Supplementary Material for: Y-Shape Spin-Separator for twodimensional Group-IV Nanoribbons based on Quantum Spin Hall
Effect
Gaurav Gupta1†, Hsin Lin2, Arun Bansil3, Mansoor Bin Abdul Jalil1, Cheng-Yi Huang4,
Wei-Feng Tsai4 and Gengchiau Liang1*
1Department
2Graphene
of Electrical and Computer Engineering, National University of Singapore, Singapore 117576
Research Centre and Department of Physics, National University of Singapore, Singapore 117542
3Department
of Physics, Northeastern University, Boston, Massachusetts 02115, USA
4Department
of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
Here we provide supporting information that would help in understanding the results
presented in the main paper. Section S1 on Local Density of States (LDOS) shows DOS
distribution (color scale is per eV) in the Y-device as a visual aid for developing an intuitive
understanding of the device operation. Section S2 on transmission helps in comprehending
results presented in Fig. 2 of the main paper. Section S3 serves as a supplementary for
visualizing the concepts presented in the main paper pertaining to effects of the buckling field
and magnetic field on the device operation by illustrating spin distribution in the device.
Section S1: Local Density of States Distribution for 14 cell wide Y-shaped Device
Figure S1.1. 5-cell wide device for the configuration of Figs. 2(a,d), i.e. Ef = 1 meV and VBA
(VCA) = 1 mV. Note stronger confinement towards the edges on all three arms, observed from
relative distribution of red region in individual plots, as the intrinsic spin orbit coupling (λSO)
strength becomes progressively stronger from (a) to (d).
Figure S1.2. Ge device corresponding to Fig. 2(c,d), i.e. 1 mV bias for two Fermi-levels (Ef).
Figure S1.3. Ge device corresponding to Fig. 2(e,f) i.e. Ef = 5 meV for two bias (VDS).
Figure S1.4. Ge device corresponding to Fig. 3, i.e. Ef = 5 meV and bias = 4 mV.
Section S2: Transmission in Equilibrium Condition
Fig. S2(a) shows T(E) for a two-terminal Ge-NR strip. The points of inflection and the
magnitude can be easily matched against the band edge and mode density, respectively. Figs.
S2(b-f), however, illustrate T(E) from Arm-A to Arm-B for a Y-shaped device where SOC is
switched off in Fig. S2(b). Bending of the channel results in interference in the propagating
modes at all energies due to the wave nature of electrons. A similar study has been performed on
a tapered nanographite channel previously1 to discuss these interference effects. Ref [2] shows
that even a small bend in the strip geometry (broken strip geometry) induces quantum
interference. Quantum interference results in bound states below the eigenvalues for a curved
strip3. These bound states interfere with the continuum states of propagating modes, i.e. the Fano
resonance4 to yield dips observed in T(E). However, in the presence of SOC, within 2 λSO energy
range, transport is confined to the edges, and waves propagate without interference. T(E)
therefore tends to unity, and again corresponds to the mode density for spin polarized bands on
each edge. Beyond λSO, T(E) joins as a continuous curve along the energy-axis for low
transmission unpolarized states. Therefore, if λSO is sufficiently strong then spin-polarized hightransmission can be obtained for a large energy range to implement a spin-separator device. Here
we would like to draw attention to the fact that as width of the nanoribbon increases, the first
bulk band edge would move closer to λSO, and therefore, for sufficiently wide ribbon the T(E)
would not decrease as we transition out of the spin-polarized edge states at higher magnitudes of
energy. This is the reason for roughly flat T(E) for Pb in Fig. S2(f).
Figure S2. Transmission (T) vs Energy (E) for different devices for transport at equilibrium. (a)
Two-terminal Ge device (with SOC). Here, T(E) corresponds to the band structure. Absence of
SOC just changes the energy value for inflections in T(E) plot (driven by the change in bandstructure). (b) Three-terminal Y-shaped Ge device without SOC. (c) Y-shaped Ge device with
SOC. (d) Y-shaped Si device with SOC. (e) Y-shaped Sn device with SOC. (f) Y-shaped Pb
device with SOC. (a-f) For various systems T(-E) = T(E). Note that SOC creates spin-polarized
edge states that drive the T(E) to one in + λSO energy range, around which T(E) declines similar
to the case without SOC (b). If device is sufficiently wide to have first bulk band edge at ~ λSO
eV, then T(E) will be roughly flat throughout + λSO energy range. The local minima of Fig. 2(b)
and local maxima of Fig. 2(c) in the main paper should be absent for such widths.
Section S3: Spin Distribution for 14 cell wide Y-shaped Device
Figure S3.1. Spin and charge distribution for the Germanene Device of Fig. 3, i.e. Ef = 5 meV
and bias = 4 mV, where color scale gives the magnnitude of Trace(σZ.Gn). Observe that Fig. 3(e)
of the main paper shows that SP should increase for Arm-C and decrease for Arm-B on applying
λV on Arm-A. However, (b) shows a reduced magnitude of color intensity. This is due to reduced
current as shown in Fig. 3(a) of the main paper. Increase in spin polarization is verified by
plotting the normalized spin distribution as shown in Fig. S3.2(b). Color scale has units of per eV.
Figure S3.2. Normalized Spin-Polarization (SP) distribution (color scale) in Germanene Device
for Fig. 3, i.e. Ef = 5 meV and bias = 4 meV. Note that color information does not convey full
information on spin flux through the device because charge may not be flowing through a certain
region. Therefore, the results should be studied together with LDOS (Section S1) and unnormalized spin-polarization (Fig. S3.1) for understanding the concepts presented in the main
paper.
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