Interplay of surface morphology, strain relief, and ... - Springer Link

Appl. Phys. A 69, 481–488 (1999) / Digital Object Identifier (DOI) 10.1007/s003399900170

Applied Physics A Materials Science & Processing  Springer-Verlag 1999

Interplay of surface morphology, strain relief, and surface stress during surfactant mediated epitaxy of Ge on Si P. Zahl, P. Kury, M. Horn von Hoegen∗ Institut für Festkörperphysik, Universität Hannover, Appelstrasse 2, D–30167 Hannover, Germany Received: 1 April 1999/Accepted: 17 August 1999/Published online: 6 October 1999

Abstract. Lattice-mismatch-induced surface or film stress has significant influence on the morphology of heteroepitaxial films. This is demonstrated using Sb surfactant-mediated epitaxy of Ge on Si(111). The surfactant forces a twodimensional growth of a continous Ge film instead of islanding. Two qualitatively different growth regimes are observed. Elastic relaxation: Prior to the generation of strain-relieving defects the Ge film grows pseudomorphically with the Si lattice constant and is under strong compressive stress. The Ge film relieves strain by forming a rough surface on a nm scale which allows partial elastic relaxation towards the Ge bulk lattice constant. The unfavorable increase of surface area is outbalanced by the large decrease of strain energy. The change of film stress and surface morphology is monitored in situ during deposition at elevated temperature with surface stress-induced optical deflection and high-resolution spot profile analysis low-energy electron diffraction. Plastic relaxation: After a critical thickness the generation of dislocations is initiated. The rough phase acts as a nucleation center for dislocations. On Si(111) those misfit dislocations are arranged in a threefold quasi periodic array at the interface that accommodate exactly the different lattice constants of Ge and Si. PACS: 68.55.-a; 68.35.-p; 61.14.Hg Nowadays a major portion of modern electronic devices are based on semiconductor heterostructures. The band gap and band discontinuity over the heterojunction can be adjusted by the material composition of the heterostructure. However, such a choice of different materials is usually accompanied by a mismatch of the lattice constants which may be utilized to tune the electronic properties: highest mobilities for two dimensional electron gases have been established in strained heterostructure devices (MODFET, HEMT [1]). The fabrication of such structures, however, is in general governed by the difference in lattice constants ∆a = aS − aF and surface free energies σS , σF of the constituents. The latter ∗ Present

Germany

address: Institut für Laser- und Plasmaphysik, Universität Essen,

determines the possibility of wetting a substrate of element S with a hetero-epitaxial film of element F. The inequality σS > σi + σF ,

(1)

with the free energy of the substrate σS , the interface free energy σi , and the surface free energy of the heteroepitaxial film σF , sets the condition for the epitaxial film F to wet the substrate S. In this case layer-by-layer growth (Frank–Van der Merwe) may occur, changing into the Stranski–Krastanov mode beyond a critical coverage if the overlayer strain is increasing with thickness [2]. If the inequality has the opposite sign, usually Volmer–Weber growth occurs: immediate islanding of the overlayer and formation of three dimensional (3D-) clusters is observed. For the Si/Ge system, Si has the higher surface free energy. Therefore Si does not wet a Ge substrate and the Si film immediately starts to island [3] as sketched in Fig. 1. Consequently Ge is able to wet a Si surface, however, after formation of a wetting layer, islanding of the Ge film occurs [3] due to the accumulation of lattice-mismatch-induced strain. During the coherent stage of growth (prior to the generation of dislocations) the surface free energy σF of the strained film increases as function of coverage ΘF : σF (ΘF ) = σF,0 + const ΘF

∆a . aS

(2)

σF,0 is the surface free energy of the surface from the strained film F and the second term is the strain energy stored in a homogeneously compressed and flat hetero-film of coverage ΘF . The inequality (1) describes the growth mode under equilibrium conditions. Utilising kinetic limitations allows one to deposit a continuous film F on a substrate S, even if this is thermodynamically unfavorable. However, growth at low temperature and/or high deposition rates usually results in poor crystal quality and rough surfaces [4, 5]. A way out of this dilemma is surfactant-mediated epitaxy: a substantial modification of the growth mode is obtained by the introduction of a third element as surfactant (surface-active-species) [6]. The surfactant allows wetting of both species and prohibits formation of 3D clusters. Smooth

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Fig. 1a–c. The growth mode under thermal equilibrium is determined by the difference in surface free energy and lattice mismatch. a Due to its higher surface free energy, Si does not wet a Ge substrate and immediately forms 3D clusters. b Ge with its lower surface free energy wets a Si substrate. A wetting layer of 3–4 ML thickness forms before islanding. The accumulation of strain in the pseudomorphic wetting layer leads to island formation. The Ge in the 3D clusters is partially relaxed towards the bulk lattice constant. c Termination of the semiconductor surface by a surfactant results in layer-by-layer growth of both Ge on Si and Si on Ge. Islanding is prohibited, smooth and continuous films can be grown. The surfactant is floating on the growth front without significant incorporation

and continuous films of element F could be grown on a substrate S and vice versa. The surfactant (usually a group-V element for group-IV semiconductors) saturates the dangling bonds of the semiconductor surface which results in a strong decrease of the surface free energy [7]. With the surface properties now mainly terminated by the surfactant layer, the difference in surface free energy of the Ge and Si is reduced, which lowers the tendency towards islanding. This reduction of surface free energy is the driving force for the strong segregation of the surfactant, which floats on the growth front without significant incorporation [8–10]. The ideal surfactant modifies quasi catalytically the surface properties of the growing film without doping of the film. This change of growth mode is demonstrated with SEM images in Fig. 2 for a Ge film grown on Si(111). For temperatures above 500 ◦ C deposition without surfactant results in the formation of 3D Ge clusters on top of a 3-ML-thick pseu-

domorphic and strained Ge film (Fig. 2a). This wetting layer is flat and smooth and relieves part of the stress by a (5 × 5) reconstruction (see Fig. 6) [11, 12]. With the help of Sb as surfactant the growth mode changes completely: the formation of 3D clusters is inhibited [13], and continuous Ge films of arbitrary thickness can be grown, as demonstrated in Fig. 2b. As an additional effect the intermixing between Ge and Si is strongly suppressed [14]. With a surfactant the first term of (2) can be manipulated. However, the accumulation of strain energy, as described by the second term of (2), is not affected (excluding strong incorporation of the surfactant in the film). Therefore the latticemismatch-induced stress is still present and has significant influence both on the surface and bulk morphology of the Ge film. Also in the presence of surfactants strain-relieving defects must be generated beyond a critical thickness [15–17]. With the surface terminated by a surfactant also the nucleation and formation of strain relieving defects is modified [6, 15, 16]. For growth on Si(111) substrates the formation of a periodic array of misfit dislocations is observed which is completely confined to the Si/Ge interface [13, 18]. The dislocations have a mean separation of 104 Å. The Ge film itself is free of threading defects. It is relaxed and shows the Ge bulk lattice constant as determined by X-ray diffraction (Fig. 3). In contrast to this, deposition without surfactant always results in an uncontrolled defect structure with densities of threading defects of the order of 1011 –1012 cm−2 . The formation of this periodic array of interfacial misfit dislocations is a direct consequence of the use of surfactants during growth: it hinders the formation of 3D clusters. This causes a roughening transition of the pseudomorphic Ge film in order to relieve compressive strain. The surface shows a pronounced roughness on a nm scale which allows partial elastic relaxation towards the Ge bulk lattice constant as sketched in Fig. 4 [19, 20]. The unfavorable increase of surface area is outbalanced by the large decrease of strain energy. A smooth and flat Ge film would allow no strain relief except an increase of the layer distance due to tetragonal distortion of the strained film (see top of Fig. 4). This change of surface morphology has been monitored in situ during deposition at elevated temperature with highresolution LEED. This process of elastic strain relief is observed in a continuous variation of the parallel and vertical lattice constant (increase of the layer distance of the Ge film) during Ge film growth [21]. On Si(111) the formation of irregular triangular Ge pyramids composed of (113)-type facets has been found [21].

Fig. 2a,b. The change of growth mode by surfactants is demonstrated with SEM top views for deposition of 50 ML Ge on Si(111). a Ge islanding without surfactant. b Smooth Ge surface with Sb as surfactant

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Fig. 3. θ − 2θ X-ray diffraction scan of the (333)-Bragg peak of a 1000-nmthick Ge film grown with Sb as surfactant. The Cu K α1 line has been used. At lower angles the Ge peak is seen. The vertical lattice constant is in perfect agreement with bulk Ge. The sharp Ge spot reflects the success of the surfactant-mediated epitaxy in growing a highly perfect and flat Ge film on Si(111)

in the coherent stage of growth with the consequence of an uncontrolled defect structure in the Ge film [22]. The formation and evolution of this network has been observed in situ by high-resolution LEED during deposition [21, 23]. The strain field of each dislocation causes a weak height undulation of the surface. The periodic arrangement of this undulation results in a spot splitting in LEED [21, 23, 24]. After deposition of only 40 Å Ge, a completely relaxed, defect-free and smooth Ge-film is observed. The LEED pattern (Fig. 5) shows a well-ordered Ge(111)/Sb-(2 × 1) reconstruction with extremely low background. The absence of Si spots reflects the formation of a continuous Ge film. Growth without surfactant results in islanding. The LEED pattern of such a “film” (Fig. 6) shows the (5 × 5) reconstruction of the 3-ML-thick pseudomorphic Ge wetting layer. The relaxed 3D Ge islands are apparant in weak spots with larger lattice constant (see inset of Fig. 6). The density of threading defects in the Ge film is below 5 × 107 cm−2 as estimated by optical microscopy. The very low doping level below 1 × 1016 cm3 for incorporation of Sb reflects the extreme efficient segregation of the surfactants [9, 10, 25, 26]. The excellent quality of such surfactantmediated epitaxy-grown 1000-nm-thick Ge films made it possible to achieve the up to now highest electron mobilities of 3150 cm2 /V s for relaxed Ge integrated on Si substrates. With this film also Ge p-channel MOSFETs have been succesfully fabricated [27]. In this paper we present the first simultaneous in situ measurements of the evolution of surface and film stress and surface and film morphology during Sb surfactant-mediated heteroepitaxial growth of Ge on Si(111).

Fig. 4. a A smooth and flat hetero-layer is highly strained and relieves stress only by tetragonal distortion: i.e. an increase of the layer distance: the Bragg condition is strongly shifted. The lateral lattice constant is determined by the substrate. b A rough and open Ge surface allows the partial elastic relaxation of the Ge film towards the bulk lattice constant. The increased surface area and surface free energy is outbalanced by the strong decrease of strain energy

Beyond a “critical” thickness, finally dislocations are generated. The troughs between the pyramids of the rough surface act as nucleation centers for the dislocations. On Si(111) those dislocations are arranged in a threefold quasi-periodic array at the interface that accommodates exactly the different lattice constants of Ge and Si. The roughening transition of the Ge film is a necessary pre-condition for the formation of this periodic array of dislocations. For surfactant-mediated growth of Ge0.5 Si0.5 alloys we do not observe a rough surface

Fig. 5. LEED pattern at 150 eV of a 50-ML Ge film grown with Sb as surfactant at 600 ◦ C. Additonal spots with different lattice constants are not present. The Ge film is smooth and continuous and shows the Sb-induced (2 × 1) reconstruction in three rotational domains. Vertical lines mark the virtual spot positions of the underlying Si substrate. The Ge spots are shifted by ≈ 4% towards the (00) spot: the Ge film is relaxed. The positions of the original Si spots are marked with bars

484

optical table

laser diode

micrometerdrive for PSD calibration

telescope

PSD

beamsplitter

external e-gun

d

sample

Fig. 6. LEED pattern at 150 eV of a 50-ML Ge film grown without surfactant at 600 ◦ C. The pattern (see magnification of the (01) spot) shows both the Si and Ge lattice constant indicating islanding of the Ge film. The Si substrate shows a Ge-induced (5 × 5) reconstruction. Vertical lines mark the position of the fundamental Si substrate spots

1 Experimental evaporators

SPA-LEED

The experiments have been performed in a UHV chamber equipped with a third-generation cone-shaped SPALEED [28] as shown in Fig. 7. The built-in electron gun is used for standard SPA-LEED measurements. A second electron gun in a RHEED-like geometry allows electron diffraction measurments during deposition at normal incidence and growth [29]. The Ge is evaporated from an electronbombardment-heated graphite crucible; the flux (typically 1 ML/min, 1 ML = 7.2 × 1014 atoms/cm2) is monitored by a quartz microbalance. The Ge flux has been calibrated using the LEED intensity curves measured during deposition. Sb is evaporated from a quartz crucible, also equipped with shutter and microbalance. To reduce the heat flux to the sample as much as possible all evaporators are mounted in water-cooled shrouds. The film or surface stress is measured by means of sample bending and optical deflection with an experimental setup sketched in Fig. 8 and following the setup from Tromp and Schell-Sorokin [30]. The optics with laser, beam splitter, and detector is mounted outside the chamber. Two NW150 windows allow optical access to the back of the sample for the two positions mentioned above. The bending of the sample is measured by a differential technique using one reference and one probe laser beam and a four-quadrant split photodiode. With this technique shifts of the deflected beam by thermal drift or the sample holder can be very efficiently eliminated from the true bending signal. The difference signal is directly recorded without lock-in technique by a computer AD-data acquisition board. The vacuum system and the optical setup are insulated by foam plates from the floor of the

Fig. 7. UHV chamber used for stress measurement, MBE and SPA-LEED experiments. The optical table is mounted at a NW150 viewport and carries the diode laser, beam-splitting and alignment optics. The position sensitive detector (PSD) is mounted on an xy table. Absolute calibration of the PSD sensitivity is realized by a x-actuator with sub-µm resolution

Fig. 8. Sketch of the sample holder, the U-shaped Si sample, and the two beams of the optical deflection detector. Size of the sample is 11 × 20 × 0.1 mm. The legs have a width of 4 mm. The sample is heated by direct current passing through the U

laboratory. Further details of this technique can be found in the literature [30]. Following Stoney [31, 32] the curvature R is given by R=

Et 2 , 6(1 − ν)∆σ

∆σ = σ1 − σ2 ,

(3)

485 Fig. 9. From the measured deflection signal, the calculated surface stress during Sb deposition and Ge growth on Si(111) at 600 ◦ C is shown. Sb adsorption (regime A) causes strong compressive stress of ≈ 2 N/m. During the coherent stage of growth (regime B) Ge deposition results in an initially linear increase of compressive film stress. The slope of 1.0 (N/m)/ML Ge compares well with the behavior expected from bulk elasticity constants. Partial strain relief by roughness causes the weaker slope of the stress signal. Dislocations are generated during stage C and the film stress is very efficiently relieved. The weak increase of the compressive stress in regime D reflects the repulsive interaction of the interfacial misfit dislocations

where t is the sample thickness, E is Young’s modulus, and ν the Poisson ratio [33, 34]. σ1 and σ2 are the stresses present in the front and back surface of the sample. ∆σ is the stress applied to one of the samples’ surface so that the other side remains unaffected. Rearranging (3) and replacing R by geometrical equivalents we obtain R=

2Ls , D

(4)

with the distance L between the sample and the detector, the separation s between the two laser beams on the sample, and the displacment D of the two spots on the detector. The variation of surface stress can now be expressed as ∆σ =

E t2 D. (1 − ν) 12Ls

(5)

The sensitivity of deflection setup is mainly determined by the thickness t of the substrate. The distance L and the separation s scales lineary to the measured stress signal ∆σ. The displacement D of the two spots on the detector is directly proportional to our stress signal and the experimentally measured quantity. By defocusing of the laser spots (on the detector) we can adjust the sensitivity of the setup (the finer the spot, the larger is the sensitivity). The sensitivity of the four-quadrant split Si photodiode was calibrated and adjusted prior to and after each bending experiment with a linear motion drive moving the detector. The vibrational noise level in the lab limits the resolution of deflection signal in time. The long-time stability is limited by the mechanical stiffness of the manipulator and sample holder. But mechanical and thermal drifts in both of the deflected beams are eleminated in the differential signal, which is electronically generated after precise gain adjustment. Using the constants given in [33] we obtain,   E N = 2.29 × 1011 2 , (6) 1 − ν Si(111) m and it is possible to determine absolute stress values from the sample bending. Without further vibrational insulation of the chamber we are able to measure the deflection induced by about 1/10 of

a ML of adsorbed Sb, i.e., a curvature larger than 10 km. With an air suspension of the whole setup the signal-to-noise ratio could be improved further. The Si substrates have been cut by laser ablation [35–39] from a 100-µm-thick well-oriented Si(111) wafer (Virgina Semiconductor, n-type, 20 mΩ cm) which has been polished on both sides. Samples have been treated by a standard cleaning process (RCA) [40] before degassing them for 12 h in the main chamber. The oxide is removed by a short flash close to the melting point of Si. The samples are U-shaped and mounted at the end of the legs by two point clamps on a sapphire plate in a transferable sample holder (see Fig. 8). This allows resistive heating of the samples from room temperature up to the melting point. Due to the thin samples the temperature homogenity over the probe area of 10 × 4 mm is better than 10 ◦ C. These point clamps ensure a mounting without distortion of the sample. Therefore changes of the sample temperature have only a small and reversible influence on the deflection signal. Thermal equilibrium after flashing of the Si samples is achieved after approximately 10 min. The bending signal is not affected by the heat flux from the evaporators. During deposition both diffraction patterns and the deflection signal have been measured simultaneously. This allows the correlation between morphological and strain information. In the following we will present first results on the strain evolution during surfactant-mediated epitaxy of Ge on Si(111) at 600 ◦ C.

2 Results Figure 9 shows the differential deflection signal during the experiment. Prior to the Ge deposition the Si(111) surface has been covered by Sb (shutter open at t = 100 s). The abrupt decrease of the bending signal is due to the Sb adsorption of the surface. The signal saturates at a compressive stress of 2 N/m (1.2 eV/(1 × 1) unit cell) after ≈ 400 s which is very likely the saturation√coverage √ of 1 ML on Si(111). The LEED pattern shows a ( 3 × 3) reconstruction. The slight decrease is √ √probably caused by improvements of the Sb-induced ( 3 × 3) surface structure [30].

486

At 500 s the Ge shutter has been opened. The sample immediately starts bending with an initially linear increasing compressive stress of 1.0 (N/m)/ML Ge. This value agrees well with that expected from Ge bulk elasticity constants 1.0 (N/m)/ML [33]. This shows, that the first 2–3 layers grow completely strained without any efficient strain relief mechanism. This behavior is in contrast to Ge deposition on Si(001) without surfactant, where intermixing and missing-dimer reconstruction very efficiently relieve stress for the first monolayers [30, 41–43]. The deflection signal deviates significantly from the linear behavior beyond 2–3 ML and shows a smaller slope up to a coverage of 8 ML. We attribute this behavior to strain relief by roughening of the Ge film. The upper layers of the Ge film break up and form small triangular pyramids which cover the entire surface. From the deviation of the linear decrease we can conclude that 45% of the strain is relieved by this mechanism compared with a continuous and flat strained Ge film. This decrease in film free energy of 2.2 eV per (1 × 1) unit cell exceeds the increase of surface free energy due to the formation of (113) facets and the increase in surface area. Around a coverage of 8 ML the deflection signal is constant and becomes smaller with increasing coverage: strain is relieved very efficiently. We attribute this decrease of film

0.0 ML Ge

stress to the formation of misfit dislocations at the Si/Ge interface. At the same time satellite spots appear surrounding all spots in the LEED pattern as shown for the (00) spot in Fig. 10: the periodic interfacial dislocation network is formed. With increasing coverage the total intensity of the satellites and the (00) spot become more intense as shown in Fig. 11. This reflects the smoothing of the Ge surface. With further increasing Ge coverage the intensity of the satellite decreases to zero. This is because the strain fields of the dislocations become wider and wider and start to overlap and to wipe out each other [23]. Beyond a Ge coverage of 15 ML the compressive film stress increases again slowly with a maximum slope of 0.15 (N/m)/ML. This behavior is not completely understood yet. It may be interpreted as residual strain, because the number of dislocations are too low to completely accommodate the Ge to the Si lattice constant. However, an Xray diffraction study reveals the bulk lattice constant 1 for the Ge film as also observed for thicker films as shown in Fig. 3 [8]. Due to the same reason also the strong incorporation of Sb can be excluded. In all growth experiments background doping levels below 1019 Sb/cm3 have always been determined [9, 10, 25, 44, 45]. We attribute this slight increase of compressive stress to the elastic interaction between the strain fields of the indi-

4.3 ML Ge

6.6 ML Ge

8.0 ML Ge

11.4 ML Ge

13.6 ML Ge

16.4 ML Ge

18.2 ML Ge

20.4 ML Ge

24.0 ML Ge

26.7 ML Ge

31.1 ML Ge

37.1 ML Ge

39.2 ML Ge

41.5 ML Ge

44.1 ML Ge

46.5 ML Ge

Si(111) - (7x7) 72eV (0,1/7)

(0,0)

Fig. 10. LEED pattern of the (00) spot during Sb-mediated Ge growth on Si(111). The top left pattern shows the initial (7 × 7) reconstruction prior to Sb adsorption (scan has a size of 50 × 50% of the surface Brillouin zone). The smaller scans (scan size is 15 × 15% of the surface Brillouin zone) show the evolution of the (00) spot after Sb adsorption and during Ge deposition. With increasing Ge coverage satellite spots arise at ≈ 10 ML, reflecting the formation of the periodic array of misfit dislocations confined to the Si/Ge interface. The satellites die out beyond 50–60 ML because the strain fields overlap more and more and cancel out each other

487

This repulsive interaction is responsible for the ordering into a periodic array. This ordering process has already been observed in a reduction of the full width at half maximum of the satellite spots with Ge film thickness [21]. We therefore expect an additional buildup of film stress due to the repulsive interaction between the dislocations for a thickness of the Ge film around anet /2 ≈ 52 Å, i.e. around 30 ML of coverage. For even higher thickness the strain fields overlap more and more and cancel out each other at a coverage of 60–80 ML.We therefore expect a smaller slope and a saturation of the bending signal. 3 Conclusion

Fig. 11a,b. Intensity of the (00) spot at 72 eV during Ge deposition. a Intergral intensity of the first-order satellite spots as function of Ge coverage. From the shape of the curve and the location of the maxium intensity the Ge coverage has been calibrated by comparison with [21, 23]. The surface height undulation smoothes out at ≈ 60 ML. b Intensity of the central spike of the array of satellites

vidual dislocations at the interface. [46, 47] Following simple elasticity theory [48] the strain fields surrounding an edge dislocation in an infinite layer can be described by the displacements u z and u x from the perfect lattice positions (Burgers vector in x, dislocation line in y and surface normal in z, √ x 2 + z 2 represents the distance from the dislocation line at the interface) B = (−b, 0, 0), uz = −

ux =

b = |B| ,

(7)

 2  x + z2 b z2 (1 − 2ν)b ln + , 2 2 8π(1 − ν) b 4π(1 − ν) x + z 2 x

b xz b − arctan 4π(1 − ν) x 2 + z 2 2π z

We have presented the first simultaneous in situ measurements of the evolution of surface and film stress and surface and film morphology during Sb surfactant-mediated heteroepitaxial growth of Ge on Si(111). The surfactant hinders 3D island formation and allows the growth of a smooth and continuous Ge film. First, prior to the formation of dislocations, the coherent Ge film undergoes a roughening transition in order to relieve strain by an elastic relaxation towards its bulk lattice constant. This transition reduces the stored strain energy strongly compared with a homogeneously flat strained Ge film. The gain in energy of 2.2 eV/(1 × 1) unit cell outbalances the increase of surface free energy due to the enlarged surface area. Second, the plastic strain relief occurs by the formation of a periodic array of misfit dislocations completely confined to the Si/Ge interface. This results in a very efficient reduction of film stress. With increasing thickness of the Ge film the strain fields of the dislocations interact by repulsive forces mediated by elastic distortion of the lattice. This results in a weak increase of film stress until the strain fields cancel out each other for a Ge film thickness above 100 Å. This repulsive interaction is also the reason for the ordering of the periodic array of dislocation. Now a relaxed and smooth Ge film can be grown without further buildup of film stress. Acknowledgements. The authors thanks R.M. Tromp, M.C. Reuter, and R. Koch for continued and helpful discussions. Finacial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. We thank Dr. C. Momma at the Laserzentrum Hannover for technical assistance with the laser cutting of the Si samples.

(8) References

,

(9)

with the Burgers vector (7) of the dislocation and the Poisson ratio ν. The strain fields are dominated by a Lorentzian-shaped distortion in x- and z-direction (the u z distortion manifests in the surface height undulation which is observed by spot splitting in LEED, but also by STM [45, 49, 50]). The full width at half maximum of the Lorentzian is twice the thickness of the film. With increasing thickness of the film the strain fields spread out further and start to interact. This becomes significant at a film thickness of half the distance between the dislocations (anet = 104 Å).

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