116 CHAPTER 3. CHARGE TRANSPORT IN MATERIALS
Ec
Ev –x 1
x 2 Ec
Ev
–x 1 0x 2
x
(a)
(b)
Available empty
states (holes) in
valence band
Electrons in
conduction band
Eg
++++++ –––––––––
Tunnel barrier
Figure 3.12: (a) A schematic showing the band profile for ap–njunction. An electron in the
conduction band can tunnel into an unoccupied state in the valence band or vice versa. (b) The
potential profile seen by the electron during the tunneling process.
The exponent for the tunneling probability is (m∗(GaAs) = 0.065 m 0 ;m∗(InAs)
∼0.02 m 0 ;Eg(GaAs) = 1.5 eV;Eg(InAs) = 0.4 eV) for GaAs
−
4 ×(2× 0. 065 × 0. 91 × 10 −^30 kg)^1 /^2 (1. 5 × 1. 6 × 10 −^19 J)^3 /^2
3 ×(1. 6 × 10 −^19 C)(1. 05 × 10 −^34 Js)(2× 107 V/m)
= − 160
The tunneling probability is exp(−160)∼=0. For InAs the exponent turns out to be−12.5
and the tunneling probability is
T= exp (− 12 .5) = 3. 7 × 10 −^6
In InAs the band-to-band tunneling will start becoming very important if the field is
∼ 2 × 105 V/cm.