The Effect of Strained Bonds on the Electronic Structure of Amorphous Silicon
Reza Vatan Meidanshahi, Payam Mehr, Stephan Marshal Goodnick
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec The Effect of Strained Bonds on the Electronic Structure of Amorphous Silicon
Reza Vatan Meidanshahi, ∗ Payam Mehr, and Stephen M. Goodnick
School of Electrical, Computer and Energy Engineering, Tempe, AZ, 85281, USA (Dated: December 23, 2020)Several amorphous silicon structures were generated using a classical molecular dynamics (MD)protocol of melting and quenching with different quenching rates. An analysis of the calculatedelectronic properties of these structures revealed that the midgap state density of a-Si which is ofinterest for solar cell and thin film transistor applications can be correlated to bond angle standarddeviation. We also found that this parameter can strongly determine the excess energy of a-Si,which is an important criteria in theoretically generating realistic atomic structures of a-Si.
PACS numbers: Valid PACS appear here
I. INTRODUCTION
Amorphous silicon (a-Si) has been extensively inves-tigated as the archetypal amorphous covalently bondedmaterial and is widely used in numerous electronic andphotovoltaic devices. These features have spurred an in-tense interest in their electronic properties over the lastfew decades. a-Si can be fabricated using different meth-ods like laser melting, ion implantation, and growth tech-niques [1–3]. The atomic structure of the a-Si obtainedusing different fabrication methods to a large extent de-pends on the preparation technique. As an example,ion implantation often gives samples with large danglingbonds (DBs) and vapor deposition results in samples withvoids. These differences in atomic structure as a resultof the fabrication method, strongly affects the electronicproperties of a-Si. Though it has been extensively inves-tigated, the exact nature of the relation, between atomicstructure and electronic properties, has been a subjectof intense contention. In addition to gaining a more ac-curate understanding of the basic characteristics of co-valently bonded amorphous materials, a better under-standing of this relation would be helpful for technologi-cal progress.One of the main electronic features which makes a-Sidifferent from its crystalline counterpart is the presenceof electronic states in its energy gap which are calledmidgap states. The density of midgap states can controlmany electronic properties of the material by affectingtrapping and recombination processes and consequentlyaffect device functionality [4]. The common belief is thatmidgap states are generated by coordination defects orDBs. However, there is ample evidence that other lessunstable structural defects might also contribute to thesestates [5–12]. Calculations of defects in a-Si with or with-out hydrogen, were mostly involved in understanding itsphysical behavior. Nevertheless, significant recent devel-opments in the theoretical analyses of the defect-midgapstate relation took place were made by investigating theelectronic structure of several configurations of a-Si or a- ∗ Electronic address: [email protected]
Si:H. Wagner et al. [8] created different amorphous struc-tures by applying Wooten-Wiener-Weaire process withthe Keating potential and studied their electronic proper-ties within a density-functional theory (DFT) approach.They observed that configurations with high strainedbonds (SBs), as the configuration with DBs, containhighly dense midgaps which are able to trap holes in a-Si.Khomyakov et al. [9] created a-Si:H model of 500 atomsby applying large-scale replica-exchange MD using DFT-derived classical potentials. Their bond-resolved den-sity of states (DOS) indicated that, contrary to commonbelief, domains with highly strained Si-Si bonds signifi-cantly contribute to midgap states density no less thanDBs. Using a first principle study of electron paramag-netic resonance (EPR), Pfanner et al. [10] showed thata strong indirect effect of network and strained bonds oncreating midgap states. In addition to a-Si, the relativeimportance of SBs and DBs in determining the midgapstate density is relevant for other amorphous materialsalso. However, except for the correlation between midgapstate and DBs, none of the previous investigations haveprovided a quantitative relation between midgap statedensity and other structural parameters.This paper presents quantitative insights on the rela-tion between the midgap state density and several struc-tural parameters. The specific aim of this paper istwofold: (i) to identify which of the structural defectsincluding dangling bonds (DBs), floating bonds (FBs),bond length average (BLA), bond length standard de-viation (BLSTD), bond angle average (BAA) and bondangle standard deviation (BASTD) are capable of accu-rately describing the nature of midgap state density (ii)to clarify the role of these structural defect on the excessenergy of a-Si, which has been recently proposed as animportant electronic property in the simulation of a-Si[13]. In our approach, the first step involves the gener-ation of 23 different large supercells of a-Si models with216 Si atoms. In this step, molecular dynamics simu-lation of melting and quenching process with differentcooling rate was applied to crystalline silicon (c-Si) su-percell while the inter-atomic interaction is described bythe Tersoff potential. In the second step, the structuresare optimized by carrying out DFT relaxation calcula-tions. In the third step, the integrated density of midgapstates, the excess energy of obtained structures are com-puted on the relaxed structures using DFT simulations.These quantities are then correlated to each of the afore-mentioned structural parameters. Finally, the calculatedresults are compared to previous computational investi-gations.
II. METHODA. Technical Details
We use a melting and quenching approach to generatea structural model of a-Si as the starting atomic struc-ture for the DFT calculations. The LAMMPS moleculardynamic code [14] is used for simulating the melting andquenching processes. In the MD simulations, the Tersoffinteratomic potential [15] was employed for describing Siatom interactions, with a cut-off radii of 2.7 ˚A (taper)and 3.0 ˚A (maximum); this potential has been widelyused for generating Si based structures [16–18]. Full ionrelaxation of the resulting structure from the MD simu-lation was performed at the DFT-level as implemented inthe Quantum Espresso 6.2.1 software package [19]. TheBFGS quasi-Newton algorithm, based on the trust radiusprocedure was used as the optimization algorithm to findthe relaxed structure. The structural analysis of the finala-Si structure was performed using the ISAACS software[20]. Both ionic relaxation and electronic structure cal-culations were performed using the Becke-Lee Yang-Parr(BLYP) exchange-correlation functional [21, 22]. Thecore and valence electron interactions were described bythe Norm-Conserving Pseudopotential function. Unlessotherwise stated, an energy cutoff of 12 Ry was selectedfor the plane-wave basis set. A 4 × × B. Generation of a-Si:H/c-Si Structures
Molecular dynamics simulations in conjunction withDFT calculations have been demonstrated to yield amor-phous material structures whose properties are commen-surate with experimental results [23, 24]. Therefore, weinitially carried out MD simulations to generate a generalform of the a-Si structure, and then relaxed the structureusing a DFT calculation to obtain experimentally com-patible structures. MD simulation of the melting andquenching process was carried out on a crystalline Sistructure in order to create an a-Si supercell containing216 Si atoms (a-Si216) with three dimensional (3D) pe-riodic boundary conditions. A diamond starting atomicstructure of crystalline Si with a lattice constant of a =5.46 ˚A was constructed using a cubic supercell with thedimension of a = b = c = 3a , which was periodically FIG. 1: The atomic structure of a typical simulated a-Si216supercell. repeated in 3-D space to generate an infinite network ofatoms. The value of a was chosen in such a way thatthe mass density of our supercell is equal to the massdensity of a-Si measured by experiments [25, 26]. Then,we simulated a 10 ps melting process at 3000 K in 0.1fs time steps, with a fixed volume and temperature en-semble (NVT).The structure was then quenched to 300K with different cooling rates ranging from 9 × to3 × K/s, and annealed for 25 ps at 300 K afterwards.Finally, the structure was optimized using a DFT relax-ation calculation.Figure 1 illustrates the atomic structures of a typicalsimulated a-Si supercell containing 216 Si atoms obtainedfrom MD simulation with a cooling rate of 10 andDFT relaxtion calculation. Based on the periodic struc-ture formed from the supercell in Figure 1, we found onedangling bond and one floating bond per supercell, withan assumed Si-Si bond length cutoff of 2.58 ˚A, which is10% longer than the experimental Si-Si bond length (2.35˚A). The structure mostly displays stable 5, 6, or 7 foldrings, and there are no large voids or holes in it. Theaverage Si-Si bond length is 2.354 ˚A with an rms value of0.049 ˚A. The average Si-Si-Si bond angle is 108.2 ◦ withan rms value of 13.7 ◦ . In the crystalline form of Si, theSi-Si bond length is 2.35 ˚A and the Si-Si-Si bond angleis 109.5 ◦ . III. RESULTSA. Midgap States Density
The crystalline form of bulk Si shows clear valenceband to conduction band energy gap without any midgapstates inside the gap. However, midgap states only existin the amorphous form of Si. The number of midgapstates is highly dependent on the atomic structure ofthe amorphous network. In order to identify the struc-
FIG. 2: Integrated midgap states vs different structural defects a) dangling bonds (DB) b) Floating Bonds (FB) c) bondlength average (BLA) d) bond length standard deviation (BLSTD) e) bond angle average (BAA) and f) bond angle standarddeviation (BASTD). tural parameter having the most significant effect on themidgap states, the total number of midgap states in theobtained a-Si supercells were calculated by integratingthe density of states of a-Si inside the energy range ofband-gap of Si in its perfect crystal form. By performingDFT calculations on perfect crystal form of Si, we foundthat the energy range of Si band gap in crystalline formis 6.6-7.5 eV.Fig. 2 illustrates the integrated density of states vari-ations versus different structural parameters. A linear fitwas applied to all the graphs and the obtained regressioncoefficients were considered as the criteria for determin- ing the accuracy of the fitted and the obtained relations.As Figure 1 denotes, it is obvious that the greatest re-gression is obtained for the relation of integrated midgapstates and bond angle standard deviation. The obtainedrelation is as follow. I = 3 . σ BAST D − .
86 (1)Where I and σ BAST D stand for integrated midgapstates and bond angle standard deviation, respectively.According this relation, the integrated midgap statesmonotonically increases as the bond angle standard devi-ation increases. Since any bond angle deviation from theideal value (109.45 ◦ ) cause bond strain, the bond anglestandard deviation is an estimation of stored strain inan amorphous structure. Therefore, the obtained depen-dency of midgap state density and bond angle standarddeviation indicates that strained bonds can significantlycause midgap states even more than dangling bond. Thisfinding is in contrast with the common belief that themidgap states are only due to dangling bonds, and isconsistent with the recent studies in the importance ofstrained bonds in creating midgap states [10, 11].The low regression values for bond length average andbond length standard deviation denote that none of thesequantities are good descriptors for midgap state densityof an amorphous structure. As seen from the graphs,bond length average and bond length standard deviationchanges is negligble from one structure to another is neg-ligible and therefore the low sensitivity of midgap statesdensity to these quantities is not unexpected. In addi-tion, the obtained regression indicate bond angle averageis not a proper quantity for describing the midgap statedensity of amorphous Si. The reason for this observationmight be due to missing the information about the nega-tive and positive bond angle deviation by canceling themeach other when they are added up.Our observation regarding the strong dependency ofmidgap states density to bond angle standard devia-tion is reasonable from a chemical bonding perspective.Bond angle standard deviation contains all the informa-tion about any deviation, regardless its sign, from idealbond angle. An ideal bond angle of 109.45 correspondsto the perfect SP hybridization which cause the bond-ing and anti-bonding orbitals locate only in the valenceband and the conduction band sides, respectively, and noatomic orbital in the band gap. However, when a bondangle associated with a specific atom deviates from itsideal value, the hybridization of that Si atom transformfrom SP to SP n where n is an integer number. Whena bond angle is greater than 109.45 ◦ , then n would beless than 3 and consequently some midgap states appearclose to the conduction band edge, due to their S-like or-bital properties. In vice versa, when a bond angle is lessthan 109.45 ◦ , then n would be greater than 3 and conse-quently some midgap states appear close to the valenceband edge, due to their S-like orbital properties.The accuracy of the integrated midgap states depen-dency on the bond angle standard deviation was checkedby calculating the integrated midgap states and bondangle standard deviation of the low strained supercellsimulated by Pedersen et. al. [13]. The mentioned su-percell was taken from the reference [13] and then was op-timized using BLYP functional and finally its electronicdensity of states was computed. We found that the inte-grated midgap states and bond angle standard deviationare 11.49 and 10.920 respectively. If we calculate inte-grated midgap states through the equation 1 using theobtained bond angle standard deviation value, an inte-grated midgap states of 13.72 will be resulted which is close to the integrated midgap states obtained from DFTcalculations (14.15). B. Excess Energy
To further progress in generating an optimal a-Simodel, Pederson et al. have recently proposed to focuson the excess energy of the amorphous structure relativeto the crystal, a quantity which can be measured fromcalorimetry experiments [13]. This parameter has beengetting considered as a critical property in simulation ofamorphous structure. Therefore, we also researched theeffect of different structural defects on the excess energystored in an amorphous network.In order to calculate the excess energy, we initially car-ried out a DFT relaxation calculation on a crystalline sil-icon (c-Si) supercell comprised of 216 Si atoms with thesame size as the a-Si supercells. Then, we subtracted theenergy of a-Si supercell from the energy of c-Si and finallythe energy difference is considered as the excess energystored in the a-Si model. Figure 3 shows the computedexcess energies versus different structural defects for thesimulated a-Si supercells. As in the midgap states den-sity calculations, a linear regression method is used to fitthe resulting points to a line and the regression coeffi-cient is considered as the criteria for the accuracy of thelinear relation. It can be concluded from Figure 3 thatthe best correlation is observed between excess energyand bond angle standard deviation with a regression co-efficient of 0.92. As seen from the figure, the obtainedequation for calculating excess energy using bond anglestandard deviation is as follow. E ex = 0 . σ BAST D − . ex and σ stand for excess energy and bondangle standard deviation, respectively. From equation, itis obvious that the excess energy linearly increases withbond angle standard deviation. The same ration as parta can be used for explaining the strong correlation be-tween E ex and σ . As mentioned before, Si atoms withbond angle deviation from ideal value gets hybridizedwith SP n instead of SP . The SP n hybridized orbitals cannot create strong equal number of chemical bonds due totheir weak overlap compared to SP orbitals. Hence, re-gardless of the number of the dangling bonds which isusually considered as the criteria for estimating the sta-bility of a-Si model, the bond angle standard deviationis more important in determining the stability of amor-phous network. This finding also shows that the bondangle standard deviation is the more fundamental prop-erty of an amorphous network than excess energy thatcan describe the quality of generated a-Si models. Sameas part A, we checked the accuracy of our obtained rela-tion using the optimal a-Si model generated by Pedersenet. al. [13]. The optimal a-Si structure was taken fromthe reference and the excess energy was computed after FIG. 3: Excess energy vs different structural defects a) dangling bonds (DB) b) Floating Bonds (FB) c) bond length average(BLA) d) bond length standard deviation (BLSTD) e) bond angle average (BAA) and f) bond angle standard deviation(BASTD). optimizing the structure by BLYP function. Calculatingthe excess energy by plugging bond angle standard devi-ation value to equation 2 results in 0.22 eV/atom, whichpresents a 13.6% error with respect to the 0.19 eV/atomresulted from the DFT simulations.
IV. CONCLUSION
Molecular dynamic simulations and DFT relaxationcalculations of various a-Si supercells comprised of 216 Siatoms revealed the strong dependency of both integrated density of midgap states and excess energy on the bondangle standard deviation. Consequently, the bond anglestandard deviation is more deterministic in the a-Si sta-bility evaluation than the conventional methods in whichthe number of the dangling bonds are being considered.
V. ACKNOWLEDGMENTS
This material is based upon work primarily supportedby the Engineering Research Center Program of the Na-tional Science Foundation and the Office of Energy Ef-ficiency and Renewable Energy of the Department ofEnergy under NSF Cooperative Agreement No.EEC-1041895. Any opinions, findings and conclusions or rec- ommendations expressed in this material are those of theauthor(s) and do not necessarily reflect those of the Na-tional Science Foundation or Department of Energy. [1] K. Laaziri, S. Kycia, S. Roorda, M. Chicoine, J. Robert-son, J. Wang, and S. Moss, Physical review letters ,3460 (1999).[2] F. Kail, J. Farjas, P. Roura, C. Secouard, O. Nos,J. Bertomeu, and P. R. i. Cabarrocas, physica status so-lidi (RRL)–Rapid Research Letters , 361 (2011).[3] Y. Fuentes-Edfuf, M. Garcia-Lechuga, D. Puerto, C. Flo-rian, A. Garcia-Leis, S. Sanchez-Cortes, J. Solis, andJ. Siegel, Applied Physics Letters , 211602 (2017).[4] P. Mehr, X. Zhang, W. Lepkowski, C. Li, and T. J.Thornton, Solid-State Electronics , 47 (2018).[5] T. Shimizu, Japanese journal of applied physics , 3257(2004).[6] T. Gotoh, S. Nonomura, M. Nishio, S. Nitta, M. Kondo,and A. Matsuda, Applied physics letters , 2978 (1998).[7] E. Stratakis, E. Spanakis, P. Tzanetakis, H. Fritzsche,S. Guha, and J. Yang, Applied physics letters , 1734(2002).[8] L. K. Wagner and J. C. Grossman, Physical review letters , 265501 (2008).[9] P. Khomyakov, W. Andreoni, N. Afify, and A. Curioni,Physical review letters , 255502 (2011).[10] G. Pfanner, C. Freysoldt, J. Neugebauer, F. Inam,D. Drabold, K. Jarolimek, and M. Zeman, Physical Re-view B , 125308 (2013).[11] R. V. Meidanshahi, S. Bowden, and S. M. Goodnick,Physical Chemistry Chemical Physics , 13248 (2019).[12] R. V. Meidanshahi, S. M. Goodnick, and D. Vasileska,arXiv preprint arXiv:2011.14158 (2020).[13] A. Pedersen, L. Pizzagalli, and H. J´onsson, New Journal of Physics , 063018 (2017).[14] S. Plimpton, Journal of computational physics , 1(1995).[15] J. Tersoff, Physical Review B , 5566 (1989).[16] T. Ohira, T. Inamuro, and T. Adachi, Solar energy ma-terials and solar cells , 565 (1994).[17] M. Nolan, M. Legesse, and G. Fagas, Physical ChemistryChemical Physics , 15173 (2012).[18] M. Legesse, M. Nolan, and G. Fagas, Journal of AppliedPhysics , 203711 (2014).[19] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-cioni, I. Dabo, et al., Journal of Physics: CondensedMatter , 395502 (2009).[20] S. Le Roux and V. Petkov, Journal of Applied Crystal-lography , 181 (2010).[21] A. D. Becke, The Journal of Chemical Physics , 5648(1993).[22] C. Lee, W. Yang, and R. G. Parr, Physical review B ,785 (1988).[23] M. Ishimaru, S. Munetoh, and T. Motooka, Physical Re-view B , 15133 (1997).[24] I. ˇStich, R. Car, and M. Parrinello, Physical Review B , 11092 (1991).[25] J. Custer, M. O. Thompson, D. Jacobson, J. Poate,S. Roorda, W. Sinke, and F. Spaepen, Applied physicsletters , 437 (1994).[26] A. Smets, W. Kessels, and M. Van de Sanden, Appliedphysics letters82