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Electronic properties of diamond clusters: self-consistent tight binding simulation
Diamond & Related Materials 13 (2004) 1826 – 1833 www.elsevier.com/locate/diamond
Electronic properties of diamond clusters: self-consistent tight binding simulation D.A. Areshkin, O.A. Shenderova, S.P. Adiga, D.W. Brenner * Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7907, USA Received 21 August 2003; received in revised form 26 April 2004; accepted 29 April 2004 Available online 28 July 2004
Abstract A self-consistent environment-dependent tight binding method is used to examine electron emission-related properties of hydrogen passivated nano-diamond (ND) particles. For sizes larger than 2.5 nm particle bandgap was found to be equal to the bandgap of bulk diamond. Coulomb potential distributions and electron affinities of clusters were found to be insensitive to the particle size if it exceeds 1.0 nm. Tunneling probabilities for homogeneous and inhomogeneous emission models were estimated. The simulation results indicate that the low emission threshold for hydrogen passivated diamond nano-clusters is due to hydrogen-assisted emission from the edges of small unpassivated islands. Essentially the same mechanism is claimed to be responsible for good emission properties of hydrogen passivated diamond films by Ristein [Diam. Relat. Mater. 9, 1129 (2000)]. D 2004 Elsevier B.V. All rights reserved. Keywords: Diamond clusters; Field emission; Electron affinity
1. Introduction Nano-diamond (ND) clusters, also called ultradispersed diamond (UDD), can be obtained from detonation products in large amounts with high uniformity and reproducibility [1,2]. After removal of an external disordered carbon shell, the sizes of nano-diamond clusters range from 2 to 10 nm, with a narrow peak in size distribution at about 5 nm [3]. Traditionally, UDD is used for galvanic coatings, polymer composites, polishing, and lubricant additives [3]. Attempts to use UDD powder for field emission enhancement [4– 7] and suggestions for other electronic applications [8] were also reported. There is experimental evidence [6] that ND coatings deposited on Si needle-shaped field emitters substantially lowers emission threshold voltages relative to that of bare Si needles and needles with micro-diamond powder coatings. ND coatings are created either by bias enhanced microwave plasma chemical vapor deposition [9] or by electrophoresis from a suspension of diamond powder in ethanol [10]. After deposition the coatings are exposed to a hydrogen plasma, * Corresponding author. E-mail address: [email protected] (D.W. Brenner). 0925-9635/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2004.04.012
which is intended to have a twofold effect. First, it etches much of the loosely bounded carbon atoms, preferentially in the outer regions of the graphene and/or amorphous carbon (a-C) layers that surround ND particles. Second, the hydrogen treatment is used to saturate free carbon bonds at the surface. TEM images of a Si needle tip with a ND coating, however, show surface roughness and irregularity that likely inhibits perfect hydrogen coverage [7], and the hydrogenated ND surface may therefore contain small, unpassivated islands. Chains of such islands would serve as conducting channels between a graphene/a-C phase backcontact and the cluster’s outer surface from which electrons are emitted. The issue addressed here is a comparison of emission probabilities from the surface and through the bulk of ND clusters. These two main field emission mechanisms that are explored are called the Inhomogeneous and Homogeneous Emission Models (IEM and HEM), respectively [11]. At the same time, the field emission threshold from a Mo tip with a single ND particle is higher compared to the threshold of bare Mo [6]. That speaks in favor of the IEM because from the HEM standpoint single particle properties are similar to the properties of ND coating. On the other hand a single particle may not have surface irregularities inherent for a ND coating, which are required for IEM. The goal of
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this paper is to explain field emission results by using selfconsistent (SC) environment-dependent tight binding (SCEDTB) [12,13] simulations. SC-EDTB is a hybrid minimum basis set TB-DFT scheme, which was designed to model moderate size (100 –1000 atoms) metallic or semiconducting systems. The method builds on a non-self-consistent environment-dependent TB model for carbon [14] with parameters added to describe hydrocarbon bonds and to account for self-consistent charge transfer. In this approach, the Hamiltonian matrix H is composed of two parts, the original EDTB [14] matrix H0, and SC corrections DH. The latter accounts for charge transfer between constituent atoms, and is supposed to be zero for periodic structures for which the original non-SC EDTB parameterization was made. DH is computed in a standard DFT manner by using a localized atomic basis set. The atomic basis functions were chosen by fitting to fully first principles molecular spectra and charge distributions. Hartree integrals are evaluated exactly, exchange integrals are evaluated approximately as a first-order expansion over an equilibrium electron density. The correlation potential and hence correlation integrals are neglected. SCEDTB produces reliable results for energy spectra and Mulliken populations for hydrocarbon molecules and hydrogenated diamond surfaces. The remainder of this paper is organized as follows. Presented in Section 2 are simulation results related to the bandstructure and electrostatic potential size dependence. We include these to demonstrate that the size effect due to the quantum confinement cannot influence the field emission mechanism. It was found that for clusters larger than 1 nm, the cluster size has no significant effect on the energy spectrum and Coulomb potential distribution. This finding is in good agreement with the recent X-ray spectroscopy studies and ab-initio simulations [15]. Section 3 contains simulation results used to compare the HEM and IEM. The calculations indicate that the low emission threshold for hydrogen passivated diamond clusters can be due to hydrogen-assisted emission from the edges of small unpassivated islands. This mechanism is essentially IEM, which was proposed as a dominant emission mechanism for hydrogenated polycrystalline diamond films containing un-hydrogenated sp2-like patches [11].
cluster. To assess the magnitude of the size effect we considered six clusters shown in Fig. 1. All but the 945 carbon atom clusters have octahedron shapes represented by (111) facets with the top and the bottom vertices cut off to produce (100) surfaces. All clusters had been hydrogenated and (001) surfaces reconstructed to reduce the surface energy. An analytical bond-order potential [16] was used to relax structures presented throughout this paper to their minimum energy configuration. Plotted in Fig. 2 is the density of states for four diamond clusters. The solid line indicates density of sates for bulk diamond. The bulk density of states is shifted by the averaged Coulomb potential experienced by carbon atoms in the cluster (cf. Fig. 3) due to the surface dipole layer. Dimensional effects are most apparent for the states in the valence band; the highest-occupied molecular orbitals for the 34 and 161 atom clusters are located respectively 2.5 and 1.25 eV below the bulk valence band edge. At the same time, even for the smallest cluster the energy of the lowestunoccupied molecular orbitals almost coincide with the bulk conduction band edge. This result can be intuitively expected because the states with higher energies and smaller wavelengths are less sensitive to the dimension of the system. Band gap size effect for diamond clusters is in sharp contrast to silicon and germanium clusters, which exhibit quantum confinement effects for clusters up to 7 nm in diameter [15,21]. Because conduction energy states in
2. Size effect It is convenient to separate the field emission related properties of ND coatings in two groups. The first group is related to the properties of a stand-alone perfect ND cluster. The second group is associated with ‘‘imperfections’’ such as the boundaries between clusters or residues of outer cluster shells, which may an contain an sp2 phase, and thus be responsible for IEM. In this section we consider the first group. Potentially, variation of the bandgap and Coulomb potential due to the dimensional effect can result in a built-in effective field, which facilitates electron extraction from the
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Fig. 1. Illustrations of the nano-diamond clusters studied.
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Fig. 2. SC-EDTB electronic density of states for the four smallest clusters illustrated in Fig. 1 (histogram) and for bulk diamond (solid line). The histogram at the bottom of pane (a) is a DFT spectrum generated by CERIUS 2; builder: DMOL3.
diamond are about 3 eV higher than in Si or Ge, they are less sensitive to quantum confinement, and therefore diamond shows a less pronounced size effect than semiconductors. As follows from Fig. 4, band gap size effect is negligible for clusters larger than 2.2 –2.5 nm. This critical cluster size for quantum confinement is in a reasonably good agreement with X-ray spectroscopic measurements and ab-initio calculations [7]. At the same time, it apparently disagrees with earlier indirect X-ray measurements [8] for CVD grown ND coating, which indicate that in 3.6 nm cluster the conduction band edge position is 1.2 eV above the conduction band edge for bulk diamond. The Coulomb potential experienced by electrons in the clusters is plotted in Fig. 3. We plot the Coulomb potential
associated with an uncompensated rather than with a total charge. The potential related to a given carbon atom is produced by the difference between the actual charge density for this atom and the charge density associated with a carbon atom in bulk diamond. Potential related to a given hydrogen atom is produced by the difference between the actual charge density and atomic hydrogen charge density. Subtraction of background charge density eliminates short-range potential oscillations associated with single atoms and makes the long-range potential component due to interatomic charge transfer clearly visible. The main feature of the potential distribution in Fig. 3 is a sharp rise at the cluster surface produced by the hydrogencarbon dipole layer. The potential size effect is significant
Fig. 3. Coulomb potential distributions for the nano-diamond clusters plotted along the h100i and h111i directions passing through the centers of mass of the clusters.
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Fig. 4. Size dependence of a band gap for ND clusters.
only for the smallest cluster. The potential rise at the boundary gives 1.45 eV electron affinity (EA) of the hydrogenated h111i surface. That value is close to the experimentally measured electron affinity [11]. The present findings indicate that the size effect for ND clusters rapidly diminishes with an increase of cluster size and is negligibly small for larger than 2.5 nm. Size effects cannot be responsible for the enhanced field emission reported in Refs. [4 –6], because the cluster size in these experiments was estimated to be about 5 nm. Also, it cannot account for suppressed electron emission from an individual diamond cluster on Mo tip [7].
3. IEM vs. HEM The IEM assumes that electrons are emitted from small, unpassivated islands on hydrogenated diamond surfaces. This mechanism was experimentally investigated and de-
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scribed by Ristein [11], and therefore only a brief outline of the model is described here. Suppose that there is an infinite hydrogen passivated diamond surface, and that the vacuum potential is equal to zero. It is convenient to view the passivated diamond surface as an infinite dipole layer. Because EA = 1.4 eV the electron potential in diamond is 1.4 V. If there is a small, unpassivated region (Fig. 5(a)) on a diamond surface it can be considered as an opening in the infinite dipole layer. From symmetry reasons it can be inferred that the potential at the middle of the opening is 0.7 V, which is the average between the potentials in diamond (1.4 V) and in vacuum (0 V) infinitely far from the interface (Fig. 5(b)). Due to the presence of k-bonding the unpassivated surface is metallic. The total potential is a sum of the dipole layer, image force, and applied potentials. Electron emission occurs from the edges of the unpassivated region along the field line as schematically shown in Fig. 5(c). Rather than overcoming a high potential barrier right in front of the unpassivated island, electrons skirt this region by inclining towards the lower potential passivated region. Hence, the potential barrier for IEM is about 1.0 eV lower than for electrons emitted from a completely unpassivated diamond surface. The HEM assumes that electrons are first injected into the diamond conduction band from a metallic or semiconducting backcontact. Injection occurs by tunneling through a triangular shaped barrier skewed by an applied field. Injected electrons are then allowed to freely leave to vacuum through the negative EA of the outer hydrogenated surface. For very thin diamond films or for sufficiently small ND clusters the Fermi energy is always below the conduction band (CB) edge (Fig. 6(b)). In such cases emission may result from direct tunneling through the diamond band gap. To make the barrier more penetrable for tunneling electrons it is desirable to decrease the energy of the CB edge until the EA becomes zero (Fig 6(c)). Further CB edge lowering
Fig. 5. (a) Schematic diagram of infinite hydrogenated diamond surface with small unpassivated metallic patch. (b) Equipotential lines for the infinite dipole layer with circular hole in it. (c) Potential experienced by an electron in vacuum in the vicinity of a round unpassivated metallic patch when a uniform external field is applied normally to the dipole layer. The force field line originating from the patch edge shows the classical trajectory for an emitted electron.
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Fig. 6. Illustration of homogeneous emission through a ND cluster. Band structure for graphite/diamond/vacuum when (a) no field is applied and (b) under applied field. The difference between the Fermi level and CB edge depends on geometry and may vary between 5.5 and 3.0 eV. The CB edge position in ta-C depends on amorphization degree. Band structures in an applied field for ta-C with zero (c) and positive (d) EA.
leads to a positive EA (Fig. 6(d)). Under non-resonance conditions the double barrier structure has a maximum transparency when the difference between the vacuum and Fermi level is split equally between the two barriers. Experimental studies [17] indicate that the degree of amorphization of tetrahedrally coordinated amorphous carbon (ta-C) and the amount of hydrogen in the bulk can be used for tuning the band gap size between 5.5 and 0.5 eV. The case that resembles Fig. 6(c) was implemented for these calculations. A zero EA eigenvalue spectrum for a hydrogen passivated ta-C cluster composed of 5– 7-membered rings is plotted in Fig. 7. The discussion of the simulation results starts with the HEM. The system illustrated in Fig. 8(a) is used to analyze the HEM for ND clusters. To make the comparison between
HEM and IEM more realistic a ta-C cluster is considered instead of a crystalline ND particle because the former has a lower emission threshold. A graphene/a-C phase backcontact connecting the ND to Si is modeled by a short metallic single-wall nano-tube (SWNT). An applied field of 0.21 V/ ˚ is directed along the SWNT axis. Plotted in Fig. 8(b) is A the Coulomb potential along the line lying at the SWNT surface. The potential is constant at the surface of the metallic nanotube, and changes linearly inside the semiconducting cluster. The dielectric permittivity of diamond is 5.7, and thus the potential change rate inside the cluster is less than in vacuum. Note, however, that due to the small cluster size the field intensity in vacuum is almost the same as inside the cluster. Fig. 8(b) is used to derive the shape of the potential barrier (Fig. 8(c)) to be overcome by tunneling
Fig. 7. Eigenvalue spectrum for a hydrogen passivated 5 – 7 membered ring ta-C cluster composed of 276 carbon and 168 hydrogen atoms. To account for an EA = 1.5 eV the bulk diamond spectrum (solid line) was shifted by 1.5 eV.
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The one-dimensional WKB approximation ( PðEÞ ¼ exp 2
Z
b a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 2m ½V ðxÞ Edx h2
ð1Þ
is used to estimate the tunneling probability through the barrier shown in Fig. 8(c) as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3:0 1:65 ¼ 0:36 exp 1:65 þ x dx 8:2 0 Z 5:1 pffiffiffiffiffiffiffiffiffi 1:65 dx ¼ 3:5 109 Z
PHEM
8:2
ð1aÞ
0
For evaluation of Eq. (1a), m is assumed to be the electron mass in vacuum.
Fig. 8. (a) System used to analyze the HEM. (b) The electron Coulomb potential along the line parallel to the SWNT and lying on its surface. (c) Schematic representation of a one-dimensional potential barrier seen by conduction electrons in SWNT. (d) Lowest energy CB wave function. The size of the circle represents MP for the given wave function for each particular atom. A cross-sectional view is shown for clarity. Distances are in ˚ , potentials in eV. A
electrons. The center of mass (point C in Fig. 8(b)) is chosen to match a zero external potential. The potential is measured with respect to the vacuum level infinitely far from the system in the plane passing through point C and normal to the applied field. The Fermi energy appears to be 1.65 eV, and the lowest CB wave function (Fig. 8(d)) lies 1.65 eV above Fermi level. This wave function is located in the flat potential region AB. Region AB in Fig. 9(b) corresponds to plateau EF on the schematic band structure diagram (Fig. 8(c)). The plateau is located 1.65 eV above the Fermi level and stands for the CB edge in this region. In the rest of the cluster the CB edge varies linearly in the same fashion as the Coulomb potential. Segment FG has the same slope as segment BD. Thus it is inferred that the barrier height at the tube – cluster contact is 3.0 eV. As follows from Fig. 8(b), the vacuum potential near the cluster surface is 1.75 eV and just 0.1 eV below Fermi level. That defines point H in the band structure diagram. Note that although the standalone ta-C cluster exhibits an almost zero EA (Fig. 7), the system composed of the ta-C cluster and SWNT has a negative EA. This is due to the edge effect, i.e. plateau EF, and because the ta-C cluster becomes negatively charged when it comes into electric contact with the metallic SWNT in an applied field. Hence its band structure is shifted towards higher energies and the EA becomes negative.
Fig. 9. (a – b) Wave functions with the energies belonging to the interval [EF, EF + 0.5 eV]. The circle sizes are proportional to the MPs in the given energy range. (b) Same view as in (a); pane (a) is the bottom view. (c) Coulomb potential along field line AD. (d) Electron density at the emitting atom along the line connecting the nucleus and point A. (e) Cross-sectional Coulomb potential plot. The cross-section plane passes through point A, and is oriented in the same way as pane (a). (f) Coulomb potential along ˚ , potentials in eV, density in A ˚ 3. line BC. Distances are in A
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To make the system shown in Fig. 9(a) suitable for an IEM analysis the hydrogen atoms are removed within a narrow strip that connects the SWNT and the left cluster facet. Other system parameters and the applied field remain the same. The unpassivated surface strip is metallic, and hence it has the same potential as the metallic SWNT. Illustrated in Fig. 9(a,b) are wave functions with energies within 0.5 eV above the Fermi level. The existence of wave functions that have energies near the Fermi level and are spread continuously between two given points implies the possibility of current flow between these points. Thus the only obstacle the emitted electrons need to overcome is the potential barrier between the surface atom(s) and the vacuum equipotential line with the energy equal to the Fermi level. To obtain the shape of that barrier, field lines are calculated that originate from the grid points on the surface of spheres centered on the surface atoms of the cluster. The tunneling probability is calculated from Eq. (1) for each field line using integration along the field line instead of a one-dimensional integration. The force line corresponding to the highest tunneling probability is selected. The grid point on the sphere surrounding the particular atom where the optimal field line originates is considered the emission site. In Fig. 9, the emission site is marked by point A and the optimal field line AD is indicated. Point D corresponds to the potential equal to the Fermi energy, i.e. electron motion along the field line beyond point D corresponds to a positive electron energy. Note that the field line has a sharp bend towards the hydrogenated surface near the emission site. This is similar to the one shown in Fig. 5(c), and is considered an IEM hallmark. Point A is not well defined; it is chosen here to lie on the ˚ radius. Plotted in Fig. 9(d) is the radial sphere with a 1.0 A density distribution for the surface atom where the optimal field line originates. The density is plotted along the line connecting point A and the nucleus of the emission atom (in Fig. 9(d) it has coordinate 1.0). As seen from the plot, the density at A is one-third of the peak density, which is assumed to be sufficient for emission. Applying Eq. (1) to the barrier shown in Fig. 9(c) results in a tunneling probability equal to 4.1 10 5, which is four orders of magnitude higher than the HEM case. If the optimal field line is started from the nucleus rather than point A, the tunneling probability will decrease by a factor of 20. That, however, does not change the prevalence of the IEM over the HEM for tunneling probability in ND coatings. The value of electron mass to be used when applying Eq. (1) to the barrier inside the ND or ta-C particle is not well determined. The electron effective mass in semiconductors can be used only in the vicinity of the CB edge because the idea of effective mass is based on the periodic nature of Bloch functions. When the electron energy is 1.6– 3.0 eV below the CB edge, instead of periodic oscillations the wave function exhibits a strong exponential decay and the concept of effective electron mass becomes invalid. That is the reason for using an effective mass equal to unity for
estimating the HEM tunneling probability. However, it is interesting to note that in this example the HEM and IEM tunneling probabilities become equal when the electron effective mass in Eq. (1) equals 0.27. The reported effective electron mass values for diamond substantially differs from those for ta-C. The experimental analysis of drift velocity in diamond [18] gives m? = 0.36 and mN = 1.4. Similar values for diamond were obtained from DF local density approximation simulations (m? = 0.26 and mN = 1.50) [19]. On the other hand the study of blue shift variations in ta-C superlattices with respect to quantum well thickness [20] gives a much smaller averaged effective mass mave = 0.067. It is necessary to keep in mind that the one-dimensional approximation may be inadequate for tunneling efficiency estimates. The total tunneling efficiency can be viewed as an integral over the cluster surface for the HEM and over all force lines originating from all surface emission sites for the IEM. Due to the integration the ratio of total HEM to IEM probabilities may substantially increase. However, this ratio decreases for small fields because barrier lowering for the IEM does not depend on applied field, while HEM tunneling strongly depends on barrier tilt. Therefore it is likely that the onset of emission current at the threshold applied voltage occurs due to inhomogeneous emission. Note that ˚ applied field, which we used solely for demona 0.21 V/A stration purposes, is difficult to achieve even with a strong field enhancement at a Si needle tip. Experimental setups [4 – 7] result in a much smaller barrier tilt than is schematically shown in Fig. 6. The chains of unpassivated islands act like atomically thin metal whiskers. Because field enhancement is proportional to aspect ratio, such whiskers may significantly increase the extracting field. This idea is illustrated by Fig. 9(f), which shows the Coulomb potential along line BC (Fig. 9(a)) that passes in close proximity to the SWNT and to the surface conducting channel. The potential of the conducting channel is the same as the potential of the SWNT. At the same time, the Coulomb potential around the channel, both in vacuum and in the semiconducting cluster, changes in a linear fashion. Strong field enhancement can be seen from Fig. 9(e). The potential at the emitting atom is much higher than the potential one interatomic distance apart. Note that the emission site in Fig. 9(e) looks like an isolated high potential region because other atoms constituting the conduction strip are below the crosssectional plane.
4. Conclusions The presented results demonstrate that perfect ND clusters do not exhibit an appreciable size effect, which may somehow influence field emission. At the same time, incomplete hydrogen passivation and/or a remaining graphite phase may create perfect conditions for hydrogen-assisted inhomogeneous emission from the nano-diamond
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aggregate. We conclude that the enhanced field emission results from the surface imperfections rather than the ‘‘bulk’’ properties of nano-diamond clusters.
Acknowledgements This work was funded by a supplement to the Office of Naval Research Contract N00014-95-1-0279 and through a Multi-University Research Initiative funded by the Office of Naval Research through a subcontract from the University of North Carolina at Chapel Hill. The work has been completed at Naval Research Laboratory, Code 6180.
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Electronic properties of diamond clusters: self-consistent tight binding simulation D.A. Areshkin, O.A. Shenderova, S.P. Adiga, D.W. Brenner * Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7907, USA Received 21 August 2003; received in revised form 26 April 2004; accepted 29 April 2004 Available online 28 July 2004
Abstract A self-consistent environment-dependent tight binding method is used to examine electron emission-related properties of hydrogen passivated nano-diamond (ND) particles. For sizes larger than 2.5 nm particle bandgap was found to be equal to the bandgap of bulk diamond. Coulomb potential distributions and electron affinities of clusters were found to be insensitive to the particle size if it exceeds 1.0 nm. Tunneling probabilities for homogeneous and inhomogeneous emission models were estimated. The simulation results indicate that the low emission threshold for hydrogen passivated diamond nano-clusters is due to hydrogen-assisted emission from the edges of small unpassivated islands. Essentially the same mechanism is claimed to be responsible for good emission properties of hydrogen passivated diamond films by Ristein [Diam. Relat. Mater. 9, 1129 (2000)]. D 2004 Elsevier B.V. All rights reserved. Keywords: Diamond clusters; Field emission; Electron affinity
1. Introduction Nano-diamond (ND) clusters, also called ultradispersed diamond (UDD), can be obtained from detonation products in large amounts with high uniformity and reproducibility [1,2]. After removal of an external disordered carbon shell, the sizes of nano-diamond clusters range from 2 to 10 nm, with a narrow peak in size distribution at about 5 nm [3]. Traditionally, UDD is used for galvanic coatings, polymer composites, polishing, and lubricant additives [3]. Attempts to use UDD powder for field emission enhancement [4– 7] and suggestions for other electronic applications [8] were also reported. There is experimental evidence [6] that ND coatings deposited on Si needle-shaped field emitters substantially lowers emission threshold voltages relative to that of bare Si needles and needles with micro-diamond powder coatings. ND coatings are created either by bias enhanced microwave plasma chemical vapor deposition [9] or by electrophoresis from a suspension of diamond powder in ethanol [10]. After deposition the coatings are exposed to a hydrogen plasma, * Corresponding author. E-mail address: [email protected] (D.W. Brenner). 0925-9635/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2004.04.012
which is intended to have a twofold effect. First, it etches much of the loosely bounded carbon atoms, preferentially in the outer regions of the graphene and/or amorphous carbon (a-C) layers that surround ND particles. Second, the hydrogen treatment is used to saturate free carbon bonds at the surface. TEM images of a Si needle tip with a ND coating, however, show surface roughness and irregularity that likely inhibits perfect hydrogen coverage [7], and the hydrogenated ND surface may therefore contain small, unpassivated islands. Chains of such islands would serve as conducting channels between a graphene/a-C phase backcontact and the cluster’s outer surface from which electrons are emitted. The issue addressed here is a comparison of emission probabilities from the surface and through the bulk of ND clusters. These two main field emission mechanisms that are explored are called the Inhomogeneous and Homogeneous Emission Models (IEM and HEM), respectively [11]. At the same time, the field emission threshold from a Mo tip with a single ND particle is higher compared to the threshold of bare Mo [6]. That speaks in favor of the IEM because from the HEM standpoint single particle properties are similar to the properties of ND coating. On the other hand a single particle may not have surface irregularities inherent for a ND coating, which are required for IEM. The goal of
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this paper is to explain field emission results by using selfconsistent (SC) environment-dependent tight binding (SCEDTB) [12,13] simulations. SC-EDTB is a hybrid minimum basis set TB-DFT scheme, which was designed to model moderate size (100 –1000 atoms) metallic or semiconducting systems. The method builds on a non-self-consistent environment-dependent TB model for carbon [14] with parameters added to describe hydrocarbon bonds and to account for self-consistent charge transfer. In this approach, the Hamiltonian matrix H is composed of two parts, the original EDTB [14] matrix H0, and SC corrections DH. The latter accounts for charge transfer between constituent atoms, and is supposed to be zero for periodic structures for which the original non-SC EDTB parameterization was made. DH is computed in a standard DFT manner by using a localized atomic basis set. The atomic basis functions were chosen by fitting to fully first principles molecular spectra and charge distributions. Hartree integrals are evaluated exactly, exchange integrals are evaluated approximately as a first-order expansion over an equilibrium electron density. The correlation potential and hence correlation integrals are neglected. SCEDTB produces reliable results for energy spectra and Mulliken populations for hydrocarbon molecules and hydrogenated diamond surfaces. The remainder of this paper is organized as follows. Presented in Section 2 are simulation results related to the bandstructure and electrostatic potential size dependence. We include these to demonstrate that the size effect due to the quantum confinement cannot influence the field emission mechanism. It was found that for clusters larger than 1 nm, the cluster size has no significant effect on the energy spectrum and Coulomb potential distribution. This finding is in good agreement with the recent X-ray spectroscopy studies and ab-initio simulations [15]. Section 3 contains simulation results used to compare the HEM and IEM. The calculations indicate that the low emission threshold for hydrogen passivated diamond clusters can be due to hydrogen-assisted emission from the edges of small unpassivated islands. This mechanism is essentially IEM, which was proposed as a dominant emission mechanism for hydrogenated polycrystalline diamond films containing un-hydrogenated sp2-like patches [11].
cluster. To assess the magnitude of the size effect we considered six clusters shown in Fig. 1. All but the 945 carbon atom clusters have octahedron shapes represented by (111) facets with the top and the bottom vertices cut off to produce (100) surfaces. All clusters had been hydrogenated and (001) surfaces reconstructed to reduce the surface energy. An analytical bond-order potential [16] was used to relax structures presented throughout this paper to their minimum energy configuration. Plotted in Fig. 2 is the density of states for four diamond clusters. The solid line indicates density of sates for bulk diamond. The bulk density of states is shifted by the averaged Coulomb potential experienced by carbon atoms in the cluster (cf. Fig. 3) due to the surface dipole layer. Dimensional effects are most apparent for the states in the valence band; the highest-occupied molecular orbitals for the 34 and 161 atom clusters are located respectively 2.5 and 1.25 eV below the bulk valence band edge. At the same time, even for the smallest cluster the energy of the lowestunoccupied molecular orbitals almost coincide with the bulk conduction band edge. This result can be intuitively expected because the states with higher energies and smaller wavelengths are less sensitive to the dimension of the system. Band gap size effect for diamond clusters is in sharp contrast to silicon and germanium clusters, which exhibit quantum confinement effects for clusters up to 7 nm in diameter [15,21]. Because conduction energy states in
2. Size effect It is convenient to separate the field emission related properties of ND coatings in two groups. The first group is related to the properties of a stand-alone perfect ND cluster. The second group is associated with ‘‘imperfections’’ such as the boundaries between clusters or residues of outer cluster shells, which may an contain an sp2 phase, and thus be responsible for IEM. In this section we consider the first group. Potentially, variation of the bandgap and Coulomb potential due to the dimensional effect can result in a built-in effective field, which facilitates electron extraction from the
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Fig. 1. Illustrations of the nano-diamond clusters studied.
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Fig. 2. SC-EDTB electronic density of states for the four smallest clusters illustrated in Fig. 1 (histogram) and for bulk diamond (solid line). The histogram at the bottom of pane (a) is a DFT spectrum generated by CERIUS 2; builder: DMOL3.
diamond are about 3 eV higher than in Si or Ge, they are less sensitive to quantum confinement, and therefore diamond shows a less pronounced size effect than semiconductors. As follows from Fig. 4, band gap size effect is negligible for clusters larger than 2.2 –2.5 nm. This critical cluster size for quantum confinement is in a reasonably good agreement with X-ray spectroscopic measurements and ab-initio calculations [7]. At the same time, it apparently disagrees with earlier indirect X-ray measurements [8] for CVD grown ND coating, which indicate that in 3.6 nm cluster the conduction band edge position is 1.2 eV above the conduction band edge for bulk diamond. The Coulomb potential experienced by electrons in the clusters is plotted in Fig. 3. We plot the Coulomb potential
associated with an uncompensated rather than with a total charge. The potential related to a given carbon atom is produced by the difference between the actual charge density for this atom and the charge density associated with a carbon atom in bulk diamond. Potential related to a given hydrogen atom is produced by the difference between the actual charge density and atomic hydrogen charge density. Subtraction of background charge density eliminates short-range potential oscillations associated with single atoms and makes the long-range potential component due to interatomic charge transfer clearly visible. The main feature of the potential distribution in Fig. 3 is a sharp rise at the cluster surface produced by the hydrogencarbon dipole layer. The potential size effect is significant
Fig. 3. Coulomb potential distributions for the nano-diamond clusters plotted along the h100i and h111i directions passing through the centers of mass of the clusters.
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Fig. 4. Size dependence of a band gap for ND clusters.
only for the smallest cluster. The potential rise at the boundary gives 1.45 eV electron affinity (EA) of the hydrogenated h111i surface. That value is close to the experimentally measured electron affinity [11]. The present findings indicate that the size effect for ND clusters rapidly diminishes with an increase of cluster size and is negligibly small for larger than 2.5 nm. Size effects cannot be responsible for the enhanced field emission reported in Refs. [4 –6], because the cluster size in these experiments was estimated to be about 5 nm. Also, it cannot account for suppressed electron emission from an individual diamond cluster on Mo tip [7].
3. IEM vs. HEM The IEM assumes that electrons are emitted from small, unpassivated islands on hydrogenated diamond surfaces. This mechanism was experimentally investigated and de-
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scribed by Ristein [11], and therefore only a brief outline of the model is described here. Suppose that there is an infinite hydrogen passivated diamond surface, and that the vacuum potential is equal to zero. It is convenient to view the passivated diamond surface as an infinite dipole layer. Because EA = 1.4 eV the electron potential in diamond is 1.4 V. If there is a small, unpassivated region (Fig. 5(a)) on a diamond surface it can be considered as an opening in the infinite dipole layer. From symmetry reasons it can be inferred that the potential at the middle of the opening is 0.7 V, which is the average between the potentials in diamond (1.4 V) and in vacuum (0 V) infinitely far from the interface (Fig. 5(b)). Due to the presence of k-bonding the unpassivated surface is metallic. The total potential is a sum of the dipole layer, image force, and applied potentials. Electron emission occurs from the edges of the unpassivated region along the field line as schematically shown in Fig. 5(c). Rather than overcoming a high potential barrier right in front of the unpassivated island, electrons skirt this region by inclining towards the lower potential passivated region. Hence, the potential barrier for IEM is about 1.0 eV lower than for electrons emitted from a completely unpassivated diamond surface. The HEM assumes that electrons are first injected into the diamond conduction band from a metallic or semiconducting backcontact. Injection occurs by tunneling through a triangular shaped barrier skewed by an applied field. Injected electrons are then allowed to freely leave to vacuum through the negative EA of the outer hydrogenated surface. For very thin diamond films or for sufficiently small ND clusters the Fermi energy is always below the conduction band (CB) edge (Fig. 6(b)). In such cases emission may result from direct tunneling through the diamond band gap. To make the barrier more penetrable for tunneling electrons it is desirable to decrease the energy of the CB edge until the EA becomes zero (Fig 6(c)). Further CB edge lowering
Fig. 5. (a) Schematic diagram of infinite hydrogenated diamond surface with small unpassivated metallic patch. (b) Equipotential lines for the infinite dipole layer with circular hole in it. (c) Potential experienced by an electron in vacuum in the vicinity of a round unpassivated metallic patch when a uniform external field is applied normally to the dipole layer. The force field line originating from the patch edge shows the classical trajectory for an emitted electron.
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Fig. 6. Illustration of homogeneous emission through a ND cluster. Band structure for graphite/diamond/vacuum when (a) no field is applied and (b) under applied field. The difference between the Fermi level and CB edge depends on geometry and may vary between 5.5 and 3.0 eV. The CB edge position in ta-C depends on amorphization degree. Band structures in an applied field for ta-C with zero (c) and positive (d) EA.
leads to a positive EA (Fig. 6(d)). Under non-resonance conditions the double barrier structure has a maximum transparency when the difference between the vacuum and Fermi level is split equally between the two barriers. Experimental studies [17] indicate that the degree of amorphization of tetrahedrally coordinated amorphous carbon (ta-C) and the amount of hydrogen in the bulk can be used for tuning the band gap size between 5.5 and 0.5 eV. The case that resembles Fig. 6(c) was implemented for these calculations. A zero EA eigenvalue spectrum for a hydrogen passivated ta-C cluster composed of 5– 7-membered rings is plotted in Fig. 7. The discussion of the simulation results starts with the HEM. The system illustrated in Fig. 8(a) is used to analyze the HEM for ND clusters. To make the comparison between
HEM and IEM more realistic a ta-C cluster is considered instead of a crystalline ND particle because the former has a lower emission threshold. A graphene/a-C phase backcontact connecting the ND to Si is modeled by a short metallic single-wall nano-tube (SWNT). An applied field of 0.21 V/ ˚ is directed along the SWNT axis. Plotted in Fig. 8(b) is A the Coulomb potential along the line lying at the SWNT surface. The potential is constant at the surface of the metallic nanotube, and changes linearly inside the semiconducting cluster. The dielectric permittivity of diamond is 5.7, and thus the potential change rate inside the cluster is less than in vacuum. Note, however, that due to the small cluster size the field intensity in vacuum is almost the same as inside the cluster. Fig. 8(b) is used to derive the shape of the potential barrier (Fig. 8(c)) to be overcome by tunneling
Fig. 7. Eigenvalue spectrum for a hydrogen passivated 5 – 7 membered ring ta-C cluster composed of 276 carbon and 168 hydrogen atoms. To account for an EA = 1.5 eV the bulk diamond spectrum (solid line) was shifted by 1.5 eV.
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The one-dimensional WKB approximation ( PðEÞ ¼ exp 2
Z
b a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 2m ½V ðxÞ Edx h2
ð1Þ
is used to estimate the tunneling probability through the barrier shown in Fig. 8(c) as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3:0 1:65 ¼ 0:36 exp 1:65 þ x dx 8:2 0 Z 5:1 pffiffiffiffiffiffiffiffiffi 1:65 dx ¼ 3:5 109 Z
PHEM
8:2
ð1aÞ
0
For evaluation of Eq. (1a), m is assumed to be the electron mass in vacuum.
Fig. 8. (a) System used to analyze the HEM. (b) The electron Coulomb potential along the line parallel to the SWNT and lying on its surface. (c) Schematic representation of a one-dimensional potential barrier seen by conduction electrons in SWNT. (d) Lowest energy CB wave function. The size of the circle represents MP for the given wave function for each particular atom. A cross-sectional view is shown for clarity. Distances are in ˚ , potentials in eV. A
electrons. The center of mass (point C in Fig. 8(b)) is chosen to match a zero external potential. The potential is measured with respect to the vacuum level infinitely far from the system in the plane passing through point C and normal to the applied field. The Fermi energy appears to be 1.65 eV, and the lowest CB wave function (Fig. 8(d)) lies 1.65 eV above Fermi level. This wave function is located in the flat potential region AB. Region AB in Fig. 9(b) corresponds to plateau EF on the schematic band structure diagram (Fig. 8(c)). The plateau is located 1.65 eV above the Fermi level and stands for the CB edge in this region. In the rest of the cluster the CB edge varies linearly in the same fashion as the Coulomb potential. Segment FG has the same slope as segment BD. Thus it is inferred that the barrier height at the tube – cluster contact is 3.0 eV. As follows from Fig. 8(b), the vacuum potential near the cluster surface is 1.75 eV and just 0.1 eV below Fermi level. That defines point H in the band structure diagram. Note that although the standalone ta-C cluster exhibits an almost zero EA (Fig. 7), the system composed of the ta-C cluster and SWNT has a negative EA. This is due to the edge effect, i.e. plateau EF, and because the ta-C cluster becomes negatively charged when it comes into electric contact with the metallic SWNT in an applied field. Hence its band structure is shifted towards higher energies and the EA becomes negative.
Fig. 9. (a – b) Wave functions with the energies belonging to the interval [EF, EF + 0.5 eV]. The circle sizes are proportional to the MPs in the given energy range. (b) Same view as in (a); pane (a) is the bottom view. (c) Coulomb potential along field line AD. (d) Electron density at the emitting atom along the line connecting the nucleus and point A. (e) Cross-sectional Coulomb potential plot. The cross-section plane passes through point A, and is oriented in the same way as pane (a). (f) Coulomb potential along ˚ , potentials in eV, density in A ˚ 3. line BC. Distances are in A
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To make the system shown in Fig. 9(a) suitable for an IEM analysis the hydrogen atoms are removed within a narrow strip that connects the SWNT and the left cluster facet. Other system parameters and the applied field remain the same. The unpassivated surface strip is metallic, and hence it has the same potential as the metallic SWNT. Illustrated in Fig. 9(a,b) are wave functions with energies within 0.5 eV above the Fermi level. The existence of wave functions that have energies near the Fermi level and are spread continuously between two given points implies the possibility of current flow between these points. Thus the only obstacle the emitted electrons need to overcome is the potential barrier between the surface atom(s) and the vacuum equipotential line with the energy equal to the Fermi level. To obtain the shape of that barrier, field lines are calculated that originate from the grid points on the surface of spheres centered on the surface atoms of the cluster. The tunneling probability is calculated from Eq. (1) for each field line using integration along the field line instead of a one-dimensional integration. The force line corresponding to the highest tunneling probability is selected. The grid point on the sphere surrounding the particular atom where the optimal field line originates is considered the emission site. In Fig. 9, the emission site is marked by point A and the optimal field line AD is indicated. Point D corresponds to the potential equal to the Fermi energy, i.e. electron motion along the field line beyond point D corresponds to a positive electron energy. Note that the field line has a sharp bend towards the hydrogenated surface near the emission site. This is similar to the one shown in Fig. 5(c), and is considered an IEM hallmark. Point A is not well defined; it is chosen here to lie on the ˚ radius. Plotted in Fig. 9(d) is the radial sphere with a 1.0 A density distribution for the surface atom where the optimal field line originates. The density is plotted along the line connecting point A and the nucleus of the emission atom (in Fig. 9(d) it has coordinate 1.0). As seen from the plot, the density at A is one-third of the peak density, which is assumed to be sufficient for emission. Applying Eq. (1) to the barrier shown in Fig. 9(c) results in a tunneling probability equal to 4.1 10 5, which is four orders of magnitude higher than the HEM case. If the optimal field line is started from the nucleus rather than point A, the tunneling probability will decrease by a factor of 20. That, however, does not change the prevalence of the IEM over the HEM for tunneling probability in ND coatings. The value of electron mass to be used when applying Eq. (1) to the barrier inside the ND or ta-C particle is not well determined. The electron effective mass in semiconductors can be used only in the vicinity of the CB edge because the idea of effective mass is based on the periodic nature of Bloch functions. When the electron energy is 1.6– 3.0 eV below the CB edge, instead of periodic oscillations the wave function exhibits a strong exponential decay and the concept of effective electron mass becomes invalid. That is the reason for using an effective mass equal to unity for
estimating the HEM tunneling probability. However, it is interesting to note that in this example the HEM and IEM tunneling probabilities become equal when the electron effective mass in Eq. (1) equals 0.27. The reported effective electron mass values for diamond substantially differs from those for ta-C. The experimental analysis of drift velocity in diamond [18] gives m? = 0.36 and mN = 1.4. Similar values for diamond were obtained from DF local density approximation simulations (m? = 0.26 and mN = 1.50) [19]. On the other hand the study of blue shift variations in ta-C superlattices with respect to quantum well thickness [20] gives a much smaller averaged effective mass mave = 0.067. It is necessary to keep in mind that the one-dimensional approximation may be inadequate for tunneling efficiency estimates. The total tunneling efficiency can be viewed as an integral over the cluster surface for the HEM and over all force lines originating from all surface emission sites for the IEM. Due to the integration the ratio of total HEM to IEM probabilities may substantially increase. However, this ratio decreases for small fields because barrier lowering for the IEM does not depend on applied field, while HEM tunneling strongly depends on barrier tilt. Therefore it is likely that the onset of emission current at the threshold applied voltage occurs due to inhomogeneous emission. Note that ˚ applied field, which we used solely for demona 0.21 V/A stration purposes, is difficult to achieve even with a strong field enhancement at a Si needle tip. Experimental setups [4 – 7] result in a much smaller barrier tilt than is schematically shown in Fig. 6. The chains of unpassivated islands act like atomically thin metal whiskers. Because field enhancement is proportional to aspect ratio, such whiskers may significantly increase the extracting field. This idea is illustrated by Fig. 9(f), which shows the Coulomb potential along line BC (Fig. 9(a)) that passes in close proximity to the SWNT and to the surface conducting channel. The potential of the conducting channel is the same as the potential of the SWNT. At the same time, the Coulomb potential around the channel, both in vacuum and in the semiconducting cluster, changes in a linear fashion. Strong field enhancement can be seen from Fig. 9(e). The potential at the emitting atom is much higher than the potential one interatomic distance apart. Note that the emission site in Fig. 9(e) looks like an isolated high potential region because other atoms constituting the conduction strip are below the crosssectional plane.
4. Conclusions The presented results demonstrate that perfect ND clusters do not exhibit an appreciable size effect, which may somehow influence field emission. At the same time, incomplete hydrogen passivation and/or a remaining graphite phase may create perfect conditions for hydrogen-assisted inhomogeneous emission from the nano-diamond
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aggregate. We conclude that the enhanced field emission results from the surface imperfections rather than the ‘‘bulk’’ properties of nano-diamond clusters.
Acknowledgements This work was funded by a supplement to the Office of Naval Research Contract N00014-95-1-0279 and through a Multi-University Research Initiative funded by the Office of Naval Research through a subcontract from the University of North Carolina at Chapel Hill. The work has been completed at Naval Research Laboratory, Code 6180.
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