Si (100) heterostructures

N

ELSEVIER

PHYSICA Physica A 239 (1997) 11-17

Kinetic critical thickness for surface wave instability vs. misfit dislocation formation in GexSil_x/Si ( 1 0 0 ) heterostructures D . D . P e r o v i d a'*, B. B a h i e r a t h a n a, H. L a f o n t a i n e b, D . C . H o u g h t o n b, D . W . M c C o m b c,1 a Departnwnt of Metalluryy and Materials Science, UniversiO, of Toronto, Toronto, Canada M5S 3E4 b Institute for Microstructural Sciences, National Research Council, Ottawa, Canada K1A OR6 c Brockhouse Institute for Materials Research, McMaster University, Hamilton, Canada L8S 4M1

Abstract

The kinetic critical thicknesses for surface wave formation and misfit dislocation generation in UHVCVD-grown GeSi/Si have been quantitatively determined using microscopical techniques. A refined morphological instability theory has been developed using a coupled continuum/atomistic treatment that incorporates a nucleation barrier to the onset of surface wave formation. The theory' accurately predicts the onset of surface wave formation as a function of thickness, composition, temperature and deposition rate. The interplay between misfit dislocation generation and surface wave formation can be elucidated from two-dimensional strain relaxation instability diagrams obtained from a 4-D parameter space.

The principal factor governing the development of metastable strained layer heterostructure-based devices is the relaxation of built-in elastic strains either during growth or following post-growth processing. Accordingly a clear understanding of the interplay between growth variables is required in order to provide the crystal grower with accurate, predictive models that define specific conditions to achieve either defect-free planar growth or vertically patterned self-assembled morphologies, which have recently been the subject of much interest [1]. Earlier research work focused on the study of elastic strain relaxation via misfit dislocation generation [2,3]. More recently, research has focused on the study of competitive mechanisms for elastic strain relief wherein planar epitaxial films undergo a * Corresponding author. 1Permanent Address: Department of Chemistry, University of Glasgow, Glasgow G12 8QQ, Scotland. 0378-4371/97/$i7.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PH S03 78-43 71 ( 97 )000 19-8

12

D.D. Perovid et al./Physica A 239 (1997) l l 17

morphological surface instability or change in growth mode to produce surface waves/ cusps or Stranski-Krastanov islands, respectively. A number of theories [1] have been developed thus far, based on either continuum or atomistic treatments, but each possesses a specific limitation for generally describing the evolution of surface morphology in a growing thin film. In this paper we quantitatively describe the interplay between the different strain relaxation mechanisms using combined continuum/atomistic kinetic theories developed in conjunction with experimental studies on GexSil-x/Si (1 0 0) heterostructures. A wide range of GexSi1_x/Si (0.03 < x < 0.5) single and multilayer heterostructures were grown by ultra-high vacuum chemical vapour deposition (UHVCVD) at 525°C in a Leybold Sirius hot wall reactor as discussed in detail elsewhere [4]. In UHVCVD the deposition rates are significantly lower relative to typical molecular beam epitaxy (MBE) conditions and, unlike MBE, the UHVCVD growth rate is a complex function of composition (i.e. Ge fraction) [4]. Moreover, the GeSi growth rate is independent of growth temperature in MBE but exponentially related to temperature in CVD growth. It will be shown that the inherent coupling of temperature, growth rate and composition in UHVCVD growth has a significant effect on strain relaxation behaviour in comparison to MBE growth. In order to accurately model the kinetic critical thickness for the onset of surface wave formation, Ge~Sil_x/Si heterostructures were grown to study the effects of increasing strained layer thickness at constant Ge fraction and increasing Ge fractions for constant layer thicknesses. Samples were cleaved and examined in cross-section and plan-view by field-emission transmission and scanning electron microscopy and atomic force microscopy. A typical example of a Ge0.4Si0.6/Si (1 0 0) multilayer structure is shown in Fig. l(a). The thinner layers near the substrate are planar and dislocation-free. Upon increasing the Ge0.4Si0.6 strained layer thickness, undulations (i.e. surface waves) are stabilized above a specific layer thickness (-~ 3.6 nm) and become more pronounced as the alloy layer thickness is increased. In addition the two thickest layers contain a high density of misfit dislocations at the lower interfaces. Although growth of sufficiently thick intermediate layers of the substrate material (i.e. Si) can planarize the surface waves, residual lattice plane distortions remain which induce vertically correlated growth of undulations (i.e. self-assembly) [5]; this has been confirmed by reciprocal space mapping X-ray diffraction [6]. Detailed analysis of the composition across the surface waves was performed using energy-dispersive X-ray (EDX) spectroscopy with a JEOL 2010F STEM (probe size = 1 nm) at 200 kV. Cross-sectional samples were prepared using a small-angle cleavage technique [7] to avoid differential sputtering effects, which are possible in standard thin foil atom milling procedures. Fig. l(b) is a Ge K~ X-ray map from the top layers in Fig. l(a) showing significant variation in lateral composition with Ge enrichment in the crests and Ge depletion in the troughs. Quantitative analysis using a GeSi standard was performed, including electron beam broadening effects, to determine the absolute lateral concentration profiles as shown in Fig. l(c). It was clear from this

D.D. P e r o v i d et a l . / P h y s i c a

8o L

t

I

I

~-

! I'-'~-Ge'~

A 239 (1997)

I

11 1 7

l

j

~.... =-KI i c t

............. i......... ~"["............ i............. i............. i............ i"'"'~":'~ ........... ............. :............. i :Li

i

i

',,!

............. i............. i......... ~i~: ...... i

. . . . . . . . . . .

:

i.'i

i

0ii Coordinate

Fig. 1. (a) High-angle annular dark-field STEM image of a UHVCVD-grown (525°C) Ge0.4Si0.6/Si heterostructure containing alloy layers of 1.2,2.5,3.6,5,7.5,10 and 12.5 nm thickness and Si spacer layers 46.3 nm in thickness. The surface undulations are bounded by {5 0 1} facets. (b) Ge K~ X-ray map of the top layers in (a) showing lateral variations in Ge concentration. (c) Quantitative profile taken across one perturbation wavelength from the top layer in (b), midway through the layer thickness; the data are not influenced by electron beam broadening.

14

D.D. Perovi( et al./Physica A 239 (1997) 11-17

and other samples that significant deviations from the nominal Ge concentration occur during growth [8]. Similar trends to that shown in Fig. 1 were obtained for lower Ge fraction alloy layer growths; the UHVCVD experimental data are summarized in Fig. 2. However, for Ge fractions below ~ 25%, it was difficult to observe the onset of surface wave generation since strain relaxation was dominated by misfit dislocation injection. Similarly, a number of previous studies [1,9-16], employing a wide range of growth techniques, have quoted various specific conditions (e.g. thickness, composition, temperature and growth rate) for surface wave formation in the GexSil_x/Si system. These results motivated us to develop a generalized description of surface wave instability that includes all relevant growth parameters. A number of continuum-based theories have been developed to describe the dislocation-free instability of thin films. In the presence of heteroepitaxial misfit strains, all planar traction-free surfaces are unstable against mass rearrangement resulting in a perturbed surface with a non-uniform stress distribution. The perturbation wavelength is governed by the destabilizing influence of elastic strain energy (mass transport by surface diffusion from regions of high stress (e.g. troughs) to low stress (i.e. crests) in competition with the stabilizing influence of surface energy (mass transport from regions of high to low curvature). Most theories employ thermodynamic or energy minimization treatments of semi-infinite, uniaxially stressed solids, using either linear stability analyses of dynamic models for surface evolution [17,18] or static energy minimization calculations [19,20]. Voorhees and coworkers [21,22] have developed the most comprehensive continuum theory which treats the dynamic case of a growing epitaxial binary alloy film deposited on a substrate with different lattice constant and elastic modulus, including compositionally generated stress effects [22] to explicitly account for the sign of the misfit stress. In view of the significant lateral compositional inhomogeneities observed (Fig. 1), it is clear that solute expansion stresses must be coupled with the substrate-epitaxial layer lattice mismatch. Alternatively, a number of non-continuum descriptions of strain-induced islanding have been developed [ 11-13,23] based on explicit treatments of either surface steps or facets, which predict the existence of an activation barrier to stress-driven surface instabilities. We initially adopted the former (or continuum) model [21,22] to calculate the kinetic critical thickness for surface wave generation as a function of various growth variables. Although a true critical thickness does not exist for any stressed film on a substrate of finite stiffness, a growing film can be kinetically stabilized by the deposition flux to yield a metastable, planar growth surface [21,22,24] up to a certain kinetic "critical" thickness. Initially, comparison of calculated results with experiment resulted in poor agreement. Since the continuum theories do not incorporate an activation barrier to the onset of surface wave formation, we introduced a surface energy barrier associated with the sequential incorporation of atoms at step edges during the nucleation of surface wave facets [11,13]. An atom diffusing from the {100} surface up the surface wave must overcome the large stress concentration field (i.e. high local surface chemical potential) at the edge of the perturbation before reaching the elastically relaxed region

D.D. Perovid et al./Physica A 239 (1997) 11-17

E" e~ 1OS

60" Dislocations (Theory) O Planar Growth (Expt.) ......... (. . . . )

Dislocated.

kPI2nar

'. "%

~)

101

~)

+ 60" Dislocations (Expt.) ....... Surface Waves (Theory)

%% ~

Planar

O

"-......~_ ......

~1o -~ o

I~ ~ l

0.1

J..

zx waves

.......... ° ......... -~-o..."22~.

0.2

0.3 0.4 Ge Fraction (x)

0.5

0.6

Fig. 2. Kinetic critical thickness for surface wave generation versus misfit dislocation formation. Data and theory represent GeSi/Si (100) heterostructuresgrown at 525°C by UHVCVD (see text for details).

at the top of the perturbation where the surface chemical potential is significantly reduced. We believe the barrier is inextricably linked to the nucleation of 11.2 ° strain stabilized {501} surface facets [11,13] beyond a critical thickness (Fig. l(a)). This barrier increases predictably with perturbation size and misfit strain [11,13] and can be at least twice as large as the barrier to surface diffusion of Ge on Si ( 1 0 0 ) [25]. Upon coupling the nucleation barrier with the activation energy for surface diffusion in the continuum theory [22], excellent agreement with experiment was attained as shown in Fig. 2. Specifically, the kinetic critical thickness for surface waves (hs) varied with misfit strain (e) as hs e( e-4, in agreement with other discrete models [11,24], and not as e-8 as predicted from continuum theory without a nucleation barrier [21,22]. Secondly, the kinetic critical thickness curve shown in Fig. 2 extends to ~ 3 monolayers for x = 1, in agreement with the Stranski-Krastanov limit for pure Ge island formation on Si (10 0) [26]. In fact quantitative comparisons of our refined theory yields excellent agreement with other experimental data [9-14,16], regardless of growth conditions or technique, and reconciles the discrepancies quoted by Guyer and Voorhees [27]. A detailed report will be published elsewhere. For completeness, the kinetic critical thickness for 60 ° misfit dislocation generation was calculated using a comprehensive misfit dislocation strain relaxation model previously developed [3,28] for planar G e x S i l _ x / S i ( 1 0 0 ) heterostructures. The dislocation kinetic critical thickness (hd) varies with misfit strain as ha (x ~-1.5, and not as he o( e-l, as is typically assumed [11], based on equilibrium considerations. The refined continuum/atomistic theory described here can be used to produce a strain relaxation stability map, which in the most general case should be

16

D.D. Perovid et al./Physica A 239 (1997) 11-17

four-dimensional in terms of thickness, temperature, composition, deposition rate. Fig. 2 provides accurate predictions of strain relaxation mechanisms for a wide range of composition, including the effects of compositionally dependent growth rate variations inherent in UHVCVD growth [4]. It should be noted that the lower bounds for the "dislocated waves" instability region are inaccurate. To the left of the crossover, dislocation generation will relax a significant fraction of the elastic strain pushing the surface wave critical thickness to higher values. To the right of the crossover, the presence of surface undulations induce stress concentrations which can significantly reduce the activation barrier for dislocation nucleation [1]. This effect has not yet been incorporated in the dislocation theory shown in Fig. 2 resulting in an overestimation of the coherent/dislocated wave transition. In summary the continuum theory of Guyer and Voorhees has been modified to include a nucleation barrier to the onset of surface wave formation. This provides an accurate predictive model for determining instability conditions as a function of all relevant growth parameters including thickness, temperature, composition and deposition rate. The availability of accurate strain relaxation instability diagrams facilitates the design of specific epitaxial morphologies such as flat, dislocation-free heterostructures for high-speed transistors or self-assembled patterned morphologies for quantum wires and dots.

References [1] For example see, Mat. Res. Soc. Bull. 21 (4) (1996) 18-54 and references therein. [2] J.H. van der Merwe, Crit. Rev. Sol. St. Mat. Sci. 17 (1991) 187. [3] D.D. Perovi6 and D.C. Houghton, Inst. Phys. Conf. Ser. 146 (1995) 117; D.D. Perovid, Rep. Prog. Phys., to be published. [4] H. Lafontaine, D.C. Houghton, D. Elliot, N.L. Rowell, J.-M. Baribeau, S. LaframboJse, G.I. Sproule and S.J. Rolfe, J. Vac. Sci. Technol. B 14 (1996) 1675. [5] Q. Xie, A. Madhukar, P. Chen and N.P. Kobayashi, Phys. Rev. Lett. 75 (1995) 2542. [6] J.-M. Baribeau, J. Crystal Growth 157 (1995) 52. [7] J.P. McCaffrey, Microsc. Res. Technol. 24 (1993) 180. [8] T. Walther, C.J. Humphreys, A.G. Cullis and D.J. Robbins, Mater. Sci. Forum 196-201 (1995) 505. [9] B. Cunningham J.O. Chu and S. Akbar, Appl. Phys Lett 59 (1991) 3574. [10] A.G. Cullis, D.J. Robbins, S.J. Barnett and A.J. Pidduck, J. Vac. Sci. Technol. A 12 (1994) 1924. [11] J. Tersoff and F.K. LeGoues, Phys. Rev. Lett. 72 (1994) 3570; 74 (1995) 4962. [12] Y.H. Xie, G.H. Gilmer, C. Roland, P.J. Silverman, S.K. Buratto, J.Y. Cheng, E.A. Fitzgerald, A.R. Kortan, S. Schuppler, M.A. Marcus and P.H. Citrin, Phys. Rev. Lett. 73 (1994) 3006; 74 (1995) 4963. [13] K.M. Chen, D.E. Jesson, S.J. Pennycook, T. Thundat and R.J. Warmack, Mat. Res. Soc. Symp. Proc. 399 (1996) 271. [14] H. Sunamura,Y. Shiraki and S. Fukatsu, Appl. Phys. Lett. 66 (1995) 953. [15] M. Albrecht, S. Christiansen, J. Michler, H.P. Strunk, P.O. Hansson and E. Bauser, Appl. Phys. Lett. 67 (1995) 1232. [16] A.B. Storm, P.W. Lukey, K. Weruer, J. Caro and S. Radelaar, J. Crystal Growth 157 (1995) 312. [17] RJ. Asaro and W.A. Tiller, Metall. Trans. 3 (1972) 1789. [18] D.J. Srolovitz, Acta. Metall. 37 (1989) 621. [19] M.A. Grinfetd, Soy. Phys. Dokl. 31 (1986) 831. [20] H. Gao, J. Mech. Phys. Sol. 42 (1994) 741.

D.D. Perovid et al./Physica A 239 (1997) 11-17

17

[21] B.J. Spencer, P.W. Voorhees and S.H. Davis, Phys. Rev. Lett. 67 (1991) 3696; J. Appl. Phys. 73 (1993) 4955. [22] J.E. Guyer and P.W. Voorhees, Phys. Rev. Lett. 74 (1995) 4031; Phys. Rev. B 54 (1996) 11710. [23] A. Madhukar, J. Crystal Growth 163 (1996) 149. [24] C.W. Snyder, J.F. Mansfield and B.G. On', Phys. Rev. B. 46 (1992) 9551. [25] M.G. Lagally, Jpn. J. Appl. Phys. 32 (i993) 1493. [26] M. Zinke-Allmang and S. Stoyanov, Jpn. J. Appl. Phys. 29 (1990) 1884. [27] J.E. Guyer and P.W. Voorhees, Mat. Res. Soc. Symp. Proc. 399 (1996) 351. [28] D.C. Houghton, J. Appl. Phys. 70 (1991) 2136.