Fundamentals on Bionanotechnologies - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 4

related by 2˛ C ˇ D 1 and are equal to ˛ D 0 and ˇ D 1 in AlGaAs compounds,
which were the first semiconductors that displayed ballistic transport.
The spatial restrictions on electron motion are expressed in the specific form
of the boundary conditions imposed on the Schrodinger equation. A structure in
which electrons are confined at the nanoscale by potential barriers along the, say, z
direction but are free to travel along the transverse x and y directions is referred to
as quantum well (QW). In a quantum well with infinite-height potential barriers, the
Schr odinger equation is accompanied by the boundary conditions ‰.x;y;0/ D ‰
.x;y;L z / D 0,whereL z is the width of the quantum well.
If V D 0, the solution of the Schrodinger equation can be written as ‰.x;y; z / D
.2=L z L x L y / 1=2 sin.k z z / exp.ik x x/exp.ik y y/,whereL x and L y are, respectively,
the dimensions of the structure along x and y. In ballistic devices, L z is comparable
to the Fermi wavelength F and L z <L x ;L y L fp ;L ph . Another effect of the
boundary conditions is a discrete spectrum for the z component of the electron
momentum k z D p=L z , which induces a discretization of the energy levels along
the direction of spatial restriction. The energy dispersion relation in the quantum
well in which the bottom of the conduction band E c is considered as reference is
given by

p
L z

2

2 2 2

2m .k x C k y /; (1.3)

E.k x ;k y ;k z / D E c C „ C „

2m .k x C k y / D E s;p C „

2m

where E s;p is the cutoff energy of the discrete subband labeled by the integer p;the
subbands are also referred to as transverse modes. The difference in energy between
adjacent subbands is greater for more confined electrons, i.e., for smaller L z .
For an arbitrary energy distribution in the k space, which takes E. k / constant
values on a k -space surface †, a spin-degenerate density of states (DOS) can be
defined as

.E/ D .2/ 3 Z

d S
jr k E j EDconst: :

(1.4)

†

Then, in the quantum well case, the DOS particularizes to

X p #.E E s;p /;

m
„

QW .E/ D (1.5)

2 L z

where # denotes the unit step function. As follows from ( 1.5 ), and as illustrated in
Fig. 1.4 , the DOS in the quantum well is discontinuous, in contrast to the case of bulk
semiconductors, where the absence of spatial constraints leads to a continuous DOS.
At equilibrium conditions at temperature T , the electrons occupy the discrete
energy levels of the quantum well according to the Fermi-Dirac distribution
function ( 1.1 ), so that the electron density per unit area at equilibrium is given by
( Ferry and Goodnick 2009 )