Geometric phase in $p-n$ junctions of helical edge states

The quantum spin Hall effect is endowed with topologically protected edge modes with a gapless Dirac spectrum. Applying a magnetic field locally along the edge leads to a gapped edge spectrum with the opposite parity for winding of spin texture for conduction and valence bands. Using Pancharatnam's prescription for the geometric phase it is shown that mismatch of this parity across a $p-n$ junction, which could be engineered into the edge by electrical gate induced doping, leads to a phase dependence in the two-terminal conductance which is quantized to either zero or $\pi$. It is further shown that application of a nonuniform magnetic field across the junction could lead to a nonquantized value of this geometric phase which is tunable between zero and $\pi$. A current asymmetry measurement which is shown to be robust against electron-electron interactions is proposed to infer the appearance of this Pancharatnam's geometric phase in transport across such junctions.

Figure 1

The right panel shows the dispersion for $n$-doped ($x<0$) and $p$-doped ($x>0$) sides of a $n-p$ junction where the black arrows show the spin texture of the dispersion. The left panel shows the spin states of incident ($i$, orange), reflected ($r$, magenta), and transmitted ($t$, red) electron wave functions at the Fermi level represented as three dots placed on the great circle contained in the $x-z$ plane on the Bloch sphere.

Figure 2

(a) Variation of the Landauer conductance ${\mathcal{G}}_{{\delta }_1,{\delta }_2}^{\varphi }$ (in units of $e^2/h$) for the case of $n-n$ (blue curve) and $n-p$ (red curve) as a function of the chemical potential $V_{{\mathcal{G}}_2}$ (or ${\delta }_2$) with $V_{{\mathcal{G}}_1}={\delta }_1=0.15$ mV, $B=50$ mT, and $\varphi =0$. The inset shows variation of the Landauer conductance as a function of the strength of the applied magnetic field $B$ with ${\delta }_1=0.15$ mV, $V_{{\mathcal{G}}_2}=0.1$ mV, and $\varphi =0$. (b) Variation of the Landauer conductance (in units of $e^2/h$) as a function of $\varphi$. (c) Variation of $\mathrm{\Delta }{\mathcal{G}}_{{\delta }_1,{\delta }_2}^{\varphi }$ (in units of $e^2/h$) and $cos\mathrm{\Omega }$ (in inset) as a function of $\varphi$. In panels (b) and (c), we have used $V_{{\mathcal{G}}_1}={\delta }_1=0.15$ mV, $V_{{\mathcal{G}}_2}=0.1$ mV, and $B=50$ mT.

Figure 3

Numerical plot of the Landauer conductance ${\mathcal{G}}_{{\delta }_1,{\delta }_2}^{\varphi }$ in units of $e^2/h$ as a function of rotation of the magnetic field $\varphi$ for different values of the junction length $l$ in the case of (a) the $n-n$ junction and (b) the $n-p$ junction. Here, we have used $V_{{\mathcal{G}}_1}=0.1$ mV, $V_{{\mathcal{G}}_2}=0.15$ mV, $B=50$ mT, and the effective Landé $g$ factor = 50. Here $l_B=1.25 \mu \mathrm{m}$.