Environmental Engineering Reference
In-Depth Information
The energy
E
R
released in the recombination is given entirely to a photon (the
quantum unit of light), according to the relation
E
R
¼
are,
respectively, the frequency and wavelength of the emitted light,
c
is the speed of light
3
h
n
¼
hc
/
l
. Here,
n
and
l
10
8
m/s, and
h
is Plancks constant
h
10
34
Js
10
15
eVs. The
¼
6.63
¼
4.136
color of the emitted light is controlled by the choice of
L
, since
E
R
¼
E
h
,
where
E
G
is the semiconductor bandgap, and the electron and hole con
nement
energies,
E
e
and
E
h
, respectively, become larger with decreasing
L
as
h
2
/8
mL
2
.
These con
nement (blueshift) energies are proportional to 1/
L
2
. Since these terms
increase the energy of the emitted photon, they act to shorten the wavelength of the
light relative to that emitted by the bulk semiconductor, an effect referred to as the
blueshift of light from the quantum dot. These nanocrystals are used in biological
research as markers for particular kinds of cells, as observed under an optical
microscope with background ultraviolet light (UV) illumination.
In these applications, the basic semiconductor QDcrystal is coatedwith a thin layer
to make it impervious to (and soluble in) the aqueous biological environment.
Another coating may then be applied that allows the QD to preferentially bond to a
speci
c biological cell or cell component of interest. The biological researcher may,
for example, see the outer cell surface illuminated in green while the surface of the
inner cell nucleus may be illuminated in red, all under illumination of the whole
E
G
þ
E
e
þ
eld
with light of a single shorter wavelength.
8.2.1
Electrons and Holes in a 3D
Quantum Dot
Equations 2.11
-
2.16 and 3.15 are applicable to the electron and hole states in
semiconductor quantum dots, which are used in biological research as color-
coded fluorescent markers. Typical semiconductors for this application are CdSe
and CdTe.
A hole (missing electron) in a full-energy band behaves very much like an
electron, except that it has a positive charge, and tends to
float to the top of the band.
That is, the energy of the hole increases opposite to the energy of an electron.
The rules of quantum mechanics that have been developed so far are also
applicable to holes in semiconductors. To create an electron
hole pair in a semi-
conductor requires an energy at least equal to the energy bandgap,
E
g
, of the
semiconductor.
We found earlier that thewavefunction for a particle in a three-dimensional in
nite
trap of volume
L
3
with impenetrable walls are given as
-
3
=
2
sin
n
ð
x
;
y
;
z
Þ¼ð
2
=
L
Þ
ð
n
x
p
x
=
L
Þ
sin
ð
n
y
p
y
=
L
Þ
sin
ð
n
z
p
z
=
L
Þ;
where n
x
¼
1
;
2
...
;
etc
:;
and
E
n
¼
h
2
8
mL
2
n
x
þ
n
z
:
n
y
þ
=
