Interference effects induced by a precessing easy-plane magnet coupled to a helical edge state

The interaction of a magnetic insulator with the helical electronic edge of a two-dimensional topological insulator has been shown to lead to many interesting phenomena. One of these is that for a suitable orientation of the magnetic anisotropy axis, the exchange coupling to an easy-plane magnet has no effect on DC electrical transport through a helical edge, despite the fact that it opens a gap in the spectrum of the helical edge [Meng et al., Phys. Rev. B 90, 205403 (2014)]. Here, we theoretically consider such a magnet embedded in an interferometer, consisting of a pair of helical edge states connected by two tunneling contacts, at which electrons can tunnel between the two edges. Using a scattering matrix approach, we show that the presence of the magnet in one of the interferometer arms gives rise to AC currents in response to an applied DC voltage. On the other hand, the DC Aharonov-Bohm effect is absent at zero temperature and small DC voltages, and only appears if the applied voltage or the temperature exceeds the magnet-induced excitation gap.

Figure 1

Schematic picture of the interferometer considered here. The interferometer can be realized by bringing helical edge channels of the same quantum spin Hall insulator sufficiently close together that they can make electrical contact (left), or by bringing the edge channels of different quantum spin Hall insulators into contact (right). Both panels show the location of eight reference points $i=1,2,...,8$ used for the calculations in the main text. The “positive” current direction for $i=1,2,3,4$ and for $i=5,6,7,8$ is to the right and to the left, respectively. The interferometer is threaded by a magnetic flux $\mathrm{\Phi }$.

Figure 2

In the theoretical description, currents are calculated for eight reference points $i=1,2,...,8$, as shown in the left panel. The “positive” current direction for $i=1,2,3,4$ and for $i=5,6,7,8$ is to the right and to the left, respectively. The two tunneling point contacts between the opposing helical edge states are described by scattering matrices $S^{\left(\mathrm{C}1\right)}$ and $S^{\left(\mathrm{C}2\right)}$. The definitions of the transmission coefficients $T_1$ and $T_1^{\prime }$ and the reflection coefficient $R_1$ for the left tunneling point contact are shown in the right panel.

Figure 3

Schematic picture of the “right point contact open” geometry for a realization in which the interfering edge channels are on the same quantum spin Hall insulator (left) and on different quantum spin Hall insulators (right). The positions labeled $i=1,2,...,8$ refer to the reference positions used for the calculations in the main text.

Figure 4

Distribution functions $f_{3+}\left({\varepsilon }_{+}\right)$ and $f_7\left({\varepsilon }_{-}\right)$ for bias voltage $eV<2{\mathrm{\Delta }}^{\prime }/(2-T_1^{\prime })$ (top panel) and $eV>2{\mathrm{\Delta }}^{\prime }/(2-T_1^{\prime })$ (bottom panel). The bias voltage $V$ is applied to lead “1” only; the (electro)chemical potentials of leads “2,” “7,” and “8” are held constant at the value $\mu =0$.

Figure 5

Schematic picture of the “right point contact closed” geometry for a realization in which the interfering edge channels are on the same quantum spin Hall insulator (left) and on different quantum spin Hall insulators (right). The positions labeled $i=1,2,...,8$ refer to the reference positions used for the calculations in the main text.

Figure 6

Contributions to the AC current amplitude $I_1\left({\omega }_{\mathrm{M}}\right)$ in lead “1” that depend on the distribution function $f_2\left(\varepsilon \right)$ in lead “2.” The panels (a)–(f) are in the same order as the terms of Eq. (41), but when attempting to match the expressions from the panels to Eq. (41) care must be taken as explained in the following. An overall minus sign that comes from the definition of the current $I_1\left(\omega \right)$ in Eq. (3) is not included. In panel (a) unitarity of $S^{\left(\mathrm{M}\right)}$ must be used to make the replacement $-r_{\mathrm{M}}^{*}\left({\varepsilon }_{-}\right)t_{\mathrm{M}}\left({\varepsilon }_{-}\right)=r_{\mathrm{M}}^{\prime }\left({\varepsilon }_{-}\right)t_{\mathrm{M}}^{\prime *}\left({\varepsilon }_{-}\right)$ to arrive at the corresponding term in Eq. (41). In comparison to Eq. (41), the contribution from panel (f) contains an extra factor of $|r_{\mathrm{M}}\left({\varepsilon }_{-}\right){|}^2$. This factor has been dropped in Eq. (41), as $\sqrt{T_1^{\prime }{R}_1}{r}_{\mathrm{M}}^{\prime }\left({\varepsilon }_{-}\right){\left|r_{\mathrm{M}}\left({\varepsilon }_{-}\right)\right|}^2\approx \sqrt{T_1^{\prime }{R}_1}{r}_{\mathrm{M}}^{\prime }\left({\varepsilon }_{-}\right)$ to the level of approximation made in Eq. (41). In panels (c) and (f) the property ${\widehat{m}}_{-}{\widehat{m}}_{+}=\mathbb{1}$ is used, as the red or blue paths reflect off both sides of the magnet. The factors of $-1, \sqrt{R_1}$, and $i\sqrt{T_1^{\prime }}$ printed along the paths come from the scattering matrix elements of $S^{\left(\mathrm{C}1\right)}$ and $S^{\left(\mathrm{C}2\right)}$ related to the paths that are shown in the limits considered.

Figure 7

From top to bottom: Precession frequency ${\omega }_{\mathrm{M}}$, DC current $I_2\left(0\right)$ at zero Aharonov-Bohm phase $\varphi , \varphi$-dependent contribution to DC current defined in Eq. (51), and the magnitude $|I_2\left({\omega }_{\mathrm{M}}\right)|$ and $|I_2\left(2{\omega }_{\mathrm{M}}\right)|$ of the AC current components at frequency ${\omega }_{\mathrm{M}}$ and $2{\omega }_{\mathrm{M}}$, as a function of the bias voltage $V$ at lead “1.” All currents are evaluated in lead “2.” The solid curves show the numerical evaluation of the full solution without the approximation of small $R_j, T_j^{\prime }, j=1,2$. The dashed curves are obtained using the analytical expressions (47, 48, 49, 50), with the exception of the second panel, where we have included all $\varphi$-independent terms up to second order in ${\left(R_j\right)}^{1/2}, {\left(T_j^{\prime }\right)}^{1/2}$, not only those of lowest nontrivial order given in $I_2\left(0\right)$ in Eqs. (48). The parameters are chosen as $T_1=0.79, R_1=0.09, T_1^{\prime }=0.12, T_2=0.78, R_2=0.06, T_1^{\prime }=0.16, k_{F}L=1$, and $k_F{L}_3=0.1$.

Figure 8

Aharonov-Bohm current $\delta I_2(0,\varphi )$ from Eq. (54) as a function of applied bias $V$ in lead “1” for several values of $k_{\mathrm{B}}T/\mathrm{\Delta }$; from bottom to top they are $0.01,0.1,0.25,0.5$, and 2. The precession frequency ${\omega }_{\mathrm{M}}$ is determined from the solution of Eq. (47) at finite $T$. The phases are chosen such that $sin\left[k_F(L_3+L)+\varphi \right]=1$.