Low energy electrons focused by the image charge interaction in carbon nanotubes

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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 Ha¨berle

c

Instituto de Fı´sica, Pontificia Universidad Cato´lica de Chile, 6904411 Santiago, Chile Instituto de Quı´mica y Bioquı´mica, Facultad de Ciencias, Universidad de Valparaı´so, Av. Gran Bretan˜a 1111, Valparaı´so, Chile Departamento de Fı´sica, Universidad Te´cnica Federico Santa Marı´a, 2390123 Valparaı´so, 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

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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

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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, Ha¨berle P. to be published.

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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]:

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D/ ¼ b

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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.)

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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.)

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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

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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

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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.

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R E F E R E N C E S

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