Page 325 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
In the other direction, from the metal to the semiconductor, there is no
dependence on applied voltage, because all carriers only see the barrier
energy, so that
Φ
2 B
J M → S = – A∗ T exp – --------- (7.187)
k T
B
Summing up (7.186) and (7.187), we obtain the total current flowing
through the junction

Φ
2 B  qV   qV 
J = A∗ T exp – --------- exp ------- – 1 = J exp ------- – 1 (7.188)
k T  kT  o  kT 
B
2
where J = A∗ T exp – [ Φ ⁄ ( k T)] .
o B B
Barrier To proceed with a theoretical model of the tunnel current, the quantum
Tunneling transmission coefficient T η() of electrons, where η measures energy
downward from the tip of the barrier, should be computed using a quan-
tum-mechanical approach, and we find that T η() ∝ exp – [ η E ] , for
⁄
0
some datum energy E . The current is then proportional to an integral
0
over the energy height of the barrier, of the products of transmission
coefficient, the occupation probability F of electrons that are available
M
for tunneling (a Fermi distribution), and the unoccupation probability of
states F that the electrons will tunnel into (another Fermi distribution).
S
These concepts are clarified by Figure 7.29. As a result of all these terms,
we find that the tunneling current has the following form [7.17]

 N 
D
J ∝ exp – 2Φ ⁄ q— ------------ (7.189)
B 
t
 ε m∗ 
S
i.e., it depends exponentially on the root of the impurity concentration.

Parameter The capacitance of a metal-semiconductor contact depends on both the
Extraction built-in potential and the doping level, and can be measured to extract
these values. The depletion capacitance can be modelled by a parallel-

322 Semiconductors for Micro and Nanosystem Technology