energetics of misfit- and threading-dislocation arrays in heteroepitaxial films

PHYSICAL REVIE%' 8 VOLUME 44, NUMBER 3 15 JULY 1991-I

Energetics of misfit- and threading-dislocation arrays in heteroepitaxial films

A. RockettThe Coordinated Science Laboratory, The Materials Research Laboratory,

and the Department ofMaterials Science and Engineering,University of Illinois, 1101 West Springjield Avenue, Urbana, Illinois 61801

C. J. KielyDepartment ofMaterials Science and Engineering, University ofLiverpool,

P.O. Box 747, Liuerpool, L69 3BX, United Kingdom(Received 15 August 1990; revised manuscript received 6 December 1990)

A theory relating the separation of misfit dislocations to lattice mismatch and film thickness inheteroepitaxial thin films is presented. From this, the energy as a function of dislocation spacing is cal-culated and is shown to include an attractive and repulsive region. The dislocation-formation energyand Peierls barrier to network ordering are shown to be estimable on the basis of measured dispersionsin dislocation spacings. The spacing is predicted to be more uniform as the mismatch increases. Ther-modynamic functions, such as the compressibility of the dislocation network, can be calculated from theenergy —dislocation-spacing relationship. A formula relating the equilibrium dislocation spacing to film

thickness, mismatch, and misfit-dislocation character is also derived. Finally, the density of threadingdislocations is calculated both at the heterojunction and at the film surface, by assuming a threading-dislocation reaction process. The results are shown to be in good agreement with the experimental datafor Si, Ge& /Si, InSb/GaAs, and In„Ga& As/GaAs structures.

I. INTRODUCTION

One of the principal limitations to progress in the ap-plication of semiconductor heterojunctions in state-of-the-art devices is control of the threading-dislocationdensity in the layers. This is especially important forfilms grown epitaxially on lattice-mismatched substrates.Improved methods for reducing the density of threadingdislocations in the film are waiting for the development ofa comprehensive fundamental model for the formationand alignment of such dislocations. Theoretical studieshave approached the problem of developing a relation-ship between the critical thickness at which dislocationsform and the bulk lattice mismatch in two ways. Thefirst involves balancing the force required to form an ar-ray of noninter acting dislocations against the strain-induced stresses in the film. ' The second method mini-mizes the total energy which is a sum of the energy ofnoninteracting dislocations and the lattice strain ener-gy. ' These theories appear to produce conflicting re-sults. However, a recent work by Willis, Jain, and Bul-lough reconciled these approaches, showing them toproduce similar results, while, at the same time, includingthe possibility of dislocation interactions. It is clear fromthe fact that ordered dislocation arrays are produced thatthe interaction energy of dislocations in a misfit arraymust be significant. This was clearly demonstrated in re-cent results in which 60'-type misfit dislocations in aSi Ge& /Si heterojunction were pushed apart by newdislocations entering between preexisting ones.

In this paper we present an alternate derivation of theenergy minimization formalism which includes the dislo-cation interaction energies. It results in an explicit rela-

tionship giving the equilibrium dislocation separation Das a function of the film thickness h and the bulk latticemismatch for any character of dislocation. The result isshown to yield a critical thickness which is qualitativelyidentical to that of Matthews and Blakeslee but is basedon the more straightforward energy minimization ap-proach. Thus we confirm the qualitative similarity of thetwo previously reported approaches. We then proceed touse the formalism developed to calculate the excess ener-

gy available for driving dislocation formation or orderingas a function of array spacing. We estimate the allowedvariations in dislocation separation in a net, and calculatethe rate of introduction of threading dislocations into thefilm as misfit dislocations are added to remove remnantstrain.

II. MODEL

In this section we derive a relationship among total en-ergy, lattice misfit, and misfit-dislocation spacing. Thederivation is based on a uniform network of dislocationswith an average spacing D in a film of thickness h. Multi-layer films are planned to be dealt with in future papers.The spacing is assumed to be the same in the two orthog-onal directions of the lattice defined by the misfit-dislocation lines. Local deviations from uniformity areconsidered in Sec. III B. The dislocations are further as-sumed to be a single type; for example, all 60' type or allpure edge type. The coordinate system used in the fol-lowing derivation is defined in Fig. 1.

It is convenient to consider a unit of surface area (inwhich the energies are calcu1ated) to be a square of sideD. Thus the unit surface area always contains a length of

1154 1991 The American Physical Society

ENERGETICS OF MISFIT- AND THREADING-DISLOCATION. . . 1155

bx g

b =a/2 [10'1](111)f(D)=f„——

D(4)

(001)

= ~ [«0] For a fractional misfit f (D), the strain energy per unitarea in the film is given by

E, =296 l f (D)]

FIG. 1. A schematic diagram giving the definition of thedislocation line direction g and Burgers vector b with respect tothe interfacial plane as well as the angles referenced in the text.

E, E, 2D Ed+D2 D2 D2

E, 2Ed+ (2)

where Ed is the energy of the two dislocations in the areaD and includes interactions between dislocations in adja-cent unit cells. The energy associated with the intersec-tion of the two orthogonal dislocations in the unit cell isignored in this treatment. For pure edge-type disloca-tions this is a reasonable assumption. However, the in-teraction energy between 60' dislocations with orthogonalline directions may be more important. The interactionsamong 60'-type dislocations are discussed in Ref. 7.

The fractional misfit between the film and substrate lat-tices, with bulk lattice constants a& and a„respectively,is defined as

2la~ —a,aI+a, a

For the current analysis the misfit is a variable which de-pends on the spacing D and Burgers vector b of misfitdislocations in the interface. The contribution to misfitrelief perpendicular to the dislocation lines is proportion-al to the projected edge component b cosP of b in theplane of the interface as viewed along the dislocation line.The fractional misfit relieved is b cosP/D and P is definedas shown in Fig. 1. The total misfit as a function of dislo-cation spacing is then given by

dislocation line 2D. The choice of boundaries is suchthat they are high-symmetry lines in the interface and thearea represents a unit cell of the surface. Furthermore,the amount of dislocation line need not be calculated in afixed area, which simplifies development of the subse-quent results. The total energy of the system for a squaresubstrate of side L, is given by the energy per cell E,multiplied by the number of cells n =L, /D squared, or

LiEEtot

D

The equilibrium dislocation spacing is found by minimiz-ing E, /D with respect to D. For a two-dimensional pla-nar misfit-dislocation net with a uniform spacing D, E, isrelated to the strain energy E, and the dislocation net en-ergy per unit-cell area by

pb (1—vcos 9) aLL ln4'(1 —v) b

The interaction energy for two parallel dislocations of ar-bitrary character is

9) D ~b 'singL ln

2m(1 —v) L 2n(1 —v)(7)

where 0 and 4 are defined in Fig. 1. The total dislocationenergy is then given by Ed =2E, +E, . From Eqs. (6) and(7) taking the length of the dislocations to be D,

Dpb 2 ar

(1—vcos 0)ln +sinOcos+2vr 1 —v

where r is the effective distance to which the strain fieldextends and o, is a correction term of order unity whichaccounts for the core energy of the dislocation. For pureedge dislocations, 4'=90. A range of values of a from 1

to 4 have been used previously. We have followed Dixonand Goodhew by selecting o.=3.

When the dislocations are separated by a large distanceD))h, the strain field is assumed to terminate at thenearest film surface, dislocation interactions areinsignificant, and r =h. The assumption of a cutoff instrain field at a distance h is reasonable if the shearmoduli of the film and substrate are nearly equal. Theimage force introduced by a dislocation array of oppositetype located outside the surface by a distance h nearlyeliminates the strain field of the interfacial dislocationsfar into the substrate. The strain field will include a con-tribution from the film and a contribution from the sub-state. Thus the value of p should be a weighted averageof the film and substrate values. For the model presentedhere the strain field deep in the substrate is assumed to be

where p is the shear modulus and v is the Poisson ratio ofthe film. This formula accounts for the deformation ofthe film normal to the interface, and assumes that all ofthe strain energy occurs in the film. A small distortion inthe substrate atom positions also occurs which is general-ly much less than in the film. For roughly equal shearmoduli in the film and substrate, the strain energy in thesubstrate is less than the strain energy in the film, to firstorder, by the ratio of h/t. Here, t is the substrate thick-ness. Thus for thin single layer films the assumption ofno elastic energy in the substrate is valid.

The energy of a dislocation line per unit length is foundby summation of the self-energies and interaction ener-gies of dislocations at a separation of D and with lengthL. The self-energy per length L for any type of disloca-tion is

1156 A. ROCKETT AND C. J. KIELY

eliminated by the image forces at the nearby surface.When the shear moduli of the film and substrate aresignificantly different the strain field does not cancel andextends to the farthest boundary of the crystal. The cal-culation to estimate the error is analogous to solvingPoisson's equation for a charge at the interface betweentwo materials of different dielectric constant. The errorenters only logarithmically and is ignored in this treat-ment.

When D is less than h the strain field interaction dom-inates the dislocation energy and the crystal dimensionterm drops out of the resulting energy expression. Equa-tion (8) remains unchanged with r =D. For a dislocationnet the interaction energy per unit length is doubled byinteraction with dislocations in both adjacent unit cells.However, only half of this energy is distributed to eachdislocation in the unit cell with the other half going tothe dislocation with which the interaction occurs. Thusthe energy per unit length of a dislocation in a regular ar-ray of spacing D «h is given bypb', oD

(1—vcos 0)ln + sinO cos%'D 2'(1 —v) b

by multiplying both sides of these equations by 2/D be-cause a length of dislocation 2D is contained in an areaD . This dislocation line energy term can be combinedwith Eq. (5) to yield the total strain-related energy of thebilayer,

E, 1+v b=2phf (D) + (1—vcos 0)lnD2 1 —v AD (1 —v) b

+ sinO cos%

(10)

where r =h for h «D and r =D for D «h. The equilib-rium dislocation spacing is found by minimizing this en-ergy per unit area with respect to D usingd(E, /D )/dD =0. Differentiating Eq. (10),

d(E, /D ) 1+vD =4phb cosP f (D)dD 1 —v

(1—v cos 0)lnp~ q orrr(1 —v)

Because there are two orthogonal dislocation nets presentin the sample the total amount of dislocation energy isdoubled [see Eq. (2)]. Equation (8) and the correspondingexpression (9) are in units of energy per length D ratherthan in energy per unit area D . The units are corrected

I

+ sinO cos%

+ pb (1—vcos 0) D dr~(1—v) r dD

Setting this equation to zero and solving for h yields

h= b (1 —vcos 0)ln(arlb)+singcos+ —(Dlr)drldD4~(1+v)cosP f(D)

D (h)b cosp

b,a /a —gp /h

where

b

4~(1+v)cosQ

(13)

(14)

For D~ co, r =h, and f (D) =kala, Eq. (12) is quali-tatively identical to that of Matthews and Blakeslee' forthe critical thickness at which strain relief occurs butrepresents an equilibrium strain relief condition. The ma-jor differences between Eq. (12) and Matthews andBlakeslee's Eq. (5) in Ref. 1 are a factor of 2 due to thedifferentiation of [f(D)] and the terms involvingsingcos%' and (Dlr)drldD. The former results fromdislocation interactions while the latter is due to theeffective cutoff of the strain field at the nearest crystalboundary. To this point we have verified the results ofWillis, Jain, and Bullough showing that the result ofMatthews and Blakeslee' is qualitatively identical tothat obtained by Frank and van der Merwe ' using a to-tal energy minimization and including dislocation in-teractions.

A more useful result is obtained by solving the energyminimization for D(h). Substituting for f (D) from Eqs.(3) and (4),

I

is a constant and

p =(1—vcos 0)ln +singcosq' ——cxr D drb r dD

(15)

Note that Eq. (13) can be rewritten as

DD(h)=

1 —h, /h(16)

where h, =gp/(ha/a) is the equilibrium critical thick-ness for the onset of strain relief [from Eq. (12) with r =hand f (D) = b,a /a]. For h —+ ~ Eq. (16) yieldsD =b cosP/(ha/a), as it should. When h issmall, D(h) diverges at the critical thickness. Note thatp is either (1—v cos 0)ln(ah /b)+sing cosqI or( 1 —v cos 0)ln( aD /b) +sing cosV —1 depending on thevalue of r. When h «D, r =h and p is increasing with h.A maximum in p occurs in the range where h -D abovewhich p decreases to in[a cosP(ba/a) ]+sing cos% —1 atlarge h. In general p will lie in the range of 1 —7 withsmaller values for higher misfit systems. Variations in pthus represent minor, slowly varying corrections to thedislocation spacing.

The expression for D(h) can be used to calculate therate at which misfit dislocations are introduced into the


film as a function of h. In a region of wafer surface of di-mension L there will be n =L/D dislocations. The rateof addition of dislocations in a length of surface L as thefilm thickness changes is given by d ( n IL ) Idh=dD '/dh = D—dD/dh. Differentiating Eq. (13)with respect to h yields

dD ( 1/h )dp /dh —p /h=b cos( b,a la —

gp /h )(17)

For h ))h, =gp/(ha/a), dpjdh =0. Substituting fromdD/dh =D d(n/L) jdh,

d (n/L) bgp cosPdh D ( h b,a /a —

gp )(18)

d (n IL)dh

g(p —1)b cosg[h —gp/(b, a/a) ]

(20)

It should be noted that these formulas are for an equilib-rium dislocation spacing. Variations from equilibriumare considered in the next section.

III. DISCUSSION

The following assumes a heterojunction grown on a(001) surface of a diamond or zinc-blende semiconductor.It does not account for the possibility of misfit disloca-tions extending out of the plane of the interface. Thiscould be driven, for example, by attraction to the surfacedue to image forces or by large difFerences in shearmodulus across the interface. The interaction betweendislocations will depend on their Burgers vectors and linedirections. When the majority of dislocations are of the60 type, a large number of possible interactions result.Hence the following analysis is valid for locally homo-geneous regions of the dislocation net.

A. Nonequilibrium dislocation spacing

As strained layer epitaxy begins with h & h, there is nophysical solution of Eq. (16) and the total energy of thesystem is increased by adding misfit dislocation at anyspacing. Several possibilities exist for misfit relief as thelayer growth proceeds beyond h =h, . When the misfit issmall and growth is two dimensional, 60'-type disloca-tions are introduced by extension of existing threadingdislocations, ' by operation of a threading-dislocationsource, or by expansion of loops by glide on [111] planesbetween the heterointerface and the surface. When themisfit is large and growth is three dimensional, perfectedge dislocations may be formed at the island perimeters.These glide along the interface to form the misfit net-work. An intermediate case must occur if the edge dislo-cation propagation process has only partially relieved theultimate misfit (D)D„)when coalescence occurs. In

The second term in the parentheses is small for h ))h, .Assuming that D b-cosPI(ha ja),

d(n IL) gpdh h ~b cosP

near the end of the strain relief process. When h ((D,dp/dh =h ', and

this case the remainder of the strain relief is carried outby 60'-type dislocations as in the low-strain case. In gen-eral, some pure edge-type dislocations and some 60 dislo-cations which are non-strain-relieving are observed. Themodel presented in Sec. II is applicable to regions of theinterface which are relatively homogeneous in dislocationtype. The introduction of, for example, edge-type dislo-cations into a 60 -type dislocation net will introduce localperturbations in the net.

We examine first the case where growth is two dimen-sional and strain relief is almost entirely by 60 -type dislo-cations. This is the case for the In Ga, As/GaAs andGe„Si& „/Si heterostructures. When h )h„therrno-dynamics favors the introduction of dislocations to re-lieve strain but a kinetic barrier for formation and/orpropagation of the dislocations remains. ' '" When h risessufficiently above h„preexisting threading dislocationscan be displaced to produce misfit-dislocation segments. '

Because the dislocation segments are present in the film,only the energy to extend the dislocation through thefilm, no formation energy, is required. The energy forsuch dislocation extension can be estimated from Eq. (10)using the initial thickness of films at which threading-dislocation extension is observed.

Experimental values for dislocations spacing as a func-tion of misfit are available for the In„Ga& As/CxaAssystem for x =0.10, 0.15, 0.20, and 0.25. We have es-timated the critical thicknesses as a function of misfitbased on these data. The procedure was as follows. Thebest fit was obtained to the data using Eq. (16) with Dcalculated based on bulk lattice parameters. The value ofh, was allowed to vary as a fitting parameter. The calcu-lated curve was found to lie below the observed disloca-tion spacings. This is due to the material being in a meta-stable excess strain state, ' D )b cosP j(b,a ja). Thefit value of h, was compared with values calculated fromEq. (12). Critical thicknesses, fitted to experimentalvalues of D (h) for two orthogonal directions in the inter-face, and calculated values are presented in Table I. Thecalculated values of h, are in good agreement with thefitted values for most of the experimental data pointsespecially in the case of the [110] direction. These criti-cal thicknesses correspond to strain energies of 0.65 and0.82 eV/atom assuming an atomic density of -3X10'atoms cm and give a measure of the excess energy re-quired to drive threading-dislocation extension.

Dislocation generation from operation of a pointsource produces misfit dislocations which must forcethemselves between those generated in previous cycles ofthe source. An example of the spreading of preexistingmisfit dislocations during the introduction of a new dislo-cation is shown in Fig. 2. As for the threading-dislocation extension case discussed above, dislocationsforcing their way between existing segments require a netexcess strain energy to drive their formation. However,for the dislocation to squeeze into the network this driv-ing force must also be sufhcient to overcome forces resist-ing spreading of the network. This is shown schernatical-ly in Fig. 3. The preexisting dislocations move apart un-.til the available excess strain energy plus the energygained by reduction of repulsive interactions is less than


TABLE I. Estimated values for critical thickness for the In Ga& As/GaAs system based on experi-mental data from Ref. 7.

Infraction Misfit

Aa /ah, ([110])

(nm)

Experimentalestimates

h, ([110])(nm)

Calculated fromEq. (12)

h,( )

0.100.150.200.25

0.0070.0100.0140.017

45351411

50401915

36.522.315.711.7

the Peierls barrier for motion of the outermost disloca-tion. To force a dislocation into a preexisting net, thestrain energy must be sufficient to drive dislocation linecreation [from Eq. (9)] and to create it in a network ofspacing equal to that at which spreading stops. Thismeans that the film must be thicker to introduce disloca-tions into a network by forcing them between preexistingsegments than would be needed for creating isolated seg-

' "" " h"- W%WW'"'iy'i

h ihhg!!).1

Q:::Th

ments. Clearly, this can only happen when dislocationnucleation is slow. When the network spreads as a dislo-cation is introduced we refer to it as being in compres-sion, since the expansion stops before it would if thePeierls barrier was zero. When the network could adddislocations without spreading the preexisting net but islimited by the dislocation nucleation rate, we refer to thenetwork as being in tension.

The energy of a uniform network as a function of Dmay be calculated from Eq. (10). Representative curvesof E,(D)/D are plotted in Fig. 4. The plots clearlyresemble the interaction between particles in a lattice.The attractive portion of the curves results from strainrelief while the repulsive portion combines the strain fieldrepulsion of the dislocations with strain over-relief. Asthe mismatch between the materials increases [Fig. 4(a)]the energy minimum becomes sharper and the minimumenergy becomes greater. A similar behavior is found asthe thickness of the film increases, as shown in Fig. 4(b).When D =D and h )D, f (D)=0 and E, does notdepend on h. Hence all E,(D, h) curves pass through acommon point at E,(D, h )D„).Note that forsufficiently thin films there is little difference in networkenergy as the network expands (there is little strain ener-gy to relieve) and a wide range of dislocation spacingsmay exist with little penalty in total system energy.

If the energy barrier to network ordering is constant,the dispersion in dislocation spacings will be smaller in

ThreadingDislocationSegment

FIG. 2. Two transmission electron micrographs obtained bydigitization of video images from Ref. 6. The micrographsshow a propagating misfit-threading-dislocation unit at astrained Gep 2Sip g/Si heterojunction forcing itself between twopreexisting segments of misfit dislocation. The dislocation linesegments in the micrographs are shown schematically in the in-sets. In (a), a misfit dislocation generated by motion of a thread-ing dislocation has interacted with two orthogonal dislocationsin passing and has begun to repel a parallel segment. (b), taken1.2 sec later, shows the parallel segment continuing to moveaway from the new misfit dislocation due to the strain field in-teractions. The scale marks are accurate to approximately+10%.

FIG. 3. A schematic diagram showing the forces controllingexpansion of a dislocation network as a new dislocation is intro-duced from a remote point source between preexisting misfitdislocations of the same type. The strain field results in a forceF, repelling surrounding dislocations while the Peierls energybarrier results in a force F~ which opposes dislocation motion.Expansion of the network continues until F, =F .

ENERGETICS OF MISFIT- AND THREADING-DISLOCATION. . . 11S9

more strongly mismatched systems or in thicker films.However, as the number of dislocations which must bedisplaced to produce ordering increases the energy bar-rier is likely to increase. Although Figs. 4(a) and 4(b) arecalculated for an infinite, well-ordered dislocation net theprimary energy terms will be relatively local. Hence theillustrated behavior is probably correct to order of mag-nitude even for very small groups of dislocations. Theenergy barrier to dislocation generation and ordering

must be less than the maximum value, from Fig. 4, cor-responding to the largest or smallest observed spacings.

The explicit form of E (L)) in Eq. (10) makes it possibleto calculate thermodynamic properties of the network.As an example we calculate the network compressibility.Defining the pressures as —dE/d V, where V =D is thenetwork area, the compressibility v = [ —V (d E /dV )] ' may be calculated by diff'erentiating Eq. (10)twice with respect to D . This yields

b oP 8+C 1 + — +b r dD r dD

2 (21)

where

1+vB =ph 1—

C= pb(1 —vcos i9)

4m ( 1 —v)cosP

(22)

this mismatch at large thickness is only 6.85b where b isthe Burgers vector. Since the actual spacing must be anintegral value, relatively large local fractional fluctuationsin D will be built into the network in order to achievenonintegral values of average D. This raises the energy ofthe real network (analogous to zero-point vibrations in aquantum oscillator). However, the majority of the ffuc-

are constants. The compressibility is positive and de-creasing with decreasing D at small spacings. This pro-vides a quantitative prediction of the amount by whichthe network is more resistant to a given amount ofcompression than to the same amount of extension. Ifthe entropy of the dislocation network is estimated, freeenergies can be determined and other thermodynamicquantities such as the equilibrium fluctuations in netspacing can be determined.

In heavily mismatched materials more efficient strainrelief is possible as pure edge-type dislocations formalong the perimeter of three-dimensional islands (pureedge-type dislocations relieve twice as much strain as60'-type dislocations). Strain relief generally occurs forvery thin layers prior to island coalescence. The intro-duction of dislocations from the film edge results in a net-work in tension. The dislocations are only added whenan excess strain energy sufficient to drive their formationat the film edge is available. This energy is expected to bevery low since the dislocation can be formed during theaddition of adatoms to the island. ' ' Thus the disloca-tion network in these materials can be expected to havenearly perfect spacing at island coalescence [the E(D)curve exhibits a sharp minimum] and to be in relativelyuniform tension.

The formation of a pure edge-type dislocation networkhas been observed for the InSb/GaAs system for whichthe bulk lattice mismatch is 14.6%.' A network withinprecoalescence islands of 30 nm height was observed tobe well ordered and at nearly the spacing required forcomplete strain relief, as shown in Fig. 5. Furthermore,the dislocation spacing is nearly perfectly uniform to theedge of the islands even though the height of the island ischanging rapidly there. The final dislocation spacing for

120MEc100—

80—C5

60.—

40C)

P2O-CD

LLI

0

I

IIIIIIII II II I

II 1

III I

IIIIIII

I lI

I

Dislocation Network Energy—

Misfit:~e~o~ Q 0]

0.05~~~~ P ]0

I

Wo ~+ ~a ~0~+»~+ega o~a me o~o ~o~ ~ ~ +~+~20 40 60 80

Dislocation Spacing D (nm)

(b) poOJ

ElI

O — )III

II' I10—,I

1Itl

V) , Ic (ICO ~ I

lg

CDcLLI

'o

I'

I'

I

Dislocation Network Energy

Film Thickness——- 50 nrn

]pp1000

I t I t I t I

20 40 60 80Dislocation Spacing D (nm)

100

FIG. 4. A plot of the calculated system energy E(D) per unitD area as a function of dislocation spacing D. The energy isnormalized by the shear modulus of the film and the Burgersvector of the dislocation. (a) shows the effect of misfit on theE(D) curve. (b) shows the effect of film thickness on E(D).


FICi. 5. Cross-sectional transmission electron micrographshowing a lattice image of a 13-nm-high InSb island on a GaAssurface. The zone axis of the image is {110)and fringes in theimage are {111)planes. The bright points along the heterojunc-tion indicate the presence of misfit dislocations. The disloca-tions are pure edge type with an average spacing of one everyseven {111)planes.

tuations in D will occur due to misalignment of the net-works in adjacent islands upon coalescence. The effectsof island coalescence on the network are planned to beconsidered in a separate publication.

To complete this section, we consider the factors deter-mining the kinetic barrier to misfit relief. In all cases,dislocation motion can be thermally activated. Thusgrowth of mismatched systems at elevated temperaturesincreases the rate of strain relief and decreases the rem-nant strain. A detailed theory relating this temperaturedependence to remnant strain has been developed byDodson and Tsao and may be found in Refs. 10 and 11.When strain relief occurs by loop nucleation, the pres-ence of stress concentrators such as steps and other de-fects reduces the energy barrier substantially. Thusgrowth of smoother surfaces and interfaces on relativelywell oriented substrates may increase the thickness atwhich strain relief would occur by loop expansion.

B. Density of misfit dislocations with film thickness

L represents the average separation of the threadingdislocations when the loop ceases to expand and is as-sumed to be much larger than D and much smaller thanthe wafer dimension. Clearly, the fewer pinning points inthe crystal available to halt loop expansion the fewerthreading dislocations will be introduced. The density ofthreading dislocations in a fixed unit of area will scaleroughly as the square of the pinning site density haltingloop expansion. It is assumed in the following treatmentthat the pinning site density is constant and loops expandto a fixed size independent of threading-dislocation densi-ty. An alternate assumption would be that the loop di-mension scales with threading-dislocation density, whichmay be more accurate at high dislocation densities nearthe interface.

The rate of addition of threading dislocations decreasesas film thickness increases. However, in the absence of amechanism for eliminating threading dislocations the to-tal population of threading dislocations in the film wouldalways increase. From transmission electron microscopyevidence it is clear that the density of these dislocationsdecreases rapidly with distance from the heterointer-face. ' For this to occur it is necessary that threadingdislocations react to be eliminated. This is equivalent, forthreading dislocations propagating to the surface, to re-quiring that dislocation trace steps on the surface be el-iminated by reaction. A dislocation loop originating atthe heterojunction but ceasing to expand prior to reach-ing the surface would lead to threading dislocations ap-pearing to terminate in the film. This could happen ei-ther spontaneously or due to a reaction with anotherloop. The order of the process of terminating threading-dislocation propagation is given by the linear density ofdislocations involved in the termination. If the order ofthe process for terminating the propagation of threadingdislocations is n, the rate of elimination of threadingdislocations per unit length d (N /L)/dh will be approx-imately

d(N+ /L)dh h b cosP

(24)

Following the initial strain relief achieved throughsources requiring low energies to operate, the relief pro-cess is thought to proceed through the formation of dislo-cation loops. These typically include a segment of misfitdislocation, a surface trace, and two threading segmentsif the loop does not reach the sample perimeter. Themisfit segments must fit into the existing network and arelikely to encounter some obstacle to extension beforereaching the edge of the crystal. Hence the number ofthreading dislocations introduced by addition of a seg-ment of misfit dislocation is roughly 2. Assuming this tobe the case, the density of threading dislocations in thefilm can be estimated as a function of thickness from Eq.(19).

The rate of addition of misfit dislocations per unitlength (orthogonal to the added dislocation lines) at largethickness is inversely proportional to h . The rate of ad-dition of threading dislocations per "unit length" I istwice the rate of addition of misfit dislocations or

d (N /L)dh

N,=RL

(25)

d(N, /L)dh h b cosP

—RL

(26)

Near the heterojunction little interaction of threading-dislocation segments has occurred and few are eliminat-ed. There can be no elimination of threading dislocationsvery near the interface as this would also necessarilyeliminate the strain relieving dislocation. Thus the rela-tive recombination probability R decreases to zero nearthe interface. Equation (26) with R =0 is exactly solubleand yields

where X, is the total number of threading dislocationsavailable to interact and R is the probability per unit filmthickness for this elimination. The net rate of change ofthreading-dislocation density at the film surface is thusd (N, /L)dh =d(N+ /L)/dh —d (N /L)/dh, or com-bining Eqs. (24) and (25), for h ))h,


1.0 0.03

o QQ8~ cu0

~ 0.6

V3

o 0.4C3C3—0.2C5

0.0 I

0

ar Heterojunction

Reaching the Surface~as ~~

l i I

10 20Relative Thickness h/hc

30

C3

&0.02CD

CD

CL

0.01CO

00

o Data from Ref. 15— Ca lculated

hc = 7.18nm2)p/bcos g = 0.520

I I

200h (nm)

FIG. 6. Threading-dislocation densities at the heterojunctionwhere no dislocation reaction has occurred and at the film sur-face calculated by solution of Eq. (26) assuming a second-orderdislocation recombination process, n =2.

FIG. 7. Shows the calculated strain relieved (which is pro-portional in this model to the threading-dislocation density atthe heterojunction) and the experimentally observed valuesfrom Ref. 15 for a Sio,Geo, /Si(001) structure.

N(h) 2'(1 —h, /h)L h, b cosP

(27)

at the interface. This threading-dislocation density al-ways increases as the film grows.

When h ))h, the rate of addition of dislocations be-comes nearly zero. In this limit, for finite R, Eq. (26)reduces to d (N, /L)/dh = —R (N, /L)", yielding

1V, =[R(h —h )(n —1)]' "L C

for n ) 1, at the film surface. When n = 1,

(28)

N(h)L

=e (29)

The threading-dislocation density at the film surfaceinitially rises as dislocations are formed but then de-creases as the rate of threading-dislocation removal be-comes greater than the rate of dislocation introduction.The dislocation distributions at the surface and near theheterojunction are plotted in Fig. 6 assuming n =2. Thedistribution between these two limits will fall between theplotted curves and will decrease according to Eq. (28) atlarge h. The progression of curves from the surface limitto the heterojunction limit (R =0) depends on the way inwhich R changes with distance from the interface andmay be affected by changes in I. There are undoubtedlya large number of mechanisms by which threading dislo-cations can interact and cease to propagate as the filmgrows. The exponent of N, in Eq. (25) depends on the or-der of the interaction while R is an empirical constantwhich accounts for all of the reactive mechanisms. Thusboth R and the exponent X, will be difficult to calculate apriori without a detailed understanding of the mecha-nisms by which threading dislocations are eliminated.However, by fitting experimental data it should be possi-ble to determine both R and the exponent n. With infor-mation concerning R and n, a dominant mechanism ofthreading-dislocation formation and removal could be de-duced.

Two variables of strain relief are easily observed inmismatched systems, the total strain relieved (by measur-ing the film lattice parameter in the interface plane), andthe etch-pit density at the surface giving the number ofthreading dislocations reaching the surface. The latter isprobably composed of a set of dislocations propagatingfrom the substrate of essentially fixed quantity plus anumber of threading dislocations introduced duringstrain relief. Hence it should be possible to estimateN(h) from the etch-pit density after subtracting a con-stant number of intrinsic defects propagating out of thesubstrate. The number of threading dislocations at theinterface will be proportional to the misfit relieved andincreases according to Eq. (27) at large film thicknesses.

Based on the misfit relieved N(h)/L at the interfacecan be estimated. To provide the proper asymptotic be-havior, the value of 2'/(h, b cosP) in Eq. (27) mustequal 1/D, which is calculable for a given system. Theremaining parameter R is determined by fitting the solu-tion of Eq. (26) to N(h) at the surface estimated frometch-pit densities. We have carried out half of this pro-cess for Geo &Sio &

for which values of fractional misfit re-lieved are available. ' The result is shown in Fig. 7. Thefit is excellent at large values of h but deviates at lowervalues for which Eq. (26) is not valid, L is probably notconstant, and D is far from D . To determine therecombination mechanism for threading dislocations itwill be necessary to have measurements of strain relievedand etch-pit density in a single set of films.

IV. CONCLUSIONS

The energy minimization formalism originally pro-posed by van der Merwe for calculating strain relief inlattice-mismatched heteroepitaxial systems has been ex-tended to include dislocation interactions. From this theenergy as a function of dislocation spacing has been cal-.culated and shown to include an attractive and repulsiveregion. This permits estimations of the dislocation for-mation energy and Peierls barrier to network orderingfrom measured dispersions in dislocation spacings. It


further explains why the dislocation network is better or-dered in highly mismatched systems than in weaklymismatched heterostructures. It has been shown thatthermodynamic functions such as the compressibility ofthe dislocation network can be calculated from theenergy —dislocation-spacing relationship. A formula re-lating the equilibrium dislocation spacing to film thick-ness, mismatch, and misfit-dislocation character is alsocalculated. Finally, the density of threading dislocationsis calculated both at the heterojunction and at the filmsurface assuming a threading-dislocation reaction pro-cess. The interface threading-dislocation behavior is

shown to be in good agreement with the fractional strainrelieved in Si„Ge& /Si structures.

ACKNOWLEDGMENTS

The authors wish to thank P. J. Goodhew for manyuseful discussions and advice and R. Hull for his com-ments and experimental data. The authors gratefully ac-knowledge the support of the Department of Energy un-der Contract No. DEAC-76-ER01198.

~J. W. Matthews and A. E. Blakeslee, J. Cryst. Growth 27, 118(1975).

2J. W. Matthews, J. Vac. Sci. Technol. 12, 126 (1975).3F. C. Frank and J. van der Merwe, Proc. R. Soc. London, Ser.

A 198, 261 (1949).4F. C. Frank and J. van der Merwe, Proc. R. Soc. London, Ser.

A 198, 2205 (1949).5J. R. Willis, Suresh C. Jain, and R. Bullough, Proc. Mat. Res.

Soc. (to be published).6R. Hull and J. C. Bean (unpublished).7R. H. Dixon and P. J. Goodhew, J. Appl. Phys. 68, 3168 (1990).SEpitaxiah Growth Part 8, edited by J. W. Matthews (Academic,

New York, 1975).Equations (6)—(9) are developed from more general forms given

in John Price Hirth and Jens Lothe, Theory of Dislocations,2nd ed. (Wiley, New York, 1982).

08rian W. Dodson and Jeffrey Y. Tsao, Appl. Phys. Lett. 51,

1325 (1987).~ Brian W. Dodson and Jeffrey Y. Tsao, Appl. Phys. Lett. 53,

2498 (1988), and references cited therein.' C. J. Kiely, J. -I. Chyi, A. Rockett, and H. Morkoq, Philos.

Mag. A 60, 321 (1989).3C. J. Kiely, A. Rockett, J. -I. Chyi, and H. Morkoq, in Atomic

Scale Structure of Interfaces, edited by R. D. Brigans, R. M.Feenstra, and J. M. Gibson, MRS Symposia Proceedings No.159 (Materials Research Society, Pittsburgh, in press) ~

See, for example, Chin-An Chang, C. M. Serrano, L. L.Chang, and L. Esaki, J. Vac. Sci. Technol. 17, 603 (1980); orG. R. Booker, J. M. Titchmarsh, J. Fletcher, D. B. Darby,M. Hockly, and M. Al-Jassim, J. Cryst. Growth 45, 407(1987).

5J. C. Bean, L. C. Feldman, A. T. Firoy, S. Nakahura, and I.K. Robinson, J. Vac. Sci. Technol. A 2, 436 (1984).

energetics of misfit- and threading-dislocation arrays in heteroepitaxial films

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