Berry phase jumps and giant nonreciprocity in Dirac quantum dots

We predict that a strong nonreciprocity in the resonance spectra of Dirac quantum dots can be induced by the Berry phase. The nonreciprocity arises in relatively weak magnetic fields and is manifest in anomalously large field-induced splittings of quantum dot resonances which are degenerate at $B=0$ due to time-reversal symmetry. This exotic behavior, which is governed by field-induced jumps in the Berry phase of confined electronic states, is unique to quantum dots in Dirac materials and is absent in conventional quantum dots. The effect is strong for gapless Dirac particles and can overwhelm the $B$-induced orbital and Zeeman splittings. A finite Dirac mass suppresses the effect. The nonreciprocity, predicted for generic two-dimensional Dirac materials, is accessible through Faraday and Kerr optical rotation measurements and scanning tunneling spectroscopy.

Figure 1

Controlling the Berry phase of confined Dirac electrons using magnetic fields. Shown are semiclassical orbits of a massless particle exhibiting topologically distinct orbital behavior corresponding to (a) $B<B_{\mathrm{c}}$ and (b) $B>B_{\mathrm{c}}$ [see critical field $B_{\mathrm{c}}$ in Eq. (7)]. The Berry phase, determined by the solid angle subtended by $\mathit{h}=(h_x,h_y,h_z)$ in Eq. (3), jumps from ${\phi }_{\mathrm{B}}=0$ to ${\phi }_{\mathrm{B}}=\pi$ at $B=B_{\mathrm{c}}$; see insets [for gapless systems $h_{x,y}=v{q}_{x,y}$ and $h_z=0$, with $q_{x,y}$ the kinetic momentum (red vectors) and $v$ the Fermi velocity]. Here we used $m=1/2$, energy $\varepsilon =1.35\hslash v/r_{*}$, with $r_{*}$ defined in Eq. (11), $B/B_{\mathrm{c}}=0.8$ for (a) and $B/B_{\mathrm{c}}=1.6$ for (b).

Figure 2

Topologically distinct mappings of $\mathit{q}$ [Eq. (6)] to the surface of a torus (a), plotted for (b) $B<B_{\mathrm{c}}$ and (c) $B>B_{\mathrm{c}}$. Indicated with blue (red) arrows is $\mathit{q}$ along the curves ${\mathcal{C}}_{\theta } \left({\mathcal{C}}_r\right)$ shown in panel (a), where dotted lines/arrows indicate a curve/vector in the bottom surface of the torus. At $B=B_{\mathrm{c}}$, there is a transition between trivial and nontrivial winding of $\mathit{q}$ along ${\mathcal{C}}_r$. This results in $B$-induced phase jumps of the Berry phase. Here we define $q_{*}={\varepsilon }_{*}/v$ and use the same parameter values as in Fig. 1.

Figure 3

Magnetic response of quantum dot resonances in a gapless Dirac system. (a) The quantum dot is defined by the circular $pn$ ring (dashed lines) induced by a radial electrostatic potential $U\left(\mathit{r}\right)$. (b) The magnetic response is dominated by the Berry-phase splitting $\mathrm{\Delta }{\varepsilon }_{\mathrm{B}}$, which is approximately half the resonance period $\mathrm{\Delta }\varepsilon$. Also indicated in the figure is the orbital splitting $\mathrm{\Delta }{\varepsilon }_{\mathrm{orb}}$. Peak splitting is calculated from Eq. (2) for $n=0,1,2,m=\pm 1/2$, and $\gamma =0.6$; $B_{\mathrm{c}}$ is calculated from Eq. (7); ${\varepsilon }_{*}$ and $B_{*}$ are defined in Eq. (11).

Figure 4

Maps of the local density of states as a function of position $r_0$ for a gapless Dirac quantum dot displaying splitting of resonances in weak magnetic fields: (a) $r_0=0$, (b) $r_0=r_{*}$, and (c) $r_0=2r_{*}$. Indicated with dotted lines is Eq. (7) for half-integer $m$. The off-centered spectral maps (b),(c) are qualitatively different from the centered case (a) which is sensitive primarily to $m=\pm 1/2$ states. Characteristic units for magnetic field, $B_{*}$, are defined in Eq. (11). Plotted with dotted lines is Eq. (7) for half-integer $m$. To enhance spectral features, we plot in both panels the derivative of the local density of states in Eq. (12).

Figure 5

Partial-$m$ contribution to the on-center density of states for quantum dots in (a) gapless and (b) gapped Dirac systems. The strong nonreciprocal effect induced by Berry phase disappears when a large gap $\mathrm{\Delta }$ is opened. As a result, resonance splitting is dominated by (a) the Berry phase jump in gapless systems, and (b) orbital effects in gapped systems. The distinct behavior between (a) and (b) is shown in the partial $m=1/2$ maps of the density of states [indicated with a dotted line is Eq. (7) with $m=1/2$; $\mathrm{\Delta }/{\varepsilon }_{*}=5$ was used in (b)].

Figure 6

(a) Schematics of the electrostatic model, showing a metallic sphere of radius $R$ separated a distance $d$ from the graphene plane. A potential bias $V_{\mathrm{b}}$ applied on the sphere induces a local variation of the carrier density, different from the carrier concentration density $n_{\infty }$ far from the sphere. (b), (c) Band structure schematics showing band alignment between the metallic sphere and graphene for (b) large separation and (c) close proximity. Here $V_{\mathrm{cpd}}$ is the contact potential difference between graphene and the metallic sphere, $\delta V_{\mathrm{b}}=V_{\mathrm{b}}-V_{\mathrm{cpd}}$, and ${\mu }_0$ is the Fermi energy under the sphere.