Unraveling the mesoscopic character of quantum dots in nanophotonics

We provide a microscopic theory for semiconductor quantum dots that explains the pronounced deviations from the prevalent point-dipole description that were recently observed in spectroscopic experiments on quantum dots in photonic nanostructures. At the microscopic level the deviations originate from structural inhomogeneities generating a large circular quantum current density that flows inside the quantum dot over mesoscopic length scales. The model is supported by the experimental data, where a strong variation of the multipolar moments across the emission spectrum of quantum dots is observed. Our work enriches the physical understanding of quantum dots and is of significance for the fields of nanophotonics, quantum photonics, and quantum-information science, where quantum dots are actively employed.

We provide a microscopic theory for semiconductor quantum dots that explains the pronounced deviations from the prevalent point-dipole description that were recently observed in spectroscopic experiments on quantum dots in photonic nanostructures. At the microscopic level the deviations originate from structural inhomogeneities generating a large circular quantum current density that flows inside the quantum dot over mesoscopic length scales. The model is supported by the experimental data, where a strong variation of the multipolar moments across the emission spectrum of quantum dots is observed. Our work enriches the physical understanding of quantum dots and is of significance for the fields of nanophotonics, quantum photonics, and quantum-information science, where quantum dots are actively employed.
Semiconductor quantum dots (QDs) are compatible with semiconductor technology and exhibit optically active transitions with high quantum efficiency, which renders them promising single-photon sources for a range of solid-state quantum-optical devices 1 . For instance, strong coupling between a QD and a cavity photon 2 or near-unity coupling to a photonic-crystal waveguide have been demonstrated 3 . Furthermore, QDs have been proposed as the enabling part of highly efficient solar cells 4 and as central nodes in future quantum-information systems with entangled stationary and flying quantum bits 5 . All of these applications require a profound understanding of their optical properties. The latter was recently challenged by the observation that QDs can break the dipole approximation and was explained by a phenomenologically defined mesoscopic moment 6 , which probes the multipolar electromagnetic vacuum field 7 . So far, the physical origin of these mesoscopic quantum effects has not been established. Here we present a microscopic theory of the QD wavefunctions that provides physical insight into the mesoscopic character of QDs. We find that the microscopic quantum current density flows along a curved path inside the QD, cf. Fig. 1(b), which generates large electric and magnetic moments leading to light-matter interaction of both electric and magnetic character 8 . The mesoscopic moment, which includes the magnetic light-matter interaction, can be tuned over orders of magnitude by controlling the size and shape of QDs. The theoretical model is supported by comparing to experimental data on time-resolved spectroscopy of QDs positioned near a dielectric interface in a Drexhage geometry 9 (cf. Fig. 1), where a strong variation of the mesoscopic moment with emission energy is found for the case of In(Ga)As self-assembled QDs.
A central quantity describing the optical transition from the excited state Ψe to the ground state Ψg of a QD is the dipole moment µ = (e/m0) Ψg|p|Ψe , where e and m0 are the elementary charge and electron mass, respectively. We consider the x-polarized dipole moment of the exciton µ = µx, as sketched in Fig. 1, wherex is the Cartesian unit vector. Until recently, µ was the only QD property used to describe the interaction with light. Recent experimental studies of spontaneous emission from QDs near a silver interface are, however, incompatible with such a description 6 and a more extended description is required. So far, the decay dynamics beyond the dipole approximation was described by a phenomenological mesoscopic moment Λ = (e/m0) Ψg|xpz|Ψe 7 . Combined with the microscopically well-understood dipole x y z Figure 1: Unraveling the mesoscopic character of QDs in the vicinity of a GaAs-air interface. The presence of the interface breaks the parity symmetry of the environment in the z-direction. Since reflections occur at the interface (the circular white arrow), the imaginary part of the electric field E(r) generated by the electric-dipole component, which triggers spontaneous emission, inherits this lack of symmetry and is curved (indicated by the green arrow). (a) In the dipole approximation, the QD microscopic current j(r) (brown arrow) perceives only the parallel component but not the out-of-plane component (the "curvature") of the electromagnetic field at its position. (b) In In(Ga)As self-assembled QDs, the current density flows along a curved path that resembles the shape of the field environment thereby exchanging energy more efficiently with it. As a consequence, the spontaneous-emission decay rate is enhanced and the photons (red arrows) are emitted at a faster rate compared to the case in (a). moment, this quantity accounts for the interaction with light caused by the extended mesoscopic nature of QDs. Previous theories have investigated mesoscopic effects at the level of the QD spatial extent and symmetry, and have discarded their atomistic nature because the unit cells are small compared to the wavelength of light [10][11][12][13][14][15][16][17] . These approaches fail to explain the large mesoscopic moment Λ observed experimentally. In the present work we show that, surprisingly, the atomistic character plays a crucial role and explains the physical origin of the mesoscopic moment. We find that the latter is caused by the change in the periodicity of the underlying crystal lattice in the QD. Since Bloch functions with different periodicities cannot remain in phase throughout the QD, this necessarily leads to a phase gradient and a resulting current in the growth direction of the QD, which gives rise to the mesoscopic moment, cf. Fig. 1   magnetic-field fluctuations simultaneously. To obtain an optimized light-matter coupling, the electromagnetic environment must be shaped similarly to the flow of the QD current, as sketched in Fig. 1(b), where we exemplify the case of a GaAsair interface. The experimental data demonstrating this effect are presented in the following. Figure 2(a) displays the measured spontaneous-emission decay rates for ensembles of QDs that are placed at different distances to a GaAs-air interface. The experimental data were published in Refs. 18,19, where the data recorded at distances above 75 nm were employed to reliably extract the dipole contribution to the spontaneous-emission process. We observe a systematic deviation from dipole theory at distances below ∼ 75 nm, which was previously speculated to be a result of enhanced loss processes at the etched interfaces 18 . These deviations can instead be explained by the contributions from the mesoscopic moment Λ to the light-matter interaction strength, and we are able to quantitatively reproduce the functional dependence observed in the experiment for all emission energies, as exemplified in Fig. 2(a), see Supplementary Information for details. The extracted mesoscopic strength Λ/µ increases with emission energy and varies from 10 to 23 nm over the inhomogeneously broadened emission spectrum, cf. Fig. 2(b), which reflects a pronounced dependence on QD size. In the following we develop a theory that quantitatively explains the experimental data of Fig. 2 Numerous band structure models have been proposed for QDs ranging from continuum approaches (e.g., the multiband k.p theory 20 ), which discard the QD atomistic nature and consider only the macroscopic potential, to atomistic models (e.g., the empirical pseudopotential theory 21 ), which simulate the contribution of every single atom comprising the QD. In the present work, it suffices to consider the simplest band structure model, i.e., the two-band effective-mass theory. We show that, by including a small extension of it, the mesoscopic nature of QDs can be explained remarkably well. In this picture, the QD single-particle wavefunction Ψ(r) can be written as a product of a periodic Bloch function u(r), which captures the properties on the length scale of the crystal unit cell, and a slowly varying envelope ψ(r) inheriting the size and symmetry of the QD potential, i.e., Ψi(r) = ui(r)ψi(r) with i = {g, e} corresponding to either the ground or excited state, respectively. We consider the x-polarized exciton with the valence-band heavyhole Bloch function ug = ux 22 inheriting the atomic px symmetry. The standard textbook approach for evaluating the transition dipole moment µ is to assume that the envelope function varies slowly over a unit cell so that µ can be written as a product of the Bloch matrix element pcv and a threedimensional overlap integral between the envelope functions, i.e., µ = (e/m0) uxψg|px|ueψe ≈ (e/m0)pcv ψg|ψe , where pcv = V −1 UC UC d 3 ru * xpx ue is given by an integral over the unit cell with VUC being the unit-cell volume. In other words, the transition dipole moment is primarily a unit-cell effect and is marginally affected by the envelope functions, as their overlap is normally close to unity 18 . Importantly, the large mesoscopic strength Λ/µ observed experimentally (see Fig. 2(b)) cannot be reproduced by a similar calculation, which leads to where <>UC≡ V −1 UC UC d 3 r denotes integration over a unit cell. The first three contributions vanish for symmetry reasons. The fourth contribution is vanishingly small and does not scale with the QD size: for Gaussian envelopes allowing for realistic mutual displacements of 1-2 nm between the electron and the hole in the growth direction (note that the integral vanishes in the absence of such a displacement) we estimate Λ/µ ∼ 10 −4 nm. This suggests that the large mesoscopic strength Λ/µ ∼ 10-20 nm observed experimentally cannot be explained solely by the envelope wavefunctions. In the following we show that structural gradients at the nanoscopic crystal-lattice length scale can explain the effect.
It is often assumed that solid-state emitters have a homogeneous chemical composition, which renders substantial simplifications in the computation of the wavefunctions. In particular, the homogeneity justifies the use of bulk-material Bloch functions, and only the slowly varying envelopes describe the properties of the nanostructure. This assumption works excellently for quantum wells and lattice-matched QDs, where the structures are either strain free or pseudomorphically grown on the substrate material. As a result, the wavefunctions are confined to a chemically homogeneous region of space. InAs QDs are grown by self-assembly induced by strain relaxation, which unavoidably leads to the generation of chemical gradients at the crystal-lattice level. In particular, large lattice-constant shifts were observed in the growth direction of QDs 23,24 . This limits the applicability of the standard envelope-function theories and, in particular, of the effectivemass formalism. A complete theory encompassing the spatial position and symmetry of every single atom comprising the QD would generally be required. Remarkably, the essential physics of the mesoscopic light-matter interaction can be captured by only a minor extension of the effective-mass theory. We assume that the lattice periodicity changes at a certain position z = zT along the QD height by an amount ∆a l = 110 pm at a central value a l = 605 pm as found experimentally in Ref. 23, see Fig. 3(a). This corresponds to a relative lattice-constant shift of 18%, which is strain induced and is substantially larger than the lattice-constant mismatch between InAs and GaAs of 7%. We note that, in general, the lattice periodicity changes twice: first it is expanded at the QD base (GaAs-In(Ga)As transition) before being shrunk back at the QD tip (In(Ga)As-GaAs transition). Since the exciton is spatially confined near the tip where the indium concentration is highest 24 , we only consider the second transition region. The Bloch functions change periodicity as well, cf. Fig. 3(b), and we model this by expanding them in a Fourier series with a position-dependent lattice wavevector k l (z) This Ansatz ensures opposite parity of the conduction-and valence-band Bloch functions along x. Furthermore, we implicitly assume the shape of the Bloch functions to remain the same, and only their periodicity to vary spatially. Now we return to the evaluation of the mesoscopic moment and separate the slowly-and rapidly-varying contributions as where Rq denotes the position of the q-th unit cell and N is the total number of unit cells in the QD. In a homogeneous region of the QD (the blue unit cell in Fig. 3(a)) the unit-cell integrand of Eq. (3) is odd in x-and z-directions, cf. Fig. 3(c), which leads to a vanishing integral. However, in the transition region around z = zT strong gradients are present, which destroy the parity of the integrand (see the pink and green unit cells in Fig. 3(a,c)) and generate a substantial contribution to Λ.
In the following, we calculate the mesoscopic moment Λ and show that its magnitude is sensitive to the QD geometry. With the Ansatz in Eq. (2) we first compute the dipole Bloch matrix element px and then evaluate Λ, which can be expressed in terms of px because the x-and z-derivatives of the Bloch functions yield similar results, cf. Eqs. (2-3). The resulting expression for Λ/µ reads We have thus been able to express a crystal-lattice effect in terms of the slowly varying envelope functions. The mesoscopic strength scales quadratically with the in-plane size of the QD, Λ/µ ∼ L 2 r , because the term ψg x 2 [∂zk l (z)] ψe contains the variance of the exciton wavefunction in the xdirection. Moreover, it increases with decreasing QD height, Λ/µ ∼ L −1 z , since in shallow QDs the relative importance of the lattice-constant transition region is increased. We use Eq. (4) to model the spectral dependence of Λ/µ, see Fig. 2(b), where only the height of QDs is assumed to vary across the spectrum while the in-plane size remains constant. This assumption is supported by studies of size and shape performed on self-assembled QDs 25 , where a small relative distribution of the in-plane QD size is observed. By mapping the quantization energy to the QD size, we are able to extract a QD height that varies from 11 nm to 3 nm across the inhomogeneously broadened spectrum, cf. Fig. 4(a), which agrees well with the values obtained from atomic-force microscopy measurements 19 . In the present work we assume zT = 0, i.e., the transition region is situated at the center of the QD slowly varying envelopes. The details of the calculation are given in the Supplementary Information. From Fig. 2(b) we conclude that QDs with larger emission energy have larger mesoscopic strengths because they are shallow so that a large part of the excitonic wavefunction is affected by the lattice inhomogeneity.
The existence of deviations from the dipole theory even in simple weakly confining dielectric structures (cf. Fig. 2) points to their relevance in any nanophotonic structure that breaks parity symmetry along the QD height. Motivated by these findings, we derive a simplified analytic relation between the QD geometry and the mesoscopic strength for in-plane rotationally symmetric Gaussian slowly varying envelopes where σz is the height (HWHM) of the electron envelope, σr the QD radius, ∆a l /a l the relative lattice-constant shift and ξz ≈ 5 is the ratio between the electron and hole effective masses 26 . We plot the mesoscopic strength as a function of the in-plane radius for three fixed heights in Fig. 4(b). The largest mesoscopic strengths are achieved by shallow and wide (diskshaped) QDs. For instance, taking a relative extreme case of a height of σz = 2 nm and a radius of σr = 30 nm yields a mesoscopic strength as large as Λ/µ = 150 nm, which is an order of magnitude larger than the values observed in experiments so far. Such QDs would constitute a mesoscopic entity in which mesoscopic effects may dominate the light-matter interaction strength. For instance, a QD with Λ/µ = 150 nm placed in front of a silver mirror would exhibit a Purcell factor that is nearly 100 times larger than the case of a point-dipole source. Aside from this, such QDs may be extremely efficient at interfacing both electric and magnetic degrees of freedom also in structures that conserve parity symmetry, such as photoniccrystal cavities and waveguides, owing to the substantial increase of second-order light-matter-interaction processes that are weak for current In(Ga)As QDs.
Knowledge about the quantum-mechanical wavefunctions allows computing the current density jQD(r) flowing through the QD. We define the latter by comparing the interaction HamiltonianĤint = (e/m0)A ·p, where A is the vector potential, to the classical particle-field interaction Hamiltonian Hint = A(r) · j(r) 9 . The quantum-mechanical current density can therefore be written as jQD(r) = (e/m0)Ψg(r)pΨe(r) or In the following, Gaussian slowly varying envelopes are used to model the current density. In QDs with a homogeneous crystal lattice and thus negligible mesoscopic moment, the current density flows only along the direction of the dipole moment because there are no gradients in the z-direction and the second term from the right-hand side of Eq. (6) vanishes (see Fig. 4(c)). Note that for simplicity we ignore the modulation of jQD by the periodic Bloch functions in Fig. 4. The presence of lattice inhomogeneities changes the flow dramatically because strong gradients in the z-direction arise. The current density flows along a curved path as illustrated in Figs. 4(d-f), conferring pronounced mesoscopic properties to QDs. The wider the QD is, the sharper the transverse oscillations of the current are and the larger Λ/µ is. This effect offers the possibility to enhance (diminish) the light-matter interaction by placing QDs in environments where the electric vacuum field exhibits gradients with the same (opposite) sign, see also Fig. 1. This opens new opportunities for designing efficient light-matter interfaces that exploit mesoscopic effects to enhance the interaction. Aside from the local light-matter coupling strength, other degrees of freedom could be potentially tailored by exploiting the mesoscopic interaction, such as the photon-emission directionality or polarization.
In conclusion, we have developed a novel microscopic model that successfully explains the large mesoscopic strengths of In(Ga)As QDs observed experimentally. We find the effect to be governed by the lack of symmetry of the nanoscopic crystal lattice and scaled by the extended mesoscopic size of the QD. The microscopic current density oscillates along a non-trivial curved path and can be expressed as a superposition between electric-dipole, magnetic-dipole, and electric-quadrupole moments. This mesoscopic current is generated at the unit-cell level in analogy to the generation of currents in macroscopic systems. Our work deepens the physical understanding of semiconductor QDs and we therefore expect it to be of significance for the active fields of solid-state quantum electrodynamics and quantum-information processing, where efficient quantum interfaces between QDs and light are exploited.

Supplementary Information
Here we outline the details of the theory used to analyze the experimental data of Fig. 2. We start with Fig. 2(a), where the spontaneous-emission decay rates as a function of the distance z0 to the air interface are plotted. The decay rate beyond the dipole approximation can be decomposed into three decay channels Γ = Γ (0) + Γ (1) + Γ (2) with 7 where Gij(r, r ) is the electromagnetic Green's tensor, which is evaluated with the help of Ref. 27, C = 2e 2 / 0 m 2 0 c 2 0 , e is the elementary charge, m0 the electron mass, 0 the vacuum permittivity, and c0 the vacuum speed of light. We set Λ and µ as free parameters and fit the experimental data with the resulting dependence plotted in Fig. 2(a). It should be mentioned that a data point observed at a distance of 20 nm from the GaAs-air interface is not shown in Fig. 2(a) and is omitted from the analysis because it shows a much higher decay rate and lower photoluminescence intensity, which is likely to be caused by nonradiative tunneling of the QD charge carriers to surface states 19 . A phenomenological distanceindependent loss rate is added to Γ to account for intrinsic nonradiative decay channels within the QD but is found to be negligibly small in the present analysis. This procedure is used independently for every emission energy resulting in the data points in Fig. 2(b). The dipole-theory fit is performed by setting Λ = 0 and excluding the first six data points from the analysis. For completeness, we plot distance-dependent decay rates for various mesoscopic strengths in Fig. 5. It can be seen that both the amplitude and the phase of the oscillations are substantially affected at distances smaller than ∼ 100 nm from the interface, which allows extracting the mesoscopic strength from the experimental data. It is interesting to note that the effect is more dramatic for QDs flipped upside down for which Λ/µ < 0 as indicated by the red curves in Fig. 5. In Ref. 6 this effect was used to demonstrate the existence of the mesoscopic moment.  In the following we present the analysis of the experimental data from Fig. 2(b) using the theoretical model developed in the paper. In the experiment we map the mesoscopic strength as a function of emission energy E. Equation (4) implicitly describes the mesoscopic strength versus the QD size, as explained in the text. We therefore convert the QD size into a quantization energy and in this regard make several assumptions. First, we consider the inhomogeneously broadened spectrum to be caused only by the random distribution of the size (and consequently of the quantization/emission energy) of QDs. Other parameters such as strain distribution or chemical composition are considered constant over the emission spectrum. Second, we assume that only the height of the QDs contributes to the quantization energy, since atomic-force microscopy measurements show that the height of self-assembled In(Ga)As is generally much smaller than the in-plane size 24 . Third, we assume a sharp transition in the lattice constant ∂zk l = ∆k l δ(z − zT ) for simplicity. This approximation is excellent because ∂zk l is multiplied with a slowly varying integrand in Eq. (4), see also Fig. 3(a). Fourth, we assume disk-shaped wavefunctions that can be decomposed into in-plane Φ and out-of-plane φ components, i.e., ψ(r) = Φ(x, y)φ(z). With this, Eq. (4) can be rewritten The first term denotes the relative change in the lattice constant while the second and third terms contain the dependence on the in-plane QD size and QD height, respectively. We find the functional dependence, f , between the third term and the quantization energy, E − E0, using a finite-potentialwell model, where E is the emission energy and E0 the bulk band gap of the QD material. We therefore obtain where S = −(∆a l /a l ) Φg x 2 Φe / Φg|Φe . The trend of the experimental data from Fig. 2(b), i.e., that Λ/µ increases with energy, agrees very well with our model (see the theory curve in the same figure), if the in-plane QD size is constant across the emission spectrum and only the height varies. This behavior has been reported in the literature in studies of QD size and shape using similar growth conditions as used here 25 . We note that if the QDs had a constant aspect ratio, the mesoscopic strength would be predicted to decrease with energy because the effect depends stronger on the in-plane QD size, cf. Eq. (5). We note, that some degree of correlation between height and width has been observed 28 , but our study suggests that such correlations are small in our sample.
We also stress that our study deals with the in-plane size of the QD wavefunctions, which generally can be different than the QD size measured by surface-profile techniques. We use S and E0 as fitting parameters and obtain a bulk band gap E0 = 1.13 eV of the QD, which yields quantization energies ranging from 40 meV up to 140 meV across the emission spectrum. The resulting curve in Fig. 2(b) agrees well with the experimental results.