CARBON
8 0 ( 2 0 1 4 ) 5 0 –5 8
Available at www.sciencedirect.com
ScienceDirect
journal homepage: www.elsevier.com/locate/carbon
Low energy electrons focused by the image charge
interaction in carbon nanotubes
Samuel A. Hevia
a
b
c
a,*
,
Rodrigo Segura b, Patricio Häberle
c
Instituto de Fı́sica, Pontificia Universidad Católica de Chile, 6904411 Santiago, Chile
Instituto de Quı́mica y Bioquı́mica, Facultad de Ciencias, Universidad de Valparaı́so, Av. Gran Bretaña 1111, Valparaı́so, Chile
Departamento de Fı́sica, Universidad Técnica Federico Santa Marı́a, 2390123 Valparaı́so, Chile
A R T I C L E I N F O
A B S T R A C T
Article history:
Electrons with energy in the range of a few eV are strongly affected by the interaction with
Received 6 March 2014
the polarization charges they induce on a surface. This report shows how this effect is rel-
Accepted 12 August 2014
evant for the data analysis of inverse photoemission spectroscopy (IPS) from carbon nano-
Available online 23 August 2014
tube (CNT) arrays. IPS from CNTs exhibit two main resonances, located around 2.5 eV and
12.5 eV above the Fermi level. The intensity of the first resonance is dependent on the average tube diameter and the second one has a distinctive spectral shape, which is related to
the graphitization level of the CNT external walls. In order to analyze the origin of these
resonances, a phenomenological reconstruction of an IPS spectrum from a CNT collection
was performed. This reconstruction successfully reproduces the spectral shape of the
12.5 eV resonance. However, the intensity is lower than the actual measurements in the
initial energy range of the spectrum. The analysis of these results suggests that the additional intensity required to reproduce the experimental data, has its origin in an electronic
focusing mechanism, induced by the CNT image charge potential. This effect is significant
for low energy electrons and small diameter tubes.
Ó 2014 Elsevier Ltd. All rights reserved.
1.
Introduction
Carbon nanotubes (CNTs) have received a widespread attention due to their interesting physical properties [1] and the
promising new applications in a variety of systems [2,3].
Precisely, the interaction of probe particles such as electrons
or photons with CNTs and other graphene based nanostructures, is of current scientific interest due mainly to their
potential applications in sensors and devices [4,5].
Electron spectroscopies have been widely used to examine
the electronic structure of CNTs below the Fermi level (EF)
[6–12], however, there are only few experimental reports
dealing with states above this energy level [2,11–13]. Some
unoccupied electronic states, which are not linked to the
* Corresponding author.
E-mail address: [email protected] (S.A. Hevia).
http://dx.doi.org/10.1016/j.carbon.2014.08.023
0008-6223/Ó 2014 Elsevier Ltd. All rights reserved.
band structure, are the image charge states [14]. These type
of surface states in CNTs are generated by the interaction
between an external electron and the polarization charges it
induces on a nanotube [15,16]. Thus, this long range potential
allows the formation of extended unoccupied electronic
states around a CNT [17–19]. Additionally, as shown in subSection 3.1, when unbound electrons approach the vicinity
of a nanotube, as they would in inverse photoemission spectroscopy (IPS), this attractive potential bends their trajectories
inducing a focusing effect around the tubes.
This report presents results from a study of the unoccupied electronic states of single wall carbon nanotubes
(SWCNTs) and multiple wall carbon nanotubes (MWCNTs)
using IPS. Samples were prepared as non-oriented CNT
CARBON
8 0 (2 0 14 ) 5 0–58
51
collections, with well defined mean diameters and different
levels of graphitization. Under electronic irradiation these
samples emit photons by the optical decay of electrons
between two originally unoccupied electronic states. Two
main resonances in the intensity of the outgoing photons
were detected in all CNT samples. A correlation between
the intensity of the peak and the mean diameter of the tubes,
was observed for the resonance at 2.5 eV above EF. In Section
3, we discuss the evidence linking the focusing effect with an
enhanced transition probability into the p bands. The
enhancement is indeed significant for small CNT diameters
and low energy incident electrons. In addition to this low
energy peak, a wide resonance located close to 12.5 eV was
also found. This feature has a distinctive spectral shape
which can be correlated with the graphitization level of the
CNTs. The origin of these two resonances has been elucidated
by performing a phenomenological reconstruction of an IPSCNT spectrum, based on measurements obtained from highly
oriented pyrolytic graphite (HOPG).
2.
Experimental
2.1.
Sample characterization
For this study eight different CNT samples have been considered. Each one of them has been labeled with an acronym
indicating some physical characteristics of the nanotubes
they contain, and if it is the case, the treatment to which they
have been subjected. In the rest of this section, we describe
the microscopic structure of these samples with the purpose
of correlating the IPS data with the main physical parameters
of each sample.
Fig. 1 displays scanning electron microscopy (SEM) images
and Fig. 2 shows transmission electron microscopy (TEM)
images from the different samples. Fig. 1(a) is a side view of
the ‘‘MWS50’’ sample which is formed by ‘‘MWCNTs’’ with a
‘‘Straight’’ profile and average diameter close to ‘‘50’’ nm
(see Fig. 2(a) and (b)). They were synthesized by pyrolysis of
Iron Phthalocyanine at 1000°C over a Si substrate [20]. These
MWCNTs grow away from the substrate up to lengths close
to 20 lm, however at the top of the sample the nanotubes
bend laterally as shown in Fig. 1(b). The sample labeled as
‘‘MWS50-IM’’ was prepared by Ion Milling of sample MWS50,
with 1 keV argon ions during one hour. The ion current on
the sample was 10 lA per cm2. Under this dose the CNTs in
the samples present extensive damage. Fig. 1(c) and (d) show
the corresponding micrographs before and after the irradiation procedure.
Samples labeled as ‘‘MWH50’’ and ‘‘MWH10’’ contain CNTs
synthesized by the catalytic decomposition of acetylene at
800 °C and 700 °C over Fe covered SiO2 substrates [21]. Average
diameters close to 50 nm and 10 nm were obtained for these
tubes (see Fig. 2(c) and (e)). As shown in Fig. 1(e) and 1(g),
the CNTs display the characteristic helical distortion common
to many tubes grown from acetylene. Notice that letters ‘‘S’’
and ‘‘H’’, in the sample labels, have been used to describe
the: ‘‘straight’’ or ‘‘helicoidal’’ shape of the MWCNTs.
A relevant characteristic of a wide diameter CNT is the
degree of graphitization of their walls. TEM was used to
Fig. 1 – Scanning electron microscopy images of different
samples used in the course of this study. (a) Is a side view
and (b) is a top view image from the sample labeled as
MWS50 (see text for sample labels). Images (c) and (d) are
respectively 45 degree micrographs of the sample MWS50
before and after an ion milling treatment. Images (e) and (f)
are a side and top views of sample MWH50. Image (g)
corresponds to a magnified side view of the tubes contained
in sample MWH50, where the helical character of the tubes
is clearly distinguished. Image (h) is a top view of sample
MWS50-F. The tubes were removed from the synthesis
substrate and dispersed on a silicon chip. Image (i)
corresponds to a top view of sample MWS65, see text for
further details. (A color version of this figure can be viewed
online.)
identify this feature together with the morphology of the
CNTs and the corresponding average diameters of the tubes
in each sample. As can be verified in the higher magnification
images of Fig. 2, the sample labeled as MWS50 contain CNTs
52
CARBON
8 0 ( 2 0 1 4 ) 5 0 –5 8
with a higher degree of graphitization than those contained in
samples MWH50 or MWH10 (see Fig. 2(b) and (d)), which were
grown by different synthesis method.
Samples labeled as ‘‘MWS65’’, ‘‘MWS65-A’’, ‘‘MWS50-F’’ and
‘‘SW1.3’’ were prepared by dropping the corresponding CNTs
on a silicon chip. The CNTs in MWS65 (Fig. 1(i)) were synthesized by the decomposition of acetylene at 650 °C, using a porous alumina membranes as template [22]. These 65 nm
diameter tubes are shown in Fig. 2(g) and (h). A fraction of
these CNTs were ‘‘annealed’’ up to 2600 °C to prepare sample
MWS65-A. This treatment greatly increases the graphitization
level of the walls while the radius of the tubes remain almost
unaltered, as shown in Fig. 2(i) and (j). Sample MWS50-F,
shown in Fig. 1(h), was prepared from the same tubes in sample MWS50, but in this case they lay mostly flat over the substrate. Finally, SWCNTs with 1.3 nm diameter,1 shown in
Fig. 2(f), were used to prepared sample SW1.3. All these samples were warmed up to 200 °C in an inert atmosphere prior to
their introduction into the spectrometer chamber.
We have also performed Raman characterization of all
samples. The information derived from the Raman spectra
is consistent with the TEM images, in particular with regards
to the graphitic character of the different samples. These data
will be reported elsewhere 2.
2.2.
Inverse photoemission spectroscopy
IPS measurements were performed in ultra high vacuum
(UHV) conditions (base pressure 1010 Torr) in a home built
isochromat spectrometer [23] with a resolution of 0.4 eV.
The value of EF was calibrated using an Al (100) crystal. All
samples were annealed up to 350 °C in UHV to remove humidity and other adsorbates, but measurements were carried out
at room temperature. The incident electron current used for
these measurements was around 0.5 lA. Since the electron
beam shines an approximate area of 1 mm2, the current density on the samples is approximately one electron per square
nanometer per second. The incident beam energy was varied
between 6 eV and 30 eV. After several measuring cycles no
signs of sample damage caused by the electronic irradiation
was observed.
Fig. 3 shows IPS spectra measured in normal incidence
from different samples. All of them display two main resonances, which have been labeled as ‘‘A’’, close to 2.5 eV, and
‘‘B’’, around 12.5 eV above EF.
The intensity of A is in general higher for the smaller tube
diameters. An absolute maximum intensity for this resonance is seen for the very narrow tubes in sample SW1.3.
The comparison of the A intensities between MWS50 and
MWS50-F, is indeed a bit puzzling at first, since both samples
contain the same tubes and they should exhibit the same
intensity for A.
Resonance B is clearly better defined in the IPS spectra
from samples: MWS50, MWS50-F, MWS65-A and SW1.3, compared to spectra from helicoidal tubes and other samples. The
four samples, mentioned above, contain more graphitic and
1
2
less amorphous carbon in the tube walls, as it has been verified by the higher resolution TEM images shown in Fig. 2. It is
interesting to notice the effect of the post synthesis annealing
treatment of the CNTs prepared in porous alumina. The spectrum from sample MWS65-A shows an improved definition of
both main resonances compared with MWS65, particularly in
resonance B. On the other hand, the in situ ion milling of
sample MWS50 induces substantial changes in the IPS spectrum. Both A and B can be hardly detected after this procedure (MWS50-IM).
Further information on the origin of the observed resonances can be obtained by comparing the IPS-CNT data with
spectra taken from graphite [24,25]. Based on measurements
performed on HOPG, the details on how to perform this comparison are developed in the following section.
3.
Discussion
An approach to correlate IPS intensities measured from CNTs,
with the underlying electronic structure of the honeycomb
carbon lattice, is to design a procedure to reconstruct a spectrum from a CNT collection based on measurements performed on HOPG. However, this process requires first
establishing the way in which the tube cylindrical shapes
modulate the spectral response by modifying the electronic
trajectories.
3.1.
Electronic focusing
Unoccupied image states in CNTs have been found experimentally and described theoretically through a model that
provides both the wave functions and energy spectra of these
resonances [15,17–19]. The interaction responsible for the
existence of these states, is the attractive image charge
potential [16], which exists between a neutral CNT and an
external charged particle. The long range character of this
interaction not only allows the emergence of these bound
unoccupied states but it can also explain how the trajectories
of nearby unbound electrons are modified.
The image potential between an electron an a CNT is given
by [26],
1 Z 1
e2 X
Im ðkRÞ 2
Vim ðqÞ ¼ K ðkqÞdk;
ð1Þ
p m¼1 0 Km ðkRÞ m
where Im ðxÞ and Km ðxÞ are the modified Bessel functions, the
capital letter R is the radius of the tube and q is the radial
coordinate of the electron.
To describe the potential near the carbon wall, both electronic exchange and correlation effects must be included.
The electric potential can be modeled as produced by a cylindrical jellium like surface barrier, similar to the standard theoretical approach used to explain the interaction of an
electron with a flat metal surface [16]. The potential far from
the surface retains the image charge form, with the corresponding cylindrical image surface plane, displaced outwards
Helix Material Solutions, Inc. 819 West Arapaho Road, Suite 24B-187 Richardson TX 75080, USA.
Hevia SA, Segura R, Häberle P. to be published.
CARBON
8 0 (2 0 14 ) 5 0–58
53
Fig. 3 – Data points and smoothed solid traces of normal
incidence IPS intensities from different CNT samples. The
characteristics of each sample have been explained in the
text. Resonances A and B can be recognized in all spectra. (A
color version of this figure can be viewed online.)
from the edge of the jellium background (radius R). The full
potential can then be defined as following:
8
Uo
>
>
;
>
>
>
< Ao exp ½Bo jq ðR þ DR=2Þj þ 1
VðqÞ ¼ R 6 q 6 R þ DR=2
ð2Þ
>
>
>
Vim ðq DR=2Þ½1 exp ½km jq ðR þ DR=2Þjm ;
>
>
:
q > R þ DR=2
Fig. 2 – Transmission electron microscopy of representative
CNTs from the samples in this study. (a) Image of CNTs
contained in sample MWS50 and (b) is a characteristic
micrograph of the wall of one of these tubes. Images (c) and
(d) are the corresponding micrographs of sample MWH50.
Images (e) and (f) are taken from CNTs contained in samples
MWH10 and SW1.3, respectively. (g) Image of CNTs
contained in sample MWS65; (h) is a micrograph of the walls
of one of these tubes. Images (i) and (j) are the
corresponding micrographs of sample MWS65-A. See text
for sample labeling and further details. (A color version of
this figure can be viewed online.)
In the expression above, R þ DR=2 is the radius of the reference cylinder for the image potential. By considering the continuity of the potential and its derivative at q ¼ R þ DR=2, the
constants Ao and Bo can be determined. The trajectory calculations presented below have been done using Uo = 15.24 eV,
DR = 0.296 nm, k = 9.6 nm1 and m = 0.94. See Zamkov et al.
[18] for a detailed discussion regarding the potential and the
definitions of the different parameters.
Our interest is to describe the trajectories of electrons
approaching a CNT. We consider first the motion perpendicular to the tube axis. In a semi-classical approach the trajectory
can be described as that of an electron moving under the
action of a cylindrical potential (VðqÞ). The electronic trajectories can then be obtained by direct numerical integration of
the following expression [27]:
54
D/ ¼ b
CARBON
Z
0
qf
qi
8 0 ( 2 0 1 4 ) 5 0 –5 8
1
dq0
B
C
@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A;
0Þ
b2
q02 1 Vðq
E
q02
ð3Þ
where, D/; qi ; qf and the impact parameter ‘‘b’’ have been
defined in Fig. 4(a). The traces of several electronic trajectories, for 1 eV electrons (dashed lines), with different impact
parameters, have been also shown in the same figure, as they
approach a 10 nm diameter CNT. This picture shows how the
electron trajectories are bending, inducing a focusing effect
around the tube.
A way to characterize the strength of this focussing effect
is by considering the magnitude of the dimensionless parameter Db=R, for a set of electronic trajectories with a common
value of their kinetic energies, incident on tubes with different radius. Db is the additional range of impact parameters
that leads to a collision of the incoming electrons with a particular CNT, when the trajectories are modified by the image
charge potential. Hence, including the image potential, effectively increases the cross sectional area of the tubes for an
incoming electron flux. In Fig. 4(b), the values of Db=R have
been tabulated for different CNT diameters and electron
kinetic energies. The main message behind this set of values
is that the focusing effect is significant for low energy electrons incident on small diameter tubes.
In the case of an electron moving with a component of its
velocity in a direction parallel to the tube axis, the situation
leads to similar conclusions. The trajectory can be analyzed
Fig. 4 – (a) Calculated trajectories of electrons traveling
perpendicular to the CNT axis. Several parameters used in
the calculations have been defined graphically. R is the tube
radius, qi and qf are the initial and final radial coordinates of
the incident electron measured with respect to the CNT axis.
Doted lines represent a set of calculated trajectories for 1 eV
electrons incident on a 10 nm CNT. (b) Values of Db=R
obtained from the trajectories as a function of the electronic
kinetic energy and tube diameters. (A color version of this
figure can be viewed online.)
from a co-moving reference frame, with a velocity parallel
to the tube. Finally, the potential VðqÞ, shown above, is strictly
valid for metallic tubes. Nevertheless, it can be used to
describe non-metallic tubes by introducing a dielectric tensor
[15]. The image potential is practically the same as the one
shown in Eq. (1). The effect of the different CNT chiralities
[1] can be included by a suitable choice for the value Uo in
the image potential (see Eq. (3)). Numerical calculations (not
shown here) have also demonstrated that the focusing mechanism persists even if the CNTs form bundles, as is usually
the case for SWNTs deposited on surfaces.
3.2.
Reconstructed IPS spectrum of a CNTs collection
This section describes the reconstruction of an IPS spectrum
from a CNT array, based on measurements performed on
graphite. The main idea behind this procedure is to recognize features in the CNT spectrum inherited from the tubes’
source material. A description of the p and r electronic
bands of graphene, as shown in Fig. 5(a), are necessary to
Fig. 5 – (a) Tight binding calculation of Graphene’s band
structure, adapted from Saito et al. [1]. 5(b) Density of states
(DOS) of the p band, with the energy scale in units of co
(energy of the p band at the M point). Inset in 5(b) is a
density plot of the p band in k-space with labels for the
high symmetry points of the BZ. Highlighted in red (online)
are the states with energies close to co . Experimental
measurements are consistent with a value of co = 2.5 eV [30].
(A color version of this figure can be viewed online.)
CARBON
8 0 (2 0 14 ) 5 0–58
undertake this procedure. The p band is particularly relevant in the discussion below, for that reason the density of
states (DOS) for this band is shown in Fig. 5(b). The energy
scale is measured in units of co , which is the energy of the
p band at the M point of the first Brillouin zone. The insert
in Fig. 5 (b), is a density plot of the p band in k-space. The
light gray areas correspond to high energy states, while solid
black to energies near EF. Highlighted in red traces (on line)
are the states with energies close to co , which induce the
sharp peak in the DOS.
For the energies used in IPS, electrons have short penetration lengths in a solid [28]. Hence, in the case of CNTs, the
photons collected are produced mainly in the external walls.
If the diameter of a particular tube is large enough, the curvature and the azimuthal quantization have an almost negligible effect on the tube’s electronic structure. In other words,
the interaction of incoming electrons with the CNT wall can
be described as equivalent to the interaction of electrons with
a locally flat graphene layer.
The geometrical aspects of the proposed spectral reconstruction can be extracted from Fig. 6. To illustrate the main
features of this reconstruction we considered a large diameter
CNT. In this case, focusing effects are negligible and the electronic trajectories can be considered as straight lines. As
shown in the diagram depicted in Fig. 6a, the incident
Fig. 6 – (a) Schematic representation of electrons incident on
a CNT. (b) Diagram describing the geometrical parameters at
the electronic collision point on a flat graphene layer. (A
color version of this figure can be viewed online.)
55
electrons are traveling in a direction perpendicular to the
tube’s axis. Depending on the angle between the tube’s local
surface normal and the electronic momentum at the collision
point, different regions of the graphene’s k-space are
explored. An IPS spectrum from a collection of these CNTs
could in principle be reproduced by averaging data collected
from a flat graphene layer, with different values for the angle
of incidence, see Fig. 6a and b. This angle depends on the
orientation of the surface normal at the collision point. For
the situation displayed in Fig. 6a, the graphene’s k-space line
explored by the electrons, as the impact parameter is
changed, is along the CM direction (see Fig. 5).
Since the sample is formed by a collection of CNTs in different orientations or the tubes have different chiralities, the
axes of the tubes are not necessarily normal to the incoming
flux. The incident electrons could then couple to states in the
complete BZ. Considering that graphite is a laminar material,
a reasonable reconstruction of the IPS spectrum from a collection of large diameter CNTs, can be obtained by a suitable
average of IPS spectra from graphite instead of graphene. If
HOPG is used instead, part of the averaging appears naturally
through the characteristic azimuthal disorder present in this
material [29]. Hence, the phenomenological reconstruction of
a normal incidence IPS spectrum, from a randomly oriented
CNT collection, should consider contributions from different
incidence angles on a flat HOPG surface.
In Fig. 7, the inset displays a set of traces (in solid black),
which are the smoothed IPS data collected from HOPG for different angles of incidence up to 70°. The top traces, in blue (on
line), for incidence angles beyond 70° and up to 90°, have been
extrapolated from the measured data. The spectra collected
up to 70° have been used to generate the curve labelled as
‘‘REC-70’’, which is the average of all HOPG spectra. Fig. 7 also
includes IPS measurements from the CNT samples labeled as
SW1.3 and MWS65-A. The reconstructed spectrum (REC-70)
agrees well with the experimental data for energies above
8 eV. This is indeed a remarkable agreement considering the
different spectral shapes of the data being averaged. Therefore, the reconstructed spectrum provides a robust prediction
for resonance B in CNTs. We can trace back the origin of this
resonance, concluding that is due to multiple transitions
mainly into r states [24,30]. On the other hand, IPS data
(Fig. 3) and TEM images of these tubes (Fig. 2) link the sharpness of resonance B, to the graphitization level of the tube’s
external walls in each sample, consistent with the proposed
modeling procedure.
Since the CNT-IPS spectrum reconstruction was done
using data collected from a flat graphitic surface and considering that this procedure does not depend of the tube diameter, it is not surprising that REC-70 is inadequate to predict the
low energy range of the CNT data. It is precisely resonance A,
with a peak value around 2.5 eV, the one that exhibits a higher
intensity in tubes with smaller diameters. If a proper comparison is made among tubes with similar levels of graphitic content, the resulting intensity of resonance A is indeed
dependent on the tube radius.
Ideally the reconstruction should include angles of incidence as close to 90° as possible. Due to limitations in our
spectrometer, we can collect data only up to 70°. The
‘‘missing’’ HOPG spectra should exhibit high intensities at
56
CARBON
8 0 ( 2 0 1 4 ) 5 0 –5 8
The comparison of the intensities of resonance A between
curves MWS50 and MWS50-F in Fig. 3, is a bit puzzling at first,
since both samples contain the same tubes. However MWS50F shows a slightly reduced A intensity when compared with
MWS50, because the tubes in both samples have, on average,
different orientations. A large fraction of the CNTs in MWS50
have their axes mostly perpendicular to the substrate. Hence,
electrons incident along the substrate normal, will collide
with the tubes mostly at high angles of incidence. Within
the model used for the spectral reconstruction, the higher
intensity of A in MWS50 agrees well with the stronger contribution from these angles.
3.3.
Fig. 7 – IPS reconstructed spectra for large diameters CNT
sample. The trace at the bottom (REC-70) correspond to the
average of all measured HOPG spectra up to 70°. To facilitate
the comparison with experiments, data from samples
SW1.3 and MWS50 have also been included in this figure.
The insert shows IPS measurements from HOPG in black
lines. The traces shown in blue (online), are obtained by
extrapolating the measured HOPG data for angles beyond
70°. Blue (online) is also used for the reconstructed spectra
which make use of the extrapolated data. See text for
further details on the reconstruction and labeling of the
different traces. (A color version of this figure can be viewed
online.)
low energies, which are indeed relevant for the reconstruction. To get an indication of the influence of these missing
spectra, they have been extrapolated taking into account
the measured dispersion of the high intensity peak at
low energies. The extrapolated data is shown by the
blue (on line) traces in the inset of Fig. 7. Including these
extrapolated spectra in the reconstruction, provides a
reasonable fit to the measured data of a wide CNT. This
statement can be confirmed by comparing the reconstructed
spectrum ‘‘R-MWS65-A’’ with the corresponding one
measured from graphitic wide tubes, the MWS65-A. The
curve R-MWS65-A was built by averaging HOPG spectra
collected up to 70° with those extrapolated for larger angles.
There is indeed an improved fit when the high incidence
angles (extrapolated spectra) are included in the spectral
reconstruction procedure. Nevertheless, the additional spectra are not enough to reproduce a spectrum from a sample
with small diameter tubes, as those from SW1.3 shown in
Fig. 7. For these very narrow tubes, the bending of the
incoming electrons’ trajectories has to be included in the
analysis to obtain an improved description of the experimental results.
Effect of electronic focusing in the IPS measurements
From the previous analysis, it is clear that for a complete
description of the low energy IPS intensity, electronic focusing
should be included. Since the image charge potential in Eq. (3)
only depends on the radial coordinate, the electrons modify
their trajectories close to the tube, but at the same time their
total energy and angular momentum with respect to the tube
axis remains constant. For electrons with an impact parameter < R, see Fig. 4(a), the actual value of the electronic parallel
momentum (hkk ) is the same with or without focussing. This
is relevant in IPS, since energy and hkk are conserved quantities in an IPS transition occurring in a periodic surface, as is
the case in graphitic CNTs. This equivalence, confirms the
validity of the non focusing spectral reconstruction in the
description of a collection of wide diameter tubes up to an
impact parameter equal to R. So then, what are the new features gained by including the focusing effect? There is an
additional electronic flux which has to be included in order
to account for the experimental results. In the non focusing
description, only electrons with an impact parameter < R collide with a CNT. In a theoretical description including focusing, electrons with b beyond R can reach the tube. This
additional flux can now contribute to the IPS intensity.
Fig. 4(a) is useful to clarify this point.
The additional electronic flux in the Db region of the diagram gives rise to an increased IPS photon intensity for the
low energy region close to EF. The values of hkk for these
electrons are larger than for electrons with an impact
parameter < R. Hence, focussing not only increases the flux
but it also increases the relevance of the large incidence
angles in the spectral reconstruction. This new contribution
is expected to be radius and energy dependent as shown in
Fig. 4(b).
The low energy IPS resonances occurring for large incidence angles in HOPG are dominated by transitions into the
p band. Graphite’s p band, in this energy range, displays a
high density of states (DOS) and an increased probability for
optical transitions. In particular to the empty states close to
the ‘‘M’’ point of the BZ (see inset in Fig. 5(b)). Previous measurements performed on a graphite single crystal [30,31] have
shown that the energy for the ‘‘M’’ point is indeed close to
2.5 eV, in coincidence with our own results.
A reconstruction of the spectrum ‘‘SW1.3’’, including
extrapolated spectra, is shown in Fig. 7 (‘‘R-SW1.3’’). It displays a much better agreement with the experimental data
when focussing effects are included. The increased weight
CARBON
8 0 (2 0 14 ) 5 0–58
factor of the larger angles was determined through the value
of Db calculated from the electronic trajectories as shown in
Fig. 4. Hence, the enhanced intensity of A, for narrow tubes,
is rather due both to a dynamic effect over the electrons colliding with the tubes, together with the intrinsic electronic
contribution of the graphene sheets forming the CNTs. It
should be noted, that in our particular spectrometer we could
never observe a direct transition to the M point when measurements are performed over flat graphite. The high density
of state around the M point can only be reached by increasing
k// in the incident electron flux. The cylindrical geometry of
the tubes and the image potential contribute exactly in this
manner, providing the additional parallel momentum
required to make the pi* resonance stronger in CNTs.
At this point, it is worth mentioning that the image state
which has been detected in CNTs with 2-photon photoemission [17], has not been found in IPS measurements in this
or other studies [12]. In the case of HOPG, see the inset in
Fig. 7, the image state can be observed as a narrow resonance
3.7 eV above EF and it is clearly distinguishable up to 15°. If the
same relation between the intensities of the image state and
the prominent resonance close to EF in HOPG is maintained in
CNTs, it would be very hard to detect it by IPS measurements
from a randomly oriented CNT sample. This difficulty comes
about while considering the proposed reconstruction
procedure, from averaging the contributions from different
incidence angles to form the reconstructed spectrum. The
high intensity resonance observed at low energies in HOPG,
starting at 50° (inset Fig. 7), traverses precisely the energy
range where the weak signal from the image state has been
detected. Hence, the low intensity feature is overwhelmed
by the strong resonance and can not be distinguished above
the background signal.
4.
Summary
Many spectroscopies such as photoemission, electron energy
loss, inverse photoemission and tunneling spectroscopies,
use low energy electrons as their main probe to examine both
the electronic and geometrical structure of systems such as
CNTs. The adequate interpretation of the spectral response
of these techniques, requires a thorough understanding of
the detailed interaction of the nanoscale objects being examined by the probing electrons. This report proposes a way to
model the IPS response of randomly oriented CNTs, as due
to a superposition of spectra from HOPG together with an
image charge induced focusing of electrons. The tube diameter dependence of the IPS spectra, in the case of low kinetic
energy electrons, can be qualitatively described through the
image charge potential. The reconstruction procedure, hereby
described, can bring out the influence of the tubular geometry
and the electronic structure of graphitic CNTs, in a wide
energy range above EF.
Acknowledgment
This research has been partially funded by: FONDECYT, Chile,
grants 11110352, 1121203, 1110935.
57
R E F E R E N C E S
[1] Saito R, Dresselhaus G, Dresselhaus MS. Physical properties
of carbon nanotubes. London: Imperial College Press; 1998.
[2] Dresselhaus MS, Dresselhaus G, Avouris Ph. Carbon
nanotubes: synthesis, structure, properties, and
applications. Heidelberg: Springer-Verlag Press; 2001.
[3] Jorio A, Dresselhaus MS, Dresselhaus G. Carbon nanotubes:
advanced topics in the synthesis, structure, properties and
applications. Heidelberg: Springer-Verlag Press; 2008.
[4] Cao G, Lee YZ, Peng R, Liu Z, Rajaram R, Calderon-Colon X,
et al. A dynamic micro-CT scanner based on a carbon
nanotube field emission X-ray source. Phys Med Biol
2009;54(8):2323–40.
[5] Merkoçi A, Pumera M, Llopis X, Pérez B, del Valle M, Alegret S.
New materials for electrochemical sensing VI: carbon
nanotubes. TrAC Trends Anal Chem 2005;24(9):826–38.
[6] Pichler T, Knupfer M, Golden MS, Fink J, Rinzler A, Smalley RE.
Localized and delocalized electronic states in single-wall
carbon nanotubes. Phys Rev Lett 1998;80(21):4729–32.
[7] Wildöer JWG, Venema LC, Rinzler AG, Smalley RE, Dekker C.
Electronic structure of atomically resolved carbon nanotubes.
Nature 1998;391(1):59–62.
[8] Ishii H, Kataura H, Shiozawa H, Yoshioka H, Otsubo H,
Takayama Y, et al. Direct observation of Tomonaga Luttingerliquid state in carbon nanotubes at low temperatures. Nature
2003;426(4):540–4.
[9] Suzuki S, Watanabe Y, Kiyokura T, Nath KG, Ogino T, Heun S,
et al. Electronic structure at carbon nanotube tips studied by
photoemission spectroscopy. Phys Rev B
2001;63(24):245418–27.
[10] Suzuki S, Watanabe Y, Ogino T, Heun S, Gregoratti L, Barinov
A, et al. Electronic structure of carbon nanotubes studied by
photoelectron spectromicroscopy. Phys Rev B
2002;66(3):035414–24.
[11] Schiessling J, Kjeldgaard, Rohmund F, Falk LKL, Campbell EEB,
Nordgren J, et al. Synchrotron radiation study of the
electronic structure of multiwalled carbon nanotubes. J Phys
Condens Matter 2003;15:6563–79.
[12] Shiozawa H, Ishii H, Kihara H, Sasaki N, Nakamura S, Yoshida
T, et al. Photoemission and inverse photoemission study of
the electronic structure of C60 fullerenes encapsulated in
single-walled carbon nanotubes. Phys Rev B
2006;73(7):075406–15.
[13] Hevia S, Ibáñez W, Segura R, Häberle P. Inverse
photoemission spectroscopy of multiwall carbon nanotubes.
Braz J Phys 2006;36(3B):894–7.
[14] Fuggle JC, Inglesfield JE. Unoccupied electronic
states. Berlin: Springer; 1992.
[15] Granger BE, Kral P, Sadeghpour HR, Shapiro M. Highly
extended image states around nanotubes. Phys Rev Lett
2002;89(13):135506–14.
[16] Jennings PJ, Jones RO, Weinert M. Surface barrier for electrons
in metals. Phys Rev B 1988;37(11):6113–20.
[17] Zamkov M, Woody N, Shan B, Chakraborty HS, Chang Z,
Thumm U, et al. Time-resolved photoimaging of imagepotential states in carbon nanotubes. Phys Rev Lett
2004;93(15):156803–4.
[18] Zamkov M, Chakraborty HS, Habib A, Woody N, Thumm U,
Richard P. Image-potential states of single- and multiwalled
carbon nanotubes. Phys Rev B 2004;70(11):115419–28.
[19] Segui S, Celedón C, Bocan GA, Gervasoni JL, Arista N. Tubular
image states: general formulation and properties for metallic
and nonmetallic nanotubes. Phys Rev B 2012;85(23):235441–7.
[20] Segura RA, Ibáñez W, Soto R, Hevia S, Häberle P. Growth
morphology and spectroscopy of multiwall carbon
nanotubes synthesized by pyrolysis of iron phthalocyanine. J
Nanosci Nanotech 2006;6(7):1945–53.
58
CARBON
8 0 ( 2 0 1 4 ) 5 0 –5 8
[21] Segura R, Flores M, Hevia S, Häberle P. Synthesis,
characterization and spectroscopy of carbon based
nanoscale materials. Microelectron J 2008;39(3–4):529–33.
[22] Hevia S, Homm P, Cortés A, Núñez V, Contreras C, Vera J,
et al. Selective growth of palladium and titanium dioxide
nanostructures inside carbon nanotube membranes. Nanosci
Res Lett 2012;7(1):342–8.
[23] Häberle P, Ibáñez W, Esparza R, Vargas P. Unoccupied
electronic states of Au(113): theory and experiment. Phys Rev
B 2001;63(23):235412–6.
[24] Schäfer I, Schläter M, Skibowski M. Conduction-band
structure of graphite studied by combined angle-resolved
inverse photoemission and target current spectroscopy. Phys
Rev B 1987;35(14):7663–70.
[25] Maeda F, Takahashi T, Ohsawa H, Suzuki S. Unoccupiedelectronic-band structure of graphite studied by
angle-resolved secondary-electron emission and inverse
photoemission. Phys Rev B 1988;37(9):4482–8.
[26] Gervasoni JL, Arista NR. Plasmon excitations in cylindrical
wires by external charged particles. Phys Rev B
2003;68(23):235302.
[27] Fetter AL, Walecka JD. Theoretical mechanics of particles and
continua. New York: Courier Dover Publications; 2003.
[28] Penn D. Electron mean-free-path calculations using a model
dielectric function. Phys Rev B 1987;35(2):482–6.
[29] Ferralis N, Pussi K, Finberg SE, Smerdon J, Lindroos M,
McGrath R, et al. Low-energy electron diffraction study of
potassium adsorbed on single-crystal graphite and highly
oriented pyrolytic graphite. Phys Rev B 2004;70(24):, 245407–6.
[30] Claessen R, Carstensen H, Skibowski M. Conduction-band
structure of graphite single crystals studied by angleresolved inverse photoemission and target-current
spectroscopy. Phys Rev B 1988;38(17):12582–8.
[31] Collins IR, Andrews PT, Law AR. Unoccupied electronic states
of single-crystal graphite by angle-resolved ultraviolet
inverse photoemission. Phys Rev B 1988;38(18):13348–54.