Atomically Thin Ordered Alloys of Transition Metal Dichalcogenides

Subscriber access provided by University of Alabama

Article

Atomically Thin Ordered Alloys of Transition Metal Dichalcogenides: Stability and Band Structures Mohnish Pandey, Karsten W. Jacobsen, and Kristian S. Thygesen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b07283 • Publication Date (Web): 15 Sep 2016 Downloaded from http://pubs.acs.org on September 16, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Atomically Thin Ordered Alloys of Transition Metal Dichalcogenides: Stability and Band Structures Mohnish Pandey,∗,† Karsten W. Jacobsen,† and Kristian S. Thygesen†,‡ Center for Atomic-scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, DK - 2800 Kongens Lyngby, Denmark, and Center for Nanostructured Graphene (CNG), Department of Physics, Technical University of Denmark, DK - 2800 Kongens Lyngby, Denmark E-mail: [email protected] Phone: +45 4525 3204

∗ To

whom correspondence should be addressed for Atomic-scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, DK - 2800 Kongens Lyngby, Denmark ‡ Center for Nanostructured Graphene (CNG), Department of Physics, Technical University of Denmark, DK - 2800 Kongens Lyngby, Denmark † Center

1

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We explore the possibility of modulating the electronic band edges of the transition metal dichalcogenides (TMDs) via alloying of different semiconductors within the same group (intra-group alloying). The stability of the ordered alloys is assessed from the calculated mixing enthalpy which is found to be close to zero for several alloys and below 20 meV/atom for all of the alloys. We explore to what extent the electronic properties like the band gap and band edge positions of the alloy can be evaluated by taking the weighted average of the corresponding properties of the pristine systems. In general, this approach works well with the only exception being Cr containing compounds. Because the calculated properties of the alloys are very similar to the weighted averages, we expect that the trends observed for the ordered alloys will also hold for more realistic disordered alloys.

Introduction The interest in the rapidly expanding class of two-dimensional (2D) materials has increased steadily since the discovery of graphene. From a fundamental point of view, the 2D materials are attractive due to their unique physical properties arising mainly, but not exclusively, from the reduced dimensionality which entails a lower degree of screening and thus enhanced manybody effects. From an engineering perspective, the 2D materials are of interest because of their easily tunable properties. For example, band structures 1–3 and dielectric properties 4 can be accurately controlled with atomic-scale precision by varying the number, type, and stacking sequence of the 2D crystals. Similarly, the chemical activity can be tuned by nanostructuring, e.g. via controlling edge termination or defect concentration, and/or by application of mechanical strain. 5,6 The transition metal dichalcogenides (TMDs) with the general chemical formula MX2 (M: metal, X: chalcogen) represent one of the major classes of the 2D materials. An extensive overview of the electronic properties of the semi-conductors from this class of materials can 2


Page 2 of 18

Page 3 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


be found in Ref. 7. Experimentally, the TMDs which have been mostly explored so far are based on metals from the titanium and molybdenum groups (group IV and VI). 8–12 One interesting aspect of these TMDs is that, depending on the structure, the dichalcogenides of these groups can be either metallic (1T phase) or semiconducting (2H phase). The metallic TMDs exhibit exotic electronic excitations leading to quantum spin Hall effects, undergo transitions to charge density wave phases at lower temperatures 11–13 and show significant catalytic activity e.g. for hydrogen evolution. 9,10,14 The semiconducting phases of the group IV and VI TMDs, which are the focus of the present work, have been used for applications within photovoltaics, light emitting diodes, and field effect transistors. 2,15–17 Despite the many possibilities for tuning the electronic properties of the 2D materials by external means (stacking, straining, etc.), it remains of interest to expand the space of basic 2D compounds. One of the primary strategies to accomplish this is via alloying. In the case of TMDs, alloying can be achieved either by alloying of the chalcogen- or the metal atoms. 18,19 Recent calculations have shed light on the alloying mechanism in the case of both ordered and disordered alloys of the group V and VII metal dichalcogenides (metal alloying) and group VI dichalcogenides (chalcogen alloying) and it has been shown that many stable alloys and a wide tunability of the band gap can be accomplished in the TMDs. 20,21 While the main focus of most of the studies has been on the band gap, the defect properties of the TMDs can also be influenced and to some extent controlled via alloying which changes the position of the valence band maximum (VBM)/conduction band minimum (CBM) thus changing the relative position of the defect levels as well. 22 These examples suggest that alloying provides an efficient and versatile means for tuning the electronic properties of the TMDs. In the present study, we explore the stability and band structures of ordered alloys of the TMDs of group IV and VI (intra-group alloys) using first-principles density functional theory (DFT) calculations. The calculated heat of mixing is close to zero for most of the alloys and never above 20 meV/atom, indicating that synthesis of stable alloys should be

3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

feasible. Most interestingly, our calculations show that the band gap and absolute band edge positions of the popular Mo- and W-based TMDs can be tuned by more than 0.5 eV by alloying with Cr. Alloying of the other compounds leads to smaller changes in band gap and band edge positions up to 0.2 eV. We find that the calculated band gap and band edge positions of the alloys can be described, at least qualitatively, by taking the weighted average of the corresponding quantities for the constituent TMDs. This indicates that the properties of the more realistic disordered alloys should be similar to those of the ordered alloys studied here.

Computational Details The electronic structure calculations are performed using the Projector Augmented Wave (PAW) 23 formalism as implemented in the GPAW code. 24 The lattice constants of the pristine systems as well as the alloys are optimized using the PBE exchange-correlation (xc) functional. 25 A kinetic energy cutoff of 800 eV is used for the expansion of the wave functions in the plane wave basis. The unit cell vectors as well as the atomic positions are relaxed until the forces are below 0.05 eV/Å. The Brillouin zone is sampled with a k-point mesh of 9×9×1 for the relaxation and 12×12×1 for the calculation of the heat of mixing and band gaps. In order to have an estimate of uncertainties in the calculation of the heat of mixing the Bayesian error estimation functional mBEEF 26 is used which has been previously shown to predict the heats of formation quite accurately. 27 The hybrid functional HSE06 is used for the calculation of the band gaps in order to circumvent the issue of underestimation of the band gaps in the generalized gradient approximation. 28,29

Results and Discussion The ordered alloys are modelled as shown in Figure 1. In the figure, the dotted rectangle represents the unit cell used for the calculations and M1 and M2 are different metal atoms 4


Page 4 of 18

Page 5 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and X represents the chalcogen species. The number of M1 and M2 in the unit cell is varied to get different composition of the mixture. The ordered configuration corresponds to a structure in which the number of M2 neighbors M1 has is equal to the ratio of M2 and M1 in the mixture. In order to assess the stability of the mixture the relevant thermodynamic quantity is the heat of mixing (∆Hmix ) which for a TMD mixture is defined as: ∆Hmix = EAy B1−y X2 − yEAX2 − (1 − y)EBX2 ,

(1)

where EAy B1−y X2 is the total energy of the alloy and yEAX2 , EBX2 are the total energies of the constituents AX2 and BX2 , respectively. The supercells used to simulate the mixtures contain 12 atoms with 4 and 8 metal and chalcogen atoms, respectively. Hence, y in the Eq. (1) can take values 0 (corresponding to the pristine BX2 ), 0.25, 0.5, 0.75 and 1 (corresponding to the pristine AX2 ). The calculated ∆Hmix are shown in the Figure 2. Clearly, most of the compounds have an enthalpy of mixing close to zero. However, the entropic effects, which are not included here, will also play a role for the stability of the mixtures at finite temperatures. Thus the enthalpies of mixing reported here only represent an upper bound on the mixing free energy. The uncertainties of the calculated mixing energies due to the approximate xc-functional can be directly evaluated with mBEEF. 26 mBEEF provides not only a single optimal functional but an ensemble of functionals. 30 The mixing energy for a particular system is then calculated using all the functionals in the ensemble and the spread of the calculated values provide an estimate of the error bar (vertical bars in Figure 2). Typically, larger uncertainties signal larger differences in the chemical environment (or bonding) in the mixture and the constituent systems. The figure clearly shows that all but the Cr containing mixtures have small uncertainties in the heat of mixing which is a signature of the Cr containing mixtures having significantly different charge density (or the chemical environment) than the constituent TMDs. As a result, the electronic structure of the chromium mixtures are expected to behave differently than the other mixtures.

5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

As mentioned earlier, the semi-local xc-functionals like the PBE significantly underestimate band gaps. The gold standard for band structure calculations of inorganic solids is the GW approximation. 31 For 2D materials, the GW method also seems to perform well, although experimental benchmark results are scarce; however, the computational cost associated with evaluating the GW self-energy is significant and for that reason we resort to the HSE06 xc-functional which generally yields decent agreement with experiments and GW. To assess the quality of the HSE06 band gaps for the monolayer TMDs, we compare to non-selfconsistent G0 W0 results for the pristine compounds, see Figure 3. For details on the G0 W0 calculations we refer to Ref. 7. As expected the PBE greatly underestimates the quasiparticle band gaps with a mean average error relative to G0 W0 of 0.92 eV. The HSE06 also underestimates the gaps compared to G0 W0 , but the mean average error is reduced to 0.27 eV. Based on this, we find it reasonable to rely on the HSE06 for calculating the band gaps and band edge positions of the alloys in the following. We stress that the GW band gaps used as references in Figure 3 represent the quasiparticle gaps as opposed to optical gaps. In particular, they do not include excitonic effects which can be rather large in two-dimensional semiconductors and thus we expect the optical gaps to be 0.2-0.5 eV lower than the quasiparticle gaps. 7 Figure 4 & Figure 5 show the position of the band edges of the pristine systems and the mixtures. The levels are aligned with respect to the vacuum by taking the asymptotic value of the Hartree potential in the vacuum region. The average (or predicted) value of the band edges is calculated by taking the weighted average of the band edges obtained for the pristine systems. The comparison of the calculated and predicted values of the levels is shown with the green and red lines, respectively. In most of the cases the predicted values are close to the calculated ones, even though the effect of alloying is rather small. The latter is due to the fact that within a specific group of metals, the electronic structure is mainly determined by the chalcogen atom. An exception from this rule occurs for the chromium containing mixtures. In this case, the predicted values are overestimated as compared to the calculated

6


Page 6 of 18

Page 7 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


values, however, even in this case the predicted trend in the variation of the level alignment is in agreement with the calculated values. One of the main reasons for chromium containing mixtures to behave differently than the others is the notably different lattice constant of the chromium dichalogenides as compared to the dichalcogenides of Mo and W (see Ref. 32 for the lattice constants). On the other hand, the lattice constants of the other TMDs of a particular group are mainly decided by the type of the chalcogen atom, for example, MoS2 and WS2 have similar lattice constants, TiS2 , ZrS2 and HfS2 which belong to group 4 have similar lattice constants as well and the same trend follows for the other chalcogen atoms (Se and Te). Therefore, the different extent of hybridization and strain effects arising from the different lattice constants of the chromium dichalcogenides might play an import role in deviating from the linear variation in the position of the band edges as observed in the other alloys. Another observation from Figure 4 & Figure 5 is that the positions of the band edges of the group-IV (Ti, Zr, Hf) dichalcogenides remain almost constant throughout the concentrations including the end points (the pristine systems). The similarity of the systems is also manifested in the heats of the mixing which are close to zero with very small uncertainties. The variation of band edges in the W/Mo alloy follows the weighted average very closely, see Figure 4(f), (l) and (r). Although the effect of alloying is small, it is possible to tune the band edge positions within 0.2 eV by varying the Mo/W concentration. As already mentioned the Cr containing alloys show the most non-trivial behaviour. As can be seen in Figure 4 the band edge positions change significantly when Mo or W is substituted for Cr with the CBM varying more slowly than the VBM and a tunability of the VBM as high as ∼0.5 eV can be achived via alloying. This indicates a large degree of interaction between the Cr and Mo d-orbitals which is also manifested in the relatively large uncertainty in the calculated enthalpy of mixing for this alloy, see Figure 2. Figure 6 shows the predicted (weighted average) and calculated band gaps of all the TMD mixtures. The trend in the predicted/calculated deviation follows that of the band

7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

edge positions. As in the case of level alignment, the largest difference between the calculated and the average values occurs for chromium containing mixtures. The trend in the variation in the band gap of the group-IV and Mo/W mixtures originates from the nearly aligned VBM and CBM (in group-IV) or nearly linear variation in the band edge positions (Mo/W mixtures). As expected, the predicted and the calculated values of the band gap differ significantly in the chromium containing mixtures because of ‘mis-alignment’ of the vacuum levels.

Conclusions We explored the effect on the electronic properties of alloying the metal atoms of the group IV and VI monolayer transition metal dichalcogenides (TMDs). The stability of the alloys are assessed via the heat of mixing along with the uncertainties calculated with the Bayesian error estimation functional. The trends in thermodynamic stability and electronic properties of the ordered alloys in our work provides a guiding principle to study random alloys as well which are futher stabilized due to entropic effects. One of our main findings is that the weighted average of band gap/band edge of the pristine systems is very close to the calculated values of all but chromium containing mixtures. The behavior of the chromium as an outlier is understood in terms of a different kind of level alignment as compared to the other class of materials. Therefore, the heurestics based on the level alignment can be further used for other class of 2D materials to engineering their electronic properties.

Acknowledgement The authors gratefully acknowledge the financial support from the Center for Nanostructured Graphene (Project No. DNRF103) financed by the Danish National Research Foundation.

8


Page 8 of 18

Page 9 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References (1) Mak, K. F.; Lee, C.; Hone, J.; Shan, J.; Heinz, T. F. Atomically Thin MoS2 : A New Direct-Gap Semiconductor. Phys. Rev. Lett. 2010, 105, 136805. (2) Splendiani, A.; Sun, L.; Zhang, Y.; Li, T.; Kim, J.; Chim, C.-Y.; Galli, G.; Wang, F. Emerging Photoluminescence in Monolayer MoS2 . Nano Lett. 2010, 10, 1271–1275. (3) Zhang, C.; Johnson, A.; Hsu, C.-L.; Li, L.-J.; Shih, C.-K. Direct Imaging of Band Profile in Single Layer MoS2 on Graphite: Quasiparticle Energy Gap, Metallic Edge States, and Edge Band Bending. Nano Lett. 2014, 14, 2443–2447. (4) Andersen, K.; Latini, S.; Thygesen, K. S. Dielectric Genome of van der Waals Heterostructures. Nano Lett. 2015, 15, 4616–4621. (5) Li, H.; Tsai, C.; Koh, A. L.; Cai, L.; Contryman, A. W.; Fragapane, A. H.; Zhao, J.; Han, H. S.; Manoharan, H. C.; A.-Pedersen, F.; Nørskov, J. K.; Zheng, X. Activating and optimizing MoS2 basal planes for hydrogen evolution through the formation of strained sulphur vacancies. Nat. Mater. 2016, 15, 48–53. (6) Tsai, C.; A.-Pedersen, F.; Nørskov, J. K. Tuning the MoS2 Edge-Site Activity for Hydrogen Evolution via Support Interactions. Nano Lett. 2014, 14, 1381–1387. (7) Rasmussen, F. A.; Thygesen, K. S. Computational 2D Materials Database: Electronic Structure of Transition-Metal Dichalcogenides and Oxides. J. Phys. Chem. C 2015, 119, 13169–13183. (8) Conley, H. J.; Wang, B.; Ziegler, J. I.; Haglund, R. F.; Pantelides, S. T.; Bolotin, K. I. Bandgap Engineering of Strained Monolayer and Bilayer MoS2 . Nano Lett. 2013, 13, 3626–3630. (9) Voiry, D.; Yamaguchi, H.; Li, J.; Silva, R.; Alves, D. C. B.; Fujita, T.; Chen, M.; Asefa, T.; Shenoy, V. B.; Eda, G.; , M. C. Enhanced Catalytic Activity in Strained 9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Chemically Exfoliated WS2 Nanosheets for Hydrogen Evolution. Nat. Mater. 2013, 12, 850–855. (10) Voiry, D.; Salehi, M.; Silva, R.; Fujita, T.; Chen, M.; Asefa, T.; Shenoy, V. B.; Eda, G.; Chhowalla, M. Conducting MoS2 Nanosheets as Catalysts for Hydrogen Evolution Reaction. Nano Lett. 2013, 13, 6222–6227. (11) Rossnage, K. On the Origin of Charge-Density Waves in Select Layered TransitionMetal Dichalcogenides. J. Phys.: Condens. Matter 2011, 23, 213001. (12) Dolui, K.; Sanvito, S. Dimensionality Driven Charge Density Wave Instability in TiS2 . arXiv preprint arXiv:1310.1866 2013, (13) Qian, X.; Liu, J.; Fu, L.; Li, J. Quantum Spin Hall Effect in Two-dimensional Transition Metal Dichalcogenides. Science 2014, 346, 1344–1347. (14) Pandey, M.; Vojvodic, A.; Thygesen, K. S.; Jacobsen, K. W. Two-Dimensional Metal Dichalcogenides and Oxides for Hydrogen Evolution: A Computational Screening Approach. J. Phys. Chem. Lett. 2015, 6, 1577–1585. (15) Bernardi, M.; Palummo, M.; Grossman, J. C. Extraordinary Sunlight Absorption and One Nanometer Thick Photovoltaics Using Two-Dimensional Monolayer Materials. Nano Lett. 2013, 13, 3664–3670. (16) Amani, M.; Lien, D.-H.; Kiriya, D.; Xiao, J.; Azcatl, A.; Noh, J.; Madhvapathy, S. R.; Addou, R.; KC, S.; Dubey, M.; Cho, K.; et al., Near-unity Photoluminescence Quantum Yield in MoS2 . Science 2015, 350, 1065–1068. (17) Radisavljevic, B.; Radenovic, A.; Brivio, J.; Giacometti, V.; Kis, A. Single-layer MoS2 Transistors. Nat. Nanotechnol. 2011, 6, 147–150. (18) Xie, L. M. Two-dimensional Transition Metal Dichalcogenide Alloys: Preparation, Characterization and Applications. Nanoscale 2015, 7, 18392–18401. 10


Page 10 of 18

Page 11 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(19) Su, S.-H.; Hsu, W.-T.; Hsu, C.-L.; Chen, C.-H.; Chiu, M.-H.; Lin, Y.-C.; Chang, W.H.; Suenaga, K.; He, J.-H.; Li, L.-J. Controllable Synthesis of Band Gap-Tunable and Monolayer Transition Metal Dichalcogenide Alloys. Front. Energy Res. 2014, 2 . (20) Tan, T. L.; Ng, M.-F.; Eda, G. Stable Monolayer Transition Metal Dichalcogenide Ordered Alloys with Tunable Electronic Properties. J. Phys. Chem. C 2016, 120, 2501–2508. (21) Komsa, H.-P.; Krasheninnikov, A. V. Two-Dimensional Transition Metal Dichalcogenide Alloys: Stability and Electronic Properties. J. Phys. Chem. Lett. 2012, 3, 3652– 3656. (22) Huang, B.; Yoon, M.; Sumpter, B. G.; Wei, S.-H.; Liu, F. Alloy Engineering of Defect Properties in Semiconductors: Suppression of Deep Levels in Transition-Metal Dichalcogenides. Phys. Rev. Lett. 2015, 115, 126806. (23) Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmentedwave Method. Phys. Rev. B 1999, 59, 1758–1775. (24) Enkovaara, J.; Rostgaard, C.; Mortensen, J. J.; Chen, J.; Dułak, M.; Ferrighi, L.; Gavnholt, J.; Glinsvad, C.; Haikola, V.; Hansen, H. A.; et al, Electronic Structure Calculations with GPAW: a Real-space Implementation of the Projector Augmentedwave Method. J. Phys.: Condens. Matter 2010, 22, 253202. (25) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. (26) Wellendorff, J.; Lundgaard, K. T.; Jacobsen, K. W.; Bligaard, T. mBEEF: An Accurate Semi-local Bayesian Error Estimation Density Functional. J. Chem. Phys. 2014, 140, 144107.

11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(27) Pandey, M.; Jacobsen, K. W. Heats of Formation of Solids with Error Estimation: The mBEEF Functional with and without Fitted Reference Energies. Phys. Rev. B 2015, 91, 235201. (28) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid Functionals Based on a Screened Coulomb Potential. J. Chem. Phys. 2003, 118, 8207–8215. (29) Paier, J.; Marsman, M.; Hummer, K.; Kresse, G.; Gerber, I. C.; Ángyán, J. G. Screened Hybrid Density Functionals Applied to Solids. J. Chem. Phys. 2006, 124, 154709. (30) Frederiksen, S. L.; Jacobsen, K. W.; Brown, K. S.; Sethna, J. P. Bayesian Ensemble Approach to Error Estimation of Interatomic Potentials. Phys. Rev. Lett. 2004, 93, 165501. (31) Hybertsen, M. S.; Louie, S. G. First-Principles Theory of Quasiparticles: Calculation of Band Gaps in Semiconductors and Insulators. Phys. Rev. Lett. 1985, 55, 1418–1421. (32) Computational Materials Repository. https://cmr.fysik.dtu.dk/, Accessed 10-May2016.

12


Page 12 of 18

Page 13 of 18


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment



Page 14 of 18

Page 15 of 18





Page 16 of 18

Page 17 of 18





Page 18 of 18

Atomically Thin Ordered Alloys of Transition Metal Dichalcogenides

Recommend Documents