# 1.2.1 John G. Simmons Formula

Now, using the Sommerfeld model (see **chapter 1.1.3**) and WKB approximation (see **chapter 1.1.2**) and assuming that *T* = 0, potential barrier is of arbitrary shape and the mass of electrons is isotropic in space, we can derive an expression for the tunneling current flowing in a metal-insulator-metal (MIM) system.

**Fig. 1. Diagram of MIM system in equilibrium.
j _{1} and j_{2} – work function of the left and right metals, respectively.**

**Fig. 2. Model of MIM system with an arbitrary shape potential
barrier. Positive potential is applied to the right metal.**

Consider two metal electrodes with an insulator of thickness L between them. If electrodes are under the same potential, the system is in thermodynamic equilibrium (see **chapter 1.1.3**) and Fermi levels of electrodes coincide **(Fig. 1)**. However, if electrodes are under different potentials, current flow between them is available. **Fig. 2** shows the energy diagram of electrodes with applied bias energy *eV*. Potential barrier width for electrons occupying the Fermi level is denoted as d_{z} = z_{2} – z_{1}. Consider that all the current flowing in the system is due to the tunneling effect.

Probability *D(E _{z})* of the electron transmission through the po-tential barrier of height

*U(z)*is given by expression (4) in

**chapter 1.1.2**. For the number of electrons

*N*tunneling through the barrier from electrode 1 into electrode 2, we can write [

_{1}**1, 2**]

(1)

where

(2)

and *E _{m}* – maximum energy of tunneling electrons.

Integration of expression (2) can be performed in polar coordinates. Because in the model under consideration , and total energy is , changing variables , , we get

(3)

Substituting (3) in (1), we obtain

(4)

The number of electrons *N _{2}* tunneling back from electrode 2 into electrode 1 is calculated in the same way. According to (4) from

**chapter 1.1.2**, the potential barrier transparency in the given case will be such as if positive voltage

*V*is applied to electrode 1 relative to electrode 2. In this case

(5)

Net electrons flow *N* through the barrier is obviously *N* = *N _{1}* –

*N*. Let us denote

_{2}(6)

Then, the tunneling current density *J* is

(7)

According **Fig. 2**, *U(z)* can be written in the form
. Then, integrating (4) from **chapter 1.1.2** and using expression (A5) from **Appendix**, we get

(8)

where
– average barrier height relative to Fermi level of the negative electrode;
;
, *b* – dimensionless factor defined in the **Appendix** (A6).

At *T* = 0 K

(9)

Introducing (8) and (9) into (7), we obtain

(10)

Integrating (10), we get

(11)

where .

Thus, expression (11) approximates the tunneling current in the MIM system for arbitrary barrier shape.

## Summary.

- The general expression (7) to calculate the tunneling current in the MIM system was derived in this chapter.
- The analytic approximate solution (11) of tunneling current in the MIM system was calculated.

## References.

- Burshtein E., Lundquist S. Tunneling phenomena in solid bodies. Mir, 1973 (in Russian)
- John G. Simmons. J. Appl. Phys. - 1963. - V. 34 1793.
- John G. Simmons. J. Appl. Phys. - 1963. - V. 34 238.