Exciton properties in zincblende InGaN-GaN quantum wells under the effects of intense laser fields
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Abstract
In this work, we study the exciton states in a zincblende InGaN/GaN quantum well using a variational technique. The system is considered under the action of intense laser fields with the incorporation of a direct current electric field as an additional external probe. The effects of these external influences as well as of the changes in the geometry of the heterostructure on the exciton binding energy are discussed in detail.
Keywords
Nitrides Excitons Intense laser field Quantum wellsBackground
InGaN-based systems have revealed a high prospect for applications in optoelectronics. Although the hexagonal (wurtzite) allotropic form is the one most commonly considered, the zincblende (ZB) III-V nitrides are also very promising materials that have been obtained with high-quality crystal structure[1, 2, 3, 4]. This is mostly due to the fact that the cubic symmetry avoids the presence of rather high spontaneous polarizations in the crystal, which are, in a greater extent, responsible for the presence of large built-in fields in wurtzite-based heterostructures, responsible for important reductions in the oscillator strength, and the optical recombination rates in that kind of systems[5, 6]. However, the ZB structure in nitrides is provided with higher carrier mobilities, larger optical gain and lower threshold current density because of its smaller effective mass, and has mirror facets compatible with substrates such as GaAs[7, 8, 9]. In consequence, the ZB nitride-based heterostructures have drawn much attention in recent times[10, 11, 12, 13].
The knowledge of exciton states is important for the correct understanding of some optical properties in the semiconducting low-dimensional systems. Investigations on excitons and related optical properties in ZB nitride low-dimensional systems have been mostly performed in quantum dots[14, 15, 16, 17, 18], but much less in quantum well (QW) heterostructures[19, 20].
Research activities on the interaction of intense laser fields (ILF) with carriers in semiconductor nanostructures have revealed interesting physical phenomena. For instance, the presence of changes in the electron density of states in QWs and quantum well wires (QWWs)[21, 22], the measurement of zero-resistance states in two-dimensional electron gases under microwave radiation[23], terahertz resonant absorption in QWs[24], and Floquet-Bloch states in single-walled carbon nanotubes[25], among others. A number of investigations on the effect of laser fields on low dimensional heterostructures have been published. The dressed atom approach was extended by Brandi et al.[26, 27] to treat the influence of the laser field upon a semiconductor system. In the model, the interaction with the laser is taken into account through the renormalization of the semiconductor effective mass. The appearance of an unexpected transition from single to double QW potential induced by ILF was revealed in a theoretical study from Lima et al.[28]. Within the laser-dressed potential model, it is found that the formation of a double-well potential for values of the laser frequencies and intensities such that the so-called laser-dressing parameter α_{0} is larger than L/2, where L is the QW width. This fact is associated with the possibility of generating resonant states into the system’s channel as well as of controlling the population inversion in QW lasers operating in the optical pumping scheme.
The present work is concerned with the theoretical study of the effects of ILF on exciton states in single ZB nitride QWs of the InGaN-GaN prototype. The research is extended to include the additional influence of an applied direct (dc) electric field oriented along the growth direction of the system. The paper is organized as follows. In the ‘Theoretical framework’ subsection in the ‘Methods’ section, we describe the theoretical framework. The ‘Results and discussion’ section is dedicated to the results and discussion, and finally, our conclusions are given in the ‘Conclusions’ section.
Methods
Theoretical framework
The electron and hole effective masses and the static dielectric constant have been considered to have the same value (the one in In_{x}Ga_{1−x}N) throughout the In_{x}Ga_{1−x}N-GaN QW.
The method for the obtention of the electron and hole states is based on the work by Xia and Fan[29].
In Equation 7, I and ω are, respectively, the average intensity and the frequency of the laser, c is the velocity of the light, and _{A 0} is the amplitude of the vector potential associated with the incident radiation. A detailed discussion on the derivation of 〈V〉(z_{i}α_{0i}) is provided in other studies[28, 33, 34, 35, 36, 37].
wherez_{eh} = z_{e}−z_{h}andα_{0} = (eA_{0})/(μcω).
where λ is the variational parameter. Besides,$\overrightarrow{{r}_{1}}=\overrightarrow{r}-(0,0,{\alpha}_{0})$ and$\overrightarrow{{r}_{2}}=\overrightarrow{r}+(0,0,{\alpha}_{0})$ with$\overrightarrow{r}=(x,y,{z}_{e}-{z}_{h})$.
whereE_{0} is the eigenvalue of the Hamiltonian in Equation 4 without the Coulomb interaction term—the last one at the right hand side—andλ_{min} is the value of the variational parameter in which the energy in Equation 11 reaches its minimum.
Results and discussion
Zincblende III-V nitride heterostructures are strained ones, given the lattice mismatch between the constituent materials. Although we are considering here a (001)-oriented In_{0.2}Ga_{0.8}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,[40, 41]). Data related with material properties and confining potential are taken from another work[42].
Considering the strain effects between the well and barrier materials, the electron and hole confinement potential have been obtained, respectively, by the following:
The effective mass parameters used in the calculations[42]
a | C_{11} | C_{12} | a_{v} | a_{c} | b | E_{g} | m_{e} | m_{h} | ε | |
---|---|---|---|---|---|---|---|---|---|---|
InN | 4.98 | 187 | 125 | -0.7 | -2.65 | -1.2 | 0.7 | 0.1 | 0.835 | 9.7 |
GaN | 4.50 | 293 | 159 | -0.69 | -6.71 | -2.0 | 3.22 | 0.19 | 0.81 |
In the fifth row (Figure1e), the evolution of the confined electron and hole levels as functions ofα_{0} 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).
The configuration chosen includes an applied laser field with intensity given, in each case, by the parameterα_{0} = 3L/4. It is seen that for the two lowest values of the well width,E_{b}is a slight decreasing function of F until a certain critical value,F_{c}, 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 < F_{c} 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 inE_{b}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 ofE_{b} 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 ±L_{∞}/2. Therefore, one may see thatE_{b}adopts a constant value when F becomes large enough.
It is worth mentioning that, for all the values of L taken into account, settingα_{0} = 3L/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 inE_{b} is manifested for smaller dc field intensities.
The curves that correspond to the two highest values of L in Figure5 show an increasing behavior for the smallest electric field amplitudes. This fact relates with the reduction in 〈ϕ|z_{e}−z_{h}|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 inE_{b}. Notice that the pronounced fall is also present when L = 150Å, but for L = 200ÅE_{b} 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.
Conclusions
The properties of heavy-hole excitons in InGaN-GaN-based quantum wells under intense laser and applied dc electric fields are studied for a set of different values of the fields intensities and the well spatial dimensions. In general, for a fixed geometry of the unperturbed system, the exciton binding energy is a decreasing function of the intense laser field parameter and of the dc electric field, although certain combinations of the two applied field intensities may lead to a rather insensitive behavior of the binding energy with respect to the application of a dc field. It is shown that the changes of the degree of carrier confinement and of the carrier polarization associated to the influence of the laser and the dc fields are mainly responsible for the exciton properties mentioned. To our knowledge, there seem to be no previous reports on exciton properties in zincblende nitride QW induced by intense laser fields. Thus, the results of the present work might be considered as a first approximation to the subject in this kind of systems.
Notes
Acknowledgements
Carlos A Duque is grateful to the Colombian Agencies CODI-Universidad de Antioquia (project: E01535-Efectos de la presión hidrostática y de los campos eléctrico y magnético sobre las propiedades ópticas no lineales de puntos, hilos y anillos cuánticos de GaAs-(Ga,Al)As y Si/SiO_{2}) and Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (CAD-exclusive dedication project 2012-2013). Miguel E Mora-Ramos thanks Mexican CONACYT for the support through 2011-2012 sabbatical grant no. 180636 and research grant CB-2008-101777. He is also grateful to the Escuela de Ingeniería de Antioquia and the Universidad de Antioquia for hospitality in his sabbatical stay.
Supplementary material
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