Two-dimensional bound excitons in the real space and Landau quantization space: a comparative study

Abstract

The Landau quantization space is based on the respective motion of the electron and hole in a magnetic field and can provide a new route to understand the bound exciton behaviors observed in the experiments. In this paper, we study the two-dimensional exciton properties of monolayer WSe$_2$ in both the real space and Landau quantization space. Focusing on the excitons of zero center-of-mass momentum, we calculate its energy spectrum in both spaces, with the results agreeing well with each other. We then obtain the diamagnetic coefficients and root-mean-square radius, which are consistent with the available $s$ state data in the experiment. More importantly, in the exciton state $nl$, we find that the dominant electron-hole pair component may shift with the magnetic field and the Coulomb interactions, and reveal that the magnetic field will drive the dominant component to be the free electron-hole pair $\{n_e=n+l-1,n_h=n-1\}$, whereas the Coulomb interactions drives it to be the pair of the lower index.

Figures (6)

• Figure 1: (Color online) (a) The exciton energy spectrum $\varepsilon_{nl}$ vs the magnetic field $B$. (b) The energy difference $\Delta\varepsilon_{nl}=\varepsilon_{n,+l}-\varepsilon_{n,-l}$ vs $B$. In both figures, the solid and dashed lines denote the results from the real space and Landau quantization space, respectively. The inset in (b) shows $\Delta\varepsilon_{nl}$ scaled by $l$, all of which collapse on the same line.
• Figure 2: (Color online) The probability $W_{nl}$ of finding the bound exciton vs the electron-hole separation $r$, with $l=0$ in (a) and $l=1,2$ in (b). In each lower and upper figure, the magnetic field is set as $B=0$ and $B=20$ T, respectively. For clarity, the probability curves under $B=20$ T are shifted vertically by $0.6$ in (a) and $0.3$ in (b).
• Figure 3: (Color online) The diamagnetic shift $\Delta\varepsilon_{nl}^{\text{dia}}$ vs the magnetic field $B$ up to 65 T. In (a), $\Delta\varepsilon_{nl}^{\text{dia}}$ is obtained from Eq. (\ref{['diam2']}). The solid and dashed lines denote the results from the real space and Landau quantization space, respectively. In (b), $\Delta\varepsilon_{nl}^{\text{dia}}$ is obtained by fitting the results in (a) through $\Delta\varepsilon_{nl}^{\text{dia}}=\sigma_{nl}B^2$, from which the diamagnetic coefficient $\sigma_{nl}$ and rms radius $r_{nl}$ can also be solved and are listed in Table \ref{['table1']}. The insets in (a) and (b) show the results of the ground state $1s$.
• Figure 4: (Color online) The weight $|\phi_{nl}^i|^2$ of the free electron-hole pair $\{n_e=i+l,n_h=i\}$ in the exciton state $nl$ vs the hole LL index $i$. In (a)-(c), the weights are shown for the $s$, $p+$ and $d+$ states, respectively. The lower and upper figures are the results under different magnetic fields. For clarity, the weights in the upper figures are shifted vertically by $0.8$. The inset in (a) shows the weight $|\phi_{ns}^i|^2$ vs the index $i$ up to $i=300$.
• Figure 5: (Color online) The phase diagrams of the dominant electron-hole pair $\{n_e=n_h=i\}$ in the parameter space spanned by the magnetic field $B$ and the relative dielectric constant $\epsilon_v$. (a)-(c) show the results of the $2s$, $3s$ and $4s$ states, respectively. The different dominant components $i$ are labeled in different colors. The two asterisks marked in each figure correspond to the parameters chosen in Fig. \ref{['Fig4']}(a).