HIGH-RESOLUTION X-RAY DIFFRACTION OF EPITAXIAL LAYERS ON VICINAL SEMICONDUCTOR SUBSTRATES
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1 Philips J. Res. 47 (1993) HIGH-RESOLUTION X-RAY DIFFRACTION OF EPITAXIAL LAYERS ON VICINAL SEMICONDUCTOR SUBSTRATES by P. VAN DER SLUIS Philips Research Laboratories, P'I), Box JA Eindhoven. The Netherlands Abstract For a misoriented (hkl) substrate crystal, the (hkl) lattice plane normal and surface normal do not coincide. Rocking curve measurements of a sample with an epitaxial layer grown on such a substrate show a variation of substrate epitaxiallayer peak distances with rotation around the surface normal of the sample. The variation in angular peak distances is frequently attributed to a relative tilt of the epitaxi al layer with respect to the substrate. For exactly oriented substrates the lattice of the epitaxial layer is tetragonally distorted because ofthe lattice parameter difference. We show that for fully strained epitaxial layers of semiconductors with the zinc blende structure the variation in peak distance is due to distortion of the tetragonal symmetry, caused by the anisotropic elasticity of these materials. The effects of misorientation are calculated quantitatively and simulated by using an effective asymmetry, consisting of the asymmetry angle of the lattice plane plus a projection of the misorientation onto the diffraction plane. Keywords: high-resolution X-ray diffraction, misorientation, monoclinic distortion, semiconductor, vicinal substrates. 1. Introduetion High-resolution X-ray diffraction can be used to determine strain, composition, thickness and interface roughness of epitaxially grown semiconductor layers on substrates':"), For simple structures (e.g. one or two layers on a substrate), the lattice mismatch can, normally, be determined by the identification of epitaxial-layer peaks and measurement of their angular distance to the substrate peak. Dynamical simulations are required for an accurate analysis of the rocking curve of a more complicated structure'). Problems arise with substrates where the surface normal does not coincide PbiUpsJournal of Research Vol.47 Nos
2 P. van der Sluis [OOI) Epitaxial layer Fig. 1. A sample with misorientation: The sample normal (N) does not coincide with the (OOI) lattice plane normal. e is the misorientation angle. with a symmetry axis of the sample (usually 001). Rocking curve measurements show that the angular distance from an epi taxi al layer peak to the substrate peak depends on the rotation around the surface normal. This effect is frequently attributed to a relative tilt ofthe epitaxiallayer with respect to the substratev"). We will show that for fully strained epitaxiallayers, lattice plane tilt (i.e. lattice deformation) is the cause of this variation. Furthermore, methods for geometrical interpretation and a method to take this effect into account in scattering simulations will be given. 2. Theory 2.1. Reciprocallattice geometry Consider a misoriented (001) semiconductor substrate crystal with a misorientation angle B overgrown with a thin lattice mismatched epitaxiallayer (fig. I). The misorientation angle s is less than a few degrees and the surface of such a crystal is called a vicinal face. The growth direction for such a substrate is not parallel to any symmetry axis. If the growth direction is exactly parallel to a symmetry axis the epitaxiallayer will have a tetragonally distorted lattice"). Semiconductors with the zinc blende structure are elastically anisotropic. Therefore, the symmetry of the epitaxiallayer grown on a vicinal substrate will be even lower than tetragonal owing to shear stresses. Hornstra and Bartels") developed a method, based on elasticity theory, to calculate this lattice deformation for epitaxiallayers grown on any low-index (hkl) face. For faces tilted in the (110) direction, such as the (113) face, the lattice deformation is monoclinic'). A vicinal face can be described with very high indices (hkl). We apply the 204 Philips Journal of Research Vol.47 Nos
3 High-resolution X-ray diffraction of epitaxial layers TABLE I Monoclinic distortion and (X) angles for III-V and IV-IV semiconductors, for a misorientation of IOaway from the (001) orientation in the [IlO] direction and a lattice mismatch of 1%. Material Deviation P from 90 0 Angle X (deg) (deg) Si Ge AISb GaP GaAs GaSb InP InAs InSb method of Hornstra and Barrels") to these high index faces. The amount of deformation depends on the magnitude of the misorientation angle and the lattice mismatch of the layer. As an example we calculated the lattice deformation of a layer with a lattice mismatch of I% on a substrate with a 10 misorientation in the [110] direction. A misorientation in the [IlO] direction is the most frequently encountered misorientation. The results for most common III-Vand IV-IV semiconductors are presented in Table 1. For all the materials the monoclinic deviation angles lie within 7% of the average deviation angle. So, although both the elasticity constants and the anisotropy of these materials vary widely"), the monoclinic distortion is almost identical for the chosen mismatch and misorientation. It can be shown that the distortion is linearly dependent on the lattice mismatch. In addition, calculations show that for small misorientation angles, the monoclinic distortion is linearly dependent on this misorientation. This means that the monoclinic distortion is almost identical for all materials of Table I, irrespective of lattice mismatch, as long as vicinal substrates are considered. The geometry of the reciprocallattice is depicted in fig. 2. The intensity distribution near the 001 substrate peak is depicted with a dot labelled 001s. Owing to the finite size of the thin epitaxial-iayer lattice, the intensity distribution near the layer peak lies on a line, extended in the direction perpendicular to the surface, which makes an angle 8 with the direction 0-00Is. The maximum of the layer intensity is depicted with a dot on the line-shaped Philips Journal of Research Vol.47 Nos
4 P. van der Sluis E i X / 1\./ 001 i..."" s [OOIJ~N o Fig. 2. Geometry of the reciprocallattice of a sample with misorientation, overgrown coherently with a lattice-mismatched epitaxial layer. Reciprocal lattice parameters are indicated by a*. Angles and reciprocal lattice points are defined in the text. intensity distribution and labelled The angle between the lines and s is f3. The angle 90 o +f3 is the monoclinic angle of the epitaxiallayer lattice (in real space). The angle X is defined as the angle that the line 001 s makes with s. From geometrical considerations it follows that tan X tan f3 = (Óa/a)j_ where (Óa/a)j_ is the perpendicular lattice mismatch. So for a given misorientation e we can calculate, using the method of Hornstra and Barrels"), the monoclinic distortion angle f3 and using eq. (1) we can calculate the angle X. In Table I we have calculated this angle X for all the materials considered and we find that X is approximately equal to e. The largest deviation is found for Si and amounts 28%. For the III-V materials the deviation is generally less than 10%. Calculation shows that this finding is independent of lattice mismatch and misorientation, as long as vicinal substrates are considered. Rearranging eq. (1) and substituting X = e we get tan f3 = (Óa/a)j_ tan e (2) This relation is identical to the empirical relation proposed by Nagai"), based on a simple geometrical model. This relation has been used extensively (I) 206 Philips Journalof Research Vol.47 Nos.3-S 1993
5 High-resolution X-ray diffraction of epitaxial layers 004 L 004 s lak [1.: 0 Fig. 3. Geometry of the reciprocallattice of an exactly oriented sample overgrown coherently with a lattice-mismatched epitaxial layer. The symmetric 004 and asymmetrie 224 lattice points are shown. to describe the behaviour of layers on vicinal substrates and is known to be accurate. 5,6,8-IO) 2.2. Simulations and geometrical interpretation of rocking curves The fact that X is approximately identical to e has important advantages for the interpretation of diffraction patterns of structures on vicinal substrates. Consider reciprocal space near the asymmetrie 224 reflection and the symmetric 004 reflection of an exactly (001)-oriented substrate overgrown with a thin epitaxiallayer (fig. 3). The substrate reflection is again indicated with a dot and the epitaxial layer reflection with a line perpendicular to the growth direction, which is now coincident with the [001] direction. The asymmetry angle of the asymmetrie lattice plane is called rp. Comparing fig. 2 and fig. 3, it is clear that under the condition that X = e, the geometry of diffraction from the vicinal sample is identical to the geometry of diffraction for the asymmetrie reflection from the exactly oriented sample: X equals rp and fj equals (j, which is the angle between 0-224, and , This means that we can consider a reflection of a lattice plane on a vicinal substrate with misorientation angle s as if it were a reflection from a lattice plane with an asymmetry angle s. If this lattice plane is already asymmetrie (like the 224lattice plane on a (OOI)-oriented substrate) it can be treated as a lattice plane with an effective asymmetry angle rp+e. This opens up the possibility of performing dynamical simulations on these types of structures with standard unadapted simulation software (see ex- Philips Journal of Research Vol.47 Nos. 3-S
6 P. van der Sluis perimental): Simply add the misorientation angle to the asymmetry angle of the lattice plane under consideration. The other way round, the misorientation angle can also be determined from simulations. When the crystal (fig. 1) is rotated around the (001) lattice plane normal by 90, the effective misorientation becomes invisible. For a rotation of 180 the sign of the contribution of /l to the effective misorientation is reversed. In general, the misorientation as a function of rotation (angle r) around the surface normal is cpeff = cpo + /lcos r (3) where cpo is the asymmetry angle of the lattice plane. For strained layers this asymmetry angle is not equal to the asymmetry angle of the substrate, but differs by an angle fie, which is strain dependent.' ) The misorientation and its direction can be determined when at least three rocking curves are measured, and simulated with the same set of parameters except the misorientation angle. Also geometrical interpretation of the diffraction pattern is now identical to the geometrical interpretation of the diffraction pattern of an asymmetrie reflection. For unrelaxed structures, the diffraction pattern of any asymmetrie reflection can be calculated once a symmetric reflection is measured. The substrate to epitaxiallayer peak distance in a rocking curve is the difference in crystal setting for Bragg reflection of the substrate and epitaxial layer. This angle is given byll) where (j is given by and el is given by tan s fiksin cp fikcos cp +2sin es /À Àfikcos cp +2sin es 2cos(j where es is the Bragg angle of the substrate, fh is the Bragg angle of the layer, cp is the asymmetry of the lattice plane, À the X-ray wavelength and fik is the distance in reciprocal space between the substrate and epitaxial layer reflection. It can be obtained from the measurement of a symmetric reflection (4) (5) (6) (7) The symmetric reflection can be measured at any orientation around the 208 Phllips Journal of Research Vol.47 Nos
7 High-resolution X-ray diffraction of epitaxial layers Fig. 4. The geometry and orientation of the epitaxiallayer lattice a) in case of the monoclinic distortion model and b) in case of the tilted layer model, in relation to the substrate lattice directions. sample normal. From eq. (3) it can be seen that in order to obtain a correct value for Sk from a misoriented sample the symmetric reflection has to be measured with the diffraction plane at 90 to the misorientation Layer lattice symmetry The substrate epitaxial layer peak distance Am can be calculated for any reflection and any misorientation once Ak is measured. Calculations show that Am as a function of rotation (r) around the surface normal varies with cos r. This is generally attributed to tilting of the layer4-6). The tilt is supposed to originate from the terracing ofthe substrate. In our model the lattice plane, not the layer, is tilted, resulting in a lower symmetry (fig. 4). Calculations with eq. (4) show that the variation of Amwith. depends on the reflection. For the low angle of incidence asymmetrie reflection a smaller variation of Am with. is calculated than for the corresponding high angle of incidence asymmetrie reflection. In the tilted layer model all lattice planes will be tilted (fig. 4). Therefore, the variation of Am with rotation. will in this model be independent of the reflection (except for the effect of projection, as described by relation (3)). This thus offers a way to find out whether the layers or the lattice planes are tilted. There is an alternative way to find out which model describes the lattice geometry. The angular separation calculated with (4) is approximately linear in Sk and 8. This means that the width of a thin-layer peak will depend on 8. For an 8 causing the layer peak to shift towards the substrate peak we expect a narrower peak width. For an 8 causing the layer peak to shift away from the substrate peak we expect a larger width. This narrowing and broadening of epitaxiallayer peaks is thus also expected for asymmetrie reflections. In the tilted layer model, the tilt of the layer will only shift the layer peak in a rocking curve. The peak width will thus be independent ofrotation. We have observed Philips Journal of Research Vol.47 Nos. 3-S
8 P. van der Sluis this frequently in samples with relaxed layers. This method is thus a very simple way to discriminate between layer tilt and lattice plane tilt (i.e. lattice deformation). 3. Experimental The diffraction curves were obtained with a Philips high-resolution diffractometer (HR-I) using a four-crystal monochromator with (110) Ge monochromator crystals. The 220 reflection and Cu Kal radiation were used. Rocking curves were recorded with an open detector, which has a receiving angle of about 4. The omega resolution is 0.9 are seconds. Simulations are carried out with the Philips HRS dynamical scattering simulation program, which is based on the Takagi- Taupin equations'<"). All plots of rocking curves and simulations are on a logarithmic intensity scale, because of the large dynamic range of the recorded intensity. Misorientations are determined by a mixed optical and XRD technique. The surface is aligned perpendicular to the rotation axis using a small He-Ne laser: the reflection of the laser has to be stationary when projected, as a function of the rotation. Then a symmetric substrate peak is recorded as a function of rotation. The amplitude of the variation of the diffraction angle equals twice the misorientation angle. 4. Results 4.1. Effect of asymmetry The test sample consists of a 135 Á thick In O 324 Ga O 676As strained quantum well and a 1240 Á thick InP capping layer on a 2 misoriented (001) InP substrate. Figure 5 shows the variation ofthe substrate diffraction angle ofthe sample aligned with a laser, as a function of rotation angle. From these measurements a misorientation angle IJ of 1.95 (± 0.07 ) in the [110] direction is found. The upper curve of fig. 6 shows the rocking curve near the 004 reflection, taken with the diffraction plane at 90 to the misorientation direction. The lower curve is a simulation. The broad peak to the right and the small broad higher order fringes come from the very thin InGaAs layer. The highfrequency interference fringes and the broadening of the lower part of the substrate peak originate from the InP top layer. From this reflection the structural parameters are obtained that are required for the calculations and simulations. As an example of an asymmetrie reflection we have chosen the 224 reflec- 210 Philips Journal of Research Vol. 47 Nos
9 High-resolution Xi-ray diffraction of epitaxial layers o T (0) Fig. 5. Variation of the substrate diffraction angle (Os) as a function of rotation (r) of a laseraligned sample with a misorientation of 1.95 (±O.07 ). The line is a fit to the data. tion. This reflection can be measured with either a high angle of incidence and a low exit angle (224 h ) or with a low angle of incidence and a high exit angle (224 1 ). In this way the asymmetry ofthe reflection changes sign. The results for the measurements and simulations for the directions perpendicular to the misorientation (B = 0) are shown in fig. 7. It shows not only the expected variation of substrate epitaxial layer peak distance, but also the predicted t I (cps) 1M 100K 10K 1K InP 100 InGaAs CJ (")-- Fig. 6. The diffracted intensity I in counts S-I versus rocking angle ((1)) in are seconds near the 004 reflection ofthe test sample. The upper curve is the measurement, the lower curve is a simulation, displaced downwards for clarity. o Philips Journal of Research Vol.47 Nos
10 P. van der Sluis t I (cps) look 10K lk 100 çp = K 10K lk o CJ(")_ Fig. 7. The diffracted intensity I in counts S-I versus rocking angle w in are seconds near the 224 reflections of the test sample. The two upper curves are for the high angle of incidence and the lower two curves for the low angle of incidence. In both cases the upper curve is the measurement and the lower curve is a simulation, displaced downwards for clarity. narrowing and broadening of the epitaxial layer peak. The results from the calculations with (4) for the directions perpendicular to the misorientation are shown in the first line of Table Il. Measurement and calculation agree within 2%. Considering the fact that this geometrical approach neglects dynamical scattering effects, the correspondence is excellent Effect of misorientation Calculations with (4) show that the effects of misorientation will be most visible for an asymmetrie reflection measured at high angle of incidence. For such a reflection the variation of the peak distance is relatively large, while the angular peak distance itself is short. We therefore show the 224 reflection at the high angle ofincidence. Because the misorientation is in the [IlO] direction we expect to find one 224 reflection with an effectivemisorientation of -1.95, two with an effective misorientation ofo and one with an effectivemisorientation of Actually these four reflections are the 224, 224, 224 and 224. The two extremes are depicted in fig. 8, together with simulations. Calculations are summarized in Table 11. For this high angle of incidence reflection, the 212 Philips Journalof Research Vol. 47 Nos. 3-S 1993
11 High-resolution X-ray diffraction of epitaxial layers TABLE IJ èalculation of the maximum variation of the substrate-iayer peak distance in a rocking curve, when rotated around the surface normal. Reflection Asymmetry Misorientation (deg) (deg) Maximum variation in dm (deg) Measured Monoclinic Layer tilt model model 004 (±) (±) l h (±) l (±) O h (±) t I (cps) look 10K lk 100 ê= K lk o W(")- Fig. 8. The diffracted intensity I in counts S-I versus rocking angle (J) in arc seconds near the 224 reflection of the test sample measured with a high angle of incidence. The higher two curves have an effective misorientation of and the lower two curves with an effective misorientation of In both cases the upper curve is the measurement and the lower curve is a simulation, displaced downwards for clarity. Philips Journalof Research Vol.47 Nos
12 P. van der Sluis effect of the misorientation is indeed very large compared with the peak separation. The effect is so pronounced that even for nominally oriented samples, where the misorientation is generally below 0.25, the effect has to be taken into account. We established that a misorientation and an asymmetry can be treated in a similar way. This means that the effects from misorientation and asymmetry should show up in a similar way in a rocking curve. Comparison of fig. 7 (variation ofthe asymmetry) and fig. 8 (variation ofthe misorientation) makes clear that this is indeed the case Lattice geometry Figure 8 shows that the peak width of the layer peak is smaller when the layer peak is nearer to the substrate peak. This follows from (4) and is also simulated (lower curve). This means that the variation of the epitaxiallayer is not due to a layer tilt. The (044), (044), (044) and (044) lattice planes are rotated 45 around the 001 lattice plane normal, when compared with the set of 224 reflections. We therefore expect two reflections with an effective misorientation ofo.5.j2 times 1.95 and two reflections with an effective misorientation of J2 times We indeed find only two different diffractograms for the four reflections, that can be simulated with the aforementioned misorientations. All measured peak distance variations show the same dependence on the reflection as calculated by our model (Table II). The layer tilt model, however, predicts no dependence on the reflection in disagreement with the measurements. The different variation predicted by the tilt model for the 044 h reflection is because it can only be measured with the diffraction plane rotated 45 away from the misorientation direction. Apparently, for our test sample the epitaxiallayer is not tilted, but the symmetry is monoclinic. 5. Conclusions Rocking curve measurements of a sample with misorientation shows a variation of angular peak distances with rotation around the sample normal. We have shown that lattice deformation, caused by the elastic anisotropy of the materials, is the cause of this variation. Our model corresponds numerically to a well-established empirical model. The effects of misorientation can be calculated quantitatively or simulated by using an effective asymmetry, consisting of the asymmetry of the lattice plane plus a projection of the misorientation onto the diffraction plane. 214 Philips Journalor Research Vol.47 Nos
13 High-resolution X-ray diffraction of epitaxial layers Acknowledgement The author is indebted to P. Thijs for growing the test sample. REFERENCES I) W.J. Bartels and W. Nijman, J. Cryst. Growth, 44,518 (1978). 2) P.F. Fewster, Philips J. Res., 41, 268 (1986). 3) P.F. Fewster and C.J. Curling, J. App!. Phys., 62, 4154 (1987). 4) Y. Kawamura and H. Okamoto, J. Appl. Phys., 50 (6), 4457 (1979). 5) H. Nagai, J. App!. Phys., 45, 3789 (1974). 6) J.E. Ayers, S.K. Ghandhi and L.J. Schowalter, J. Cryst. Growth, 113,430 (1991). 7) J. Hornstra and W.J. Bartels, J. Cryst. Growth, 44,513 (1978). 8) P. Auvray, M. Baudet and A. Regreny, J. Cryst. Growth, 95, 288 (1989). 9) P. Maigne and A.P. Roth, J. Cryst. Growth, 118, 117 (1992). 10) A. Pesek, K. Hingerl, F. Riesz and K. Liscka, Semicond. Sci. Techno!., 6, 705 (1991). ") P. van der Sluis, unpublished (1992). 12) S. Takagi, Acta Crystallogr., 15, 1311 (1962). 13) S. Takagi, J. Phys. Soc. Jpn, 26, 1239 (1969). 14) D. Taupin, Bull. Soc. Franç. Minér. Cryst., 87, 469 (1964). Author P. van der Sluis; Drs. degree (chemistry), University of Utrecht, The Netherlands, 1985; Ph.D., University of Utrecht, The Netherlands, 1989.Post-doctoral appointment, University of Utrecht, The Netherlands, 1989; Philips Research Laboratories, Eindhoven His thesis work and post-doctoral appointment concerned crystal growth and crystallography for single-crystal structure determination. At Phi1ipshis work is in the field ofhigh-resolution X-ray diffraction, mainly on semiconductor materials. Philips Journalof Research Vol.47 Nos
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