Misfit Strain and Accommodation in SiGe Heterostructures

SEMICONDUCTORS AND SEMIMETALS.VOL. 56

CHAPTER3

Misfit Strain and Accommodation in SiGe Heterostructures R . Hull DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING OF VIRGINIA UNIVERSITY

CHARLOTTESVILLE. VIRGINIA

I . ORIGINOF STRAIN IN HETEROEPITAXY. . . . . . . . . . . . . . . . . . . . . . . . . . . 11. ACCOMMODATION OF S T R A I N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Elastic Distortion of Atomic Bonds in the Epitaxial Layer . . . . . . . . . . . . . . . . . 2. Roughening of the Epitaxial Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Interdifision across the Epilayer/Substrate Interface . . . . . . . . . . . . . . . . . . . . 4 . Plastic Relaxation ofstrain by Misfit Dislocations . . . . . . . . . . . . . . . . . . . . . 5. Competition Between Different Strain Relief Mechanisms . . . . . . . . . . . . . . . . . . 111. REVIEWOF BASICDISLOCATION THEORY. . . . . . . . . . . . . . . . . . . . . . . . . . 1. Definition and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Energy of Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Forces on Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. GlideandClimb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Geometry of Interfacial Misfit Dislocation Arrays . . . . . . . . . . . . . . . . . . . . . . 6. Motion of Dislocations: Kinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7. Dislocation Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Partial versus Total Misfit Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. EXCESSS T R E S S , EQUILIBRIUM S T R A I N A N D CRITICAL THICKNESS ............ I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Matthews-Blakeslee Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Accuracy of the MB Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Other Critical Thickness Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Extension to Partial Misfit Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Models for Critical Thickness in Multilayer Structures . . . . . . . . . . . . . . . . . . . V. METASTABILITY A N D MISFITDISLOCATION KINETICS. . . . . . . . . . . . . . . . . . . 1. Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Misfit Dislocation Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Misfit Dislocation Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Misfit Dislocation Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Kinetic Modeling of Strain Relaxation by Misfit Dislocations . . . . . . . . . . . . . . . .

102 103 103 105 105 107 108

109 109 111 112 112 113 116 117 119

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133 144 149 152

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VI. MISFIT A N D THREADING DISLOCATION REDUCTION TECHNIQUES. . . . . . . . . . . . I . Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. BufferLayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Threading Dislocation Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES ..........................................

155 155 156 151

161 164

I. Origin of Strain in Heteroepitaxy The fundamental origins of strain in heteroepitaxy are from two sources: (i) The difference in lattice parameter between epitaxial layer(s) and substrate. (ii) Differential thermal expansion coefficients between epitaxial layer(s) and substrate. Consider a thin epitaxial layer with free lattice parameter a, deposited upon an infinite substrate with lattice parameter n,. If the lattice parameter difference between substrate and epitaxial layer is accommodated entirely elastically, as in Fig. l(a), and assuming for the moment that the thermal expansion coefficients of the two materials are equal, then the elastic strain in the epitaxial layer is given by’:

For a, > a,, the strain in the epitaxial layer is compressive and e is positive; for a, < a, the strain in the epitaxial layer is tensile and E is negative. The lattice constants of Si and Ge at room temperature are 0.5431 and 0.5658 nm, respectively; Ge,Sil-, lattice parameters, a@), have been tabulated by Dismukes et al. (1964). They observed slight deviations from Vegard’s law (Le., slight deviations from the relationship a(x) = a ~ i X ( U G ~- asi)). If this deviation from linearity is defined by A = a(x) - [asi +X(UG, - asi) ] , maximum values of A -0.007 nm were found at x 0.5. A parabolic fitting to the data of Dismukes et al. at room temperature (Herzog, 1995) yields

-

+

a ( x ) = 0.5431

-

+ 0.01992~+ 0 . 0 0 2 7 3 3 ~nm~

(2)

In practice, relatively little error (of order 0.1% error in a(x) and 7% error in ~ ( x )is) involved if the room temperature value of a(x) is assumed to follow the linear form

a(x)= 0.5431

+ 0.0227~nm

(3)

‘Note that we are assuming an infinite substrate thickness here, such that all lattice mismatch strain is accommodated by the epitaxial layer. For substrate thicknesses that are not effectively infinite with respect to the epitaxial layer, strain will be partitioned between substrate and epitaxial layer. In practice, typical Si substrate thicknesses are several hundred microns, which is effectively infinite with respect to any reasonable epilayer thickness.

3 MISFIT STRAINAND ACCOMMODATION

whence, from Eq. (1)

103

= 0.0409~

(4) The simplified relations of Eqs. (3) and (4) will be used in the remainder of this chapter. In general, materials have temperature-dependent lattice parameters via thermal expansion coefficients (which are themselves temperature dependent). The linear thermal expansion coefficients, a ( x , T ) have been measured as functions of temperature for Si, Ge and Ge,Sil-, alloys (Wang and Zheng, 1995). The values of a(x,T ) are somewhat nonlinear with Ge fraction and temperature, but for the range 0-800 "C, within accuracies of order 10% EO(X)

+

= (2.7 0.0026T) x low6 a~~ = (5.9+0.0021T) x asi

Linear interpolation for intermediate Ge, Si 1 compositions will generally overestimate a ( x , T ) , especially for higher x and T . Nevertheless a linear interpolation of Eqs. 5(a) and (b) will yield estimates for a @ ,T ) which are accurate to order 30% a ( x , T ) = [(2.7

+ 3 . 2 ~ +) (0.0026 - 0.0005x)TI x

(6)

The thermal mismatch stress, E T S , induced as a function of change in temperature, A T , with respect to a substrate with thermal expansion coefficient, asub(T),is thus

-

For example, considering a Geo.2Sio.s layer grown upon a Si substrate, at 550 O C (a typical temperature for molecular beam epitaxy, MBE, growth), we have E T S 2x that is, just a few percent of the lattice mismatch strain EO. Thermal m i s match strains are thus relatively small compared to lattice mismatch strains. They can, however, become important at large epitaxial layer thicknesses where lattice mismatch strain is largely accommodated by misfit dislocation generation at the crystal growth temperature, and thermal stresses are induced during cooling to room temperature, as will be discussed later in this chapter.

11. Accommodation of Strain

In strained layer epitaxy, the lattice parameter difference between a thin epitaxial layer and an epitaxial substrate can be accommodated by several mechanisms. 1.

ELASTICDISTORTION OF ATOMICBONDS I N THE EPITAXIAL LAYER

This configuration is shown in Fig. l(a). The in-plane lattice parameter of the epitaxial layer uep is distorted to that of the substrate a,. The epilayer lattice parameter

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104

a

b

C

FIG. 1. Schematic illustration of mechanisms for accommodation of lattice mismatch strain: (a) elastic distortion of epitaxial layer; (b) roughening of epitaxial layer; (c) interdiffusion; (d) plastic relaxation via misfit dislocations.

perpendicular to the interface, sen, then relaxes along the interface normal, to produce a tetragonal distortion of the unit cell (recall that in their unconstrained states, Si, Ge and Ge,Sil-, have diamond cubic lattices). The magnitude of the tetragonal distortion is given by aen/aep 1 ~ ( 1 ~)/(l - v) (8 1 ‘V

+ +

Here, E is the epilayer strain (which may have relaxed from 60 if any of the relaxation mechanisms in Sections 11.2, 3, and 4 have operated), and u is the Poisson ratio of the epilayer material. For an elastically isotropic cubic crystal, u is derived from the . of c1 1 and c12 appropriate elastic constants by the relation u = c12/(c11 ~ 1 2 ) Values (Landholt-Bornstein, 1982) for Ge and Si then yield u = 0.273 and 0.277 for Ge and Si, respectively. The generally quoted value for u in Ge,Sil-, in the literature is 0.28, which we shall use henceforth. The sense of the tetragonal distortion in Eq. (8) is positive (aefl> aep)for a, > a,, that is, the epilayer lattice relaxes outwards along the interface normal. For a, < u , ~ , the tetragonal distortion is negative (aen < a,), and the epilayer relaxes inwards along the interface normal. The tetragonally distorted epitaxial layer stores an enormous elastic strain energy (of the order 2 x 107Jm-3 for a lattice mismatch strain of 0.01). The stored elastic strain energy in the epitaxial layer, per unit interfacial area, is given by continuum elasticity theory as2 E,i = 2 G 4 ( 1 + u)h/(l - u ) (9)

+

Here h is the epilayer thickness and G is the epilayer shear modulus. As Si and Ge are relatively anisotropic elastic materials, average shear moduli are generally used 2Equation (9) assumes that all strain accommodation is elastic. If any of the lattice mismatch strain is relaxed by the mechanisms discussed in Sections II.2., 3., and 4.. then E, should be replaced by F , the remaining elastic strain.

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in strained layer theory, derived by averaging over the appropriate elastic constants, or elastic compliances, denoted, respectively, by the Voigt and Reuss average moduli (Hirth and Lothe, 1982). The Voigt and Reuss methods give for G in Si, Ge values of 6533 GPa and 68,56 GPa, respectively. This use of isotropic elasticity theory combined with these average elastic moduli is a significant approximation, compared to a rigorous application of anisotropic theory. However, the mathematical complexity in calculating the strain fields of dislocations in anisotropic materials is daunting (Hirth and Lothe, 1982), so less precise isotropic theory is generally used. This can lead to errors of the order 20% in subsequent calculations.

2.

ROUGHENING OF THE EPITAXIAL LAYER

As shown schematically in Fig. l(b), roughening of the epitaxial layer surface allows atomic bonds near the surface to relax towards their equilibrium length and orientation. The basic energetic competition in this process is between the surface energy of the epitaxial layer (representing an increase in the system energy as surface roughening increases the total surface area, and hence surface energy, of the epitaxial layer) and elastic energy (which is reduced by roughening, therefore representing a decrease in the system energy). This mechanism will not be treated further in this chapter, as it is dealt with in detail in the chapter by Savage in this volume. INTERFACE 3. INTERDIFFUSION ACROSS THE EPILAYER/~UBSTRATE

As the areal strain energy density, given by E q . (9), varies as E:, lowering of the average strain by interdiffusion will reduce the elastic strain energy stored in a heteroepitaxial system. For example, consider the highly simplified configuration where an initially abrupt interface between a Ge,Sil-, epilayer of thickness h nm and an infinite Si substrate undergoes an interdiffusionalprocess, such that the Ge,Sil -, layer redistributes itself into a layer of thickness h h~ nm, and a uniform Ge fraction of x h / ( h h ~ ) . The initial areal energy density will reduce from k'x2h to k'x2h2/(h h ~ )where , k' is the relevant constant of proportionality from Eq. (9). In the limit that h~ tends to infinity, the strain energy tends to zero. Of course, the abrupt diffusion profile implied by this theoretical experiment is highly unphysical, but the principle of strain energy reduction via interdiffusion will hold for any diffusion profile with a monotonic decay of Ge fraction. Interdiffusion at Ge,Sil-,/Si and Ge/Si interfaces has been studied by several authors. Early studies by Fiory et al. (1985) established the presence of significant interdiffusion at temperaturesof 800 "C and higher in a 10 nm Si/14 nm Ge0.24Si0.76/Si(1~) structure via HeC ion channeling and backscattering studies. As this structure is below the critical thickness criterion for relaxation of strain via formation of a misfit dislocation array, relaxation of strain in this structure was determined to occur via in-

+

+

+

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FIG. 2. Measurements of interdiffusion at Ge,Sil -,/Si interfaces: Arrhenius plot of the diffusion coefficient of Ge in strained Si/Ge,Sil-,/Si structures for x = 0.07 (circles), x = 0.16 (squares); x = 0.33 (triangles), derived from diffusion in the tails of the diffusion profile (filled symbols) and at the peak of the diffusion profile (open symbols). Solid line is an extrapolation of Ge tracer diffusion in bulk Si (McVay and Ducharme, 1974). Reprinted from Thin Solid Films 183, G.F.A. Van de Walle, L.J. Van Ijzendoorn. A.A. Van Gorkum, R.A. Van den Heuvel, A.M.L. Theunissed and D.J. Gravestein, “Germanium Diflususion and Strain Relaxation in Si/Sil-,Ge,/Si Structures,” pgs. 183-190. 1990, with permission from Elsevier Science, The Boulevard, Langford Lane, Kidlington OX5 IGB, UK.

terdiffusion. A more extensive study by Van de Walle et al. (1990) derived diffusion coefficients for Ge in Si/GexSil-,/Si structures. A summary of their data is shown in Fig. 2. The measured diffusion constant appears to be relatively independent of Ge fraction in the range x = 0.07-0.33. This is in contrast to earlier measurements of the diffusion coefficient measured for Ge in large-grain polycrystalline Ge, Si 1--x material (McVay and DuCharme, 1974), where diffusion was noted to increase rapidly with x up to a maximum at x 0.8. Van de Walle et al. observed that diffusion in the tails of the Ge concentration profiles correlated well with measured diffusion coefficients for Ge diffusion in bulk Si, whereas measured diffusion constants in the peaks of the Ge concentration were an order of magnitude greater. The measured diffusion constants suggest that diffusion lengths of the order of one monolayer at 800°C would take 1 hr in the Ge tail. Thus strain relaxation by interdiffusion is not expected to be a significant factor during growth of relatively dilute (i.e., low Ge fraction) and thick

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(compared to one monolayer) Ge,Sil-, layers on Si substrates, for techniques such as MBE and ultrahigh-vacuum chemical vapor deposition (UHV-CVD) where growth temperatures are typically less than 800 "C. A configuration where interdiffusion can represent a significant strain relaxation mechanism, even at typical MBE growth temperatures, are the monolayer-scale GeSi superlattices which at one time were widely fabricated in pursuit of efficient optical emission (see chapters by Cerdeira, Campbell, and Shaw and Jaros in this volume). Chang et al. (1990) reported diffusion coefficients in monolayer-scale Ge-Si superlattices that were consistent with the diffusion constants measured for Ge in pure Si, and by Van de Walle ef al. for tail diffusion. Baribeau (1993) and Lockwood et al. (1992) studied similar systems and concluded that for atomically abrupt interfaces, diffusion was indeed of the order of that associated with Ge diffusion in bulk Si, but that existing intermixing or segregation at the interfaces could dramatically enhance diffusion, due to strong increase in Ge diffusivity with increasing x in Ge, Sil Under such conditions, substantial diffusion (enough to homogenize a multilayer structure consisting of alternating layers of 4.5 monolayers of Si and 3.5 monolayers of Ge) was observed for a 20-s anneal at 700 "C.

4.

PLASTIC

RELAXATION OF STRAIN B Y MISFITDISLOCATIONS

As shown schematically in Fig. l(d), another mechanism for strain relief is via generation of an interfacial dislocation array, known as a misfit dislocation array, which allows the epitaxial layer to relax towards its bulk lattice parameter. This is a very prevalent mechanism for strain relief in Ge, Si 1-,-based heterostructures. It is experimentally observed, and theoretically predicted, that the misfit dislocation array forms only in epitaxial layers thicker than a minimum thickness, known as the critical thickness h,. The interfacial dislocation array relaxes elastic strain energy by allowing the average lattice parameter of the epitaxial layer to relax towards its unconstrained value. However, the dislocations have a self-energy &is, manifested by the stress field they produce in the surrounding crystal. Thus the energy of a heteroepitaxial system containing a total line length L of interfacial dislocation is (ignoring surface roughening and interdiffusion) Etot = Eel(L, h ) Edis(L, h ) (10) We know from Eq. (9) that E,) varies linearly with h . As each unit length of dislocation will reduce a finite amount of strain in the epitaxial layer (this will be discussed more fully in the next section of this chapter), we can also deduce that F decreases linearly with L . The energy of the dislocation array will increase with L , and at relatively low values of L (specifically such that the average separation of the interfacial dislocations is less than the epilayer thickness), this increase will be linear as will be discussed in Section 111 of this chapter. At higher dislocation densities (i.e., greater L ) , the increase will no longer be linear, due to the importance of dislocation interactions.

+

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R. HULL

Finally, as we shall also see in Section 111, the self-energy of an individual dislocation varies logarithmically with its distance from a free surface. Thus for low dislocation densities Etot= k l h ( ~ ok3L ln(k4h) (11)

+

In this equation kl through k4 are constants whose magnitudes will be derived in later sections of this chapter. Comparing Eq. (1 1) to Eq. (9), the change in total energy as a result of the introduction of the dislocation array is

AE,,, = k3L ln(k4h)

+ klk?jhL2 - 2 k l k 2 h ~ ~ L

(12)

For h > hc, we would of course expect that AEtot < 0, namely, that introduction of the dislocation array lowers the energy of the system. For h < h,, A EtOt > 0, that is, the dislocation array is not energetically favored. The critical thickness is found by finding that value of h for which the change in total energy A Et,,, for introduction of a single misjit dislocation is zero. Thus we take the limit of Eq. (12) as L --f 0, allowing solution for h = h, h, = k3 ln(k4hc)/2klk2~o (13) For a given epitaxial layer thickness h > h, and residual lattice-mismatch strain E =

(eo - k2L), the minimum energy configuration will correspond to (GEtot/GL)h,s= 0,

allowing solution for the equilibrium value of L from Eq. 12:3 k3 ln(k4h) = 2hklkz[eo - k2L]

(14)

Before a more rigorous development of models for the critical thickness and equilibrium misfit dislocation densities (i.e., evaluation of the constants ki in the foregoing equations), we will briefly review the relevant aspects of dislocation theory.

5 . COMPETITION BETWEENDIFFERENT STRAINRELIEFMECHANISMS Of the strain relief mechanisms discussed in the preceding four sections, interdiffusion is significant only at growth or annealing temperaturehime cycles of order 800 "C/ 1 hr or greater (except for the relatively specialized configuration of monolayer scale Si:Ge superlattice structures). Thus, interdiffusion is not a significant mechanism at the growth temperatures typically used during MBE or UHV-CVD growth, although it could conceivably be significant during high-temperature post-growth processing (e.g., implant activation or oxidation processes). Relaxation via surface roughening can occur for any epitaxial layer thickness (see chapter by Savage in this volume). Strain relaxation by misfit dislocations occurs only for layer thicknesses greater than the critical thickness. Both processes are kinetically 3Note that substitution of L = 0 into Eq. (14) reduces this expression to Eq. (13) as expected.

3 MISFIT S T R A I N AND ACCOMMODATION

109

limited, either by surface diffusion lengths in the case of surface roughening, or by nucleatiodpropagation barriers in the case of misfit dislocations. These two processes may be competitive, such that the strain relaxed by surface roughening (which may occur before a sufficient epilayer has been deposited for misfit dislocations to be energetically favored) reduces or eliminates the driving force for dislocation introduction, or they may be cooperative, as in the observed reduction of energetic barriers for dislocation nucleation associated with surface morphology (Cullis et al., 1994; Jesson et al., 1993, 1995), and surface morphology induced by misfit dislocations (Hsu et al., 1994; Fitzgerald and Samavedam, 1997). Tersoff and LeGoues (1994) modeled the introduction of misfit dislocations into planar epilayers, and into surface islands or pits. They demonstrated that if the energy barrier is essentially zero for dislocation introduction into islands or pits (this should be regarded very much as an approximation, although it is reasonable that the activation barrier is greatly reduced with respect to a planar surface), a temperature dependent critical strain (of order 0.01) exists above which strain relief is dominated by dislocation injection into roughened surfaces, and below which dislocation injection into planar surfaces is favored. This is consistent with general experimental observations that roughening of Ge,Sil-, epilayers is greater for higher Ge concentrations and temperatures. The detailed balance between roughening and dislocation generation, however, is still a topic of active experimental research and simulation.

111. Review of Basic Dislocation Theory We now briefly review the salient properties of dislocations pertinent to understanding their role in relieving lattice mismatch. For a full description of dislocation theory, the reader is referred to the volume by Hirth and Lothe (1982).

1.

DEFINITIONAND GEOMETRY

A perfect or total dislocation is a line defect bounding a slipped region of crystal. A circuit drawn round atoms enclosing this line will have a closure failure as illustrated in Fig. 3(a); this closure failure is known as the Burgers vector of the dislocation. For a perfect or total dislocation, the Burgers vector is a lattice translation vector. Although the line direction of a given dislocation may vary arbitrarily, its Burgers vector is con~ t a n t .A~ total dislocation cannot end within the bulk of a crystal-it must terminate 4Apart from a possible difference in its sign, depending upon the convention used. A widely accepted convention for determining the sign of the Burgers vector is to draw the circuit from start S to finish F in the direction of a right-handed screw, R H-the so-called F S I R H convention (Bilby et al., 1955). Under this convention, the opposite sides of a dislocation loop have opposite Burgers vectors’ signs.

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R. HULL

FIG.3 . (a) Illustration of the Burgers vector of a total dislocation (in this example an edge dislocation, with the line direction u running perpendicular to the page). The dislocation core is at X. A circuit of 5 atoms square, which would close in perfect material, demonstrates a closure failure, the Burgers vector b. when drawn around the dislocation core. Reprinted from R. Hull and J.C. Bean, Chapter 1, Semiconductors and Semimetals Vol. 33, ed. T.P. Pearsall, (Academic Press, Orlando, FL, 1990). (b) Illustration of the misfit dislocation (AB) I threading dislocation (BC) geometry at a Ge,Sil-,/Si interface. Reprinted with permission from R. Hull and J.C. Bean, Critical Review in Solid State and Materials Science 17. 507-546 (1992). Copyright CRC Press, Boca Raton, Florida.

at an interface with noncrystal (generally a free surface, but possibly also an interface with amorphous material), at a node with another defect, or upon itself to form a loop. The geometrical requirement that a dislocation must terminate at another dislocation, upon itself, or at a free surface means that something has to happen with the ends of misfit dislocations-they cannot simply terminate within the interface. If the defect density is relatively low and dislocation interactions thus unlikely, the most obvious place for the misfit dislocation to terminate is at the nearest free surface, which will in general be the growth (i.e., epilayer or cap) surface. This requires threading dislocations, which traverse the epitaxial layer from interface to surface, as illustrated in Fig. 3(b), and in general each misfit dislocation will be associated with a threading defect at each end, unless the length of the misfit dislocation grows sufficiently that it can terminate at the wafer or feature (e.g., mesa) edge, or at a node with another defect. Propagation of misfit dislocations occurs by lateral propagation of the threading arms. These threading dislocations are extremely deleterious to practical application of strained layer epitaxy. For many potential device applications, a high interfacial misfit

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111

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dislocation density is tolerable if the epilayer quality is sufficiently high at some distance from the interface (say, 1 wm or so). Threading dislocations compromise this possibility. Techniques for reducing these threading defect densities will be discussed in Section VI of this chapter. The character of a dislocation is defined by the relationship between its line direction u and its Burgers vector, b. If b is parallel or antiparallel to u, the dislocation is said to be of screw character. If b is perpendicular to u, the dislocation is said to be of edge character. In intermediate configurations, the dislocation is said to be of mixed character. As u may vary along a dislocation but b may not, a nonstraight dislocation will vary in character along its length.

2.

ENERGY OF DISLOCATIONS

A dislocation has a self-energy arising from the distortions it produces in the surrounding medium. This energy may be divided into two contributions, those arising from inside and those arising from outside the dislocation core. The distortions of atomic positions inside the core are sufficiently high that linear elasticity theory does not apply (dangling bonds may also exist, producing electronic contributions to the energy). The dislocation core energy is not well known, but depends upon both the material type (predominantly the nature of the interatomic bonding) and the dislocation Burgers vector and character. The dimensions of the core are also uncertain, but theoretical and experimental estimates of its diameter are of the order of the magnitude of the Burgers vector for covalent semiconductors and several Burgers vectors for metals (Hirth and Lothe, 1982). Outside the core, atomic distortions may be modeled using linear elasticity theory and exact expressions for this energy can be derived. The self-energy per unit length of an infinitely long dislocation parallel to a free surface a distance R away is5

In this equation, 8 is the angle between b and u and a is a factor intended to account for the dislocation core energy (a is generally estimated to be in the range 1 4 in semiconductors). The dislocation self-energy thus varies as the square of the magnitude of its Burgers vector. This strongly encourages the Burgers vector of a total dislocation to be the minimum lattice translation vector in a given class of crystal structure. For the diamond cubic (dc) structure of Ge,Sil-,, Si and Ge, this minimum vector is a/2(011). This is indeed the Burgers vector almost invariably observed for total dislocations in these structures. The value of R in Eq. (15) pertinent to calculations of interfacial mis5Strictly speaking this formula is derived for a dislocation within a cylindrical volume of inside radius b/u and outside radius R, but application to a planar free surface a distance R from the dislocation is generally a good approximation. We also continue to use the asumption of isotropic elasticity in this treatment.

112

R.HULL

fit dislocation energies for uncapped strained epilayers is generally the epitaxial film thickness h . For interfaces with very high defect densities, a cut-off parameter R , corresponding to the average distance between defects p (if this is less than the distance to the epilayer surface), is more appropriate. Thus if we consider the total energy of an interfacial misfit dislocation array, the energy of the m a y will increase approximately linearly with dislocation density for p > h , as each dislocation will have essentially the same self-energy given by Eq. (15) with R = h. For p 5 h , however, the array energy will increase sublinearly with density, as the energy of each dislocation will be given by Eq. (15) with R p ; the energy per dislocation will then decrease approximately logarithmically as p decreases.

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3.

FORCESON DISLOCATIONS

The self-energy of a dislocation produces a virtual force pulling it towards an image dislocation on the opposite side of a free surface. The simplest case is for an infinite straight screw dislocation running parallel to a free surface whose image is a dislocation of opposite Burgers vector in vacuu, equidistant from the surface. Other configurations have more complex image constructions. Image effects also exist across internal interfaces between materials with different shear moduli. The stress field around a dislocation produces an interaction force, Fij, between two separate dislocations. The magnitude of this force is again configuration dependent. For the simplest case of parallel dislocation segments, the interaction force per unit length is given by Fj,i = Gkj, (bj . bj)/ R' (16) where R' is the separation of the two segments and kin is the constant of proportionality (equal to 1/2 n for parallel screw segments). This force is thus maximally attractive for antiparallel Burgers vectors, maximally repulsive for parallel Burgers vectors and zero for orthogonal Burgers vectors. The general expressions for FiJ are more complex, but are relatively straightforward to derive (Hirth and Lothe, 1982).

4.

GLIDEA N D CLIMB

Motion of dislocations occurs most easily within their glide planes, which are the planes containing their line direction and Burgers vectors. For a screw dislocation, u is parallel to b and thus any plane is a potential glide plane. For mixed or edge dislocations, there is only one unique glide plane whose normal is given by b x u. Glide also occurs by far the most easily on the widest spaced planes in a given system because the Peierls stress (Peierls, 1940) resisting dislocation motion decreases with increasing planar separation. For diamond cubic, zinc blende and face-centered cubic crystals this corresponds to the { 111] sets of planes, and these are the almost ubiquitously

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observed glide planes in these structures. Glide occurs by reconfiguration of bonds at the dislocation core to effectively move the core one atomic spacing; this process is further aided by nucleation and motion of atomic-scale kinks, which will be discussed shortly. No mass transport of point defects is required in the glide process. Motion out of the glide plane, called climb, however, has to occur by extension or shrinkage of the half plane terminating at the dislocation core, requiring mass transport of point defects. Such diffusion processes are generally very much slower than glide processes in most temperature regimes.

5 . GEOMETRYOF INTERFACIAL MISFITDISLOCATION ARRAYS The resolved lattice mismatch stress a, acting on a misfit dislocation with Burgers vector b is given by the Schmid factor S (Schrnid, 1931) a, = aos = 0 0 cos h cos fp

(17)

where h is the angle between b and that direction in the epilayerhubstrate interface perpendicular to the misfit dislocation line direction, fp is the angle between the glide plane and the interface normal, and 00 is the lattice mismatch stress, given by standard isotropic elasticity theory as 00

+ ~ ) / ( -l U )

= 2C&(l

-

(18)

In the Ge,Sil-,/Si system, a0 9 . 4 ~GPa, an enormous stress! The effective strain relieving component of the misfit dislocation is given by beR = b cos h

(19)

Equation (17) shows that only dislocations gliding on planes inclined to the interface will experience a resolved stress (for fp = 90", cosfp = 0; for fp = 0", h = 90" and cos h = 0). For the Ge,Sil-,/Si(100) orientation, the four possible inclined { 111) glide planes intersect the (100) interface along orthogonal in-plane [Oil] and [Ol-I] directions, with a pair of glide planes intersecting along each direction. The orientation of one of these glide planes, and the accompanying possible a/2( 110) dislocations are shown in Fig. 4. The intersectionsof the glide planes with the interface thus produces a square mesh of interfacial dislocations, as shown experimentally in Fig. 5. In general, only dislocations with Burgers vectors lying within these { 111) planes will be able to move by glide. For a given glide plane three such Burgers vectors exist, for example, for the (-111) plane in Fig. 4, b = a/2[101], a/2[110] or a/2[01-1]. Of these three Burgers vectors, the last is a screw dislocation and will not experience any resolved lattice mismatch stress as cos h = 0. The first two are of mixed edge and screw character and are known as 60" dislocations, corresponding to the angle between b and U. From Eq. (19), only 50% of the magnitude of their Burgers vectors projects onto the inter-

interface. FIG. 4. Schematic illustration of the geometry of misfit dislocations at a Ge,Sil-,/Si(100) One inclined (-1 11) glide plane and consequent interfacial [Ol-11 dislocation direction are shown, together with the possible a/2(110) Burgers vectors orientations, where bl is of edge type and h2 and b3 are of 60" (glide) type. Reprinted with permission from R. Hull and J.C. Bean, Critical Review in Solid Srare and Materials Science 17,507-546 (1992). Copyright CRC Press, Boca Raton, Florida.

11:

1Bc3

FIG. 5 . Schematic illustrations of the symmetries of interfacial misfit dislocations at Ge,Sil-,/Si(lOO), (110) and ( 1 11) interfaces. Solid straight lines show the interfacial misfit dislocations; dashed lines outline intersecting ( 111) glide planes. Also shown are experimental verifications of these interfacial dislocation geometries from plan view (electron beam perpendicular to Ge, Sil -.r/Si interface) TEM.

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facial plane, thus they are only 50% effective at relieving lattice mismatch. Because of their ability to propagate rapidly by glide, these are the general total dislocations associated with strain relief in Ge,Sil -,-based heterostructures. The final possibility to consider is that of edge dislocations, for example, for the configuration of Fig. 4, b = a/2[0-1-1]. Such dislocations have their Burgers vectors lying within the interfacial plane and are 100% effective at removing lattice mismatch. The Burgers vectors do not, however, lie within any glide plane, and thus these defects must move by far slower climb processes. Such edge dislocations are, however, frequently observed at high strains (of order 2% or greater) in the Ge,Sil-,/Si (Hull and Bean, 1989a; Kvam et al., 1990; Narayan and Sharan, 1991) and many other lattice-mismatched heteroepitaxial semiconductor systems. It is somewhat mysterious that such dislocations can be prevalent, because of the requirement for motion by climb. It has been suggested that they are formed by reaction of 60" dislocations on different glide planes (Kvam et al., 1990; Narayan and Sharan, 1991); for example, for an interfacial misfit dislocation line direction u = [0111, an edge dislocation of Burgers vector a/2[01-1] could be formed by the reaction a/2[10-1](1-11) a/2[-110](11-1) (in this reaction the square brackets refer to the dislocation Burgers vector and the curved brackets to the glide plane). Because they are prevalent at low and moderate strains, subsequent discussion will concentrate primarily on 6Ooa/2(101) glide dislocations. The geometry of misfit dislocations will be different on surfaces other than (100). If the dislocations glide on inclined { 11l } planes, the interfacial misfit dislocation line directions will be defined by the intersection of these planes with the relevant interface. The expected geometries for (loo), (110) and (1 11) interfaces are illustrated schematically and experimentally in Fig. 5. At the enormously high lattice mismatch stresses that can exist in strained layer systems, the presumption of dislocation glide on the widest spaced planes in the structure, that is, { 111) planes for the diamond cubic (dc), zinc blende (zb) and face-centered cubic (fcc) structures, may be overcome. As was first observed in highly strained (EO 0.03; a0 5 GPa) (Al)GaAs/Ino.4Gao.6As/GaAs(lOO) zb structures by Bonar et al. (1992), secondary slip systems may operate at these enormous stresses. In the work of Bonar et al., a/2(101) dislocations were observed gliding on { 110) planes, producing (010)misfit dislocation directions in the (100) interface. The same slip system has since been observed in Ge0.86Si0.~4/Si(100)(EO 0.035,ao 8 GPa) structures by Albrecht et al. (1993). The observations of this secondary slip system at these enormous lattice-mismatch stresses may be related to a more efficient Schmid factor, and, therefore, higher a, for a given ao.For the (lOl){lll}slip system, Sill = 0.42. For the (101){110}system, Silo = 0.50. Therefore the extra resolved stress on the 0.08~0 (lOl){llO}system compared to the (101){111}system is cro(S110 - S111) or 0.4 - 0.6 GPa for the structures described by Bonar et al. and Albrecht et al. This more efficiently resolved applied stress may allow a secondary slip system with higher Peierls stress to operate. At lower strains (< 0.03), however, the (1lo){11I} slip system appears to operate ubiquitously.

+

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116 6.

MOTION OF DISLOCATIONS: KINKS

The glide process is facilitated by nucleation of atomic-scale lunks along the propagating dislocation line, with lateral motion of the kink arms transverse to the dislocation line effectively moving the entire dislocation length. In the absence of kinks, dislocations in semiconductors typically lie along well-defined crystallographic directions corresponding to minima in the crystal potential (known as Peierls valleys). These are the (01I ) directions for the dc structure of Ge, Sil -, . Formation of a kink requires motion of a small length of dislocation line across the potential, or Peierls, barrier between valleys. Subsequent lateral motion of the kink arms also involves motion over a secondary Peierls barrier. This process is illustrated in Fig. 6. In metals the energy required to form and move these kinks is relatively low-typically less than of the order 0.1 eV-because of the low Peierls barriers resulting from the relatively weak metallic bonding. This low kink energy in metals means that the dislocations are not strongly constrained to lying within Peierls valleys, and thus are not very straight. Dislocation motion and configurations in semiconductors, however, are typically dominated by the Peierls barriers. In materials that are purely covalently bonded such as Ge and Si, the glide activation energies are particularly high, for example, 1.6 eV in Ge and 2.2 eV in Si for stresses in the tens to hundreds of MPa regime (Alexander and Haasen, 1968; Patel and Chaudhuri, 1966; Imai and Sumino, 1983; George and Rabier, 1987). These glide activation energies arise from the Peierls barriers that have to be overcome in

I

I

J

c

FIG. 6. Motion of dislocations by kink pairs in a semiconductor crystal. A small length s of a straight dislocation line jumps an interatomic distance q transverse to the line, to from a kink pair. The kinks then run parallel to the dislocation line by successive jumps of interatomic distance a, thereby effectively moving the entire dislocation line a distance q to the right.

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forming and moving kinks. The details of the kink nucleation and propagation process have been derived in some detail by Hirth and Lothe (1982) and corresponding dislocation velocities u as functions of temperature and applied stress calculated. These lead to an expression of the form u = UOD," exp[-E,(a)/kT]

where uo is a prefactor containing an attempt frequency (commonly taken to be of the order of the Debye frequency), kink jump distances and an inverse dependence upon temperature and E , (a)is the glide activation energy. Based upon the double kink theory (Hirth and Lothe, 1982), the prefactor uo and the activation energy &(a) depend upon the propagating dislocation length. (This essentially depends upon whether kinks collide with each other before reaching the end of the propagating dislocation segment. It is generally assumed that they do in bulk samples, whereas for very short propagating dislocation lengths, as may occur in thin epilayers, kinks may reach the end of the propagating segment before colliding with each other.) The activation energy is also predicted to exhibit a stress dependence at applied stresses that are a significant fraction of the Peierl's stress. Consistent with the double kink theory (Hirth and Lothe, 1982), the pre-exponential power m is found experimentally to be of the oriler 1.0 at stresses of the order tens of MPa in very pure Si (Imai and Sumino, 1983). The kink model for dislocation motion is widely accepted, and experimental results are generally well described by Eq. (20). There is some variation in the measured magnitude of m , which is found to vary in the range 1.0-2.0 (Alexander and Haasen, 1968); this variation may arise partly from difficulties in separating pre-exponential and exponential stress dependence of the measured velocities. Also, the prefactors derived from measurements on bulk semiconductors typically are 2-3 orders of magnitude lower than the Hirth-Lothe theoretical predictions (Imai and Sumino, 1983). This may be due to obstacles to dislocation motion (Moller, 1978; Nititenko et al., 1988, Kolar et al., 1996) such as point defects, impurities, inhomogeneities in the dislocation core structure etc., or to entropic effects in kink formation or migration (Marklund, 1985), or to charge states associated with dangling bonds at kinks (this latter model is supported by observations of a doping dependence of dislocation velocity [Hirsch, 19811). 7.

DISLOCATION DISSOCIATION

A final major consideration is that total dislocations may be dissociated into partial dislocations. A partial dislocation is a dislocation whose Burgers vector is not a lattice vector. In fcc, zb and dc materials a very common partial dislocation corresponds to a stacking fault in the cubic ABC stacking sequence of atoms along (111) directions. The partial dislocations bounding these stacking faults have Burgers vectors of a / 6 ( 112) or

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a / 3 ( 111) and are called Shockley and Frank partials, respectively. Only the Shockley partial can glide within (11I} planes. Total dislocations can dissociate if they can lower their energy according to the requirement that C b 2 is less after the dissociation than before (recall from Eq. (15) that the self-energy of a dislocation is proportional to b2). For example, the reaction

+

~ / 2 [ 1 1 0 ]= ~/6[121] ~/6[21-1]

(21)

is energetically favorable. The resulting partial dislocations mutually repel each other and glide apart on the (-1 1-1) plane. This produces a ribbon of stacking fault between them, and at some point the extra cost in energy from the stacking fault balances the interaction energy of the two partials. Typical dissociation widths in unstressed Si and Ge are of the order of a few nm, implying stacking fault energies y of the order 50-80 mJ mP2 (Gomez et al., 1975; Cockayne and Hons, 1979; Bourret and Desseaux, 1979). Similar stacking fault energies have been measured in epitaxial Ge,Sil --I layers (Steinkamp and Jager, 1992; Hull et al., 1993). A high-resolution electron microscope lattice image of the dissociation reaction of Eq. (21) at a Geo,7sSio,25/Si(100) interface is shown in Fig. 7. The existence of partial dislocations, and the possibility of dissociation, can significantly affect the energetics of dislocation motion. For example, the partials may have different core structures and charge states from each other and from the undissociated dislocation. The Peierls barriers for motion of partial dislocations may be different than for motion of a total dislocation, particularly if kink formation and motion on the two

FIG.7. Cross-sectional (electron beam parallel to a Geo,7sSi0,25/Si( 100) interface) TEM lattice image of dissociation of a 60°b = a/2(101) dislocation into h = a/6(21I ) Shockley partials, separated by a region

of stacking fault.

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partials are correlated (Heggie and Jones, 1987). The existence of partial and dissociated dislocations may also be of significance in dislocation nucleation and motion, as will be discussed in Section V. 8. PARTIAL VERSUS TOTALMISFITDISLOCATIONS

The dissociation reaction described by Eq. (21) also has significant implications for the microstructure of misfit dislocations. We characterize the different partials in this equation according to the angle 8 between their line directions u and Burgers vectors b. For dissociation from the 8 = 6Ooa/2(101) total dislocation, the two a/6(211) partials have 8 = 30" and 8 = 90", respectively. The resolved lattice mismatch stress, from Eq. (17), on these two partials is very different. In general, the Schmid factors for the three types of dislocations are ordered

The order in which the partials move as the dissociated a / 2 (101) defect propagates is determined by a geometrical construction known as the Thompson tetrahedron construction (Thompson, 1953). This construction is based upon the requirement that the stacking fault bounded by the two partials produces only second-nearest stacking violation in the cubic lattice, that is, faults of the type ABCABCBCABCA, which is a relatively lowcmergy planar fault (-- 50-80 mJ mW2in Si, Ge and Ge,Sil-,, as described in Section 111.7), as opposed to nearest neighbor stacking violation, ABCABCKABCAB, which is a much higher energy planar fault. For compressively strained layers grown on a (100) surface, such as Ge,Sil-,/ Si( loo), the Thompson tetrahedron construction shows that for creation of an intrinsic stacking fault the 30" partial leads and the 90" partial trails as the dissociated a / 2 ( 101) dislocation propagates through the lattice. From Eq. (22), the trailing partial then experiences a greater lattice mismatch stress than the leading partial. The two partials are therefore compressed more closely than their zero-stress equilibrium separation. (In the limit of very high applied stresses, it may not be energetically favorable for the a / 2 ( 101) dislocation to dissociate at all in this configuration.) In this condition, the narrowly dissociated defect can be accurately approximated by a single 60"a/2( 101) dislocation. For other interfacial configurations, for example, tensile strain layers on (100) interfaces (Ge,Sil-,/Ge(100)), or compressively strained layers on (110) or (I 11) interfaces (GexSil-x/Si(l 10) and Ge,Sil-,/Si(ll 1)), the 90" partial leads. In these configurations, therefore, the leading partial experiences a greater resolved lattice mismatch stress than the trailing partial. This will increase the equilibrium separation, and if the net stress on the leading 90" partial is sufficiently greater than on the trailing 30" partial, the restoring stress due to the stacking fault energy between the partials can be

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FIG. 8. Cross-sectional TEM image ([220] bright field near the [I-101 pole) of a 41 nm Ge0,33Si0,67/Si(lIO)structure, showing stacking faults produced by passage of a/6(211) partial misfit dislocations. Reprinted with permission from R. Hull er ul. Appl. Phys. Left. 59, 964, Figure 3(a). Copyright 1991 American Institute of Physics.

overcome, and the dissociation width becomes infinite. The 90°a/6(21 1) partial will then effectively propagate as an isolated partial misfit dislocation, leaving a stacking fault in the lattice as it propagates (Fig. 8). In Section IV we will analyze the conditions under which this can occur, and find solutions in ( h ,x) space for which the partial dislocation is favored.

IV. Excess Stress, Equilibrium Strain and Critical Thickness 1 . INTRODUCTION We will now analyze more rigorously the conditions that define whether strain in a lattice-mismatched heterostructure is accommodated elastically or by misfit dislocations (in the limits where there is no interdiffusion across the epilayer/substrate interface, and where the epilayer surface is planar, that is, we ignore the relaxation mechanisms discussed in Sections 11.2 and 11.3). In particular, we will derive quantitative expressions for the critical thickness for the onset of misfit dislocation relaxation, the equilibrium amount of relaxation for 12 > h,, and the net or effective driving stress acting on a misfit dislocation.

2.

MATTHEWS-BLAKESLEE FRAMEWORK

Perhaps the most intuitive and general framework for analyzing these quantities is that originally developed by Matthews and Blakeslee (MB) (Matthews and Blakeslee, 1974, 1975, 1976; Matthews, 1975), and illustrated in Fig. 9. The original MB treat-

3

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Schematic illustration of the Matthews-Blakeslee model of critical thickness

ment analyzed the net force on a threading dislocation. We will derive an equivalent analysis in terms of stresses, which enables extension to more general results. The primary stresses acting on the dislocation are shown in Fig. 9. The resolved lattice mismatch stress a, drives the growth of misfit dislocations by lateral propagation of the threading arm. This is because growth of misfit dislocations (up to an equilibrium density) relaxes elastic strain by allowing the epitaxial layer to relax towards its free lattice parameter. The magnitude of a, has already been given in Eqs. (17) and (18). We also know from Section I11 that misfit dislocations have a self-energy, arising from their strain fields in the surrounding crystal. This produces a restoring stress ar (MB referred to the corresponding force as the “line tension” of the dislocation), which acts so as to inhibit growth of the misfit dislocation. The magnitude of this stress is easily derived from the equation for the self-energy per unit length of dislocation, Eq. (1 5). In addition, for partial misfit dislocations, there is a restoring force due to the energy of the stacking fault created by passage of the defect vsf. The net stress (or following the nomenclature of Dodson and Tsao (1987) the “excess stress” ue,) is thus given by

Here, E is the residual elastic strain in the system following partial plastic dislocation relaxation, defined by E = EO - [(bcosh)/p]. Other parameters in Eq. (23) have been previously defined in the text. Note that the generally quoted form of the MB model includes only the forces corresponding to a, and O T . Also, in the original MB framework, the quantity a was generally taken to be of magnitude e , such that they rewrite ln(ah/b) as [ln(h/b) I]. Figure lO(a) shows the dependence of a,, upon epilayer thickness h, using standard values (which we will assume in calculations henceforth, unless otherwise quoted) for (100) epitaxy of u = 0.28, cos8 = 0.5, cosh = 0.5, b = 3.9 nm, G = 64 GPa, E =0.41~ (where x is the Ge fraction in Ge,Sil-,), and a = 2. At very low values of

+

R. HULL

122 a

FIG. 10. Variation of (a) a,, and (b) eey with epilayer thickness h for a Ge0,25Si0,75/Si(IOO)structure. Reprinted with permission from R. Hull and J.C. Bean, Critical Review in Solid Stare andMateriais Science, 17, 507-546 (1992). Copyright CRC Press, Boca Raton, Florida

h , vexis large and negative. This corresponds to the regime where introduction of misfit dislocations increases the energy of the system. With increasing h, ae, becomes less negative, until at h = h, it becomes equal to zero. This defines the critical thickness of the system. For increasing h > h,, a,, becomes increasingly positive (in the absence of plastic relaxation), up to an asymptotic limit of a, - O , ~ Jindicating , that the misfit dislocation array is increasingly energetically favored in the structure. The equilibrium configuration of the system for any h > h , is that aeX= 0. The magnitude of the critical thickness is found by solving Eq. (23) for a,, = 0 and h = h,. For total dislocations (i.e. b = a / 2 ( 101)) where y = 0, this yields6 h , = b( 1 - u cos2 0) 1n(ahC/b)/[8n(1

+ u)s cos h]

(24)

At increasing h > h,, increasingly larger densities of interfacial misfit dislocations are favored, and correspondingly smaller amounts of residual elastic strain E . For h > 12, , the equilibrium residual elastic strain seq is found by solving Eq. (23) for a,, = 0 and E = E , ~ . The variation of E , ~with h is plotted in Fig. lO(b). 'Equation (24) thus effectively yields the values of kl , k z . k i , and k4 in Eq. ( I 3)

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FIG. 11. Predictions of the MB theory for the critical thickness h, in Ge,Sil-,/Si(100) structures for different values of the core energy parameter a. Also shown are experimental measurements of h , for different growthhnealing temperatures from the work of (a) Bean et al. (1984), (b) Kasper et al. (1973, (c) Green et al. (1991), and (d) Houghton etul. (1990).

Equation (24) does not have analytical solutions, but is simple to solve numerically. Solutions to this equation for the Ge,Sil-,/Si( 100) system are shown in Fig. 11. It is seen that the equilibrium critical thickness decreases rapidly with increasing Ge concentration, as the volumetric strain energy density increases as e2. Also shown in Fig. 11 are experimental measurements of critical layer thickness in the Ge,Sil-,/Si( 100) system from different groups, at several different growth temperatures. In the limit of high-growth temperatures (relative to the melting points of the materials, T, = 1412 "C for Si and 940 "C for Ge, with close to linear interpolation for melting temperatures of intermediate alloys [Stohr and Klemm, 1939]), experiment and equilibrium prediction agree well. At lower growth temperatures, experiment and theory are seen to diverge increasingly, in that increasingly larger critical thicknesses are measured experimentally as the growth temperature decreases. We shall discuss in Section V of this chapter that this divergence is due to the thermally activated kinetics of generation of the misfit dislocation array. In brief, as the equilibrium critical thickness is exceeded, plastic relaxation by generation of interfacial misfit dislocations is favored, but activation barriers exist to the nucleation and propagation of misfit dislocations. With decreasing temperature, the nucleation and propagation rates decrease, and plastic relaxation by misfit dislocations lags increasingly behind the equilibrium limit. The "experimental" critical thickness at a given growth temperature then becomes that epilayer thickness at which strain relaxation, or dislocation generation, is first experimentally detected (Fritz, 1987). Dodson and Tsao (1987) first demonstrated convincingly how kinetic modeling of this minimum detectable dislocation density process could accurately predict a "temperature-dependent'' critical thickness.

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ACCURACYOF THE MB MODEL

How accurate are the predictions of Eq. (24), in light of the perspective that there have been many refinements or recalculations of the critical thickness since MB originally developed their model? First, there are several approximations in the original formulation of this equation by MB. It assumes isotropic elasticity theory, whereas as we have already discussed, Si and Ge are relatively elastically anisotropic. The treatment of the dislocation core energy, where linear elasticity theory can no longer be applied, is very oversimplified. This core energy is approximated by assuming that we can apply standard elasticity theory outside of a “core radius” r, = ba, and accounting for the core energy inside a cylinder of radius r, by simply adding on a fixed energy per unit length of dislocation, Ecore: E,,,

= [Gb2(1 - u COS* 0)/471( 1 - v ) ]ln(a)

(25)

We can only ascribe a if we know EcOre. Accurate estimates of this quantity require sophisticated a b initio calculations of the core structure (Nandekhar and Narayan, 1990; Jones et al., 1993). In addition, E,,,,, and therefore a , are invariant with h only for h >> b, and will vary with the character of the dislocation, that is, the value of 8 . Estimates of cy in the literature vary from 0.5 - 4 (Nandekhar and Narayan, 1990; Hirth and Lothe, 1982; Perovic and Houghton, 1992; Beltz and Freund, 1994; Ichimura and Narayan, 1995), with the more recent total energy calculations generally suggesting a 1. We assume a value a = 2 in subsequent calculations. This variation in a affects the critical thickness calculation most significantly at higher E (higher x in Ge,Sil-,/Si), and lower h,. For example, if h = 1 nm (corresponding to h, for EO 0.04), In(ah,/b) varies between 0.9 and 2.3 as a varies between I and 4, a O.Ol), the variation of 160%. However, if h = 10 nm (corresponding to h, for 60 corresponding variation of ln(ah,/b) is between 3.2 and 4.6, or a potential error of 40%. In summary, the MB model in the form of Eqs. (23) and (24) should be regarded as approximate, to an accuracy perhaps 20-50% for critical thicknesses greater than a few nanometers. This makes it a very useful model.

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4. OTHERCRITICALTHICKNESS MODELS Our initial discussion of strain relaxation by misfit dislocations in Section 11.4 introduced the concepts of misfit dislocations and critical thickness qualitatively using energetic arguments. This was the basis of the earliest models of critical thickness developed by Frank and van der Merwe (1949a,b,c), who used a one-dimensional Fourier series to represent interactions between atoms in the epilayer and substrate, and demonstrated the concept of critical thickness. Subsequent development by Van der Merwe and co-workers (Van der Merwe, 1963; Van der Merwe and Ball, 1975; Van der Merwe

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and Jesser, 1988) developed the theme of calculation of the energy of a misfit dislocation array, including dislocation interaction energies, and comparison to strain energy relaxed within the epitaxial layer. Many of these models were mathematically complex, and had analytical solutions only in the limits of very thin and very thick epitaxial layers. Nevertheless, the energy balance approach derived in these papers set the foundations for subsequent development in this field, and enjoyed considerable success in predicting the critical thickness in body-centered cubic metal systems. Formally, the evaluation of critical thickness should be equivalent using either energetic or mechanical constructions, providing equivalent approximations are made in the two approaches (Willis et al., 1990). We now summarize the salient features of some other critical thickness models which have been developed as refinements of, or alternatives to, the MB framework:

(a) The model of People and Bean (1985, 1986), which attempted to reconcile theory with the early MBE measurements of h, in the Ge,Sil-,/Si(100) system (Kasper et al., 1975; Bean et al., 1984). This model assumed the self-energy of the dislocation to be localized within a certain region (of order five times the Burgers vector magnitude) centered on the dislocation core. This is in contrast to the usual elasticity treatment of dislocation self energy, and to subsequent verification that the MB theory describes the Ge,Sil-,/Si(100) system well in the limits of highly sensitive experimental techniques or very high temperatures (Houghton et al., 1990; Green et al., 1991).

(b) The model of Cammarata and Sieradzki (1989), who incorporated the concept of surface stress into the MB framework. For the Ge,Sil-,/Si system they argued that surface stress will be inward along the surface normal, and thus reduce the tetragonal distortion of the Ge,Sil-, epilayer and increase h,. These effects are significant only in the range of high Ge concentration where h, becomes relatively small. (c) The work of Willis et al. (1990), who derived a more precise expression for the energy of an array of misfit dislocations than the original Van der Merwe formulations. Subsequent development of this work using both energy minimization and force balance analyses (Gosling et al., 1992) has enabled refinement of both the Matthews-Blakeslee and Van der Merwe models. (d) The work of Chidambarro et al. (1990) who considered the effect of the orientation of the threading arm within the glide plane, and also analyzed a quasi-static Peierls force on the misfit dislocation. Certain orientations (e.g., screw) of the threading arm were determined to enhance the predicted critical thickness significantly.

(e) Fox and Jesser (1990) have invoked a static Peierls stress as an additional restoring stress upon a misfitkhreading dislocation, thereby increasing the critical thickness. It seems to the present author, however, that the effect of the Peierl’s barrier enters

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into dislocation kinetics, as discussed in a later section, rather than for the static case of the equilibrium critical thickness. (f) Shintani and Fujita (1994) have performed a calculation for critical thickness based upon anisotropic elasticity theory. Of the forementioned treatments, models (c), (d) and ( f ) represent refinements/ improvements of the original MB formulation, but do not radically alter the predicted magnitudes of critical thickness or excess stress. Models (b) and (e) represent additional terms to be considered in the MB analysis; of these models, (b) is significant only at relatively small critical thicknesses (of order a few nanometers or less), and (e) does not appear appropriate to the static critical thickness configuration. Model (a) is radically different from the MB model, but does not have the physical plausibility of the MB framework. In summary, the MB model remains an extremely useful general framework for predicting equilibrium relaxation by misfit dislocations in strained layer heterostructures. TO PARTIAL MISFITDISLOCATIONS 5 . EXTENSION

So far, our discussion of critical thickness has been restricted to total (b = a / 2 ( 101)) misfit dislocations, which represent the appropriate misfit dislocation microstructure for interfacial configurations where the 30"a/6(21 1) partial leads in the dissociated total dislocation. This corresponds to the Ge,Sil-,v/Si(lOO), Ge,Sil-,/Ge( 110) and Ge,Sil-,/Ge( 1 11) configurations among low index planes. For the interfacial configurations where the 90" partial leads, that is, Ge,Sil-,/Ge(100), Ge,Sil-,/Si( 110) and Ge,Si~-,/Si(lll), we should analyze the excess stress and critical thickness appropriate to the 90°a/6(21 1) misfit dislocation, and compare to the 60°a/2(101) misfit dislocation configuration. To simplify the calculations, we shall consider the 60"a/2(101) misfit dislocation in its undissociated state. (Consideration of the dissociated 60"a/2( 101) configuration with the 30" partial leading will generally lead to lower energies than for the undissociated configuration. Our simplified analysis, therefore, will tend to overestimate slightly the regimes in which the 90" partial is preferred over the 60" total dislocation.) The relevant descriptions of excess stress and critical thickness for the undissociated 60" total dislocation have already been given in Eqs. (23) and (24). For the 90" partial, we apply Eq. (23) with the relevant value of y , and modify Eq. (24) according to h , = Gb2cos@(1-ucos2B)ln(ah,/b)/[(8rrbG(1+

W ) E C O S ~ C O S @ ) - ( ~ ~ ~ ~ ( ~ - U ) ) ]

(26) By comparison of Eqs. (23), (24), and (26), we can then determine which is the favored dislocation type, 60" total, or 90" partial. The magnitudes of the critical thickness and the excess stress for the 90" partial dislocation are very sensitive to the stacking

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fault energy. Here we use the value of y = 65 mJ m-’ that we have previously determined for Ge,Sil-, alloys (Hull et al., 1993). A comparison of critical thicknesses for 90”a/6(211) and 6O0a/2(101) dislocations in the Ge,Sil-,/Si(IlO) system is shown in Fig. 12(a). It is observed that the 90” partial dislocation has a lower critical thickness at higher Ge concentrations, but the 60” total dislocation has the lower critical thickness at lower Ge concentrations. This transition is essentially due to the stacking fault energy associated with the partial dislocation: at zero stacking fault energy the 90” partial dislocation would always have the lower critical thickness, but with increasing

n

E 0

Ob!o

0.1

0.2

J

0.3

0.4

X

’

K

n

E 0,

+-.

0

3

1

60

v)

0)

2

v

c

NONE I

O’

‘

Oll

1

I

0.2

0.3

w I

0.4

I

x

FIG. 12. (a) Comparison of critical thicknesses for 90°a/6(21 1) and 60°a/2(101) dislocations in the Ge,SiI-,/Si(llO) system, assuming CY = 2 and y = 65 m J m-’. (b) Predicted dislocation microstructure in the Ge,Sil~,/Si(llO) system as functions of epilayer thickness h and Ge concentration x assuming CY = 2 and y = 65 mJ m-2. Regions labeled “NONE,” “60” and “90,” respectively, refer to regions where no dislocations are expected, regions where 60”a/2(101) dislocations are most favored, and regions where 90°a/6(21 1) defects are most favored.

128

R. HULL

stacking fault energy the critical thickness for the partial dislocation increases. The effect of the stacking fault energy is increasingly significant in thicker layers (because the total energy of the stacking fault per unit interfacial misfit dislocation length increases). Thus at lower Ge concentrations, where the critical thickness is higher, the effect of the stacking fault energy upon h,(90) is maximized, and the 60" partial dislocation has the lower critical thickness. Thus below a critical Ge concentration, x, hc(60) < h,(90), whereas for x > x, hc(60) > hc(90). These trends are also evident in analysis of the energetically favored dislocation type as functions of epilayer thickness h and Ge concentration x. This is defined by which dislocation has the greater excess stress acting upon it as a function of ( h ,x). The results of this analysis for the Ge,Sil-,/Si(lIO) system are shown in Fig. 12(b). For x < x, the 60" total dislocation is favored for all h. The curve AB corresponds to the plot of hc(60) in this composition range. For x > x,, the 90" partial dislocation is favored for lower epilayer thicknesses (and the curve BC corresponds to the plot of h,(90) in this composition range), but as h increases, the stacking fault energy associated with the partial dislocation increases and at a thickness h,, (x) defined by the curve BD, the 60" total dislocation again becomes energetically favored. The locus of BD is extremely sensitive to the value of y , and we have used the sensitivity of this transition to experimentally determine y (Hull et al., 1993). Thus, there exists a defined area of ( h ,x) space where the 90" partial dislocation is energetically favored. The trends of increasing tendency for 90"a/6(21 1) partial dislocations with increasing Ge concentration x and decreasing epilayer thickness have been experimentally verified in Ge,Sil-,/Si(llO) heterostructures (Hull et al., 1993). 6.

IN MULTILAYER STRUCTURES MODELSFOR CRITICAL THICKNESS

A very common configuration for strained Ge,Sil-, layers is to be confined between a Si substrate and a Si cap. For this geometry, the misfit dislocation configuration may involve interfacial segments either at both interfaces or just the bottom interface, as indicated schematically in Fig. 13. The double interface configuration is more general, and is demonstrated experimentally in Fig. 14. The segment of the dislocation loop at the lower Ge,Sil-,/Si interface allows the lattice parameter of the Ge,Si]-, to be relaxed toward its equilibrium value by increasing the average lattice parameter of the material above this dislocation segment. However, in the absence of the misfit dislocation segment at the upper Ge,Sil-,/Si interface, the Si cap would also have its average lattice parameter increased, thus straining it away from its equilibrium value. The upper misfit dislocation segment thus adjusts the lattice parameter in the Si cap back to its original, unstrained value. The extra interfacial segment increases the dislocation self-energy, but only in relatively thin Si capping layers is this energy contribution greater than the strain energy which would otherwise exist in the Si cap. The ranges of Ge concentrations, epilayer thicknesses and cap thicknesses h,,, which define the

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a

CAP EPI SUB FIG. 13. Schematic illustration of misfit dislocation configurations at capped Si/Ge,Sil-,/Si tures: (a) single interfacial dislocation; (b) double interfacial dislocation.

struc-

two configurations of Fig. 13, have been studied by several groups (Tsao and Dodson, 1988; Twigg, 1990; Gosling et al., 1993). For the double misfit segment configuration, the extra interfacial segment modifies the dislocation self-energy and critical thickness expressions, Eqs. (15) and (24). The self-energy per unit interfacial length of the dislocation dipole is given by the separate energies of the top and bottom dislocations minus the interaction energy between them. For a capping layer thickness hcap and a strained

FIG. 14. Plan-view TEM image of a 300-nm Si/100 nm Geo.2SiO.S/Si(lOO) structure. Closely spaced interfacial dislocation pairs correspond to segments of the same dislocation loop at top and bottom Ge,Sil-,/Si interfaces as illustrated schematically in Fig. 13(b). Reprinted with permission from R. Hull and J.C. Bean ,Appl. Phys. Lett. 54,92, Figure 2. Copyright 1989 American Institute of Physics.

R. HULL

layer thickness h , this yields

The last pair of logarithmic terms in this equation is the interaction energy between the two interfacial dislocations (Hirth and Lothe, 1982). For the limit heap >> h , the entire logarithmic expression simplifies to 2 In(ah/b), or simply twice the energy of the single interfacial dislocation at an uncapped epilayer. The restoring stress for the buried layer q - b is then a factor of two higher than for the single layer (as given by Eq. (23)), and the critical thickness for h, >> b is also approximately a factor of two higher. In fact this approximation is reasonably accurate for all hcap > h if h >> b. This factor of approximately two in critical thickness from capping of strained layers is of great benefit in post-growth processing of practical strained layer devices. Extension of critical thickness concepts to superlattice structures i s complex due to the larger number of degrees of freedom involved (individual layer strains and dimensions, total number of layers, total multilayer thickness, etc.). Experimentally, it is generally observed that providing each of the individual strained layers within the multilayer structure is thinner than the critical thickness for that particular layer grown directly onto the substrate, then the great majority of the strain relaxation occurs via a misfit dislocation network at the interface between the substrate and the first strained multilayer constituent (Hull et al., 1986). If individual strained layers within the multilayer do exceed the relevant single-layer critical thickness, then substantial misfit dislocation densities will generally be observed at intermediate interfaces within the multilayer. Thus, for the configuration where individual layers do not exceed the relevant single-layer critical thickness, the relaxation may be regarded as occurring primarily between the substrate and the multilayer structure taken as a unit. A simple energetic model has been developed (Hull et al., 1986),based upon reduction of the multilayer to an equivalent single strained layer. In this model, a multilayer structure is considered consisting of n periods of bilayers A and B , of thickness dA and d B , compound elastic constants k A and k ~ and , lattice mismatch strains with respect to the substrate of E A and E B , respectively. The amount of strain energy which may be relaxed by a misfit dislocation array at the substratehperlattice interface is found to be:

-

The form of this relaxable energy is equivalent to the average strain energy in the superlattice, weighted over the appropriate elastic constants and layer thicknesses. For k B and that Vegard’s Law applies (i.e., the Ge,Sil-,/Si system, if we assume k A that the strain of Ge,Sil-, with respect to Si varies linearly with x), then the relaxable strain energy simplifies to that in an equivalent uniform layer of the same total thickness

-

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+

as the superlattice = n(dGeSi dsi), and the average superlattice composition x, = x&esi/(&esi &i). The preceding analysis provides an equilibrium framework for analysis of critical dimensions for the onset of misfit dislocation generation in GeSi/Si superlattice structures. In practice, kinetic effects and dislocation interactions are likely to be very significant in these more complex structures (Hauenstein et al., 1989). It has been found, however, that relaxation kinetics of equivalent single layers at MBE growth temperatures of 550 "Cprovide an approximate guide to kinetic relaxation rates in superlattice structures (Hull et al., 1986).

+

V. Metastability and Misfit Dislocation Kinetics 1. BASICCONCEPTS

a. Introduction To date, we have considered only equilibrium descriptions of the misfit dislocation array. However, it is clear from the comparison of experimental data with equilibrium modeling of critical thickness in the GexSi~-,/Si(1O0) system, as illustrated in Fig. 11, that there are significant kinetic effects associated with generation of misfit dislocation arrays. These arise primarily from the substantial energetic barriers generally associated with dislocation nucleation and propagation, which have to be overcome by thermal activation, and with interactions between different dislocations in the array. In this section, we will consider each of these processes in turn. The primary effect of kinetic barriers in the development of the interfacial dislocation array is that the magnitude of plastic relaxation will lag behind the equilibrium condition. This gives rise to nzetastably strained structures. The definition of a metastably strained structure is that the excess stress on the operative dislocation type aeXis greater than zero. Increasing excess stress corresponds to increasing metastability of the structure. The definition of equilibrium is that aeX= 0. Experimentally, we describe the degree of metastability by a slight modification of Eq. (23) ~

c

+

= x 2GS[&o- (bcosh/p)](l ~ ) / ( l- U ) - [Gbcos@(l - ucos28)/4nh(l - u)]ln(ah/b) - y/b

(29)

Here, as discussed in Section IV, we have transformed E in Eq. (23) to [ ~ - ( cos b Alp)], where EO is the initial lattice mismatch strain (a, - a,)/a,, and (bcos A l p ) is the amount of strain relaxed by the misfit dislocation array. The maximum degree of metastability in a given structure corresponds to an infinite spacing of the misfit dislocations. As development of the misfit dislocation array proceeds, p and a,, decrease

-

132

R. HULL

until equilibrium is reached at a,, = 0 and p = pmjn.The value of pmill is found by solving Eq. (29) for oeex = 0. Kinetic effects are very substantial in GeSi-based heterostructures, as the covalent bonds of Ge and Si lattices are relatively strong. Thus, as described in Section 111, the activation energies for dislocation motion in bulk Ge and bulk Si are of order 1.6 eV and 2.2 eV, respectively. These values are very much greater than kT at any reasonable crystal growth temperature. Dislocation nucleation is also generally associated with significant activation barriers. Thus, particularly at moderate growth temperatures, the actual development of the misfit dislocation array may lag orders of magnitude behind the equilibrium configuration, and substantial excess stresses (up to 1.0 GPa) may develop. This is the essential reason for the ability to grow low misfit dislocation densities far beyond the equilibrium critical thickness at lower growth temperatures, as illustrated in Fig. 11. 6. An Idea of the Numbers Involved in Relaxation by MisJit Dislocations What kinds of numbers of total misfit dislocation length, interfacial spacings, and numbers of individual dislocation loops are we considering in strained layer relaxation? Consider a 10 cm diameter Si(100) substrate. Let a Ge,Sil-, epilayer be grown upon it with a lattice mismatch of 1% with respect to the substrate. Let the residual strain in the structure be close to zero (this will be true for a sufficiently thick epilayer grown at a sufficiently high temperaturc). Let the interfacial misfit dislocations be of the 60" a/2(110) type, such that the Burgers vector magnitude is 0.39 nm. The required interfacial misfit dislocation spacing will be given by the relation EO = b cos A l p , as previously discussed. This yields an interfacial misfit dislocation spacing 20 nm. The equilibrium misfit dislocation density is thus very high for epitaxial layer thicknesses that are significantly greater than the critical layer thickness. This means that dislocation interactions become very important as the structure relaxes plastically towards equilibrium. An average dislocation spacing of 20 nm across a 10 cm wafer requires a total length of the orthogonal interfacial misfit dislocation grid of the order lo6 m (recall that two separate, orthogonal, dislocation arrays have to be created). Thus f o r complete relaxation enormous line lengths of dislocation have to be created (in this example, of order 1000 km over a 10 cm wafer!). To create this line length in a finite time interval propagation and nucleation rates will need to be relatively rapid. So how many misfit dislocations are needed to create this array? Even if each dislocation makes a chord from edge to edge of the wafer, lo7 separate dislocations will be required, or of the order lo5 cm-2 of substrate area. Thus high densities of misfit dislocation nucleation sources are required. However, this estimate should be regarded as an absolute minimum required source density. Let us now consider the implications of finite dislocation propagation velocities (in general, there will not be sufficient time at temperature for the dislocation to grow to be sufficiently long to traverse the entire

-

-

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wafer-unless the growth temperature is very high, for example, the growth by CVD of Ge,Sl-,/Si(lOO) with x 5 0.05 at 1120 "C [Rozgonyi et al., 19871). As will be discussed later in this section, typical misfit dislocation velocities in GeSi-based heterostructures at strains of order 1% are of the order 0.1 - l p m s-l at 550 "C. At a typical MBE growth rate of 0.lnm s-l, it takes 1000 s to grow a 100-nm epilayer, by 0.01 is typically > 10% relaxed. Based upon the which time an initial strain of EO foregoing analysis, a 10% relaxation of the original strain requires a total interfacial dislocation length lo5 m. The maximum length of individual dislocations, based upon their propagation velocity, is 0.1-1 mm (the average will obviously be significantly less than this). Thus a minimum total number of 108-109 dislocation nucleation events are required, corresponding to an areal density 106-107 cmP2, and a nucleation rate 103-104 cmP2 sP1. (These numbers again represent a lower limit to the required source density as not all dislocations will nucleate at the start of the relaxation process, propagation velocities will be substantially lower than the number quoted above until the critical thickness is significantly exceeded, and misfit dislocation interactions may also slow propagation.) Identification of plausible mechanisms for generating such dislocation source densities is a critical question in our current understanding of strained layer relaxation.

-

-

-

2. MISFITDISLOCATION NUCLEATION a. Introduction The precise mechanisms of nucleation of misfit dislocations in strained heteroepitaxial systems remain somewhat elusive and controversial. There are many papers on this topic in the literature, and in this section we will summarize this literature with respect to nucleation in GeSi-based systems. The generic candidates for misfit dislocation nucleation sources are:

(a) Heterogeneous nucleation at specific local strain concentrations, due for example to

growth artifacts or pre-existing substrate defects. (b) Homogeneous or spontaneous nucleation of dislocation loops or half-loops. (c) Multiplication mechanisms arising from dislocation pinning and/or interaction processes. These are generally extensions of classic mechanisms such as the FrankRead source in bulk crystals (Frank and Read, 1950). We will now consider each of these generic mechanisms in more detail. b. Heterogeneous Nucleation Sources By heterogeneous nucleation, we mean nucleation of dislocations from sources that are not native to or inherent in the structure.

134

R. HULL

The original formulations of Matthews and Blakeslee generally assumed that the required density of defect sources were generated by existing dislocations in the substrate (although they did consider other potential nucleation mechanisms). This may have been a plausible mechanism for the GaAs substrates of the 1970s that they were considering, but contemporary commercial Si substrates typically have defect densities in the range 1-lo2 cm-2. Clearly, these substrates themselves cannot provide sufficient densities of defect sources, based upon the required source densities discussed in Section V. 1. Other heterogeneous features that may develop during epitaxial growth, or which arise from incomplete cleaning of the substrate surface prior to growth, can act as misfit dislocation nucleation sources if there are high local stresses and strains associated with them. Examples include residual surface oxide or carbide after substrate cleaning, particulates on the substrate surface or included during growth, source “spitting,” contaminants, transition metal precipitation, etc. For example, in our experience of Ge,Sil-, growth upon Si substrates by molecular beam epitaxy (MBE), the most generic heterogeneous source we observe are polycrystalline Si inclusions, of order a few hundred nanometers in size, and present at densities of the order lo3 cmP2, as illustrated in Fig. 15. We believe that these sources are associated with flaking of polycrystalline Si from deposits built up on the walls of the growth chamber. Other heterogeneous sources that have been documented in GeSi/Si heteroepitaxy include surface nucleation mechanisms associated with trace impurities (-- l O I 4 cmP3) of Cu (Higgs el al., 1991), and heterogeneous distributions of 1/6(114) diamondshaped stacking faults (Eaglesham et al., 1989) that have dimensions of order 100 nm

FIG. 15. Example of a heterogeneous dislocation source (an inclusion of polycrystalline Si that arises from flaking from the walls of the MBE growth chamber) in a 200-nm Geo.1 5 S i o . ~ ~ / S100) i ( heterostructure. Photo courtesy of F. M. Russ.

3 MISFITSTRAIN AND ACCOMMODATION

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and act as sources for 60°a/2(110) misfit dislocations. (Defects with similar geometries have been correlated with precipitation of metallic impurities, for example, Fe in Si [DeCoteau et al., 19921). Each of the features listed here certainly can provide sufficient source densities if the growth quality is poor enough (i.e., feature density and, therefore, source density is high enough), but each should be controllable in high-quality crystal growth to lo3 cm-2, or better. Of course, a single source may emit more than one misfit dislocation, but each source will only be able to relax strain in the heterostructure over dimensions comparable to the source size (which for isotropic source dimensions must be no greater than the epitaxial growth thickness). In summary, it is difficult to imagine that any of these heterogeneous nucleation mechanisms can generate the high densities of dislocations consistent with the observed relaxation of moderately or highly strained structures.

-

c. Homogeneous Nucleation Sources By homogeneous nucleation, we mean nucleation of dislocations from sources that are inherent in the structure. The most obvious homogeneous nucleation source in lattice-mismatched heteroepitaxy is the elastic strain, which if sufficiently high can lead to spontaneous (i.e., zero or negative activation energy) nucleation of dislocation loops or surface half-loops. At lower strains, homogeneous dislocation nucleation is associated with an energy barrier. This situation has been modeled by several authors, for example, Cherns and Stowell (1975, 1976); Matthews et al. (1976); Fitzgerald et al. (1989); Eaglesham et al. (1989); Hull and Bean (1989a); Kamat and Hirth (1990); Dregia and Hirth (1991); Perovic and Houghton (1992); and Jain et al. (1995). The general principle underlying these models is that a growing dislocation loop of appropriate Burgers vector relaxes strain energy within the epilayer Est, but balancing this is the self-energy of the dislocation loop itself El (this is closely analogous to the Matthews-Blakeslee critical thickness criterion for growth of interfacial misfit dislocations). Other energy terms that need to be considered are the energies of steps created or removed in the nucleation process ESP(a 60" a/2(101) dislocation, for example, has a Burgers vector component normal to the surface and must thus always be associated with step creation or removal), and the energy of any stacking fault created E,f either as a result of dislocation dissociation, or as a result of separate nucleation of partial dislocations. This produces a total energy of the form Er = El - Est i= E,, f Esf (30) Note that E,f may either increase or decrease the total energy: a stacking fault clearly increases the system energy, but a 30" partial dislocation, for example, could nucleate along the path of an existing 90" partial dislocation to remove the existing stacking fault and generate a total misfit dislocation. The total system energy will in general pass

R . HULL

136

through a maximum value A E (which represents an activation barrier for homogeneous dislocation nucleation), at a critical loop radius R,. The magnitude of A E as a function of strain depends sensitively upon the Burgers vector of the dislocation, the magnitude of the dislocation core energy assumed, and whether or not ESt and E,f are included in the calculation. Variations between these different terms cause a range of predicted activation barriers at a given strain in calculations by different authors, but consensus exists that activation barriers approach a physically attainable limit (of order a few electron volts or less) for strains of the order 2 4 % (corresponding to x 2 0.5 in Ge,Sil-,) , and are unphysically high (100 eV or greater) at strains below about 1% (corresponding to x 5 0.25 in Ge,Sil-,). A common expression used for the loop self-energy (Bacon and Crocker, 1965) for a complete loop of radius R is El = [Gb2R/2(1

-

+ (1 - u/2)(1 - b:/b2)][ln(2nRa/b)

u)][b:/b2

-

1.7581 (31)

Here b, is the component of the dislocation Burgers vector normal to the loop. The original Bacon and Crocker expression also contained an extra term due to surface tractions; the magnitude of this extra expression is relatively low and is generally omitted by most authors. The strain energy relaxed by the loop is given by Est = [2nR2G( 1

+ u ) E / ( ~- ~ ) ] ( bcos, q5 cos A + b, cos2 q5)

(32)

Here b, and be are the glide and climb components, respectively, of the dislocation Burgers vector. Considering the energy terms in Eqs. (31j and (32) only, the critical radius is found by setting 6Et/6R = 0, yielding R, =

{b2tb;/b2

+ (1 - u/2)(1 - b;/b2)l[ln(2nRca/b) Sn(l+

- 1.758

+ 2na/b]}

u)E(6~COSq5COSh+b,cos~~)

(33)

Assuming the most common configuration of 6Ooa/2(101) dislocations moving on (11 1) glide planes with a (100) interface, we have b, = be = 0, b, = b = 0.39 nm, simplifying Eq. (33) to cos h = 0.5, cos q5 =

m,

R, =

&b(l - u/2)[1n(2naRc/b) - 1.758 - 2na/b] 4&(1

+ VIE

(34)

This expression may be solved numerically for R, as a function of E and the value substituted back into Eq. (30) to yield A E For a dislocation half-loop nucleating at the epilayer free surface, ESt is halved relative to a full loop, El is approximately halved (this is actually an involved calculation if done rigorously because of the complexity of the image interaction calculation; these

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complexities are generally ignored in the literature, and a factor of a half used), R, is approximately unchanged, El is approximately halved and A E is approximately halved. This indicates why surface half-loop nucleation is generally assumed to be greatly favored over full-loop nucleation within the epitaxial layer. The value of R, for the surface half-loop is closely analogous to the Matthews-Blakeslee critical thickness for an uncapped epilayer. The small quantitative differences between R, and h, arise because the dislocation configuration is somewhat different in the two cases, thus modifying the expression for dislocation self-energy, and because R, is measured in an inclined { 111) plane, while h, is measured along the interface normal. Evaluation of the activation barrier A E (for either half- or full-loops) shows that it increases approximately as the magnitude of the Burgers vector cubed (this strongly reduces the activation barrier for partial vs total dislocations [Wegscheider et al., 1990]), increases strongly with the magnitude of the core energy parameter a and is approximately inversely proportional to the strain. In Fig. 16 we show the variation of A E with x for surface half-loop nucleation in the Ge,Si~-,/Si(IOO) system for 60°a/2(110) dislocations with removal and creation of surface steps as they nucleate; EStis approximated from the relation ESt= 2Rbx sin3!, (35) where j3 is the angle between the dislocation Burgers vector and the free surface, and x is the areal epilayer surface energy, which, following Cherns and Stowell (19759, we approximate by x Gb/8. Note the great sensitivity of the results for A E in Fig. 16 to the value of a. The foregoing calculations suggest physically accessible activation barriers (of order 5 eV)7 for strains of order 3 4 % or greater. However, quasi-homogeneous processes can provide substantial densities of sites at the epilayer surface, which significantly reduce the homogeneous nucleation barrier. By quasi-homogeneous, we mean features in the crystal that, although arising from inherent physical processes in the crystal growth, represent perturbations in the average crystal structure. For example, we have demonstrated that statistical fluctuations in the Ge concentration of a Ge,Sil-, alloy can produce significant densities of local volumes where the Ge concentration is significantly higher than the average matrix composition (Hull and Bean, 1989a). Local Ge-rich regions have also been implicated by Perovic and Houghton (1992) in dislocation nucleation. Another quasi-homogeneous process that can locally reduce activation barriers for dislocation nucleation is roughening of the epilayer growth surface. We have already considered in the analysis of Eqs. (30)-(35) how removal of monolayer surface steps

-

-

7Consider the number of surface atomic sites, IOl5 cm-2, available for nucleation, and a nucleation s-’ in Si. We thus have a attempt frequency, which we will approximate by the Debye frequency, lo2* ~ r n - ~ s - ’ .For a crystal temperature of 1000 K, kT 0.086 eV, and for a nucleation prefactor of lo-”, which when multiplied by the prefactor 5 eV activation energy, the Boltzmann factor yields a significant nucleation rate.

-

-

-

R. HULL

138

%

U

w Q

FIG. 16. Homogeneous activation barriers A E for nucleation of 60°n/2(101) half-loops at Ge,Sil-,/Si( 100) surfaces, for different values of the dislocation core energy parameter a with removal of surface steps (RS) or creation of steps (CS).

can reduce activation energies. Local stresses associated with surface steps and facets (particularly if they are several monolayers, or more, high) can also significantly enhance nucleation locally. For example, theoretical (Sorolovitz, 1989; Grinfeld and Sorolovitz, 1995) and experimental (Jesson et al., 1993, 1995) studies have postulated and demonstrated the formation of surface cusps during strained epilayer evolution. These cusps are associated with substantial local stress concentrations, and reduced barriers for dislocation nucleation. Similar observations of dislocation injection associated with troughs in surface roughness in the InGaAs/GaAs system have been described by Cullis et al. (1995). In general, the boundary between “homogeneous” and “heterogeneous” source mechanisms can become indistinct, and to some extent no nucleation event will be truly homogeneous within the strained layer: loop nucleation will always be energetically favored at some nonperiodic event, such as a surface step, cusp, or locally enhanced Ge concentration. This may produce a “hierarchy” of misfit dislocation nucleation sources, with the lowest activation energy processes exhausted first, and subsequent activation of increasingly higher activation energy events. d. Dislocation Multiplication

The most intuitively appealing generic candidate for a source mechanism producing high densities of misfit dislocations is a multiplication mechanism, by analogy to deformation experiments in bulk semiconductors. The idea of a regenerative dislocation source goes as far back as the Frank-Read source (Frank and Read, 1950). Probably the first application of this concept to semiconductor strained layer epitaxy was reported by Hagen and Strunk (1978) for the growth of Ge on GaAs. Their proposed mecha-

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FIG. 17. Schematic illustration of the Hagen-Strunk mechanism of dislocation multiplication, (a)-(d). (e) shows the expected configuration of intersecting dislocations without the Hagen-Strunk mechanism operating; ( f )shows the expected configuration after operation of the Hagen-Strunk mechanism. Reprinted with permission from R. Hull and J.C. Bean, Critical Reviews in Solid State and Materials Science 17, 507-546 (1992). Copyright CRC Press, Boca Raton, Florida.

nism, illustrated in Fig. 17, relies upon intersection of dislocations with equal Burgers vectors along orthogonal interfacial (011) directions within a (100) interface. The very high energy configuration associated with the intersection is lowered by formation of two interfacial segments with localized climb producing rounding near the original intersection, as shown in Fig. 17(b). This reduces the very high radius of curvature at the original intersection, and lowers the dislocation configurational energy. Local stresses near the dislocation intersection now modify the local energy balance for the dislocation, causing one of the rounded segments of dislocation to move toward the surface, as shown in Fig. 17(c). When the tip intersects the surface, Fig. 17(d), a new dislocation segment is formed, which can act as a new misfit dislocation. This process may repeat many times, producing a regenerative dislocation source. Several authors have since invoked this mechanism as a dislocation source in Ge,Sil-,/Si strained layer epitaxy (e.g., Rajan and Denhoff, 1987; Kvam el aE., 1988). However, the experimental evidence for the Hagen-Stmnk source is often based upon dislocation configurations as shown schematically in Fig. 17(e), where dislocation segments align along (011) directions across the original intersection event. The true “footprint” of the Hagen-Strunk source is as shown in Fig. 17(f) where alignment across the intersection does not occur. The configuration of Fig. 17(e) is simply that of Fig. 17(b) where many orthogonal dislocation intersection events have occurred (Dixon and Goodhew, 1990).

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Direct evidence for different multiplication mechanisms has been reported by several other authors. Tuppen et al. (1989, 1990) used Nomarski microscopy of chem(x 5 0.15, epilayer thickness 0.9pm) structures ically etched Ge,Sil-,/Si(100) to demonstrate greatly enhanced dislocation nucleation activity associated with dislocation intersections. Two distinct Frank-Read type and cross-slip mechanisms for the multiplication process were proposed. An important requirement for these specific mechanisms, and for most other multiplication mechanisms, is the existence of two pinning points along that segment of dislocation line that acts as the regenerative source. The segment then grows by bowing between the pinning points, and may eventually configure into re-entrant and re-generative geometries, similar to the FrankRead source. In the models described by Tuppen et al. (1989, 1990), dislocation intersections act as these pinning points. Capano (1992) described several multiplicative configurations, not involving dislocation intersections, for a single isolated threading dislocation. These mechanisms generally involved dislocation cross-slip following pinning of segments of the dislocation by inherent defects in the dislocation or host crystal (e.g., constrictions in the partial dissociation along the dislocation line). Capano measured a minimum “multiplication thickness” for these processes of 0.67 p m for Ge0,13Si0,87/Si(100) structures. This is very consistent with the experiments of Tuppen et al. (1989, 1990), who studied 0.5 pm, 0.7 p m and 0.9 p m thick epilayers with Ge concentration x = 0.13, and first observed evidence of multiplication processes in the 0.7 p m layer. Capano also modeled the minimum epilayer thickness required to accommodate the intermediate cross-slip configurations in the multiplication processes he described, and derived minimum epilayer thicknesses an order of magnitude lower than he experimentally observed. The discrepancy was attributed to the density of available pinning points along the dislocation line. A very complete description of a Frank-Read type source in Ge,Sil-,/Si epitaxy has been provided by LeGoues et al. (1991,1992). This mechanism involves dislocation interactions at the interface to provide the required pinning points, and was most generally observed in graded composition layers. Similar mechanisms have also been reported by Lefebvre et al. (1991) for the strained InGaAs/GaAs system. Beanland (1995) described additional multiplication sources in InGaAs/GaAs structures, involving operation of spiral sources operating either from dislocation interactions, or from the tips of pre-existing edge dislocations. Albrecht et al. (1995) demonstrated that dislocation multiplication could be associated with surface roughening (see discussion in Section V.2.c). They used finite element modeling to model stress concentrations at ripple troughs in relatively low Ge concentration (x = 0.03) Ge,Sil-,/Si( 100) heterostructures, which were demonstrated to lead to dislocation injection into the substrate and subsequent Frank-Read type multiplication sources. In summary, it appears that dislocation multiplication mechanisms do occur in strained GeSi layers. However, a significant density of dislocation pinning points is required for these mechanisms to operate efficiently. These pinning points can either

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141

occur from dislocation intersections (which requires a significant source dislocation density), or from inherent features of the dislocation or host crystal (constrictions,point defects, etc.) It also appears that minimum epilayer thicknesses are required for multiplication, as intermediate cross-slip or loop configurations have to be accommodated during the regeneration process. Tuppen et al. (1989, 1990) and Capano (1992) measured minimum epilayer thicknesses > 0.5 p m for Ge concentrations of x = 0.13, but LeGoues et al. (1991, 1992) did observe multiplication in graded structures with thicknesses as small as 200-300 nm. e. Experimental Measurements of Dislocation Nucleation Rates

A powerful insight can be provided into the epilayer thickness and composition ( h ,x) regimes in which different dislocation mechanisms dominate by experimental study of dislocation nucleation rates and activation energies. Unfortunately, relatively little experimental data have been reported for the Ge,Sil-,, or any other strained layer system. The most extensive experimental work to date has been by Houghton (1991), who used direct counting of dislocation source densities via optical microscopy of etched (Si)/Ge,Sil-,/Si(lOO) structures following post-growth annealing. Only the early stages of strain relaxation were studied (due to the resolution limits inherent to conventional optical microscopy), where less than about 0.1% of the initial lattice mismatch strain was relaxed. The number of observed dislocations followed the trend

In this equation B is a constant of the order loi8 s-’. The activation energy for dislocation nucleation, En is of the order 2.5 eV for 0.0 < x < 0.3, and epilayer thicknesses in the range 100-3500 nm. Measured nucleation rates were in the range 10-’-105 cm-*s-l. In the Houghton formulation, NO is an adjustable parameter (determined by experimental inspection) for each structure and represents the number of pre-existing heterogeneous nucleation sites before the anneal is started. Measured values for NO reported by Houghton were in the range 103-105 cm-* and generally increase for higher x. Equation (36) implies that, at least in the early stages of relaxation, a12 misfit dislocations are nucleated from heterogeneous sources. As discussed in Section V. 1 of this chapter, this appears implausible during later stages of dislocation-mediated strain relief. Also note that Perovic and Houghton (1992) later reinterpreted Eq. (36) in terms of “barrierless” nucleation of dislocations, where it was assumed that the activation energy of 2.5 eV measured for dislocation nucleation was essentially the 2.2-2.3 eV activation energy associated with dislocation growth by glide (see Section V.3 of this chapter), that is, the activation energy of Eq. (36) corresponded to the rate at which dislocations grew to be significantly large to be detected by optical microscopy.

142

R . HULL

Some other data for dislocation nucleation rates in GeSi/Si heterostructures have been reported by Hull et al. (1989a) using in situ TEM measurements (these experiments correspond to much later stages of the relaxation process than those reported by Houghton, as TEM imaging covers a much smaller field of view than optical microscopy). Nucleation activation energies of the order 0.3 eV were reported for a 35 nm Geo.25Si0.7s/Si(100)structure, with a prefactor of 2.2 x lo6 cm-2.y-l. The observed dislocation densities were relatively high (-- 107-108 cm-’), therefore strongly suggesting quasi-homogeneous nucleation in these experiments. Although this activation energy is very different to that reported by Houghton, overall nucleation rates at comparable excess stress are of the same order of magnitude (e.g., at a temperature of 650 “C in a 100 nm x = 0.23 structure, Houghton reports a nucleation rate 3 x lo4 cm-’s-l, whereas for the 35 nm, x = 0.25 structure, Hull et al. (1989a) reported a nucleation rate 5 x lo4 cm-2s-’). In capped Si/Ge,Sil-,/Si(100) structures (0.10 < x < 0.30) we have measured activation energies in the range 0.5-1.0 eV (Hull et al., 1997). Nucleation data from the work of Houghton and of Hull et al. is shown in Fig. 18. Other quantifications of misfit dislocation nucleation in the Ge,Sil-,/Si system are the work of LeGoues et al. (1993), relating to the Frank-Read-like multiplication mechanism proposed by that group, and Wickenhauser et al. (1 997), relating to heterogeneous nucleation in Geo.16Sio.84layers. LeGoues et al. (1993) inferred an activation 5 eV, and a total dislocation nucleation rate energy for the former mechanism of l the amount of crystallographic tilt of the epilayer produced of 6 x 10’ ~ m - ~ s -from by differential dislocation densities on different glide planes in Ge,Sil-, layers grown on misoriented Si( 100) substrates. Wickenhauser er al. (1997) determined a heterogeneous source (the physical nature of the source was not identified) activation energy of 2.8 eV.

-

-

-

-

Summary of Misfit Dislocation Nucleation From the discussion of the preceding few subsections it is clear that there is not a universal model for dislocation nucleation in GeSi-based heterostructures, and, further, that we should not expect such a universal model. The very high dislocation nucleation rates associated with strain relaxation in higherstrained systems are consistent with homogeneous, or quasi-homogeneous processes (such as alloy clustering, or epilayer roughening) in this strain regime. Although calculations of true homogeneous nucleation predict that strains of order 3 4 % or higher are required for appreciable nucleation rates, local stresses associated with quasi-homogeneous processes appear to lower this strain threshold down to 2% or even 1%. At strains below this level, it appears that a combination of heterogeneous nucleation and multiplication dominates. The initial heterogeneous stage is severely source limited, thus the metastable growth regime at low strain and low temperature is very large, as indicated in Fig. 11. As the network of dislocations from heterogeneous

3

MISFITSTRAIN AND ACCOMMODATION

a

.

10'

-

nn

143

10'.

"

u- 10' t-

4 U

*

102

0 I$

lo'

2

100

d

10'9

06

I .6

I 1

INVERSE TEMPERATURE, looo/T K - I

1

i

I lo2 1

L

1 OI6

J

A

m A

RHx=OZS,h=35nm R H x=O 30, h=30 nm DCH xdJ.23, h=l OOnm

7

8 l/kT

A10

9

(J-lxlO19)

FIG. 18. (a) Measured misfit dislocation nucleation rates during annealing of (Si)/Ge,Si*-,/Si( 100) heterostructures; A = 20 period superlattice of 32 nm .%/lo nm G ~ o . ~ o S ~ D O .=~ 500 O ; nm Geo.035Sio.965; E = 3000 nm Ge0.035Si0.965; F = 190 nm Ge0.17Si0.83; G = 100 nm Ge0.23Si0.77. Reprinted with permission from D.C. Houghton, Appl. Phys. Lett. 55, 2124, Figure 2. Copyright 1990 American Institute of Physics. (h) Comparison of nucleation rates measured by Houghton (1991) (DCH) and Hull etal. (1989a, 1997) (RH) for Ge,Sii-,/Si(100) structures with similar composition (x 0.2-0.3) and excess stress.

-

sources eventually develops, the required dislocation intersection events generally assumed to be required to fuel multiplication processes can occur. We can thus tentatively map out three broad nucleation regimes: in high strain systems (> 1-2%), relaxation is by homogeneous and quasi-homogeneous nucleation processes. In low strain systems (- 1% or less), heterogeneous sources provide the initial background dislocation density in the low epilayer thickness regime of the order of a few hundred nanometers or less. Multiplication mechanisms then become dominant in the low strain, high thickness regime.

R. HULL

144

Experimental mapping of dislocation nucleation rates and activation barriers have so far provided little additional insight into delineating these different nucleation regimes. The only extensive existing data set from Houghton is apparently consistent with heterogeneous sources in the low strain regime.

3.

MISFITDISLOCATION PROPAGATION

Experimental descriptions of dislocation motion are relatively well developed, and in general follow straightforwardly from treatments of dislocation glide in bulk Si and Ge crystals, as reviewed in Section 111and described by Eq. (20) u = ugmm exp[-E,,(a)/kT]

For misfit dislocations in Ge,Sil-, heterostructures, we shall also use this equation to describe dislocation velocity, using the excess stress for CJ in the prefactor. In the low stress (10-100 MPa) regime for intrinsic, pure crystals, EL, I .6 eV in Ge and 2.2 eV in Si. The prefactors are very similar for the two materials ( u g 3 x 10-'m2Kg-'s) (Alexander and Haasen, 1968) and thus dislocations glide a lot 5000 at 550 "C). Thus we should expect that for faster in Ge than Si (by a factor Ge,Sil-, alloys the glide activation energy will decrease, and the glide velocity will increase, with increasing x. Unfortunately, the dependence of E , ( x ) from bulk crystals is not known (apart from recent measurements very near x = 1.0 (Yonenaga and Sumino, 1996), which scaled well from measurements on bulk Ge), and in general we are restricted to linear interpolation of the values for bulk Si and Ge. The composition dependence of E,(x) in Ge,Sil-, alloys is further complicated by the relationship between x and o:it is predicted that kink nucleation energies (and hence glide activation energies) are reduced at sufficiently high applied stresses (Seeger and Schiller, 1962; Hirth and Lothe, 1982), such as are typically encountered in Ge,Sil-,/Si heteroepitaxy for all but the most dilute Ge concentrations. The stress dependence of the dislocation velocity is still somewhat uncertain in bulk Si and Ge crystals, let alone Ge,Sil-, alloys; undoubtedly part of this uncertainty depends upon how the stress dependence is apportioned between the prefactor and the exponential in Eq. (20). However, there is strong evidence from bulk measurements in Si that the activation energy E , is stressdependent for (T of the order of a few hundred MPa or more (Kusters and Alexander, 1983). In general, these stress effects will also need to be deconvoluted in glide activation energy measurements at the enormous stresses (of order 1 GPa), which can be present in Ge,Sil-, heterostructures. An extensive set of measurements by Imai and Sumino (1983) has yielded a linear stress dependence, nz = 1 .O in Eq. (20), at stresses of the order tens of MPa in Si, and this is also the dependence predicted by the generally accepted microscopic model of dislocation motion, the Hirth-Lothe diffusive double kink model (Hirth and Lothe, 1982).

-

- -

3 MISFITS T R A I N A N D ACCOMMODATION

145

Misfit dislocation propagation velocities in (Si)/Ge, Sil --x /Si( 100) heterostructures have been studied by several groups, using either in situ TEM observations (Hull et al., 1989a, 1991a; Nix et al., 1990) or chemical etching and optical microscopy (Tuppen and Gibbings, 1990; Houghton 1991; Yamashita et al., 1993). In Fig. 19, we plot measured dislocation propagation velocities vs excess stress a,, from these different groups for a temperature of 550 "C (interpolated where necessary from measurements at other temperatures). The data between different groups and different techniques agree relatively well. Note that even the relatively limited vertical scatter of data in this plot is not necessarily due to experimental error, as there are factors other than just excess stress (primarily Ge concentration in the Ge,Sil-, alloy, as discussed previously) which determine dislocation velocity. In our own work we have made hundreds of measurements of misfit dislocation velocities from a wide range of (Si)/Ge,Sil -,/Si( 100) heterostructures (see, for example, Hull et al., 1991a; Hull and Bean, 1993). The data from different structures can be effectively normalized to each other by plotting the logarithm of the quantity v* inverse temperature, where: u* = [v,e-

1/sex

[0.6x(eV)/kT]

(37)

In this expression, v, is the measured dislocation velocity and the quantity e-[0.6x(eV)lkT1 accounts for the 0.6 eV glide activation energy difference between Ge and Si, such that we are assuming E , ( x ) = 2.2 - 0 . 6 ~eV in Ge,Sil-,. Equation (37) therefore effectively normalizes the measured dislocation velocity to an equivalent velocity at an excess stress of 1 Pa in pure Si. In Fig. 20, we plot our measurements of misfit dislocation velocities from a wide range of (Si)/Ge,Sil -x/Si( 100) heterostructures (Hull et al., 1991a; Hull and Bean, 1993), normalized according to Eq. (37).

1

-,

)4

/Si(lOO) heterostructures vs FIG. 19. Measured dislocation Propagation velocities in (Si)/Ge,SlI excess stress oex at 550 "C from Houghton (1990), Hull et al. (1991a, 1993), Tuppen and Gibbings (1990), Nix et al. (1990), and Yamashita et al. (1993).

R. HULL

146

-

-

M -

Y f

v

3

B

-

L

-35

.

\4

-33

h

1

~

-31 -39-

-41 -43 -45

~

-

' 7

"*=

8

9 l/kT (J-1x1019)

10

1

FIG. 20. Normalized dislocation velocities for uncapped and capped (300 nm Si)Ge, Sil -,/Si(lOO) structures. Faint dashed lines either side of the main regression lines correspond to 0.7 confidence bounds.

Note that there is a systematic difference in normalized velocity for capped vs uncapped Ge,Sil --x layers. We have ascribed this difference to different microscopic kink mechanisms for dislocation propagation motion in these two geometries. The regression line fits to the data in Fig. 20 are given for uncapped structures (a) and capped structures (b) by u* = ,-(7.8+l.4),-(2.03+0.10

*=

,-

eV)/kTm2S~g-I

,-1.93+0.10 eV)/kT m 2 s ~ g - I

( 10.4fl.4)

(

(a) (b)

(38)

To calculate the misfit dislocation velocity at a given temperature in a given (Si)/ Ge,Sil -x/Si( 100) heterostnicture, therefore, one simply multiplies the relevant u* by

a , , ,iO.WeV)lkTI

Note that Eqs. 38(a) and (b) are still largely empirical, as they ignore further microscopic details of the diffusive double kink model such as the dependence in some regimes of the velocity upon the length of the propagating dislocation line (see Section 111.6) and single vs double kink dynamics (Hirth and Lothe, 1982; Hull et al., 1991a; Hull and Bean, 1993). A stress-independent activation energy is also implied by the form of Eqs. 38(a) and (b), although note that the regression calculations give activation energies 1.9-2.0 eV, which is less than the 2.2 eV activation energy that would be expected from the normalization to equivalent velocities in Si. (The lower value of 1.9-2.0 eV corresponds to an average 0.2-0.3 eV reduction in activation energy due to the stress dependence.) However, from the regression coefficients of Eqs. 38(a) and (b) ( R = 0.93 and 0.96, respectively), it is apparent that the equations offer good semiempirical descriptions of misfit dislocation velocities. There have been other attempts to fit measured dislocation propagation velocities systematically to an empirical model. Houghton (1991) modeled his measurements

3

MISFIT S T R A I N A N D ACCOMMODATION

147

of misfit dislocation velocities using a slightly modified version of Eq. (20): u = uo(oex/G)me(-EL,/kT) with m = 2, uo = (4 i 2) x 10" ms-', and a constant activation energy of E , = 2.25 f 0.05 eV for 0.0 < x < 0.3 in (Si)/GeXSil-,/Si(100) structures. The higher pre-exponential factor, m, may be accounted for by the assump-

tion of the constant activation energy: the higher Ge concentration structures (which in general will correspond to the structures with higher excess stresses) will have enhanced velocities due to the lower dislocation glide activation energy in Ge than in Si. Assumption of a constant value for E , will therefore tend to overestimate the pre-exponential stress dependence of dislocation velocities in higher Ge concentration films, producing an artificially high value of rn. Tuppen and Gibbings (1990) studied misfit dislocation velocities in (Si)/GexSil-,/Si(lOO) structures with relatively low Ge concentrations (typically x i 0.15). They observed a linear pre-exponential dependence of velocity upon excess stress, m = 1 in Eq. (20), and a prefactor consistent with bulk Si and Ge measurements (UO = 2.81 x m2Kg-'s). The measured glide activation energy was E , ( x ) = 2.156 - 0 . 7 ~ eV, which is somewhat lower than predicted from bulk values (perhaps due to the theoretically predicted stress dependence of the activation energy). They also observed in certain structures a dependence of the dislocation velocity upon the threading dislocation length L f i h (this relationship arises from the orientation of the threading arms within the inclined { 111) glide planes), where h is the Ge,Sil-, epilayer thickness, consistent with the predictions of the double kink model (Hirth and Lothe, 1982) for dislocation lengths, L << X, with X the average distance between kinks. (Note that the double kink model also predicts an increase in activation energy in this regime, compared to the bulk regime where L >> X.This was not observed by Tuppen and Gibbings, however.) A similar length dependence has been reported by Yamashita et al. (1993). In Fig. 21 we show data from misfit dislocation velocities in GeXSil-,/Ge(100) structures (Hull et al., 1994). These structures are in the low strain (x > 0.8) regime, such that the preferred dislocation type is b = a/2(101) (total dislocation), rather than the b = a/6(211) (partial dislocation) configuration, which would be expected at higher strains (see Section 111.8). Figure 21(a) shows that for comparably strained Ge,Sil-, layers ( E 0.008) grown on Ge(100) and Si(100) substrates, dislocation velocities are much higher for the Ge-rich alloy grown on the Ge substrate than for the Si-rich alloy grown on the Si substrate, as expected from the lower activation energy for dislocation glide in Ge than Si. In Fig. 21 (b) we show how these measurements of dislocation velocities in Ge,Sil-,/Ge( 100) structures follow the same scaling laws as the structures grown on Si substrates, as defined by Eq. 38(a). Studies of misfit dislocation velocities in GexSi~-,/Si(ll0) structures (Hull and Bean, 1993) have enabled comparison of the velocities of b = a/6(211) and b = a/2(101) Burgers vector misfit dislocations. As illustrated in Fig. 22, the partial dislocations actually have lower velocities than the total dislocations. This is somewhat surprising, as we would expect the motion of the total a/2(101) dislocation to be con-

-

-

R. HULL

148

13 1

i 1

.-... -.

-t

-1

685A x=O 18, Si(100) 750A x=0.80, Ge(100)

____

A

9

8

-34~

_ _ _ _

l/kT

i

10

11 (J-lxlO19)

.

1

I _J

12

I

FIG. 2 1. (a) Comparison of measured dislocation velocities in Ge,Sil-, layers with comparable atrain and excess stress grown on Si(100) and Ge(100) substrates. (b) Normalized dislocation velocities in Ge,Sil-, layers grown on Si(100) and Ge(100) substrates. Reprinted with permission from R. Hull er a/. Appl. Phys. Leu. 65, 327, Figures I and 3. Copyright 1994 American Institute of Physics.

G

-34 -36 -

-3 8 -40 -

-42 -

-44-46;

9 10 l/kT (J-1x1019)

FrG. 22. Experimental measurements of dislocation velocities for 60"a/2( 101) and 9Oou/6(211) misfit dislocations in Ge, Sil-, /Si( 110) heterostructures. Velocities are normalized according to Eq. (37).

3

MISFITS T R A I N A N D ACCOMMOI)I\TION

149

trolled by the slower of the two (i/6(211) partials into which it is dissociated, that is, the maximum velocity of the total dislocation would be limited to that of the slower of the two partials. Hull and Bean (1993) suggested that this apparent dichotomy can be resolved by consideration of details of the kink nucleation process (essentially the effect of stacking fault energies within the critical kink configurations). The data o n dislocation glide in bulk Si and Ge also reveals strong dependence upon dopant and other impurities (Hirsch 1981; Imai and Sumino, 1983; George and Rabier, 1987). Some measurements of dopant and impurity effects upon dislocation glide in Ge,Sil-, heterostructures have also been made. Gibbings et rtl. (1992) studied the effect of both p-type (€3) and n-type (As) doping upon dislocation velocities in strained Ge,Sil .-, layers grown upon Si( 100) substrates. Consistent with studies of doping effects upon dislocation velocities in bulk Si, it was found that high n-type doping (1 l o i 7 cm-j) could enhance dislocation motion in the Ge,Sil-., layer, while p-type doping had relatively little effect. The effects of very high (- 10'9-102"~ m - oxygen ~ ) concentrations upon dislocation glide velocities in strained Ge,Sil -.r layer have also been studied (Nix et al., 1990; Noble et al., 1991; Hull et al., 1991b). These studies concluded that misfit dislocation velocities were reduced and critical thicknesses for dislocation introduction increased, presumably by locking effects of the oxygen upon dislocation motion. Finally, Pethukov (1995) has considered the effects of compositional fluctuations within the GeSi alloy upon dislocation propagation.

4.

MIsm

DISLOCATION IKTERACTIONS

Interaction between misfit dislocations is a critical process in the later stages of plastic relaxation. The primary interaction mechanism arises from the force between two dislocation segments, which results from the interaction of the strain fields around them. As described by Eq. (16), this force is generally inversely proportional to the segment separation, and proportional to the dot product of the two dislocations Burgers vectors. Calculation of the exact magnitude, components and spatial variation of this force is complex for the general configuration, but relatively straightforward for simplified configurations (e.g., for parallel screw dislocations, the force per unit length, F / L = Gh2/2n). The most dramatic effect of misfit dislocation interactions is that they can pin motion of propagating dislocations, effectively leading to effects that are analogous to workhardening in metals (Dodson 1988, Hull and Bean, 1989b; Freund, 1990. Gosling etal., 1994; Fischer and Richter, 1994; Gillard and Nix, 1995; Schwarz. 1997), as illustrated experimentally in Fig. 23. To understand this, consider the case of a total dislocation propagating along an (011 ) direction in a (100) interface. and about to intersect a pre-

150

R. HULL

FIG. 23. Experimental demonstration of misfit dislocation blocking events in a 300 nm S1/59 nm Ge0,18Si0,82/Si(100) heterostructure.

existing orthogonal interfacial dislocation, as illustrated schematically in Fig. 24. The excess stress driving dislocation motion is o,, = a, - 07. If the Burgers vectors of the two dislocations are parallel, there will be a repulsive interdislocation stress between them, which will act against the excess stress, and whose magnitude will depend upon the longitudinal coordinate along the threading arm (it will be highest at the points of closest approach near the interface, and lowest near the epilayer surface). As the two dislocations come closer, this repulsive stress will exceed the excess stress along greater fractions of the propagating threading dislocation, pinning motion of those segments of the threading arm. If enough of the threading arm is pinned, it will be unable to move past the orthogonal dislocation. This is clearly more likely to occur (because of the l / r dependence of the inter-dislocation force) in thinner epitaxial layers than thicker epitaxial layers. (Note that although the preceding discussion assumes a repulsive interaction, that is, parallel components of Burgers vectors, the discussion is

FIG. 24. Schematic illustration of the forces acting on a propagating threading dislocation ( B C ) when it encounters a pre-existing orthogonal misfit dislocation (D); cr, and oy are the Matthews-Blakeslee lattice mismatch and line tension stresses, respectively; OD is the horizontal component of the inter dislocation stress between D and B C .

3 MISFITS T R A I N A N D ACCOMMODATION

151

quantitatively the same for attractive interactions, that is, anti-parallel components of Burgers vectors, as the net stress is then reduced as the propagating dislocation attempts to pull away from the intersection event. For simplicity, we will assume repulsive dislocation interactions in subsequent discussion, but the concepts will be the same for attractive interactions.) Thus, dislocation pinning events are most likely in thinner layers. Substantial dislocation densities in thin layers occur more readily in structures that are initially more highly strained (because the critical thickness is lower and dislocation nucleation and propagation kinetics are more rapid than lower strained structures). Thus these pinning events tend to be more critical in growth of higher strain, lower thickness structures. This is one reason why, for a given amount of relaxation, higher strain systems have higher threading dislocation densities than lower strain systems (Hull and Bean, 1989b; Kvam 1990). The pinning process has been modeled in detail by Freund (1 990), Gosling et al. (1994) and Schwarz (1997). These authors evaluated the elastic integrals for this configuration, and modeled the regimes of epilayer thickness and strain ( h ,E ) where this blocking occurs. Results from the work of Freund are shown in Fig. 25. The formula-

-E

0

case

A

A

case

B

C

W J

FIG.25. Plot of minimum strain required for passage of a 60°a/2(101) misfit dislocation past a preexisting orthogonal 60"a/2(101) misfit dislocation at a Ge,Si~-,/Si(100) interface vs epitaxial layer thickness. The solid line marked A corresponds to the case where the misfit dislocations Burgers vectors are inclined at 90' to each other and the solid line marked B corresponds to the case where the misfit dislocations Burgers vectors are parallel to each other. The dashed line corresponds to the minimum strain analog to the critical thickness for misfit dislocation introduction. Reprinted with permission from L.B. Freund, J. Appl. Phys. 68, 2073, Figure 11. Copyright 1990 American Institute of Physics.

152

R . HULL

tion of Gosling et al. (1995) is relatively amenable to analytical approximations, and to evaluation of a more general calculation, the mean position-dependent stress field om(I-) experienced by a propagating dislocation. Inspection of Fig. 25 shows that there is a region of ( h ,6) space for which dislocation blocking is possible. The lower bound of this region is defined by the critical , which dislocation motion is not energetically favored (indethickness h , ( ~ ) below pendent of any interaction processes). The upper bound is defined by the curve hb(&) (note that the locus of this curve depends upon the angle between the Burgers vectors of the intersecting dislocations). For intermediate thicknesses hc(&) < h < h b ( & ) , pinning of propagating dislocations will occur. For thicknesses h > hb(&),blocking will not occur, but the propagating dislocation will be slowed during approach to the intersection event (for a repulsive interaction stress), as the local net stress is reduced by the interaction stress. Conversely, as the propagating dislocation moves away from the interaction event, the net stress is increased by the repulsive interaction. Dislocation interactions also inhibit complete strain relaxation in any thickness of strained epilayer (this will have significant ramifications for the reduction of threading dislocation densities, as will be discussed in Section VI of this chapter). This is because as the residual elastic strain tends to zero, the blocking thickness hb(&)tends to infinity. Thus, in the latest stages of strain relaxation, dislocations will find it difficult to propagate past each other in any epitaxial layer thickness, leaving residual threading dislocation densities.

5 . KINETICMODELING OF STRAIN RELAXATION B Y MISFITDISLOCATIONS The processes of dislocation nucleation, propagation, and interaction define the kinetics of relaxation by misfit dislocations. There have been several attempts to combine these processes into predictive models of strain relaxation. Dodson and Tsao (1987) published the first comprehensive kinetic description of relaxation by misfit dislocations in Ge,Sil -n /Si heterostructures. They combined the concepts of excess stress, bulk parameters for dislocation propagation, and dislocation multiplication arising from a pre-existing dislocation source density, into a theory of plastic flow, thereby producing a predictive equation for strain (and hence dislocation density) for finite time at temperature

+

d [ A ~ ( t ) ] / d=r CG2(so- A & ( t )- Ecq)2(Ae(r) E,)

(39)

Here, A s ( t ) is the amount of strain relieved by misfit dislocations, EO is the initial lattice mismatch strain, E, arises from an initial “source” density of dislocations from which multiplication proceeds, eeq is the equilibrium strain in the structure, predicted by solution for E of Eq. (23) with oeX= 0, and C is a constant (C and E, were found to be 30.1 and lop4, respectively, from fitting to available data). Using this model, Dodson and Tsao were able to predict a wide range of experimental measurements of strain relaxation in GeSi heterostructures. In particular, they were the first to demon-

3

MISFITS T R A I N AND ACCOMMODATION

K I N E T I C MODEL ( 5 5 0 0I~

x

.

BEAN

153

E r AL

550'C 0 \

0

s 3 v lo -

-

KASPER

ET AL

750°C

K I N E T I C MODEL ( 7 5 0 ~ ~

0

0.01

0.02

0.03

LATTICE MISMATCH

FIG. 26. Predictions of temperature-dependent "critical thickness" (as defined by the Ge,Sil --x layer thickness at which misfit dislocations are first experimentally detected) from the Dodson-Tsao model and corresponding experimental data in the Ge,Sil -x/Si(lOO) system from Bean et al. (1984), Tg = 550 "C; and Kasper et al. (1975), Tg = 750OC. Reprinted with permission from B.W. Dodson and J.Y. Tsao, Appl. Phys. Lett. 51, 1325, Figure 3. Copyright 1987 American Institute of Physics.

strate convincingly that kinetic modeling of the minimum detectable strain relaxation (or equivalently, minimum misfit dislocation density) accurately predicted temperaturedependent critical thicknesses, as illustrated in Fig. 26. In subsequent work, Tsao et al. (1988) constructed a framework for describing relaxation kinetics by misfit dislocations in Ge,Sil -,/Ge( 100) heterostructures in terms of deformation mechanism maps (Frost and Ashby, 1982). Experimentally, it was found that temperature and excess stress were the critical parameters in determining plastic relaxation rates, consistent with thermal activation of dislocation nucleation and motion, as discussed previously in this section. For example, it was determined that the onset of detectable strain reiaxation (using Rutherford backscattering spectroscopy, with a detectable strain relaxation of 1 part in lo3) occurred at excess stresses of 0.024 G and 0.0085 G at 494 "C and 568 "C, respectively. This is consistent with much more rapid relaxation kinetics at the higher temperature. Dodson and Tsao (1988) subsequently developed scaling relations for relaxation by misfit dislocations in GeSi heterostructures as functions of temperature and excess stress

-

As(gex1,

Ti)/A&(oe,o, To) = (a,-xi/a,,o)2elE(g,exo)/kTo - E ( O ~ I ) / ~ T I(40) I

where A&(ce,1, T I )represents the degree of plastic relaxation for a structure with excess stress u , , ~at a temperature Ti during a growth or annealing cycle, and As(cr,,o, To) represents a relaxation standard for a structure with excess stress ae,0 and temperature

154

R. HULL

To. The effective activation energy E(o,,J was determined to be given by (Dodson and Tsao, 1988) where T, is an effective alloy melting temperature obtained by a linear weighting of the melting temperatures of the pure components. Hull et al. (1989b) attempted to model the strain relaxation process in GeSi heterostructures by direct measurement of the fundamental parameters defining misfit dislocation kinetics, and subsequent incorporation into a kinetic model. The relevant measurements were made by direct in situ TEM observations and quantification of misfit dislocation nucleation, propagation and interaction processes, followed by incorporation into the equation'

where A&(t), L ( t ) , N ( t ) and u ( t ) are the degree of strain relaxation by misfit dislocations, total interfacial misfit dislocation length per unit area of interface, number of growing dislocations per unit area of interface and average dislocation velocity at time t , respectively. The integral is evaluated over all time for which a,, > 0, for either growth or post-growth annealing. Experimental descriptions are used for N ( t ) and v(r). The effects of misfit dislocation interactions are incorporated by reducing N ( t ) by the number of dislocations pinned according to the previous discussion in this section, and by incorporating an empirical velocity-stress-temperature relation in a regime where dislocation interactions strongly affect propagation rates (Hull et al., 1989a,b). Comparisons of this model with experiment are shown in Fig. 27. Subsequent models have modified and developed the concepts embodied in Eq. (42); Houghton (1991) applied a version of Eq. (42) to the initial stages of relaxation. In this regime, where misfit dislocation densities are low, dislocation interactions are relatively unimportant, and the integral in Eq. (42) simplifies to the product of the expressions for dislocation nucleation and propagation, which were directly measured by Houghton, and represented by the e q ~ a t i o n : ~

where NO is the initial source density of misfit dislocations at time t = 0. Typical measured values of NO were in the range 103-105 cm-*. Gosling et al. (1994) further *The exact relationship between A s and L ( t ) depends upon the shape of the substrate, and Eq. (42) is strictly true only for a square or rectangular substrate. For a circular substrate, a constant of proportionality 4/71 should be included in the middle expression in Eq. (42). 'Note that the equation's presentation here is identical to its presentation in work by Houghton (1991). The reason that it does not appear to be dimensionally correct is that various constants (e.g., a factor G-"') are combined into the perfactor term of I .9 x lo4.

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FIG.27. Comparison of the predictive strain relaxation model of Hull et al. (1989b) with experimental data for a 35 nm Ge0.2sSi0.75 layer annealed for 4 min at successively higher temperatures: (a) average distance between misfit dislocations p ; (b) areal density of threading dislocations N ; and (c) average misfit dislocation length 1. Reprinted with permission from R. Hull et al., J. Appl. Phys. 66, 5837, Figure 6. Copyright 1989 American Institute of Physics.

developed the concepts of Eq. (42), using more complete descriptions of misfit dislocation interactions, and a fittable form for dislocation nucleation. They were able to successfully reproduce the experimental data of Fig. 27. In summary, although the different models developed to date for predicting relaxation rates by misfit dislocations in Ge,Sil-, heterostructures may at first sight appear to have different forms, they generally attempt to combine the concepts of misfit dislocation nucleation, propagation, and interaction to construct a framework for predicting relaxation kinetics. In general, the framework of Eq. (42) has been adopted for this kinetic modeling. Quantitative implementation of this framework, however, will depend upon the strain and relaxation regimes under consideration. In general, a universal model for misfit dislocation relaxation kinetics is unlikely to have a simple analytical form due to existence of different mechanistic regimes, and due to the complexity of some of the constituent processes (e.g., interactions, multiplication).

VI. Misfit and Threading Dislocation Reduction Techniques 1.

INTRODUCTION

Silicon substrate wafers can now be routinely grown with fewer than 10 dislocations per cm’, representing a remarkable degree of structural perfection. Homoepitax-

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ial growth of Si can compete with this level of structural quality. If Ge,Sil-,-based strained layer epitaxy is to compete with existing device technologies, either it must offer unique and overwhelming advantages over bulk or homoepitaxial structures, or it must be of sufficient structural perfection to be compatible with existing processing technologies. In this section, we will discuss prospects and techniques for attaining Ge, Sil-, layers (especially supercritical structures) of the required structural quality. The magnitude of defect densities tolerable in device and circuit manufacture is a somewhat subjective (and controversial) topic, but maximum permissible densities of order lO5cmP2 in majority carrier devices, lo3 cmP2 in discrete minority carrier devices, and I0 cmP2 for Si integrated circuit technology are the ranges of numbers typically quoted. The only reliable way to avoid misfit and threading dislocations in strained layer epitaxy is to remain below the equilibrium critical thickness, and many technological applications of Ge,Si 1 -,are evolving towards subcritical layers. This, however, places extremely severe limitations on layer thicknesses, as demonstrated by Fig. 1 1, which will have ramifications for device growth, processing, and doping. An example where such layer dimensions are compatible with these constraints, however, is the Si/Ge,Sil-,/Si heterojunction bipolar transistor (Harame et al., 1995a,b), where a fortuitous combination of strain-induced bandgap lowering and a high valence band offset have allowed useful bandgap variation at relatively low x , thus enabling Ge,Sil -, base layers which can be of practical use, while remaining below the critical thickness. Tolerably low defect levels may also in principle be achieved by growth in the metastable regime, that is, at layer thicknesses greater than the Matthews-Blakeslee equilibrium critical thickness, if the structure is grown at sufficiently low temperatures, strains, and excess stresses that the misfit defect density is still relatively low. Post-growth thermal exposure during device processing will cause further nucleation and propagation of dislocations, however. To prevent further significant degradation, therefore, each time-temperature cycle during processing would typically be restricted to less than that of the original growth cycle. This may be an impractical restraint. In addition, device fabrication processes may enhance plastic relaxation. For example, Hull et al. (1990) have shown that contact implantation and thermal activation in a Si/Ge,Sil-, /Si heterostructure significantly enhances misfit dislocation generation rates compared to unimplanted structures, via formation of high densities of misfit dislocation sources from condensation of point defects into dislocation loops during implant activation.

2.

BUFFERLAYERS

An important technique for attempting to separate device quality material regions spatially from dislocated interfaces is incorporation of sacrificial buffer layers (e.g. Kasper and Schaffler, 1991). In this concept, a thick (of order 1 ,urn) Ge,Sil-, layer is

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grown upon the Si substrate, such that almost complete relaxation of lattice mismatch strain occurs via misfit dislocation generation. This produces a fully developed interfacial misfit dislocation array, with a residual density of threading dislocations traversing the buffer layer to the growth surface. The device-quality material is then grown upon the buffer layer (and generally, closely lattice-matched to it). In this configuration, the density of threading dislocations extending through to the device will be the important metric for structural quality. Several techniques have been developed for filtering these threading dislocations, to reduce their density to acceptable levels.

3.

THREADING DISLOCATION FILTERING

a. Introduction One mechanism for filtering of threading dislocations is for the misfit segments to grow sufficiently long that they terminate at the edges of the wafer. The ideal relaxed epilayer configuration would then correspond to two orthogonal sets of parallel, equally spaced (011) dislocations running across the entire (100) wafer. This would result in no threading dislocations propagating through the buffer layer. In practice, finite dislocation propagation rates and blocking via dislocation interactions prevent this (see Section V of this chapter). Dislocation propagation can be enhanced either by growing at higher temperatures, or by post-growth annealing (the latter may be preferred due to likely surface morphology problems in strained layer growth at higher temperatures). Lateral motion to the edges of a standard Si wafer (i.e., diameter 10-20 cm), however, is still highly improbable because as relaxation by dislocations proceeds towards its equilibrium limit, the excess stress becomes increasingly low causing (i) dislocation blocking to become increasingly prevalent and (ii) propagation velocities to become increasingly low. Dislocation interactions, however, can be exploited to advantage if dislocation annihilation processes can be encouraged. As illustrated in Fig. 28, threading dislocations of opposite Burgers vectors can attract each other and annihilate, transforming two dislocation loops into one, and removing two threading dislocations from the structure. Both thermal annealing (either during growth or post-growth), and growing the buffer layer to greater thicknesses will enhance the total annihilation probability. These processes can be extremely effective at high threading dislocation densities, where such interaction events are statistically likely, but as the density decreases, so does the probability of further dislocation interactions and further defect reduction. This renders the annihilation process virtually ineffective at lower threading defect densities. Typical threading defect densities in uniform buffer layers grown with an abrupt interface to 1 p m for strains > 1% the substrate are of order 107-109 cm-2 after growth of (Kasper and Schaffler, 1991). Lower strain systems may allow perhaps one order of magnitude reduction in this density because of reduced dislocation pinning events.

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FIG. 28. Schematic illustrationa of possible mechanisms for dislocation threading arm interactiodannthilation events in a strained layer superlattice. Reprinted from R. Hull and J.C. Bean, Chapter 1 , Semiconductors und Semimetals, Vol. 33, ed. T.P. Pearsall, (Academic Press, Orlando, Florida, 1990).

Note that differential thermal expansion coefficients are significant in this geometry, as they may cause substantial defect generation in structures that are fully relaxed at the growth temperature. Thus in the Ge,Sil-,/Si( 100) system, complete relaxation of the compressive lattice mismatch stress at the growth temperature will result in a tensile strain during cooldown due to the higher thermal expansion coefficient of Ge, Sil --x than Si. This can generate new misfit dislocations or cause extended splitting of the a/6(211) partial dislocations, which constitute a dissociated a/2( 101)total dislocation, as discussed in Section 111.7, leaving a 90" a/6(112) partial at the (100) interface. b.

Graded Layers

A promising technique for minimizing dislocation blocking processes in buffer layers is continuous grading of strain in the buffer layer. Compositional grading of buffer layers in the Ge,Sil-,/Si system (e.g. Fitzgerald et ~ l . 1991; , Tuppen et nl., 1991) has yielded threading defect densities in the range 105-106 cm-* for lattice mismatch strains as high as 3% and buffer layer thicknesses in the range 1-10 pm. The main benefit of such compositional grading is that instead of the misfit dislocations being confined to a single Ge,Sil-,/Si interface, there will be a distribution through the epilayer to compensate for the continuously varying strain field. This provides an extra degree of freedom for misfit dislocations to propagate past each other (as they may be at different heights in the structure) and thus minimize pinning events. The vertical distribution of misfit dislocations can also vary during specimen cooldown after growth, minimizing the effects of differential thermal expansion coefficients. There has been considerable work on modeling and measurement of strain relaxation in such compositionally graded layers (Tersoff, 1993; Shiryaev, 1993; Tongyi, 1995; Mooney et al., 1995; Li et al., 1995). In particular, it is observed experimentally (and predicted theoretically) that dislocation pileups within the graded layers can cause significant surface

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morphology (Shiryaev et al., 1994; Samavedam and Fitzgerald, 1997), with significant ramifications for technological applications of these structures. Such effects can be ameliorated by growing on off-axis substrates (Fitzgerald and Samavedam, 1997) and by intermediate chemical-mechanical polishing stages (Currie et al., 1998).

c. Strained Layer Superlattices The probability of threading dislocation annihilation processes can be substantially enhanced by providing specific directions for threading dislocation motion. This is the underlying concept of strained layer superlatrice filtering, originally proposed by Matthews and Blakeslee (1974, 1975, 1976). This technique consists of growth of a stack of strained layers, generally on top of the relaxed buffer layer. The thickness and strain of each individual layer within the stack is designed to be insufficient to allow significant nucleation of additional dislocations, but sufficient to deflect threading dislocations into being misfit dislocations at the superlattice interfaces (this translates effectively into the criterion that the MB critical thickness be exceeded, but not by too much!). For a (100) interface the interfacial misfit dislocations are constrained to move along the interfacial (011) directions, thereby increasing the probability of their meeting, interacting, and annihilating (compared to random motion of threading dislocations within their slip planes in the absence of the superlattice). Many groups have since claimed successful application of this technique, in a range of different strained layer semiconductor systems e.g. (Olsen el al., 1975; Dupuis et al., 1986; Liliental-Weber et al., 1987). Typically, threading dislocation density reductions from the range 108-109 cm-2 down to 106-107 cm-2 were claimed. However, again as the threading dislocation densities decrease, the probability of interaction and annihilation also decreases. Quantitative analysis of these probabilities (Hull et al., 1989c) suggests that defect densities much below lo6 cm-= are unlikely to be achieved. d.

Limited Area Growth

A very promising approach to reducing threading dislocation densities is limited area epitaxy, that is, growth on mesas or in windows, typically with dimensions in the range 10-100 pm. One potential advantage of the limited area is that the number of heterogeneous dislocation nucleation sites within this reduced growth area may be vanishingly small, thus inhibiting generation of misfit dislocations altogether (at sufficiently high strains, however, quasihomogeneous nucleation at the mesa edges is likely to operate). The most important advantage, however, is in reduction of threading dislocation densities, as the dislocation now only has to propagate a far more limited distance to reach the mesa edge than it would have to reach the wafer edge. The classic original demonstration of the potential of this potential of this technique is that by Fitzgerald et al. (1988, 1989) in the In,Ga~-,As/GaAs(100) system, as illustrated in Fig. 29, and

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3000

I

100

0

200

300

400

500

Circle Diometer (microns)

3000

.-. f

>

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b

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2000

.

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D

2 v

-g

L 0

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I

O[110]

0

.t

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_I

0 0

I00

200

300

400

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circle Diameter (misrona)

FIG. 29. Illustration of the benefits of reduced area growth: linear density of interfacial misfit dislocations vs mesa diameter for 350 nm Ino.osGag.9sAs layers grown onto GaAs substrates with (a) 1.5 x lo5 cm-' and (b) lo4 cmP2 preexisting dislocations in the substrate. Reprinted with permission from E.A. Fitzgerald et al., J. Appl. Phys. 65, 2220, Figures 4(a) and 4(b). Copyright 1989 American Institute of Physics.

subsequently developed by several other groups in a range of strained semiconductor systems (Matyi et al., 1988; Lee et al., 1988; Guha et al., 1990; Noble et al., 1990; Knall et al., 1994). The primary disadvantages of the limited growth area technique are the reduced dimensions for device processing, and in configurations such as mesa growth, a highly nonplanar geometry, which may be incompatible with some processing steps. One at-

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tempt to address these shortcomings was described by Hull et al. (1991~)who devised a substrate patterning technique involving a two-dimensional array of oxide dots, in which neighboring dots were slightly offset from each other with respect to the interfacial (011) directions. Thus any misfit dislocation traveling along an interfacial (011) direction must intercept a dot within a path length, A given by simple geometrical analysis as A = L 2 / s , where L is the interdot spacing and s is the offset of neighboring dots along the interfacial (011) directions. The misfit dislocation can then terminate at the crystal/amorphous interface at the dot, thereby annihilating the threading dislocation. Thus the advantages of the limited growth area approach (finite misfit dislocation propagation lengths for threading dislocation annihilation) are retained in this dot array, while allowing connectivity of the structure across the entire wafer, and planar geometries (if the epitaxial layer thickness is approximately equal to the oxide dot thickness). Residual threading defect densities lo5 cm-* were reported for MBEgrown Ge,Sil-, layers 200-800 nm thick with x = 0.15-0.20 on the patterned wafers, whereas growth on unpatterned wafers typically yielded threading dislocation densities one to two orders of magnitude higher. Post-growth annealing of the patterned structures to temperatures as high as 900 “Ccaused the structure to relax almost completely to equilibrium, dramatically enhancing interfacial misfit dislocation densities, but with threading dislocation densities typically remaining < lo6 cmP2.

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VII. Conclusions The following are summaries of the salient points of each preceding section: The lattice constants of Ge,Sil-, may be estimated with reasonable accuracy by linear interpolation of the lattice parameters of Ge and Si. The strain in a Ge,Sil-,based heterostructure arises primarily from lattice parameter differences. Differential thermal expansion coefficients produce a second-order effect. There are four primary mechanisms for lattice-mismatch strain accommodation: (a) elastic distortion of the epilayer; (b) roughening of the epilayer; (c) interdiffusion; and (d) plastic relaxation by misfit dislocations. Of these mechanisms, (c) is only significant at very high-growth temperatures, or for ultrathin layers. Mechanism (d) requires a minimum layer thickness (the “critical thickness”) to operate. Mechanism (b) is prevalent, especially at higher temperatures and strains. Strain not accommodated by mechanisms (b)-(d) is accommodated by mechanism (a). There are extensive and rigorous elastic descriptions of dislocation structure and properties. Atomistic details at the dislocation core are not so well understood. Depending on the predictions of the Thompson tetrahedron construction, we need to consider the possibility of b = a/6(211) partial misfit dislocations, as well as conventional b = a/2( 101) total misfit dislocations in the diamond cubic Ge,Sil-, lattice.

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(IV) The Matthews-Blakeslee model provides an excellent framework for the critical thickness for relaxation by misfit dislocations. Other models generally either fall into the category of a broadly equivalent energetic approach, or into the category of developments/refinements of the MB theory. Accuracies within a factor two (better at lower strains) should be expected for application of the basic MB model. For configurations where the Thompson tetrahedron construction predicts that the lattice-mismatch stress acts so as to increase dissociation of a / 2 ( 101) total dislocations into a/6(211) partials, the partial misfit dislocation configuration will be favored at higher strains and lower epilayer thicknesses.

(V) Finite misfit dislocation nucleation and propagation rates prevent strain relaxation from keeping pace with the equilibrium condition for typical growth temperatures and times. This produces metastable (defined as a,, > 0) regimes of growth in Ge,Si 1 heterostructurep, which become increasingly broad with decreasing growth temperature. Consequently, relaxation by misfit dislocations is often experimentally detected only in Ge,Sil -, layers substantially thicker than predicted by the MB equilibrium theory, the discrepancy increasing with decreasing strain and temperature. Analysis of observed relaxation processes shows that enormous lengths of misfit dislocation need to be generated, and very high source densities are required. Considerable uncertainty remains about the dominant misfit dislocation nucleation mechanisms in different regimes of strain, epilayer thicknesses and temperature in the Ge,Sil-,/Si system. The generic mechanisms for dislocation sources are homogeneous, heterogeneous, and multiplication. True homogeneous nucleation requires high lattice mismatch strains of order 3 4 % , or greater. “Quasihomogeneous” nucleation, that is, nucleation aided by inherent physical properties of the structure, such as random concentrational clustering or surface roughening during growth, can lower this minimum strain threshold down to 1%. Below this threshold, heterogeneous or multiplication mechanisms operate. Heterogeneous sources are necessarily limited in high quality epitaxial growth, and plastic relaxation that relies solely upon heterogeneous nucleation will be very sluggish. Multiplication mechanisms can act as efficient dislocation sources, apparently in any strain regime, but generally require minimum layer dimensions (of order hundreds of nanometers) to operate, and typically require precursor heterogeneous sources. In general, relaxation at low strains is nucleation-limited. Experimentally, there is a paucity of nucleation rate data in the Ge,Sil-. system. Good experimental descriptions of 60”a/2( 101) misfit dislocation propagation velocities exist for the (Si)/Ge,Sil -,/Si( 100) system. Observed velocities follow broadly the same trends as those determined from bulk deformation experiments of Ge and Si. Dislocation velocities depend upon strain, epilayer thickness, and Ge concentration, and monotonically increase with these parameters. Again, the propagation component of relaxation by dislocations is much more rapid in the high

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strain regime. Observations in the Ge,Sil-,/Si( 1 10) system allow propagation velocities of a/2(101) and a/6(211) dislocations to be separately determined. Dislocation interactions are very important in plastic relaxation, particularly in thinner films and in the terminal stages of strain relief, where orthogonal interfacial dislocations can pin each other’s motion. Such pinning processes can stabilize high threading dislocation densities in the structure. Kinetic models of strain relaxation rates, incorporating the concepts of dislocation nucleation, propagation, and interaction rates have been developed, and show reasonable predictive capability. These models are generally based upon the concepts of excess stress, thermal activation of dislocation nucleation and propagation, and interaction blocking events. VI) Many techniques have been developed to reduce threading dislocation densities in Ge,Sil -,/Si structures. These techniques rely on removing threading dislocations by annihilation either with other defects or by propagation to the edge of the epitaxial growth area. Strained layer superlattice filtering relies on threading defects annihilating through interaction, but this interaction probability decreases as the threading defect population decreases, running out of steam for areal densities less than about lo6 cmP2. Thick, compositionally graded layers appear to be promising in producing low threading dislocation densities, probably because the threading dislocations are less subject to interaction pinning events as their accompanying misfit segments are distributed vertically through the graded structure. Substantial surface morphology can develop from dislocation pileups, however. Reduced area (e.g., mesa) growth schemes result in lower threading dislocation densities both because of reduced probability of dislocation nucleation within the reduced area, and because threading dislocations have to propagate much smaller distances to terminate at the edge of the growth area.

ACKNOWLEDGMENTS The author would like to thank a large number of colleagues, past and present, for collaborations and contributions to my understanding of this field. These include: J. Bean, D. Bahnck, J. Bonar, D. Eaglesham, G. Fitzgerald, L. Feldman, M. Green, M. Gibson, G. Higashi, Y. Fen Hsieh, T. Pearsall, R. People, L. Peticolas, F. Ross, K. Short, B. Weir, A. White, Y. Hong Xie (with all of whom I collaborated during my tenure at Bell Laboratories); E. Kvam (Purdue University); B. Dodson and J. Tsao (Sandia); J. Hirth (Washington State University); M. Albrecht and H. Strunk (University of Erlangen, Germany); D. Perovic (University of Toronto); D. Houghton (National Research Council of Canada); C. Tuppen (British Telecomm); D. Noble (Stanford University); B. Freund (Brown University); F. LeGoues, B. Meyerson, K. Schonenberg, D. Harame, R. Tromp and M. Reuter (IBM); P. Pirouz (Case Western); and W. Jesser, E. Stach, J. Demerast and Y. Quan (University of Virginia).

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REFERENCES Albrecht, M., Strunk, H.P., Hull R. and Bonar, J.M. (1993). Appl. Phys. Lett., 62 2206. Albrecht, M., Christiansen S., and Strunk, H.P. (1995). Phys. Stat. Sol. (a), 150,453. Alexander H. and Haasen P. (1968). In Solid State Physics, V. 22. Bacon D.J., and Crocker, A.G. (1965). Phil. Mag., 12, 195. Baribeau, J.M. (1993). J. Appl. Phys., 74,3805. Bean, J.C., Feldman, L.C., Fiory, A.T., Nakahara S., and Robinson, I.K. (1984). J. Vac. Sci. Technol.. A2, 436. Beanland, R. (1995).J. Appl. Phys., 77,6217. Beltz G.B. and Freund, L.B. (1994). Philos. Mag., A69, 183. Bilby, B.A., Bullough R., and Smith, E. (1955). Proc. Roy. Soc., A231,263. Bonar, J.M., Hull, R., Walker J.F., and Malik, R. (1992). Appl. Phys. Lett., 60 1327. Bourret A. and Desseaux, J. (1979). Phil. Mag., A39,405. Cammarata R.C. and Sieradzki, K. (1989). J. Appl. Phys. Lett., 55, 1197. Capano, M.A. (1992). Phys. Rev., B45, 11768. Chang, S.J., Arbet, V., Wang, K.L., Bowman, Jr, R.C., Adams, P.M., Nayak, D., and Woo, J.C.C. (1990). J. Elec. Mat., 19, 125. Cherns D. and Stowell, M.J. (1975). Thin Solid Films, 29, 107, 127; (1976). 37, 249. Chidambamo, D., Srinivasan, G.R., Cunningham, B., and Murthy, C.S. (1990). Appl. Phys. Lett., 57, 1001. Cockayne D.J.H. and Hons, A. (1979). J. Physique Colloq., 40, C6. Cullis, A.G., Robbins, D.J., Barnett S.J., and Pidduck, A.J. (1994). J. Vuc. Sci. Technol., A12. 1924. Cullis, A.G., Pidduck A.J., andEmery, M.T. (1995). Phys. Rev. Lett., 75,2368. Cume, M.T., Samavedam, S.B., Langdo, T.A., Leitz, C.W., and Fitzgerald, E.A. (1998). Appl. Phys. Lett., 62, 1718. DeCoteau, M.D., Wilshaw, P.R.. and Faker, R. (1992). Proc. of 16th International Conference on Defects in Semiconductors, Materials Science Forum, 83-87, 185. Dismukes, J.P., Ekstrom, L., and Paff, R.J. (1964). J. Phys. Chem., 68, 3021. Dixon R.H. and Goodhew, P.J. (1990). J. Appl. Phys., 68,3163. Dodson, B.W. (1988).Appl. Phys. Lett., 53, 37. Dods0nB.W. andTsao,J.Y. (1987).Appl. Phys. Lett., 51, 1325. Dodson B.W. and Tsao, J.Y. (1988).Appl. Phys. Lett., 53,2498. Dregia S.A. and Hirth, J.P. (1991). J. Appl. Phys., 69, 2169. Dupuis, R.D., Bean, J.C., Brown, J.M., Macrander, A.T., Miller R.C., and Hopkins, L.C. (1986). J. Elrc. Mat., 16, 69. Eaglesham, D.J., Kvam, E.P., Maher, D.M., Humphreys, C.J., and Bean, J.C. ( I 989). Phil. Mug., A59, 1059. Fiory, A.T., Bean, J.C., Hull, R., and Nakahara, S. (1985). Phys. Reti..,B31,4063. Fischer A. and Richter, H. (1994).Appl. Phys. Lett., 64, 12 18. Fitzgerald, E.A., Kirchner, P.D., Proano, R., Petit, G.D., Woodall, J.M., and Ast, D.G. (1988). Appl. Phys. Lett., 52. 1496. Fitzgerald, E.A., Watson, G.P., Proano, R.E., Ast, D.G., Kirchner, P.D., Pettit, G.D., and Woodall, J.M. (1989). J. Appl. Phys., 65,2220. Fitzgerald, E.A., Xie, Y.H., Green, M.L., Brasen, D., Kortan, A.R., Michel, J., Mie, Y.J., and Weir, B.E. (1991). Appl. Phys. Lett., 59, 81 1. Fitzgerald E.A. and Samavedam, S.B. (1997). Thin Solid Films, 294,3. Fox B.A. and Jesser, W.A. (1990) J. Appl. Phys., 68,2801. Frank F.C. and Van der Menve, J.H. (1949a). Proc. Roy. Soc., A198,205; (1949b). A198,216. Frank, F.C. and Van der Merwe. J.H. (1949~).Proc. Roy. Soc., A200, 125. Frank F.C. and Read W.T. (1950). In Symposium on Plastic Deformation of Crystalline Solids, Pittsburgh: Carnegie Institute of Technology, p. 44.

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