the height difference between the mass center of NH3 and top S layer of MoS2. The inset figure shows the most stable adsorption configuration among all possible configurations listed in Supplementary Fig. 6. The H, N, S, and Mo atoms are shown as white, blue, yellow, and green balls, respectively. b The NH3 adsorption density on bilayer MoS2 under different NH3 concentrations. c The illustrative representation of the Fermi level shift induced by NH3 adsorption on the bilayer MoS2. d The comparison between the carrier concentrations calculated by theoretical method and the corresponding interpolated values. e The carrier concentrations of bilayer MoS2 under different NH3 concentrations calculated by charge-transfer-based (CTB) method (orange points, see detail in Supplementary Note 7) and our method (blue points), respectively.

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Carrier mobility module

For the determination of mobility, we prioritize the phonon and ionized impurity scattering in this work. Due to the extensive computational requirements of carrier mobility module (highlighted using red color in Fig. 1c), particularly those involving DFPT, we employ the bilayer MoS2 unit cell for our carrier mobility computations. The calculated band structure in Fig. 3a demonstrates that the bilayer MoS2 has an indirect band gap of 1.65 eV, aligning with previously reported theoretical values31. Figure 3b depicts the phonon dispersion of bilayer MoS2, where the absence of imaginary frequencies confirms the dynamical stability of its structure. We also examined the band structure and phonon dispersion of monolayer MoS2 (Supplementary Fig. 2 and Supplementary Fig. 3). Finally, the calculated carrier mobilities of monolayer and bilayer MoS2 are 111.64 and 158.71 cm2 V−1 s−1, respectively. Notably, for the extensively studied monolayer MoS2, our computed mobility is in good consistent with other computational values32,33,34.

Fig. 3: The carrier mobility and scattering rate of bilayer MoS2 under varying NH3 concentrations.
figure 3

a The band structure of bilayer MoS2 unit cell. b The phonon dispersion of bilayer MoS2 unit cell. c The logarithm of the electron–phonon limited mobility (blue points), electron-ionized impurity limited mobility (red points), and total mobility (orange points) of MoS2 as a function of NH3 concentrations. The total mobility can be determined by Matthiessen’s rule: \(\frac{1}{{\mu }_{{{{\rm{total}}}}}}=\frac{1}{{\mu }_{{{{\rm{e}}}}-{{{\rm{ph}}}}}}+\frac{1}{{\mu }_{{{{\rm{e}}}}-{{{\rm{imp}}}}}}\). d The electron-phonon scattering rate (blue pints), electron-ionized impurity scattering rate (red points) and total scattering rate (orange points) of MoS2 as a function of electron energy at 1000 ppm NH3 concentration. Zero points are at the conduction band minimum. The total scattering rate (i.e. the reciprocal of relaxation time) can be determined via: \(\frac{1}{{\tau }_{{{{\rm{total}}}}}}=\frac{1}{{\tau }_{{{{\rm{e}}}}-{{{\rm{ph}}}}}}+\frac{1}{{\tau }_{{{{\rm{e}}}}-{{{\rm{imp}}}}}}\).

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Furthermore, we probe into the mobility of bilayer MoS2, which is limited by electron-phonon scattering and ionized impurity scattering across varying NH3 concentrations. As depicted in Fig. 3c, it is evident that electron-phonon scattering is largely invariant to fluctuations in NH3 concentrations. This finding aligns with the research conducted by Liu group32, who utilized a more precise methodology involving quadrupole scattering to reveal the insensitivity of monolayer MoS2 to carrier concentrations. It is widely recognized that the electron-phonon scattering is chiefly tied to lattice thermal vibrations, and has a pronounced temperature dependence33,35,36,37. In contrast, the ionized impurity scattering distinctly depends on carrier concentrations38. As further corroborated in Fig. 3c, the ionized impurity scattering intensifies at high carrier concentrations, which corresponds to high NH3 concentrations, subsequently reducing the carrier mobility. Even so, the total mobility of bilayer MoS2 remains predominantly determined by electron-phonon scattering across all NH3 concentrations examined in this study. Figure 3d details the scattering rate, limited by electron-phonon and ionized impurity scatterings, for an electron within the conduction band spanning an energy range of 0.1 eV at 1000 ppm NH3 concentration. This illustration confirms that the electron-phonon interaction is the primary contributor to the total scattering rate, resulting in the total mobility of 2D MoS2 being dependent on the electron-phonon limited mobility, as demonstrated in Fig. 3c.

Comparison between theoretical and experimental results

Combining the carrier concentration and mobility, we derived the response of 2D MoS2 across various NH3 concentrations, as illustrated in Fig. 4a. Notably, the response predicted by our scheme aligns very well with experimental data7,26. We have also compared with the charge-transfer-based method (orange squares), using the same carrier mobility. Figure 4a shows that charge-transfer-based method tends to overestimate the response, potentially leading to an underestimated LOD. A closer look for the response of 2D MoS2 at low NH3 concentrations (10−2–102 ppm) is also shown in Fig. 4b. Our results indicate a discernible response (2.75%) only starting from 10 ppm. This observation is consistent with experimental observations that pinpoint an LOD around 30 ppm for 2D MoS2 in response to NH326. While the results of charge-transfer-based method suggest that 2D MoS2 exhibits a detectable response (3.50%) even at 10−1 ppm NH3, which is a value 2 orders of magnitude higher than both experimental results and our predictions.

Fig. 4: The comparison between experimental results and computational predictions.
figure 4

a The response of MoS2 to different NH3 concentrations obtained by experiments7,26 (red points), charge-transfer-based (CTB) method (orange points, see detail in Supplementary Note 7) and our method illustrated in this work (blue points). b The comparison of LOD between measured in experiments (red points), predicted by charge-transfer-based method (orange points) and our method (blue points). c The conductivity (gray points), electron concentration (purple points), and mobility (cyan points) of MoS2 to different NH3 concentrations. d The comparison between the response (S) calculated by conductivity (gray columns, \(S=\frac{| \frac{1}{{\sigma }_{0}}-\frac{1}{{\sigma }_{{{{{\rm{NH}}}}}_{3}}}| }{\frac{1}{{\sigma }_{0}}}\times 100 \%\)) and electron concentration (purple columns, \(S=\frac{| \frac{1}{{n}_{0}}-\frac{1}{{n}_{{{{{\rm{NH}}}}}_{3}}}| }{\frac{1}{{n}_{0}}}\times 100 \%\)), respectively. σ0 and n0 are the conductivity and carrier concentration of pure MoS2, respectively. \({\sigma }_{{{{{\rm{NH}}}}}_{3}}\) and \({n}_{{{{{\rm{NH}}}}}_{3}}\) are the conductivity and carrier concentration of MoS2 with NH3 adsorption, respectively.

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To obtain a quantitative insight of carrier concentration and mobility contributions to the overall response of 2D MoS2 to NH3, we decouple these two factors from MoS2’s conductivity (Fig. 4c). It indicates that as NH3 concentration rises, the conductivity of 2D MoS2 also increases, a trend attributable to the escalating carrier concentrations. This congruence suggests that 2D MoS2 is a carrier concentration-dominated sensing material. Specifically, at low NH3 concentrations (<102 ppm), the carrier mobility of 2D MoS2 remains relatively unchanged, indicating that the response of 2D MoS2 to NH3 can be reliably estimated based solely on carrier concentration. However, at higher NH3 concentrations (>102 ppm), the diminishing carrier mobility tempers the conductivity growth of 2D MoS2, resulting in a subdued response. This trade-off between carrier concentration and mobility at high NH3 concentrations implies that the overall response of 2D MoS2 cannot be simply determined by carrier concentration under these circumstances. These observations lead us to consider two potential culprits behind the overestimated response of the charge-transfer-based method: (1) an overestimation of the carrier concentration changes (Fig. 2e), or (2) the oversight regarding the decrease in carrier mobility (Fig. 3c). To clarify this issue, we computed the carrier concentration-based response of MoS2, which only considers carrier concentration variations (depicted by the purple columns in Fig. 4d). It suggests that a reasonable response of 2D MoS2 can be estimated solely from carrier concentration. Therefore, the overestimated variation in carrier concentrations ought to be the key factor that results in overestimating the response of charge-transfer-based method, not the neglecting of the decrease in carrier mobility. This overestimation arises from the fact that the calculated charges transferred from NH3 to MoS2 may not fully become mobile-free carriers within MoS2. Instead, these transferred charges might either be trapped at specific sites near the interface or recombine with holes in MoS2. Moreover, it is vital to recognize that the determination of transferred charges by different computational methods, such as Bader charge and Mulliken charge, usually manifests notable disparities39, making the accurate prediction of carrier concentration variations by charge-transfer-based method even more challenging.

The above analysis demonstrates that charge transfer can serve as a valuable qualitative assessment of gas sensing performance, particularly for carrier concentration-dominated materials such as MoS2. More importantly, it reveals the potential of carrier mobility engineering as a novel scheme for designing and optimizing sensing materials.

Discussion

In summary, we propose a first-principles framework to assess the gas response of 2D sensing materials, featuring two pivotal modules: (1) a carrier concentration module, designed to deliver more accurate carrier concentrations by probing the electronic structure around Fermi level, and (2) a carrier mobility module, aimed at calculating carrier mobility with DFPT and Wannier interpolation, considering electron-phonon and ionized impurity scattering. As a proof of principle, we take 2D MoS2 as a prototype to evaluate our method. The comparative analysis with experimental results indicates that our method can provide an accurate prediction of response and LOD for 2D MoS2 to NH3, avoiding the overestimated response and underestimated LOD as encountered in charge-transfer-based method. Then, by decoupling carrier concentration and mobility from the conductivity of MoS2, our analysis provides a quantitative insight into their respective contributions to the overall response of 2D MoS2 to NH3, demonstrating that its gas sensing mechanism is primarily dominated by carrier concentration. Finally, we find that the overestimated response or underestimated LOD in charge-transfer-based method are attributed to overestimating the changes in carrier concentrations of MoS2 induced by NH3 adsorption. Our method not only confirms the charge-transfer-based method can provide a reasonable qualitative prediction for the gas response of carrier concentration-dominated materials like MoS2, but also opens exciting opportunities to explore carrier mobility-dominated materials such as SnO2 and alloyed MoS2−xSex15,28. Therefore, the first-principles method presented in this work enables us to directly screen the promising 2D materials with high gas sensing performance, and also provides a deeper understanding of their sensing mechanism, thereby paving a critical step toward the development of 2D gas sensing materials.

Methods

First-principles calculations of unit cell

All first-principles calculations are performed via the Quantum ESPRESSO package40 with the Norm-conserving pseudopotentials and the Perdew-Burke-Ernzerhof41 (PBE) exchange-correlation functional. A cutoff energy of 60 Ry and a 36 × 36 × 1 k-mesh42 are used to determine the equilibrium lattice constant until the force on each atom becomes less than 0.0001 Ry/Bohr. It has been reported that the analytic expression for Fröhlich electron-phonon matrix elements is strictly valid only in the long-wavelength limit (q → 0)37,43. To address this point, the equilibrium properties of phonons and the electron-phonon interaction matrices are calculated by density functional perturbation theory44 for a corse 12 × 12 × 2 k-mesh and a 12 × 12 × 2 q-mesh. We then use the EPW software package45 to interpolate the electronic information and phonon information, as well as the electron-phonon coupling matrices to a fine mesh using the Wannier interpolation method46. Based on the fat-band analysis (Supplementary Fig. 4 and Supplementary Fig. 5), the 5 d orbitals of Mo atom and 3 p orbitals of S atom are chosen as the initial guess to obtain the maximally localized Wannier Functions (MLWFs). Moreover, we compare the electronic band structures of monolayer MoS2 using first-principles calculations and Wannier interpolation (Supplementary Fig. 9). Additionally, the similar comparison for the phonon dispersion of monolayer MoS2 is presented in Supplementary Fig. 10. The good agreement and low spread of Wannier Functions (Supplementary Table 1 and Supplementary Table 2) ensure the reliability of our calculation. Our test results show that the calculation on a 600 × 600 × 4 k-mesh associated with a 100 × 100 × 4 q-mesh achieves the best balance between mobility convergence and computational efficiency for monolayer MoS2 (Supplementary Fig. 1). Additionally, both Li et al.47 and Late et al.7 found that multilayer MoS2 displays enhanced stability and sensitivity to target gases over its monolayer counterpart. We thus use this optimized mesh to compute the transport properties of bilayer MoS2 in this work. We have included semi-empirical dispersive van der Waals (vdW) interaction using Grimme-D3 parametrization48, which gives the closest interlayer distance (0.696 nm) of bilayer MoS2 to experimental results (0.70 nm)49.

First-principles calculations of slab

For calculating the adsorption density of NH3 on the bilayer MoS2 surface, we build a large MoS2 slab using a bilayer 4 × 4 supercell of MoS2, to avoid the interaction between adsorbed NH3 molecules (the distance between adsorbed NH3 molecules is larger than 10 Å). A cutoff energy of 30 Ry and a 3 × 3 × 1 k-mesh are used to relax the system until the force on each atom becomes less than 0.0005 Ry/Bohr.