Zitterbewegung and quantum revivals in monolayer graphene quantum dots in magnetic fields

The wave-packet evolution in graphene quantum dots in magnetic fields has been theoretically studied. By analyzing an effective Hamiltonian model we show the wave-packet dynamics exhibits three types of periodicities (Zitterbewegung, classical, and revival times). The influence of the size of the quantum dot and the strength of the external magnetic field in these periodicities has been considered. In addition, we have found that valley degeneracy breaking is shown by both classical and revival times.

Figure 1

Time dependence of the autocorrelation function ${\left|A\left(t\right)\right|}^2$ in a graphene monolayer quantum dot for $B=4$ T, $n_0=7$, $\sigma =1$, and $R=70$ nm in the $K$ valley. ZB of the electrons with $T_{\mathrm{Zb}}\simeq 9.37$ fs (top). First classical periods of motion with $T_{\mathrm{Cl}}\simeq 0.157$ ps (middle). Long-time dependence with $T_{\mathrm{Re}}\simeq 8.74$ ps (bottom). The Zitterbewegung, classical, and main fractional revival periods are indicated by vertical dotted lines.

Figure 2

Time dependence of electric current $j_{\varphi }$ and $j_r$ (in pAm) in a graphene monolayer quantum dot for $B=4$ T, $n_0=7$, $\sigma =1$, and $R=70$ nm in the $K$ valley. (a), (b) ZB of the electrons with $T_{\mathrm{Zb}}\simeq 9.37$ fs. (c), (d) First classical periods of motion with $T_{\mathrm{Cl}}\simeq 0.157$ ps. (e), (f) Long-time dependence with $T_{\mathrm{Re}}\simeq 8.74$ ps. The Zitterbewegung, classical, and main fractional revival periods are indicated by vertical dotted lines.

Figure 3

Time dependence of $\langle r\rangle$ in a graphene monolayer quantum dot for $B=4$ T, $n_0=7$, $\sigma =1$, and $R=70$ nm in the $K$ valley. (a) ZB of the electrons with $T_{\mathrm{Zb}}\simeq 9.37$ fs. (b) First classical periods of motion with $T_{\mathrm{Cl}}\simeq 0.157$ ps. (c) Long-time dependence with $T_{\mathrm{Re}}\simeq 8.74$ ps. The Zitterbewegung, classical, and main fractional revival periods are indicated by vertical dotted lines.

Figure 4

Wave-packet regeneration times for a graphene monolayer quantum dot with $R=70$ nm in a perpendicular magnetic field as a function of the field intensity. The Zitterbewegung time is shown in the left panel, the classical time in the central panel, and the revival time in the right panel.

Figure 5

Same as Fig. 4 but for a graphene monolayer quantum dot with a radius $R=140$ nm.

Figure 6

Wave-packet regeneration times for a graphene monolayer quantum dot in a perpendicular magnetic field $B=10$ T as a function of the dot radius. The Zitterbewegung time is shown in the left panel, the classical time in the central panel, and the revival time in the right panel.