Polarization anisotropy of the emission from type-II quantum dots

We study the polarization response of the emission from type-II GaAsSb capped InAs quantum dots. The theoretical prediction based on the calculations of the overlap integrals of the single-particle states obtained in the k.p framework is given. This is verified experimentally by polarization resolved photoluminescence measurements on samples with the type-II confinement. We show that the polarization anisotropy might be utilized to find the vertical position of the hole wavefunction and its orientation with respect to crystallographic axes of the sample. A proposition for usage in the information technology as a room temperature photonic gate operating at the communication wavelengths as well as a simple model to estimate the energy of fine-structure splitting for type-II GaAsSb capped InAs QDs are given.

The ground state wavefunction of holes in type-II InAs quantum dots (QDs) with GaAs 1−y Sb y capping layer (CL) resides outside of the dot volume and in general has the form of two mutually perpendicular pairs of segments [1,2]. Remarkably, the pair oriented along [110] crystallographic direction is positioned close to the QD base, while the other, oriented along [1][2][3][4][5][6][7][8][9][10], is located above the dot [1][2][3][4][5]. Both the vertical position and the orientation of the hole wavefunction have been recently discussed in the literature [3][4][5][6][7][8]. It has been shown elsewhere [1,5] that the potential minimum for holes in this system results from a delicate interplay between quantum size effect and the piezoelectric potential and is, thus, sensitive to the thickness d of CL [3,5]. In this work, we utilize the d-dependence of the orientation of the hole wavefunction to show that the anisotropy of the polarization of the photoluminescence (PL) from this system can be used to determine the vertical position of the hole, as a gated source of radiation with defined polarization, or to estimate the fine-structure splitting energy of the exciton in type-II heterostructure. Theory.-To assess the properties of PL from type-II QDs, we have first calculated the one-particle wavefunctions of the InAs QDs capped by the GaAs 1−y Sb y CL as a function of d by the envelope function approach using the nextnano++ simulation suite [9] with the inclusion of the elastic strain and piezoelectricity. Secondly, the oscillator strengths I ab of the transitions between states a and b were calculated by a customarily built code based on the approach outlined in Refs. 10 and 11, where ψ a and ψ b are the wavefunctions of the states a and b, respectively, and ψ = 8 i=1 φ i u i where φ i and u i are the envelope and Bloch functions for the band i, re-spectively. The other quantities are as follows: e denotes the polarization vector, E ab the difference between the eigenenergies of the states a and b,P ( r) the momentum operator, m 0 is the free electron mass, and the reduced Planck constant. Furthermore, because of faster spatial variations of u compared to φ we make the following ap-proximationP ( r)ψ = (P ( r)u)φ + u(P ( r)φ) ≈ (P ( r)u)φ. For more details see Refs. 10-12. Results.-We focus on the investigation of the polarization anisotropy of interband transitions, i.e {a, b} = {c, v}, where c (v) is the conduction (valence) band state. Including spin, the exciton transitions are formed by nearly degenerate quadruplets (bright and dark doublets) [13,14] which were, however, not resolved in our experiment. Thus, we calculate I = 1/4 {↑,↓}⊗{↑,↓} I eh only. In our experiments, both the excitation and the detected PL radiation propagate perpendicularly to the sample surface and PL is thus polarized parallel to that; the angle between the crystallographic direction [110] and the polarization vector is denoted α. We plot our theoretical and experimental results in terms of the degree of polarization C(α) = (I(α) − I min )/(I max + I min ) where I min and I max are the extremal values of I(α). Note that for the angle α max such that I(α max ) = I max the previous relation gives C(α max ) ≡ C max , the maximum obtained degree of polarization. The composition of GaAs 1−y Sb y CL was kept spatially constant with Sb content of y = 0.24 in all calculations and only d was varied. Notice that we have chosen the particular value of y to ensure that, except for the thinnest CLs, the transition would be purely of type-II [1,3]. We have calculated the polarization properties for several dot geometries and In composition profiles, see Tab. I.  The choice of the QD shapes was motivated by the results of Refs. 1-3, 5, 15-17. Except for the structure with trumpet-like profile [1,2,16,17], the values of constant In content in the QDs were chosen so as the ground state emission energy lies within the range of the communication bands, 1.3-1.55 µm. Based on the predicted properties of these structures, the technology of QD sample preparation was subsequently optimized to obtain QDs with desired properties, i.e. the long emission wavelength and type-II band alignment.
Typical results of our calculations are shown in Fig. 1 for the lens-shaped QD with the base diameter and height of 12 and 4 nm, respectively, and two CL thicknesses of d = 3 nm, and d = 12 nm. Notably, the emitted light is preferentially polarized along [110] ([1-10]) crystallographic direction for thin (thick) CL. The dominant contribution to I(α) in QDs comes from the areas with the largest overlap between electrons and holes. The electron is firmly bound in the body of type-II QDs for all Sb contents and CL thicknesses [1,3]. Thus, also due to the larger effective mass of holes the overlap and the resulting polarization anisotropy of I(α) is dictated by those parts of the hole wavefunction which are located closest to the dot, see panel (a) of Fig. 1. We show in Fig. 2 (a) that with d increasing, the vertical position of the hole wave function is first shifted upwards and the average of the vertical hole coordinate measured from the top of the QD, denotedz hole , increases. At the same time C max is reduced [ Fig. 2(b)] until, for certain critical CL thickness d c , the emission becomes isotropic (C max = 0) and the average hole vertical position coincides with the top of the QD (z hole = 0). During this period also α max = 0 • holds and the hole segments are oriented along [110] direction. However, further increase of d results in increase ofz hole and C max . Also a flip of the orientation α max to [1][2][3][4][5][6][7][8][9][10] direction for d c is observed and this is attained for even larger values of d. While α max is identical with the orientation of the dominant pair of segments of the hole wavefunction, i.e. those with the largest overlap with the electrons, C max corresponds to the ratio of overlaps of both pairs with them, see also Fig. 1 (a). The correspondence of C max is, however, not exact because it is influenced also by a slight elongation of the electron wavefunction, occurring along [1][2][3][4][5][6][7][8][9][10]. The overlap of electrons and holes is rather small in type-II QDs [1,2,19]. However, particularly for GaAsSb capped InAs QDs we can take advantage of that to estimate the energy of the fine-structure splitting (∆E FSS ), i.e. the energy separation of the bright excitonic doublet, from C max . Motivated by the results of Ref. 5 we estimate ∆E FSS as where e is the elementary charge, ε the permittivity, and l QD the mean diameter of the QD base. The lengthl QD replaces | r e − r h |, where r e ( r h ) is the position of the electron (hole). In constructingl QD we have noted a typical situation for type-II GaAsSb capped InAs QDs where the maximum of probability densities for electron and hole is positioned at similar distances from the edge of the dot. To make the estimation even easier we considered the QD to have a shape of a hemisphere with electron located in the center of its circular base. Hencē l QD is in turn equal to the diameter of the hemisphere's base. Equation 2 provides a simple, rough estimate of ∆E FSS from the measured values of C max and the structural parameters of QDs. The latter can be obtained by standard structural characterization tools. The comparison of Eq. 2 to more elaborate calculations of ∆E FSS , see Ref. 5 for details, is shown in Fig. 2 (a). The error is as large as an order of magnitude. Evidently, Eq. 2 gives most precise estimates of ∆E FSS for small values of C max . The value of α max is identical with that for the lower energy exciton of the corresponding exciton doublet.
We have calculated α max and C max also for other QD shapes and In composition from Tab. I and confirmed the existence of d c associated with the vertical position of the (ground state) hole wavefunction reaching the QD apex also for pyramidal QDs. Only in truncated shapes the hole segments are oriented along [110] for all d values in agreement with the results of Ref. 3. We note that the error of the estimate of ∆E FSS was the same for all studied QDs listed in Tab. I. It is the correspondence between the orientation of its segments and the vertical position of the hole wavefunction in respect to the QD volume that enables us to determine its position for untruncated QD shapes in real experiments simply by measuring the direction of the inplane PL polarization anisotropy. The behavior just described is closely connected to the piezoelectricity induced by the shear stress as can be seen from the dependence of C max on d being more than hun-dred times weaker when the piezoelectric term e 14 is set to zero in the calculations, see the dash-dotted curve in Fig. 2 (b). We note that the effect of the shear elements of the Bir-Pikus strain Hamiltonian [20] via the deformation potentials is negligible as well. The quantities C max and α max have an intimate relation to the mixing of heavy (HH) and light (LH) hole states as that is responsible for the shape and orientation of the envelope wavefunction of the hole [21,22]. For our dots the phase of HH-LH mixing determines α max , and its amplitude C max . Both parameters can be determined using the results of Refs. 23  The measurements of the polarization anisotropy of PL were performed using the NT-MDT Ntegra-Spectra spectrometer. The samples were positioned in the cryostat and cooled to liquid nitrogen temperature (LN2) (for room temperature PL results see Ref. 18) and were pumped by the solid-state laser with the wavelength of 785 nm and maximum power on the sample surface of 30 mW which was varied by a tunable neutral density (ND) filter. In every experiment we have collected PL light from large number of QDs (∼2000) in order to damp the deviations from the mean properties of the ensemble of dots. The polarization of PL was analyzed by a rotating half-wave plate followed by a fixed linear polarizer. Finally, the PL signal was dispersed by a 150 grooves/mm ruled grating and detected by the InGaAs line-CCD camera, cooled to minus 90 • C. The experimental procedure was as follows. First, the type of confinement was determined by measuring the pumping dependency of the PL spectra. Either a blueshift of the whole spectrum was observed, indicating type-II confinement [6,19], or the spectrum only increased in magnitude without any visible spectral shift, and we considered it as type I. For a typical example of the results obtained on sample C see Fig. 3 (a). Consequently, the PL spectra for 37 angular positions of λ/2plate at fixed laser pumping power were acquired and fitted by a sum of Gauss-Lorentz (GL) profiles [26]. As our calculations described above are most accurate for the ground-state transition, we consider here the inten-  sity of the GL band with the lowest energy only. Examples are shown in Fig. 3 (b) and (c) for type-II samples A and C. Note that the dominant polarization direction is clearly oriented along [110] in (b), and close to [1][2][3][4][5][6][7][8][9][10] in (c). We stress that due to expected small values of C max we paid a particular attention to (i) the reduction of the residual polarization of the whole setup, see red circles in Fig. 3 (c), and (ii) proper fitting of the spectra. The resulting α max for all samples and different positions on them are summarized in Fig. 4 (a). For type II we have observed α max corresponding to both [110] and [1][2][3][4][5][6][7][8][9][10] crystallographic directions, the former being more frequent. This in turn means that holes were located preferentially close to the base of the QD. Furthermore, for type I α max corresponded to [110] direction, in agreement with the results on InAs/GaAs QDs [27]. We note, however, that the spread of the obtained values of α max might be as large as 30 degrees. We assume that this is an effect of the QD and CL irregularities which were still not averaged out and also due to remaining imperfections of the measurement procedure.
The measured values of C max are given in Fig. 4 (b) and are varying from 0.03 to 0.12. Clearly, type-II GaAsSb capped InAs QDs might possess a significant polarization anisotropy. Finally, the nature of the studied effect opens a number of possibilities to tune α max and C max by external fields ,   IIA1 IIA2 IIA3 IIA4 IIB1 IIB2 IIC1 IIC2 IIC3 IIC4 IIC5 IIC6 IC7   e.g. electric or strain [1,21,28]. In the information technology, this could enable the two perpendicular polarizations of the emitted photons to serve as a low-drain photonic realization of "zeros" and "ones", with the bonus of operation at communication wavelengths. In addition, this behavior can be reached for large ensembles of QDs, which was demonstrated here by measurements on the MOVPE-prepared samples, and even at room temperature.
Conclusions.-We have theoretically predicted, and for the first time also experimentally observed two perpendicular PL polarizations of type-II GaAsSb capped InAs QDs and explained it as an almost exclusive effect of the piezoelectricity. The measurement of polarization anisotropy enables the determination of the vertical position of the holes and their orientation in the studied system. Furthermore, a simple relation to estimate the energy of FSS in type II QD structures and the utilization of the polarization anisotropy as a room-temperature gate based on photons with energy in the communication bands and defined polarization state were proposed.