Journal of Non-Crystalline Solids 293±295 (2001) 458±463
www.elsevier.com/locate/jnoncrysol
Atomic structure of amorphous solids from high resolution
electron microscopy ± a technique for the new millennium?
Gavin Mountjoy *
School of Physical Sciences, University of Kent at Canterbury, Canterbury CT2 7NR, UK
Abstract
A highP
performance electron lens can distinguish scattering from di€erent regions of a sample, i.e., atomic density
± information which is not available from any other
qatm …r† ˆ i d…r Ri †, with a resolution approaching 1 A
technique. Early high resolution electron microscopy (HREM) studies of amorphous solids faced scepticism due to the
association of `amorphous' with `random'. This paper demonstrates two ways in which HREM gives information about
atomic structure,
which is not available from di€raction. Firstly, while di€raction depends on pair correlations, HREM
R
depends on qatm …r† dz, and hence provides a probe for triplet correlations (for example). Secondly, while di€raction
depends on jA…k†j2 (where A…k† ˆ FTfqatm …r†g is the scattering amplitude), HREM depends on A…k†. This provides a
probe for locally anisotropic medium range order (MRO), such as in the quasi-Bragg plane model for MRO in a-SiO2
by Gaskell et al. These relationships are demonstrated using computer simulations on models of ta-C. Ó 2001 Elsevier
Science B.V. All rights reserved.
PACS: 61.14.Dc; 61.16.Bq; 61.43.Bn
1. Introduction
In transmission electron microscopy (TEM), a
beam of high energy electrons (hundreds of keV,
passes through and is scattered by a
k 0:1 A)
very thin sample (thickness t 1 lm). This is
analogous to X-ray and neutron di€raction, but
for electron di€raction of amorphous materials
inelastic and multiple scattering corrections are
more dicult due to the strong interactions of
electrons [1]. However, the key ingredient of
TEM is the electron lens which can focus or
magnify the probe electron distribution. The lens
may be used to focus the incident electron beam
*
Tel.: +44-1227 764 000 ext. 3776; fax: +44-1227 827 558.
E-mail address: [email protected] (G. Mountjoy).
onto a nanometer-sized region of the sample, as
in the microdi€raction technique [2]. Alternatively, the lens may be used to magnify the
spatial distribution of probe electrons at the exit
surface of the sample, as in the high resolution electron microscopy (HREM) technique [3].
In the ¯uctuation coherence microscopy
technique, both the incident angular distribution
and the exiting spatial distribution are measured
[4].
This paper presents two ways in which HREM
can probe atomic structure of amorphous materials, which are not possible using di€raction. An
HREM image depends on the projection (see
Section 2.1) of the atomic structure
or particle
P
distribution function q…r† ˆ i d…r Ri † where
r ˆ …x; y; z† and ri are atomic positions (see Fig.
1(a)). The resolution of the image is limited (see
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 6 9 5 - 0
G. Mountjoy / Journal of Non-Crystalline Solids 293±295 (2001) 458±463
459
The model has been
Fig. 1. Quasi-Bragg plane MRO in 500 atom central region of 1000 atom model of ta-C (diameter ˆ 18:1 A).
1 . (a) 2d
^1 is horizontal in the page, where A…Q1 k
^1 † is the maximum in A…Q1 k†
^ for varying directions k
^ and Q1 ˆ 2:9 A
rotated so that k
projection and (b) simulated HREM image (400 keV, Drmin 1:7 A).
Fig. 1(b)) due to the lens contrast transfer function
CTF…k† [3], where k ˆ jkj and k ˆ …kx ; ky ; kz † is the
scattering vector. This has a similar e€ect to a
modi®cation function in di€raction, with
jCTF…k†j 11 for 0 < k < kmax (see Fig. 2), but
is small due to lens aberrations, and
kmax 6 4 A
this gives a poor spatial resolution Drmin P 1:5 A
Fig. 2. Electron lens contrast transfer function CTF…k† for
conditions similar to those used in Fig. 1(b) (400 keV,
Arrow indicates position of FSDP in ta-C,
Drmin 1:7 A).
1.
Q1 ˆ 2:9 A
(depending on microscope parameters). However,
there have been recently a number of technical
breakthroughs which raise the future possibility of
resolution images (see [5]). Hence
obtaining sub-A
it is worthwhile to examine the information content of projections of amorphous structures, and
this paper draws on selected results from a previous study [5] using models of tetrahedral amorphous C (ta-C).
Whereas as the spatial dependence of an
HREM image re¯ects the projection of atomic
structure, the k-dependence of an HREM image
re¯ects the spatial frequencies in the atomic
structure which are perpendicular to the direction
of projection (see Section 2.3). In this way, the kdependence of scattering can be obtained from an
HREM image for di€erent regions within a sample, and regions with characteristic spatial frequencies may be identi®ed from features in
HREM images. In the quasi-Bragg plane model of
medium range order (MRO) in a-SiO2 [6], there
are oscillations in q…r† with period 2p=Q1 , where
Q1 is the position of the ®rst sharp di€raction peak
^1 (perpendicular to the
(FSDP), and direction k
oscillations). Hence these are characterised by a
^1 . HREM is well
prominent spatial frequency Q1 k
suited for investigating such MRO, which may be
present in other amorphous materials, but which
460
G. Mountjoy / Journal of Non-Crystalline Solids 293±295 (2001) 458±463
cannot be directly observed using di€raction. This
is illustrated using models of ta-C. Note that ta-C
can be expected to have similar structural features
to a-Si and a-Ge, and these materials have frequently been studied using TEM, in particular
HREM.
2. Theory
2.1. Theory of HREM ± information about
R
q…r† dz
For HREM of amorphous materials, the weak
phase objection approximation for scattering amplitude is [3]
Z
Z
A…kx ; ky † ˆ i/FT2d
Vatom …r†dz 2d q…r†dz ;
…1†
where / is an interaction constant, Vatom is an
atomic potential, and is the convolution operator. This approximation assumes that the sample
(depending on
is extremely thin, e.g., 6 100 A
element). Applying the projection operation to
q…r† on the RHS of Eq. (1) is equivalent to restricting k to two dimensions (2d) on the LHS. In
fact, for k ˆ …kx ; ky ; 0† Eq. (1) is consistent with
the usual expression for di€raction amplitude,
i.e.,
2.2. Projection distribution functions
As shown in Section 2.1, the fundamental
structural information underlying an HREM image
particle distribution function
R is the projected
P
q…r† dz ˆ i d……x; y† …xi ; yi ††, where …xi ; yi † are
the projected atom positions. The previous study
[5] showed that distribution functions of 2d projections are directly dependent on those of the
corresponding 3d amorphous structures. For pair
distribution functions, this is clear from the
equivalence of Eqs. (1) and (2). Only the result
for triplet distribution functions is shown here,
i.e.,
g3P …~
r12 ;~r13 ;/23 †
Z Z
1 t t
z12 z13 2
1
1
t
t
t
t t
0:5
0:5
2
2
‡ z213 † ;h23 dz12 dz13 ; …4†
‡ z212 † ;…~r13
g3 …r12
where g3…r12 ; r13 ; h23 † is the relative probability of
®nding atoms in the 3d structure with separations
r12 ; r13 and angle h23 , g3P is the equivalent function
for the 2d projection, ~r ˆ …x; y† and t is thickness.
Note these functions are isotropic averages over all
particles.
2.3. Theory of HREM ± information about A…k†
However, HREM preserves information
R about
the phase of A…kx ; ky †, and hence about q…r† dz.
The lens forms an image by combining scattered
and unscattered electrons, with a phase shift of
‡p=2 applied to A…kx ; ky † to cause interference
with d…k†. This results in an HREM image with
intensity
Z
I…x; y† 1 ‡ 2/ Vatom …r† dz
…3†
Z
2d q…r† dz 2d FT2d fCTF…k†g:
Comparison of Eqs. (1) and (3) shows that the
FT2d of I…x; y† has the same …kx ; ky †-dependence as
that of A…kx ; ky †, except for the factor CTF…k†
which e€ectively limits k < kmax . However, whereas
di€raction gives A…k† for the whole sample, an
HREM image gives A…kx ; ky † for di€erent regions
of the sample. For the region of an HREM image
centered at …X1 ; Y1 † with j…x; y† …X1 ; Y1 †j < R1 , the
image intensity is
Z
I1 …x; y† 1 ‡ 2/ Vatom …r† dz
Z
…5†
2d q1 …r† dz 2d FT2d fCTF…k†g
As discussed in Section 1, CTF…k† is an e€ective
modi®cation function due to the electron lens (see
Fig. 2).
where
q1 …r†
has
provided
R1 Drmin ,
j…x; y† …X1 ; Y1 †j < R1 . Then the FT2d of I1 …x; y†
has a …kx ; ky †-dependence that is given by
A0 …k† / FT3d fVatom …r†gFT3d fq…r†g:
…2†
G. Mountjoy / Journal of Non-Crystalline Solids 293±295 (2001) 458±463
Z
A1 …kx ; ky † 2/FT2d
Vatom …r† dz
Z
q1 …r† dz CTF…k†:
FT2d
…6†
Applying the projection operation to q1 …r† on the
RHS of Eq. (6) is equivalent to restricting k to 2d
on the LHS. In fact, for k ˆ …kx ; ky ; 0† Eq. (6) is
consistent with the expression for di€raction from
a small, cylindrical volume of the sample,
A01 …k† / FT3d fVatom …r†gFT3d fq1 …r†g
…7†
except for the additional factor CTF…k† in Eq. (6).
Hence HREM is useful for evaluating of local
variations in A01 …kx ; ky † due to variations in MRO. In
fact, the principles outlined in this section have been
used in a study of small angle scattering features in
A1 …kx ; ky † due voids and defects in a-SiO2 [7]. In that
case the electron lens parameters were chosen to
1 < k < 0:9 A
1.
give jCTF…k†j 1 for 0:3 A
2.4. Quasi-Bragg plane model of MRO
In this section details of the quasi-Bragg plane
model [6] are reviewed. It is proposed that in some
glasses, e.g., a-SiO2 , the FSDP at k ˆ Q1 is related
to regions with planar-like oscillations in q…r† with
period 2p=Q1 , which are analogous to Bragg
planes in compositionally equivalent crystals. In
^1 perpendicular to the oscillations
the direction k
there is signi®cantly large scattering amplitude
^1 †j compared to the average jA…Q1 k†j
^ over
jA…Q1 k
^ Previously developed predictions
all directions k.
for random variations in isotropic scattering [8]
^ denoted
can be used to test maxima in A…Q1 k†,
^1 †, and hence to identify regions with quasiA…Q1 k
Bragg plane MRO. The evaluation depends on
^1 †j=jA…Q1 k†j
^ ˆ 1 for
sample volume, and jA…Q1 k
large volumes, as in di€raction. However, in Section 2.3 it was shown that HREM provides a
^1 † for cylindrical volmeans of measuring A…Q1 k
umes of sample with radius R1 Drmin P 1:5 A,
1
. These conditions were
provided k < kmax 6 4 A
satis®ed in the original study of a-SiO2 [6], which
1 and R1 ˆ 27 A.
That study
had Q1 ˆ 1:5 A
showed that a region with planar-like oscillations
461
can cause fringes in simulated HREM images,
^1 .
when the projection is perpendicular to k
3. Results from calculations
Two di€erent models of ta-C have been used.
They
have
cubic
boundary
conditions,
3
q0 ˆ 0:17 atm A , box lengths of 18.1 and 43.3
and 1000 and 13810 atoms, respectively. These
A,
models were made available for the present study
by Gilkes who developed them by adapting earlier
models of a-Si [9,10]. The original construction for
the 1000 atom model used Monte Carlo bondswitching with a Keating potential [10], and for the
13810 atom model used molecular dynamics with a
Stillinger±Weber potential [9]. There was no deliberate attempt to introduce MRO during the
construction. Both models reproduce the experimental FSDP of ta-C.
To test for regions of quasi-Bragg plane MRO,
^ with varying direction k
^ and
the values of A…Q1 k†
1
Q1 ˆ 2:9 A (the position of the FSDP in ta-C)
were calculated for spherical regions containing
Such regions were
500 atoms (diameter 18.1 A).
taken from the centre of the 1000 atom model, and
the 8 quadrants of the 13 810 atom model. Two of
the nine regions tested were found to have maxi^1 † with ratio jA…Q1 k
^1 †j=jA…Q1 k†j
^ exmum A…Q1 k
ceeding the predicted 5% con®dence limit of 3.6
for isotropic scattering. This included the central
region of 1000 atom model with ratio ˆ 4:7 (i.e.,
signi®cantly anisotropic). Fig. 3 shows the fre^ for varying direcquency distribution of jA…Q1 k†j
^
tion k observed for the central region of the 1000
atom model, for a region of the 13 810 atom model
with ratio ˆ 2:9 (i.e., not signi®cantly anisotropic),
and predicted for an isotropic model. The central
region of the 1000 atom model was rotated so that
^1 is perpendicular to the projection. Fig. 1(a)
k
shows the projection of atomic positions, and Fig.
1(b) shows a simulated HREM image with fringes.
The simulated HREM image was obtained from
the CERIUS 2.0 HREM software package (which
uses the multislice algorithm), with parameters
corresponding to a JEOL4000EX-II HREM instrument.
462
G. Mountjoy / Journal of Non-Crystalline Solids 293±295 (2001) 458±463
g3P . These are functions of three variables. Fig. 4
shows the results for r12 (or r~12 ) ®xed approximately equal to the nearest neighbour distance
with r13 and h23 free to vary in polar
(i.e., 1.52 A)
coordinates. The result for the 2d projection is in
fact the average for projections along x, y and z
axes. As expected, the order in the 2d projection
re¯ects that in the 3d structure. Both show peaks
h23 110° and r13 2:35 A,
at r13 1:5 A,
h23 45° corresponding to three carbon atoms
forming a tetrahedral bond. The result for the 2d
projection also shows an artefact at r12 1:4 A
and /23 55°, which is due to the projection of
second nearest neighbour atoms (see [5]).
^ with varying diFig. 3. Frequency distribution of jA…Q1 k†j,
1 , for 500 atom regions of models of
^ and Q1 ˆ 2:9 A
rections k
ta-C. Regions (o) without and, …† with quasi-Bragg plane
MRO (corresponding to Fig. 1), and (solid line) prediction for
isotropic scattering [8].
Also presented for the 1000 atom model of ta-C
are the triplet distribution functions of the 3d
structure, g3…r12 ; r13 ; h23 †, and of the 2d projection,
4. Discussion
Models of ta-C have been used to demonstrate
the relationship between fringes in HREM images
and quasi-Bragg plane MRO. The simulated
HREM image in Fig. 1(b) is based on a spherical
Fig. 4. Triplet distribution functions of (a) 3d structure, g3…r12 ; r13 ; h23 †, and (b) 2d projection, g3P , for 1000 atom model of ta-C
Contours indicate values of functions (solid lines) >1 and (dashed lines) <1, for varying r13 and h23 (with r12 1:5 A).
…t ˆ 18:1 A†.
G. Mountjoy / Journal of Non-Crystalline Solids 293±295 (2001) 458±463
region from a model of ta-C. A real HREM
sample has a rectangular geometry, and regions of
quasi-Bragg plane MRO could be surrounded by
regions without such MRO, or with such MRO
but in di€erent orientations. This would certainly
diminish the visibility of fringes from any one region of quasi-Bragg plane MRO. However, simulating this e€ect requires larger models (note that
cubic boundary conditions become incongruent
when models are rotated).
Obviously, it was necessary to use models which
contained regions with quasi-Bragg plane MRO,
and it was fortunate that the models available for
^1 †
the study contained regions with maxima A…Q1 k
^1 †j=jA…Q1 k†j
^ > 3:6. The purhaving ratios jA…Q1 k
pose of this study is not to argue that ta-C contains
regions of quasi-Bragg plane MRO, but rather to
show that such MRO could in principle be investigated using HREM. However, it is noteworthy
that HREM studies of ta-C [11] have identi®ed
non-random features and fringes having periodic (compared to 2p=Q1 ˆ 2:2 A).
ity 2.1 A
In using HREM to obtain information about
triplet distribution functions (for example), the
present experimental obstacle is the limited spatial
resolution
of
electron
lenses
(currently
While this is a substantial obstacle
Drmin P 1:5 A).
for ta-C (see Fig. 1), the situation is more favourable for a-Si and a-Ge, which should have
similar structural features to ta-C, but which have
a larger scale of short range order (nearest neigh Considering the likely
bour distances of 2.4 A).
improvements to resolution in coming years (to
there is cause to be optimistic about future
sub-A),
developments in this area.
5. Conclusions
TEM provides scattering geometries which are
not available in conventional di€raction. In particular, HREM can provide information about
variations in A…k† for di€erent regions of the
463
sample. This is well suited for investigating the
quasi-Bragg plane model of MRO, and the results
presented here for models of ta-C supplement
previous results for a-SiO2 , and show that quasiBragg MRO causes fringes in simulated HREM
images. In addition, HREM is based on the projection of atomic structure, and hence incorporates
information about higher order correlations. For
example, the triplet distribution function of a 2d
projection shows similar features to that of the
corresponding 3d structure, as demonstrated here
using models of ta-C. Future improvements to
HREM resolution should reduce the barriers to
experimental observation of such higher order
correlations.
Acknowledgements
Thanks to K.W.R. Gilkes for making available
models of ta-C, and J.M. Rodenburg for useful
discussions. Part of this work was carried out
during a PhD at University of Cambridge, under
the supervision of P.H. Gaskell and the sponsorship of the Association of Commonwealth Universities.
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