Zincblende III-V nitride heterostructures are strained ones, given the lattice mismatch between the constituent materials. Although we are considering here a (001)-oriented In0.2
N-GaN QW configuration, the small indium content does not prevent from taking strain effects into account. In particular, there is a breaking of the degeneracy of heavy and light hole valence bands at the center of the two-dimensional Brillouin zone. In this work, we are including strain effects in the most simple way, that is, by incorporating the strain-induced shifts of the conduction and valence band edges in the unperturbed potential profile configuration for both electrons and holes (see, for instance,
]). Data related with material properties and confining potential are taken from another work
Considering the strain effects between the well and barrier materials, the electron and hole confinement potential have been obtained, respectively, by the following:
, where Q
, and e11
. The super-index w
refer to the well (Inx
N) and barrier (GaN) materials.
, the main used parameters are reported. Here,
m0 is the free electron mass. The parameters of the Inx
Ga1−xN material have been obtained by linear interpolation between InN and GaN.
Table 1 The effective mass parameters used in the calculations
The potential responsible for the confinement of electrons and heavy holes in the QW is depicted in Figure
for several values of the ILF parameter (Figure
a,b,c,d). The column at the left-hand side contains the graphics that correspond to the conduction band profile, while the corresponding valence band bendings are shown in the column at the right. It is possible to observe the evolution of the QW shape associated with the change in the laser intensity—without applied dc field—by going through rows one to four in the picture. The transition from a single to a double QW potential is detected in the figures of the fourth road. We consider, of interest, to highlight that the confining potential for holes in a In0.2
N-GaN QW also experiences that kind of single-to-double QW transition at the value of
reported in the current work. This is because such a feature is not present in the case, for instance, of a Ga0.7
As-GaAs QW, in which, for the same value of the ILF parameter, the shape of the conduction band profile is very similar with that of Figure
]. Despite the greater value of the hole effective mass in the present system compared with that of the arsenide-based one, the main reason of such a difference lies in the height of the valence band confining barrier, which in the latter case is almost three times larger than the one formed in the nitride-based heterostructure studied here.
Figure 1 Confinement potential andz-dependent amplitude of probability for the first two electron and ground hole confined states in aIn0.2Ga0.8N-GaN QW. The results are for L=200 Å and have been considered several values of the ILF-parameter:
α (more ...)
In the fifth row (Figure
e), the evolution of the confined electron and hole levels as functions of
α0 clearly show the growth in the energy values that resulted from the laser-induced deformation of the conduction and valence band potential profiles. Such modification in the QW shape involves a significant rise of the well bottom which acts by pushing up the energy levels. In the valence band, the original depth of the QW is only enough to accommodate a single heavy-hole level and, according to the basic properties of the confined one-dimensional motion, there will always be one energy level in the hole subsystem. In the conduction band, for sufficiently large laser field intensities, the first excited state is expelled from the QW, and there only remains a single confined level (the ground state one).
contains our results for the heavy-hole exciton binding energy as a function of the QW width, without the application of any dc electric field and taking several values of the
as a parameter. The shape of the curves is typical in the case of a zincblende QW. Independent of the laser intensity, there is initially a growth in
associated to the transition from a purely two-dimensional exciton to a quasi-two-dimensional one, that is, for the lower values of the well width, it favored the overlap between the confined electron and hole densities of probability, making that the expected values of the inter-carrier distance, ϕ
to be smaller, thus provoking the strengthening of the Coulombic interaction between them. As long as the QW widens, this expected value becomes larger, and the electrostatic interaction weakens, with the consequent reduction in the exciton binding energies. The decrease in
for a fixed well width, L
, observed when going from a zero laser field to a more intense one is also due to a decrease in the Coulombic correlation between both types of carriers. In fact, as can be seen from Figure
, augmenting the laser intensity makes the allowed confined energy states to shift upwards. Therefore, the corresponding wave functions will spread over a wider interval of the coordinate, and the values of ϕ
will be larger. The kind of convergence exhibited by the curves for larger L
reflects the increasing effect of the rigid barriers located at ±
). This means that in all cases, the curves are tending toward the the exciton binding energy of an infinite barrier QW of width Ł∞
, with or without a laser effect.
If an intense laser field is applied taking the QW geometry as a varying parameter, the results obtained for the heavy-hole exciton binding energy as a function of
α0 are those shown in the Figure
. They are consistent with the explanation given above regarding the weakening in the strength of the electron-hole interaction associated with the loss of confinement induced either by the increment in the laser intensity or by the enlargement of the QW size.
If a dc electric field of increasing intensity is applied to the system, keeping fixed its dimension, the heavy-hole exciton binding energy evolves as observed in the Figure
. Once again, the value of the laser field strength appears parameterizing the different curves in the graphics. In the case of zero laser field, the variation of Eb
)corresponds to an all the way decreasing function, the dc electric field amplitude. It is known that the dc field effect is mainly that of augmenting the polarization by pushing apart, spatially speaking, the carriers of opposite sign. At the same time, the rectangular QW potential profile transforms in a way that reduced the degree of carrier localization inside the well region. All this has the consequence of increasing the value of ϕ
and the corresponding fall in the Coulomb interaction. However, this particular evolution of the binding energy seem to practically disappear for the two intermediate values of the ILF parameter considered. One notices from Figure
that a very slight decrease in
is obtained when the value of F
goes from zero to 20 kV/cm, if
is a quarter of the QW width. At the same time, what we see when
is equal to the half of the well width is, even, a slight increase in
over almost the entire interval of F
considered, though for the largest values of the dc field amplitude, that quantity starts showing a decreasing behavior. Hence, what is happening here is a phenomenon of compensation of the progressive augmenting of the electron-hole expected distance via the deformation of the QW potential profile obtained when combining the effects of the two kinds of externally applied fields, that is, if the effect of the dc field is to push the electronic wave function towards the left-hand side of the QW, given that the height of the barrier for electrons is significantly bigger than the one corresponding to the valence band, the displacement of the electron wave function is counteracted by the barrier repulsion (one must keep in mind that the dc field strength values considered here are not very high). On the other hand, the electric field will induce a displacement of the heavy hole towards the right. However, the QW barrier height is so small here that, thanks to the ILF-induced pushing-up effect of the energy level position, the hole density of probability can penetrate further to the left, with the consequent increment in the overlap between electron and hole wave functions. As a result of this, the expected electron-hole distance diminishes. This is the cause of the compensating effect and the apparent insensitivity of
with respect to F
for such modified QW shapes associated to such particular values of
. Once the laser field intensity is sufficiently high (lower curve in Figure
), the heavy-hole exciton binding energy recovers its decreasing variation as a function of the dc field strength (again due to the fall in the carrier localization), with the exception of a very slight increment noticed for very small values of F
. Here, the combination of the slow linear change of |eFz
|with the ILF-induced double QW shape of the confining potential (Figure
d) leads to the kind of compensating effect mentioned above. In this case, it leads to a small reduction in ϕ
and the observed little increase in
in that region.
Figure 4 Binding energy of heavy-hole exciton in a In0.2Ga0.8N-GaN QW. As a function of the applied electric field with L=200Å and several values of the ILF-parameter.
shows the variation of the heavy-hole exciton binding energy as a consequence of the increment in the intensity of the z-oriented applied dc electric field. In this situation, the width of the QW appears and is considered as the parameter that differentiates between the curves depicted.
Figure 5 Binding energy of heavy-hole exciton in a In0.2Ga0.8N-GaN QW. As a function of the applied electric field for several values of the quantum well-width (L) with
The configuration chosen includes an applied laser field with intensity given, in each case, by the parameter
/4. It is seen that for the two lowest values of the well width,
is a slight decreasing function of F
until a certain critical value,
, of the dc field strength at which initiates an abrupt fall that leads to a constant, limit value, that remains for the rest of the increasing range of the amplitude F
. The decrease occurring while F
is justified along the same arguments expressed above with regard to the progressive enlargement of the inter-carrier average distance that associates with the loss in electron and hole confinement. The abrupt descent in
has to do with the escape of one (electron or hole) of the wave functions away from the QW region, towards the infinite barrier on the side it was pushed to by the electric field. The value of the expected electron-hole distance then suffers a sudden rise which reflects in the drop of
observed. Augmenting further the dc field strength will function to cause the same effect on the other wave function in such a way that the increase in F
will not have any other influence on the polarization because the carriers will remain confined by the infinite barriers at ±
/2. Therefore, one may see that
adopts a constant value when F
becomes large enough.
It is worth mentioning that, for all the values of L
taken into account, setting
/4implies a great modification of the confining potential profile which, as one of the main features, presents a significantly reduced effective well depth. At the same time, the effect of confinement reduction on the carrier wave functions is more pronounced for narrower QWs, for the allowed energy levels are, initially, placed at higher energy positions. Thus, the application of the not so intense dc fields readily leads to the mentioned wave function escape. This explains why the phenomenon of abrupt change in
is manifested for smaller dc field intensities.
The curves that correspond to the two highest values of L
show an increasing behavior for the smallest electric field amplitudes. This fact relates with the reduction in ϕ
obtained as a result of the combination of the laser and dc fields on the confinement of the carriers. A small F
associates with a slight linear deformation of the already modified (by the laser effect) potential profile. The electron and hole densities of probability are pushed in opposite directions, but the potential well barriers, not so deformed, repel them away. This has the consequence of bringing the two particles a little bit closer and, therefore, of augmenting the strength of their Coulombic interaction. However, when the dc field is augmented, the dominant influence is that causing the spatial spreading of the carrier wave functions, which leads to the decrease in
. Notice that the pronounced fall is also present when L
150Å, but for L
is a rather smooth monotonically decreasing function of F
, without any abrupt change. This is because the QW width is large enough to avoid the sudden escape of the wave functions and also because the limiting infinite barriers are much closer to the inner well ones.