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Analysis of crystal defects by transmission electron microscopy
Materials Chemistry 4 (1979) 453 - 472 © CENFOR S.R.L. - Printed in Italy
A N A L Y S I S OF CRYSTAL DEFECTS BY TRANSMISSION ELECTRON MICROSCOPY *
A. ARMIGLIATO Laboratorio
LAMEL
del C.N.R.
- Via Castagnoli,
1 - 40126
BOLOGNA
- Italy.
Summary -- In the present paper the basic concepts of the transmission electron microscopy are outlined. In particular, the relevant aspects of both the kinematical and the dynamical theory of contrast, and their usefulness in the interpretation of the electron micrographs and of the diffractior~ patterns, are shortly discussed. Among the various examples of application of this technique to the materials science reported in literature, the methods currently employed to characterize, from the electron micrographs, lattice defects in crystalline materials, such as dislocations, loops, stacking faults and twins, are emphasized in this article.
INTRODUCTION
The role of the transmission electron microscopy in the different fields of the scientific research is of increasing importance. This technique has proved to be a powerful tool in the study of lattice defects in crystalline materials, like metals, semiconductors, minerals, glasses and ceramics. To analyze defects by electron microscopy, a knowledge of the principles of electron diffraction as well as the theory of image con-
* Presented at the IV Scientific Meeting of the Italian Association for Crystal Growth (AICC), Parma, Italy, 26-28 February 1979.
454 trast is required. Although a detailed account of the kinematical and dynamical theories of contrast is beyond the scope of the present article, some introductory remarks will be given. Among the typical problems encountered by the materials scientist, it will be shown how the study of the electron micrographs and the corresponding diffraction patterns enables one to determine the Burgers vector of a dislocation, the displacement vector of a stacking fault and the twinning planes in a crystal.
FORMATION OF THE IMAGE AND OF THE DIFFRACTION PATTERN As it is well known, the objective lens of a transmission electron microscope (TEM) focuses an electron beam transmitted by a crystalline thin film in its back focal plane, giving rise to a diffraction pattern; in the conjugate plane of the object plane, an inverted image takes place (the so-called 'first image'). Therefore, if the subsequent lens (first intermediate lens) is energized so that its object plane is the objective back focal plane, the diffraction pattern will be imaged onto the final screen; the overall magnification of the pattern depends on the product of the objective focal length and the magnification of the intermediates and the projector lenses. Instead, if the object plane of the first intermediate lens is the first image plane, a magnified image of the specimen will be obtained. The image and the diffraction pattern are not aligned: in fact, since the magnetic lenses rotate the plane of motion of the electrons as a function of their excitation, there will be a relative rotation of the diffraction pattern with respect to the image. In addition, as the objective lens invertes the image and not the diffraction pattern, an extra inversion will take place, which is not modified by all the subsequent lenses, because they act in the same way on both the image and the diffraction pattern. Such an extra inversion always occurs in a five lenses TEM , whereas in a six lenses instrument it depends on the excitation of the magnifying lenses. As will be seen in the next section, the knowledge of the im-
455 age-to-diffraction pattern rotation is of fundamental importance in the analysis of the defects and of the crystallographic directions. The ray paths corresponding to an image and a diffraction pattern in a six lenses TEM are sketched in Fig. 1, both in the case where the inversion is present (Fig. la) and absent (Fig. lb) I
specimen
objective lens
objective lens
image
- -
f
~..... intermedi~ate dif fraction lens
~ q
- -
Final image
- ~
Final diffroctlon potlern
(a)
Ist image
diffraction lens
intermed~te lens
-~
projector lens
./
Final diffrachon pattern
Final enoge
(b)
Fig. 1 - Ray paths for the formation o f an image and o f the corresponding diffraction pattern in a six-lenses TEM, both in case o f inversion (a) and non inversion (b) (After Loretto and Smallmanl ).
If an aperture is inserted at the level of the first image, the diffraction pattern of a selected area of the specimen (SAD) is obtained. The correspondence between the area included in the aperture and the area which actually diffracts depends on both the spherical aberration and the focusing errors and is not better than about 2000 A at 100 kV. An image obtained with the direct and the diffracted waves exhibits a poor contrast. Instead, if only one beam is selected by means
456 of an aperture inserted in the objective back focal plane (objective aperture), the contrast will be greatly improved. This mode of operation is called 'diffraction contrast', and is currently used in the analysis of crystal defects. In particular, if the selected beam is the transmitted one, a bright-field image will be obtained, whereas if a diffracted beam is chosen, a dark-field image will be produced.
ELECTRON DIFFRACTION PATTERNS The scattering of a monochromatic radiation by a crystal leads to the formation of diffracted beams which obey the Bragg law: (1)
n X = 2 d sin0
where 0 i s the angle between the incident beam and the planes of the crystal of spacing d, X is the wavelength of the electrons (X (A) =X/150/[1 + 10"6E(volts)] ), n is the diffraction order and E is the accelerating voltage. In the case of a single crystal, the geometrical formation of diffraction patterns is more easily represented by the reciprocal lattice and Ewald sphere construction. The incident beam direction is identified by the vector of modulus 1IX which joins the centre of the sphere to the centre of the reciprocal lattice; if another point of this lattice lies on the sphere, a diffracted beam will be originated at an angle 20, according to the Bragg law. The g--vector is the normal to the diffracting planes and its modulus is [ g l = 1/d. At 100 kV the radius of the Ewald sphere is about 50 times the spacing of the low indices lattice planes of the most c o m m o n materials and thus an electron diffraction pattern reproduces an indistorted section of the reciprocal lattice perpendicular to the beam direction. Instead, in the case of X-ray diffraction, where 1]X is of the same order of magnitude of [ ~ [, a distorted image of the corresponding section of the reciprocal lattice will be formed, unless special geometries are used, as in the precession method. In addition, due to the short wavelength of the electrons us-
457
ed in a TEM (X = 0.037 )~ at 100 kV), the Bragg angle 20 is of the order of 1 ° for low indices reflections and then only the lattice planes nearly parallel to the incident beam can be diffracted. The orientation of a crystal of known structure, with respect to the electron beam, can be accomplished by comparison with the crystallographic projections, which correspond to the different sections of the reciprocal lattice. The prominent cross-grating patterns for the face centered cubic lattice (fcc) are reported in Fig. 21 . The relative orientation of the different projections is easily obtained by means of a
• .Jo."
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• "•' 00(I " ~•" )07 "
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.
.
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[001]
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[103]
[213l
Fig. 2 - Prominent cross-grating patterns for fcc lattices (After Loretto and Smallmanl ).
stereographic projection, which is very useful in the analysis of crystal defects, as it enables one to identify, e.g. the plane of a dislocation loop or of a stacking fault, as well as the orientation relationships between a precipitate and the matrix. In Fig. 3 2 is reported the (110) stereographic projection of a cubic lattice: the angles between the planes can be deduced by super-
458 +i?
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Fig. 3 - (110) stereographic projection for a cubic lattice (After Taylor2).
imposing a Wulff net o n t o the projection. The electron waves can be diffracted b y planes which are n o t in the exact Bragg orientation. The corresponding p o i n t o n the reciprocal lattice is n o t l o c a t e d on the Ewald sphere. This deviation from the Bragg angle is represented b y a small v e c t o r "s, according to the con-
{0
9 Fig. 4 - Construction o f the deviation parameter s (After Smallman and Ashbee3).
459
struction of Fig. 4 3 . The parameter "s is assumed to be positive when the reciprocal lattice point is inside the Ewald sphere, and negative when it is outside it. As the crystal thickness increases, the appearance in the diffraction patterns of pairs of black and white lines (Kikuchi lines), which are due to the inelastic scattering of the electrons, is noticed; their formation is illustrated in Fig. 5 3 . The electron beam is scattered incoherently by the crystal, giving rise to a background e n h a n c e m e n t as !Incident Z~
/ / / /
f
/ / / I D
Photo~lraphic plate 0
L
Fig. 5 - The mechanism of formation of the Kikuchi lines (After Smallman and Ashbee3).
well as to waves diffracted by the crystal planes. Thus electrons 1 and 2, which are b o t h at the Bragg angle to the crystal plane P, are scattered to points L and D on the photographic plate, respectively. However, because the incoherent scattering cross section is peaked in the forward direction, the point D receives an excess of electrons, and L a deficit. Thus a dark line appears at D and a white line at L on the photographic plate. Strictly, the lines should take the form of hyperbolae, but, due to the large size of the Ewald sphere, they are practically straight lines. Moreover, as the Kikuchi lines are rigidly fixed to the specimen, they will run in a direction perpendicular to ~ as ~- varies, so allowing one to determine the value and the sign of ~, and hence the orientation of a crystal more accurately than with the analysis of the spot pattern.
460 FORMATION OF IMAGE CONTRAST As m e n t i o n e d before, a crystal may be misoriented with respect to the exact reflecting position and yet still diffract. This property of the electron diffraction can be explained according to the kinematical theory of contrast. The basic assumption of this theory are that the specimen is a thin film oriented so that only one set of lattice planes are near the Bragg position ( ' t w o beam approximation'), the deviation parameter "s is large and no interference between the primary and the diffracted beams takes place. According to the kinematical theory, the diffracted amplitude ~g by a perfect crystal is given by the following expression:
(2)
izr Cg = ~g
and the diffracted intensity
sinrr ts exp ( - ~' ist) rrs
Ig: 7r~
(3)
Ig = ~
sin s(~r ts)
(Trs)2
where t is the film thickness and ~g is the extinction distance, i.e. a length proportional to the n u m b e r of planes that the beam needs to travel in order to be completely diffracted away. This n u m b e r is of the order of some tens in electron diffraction, whereas about 10 4 planes are needed in the case of X-ray diffraction. F r o m (3) it comes out that the diffracted intensity is significant in a range of orientations A0 ~ d/t. By assuming t = 1 0 0 / ~ and d = = 1 A, one gets A0 -- 10 .2 rad, which is of the same order of magnitude of the Bragg angle. This contrasts with A0 ~-- 1 0 -4 in the case of X-ray diffraction. If the crystal is distorted, e.g. due to the presence of lattice defects, a given n u m b e r of atoms will be displaced from their ideal position. Consequently, if an atom located at a position r- in the perfect crystal is displaced by a vector R, the amplitude of the diffracted
461 wave ¢g becomes: Og = iTr f0texp [-27ri ( ~ - + ~ ) ( ~ + R ) ]
dz=
(4) iTr t = ~g f0 exp [ - 27ri (sz + ~ . P,)] dz
(since g • r = 1 and ~ • R is negligible). Thus the presence of lattice defects induces an additional phase factor ot = 2~rg • R, which is responsible for the image contrast, as will be shown in the next section. According to the kinematical theory, the bright-field image is c o m p l e m e n t a r y to the dark-field image, but this is not always true: the reason for this discrepancy is that the kinematical theory neglects the absorption as well as the interaction between the incident and the diffracted beams. These aspects are taken into account by the dynamical theory of contrast, which gives an explanation, in terms of s t a n d ing waves (Bloch waves) of the so-called anomalous absorption effect, which is due to the different propagation of these standing waves in the crystal. This effect destroys the complementary nature of the bright- and dark-field images and allows one to explain, e.g. the intensity distribution of the fringes at a stacking fault (see below).
ANALYSIS OF LATTICE DEFECTS From the previous discussion it descends that the presence of lattice defects gives rise to the additional phase factor 0t = 27r~ • R in the equation for the diffracted amplitude ~bg and that, due to the anomalous absorption, the bright-field and the dark-field images are not c o m p l e m e n t a r y . These two results are of fundamental importance in the analysis of lattice defects.
462 DISLOCATIONS In the case of dislocations the displacement vector R is merely the Burgers vector'b: atoms near the core of the dislocation are displaced mainly along a direction parallel to b. The tilting of the lattice planes near an edge and a screw dislocation are schematized in Fig. 6 ~ . No contrast will occur when ~ = 27r~ • b = 0; this occurs when
I I 11
I 1 ::;v'
"
lttttt b
(01
b
(b}
Fig. 6 - Schematic diagram showing the tilting o f the lattice planes near an edge (a) and a screw (b) dislocation (After Smallman and Ashbee3 ).
is perpendicular to ~, i.e. parallel to the reflecting planes. This holds strictly in the case of a screw dislocation (b parallel to the dislocation line), as can be seen in Fig. 74. Dislocation B is out of contrast in Fig. 7b and, since ~ is perpendicular to b, the Burgers vector can be easily deduced. In the case of edge dislocations there is also a small atomic displacement along a direction perpendicular to b, which gives rise to a residual contrast; in practice, the determination of b requires that g" b A ~ = 0, where ~ is the unit vector along the positive direction of the dislocation line. Generally it is necessary to take micrographs with at least two different operating reflections, for w h i c h ~ • b = 0. The modulus and the direction of b so obtained allows one to deduce the character of the dislocation.
463
5000
Fig. 7 - Electron micrographs showing a screw dislocation (B) in contra.:t (a) and out o f contrast (b). From the directions o f the g-vectors reported in the inserts, it comes aout that in (b) g is perpendicular to the dislocation line and then to
(~. g = o) (After mrsch et
al.*).
Partial dislocations can be characterized according to the same • b rule, but in this case the problem is somewhat more complicated; anyway, additional rules on the values and the sign of the product make the identification quite feasible. As to the width w of the image of a dislocation, it is about ~g/3, when the contrast is at its maximum ( ; ~ 0)g for example, w = 200 A for dislocations in silicon, imaged at 100 kV with g = 111. If a particular dark-field technique ('weak-beam technique') is employed, the value of w can be reduced down to 20-40 A, thus allowing the dissociation of a unit dislocation into partials to be observed. TherefDre the 'contrast resolution' of dislocations in electron microscopy is m u c h better than the one obtainable by X-ray topography (w > 1 /am); thus the m a x i m u m density of dislocations observable in
464 an electron micrograph can be as high as 10 x 2 cm-2, whereas it is only 10 s cm -2 in an X-ray topograph. An interesting feature of the dislocation contrast, which allows one to characterize the dislocation loops, is shown in Fig. 8 s . The planes near the core of a dislocation (a pure edge in this figure) are
m~
~
~ - - s>O
I
s,o
J--.- s
-I-
,
I
Fig. 8 - Schematic diagram showing the position of the image of an edge disloca. tion and its actual position in the crystal for various combinations of the senses of'g and s (After DupuyS).
inclined towards the exact reflecting position on one side and outwards it on the opposite side. Consequently, the image will be displaced from its actual position in the crystal, and this displacement is inverted if the sense of either g or s- is changed (but not both together). According to the convention of the signs used in Fig. 8, the image will result to be displaced toward right if (g • b) s > 0 and toward left if (g" b) s <~ 0. Thus it is possible to determine the sense of b unambiguously, i.e. to distinguish b e t w e e n dislocation with Burgers vectors b and - b .
DISLOCATION LOOPS It is well k n o w n that dislocation loops can be of two types: in, terstitial, when are generated b y insertion of an extra plane into the
465
lattice, and vacancy if a layer of ~itoms is removed. The possibility of distinguishing between the two types _and hence of identifying a loop relies on the sign of the product (g - b) s, as discussed above. The displacement of the image will result in a variation of the apparent loop diameter. It is to be pointed out that it is necessary to know the inclination of the loop with respect to the crystal surface because, for example, an interstitial loop lying in a given plane gives rise to the same contrast in the image as a vacancy loop inclined in the opposite sense. This ambiguity is clearly illustrated in Fig. 9 6 , which also
[~ bl,-o
~.
(il))
(( }
,~ ))
(:i))
([))
(())
~i ))
~{I::
a
b
c
d
f re~lect~g ~ p ~ t
~,
le n l . , o
Fig. 9 - Schematic diagram showing the variation in the width of the image of an interstitial (a, b) and vacancy (c, d) dislocation loop as a function o f the sign o f (g" -b) s (Adapted from Maher and Eyre6).
shows that ( ~ . b) s > 0 when the loop image lies outside the loop, whereas ( g . b) s < 0 when the image is inside the loop. In Fig. 1 0 7 is reported a pair of micrographs of loops in silicon obtained by changing the operating vector g in its sense. Finally, it is well to remember that the dislocations which form a loop can be unit or partials; in the latter case, the loop includes a stacking fault.
466
Fig. 10 - Electron micrographs showing the effect of the sens of g on the image width of dislocation loops in silicon (s ~ 0 in both the micrographs) 7.
STACKING FAULTS A stacking fault may be generated either by shear or by a collapse of interstitial atoms (extrinsic faults) or of vacancies ~intrinsic faults). They occur in the close packed planes of cubic or hexagonal structures. When the faults are inclined with respect to the beam direction, their image consists of alternately light and dark fringes, running parallel to the intersection between the plane of the fault and the specimen surface. In cubic crystals they lie in the [111] planes and hence the displacement vector is R = + 1/3 < 111 >. A schematic diagram of both an extrinsic and an intrinsic fault is reported in Fig. 118 , which shows the failure of the stacking sequence of the planes (111). If the fault is entirely confined within the specimen, it is possible to image also the dislocation loop surrounding it; in this case, the determination of the sign of R is performed according to the criterion described in the previous paragraph. Instead, if the fault has been truncated by the thinning process, its identification will be based on the anomalQus absorption effect mentioned in the previous section. According to this, the bright-field image will be symmetrical about the centre of the fault, whereas the dark-field image will be asymme-
467 A
A
C
C
D
~
B
C
~
C
C
C
D
B
7 J
~ -
O
B
A
A
fa)
C '
~
C
D
B
A
A
(b)
Fig. 11 - Schematic diagram showing the failure in the stacking sequence o f planes ABC due to the presence o f an instrinsic (a) and an extrinsic (b) stacking fault (After Weertman and Weertman S).
trical, tending to be similar to the bright-field at one edge and pseudocomplementary at the other. In Fig. 12 7 are reported a bright-field and a dark-field image of a stacking fault in silicon; the fringes at the edges are both dark in Fig. 12a, whereas in Fig. 12b there is a light fringe at one edge and a
Fig. 12 - Bright-field (BF) a n d dark-field (DF) electron micrographs o f an extrinsic stacking fault in silicon ~.
468 dark one at the other. From the contrast theory it comes out that the edge-fringes are similar at the top of the foil and pseudo-complementary at the bottom, thus enabling one to determine the plane of the fault. In addition, for the type of operating reflection used in Fig. 12 (g = 032) the fault is extrinsic if g points towards the light fringe in a dark-field image, whereas it will be intrinsic if g points away from it. From Fig. 12b it is possible to conclude that the fault was extrinsic, i.e. generated by the insertion of an extra [111} plane.
TWINS According to the twinning theory, the orientation of a lattice after twinning is given by a rotation of 180" about either the shear direction or the normal to the twinning plane, depending on the kind of twinning involved. Twinning is produced by plastic deformation and can occur, e.g. in the epitaxial regrowth of films under a crystalline substrate. (L/I~plone f
~'~iT] "OTO
15~ i5o ,5,
plane
L..
(a)
Fig. 13 a) (570) stereographic projection of a cubic lattice, on wt~,ct~ ,s reported the reflection of poles (111) and (117) after twinning on the (171) and (171-) planes, respectively (After ]ohari and Thomasg); b) diffraction pattern of the twins reported in Fig. 14, -
-
469
The stereographic prqiections are very useful in the analysis of twins; an example of application of this method to the twinning on the (1]-1) and (11]-) planes of a fcc crystal is given in Fig. 13a 9 . The projection chosen is the (510); the poles on one side of these planes are reflected on the other sidealong the latitude circle. Thus pole (111) after twinning on the (111)plane supe_ri_mposes on pole (151), whereas pole (11]-) after twinning on the (111) plane superimposes on pole (15]-). In the corresponding diffraction pattern (Fig. 13b) the spot (111) of a twin appears along the same direction as (151), with (333) coinciding with (151), as d333 = dl s l in cubic crystals; likewise, the spot (333) will coincide with (15]-) after twinning on the (111) plane. In this pattern it is present also a second set of spots, which are due to double diffraction. The twins that give rise to the pattern of Fig. 13b are visible in the micrographs reported in Fig. 14. The image in Fig. 14b is a darkfield image obtained by allow!ng both the (111) and the (1!1) diffracted beams to pass through the objective aperture. By properly
Fig. 14 - Bright-field (a) and dark-field (b) electron micrographs of twins with the (171) (twin A) and (171-) (twin B) composition planes.
470 aligning the micrographs, diffractioia patterns and stereographic proj e c t i o n a n d c o m p a r i n g t h e t r a c e s , i t is p o s s i b l e t o u n i q u e l y i d e n t i f y the twin planes.
Acknowledgement The author is indebted to Dr. M. Servidori and Mr. G. Ruffini.
REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
M.H. LORETTO and R.E. SMALLMAN - Defects analysis in electron microscopy, Chapman and Hall, London, 1975. A. TAYLOR -- X-ray metallography, J o h n Wiley & Sons, Inc., New York, 1961. R.E. SMALLMAN and K.H.G. ASHBEE -- Modern metallography, Pergamon Press Ltd., Oxford, 1966. P.B. HIRSCH, A. HOWIE and M.J. WHELAN -- Phil. Trans. Roy. Soc., A252, 499, 1970. M. DUPUY -- in Comptes Rendus du Colloque: Caracterisation des materiaux et technologies serniconducteurs, L.E.T.I., Grenoble, 1972. D.M. MAHER and B.L. EYRE -- Phil. Mag., 23, 409, 1971. A. ARMIGLIATO -- in Tecniche di indagine nel campo dei materiali, (P. Mazzoldi Ed.), University of Padova, Institute o f Physics, 1978. J. WEERTMAN and J.R. WEERTMAN -- Elementary dislocation theory, The McMillan Company, New Yorl~, 1964. O. J O H A R I and G. THOMAS -- The stereographic projection and its applications, Vol. IIA in: Techniques of metals research, (R.F. Bunshah Ed.), J o h n Wiley & Sons, Inc., New York, 1969.
FURTHER READING
P.B. HIRSCH, A. HOWIE, R.B. NICHOLSON, D.W. PASHLEY and M J . WHELAN --Electron microscopy of thin crystals, Butterworths, London, 1965. C.E. HALL -- Introduction to electron microscopy, Mc Graw-Hill, New York, 1966. R.D. HEIDENREICH -- Fundamentals of transmission electron microscopy, J o h n Wiley & Sons, Inc., New York, 1964.
471 P.W. HAWKES -- Electron optics and electron microscopy, Taylor and Francis, London, 1972. G. THOMAS -- Transmission electron microscopy of metals, J o h n Wiley & Sons, Inc., New York, 1966. D. KAY (Ed.) -- Techniques for electron microscopy, Blackwell, Oxford, 1967. A.M. G L A U E R T (Ed.) -- Practical methods in electron microscopy, 2 vol., North Holland, Amsterdam, 1972. B. J O U F F R E Y (Ed.) -- Mdthodes et techniques nouvelles d'observation en metallurgie physique, Soci6t6 Franqaise de Microscopie Electronique, Paris, 1972. U. VALDRE (Ed.) -- Electron microscopy in materials science, Academic Press, New York, 1971. S. AMELINCKX, R. GEVERS, G. REMAUT and J.Van LANDUYT (Eds.) -Modern diffraction and imaging techniques in materials science, N.orth Holland, Amsterdam, 1970. H. MODIN and S. MODIN --Metallurgical microscopy, Butterworths, London, 1973. U. VALDRI~ and E. RUEDL (Eds.) - Electron,microscopy in materials science, Commission of the European Communities, Brussels and Luxembourg, 1976.
A N A L Y S I S OF CRYSTAL DEFECTS BY TRANSMISSION ELECTRON MICROSCOPY *
A. ARMIGLIATO Laboratorio
LAMEL
del C.N.R.
- Via Castagnoli,
1 - 40126
BOLOGNA
- Italy.
Summary -- In the present paper the basic concepts of the transmission electron microscopy are outlined. In particular, the relevant aspects of both the kinematical and the dynamical theory of contrast, and their usefulness in the interpretation of the electron micrographs and of the diffractior~ patterns, are shortly discussed. Among the various examples of application of this technique to the materials science reported in literature, the methods currently employed to characterize, from the electron micrographs, lattice defects in crystalline materials, such as dislocations, loops, stacking faults and twins, are emphasized in this article.
INTRODUCTION
The role of the transmission electron microscopy in the different fields of the scientific research is of increasing importance. This technique has proved to be a powerful tool in the study of lattice defects in crystalline materials, like metals, semiconductors, minerals, glasses and ceramics. To analyze defects by electron microscopy, a knowledge of the principles of electron diffraction as well as the theory of image con-
* Presented at the IV Scientific Meeting of the Italian Association for Crystal Growth (AICC), Parma, Italy, 26-28 February 1979.
454 trast is required. Although a detailed account of the kinematical and dynamical theories of contrast is beyond the scope of the present article, some introductory remarks will be given. Among the typical problems encountered by the materials scientist, it will be shown how the study of the electron micrographs and the corresponding diffraction patterns enables one to determine the Burgers vector of a dislocation, the displacement vector of a stacking fault and the twinning planes in a crystal.
FORMATION OF THE IMAGE AND OF THE DIFFRACTION PATTERN As it is well known, the objective lens of a transmission electron microscope (TEM) focuses an electron beam transmitted by a crystalline thin film in its back focal plane, giving rise to a diffraction pattern; in the conjugate plane of the object plane, an inverted image takes place (the so-called 'first image'). Therefore, if the subsequent lens (first intermediate lens) is energized so that its object plane is the objective back focal plane, the diffraction pattern will be imaged onto the final screen; the overall magnification of the pattern depends on the product of the objective focal length and the magnification of the intermediates and the projector lenses. Instead, if the object plane of the first intermediate lens is the first image plane, a magnified image of the specimen will be obtained. The image and the diffraction pattern are not aligned: in fact, since the magnetic lenses rotate the plane of motion of the electrons as a function of their excitation, there will be a relative rotation of the diffraction pattern with respect to the image. In addition, as the objective lens invertes the image and not the diffraction pattern, an extra inversion will take place, which is not modified by all the subsequent lenses, because they act in the same way on both the image and the diffraction pattern. Such an extra inversion always occurs in a five lenses TEM , whereas in a six lenses instrument it depends on the excitation of the magnifying lenses. As will be seen in the next section, the knowledge of the im-
455 age-to-diffraction pattern rotation is of fundamental importance in the analysis of the defects and of the crystallographic directions. The ray paths corresponding to an image and a diffraction pattern in a six lenses TEM are sketched in Fig. 1, both in the case where the inversion is present (Fig. la) and absent (Fig. lb) I
specimen
objective lens
objective lens
image
- -
f
~..... intermedi~ate dif fraction lens
~ q
- -
Final image
- ~
Final diffroctlon potlern
(a)
Ist image
diffraction lens
intermed~te lens
-~
projector lens
./
Final diffrachon pattern
Final enoge
(b)
Fig. 1 - Ray paths for the formation o f an image and o f the corresponding diffraction pattern in a six-lenses TEM, both in case o f inversion (a) and non inversion (b) (After Loretto and Smallmanl ).
If an aperture is inserted at the level of the first image, the diffraction pattern of a selected area of the specimen (SAD) is obtained. The correspondence between the area included in the aperture and the area which actually diffracts depends on both the spherical aberration and the focusing errors and is not better than about 2000 A at 100 kV. An image obtained with the direct and the diffracted waves exhibits a poor contrast. Instead, if only one beam is selected by means
456 of an aperture inserted in the objective back focal plane (objective aperture), the contrast will be greatly improved. This mode of operation is called 'diffraction contrast', and is currently used in the analysis of crystal defects. In particular, if the selected beam is the transmitted one, a bright-field image will be obtained, whereas if a diffracted beam is chosen, a dark-field image will be produced.
ELECTRON DIFFRACTION PATTERNS The scattering of a monochromatic radiation by a crystal leads to the formation of diffracted beams which obey the Bragg law: (1)
n X = 2 d sin0
where 0 i s the angle between the incident beam and the planes of the crystal of spacing d, X is the wavelength of the electrons (X (A) =X/150/[1 + 10"6E(volts)] ), n is the diffraction order and E is the accelerating voltage. In the case of a single crystal, the geometrical formation of diffraction patterns is more easily represented by the reciprocal lattice and Ewald sphere construction. The incident beam direction is identified by the vector of modulus 1IX which joins the centre of the sphere to the centre of the reciprocal lattice; if another point of this lattice lies on the sphere, a diffracted beam will be originated at an angle 20, according to the Bragg law. The g--vector is the normal to the diffracting planes and its modulus is [ g l = 1/d. At 100 kV the radius of the Ewald sphere is about 50 times the spacing of the low indices lattice planes of the most c o m m o n materials and thus an electron diffraction pattern reproduces an indistorted section of the reciprocal lattice perpendicular to the beam direction. Instead, in the case of X-ray diffraction, where 1]X is of the same order of magnitude of [ ~ [, a distorted image of the corresponding section of the reciprocal lattice will be formed, unless special geometries are used, as in the precession method. In addition, due to the short wavelength of the electrons us-
457
ed in a TEM (X = 0.037 )~ at 100 kV), the Bragg angle 20 is of the order of 1 ° for low indices reflections and then only the lattice planes nearly parallel to the incident beam can be diffracted. The orientation of a crystal of known structure, with respect to the electron beam, can be accomplished by comparison with the crystallographic projections, which correspond to the different sections of the reciprocal lattice. The prominent cross-grating patterns for the face centered cubic lattice (fcc) are reported in Fig. 21 . The relative orientation of the different projections is easily obtained by means of a
• .Jo."
" ~ot2o
• "•' 00(I " ~•" )07 "
"
(1011
.
.
.
~'~ ~?~
•
o8o
[111]
~'-o
[001]
•
[112]
~
~%
.
~
"
~
•
" &o%"
[103]
[213l
Fig. 2 - Prominent cross-grating patterns for fcc lattices (After Loretto and Smallmanl ).
stereographic projection, which is very useful in the analysis of crystal defects, as it enables one to identify, e.g. the plane of a dislocation loop or of a stacking fault, as well as the orientation relationships between a precipitate and the matrix. In Fig. 3 2 is reported the (110) stereographic projection of a cubic lattice: the angles between the planes can be deduced by super-
458 +i?
~-
OOi
tl? T
iT3
~3
3~5
,03 2T3 315 •
•;
•
,3. . . . . \
- .
•
e,3~
e023 75~
,,5
osIT
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l-i
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•
-
23
~ •
3~ 1 3 ]
•
-
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-
,tz •
iii
-.
,,+'.°.~+ ,+f,+-,,+..o: _
.
-
~I
+
..+,+ t:5
"
+ +
.3"~ :+o~, +'~ • "' -..,3~ _.o~.z . ; ~ .. • • • o,, - ~+ =, 5",( =+,++ 5++ .+'+ . ? " -+,+ +_=
..... 3~
T35 T2
0113 e315
•
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1211
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COl2
-
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o;3
~ i ~5
IOz
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.
--
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212
.
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''+ +,03
.-
e o,~
+,2
"--" ...... ll?q
-
.3~s
/ /
. . . . . "0,)
33g T3
lit
001
Fig. 3 - (110) stereographic projection for a cubic lattice (After Taylor2).
imposing a Wulff net o n t o the projection. The electron waves can be diffracted b y planes which are n o t in the exact Bragg orientation. The corresponding p o i n t o n the reciprocal lattice is n o t l o c a t e d on the Ewald sphere. This deviation from the Bragg angle is represented b y a small v e c t o r "s, according to the con-
{0
9 Fig. 4 - Construction o f the deviation parameter s (After Smallman and Ashbee3).
459
struction of Fig. 4 3 . The parameter "s is assumed to be positive when the reciprocal lattice point is inside the Ewald sphere, and negative when it is outside it. As the crystal thickness increases, the appearance in the diffraction patterns of pairs of black and white lines (Kikuchi lines), which are due to the inelastic scattering of the electrons, is noticed; their formation is illustrated in Fig. 5 3 . The electron beam is scattered incoherently by the crystal, giving rise to a background e n h a n c e m e n t as !Incident Z~
/ / / /
f
/ / / I D
Photo~lraphic plate 0
L
Fig. 5 - The mechanism of formation of the Kikuchi lines (After Smallman and Ashbee3).
well as to waves diffracted by the crystal planes. Thus electrons 1 and 2, which are b o t h at the Bragg angle to the crystal plane P, are scattered to points L and D on the photographic plate, respectively. However, because the incoherent scattering cross section is peaked in the forward direction, the point D receives an excess of electrons, and L a deficit. Thus a dark line appears at D and a white line at L on the photographic plate. Strictly, the lines should take the form of hyperbolae, but, due to the large size of the Ewald sphere, they are practically straight lines. Moreover, as the Kikuchi lines are rigidly fixed to the specimen, they will run in a direction perpendicular to ~ as ~- varies, so allowing one to determine the value and the sign of ~, and hence the orientation of a crystal more accurately than with the analysis of the spot pattern.
460 FORMATION OF IMAGE CONTRAST As m e n t i o n e d before, a crystal may be misoriented with respect to the exact reflecting position and yet still diffract. This property of the electron diffraction can be explained according to the kinematical theory of contrast. The basic assumption of this theory are that the specimen is a thin film oriented so that only one set of lattice planes are near the Bragg position ( ' t w o beam approximation'), the deviation parameter "s is large and no interference between the primary and the diffracted beams takes place. According to the kinematical theory, the diffracted amplitude ~g by a perfect crystal is given by the following expression:
(2)
izr Cg = ~g
and the diffracted intensity
sinrr ts exp ( - ~' ist) rrs
Ig: 7r~
(3)
Ig = ~
sin s(~r ts)
(Trs)2
where t is the film thickness and ~g is the extinction distance, i.e. a length proportional to the n u m b e r of planes that the beam needs to travel in order to be completely diffracted away. This n u m b e r is of the order of some tens in electron diffraction, whereas about 10 4 planes are needed in the case of X-ray diffraction. F r o m (3) it comes out that the diffracted intensity is significant in a range of orientations A0 ~ d/t. By assuming t = 1 0 0 / ~ and d = = 1 A, one gets A0 -- 10 .2 rad, which is of the same order of magnitude of the Bragg angle. This contrasts with A0 ~-- 1 0 -4 in the case of X-ray diffraction. If the crystal is distorted, e.g. due to the presence of lattice defects, a given n u m b e r of atoms will be displaced from their ideal position. Consequently, if an atom located at a position r- in the perfect crystal is displaced by a vector R, the amplitude of the diffracted
461 wave ¢g becomes: Og = iTr f0texp [-27ri ( ~ - + ~ ) ( ~ + R ) ]
dz=
(4) iTr t = ~g f0 exp [ - 27ri (sz + ~ . P,)] dz
(since g • r = 1 and ~ • R is negligible). Thus the presence of lattice defects induces an additional phase factor ot = 2~rg • R, which is responsible for the image contrast, as will be shown in the next section. According to the kinematical theory, the bright-field image is c o m p l e m e n t a r y to the dark-field image, but this is not always true: the reason for this discrepancy is that the kinematical theory neglects the absorption as well as the interaction between the incident and the diffracted beams. These aspects are taken into account by the dynamical theory of contrast, which gives an explanation, in terms of s t a n d ing waves (Bloch waves) of the so-called anomalous absorption effect, which is due to the different propagation of these standing waves in the crystal. This effect destroys the complementary nature of the bright- and dark-field images and allows one to explain, e.g. the intensity distribution of the fringes at a stacking fault (see below).
ANALYSIS OF LATTICE DEFECTS From the previous discussion it descends that the presence of lattice defects gives rise to the additional phase factor 0t = 27r~ • R in the equation for the diffracted amplitude ~bg and that, due to the anomalous absorption, the bright-field and the dark-field images are not c o m p l e m e n t a r y . These two results are of fundamental importance in the analysis of lattice defects.
462 DISLOCATIONS In the case of dislocations the displacement vector R is merely the Burgers vector'b: atoms near the core of the dislocation are displaced mainly along a direction parallel to b. The tilting of the lattice planes near an edge and a screw dislocation are schematized in Fig. 6 ~ . No contrast will occur when ~ = 27r~ • b = 0; this occurs when
I I 11
I 1 ::;v'
"
lttttt b
(01
b
(b}
Fig. 6 - Schematic diagram showing the tilting o f the lattice planes near an edge (a) and a screw (b) dislocation (After Smallman and Ashbee3 ).
is perpendicular to ~, i.e. parallel to the reflecting planes. This holds strictly in the case of a screw dislocation (b parallel to the dislocation line), as can be seen in Fig. 74. Dislocation B is out of contrast in Fig. 7b and, since ~ is perpendicular to b, the Burgers vector can be easily deduced. In the case of edge dislocations there is also a small atomic displacement along a direction perpendicular to b, which gives rise to a residual contrast; in practice, the determination of b requires that g" b A ~ = 0, where ~ is the unit vector along the positive direction of the dislocation line. Generally it is necessary to take micrographs with at least two different operating reflections, for w h i c h ~ • b = 0. The modulus and the direction of b so obtained allows one to deduce the character of the dislocation.
463
5000
Fig. 7 - Electron micrographs showing a screw dislocation (B) in contra.:t (a) and out o f contrast (b). From the directions o f the g-vectors reported in the inserts, it comes aout that in (b) g is perpendicular to the dislocation line and then to
(~. g = o) (After mrsch et
al.*).
Partial dislocations can be characterized according to the same • b rule, but in this case the problem is somewhat more complicated; anyway, additional rules on the values and the sign of the product make the identification quite feasible. As to the width w of the image of a dislocation, it is about ~g/3, when the contrast is at its maximum ( ; ~ 0)g for example, w = 200 A for dislocations in silicon, imaged at 100 kV with g = 111. If a particular dark-field technique ('weak-beam technique') is employed, the value of w can be reduced down to 20-40 A, thus allowing the dissociation of a unit dislocation into partials to be observed. TherefDre the 'contrast resolution' of dislocations in electron microscopy is m u c h better than the one obtainable by X-ray topography (w > 1 /am); thus the m a x i m u m density of dislocations observable in
464 an electron micrograph can be as high as 10 x 2 cm-2, whereas it is only 10 s cm -2 in an X-ray topograph. An interesting feature of the dislocation contrast, which allows one to characterize the dislocation loops, is shown in Fig. 8 s . The planes near the core of a dislocation (a pure edge in this figure) are
m~
~
~ - - s>O
I
s,o
J--.- s
-I-
,
I
Fig. 8 - Schematic diagram showing the position of the image of an edge disloca. tion and its actual position in the crystal for various combinations of the senses of'g and s (After DupuyS).
inclined towards the exact reflecting position on one side and outwards it on the opposite side. Consequently, the image will be displaced from its actual position in the crystal, and this displacement is inverted if the sense of either g or s- is changed (but not both together). According to the convention of the signs used in Fig. 8, the image will result to be displaced toward right if (g • b) s > 0 and toward left if (g" b) s <~ 0. Thus it is possible to determine the sense of b unambiguously, i.e. to distinguish b e t w e e n dislocation with Burgers vectors b and - b .
DISLOCATION LOOPS It is well k n o w n that dislocation loops can be of two types: in, terstitial, when are generated b y insertion of an extra plane into the
465
lattice, and vacancy if a layer of ~itoms is removed. The possibility of distinguishing between the two types _and hence of identifying a loop relies on the sign of the product (g - b) s, as discussed above. The displacement of the image will result in a variation of the apparent loop diameter. It is to be pointed out that it is necessary to know the inclination of the loop with respect to the crystal surface because, for example, an interstitial loop lying in a given plane gives rise to the same contrast in the image as a vacancy loop inclined in the opposite sense. This ambiguity is clearly illustrated in Fig. 9 6 , which also
[~ bl,-o
~.
(il))
(( }
,~ ))
(:i))
([))
(())
~i ))
~{I::
a
b
c
d
f re~lect~g ~ p ~ t
~,
le n l . , o
Fig. 9 - Schematic diagram showing the variation in the width of the image of an interstitial (a, b) and vacancy (c, d) dislocation loop as a function o f the sign o f (g" -b) s (Adapted from Maher and Eyre6).
shows that ( ~ . b) s > 0 when the loop image lies outside the loop, whereas ( g . b) s < 0 when the image is inside the loop. In Fig. 1 0 7 is reported a pair of micrographs of loops in silicon obtained by changing the operating vector g in its sense. Finally, it is well to remember that the dislocations which form a loop can be unit or partials; in the latter case, the loop includes a stacking fault.
466
Fig. 10 - Electron micrographs showing the effect of the sens of g on the image width of dislocation loops in silicon (s ~ 0 in both the micrographs) 7.
STACKING FAULTS A stacking fault may be generated either by shear or by a collapse of interstitial atoms (extrinsic faults) or of vacancies ~intrinsic faults). They occur in the close packed planes of cubic or hexagonal structures. When the faults are inclined with respect to the beam direction, their image consists of alternately light and dark fringes, running parallel to the intersection between the plane of the fault and the specimen surface. In cubic crystals they lie in the [111] planes and hence the displacement vector is R = + 1/3 < 111 >. A schematic diagram of both an extrinsic and an intrinsic fault is reported in Fig. 118 , which shows the failure of the stacking sequence of the planes (111). If the fault is entirely confined within the specimen, it is possible to image also the dislocation loop surrounding it; in this case, the determination of the sign of R is performed according to the criterion described in the previous paragraph. Instead, if the fault has been truncated by the thinning process, its identification will be based on the anomalQus absorption effect mentioned in the previous section. According to this, the bright-field image will be symmetrical about the centre of the fault, whereas the dark-field image will be asymme-
467 A
A
C
C
D
~
B
C
~
C
C
C
D
B
7 J
~ -
O
B
A
A
fa)
C '
~
C
D
B
A
A
(b)
Fig. 11 - Schematic diagram showing the failure in the stacking sequence o f planes ABC due to the presence o f an instrinsic (a) and an extrinsic (b) stacking fault (After Weertman and Weertman S).
trical, tending to be similar to the bright-field at one edge and pseudocomplementary at the other. In Fig. 12 7 are reported a bright-field and a dark-field image of a stacking fault in silicon; the fringes at the edges are both dark in Fig. 12a, whereas in Fig. 12b there is a light fringe at one edge and a
Fig. 12 - Bright-field (BF) a n d dark-field (DF) electron micrographs o f an extrinsic stacking fault in silicon ~.
468 dark one at the other. From the contrast theory it comes out that the edge-fringes are similar at the top of the foil and pseudo-complementary at the bottom, thus enabling one to determine the plane of the fault. In addition, for the type of operating reflection used in Fig. 12 (g = 032) the fault is extrinsic if g points towards the light fringe in a dark-field image, whereas it will be intrinsic if g points away from it. From Fig. 12b it is possible to conclude that the fault was extrinsic, i.e. generated by the insertion of an extra [111} plane.
TWINS According to the twinning theory, the orientation of a lattice after twinning is given by a rotation of 180" about either the shear direction or the normal to the twinning plane, depending on the kind of twinning involved. Twinning is produced by plastic deformation and can occur, e.g. in the epitaxial regrowth of films under a crystalline substrate. (L/I~plone f
~'~iT] "OTO
15~ i5o ,5,
plane
L..
(a)
Fig. 13 a) (570) stereographic projection of a cubic lattice, on wt~,ct~ ,s reported the reflection of poles (111) and (117) after twinning on the (171) and (171-) planes, respectively (After ]ohari and Thomasg); b) diffraction pattern of the twins reported in Fig. 14, -
-
469
The stereographic prqiections are very useful in the analysis of twins; an example of application of this method to the twinning on the (1]-1) and (11]-) planes of a fcc crystal is given in Fig. 13a 9 . The projection chosen is the (510); the poles on one side of these planes are reflected on the other sidealong the latitude circle. Thus pole (111) after twinning on the (111)plane supe_ri_mposes on pole (151), whereas pole (11]-) after twinning on the (111) plane superimposes on pole (15]-). In the corresponding diffraction pattern (Fig. 13b) the spot (111) of a twin appears along the same direction as (151), with (333) coinciding with (151), as d333 = dl s l in cubic crystals; likewise, the spot (333) will coincide with (15]-) after twinning on the (111) plane. In this pattern it is present also a second set of spots, which are due to double diffraction. The twins that give rise to the pattern of Fig. 13b are visible in the micrographs reported in Fig. 14. The image in Fig. 14b is a darkfield image obtained by allow!ng both the (111) and the (1!1) diffracted beams to pass through the objective aperture. By properly
Fig. 14 - Bright-field (a) and dark-field (b) electron micrographs of twins with the (171) (twin A) and (171-) (twin B) composition planes.
470 aligning the micrographs, diffractioia patterns and stereographic proj e c t i o n a n d c o m p a r i n g t h e t r a c e s , i t is p o s s i b l e t o u n i q u e l y i d e n t i f y the twin planes.
Acknowledgement The author is indebted to Dr. M. Servidori and Mr. G. Ruffini.
REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
M.H. LORETTO and R.E. SMALLMAN - Defects analysis in electron microscopy, Chapman and Hall, London, 1975. A. TAYLOR -- X-ray metallography, J o h n Wiley & Sons, Inc., New York, 1961. R.E. SMALLMAN and K.H.G. ASHBEE -- Modern metallography, Pergamon Press Ltd., Oxford, 1966. P.B. HIRSCH, A. HOWIE and M.J. WHELAN -- Phil. Trans. Roy. Soc., A252, 499, 1970. M. DUPUY -- in Comptes Rendus du Colloque: Caracterisation des materiaux et technologies serniconducteurs, L.E.T.I., Grenoble, 1972. D.M. MAHER and B.L. EYRE -- Phil. Mag., 23, 409, 1971. A. ARMIGLIATO -- in Tecniche di indagine nel campo dei materiali, (P. Mazzoldi Ed.), University of Padova, Institute o f Physics, 1978. J. WEERTMAN and J.R. WEERTMAN -- Elementary dislocation theory, The McMillan Company, New Yorl~, 1964. O. J O H A R I and G. THOMAS -- The stereographic projection and its applications, Vol. IIA in: Techniques of metals research, (R.F. Bunshah Ed.), J o h n Wiley & Sons, Inc., New York, 1969.
FURTHER READING
P.B. HIRSCH, A. HOWIE, R.B. NICHOLSON, D.W. PASHLEY and M J . WHELAN --Electron microscopy of thin crystals, Butterworths, London, 1965. C.E. HALL -- Introduction to electron microscopy, Mc Graw-Hill, New York, 1966. R.D. HEIDENREICH -- Fundamentals of transmission electron microscopy, J o h n Wiley & Sons, Inc., New York, 1964.
471 P.W. HAWKES -- Electron optics and electron microscopy, Taylor and Francis, London, 1972. G. THOMAS -- Transmission electron microscopy of metals, J o h n Wiley & Sons, Inc., New York, 1966. D. KAY (Ed.) -- Techniques for electron microscopy, Blackwell, Oxford, 1967. A.M. G L A U E R T (Ed.) -- Practical methods in electron microscopy, 2 vol., North Holland, Amsterdam, 1972. B. J O U F F R E Y (Ed.) -- Mdthodes et techniques nouvelles d'observation en metallurgie physique, Soci6t6 Franqaise de Microscopie Electronique, Paris, 1972. U. VALDRE (Ed.) -- Electron microscopy in materials science, Academic Press, New York, 1971. S. AMELINCKX, R. GEVERS, G. REMAUT and J.Van LANDUYT (Eds.) -Modern diffraction and imaging techniques in materials science, N.orth Holland, Amsterdam, 1970. H. MODIN and S. MODIN --Metallurgical microscopy, Butterworths, London, 1973. U. VALDRI~ and E. RUEDL (Eds.) - Electron,microscopy in materials science, Commission of the European Communities, Brussels and Luxembourg, 1976.