# 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} m^{2}.

**Fig. 4. Current-voltage characteristic for carbon electrode 1 and platinum electrode 2 at
= 5 Å and contact area 10 ^{-17} m^{2}. 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)