Optical signatures of valence-band mixing in positive trion recombination spectra of double quantum dots

We consider optical signatures of valence band mixing in positive trion and exciton complexes in vertically stacked InGaAs quantum dots. We use the configuration interaction method and an axially symmetric four-band Luttinger-Kohn Hamiltonian (KL) that allows for heavy-hole and light-hole band mixing due to spin-orbit interaction. A scalar effective hole mass model is also included for comparison. We found essential differences (i.e., different recombination patterns) between the KL and separated-bands model spectra. In the weak-coupling regime for KL model, we obtained a good agreement with experimentally observed X patterns in contrast to the scalar effective mass model.

Figure 1

(a) Definition of terms united, dissociated, and disrupted that are used in the text for characterization of the trion initial states and recombination lines. (b) Color line sequence used in spectra figures. Dash-dotted lines denote the initial trion state (singlets or triplets), dash lines denote the final hole states.

Figure 2

Energy spectrum and recombination probabilities (marked as line width) for positive trion and exciton vs electric field intensity parallel to $z$ axis. The case of asymmetric dots with barrier width $D=4.1$ nm: (a) for the KL Hamiltonian and (b) for the unmixed HH and LH bands Hamiltonian. Red color indicates the exciton recombination probability and black is for the trion recombination to hole ground state. Particle localization diagrams are described in the text. A letter “s” denotes a singlet initial state, “t” denotes a triplet state.

Figure 3

Energy difference between two lowest-energy singlet states (green line) and between two lowest-energy nondegenerated hole final states (red line) for an asymmetric system, 4.1 nm barrier. Each blue line marks a tunneling interval of the corresponding particle. (a) and (b) For the KL Hamiltonian. (c) and (d) For the unmixed HH and LH band Hamiltonians. See text for details.

Figure 4

Fragments of the corresponding spectra from Fig. 2 with tunneling intervals from Fig. 3 marked by vertical lines: green for trion initial state hole, red for final state hole.

Figure 5

Same as Fig. 2 only for interdot barrier of $7.1$ nm. Blue color is for trion recombination to a hole first excited state and the other colors are for successive excited final hole states.

Figure 6

Same as Fig. 3 only for interdot barrier of 7.1 nm.

Figure 7

Fragments of corresponding spectra from Fig. 5 with tunneling intervals from Fig. 6 marked by vertical lines: green for trion state hole, red for final state hole.

Figure 8

Same as Fig. 2 only for interdot barrier of 10.1 nm. Blue color is for a trion recombination to a hole first excited state and the other colors are for successive excited final hole states. The dotted line indicates the region presented in Fig. 10.

Figure 9

Same as Fig. 3 only for interdot barrier of 10.1 nm. The top scale in (b) is for the green line and the bottom scale is for the red line.

Figure 10

(a)–(c) Fragments of the corresponding spectra from Fig. 8 with tunneling intervals from Fig. 9 marked by vertical lines: green for a trion state hole, red for a final state hole. (a) is for KL, (b) and (c) are for separated-bands models. Left and right energy scales in (b) and (c) describe bottom and top halves of the relevant picture, respectively. The linewidth is proportional to the square root of the recombination probability. (d) A fragment of the spectrum in Fig. 8 without recombination probabilities.

Figure 11

Energy spectrum and recombination probabilities (marked as linewidth) for positive trion (black) and exciton (red lines) vs electric field intensity parallel to $z$ axis. The case of symmetric dots with barrier width $D=4.1$ nm. (a) For KL Hamiltonian and (b) for unmixed HH and LH bands Hamiltonian. Red color indicates the exciton recombination, black is for trion recombination to hole ground state. Particle localization diagrams are described in the text. “S” denotes a singlet state, “t” denotes a triplet state.

Figure 12

Same as Fig. 11 only for interdot barrier of 7.1 nm.

Figure 13

Same as Fig. 11 only for interdot barrier of 10.1 nm.