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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]
Energy diagram of the MIM system then is shown on Fig. 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 .
 |
Fig. 1. Potential barrier in the MIM system then ~ 0.
 and  – work function of the left and right metals, respectively. |
In this case, as shown on Fig. 1,  and 
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.
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| 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)
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