Fundamentals on Bionanotechnologies - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 10

Fig. 1.9 Forward- and
backward-propagating
components of the electron
wavefunction at an interface
between adjacent layers i and
i

x

A i A i +1

B i +1

B i

C 1

m i , V i , k i m i +1 , V i +1 , k i +1

x = x i

A i exp. ik i x i /
B i exp. ik i x i /

.1 C v iC1 = v i /.1 v iC1 = v i /
.1 v iC1 = v i /.1 C v iC1 = v i /

A iC1 exp. ik iC1 x i /
B iC1 exp. ik iC1 x i /

;

1
2

D

(1.15)

where v i D„ k i =m i is the electron velocity in layer i. Analogously, the 2 2
transfer matrix for free propagation across the ith layer between planes situated at
x D x i1 and x D x i is diagonal, with nonvanishing elements expΠik i .x i x i1 /
and expΠik i .x i x i1 /. As a result, the transmission probability for a structure
composed of a succession of N layers is given by

2 =. v 1 j A 1 j 2 / D v N =. v 1 j M 11 j 2 /

T D v N j A N j (1.16)

and is directly dependent on the element M 11 of the total transfer matrix with
elements M pq , p,q D 1;2, determined by multiplying the matrices corresponding
to each interface and each layer. The transmission probability is the ratio between
transmitted and incident electron probability currents, defined in any layer j as
J j D . „ =2m j i/.‰ j @ x ‰ j ‰ j @ x ‰ j /,where signifies complex conjugation and
@ x is a shorthand notation for partial derivation with respect to x.
When some of the wavenumbers k i are imaginary, we encounter the tunneling
phenomenon, widely used in nanodevices, and which can be modeled using the
matrix formalism described above. The wavenumber is imaginary only if the
electron energy is smaller than the potential energy, situation in which electron
propagation is not allowed from a classical point of view. Therefore, tunneling
is a quantum phenomenon. The layer with an imaginary wavenumber is a barrier
for electron propagation, and the electron wavefunction decays exponentially
inside it, analogous to evanescent electromagnetic waves. Because the transmission
probability across a barrier layer becomes zero unless it is narrow enough, tunneling
of electrons propagating with constant energy E occurs only through thin potential
barriers or a succession of such barriers surrounded by quantum wells, i.e., regions
with real wavenumbers.
However, classical transport across a potential barrier is permitted if the electron
acquires extra energy. When this additional energy is thermal, the process is called
thermionic emission. At finite temperatures, thermionic emission accompanies the
tunneling process and becomes the major electron transport mechanism at high
temperatures ( Appenzeller et al. 2004 ). The thermionic emission contributes to
the net current across a rectangular barrier with a temperature-dependent term
I / T 2 exp. =k B T/,where is the barrier height.