Numerical study of PbTe-Pb hybrid nanowires for engineering Majorana zero modes

Epitaxial semiconductor-superconductor (SM-SC) hybrid nanowires are potential candidates for implementing Majorana qubits. Recent experimental and theoretical works show that charged impurities in SM remain a major problem in all existing hybrid nanowires, in which the SM is either InAs or InSb while the SC is mainly Al. Here, we theoretically validate the recently proposed PbTe-Pb hybrid nanowire as a potential candidate for Majorana devices. By studying the electrostatic and electronic properties of PbTe nanowires, we demonstrate that the huge dielectric constant of PbTe endows itself a high tolerance of charged impurity, which is a potential advantage over InAs and InSb nanowires. Moreover, we find that the effective axial Landé $g$ factor and Rashba spin-orbit coupling strength of PbTe nanowires are comparable to those of InAs nanowires. The conceivable merits of using Pb as the SC are (i) Pb has a larger superconducting gap, higher critical temperature, and higher parallel critical magnetic field than those of Al; (ii) a superconducting gap comparable with those of InAs-Al and InSb-Al can be induced in PbTe-Pb even by a weak coupling between Pb and PbTe, which simultaneously relieves the adverse renormalization and induced disorder effects on SM from SC; and (iii) Pb film can be grown on PbTe with a thin buffer CdTe layer in between, solving the lattice mismatch problem as an important source of disorder. In the presence of a parallel magnetic field, we show that the typical BdG energy spectrum and tunneling spectroscopy of PbTe-Pb resemble those of InAs and InSb based hybrid nanowires exposed to a tilting magnetic field, as a result of the highly anisotropic Landé $g$ factors of PbTe nanowires. The calculated topological phase diagrams of PbTe-Pb indicate that the multivalley character of PbTe makes it easier than InAs and InSb to access topological superconducting phases. Our results could facilitate the experimental realization of PbTe-Pb hybrid nanowires and inspire further theoretical works.

Figure 1

(a) Schematic view of a squared PbTe-Pb hybrid nanowire placed on a ${\text{HfO}}_2$ dielectric layer, together with a laboratory coordinate system whose $z$ axis is along the wire's axial direction and the $x$ and $y$ axes are perpendicular to the wire's side facets. The device is gated from below by a back-gate voltage $V_{bg}$. (b) Transverse profile of the device enclosed by a large enough squared box. The relevant length scales are indicated with double arrows. (c) Longitudinal axes (colored solid lines) of the four L valleys of a particular PbTe nanowire grown along [100] direction. The orientation of a valley is defined by azimuth angles $(\alpha ,\beta )$. (d) The $\mathbf{k}·\mathbf{p}$ Hamiltonian (1, 2, 3, 4) for a valley, e.g., $\left[1\overline{1}1\right]$, of PbTe is established in a local coordinate system $x^{\prime }\text{−}{y}^{\prime }\text{−}{z}^{\prime }$ whose $z^{\prime }$ axis is along the longitudinal axis of the valley. The anisotropic effective electron masses of PbTe manifest as an ellipsoid constant energy surface (in gray) around the $L$ point.

Figure 2

View of the longitudinal axes (colored solid lines) of the four valleys in the laboratory coordinate system $x\text{−}y\text{−}z$ of PbTe nanowires with nine different growth orientations. The dashed lines are shown to aid the 3D visualization. By molecular beam epitaxy technique, wires (I)-(III) can be grown on CdTe (001) substrate, wires (IV)-(VII) on CdTe (110) substrate, and wires (VIII)-(IX) on CdTe (111) substrate.

Figure 3

(a) Conduction band edge $E_c(x,y)$ of a squared PbTe nanowire with side length $l=100$ nm at back-gate voltage $V_{bg}=-8$ V. [(b) and (c)] Horizontal line cuts of conduction ($E_c$) and valence ($E_v$) band edges, respectively, along the dotted line in (a) at different $V_{bg}$ that are uniformly varied. The dashed lines mark the Fermi level. [(d) and (f)] Results calculated with the same parameters as those of (a)–(c) except that the dielectric constant of the nanowire is changed to 15 that is realistic for InAs and InSb. The associated values of $V_{bg}$ are indicated in each figure.

Figure 4

Dispersions of the nine PbTe nanowires shown in Fig. 2 with side length $l=100$ nm at $V_{bg}=-8$ V and $B_z=0$. Valleys in (a) and (d) are doubly degenerate due to a symmetry protection. Whereas, valley degeneracy is partly lifted in (b), (e), and (h) and is completely lifted in the remaining figures.

Figure 5

The left and right panels show the changes of the conduction band edge $\delta E_c(x,y)$ with respect to Figs. 3 and 3, respectively, induced by a uniform and infinitely long cylinder charged impurity with charge density of ${\rho }_{\mathrm{imp}}={10}^{19}e/{\text{cm}}^3$ and radius of 2 nm centered at different position $(x_{\mathrm{imp}},y_{\mathrm{imp}})$. [(a) and (d)] $(x_{\mathrm{imp}},y_{\mathrm{imp}})=(-40,0)\mathrm{nm}$, [(b) and (e)] $(x_{\mathrm{imp}},y_{\mathrm{imp}})=(-20,0)\mathrm{nm}$, and [(c) and (f)] $(x_{\mathrm{imp}},y_{\mathrm{imp}})=(0,0)\mathrm{nm}$. Notably, $\delta E_c$ in the left panel are about 50 times smaller than those in the right, indicating that the huge dielectric constant of PbTe nanowire makes it less sensitive to charged impurity compared to InAs and InSb nanowires.

Figure 6

Changes of the dispersion at $k_z=0$ of wire case II, which result from the changes of the conduction band edge shown in Fig. 5. Different symbols denote the subbands belonging to the four valleys indicated in (a). Results of valley $\left[\overline{1}\overline{1}1\right]$ and [111] coincide as these two valleys are degenerate.

Figure 7

Dependence of the SOC strength $\alpha$ of the four valleys of PbTe nanowires grown along the nine different orientations shown in Fig. 2 on the ratio of uniform electric fields $E_x$ and $E_y$ with the modulus $E={(E_x^2+E_y^2)}^{1/2}$ being fixed at 1 mV/nm. $\alpha ={({\alpha }_x^2+{\alpha }_y^2+{\alpha }_z^2)}^{1/2}$ is calculated from Eqs. (12) and (19) and it scales linearly with $E$. These results are independent of the device geometry. In each figure, the axial effective electron masses $m_z^{*}$ of valley $\left[1\overline{1}1\right]$, $\left[\overline{1}11\right]$, $\left[\overline{1}\overline{1}1\right]$, [111] are indicated from top to bottom.

Figure 8

Dependence of $|cos\theta |$, with $\theta$ being the angle between the vectors of SOC and Zeeman fields, of the four valleys of PbTe nanowires grown along the nine different orientations shown in Fig. 2 on the ratio of uniform electric fields $E_x$ and $E_y$. These results are independent of the device geometry. The markers in each figure are the same as those in Fig. 7.

Figure 9

(a) Lower bound of the back-gate voltage $V_{bg}^{lb}$ as a function of the side length $l$ of a squared PbTe nanowire. For $V_{bg}$ above $V_{bg}^{lb}$ the entire valence band edge is below the Fermi level such that no hole states are populated. (b)-(j) Side length dependence of the accessible maximum SOC strength of the four valleys of PbTe nanowires grown along the nine different orientations shown in Fig. 2. These results are for the particular device shown in Fig. 1 with $V_{bg}=V_{bg}^{lb}$. The markers in each figure are the same as those in Fig. 7.

Figure 10

Typical BdG energy spectra of the 1D effective Hamiltonian (18) of PbTe-Pb hybrid nanowire with different parameters. The effective Landé factors and electron masses are used as those of valley $\left[1\overline{1}1\right]$ of (a) wire case II and (b)–(d) wire case III as listed in Table 4, while the chemical potentials are indicated in each figure. Other parameters used are ${\alpha }_x=5$ meVnm, ${\alpha }_y=25$ meV nm, ${\alpha }_z=0$, and ${\mathrm{\Delta }}_{\mathrm{ind}}=0.3\mathrm{meV}$. $B_{c1}$ and $B_{c2}$ are two critical magnetic fields defined in Eqs. (23) and (24), respectively. In (a)–(c), a pair of MZMs exist in the topological superconducting region $B_{c1}<B_z<B_{c2}$. No MZMs exists in (d) because $B_{c2}<B_{c1}$. (e)–(h) are calculated with the same parameters as (a)–(d) except that the wire length is decreased from 5 to $1.5\mu \text{m}$. Due to the finite size effects, no gap closing is observed at $B_z=B_{c2}$ in (e)–(g).

Figure 11

BdG energy spectra (left panel) and tunneling conductance maps (right panel) for a PbTe-Pb hybrid nanowire made up of wire case V with a finite length of $2\mu m$. The results are for the particular device geometry shown in Fig. 1 with $l=40$ nm and $V_{bg}=-10.5$ V. (a)–(c) and (e)–(g) show valley-resolved results while (d) and (h) take all valleys into account. Note that valleys $\left[1\overline{1}1\right]$ and $\left[\overline{1}11\right]$ are degenerate as indicated on top of (a). The MZMs shown in (b) and (d) manifest themselves as sharp quantized zero-bias peaks in (f) and (h), respectively, in a wide scope of $B_z$ ranging from 0.7 to 1.5 T.

Figure 12

Phase diagrams of PbTe-Pb hybrid nanowires made up of PbTe nanowires grown along seven different orientations as indicated. The results are for the particular device geometry shown in Fig. 1 with the side length $l=40$ nm. The gray and white regions represent metallic phase and trivial superconducting phase, respectively, while the other colored regions represent topological superconducting phases harboring $N$ pairs of MZMs as indicated in the right bottom. For the $N=1$ phases, the valley from which the MZMs are created is distinguished by different colors. The dashed line in (c) marks the parameter regime explored in Fig. 11.