Electronic properties of bilayer graphene with magnetic quantum structures studied using the Dirac equation

In this subsection, we discuss the electronic properties of BLG with an MQD. In Figs. 2(a) and 2(b), the eigenenergies $\epsilon_{nm}$ are plotted as functions of $\alpha_{2}$ and $\alpha_{1}$, respectively, for two different MQD cases. The radius $r_{0}$ of the MQD is $5$ in dimensionless units. In the quantum state $(n,m)$, $n$ is the number of nodes in the radial wavefunction and $m$ is the angular momentum quantum number.

First of all, to allow comparisons with the typical MQDs considered in previous work based on a 2DEG model, we consider the case in which the inside magnetic field $\alpha_{1}$ of MQD is zero and the outside magnetic field $\alpha_{2}$ varies. The results are shown in Fig. 2(a). There is a noticeable difference in that we have zero-energy states for the MQD in BLG. These zero-energy states are also present for an MQD in monolayer graphene. The red dashed lines in the figure represent the dimensionless Landau levels (LLs) for BLG [33], i.e.,

where $N$ is the Landau index corresponding to $n+(m+|m|)/2$ in our calculation. For a magnetic field that is weak compared with $t$, the LLs are approximately proportional to $B$. Except for these features, the energy spectra of BLG with a typical MQD show the general characteristics of an MQD in the 2DEG case [7, 34, 35]. For a small outer magnetic field $\alpha_{2}$, the wavefunctions are mostly distributed outside the MQD, as a consequence of which the energies are close to the LLs of BLG. On the other hand, as $\alpha_{2}$ increases, the cyclotron radius of the electrons becomes smaller than the radius of the dot, and the energies start to deviate from the LLs. Because the Lorentz force leads to strong localization of the electrons within MQD, the energy evolution is very similar to that of an MQD in the 2DEG case. These energy changes are also the same as for a conventional circular dot that is electrostatically confined by hard walls without magnetic fields.

In Fig. 2(b), the inside magnetic field $\alpha_{1}$ varies, while the outside magnetic field is fixed at $\alpha_{2}=1$. The LLs inside the MQD (red dashed lines) are fan-shaped because $\alpha_{1}$ increases, and the LLs outside the MQD (black dashed lines) are flat because $\alpha_{2}$ remains constant. When $\alpha_{1}$ increases in the positive direction, the wavefunctions are localized within the MQD. Thus, the eigenenergies approach the LLs inside the MQD. However, when $\alpha_{1}$ becomes negative, the evolution of the eigenenergy exhibits two peculiar characteristics: (i) the eigenenergies converge to the LLs outside the dot in a stepwise manner through energy anticrossing; (ii) the eigenenergies of some quantum states approach zero energy. We now consider these characteristics in greater detail.

To explain the stepped shape of the energy evolution for a negative inner magnetic field $\alpha_{1}$, in Fig. 3 we plot $\epsilon_{nm}$ as functions of negative $\alpha_{1}$, the effective potential of MQD for $m=0$, and the probability density of a given quantum state $(n,m)$.

In Fig. 3(a), the red dashed lines and black dashed lines indicate the LLs inside and outside the MQD, respectively. The $(0,0)$ state has zero energy regardless of $|\alpha_{1}|$, and the $(1,0)$ state tends to zero energy as $|\alpha_{1}|$ increases. The eigenenergies $(n\geq 2,0)$ are determined by one of the LLs inside or outside the MQD, depending on the value of $\alpha_{1}$. For small $\alpha_{1}$ near zero, the wavefunctions are located mainly at the boundary of the MQD, and the eigenenergies deviate from the LLs both inside and outside the MQD. However, as $|\alpha_{1}|$ starts to increase, the wavefunctions move to the inside of the MQD and are affected by the inside magnetic field $\alpha_{1}$, and the energies start to follow the LLs inside the MQD. When an LL inside the MQD becomes equal to an LL outside the MQD, a state following an LL inside the MQD transitions to an LL outside the MQD, and vice versa. At such a time, energy anticrossing occurs, as shown in the inset in Fig. 3(a). These anticrossing phenomena are also found in electrostatically confined quantum dots and rings in monolayer graphene and BLG [20, 36, 17].

Figure 3(b) is a qualitative representation of the effective potential for the $m=0$ state of BLG with an MQD. We shall explain these anticrossing phenomena in terms of the effective potential. If whole wavefunctions can be localized either inside or outside the MQD, we can simply assume that the system energies are determined only by the LLs in the respective region. As the magnetic field $\alpha_{1}$ increases, the energy-level spacing inside the MQD becomes wider, and the energy levels start to become equal to the LLs outside the MQD. When the first LL inside the MQD becomes higher than the first LL outside the MQD, to maintain the energy level sequence for a fixed $m$, the state with the first LL inside the MQD moves to the first LL outside the MQD. Therefore, energy anticrossing occurs whenever LLs in the two different regions become the same. As the magnetic field increases, the wavefunction moves back and forth between the outside and inside of the MQD until the first inside LL becomes larger than the other fixed outside LLs, and, as a result, the wavefunctions of all states are located outside the MQD.

The $(2,0)$ state experiences energy anticrossing once and the $(3,0)$ state three times in the region $0<|\alpha_{1}|<3$, as shown in Fig. 3(a). To explain the alternating movement of wavefunctions as the magnetic field inside the MQD varies, we show the probabilities densities of the wavefunctions for the $(2,0)$ and $(3,0)$ states in Fig.s 3(c) and 3(d), respectively. For the $(2,0)$ state, the wavefunction moves from inside the MQD at $\alpha_{1}=-0.81$ to outside it at $\alpha_{1}=-1.45$ [Fig. 3(c)]. For the $(3,0)$ state, the wavefunction moves from outside the MQD at $\alpha_{1}=-0.81$ to inside it at $\alpha_{1}=-1.45$, and then moves out again at $\alpha_{1}=-1.85$ [Fig. 3(d)]. Before and after energy anticrossing occurs, the probability densities are located alternately inside and outside the MQD.

It is known that in monolayer graphene and BLG in a uniform magnetic field, wavefunctions of states at the zero Landau energy are completely localized in a particular subatom [13]. For the opposite magnetic field direction, the wavefunctions are all localized in the other subatom. In the present study, we consider a nonuniform magnetic field, and the magnetic field inside the MQD is opposite to that outside it. We would like to check if wavefunction separation occurs in this case as well. In Fig. 4, we plot the probability density of different subatoms for $\alpha_{1}=-0.25$, $0.00$, and $+0.61$. For $\alpha_{1}=+0.61$ (green lines), the $(0,1)$ state has nonzero eigenenergy, and the probability densities of all subatoms are located within the MQD in Figs. 4(a)–4(d). There is no wavefunction separation. For $\alpha_{1}=0.00$ (orange lines), in Figs. 4(a) and 4(b), the probability densities of the $A$ and $A^{\prime}$ subatoms are distributed over the MQD boundary, while in Figs. 4(c) and 4(d), the wavefunctions of the $B$ and $B^{\prime}$ subatoms are localized inside the MQD. There is still no clear wavefunction separation. Now, for $\alpha_{1}=-0.21$ (blue lines), this state approaches zero energy, and the wavefunctions of the $A$ and $A^{\prime}$ subatoms are mainly located outside the MQD, as shown in Figs. 4(a) and 4(b), while the wavefunctions of the $B$ and $B^{\prime}$ subatoms are more strongly localized within the MQD than those for $\alpha_{1}=0.00$, as shown in Figs. 4(c) and 4(d). As a result, $|\phi_{A}|^{2}$ and $|\phi_{A^{\prime}}|^{2}$ are distributed in a ring shape along the MQD boundary, while $|\phi_{B}|^{2}$ and $|\phi_{B^{\prime}}|$ are distributed inside the MQD. The $(n,m\geq 0)$ states, where $n=0,1$, tend toward zero energy, because the wavefunctions of each subatom start to be separated when $\alpha_{1}$ increases in the negative direction. From these results, we can infer that wavefunction separation also occurs for zero-eigenenergy states in BLG, even with a nonuniform magnetic field distribution.

In this subsection, we discuss the electronic properties of BLG with an MQR. For a fixed ring area with parallel magnetic fields ($\alpha_{1}=\alpha_{2}$ ), the results for the eigenenergy spectra show nothing unusual compared with the cases of 2DEG and monolayer graphene [34, 37, 38, 39]. Here, we investigate BLG with an MQR from a different perspective.

First, we investigate the eigenenergies by varying the inner radius $r_{1}$ of the MQR with parallel magnetic fields ($\alpha_{1}=\alpha_{2}$ ). We fix the outer radius of the MQR as $r_{2}=5$. The results are presented in Fig. 5. In Fig. 5(a), the eigenenergies $\epsilon_{nm}$ are shown as functions of $r_{1}$. The eigenenergy of the $(1,0)$ state (blue line) increases until it converges to the LL ($n=1$) inside the ring as $r_{1}$ increases. The eigenenergy of the $(2,0)$ state (orange line) exhibits a similar tendency, and it converge to the LL ($n=2$) inside the ring through energy anticrossing near the LL ($n=1$), as shown in Fig. 5(a) The average kinetic energy of an electron in a magnetic field is higher than in the absence of the field. Increasing the inner radius $r_{1}$ causes a decrease in the missing magnetic flux in the ring area, and so the eigenenergies of the BLG increase.

To explain the wavefunction movement near anticrossing, in Fig. 5(b), we plot the probability density distribution of the $(1,0)$ state as a function of $r$ for different values of the inner ring radius $r_{1}$ For $r_{1}=2.93$ (blue lines) before anticrossing, the wavefunction is distributed widely over the MQR. For $r_{1}=3.09$ (green lines) after anticrossing, the wavefunction is localized inside the inner circle of the MQR. The energy converges to the LL ($n=1$) inside the ring when $r_{1}>3.04$. This anticrossing can be explained using the same arguments as in the MQD case. At anticrossing, the size of the inner magnetic field region is large enough to encompass the wavefunctions of all subatoms for a lower energy. The wavefunctions within the ring move to the inner magnetic field region ($r_{1}<r$), while a part of the wavefunction for higher energy moves to the ring region $r_{1}<r<r_{2}$. Before and after energy anticrossing occurs, the wavefunctions are alternately located in regions with $B=0$ and $B\neq 0$ (inside the inner circle or outside the outer circle) of the MQR.

To study the angular momentum transition as the inner radius of the MQR varies, in Fig. 5(c), we plot $\epsilon_{nm}$ as functions of $r_{1}$ in the region $\epsilon_{nm}<0.5$ for the $(1,m)$ states. For an MQR with a finite width and for a conventional quantum ring confined by an electrostatic potential, an angular momentum transition is observed with increasing magnetic field [18, 34, 37]. When the size of the ring is fixed, this transition is due mainly to the missing flux quanta in the ring area. However, in the present study, an angular momentum transition is observed with increasing inner ring radius $r_{1}$ for a fixed magnetic field. As the inner ring radius increases, the number of missing flux quanta in the ring area ($r_{1}<r<r_{2}$) decreases, but the number of flux quanta within the inner ring radius increases. The small value of $m$ leads to rapid recovery of the LL as the inner circle radius $r_{1}$ increases. From the existence of this phenomenon, we can infer that the angular momentum transition depends more on the inner circle flux quanta than on the missing flux quanta in the ring area. The argument based on missing flux quanta is only valid when the ring area remains constant.

In Fig. 6, we show the eigenenergies $\epsilon_{nm}$ of BLG with an MQR as functions of $\alpha_{1}$ for fixed $r_{1}=3$, $r_{2}=5$, and $\alpha_{2}=1$. Here, we change the magnetic field $\alpha_{1}$ of the inner circle. As we vary $\alpha_{1}$ from $-1.5$ to $+1.5$, we can see the energy spectra of the MQR in parallel magnetic fields as well as in antiparallel fields. The energies of the $(0,m\geq 0)$ and $(1,m\geq 0)$ states of the MQR approach zero, similarly to what happens for the MQD, although the states of the MQR reach zero energy at a larger $|\alpha_{1}|$ value than those of the MQD. Geometrically, the difference between an MQD and an MQR is the presence in the latter of the zero-magnetic-field region $r_{1}<r<r_{2}$. This difference is responsible for the very different behavior of $\epsilon_{nm}$.

For conventional quantum rings (CQRs) formed by an electrostatic potential or by an etched boundary and based on a 2DEG and monolayer graphene, the Aharonov–Bohm (AB) effect has been studied both theoretically and experimentally [14, 40, 41, 42, 43, 44]. On the other hand, little attention has been paid to the AB effect for an MQR in a 2DEG. In a CQR, the energy of an electron bound in the ring oscillates as a function of the magnetic flux $\phi$ inside the ring. We find that the energy dispersion of BLG with an MQR also exhibits this effect for both large positive and large negative values of $|\alpha_{1}|$. To investigate the AB effect for an MQR, we calculate the period of the magnetic flux $\Delta\phi$ of the MQR numerically, and we find that this period is close to $\phi_{0}$: for $\alpha_{1}<0$, it is $1.01\phi_{0}$, and for $\alpha_{1}>0$, it is $0.97\phi_{0}$. For small $\alpha_{1}$, the wavefunctions are not well localized in the ring region. Thus, the AB effect is not clearly seen in the range between $\alpha_{1}=-0.5$ and $\alpha_{1}=0.5$. However, as $|\alpha_{1}|$ increases, the quantum states start to be strongly localized in the ring region, and exhibit an obvious AB effect. Unlike CQRs, whose energy has symmetry with regard to the direction of the magnetic flux inside the ring [14], the energy in our case does not possess such symmetry, because our MQR is formed by a nonuniform magnetic field. We believe that this AB effect should also be seen in MQRs based on a 2DEG or monolayer graphene.