Exciton states in a circular graphene quantum dot: Magnetic field induced intravalley to intervalley transition

The magnetic-field dependence of the energy spectrum, wave function, binding energy, and oscillator strength of exciton states confined in a circular graphene quantum dot (CGQD) is obtained within the configuration interaction method. We predict that (i) excitonic effects are very significant in the CGQD as a consequence of a combination of geometric confinement, magnetic confinement, and reduced screening; (ii) two types of excitons (intravalley and intervalley) are present in the CGQD because of the valley degree of freedom in graphene; (iii) the intravalley and intervalley exciton states display different magnetic-field dependencies due to the different electron-hole symmetries of the single-particle energy spectra; (iv) with increasing magnetic field, the exciton ground state in the CGQD undergoes an intravalley to intervalley transition accompanied by a change of angular momentum; (v) the exciton binding energy does not increase monotonically with the magnetic field due to the competition between geometric and magnetic confinements; and (vi) the optical transitions of the intervalley and intravalley excitons can be tuned by the magnetic field, and valley-dependent excitonic transitions can be realized in a CGQD.

Figure 1

Model system considered in the present work: (a) a circular graphene quantum dot (CGQD) of radius $R$ in the presence of a perpendicular magnetic field $B$, and (b) an exciton ($X$) formed in the CGQD by a conduction electron ($e$) and a valence hole ($h$) via the attractive Coulomb interaction.

Figure 2

Single-particle energy spectrum in a CGQD with $R=50$ nm in the presence of a perpendicular magnetic field. Only the six lowest electron and hole states are shown for the angular quantum number $-6\le m\le 6$. The blue solid and red dashed curves denote the results for the $K$ and $K^{\prime }$ valleys, respectively, as indicated.

Figure 3

Magnetic energy spectra of noninteracting states (a) and exciton states (b) for the same CGQD as in Fig. 2. Energy levels with the same total valley index $T$ are plotted in the same type of curve: the red curves for $T=\pm 2$ with ${\tau }_e={\tau }_h=\pm 1$, the green curves for $T=0$ with ${\tau }_e=-{\tau }_h=1$, and the blue curves also for $T=0$ but with ${\tau }_e=-{\tau }_h=-1$. Note that the energy levels for $T=\pm 2$ are degenerate due to the same electron-hole symmetry and thus are plotted with the same red color.

Figure 4

Magnetic-field dependencies of noninteracting ground-state energy and exciton ground-state energy for two dot radii $R$'s as indicated. Here, $(T,M)$ are a pair of quantum numbers: the total valley index and the total angular momentum. The black arrows indicate the critical magnetic fields $B_c^1=1.3$ T for $R=50$ nm and $B_c^2=3.6$ T for $R=30$ nm. At these magnetic fields, the exciton ground state in the CGQD undergoes an intravalley to intervalley transition accompanied by a change of angular momentum, i.e., a combined valley and angular momentum transition.

Figure 5

Magnetic-field dependence of the binding energy $E_B$ of the exciton ground state for the same CGQDs as in Fig. 4. The black arrows indicate the critical magnetic fields $B_c^1=1.3$ T for $R=50$ nm and $B_c^2=3.6$ T for $R=30$ nm. At these magnetic fields, $E_B$ changes abruptly.

Figure 6

Conditional probability densities (CPDs) and electron-hole pair densities (EHPDs) of the exciton ground state for the same CGQD as Fig. 2 for different magnetic fields: (a) the CPD at $B=2$ T; (b) the CPD at $B=5$ T; (c) the CPD at $B=10$ T with fixed electron position at $\mathbf{r}=(R/2,0)$; and (d) the EHPDs for these magnetic fields.

Figure 7

$B-R$ phase diagram for the intravalley to intervalley exciton transition in the CGQD. The black dashed curve represents the dependence relation of the critical magnetic field $B_c$ with the dot radius $R$.

Figure 8

Excitonic transition energies and strengths for the same CGQD as in Fig. 2. The red circles denote the results for the intravalley exciton states with $T=\pm 2$, and the green/blue circles denote the results for the intervalley exciton states with $T=0$. The size of the solid circle indicates the excitonic transition oscillator strength.