Controlling spin polarization of a quantum dot via a helical edge state

We investigate a Zeeman-split quantum dot (QD) containing a single spin $1/2$ weakly coupled to a helical Luttinger liquid (HLL) within a generalized master equation approach. The HLL induces a tunable magnetization direction on the QD controlled by an applied bias voltage when the quantization axes of the QD and the HLL are noncollinear. The backscattering conductance (BSC) in the HLL is finite and shows a resonance feature when the bias voltage equals the Zeeman energy in magnitude. The observed BSC asymmetry in bias voltage directly reflects the quantization axis of the HLL spin.

Figure 1

Setup considered in this paper (a) and construction of the effective QD Hamiltonian Eq. (5) (b). An HLL is coupled to a QD described by a Kondo Hamiltonian $H_K$. A magnetic field $\vec{B}$ is applied to the QD, which is tilted with respect to the quantization axis $\widehat{z}$ of the helical edge state by an angle ${\theta }_Z$. The effective system Hamiltonian for the QD is the sum of a Zeeman term $g{\mu }_B\vec{B}·\vec{S}$ and an induced part ${\mathrm{\Delta }}_V{S}^z$ that corresponds to the spin polarization of the HLL driven by a bias voltage $V$. The resulting effective field points along $\vec{n}$ with tilt angle $\theta$ and has strength ${\mathrm{\Delta }}_S$.

Figure 2

Backscattering current and QD spin polarization (SP) in $y$ and $z$ direction for $J/v_F=0.1, \alpha /\beta v_F={10}^{-3}, {\theta }_Z=\pi /6, {\mathrm{\Delta }}_Z=15k_{\mathrm{B}}T$ (blue solid), ${\mathrm{\Delta }}_Z=0.5k_{\mathrm{B}}T$ (red dashed), and ${\mathrm{\Delta }}_Z=0$ (black dash dotted) as a function of the bias $V$ and for $K=1$ is shown. The inset shows the SP in the $z-y$ plane for the same bias range. For illustration, the Bloch sphere and the direction of the Zeeman field are added in gray. When $\left|V\right|$ is comparable to ${\mathrm{\Delta }}_Z$, the SP is along $\vec{B}$ whereas it is along the HLL quantization axis $\widehat{z}$ when $|{\mathrm{\Delta }}_V|$ is larger than ${\mathrm{\Delta }}_Z$. The spin flip along the magnetic field direction and the peculiar change of the SP within the $z-y$ plane shows clear signatures in the backscattering current. For better comparability we restored units in the plot.

Figure 3

Differential conductance $G_{\mathrm{bs}}$ as a function of bias voltage $V$ for different tilt-angles: ${\theta }_Z=\pi /2$ (green dash dotted), $\pi /3$ (orange dashed), $\pi /6$ (blue solid), ${\mathrm{\Delta }}_Z=15k_{\mathrm{B}}T, J/v_F=0.1, \alpha /\beta v_F={10}^{-3}$, and $K=1$. $G_{\mathrm{bs}}$ shows a peak when $\left|V\right|\approx {\mathrm{\Delta }}_Z$ with an asymmetry between $V>0$ and $V<0$ that vanishes when the Zeeman field is perpendicular to the lead quantization axis (${\theta }_Z=\pi /2$). The units have been restored in the plots for better comparability.

Figure 4

Renormalized differential conductance as a function of bias voltage $V$ for different HLL interaction strengths: $K=0.6$ (black dash dotted), $K=0.8$ (red dashed) and $K=1$ (blue solid), ${\mathrm{\Delta }}_Z=15k_{\mathrm{B}}T, J/v_F=0.1, \alpha /\beta v_F={10}^{-3}$, and ${\theta }_Z=\pi /6$. The inset shows the renormalized differential conductance for the parameters from Fig. 3 with $K=0.8$.