1.2.2 John G. Simmons Formula in a Case of Small, Intermediate and High Voltage (Field Emission Mode).

According chapter 1.2.1, the approximate expression for the tunneling current in the MIM system can be written as [1]:

(1)

where , , – average barrier height, – barrier width, – voltage between electrodes.

Small voltage.

At low voltages , expression (1) can be simplified [1]

(2)

where . Since , we can consider that doesn't depend on . Thus, in the case of small applied voltage, the tunneling current proportionate to . Energy diagram of the MIM system then is shown on Fig. 1.

Fig. 1. Potential barrier in the MIM system then ~ 0.
and – work function of the left and right metals, respectively.

Intermediate voltage.

If , then and (Fig. 2).

Fig. 2. Potential barrier in the MIM system then .
and – work function of the left and right metals, respectively.

In [2] it is shown, that for this case the tunneling current-voltage relation is given by

(3)

where .

High voltage – Field emission mode.

The case when corresponds to energy diagram shown in Fig. 3 and to the following , .

Fig. 3. Potential barrier in the MIM system then .
and – work function of the left and right metals, respectively.

Substituting and into equation (1), we obtain

(4)

where – electric field strength.

At high applied voltage ( ) the Fermi level of electrode 2 is lower than the conduction band bottom of electrode 1. Under such conditions, electrons can not tunnel from electrode 2 into electrode 1 because of lack of empty states. An inverse situation is for electrons tunneling from electrode 1 into empty states of electrode 2. This process is similar to autoelectronic emission from a metal into vacuum. Thus, since , the second summand in (4) can be neglected and for the current we get

(5)

where coefficient b = 23/24. This result agrees qualitatively with an analytical expression for the field emission current density [3].

Thus, using formulas (2)–(5), we can compute the tunnel current at given system parameters and plot current-voltage characteristics. Fig. 4 shows theoretical tunneling current-applied voltage plot in case of carbon electrode 1 ( = 4,7 эВ) and platinum electrode 2 ( = 5,3 эВ) at = 5 Å and contact area S = 10^-17 m².

Fig. 4. Current-voltage characteristic for carbon electrode 1 and platinum electrode 2 at = 5 Å and contact area 10^-17 m². Parts of J(V) curve correspond to the following expressions: AB – (22), BC – (23), CD – (24), DE – (25).

Summary.

Depend upon magnitude of applied voltage, formula (1) can be simplified (2)–(5).
It is possible to describe the experimental tunneling current dependences by approximated expressions (2)–(5) in accordance with magnitude of applied voltage.

References.

John G. Simmons. J. Appl. Phys. - 1963. - V. 34 1793.
John G. Simmons. J. Appl. Phys. - 1963. - V. 34 238.
Dobretzov L.N., Gomounova M.V. Emission electronics. Nauka, 1966 (in Russian)