Fundamentals on Bionanotechnologies - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 5

Fig. 1.4 Density of states in
a quantum well

r QW

z

E F

y

x

E

E S,1 E S,2 E S,3

Z

2 X
p

QW .E/f.E/dE D k B T m
„

n D L z lnŒ1 C exp.E F E s;p /=k B T: (1.6)

0

In the degenerate limit or at low temperatures, when k B T E F , the Fermi-Dirac
distribution function is proportional to #.E F E/, so that all electron subbands
below the Fermi energy are filled with electrons, and all subbands above it are
empty. At low temperatures, the electrons with energy E reside in a number of
subbands M.E/, which can be determined by counting the transverse modes with
cutoff energies below E.
When the Fermi energy level in a quantum well is positioned between the
first and the second energy subband, as displayed in Fig. 1.4 ,wehaveatwo-
dimensional electron gas (2DEG), which has a metallic behavior because E F is
inside the conduction band. In this case, the Fermi wavenumber k F , determined
from the electron kinetic energy as E kin D E F E s;1 D„

2 k F =2m, is correlated

2 /.E F E s;1 / through the formula

to the electron density per unit area n D .m= „

( Ferry and Goodnick 2009 )

k F D .2n/ 1=2 : (1.7)

The Fermi wavelength is defined as F D 2=k F .
A nanoscale structure is called quantum wire (QWR) if the electron motion is
spatially restricted by energy potentials in regions of widths L y and L z along two
directions: y and z , but the electron can move freely along x. If the constrain-
ing potentials have infinite heights, the electron wavefuntion has the expression
‰.x;y; z / D Œ2=.L y L z L x / 1=2 sin.k y L y / sin.k z L z / exp.ik x x/, and boundary con-
ditions similar to those in the quantum well case imply that k y D p=L y ;k z D
q=L z , with p, q integer numbers. The energy dispersion relation in QWR is then
given by

p
L y

2 q

L z

2

2 2 2 k x 2 k x

2m ; (1.8)

E.k x ;k y ;k z / D E c C „ C „ C „

2m D E s;pq C „ 2m 2m