Isolating Majorana fermions with finite Kitaev nanowires and temperature: Universality of the zero-bias conductance

The zero-bias peak (ZBP) is understood as the definite signature of a Majorana bound state (MBS) when attached to a semi-infinite Kitaev nanowire (KNW) nearby zero temperature. However, such characteristics concerning the realization of the KNW constitute a profound experimental challenge. We explore theoretically a QD connected to a topological KNW of finite size at nonzero temperatures and show that overlapped MBSs of the wire edges can become effectively decoupled from each other and the characteristic ZBP can be fully recovered if one tunes the system into the leaked Majorana fermion fixed point. At very low temperatures, the MBSs become strongly coupled. We derive universal features of the conductance as a function of the temperature and the relevant crossover temperatures. Our findings offer additional guides to identify signatures of MBSs in solid state setups.

Figure 1

Sketch of the setup here explored. A piece of semiconductor nanowire is placed on an s-wave superconductor (SC) material. In this semiconductor-superconductor segment, by considering suitable Zeeman field and spin-orbit coupling, a topological KNW emerges, giving rise to overlapped (${\varepsilon }_m$) MBSs ${\eta }_1$ and ${\eta }_2$ at their edges. A QD (gate-tunable energy ${\epsilon }_d$) between two metallic leads (coupling $V$) is built in the semiconducting segment, where ${\eta }_1$ leaks into (coupling $\lambda$) and can be detected as a ZBP.

Figure 2

Zero-bias conductance as a function of the temperature in logarithmic scale for ${\epsilon }_d=0.1D, \mathrm{\Gamma }=0.005\pi D, \lambda =0.0008D, {\epsilon }_m=5\times {10}^{-8}D$. The model parameters were carefully chosen to make the several fixed points very clear. Continuous lines are given by Eqs. (23) and (28) with crossover temperatures $T_1$ [Eq. (20)] and $T_2$ [Eq. (21)]. The inset pictorially shows the Majorana representation of the problem in the leaked Majorana (LM) fixed point, considering a tight-binding description of the conduction band, comprising two uncoupled zigzag chains of Majorana fermions (blue circles). Majorana fermions ${\eta }_1$ and ${\zeta }_2$ are strongly coupled and removed from the system (see the horizontal arrow). One of the Majorana fermion chains gives rise to half of the single-particle excitations of the free conduction band, while the other chain, coupled to the Majorana fermion ${\zeta }_1$, gives rise to half of the single-particle excitations of the ${\epsilon }_d=0$ resonant level model. This implies a conductance equal to half of that in such resonant level model, explaining the characteristic value 0.5 $e^2/h$ in the LM fixed point.

Figure 3

NRG flow for the lowest 20 rescaled energies. The model parameters are the same as in Fig. 2, $\mathrm{\Lambda }=4.0$ and $z=1.0$. The fixed points appearing in the conductance correspond to the intervals in the number of iterations $N$ where the rescaled energies also become approximately constant. The columns of horizontal lines on the left and on the right correspond to many-body excitation energies formed by combining single-particle excitations of the resonant level model with ${\epsilon }_d=0.1D$, whereas the central column of horizontal lines correspond to many-body excitation energies formed combining single-particle excitations of the free conduction band with single-particle excitations of the ${\epsilon }_d=0$ resonant level model. The agreement between the energies in the flow and those lines confirms the discussion in Fig. 2 and in the main text.

Figure 4

(a) Zero-bias conductance as a function of the temperature in logarithmic scale for 11 different model parameter pairs $(\lambda ,{\epsilon }_m)$. In all curves, ${\epsilon }_d=0$ and $\mathrm{\Gamma }=\pi 0.005D$. The inset shows the distribution of the corresponding crossover temperatures $T_1$ and $T_2$ [Eqs. (20) and (21)]. Curves 1–5 have the same $\lambda$ and the same $T_1$. As $T_2$ increases (with ${\epsilon }_m^2$), approaching $T_1$, the crossovers will merge at some point, and the LM fixed point ceases to be achieved. Curves 6–11 have the same ${\epsilon }_m$, and their minima between the crossovers are approximately at the same point [Eq. (22)]. As $\lambda$ increases, $T_1$ rises and $T_2$ lowers, broadening the LM fixed point plateau. (b) and (c) Conductance as a function of $T/T_2$ and $T/T_1$, respectively. The points collapse into the continuous lines in Eq. (23) with $\mathcal{H}\left(t\right)$ in Eq. (28). In (b), when $T_1$ is not much higher than $T_2$, deviations from the universal curve will start at low temperatures. In (c), deviations at $T<T_1$ occur when $T_2$ is not much lower than $T_1$ while deviations at $T>T_1$ occur when $T_1$ is not much smaller than $\mathrm{\Gamma }/k_B$.

Figure 5

(a) Conductance as a function of the temperature for different values of the QD energy. Experimentally accessible values were used for the other model parameters (see main text): $D=20$ meV, $\mathrm{\Gamma }=0.1$ meV, ${\epsilon }_m=0.001$ meV, $\lambda =0.02236$ meV. From top to bottom, we have ${\epsilon }_d=0.0,0.1,0.164,0.244$ and $0.4\text{meV}$, respectively. The continuous lines represent the calculated conductances for each ${\epsilon }_d$, while the points represent the universal expression [Eq. (23)] for both $T^{★}=T_1$ and $T^{★}=T_2$ with $\mathcal{H}\left(t\right)$ given by Eq. (28). In the highlighted range ($0.03\mathrm{K}<\mathrm{T}<0.3\mathrm{K}$), we see that the universal expression (with $T^{★}=T_1$) describes accurately the conductance behavior. This also happens for $T\sim T_2$, but, in this case, the temperatures are more difficult to be achieved experimentally. (b) The continuous line represents the conductance calculated from Eq. (23) at $T=0.1$ K and the points are the exactly computed conductance at this temperature for the values of ${\epsilon }_d$ used in panel (a).