InN for Photovoltaic - ACS Publications - American Chemical Society

Article pubs.acs.org/JPCC

Elucidating the Optical Properties of Novel Heterolayered Materials Based on MoTe2−InN for Photovoltaic Applications Cesar E. P. Villegas and A. R. Rocha* Instituto de Física Teórica, Universidade Estadual Paulista (UNESP), Rua Dr. Bento T. Ferraz, 271, São Paulo SP 01140-070, Brazil S Supporting Information *

ABSTRACT: Efficient excitonic solar cells preferably require materials with an optical gap in the near-infrared region and high absorption coefficients. Additionally, it is well-known that heterostructures open the possibility of tailoring device properties by taking advantage of the characteristics of individual materials, so that new practical applications can arise. Regarding these ingredients, we propose that the recent synthesized monolayer MoTe2 and the InN compound seem to favorably fit into this category. We carry out ab initio density functional theory calculations to study the electronic and optical properties of heterostructures based on MoTe2 and InN monolayers. Our results indicate that one of the most stable heterostructures presents type-II band alignments and photoexcited states in the energy range of 1.1−1.3 eV, where power conversion efficiency reaches its maximum. We also propose a prototypical device based on these materials and study their potential as excitonic solar cells. In doing so, we show that heterostructures based on MoTe2−InN are able to combine the nearinfrared absorption of MoTe2 together with the low refractive index and high absorbance of InN to give rise to improved optical properties such as the formation of photoexcited states with lower binding energies, when compared with the individual monolayers, long exciton lifetimes in the nanosecond scale, as well as high power density ratios. These overall results point toward the potential of MoTe2−InN heterostructures for photovoltaic applications.

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Eg, ranges from 1.6 to 2.0 eV;22 thus, a solar cell based on them could not achieve the optimal efficiency which has been predicted to lie in the Eg ≈ 1.1 eV region.23 Recently, Ruppert et al.24 have paved the route toward the synthesis and further development of new devices based on MoTe2. They successfully synthesized the monolayer form of this TMDC, and estimated, by photoluminescence, its optical gap, which is approximately 1.1 eV. As theoretically predicted, this is the material with the narrowest band gap among the dichalcogenides.16 At the same time, there has been a proposal for a transistor based on bilayer MoTe2 that shows similar electronic properties compared to its MoS2 counterpart25 as well as studies that target heterostructures based on the vertical stacking of MoS2−MoTe2 to pursue possible technological applications.26,27 Consequently, few-layer and, ultimately, monolayer MoTe2 seems a promising candidate for a wide range of devices including photovoltaic ones. In the same token, group III−V semiconductor thin films have been successfully used in the design of highly efficient multijunction solar cells due to their direct band gap, which spans the infrared and visible range.28,29 In particular, among the (Al,Ga,In)N semiconductor compounds, InN has the narrowest band gap lying in the infrared region and the highest

INTRODUCTION The successful realization of graphene1 has led to the synthesis of a number of atomically thin two-dimensional (2D) materials2−4 and to the theoretical prediction of several compounds based on group III−V5 and indium chalcogenides6 to name but a few.7,8 Most recently, transition-metal dichalcogenides (TMDC) are being intensively studied for their potential in the design of transistors,9−12 lithium batteries,13 memories,14 and photovoltaic cells.15 In the latter case, this is a direct consequence of these materials’ optical band gap lying in the visible spectrum and their strong lightmatter coupling.16 Together, these features make them ideal candidates for atomically thin optoelectronic applications. In designing excitonic solar cells, besides the presence of a band gap preferably coinciding with the near-infrared spectrum, there are a number of key ingredients which are important for efficiently harvesting solar energy. In particular, the generation of excitons and their subsequent efficient dissociation into free charges due to type-II band alignments are desirable features.17−19 In fact, following these requirements, novel heterostructures based on the vertical stacking of different hexagonal 2D crystals have been recently proposed, pointing toward a new route for designing highly efficient photovoltaic devices.15,20,21 In the TMDC family MoS2, MoSe2, and WS2 monolayers have been widely studied since their realization in the laboratory. Experimental results show that their optical gap, © 2015 American Chemical Society

Received: December 9, 2014 Revised: May 1, 2015 Published: May 4, 2015 11886

DOI: 10.1021/jp5122596 J. Phys. Chem. C 2015, 119, 11886−11895

Article

The Journal of Physical Chemistry C

Figure 1. Top and side view representation of the fully relaxed MoTe2/InN heterostructures labeled as (a) AA, (b) C27, and (c) T. The different stackings are obtained as a result of different rotations and translations over the heterostructure AA.

carrier mobility, drift velocity, and absorption coefficient,30,31 which make it an attractive material for optoelectronic applications. 32 In fact, there have been proposals of incorporating nanoparticles and few layers of wurtzite InN to enhance the efficiency of dye-sensitized solar cells.33,34 However, due to the difficulties of synthesizing it in bulk form,30 which is strongly related to the substrate lattice mismatch, it has been the least studied among the nitrides. To date, the development of new experimental techniques regarding control in growth and fabrication of nanostructures has allowed for synthesized hexagonal wurtzite InN thin films35−37 and down to its bilayer form.38 Hence, by combining such achievements with exfoliation techniques, one can foresee a favorable scenario toward realization of single layer h-InN, which have been theoretically predicted5 to be as stable as other already synthesized few-layer III−V semiconductors such as hBN2 and, more recently, h-AlN.39 Regarding a practical application in photovoltaics, one can notice that the aforementioned crystals individually possess some of the key ingredients for designing efficient solar cells. Therefore, one can envision that by combining them into an heterostructure, new interesting properties that indeed are worthy of exploration could arise. In that sense, theoretical predictions of the optical properties of these heterostructures might provide fruitful insights toward the design of a new generation of atomically thin solar cells. Thus, here we propose a heterostructure based on monolayers of MoTe2 and InN. In addition to the properties mentioned above, our interest in these materials come from the small lattice mismatch between few-layer MoTe2 and h-InN, which in principle, might benefit the growth of heterostructures in laboratory. In this work, we carry out ab initio density functional theory (DFT) calculations combined with many-body perturbation corrections within the G0W0 approximation. The subsequent solution of the Bethe-Salpeter equation (BSE) is used to describe the optical properties including excitonic effects. Our results indicate that the energy alignment on the hetero-

structures based on MoTe2 and InN are, for some cases, of type-II. In addition, photoexcited excitons are generated in the energy range of 1.1−1.3 eV, for which the power conversion efficiency reaches its maximum value.23 We also estimate the power density as being up to 1 order of magnitude higher than those for multijunction solar cells. Finally, we show that by constructing heterostructures based on MoTe2−InN one is able to combine the near-infrared absorption of MoTe2 with the low refractive index and high absorbance of h-InN to produce improved optical properties such as excitonic states with lower binding energies (with respect to the monolayers), and large exciton lifetimes reaching the nanosecond scale. Ultimately, our results show that MoTe2−InN heterostructures possess the potential for photovoltaic applications.

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THEORY AND METHODOLOGY This work initially considers pristine monolayers of MoTe2 and InN. The first consists of a nonplanar honeycomb lattice where one Mo atom is bonded to two superimposed Te atoms (see Figure 1), an arrangement which is similar to most dichalcogenides.40 In contrast, the pristine InN monolayer forms a planar honeycomb structure where one In atom is bonded to three N atoms resembling h-BN. In order to predict the dynamical stability of h-InN monolayer, we carried out a detailed study of the phonon dispersion (see the Supporting Information). Our results show real frequencies along the highsymmetry points of the Brillouin zone. This is in agreement with a previous theoretical study that also predicted the stability of monolayer h-InN.5 Next, we constructed the MoTe2/InN heterostructures for which three different stackings have been studied (shown in Figure 1a−c). The structures are labeled following the notation used in previous works on TMDCs,41 and resulted from a series of translations and rotations of the pristine InN monolayer with respect to the MoTe2 one. We optimized six different geometric arrangements where the In and N atoms were located at hollow, top, and bridge positions with respect to the 11887


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Table 1. Structural Parameters, Binding Energies and Electronic Band Gaps of Monolayers MoTe2 and InN and stacking AA, C27, and T Calculated for Different Theory Levels and Functionalsa structural parameters (Å) a MoTe2 InN AA T C27

3.52 3.53 3.52 3.52 3.52

dLDA

3.37 3.76 3.20

binding energy (eV) dPBE‑D

LDA

3.50 3.50 3.19

12.0b 10.9c 0.18 0.09 0.18

energy gap (eV)

PBE-D

LDA

HSE

G0W0

BSE

0.37 0.27 0.38

1.12 0.68 0.87 0.8 1.12

1.76 1.97 1.58 1.42 1.83

1.89 2.12 1.76 1.48 2.08

1.14 1.30 1.17 1.12 0.90

a

The single and quasiparticle band gaps for MoTe2, InN, and C27 structures are all direct. bCohesive energy of MoTe2 monolayer extracted from ref 48. cCohesive energy of InN monolayer extracted from ref 5.

Figure 2. Band structures of (a) MoTe2 and (b) InN monolayers. The corresponding density of states are presented in (c) and (d−f), respectively. (e) Variation of the band gap as a function of tensile strain. The HSE hybrid functional was employed for these calculations.

Simulation Package (VASP)47 code, with a mesh of 8 × 8 × 1 and an energy cut off of 60 Ry, was used on top of the fully relaxed structures. The interaction between monolayers of MoTe2 and InN was studied by calculating the binding energy per atom, n, of the system

MoTe2 monolayer. However, the other three possible configurations presented, after relaxation, similar electronic and structural parameters. The geometric and electronic structures of all configurations are shown in Figure S2 (Supporting Information). In all cases, plane-wave (PW) DFT is employed to obtain the electronic ground state for different exchange-correlation functionals, including local density approximation (LDA)42 and the Perdew−Burke−Ernzerhof (PBE)43 generalized gradient currently implemented in the Quantum-Espresso package.44 van der Waals corrections within the semiempirical dispersion scheme (PBE-D) proposed by Grimme45 were used on top of PBE for the geometry optimization of the bilayer heterostructures. Norm-conserving pseudopotentials with 4s4p and 4d semicore states were adopted to describe electronic states of molybdenum and indium, respectively. The calculations were performed with a k-sampling grid within the Monkhorst−Pack scheme of 12 × 12 × 1 and kinetic energy cutoff of 110 Ry. The structures were fully optimized to their equilibrium position with forces smaller than 0.01 eV/Å for a supercell (5 atoms per unit cell) with a vacuum region of 18 Å in the transverse direction. Finally, we should mention that the Heyd−Scuseria− Ernzerhof (HSE)46 hybrid functional was also employed to describe the electronic structure of the monolayers and heterostructures. For this purpose, the Vienna Ab Initio

E b = [E MoTe2 /InN − (E MoTe2 + E InN)]/n

(1)

where EMoTe2/InN, EMoTe2, and EInN are the total energies of the heterostructure and the individual monolayer, respectively, and n corresponds to the total number of sites in the unit cell. In order to obtain the optical properties, the calculations were carried out in three steps. First, we used the PW−DFT results from our relaxed structures. Next, within the G0W0 approximation, the quasiparticle energies were obtained considering the corresponding Khon−Sham eigenstates and eigenvalues as a starting point EnQP = EnKS + ⟨ΨnKS|Σ(EnQP) − VXC|ΨnKS⟩

(2)

Here, VXC is the exchange correlation potential at the DFT level and Σ(EQP n ) is the self-energy operator. The screened Coulomb potential W0 is calculated within the Plasmon−Pole approach with an energy cutoff of 24 Ry and including 2600 unoccupied bands. In addition, we used a truncated screened Coulomb interaction to avoid image effects between periodic cells. Finally, the electron−hole interactions are included by solving the Bethe−Salpeter equation49 11888


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Figure 3. k-resolved spectral function along the path Γ-M-K-Γ for (a) AA, (b) T, and (c) C27 heterostructures. Bright (dark) color indicates the regions where exits high (low) density of electronic states. (d) Schematic representation of the band alignments in heterostructures AA, C27, and T. The relative energy positions for AA, T, and C27 were taken from the quasiparticle band structures. The white arrows indicate the lowest optical transitions taking place in the systems. QP S S S ∑ ⟨vc k|K eh|v′c′k′⟩ + (EcQP k − Ev k )A vc k = Ω A vc k v′c′k′

states (PDOS) of MoTe2 (shown in Figure 2c) indicates the strong contribution of the Mo atom to both the valence band maximum (VBM) and conduction band minimum (CBM), whereas the Te atom gives a small contribution to the CBM. This is similar to other results showing that 4d and 5p orbitals of Mo and Te, respectively, give the largest contribution to the states around the Fermi level.40 For the case of h-InN, parts d and f of Figure 2 show that the VBM is mostly nitrogen-2pdominated, whereas the CBM is composed essentially of In 4s and N 2s orbitals. We proceed by analyzing the tensile strain effects on the isolated monolayers. To this purpose, the optimized lattice parameter is varied within a range of −2% to +2%. For MoTe2, two different effects can be observed from Figure 2e: (i) an increase of the direct band gap ΔK as the lattice constant decreases and (ii) the opposite effect for the indirect band gap ΔKΣ. This can be understood as follows: at K and Σ valleys at the bottom-most conduction band, dz2 and dx2−y2 orbitals, respectively, have the largest contribution. Thus, by compressing the monolayer, the distance between Mo−Te atoms reduces. Consequently, at the K point, the coupling between out-of-plane dz2 and p orbitals becomes stronger, causing the increase of the direct band gap. In contrast, at the Σ valley, the overlap between in-plane dx2−y2 and p orbitals becomes weaker, yielding the opposite trend in the ΔKΣ band gap. These effects are in agreement with previous studies concerning strain effects in TMDCs.53 For InN, the reduction of the lattice parameter leads to a bigger overlap between N 2p and In 4s orbitals around the Fermi level, which results in the increase of the gap. In contrast to the MoTe2 case, the predominant contribution of N 2p orbitals at Γ (K) guarantees that both direct (ΔΓ) and indirect (ΔKΓ) band gaps increase (decrease) simultaneously as the strain is varied. On the one hand, these results indicate that

(3)

where ASvck, ΩS are the exciton eigenfunction, and eigenvalues for the Sth exciton respectively, and Keh is the electron−hole interaction kernel. Within the Tamm−Dancoff approximation,50 five valence bands and five conduction bands are included to solve the BSE. Once the excitonic eigenvalues and eigenfunctions are obtained, one can calculate the imaginary part of the dielectric function ϵ2(ω) =

16π 2e 2 ω2

∑ |e⟨0|v|S⟩|2 δ(ω − ΩS) S

(4)

where v corresponds to the velocity operator and e is the direction of polarization of the incident light; chosen here to be on the plane of the monolayers. A finer k-grid sample of 60 × 60 × 1 was used during the BSE procedure to better describe the optical transitions. The G0W0 and BSE calculations were performed using the BerkeleyGW package.51

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RESULTS Electronic Structure. First, we discuss the electronic structure of the individual MoTe2 and InN monolayers. The relevant structural and electronic parameters are presented in Table 1. The fully relaxed lattice parameters for MoTe2 and hInN are 3.52 and 3.53 Å, respectively. These values are in good agreement with the experimental ones for the few-layer form of the crystals,36,52 indicating a mismatch of 0.3%. The band structures shown in Figure 2a,b calculated using HSE reveal that both MoTe2 and InN possess a direct band gap at the K (ΔK = 1.76 eV) and Γ (ΔΓ = 1.97 eV) symmetry points. We note that the overall effect of the hybrid functional was to rigidly shift the PBE-calculated bands. The projected density of 11889


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Figure 4. (a) Top to bottom, absorption spectra of MoTe2, InN, AA, T and C27 systems. The black-dashed lines represent the sum of the individual monolayer spectra. (b) Quasiparticle band structure of MoTe2 (left) and InN (right) monolayers, indicating the first optical transitions. (c) Schematic representation of the possible mechanisms of exciton generation (I), charge separation (II), and recombination (III) in heterostructures AA and T. The mechanisms involved in the exciton dissociation are very similar to the one occurring in organic solar cells.19 Artificial Gaussian broadening of 0.03 eV was used in all cases.

Although there is a high density of nitrogen and Mo states at K, the probability of such states becoming photoactivated and giving rise to direct optical transitions is negligible since they correspond to different layers. We will later see, from the analysis of the absorption spectrum, that indeed these kinds of transitions are forbidden. Furthermore, the T stacking also possesses an indirect band gap of 1.42 eV whose VBM and CBM are located at the K and Γ points, respectively. For this configuration, the Mo atom mostly contributes with the VBM, whereas the In atom has the largest weight on the CBM. In fact, this configuration also provides a type-II heterojunction in which the MoTe 2 monolayer acts as a donor and the InN monolayer plays the role of an acceptor (see Figure 3c). In contrast to the two previously studied cases, the C27 arrangement presents a direct gap of 1.83 eV at K, where both the VBM and CBM states are mainly governed by Mo atoms. Thus, this stacking forms a type-I heterojunction which does not provide the required band alignment for efficient exciton dissociation. Although the band alignment of this heterostructure is not ideal for excitonic solar cell applications, the fact that it possesses a direct band gap and its slightly lower energy with respect to the AA configuration might be interesting for further electronic applications. Finally, our calculations using the G0W0 approach result in a similar electronic structure for all systems. In fact, as can be seen in Figure S3 (Supporting Information), the gap using both hybrids as well as GW essentially give rise to a rigid shift compared to the GGA bands. In addition, when comparing the energy gaps shown in Table 1, we note that the quasiparticle corrections for all cases are slightly larger than those for the hybrid functional corrections.

relatively modest strain values can lead, in both cases, to direct and indirect gap transitions as the strain goes from negative to positive. On the other hand, it leads us to conclude that the strain induced by placing one monolayer on top of the other will not dramatically alter the order of the gaps. Concerning the energetics of the heterostructures, Table 1 summarizes the binding energy and the interlayer distance, d, for the different geometries. Both the LDA and PBE-D functionals show that configuration T is the least energetically favorable, whereas the C27 and AA are the most stable structures. Note that by organizing the different stackings one is able to tune the band gap in a wide range of values including higher, lower, and in between for the gaps of the isolated structures. In Figure 3, we present the k-resolved spectral function projected onto the states of the constituent atom of each monolayer for the different stackings. The electronic structures of the three configurations are essentially composed of the superposition of the electronic states of the individual monolayers with slight changes on the VBM and CBM states. Their origin stems from the vdW interaction between the monolayers and the rippling induced on InN after forming the heterostructure (see Table S1, Supporting Information). The band structure of heterostructure AA presented in Figure 3a shows an indirect band gap of 1.58 eV with the VBM and CBM located at Γ and K, respectively. In addition, the states located immediately below and above the Fermi energy are mainly governed by N and Mo atoms. Because of this clear separation, from the band structure one can construct the relative band alignment for each heterostructure, which is shown in Figure 3d. In doing so, it is possible to see that the AA stacking produces a type-II heterojunction. 11890


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The Journal of Physical Chemistry C Optical Properties. Before starting with the study of the optical properties, it is worth mentioning that the supercell method to describe the screening of 2D systems leads to an inverse dependence of the dielectric constant with the vacuum region.54 Therefore, a vacuum-independent physical observable to quantify the optical spectrum can be obtained by computing the imaginary part of the polarizability per unit area, α2(ω) = Lz[ϵ(ω) − 1]/4π, where Lz and ϵ(ω) are the distance between periodic images and dielectric constant, respectively. Moreover, the thickness-independent absorbance is related to the polarizability per unit area through the equation A(ω) = 4πωImα2(ω)/c.55 Hereafter, we will use the mentioned optical properties to describe our systems and avoid the supercell vacuum dependence of the dielectric constant. As previously mentioned the absorption spectrum was obtained by solving the Bethe−Salpeter equation to include the electron−hole interactions. Figure 4a shows the absorption for MoTe 2 and InN monolayers as well as for the heterojunctions. We observed that the photoexcited states for the heterostructures are dominated by direct transitions around K and Γ points. Similar trends have been predicted previously in heterostructures with indirect gap due to small values for the oscillator strength related to those transitions.26,56 The optical spectrum of the MoTe2 monolayer presents three intense peaks, which are extended around an energy range going from 1.0 to 1.65 eV. The lowest bright exciton E11 0 = 1.14 eV presents a large binding energy of 0.75 eV and arises due to direct transitions involving the VBM and CBM at K. The 11 second and third peaks, E11 1 = 1.39 eV and E2 = 1.65 eV, correspond to the first and second excited-state of exciton E11 0 , respectively. These two excited states become photoactivated at wavevectors values along KΓ. The exact positions of these transitions in the Brillouin zone are schematically represented in Figure 4b. We note that the nature of E11 0 and its excited state qualitatively resembles the excitonic peaks predicted for MoS2 monolayers.57 Moreover, the position of the first excitonic peak is in excellent agreement with recent photoluminescence experiments.24 The absorption of the h-InN monolayer presents two intense peaks in the energy window shown. The first peak E11 0 = 1.3 eV results from direct transitions at Γ involving the VBM and CBM states and yields exciton binding energy of 0.82 eV. The second peak, E11 1 = 1.9 eV, arises from direct transitions around Γ point and corresponds to the first excited-state of exciton E11 0 . It should be noted that at frequency ℏω = 4.9 eV (see the Supporting Information) there is also another intense peak that corresponds to the second excited state of E11 0 . These optical transitions are shown in Figure 4b. Concerning the stacked monolayers, for the AA case, the first peak E21 0 = 1.17 eV results from direct transitions at K point and involves the second VB and CBM states. The exciton binding energy corresponding to this structure is ∼0.93 eV. The second peak E11 0 = 1.27 eV is constituted by transitions taking place around Γ and involve only electronic orbitals from the InN monolayer. On the other hand, the third and fourth peaks comprised transitions taking place along the KΓ path and correspond to the first and second excited-state of exciton E21 0 . For T stacking, the first E11 0 = 1.12 eV peak originates from direct transitions from the topmost VB to the CBM taking place at Γ. The second peak E11 1 = 1.22 eV occurs at the K point, and its photoactivation is purely related to the MoTe2 monolayer. Notice that the binding energy for the first excitonic state is 0.6 eV, much smaller than those related to the isolated

monolayers. Similarly, the third excitonic peak arises at almost the same wave-vector value as the third bright exciton of AA stacking; however, in this case, the transition involves the topmost VB and bottom-most CB states. In addition, the C27 heterostructure possesses similar features compared to T stacking, especially regarding the bands involved in the relevant optical transition and their wavevector position in the Brillouin zone. Furthermore, the first excitonic state arises at an energy E11 0 = 0.9 eV, which yields higher binding energies of ∼1.1 eV when compared with other heterojunctions. We stress that due to the direct band gap of C27 stacking the first excitonic peak arises slightly red-shifted with respect the other heterostructures. This is because configurations AA and T are indirect band gap materials whose optical transitions possess small oscillator strength. This prevents them from presenting peaks at energies lower than 1.1 eV. We observe that the optical spectra of heterostructures AA, T, and C27 do not reproduce the curve resulting from the sum of the optical spectrum of MoTe2 and InN monolayers (presented as black-dashed lines in Figure 4a). The main features of the summed curve are present in the three stackings, but the positions of the peaks and intensities are slightly shifted. These changes in the optical absorption could arise during the formation of the heterostructures as a result of orbital hybridization between monolayers due to van der Waals interactions. The thickness independent absorbance for 2D materials20 is given by A(ω) =

ωϵ2(ω)Lz c

(5)

In Figure 5, we present the curves for the intensity of the first and second peaks. Clearly, InN is twice the value found for

Figure 5. Absorbance for MoTe2, InN, AA, and T systems. Inset shows the refractive index as a function of the photon energy calculated by eq 6.

MoTe2. Furthermore, the first absorbance peak of the heterostructures displays an enhancement of 15% with respect to the MoTe2 peak. Although we do not observe an enhancement in the intensities of the heterojunctions with respect to the InN monolayer, the benefit of forming the heterostructures is to induce a red-shift in the peak position of pristine InN. The inset of Figure 5 shows the refractive index 11891


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The Journal of Physical Chemistry C n(ω) =

1 ⎡ ⎢ϵ1 + 2⎣

⎤1/2 ϵ12 + ϵ22 ⎥ ⎦

the information related to the quality of the device such as the carrier resistance and recombination processes,60

(6)

D A ⎤ Voc = ⎡⎣Eg − (E LUMO − E LUMO ) − 0.3⎦ /e

calculated from the value of the imaginary part of the dielectric constant, ϵ2, and by making use of the Kramers−Kronig relations to obtain the real part, ϵ1. InN possesses the minimum value for the refractive index within the energy region from 0.5 to 1.7 eV, indicating that this material offers the minimum resistance to absorb sunlight when compared with the other structures. However, the refractive index for both arrangements AA and T allows us to obtain an intermediate value for this index, reducing, in this way, the effective refractive index of MoTe2. Notice that although the absolute value of the refractive index depends on the supercell vacuum we are able to make reasonable predictions based on this observable, since the vacuum region imposed for all the systems is exactly the same and only the relative difference between the systems is of interest here. As mentioned previously, the main advantage of building these heterostructures is to enhance the optical properties of the isolated monolayers. For instance, arrangements AA and T lower the energy of the first excitonic peak of the InN monolayer, setting it close to the 1.1 eV region. Furthermore, the heterostructure T significantly reduces the first excitonbinding energies of both isolated monolayers to values ∼0.6 eV. This reduction on the excitons’ binding energies should, in principle, benefit their dissociation into free charges to produce photocurrent. Moreover, as we will later see, the exciton lifetime for the heterojunctions show a larger value than those for the individual monolayers. Therefore, the overall optical properties reveal the benefits of building these heterojunctions. Photovoltaic Potential. In order to further explore the possible use of the heterostructures for photovoltaic applications, we estimate the upper limit for the short-circuit current density in a hypothetical device by considering JSC = e

∫E

is the open circuit voltage, and Ii = 100 mW/cm2 is the total solar irradiance. We adopt a moderate value for the fill factor, FF = 0.6, which represents the average of the fill factors related to several devices in which carrier transport is observed.59 The value of the open circuit voltage Voc is extracted from the quasiparticle band alignments. We find power conversion efficiencies of 2.2% and 1.9% for AA and T stacking, respectively. Although the values of maximum efficiencies calculated here are not comparable with the efficiencies provided by GaAs-based solar cells (which are typically much thicker), it should be noticed that the power densities (the power generated per unit of volume) found in our heterostructures are 366 and 259 kW/kg, respectively, which is 1 order of magnitude higher than those found in highefficiency multijunction solar cells.61,62 Exciton Lifetimes. Finally, we calculate the radiative recombination rates and the exciton lifetimes. We adapt the Shockley−van Roosbroeck model63 used for describing exciton lifetimes in three-dimensional electron gases to a 2D electron gas (see the Supporting Information). By doing this, the supercell vacuum-independent 2D spontaneous emission can be written as .(ω) =

A(ω)Iph(ω)dω

(7)

where Iph(ω) is the photon spectral irradiance of the incident AM1.5 solar spectrum,58 e is the electron charge, and A(ω) is the thickness-independent absorbance. Notice that the lower limit of the integral refers to the optical band gap. The short-circuit current densities for monolayer MoTe2, InN, and AA and T heterostructures are found to be 2.4, 1.07, 2.5, and 2.65 mA/cm2, respectively. The value for T is approximately 60% larger than that for monolayer InN and almost 10% greater than that for monolayer MoTe2. These results strengthen the idea that by forming a heterostructure one can effectively tailor the absorption curve to energy values close to 1.1 eV and, consequently, increase the short-circuit current and power conversion efficiency of the device. In addition, this result is in agreement with the Shockley− Queisser limit, which states that the most efficient photovoltaic devices should possess optical gaps in the mentioned energy region.23 We also estimated the power conversion efficiency by evaluating59 PCE = FF × JSC × Voc/Ii

ωϵ2(ω) 16πω Lz ℏc[exp(ℏω/kBT ) − 1] c

(10)

where ℏ is Planck’s constant, c is the speed of light, kB is Boltzmann’s constant, and Lz is the supercell vacuum region. The total radiative recombination rate, G, is calculated by integrating eq 10, and then the radiative lifetime can be obtained as τ = ΔρΔz/G, where Δρ represents the excess carrier density and Δz is the with of the materials. This way, we consider only the region in the cell where it is effectively charge concentrated. In order to obtain representative exciton lifetimes for our materials, we adopted experimental values for the excess carrier density reported previously in the measurement of the exciton lifetime for MoS2 monolayers,64 which is of the order of 1011 cm−2. The results are summarized in Table 2. It can be observed that the exciton lifetime of MoTe2 is almost six times the value

∞

gap / ℏ

(9)

Table 2. First and Second Column Shows the Total Recombination Time, Exciton Lifetime for MoTe2, InN, AA, and T Systems calculated at T = 300 K and for a Carrier Density Δρ = 3 × 1011 cm−2a effective mass (me) 20

G (10 cm MoTe2 InN AA T

−2

3.15 19.80 1.94 2.22

−1

s )

τ (ns)

me

mh

0.95 0.15 1.54 1.35

0.34 0.21 0.30 0.22

0.48 0.42 0.43 0.47

Δz for MoTe2 and InN monolayers are 0.65 and 0.42 nm, respectively. These values were extracted from experimental data for the average thickness in 2D materials.24,65 For heterostructures, we assumed a conservative value of 1.1 nm. In the third column, the effective mass which is calculated from the quasiparticle band structures. a

(8)

which is commonly used in determining the efficiency of excitonic solar cells. Here, FF is the fill factor, which comprises 11892



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obtained for InN. Moreover, the exciton lifetime in the heterostructures are almost 1 order of magnitude higher with respect InN monolayer. It is worth pointing out that although between the monolayers InN possesses the strongest exciton binding energy (0.82 eV), it does not prevent the material from eventual exciton recombination. In fact, our exciton lifetime model shows that heterostructure T, the one with the lowest exciton binding energy (0.6 eV), provides one of the largest exciton lifetimes. Finally, we stress that the estimated radiative recombination exciton lifetime for the MoTe2 monolayer provides reasonable values that are in good agreement with previous theoretical studies based on effective models for the MoS2 monolayer.66 Notice that if nonradiative recombination mechanisms, due to defects for example, were included in our model, the values reported here should become smaller. This statement is supported by recent experimental measurements of nonradiative electron−hole lifetimes in MoS2 monolayers67 that reveal shorter values when compared to the nanosecond time scale observed in our model. Furthermore, a recent experimental study by Rivera et al.68 on exciton lifetimes in MoSe2−WSe2 heterostructures reported long-lived excitons in the nanosecond scale, similar to the values found in this work.

CONCLUSIONS In conclusion, we have carried out DFT calculations combined with many-body perturbation theory to study the electronic and optical properties of novel atomically thin systems based on hInN and MoTe2. We found that the energy alignment on these heterostructures is, for some cases, of type II and provides excitons that are photoexcited at energies ranging from 1.1 to 1.3 eV. We have also estimated the power density of these heterostructures and shown that they can reach values up to 1 order of magnitude higher than multijunction solar cells. We also show that forming a heterostructure of MoTe2−InN lets us us combine the near-infrared absorption of MoTe2 with the low refractive index and high absorbance of h-InN, leading to improved optical properties with respect to the precursor materials such as photoexcited states with lower exciton binding energies and long exciton lifetimes in the nanosecond scale. The overall results indicate that these systems hold promising potential for photovoltaic applications. ASSOCIATED CONTENT

S Supporting Information *

Detailed study of the dynamic stability of h-InN, the quasiparticle electronic structure, and the exciton recombination model for slablike structures. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/jp5122596.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +55-11-3393-7804. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support from the Brazilian agency FAPESP and computational support from GRID-UNESP and CENAPAD/SP. 11893


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InN for Photovoltaic - ACS Publications - American Chemical Society

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