Comment on “Electric field effect on the second-order nonlinear optical properties of parabolic
and semiparabolic quantum wells”
İbrahim Karabulut,* Ülfet Atav, and Haluk Şafak
Physics Department, Faculty of Arts and Science, Selcuk University, Campus 42075, Konya, Turkey
Received 15 February 2005; revised manuscript received 28 April 2005; published 23 November 2005
Zhang and Xie Phys. Rev. B 68, 235315 2003 presented their results for the second harmonic generation
SHG susceptibility in electric-field-biased parabolic and semiparabolic quantum wells QW’s. In this Com-
ment, we demonstrate that the results presented for the SHG coefficient in a parabolic QW with an applied
electric field are not physically sound and they contradict with well-established properties of the Hermite
polynomials which just happen to be the wave functions describing such a system. It is shown that in a
parabolic QW with an applied electric field intersubband transitions contribute to neither second nor any higher
order harmonic generation.
DOI: 10.1103/PhysRevB.72.207301 PACS numbers: 42.65.Ky, 42.79.Nv, 73.21.Fg, 78.66.Fd
In a recent study, Zhang and Xie,
1
attempted to calculate
the contribution of intersubband transitions to the second
harmonic generation SHG susceptibility in parabolic and
semiparabolic quantum well QW structures with an exter-
nally applied electric field
1
by using the compact density
matrix approach and iterative procedure.
2,3
The nonlinear optical properties of a quantum well struc-
ture have contributions from the bulk susceptibility of the
material, the interband transitions and the intersubband tran-
sitions. The bulk susceptibility arises from the self-asym-
metry of the crystal structure and it is not very large. Also the
contributions of interband and intersubband transitions are
zero for a symmetrical quantum well structure,
6
but as the
symmetry is broken, nonvanishing contributions to second
order nonlinear optical susceptibilities are expected to ap-
pear. Consequently, either quantum well structures are pro-
duced with a built-in asymmetry
6,7
or externally applied elec-
tric fields are used to remove the symmetry.
The energy levels of electrons and holes move in opposite
directions under the influence of an externally applied elec-
tric field; also the wave functions are displaced in opposite
directions. Therefore the application of an external electric
field is expected to have a strong influence on the excitonic
effects or the interband transitions, resulting in a contribution
to the second order nonlinear optical susceptibility. The mag-
nitude of these contributions are mostly determined by the
dipole matrix elements. The dipole matrix elements of inter-
subband transitions are orders of magnitude larger than those
of interband transitions,
6
and if the breaking of the symmetry
can be achieved, the intersubband transitions give the major
contribution to the second order nonlinear optical suscepti-
bilities. So, it is very desirable to have contributions from the
intersubband transitions to obtain large second order nonlin-
ear optical susceptibilities.
Zhang and Xie
1
have considered only the contributions of
intersubband transitions to the SHG susceptibility. They re-
port nonzero contributions to SHG susceptibilities for both
parabolic and semiparabolic QW structures and they con-
clude that “the SHG susceptibility in semiparabolic QW is
larger than that in parabolic QW due to the self-asymmetry
of the semiparabolic QW.” Although the results presented in
the above-mentioned work might be correct for semipara-
bolic QWs, those pertaining to parabolic QWs are com-
pletely wrong and contradicts with well-established litera-
ture.
4,5
In this Comment, we consider the contribution of
intersubband transitions to the SHG susceptibility of a para-
bolic QW under the influence of an externally applied elec-
tric field. We demonstrate that no contribution to SHG can be
obtained in a parabolic QW whether an electric field is ap-
plied or not. For the sake of consistency, we use the notation
of Zhang and Xie in our analysis.
From basic symmetry considerations, one arrives at the
conclusion that contributions to even order susceptibilities
should vanish in structures with inversion symmetry; there-
fore no contribution to SHG susceptibility should be ob-
served in a parabolic QW. In a symmetric QW structure,
contribution to SHG can only be observed if the symmetry of
the conduction band potential is broken through either by the
advanced material growing technology or by the application
of an external bias field. However, we should note that the
application of an external electric field does not remove the
symmetry of the conduction band potential in a parabolic
QW. In this respect, parabolic QW structures are an excep-
tion.
The compact density matrix approach is extensively used
in the analysis of nonlinear responses of QW structures.
Zhang and Xie have applied this approach to parabolic and
semiparabolic QWs. Within this approach and under the as-
sumption of two photon resonance, i.e., E
21
E
32
, the formula for the contribution to SHG susceptibility
per unit volume is given by Eq. 20 in Ref. 1.
2
2
=
q
3
12
23
31
s
0
1
E
31
-2 + i
0
E
21
- + i
0
. 1
The volume SHG susceptibility has a resonant peak value for
= given by
PHYSICAL REVIEW B 72, 207301 2005
1098-0121/2005/7220/2073012/$23.00 ©2005 The American Physical Society 207301-1