10.3. RESONANT TUNNELING 495
ψ = A 2 eik z^2 + B 2 e–ik z ψ = A 3 eik z^3 + B 3 e–ik z 3
ψ = A 1 eik z 1
+B 1 e–ik z^1
z 1 z 2 z 3 z 4
W
2Tunneling has
resonancesFigure 10.5: Typical resonant tunneling structure with two barriers. The wavefunction in each
region has a general form shown. By matching the wavefunctions and their derivatives at the
boundaries one can obtain tunneling probabilities.
A simple application of this formalism is the tunneling of electrons through a single barrier of
heightV 0 and widtha. The tunneling probability is given by
T 1 B(E)=
∣∣
∣
∣
A 3
A 1
∣∣
∣
∣
2=
4 E(V 0 −E)
V 02 sinh^2 (γa)+4E(V 0 −E)(10.3.1)
with
γ=1
√
2 m(V 0 −E) (10.3.2)If we have two barriers as shown in figure 10.4, the tunneling through the double barrier is
given by
T 2 B=[
1+
4 R 1 B
T 12 B
sin^2 (k 1 W−θ)]− 1
(10.3.3)
whereR 1 Bis the reflection probability from a single barrier
R 1 B=
V 02 sinh^2 γa
V 02 sinh^2 γa+4E(V 0 −E)(10.3.4)
andθis given by
tanθ=2 k 1 γcoshγa
(k 12 −γ^2 )sinhγa