Synthetic Rashba spin–orbit system using a silicon metal-oxide semiconductor

The spin–orbit interaction (SOI), mainly manifesting itself in heavy elements and compound materials, has been attracting much attention as a means of manipulating and/or converting a spin degree of freedom. Here, we show that a Si metal-oxide- semiconductor (MOS) heterostructure possesses Rashba-type SOI, although Si is a light element and has lattice inversion symmetry resulting in inherently negligible SOI in bulk form. When a strong gate electric field is applied to the Si MOS, we observe spin lifetime anisotropy of propagating spins in the Si through the formation of an emergent effective magnetic field due to the SOI. Furthermore, the Rashba parameter α in the system increases linearly up to 9.8 × 10−16 eV m for a gate electric field of 0.5 V nm−1; that is, it is gate tuneable and the spin splitting of 0.6 μeV is relatively large. Our finding establishes a family of spin–orbit systems. Silicon is a light element with high lattice inversion symmetry, and so is not expected to possess a substantial spin–orbit interaction (SOI), which is desirable for spintronics. Here, a silicon-based heterostructure is demonstrated to have a gate-tuneable Rashba-type SOI.

T he spin-orbit interaction (SOI) gives rise to a wide variety of condensed matter physics, and it has played a central part in a wide variety of physical phenomena such as spin manipulation without magnetic fields 1,2 , ordinal and inverse spin Hall effects enabling spin conversion [3][4][5][6][7] , the spin-galvanic effect and its inverse effect 8,9 , giant spin splitting at an interface and in bulk 10,11 and Ising-Cooper pairing superconductivity 12 . Breaking the bulk inversion symmetry and/or structural inversion symmetry is one of the pivotal approaches to generate the SOI. For example, bulk inversion symmetry breaking in strained GaAs allows coherent spin manipulation without an external magnetic field 1 , and structural inversion symmetry breaking induces giant spin splitting at the Bi-Ag interface 10 and in the BiTeI bulk 11 . The other important material with strong SOI is a single heavy element, where the SOI magnitude is roughly proportional to the fourth power of the atomic number [13][14][15] . Because of this background, materials with high inversion symmetry and light elements have been outside the scope of SOI physics. Single-layer graphene (SLG) with a transition metal dichalcogenide (van der Waals heterostructure) [16][17][18][19] , bilayer graphene 20,21 and a subsurface state of p-Ge(111) 22,23 are limited examples, and it is noteworthy that the magnitude of the SOI is not tuneable and/or a specific system is necessary for generating a SOI in them. Hence, it would be surprising but important to realize a tuneable synthetic SOI system using a light element with high inversion symmetry because such a success would dramatically expand the horizons of SOI physics.
Si has been believed to be an unsuitable material for SOI generation because of its lightness and bulk inversion symmetry, and Si has attracted attention mainly because of its good spin coherence. However, veiled physical properties can appear when a strong gate electric field is applied. Indeed, strong electric field application via solid and ionic gating materials allowed pioneering new physics in condensed matter, such as spin manipulation in III-V compound semiconductor heterostructures 1 , superconductivity in oxide insulators 24 , strong modulation of the Curie temperature in nanometre-thick Co 25 and gate-tuneable SOI resulting in the tuneable inverse spin Hall effect in nanometre-thick Pt 26 . Thus, while a gate electric field merely plays a part in modulating the conductivity of Si in conventional Si-based electronic devices, the spin degree of freedom can be inherently modulated by a sufficiently strong gate electric field even in Si, creating a synthetic Rashba spin-orbit system. To test this concept, a thin Si metal-oxide-semiconductor (MOS) spin channel with a gate insulator can be a model system because the spin propagation physics is well understood in Si MOS.

Experimental concept and device structure
When a Rashba field, that is, an emergent effective magnetic field, is successfully generated in a Si MOS channel, spin lifetime anisotropy arises between the lifetime of parallel and perpendicular spins to a channel plane because the emergent effective magnetic field gives rise to additional spin precession around the field direction or spin-locking along the field direction in a spin-propagating-media with the Rashba field. Thus, to identify whether the synthetic Rashba SOI is generated, the spin lifetime anisotropy as a function of the gate voltage amplitude, V g , and angle of an external magnetic field, B ex , should be examined. Unless a gate electric field is applied in a Si spin channel, B ex is the sole source of spin precession. Meanwhile, application of a gate electric field gives rise to an emergent effective magnetic field perpendicular to the k-vector of the propagating spins as γℏB eff = 2α(k × z), where γ is the gyromagnetic ratio, ℏ is the Dirac constant, B eff is the emergent effective magnetic field and z is the unit vector along the direction of the gate electric field. Since B eff provides an additional contribution to spin precession, the spin lifetimes parallel (τ ∥ s ) and perpendicular (τ ⊥ s ) to the spin channel plane exhibit anisotropy. The most appropriate approach to substantiate the anisotropy is the method established for testing the spin lifetime anisotropy in SLG 27 and applied to bilayer graphene 20,21 . Raes et al. established the approach, where they applied out-of-plane (oblique) magnetic fields and implemented spin precession measurements under oblique magnetic fields to generate an out-of-plane spin Synthetic Rashba spin-orbit system using a silicon metal-oxide semiconductor Soobeom Lee 1 , Hayato Koike 2 , Minori Goto 3 , Shinji Miwa 3,5 , Yoshishige Suzuki 3 , Naoto Yamashita 1 , Ryo Ohshima 1 , Ei Shigematsu 1 , Yuichiro Ando 1,4 and Masashi Shiraishi 1 ✉ The spin-orbit interaction (SOI), mainly manifesting itself in heavy elements and compound materials, has been attracting much attention as a means of manipulating and/or converting a spin degree of freedom. Here, we show that a Si metal-oxidesemiconductor (MOS) heterostructure possesses Rashba-type SOI, although Si is a light element and has lattice inversion symmetry resulting in inherently negligible SOI in bulk form. When a strong gate electric field is applied to the Si MOS, we observe spin lifetime anisotropy of propagating spins in the Si through the formation of an emergent effective magnetic field due to the SOI. Furthermore, the Rashba parameter α in the system increases linearly up to 9.8 × 10 −16 eV m for a gate electric field of 0.5 V nm −1 ; that is, it is gate tuneable and the spin splitting of 0.6 μeV is relatively large. Our finding establishes a family of spin-orbit systems.
population. The superiority of this method is that it provides reliable results for both low-and high-carrier densities and thus can be used for gate-tuneable systems such as Si MOS. Figure 1a shows the structure of a synthetic Rashba device consisting of a 100-nm-thick n-type Si channel (the carrier concentration, n, was 5 × 10 16 cm −3 , that is, the Si was non-degenerate) on a 200-nm-thick SiO 2 gate insulator (Si MOS) and the setup for measurement of the non-local four-terminal magnetoresistance (NL4T-MR) and spin lifetime anisotropy in the Si MOS under the application of V g (Methods). All measurements were implemented at 300 K. V g was applied from the backside of the device and was varied from 0 to 100 V; that is, an electric field was applied to the Si from 0 to 0.5 V nm −1 via SiO 2 . Modulation of the conductivity, σ Si , of the Si MOS was realized as shown in Fig. 1b, which indicates that the back-gate voltages were efficiently applied to the Si MOS channel. The NL4T-MR from the Si MOS under V g = 0 V with sweeping of the in-plane external magnetic field is shown in Fig. 1c, and clear hysteresis in spin signals as a manifestation of successful spin transport in the Si MOS is observed. To clarify the spin lifetime anisotropy, B ex was applied with changing applied angle β, as shown in Fig. 1a, where β was varied from 10° to 90°.

Spin lifetime anisotropy in Si MOS by gating
Spin precession signals for various β values (as representatives, 10°, 30°, 45°, 60° and 90°) as a function of V g (as representatives, 0, 10, 60 and 100 V) are depicted in Fig. 2a-d. The signals were nicely reproduced by a conventional one-dimensional fitting function [28][29][30] , where the spin lifetime at β = 90°, that is, τ ∥ s under V g = 0 V was estimated to be 2.5 ± 0.1 ns, and the precessional motion of the spins dephases at B ex ≅ 40 mT (Supplementary Information). For B ex > 40 mT, the spin voltage, V NL4T , is determined by the remanent non-precessional spin component that lies along the magnetic field direction. The angular dependence of normalized V NL4T is described as where L is the centre-to-centre distance between two ferromagnetic contacts, D is the diffusion constant and ζ is the spin lifetime anisotropy ratio, ζ = τ ⊥ s /τ ∥ s (ref. 27 ). The ratio ζ is a fingerprint of the anisotropy, where ζ < 1 is expected when spin-orbit fields inducing spin relaxation are preferential in the Si MOS plane and ζ = 1 is observed for an isotropic spin lifetime. Thus, ζ is quantitatively characterized by plotting the normalized V NL4T at β = 0° and β ≠ 0°. Figure 2e-h shows the normalized V NL4T as a function of cos 2 β, where the magnitudes of ζ estimated by using equation (1) are also shown, where error bars shown in Fig. 2e-h were estimated by using the rules for error propagation and these error bars were considered in the fitting to extract ζ. ζ changes with V g and is close to 1 when V g = 10 V. The complete dataset of ζ as a function of V g is shown in Fig. 3, and a prominent V g dependence of ζ is seen. Indeed, ζ monotonically and prominently decreases when V g > 10 V, which is unlike SLG 27 . As mentioned above, ζ is a fingerprint of the spin-orbit fields in Si, and the observed V g dependence of ζ unequivocally shows that a spin-orbit field is generated in the Si MOS and that spin lifetime anisotropy arises from emergent effective magnetic fields that are tuneable by V g , while the amplitudes of ζ are not as large as those observed in a system with intrinsically large SOI, such as WS 2 /SLG 17 . However, notably, the experimental result counters the conventional understanding that Si has negligibly small SOI, and we have successfully is applied between one ferromagnetic contact and one non-magnetic contact, and NL4T voltages (V NL4T ) are measured in the separated circuit including a ferromagnetic contact and a non-magnetic contact: non-local alignment. An external magnetic field (B ex ) is applied in the yz plane with a tilt angle of β. b, Conductivity of the Si mOS (σ Si ) as a function of gate voltage (V g ). σ Si was measured using a conventional four-terminal method. c, Typical non-local magnetoresistance between parallel and antiparallel magnetization alignments as a manifestation of successful spin propagation in the Si mOS. An in-plane (β = 0°) external magnetic field was swept upwards (red solid line) and downwards (black solid line). demonstrated not only generation but also manipulation of SOI in the Si MOS by gating. As mentioned above, the Rashba field generates an emergent effective magnetic field. In our setup, the gate electric field is applied perpendicular to the Si plane (along the z direction, as shown in Fig. 1a) and, thus, the effective magnetic field is generated in-plane along the (−y) direction because the motion of the propagating spins is along the x direction (Fig. 4a). The in-plane emergent effective magnetic field further helps spin relaxation of the propagating spins, which enhances the spin lifetime anisotropy. ζ is close to 1, not at V g = 0 V but at V g = 10 V; that is, the spin lifetime is isotropic at V g = 10 V. This suggests that a built-in and non-negligible electric field, whose direction is opposite to that of the positive external gate electric field, is generated in the Si spin channel. Thus, the SOI in the Si MOS is ascribable to the thin (100 nm) channel and/or the Si-SiO 2 interface (see also Supplemental Information for a brief discussion on where the Rashba field exists).

Physics behind of the spin lifetime anisotropy
How spins precess under magnetic fields is described by equation (2) in the simplest model 31 : where ω is the angular frequency of spin precession, Δθ is the spin precession angle, t is the dwell time of the spins propagating between two electrodes, γ is the gyromagnetic ratio, B is the magnetic field that spins sense, k x (= (3π 2 n) 1 3 ) is the wavenumber of the propagating spins diffusing along the x direction and α is the Rashba parameter (Fig. 4a). Thus, the Rashba field strength that determines the strength of the emergent effective magnetic field is quantitatively described by α. The upper and lower figures in Fig. 4b show enlarged views of the spin precession signals at β = 90° for V g = 10 V (ζ = 0.99, isotropic spin lifetime) and 100 V (ζ = 0.75, anisotropic spin lifetime), respectively, and the dashed lines show the external magnetic field, where the averaged spin precession angle is π and −π. We postulate that the Rashba SOI at V g = 10 V is negligible and no emergent effective field is generated in the Si MOS, because the external magnetic field for π-rotation is the smallest when V g = 10 V (see also the following discussion). Here, a B ex of 6.6 mT is necessary for π-rotation at V g = 10 V. Nevertheless, a greater B ex of 10.6 mT is needed for π rotation at V g = 100 V. The enhancement of the external magnetic field, B ex , needed for π rotation of the spins, helps in understanding the underlying physics. Since we exerted a non-local four-terminal scheme for spin transport, only spin diffusion occurs. Here, D is independent of V g (Supplemental Information), which shows that mobility μ is unchanged by the gating because we exerted non-degenerate inorganic semiconductor and thus Einstein's relation, eD = μk B T, holds in our system (k B is the Boltzmann constant and T is temperature). Furthermore, since mobility μ is described as μ = eτ e /m * (τ e is momentum relaxation time and m * is the effective mass), unchanged mobility indicates unchanged momentum relaxation time. Thus, our experimental finding of the unchanged D by the gating is compelling evidence for the unchanged Fermi velocity v F for the gating, resulting in the fact that the dwell time t for π rotation of the spins in the Si MOS does not vary with the gate voltages, since D is described as (1/3)v F 2 τ e . At V g = 10 V, the B that spins sense can be regarded as merely B ex because no B eff arises, and the spin precession angle is determined to be Δθ = γ|Bex|t. Meanwhile, at V g = 100 V, the B that spins sense is the coupled magnetic field of B ex and B eff , that is, Furthermore, the spin precession angle in the anisotropy measurement setup is determined by the difference between √ B 2 ex + B 2 eff and B eff because the averaged spin precession angle in the measurement setup is determined by the difference in the magnetic fields with and without the external magnetic field, B ex . Consequently, the averaged spin precession angle at V g = 100 V is described as Δθ = γ B 2 ex + B 2 eff − B eff t. Given that the dwell times for π rotation are unchanged at V g = 10 and 100 V and the B ex values for π rotation at V g = 10 and 100 V are 6.6 and 10.6 mT, respectively, the B eff at V g = 100 V (the electric field of 0.5 V nm −1 ) is estimated to be 5.3 mT, resulting in a Rashba parameter, α, of 9.8 × 10 −16 eV m. α is continuously tuned as a function of V g as shown in Fig. 4(d), which reflects the compelling result of a tuneable Rashba SOI in the Si MOS. The spin splitting, Δ 0 (=gμ B B eff , where g is the g factor, μ B is the Bohr magneton and B eff is the emergent effective magnetic field) at V g = 100 V for the Si MOS is calculated to be 0.6 μeV, which is much smaller than that in III-V heterostructures 1 but is comparable to that in strained GaAs 2 . Although the SOI in GaAs is inherently much greater, the g factor of Si (1.9979 in experiment and 1.99875 in theory at room temperature 32 ) is approximately five-fold greater than that in GaAs (0.441, ref. 2 ), which is the reason why comparable spin splitting is realized in the Si MOS despite the small α. Notably, 20-fold enhancement of α can be realized by replacing a thin gate insulator with a greater dielectric constant material, such as 50 nm HfO 2 (ref. 25 ), and most likely, much greater enhancement may be achieved by using ionic gating, where a strong electric field is applicable 26 . The enhancement would allow larger spin splitting in the Si MOS, as in III-V heterostructures 1 , enabling other spin manipulation functions in Si.

Outlook
The finding in this study of successful creation of a synthetic Rashba system using a ubiquitous light element counters the conventional understanding in SOI physics that the SOI in Si is negligibly small and a synthetic SOI as realized in van der Waals heterostructures cannot be created in Si. Thus, this report pioneers an approach to creating a material system with a sizeable SOI. Since the created SOI in Si MOS is gate tuneable and the spin splitting Δ 0 is comparable to that in GaAs strained under an application of gate voltages via 200-nm-thick SiO 2 , further enhancement of α by introducing insulating gate materials with larger dielectric constants could pave the way to more efficient spin manipulation systems using Si. A gate electric field is applied perpendicular to the Si mOS plane (E z ), and the spin momentum is parallel to the plane (ℏk x ). Consequently, an emergent effective magnetic field, B eff , is generated along the −y direction due to the Rashba SOI. An external magnetic field is applied perpendicular to the plane (B ex ), which allows spin precession of the in-plane spins. Fm, ferromagnetic. b, Spin precession signals at β = 90°, where V g is set to 10 and 100 V. Grey dots are experimental data, and black solid lines are the fitting results. blue dashed lines indicate the magnetic fields for averaged π and −π rotation of spins accumulated below the detector. c, V g dependence of the Rashba parameter (α). Given that ζ is almost 1 (isotropic spin lifetime) at V g = 10 V, we postulate that the Rashba SOI at V g = 10 V is negligible in the estimation of α.