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J. Phys. Chem. C 2007, 111, 6690-6693
Field Emission Signature of Pentagons at Carbon Nanotube Caps Mohammad Khazaei,*,† Kenneth A. Dean,‡ Amir A. Farajian,§ and Yoshiyuki Kawazoe† Institute for Materials Research, Tohoku UniVersity, Sendai 980-8577, Japan, Embedded Systems Research, Motorola, Inc., 2100 East Elliot Road, Tempe, Arizona 85284, and Department of Mechanical Engineering and Materials Science, Rice UniVersity, Houston, Texas 77005 ReceiVed: December 11, 2006; In Final Form: March 6, 2007
Localization of emitting states and their tunneling probabilities cause the nanotube cap geometry to have decisive impact on field emission patterns and currents. We show how different arrangements of pentagon rings at the tip can create specific field emission features, utilizing a method based on first principles calculations. The results give an explanation for different field emission patterns observed in experiments, and provide a feasible way to distinguish different cap structures from experimental results. A set of general rules is deduced to infer the tip configuration through the experimental field emission patterns. The calculations agree very well with our experimental results, and are of fundamental interest in characterization and design of carbon nanotube emitters and probes.
Capped carbon nanotubes, with superior chemical stability even under strong external electric field, are one of the most promising materials for making field emission devices1 and scanning probes.2 Electron holography experiments on fieldemitting carbon nanotubes show that the local electric field and hence the associated emission current are concentrated precisely at the tips of carbon nanotubes and not at sidewalls.3,4 The morphology and sharpness of the tip of capped carbon nanotubes are controlled by the number of pentagon rings and their relative positions at the tip. According to Euler’s rule, the presence of 12 pentagon rings is enough to make a carbon cage. Consequently, a carbon nanotube can be closed by the presence of six pentagon rings. Depending on the diameter of the nanotubes, there are many different cap geometries resulting from different possibilities of the pentagon rings arrangements.5,6 In recent experiments using in situ transmission electron microscopy (TEM), it is shown that the work function is sensitive to atomic structure and the surface specification of the emitting tips.7 Theoretically, it is observed that different cap geometries have different electronic structures. In other words, the topology of the cap affects the strength and position of the peaks observed in density of states spectra near the Fermi energy.8-11 This can be used to distinguish different tip structures. However, due to large curvature of the nanotubes at the cap region, scanning tunneling microscopy (STM) or TEM techniques using coneshaped probes cannot resolve the atomic configuration of the cap precisely. In a previous work, we introduced a general formalism to obtain field emission properties of any kind of nanostructures, based on first principles local density of states (LDOS) and effective potentials.1 We applied the method to pristine and Cs doped capped carbon nanotubes. The method could excellently reproduce the field emission patterns of capped carbon nanotubes observed in experiments.12,13 It revealed that the experimental field emission images are the shapes of the local density of states weighted by probabilities of electron tunneling at the * Corresponding author. E-mail:
[email protected]. † Tohoku University. ‡ Motorola, Inc. § Rice University.
structure-vacuum barrier.1 There are many different experimental field emission patterns of capped carbon nanotubes available in literatures.12-14 In the present work, we explain how these different field emission patterns result from different distribution of pentagon rings at the caps. From our calculations a set of simple general rules are inferred to predict the precise atomic coordinates of the caps through experimental field emission patterns. Our recent experimental results supplement the calculations and support their validity. To understand the relation between the confinements of the pentagon rings and field emission patterns, we have created six different capped (10,0) carbon nanotubes,5 as seen in Figure 1. All the structures have six different pentagon rings distributed at their tips. Each cap structure totally consists of 110-122 carbon atoms. We should note that there might be more than 19 different cap geometries for a (10,0) nanotube.6 However, we have arbitrarily selected only six of them. In order to mimic a long nanotube, two carbon rings at the open end are kept fixed and the structure is optimized. All unconstrained carbon atoms are relaxed until the maximum force acting on them becomes less than 0.04 eV/Å. To avoid the unwanted effects of the carbon-hydrogen dipoles on field emission, the ending layers are not saturated with hydrogen. Instead, we add a single armchair like layer to the ending zigzag layer of the nanotubes. The ending C-C bonds are saturated by reducing their lengths to 1.24 A. All computations including relaxations and electronic structure calculations are ab initio based on generalized gradient approximation in density functional theory. The optimization calculations are done in a 25 × 25 × 25 Å3 supercell, while the electronic structure calculations are done with a larger 25 × 25 × 40 Å3 supercell (due to applying an external field). All the calculations are carried out using Gamma point with a mesh cutoff energy of 150 Ry. The calculations are performed using SIESTA code.15-17 Details and justifications are explained in refs 1, 18, and 19. Within the methodology that we use, the emission current is proportional to the probability of electron tunneling from the surface to vacuum, local density of states at left turning points, and a slowly varying function of energy called λ. Left turning
10.1021/jp068491i CCC: $37.00 © 2007 American Chemical Society Published on Web 04/13/2007
Pentagons at Carbon Nanotube Caps
Figure 1. From left to right: the first column contains the top views of different capped (10,0) nanotubes with different distributions of pentagon rings at the cap. The second to forth columns show total tunneling probabilities, local density of states (LDOS) at left turning points, and total emission current patterns, calculated when the caps are under an external electric field 0.5 V/Å. The fifth column shows experimental field emission patterns which match our calculated ones.
points are where the energy of emitting state intersects the potential energy barrier at the vacuum-structure boundary. The function λ results from the asymptotic matching of the wave function of the emitting state at a left turning point with the WKB (Wentzel-Kramers-Brillouin approximation) wave function inside the barrier. More details of calculations can be found in refs 1, 20, and 21. The sub-Angstrom resolution of the calculated emission patterns allows accurate comparisons to be made with experimental results. The experimental field emission measurements are performed in a field emission microscope system with a point-to-plane electrode geometry and a nominal emitter to anode separation of 2 cm. Field emission measurements are made under continuous (nonpulsed) bias conditions. During field emission measurements, nanotube field emitters are maintained at ground potential and a positive bias is applied on the phosphor-coated anode plate. Field emission images, produced by electrons incident on the phosphor screen, are collected using a video camera and are recorded continuously onto video tape. Nanotubes are attached to the end of a tungsten heater filament, which allowed for rapidly heating the nanotube during the experiments to remove surface adsorbates. During image collection, the clean nanotube surface is monitored continually in the field emission microscope. If adsorbates land on the nanotube, thus distorting the image of the clean nanotube surface, the nanotube temperature is rapidly raised and lowered, thereby thermally desorbing the adsorbate. After obtaining the optimized tip structures, a set of total energy calculations are carried out to determine the stability of different tips. Due to different number of carbon atoms constructing tip geometries shown in the first column of Figure
J. Phys. Chem. C, Vol. 111, No. 18, 2007 6691 1, total energy per carbon atoms is presented instead of the total energy in Table 1. The results indicate that all capped structures used in this article have similar stability within 0.03 eV. Therefore, we can expect that all these structures could be possibly created during nanotube growth or field emission process at high current conditions.22 Through electronic structure calculations, the work function of any structure can be estimated. Theoretically, the work function is defined as the energy difference between highest occupied molecular orbital (HOMO) energy level and vacuum or zero energy. From Table 1, it is observed that different cap geometries have different work functions. Our calculations show that the work functions of carbon nanotubes depend on the topology of their tips, as has already been reported in recent experiments.7 Therefore beside the effect of impurities such as Hydrogen adsorption on work function,23 this can be another reason for the existence of various reported values for work functions of carbon nanotubes, from 4.51 to 5.30 eV,7,24 in the literature. Table 1 shows the calculated current emission from HOMO, and from the states degenerate with it, of different cap structures when the capped nanotubes are under an external electric field 0.5 V/Å. Table 1 also shows the calculated total emitted currents for various cap geometries. The total currents are calculated by summing the currents from all relevant individual states below HOMO, i.e., the states for which LDOS and tunneling probabilities give rise to appreciable currents. We observed that the total emission current and the current emitted from HOMO of different cap structures differ from one structure to another. It means that there is no equal current emission from the nanotubes with different cap morphologies even if they have the same length and chirality. Also, quite interestingly, the calculations show that the nanotube caps with the largest current from HOMO states do not have the highest overall emission current. In our recent experiments,14 using field evaporating technique, 13 different cap geometries are produced from an individual capped nanotube without any significant changes in nanotube length. After each cap creation, the current emission is estimated at V ) 900 V. The measured current varies by about 180%. This strongly supports our theoretical results of 170% variation in current values, as is seen in Table 1. Depending on the applied electric field, the extracted current from a carbon nanotube is changed. The maximum current taken from an individual nanotube without any catastrophic failure during emission is reported to be within the range of 100 nA-15 µA.7,22,25 Extracting very large field emission currents from carbon nanotubes degrades the current-voltage behavior and changes the nanotube structure.22 Here, however, we do not consider the stability of the caps at high current emission conditions. Our calculations show that Cap4, the capped nanotube structure with 5-fold symmetric distribution of the pentagon rings, has the largest total field emission current among all the structures. This is due to larger contribution of its tip pentagon ring in electron localization and electron tunneling as we observe from Figure 1. Comparing the total current emission of each capped nanotube with its emitted current from HOMO, it is observed that there are significant contributions from energy levels deep below HOMO in total currents. Under strong applied electric fields, the structure-vacuum barrier height decreases at very short distances near the nanotube surface.26,27 Consequently, the tunneling probability of the electronic states located deeply below HOMO increases, resulting in a large contribution of deep electronic states in total current. This also explains the broadening of energy spectra of emitted electrons to very deep energies
6692 J. Phys. Chem. C, Vol. 111, No. 18, 2007
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TABLE 1: Stabilities and Field Emission Properties of Various Tip Geometries Shown in Figure 1a
geometry
constituting number of carbon atoms
total energy per atom (eV)
work function (eV)
Cap1 Cap2 Cap3 Cap4 Cap5 Cap6
122 118 116 110 112 112
-193.01 -192.99 -192.99 -192.98 -192.98 -192.99
5.07 4.83 5.13 5.04 5.13 4.94
HOMO and the states degenerate with it
current from degenerate HOMO (µA)
total emission current (µA)
(HOMO-1) + HOMO HOMO + LU MO + (LUM O + 1) (HOMO-2) + (HOMO-1) + HOMO HOMO + LU MO + (LUM O + 1) + (LUM O + 2) (HOMO-2) + (HOMO-1) + HOMO HOMO
5.1 2.7 12.1 3.6 5.0 2.5
46.7 40.4 46.3 60.9 37.7 35.7
a The stabilities are determined by calculating total energy per carbon atom for each structure. The work function is defined as the difference between the HOMO energy level and the vacuum energy. The emission currents are calculated when the tip structures are under an external electric field 0.5 V/Å. HOMO and LUMO refer to highest occupied and lowest unoccupied molecular orbitals, respectively.
at strong electric fields.27 Considering the large contribution of the electronic states below HOMO to total current emission, the total tunneling probability, total LDOS at left turning points, and total current emission patterns are presented in Figure 1 instead of the patterns corresponding only to HOMO.1 The total patterns are calculated by summing individual energy-resolved patterns (tunneling, LDOS, and current) of all contributing states. The second column of Figure 1 shows the total tunneling probability patterns of electrons from different capped carbon nanotubes under an external electric field 0.5 V/Å. The tunneling probability is a function of effective potential at structurevacuum boundaries.1 The height and behavior of the effective potential barrier near the surface are different for various tips. Hence, the tunneling patterns are different for different cap geometries. For each tunneling pattern, the bright spots correspond to sharp areas of the cap geometry; the topmost C-C bond on Cap1, 2, 3, and 5, the topmost pentagon ring on Cap4 and the topmost hexagon ring on Cap6, all show the largest electron tunneling probability areas of the caps. One should note that it is not necessary to have large emission currents from the sharp regions of the caps. Except for tunneling probability, the current emission patterns are affected by how emitting states are localized on the caps.1 From the tunneling patterns, it can also be clearly observed that the cap of nanotubes cannot be modeled by a metallic hemisphere with a fixed barrier height.28 The third column of Figure 1 shows the total LDOS patterns calculated at left turning points for different structures. The forth column shows the calculated total current emission patterns. The λ function (not shown here) has nearly the same value for all the surface points and does not affect the emission patterns. From the total LDOS patterns, it is observed that the localization of states at the pentagon rings and/or bonds connected to pentagon rings are larger than other atomic sites, similar to what Hou et al. observe in their STM image simulations for an adsorbed C60 on a silicon surface.29 Comparing the created emission pattern of each individual electronic state with its corresponding tunneling and LDOS patterns (not shown), we observe that the emission patterns look like the LDOS patterns weighted by the tunneling probabilities.1 Therefore in nanometric systems, the experimental field emission patterns at low electric fields (where few energy states near HOMO contribute in total emission current) can provide unique information about electronic structures which are hard to obtain by other methods, especially for curved surfaces. From the total current patterns, it is observed that at a high electric field, for example at 0.5 V/A, the emission patterns resemble the precise configuration of the tip structures. Different electronic states are localized at different parts of the cap or stem of the nanotubes. At high electric fields, many electronic states contribute to total current. This causes electrons to be emitted from different positions of the cap, and of course they
are emitted with different tunneling probabilities. At low electric fields close to threshold voltages, on the other hand, most of emission current comes from HOMO and very few energy levels below HOMO. These energy levels are not usually distributed over the whole of the cap. Hence, in this case, the emission patterns do not resemble precisely the configuration of the cap. The fifth column in Figure 1 shows some of the experimental field emission patterns obtained from single/multiwalled capped carbon nanotubes, which excellently match our computed images. Our calculations show that it is feasible to guess the cap configurations of nanotubes through the experimental field emission patterns, by comparing the theoretical patterns with experimental ones. From the total current patterns, one can observe how different distributions of pentagon rings at the caps create different unique patterns of emission, by creating different surface electronic structures. We mentioned above that tunneling probabilities at sharp points of the caps are larger than the tunneling probabilities from other points. Moreover, from the LDOS patterns, it is observed that the LDOS reaches a maximum at the pentagon rings and/or the bonds connected to the pentagon rings. The cap regions for which both the tunneling probability and the LDOS are at their maximums indicate the regions with largest emission current. Therefore, the fingerprints of pentagon rings can be recognized in the total emission current patterns by a set of simple rules: Consider the topmost pentagons (that is the pentagons closest to the tip of the cap). Then consider the topmost bonds that either belong to these pentagons or have one of their ends on these pentagons. These bonds are the regions with largest tunneling probability and LDOS, as explained before. Thus, the total current emission pattern peaks out at these bonds and their corresponding ending atoms. The relative intensities of the emission pattern peaks increase if (a) the topmost bonds have corresponding ending atoms belonging to more than one pentagon and (b) the topmost bonds/atoms are closer to the tip of the cap, where there is maximum enhancement of the electric field. From these simple rules, the distribution of the cap pentagons can be derived from their fingerprints in the total current emission pattern. It should be noticed that using the above-mentioned set of rules results in considerable reduction of computation efforts for determining the cap structures, especially for nanotubes with large diameters and cap areas. For example, there are at least 677 possible cap configurations for a (13,2) nanotube.6 Utilizing our rules, we do not need to calculate emission patterns for all the possible configurations. We can choose some of them for further considerations. In this regard, by means of the above rules, we assign a cap configuration to each of our other experimental filed emission patterns shown in Figure 2. The possibility of existence for such cap structures is already considered by O h sawa et al.5,6 In Figure 2, the axes of the capped nanotubes deviate from the direction
Pentagons at Carbon Nanotube Caps
J. Phys. Chem. C, Vol. 111, No. 18, 2007 6693 Acknowledgment. The authors sincerely thank the crew of the Center for Computational Materials Science of the Institute for Materials Research, Tohoku University, for their continuous support of the supercomputing facilities. We are also grateful to Olga V. Pupysheva for helpful comments. Supporting Information Available: Some of the commands set for the electronic structure calculations with Siesta + atomic coordinates of the caps. This material is available free of charge via the Internet at http://pubs.acs.org.
Figure 2. Suggested cap geometries for different experimental field emission patterns based on our general set of rules. The axes of the capped nanotubes deviate from the normal direction to the plane of the figure, along which we believe the external electric field has been applied.
normal to the plane of the figure, along which we believe the external electric filed should have been applied in the experiments. In the experiments the direction of the applied electric field is not always in parallel to nanotube axis. However, we should note that in Figure 1, the axis of the capped nanotubes do not deviate from the normal to the figure plane. In all our calculations corresponding to this figure, the external electric field is applied in parallel to the nanotube axis. Although in the present study we did not consider various chiral nanotubes in our calculations, we believe that our results are quite general. The chirality of nanotubes might affect the field emission process of long nanotubes where the extended states start to contribute in current emission simultaneously with localized states. However, S. Han and J. Ihm30 find that the emission current from the states localized at the tip is more than ten times larger than direct contribution from extended states. In our simulations the nanotubes are short. Hence, all the states are localized. Our field emission patterns are mainly generated by the localized states on the caps, not by the ones localized on the stems. For a simple test calculation, we considered the emission pattern of a (5,5) stem nanotube with a cap which has the same symmetry as Cap4 of the present study. There is almost no difference between the calculated emission patterns of capped (5,5) and capped (10,0). To conclude, our calculations show that each cap structure has a specific field emission pattern, and a unique work function, depending on the particular arrangement of the six cap pentagons. While there are many high-resolution images from the bulk lattice of carbon nanotubes, there is no clear image from the cap regions. On the basis of our calculations and experimental results, we suggest a set of simple rules to determine the configuration of the caps through experimental field emission patterns. There are very good agreements between our calculated patterns and the ones observed in experiments. This indicates the efficiency of using field-emission microscopy for determining the structure of curved nanoscale surfaces, where conventional transmission electron microscopy (TEM) and scanning tunneling microscopy (STM) methods have limited abilities.
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