Structural, electronic and optical properties of tetrahedral S i x G e 47−x : H 60 nanocrystals: A Density Functional study
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Structural, electronic and optical properties of tetrahedral
S i x Ge − x : H nanocrystals: A Density Functionalstudy C. S. GaroufalisDepartment of Physics, University of Patras, GREECE, 26504, PatrasNovember 8, 2018
The structural, cohesive, electronic and optical properties of mixed SiGe:H quantum dots arestudied by Density Functional Theory (DFT) calculations on a representative ensemble ofmedium size nanoparticles of the form Si x Ge − x : H . The calculations have been performedin the framework of the hybrid non-local exchange-correlation functional of Becke, Lee, Parrand Yang (B3LYP). Besides the ground state DFT/B3LYP values we provide reliable resultfor the lowest spin and symmetry allowed electronic transition based on Time Dependent DFT(TDDFT/B3LYP) calculations. Our results show that the optical gap depends not only on therelative concentrations of silicon, germanium and hydrogen, but also on the relative positionof the silicon and germanium shells relative to the surface of the nanocrystal. This is also truefor the structural, cohesive and electronic properties allowing for possible electronic and opticalgap engineering. Moreover, it is found that for the cases of nanoparticles with pure Ge or Sicore, the optical properties are mainly determined by the Ge part of the nanoparticle, whilesilicon seem to act as a passivant. The possibility of tunable photoluminescence (PL) from silicon and silicon-like (e.g. germa-nium) quantum dots (and nanowires), has stimulated intensive research on this type of materialsover the last decade [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Until recently, silicon nanocrystalshave practically ”monopolized” the interest of the researchers. A large portion of this type ofwork has been devoted to understanding the visible photoluminescence of these materials andits dependence on the diameter of the nanoparticles. It is widely accepted and well establishedby now (see for instance refs. 12-15) that the luminescence of oxygen-free Si nanocrystals1of well defined diameter) is mainly due to quantum confinement (QC) of the correspondingnanoparticles. This is also true for Ge nanoparticles[7, 8]. It is known that the effect of quan-tum confinement is even more pronounced for the case of Ge nanoparticles. This can be easilyunderstood by comparing the electron and hole effective masses and dielectric constants of Siand Ge. In particular, the smaller electron and hole effective masses of Ge along with the largerdielectric constant (compared to Si) result in a larger exciton Bohr radius for Ge. Consequently,it might be expected that the effect of QC on the optical properties of Ge nanoparticles willbe more pronounced. The PL properties of such nanocrystals (Si or Ge) are mainly controlledby suitably regulating the size of the nanocrystals and in many cases, their surface passiva-tion. The possibility of combining the advantages of Si (in the electronic properties) with thoseof Ge (especially structural and mechanical properties) appears to be a natural extension ofscientific interest and an intriguing and potentially promising field for the development of op-toelectronic nanodevices. It has been demonstrated by both experimental observation[13] andtheoretical calculations[13, 14] that the lattice mismatch of Si and Ge has a significant effecton the electronic properties of Si − x Ge x alloys. The induced strain affects mainly the tail ofthe conduction band which results in an almost linear decrease of the indirect band gap. Inthis sense, it may be expected that a similar behavior may introduce interesting optical fea-tures in Si x Ge y : H z nanocrystals. Several of these issues have been recently addressed byMing Yu et. al.[20] in the framework of Density Functional Theory. In particular, they haveperformed DFT/LDA molecular dynamics calculations on medium size Si x Ge y and Si x Ge y : H ( x + y = 71) mixed nanoparticles. Especially for the case of hydrogenated nanocrystals theyfound that the dependence of the single particle HOMO-LUMO gap on the relative composi-tion of the clusters exhibits many similarities with the corresponding one of the bulk Si − x Ge x alloys. At this point it should be noted that the structure of the specific nanocrystals has beenfully relaxed through a molecular dynamics procedure with an initial temperature of 1000 K.As a result a large portion of the strain induced by the Si/Ge mismatch (in the initial geo-metric configurations) has been largely relieved. However, it would be interesting to expandthe investigation for the case of mixed Si/Ge nanocrystal which have not undergone such anannealing procedure. In this case, the aforementioned strain can not be fully relieved sincethe individual atoms are only allowed to a local relaxation around their original position (i.ethey are not allowed to diffuse through the shells). With this in mind, we have examined theoptical and electronic properties of mixed nanocrystals of the form Si x Ge − x : H . We havestudied in detail the variation of the cohesive, electronic and optical properties as a functionof x . Moreover, we have examined the dependence of these properties on the position of eachatomic species relative to the nanocrystal’s surface.2 Technical details of the calculations
All ground state calculations in this work are based on Density Functional Theory (DFT), whileall excited state calculations are based on TDDFT. In both cases we employed the nonlocalexchange-correlation functional of Becke, Lee, Yang and Parr (B3LYP) [18]. The accuracy ofthese calculations (TDDFT/B3LYP) for the optical gap has been tested before by by compar-ison with high level multireference second-order perturbation theory (MR-MP2) calculationsfor the case of Si nanocrystals[5]. The size of the Si x Ge − x : H nanocrystals consideredhere is approximately 10-12˚A. The symmetry of the nanocrystals is T d and their geometrieshave been fully optimized within this symmetry constrain using the hybrid B3LYP functional.To preserve the T d symmetry, we substituted shells of silicon (rather than isolated atoms) byequivalent germanium shells. This choice introduces an additional restriction on the variationof Si concentration. This procedure imposes some constrains in the relaxation of the inter-atomic forces. In particular, although bond lengths an angles are allowed to relax, the atomsare not allowed to change their relative position in the nanoparticle. As a result, migrationfrom the inner core to the surface (or vice versa) is not possible (such migrations were bothallowed an observed in the MD calculations of Ming Yu et al[20]). We have examined in de-tail most of the structural (bond length distribution), cohesive (binding energies), electronic(DOS, electronic gaps) and optical properties as a function of the concentration x . The opticalgap is defined as the energy of the lowest spin and symmetry allowed excitation calculatedby the TDDFT/B3LYP method. Moreover, for the same concentration x we have consideredalternative ways of substitution of the shells of silicon atoms by germanium. The bulk of ourcalculations were performed with the TURBOMOLE [15] suite of programs using Gaussianatomic orbital basis sets of split valence [SV(P)]: [4s3p1d]/[2s] quality [16]. Test calculationswith the larger TZVP basis set revealed only marginal deviations from the corresponding SV(P)results. Representative geometries of Si x Ge − x : H nanoclusters are shown in figure 1 for variousconcentrations and substitutions. The bonding characteristics of the various structures canbe easily visualized and described graphically in a synoptic way, through the bond-lengthdistributions, which is presented if figure 2. All graphs in figure 2 correspond to nanocrystalswith the silicon atoms concentrated in the inner core of the nanoparticle. As we can see theSi-Si distribution has a peak around 2.48 ˚A for the first shell of neighbors (connected to thecentral atom) and a second peak around the 2.37 ˚Afor the rest of the silicon atoms. This secondpeak, corresponding to shorter bond-lengths by 0.1 ˚A, is more or less constant, with a tendency3o approach the bulk value of 2.36 ˚Afor larger nanocrystals. This is also true Ge-Ge bonds.Comparing figures 2(a) and 2(b) we can see that in both cases the bonds of the central atomwith the first shell of neighbors are longer by 0.1 ˚A. We also observe in figs. 2(e) and 2(f)that there are no Ge-Ge bonds, although there is a significant amount of germanium atoms.Such bonding characteristics are found to be directly related to both the electronic and opticalproperties of the nanoparticles. As was explained earlier, with the same concentrations ( x )more than one nanocrystals can be constructed. Moreover, since the Ge substitutions in thepresent work deal with spherical shells of neighbors rather than with individual atoms, we candistinguish two classes of nanocrystals with similar concentrations; Those with the Ge atomsin the inner core, and those with the Ge atoms in the outer shells (”surface”). The structuraland cohesive characteristics are different in the two cases. As we can see in figure 3, we havetwo distinct curves depending on the exact location of the Ge layer relatively to the surfaceof the nanocrystals. It is clear from this plot that it is preferable to have the Ge atoms inthe ”inner” part of the nanocrystal. This tendency is directly related to the effect of surfacehydrogen atoms and it can be quantified by considering the binding energy of the independentSi-H( BE Si − H ) and Ge-H( BE Ge − H ) bonds. This can be approximated by the formulae BE Si − H = BE SiH , BE Ge − H = BE GeH BE SiH and BE GeH are the corresponding binding energies of the SiH and GeH molecules).In this way we can define the surface energy of the nanoparticle as SE = N Si − H · BE Si − H + N Ge − H · BE Ge − H (where N Si − H and N Ge − H are the number of Si-H and Ge-H bonds respectively). The depen-dence of surface energy on the composition of the nanocrystals and the position of Si and Geatoms relative to the surface is shown in figure 3b. It becomes evident from this figure thatthe stability of the hydrogenated clusters is largely determined by their surface. Almost 63%of the total binding energy of the nanocrystals is attributed to the surface Si-H/Ge-H bonds.As a result,the large differences is surface energy between Ge(core) and Si(core) nanoparticles(fig.3b) is responsible for the shape and energetic ordering of the total binding energies shownin fig. 3a. However, it should be noted that without the hydrogen passivation of the danglingbonds, it would be natural (energetically favored) for the Ge atoms to segregate onto the sur-face in order to minimize the cost of the dangling bonds. Indeed, as was stated by Tarus etal [19], for hydrogen-free SiGe nanoclusters, germanium tends to segregate onto the surface.The above conclusions, are in agreement with recent theoretical calculation (LDA) of Ramoset. al[21] . Moreover, the observed trends, are consistent with a series of experimental data(ree ref,21-26 of Ramos et. al[21]) 4 .2 Electronic and optical properties In figure 4 we have plotted the total and the projected density of states (DOS and PDOS) forfour typical nanocrystals. The DOS curves were generated from the eigenstates of the groundstate calculations with a suitable gaussian broadening[22]. The largest variation with the Geconcentration occurs in the valence band edges, while the conduction band edge is relativelyless sensitive. From these diagrams it can be seen that the hydrogen contribution lies deep inthe valence band (in the energy region between -10.5 eV to 9.0 eV) leaving Si and/or Ge todominate the character of the band edges. For The cases of Si Ge : H and Si Ge : H nanoparticles (Si in the core and Ge in the surface) this hydrogen related peak appears to beslightly broadened. This is probably due to the looser binding of the hydrogen atoms with thesurface Ge atoms (looser as compared with the corresponding Si-H binding). An interestingconclusion which can be drawn from the DOS diagrams is related to the character of theconduction band edges.It appears that when the core (Si or Ge) is adequately large (i.e. ∼ Si x Ge y : H z nanocrystals have alsobeen recently performed by Yu et al. [20], specifically for nanocrystals with a total numberof Si and Ge atoms of 71 (x+y=71). These calculations were based on density-functionaltheory (DFT) in the local-density approximation (LDA). The resulting HOMO-LUMO gapsrange from 3.3 - 4.1 eV corresponding to the pure Ge and pure Si nanocrystals. In order tocompare our calculation with the results of Yu et al.[20] we performed similar DFT/B3LYPcalculations for the pure Si H Ge H nanoparticles. Our values of 4.0 eV for the pureGe nanocrystal and 4.6 eV for the pure Si nanocrystal are in very good agreement with thevalues of Yu et al. if one considers the inherent tendency of LDA[5] to underestimate thesingle particle HOMO-LUMO gap by approximately 0.6-0.7 eV. A striking difference in thework of Yu et. al.[20] is that instead of shells of atomic neighbors, used in the present work(strained nanocrystals), the Ge atoms in ref [20] are distributed more homogeneously, and theyare allowed to diffuse through the shells (complete relaxation of strain). As a result the gapdependence on Ge concentration appears to be practically linear. As a means to provide a moreaccurate and detailed account of the optical properties of these nanocrystals we employed theTDDFT/B3LYP combination in order to calculated their optical gap (i.e. lowest symmetryand spin allowed electronic excitation). The results are shown in table 1.The first commentthat can be made by inspecting these values concerns the nature of the transitions.It is evidentthat for the nanoparticles in which the Ge atoms reside in the inner core, the lowest allowedtransition is always between the HOMO- and LUMO orbitals. Moreover these transition appearto have relatively larger oscillator strengths. On the contrary, when there is a silicon inner core,the oscillator strengths are smaller, while the nature of the transitions becomes more complex.5or example, we can see a non negligible degeneracy concerning the fundamental optical gaptogether with an increase on the multireference character of the transitions. In figure 5 we showa graphic representation of the variation of the optical gap as a function of the number of Siatoms ( x ) contained in the nanocrystal. Both types of nanoparticles(Si(core) and Ge(core)) areincluded. We can clearly distinguish two sets of points (disjoint curves) which correspond to thetwo different types of clusters (Si(core) and Ge(core)). An analogous variation is also observedfor the HOMO-LUMO gap (i.e. we have an upper and a lower curve). Surprisingly enough, thelarger optical (and HOMO-LUMO) gaps correspond to germanium atoms lying in the surfaceregion, which as we have seen in figure 3 is not energetically as stable as the opposite case.Usually, the most stable structures are the ones which exhibit the largest gap. However, thisrule of thumb seems not to be applicable in this case. It is interesting to point out that for thecase Ge(core) nanoparticles the gap decreases as the size of the core increases.This is a commonquantum confinement behavior (see for example ref [4, 5, 7]). As a result, it may be alleged thatthe Ge(core) Si x Ge y : H z behave as Ge nanoparticles which are passivated by a Si : H layer.Comparing the optical gap of Ge(core) Si Ge : H nanoparticle with the corresponding oneof Ge : H cluster we find that it is smaller by 2.47 eV ( the optical gaps are 3.73 eV for Si Ge : H and 6.2 eV for Si : H ). This large difference may originate from a less effectivepassivation of the Ge core by the Si passivants (less effective compared to Ge passivation by Hatoms). To check this hypothesis we followed a simple line of argument which goes as follows.By simple calculations on SiH , GeH , and H Si − GeH molecules we can find the bindingenergies for the the Ge − H , and Si − Ge bonds. In particular, we find (as expected) that BE Ge − H > BE Si − Ge . Next, we modify the Ge − H bond length in GeH molecule in order toequate the resulting BE ∗ Ge − H to BE Ge − Si . This is achieved when the Ge − H bond in GeH molecule is elongated to ∼ . Ge ∗ − H bond distance for the passivationof the Ge H cluster and calculated again its optical gap. The new value is now 3.7 eV andpractically coincides with the 3.73 eV of the Ge(core) Si Ge : H nanoparticle. This result,although it does not prove the aforementioned hypothesis, is highly suggestive of its validity.The only other Si x Ge − x : H nanoparticle with a Ge core fully capped (passivated) by Siatoms, suitable for extending the test of our hypothesis, is Si Ge : H (which should becompared to Ge H ). However, the Ge − H bond elongation to ∼ . Ge H clusterappears problematic since it brings the hydrogen passivants too close to each other inducingsignificant H − H interactions.On the other hand the variation of the optical gap for the case of Si(core) Si x Ge − x : H nanoparticles as a function of the Si core size (upper curve in fig.5) appears to be unexpected(as the Si core increases the gap also increases). This behavior suggest that the hypothesis ofquantum confinement is not applicable here. In other words, the specific behavior can not beexplained by considering that the Germanium atoms passivate the inner Si core. Surprisingly,an explanation can be obtained again by considering the Si atoms of the inner core to passivate“internally”the outer shell of Ge atoms (see fig.6). In order to test this we performed additional6alculation on modified versions of the Si Ge : H and Si Ge : H nanoparticles. Inparticular, we removed the inner Si atoms and we passivated the created internal Ge danglingbonds with hydrogen. The results indeed show an increase of the optical gap as we go from H Ge : H to H Ge : H . At this point it should be noted that the “internal”hydrogenpassivation is (again) more effective than the passivation by Si ( BE Ge − H > BE Si − Ge ). Asa result these calculation could only reproduce the trend of gap increase and not the actualvalues. Unfortunately, the Ge − H bond elongation to ∼ . We have shown that, indeed, the mixed SiGenanocrystals have optical and electronic properties intermediate between those of pure Si andGe nanocrystals. The large variety of optical and band gaps depends, not only on the size ofthe nanocrystals and the relative concentrations of Si and Ge, but also on the relative spatialdistribution of the Ge atoms with respect to the surface of the nanocrystals. The stability ofthe structures is largely define by the hydrogen surface passivation. As a result, the most stablenanoparticle are those with the silicon atoms on the surface (mostly due to the larger bindingenergy of the Si-H bonds). The optical properties of Si(core) and Ge(core) nanoparticles arefound to exhibit significant differences. For the Ge(core) nanocrystals the lowest spin andsymmetry transition are always between the HOMO and LUMO orbitals, while for the Si(core)ones both HOMO-1 and LUMO+1 contributions are important (in this case the transitionsexhibit a more pronounced multireference character). The variation of the optical gap as afunction of the core size (Si or Ge) depends drastically on the nature of the core (Si or Ge).However, for both cases the optical gap variation can be rationalized by considering that thesilicon atoms behave as simple passivants of the Ge cluster.These additional degrees of freedom with regard to the properties of mixed SiGe:H nanoparticlesmay be important in the future design of such (and similar) systems, allowing for possibleelectronic and optical gap engineering.
We thank the European Social Fund (ESF), Operational Program for Educational and Voca-tional Training II (EPEAEK II), and particularly the Program PYTHAGORAS, for funding7he above work 8 eferences [1] L. T. Canham,
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Phys. Rev. B , 2003, Phys. Rev. B , 2005, , 045351[22] N. M. O’Boyle, GaussSum 2.0, 2006. Available at http://gausssum.sf.net10able 1: Lowest Spin and symmetry allowed electronic transitionsnanoparticle core excitation Oscillator Dominantenergy (eV) Strength Contributions Si Ge : H Ge 3.80 0.086 H → L (98%) Si Ge : H Ge 3.73 0.155 H → L (98%) Si Ge : H Ge 3.69 0.261 H → L (97%) Si Ge : H Ge 3.54 0.201 H → L (98%) Si Ge : H Ge 3.52 0.177 H → L (98%) Si Ge : H Ge 3.46 0.254 H → L (98%) Si Ge : H Si 3.74 0.205 H → L (97%) Si Ge : H Si 3.88 0.017 H → L+1 (57%), H-1 → L (35%)3.92 0.110 H-1 → L (60%), H → L+1 (38%) Si Ge : H Si 3.91 0.067 H-1 → L (93%)4.0 0.036 H → L+1 (81%), H-1 → L+1 (17%)4.0 0.030 H-1 → L+1 (15%), H → L+1 (79%) Si Ge : H Si 3.97 0.079 H → L (96%)4.03 0.083 H → L+1 (97%) Si Ge : H Si 3.99 0.121 H → L (88%), H → L+1 (8%) Si Ge : H Si 4.02 0.116 H → L (71%), H → L+1 (26%)11 ig. 1
Typical Si x Ge − x : H nanocrystal (a) Si Ge : H (b) Si Ge : H , (c) Si Ge : H (d) Si Ge : H ). The Ge atoms are shown with green color,while Si atoms are blue. Fig. 2
Bond distribution in Si x Ge − x : H for x=0, 1, 5, 17, 29, 35, 47. The Ge-Ge,Si-Ge and Si-Si bond distributions are shown separately. The constant number ofthe hydrogen atoms (60) is not shown in the graphs. Fig. 3 (a) Total binding energy as a function of the number of silicon atoms (b) Surfaceenergy
Fig. 4
Projected and total Density density of states (PDOS and DOS) of 4 representativenanocrystals.
Fig. 5
The variation of the optical gap as a function of the number of silicon atoms, forthe two categories (Ge(core) and Si(core)) of Si x Ge − x ; H nanocrystals Fig. 6
Schematic representation of (a) Ge)core nanoparticle, which behaves as a Ge nanopar-ticle passivated by silicon, and (b) Si(core) nanoparticle which behaves as a hollowGe nanoparticle with surface hydrogen passivation and internal Si passivation.12igure 1: Typical Si x Ge − x : H nanocrystal (a) Si Ge : H (b) Si Ge : H , (c) Si Ge : H (d) Si Ge : H ). The Ge atoms are shown with green color, while Si atomsare blue. 13 .3 2.4 2.5 2.6 Bond Length Si Ge ) Tolat bond distributionGe-Ge bond distributionSi-Ge bond distributionSi-Si bond distribution
Si atoms in the inner core (g)
Bond Length Si Ge (a) Bond Length Si Ge (b) Bond Length Si Ge (c) B ond D i s t r i bu t i on Bond Length Si Ge (d) Bond Length Si Ge (e) Bond Length Si Ge (f) Figure 2: Bond distribution in Si x Ge − x : H for x=0, 1, 5, 17, 29, 35, 47. The Ge-Ge, Si-Geand Si-Si bond distributions are shown separately. The constant number of the hydrogen atoms(60) is not shown in the graphs. 14 (b) Ge inner core Si inner core B i nd i ng E n e r g y ( e V ) Number of Si atoms
Ge inner core Si inner core B i nd i ng E n e r g y ( e V ) Binding Energies of mixed Si x Ge :H (a) Figure 3: (a) Total binding energy as a function of the number of silicon atoms (b) Surfaceenergy 15
12 -10 -8 -6 -4 -2 0 2 4 D e n s i t y o f S t a t es ( a r b . un i t s ) Energy (ev) Si Ge (Ge core)Si Ge (Ge core)Si Ge (Si core) Ge PDOS
Si PDOS
Total DOS Si Ge (Si core) Figure 4: Projected and total Density density of states (PDOS and DOS) of 4 representativenanocrystals. 16 Ge Si Ge Si Ge Si Ge Si Ge Si Ge Si Ge Si Ge Si inner core Ge inner core O p t i ca l G a p ( e V ) Number of Si atomsFrom (T d ) Ge to (T d ) Si Si Ge Si Ge Si Ge Si Ge Figure 5: The variation of the optical gap as a function of the number of silicon atoms, for thetwo categories (Ge(core) and Si(core)) of Si x Ge − x ; H60