Download Applying an information transmission approach to extract valence

Ultramicroscopy 111 (2011) 912–919
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Ultramicroscopy
journal homepage: www.elsevier.com/locate/ultramic
Applying an information transmission approach to extract valence electron
information from reconstructed exit waves
Qiang Xu a,b,n, Henny W. Zandbergen b, Dirk Van Dyck a,c
a
EMAT, University of Antwerp, 2020 Antwerp Groenenborgerlaan, 171, U316, Belgium
National Centre for HREM, Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Lorentzweg 1, Delft, The Netherlands
c
Vison Vision Lab, University of Antwerp, 2020 Antwerp Groenenborgerlaan, 171, U316, Belgium
b
a r t i c l e i n f o
a b s t r a c t
Available online 1 February 2011
The knowledge of the valence electron distribution is essential for understanding the properties of
materials. However this information is difficult to obtain from HREM images because it is easily
obscured by the large scattering contribution of core electrons and by the strong dynamical scattering
process. In order to develop a sensitive method to extract the information of valence electrons, we have
used an information transmission approach to describe the electron interaction with the object. The
scattered electron wave is decomposed in a set of basic functions, which are the eigen functions of the
Hamiltonian of the projected electrostatic object potential. Each basic function behaves as a communication channel that transfers the information of the object with its own transmission characteristic.
By properly combining the components of the different channels, it is possible to design a scheme to
extract the information of valence electron distribution from a series of exit waves. The method is
described theoretically and demonstrated by means of computer simulations.
& 2011 Elsevier B.V. All rights reserved.
Keywords:
Dynamical scattering
Electron distribution
Information transmission
1. Introduction
High resolution electron transmission microscopy (HRTEM) has
proven its ability to provide structure analysis down to the atomic
scale. But apart from atomic positions, the distribution of electrons,
especially the distribution of valence electrons, is also essential for
understanding the properties of materials and the experimental
measurement of the valence electron distribution would be hence
of considerable importance. Whereas electron diffraction has been
successfully explored to obtain the averaged valence electron distribution of crystals [1,2], direct imaging of valence electron distributions in real space would give more details about the local structure,
for instance, interfaces, nanoparticles, defects, etc. The recent development of aberration correctors improves the resolution of TEM to
the sub-angstrom regime, comparable to that of electron diffraction [3–5]. This suggests that the imaging resolution does not put
limit any more on the direct imaging of valence electron distribution
in real space. However, the task is not easy and has not been possible
till now for two main reasons: (1) The images are formed from the
scattered incoming electron wave by all charges, nucleus and all
electrons (including core electrons and valence electrons). Among the
total charge densities, valence electrons contribute only a tiny
fraction. (2) The interaction of the incident wave with the total
n
Corresponding author at: National Centre for HREM, Kavli Institute of
Nanoscience, Delft University of Technology, 2628 CJ Lorentzweg 1, Delft, The
Netherlands
E-mail address: [email protected] (Q. Xu).
0304-3991/$ - see front matter & 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.ultramic.2011.01.032
charge densities is so strong that the scattering is a dynamical
process, making the HRTEM image not intuitive to interpret.
In this paper, we will show that it is in principle possible to
reconstruct an image with enhanced valence electron distribution
information by combining a series of exit waves (that is the high
energy complex electron waves at back surface), based on an
interpretation of dynamical scattering using an information transmission approach. To facilitate the understanding, the paper is
structured as follows: after briefly introducing the information
transmission theory, the problem of obtaining the electron distribution from HREM is described as an inverse problem of
information transmission, which can be classified into two subinverse problems: (1) retrieval of exit waves from images and
(2) retrieval of the object structure from the exit waves. We first
reformulate these two inverse problems in a general term of
information transmission to show that the success of solving one
problem can be applied to the other. Then we will focus on the
possible reconstruction of the electron distribution from the exit
waves. And finally based on the developed interpretation, we will
give one example of a designed scheme which shows the possibility of combining a thickness series of exit waves to reconstruct
a rescaled electron density with enhanced contribution of valence
electron distribution and test it by image simulations.
2. General information transmission theory for HRTEM
The image formation in an electron microscope can be considered as an information transmission process conveying the
Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919
913
retrieve the electron distribution of the object, two inverse
problems need to be solved successively: (1) restoration of the
exit wave from recorded HREM images and (2) restoration of the
electron distribution (object structure) from the exit waves. We
will firstly investigate proper channel models to understand the
whole transmission process.
In both transmission sub-processes, the propagation of a fast
electron, neglecting back scattering, is accurately described by the
high energy equation which is formally equivalent to a time
dependent Schrodinger equation, in which the time can be
replaced by the distance z along the incident beam direction, as
if the electron propagate with a constant speed
,
_@c ,
ðR ,tÞ ¼ HcðR ,tÞ
i@t
ð4Þ
with
Fig. 1. Illustration of a simple channel model of interpreting of information
transmission. ci gives the transmission characteristic properties of the channel i.
information of the investigated object to the image. According to
information transmission theory, the transmission process is
decomposed in different communication channels or simply
channels. A channel can refer to either a physical transmission
medium such as a wire, the electron optics system of the
microscope, etc, or the quantum mechanical models which
describe the physical interaction of the electron with the object.
Every channel has its own specific transmission characteristics.
And since the characteristics of the channels can differ, the
combination of the different components at the end of the
transmission process may dramatically differ from the input. In
order to understand the whole information transmission process,
the deformation and the quality of the transmitted information,
one has to investigate the transmission characteristics of the
different channels. This is illustrated in Fig. 1, which shows a
theoretical single source of channel model to be used in the
article. As stated above, the information of the source Is can
always be decomposed into different components si:
X
Is ¼
si
ð1Þ
i
,
_
DeVðR ,tÞ
2m
ð5Þ
,
where H is taken in the x–y plane perpendicular to the z axis. R
denots a two dimensional vector in the x–y plane; D is the
Laplacian
operator in the x–y plane; e is the charge of one electron
,
and VðR ,tÞ is the potential acted on incident electron at time t. The
propagation in vacuum through the magnetic lenses and the
scattering in the crystal are described by two different
Hamiltonians.
3.1. Channel model I (from exit waves to images)
Between the exit face of the object and the image plane,
the electron wave senses the effect of the magnetic field of the
imaging lenses. Since the magnetic field does not change
the potential energy of the electron (ignoring spin effects), the
corresponding eigen states of the Hamiltonian are plane waves,
representing only the kinetic energy. As shown by Scherzer, the
magnetic field of the objective lens affects the phases of the
different plane wave component. The information transmission
from exit wave to image wave is described in the reciprocal space
by the following formulas
X
Cex ¼
fg ,
ð6Þ
g
Each of them transfers through a different channel. During the
transmission, they are modulated by the corresponding channels
so that the information received by the destination Id is
X
Id ¼
wi si
ð2Þ
i
and
wi ¼ fi ðp1 ,p2 ,:::Þ
H¼
ð3Þ
where wi gives the transmission characteristic properties of the
channel i, called its transmission rate; p are those parameters that
influenced the transmission properties. This information transmission model is generally applied in quantum mechanics. In this
paper, we will use it to describe the information transmission
in TEM.
3. Information transmission in TEM
The information transmission process in TEM is accomplished
in two steps: firstly, the electron interacts with the specimen
through dynamical scattering, resulting in a complex electron
wave at the exit surface of the sample (simply called exit wave).
Secondly, the exit wave is transferred through the electron optics
system of the microscope to the image plane where it is recorded.
In both steps, the information is distorted. Therefore, in order to
X
Cim ¼
wg fg ,
ð7Þ
g
3
wg 6expðiðpDf lg 2 þ 12pCs l g 4 þ:::ÞÞ
ð8Þ
where wg describes the transmission characteristic of the channel,
which is well known as the phase transfer function or (for weak
objects) the contrast transfer function (CTF) [6]; Cs is the spherical
aberration of the objective lens, (other instrumental parameters
Cc might be also important, but are ignored here for simplicity), Df
is the defocus value, describing the imaging condition; l is the
wavelength of the incident beam, g denotes the spatial frequency
of the plane wave.
According to formulas (6)–(8), the information transmission
process from exit wave to image wave can be described by
decomposing the signal into different communication channels
of plane waves, which are the eigen states of the Hamiltonian
representing the kinematical energy. The electron optic system
thus behaves as a spatial frequency filter, which modulates the
phases of the transmitted plane waves according to the spatial
frequencies. Note that the retrieval of the source information
(that is the exit wave in this step) has been successfully solved
based on the interpretation of image formation as an information
transmission. One can use appropriate combinations of the content of the channels to reconstruct the exit wave by varying the
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Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919
defocus or the wavelength (.e.g. focal series reconstruction)
[5,7–10].
3.2. Channel model II (from object to exit waves)
Inside the sample, the electron senses the crystal potential of
the object, and thus carries this information of the object to exit
wave. This process can be also described in the high energy
Eqs. (4)–(5) which can be regarded as a time-dependent Schrödinger
equation in which the time is replaced by the depth z (sample
thickness) using t ¼ mz=_k and the crystal
potential is simplified
,
as an averaged projected potential UðR Þ along the z direction with
Z
,
,
1 2m jd
VðR ,zÞdz
ð9Þ
UðR Þ ¼
2
d _ ðj1Þd
where d is the thickness of one repeated unit along the z direction;
Thus, the Eq. (5) is simplified as
,
,
@cðR ,zÞ
i
¼
HcðR ,zÞ
@z
4pk
ð10Þ
where H is
H¼
_
DeUðRÞ
2m
ð11Þ
,
The solution of cðR ,zÞ denotes the high energy electron wave
at exit surface after penetrating a sample of thickness z.
Inspired, by the usual quantum mechanics approach, we
expand cðR ,zÞ using the complete set of eigen functions of the
Hamiltonian (11) [11].
X
!
cðR ,zÞ ¼
wn ðzÞfn ð R Þ
,
ð12Þ
n
with
En z
wn ðzÞ ¼ Cn exp ip
El
ð13Þ
the energy and wavelength of the
where E, and l are, respectively,
,
incident electrons; fn ðR Þ, and En are, respectively, the eigenfunction and the corresponding eigenvalue of the Hamiltonian, which
can be obtained by solving the eigen equation
,
,
Hfn ðR Þ ¼ En fn ðR Þ
the eigen states to the exit wave. Unlike the thickness dependency is addressed in the normal understanding of dynamical
scattering, we would like to point out here the importance of the
En dependence of dynamical scattering, especially on imaging the
electron distribution.
HREM imaging is usually taken,along a low order zone axis
orientation. The crystal potential UðR Þ can therefore be considered
as a two-dimensional assembly of potential wells, corresponding
to the different projected atom columns
,
UðR Þ ¼
X , ,
UðR R i Þ
ð16Þ
i
,
The eigen-state function fn ðR Þ behave like molecular-orbitals
in two-dimensions. The deepest states are highly localized, bound
to the core of projected atom columns, and provide the atomic
information (the nucleus). They are similar to the 1s states of
atoms, but only in two dimension projection plane; whereas the
higher states are delocalized among neighboring atom columns,
and more sensitive to the valence electrons. In (14) and (15), we
only need to consider those bound states (En o 0), since the
unbound states with En Z0 are not standing waves, similar to
plane waves and provide only nearly constant background and
gives no effective information of the object.
When the sample is thin ,
(z 5 El=pjEn j for all En), the transmission rate of the channel fn ðR Þ (see (13)) can be written as
wn i
pz
El
Cn En
ð17Þ
One can see that all the states are transferred in the same
phase, but only the deepest states with large 9En9 have a high
transmission rate (see Fig. 2). Thus, the exit wave highlights only
the nucleus information, for instance the atom positions or
atomic types. In contrast, the high energy states, associated with
the valence electron distribution, do not effectively contribute to
the exit wave because of their small 9En9. When the sample is
thick, the transmission rate wn depends nonlinearly on the energy
(see Fig. 2), the information of some particular valence states can
ð14Þ
with H is given in (11). The coefficients Cn will be determined
from the boundary condition
,
Note that the eigen
functions fn ðR Þ are only related to the
,
projected potential UðR Þ, not dependent to the thickness. The total
!
density rð R Þ of all the eigen states of the projected potential
X
X
!
!
!
rð R Þ ¼ rn ð R Þ ¼ 9fn ð R Þ92
ð15Þ
n
n
provides the projected electron distribution of the object, and
thus is an
intrinsic property of the object. In contrast, the exit
,
wave cðR ,zÞ is strongly dependent on the sample thickness z, due
to the dynamical scattering. The thickness dependency of the
different channels is explicitly given by (12) and (13). Comparing (12) and (13) to (7) and (.8), one can notice the similarity
between the formulas used to describe the two different information transmission steps. This similarity suggests that the eigen
state fn of the Hamiltonian can be regarded as a communication
channel that transfers the information of the object to the exit
wave. The transmission characteristic of each channel fn is given
in (13), which can be considered as a ‘‘transfer function’’ for the
dynamical scattering. The channels are labeled by n, the quantum
number of the electron state or labeled by En, and the eigen
energy of the state n. Thus, the dynamical scattering can be then
understood as an Eigen state filter, modulating the contribution of
Fig. 2. Plot of imaginary part of wn as the function of En. The electron eigen functions
act as channels transferring the information of the objet to the exit waves. The exit
wave conserves the information of all the occupied two dimensional electron
states, but modulated by Eigen energy dependent transmission rate wn. When the
specimen is thin, only the deepest states (core state) have enough transmission
rates. As a result, the exit wave of thin specimen highlights the information of the
core state, indicating the atom positions and atomic type; however, it damps the
information of high energy states, thus hardly showing valence electron distribution (see Fig. 4(a) and (g)). When sample is thick, the transmission rate is
nonlinearly dependent on the energy, some states are highlighted and some not.
The exit wave can be hardly interpreted without known sample thickness.
Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919
be more effectively transmitted into the exit wave than those core
states. However, without knowing the sample thickness, it is
impossible to determine which states are enhanced and how
much they are enhanced. The information of valence electron is
scrambled and needs to be restored.
3.3. Channel filter of eigen states
M
X
N
X
win fn
ð19Þ
n¼1
Bringing (19) into (18), the channel filter is then described as
M X
N
X
xin win fn
ð20Þ
ð25Þ
ð26Þ
Once each eigen-state has been reconstructed using the
channel filter given in (20), the total electron distribution can
be then calculated from (15)
3.4. A simple filter to enhance the valence electron distribution
The channel filter method, although theoretically possible, is
not exactly needed for the retrieval of the valence electron
distribution. One of the main reasons is that the retrieval of the
electron distribution only
2 requires obtaining the sum of all
density functions of fn , instead of the details of each Eigen
wave function fn . Other methods can be designed by following
the idea of channel filtering. Here we propose a simpler method
as an example.
One possible filter can be designed by calculating the amplitude square of the standard deviation of a series of exit waves
with different sample thicknesses (named as STD method),
mathematically written as
2
M
1 X
2
Istd ¼ 9stdðcz1 , cz2 , cz3 ,. . ., czM Þ9 ¼ ðczi cÞ
ð27Þ
M
i¼1
M
1 X
c
M i ¼ 1 zi
x2n
ð28Þ
where czi denotes the exit wave of the thickness zi and M denotes
the total number of the exit waves used for the calculation. For a
large number of randomly chosen data sets, Istd will converge to
its expectation value Iev,
R
2
zu
zl ðcðzÞcðzÞÞdz
2
ð29Þ
Istd Iev ¼ 9stdðcðzÞÞ9 ¼
R zu
zl dz
with
cðzÞ N
X
wn fn ,
ð30Þ
n¼1
Eq. (20) can also be expressed in a simple matrix calculus. Let
W, Xn, and Yn denote an M*N matrix, an M order vector and an N
order vector, respectively, with the elements listed as follows:
2 1
3
w1 w21 wM
1
6 1
7
h i
6 w2 w22 wM
2 7
7,
W ¼ win
ð21Þ
¼6
6
^
&
^ 7
MN
4 ^
5
1
2
M
wN wN wN
Yn ¼ d1n
ð24Þ
Xn ¼ W 1 Yn
i¼1n¼1
i¼j
Thus, the vector of weighting factors Xn is given by
c¼
the ith exit wave in the series; M denotes the total
where
number of the exit waves and xin is the weighting factor of the ith
exit wave that is used to create the filter. The weighting factor xin
can be derived as follows:
From (12), one can write Ciex as
h
1
Xn ¼ xn
ia j
1
Then the desired xin for n state channel filter can be obtained by
solving the linear equation
ð18Þ
Ciex is
fn ¼
0
with
xin Ciex
i¼1
Ciex ¼
dij ¼
WXn ¼ Yn
As stated above, the dynamical scattering modulates the
information of all electron states according to their energy levels.
The way of modulation can be described in the formulas similar to
that used for interpreting image formations. Thus, the restoration
of the exit wave from HREM images and the restoration of the
electron density from the exit waves are actually similar problems. The success of solving one of them can be then exampled
to the other.
One possible way would be to reconstruct each electron state
by designing a channel filter. In the information transmission
interpretation of the electron dynamical scattering, each electron
state behaves as a single channel, transferring into the exit waves
with its own predicated character. Thus, in principle one can
construct an image representing only the component of one
specifically chosen channel by combining a number of exit waves
obtained at different modulation conditions with properly chosen
weighting factors. These factors can be even complex numbers.
Similar example can be found in the retrieval of the exit wave
from HREM images taken at different imaging conditions (varying
focus or wavelength).
The thickness of the object is one of the parameters that
influences the characteristic of the channels and free to change.
Therefore, the channel filter of an eigen state can be constructed
by combining a thickness series of exit waves. This filtering
process can be mathematically described as follows.
We want to reconstruct the content of the channel n in a linear
combination
fn ¼
where
(
915
xM
n
xin d2n dNn
T
iT
,
ð22Þ
En z
wn ¼ Cn exp ip
,
El
cðzÞ ¼
Z
zu
cðzÞdz
ð32Þ
zl
where zl and zu are the lower and upper integral limit of the
thickness, set by the minimum and maximum thicknesses of the
series, respectively. Note that wn behaves as a sine wave, following the orthogonality property
R z2
ð23Þ
ð31Þ
lim
ðz1 z2 Þ-1
z1
z
i expðikn z þ yÞ9z21
expðikn z þ yÞdz
¼0
¼ lim
Rz
ðz1 z2 Þ-1
kn ðz2 z1 Þ
0 dz
ð33Þ
916
Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919
Bringing (30), (31), (34) and (33) into (29), one can easily get
Istd N
X
2
9Cn 9 9fn 9
2
ð34Þ
n¼1
for
zu zl Z
2p
kmin
ð35Þ
with
kmin ¼ pEN
ð36Þ
El
where EN denotes the highest eigen energy of the bound state
need to be investigated.
Thus, Eq. (34) provides the physical meaning of the STD image.
Comparing (34) to (15), one can see that the image Istd resembles
the total projected electron distribution and can be interpreted as
one kind of weighted projected electron distribution with the
weighting factor given by jCn j2 for the density of the n state. In
another way, by taking the Istd as the final received information
!
and taking the projected electron distribution rð R Þ as the input
information, one can also build an information transmission
channel model to interpret the STD image: the total electron
!
distribution rð R Þ is composed of the electron density of all the
2
eigen states 9fn 9 ; each of them transfers into STD images
through the channel n with the transmission characteristic
2
described by 9Cn 9 .
2
Note that two properties of the transmission rate 9Cn 9 provide
STD images suitable to investigate the valence electron distribu2
tions of samples. One is that 9Cn 9 is only determined by the
incident boundary condition, and is not dependent on the sample
thickness. Thus the ideal STD image would be free of the influence
of the thickness. Secondly, for the condition of plane wave
2
incidence, it has been shown that 9Cn 9 is inversely proportional
to the corresponding absolute eigen energy [12],
1
2
9Cn 9 p ð37Þ
En
This energy dependence is very important for imaging valence
electrons: valence electrons constitute only a small fraction of the
electrons, and they are delocalized between atoms and distributed in a large space. Thus, the density of valence electron is
much smaller than that of the core electrons. An image presenting
!
an exact electron density map rð R Þ will only indicate the
information of core electron, not give a clear view of the valence
electrons, especially for heavy atoms. Thus, the enhancing of the
transmission rate of the valence electron information, like what
achieved by the standard deviation method, is essential for
viewing valence electron distributions. This feature will be further
demonstrated in the simulation part.
A typical TEM sample is usually wedge shaped, providing a
certain thickness variation. For a crystal material, the reconstructed exit waves contain information from an area of hundreds
of unit cells. The sub-image with the size of one unit cell
corresponds to the exit wave of a certain thickness. One may
easily get a series of exit waves with different thickness for the
application of the standard deviation method. When investigating
the local electronic structure, for instance the electron distribution of near an interface or defects, one may not easily obtain such
a thickness series of the exit waves. In this case, one need vary the
wavelength of the incident beam to get the series of exit waves
with proper modulations, which requires a further research.
2
Note that for different incidence conditions 9Cn 9 will have
different energy dependencies. This provides us the flexibility of
designing various transmission filters to image the desired
Fig. 3. Comparison plot of the transmission rate of electron distributions as the
function of eigen energy. The transmission rate of the STD image shows the feature
that the electron distributions of the higher energy states have relatively the
higher transmission rate. This helps STD image to highlight the valence electron
distributions (see Fig. 4(p) and (v)), comparing to the image with the undistorted
transmission rate (See Fig. 4(q) and (w)), which gives exact total projected
electron distribution map. If the transmission rate behalves close to a delta
function, the resulted image will show the electron distribution at one energy
level only. The transmission rate of the image obtained at kinematical scattering
condition (thin sample) is also included, which determines that the image take at
this condition cannot be used to visualize the valence electron distributions.
electron density by combining a series of STD images taken with
different incidence conditions. For instance, if the electron density
of one particular state needs to be highlighted (for instance:
O 2p), one can design a channel filter, that only allows the
electron density of a particular energy state to pass, as described
in the Eqs. (20)–(26). For this purpose, the characteristic transmission rate should approximate a delta function (see Fig. 3).
Although the reconstructed images generally provide only the
two dimensional projected information, one could apply a tomography technique to obtain the complete three dimensional
information, since the reconstructed images are ideally not
dependent on the sample thickness. We will not further elaborate
all these possibilities in the current paper and leave them in the
future work.
4. Simulations
In order to test if the STD method follows the design or not, an
image simulation has been carried on two structure models of
SrTiO3 with different electron distribution configurations. In the
first configuration all atoms are regarded as isolated neutral
atoms, ignoring valence electron redistribution induced by bonding. In the second configuration bonding effects are taken into
account using a multi-pole model to describe the electron transfer
and redistribution [13,14]. Thus, these two structure models are
mostly same, except slight difference of the valence electron
distribution. The exit waves, projected potential and STD images
are simulated for these two configurations along [1 0 0] direction
and respectively shown in Fig. 4.
In Fig. 4, one can hardly see the contrast difference from the
kinematical images of the two structure models, for instance, the
exit waves acquired at thin sample thickness, Fig. 4(a) and (g) or
the projected potential images, Fig. 4(f) and (l). This is expected
from Eq. (17) as the kinematical images highlight only the core
electron information and the two structure models have nearly
same core electron distributions. From the exit waves of the thick
samples, the contrast difference between the images of the two
structure models can be visualized due to the dynamical
Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919
917
Fig. 4. Comparison of Exit Waves, Projected Potential and STD images for two different electron distribution models of SrTiO3: (a)–(e) shows the exit waves of the isolated neutral
atom model at the thickness 5, 50, 150, 250, 350 Å along [1 0 0], respectively, and (f) shows the projected potential of the neutral model. A square frame outline the unit
cell and the atomic types column are only indicated at its projected position in (a) for simplicity. (g)–(k) shows the exit waves of the multipole model at the thickness 5, 50,
150, 250, 350 Å, respectively, and (l) gives the projected potential of the multipole model. (m)–(o) show the color map of STD images of the neutral model constructed from
different ways and (m) is build from a thickness series containing 500 simulated exit waves with thickness range from 1 to 500 Å (step size 1 Å) based on the Eq. (27),
representing the ideally expected STD image based on (29); (n) and (p) are constructed from 50 exit waves with the thickness randomly chosen in the range of 1–500 Å,
representing the image experimentally available. The STD image (m) is also shown in contour map (p), compared with the contour map (q) of the total electron density and
a rescaled total electron density (r), in which the core electron density is rescaled into logarithm scale and the valence electron density keeps normal linear scale.
Accordingly the STD images of multipole model are shown in (s)–(x), created using the same way to that of the neutral atom model. (colar images are used for better
viewing the details. They are originally all 8 bit gray images, with the gray level translated into the corresponding color (rainbow color map) by using in Digitalmicrograph.
The gamma is always set as 0.67 for better viewing the details.).(For interpretation of the references to color in this figure legend, the reader is referred to the web version
of this article.)
scattering, for instance, Fig. 4(d) and (j); Fig. 4(e) and (k), etc.
However, the contrast difference varies with the sample thickness
and hardly interpretable.
Among these images, the STD images can provide easily
interpretable maps of the two different valence electron distributions. For the neutral atom structure model, one can see that all
the atoms in the STD image (Fig. 4(m)–(p)) exhibits the nearly
spherical symmetry, which is the feature of isolated neutral
atoms; whereas the STD image for the multi-pole model
(Fig. 4(s)–(v)) indicates more asymmetry feature caused by the
bonding effects; for instance the asymmetry effect caused by the
dipole of the p orbital for O and the quadrupole pole of the d
orbital for Ti atom column can be seen. Note that comparing the
intensity level at the center positions of the three atom columns
(Fig. 4(m) and (s)), one can see that in the STD images the center
peak of the oxygen column is as high as the center peak of the Sr
column or the center peak of Ti+ O column; whereas in the
projected potential images the center peak of the oxygen is much
weaker than that of the others. This also implies that STD images
enhance the ‘‘weak’’ information of the light atom signal. The
enhancement of oxygen center peak can be also understood from
Eqs. (34) and (37), which provide the physical meaning of STD
image and its energy related transmission characteristic. To
briefly explain here, the center peak of each atom, can be
described by the 1s state density function of the corresponding
atom column, but modified by multiplying a weighting factor
(note this 1s state function used here is a two-dimensional eigen
function of the projected potential, close to the Bessel function,
not exactly same to the conventional 1s state of H atom). The
weighting factor is related to the eigen energy of the corresponding 1s state and influences the contrast contribution of the 1s
state to the image. Without considering the weighting factor, the
1s state density function of a heavier atom column is sharper and
higher than the 1s state function of a lighter atom column. One
could expect a higher central peak for heavier atom columns.
However, in the STD image the weighting factor is inversely
proportional to the absolute value of the eigen energy of 1s states.
It gives a larger weighting factor for the 1s state of the lighter
atom column. By taking this effect into account, the final peak
height of light atom columns in STD is adjusted to the same level
of heavy atom columns.
Note that the STD images created from the same structure
model resemble each other; despite they are calculated from
different thickness series of exit waves (see Fig. 4(m)–(o) for the
neutral model and Fig. 4(s)–(u) for the multipole model). The
corresponding contour maps are so similar that only one of them
is given for each model in the Fig. 4.
One can compare the STD images to the corresponding total
electron density images (Fig. 4(q) and (w)). From the total
electron density images, it is hard to distinguish the two different
valence electron configurations, even though three times more
contour lines are used in Fig. 4(q) and (w). The similarity of the
total electron density images is caused by that the contrast of the
image is dominated by the core electron density and the core
electron density has nearly no difference in the two structure
models. Therefore, the enhancement of high energy states information is crucial for STD images for visualizing the information of
valence electrons. There are many ways to enhance the information of the valence electrons, for instance, Fig. 4(r) and (x) shows
the contour maps of the rescaled total electron density images of
the corresponding structure models. Each rescaled electron density map is created from summing up an image representing the
logarithm of the core electron density rc and an image representing the valence electron density rv of the corresponding structure
918
Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919
model
¼
tot
Irescaled
¼ Iðlogðrc ÞÞ þ Iðrv Þ
ð38Þ
Obviously, the way of the rescaling in (38) improves the
contrast contribution of valence electrons. Thus, in Fig. 4(r) and
(x), one can see the difference of the two valence electron
distribution configurations. Note that they resemble to the
corresponding contour maps of the STD images Fig. 4(p) and (v),
though not exactly same because of the different imaging
enhancement mechanics.
From the above comparison, one may see that STD provides a
way of rescaling the density of electrons for a better visualization
of valence electron information. This rescaling is necessary but
not unique. To reach the convergence, STD requires a thickness
series with a large thickness range (in our simulation, thickness
range around 40 nm has been tested). This might hurdle its
practical application since the exit wave of thick sample
( 420 nm) can be hardly reconstructed so far. However, this is
the current limitation of exit wave reconstruction. Holography
and phase plate methods are developing. Furthermore other
better rescaling method may be designed, requiring less number
of exit waves and less thickness ranges. Towards this direction,
more efforts and investigation are required.
5. Discussion
From the high energy equation, we have built up an information transmission approach to interpret the dynamical scattering
and proposed a scheme to retrieve the information about valence
electron distribution. In order to make our approach physically
easy accessible, we have only considered the elastic dynamical
scattering and ignored absorption effects. However, all these type
of simplifications will not influence the major validity of the
information transmission approach, since decomposing the exit
waves into a series of eigen functions of the Hamiltonian is
always allowed by the general quantum mechanics and furthermore, the set of eigen functions of the Hamiltonian always
represent the intrinsic properties of the object, not thickness
dependent. It is possible to include other factors, such as inelastic
scattering. This lead to more complicated mathematics, and more
complicated way to design the filters without losing the essence.
For instance, one can consider those inelastically scattered
electrons as a kind of pseudo absorption effects [15,16], thus the
real potential U in the Hamiltonian in (11) is replaced by a
complex potential Uu. In the first order approximation, one can
write
Uu ¼ bU
ð39Þ
with
b ¼ 1 þ ia
ð40Þ
where the imaginary part a describes the absorption factor and
a 51.
It can be proved that
Enu ¼ bEn ,
ð41Þ
fnu ðRuÞ ¼ fn ðRÞ
ð42Þ
with
R
Ru ¼ pffiffiffi
b
ð43Þ
The corresponding exit wave cex
u ðzÞ, including absorption, can
then be written as
X
bEn z
cex
u ðzÞ ¼
Cn exp ip
fnu
El
n
X
En z
aEn z
Cn exp ip
fnu
exp p
El
El
n
Since a 51, the exit wave can be simplified as
X
En z
paz X
cex
u ðzÞ ¼
Cn exp ip
fnu þ
Cn En fnu
E
El n
l
n
X
En z
paz
¼
Cn exp ip
fnu þ
Uu
E
El
l
n
ð44Þ
ð45Þ
where we have applied the boundary condition at the condition of
a plane wave incidence [12].
X
Cn En fnu ¼ Uu
ð46Þ
n
From (45), one can similarly obtain the expectation value of
the corresponding standard deviation image by taking use of the
orthogonality of sine wave:
R
2
zu
zl ðcuðzÞcuðzÞÞdz
2
Istd
u Iev ¼ 9stdðcex
u ðzÞÞ9 ¼
R zu
zl dz
X
1 paðzu zl ÞUu 2
2
2
9Cn 9 9fnu 9 þ
ð47Þ
12
El
n
where zl and zu are the lower and upper integral limit of the
thickness, set by the minimum and maximum thicknesses of the
series. From (47), one can see that absorption will cause an extra
correction term in the standard deviation image. But the term is
in the second order of absorption factor a(a 5 1), thus, can be
ignored in the first order approximation.
6. Summary
We have introduced an information transmission approach to
describe the image formation in an electron microscope, both for
the lenses and for the dynamical scattering in the object. Based on
this interpretation, there is no inherent difference between the
interaction of high energy electron with the specimen and the
interaction of the electrons with the electron optics. Both of them
can be considered as information transmission process, through
the communication channels, which are the eigen functions of the
Hamiltonian. The description of the information transmission in
terms of the decomposition in different information channels,
each with its own characteristic, also enable to design a particular
filter to enhance the content of a particular channel. Based on this
interpretation, we have extended the idea of retrieving the exit
wave from HREM images to retrieving the object from the exit
wave. By applying a so called ‘‘standard deviation method’’, the
distribution of valence electrons, though scrambled in the strong
dynamical scattering, can be enhanced in the image reconstructed
from a thickness series of exit waves. This approach is not limited
to the application of imaging the electron distribution. It can be
also extended to obtain other desirable properties of the object as
well as to design various suitable methods for acquiring them.
Acknowledgement
The authors are grateful for the financial support by FOMprogram 08IP05 in Netherlands and FWO-project G.0188.08 in
Belgium.
References
[1] J.M. Zuo, M. Kim, M. O’Keeffe, J.C.H. Spence, Nature 40 (1999) 49.
[2] J. Yimei Zhu, Tafto Philos. Mag. B 75 (1997) 785–791.
Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919
[3] M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius, K. Urban, Nature 392
(1998) 768.
[4] K.W. Urban, Science 321 (2008) 506–510.
[5] S.J. Haigh, H. Sawada, A.I. Kirkland, Phys .Rev. Lett. 103 (2009) 126101.
[6] D.B. Williams, C.B. Carter, Transmission Electron Microscopy, 2nd, Plenum
Press, New York, 2009.
[7] W. Coene, G. Janssen, M. Op de Beeck, D. van Dyck, Phys. Rev. Lett. 69 (1992)
3743.
[8] W.D. Orchowski, Rau, H. Lichte, Phy. Rev. Lett. 74 (1995) 399.
[9]
[10]
[11]
[12]
[13]
[14]
919
L. Jia, A. Thust, Phys. Rev. Lett. 82 (1999) 5052.
J.H. Chen, E. Costan, M.A.V. Huis, Q. Xu, H.W. Zandbergen, Science 21 (2006) 312.
D.Van Dyck and Mop de Beeck, Ultramicroscopy, 64, 99 (19996).
D. Van Dyck, J.H. Chen, Solid State Commun. 109 (1999) 501.
N.K. Hansen, P. Coppens, Acta Cryst. A34 (1978) 909–921.
T. Lippmann, P. Blaha, N.H. Andersen, H.F. Poulsen, T. Wolf, J.R. Schneider,
K.H. Schwarz, Acta Cryst. A59 (2003) 437–451.
[15] H. Yoshioka, J. Phys. Soc. Jpn. 12 (1957) 618.
[16] D. Van Dyck, H. Lichte, J.C.H. Spence, Ultramicroscopy 81 (2000) 187–194.