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Variation of Pendellösung fringes in elastically deformed silicon single crystals
Volume 2,number 4B
VARIATION
MATERIALSLETTERS
OF PENDELLhUNG
IN ELASTICALLY
May 1984
FRINGES
DEFORMED SILICON SINGLE CRYSTALS
G.E. WHITE and Haydn CHEN Department of Metallurgy and Mining Engineering and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Received 29 March 1984
obtained from x-ray diffraction section topographs are sensitive to elastic Pendellosung frirqs The strain sensitivity is higher for topographs taken with a diffraction strain lo~r than 10 . This effect is demonstrated in two types of vector closer to the lattice displacement vector. samples: (1) a Si(ll1) wafer with an epitaxial Pd2Si thin film and naturally present strain, and (2) a triangular shaped Si(ll1) crystal with an artifical strain field produced by one point bending.
1.
INTRODUCTION
It is a well known fact [l] that Pendellosung fringes in section topographs are useful in the detection of elastic strain in single crystals. Pendellosung fringes might be defined as the existence of definite phase betwen waves relations electromagnetic travelling within the Borrmann triangle during transmission a diffraction event in the geometry (Fig. 1). As one travels along the
base of the Borrmann triangle one would go points wavefields are through where tm successively in and out of phase, arriving thereby resulting in fringes across a section These fringes are observable when topograph. divergent width of the spherically the with the incidence beam, a, is small compared width of the Borrmann fan [II, i.e. a << when the crystal is highly 2X*tanHB, and (A typical fringe pattern is shown perfect. in Figure 4a for a perfect Si single crystal relative However, the phase sample.) relationship of these interference fringes can be disrupted if strain is introduced into the gradient, thermal sample (e.g. by deformation, imperfections, or mechanical In this study section etc.). topography, taken on a Lang camera, was employed on tm under the types of Si(111) single crystals influence of elastic strain. The purpose of this work is to demonstrate the effect of an strain elastic field, whether generated means, on the artifical naturally or by Pendellosung fringes.
2.
Fig. 1. Eorrmann triangle (bound by dashed lines and spec$en surfaces) for an incident spherical wave S . Surfaces of constant phase of interfering xcray waves within the Borrmann triangle are indicated by hyperbolic curves.
0 167-557x/84/$ 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
MPERIMENTAL
Two types of samples ware used in the study. lhe first specimen was a Si(ll1) single crystal wafer one inch in diameter, and covered with an 1OOOA thick Pd film prepared by means of electron beam evaporation. It was then annealed at 250°C for one hour in a pure He atmosphere in order to form a layer of epitaxial Pd2Si film. The second sample was a float-zone grown Si(ll1) crystal which was cut into a triangular and deformed shape
347
Volume 2, number 4B
MATERIALS
LETTERS
May 1984
interference fringes properly referred to as Pendell6sung f rlnges, the appearance of which varied d,ue to selection of the diffraction All topographs were recorded on vector, g. Ilford L4 emulsion nuclear plates of 50pm thickness.
3.
Fig. 2. Side view of a miniature one-point bending device. A pivoting arm can be displaced by a micrometer at position “a”, thereby causing a displacement (6) at the tip of a triangular specimen in contact with the pivoting arm at position “c”.
.
elastically by a one point bending mechanism in order to generate a nearly one-dimensional curvature (Fig. 2). The first sample possesses strain which is originated from the lattice and/or thermal mismatches between the epitaxial Pd2Si thin film and the underlying Si substrates [2,3]. The second sample strain contains that is generated by an external bending. The substrate thickness of the first sample is O.hm while the second sample is 0.5mm. The Lang topographic method was used t,o produce section topographs. The crystals were set to diffract characteristic MoKa radiation in the Laue mode with a narrow ri b bon x-ray beam widcb of - 2Opm and a beam divergence of ” 1x10 radians. Roth and symmetric asymmetric reflections were utilized for this study. The x-ray energy flow inside the crystal occurs throughout the triangular prism bounded by the planes containing the directions S and z drawn through the line of incidence a:d the gx-ray exit surface of the crystal. At the exit surface the energy flow splits into tm beams, one almost garallet to the diffracted one beam direction S and parallel to the incident beam dir&ion 8 . These twn projections of energy flow ‘so obtained are called the diffracted beam section topograph and the direct beam section topograph, respectively. In this study we are concerned only beam with the diffracted section to,pograph. The section pattern thus obtained from the samples contained a set of
348
FESULTS
During the formation of the Pd2Si film upon annealing the Si crystal covered with Pd overlayer, the underlying Si substrate was elastically bent 12-41. The bending radius of the particular Si crystal studied was fouqd to be 25 meters by an x-ray method similar in principle to the Newton ring technique 131. The effect of this naturally produced strain on the PendellBsung fringes is demonstrated in Figures 3a to 3c. Ffgure_3a shows a section topograph taken asymmetric with g = [ill]. mis is an reflection with its diffraction vector located from the specimen surface normal which 70.5” is [Ill]. There is a uniform contrast across image and no Pendellosung fringes are seen. The +$econd_ topograph, Figure 3b, was taken with g = [2201, a syrometrie reflection. This diffraction vector is 90’ from the surface normal and fringes are clearly visible. Figurz 3c_ shows a third topograph taken with g = 14401 +&ich_ is the second order reflection of g = [220]. A nice undistorted fringe pattern is observed. Figures 4 and 5 are section topographs taken from a triangular Si crystal with various degrees of bending for the 111 and i;22 reflections, respectively. Figures 4a and 5a show section topographs taken at zero displacement, i.e. free of strain. The 111 reflection is asymmetrical while the 422 reflection is symmetrical. In both cases fringes are highly visible. Figures 4b and 5b are taken with a 50pm displacement
lmm
a
b
c
Section topographs taken+from an Fig. 3. anne@ed Ed-Si specimen+with_(a) g = [ill] , (b) g = 12201, and Cc) g = f4401.
Volume 2, number 4B
MATERIALS
tmm
4.
from a Fig. 4. Section topographs+taken triangular Si crystal with g = [ill] at various displacements (6). (a) 6 = 0 (i.e. strain free), (b) 6 = 50um, and (c) S = 100nm.
corresponding to a bending radius of 14.2m. _The number of Pendellosung fringes for the 111 reflection has dscreased significantly, as compared to the 422 reflection. Close examination of the 422 topograph reveals a* apparent phase shift in the fringe pattern in which a bright band has now appeared in the center of the image, replacing a dark band which originally existed in Figure 5a. In examining the other set of topographs with a displacement of 1OOpm (Figures 4c and 5c) one can the
_see that the 111 reflection.
fringes
have
diminished
for
Fringe-s can still be observed, however, in the 422 reflection in spite of an obvious shift that has phase occurred in the latter topograph as compared with Figure 5b. In fact, Pendellosung fringes were observed for the 422 reflection with a displacement
as
large
as
200pm.
lmm
a
b
LETTERS
C
Fig. 5. Section topographs+take_n from a triangular Si crystal with g = [422] at various displacements (a) 6 = 0, (b) 6 = 50 urn, and (c) 6 = lOOurn.
May 1984
DISCIJSSION
The appearance of Pendellasung fringes is a effects in scattering result of dynamical perfect single crystals. As sketched in Figure 1 the surfaces of constant phase of propagating x-ray waves are on hyperbolic sided cylinders and thus, in a parallel specimen, the exit surface cuts these cylinders in straight lines; hence the parallel sided fringes observed in section These fringes, however, will be topographs. interrupted when imperfections such as strain degree of fields are present. The interruption is. gositivel-y dependent upon the magnitude o,f g*u where g is the diffraction vector and u is the displacement vector of the For the tm types of samples lattices [1,5]. studied, bending of the crystal produces shear strain and is responsible for the distortion The bending of of the Pendellosung fringes. displacement results in a the Srystal field u which is parallel to the normal of the sample section surface. Consequently topographs taken with an inclined diffraction vector with +respect surface to the sample plane (e.g. g = [ill]) will reveal a higher level of interruption due to the bending of the crystal a0 compared with the topograph taken with a+g vector in the plane of the This is indeed what sample (e.g. g = [4221). one has seen in Figures 3, 4 and 5. For
instance, the Pendellosung fringes nearly of displacement disappeareq a -at JOOpm_for g = [ill] (Fig. 4b) but not for g = [422] (Fig. 5b). While the fringes gre s-till present in topographs taken with g = [422] at various degrees of bending one could also observe the variation of fringes, along with a progressive reduction in contrast as a function of increasing strain. The variation of the fringes described above is due to the curving of the trajectories of the propagating x-ray waves along the strained lattice planes within crystal. the Thus, the relative phase relationship between two contributing waves arriving at a fixed position on the base of the Borrmann triangle varies with the strain
any deviation of the mechanical surface normal from a perfect direction crystallographic (1111 in this study) would vary the (e.g. fringes pattern and the contrast in much the same +wa,y as asymmetric reflections any since g*u no longer vanishes in this case. The degree of sensitivity of Pendellosung
349
Volume 2,number 4B
MATERIALSLETTERS
fringes to the strain magnitude may be estimated as follows. The amount of shear strain generated by the one point bending of a triangular crystal may be calculated using elementary beam theory [6]. Relating our crystal geometry to that of a cantilever loaded at one end, the following equation for of the strain in relation to the amount deflection or displacement, 6, is obtained: E
=I
6t
where t is the thickness of the beam (i.e., Si crystal) and L is the length of beam from its clamped position to the point of the applied In Fig. 5, a complete reversal in load. contrast at the center band of a section topograph is observed for every additional displacement of 50ym (i.e., A6 = 50km). IJsing equation (1) with t = 0.5mm and R = 4cm, this incremzyt in implies that a small can be strain, AE, at a level as low as - 10 readily detected even when ,a lea_st sensitive diffraction vector this (e.g., g = [422] in ynder more favored conditions case) is used. when an inclined g vector, with respect to the specimen surface plane, is utilized in conjunction with a better optical arrangement narrower incident beam, shorter (e.g. an improvement in strain wavelength etc.), sensitivity by an order of magnitude can be comfortably expected. In summary, the present study demonstrated the effect of elastic strain field produced by bending on Pendellosung fringes. A the shift of the fringes in conjunction gradual with a progressive reduction in contrast is
350
May 1984
shown to be related to the increasing strain field. The sensitivity of these fringes to strain $s higher for those topographs taken with a g vector close to the sample surface normal as expected from dynamical diffraction theory.as lo% small presence of elastic strain as or less can be readily detected. ACKNOWLEDGEMENTS This mrk was supported by the Materials Science Division of the U.S. Department of Energy under contract #DE-AUJ2-76ERC1198. One of the authors (GEW) would like to acknowledge support through an IBM graduate fellowship. Experiments were performed at the University of Illinois Materials Research Laboratory's Center for Microanalysis of Materials which is supported by the Materials Science Division of the Department of bergy.
REFERENCES [l]
[2] [3] [4]
[5] [6]
B. K. Tanner, X-ray Diffraction Topography (Pergamon Press, U.K., 1976) and references therein. Y. Chen. G. E. White, S. R. Stock and P. S. Ho, Thin Solid Films 93 (1982) 161. G. E. White, Masters lhesis, University of Illinois, Urbana, IL (1982). J. Angilello, F. d'Huerle, S. Petersson and A. Segmuller, J. Vat. Sci. Tech. 16 (1980) 470. M. Hart, Zeitschrift fiir Physik 189 (1966) 269. S. Timoshenko and J. N. Goodier, Theory of Applied Elasticity (McGraw Hill, N.Y. 951) p. 35.
VARIATION
MATERIALSLETTERS
OF PENDELLhUNG
IN ELASTICALLY
May 1984
FRINGES
DEFORMED SILICON SINGLE CRYSTALS
G.E. WHITE and Haydn CHEN Department of Metallurgy and Mining Engineering and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Received 29 March 1984
obtained from x-ray diffraction section topographs are sensitive to elastic Pendellosung frirqs The strain sensitivity is higher for topographs taken with a diffraction strain lo~r than 10 . This effect is demonstrated in two types of vector closer to the lattice displacement vector. samples: (1) a Si(ll1) wafer with an epitaxial Pd2Si thin film and naturally present strain, and (2) a triangular shaped Si(ll1) crystal with an artifical strain field produced by one point bending.
1.
INTRODUCTION
It is a well known fact [l] that Pendellosung fringes in section topographs are useful in the detection of elastic strain in single crystals. Pendellosung fringes might be defined as the existence of definite phase betwen waves relations electromagnetic travelling within the Borrmann triangle during transmission a diffraction event in the geometry (Fig. 1). As one travels along the
base of the Borrmann triangle one would go points wavefields are through where tm successively in and out of phase, arriving thereby resulting in fringes across a section These fringes are observable when topograph. divergent width of the spherically the with the incidence beam, a, is small compared width of the Borrmann fan [II, i.e. a << when the crystal is highly 2X*tanHB, and (A typical fringe pattern is shown perfect. in Figure 4a for a perfect Si single crystal relative However, the phase sample.) relationship of these interference fringes can be disrupted if strain is introduced into the gradient, thermal sample (e.g. by deformation, imperfections, or mechanical In this study section etc.). topography, taken on a Lang camera, was employed on tm under the types of Si(111) single crystals influence of elastic strain. The purpose of this work is to demonstrate the effect of an strain elastic field, whether generated means, on the artifical naturally or by Pendellosung fringes.
2.
Fig. 1. Eorrmann triangle (bound by dashed lines and spec$en surfaces) for an incident spherical wave S . Surfaces of constant phase of interfering xcray waves within the Borrmann triangle are indicated by hyperbolic curves.
0 167-557x/84/$ 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
MPERIMENTAL
Two types of samples ware used in the study. lhe first specimen was a Si(ll1) single crystal wafer one inch in diameter, and covered with an 1OOOA thick Pd film prepared by means of electron beam evaporation. It was then annealed at 250°C for one hour in a pure He atmosphere in order to form a layer of epitaxial Pd2Si film. The second sample was a float-zone grown Si(ll1) crystal which was cut into a triangular and deformed shape
347
Volume 2, number 4B
MATERIALS
LETTERS
May 1984
interference fringes properly referred to as Pendell6sung f rlnges, the appearance of which varied d,ue to selection of the diffraction All topographs were recorded on vector, g. Ilford L4 emulsion nuclear plates of 50pm thickness.
3.
Fig. 2. Side view of a miniature one-point bending device. A pivoting arm can be displaced by a micrometer at position “a”, thereby causing a displacement (6) at the tip of a triangular specimen in contact with the pivoting arm at position “c”.
.
elastically by a one point bending mechanism in order to generate a nearly one-dimensional curvature (Fig. 2). The first sample possesses strain which is originated from the lattice and/or thermal mismatches between the epitaxial Pd2Si thin film and the underlying Si substrates [2,3]. The second sample strain contains that is generated by an external bending. The substrate thickness of the first sample is O.hm while the second sample is 0.5mm. The Lang topographic method was used t,o produce section topographs. The crystals were set to diffract characteristic MoKa radiation in the Laue mode with a narrow ri b bon x-ray beam widcb of - 2Opm and a beam divergence of ” 1x10 radians. Roth and symmetric asymmetric reflections were utilized for this study. The x-ray energy flow inside the crystal occurs throughout the triangular prism bounded by the planes containing the directions S and z drawn through the line of incidence a:d the gx-ray exit surface of the crystal. At the exit surface the energy flow splits into tm beams, one almost garallet to the diffracted one beam direction S and parallel to the incident beam dir&ion 8 . These twn projections of energy flow ‘so obtained are called the diffracted beam section topograph and the direct beam section topograph, respectively. In this study we are concerned only beam with the diffracted section to,pograph. The section pattern thus obtained from the samples contained a set of
348
FESULTS
During the formation of the Pd2Si film upon annealing the Si crystal covered with Pd overlayer, the underlying Si substrate was elastically bent 12-41. The bending radius of the particular Si crystal studied was fouqd to be 25 meters by an x-ray method similar in principle to the Newton ring technique 131. The effect of this naturally produced strain on the PendellBsung fringes is demonstrated in Figures 3a to 3c. Ffgure_3a shows a section topograph taken asymmetric with g = [ill]. mis is an reflection with its diffraction vector located from the specimen surface normal which 70.5” is [Ill]. There is a uniform contrast across image and no Pendellosung fringes are seen. The +$econd_ topograph, Figure 3b, was taken with g = [2201, a syrometrie reflection. This diffraction vector is 90’ from the surface normal and fringes are clearly visible. Figurz 3c_ shows a third topograph taken with g = 14401 +&ich_ is the second order reflection of g = [220]. A nice undistorted fringe pattern is observed. Figures 4 and 5 are section topographs taken from a triangular Si crystal with various degrees of bending for the 111 and i;22 reflections, respectively. Figures 4a and 5a show section topographs taken at zero displacement, i.e. free of strain. The 111 reflection is asymmetrical while the 422 reflection is symmetrical. In both cases fringes are highly visible. Figures 4b and 5b are taken with a 50pm displacement
lmm
a
b
c
Section topographs taken+from an Fig. 3. anne@ed Ed-Si specimen+with_(a) g = [ill] , (b) g = 12201, and Cc) g = f4401.
Volume 2, number 4B
MATERIALS
tmm
4.
from a Fig. 4. Section topographs+taken triangular Si crystal with g = [ill] at various displacements (6). (a) 6 = 0 (i.e. strain free), (b) 6 = 50um, and (c) S = 100nm.
corresponding to a bending radius of 14.2m. _The number of Pendellosung fringes for the 111 reflection has dscreased significantly, as compared to the 422 reflection. Close examination of the 422 topograph reveals a* apparent phase shift in the fringe pattern in which a bright band has now appeared in the center of the image, replacing a dark band which originally existed in Figure 5a. In examining the other set of topographs with a displacement of 1OOpm (Figures 4c and 5c) one can the
_see that the 111 reflection.
fringes
have
diminished
for
Fringe-s can still be observed, however, in the 422 reflection in spite of an obvious shift that has phase occurred in the latter topograph as compared with Figure 5b. In fact, Pendellosung fringes were observed for the 422 reflection with a displacement
as
large
as
200pm.
lmm
a
b
LETTERS
C
Fig. 5. Section topographs+take_n from a triangular Si crystal with g = [422] at various displacements (a) 6 = 0, (b) 6 = 50 urn, and (c) 6 = lOOurn.
May 1984
DISCIJSSION
The appearance of Pendellasung fringes is a effects in scattering result of dynamical perfect single crystals. As sketched in Figure 1 the surfaces of constant phase of propagating x-ray waves are on hyperbolic sided cylinders and thus, in a parallel specimen, the exit surface cuts these cylinders in straight lines; hence the parallel sided fringes observed in section These fringes, however, will be topographs. interrupted when imperfections such as strain degree of fields are present. The interruption is. gositivel-y dependent upon the magnitude o,f g*u where g is the diffraction vector and u is the displacement vector of the For the tm types of samples lattices [1,5]. studied, bending of the crystal produces shear strain and is responsible for the distortion The bending of of the Pendellosung fringes. displacement results in a the Srystal field u which is parallel to the normal of the sample section surface. Consequently topographs taken with an inclined diffraction vector with +respect surface to the sample plane (e.g. g = [ill]) will reveal a higher level of interruption due to the bending of the crystal a0 compared with the topograph taken with a+g vector in the plane of the This is indeed what sample (e.g. g = [4221). one has seen in Figures 3, 4 and 5. For
instance, the Pendellosung fringes nearly of displacement disappeareq a -at JOOpm_for g = [ill] (Fig. 4b) but not for g = [422] (Fig. 5b). While the fringes gre s-till present in topographs taken with g = [422] at various degrees of bending one could also observe the variation of fringes, along with a progressive reduction in contrast as a function of increasing strain. The variation of the fringes described above is due to the curving of the trajectories of the propagating x-ray waves along the strained lattice planes within crystal. the Thus, the relative phase relationship between two contributing waves arriving at a fixed position on the base of the Borrmann triangle varies with the strain
any deviation of the mechanical surface normal from a perfect direction crystallographic (1111 in this study) would vary the (e.g. fringes pattern and the contrast in much the same +wa,y as asymmetric reflections any since g*u no longer vanishes in this case. The degree of sensitivity of Pendellosung
349
Volume 2,number 4B
MATERIALSLETTERS
fringes to the strain magnitude may be estimated as follows. The amount of shear strain generated by the one point bending of a triangular crystal may be calculated using elementary beam theory [6]. Relating our crystal geometry to that of a cantilever loaded at one end, the following equation for of the strain in relation to the amount deflection or displacement, 6, is obtained: E
=I
6t
where t is the thickness of the beam (i.e., Si crystal) and L is the length of beam from its clamped position to the point of the applied In Fig. 5, a complete reversal in load. contrast at the center band of a section topograph is observed for every additional displacement of 50ym (i.e., A6 = 50km). IJsing equation (1) with t = 0.5mm and R = 4cm, this incremzyt in implies that a small can be strain, AE, at a level as low as - 10 readily detected even when ,a lea_st sensitive diffraction vector this (e.g., g = [422] in ynder more favored conditions case) is used. when an inclined g vector, with respect to the specimen surface plane, is utilized in conjunction with a better optical arrangement narrower incident beam, shorter (e.g. an improvement in strain wavelength etc.), sensitivity by an order of magnitude can be comfortably expected. In summary, the present study demonstrated the effect of elastic strain field produced by bending on Pendellosung fringes. A the shift of the fringes in conjunction gradual with a progressive reduction in contrast is
350
May 1984
shown to be related to the increasing strain field. The sensitivity of these fringes to strain $s higher for those topographs taken with a g vector close to the sample surface normal as expected from dynamical diffraction theory.as lo% small presence of elastic strain as or less can be readily detected. ACKNOWLEDGEMENTS This mrk was supported by the Materials Science Division of the U.S. Department of Energy under contract #DE-AUJ2-76ERC1198. One of the authors (GEW) would like to acknowledge support through an IBM graduate fellowship. Experiments were performed at the University of Illinois Materials Research Laboratory's Center for Microanalysis of Materials which is supported by the Materials Science Division of the Department of bergy.
REFERENCES [l]
[2] [3] [4]
[5] [6]
B. K. Tanner, X-ray Diffraction Topography (Pergamon Press, U.K., 1976) and references therein. Y. Chen. G. E. White, S. R. Stock and P. S. Ho, Thin Solid Films 93 (1982) 161. G. E. White, Masters lhesis, University of Illinois, Urbana, IL (1982). J. Angilello, F. d'Huerle, S. Petersson and A. Segmuller, J. Vat. Sci. Tech. 16 (1980) 470. M. Hart, Zeitschrift fiir Physik 189 (1966) 269. S. Timoshenko and J. N. Goodier, Theory of Applied Elasticity (McGraw Hill, N.Y. 951) p. 35.