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Nanotechnology and Nanomaterials » "Carbon Nanotubes", book edited by Jose Mauricio Marulanda, ISBN 978-953-307-054-4, Published: March 1, 2010 under CC BY-NC-SA 3.0 license. © The Author(s).

Chapter 17

Carbon Nanotube Field Emitters

By Alexander Zhbanov, Evgeny Pogorelov and Yia-Chung Chang
DOI: 10.5772/39432

Article top

Overview

Photograph of strips of oriented CNTs synthesized on a substrate. The strip width is 20 μm and the gap between the strips is 5 μm (Zhbanov et al., 2004).
Figure 1. Photograph of strips of oriented CNTs synthesized on a substrate. The strip width is 20 μm and the gap between the strips is 5 μm (Zhbanov et al., 2004).
Nanotube growth for various concentrations of catalytic ink used for the precursor application. Catalyst concentration in the solution was 1 mM (a), 5 mM (b), 40 mM (c), and 50 mM (d). The figure is taken from (Bonard et al., 2001b).
Figure 2. Nanotube growth for various concentrations of catalytic ink used for the precursor application. Catalyst concentration in the solution was 1 mM (a), 5 mM (b), 40 mM (c), and 50 mM (d). The figure is taken from (Bonard et al., 2001b).
High-resolution electron microscope image of nanotubes with conical layers. Graphene layers are marked by arrows with points, and the CNT growth direction is marked by large arrows (Musatov et al., 2001).
Figure 3. High-resolution electron microscope image of nanotubes with conical layers. Graphene layers are marked by arrows with points, and the CNT growth direction is marked by large arrows (Musatov et al., 2001).
Scheme of aligned nanotube film, the model of a hemisphere on a post: (a) side view; (b) top view.
Figure 4. Scheme of aligned nanotube film, the model of a hemisphere on a post: (a) side view; (b) top view.
Schemes of simplest models for field enhancement factor estimation: (a) hyperboloid near a plate; (b) hemisphere on a plane; (c) floating sphere at emitter-plane potential, and (d) hemi-ellipsoid on a plane.
Figure 5. Schemes of simplest models for field enhancement factor estimation: (a) hyperboloid near a plate; (b) hemisphere on a plane; (c) floating sphere at emitter-plane potential, and (d) hemi-ellipsoid on a plane.
Hyperboloid near a plate in prolate spheroidal coordinates.
Figure 6. Hyperboloid near a plate in prolate spheroidal coordinates.
Two conducting spheres of radius  at cathode potential in uniform electric E
								0.
Figure 7. Two conducting spheres of radius at cathode potential in uniform electric E 0.
Honeycomb structure, distance between spheres is D, sphere radius is .
Figure 8. Honeycomb structure, distance between spheres is D, sphere radius is .
Modeling of screening effect for floating spheres.
Figure 9. Modeling of screening effect for floating spheres.
Anode current density versus dimensionless sizes: (a) optimal distance between emitters if the height is fixed; (b) influence of emitter height on anode current if the density of emitting centers is constant.
Figure 10. Anode current density versus dimensionless sizes: (a) optimal distance between emitters if the height is fixed; (b) influence of emitter height on anode current if the density of emitting centers is constant.
Geometrical model for the limited distance, L between cathode and anode.
Figure 11. Geometrical model for the limited distance, L between cathode and anode.
Linearly charged thread in a uniform electric field along z.
Figure 12. Linearly charged thread in a uniform electric field along z.
Geometry for the calculation of ponderomotive force acting on the belt between ra
								 = A and ra
								 = B.
Figure 13. Geometry for the calculation of ponderomotive force acting on the belt between ra = A and ra = B.
Distribution of force and total force isolines. (a) Distribution of relative force 
									
										
											
												
													F
													z
												
												(
												H
												
												z
												)
												/
												
													F
													
														t
														o
														t
														a
														l
													
												
											
										
									
									
								over the axis of needle. (b) Isolines of total force on the 
									
										
											
												(
												ρ
												
												
												H
												/
												ρ
												)
											
										
									
									
								 plane.
Figure 14. Distribution of force and total force isolines. (a) Distribution of relative force F z ( H z ) / F t o t a l over the axis of needle. (b) Isolines of total force on the ( ρ H / ρ ) plane.
Distribution of emission current and total emission current isolines for work function φ=4.8 eV. (a) Distribution of relative emission current over Θ. (b) Total emission current isolines on the (βE
								0ρ) plane.
Figure 15. Distribution of emission current and total emission current isolines for work function φ=4.8 eV. (a) Distribution of relative emission current over Θ. (b) Total emission current isolines on the (βE 0ρ) plane.
Infinite set of “image” charges for the simulation of a planar anode.
Figure 16. Infinite set of “image” charges for the simulation of a planar anode.
Emission current as a function of the anode-cathode distance parameter 
									
										
											
												μ
												=
												L
												/
												H
											
										
									
									
								 for constant electric field 
									
										
											
												
													E
													0
												
												≡
												V
												/
												L
											
										
									
									
								 (red line) or 
									
										
											
												
													
														E
														′
													
													0
												
												≡
												V
												/
												(
												L
												−
												H
												)
											
										
									
									
								 (blue line). Radius of curvature is fixed at
									
										
											
												ρ
												=
												5
											
										
									
									
								nm. For solid curves, 
									
										
											
												H
												/
												ρ
												=
												5800
											
										
									
									
								and for dash curves,
									
										
											
												H
												/
												ρ
												=
												10000
											
										
									
									
								. For red and blue solid curves
									
										
											
												
													E
													0
												
												=
												5
												
												V/μ
												
												m
											
										
									
									
								, for red and blue dash curves
									
										
											
												
													E
													0
												
												=
												3
												
												V/μ
												
												m
Figure 17. Emission current as a function of the anode-cathode distance parameter μ = L / H for constant electric field E 0 ≡ V / L (red line) or E ′ 0 ≡ V / ( L − H ) (blue line). Radius of curvature is fixed at ρ = 5 nm. For solid curves, H / ρ = 5800 and for dash curves, H / ρ = 10000 . For red and blue solid curves E 0 = 5 V/μ m , for red and blue dash curves E 0 = 3 V/μ m
Comparison the field enhancement factors for three models: floating sphere (green line), hemi-ellipsoid (red line), and hemisphere on a post (blue line).
Figure 18. Comparison the field enhancement factors for three models: floating sphere (green line), hemi-ellipsoid (red line), and hemisphere on a post (blue line).
Trajectory analysis for the flat-panel display produced by the limited liability company “Volga-Svet” and “CopyTele Inc.” (Zhbanov et al., 2004).
Figure 19. Trajectory analysis for the flat-panel display produced by the limited liability company “Volga-Svet” and “CopyTele Inc.” (Zhbanov et al., 2004).
Compact X-ray tube. The figure is taken from (Musatov et al., 2007).
Figure 20. Compact X-ray tube. The figure is taken from (Musatov et al., 2007).
Schematic of the electron gun for an amplifier. Magnetic field B=0.08 T, voltage V=400 V, and current I=1 A/cm2. The electron gun (1) consists of edge or cylindrical field emitters (2) covered with CNT film, a control electrode (3), and three anodes (4) (Zakharchenko et al., 1996).
Figure 21. Schematic of the electron gun for an amplifier. Magnetic field B=0.08 T, voltage V=400 V, and current I=1 A/cm2. The electron gun (1) consists of edge or cylindrical field emitters (2) covered with CNT film, a control electrode (3), and three anodes (4) (Zakharchenko et al., 1996).

Carbon nanotube field emitters

Alexander Zhbanov1, 2, Evgeny Pogorelov1 and Yia-Chung Chang1

1. Introduction

Application of various one-dimensional nanostructure materials as field emission sources has attracted extensive scientific efforts. Elongated structures are suitable for achieving high field-emission-current density at a low electric field because of their high aspect ratio. Area of its application includes a wide range of field-emission-based devices such as flat-panel displays, electron microscopes, vacuum microwave amplifiers, X-ray tube sources, cathode-ray lamps, nanolithography systems, gas detectors, mass spectrometers etc.

Since the discovery of carbon nanotubes (CNTs) (Iijima, 1991; Iijima & Ichihashi, 1993; Bethune et al., 1993) and experimental observations of their remarkable field emission characteristics (Rinzler et al., 1995; de Heer et al., 1995; Chernazatonskii et al., 1995), significant efforts have been devoted to the application of using CNTs for electron sources.

One of the main problems for design such field emission emitter is the difficulties in estimation of the electric field on the apex of nanotubes. Only a few works considered forces acting on nanoemitters under electric field. Thus far, there is no analytical formula which provides a good approximation to the total current generated by the nanoscale field emitter. In this chapter, we theoretically consider the electric field strength, field enhancement factor, ponderomotive forces, and total current of a metallic elliptical needle in the form of hemi- ellipsoid in the presence of a flat anode. Also we shortly review the history CNT cold emitters and technology of their fabrication. Furthermore we consider the application areas of CNT electron sources.

2. Historical preview

Field emission is an emission of electrons from a solid surface under action of external high electric field E. Field emission was experimentally discovered in 1987 by R.W. Wood (Wood, 1897). In 1929 R.A. Millikan and C.C. Lauritsen established linear dependence of the logarithm of current density on 1/E (Millikan & Lauritsen, 1929). Field emission was explained by quantum tunneling of electrons through the surface potential barrier. This theory was developed by R.H. Fowler and L.W. Nordheim in 1928 (Fowler & Nordheim, 1928).

According to the Fowler–Nordheim theory, the current density of the field emission j is determined by the following expression

jC1E2φexp(C2φ3/2E)
(1)

where j denotes the emission current density in Acm-2, E is local electric field at the emitting surface in Vcm-1, is work function in eV, and the first and second Fowler–Nordheim constants are C 1 = 1.56 × 10-6 AeVV-2, C2 = 6.83 × 107 VeV-3/2cm-1, respectively. The electric field E at the CNT tip increases compared with the average field E 0. Substituting in Eq. (1) the expression E = E 0, where is a field enhancement factor, we shall write the Fowler-Nordheim dependence in the form:

jC1(βE0)2φexp(C2φ3/2βE0)
(2)

Thus current-voltage characteristics of the field electron emission in the Fowler-Nordheim coordinates (logj/E021/E0) are presented by straight lines. It was assumed that = 4.8 eV for nanotubes. The field enhancement factor β varied from 300 to 3000 depending on the tube size.

Theory of field emission considered in details in recent books (Fursey, 2005; Ducastelle et al., 2006).

Strong electric fields (E~107 Vcm-1) near a surface are necessary to obtain the appreciable field emission current from pure metals. Therefore, the emitters in early investigations were produced in the form of thin spike with radiuses of curvature on the ends about 1 micron.

Development of lithographic techniques allowed fabricating so called “Spindt tips” in which the field emitters are small sharp molybdenum microcones. One of the first papers describing such technology has appeared in 1968 (Spindt, 1968). Essential efforts have been spent by several companies for development of the Spind-type field emission display, but no large-screen production has been forthcoming.

The new potential in designing field emitters and devices on their basis has appeared after discovery of carbon nanotubes.

Field emission of carbon nanotubes was for the first time reported by Fishbine (Phillips Lab.) (Fishbine et al., 1994), Gulyaev (Institute of Radio-engineering and Electronics, Russia) (Gulyaev et al., 1994), and Rinzler (Rice University) (Rinzler et al., 1994) in 1994.

Four first journal papers (Gulyaev et al., 1995; Chernozatonskii et al., 1995; Rinzler et al., 1995; de Heer et al., 1995) dedicated to this problem were published in 1995. As is known, several papers have appeared in the two subsequent years: two works (Chernozatonskii et al., 1996; Collins & Zettl, 1996) were published in 1996 and seven works (Collins & Zettl, 1997; Gulyaev et al., 1997; Sinitsyn et al., 1997; de Heer et al., 1997; Saito et al., 1997a; Saito et al., 1997b; Lee et al., 1997) were published in 1997. Starting from 1998, interest in field-emission properties of CNT was increasing explosively all over the world. Today we can speak of thousands of published papers.

Recently, field emission from metals (Lee et al., 2002), metal oxides (Li et al., 2006; Banerjee et al., 2004; Jo et al., 2003; Seelaboyina et al., 2006), metal carbides (Charbonnier et al., 2001), and other elongated nanostructures have also been explored. It is now possible to control the diameter, height, radius of curvature of the tip, and basic form of emitters during growth. Elongated structures of different shapes such as nanotubes, nanocones, nanofibers, nanowires, nanoneedles, and nanorods have been successfully grown (Li et al., 2006; Li et al., 2007; Jang et al., 2005; Hu & Huang, 2003).

Promising new materials for field-emission sources are B- and N-doped CNTs. Terrones et al. (Terrones et al., 2004; Terrones et al., 2008) have reviewed the field emission properties of B- and N-doped CNTs and nanofibres. B-doped multi-wall CNTs could exhibit enhanced field emission (turn on voltages of ~1.4 V/μm) when compared to pristine multi-wall CNTs (turn on voltages of ~3 V/μm). N-doped CNTs are able to emit electrons at relatively low turn-on voltages (2 V/μm). This phenomenon arises from the presence of B atoms (holes) or N atoms (donors) at the nanotube tips.

3. Carbon nanotubes field emitters

3.1. Physical properties of carbon nanotubes suitable for cold emission

From the practical application point of view CNTs are preferable field emitters due to their low threshold voltage, good emission stability and long emitter lifetime.

CNTs possess these advantages due to the large aspect ratio, high electric and thermal conductivity, highest flexibility, elasticity, and Young’s modulus. Their strong covalent bonding makes them chemically inert to poisoning and physically inert to sputtering during field emission. They can also carry a very high current density of order 109 A cm-2 before electromigration. Nanotubes have a high melting point and preserve their high aspect ratio over time. CNTs emits electrons under conditions of technical vacuum. They are chemically inert to poisoning due to strong covalent bonding. Measuring of field emission properties (Kung et al., 2002) and theoretical ab-initio calculations (Park et al., 2001) shows that emission currents are significantly enhanced when oxygen is adsorbed at the tip of carbon nanotubes.

3.2. Manufacturing techniques for CNT-based field-emission cathodes

Many technologies for fabrication of CNT-based field-emission cathodes were offered. We shall consider only some of them.

Individual CNT field emitters have a large potential for application in electron guns for scanning electron microscopes. To investigate the emission properties of individual CNTs de Jonge et al (de Jonge & Bonard, 2004) improved the mounting method using a piezo-driven nanomanipulator. For the mounting of an individual CNT on a tungsten tip, a tungsten wire was fixed by laser-welding on a titanium (or tungsten) filament.

Field emission CNT-based cathodes are manufactured either as a bulk solid containing nanotubes or as a film with thickness from hundreds of nanometers to tens of microns.

Bulk cathodes are known to be manufactured by two methods. The Alex Zettl team from California University, Berkley, USA used a technology in accordance with which the ready material of unsorted randomly aligned nanotubes is mixed into a compound, baked, and surface ground. Flexible and elastic nanotubes are not broken during the grinding. In accordance with the technology used by the Yahachi Saito team of Mie University (Japan), the graphite-electrode material processed by an electric arc is cut to pellets and glued to a stainless-steel plate by silver paste.

Film technologies are used in all other cases. Film cathodes are basically manufactured by two methods: either preliminary synthesized tubes are attached to a substrate or the tubes are grown directly on the substrate.

In the two methods, different technologies yield films of both well oriented and strongly entangled tubes.

N.I. Sinitsyn group from Institute of Radio-engineering and Electronics (Saratov, Russia) used CVD methods for synthesizing films of both regularly grown nanotubes (Figure 1) and “felt” of entangled fibers. Strips were obtained using a catalyst deposited through a template (Zhbanov et al., 2004).

media/image4.jpg

Figure 1.

Photograph of strips of oriented CNTs synthesized on a substrate. The strip width is 20 μm and the gap between the strips is 5 μm (Zhbanov et al., 2004).

The Jean-Marc Bonard team of Lousanne Polytechnical School (Switzerland) developed the technology of microcontact printing of catalytic precursor for growing oriented tubes arranged in accordance with a specified pattern on a substrate (Bonard et al., 2001b). The catalyst, the so-called “ink”, was applied to the stamp surface. The ink was a solution containing from 1 to 50 mM of Fe(NO3)3 9H2O. The duration of contact during the printing was 3 s. Nanotube deposition was by the CVD method in a standard flow reactor at a temperature of 720 C.

In the case of low concentration of catalyst (1 mM, Fig. 2 a), several single nanotubes are randomly distributed over the printing region. The catalyst-concentration growth is accompanied by formation of films of entangled tubes, as is shown in Figs. 2 b and 2 c. For concentrations about 50 mM, clusters of nanotubes oriented normally to the surface are formed. Figure 2 d shows that the sides of the walls are flat, and not a single tube is hanging outward. For concentrations above 60 mM, growth of nanotubes is retarded, and the printed template is covered by amorphous carbon particles.

media/image5.jpg

Figure 2.

Nanotube growth for various concentrations of catalytic ink used for the precursor application. Catalyst concentration in the solution was 1 mM (a), 5 mM (b), 40 mM (c), and 50 mM (d). The figure is taken from (Bonard et al., 2001b).

Hongjie Dai team of Stanford University, USA, used the following technology for obtaining arrays of well-oriented carbon nanotubes. First, porous silicon was formed on the surface of a silicon substrate by anode etching and then the ferrum film was deposited on the latter through the shadow mask by electron-beam evaporation (Fan et al., 1999). Then nanotubes were grown as a result of acetylene decomposition in argon flow at 700 C.

E. F.Kukovitsky team of the Kazan Physics-Technical Institute (Russia) developed the technology of synthesis of oriented nanotubes with conical layers (Figure 3) (Musatov et al., 2001; Kukovitsky et al., 2003). The first stage of the process involves polyethylene pyrolysis in the first oven at a temperature of 600 C. Then, by the helium flow, the gaseous products of pyrolysis are transferred to the second oven where nanotubes grow on the nickel foil catalyst at a temperature of 800 to 900 ◦C. For the obtained specimens, the current density was 10 mA/cm2 for the electric field from 4 to 4.5 V/µm.

media/image6.jpg

Figure 3.

High-resolution electron microscope image of nanotubes with conical layers. Graphene layers are marked by arrows with points, and the CNT growth direction is marked by large arrows (Musatov et al., 2001).

As is obvious from the literature analysis, almost all CNT-based cathodes show high emission irrespective of the fact whether the tubes are multi-wall or single-wall, well-oriented or entangled. Bamboo-shaped aligned carbon nanotubes (Srivastava et al., 2006; Ghosh et al., 2008) as well as carbon nanocones (Yudasaka et al., 2008) demonstrate high field electron emission.

Let's note, that not only elongated carbon nanotubes, but also pyramids from fullerenes are used as cold cathodes. Formation and characteristics of fullerene coatings on the surface of tungsten tip field emitters and emitters with ribbed crystals formed on their surface are studied by group of Sominskii from St. Petersburg State Technical University (Russia) (Tumareva et al., 2002; Tumareva et al., 2008). Methods of creating microprotusions on the surface of the coatings that considerably enhance the electric field have been developed and tested. Emitters with a single microprotrusion demonstrated emission current densities up to 106–107 A/cm2. It was shown that single micron-sized emitters can stably operate at currents up to 100 A.

3.3. Electric field and field enhancement factor in diode configuration

The field enhancement factor is very important parameter for characterization of CNT emitters.

The model of a hemisphere on a post for CNT emitters is widely used in analytical approximations and numerical simulations (Figure 4). To calculate the electric-field intensity and the field enhancement factor on the nanotube tips, the following assumptions are usually done:

1) Nanotubes are regularly located on a flat substrate in a “honeycomb-like” order. A nanotube is a cylinder with height h and Diagram 2 capped by a hemisphere of radius. Total height of closed nanotube is H, the distance from cathode to anode is L, the gap between anode and nanotube tip is l, and the distance between the nearest neighbors is D.

2) A nanotube obeys the laws of continuous medium, is perfectly conducting, and the cathode potential is maintained on its entire surface.

media/image7.jpg

Figure 4.

Scheme of aligned nanotube film, the model of a hemisphere on a post: (a) side view; (b) top view.

Let us introduce dimensionless parameters for the geometrical characterization of model. The dimensionless height of emitter, the dimensionless gap between anode and emitter tip, and the dimensionless distance between individual emitter are the following:

η=ρhλ=ρlδ=ρD
(3)

Until now the analytical solution for the model of a hemisphere on a post is unknown. There is no even a solution for the individual cylindrical nanotube closed by hemispherical cap in a uniform electric field.

media/image9.jpg

Figure 5.

Schemes of simplest models for field enhancement factor estimation: (a) hyperboloid near a plate; (b) hemisphere on a plane; (c) floating sphere at emitter-plane potential, and (d) hemi-ellipsoid on a plane.

Numerical simulations were reported in many papers (Edgcombe & Valdrè, 2001; Edgcombe & Valdrè, 2002; Read & Bowring, 2004 ; Musatov et al., 2001). Calculation difficulties in these numerical methods arise due to the large nanotube aspect ratio and very long distance between cathode and anode in comparison with emitter height. Usually, these numerical results were generalized and simple fitting formulas of field enhancement factor for individual nanotube (Edgcombe & Valdrè, 2001; Edgcombe & Valdrè, 2002; Read & Bowring, 2004 ; Shang et al., 2007), for nanotube in space between parallel cathode and anode planes (Bonard et al., 2002a; Filip et al., 2001; Nilsson et al, 2002; Smith et al., 2005), and for a nanotube surrounded by neighboring nanotubes with a screening effect (Jo et al., 2003; Glukhova et al., 2003; Nilsson et al., 2000; Read & Bowring, 2004 ; Wang et al., 2005) were suggested. The main problem for such algebraic fitting formulas is the lack of a definitive proof of their accuracy.

Four of the simplest models are the “hyperboloid near a plate” model, the “hemisphere on a plane” model, the “floating sphere at emitter-plane potential” model, and the “hemi-ellipsoid on plane” model. We follow to the classification suggested by Forbes et al. (Forbes et al., 2003).

These models allows analytical solutions, they are illustrated in Figure 5. We will use dimensionless parameters Eq. (3) to define geometry of these models.

3.3.1. Hyperboloid near a plate model

We introduce the prolate spheroidal coordinates and to consider the model of a hyperboloid near a plate (Figure 6).

media/image10.jpg

Figure 6.

Hyperboloid near a plate in prolate spheroidal coordinates.

The equation of prolate spheroid is:

r2a2(σ21)+z2a2σ2=1;σ1
(4)

The equation of hyperboloid of two sheets is:

r2a2(τ21)+z2a2τ2=1;1τ1
(5)

Points F (0;-a) and F’ (0; a) are the foci of the hyperboloids and spheroids. The cathode represents a hyperboloid of revolution 0 = const and the anode is a plane = 0. They are show in Figure 6 by solid red lines. The radius of hyperboloid curvature of the tip is .

The electric field is calculated according to the formula:

E=Va(σ2τ2)(1τ2)arctanh(l/a)
(6)

where function arctanh is inverse hyperbolic tangent, V is the voltage applied across a gap between anode and cathode.

The model of a hyperboloid near a plate is suitable to describe the interaction of individual CNT field emitter with surface in scanning electron microscopes. Usually in cases important for practice we have << l and l<< H.

If << l the maximal value of the module of intensity is approximated by the formula (Drechsler & Müller, 1953):

EtopVρln(4l/ρ)
(7)

If we define the macroscopic field by E 0 = V/l then we can write the field enhancement factor

β=1+λλarctanh1/(1+λ)
(8)

Let us estimate the electric force acting on the surface of ellipsoid. The electrostatic force acting on the elementary area, s of the external surface is given by

F=Sε02E2nds
(9)

where 0 is the electric constant, n is a vector normal to the surface.

Taking into account that the infinitesimal surface element is ds=2πa2(1τ02)(σ2τ02)dσ , we can analytically integrate the force acting on the top of a hyperboloid surface of height H (see Figure 6). It is clear that r-component of force equals to zero, F r = 0. For the z-component we have

Fz(H)=πε0V22arctanh21/(1+λ)ln(1+2Hρ+2Hl+H2lρ+H2l2)
(10)

The total current is calculated by integration of current density from Eq. (1) over a hyperboloid surface

Jtotaljds
(11)

We note that the exchange and correlation effect is ignored in the basic equation (1). Thus the Fowler–Nordheim theory is suitable only for approximate calculations. Nevertheless this theory is widely used for analysis of field emission current from elongated nanostructures.

After substitution of field distribution over the sphere surface and an infinitesimal surface element we have

jds=2πC1V2φ1τ02arctanh2τ01exp(C2V1φ3/2a(1τ02)(σ2τ02)arctanhτ0)σ2τ02dσ
(12)

If << l then 0 ≈ 1 and σ2τ02σ212(σ1) . This approximation allows us to reduce our integral to another one 1exp(ax1)x1dx=2a

Thus the total field emission current is

Jtotal2πC1V31+λC2φ5/2ρarctanh31/(1+λ)
(13)

where the total current, Jtotal is measured in A; the radius of curvature, is measured in cm.

3.3.2. Hemisphere on a plane

The metallic sphere in a uniform electric field E 0 (Figure 5(b)) was considered in many papers (for example Refs. (Forbes et al., 2003; Wang et al., 2004; Pogorelov et al., 2009)). We can replace the sphere by point electric dipole. If the electric dipole moment is p 0 then the dipole potential is

ϕdip=p04πε0z(z2+r2)3/2
(14)

Equation of circle is ϕdip+zE0=0 . From this equation we can find the relation between the electric dipole moment and the sphere radius: p 0 = 4 0 E 0 3. The electric field on the top of hemisphere reaches Etop=p0/2πε0ρ3+E0=3E0 . The field enhancement factor is β=Etop/E0=3 . The field distribution over the sphere surface have the form E=3E0cosθ , where is polar angle.

Pogorelov et al. (Pogorelov et al., 2009) have shown that the total current emitted from the hemisphere surface is

Jtotal=2πρ2C1φ7/2C233E0[E1(C3)6+(13C3316C32+16C3)exp(C3)]
(15)

where C3=C2φ3/2/3E0 and E1(x)exp(xt)/tdt is the exponential integral.

Due to small β for the hemisphere we need to use very strong electrical field to produce slightly visible current in experiment.

3.3.3. Floating sphere at emitter-plane potential

The “floating sphere at emitter-plane potential” model has no “body” of the field emitter and possesses only its “head”. This model gives too high estimation of electric field on the apex of nanotube but plausibly reproduce tendencies of change of the field enhancement factor. Approximate analytical solution for the “floating sphere at emitter-plane potential” model is well known (for example Refs. (Forbes et al., 2003; Wang et al., 2004)). To solve this problem the method of images (Jackson, 1999) is usually used.

media/image34.jpg

Figure 7.

Two conducting spheres of radius at cathode potential in uniform electric E 0.

The charge -q 0 = -4 0 hE 0 and the electric dipole p 0 = 4 0 E 0 3 placed at point A (Figure 7) create a sphere of radius and potential = 0 in uniform external electric field. The charge q 0 and dipole p 0 cause a potential variation across the emitter plane. To correct this we have to place an image-charge q 0 and image-dipole p 0 at point A’ behind the emitter plane. The image-charge and image-dipole will distort the surface of sphere. To restore the shape we should place additional charge -q 1 = -q 0 /2h and dipole p 1 = p 0 3/8h 3 at point B on the distance s 1 = 2/2h from the center of sphere (see Figure 7).

Next we have to put q 1 and p 1 at point B’, after to put -q 2 and p 2 at C and so on. Neglecting terms of higher smallness in this series of approximation we find the electric field on the top of floating sphere

Etop=14πε0q0ρ2+2p04πε01ρ3+14πε0q1(s1+ρ)2+E0E0(hρ+3.5)
(16)

Thus the field enhancement factor is

β=1η+3.5=Hρ+2.5
(17)

We can provide more accurate calculations. Recurring formulas for the distance s i+1, the charge q i+1, and the dipole moment p i+1 through s i, q i, and p i are the following

si+1=ρ22hsi pi+1=piρ3(2hsi)3 , and qi+1=qiρ2hsipiρ(2hsi)2 , where the initial distance is zero: s 0 = 0

Series expansion of the field enhancement factor is

β=η1+7212η+18η2+716η32532η4+2532η5+O(η6)
(18)

As the next step of approaching to CNT film, consider an assembly of floating spheres and a screening of the individual emitter by neighbors. The view from above of the sphere surrounded by another one is shown in Figure 8. Large red circles in this picture are the floating spheres. Small black circles mark places where charges are located. Numbers “0” show initial charges in the center of balls. Numbers “1” specify image charges induced only by nearest neighbors. Numbers “2” concern to secondary image charges.

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Figure 8.

Honeycomb structure, distance between spheres is D, sphere radius is .

If the distance between spheres, D is large enough (D>>) then all image charges collect on small area around the center of sphere. In that case we can combine all charges inside the ball into its center. Also we will neglect influence of image dipoles.

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Figure 9.

Modeling of screening effect for floating spheres.

The set of floating spheres produces an idealized surface charge density σ=2q/3D2 . Positively and negatively charged surfaces form the parallel plate capacitor (Figure 9). The electric field between two large parallel plates is given by E'=σ/ε0

From the equation h(E023ε0qD2)=14πε0qρ we can find the total charge in the center of each sphere q=ε0hE0(14πρ+2h3D2)1

Thus we can find the maximal field on the surface of floating sphere

Etop=hE0ρ3D2(3D2+8πρh)+E0
(19)

and the field enhancement factor

β=33η+8πδ2+1
(20)

More accurate approximation

ϕσ+=2πxXσ4πε0r(2h+ρ)2+r2drϕσ=2πxXσ4πε0rρ2+r2drx=32πD
(21)
ϕP"=limX(ϕσ++ϕσ)+14πε0(q'2h+ρq'ρ)
(22)

Solving equation ϕP"=E0h we find

β=6(1+η)246πδ(1+η)(2πδ2(2+η)2+3η2η2πδ2+3)+3η(2+η)
(23)

Eq. (23) is transformed to Eq. (20) after neglect in values of higher order of smallness.

On the one hand the field enhancement factor and the current density on nanotube apex reach its maximum if the distance between emitters is very large. On the over hand in this case the current density on the anode will be very small. Clearly we can find optimum distance between emitters. As an approximation, assume that the emitting surface of each sphere equals 2 and that the electric field is a constant on this surface. The anode current density takes the form

janode2π3δ2C1(βE0)2φexp(C2φ3/2βE0)
(24)

If h>> and D>> then β3/(3η+8πδ2) . Let’s use this relation for the simplicity.

Solving the equation

janodeδ=0=24πE0δ233E0η+24πφ3/2δ2C2η+643π2C2φ3/2δ4
(25)

we find the optimal dimensionless distance between emitters in honeycomb structure

δopt=(3E0η4π(3E0+3C2φ3/2η+3E02+18C2E0φ3/2η+3C22φ3η2))1/2
(26)

After neglect terms of higher smallness we can write the simplification

δopt3E0η4π(E0+2C2φ3/2η)
(27)

Figure 10 illustrates the dependence of anode current density from geometrical parameters of emitter. We have assumed that the work function is = 4.8 eV, the external field is E 0 = 60000 Vcm-1, the dimensionless height is = 0.001 (for Fig. 10a), and the dimensionless distance between emitters is = 0.002 (for Fig. 10b).

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Figure 10.

Anode current density versus dimensionless sizes: (a) optimal distance between emitters if the height is fixed; (b) influence of emitter height on anode current if the density of emitting centers is constant.

Let’s consider influence of the limited anode-cathode distance (Figure 11.) on the field enhancement factor.

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Figure 11.

Geometrical model for the limited distance, L between cathode and anode.

The cathode has zero potential ϕc=0 ; ϕa=E0L is the anode potential.

As before, assume that average charge per each conductive ball, q is concentrated at its center. Equation for average potential ϕb on plane with grounded conductive balls is

ε0hϕc+(ε0h+ε0l)ϕbε0lϕa=σ
(28)

Solving the equation 2hl3Lε0D2q+E0h=14πε0ρq , we find the charge in the center of ball

q=43πε0E0hρD2L3D2L+8πhρL8πh2ρ
(29)

Thus the maximal field on the surface of floating sphere is

Etop=E0hρ[1+8π3hρD2(1hL)]1
(30)

The field enhancement factor is

β=[ρh+8π3(ρD)2(1hL)]1=[η+8π3δ2(1λη)]1
(31)

Let us note here that the model of floating sphere and the method of images allow considering field emission not only on flat anode but also on spherical anode.

3.3.4. Hemi-ellipsoid on a plane

Consider a prolate metallic spheroid in a uniform electric field. We can replace the spheroid by a linearly charged thread as we show in our recent paper (Pogorelov et al., 2009). The thread is a green line in Figure 12 and the linear charge distribution is represented by a red line. The length of a thread is 2h. The electrostatic potential produced by the charged thread is

ϕ(zr)=hh14πε0τz'dz'(z'z)2+r2
(32)

where (r; z) denotes the in-plane radial and z coordinates, τz is the linear charge density at point (0; z), h is half of the thread length. The solution is independent of the azimuthal angle. We assume the coefficient of linear charge density, to be positive.

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Figure 12.

Linearly charged thread in a uniform electric field along z.

The shape of metallic spheroid is given by the solution to the equation.

Using coordinates on the spheroid surface ra=(zh)2+r2 and rb=(z+h)2+r2 and a dimensionless parameter, the eccentricity ξ=2hra+rb(0ξ1) , we can rewrite Eq. (33) in the form

4hra+rbln(ra+rb+2hra+rb2h)=CwhereC=4πε0E0τ=ln(1+ξ1ξ)2ξ
(34)

The zero equipotential which represents the metallic hemi-ellipsoidal cathode on a plate is shown in Figure 12 by solid blue line. Points (0, -h) and (0, h) are the foci of the ellipse, ra and rb are distances between (r; z) and the two foci.

If ξ is close to 1, the ellipse becomes elongated. If ξ0 the ellipse turns into a circle. Therefore, by changing the coefficient of linear charge density, we may modify the shape of the ellipse.

We can also adjust other geometrical parameters of the ellipse: the length of semi-major axis or height H; the length of semi-minor axis or base radius, R at z = 0; and radius of curvature, at point (0, H) (see Figure 13).

H=hξ=ra+rb2R=h1ξ2ξρ=R2H
(35)

We can calculate components of the electric field on the surface of the metallic spheroid:

Ez=E0Ch(rbra)2rarb(rb+ra)Er=E0Ch(rbra)4h2(rbra)2rarb(rb+ra)24h2
(36)

Thus the modulus of the electric field is

E=Ez2+Er2=E0C4h2(rbra)(rb+ra)rarb[(rb+ra)24h2]
(37)

Eqs. (36), (37) allow determining the electric field strength on the surface of the half ellipsoid at an arbitrary point. The field enhancement factor at the apex of the ellipsoid is as follows:

β=2ξ3(1ξ2)C=2ξ3(1ξ2)(ln1+ξ1ξ2ξ)
(38)

Analytical expressions for field strength on the z-axis and for field enhancement factor on the tip of the half ellipsoid obtained previously (Forbes et al., 2003; Kosmahl, 1991; Latham, 1981; Latham, 1995) are in agreement with our result. Here, by taking gradient of Eq. (33) we can obtain the field strength at any point we desire. In the limit ξ0 we have a metallic half sphere and the field enhancement factor = 3. If ξ1 then for the elongated metallic needle, we have

β2Hρ1ln(4H/ρ)2
(39)

Ponderomotive forces. Let us estimate the electric force acting on the surface of ellipsoid. We can calculate the force acting on the spheroid surface between circles ra = A and ra = B (see Figure 13).

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Figure 13.

Geometry for the calculation of ponderomotive force acting on the belt between ra = A and ra = B.

It is obvious that r-component of force on that surface is equal to zero. For the z-component, after routine operations we obtain

Fz(zAzB)=πρ2ε0(βE0)22(1ρH)2[1(1ρH)z2H2ln{1(1ρH)z2H2}]|zAzB
(40)

where z varies from zA to zB (see Figure 13). It gives us the value of net detaching force acting on the surface of the spheroid integrated over the surface between planes z=zA and z=zB . The total detaching force acting on the ellipsoidal needle is

Ftotal=πρ2ε0(βE0)22(1ρH)2(ρH1lnρH)
(41)

In Fig. 14a we show the relative detaching force Fz(Hz)/Ftotal as a function of coordinate z, where Fz(Hz) is the force acting on the surface between plane z=z and the tip of the spheroid (z=H) . Distribution depends only on the parameter H/ρ . The major part of the detaching force is concentrated near the tip when H/ρ is large. Fig. 14b shows the total force isolines on the ( ρH/ρ ) plane. When we chose logarithmic scale for ρ and H/ρ with logarithmic steps we obtain a set of nearly straight isolines with equal distances in the plot.

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Figure 14.

Distribution of force and total force isolines. (a) Distribution of relative force Fz(Hz)/Ftotal over the axis of needle. (b) Isolines of total force on the (ρH/ρ) plane.

We think that under the action of ponderomotive forces in the external electric field, carbon nanotubes that are even chaotically located on the substrate straighten and become oriented (Musatov et al., 2001; Glukhova et al., 2003).

Field emission from individual needle. The current emitted from the surface bounded by θ(0,Θ) can be expressed in terms of the E1-function

J(Θ)=2πρ2C1(βE0)2φE1[C22φ3/2βE01+λ]|λ=cosΘ1
(42)

where λ=cosθ θ is angle between axis z and ra as shown in Figure 12.

By comparing with accurate numerical integration of (1) we find that formula (42) is accurate up to the first four digits for H/ρ100 . So formula (42) is a good approximation for investigating the current emitted from an area of surface depending on angle Θ . The total current with similar accuracy can be written approximately as

Jtotal=2πρ2C1(βE0)2φE1[C22φ3/2βE01+λ]|ξ12πρ2C1(βE0)2φE1[C2φ3/2βE0]
(43)
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Figure 15.

Distribution of emission current and total emission current isolines for work function φ=4.8 eV. (a) Distribution of relative emission current over Θ. (b) Total emission current isolines on the (βE 0ρ) plane.

The emission current has much sharper distribution on the tip than the detaching force due to exponential dependence of Fowler-Nordheim formula (1). In Fig. 15a we plot the relative emission current, J(Θ)/Jtotal as a function of Θ (angle between ra and axis z), where J(Θ) is emission from the tip on the surface of the ellipsoidal needle. From (43) we see that total emission current depends only on the area of the hemisphere, 2πρ 2, work function, φ, and the electric field on the tip, βE 0. So in Fig. 15b we show the total emission current isolines within a reasonable range (from 104 to 104 nA) with logarithmic steps on the (βE 0 ρ) plane, where the work function φ=4.8 eV is assumed. If the electric field is E 0≈1..5 V/μm, then we have to use an enhancement factor β≈103..104 to get an appreciable current.

Simulation of planar anode by “image” charges. The presence of a flat anode placed at a distances comparable with the length of the nanoneedle has strong influence on the emission current and the detaching force. The basic idea of calculation is to replace the cathode by a linearly charged thread in a uniform electric field and to use a set of “image” charges to reproduce the anode as shown in Figure 16. We put infinite set of “images” of the linearly charged thread with the same spacing, 2L ( LH ). It is clear that planes z=0 and z=L are planes of symmetry for distributed charges and will have potentials 0 on cathode and V=E0L on anode.

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Figure 16.

Infinite set of “image” charges for the simulation of a planar anode.

With the same τ we will get different geometry of cathode due to additional potential of “images”. We assume that the new form of the elongated needle will be approximately described by ellipsoid especially near the tip. On the surface of the thin ellipsoid, we have rh and zHL . So, we can describe the image potential ϕi as a function of z

ϕi(z)=E0C*n=1[(z2nL)ln2nLz+h2nLzh+(z+2nL)ln2nL+z+h2nL+zh]
(44)

where C*4πε0E0τ*L=μHμ1 . The thread potential is

ϕ=zE0C[4hra+rbln(ra+rb+2hra+rb2h)]
(45)

Near the tip zH we have the following equation for describing the shape of the ellipsoid:

ϕ(rz)+E0z+ϕi(z)ϕ(rz)+E0z+zHϕi(H)=0
(46)

So we have to change constant C on constant C * as following

C*=ln(1+ξ1ξ)2ξP
(47)

where

P=n=1[(2nμ1)ln2nμ1+ξ2nμ1ξ(2nμ+1)ln2nμ+1+ξ2nμ+1ξ]
(48)

Therefore we have the field emission factor in the form

β(L/HH/ρ)=2ξ3(1ξ2)C*
(49)

Finally for the force and emission current we may use the above derived formulas. Instead of (38) we should use (47), (48), and (49). So, the parameter μ=L/H in the force and emission current formulas (40), (41), (42), and (43) should be included only through the field enhancement factor β=β(L/HH/ρ)

What distance between the anode and cathode is large enough to assert that the elongated metallic spheroid is placed in a uniform electric field? In experiment we may measure distance L between the anode and cathode and the distance, LH between the anode and the needle apex. We can determine the applied electric field both by E0=V/L and by E0=V/(LH) . In our calculations we move the anode and increase the voltage V so as to keep E0 or E0 immutable. It is clear that for a large distance (in the limit L→∞) the difference between definitions of E0 and E0 disappears. Figure 17 shows the total current emitted from the needle versus the anode-cathode distance.

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Figure 17.

Emission current as a function of the anode-cathode distance parameter μ=L/H for constant electric field E0V/L (red line) or E0V/(LH) (blue line). Radius of curvature is fixed at ρ=5 nm. For solid curves, H/ρ=5800 and for dash curves, H/ρ=10000 . For red and blue solid curves E0=5V/μm , for red and blue dash curves E0=3V/μm

We set for blue lines E0=const and for red lines E0=const . With E0=E0=const the total currents have the same limit for μ=L/H . But currents with E0=const tend to approach the limit much slower than the corresponding currents with E0=constant . If we define the applied electric field as E0 and the anode-cathode distance is ten times more than the needle height, then we can neglect the influence of the anode and assume that the metallic needle is placed in a uniform field. In contrast, if the applied electric field is defined as E0 , then the anode-cathode distance must exceed one thousand times of the needle height for the above statement to be valid.

3.3.5. The model of a hemisphere on a post. Numerical approximations

The model of a hemisphere on a post allows only the numerical solution. There are many numerical results obtained by various researchers which have been generalized by simple algebraic formulas of field enhancement factor for an individual nanotube and assembly of nanotubes.

From our point of view the most accurate formula belongs to Edgcombe et. al. (Edgcombe & Valdrè, 2001; Edgcombe & Valdrè, 2002; Forbes et al., 2003)

β0=1.2(2.5+Hρ)0.9
(50)

This formula accurately describes the field enhancement factor of individual nanotube in a uniform electric field. Comparison of field enhancement factors for the “floating sphere at emitter-plane potential” model (green line), the “hemi-ellipsoid on plane” model (red line), and fitting formula for the “hemisphere on a post” model (blue line) is shown in Figure 18.

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Figure 18.

Comparison the field enhancement factors for three models: floating sphere (green line), hemi-ellipsoid (red line), and hemisphere on a post (blue line).

For a nanotube in space between parallel cathode and anode planes we prefer approximation (Bonard et al., 2002)

β=β0[1+0.013(LHL)10.033(LHL)]
(51)

For a nanotube surrounded by neighboring nanotubes with a screening effect we recommend (Jo et al., 2003)

β=β0[1exp(2.3172DH)]
(52)

3.4. Further investigations

Let us shortly mention the further work that should now be carried out on the field-emission properties of CNTs.

1) Developing of field emission theory for CNT emitters

2) Temperature effects in CNTs: Joule heating; Peltier and Nottingham effects; carbon nanotube heat radiation; thermo-field emission from nanotubes

3) Action of electrostatic forces on CNT field emitters: elongation and straightening of nanotube; pulling out of nanotubes in strong electric field.

4) Reliability of CNT field emitters: degradation mechanism of field emission; lifetime of nanotube emitters.

4. Applications

Field emission is the most promising application areas of carbon nanotubes. We shall consider only a few samples of working devices.

4.1. Field emission displays

Flat-panel displays with CNT-based cathodes are proposed as alternative to other displays with film emitters. The first diode-type display consisting of 32 32 matrix-addressable pixels was manufactured by Wang et al. in 1998 (Wang et al., 1998). At present, flat-panel displays based on CNT field-emission cathodes are developed in hundreds of laboratories, and engineering samples have been already manufactured.

A 4.5-inch full-color CNT display developed by Choi et al. (Choi et al., 1999) from Samsung Company was demonstrated at several exhibitions.

A full-color addressable display developed jointly by the limited liability company “Volga-Svet” (Saratov, Russia) and “CopyTele Inc.} (New York, USA) (Abanshin et al., 2002) was demonstrated at the conference IveSC’02. Emission in the proposed design was from thin edges of carbon nanocluster films, which are slightly hanging from the supporting pedestals (Figure 19). The anode covered by a phosphor layer is shown on the top in Figure 19. Gating/blocking of emission current is performed by a metal control electrode located on the substrate between the pedestals. The figure shows the results of the trajectory analysis for an option of the structure (Zhbanov et al., 2004).

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Figure 19.

Trajectory analysis for the flat-panel display produced by the limited liability company “Volga-Svet” and “CopyTele Inc.” (Zhbanov et al., 2004).

4.2. X-ray tubes

A compact X-ray tube with a field emitter based on carbon nanotubes was developed by Musatov’s group from Institute of Radio-engineering and Electronics (Russia) (Musatov et al., 2007). Over a long time interval, the X-ray tube maintains an anode current of 300 A, an anode voltage of 10 kV, and the stable characteristics of the field emitter (Figure 20).

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Figure 20.

Compact X-ray tube. The figure is taken from (Musatov et al., 2007).

4.3. Light elements

The Bonard team developed the cathodoluminescent light-emitting element of cylindrical geometry (Bonard et al., 2001a). The cathode in the form of a metal rod with deposited CNTs is located on the axis of a glass tube covered by phosphor from the inside. The operating voltage is 7.5 kV, the current density on the cathode is 0.25 mA/cm2, the current density on the anode is 0.03 mA/cm2, and the lamp luminance is 104 cd/m2.

CNT light-emitting elements of various colors in the form of a filamentary electric lamp are proposed by Saito team in (Bonard et al., 2002b). The operating grid voltage is from 0.2 to 1.2 kV, the current density at the cathode with an area of 2 mm2 is 10 mA/cm2, the average electric field intensity is 1.5 V/µm, and the luminance of elements of various colors ranges from 1.5 104 to 6.3 104 cd/m2.

Samples of the light-emitting elements were demonstrated by A.N. Obraztsov (Obraztsov et al., 2002) and E. P. Sheshin (Leshukov et al., 2002) at the Saratov 4th International Conference on Vacuum Electrons Sources (IveSC’02).

4.4. Microwave devices

The electron gun consisting of edge or cylindrical emitters with flat faces; a grid control electrode placed above the emitters; and a three-anode focusing system which is common for all the emitters (Figure 21) was designed by Zakharchenko’s group (Zakharchenko et al., 1996). The flat face surfaces of the edge emitter are covered with a film made of carbon nanotubes of Diagram 30–100 Å. A substantial density of emitting centers being as great as 108–1010 tips/cm2 provides a high current density at low accelerating voltages.

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Figure 21.

Schematic of the electron gun for an amplifier. Magnetic field B=0.08 T, voltage V=400 V, and current I=1 A/cm2. The electron gun (1) consists of edge or cylindrical field emitters (2) covered with CNT film, a control electrode (3), and three anodes (4) (Zakharchenko et al., 1996).

5. Conclusions and Acknowledgements

In this chapter we theoretically investigate the field emission from carbon nanotube field emitters in diode configuration between a flat anode and cathode. Exact analytical formulas of the electrical field, field enhancement factor, ponderomotive force, and field emission current are found. Applied voltage, height of the needle, radius of curvature on its top, and the work function are the parameters at our disposal. The field enhancement factor, total force and emission current, as well as their distributions on the top of the needle for a wide range of parameters have been calculated and analyzed. Also we review the technology of fabrication and the application areas of CNT electron sources.

We have right to conclude that carbon nanotubes are excellent field emitters. Now CNTs conquer appreciable positions as electron sources in compact X-ray tubes, lighting elements, scanning electron microscopy electron guns, and electron guns for microwave devices. We think CNTs have good potential in the field emission display market.

We gratefully acknowledge support through the National Science Council of Taiwan, Republic of China, through the project NSC 95-2112-M-001-068-MY3.

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