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Strain relaxation in Si−xGex layers on Si(001)
j. . . . . . . . C R Y S T A L GROWTH
Journal of Crystal Growth 116 (1992) 260-270 North-Holland
Strain relaxation in Sil_xGe x layers on Si(001) M.A. Capano 1 Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
L. Hart, D.K. Bowen, D. Gordon-Smith, C.R. Thomas Department of Engineering, UniL,ersityof Warwick, Coventry, UK
C.J. Gibbings, M.A.G. Halliwell British Telecom Laboratories, Martlesham Heath, Ipswich IP5 7RE, UK
and
L.W. H o b b s Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received 20 August 1991
The lattice relaxation of strained Sil_xGe X layers on Si (001) substrates has been examined. Three specimens consisting of a single Si I_xGex layer were grown by molecular beam epitaxy. All layers were grown with a nominal composition of x = 0.14 to thicknesses of 0.5, 1.0 or 1.5 p.m. Double-crystal and white-radiation topographic methods were used to reveal the misfit dislocation structure and distribution. The misfit dislocations were shown to extend from heterogeneous nucleation sites along the (110) directions in the plane of the interface. A symmetric distribution of dislocations between the orthogonal (110) directions was observed. The Burgers vectors of the misfit dislocation array were evenly distributed amongst the available 60°-type candidates. Double-crystal X-ray diffractometry showed the 0.5 and 1.0/zm layers to be fully strained to within the experimental uncertainty. Secondary branching of misfit dislocations was observed in the 1.0/xm layer which indicated cross-slip of the threading dislocation segments.
1. Introduction The relaxation of strained layer SiGe/Si heterojunctions by the generation of mismatch dislocations is a subject of great interest [1], especially for devices such as the heterojunction bipolar transistor (HBT), in which the influence of such
l Current address: WL/MLBM, Wright-Patterson AFB, Ohio 45433-6533, USA.
dislocations at the base-emitter or base-collector junction may be deleterious. A specific question of current interest is determining how the misfit dislocation network develops in the initial stages of strain relaxation. Finding a solution to this question, though, mandates tradeoffs between experimental techniques with varying abilities to resolve the relaxation process [2,3]. There are, in fact, a number of causes which limit experimental resolution, and these may be divided into two broad categories. Sampling-specific causes
0022-0248/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
M.A. Capano et al. / Strain relaxation in Si z xGex layers on Si(O01)
include small sampling volumes, poor spatial resolution, poor strain sensitivity, adverse probespecimen interactions and specimen-preparation effects. In addition to these factors, specimenspecific causes such as uncertainties in layer thickness and composition, inaccuracies in growth and annealing temperatures, and specimen nonuniformities, among other problems, conspire to confuse the interpretation of strain relaxation processes. Most research of strain relaxation has been based either on transmission electron microscopy (TEM), or on light-optical microscopy of suitably etched specimens. The former possesses high spatial resolution, but suffers from poor strain sensitivity and limited sampling area. The latter is capable of sampling relatively larger areas and partially complements TEM. The true contiguity of the defect structure, though, can only be indirectly inferred from the etch-pit distribution, and directional information, such as Burgers vector information obtained from diffraction data, is not available. Thus, neither technique is completely satisfactory for examining the evolution of the dislocation structure over large areas in the early stages of strain relaxation. X-ray topography is complementary to both large scale dislocation imaging techniques such as chemical etching, and to the high spatial resolution of TEM. When performed at synchrotronradiation facilities, X-ray topography has high strain sensitivity, allows for the identification of dislocation Burgers vectors, and can sample large specimen areas while maintaining adequate spatial resolution. In this paper the evolution of the misfit dislocation structure in the earliest stages of strain relaxation is monitored. The samples studied here have already been studied by chemical etching and were shown to have patchy relaxation as the metastable critical thickness is approached [4]. Quantitative measurement of the lattice relaxation is also obtained using double crystal diffractometry. The goal of the diffractometry experiments is to measure the layer lattice parameters in directions both parallel and perpendicular to the interface, and to determine accurately the layer composition and average relaxation.
261
2. Experimental procedure Three specimens, each consisting of a single Si~_xGex layer on a Si(001) substrate, were grown in a VG Semicon V80 MBE system. Layer thicknesses of 0.5, 1.0 or 1.5/.tm were chosen for this study, and all layers had a nominal composition of x = 0.14. The growth rate and growth temperature used were approximately 3 /zm/h and 550°C, respectively. No post-growth thermal treatments were applied to these specimens. White-radiation and double-crystal topography experiments were performed at the Synchrotron Radiation Source in Daresbury, UK. For the white-radiation method shown in fig. 1, a polychromatic beam, with cross-sectional area defined by a pair of orthogonal beam slits, irradiates a stationary specimen. An exposed nuclear emulsion plate placed beyond the specimen exhibits a pattern of diffraction spots corresponding to a transmission Laue photograph, the anglewavelength relationship for each spot being described by Bragg's law. Contained within each spot is topological information about the atomic planes from which diffraction occurs. In the present case, Laue patterns were recorded on Ilford L4 nuclear emulsion plates with the incident beam normal to the specimen and a beam cross-section
Fig. 1. The pertinent components needed for white-radiation topography. The synchrotron beam irradiates a stationary specimen, producing a transmission Laue pattern on the exposed plate. Multiple topographs are collected in one exposure.
262
M.A. Capano et al. / Strain relaxation in Si I xGex layers on Si(O01)
of 10 mm x 10 mm. With this configuration, four {331} topographs were recorded simultaneously. The incidence angle for the {331} planes was 13.26 °, corresponding to a beam energy (wavelength) of 21.52 keV (0.057 nm). Higher-order harmonic contributions to the data were insignificant since the spectral intensity drops off rapidly above about 24 keV. The (400) topographs were recorded by rotating the specimen 15° about the goniometer axis, along with a corresponding repositioning of the plate. The incidence angle of 15° corresponds to an energy of 17.6 keV and significant harmonic contamination of the data is not expected. White-radiation topography has the advantage, then, of enabling rapid surveys of the defect network over large areas while providing ample data for a complete Burgers vector analysis. A drawback of the technique is the projection of data through the thickness of the wafer, which makes specifying the defect depth insuperably difficult. To investigate the dislocation network near the surface and interface region, (224) and (004) double-crystal topographs were recorded in reflection. For this technique, shown in fig. 2, the specimen was irradiated with a beam which was first conditioned by a stationary reference crystal. A (004) reflection from a Si monochromating crystal was used to condition the beam for all double-crystal topographs. The diffracted intensity from a single reflection was recorded on the topographic plate. However, since the specimens stal Incidentbeam Topographic plate
Specimen Fig. 2. The components needed for double-crystal topography. The synchrotron beam is first conditioned by a Si(004) reflection before irradiating the specimen. A single topograph from diffraction planes (hkl) is collected in one exposure. Curvature effects were eliminated by scanning the specimen about an axis (not shown) perpendicular to the plane of the page.
possessed a curvature induced by the lattice mismatch, only a narrow band of intensity was seen on the plate for a fixed angular orientation. This narrow band corresponds to the portion of the specimen lying within the angular acceptance for Bragg diffraction. This problem was obviated by scanning the specimen through the substrate or layer rocking-curve peak in 2 arc sec steps. Although the resolution in a direction normal to the goniometer axis was reduced by scanning, the topograph contained useful information over an area on the order of a centimeter. Exposure times were 1-2 s for the white-radiation topographs and considerably longer (about 1 h) for the double-crystal topographs. The exposed Ilford L4 nuclear emulsion plates were developed in a 1:3 solution of Kodak D19 developer and deionized water at room temperature. The plates were then fixed in a 1 : 7 mixture of Kodak Rapid Fixer and deionized water, followed by a washing in flowing water for about an hour or more. Magnified images were taken with a 35 mm camera attached to a Reichart-Jong light-optical microscope using reflected light. The final magnification for most prints after photographic enlarging was nearly 100 x . Enlarged areas from the original plates were randomly selected. Standard metallographic techniques were used to obtain statistical information about the misfit dislocation network. Quantification of strain relaxation as a function of layer thickness was accomplished using double-crystal X-ray diffractometry. Surface-symmetric (004) reflections were used to obtain the overlayer lattice parameter normal to the interface. In-plane lattice parameters were measured using the four {115} reflections in both the glancing-incidence and glancing-exit configurations. Conditioning of the Cu K a radiation from a laboratory generator was achieved with the (115) reflection from a GaAs crystal. Data were collected with a Bede Model 150 double-crystal diffractometer equipped with a rotary specimen stage. The diffracted intensities were measured with a wide-area NaI(T1) scintillation detector. Four complete sets of (004) rocking curves and {115} glancing-exit/glancing-incidence rocking curve pairs were collected, each set corresponding to a
M.A. Capano et aL / Strain relaxation in S i
90° rotation of the specimen about its surface normal. Twelve rocking curves were collected for each specimen. Peak splitting values from the (004) reflection, needed to calculate the overlayer lattice parameter perpendicular to the interface, were averaged over the four experimental values. Peak locations were determined from the position of maximum intensity: peak splitting values were accurate to within 7 arc sec. The lattice parameter of the layer in the interface plane was determined by extracting the peak splitting component from the {115} rocking curve pairs due solely to strain ( A d / d ) , averaging with the data set corresponding to a 180° rotation of the specimen, and evaluating the in-plane parameter along the [110] direction from the equation
Ad d
1 h2+k2+l
2
[
X ( h 2 + k 2)
aas _
as
+ l 2c
as
,]
,
(1)
where a and c refer to the lattice parameters of the elastically-distorted overlayer unit cell parallel and perpendicular to the interface, respectively, and a S is the lattice parameter of the substrate. Similar calculations for the orthogonal (110) direction were performed to assess the symmetry of strain relief in the interfacial plane. With these lattice parameters the percentage of relaxation, R, was defined as a -a s R- - × 100%. c -a s
(2)
The equivalent lattice parameter, aeq , for the fully-relaxed, cubic unit cell with Ge composition, x, was obtained from the equation [ - 2 v / ( 1 - v)] (a - aeq ) = c - aeq,
(3)
I _
xGex layers on Si(O01)
263
3. Results and discussion
3.1. Double-crystal diffractometry A first measure of strain relaxation in the series of SiGe/Si specimens is seen from the rocking-curve data presented in table 1. The data from the 0.5 and 1.0 /xm layers are essentially identical. A measured zero-percent relaxation is expected from single epilayer specimens unless the misfit dislocation density is greater than about 1 0 4 c m - 2 [5], because of the limited sensitivity of double-crystal diffractometry to the onset of dislocation generation [2,3]. The absence of strain relaxation [6] as defined by eq. (2) does not mean that the misfit density is zero, but that it is less than ~ 10 4 cm -2, a rather large value which serves to underscore the differences between diffractometry and topography. Double-crystal X-ray diffractometry measures relaxations by shifts in diffraction peaks caused by a concomitant reduction in the Poisson expansion as misfit dislocations form at the interface. Since the Poisson expansion is averaged over the incident beam diameter, many dislocations are required before an actual peak shift occurs. The dislocations do affect the shape of the peak, however. X-ray topography is sensitive to the strain field around a dislocation. So, provided the width of the dislocation image and the length of the dislocation exceed a practical resolution limit (1-20 /xm), single dislocations become visible and linear densities of 1 mm-1 or less are discernible. Slight variations in the layer composition are also noted in table 1 between the two thinnest overlayers. More significant variations in both composition and relaxation are present in the 1.5 /xm layer compared with the two thinner overlayers. The high degree of lattice relaxation in the 1.5 /zm layer specimen precludes any statistical
where the relationship between x and aeq is given by
Table 1 Structural data for the SiGe/Si specimens
x = (aeq - as~)/(a~e - as, ).
t (/zm)
a (,~)
c (,~)
R (%)
aeq (,~)
x
0.5 1.0 1.5
5.4307 5.4307 5.4468
5.4815 5.4823 5.4791
0 (3) 0 (3) 33 (3)
5.4595 5.4599 5.4651
0.127 (4) 0.129 (4) 0.152 (4)
(4)
In eq. (4), age and as~ are the bulk lattice parameters of Ge and Si, respectively.
264
M.A. Capano et al. / Strain relaxation in Si l _ xGe x layers on Si(O01)
Fig. 3. White-radiation (400) topograph of the 0.5 p.m layer specimen. The scale marker equals 0.5 ram.
evaluation of the strain relaxation or misfit dislocation distribution because the overlapping strain fields of the closely spaced dislocations cannot be resolved. Although not explicitly presented in the table, the strain relaxation in orthogonal directions is symmetric to within experimental error, resulting in an isotropic strain state in the interfacial plane. The actual compositions of the 0.5 and 1.0 ~m layers are about l% less than 14% Ge and the composition of the 1.5 ~m layer is about 1% greater than 14% Ge, nominally believed to be the composition of this series.
3.2. X-ray topography As outlined in the experimental section, white-radiation topography provides a general description of the misfit dislocation network over large specimen areas. An example of such a topograph is shown in fig. 3, taken from the 0.5 /zm layer specimen. The linear misfit dislocation density and the average dislocation length after
growth of the 0.5 /zm thick layer are 2.1 mm -1 and 260 /zm, respectively. Comparison of the measured dislocation densities from white-radiation topographs with densities measured from double-crystal topographs show that virtually all dislocation images are localized to the interface region and that the initial substrate material is nearly dislocation free. The images in fig. 3, then, are considered as misfit dislocation segments lying at the substrate/epilayer interface. The most obvious and significant observation, though, are the dislocation crosses seen in fig. 3. Threading dislocations are immediately ruled out as the primary dislocation source because of the observed dislocation arrangements. It would be difficult to imagine how a threading dislocation, presumably located at the center of a cross, could produce the furcation patterns observed. The possibility that a threading dislocation, lying at the end of a single branch, undergoes slip to generate a misfit segment in the plane of the interface, and then undergoes a cross-slip event to form the orthogonal branch, is unlikely since most of the crosses possess two-fold rotational symmetry. Random cross-slip events would not yield crosses with rotational symmetry. However, since orthogonal dislocation segments are so highly correlated, homogeneous nucleation of dislocation half-loops at the surface is not a primary dislocation source, either. If half-loop nucleation were truly homogeneous, orthogonal misfit segments would be randomly distributed and the observed correlation would not exist. Homogeneous nucleation is further discounted because the activation energy required to nucleate a single half-loop for the relatively small strains of current interest (x < 0.5) is unreasonably large [7,81. These crosses are believed to arise from heterogeneous nucleation of dislocation loops lying along orthogonal (110) directions. Heterogeneous sites may possibly result from particulates from the MBE chamber walls [9], but the exact nature of the source is not known. The site density of 3.4 mm -2 measured in the present case is in good agreement with the findings of other researchers who have studied this growth perturbation [9,10]. Fig. 4 illustrates how the dislocation
M.A. Capano et al. / Strain relaxation in (a)
Nucleationsite
Si 1 _
xGex layers on Si(O01)
265
areas, the relaxation becomes symmetric. This observation is fully corroborated by the doublecrystal X-ray diffractometry experiments described earlier. Burgers vector determination for individual dislocations is possible by analyzing the contrast on the (400) and the four {331} white-radiation topographs. Typical contrast variations from orthogonal dislocation loops are presented in fig. 5. This analysis shows that the misfit segments are 60°-type dislocations. Possible directions for the Burgers vectors, assuming line vectors along [110] and [110], are presented in fig. 6. Comparison of the measured and theoretical [11] contrast widths of the direct dislocation images shows these widths to be nearly equal, suggesting that only a single dislocation is responsible for the contrast along any one direction of the cross. In all but a few cases, dislocations of a single cross have different Burgers vectors. This has two important implications. First, in III-V epitaxial systems, dislocations defined by either the group III or the
iiiiiiiiiiiiiiiiiiiiiij!iiiij iiiiiii!iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!!ii (~
~WJ
~
"
/
Epilayer
Fig. 4. Schematic illustration of the cross-pattern formaiion mechanism. In (a), the nucleation of a half-loop, its expansion, and the creation of a misfit segment are represented in two dimensions. The projected view in (b) illustrates how loops expand in orthogonal directions. No attempt has been made to make the threading segments appear as though they lie on {111} glide planes.
crosses are formed. A dislocation half-loop nucleates at a heterogeneous site, and expands under the influence of the misfit stress leaving a segment of misfit dislocation at the interface. This process occurs in orthogonal directions and, when projected onto the interface, the dislocations appear as crosses. The inclination of the threading segments cannot be seen because the spatial resolution of the technique (about 1 ~m) is inadequate. The ratio of dislocation lengths in the orthogonal (110) directions was 1.01, indicative of uniform strain relaxation in these directions. A standard deviation of 0.4 for this ratio indicates there is generally a disparity in the half-loop diameters extending from one source. This implies that nucleation of the individual loops is not simultaneous. Strain relaxation may therefore be slightly asymmetric locally but, when averaged over large
iili e ii;
Fig. 5. A {331} white-radiation topograph of the 0.5/xm layer specimen. The scale marker equals 0.5 mm.
266
M.A. Capano et al. / Strain relaxation in Si l _ xGex layers on Si(O01)
[001]
[001]
I
~
- - ~ [010]
"..,.
[lOO]
[110]
[110] line direction
[TIO] 40101
.i
[1001 [110] line direction
Fig. 6. The possible Burgers vector directions for orthogonal sets of misfit dislocations assuming a line sense in the directions shown in the figure. Of course, the Burgers vector is found by multiplying these vectors by ½a.
group V sublattice (so-called a and /3 dislocations) exhibit different nucleation barriers, which leads to an asymmetric distribution of Burgers vectors and, therefore, to asymmetric strain relaxation [6]. In SiGe on Si, the Burgers vectors are equally distributed amongst all of the possible (a)
directions shown in fig. 6, indicating once again that strain relaxation is symmetrical. Second, the presence of long-range shear stresses resulting from unbalanced screw components of a misfit dislocation array, as shown in fig. 7a, destabilizes the misfit array. It has been suggested [12,13] that this instability induces a second array of disloca-
(b)
VVV PPI []10] (c)
y\
[110]
Fig. 7. (a) An array of 60°-type dislocations with unbalanced screw components. A projected view of the Burgers vectors are denoted by arrows. (b) A second array with Burgers vectors complementary to the first set. (c) The typical arrangement of dislocation Burgers vectors found in this study. The orthogonal arms of a cross have Burgers vectors which are perpendicular to each other when projected onto the interface plane.
Fig. 8. White-radiation (400) topograph of the 1.0 /~m layer specimen. The scale marker equals 0.5 mm.
M.A. Capano et al. / Strain relaxation in Si I xGe~ layers on Si(OO1)
tions which compensate these long-range shear stresses in both III-V and SiGe systems. This second array, having Burgers vectors complementary to the first set as shown in fig. 7b, would then enable dislocation reactions to occur forming Lomer-type dislocations near the interface. However, most dislocation crosses examined here have Burgers vectors which, after projection onto the interface plane, appear as in fig. 7c. Thus, the shear stresses are balanced immediately in most cases and, when averaged over the entire specimen area, instabilities in the dislocation array are avoided. Subsequent defect nucleation is determined by the local strain state and not in response to this instability. The identity of individual dislocation crosses is lost somewhat when the layer thickness is increased to 1.0 ~m because of dislocation intersections from different sources. An example of the increased misfit density in this thicker layer is
267
presented in fig. 8. The linear misfit dislocation density and average dislocation length for the 1.0 ~m layer specimen are 11.1 mm -~ and 550 ~m, respectively. A precise measurement of the source density for this specimen could not be made, but it was estimated to be greater than the density measured in the 0.5 p.m layer specimen. This would be the case for continuous surface nucleation. The length ratio along orthogonal (110) directions was 1.07, implying uniform strain relaxation in these directions. One interesting characteristic from the 1.0 ~m layer are the triangular or wedge-shaped features seen in fig. 9. Close examination of the doublecrystal topograph printed at higher magnification, fig. 10, reveals closely spaced dislocations within these triangles which appear to branch off from the primary dislocation segments. This secondary branching is unlikely to result from the primary heterogeneous nucleation events described above
Fig. 9. Double-crystal (004) topograph from the 1.0 ~m layer specimen. Selected triangular features are denoted by arrows. The scale marker equals 300 p~m.
268
M.A. Capano et aL / Strain relaxation in Si I _ xGex layers on Si(O01)
since the monotonically decreasing dislocation lengths within a triangle suggest a sequential process. If heterogeneous nucleation is responsible for secondary branching, the pattern would be irregularly shaped because the dislocation lengths which define the pattern would vary randomly. The secondary branches are believed to be remnants of dislocation loops which form in the vicinity of the primary misfit dislocation arm, expand and form secondary misfit segments. The exact mechanism by which these loops are generated is unknown at this time but may possibly result from cross slip of a portion of the glissile threading segment. If the portion undergoing cross slip in fig. l l a is lengthy enough, a Frank-Read source becomes active (fig. 1lb) which eventually produces a separate dislocation loop and the original threading segment. The process repeats itself, leaving a series of loops generated at different times. When projected onto the interface plane, the triangular pattern depicted in fig. llc
emerges which has monotonically decreasing location lengths. The overlap of dislocation strain fields in highly relaxed 1.5/zm layer (fig. 12) precludes measurement of dislocation statistics from specimen.
disthe any this
4. Conclusions
X-ray topography has been successfully used to examine the misfit dislocation generation in the early stages of strain relief. Strain relaxation in low-Ge-composition Sil_xGe x on Si(001) has been shown to be initiated by heterogeneous nucleation of dislocation half-loops which propagate to the interface forming misfit segments. These primary loop-nucleation events occur in orthogonal (ll0)-type directions as evinced by the dislocation cross-hatch patterns observed on whiteradiation topographs. As relaxation continues, the
Fig. 10. Double-crystal (224) topograph from the 1.0 p,m layer specimen. The scale marker equals 300 p,m.
M.A. Capano et al. / Strain relaxation in Si 1 _ xGex layers on Si(O01)
269
misfit dislocation network is further developed by continued heterogeneous nucleation and the formation of secondary branches in the dislocation pattern. These secondary branches, which are assumed to be the projection of dislocation halfloops onto the interface, expand under the imposed misfit stress, thereby accelerating the relaxation process and increasing the coverage of dislocations over the interfacial plane.
Acknowledgments M.A.C. and L.W.H. acknowledge financial support provided by the Center for Materials Science and Engineering at MIT under contract number N S F / M R L DMR87-19217, and permission from the United States Air Force for M.A.C. to engage in this research. Synchrotron radiation
(a) /
.,/'
(111)
Fig. 12. White-radiation (400) topograph of the 1.5/xm layer specimen. The scale marker equals 0.5 mm.
facilities were provided by the UK Science and Engineering Research Council. (b)
(1T1)
(c)
/
Secondarydislocationloops
/ Heterogeneousnucleationsite ~ " - ~ Primarydislocations Fig. 11. (a) The portion of the threading dislocation lying at the intersection of two {111} planes has pure screw character and can cross-slip. (b) A Frank-Read source becomes active producing a separate dislocation loop and the original threading segment. (c) After several repetitions, a series of loops are generated. When these are projected onto the interface plane, a triangular pattern emerges.
References [1] See, for example, Layered Structures - Heteroepitaxy, Superlattices, Strain, and Metastability, Mater. Res. Soc. Symp. Proc., Vol. 160 (Mater. Res. Soc., Pittsburgh, PA, 1990). [2] I.J. Fritz, Appl. Phys. Letters 51 (1987) 1080. [3] P.L. Gourley, I.J. Fritz and L.R. Dawson, Appl. Phys. Letters 52 (1988) 377. [4] C.G. Tuppen, C.J. Gibbings and M. Hockly, J. Crystal Growth 94 (1989) 392. [5] K.L. Kavanagh, M.A. Capano, LW. Hobbs, J.C. Barbour, P.M.J. Maree, W. Schaff, J.W. Mayer, D. Pettit, J.M. Woodall, J.A. Stroscio and R.M. Feenstra, J. Appl. Phys 64 (1988) 4843. [6] In the present case, small negative values, well within experimental error, were actually calculated for the relaxation of the two thinnest overlayers but were rounded up to the physically reasonable value of zero. [7] D.J. Eaglesbam, D.M. Maher, E.P. Kvam, C.J. Humphreys and J.C. Bean, Phys. Rev. Letters 62 (1989) 187.
270
M.A. Capano et al. / Strain relaxation in Si 1 _ xGex layers on Si(O01)
[8] D.J. Eaglesham, E.P. Kvam, D.M. Maher, C.J. Humphreys and J.C. Bean, Phil. Mag. A 59 (1989) 1059. [9] J.C. Bean, J. Electron. Mater. 19 (1990) 1055. [10] D. Bellevance, in: Silicon Molecular Beam Epitaxy, Vol. II, Eds. E. Kasper and J.C. Bean (CRC Press, Boca Raton, FL, 1988) ch. 13.
[11] A.R. Lang and M. Polcarova, Proc. Roy. Soc. (London) A 285 (1965) 297. [12] E.P. Kvam, D.M. Maher and C.J. Humphreys, Mater. Res. Soc. Symp. Proc. 160 (1990) 71. [13] E.A. Fitzgerald, D.G. Ast, P.D. Kirchner, G.D. Pettit and J.M. Woodall, J. Appl. Phys. 63 (1988) 693.
Journal of Crystal Growth 116 (1992) 260-270 North-Holland
Strain relaxation in Sil_xGe x layers on Si(001) M.A. Capano 1 Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
L. Hart, D.K. Bowen, D. Gordon-Smith, C.R. Thomas Department of Engineering, UniL,ersityof Warwick, Coventry, UK
C.J. Gibbings, M.A.G. Halliwell British Telecom Laboratories, Martlesham Heath, Ipswich IP5 7RE, UK
and
L.W. H o b b s Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received 20 August 1991
The lattice relaxation of strained Sil_xGe X layers on Si (001) substrates has been examined. Three specimens consisting of a single Si I_xGex layer were grown by molecular beam epitaxy. All layers were grown with a nominal composition of x = 0.14 to thicknesses of 0.5, 1.0 or 1.5 p.m. Double-crystal and white-radiation topographic methods were used to reveal the misfit dislocation structure and distribution. The misfit dislocations were shown to extend from heterogeneous nucleation sites along the (110) directions in the plane of the interface. A symmetric distribution of dislocations between the orthogonal (110) directions was observed. The Burgers vectors of the misfit dislocation array were evenly distributed amongst the available 60°-type candidates. Double-crystal X-ray diffractometry showed the 0.5 and 1.0/zm layers to be fully strained to within the experimental uncertainty. Secondary branching of misfit dislocations was observed in the 1.0/xm layer which indicated cross-slip of the threading dislocation segments.
1. Introduction The relaxation of strained layer SiGe/Si heterojunctions by the generation of mismatch dislocations is a subject of great interest [1], especially for devices such as the heterojunction bipolar transistor (HBT), in which the influence of such
l Current address: WL/MLBM, Wright-Patterson AFB, Ohio 45433-6533, USA.
dislocations at the base-emitter or base-collector junction may be deleterious. A specific question of current interest is determining how the misfit dislocation network develops in the initial stages of strain relaxation. Finding a solution to this question, though, mandates tradeoffs between experimental techniques with varying abilities to resolve the relaxation process [2,3]. There are, in fact, a number of causes which limit experimental resolution, and these may be divided into two broad categories. Sampling-specific causes
0022-0248/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
M.A. Capano et al. / Strain relaxation in Si z xGex layers on Si(O01)
include small sampling volumes, poor spatial resolution, poor strain sensitivity, adverse probespecimen interactions and specimen-preparation effects. In addition to these factors, specimenspecific causes such as uncertainties in layer thickness and composition, inaccuracies in growth and annealing temperatures, and specimen nonuniformities, among other problems, conspire to confuse the interpretation of strain relaxation processes. Most research of strain relaxation has been based either on transmission electron microscopy (TEM), or on light-optical microscopy of suitably etched specimens. The former possesses high spatial resolution, but suffers from poor strain sensitivity and limited sampling area. The latter is capable of sampling relatively larger areas and partially complements TEM. The true contiguity of the defect structure, though, can only be indirectly inferred from the etch-pit distribution, and directional information, such as Burgers vector information obtained from diffraction data, is not available. Thus, neither technique is completely satisfactory for examining the evolution of the dislocation structure over large areas in the early stages of strain relaxation. X-ray topography is complementary to both large scale dislocation imaging techniques such as chemical etching, and to the high spatial resolution of TEM. When performed at synchrotronradiation facilities, X-ray topography has high strain sensitivity, allows for the identification of dislocation Burgers vectors, and can sample large specimen areas while maintaining adequate spatial resolution. In this paper the evolution of the misfit dislocation structure in the earliest stages of strain relaxation is monitored. The samples studied here have already been studied by chemical etching and were shown to have patchy relaxation as the metastable critical thickness is approached [4]. Quantitative measurement of the lattice relaxation is also obtained using double crystal diffractometry. The goal of the diffractometry experiments is to measure the layer lattice parameters in directions both parallel and perpendicular to the interface, and to determine accurately the layer composition and average relaxation.
261
2. Experimental procedure Three specimens, each consisting of a single Si~_xGex layer on a Si(001) substrate, were grown in a VG Semicon V80 MBE system. Layer thicknesses of 0.5, 1.0 or 1.5/.tm were chosen for this study, and all layers had a nominal composition of x = 0.14. The growth rate and growth temperature used were approximately 3 /zm/h and 550°C, respectively. No post-growth thermal treatments were applied to these specimens. White-radiation and double-crystal topography experiments were performed at the Synchrotron Radiation Source in Daresbury, UK. For the white-radiation method shown in fig. 1, a polychromatic beam, with cross-sectional area defined by a pair of orthogonal beam slits, irradiates a stationary specimen. An exposed nuclear emulsion plate placed beyond the specimen exhibits a pattern of diffraction spots corresponding to a transmission Laue photograph, the anglewavelength relationship for each spot being described by Bragg's law. Contained within each spot is topological information about the atomic planes from which diffraction occurs. In the present case, Laue patterns were recorded on Ilford L4 nuclear emulsion plates with the incident beam normal to the specimen and a beam cross-section
Fig. 1. The pertinent components needed for white-radiation topography. The synchrotron beam irradiates a stationary specimen, producing a transmission Laue pattern on the exposed plate. Multiple topographs are collected in one exposure.
262
M.A. Capano et al. / Strain relaxation in Si I xGex layers on Si(O01)
of 10 mm x 10 mm. With this configuration, four {331} topographs were recorded simultaneously. The incidence angle for the {331} planes was 13.26 °, corresponding to a beam energy (wavelength) of 21.52 keV (0.057 nm). Higher-order harmonic contributions to the data were insignificant since the spectral intensity drops off rapidly above about 24 keV. The (400) topographs were recorded by rotating the specimen 15° about the goniometer axis, along with a corresponding repositioning of the plate. The incidence angle of 15° corresponds to an energy of 17.6 keV and significant harmonic contamination of the data is not expected. White-radiation topography has the advantage, then, of enabling rapid surveys of the defect network over large areas while providing ample data for a complete Burgers vector analysis. A drawback of the technique is the projection of data through the thickness of the wafer, which makes specifying the defect depth insuperably difficult. To investigate the dislocation network near the surface and interface region, (224) and (004) double-crystal topographs were recorded in reflection. For this technique, shown in fig. 2, the specimen was irradiated with a beam which was first conditioned by a stationary reference crystal. A (004) reflection from a Si monochromating crystal was used to condition the beam for all double-crystal topographs. The diffracted intensity from a single reflection was recorded on the topographic plate. However, since the specimens stal Incidentbeam Topographic plate
Specimen Fig. 2. The components needed for double-crystal topography. The synchrotron beam is first conditioned by a Si(004) reflection before irradiating the specimen. A single topograph from diffraction planes (hkl) is collected in one exposure. Curvature effects were eliminated by scanning the specimen about an axis (not shown) perpendicular to the plane of the page.
possessed a curvature induced by the lattice mismatch, only a narrow band of intensity was seen on the plate for a fixed angular orientation. This narrow band corresponds to the portion of the specimen lying within the angular acceptance for Bragg diffraction. This problem was obviated by scanning the specimen through the substrate or layer rocking-curve peak in 2 arc sec steps. Although the resolution in a direction normal to the goniometer axis was reduced by scanning, the topograph contained useful information over an area on the order of a centimeter. Exposure times were 1-2 s for the white-radiation topographs and considerably longer (about 1 h) for the double-crystal topographs. The exposed Ilford L4 nuclear emulsion plates were developed in a 1:3 solution of Kodak D19 developer and deionized water at room temperature. The plates were then fixed in a 1 : 7 mixture of Kodak Rapid Fixer and deionized water, followed by a washing in flowing water for about an hour or more. Magnified images were taken with a 35 mm camera attached to a Reichart-Jong light-optical microscope using reflected light. The final magnification for most prints after photographic enlarging was nearly 100 x . Enlarged areas from the original plates were randomly selected. Standard metallographic techniques were used to obtain statistical information about the misfit dislocation network. Quantification of strain relaxation as a function of layer thickness was accomplished using double-crystal X-ray diffractometry. Surface-symmetric (004) reflections were used to obtain the overlayer lattice parameter normal to the interface. In-plane lattice parameters were measured using the four {115} reflections in both the glancing-incidence and glancing-exit configurations. Conditioning of the Cu K a radiation from a laboratory generator was achieved with the (115) reflection from a GaAs crystal. Data were collected with a Bede Model 150 double-crystal diffractometer equipped with a rotary specimen stage. The diffracted intensities were measured with a wide-area NaI(T1) scintillation detector. Four complete sets of (004) rocking curves and {115} glancing-exit/glancing-incidence rocking curve pairs were collected, each set corresponding to a
M.A. Capano et aL / Strain relaxation in S i
90° rotation of the specimen about its surface normal. Twelve rocking curves were collected for each specimen. Peak splitting values from the (004) reflection, needed to calculate the overlayer lattice parameter perpendicular to the interface, were averaged over the four experimental values. Peak locations were determined from the position of maximum intensity: peak splitting values were accurate to within 7 arc sec. The lattice parameter of the layer in the interface plane was determined by extracting the peak splitting component from the {115} rocking curve pairs due solely to strain ( A d / d ) , averaging with the data set corresponding to a 180° rotation of the specimen, and evaluating the in-plane parameter along the [110] direction from the equation
Ad d
1 h2+k2+l
2
[
X ( h 2 + k 2)
aas _
as
+ l 2c
as
,]
,
(1)
where a and c refer to the lattice parameters of the elastically-distorted overlayer unit cell parallel and perpendicular to the interface, respectively, and a S is the lattice parameter of the substrate. Similar calculations for the orthogonal (110) direction were performed to assess the symmetry of strain relief in the interfacial plane. With these lattice parameters the percentage of relaxation, R, was defined as a -a s R- - × 100%. c -a s
(2)
The equivalent lattice parameter, aeq , for the fully-relaxed, cubic unit cell with Ge composition, x, was obtained from the equation [ - 2 v / ( 1 - v)] (a - aeq ) = c - aeq,
(3)
I _
xGex layers on Si(O01)
263
3. Results and discussion
3.1. Double-crystal diffractometry A first measure of strain relaxation in the series of SiGe/Si specimens is seen from the rocking-curve data presented in table 1. The data from the 0.5 and 1.0 /xm layers are essentially identical. A measured zero-percent relaxation is expected from single epilayer specimens unless the misfit dislocation density is greater than about 1 0 4 c m - 2 [5], because of the limited sensitivity of double-crystal diffractometry to the onset of dislocation generation [2,3]. The absence of strain relaxation [6] as defined by eq. (2) does not mean that the misfit density is zero, but that it is less than ~ 10 4 cm -2, a rather large value which serves to underscore the differences between diffractometry and topography. Double-crystal X-ray diffractometry measures relaxations by shifts in diffraction peaks caused by a concomitant reduction in the Poisson expansion as misfit dislocations form at the interface. Since the Poisson expansion is averaged over the incident beam diameter, many dislocations are required before an actual peak shift occurs. The dislocations do affect the shape of the peak, however. X-ray topography is sensitive to the strain field around a dislocation. So, provided the width of the dislocation image and the length of the dislocation exceed a practical resolution limit (1-20 /xm), single dislocations become visible and linear densities of 1 mm-1 or less are discernible. Slight variations in the layer composition are also noted in table 1 between the two thinnest overlayers. More significant variations in both composition and relaxation are present in the 1.5 /xm layer compared with the two thinner overlayers. The high degree of lattice relaxation in the 1.5 /zm layer specimen precludes any statistical
where the relationship between x and aeq is given by
Table 1 Structural data for the SiGe/Si specimens
x = (aeq - as~)/(a~e - as, ).
t (/zm)
a (,~)
c (,~)
R (%)
aeq (,~)
x
0.5 1.0 1.5
5.4307 5.4307 5.4468
5.4815 5.4823 5.4791
0 (3) 0 (3) 33 (3)
5.4595 5.4599 5.4651
0.127 (4) 0.129 (4) 0.152 (4)
(4)
In eq. (4), age and as~ are the bulk lattice parameters of Ge and Si, respectively.
264
M.A. Capano et al. / Strain relaxation in Si l _ xGe x layers on Si(O01)
Fig. 3. White-radiation (400) topograph of the 0.5 p.m layer specimen. The scale marker equals 0.5 ram.
evaluation of the strain relaxation or misfit dislocation distribution because the overlapping strain fields of the closely spaced dislocations cannot be resolved. Although not explicitly presented in the table, the strain relaxation in orthogonal directions is symmetric to within experimental error, resulting in an isotropic strain state in the interfacial plane. The actual compositions of the 0.5 and 1.0 ~m layers are about l% less than 14% Ge and the composition of the 1.5 ~m layer is about 1% greater than 14% Ge, nominally believed to be the composition of this series.
3.2. X-ray topography As outlined in the experimental section, white-radiation topography provides a general description of the misfit dislocation network over large specimen areas. An example of such a topograph is shown in fig. 3, taken from the 0.5 /zm layer specimen. The linear misfit dislocation density and the average dislocation length after
growth of the 0.5 /zm thick layer are 2.1 mm -1 and 260 /zm, respectively. Comparison of the measured dislocation densities from white-radiation topographs with densities measured from double-crystal topographs show that virtually all dislocation images are localized to the interface region and that the initial substrate material is nearly dislocation free. The images in fig. 3, then, are considered as misfit dislocation segments lying at the substrate/epilayer interface. The most obvious and significant observation, though, are the dislocation crosses seen in fig. 3. Threading dislocations are immediately ruled out as the primary dislocation source because of the observed dislocation arrangements. It would be difficult to imagine how a threading dislocation, presumably located at the center of a cross, could produce the furcation patterns observed. The possibility that a threading dislocation, lying at the end of a single branch, undergoes slip to generate a misfit segment in the plane of the interface, and then undergoes a cross-slip event to form the orthogonal branch, is unlikely since most of the crosses possess two-fold rotational symmetry. Random cross-slip events would not yield crosses with rotational symmetry. However, since orthogonal dislocation segments are so highly correlated, homogeneous nucleation of dislocation half-loops at the surface is not a primary dislocation source, either. If half-loop nucleation were truly homogeneous, orthogonal misfit segments would be randomly distributed and the observed correlation would not exist. Homogeneous nucleation is further discounted because the activation energy required to nucleate a single half-loop for the relatively small strains of current interest (x < 0.5) is unreasonably large [7,81. These crosses are believed to arise from heterogeneous nucleation of dislocation loops lying along orthogonal (110) directions. Heterogeneous sites may possibly result from particulates from the MBE chamber walls [9], but the exact nature of the source is not known. The site density of 3.4 mm -2 measured in the present case is in good agreement with the findings of other researchers who have studied this growth perturbation [9,10]. Fig. 4 illustrates how the dislocation
M.A. Capano et al. / Strain relaxation in (a)
Nucleationsite
Si 1 _
xGex layers on Si(O01)
265
areas, the relaxation becomes symmetric. This observation is fully corroborated by the doublecrystal X-ray diffractometry experiments described earlier. Burgers vector determination for individual dislocations is possible by analyzing the contrast on the (400) and the four {331} white-radiation topographs. Typical contrast variations from orthogonal dislocation loops are presented in fig. 5. This analysis shows that the misfit segments are 60°-type dislocations. Possible directions for the Burgers vectors, assuming line vectors along [110] and [110], are presented in fig. 6. Comparison of the measured and theoretical [11] contrast widths of the direct dislocation images shows these widths to be nearly equal, suggesting that only a single dislocation is responsible for the contrast along any one direction of the cross. In all but a few cases, dislocations of a single cross have different Burgers vectors. This has two important implications. First, in III-V epitaxial systems, dislocations defined by either the group III or the
iiiiiiiiiiiiiiiiiiiiiij!iiiij iiiiiii!iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!!ii (~
~WJ
~
"
/
Epilayer
Fig. 4. Schematic illustration of the cross-pattern formaiion mechanism. In (a), the nucleation of a half-loop, its expansion, and the creation of a misfit segment are represented in two dimensions. The projected view in (b) illustrates how loops expand in orthogonal directions. No attempt has been made to make the threading segments appear as though they lie on {111} glide planes.
crosses are formed. A dislocation half-loop nucleates at a heterogeneous site, and expands under the influence of the misfit stress leaving a segment of misfit dislocation at the interface. This process occurs in orthogonal directions and, when projected onto the interface, the dislocations appear as crosses. The inclination of the threading segments cannot be seen because the spatial resolution of the technique (about 1 ~m) is inadequate. The ratio of dislocation lengths in the orthogonal (110) directions was 1.01, indicative of uniform strain relaxation in these directions. A standard deviation of 0.4 for this ratio indicates there is generally a disparity in the half-loop diameters extending from one source. This implies that nucleation of the individual loops is not simultaneous. Strain relaxation may therefore be slightly asymmetric locally but, when averaged over large
iili e ii;
Fig. 5. A {331} white-radiation topograph of the 0.5/xm layer specimen. The scale marker equals 0.5 mm.
266
M.A. Capano et al. / Strain relaxation in Si l _ xGex layers on Si(O01)
[001]
[001]
I
~
- - ~ [010]
"..,.
[lOO]
[110]
[110] line direction
[TIO] 40101
.i
[1001 [110] line direction
Fig. 6. The possible Burgers vector directions for orthogonal sets of misfit dislocations assuming a line sense in the directions shown in the figure. Of course, the Burgers vector is found by multiplying these vectors by ½a.
group V sublattice (so-called a and /3 dislocations) exhibit different nucleation barriers, which leads to an asymmetric distribution of Burgers vectors and, therefore, to asymmetric strain relaxation [6]. In SiGe on Si, the Burgers vectors are equally distributed amongst all of the possible (a)
directions shown in fig. 6, indicating once again that strain relaxation is symmetrical. Second, the presence of long-range shear stresses resulting from unbalanced screw components of a misfit dislocation array, as shown in fig. 7a, destabilizes the misfit array. It has been suggested [12,13] that this instability induces a second array of disloca-
(b)
VVV PPI []10] (c)
y\
[110]
Fig. 7. (a) An array of 60°-type dislocations with unbalanced screw components. A projected view of the Burgers vectors are denoted by arrows. (b) A second array with Burgers vectors complementary to the first set. (c) The typical arrangement of dislocation Burgers vectors found in this study. The orthogonal arms of a cross have Burgers vectors which are perpendicular to each other when projected onto the interface plane.
Fig. 8. White-radiation (400) topograph of the 1.0 /~m layer specimen. The scale marker equals 0.5 mm.
M.A. Capano et al. / Strain relaxation in Si I xGe~ layers on Si(OO1)
tions which compensate these long-range shear stresses in both III-V and SiGe systems. This second array, having Burgers vectors complementary to the first set as shown in fig. 7b, would then enable dislocation reactions to occur forming Lomer-type dislocations near the interface. However, most dislocation crosses examined here have Burgers vectors which, after projection onto the interface plane, appear as in fig. 7c. Thus, the shear stresses are balanced immediately in most cases and, when averaged over the entire specimen area, instabilities in the dislocation array are avoided. Subsequent defect nucleation is determined by the local strain state and not in response to this instability. The identity of individual dislocation crosses is lost somewhat when the layer thickness is increased to 1.0 ~m because of dislocation intersections from different sources. An example of the increased misfit density in this thicker layer is
267
presented in fig. 8. The linear misfit dislocation density and average dislocation length for the 1.0 ~m layer specimen are 11.1 mm -~ and 550 ~m, respectively. A precise measurement of the source density for this specimen could not be made, but it was estimated to be greater than the density measured in the 0.5 p.m layer specimen. This would be the case for continuous surface nucleation. The length ratio along orthogonal (110) directions was 1.07, implying uniform strain relaxation in these directions. One interesting characteristic from the 1.0 ~m layer are the triangular or wedge-shaped features seen in fig. 9. Close examination of the doublecrystal topograph printed at higher magnification, fig. 10, reveals closely spaced dislocations within these triangles which appear to branch off from the primary dislocation segments. This secondary branching is unlikely to result from the primary heterogeneous nucleation events described above
Fig. 9. Double-crystal (004) topograph from the 1.0 ~m layer specimen. Selected triangular features are denoted by arrows. The scale marker equals 300 p~m.
268
M.A. Capano et aL / Strain relaxation in Si I _ xGex layers on Si(O01)
since the monotonically decreasing dislocation lengths within a triangle suggest a sequential process. If heterogeneous nucleation is responsible for secondary branching, the pattern would be irregularly shaped because the dislocation lengths which define the pattern would vary randomly. The secondary branches are believed to be remnants of dislocation loops which form in the vicinity of the primary misfit dislocation arm, expand and form secondary misfit segments. The exact mechanism by which these loops are generated is unknown at this time but may possibly result from cross slip of a portion of the glissile threading segment. If the portion undergoing cross slip in fig. l l a is lengthy enough, a Frank-Read source becomes active (fig. 1lb) which eventually produces a separate dislocation loop and the original threading segment. The process repeats itself, leaving a series of loops generated at different times. When projected onto the interface plane, the triangular pattern depicted in fig. llc
emerges which has monotonically decreasing location lengths. The overlap of dislocation strain fields in highly relaxed 1.5/zm layer (fig. 12) precludes measurement of dislocation statistics from specimen.
disthe any this
4. Conclusions
X-ray topography has been successfully used to examine the misfit dislocation generation in the early stages of strain relief. Strain relaxation in low-Ge-composition Sil_xGe x on Si(001) has been shown to be initiated by heterogeneous nucleation of dislocation half-loops which propagate to the interface forming misfit segments. These primary loop-nucleation events occur in orthogonal (ll0)-type directions as evinced by the dislocation cross-hatch patterns observed on whiteradiation topographs. As relaxation continues, the
Fig. 10. Double-crystal (224) topograph from the 1.0 p,m layer specimen. The scale marker equals 300 p,m.
M.A. Capano et al. / Strain relaxation in Si 1 _ xGex layers on Si(O01)
269
misfit dislocation network is further developed by continued heterogeneous nucleation and the formation of secondary branches in the dislocation pattern. These secondary branches, which are assumed to be the projection of dislocation halfloops onto the interface, expand under the imposed misfit stress, thereby accelerating the relaxation process and increasing the coverage of dislocations over the interfacial plane.
Acknowledgments M.A.C. and L.W.H. acknowledge financial support provided by the Center for Materials Science and Engineering at MIT under contract number N S F / M R L DMR87-19217, and permission from the United States Air Force for M.A.C. to engage in this research. Synchrotron radiation
(a) /
.,/'
(111)
Fig. 12. White-radiation (400) topograph of the 1.5/xm layer specimen. The scale marker equals 0.5 mm.
facilities were provided by the UK Science and Engineering Research Council. (b)
(1T1)
(c)
/
Secondarydislocationloops
/ Heterogeneousnucleationsite ~ " - ~ Primarydislocations Fig. 11. (a) The portion of the threading dislocation lying at the intersection of two {111} planes has pure screw character and can cross-slip. (b) A Frank-Read source becomes active producing a separate dislocation loop and the original threading segment. (c) After several repetitions, a series of loops are generated. When these are projected onto the interface plane, a triangular pattern emerges.
References [1] See, for example, Layered Structures - Heteroepitaxy, Superlattices, Strain, and Metastability, Mater. Res. Soc. Symp. Proc., Vol. 160 (Mater. Res. Soc., Pittsburgh, PA, 1990). [2] I.J. Fritz, Appl. Phys. Letters 51 (1987) 1080. [3] P.L. Gourley, I.J. Fritz and L.R. Dawson, Appl. Phys. Letters 52 (1988) 377. [4] C.G. Tuppen, C.J. Gibbings and M. Hockly, J. Crystal Growth 94 (1989) 392. [5] K.L. Kavanagh, M.A. Capano, LW. Hobbs, J.C. Barbour, P.M.J. Maree, W. Schaff, J.W. Mayer, D. Pettit, J.M. Woodall, J.A. Stroscio and R.M. Feenstra, J. Appl. Phys 64 (1988) 4843. [6] In the present case, small negative values, well within experimental error, were actually calculated for the relaxation of the two thinnest overlayers but were rounded up to the physically reasonable value of zero. [7] D.J. Eaglesbam, D.M. Maher, E.P. Kvam, C.J. Humphreys and J.C. Bean, Phys. Rev. Letters 62 (1989) 187.
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M.A. Capano et al. / Strain relaxation in Si 1 _ xGex layers on Si(O01)
[8] D.J. Eaglesham, E.P. Kvam, D.M. Maher, C.J. Humphreys and J.C. Bean, Phil. Mag. A 59 (1989) 1059. [9] J.C. Bean, J. Electron. Mater. 19 (1990) 1055. [10] D. Bellevance, in: Silicon Molecular Beam Epitaxy, Vol. II, Eds. E. Kasper and J.C. Bean (CRC Press, Boca Raton, FL, 1988) ch. 13.
[11] A.R. Lang and M. Polcarova, Proc. Roy. Soc. (London) A 285 (1965) 297. [12] E.P. Kvam, D.M. Maher and C.J. Humphreys, Mater. Res. Soc. Symp. Proc. 160 (1990) 71. [13] E.A. Fitzgerald, D.G. Ast, P.D. Kirchner, G.D. Pettit and J.M. Woodall, J. Appl. Phys. 63 (1988) 693.