- Email: [email protected]
X-ray investigations of III–V compounds: layers, nanostructures, surfaces
Materials Science and Engineering B80 (2001) 81 – 86 www.elsevier.com/locate/mseb
X-ray investigations of III–V compounds: layers, nanostructures, surfaces B. Jenichen * Paul-Drude-Institut fuer Festkoerperforschung, Haus6ogteiplatz 5 – 7, 10117 Berlin, Germany
Abstract Some recent results of X-ray investigations of epitaxial layer systems are reviewed. The interest is directed to correlation phenomena in the distribution of misfit dislocations, to the coexistence of two phases of MnAs in a large range of temperatures, to the stress relaxation in quantum wire structures on InP and to the determination of phases of surface reflections. © 2001 Elsevier Science B.V. All rights reserved. Keywords: X-ray; Misfit dislocations; Phase coexistence; Quantum wires; Surface crystallography
1. Introduction The effort to extend the range of available material parameters (e.g. the tuning of the lasing wavelength of a semiconductor laser) has led to new trends in the field of the epitaxy of III – V-compounds. Attention has moved from unstrained to strained or strain compensated systems — such as InGaAsP on InP or GaInNAs on GaAs — even to systems with very large misfit (AlSb or GaN on GaAs) and to the epitaxy of dissimilar materials (i.e. the crystal structure of the epitaxial layer and the substrate are completely different, such as MnAs on GaAs). There is an increased interest in the in-situ investigation of very thin strained layers (for active layers of laser structures). Furthermore, in order to obtain quantum effects in devices, the dimensions have to be reduced to the nanometer range. This is also a challenge for the characterisation methods. The amount of material available for investigation is reduced by lateral structuring. X-ray reciprocal space mapping [1,2] is well-suited for the measurement of strain in heteroepitaxial structures and can be useful for the determination of composition and stress relaxation phenomena. The combination with other methods — such as X-ray topography [3], secondary ion mass spectometry and photoluminescence measurements — extends the range * Tel.: +49-30-20377324; fax: +49-30-20377201. E-mail address: [email protected] (B. Jenichen).
of possible applications. Due to the application of intense synchrotron sources and/or sophisticated X-ray optics, the X-ray analysis of layer systems down to a few monolayers can be performed [4]. It is also possible to investigate samples in UHV environment and to study the epitaxial growth in-situ [5]. An influence of the surface reconstruction on the ordering inside thin epitaxial layers has recently been measured [6]. The methods of surface diffraction are now applied to the characterisation of thin epitaxial systems suitable for devices [2]. In the present paper, some recent results in the field of X-ray investigations of epitaxial layer systems and surface diffraction are reviewed. First systems with large lattice mismatch and misfit dislocations (AlSb and MnAs on GaAs) are considered, then the strained system InGaAsP/InP with elastic stress relaxation via lateral patterning, and finally, a reconstructed Germanium surface as a model system for the application of a new method to determine the phase of surface reflections.
2. Correlated arrangement of misfit dislocations between AlSb layer and the GaAs substrate — a reduction of strain inhomogeneities The difference between the lattice parameters of an epitaxial layer and the substrate crystal gives rise to elastic strain, which is relaxed by discrete portions, the
0921-5107/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 0 ) 0 0 5 9 4 - 8
82
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
dislocations. The non-uniformity of the strain concentrated at the dislocation lines gives rise to X-ray diffuse scattering. The position of this diffuse peak is governed by the mean distortions. The peak width is a meansquare effect due to non-uniformity of the strain and depends on the Burger’s vectors of the individual dislocations. The peak shapes are close to anisotropic Gaussians extended in the direction perpendicular to the diffraction vector. The peak width for spatially uncorrelated dislocations is proportional to the square root of the dislocation density. Deviations of the observed peak width from the calculated peak width are due to correlations in the dislocation positions [7]. An InAs/ AlSb superlattice with 50 periods of six monolayers of InAs and six monolayers of AlSb (overall thickness of 0.2 mm) on top of a 1-mm-thick relaxed AlSb buffer layer grown on a GaAs substrate was investigated. The mean distance of misfit dislocations between the buffer and the substrate is 2.5 nm. The observed (004) peak of the AlSb buffer layer (Fig. 1) is more than five times narrower than the one calculated under the assumption of uncorrelated dislocations. The conclusion is therefore, that the misfit dislocations are positionally correlated, in agreement with the TEM observations of periodic arrays of misfit dislocations [8]. The non-uniformity of the topmost part of the layer is readily given by the peak half-width of the whole layer [7]. Thus, the
Fig. 1. Comparison between measured and calculated X-ray-diffraction peak profiles (004 reflection) of the fully relaxed AlSb buffer layer (thickness 1 mm) on a GaAs substrate. The measured intensity (full lines) agrees with the calculations for the correlation parameter [7] k=0.03 (broken lines). The calculation of the qx -scan for uncorrelated dislocations (dotted line, k = 1) is shown for comparison.
Fig. 2. Scheme of the epitaxy of MnAs on GaAs (001). Hexagonal (full line) and orthorhombic (broken line) cells are sketched. The epitaxial relationship was established with the c-axis of the hexagonal phase parallel to the GaAs surface.
non-uniformity of the strain near the surface is less than expected for an uncorrelated dislocation distribution. This finding is experimentally proven by the peak width of the average lattice of the thin superlattice on top of the buffer layer.
3. Magnetic MnAs on GaAs — coexistence of two phases in an epitaxial layer As an example for the epitaxy of dissimilar materials, results of diffraction measurements of heteroepitaxial films of MnAs (hexagonal and orthorhombic phase) on GaAs (001) are presented. We found experimental evidence of an equilibrium phase coexistence in heteroepitaxial films over a wide temperature interval. In films of MnAs grown epitaxially on GaAs, we observe the coexistence of two structurally distinct phases, hexagonal a-MnAs and orthorhombic b-MnAs, in a range from the bulk phase transition temperature to at least 20°C below it [9]. For a usual bulk system, the Gibbs phase rule limits the coexistence between phases with the same chemical content to a single temperature. The bulk MnAs crystal experiences, at 40°C, a first-order phase transition from the ferromagnetic hexagonal a-MnAs phase (below the transition temperature) to the paramagnetic orthorhombic phase b-MnAs (above the transition temperature) [10]; the hexagon anisotropically shrinks in both directions (Fig. 2). Epitaxial layers grow with the c-axis of the hexagonal phase parallel to the GaAs surface [11]. Note that the phase coexistence is possible, according to the Gibbs phase rule, only at the transition temperature. The temperature-dependent X-ray diffraction measurements on 250 nm thick MnAs heteroepitaxial layers shown in Fig. 3 clearly reveal the peak of the GaAs substrate and the peaks of the a- and b-MnAs phases. When the temperature is changed, the intensities of the two MnAs film peaks also change. The measurements were performed in two thermal cycles of stepwise cooling and heating between 45 and 27°C. The
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
83
deeper free energy minimum and makes a uniform phase transformation in the whole film unfavorable. Calculations show that the free energy minimum is reached by a phase transformation in part of the crystal [9]. The relative fractions of the two phases depend on temperature and the phase coexistence takes place over a finite temperature range.
4. Compressively strained InGaAsP quantum wires on InP — elastic stress relaxation even in overgrown wires
Fig. 3. Diffraction curve ( –2[-scan) near the GaAs (002) reflection measured at a temperature of 30°C (CuKa1 radiation). The a-MnAs (− 1100) and the b-MnAs (020) reflections are clearly distinguished.
ratio of the integrated intensities does not show any hysteresis and does not change from one cycle to another, which points to an equilibrium coexistence of the two phases. The ratio of the volume fractions of the phases in the film, determined from the integrated intensities of the peaks, depends almost linearly on the temperature, as shown in Fig. 4. Any equilibrium first-order phase transition proceeds as a discontinuous motion of the system from one minimum of its free energy to the other, which becomes deeper at the transition temperature. In the crystalline system under investigation, the shape and size of the unit cell changes at the transition. Since the linear dimensions of the film are fixed by the epitaxy, the strain due to phase transformation cannot be released and gives rise to elastic strain. Near the transition temperature of the free crystal, the elastic energy exceeds the free energy gain due to the transition into the
Fig. 4. Fraction of a-MnAs calculated from the integrated intensities of the observed MnAs reflections.
An improvement in the performance of semiconductor lasers is expected from the introduction of quantum wire structures into the active region [12]. Quantum wire lasers with low threshold and high differential quantum efficiency have been demonstrated [13]. Most of the X-ray investigations on surface gratings including wire structures were concerned with lateral periodicities, which are much larger than the dimensions of the quantization [14]. Smaller periodicities of the quantum wire gratings have been measured by utilizing high-intensity synchrotron sources [15]. An elastic relaxation of the wire structures near the free surface was observed [15,16]. In Ref. [17], we have applied X-ray diffractometry to investigate the elastic stress relaxation in Ga0.22In0.78As0.80P0.20 quantum wire structures (1% compressively strained) with a typical wire width of 35 nm and a thickness of 10 nm. Similar wire structures were also applied for laser device structures. The influence of epitaxial overgrowth on the distribution of the diffracted (or diffusely scattered) X-ray intensities and the shift of the photoluminescence lines was studied. The abrupt termination of the crystal structure at smooth sample surfaces leads to crystal truncation rods in the diffraction pattern. Lateral structures, such as surface reconstructions or surface gratings, cause satellite rods. These additional rods are sensitive to the properties of lateral structures with reduced influence of the bulk signal. Reciprocal space maps near the (224) reflection (grazing exit) of quantum wire structures based on the (In,Ga)(As,P) materials system were recorded. The results [17] were obtained recently at the ESRF (Troika II undulator beamline, wavelength 0.156 nm, Du/u=4× 10 − 5). The wires are sufficiently small to show quantum effects. Their geometry is sketched in Fig. 5(a). Fig. 5(b) shows the reciprocal space map of the freestanding wire structures. The regular pattern of the vertical grating truncation rods originates from the lateral periodicity. The trapezoidal shape of the grating ridges is obvious from the inclined diffuse rods perpendicular to the surface of the sidewalls. Fig. 5(c) demonstrates a map of the same wire structure overgrown at 600°C, where these inclined rods have vanished. Dis-
84
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
wires. We observe a larger photoluminescence line shift between patterned and unpatterned regions for the covered wire structures [sample overgrown at T= 600°C: Du = (− 2793) nm, sample overgrown at T= 650°C: Du = (− 429 3) nm] than for the free standing wires [Du = (− 691) nm]. This large difference cannot be explained by average strain, it is probably still caused by the unintentional misfit of the cover layer [17] and inhomogeneities in the strain field which influence the confinement of the carriers. The measurements at the ESRF showed, that for the study of quantum wire arrays, with wire width of 35 nm and thickness of 10 nm, the high intensity source is really inevitable. An improvement of the signal-to-noise ratio is expected from stacking several wire structures in the vertical direction.
5. The determination of the phase of surface reflections — a support for direct methods in surface structure determination
Fig. 5. Schematic view of the quantum wire geometry (a). Reciprocal space maps near the asymmetric (224) InP reflection (grazing exit) taken at a wavelength of 0.155 nm of the samples with free standing wires (b) and buried (c), with nominally 100 nm lattice matched InGaAsP at a growth temperature of 600°C.
tinct wire regions are visible in Fig. 5(b,c) near Qz 4.22 A, − 1 and Qx 3.02 A, − 1. These regions show little changes due to the overgrowth. The nominally 100-nm thick cover layer above the wire gratings still shows some lattice mismatch with respect to the InP substrate. We speculate that this mismatch is responsible for the non-vanishing relaxation of the quantum
The phase problem is the fundamental problem of X-ray crystallography; a diffraction experiment commonly provides, from the measured intensities, the amplitudes Fh of the structure factors Fh = Fh exp(ihh), while the phases hh remain unknown. Surface crystallography encounters the same problem. The vast majority of surface structures are solved by the classical methods of crystallography [4], involving the construction of structure models on the basis of Patterson maps and subsequent refinement using structure factor amplitudes. Modern developments apply ‘direct methods’ which derive phases from amplitudes using plausible assumptions [19]. Knowing the phases of some strong reflections enhances the applicability of direct methods, which are more powerful in extending phase information from a limited set of reflections to additional shells of reflections. The direct determination of the structure factor phases in a diffraction experiment has to involve the interference between different beams. Surface diffraction has an advantage due to the presence of the bulk crystal underneath, which can be used to produce a reference beam. We excite a strong bulk reflection while measuring a surface reflection far from it. We choose diffraction conditions so that a surface reflection h and a bulk reflection G are simultaneously excited. The bulk-diffracted wave, whose amplitude is comparable with the amplitude of the incident wave, excites another surface reflection h%= h–G. The diffraction signal is due to the interference between the incident wave scattered on h and the bulk-diffracted wave scattered on h%. The conditions for interference can be varied by the angular deviation of the incident wave from the Bragg position for the reflection G.
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
The new method is somewhat similar to standing wave techniques, in the sense that diffraction from the bulk crystal is used to produce the second coherent wave for illumination of the surface atoms. In this method [18], the bulk-diffracted wave experiences, in turn, further diffraction by the surface structure. The multibeam-diffraction requirement that all relevant reciprocal lattice points be on the Ewald sphere is replaced by a much softer condition: the Ewald sphere should contain the points 0 and G and intersect the Bragg rod of the surface reflection h, as shown in Fig. 6. Therefore, a precise adjustment of the wavelength is not needed. An experimental observation of the interference signal has also been presented in Ref. [18]. The measurements were performed at the W1 wiggler beamline of HASYLAB (DESY, Hamburg), on a Ge(113)(3× 1) reconstructed surface. We used a strong bulk reflection parallel to the surface, G =[6 0 0]surf =[2 −2 0]cubic and the directly excited surface reflection h =[11 5 L]; hence, the reflection h% =[5 5 L] is the Umweganregung excitation. The intensity of the bulk reflection was measured first. After that, the detector was moved to the position corresponding the surface reflection h and the intensity was measured in the same interval of the sample rotation angles for several values of L (see symbols in Fig. 7). The solid lines in Fig. 7 are calculated taking into account the interference of both surface reflections [18]. The structure factors were taken from the known surface structure [20]. The surface peaks clearly reveal dips, whose position and width do not depend on L and coincide with those of the bulk diffraction peak. The good agreement between the measurements and the calculations demonstrate the viability of Umweganregung surface diffraction to determine structure factor phases.
85
Fig. 7. Intensities of the G=[6 0 0]surf =[2 −20]bulk bulk reflection and [11, 5, L] surface reflections of the Ge(113) (3 ×1) reconstructed surface measured at the interference conditions with u =0.1353 nm as a function of the deviation =[ – [B from the Bragg angle of the bulk reflection [B.
6. Summary New approaches to current problems in X-ray analysis of epitaxial growth are reviewed: the diffuse scattering by misfit dislocations located near the interface, analysis of stress relaxation in small period quantum gratings prepared from epitaxial structures and the strain mediated phase coexistence in epitaxial layers. The method of phase determination of surface reflections will support the in-situ analysis of epitaxial growth in the future.
Acknowledgements Fig. 6. Three-beam Bragg diffraction involving bulk and surface reflections. The Ewald sphere of radius k= 2y/u passes through the origin and the point G of reciprocal space and crosses the Bragg rod of the surface reflection h. The diffracted wave at h is produced by direct scattering of the incident wave and, in addition, by scattering of the bulk-diffracted wave on the surface reflection h%= h − G.
The author thanks V. Kaganer, D. Luebbert, M. Albrecht, W. Moritz, W. Braun, F. Schippan, G. Paris, K. Kojima, P. Mikulik, B. Brar and J. Spitzer for excellent cooperation, L. Da¨weritz, H.T. Grahn, K.H. Ploog, S. Arai, H. Kroemer and T. Baumbach for their
86
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
encouragement and support of the works and the critical reading of the manuscript. [12]
References
[13]
[1] A. Iida, K. Kohra, Phys. Stat. Sol. A51 (1979) 533; P.F. Fewster, Crit. Rev. Sol. State Mater. Sci. 22 (2) (1997) 69. [2] V. Holy, U. Pietsch, T. Baumbach, High-resolution X-ray scattering from thin films and multilayers, in: G. Ho¨hler (Ed.), Tracts in Modern Physics, vol. 149, Springer, Berlin, 1999. [3] B. Jenichen, R. Hey, S. Westphal, V. Kaganer, R. Ko¨hler, Mater. Sci. Eng. B28 (1994) 520. [4] R. Feidenhans%l, Surf. Sci. Rep. 10 (1989) 105; I.K. Robinson, D.J. Tweet, Rep. Prog. Phys. 55 (1992) 599. [5] E. Vlieg, A.W. Denier van der Gon, J.F. van der Veen, J.E. MacDonald, C. Norris, Phys. Rev. Lett. 61 (1988) 2241; P.H. Fuoss, D.W. Kisker, F.J. Lamelas, G.B. Stephenson, P. Imperatori, S. Brennan, Phys. Rev. Lett. 69 (1992) 2791. [6] K.L. Whiteaker, I.K. Robinson, J.E. Van Nostrand, D.G. Cahill, Phys. Rev. B57 (1998) 12410. [7] V.M. Kaganer, R. Ko¨hler, M. Schmidbauer, R. Opitz, B. Jenichen, Phys. Rev. B55 (1997) 1793. [8] J. Spitzer, A. Ho¨pner, M. Kuball, M. Cardona, B. Jenichen, H. Neuroth, B. Brar, H. Kroemer, J. Appl. Phys. 77 (1995) 811. [9] V.M. Kaganer, B. Jenichen, F. Schippan, W. Braun, L. Da¨weritz, K.H. Ploog, Phys. Rev. Lett. 85 (2000) 341. [10] R.H. Wilson, J.S. Kasper, Acta Cryst. 17 (1964) 95; B.T.M. Willis, H.P. Rooksby, Proc. Phys. Soc. (Lond.) B67 (1954) 290. [11] M. Tanaka, J.P. Harbison, T. Sands, T.L. Cheeks, V.G. Keramidas, G.M. Rothberg, Vac. Sci. Technol. J. B12 (1994) 1091; M. Tanaka, J.P. Harbison, M.C. Park, Y.S. Park, T. Shin, G.M.
.
[14]
[15]
[16] [17]
[18]
[19]
[20]
Rothberg, J. Appl. Phys. 76 (1994) 6278; M. Tanaka, Physica E2 (1998) 372; F. Schippan, A. Trampert, L. Daeweritz, K.H. Ploog, J. Vac. Sci. Technol. B17 (1999) 1716. Y. Arakawa, H. Sakaki, Appl. Phys. Lett. 40 (1982) 939; Y. Arakawa, A. Yariv, IEEE J. Quant. Electron. QE-22 (1986) 1887; Y. Myake, M. Asada, Jpn. J. Appl. Phys. 28 (1989) 1280. T. Kojima, M. Tamura, H. Nakaya, S. Tanaka, S. Tamura, S. Arai, Jpn. J. Appl. Phys. 37 (1998) 261. A.T. Macrander, S.E.G. Slusky, Appl. Phys. Lett. 56 (1990) 443; L. Tapfer, P. Grambow, Appl. Phys. A50 (1990) 3; A.A. Darhuber, E. Koppensteiner, H. Straub, G. Brunthaler, W. Faschinger, G. Bauer, J. Appl. Phys. 76 (1994) 7816; G.T. Baumbach, M. Gailhanou, J. Phys. D 28 (1995) 2321; L. Leprince, G.T. Baumbach, A. Talneu, M. Gailhanou, J. Schneck, Appl. Phys. Lett. 71 (1997) 3227; A. Ulyanenkov, N. Darowski, J. Grenzer, U. Pietsch, K.H. Wang, A. Forchel, Phys. Rev. B60 (1999) 16701. Q. Shen, S. Kycia, E.S. Tentarelli, W.J. Schaff, L.F. Eastman, Phys. Rev. B54 (1996) 16381; Q. Shen, S. Kycia, Phys. Rev. B55 (1997) 15791. B. Jenichen, H.T. Grahn, T. Kojima, S. Arai, J. Appl. Phys. 83 (1998) 5810. D. Lu¨bbert, B. Jenichen, T. Baumbach, H.T. Grahn, G. Paris, A. Mazuelas, T. Kojima, S. Arai, J. Phys. D.: Appl. Phys. 32 (1999) A21. V.M. Kaganer, M. Albrecht, A. Hirnet, M. Gierer, W. Moritz, B. Jenichen, K.H. Ploog, Phys. Rev. B, Phys. Rev. B61 (2000) 16355. L.D. Marks, E. Bengu, C. Collazo-Davila, D. Goneza, E. Landree, C. Leslie, W. Simkler, Surf. Rev. Letts. 5 (1998) 1087; L.D. Marks, E. Landree, Acta Cryst. A54 (1998) 296; L.D. Marks, Phys. Rev. B59 (1999) 11109. H. Vogler, A. Iglesias, W. Moritz, H. Over, Phys. Rev. B 57 (1998) 2315.
X-ray investigations of III–V compounds: layers, nanostructures, surfaces B. Jenichen * Paul-Drude-Institut fuer Festkoerperforschung, Haus6ogteiplatz 5 – 7, 10117 Berlin, Germany
Abstract Some recent results of X-ray investigations of epitaxial layer systems are reviewed. The interest is directed to correlation phenomena in the distribution of misfit dislocations, to the coexistence of two phases of MnAs in a large range of temperatures, to the stress relaxation in quantum wire structures on InP and to the determination of phases of surface reflections. © 2001 Elsevier Science B.V. All rights reserved. Keywords: X-ray; Misfit dislocations; Phase coexistence; Quantum wires; Surface crystallography
1. Introduction The effort to extend the range of available material parameters (e.g. the tuning of the lasing wavelength of a semiconductor laser) has led to new trends in the field of the epitaxy of III – V-compounds. Attention has moved from unstrained to strained or strain compensated systems — such as InGaAsP on InP or GaInNAs on GaAs — even to systems with very large misfit (AlSb or GaN on GaAs) and to the epitaxy of dissimilar materials (i.e. the crystal structure of the epitaxial layer and the substrate are completely different, such as MnAs on GaAs). There is an increased interest in the in-situ investigation of very thin strained layers (for active layers of laser structures). Furthermore, in order to obtain quantum effects in devices, the dimensions have to be reduced to the nanometer range. This is also a challenge for the characterisation methods. The amount of material available for investigation is reduced by lateral structuring. X-ray reciprocal space mapping [1,2] is well-suited for the measurement of strain in heteroepitaxial structures and can be useful for the determination of composition and stress relaxation phenomena. The combination with other methods — such as X-ray topography [3], secondary ion mass spectometry and photoluminescence measurements — extends the range * Tel.: +49-30-20377324; fax: +49-30-20377201. E-mail address: [email protected] (B. Jenichen).
of possible applications. Due to the application of intense synchrotron sources and/or sophisticated X-ray optics, the X-ray analysis of layer systems down to a few monolayers can be performed [4]. It is also possible to investigate samples in UHV environment and to study the epitaxial growth in-situ [5]. An influence of the surface reconstruction on the ordering inside thin epitaxial layers has recently been measured [6]. The methods of surface diffraction are now applied to the characterisation of thin epitaxial systems suitable for devices [2]. In the present paper, some recent results in the field of X-ray investigations of epitaxial layer systems and surface diffraction are reviewed. First systems with large lattice mismatch and misfit dislocations (AlSb and MnAs on GaAs) are considered, then the strained system InGaAsP/InP with elastic stress relaxation via lateral patterning, and finally, a reconstructed Germanium surface as a model system for the application of a new method to determine the phase of surface reflections.
2. Correlated arrangement of misfit dislocations between AlSb layer and the GaAs substrate — a reduction of strain inhomogeneities The difference between the lattice parameters of an epitaxial layer and the substrate crystal gives rise to elastic strain, which is relaxed by discrete portions, the
0921-5107/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 0 ) 0 0 5 9 4 - 8
82
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
dislocations. The non-uniformity of the strain concentrated at the dislocation lines gives rise to X-ray diffuse scattering. The position of this diffuse peak is governed by the mean distortions. The peak width is a meansquare effect due to non-uniformity of the strain and depends on the Burger’s vectors of the individual dislocations. The peak shapes are close to anisotropic Gaussians extended in the direction perpendicular to the diffraction vector. The peak width for spatially uncorrelated dislocations is proportional to the square root of the dislocation density. Deviations of the observed peak width from the calculated peak width are due to correlations in the dislocation positions [7]. An InAs/ AlSb superlattice with 50 periods of six monolayers of InAs and six monolayers of AlSb (overall thickness of 0.2 mm) on top of a 1-mm-thick relaxed AlSb buffer layer grown on a GaAs substrate was investigated. The mean distance of misfit dislocations between the buffer and the substrate is 2.5 nm. The observed (004) peak of the AlSb buffer layer (Fig. 1) is more than five times narrower than the one calculated under the assumption of uncorrelated dislocations. The conclusion is therefore, that the misfit dislocations are positionally correlated, in agreement with the TEM observations of periodic arrays of misfit dislocations [8]. The non-uniformity of the topmost part of the layer is readily given by the peak half-width of the whole layer [7]. Thus, the
Fig. 1. Comparison between measured and calculated X-ray-diffraction peak profiles (004 reflection) of the fully relaxed AlSb buffer layer (thickness 1 mm) on a GaAs substrate. The measured intensity (full lines) agrees with the calculations for the correlation parameter [7] k=0.03 (broken lines). The calculation of the qx -scan for uncorrelated dislocations (dotted line, k = 1) is shown for comparison.
Fig. 2. Scheme of the epitaxy of MnAs on GaAs (001). Hexagonal (full line) and orthorhombic (broken line) cells are sketched. The epitaxial relationship was established with the c-axis of the hexagonal phase parallel to the GaAs surface.
non-uniformity of the strain near the surface is less than expected for an uncorrelated dislocation distribution. This finding is experimentally proven by the peak width of the average lattice of the thin superlattice on top of the buffer layer.
3. Magnetic MnAs on GaAs — coexistence of two phases in an epitaxial layer As an example for the epitaxy of dissimilar materials, results of diffraction measurements of heteroepitaxial films of MnAs (hexagonal and orthorhombic phase) on GaAs (001) are presented. We found experimental evidence of an equilibrium phase coexistence in heteroepitaxial films over a wide temperature interval. In films of MnAs grown epitaxially on GaAs, we observe the coexistence of two structurally distinct phases, hexagonal a-MnAs and orthorhombic b-MnAs, in a range from the bulk phase transition temperature to at least 20°C below it [9]. For a usual bulk system, the Gibbs phase rule limits the coexistence between phases with the same chemical content to a single temperature. The bulk MnAs crystal experiences, at 40°C, a first-order phase transition from the ferromagnetic hexagonal a-MnAs phase (below the transition temperature) to the paramagnetic orthorhombic phase b-MnAs (above the transition temperature) [10]; the hexagon anisotropically shrinks in both directions (Fig. 2). Epitaxial layers grow with the c-axis of the hexagonal phase parallel to the GaAs surface [11]. Note that the phase coexistence is possible, according to the Gibbs phase rule, only at the transition temperature. The temperature-dependent X-ray diffraction measurements on 250 nm thick MnAs heteroepitaxial layers shown in Fig. 3 clearly reveal the peak of the GaAs substrate and the peaks of the a- and b-MnAs phases. When the temperature is changed, the intensities of the two MnAs film peaks also change. The measurements were performed in two thermal cycles of stepwise cooling and heating between 45 and 27°C. The
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
83
deeper free energy minimum and makes a uniform phase transformation in the whole film unfavorable. Calculations show that the free energy minimum is reached by a phase transformation in part of the crystal [9]. The relative fractions of the two phases depend on temperature and the phase coexistence takes place over a finite temperature range.
4. Compressively strained InGaAsP quantum wires on InP — elastic stress relaxation even in overgrown wires
Fig. 3. Diffraction curve ( –2[-scan) near the GaAs (002) reflection measured at a temperature of 30°C (CuKa1 radiation). The a-MnAs (− 1100) and the b-MnAs (020) reflections are clearly distinguished.
ratio of the integrated intensities does not show any hysteresis and does not change from one cycle to another, which points to an equilibrium coexistence of the two phases. The ratio of the volume fractions of the phases in the film, determined from the integrated intensities of the peaks, depends almost linearly on the temperature, as shown in Fig. 4. Any equilibrium first-order phase transition proceeds as a discontinuous motion of the system from one minimum of its free energy to the other, which becomes deeper at the transition temperature. In the crystalline system under investigation, the shape and size of the unit cell changes at the transition. Since the linear dimensions of the film are fixed by the epitaxy, the strain due to phase transformation cannot be released and gives rise to elastic strain. Near the transition temperature of the free crystal, the elastic energy exceeds the free energy gain due to the transition into the
Fig. 4. Fraction of a-MnAs calculated from the integrated intensities of the observed MnAs reflections.
An improvement in the performance of semiconductor lasers is expected from the introduction of quantum wire structures into the active region [12]. Quantum wire lasers with low threshold and high differential quantum efficiency have been demonstrated [13]. Most of the X-ray investigations on surface gratings including wire structures were concerned with lateral periodicities, which are much larger than the dimensions of the quantization [14]. Smaller periodicities of the quantum wire gratings have been measured by utilizing high-intensity synchrotron sources [15]. An elastic relaxation of the wire structures near the free surface was observed [15,16]. In Ref. [17], we have applied X-ray diffractometry to investigate the elastic stress relaxation in Ga0.22In0.78As0.80P0.20 quantum wire structures (1% compressively strained) with a typical wire width of 35 nm and a thickness of 10 nm. Similar wire structures were also applied for laser device structures. The influence of epitaxial overgrowth on the distribution of the diffracted (or diffusely scattered) X-ray intensities and the shift of the photoluminescence lines was studied. The abrupt termination of the crystal structure at smooth sample surfaces leads to crystal truncation rods in the diffraction pattern. Lateral structures, such as surface reconstructions or surface gratings, cause satellite rods. These additional rods are sensitive to the properties of lateral structures with reduced influence of the bulk signal. Reciprocal space maps near the (224) reflection (grazing exit) of quantum wire structures based on the (In,Ga)(As,P) materials system were recorded. The results [17] were obtained recently at the ESRF (Troika II undulator beamline, wavelength 0.156 nm, Du/u=4× 10 − 5). The wires are sufficiently small to show quantum effects. Their geometry is sketched in Fig. 5(a). Fig. 5(b) shows the reciprocal space map of the freestanding wire structures. The regular pattern of the vertical grating truncation rods originates from the lateral periodicity. The trapezoidal shape of the grating ridges is obvious from the inclined diffuse rods perpendicular to the surface of the sidewalls. Fig. 5(c) demonstrates a map of the same wire structure overgrown at 600°C, where these inclined rods have vanished. Dis-
84
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
wires. We observe a larger photoluminescence line shift between patterned and unpatterned regions for the covered wire structures [sample overgrown at T= 600°C: Du = (− 2793) nm, sample overgrown at T= 650°C: Du = (− 429 3) nm] than for the free standing wires [Du = (− 691) nm]. This large difference cannot be explained by average strain, it is probably still caused by the unintentional misfit of the cover layer [17] and inhomogeneities in the strain field which influence the confinement of the carriers. The measurements at the ESRF showed, that for the study of quantum wire arrays, with wire width of 35 nm and thickness of 10 nm, the high intensity source is really inevitable. An improvement of the signal-to-noise ratio is expected from stacking several wire structures in the vertical direction.
5. The determination of the phase of surface reflections — a support for direct methods in surface structure determination
Fig. 5. Schematic view of the quantum wire geometry (a). Reciprocal space maps near the asymmetric (224) InP reflection (grazing exit) taken at a wavelength of 0.155 nm of the samples with free standing wires (b) and buried (c), with nominally 100 nm lattice matched InGaAsP at a growth temperature of 600°C.
tinct wire regions are visible in Fig. 5(b,c) near Qz 4.22 A, − 1 and Qx 3.02 A, − 1. These regions show little changes due to the overgrowth. The nominally 100-nm thick cover layer above the wire gratings still shows some lattice mismatch with respect to the InP substrate. We speculate that this mismatch is responsible for the non-vanishing relaxation of the quantum
The phase problem is the fundamental problem of X-ray crystallography; a diffraction experiment commonly provides, from the measured intensities, the amplitudes Fh of the structure factors Fh = Fh exp(ihh), while the phases hh remain unknown. Surface crystallography encounters the same problem. The vast majority of surface structures are solved by the classical methods of crystallography [4], involving the construction of structure models on the basis of Patterson maps and subsequent refinement using structure factor amplitudes. Modern developments apply ‘direct methods’ which derive phases from amplitudes using plausible assumptions [19]. Knowing the phases of some strong reflections enhances the applicability of direct methods, which are more powerful in extending phase information from a limited set of reflections to additional shells of reflections. The direct determination of the structure factor phases in a diffraction experiment has to involve the interference between different beams. Surface diffraction has an advantage due to the presence of the bulk crystal underneath, which can be used to produce a reference beam. We excite a strong bulk reflection while measuring a surface reflection far from it. We choose diffraction conditions so that a surface reflection h and a bulk reflection G are simultaneously excited. The bulk-diffracted wave, whose amplitude is comparable with the amplitude of the incident wave, excites another surface reflection h%= h–G. The diffraction signal is due to the interference between the incident wave scattered on h and the bulk-diffracted wave scattered on h%. The conditions for interference can be varied by the angular deviation of the incident wave from the Bragg position for the reflection G.
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
The new method is somewhat similar to standing wave techniques, in the sense that diffraction from the bulk crystal is used to produce the second coherent wave for illumination of the surface atoms. In this method [18], the bulk-diffracted wave experiences, in turn, further diffraction by the surface structure. The multibeam-diffraction requirement that all relevant reciprocal lattice points be on the Ewald sphere is replaced by a much softer condition: the Ewald sphere should contain the points 0 and G and intersect the Bragg rod of the surface reflection h, as shown in Fig. 6. Therefore, a precise adjustment of the wavelength is not needed. An experimental observation of the interference signal has also been presented in Ref. [18]. The measurements were performed at the W1 wiggler beamline of HASYLAB (DESY, Hamburg), on a Ge(113)(3× 1) reconstructed surface. We used a strong bulk reflection parallel to the surface, G =[6 0 0]surf =[2 −2 0]cubic and the directly excited surface reflection h =[11 5 L]; hence, the reflection h% =[5 5 L] is the Umweganregung excitation. The intensity of the bulk reflection was measured first. After that, the detector was moved to the position corresponding the surface reflection h and the intensity was measured in the same interval of the sample rotation angles for several values of L (see symbols in Fig. 7). The solid lines in Fig. 7 are calculated taking into account the interference of both surface reflections [18]. The structure factors were taken from the known surface structure [20]. The surface peaks clearly reveal dips, whose position and width do not depend on L and coincide with those of the bulk diffraction peak. The good agreement between the measurements and the calculations demonstrate the viability of Umweganregung surface diffraction to determine structure factor phases.
85
Fig. 7. Intensities of the G=[6 0 0]surf =[2 −20]bulk bulk reflection and [11, 5, L] surface reflections of the Ge(113) (3 ×1) reconstructed surface measured at the interference conditions with u =0.1353 nm as a function of the deviation =[ – [B from the Bragg angle of the bulk reflection [B.
6. Summary New approaches to current problems in X-ray analysis of epitaxial growth are reviewed: the diffuse scattering by misfit dislocations located near the interface, analysis of stress relaxation in small period quantum gratings prepared from epitaxial structures and the strain mediated phase coexistence in epitaxial layers. The method of phase determination of surface reflections will support the in-situ analysis of epitaxial growth in the future.
Acknowledgements Fig. 6. Three-beam Bragg diffraction involving bulk and surface reflections. The Ewald sphere of radius k= 2y/u passes through the origin and the point G of reciprocal space and crosses the Bragg rod of the surface reflection h. The diffracted wave at h is produced by direct scattering of the incident wave and, in addition, by scattering of the bulk-diffracted wave on the surface reflection h%= h − G.
The author thanks V. Kaganer, D. Luebbert, M. Albrecht, W. Moritz, W. Braun, F. Schippan, G. Paris, K. Kojima, P. Mikulik, B. Brar and J. Spitzer for excellent cooperation, L. Da¨weritz, H.T. Grahn, K.H. Ploog, S. Arai, H. Kroemer and T. Baumbach for their
86
B. Jenichen / Materials Science and Engineering B80 (2001) 81–86
encouragement and support of the works and the critical reading of the manuscript. [12]
References
[13]
[1] A. Iida, K. Kohra, Phys. Stat. Sol. A51 (1979) 533; P.F. Fewster, Crit. Rev. Sol. State Mater. Sci. 22 (2) (1997) 69. [2] V. Holy, U. Pietsch, T. Baumbach, High-resolution X-ray scattering from thin films and multilayers, in: G. Ho¨hler (Ed.), Tracts in Modern Physics, vol. 149, Springer, Berlin, 1999. [3] B. Jenichen, R. Hey, S. Westphal, V. Kaganer, R. Ko¨hler, Mater. Sci. Eng. B28 (1994) 520. [4] R. Feidenhans%l, Surf. Sci. Rep. 10 (1989) 105; I.K. Robinson, D.J. Tweet, Rep. Prog. Phys. 55 (1992) 599. [5] E. Vlieg, A.W. Denier van der Gon, J.F. van der Veen, J.E. MacDonald, C. Norris, Phys. Rev. Lett. 61 (1988) 2241; P.H. Fuoss, D.W. Kisker, F.J. Lamelas, G.B. Stephenson, P. Imperatori, S. Brennan, Phys. Rev. Lett. 69 (1992) 2791. [6] K.L. Whiteaker, I.K. Robinson, J.E. Van Nostrand, D.G. Cahill, Phys. Rev. B57 (1998) 12410. [7] V.M. Kaganer, R. Ko¨hler, M. Schmidbauer, R. Opitz, B. Jenichen, Phys. Rev. B55 (1997) 1793. [8] J. Spitzer, A. Ho¨pner, M. Kuball, M. Cardona, B. Jenichen, H. Neuroth, B. Brar, H. Kroemer, J. Appl. Phys. 77 (1995) 811. [9] V.M. Kaganer, B. Jenichen, F. Schippan, W. Braun, L. Da¨weritz, K.H. Ploog, Phys. Rev. Lett. 85 (2000) 341. [10] R.H. Wilson, J.S. Kasper, Acta Cryst. 17 (1964) 95; B.T.M. Willis, H.P. Rooksby, Proc. Phys. Soc. (Lond.) B67 (1954) 290. [11] M. Tanaka, J.P. Harbison, T. Sands, T.L. Cheeks, V.G. Keramidas, G.M. Rothberg, Vac. Sci. Technol. J. B12 (1994) 1091; M. Tanaka, J.P. Harbison, M.C. Park, Y.S. Park, T. Shin, G.M.
.
[14]
[15]
[16] [17]
[18]
[19]
[20]
Rothberg, J. Appl. Phys. 76 (1994) 6278; M. Tanaka, Physica E2 (1998) 372; F. Schippan, A. Trampert, L. Daeweritz, K.H. Ploog, J. Vac. Sci. Technol. B17 (1999) 1716. Y. Arakawa, H. Sakaki, Appl. Phys. Lett. 40 (1982) 939; Y. Arakawa, A. Yariv, IEEE J. Quant. Electron. QE-22 (1986) 1887; Y. Myake, M. Asada, Jpn. J. Appl. Phys. 28 (1989) 1280. T. Kojima, M. Tamura, H. Nakaya, S. Tanaka, S. Tamura, S. Arai, Jpn. J. Appl. Phys. 37 (1998) 261. A.T. Macrander, S.E.G. Slusky, Appl. Phys. Lett. 56 (1990) 443; L. Tapfer, P. Grambow, Appl. Phys. A50 (1990) 3; A.A. Darhuber, E. Koppensteiner, H. Straub, G. Brunthaler, W. Faschinger, G. Bauer, J. Appl. Phys. 76 (1994) 7816; G.T. Baumbach, M. Gailhanou, J. Phys. D 28 (1995) 2321; L. Leprince, G.T. Baumbach, A. Talneu, M. Gailhanou, J. Schneck, Appl. Phys. Lett. 71 (1997) 3227; A. Ulyanenkov, N. Darowski, J. Grenzer, U. Pietsch, K.H. Wang, A. Forchel, Phys. Rev. B60 (1999) 16701. Q. Shen, S. Kycia, E.S. Tentarelli, W.J. Schaff, L.F. Eastman, Phys. Rev. B54 (1996) 16381; Q. Shen, S. Kycia, Phys. Rev. B55 (1997) 15791. B. Jenichen, H.T. Grahn, T. Kojima, S. Arai, J. Appl. Phys. 83 (1998) 5810. D. Lu¨bbert, B. Jenichen, T. Baumbach, H.T. Grahn, G. Paris, A. Mazuelas, T. Kojima, S. Arai, J. Phys. D.: Appl. Phys. 32 (1999) A21. V.M. Kaganer, M. Albrecht, A. Hirnet, M. Gierer, W. Moritz, B. Jenichen, K.H. Ploog, Phys. Rev. B, Phys. Rev. B61 (2000) 16355. L.D. Marks, E. Bengu, C. Collazo-Davila, D. Goneza, E. Landree, C. Leslie, W. Simkler, Surf. Rev. Letts. 5 (1998) 1087; L.D. Marks, E. Landree, Acta Cryst. A54 (1998) 296; L.D. Marks, Phys. Rev. B59 (1999) 11109. H. Vogler, A. Iglesias, W. Moritz, H. Over, Phys. Rev. B 57 (1998) 2315.