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Localization effects on quantification in axial and planar ALCHEMI
Ultramicroscopy 26 (1988) 103-112 North-Holland, Amsterdam
103
LOCALIZATION EFFECTS ON QUANTIFICATION IN AXIAL AND PLANAR ALCHEM! J.C.H. SPENCE, M. KUWABARA and Y. KIM Department of Physics. Arizona State Unit'ersit); Tempe, Arizona 85287-1504. USA Received 11 April 1988, presented at Workshop Januat'y 1988
The effects of wave-function dimensionality and inelastic localization on ALCHEMI (Atom Location oy tYhannetmg-r_nhanced Microanalysis) are studied. The original ALCHEM! equations are shown to hold for the case of partially delocalized excitations, provided that the thickness-averaged dynamical electron wavefield varies slowly over the localization volume. Experimental and theoretical comparisons of electron channeling in GaAs show that characteristic X-ray emission intensities in the axial orientation are more sensitive to variations in X-ray energy than are those in the planar Isystematics) geometD'. The effects of variations in localization are elucidated in a two-beam analysis, and methods for extending ALCHEMI ~o low-energy X-ray emission are discussed.
1. Introduction Over the past five years more than forty papers have been published studying the dependence of characteristic X-ray emission on electron diffraction conditions in thin crystals. Many of these have aimed to determine the fractions of a substitutional impurity species which lies on particular crystallographic sites. For a review of work until early 1986, see ref. [1]. All this work has its roots in studies of electron channeling effects on X-ray microanalysis in monatomic metals [2] and in earlier work on the Borrmann effect in X-ray diffraction. (For a historical review and an outline of the method, see refs. [3,6].) However, it was the realization [3-5] that, in polyatomic crystals, the emission from host atoms on known sites could be used to provide a reference signal which provided the basis for the ratio method known as ALCHEMI [3]. By comparing the orientation dependence of X-ray emission from substitutional irapurities with that of certain reference atoms on known planes (or atomic columns), it becomes possible to make quantitative estimates of the amounts of the impurity on each of several candidate sites. Since the site occupancies are given in terms of measured X-ray counts alone, the method appears to contain no adjustable
parameters. The choice of orientations is not critical, the specimen thickness need not be known, and no calculations of the dynan'ficai electron intensity are required. In particular, the ALCHEMI equations (see eqs. (13)-(15) below) do not require that the electron beam intensity remains the same for different spectra. For tutorial reviews of the method, see refs. [3,6]. In the last two years the method has been applied to the new superconductors [7], to Fe and Mn in samarium--cobalt magnets [8], to various elements in Ti/A1 alloys [9-11], to St, Zr and U in perovskite [12], to various cations in garnet [13], and to AI and Si in feldspars [14]. Additional studies have investigated the effect of the helical ray paths in immersion lenses on ALCHEMI [15], with the surprising finding that the smallest probe gives the least "'divergence". There have also been attempts to use the channeling effect to determine the displacement of atoms from their exact substitutional sites [16]. To summarize all the published work of recent years, it appears that if a strong channeling effect can be found for ,;ome set of planes ~and this is highly material and thickness dependent), then site occupancies can usually be determined for those planes. Several authors have pointed out that the original ratio argument, which was applied to crystal
0304-3991/88/$03.50:2; Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
104
J.C.H. Spence et al. / Localization effects on quantification in axial and planar A L C H E M I
planes in the systematics or p l a n a r channeling geometry, can also be applied to atomic columns in the axial orientation. Attempts to use "Axial A L C H E M I " to obtain quantitative site occupancies have, however, met with mixed success [10,11,13,17] leading in some cases to quite unphysical results, such as negative occupancies [18], but to consistent and useful results in others [12]. It has been suggested that these effects are due to wriations in the inelastic localization [12,17]. (This refers to the width of the inelastic scattering potential or impact parameter and, for processes other than p h o n o n scattering, is inversely related to both the mean angle of inelastic scattering associated with the ionization event and to the emitted x-ray energy.) A detailed analysis of axial A L C H E M I has been given, based on the Blochwave dynamical theory of inelastic scattering in crystals [19]. There have also been several applications of similar principles to channeling effects observed on inner shell excitation edges in energy loss spectroscopy [20,21]. The purpose of this paper is to compare the sensitivities of the axial and planar orientations to variations in inelastic localization, and to suggest ways of either measuring or eliminating the dependence of A L C H E M I on variations in localization.
2. Quantitative A L C H E M I with partial localization In simplest form the A L C H E M I problem consists of a determination of the fraction C of an impurity a t o m X which lies on one of two possible candidate substitutional sites in a crystal. Let a fraction C lie on the B site; the remaining fraction ( 1 - C) then lies on the A site. The ordering of minor constituents over the M1 and M2 cation ~ ; t ~ c i n r ~ l l x z ; ~ ~rraxriA~c
cTrv~rt
same orientation dependence. This follows because the two a t o m s lie on "equivalent" sites on the two-dimensional structure (projected in the b e a m direction), a n d the total thickness-averaged dynamical electron-beam intensity in the crystal must respect the two-dimensional translational symmetry of the crystal. It is clear that the method cannot distinguish sites which differ in their coordinate in the b e a m direction. Chemical shifts may sometim~:s by used to distinguish the same atom when it occurs on different sites [23]. The preceding statements, however, make several assumptions, the most i m p o r t a n t of which are: (1) the p e r t u r b a t i o n of the dynamical wavefield by the i m p u r i t y itself is small. This has been confirmed by detailed numerical calculations for the planar case. (2) X-ray absorption effects for the various species do not have different dependencies on orientation. (3) all species have negligible inelastic delocalization. That is, the impact parameter for the ionizing collision is less than the thermal vibration amplitude. We consider this last approximation in this paper. In practice, the A L C H E M I m e t h o d has been applied to both the axial and the p l a n a r channeling geometries. This reduces the three-dimensional site determination p r o b l e m to two and one dimension respectively. ( F o r a study of the very weak dependence of c h a n n e l i n g effects on the excitation of higher-order Laue-zone reflections, see ref. [24].) In the axial case it is necessary to find an orientation which separates the possible substitutional sites for the i m p u r i t y onto different atomic columns in the b e a m direction, each containing a distinct refcrence atoms. The orientation dependence of the i m p u r i t y is then compared with that of each of the reference atom. A similar procedure is followed for the planar case, choosing planes
mTw~l~ r o l l
The method relies on the fact that the characteristic X-ray production from a thin c~'stal traversed by a kilovolt electron beam is proportional to the dynamical electron-beam intensity averaged over thickness at the site of the emitting atom [2]. If the concentration of X is small and the impurity is uniformly distributed throughout the crystal, then, for C = 1, the characteristic X-ray emission from species X and B must have the
which contain the likely substitutional sites (separated onto different planes) and also suitable reference atoms whose sites are known. By repeating the procedure for different sets of planes, a crystal site may be identified. Using the quantitative method outlined below, it is also possible, in principle, to d e t e r m i n e the fractional occupancy C in both the axial a n d planar case if the excitations are perfectly localized. For incompletely localized
J. C.H. Spence et al. / Localization effects on quantification in axial and p/anar A L C H E M I
excitations no quantitative method of fractional occupancy determination has been given; however, a method has been described for determining the site of a partially localized species which lies entirely on one site (C = 1) [12]. There are three cases in the literature in which quantitative measurements of fractional occupancy C made by A L C H E M I have been compared with measurements made by other methods [14,25,26]. All are for the planar case, and all find good agreement between the two methods within experimental error. The axial case has the important advantage of providing a stronger channeling effect; however, attempts to obtain quantitative results by this method have met with mixed success [11,13,27]. This has been ascribed to localization effects [19,27], and it has been suggested that a correction factor may be applied for lowenergy X-ray. We now wish to compare the importance of variations in inelastic localization in the axial and planar geometries. In order to do so, we re-derive the original ALCHEMI equations, including this effect. This may be done using the Bloch wave dynamical theory of inelastic scattering (for a recent treatment for inner shell processes, see ref. [28]). For our purposes we can assume that the dynamical elastic wavefunction g'(r) is known, and use the result [29] that the rate of ionization in a thin crystal traversed by a kilovolt electron beam is proportional to
z=flq,(r)l 2 V(r)
dr.
(1)
Here V(r) is the portion of the imaginary part of the optical potential which describes inner shell ionization, resulting in characteristic X-ray emission. This imaginary potential is used in the SchrSdinger equation to describe the depletion of the elastic wavefield by inelastic scattering processes (see, for example, ref. [30]). We assume that the intensity of emitted characteristic X-rays is proportional to the ionization rate. For small-angle inelastic scattering, the maximum component of momentum transfer in the beam direction is q,(max) = 0.11 q(max) I. We therefore
105
assume (from the uncertainty principle) complete delocalization in the beam direction if the loss is localized laterally. We define
l,(r)= f l4,(r)l 2 d : to be the thickness-averaged intensity in the neighborhood of atom A. The number of X-rays emitted per unit time from atomic species A is then U~=
K,f,,(,.)
VA(r ) dr,
(2)
where K A is a constant which takes account of differences in fluorescent yield and in the differing concentration of the atoms on the various planes (in the planar case) or atomic columns (in the axial case). We will write eq. (2) for compactness as N A = K A I A V A.
(3)
Detailed dynamical calculations for the form of the inelastic interaction potential have recently been made for inner shell excitations [19]. These depend on the cross-section for atomic ionization, and, if averaged over all directions of core electron ejection, show a peaked function which we approximate by a Gaussian in this paper. All previous planar A L C H E M I applications have assumed perfectly localized excitation, which corresponds to a delta function form for VA(r). Then
NA = KA/A"
(4)
From V(r) a mean-square impact parameter may be deduced, which is a measure of the inelastic localization. Here we define the localization L as the full width at half maximum height of a Gaussian V(r). While there is considerable disagreement in the literature as to the magnitude of L [19,27,31,32] it is clear from simple arguments based on the uncertainty principle that for inner shell ionization L varies inversely with the mean energs: loss responsible for ionization. This is about twice the emitted X-ray energy ~'l~,-'~-j-n [ ~'~1 It. ~heref,:,re varies by a factor of ten between the K and L fines used in our experiments with GaAs. (The energies are given in fig. 3.) L is believed to be comparable
J.C.H. Spence et al. / Localization effects on quantification in axial and planar A L C H E M I
106
with the m e a n t h e r m a l vibration a m p l i t u d e ( U 2) for low-energy ( 1 - 2 kV) X-rays. F o r G a A s at 300 K,
GKA IoVA =
KAIofVA(r)d r ,
If a third species X is introduced w i t h conc e n t r a t i o n C on the B (As) planes a n d c o n c e n t r a t i o n (1 - C ) o n t h e A ( G a ) p l a n e s t h e n w e h a v e N x = K x C l a V x + K x ( 1 -- C ) I A V x a n d for the n o n - c h a n n e l i n g orientation N ° = GKxC IoV x + GKx(1 - C) IoV x = G K x loVx =
[IA(r)+IB(r)]
dr=l,
(9)
T h e p r o b l e m n o w is to d e t e r m i n e C f r o m a k n o w l e d g e of the o b s e r v a b l e X - r a y counts alone. Add i t i o n a l i n f o r m a t i o n is available if m o r e t h a n one e m i s s i o n line is o b t a i n a b l e from each species. N o t i c e that it is n o t a s s u m e d that t l ~" b e a m curr e n t is the same for t h e c h a n n e l i n g a n d n o n - c h a n n e l i n g spectra. U s i n g (8) to p r o v i d e an expression for C, rep l a c i n g K x using (9), m u l t i p l y i n g I A V x b y N x Io~"x N°
N° Nx
IoVx
=1,
dividing by N x l - ~ / N
C
~
10Vx
IoVx
° and r e a r r a n g i n g gives = IAVx( N° Io g x Nx
1).
(10) N o w if all the e m i s s i o n lines are perfectly localized (high energy X - r a y s ) Uv×/
f~ I o ( r ) d r = fc
GKxlofVx(r)d r .
(5)
since I 0 is a s s u m e d constant w i t h i n the unit cell. The n o r m a l i z a t i o n is
(8)
oV× = N A / N . ° .
(11)
with VB the inelastic localization for the As atom emission line. T h e width of this function will differ for the K a n d L shells by a factor of about ten. The emission from the As a t o m s in the nonchanneling orientation is
Physically, this r e q u i r e s that the s t r e n g t h of the c h a n n e l i n g effect r A = NA/N°a be i n d e p e n d e n t of X - r a y energy. H e r e r~, is the ratio of the X-ray emission intensities for the c h a n n e l i n g a n d nonc h a n n e l i n g o r i e n t a t i o n s in an idealized e x p e r i m e n t with constant b e a m c u r r e n t for a single emission line. We use the ratio R ( A / X ) = r~,/r x to indicate the extent to which t w o species A a n d X s h o w the s a m e orientation d e p e n d e n c e , R = 1 i n d i c a t i n g either that there is n o c h a n n e l i n g effect, or that b o t h species show effects of equal strength, and are therefore on the s a m e site. For localized excitation of the B site, similarly.
N ~ = G K B I o V s.
i
ell
ell
where the factor G allows for c h a n g e s in b e a m current b e t w e e n spectra. The emission from the As a t o m s in the channeling orientation is N B = K B lBVa,
(6)
(7)
v×/1ov
=
(12)
J. C H. Spe,lce et aL / Localization effects o,1 quantification i,1 axial and planar ,4 L C H E M I
Using (11) and (12) in (10) we obtain R(A/X)(1 - fl)C= R(A/X)-
1.
(13)
This is the original ALCHEMI expression, which allows C to be determined from measured X-ray counts alone. Here
R(A/X)=(NA/Nx)/(N°A/N~)=rA/rx
(14)
and
fl=NBN°/NA N°.
(15)
It will be noted that the effects of any changes in beam current between the channeling and nonchanneling spectra are eliminated when the ratio rA/r x is taken.
3. Approximations in ALCHEMI Eqs. (11) and (12) express the key localization approximations of ALCHFMI (eq. (13)). We note that either of the following conditions will ensure that eqs. (11) and (12) are satisfied: (1) if the excitations are highly localized, or (2) if the intensity IA(r) varies slowly over the inelastic localization volume. In the second case IA(r) may be taken outside the integral in eq. (2) and, using (3) and (5), eq. (11) follows. The question then arises as to how rapidly this intensity varies over the localization volume for each of the planar and axial geometries. We now, therefore, compare the failure conditions of this approximation for the two geometries. This may be done by comparing the strength of the channeling effect for, say, the Ga K line r(Ga, K, planar) with that of the L line r(Ga, L, planar). If this is not independent of X-ray energy, we will have r(Ga, L, planar}/r(Ga, K, planar)
107
for p = 1 can the ALCHEMI method (eq. (13)) be expected to work.. It is important, however, to ensure that a strong channeling effect is actually observed in these experiments, and it is desirable to do this in such a way that changes in incident beam current are allowed for. Thus, either the absence of any channeling effect (N L = Nt°) or a strong X-ray energyindependent channeling effect would give p = 1 in (16) above. To distinguish these possibilities, an independent monitor of the strength of the channeling effect is required from the same experimental data. This can be done by comparing the emission from different species on different planes, using the same data. Thus we make the above measurements of p under experimental conditions such that the Ga-to-As intensity ratio changes appreciably between orientations. We then require:
R-r =
N(Ga, K, c h ) / N ( A s , K, ch) N(Ga, K, n c ) / N ( A s , K. nc)
N~,. N °a NB N~~
t~,x rB
(17)
to be as large as possible. This demonstrates a redistribution of normalized intensity over the unit cell. Since the K lines are highly localized and the Ga and As atoms lie on different planes, R T provides an independent monitor of the strengt': of the channeling effect which is independent of changes in the beam current. For the particular case of [100] axial channeling in GaAs, however, this test fails, since the small difference in atomic number leads to a rather small difference between the average electron intensity on the Ga and As columns. However, our calculations (see fig. 2) show a large difference between the normalized average intensities for the two orientations on one site.
= p not equal to 1
= ( NL/N ° ) / ( NK/N° ) = R ( L / K . Ga)p,,m:..
(16)
4. Two-beam results including delocalization and absorption for two species
The experimental results given in section 6 report values of p for both the planar and axial [100] cases for all the Ga and As L and K lines. Only
Variations in localization can be taken into account in ALCHEMI analysis if the effects of
d.C.H. Spence et al. / Localization effects on quantification in axial and planar A L C H E M I
108
localization c a n be measured against some convenient experimental parameter. Since the localization L is given approximately [33] t~y
L = h/~ge,
(18)
where @ E = ( A E / E o ) ( 1 + "t -~) is the (relativistic) characteristic inelastic scattering angle for energy loss A E, variations in the accelerating voltage E 0 might be considered, useful. However, it has been shown that with small modifications to m o d e m instruments, X-ray emission spectra can be obtained as a continuous function of incident b e a m angle, using the double-rocking apparatus of Christenson a n d Eades [15]. We n o w show that it is possible to obtain i n f o r m a t i o n on the localization function f r o m measurements of X-ray emission as a f u n c t i o n of orientation a r o u n d several Bragg positions. In the i n d e p e n d e n t Bloch-wave a p p r o x i m a t i o n the thickness-independent electron intensity at r inside a crystal is given by 2
I(r)= Y'lcdl21y'C~exp(2vrig.r)l j
(19)
g
Evidence for the failure of this a p p r o x i m a t i o n in very thin crystals is given in ref. [2]. It agrees with the exact result only for an infinitely thick crystal without absorption. However, it has been found to give good qualitative agreement with experiment for experiments such as these for GaAs. (The neglect of the unavoidably strong (400) reflection is a more severe a p p r o x i m a t i o n in the following than the i n d e p e n d e n t Bloch-wave approximation.) In the t w o - b e a m a p p r o x i m a t i o n for a non-centrosymmetric crystal with absorption and a localization function V(r) we then have, from (1) and (19), an ionization rate p r o p o r t i o n a l to
[ ,,.. : ='-
x fv(r)
,
,,f w +"t 1
cos(2wg, r) dr.
details, and plots of z(s~) for delta f u n c t i o n V(r) can be found in ref. [1]. In the absence of absorption, A(w, g ) = 0 a n d B = t. The first t e r m gives the emission f r o m a non-crystalline sample. The second term expresses the o r i e n t a t i o n - d e p e n d e n t c o n t r i b u t i o n due to a n o m a l o u s a b s o r p t i o n . Here (21)
A ( w , g ) = ~ ' t 2 / ( 1 + wZ)/d~,
where /dgP is the a n o m a l o u s absorption. T h e third t e r m describes the o r i e n t a t i o n - i n d e p e n d e n t average absorption effect (22)
B = t - ~rt2//do
where /d0 is the m e a n absorption distance [30]. In deriving eq. (20) we have assumed t h a t ~5~> ~50 a n d that t
P
f
z
= 1-
() 1
t(b/~/-~)
~rt
w
exp
21;o 1 + w 2
( 292) --
.
b2 (23)
The first m i n u s sign becomes a plus for the Ga emission. The w i d t h of the localization function in real space is 1/b. Fig. 1 shows a plot of this function for b >> g and b = ,~g with t/G~ = 0.1. The a m p l i t u d e of the channeling effect shown on the d i a g r a m is P
(20)
Here w = s ~ e (sx is the excitation error for reflection g, positive for g inside the Ewald sphere), and the origin ( r = 0) has been taken at the heavier (As) atom. (The Ga atom is at r = 1 / ( 2 g ) . ) More
AZ
=
,-, (1 --eft~go) exp(--rr2g
2/b2
).
(24)
We see that for t/ldo -- 0.1 and a fairly delocalized excitation ( 1 / b = 0.05 nm, say) the strength of the channeling effect is limited by a b s o r p t i o n effects
J. C.H. Spence et a L / Localization effects on quantification in axial and planar A LCHEMI
109
vide a refinement of the inelastic scattering potential.
14
12
5. Dynamical calculations I0
AZ Q,s
0B"N
04
!
-I
-2
i
W
i
o
q
Fig. 1. The form of eq. (23) for t / ~ = 0.1 and two different values of the localization parameter b = ,,z-g (curve a) and b >> g (curve b). The amplitude of the channeling effect A Z is marked. This behavior is found in experimental spectra [15].
for the low-order reflections (g < b), but by localization for the high-order reflections. This is to be expected as the unit cell is subdivided more finely when using higher-order reflections, and is consistent with the principle that strong channeling effects are seen for b > g. These qualitative arguments suggest the trends and sensitivities to be expected from detailed dynamical calculations. Thus a comparison of experimental X-ray emission as a continuous function of orientation with dynamical calculations (including absorption) could be expected to pro-
The differences between the planar and axial geometries in channeling experiments and the differing effects of localization may be clarified with the aid of dynamical electron diffraction calculations for the two-dimensional, thickness-averaged electron intensity l(r). This is shown for the axial and planar cases in GaAs in fig. 2 at 100 kV. A beam divergence of 3 mrad has been assumed and the crystal thickness is 100 nm. The excitation error for the (200) reflection in the planar case is Sg = - 1 / ~ ( 2 0 o ) ; however, the calculation includes all ZOLZ reflections for a Laue circle centered at Kx = 0.11 A - l and K r = 4.83 ~,- 1 The results of other workers [32] have been used to approximate the width of a Gaussian function for V(r), whose FWHM is indicated by the circle in fig. 2. We see from fig. 2c that it is difficult to obtain "'non-channeling" conditions in practice. These calculations were used to provide the entries under '" theory" in table 2.
6. Experiment TEM specimens of G;~a_s were prepared by mechanical dimple-grinding and ion-beam thin-
©
Fig. 2. Panel (a) shows the thickness-averaged electron intensity for GaAs in the [100] zone axis orientation, in v,'hich each atom column contains only one species. Circles indicate very rough localization widths, which are inversely related to the angular width of the inelastic scattering distribution. Panel (b) shows the planar case, while panel (c) shows the best "non-channeling'" orientation which could be found, using these two-dimensional calculations.
J.C.H. Spence et a L /
110
Localization effects on quantification in axial and planar A L C H E M I
a
°
b
AsKe, ,~Ket
Gat'a
GaL.
b
"
.(
lb
Energy
13
"
io
6
'
g
keV
'
~b
Energy
'
~'s
'
~o
key
Fig. 3. X-ray emission spectra from GaAs in the planar (a) and n o n - c h a n n e l i n g (b) orientations. T h e line energies are G a K~x = 9.251 kV, G a L~x -- 1.096 kV, As K a = 10.543 kV and As Lcx = 1.282 kV.
ning. Samples with a [110] surface normal were used. These were then m o u n t e d in a beryllium double-tilt side-entry holder and tilted to the [100] orientation in such a way that they were inclined towards the X-ray detector. Three spectra were recorded from each of several regions, a [00h] systematics spectrum, a [100] zone-axis spectrum and a "'non-channeling" spectrum in which the excitation of Bragg beams was minimized. By observing the T E M image it was possible to ensure that all three spectra were o b t a i n e d from the
same region of s a m p l e (i.e. the same thickness). T h e planar spectra were recorded with a small negative excitation error for the (200) reflection, for which the (200) K i k u c h i line lies slightly inside the (200) reflection. (In two-beam theory this enhances the emission from the heavier atom.) We note that the (200) extinction distance ( a b o u t 700 nm) is much larger t h a n our specimen thicknesses of about 100 nm. A typical pair of spectra is shown in fig. 3. The b e a m divergence used in these experiments can be j u d g e d from fig. 4, which shows the e x p e r i m e n t a l diffraction c o n d i t i o n s under which the p l a n a r spectra were recorded. An electron p r o b e size of a b o u t 100 n m was used, and the spectra were r e c o r d e d at 100 kV on a Philips
Table 1 Intensity ratios and the factor Ra- given in eq. (17) obtained from spectra o b t a i n e d from two different areas; a significant deviation from 1 in R I indicates strong c h a n n e l i n g Orientation [!00] zone axis
Fig. 4. Electron diffraction pattern from G a A s in the [h00] systematics orientation, showing the b e a m divergence and diffraction conditions used.
Region Region [ hO0] planar Region Region N o n c h a n n e l i n g Region Region
1 2 1 2 I 2
(3a K / AsK
Rx
Theory (R-r)
1.2423 1,1972 1.1273 0.9761 1.2705 1.1928
0.9778 1,004 0.8873 0.88183
1.052 0.715
J. C.H. Spence et al. / Localization effects on quantification m axial and planar .4 L C H E M !
Table 2 The factor p given in eq. (16} evaluated from the same spectra used for table 1: here p indicates the sensitivity of the channeling effect to variations in X-ray energy: ALCHEMI requires p = 1; the planar and axial geometries are compared Orientation
Planar Region Region Axial Region Region
p(Oa)
1 2 1 2
1.015 0.894 0.847 0.542
p(As)
0.962 0.919 0.852 0.608
Theory p(Ga)
p(As}
0.894
0.916
0.565
0.695
EM400 instrument. Tables 1 and 2 summarize the results, and give peak height ratios for several spectra. Only ratios between peaks in the same spectrum are used, to eliminate effects due to variations in electron beam intensity.
7. Discussion and conclusion The spectra summarized in tables 1 and 2 were selected from many according to the criterion that R x be as small as possible for the planar case. Since R-r depends on the values of the thicknessaveraged wave-function at two points in the unit cell, and since these depend on thickness (if this is much less than the (200) extinction distance) we do not expect to obtain the same value of R1 from every region. For p, however, the result should depend only on the dimensionality of the wavefunction and the localization. The channeling effect is noticeably weaker in the axial orientation. We attribute this to the small atomic-number difference between Ga and As. However, the results of the calculations (see fig. 2) and the small value of p obtained (see below~ allows conclusions to be drawn from this case. The existence of the unresolved Ga K,8 contribution to the As K~ line results in a small error whose magnitude can be checked by using the well resolved As K,8 line in place ,~f the As Ks. The quantity p may be interpreted as a measure of the sensitivity of the channeling effect to both variations in X-ray energy and to the dimensionality of the wavefunction. From table 2 we see that the planar orientation confers a greater ira-
111
munity to delocalization effects than does the axial orientation It should therefore be used in ALCHEMI work involving low X-ray energies (0-3 kV). The channeling effect must be expected to disappear completely at energies below about 1 kV. It has been suggested that correction factors may be applied to allow for variations in localization [27]. While these may be useful in particular cases, we note that the inelastic scattering potential which occurs in the theory of dynamical inelastic scattering [19,28] depends on the particular crystal structure, and not just the atom type. Finally. the question arises as to how localization may be assured at low energies, in order to apply the ALCHEMI method to light elements or Auger electron emission [341. Since localization depends on both the average ionization energy and the mean momentum transfer to the crystal, it would appear possible, at least in principle, to perform coincidence experiments in which both the energy loss electrons and the corresponding X-rays were detected. Such experiments have been successfully' reported for energy' loss events and X-rays [35] and for cathodoluminescence and energy loss [36]. Since the magnitude of the momentuna transfer (and hence the localization) may be fixed in such an experiment, this would appear to provide a satisfactory method. Experimental evidence for the dependence of localization on inelastic scattering angle in channeling experiments on energy loss spectra is given elsewhere [20,21,23]. While such a coincidence technique avoids the post-ionization multiple elastic scattering problems of ALCHEMI using energy loss spectroscopy [20], the count rate to be expected is low and contains an unavoidable background of false coincidences.
Acknowledgements We are grateful to Dr. J. Tafto and Dr. A. Eades for helpful discussions. This work was supported by NSF grant number DMR85-12784 and the facilities of the A S U / N S F National Center for High Resolution Electron Microscopy at Arizona State University.
112
J.C.H. Spence et al. / Localization effects on quantification in axial and planar A L C H E M I
References [1] J.C.H. Spence, R.G. Graham and D. Shindo, Mater. Res. Soc. Symp. Proc. 62 (1986) 153. [2] D. Cherns, A. Howie and M.H. Jacobs, Z. Naturforsch. 28a (1973) 565. [3] J.C.H. Spence and J. Tafto, J. Microscopy 130 (1983) 147. [4] J. Tafto and Z. Liliental, J. Appl. Cryst. 15 (1982) 260. [5] J. Tafto, J. Appl. Cryst. 15 (1982) 378. [6] K. Krishnan, Ultramicroscopy 24 (1988) 125. [7] D. Shindo, K. Hiraga, M. Hirabayashi, A. Tokiwa, M. Kikuchi, Y. Syono, O. Nakatsu, N. Kobayashi, Y. Muto and E. Aoyagi, Japan. J. Appl. Phys., in press. [81 K.M. Krishnan, L. Rabenberg, R.K. Mishra and G. Thomas, J. Appl. Phys. 55 (1984) 2058. [91 D. Shindo, M. Hirabayashi, T. Kawabata and M. Kikuchi, J. Electron Microsc. 35 (1986) 409. [101 J. Bentley, in: Proc. 44th Annual EMSA Meeting, Albuquerque, N M , 1986, Ed. G.W. Bailey (San Francisco Press, San Francisco, 1986) p. 7134. [111 D.G. Konitzer, I.P. Jones and H.L. Fraser, Scripta Met. 20 (1986) 265. [121 C.J. Rossouw, P.S. Turner and T.J. White, Phil. Mag. B, in press. [131 M.T. Otten and P.R. Buseck, Ultramicroscopy 23 (1987) 151. [141 A. McLaren and A. Fitzgerald, Physics and Chemistry of Minerals, in press. [151 K.K. Christenson and J.A. Eades, in: Proc. 44th Annual EMSA Meeting, Albuquerque, NM, 1986, Ed. G.W. Bailey (San Francisco Press, San Francisco, 1986) p. 622. [16] P. Hence and F. Glas, Phil. Mag. B, in press. [17] S.J. Pennycook and J. Narayan, Phys. Rev. Letters 54 (1985) 1542. [18] J. Bentley, personal communication, 1987. [19] C.J. Rossouw and V.W. Maslen, Ultramicroscopy 21 (1987) 277.
[201 J. Tafto and O.L. Krivanek, Nucl. Instr. Methods 194 (1982) 153. [211 J.C.H. Spence, O.L. Krivanek and M. Disko, in: Electron Microscopy and Analysis 1981, Inst. Phys. Conf. Set. 61, Ed. M.J. Goringe (Inst. Phys., London-Bristol, 1982) p. 253. J. Tafto and J.C.H. Spence, Science 218 (1982) 49. [22] [23] J. Tafte and O.L. Krivanek, Phys. Rev. Letters 48 (1982) 560. [24] D. Lynch and C.J. Rossouw, Ultramicroscopy 21 (1987) 69. J. Tafto and P.R. Buseck, Am. Mineralogist 68 (1983) 944. [25] [26] J. Smyth and J. Tafto, Geophys. Rev. Letters 9 (1982) 1113. [27] S.J. Pennycook, in: Scanning Electron Microscopy/1987, Ed. O. Johari (SEM, A M F O'Hare, IL, 1988). [28] D. Saldin and P. Rez, Phil. Mag. B55 (1987) 481. [29] R.D. Heidenreich, J. Appl. Phys. 33 (1962) 2321. [30] L. Reimer, Transmission Electron Microscopy (Springer, Berlin, 1984). [31] P.G. Self and P.R. Buseck, Phil. Mag. A48 (1983) 121. [32] A.J. Bourdillon, P.G. Self and W.M. Stobbs, Phil. Mag. A44 (1981) 1335; see also A.J. Bourdillon, Phil. Mag. A50 (1984) 839. [33] J.C.H. Spence, in: High Resolution Electron Microscopy, Eds. P. Buseck, L. Eyring and J.M. Cowley (Oxford Univ. Press, London, 1988). [34] J.C.H. Sper, ce and Y. Kim, in: Proc. N A T O Summer School on R H E E D , Veldhoven, 1987. Ed. P. Dobson (Plenum, New York, 1988). [35] P. Kruit, H. Schuman and A.P. Somlyo, Ultramicroscopy 13 (1984) 205. [36] R. Graham, J.C.H. Spence and H. Alexander, Mater. Res. Soc. Syrup. Proc. 82 (1986) 235.
103
LOCALIZATION EFFECTS ON QUANTIFICATION IN AXIAL AND PLANAR ALCHEM! J.C.H. SPENCE, M. KUWABARA and Y. KIM Department of Physics. Arizona State Unit'ersit); Tempe, Arizona 85287-1504. USA Received 11 April 1988, presented at Workshop Januat'y 1988
The effects of wave-function dimensionality and inelastic localization on ALCHEMI (Atom Location oy tYhannetmg-r_nhanced Microanalysis) are studied. The original ALCHEM! equations are shown to hold for the case of partially delocalized excitations, provided that the thickness-averaged dynamical electron wavefield varies slowly over the localization volume. Experimental and theoretical comparisons of electron channeling in GaAs show that characteristic X-ray emission intensities in the axial orientation are more sensitive to variations in X-ray energy than are those in the planar Isystematics) geometD'. The effects of variations in localization are elucidated in a two-beam analysis, and methods for extending ALCHEMI ~o low-energy X-ray emission are discussed.
1. Introduction Over the past five years more than forty papers have been published studying the dependence of characteristic X-ray emission on electron diffraction conditions in thin crystals. Many of these have aimed to determine the fractions of a substitutional impurity species which lies on particular crystallographic sites. For a review of work until early 1986, see ref. [1]. All this work has its roots in studies of electron channeling effects on X-ray microanalysis in monatomic metals [2] and in earlier work on the Borrmann effect in X-ray diffraction. (For a historical review and an outline of the method, see refs. [3,6].) However, it was the realization [3-5] that, in polyatomic crystals, the emission from host atoms on known sites could be used to provide a reference signal which provided the basis for the ratio method known as ALCHEMI [3]. By comparing the orientation dependence of X-ray emission from substitutional irapurities with that of certain reference atoms on known planes (or atomic columns), it becomes possible to make quantitative estimates of the amounts of the impurity on each of several candidate sites. Since the site occupancies are given in terms of measured X-ray counts alone, the method appears to contain no adjustable
parameters. The choice of orientations is not critical, the specimen thickness need not be known, and no calculations of the dynan'ficai electron intensity are required. In particular, the ALCHEMI equations (see eqs. (13)-(15) below) do not require that the electron beam intensity remains the same for different spectra. For tutorial reviews of the method, see refs. [3,6]. In the last two years the method has been applied to the new superconductors [7], to Fe and Mn in samarium--cobalt magnets [8], to various elements in Ti/A1 alloys [9-11], to St, Zr and U in perovskite [12], to various cations in garnet [13], and to AI and Si in feldspars [14]. Additional studies have investigated the effect of the helical ray paths in immersion lenses on ALCHEMI [15], with the surprising finding that the smallest probe gives the least "'divergence". There have also been attempts to use the channeling effect to determine the displacement of atoms from their exact substitutional sites [16]. To summarize all the published work of recent years, it appears that if a strong channeling effect can be found for ,;ome set of planes ~and this is highly material and thickness dependent), then site occupancies can usually be determined for those planes. Several authors have pointed out that the original ratio argument, which was applied to crystal
0304-3991/88/$03.50:2; Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
104
J.C.H. Spence et al. / Localization effects on quantification in axial and planar A L C H E M I
planes in the systematics or p l a n a r channeling geometry, can also be applied to atomic columns in the axial orientation. Attempts to use "Axial A L C H E M I " to obtain quantitative site occupancies have, however, met with mixed success [10,11,13,17] leading in some cases to quite unphysical results, such as negative occupancies [18], but to consistent and useful results in others [12]. It has been suggested that these effects are due to wriations in the inelastic localization [12,17]. (This refers to the width of the inelastic scattering potential or impact parameter and, for processes other than p h o n o n scattering, is inversely related to both the mean angle of inelastic scattering associated with the ionization event and to the emitted x-ray energy.) A detailed analysis of axial A L C H E M I has been given, based on the Blochwave dynamical theory of inelastic scattering in crystals [19]. There have also been several applications of similar principles to channeling effects observed on inner shell excitation edges in energy loss spectroscopy [20,21]. The purpose of this paper is to compare the sensitivities of the axial and planar orientations to variations in inelastic localization, and to suggest ways of either measuring or eliminating the dependence of A L C H E M I on variations in localization.
2. Quantitative A L C H E M I with partial localization In simplest form the A L C H E M I problem consists of a determination of the fraction C of an impurity a t o m X which lies on one of two possible candidate substitutional sites in a crystal. Let a fraction C lie on the B site; the remaining fraction ( 1 - C) then lies on the A site. The ordering of minor constituents over the M1 and M2 cation ~ ; t ~ c i n r ~ l l x z ; ~ ~rraxriA~c
cTrv~rt
same orientation dependence. This follows because the two a t o m s lie on "equivalent" sites on the two-dimensional structure (projected in the b e a m direction), a n d the total thickness-averaged dynamical electron-beam intensity in the crystal must respect the two-dimensional translational symmetry of the crystal. It is clear that the method cannot distinguish sites which differ in their coordinate in the b e a m direction. Chemical shifts may sometim~:s by used to distinguish the same atom when it occurs on different sites [23]. The preceding statements, however, make several assumptions, the most i m p o r t a n t of which are: (1) the p e r t u r b a t i o n of the dynamical wavefield by the i m p u r i t y itself is small. This has been confirmed by detailed numerical calculations for the planar case. (2) X-ray absorption effects for the various species do not have different dependencies on orientation. (3) all species have negligible inelastic delocalization. That is, the impact parameter for the ionizing collision is less than the thermal vibration amplitude. We consider this last approximation in this paper. In practice, the A L C H E M I m e t h o d has been applied to both the axial and the p l a n a r channeling geometries. This reduces the three-dimensional site determination p r o b l e m to two and one dimension respectively. ( F o r a study of the very weak dependence of c h a n n e l i n g effects on the excitation of higher-order Laue-zone reflections, see ref. [24].) In the axial case it is necessary to find an orientation which separates the possible substitutional sites for the i m p u r i t y onto different atomic columns in the b e a m direction, each containing a distinct refcrence atoms. The orientation dependence of the i m p u r i t y is then compared with that of each of the reference atom. A similar procedure is followed for the planar case, choosing planes
mTw~l~ r o l l
The method relies on the fact that the characteristic X-ray production from a thin c~'stal traversed by a kilovolt electron beam is proportional to the dynamical electron-beam intensity averaged over thickness at the site of the emitting atom [2]. If the concentration of X is small and the impurity is uniformly distributed throughout the crystal, then, for C = 1, the characteristic X-ray emission from species X and B must have the
which contain the likely substitutional sites (separated onto different planes) and also suitable reference atoms whose sites are known. By repeating the procedure for different sets of planes, a crystal site may be identified. Using the quantitative method outlined below, it is also possible, in principle, to d e t e r m i n e the fractional occupancy C in both the axial a n d planar case if the excitations are perfectly localized. For incompletely localized
J. C.H. Spence et al. / Localization effects on quantification in axial and p/anar A L C H E M I
excitations no quantitative method of fractional occupancy determination has been given; however, a method has been described for determining the site of a partially localized species which lies entirely on one site (C = 1) [12]. There are three cases in the literature in which quantitative measurements of fractional occupancy C made by A L C H E M I have been compared with measurements made by other methods [14,25,26]. All are for the planar case, and all find good agreement between the two methods within experimental error. The axial case has the important advantage of providing a stronger channeling effect; however, attempts to obtain quantitative results by this method have met with mixed success [11,13,27]. This has been ascribed to localization effects [19,27], and it has been suggested that a correction factor may be applied for lowenergy X-ray. We now wish to compare the importance of variations in inelastic localization in the axial and planar geometries. In order to do so, we re-derive the original ALCHEMI equations, including this effect. This may be done using the Bloch wave dynamical theory of inelastic scattering (for a recent treatment for inner shell processes, see ref. [28]). For our purposes we can assume that the dynamical elastic wavefunction g'(r) is known, and use the result [29] that the rate of ionization in a thin crystal traversed by a kilovolt electron beam is proportional to
z=flq,(r)l 2 V(r)
dr.
(1)
Here V(r) is the portion of the imaginary part of the optical potential which describes inner shell ionization, resulting in characteristic X-ray emission. This imaginary potential is used in the SchrSdinger equation to describe the depletion of the elastic wavefield by inelastic scattering processes (see, for example, ref. [30]). We assume that the intensity of emitted characteristic X-rays is proportional to the ionization rate. For small-angle inelastic scattering, the maximum component of momentum transfer in the beam direction is q,(max) = 0.11 q(max) I. We therefore
105
assume (from the uncertainty principle) complete delocalization in the beam direction if the loss is localized laterally. We define
l,(r)= f l4,(r)l 2 d : to be the thickness-averaged intensity in the neighborhood of atom A. The number of X-rays emitted per unit time from atomic species A is then U~=
K,f,,(,.)
VA(r ) dr,
(2)
where K A is a constant which takes account of differences in fluorescent yield and in the differing concentration of the atoms on the various planes (in the planar case) or atomic columns (in the axial case). We will write eq. (2) for compactness as N A = K A I A V A.
(3)
Detailed dynamical calculations for the form of the inelastic interaction potential have recently been made for inner shell excitations [19]. These depend on the cross-section for atomic ionization, and, if averaged over all directions of core electron ejection, show a peaked function which we approximate by a Gaussian in this paper. All previous planar A L C H E M I applications have assumed perfectly localized excitation, which corresponds to a delta function form for VA(r). Then
NA = KA/A"
(4)
From V(r) a mean-square impact parameter may be deduced, which is a measure of the inelastic localization. Here we define the localization L as the full width at half maximum height of a Gaussian V(r). While there is considerable disagreement in the literature as to the magnitude of L [19,27,31,32] it is clear from simple arguments based on the uncertainty principle that for inner shell ionization L varies inversely with the mean energs: loss responsible for ionization. This is about twice the emitted X-ray energy ~'l~,-'~-j-n [ ~'~1 It. ~heref,:,re varies by a factor of ten between the K and L fines used in our experiments with GaAs. (The energies are given in fig. 3.) L is believed to be comparable
J.C.H. Spence et al. / Localization effects on quantification in axial and planar A L C H E M I
106
with the m e a n t h e r m a l vibration a m p l i t u d e ( U 2) for low-energy ( 1 - 2 kV) X-rays. F o r G a A s at 300 K,
GKA IoVA =
KAIofVA(r)d r ,
If a third species X is introduced w i t h conc e n t r a t i o n C on the B (As) planes a n d c o n c e n t r a t i o n (1 - C ) o n t h e A ( G a ) p l a n e s t h e n w e h a v e N x = K x C l a V x + K x ( 1 -- C ) I A V x a n d for the n o n - c h a n n e l i n g orientation N ° = GKxC IoV x + GKx(1 - C) IoV x = G K x loVx =
[IA(r)+IB(r)]
dr=l,
(9)
T h e p r o b l e m n o w is to d e t e r m i n e C f r o m a k n o w l e d g e of the o b s e r v a b l e X - r a y counts alone. Add i t i o n a l i n f o r m a t i o n is available if m o r e t h a n one e m i s s i o n line is o b t a i n a b l e from each species. N o t i c e that it is n o t a s s u m e d that t l ~" b e a m curr e n t is the same for t h e c h a n n e l i n g a n d n o n - c h a n n e l i n g spectra. U s i n g (8) to p r o v i d e an expression for C, rep l a c i n g K x using (9), m u l t i p l y i n g I A V x b y N x Io~"x N°
N° Nx
IoVx
=1,
dividing by N x l - ~ / N
C
~
10Vx
IoVx
° and r e a r r a n g i n g gives = IAVx( N° Io g x Nx
1).
(10) N o w if all the e m i s s i o n lines are perfectly localized (high energy X - r a y s ) Uv×/
f~ I o ( r ) d r = fc
GKxlofVx(r)d r .
(5)
since I 0 is a s s u m e d constant w i t h i n the unit cell. The n o r m a l i z a t i o n is
(8)
oV× = N A / N . ° .
(11)
with VB the inelastic localization for the As atom emission line. T h e width of this function will differ for the K a n d L shells by a factor of about ten. The emission from the As a t o m s in the nonchanneling orientation is
Physically, this r e q u i r e s that the s t r e n g t h of the c h a n n e l i n g effect r A = NA/N°a be i n d e p e n d e n t of X - r a y energy. H e r e r~, is the ratio of the X-ray emission intensities for the c h a n n e l i n g a n d nonc h a n n e l i n g o r i e n t a t i o n s in an idealized e x p e r i m e n t with constant b e a m c u r r e n t for a single emission line. We use the ratio R ( A / X ) = r~,/r x to indicate the extent to which t w o species A a n d X s h o w the s a m e orientation d e p e n d e n c e , R = 1 i n d i c a t i n g either that there is n o c h a n n e l i n g effect, or that b o t h species show effects of equal strength, and are therefore on the s a m e site. For localized excitation of the B site, similarly.
N ~ = G K B I o V s.
i
ell
ell
where the factor G allows for c h a n g e s in b e a m current b e t w e e n spectra. The emission from the As a t o m s in the channeling orientation is N B = K B lBVa,
(6)
(7)
v×/1ov
=
(12)
J. C H. Spe,lce et aL / Localization effects o,1 quantification i,1 axial and planar ,4 L C H E M I
Using (11) and (12) in (10) we obtain R(A/X)(1 - fl)C= R(A/X)-
1.
(13)
This is the original ALCHEMI expression, which allows C to be determined from measured X-ray counts alone. Here
R(A/X)=(NA/Nx)/(N°A/N~)=rA/rx
(14)
and
fl=NBN°/NA N°.
(15)
It will be noted that the effects of any changes in beam current between the channeling and nonchanneling spectra are eliminated when the ratio rA/r x is taken.
3. Approximations in ALCHEMI Eqs. (11) and (12) express the key localization approximations of ALCHFMI (eq. (13)). We note that either of the following conditions will ensure that eqs. (11) and (12) are satisfied: (1) if the excitations are highly localized, or (2) if the intensity IA(r) varies slowly over the inelastic localization volume. In the second case IA(r) may be taken outside the integral in eq. (2) and, using (3) and (5), eq. (11) follows. The question then arises as to how rapidly this intensity varies over the localization volume for each of the planar and axial geometries. We now, therefore, compare the failure conditions of this approximation for the two geometries. This may be done by comparing the strength of the channeling effect for, say, the Ga K line r(Ga, K, planar) with that of the L line r(Ga, L, planar). If this is not independent of X-ray energy, we will have r(Ga, L, planar}/r(Ga, K, planar)
107
for p = 1 can the ALCHEMI method (eq. (13)) be expected to work.. It is important, however, to ensure that a strong channeling effect is actually observed in these experiments, and it is desirable to do this in such a way that changes in incident beam current are allowed for. Thus, either the absence of any channeling effect (N L = Nt°) or a strong X-ray energyindependent channeling effect would give p = 1 in (16) above. To distinguish these possibilities, an independent monitor of the strength of the channeling effect is required from the same experimental data. This can be done by comparing the emission from different species on different planes, using the same data. Thus we make the above measurements of p under experimental conditions such that the Ga-to-As intensity ratio changes appreciably between orientations. We then require:
R-r =
N(Ga, K, c h ) / N ( A s , K, ch) N(Ga, K, n c ) / N ( A s , K. nc)
N~,. N °a NB N~~
t~,x rB
(17)
to be as large as possible. This demonstrates a redistribution of normalized intensity over the unit cell. Since the K lines are highly localized and the Ga and As atoms lie on different planes, R T provides an independent monitor of the strengt': of the channeling effect which is independent of changes in the beam current. For the particular case of [100] axial channeling in GaAs, however, this test fails, since the small difference in atomic number leads to a rather small difference between the average electron intensity on the Ga and As columns. However, our calculations (see fig. 2) show a large difference between the normalized average intensities for the two orientations on one site.
= p not equal to 1
= ( NL/N ° ) / ( NK/N° ) = R ( L / K . Ga)p,,m:..
(16)
4. Two-beam results including delocalization and absorption for two species
The experimental results given in section 6 report values of p for both the planar and axial [100] cases for all the Ga and As L and K lines. Only
Variations in localization can be taken into account in ALCHEMI analysis if the effects of
d.C.H. Spence et al. / Localization effects on quantification in axial and planar A L C H E M I
108
localization c a n be measured against some convenient experimental parameter. Since the localization L is given approximately [33] t~y
L = h/~ge,
(18)
where @ E = ( A E / E o ) ( 1 + "t -~) is the (relativistic) characteristic inelastic scattering angle for energy loss A E, variations in the accelerating voltage E 0 might be considered, useful. However, it has been shown that with small modifications to m o d e m instruments, X-ray emission spectra can be obtained as a continuous function of incident b e a m angle, using the double-rocking apparatus of Christenson a n d Eades [15]. We n o w show that it is possible to obtain i n f o r m a t i o n on the localization function f r o m measurements of X-ray emission as a f u n c t i o n of orientation a r o u n d several Bragg positions. In the i n d e p e n d e n t Bloch-wave a p p r o x i m a t i o n the thickness-independent electron intensity at r inside a crystal is given by 2
I(r)= Y'lcdl21y'C~exp(2vrig.r)l j
(19)
g
Evidence for the failure of this a p p r o x i m a t i o n in very thin crystals is given in ref. [2]. It agrees with the exact result only for an infinitely thick crystal without absorption. However, it has been found to give good qualitative agreement with experiment for experiments such as these for GaAs. (The neglect of the unavoidably strong (400) reflection is a more severe a p p r o x i m a t i o n in the following than the i n d e p e n d e n t Bloch-wave approximation.) In the t w o - b e a m a p p r o x i m a t i o n for a non-centrosymmetric crystal with absorption and a localization function V(r) we then have, from (1) and (19), an ionization rate p r o p o r t i o n a l to
[ ,,.. : ='-
x fv(r)
,
,,f w +"t 1
cos(2wg, r) dr.
details, and plots of z(s~) for delta f u n c t i o n V(r) can be found in ref. [1]. In the absence of absorption, A(w, g ) = 0 a n d B = t. The first t e r m gives the emission f r o m a non-crystalline sample. The second term expresses the o r i e n t a t i o n - d e p e n d e n t c o n t r i b u t i o n due to a n o m a l o u s a b s o r p t i o n . Here (21)
A ( w , g ) = ~ ' t 2 / ( 1 + wZ)/d~,
where /dgP is the a n o m a l o u s absorption. T h e third t e r m describes the o r i e n t a t i o n - i n d e p e n d e n t average absorption effect (22)
B = t - ~rt2//do
where /d0 is the m e a n absorption distance [30]. In deriving eq. (20) we have assumed t h a t ~5~> ~50 a n d that t
P
f
z
= 1-
() 1
t(b/~/-~)
~rt
w
exp
21;o 1 + w 2
( 292) --
.
b2 (23)
The first m i n u s sign becomes a plus for the Ga emission. The w i d t h of the localization function in real space is 1/b. Fig. 1 shows a plot of this function for b >> g and b = ,~g with t/G~ = 0.1. The a m p l i t u d e of the channeling effect shown on the d i a g r a m is P
(20)
Here w = s ~ e (sx is the excitation error for reflection g, positive for g inside the Ewald sphere), and the origin ( r = 0) has been taken at the heavier (As) atom. (The Ga atom is at r = 1 / ( 2 g ) . ) More
AZ
=
,-, (1 --eft~go) exp(--rr2g
2/b2
).
(24)
We see that for t/ldo -- 0.1 and a fairly delocalized excitation ( 1 / b = 0.05 nm, say) the strength of the channeling effect is limited by a b s o r p t i o n effects
J. C.H. Spence et a L / Localization effects on quantification in axial and planar A LCHEMI
109
vide a refinement of the inelastic scattering potential.
14
12
5. Dynamical calculations I0
AZ Q,s
0B"N
04
!
-I
-2
i
W
i
o
q
Fig. 1. The form of eq. (23) for t / ~ = 0.1 and two different values of the localization parameter b = ,,z-g (curve a) and b >> g (curve b). The amplitude of the channeling effect A Z is marked. This behavior is found in experimental spectra [15].
for the low-order reflections (g < b), but by localization for the high-order reflections. This is to be expected as the unit cell is subdivided more finely when using higher-order reflections, and is consistent with the principle that strong channeling effects are seen for b > g. These qualitative arguments suggest the trends and sensitivities to be expected from detailed dynamical calculations. Thus a comparison of experimental X-ray emission as a continuous function of orientation with dynamical calculations (including absorption) could be expected to pro-
The differences between the planar and axial geometries in channeling experiments and the differing effects of localization may be clarified with the aid of dynamical electron diffraction calculations for the two-dimensional, thickness-averaged electron intensity l(r). This is shown for the axial and planar cases in GaAs in fig. 2 at 100 kV. A beam divergence of 3 mrad has been assumed and the crystal thickness is 100 nm. The excitation error for the (200) reflection in the planar case is Sg = - 1 / ~ ( 2 0 o ) ; however, the calculation includes all ZOLZ reflections for a Laue circle centered at Kx = 0.11 A - l and K r = 4.83 ~,- 1 The results of other workers [32] have been used to approximate the width of a Gaussian function for V(r), whose FWHM is indicated by the circle in fig. 2. We see from fig. 2c that it is difficult to obtain "'non-channeling" conditions in practice. These calculations were used to provide the entries under '" theory" in table 2.
6. Experiment TEM specimens of G;~a_s were prepared by mechanical dimple-grinding and ion-beam thin-
©
Fig. 2. Panel (a) shows the thickness-averaged electron intensity for GaAs in the [100] zone axis orientation, in v,'hich each atom column contains only one species. Circles indicate very rough localization widths, which are inversely related to the angular width of the inelastic scattering distribution. Panel (b) shows the planar case, while panel (c) shows the best "non-channeling'" orientation which could be found, using these two-dimensional calculations.
J.C.H. Spence et a L /
110
Localization effects on quantification in axial and planar A L C H E M I
a
°
b
AsKe, ,~Ket
Gat'a
GaL.
b
"
.(
lb
Energy
13
"
io
6
'
g
keV
'
~b
Energy
'
~'s
'
~o
key
Fig. 3. X-ray emission spectra from GaAs in the planar (a) and n o n - c h a n n e l i n g (b) orientations. T h e line energies are G a K~x = 9.251 kV, G a L~x -- 1.096 kV, As K a = 10.543 kV and As Lcx = 1.282 kV.
ning. Samples with a [110] surface normal were used. These were then m o u n t e d in a beryllium double-tilt side-entry holder and tilted to the [100] orientation in such a way that they were inclined towards the X-ray detector. Three spectra were recorded from each of several regions, a [00h] systematics spectrum, a [100] zone-axis spectrum and a "'non-channeling" spectrum in which the excitation of Bragg beams was minimized. By observing the T E M image it was possible to ensure that all three spectra were o b t a i n e d from the
same region of s a m p l e (i.e. the same thickness). T h e planar spectra were recorded with a small negative excitation error for the (200) reflection, for which the (200) K i k u c h i line lies slightly inside the (200) reflection. (In two-beam theory this enhances the emission from the heavier atom.) We note that the (200) extinction distance ( a b o u t 700 nm) is much larger t h a n our specimen thicknesses of about 100 nm. A typical pair of spectra is shown in fig. 3. The b e a m divergence used in these experiments can be j u d g e d from fig. 4, which shows the e x p e r i m e n t a l diffraction c o n d i t i o n s under which the p l a n a r spectra were recorded. An electron p r o b e size of a b o u t 100 n m was used, and the spectra were r e c o r d e d at 100 kV on a Philips
Table 1 Intensity ratios and the factor Ra- given in eq. (17) obtained from spectra o b t a i n e d from two different areas; a significant deviation from 1 in R I indicates strong c h a n n e l i n g Orientation [!00] zone axis
Fig. 4. Electron diffraction pattern from G a A s in the [h00] systematics orientation, showing the b e a m divergence and diffraction conditions used.
Region Region [ hO0] planar Region Region N o n c h a n n e l i n g Region Region
1 2 1 2 I 2
(3a K / AsK
Rx
Theory (R-r)
1.2423 1,1972 1.1273 0.9761 1.2705 1.1928
0.9778 1,004 0.8873 0.88183
1.052 0.715
J. C.H. Spence et al. / Localization effects on quantification m axial and planar .4 L C H E M !
Table 2 The factor p given in eq. (16} evaluated from the same spectra used for table 1: here p indicates the sensitivity of the channeling effect to variations in X-ray energy: ALCHEMI requires p = 1; the planar and axial geometries are compared Orientation
Planar Region Region Axial Region Region
p(Oa)
1 2 1 2
1.015 0.894 0.847 0.542
p(As)
0.962 0.919 0.852 0.608
Theory p(Ga)
p(As}
0.894
0.916
0.565
0.695
EM400 instrument. Tables 1 and 2 summarize the results, and give peak height ratios for several spectra. Only ratios between peaks in the same spectrum are used, to eliminate effects due to variations in electron beam intensity.
7. Discussion and conclusion The spectra summarized in tables 1 and 2 were selected from many according to the criterion that R x be as small as possible for the planar case. Since R-r depends on the values of the thicknessaveraged wave-function at two points in the unit cell, and since these depend on thickness (if this is much less than the (200) extinction distance) we do not expect to obtain the same value of R1 from every region. For p, however, the result should depend only on the dimensionality of the wavefunction and the localization. The channeling effect is noticeably weaker in the axial orientation. We attribute this to the small atomic-number difference between Ga and As. However, the results of the calculations (see fig. 2) and the small value of p obtained (see below~ allows conclusions to be drawn from this case. The existence of the unresolved Ga K,8 contribution to the As K~ line results in a small error whose magnitude can be checked by using the well resolved As K,8 line in place ,~f the As Ks. The quantity p may be interpreted as a measure of the sensitivity of the channeling effect to both variations in X-ray energy and to the dimensionality of the wavefunction. From table 2 we see that the planar orientation confers a greater ira-
111
munity to delocalization effects than does the axial orientation It should therefore be used in ALCHEMI work involving low X-ray energies (0-3 kV). The channeling effect must be expected to disappear completely at energies below about 1 kV. It has been suggested that correction factors may be applied to allow for variations in localization [27]. While these may be useful in particular cases, we note that the inelastic scattering potential which occurs in the theory of dynamical inelastic scattering [19,28] depends on the particular crystal structure, and not just the atom type. Finally. the question arises as to how localization may be assured at low energies, in order to apply the ALCHEMI method to light elements or Auger electron emission [341. Since localization depends on both the average ionization energy and the mean momentum transfer to the crystal, it would appear possible, at least in principle, to perform coincidence experiments in which both the energy loss electrons and the corresponding X-rays were detected. Such experiments have been successfully' reported for energy' loss events and X-rays [35] and for cathodoluminescence and energy loss [36]. Since the magnitude of the momentuna transfer (and hence the localization) may be fixed in such an experiment, this would appear to provide a satisfactory method. Experimental evidence for the dependence of localization on inelastic scattering angle in channeling experiments on energy loss spectra is given elsewhere [20,21,23]. While such a coincidence technique avoids the post-ionization multiple elastic scattering problems of ALCHEMI using energy loss spectroscopy [20], the count rate to be expected is low and contains an unavoidable background of false coincidences.
Acknowledgements We are grateful to Dr. J. Tafto and Dr. A. Eades for helpful discussions. This work was supported by NSF grant number DMR85-12784 and the facilities of the A S U / N S F National Center for High Resolution Electron Microscopy at Arizona State University.
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J.C.H. Spence et al. / Localization effects on quantification in axial and planar A L C H E M I
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[201 J. Tafto and O.L. Krivanek, Nucl. Instr. Methods 194 (1982) 153. [211 J.C.H. Spence, O.L. Krivanek and M. Disko, in: Electron Microscopy and Analysis 1981, Inst. Phys. Conf. Set. 61, Ed. M.J. Goringe (Inst. Phys., London-Bristol, 1982) p. 253. J. Tafto and J.C.H. Spence, Science 218 (1982) 49. [22] [23] J. Tafte and O.L. Krivanek, Phys. Rev. Letters 48 (1982) 560. [24] D. Lynch and C.J. Rossouw, Ultramicroscopy 21 (1987) 69. J. Tafto and P.R. Buseck, Am. Mineralogist 68 (1983) 944. [25] [26] J. Smyth and J. Tafto, Geophys. Rev. Letters 9 (1982) 1113. [27] S.J. Pennycook, in: Scanning Electron Microscopy/1987, Ed. O. Johari (SEM, A M F O'Hare, IL, 1988). [28] D. Saldin and P. Rez, Phil. Mag. B55 (1987) 481. [29] R.D. Heidenreich, J. Appl. Phys. 33 (1962) 2321. [30] L. Reimer, Transmission Electron Microscopy (Springer, Berlin, 1984). [31] P.G. Self and P.R. Buseck, Phil. Mag. A48 (1983) 121. [32] A.J. Bourdillon, P.G. Self and W.M. Stobbs, Phil. Mag. A44 (1981) 1335; see also A.J. Bourdillon, Phil. Mag. A50 (1984) 839. [33] J.C.H. Spence, in: High Resolution Electron Microscopy, Eds. P. Buseck, L. Eyring and J.M. Cowley (Oxford Univ. Press, London, 1988). [34] J.C.H. Sper, ce and Y. Kim, in: Proc. N A T O Summer School on R H E E D , Veldhoven, 1987. Ed. P. Dobson (Plenum, New York, 1988). [35] P. Kruit, H. Schuman and A.P. Somlyo, Ultramicroscopy 13 (1984) 205. [36] R. Graham, J.C.H. Spence and H. Alexander, Mater. Res. Soc. Syrup. Proc. 82 (1986) 235.