Field emission in an array of homogeneous identical nanometer-long nanotubes

We consider the problem of field emission based on carbon nanotubes. The length of these nanotubes differs from several nanometers to several hundreds of nanometers. We obtain the particle transmission function, considering the difference of potentials on the ends of nanotubes to be U=2÷3.5V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U = 2 \div 3.5{\text{ V}}$$\end{document}. Basing on the transmission function we calculate emission current. We establish the dependence of the Nordheim function on the length of nanotubes. The limiting transition is considered for the transmission coefficient for field emission from the cathode surface in the absence of nanotubes on it. We establish the linear dependence of current I on the field strength W and the linear dependence of function I/W2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I/W^{2}$$\end{document} on the inverse value of field strength 1/W. Qualitative matching to experimental results of emission current is obtained when the voltage was kept less than the threshold value of 260 V and the distance between cathode and anode was kept constant, ≈5mm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 5 {\text{mm}}$$\end{document}.


Introduction
Field electron emission sources are in demand for many practical applications: X-ray devices, electron sources for microwave equipment, generators and terahertz radiation detectors, compact mass spectrometers, field emission (FE) indicators and displays, as well as for other products in vacuum nanoelectronics [1][2][3].
In the case of field electron emission sources large values of the electric field strength are required (in the order of E = 10 9 ÷ 10 10 V/m). Since the field strength approximately is equal to the ratio of the applied voltage at the cathode tip to the radius of the tip, the electric field strength dramatically increases as the radius of the cathode tip decreases. The application of such high fields provides the high electrical voltage on the end of a needle; thus, the needle material must be mechanically strong to become undamaged. The examples of such unique materials can be extended carbines that are 1D (cumulene and polyyne chains) [4][5][6][7] as well as 2D structures (nanotubes and nanoribbons) [8][9][10]. These structures are described using concepts such as active resistance per unit length R a , kinetic inductanceL k and quantum capacitance C Q [11], which are obviously of quantum origin where υ F is the Fermi velocity, h is the Planck's constant, and e is the electron charge. Usually for manufacturing an isolated single-walled carbon nanotube (SWCNT) the length is less than the mean free path of electrons in a CNT (λ CNT ), which is typically > 1 μm. Due to spin degeneracy and sublattice degeneracy of electrons in graphene, each nanotube has four parallel one-dimensional conducting channels. Thus, the maximum conductivity of an isolated ballistic single-walled CNT (SWCNT) with ideal coupling with two metal contacts, determined by the Landauer-Büttiker formula, is 1/R a = 4e 2 /h = 155μS [12]. In other words, the resistance of carbon nanotube, the length of which is L < λ CNT , with ideal anchoring with two metal contacts at its ends is defined by expression R a from (1) [13]. The effective kinetic inductanceL k /4 is calculated in [13,14] by equating the kinetic energy stored in each conducting channel of a CNT to the equivalent inductance. Four parallel conducting channels in CNTs give an effective kinetic inductanceL k /4 per unit length. The quantum capacitance C Q per unit length accounts for the quantum electrostatic energy stored in the nanotube when it carries current. Since the CNT has four conducting channels as described in the previous subsection, the effective quantum capacitance obtained from four parallel capacitances C Q is determined as 4C Q . Quantum capacitance is used to model the energy needed to add an electron at an available quantum state above the Fermi level. Quantum capacitance (per unit length) can be obtained by equating the quantum of electrostatic energy to the energy of effective capacitance [14,15].
The transverse sizes of extended carbines can be 3-4 orders smaller than the longitudinal sizes, which leads to the small values of the depolarization factor along nanotubes n 3 . This regularity is important for the field emission effect, where large values of the electric field strength are required [16]. The value of the depolarization factor in the case of extended nanotubes can be calculated, if the nanotubes are approximated by the axisymmetric elongated spheroids. Wherein the greater semiaxis equals to the half of nanotube height c = L/2, and transversal sizes of spheroids a = b are taken due to the equality condition of the volumes of a particle and spheroid (see [17,18] or [19,20]). To define the depolarization factor, one should know the Newtonian potentials where >> a = b. Field emission properties of oriented bundles of the monolayer CNTs decorated by nanodiamonds were investigated in [21]. Modern theories of field electron emission have their origin in the work by Fowler and Nordheim (FN) [22,23]. Today, the FE theory is described by a whole family of the Fowler-Nordheim equations different forms [24]. In recent years, there has been a serious revision of approaches to the derivation of the basic FE equations, including the definition of the special functions [25,26]. There is a comparison of theoretical and experimental approaches of investigation of large-area electron emitters in paper [26].
In paper [27] (see also [26,28]) the main expression for local emission current density (ECD) j derived in terms of local work function ϕ and characteristic local electrostatic field value on the emitter surface W is determined by Fowler-Nordheim constants; ν GB is a "barrier form correction factor," and λ GB is a pre-exponential correction factor. The general form of a Fowler-Nordheim equation from the assumption of the experimental data correspondence to the "pure" field emission regime is derived in [29] (see also [26,30]) where Y is a dependent variable (current density, macroscopic current density or total emission current and X is an independent variable (local field, macroscopic field, applied voltage or dimensionless field), C XY is a pre-exponential function that depends weakly on X and Y the function B el X = bϕ 3/2 /c X , where F C = c X X, where F C is the local barrier field at some characteristic point «C » on the emitter surface.
However, since the uncertainties associated with parameter λ are difficult to deal with, analytical work in FE often continues to be based on the older (1956) theory of Murphy and Good, [31] in which (in its zerotemperature form) λ GB from (3) is replaced by t −2 F , where t F is the appropriate particular value of a special elliptic functions [32].
Taking into account the field emission effect, the actual task is to define the surface emitter density on the nanotube films. In such a case we can see the mutual effect of nanotubes. If CNTs are grown so that the distance between neighboring CNTs is equal to the several CNT lengths, this corresponds to the minimal influence of the surroundings on radiation, emitted by an individual CNT, and gives the strongest emission of the cathode current. However low surface density of CNTs gives low total surface current. If CNTs are grown so that the distance between neighboring CNTs is significantly less than the length of CNTs, this corresponds to the maximum influence of the surroundings on radiation, emitted by an individual CNT, but at the same time electrostatic field shielding gives a small emission of the cathode current. That is why the surface density of CNTs gives small total surface current. Typically, a film has a nanotube density of 10 8 ÷10 9 cm −2 . The effective number of emitting sites, however, is quite lower. The typical densities of 10 3 ÷ 10 4 emitters/cm 2 were reported at the onset of emission [33][34][35][36]. By using an optical microscope combined with a phosphor screen, we were able to increase the resolution of the measurement and the reported densities of 10 7 ÷ 10 8 cm −2 [37]. In [38] based on the continuum model of kp-type, we proposed an approach for the approximate calculation of the electromagnetic guided waves characteristics on the almost circular, closely packed bundles of parallel, identical and metallic carbon nanotubes. For the real parts of slow-wave coefficients Reβ Mn for azimuthally symmetric guided-wave propagation that increases with the number of metallic CNTs in the bundle and tends in such bundles to unity, which is a characteristic of macroscopic metallic wires. From [38] (see Fig. 4) it follows that the real and imaginary parts of the slowwave coefficient at d/l = 0.1 asymptotically approach stable values and are estimated as Reβ Mn = 0.072, Imβ Mn = −0.011, where L is the length of CNTs and d is the nearest distance between CNTs in the bundle. In [38] the number of parallel, identical, infinitely long, single-wall, metallic, zigzag (21,0) CNTs, arranged on a triangular lattice is N = 10 4 .
In this paper on the basis that the particle transmission function is got taking into account the theoretical and numerical results, we calculate the magnitude of current through a single nanotube with a metallic surface.. The obtained results may be applied to the aligned not closely packed metallic CNTs bundles. For this case, the distance between neighboring CNTs must be equal to or larger than the nanotube length. Such materials are currently used. For example, in paper [39] the authors constructed a microwave diode in which the carbon nanotube field emission source was directly driven at gigahertz frequencies. The matrix of the carbon nanotube bundle consisted of uniform individual carbon nanotubes, spaced at a distance, which roughly doubled their height in order to minimize electrostatic field shielding from adjacent emitters and get the maximum emission of cathode current. In the paper investigation the range of potential difference on the nanotube ends at which the apex of triangular potential, corresponding to the maximum of potential, exceeds the energy of the tunneling of electrons. In such a mode of field emission current the applied voltage between the anode and cathode U˜is less than the threshold voltage. This condition is met, when we choose the range of potential difference on the nanotube ends from 2 to 3.5 V. The paper investigates the range of nanotubes lengths from 2 to 112 nm, when the electron transport in nanotubes is determined by the ballistic mechanism, i.e., the length of the nanotube must be less than the mean free path of an electron. In this case, current I must be calculated with the use of the Landauer-Büttiker formula (see (A.1).

Transmission coefficient of field emission from the tips of nanotubes
To determine the current in the case of FE from the tips of nanotubes using formula (A.1), we must know the transmission function of a particle from the anode to the cathode. Let a nanotube be on the surface of a metallic cathode with Fermi level U = U 1 = E F (in Fig. 1 left domain < 0, where z is a longitudinal coordinate; Fig. 1 is taken from [40]). Neglecting the classical potential energy of the free electron image, potential energy U (z) in the presence of electric field W = 0 in domain z ≥ 0 also contains the potential energy of the external potential field, U ext = −|e|W z (see dashed line curve in Fig. 1a). Within the metallic cathode z < 0, the electric field is absent and, hence, the potential is constant in this region. To simplify further calculations of field emission, we approximate the potential energy in the nanotube region depicted in Fig. 1b by the potential shown by the solid line in Fig. 1c Such a potential energy approximation indicates that we must approximate the wave function in a nanotube by the wave function in a shallow rectangular potential well and use the dispersion relation for free electrons.
Based on the potential of charge carriers, we obtain the analytic solution for the transmission function in the quasi-classical approximation. To do this, we express the solution to the time-independent Schrodinger equation in the form of plane waves in the region of the potential, shown in Fig. 1c (a solid line curve) where Function χ(z) from formula (5) can be expressed in terms of the Airy functions ( [41], p. 116]). Proceeding from the continuity condition for the wave function ψ(z) and the continuity of flux density S z = [i /(2m)](ψ∂ψ * /∂z-ψ * ∂/∂z) for charge carriers at discontinuity boundaries for potential energy (6), we can write the boundary conditions in explicit form where ψ * in the expression for flux density is the complex conjugated of the wave function ψ; m is the charge carrier mass. The Airy function is the solution to equation which can be expressed in terms of an integral where ξ = 2m|e|W In (9) γ determines the form of the specific solution to Eq. (8).
In further analysis, we will require solution (9) to Schrodinger Eq. (8) in domain z ≥ L 2 in uniform electric field (6), which has asymptotic form χ ∼ exp i2|ξ| 3/2 /3 . Such an approximate solution near point z ≥ L 2 can be obtained by integrating expression (9) over the imaginary axis from +∞ to 0 and over the real axis from 0 to +∞ [41]   In Fig. 2 we show the dependence of the solution to Eq. (8) on a variable ξ. The solid blue curve corresponds to analytical solution (11); the dashed line curve (a "circle" marker) corresponds to the case, when solution χ(ξ) = Bi(ξ) + iAi(ξ) to Schrodinger Eq. (9) is a linear combination of the Airy functions (the first Ai(ξ) and the second Bi(ξ) kinds).
Using relation (5) with account for boundary conditions (7) at interfaces z = 0 and z = L 1 for the potential, we obtain the following system of four equations where This system of Eq. (12) makes it possible to obtain the transmission coefficient as a function of the particle energy. Let a plane wave with amplitude A 1 = 1 be incident from the left from domain z < 0, while in the right part, away from singular point z = L 2 (domain z > L 2 ; i.e., ξ < 0 in relation (11)), only a plane wave traveling to the right is propagating. As a result, away from singular point z > L 2 , we obtain from Eqs. (12) the following expression for the transmission coefficient where η = 2|ξ| 2/3 /3; in deriving the expression for transmission coefficient τ, we have assumed that flux density S z in domain z > L 2 is defined by the equality ( [42], p. 80]) (15) and the flux density of the incident wave is defined as (14) can be calculated using the Kramer formula From (16) it follows that When conditions ξ >> 1 and z → L 1 are hold, we obtain from expression (16) with account for (14) the following expressions where we have introduced new notation With account for relations (18), we can transform equality (17) to Finally, from expression (14) with account for (20), we obtain the following expression for the particle transmission function when we obtain expression (21), the Airy function is approximated by asymptotic solution (11). We also take into account equality For L 2 >> L 1 , the exponent in expression (21) with account for approximate equality |e|W (L 2 − L 1 ) obeys the dependence analogous to the Fowler-Nordheim dependence [22]

The influence of nanotube length at field emission on the Nordheim function
Without the field, the surface barrier seen by an escaping Fermi-level electron has height equal to the local work function|e|'. The electric field lowers the surface barrier by an amount Δ(|e|ϕ) = |e| 3 W and increases the emission current in correspondence with the Schottky effect [16] (see also [28, p. 588]). When the field emission from the surface of a metal takes place (nan-  (23) includes Nordheim function θ(y) [16].
in which argument y is the relative decrease in work function Δ(|e|ϕ) = |e| 3 W , where the work function is equal to the field electrical voltage on the ends of a nanotube defined by the Schottky effect where L 2 = −E/(|e|W ); z max = 2 −1 |e|/W is defined from the maximum of potential energy U = −|e|W z − e 2 /(4z); 2z is the distance from an electron to the free electron image toward the metallic surface.
According to (21) if there is a nanotube, the transparency coefficient is also defined by equality (24); however, instead of Nordheim function θ(y) we use function Θ(y). Finally, we obtain an expression, which was derived in theoretical paper [40]

Calculation results
The transmission function from Fig. 3a and Fig. 4a is calculated, when the Airy function is approximated by asymptotic solution (11) (curve 1 in Fig. 2). The transmission function from Fig. 3b i Fig. 4b is calculated, when the solution to the Airy function corresponds to curve 2 in Fig. 2. In calculations shown in Fig. 3a and b we assume that U 1 = −4eV, U 2 = −6.4eV). In this work values U 2 = ε C − 6.4eV were taken from [43,44] by analogy with (6), where ε C is the Coulomb integral. Value U 1 = −4eV was chosen from the condition, at which the apex of triangular potential, corresponding to the maximum of potential, exceeds the energy of the tunneling of electrons. Figure 3a and b shows the dependence of the transmission function on the electron energy for zigzag nanotubes with the metal-type conductivity, when W = 10 9 V/m, W = 0.5 × 10 9 V/m, W = 0.25 × 10 9 V/m. Solid curves correspond to the nanotubes of length L 1 = 2.0, 2.5, 3.0, 3.5 nm, when W = 10 9 V/m; dashed curves correspond to the nanotubes of length L 1 = 4, 5, 6, 7nm, when W = 0.5 × 10 9 V/m; dotted curves correspond to the nanotubes of length L 1 = 8, 10, 12, 14nm, when W = 0.25 × 10 9 V/m. Figure 4a and b shows the dependence of the transmission function on the electron energy for zigzag nanotubes with the metal-type conductivity, when the field strength is as follows W = 0.125 × 10 9 V/m, W = 0.0625 × 10 9 V/m, W = 0.03125 × 10 9 V/m. Solid curves correspond to the nanotubes of length L 1 = 16, 20, 24, 28nm, when W = 0.125 × 10 9 V/m; dashed curves correspond to  Fig. 4a is calculated, when the Airy function is approximated by asymptotic solution (11) (curve 1 in Fig. 2). The transmission function from Fig. 4b is calculated, when the solution to the Airy function corresponds to curve 2 in Fig. 2.
The electron transport in nanomaterials is determined by the ballistic mechanism when the nanotube length is much smaller than the electron mean free path. In this case, the current I can be calculated using Landauer-Büttiker formula (A.1), taking account transmissions functions shown in Fig. 3 and Fig. 4 (see also [45,46]).
In Fig. 5 there is the dependence of the envelopes of oscillating currents. I in a separate nanotube on field strength W for voltage constant value U on the ends of nanotubes (see Fig. 1), where the domain of field strength is 5 · 10 7 ≤ W ≤ 10 9 V/m, U = −W L 1 ; L 1 is the length of a nanotube. In calculations the voltage on the ends of nanotubes is U = 3.5, 3.0, 2.5, 2.0 V. In Fig. 5 (see solid curves) we can see the dependence of the envelopes of oscillating currents I in a separate nanotube on the value of field strength is obtained, when the Airy function is approximated by asymptotic solution (11). I on the value of field strength is obtained, when the solution to the Airy function corresponds to curve 2 from Fig. 2. In Fig. 5 "a" labeled lines mean the upper envelope of oscillating currents; "b" labeled lines are used for the lower envelope of oscillating currents. In Fig. 5a curves 1a, 1b (see the solid green curves) and curves 2a, 2b (see the dashed green curves) are the envelopes of oscillating currents I. They correspond to U = 2.5V on the ends of nanotubes. Curves 3a, 3b (see the solid pink curves) and 4a, 4 b (see the dashed pink curves) correspond to voltage U = 2.0V. In the inset shown in Fig. 5a we can see the analogous dependence of the current on the field strength in the interval of 10 8 ≤ W ≤ 1.1 × 10 8 V/m; curves 1, 2, 3, 4 correspond to the envelopes of oscillating currents (1a, 1b), (2a, 2b), (3a, 3b), (4a, 4b).
In Fig. 6a we can see the dependence of function I/W 2 on the invert value of field strength 1/W in the interval of 10 −9 ≤ 1/W ≤ 10 −8 m/V. In the inset there is an analogs dependence in the interval of 0.91×10 −8 ≤ 1/W ≤ 10 −8 m/V, where voltage on the ends of nanotubes is U = 2.0 V. From Fig. 6a it follows that the envelopes of oscillating currents in the figure and in the inset have practically linear dependence on the invert value of field strength 1/W. Such dependency is discussed in "Discussion of results" (see (29) and (30)). There is analogous dependence for the case, when voltage on the ends of nanotubes is U = 2.0; 2.5; 3.0; 3.5 V. The solid blue curves 5a, 5b and the dashed blue curves 6a, 6b are the envelopes of oscillating currents and correspond to U = 3.5 V; the solid red curves 7a, 7b and the dashed red curves 8a, 8b correspond to U = 3.0 V. In the inset curves 5, 6, 7, 8 correspond to the envelopes of oscillating currents (5a, 5b), (6a, 6b), (7a, 7b), (8a, 8b) In Fig. 6b we can see the dependence of current I on field strength W in the interval of 10 8 ≤ W ≤ 10 9 V/m. In the inset there is the analogous dependence of current on field strength in the interval 10 8 ≤ W ≤ 1.1×10 8 V/m. The analogous qualitative dependence of current on the field strength will be for the envelopes of oscillating currents, shown in Fig. 5.
The calculation results qualitatively match the experimental article results [47], where the emission current dependence of voltage is depicted. Figure 7 shows the plot of ln I/Ũ 2 versus 1/Ũ , whereŨ is the applied voltage between anode and cathode and theŨ is less than the threshold voltage 260 V. If we consider the field enhancement factor β = 1.4 × 10 4 , the distance between anode and cathode d ≈ 5mm, the voltagẽ U = 50, 100, 200V, then the electric field strength at the cathode apex W = βŨ/d will be, respectively, W = 1.4×10 8 , 2.8×10 8 , 5.6×10 8 V/m. The same values of the field strength are shown in Figs. 5, 6. In addition, the invert value of the voltage between anode and cathode is 1/Ũ = 5× 10 −3 , 10 −2 , 2 × 10 −2 V −1 . Figure 7 shows our calculation results and data that are depicted in the experimental article [47].

Discussion of results
With field emission from a metallic surface (there are no nanotubes) transparency coefficient D (24) contains Nordheim function θ(y) [16], in which argument y is the relative decrease in work function Δ(|e|ϕ) = |e| 3 W (CGS) and is equal to y = z max /L 2 , where z max is the point of maximum potential. The decrease of potential surface barrier Δ(|e|ϕ) increases the emission current in accordance with the Schottky effect [16] (see [28], p. 588]). The paper shows that in the presence of nanotubes on a metallic surface we can see the relative decrease in the work function by (see section "The influence of nanotube length at field. . . "). Δ(|e|ϕ)/(|e|ϕ) = 1 − (L 2 − L 1 )/L 2 = L 1 /L 2 ,where L 2 = −E/(|e|W ). Taking into account that in Fig. 1c pointz max = L 1 corresponds to the apex of triangular potential (corresponds to the maximum of potential), by analogy with mathematical expression (21) we introduce argument y for nanotubes  Fig. 7 Dependence of ln I/Ũ 2 from 1/Ũ , whereŨ is the applied voltage between anode and cathode and the volt-ageŨ is less than the threshold voltageŨ = 260 V. Circle markers correspond to calculation results, and square markers correspond to experimental data from paper [47] Based on (27) we get function Θ(y) = (1 − y) for the transparency coefficient from (26). In Table 1, which was obtained in article [40], the dependence of Nordheim function θ(y) and function Θ(y) is shown, obtained in agreement with expression (26). From (25) and (27) it follows that at field electron emission the presence of a nanotube on the surface of a cathode is equivalent to the increase of z max = 2 −1 |e|/W in formula (25). The presence of it is due to the free electron image situated toward the metallic surface. What is more, the length of nanotube L 1 plays the role of z max . The work function is decreased significantly due to it. Truly, if W = 10 9 V/m, from (25) it follows that z max ≈ 0.6 nm. It corresponds to z max < L 1 =z max , where in our calculations L 1 = 3.5; 3.0; 2.5; 2.0 nm. If W = 10 8 V/m, in our calculations L 1 = 35; 30; 25; 20 nm, so z max <<z max . The maximum value of the transmission function of electrons energy in nanotubes (see Fig. 3 and Fig. 4), obtained by numerical calculations, is lower than unity in contrast to the results of [40]. Such a dependence can be explained that to input κ from (19) we must know the derivative of function χ(ξ) from (11). In the paper Table 1 The dependence of function Θ(y) [40] and Nordheim function θ(y) [28,48] [40] we neglect the pre-exponential dependence of function χ(ξ) and consider the quickly oscillating exponential part of function χ(ξ) to make great contribution.
The last one is possible if only ξ >> 1. In this paper we consider the pre-exponential dependence of function χ(ξ) on ξ in (19). Taking into account (26) the transmission function from (21) can be expressed as follows where L 1 = U 1 /W . From (28) it follows that at the constant value of voltage U 1 = const on the ends of CNTs phase shift Δϕ = k 2 L 1 = k 2 U 1 /W decreases as field strength W increases (when L 1 of CNTs decreases). So the oscillation current period I increases as the value of field strength W increases. It is the exact dependence that we can see in Fig. 6b for the dependence of current on field strength W , where current is calculated by Landauer-Büttiker formula (A.1). From (28) it also follows that in the cases of the dependence of the current in Fowler-Nordheim coordinates (the dependence of current on the inverse value of field strength 1/W , where in calculations U 1 = const) the period of oscillation does not change, because phase shift Δϕ = k 2 U 1 /W in Eq. (28) has the linear dependence on variable 1 /W . It is this dependence which we show in Fig. 6a. The linear dependence of current I on field strength W in Fig. 5a, b (see also Fig. 6b) and the linear dependence of function I/W 2 on the inverse value of field strength 1/W in Fig. 6a mean that the dependence of current on field strength is It is equivalent to dependence where f (W ) is an oscillating function with a constant amplitude; value A from obtained calculation results is equal to A = I/W ≈ 10 −14 m/Ω (see Fig. 6b). It is known that in the case of the ballistic transport of a charge in nanomaterials the electrical conductivity depends on active resistance [11] R a = h/ 2e 2 ≈ 12.9kΩ. In such a case value A a = I a /W = U 1 /(W R a ) ≈ 1.56 × 10 −13 m/Ω, which is analogous value A from (29) and (30) We can see that A/A a = 0.064. It happens, because in contrast to point I a , which appears due to the ballistic transport of a charge in nanotubes, the current in (29) first of all depends on function D(y) from (26), which, in its turn, is an analog to the transparency coefficient. Since electrical voltage U = L 1 W = const on the ends of nanotubes is constant, we obtain It should be noted that the phase is an important quantity in physical processes. For example, the interaction of two solitons is characterized by the phase shift; the cross section of elastic and inelastic scattering of nucleons from a nucleus and of photons from the scattering centers also depends on the phase shift; and the phase shift is connected with the adiabatic invariant (Bohr-Sommerfeld quantization principle). In our paper the transmission function corresponds to expression (21) and depends on phase shift Δϕ = k 2 L 1 along a zigzag nanotube with the metal-type conductivity. The highest value of the transmission function is attained when sin 2 (k 2 L 1 −ϕ 0 ) = 0, i.e., for the phase shift Δϕ = k 2 L 1 = ϕ 0 + πn, n = 1, 2, 3, . . . , (32) where wavenumber k 2 and phase shift Δϕ were obtained proceeding from the dispersion equation for p− electrons with a rectangular potential energy profile. In [49][50][51] phase shift Δϕ and energy levels E n for stationary states of charged particles in a zigzag nanotube and armchair nanoribbons in the presence of a longitudinal constant electric field were obtained based on the two-and four-point unit cell models (the continuum model of kp-type [51]) where ϕ n,2 = ϕ(z = L), ϕ n,1 = ϕ(z = 0), L is the length of a zigzag nanotube and armchair nanoribbon, υ F is the Fermi velocity. Equidistant energy levels E n are inversely proportional to the nanotube length and coincide with the spectrum of a 1D oscillator. From (33) it follows that in the presence of a longitudinal constant electric field the functional dependence of charge carriers phase shift Δϕ and the functional dependence for the equidistant energy levels on quantum number n are the same. The value of phase shift Δϕ is independent of the nanotube length and, for large values of n, coincides with the adiabatic invariant [42] I(E) = 1 2π pdx = Δφ π = (1 + 2n) (34) where the contour integral is taken over the total period of the classical motion of a particle. Condition (34) corresponds to the Bohr-Sommerfeld quantization principle of the old quantum theory and determines in the semiclassical case of the stationary states of the particle. From (32) and (33) it follows that the value of phase shift Δϕ, which was obtained proceeding from the dispersion equation for p− electrons with a rectangular potential energy profile U , and the value of phase shift Δϕ, obtained from the continuum model of kptype, are the same, if ϕ 0 = π/2. In spite of this, in the continuum model of kp-type the energy levels are the equidistant energy levels. It, possibly, will lead to quantitative differences, obtained in this paper. To do this, we need to use the results obtained in [49][50][51] in particular-the solution to a wave function, but this task goes beyond our paper.
From the calculation results (see Figs. 3 and 4) it follows that there is a good quantitative correlation for transmission functions that are calculated from two approximations. Taking into these transmission functions, when we calculate the function of current (see Fig. 5a i b) in correspondence with Landauer-Büttiker formula (A.1) there is a linear dependence for the envelopes of oscillating currents on the field strength in the interval of 0.5 × 10 8 ≤ W ≤ 0.5 × 10 9 V/m. It corresponds to functional dependence (29). At the high values of the field strength we see the deviation from the linear dependence. The deviation from the linear dependence for lines 5b and 6b in Fig. 5b takes place, because lines 5b and 6b in Fig. 5b correspond to higher-voltage values on the ends of nanotubes than ones in Fig. 5a. Thus, the condition, under which the apex of triangular potential exceeds the energy of the tunneling of electrons, is no longer met. In such a mode of field emission current the applied voltage between the anode and cathodeŨ is commensurate with the threshold voltage [47]. In this domain possibly begins the transition to the current linear dependence on voltage in Fowler-Nordheim coordinates. Figure 6a shows the dependence of current on the fields strength in Fowler-Nordheim coordinates I/W 2 and the inverse value of field strength 1/W in the interval of 10 −9 ≤ 1/W ≤ 10 −8 m/V, where in calculations the value of voltage on the ends of nanotubes is constant. In such a case there also exists a linear dependence corresponding to functional dependence (30). From Fig. 6a it follows that in (30) the period of function f (W ) oscillation toward variable 1/W has a constant value. Figure 6b shows the dependence of current I on field strength W in the interval 10 8 ≤ W ≤ 10 9 V/m. In such a case the period of current oscillation increases as field strength W grows. In this paper the length of nanotubes is in the interval of 2 ≤ L 1 ≤ 112nm. Further increase of the length of nanotubes L 1 ≤ 500nm at constant voltage on the ends of nanotubes makes current in Fowler-Nordheim coordinates have stationary value. It can be that at the length of nanotubes L 1 > 1μm current from the field strength in Fowler-Nordheim coordinates will linearly decrease; however, in such a case the ballistic mechanism condition would not be fulfilled. Thus, we should use the Boltzmann kinetic equation in the relaxation time approximation.
In this paper we have obtained a qualitative agreement of the results of calculations with the experimental results [47] (see Fig. 7); however, we got a large quantitative discrepancy: the results of calculations are three orders of magnitude greater than the experimental results. This is primarily due to the fact that if field voltageŨ = 200 V and the distance between an anode and a cathode, d = 5mm the field strength is W = 4 · 10 4 V/m. If the voltage on the ends of nanotubes U = 2.0 V, then the length of nanotubes L 1 = U/W = 50μm. With such a length of nanotubes the electron transport in nanomaterials, determined by the ballistic mechanism, is not implemented, i.e., we cannot use the Landauer-Büttiker formula [45,46]. The ballistic mechanism can be applied on length L 1 ≤ 1μm. In this paper the range of nanotubes length change is 2 ≤ L 1 ≤ 112nm. Another reason for the large quantitative discrepancy between the calculation and experimental results may be that we approximated the wave function in the nanotube by the wave function in a shallow rectangular potential well (see Fig. 1). In general basing on the continuum model of kp-type [51], we should use the wave function, obtained from [38,49] in a zigzag nanotube and armchair nanoribbons in the presence of a longitudinal constant electric field.

Conclusion
In our work we obtained qualitative match of emission current calculations result (see Fig. 7) and experiment results [47], in case ofŨ is less than threshold value of 260 V, whereŨ is the voltage between anode and cathode and the distance between them is kept constant, d ≈ 5mm.
Electron transition in nanomaterials is determined by ballistic transport, when the nanotube length is many times less than the electron mean free path and the current value in case of FE from nanotube tip is determined using the Landauer-Büttiker formula. Emission current in Landauer-Büttiker formula depends on transmission function (28), which can be derived using Θ(y) that is an analog for the transparency function D as for Nordheim function θ(y) [16].
The results of field emission, obtained proceeding from the dispersion law of p − electrons with a rectangular potential energy profile, can be achieved, if we use the model of a two-point and four-point elementary phase cells (the continuum model of kp-type [49][50][51]) in order to get the transition function. In particular in the case of a two-point unit cell for zigzag carbon nanotubes and armchair-edge nanoribbons we ca use the wave functions expressed by the Hermite functions [50] (or results [49]). In the case of a four-point unit cell we can use the wave functions for Majorana and Dirac fermions [52].
The calculations results show that in the case of field emission in nanotubes in the interval from several nanometers to one hundred of nanometers there is a linear dependence of the envelopes of oscillating currents on field strength W in the interval of 0.5 × 10 8 ≤ W ≤ 0.5×10 9 V/m. It corresponds to functional dependence (29), where W = U 1 /L 1 , U 1 = const is strength on the ends of nanotubes. The linear function is also in the case of the dependence of current on field strength in Fowler-Nordheim coordinates of function I/W 2 of the field strength inverse value 1/W in the interval of 10 −9 ≤ 1/W ≤ 10 −8 m/V. Such a dependence takes place, because L 1 < 1μm, when the ballistic mechanism works during electron transfer in nanomaterials.