7. Fast switching

When the dip of the resonances is very narrow, the gain is very high; however, in this case, the transistor response is very slow, because it takes a substantial

amount of time to establish the resonance. In fact, the gain is proportional to the transistor's time response τ, i.e.,

$$gain \propto \left(4\pi^2/w^2 - E\right)^{-1} \propto \tau. \tag{47}$$

However, the value of both can be controlled by changing the defect's parameters. Since

$$\rho\_0 \cong 4\varepsilon \exp\left[\frac{\chi}{2} + 2\pi \frac{\sin^2(2\pi\varepsilon/w)}{w\sqrt{4\pi^2/w^2 - E}}\right],\tag{48}$$

the parameter ρ<sup>0</sup> can be chosen to place the resonance dip at any point in the regime π2=w<sup>2</sup> < E < 4π2=w<sup>2</sup> and thus to determine the transistor time response

$$\rho\_0 \cong 4\varepsilon \exp\left[\frac{\chi}{2} + \frac{2\pi}{w} \sin^2(2\pi\varepsilon/w)\sqrt{\pi}\right].\tag{49}$$

Therefore, the transistor with the quickest response is the one with a surface defect with

$$
\rho\_0 \cong \mathfrak{A}\mathfrak{e} \exp\left(\boldsymbol{\chi}/\mathfrak{Z}\right). \tag{50}
$$

In this case, the transistor time response is determined by the wire's width, i.e.,

$$
\pi \sim \left( w/\pi \right)^2,\tag{51}
$$

<sup>2</sup> which in ordinary physical units is <sup>τ</sup> � m w<sup>ð</sup> <sup>=</sup>h<sup>Þ</sup> , i.e., the narrower the wire, the shorter the transistor's time response is.

Eq. (50) teaches that such a single-atom nanotransistor can be faster than any of the cutting-edge available transistors.

It should be emphasized that the point defect does not necessarily have to be an atom. It could be a molecule or any quantum dot that can be designed of having the necessary de-Broglie wavelength ρ0.
