Electrostatic built-in fields in nitride based QDs
To address the role of the polarization potential in nitride-based QDs we apply a real-space surface integral approach developed by in our group [5]. This method admits analytic solutions in certain cases and provides an extremely useful insight into the parameters that influence the magnitude and the shape of the polarization potential. The total built-in polarization in a nitride-based nanostructure with a wurtzite crystal structure is given by P^{tot}=P^{spon}+P^{piezo}. The first contribution P^{spon} refers to the strain-induced piezoelectric polarization while the second term P^{piezo} denotes the spontaneous polarization which arises from the lack of inversion symmetry along the c-axis in a wurtzite lattice. Since the spontaneous polarization in nitride materials is a constant directed along the [0001]-direction, the surface integral expression for the induced spontaneous potential can be easily obtained from the standard electromagnetism result for a constant polarization field, where the surface integral in this case is taken over the surface of the QD. Turning to the piezoelectric contribution, the polarization vector P^{piezo} can have nonzero components in all three directions in the dot. In addition,P^{piezo} is no longer a constant, but depends on the strain field. By using integral expressions for the strain field, in combination with Maxwell's equations, one can derive equations which provide an approach to determine the three dimensional piezoelectric polarization potential Φ^{piezo}(r) by evaluating two 2-D integrals over the surface of the QD. A detailed description of the applied approach is given in Ref. [5].
The calculated total built-in potential Φ^{tot} (spontaneous + piezoelectric contribution) of a lens-shaped polar (c-plane) and non-polar (a-plane) InN QD with a diameter of d=18 nm and a height of h=2 nm embedded in GaN matrix is shown Fig.1 [2]. It can be seen that the magnitude of is significantly reduced in the case of the non-polar system.
Fig. 1: Contour plot of the total (spontaneous + piezoelectric) built-in potential for a) a polar and b) non-polar lens-shaped InN/GaN QD. The potential is shown for a slice through the QD centre. The z-axis is parallel to the [0001]-direction [2]. |
The analysis of the built-in potential has been carried out for InGaN/GaN QDs [2,6] as well as for GaN/AlN systems [4,6]. In the framework of the surface integral technique we have performed a detailed and systematic study of the polarization potential, by calculating the different contributions to the total (spontaneous + piezoelectric) built-in potential separately. Figure 2 compares the different contributions to the potential in a) polar and b) a non-polar cuboid shaped GaN/AlN QD [4].
Fig. 2: The different contributions to the total (spontaneous + piezoelectric) built-in potential Φ^{p} for a) polar and b) non-polar GaN/AlN QDs. The spontaneous polarization contribution is denoted by Φ^{sp}. The different components of piezoelectric potential are given by the axial Φ^{ax} and the shear strain related part Φ^{15}. The index ± indicates the sign of the piezoelectric constant. The total piezoelectric potential Φ^{p} (black solid line) assumes a negative e_{15}[4]. |
Our analysis reveals that the magnitude of the axial contribution (solid blue diamond lines), which is related to the diagonal terms of the strain tensor, to the total built-in potential is significantly reduced in the non-polar case. This fact can be attributed to the change in the strain field when going from polar to a non-polar system [4,6]. Turning to the shear strain component, there is a large degree of uncertainty in the shear piezoelectric constant, [7]. In particular there is uncertainty not only in the magnitude of , but also conflicting evidence as to its sign. Our investigations [4] show that only a negative value of can give a strong reduction of the built-in field in non-polar GaN/AlN QDs, as observed experimentally. While the results on non-polar GaN/AlN QDs indicate that only a negative leads to significant reduction in the built-in field, we find that for /GaN QDs it is still likely to remain significant in systems with high indium content, independent of the sign of the piezoelectric constant [6]. However, the residual field can be nearly eliminated in non-polar QD systems with lower (x<0.35) indium concentration, opening the possibility of using such systems as efficient optical sources.
In the case of polar c-plane nitride-based QDs, we have used the calculated built-in potential as an input for the investigation of excitonic complexes in these systems. These calculations were performed in the framework of a self-consistent Hartree approximation [7, 8]. Experimentally determined transition energies are well explained by our theoretical model, confirming that spectroscopic studies of exciton complexes provide a useful probe of the properties of nitride QDs. In addition, we have also performed first calculations on the single particle states in non-polar GaN/AlN QDs with a truncated pyramidal shape [4]. Despite the reduced built-in potential, we find that the overlap of the electron and hole ground state wave functions can be even smaller than that found in comparable c-plane QDs. Further investigation is now required to confirm the role of the built-in polarization potential in non-polar wurtzite QDs.
References
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[3] Excitation-induced energy shifts in the optical gain spectra of InN quantum dots M. Lorke, J. Seebeck, P. Gartner, F. Jahnke and S. Schulz, Applied Physics Letters 95, 081108 (2009).
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