Radiative properties of multi-carrier bound excitons in GaAs

The radiative properties of excitons in semiconductors are of fundamental interest in current semiconductor physics as well as of technological interest due to their impact on optoelectronic device performance Chuang (2009). While the optical selection rules for the recombination of a conduction electron and valence hole are well understood Meier and Zakharchenia (1984); Chuang (2009), the selection rules for excitonic complexes with more than two carriers are complicated due to the multiple spin and angular degrees of freedom. In high symmetry environments, exciton lifetimes can be modified by interference between different recombination pathways Efros et al. (1996). This effect in quantum dots, for example, is responsible for the dark exciton, a radiative bottleneck in applications requiring bright sources de Mello Donegá et al. (2006) or alternatively a possible long-lived storage state for quantum information applications Poem et al. (2010). Exciton lifetimes are also modified by reduced symmetry environments, which can remove the possibility of interference by energetically separating the excitonic states.

In this work, we provide a convenient and general framework for describing the optical properties of arbitrary $k=0$ excitonic complexes. We use the second quantization formalism Kira and Koch (2012) for calculating dipole matrix elements of an excitonic complex with an arbitrary number of electrons and holes in a III-V direct band gap semiconductor. Using a generalized Weisskopf-Wigner theory, we show how special spontaneous emission eigenstates and multiple radiative lifetimes may emerge. We predict three radiative lifetimes of the neutral acceptor bound-exciton (A${}^0$X) in bulk GaAs. We confirm the theory by performing polarization dependent and time-resolved photoluminescence experiments on the A${}^0$X system.

We describe the radiative lifetimes of excitons using a generalized Weisskopf-Wigner theory. The spontaneous emission rates from a set of degenerate excited states to a set of degenerate ground states are the eigenvalues of $\alpha S$, where $S=p^{†}\cdot p$ A0X (). Here $p$ is a matrix of the vector dipole matrix elements $p_{ij}=⟨g,i|\mu |e,j⟩$ between ground state $i$ and excited state $j$, $\alpha =\left(1/4\pi ϵ\right)\left(4{\omega }^3{n}^3/3\hslash c^3\right)$, $\omega$ the frequency of the transition, $ϵ$ the permittivity of the material, $n$ the index of refraction, and $c$ the speed of light. The time dependence of the excited state probability amplitudes $b$ satisfy

corresponding to exponential decay. Physically, Eq. 2 implies that radiative lifetimes are modified by constructive or destructive interference between different recombination pathways. In addition, it highlights how exciton states organize into spontaneous emission eigenstates (eigenvectors of $S$) with decay rates given by the eigenvalues of $\alpha S$.

Before applying this formalism to the three-carrier acceptor bound-exciton system, as an example we treat the simpler two-carrier light-hole exciton. Light-hole excitons, consisting of an $m_j=\pm \frac{1}{2}$ valence hole and a conduction electron, split from heavy-hole excitons ($m_j=\pm \frac{3}{2}$) in reduced symmetry environments such as quantum dots, quantum wells and strained GaAs. The formalism yields the expected four spontaneous emission (SE) recombination rates Efros et al. (1996):

The aligned spin $e_{\frac{1}{2}}^{†}{h}_{\frac{1}{2}}^{†}$ and $e_{-\frac{1}{2}}^{†}{h}_{-\frac{1}{2}}^{†}$ excitons decay independently because of the orthogonal polarizations (${\sigma }^{+}$ and ${\sigma }^{-}$) of the transitions. On the other hand, constructive and destructive interference of the $\hat{z}$ recombination pathway leads to a bright and dark exciton. While it is experimentally challenging to observe the brightest light-hole exciton due its $\hat{z}$ polarization, this exciton has recently been observed using magnetic-field measurements in strain-engineered quantum dots Huo et al. (2014). We note that identification could alternatively be made through lifetime measurements at zero magnetic field.

We now turn to the neutral acceptor-bound exciton (A${}^0$X), consisting of two $j=\frac{3}{2}$ holes and one $j=\frac{1}{2}$ electron bound to a substitutional acceptor impurity Bogardus and Bebb (1968); Haynes (1960). By recombination of the electron with one of the holes, A${}^0$X decays radiatively to a neutral acceptor (A${}^0$, a hole bound to an acceptor). Effective mass theory can be used to show that A${}^0$ has hydrogenic levels $1s$, $2s$, etc. In high-purity p-type GaAs, A${}^0$X to A${}^0$ 1s, 2s, etc. photoluminescence (PL) is readily observed and provides a useful probe for resonant excitation, as shown in Fig. 1. Remarkably, the ensemble transition linewidths of this solid-state ensemble system are less than 40 $\mu$eV (spectral linewidths in Fig. 1 are limited by the instrument resolution) A0X ().

Though the origin of the A${}^0$X fine structure was once a controversy, strain experiments support hole-hole and crystal field coupling as the dominant mechanisms for splitting the 12 fold degenerate A${}^0$X Mathieu et al. (1984); Karasyuk et al. (1998). In this scheme, the two holes lie in antisymmetric spin states with total spin 0 and 2 Mathieu et al. (1984). Hole-hole coupling splits the $j=0$ states from $j=2$ states. In zinc-blende semiconductors which possess crystal fields with $T_d$ symmetry, the $j=2$ states further split into two manifolds: ${\Gamma }_5$ with multiplicity 3 and ${\Gamma }_3$ with multiplicity 2. The full specification of A${}^0$X also includes the spin of the electron, denoted as $\uparrow$ or $\downarrow$ (Fig. 2).

Our theory and experiments show that A${}^0$X has multiple radiative lifetimes. To find the A${}^0$X radiative lifetimes, we first compute the dipole matrix elements $p_{ij}=⟨{\text{A}}^0,i|\mu |{\text{A}}^0\text{X},j⟩$ for the different A${}^0$ and A${}^0$X states A0X (). The eigenvalues of $p^{†}\cdot p$ are the radiative recombination rates of A${}^0$X. An energy splitting between excited states causes a fast oscillation in the Weisskopf-Wigner theory which destroys the coupling between non-degenerate states A0X (). As such, only degenerate excited states are included in the dipole matrix when calculating the eigenvalues of $p^{†}\cdot p$.

We measure the polarization dependence of the ${\text{A}}^0-{\text{A}}^0\text{X}$ transition using resonant continuous-wave (CW) excitation. Experiments were performed on a p-type GaAs crystal (6 $\mu$m GaAs grown by molecular beam epitaxy on a GaAs substrate, $N_a=1.2\cdot {10}^{14}{\text{cm}}^{-3}$). The sample was mounted without strain in pumped liquid He (1.9 K) and excited with a Ti:Sapphire laser. Fig. 3a shows the polarization dependence of $|{\text{A}}^0\text{X}⟩\to |{\text{A}}^0,2s⟩$ emission under resonant excitation of $|{\text{A}}^0,1s⟩\to |{\text{A}}^0\text{X}⟩$ with $\hat{y}$ and ${45}^{\circ }$ polarized excitation. The polarization visibility $C=I_{\text{max}}/I_{\text{min}}$ observed is somewhat less than would be expected from the ideal theory. The difference can be explained by relaxation between A${}^0$X spin states.

We investigate the effect of inter-level relaxation on the diminished polarization visibility with time- and polarization-resolved measurements. Either the ${\Gamma }_5$ or ${\Gamma }_3$ transitions were excited resonantly with 2 ps Ti:Sapphire pulses, spectrally filtered to obtain 16 ps pulses with 0.03 nm bandwidth. Photoluminescence to $|{\text{A}}^0,2s⟩$ was collected and imaged using a combined spectrometer/streak camera setup with a timing resolution of 27 ps. Four excitation conditions were studied, resonant excitation of ${\Gamma }_5$ or ${\Gamma }_3$ with $\hat{y}$ or 45${}^{\circ }$ linearly polarized light. PL polarized parallel and perpendicular to the excitation polarization was collected. The complete time-resolved data set is shown in Fig. 3b.

We observe a strong initial polarization visibility at $t=0$ that later decays because of inter-level relaxation. The initial polarization visibility of $C=7.2\pm 1$ for ${\Gamma }_5-y$ excitation is close to the ideal value 9 for no excited state relaxation. (The uncertainty here is due to the uncertainty in the $t=0$ time.) The decay of polarization visibility indicates the existence of spin flip processes on the same timescale as the radiative lifetime.

We use the time-resolved data to obtain estimates of the inter- and intra-level relaxation rates in the exciton system (Fig. 3b). The time resolved data were fit to a 12 state density matrix model including inter-state relaxation A0X (). In the model, an optical pulse of a given polarization coherently creates an excited state density matrix given by Eq. 16. The subsequent time evolution of the excited state density matrix $\rho$ satisfies

where $H_o$ is a diagonal matrix of the excited state energies, the second term describes radiative recombination and $L\left(\rho \right)$ is the Linbladian operator describing phenomenological relaxation between excited states A0X (). From the solution $\rho \left(t\right)$ we calculate the relative PL intensity emitted into different polarizations using Eq. 4.

This model gives a good fit to the observed time dependence of A${}^0$X emission (Fig. 3b). The 16 curves in Fig. 3b are fit simultaneously using 6 fit parameters: overall spontaneous emission rate (1.48 ns${}^{-1}$), inter-level relaxation (0.89 ns${}^{-1}$), intralevel ${\Gamma }_3$ relaxation (3.6 ns${}^{-1}$), intralevel ${\Gamma }_5$ relaxation (1.8 ns${}^{-1}$), temperature (4.7 K) and overall intensity normalization (5100 counts). These relaxation rates are shown schematically in Fig 2. The resulting A${}^0$X radiative lifetimes are $T_o(1,3,\frac{3}{4})$ where $T_o$ lies in the range 0.49 to 0.74 ns. A detailed error analysis found the main source of uncertainty in the spontaneous emission rate to be due to an ambiguity in the choice of the background level A0X (). Since hole spin flips are predicted to be much faster than electron spin flips Fu et al. (2006); Kroutvar et al. (2004), we do not include electron spin flip processes in the model (shown schematically in Fig. 2). Temperature was included as a fit parameter because the effective temperatures for bound excitons can be larger than the bath temperature Rühle and Klingenstein (1978).

Using the best fit model parameters from the time resolved experiment, we are now able to predict the polarization dependence of PL in resonant CW excitation in the presence of spin relaxation. These curves are shown in Fig. 3a as “theory,” and agree well with the experimental data.

In conclusion, we presented a convenient and general formalism for calculating the optical properties of $k=0$ excitons in III-V semiconductors with an arbitrary number of carriers. We used this formalism to derive a model of the optical properties of A${}^0$X in strain-free bulk GaAs which predicts 3 distinct radiative lifetimes. The model was confirmed using polarization and time-resolved experiments. The results are in contrast to previous reports for this system and highlight the importance of a unified treatment of all recombination pathways when deriving the radiative properties of multi-carrier excitons.

This material is based upon work supported by the National Science Foundation under Grant No. 1150647, DGE-0718124 and DGE-1256082. We would like to thank T. Saku for growing the material in NTT. YH acknowledges support from SORST and ERATO programs by JST.