Electrostatically confined trilayer graphene quantum dots

Electrically gating of trilayer graphene (TLG) opens a band gap offering the possibility to electrically engineer TLG quantum dots. We study the energy levels of such quantum dots and investigate their dependence on a perpendicular magnetic field $B$ and different types of stacking of the graphene layers. The dots are modeled as circular and confined by a truncated parabolic potential which can be realized by nanostructured gates or position-dependent doping. The energy spectra exhibit the intervalley symmetry $E_K^e\left(m\right)=-E_{K^{\prime }}^h\left(m\right)$ for the electron $\left(e\right)$ and hole $\left(h\right)$ states, where $m$ is the angular momentum quantum number and $K$ and $K^{\prime }$ label the two valleys. The electron and hole spectra for $B=0$ are twofold degenerate due to the intervalley symmetry $E_K\left(m\right)=E_{K^{\prime }}[-(m+1)]$. For both ABC $[\alpha =1.5$ (1.2) for large (small) $R]$ and ABA $\left(\alpha =1\right)$ stackings, the lowest-energy levels show approximately a $R^{-\alpha }$ dependence on the dot radius $R$ in contrast with the $1/R^3$ one for ABC-stacked dots with infinite-mass boundary. As functions of the field $B$, the oscillator strengths for dipole-allowed transitions differ drastically for the two types of stackings.

Figure 1

(Upper panels) Schematics of interlayer couplings in TLG for (a) ABC stacking and (b) ABA stacking. $A_i$ and $B_i,i=1,2,3$, are the layer sublattices, $t\approx 400$ meV the interlayer coupling, and $U_i$ $\left(i=1,2,3\right)$ the potentials applied to the layers. (Lower panels) Low-energy band structures of (c) ABC- and (d) ABA-stacked TLG in the absence of bias $(U_i=0$, dashed curves) or in the presence of bias (solid curves) as specified in the upper panels with $U_0=50$ meV.

Figure 2

(Upper panels) Lowest-energy states of an ABC-stacked TLG QD as a function of the dot radius $R$, for $m=-1$ (blue), $m=0$ (green), and $m=1$ (red), at the valleys $K$ (solid) and $K^{\prime }$ (dashed) with (a) $B=0$ and (b) $B=10$ T. The well depth is $U_0=50$ meV. The spectrum for $B=0$ is twofold degenerate due to the symmetry $E_K\left(m\right)=E_{K^{\prime }}[-(m+1)]$. Both spectra also show the symmetry $E_K^e\left(m\right)=-E_{K^{\prime }}^h\left(m\right)$. Lower panels (c) and (d) are the amplitude of the wave functions corresponding to, respectively, the points labeled by 1 and 2 in the energy spectrum of Fig. 2. Layer 1 (2, 3) is represented by the blue (red, green) curves. Sublattices $A_i$, $\left(B_i\right)$ are represented by solid (dashed) curves.

Figure 3

Energy levels of an ABC-stacked TLG QD as a function of the well depth $U_0$ for $m=-2$ (black), $m=-1$ (blue), $m=0$ (green), $m=1$ (red), and $m=2$ (purple) at the valleys $K$ (solid) and $K^{\prime }$ (dashed) with (a) $B=0$ and (b) $B=10$ T. The dot radius is $R=25$ nm. Due to the symmetry of $E_K\left(m\right)=E_{K^{\prime }}[-(m+1)]$ the spectrum for $B=0$ is twofold degenerate and both spectra show the symmetry $E_K^e\left(m\right)=-E_{K^{\prime }}^h\left(m\right)$. The yellow region in (a) indicates the continuous spectrum for $B=0$.

Figure 4

(a) The lowest-energy levels of an ABC-stacked TLG QD as a function of the magnetic field $B$, with $U_0=50$ meV and dot radius $R=25$ nm for the $K$ valley. The angular momenta are $m=-2$ (black), $m=-1$ (blue), $m=0$ (green), $m=1$ (red), and $m=2$ (purple). The dotted curves show the Landau levels of an unbiased ABC-stacked TLG. (b) As in (a) for $m=-1$ and $m=0$ at the two valleys $K$ and $K^{\prime }$.

Figure 5

Energy levels of ABA-stacking TLG QD as a function of dot radius $R$, for the angular momenta $m=-1$ (blue), $m=0$ (green), and $m=1$ (red), with $U_0=50$ meV at the two valleys $K$ (solid) and $K^{\prime }$ (dashed) when (a) $B=0$ and (b) $B=10$ T. For $B=0$, the spectrum is twofold degenerate due to the symmetry $E_K\left(m\right)=E_{K^{\prime }}[-(m-1)]$. The inset shows examples of anticrossings due to the hybridization between the gap- and magnetic-field-induced states at low magnetic fields, e.g., $B=1$ T, for $m=-1$ at the $K^{\prime }$ valley.

Figure 6

Energy spectrum of an ABA-stacked TLG QD, of radius $R=25$ nm, as a function of the well depth $U_0$ for $m=-1$ (blue), $m=0$ (green), and $m=1$ (red), at the valleys $K$ (solid) and $K^{\prime }$ (dashed), with (a) $B=0$ and (b) $B=10$ T. The yellow region in (a) shows the continuous $B=0$ spectrum.

Figure 7

Energy spectrum of an ABA-stacked TLG QD as a function of the magnetic field $B$, with $U_0=50$ meV and dot radius $R=25$ nm at the $K$ (solid curve) and $K^{\prime }$ (dashed curve) valleys. The blue, green, and red curves are for angular momenta $m=-1$, 0, and 1, respectively. The black dotted curves show the Landau levels of an ABA-stacked TLG sheet. The inset shows an enlarged view of the spectrum, showing the anticrossings for angular momentum $m=-1$ and the $K^{\prime }$ valley.

Figure 8

The wave functions corresponding to the energies marked by (a)–(h) in Fig. 7. Layer 1 (2, 3) is represented by the blue (red, green) curves. Sublattices $A_i$ $\left(B_i\right)$ are represented by solid (dashed) curves.

Figure 9

Lowest-electron-energy levels $(m=1$ at the $K$ valley) for an ABC-, ABA-stacked TLG QD, and for a AB-stacked BLG QD (blue filled, red open circles, and black open triangles, respectively). The fitting (solid curves) shows a $\sim a/R$ dependence for the ABA-stacked TLG and AB-stacked BLG dots. For the ABC-stacked TLG dot, the fitting shows a $\sim a/R^{1.2}$ (blue solid curve) and $\sim a/R^{1.5}$ (blue dashed curve) dependence at small and large radii, respectively. The well depth is $U_0=50$ meV and the corresponding fitting parameter $a$, in units of meV $\times$ nm, is equal to 283.7 (ABA), 279.9 (BLG), 280.2 (ABC, $R\lesssim 40$ nm), and 854.6 (ABC, $R\gtrsim 40$ nm).

Figure 10

Oscillator strengths for the dipole-allowed transitions $\left(-1,0\right)$ (blue), (1,0) (red), and (1,2) (green) as a function of magnetic field $B$, in (a) ABC- and (b) ABA-stacked TLG QD. The solid and dashed curves pertain to the $K$ and $K^{\prime }$ valleys, respectively.