Image formation mechanisms in scanning electron microscopy of carbon nanotubes, and retrieval of their intrinsic dimensions.
01 Jan 2013-Ultramicroscopy (Elsevier)-Vol. 124, pp 35-39
TL;DR: It is shown how SEM images can be modelled by accounting for surface enhancement effects together with the absorption coefficient for secondary electrons, and the electron-probe shape, enabling retrieval of the intrinsic nanotube dimensions.
Abstract: We present a detailed analysis of the image formation mechanisms that are involved in the imaging of carbon nanotubes with scanning electron microscopy (SEM). We show how SEM images can be modelled ...
This is the submitted version of a paper published in Ultramicroscopy.
Citation for the original published paper (version of record):
Jackman, H., Krakhmalev, P., Svensson, K. (2013)
Image formation mechanisms in scanning electron microscopy of carbon nanotubes,and retrieval
of their intrinsic dimensions..
Ultramicroscopy, 124: 35-39
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N.B. When citing this work, cite the original published paper.
Permanent link to this version:
Image formation mechanisms in scanning electron microscopy of carbon nanotubes,
and retrieval of their intrinsic dimensions.
, P. Krakhmalev
, K. Svensson
Department of Physics and Electrical Engineering, Karlstad University, SE-651 88 Karlstad, Sweden
Department of Mechanical and Materials Engineering, Karlstad University, SE-651 88 Karlstad, Sweden
We present a detailed analysis of the image formation mechanisms that are involved in the imaging of carbon nanotubes
with scanning electron microscopy (SEM). We show how SEM images can be modelled by accounting for surface enhance-
ment eﬀects together with the absorption coeﬃcient for secondary electrons, and the electron-probe shape. Images c an
then be deconvoluted, enabling retrieval of the intrinsic nanotube dimensions. Accurate estimates of their dimensions
can thereby be obtained even for structur es that are compa rable to the electr on-probe size (on the order of 2 nm). We
also present a simple and robust mo del for obtaining the outer diameter of nanotubes without any detailed knowledge
about the electron-probe shape .
The development of scanning probe microscopy (SPM)
instruments combined with sca nning electron microscopy
(SEM) and transmission electron microscopy (TEM) en-
ables free standing nanos tructures (such as carbon nan-
otub es) to be studied in terms of their mechanical and
electrical properties. The microscope is mainly used to ob-
tain structural information while the SPM instruments are
used to perform the material analysis by applying forces or
electrical currents . The space available inside SEM in-
struments is comparatively large and several probes can
be used . In TEM the space is much more limited
and usually only one SPM probe is ﬁtted , while two
probes have also recently been demonstrated . One ma-
jor drawback with the SEM is that the resolution is limited
by the electron-probe shape and the imaging mechanisms.
Nanoscale structures are thereby substantially broadened
An accurate measure of the nanotube dimensions is
crucial for the analys is of mechanical, and electrical, prop-
erties. One route is then to combine the SEM in-situ anal-
ysis with regular TEM imaging [4, 5]. The material test-
ing would then ﬁrst be performed inside an SEM, and the
sample of interest subsequently transferred into a TEM
to obtain the detailed structural informatio n. Apart from
providing only post-mortem information, the method is
also diﬃcult, as the exact sample loc ation and orienta-
tion should prefera bly be analyzed ins ide both the SEM
and TEM. This r equires a transfer o f the whole sample,
or SPM probe, with remnant carbon nanotubes on it .
Email address: firstname.lastname@example.org (H. Jackman)
Alternatively one can also tr ansfer parts of the analyzed
carbon nanotube onto a regular TEM-grid for subsequent
TEM analysis . It would instead be preferable to do all
of the analysis inside an SEM instrument .
The smallest electron-probe size in a modern, ﬁeld emis-
sion gun (FEG) SEM can be of the order of 1 nm a t
optimum settings , and individual car bon nanotubes
can thereby fairly easy be imaged, but the SEM shows
a much broader structure tha n the true one (as obtained
from TEM imaging). The problem is illustrated in Fig. 1
where the same carbon nanotube has be e n imaged both
in an SEM (Fig. 1 (a)) and a TEM (Fig. 1 (b)). If the
broadening in the SEM imag e ca n be taken accounted for,
then the whole analysis could be done from o nly in-situ
SEM expe riments.
In this work we show how the SEM image formation
can be fully modelled for nanoﬁbres , such as carbon nan-
otub es, by a convolution of the secondary electron yield
with the electron-probe shape. With a known electron-
probe shape the SEM image can then b e deconvoluted and
reveal more details. We also present an easy way to obtain
the outer diameter of carbon nanotubes ima ged in SEM,
without doing the full dec onvolution. The simple model
is fairly robust with respect to the detailed electron-probe
shape and enables an accurate estimation of the nanotube
outer diameter. A full investigation can thereby be per-
formed inside the SEM without the need for complemen-
tary TEM imaging.
2. Materials and Methods
2.1. Sample preparation
Two types of commercial CNTs have been studied:
NC2100 and NC2101, obtained from Nanocyl. Both types
Preprint submitted to Elsevier April 20, 2012
−6 −4 −2 0 2 4 6
Position, r [nm]
Figure 1: Images of the same CNT acquired by (a) SEM and (b)
TEM. (c) Shows the integrated intensity proﬁles from the two rect-
angles in (a) and (b), and the diameter estimated from the TEM
proﬁle is indicated.
were produced by catalytic chemical vapor deposition (CCVD)
with the diﬀerence being that NC2101 was functionalized
with a carboxyl group (-COOH) to reduce bundling. The
CNTs are marketed as double-walled with an average di-
ameter of 3.5 nm and lengths varying between 1-10 µm.
As received CNTs were ﬁrst dispersed in ethano l and then
sonicated for 15 minutes to reduce bundling. Sonication
for lo nger periods of time can introduce de fects so that
the concentric cylinder structure is broken . No dam-
ages could be seen in TEM images of the CNTs due to the
The solution containing the CNTs was dro p-casted onto
a holey carbon support ﬁlm for TEM (R 2/1 produced by
Quantifoil) and then allowed to dry. CNTs were in this
way distributed uniformly over the ﬁlm enabling imaging
of single CNTs without any underlying substrate.
The samples were ﬁrst studied in a JEOL (JEM 210 0)
TEM equipped with a LaB
cathode and a digital camera
from Gatan (SC1000 Orius). Single CNTs, with well de-
ﬁned concentric cylinder structure and with no amorphous
carbon deposits, were chosen and imaged. The acceler-
ation voltage used was 100 kV, since imag ing at higher
voltages can c ause dama ge to the CNTs even if they a re
imaged for short periods of time . Even at 100 kV the
CNTs ca n b ecome damag ed if subjected to a high current
density beam for longer times. Therefore the current den-
sity of the beam and the time of exposure to the beam
were minimized so that the CNTs did not g et damaged.
The best performance of the TEM is obtained at the
Scherzer defocus . One result of this defocus is that
bright Fres ne l fringes appear at the edges o f the CNT,
as shown in Fig. 1 (b) and (c). From intensity proﬁles,
obtained from TEM images, the diameter, d
, was esti-
mated by taking the distance between the points where the
bright Fresnel contrast returns to background intensity, as
shown in Fig. 1 (c). The width of the dark regions varies
with focus, but for small deviations from the Scher zer de-
focus they stay approximately constant . Estimating
the diameter this way yields the width of the CNT that
interacts with the electron beam, both in SEM and TEM,
i.e the outer diameter. The error whe n estimating the di-
ameter o f a CNT from TEM images is less than 10% for
diameters larger than 1 nm . Since all CNTs used in
this study had d > 1 nm, the diameters measured in the
TEM should deviate less tha n 10 % from the true diameter.
After the CNTs had been imaged in the TEM they
were imaged in a LEO 1530 FEG-SEM. The samples were
mounted in a custom ma de scanning transmission electron
microscopy holder enabling imaging of the CNTs with-
out any contribution from an underlying substrate. T he
electron-probe size and the secondary electron yield bo th
decrease with increasing acceleration voltage . A higher
acceleration voltage thereby improves the resolution but
decreases the signal-to-noise ration in the SEM image. An
acceleration voltage of 12 kV was found to be a good trade-
oﬀ between resolutio n and signal-to-noise ratio and was
used when acquiring all images in this study.
Electron beam-induced deposition (EBID) can be a
problem during SEM analysis and it will increase the di-
ameter of the CNTs . Such eﬀects were minimized
here by leaving the samples in the SEM chamber for at
least 10 hours, r e sulting in a chamber pressure down to
mbar, befor e exposing them to the electron beam.
To check if EBID had occurred in the SEM, all CNTs were
imaged one more time in the TEM and the EBID was
found to be ne gligible.
An SEM image can be thought of as the detected sec-
ondary electron (SE) intensity as a function of the lateral
coordinates, I(r). This can be described as a convolution
of the SE yield at each sample position, δ(r), with the
electron-probe shape, i(r):
I(r) = [δ ∗ i](r) =
)i(r − r
At low magniﬁcations, the inﬂuence of i(r) on the recorded
image can be neglected, while at higher magniﬁcations i(r)
will smooth out any sharp sample details. The exact shape
of i(r) is g enerally unknown and depends on a number of
parameters in the micros cope. Estimation of CNT diame-
ters can be done on integrated intensity proﬁles, as shown
in Fig. 1 (c). In these intensity proﬁles, a radial s ymmet-
ric i(r) can be described as one dimensional and Gaussian
and Lorentzian functions can be used to describ e i(r). An
area normalised Gaussian function is given by:
where the full width at half maximum (Γ) is equal to Γ
2 ln 2σ , while a n area normalised Lorentzian function is
is the full width at half maximum.
The SE yield from a homogeneous material depends on
the generation and scattering of SE. Three types of SE are
generated inside an SEM: SE1 , SE2 and SE3 . Electrons
labelled SE1 are excited directly by the primary beam,
SE2 are excited by backscattered electrons (BSE) near the
specimen surface, and SE3 are excited by backscattered
electrons outside the specimen. When the e xperimental
SEM images wer e obtained in this study an in-lens de-
tector was used which eﬀectively excludes SE3 , and
the contribution from SE3 was neglected when modelling
I(r). In order for SE2 to be emitted, there must be a
considerable amount of backscattered electrons from the
incoming beam. Considering CNTs with a diameter up
to about 10 nm, an insigniﬁcant frac tion of the incident
beam will be backscattered, and the SE2 contribution was
also neg le cted here. The generation of SE1 is assumed to
be propo rtional to the energy loss of the incoming elec-
trons [14, 7]. Having a thin specimen with a low average
atomic number, such as a CNT, the incident electrons will
lose a negligible amount of energy travelling thro ugh the
specimen. This makes the probability o f generating SE
approximately constant throughout the thickness o f the
For SE to be detected they ne e d to escape the speci-
men and reach the detector. Because of their low energy,
the SE are more easily absorbed than the incoming high-
energy electrons. The probability of escaping a specimen
decreases exp onentially with the distance travelled in the
with α being the abso rption coeﬃcient fo r SE of the speci-
men material. Combining the relations for ge neration and
escape of the SE an expression for the SE yield, as a func-
tion of the thickness t , was derived in :
1 − e
(1 − αt) − (αt)
where the thickness can be a function of the lateral coor-
dinates, t = t(r).
For a ﬂat specimen of constant tilt ang le , φ, Eq. 5
describe the SE yie ld. However for a CNT, φ w ill vary
across its surface and δ incre ases with increasing φ. The
relation between δ and φ is given approximately by :
∝ sec φ (6)
where the tilt angle can be a function of the lateral co-
ordinates, φ = φ(r). For a spe cimens with low average
atomic number, Z, the dependence on φ becomes more
rapid than sec φ . On the other hand, the dependence
also becomes slower with lower V
, so although carbon
has a low Z, Eq. 6 is a reasonable approximation of the
depe ndence of δ on φ since we us e d V
= 12 kV in this
By combining Eq. 5 and 6, the total SE yield is given
(r) = δ
−6 −4 −2 0 2 4 6
Position, r [nm]
−6 −4 −2 0 2 4 6
Position, r [nm]
Figure 2: (a) The total SE yi el d, δ
, for two diﬀerent α-values
(solid lines), calculated for the CNT dimensions indicated by the
dashed lines. (b) Two simulated intensity proﬁles, I
, for the CNT
dimensions in (a) using i
nm (for α = 1/20 nm
To obtain δ
(r) for a carbon nanotube we need a
value for the absorption co eﬃcient α. Experimental data
i scarce, but α may be as low as 1/20 nm
and around 1/10 nm
for polymers . In Fig. 2 (a) we
show the obtained δ
for both values of α. T he inﬂuence
of α on δ
is small and we have used the value fo r carbo n
) for the remaining analysis.
Simulated intensity proﬁles, I
(r), were obtained through
a convolution of δ
, where linear combinations
were used to describe the probe shape i
The convolution was done by sweeping a function of the
probe shape, i
), over the total SE yield, δ
and taking the integral of the product of these two func-
(r) = [δ
Ultimately the integration above s hould be performed in
the interval [−∞, ∞], but since the equations describing
i(r) rapidly goes to zero we used the interval [r−5Γ, r+5Γ]
to reduce the computational intensity. The integration in
Eq. 8 was performed numeric ally in MATLAB, were both
) and δ
) were divided into small elements.
Pair of e le ments were multiplied and the products summed
together in each step. An example of two simulated inten-
sity proﬁles is shown in Fig. 2 (b), for purely Gauss ian
and Lorentzian probe shapes.
This method is not limited to CNTs, as it can be ap-
plied on any homogeneous nanoﬁber structure. In the case
of ﬁlled nanotubes it can easily be used by adding the δ
contribution of the ﬁller to Eq. 7.
3. Results and Discussion
The inner and outer diameter of the CNTs obtained
from the TEM images was used as parameters when sim-
ulating intensity proﬁles, I
. Good agreement between
and experimental SEM intensity proﬁles (I
obtained whe n using a c ombination of i
the probe shape (with Γ
= Γ). On average a goo d
agreement was fo und when:
A result of such a simulation is shown in Fig. 3 (c), where
the SEM proﬁle of the CNT seen in Fig. 3 (b) was simu-
lated. The average Γ from all analysed proﬁles was found
to be 2.05 ± 0.05 nm.
The probe shape that worked best for reproducing I
should not be seen as an instrumental constant, since i(r)
depe nd on a number of SEM parameters. Therefore i(r)
likely varies between diﬀer ent SEM instruments and even
for a certain SEM as the experimental conditions are al-
tered, by changes in foc us and a stigmatic corr e ctions. Know-
ing the exact probe shape, one can deconvolute a full SEM
image to obtain the SE yield, and in turn from that re-
trieve the intrinsic na notube dimensions. We have used
the two dimensional version of Eq. 9 as the input point
spread function in the MATLAB-function deconvblind to
deconvolute SEM images. The r e sult o f one such decon-
volution is shown in Fig. 4, where Fig. 1 (a) was used as
the raw SEM-image.
Even though the deconvoluted image in Fig. 4 (b)
contains more deta il, it is still quite far from the sharp
function describing δ in Eq. 7 (as can be seen in Fig. 4 (c)).
One reason that the deconvolution cannot fully retrieve
the underlying str uctures, is the image no ise in the shape
−6 −4 −2 0 2 4 6
Position, r [nm]
Figure 3: Images of the same CNT acquired by (a) TEM and (b)
SEM. In (c) the SEM proﬁle, I
(·), is plotted along with three
simulated intensity proﬁles, obtained by using diﬀerent probe shapes
= 2.0 nm).
of streaks. This may be the res ult of vibrations or other
sources of instabilities, s uch as external electromagnetic
ﬁelds. Another reason is that Eq. 9 is an approximation
of the exact probe shape i (r). Other attempts to describe
the probe shape assumes a Gaussian probe shape and fails
to determine probe sizes below than 2 nm .
Instead of having to rely on the exact shape of i(r),
we have looked for an ea sier way to extract the outer di-
ameter of the nanotubes which is an important parameter
when studying the mechanical properties. We have pr evi-
ously shown, from empirical studies, that the width of the
SEM proﬁle can be used to estimate the outer diameter
of the nanotubes . Here we investigate this further by
using our knowledge of the image formation mechanisms.
The electron-yield from a nanotube has a sharp step at
the positions of the outer edges (see Fig. 2 (a)) which are
the positions that will give us the outer diameter. If we
look at the convolution of a step function, H(r − r
an arbitrary symmetric probe shape with a distinct maxi-
(r) = [H ∗ i](r), the second derivative,
becomes zero when the maximum of i(r) meets the edge
of the step function at r
. If the SE yield of a CNT would
be constant across the width, then the diameter would
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