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2D Heterojunctions from Non-local Manipulations of the Interactions Malte Rösner, Christina Steinke, Michael Lorke, Christopher Gies, Frank Jahnke, and Tim Oliver Wehling Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b05009 • Publication Date (Web): 26 Feb 2016 Downloaded from http://pubs.acs.org on February 28, 2016
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2D Heterojunctions from Non-local Manipulations of the Interactions M. Rösner,∗,†,‡ C. Steinke,†,‡ M. Lorke,† C. Gies,† F. Jahnke,† and T. O. Wehling†,‡ Institut für Theoretische Physik, Universität Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany, and Bremen Center for Computational Materials Science, Universität Bremen, Am Fallturm 1a, 28359 Bremen, Germany E-mail:
[email protected] Abstract
with different band gaps, are central building blocks of various applications 1,2 . Apart from planar junctions, which are the basis of light-emitting diodes and solar cells, more complex structures such as quantum wells 3 or quantum dots 4 hold promises in the context of quantum information processing. In the bulk, e.g. in GaAs / InGaAs material systems, heterojunctions are often fabricated by epitaxy, which can be employed up to industrial scales. In addition to bulk crystals, also monolayer thin two-dimensional (2d) materials 5 including semiconducting transition metal dichalcogenides (TMDC) have been assembled into structures like vertical 6–12 or lateral heterojunctions 13–16 . All of these systems rely on interfaces of different materials in order to gain spatial band-gap modulations. The epitaxial fabrication of well defined interfaces with the desired electronic properties underlies constraints due to available materials, and can in practice be very challenging. In 2d semiconductors the Coulomb interaction can modify band gaps on an eV scale 17–19 . Furthermore, in atomically thin layers, the Coulomb interaction can be drastically manipulated by external screening 20–28 , allowing to control the band gap by its dielectric surrounding. In this letter, we propose a scheme to build heterojunctions within a single homogeneous layer of a 2d material based on non-local manipulations of the Coulomb interaction, that is the
We propose to create lateral heterojunctions in two-dimensional materials based on non-local manipulations of the Coulomb interaction using structured dielectric environments. By means of ab initio calculations for MoS2 as well as generic semiconductor models, we show that the Coulomb-interaction induced self-energy corrections in real space are sufficiently nonlocal to be manipulated externally, but still local enough to induce spatially sharp interfaces within a single homogeneous monolayer to form heterojunctions. We find a type-II heterojunction band scheme promoted by a laterally structured dielectric environment, which exhibits a sharp band-gap crossover within less then 5 unit cells.
Keywords heterojunction, 2d materials, Coulomb interaction, substrate, non-local manipulation Introduction: Semiconductors play a major role in modern optoelectronics. Particularly heterojunctions, i.e. interfaces of materials ∗
To whom correspondence should be addressed Institut für Theoretische Physik, Universität Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany ‡ Bremen Center for Computational Materials Science, Universität Bremen, Am Fallturm 1a, 28359 Bremen, Germany †
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controlled manipulation of the long-range characteristics of the Coulomb interaction within the layered material. By placing a 2d semiconductor into a laterally structured environment (e.g. a substrate with laterally varying dielectric constants as depicted in Fig. 1), the Coulomb interaction within the 2d material changes spatially and with it, the local band gaps are modulated as well. Thus, band-gap variations like in a heterojunction can be induced externally in a homogeneous monolayer by an appropriately structured dielectric environment.
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indeed limited to the length scale of a few unit cells. In order to study heterogeneous systems as shown in Fig. 1, we switch to a generic 2d semiconductor model that is quantitatively based on our ab-intio GW results. Using this model, we demonstrate that a spatially inhomogeneous environment allows to induce externally a heterojunction of type-II in MoS2 . Locality of the Coulomb interaction and the origin of the band gap in MoS2 : The single-particle low-energy band structure in MoS2 is formed by the highest valence and the two lowest conduction bands which have predominantly molybdenum dz 2 , dxy and dx2 −y2 character. The band gap is opened due to substantial hybridization effects between the dz 2 and the {dxy , dx2 −y2 } states 29,30 . These effects are included already on the level of density functional theory (DFT), leading to a band gap of ≈ 2.0 eV 31 , To investigate the effects to this hybridization mechanism and the locality of the Coulomb interaction in MoS2 , we perform G0 W0 (GW ) calculations which take Coulomb interaction induced screened exchange effects into account. We derive the corresponding self-energy in real space from a comparison of DFT and these G0 W0 calculations 32 utilizing the Dyson equation
Figure 1: (Color online) Sketches of a monolayer (blue) in different heterogeneous dielectric environments. (a), (c) and (d) show situations with structured dielectric substrates while in (b) adsorbated polarizable molecules are responsible for the heterogeneous dielectric environment.
−1 Σ = G−1 DFT − GGW ,
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where GDFT and GGW are the electronic Green functions obtained from corresponding calculations. Within the quasi-particle approximation, the Green functions G−1 (2) GW/DFT = z + µ − HGW/DFT ,
The central quantities that define the possible technical relevance of such externally induced heterojunctions are (i) the size of realistically achievable band-gap modulations and (ii) the length scale over which these modulations take place. In the following we show that changes in the Coulomb interaction can induce band-gap modulations in the range of several 100 meV on the length scale of a few lattice spacings in homogeneous MoS2 . To this end, we consider in a first step a free standing MoS2 monolayer and analyze Coulomb interaction effects as manifesting in the electronic self-energy in real space. Based on GW calculations, we demonstrate that the dominant self-energy terms are
follow from the Wannier Hamiltonians HGW and HDFT describing the DFT and G0 W0 band structures, respectively: X GW/DFT GW/DFT Hαβ (k) = eikR tαβ (R). (3) R
The quantum numbers α, β ∈ {dz 2 , dxy , dx2−y2 } denote the dominating orbital characters of the wave functions in the Wannier basis 33 . Finally,
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the real-space self-energy is approximated as
the cells around the central unit cells in each panel. The strongest contributions to the self-energy are non-local inter-orbital exchange terms, which directly increase the hybridization and the resulting band gap. The enhanced band gap in GW calculations therefore results from nonlocal inter-orbital contributions of the Coulomb interaction. In general, we find non-local contributions to Σ, as one expects from the long range character of the Coulomb interaction in 2d semiconductors. Nevertheless, the most sizable contributions are clearly localized within a radius of less than three unit cells. Hence despite the self-energy being non-local, it could still facilitate sharp band-gap modulations in the case of structured dielectric environments, which will be discussed in the following section. The real-space structure of ΣGW αβ (R) can be understood by considering the corresponding orbital symmetries. The dz 2 orbital is invariant under the operations of the threefold rotation symmetry of the MoS2 lattice, which is reflected in the dz 2 /dz 2 panel of Fig. 2 showing the full symmetry of the lattice. The dxy and dx2 −y2 orbitals belong to a two-dimensional representation of the crystal symmetry point group, which leads to a more complex real-space structure of the corresponding self-energy terms, as seen in the dxy and dx2 −y2 panels in Fig. 2.
GW DFT ΣGW αβ (R) = tαβ (R) − tαβ (R) − ∆µ δR0 δαβ , (4)
where ∆µ aligns the Fermi energies between the DFT and G0 W0 calculations 34 . Σ (eV) −0.16 −0.12 −0.08 −0.04 0.00
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Figure 2: (Color online) Real-space representation of the GW self-energy ΣGW αβ (R) of MoS2 within the minimal Mo d basis. The grey lines mark the hexagonal Wigner-Seitz unit cells of the MoS2 lattice, where the Mo atoms are assumed to be in the center of the cells.
Heterostructures made by heterogeneous dielectric environments: Now we turn to structured dielectric environments as depicted in Fig. 1. Here the broken translational symmetry makes GW calculations numerically extremely demanding and practically unfeasible. As an alternative, we switch to a model system that mimics the essential gap-opening mechanisms and interaction effects present in semiconducting TMDCs and, at the same time allows us to study the influences of a structured dielectric environment on the local density of states (LDOS) and the resulting spatial variation of the band gap. We consider a two-band semiconductor model which is in momentum (k) space described by
Fig. 2 shows the resulting map of ΣGW αβ (R), which visualizes the R dependent renormalizations of the tight-binding hopping matrixelements due to the Coulomb interaction. These elements can be separated into intraand inter-orbital contributions arising from Coulomb-interaction induced changes in the intra- (α ↔ α) and inter-orbital (α ↔ β) hoppings. In more detail, there are local renormalizations Σαβ (R = 0), and, more importantly, non-local self-enery terms Σαβ (R 6= 0) which arise from the non-local character of the Coulomb interaction. These non-local terms in ΣGW αβ (R) can be found in the maps of Fig. 2 in
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a single-particle Hamiltonian of the form ǫ|| (k) t⊥ , HSP (k) = t⊥ −ǫ|| (k)
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modifies the interaction matrix elements Uij , Z Uij = ε−1 (ri , rl )v(rl , rj )d3 rl . (8)
(5)
For heterogeneous environments, as illustrated in Fig. 1, the corresponding backgroundscreened Coulomb interaction can be obtained from a solution of the resulting Poisson equation. In general, numerical schemes need to be employed for this purpose, although analytical results exist for simplified situations (see Supporting Information). The Hartree-Fock real-space self-energy resulting from Eq. (6) and the electron-ion attraction (see Supporting Information) lead to a Coulomb-interaction induced correction of the single-particle Hamiltonian according to 36 X Σij = δij 2Uil δnl −Uij hc†j ci i, (9) | {z } l Fock {z } |
where ±ǫ|| (k) is the tight-binding dispersion of a pair of 2d lattices with nearest-neighbor inplane hopping ±t (see Supporting Information for a graphical illustration and a detailled discussion of the model). The gap arising in this model is a hybridization gap opened due to t⊥ which is the same mechanism we have discussed in the previous section for semiconducting TMDC materials. The hopping parameters are chosen to reproduce the band width (Wk ≈ 1.0 eV) and the DFT band gap (∆ ≈ 2.0 eV) of MoS2 , which is similar to other semiconducting TMDCs 31 . In fact, Eq. (5) is the most simple description of a 2d semiconductor and thus more generic. The Coulomb interaction gives rise to electron-electron, electron-ion and ion-ion interaction terms: HCoulomb = Hee + Hei + Hii . The ions are assumed to have a fixed positive charge Ze = +1e to ensure charge neutrality of the whole system, i.e. Z = 2¯ n, where n ¯ is the average electron occupation per spin and orbital. The ionic positions are assumed to be fixed. Thus, Hii leads to a constant shift of the total energy, which will be neglected in the following. The remaining Coulomb terms read 35,36 1X (6) Uij c†iσ c†jσ′ cjσ′ ciσ Hee = 2 ′ ijσσ X Hei = − Uij n ˆ iσ Z, (7)
Hartree
where δnl = hˆ nl i − n ¯ is the deviation from the average occupation n ¯ and spin indices are suppressed as Σij is spin diagonal. Using this Σij we are now able to calculate the LDOS for arbitrary dielectric environments. 37 In Fig. 3 we present a direct visual account of the band-gap variation induced by a laterally changing dielectric environment, as indicated by the sketch of the heterostructure. We have considered a 2d layer of zero height surrounded by two different dielectric half spaces with ε1 = 5 and ε2 = 15. 38 The overall variation of the band gaps along the spatial direction is reminiscent of a heterojunction band diagram of type-II. In both regions, two main characteristics can be clearly seen: The van-Hove singularities (vHS) as maxima in the LDOS, and spatially dependent band gaps Egap (r) as energy ranges where the LDOS vanishes between the singularities 39 . Egap (r) is clearly reduced in the ε2 region (Egap ≈ 1.9 eV) compared to the ε1 area (Egap ≈ 2.9 eV) as a result of stronger external screening effects of the ε2 substrate and correspondingly reduced Coulomb interaction. For the given ε2 /ε1 ratio and also for smaller ratios, we find a nearly vanishing conductionband offset (CBO), while the different band
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where Uij = U(ri , rj ) is the interaction energy between electrons or ions at sites ri and rj , σ labels the electron spin, c†iσ (ciσ ) are the corresponding electronic creation (annihilation) operators, and n ˆ iσ = c†iσ ciσ are electron occupation operators. The environment of the material screens the bare Coulomb potentials vij via the dielectric function εij and correspondingly
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mation. In lateral heterojunctions made from stitching together different TMDCs, a comparable length-scale has been reported 14 . Thus, we find a similar behavior with the difference, that here the heterojunction does not arise from different materials, but is induced externally by structuring the dielectric substrate. There are intrinsic and extrinsic factors limiting the length scale over which band-gap variations in dielectrically induced heterojunctions can be realized. The major extrinsic factor determining the sharpness of the induced bandgap variation, is the length-scale on which the dielectric environment changes, which depends on experimental substrate or adsorbate preparation procedures. There are several experimental ways to realize nearly atomically sharp variations of the dielectric polarizability of the environment of a 2d material. Examples range from the extreme case of substrates containing holes 42–44 , patterned adsorption of polarizable molecules 45–48 , and intercalation or adsorption of atoms 49,50 to self-organized growth of structured dielectrics by epitaxial means 51–54 . A lower intrinsic bound for the length scale, on which the band-gap variation takes place is defined by the spatial extend of the selfenergy which can be deduced qualitatively from the underlying model. According to Eq. (9) the range of the self-energy is limited by the real-space decay range of the correlation functions hc†j ci i. The latter is determined by the ratio of hybridization t⊥ to the band width Wk ∝ |t| of the in-plane dispersion, as can be understood from the Wannier Hamiltonian in k-space representation, Eq. (5). There is significant hybridization between the two layers if |ǫk (k)| . t⊥ , i.e. in a region extending about δk = t⊥ /vf ∼ t⊥ /(Wk a) around the ǫk (k) = 0 line in k-space, where vf = |∂k ǫk (k)| ∼ Wk a is the group velocity associated with the inplane dispersion. By the uncertainty principle, the momentum-space extent δk translates into a range δr ∼ 1/δk ≈ Wk a/t⊥ of the correlation functions hc†j ci i in real space. As a consequence, Σij is generically limited to the scale of a few unit cells as long as hybridization (t⊥ ) and band width (Wk ) are similar in size. This finding is reflected in the numerical data
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Figure 3: (Color online) Local density of states for different unit cells along a line perpendicular to the dielectric interface of the substrate (left: ε1 = 5, right: ε2 = 15). The model parameters are set to t = 0.26 eV, t⊥ = 0.85 eV, a = 3.18 Å (in-plane lattices constant) and c = a/4 (layer separation), see Supporting Information for more details. gaps can be tuned precisely between both regions, as we show in the Supporting Information. Thus the ratio between the CBO and the band gaps can be controlled in this kind of heterostructures, allowing e.g. for optimized solar cell setups 40,41 . The kind of band diagram shown in Fig. 3 will arise in all systems shown in Fig. 1, although the effect of the structured environment is strongest in the setups corresponding to the panels Fig. 1 (a) and Fig. 1 (c). To have a strong effect in the situation of Fig. 1 (b) one should use a substrate which is significantly less polarizable than the adsorbed molecules. Similarly, in the setup of Fig. 1 (d) one would expect the strongest effect if the capping layer has a small polarizability (e.g. ε3 < ε1/2 ). Most importantly for electronic functionalities and particularly regarding electronic transport in these heterojunctions, the band-gap changes within less than 5 unit cells around the interface, which holds for a wide range of ε2 /ε1 ratios as we show in the Supporting Infor-
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any effect on the band structure. Especially the non-local Fock terms ΣF , as we show in more detail in the Supporting Information, increase the band gap by modifying the hybridization (as seen in ΣAB /ΣBA in Fig. 4). This tendency is independent of the dielectric constant and inherent to all semiconducting 2d materials where band gaps result from hybridization effects. Consequently, in all materials of this kind, heterojunctions can be induced by an external manipulation of the Coulomb interaction. For heterojunctions obtained from non-local manipulations of interactions we expect that screening and exchange-interaction induced confinement potentials affect uncorrelated electrons and holes quite differently compared to correlated electron-hole pairs. For instance, optical absorption energies related to the excitation of correlated electron-hole pairs (i.e. excitons) depend on the quasi-particle band gaps but also on the excitonic binding energies which are both decreased by a highly polarizable dielectric environment. Hence, the excitonic absorption energies will change less by external manipulations of the Coulomb interaction 25 than the single-particle properties, which are most relevant for electronic transport. As a consequence, the relation between optical and transport properties in the kind of heterojunctions proposed here will likely differ from heterojunctions created by stitching different materials together.
for the non-local real-space self-energy Σij depicted in Fig. 4. In analogy to the discussion of the MoS2 GW self-energy in the homogeneous case, we show in Fig. 4 the self-energy in the middle of the ε1 = 5 area of the heterostructure (labeled as “−10” in Fig. 3). Here, the local dielectric environment is essentially homogeneous and thus comparable to the fully homogeneous case. The off-diagonal self-energy terms shown in Fig. 4 are by definition nonlocal, as they describe modulations of interlayer couplings (separated by Rz = c), but significant contributions are limited to a single unit cell. For orbitals in the same layer (diagonal panels in Fig. 4), the self-energy is smaller, and substantial contributions are limited to about two unit cells. Hence the real-space structure of the model self-energy in this homogeneous-like area of the system is quite similar to the self-energy in MoS2 obtained from full ab initio calculations (see Fig. 2), Σ (eV) −0.6
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Conclusions: We have demonstrated that in MoS2 , as a typical 2d TMDC semiconductor, heterostructures can be formed by means of spatially structured dielectric environments. For this purpose, we have used ab initio calculations and a generic 2d semiconductor model, to show, that the external manipulation of the Coulomb interaction allows for sharp, spatially modulated band gaps. Hence new kinds of heterojunctions can be constructed by placing semiconducting 2d materials on appropriately structured substrates. Similarly, polarizable molecules could be deposited on top of 2d materials to cover parts of the surface to form heterojunctions. Such heterojunctions bring the ad-
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Figure 4: (Color online) Real-space representation of the Hartree-Fock self-energy Σij in the ε1 = 5 area of the system shown in Fig. 3 (unit cell “−10”). The grey lines mark the hexagonal Wigner-Seitz unit cells. Note, the diagonal elements were enhanced by a factor of 5. More specifically, the Hartree contribution of the self-energy ΣH is diagonal and has hardly
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(5) Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V.; Geim, A. K. Proc. Natl. Acad. Sci. U. S. A. 2005, 102, 10451–10453.
vantage that only the environment of the active material but not the material itself needs to be structured during the fabrication process. One could thus add the active 2d semiconducting layer to independently pre-structured dielectric layers which is very attractive from a fabrication point of view. While we have considered a single interface in this letter, our findings can be generalized to more complex structures. One can for instance use two parallel interfaces to realize a quantum wire-like structure. Finally, with four interfaces (two in the x- and two in the y-direction) or also partial coverage of finite areas with adsorbates quantum dots could be externally induced in monolayers of homogeneous 2d materials.
(6) Radisavljevic, B.; Radenovic, A.; Brivio, J.; Giacometti, V.; Kis, A. Nat. Nanotechnol. 2011, 6, 147–150. (7) Britnell, L.; Gorbachev, R. V.; Jalil, R.; Belle, B. D.; Schedin, F.; Mishchenko, A.; Georgiou, T.; Katsnelson, M. I.; Eaves, L.; Morozov, S. V.; Peres, N. M. R.; Leist, J.; Geim, A. K.; Novoselov, K. S.; Ponomarenko, L. A. Science 2012, 335, 947– 950. (8) Britnell, L.; Ribeiro, R. M.; Eckmann, A.; Jalil, R.; Belle, B. D.; Mishchenko, A.; Kim, Y.-J.; Gorbachev, R. V.; Georgiou, T.; Morozov, S. V.; Grigorenko, A. N.; Geim, A. K.; Casiraghi, C.; Neto, A. H. C.; Novoselov, K. S. Science 2013, 340, 1311–1314.
Acknowledgement We thank Knut MüllerCaspary for insightful discussions on the epitaxial growth of heterostructures. This work was supported by the European Graphene Flagship and DFG via SPP 1459, and through computing resources at the North German Supercomputing Alliance (HLRN).
(9) Georgiou, T.; Jalil, R.; Belle, B. D.; Britnell, L.; Gorbachev, R. V.; Morozov, S. V.; Kim, Y.-J.; Gholinia, A.; Haigh, S. J.; Makarovsky, O.; Eaves, L.; Ponomarenko, L. A.; Geim, A. K.; Novoselov, K. S.; Mishchenko, A. Nat. Nanotechnol. 2013, 8, 100–103.
Supporting Information Available: Details on the generic model, including derivations of the Hartree-Fock self-energy and the screened Coulomb interaction, Hartree and Fock contributions to the band gap and local density of states for different heterogeneous substrates. This material is available free of charge via the Internet at http://pubs.acs. org/.
(10) Yu, W. J.; Liu, Y.; Zhou, H.; Yin, A.; Li, Z.; Huang, Y.; Duan, X. Nat. Nanotechnol. 2013, 8, 952–958.
Notes: The authors declare no competing financial interests.
(11) Jariwala, D.; Sangwan, V. K.; Lauhon, L. J.; Marks, T. J.; Hersam, M. C. ACS Nano 2014, 8, 1102– 1120.
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(32) The corresponding calculations are performed with the Vienna ab initio simulation package (VASP) 55,56 . The DFT calculations are carried out in the GGA (PBE) 57 using a 18×18 k-mesh and an energy cut-off of 280 eV. The lattice constant is set to 3.18 Å, the sulfur z-positions are relaxed to ±1.57 Å and an interlayer separation of 55 Å is chosen to minimize artificial self-interactions within the supercell approach. For the evaluation of the polarization in the G0 W0 calculations 192 bands together with an energy cut-off of 150 eV are used.
(20) Keldysh L.V., 1979, 29, 658. (21) Park, C.-H.; Giustino, F.; Spataru, C. D.; Cohen, M. L.; Louie, S. G. Nano Lett. 2009, 9, 4234–4239. (22) Park, C.-H.; Giustino, F.; Spataru, C. D.; Cohen, M. L.; Louie, S. G. Phys. Rev. Lett. 2009, 102, 076803. (23) Park, C.-H.; Louie, S. G. Nano Lett. 2010, 10, 426–431. (24) Chernikov, A.; Berkelbach, T. C.; Hill, H. M.; Rigosi, A.; Li, Y.; Aslan, O. B.; Reichman, D. R.; Hybertsen, M. S.; Heinz, T. F. Phys. Rev. Lett. 2014, 113, 076802.
(33) Hereby we aim to describe the highest valence band and the two lowest conduction bands. All of those three bands are actually entangled and hybridized with further molybdenum d orbitals as well as sulfur p orbitals. To get well defined electronic bands, we use the Wannier90 package 58 in order to properly disentangle the bands. Thereby we use a “frozen” or “inner” energy window, which fixes the valenceband’s energy as well as most parts of the
(25) Lin, Y.; Ling, X.; Yu, L.; Huang, S.; Hsu, A. L.; Lee, Y.-H.; Kong, J.; Dresselhaus, M. S.; Palacios, T. Nano Lett. 2014, 14, 5569–5576.
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lowest conduction band. Afterwards we stay with the resulting projections without performing a maximal localization.
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(34) This is realized by choosing ∆µ such that Tr ΣGW (0) = 0.
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(37) We use non-primitive rectangular unit cells, which involve 4 atoms. All supercells consist of 50 × 30 non-primitive unit cells. We use periodic boundary conditions in the y-direction and fixed boundaries in the x-direction. In the case of the heterostructures, the plane, which separates the different dielectric areas from each other is chosen to be parallel to the y-axis and is placed between the 25th and 26th unit cells on the x-axis.
(47) De Wild, M.; Berner, S.; Suzuki, H.; Ramoino, L.; Baratoff, A.; Junga, T. A. Ann. N. Y. Acad. Sci. 2003, 1006, 291– 305. (48) Suto, K.; Yoshimoto, S.; Itaya, K. J. Am. Chem. Soc. 2003, 125, 14976–14977. (49) Petrović, M.; Šrut Rakić, I.; Runte, S.; Busse, C.; Sadowski, J. T.; Lazić, P.; Pletikosić, I.; Pan, Z.-H.; Milun, M.; Pervan, P. et al. Nat. Commun. 2013, 4, 2772.
(38) For this scenario the corresponding Coulomb interaction Uij can be obtained analytically, as shown in the Supporting Information.
(50) Ulstrup, S.; Andersen, M.; Bianchi, M.; Barreto, L.; Hammer, B.; Hornekær, L.; Hofmann, P. 2D Mater. 2014, 1, 025002.
(39) Since an artificial broadening δ = 5 meV is involved in the evaluation of the LDOS it never vanishes completely. Therefore we consider values smaller than 0.02 as zero.
(51) Klie, R. F.; Zhu, Y.; Altman, E. I.; Liang, Y. Appl. Phys. Lett. 2005, 87, 143106.
(40) Janssen, R. A. J.; Nelson, J. Adv. Mater. 2013, 25, 1847–1858. (41) Sharbati, S.; Sites, J. IEEE J. Photovolt. 2014, 4, 697–702.
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(53) Hudait, M. K.; Clavel, M.; Zhu, Y.; Goley, P. S.; Kundu, S.; Maurya, D.; Priya, S. ACS Appl. Mater. Interfaces 2015, 7, 5471–5479.
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(55) Kresse, G.; Hafner, J. Phys. Rev. B 1993, 47, 558–561. (56) Kresse, G.; Furthmüller, J. Comput. Mater. Sci. 1996, 6, 15–50. (57) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865–3868. (58) Mostofi, A. A.; Yates, J. R.; Lee, Y.S.; Souza, I.; Vanderbilt, D.; Marzari, N. Comput. Phys. Commun. 2008, 178, 685– 699.
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