- Email: [email protected]
Quantitative analysis of strain field in thin films from HRTEM micrographs
Thin Solid Films 319 Ž1998. 157–162
Quantitative analysis of strain field in thin films from HRTEM micrographs E. Snoeck
a,)
, B. Warot a , H. Ardhuin a , A. Rocher a , M.J. Casanove a , R. Kilaas b, M.J. Hytch ¨ c a
CEMES-CNRS BP4347, F-31055 Toulouse Cedex 5, France b NCEMr LBL, Bldg. 72, Berkeley, CA 94720, USA c CECM 15, rue G. Urbain, F-94407 Vitry, France
Abstract A method for measuring and mapping displacement fields and strain fields has been developed. It is based on the Fourier analysis of a HRTEM lattice image selecting a strong Bragg reflection and performing an inverse Fourier transform. The phase component of the resulting complex image gives information on local displacements of atomic planes. The two-dimensional Ž2D. displacement field can then be derived by applying the method to two non-collinear Fourier components. Local strain tensor components ´ x x Ž™ r ., ´ y y Ž™ r ., ´ x y Ž™ r. ™ and ´ y x Ž r . can be obtained by analysing the derivative of the displacement field. Applied to HRTEM images of thin films, this technique gives quantitative information on the variations of the strain relaxations in the different layers. The method is illustrated studying a fully relaxed GaSbrGaAs interface. q 1998 Elsevier Science S.A. Keywords: Displacement; Strain; GaSbrGaAs interface; HREM
1. Introduction In general, the HRTEM image contrast is not a simple function of the position of individual atoms in the crystal, and the determination of the crystal structure often requires a comparison with computer calculations. However, in many cases, atomic columns will appear as nearly Gaussian shaped regions of contrast, appearing either dark or bright depending on the crystal thickness and the objective lens defocus. Such HRTEM images can, therefore, be used to determine local distortions of the lattice. Techniques based upon locating the peaks in image contrast and associating them with atomic columns have been used by many authors in order to analyze local variations in structural parameters w1x, such as lattice parameters in epitaxially strained layers w2,3x, displacement fields across ferroelectric domain boundaries and rigid body displacements w4x. These methods rely on fitting a 2D reference lattice to
)
Corresponding author. E-mail: [email protected].
0040-6090r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved. PII S 0 0 4 0 - 6 0 9 0 Ž 9 7 . 0 1 1 1 3 - 9
a subset of peaks associated with a non-distorted region of the image and measuring local deviations from lattice positions. The method discussed in this paper is based upon the analysis of phase components of two non-collinear spatial frequencies describing the HRTEM image.
2. Method A 1-dimensional perfectly periodic set of fringes can be written as a cosine function with a frequency g 0 equal to the inverse of the fringe periodicity and a constant phase value f 0 , which can be chosen equal to zero: I Ž x . s AcosŽ2p g 0 x q f 0 .. Slight spatial variations of the fringe periodicity can be included by introducing a spatially varying phase term, which is constant in the regions where the set of fringes is perfectly periodic but not when variations of this periodicity occur: I Ž x . s AcosŽ2p g 0 x q f Ž x ... This phase term can also be rewritten as a function
158
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
of the frequency g 0 and the displacement of the non-regular fringes from their ideal periodic position uŽ x .: f Ž x . s 2p g 0 uŽ x .. The displacements of the fringes from their periodic position can then be written: uŽ x . s f Ž x .r2p g 0 and can be easily determined from the phase term and the frequency of the reference set of fringes. Furthermore, the Eu Ž x . E fŽ x. deformation ´ x Ž™ r. s s can also be Ex E x 2p g 0 deduced. Such considerations applied on a periodic HRTEM micrograph can be used to measure the displacements of the atoms from their periodic positions and the strain field associated with these displacements. The method is further discussed by Hytch ¨ w5,6x and Hytch ¨ et al. w7x and previous analyses can be found in Refs. w7–9x.
ž /
3. HRTEM images and Fourier filtering Fig. 1 is an HRTEM micrograph of a GaSbrGaAs interface observed along the w110x zone axis with its Fourier transform in inset. GaSb and GaAs have the same
diamond structure with a lattice mismatch of 8%. Periodic misfit dislocations around which a Burger circuit is drawn show up at the GaSbrGaAs interface. Such a micrograph is a discrete periodic image that can be written as a Fourier series over a set of discrete frequency components corresponding to the Bragg reflections. Placing an aperture around one of these frequencies Ž ™ g 0 . such that no other Bragg frequency is included and performing an inverse Fourier transform will result in a complex image: ™™
HgX 0Ž ™ r . s A g 0 e 2p i g 0 r
Ž 1.
with a phase term that oscillates with a frequency < g 0 < s 1rd 0 , where d 0 corresponds to the distance between the planes normal to ™ g 0 . If there are slight variations in the spacing of these planes in the field of view, the apertured Fourier transform will contain additional information, D™ g Ž™ r ., centered around the vector ™ g 0 which measures variations in spacing and orientation of the chosen lattice planes across the image: ™ g Ž™ r . s™ g0 q D ™ g Ž™ r .. In the same ™ way, the amplitude term, A g Ž r ., will be space-dependent.
Fig. 1. HRTEM micrograph of a GaSbrGaAs interface observed along the w110x zone axis with its FFT inset.
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
159
The resulting inverse Fourier transform will now give an image: BXg Ž ™ r . s 2 Ag Ž™ r . cos Ž 2p ™ g Ž™ r.. s 2 Ag Ž™ r . cos 2p ™ g 0™ rqD™ g Ž™ r .™ r
ž
/
Ž 2.
Such an image is shown in Fig. 2, where the frequency corresponding to the Ž111. Bragg reflection of GaSb has been selected. On the image one clearly sees the additional Ž111. planes in the GaAs region. The space dependent phase term D ™ g Ž™ r .™ r can be rewritten as a function of the shift in lattice positions Žplanes., ™ uŽ™ r ., with respect to the reference lattice of GaSb given by 1rg 0 . Eq. Ž2. becomes: BXg Ž ™ r . s 2 Ag Ž™ r . cos2p ™ g 0™ rqD™ g Ž™ r .™ r s2 Ag s2 Ag
ž Ž r . cos2p ž g r y g Ž r . cos ž 2p g r q P
/
™
™™ 0
™ ™ 0u r
™
™™ 0
g0
Ž ./
r. / Ž™
Ž 3.
with ™
P™ g 0Ž r . s y2p
™™ ™ g0 u r
Fig. 3. Phase image wyp ,p x evidencing the 2p phase shift corresponding to the Ž111. supplementary planes in GaAs.
Ž .
Thus, the inverse Fourier transform will consist of two terms: an amplitude, A g Ž™ r ., and a phase, both given as a function of position, each can be displayed individually. Ž™. Ž™. The phase 2p ™ g 0™ r q P™ g 0 r can be imaged and the P™ g0 r term can be found by subtracting from the image the ramp 2p ™ g 0™ r component. The vector g 0 determines a set of reference planes that can be though of as extending throughout the image. Thus, it is important to decide what vector to select as the reference vector g 0 since the remaining analysis depends on it. In practice, one chooses
a reference area in the image where the lattice parameter is believed to be constant. One can refine the g-vector by selecting a reference region in the phase image and fitting a surface to the phase within that area in order to subtract off a possible ramp due to a slight error in the value for g 0 originally chosen, so that the g 0 vector used for the analysis corresponds to the mean plane spacing for that reference region. Such calculations were performed on our example and the resulting phase image is shown in Fig. 3. One clearly sees a constant contrast region in the GaSb area, which has been chosen as reference region, and phase shifts on the GaAs side. Each 2p phase shift corresponds to one additional Ž111. plane.
4. Determination of the displacement field As discussed in the Section 2, the phase image related to g 0 can give information about the displacements of the corresponding planes relative to the reference as a function of position within the image. By choosing two non-collinear ™ g i and ™ g j vectors, which define a 2D lattice, one can derive the displacement field with respect to the chosen reference lattice from the resulting phase images as follows: Pg iŽ ™ r . s y2p ™™ gi uŽ™ r . s y2p Ž g i x u x Ž ™ r . q gi y u y Ž™ r. .
Ž 4.
Fig. 2. Inverse Fourier transform of the image after selecting the Ž111. reflection showing the presence of extra Ž111. planes in the GaAs.
where g i x and g i y are the x and y components of the vectors ™ g i and u x (™ r ., u y (™ r . are the x and y components of the displacement field at the position r in the image. Provided that the same area has been selected for refining
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
160
™
Fig. 4. Images of the displacement components: Ža. u x Ž™ r .; Žb. u y Ž™ r .; Žc. displacement field UŽ™ r .; Žd. enlargement of the displacement field around a dislocation line.
the two non-collinear vectors ™ g 1 and ™ g 2 , the displacement field with respect to the mean 2D lattice can be calculated from Eq. Ž4. and is given by: ™
uxŽ r. sy ™
uyŽ r. sy
1 2p 1 2p
ž ž
Pg 1Ž ™ r . g 2 y y Pg 2Ž ™ r . g1 y g1 x g2 y y g1 y g2 x Pg 2Ž ™ r . g 1 x y Pg 1Ž ™ r . g2 x g1 x g2 y y g1 y g2 x
/ /
Ž 5a . Ž 5b .
Eqs. Ž5a. and Ž5b. represent two images containing the x and y components of the displacement field. This was performed in our example by selecting the Ž111. and Ž111.
Bragg reflections of GaSb as ™ g 1 and ™ g 2 . The u x Ž™ r . and ™ Ž . u y r images are shown in Fig. 4a and b, respectively, where the x direction is perpendicular to the interface and the y direction parallel to it. These images can be combined into a vector plot showing the displacement field as a function of position in the image. The resulting displacement field of GaAs atomic positions relative to the GaSb one is shown in Fig. 4c. When the relative atomic displacement in GaAs corresponds to a integral number of GaSb lattice spacings, one finds an equivalent zero shift which defines the so-called coincidence lattice. Such an image is the equivalent of the moire´ fringes observed when
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
161
Fig. 5. Images of the strain components: Ža. ´ x x Ž™ r .; Žb. ´ y y Ž™ r .; Žc. and Žd. projection of the ´ x x Ž™ r . and ´ y y Ž™ r . strain image perpendicularly to the interface.
two different lattices are superimposed. Moreover, such a displacement map permits us to visualize the atomic displacements close to the interface in the vicinity of the dislocations as shown in Fig. 4d. 5. Determination of the strain field The 2 = 2 local distortion tensor of the lattice relative to the reference region can be deduced from the derivative of the displacement field:
E™ ui Ž™ r.
s Ei j Ž ™ r . with i , j s x , y
Ej In order to avoid cumulative errors in successive calculations of the displacements u x Ž™ r . and u y Ž™ r . ŽEqs. Ž5a.
and Ž5b.. and their derivatives, the strain tensor can be obtained more accurately from the phase images by calculating the derivative of Eq. Ž4.. One gets:
E Ex
ž
Pg iŽ ™ r . s y2p ™ gi
/
E™ uŽ™ r. Ex
ž
s y2p g i x
E Ey
žP
gi
™
Ž r . / s y2p
™ gi
ž
E u x Ž™ r. Ex
q gi y
E u y Ž™ r. Ex
/
Ž 6a .
/
Ž 6b .
E™ uŽ™ r. Ey
s y2p g i x
E u x Ž™ r. Ey
q gi y
E u y Ž™ r. Ey
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
162
giving
E ™
Ex xŽ r. sy
1 2p
žP Ex
r . / g2 y y Ž™
E
P Ž™ r . g1 y E x g2 g1 x g2 y y g1 y g2 x
g1
ž
/
0
Ž 7a . E E y y Ž™ r. sy
E x y Ž™ r. sy
™
EyxŽ r. sy
1 2p
1 2p
1 2p
Ey
žP
g2
r . / g1 x y Ž™
žP Ey
P Ž™ r . g2 x E y g1 g1 x g2 y y g1 y g2 x
r . / g2 y y Ž™
g1
ž
/
0 0 0
E
P Ž™ r . g1 y E y g2 g1 x g2 y y g1 y g2 x
ž
/
Ž 7c . E
žP Ex
g2
r . / g1 x y Ž™
6. Conclusions
E
Ž 7b . E
perfect Lomer dislocation with an edge Burgers vector parallel to the interface. Such dislocations regularly distributed along the interface were shown to allow the complete relaxation of the GaSb layer.
E
P Ž™ r . g2 x E x g1 g1 x g2 y y g1 y g2 x
ž
/
Ž 7d . The local distortion components, E i j Ž™ r ., can then be found directly by applying Eqs. Ž7a., Ž7b., Ž7c. and Ž7d.. The strain tensor ´ i j Ž™ r . and the local rigid rotation v i j Ž™ r ., can be obtained by separating the E i j Ž™ r . matrix into symmetric and antisymmetric terms: ´ i j Ž™ r . s 1r2wE i j Ž™ r. ™T ™ ™ q E i j Ž r . x and v i j Ž r . s 1r2wE i j Ž r . y E i j Ž™ r . T x where T denotes the transpose of the matrix. These calculations were performed on our example and the resulting ´ x x Ž™ r. and ´ y y Ž™ r . images are reproduced in Fig. 5a and b, respectively. One has to keep in mind that the strain tensor calculated in this way corresponds to the relative distortions of one lattice relative to the reference one Žin our case Ž d GaSb y d GaAs .rd GaSb ., which is different than the usual strain tensor: Ž d GaAs-strained y d GaAs-bulk .rd GaAs-bulk .. On these images two different flat grey levels appear in the two layers with very sharp contrast located around the dislocation cores where the strain is maximal. The mean values of the ´ x x Ž™ r . and ´ y y Ž™ r . strain images projected parallel to the interface and between the dislocation cores are plotted in Fig. 5c and d, respectively. The strain values of both ´ x x Ž™ r . and ´ y y Ž™ r . in the GaAs substrate relatively to the GaSb epilayer far from the interface reach the expected value of y8%, which exactly corresponds to the misfit and indicates that the layer is completely relaxed. Moreover, such curves indicate that the interface is very ˚ Previous results w10x sharp with a thickness of about 15 A. have shown that the two interface dislocations observed in Fig. 1 are 608 dislocations with two complementary Burger vectors 1r2 w101x and 1r2 w011x, the result of which is one
The method described here, illustrated on the well known GaSbrGaAs system, permits local variations of atomic positions to be determined from HRTEM micrographs without establishing the peak positions in the image. It is, hence, less dependent on imaging conditions than the peak finding method. Quantitative information on the strain field can then be obtained as shown by the ˚. results obtained on epitaxial Co ultrathin films Ž8 A ˚ Ž . w x Ž . deposited on Au 111 11 , on Fe 15 A epitaxially deposited on MgOŽ001. w12x, and on the fine structure of ferroelastic domain walls in PbTiO 3 w7,13,14x. However, like the peak finding procedure w1x, this method requires some precautions: the HRTEM micrographs have to present flat background contrast: if contrast changes occur on a large scale Ždue for instance to thickness variations., one would get additional phase shifts, which will give large artefacts both on the displacement field and the strain one. Moreover, the HRTEM images should not be noisy: the noise can be described by the superimposition of additional non-negligible g vectors, resulting in a low resolution displacement field image and a highly perturbed nonsignificant strain map.
References w1x R. Bierwolf, M. Hohenstein, F. Phillipp, O. Brandt, G.E. Crook, K. Ploog, Ultramicroscopy 49 Ž1993. 273. w2x P. Bayle, T. Deutsch, B. Gilles, F. Lancon, A. Marty, J. Thibault, Ultramicroscopy 56 Ž1993. 94. w3x P.H. Jouneau, A. Tardot, B. Feuillet, H. Mariette, J. Cibert, J. Appl. Phys. 75 Ž1994. 7310. w4x S. Stemmer, S.K. Streiffer, F. Ernst, M. Ruhle, W.Y. Hsu, R. Raj, ¨ Solid State Ionics 75 Ž1995. 43. w5x M.J. Hytch, in: P. Hawkes ŽEd.., Scanning Microscopy Suppl. 10, ¨ Signal and Image in Microscopy and Microanalyses, 1997. w6x M.J. Hytch, Microstructures, Microscopy Microanalyses, 1997. ¨ w7x M.J. Hytch, E. Snoeck, R. Kilaas, submitted to Ultramicroscopy. ¨ w8x M.J. Hytch, M. Gandais, Phil. Mag. A 72 Ž1995. 619. ¨ w9x M.J. Hytch, P. Bayle, Proc. ICEM 13th ŽEditions de Physique, Paris. ¨ A 2 Ž1994. 129. w10x A. Rocher, J.M. Kang, Inst. Phys. Conf. Ser. 146 Ž1995. 135–142. w11x H. Ardhuin, E. Snoeck, M.J. Casanove, submitted to the J. Cryst. Growth. w12x B. Warot, E. Snoeck, L. Ressier, J.P. Peyrade, Louis Neel ´ Coll., Banyuls, 5–7 June 1997. w13x F.M. Ross, R. Kilaas, E. Snoeck, M. Hytch, A. Thorel, L. Normand, ¨ Proceeding of the MRS-Meeting Boston 1996, in press. w14x L. Normand, R. Kilaas, Y. Montardi, A. Thorel, Fourth Euro ceramics, Electroceramics 5 Ž1995. 241.
Quantitative analysis of strain field in thin films from HRTEM micrographs E. Snoeck
a,)
, B. Warot a , H. Ardhuin a , A. Rocher a , M.J. Casanove a , R. Kilaas b, M.J. Hytch ¨ c a
CEMES-CNRS BP4347, F-31055 Toulouse Cedex 5, France b NCEMr LBL, Bldg. 72, Berkeley, CA 94720, USA c CECM 15, rue G. Urbain, F-94407 Vitry, France
Abstract A method for measuring and mapping displacement fields and strain fields has been developed. It is based on the Fourier analysis of a HRTEM lattice image selecting a strong Bragg reflection and performing an inverse Fourier transform. The phase component of the resulting complex image gives information on local displacements of atomic planes. The two-dimensional Ž2D. displacement field can then be derived by applying the method to two non-collinear Fourier components. Local strain tensor components ´ x x Ž™ r ., ´ y y Ž™ r ., ´ x y Ž™ r. ™ and ´ y x Ž r . can be obtained by analysing the derivative of the displacement field. Applied to HRTEM images of thin films, this technique gives quantitative information on the variations of the strain relaxations in the different layers. The method is illustrated studying a fully relaxed GaSbrGaAs interface. q 1998 Elsevier Science S.A. Keywords: Displacement; Strain; GaSbrGaAs interface; HREM
1. Introduction In general, the HRTEM image contrast is not a simple function of the position of individual atoms in the crystal, and the determination of the crystal structure often requires a comparison with computer calculations. However, in many cases, atomic columns will appear as nearly Gaussian shaped regions of contrast, appearing either dark or bright depending on the crystal thickness and the objective lens defocus. Such HRTEM images can, therefore, be used to determine local distortions of the lattice. Techniques based upon locating the peaks in image contrast and associating them with atomic columns have been used by many authors in order to analyze local variations in structural parameters w1x, such as lattice parameters in epitaxially strained layers w2,3x, displacement fields across ferroelectric domain boundaries and rigid body displacements w4x. These methods rely on fitting a 2D reference lattice to
)
Corresponding author. E-mail: [email protected].
0040-6090r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved. PII S 0 0 4 0 - 6 0 9 0 Ž 9 7 . 0 1 1 1 3 - 9
a subset of peaks associated with a non-distorted region of the image and measuring local deviations from lattice positions. The method discussed in this paper is based upon the analysis of phase components of two non-collinear spatial frequencies describing the HRTEM image.
2. Method A 1-dimensional perfectly periodic set of fringes can be written as a cosine function with a frequency g 0 equal to the inverse of the fringe periodicity and a constant phase value f 0 , which can be chosen equal to zero: I Ž x . s AcosŽ2p g 0 x q f 0 .. Slight spatial variations of the fringe periodicity can be included by introducing a spatially varying phase term, which is constant in the regions where the set of fringes is perfectly periodic but not when variations of this periodicity occur: I Ž x . s AcosŽ2p g 0 x q f Ž x ... This phase term can also be rewritten as a function
158
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
of the frequency g 0 and the displacement of the non-regular fringes from their ideal periodic position uŽ x .: f Ž x . s 2p g 0 uŽ x .. The displacements of the fringes from their periodic position can then be written: uŽ x . s f Ž x .r2p g 0 and can be easily determined from the phase term and the frequency of the reference set of fringes. Furthermore, the Eu Ž x . E fŽ x. deformation ´ x Ž™ r. s s can also be Ex E x 2p g 0 deduced. Such considerations applied on a periodic HRTEM micrograph can be used to measure the displacements of the atoms from their periodic positions and the strain field associated with these displacements. The method is further discussed by Hytch ¨ w5,6x and Hytch ¨ et al. w7x and previous analyses can be found in Refs. w7–9x.
ž /
3. HRTEM images and Fourier filtering Fig. 1 is an HRTEM micrograph of a GaSbrGaAs interface observed along the w110x zone axis with its Fourier transform in inset. GaSb and GaAs have the same
diamond structure with a lattice mismatch of 8%. Periodic misfit dislocations around which a Burger circuit is drawn show up at the GaSbrGaAs interface. Such a micrograph is a discrete periodic image that can be written as a Fourier series over a set of discrete frequency components corresponding to the Bragg reflections. Placing an aperture around one of these frequencies Ž ™ g 0 . such that no other Bragg frequency is included and performing an inverse Fourier transform will result in a complex image: ™™
HgX 0Ž ™ r . s A g 0 e 2p i g 0 r
Ž 1.
with a phase term that oscillates with a frequency < g 0 < s 1rd 0 , where d 0 corresponds to the distance between the planes normal to ™ g 0 . If there are slight variations in the spacing of these planes in the field of view, the apertured Fourier transform will contain additional information, D™ g Ž™ r ., centered around the vector ™ g 0 which measures variations in spacing and orientation of the chosen lattice planes across the image: ™ g Ž™ r . s™ g0 q D ™ g Ž™ r .. In the same ™ way, the amplitude term, A g Ž r ., will be space-dependent.
Fig. 1. HRTEM micrograph of a GaSbrGaAs interface observed along the w110x zone axis with its FFT inset.
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
159
The resulting inverse Fourier transform will now give an image: BXg Ž ™ r . s 2 Ag Ž™ r . cos Ž 2p ™ g Ž™ r.. s 2 Ag Ž™ r . cos 2p ™ g 0™ rqD™ g Ž™ r .™ r
ž
/
Ž 2.
Such an image is shown in Fig. 2, where the frequency corresponding to the Ž111. Bragg reflection of GaSb has been selected. On the image one clearly sees the additional Ž111. planes in the GaAs region. The space dependent phase term D ™ g Ž™ r .™ r can be rewritten as a function of the shift in lattice positions Žplanes., ™ uŽ™ r ., with respect to the reference lattice of GaSb given by 1rg 0 . Eq. Ž2. becomes: BXg Ž ™ r . s 2 Ag Ž™ r . cos2p ™ g 0™ rqD™ g Ž™ r .™ r s2 Ag s2 Ag
ž Ž r . cos2p ž g r y g Ž r . cos ž 2p g r q P
/
™
™™ 0
™ ™ 0u r
™
™™ 0
g0
Ž ./
r. / Ž™
Ž 3.
with ™
P™ g 0Ž r . s y2p
™™ ™ g0 u r
Fig. 3. Phase image wyp ,p x evidencing the 2p phase shift corresponding to the Ž111. supplementary planes in GaAs.
Ž .
Thus, the inverse Fourier transform will consist of two terms: an amplitude, A g Ž™ r ., and a phase, both given as a function of position, each can be displayed individually. Ž™. Ž™. The phase 2p ™ g 0™ r q P™ g 0 r can be imaged and the P™ g0 r term can be found by subtracting from the image the ramp 2p ™ g 0™ r component. The vector g 0 determines a set of reference planes that can be though of as extending throughout the image. Thus, it is important to decide what vector to select as the reference vector g 0 since the remaining analysis depends on it. In practice, one chooses
a reference area in the image where the lattice parameter is believed to be constant. One can refine the g-vector by selecting a reference region in the phase image and fitting a surface to the phase within that area in order to subtract off a possible ramp due to a slight error in the value for g 0 originally chosen, so that the g 0 vector used for the analysis corresponds to the mean plane spacing for that reference region. Such calculations were performed on our example and the resulting phase image is shown in Fig. 3. One clearly sees a constant contrast region in the GaSb area, which has been chosen as reference region, and phase shifts on the GaAs side. Each 2p phase shift corresponds to one additional Ž111. plane.
4. Determination of the displacement field As discussed in the Section 2, the phase image related to g 0 can give information about the displacements of the corresponding planes relative to the reference as a function of position within the image. By choosing two non-collinear ™ g i and ™ g j vectors, which define a 2D lattice, one can derive the displacement field with respect to the chosen reference lattice from the resulting phase images as follows: Pg iŽ ™ r . s y2p ™™ gi uŽ™ r . s y2p Ž g i x u x Ž ™ r . q gi y u y Ž™ r. .
Ž 4.
Fig. 2. Inverse Fourier transform of the image after selecting the Ž111. reflection showing the presence of extra Ž111. planes in the GaAs.
where g i x and g i y are the x and y components of the vectors ™ g i and u x (™ r ., u y (™ r . are the x and y components of the displacement field at the position r in the image. Provided that the same area has been selected for refining
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
160
™
Fig. 4. Images of the displacement components: Ža. u x Ž™ r .; Žb. u y Ž™ r .; Žc. displacement field UŽ™ r .; Žd. enlargement of the displacement field around a dislocation line.
the two non-collinear vectors ™ g 1 and ™ g 2 , the displacement field with respect to the mean 2D lattice can be calculated from Eq. Ž4. and is given by: ™
uxŽ r. sy ™
uyŽ r. sy
1 2p 1 2p
ž ž
Pg 1Ž ™ r . g 2 y y Pg 2Ž ™ r . g1 y g1 x g2 y y g1 y g2 x Pg 2Ž ™ r . g 1 x y Pg 1Ž ™ r . g2 x g1 x g2 y y g1 y g2 x
/ /
Ž 5a . Ž 5b .
Eqs. Ž5a. and Ž5b. represent two images containing the x and y components of the displacement field. This was performed in our example by selecting the Ž111. and Ž111.
Bragg reflections of GaSb as ™ g 1 and ™ g 2 . The u x Ž™ r . and ™ Ž . u y r images are shown in Fig. 4a and b, respectively, where the x direction is perpendicular to the interface and the y direction parallel to it. These images can be combined into a vector plot showing the displacement field as a function of position in the image. The resulting displacement field of GaAs atomic positions relative to the GaSb one is shown in Fig. 4c. When the relative atomic displacement in GaAs corresponds to a integral number of GaSb lattice spacings, one finds an equivalent zero shift which defines the so-called coincidence lattice. Such an image is the equivalent of the moire´ fringes observed when
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
161
Fig. 5. Images of the strain components: Ža. ´ x x Ž™ r .; Žb. ´ y y Ž™ r .; Žc. and Žd. projection of the ´ x x Ž™ r . and ´ y y Ž™ r . strain image perpendicularly to the interface.
two different lattices are superimposed. Moreover, such a displacement map permits us to visualize the atomic displacements close to the interface in the vicinity of the dislocations as shown in Fig. 4d. 5. Determination of the strain field The 2 = 2 local distortion tensor of the lattice relative to the reference region can be deduced from the derivative of the displacement field:
E™ ui Ž™ r.
s Ei j Ž ™ r . with i , j s x , y
Ej In order to avoid cumulative errors in successive calculations of the displacements u x Ž™ r . and u y Ž™ r . ŽEqs. Ž5a.
and Ž5b.. and their derivatives, the strain tensor can be obtained more accurately from the phase images by calculating the derivative of Eq. Ž4.. One gets:
E Ex
ž
Pg iŽ ™ r . s y2p ™ gi
/
E™ uŽ™ r. Ex
ž
s y2p g i x
E Ey
žP
gi
™
Ž r . / s y2p
™ gi
ž
E u x Ž™ r. Ex
q gi y
E u y Ž™ r. Ex
/
Ž 6a .
/
Ž 6b .
E™ uŽ™ r. Ey
s y2p g i x
E u x Ž™ r. Ey
q gi y
E u y Ž™ r. Ey
E. Snoeck et al.r Thin Solid Films 319 (1998) 157–162
162
giving
E ™
Ex xŽ r. sy
1 2p
žP Ex
r . / g2 y y Ž™
E
P Ž™ r . g1 y E x g2 g1 x g2 y y g1 y g2 x
g1
ž
/
0
Ž 7a . E E y y Ž™ r. sy
E x y Ž™ r. sy
™
EyxŽ r. sy
1 2p
1 2p
1 2p
Ey
žP
g2
r . / g1 x y Ž™
žP Ey
P Ž™ r . g2 x E y g1 g1 x g2 y y g1 y g2 x
r . / g2 y y Ž™
g1
ž
/
0 0 0
E
P Ž™ r . g1 y E y g2 g1 x g2 y y g1 y g2 x
ž
/
Ž 7c . E
žP Ex
g2
r . / g1 x y Ž™
6. Conclusions
E
Ž 7b . E
perfect Lomer dislocation with an edge Burgers vector parallel to the interface. Such dislocations regularly distributed along the interface were shown to allow the complete relaxation of the GaSb layer.
E
P Ž™ r . g2 x E x g1 g1 x g2 y y g1 y g2 x
ž
/
Ž 7d . The local distortion components, E i j Ž™ r ., can then be found directly by applying Eqs. Ž7a., Ž7b., Ž7c. and Ž7d.. The strain tensor ´ i j Ž™ r . and the local rigid rotation v i j Ž™ r ., can be obtained by separating the E i j Ž™ r . matrix into symmetric and antisymmetric terms: ´ i j Ž™ r . s 1r2wE i j Ž™ r. ™T ™ ™ q E i j Ž r . x and v i j Ž r . s 1r2wE i j Ž r . y E i j Ž™ r . T x where T denotes the transpose of the matrix. These calculations were performed on our example and the resulting ´ x x Ž™ r. and ´ y y Ž™ r . images are reproduced in Fig. 5a and b, respectively. One has to keep in mind that the strain tensor calculated in this way corresponds to the relative distortions of one lattice relative to the reference one Žin our case Ž d GaSb y d GaAs .rd GaSb ., which is different than the usual strain tensor: Ž d GaAs-strained y d GaAs-bulk .rd GaAs-bulk .. On these images two different flat grey levels appear in the two layers with very sharp contrast located around the dislocation cores where the strain is maximal. The mean values of the ´ x x Ž™ r . and ´ y y Ž™ r . strain images projected parallel to the interface and between the dislocation cores are plotted in Fig. 5c and d, respectively. The strain values of both ´ x x Ž™ r . and ´ y y Ž™ r . in the GaAs substrate relatively to the GaSb epilayer far from the interface reach the expected value of y8%, which exactly corresponds to the misfit and indicates that the layer is completely relaxed. Moreover, such curves indicate that the interface is very ˚ Previous results w10x sharp with a thickness of about 15 A. have shown that the two interface dislocations observed in Fig. 1 are 608 dislocations with two complementary Burger vectors 1r2 w101x and 1r2 w011x, the result of which is one
The method described here, illustrated on the well known GaSbrGaAs system, permits local variations of atomic positions to be determined from HRTEM micrographs without establishing the peak positions in the image. It is, hence, less dependent on imaging conditions than the peak finding method. Quantitative information on the strain field can then be obtained as shown by the ˚. results obtained on epitaxial Co ultrathin films Ž8 A ˚ Ž . w x Ž . deposited on Au 111 11 , on Fe 15 A epitaxially deposited on MgOŽ001. w12x, and on the fine structure of ferroelastic domain walls in PbTiO 3 w7,13,14x. However, like the peak finding procedure w1x, this method requires some precautions: the HRTEM micrographs have to present flat background contrast: if contrast changes occur on a large scale Ždue for instance to thickness variations., one would get additional phase shifts, which will give large artefacts both on the displacement field and the strain one. Moreover, the HRTEM images should not be noisy: the noise can be described by the superimposition of additional non-negligible g vectors, resulting in a low resolution displacement field image and a highly perturbed nonsignificant strain map.
References w1x R. Bierwolf, M. Hohenstein, F. Phillipp, O. Brandt, G.E. Crook, K. Ploog, Ultramicroscopy 49 Ž1993. 273. w2x P. Bayle, T. Deutsch, B. Gilles, F. Lancon, A. Marty, J. Thibault, Ultramicroscopy 56 Ž1993. 94. w3x P.H. Jouneau, A. Tardot, B. Feuillet, H. Mariette, J. Cibert, J. Appl. Phys. 75 Ž1994. 7310. w4x S. Stemmer, S.K. Streiffer, F. Ernst, M. Ruhle, W.Y. Hsu, R. Raj, ¨ Solid State Ionics 75 Ž1995. 43. w5x M.J. Hytch, in: P. Hawkes ŽEd.., Scanning Microscopy Suppl. 10, ¨ Signal and Image in Microscopy and Microanalyses, 1997. w6x M.J. Hytch, Microstructures, Microscopy Microanalyses, 1997. ¨ w7x M.J. Hytch, E. Snoeck, R. Kilaas, submitted to Ultramicroscopy. ¨ w8x M.J. Hytch, M. Gandais, Phil. Mag. A 72 Ž1995. 619. ¨ w9x M.J. Hytch, P. Bayle, Proc. ICEM 13th ŽEditions de Physique, Paris. ¨ A 2 Ž1994. 129. w10x A. Rocher, J.M. Kang, Inst. Phys. Conf. Ser. 146 Ž1995. 135–142. w11x H. Ardhuin, E. Snoeck, M.J. Casanove, submitted to the J. Cryst. Growth. w12x B. Warot, E. Snoeck, L. Ressier, J.P. Peyrade, Louis Neel ´ Coll., Banyuls, 5–7 June 1997. w13x F.M. Ross, R. Kilaas, E. Snoeck, M. Hytch, A. Thorel, L. Normand, ¨ Proceeding of the MRS-Meeting Boston 1996, in press. w14x L. Normand, R. Kilaas, Y. Montardi, A. Thorel, Fourth Euro ceramics, Electroceramics 5 Ž1995. 241.