Renormalization of the quantum dot $g$-factor in superconducting Rashba nanowires

We study analytically and numerically the renormalization of the $g$-factor in semiconducting Rashba nanowires (NWs), consisting of a normal and superconducting section. If the potential barrier between the sections is high, a quantum dot (QD) is formed in the normal section. For harmonic (hard-wall) confinement, the effective $g$-factor of all QD levels is suppressed exponentially (power law) in the product of the spin-orbit interaction (SOI) wave vector and the QD length. If the barrier between the two sections is removed, the $g$-factor of the emerging Andreev bound states is suppressed less strongly. In the strong SOI regime, and if the chemical potential is tuned to the SOI energy in both sections, the $g$-factor saturates to a universal constant. Remarkably, the effective $g$-factor shows a pronounced peak at the SOI energy as function of the chemical potentials. In addition, if the SOI is uniform, the $g$-factor renormalization as a function of the chemical potential is given by a universal dependence, which is independent of the QD size. This prediction provides a powerful tool to determine experimentally whether the SOI in the whole NW is uniform and, moreover, gives direct access to the SOI strengths of the NW via $g$-factor measurements. In addition, it allows one to find the optimum position of the chemical potential for bringing the NW into the topological phase at large magnetic fields.

Figure 1

Schematic setup of the system: a one-dimensional NW is aligned along the $x$ axis in the presence of an external magnetic field $\mathbf{B}$ applied parallel to the NW and perpendicular to the Rashba SOI vector, which points along the $z$ direction. The NW consists of two sections: In the normal section (blue), hosting a QD, the chemical potential is controlled by the external side gate (orange). The tunnel gate (violet) controls the height of the barrier potential ${\mu }_b$ and separates the normal section from the superconducting section (green) with proximity-induced superconductivity.

Figure 2

Effective Zeeman splitting $\delta E/t$ between the two lowest energy states for a normal QD with hard-wall confinement as a function of Zeeman energy $V_Z/t$ (a) in the regime of weak SOI, $\overline{\alpha }/t=0.05$, and (b) in the regime of strong SOI, $\overline{\alpha }/t=0.4$. The results are obtained numerically (black dotted lines) by exact diagonalization of Eq. (1). The Zeeman splitting is linearly proportional to the Zeeman energy for small values of $V_Z$ that do not exceed the QD level spacing ${\delta }_{n=1}$ (blue dashed line). The red solid line corresponds to the linear fit giving the renormalized values for the $g$-factor: (a) $g^{*}/g=0.13$ and (b) $g^{*}/g=1.2\times {10}^{-4}$. The parameters are chosen as $N_n=100$ and ${\mu }_j/t=-2$.

Figure 3

Renormalized $g$-factor $g^{*}/g$ as a function of $k_{\text{so}}L$ for a QD with hard-wall confinement for the four lowest levels (a) $n=1$, (b) $n=2$, (c) $n=3$, and (d) $n=4$. Here, $L$ is the length of the QD and $k_{\text{so}}=m\alpha /{\hslash }^2$ the SOI wave vector. The stronger the SOI or the longer the QD is, the more pronounced is the $g$-factor suppression. We note that the $g$-factor has zeros. The results are obtained numerically (black solid line) by exact diagonalization of Eq. (1) and analytically (red dashed line) by making use of Eq. (9). There is excellent agreement between the two approaches. The parameters are chosen as $N_n=99$ and $V_Z/t={10}^{-4}$.

Figure 4

Renormalized $g$-factor $g^{*}/g$ as a function of $k_{\text{so}}l$ for a QD with a harmonic confinement potential for the same set of parameters as in Fig. 3. Here, $l$ is the effective QD size and $k_{\text{so}}$ is the SOI wave vector. Again, there is an excellent agreement between numerical (black solid line) and analytical (red dashed line) results. The renormalized $g$-factor is suppressed exponentially. For higher orbital levels, the oscillations in $g^{*}$ allows one to fine-tune to the regime in which $g^{*}=0$. Numerical results are obtained for $\hslash {\omega }_0/t=0.02$.

Figure 5

(a) Energy spectrum, $E/\mathrm{\Delta }$, and (b) effective $g$-factor, $g^{*}/g$, of the subgap states as a function of the dot size $N_n$ in the presence of a high (${\mu }_b=-t$) and sharp ($W_b=1$) barrier. In this configuration, we recover results obtained for an isolated QD with hard-wall confinement. The first ($m=1$, red), the second ($m=2$, blue), and the third ($m=3$, green) dot levels appear below the gap at different values of the dot size. Here, we label QD levels according to their energy distance to the chemical potential. (b) The effective $g$-factor, ${(g^{*}/g)}_m$, found numerically (colored dots) can be described well by analytical expressions obtained for hard-wall confinement [see Eq. (9)], where we identify the three lowest orbital levels of the QD with $n=1$ (solid black line), $n=2$ (dashed black line), and $n=3$ (dot-dashed black line). We note that numerically we can determine only the $g$-factors for the QD levels with energies below the gap. The parameters are chosen as $\overline{\alpha }/t=0.08$ ($k_{\text{so}}a=0.08$), ${\mu }_s/t={\mu }_n/t=-2,\mathrm{\Delta }/t=0.01$, and $V_Z/t={10}^{-4}$.

Figure 6

The same as in Fig. 5, however, in the absence of a barrier, ${\mu }_b=0$. (a) As the barrier is removed, the wave functions corresponding to the lowest QD levels become ABSs which extend also into the superconducting section. As a result, anticrossings between electron and hole ABS levels are induced by the superconducting pairing. The renormalization of the effective $g$-factor is still substantial, however, its functional form deviates from one obtained for an isolated normal QD. Generally, the $g$-factor suppression is less pronounced for large QDs.

Figure 7

Effective $g$-factor of the lowest ABS as a function of the dot size $N_n$ for different values of SOI: weak SOI regime with $\overline{\alpha }/t=0.08$ (red solid line), moderate SOI regime with $\overline{\alpha }/t=0.2$ (blue dashed line), and strong SOI regime with $\overline{\alpha }/t=0.4$ (green dot-dashed line). The dependence on the SOI strength is less pronounced compared to the case of a normal isolated QD. The oscillations with the period set by $k_{\text{so}}$ are most pronounced in the weak SOI regime. The parameters are chosen as $\mathrm{\Delta }/t=0.01,{\mu }_n/t={\mu }_s/t=-2$, and $V_Z/t={10}^{-4}$.

Figure 8

Effective $g$-factor of the lowest ABS as a function of dot size $N_n$ for different gap values: $\mathrm{\Delta }/t=0.2$ (red solid line), $\mathrm{\Delta }/t=0.02$ (blue dashed line), and $\mathrm{\Delta }/t=0.002$ (green dot-dashed line). The oscillations of the effective $g$-factor almost disappear and $g^{*}/g$ becomes constant with decreasing $\mathrm{\Delta }$. If the dot size is much smaller than the SC localization length, the spatial support of the ABS is shifted to the superconducting section, and, thus, there is only a weak dependence on the parameters of the normal section. The parameters are chosen as $\overline{\alpha }/t=0.08,{\mu }_n/t={\mu }_s/t=-2$, and $V_Z/t={10}^{-4}$.

Figure 9

(a) Energy $\epsilon /\mathrm{\Delta }$ of the lowest ABS as function of dot size $N_n$ in the absence of magnetic field, $V_Z=0$. As $N_n$ is increased, the bound state energy decreases and the ABS moves close to the chemical potential. The black line is obtained by numerical diagonalization of the tight-binding model [see Eq. (15)], the red dotted line represents the numerical solution of the transcendental equation for the energy spectrum obtained analytically [see Eq. (A10)], and the blue dashed line represents an approximate analytical solution [see Eq. (16)]. (b) Effective $g$-factor of the lowest ABS as a function of dot size $N_n$. The $g$-factor exhibits only weak dependence on $N_n$. For large QDs, it saturates to the length-independent value $g^{*}/g=1/\pi$. The black line is obtained by numerical diagonalization of the tight-binding model [see Eq. (15)], the red dotted line represents the numerical solution obtained from the transcendental equation for the spectrum [see Eqs. (A10) and (A19)], and the blue dashed line represents an approximate analytical solution [see Eqs. (16) and (18)]. The parameters are chosen as $V_Z/t={10}^{-4},\overline{\alpha }/t=0.4,\mathrm{\Delta }/t=0.01,{\mu }_b=0,{\mu }_s/t={\mu }_n/t=-2$, and $N_s=500$.

Figure 10

Effective $g$-factor of the lowest ABS as function of dot size $N_n$ for different values of ${\mu }_n$ (a) in weak SOI regime, $\overline{\alpha }/t=0.08$, and (b) in strong SOI regime, $\overline{\alpha }/t=0.4$. In panel (a) [(b)], the red solid line corresponds to ${\mu }_n/t=-2$ (${\mu }_n/t=-2$), blue dashed line to ${\mu }_n/t=-1.98$ (${\mu }_n/t=-1.8$), and green dot-dashed line to ${\mu }_n/t=-1.96$ (${\mu }_n/t=-2.1$). The other parameters are chosen as $\mathrm{\Delta }/t=0.01,{\mu }_s/t=-2$, and $V_Z/t={10}^{-4}$. In the strong SOI regime, the $g$-factor is maximal if the chemical potential is tuned to the SOI energy, which corresponds to ${\mu }_n/t=-2$ in our model. Shifting away from this fine-tuned point results in strong suppression of $g^{*}$.

Figure 11

Effective $g$-factor of the lowest ABS as function of chemical potential ${\mu }_n/t$ in the normal section for different dot sizes: $N_n=40$ (red solid line), $N_n=70$ (blue dashed line), and $N_n=100$ (green dot-dashed line). In panel (a), the chemical potential in the SC section is fixed to the SOI energy ${\mu }_s/t=-2$, whereas in panel (b) we keep the chemical potential uniform in the whole NW length, ${\mu }_n={\mu }_s$. We note that in the latter case, the $g$-factor dependence on ${\mu }_n$ is described by the universal curve independent of the QD size. In both panels, $g^{*}/g$ has a maximum at ${\mu }_n/t=-2$, corresponding to tuning the chemical potential to the SOI energy. The parameters are chosen as $\overline{\alpha }/t=0.4,\mathrm{\Delta }/t=0.01$, and $N_s=300$.

Figure 12

Effective $g$-factor of the lowest ABS as function of chemical potential ${\mu }_n={\mu }_s$ for different dot sizes: $N_n=40$ (red solid line), $N_n=70$ (blue dashed line), and $N_n=100$ (green dot-dashed line). The SOI is chosen to be stronger in the normal section, ${\overline{\alpha }}_n/t=0.4$, than in the superconducting section, ${\overline{\alpha }}_s/t=0.08$. Again, the $g$-factor reaches a maximum if both chemical potentials are tuned to their corresponding SOI energies. The superconducting gap is fixed to $\mathrm{\Delta }/t=0.01$.