SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

10.4. QUANTUM INTERFERENCE EFFECTS 497

where the longitudinal velocity is

v=

1



∂E

∂k

and the net current is due to the electrons going from the left-hand side with energyEand from
the right-hand side with energyE

′
=E+e|E|l=E+eVwhere|E|is the electric field andl
is the distance between the contacts on the two sides.

J =

e

4 π^3 

∫

dkT(E)

∂E

∂k

∫

d^2 kt

[

1

exp [(Et+E−EF)/kBT]+1

−

1

exp [(Et+E+eV−EF)/kBT]+1

]

The transverse momentum integral can be simplified by noting that

d^2 kt = ktdktdφ

=

m∗dEtdφ

^2

This gives

J =

em∗

2 π^2 ^3

∫

dET(E)

∫∞

0

dEt

[

1

exp [(Et+E−EF)/kBT]+1

−

1

exp [(Et+E+eV−EF)/kBT]+1

]

=

em∗

2 π^2 ^3

∫∞

0

T(E)ln

[

1+exp[(EF−E)/kBT]

1+exp[(EF−E−eV)/kBT]

]

dE (10.3.7)

In figure 10.7 we show typical current-voltage characteristics measured in resonant double
barrier structures. The results show are for a InGaAs/AlAs structure with parameters shown. As
can be seen a large peak to valley current ration can be obtained at room temperature. There is
a region of negative resistance as expected from simple arguments. The negative resistance can
be exploited for microwave devices or for digital applications.

10.4 QUANTUM INTERFERENCE EFFECTS ....................

In a perfectly periodic potential the electron wavefunction has the form

ψk(r)=uk(r)eik·r

and the electron maintains its phase coherence as it propagates in the structure. However, in
a real material electrons scatter from a variety of sources. In high-quality semiconductors (the
material of choice for most information-processing devices) the mean free path is∼ 100 Aat ̊
room temperature and∼ 1000 A at liquid helium. For sub-micron devices it is possible to see ̊
quantum interference effects at very low temperatures in semiconductor devices. These effects