Electronic structure of a model nanocrystalline/amorphous mixed-phase silicon

Shintaro Nomura,* Xinwei Zhao, Yoshinobu Aoyagi, and Takuo SuganoFrontier Research Program, The Institute of Physical and Chemical Research (RIKEN), Hirosawa, Wako-shi, Saitama, Japan

~Received 11 March 1996!

Electronic states of a model of nanocrystalline phase Si surrounded by amorphous phase Si are studied. AMonte Carlo calculation followed by an empirical pseudopotential calculation is performed for obtaining theelectronic states of the model structure containing up to 1000 atoms. The ratios of the local density of stateswithin the nanocrystalline phase to the total density of states are found to be inhomogeneous depending on theenergy and are larger than the ratio of the number of atoms in the nanocrystalline phase in the energy regionbetween 21 and 24 eV, and 2 and 5 eV, where the optical transitions mainly take place.@S0163-1829~96!01743-2#

I. INTRODUCTION

A shift of energy levels with a change in size of nanoscalesemiconductors is often referred to as a quantum size effect.One such effect is the blue shift of energy levels with reduc-tion of the size of a confining potential in real space bygaining the kinetic energy. This effect is understood well indirect-gap semiconductor quantum dots.1,2 The size effect inindirect-gap semiconductors is not as simple as direct-gapsemiconductors when the optical dipole transitions betweenconduction and valence levels are argued. The optical dipoletransition is dominated by a phonon-assisted transition for alarge size of the confining potential. The radiative transitionwithout phonons becomes more dominant with decrease insize because of the overlap of wave functions of the conduc-tion electron and the valence hole in a momentum space. Theelectronic states and optical properties of Si quantum dotsstarted attracting interest after reports of light emission fromthese indirect-gap systems.3–7 Models are considered for Siclusters,8 crystalline Si capped with H atoms,9,10 crystallineSi covered with CaF2 compounds,

11 and Si crystalline sur-rounded by artificial potential barrier assumeda priori.12,13

In all of these models, quantum size effects have originatedfrom confining electrons into the crystalline phase by energypotential barriers.

The blueshift of absorption spectra with a decrease in sizehas been clearly observed in Si nanocrystallites prepared inamorphous Si films by a modulation spectroscopy in mag-netic fields.14 The samples were fabricated by crystallizationof amorphous Si to form nm-sized Si crystals in amorphousSi. Blue photoluminescence was observed from the samesamples at room temperature.7 This is rather surprising sincea smaller band gap of the amorphous Si matrix than the Sinanocrystallites creates no apparent potential barrier whichconfines electrons and holes within the Si nanocrystallites. Itis evident that the conventional confinement effect by a po-tential barrier cannot account for the observed blueshift ofthe absorption spectra.

Motivated by the above experimental result, we propose amodel for a nanoscale structure of crystalline phase Si sur-rounded by amorphous phase Si. This model is quite differ-ent from models considered before. No potential barrier be-tween the crystalline phase and the amorphous phase is

assumeda priori. Thus one might consider naively that theelectrons can diffuse between two phases freely and theremight be no size effect to be observed. Instead we are con-sidering a more fundamental question: is there any size ef-fect when one changes the size of the ordered phase sur-rounded by the disordered phase without any externalpotential barrier between them? This is purely quantum me-chanical because the interference of the wave functionsshould play an important role, which is not understood by theclassical diffusive motion of electrons.

In this paper, electronic states of the crystalline phase Siin the amorphous phase Si are studied. The empiricalpotential-energy function proposed by Stillinger and Weber15

and the empirical pseudopotential method10 are used in thecalculation since it is still not practical to apply the first-principle local-density approximation~LDA ! method to asystem of 1000 atoms because of computational require-ments. Details of the calculation are described in Sec. II. Theobtained structure factors and density of states of the modelstructures are presented in Secs. III A and III B, followed bydiscussions in Sec. III C.

II. CALCULATION METHOD OF A MODELNANOCRYSTALLINE PHASE/AMORPHOUS PHASE

SILICON STRUCTURE

To model a nanocrystalline phase/amorphous phase Sistructure, we assume cubic periodically repeated supercellsconsisting of nanocrystalline phase and amorphous phase re-gions with ~100! interface, as shown in Fig. 1. This type ofstructure is called as a ‘‘dispersion microstructure,’’ whereone component is dispersed in the matrix of the other. Theformer and the latter correspond to the nanocrystalline phaseand the amorphous phase, respectively.

The dimension of the supercellL is integer multiples ofthe bulk lattice constanta0. The periodic boundary condi-tions are maintained. A nanocrystalline phase region is de-fined by a cubic regionL c

3, which is also integer multiples ofa0. An amorphous phase region fills in the region betweenthe cubic regionsL3 andL c

3. In the largest model constructedin the present study,L andLc are taken to be 2.72 and 2.17nm, respectively, and the total number of Si atoms is 1000,out of which 512 atoms belong to nanocrystalline phase re-

PHYSICAL REVIEW B 15 NOVEMBER 1996-IVOLUME 54, NUMBER 19

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gion. Three model structures are studied by varying the sizeof the system by keepingL-Lc to be a constanta0. Withinthe nanocrystalline phase region, the perfect tetrahedrallybonded crystal structure is maintained. The density of Si at-oms is assumed to be the same as the bulk crystal Si for asimplicity.

Note here that no hydrogen atoms are added to the modelstructure for termination of dangling bonds. We intend tomodel samples in which dangling bonds are intentionally leftunsaturated. This differs from some available electronicstructure calculations for modeling hydrogenated amorphoussilicon.16,17 A Fourier transformed far infrared measurementwas performed for the samples in the literature14 and no hy-drogen related absorption was observed in comparison tohydrogen saturated samples.7 Thus our model is consideredto be a reasonable modeling of the samples investigated inthe literature.14

A model potential-energy function proposed by Stillingerand Weber15 is used to construct model amorphous Si andnanocrystalline phase/amorphous phase Si structures. Thepotential-energy function is expanded up to three-body termsas follows:

F5(i

n1~ i !1(i , ji, j

n2~ i , j !1 (i , j ,ki, j,k

n3~ i , j ,k!, ~1!

where the one-body termn1 is absent in our case,

n2~ i , j !5A~Bri j2p21!expF b

r i j2a G , ~2!

and

n3~ i , j ,k!5hki j1hi jk1hjki ,

where

hki j5l expF g

r i j2a GexpF g

r ik2a G S cosvki j11

3D2

~3!

andui jk is the angle betweenr i j andr ik . The parametersA,B, p, q, a, l, and g are given in Ref. 15. This modelpotential-energy function is known to describe interactions inliquid phase as well as amorphous phase Si.18

A Monte Carlo method with the Metropolis algorithm foroptimization of the atomic positions is applied to lower thetotal energy of the system. This method was first applied tothe construction of an amorphous Si structure by Wooten,Winer, and Weaire.19 Atomic positions are modified at eachstep according to the Metropolis algorithm. A new atomicposition is adopted if the potential is lowered by the newposition or at the possibility of exp~2E/kT! if the potentialis raised. This scheme is necessary to escape from metastablestates. First, the temperatureT is kept at 19 100 K to suffi-ciently randomize the structure, and then is lowered to 1910K, which is about 10% above the melting point of Si, fol-lowed by linear cooling of the system to 0 K by 21 steps.During these processes, only the atoms in the amorphousphase region are moved, while the atoms in the nanocrystal-line phase region are fixed. The atoms in the amorphousphase region are not permitted to enter the nanocrystallineregion. The same parameters for model construction se-quence are used for both the amorphous structures and thenanocrystalline phase/amorphous phase structures. The meanthree-body energy per atom, which is a measure of deviationfrom the perfect tetrahedral structure, is 0.146 and 0.139 inunits of 3.47310212 erg for the amorphous structures and thenanocrystalline phase/amorphous phase structures, respec-tively, in agreement with the result of a molecular-dynamicsstudy for amorphous Si.18

The empirical pseudopotential method is used for theelectronic states calculations. Since it is not practical to ap-ply ab initio molecular dynamics calculations at present to asystem size of 1000 atoms or more,20 we choose to keep thecomputational requirement of a calculation small to increasethe system size. Calculations of amorphous Si were com-pared with non-self-consistent and self-consistent methods,and both results were found to be in excellent agreement.17

Thus the empirical pseudopotential method is considered tobe sufficient for the present investigations. We use theWang-Zunger form for the continuous pseudopotential for aSi atom.10 The cut-off energy is taken to be 4.5 Ry, thepreconditioned conjugate method utilized for obtaining theeigenstates. The eigenstates are calculated at eight pointswithin the Brillouin zone for models with 216 atoms, and ata single special point for models with 512 or larger numberof atoms.

III. RESULTS AND DISCUSSION

A. Structure factor

Structural models studied are designated in the followingas n NC or n A, where n denotes the total number of Siatoms in the unit cell, and NC or A denotes nanocrystalline-amorphous phase and amorphous models, respectively. A to-tal of six models, i.e., 216NC, 512NC, and 1000NC, 216A,512A, and 1000A, are studied for nanocrystalline phase/amorphous phase and amorphous phase Si. Figure 2 showsthe structure factors for 1000A and 1000NC. The structurefactor for 1000NC has characteristic features of both theamorphous and crystal phase. Distinguishing points with

FIG. 1. A schematic description of a model nanocrystallinephase/amorphous phase Si structure studied. A cubic nanocrystal-line phase region defined byLc is surrounded by an amorphousphase region defined byL. The periodic boundary conditions areapplied to the system.L-Lc is kept to be a constanta0 on changingthe size of the system.

54 13 975ELECTRONIC STRUCTURE OF A MODEL . . .

S(q) larger than about 2, whereq5A332p/a0,A832p/a0 , A1132p/a0 , . . . , reflect the diamond crystalstructure in the nanocrystalline phase. Characteristic featuresof the amorphous phase can be seen as the oscillating struc-ture in the region ofS(q) smaller than 1. This oscillatingstructure confirms that not all of the region is crystallized bythe annealing processes albeit the large area of thecrystalline-amorphous interface which promotes the crystal-lization acting as a role of ‘‘seed crystal.’’ Note here thatthere are points in the region ofS(q) between 1 and 2, whichoriginate from the interface region. Thus the structure factorshows that the model nanocrystalline-amorphous phase Sistructure is not simply a mixture of amorphous and crystal-line phases.

B. Density of states

The results of the total density of states of 1000A and1000NC are shown in Figs. 3~a! and 3~b!, respectively, av-eraged over 0.1 eV energy windows. Figure 3~a! shows twobroad peaks at22 and210 eV, characterizing the amor-phous phase. The fundamental gap is filled by the states con-

cerned with dangling bonds with the density of the order of1020 cm23. The dangling bonds are also responsible for thestates lower than212 eV. As we will see later, these statesare highly localized.21

In contrast to the amorphous phase, three distinct peakscan be discerned in the total density of states of 1000NC asshown in Fig. 3~b! at 23, 27, and210 eV, dividing thevalence bands into three regions withp-like character,s- andp-hybridized character, ands-like character. These struc-tures, especially the sharp peak at27 eV, are similar to thecrystal Si. The electron density at the Fermi energy (Ef) of1000NC is smaller than 1000A. The electron density atEf isroughly proportional to the number of dangling bonds percell. Note here the similarity of the density of states of1000NC to bulk Si as compared to the Si cluster with 1315Si atoms capped with H atoms,22 which shows broader peaksthan Fig. 3~b!. The surrounding amorphous phase Si affectsthe interior electron states much less than H atoms.

The local density of states of 216NC, 512NC, and1000NC is shown in Fig. 4 to make a comparison betweenthree model structures. The local density of states is definedby the density of states within the square crystalline phase

FIG. 2. The structure factors of~a! amorphous Si and~b! nanoc-rystalline phase/amorphous phase Si with 1000 atoms.~Inset! Thesame figure with a magnifiedy axis.

FIG. 3. Total density of states in~a! amorphous Si and~b!nanocrystalline phase/amorphous phase Si with 1000 atoms.Ef isshown by the arrows.

13 976 54NOMURA, ZHAO, AOYAGI, AND SUGANO

region L c3, as shown in Fig. 1, which is normalized toa0

3.Peaks at23, 27, and210 eV become sharper with an in-crease in the number of atoms in the crystalline phase, cor-responding toX4, L1, andL28 critical points, respectively, inbulk Si. The local density of states at the Fermi energy islower for larger number of atoms in the crystalline phase.

The ratios of the local density of states in the crystallinephase region and the total density of states in nanocrystallinephase/amorphous phase Si is depicted in Fig. 5. The dashedline shows the ratio of the number of atoms in the crystallinephase and the total number of atoms, i.e., 512/1000 for the1000NC case. The electron density within the square crystalregion is larger than the number of atoms in the energy suchas between21 and24 eV and between 2 and 5 eV. In thisenergy region, it is energetically favorable for the electronsto be in the crystalline region.

The states in the vicinity of the pseudogap and in theenergy lower than212 eV tend to be localized in the amor-phous phase region. The electrons with the energy within thepseudogap are inhibited to propagate and the amplitude ofthe wave function decays exponentially in the crystalline re-gion. Thus these electrons are loosely confined in the amor-phous phase. The existence of dangling bonds in the amor-phous phase or in the interface region enhances thislocalization character further.

The degree of localization of the electron states is charac-terized by the inverse participation ratio defined by

pi5E uC i* ~r !C i~r !u2dr , ~4!

whereCi~r ! is the wave function of the corresponding state.The inverse participation ratio is smallest for a plane-wave-

FIG. 4. Local density of states in nanocrystalline phase/amorphous phase Si with~a! 216, ~b! 512, and~c! 1000 atoms.

54 13 977ELECTRONIC STRUCTURE OF A MODEL . . .

like delocalized state and largest for a localized state at anatom. Figures 6~a! and 6~b! clearly show that the states in thepseudogap aroundEf is more localized in 1000NC than in1000A, suggesting that more dangling bonds are present in1000NC than in 1000A. The relaxation of atomic positions ismore difficult at the border of crystal/amorphous phases be-cause of the rigid atomic positions in the crystal phase. Thisresult agrees with the defect density measurements.23

It is also shown in Figs. 6~a! and 6~b! that the states below212 eV are more localized than others, which are seen com-mon to the 1000A and 1000NC cases. The inverse participa-tion ratios for the 1000A is rather featureless except for theselocalized states below 12 eV and a small and broad peak atEf corresponding to the pseudogap states. Small peaks canbe seen in Fig. 6~b! at28,27, and25 eV for 1000NC. Thefirst and second peaks correspond to the dips at28 and25eV in Fig. 5, where less electron densities are present in thecrystal phase.

C. Discussion

One of the most extreme consequences of a dispersionmicrostructure of the model nanocrystalline/amorphousphase structure as seen in Fig. 1 is that the structure does nothave translational symmetry except for the artificially im-posed periodicity of the model, of which effects are small formodels with more than 512 atoms. The wave vectork largerthan 2p/L is undefined in the amorphous phase region and isonly approximately defined in the nanocrystalline phase re-gion. Therefore, the wave functions are spread ink space, asin the case of confined wave functions in Si crystallites ter-minated with H atoms. The selection rules for the opticaltransitions are relaxed, and indirect transitions in bulk be-come partially optically active in nanocrystalline phase sur-rounded by amorphous phase. One important difference ofthe electron states in the model nanocrystalline/amorphousphase structure is that the electron states are continuous, asseen in Fig. 3~b!, which can be compared to discrete states inSi confined structures. Since the phase of the wave functionsat the crystal-amorphous boundaries is not fixed by anyboundary condition, instead it varies depending on positions,the eigenenergies of the electrons are nearly continuous.

Two extreme cases are considered for the distribution ofelectrons: One is the case of a plane-wave-like delocalizedstate where the electrons distribute homogeneously, and theother is the case of a confined state where the electrons dis-tribute only in a spatially restricted region within an energywindow. The latter case is typically observed in Si clusterscapped with hydrogen atoms.

Our nanocrystalline/amorphous Si model is in betweenthe above two extreme cases, as shown in Fig. 5. The spatialdistribution of an electron depends on the energy of the elec-tron. About 25% of electron density is in the crystal phase inthe pseudogap, while nearly 70% is in the crystal phase atthe27 eV peak of thes-p hybridized state.

In the pseudogap states, the propagation of the electronwave functions in the crystal phase is prohibited. The elec-tron wave functions penetrate only in the vicinity of the in-terface of the crystal/amorphous phase and decay exponen-tially as evanescent waves. Size dependence of the localdensity of states as shown in Fig. 4 shows that local densityof states within the pseudogap is smaller for larger numberof atoms in the crystal phase.

FIG. 5. Ratios of the local density of states in the nanocrystal-line region and the total density of states in nanocrystalline/amorphous Si.Ef is shown by the arrows.

FIG. 6. Inverse participation ratios of~a! amorphous Si and~b!nanocrystalline phase/amorphous phase Si with 1000 atoms.Ef isshown by the arrows.

13 978 54NOMURA, ZHAO, AOYAGI, AND SUGANO

On the other hand, in the energy between21 and24 eV,between 2 and 5 eV, and especially at the peak of27 eV,more density of electrons are in the crystal phase than theratio of the number of atoms. In these regions, it is energeti-cally more favorable for the electron density to be in thecrystal phase because of the resonance of the electron wavefunctions with the crystal structure. It is important to noticehere that the states in the energy between21 and24 eV andbetween 2 and 5 eV are more delocalized as seen in Fig. 6~b!than the states in the pseudogap. The optical transitions areconsidered to be larger for the transition between these de-localized states, the majority of which take place within thecrystal phase.

Here, we draw attention that the above calculations havebeen done for model structures withL andLc smaller than2.72 and 2.17 nm, respectively. These are smaller than thesize of the crystallites in samples studied in the literature,14

which was in the range of 3 to 5 nm. Thus the size depen-dence of the local density of states shown in Fig. 4 is notdirectly comparable to the experimental result. The existenceof the localized states in the pseudogap originated from dan-

gling bonds also obscures the size effect and makes it diffi-cult to define ‘‘band gap’’ of these systems. Still, an increasein the local density of states concerning 3 to 4 eV opticaltransitions between delocalized states can be observed in Fig.4.

In summary, the electronic states calculation of a pro-posed model nanocrystalline phase/amorphous phase Sistructure is described in this paper. Distribution and localiza-tion of the wave functions in the nanocrystalline phase andamorphous phase are studied and the ratio of the local den-sity of states within the nanocrystalline phase is found to belarger than the ratio of the number of atoms in the energybetween21 and24 eV and 2 and 5 eV.

ACKNOWLEDGMENTS

We acknowledge the Computation Center of RIKEN forproviding CPU time of VPP-500. This work is partially sup-ported by the Grant-in-Aid for Science Research~Grant Nos.08455152 and 08650415! from the Ministry of Education,Science, and Culture, Japan.

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54 13 979ELECTRONIC STRUCTURE OF A MODEL . . .