Graphene-Based Heterojunction between Two Topological Insulators

Quantum Hall (QH) and quantum spin Hall (QSH) phases have very different edge states and, when going from one phase to the other, the direction of one edge state must be reversed. We study this phenomenon in graphene in the presence of a strong perpendicular magnetic field on top of a spin-orbit (SO)-induced QSH phase. We show that, below the SO gap, the QSH phase is virtually unaffected by the presence of the magnetic field. Above the SO gap, the QH phase is restored. An electrostatic gate placed on top of the system allows the creation of a QSH-QH junction which is characterized by the existence of a spin-polarized chiral state, propagating along the topological interface. We find that such a setup naturally provides an extremely sensitive spin-polarized current switch which could pave the way to novel spin-based electronic devices.

Popular Summary

The quantum Hall states and the quantum spin Hall states of electrically driven solid-state materials are two paradigmatic examples of topological insulators. They share an important feature: There is an electronic traffic along the edges of the material samples that are electrically insulating in their interior. In the former state, a high magnetic field applied perpendicular to a two-dimensional electron gas in a semiconductor at low temperature splits moving electrons into those traveling in one direction along one edge and those in the opposite direction along the opposite edge. But each one-lane traffic flow consists of electrons with both “up” spins and “down” spins. In the latter state, however, the mechanism of the separation comes from within—the so-called fundamental spin-orbital coupling between an electron’s spin and its motion in space. This mechanism leads to a more complex pattern of separation: Along each edge of the two-dimensional electronic system, two-lane traffic appears, with one of the lanes involving only spin-up electrons moving in one direction, and the other only spin-down electrons in the opposite direction. No systems have been found experimentally where both states coexist. In this theoretical paper, we focus on graphene, propose an experimental setup with graphene that can in principle host both states at the same time, and reveal a new and particularly interesting state localized at the interface between the two coexisting states.

Graphene has been experimentally shown to exhibit a quantum Hall phase, and a recent promising proposal for engineering quantum spin Hall states by depositing heavy adatoms on a graphene monolayer and thus increasing the graphene’s spin-orbital coupling is gaining momentum. But, the need of a high magnetic field to generate the quantum Hall phase was expected to destroy the quantum spin Hall one. Surprisingly, our theoretical study shows that the quantum spin Hall phase survives the application of a strong magnetic field, a feat which can be traced back to the “relativistic” Dirac-like nature of the charge carriers in graphene. In fact, we find that a quantum Hall state can be reversibly switched to a quantum spin Hall one by simply tuning the chemical potential in the graphene sample using a standard four-terminal setup for electron transport. Applying a top voltage gate to part of the graphene then allows for a heterojunction to be created between the quantum Hall and the quantum spin Hall phases in a controlled way. Most interesting, we find that a robust state of spin-polarized current exists at the heterojunction.

We see a number of questions for further exploration and development. For example, what is the precise nature of the current state localized at the heterojunction? Does it bear any fundamental connections to charge-current states observed in quantum Hall $n\mathrm{\text{−}}p$ junctions? Can the tunable transition between a quantum Hall state and a quantum spin Hall state arise in bilayer graphene? Our proposed four-terminal setup may also be developed into an extremely sensitive switch for spin-polarized current.

Figure 1

Schematics for the edge states in a four-terminal Z-shape geometry when the system is in the (a) QSH phase and the (b) QH phase. In (a), opposite spins (thick blue lines and red dotted lines) propagate in opposite directions on a given edge, while in (b), they propagate in the same direction. (c) Topological heterojunction, with QSH edge states on the left and QH edge states on the right. This junction can be achieved experimentally by applying a top gate (shaded region) on the right half of the sample. While one of the spin species (thick blue line) can propagate through this junction, the other one (red dotted line) cannot, giving rise to a chiral state, localized at the interface between QH and QSH phases, which connects both edges.

Figure 2

Band structure of a (semimetallic armchair) graphene ribbon in the (a) QH and (b) QSH phase. When both magnetic-field coupling and SO coupling are present (c), the resulting band structure leads to a QSH phase for $|E_{\mathrm{F}}|<{\Delta }_{\mathrm{so}}$ (shaded region) and a QH phase for $|E_{\mathrm{F}}|>{\Delta }_{\mathrm{so}}$. Compared to the pure QH and QSH cases, the spin degeneracy is lifted (blue thick lines and red dotted lines), seen as particularly prominent in the lowest band which consists of spin-polarized branches at $E=\pm {\Delta }_{\mathrm{so}}$. As the Fermi energy crosses the SO gap, the localization of the unhappy spin (red dotted lines) shifts from one edge to the other, while it is fully localized in the bulk when $E_{\mathrm{F}}={\Delta }_{\mathrm{so}}$. This is illustrated in the corresponding current-density plots (d). On the other hand, the happy spin (thick blue) gets increasingly localized on the same edge as the Fermi energy crosses the transition region (not shown). The parameters used are ${\lambda }_{\mathrm{so}}=0.02$, $l_B\simeq 8$, and a ribbon width $W=40\sqrt{3}$.

Figure 3

QSH-QH transition in a four-terminal $\Psi$-shape geometry. (a) Dimensionless differential conductance from lead 0, where the current is injected, to outgoing leads 1, 2, and 3 in the geometry depicted in the inset, as a function of the Fermi energy (${\lambda }_{\mathrm{so}}=0.02$, $l_B\simeq 8$, and lead widths $W_0=64\sqrt{3}$, $W_1=22\sqrt{3}$). Below the SO gap, current is carried by QSH edge states, which propagate on different edges (left inset), while above the SO gap it is carried by QH edge states, which propagate on the same edge (right inset). (b) Dimensionless differential conductance from lead 0 to lead 2 as a function of the classical cyclotron radius $R_{\mathrm{c}}$ [same values of ${\lambda }_{\mathrm{so}}$, $W_0$, and $W_1$ as in (a)]. Circular symbols correspond to data with $E_{\mathrm{F}}<{\Delta }_{\mathrm{so}}$, while other symbols (dashed lines) correspond to data with $E_{\mathrm{F}}>{\Delta }_{\mathrm{so}}$: respectively, $E_F/{\Delta }_{\mathrm{so}}=1.06$ (blue diamonds); $E_{\mathrm{F}}/{\Delta }_{\mathrm{so}}=1.25$ (violet triangles); and $E_{\mathrm{F}}/{\Delta }_{\mathrm{so}}=1.44$ (red squares). While this direct transmission is vanishingly small and independent of the magnetic field below the SO gap, it increases with both $R_{\mathrm{c}}$ and $E_{\mathrm{F}}$ above the SO gap. The latter situation arises because the QH phase is destroyed as $2R_{\mathrm{c}}\gtrsim W_1$, and because the number of transmitting channels increases with $l_B$ and $E_{\mathrm{F}}$.

Figure 4

Effect of very strong disorder on the QSH phase in the four-terminal $\Psi$-shape geometry. We plot the dimensionless differential conductance from lead 0 to leads 1 and 2 as a function of disorder strength for ${\lambda }_{\mathrm{so}}=0.02$, $l_B\simeq 8$, $E_{\mathrm{F}}=0.05$, and otherwise the same system as in Fig. 3. Circular symbols stand for $T_{10}$ and triangular symbols for $T_{20}$. The behavior for $T_{30}$ (not shown) is the same as for $T_{10}$. Dashed black lines correspond to purely scalar disorder ($V_{\mathrm{s}}$), dotted (red) lines correspond to purely Zeeman-like disorder ($V_{\mathrm{z}}$), and filled blue symbols correspond to purely $S_{\mathrm{z}}$ nonconserving disorder ($V_{\mathrm{m}}$). The latter is clearly the dominant effect on $T_{10}$, although unrealistically large values are required to quantitatively affect the edge-state transmission. (See main text for a mean-free-path estimate.) Each data point has been averaged over 48 disorder configurations. Unless shown, error bars are smaller than symbol sizes. Full black line: fit $y=1-3.0x^2$.

Figure 5

Heterostructure in a four-terminal Z-shape sample as depicted in Fig. 1c. Transmission probabilities from lead 0, where the current is injected, to outgoing leads 1 (dotted lines), 2 (dashed lines), and 3 (solid lines) as a function of the top gate voltage $V_{\mathrm{g}}$. When $V_{\mathrm{g}}<({\Delta }_{\mathrm{so}}-E_{\mathrm{F}})$, such that left and right regions are in the QSH phase, current is perfectly transmitted by the QSH edge states, as shown in the current-density plot in the left inset. When $V_{\mathrm{g}}$ is high enough [$V_{\mathrm{g}}>({\Delta }_{\mathrm{so}}-E_{\mathrm{F}})$] that the right part of the sample enters the QH phase, a QSH-QH junction is created, characterized by a chiral state propagating along the interface. This is illustrated in the current-density plot shown in the right inset. Once it has reached the opposite edge, this chiral state is partially transmitted in lead 2 and partially transmitted in lead 3, with proportions shown in the main plot. The light red curves correspond to an abrupt voltage change across the junction region while the dark blue curves correspond to a smooth transition. Parameters are $E_{\mathrm{F}}/{\Delta }_{\mathrm{so}}=0.58$ (in the left half of the sample), ${\lambda }_{\mathrm{so}}=0.02$, $l_B\simeq 8$, and widths $W_0=W_{\mathrm{c}}=40\sqrt{3}$ for the leads and central region.