Variational quantum Monte Carlo study of charged excitons in fractional dimensional space

In this article we study excitons and trions in fractional dimensional spaces using the model suggested by C. Palmer [J. Phys. A: Math. Gen. 37, 6987 (2004)] through variational quantum Monte Carlo. We present a direct approach for estimating the exciton binding energy and discuss the von Neumann rejection- and Metropolis sampling methods. A simple variational estimate of trions is presented which shows good agreement with previous calculations done within the fractional dimensional model presented by D. R. Herrick and F. H. Stillinger [Phys. Rev. A 11, 42 (1975) and J. Math. Phys. 18, 1224 (1977)]. We explain the spatial physics of the positive and negative trions by investigating angular and inter-atomic distances. We then examine the wave function and explain the differences between the positive and negative trions with heavy holes. As applications of the fractional dimensional model we study three systems: First we apply the model to estimate the energy of the hydrogen molecular ion H${}_2^{+}$. Then we estimate trion binding energies in GaAs-based quantum wells and we demonstrate a good agreement with other theoretical work as well as experimentally observed binding energies. Finally, we apply the results to carbon nanotubes. We find good agreement with recently observed binding energies of the positively charged trion.

Figure 1

Exciton binding energies calculated using VQMC with $\Psi =e^{-ar}$ for different $a$’s. In a, c, and e the binding energies $\langle \Psi \left|\widehat{H}\right|\Psi \rangle /\langle \Psi |\Psi \rangle$ are calculated for $a=2/(D-1)$, $a=1/(D-1)$ and $a=3/(D-1)$, respectively. Four curves are presented: Four curves are presented: (I) a Metropolis calculation, (II) a von Neumann calculation, and, (III) a calculation based on Eq. (5) and the distribution generated by Eq. (7), and the solid line (Exact) is the exact analytical expression for the expectation value Eq. (12). In b, d, and f the local energies and exact probability distributions are given for the three cases: $a=2/(D-1)$, $a=1/(D-1)$and $a=3/(D-1)$. Additionally in b, we have given the probability distribution obtained from our Metropolis algorithm. The illustrations as well as calculations in b, d, and f were made for $D=1.7$.

Figure 2

Trion binding energy for a) $S_{1.0}^{\pm }$, b) $S_{0.0}^{-}$ and c) $S_{0.0}^{+}$ trion using VQMC and a basis expansion. The basis expansion was carried out as done in Ref. 19 within the Herrick-Stillinger framework. The VQMC binding energy was found using the Palmer model[23] with collapse of one axis at a time, first $z$ and then $y$. Inset: The lowest eigenvalues $E_T$ of the respective Hamiltonians.

Figure 3

Trion binding energies $E_X-E_T$ for $S_{\sigma }^{+}$ and $S_{\sigma }^{-}$ trions using VQMC for the wave function in Eq. (10) and the basis expansion given in Ref. 19. In (a) we give the binding energies for $D=1.7$, and (b) for $D=2$.

Figure 4

Trion binding energy for $S_{0.0}^{-}$ trion versus the dimension $D=\alpha +\beta +\gamma$ for different collapse orders. Collapse order 1, first we collapse the $z$-axis and then the $y$-axis. Collapse order 2, all collapse parameters were set to $D/3$. Collapse order 3, we kept $\alpha =1$ and set $\beta =\gamma =(D-1)/2$. The small deviation between the different ways of collapsing the axes is ascribed to the sampling of the singularities as illustrated in Fig. 1.

Figure 5

Expectation values of $r_{12}$, $r_{13}$,$r_{23}$, and ${\theta }_{23}$ as a function of dimension $D$ for a) $S_{1.0}^{+/-}$, b) $S_{0.0}^{-}$, and c) $S_{0.0}^{+}$.

Figure 6

Radial probability distribution ${\rho }_r$ (Eq. (16)) for the $S_{0.0}^{-}$ state for different dimensions $D$: a) $D=3.0$, b) $D=2.5$, c) $D=2.0$, and d) $D=1.7$.

Figure 7

Radial probability distribution for the positively charged heavy hole trion $S_{0.0}^{+}$ for a) $D=3.0$, b) $D=2.5$, c) $D=2.0$, and d) $D=1.7$. In comparison with Fig. 7 these distributions are much more localized. One very interesting feature is the local maximum along the line $r_2=r_1+k$ where $k\approx 1a_B^{*}$. This shows that the electron with high probability can be found near one of the two holes.

Figure 8

Probability distribution ${\rho }_{\theta }$ for $S_{0.0}^{-}$ as function of angles ${\theta }_{12}$ and ${\theta }_{13}$ for a) $D=3.0$, b) $D=2.5$, c) $D=2.0$, and d) $D=1.7$. The definition of ${\rho }_{\theta }$ is given in Eq. (17). Notice the pattern along the line ${\theta }_{13}={\theta }_{12}$ as well as the local minimum. This is a result of electron-electron repulsion and it is responsible for the expectation value $\langle S_{0.0}^{-}\left|{\theta }_{23}\right|S_{0.0}^{-}\rangle$ being slightly larger than $\pi /2$.

Figure 9

Probability distribution ${\rho }_{\theta }$ for the $S_{0.0}^{+}$ trion and ${\theta }_{13}$ for a) $D=3.0$, b) $D=2.5$, c) $D=2.0$, and d) $D=1.7$. In comparison with Fig. 8 these distributions are much more localized near the line ${\theta }_{13}={\theta }_{12}$.

Figure 10

Trion binding energy in GaAs-based quantum wells for the positive and negative trion.

Figure 11

Comparison between calculated trion binding energies (black squares) and experimentally observed values (red circles). The energies were calculated by assuming that CNTs effectively are $1.71$ dimensional nano structures[19] and assuming that[20] $E_{\sigma }^{+}\approx E_{1.0}^{+/-}$. The effective energies were then converted into physical energies by calculating the effective $R{y}^{*}$, as explained in the text, using a nearest neighbor tight binding model.