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Chapter 1 Atomic Resolution Transmission Electron Microscopy 55
ψ
λ
π
σψxyz z
i
ziVxyzxyz
xy z
,, ,, ,,+
(
)
=∇+
(
)
(
)
∆∆
∆
exp
4
2
(66)
where V
∆z
is the projected specimen potential between z and z + ∆z:
Vxyz Vxyz dz
z
z
zz
∆
∆
,, ,,
(
)
=
′
(
)
′
+
∫
(67)
The two operators in the exponent in Eq. (66) do not commute
and therefore this function cannot be rewritten as a product of two
exponential functions. However, as both operators are small (to the
order of ∆z), the approximation
eee BA ee
AB AB ABεε ε ε εε
ε
εε
+
=+ +=+
2
32
2
[]() (), OO
(68)
can be used.
This leaves the expression
ψ
λ
π
ψxyz z
iz
txyz xyz
xy
,, ,, ,,+
(
)
=∇
(
)
(
)
(
)
∆
∆
exp
4
2
(69)
where t(x, y, z) is the transmission function for the specimen slice
between z and z + ∆z:
t(x, y, z) = exp[iσV
∆z
(x, y, z)] (70)
The fi rst operator can be further separated into components for x and
y coordinates as
exp exp exp
iz
i
x
i
y
xy
λ
π
αα
∆
4
2
2
2
2
2
∇
(
)
=
∂
∂
∂
∂
(71)
where α = λ∆z/4π.
Considering the Fourier transform of the operator for the x direction
applied to a function f(x):
FT i
x
fx e i
x
fxdx
kx
exp expαα
π
∂
∂
()
=
∂
∂
()
−
∫
2
2
2
2
2
(72)
=
∑
∫
−
()
!
()
()
i
n
efxdx
n
kx n
α
π22
(73)
=−
∑
∫
−
()
!
() ()
i
n
kefxdx
n
nkx
α
π
π
2
22
(74)
= e
−4iπ
2
k
2
α
FT[f(x)] (75)
The above can be repeated for the y direction, fi nally yielding
FT
iz
fxy i zk k FTfxy
xy x y
exp exp
λ
π
πλ
∆
∆
4
222
∇
(
)
(
)
=− +
()
[]
(
)
,,
[[]
(76)
Using this result a single multislice step can be written as
ψ(x, y, z + ∆z) = FT
−1
{P(k
x
, k
y
)FT [t(x, y, z)ψ(x, y, z)]} (77)
where the real-space transmission function, t(x, y, z), and the Fourier
space propagation function, P(k
x
, k
y
), are given by
56
t(x, y, z) = exp[iσV
∆z
(x, y, z)] (78)
P(k
x
, k
y
) = exp[−iπλ∆z(k
2
x
+ k
2
y
)] (79)
It should be noted that the propagator has the same form as a wave
aberration with a defocus of ∆z.
The algorithm can be implemented very effi ciently using FFTs and
is particularly suitable for simulations of crystals, as only one unit
cell has to be calculated and the periodic boundary conditions
necessary for the Fourier transforms are automatically fulfi lled. The
density of sampling points in the unit cell determines the maximum
spatial frequency in the Fourier transform and therefore the maximum
number of diffracted beams included in the simulation. To avoid
aliasing artifacts, it is also necessary to exclude all beams above a limit
less than the Nyquist frequency in the propagation step (Kirkland,
1998).
Although the multislice method provides an accurate and computa-
tionally tractable method for calculating HRTEM images the approxi-
mations made in the above derivation can, under certain conditions,
limit the accuracy achieved.
The number of beams included in the simulation is ultimately limited
by the discrete sampling of the unit cell. Unlike the Bloch wave approach
(Bethe, 1928), which gives self-consistent results even with a relatively
small number of beams, the multislice method is reliable only when
scattering into beams that are not included in the simulation is
negligible. For any given simulation, this latter effect can be tested
by verifying that the total intensity of the wavefunction does not
decrease signifi cantly as the wave propagates through the specimen.
A decrease of 5% over the complete specimen is considered acceptable
(Kirkland, 1998), but with modern computers it is possible to choose a
suffi ciently large number of beams to give a loss of intensity smaller
than 0.1%.
The paraxial approximation implies the neglect of the term
i
z
λ
π
ψ
4
2
2
∂
∂
(80)
A wave scattered by an angle, θ, toward the x direction can be
written as
ψ = e
2πk[sin θx+(cos θ−1)z]
(81)
with k = 1/λ, hence
∂
∂
<
2
2
224
ψ
πθ
z
k
max
(82)
Therefore, the error due to the paraxial approximation accumulated
over a specimen thickness, t, can be estimated as
∆ψ
λ
π
ψπ
λ
θ<
∂
∂
<
t
z
t
44
2
2
4
max
max
(83)
This term is vanishingly small for typical situations in HRTEM
(e.g., E = 300 keV, λ = 2 pm, t = 20 nm, θ
max
= 20 mrad yields ∆ψ =
A.I. Kirkland et al.
Chapter 1 Atomic Resolution Transmission Electron Microscopy 57
0.16%) and therefore any error introduced by the paraxial approxima-
tion can generally be neglected.
In principle, the error due to the paraxial approximation can be made
arbitrarily small by making the slices suffi ciently thin. However, as the
potential of each atom is usually projected into the slice that contains
the atom center, the accuracy is still limited when propagation over the
range of the atomic potential has a noticeable effect. In this case, the
atomic potential has to be divided across several slices.
A useful feature of the exit-plane wavefunction calculated using the
multislice method is that it depends qualitatively on the local potential
of the crystal within a small cylinder whose axis is aligned to the inci-
dent beam direction with a diameter of typically a few nanometers for
HRTEM imaging conditions. This suggests that HRTEM images of
defects within a crystalline supercell can be simulated without per-
fectly smooth periodic continuation of the crystal potential at the
supercell boundaries, as any abrupt changes in the potential will infl u-
ence the exit-plane wavefunction only laterally, within the radius of the
above cylinder. Thus, simulated images of large scale defects can be
separately computed in a montage of “patches” and subsequently
joined (Olsen and Spence, 1981) provided the small allowance described
is made at the borders for discontinuities in the potential. Signifi cantly
this is not the case for simulations of dynamic electron diffraction pat-
terns from equivalent structures and these cannot be calculated using
this “patching” approach.
Finally we observe that the derivation of the multislice algorithm
described here is based on the Schrödinger equation, whereas a fully
relativistic treatment would require the use of the Dirac equation.
However, it has been shown (Fujiwara, 1962; Moodie et al., 2001) that
provided relativistically correct expressions are used for both electron
mass and wavelength, the expressions derived from the Schrödinger
equation are an extremely good approximation of those derived from
the Dirac equation.
6 Conclusions and Future Prospects
The current status of HRTEM is such that instrumentation, theory, and
computational tools are available for structural determination of a
wide range of technologically important materials at atomic resolution.
However, there still remain a number of signifi cant experimental and
theoretical challenges within this fi eld.
From a theoretical standpoint the majority of HRTEM images are
still matched with simulations qualitatively, and even in instances in
which quantitative metrics are applied a relatively poorly understood
contrast mismatch remains (Boothroyd, 1998). There are also the con-
tinuing requirements for refi nement and extension of current theoreti-
cal frameworks to accommodate the increasing resolution and precision
in the data available. Similarly, almost all HRTEM data are recorded
for a limited number of projections and therefore access to full three-
dimensional information is restricted.
58
Instrumentation for HRTEM continues to develop and the relatively
recent development of monochromators, aberration correctors, and
energy fi lters represent substantial advances. However, there remains
considerable scope for improvements in detectors and the correction
of chromatic aberration has not been realized experimentally at the
time of writing.
Therefore, at present, the “grand challenge” is still the fully quantita-
tive determination of an unknown structure from HRTEM data, quan-
titatively matched to accurate image simulations and with suffi cient
resolution to determine the coordinates of all atom species in three
dimensions.
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