A three-dimensional model for thermionic emission from graphene and carbon nanotube

Graphene and carbon nanotube (CNT) have been proposed to be good materials for thermionic energy converter (TEC). For accurate simulation of performance of TEC, it is important to know the correct equation for temperature dependence of thermionic emission current density (J) from graphene and carbon nanotube. In this paper we first consider the existing theory of electron energy dispersion relation in graphene to reconsider the relations between Fermi energy and Fermi velocity in relation to some form of electron mass in graphene. We then consider existing various models of temperature dependence of J versus T (J(T)) and their applicability to nano-materials. We find that no model exists to date that fully conforms to the available experimental data on J(T) of nano materials. By providing justifications for three components of electron momentum vector during thermionic emission from graphene, we then find a three-dimensional model that fits the experimental thermionic emission data from graphene and carbon nanotube far better than any existing model. We present a detailed comparison of our model with existing models of thermionic emission. The work function determined using our model also agrees very well with independent experimental results. This model is expected to be very effective in modelling TEC with graphene or CNT.


Introduction
Thermionic emission of electrons played a key role in advancement of Physics, electronics and many areas of modern science and technology. The equation for the current density (J) of emitted electrons at temperature T is so far given by the well-known Richardson-Dushman Equation: A 0 is the well-known Richardson-Dushman constant is´-Am K 1.2 10 6 2 2 where W is the work function of the material and k B is the Boltzmann constant. The equation known as RD law can be derived following Sommerfield's model [1] of three-dimensional free electron gas in a metal.
In recent years, thermionic electron emission and field electron emission from low-dimensional nanostructures, e.g. carbon nanotubes (CNTs), graphenes, etc, have been widely studied experimentally [2][3][4][5][6][7][8][9]. To interpret the measured emission current density from low-dimensional nanostructures, the macroscopic Richardson-Dushman law and Fowler-Nordheim law were usually directly applied in previous reports [3,4,[6][7][8]. Like carbon nanotube (CNT) [4][5][6] the electron emission processes (field and photo-assisted over-barrier electron emission) from graphene [7,8] seem to require revisions of existing law (Richardson-Dushman (RD)) to account for unique properties of graphene. Some recent works [2,3] question the validity of the RD law [9] in graphene [2,3,10] and CNT [11,12] while some indicate the validity [13]. In recent years thermionic emission from carbon nanotubes and graphene has received special attention [14][15][16][17][18][19][20] still questions remain about the validity of the RD law in these materials. Wei et al. [21] carried out very careful measurements of thermionic emission current density from many individual CNTs and examined the validity of the RD law in details for each showed an upward bending instead of giving a straight line as required by the RD law. Thus, they concluded that RD law breaks down for the thermionic emission from individual CNTs. Sherehiy [22] carried out studies on 'Thermionic emission properties of novel carbon nano-structures' and determined work functions of different nano-structures from the data on thermionic emission current density at different temperatures, reduced to zero Schottky effect. They had an interesting discoverynanostructures with lower surface charge density have higher work function-which was qualitatively explained to be due to different density of states.
Since the discovery of exfoliated mono-layer graphene [23] in 2004, graphene has exhibited many unique properties such as: linear band structure [24], ultra-high mobility > -( ( ) ) Vs 40000 cm , 2 1 high thermal and electrical conductivity [24]. Graphene is typically referred to a single atom thick layer of carbon, although sometimes bilayer or trilayer graphene are also mentioned. Graphene is a two-dimensional (2D) material, formed of a hexagonal lattice of carbon atoms which are covalently bonded in the plane by sp 2 bonds between adjacent carbon atoms. The bonding energy (approximately 5.9 eV) between adjacent carbon atoms is among the highest in nature (slightly higher than the sp 3 bonds in diamond) [25]. With a single-atom-thick sheet of sp 2 -hybridized carbon atoms, graphene exhibits great promises for future applications in energy storage [26], nanoelectronics [27,28], and composites [29]. With the high bond strength among the adjacent in-plane carbon atoms thus graphene is a material for high temperature devices [30] (operating in vacuum) and thus has a potential for a suitable candidate as an emitter in a thermoelectronic [with no ions involved] energy converter [TEC].
In a TEC [31][32][33][34] with the work function of the emitter, > W W e c of the anode (collector) and when the emitter and collector is connected through a load, the output power is = - where I e and I c are the emitter and collector emission currents. The latter quantities are primarily controlled by temperature and work functions of emitter and collector T T W and W , , , e c e c the emitter collector configuration (i.e., the space between them), and arrangements that control space charge. Recently Yuan et al. (2017) discussed new method of controlling space charge using magnetic field and gate voltage [34]. To model a TEC, specially, with graphene as an emitter, it is very important to obtain the accurate model of temperature dependence of W W I and I , , .
e c e c It may be mentioned that thermionic energy converter (TEC) once perfected can store the electrical energy by charging a battery with circuits similar to that of solar panel. It will reduce the dependence on silicon.
Liang and Ang [2,3] have fitted J versus T data from monolayer two dimensional graphene [35] by developing a thermoelectronic emission equation, different from that of RD law, assuming massless Dirac electron inside graphene. In their theory they have considered Fermi energy and Fermi velocity of electrons in graphene to be two independent parameters. As shown below this concept contradicts the current understanding of electron dynamics in graphene. Apart from this their theory has many errors from physics and mathematics point of view as pointed out in [36].
Moreover, they did not consider the variation of work function with temperature (T) in the thermionic emission equation, which our present work shows to be very important, specially, when the Fermi energy at 0 K, ( ) E F0 is low, as it is in graphene. It has not been known whether their theory applies for the CNT. Moreover, in the light of experimental discovery [37] of finite dynamic electron mass and the attribution of some form of mass to electrons in graphene based on the current concept of electron dynamics as given below, their model becomes questionable. Thermionic emission involves electron dynamics and hence dynamic electron mass or some form electron mass in graphene should apply. Otherwise, the fundamental question how electrons acquire mass when emitted from graphene, if their effective mass is zero in graphene. Moreover, in fitting the experimental J versus T data for monolayer graphene they had assumed the Fermi group velocity, V F to be´-1 10 m s 6 1 while more recent experimental works show it to lie in the range 1.73 to´-2.49 10 m s . 6 1 For monolayer graphene the latter value applies. As discussed later V F affects significantly the fitting of the experimental data.
There are thus challenges in a correct formulation of the theory of electron emission from graphene and CNT. Below we consider a few possible models for thermionic emission from graphene briefly without giving the elaborate derivations of the models because of space. Then we provide first brief outline of the existing theory of electrons in graphene. This shows that electrons are not truly massless as conjectured by many earlier. Then we present our own simple model which is seen to fit the experimental data of thermionic emission from both graphene and carbon nanotube.

Brief outline of different possible models of thermionic emission from graphene
In graphene, if one considers emission from graphene edge, the emission may be treated like that of Somerfield model except the fact that the density of states in a two-dimensional material is independent of energy. Such treatment has been recently conducted by Wei et al. [11]. Moreover, as the expected net current from edge is quite low, edge emission may not be of quite practical interest. However, while treating the emission perpendicular to the graphene sheet (i.e., along the z-direction from the surface) one must consider the quantum confinement effect along z, which can be done in the line of the following two models, considering electrons to have finite non-zero effective mass: (i) The electrons along z have energy levels like a particle in an infinite square well potential with non-zero mass but the vacuum level is separated from the Fermi level by the work function, W .
(ii) The electron energy levels are guided by finite barrier ( ) V r potential well and the emission takes place when the tunneling electrons have sufficient energy to overcome the work function and the emission needs to be treated with tunneling probability. Our investigation with model (i) shows that while close form expression for J is derivable based on parameters that can be either determined or evaluated theoretically, the model fails to correlate with experimental observations on both electron density and current density. While Wei et al. [11] have investigated model (ii) based on tunneling at different quantized energy levels with different tunneling times (knowledge of which is quite uncertain), it is not possible to arrive at a close form expression for J versus T and it is extremely hard to apply such model to J versus T relations in graphene and CNT and to simulate performance evaluation of TEC based on these nano materials.
3. Brief outline of electron dynamics in graphene 3.1. Ideas behind massless nature of electrons in graphene To understand the initial ideas of massless nature of electrons in graphene we follow initially the simple treatment of Zhang (2006) [38] and that of Neto et al. [39] to show the linear energy dispersion that is possible only for massless particle. Then we argue that for thermionic emission finite mass electron is necessary and it is supported by experimental and also has theoretical backing.

Linear energy dispersion in graphene
Graphene is single atomic sheet of carbon atoms that are arranged into a honeycomb lattice with two c atoms in a unit cell (figure 1). Crystal structure of graphene is shown in figure 1. The lattice vectors can be written as: Of particular importance for the electron dynamics in graphene are the two points K and K′ at the corners of the graphene Brillouin zone (figure 1). These are:  A carbon atom has 6 electrons. Two electrons are in 1S 2 shell. Four are in 2S 2 and 2p 2 sub shells. The first two formed filled energy levels. The three of the four electrons of carbon atom 1 form three σ bonds with the three of four electrons of carbon atom 2 in the plane of the graphene sheet. The σ bonds are localized and form filled energy bands and thus they do not contribute to electron conduction.
The fourth electron is in 2p z orbital which oriented normal to the graphene plane. The fourth electron of carbon atom 1 form π bond with the corresponding electron of atom 2. There are two π band energyone bonding (valence band-filled-π) and another antibonding (conduction band-unfilled-π′). The idea of massless nature of electrons in the conduction band of graphene emanate from the energy dispersion relations calculated on the basis of tight binding approximation and it is given as follows. Let us call the two 2p z orbitals of the fourth electron in carbon atoms 1 and 2 as Φ 1 and Φ 2 respectively [each of which is normalized and Φ 1 and Φ 2 are orthogonal]. The total wave function of the two electrons can be given as (with linear combination of atomic orbitals) F = F + F ( ) c c 2 In the graphene lattice the net wave function must follow the periodicity of the lattice R as given by Block wave function: Considering a single electron to be described by wave function Y( ) r and its motion is governed by the Hamiltonian H, which is given by the potential of all the carbon atoms as The wave function Y( ) r satisfies the Schrodinger equation: The Hamiltonian H can be written as: To get the energy eigenvalues of equation (6) we project it on Φ 1 and Φ 2 and the two equations áF Yñ = áF Yñ = | | | ( ) j H E j , j 1, 2 10 To evaluate equation (10), we consider only nearest neighbor products considering only R=0, a 1 and a 2 in equation (4). Then i 2 We set these equal energies to be zero. Then ik a ik a . . 1 1 [40]. As an approximation β can be neglected. Equation (17) is further simplified if we exploit the fact that γ 0 is small. Then one gets by solving where k y and k x are the y and x component of the wave vector k in graphene plane. The±sign refers to upper (π * ) band and lower (π) band respectively. On neglect of β the energy dispersion is symmetric around zero energy in k-space. The dispersion close to one of the Dirac points (at the K and K′ points in the Brillouin zone, gives in the first order approximation, energy of electrons in graphene as proportional to absolute value of momentum. | | q defines absolute value of momentum relative to the Dirac point. This reminds us of energy dispersion relation for massless relativistic particles (photons, neutrinos etc) that comes from Einstein's relation with analogy that V , F being the maximum electron velocity in graphene, replaces c. Even though the first approximation equation (19) gives massless (energy proportional to momentum) nature of electrons in graphene, a cyclotron mass can be attributed to it from the following consideration: within semiclassical approximation the cyclotron mass is given by [Ashcroft and Marmin (1976) (20) and (19) then one gets, The cyclotron mass determined for electrons in graphene is shown theoretically and found to depend on carrier concentration. It varies from very close to m 0.025 0 for low concentration to about m 0.07 0 (rest mass of electron) for carrier concentration, n of about´-7 10 cm 12 2 [39,41]. The equation (22) thus shows clearly that E F and V F are related and definitely not two independent quantities as treated in the recent formula for thermionic emission current density derived by Liang and Ang [10].
Equations (21) and (22) become somewhat complicated if one takes into account the second order term in equation (18) that gives the dependence of energy on | | q . 2 This term reminds us that graphene electron is not truly massless, as conjectured from equation (19) and neither it is relativistic in the true sense of term (velocity ∼ 0.1 c or higher). Energy proportional to | | q 2 reminds some form of mass attribution to electrons in graphene. Yoon et al. [37] measured kinetic inductance of electrons at microwave frequencies. From such measurements they extracted dynamical electron mass to be in the range m 0.01 e to m 0.024 . e Thus, it is not correct to assume electrons to behave completely massless [as per equation (19)] for thermionic emission, as has been done in the treatment of thermionic emission from graphene by Liang and Ang [10]. If we do, the question is how do the electrons acquire mass when ejected out of graphene? It is not possible to answer that by the theory of Liang and Ang. Moreover, the theory has many faults as mentioned earlier [36]. Using equation (19) the density of states, one can easily see that r ( ) E becomes proportional to the energy E. This was used in the Liang and Ang model. When we take equation (18) the r ( ) E becomes a complicated function of energy E. The derivation of thermionic current density J T versus relation becomes complicated. Further complication arises from the fact that for thermionic emission the electron must have a component of momentum, k z normal to the graphene plane in addition to components of q in the x-y plane of graphene. A proper model must take the energy quantization along z into account in addition to E(q). Then preliminary investigation shows that it is difficult to get a close form of expression for J T versus . For proper performance evaluation of thermionic energy converter with graphene complicated J versus T relation is not helpful and one needs a close form expression.
Based on above discussions, for proper performance evaluation of thermionic energy converter with graphene, in this paper, we have sought an alternative approach to derive the correct J T versus relation that can apply to both graphene and carbon nanotube. In this new approach we have considered the temperature dependence of the Fermi energy ( ) E T , F work function W, along with thermal expansion of materials. We have assumed three-dimensional density of states. During thermionic emission from the surface of monolayer graphene, electron motion cannot be considered as being restricted in two-dimensional plane of graphene (as in the case of electrical current conduction in the plane of graphene or along the length of a CNT) because, for emission perpendicular to the graphene surface, electrons must have in-plane motion as well and thus thermionic emission involves three-dimensional motion of electrons with three components of the wave vector. With wave vectors only along the graphene plane it is not possible to treat or model the thermionic emission perpendicular to the graphene plane. The thermionic electrons (from the graphene surface) will have wave vector components along the direction of emission ( ) z as well as in plane ( ) x y , .The emitted electrons leave vacant energy states within graphene after emission. When the electrons are returned to the emitter via the lead current from the collector (anode) [the anode and emitter are electrically connected] they need to be redistributed to fill those vacant energy states again. This involves an in-plane motion of electron within graphene (emitter) and as a result such electrons must have also wave vector components within the plane of graphene ( ) xy . Thus, for thermionic emission of electrons even from a monolayer graphene the wave vectors of electrons have three finite components ( ) k k and k , x y z of the wave vector and the three-dimensional model for thermionic emission should be valid even for graphene, a two-dimensional material. It is only for thermionic emission from the graphene edges a two -dimensional model would be appropriate but the net current is too insignificant for it to have any practical application. For thermionic emission from the cylindrical surface of a hot CNT the electrons are also expected to have finite non-zero x y z , , components of the wave vectors and hence a three-dimensional density of states should be applicable also for thermionic emission from CNT.
For thermionic emission out of graphene plane electrons must have z-component of momentum. How is this possible in an ideal 2D monolayer graphene? The dispersion relation discussed above (equation (18)) contains in plane wave vector k, that has both x y , components, k k x y , but not z (normal to the plane) component. A graphene surface from which thermionic emission generally takes place is macroscopic (usually the area is more than 1 cm × 1 cm) and not nanoscopic. Therefore, electron, in all practical reality, follow Bloch wave function guided by periodicity in the plane of graphene and hence these momentum components assume practically continuous values (not discrete). However, for the z-component of motion as the electron is quantum-confined in z-direction (one atom layer in graphene), a z-component of momentum  ( ) /a can arise (even from uncertainty principle). This z-momentum is intrinsic in the sense that it is there even at 0 K. It does not cause electron emission. There are discrete excited energy levels higher than  ma 2 2 2 and Fermi energy, E F by definition is the highest occupied energy level at 0 K. At finite temperature electrons start occupying levels higher than E F , the general consensus for thermionic emission along z-direction is that, It is an open question how electron in 2D graphene acquire this z-component of momentum and some of the possible ways this can happen are discussed in the appendix B.
Using the three-dimensional density of states and considering thermal expansion, we have obtained an equation W(T) containing terms up to fifth power of T and a final equation for J T versus using the W(T). It provides an excellent fit between our theory and the experimental data of J versus T for both graphene and carbon nanotube. The fits are seen to be far better than those of any existing model. From the best fit of experimental data, we have estimated the variation of ( ) E T F and W(T) within the given temperature range of the data for both graphene and CNT. Using the new model, we have estimated the effective thermionic mass of electron in graphene for the first time and found to be in excellent agreement with the measurements of dynamic electron mass in literature. The estimated work function at 300 K is also in good agreement with experimental values. These indicate the success of our model. Other existing models fail to determine the effective thermionic mass of electrons from J T versus data of graphene. After making detailed comparison of the existing models of thermionic emission we find that the new model explains not only the J T versus data for graphene and CNT but also experimental values of some important physical parameters. The justification for three-dimensional model of thermionic emission for graphene can be understood from the discussion in appendix A and the following.
During thermionic emission from monolayer graphene, electron motion cannot be considered as a restricted two dimensional motion (as in the case of electrical current conduction in the plane of graphene) because, for emission perpendicular to the graphene surface, electrons must have motion in plane as well and thus thermionic emission involves three dimensional motion of electrons with three components of the wave vector and hence it is not surprising that a good three dimensional model with three dimensional density of states as presented here, could excellently fit the thermionic emission from graphene as seen in the results described below. Even in the theory of Liang and Ang [2,10] which is claimed to be a two-dimensional model, wave vectors both in graphene plane and perpendicular to the plane have been considered. With wave vectors only in the graphene plane it is not possible to treat or model the thermionic emission perpendicular to the graphene plane. The thermionic emission of electrons from the cylindrical surface of a single or multi-walled carbon nanotube must also involve three components of electron wave vector.
Thus, a three-dimensional model of thermionic emission as presented in this work should be valid for graphene and carbon-nanotube. The agreement with experiments as mentioned below for these materials support the model presented in this work.

Variation of work function (W) of a metal with temperature
For nano-materials both the surface density and volume density of free electrons are far less than those in metals and accordingly the Fermi energy is expected to be much less. Its temperature dependence is expected to have greater influence on W than for metals. The relation of W with T depends primarily on the change of E F with T. To work out this change we rely on the fact that the total number of electrons ( ) N T in a given piece of metal at = T 0 K is the same at = T T K. (It is assumed that during thermionic emission in a TEC, electrons will be replaced at the emitter through load current). Now at  (25) is called chemical potential in many books. We call it temperature dependent Fermi energy level. Considering thermal expansion of the metal, where α is linear thermal expansion coefficient. Thermionic emission takes place at high temperature > ( ) 1000 K . The work function ( ) W is defined as: where E v is the vacuum level. E F is dependent on temperature. Change of E F with T will change the work function with T and this in turn will affect the thermionic emission current density from that given by RD Equation. This is going to play a special role in nano-materials where E F is low at ambient temperature. Our primary objective now is to obtain E F and hence W , as a function of T from the equations (23)- (27) to (35). The 2nd objective is to see how this affects the RD equation (1) and how the modified RD equation fits the experimental results of graphene.
as discussed above, from equations (23) to (26) ò ò The LHS of equation (29) is independent of T while the RHS is dependent on T. Solution of this equation (29) for E F as a function of T is a non-trivial problem. There are two ways it can be accomplished as described below:

Method 1
Expanding the RHS of equation (29) we obtain equations (30)-(33) To obtain E F as a function of E F0 and T, we put = E E F F 0 as a first approximation in the RHS of Equation (31) and we get equation (32) a a a = - Using equation (27) we then obtain W 0 is the work function of the material at = T 0 K.

Second method
The second method of obtaining E F as a function of T for a given E F0 involves numerical integration of the RHS of equation (29) using the value of α for the material. To obtain E F at a value of T for a given value of E F0 we have carried out the numerical integration (equation (29)) using, D = / E E 10000 F0 and the upper limit of the RHS as E 100 .
F Then for a given E F0 and T, we find the value of E F that gives close agreement between the LHS and RHS of equation (29) such that the absolute value of the difference between the two lies within 0.1% of the value of the LHS. By changing T we get another value of E F for the same E F0 . Thus one can obtain E F as a function of T for a given E F0 . Using these ( ) E T F one can obtain the changes in work function relative to its value at absolute zero, W 0 . These in turn can be used to obtain J T versus for graphene for a given value of E F0 and W 0 using the equation (34).
By finding the best fit between the experimental values of J versus T curve for graphene and carbon nanotube one then can obtain the values of W 0, E F0 for the materials. It can be seen from computations that the changes in ( )/ W T W 0 i.e., D ( )/ W T W 0 increases as E F0 decreases. This second approach though ideal for nano material in this model, is quite time consuming to generate the J T versus curves to find the best fit with experimental values.
In equation (

New modified Richardson-Dushman equation
As explained earlier, thermionic emission even from graphene (2D) and carbon nanotube (1D) involves electron motions with three finite components of wave vectors. Hence a three-dimensional model of thermionic emission current density should be applicable for both graphene and carbon nanotube. Considering the corresponding thermal expansion coefficient, and using equation (33), the modified Richardson-Dushman Equation (MRDE) for thermoelectron current density (emitted along z direction) is given by: Where W(T) is given by equation (33). Equation (35a) gives us new thermionic emission equation that should be applied for materials like graphene, carbon nanotubes which have low E F0 and for temperatures less than / E k .
For 2D graphene, = r 2 and for 1D CNT, = r 1 in equation (35b) where, it can be easily seen that the effective RD constant is given by The E F0 is related to the free electron concentration by the following equation:

Application of modified Richardson-Dushman equation to thermoelectron emission from graphene
In this study, we investigated if the MRDE can be applied to thermoelectron emission from graphene which has exhibited metallic properties [42][43][44]. Experimental data of J for a mono atomic layer of suspended graphene has been extracted from figure 9(a) of [2]. Using equation (35) and the experimental value of thermal expansion coefficient -´--8 10 K     tables 1 and 2). By comparing the best MRD law fit (figure 2) with the best RD law fit (figure 2), we can visually also see that the fit is better with MRD law than with RD law. This is reflected in the S values also (tables 1 and 2). This shows the usefulness of our equation (35) for thermoelectron emission current density J. Using equation (35) and the values of W 0 and E F0 corresponding to the best fit [figures 2 and 3] of the thermionic data we find that the work function of monolayer graphene increases from 4.692 eV at 1620 K    to 4.725 eV at 1795 K ( figure 4). Surprisingly, like that in CNT, the work function (4.72 eV) corresponding to the best RD law fit of J T versus data of graphene falls within this range. For nanomaterials E F0 is quite small compared to metals and our equations (33) and (34) should be used to model ( ) W T and thermoelectron emission current density J from nanomaterials. The equations should be used for performance evaluation of TEC using graphene, carbon nanotubes etc as emitter and collector.
There are various reports on work function of graphene. Song et al. and Seo et al. [47,48] claimed to have carried out accurate determination of work function of graphene through a detailed analysis of the capacitancevoltage characteristics of a metal-graphene-oxide-semiconductor (MGOS) capacitor structure. They found a value 4.62 eV for the work function of graphene. Lee et al. [49] measured by scanning kelvin probe microscopy the unbiased work function of graphene to be 4.56 eV and found that it is influenced by electric field due to bias gate. In our model, the work function of graphene increases with temperature. At 300 K, it is 4.593 eV. It is to be noted that the thermionic emission data that has been fitted with our MRD model correspond to that of monolayer suspended graphene of Liang and Ang. We can assume the layer to be pure, i.e., without contamination. The experimentally measured value of pristine graphene is 4.56 eV [50]. Thus, we see that our estimate of work function of single layer graphene is in close agreement with the experimental values of work function of graphene reported in literature. Our model offers very accurate determination of work function if the thermionic data can be very accurate. To our knowledge, there is no experimental data on temperature dependent measurement of work function of pure monolayer graphene to confirm our theoretical prediction of increase of work function with temperature. Such data could have lent additional credence to the model.

Thermionic emission from carbon nanotube
After the initial success of MRDE model on thermionic emission from graphene (section 6.1), we apply equation (35a) the modified Richardson-Dushman equation (MRDE) to thermionic emission from CNT. For nanomaterials, because of low density of free electrons compared to metals, Fermi energy (equation (36)) is much lower than that of metals. As a result, the equation (35) should be applicable to nano-materials in its entirety. For metals which has E FO ∼10 eV, it can be shown that equation (35a) essentially reverts back to RD law with effective thermionic constant given by equation (35c).
We rely on the data kindly supplied to us by Wei [11]. Wei et al. [11] performed very careful and delicate studies on individual hot CNTs where simultaneous measurements of J and T have been performed. To simulate the J versus T curve using equation (35) and to study the best fit of the experimental data, we used the value of thermal expansion co-efficient a =´--2 10 K 5 1 as obtained by Deng et al. [46] for both single and double walled CNT. We have given systematic wide range of variation by 0.002 eV to both W 0 and E F0 . W 0 and E F0 are the only two parameters which are varied to obtain the best fit in this paper. Figure 5 shows the theoretical fit   E F0 (table 4). The lower the value of S the better is expected to be the overall fitting. Figure 5 also shows the fit with values of W 0 and E F0 slightly different from those of the best fit values. We see that as S changes with values of W 0 and E F0 [table 4] so does the visual fit of the theoretical curve with the experimental data (figure 5) Thus, with MRDE, figure 5 gives the best fit with value of / S N (N=number of the data points) well within the mean experimental error [11]. The variation of work function with temperature for the best fit values of W 0 and E F0 and some other values are shown in figure 6 for CNT. The  found for CNT ( figure 5). Anantram and Leonard [51] reported that electrons in the crossing sub-bands have a large velocity of -8 10 m s 5 1 at the Fermi energy of CNT. If we assume that this is the Fermi velocity than the Fermi energy turns out to be 1.82 eV, which is surprisingly in close agreement with the value 1.87 eV, obtained from the fitting. Apart from good fitting of J T versus data for CNT, this agreement lends additional credence to the validity of our model for CNT.
We next study the comparison of our MRD model with RD law. In RD law, the work function W is temperature independent. To study the fit of the J T versus experimental data with the values obtained by using RD law, we give a systematic variation of W, the work function of CNT, from 3.88 to 3.93 by 0.001 eV. We find that the best fit (figure 7) with RD law is obtained at =    The emission current density J in that theory is shown [2] to be given by Liang and Ang [2] used equation (37) and obtained their best fit of the experimental data of J versus T on monolayer suspended graphene for f = 4.514 eV, = E 0.083 eV    density in graphene far better than that of Liang and Ang's theory and that by RD law. Figure 8 shows the comparison of the three models (MRDE, RD and Liang and Ang) in one plot.
In the light of discovery of finite dynamic mass or effective mass of electron in graphene, the theory raises questions as mentioned earlier. However, we have examined to see if the theory can fit the experimental data in graphene better than our MRD equation. As explained earlier in section 3.1. some form of mass (though smaller than free electron mass) needs to be attributed to electrons in graphene.

Temperature dependence of work functions for graphene and CNT
We have included the plot of temperature variation of work function for graphene and CNT. We have shown in dT for these metals is smaller than that of graphene and CNT by a factor 7.8 and 6.2 respectively. Thus even though the temperature dependence of J from metals can be roughly described by for graphene and CNT, temperature dependence of J should be described by equation (35a).

Conclusion
We have considered first the existing theory of electron energy dispersion in 2D monolayer graphene to reconsider the massless nature of electron in graphene and the relations between Fermi velocity and Fermi energy of electrons in graphene as these are important for any new models on thermionic emission from graphene. We then consider various existing models for thermionic emission from nano materials. We find that there exists none that truly explains all the features of thermionic emission from such materials. We then give justifications that a three-dimensional model should be applicable for thermionic emission from nano materials such as CNT and graphene. We have presented briefly the physics of graphene as existing already in literature. We see that Fermi energy and Fermi velocity are not two independent quantities but rather related. Moreover, we see that the electrons in graphene are not truly massless as often is conjectured. The presence of terms with both | | q and | | q 2 in energy [equation (19)] makes the expression for density of states complicated for a thermionic emission current density (J) versus T relation to be smoothly applicable for performance evaluation of a graphene based TEC. Further complication is added when the terms γ 0 and β are included and not neglected, as before, in the energy dispersion relation. Moreover, as explained earlier, the graphene electron mass cannot be said to be truly massless when these terms are considered (as ideally it should be) in the energy dispersion relation. The observed finite dynamic mass supports this view.
In the light of the above, thus, for proper performance evaluation of TEC with graphene we have considered a simple model of temperature variation of work function of metals considering thermal expansion and In nanomaterials like graphene and carbon nanotube the electron density and consequently the Fermi energy, E F0 is much lower than that of metals. Thus, it should be applicable for nano-materials. We have seen that our new thermionic emission current density equation fits excellently well the experimental data for monolayer graphene (without the substrate effect) and much better than any existing models. We have determined this by studying the variation of S = å -(( ) J J i th i exp 2 for wide range of variables for the different existing models and obtaining the minimum values of S with different models and comparing the minimum values and using the visual eye fit which agrees completely with minimum S as expected. Moreover, the advantage of the new model unlike any other model, is that it provides unique and accurate method of determination of E F over the entire temperature range starting from 0 K, which RD law and other models cannot provide. The work function of suspended monolayer graphene is seen to change from 4.693 to 4.725 eV as temperature changes from 1618 K to 1795 K and E F from 0.203 to 0.070 eV in the temperature range 0 to K 1795 . This change in work function with temperature influences the thermionic emission in accordance with equation (35b) in graphene, a fact that has not been considered before in literature. The model fits the J T versus data for CNT very well and better than RD law. The values of work function determined with our model for graphene also agrees very well experimental value [50]. The Fermi energy determined from the model for CNT agrees fairly well with independent estimate of Fermi velocity. These excellent agreements with independent experimental results show strong support for the model of thermionic emission current density from graphene and CNT presented in this work.
The model is expected to help accurate simulation of performance of thermionic energy converter with graphene or carbon nanotube. Our models lend an estimate of temperature dependent Fermi energy of graphene from the J T versus data of thermionic emission. We have shown that the MRDE model fits the J T versus data of graphene better than the Liang and Ang model.
Nanomaterials have been proposed to be good candidates for thermionic energy converters, specially solar energy. Our above equation should be used to simulate accurately the performance of such TEC, instead of ordinary RD law. Moreover, for electric field emission of electrons from nano materials at finite temperature, the temperature dependence of work function should play an important role and for modification of Fowler-Nordheim law for nano material, the temperature dependence of work function as mentioned above should be important [54]. W=work function. The lattice phonons in graphene can also have three components of momentum because the atoms in graphene can vibrate in all three directions (x, y, z) directions. The vibrations in the z-directions could be in a transverse wave form while the vibrations in x-y plane are longitudinal. If the free electrons are impacted by phonons which have momentum components in x, y, z directions, the electrons can receive the z-component of momentum from conservation principle that hold during collision.
In the above scenarios one may argue that only high energy spectrum of thermal photons or lattice phonons would be able to give enough momentum to electron for emission. In such case the efficiency of thermionic emission would be very low. However, multiphotonic or multi-phonon collisions with electrons will be needed for the required momentum transfer.
We feel that at a given temperature, the conservation of momentum must hold for electron motions in the plane of graphene (in absence of any external field), if we neglect the edge thermionic emissions. The huge uncertainty in z-component of electron momentum which is present in graphene even at 0 K does not take part (either positively or negatively) in thermionic emission. If it did, the emission would have been seen at 0 K even. Emission along z-direction takes place at elevated temperature and as explained above it is possible only when the phonons or photons transfer to electrons the z-component of momentum sufficient for emission to take place. In our opinion there can't be any cross momentum transfer-i.e., the in-plane components of momentum being converted to z-component of momentum. This is not possible. This latter part, if at all would be possible, would require a new physics.