- Email: [email protected]
Si interface
Ultramicroscopy North-Holland
32 (1990) 275-289
275
DYNAMIC THEORY OF HIGH-ANGLE FOR A Ge/Si INTERFACE Z.L. WANG Department Received
ANNULAR-DARK-FIELD
STEM LA’ITICE IMAGES
* and J.M. COWLEY
of Physics, Arizona State University,
15 November
Tempe, AZ 85287-1504,
USA
1989
A dynamical multislice approach for simulating the annular-dark-field (ADF) scanning transmission electron microscopy (STEM) images is presented. The ADF imaging contrast is dominated by thermal diffuse scattering (TDS) if the diffraction contrast from the high-order Laue zones (HOLZ) is excluded. The simulation of ADF images can be greatly simplified if the assumption is made of incoherent imaging theory such as is used in optics, that is, the image intensity is a thickness integration of the convolution of the electron probe intensity distribution at depth z with the localized inelastic generation function 1G 12. This result may also be derived on the basis of coherent elastic scattering theory. The calculation according to the simplified theory can give almost exactly the same contrast as obtained from full dynamical inelastic calculations for thin specimens. The TDS can give atomic resolution lattice images if a small probe, less than the size of the unit cell, is used. A programming flow-chart for simulating ADF images is given. Calculations for Ge/Si(lOO) interfaces have shown that the ADF imaging contrast of the atoms is not affected by correlations of vibrations of neighboring atoms. It is pointed out that the contribution from elastic large-angle scattering will become important only if the atomic vibration amplitudes are less than about 0.03 A. The ADF image production may not be considered as incoherent imaging if this is the case.
1. Introduction Recently, Wang and Cowley [l] have applied the generalized multiple-elastic and -inelastic multislice theory, proposed by Wang [2], to simulate the atomic resolution high-angle annular-dark-field (ADF) scanning transmission electron microscopy (STEM) images, taking into account the thermal diffuse scattering (TDS). In that paper, we have shown that the TDS may play a dominant role in determining the ADF imaging contrast, depending strongly on the atomic number, the sizes (or shapes) of the atoms and their relative vibration amplitudes. It has also been shown by Wang and Cowley [l] that the ADF image has the following characteristics if the first-order Laue zone (FOLZ) (or high-order Laue zones (HOLZ)) is excluded: (1) there is no contrast reversal with change of focus;
* To whom correspondence should be addressed; Metals and Ceramics Division, Oak Ridge National tory, Oak Ridge, TN 37831-6376, USA. 0304-3991/X1/$03.50
0 1990 - Elsevier
now at: Labora-
Science Publishers
(2) there is no contrast reversal with change of specimen thickness and (3) the ADF image shows highly localized scattering from each atomic column and can provide some chemical information for a specimen having two or more atoms with very different atomic numbers. These conclusions agree with the experimental observations [3]. The HOLZ can introduce strong diffraction contrast in the ADF image and possibly give some misleading results in the image interpretations. In this paper, as a continuation of our previous work, we report the further theoretical investigations of ADF imaging formation in STEM. Several approximate approaches for simulating ADF images will be derived from dynamical inelastic scattering theory of thermal diffuse scattering. It will be shown that the incoherent imaging theory can be used to simulate ADF images and quantitative results are expected for thin specimens. Atomic resolution lattice images can be formed by these incoherent inelastically scattered electrons. It is concluded that the relative contrast of an atom is almost independent of the thermal vibration status of its neighbors. Hence, the simulation
B.V. (North-Holland)
276
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle
of ADF images will not be affected to a significant extent by assuming either the Einstein model, the Debye model or any other model.
2. TDS in electron diffraction As pointed out in our first paper [l], two inelastic processes may be important to ADF imaging formation besides the elastic contribution of the HOLZ. The first process is the electron core shell excitation in the energy-loss range of several hundreds to several thousands eV, with an angular spreading of an order 6, = A E/2 E,, where A E is the electron energy-loss and E, is the incident electron energy. Although large momentum transfer may be involved in this process, the probability of exciting the core shells is very small if the specimen is thin. Shown in fig. 1 are two electron diffraction patterns of different beam convergences from Si(100) having a thickness of more than a thousand A. The Kikuchi lines disappear if the sample is thin. Then we can neglect the contribution of core shell losses in the ADF image simulations for thin specimens, such as those about 200-500 A in thickness.
ADF STEM
lattice images
Thermal diffuse scattering (TDS) is the other process which contributes to ADF imaging. This process does not introduce any significant energyloss but produces large momentum transfer, which can scatter the electrons to the angular range of the ADF detector. As arrowed in fig. la, the TDS streaks can be seen at higher angles. Since the ADF images are formed by the electrons which have been scattered far from the central transmitted spot for angles larger than about 40 mrad, the contribution of TDS may be of crucial importance even if the specimen is thin. When the beam convergence is getting large, the thermal diffuse scattered electrons are distributed almost uniformly in the diffraction pattern (fig. lb). As indicated in fig. lb, the detection angular range of an ADF detector may be characterized by ( u2 ui) in the reciprocal space. The electrons scattered into this angular range may be due to thermal diffuse scattering rather than large-angle elastic scattering. As indicated in fig. 1, the FOLZ appears in the diffraction patterns. We have shown that the detection of the FOLZ in ADF imaging may introduce strong diffraction contrast [l]. It is suggested that FOLZ or high-order Laue zones (HOLZ)
Fig. 1. Convergent beam electron diffraction patterns from Si(100) for different beam convergences. These patterns were taken in a JEOL 2000FX TEM microscope, operating at 200 kV.
Z.L. Wang, J.M. Cowley / Dynamic
theory of high-angle ADF STEM
should be excluded in forming the ADF images in order to avoid difficulties in image interpretation.
coefficient p(b,
lattice images
217
is
z) =u2
A,&(b-Rj,
Az)[’
3. Dynamical TDS theory of ADF imaging /AZ,
3.1. General theory In our first paper, we have presented the full dynamic multislice theory derived from quantum mechanics for including TDS in electron imaging simulation [l]. Here we use the final results of the theory for the further analysis. Based on the generalized multiple-elastic and -inelastic multislice theory by Wang [2], Wang and Cowley have given a multislice formula for TDS before (at z = zO) and after (at z = z) transmitting through a thin crystal slice of thickness AZ at position b = (x, y) in the slice plane. For the elastic scattered wave after penetrating through the n th slice,
(3) the TDS generation G(b,
z) exp(-p(b,
z) Az/2)+,-,(b,
* P(b),
= ioc
A,
exp(icui)
= { q(b,
z) exp( -p(b,
xu;(b-R,, and the propagation
A is the wavelength
(Ia)
u = e/Au.
x [6-i@,
z)Az/2)
z,,) + G(b> z)&-,(b,
*P(b).
(lb)
With a slice phase grating q(b,
z) = exp[iau’(b,
u’(b,
AZ) = /‘dz&/(b 20 i
z,,)]}
function,
AZ)], - Rj, z),
(24 (2b)
and u’ is the thermally modified atomic potential (i.e. add a Debye-Waller factor in the atomic scattering factor), locating at Rj. The absorption
&
+ A,, exp(i$)
&]
J
~a>]
where P(b) is the propagation function and the symbol * indicates convolution. Note different notations are used here other than the ones in ref. [l] for the convenience of discussion. The symbol +n means the elastic wave at the n th slice. For the TDS wave,
is
z)
P(b)=&exp = [q(b,
function
AZ) function
[ 1.
(4 is
(5)
i&
of the electron
and (6)
Here uj(b - Rj, AZ) is the projection of the atomic potential peak located at R, in the (x, y) plane within the slice, A, and A, indicate the mean displacements of the jth atom in the x and y directions respectively; ai and (Y: are the time-dependent phases of the x and y vibrations of the jth atom, which means that the vibrations in the x and y directions need to be treated independently and the vibration of each atom needs to be treated incoherently if there is no vibrational coupling. Phase relationship for the atomic vibrations could be included in the calculation by assuming that a’, and a$ depend on the atomic positions for the particular assumed phonon excitation model. It is also important to note that the non-symmetric vibration of the atoms in the x and y directions can be introduced in the calculations. This may be important for simulating reflection high energy electron diffraction (RHEED), because the atoms at the surface may vibrate anisotropically in the directions parallel and perpendicular to the surface.
Z.L. Wang, J.M. Cowley / Dynamic
278
theory of high-angle ADF STEM
lattice images
The atoms in different crystal slices may vibrate completely randomly or with some kind of phase correlation. To include these possibilities in the dynamical calculation, it may be an approximation to introduce arbitrary phases ( CY,for the n th slice) to the TDS wave generated from different crystal slices. A large number of random phases can effectively describe the incoherent addition (due to thermal statistical average) of the TDS waves from different slices. For the case that there is phase correlation between the neighboring atoms, a proper choice of the added phases can represent the coupling of atomic vibrations. Then we can write eq. (lb) as
where A& is the TDS wave generated from the n th slice (at depth z, = n AZ) after converting to the reciprocal space, u = ( ux, uY). The summation of n means the incoherent adding of intensities of the TDS waves generated from different slices, i.e. the intensity integration over crystal thickness. From eq. (lb),
+T(b,
D(u)
z) = { q(b,
z) exp( -p(b,
X [+T-i(b, X@%-i(b,
2) AZ)
zO> + exp(ia,)G(b, %)I}
*P(b).
theories
under
z) (7)
certain
$2 zn)
= [G(u, and D detector,
z) * $J,&
is the
1
=
function
of the
ADF
(10)
otherwise.
+,)
= x/a.(jdq n X
j
G*(q,
db +Z(~-$,,
zn>
z,z)
Xexp[-i2?r(u-q)=b] X jdq’
(8)
detection
(9)
+n is the probe shape at depth z, determined by eq. (la); ui and u2 are the inner and outer radii of the ADF detector in reciprocal space. Put eq. (9) into (8) and use the Fourier transforms of G and c$~,,and also 1P(u) 1 = 1; one has
ap-
3.2.1. Case I: no elastic scattering after the TDS Eq. (1) can be simplified if the elastic scattering of the electrons after being inelastically scattered is neglected. This is a good approximation for thin specimens. In this case, the TDS wave generated from each slice should be added incoherently and the change of the electron angular distribution after TDS, if any, is limited to the ADF detector range. This is a reasonable approximation when the signal detected is given by integration of intensities over a wide-angle annular detector. Then the intensity detected by the ADF detector for a probe, centered at br, = (x,, yr), may be written approximately as eq. (8) in the single phonon excitation model:
z,,r b,)] P(u),
for u1
10
Full dynamical calculation following eqs. (la) and (7) for inelastic incoherent processes, including multiple elastic scattering after the inelastic process, involves huge amounts of calculations. It is necessary, in practice, to seek some simplified approaches which may give reasonable results under certain approximations. This will be the purpose of the following sections. 3.2. Some simplified proximations
A@,
G(q’,
Xexp[i2p(U
a,,) jdb’
+,@-$,,
zn>
- q’) -b’] D(u)),
(11)
where q and q’ are the two-dimensional vectors in the reciprocal space. By introducing the inverse Fourier transform of the detector function D and integrating over U, eq. (11) becomes I($,)
= c jdb n x
j
G*@,
db’ G(b’
xD(b
- b’).
z,>@(b-b,,
z,&,@‘-
z,,)
br, zn) (12)
This is the detected intensity of the ADF detector for an incident probe centered at $,. It depends on the elastic probe function, the localized
oniy
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
inelastic generation function and the detector shape function. It is expected that eq. (12) may give reasonable results if the sample is thin. For thicker specimens, the later elastic scattering of the electrons after inelastic scattering could deflect the electrons out of the ADF detecting angular range into large-angle Kikuchi patterns. Then a full dynamic calculation following eqs. (la) and (7) may be required. In practice, if the ADF detector is large enough so that the scattering of the electrons is still within the detector plane, then (12) is still a good approximation. 3.2.2. Case II: total detection of the inelastic scattered electrons Eq. (12) can be further simplified if all the TDS electrons are treated incoherently and totally collected by the detector or if the signal collected by the detector is assumed to be proportional to the total TDS scattering. Under this assumption, the elastic Bragg scattering of the electrons after the inelastic scattering (TDS) will not have any effect on the STEM image simulation. Taking the detection function as a unity, i.e. D(b - b’) = S(b - 6’), then from eq. (12)
To compare eq. (13) with the theory given by Pennycook et al. [S], we assume that the inelastic scattering is ideally localized, that is, IG12=&S(bp-Rj),
where u, is the inelastic scattering cross section the jth atom; then eq. (13) becomes
of
(14) n i This is the result given by Pennycook et al. [5] based on the Block wave approach, which is an approximate result of eq. (13) particular for coreshell losses. However, the 6 function approximation is not a good approximation for TDS, because the TDS generation function I G I 2 is zero at the nuclear position of each atom, because auf/ax = 0 and av’/ay=O at b= Rj (i.e. the nuclear positions). Then it is suggested that eq. (14) cannot give quantitative results of ADF imaging contrast. For a simple case, I G I 2 can be approximately taken as
I GTDS I 2z
This result is exact if the ADF detector detects all the inelastic scattered electrons even for thick crystals. I G I 2 is effectively the scattering cross section of the inelastic process, and I C#J~ I 2 is the distribution of electron current density at depth z. For very thin objects, eq. (13) can be connected with incoherent imaging theory in optics. I $n I 2 can then be equated to the modulus square of the impulse response of the lens (i.e. the intensity distribution of the image of a point object), and G is the transmission function of the object [4]. Eq. (13) still holds if there is elastic scattering of the inelastic waves (i.e. Kikuchi lines). In practical ADF STEM imaging, the collection angular range of the ADF detector is limited to about 40-100 mrad, which of course does not collect all the inelastic scattered electrons. Hence it is necessary the check the valid&y of eq. (13).
279
CujS( I b - Rj I - 4/2),
i
where ad is approximately the half-width of the jth atom. Then the convolution of the probe intensity distribution with ] G,,, ] 2 will give a peak at the nuclear sites, which comes from the circular addition of the intensity tails of the ring-shape ] G ] 2 function, but its height is not proportional to the uj. Then the intensity at an atomic site is determined by the shape of the atom (ai), the scattering power of the atom (2) and the probe shape at depth z. If the electron probe were a point, then the observed ADF image of the atoms would be a ring-shape intensity with minimum at the nuclear position for TDS. It is the TDS which destroys the Z2 contrast rule of Rutherford scattering. Eq. (12) can be approximately written in the form of eq. (13) if the inner radius of the ADF detector is much larger than the Bragg angles around the center spot. In this case, D(b) is very close to a 6 function. In practice, this is a very
Z.L. Wang, J.M. Cowley / Dynamic
280
good approximation. section 5.5.
This point
will be proved
theory of high-angle ADF STEM.lattice
in
3.2.3. A modified generation function method To make a better approximation taking into account the size of the detector, we try to introduce a modified generation function, which is defined by (15) [6]. In the reciprocal space G(u, Gnew(~, z) =
z)
0
for ui
05)
This newly defined inelastic generation function drops the Fourier components falling out of the angular range of the ADF detector in the reciprocal space and picks up those left. Then the detected intensity can be approximately written as
images
Eq. (17) can be generalized to consider the contributions of several inelastic processes. Then the G function can be replaced by IGw,
I2 =
IGTDS
I2 +
I %x,-loss I 2 + . . . .
The calculation of IG,,,,_ Loss I' can be related with the positionally dependent excitation probability of the atomic inner shells [8]. For TDS process, after making the time average, from eq. (4)
I%&~
z) I 2 2
4
xexp(-p(b) AZ), * +?J U? z,, Q]
p(u)
I 2.
(16)
Compared with eq. (8), we have dropped the detection function D, which is contained in the defined new generation function (eq. (15)). Following the analysis similar to that in the last section, one has
(19)
where u” means the atomic potential centered at position R, after taking off the Fourier components out of the angular range of the detector. The accuracy of simulating ADF images using eq. (17) will be examined in section 5.5. 3.3. Coherent images
Compared with eq. (12), eq. (17) may significantly simplify the numerical calculation. Eq. (17) still holds if the inelastically scattered electrons are elastically scattered to produce Kikuchi patterns within the detector angular range. It is important to note that the STEM imaging intensity is a convolution of the probe intensity with the square of the inelastic generation function, which is proportional to the scattering cross section of the inelastic process. Then the phase of the generation function drops out. This will provide a great advantage for practical numerical calculations, because the phase of an inelastic transition matrix element is always unknown either theoretically or experimentally.
(18)
elastic
scattering
theory
for
ADF
Although it is convenient to discuss high-angle ADF imaging in terms of inelastic scattering and incoherent imaging, it should be noted that the imaging theory for TDS is based on a coherent elastic scattering formulation. Because the energy loss for TDS is unobservably small, the image produced by TDS can be described as the sum of a large number of instantaneous images produced by elastic scattering from various configurations of static displaced atoms. It is convenient that, in the average over time, some of the terms in the intensity expression average to zero. The same terms will, in any case, be small if the atom displacements at one time vary randomly along the columns of atoms parallel to the incident beam direction. It follows that the ADF image intensity will be much the same as for TDS if the atoms in a crystal have static displacements from their mean
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
position which are not correlated over large distances. This is the case, for example, when the atom displacements are generated by point defects, impurity atoms or short-range ordering of the atomic species on the lattice sites. For purely static displacements, the ADF imaging theory is then based on purely elastic, coherent imaging theory. For ideally perfect non-vibrating crystals, the correlation in atomic positions between slices concentrates the high-angle scattering into the sharp reflections in the HOLZ rings of the diffraction pattern and this gives the characteristic coherentdiffraction effects in the image [l]. Deviations from perfect periodicity of the atomic arrangement, and especially deviations involving atom displacement, give diffuse high-angle scattering. If the correlation range of the deviations from perfect periodicity is small in comparison with the crystal thickness, it is possible to formulate an “effective incoherent imaging scattering function” which may then be used, as in the case of TDS, to simplify the calculations of high-angle ADF image intensity. For very thin specimens, for which a phase-object approximation applies, it was shown by Cowley [7] that the effective incoherent imaging scattering function is [l - cos(a~(b))]
= a2$(b)/2,
where I#I(~) is the projection of the specimen potential distribution, and convolution of this function with the intensity distribution of the incident beam gives the ADF image intensity. For greater thickness, the incident wave function for each slice is modified by diffraction in the specimen and the correlation of atomic positions in successive slices introduces the three-dimensional coherent diffraction effects which limits the validity of the incoherent imaging approximation.
4. Computation method The detailed computation method for ADF imaging simulation has been given elsewhere [l]. In the following calculations in sections 5.1-5.4,
281
we employed the full dynamical theory described in eqs. (la) and (7) to simulate the contribution of TDS electrons to the ADF image for different vibrational phase couplings. The accuracy of using eqs. (13) and (17) for predicting image contrast will be examined in section 5.5. In this study, we use the Ge/Si (100) interface as an example to demonstrate the simulation of ADF images. The unit Ocell is constructed in a dimension of 20 A X 20 A and separated by 128 X 128 pixels in numerical calculations. The slice thickness is chosen as 5.43 A and all the computations are carried for a slab of thickness 108.6 A. The angular detection range of the ADF detector is limited to 37-111 mrad, which excludes the diffraction contrast effect arising from the FOLZ, located at 115 mrad. STEM images are formed by scanning a very small electron probe across the specimen. The probe is formed by an objective aperture of 0.5 A-’ in radius, the spherical aberration C, is taken as 0.8 mm, the lens defocus is taken as 800 A and V= 100 kV. The generated electron probe is shown in fig. 2. This coherent probe is small enough to resolve the 2 A lattice spacing between Si and Si or Si and Ge atoms. To reduce the interacting CPU time, one calculates only a line scan of the electron probe across the Ge/Si boundary through the center positions of the atomic rows. For simplicity of the calculation, the Ge and Si lattices are assumed to be matched coherently at the boundary and both of the materials have the same lattice constant. The beam
4 Fig. 2. STEM electron probe generated by an objective aperture of radius 0.5 A-’ (corresponding to 18.5 mrad in semiconvergence angle), C, = 0.8 mm and Af = 800 A for 100 kV electrons.
2. L. Wang, J. M. Cowley / Dynamic theory of high-angle ADF STEM
282
lattice images o Elastic
incident azimuth was chosen as [loo]. The atomic scattering factor is taken in the form of Mott formula, f(s)
= 0.023934[ Z - F,(s)]/s2,
Elastic
& TDS
?.I
SI
(20)
where s = u/2, Z is the atomic number and fx is the X-ray scattering factor for this atomic number. Then the scattered intensity from a single atom is proportional to Z2 and inversely proportional to s4 at very large scattering angle. This is the result for Rutherford scattering from a single atom and should be included in the calculations automatically. Also a proper Debye-Waller factor has to be added to the scattering factors of each atom according to the assumed mean displacements. This can significantly affect the relative contrast of each atom in the ADF images.
5. Calculation results In this section we present the simulated ADF images of the Ge/Si semiconductor interface using the elastic and inelastic scattering theories. Before presenting the results, it is important to point out that the arbitrary phases added to the TDS waves generated from different slices should be the same set for different scanning probe positions, otherwise some artifacts will be introduced in the calculation. 5. I. In-phase
n
vibration
Fig. 3 shows a simulated Ge/Si interface ADF image by assuming that all the atoms in all slices are vibrating in-phase (i.e. crystal is vibrating as a whole). The elastic scattering then comes from the time-averaged structure (with a Debey-Waller factor on the scattering amplitudes) and the “TDS” gives the time-average image for displaced crystals of non-vibrating atoms. The coefficients are the mean vibration amplitudes of Asi and A, the Si and Ge atoms. The large-angle elastic scattering and the TDS give about the same contribution to the ADF imaging contrast. The atomic positions, separated by 2 A, are indicated along the horizontal axis. The intensities are plotted on a
0
G.2
Ge
Ge
Ge
SI
Fig. 3. Simulated ADF imaging profile across the Ge/Si boundary according to eq. (1) by assuming that all the atoms within a same slice and those in different slices are vibrating in-phase. The thermal vibration amplitudes of the Ge and Si atoms are taken as 0.1 A.
normalized scale along the vertical axis so that the numbers for different simulations can be compared with each other. The interface is seen to be sharp and the Ge and Si atoms at the boundary show very distinct contrast. The relative intensity of Ioe/lsi is about 2.2 for both the TDS electrons and the elastically scattered electrons. The summation of the two also gives the contrast of about 2.2. In disagreement with the extreme classical model of single-atom Rutherford scattering, the contrast variation is 2.2 for this particular ADF collecting angular range rather than Z,‘/Z,” = 5.2, which would follow from Z,/Z, = 2.28. This difference may come from the different origins of TDS and Rutherford scattering. Hence, it should be more proper to claim that ADF image contrast is the contrast coming from TDS rather than large-angle Rutherford scattering. The result in fig. 3 is obtained by assuming that both the Ge and Si atoms are vibrating with the same ampli-
.
0
Ge
Ge
Elastic
& TDS
-::::::I
Go
Ge
SI
SI
SI
Fig. 4. Same as fig. 3 except that the thermal vibration tudes of the Ge and Si atoms are taken as 0.1 and respectively.
ampli0.2 A,
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
tudes. The situation may change if the two atoms have different thermal vibration amplitudes. Shown in fig. 4 is the same calculation as fig. 3 except that the thermal vibration amplitude of the Si atoms is doubled. It is obvious that Si then shows stronger intensity and the final imaging contrast is about ~oce/lsi = 4/3. This shows that the TDS may increase the visibility of light atoms and the ADF image contrast cannot be simply predicted using the 2’ rule [l]. 5.2. Contrast due to random atomic vibrations In practical phonon excitations, the atoms in the crystal will not vibrate in the same phase. The random vibration of the atoms will statistically introduce a diffuse scattering background in electron diffraction patterns. Fig. 5 shows a comparison of the calculated ADF image contrast across the Ge/Si interface with (curve B) and without (curve A) considering the randomness of the atomic vibration in different slices. Note curve B was calculated by adding a random phase to the TDS waves generated from different slices but not to the atoms within a same slice. The first change we see is the relative magnitude of the ADF imaging intensity between the curve B and A (bottom blank circles). The random vibration of the atoms in different slices introduces a diffuse
I
1
0
.opA
0
Ge
Ge
Ge
Ge
S,
SI
SI
Fig. 5. Comparison of the simulated ADF imaging profile arising from pure TDS across the Ge/Si boundary with (curve B) and without (curve A) adding random phases to the TDS waves generated from different slices. Note curve A is magnified by 18.66 times to compare with curve B. The thermal vibration amplitudes of the Ge and Si atoms are taken as 0.1 A.
Ge
Ge
Ge
Ga
Si
283
SI
SI
Fig. 6. Comparison of the simulated ADF imaging profile arising from pure TDS across the Ge/Si boundary with (curve B) and without (curve A) adding a random phase to the TDS waves generated from different slices. Note curve A is magnified 18.4 times to compare with curve B. The thermal vibration amplitudes of the Ge and Si atoms are 0.1 and 0.2 A, respectively.
scattered background in the diffraction pattern particularly at the collecting angular range of the ADF detector, then the ADF imaging intensity from TDS is increased about 19 times compared to the calculations for in-phase vibrations. Comparing the vertical scale of the curve B with that of fig. 3, it can be positively concluded that the TDS dominates the image formation in ADF STEM rather than elastic scattering if the thermal vibration amplitudes of the Ge and Si atoms are close to 0.1 A. In order to see the relative contrast variation after introducing the vibration randomness, we magnified the curve A for 18.66 times to match the intensity of curve B at the Ge sites. The relative visibility of the Ge atoms with respect to the Si has been changed slightly. Also the peak intensity at the Ge sites is increased compared to the valley part between the atoms in curve B. This contrast deviation becomes more significant with the increase of the vibration amplitude of the Si atoms. Shown in fig. 6 is another comparison similar to that of fig. 5 except that the thermal vibration amplitude of the Si atoms is doubled. The in-phase calculation gives stronger intensity at the Si sites but the random phase calculation gives the opposite result.
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
284
5.3. Dependence of ADF imaging intensity and contrast on the correlation of vibrations of neighboring atoms in the same crystal slice ADF imaging contrast of a row of atoms may not depend on correlations with the thermal vibrations of other rows of atoms if the specimen is thin. In the STEM case, this statement may be true if the probe size is less than the spacing between the atoms. Fig. 7 shows a comparison of the calculated ADF imaging contrast of the Ge/Si interface for the following cases: (A) the atoms within the same slice are vibrating in-phase but with random phases for different slices; (B) the atoms are vibrating with random phases within the same slice and also randomly for different slices; (C) neighboring atoms within the same slice are vibrating with a phase difference of 2n/5 in the x direction but in-phase in the y direction, the atomic vibration is treated randomly for different slices; and (D) neighboring atoms within the same slice are vibrating with a phase difference of 277/5 in the y direction but in-phase in the x direction, the atomic vibration is treated randomly for different slices. Note in these calculations the random phases added to the TDS waves generated from different slices are the same set to avoid statistical fluctuations. It is seen that the ADF imaging intensity profiles not only show exactly the same intensity, but also the relative contrast of
Ge
Ge
Ge
Ga
S,
SI
SI
Fig. I. Dependence of ADF imaging intensity and contrast arising from pure TDS on the vibration status of the atoms in the same slice. This figure compares the simulated ADF imaging profile arising from TDS across the Ge/Si boundary for different vibration phase couplings of the atoms within a slice (see text for details). The thermal vibration amplitudes of the Ge and Si atoms are taken as 0.1 A.
Ge
Ge
Ge
Ge
SI
SI
SI
Fig. 8. Dependence of ADF imaging contrast arising from TDS on the vibration status of the atoms in different slices. This figure compares the simulated ADF imaging profile arising from TDS across the Ge/Si boundary for different vibration phase couplings of the atoms in different slices (see text for details). The thermal vibration amplitudes of the Ge and Si atoms are 0.1 and 0.2 A, respectively.
the Ge and Si atoms is almost identical. This shows that the ADF imaging contrast is independent of the vibration status of its neighbor atoms within the same slice. In other words, whether the thermal vibrations of the atoms are assumed to obey the Einstein model, Debye model or any other model will not affect the ADF imaging contrast if the probe size is small. These calculation results are in agreement with Cowley’s conclusion [9]. 5.4. Dependence of ADF imaging contrast on the correlation of vibrations for neighboring atoms in different crystal slices It was shown in the last section that the ADF imaging contrast is independent of the thermal correlation of vibrations of the neighbor atoms within the same slice. Then the imaging simulation can be simply done by assuming that all the atoms within the same slice are vibrating in-phase. It is necessary to find out the dependence of the imaging contrast on the atomic vibrations in different crystal slices. Shown in fig. 8 is a comparison of the calculated ADF imaging intensity profiles across the Ge/Si boundary for the following three cases: (A) random phases are added to each slice; (B) designated phase sequence of lr/2, 0, r/2, 0, . . . , are added to successive slice; and (C) designated phase
Z.L. Wang, J.M. Cowley / Dynamic theory
sequence of r/2, -r/2, n/2, -r/2, . . . , are added to successive slices. The thermal vibration amplitude of the Si atoms is doubled compared with that of fig. 7. Some differences in the relative magnitudes of the ADF images are seen, but the contrast profiles for these cases follow almost exactly the same curve after being normalized at the Ge sites (curve B and C are magnified for 2.03 and 1.093 times respectively). This shows that the vibration status of the atoms in difference slices may introduce some difference in the intensity magnitudes but very little change in the relative contrast of the Ge and Si atoms which is the quantity usually observed in experiments. It thus can be concluded that the simulation of the ADF image contrast arising from TDS will not be affected by the relative vibrations of the neighboring atoms, but the absolute intensity of the image varies somewhat with the coupling of vibrations of the atoms in different layers. Then for absolute quantitative (i.e. absolute intensity distribution rather than relative contrast) analysis of the ADF image it is necessary to consider the vibrational coupling among the atoms. Fairly reasonable contrast information can be obtained- by assuming arbitrary vibrations. This may provide a general guide for ADF imaging simulations.
of high-angle ADF STEM lattice images
285
dynamic scattering theory of eqs. (la) and (7), which should apply to crystals of any thickness if only single-phonon processes are considered. The simplified methods, described in eqs. (13) and (17) can greatly reduce the amount of calculation, but the accuracy of this treatment has to be examined. Shown in fig. 9a is a comparison of the calculated ADF image across the Ge/Si boundary using the theory of full dynamic incoherent scattering (described in the last paragraph) (curve A), eq. (17) (curve B) and eq. (13) (curve C). Different absolute intensities are obtained for the different cases, but each of the curves shows almost the same variation tendency. To see the relative contrast variation for the Ge and the Si atoms, the three curves are normalized at the Ge site and compared in fig. 9b. It is surprising to note that the three curves show almost the same contrast variations. The contrast difference between them is less than 5%. This gives the answer that the simulation based on either eq. (13) or eq. aTo,
5.5. Validity of the simplified incoherent theory (eqs. (13) and (I 7)) for ADF image simulations 04 ::::
The quantitative simulation of ADF images involves the assumption of incoherence of the TDS (due to thermal average) waves generated from different crystal slices if all the atoms are vibrating completely randomly. In this case we do not have to consider the phase correlation among the atoms within the same slice, as shown in section 5.3. To process this calculation accurately, one has to calculate the TDS wave generated from different slices and let them be scattered elastically through the rest of the crystal thickness individually, then add the intensities of each individual TDS components generated from different slices after penetrating through the crystal. This calculation process considers the incoherency of the inelastic scattering precisely but involves huge amounts of calculations. This is the exact result of
: ::::
: ::::::::::::::
_-
:
:::-I
Ge Ge si I::::::::::::::::::::::::::::t Si
Fig. 9. (a) Comparison of the simulated ADF image intensity across the Ge/Si interface using the full dynamical incoherent scattering theory (i.e. consider the elastic scattering of the electrons after being inelastic scattered) (curve A), simplified incoherent theory eq. (17) (curve B) and eq. (13) (curve ,C). (b) Comparison of the three curves shown in (a) after being normalized at the Ge sites. This is to test the accuracy of simulated imaging contrast using the simplified theory.
286
2.L.
Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM
(17) can give almost the accurate contrast distribution of the ADF image for a crystal slab of about 100 A in thickness. This important result shows that the incoherent imaging theory can be used to simulate the ADF imaging contrast quantitatively if the image intensity is dominated by TDS. It may be noted that the Si atom closest to the Ge boundary has higher contrast than the next Si atom. This may come from the contribution of the tail of the probe scattering from the Ge or from a spreading of the probe in the crystal due to the discontinuity at the interface. Then it is possible that an error of one monolayer may be made in the chemical identifications. Eq. (13) was derived by assuming that all the inelastically scattered electrons are collected by the detector. The image simulation based on this assumption still gives the correct contrast if the inner radius of the ADF detector is large enough to avoid the diffraction contrast and its outer radius is large enough to collect most of the largeangle inelastic scattering. This correctness indicates that the portion of electrons scattered to the detector depends only on the localized scattering power of the individual atoms if the sample is thin. It is important to emphasize that the incoherent inelastic scattering can give atomic resolution lattice images. The mechanism is obviously different from the coherent TEM or STEM imaging theory. The incoherent images can be simulated easily because there is little dependence on focus and thickness. It must be pointed out that the above calculations in section 5 were carried out by using a small probe of about 1.2 A in half width. Also the thermal vibration amplitudes of the atoms were assumed to be around 0.1 A. The conclusions are drawn on the basis of these pre-defined parameters. If the thermal vibration amplitude of an atom is reduced to about 0.01 A, then the contribution of the TDS scattering to the ADF image is reduced by about 40-100 times. In this case, the contribution from pure elastic scattering will play an important role (see the result in fig. 3). In this case the calculation based purely on eqs. (13) and (17) may not give reasonable results. This situation may not occur because the mean vibration
lattice images
amplitude of a Si atom is approximately 0.045 A even at 0 K. It is expected that the incoherent imaging theory can accurately predict the contrast of ADF images obtained at high temperatures.
6. Programming procedures for simulating ADF images based on a multislice program for HREM As a summary of the above calculations, we give simple programming procedures for simulating ADF imaging contrast in the multislice algorithm for thin specimens, which can be easily modified and is based on the standard high-resolution electron microscopy image-simulation program. 1. Follow the routine procedure for generating projected atomic potentials for a slice of thickness AZ:
(21) A proper Debye-Waller factor has to be added to each atomic scattering factor according to the assumed mean vibration amplitudes of each atom. 2. Calculate the absorption function of the slice:
p(b, 2) = cJ2
+C
Ajy&U;(b-ni)J’]/Az.
(22)
i 3. Turn u’ to reciprocal transformation:
space
using
Fourier
U;(U) =.%=(u,(<)). 4. Take off the Fourier components falling outside the detector angular range of ui < u < u2: U;‘(U) = 5. Turn u;‘(b)
U,‘(U)
for u,
0
otherwise.
(23)
u” back to real space:
=.P-‘(u;‘(U)).
(24)
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
6. Calculate IG,,,(b,
the TDS generation
287
slice:
function:
q(b) = exp[iaV’(b)],
z) I *
@a)
with V’(b)=
Xexp[ -p(b)
AZ].
(25)
It is possible to add the contribution of other inelastic scattering processes, such as atomic core shell losses, to the ADF image at this stage. 7. Calculate the transmission function of the
xo’(b-Rj).
(26’4
Multislice calculation of the imaging intensity for probe position bp. The shape of the incident probe at depth z, is determined by
h,(b-bp, zn) G)AZPI x+n-,(b - bp,z,,)} * f’(b).
= { q(b,
z,) exp[ -p(b,
(27)
A CYCLE Of=PROBE SCANNING IN STEM
ELASTIC ADF
-
I Elas
GENERATED TDS A1TDS
MULTISLICE INTERACTION CYCLE
TDS ADF ‘TDS
1 STOP Fig. 10. Flow-chart of an ADF imaging simulation program, modified based on the HREM imaging simulation program initiated by Isbizuka [lo], according to the theory given by eq. (17) or eq. (13). The procedures introduced in section 6 are contained in this program (please contact Z.L. Wang if any one is interested in this program).
Z.L.. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
288
9. TDS contribution to the image intensity from the nth slice: &&rJn =
J
db ]G,,,,(b,
= ]G,,,(b,,
zn) I2 I%n(b-bp,
zn) 12*
I$&,
zn> I2
zn>12.
(28)
10. Add all the generated TDS inter& .y from all the slices (total N) for probe position bp: (29) 11. Calculate elastic contribution to an ADF image after penetrating the crystral (the Nth slice):
where D is the defined detector
function in eq.
(IO). 12. Add the total contribution inelastic scattering:
of elastic and
I($) =bs(bp) +Zm&p).
(31)
13. Repeat procedures (8) to (12) for another scanning position. Fig. 10 shows a flow-chart of the ADF image simulation program according to eq. (17) (or eq. (13)). This program was modified based on the HREM imaging simulation program initiated by Ishizuka [lo]. The flow-chart given in our first paper [l] deals with the ADF image simulations following eqs. (la) and (7), considering the vibrational coupling among the atoms. The calculation following fig. IO should give an easy and quick method for image interpretation.
7. Conclusions Dynamical calculations have shown that the contrast of annular-dark-field (ADF) scanning transmission electron microscopy (STEM) images is dominated by thermal diffuse scattering (TDS). The simulation of the ADF image can simply follow the incoherent imaging theory, that is, the
image intensity is a thickness integration of the convolution of the electron probe intensity distribution at depth z with the inelastic generation I G ( 2. This can greatly reduce the function numerical computations compared to the exact calculations following the full dynamical elastic and inelastic scattering theory (i.e. considering the elastic scattering of the new generated inelastic wave from each slice). This ADF imaging theory is the same as the incoherent imaging theory in optics. Similar results can also be derived from coherent elastic scattering theory, as described in section 3.3. Then it will not make any difference whether the ADF imaging is described as coherent or incoherent. It is the result of a combination of dynamical elastic and inelastic scattering. Its contrast cannot be predicted using any simple model like single-atom Rutherford scattering. The calculation according to this simplified theory can give almost the exact contrast as obtained from the full dynamical inelastic incoherent calculations but not the absolute intensity. The former is the most wanted quantity in practical analysis. The relative contrast error is estimated to be less than 5% and the CPU time can be cut down by N/2 times compared to the accurate full dynamic calculation, where N is the total number of slices. It is obvious that the simulation for thick crystals, i.e. large N, becomes very time consuming if using the full theory. Then the introduction of eqs. (13) and (17) makes the ADF imaging simulation feasible. A routine multislice image-simulation program for high-resolution electron microscopy (HREM) can be easily modified for simulating ADF images following the procedures given in section 6. As claimed in section 5.4, eq. (17) is almost exact if the specimen is thin. This thin specimen can be up to 500 A or thicker and satisfactory accuracy of imaging contrast can still be obtained. This can generally meet the specimen thicknesses in most high-resolution lattice images. The incoherent TDS can produce atomic resolution lattice images if a small electron probe, smaller than the unit cell, is provided. This image is produced by the localized elastic and inelastic scattering from the atomic sites. Channeling effect can enhance the ADF image contrast but is not necessary to form the Z-contrast image. The ADF
Z.L. Wang, J.M. Cowley / Dynamic theov
image contrast is found to be Z-dependent and is capable of providing localized specimen chemical information at sharp atomic interfaces, but a monolayer uncertainty may be introduced due to the scattering of the probe tail or the spreading of the probe. Dynamical calculations have shown that the ADF imaging contrast of an atom is almost independent of the vibration status of its neighbors, so that the image simulation will not be affected by assuming either Einstein model, Debye model or any other models. This can greatly simplify the theoretical analysis of ADF images and expands its applications in material science. Quantitative simulation can be obtained if the mean vibration amplitude of each atom in the specified specimen is known. It must be emphasized that the domination of TDS process to ADF imaging contrast reduces if the thermal mean displacements of the atoms decreases. It is expected that the elastic contribution will become important if the atomic thermal vibration amplitudes are less than about 0.03 A, i.e. for only a few substances at temperature close to 0 K.
of high-angle ADF STEM lattice images
289
Acknowledgements
The authors thank Dr. J.C.H. Spence for this encouragement, discussions and detailed comments on the manuscript. Thanks also go to Drs. P. Rez and P. Perkes for maintenance of and assistance with the computer system. This work was supported by NSF grant DMR 88-10238, and made use of the ASU Facility for HREM, supported by NSF grand DMR 8611609. References [l] Z.L. Wang and J.M. Cowley, Ultramicroscopy 31 (1989) 431. [Z] Z.L. Wang, Acta Cryst. A 45 (1989) 636. [3] S.J. Pennycook, EMSA Bull. 19 (1989) 67. [4] See, for example, J.C.H. Spence, in: Experimental High Resolution Electron Microscopy, 2nd ed. (Oxford Press, Oxford, 1989). [5] S.J. Pennycook, D.E. Jesson and M.F. Chisholm, Inst. Phys. Conf. Ser. 100, section 1 (1989) 51. [6] A. Howie, private communication. [7] J.M. Cowley, Ultramicroscopy 2 (1976) 3. [8] R.H. Ritchie, Phil. Mag. A 44 (1981) 931. [9] J.M. Cowley, Acta Cryst. A 44 (1988) 847. [lo] K. Ishizuka and N. Uyeda, Acta Cryst. A 33 (1977) 740.
32 (1990) 275-289
275
DYNAMIC THEORY OF HIGH-ANGLE FOR A Ge/Si INTERFACE Z.L. WANG Department Received
ANNULAR-DARK-FIELD
STEM LA’ITICE IMAGES
* and J.M. COWLEY
of Physics, Arizona State University,
15 November
Tempe, AZ 85287-1504,
USA
1989
A dynamical multislice approach for simulating the annular-dark-field (ADF) scanning transmission electron microscopy (STEM) images is presented. The ADF imaging contrast is dominated by thermal diffuse scattering (TDS) if the diffraction contrast from the high-order Laue zones (HOLZ) is excluded. The simulation of ADF images can be greatly simplified if the assumption is made of incoherent imaging theory such as is used in optics, that is, the image intensity is a thickness integration of the convolution of the electron probe intensity distribution at depth z with the localized inelastic generation function 1G 12. This result may also be derived on the basis of coherent elastic scattering theory. The calculation according to the simplified theory can give almost exactly the same contrast as obtained from full dynamical inelastic calculations for thin specimens. The TDS can give atomic resolution lattice images if a small probe, less than the size of the unit cell, is used. A programming flow-chart for simulating ADF images is given. Calculations for Ge/Si(lOO) interfaces have shown that the ADF imaging contrast of the atoms is not affected by correlations of vibrations of neighboring atoms. It is pointed out that the contribution from elastic large-angle scattering will become important only if the atomic vibration amplitudes are less than about 0.03 A. The ADF image production may not be considered as incoherent imaging if this is the case.
1. Introduction Recently, Wang and Cowley [l] have applied the generalized multiple-elastic and -inelastic multislice theory, proposed by Wang [2], to simulate the atomic resolution high-angle annular-dark-field (ADF) scanning transmission electron microscopy (STEM) images, taking into account the thermal diffuse scattering (TDS). In that paper, we have shown that the TDS may play a dominant role in determining the ADF imaging contrast, depending strongly on the atomic number, the sizes (or shapes) of the atoms and their relative vibration amplitudes. It has also been shown by Wang and Cowley [l] that the ADF image has the following characteristics if the first-order Laue zone (FOLZ) (or high-order Laue zones (HOLZ)) is excluded: (1) there is no contrast reversal with change of focus;
* To whom correspondence should be addressed; Metals and Ceramics Division, Oak Ridge National tory, Oak Ridge, TN 37831-6376, USA. 0304-3991/X1/$03.50
0 1990 - Elsevier
now at: Labora-
Science Publishers
(2) there is no contrast reversal with change of specimen thickness and (3) the ADF image shows highly localized scattering from each atomic column and can provide some chemical information for a specimen having two or more atoms with very different atomic numbers. These conclusions agree with the experimental observations [3]. The HOLZ can introduce strong diffraction contrast in the ADF image and possibly give some misleading results in the image interpretations. In this paper, as a continuation of our previous work, we report the further theoretical investigations of ADF imaging formation in STEM. Several approximate approaches for simulating ADF images will be derived from dynamical inelastic scattering theory of thermal diffuse scattering. It will be shown that the incoherent imaging theory can be used to simulate ADF images and quantitative results are expected for thin specimens. Atomic resolution lattice images can be formed by these incoherent inelastically scattered electrons. It is concluded that the relative contrast of an atom is almost independent of the thermal vibration status of its neighbors. Hence, the simulation
B.V. (North-Holland)
276
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle
of ADF images will not be affected to a significant extent by assuming either the Einstein model, the Debye model or any other model.
2. TDS in electron diffraction As pointed out in our first paper [l], two inelastic processes may be important to ADF imaging formation besides the elastic contribution of the HOLZ. The first process is the electron core shell excitation in the energy-loss range of several hundreds to several thousands eV, with an angular spreading of an order 6, = A E/2 E,, where A E is the electron energy-loss and E, is the incident electron energy. Although large momentum transfer may be involved in this process, the probability of exciting the core shells is very small if the specimen is thin. Shown in fig. 1 are two electron diffraction patterns of different beam convergences from Si(100) having a thickness of more than a thousand A. The Kikuchi lines disappear if the sample is thin. Then we can neglect the contribution of core shell losses in the ADF image simulations for thin specimens, such as those about 200-500 A in thickness.
ADF STEM
lattice images
Thermal diffuse scattering (TDS) is the other process which contributes to ADF imaging. This process does not introduce any significant energyloss but produces large momentum transfer, which can scatter the electrons to the angular range of the ADF detector. As arrowed in fig. la, the TDS streaks can be seen at higher angles. Since the ADF images are formed by the electrons which have been scattered far from the central transmitted spot for angles larger than about 40 mrad, the contribution of TDS may be of crucial importance even if the specimen is thin. When the beam convergence is getting large, the thermal diffuse scattered electrons are distributed almost uniformly in the diffraction pattern (fig. lb). As indicated in fig. lb, the detection angular range of an ADF detector may be characterized by ( u2 ui) in the reciprocal space. The electrons scattered into this angular range may be due to thermal diffuse scattering rather than large-angle elastic scattering. As indicated in fig. 1, the FOLZ appears in the diffraction patterns. We have shown that the detection of the FOLZ in ADF imaging may introduce strong diffraction contrast [l]. It is suggested that FOLZ or high-order Laue zones (HOLZ)
Fig. 1. Convergent beam electron diffraction patterns from Si(100) for different beam convergences. These patterns were taken in a JEOL 2000FX TEM microscope, operating at 200 kV.
Z.L. Wang, J.M. Cowley / Dynamic
theory of high-angle ADF STEM
should be excluded in forming the ADF images in order to avoid difficulties in image interpretation.
coefficient p(b,
lattice images
217
is
z) =u2
A,&(b-Rj,
Az)[’
3. Dynamical TDS theory of ADF imaging /AZ,
3.1. General theory In our first paper, we have presented the full dynamic multislice theory derived from quantum mechanics for including TDS in electron imaging simulation [l]. Here we use the final results of the theory for the further analysis. Based on the generalized multiple-elastic and -inelastic multislice theory by Wang [2], Wang and Cowley have given a multislice formula for TDS before (at z = zO) and after (at z = z) transmitting through a thin crystal slice of thickness AZ at position b = (x, y) in the slice plane. For the elastic scattered wave after penetrating through the n th slice,
(3) the TDS generation G(b,
z) exp(-p(b,
z) Az/2)+,-,(b,
* P(b),
= ioc
A,
exp(icui)
= { q(b,
z) exp( -p(b,
xu;(b-R,, and the propagation
A is the wavelength
(Ia)
u = e/Au.
x [6-i@,
z)Az/2)
z,,) + G(b> z)&-,(b,
*P(b).
(lb)
With a slice phase grating q(b,
z) = exp[iau’(b,
u’(b,
AZ) = /‘dz&/(b 20 i
z,,)]}
function,
AZ)], - Rj, z),
(24 (2b)
and u’ is the thermally modified atomic potential (i.e. add a Debye-Waller factor in the atomic scattering factor), locating at Rj. The absorption
&
+ A,, exp(i$)
&]
J
~a>]
where P(b) is the propagation function and the symbol * indicates convolution. Note different notations are used here other than the ones in ref. [l] for the convenience of discussion. The symbol +n means the elastic wave at the n th slice. For the TDS wave,
is
z)
P(b)=&exp = [q(b,
function
AZ) function
[ 1.
(4 is
(5)
i&
of the electron
and (6)
Here uj(b - Rj, AZ) is the projection of the atomic potential peak located at R, in the (x, y) plane within the slice, A, and A, indicate the mean displacements of the jth atom in the x and y directions respectively; ai and (Y: are the time-dependent phases of the x and y vibrations of the jth atom, which means that the vibrations in the x and y directions need to be treated independently and the vibration of each atom needs to be treated incoherently if there is no vibrational coupling. Phase relationship for the atomic vibrations could be included in the calculation by assuming that a’, and a$ depend on the atomic positions for the particular assumed phonon excitation model. It is also important to note that the non-symmetric vibration of the atoms in the x and y directions can be introduced in the calculations. This may be important for simulating reflection high energy electron diffraction (RHEED), because the atoms at the surface may vibrate anisotropically in the directions parallel and perpendicular to the surface.
Z.L. Wang, J.M. Cowley / Dynamic
278
theory of high-angle ADF STEM
lattice images
The atoms in different crystal slices may vibrate completely randomly or with some kind of phase correlation. To include these possibilities in the dynamical calculation, it may be an approximation to introduce arbitrary phases ( CY,for the n th slice) to the TDS wave generated from different crystal slices. A large number of random phases can effectively describe the incoherent addition (due to thermal statistical average) of the TDS waves from different slices. For the case that there is phase correlation between the neighboring atoms, a proper choice of the added phases can represent the coupling of atomic vibrations. Then we can write eq. (lb) as
where A& is the TDS wave generated from the n th slice (at depth z, = n AZ) after converting to the reciprocal space, u = ( ux, uY). The summation of n means the incoherent adding of intensities of the TDS waves generated from different slices, i.e. the intensity integration over crystal thickness. From eq. (lb),
+T(b,
D(u)
z) = { q(b,
z) exp( -p(b,
X [+T-i(b, X@%-i(b,
2) AZ)
zO> + exp(ia,)G(b, %)I}
*P(b).
theories
under
z) (7)
certain
$2 zn)
= [G(u, and D detector,
z) * $J,&
is the
1
=
function
of the
ADF
(10)
otherwise.
+,)
= x/a.(jdq n X
j
G*(q,
db +Z(~-$,,
zn>
z,z)
Xexp[-i2?r(u-q)=b] X jdq’
(8)
detection
(9)
+n is the probe shape at depth z, determined by eq. (la); ui and u2 are the inner and outer radii of the ADF detector in reciprocal space. Put eq. (9) into (8) and use the Fourier transforms of G and c$~,,and also 1P(u) 1 = 1; one has
ap-
3.2.1. Case I: no elastic scattering after the TDS Eq. (1) can be simplified if the elastic scattering of the electrons after being inelastically scattered is neglected. This is a good approximation for thin specimens. In this case, the TDS wave generated from each slice should be added incoherently and the change of the electron angular distribution after TDS, if any, is limited to the ADF detector range. This is a reasonable approximation when the signal detected is given by integration of intensities over a wide-angle annular detector. Then the intensity detected by the ADF detector for a probe, centered at br, = (x,, yr), may be written approximately as eq. (8) in the single phonon excitation model:
z,,r b,)] P(u),
for u1
10
Full dynamical calculation following eqs. (la) and (7) for inelastic incoherent processes, including multiple elastic scattering after the inelastic process, involves huge amounts of calculations. It is necessary, in practice, to seek some simplified approaches which may give reasonable results under certain approximations. This will be the purpose of the following sections. 3.2. Some simplified proximations
A@,
G(q’,
Xexp[i2p(U
a,,) jdb’
+,@-$,,
zn>
- q’) -b’] D(u)),
(11)
where q and q’ are the two-dimensional vectors in the reciprocal space. By introducing the inverse Fourier transform of the detector function D and integrating over U, eq. (11) becomes I($,)
= c jdb n x
j
G*@,
db’ G(b’
xD(b
- b’).
z,>@(b-b,,
z,&,@‘-
z,,)
br, zn) (12)
This is the detected intensity of the ADF detector for an incident probe centered at $,. It depends on the elastic probe function, the localized
oniy
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
inelastic generation function and the detector shape function. It is expected that eq. (12) may give reasonable results if the sample is thin. For thicker specimens, the later elastic scattering of the electrons after inelastic scattering could deflect the electrons out of the ADF detecting angular range into large-angle Kikuchi patterns. Then a full dynamic calculation following eqs. (la) and (7) may be required. In practice, if the ADF detector is large enough so that the scattering of the electrons is still within the detector plane, then (12) is still a good approximation. 3.2.2. Case II: total detection of the inelastic scattered electrons Eq. (12) can be further simplified if all the TDS electrons are treated incoherently and totally collected by the detector or if the signal collected by the detector is assumed to be proportional to the total TDS scattering. Under this assumption, the elastic Bragg scattering of the electrons after the inelastic scattering (TDS) will not have any effect on the STEM image simulation. Taking the detection function as a unity, i.e. D(b - b’) = S(b - 6’), then from eq. (12)
To compare eq. (13) with the theory given by Pennycook et al. [S], we assume that the inelastic scattering is ideally localized, that is, IG12=&S(bp-Rj),
where u, is the inelastic scattering cross section the jth atom; then eq. (13) becomes
of
(14) n i This is the result given by Pennycook et al. [5] based on the Block wave approach, which is an approximate result of eq. (13) particular for coreshell losses. However, the 6 function approximation is not a good approximation for TDS, because the TDS generation function I G I 2 is zero at the nuclear position of each atom, because auf/ax = 0 and av’/ay=O at b= Rj (i.e. the nuclear positions). Then it is suggested that eq. (14) cannot give quantitative results of ADF imaging contrast. For a simple case, I G I 2 can be approximately taken as
I GTDS I 2z
This result is exact if the ADF detector detects all the inelastic scattered electrons even for thick crystals. I G I 2 is effectively the scattering cross section of the inelastic process, and I C#J~ I 2 is the distribution of electron current density at depth z. For very thin objects, eq. (13) can be connected with incoherent imaging theory in optics. I $n I 2 can then be equated to the modulus square of the impulse response of the lens (i.e. the intensity distribution of the image of a point object), and G is the transmission function of the object [4]. Eq. (13) still holds if there is elastic scattering of the inelastic waves (i.e. Kikuchi lines). In practical ADF STEM imaging, the collection angular range of the ADF detector is limited to about 40-100 mrad, which of course does not collect all the inelastic scattered electrons. Hence it is necessary the check the valid&y of eq. (13).
279
CujS( I b - Rj I - 4/2),
i
where ad is approximately the half-width of the jth atom. Then the convolution of the probe intensity distribution with ] G,,, ] 2 will give a peak at the nuclear sites, which comes from the circular addition of the intensity tails of the ring-shape ] G ] 2 function, but its height is not proportional to the uj. Then the intensity at an atomic site is determined by the shape of the atom (ai), the scattering power of the atom (2) and the probe shape at depth z. If the electron probe were a point, then the observed ADF image of the atoms would be a ring-shape intensity with minimum at the nuclear position for TDS. It is the TDS which destroys the Z2 contrast rule of Rutherford scattering. Eq. (12) can be approximately written in the form of eq. (13) if the inner radius of the ADF detector is much larger than the Bragg angles around the center spot. In this case, D(b) is very close to a 6 function. In practice, this is a very
Z.L. Wang, J.M. Cowley / Dynamic
280
good approximation. section 5.5.
This point
will be proved
theory of high-angle ADF STEM.lattice
in
3.2.3. A modified generation function method To make a better approximation taking into account the size of the detector, we try to introduce a modified generation function, which is defined by (15) [6]. In the reciprocal space G(u, Gnew(~, z) =
z)
0
for ui
05)
This newly defined inelastic generation function drops the Fourier components falling out of the angular range of the ADF detector in the reciprocal space and picks up those left. Then the detected intensity can be approximately written as
images
Eq. (17) can be generalized to consider the contributions of several inelastic processes. Then the G function can be replaced by IGw,
I2 =
IGTDS
I2 +
I %x,-loss I 2 + . . . .
The calculation of IG,,,,_ Loss I' can be related with the positionally dependent excitation probability of the atomic inner shells [8]. For TDS process, after making the time average, from eq. (4)
I%&~
z) I 2 2
4
xexp(-p(b) AZ), * +?J U? z,, Q]
p(u)
I 2.
(16)
Compared with eq. (8), we have dropped the detection function D, which is contained in the defined new generation function (eq. (15)). Following the analysis similar to that in the last section, one has
(19)
where u” means the atomic potential centered at position R, after taking off the Fourier components out of the angular range of the detector. The accuracy of simulating ADF images using eq. (17) will be examined in section 5.5. 3.3. Coherent images
Compared with eq. (12), eq. (17) may significantly simplify the numerical calculation. Eq. (17) still holds if the inelastically scattered electrons are elastically scattered to produce Kikuchi patterns within the detector angular range. It is important to note that the STEM imaging intensity is a convolution of the probe intensity with the square of the inelastic generation function, which is proportional to the scattering cross section of the inelastic process. Then the phase of the generation function drops out. This will provide a great advantage for practical numerical calculations, because the phase of an inelastic transition matrix element is always unknown either theoretically or experimentally.
(18)
elastic
scattering
theory
for
ADF
Although it is convenient to discuss high-angle ADF imaging in terms of inelastic scattering and incoherent imaging, it should be noted that the imaging theory for TDS is based on a coherent elastic scattering formulation. Because the energy loss for TDS is unobservably small, the image produced by TDS can be described as the sum of a large number of instantaneous images produced by elastic scattering from various configurations of static displaced atoms. It is convenient that, in the average over time, some of the terms in the intensity expression average to zero. The same terms will, in any case, be small if the atom displacements at one time vary randomly along the columns of atoms parallel to the incident beam direction. It follows that the ADF image intensity will be much the same as for TDS if the atoms in a crystal have static displacements from their mean
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
position which are not correlated over large distances. This is the case, for example, when the atom displacements are generated by point defects, impurity atoms or short-range ordering of the atomic species on the lattice sites. For purely static displacements, the ADF imaging theory is then based on purely elastic, coherent imaging theory. For ideally perfect non-vibrating crystals, the correlation in atomic positions between slices concentrates the high-angle scattering into the sharp reflections in the HOLZ rings of the diffraction pattern and this gives the characteristic coherentdiffraction effects in the image [l]. Deviations from perfect periodicity of the atomic arrangement, and especially deviations involving atom displacement, give diffuse high-angle scattering. If the correlation range of the deviations from perfect periodicity is small in comparison with the crystal thickness, it is possible to formulate an “effective incoherent imaging scattering function” which may then be used, as in the case of TDS, to simplify the calculations of high-angle ADF image intensity. For very thin specimens, for which a phase-object approximation applies, it was shown by Cowley [7] that the effective incoherent imaging scattering function is [l - cos(a~(b))]
= a2$(b)/2,
where I#I(~) is the projection of the specimen potential distribution, and convolution of this function with the intensity distribution of the incident beam gives the ADF image intensity. For greater thickness, the incident wave function for each slice is modified by diffraction in the specimen and the correlation of atomic positions in successive slices introduces the three-dimensional coherent diffraction effects which limits the validity of the incoherent imaging approximation.
4. Computation method The detailed computation method for ADF imaging simulation has been given elsewhere [l]. In the following calculations in sections 5.1-5.4,
281
we employed the full dynamical theory described in eqs. (la) and (7) to simulate the contribution of TDS electrons to the ADF image for different vibrational phase couplings. The accuracy of using eqs. (13) and (17) for predicting image contrast will be examined in section 5.5. In this study, we use the Ge/Si (100) interface as an example to demonstrate the simulation of ADF images. The unit Ocell is constructed in a dimension of 20 A X 20 A and separated by 128 X 128 pixels in numerical calculations. The slice thickness is chosen as 5.43 A and all the computations are carried for a slab of thickness 108.6 A. The angular detection range of the ADF detector is limited to 37-111 mrad, which excludes the diffraction contrast effect arising from the FOLZ, located at 115 mrad. STEM images are formed by scanning a very small electron probe across the specimen. The probe is formed by an objective aperture of 0.5 A-’ in radius, the spherical aberration C, is taken as 0.8 mm, the lens defocus is taken as 800 A and V= 100 kV. The generated electron probe is shown in fig. 2. This coherent probe is small enough to resolve the 2 A lattice spacing between Si and Si or Si and Ge atoms. To reduce the interacting CPU time, one calculates only a line scan of the electron probe across the Ge/Si boundary through the center positions of the atomic rows. For simplicity of the calculation, the Ge and Si lattices are assumed to be matched coherently at the boundary and both of the materials have the same lattice constant. The beam
4 Fig. 2. STEM electron probe generated by an objective aperture of radius 0.5 A-’ (corresponding to 18.5 mrad in semiconvergence angle), C, = 0.8 mm and Af = 800 A for 100 kV electrons.
2. L. Wang, J. M. Cowley / Dynamic theory of high-angle ADF STEM
282
lattice images o Elastic
incident azimuth was chosen as [loo]. The atomic scattering factor is taken in the form of Mott formula, f(s)
= 0.023934[ Z - F,(s)]/s2,
Elastic
& TDS
?.I
SI
(20)
where s = u/2, Z is the atomic number and fx is the X-ray scattering factor for this atomic number. Then the scattered intensity from a single atom is proportional to Z2 and inversely proportional to s4 at very large scattering angle. This is the result for Rutherford scattering from a single atom and should be included in the calculations automatically. Also a proper Debye-Waller factor has to be added to the scattering factors of each atom according to the assumed mean displacements. This can significantly affect the relative contrast of each atom in the ADF images.
5. Calculation results In this section we present the simulated ADF images of the Ge/Si semiconductor interface using the elastic and inelastic scattering theories. Before presenting the results, it is important to point out that the arbitrary phases added to the TDS waves generated from different slices should be the same set for different scanning probe positions, otherwise some artifacts will be introduced in the calculation. 5. I. In-phase
n
vibration
Fig. 3 shows a simulated Ge/Si interface ADF image by assuming that all the atoms in all slices are vibrating in-phase (i.e. crystal is vibrating as a whole). The elastic scattering then comes from the time-averaged structure (with a Debey-Waller factor on the scattering amplitudes) and the “TDS” gives the time-average image for displaced crystals of non-vibrating atoms. The coefficients are the mean vibration amplitudes of Asi and A, the Si and Ge atoms. The large-angle elastic scattering and the TDS give about the same contribution to the ADF imaging contrast. The atomic positions, separated by 2 A, are indicated along the horizontal axis. The intensities are plotted on a
0
G.2
Ge
Ge
Ge
SI
Fig. 3. Simulated ADF imaging profile across the Ge/Si boundary according to eq. (1) by assuming that all the atoms within a same slice and those in different slices are vibrating in-phase. The thermal vibration amplitudes of the Ge and Si atoms are taken as 0.1 A.
normalized scale along the vertical axis so that the numbers for different simulations can be compared with each other. The interface is seen to be sharp and the Ge and Si atoms at the boundary show very distinct contrast. The relative intensity of Ioe/lsi is about 2.2 for both the TDS electrons and the elastically scattered electrons. The summation of the two also gives the contrast of about 2.2. In disagreement with the extreme classical model of single-atom Rutherford scattering, the contrast variation is 2.2 for this particular ADF collecting angular range rather than Z,‘/Z,” = 5.2, which would follow from Z,/Z, = 2.28. This difference may come from the different origins of TDS and Rutherford scattering. Hence, it should be more proper to claim that ADF image contrast is the contrast coming from TDS rather than large-angle Rutherford scattering. The result in fig. 3 is obtained by assuming that both the Ge and Si atoms are vibrating with the same ampli-
.
0
Ge
Ge
Elastic
& TDS
-::::::I
Go
Ge
SI
SI
SI
Fig. 4. Same as fig. 3 except that the thermal vibration tudes of the Ge and Si atoms are taken as 0.1 and respectively.
ampli0.2 A,
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
tudes. The situation may change if the two atoms have different thermal vibration amplitudes. Shown in fig. 4 is the same calculation as fig. 3 except that the thermal vibration amplitude of the Si atoms is doubled. It is obvious that Si then shows stronger intensity and the final imaging contrast is about ~oce/lsi = 4/3. This shows that the TDS may increase the visibility of light atoms and the ADF image contrast cannot be simply predicted using the 2’ rule [l]. 5.2. Contrast due to random atomic vibrations In practical phonon excitations, the atoms in the crystal will not vibrate in the same phase. The random vibration of the atoms will statistically introduce a diffuse scattering background in electron diffraction patterns. Fig. 5 shows a comparison of the calculated ADF image contrast across the Ge/Si interface with (curve B) and without (curve A) considering the randomness of the atomic vibration in different slices. Note curve B was calculated by adding a random phase to the TDS waves generated from different slices but not to the atoms within a same slice. The first change we see is the relative magnitude of the ADF imaging intensity between the curve B and A (bottom blank circles). The random vibration of the atoms in different slices introduces a diffuse
I
1
0
.opA
0
Ge
Ge
Ge
Ge
S,
SI
SI
Fig. 5. Comparison of the simulated ADF imaging profile arising from pure TDS across the Ge/Si boundary with (curve B) and without (curve A) adding random phases to the TDS waves generated from different slices. Note curve A is magnified by 18.66 times to compare with curve B. The thermal vibration amplitudes of the Ge and Si atoms are taken as 0.1 A.
Ge
Ge
Ge
Ga
Si
283
SI
SI
Fig. 6. Comparison of the simulated ADF imaging profile arising from pure TDS across the Ge/Si boundary with (curve B) and without (curve A) adding a random phase to the TDS waves generated from different slices. Note curve A is magnified 18.4 times to compare with curve B. The thermal vibration amplitudes of the Ge and Si atoms are 0.1 and 0.2 A, respectively.
scattered background in the diffraction pattern particularly at the collecting angular range of the ADF detector, then the ADF imaging intensity from TDS is increased about 19 times compared to the calculations for in-phase vibrations. Comparing the vertical scale of the curve B with that of fig. 3, it can be positively concluded that the TDS dominates the image formation in ADF STEM rather than elastic scattering if the thermal vibration amplitudes of the Ge and Si atoms are close to 0.1 A. In order to see the relative contrast variation after introducing the vibration randomness, we magnified the curve A for 18.66 times to match the intensity of curve B at the Ge sites. The relative visibility of the Ge atoms with respect to the Si has been changed slightly. Also the peak intensity at the Ge sites is increased compared to the valley part between the atoms in curve B. This contrast deviation becomes more significant with the increase of the vibration amplitude of the Si atoms. Shown in fig. 6 is another comparison similar to that of fig. 5 except that the thermal vibration amplitude of the Si atoms is doubled. The in-phase calculation gives stronger intensity at the Si sites but the random phase calculation gives the opposite result.
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
284
5.3. Dependence of ADF imaging intensity and contrast on the correlation of vibrations of neighboring atoms in the same crystal slice ADF imaging contrast of a row of atoms may not depend on correlations with the thermal vibrations of other rows of atoms if the specimen is thin. In the STEM case, this statement may be true if the probe size is less than the spacing between the atoms. Fig. 7 shows a comparison of the calculated ADF imaging contrast of the Ge/Si interface for the following cases: (A) the atoms within the same slice are vibrating in-phase but with random phases for different slices; (B) the atoms are vibrating with random phases within the same slice and also randomly for different slices; (C) neighboring atoms within the same slice are vibrating with a phase difference of 2n/5 in the x direction but in-phase in the y direction, the atomic vibration is treated randomly for different slices; and (D) neighboring atoms within the same slice are vibrating with a phase difference of 277/5 in the y direction but in-phase in the x direction, the atomic vibration is treated randomly for different slices. Note in these calculations the random phases added to the TDS waves generated from different slices are the same set to avoid statistical fluctuations. It is seen that the ADF imaging intensity profiles not only show exactly the same intensity, but also the relative contrast of
Ge
Ge
Ge
Ga
S,
SI
SI
Fig. I. Dependence of ADF imaging intensity and contrast arising from pure TDS on the vibration status of the atoms in the same slice. This figure compares the simulated ADF imaging profile arising from TDS across the Ge/Si boundary for different vibration phase couplings of the atoms within a slice (see text for details). The thermal vibration amplitudes of the Ge and Si atoms are taken as 0.1 A.
Ge
Ge
Ge
Ge
SI
SI
SI
Fig. 8. Dependence of ADF imaging contrast arising from TDS on the vibration status of the atoms in different slices. This figure compares the simulated ADF imaging profile arising from TDS across the Ge/Si boundary for different vibration phase couplings of the atoms in different slices (see text for details). The thermal vibration amplitudes of the Ge and Si atoms are 0.1 and 0.2 A, respectively.
the Ge and Si atoms is almost identical. This shows that the ADF imaging contrast is independent of the vibration status of its neighbor atoms within the same slice. In other words, whether the thermal vibrations of the atoms are assumed to obey the Einstein model, Debye model or any other model will not affect the ADF imaging contrast if the probe size is small. These calculation results are in agreement with Cowley’s conclusion [9]. 5.4. Dependence of ADF imaging contrast on the correlation of vibrations for neighboring atoms in different crystal slices It was shown in the last section that the ADF imaging contrast is independent of the thermal correlation of vibrations of the neighbor atoms within the same slice. Then the imaging simulation can be simply done by assuming that all the atoms within the same slice are vibrating in-phase. It is necessary to find out the dependence of the imaging contrast on the atomic vibrations in different crystal slices. Shown in fig. 8 is a comparison of the calculated ADF imaging intensity profiles across the Ge/Si boundary for the following three cases: (A) random phases are added to each slice; (B) designated phase sequence of lr/2, 0, r/2, 0, . . . , are added to successive slice; and (C) designated phase
Z.L. Wang, J.M. Cowley / Dynamic theory
sequence of r/2, -r/2, n/2, -r/2, . . . , are added to successive slices. The thermal vibration amplitude of the Si atoms is doubled compared with that of fig. 7. Some differences in the relative magnitudes of the ADF images are seen, but the contrast profiles for these cases follow almost exactly the same curve after being normalized at the Ge sites (curve B and C are magnified for 2.03 and 1.093 times respectively). This shows that the vibration status of the atoms in difference slices may introduce some difference in the intensity magnitudes but very little change in the relative contrast of the Ge and Si atoms which is the quantity usually observed in experiments. It thus can be concluded that the simulation of the ADF image contrast arising from TDS will not be affected by the relative vibrations of the neighboring atoms, but the absolute intensity of the image varies somewhat with the coupling of vibrations of the atoms in different layers. Then for absolute quantitative (i.e. absolute intensity distribution rather than relative contrast) analysis of the ADF image it is necessary to consider the vibrational coupling among the atoms. Fairly reasonable contrast information can be obtained- by assuming arbitrary vibrations. This may provide a general guide for ADF imaging simulations.
of high-angle ADF STEM lattice images
285
dynamic scattering theory of eqs. (la) and (7), which should apply to crystals of any thickness if only single-phonon processes are considered. The simplified methods, described in eqs. (13) and (17) can greatly reduce the amount of calculation, but the accuracy of this treatment has to be examined. Shown in fig. 9a is a comparison of the calculated ADF image across the Ge/Si boundary using the theory of full dynamic incoherent scattering (described in the last paragraph) (curve A), eq. (17) (curve B) and eq. (13) (curve C). Different absolute intensities are obtained for the different cases, but each of the curves shows almost the same variation tendency. To see the relative contrast variation for the Ge and the Si atoms, the three curves are normalized at the Ge site and compared in fig. 9b. It is surprising to note that the three curves show almost the same contrast variations. The contrast difference between them is less than 5%. This gives the answer that the simulation based on either eq. (13) or eq. aTo,
5.5. Validity of the simplified incoherent theory (eqs. (13) and (I 7)) for ADF image simulations 04 ::::
The quantitative simulation of ADF images involves the assumption of incoherence of the TDS (due to thermal average) waves generated from different crystal slices if all the atoms are vibrating completely randomly. In this case we do not have to consider the phase correlation among the atoms within the same slice, as shown in section 5.3. To process this calculation accurately, one has to calculate the TDS wave generated from different slices and let them be scattered elastically through the rest of the crystal thickness individually, then add the intensities of each individual TDS components generated from different slices after penetrating through the crystal. This calculation process considers the incoherency of the inelastic scattering precisely but involves huge amounts of calculations. This is the exact result of
: ::::
: ::::::::::::::
_-
:
:::-I
Ge Ge si I::::::::::::::::::::::::::::t Si
Fig. 9. (a) Comparison of the simulated ADF image intensity across the Ge/Si interface using the full dynamical incoherent scattering theory (i.e. consider the elastic scattering of the electrons after being inelastic scattered) (curve A), simplified incoherent theory eq. (17) (curve B) and eq. (13) (curve ,C). (b) Comparison of the three curves shown in (a) after being normalized at the Ge sites. This is to test the accuracy of simulated imaging contrast using the simplified theory.
286
2.L.
Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM
(17) can give almost the accurate contrast distribution of the ADF image for a crystal slab of about 100 A in thickness. This important result shows that the incoherent imaging theory can be used to simulate the ADF imaging contrast quantitatively if the image intensity is dominated by TDS. It may be noted that the Si atom closest to the Ge boundary has higher contrast than the next Si atom. This may come from the contribution of the tail of the probe scattering from the Ge or from a spreading of the probe in the crystal due to the discontinuity at the interface. Then it is possible that an error of one monolayer may be made in the chemical identifications. Eq. (13) was derived by assuming that all the inelastically scattered electrons are collected by the detector. The image simulation based on this assumption still gives the correct contrast if the inner radius of the ADF detector is large enough to avoid the diffraction contrast and its outer radius is large enough to collect most of the largeangle inelastic scattering. This correctness indicates that the portion of electrons scattered to the detector depends only on the localized scattering power of the individual atoms if the sample is thin. It is important to emphasize that the incoherent inelastic scattering can give atomic resolution lattice images. The mechanism is obviously different from the coherent TEM or STEM imaging theory. The incoherent images can be simulated easily because there is little dependence on focus and thickness. It must be pointed out that the above calculations in section 5 were carried out by using a small probe of about 1.2 A in half width. Also the thermal vibration amplitudes of the atoms were assumed to be around 0.1 A. The conclusions are drawn on the basis of these pre-defined parameters. If the thermal vibration amplitude of an atom is reduced to about 0.01 A, then the contribution of the TDS scattering to the ADF image is reduced by about 40-100 times. In this case, the contribution from pure elastic scattering will play an important role (see the result in fig. 3). In this case the calculation based purely on eqs. (13) and (17) may not give reasonable results. This situation may not occur because the mean vibration
lattice images
amplitude of a Si atom is approximately 0.045 A even at 0 K. It is expected that the incoherent imaging theory can accurately predict the contrast of ADF images obtained at high temperatures.
6. Programming procedures for simulating ADF images based on a multislice program for HREM As a summary of the above calculations, we give simple programming procedures for simulating ADF imaging contrast in the multislice algorithm for thin specimens, which can be easily modified and is based on the standard high-resolution electron microscopy image-simulation program. 1. Follow the routine procedure for generating projected atomic potentials for a slice of thickness AZ:
(21) A proper Debye-Waller factor has to be added to each atomic scattering factor according to the assumed mean vibration amplitudes of each atom. 2. Calculate the absorption function of the slice:
p(b, 2) = cJ2
+C
Ajy&U;(b-ni)J’]/Az.
(22)
i 3. Turn u’ to reciprocal transformation:
space
using
Fourier
U;(U) =.%=(u,(<)). 4. Take off the Fourier components falling outside the detector angular range of ui < u < u2: U;‘(U) = 5. Turn u;‘(b)
U,‘(U)
for u,
0
otherwise.
(23)
u” back to real space:
=.P-‘(u;‘(U)).
(24)
Z.L. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
6. Calculate IG,,,(b,
the TDS generation
287
slice:
function:
q(b) = exp[iaV’(b)],
z) I *
@a)
with V’(b)=
Xexp[ -p(b)
AZ].
(25)
It is possible to add the contribution of other inelastic scattering processes, such as atomic core shell losses, to the ADF image at this stage. 7. Calculate the transmission function of the
xo’(b-Rj).
(26’4
Multislice calculation of the imaging intensity for probe position bp. The shape of the incident probe at depth z, is determined by
h,(b-bp, zn) G)AZPI x+n-,(b - bp,z,,)} * f’(b).
= { q(b,
z,) exp[ -p(b,
(27)
A CYCLE Of=PROBE SCANNING IN STEM
ELASTIC ADF
-
I Elas
GENERATED TDS A1TDS
MULTISLICE INTERACTION CYCLE
TDS ADF ‘TDS
1 STOP Fig. 10. Flow-chart of an ADF imaging simulation program, modified based on the HREM imaging simulation program initiated by Isbizuka [lo], according to the theory given by eq. (17) or eq. (13). The procedures introduced in section 6 are contained in this program (please contact Z.L. Wang if any one is interested in this program).
Z.L.. Wang, J.M. Cowley / Dynamic theory of high-angle ADF STEM lattice images
288
9. TDS contribution to the image intensity from the nth slice: &&rJn =
J
db ]G,,,,(b,
= ]G,,,(b,,
zn) I2 I%n(b-bp,
zn) 12*
I$&,
zn> I2
zn>12.
(28)
10. Add all the generated TDS inter& .y from all the slices (total N) for probe position bp: (29) 11. Calculate elastic contribution to an ADF image after penetrating the crystral (the Nth slice):
where D is the defined detector
function in eq.
(IO). 12. Add the total contribution inelastic scattering:
of elastic and
I($) =bs(bp) +Zm&p).
(31)
13. Repeat procedures (8) to (12) for another scanning position. Fig. 10 shows a flow-chart of the ADF image simulation program according to eq. (17) (or eq. (13)). This program was modified based on the HREM imaging simulation program initiated by Ishizuka [lo]. The flow-chart given in our first paper [l] deals with the ADF image simulations following eqs. (la) and (7), considering the vibrational coupling among the atoms. The calculation following fig. IO should give an easy and quick method for image interpretation.
7. Conclusions Dynamical calculations have shown that the contrast of annular-dark-field (ADF) scanning transmission electron microscopy (STEM) images is dominated by thermal diffuse scattering (TDS). The simulation of the ADF image can simply follow the incoherent imaging theory, that is, the
image intensity is a thickness integration of the convolution of the electron probe intensity distribution at depth z with the inelastic generation I G ( 2. This can greatly reduce the function numerical computations compared to the exact calculations following the full dynamical elastic and inelastic scattering theory (i.e. considering the elastic scattering of the new generated inelastic wave from each slice). This ADF imaging theory is the same as the incoherent imaging theory in optics. Similar results can also be derived from coherent elastic scattering theory, as described in section 3.3. Then it will not make any difference whether the ADF imaging is described as coherent or incoherent. It is the result of a combination of dynamical elastic and inelastic scattering. Its contrast cannot be predicted using any simple model like single-atom Rutherford scattering. The calculation according to this simplified theory can give almost the exact contrast as obtained from the full dynamical inelastic incoherent calculations but not the absolute intensity. The former is the most wanted quantity in practical analysis. The relative contrast error is estimated to be less than 5% and the CPU time can be cut down by N/2 times compared to the accurate full dynamic calculation, where N is the total number of slices. It is obvious that the simulation for thick crystals, i.e. large N, becomes very time consuming if using the full theory. Then the introduction of eqs. (13) and (17) makes the ADF imaging simulation feasible. A routine multislice image-simulation program for high-resolution electron microscopy (HREM) can be easily modified for simulating ADF images following the procedures given in section 6. As claimed in section 5.4, eq. (17) is almost exact if the specimen is thin. This thin specimen can be up to 500 A or thicker and satisfactory accuracy of imaging contrast can still be obtained. This can generally meet the specimen thicknesses in most high-resolution lattice images. The incoherent TDS can produce atomic resolution lattice images if a small electron probe, smaller than the unit cell, is provided. This image is produced by the localized elastic and inelastic scattering from the atomic sites. Channeling effect can enhance the ADF image contrast but is not necessary to form the Z-contrast image. The ADF
Z.L. Wang, J.M. Cowley / Dynamic theov
image contrast is found to be Z-dependent and is capable of providing localized specimen chemical information at sharp atomic interfaces, but a monolayer uncertainty may be introduced due to the scattering of the probe tail or the spreading of the probe. Dynamical calculations have shown that the ADF imaging contrast of an atom is almost independent of the vibration status of its neighbors, so that the image simulation will not be affected by assuming either Einstein model, Debye model or any other models. This can greatly simplify the theoretical analysis of ADF images and expands its applications in material science. Quantitative simulation can be obtained if the mean vibration amplitude of each atom in the specified specimen is known. It must be emphasized that the domination of TDS process to ADF imaging contrast reduces if the thermal mean displacements of the atoms decreases. It is expected that the elastic contribution will become important if the atomic thermal vibration amplitudes are less than about 0.03 A, i.e. for only a few substances at temperature close to 0 K.
of high-angle ADF STEM lattice images
289
Acknowledgements
The authors thank Dr. J.C.H. Spence for this encouragement, discussions and detailed comments on the manuscript. Thanks also go to Drs. P. Rez and P. Perkes for maintenance of and assistance with the computer system. This work was supported by NSF grant DMR 88-10238, and made use of the ASU Facility for HREM, supported by NSF grand DMR 8611609. References [l] Z.L. Wang and J.M. Cowley, Ultramicroscopy 31 (1989) 431. [Z] Z.L. Wang, Acta Cryst. A 45 (1989) 636. [3] S.J. Pennycook, EMSA Bull. 19 (1989) 67. [4] See, for example, J.C.H. Spence, in: Experimental High Resolution Electron Microscopy, 2nd ed. (Oxford Press, Oxford, 1989). [5] S.J. Pennycook, D.E. Jesson and M.F. Chisholm, Inst. Phys. Conf. Ser. 100, section 1 (1989) 51. [6] A. Howie, private communication. [7] J.M. Cowley, Ultramicroscopy 2 (1976) 3. [8] R.H. Ritchie, Phil. Mag. A 44 (1981) 931. [9] J.M. Cowley, Acta Cryst. A 44 (1988) 847. [lo] K. Ishizuka and N. Uyeda, Acta Cryst. A 33 (1977) 740.