Biexciton spin and angular momentum transitions in vertically coupled quantum dots

An “exact” numerical calculation is presented for the energy spectrum of biexcitons, with spatially separated electrons and holes, in coupled quantum dots in the presence of a perpendicular magnetic field. When the interdot distance is small the two electrons and two holes form a strongly correlated complex whose spin state undergoes characteristic transitions with increasing interdot separation and magnetic field. For the smallest distances and magnetic fields the biexciton is bound with respect to dissociation into two excitons. At larger distances we observe angular momentum transitions with sawtooth-shaped phase boundaries. The biexciton diamagnetic shift shows a sign reversal as a function of the interdot distance.

Figure 1

Magnetic-field evolution of four possible spin configurations in the subspaces of the total angular momentum $L=0$ (a) and $L=1$ (b) in the absence of the Zeeman interaction $(g_e^{*}=g_h^{*}=0)$. The different states are identified by a pair of numbers denoting singlet (“1”) and triplet (“3”) states of, respectively, electrons and holes. The inset shows the schemes of level rearrangement. The ground-state configuration switches from 11 to 33 at around $2.2\mathrm{T}$ while in the subspace $L=1$ there is a transition from 13 to 31 at $4.3\mathrm{T}$.

Figure 2

Phase diagram of biexciton spin-multiplicity transitions in the absence of the Zeeman term. The full lines correspond to the subspace $L=0$, and these transitions are the actual ground-state transitions. With increasing magnetic field and/or interdot distance, at the lower boundary the configuration switches from 11 to 33 and switches back to 11 at the higher boundary. The dotted lines depict the transitions from state 13 to 31 and vice versa in the $L=1$ subspace. The hash-marked area close to the origin indicates the parameter range where the biexciton binding energy is positive.

Figure 3

The probabilities $p_m$ to find the holes in a state of total angular momentum $m=0,\dots ,5$ for the biexciton state $L=0$. The electrons are found in states of total angular momentum $-m$ with the same probabilities. (a) corresponds to the spin configuration 11, and (b) is plotted for the configuration 33. Note that only at the lowest magnetic fields a predominant value of angular momentum can be identified. The interdot distance is $d=0.7a_B^{*}$.

Figure 4

The same as in Fig. 1 but with the effects of the Zeeman interaction. The most conspicuous transition $11\to 33$ survives and is slightly shifted towards lower fields.

Figure 5

Electron (full line) and hole (dashed line) charge density distribution in the bound ground state at the interdot separation $d=0.1a_B^{*}$ and magnetic field $B=3\mathrm{T}$. Stronger lateral binding of the holes is apparent. For comparison, we show also the charge distribution in decoupled $(d=100a_B^{*})$ dots at the same magnetic field.

Figure 6

Electron-hole correlation function—the conditional probability to find an electron when the two holes are fixed opposite to each other at a distance (a) $0.5a_B^{*}$, (b) $a_B^{*}$, (c) $1.5a_B^{*}$, and (d) $2.5a_B^{*}$ from the center. The positions of holes are marked with white crosses.

Figure 7

The ground-state angular momentum phase diagram. The full lines denote the boundaries between the states of different angular momenta given by numbers. Areas corresponding to odd angular momenta are indicated by hash marks for clarity.

Figure 8

The ground-state angular momentum phase diagram obtained from a simplified model. The full (dotted) lines denote the successive phase boundaries at which the angular momentum of the electron (hole) dot switches to a higher negative (positive) value.

Figure 9

The pattern of the angular momenta at the intersection of electron and hole phase boundaries. (a) shows the independent dot result, which is modified to that schematically shown in (b) by a more careful inclusion of interparticle interactions.

Figure 10

The same as in Fig. 7 but now with the Zeeman interaction included. Regions corresponding to the angular momenta $L=1$ and $L=4$ are distinguished by hash marks.

Figure 11

The diamagnetic shift (full line, left axis) and the zero-magnetic-field binding energy (dashed line, right axis) of a biexciton confined in a double quantum dot. At low interdot distances the biexciton is bound and the diamagnetic shift becomes negative.