Neutral and charged electron-hole complexes in artificial molecules: Quantum transitions induced by the in-plane magnetic field

We theoretically investigate the properties of neutral and charged excitons and of the biexciton in vertically coupled quantum dots, as a function of the in-plane magnetic field $B_{\Vert }$. The main effect of the field consists in the suppression of the bonding-antibonding splitting, and in the resulting enhancement of the interdot correlations. As a consequence, the excitons form with the additional carrier a bound or an unbound complex depending on the sign of the charging, whereas the biexciton undergoes a transition between different quantum states with increasing $B_{\Vert }$. The discussed behaviors and transitions show up in the field dependence of experimentally accessible quantities, such as the charged-exciton and biexciton binding energies.

Figure 1

Single-particle energies for electrons (left axis, black lines) and holes (right axis, gray lines) in a symmetric double QD, as a function of the in-plane magnetic field. The curves correspond to the $B$ (solid lines) and $\mathit{AB}$ (dashed lines) eigenstates. The insets show the square of the electron (a) and hole (b) $B$ wave functions along $z$ for two selected values of $B_{\Vert }$, namely $0\mathrm{T}$ (solid lines) and $15\mathrm{T}$ (dashed lines), for $x=y=0$. In the lower insets we plot the electron charge density integrated in the interdot barrier for the above confinement potential [panel (c)], as well as for a double-well (d) profile in the growth direction (solid lines, left axes), respectively; the magnetic-field dependence of ${\Delta }_{B\mathit{AB}}$ is also shown (dotted lines, right axes). In the former case we identify the barrier with the region $\mid z\mid <a∕\sqrt{3}$; in the latter case, the width of each well is $l=12\mathrm{nm}$, the width and height of the interdot barrier are $d=4\mathrm{nm}$ and $V_0^e=400\mathrm{meV}$, respectively.

Figure 2

Expectation values of the field-dependent terms in the single-electron Hamiltonian of Eq. (1): The solid (dashed) lines correspond to the $B$ $\left(\mathit{AB}\right)$ states; circles, squares, and triangles stand for the diamagnetic part, the paramagnetic one, and the overall expectation value, respectively.

Figure 3

The binding energies of the neutral $\left(X\right)$ and charged ($X^{-}$ and $X^{+}$) excitons, and of the biexciton $B$ as a function of the in-plane magnetic field $B_{\Vert }$ are denoted by squares, circles, upward- and downward-oriented triangles, respectively.

Figure 4

Projection along $z$ of the pair-correlation functions, ${\overline{g}}_{\chi {\chi }^{\prime }}(0,0,z)$, of the neutral and charged excitons, for two selected values of the in-plane magnetic field, namely $B_{\Vert }=0\mathrm{T}$ and $B_{\Vert }=15\mathrm{T}$ (solid and dotted lines, respectively). In detail, In panels (a)–(c) $\chi =e$ and ${\chi }^{\prime }=h$, whereas $\chi ={\chi }^{\prime }=h$ in (d) and $\chi ={\chi }^{\prime }=e$ in (e); panels (a) and (d) refer to $X^{+}$, (c) and (e) to $X^{-}$ and (b) to $X$.

Figure 5

Projection along $z$ of the hole-hole [panel (a)], electron-hole (b), and electron-electron (c) pair-correlation functions for the biexciton ground state $\mid B⟩$, for $B_{\Vert }=0\mathrm{T}$ and $B_{\Vert }=15\mathrm{T}$ (solid and dotted lines, respectively).

Figure 6

Average values of the Coulomb terms $⟨B_i\mid V_{\chi {\chi }^{\prime }}\mid B_i⟩$ [panels ((a)–(c) correspond to $\chi {\chi }^{\prime }=eh,ee,hh$] and of $⟨B_i\mid {\mathcal{H}}_{\mathit{sp}}\mid B_i⟩$ [panel (d)] for the biexciton ground and first excited-states $\mid B_i⟩$; the solid and dotted gray (black) lines correspond to the states $\mid B_0⟩$ and $\mid B_1⟩$ ($\mid B_3⟩$ and $\mid B_2⟩$), respectively.

Figure 7

Optical transitions as a function of the magnetic field $B_{\Vert }$. (a) Emission energies from the groundstates of the neutral and charged excitons $X$, $X^{+}$, and $X^{-}$ (dotted, dashed, and solid lines, respectively). (b) Emission energies from the biexciton states $\mid B_0⟩$ (squares) and $\mid B_3⟩$ (downwards-oriented triangles): The two lines anticross at $B_{\Vert }\sim 8\mathrm{T}$. Inset, crossing between the lines corresponding to the biexciton states $\mid B_1⟩$ (upwards-oriented triangles) and $\mid B_2⟩$ (circles). (c) Squared moduli of the amplitudes ${\mathcal{A}}_{i\to f}$; the initial states correspond to $\mid B_i⟩$, with $i=0,1,2,3$, and the same conventions as above for the symbols; the final states are the exciton ground- (decay from $\mid B_0⟩$ and $\mid B_3⟩$) and first excited-states ( $\mid B_1⟩$ and $\mid B_2⟩$).