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

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.

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:

$$\begin{eqnarray*}&&{a}_{1}=a\left(1/2,\sqrt{3}/2\right)\end{eqnarray*}$$

$$\begin{eqnarray*}&&{a}_{2}=a\left(1/2,-\sqrt{3}/2\right)\end{eqnarray*}$$

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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:

$K\,=\,\left(\tfrac{2\pi }{3a},\tfrac{2\pi }{3\sqrt{3}a}\right),K^{\prime} =\left(\tfrac{2\pi }{3a},-\tfrac{2\pi }{3\sqrt{3}a}\right)$ which are known as Dirac points.

A carbon atom has 6 electrons. Two electrons are in 1S² shell. Four are in 2S² and 2p² 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 energy – one 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 Φ₁and Φ₂respectively [each of which is normalized and Φ₁and Φ₂ are orthogonal]. The total wave function of the two electrons can be given as (with linear combination of atomic orbitals)

$$\begin{eqnarray}&&{\rm{\Phi }}={c}_{1}{{\rm{\Phi }}}_{1}+{c}_{2}{{\rm{\Phi }}}_{2}\end{eqnarray} \tag{ 2 }$$

b₁ and b₂ are the overlapping constants and satisfy the condition

$$\begin{eqnarray}&&{c}_{1}^{2}+{c}_{2\,}^{2}=1\end{eqnarray} \tag{ 3 }$$

In the graphene lattice the net wave function must follow the periodicity of the lattice R as given by Block wave function:

$$\begin{eqnarray}&&{\rm{\Psi }}\left(r\right)=\displaystyle \sum _{R}{e}^{ik.R}{\rm{\Phi }}\left({\rm{r}}-{\rm{R}}\right)\end{eqnarray} \tag{ 4 }$$

Considering a single electron to be described by wave function ${\rm{\Psi }}\left(r\right)$ and its motion is governed by the Hamiltonian H, which is given by the potential of all the carbon atoms as

$$\begin{eqnarray}&&H\,=-\displaystyle \frac{{\hslash }^{2}}{2m}{{\rm{\nabla }}}^{2}+\displaystyle \sum _{R}\left[{\rm{V}}\left({\rm{r}}-{r}_{1}-{\rm{R}}\right)+\,V\left({\rm{r}}-{r}_{2}-{\rm{R}}\right)\right]\end{eqnarray} \tag{ 5 }$$

The wave function ${\rm{\Psi }}\left(r\right)$ satisfies the Schrodinger equation:

$$\begin{eqnarray}&&H{\rm{\Psi }}\left(r\right)={\rm{E}}{\rm{\Psi }}\left(r\right)\end{eqnarray} \tag{ 6 }$$

The Hamiltonian $H$ can be written as:

$$\begin{eqnarray}&&H={H}_{1}+{\rm{\Delta }}{H}_{2}={H}_{2}+{\rm{\Delta }}{H}_{1}\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}&&{H}_{1}=-\displaystyle \frac{{\hslash }^{2}}{2m}{{\rm{\nabla }}}^{2}\,+\displaystyle \sum _{R}{\rm{V}}\left({\rm{r}}-{r}_{1}-{\rm{R}}\right)\end{eqnarray} \tag{ 8 }$$

And

$$\begin{eqnarray}&&{\rm{\Delta }}{H}_{2}=\displaystyle \sum _{R}{\rm{V}}\left({\rm{r}}-{r}_{2}-{\rm{R}}\right)\end{eqnarray} \tag{ 9 }$$

To get the energy eigenvalues of equation (6) we project it on Φ₁ and Φ₂ and the two equations

$$\begin{eqnarray}&&\left\langle {\rm{\Phi }}{\rm{j}}\left|{\rm{H}}\right|\,{\rm{\Psi }}\right\rangle ={\rm{E}}\left\langle {\rm{\Phi }}{\rm{j}}\left|{\rm{\Psi }}\right.\right\rangle ,\,{\rm{j}}=1,2\end{eqnarray} \tag{ 10 }$$

To evaluate equation (10), we consider only nearest neighbor products considering only R = 0, a₁ and a₂ in equation (4). Then

$$\begin{eqnarray}&&{\rm{Then}},\left\langle {{\rm{\Phi }}}_{1}\right|{\rm{\Psi }}\geqslant \,=\,{c}_{1}+{c}_{2}\left\langle \left.{{\rm{\Phi }}}_{1}\right|{{\rm{\Phi }}}_{2}\right\rangle \left[1+{e}^{-ik.{a}_{1}}+{e}^{-ik.{a}_{2}}\right]\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}&&\left\langle {{\rm{\Phi }}}_{2}\right|{\rm{\Psi }}\geqslant \,=\,{c}_{2}+{c}_{1}\left\langle \left.{{\rm{\Phi }}}_{2}\right|{{\rm{\Phi }}}_{1}\right\rangle \left[1+{e}^{-ik.{a}_{1}}+{e}^{-ik.{a}_{2}}\right]\end{eqnarray} \tag{ 11b }$$

In equation (10)

$$\begin{eqnarray}&&\left\langle {{\rm{\Phi }}}_{1}\right|{{\rm{\Phi }}}_{2}\geqslant \,=\,\left\langle \left.{{\rm{\Phi }}}_{2}\right|{{\rm{\Phi }}}_{1}\right\rangle =A\,{\rm{real}}\,{\rm{term}}\,{\rm{called}}\,{\gamma }_{0}\end{eqnarray} \tag{ 12 }$$

To evaluate LHS of equation (11) we note that

${H}_{1}$ and ${H}_{2}$ satisfy,

$$\begin{eqnarray}&&{H}_{i}{{\rm{\Phi }}}_{j}={\partial }_{ij}{\varepsilon }_{i}{{\rm{\Phi }}}_{j}\end{eqnarray} \tag{ 13 }$$

${\varepsilon }_{i}$ is the energy of the $2{p}_{z}$ electron of atom i. Obviously ${\varepsilon }_{i}={\varepsilon }_{2}.$ We set these equal energies to be zero.

Then

$$\begin{eqnarray}&&\left\langle {{\rm{\Phi }}}_{1}\left|{\rm{H}}\right|\,{\rm{\Psi }}\right\rangle =\,\left\langle {{\rm{\Phi }}}_{1}\left|{\rm{\Delta }}{{\rm{H}}}_{1}\right|{\rm{\Psi }}\right\rangle ={b}_{1}\beta +{b}_{2}{\gamma }_{1}f* \left(k\right)\end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray}&&\left\langle {{\rm{\Phi }}}_{2}\left|{\rm{H}}\right|{\rm{\Psi }}\right\rangle =\,\left\langle {{\rm{\Phi }}}_{2}\left|{\rm{\Delta }}{{\rm{H}}}_{2}\right|{\rm{\Psi }}\right\rangle ={b}_{2}\beta +{b}_{1}{\gamma }_{1}f\left(k\right)\end{eqnarray} \tag{ 15 }$$

where

$$\begin{eqnarray}&&f\left(k\right)=1+{e}^{ik.{a}_{1}}+{e}^{ik.{a}_{2}}\end{eqnarray} \tag{ 16 }$$

and $\beta =\,\left\langle {{\rm{\Phi }}}_{1}\left|{\rm{\Delta }}{{\rm{H}}}_{1}\right|{{\rm{\Phi }}}_{1}\right\rangle$

$$\begin{eqnarray*}&&{\gamma }_{1}=\left\langle {{\rm{\Phi }}}_{1}\left|{\rm{\Delta }}{{\rm{H}}}_{1}\right|{{\rm{\Phi }}}_{2}\right\rangle \,=\left\langle {{\rm{\Phi }}}_{2}\left|{\rm{\Delta }}{{\rm{H}}}_{2}\right|{{\rm{\Phi }}}_{1}\right\rangle \end{eqnarray*}$$

Using equations (14)–(16), (11a) and (11b), equation (10) becomes, upon neglect of ${\gamma }_{0},$ which is quite small,

$$\begin{eqnarray}\left[\begin{array}{cc}\beta & {\gamma }_{1}f\,* \,\left(k\right)\\ f\left(k\right){\gamma }_{1} & \beta \end{array}\right]\left(\begin{array}{c}{c}_{1}\\ {c}_{2}\end{array}\right)\,=\,{\rm{E}}\left(\begin{array}{c}{c}_{1}\\ {c}_{2}\end{array}\right)\end{eqnarray} \tag{ 17 }$$

β describes the variation of energy of the ${p}_{z}$ atomic orbital induced by electrons of all other carbon atoms in the graphene plane. It corresponds to a small rigid shift of the energy band. It is Reich et al. (2002) that 0.02${\gamma }_{1}\leqslant \beta \leqslant 0.2{\gamma }_{1}$ [40]. As an approximation β can be neglected. Equation (17) is further simplified if we exploit the fact that γ₀ is small. Then one gets by solving

$$\begin{eqnarray}&&E\left(k\right)=\pm {\gamma }_{1}\sqrt{3+4\,\cos \left(\sqrt{3}{k}_{y}a/2\right)\cos \left(\displaystyle \frac{3}{2}{k}_{x}a\right)+2\,\cos \left(\sqrt{3}{k}_{y}a\right)}\end{eqnarray} \tag{ 18a }$$

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, figure 1) using

$$\begin{eqnarray}&&k=K+q\,E\left(q\right)=\pm {V}_{F}\left|q\right|\,\mp \displaystyle \frac{3{\gamma }_{1}{a}^{2}}{8}\,\sin \left(3{\theta }_{q}\right){\left|q\right|}^{2}\end{eqnarray} \tag{ 18b }$$

where ${\theta }_{q}$ = arc tan $\left(\tfrac{{k}_{x}}{{k}_{y}}\right)$ and ${V}_{F}\,=\,3{\gamma }_{1}a/2$

If we neglect the ${\left|q\right|}^{2}$ term then from (18)

$$\begin{eqnarray}&&E\left(q\right)=\pm {V}_{F}\left|q\right|\end{eqnarray} \tag{ 19 }$$

Equation (19) gives in the first order approximation, energy of electrons in graphene as proportional to absolute value of momentum. $\left|q\right|$ 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

$$\begin{eqnarray*}&&{E}^{2}={p}^{2}{c}^{2}+{m}_{0}^{2}{c}^{4}\end{eqnarray*}$$

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) [1]]

$$\begin{eqnarray}&&{m}_{c}=\displaystyle \frac{1}{2\pi }{\left[\displaystyle \frac{\partial A\left(E\right)}{\partial E}\right]}_{E={E}_{F}}\end{eqnarray} \tag{ 20 }$$

where $A\left(E\right)=\pi q{\left(E\right)}^{2}$

Using equations (20) and (19) then one gets,

$$\begin{eqnarray}&&{m}_{c}=\,\displaystyle \frac{{E}_{F}}{{V}_{F}^{2}}\end{eqnarray} \tag{ 21 }$$

Equation (21) clearly says that

$$\begin{eqnarray}&&{E}_{F}={m}_{c}{V}_{F}^{2}\end{eqnarray} \tag{ 22 }$$

The cyclotron mass determined for electrons in graphene is shown theoretically and found to depend on carrier concentration. It varies from very close to $0.025{m}_{0}$ for low concentration to about $0.07{m}_{0}$ (rest mass of electron) for carrier concentration, n of about $7\times {10}^{12}\,{{\rm{cm}}}^{-2}$ [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 ${\left|q\right|}^{2}.$ 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 ${\left|q\right|}^{2}$ 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 $0.01{m}_{e}$ to $0.024{m}_{e}.$ 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 $\rho \left(E\right)$ becomes proportional to the energy E. This was used in the Liang and Ang model. When we take equation (18) the $\rho \left(E\right)$ becomes a complicated function of energy E. The derivation of thermionic current density $J\,{\rm{versus}}\,T$ 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 $J\,{\rm{versus}}\,T.$

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 $J\,{\rm{versus}}\,T$ relation that can apply to both graphene and carbon nanotube. In this new approach we have considered the temperature dependence of the Fermi energy ${E}_{F}\left(T\right),$ 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 $\left(z\right)$ as well as in plane $\left(x,y\right).$ 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 $\left(xy\right).$ Thus, for thermionic emission of electrons even from a monolayer graphene the wave vectors of electrons have three finite components $\left({k}_{x},\,{k}_{y}\,and\,{k}_{z}\right)$ 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, ${k}_{x,}\,{k}_{y}$ 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 $\left(\hslash /a\right)$ 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 $\tfrac{{\hslash }^{2}}{2m{a}^{2}}$ 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, $W+{E}_{F}\leqslant \left|\tfrac{{p}_{z}^{2}}{2m}\right|\leqslant \infty .$ 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 $J\,{\rm{versus}}\,T$ 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 ${E}_{F}\left(T\right)$ 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 $J\,{\rm{versus}}\,T$ data of graphene. After making detailed comparison of the existing models of thermionic emission we find that the new model explains not only the $J\,{\rm{versus}}\,T$ 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.

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 $N\left(T\right)$ in a given piece of metal at $T=0\,{\rm{K}}$ 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 $T=0\,K,\,E\leqslant {E}_{F0}$ and the Fermi function $f\left(E\right)=1.$

Using the concept of density of states $g\left(E\right),$ the total number, N of free electrons in the metal are given by: Temperature 0 K,

$$\begin{eqnarray}&&N\left(T=0\,K\right)={V}_{0}\displaystyle {\int }_{0}^{{E}_{F0}}g\left(E\right)dE\,\end{eqnarray} \tag{ 23 }$$

Finite temperature T,

$$\begin{eqnarray}&&{\rm{where}}\,N\left(T\right)=V\displaystyle {\int }_{0}^{\infty }g\left(E\right)f\left(E\right)dE\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&f\left(E\right)=\displaystyle \frac{1}{1+\exp \left(\tfrac{E-{E}_{F}}{{k}_{B}T}\right)}\end{eqnarray} \tag{ 25 }$$

f(E) is the well-known Fermi-function. (E_F(T)) in equation (25) is called chemical potential in many books. We call it temperature dependent Fermi energy level. Considering thermal expansion of the metal,

$$\begin{eqnarray}&&V={V}_{0}\left(1+r\alpha T\right)\end{eqnarray} \tag{ 26 }$$

where α is linear thermal expansion coefficient. Thermionic emission takes place at high temperature $\left(\gt 1000\,{\rm{K}}\right).$ The work function $\left(W\right)$ is defined as:

$$\begin{eqnarray}&&W\left(T\right)={E}_{v}-{E}_{F}\left(T\right)\end{eqnarray} \tag{ 27 }$$

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.

Since $N\left(T=0\right)=N\left(T\right)$ as discussed above, from equations (23) to (26)

$$\begin{eqnarray}&&{V}_{0}\displaystyle {\int }_{0}^{{E}_{F0}}g\left(E\right)dE={V}_{0}\left(1+r\alpha T\right)\displaystyle {\int }_{0}^{\infty }g\left(E\right)f\left(E\right)dE\end{eqnarray} \tag{ 28 }$$

Using the expressions for$g{\left(E\right)}^{\left(1\right)}$ and $f\left(E\right)$ equation (27) becomes

$$\begin{eqnarray}&&\displaystyle {\int }_{0}^{{E}_{F0}}{E}^{1/2}dE=\left(1+r\alpha T\right)\displaystyle {\int }_{0}^{\infty }\left(\displaystyle \frac{{E}^{1/2}}{1+\exp \left(\tfrac{E-{E}_{F}}{{k}_{B}T}\right)}\right)dE\end{eqnarray} \tag{ 29 }$$

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:

Expanding the RHS of equation (29) we obtain equations (30)–(33)

$$\begin{eqnarray}&&{E}_{F0}^{3/2}={E}_{F}^{1/2}\left(1+r\alpha T\right)\left[{E}_{F}+\displaystyle \frac{{\pi }^{2}{\left({k}_{B}T\right)}^{2}}{12{E}_{F}}+\displaystyle \frac{7{\pi }^{4}{\left({k}_{B}T\right)}^{4}}{960{E}_{{E}_{F}}^{3}}\right]\end{eqnarray} \tag{ 30 }$$

Equation (30) after rearrangements becomes

$$\begin{eqnarray}&&\,\begin{array}{l}{E}_{F}=\Space{0ex}{0.25ex}{0ex}\left(\frac{{E}_{F0}^{3/2}}{{E}_{F}^{1/2}}\right)-r\alpha T{E}_{F}-(1+r\alpha T)\left(\frac{{\pi }^{2}}{12}\right){\left(\frac{{k}_{B}T}{{E}_{F}}\right)}^{2}\left({E}_{F})-\,(1+r\alpha T)\left(\frac{7{\pi }^{4}}{960}\right){\left(\frac{{k}_{B}T}{{E}_{F}}\right)}^{4}({E}_{F})\right)\\ \end{array}\,\end{eqnarray} \tag{ 31 }$$

To obtain ${E}_{F}$ as a function of ${E}_{F0}$ and T, we put${E}_{F}={E}_{F0}$ as a first approximation in the RHS of

Equation (31) and we get equation (32)

$$\begin{eqnarray}&&{E}_{F}={E}_{F0}-\left[\begin{array}{c}r\alpha T{E}_{F0}+(1+r\alpha T)\left(\frac{{\pi }^{2}}{12}\right){\left(\frac{{k}_{B}T}{{E}_{F0}}\right)}^{2}({E}_{F0})+(1+r\alpha T)\left(\frac{7{\pi }^{4}}{960}\right){\left(\frac{{k}_{B}T}{{E}_{F0}}\right)}^{4}({E}_{F0})\\ \end{array}\right]\end{eqnarray} \tag{ 32 }$$

Using equation (27) we then obtain ${W}_{0}$ is the work function of the material at $T=0\,{\rm{K}}.$

$$\begin{eqnarray}&&\,\begin{array}{l}W(T)=\,{W}_{0}+\left[\,r\alpha T{E}_{F0}+(1+r\alpha T)\left(\frac{{\pi }^{2}}{12}\right){\left(\frac{{k}_{B}T}{{E}_{F0}}\right)}^{2}{E}_{F0}+(1+r\alpha T)\left(\frac{7{\pi }^{4}}{960}\right){\left(\frac{{k}_{B}T}{{E}_{F0}}\right)}^{4}{E}_{F0}\right]\\ \end{array}\,\end{eqnarray} \tag{ 33 }$$

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, ${\rm{\Delta }}E={E}_{F0}/10000$ and the upper limit of the RHS as $100{E}_{F}.$ 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 ${E}_{F}\left(T\right)$ one can obtain the changes in work function relative to its value at absolute zero, W₀. These in turn can be used to obtain $J\,{\rm{versus}}\,T$ for graphene for a given value of ${E}_{F0}$ and W₀ using the equation (34).

$$\begin{eqnarray}&&J={A}_{0}{T}^{2}\exp \left(-W\left(T\right)/{k}_{B}T\right)\end{eqnarray} \tag{ 34 }$$

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 $W\left(T\right)/{W}_{0}$ i.e., ${\rm{\Delta }}W\left(T\right)/{W}_{0}$ increases as ${E}_{F0}$ decreases. This second approach though ideal for nano material in this model, is quite time consuming to generate the $J\,{\rm{versus}}\,T$ curves to find the best fit with experimental values.

In equation (33) W₀ corresponds to $T=0\,{\rm{K}}.$ The numerical coefficients for successive terms after the fourth power of T in equation (33) get smaller and smaller than unity. Moreover, the term${\left(\tfrac{{k}_{BT}}{{E}_{F0}}\right)}^{n}$ decreases faster for $n\gt 4,$ with temperature T such that ${k}_{B}T\lt {E}_{F},{E}_{F0}.$ Then terms up to the fourth power of T are sufficient in equation (33). We have verified by actual numerical computation of the equation (29) that $\left(\left({E}_{F}-{E}_{F0}\right)/{E}_{F0}\right)$ calculated from equation (32) is within 0.3% of the value obtained by actual numerical computation when E_F0 is up to $0.7\,{\rm{eV}}$ and temperature range 300–2500 K. For nano materials like graphene and CNT operating at temperatures less than ${E}_{F0}/{k}_{B},$ equation (32) or equation (33) is thus sufficient but at temperatures greater than ${E}_{F0}/{k}_{B},$ sixth or higher power terms would be necessary.

In this study, we investigated if the MRDE can be applied to thermoelectron emission from graphene which has exhibited metallic properties [42–44]. 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 co-efficient $-8\times {10}^{-6}\,{{\rm{K}}}^{-1}$ [45, 46] and different values of W₀ and ${E}_{F0},$ each step of 0.001, we have examined how the equation fits the experimental values of J versus T for monolayer suspended graphene² in the temperature range $1620-1795\,{\rm{K}}.$ We have used visual eye best fitting along with minimum value of $S=\displaystyle {\sum }_{i}{\left(\left({J}_{th}-{J}_{\exp }\right)\right.}_{i}^{2}$ to ascertain the best fit values of W0 and E_F0. The various fits are shown in figures 2 and 3. To obtain the best fit from apparent many good fit (figure 2) we also rely on the value of S (tables 1 and 2). The values of the parameters ${W}_{0},{E}_{F0}\left(4.52\,{\rm{eV}},\,0.203\,{\rm{eV}}\right),$ that give the minimum value of S for the experimental data are accepted as the correct value for monolayer graphene. Figure 3 shows the shift from best fit with change of the values of ${W}_{0},{E}_{F0}$ in MRDE model. We see that the minimum value of $S=0.000171\,{{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}$ is obtained for ${W}_{0}=4.592\,{\rm{eV}}$ and ${E}_{F0}=0.203\,{\rm{eV}}$ and the fit is excellent. For other values of W₀ and ${E}_{F0}$ we can easily see from figures 2 and 3, that the fit is not as good as that for ${W}_{0}=4.592\,{\rm{eV}}$ and ${E}_{F0}=0.203\,{\rm{eV}}$ and the value of S (tables 1 and 2) is significantly higher than the minimum value of $0.000171\,{{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}.$in the entire temperature range $1620-1795\,{\rm{K}}.$ By giving a wide variation in the work function of graphene we have examined in detail how the RD law fits (figure 2) the experimental data of $J\,{\rm{versus}}\,T$ of graphene in the above temperature range. The best RD law fit is shown in figure 2 for $W=4.722\,{\rm{eV}}.$ This gives the value of $S=\displaystyle {\sum }_{i}{\left({J}_{th}-{J}_{\exp }\right)}_{i}^{2}$ = $0.000553\,{{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}$ (table 2) nearly three times the best fit value as obtained above with MRD. For other values of W the RD law the fit is much worse (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 W₀ 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 $4.692\,{\rm{eV}}$ at $1620\,{\rm{K}}$ to $4.725\,{\rm{eV}}$ at $1795\,{\rm{K}}$ (figure 4). Surprisingly, like that in CNT, the work function (4.72 eV) corresponding to the best RD law fit of $J\,{\rm{versus}}\,T$ 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 $W\left(T\right)$ 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.

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Table 1.  Values of S for different values of W₀ and ${E}_{F0}$ in MRDE model of Graphene.

	MRDE GRAPHENE		RD GRAPHENE
${{\rm{W}}}_{0}\,\left({\rm{eV}}\right)$	${{\rm{E}}}_{{\rm{F}}0}\,\left({\rm{eV}}\right)$	S = $\left({{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}\right)$	${{\rm{W}}}_{0}\,\left({\rm{eV}}\right)$	S = $\left({{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}\right)$
4.592	0.203	0.000 171	4.592	0.531
			4.71	0.0020
			4.72	0.000 553
			4.73	0.001 70
			4.74	0.0053
			4.68	0.0285
			4.77	0.0245

Table 2.  Comparison of S for three different models of Graphene: MRDE, RD and Liang and Ang model.

	MRDE GRAPHENE		RD		Liang and Ang
${{\rm{W}}}_{0}\left({\rm{eV}}\right)$	${{\rm{E}}}_{{\rm{F}}0}\left({\rm{eV}}\right)$	S = $\left({{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}\right)$	${{\rm{W}}}_{0}\left({\rm{eV}}\right)$	S = $\left({{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}\right)$	${{\rm{W}}}_{0}\left({\rm{eV}}\right)$	${{\rm{E}}}_{{\rm{F}}0}$ $\left({\rm{eV}}\right)$	${{\rm{V}}}_{{\rm{F}}}\left({\rm{m}}\,{{\rm{s}}}^{-1}\right)$	S = $\left({{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}\right)$
4.592	0.190	0.003 71	4.592	0.531	4.592	0.190	2.49E6	0.1911
4.592	0.210	0.000 941	4.514	2.49	4.592	0.201	2.49E6	0.1818
4.592	0.203	0.000 171	4.72	0.000 553	4.592	0.21	2.49E6	0.1765
					4.514	0.083	2.49E6	0.2101
					4.514	0.083	1E6	0.0056

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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 capacitance-voltage characteristics of a metal–graphene-oxide-semiconductor (MGOS) capacitor structure. They found a value $4.62\,{\rm{eV}}$ for the work function of graphene. Lee et al. [49] measured by scanning kelvin probe microscopy the unbiased work function of graphene to be $4.56\,{\rm{eV}}$ 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 $300\,{\rm{K}},$ it is $4.593\,{\rm{eV}}.$ 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 $4.56\,{\rm{eV}}$ [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.

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 $\alpha =2\times {10}^{-5}\,{{\rm{K}}}^{-1}$ as obtained by Deng et al. [46] for both single and double walled CNT. We have given systematic wide range of variation by $0.002\,{\rm{eV}}$ to both W₀ and E_F0. W₀ 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 (solid line) with experimental data (dots) using equation (35) with ${W}_{0}=3.67\,{\rm{eV}}$ and ${E}_{F0}=1.87\,{\rm{eV}}.$ The fitting seems good visually. The value of $S=\displaystyle {\sum }_{i}{\left({J}_{th}-{J}_{\exp }\right)}_{i}^{2}=9870\tfrac{{A}^{2}}{{m}^{4}}$ for this fit. It gives the lowest value of S as seen by us with wide range of variation of W₀ and ${E}_{F0}$ (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 W₀ and E_F0 slightly different from those of the best fit values. We see that as S changes with values of W₀ 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 $\sqrt{S/N}$ (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 W₀ and E_F0 and some other values are shown in figure 6 for CNT. The work function increases at an average rate of $1.33\times {10}^{-4}\,{\rm{eV}}\,{{\rm{K}}}^{-1}$ for CNT. One very interesting result that emerges from the fitting with this new model is the best value of ${E}_{F0}\left(=1.87\,{\rm{eV}}\right)$ found for CNT (figure 5). Anantram and Leonard [51] reported that electrons in the crossing sub-bands have a large velocity of $8\times {10}^{5}\,{\rm{m}}\,{{\rm{s}}}^{-1}$ at the Fermi energy of CNT. If we assume that this is the Fermi velocity than the Fermi energy turns out to be $1.82\,{\rm{eV}},$ which is surprisingly in close agreement with the value $1.87\,{\rm{eV}},$ obtained from the fitting. Apart from good fitting of $J\,{\rm{versus}}\,T$ data for CNT, this agreement lends additional credence to the validity of our model for CNT.

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Table 3.  Comparison of S for CNT with MRDE and RD models.

	MRDE CNT		RD CNT
${{\rm{W}}}_{0}\left({\rm{eV}}\right)$	${{\rm{E}}}_{{\rm{F}}0}\left({\rm{eV}}\right)$	S = $\left({{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}\right)$	${{\rm{W}}}_{0}\left({\rm{eV}}\right)$	S = $\left({{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}\right)$
3.67	1.87	9870	3.67	4118 700
			3.69	3006 500
			3.87	324 970
			3.91	23 366
			3.93	81 022

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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 $J\,{\rm{versus}}\,T$ 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 $0.001\,{\rm{eV}}.$ We find that the best fit (figure 7) with RD law is obtained at $W=3.91\,{\rm{eV}}$ for which the value of $S=23366\,{{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}.$ The RD law fit gets worse (figure 7), with high values of S (table 4) when W is changed by more than 0.01 on either side of $W=3.91\,{\rm{eV}}.$ Now comparing the best MRD fit (figures 7 and 5) of $J\,{\rm{versus}}\,T$ experimental values of CNT with the best RD law fit (figure 7), obviously the MRD law gives better fit in terms of both lowest value of S (tables 3 and 4) and the visual eye fitting (figure 7). Thus, MRD is superior to the RD law for CNT. For the best MRDE law in table (4), the W varies from $3.865\,{\rm{eV}}$ to $3.904\,{\rm{eV}}$ (figure 6) and surprisingly the work function ($3.91\,{\rm{eV}}$) for the best RD law fit figure (7) is closed to these values.

Table 4.  Comparison of S for CNT for different values of W₀ and E_F0 in MRDE model.

	MRDE CNT
${{\rm{W}}}_{0}\left({\rm{eV}}\right)$	${{\rm{E}}}_{{\rm{F}}0}\left({\rm{eV}}\right)$	S = $\left({{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}\right)$
3.67	1.87	9870
3.69	1.87	63 471
3.67	1.89	10 360
3.69	1.89	75 407
3.67	1.85	10 930
3.69	1.85	52 633

Liang and Ang [2] developed a theory of thermionic emission from graphene based on massless Dirac electrons in graphene. They had seen good agreement with their theory for graphene.

The emission current density J in that theory is shown [2] to be given by

$$\begin{eqnarray}&&J=\displaystyle \frac{e{T}^{3}{k}_{B}^{3}}{\pi {v}_{F}^{2}{\hslash }^{3}}\exp \left(-\displaystyle \frac{\phi -{E}_{F}}{{k}_{B}T}\right)\end{eqnarray} \tag{ 37 }$$

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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 $\phi =4.514\,{\rm{eV}},$ ${E}_{F}=0.083\,{\rm{eV}}$ and ${V}_{F}={10}^{6}\,{\rm{m}}\,{{\rm{s}}}^{-1}.$ For these values we obtain S =$\displaystyle \sum _{i}\left({\left({J}_{th}-{J}_{\exp }\right)}_{i}^{2}\right.=0.00383\,{{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}$ which is nearly 20 times the values of S obtained for the best fit by MRDE model. It should be seen that ϕ − E_F should be considered as a single variable in equation (37), instead of two independent variables ϕ and E_F, since it is not possible to relate E_F and V_F in the above equation. We have explained this situation in section 3.1. The lower the value of S the better is the fit expected. The Fermi velocity, ${V}_{F}$ of electrons in graphene are measured [52] to lie between $0.85\times {10}^{6}\,{\rm{m}}\,{{\rm{s}}}^{-1}$ and $2.49\times {10}^{6}\,{\rm{m}}\,{{\rm{s}}}^{-1}$ depending on the substrate in contact and the environment surrounding the graphene. For suspended graphene V_F is measured to be as high as $3\times {10}^{6}\,{\rm{m}}\,{{\rm{s}}}^{-1}.$ It is found to be inversely proportional to the dielectric constant of the substrate in contact. For monolayer graphene ${V}_{F}=2.49\times {10}^{6}\,{\rm{m}}\,{{\rm{s}}}^{-1}$ [52] is more appropriate than the value ${10}^{6}\,{\rm{m}}\,{{\rm{s}}}^{-1}$ used by Liang and Ang. Now if we use the exact value of ${V}_{F}=2.49\times {10}^{6}\,{\rm{m}}\,{{\rm{s}}}^{-1}$ as obtained experimentally for monolayer graphene, the value of S becomes $0.212\,{{\rm{A}}}^{2}\,{{\rm{m}}}^{-4}$ which is 1254 times the value obtained with MRD. If we use ${V}_{F}=1.73\times {10}^{6}\,{\rm{m}}\,{{\rm{s}}}^{-1},$ then S is $0.1251\,{{\rm{A}}}^{2}\,{{\rm{m}}}^{-4},$ which is 676 times the minimum value of S obtained with our model. Thus, our MRD equation fits the thermionic emission current 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.

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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.

We have included the plot of temperature variation of work function for graphene and CNT. We have shown in figure 6 the variation of work functions for slightly different values of E_F0. The graphs show the average $\tfrac{dW}{dT}$ for graphene is $1.85\times {10}^{-4}\,{\rm{eV}}\,{{\rm{K}}}^{-1}$ and $1.35\times {10}^{-4}\,{\rm{eV}}\,{{\rm{K}}}^{-1}$ for CNT.

We do not have experimental data for temperature coefficient, $\mu =\tfrac{dW}{dT}$ of work function of all metals. Moreover, all metals are not expected to behave in the same way. We refer to the work of Ibragimov and Korol'kov (2001) [53]. From there we see that experimental μ for Ga, In, Sn, Bi, Tl and Pb are (in units of ${10}^{-6}\,{\rm{eV}}\,{{\rm{K}}}^{-1}$): 23.1, 25.3, 23.4, 13, 30 and 28 respectively. Thus, we see that $\mu =\tfrac{dW}{dT}$ 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 ${T}^{2}\exp \left(-W/{k}_{B}T\right),$ for graphene and CNT, temperature dependence of J should be described by equation (35a).