Journal of Nanoparticle Research 1: 329347, 1999. 1999 Kluwer Academic Publishers. Printed in the Netherlands.

Invited paper

Strain directed assembly of nanoparticle arrays within a semiconductor

C.-Y. Hung1, A.F. Marshall2, D.-K. Kim1, W.D. Nix3, J.S. Harris, Jr.1 and R.A. Kiehl11Solid State and Photonics Laboratory, 2Center for Materials Research,3Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA

Received 7 December 1998; accepted in revised form 15 April 1999

Key words: nanoparticles, self assembly, nanotechnology, precipitation, elastic properties

Abstract

The use of strain to direct the assembly of nanoparticle arrays in a semiconductor is investigated experimentally andtheoretically. The process uses crystal strain produced by a surface structure and variations in layer compositionto guide the formation of arsenic precipitates in a GaAs-based structure grown at low temperature by molecularbeam epitaxy. Remarkable patterning effects, including the formation of single and double one-dimensional arrayswith completely clear fields are achieved for particles in the 10-nm size regime at a depth of about 50 nm from thesemiconductor surface. Experimental results on the time dependence of the strain patterning indicates that straincontrols the late stage of the coarsening process, rather than the precipitate nucleation. Comparison of the observedparticle distributions with theoretical calculations of the stress and strain distributions reveals that the precipitatesform in regions of maximum strain energy, rather than near extremum points of hydrostatic stress or dilatationstrain. It is therefore concluded that the patterning results from modulus differences between the particle and matrixmaterials rather than from other strain related effects. The results presented here should be useful for extendingstrain directed assembly to other materials systems and to other configurations of particles.

Introduction

A process for directing the assembly of nanoparticleswithin a solid so as to control the spatial distributionsin one, two, or three dimensions is of potential interestfor a variety of applications. The ability to preciselycontrol the periodicity in particle arrays or the couplingbetween sets of particles could lead to materials withinteresting optical or electronic properties. Moreover,if the positional control could be made to originate fromstructures that also provide electrical or optical accessto the particles, this could open new possibilities fornanostructure devices and circuits.

Nanoparticles can be formed within a solid by theprocess of precipitation. The formation of nanocrystalswithin a semiconductor is of particular interest becausethis allows the electrical and optical properties of the

particle and the matrix to be tailored over wide ranges.In this paper we examine a process in which strain isused to direct the assembly of nanocrystals within asemiconductor by controlled precipitation (Kiehl et al.,1995, 1996; Hung et al., 1997, 1998). The processuses crystal strain and layer composition to guide theformation of arsenic precipitates in a GaAs-based struc-ture grown at low temperature by molecular beam epi-taxy (MBE). The inhomogeneous strain field producedby stressors fabricated on the surface of the epitaxialstructure is used to control the lateral position of theparticles, while the composition of the epitaxial layerscontrols the vertical position of the particles.

This paper is designed to be a comprehensive treat-ment of this topic in which essential backgroundmaterial and key prior results are briefly presentedtogether with results from our recent experimental and

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theoretical studies. We will begin by discussing theprecipitation process for hexagonal arsenic crystals innon-stoichiometric GaAs grown at low temperatures bymolecular beam epitaxy. The control of precipitate sizeand uniformity is discussed and the use of composi-tional variations to produce preferential precipitation inselected epitaxial layers is described. We then discussresults on strain directed assembly of one-dimensionalparticle arrays. Experimental results on the time depen-dence of the patterning effects during annealing aregiven first. Results on the observed dependence of thestrain patterning on stressor geometry are presentednext. The theoretical stress and strain distributions inour structure are then examined using the finite elementmethod and compared with the observed particle dis-tributions in the structures. As summarized in the con-clusion of this paper, the theoretical and experimentalresults provide a detailed understanding of the pattern-ing mechanism, which should be useful for extendingthis process to other materials systems and particleconfigurations.

Arsenic precipitation in non-stoichiometricGaAs

Growth of non-stoichiometric material

Molecular beam epitaxial growth of high-quality stoi-chiometric GaAs is typically done at a substrate tem-perature of 600C, and an As/Ga flux ratio in the rangeof 1520. If such a V/III flux ratio is maintained, butthe substrate temperature is lowered to 200250C, thevapor pressure of As will be lower than the impingingAs pressure and excess arsenic will tend to accumu-late on the surface. As a result, as much as 1.5% excessAs will be incorporated into the GaAs film, whichis called non-stoichiometric (NS) or low-temperature-grown (LTG) GaAs (Melloch et al., 1992; Yu et al.,1992). Similar techniques can also be applied to growNS-AlGaAs, NS-InGaAs and other non-stoichiometricarsenide layers.

The excess As incorporated into NS-GaAs resultsin various forms of point defects, such as arsenic anti-sites AsGa, arsenic interstitials AsI and gallium vacan-cies VGa (Kaminska et al., 1989; Look et al., 1990), asshown schematically in Figure 1. The concentration ofarsenic interstitials AsI appears to be negligible com-pared with AsGa and VGa according to studies by Liuet al. (1995). Arsenic anti-site defects are found withconcentrations in the range of 21019 to 11020 cm3

Figure 1. Schematic illustrating the point defects in arsenic richnon-stoichiometric GaAs prior to annealing. Non-stoichiometricGaAs is grown at low temperatures (e.g., 200C) by molecularbeam epitaxy.

in as-grown layers, depending on the growth tempera-ture. The concentration of VGa in as-grown NS-GaAsis approximately 1019 cm3 (Look et al., 1990).

The amount of excess arsenic incorporation isa strong function of the substrate temperature andincreases as the growth temperature is decreased. Thelattice constant of InGaAs is larger than that of GaAsbecause the As atom is larger than the Ga atom. Asa result of this lattice mismatch, there is a limit tothe amount of excess arsenic that can be incorporatedinto the layer by lowering the substrate temperature(Melloch et al., 1992). For a given amount of excessAs, there will be a critical thickness beyond which thefilm turns polycrystalline or even amorphous.

Formation of arsenic precipitates during annealing

When NS-GaAs epilayers are annealed above about400C, the excess As atoms cluster to form precipi-tates (Melloch et al., 1992). A precipitation process isa solidsolid transformation and can be expressed as:

0 D C

where 0 is the metastable supersaturated solid solu-tion, is a stable or metastable precipitate, and is amore stable solid solution with the same crystal struc-ture as 0 but with a composition closer to equilibrium.In general, the approach to equilibrium for the supersat-urated phase 0 can occur by at least two mechanisms:(1) nucleation and growth and (2) spinodal decompo-sition. The precipitation of excess arsenic is believedto occur by nucleation and growth.

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Nucleation occurs at the beginning of the precipita-tion process. The free energy difference between the 0matrix and the0matrix with-phase precipitates is thedriving force for the nucleation of precipitates. In thenucleation of -phase in a matrix of supersaturated 0-phase, B-atoms within the 0 matrix must first diffusetogether to form a small volume with the composi-tion. An interface must be created during this processand this also leads to a change in free energy. The totalfree energy change associated with the nucleation pro-cess will have three contributions

1GT D N.1ga C1gel/C .N/2=3

where 1ga is the difference in free energy per atombetween the supersaturated 0-phase and the -phase,1gel is the elastic strain energy per atom, is the inter-face free energy, and is the shape factor. The elas-tic energy here results from a change in volume orshape during the precipitation process, which will bediscussed in more detail later.

The first term in the above equation is usually nega-tive, and the free energy of the system will decrease asB-atoms are added to the precipitate (a driving force fornucleation). However, the interfacial free energy andelastic energy will increase as the nucleus grows. Thus,if the total free energy change is plotted as a functionof the nucleus radius, it will initially increase, reacha maximum point, and then decrease as the nucleusradius increases. The maximum (positive) change inthe free energy in this transformation is called thenucleation barrier.

The precipitate shape satisfying the above criterion isthat which minimizes the total interfacial free energy. Inthe simplest case, the particles tend to have a sphericalshape since this results in the minimum surface area fora fixed volume. This is the case of As precipitates inGaAs, the particles are observed to be nearly spherical.

(a) (b)850C 600C

Figure 2. High resolution (110) cross-sectional TEM images, showing arsenic precipitates formed in GaAs after annealing at (a) 850C(b) 600C.

Thus far, we have discussed the kinetics of an iso-lated precipitate and its surrounding matrix. If we nowconsider the situation with more than one precipitate inthe matrix, one would expect that the growth of eachprecipitate will influence the others so as to minimizethe total free energy of the whole system. This processis important in the late stages of precipitate growth, andis referred to as coarsening (Porter & Easterling, 1991).A high concentration of small precipitates will tend tocoarsen into a lower concentration of large particleswith a smaller total interfacial area in order to min-imize surface energy. Coarsening dominated by sur-face energy in this way is known as Ostwald ripening(Lifshitz & Slyozov, 1961).

In any two-phase system, including a matrix and pre-cipitate, there will be a range of particle sizes due to dif-ferences in the time of nucleation and the rate of growth.In the case of two adjacent spherical precipitates withdifferent diameters, the free energy per atom of thelarge precipitate is smaller than that of the small precip-itate. According to the well known method of commontangents (Porter & Easterling, 1991), one finds that thesolute (atom-B) concentration in the matrix adjacentto a particle is higher when the precipitate is smaller.Therefore, there will be a concentration gradient in thematrix which causes the solute (B atoms) to diffuse inthe direction from the smaller particle toward the largerparticle, so that the small particles shrink, and eventu-ally disappear, while large particles grow. The overallresult is that the total number of particles decreases andthe mean radius increases with time.

Precipitate size and structure

Figure 2(a) and (b) are (110) cross-sectional trans-mission electron micrograph (TEM) images of Asprecipitates in GaAs samples annealed under different

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thermal cycles. At the lower anneal temperature theprecipitates are small and very dense. At higher annealtemperatures, they are much larger and less dense, pre-serving the total arsenic volume.

The influence of the anneal temperature on the for-mation of As precipitates has been investigated byMelloch et al. using rapid thermal anneals (RTA) andfurnace anneals (Melloch et al., 1992). Figure 3 sum-marizes the average sizes and spacings of precipitatesas a function of temperature for a 30 s isochronal annealfor four different samples with various amounts ofexcess As in the range 0.20.9%. The data in Figure 3shows that the size of the precipitates can be designedto be in the range of about 220 nm over this tempera-ture range.

TEM images of large precipitates formed at 850Cand small precipitates formed at 600C are shown inFigure 4. The images of the large precipitates show setsof lattice planes that are different from those of the sur-rounding GaAs, indicating a distinct crystal structureand that the lattices of the precipitate and the matrix aresemicoherent (coherent in some directions and inco-herent in others). In directions where the planes of theprecipitate align with the matrix, misfit dislocations cansometimes be identified. The crystal structure of largeprecipitates has been determined to be hexagonal with(0003) planes oriented almost parallel to f111g planesof GaAs (Claverie & Weber, 1992). The image for thesmaller precipitate in Figure 4 appears to be coher-ent with the surrounding GaAs. This is typical of whathas been reported for As precipitates in the range of

Figure 3. Average sizes and spacing of As precipitates in GaAsand AlGaAs as a function of the anneal temperature (AfterMelloch et al., 1992). The calibration samples from our experi-ments are shown as black circles.

(a)

(b)

Figure 4. High resolution TEM image showing precipitatesfound in GaAs after annealing at (a) 600C for 30,000 s and (b)850C for 30 s.

24 nm. The crystal structure of such small As precip-itates, which has sometimes been referred to as pseu-docubic (Liliental-Weber et al., 1991), is not yet aswell established.

Bulk hexagonal As is a semimetal and there is exper-imental evidence that As precipitates behave as buriedmetallic Schottky barriers (Mahalingam et al., 1992;Ibbetson et al., 1993). The electronic properties ofsemiconductor nanocrystals in the size regime wherethe particle contains only a few hundred or a fewthousand atoms is strongly size dependent (Alivisatos,1996), and some size dependence can also be expectedin the case of semimetallic As precipitates. While theelectronic properties are critical for potential nanoelec-tronic applications, this subject is beyond the scope ofthe present study, which is concerned with the pattern-ing of the particles.

Preferential precipitation inheterostructure layers

In uniform layers of NS-GaAs, the precipitates nucle-ate homogeneously and are distributed uniformly in the

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layers. In a NS-AlGaAs/GaAs heterostructure, how-ever, precipitates form preferentially in the GaAs lay-ers (Mahalingam et al., 1992). This effect has beenattributed to the differences between the interfacialenergies associated with the arsenic precipitates in theGaAs and the AlGaAs matrix. The interfacial energy ofprecipitates in GaAs is lower than in AlGaAs becausethe GaAs bond is weaker than the AlAs bond. Fromthis, one would expect a diffusion of AsGa point defectsfrom the AlGaAs layers to the GaAs layers so that thetotal free energy of the system is reduced, similar toOstwald ripening. Thus, the preferential precipitationof As in GaAs is believed to be a coarsening processin which precipitates nucleate throughout the AlGaAsand GaAs layers but preferentially grow in the GaAslayers with continued annealing.

The above picture is supported by experiments onthe time evolution of the precipitate distribution shownin Figure 5. The samples were annealed at 600C forthree different times. At this relatively low annealingtemperature, the diffusion of atomic arsenic throughthe lattice is relatively slow and the buildup of particledensity within the GaAs well can be clearly seen.

In NS-GaAs, a high concentration of vacancies(1019 cm3) exists in the as-grown layers. The pres-ence of vacancies is important in the precipitation pro-cess since the diffusion of AsGa anti-sites is vacancyassisted, as discussed by Bliss et al. (1992). Duringannealing, vacancies will diffuse to vacancy sinks, suchas defect complexes or interfaces which include thesample surface, the layer interface, and the arsenic pre-cipitate interface. Due to the annihilation of vacanciesduring annealing, the supersaturated vacancy densitydecreases with time. Therefore, there is a strongdecrease in diffusion during annealing. Such detailsconcerning the time dependence of the point defectconcentrations are important to the preferential precip-itation in epitaxial layers (Hung et al., 1998) and couldalso play a key role in the strain patterning effects ofinterest here.

Strain directed assembly of 1D nanoparticlearrays

Kiehl and coworkers (1995) were first to demonstratethat the lateral position of arsenic precipitates in GaAscan be controlled by a stress structure (stressor). Theiroriginal idea was based on the notion that an inten-tional modulation of strain in the epitaxial structurecould be effective in determining the most favorable

Figure 5. Histograms showing the distribution of arsenic precip-itates in the growth direction for (a) 600C/30 s, (b) 600C/300 s,and (c) 600C/30,000 s annealing cycles.

nucleation and equilibrium positions for arsenic pre-cipitation in NS-GaAs. Initial experiments (Kiehl et al.,1995) showed a correlation between the position ofnear-surface precipitates in a homogeneous NS-GaAsstructure and line stressors fabricated on the surface.Control of both the lateral and vertical positions ofthe precipitates was then demonstrated by combin-ing strain patterning with preferential precipitation inan AlGaAs/GaAs heterostructure (Kiehl et al., 1996).Composition controls the vertical position of the parti-cles, while strain controls the lateral position, as illus-trated in Figure 6.

The structure used in these earlier experiments wasbased on a non-stoichiometric AlGaAs/GaAs/AlGaAssandwich that was capped with a thin InGaAs-basedlayer. The GaAs and AlGaAs layers in the sandwich

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Figure 6. Schematic diagram illustrating the use of strain andcomposition to control the position of As precipitates in anAlGaAs/GaAs heterostructure.

Figure 7. Cross-sectional SEM showing the strain directedassembly As precipitates in an AlGaAs/GaAs heterostructurewith line stressors fabricated on the surface. The precipitate diam-eter and stressor width are approximately 20 and 200 nm, respec-tively. The precipitates have formed in a 10-nm GaAs layer (light-gray region). (After Kiehl et al., 1996).

were approximately 10 and 50 nm, respectively. TheInGaAs layer in the structure is strained as a resultof its larger lattice constant. After the growth of thesandwich, the InGaAs cap layer was patterned bye-beam lithography, covered by a SiN dielectric film,and annealed by rapid thermal annealing. The epitaxialstructure, sample growth, and preparation proceduresused in the present work are similar to those used inthe earlier studies. Further details will be given later inthis paper.

0.1 m

Figure 8. Plan view TEM showing the formation of 1D arsenicparticle arrays beneath line stressors. The stressors are 90-nmwide and spaced with a 290-nm pitch. The particles beneath thestressors have an average diameter of 16 nm and an average edge-to-edge spacing of 23 nm. (After Kiehl et al., 1996).

Figure 7 shows a cross-sectional high resolutionscanning electron micrograph (SEM) image of thistype of sample after annealing. A single line of parti-cles approximately 20 nm in diameter can be seen cen-tered under each stressor. Particles are also found inthe spaces between the stressors. The particles are ver-tically positioned at the plane of the GaAs layer (thelight gray band in the image) at a depth of 45 nm fromthe etched AlGaAs surface.

In order to observe the distribution of particles alongthe stressor lines, the samples were also examined byplan-view (TEM). A plan-view TEM micrograph isshown in Figure 8 for a sample having approximately90-nm wide stressor lines set at a 290-nm pitch. It isseen that 1D arrays of particles with an average diame-ter of 16 nm and average edge-to-edge spacing of 23 nmare formed beneath the stressors.

The above results show that the particles (1) form ina single line beneath each stressor, (2) form a disperseddistribution centered midway between stressors, and(3) are almost completely absent within wide bandsrunning along each stressor edge.

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Time and geometry dependencies ofstrain patterning

Materials growth and sample preparation

In order to investigate the extent of the strain patterningand to gain a better understanding of the detailed mech-anism behind this effect, we examined the dependen-cies of the patterning on annealing cycle and the stres-sor geometry. The samples were grown in a Varian GENII MBE system on a two-inch diameter substrate. Thegrowth rates were typically 0.15m/h and 0.27m/hfor GaAs and AlGaAs, respectively, with a ratio ofgroup V to group III beam equivalent pressures equalto 36.

A 200 nm GaAs buffer layer was first grown on asemi-insulating substrate to eliminate defects from thesubstrate. This was followed by a 50-nm Al0:4Ga0:6Aslayer, which serves as a blocking layer for the diffusionof excess arsenic from the upper layers. These first twolayers were stoichiometric layers grown at 600C. Thesubstrate temperature was then lowered to 200C andthree non-stoichiometric layers were grown: 200 nm

(a) (b)

Figure 9. Plan-view TEM image and schematic illustrating the patterning for a 850C/30 s anneal.

of NS-Al0:4Ga0:6As, 10 nm of NS-GaAs, and 50 nm ofNS-Al0:4Ga0:6As. Finally, the substrate temperature wasraised to 450C and the following layers were grown:2 nm of GaAs, 12 nm of In0:3Ga0:7As, and a 4 nm GaAscap.

The top three layers of the structure are used in thefabrication of surface stressors. InGaAs has a largerlattice constant than GaAs, with a lattice mismatch of7.1% for pure InAs. When grown epitaxially on GaAs,the InGaAs lattice strains to match the in-plane latticeconstant of the underlying GaAs substrate. Strainedlayers that are free of misfit dislocations can be grown,provided that the layer is below a composition depen-dent critical thickness.

Surface stressors consisting of narrow lines formedin the strained top layers of the structure were fab-ricated by patterning the surface by e-beam lithogra-phy and etching through to the NS-Al0:4Ga0:6As layerwith a selective wet chemical etch. To prevent arsenicloss through the surface during the anneals, a Si3N4dielectric film was then deposited by plasma enhancedchemical vapor deposition at 350C. Different anneal-ing temperatures were employed in an ambient of

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(a) (b)

Figure 10. Plan-view TEM image and schematic illustrating the patterning for a 900C/15 min anneal.

Figure 11. Plan-view TEM image and schematic illustrating the patterning for a 900C/1 h anneal.

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nitrogen and forming gas for three different annealtimes, 30, 300 and 30,000 s. Rapid thermal anneal-ing (RTA) was used for the two shorter anneals and afurnace (quartz tube) anneal was used for the longestanneal. The distributions and sizes of arsenic precipi-tates in the structure were examined by TEM incross section.

Time dependence

Figures 9, 10, and 11 are the bright-field plan-viewTEM micrographs under the (001) zone-axis condition,showing the lateral distributions of arsenic precipitatesalong the InGaAs stressor lines in the annealed samplesfor the three annealing cycles. The main variable in thisset of experiments is the annealing time.

Figure 9(a), the annealing case of 850C/30 s, showsthat arsenic precipitates were uniformly distributedacross the sample with no evidence of strain patterning.Cross-sectional TEM analysis of this sample showedthat a preferential coarsening of arsenic precipitateshad occurred in the NS-GaAs layer so that most of theprecipitates were confined in this plane although theywere not strain patterned laterally, as illustrated in theFigure 9(b).

When the anneal time was increased to 15 min, anoticeable strain patterning appeared. Figure 10 showsa higher precipitate density beneath the InGaAs stres-sors. Close inspection of the micrographs indicated thatthe particle density beneath the stressors was higherthan in the spaces by a factor of about 2.4. We alsoobserved that the arsenic precipitates underneath thestressors are larger than those in between. These resultsshow that the strain patterning involves the diffusionof excess arsenic from the spaces to the region beneaththe stressors, as illustrated in Figure 10(b). Figure 11presents the case of the longest anneal, one hour. Thearch-shaped contrast and wide dark bands around andacross the InGaAs stressor lines are diffraction con-trast from the bending of the thin TEM specimen. Notethat, because of this influence, some arsenic particleswere either hidden or showed a gray contrast in thedarker regions. Three interesting results are shown inthis image: (1) a single line of arsenic particles with anaverage size of 14.4 nm beneath each InGaAs stressor,(2) a field clear of arsenic particles in the spacing regionbetween the stressors, and (3) a more uniform parti-cle spacing (90 nm in average) underneath the stressorscompared to those seen in Figures 9 and 10.

The formation of a single line of precipitates withcompletely clear fields shown in Figure 11 is quitedramatic. Moreover, the trend in the results of Figures 9through 11 provide information that is useful in deter-mining the details of the strain patterning mechanism.One possible scenario has been that strain modulatesthe nucleation barrier, so that precipitates nucleate ear-lier beneath the stressor. These larger particles wouldtend to grow faster than the smaller particles in thespaces, as in the case of normal Ostwald ripening. Ourresults show, however, that the size distribution of par-ticles in the early stage of growth is uniform with nonoticeable spatial dependence. Thus, it appears that theeffect of strain on the nucleation barrier does not playan important role in the patterning. Instead, it appears

(a)

(b)

Figure 12. (a) Plan-view TEM image and (b) histogram of thearsenic distribution beneath the stressors for a sample with 135 nmstressors. The annealing cycle was 900C for 1 h.

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(a)

(b)

Figure 13. (a) Plan-view TEM image and (b) histogram of thearsenic distribution beneath the stressors for a sample with 200 nmstressors. The annealing cycle was 900C for 40 min.

that the strain patterning is a coarsening effect, similarto what is observed for preferential precipitation dueto composition. In other words, rather than controllingthe position where particles initially form, the strainmodulates the redistribution of atomic arsenic betweenthe precipitates during the late stages of coarsening.As a result, the arsenic particles beneath the InGaAsstressor grow while others shrink and eventuallydisappear.

Geometry dependence

We have also experimentally examined the dependenceof the strain patterning on the width of the stressor.Figure 12 shows a plan-view TEM image for the same

(a)

(b)

Figure 14. (a) Plan-view TEM image and (b) histogram of thearsenic distribution beneath the stressors for a sample with 410 nmstressors. The annealing cycle was 900C for 40 min.

sample as in Figure 11, but over a larger area. The widthof the stressors in this case is 135 nm. A histogramshowing the lateral distribution of precipitates beneaththe stressors is shown in Figure 12(b). The distributionis sharply peaked in the middle of the stressor. (Notethat the extent of the horizontal axis in the histogramcorresponds to the stressor width.)

Figure 13 shows the plan-view TEM image and his-togram for a wider stressor having a width of 200 nm. Inthis case we observe that precipitates deviate from themiddle point of the stressor, i.e., the distribution broad-ens. Figure 14 shows the TEM image and histogram fora 410 nm wide stressor. At a width of 410 nm, arsenicprecipitates no longer form along the center line ofthe stressor. Instead, they form in narrow regions justinside of the two edges of the stressor, thereby forminga double-line array of particles. We will discuss thisdependence of the distribution on stressor geometry inthe next section.

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Stress and strain calculation

In order to gain a better understanding of the detailedmechanism of the strain patterning observed in ourexperiments, an analysis was made of the stress andstrain fields in our structure, which is shown schemat-ically in Figure 15. The sources of strain are (1) theepitaxial mismatch strain between InGaAs and GaAslayers, (2) the thermal strain during annealing dueto different thermal expansion coefficients amongInGaAs, GaAs, AlGaAs, and Si3N4, and (3) the strainproduced by the stress in the dielectric cap on the pat-terned semiconductor structure. The various param-eters include the composition of the InGaAs layer, thewidth (w) and thickness (t) of the stressor, and the com-position and thickness of the dielectric cap.

The origins of strain in a thin film grown on a sub-strate usually include thermal strain, intrinsic strainand epitaxial strain (Baker & Nix, 1990). These threestrains are described by:

Thermal : "misfit D .f s/.T T0/Intrinsic : "misfit D 1transformation=3

Epitaxial : "misfit D .df ds/=dswhere T0 is the growth temperature, is the ther-mal expansion coefficient of the material, and d is the

Figure 15. Schematic showing the geometry of the patterned epitaxial structure with the Si3N4 dielectric cap.

lattice constant of the material. 1transformation is the vol-ume change of a thin film if a phase transformationoccurs during the thermal cycle, the subscript f indi-cates thin film, and s stands for substrate.

The coefficients of thermal expansion and latticeconstants of GaAs, AlAs, InAs and SiN were takenfrom the literature (Neuberger, 1971; Walle, 1989),assuming a linear relationship with composition in thealloy systems. This gives an epitaxial strain betweenthe In0:25Ga0:75As and Al0:4Ga0:6As layers used in ourstructure equal to 0.0176. The thermal strains betweenthe In0:25Ga0:75As, Al0:4Ga0:6As, GaAs and Si3N4 lay-ers are usually two or three orders of magnitude lowerthan the epitaxial strain, even for temperatures severalhundred of degrees away from the growth or depositiontemperature.

We are interested in the stress and strain distributionsin the AlGaAs/GaAs/AlGaAs epitaxial layers wherethe precipitates form. The InGaAs layer and Si3N4 capcan be considered as surface features producing thestress and strain. The rest of the structure is taken to be anon-rigid GaAs substrate since the mechanical proper-ties of AlGaAs and GaAs are nearly the same. A generalpurpose finite element code, MARC (1988), was usedto solve the strains/stresses for the geometry involved.Because this code cannot directly model the epitax-ial strain, an effective thermal mismatch was used to

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approximate the epitaxial strain (Xu & Petroff, 1991).This was done by using an effective coefficient of ther-mal expansion (CTE) for the InGaAs layer, based onthe layer mismatch and the annealing temperature. Thesame method was used to model the stress conditionsof the Si3N4 films.

Figure 16 shows a diagram of the geometry usedin the simulation, including boundary conditions. Thethickness of the Si3N4 passivation over the stressor isthe same over the stressor and the substrate. The stres-sor, the dielectric, and the substrate are assumed to beperfectly bonded. The structure is constrained so thatit is not displaced in the x-direction at the nodes alongthe y-axis, as indicated in the figure. Moreover, thesubstrate is assumed to be much larger than the stressorin each direction and constrained not to move in eitherthe x or the y directions at the lower left node. Sinceour stressors are very long compared to their width andheight, the boundary conditions of the plane strain case

Figure 16. Geometry and boundary conditions for the structuremodeled in the finite element method calculations.

are used to reduce the situation to the two dimensionalcase.

Figure 17 shows the two-dimensional contour plotsof the FEM results for the strain distributions in thecase of a stressor thickness t and width w equal to 15and 100 nm, respectively. The Si3N4 layer was taken as70 nm. These values are close to the experimental val-ues for the narrow stressor. Figure 17(a), (b) and (c)show the "xx; "yy , and "xy fields, respectively. The rect-angular block of concentrated contours in Figure 17(a)shows the position of the InGaAs stressor. The regionto the right and above this block is the Si3N4 dielec-tric (which follows the contour of the patterned semi-conductor) and the region below the block representsthe epitaxial AlGaAs/GaAs layers and a portion of theGaAs substrate. Figure 17(a) shows that the InGaAsstressor is under compression in the x-direction. Thepatterning of the InGaAs layer allows the strain to relaxat its edges, thereby generating a tensile strain field inthe x-direction beneath the stressor. Figure 17(b) showsthat the strain field beneath the stressor is compres-sive in the y-direction as a result of the Poisson effect.Figure 17(c) shows a large shear strain is generated atthe edges of the stressor.

The experimental results described earlier in thispaper indicate that the particle distribution builds upin the buried GaAs layer due to compositional differ-ences prior to the effect of strain patterning. Thus, thestrain and stress distributions within the GaAs layercentered 50 nm below the surface should be the mostimportant for the strain patterning. In Figures 18 and19, we compare the calculated components of the strainand stress fields at a depth corresponding to the cen-ter of the GaAs layer for two different stressor widths,representing the narrowest and widest stressors exam-ined in the experiments.

Figure 18 shows results for a stressor width w equalto 100 nm. In this case, a transverse tensile strain "xxand a normal compressive strain "yy are produced in theGaAs well mainly by the strain relaxation at the edge ofthe stressor. These components have extremum valuesbeneath the center of the InGaAs stressor. The shearstrain, "xy is positive and its maximum appears under-neath the edge of the stressor. The dilatation strain, 8,which is defined by the sum "xx , "yy and "zz, is plottedin the same figure as well. Figure 18(b) is the stress plotfor the same stressor geometry. Each component of thestress shows a trend similar to its corresponding strain.xx in the GaAs well has a maximum value of 0.15 GPa

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Figure 17. Two-dimensional contour plots of the FEM results for transverse strain "xx , normal strain "yy , and shear strain "xy .

in tension. This is about one-tenth of the biaxial stressin a uniform In0:25Ga0:75As film on a GaAs substrate.The hydrostatic stress, 1T , which is equal to the sum1=3.xx C yy C zz/, is also shown. Note that neitherthe dilatation strain nor the hydrostatic stress exhibit astrongly peaked behavior and that the values of thesecomponents beneath the stressors are comparable tothose between the stressors.

The calculated strain and stress distributions at thedepth of the GaAs layer are plotted in Figure 19 fora large stressor width of 500 nm. The "xx and "yy

values beneath the stressor are tensile and compres-sive, respectively, as seen in the previous case of thenarrow stressor. However, the peak positions of bothstrain components shift from the center to a posi-tion just inside the edge of the stressor. The shearstrain is zero at the center and goes through negativevalues before reaching its maximum at the edge ofthe stressor. Note that the dilatation strain and hydro-static stress distributions are again relatively flat andtake on comparable values beneath and outside of thestressor.

342

Figure 18. The (a) strain and (b) stress fields in the NS-GaAslayer along the x-direction for t D 15 nm, w D 100 nm, and atensile Si3N4 cap.

Strain patterning mechanism

A reduction of elastic energy can be the driving forcefor the migration of excess arsenic in an inhomoge-neous strain field. The reduction in the free energyof the whole system can come from three sources:changes in particle shape due to diffusion, volumechanges of particles under stress, and modulus differ-ences between the particle and matrix materials. Weexamine these in the following.

Effects due to particle shape

When a precipitate is under a non-uniform stress field orthe applied stresses at the interface between the precip-itate and the matrix are not symmetrical, the precipitateis loaded with a stress state, which includes a pure shearstress. This situation is illustrated in Figure 20(a). The

Figure 19. The (a) strain and (b) stress fields in the NS-GaAs layeralong the x-direction for the same parameters as in Figure 18,except w D 500 nm.

presence of the shear will create normal traction on thesurface of the precipitate such that two diametricallyopposed points on the sphere are positions of maxi-mum compression, and perpendicular to this diameterare two diametrically opposed points which are cen-ters of maximum tension. As a result, the shape of theprecipitate can change from a sphere to an ellipsoidwith an elongation in the original direction of maxi-mum tension until the stresses at the interface becomesymmetric, as shown in Figure 20(a).

The diffusion relaxation of the shear is accompaniedby a decrease in the elastic interaction energy, which isusually proportional to the square of the shear stress.However, the magnitude of this effect is usually muchsmaller than those from the effects of size and moduluschange. Moreover, the migration of the solute atoms inthis case usually occurs locally in each precipitate, notamong the precipitates. Thus while this effect might

343

Figure 20. Schematic showing the effect of (a) diffusion, (b) vol-ume change, and (c) shear modulus differences on the elasticinteraction.

lead to a slight modulation in the shape of precipitatesin our structure, it is unlikely that this effect plays animportant role in the strain patterning.

Effects due to volume differences

If we regard the precipitates as elastic spheres, theenergy of a precipitate in a crystal can be considered asthe strain energy created in the system by the expansionor contraction of a cluster of solvent atoms to a sizeequal to that of the same number of solute atoms. Inthe present case, the excess arsenic atoms are consid-ered as the solute and the GaAs matrix as the solvent.The strain energy of an arsenic precipitate in the GaAsmatrix can be simply estimated within this picture ifwe consider that the formation of an As precipitate isaccomplished by replacing Ga atoms originally locatedin this volume with As atoms. For the case of a stress-free medium, this strain energy is due entirely to theelastic resistance of the material (GaAs) surrounding

the (As) precipitate. However, if the expansion (or con-traction) occurs in a region already subjected to a stress(strain) field, extra work will be done against the forcesacting on the precipitate from this field. This will alterthe total strain energy of the system by an amount thatis called the energy of interaction of the precipitate andthe stress field.

When the stress field is non-uniform, the work doneby expanding a precipitate against the field will varywith the position of the precipitate in the field. Becauseall systems have a tendency to decrease their freeenergy (the strain energy in this case), atoms in theprecipitate will experience a force driving them to theregion where the precipitates can have a lower inter-action energy. The volume of each precipitate changescorrespondingly.

Figure 20(b) is a schematic diagram showing thechanges in volume among particles. The variation of theinteraction energy due to the volume change is given by

1U volumeint .xi; yi; zi/ D .Tint1V /i D .Tint V 2/iwhere Tint is the hydrostatic stress, 2 is the dilatationand V is the original volume of the precipitate formedby the solutes. We see that this quantity depends uponthe dilatation and the hydrostatic stress in the neigh-borhood of the precipitate.

The finite element calculations show that the spatialdependence of the dilatation and hydrostatic stress areabout the same for the wide and narrow stressor geome-tries. Thus, the trends in these quantities cannot explainwhy the precipitates form a single-line array beneath anarrow stressor and double-line array beneath a widestressor, and it is clear that the strain patterning doesnot correspond to a minimization of the energy of inter-action related to the volume differences between thehexagonal As and GaAs crystals.

Effects due to modulus differences

The atoms in the precipitates and the atoms in thematrix may have the same size but different stiffnesses,one being elastically softer than the other. The interac-tion energy is also affected by the difference betweenthe stiffnesses of the materials (Fleischer, 1961). Suchan elastic non-uniformity interacts with the stress field.Under a uniform stress field, an additional interactionenergy will be generated and added to the total elasticenergy of the whole system. If the stress field is non-uniform, the interaction energy of each particle willvary with its position in the field. Thus, there will be

344

a force driving the migration of solute atoms betweenprecipitates so as to reduce the total elastic energywith the result that some precipitates grow while othersshrink.

Figure 20(c) is a schematic diagram showing theeffect of modulus change on the elastic interaction.Consider a homogenous, finite, linear elastic solidwhose external boundary Sext is subjected to givendisplacements UA (strains) and given tractions T A(stresses) on the portion of Sext denoted as Su. Nowimagine that a volume V0 with bounding surface S0 hasits elastic stiffnesses changed from their original con-stant values Cijkl to new constant values given by Cijklwhile the original boundary conditions onSext are main-tained, as illustrated in Figure 20(c). The change of theinteraction energy produced by this modulus differenceis given by

1Umodulusint .xi; yi; zi/ D 1=2.Cijkl Cijkl/ZV0;i

eAije0kl

dV

where eAij

and e0ij

represent the strain states before andafter the elastic constants are changed.

In the present case of hexagonal-As and cubic GaAs,the particle and matrix have very different elastic con-stants. GaAs has a zinc-blend structure with stiffnesscoefficients C11 D 118 GPa, C12 D 53 GPa and C44 D59 GPa; arsenic has a hexagonal structure with the stiff-ness coefficients C11 D 130 GPa, C33 D 58:7 GPa,C44 D 22:5 GPa, C12 D 30:3 GPa, C13 D 64:3 GPa andC14 D 3:7 GPa. Comparing C44 for the two mate-rials, we see that hexagonal-As is elastically softerthan GaAs. Therefore, the effect of modulus differ-ences on the elastic interaction could play an impor-tant role in the relaxation of elastic strain energy inour case. In the following section, we examine thispossibility.

It should be mentioned that the As precipitate andGaAs matrix lattices are semicoherent in the particlesize regime of these experiments, as discussed earlier.However, even in the case of incoherent particles, therelaxation of strain will, in general, depend on the rela-tive elastic properties of the particle and matrix. Only inthe special case of pure shear strain can the relaxationbe independent of the elastic properties of the particle.In the general case, where there are hydrostatic com-ponents to the strain state (as seen in Figure 19), therelative elastic properties will play a role even if theinterfaces can freely slide and allow easy diffusion.

Strain energy distributions

To examine the effect of modulus difference on strainpatterning, we make two simplifying approximations.First, the GaAs substrate and arsenic precipitates aretaken to be elastically isotropic materials. Second,since the difference in the strains before and after themodulus change are generally very small, we assumee0ijD eA

ij. Thus, the above equation can be simplified to

be (Timoshenko & Goodier, 1970)

1Umodulusint .xi; yi; zi/ D 1=2.E E/ZV0;i

eAijeAkl

dV

D 1=21EXjk

"2jk

D 1=21EEEXjk

"2jk

.j or k D x; y; z/D fUi

whereE andE are the Youngs modulus of arsenic andGaAs, respectively, since the stiffness in any directionis equal to the Youngs modulus for an isotropic mate-rial. The "jk are the components of the strain tensor andUi is the strain energy originally stored in the system,such as that induced by the stressors, and is given by

Ui DXjk

1=2E"2jk:

The factor f represents the ratio of the modulus dif-ference to Youngs modulus of GaAs. Since arsenic iselastically softer than GaAs and most of its stiffnesscoefficientsCijkl are smaller than those of GaAs,EEand, in turn, the factor f have negative values in ourcase. Since the strain energy is always positive, weknow from the equations above that the change of mod-ulus from E(GaAs) to E(As) in the volume of V0results in a reduction of the interaction energy (or thetotal elastic energy of the system). This will be a forcedriving the migration of excess arsenic as discussedabove.

The importance of the above is that it shows that thechange in the interaction energy that occurs when a par-ticle is introduced at a given position is simply propor-tional to the original strain energy at that position. Theinteraction energy is reduced when the particle is softerthan the lattice. Thus, due to the modulus differencesfor our materials, the arsenic precipitates should tendto grow at positions of larger strain energy, since thiswill reduce the total free energy for the whole system.

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Figure 21. Comparison of (a) the calculated strain energy distri-bution and (b) the experimental particle distribution for a 100 nmwide stressor. The strain energy at the depth of the confining GaAslayer (the particle plane).

Plots of the calculated strain energy in the GaAslayer are shown for a narrow stressor in Figure 21and a wide stressor in Figure 22. Histograms of theexperimental particle distributions for narrow and widestressors are also shown in the figures. It can be seenthat the experimental particle distributions are in excel-lent agreement with the peaks in the calculated strainenergy plots. Therefore, we conclude that the reduc-tion in strain energy related to the difference betweenthe modulus of the As precipitates and the GaAs matrixis responsible for the strain patterning observed in ourexperiments.

From the above, one can roughly explain the trendsin the lateral patterning effects of the precipitatesobserved in our experiments as follows. The inhomo-geneous strain fields in the epitaxial AlGaAs/GaAs lay-ers are mainly caused by the strain relaxation at theedges of the InGaAs stressors. When the stressor is verywide, the strain is maximum near its edges and smallnear its center. As the width of the stressor decreases,the peak of strain moves towards the center from both

Figure 22. Comparison of (a) the calculated strain energy distri-bution and (b) the experimental particle distribution for a 410 nmwide stressor. The strain energy at the depth of the confining GaAslayer (the particle plane).

edges, finally merging and becoming a single peak atthe center.

To exploit the strain patterning effect, the stressorshould be designed to produce a maximum in strainenergy at the desired particle position. In structuressuch as ours where layer composition is used to con-trol the vertical position of the particles, it is the strainenergy in the plane where the particles will be confinedthat is most important. In the simple case where it isdesired to form particles in a single line beneath thestressor, narrowing the stressor leads to a single peak inthe strain energy distribution. However, since the pene-tration depth of the strain field decreases in proportionto the stressor width, narrowing the stressor reduces thestrain at a given depth. Thus there is an optimum widthfor strain patterning. This is illustrated in Figure 23,which shows the calculated strain energy in the confin-ing GaAs layer as a function of stressor width.

Strain energy calculations such as those describedhere should be useful for designing stressors for assem-bling other configurations of particles. For example,

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Figure 23. Calculated strain energy distribution for various stres-sor widths. (The width w is shown in .)

designs optimized for strain release at the corners of asquare or polygon-shaped stressor could be used to fab-ricate assemblies of closely spaced particles in nearlyarbitrary configurations.

Conclusion

The use of strain to direct the assembly of nanoparticlearrays in a semiconductor was investigated experimen-tally and theoretically in this paper. Remarkable pat-terning effects, including the formation of single anddouble one-dimensional arrays with completely clearfields were achieved for 10-nm particles at a depth of50 nm from the surface. Experimental results on thetime dependence of the strain patterning indicate thatstrain controls the late stage of the coarsening process,rather than the precipitate nucleation stage. Compari-son of the observed spatial particle distributions withtheoretical calculations of the stress and strain distri-butions in the structure reveals that the precipitatesform in regions of maximum strain energy, rather thannear extremum points of hydrostatic stress or dilatationstrain. Based on these results, we conclude that the pat-terning results from modulus differences between theparticle and matrix materials, rather than from volumedifferences or other strain related effects. Thus, it isthe relative softness of the As precipitates compared tothe GaAs matrix that is the key to the strain patterningobserved in our experiments.

The results presented here should be useful forextending strain directed assembly to other materi-als systems and to other configurations of particles.

An important feature of this process is that, in addi-tion to providing a means for assembling arrays andarbitrary configurations of particles, strain patterningalso provides a means for self-aligning particles to anoptical or electrical access structure, since this struc-ture could be the stressor itself. Thus, strain directedassembly could provide an attractive high throughputfabrication technology for nanoelectronic circuits andother applications. One interesting example is single-electron tunneling circuitry (Likharev, 1999), which isbased on electron tunneling in assemblies of conduc-tive nanometer-scale particles.

Acknowledgements

The authors would like to acknowledge technical con-tributions from T. Cuk. R. A. Kiehl would also like tothank his coworkers at Fujitsu Laboratories, who col-laborated in the early experiments on this topic. Thiswork was supported by the Air Force through contractF49620-97-1-0444 and by ONR/DARPA through con-tract N00014-96-1-0983. Support from Fujitsu Labo-ratories, Ltd. is also gratefully acknowledged.

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