Schottky effect

The Schottky effect causes the reduction of the work function for electrons on a metal surface by a high electric field strength in the outer space. This effect occurs with hot cathodes (metal- vacuum interface) and Schottky contacts (metal- semiconductor contacts ) such as Schottky diodes . The effect was named after the German physicist Walter Schottky .

Explanation

Reduced work function due to the Schottky effect. The upper curve is the image force potential asymptotically approaching the vacuum level. The distance between the vacuum potential and the Fermi level E _F in the metal is the original work function. The lower curve is the sum of the image force potential and the linearly decreasing potential of an external homogeneous field. The height of the resulting maximum above E _F is the reduced work function.

For the sake of simplicity, a metal surface is first considered in a vacuum. An electron at a distance induces a positive charge on the metal surface. The attractive force between the induced charge and the electron is exactly the force between the electron and an equally large positive image charge in and mirror or image force mentioned. ${\displaystyle x}$ ${\displaystyle -x}$

${\displaystyle F(x)={\frac {-e^{2}}{4\pi \varepsilon _{0}(2x)^{2}}}}$

With the dielectric constant . The potential energy of the electron results from the work that has to be done to bring the electron from to : ${\displaystyle \varepsilon _{0}}$ ${\displaystyle W}$ ${\displaystyle \infty }$ ${\displaystyle x}$

${\displaystyle E_{\text{Bild}}(x)=W(x)=-\int _{\infty }^{x}{F(x')dx'}=-{\frac {e^{2}}{16\pi \varepsilon _{0}}}\cdot {\frac {1}{x}}}$

(The work is negative, because the attractive force between the electron and the mirror charge acts in the direction of integration.)

A linear potential curve from a homogeneous field in the outer space is superimposed on the image force potential ${\displaystyle E}$

${\displaystyle E_{\mathrm {pot} }(x)=-{\frac {e^{2}}{16\pi \varepsilon _{0}}}\cdot {\frac {1}{x}}-eEx}$.

If the external field is very strong, it leads to a lowering of the potential within the short range of the mirror force. With increasing field strength, the maximum of the potential curve moves closer to the surface,

${\displaystyle x_{\mathrm {max} }={\sqrt {\frac {e}{16\pi \varepsilon _{0}E}}}\,,}$

and sinks in the process

${\displaystyle \Delta W={\sqrt {\frac {e^{3}E}{4\pi \varepsilon _{0}}}}\,.}$

For an electric field of strength, this results in and , which would roughly quadruple the current strength of the Schottky emission (see Edison-Richardson effect ) at 1000 Kelvin. In general, the Schottky effect increases the current intensity of the glow emission by the factor . ${\displaystyle E=10^{7}\,\mathrm {V} /\mathrm {m} }$ ${\displaystyle x_{\mathrm {max} }\approx 6\,\mathrm {nm} }$ ${\displaystyle \Delta W\approx 0{,}12\,\mathrm {eV} }$ ${\displaystyle \mathrm {e} ^{\Delta W/(k_{\mathrm {B} }T)}}$

For higher field strengths than , the tunnel effect must be taken into account, because the width of the barrier is then no longer large compared to the wavelength of the electrons. With the tunnel current is considerable even with a cold electrode, see field emission . ${\displaystyle E=10^{8}\,\mathrm {V} /\mathrm {m} }$ ${\displaystyle E=10^{9}\,\mathrm {V} /\mathrm {m} }$

The above principle also applies to metal-semiconductor interface layers. In this case, the “external” field exists even with short-circuited connections, namely through the space charge zone in the semiconductor material (whose dielectric constant must be taken into account in the formulas).