The generalized Schottky model of thermionic emission in complex metal with nonextensive quantum statistics

Abstract

In this paper, by use of the new version of nonextensive quantum statistics, established on basis of a technique called parameter transformation, we study the thermionic emission phenomenon in the complex metals, which contains long rang interactions or correlations. Employing the free electron model, one can obtain the analytical expression of the emission current density with zero field. It is interesting that, due to the Tsallis correction related to the parameter ν, the emission current density with zero field exhibits complicated dependence on temperature. By generalizing the Schottky effect/model, the thermionic emission expression with external field is also obtained. The numerical analysis shows that the emission current with external field is increasing sensitively with the nonextensive parameter ν. These results obtained in the present work might be suitable in metal materials research such as the metal containing impurities, the one whose electron density is much larger, the transition metal, the system with heavy Fermion and the one whose lattice oscillation contains non-nearest interactions due to some pressure from outside.

Appendix A

In order to prove (14), we need to calculate the following integral,

$$\iiint {a^{ - \beta (\varepsilon - \mu )} \text{d}p_{x} \text{d}p_{y} \text{d}p_{z} } = a^{\beta \mu } \iiint {a^{{ - \beta \frac{{p_{x}^{2} + p_{y}^{2} + p_{z}^{2} }}{2m}}} \text{d}p_{x} \text{d}p_{y} \text{d}p_{z} }.$$

(41)

Notice that the lower limit of the integral along x direction is χ. The integral of (41) in the p_y − p_z plane is

$$I = a^{\beta \mu } \left( {\frac{2m}{{\beta \ln a}}} \right)^{{\tfrac{3}{2}}} \sqrt \pi \int_{{x_{0} }}^{ + \infty } {e^{{ - x^{2} }} dx} ,$$

(42)

where

$$x_{0} = \sqrt {\beta \chi \ln a} ,\begin{array}{*{20}c} {} & {x^2 = \beta \frac{{p_{x}^{2} }}{2m}} \\ \end{array} \ln a.$$

(43)

Considering the variable substitution

$$x^{2} - x_{0}^{2} = t^{2} ,$$

(44)

with t > 0, one has

$$I = a^{\beta \mu } \left( {\frac{2m}{{\beta \ln a}}} \right)^{{\tfrac{3}{2}}} \sqrt \pi e^{{ - x_{0}^{2} }} \int_{0}^{ + \infty } {e^{{ - t^{2} }} \frac{t}{{\sqrt {x_{0}^{2} + t^{2} } }}dt} .$$

(45)

It is noticeable that the function

$$f(t) = e^{{ - t^{2} }} t,$$

(46)

has maximum at $1/\sqrt 2$. The integral in (45) is mainly around this maximum. On the other hand, there is

$$x_{0}^{2} = \beta \chi \ln a \gg 1.$$

(47)

So (45) is changed into

$$\begin{gathered} I \approx a^{\beta \mu } \left( {\frac{2m}{{\beta \ln a}}} \right)^{{\tfrac{3}{2}}} \sqrt \pi e^{{ - x_{0}^{2} }} \int_{0}^{ + \infty } {e^{{ - t^{2} }} \frac{t}{{x_{0} }}dt} \hfill \\ = a^{\beta \mu } \left( {\frac{2m}{{\beta \ln a}}} \right)^{{\tfrac{3}{2}}} \sqrt \pi e^{{ - x_{0}^{2} }} \frac{1}{{2x_{0} }} = a^{\beta (\mu - \chi )} \left( {\frac{2m}{{\ln a}}} \right)^{{\tfrac{3}{2}}} \frac{\sqrt \pi }{{2\sqrt \chi }}\beta^{ - 2} . \hfill \\ \end{gathered}$$

(48)

Therefore,

$$\begin{gathered} - \frac{\partial }{\ln a\partial \beta }\ln \iiint {a^{ - (\varepsilon - \mu )/kT} \text{d}p_{x} \text{d}p_{y} \text{d}p_{z} } \hfill \\ = - \frac{\partial }{\ln a\partial \beta }\ln [e^{\beta (\mu - \chi )\ln a} \beta^{ - 2} ] \hfill \\ = W + \frac{2kT}{{\ln a}}. \hfill \\ \end{gathered}$$

(49)

Then, (14) is verified.