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Nanometer undulations on CaF2 cleaved surfaces observed by Atomic Force Microscopy
ScriptaMaterialia, Vol. 34, No. 11, pp. 1673-1678,1996 Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Rinted in the USA. All rights reserved 1359-6462196 $12.00 + .OO
Pergamon PII 81359-6462(96)00058-9
NANOMETER UNDULATIONS ON CaF, CLEAVED SURFACES OBSERVED BY ATOMIC FORCE MICROSCOPY C. Coupeau, M.K. Small, N. Junqua and J. Grilhe Laboratoire de Metallurgie Physique-IX4 CNRS 13 1 Universite de Poitiers-Faculte des Sciences Bat. S.P.2.M.I Bd 3, Teleport 2, BP 179,86960 Futuroscope Cedex, France (Received October 23, 1995) (Revised November 22, 1995) Introduction
Cleaved surfaces of CaFz single crystals have been examined by Atomic Force Microscopy (AFM) (1). In a few places the surfaces exhibit smooth undulations in addition to abrupt atomic steps. The undulations are readily distinguished from straight cleavage steps caused by slight surface shifts from the ideal cleavage plane. Experimental data is presented for the existence of this surface feature and a model is proposed to explain this topography. ExDerimental
Procedure
Single crystals Iof calcium fluoride were cleaved in air and the cleavage plane (111) was examined immediately by Ihe AFM. Characterization was performed using an AFM with an interferometric deflection detector with a 18 pm scan range. Microfabricated silicon nitride cantilevers were used with a spring constant of 0.1 N/m and the applied force was of the order of 10 nN. The images shown here were taken from top to bottom unfiltered unless otherwise specified, but were treated to remove uniform curvature and slope. Results
As expected, the specimen showed cleavage steps running in preferred crystallographic directions (Figure la). Two types can be distinguished: there is a river pattern of cleavage steps in the top left of the image (Part A) and flat widely spaced terraces in part B. The perfectly straight cleavage steps in the lower part cannot be confused with the slightly wavy undulation (Part C) (enlargement Fig. lc-d) which is only consistent with im elastic deformation of the surface. The shape ofthe undulations varies somewhat from one area to another. The wavelength ranges from 550 to 700 pm of amplitude up to 2 nm, though generally closer to 1 nm. For instance, the cross section 1673
1674
CLEAVED SURFACES BY AFM
a
C
Vol. 34, No. 11
b
d
Figure 1. SAAFM CaF, images showing cleavage steps (A and B) and wavy undulation (C) with (a) scan range to 18 pm and scan angle to 00, (b) scan range to 18 pm and scan angle to 90’, (c) scan range to 9 pm, (d) scan range to 6 pm.
of the CaF, surface (line D in Fig. la) is seen in Figure 2. This undulation has a periodicity of 654 nm and an amplitude of 1.3 nm. The undulations often follow preferred directions though less precisely than the surface steps and seem to coincide with the intersection of the cleavage and the crystal growth planes. It is necessary to demonstrate that this effect is not an artifact of the microscopy, and several precautions have been taken to eliminate this possibility. Firstly, the scans were taken at different angles to see whether the undulations retain their orientation relative to the specimen (Figure lb). It must be noticed that this sample has been slightly deformed in situ at room temperature (2) to see the emergence of slip lines which correspond to the vertical lines in Figure 1b; they are not clearly visible in Figure 1a, because of the scan direction.
Figure 2. CaF2 image cress section (Line D in Fig. 1) which gives a spectral period of 654 nm.
Figure 3. Schematic of an array of subgrain boundaries in a semi-infinite medium.
Vol. 34, No. 11
CLEAVEDSURFACESBY AFM
1675
Figure 4. An edge dislocation at x = 0 and z = -d in an elastic isotropic halfspace.
Furthermore the AFM employed for these measurements is a special design (Stand Alone AFM) (2), which excludes the possible artifact due to interference effects between the laser beam reflected by the cantilever and the partial transmitted beam reflected by the substrate; this might occur in a standard AFM due to the curvature of the piezoelectric transducer capable of large scans. Moreover, if this interference occurs, it would occur everywhere on the scan area, which is not the case. To conclude, Figures 1c and Id show that the same surface undulation wavelength is obtained with a smaller scan area, which excludes the possibility of an artifact due to interference effects and confirms that the undulations are a feature of the cleaved surface since their presence or absence is completely reproducible. Discussion From these AFM observations of cleaved surfaces, it is evident that the undulations are not cleavage steps, nor are they artifacts caused by vibrations or electronic noise in the microscope. The surface undulations are almost certainly due to elastic deformation of the surface induced by internal defects, such as precipitates or dislocations. The measured amplitudes are too large to be caused by single dislocations, which can produce surface displacement no more than one Burgers vector. Larger effects would have to be due to the colmbined influence of several dislocations. A random arrangement of dislocations would not produce such a pattern; periodics arrays of dislocations can be accounted for by the presence of subgrain boundaries in the crystal. Calculations are presented for the surface displacement resulting from periodic arrangements of lowangle tilt boundaries modeled as an array of evenly spaced edge dislocations with Burgers vectors parallel to the surface of an elastic isotropic half-space. In a real crystal the subgrains will probably not be exactly perpendicular to the surface. In this case, dislocations very near the surface may glide to it and be armihilated, thus creating a dislocation-free zone to a depth greater than the dislocation spacing. However, for the sake of simplicity we will assume that the dislocations are futed in the crystal and that the surface falls at an arbitrary distance between two dislocations in the subgrain. Subgrain boundaries made up of dislocations of opposite sign are separated by a distance 1.12and dislocations in each array are separated by a distance c. This configuration is shown in Figure 3. By the principle of superposition, the elastic deformation at the surfa.ce can be calculated from the sum of the displacements due to the strain fields of each dislocation. The lateral elastic displacement is always negligible compared to the oscillation wavelength and only the vertical one is relevant to characterize the surface shape. The vertical displacement of the surface induced by a single dislocation is obtained by integration of the strain field components (3) of dislocation near a free surface (4). The strains are defined in the (x,z) coordinate system with the dislocation located at x = 0 and z = -d, and the surface at z = 0, as shown in Fig. 4.
CLEAVED
1676
SURFACES
BY AFM
Vol. 34, No. 11
4 Normalned surface displacement as a fimtion of dislocation spacing
3 2 I 0 UJb
-1 -2 -3 -0.5
0.5
0.0
1.0
1.5
XIX Figure 5. Normalized surface displacement of c/A of 0.05, 0.1, and 0.2. The distance to the first dislocation h, is equal to the dislocation spacing.
We obtain: u,(x,d)
= b x
dz (d’
(1)
+ x2)
The vertical displacement of a surface due to a single tilt boundary is found by summing the displacements due to each dislocation in the boundary. We also introduce the factor h into the calculation, which is the distance from the surface to the first dislocation. We obtain for a positive tilt boundary: U&
= c
Uz(x,pc
+ h)
p=o
(2)
The displacement due to a negative tilt boundary is the reverse of Eq.(2), i.e. U& = -U,,,:. Finally, the surface undulation is calculated by summing over the displacements due to an infinite array of alternating positive and negative tilt boundaries positioned at ql and A(q + l/2) respectively: + qlL) + U&(x + A(q + l/2))]
(3)
Equation (3) can be solved analytically using summation solutions in ref. (3). To simplify the expression, c, h, and x are replaced by the dimensionless constants k = 2&l, q = 27Nl and 5 = 27&k and the total displacement, UZDt,is normalized by the Burgers vector, b. The result is as follows: tot UZ _= b
-f
$
cosEsinh(pk + ‘1) (Pk
+ rl) cos2s
-
cosh2(pk + ‘I)
(4)
The calculation in the previous section yields an expression for the deflection of a surface as a function of the distance between dislocations in a tilt boundary, the spacing between boundaries, and the distance from the surface to the first dislocation. It leads to a local dislocation density of around lo9 cm-*. This value is higher than the typical vilue for such crystals. However as indicated previously, the phenomenon has been observed on only few areas and should not change the average density. The simplest case for the depth of the frst dislocation taken as the same as the dislocation spacing (h = c) is plotted in Fig. 5 for c/J. values of 0.05,O.l and 0.2. Equation (4) predicts a smoothly varying surface displacement whose amplitude rapidly attains 2 to 4 times the Burgers vector or roughly 0.5 to 1.5
Vol. 34, No. 11
-0.12
CLEAVED SURFACES BY AFM
-0.1
-0.08
-0.06
.0,01
+,.02
0
0.02
0.04
0.06
D.08
1677
0.1
0.92
Figure 6. Rocking curve relative to CaF, various areas.
run, depending on b. For small angles, the spacing between dislocations is related to the tilt angle, 8, by the expression 0 = b/c. In the case of CaF,, the Burgers vector is 0.39 nm (for a/2[ 1lo] dislocation). If the undulation wavelength is 0.65 pm then the values of c/A in Fig. 5 correspond to tilt angles of 0.68, 0.34 and 0.17 deg. The calculation shows that even very small tilt angles can produce several Angstroms of surface displacement, which is consistent with the experimental observations. The model of a series of tilt angles of alternating sign in the crystal can explain in both magnitude and form the observed phenomenon of nmscale surface undulations on cleaved single crystal surfaces. Low angles tilt boundaries have been often observed (5,6,7) in single crystals by various techniques (X-Ray topography, TEM, etching..), but they generally correspond to grains of several micrometers in size. It is worth noting that the observational area of an AFM with high vertical resolution is limited to a few micrometers, and hence periodic structures which may correspond to larger grain size cannot be observed with this apparatus. The present ‘observations only show the presence in some parts of the specimen, of grains smaller than a micrometer in size. Other experimental techniques previously reported are not capable of observing such features. Thus emh pitting studies are not suitable for such small dimensions; X-ray topography is limited by the size of the probe; TEM investigations are technically difficult to realize on such specimens because of irradiation problems. However an X-ray rocking curve analysis of the specimen that shows the surface undulation in Fi,g. 1 has been carried out and confirms the presence of low angle tilt boundaries, in agreement with the present model. In Figure 6, the dotted line represents a reference peak giving the experimental resolution; the other curves are experimented from various regions of the CaF, specimen. The incidence angle 0 has here no physical signific:ance. Each experimental curve has been centered artificially and normalized in order to compare rocking curves widths which characterize the diffracting planes misorient&ion. It shows unambiguously a slight misorientation of the crystallographic planes in agreement with the present assumptions.
1678
CLEAVED SURFACES BY AFM
Vol. 34, No. 11
Conclusion Surface undulations have been observed in a few regions of an unstressed single crystal of CaF2. This feature is clearly distinguishable from cleavage steps which are characterized by an abrupt change in height and appear as line boundaries between two areas of constant height in the images. The presence of an array of dislocations in the form of low angle tilt boundaries may provide an explanation for the appearance of an undulatory surface. X-ray rocking curve analysis agrees with this hypothesis. Low angles tilt boundaries have been often formed in several micrometer distances during crystal growth and can be accounted for by temperature and impurity concentrations fluctuations. The present observations show the presence in some parts of the sample, of grains smaller than a micrometer in size. Acknowledgment The authors wish to thank A. Declemy for carrying out the X-ray diffraction study and fruitful discussions and L. Cartz for useful suggestions. The work presented here was supported by the DRET agency. One of authors (M.K. Small) wishes to acknowledge the financial support of Region Poitou-Charentes. References I. 2. 3. 4. 5. 6. 7.
G. Binnig, C.F. Quate, Ch. Gerber, Whys.Rev. Lett. 56(9), p. 930 (1986). M.K. Small, C. Coupeau and J. Grilhe, Scr&~tuM&II. Muter. 32(10), 1573-1578 (1995). S. Timoshenko and J. N. Goodier, 7?reory ofelasfici@, McGraw-Hill (1951). J.P. Hirth and J. Lothe, Theory ofDislocations (John Wiley & Sons, New York, 1982) p. 733. S. Amelinckx and W. Dekeyser, The structure andproperties ofgruin boundaries, SolidState Physics, 8,325 (1959). C. Elbaum, Progress in Metal Physics (Pergamon Press, New York), 8, pp 203-253 (1959). S. Amelinckx, Actu Metallurgica, 2 (1954).
Pergamon PII 81359-6462(96)00058-9
NANOMETER UNDULATIONS ON CaF, CLEAVED SURFACES OBSERVED BY ATOMIC FORCE MICROSCOPY C. Coupeau, M.K. Small, N. Junqua and J. Grilhe Laboratoire de Metallurgie Physique-IX4 CNRS 13 1 Universite de Poitiers-Faculte des Sciences Bat. S.P.2.M.I Bd 3, Teleport 2, BP 179,86960 Futuroscope Cedex, France (Received October 23, 1995) (Revised November 22, 1995) Introduction
Cleaved surfaces of CaFz single crystals have been examined by Atomic Force Microscopy (AFM) (1). In a few places the surfaces exhibit smooth undulations in addition to abrupt atomic steps. The undulations are readily distinguished from straight cleavage steps caused by slight surface shifts from the ideal cleavage plane. Experimental data is presented for the existence of this surface feature and a model is proposed to explain this topography. ExDerimental
Procedure
Single crystals Iof calcium fluoride were cleaved in air and the cleavage plane (111) was examined immediately by Ihe AFM. Characterization was performed using an AFM with an interferometric deflection detector with a 18 pm scan range. Microfabricated silicon nitride cantilevers were used with a spring constant of 0.1 N/m and the applied force was of the order of 10 nN. The images shown here were taken from top to bottom unfiltered unless otherwise specified, but were treated to remove uniform curvature and slope. Results
As expected, the specimen showed cleavage steps running in preferred crystallographic directions (Figure la). Two types can be distinguished: there is a river pattern of cleavage steps in the top left of the image (Part A) and flat widely spaced terraces in part B. The perfectly straight cleavage steps in the lower part cannot be confused with the slightly wavy undulation (Part C) (enlargement Fig. lc-d) which is only consistent with im elastic deformation of the surface. The shape ofthe undulations varies somewhat from one area to another. The wavelength ranges from 550 to 700 pm of amplitude up to 2 nm, though generally closer to 1 nm. For instance, the cross section 1673
1674
CLEAVED SURFACES BY AFM
a
C
Vol. 34, No. 11
b
d
Figure 1. SAAFM CaF, images showing cleavage steps (A and B) and wavy undulation (C) with (a) scan range to 18 pm and scan angle to 00, (b) scan range to 18 pm and scan angle to 90’, (c) scan range to 9 pm, (d) scan range to 6 pm.
of the CaF, surface (line D in Fig. la) is seen in Figure 2. This undulation has a periodicity of 654 nm and an amplitude of 1.3 nm. The undulations often follow preferred directions though less precisely than the surface steps and seem to coincide with the intersection of the cleavage and the crystal growth planes. It is necessary to demonstrate that this effect is not an artifact of the microscopy, and several precautions have been taken to eliminate this possibility. Firstly, the scans were taken at different angles to see whether the undulations retain their orientation relative to the specimen (Figure lb). It must be noticed that this sample has been slightly deformed in situ at room temperature (2) to see the emergence of slip lines which correspond to the vertical lines in Figure 1b; they are not clearly visible in Figure 1a, because of the scan direction.
Figure 2. CaF2 image cress section (Line D in Fig. 1) which gives a spectral period of 654 nm.
Figure 3. Schematic of an array of subgrain boundaries in a semi-infinite medium.
Vol. 34, No. 11
CLEAVEDSURFACESBY AFM
1675
Figure 4. An edge dislocation at x = 0 and z = -d in an elastic isotropic halfspace.
Furthermore the AFM employed for these measurements is a special design (Stand Alone AFM) (2), which excludes the possible artifact due to interference effects between the laser beam reflected by the cantilever and the partial transmitted beam reflected by the substrate; this might occur in a standard AFM due to the curvature of the piezoelectric transducer capable of large scans. Moreover, if this interference occurs, it would occur everywhere on the scan area, which is not the case. To conclude, Figures 1c and Id show that the same surface undulation wavelength is obtained with a smaller scan area, which excludes the possibility of an artifact due to interference effects and confirms that the undulations are a feature of the cleaved surface since their presence or absence is completely reproducible. Discussion From these AFM observations of cleaved surfaces, it is evident that the undulations are not cleavage steps, nor are they artifacts caused by vibrations or electronic noise in the microscope. The surface undulations are almost certainly due to elastic deformation of the surface induced by internal defects, such as precipitates or dislocations. The measured amplitudes are too large to be caused by single dislocations, which can produce surface displacement no more than one Burgers vector. Larger effects would have to be due to the colmbined influence of several dislocations. A random arrangement of dislocations would not produce such a pattern; periodics arrays of dislocations can be accounted for by the presence of subgrain boundaries in the crystal. Calculations are presented for the surface displacement resulting from periodic arrangements of lowangle tilt boundaries modeled as an array of evenly spaced edge dislocations with Burgers vectors parallel to the surface of an elastic isotropic half-space. In a real crystal the subgrains will probably not be exactly perpendicular to the surface. In this case, dislocations very near the surface may glide to it and be armihilated, thus creating a dislocation-free zone to a depth greater than the dislocation spacing. However, for the sake of simplicity we will assume that the dislocations are futed in the crystal and that the surface falls at an arbitrary distance between two dislocations in the subgrain. Subgrain boundaries made up of dislocations of opposite sign are separated by a distance 1.12and dislocations in each array are separated by a distance c. This configuration is shown in Figure 3. By the principle of superposition, the elastic deformation at the surfa.ce can be calculated from the sum of the displacements due to the strain fields of each dislocation. The lateral elastic displacement is always negligible compared to the oscillation wavelength and only the vertical one is relevant to characterize the surface shape. The vertical displacement of the surface induced by a single dislocation is obtained by integration of the strain field components (3) of dislocation near a free surface (4). The strains are defined in the (x,z) coordinate system with the dislocation located at x = 0 and z = -d, and the surface at z = 0, as shown in Fig. 4.
CLEAVED
1676
SURFACES
BY AFM
Vol. 34, No. 11
4 Normalned surface displacement as a fimtion of dislocation spacing
3 2 I 0 UJb
-1 -2 -3 -0.5
0.5
0.0
1.0
1.5
XIX Figure 5. Normalized surface displacement of c/A of 0.05, 0.1, and 0.2. The distance to the first dislocation h, is equal to the dislocation spacing.
We obtain: u,(x,d)
= b x
dz (d’
(1)
+ x2)
The vertical displacement of a surface due to a single tilt boundary is found by summing the displacements due to each dislocation in the boundary. We also introduce the factor h into the calculation, which is the distance from the surface to the first dislocation. We obtain for a positive tilt boundary: U&
= c
Uz(x,pc
+ h)
p=o
(2)
The displacement due to a negative tilt boundary is the reverse of Eq.(2), i.e. U& = -U,,,:. Finally, the surface undulation is calculated by summing over the displacements due to an infinite array of alternating positive and negative tilt boundaries positioned at ql and A(q + l/2) respectively: + qlL) + U&(x + A(q + l/2))]
(3)
Equation (3) can be solved analytically using summation solutions in ref. (3). To simplify the expression, c, h, and x are replaced by the dimensionless constants k = 2&l, q = 27Nl and 5 = 27&k and the total displacement, UZDt,is normalized by the Burgers vector, b. The result is as follows: tot UZ _= b
-f
$
cosEsinh(pk + ‘1) (Pk
+ rl) cos2s
-
cosh2(pk + ‘I)
(4)
The calculation in the previous section yields an expression for the deflection of a surface as a function of the distance between dislocations in a tilt boundary, the spacing between boundaries, and the distance from the surface to the first dislocation. It leads to a local dislocation density of around lo9 cm-*. This value is higher than the typical vilue for such crystals. However as indicated previously, the phenomenon has been observed on only few areas and should not change the average density. The simplest case for the depth of the frst dislocation taken as the same as the dislocation spacing (h = c) is plotted in Fig. 5 for c/J. values of 0.05,O.l and 0.2. Equation (4) predicts a smoothly varying surface displacement whose amplitude rapidly attains 2 to 4 times the Burgers vector or roughly 0.5 to 1.5
Vol. 34, No. 11
-0.12
CLEAVED SURFACES BY AFM
-0.1
-0.08
-0.06
.0,01
+,.02
0
0.02
0.04
0.06
D.08
1677
0.1
0.92
Figure 6. Rocking curve relative to CaF, various areas.
run, depending on b. For small angles, the spacing between dislocations is related to the tilt angle, 8, by the expression 0 = b/c. In the case of CaF,, the Burgers vector is 0.39 nm (for a/2[ 1lo] dislocation). If the undulation wavelength is 0.65 pm then the values of c/A in Fig. 5 correspond to tilt angles of 0.68, 0.34 and 0.17 deg. The calculation shows that even very small tilt angles can produce several Angstroms of surface displacement, which is consistent with the experimental observations. The model of a series of tilt angles of alternating sign in the crystal can explain in both magnitude and form the observed phenomenon of nmscale surface undulations on cleaved single crystal surfaces. Low angles tilt boundaries have been often observed (5,6,7) in single crystals by various techniques (X-Ray topography, TEM, etching..), but they generally correspond to grains of several micrometers in size. It is worth noting that the observational area of an AFM with high vertical resolution is limited to a few micrometers, and hence periodic structures which may correspond to larger grain size cannot be observed with this apparatus. The present ‘observations only show the presence in some parts of the specimen, of grains smaller than a micrometer in size. Other experimental techniques previously reported are not capable of observing such features. Thus emh pitting studies are not suitable for such small dimensions; X-ray topography is limited by the size of the probe; TEM investigations are technically difficult to realize on such specimens because of irradiation problems. However an X-ray rocking curve analysis of the specimen that shows the surface undulation in Fi,g. 1 has been carried out and confirms the presence of low angle tilt boundaries, in agreement with the present model. In Figure 6, the dotted line represents a reference peak giving the experimental resolution; the other curves are experimented from various regions of the CaF, specimen. The incidence angle 0 has here no physical signific:ance. Each experimental curve has been centered artificially and normalized in order to compare rocking curves widths which characterize the diffracting planes misorient&ion. It shows unambiguously a slight misorientation of the crystallographic planes in agreement with the present assumptions.
1678
CLEAVED SURFACES BY AFM
Vol. 34, No. 11
Conclusion Surface undulations have been observed in a few regions of an unstressed single crystal of CaF2. This feature is clearly distinguishable from cleavage steps which are characterized by an abrupt change in height and appear as line boundaries between two areas of constant height in the images. The presence of an array of dislocations in the form of low angle tilt boundaries may provide an explanation for the appearance of an undulatory surface. X-ray rocking curve analysis agrees with this hypothesis. Low angles tilt boundaries have been often formed in several micrometer distances during crystal growth and can be accounted for by temperature and impurity concentrations fluctuations. The present observations show the presence in some parts of the sample, of grains smaller than a micrometer in size. Acknowledgment The authors wish to thank A. Declemy for carrying out the X-ray diffraction study and fruitful discussions and L. Cartz for useful suggestions. The work presented here was supported by the DRET agency. One of authors (M.K. Small) wishes to acknowledge the financial support of Region Poitou-Charentes. References I. 2. 3. 4. 5. 6. 7.
G. Binnig, C.F. Quate, Ch. Gerber, Whys.Rev. Lett. 56(9), p. 930 (1986). M.K. Small, C. Coupeau and J. Grilhe, Scr&~tuM&II. Muter. 32(10), 1573-1578 (1995). S. Timoshenko and J. N. Goodier, 7?reory ofelasfici@, McGraw-Hill (1951). J.P. Hirth and J. Lothe, Theory ofDislocations (John Wiley & Sons, New York, 1982) p. 733. S. Amelinckx and W. Dekeyser, The structure andproperties ofgruin boundaries, SolidState Physics, 8,325 (1959). C. Elbaum, Progress in Metal Physics (Pergamon Press, New York), 8, pp 203-253 (1959). S. Amelinckx, Actu Metallurgica, 2 (1954).