Abstract
Spin–momentum locking in the Dirac surface state of a topological insulator (TI)^{1,2,3,4,5,6} offers a distinct possibility for highly efficient chargetospin current (C–S) conversion compared with spin Hall effects in conventional paramagnetic metals^{7,8,9,10,11,12,13}. For the development of TIbased spin current devices, it is essential to evaluate this conversion efficiency quantitatively as a function of the Fermi level position E_{F}. Here we introduce a coefficient q_{ICS} to characterize the interface C–S conversion effect by means of the spin torque ferromagnetic resonance (STFMR) for (Bi_{1−x}Sb_{x})_{2}Te_{3} thin films as E_{F} is tuned across the bandgap. In bulk insulating conditions, the interface C–S conversion effect via the Dirac surface state is evaluated as having large, nearly constant values of q_{ICS}, reflecting that q_{ICS} is inversely proportional to the Fermi velocity v_{F}, which is almost constant. However, when E_{F} traverses through the Dirac point, the q_{ICS} is remarkably reduced, possibly due to inhomogeneity of k_{F} and/or instability of the helical spin structure. These results demonstrate that fine tuning of E_{F} in TIbased heterostructures is critical in maximizing the efficiency using the spin–momentum locking mechanism.
Main
Threedimensional topological insulators (TIs) possess metallic surface states in which the spins of carriers are locked orthogonal to their momenta as a result of the timereversal invariant. This feature is called ‘spin–momentum locking’, and has been employed as the principal mechanism to induce spin accumulation in the surface states of TIs (refs 1,3,4,5,14,15,16,17). Conceptually, the charge current can fully contribute to the spin current via spin–momentum locking; a C–S conversion efficiency θ_{CS} of 100% is expected at the nonTI/TI heterointerface. This highly efficient C–S conversion can be widely applicable to spintronic devices. However, the C–S conversion efficiency deduced from the spin torque measurement can exceed 100% for TIs with E_{F} located in the bulk band, leading to mixed contributions from the surface and bulk bands^{3,4}, when the efficiency is defined as θ_{CS} = J_{S}/J_{C}, where J_{S} is the spin current density (A m^{−2}) and J_{C} is the charge current density (A m^{−2}) in the entire TI layer. Here we isolate the contribution of the Dirac electrons in the C–S conversion process and clarify the role of the Fermi level E_{F} and the Fermi velocity v_{F} by employing TI samples with various E_{F} positions. Accordingly, we define the interface C–S conversion coefficient q_{ICS} as q_{ICS} = J_{S}/j_{C}, where j_{C} is the surface charge current density (A m^{−1}). Based on the concept of spin–momentum locking, the magnitude of J_{S} is governed by that of j_{C}, which is linked with the conductivity of the surface states on the TI layer depending on the Fermi energy and Dirac dispersion: the Fermi velocity v_{F} and the Fermi wavevector k_{F} (refs 15,16,17). In this study, we quantitatively evaluate the interface C–S conversion effect by means of the STFMR technique for 8nm (Bi_{1−x}Sb_{x})_{2}Te_{3}/8nm Cu/10nm Ni_{80}Fe_{20} (Py) trilayer films, as shown in Fig. 1a. Systematic control of the Fermi levels by varying x in (Bi_{1−x}Sb_{x})_{2}Te_{3} (BST) thin films enables us to investigate the relationship between q_{ICS} and the transport properties at the surface state.
The STFMR technique has been routinely employed to evaluate the spin current induced via the spin Hall effect in paramagnetic metals^{10}. Here we apply this technique to characterize quantitatively the interface C–S conversion effect due to spin–momentum locking in Cuinserted TIbased trilayer heterostructures, as shown in the top schematic of Fig. 1a. On insertion of a Cu layer between the TI and ferromagnet layers, spin accumulation at the surface states can be separately evaluated owing to suppression of the exchange coupling between the ferromagnet and the surface states of TI (refs 18,19,20). In addition, the deposition of Cu on BST probably plays a minor role in varying the surface state condition, such as through an energy shift of the Dirac point and valence band maximum^{21}, owing to a similar magnitude between work function of Cu and the electron negativity of BST. A photo of the device and measurement circuit is shown at the bottom of Fig. 1a. To evaluate q_{ICS} = J_{S}/j_{C} by means of STFMR, the charge current distribution in the trilayer should be clarified numerically, because j_{C} in the TI layer is one of the dominant factors in this evaluation technique. When a radiofrequency (rf) current flows in the trilayer film, FMR is excited in the top Py layer under an external static magnetic field H_{ext}. Owing to the presence of the highly conductive Cu layer, the peak of the current density is located towards the outside of the Py layer so that homogeneous rf fields (H_{rf}) can be applied to the Py layer (see Supplementary Information 1), providing better conditions for characterizing the C–S conversion effect by means of STFMR^{22}. Accumulation of spins takes place simultaneously in the surface state of the TI; these accumulated spins generate a spin current J_{S} in the orthogonal direction, diffusing into both Cu and Py layers, and thus exert a spin torque on the Py layer (white arrow in Fig. 1a). Note that the spin pumping effect, as an inverse effect to the chargetospin conversion, provides a small contribution in the evaluation of q_{ICS} (Supplementary Information 8). A typical STFMR spectrum is shown in Fig. 1b: the symmetric voltage V^{Sym} is attributable to the spin torque τ_{∥}/ corresponding to the spin current density J_{S} (details discussed later). By quantitative evaluation of V^{Sym}, we can deduce the interface C–S conversion coefficient q_{ICS}.
Figure 2a shows the Hall coefficients R_{H} obtained for seven films having different Sb composition x at 10 K. The value of R_{H} is negative for x = 0, with an increasing magnitude as x is increased to 0.82, indicating ntype conduction and a reduction in the electron density. The polarity of R_{H} abruptly reverses its sign when x reaches a value of approximately 0.84, revealing that the Fermi energy traverses the Dirac point (DP). In the range 0.88 ≤ x ≤ 1, the polarity of R_{H} is positive, indicating ptype conduction. The charge carrier densities and mobilities shown in Fig. 2b are estimated from the R_{H} values. Compared to the previous studies^{3,23}, the charge carrier density in the surface state reaches as low as 10^{12} cm^{−2}, indicating that E_{F} is finely tuned close to the Dirac point. The mobility (μ) in BST films with x = 0.88 reaches a maximum value of 1,900 cm^{2} V^{−1} s^{−1}, which is comparable to the previous results^{24}. These transport properties and the temperature dependence of the resistivity (Supplementary Information 10) ensure that the Fermi level of BST is systematically varied from n to ptype across the DP in a controlled manner, as shown in Fig. 2d.
Figure 2c shows the dependence of V^{Sym} on the Sb composition x obtained from an STFMR spectrum measured at an rf power of 8 mW. We confirmed that an rf power of 8 mW is low enough to ensure a linear response, with suppression of heating effects (Supplementary Information 5). The sign of V^{Sym} indicates the spin polarization direction of the spin current. We found a positive V^{Sym} in both n and ptype BST films, which is an ideal feature of the C–S conversion via spin–momentum locking^{1,2,3,4,5,6}. In the Dirac dispersion shown in Fig. 2e, f, spins on the Fermi circles of n and ptype surface states of BSTs rotate clockwise and anticlockwise, respectively^{24}. When the electric field E_{x} is applied in the −xdirection, the Fermi circle with the chiral spin structure is shifted from the dashed circles to the solid circles by an amount proportional to E_{x} along k_{x}, as shown in Fig. 2e. When the Fermi level E_{F} is above the DP, the surface state of BST films has a higher population of down spins, generating spin polarization of the spin current along the −ydirection. When the Fermi level E_{F} is in the valence band of the Dirac dispersion, up spins with momenta along +k_{x} are fewer in number than down spins with −k_{x}. Thus the accumulated spin is oriented along the same direction for both n and ptype BST films. Note that these results are different from the case of a typical semiconductor such as GaAs^{25}, whose spin Hall effect exhibits a different sign, depending on the carrier type.
The values of q_{ICS} and the spin current conductivity σ_{S} of BST films are summarized as a function of x in Fig. 3. The value of q_{ICS} can be experimentally evaluated from the ratio of V^{Sym} to V^{Anti} in the STFMR spectrum. By using the conventional evaluation term θ_{CS} = J_{S}/J_{C}, with assumption of a uniform J_{C} in the BST film regardless of E_{F} position, large values of θ_{CS} are obtained for x = 0.5, 0.7 and 0.9, consistent with previous studies^{3} (Supplementary Information 9). Here, we propose a scheme for evaluation of q_{ICS} making use of j_{c}. In the STFMR process, the values of V^{Sym} and V^{Anti} correspond, respectively, to the spininduced torque τ_{∥} and the Oerstedfieldinduced torque τ_{⊥} generated by charge current flow. These two torques per unit moment on the Py are respectively expressed as τ_{∥} = ℏJ_{S}/(2eμ_{0}M_{S}t_{Py}) and τ_{⊥} = ξ{J_{C}^{Cu}t_{Cu}/2 + j_{C}/2}, where M_{S}, t and ξ are the saturation magnetization, film thickness, and reduction factor of the rf field. Note that V^{Anti} shows a sin2θcosθ dependence on the rotation angle of the applied magnetic field (Supplementary Information 6), indicating that V^{Anti} originates purely from the Oersted field. The value of ξ is calculated numerically by means of a finite element method (see Supplementary Information 1). The value of q_{ICS} can thus be given by
where a is the ratio of J_{C}^{Cu} (A m^{−2}) to j_{C}. The spin current density into Py J_{S}^{Py} (A m^{−2}) is proportional to the spin accumulation at the surface state of the TI, 〈δS_{0}〉, which is expressed as
where k_{F} is the Fermi wavenumber, δk_{x} is the shift of Fermi circle, and τ is the relaxation time. In the twodimensional system, k_{F}^{2} is proportional to the carrier density. Therefore, 〈δS_{0}〉 reduces to ℏj_{C}/2ev_{F} and q_{ICS} = J_{S}^{Py}/j_{C} ∝ v_{F}^{−1} is obtained. According to angleresolved photoemission spectra, v_{F} in BST increases slightly from 3.6 to 3.9 × 10^{5} m s^{−1} as the Sb composition x increases from 0.5 to 0.9 (ref. 26), suggesting that q_{ICS} is almost constant in this composition range. For bulk insulating BST films with x = 0.5, 0.7 and 0.9 (apart from two samples in the vicinity of the DP) we observed that the values of q_{ICS} indeed lie within the range 0.45–0.57 nm^{−1}, similar in magnitude to the previous results of STFMR measurements in Py/Bi_{2}Se_{3} bilayer film^{3}. Of particular interest is the case x = 0.9; although its V^{Sym} value is nearly half those of x = 0.5 and 0.7, as shown in Fig. 2c, the q_{ICS} value is still comparable owing to the correction by the small j_{C}. Since we assume the surface thickness responsible for j_{C} to be 1 nm, the value of θ_{CS} is estimated to be 45 to 57% for these BST films, yielding a much higher conversion efficiency than those in typical transition metals such as βTa (15%) and βW (33%) (refs 10,11,12,13). Consequently, the interface C–S conversion effect via spin–momentum locking at the surface state is evaluated fairly well using q_{ICS}, and consistent with the naive expectation of an at most 100% conversion efficiency. Note that the estimated conversion efficiency θ_{CS} is proportional to the conducting channel thickness, which contributes to J_{C}; therefore, the large values of θ_{CS} claimed for high conversion efficiencies in previous studies^{3,4} may be overestimated by assuming a thicker conducting layer. Our estimated values of q_{ICS} less than unity appears consistent with the theoretical evaluation, which has proposed a reduction of the inplane spin polarization due to spin–orbit entanglement in such a material with a strong spin–orbit interaction^{27}. We would also like to note that the product of q_{ICS} and the inverse conversion coefficient is expected to be approximately unity for the ideal case where there is no reduction of the inplane spin polarization in the surface state of the TI^{28}.
We now discuss the Fermilevel dependence of q_{ICS} shown in Fig. 3a. First, in contrast to the almost constant q_{ICS} for bulk insulating BST films with x = 0.5, 0.7 and 0.9, the values of q_{ICS} show a sharp dip around DP x ∼ 0.82 and 0.88; the value of q_{ICS} decreases dramatically when E_{F} is located close to the DP, or equivalently k_{F} becomes approximately zero, originating from the almost zero V^{Sym} in our experiments (see Fig. 2c, d). Evaluated from the lowest charge carrier density of approximately 10^{12} cm^{−2}, the E_{F} position is located within ±60 meV of the DP for x = 0.82 and 0.88, resulting in a small 〈δS_{0}〉 due to the small δk_{x}. In such a situation, with E_{F} close to the DP, a finite amount of scattering may reduce the generated spin polarization, as reported in experiments with spinresolved angulardependent photoemission spectroscopy (ARPES)^{29,30,31} and scanning tunnelling spectroscopy^{32}. Here we give possible reasons for the reduction in the spin polarization. First, when there are inhomogeneities, such as in the Bi/Sb composition, which can be regarded as analogy to electron–hole puddles in graphene^{33}, in the surface state of BST around the Dirac point, charge current can flow in directions other than the electric field direction. As a result, 〈δS_{0}〉 with various spin directions will occur in the surface state of the TI, indicating that the 〈δS_{0}〉 in the y direction will decrease. Furthermore, if there is an additional surface state due to Cu deposition on the TIs, the generated spin polarization will be decreased (Supplementary Information 4). In this situation, it is likely that an ideal constant value of q_{ICS} cannot be recovered due to the almost zero V^{sym} with finite j_{C}. Second, for bulk conductive BST films with x = 0 (Bi_{2}Te_{3}) and x = 1 (Sb_{2}Te_{3}), when we apply the same analytical method as used for bulk insulating films with an assumption of a 1 nm conducting surface layer, the estimated values of q_{ICS} are found to be roughly equal to, or even twice as much as, those for the bulk insulating BST films. However, a quantitative evaluation of q_{ICS} and θ_{CS} in bulk conducting samples is fairly difficult for the following reasons. The first reason is estimation of the effects of parasitic currents: here we assume that the surface is as conductive as the bulk (see Supplementary Information 1). Considering that the surface is expected to be more conductive, the present value of q_{ICS} might be overestimated. Although the assumption of a completely insulating bulk is simplistic, a small contribution from a bulk current provides a small perturbation to the present results; we have shown an insignificant effect of the current ratio between the surface and the bulk on the evaluated values of q_{ICS} (Supplementary Information 2). The second reason is the Rashba effect: if the charge current in the bulk band also contributes to J_{S} via a Rashbasplit band, as in Bi_{2}Se_{3} (ref. 20), the opposite spin polarization may cancel part of the surface spin accumulation (see Supplementary Information 4). The final reason is the bulk spin Hall effect: the bulk charge current can also contribute to J_{S} via the ordinary spin Hall effect, of which the sign is as yet unclear.
Finally we show the spin current conductivity σ_{S}, defined as σ_{S} = q_{ICS}σ^{surf}, where σ^{surf} is the conductivity of the surface state of the TI, as a function of the Sb composition x in Fig. 3b. The values of σ_{S} in bulk insulating BST (x = 0.5 and 0.7), excluding the DP and bulk conductive BST, take values close to 1.8 × 10^{5} Ω^{−1} m^{−1}, which are comparable to those reported for threedimensional processes originating from the spin Hall effect in paramagnetic metals such as Pt (3.4 × 10^{5} Ω^{−1} m^{−1}) (ref. 10) and βW (1.3 × 10^{5} Ω^{−1} m^{−1}) (ref. 11). This high value of σ_{S} is certainly beneficial, not only for realizing highly efficient magnetization switching, but also for realizing nonvolatile spin switching for Boolean and nonBoolean logic initially based on metal spin Hall effects^{34}.
Methods
Sample fabrication.
We grew 8nmthick BST films on semiinsulating InP(111) substrates by molecular beam epitaxy. The detailed growth conditions are described in a previous paper^{23}. The Bi/Sb ratio was tuned by adjusting the ratio of the beam equivalent pressures of Bi and Sb. Resistivity and Hall effect measurements were carried out using small chips derived from the same samples as used for the STFMR measurements (see Supplementary Information 2). Thin films of 8nm Cu/10nm Ni_{80}Fe_{20} (Py)/5nm Al_{2}O_{3} were grown on the BST films by ebeam evaporation at a pressure of 5 × 10^{−5} Pa. Al_{2}O_{3} is used as an insulating capping layer. The resistivities of Cu and Py are measured to be 10 and 60 μΩ cm at 10 K. The BST/Cu/Py trilayer films were patterned into rectangular elements (10 × 30, 15 × 45, 20 × 60, 30 × 90, 40 × 120 μm^{2}) using optical lithography and an Arion etching technique. A coplanar waveguide of 5nm Ti/200nm Au was deposited on both sides of the rectangular elements.
STFMR measurement setup.
An rf current with an input power of 8 mW is applied along the long edge of the rectangle by means of a microwave analog signal generator (Keysight: MXG N5183A). An external static magnetic field H_{ext} in the range from 0 to 2.0 kOe is also applied in the film plane at an angle of θ = 45 ° with respect to the current flow direction. We demonstrated the rf power dependence of V^{Sym}, the heating effect, the dc current dependence of the resonance field, the frequency dependence of the halfwidth at halfmaximum of Δ for V^{Sym} and the dependence on magnetic field angle of V^{Anti}, and concluded that the detected V^{Sym} and V^{Anti} are primarily due to the chargetospin conversion effect (Supplementary Information). All the experiments were performed at 10 K to measure the surfacedominant properties of TI.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge fruitful discussions with K. Nomura. This work was supported by GrantinAid for Scientific Research on the Innovative Area, ‘Nano Spin Conversion Science’ (Grant No. 26103002). R.Y. is supported by the Japan Society for the Promotion of Science (JSPS) through a research fellowship for young scientists. This research was supported by the Japan Society for the Promotion of Science through the Funding Program for WorldLeading Innovative R & D on Science and Technology (FIRST Program) on ‘Quantum Science on Strong Correlation’ initiated by the Council for Science and Technology Policy.
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Y.O. and Y.T. conceived the project. K.K. made the devices and performed the spin torque ferromagnetic resonance measurements. R.Y. grew the topological insulator thin films and performed Hall measurements. K.K. analysed the data and wrote the manuscript with contributions from all authors. A.T., Y.F., K.S.T., J.M., M.K., Y.T. and Y.O. jointly discussed the results.
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Kondou, K., Yoshimi, R., Tsukazaki, A. et al. Fermileveldependent chargetospin current conversion by Dirac surface states of topological insulators. Nature Phys 12, 1027–1031 (2016). https://doi.org/10.1038/nphys3833
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DOI: https://doi.org/10.1038/nphys3833
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