Interband, intraband and excited-state direct photon absorption of silicon and germanium nanocrystals embedded in a wide band-gap lattice
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Interband, intraband and excited-state direct photon absorption of silicon andgermanium nanocrystals embedded in a wide band-gap lattice
C. Bulutay ∗ Department of Physics and UNAM - Institute of Materials Science and Nanotechnology,Bilkent University, Bilkent, Ankara, 06800, Turkey (Dated: October 24, 2018)Embedded Si and Ge nanocrystals (NCs) in wide band-gap matrices are studied theoretically usingan atomistic pseudopotential approach. From small clusters to large NCs containing on the orderof several thousand atoms are considered. Effective band-gap values as a function of NC diameterreproduce very well the available experimental and theoretical data. It is observed that the highestoccupied molecular orbital for both Si and Ge NCs and the lowest unoccupied molecular orbital forSi NCs display oscillations with respect to size among the different irreducible representations ofthe C v point group to which these spherical NCs belong. Based on this electronic structure, firstthe interband absorption is thoroughly studied which shows the importance of surface polarizationeffects that significantly reduce the absorption when included. This reduction is found to increasewith decreasing NC size or with increasing permittivity mismatch between the NC core and thehost matrix. Reasonable agreement is observed with the experimental absorption spectra whereavailable. The deformation of spherical NCs into prolate or oblate ellipsoids are seen to introduceno pronounced effects for the absorption spectra. Next, intraconduction and intravalence bandabsorption coefficients are obtained in the wavelength range from far-infrared to visible region.These results can be valuable for the infrared photodetection prospects of these NC arrays. Finally,excited-state absorption at three different optical pump wavelengths, 532 nm, 355 nm and 266 nmare studied for 3- and 4 nm-diameter NCs. This reveals strong absorption windows in the caseof holes and a broad spectrum in the case of electrons which can especially be relevant for thediscussions on achieving gain in these structures. PACS numbers: 73.22.-f, 78.67.Bf, 78.40.-q
I. INTRODUCTION
The field of silicon and germanium nanocrystals (NCs)is very active due to important technological achieve-ments and prospects particularly in connection with op-tics such as light emitting diodes and lasers.
Twofundamental processes describing the interaction of lightwith matter are the photon absorption and emission.In the context of NCs, it has been shown both exper-imentally and theoretically that the interface prop-erties have dramatic effects on the emission properties.On the other hand, absorption measurements are lesssensitive to surface quality and allow for a more di-rect characterization of the intrinsic structure of NCs. Therefore, the study of the direct photon absorption inNCs can provide a clear physical understanding. More-over, with the ever-growing importance of renewable en-ergy resources, the research on the new-generation pho-tovoltaics has gained momentum and hence the sub-ject of direct photon absorption in nanocrystalline sili-con ( nc -Si). However, both experimentally and theoretically, researchers till nowhave predominantly focused on the interband absorptionprocess. This is the only optical absorption possibilityfor an intrinsic semiconductor NC under equilibrium. Byrelaxing these two constraints we can introduce other ab-sorption channels, mainly through carrier injection or op-tical pumping. The associated absorption in either caseis sometimes referred to as “free” carrier absorption de-spite the carrier confinement in NCs. In our work we discriminate between the two. The electrical injection ordoping gives rise to intraband absorption, also termedas intersubband absorption which has practical impor-tance for mid- and near-infrared photodetectors. Theoptical pumping which is usually well above the effectiveband gap leads to excited-state absorption (also termedas photoinduced absorption) which is an undesired effectthat can inhibit the development of optical gain. Recentexperiments on excited-state absorption concluded thatmore attention should be devoted to the role of the exci-tation conditions in the quest for the silicon laser.
Therefore, the aim of this work is to provide a compre-hensive theoretical account of all of these direct photonabsorption mechanisms in Si and Ge NCs under varioussize, shape and excitation conditions. This provides acomplementary track to the existing experimental effortswhere the size and shape control are currently major ob-stacles.The absorption coefficient of the semiconductor NCsdepends on the product of the optical transition os-cillator strength and their joint density of states aswell as to their volume filling factor within the ma-trix. Therefore, the essential decision on a theoreticalstudy is the sophistication level of the electronic struc-ture. The usual trade off between the computationalcost and accuracy is operational. The constraints onthe former are quite stringent as a NC including theactive region of the matrix surrounding itself can con-tain on the order of ten thousand atoms. As for thelatter, not only the accuracy but also the validity of achosen approach can become questionable. Computa-tionally low-cost approaches like the envelope function inconjunction with 8-band k · p are not as accurate for thistask and furthermore, they miss some critical symme-tries of the underlying lattice. On the other extreme,there lies the density functional theory-based ab initio codes which have been applied to smaller NCs con-taining less than 1000 atoms which still require very de-manding computational resources. The ab initio analysis of larger NCs of sizes between 3-10 nm is practically not possible with the current com-puter power. While this technological hurdle will begradually overcome in the years to come, there existsother atomistic approaches that can be employed forNC research which can be run on modest platforms andare much simpler to develop, such as the tight bindingtechnique which has been successfully employed by sev-eral groups.
On the pseudopotential-based ap-proaches, two new recipes were proposed by Wang andZunger over the last decade.
The folded spectrummethod relies on standard plane wave basis and di-rect diagonalization; its speed is granted from being fo-cused on relatively few targetted states. For the studyof excitons this approach becomes very suitable whereasfor the optical absorption spectra where a large numberof states contribute it loses its advantage. Their otherrecipe is the so-called linear combination of bulk bands(LCBB). As a matter of fact, the idea of usingbulk Bloch states in confined systems goes back to ear-lier times, one of its first implementations being the stud-ies of Ninno et al.
Up to now, it has been used forself-assembled quantum dots, superlattices, andhigh-electron mobility transistors, and very recently onthe nc -Si aggregation stages. In this work, we applyLCBB to the electronic structure and absorption spectraof Si and Ge NCs. An important feature of this work, incontrast to commonly studied hydrogen-passivated NCsis that we consider NCs embedded in a wide band-gapmatrix which is usually silica. In principle, other matri-ces such as alumina or silicon nitride can be investigatedalong the same lines.The organization of the paper is as follows: in Sec-tion II we describe the theoretical framework which in-cludes some brief information on the LCBB techniqueand the absorption expressions. A self-critique of the the-oretical model is done in Section III. Section IV presentsthe results and discussions on the band edge electronicstructure, interband, intraband, and excited-state ab-sorptions followed by our conclusions in Section V. Ap-pendix section contains technical details on the employedpseudopotential form factors and our LCBB implemen-tation.
II. THEORY
For the electronic structure of large-scale atomisticsystems Wang and Zunger have developed the LCBB method which is particularly convenient for embeddedNCs containing several thousand atoms.
The factthat it is a pseudopotential-based method makes it morepreferable over the empirical tight binding technique forthe study of optical properties as aimed in this work. Inthis technique the NC wavefunction with a state label j is expanded in terms of the bulk Bloch bands of the con-stituent core and/or embedding medium (matrix) mate-rials ψ j ( ~r ) = 1 √ N X n,~k,σ C σn,~k,j e i~k · ~r u σn,~k ( ~r ) , (1)where N is the number of primitive cells within the com-putational supercell, C σn,~k,j is the expansion coefficientset to be determined and σ is the constituent bulk mate-rial label pointing to the NC core or embedding medium. u σn,~k ( ~r ) is the cell-periodic part of the Bloch states whichcan be expanded in terms of the reciprocal lattice vectors { ~G } as u σn,~k ( ~r ) = 1Ω X ~G B σn~k (cid:16) ~G (cid:17) e i ~G · ~r , (2)where Ω is the volume of the primitive cell. The atom-istic Hamiltonian for the system is given byˆ H = − ~ ∇ m + X σ, ~R j ,α W σα ( ~R j ) υ σα (cid:16) ~r − ~R j − ~d σα (cid:17) , (3)where W σα ( ~R j ) is the weight function that takes values0 or 1 depending on the type of atom at the position ~R j − ~d σα , and υ σα is the screened spherical pseudopoten-tial of atom α of the material σ . We use semiempiricalpseudopotentials for Si and Ge developed particularly forstrained Si/Ge superlattices which reproduces a large va-riety of measured physical data such as bulk band struc-tures, deformation potentials, electron-phonon matrix el-ements, and heterostructure valence band offsets. Withsuch a choice, this approach benefits from the empiricalpseudopotential method (EPM), which in addition to itssimplicity has another advantage over the more accuratedensity functional ab initio techniques that run into well-known band-gap problem which is a disadvantage forthe correct prediction of the excitation energies.The formulation can be cast into the following gener-alized eigenvalue equation: X n,~k,σ H n ′ ~k ′ σ ′ ,n~kσ C σn,~k = E X n,~k,σ S n ′ ~k ′ σ ′ ,n~kσ C σn,~k , (4)where H n ′ ~k ′ σ ′ ,n~kσ ≡ D n ′ ~k ′ σ ′ | ˆ T + ˆ V xtal | n~kσ E , D n ′ ~k ′ σ ′ | ˆ T | n~kσ E = δ ~k ′ ,~k X ~G ~ m (cid:12)(cid:12)(cid:12) ~G + ~k (cid:12)(cid:12)(cid:12) B σ ′ n ′ ~k ′ (cid:16) ~G (cid:17) ∗ B σn~k (cid:16) ~G (cid:17) , D n ′ ~k ′ σ ′ | ˆ V xtal | n~kσ E = X ~G, ~G ′ B σ ′ n ′ ~k ′ (cid:16) ~G (cid:17) ∗ B σn~k (cid:16) ~G (cid:17) × X σ ′′ ,α V σ ′′ α (cid:18)(cid:12)(cid:12)(cid:12) ~G + ~k − ~G ′ − ~k ′ (cid:12)(cid:12)(cid:12) (cid:19) × W σ ′′ α (cid:16) ~k − ~k ′ (cid:17) e i ( ~G + ~k − ~G ′ − ~k ′ ) · ~d σ ′′ α ,S n ′ ~k ′ σ ′ ,n~kσ ≡ D n ′ ~k ′ σ ′ | n~kσ E . Here, the atoms are on regular sites of the underlyingBravais lattice: ~R n ,n ,n = n ~a + n ~a + n ~a where { ~a i } are its direct lattice vectors of the Bravais lat-tice. Both the NC and the host matrix are assumed topossess the same lattice constant and the whole struc-ture is within a supercell which imposes the periodic-ity condition W (cid:16) ~R n ,n ,n + N i ~a i (cid:17) = W (cid:16) ~R n ,n ,n (cid:17) ,recalling its Fourier representation W (cid:16) ~R n ,n ,n (cid:17) → P ˜ W ( q ) e i~q · ~R n ,n ,n , implies e i~q · N i ~a i = 1, so that ~q → ~q m ,m ,m = ~b m N + ~b m N + ~b m N , where { ~b i } are thereciprocal lattice vectors of the bulk material. Thus thereciprocal space of the supercell arrangement is not acontinuum but is of the grid form composed of points { ~q m ,m ,m } , where m i = 0 , , . . . , N i − However, the pseudopotentialfor oxygen is nontrivial in the case of EPM and further-more, lattice constant of SiO is not matched to either ofthe core materials introducing strain effects. Therefore,we embed the Si and Ge NCs into an artificial wide band-gap medium which for the former reproduces the properband alignment of the Si/SiO system. To circumvent thestrain effects which are indeed present in the actual sam-ples, we set the lattice constant and crystal structure ofthe matrix equal to that of the core material. The pseu-dopotential form factors of the wide band-gap matricesfor Si and Ge can easily be produced starting from thoseof the core materials. More details are provided in theAppendix section. The resultant bulk band structuresfor Si and Ge and their host wide band-gap matrices areshown in Fig. 1. With the use of such a lattice-matchedmatrix providing the perfect termination of the surfacebonds of the NC core atoms lead to the removal of all gapstates as can be observed in Fig. 2. In these plots, theevolution of the effective band-gaps towards their bulkvalues (marked by dashed lines) is clearly seen as thediameter increases.Once the electronic wavefunctions of the NCs are avail-able, their linear optical properties can be readily com-puted. The three different types of direct (zero-phonon) -15-10-50510-15-10-50510 Si KLWX E ne r g y ( e V ) Ge b)a) KLWX E ne r g y ( e V ) FIG. 1: (Color online) EPM band structures for bulk (a)Si, (b) Ge together with their wide band-gap matrices (thicklines) which for the former reproduces the band line-up of theSi/SiO interface. photon absorption processes considered in this work areillustrated in Fig. 3. These are interband, intraband andexcited-state absorptions. In the latter, the blue (dark-colored) arrow represents optical pumping and follow-ing carrier relaxation, the downward green arrow corre-sponds to luminescence which can be to a final interfacestate (dashed line). For all these processes, the relevantquantity is the imaginary part of the dielectric function.Within the independent-particle approximation and theartificial supercell framework it becomesIm { ǫ aa ( ω ) } = (2 πe ~ ) m V SC X c,v f aacv E c − E v × Γ / (2 π )[ E c − E v − ~ ω ] + (Γ / , (5)where, a = x, y, z denotes the cartesian components ofthe dielectric tensor and f aacv = 2 m (cid:12)(cid:12)(cid:12) h c (cid:12)(cid:12)(cid:12) p a m (cid:12)(cid:12)(cid:12) v i (cid:12)(cid:12)(cid:12) E c − E v , (6)is the oscillator strength of the transition. In these ex-pressions m is the free electron mass, e is the magnitude FIG. 2: (Color online) The variation of NC states with respectto diameter for Si and Ge NCs. The bulk band edges aremarked with a dashed line for comparison.FIG. 3: (Color online) Illustration for the three different ab-sorption processes in NCs considered in this work: interband,intraband and excited-state absorption. The yellow (light-colored) arrows indicate the direct photon absorption transi-tions, the blue (dark-colored) arrow represents optical pump-ing and the downward green arrow corresponds to lumines-cence which can be to a interface state (dashed line). of the electronic charge, and Γ is the full-width at halfmaximum value of the Lorentzian broadening. The label v ( c ) correspond to occupied (empty) valence (conduc-tion) states referring only to their orbital parts in theabsence of spin-orbit coupling; the spin summation termis already accounted in the prefactor of Eq. 5. Finally, V SC is the volume of the supercell which is a fixed valuechosen conveniently large to accommodate the NCs ofvarying diameters, however, if one uses instead, that ofthe NC, V NC , this corresponds calculating Im { ǫ aa } /f v where f v = V NC /V SC is the volume filling ratio of theNC. For the sake of generality, this is the form we shallbe presenting our results. The electromagnetic intensity absorption coefficient α ( ω ) is related to the imaginarypart of the dielectric function through Im { ǫ aa ( ω ) } = n r cω α aa ( ω ) , (7)where n r is the index of refraction and c is the speed oflight.In the case of intraband absorption, its rate dependson the amount of excited carriers. Therefore, we considerthe absorption rate for one excited electron or hole thatlies at an initial state i with energy E i . As there are anumber of closely spaced such states, we perform a Boltz-mann averaging over these states as e − βE i / P j e − βE j .We further assume that the final states have no occu-pancy restriction, which can easily be relaxed if needed.The expression for absorption rate per an excited carrierin each NC becomes α aa f v = πe m cn r ωV NC X i,f e − βE i P j e − βE j f aafi [ E f − E i ] × Γ / (2 π )[ E f − E i − ~ ω ] + (Γ / , (8)where again a is the light polarization direction.Finally, we include the surface polarization effects, alsocalled local field effects (LFE) using a simple semiclassi-cal model which agrees remarkably well with more rigor-ous treatments. We give a brief description of its im-plementation. First, using the expression ǫ SC = f v ǫ NC + (1 − f v ) ǫ matrix , (9)we extract (i.e., de-embed) the size-dependent NC di-electric function, ǫ NC , where ǫ SC corresponds to Eq. 5,suppressing the cartesian indices. ǫ matrix is the dielec-tric function of the host matrix; for simplicity we setit to the permittivity value of SiO , i.e., ǫ matrix = 4.Since the wide band-gap matrix introduces no absorp-tion up to an energy of about 9 eV, we can approxi-mate Im { ǫ NC } = Im { ǫ SC } /f v . One can similarly ob-tain the Re { ǫ NC } within the random-phase approxima-tion, hence get the full complex dielectric function ǫ NC .According to the classical Clausius-Mossotti approach,which is shown to work also for NCs, the dielectricfunction of the NC is modified as ǫ NC,LFE = ǫ matrix (cid:20) ǫ NC − ǫ matrix ǫ NC + 2 ǫ matrix (cid:21) , (10)to account for LFE. The corresponding supercell dielec-tric function, ǫ SC,LFE follows using Eq. 9. Similarly, theintensity absorption coefficients are also modified due tosurface polarization effects, cf. Eq. 7. Its consequenceswill be reported in Section IV.
III. A SELF-CRITIQUE OF THETHEORETICAL MODEL
The most crucial simplification of our model is the factthat strain-related effects are avoided, a route which isshared by other theoretical works.
For largeNCs this may not be critical, however, for very smallsizes this simplification is questionable. An importantsupport for our act is that Weissker and coworkers haveconcluded that while there is some shift and possibly a re-distribution of oscillator strengths after ionic relaxation,the overall appearance of the absorption spectra does notchange strongly. We should mention that Wang andZunger have offered a recipe for including strain withinthe LCBB framework, however, this is considerably moreinvolved. Another widespread simplification on Si andGe NCs is the omission of the spin-orbit coupling andthe nonlocal (angular momentum-dependent) pseudopo-tential terms in the electronic structure Hamiltonian. Es-pecially the former is not significant for Si which is a lightatom but it can have a quantitative impact on the valencestates of Ge NCs; such a treatment is available in Ref. 53.On the dielectric response, there are much more sophis-ticated and involved treatments whereas ours is equiv-alent to the independent particle random phase approxi-mation of the macroscopic dielectric function with thesurface polarization effects included within the classicalClausius-Mossotti model. The contribution of the ex-cluded excitonic and other many-body effects beyond themean-field level can be assessed a posteriori by compar-ing with the available experimental data. However, itis certain that the precedence should be given to classi-cal electrostatics for properly describing the backgrounddielectric mismatch between the core and the wide band-gap matrix. In our treatment this is implemented at anatomistic level.Another effect not accounted in this work is the roleof the interface region. Our wide band-gap matrix canreproduce the proper band alignment and dielectric con-finement of an SiO matrix, however, the interface chem-istry such as silicon-oxygen double bonds are not repre-sented. These were shown to be much more effective onthe emission spectra. Nevertheless, our results can betaken as the benchmark for the performance of the atom-istic quantum and dielectric confinement with a clean andinert interface. Finally, we do not consider the phonon-assisted or nonlinear absorption. The list of these ma-jor simplifications also suggest possible improvements ofthis work. IV. RESULTS AND DISCUSSIONS
In this section we present our theoretical investigationof the linear optical properties of Si and Ge NCs. Threedifferent direct photon absorption processes are consid-ered as illustrated in Fig. 3 each of which can serve fortechnological applications as well as to our basic under- standing. However, we first begin with the dependenceof the optical gap on the NC size, mainly as a check ofour general framework. There exist two different atomicarrangements of a spherical NC depending on whetherthe center of the NC is an atomic position or a tetrahe-dral interstitial location; under no ionic relaxation, Del-ley and Steigmeier have treated both of these classes ashaving the T d point symmetry. However, the tetrahe-dral interstitial-centered arrangement should rather havethe lower point symmetry of C v and it is the arrange-ment that we construct our NCs. This leads to evennumber of NC core atoms, whereas it becomes an oddnumber with the T d point symmetry. We identify the ir-reducible representation of a chosen NC state by checkingits projection to the subspace of each representation. For the C v point group these are denoted by A , A ,and E . We utilize this group-theoretic analysis in thenext subsections. This Work Furukawa (cid:214)g(cid:252)t Vasiliev Garoufalisnc-Si O p t i c a l G ap ( e V ) Diameter (nm) This Work Niquet Tsolakidis Takeoda Kanemitsunc-Ge O p t i c a l G ap ( e V ) Diameter (nm)
FIG. 4: (Color online) Comparison of optical gap as a func-tion of NC diameter of this work with previous experimentaland theoretical data: Furukawa, Kanemitsu, Takeoda, ¨O˘g¨ut, Vasiliev, Garoufalis, Niquet, Tsolakidis. A. Effective optical gap
The hallmark of quantum size effect in NCs has beenthe effective optical gap with quite a number of theoret-ical and experimental studies per-formed within the last decade. Figure 4 contains a com-pilation of some representative results. For Si NCs, it canbe observed that there is a good agreement among theexisting data, including ours. On the other hand, for thecase of Ge NCs there is a large spread between the exper-imental data whereas our theoretical results are in verygood agreement with both ab initio and tight bindingresults. In our approach the optical gap directly corre-sponds to the LUMO-HOMO energy difference, as calcu-lated by the single-particle Hamiltonian in Eq. (3). Thissimplicity relies on the finding of Delerue and coworkersthat the self-energy and Coulomb corrections almost ex-actly cancel each other for Si NCs larger than a diameterof 1.2 nm. HOMOLUMO nc-Si E ne r g y ( e V ) Diameter (nm)HOMOLUMO nc-Ge E ne r g y ( e V ) Diameter (nm)
FIG. 5: (Color online) The variation of HOMO and LUMOenergies with respect to NC diameter for Si and Ge NCs thatbelong to C v point group. B. HOMO and LUMO oscillations with respect tosize
When we plot the variation of individual LUMO andHOMO levels as in Fig. 5 we observe with the exception of nc -Ge LUMO curve some non-smooth behavior that getspronounced towards smaller sizes. The triple degeneracyin the absence of spin-orbit coupling of the valence bandmaximum in bulk Si and Ge is lifted into two degener-ate and one nondegenerate states. The energy differencebetween these two set of states is observed to displayan oscillatory behavior as the NC size gets smaller asshown in Fig. 6 (a). Using the C v point group symme-try operations we identify the doubly degenerate statesto belong to E representation and nondegenerate state to nc -Ge nc -Si E ( D oub l e t - S i ng l e t ) ( m e V ) Diameter (in a units) (b) (a) nc -Si E ne r g y ( e V ) Diameter (nm) A E FIG. 6: (Color online) (a) The energy difference between dou-bly degenerate and nondegenerate states, one of which be-comes the HOMO with respect to diameter in Si and Ge NCsthat belong to C v point group; solid lines are for guiding theeyes; a is the lattice constant for Si or Ge NC. (b) The low-est three conduction states, one of which becomes the LUMOwith respect to diameter in Si NCs. A or A . Furthermore, we observe a similar oscillationin the LUMO region of Si NCs as shown in Fig. 6(b).The low-lying conduction states of Si NCs form six-packgroups which is inhereted from the six equivalent 0.85Xconduction band minima of bulk Si. The confinementmarginally lifts the degeneracy by sampling contributionsfrom other parts of the Brillouin zone. This trend is ob-served in Fig. 6(b) as the NC size gets smaller. On theother hand, for Ge NCs all LUMO states belong to thesame A representation and therefore shows no oscilla-tions (cf. Fig. 5). Ultimately, the source of these oscilla-tions is the variation of the asphericity of the NCs of C v point symmetry with respect to size, which can energeti-cally favor one of the closely spaced states. In the case ofthe LUMO state of Ge NCs, there is a substantial energygap between LUMO and the next higher-lying state.For further insight, we display in Fig. 7 the isosur-face plots of the envelope of the six highest states up toHOMO for a Si NC of diameter 2.16 nm. Point grouprepresentation of each state is also indicated. For thisparticular diameter, HOMO has E representation whichis twofold degenerate. The nondegenerate A state alsobecomes the HOMO for different diameters. This is illus-trated in Table I which shows the evolution of the HOMOand LUMO symmetries as a function of diameter for Siand Ge NCs. There, it can be observed that for the latterthe HOMO can also acquire the A for larger diameters. FIG. 7: (Color online) The isosurfaces of the envelopes of thewavefunctions of the highest-lying six valence states up toHOMO for a Si NC of diameter 2.16 nm. The isosurfaces aredrawn for the 95% of the peak value of the envelope wavefunc-tions. The C v point group representations are indicated foreach wavefunction. Note that some of the plots are rotatedwith respect to others for best viewing angle. C. Interband absorption
The interband absorptions of Si and Ge NCs for a va-riety of diameters are shown in Fig. 8. For a fair com-parison, all different size NCs should possess the samevolume filling factor. Therefore, we display the results atunity volume filling or equivalently per f v . The left andright panels display the cases without and with surfacepolarization effects (or LFE), respectively. There existsremarkable differences between the two for both Si andGe NCs. For instance, even though Ge NCs do not showsignificant size dependence without LFE, this is not thecase when LFE is included. From the ratio of both pan-els, the so-called local field absorption reduction factorcan be extracted as shown in Fig. 9. It can be observed that its size dependence is much stronger than the energydependence. This reduction in the absorption due to LFEcan become a major concern for solar cell applications.It needs to be mentioned that this effect is highly sensi-tive to the permittivity mismatch between the core andmatrix media. To illustrate this point, in Fig. 9 the casefor Al O matrix (having a permittivity of 9.1) is alsodisplayed for 1.41 nm Si NC, where it can be seen thatcompared to SiO (with a permittivity of about 4) thereduction in absorption due to LFE is much less. Basedon this finding, we employ these size-dependent absorp-tion reduction factors in the results to follow includingthe intraband and excited-state cases. nc-Si, w/o LFE I m {} / f v Photon Energy (eV)1.47 nm2 nmD=4 nmnc-Ge, w/o LFE I m {} / f v Photon Energy (eV) nc-Si, with LFE 1.41 nm1.8 nmD=4 nm2.45 nm3 nm I m {} / f v Photon Energy (eV)D=4 nm 2 nm 1.47 nmnc-Ge, with LFE I m {} / f v Photon Energy (eV)
FIG. 8: (Color online) The imaginary part of dielectric func-tion for unity volume filling factor, f v for Si and Ge NCs atdifferent diameters with (right panel) and without (left panel)local field effects. A Lorentzian broadening energy full widthof 200 meV is used. In Fig. 10 we compare our results with the experimen-tal data of Wilcoxon et al. for Si NCs and Ge NCs. There is a good overall agreement in both cases espe-cially with LFE, however, for the case of Si NCs this ismuch more satisfactory. The major discrepancies can beattributed to excitonic effects that are not included inour work. In the case of Si NCs (Fig. 10(a)), we alsodisplay the tight binding result of Trani et al. which alsoincludes LFE. An issue of practical concern is the effect of deviationfrom the spherical shape of the NCs depending on thegrowth conditions. At this point we would like to in-vestigate the effect of shape anisotropy on the interbandabsorption. Starting from a spherical 2.54 nm diameter
TABLE I: C v irreducible representations of the HOMO and LUMO in Si and Ge NCs of various diameters ( D ). nc -Si nc -Ge N core D (nm) HOMO LUMO D (nm) HOMO LUMO32 1.06 A A A A
38 1.13 A A A A
56 1.29 A A A A
74 1.41
E E
E A
86 1.49
E E
E A
116 1.64 A E A A
130 1.71
E A A A
136 1.73 A E A A
190 1.94 A A A A
264 2.16
E A E A
384 2.45 A E A A
522 2.71
E E
E A
690 2.98 A A A A
768 3.08 A A E A E E A A Al O SiO nc-Si D=1.41 nm Energy (eV)nc-Ge 1.47 nm 4 nm2 nm Lo c a l F i e l d A b s o r p t i on R edu c t i on F a c t o r FIG. 9: (Color online) The local field absorption factor ex-tracted from the previous figure.
Ge NC, we form prolate and oblate ellipsoidal NCs withellipticities e = − . zz components of the imaginary part of the dielectric ten-sor for three different ellipticities. It is observed that theeffect on the interband absorption is not significant; thedifference is even less for the Si NCs (not shown). D. Intraband absorption
Unlike the interband case, for the intraband absorptionwe need to introduce electrons to the conduction statesor holes to the valence states by an injection mechanism.We assume that after injection these carriers relax totheir respective band edges and attain a thermal distri-bution. Therefore, we perform a Boltzmann averaging atroom temperature (300 K) over the initial states aroundLUMO (HOMO) for electrons (holes). The absorptioncoefficients to be presented are for unity volume fillingfactors and for one carrier per NC; they can easily bescaled to a different average number of injected carriersand volume filling factors. In Fig. 12 the Si NCs of differ-ent diameters are compared. The intraband absorptionis observed to be enhanced as the NC size grows up toabout 3 nm followed by a drastic fall for larger sizes. Forboth holes and electrons very large number of absorp-tion peaks are observed from 0.5 eV to 2 eV. Recently,de Sousa et al. have also considered the intraband ab-sorption in Si NCs using the effective mass approximationand taking into account the multi-valley anisotropic bandstructure of Si. However, their absorption spectra lacksmuch of the features seen in Fig. 12. Turning to Ge NCs,shown in Fig. 13 the intravalence band absorption profileis very similar to that of Si NCs, however in this case theintraconduction band absorption is much weaker.Mimura et al. have measured the optical absorptionin heavily phosphorus doped Si NCs of a diameter of4.7 nm. This provides us an opportunity to compare our b)a) TB+LF-TraniThis workWilcoxon This work+LFE nc-Si, D=1.8 nm A b s o r ban c e ( A r b i t r a r y U n i t s ) Photon Energy (eV)This work+LFE nc-Ge, D=4 nm
This work Wilcoxon A b s o r ban c e ( a r b i t r a r y un i t s ) Photon Energy (eV)
FIG. 10: (Color online) Comparison of our absorbance re-sults with the available data: for Si, experimental work ofWilcoxon and the theoretical tight binding results of Trani and for Ge, the experimental work of Wilcoxon . For ourspectra a Lorentzian broadening energy full width of 200 meVis used. results on the intraconduction band absorption in Si NCs.There is a good order-of-magnitude agreement. However,in contrast to our spectra in Fig. 12 which contains well-resolved peaks, they have registered a smooth spectrumwhich has been attributed by the authors to the smearingout due to size and shape distribution within their NCensemble. E. Excited-state absorption
Finally, we consider another intraband absorption pro-cess where the system is under a continuous interband
FIG. 11: (Color online) The effect of ellipticity, e on theIm { ǫ zz } /f v for a Ge NC with a diameter of 2.54 nm. Theinsets show NC core atoms of the prolate ( e = − .
6) andoblate ( e = 0 .
6) shapes; the z -direction is also indicated. ALorentzian broadening energy full width of 200 meV is used. optical pumping that creates electrons and holes with ex-cess energy. We consider three different excitation wave-lengths: 532 nm, 355 nm and 266 nm which respectivelycorrespond to the second-, third- and fourth-harmonic ofthe Nd-YAG laser at 1064 nm. The initial states of thecarriers after optical pumping are chosen to be at the pairof states with the maximum oscillator strength amonginterband transitions under the chosen excitation. Thedetermined energies of these states are tabulated in Ta-ble II where it can be observed that in general the excessenergy is unevenly partitioned, mainly in favor of theconduction states. Once again a Boltzmann averaging isused to get the contribution of states within the thermalenergy neighborhood.Considering 3 nm and 4 nm diameters, the results areshown in Figs. 14 and 15 for Si and Ge NCs, respectively.Note that the 532 nm excitation results are qualitativelysimilar to those in intraband absorption, cf. Figs. 12and 13. This is expected on the grounds of small ex-cess energy for this case. Some general trends can beextracted from these results. First of all, the conductionband absorption is in general smooth over a wide en-ergy range. On the other hand the valence band absorp-0 ConductionValence nc-Si, D=1.63 nm /f v ( c m - ) Photon Energy (eV)
ConductionValence nc-Si, D=2.16 nm
Photon Energy (eV)
ConductionValence nc-Si, D=4 nm
Photon Energy (eV)
ConductionValence nc-Si, D=3 nm /f v ( c m - ) Photon Energy (eV)
FIG. 12: (Color online) Intravalence and intraconductionstate absorption coefficients in Si NCs of different diametersper excited carrier and at unity filling factor. A Lorentzianbroadening energy full width of 30 meV is used. Mind thechange in the vertical scale for 4 nm diameter case. tion contains pronounced absorption at several narrowenergy windows mainly below 1 eV and they get muchweaker than the conduction band absorption in the re-maining energies. As the excitation energy increases theabsorption coefficient per excited carrier in general de-creases. In connection to silicon photonics, we shouldpoint out that the excited-state absorption is substantialincluding the important 1.55 µ m fiber optics communi-cation wavelength. These results provide a more com-prehensive picture than the reported experimental mea-surements which are usually obtained at a singleenergy of the probe beam. Finally, it needs to be men-tioned that for both intraband and excited-state absorp-tions displayed in Figs. 12 to 15, the high energy partswill be masked by the interband transition whenever itbecomes energetically possible. V. CONCLUSIONS
The subject of Si and Ge NCs has become an estab-lished research field. A fundamental process such the ConductionValencenc-Ge, D=1.7 nm /f v ( c m - ) Photon Energy (eV) ConductionValencenc-Ge, D=2.26 nmPhoton Energy (eV)ConductionValencenc-Ge, D=2.83 nm /f v ( c m - ) Photon Energy (eV) ConductionValencenc-Ge, D=4 nmPhoton Energy (eV)
FIG. 13: (Color online) Same as Fig. 12 but for Ge NCs. direct photon absorption deserves further investigationfrom a number of perspectives. In this theoretical study,we consider the interband, intraband and excited-stateabsorption in embedded Si and Ge NCs of various sizes.For this purpose, we developed an atomistic pseudopo-tential electronic structure tool, the results of which agreevery well with the published data. It is further observedthat the HOMO for both Si and Ge NCs and the LUMOfor Si NCs display oscillations with respect to size amongdifferent representations of the C v point group to whichthese spherical NCs belong. Our detailed investigationof the interband absorption reveals the importance ofsurface polarization effects that significantly reduce theabsorption when included. This reduction is found toincrease with decreasing NC size or with increasing per-mittivity mismatch between the NC core and the hostmatrix. These findings should be taken into accountfor applications where the absorption is desired to beeither enhanced or reduced. For both NC types the de-viation from sphericity shows no pronounced effect onthe interband absorption. Next, the intraband process isconsidered which has potential applications on mid- andnear-infrared photodetection. The intravalence band ab-sorption is stronger compared to intraconduction bandespecially below 1 eV. Finally, we demonstrate that op-tical pumping introduces a new degrees of freedom tointraband absorption. This is studied under the title of1 TABLE II: The excited-state energies of the carriers within the valence and conduction states under three different interbandpump energies for nc -Si and nc -Ge. The energies are given in eV and measured from the HOMO and LUMO, respectively. nc -Si nc -GePump D = 3 nm D = 4 nm D = 3 nm D = 4 nm2.33 0.197, 0.021 0.021, 0.551 0.103, 0.643 0.211, 0.6633.50 0.000, 1.432 0.316, 1.440 0.400, 1.508 1.141, 0.8924.66 0.188, 2.414 0.713, 2.218 0.511, 2.551 1.360, 1.853 Pump Energy=355 nm nc -Si, D=4 nmConductionValencePhoton Energy (eV)ConductionValencePump Energy=355 nm nc -Si, D=3 nm /f v ( c m - ) Photon Energy (eV)
ConductionValence nc -Si, D=4 nmPump Energy=266 nmPhoton Energy (eV)ValenceConduction nc -Si, D=4 nmPump Energy=532 nmPhoton Energy (eV) Valence Conduction nc -Si, D=3 nm Pump Energy=532 nm /f v ( c m - ) Photon Energy (eV) ConductionValence nc -Si, D=3 nmPump Energy=266 nm /f v ( c m - ) Photon Energy (eV)
FIG. 14: (Color online) Excited-state absorption within va-lence and conduction states of Si NCs per excited carrier andat unity filling factor under three different optical pumpingwavelengths of 532 nm, 355 nm and 266 nm. Dotted linesin black color refer to total absorption coefficients. Two dif-ferent diameters are considered, 3 and 4 nm. A Lorentzianbroadening energy full width of 30 meV is used. excited-state absorption. Our major finding is that theexcited-state absorption is substantial including the im-portant 1.55 µ m fiber optics communication wavelength.Within the context of achieving gain and lasing in theseNCs, excited-state absorption is a parasitic process, how-ever, it can acquire a positive role in a different applica-tion. ValenceConduction nc -Ge, D=4 nmPump Energy=355 nmPhoton Energy (eV)ValenceConduction nc -Ge, D=4 nmPump Energy=532 nmPhoton Energy (eV)Valence Conduction nc -Ge, D=4 nmPump Energy=266 nmPhoton Energy (eV)Valence Conduction nc -Ge, D=3 nmPump Energy=266 nm / f v ( c m - ) Photon Energy (eV) / f v ( c m - ) ValenceConduction nc -Ge, D=3 nmPump Energy=532 nmPhoton Energy (eV)Valence Conduction nc -Ge, D=3 nmPump Energy=355 nm / f v ( c m - ) Photon Energy (eV)
FIG. 15: (Color online) Same as Fig. 14 but for Ge NCs.
Acknowledgments
The author is grateful to Aykutlu Dˆana for his sugges-tion of the intersubband absorption and its photodetectorapplications. This work has been supported by the Eu-ropean FP6 Project SEMINANO with the contract num-ber NMP4 CT2004 505285 and by the Turkish Scientificand Technical Council T ¨UB˙ITAK with the project num-ber 106T048. The computational resources are suppliedin part by T ¨UB˙ITAK through TR-Grid e-InfrastructureProject.2
TABLE III: Parameters of the pseudopotential form factors of Si, Ge and their wide band-gap matrices. a is the latticeconstant. See text for the units. a (˚A) a a a a a a Si 5.43 1.5708 2.2278 0.606 -1.972 5.0 0.3Matrix-Si 5.43 1.5708 2.5 0.135 -13.2 6.0 0.3Ge 5.65 0.7158 2.3592 0.74 -0.38 5.0 0.3Matrix-Ge 5.65 0.4101 2.7 0.07 -2.2 5.0 0.3
APPENDIX: SOME TECHNICAL DETAILS ONTHE LCBB IMPLEMENTATION
In this section we first provide the details on the pseu-dopotential form factors of the bulk Si, Ge and their as-sociated wide band-gap matrices to substitute for SiO .We use the local empirical pseudopotentials for Si andGe developed by Friedel, Hybertsen and Schl¨uter. Theyuse the following functional form for the pseudopotentialform factor at a general wave number q : V PP ( q ) = a (cid:0) q − a (cid:1) e a ( q − a ) + 1 (cid:20)
12 tanh (cid:18) a − q a (cid:19) + 12 (cid:21) . Using the parameters supplied in Table III, the pseudopo-tential form factors come out in Rydbergs and the wavenumber in the above equation should be taken in atomicunits (1/Bohr radius). Another important technical re-mark is about the EPM cut off energies. We observed that even though the EPM band energies (i.e., eigen-values) converge reasonably well with cut off energies aslow as 5-10 Ry, the corresponding Bloch functions (i.e.,eigenvectors) require substantially higher values to con-verge. The results in this study are obtained using 14and 16 Ry for Si, Ge, respectively.Finally, some comments on the LCBB basis set con-struction is in order. We only employ the bulk bands ofthe core material. The bulk band indices are chosen fromthe four valence bands an the lowest three to four con-duction bands; usually these are not used in conjunctionbut separately for the NC valence and conduction states,respectively. The basis set is formed from a samplingover a three-dimensional rectangular grid in the recip-rocal space centered around the Γ-point. Its extend isdetermined by the full coverage of the significant bandextrema, such as conduction band minima of Si at thesix equivalent 0.85 X points. The final LCBB basis settypically contains some ten thousand members. ∗ Electronic address: [email protected] S. Ossicini, L. Pavesi, and F. Priolo,
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