Biomedical Engineering Reference
In-Depth Information
5.3.2
HH-LH Intermixing
The negligible change in the magnitude of the hydrostatic strain (as SQD
QDM-
4) implies that the lowest conduction band edge will experience very small change
as it is only affected by the hydrostatic component [
34
]. The valence band edges are
modified by both the hydrostatic strain and the biaxial strain. The impact of strain
on the highest two valence band edges, HH and LH, is analytically expressed as:
→
b
v
∈
B
2
δ
E
HH
=
a
v
∈
H
+
(5.4)
b
v
∈
B
2
δ
E
LH
=
a
v
∈
H
−
(5.5)
Here
a
v
and
b
v
are the valence band deformation potential constants. The values for
these constants for InAs systems are
a
v
=
8 eV, respectively
[
34
]. From the Eqs. (
5.4
)and(
5.5
), it is evident that the magnitude of the
1
.
0eVand
b
v
=
−
1
.
∈
B
determines the HH-LH splitting. For the single QD layer (SQD), due to a large
negative value of
∈
B
around the center of the QD, the HH and LH band edges
will be considerably separated inside the QD region. This will induce dominant HH
character in the highest few valence band states which will be closer to the HH band
edge. As the magnitude of
∈
B
decreases with the increasing size of the QDMs, the
HH-LH splitting reduces, increasing the LH component in the valence band states.
For the QDM-4 system, the nearly zero magnitude of the
∈
B
around the center of
the molecule implies that the HH and LH bands will be nearly degenerate around
the center of the QDM. The valence band states will therefore be of highly mixed
character, consisting of contributions from both the HH and the LH bands.
Figure
5.3
a-e plots the highest two local valence band edges, HH and LH, for
all of the QD systems under study along the [001] direction through the center of
the QDs. Highly negative biaxial strain in the SQD results in
152 meV splitting
of the HH and LH bands within the QD region. As the biaxial strain around the
center of the QDMs decreases (approaching towards zero for the QDM-4), the HH-
LH splitting around the center of the molecule also decreases to
≈
≈
117 meV,
≈
74 meV,
≈
32 meV, and
≤
28 meV for the QDM-2, QDM-3, and QDM-4 systems,
respectively.
The HH and LH character of a particular valence band state in the tight binding
formulation can be estimated as follows: If the amplitudes of the
p
x
,
p
y
,and
p
z
orbitals at any atomic site are
a
x
,
u
/
d
,
a
y
,
u
/
d
,and
a
z
,
u
/
d
, respectively (where
the subscripts
u
and
d
refer to up and down spin), then the HH contribution is
approximately proportional to
2
summed over all the
atomic sites. The LH contribution is approximately proportional to
2
+
|
a
x
,
u
−
ia
y
,
u
|
|
a
x
,
d
+
ia
y
,
d
|
2
summed over all the atomic sites. By using these expressions, we estimate that
the
2
+
|
a
z
,
u
|
|
a
z
,
d
|
HH
LH
ratio for the highest valence band state (
H
1
) decreases from
≈
108 for
the SQD to
10.6 for the QDM-2, QDM-3, and QDM-4
systems, respectively. This clearly points towards an increasing LH character in
≈
15.8,
≈
12.3, and
≈
