5. Zero transmission point

The current can vanish only when the Fermi energy is within the energy range π<sup>2</sup> <sup>2</sup> =w<sup>2</sup> < EF < 4π<sup>2</sup>=w , in which case only the first mode is propagating. The transmission of the first mode is

$$t\_{1,1} = 1 - \left\{ 1 + \frac{i\sqrt{E\_2 - E\_1}}{\left| \chi\_1(\varepsilon) \right|^2} \left[ \frac{\ln \left( \rho\_0/R(\varepsilon) \right)}{\pi} - \frac{\left| \chi\_2(\varepsilon) \right|^2}{\sqrt{E\_2 - E}} \right] \right\}^{-1},\tag{35}$$

in which case the zero-current energy is approximately

$$E\_R \cong E\_2 - \frac{\pi^2 |\chi\_2(\varepsilon)|^4}{\ln^2(\rho\_0/R(\varepsilon))}\tag{36}$$

and in the case of weak fields, it can be written

$$t\_{1,1} = 1 - \left\{ 1 + \frac{i\pi\sqrt{3}}{2\sin^2(\pi e/w)} \left[ \frac{\ln\left(\rho\_0/\mathbb{R}(e)\right)}{\pi} - \frac{2\sin^2(2\pi e/w)}{w\sqrt{\left(2\pi/w\right)^2 + \frac{1}{2}Fw - E}} \right] \right\}^{-1} \tag{37}$$

Single-Atom Field-Effect Transistor DOI: http://dx.doi.org/10.5772/intechopen.81526

### Figure 4.

The dotted curve stands for Fw<sup>3</sup> <sup>¼</sup> <sup>0</sup>; the solid curve stands for Fw<sup>3</sup> <sup>¼</sup> <sup>5</sup>; and the dashed curve stands for Fw<sup>3</sup> <sup>¼</sup> <sup>10</sup>.

with the zero-current (zero transmission) energy of

$$E\_R \cong \left(\frac{2\pi}{w}\right)^2 + \frac{1}{2}Fw - \frac{4\pi^2\sin^4(2\pi\varepsilon/w)}{w^2\ln^2(\rho\_0/\mathcal{R}(\varepsilon))}.\tag{38}$$

In the case of a surface defect, i.e., when the atom is close to the wire's boundary (see Appendix A),

$$R(\varepsilon) \cong 4\varepsilon \exp\left(\chi/2\right),\tag{39}$$

and then the zero-current energy is approximately

$$E\_R \cong \left(\frac{2\pi}{w}\right)^2 + \frac{1}{2}Fw - \frac{\varepsilon^4 (2\pi/w)^6}{\ln^2(\rho\_0 \exp\left(-\gamma/2\right)/4\epsilon)}.\tag{40}$$

Therefore, the zero-current energy has a linear dependence on the electric field (and thus on the applied external voltage). In Figure 4 this property is presented by plotting the conductance for three different transverse electric fields.
