Determining the Electronic Structure and Thermoelectric Properties of MoS 2 /MoSe 2 Type-I Heterojunction by DFT and the Landauer Approach

X = S, Se) Van der Waals heterojunctions are reported, with the intention of motivating the design of electronic devices using such materials. Calculations indicate the proposed heterojunctions are thermodynamically stable and present a band gap reduction from 1.8 eV to 0.8 eV. The latter effect is highly related to interactions between metallic d-character orbitals and chalcogen p-character orbitals. The theoretical approach allows to predict a transition from semi-conducting to semi-metallic behavior. The band alignment indicates a type-I hetero junction and band offsets of 0.2 eV. Transport properties show clear n-type nature and a high Seebeck coefficient at 300 K, along with conductivity values ( σ / τ ) in the order of 10 20 . Lastly, using the Landauer approach and ballistic transport, the proposed heterojunctions can be modeled as a channel material for a typical one-gate transistor configuration predicting subthreshold values of ≈ 60 mV dec − 1 and field–effect mobilities of ≈ 160 cm − 2 V − 1 s − 1 .


Introduction
Molybdenum disulfide (MoS 2 ) and layered transition metal dichalcogenides (TMD) are promising candidates for developing next-generation electronic, spintronic, and optoelectronic devices.This is because of several advantages as semiconductors, like a transition from indirect to direct band gap (in MoS 2 occurs ≈1.1 eV to ≈1.9 eV), [1][2][3] excellent mechanical properties, [4] different coupling conditions, which ultimately show a potential route for band gap engineering.Similarly, Zheng et al. proved by DFT calculations that charge transfer between layers depends on temperature and valence band maximum location on MoS 2 /WS 2 heterojunctions. [21]Recently, Bellus et al. investigated the band alignment in MoS 2 /ReS 2 heterojunctions using DFT calculations. [22]Their analysis reveals a type-I band alignment.Furthermore, the orbital projection indicates that both the valence band maximum (VBM) and the conduction band minimum (CBM) locate at the ReS 2 layer.This results in an incoherent charge transfer due to the lattice mismatch in the MoS 2 /ReS 2 heterostructure.Terrones et al. showed that interlayer alignment and stacking order influenced the properties of Van der Waals heterojunctions, predicting indirect to direct band gap transition even at multilayer arrangements. [23][26] For this, we argue that it is essential further investigation of the different coupling conditions and arrangements between MoS 2 and MoSe 2 , thus, being able to predict their electronic structure, resembling the conditions taking place experimentally.
This work aims to determine the change in the electronic structure with coupling conditions of Van der Waals heterojunction between MoS 2 and MoSe 2 under a dispersion corrected DFT (DFT-D2) scheme.With this, we can determine if the proposed arrangement is feasible to implement as a channel material in, e.g., a field-effect transistor or solar cells.We predicted their thermodynamic stability and the resulting band alignment, either a type-I, -II, or -III heterojunction according to the literature.We examined two alignment conditions and two lattice matches between MoS 2 and MoSe 2 .We found that after contact, the band gap of the interface reduced to ≈0.8 eV, which is comparably lower than MoS 2 and MoSe 2 intrinsic band gap.Thermoelectric calculations indicate that the heterojunction has an n-type nature, suitable for applications in electronics and photo-electronics.The numerical modeling by the Landauer approach reveals that the proposed heterojunction can achieve excellent behavior as channel material, providing valuable information to pave the research in new TMD Van der Waals devices and applications.

Geometrical Optimization of the Heterojunctions
As a comparison point, the resulting interlayer distance for MoS 2 (d i ) is 0.303 nm and for MoSe 2 is 0.313 nm, both in agreement with the reported values of 0.309 nm and 0.32 nm, [4,20] indicating an excellent description of the material by our choice of parameters; the complete parameters are listed in Table 1.MoS 2 and MoSe 2 optimized surfaces were used to generate two types of heterojunctions considering lattice match and alignment between MoS 2 and MoSe 2 .We considered the alignment to occur in a zigzag or a chalcogen-chalcogen order, labeled as AB or AA, respectively (Figure 1).Due to lattice mismatch occurring between MoS 2 and MoSe 2 , a tensile or compressive strain is naturally present at the heterojunction and is possible to quantify as a strain index being a 2 the lattice parameter of the material on top and a 1 the lattice parameter of the material at the bottom; x Table 1.Obtained lattice parameters (a and c), Van der Waals bond distance (d vdw ), binding energy (E b ), band gap (E g ), work function (W), dielectric constant (ε/ε 0 ), and conduction band alignment (ΔE c ) for all the heterojunction configurations, bulk-like, and single layer (SL) models.
Model is the ratio of layers between material at the bottom and the total number of layers composing the heterojunction.Hence, a compression strain value of −1.96% is present at the MoSe 2 /MoS 2 heterojunctions, while a tensile strain value of 2.04% is present at MoS 2 /MoSe 2 configurations.In the multi/single-layer circumstance, a compression strain value of −0.33% and a strain value of 0.34% are present for MoSe 2 /MoS 2 and MoS 2 / MoSe 2 models, respectively.In terms of bonding distance between materials, AA alignment leads to larger values of Van der Waals binding distance (d vdW ) compared to AB alignments (Table 1); except for the case of (MoS 2 /MoSe 2 ) mAA and (MoSe 2 /MoS 2 ) mAB with the zigzag arrangement, all binding distance values resulted above 0.3 nm (Figure S1, Supporting Information).
The binding energy (E b ) helped to determine the thermodynamic stability of the heterojunction model, and it was computed using the following expression, in units of eV atom −1 , being E ht , E MoS , E MoSe , and n the total energy of the heterostructure, the total energy of MoS 2 surface, the total energy of MoSe 2 surface, and the number of atoms in the structure, respectively.Six of our eight models resulted in endothermic values of E b , between 6.48 × 10 −3 eV atom −1 and 1.03 × 10 −2 eV atom −1 (see also Table 1) with linear trend between multilayer and multi/ single-layer arrangements (Figure S2, Supporting Information); these values compare to previous reports on similar heterojunction theoretical models. [27]Endothermic values of binding energy do not necessarily indicate unfavorable or unstable structures.The order of magnitude of this result implies the presence of weak Van der Waals contacts, possibly by electrostatic interactions.This also suggests that energy demand is required to achieve the desired structure interpreted as a degree of control for experimental fabrication of Van der Waals heterostructures; it is possible to supply such an amount of energy by annealing process or photo-induced annealing as reported elsewhere. [28,29]  with similar reports in the literature.[24,25,30] We attribute the reduction of the band gap to d-orbital contributions which are displaced to energy levels just inside the band gap (Figure 1) because of lattice constraint.According to our results, the highest band gap value was achieved by (MoS 2 /MoSe 2 ) mAB heterojunction configuration, being E g = 0.805 eV (indirect band gap at Γ-Κ). On the othr hand, (MoSe 2 /MoS 2 ) AA and (MoSe 2 /MoS 2 ) mAA models do not present any band gap, while (MoSe 2 /MoS 2 ) AB and (MoSe 2 /MoS 2 ) mAB cases yield a reduced band gap of 0.022 eV and 0.085 eV, respectively (Figure S3, Supporting Information).We observe that in most MoSe 2 /MoS 2 heterojunction configurations, the CBM becomes noticeable at the K point after the interface formation.This phenomenon causes an intercrossing with the Fermi level and, hence, a semi-metallic character, as reported in bent heterostructures.[11,31,32] In contrast, MoS 2 / MoSe 2 configuration has lesser metallic d-orbital contribution around the conduction band, translated as a lower disturbance of molybdenum d orbitals by interlayer interactions as previ-ously suggested.[11,33] Also, chalcogen orbitals' distribution, specifically 3p orbitals from sulfur and 4p from selenium, primarily contributing to the valence band, increase more pronounced at the MoSe 2 /MoS 2 heterojunction, getting dispersed to higher energy values and causing band gap reduction, supporting our premise on lattice constraint and band gap reduction.

Band Offset and Band Alignment Calculation
In this work, we considered the position of the macroscopic average electrostatic potential of both materials, MoS 2 and MoSe 2 , throughout the interface and used it as a reference point to locate the bulk VBM.With this, we estimated the band alignment in the MoS 2 /MoSe 2 heterojunction as done in other reports. [34,35]This macroscopic potential generates a potential band offset determined by the difference in the electrostatic potential between the two regions (in the xy plane) of the heterojunction denoted as ΔV (Figure 2).Electrostatic www.advmatinterfaces.depotential plots for the MoSe 2 /MoS 2 heterojunction configuration are depicted in Figure S4 (Supporting Information).Then, the valence band offset is computed as where E V top and E V bottom are the location of the valence band maximum of the material atop and at the bottom in each of our proposed heterojunction models concerning the average electrostatic potential.Finally, the conduction band offset is found as where E g top and E g bottom are the band gap values of the material placed on top and at the bottom in the heterostructure models.Computation gives ∆E c = −0.217eV for MoS 2 /MoSe 2 heterojunctions, while ∆E c = 0.217 eV resulted for MoSe 2 /MoS 2 (Table 1) corresponding to a type-I heterostructure and represented in Figure 3.Our results on ΔE c are comparable to those reported in polar and nonpolar heterostructures using ZnO, GaN, and MgGeN 2 ; [34] Quan et al. estimated experimentally a ∆E c of 0.46 eV in a MoTe 2 /MoS 2 nanocomposite film, [36] comparable to our estimation.Thus, this indicates the potential use of MoS 2 /MoSe 2 heterojunction in electronic devices.In this case, charge carriers (electrons or holes) get confined at the material with the lower band gap due to this band alignment.The effect of charge confinement has been reported in Si/MoS 2 heterostructures, [37] helping to increase charge injection yield.This result could be used as theoretical insights in designing metal-oxide field-effect transistors (MOSFET) and improved solar cells devices using TMD heterojunctions as channel materials.
In MoSe 2 /MoS 2 heterojunction configurations, theoretically, an inverted version of the band alignment described in Figure 3 should describe the behavior of the bands after contact, but as presented by band structure calculations, a semimetallic nature indicates a much more complicated band alignment, possibly with induced gap states and dipole formation at the interface.

Thermoelectric Properties
For further analysis, the Seebeck coefficient (S) allows us to predict the behavior of charge carriers along energy levels near the Fermi level.From our calculations of the Seebeck coefficient, in all heterojunctions the crossover from hole-controlled to electron-controlled conduction, i.e., going from positive to negative values, takes place before μ − E F = 0 (Figure 4), indicating that electron-controlled conduction dominates. [38]Seebeck coefficient is symmetric around the crossover point in MoS 2 /MoSe 2 heterojunction configurations; however, the MoSe 2 /MoS 2 configurations are slightly asymmetrical just around the crossover point.

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which can be associated with the low band gap and semimetallic behavior observed in the band structure plots.
Calculations on conductivity per relaxation time (σ/τ) show a pronounced n-type character for MoS 2 /MoSe 2 heterojunctions (Figure 4).In contrast, MoSe 2 /MoS 2 confirms our previous suggestions where MoSe 2 /MoS 2 heterojunctions have a semimetallic behavior due to metallic d-orbital reallocation.Thermoelectric properties for heterojunctions in the AA configuration have the same trend as the AB configuration, indicating little dependence on this alignment condition (Figure 4 and Figure S5, Supporting Information).In general, MoS 2 /MoSe 2 heterojunctions in the multi/single configuration have improved transport properties than the other configurations considered in this work, with higher Seebeck coefficient and values of σ/τ around the band gap in the order of 10 20 .This behavior follows previous work suggesting that multilayer configurations of MoS 2 have enhanced light adsorption. [42]In terms of the power factor (S 2 σ/τ), MoS 2 /MoSe 2 heterojunctions are superior to MoSe 2 / MoS 2 ones due to higher values of the Seebeck coefficient.

Heterojunction Model in a One-Gate Configuration
As a further examination, we model our heterojunction as a one-gate device resembling a transistor configuration.For this, we consider the Landauer approach, the ballistic transport regime, and non-degenerate statistics.We computed the current versus drain-to-source voltage (I-V) characteristic using the following expression which is a function of the surface charge (Q n ) that varies with respect to the drain to source voltage (V ds ) and gate voltage (V gs ).v t represents the charge thermal velocity in the ballistic regime (and non-degenerate statistics) and w is the width of the device.A more detailed description of this approach is provided in the Supporting Information.We aimed to test the material's capabilities for electronic applications, benefiting from the resulting type-I alignment.The only heterojunctions considered are (MoS 2 /MoSe 2 ) AB and (MoS 2 /MoSe 2 ) mAB based on our previous results on band structure and thermoelectric calculations.First, the surface charge needed to be computed, denoted by Q n , for each of the considered heterojunction models.This was by using a semi-empirical expression developed by Wright et al., [43] which states that where m is the body effect parameter (≈1 for thin oxide layers), C g is the gate capacitance, V t is the threshold voltage, and k b is the Boltzmann constant.To evaluate C g we considered HfO 2 as dielectric material-high dielectric constant of 25-with a thickness (t ox ) of 10 nm (see Supporting Information).Similarly, we employed the resulting dielectric constant of 13.6 for the MoS 2 /MoSe 2 multilayer heterojunction configuration and a dielectric constant of 11.8 for the MoS 2 /MoSe 2 multi/singlelayer heterojunction (Table 1).We extracted these values at the limit where the dielectric function of the heterojunctions approaches zero (ε(E) → 0).The semiconductor thickness (t s ) was set to 7.2 nm which is approximately the thickness of our heterojunction models.Second, we input the following parameters and function Q n into Equation ( 4 [20,44] being 0.46m 0 ; a width (w) of the one-gate device of 400 nm, and a channel length of 10 nm. [45]Finally, the drain-to-source voltage was set to −100 mV, while the threshold voltage was −10 V, −5 V, and −1 V. We first validate our model by comparing theoretical I-V characteristics utilizing the parameters of devices as reported in the literature.We found an excellent agreement of our I-V approach with experimentally obtained I-V curves (see Figure S5, Supporting Information), and after that, we proceeded with our parameters as mentioned above.
From the resulting I-V relations of our modelled one-gate device (Figure 5), we estimated a subthreshold swing (SS) of 60 mV dec −1 , field-effect mobility (μ f ) of 160 cm 2 V −1 s −1 , and an on/off ratio of the order of 10 5 using a threshold voltage of −1 V at −100 mV drain-to-source voltage and a MoS 2 /MoSe 2 heterojunctions in multilayer configuration (Figure 5a).MoS 2 /MoSe 2 heterojunctions in multilayer configuration output higher current compared to multi/single-layer ones (Figure 5b), which agrees with the calculations of σ/τ presented earlier.The order of magnitude of the output current agrees with experimental data available (see Supporting Information), confirming that our methodology can potentially assist in designing electronic and opto-electronics devices integrating TMD.

Conclusions
Our results from DFT allowed us to determine the electronic structure and transport properties of Van der Waals multilayer and multi/single-layer of MoS 2 /MoSe 2 and MoSe 2 /MoS 2 heterostructures with a zigzag (AB) and chalcogen-chalcogen (AA) interlayer alignment.After the formation of the interface, the resulting band gap is around 0.8 eV with a band offset of about 0.2 eV.The MoS 2 /MoSe 2 heterojunction preserves their semiconducting characteristics.However, MoSe 2 /MoS 2 configurations have semi-metallic behavior.This shift is strongly related to the interlayer coupling due to induced strain and subsequent reallocation of metallic d-orbital and chalcogen orbitals in energy levels inside the band gap.The resulting band alignment of MoS 2 /MoSe 2 (in AA and AB alignment configurations) indicates a type-I heterojunction, especially useful in designing optoelectronic and electronic devices.Moreover, the heterojunction presents an excellent Seebeck coefficient of ≈1000 µV K −1 and conductivity values (σ/τ) around the band gap in the order of 10 20 .Numerical modeling of our material in a one-gate transistor indicates excellent behavior as channel material with subthreshold swing values of 60 mV dec −1 and field-effect mobilities of about 160 cm −2 V −1 s −1 .As presented here, these heterojunctions have promising applications in electronic and optoelectronics applications, owing to their remarkable electronic and thermoelectric properties.
Molecular Models of MoS 2 /MoSe 2 Heterojunction: The optimized MoS 2 and MoSe 2 unit cells (space group P63/mmc) had lattice parameters of a = b = 0.319 nm and c = 1.27 nm and a = b = 0.332 nm and c = 1.29 nm, respectively.Using these models, the MoS 2 and MoSe 2 surfaces were constructed in the 〈002〉 direction, setting a thickness of ≈2.0 nm for each.This surface model consists of six Mo atoms and twelve chalcogen atoms (either S or Se), and each of these surfaces resembles a bulklike material.The heterojunction models were assembled as mentioned previously in the text.A vacuum space of ≈1.5 nm in the z direction separates periodic images, avoiding undesired contacts and interactions.These surface and heterojunction models were subjected to geometry optimization using the abovementioned parameters.

Figure 1 .
Figure 1.a-d) Resulted band structures and density of states for the multilayer heterojunctions in the AB and AA configuration, e) (MoS 2 /MoSe 2 ) AB , and f) (MoS 2 /MoSe 2 ) AA heterojunction models where cyan balls represent molybdenum ions, yellow balls represent sulfur ions and orange selenium ones; the cell is extended in the z-x plane.Dashed lines in (e) and (f) represent the boundaries of the cell; orange arrows indicate the alignment between chalcogen ions.g) The Brillouin Zone and the Γ-M-Κ-Γ path used for the band structure calculation and the model extended in the y-x plane.

Figure 2 .
Figure 2. Averaged electrostatic potentials plots (in the xy plane) of MoS 2 /MoSe 2 heterojunction configurations; green line indicates the point at which the interface is located, red line indicates the position of vacuum level.Dashed-dotted lines represent the average electrostatic potential of both materials and ΔV is the difference in the electrostatic potential energy.

Figure 3 .
Figure 3. Schematic of band alignment between MoS 2 /MoSe 2 heterojunction scenarios, where is possible to observe the ΔE c and ΔE v , which are the energy difference of conduction and valence bands, respectively.

Figure 4 .
Figure 4. Seebeck coefficient (S), conductivity (σ/τ) and power factor (S 2 σ/τ) calculation for AB heterojunction configuration at 300 K as function of μ − E 0 .Seebeck coefficient values of MoS 2 /MoSe 2 indicate a semiconductor behavior around the Fermi level.Power factor (S 2 σ/τ) values are higher at the MoS 2 /MoSe 2 heterojunctions because of their higher values of Seebeck coefficient.
) to compute the I-V characteristics of the modeled one-gate device: a thermal velocity v an electron effective mass (m*) of MoS 2 /MoSe 2 approximately equal to the effective mass of MoS 2 ,

Figure 5 .
Figure 5. a) Drain-to-source current (I ds ) as function of applied gate voltage (V g ) at a constant drain-to-source voltage (V ds ) of −100 mV and setting three threshold voltages (V T ): −10, −5, and −1 V. Dashed lines correspond to the calculations using the MoS 2 /MoSe 2 multilayer heterojunctions configuration with an estimated dielectric constant of 13.6; solid lines correspond to the MoS 2 /MoSe 2 multi/single-layer heterojunctions with a dielectric constant of 11.8.b) I ds as a function of V ds with V g = 1 V at different values of V T.
The lattice match between MoS 2 and MoSe 2 is as follows: first, we have MoS 2 used as a substrate to support MoSe 2 , the latter constrained to the MoS 2 lattice parameters.With this, we had a multilayer MoSe 2 over MoS 2 with the two AB and AA alignments, having (MoS 2 / MoSe 2 ) AB and (MoS 2 /MoSe 2 ) AA , respectively.For a multi/singlelayer situation, we set a single layer of MoSe 2 over multilayer MoS 2 , also in AB and AA, having (MoS 2 /MoSe 2 ) mAB and (MoS 2 / MoSe 2 ) mAA , respectively.Second, we placed multilayer MoS 2 on top of a multilayer MoSe 2 , where the former is constrained to the MoSe 2 lattice parameters.Including the multi/single-layer configurations, labels are (MoSe 2 /MoS 2 ) AB , (MoSe 2 /MoS 2 ) AA , (MoSe 2 /MoS 2 ) mAB , and (MoSe 2 /MoS 2 ) mAA , respectively, having in total eight possibilities of MoS 2 /MoSe 2 assembly.

2. Electronic Structure of MoS 2 /MoSe 2 Heterojunctions All
MoS 2 /MoSe 2 models, i.e., MoSe 2 is constrained to MoS 2 lattice, resulted in higher band gap values than MoSe 2 /MoS 2 models; similarly, models with AB alignment resulted also with higher band gap values than models with AA alignment (Table1).MoSe 2 /MoS 2 heterojunctions resulted in close-to-zero band gap values.This fact cannot be attributed to the exchangecorrelation functional because our computations for bulk-like and single layer MoS 2 and MoSe 2 are in complete agreement