Relative stability of negative and positive trions in model symmetric quantum wires

Binding energies of negative $\left(X^{-}\right)$ and positive trions $\left(X^{+}\right)$ in quantum wires are studied for strong quantum confinement of carriers which results in a numerical exactly solvable model. The relative electron and hole confinement have a strong effect on the stability of trions. For equal hole and electron confinement, $X^{+}$ is more stable but a small imbalance of the particle confinement towards a stronger hole confinement, e.g., due to its larger effective mass, leads to the interchange of $X^{-}$ and $X^{+}$ recombination lines in the photoluminescent spectrum as was recently observed experimentally. In case of larger $X^{-}$ stability, a magnetic field oriented parallel to the wire axis leads to a stronger increase of the $X^{+}$ binding energy resulting in a crossing of the $X^{+}$ and $X^{-}$ lines.

Figure 1

(a) (Color online) Contour plot of the interaction potential $V=V^{\mathrm{ef}}(L;z_{h1}-z_{h2})-V^{\mathrm{ef}}(L;z_{h1})-V^{\mathrm{ef}}(L;z_{h2})$ as function of the interparticle distances with lateral confinement length $L=l_e=l_h=1$. Distances and energies are given in donor units. The dashed-dotted line corresponds to $V=0$. (b) The interaction potential plotted for $L=1$ along the lines $z_{h2}=-z_{h1}$, $z_{h2}=-z_{h1}-1.5$, and $z_{h2}=-z_{h1}-3$ marked in (a) with (thick) solid, dotted and dashed lines respectively, as function of the interelectron $\left(X^{-}\right)$ or interhole $\left(X^{+}\right)$ distance $z_{12}=z_{h1}-z_{h2}$. Thin solid line shows the $-3∕\mid z_{12}\mid$ asymptotic.

Figure 2

Wave functions for negative (a,d,g,j) and positive (b,c,e,f,h,i,k,l) trions for $l_e=l_h=L=0.2$ in $z_{h1}$ and $z_{h2}$ coordinates (horizontal and vertical axis, respectively) for different values of the mass ratio $\sigma$. Plots (c,f,i,l) show the wave functions of the excited $X^{+}$ state antisymmetric with respect to the interchange of the holes. The dashed line in (f,i,l) shows the node of the wave functions. Plot (c) corresponds to an unbound state, for other plots the computational box is larger than the fragment displayed and the states are bound.

Figure 3

Electron-hole (solid lines), electron-electron, hole-hole (dashed lines) pair correlation functions plots for $X^{-}$ and $X^{+}$ (lines marked by black squares) at $\sigma =6.72$ and $l_e=l_h=0.2$.

Figure 4

Binding energies of the negative (dashed lines) and positive trion (solid lines) states for $L=0.2$ as function of the mass ratio $\sigma$. Higher solid curve corresponds to the $X^{+}$ state antisymmetric with respect to the interchange of electrons and holes, i.e., it is the first excited state. Thin vertical lines show the values of $\sigma =1$, 1.98, 6.72, and 15.2. Dotted curve, referred to the right axis, shows the exciton ground-state eigenvalue. Energies and lengths are in donor units.

Figure 5

(Color online) Binding energies of the trions as functions of the length of the lateral confinement. Lines for $\sigma =1$, 1.98, 6.72, and 15.2 plotted with black, blue, red, and green colors, respectively. For $\sigma =1$ binding energies of $X^{-}$ and $X^{+}$ are equal. The dashed lines are the energies of the excited $X^{+}$ states antisymmetric with respect to the interchange of the holes ($X^{-}$ does not possesses a bound excited state for $\sigma >1$). Lines for $X^{-}$ at $\sigma =1.98$ and 6.72 have been omitted for clarity—they are situated between the $\sigma =1$ line and $\sigma =15.2$ line for $X^{-}$ (see Fig. 4). Inset: energy eigenvalues for neutral exciton and charged trions for $\sigma =15.2$. Energies and lengths are given in donor units.

Figure 6

(Color online) Contour plot of the interaction potential for (a) negative trion $V=V^{\mathrm{ef}}(l_e;z_{h1}-z_{h2})-V^{\mathrm{ef}}(l_{eh};z_{h1})-V^{\mathrm{ef}}(l_{eh};z_{h2})$ and (b) positive trion $V=V^{\mathrm{ef}}(l_h;z_{h1}-z_{h2})-V^{\mathrm{ef}}(l_{eh};z_{h1})-V^{\mathrm{ef}}(l_{eh};z_{h2})$ as function of the interparticle distances for the lateral confinement lengths $l_e=1$ and $l_h=0.5$. Distances and energies are given in donor units.

Figure 7

(Color online) Shifts of the trion recombination PL lines with respect to the PL exciton line as function of the hole confinement length $\left(l_h\right)$ for GaAs. Different values of the electron confinement length are plotted with different colors.

Figure 8

(Color online) Difference of the positive and negative trion binding energies (in milli-electron-volts) as function of the electron and hole confinement lengths for GaAs material parameters. Blue (red) regions correspond to more stable negative (positive) trion. Above the dashed-dotted line the negative trion is unbound. The green line corresponds to $E_B\left(X^{-}\right)=4.2\mathrm{meV}$ and the yellow line to $E_B\left(X^{+}\right)=2.9\mathrm{meV}$.

Figure 9

Interparticle distances for the negative (solid lines) and positive trions (dashed lines) and for the exciton (dotted line) for GaAs material parameters and $l_e=2.95\mathrm{nm}$. $\left(e\text{−}e\right)$, $\left(e\text{−}h\right)$, and $\left(h\text{−}h\right)$ stand for the electron-electron, the electron-hole and the hole-hole distance.

Figure 10

(Color online) Magnetic field dependence of the trion binding energies in GaAs for different values of the electron and hole oscillator lengths.