Abstract
Coulomb interactions between electrons lead to the observed multiplet structure and breakdown of the Aufbau principle for atomic d and f shells^{1}. Nevertheless, these effects can disappear in extended systems. For instance, the multiplet structure of atomic carbon is not a feature of graphite or diamond. A quantum dot is an extended system containing ∼10^{6} atoms for which electron–electron interactions do survive and the interplay between the Coulomb energy, J, and the quantization energy, ΔE, is crucial to Coulomb blockade^{2,3,4,5}. We have discovered consequences of Coulomb interactions in selfassembled quantum dots by interpreting experimental spectra with an atomistic calculation. The Coulomb effects, evident in the photon emission process, are tunable in situ by controlling the quantum dot charge from +6e to −6e. The same dot shows two regimes: J≤ΔE for electron charging yet J≃ΔE for hole charging. We find a breakdown of the Aufbau principle for holes; clear proof of nonperturbative hole–hole interactions; promotion–demotion processes in the final state of the emission process, effects first predicted a decade ago^{6}; and pronounced configuration hybridizations in the initial state. The level of charge control and the energy scales result in Coulomb effects with no obvious analogues in atomic physics.
Main
We use InAs/GaAs dots (bound s,p and d orbitals) in tunnel contact with an electron reservoir that allows controlled charging^{4,7}. Figure 1 shows optical charging of quantum dot A. There are abrupt changes in the photoluminescence at particular voltages signifying charging events. The charge in each plateau is known by identifying the singly charged exciton, X^{1−}, which has no fine structure^{8} and a large extent in gate voltage, V_{g} (ref. 7). Figure 1 demonstrates charging in discrete steps from X^{6+} up to X^{6−}.
Underpinning the experiment is Coulomb blockade: the configuration of electrons with lowest energy is established by a tunnelling interaction with the electron reservoir^{4,7}. For positive charging at large negative voltages, holes accumulate in the quantum dot from the optical excitation. At each voltage, a certain number of holes must accumulate before it becomes energetically favourable for an electron to tunnel into the dot from the Fermi sea, at which point recombination yields a photon. The number of holes required increases stepwise with increasing negative voltage. The transitions between the positively charged excitons are abrupt in voltage because the device does not rely on slow hole tunnelling^{9}, exploiting instead fast electron tunnelling.
We have calculated the optical properties of an InAs dot. In the singleparticle step, the crystal potential is a superposition of atomic screened potentials v_{α} of atom type α at each relaxed atomic site R_{α n}, where n is the lattice site index: . The pseudopotentials are adjusted so that the quasiparticle energies fit experimental data on bulk InAs and GaAs (ref. 10) thus representing valence electron correlations. The Schrödinger equation is solved in a basis set consisting of strained Bloch functions of the underlying bulk semiconductor^{10}. This approach captures the multiband, intervalley and spin–orbit interactions and also includes first and secondorder piezoelectric effects^{11}. All of the valence singleparticle levels of the dots are represented, for example, for an Natom dot there are 2N occupied singleparticle levels (N∼10^{6}). These levels have mixed angular momenta, mixed heavy hole–light hole and complex nodal structure, much like the states of a giant molecule containing a comparable number of atoms. This is different to effectivemass models^{6,12} where the orbitals are hydrogeniclike. In the manybody step, we use a configuration interaction approach where all of the Slater determinants constructed from 12 electron and 12 hole singleparticle states (counting spin) interact. The interaction consists of Coulomb and exchange integrals calculated from the atomistic singleparticle wavefunctions. The screening function for these integrals is calculated following the microscopic model of Resta^{13} and exhibits a smooth transition from unscreened at short range to screened at long range. This naturally includes both long and shortrange exchange^{14}. Using atomistic orbitals to calculate manyparticle interactions is different to most calculations of correlation in quantum nanostructures where the basis of singleparticle states is taken from effectivemass models. That the interelectronic integrals computed from effectivemass wavefunctions are very different to those computed from atomistic wavefunctions can be dramatically seen by the different ensuing symmetries of the charged exciton ground states^{15}. Our method has provided a quantitative understanding of the unusual X^{2−} and X^{2+} fine structure^{16}.
Calculations for a cylindrical dot are shown in Fig. 2 where the dominant initialstate configuration is given for each charge state with its weight. The finalstate configurations are shown for selected transitions. We interpret the experimental results with the model without fitting any of the nanostructure parameters. This approach is limited only by the imperfect knowledge of the real dots’ morphology. The experiment does not measure directly the composition of the groundstate configuration. Nevertheless, information can be deduced on the basis of the sensitivity of the photoluminescence spectrum to the initialstate configuration, closed and open shells having very different optical signatures; and to the initialstate spin through fine structure, an electron–hole exchange interaction which leads to polarizationdependent energy splittings^{8}. Only an unpaired hole (electron) in the positively (negatively) charged initial state leads to fine structure allowing an experimental discrimination between zero and finite spin. We find an excellent correspondence between our measured and calculated photoluminescence spectra, allowing us to uncover a number of physical effects.
(1) Anomalous ground states. We determine the groundstate configurations of charged excitons by minimizing the manybody energy as a function of the distribution of electrons and holes in the singleparticle levels. Figure 3 shows that a system consisting of one hole and two, three, five, six, seven electrons and a system consisting of one electron and two, three, four, five holes follows the Aufbau principle whereas the others, X^{3−}, X^{5+} and X^{6+}, violate the Aufbau principle in that one lowenergy level remains empty whereas a higherenergy level is filled. The calculations for X^{3−} predict an openshell initial configuration (Fig. 2) that recombines into two final states split by 16 meV through electron–electron exchange. The initial state violates Aufbau in that the p_{1−} level remains empty whereas the higherenergy p_{2+} is filled: for dots of small height, the loss in singleparticle energy on promoting ɛ(p_{1−}) to ɛ(p_{2+}) is more than compensated by the gain in exchange energy associated with forming a higher total spin^{17}. Experimentally, the dot A X^{3−} photoluminescence is dominated by the openshell lines split by 14 meV (Fig. 4), in excellent agreement with the theoretical prediction; the two closedshell lines (green circles in Fig. 4) can be made out and arise from thermal occupation of the higherenergy initial state. Furthermore, the upper X^{3−} photoluminescence line has a clear fine structure, as predicted. For dot B (see Supplementary Information, Fig. S1), the openshell and closedshell X^{3−} lines have similar intensities—the two configurations are close to degenerate in this case; for dot C (see Supplementary Information, Fig. S2), the closedshell X^{3−} photoluminescence lines dominate. These observations point to a slight decrease in rotational symmetry from dot A to B to C.
Theory predicts a very simple spectrum for X^{5−} that originates from a strongly dominant closedshell initial configuration with no fine structure. In a calculated elongated dot, the lower X^{5−} peak weakens and a new lowerenergy peak emerges (see Supplementary Information, Fig. S3). Experimentally, there is clear shellfilling for X^{5−}: dots A, B and C all have a largeV_{g} plateau extent and an absence of fine structure. Dot A shows one lower photoluminescence peak, consistent with the theory for the symmetric dot; dots B and C show two lower peaks, consistent with the theory for the elongated dot.
The openshell ground state of X^{2+} predicted by the theory is confirmed by experiment. A polarization dependence in the transitions is found in both theory and experiment. Theoretically, X^{3+} has a closedshell ground state with no polarization dependence in the photoluminescence. Likewise, the experimental photoluminescence has no measurable polarization dependence. In the experiment, the principal X^{3+} photoluminescence peak is slightly blueshifted relative to the principal X^{2+} peak, closer to the theoretical prediction for the closedshell X^{3+} (redshift of 3 meV) than the openshell X^{3+} (blueshift of 6 meV). The lines in the experimental X^{3+} photoluminescence above the principal peak from dot A are not reproduced in the theory but these features are of lower relative intensity from dots B and C. The openshell X^{4+} configuration with finestructure splitting is confirmed experimentally through the strong polarization dependence of the transitions. X^{5+} has a most fascinating behaviour. In contrast to X^{5−}, X^{5+} is predicted to violate Aufbau in that the first d state is occupied before the second p state is filled. Furthermore, we predict that the hole charging sequence is perturbed by the presence of the electron: without the electron, the second p state is not occupied at all^{15}. Curiously, the predicted initial configuration is open shell, yet the photoluminescence is almost unpolarized, both in the experiment and in the theory, signifying a zerospinstate coupling of the unpaired holes. Small finestructure effects are still present in the theoretical results, originating from the admixed (27%) configurations in the initial state; these effects are beyond the experimental resolution. Theoretically, the signature of the openshell X^{5+} is the presence of a multitude of peaks with comparable intensity, whereas the closedshell X^{5+} configuration has one strong peak accompanied by many very weak transitions. Experimentally, there are several strong photoluminescence lines, strongly supporting the openshell configuration. The nonAufbau filling of hole states continues for X^{6+} where p_{2} is left half empty. A polarized experimental spectrum (dots A and B) with a few peaks agrees with the theoretical prediction but the X^{6+} photoluminescence is very weak.
(2) Nonperturbative Coulomb interactions. A perturbation treatment of the Coulomb interactions predicts a blueshifted X^{1+} on the basis of a redshifted X^{1−} (ref. 18). Indeed, our calculated Coulomb energies^{17} J_{hh}=25.9 meV>J_{eh}=25.3 meV>J_{ee}=24.9 meV lead to a J_{hh}−J_{eh}=0.6 meV blueshift of X^{1+} with respect to X^{0}. However, this effect is countered by the nonperturbative mixing of the h_{S}^{2}e_{S}^{1} configuration with other configurations, a mixing that produces an overall redshift of X^{1+}, a clear feature in both experiment (Fig. 4) and theory (Fig. 2). This simple fact proves that hole–hole Coulomb interactions cannot be treated perturbatively.
(3) Exchangesplit final states. In general, charged exciton states with an openshell initial configuration decay into final states split by electron–electron or hole–hole exchange. For instance, X^{2−}, X^{3−}, X^{4−} and X^{6−} each have two photoluminescence lines arising from two possible final spin states, Fig. 2. The openshell X^{2+} decays into three states, not two as for X^{2−}. The two holes in the X^{2+} final state are predominantly heavy hole in character and therefore follow the rules for spin3/2 particles: there is a twofold degenerate state with J=3 and two singlets with J=0 and J=2. The X^{2+} initial state is split by the electron–hole exchange interaction and has total angular momentum f=2 and f=1 resulting in three transitions: f=1→J=0, f=1→J=2 and f=2→J=3. For all exciton charges, the exchange splittings in experiment and theory are in perfect correspondence with good quantitative agreement.
(4) Correlationinduced ‘promotion–demotion’ exciton lines. The X^{4−} photoluminescence consists of more than the two peaks expected from electron exchange. This observation heralds a new consequence of Coulomb interactions. When the number of carriers after recombination exceeds two, a final configuration can be generated by demoting one particle to a lower level (for example, p to s) and promoting another to a higher level (for example, p to d)^{6,19}. These Augerlike configurations are themselves optically dark but admixture with bright configurations has a powerful effect on the photoluminescence spectrum. For X^{4−}, we find two further photoluminescence lines related to promotion–demotion where the electron d orbitals are occupied in the final states even though the d orbitals are empty in the initial state (Fig. 2). The original effectivemass model with parabolic confinement predicts six bright photoluminescence lines for X^{4−} (ref. 6) but this is artificially high owing to the high degeneracies in the singleparticle states. Our calculations predict instead four X^{4−} bright lines. The inner two have significant promotion–demotion character and would not appear without the d shell. The energies of the innermost transitions depend on the symmetry—the two peaks move further apart in an elongated dot (see Supplementary Information, Fig. S3). In the experiment, Fig. 4 shows two weak peaks at low energy, a strong peak at high energy and two further peaks intermediate in energy with strange line shapes. Analysis of dots B and C reveals in each case three peaks at low energy again with some peculiar structure around the peak at high energy. Our conclusion is that the experiment matches the theory well but there are further peaks at high energy in the measured photoluminescence, arising perhaps from an excited initialstate configuration. X^{5−} has a closed initial shell and does not have exchangesplit final states. Nevertheless, both the measured and calculated X^{5−} spectra exhibit a splitting into two dominant lines, a direct consequence of promotion–demotion. The large calculated mixing of Auger configurations in the final states (40% and 60% in the high and low peaks, respectively) gives rise to the large ≃15 meV splitting of the two peaks. For the elongated dot, theory predicts that the lowerenergy peak splits into two (see Supplementary Information, Fig. S3). This is exactly what we observe for dots B and C (see Supplementary Information, Figs S1, S2).
(5) Configuration mixing in the initial states. For the highly positively charged excitons, the likelihood of finding several symmetrycompatible manybody configurations is large. We obtain theoretically such a strong initialstate mixing especially for X^{4+} and X^{5+} where the majority configuration constitutes only 60% and 73%, respectively, of the initial state. The minority configurations open new recombination pathways (see also Supplementary Information, Fig. S4). Especially intriguing are photoluminescence transitions in both the experiment and the theory for X^{4+} and X^{5+} a few meV above the principal peak. An admixed configuration in the initial state with nonAufbau character is responsible for these photoluminescence lines. For X^{4+}, it is the configuration where the fifth hole occupies the d and not the p_{2} level. This opens pathways to a final configuration with an empty p_{2} state, as shown in Fig. 2. Together with hole–hole exchange and promotion–demotion effects, this leads to very complex X^{4+} and X^{5+} photoluminescence spectra. The general feature of a principal line at high energy with a multitude of lines in a bandwidth of ∼10 meV is a feature in both experiment and theory. There is clearly a very strong dot dependence: in the theory, elongating the dot induces a considerable change to the X^{4+} and X^{5+} spectra. Similarly, dots A, B, C show marked differences for X^{4+} and X^{5+}. A more detailed understanding of X^{4+} and X^{5+} will probably be possible only with a more detailed picture of the real dot morphology.
Methods
The quantum dots are selfassembled by molecular beam epitaxy and are separated by 17 nm of intrinsic GaAs from an n^{+} GaAs layer. The dots are capped with 10 nm of undoped GaAs followed by a 105 nm AlAs/GaAs superlattice. Ohmic contacts are prepared to the back contact, the earth, after which a 5nmthick NiCr Schottky barrier is evaporated onto the sample surface. Voltage V_{g} is applied to the NiCr Schottky contact. Photoluminescence is excited nonresonantly at 850 nm, collected with a lowtemperature confocal microscope (5 K) and detected with a spectrometer–InGaAs photodiode array setup with a spectral resolution of 120 μeV.
Some photoluminescence lines extend in voltage beyond the host plateau. It is important to make an assignment of the photoluminescence lines to a particular exciton charge. We find that some photoluminescence lines extend beyond the plateau to more positive V_{g} but not to more negative V_{g}. (This is the consequence of a cascade: X^{n−} emission leaves behind n electrons, which then capture a hole and emit as X^{(n−1)−} before the true groundstate configuration, X^{n−}, is reestablished.) In addition, the photoluminescence intensity from a particular exciton decreases abruptly with increasing V_{g} once the exciton no longer forms the ground state. We can determine the V_{g} boundaries of each photoluminescence line to ∼10 mV, enabling us to assign each line to a particular charge without ambiguity, Fig. 4. In Fig. 1 there are some weak biexcitonrelated features but these can be spotted easily through their dependence on pump power.
Tunnelling does not shift the photoluminescence lines appreciably: even at the plateau edges, energy shifts in the photoluminescence through a tunnel hybridization cannot be made out in Fig. 1. However, the photoluminescence energies depend on the electric field, a Stark effect, and we find that the corresponding dipole moment and polarizability depend on charge, particularly for the positively charged excitons^{9}. To facilitate comparison with our theory, we extrapolate in Fig. 4 the photoluminescence spectra to zero electric field. This is crucial as it reveals a redshifted X^{1+}, not a blueshifted X^{1+} as a cursory inspection of Fig. 1 might suggest. None of the splittings in the experimental photoluminescence depend on electric field demonstrating the validity of this method.
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Acknowledgements
The work was supported by EPSRC (UK), US Department of Energy SCBES under Contract No. DEAC3699GO10337 LAB0317, DAAD and SFB 631 (Germany) and SANDiE (EU).
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M.E. carried out the experimental work under the supervision of R.J.W. and K.K; G.B. carried out the theoretical work in A.Z.’s group; A.B. fabricated the heterostructure in P.M.P.’s group. M.E., G.B., K.K., A.Z. and R.J.W. worked jointly on the interpretation of the results.
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Ediger, M., Bester, G., Badolato, A. et al. Peculiar manybody effects revealed in the spectroscopy of highly charged quantum dots. Nature Phys 3, 774–779 (2007). https://doi.org/10.1038/nphys748
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DOI: https://doi.org/10.1038/nphys748
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