Transport Discovery of Emerging Robust Helical Surface States in $Z_2=0$ Systems

We study the possibility of realizing robust helical surface states in $Z_2=0$ systems. We find that the combination of anisotropy and finite-size confinement leads to the emergence of robust helical edge states in both two-dimensional and three-dimensional $Z_2=0$ systems. By investigating an anisotropic Bernevig-Hughes-Zhang model in a finite sample, we demonstrate that the transport manifestation of the surface states is robust against nonmagnetic disorder, resembling that of a $Z_2=1$ phase. Notably, the effective energy gap of the robust helical states can be efficiently engineered, allowing for potential applications as valley filters and valley valves. The realization of emerging robust helical surface states in realistic materials is also discussed.

Figure 1

[(a)– (c)] Schematic plots of helical edge modes in the anisotropic BHZ stripe with y termination, for $m=2.00$, 1.64, 1.56. The vertical arrows represent electron spin. In subplot (b), the helical edge channels around $k_x=\pi$ are hybridized due to finite size confinement, leaving one pair of helical edge channels around $k_x=0$. Thus, these conducting channels resemble subplot (c).

Figure 2

[(a),(e)] Schematic plot of two-terminal and $\pi$-bar devices . The disorder is considered in the central (red) region. The size parameters are $L_x=120$, $W_x=120$, $W_y=60$. [(b)–(d)] Two-terminal conductance values $G_{12,12}$ of device (a); [(f)–(h)] nonlocal conductance values $G_{14,23}$ of device (e) versus Fermi energy ${\epsilon }_F$ for different disorder strengths $W$ at various $m=2.00$ [(b),(f)], 1.64 [(c),(g)], 1.56 [(d),(h)]. The error bars denote the conductance fluctuation.

Figure 3

[(a),(c)] The two-terminal conductance $G_{12,12}$ versus Fermi energy ${\epsilon }_F$ for different sample widths $W_y$ at various $m=1.64$ (a), 1.56 (c), with $L_x=120$ and $W=1.8$. [(b),(d)] Finite size energy gap $\mathrm{\Delta }$ of $k_x=0(\pi )$ as a function of $W_y$ at fixed $m=1.64$ (b) and 1.56 (d).

Figure 4

[(a)–(c)] The surface energy bands of anisotropic WTI film with parameters (a) $m_x=0.9$, $m_y=1.1$, $N_z=120$; (b) $m_x=0.9$, $m_y=1.1$, $N_z=20$; (c), $m_x=1.1$, $m_y=0.9$, $N_z=20$. $N_z$ is the sample thickness. (d) Schematic diagram of the valley filter and valley valve device. The Fermi energy in region II (IV) can be tuned by the attached gates. The black arrows with in (out) direction represent the tensile (compressive) strain. [(e)–(h)] Schematic plots of the working mechanisms for the device. Open (filled) circles denote the electrons in $X(Y)$ valleys.