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]:
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula_in_different_modes/img01.gif)
(1)
where
,
,
– average barrier height,
– barrier width,
– voltage between electrodes.
Small voltage.
At low voltages
, expression (1) can be simplified [1]
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula_in_different_modes/img08.gif)
(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.
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula_in_different_modes/img10.gif)
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).
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula_in_different_modes/img27.gif)
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
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula_in_different_modes/img18.gif)
(3)
where
.
High voltage – Field emission mode.
The case when
corresponds to energy diagram shown in Fig. 3 and to the following
,
.
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula_in_different_modes/img28.gif)
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
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula_in_different_modes/img23.gif)
(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
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula_in_different_modes/img26.gif)
(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.
![](/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/tunnel_current_in_mim_system/john_simmons_formula_in_different_modes/img29.gif)
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)