Makoto Kohda, Shuji Nakamura, Yoshitaka Nishihara, Kensuke Kobayashi, Teruo Ono,. Jun-ichiro Ohe, Yasuhiro Tokura, Taiki Mineno, and Junsaku Nitta ...
Spin-orbit induced electronic spin separation in semiconductor nanostructures
Makoto Kohda, Shuji Nakamura, Yoshitaka Nishihara, Kensuke Kobayashi, Teruo Ono, Jun-ichiro Ohe, Yasuhiro Tokura, Taiki Mineno, and Junsaku Nitta
1
(b) (a)
(c)
Supplementary Figure S1. Lateral spin orbit interaction induced by the asymmetric lateral potential confinement. (a)Schematic picture of the quantum point contact, (b) the lateral confinement potential along y direction (dashed line in (a)) with symmetric and asymmetric potential cases, and (c) induced lateral effective magnetic fields at +y and –y edges for symmetric and asymmetric potentials.
2
Supplementary Discussion Lateral spin orbit interaction in a quantum point contact: Supplementary Figures S1(a) and S1(b) show the schematic pictures of the quantum point contact and the lateral confinement potential energy U along the y direction (the dashed line). The Hamiltonian of the lateral SOI is described by23,
(
)
r r r r r H LSOI = βσ ⋅ k × ∇U = gµ Bσ ⋅ BLSOI S1 where β is the intrinsic SOI parameter, σ is the vector of Pauli-spin matrices, k is the wave vector, and U is the lateral confinement potential of the QPC, g is the electron g-factor, µB is the Bohr magneton, and BLSOI is the effective magnetic field induced by the lateral SOI. The direction of induced effective magnetic field is perpendicular both to the electron momentum and the potential gradient directions. Recently, P. Debray et al. presented the 0.5(2e2/h) plateau by applying the asymmetric gate bias voltages to the side gates in the QPC and explained the lateral SOI as the origin of the 0.5(2e2/h) plateau22. According to the Debray’s interpretation, moving electrons with opposite spins experience opposite lateral SOI forces that leads to an accumulation of opposite spins at the opposite transverse edges. In the case of symmetric potential gradient shown with the blue line in Supplementary Figure S1(b), which is enabled by the application of the symmetric side gate bias voltage, the induced effective magnetic fields become the same magnitude but opposite directions at the opposite transverse edges of the lateral confinement potential (-y and +y edges) shown in Supplementary Figure S1(c). Resulted spin polarization at the opposite edges cancel each other, giving zero net spin polarization. However, in the asymmetric lateral potential shown with the red line in Supplementary Figure S1(b), since the magnitude of the effective magnetic fields are different at the transverse edges due to the difference of the potential gradient shown in Supplementary
3
Figure S1(c), the spin polarization is induced. In the Debray’s experiment22, the appearance/disappearance of 0.5(2e2/h) plateau is controlled by the voltage difference between the two side gates of the QPC. In our experiment, since we always applied the identical voltage to two side gates, we expect this asymmetric lateral confinement to be negligible. To be sure that the lateral SOI is not the origin in our case, we applied the asymmetric gate bias voltage dependence. Since the modulation of a lateral electric field by the asymmetric side gates does not play a role in the appearance or disappearance of 0.5(2e2/h), the lateral SOI can be ruled out as the origin of 0.5(2e2/h) observed in the present work.
Supplementary Methods Spatial modulation of Rashba SOI for the spin polarization in InGaAs QPC: We consider a 2DEG in the x-y plane and the current flows in the x direction, while a fixed boundary condition and spatial modulation of the Rashba SOI due to the lateral confinement of the QPC are imposed in the y direction. To explore the electron transport under the spatial modulation of the Rashba SOI, we calculate the time evolution of the wave packet by the equation-of-motion method based on the exponential product formula20. We consider the tight-binding model with a lattice spacing a and the hopping energy Vh = h 2 / 2m * a 2 , where m * is the effective electron mass. The system size is L x × L y = 160 × 30 . Since we set a = 30 nm, total device size is L x × L y = 4.8 × 0.9 µm2,
which is comparable for the present QPC. The origin is set to the centre of the system. The initial wave packet with spin σ is assumed to be,
4
2 πy exp ik x − δk x Ψσ (t = 0 ) = A cos L +1 4 y
χ σ S2
with
χ↑ =
1 1 1 1 , χ ↓ = . S3 2 i 2 − i
We set k x = 0.8 and δk x = 0.2 . This wave packet is entered to the spatially-modulated Rashba SOI region in the QPC, where the QPC potential is defined by,
x 2 V ( x, y ) = VQPC exp − 2 ξ x
( y − L y / 2)2 exp − ξ y2
2 + exp − x ξ2 x
( y + L y / 2)2 , S4 exp − 2 ξ y
where VQPC = 1.0t , ξ x = 5.0a , and ξ y = 10.0a . The Rashba SOI is given by
α (y) =
α R 2
1 + cos(
2πy ) , S5 L y
where α R = 3.2 × 10 −12 eVm, which is close to the Rashba SOI strength of QPC device C at VTG = 0.3 V in the experiment.
Calculation of spin resolved conductance in the one-dimensional channel: To describe the one dimensional potential landscape, we use a saddle point potential described by, 1 1 V ( x, y ) = V0 − m *ω x2 x 2 + m *ω y2 y 2 , S6 2 2 where V0 is the bottom of the saddle-point potential, m* is an effective mass for electron, and ωx and ωy describe the curvatures of the inverted parabolic potential through which the electron tunnels as it traverses the QPC. In the direction transverse to electron motion,
5
1 the subbands have energies with E n = hω y n + , where ħ is Planck’s constant and n 2 represents integer numbers. Conductance in the one-dimensional channel at zero temperature is given by
2e 2 G= h
∑ {βT (E n
F
+ βeVsd ) + (1 − β )Tn (E F − (1 − β )eVsd )}, S7
n
where e is the electron charge, EF is the Fermi energy, and β defines the potential drop asymmetricity between source and drain chemical potentials. When we apply the bias voltage Vsd, the source and drain chemical potential changes to E F + βeVsd and E F − (1 − β)eVsd , respectively. The transmission probability Tn(E) for each subband n in the QPC potential is described by Tn ( E ) =
1 S8 1 + exp{− πε n }
2
[E − E n − V0 ] . S9
and
εn =
hω x
Here, εn is the dimensionless parameter of the energy for guiding the centre motion relative to the potential landscape in the QPC. In the calculation, we set ħωx = 2.0 meV and ħωy = 7.8 meV to reproduce the experimental results. We introduce additional energy terms in Supplementary Equation S9: momentum-dependent spin splitting energy ± (m )1 2 E s , where the sign change corresponds to m Vsd directions, and Zeeman energy ± 1 2 gµ B Bex . Then, εn changes to
ε mn =
2 1 1 E − E n − V0 ± (m ) E s ± gµ B Bex . S10 hω x 2 2
ε−n and ε+n correspond to ± 1 2 E s (-Vsd direction) and m 1 2 E s (+Vsd direction), respectively, depending on the Vsd bias directions. By introducing Supplementary Equation S10 into Supplementary Equations S7 and S8, we can calculate a spin resolved
6
conductance with different Vsd by taking into account the above two energies: momentum-dependent spin splitting energy and Zeeman energy. Finally, the differential conductance is derived by dG/dE. We set the values of parameter according to the experimental results: Es = 5 meV, gµBBex = 1.95 (0.47) meV for -Vsd (+Vsd), and β = 0.5.
7
