Analysis of Strain Fields in Silicon Nanocrystals
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Analysis of Strain Fields in Silicon Nanocrystals
D¨undar E. Yılmaz ∗ and Ceyhun Bulutay † Department of Physics, Bilkent University, Ankara, 06800, Turkey
Tahir C¸ a˘gın ‡ Texas A & M University, Artie McFerrin Department of Chemical Engineering,Jack E. Brown Engineering Building, 3122 TAMU, College Station, TX 77843-3122, USA
Strain has a crucial effect on the optical and electronic properties of nanostructures. We calculatethe atomistic strain distribution in silicon nanocrystals up to a diameter of 3.2 nm embedded inan amorphous silicon dioxide matrix. A seemingly conflicting picture arises when the strain fieldis expressed in terms of bond lengths versus volumetric strain. The strain profile in either caseshows uniform behavior in the core, however it becomes nonuniform within 2-3 ˚A distance to thenanocrystal surface: tensile for bond lengths whereas compressive for volumetric strain. We reconsiletheir coexistence by an atomistic strain analysis.
The low dimensional forms of silicon embedded insilica have strong potential as an optical material.[1]Such heterogeneous structures inherently introduce thestrain as a degree of freedom for optimizing their opto-electronic properties. It was realized earlier that straincan be utilized to improve the carrier mobility in bulksilicon based structures.[2] This trend has been rapidlytranscribed to lower dimesional structures, starting withtwo-dimensional silicon structures.[3] Recently for siliconnanowires, there have been a number of attempts to tailortheir optical properties through manipulating strain.[4, 5]Futhermore, recent studies have revealed that the straincan become the major factor restricting the crystalliza-tion of the nanolayers.[6, 7] It depends on several factors,most important of which are the lattice mismatch be-tween the constituents, size of the NCs, and the growthconditions, such as the details of the growth procedure.[8]In summary, for improving the optical and electronicproperties of nanocrystals (NCs), the strain engineeringhas become an effective tool to be exploited.[9, 10, 11] Acritical challenge in this regard is to analyze the strainstate of the Si NCs embedded in silica.The close relations between strain and optical or elec-tronic properties in Si NCs have very recently becomethe center of attention.[10, 12, 13] There still remainsmuch to be done in order to understand strain in nanos-tructures at the atomistic level. As pioneered by Tsu etal.
Raman spectroscopy can be an effective experimentaltool for determining the strain state of the Si NCs.[14]Specifically, recent Raman studies reported that the SiNCs may be under a thermal residual strain and this canbe reduced by overall annealing at high temperatures[8]or by local laser annealing.[15] Due to small density dif-ference between Si NC and the surrounding a-SiO , alimited information can be gathered about its structureusing transmission electron microscopy (TEM) or even ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] high resolution TEM techniques.[16] Especially, molecu-lar dynamics simulations with realistic interaction poten-tials present an opportunity, by providing more detailedcritical information then the best imaging techniques cur-rently available and clarify the analysis of experimentalresults. Along this direction, previously [17] we focusedon Si-Si bond length distribution and reported that Si-Sibond lengths are stretched upto 3% just below the sur-face of Si NCs embedded in amorphous SiO which hasalso been very recently confirmed.[18]In this Letter, we analyze the volumetric and bondlength strain distributions in Si NCs, in particulardemonstrate that both compressive volumetric strain andtensile bond length strain coexist within the same SiNC. We accomplish this by performing trajectory anal-ysis on model samples (with ca. 5000 atoms) simulatedvia molecular dynamics using a reliable and accurate aswell as reactive force field.[19] The simulation details aresimilar to our previous work,[17] except the way we con-struct the Si NC in glass matrix. Instead of deletingall glass atoms within a predetermined radius, we re-move the glass atoms after rigorously defining the sur-face of the nanocrystal through the Delaunay triangula-tion method.[20] In this way, we have constructed NCsembedded in glass matrix with diameters ranging from2.2 nm to 3.2 nm without introducing built-in strain tothe system. In this diameter range we observe similartrends in strain, volumetric strain, and bond length dis-tribution etc., therefore, we present only the figures ofthe system for a typical NC of radius 2.6 nm.In the language of geometry, strain is defined throughan affine transformation that maps the undeformed stateto deformed state, which is called deformation gradient.Several methods to derive discrete form of deformationgradient from atomic positions are reported.[21, 22, 23]In the method proposed by Pryor et al. , the atomisticstrain tensor is derived from local transformation ma-trix that transforms nearest neighbors of a certain atomfrom its undeformed state to the deformed one. From theMD simulations, using positions of NC atoms, we firstextract each atoms displacement vector from its unde- S t r a i n ( % ) Distance to the Surface (Å)
FIG. 1: (Color online) Variation of, Si-Si bond lengths(squares), hydrostatic strain (diamonds), and the volumet-ric strain (triangles) as a function of distance to nanocrystalsurface (see text). Dashed, dotted and solid lines are guidesto the eye for the respective data set. The data for 2.6 nm di-ameter NC is used. Inset: Other NC diameters ranging from2.2 nm to 3.2 nm are also shown. formed site which is determined by positioning an idealtetrahedron to the local environment. Using these dis-placement vectors, we construct deformation matrix andderive the atomistic strain tensor from this local defor-mation tensor.[23] The first invariant of strain tensor cor-responds to the hydrostatic strain.[24] As an alternativemeasure to hydrostatic strain, we calculate volumetricstrain by considering volume change of a tetrahedronfrom its undeformed counterpart. A third measure as wehave used in our previous report,[17] is the bond lengthstrains.To verify our results we have calculated strain distri-bution in NC region for all mentioned measures. We haveplotted all three of them in Fig. 1. The results of volu-metric strain are very close to hydrostatic strain whichis the trace of strain tensor calculated with aferomen-tioned technique.[23] In these results, we observe a netcompressive behavior of strain just under the surface anda uniform tensile strain of about 1% at the core of NC.Si-Si bonds are stretched by about 1% in the core regionin agreement with the hydrostatic and volumetric strains,however, just under the surface, Si-Si bonds are stretchedup to 3% where hydrostatic and volumetric strain resultsindicate compressive strain state. The bond-stretch in Si-Si bonds due to oxidation has been shown earlier by ususing molecular dynamics simulations[17] which was alsoconfirmed by other approaches.[18] Occurrence of com-pressive volumetric strain and stretched bond lengths inthe same outer region may initially seem contradicting.However, stretching of bonds does not imply that thesystem is under tensile hydrostatic strain as well. Con-sider a tetrahedron formed by a Si atom and its fourSi neighbors (A, B, C, D) as shown in upper inset ofFig. 2. In the ideal case, the solid angle (Ω) subtended by
FIG. 2: (Color online) Dependence of solid angle subtendedby tetrahedron face to the angle between tetrahedron faceand nanocrystal surface. Illustration of solid angle subtendedby tetrahedron face (top left inset) and the angle betweentetrahedron face and NC surface (bottom right inset). each triangular face of this tetrahedron should be equalto 180 ◦ . Under a uniform deformation, bond lengthswill also be stretched, while the solid angles remain un-changed. However, under a nonuniform deformation, thechange in three solid angles causes a decrease in the vol-ume of the tetrahedron while increasing or preserving thebond lengths. Hence, a combination of stretched bondlengths with deformed solid angles may end up with anoverall reduction of the volume of the tetrahedron. Thisexplains the coexistance of compressive volumetric strainand stretched bond lengths at the region just below thesurface of NCs.To better visualize the nature of the deformation ofthe Si NCs, we consider the orientational variation of thesolid angles of the tetrahedral planes. As illustrated inthe lower inset of Fig. 2, the two important directions arethe unit normal (ˆ n S ) of the tetrahedron face subtendingthe solid angle under consideration, and the local out-ward surface normal (ˆ n NC ) of the NC. It is clearly seenfrom Fig. 2 that solid angles subtended by tetrahedrafaces oriented outward to the NC surface are increasedup to 220 ◦ , whereas those facing inward to the NC coreare decreased down to 160 ◦ . This dependence is a clearevidence of how oxidation affects strain distribution closeto the interface.To further quantify the atomistic strain in the highlycritical region within 3 ˚A distance to the interface, weclassify the average bond length and hydrostatic strainbehaviors into three categories. Figure 3 displays thepercentage as well as the bonding details of each cate-gory. In top-left, we illustrate most common type with ashare of 53.0% which is responsible for the opposite be-havior in Fig. 1 where average bond lengths of center Siatoms to its four nearest neighbors are stretched but netatomistic strain at this atom is compressive. In this case FIG. 3: (Color online) Illustrations of oxidation effects onstrain in three categories with their percentage of occurrences:Si-Si bonds are stretched and system is under compressivestrain (upper left). Si-Si bonds are stretched and system isunder tensile strain (upper right). Si-Si bonds are shortenedand the Si atom at the center is under compressive strain (bot-tom). Large spheres (gold) and small spheres (red) representsSi and O atoms, respectively. solid angles facing toward the oxide region is increasedto 270 ◦ due to oxygen bonds of Si neighbors. Althoughthese oxygen bonds stretched Si-Si bonds to 2 .
41 ˚A, netstrain on center Si atom is -2.7%. In the top-right partof the Fig. 3 we illustrate second most often case witha percentage of 42.0%, where average bond lengths andatomistic strain show similar behavior; bond lengths are stretched and net hydrostatic strain is tensile. In thiscase oxidation somewhat uniformly deforms the bondsso that solid angles are still around 180 ◦ which is thevalue for the unstrained case. Finally, as shown at bot-tom of Fig. 3, a very small percentage of atoms (5.0%)in the region beneath the surface have shortened bondlengths and compressive atomistic strain.In summary, we study the strain state of Si NCs in sil-ica matrix with diameters in 2 to 3.2 nm. The structure isassumed to be free from any thermal built-in strains. Thecore region of the NC is observed to be under a uniform1% tensile strain, where both bond length and volumet-ric strain measures are in agreement. However, towardsthe NC interface, while the Si-Si bonds become morestretched, the hydrostatic strain changes in the compres-sive direction. In the interpretation of the indirect strainmeasurements eg. from spectroscopy, this dual characterneeds to be taken into consideration. We explain thesetwo behaviors using the solid angle deformation of thetetrahedral-bonded Si atoms, and demonstrate that it isultimately caused by the oxygen atoms at the interface.An equally important finding is that the overall strainprofile within the Si NCs is quite nonuniform. As veryrecently emphasized, within the context of centrosym-metric materials, like silicon, such strain gradients locallybreak the inversion symmetry and may lead to profoundphysical consequences.[25]This work has been supported by the Turkish Scien-tific and Technical Council T ¨UB˙ITAK with the ProjectNo. 106T048 and by the European FP6 Project SEM-INANO with the Contract No. NMP4 CT2004 505285.The visit of Tahir C¸ a˘gın to Bilkent University was facili-tated by the T ¨UB˙ITAK B˙IDEB-2221 program. TC alsoacknowledges the support of NSF-IGERT. [1] V. Kumar (Editor), Nano Silicon , (Elsevier, London,2007).[2] H. M. Manasevit, I. S. Gergis, and A. B. Jones, Appl.Phys. Lett. , 464 (1982).[3] R. People, J. C. Bean, D. V. Lang, A. M. Sergent, H. L.Stormer, K. W. Wecht, R. T. Lynch, and K. Baldwin,Appl. Phys. Lett. , 1231 (1984).[4] D. M. Lyons, K. M. Ryan, M. A. Morris, and J. D.Holmes, Nano Lett. , 811 (2002).[5] K.-H. Hong, J. Kim, S.-H. Lee, and J. K. Shin, NanoLett. , 1335 (2008).[6] M. Zacharias and P. Streitenberger, Phys. Rev. B. ,8391 (2000).[7] S. Hernandez, A. Martinez, P. Parinello, Y. Lebour, B.Garrido, E. Jordana, and J. M. Fedeli, J. Appl. Phys. ,2614 (1999).[9] A. Thean and J. P. Leburton, Appl. Phys. Lett. , 1030(2001). [10] X. H. Peng, S. Ganti, A. Alizadeh, P. Sharma, S. K. Ku-mar, and S. K. Nayak, Phys. Rev. B , 035339 (2006).[11] A. M. Smith, A. M. Mohs, and S. Nie, Nature Nanotech-nol.
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