Abstract
In this paper, we use an atomistic approach to investigate strain distributions, single particle and many body electronic properties of InAs/InP nanowire quantum dots with substrate orientation varying from [111] to highindex [119], and compared with [001] case. We show that single particle gap for highindex [11k] substrates is increased with respect to [111] and [001] cases, and oscillates with the substrate index due to faceting effects. Surprisingly, the overall shelllike structure of single particle states is preserved even for highly facetted, highindex substrates. On the contrary, we demonstrate that besides two limiting highsymmetry cases, [001] and [111], the bright exciton splitting varies strongly with substrate orientation. For [112]oriented substrate, the fine structure splitting reaches maximum due to crystal lattice anisotropy despite fully cylindrical isotropic shape of nanowire quantum dot.
Keywords:
Nanowire quantum dots; Excitons; Fine structure splitting; Substrate orientationBackground
Nanowire quantum dots [1,2] grown by vaporliquidsolid (VLS) lithography [3,4] have gained a considerable attention over the last few years. VLS growth is a bottomup process that offers a possibility of tailoring quantum dot diameter, height and the composition by tuning growth conditions. For example, by selecting certain size of gold seed (catalyst) particle, one can control quantum dot diameter with nanometer precision [5] and obtain welldefined, cylindrical shape of the quantum dot. With the use of patterned substrates [5], the location of the catalyst particle itself can be precisely controlled, allowing for the quantum dot positioning and offering, thus, a substantial advantage over traditional StranskiKrastanov growth mode of selfassembled quantum dots. Further cladding process [6] reduces the surface recombination resulting in good optical properties of nanowire quantum dots; thus, such structures have been analyzed for their potential applications as efficient single photon source [7].
Nanowire quantum dots are typically grown on [111] substrates [4], while the crystal phase can vary between zincblende and wurtzite [8,9]. It has been recently shown that, regardless of the crystal phase, the high symmetry of nanowire quantum dots leads to significant reduction of their excitonic finestructure splitting [10,11], opening possible route for quantum dotbased entangled photon sources. InAs/InP quantum dots emitting at telecommunication relevant wavelengths [12] are particularly interesting for such applications.
In this paper, we use an atomistic approach [1315] to calculate electronic and optical properties of nanowire InAs/InP quantum dots. Apart from typical [111] substrate growth, we performed our calculations for quantum dots grown on highindex substrates [16] varying from [112] to [119] and compared obtained results with that of [001] substrate case. For this family of ten quantum dots, we have calculated strain distributions, single particle energies, Coulomb integrals, biexciton and trion binding energies [17] and the excitonic (bright and dark splittings) fine structure [18].
Methods
We performed our calculations for InAs/InP disklike quantum dot of 2.4nm height and 18nm diameter corresponding to typical [5] quantum dot dimensions obtained in VLS lithography. The InAs quantum dot is embedded into the center of InP nanowire of 80nm diameter and 120nm height. The nanowire and quantum dots are placed on differently oriented substrates: [001], [111] and highindex substrates [11k, with k=2,3,…9. The size of the computational domain for strain calculation reaches 32×1^{06}atoms resulting in a significant numerical challenge even on modern parallel computers. Such InP buffer thickness and height guarantee convergence of strain distribution and single particle energies well below 1 meV [19]. For calculation of strained atomic positions, we use Keating’s valence force field (VFF) model [13,20]. There are two empirical force constants (α and β) used in the VFF approach that are fit to reproduce bulk elastic properties (_{C11}_{C12} and C_{44} bulk elastic constants). At least two fitting schemes and, thus, two VFF parameterizations schemes are possible. In a more traditional [20,21] approach, two αβparameters are obtained directly from bulk elastic C_{11} and C_{12} constants only, while C_{44} constant is not a fitting target but rather comes as an output of the ‘fitting’ process, sometimes resulting in an substantial error of C_{44}[22]. Such an approach can be, however, well justified for [001] growth where strain properties are dominated by hydrostatic and biaxial strains connected to C_{11} and C_{12}. These constants should be well reproduced, while shear (offdiagonal) strains related to (sometimes poorly described) C_{44} are negligible. The latter is unfortunately not true for [111] growth. To overcome this difficulty, recently, a scheme which fits VFF parameters to all tree bulk constants on equal footing has been proposed [23]. We compared and used both of these approaches and obtained identical trends with respect to the substrate index. Hence, in this work, we use former well established method and present results for one of the parameterizations only. We leave more quantitative study of the differences between two methods for a future work [24].
Once strained atomic positions are obtained, we use them to calculate single particle energies with an empirical tightbinding model under sp_{3}d^{5}s^{*} parametrization [25] that accounts for both d orbitals and spinorbit interaction. This model incorporates onsite matrix element correction in a form suitable for nonbulk nanosystem calculation [24] and accounts for atomistic effects such as material interfaces, faceting and crystal lattice symmetry. Finally, once singleparticle energy states are found, we calculate electronelectron, electronhole and holehole Coulomb integrals, and the next step is the calculation of many body states using configuration interaction approach [15].
Results and discussion
Strain distribution
Upper row of Figure 1 shows the hydrostatic strain (the trace of strain tensor) distribution calculated for quantum dots located on differently oriented substrates. The strain is calculated on a plane crossing the quantum dot center and parallel to the quantum dot substrate. Hydrostatic strain distribution is very similar for all studied dots with maximum strain in dot center reaching ≈3.5%, which is consistent with InAs/InP bulk lattice constant mismatch. Despite the presence of low symmetry (zincblende) crystal lattice, the overall hydrostatic strain distribution reproduces the disklike symmetry of the quantum dot shape.
Figure 1. Hydrostatic strain distribution. The trace of the strain tensor distribution for InAs/InP nanowire quantum dot (h = 2.4 nm, d = 18 nm) as a function of substrate orientation. Upper row shows lateral (inplane) projection through quantum dot center, while lower row projection was calculated for perpendicular [1
 1
For higher index substrate ([112]), faceting effects become visible on the edges of quantum dots and are very well pronounced for high[118] index substrate. The characteristic steps in quantum dot strain distribution due to faceting effects are also well visible in Figure 1 (lower row) where hydrostatic strain is plotted on a [1
 1
Figure 2 shows different functions of strain calculated at the geometric dot center. The absolute of (a) the hydrostatic strain Tr(ε)=_{εxx} + _{εyy} + _{εzz}for [111] substrate (3.8%) is noticeably larger than [001] substrate (3.1%). We speculate that one could expect even larger difference between [001] and [111] substrates for highly strained InAs/GaAs quantum dots. With increasing substrate index, the magnitude of the hydrostatic strain goes down reaching, for [119] case, the value comparable to that of nontilted [001] system. Similar trend is observed for (b) biaxial strain defined as , where x,y, and z correspond to directions defined by crystal axes [100], [010] and [001], respectively. Biaxial strain defined as B(ε) is largest for flat and disklike quantum dot located on [001] substrate, while it is exactly zero (by definition and the quantum dot symmetry) for [111] case. On the contrary, function of shear (offdiagonal) strain defined as is exactly zero (at the dot center) for [001] substrate quantum dot and reaches maximum for [111], demonstrating the significant role of the [111] biaxial strain for systems grown on [111] substrate.
Figure 2. Different functions of strain at the geometric dot center. (a) The hydrostatic strain Tr(ε)=_{εxx} + _{εyy} + _{εzz}, (b) the biaxial and the average shear strain of InAs/InP nanowire quantum dot (h = 2.4 nm, d = 18 nm) as a function of substrate orientation.
The spatial distribution of B(ε) or S(ε) over entire quantum dot spatial domain is more important than one particular value at the quantum dot center. As shown in Figure 3 (upper row) for [001] and [111] substrate orientations, the spatial distribution of B(ε) is nearly cylindrical, determined by the shape symmetry of the quantum dot, reaching maximum at the dot center for [001] case and highly varying at the quantum dotmatrix interface. On the contrary, it is only the quantum dotsurrounding matrix interface where B(ε) reaches nonnegligible values for [111] case.
Figure 3. Lateral (inplane) projection through quantum dot center for two functions of strain. The biaxial strain (upper row) and the average shear strain (lower row) of the InAs/InP nanowire quantum dot (h = 2.4 nm, d = 18 nm) as functions of substrate orientation.
There is a strong quantitative difference of biaxial strain distribution between highly symmetric [001],[111] dots and that grown on [11k] (k=2,3,…9) substrate, where combination of the quantum dot and lattice symmetry results in strong anisotropy of the strain distribution as visible in Figure 3. For such low symmetry system, we may expect large fine structure splitting (FSS) for both bright and dark excitons despite nominally cylindrical symmetry of quantum dot shape.
It must be pointed here that for quantum dots grown on [001] substrate there are also two nonequivalent axes: [110] and [1
 1
By similar symmetry arguments, one can deduce that [111] substrate quantum dot has C^{3v} symmetry, and indeed, such symmetry (triangularlike) is well pronounced on biaxial strain plot in Figure 3. System with C^{3v} symmetry will have zero fine structure splitting both for bright and dark exciton states. We will verify this general analysis by more strict numerical, tightbinding calculation in the next section.
Single particle states
Once strained atomic positions are obtained, we use them to calculate single particle energies with an empirical tightbinding model [24] accounting for dorbitals and spinorbit interaction. Figure 4a,b shows several lowest electron and hole states as functions of substrate lattice orientation. Surprisingly, despite strong faceting effects visible in strain distribution (Figure 1), the spectra of confined electron states reveal robust shell structure with pshell splitting on the order of 1 meV and sp level spacing varying between 47 and 52 meV. pd level spacing is systematically larger (by 6 meV) than that of sp levels, while the third of d states is split from two other by ≈14 meV, a hallmark of disklike [26], not of lenstype (harmonic oscillatorlike) confinement.
Figure 4. Single particle energies as functions of substrate orientation. Several lowest (ground state, black/boxes; first excited state, red/circles; second excited state, blue/triangles, etc.) single particle electron (a) and hole (b) energies and single particle gap (c) of InAs/InP nanowire quantum dot (h = 2.4 nm, d = 18 nm).
The spectrum of confined hole states is more susceptible to a choice of substrate than that of electron states, yet wellvisible shell structure is still present as seen in Figure 4b. The splitting of hole p shell varies from ≈7 meV for [111] substrate and goes down to 2 meV for highindex and [001] substrates. sp hole level spacing is in the order of 20 meV, while pd level spacing is about 15 meV, both smaller than that of electrons due to larger hole effective mass. Splittings within p and d shells reach maximum for the most anisotropic quantum dot grown on [112] substrate and are generally systematically larger than for electrons, as holes are more affected by anisotropy of confining potential through biaxial strain terms in TB Hamiltonian [25]. Energies corresponding to electron levels, being formed predominately from atomic s levels, follow similar trends as hydrostatic strain in Figure 2a that can be understood in terms of simple BirPikus model [22] and single a^{c} deformation potential. Holes, on the other hand, are build mostly from atomic p orbitals and are susceptible to both hydrostatic and biaxial [001] and [111] strains that enter Hamiltonian via different deformation potentials (a^{v}b and d).
Figure 4 shows that, when going from [001] to [111] substrate, the ground hole state is energetically shifted up by 44 meV, while electron ground state is shifted by a smaller amount of about 37 meV only. This difference results in overall reduction of the single particle gap by 7 meV (Figure 4c). Our recent abinitio calculations of band deformation potentials [27] suggest gap reduction for [111]grown quantum dots compared with [001] case, yet the effect should be more pronounced for highly strained InAs/GaAs quantum dot.
Increasing substrate index from [111] to [119] reduces the single particle gap as seen in Figure 4, but even for [119] case, the gap is far from being converged and is larger by about 25 meV compared to [001] system. Interestingly, the gap value reveals oscillations due to faceting effects that were not well pronounced in the electron and hole spectra separately.
In this work, we neglect effects of piezoelectricity. Such approach is well justified [28] for InAs/InP systems due to small strain magnitude as compared to InAs/GaAs systems and partial cancelation of first order piezoelectric terms by second order contributions [29]. More importantly, contrary to straightforward k.p approach [26], piezoelectricity would not alter symmetry of the Hamiltonian which is already well defined by atomistic strained positions entering the TB calculation.
Coulomb integrals and binding energies
Figure 5a shows electronelectron , electronhole and holehole Coulomb integrals calculated for electron and hole occupying their ground s states [17]. There are no faceting effects visible in Figure ??a but rather a smooth change with the increasing substrate index: a manifestation of longrange character of direct Coulomb interaction. Electronelectron and holehole interactions are decreased and increased, respectively, for [111] system compared with [001] case; this is consistent with the decreased confinement of electron and increased confinement of hole states (shown above for single particle spectra in Figure 4). One can use J^{ss} integrals to estimated biexciton (XX) and trion (X^{}X^{+}) binding energies at the HartreeFock (HF) level [17]:
Figure 5. Coulomb intervals and binding energies. (a) Electronelectron , electronhole and holehole Coulomb integrals for electron and hole occupying their ground sstates of InAs/InP nanowire quantum dot (h=2.4 nm, d=18 nm) as functions of substrate orientation. (b) Biexciton (XX) and trion (^{X−},^{X + }) binding energies calculated by perturbative approach (HartreeFock approximation). (c) Biexciton (XX) and trion (^{X−}, X^{ + }) binding energies calculated by configuration interaction approach (including Coulomb scattering terms up to the d shell). _{EHF}, binding energy by HartreeFock approximation; _{ECI}, binding energy by configuration interaction approach.
Figure 5b shows XXX^{} and X^{+} binding energies calculated according to above formulas, suggesting that both X^{+} and XX are unbound (have positive binding energy). In HF picture, the X^{+} binding energy can be roughly estimated to vary between 4 and 6 meV; XX binding energies are also positive and reached up to 4 meV, while only X^{} is bound with bounding energy ≈−3 meV. This characteristic ordering of emission lines with increasing energy: X^{}XXX and X^{+} is analogous to that reported in the study of Gong et al. [17] for InAs/InP lensshaped quantum dot.
However, when correlation effects due to the configuration interaction (mixing) with higher shells (pd) are included [15,30,31], the binding energies are shifted towards lower energies as shown in Figure 5c. In particular, X^{} binding energy now reaches ≈−4 meV, while X^{+} binding energy is significantly reduced by a semirigid shift (≈3.5 meV) for all considered substrate indices. Most importantly, similar correlation correction (≈3.5 meV) results in binding of XX complex with binding energy of ≈−1.1 meV for [001] substrate and, interestingly, very small (0.12 meV) binding energy for quantum dot grown on [111] substrate. With correlation effects accounted for emission lines show following order: X^{}XXX and X^{+}. It must be pointed here that detailed ordering of these levels may depend on dot diameter and height [16,17] and should be subject to further studies. The hidden correlation parameter (a measure of correlation effect) [17], defined as for nanowire InAs/InP quantum dot studied in this paper, varies from 1.2 to 1.5 meV, a much larger value than 0.70.9 meV reported for selfassembled InAs/InP quantum dots, proving the necessity of the full configuration interaction [15,31] approach for the studies of excitonic complexes in nanowire quantum dots. We point here that small XX binding energy for [111]grown system can be advantageous for certain entangled photon pair generation schemes based on tuning biexciton and exciton energies to resonance [32].
Fine structure splitting
Finally, we show the exciton fine structure splitting calculated for nanowire quantum dots as a function of substrate orientation. As shown in Figure 6a, bright exciton structure splitting is exactly zero for quantum dots on [001] and [111] substrates, in agreement with previously mentioned grouptheoretical arguments. However, for quantum dot on [112] substrate, there is large bright exciton splitting (60 meV) despite nominally cylindrical shape symmetry of quantum dots. There are no faceting oscillation effects visible on the evolution bright exciton splitting. The large splitting can be attributed to strong inplane anisotropy of confining potential, as shown earlier on biaxial strain distribution on plot in Figure 2, with and quantum dot axes corresponding to nonequivalent crystal axes. With increasing substrate index bright exciton FSS is reduced, reaching a minimum of ≈0.7μeV for quantum dot on [117] substrate, then interestingly, it slowly increases reaching up to 4.3 μeV for higher index system ([11k] k=20, not shown on the plot). We have checked that alloying effects will reduce bright exciton splitting; however, splitting on the order of 20 μeV still exists for cylindrical _{InP0.8}As_{0.2}/InP quantum dot grown on [112] substrate. We leave detailed analysis of alloying effects for future research.
Figure 6. Exciton fine structure splittings of InAs/InP nanowire quantum dot. Bright (a) and dark (b) exciton fine structure splittings of InAs/InP nanowire quantum dot (h = 2.4 nm, d = 18 nm) as functions of substrate orientation.
Interestingly, splitting of electron and hole pshell cannot be used as simple measure of quantum dot anisotropy and for the straightforward estimation of the fine structure splitting. As mentioned before, electron pshell splitting does not change significantly with substrate index in contrast to bright exciton splitting. The hole pshell changes are more pronounced, and indeed, this splitting reaches a maximum (8 meV) for quantum dot grown on [112] substrate (Figure 3b). Yet, both electron and hole pshell splittings are nonzero for quantum dots on [001] and [111] substrates which have zero bright exciton splitting. Thus a more complicated character the exciton fine structure splitting is revealed, rather than a simple shape anisotropy and the splitting of single particle states.
Last but not least, in Figure 6b, we present dark exciton fine structure splitting as a function of substrate orientation. There is quantitative difference between dark and bright FSS. As predicted by symmetry analysis, dark exciton splitting is exactly zero for C^{3v} quantum dot on [111] substrate, but nonzero (0.1 μeV) for D_{2d} quantum dot on [001] substrate. In analogy to bright FSS, the dark FSS reaches maximum (0.5 μeV) for quantum dot on [112] substrate; however, it varies rapidly and stabilizes for highindex substrate at the value (0.15 μeV) similar to [001] case. We note that dark excitons gains nonnegligible oscillator strengths for quantum dots on [112][115] substrates, but the detailed analysis of dark exciton lifetimes goes beyond the scope of this work.
Conclusions
We have studied the effects of the substrate orientation on single particle and many body properties of InAs/InP nanowire disklike quantum dots (h=2.4 nm, d=18 nm). We have shown that, for highindex substrate, there are faceting effects visible in the spatial strain distributions and pronounced in the single particle energy gap. Both electron and hole energies depend on the choice of substrate index, yet the overall shelllike structure is well preserved over wide range of substrate orientations. We calculated the many body properties of nanowire quantum dots, including biexciton and trion binding energy, and concluded that the full configuration interaction treatment is necessary for accurate estimation of excitonic complex binding energies. For disklike InAs/InP quantum dot on [111] substrate, XX binding energy is very small (0.12 meV) that can be advantageous for the possible generation of entangled photon pairs via the recently proposed ‘time reordering’ scheme [32]. Finally, we calculated exciton fine structure splitting and demonstrated that besides two high symmetry cases, [001] and [111], the bright exciton structure splitting varies strongly with substrate orientation. Large bright exciton splitting (60 meV) is predicted for quantum dot grown on [112] substrate despite fully cylindrical geometry of nanowire quantum dot.
We point here that general conclusions for nanowire quantum dot systems should be made after thorough study of many different systems varying with heights and diameters [16] and including alloying effects. We leave this numerically very complex problem for a subject of our future research.
Competing interests
The author declares that he has no competing interests.
Author’s information
MZ received a Ph.D. degree in physics from Nicolaus Copernicus University, Torun, Poland in 2006. He was a postdoctoral fellow at the Institute for Microstructural Sciences, National Research Council of Canada, Ottawa for almost 3 years where he performed theoretical research in the area of atomistic, many body calculations of the electronic and optical properties of quantum dots. Since 2009, he has become an assistant professor at the Instytut Fizyki, UMK, Torun, Poland.
Acknowledgements
This work was supported by Iuventus Plus program of the Polish Ministry of Science and Higher Education and by the Foundation for Polish Science, Homing Plus Programme cofinanced by the European Union within the European Regional Development Fund.
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