# Generalized Electron Emission Theory for One-Dimensional Conducting Materials
## 3 One-Dimensional Thermionic Emission  
At zero temperature, electrons within a conductor occupy energy levels up to the Fermi energy. As the temperature increases, some electrons are excited to energy levels exceeding the work function. Electrons thermally emitted in this process, possessing energy surpassing the vacuum energy, participate in thermionic emissions.  
The one-dimensional kinetic energy of an electron is given by  
\[E_{1D}=\frac{1}{2}mv_{x}^{2}, \tag{10}\]  
where \(m\) denotes the electron mass, and \(v_{x}\) denotes the electron velocity in the \(x\)-direction. This Equation (10) can also be represented as \(E_{1D}(k_{x})=\frac{h^{2}k_{B}^{2}}{2m}\) when the wave number is incorporated, given that the electron's velocity is significantly lower than the speed of light. The electron velocity can be expressed as \(v_{x}=\frac{\hbar k}{m}\) by considering \(v_{x}=\frac{1}{h}\
abla_{k_{x}E}(k_{x})\).  
The one-dimensional current density associated with thermionic emission becomes  
\[J_{1D}=e\int_{0}^{\infty}dv_{x}v_{x}n(v_{x})D(E_{x}), \tag{11}\]  
where \(n\) signifies the density of electrons, \(D\) represents the tunneling probability, and \(e\) denotes the electric charge. The electron density, represented as \(n(E)=g(E)f(E)\), incorporates the density of states \(g(E)\) and the Fermi factor \(f(E)\), indicating the probability of a specific state having energy \(E\). When the electron energy exceeds the vacuum energy, the tunneling probability is 1. Utilizing Equation (11), the thermionic current density in one dimension is derived as [8]  
\[J_{1D}=e\int_{0}^{\infty}{(\frac{dE_{x}}{m})(2\frac{m}{h})\frac{D(E_{x})}{ \exp\left(\frac{E_{x}-E_{x}}{k_{B}T}\right)+1}}, \tag{12}\]where \(E_{F}\) represents the Fermi energy and \(k_{B}\) denotes the Boltzmann constant. Taking into account the vacuum energy, Equation (12) is rewritten as [8]  
\[J_{1D}=\frac{2 ek_{B}T}{h}{\int_{\Phi_{w}+E_{F}}^{\infty}\frac{dE_{x}/k_{B}T}{\exp \left(\frac{E_{x}-E_{F}}{k_{B}T}\right)+1}}, \tag{13}\]  
where \(\Phi_{W}\) denotes the work function of the one-dimensional material. By introducing \(x=\frac{E_{x}-E_{F}}{k_{B}T}\), Equation (13) can be expressed as  
\[J_{1D}=\frac{2 ek_{B}T}{h}{\int_{\frac{\Phi_{w}}{k_{B}T}}^{\infty}\frac{dx}{\exp (x)+1}}. \tag{14}\]  
Integration of Equation (14) yields [10]  
\[J_{1D}=\frac{2 ek_{B}T}{h}[\ln(1+e^{\frac{\Phi_{w}}{k_{B}T}})-\frac{\Phi_{w}}{k _{B}T}]. \tag{15}\]  
Equation (15) can alternatively be written as  
\[J_{1D}=\frac{2 ek_{B}T}{h}[\ln(1+e^{-\frac{\Phi_{w}}{k_{B}T}})]. \tag{16}\]  
Equation (16) represents the thermionic current density in one-dimensional materials. The current density given by Equation (16) is applicable across varying work functions and temperatures.  
When the work function \(\Phi_{W}\) is much bigger than the thermal energy \(k_{B}T\), Equation (14) simplifies to [8]  
\[J_{1D}=\frac{2e}{h}k_{B}T{\int_{\frac{\Phi_{w}}{k_{B}T}}^{\infty}\exp(-x)dx}. \tag{17}\]  
From Equation (17), the thermionic current density for one-dimensional materials is obtained as  
\[J_{1D}=\frac{2 ek_{B}T}{h}e^{-\frac{\Phi_{w}}{k_{B}T}}. \tag{18}\]  
This Equation (18) represents the one-dimensional current density for thermionic emission.  
When thermally excited electrons have energy exceeding the vacuum energy, they might undergo reflection at the boundary between the metal and vacuum. The generalized one-dimensional current density for thermally emitted electrons is then described as [8]  
\[J_{1D}=(1-r)\frac{2 ek_{B}T}{h}e^{-\frac{\Phi_{w}}{k_{B}T}}, \tag{19}\]  
where \(r\) denotes the probability of reflection for the electrons.