Frequency dependence of the radiative decay rate of excitons in self-assembled quantum dots: Experiment and theory

We analyze time-resolved spontaneous emission from excitons confined in self-assembled InAs quantum dots placed at various distances to a semiconductor-air interface. The modification of the local density of optical states due to the proximity of the interface enables unambiguous determination of the radiative and nonradiative decay rates of the excitons. From measurements at various emission energies, we obtain the frequency dependence of the radiative decay rate, which is only revealed due to the separation of the radiative and nonradiative parts. It contains detailed information about the dependence of the exciton wave function on quantum dot size. We derive the quantum optics theory of a solid-state emitter in an inhomogeneous environment and compare this theory to our experimental results. Using this model, we extract the frequency dependence of the overlap between the electron and hole wave functions. We furthermore discuss three models of quantum dot strain and compare the measured wave-function overlap to these models. The observed frequency dependence of the wave-function overlap can be understood qualitatively in terms of the different compressibility of electrons and holes originating from their different effective masses and binding energies.

Figure 1

(a) Schematic band diagram illustrating the spontaneous emission process in a QD. An electron is excited optically from a valence band to a conduction-band wetting-layer (WL) state and the generated electron and hole relax to the lowest-energy QD state on a picosecond time scale. The electron can subsequently decay by either radiative or nonradiative recombination with rates ${\Gamma }_{\text{rad}}$ and ${\Gamma }_{\text{nrad}}$, respectively. (b) Schematic illustration of the sample under investigation. InAs QDs are embedded in GaAs and positioned at different distances $z$ to the GaAs-air interface. (c) The LDOS as a function of distance $z$ to a GaAs-air interface for a dipole orientation parallel (solid curve) or perpendicular (dashed curve) to the interface.

Figure 2

(Color online) (a) Topographic atomic force micrograph depicting a surface area of $1\times 1$ μ${\text{m}}^2$ of uncapped QDs on the unprocessed wafer. The color scale runs from 0 to 20 nm. (b) Histogram of the QD height measured by analysis of the AFM data. The blue line is a fit to the histogram data using a log-normal distribution as discussed in the text.

Figure 3

Illustration of the optical measurement setup. The sample is kept at 14 K in a cryostat and illuminated by a pulsed laser. The spontaneously emitted light is collected and can be directed either to a charge-coupled device camera for sample alignment or to a spectrometer equipped with a fast single-photon counting avalanche photodiode (APD) for time-resolved measurements.

Figure 4

(Color online) (a) Measured fast decay rate (black and red squares) at an emission energy of 1.204 eV versus the normalized LDOS for a dipole orientation parallel to the semiconductor-air interface. The red line shows a linear fit, where only the black points have been included. The red points have been omitted because at positions close to the interface a systematic deviation from theory was observed, as quantified by the data in the lower panel. The slow decay rate (black triangles) is independent of the LDOS and is therefore dominated by nonradiative decay. The blue horizontal line shows the average nonradiative rate ${\Gamma }_{\text{nrad}}=0.096{\text{ns}}^{-1}$. (b) Correlation between data and theory for the linear regression of the radiative rate when systematically excluding the points nearest the interface in the fit. The correlation parameter converges to a value close to unity (blue line) when seven points (marked in red) have been excluded.

Figure 5

(Color online) Decay rates as a function of distance to the interface for six different emission energies. Each curve has been vertically offset by $0.1{\text{ns}}^{-1}$ for visual clarity. The fits have been obtained using a systematic exclusion of data points near the interface [cf. Fig. 4b]. The emission energies are 1.170 eV (solid black squares), 1.187 eV (open black squares), 1.204 eV (solid red circles), 1.216 eV (open red circles), 1.252 eV (solid blue diamonds), and 1.272 eV (open blue diamonds). For all emission energies, we note the excellent agreement with theory for $z>75\text{nm}$.

Figure 6

(Color online) The total decay rate (red squares), radiative decay rate (black triangles), and nonradiative decay rate (blue circles) as a function of QD emission energy. The radiative rate decreases with energy but the nonradiative rate simultaneously increases so that the total (measured) decay rate increases with increasing energy. The solid black line is the result of the theoretical model of the radiative decay rate using an aspect ratio of 1/2, omitting the wetting layer, and for an indium mole fraction of 0.95. The dashed blue curve shows the theoretical model of the nonradiative decay. The dash-dotted red line shows the total decay rate given by the sum of the two theoretical curves. The top scale shows the heights used in the calculations. The details of the calculation of the nonradiative rate are discussed in the text and the calculation of the radiative rate is discussed in Sec. 5.

Figure 7

(Color online) (a) The fast decay rate obtained from measurements on the unprocessed wafer (solid black squares) and the total decay rate in a homogenous medium (open red circles) obtained using the rigorous separation of radiative and nonradiative homogeneous decay rates. Evidently, the frequency dependence of the normalized LDOS (solid blue line) for the unprocessed wafer $\left(z=302\text{nm}\right)$ results in faster decay rates for QDs on the unprocessed wafer than would be the case in a homogeneous medium. (b) Normalized inhomogeneously broadened emission spectrum of the QDs obtained under weak excitation.

Figure 8

Schematic illustration of the QD geometry used to calculate the envelope wave functions. We consider a lens-shaped QD with lateral base diameter $D$ and height $h$ consisting of ${\text{In}}_c{\text{Ga}}_{1-c}\text{As}$ on a WL. The symmetry axis $z$ is indicated along with the radial directions $x,y$.

Figure 9

(Color online) (a) Measured spontaneous emission spectrum (black curve, normalized) and calculated spectrum (dashed red curve) using the height distribution of the QDs from Fig. 2b and implementing strain model 1. The resulting parameters have been optimized to fit the spectrum, and we find an aspect ratio of 1/6 and an indium mole fraction in the QD of 39%. (b) Measured electron-hole wave-function overlap (black squares) and the theoretical calculation (solid red line) using the same parameters as in (a).

Figure 10

(Color online) Calculated energy dependence of the squared electron and hole wave-function overlap using the three different strain models discussed in the text. The QD aspect ratio is 1/2 and no wetting layer was included. The curves are calculated for strain model 1 with $c=0.51$ (dash-dotted blue line), strain model 2 with $c=0.95$ (solid red line), and strain model 3 with $c=0.46$ (dashed black line). The data points show the experimentally determined wave-function overlap for these three indium mole fractions using the same color coding. Clearly, strain model 1 does not describe the experimental data well, while both models 2 and 3 lead to very good agreement. The inset shows the dependence of the QD emission energy on height for the three different models using the same color coding as in the main figure.

Figure 11

(Color online) The effect of the thickness of the wetting layer on the calculated electron-hole wave-function overlap when varying between 0 and 6 ML in steps of 1 ML. Here we have omitted strain and otherwise used the same parameters as in Fig. 10. The experimental data for $c=0.46$ are shown as black squares.

Figure 12

(Color online) Contour plots in the radial plane of the amplitude of the wave function for electrons and holes. The parameters corresponding to the calculation for strain model 2 in Fig. 10 were used. (a) Electron wave function for a QD height of 4 nm. (b) Hole wave function for a QD height of 4 nm. (c) Electron wave function for a QD height of 8 nm. (d) Hole wave function for a QD height of 8 nm.

Figure 13

Schematic illustration of the effect of strain on the band energies at $\mathbf{q}=\mathbf{0}$. (a) No strain and (b) biaxial strain. Biaxial strain shifts both the conduction band and the heavy-hole valence band closer to the continuum. The net effect is an increase in the band gap and therefore in the transition energy. Importantly, the degeneracy of the light and heavy-hole states is lifted and the lowest-energy transition involves heavy holes only.