Investigating the spin-orbit interaction in van der Waals heterostructures by means of the spin relaxation anisotropy Cite as: APL Mater. 7, 120701 (2019); https://doi.org/10.1063/1.5124894 Submitted: 18 August 2019 . Accepted: 20 November 2019 . Published Online: 10 December 2019 L. Antonio Benítez , Juan F. Sierra , Williams Savero Torres , Matias Timmermans , Marius V. Costache , and Sergio O. Valenzuela ARTICLES YOU MAY BE INTERESTED IN The rise of 2D dielectrics/ferroelectrics APL Materials 7, 120902 (2019); https://doi.org/10.1063/1.5129447 Inorganic halide perovskite materials and solar cells APL Materials 7, 120702 (2019); https://doi.org/10.1063/1.5117306 Magnetic skyrmions in nanostructures of non-centrosymmetric materials APL Materials 7, 120703 (2019); https://doi.org/10.1063/1.5130423 APL Mater. 7, 120701 (2019); https://doi.org/10.1063/1.5124894 7, 120701 © 2019 Author(s). APL Materials RESEARCH UPDATE scitation.org/journal/apm Investigating the spin-orbit interaction in van der Waals heterostructures by means of the spin relaxation anisotropy Cite as: APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 Submitted: 18 August 2019 • Accepted: 20 November 2019 • Published Online: 10 December 2019 L. Antonio Benítez,1,2,a) Juan F. Sierra,1,b) Williams Savero Torres,1 Matias Timmermans,1 Marius V. Costache,1 and Sergio O. Valenzuela1,3,c) AFFILIATIONS 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra 08193, Barcelona, Spain 2 Universitat Autònoma de Barcelona, Bellaterra 08193, Barcelona, Spain 3 ICREA—Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain a) an.benitez7@gmail.com b) juan.sierra@icn2.cat c) SOV@icrea.cat ABSTRACT Graphene offers long spin propagation and, at the same time, a versatile platform to engineer its physical properties. Proximity-induced phenomena, taking advantage of materials with large spin-orbit coupling or that are magnetic, can be used to imprint graphene with large spin-orbit coupling and magnetic correlations. However, full understanding of the proximitized graphene and the consequences on the spin transport dynamics requires the development of unconventional experimental approaches. The investigation of the spin relaxation anisotropy, defined as the ratio of lifetimes for spins pointing out of and in the graphene plane, is an important step in this direction. This review discusses various methods for extracting the spin relaxation anisotropy in graphene-based devices. Within the experimental framework, current understanding on spin transport dynamics in single-layer and bilayer graphene is presented. Due to increasing interest, experimental results in graphene in proximity with high spin-orbit layered materials are also reviewed. © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5124894., s I. INTRODUCTION long distances.6–8 Such an intrinsic property has motivated many experimental and theoretical studies over the last ten years. Early Two-dimensional (2D) materials are envisioned as fundamen- numerical calculations by Ertler et al.9 showed spin lifetimes tal building blocks for next generation nanoelectronic devices, offer- between micro- and milliseconds for single-layer graphene, in stark ing promising prospects and a vast number of potential applica- contrast with the first experimental observation of 100 ps.10 Since tions.1 Among 2D materials, graphene, a material made of carbon then, experimental efforts have focused on improving the qual- atoms arranged in a honeycomb lattice, is particularly relevant ity of the devices and then enhancing the spin relaxation param- due to its structural stability and superior electronic proper- eters. Substrate effects in spin transport were suppressed by sus- ties.2,3 Graphene is also promising for spin-based devices in which pending graphene, resulting in spin relaxation lengths of up to the electron spin degree of freedom, as opposed to its charge, 5 μm while spin relaxation times stayed in the 100 s picosec- plays a central role.4,5 The weak intrinsic spin-orbit coupling ond range.11,12 Subsequently, hexagonal boron nitride (hBN) was (SOC) in graphene (12 μeV) and the lack of hyperfine interac- used to fully encapsulate graphene, resulting in spin lifetimes tion (99% of 12 C nuclei in which the nuclear spin is zero) ensure exceeding 10 ns, but still orders of magnitude below original that spins propagate coherently through the crystal lattice over predictions.13 APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 7, 120701-1 © Author(s) 2019 APL Materials RESEARCH UPDATE scitation.org/journal/apm Recent theoretical models attribute the short experimental spin II. SPIN INJECTION AND DETECTION IN GRAPHENE lifetimes to spin-pseudospin entanglement driven by SOC or to res- LATERAL DEVICES onant scattering by local magnetic moments.6,14–18 The spin lifetime A. Nonlocal detection technique anisotropy, ζ, a parameter that quantifies the ratio between spin life- ∥ Spin transport properties in graphene are typically studied in times for spins lying in the graphene plane (τs ) and pointing out of it (τs⊥ ), can help discriminate between these relaxation mecha- lateral devices using nonlocal techniques. The basic concept dates back to 1985 when Johnson and Silsbee experimentally demon- nisms. In particular, ζ is highly sensitive to the magnitude and ori- strated that an unequal density of spin-up and spin-down electrons entation of existing spin-orbit fields (SOFs), if they drive the spin can be generated and detected in nonmagnetic materials using fer- relaxation. For instance, for SOFs oriented in the graphene plane romagnets.21,22 A typical graphene-based lateral spin device consists derived from Rashba SOC, out-of-plane spins relax faster leading ∥ of four electrodes lithographically designed on top of graphene [see to τs⊥ < τs , i.e., ζ < 1.9,19 The relation is expected to change to Fig. 1(a)]. The inner electrodes (F1 and F2) act as the spin injec- ⊥ ∥ τs > τs , and consequently, ζ > 1 for spin relaxation driven by flex- tor and spin detector, respectively, and are made of ferromagnetic ural distortions.20 When there is no preferential direction in the metals (F), typically Co. The distance L between F1 and F2 defines spin relaxation, as in the case of spin relaxation driven by param- the spin channel length. The outer electrodes (M1, M2) are ideally agnetic impurities or (random) gauge fields, the relaxation becomes made of a nonmagnetic metal so that they do not participate in ∥ isotropic with τs⊥ = τs , so ζ = 1.15,16 Given that the SOFs in graphene spin injection and detection.23 A resistive (tunnel) barrier is intro- can be altered using compounds with large SOC, spin-relaxation duced between the F metal electrode and the graphene channel. anisotropy is a crucial parameter to investigate spin-orbit proximity This barrier prevents the carriers from losing their spin informa- effects. tion and enhances the spin polarization.24,25 Typical barriers include The aim of this Research Update is to provide a comprehen- thin oxide layers such as AlOx , MgO, TiOx , or SrO,10,26–28 layered sive overview of the current experimental progress toward under- dielectrics such as hBN,29,30 and amorphous carbon layers or flu- standing spin relaxation phenomena in graphene-based devices orinated graphene.31,32 Generally, graphene lateral spin devices are by means of spin relaxation anisotropy experiments. The pre- fabricated on an insulating layer on a highly conducting substrate sented experimental methods are general and can, in principle, (e.g., SiO2 or hBN on p-doped Si). The graphene carrier density n be applied to other systems, such as transition metal dichalco- can then be controlled by applying a back-gate voltage between the genides (TMDCs), phosphorene, germanene, or silicene, as long graphene and the conducting substrate. as the corresponding spin relaxation lengths allow the fabri- In the nonlocal scheme, a current source I is applied between cation of suitable devices. This manuscript first introduces the the electrodes F1 and M1. Because of the spin-splitting of the bands general concept of spin injection and detection in lateral spin in F1, the injected current is spin-polarized, creating a spin accu- devices, followed by a detailed explanation of the different meth- mulation underneath the contact. This nonequilibrium spin density ods allowing the determination of the spin relaxation anisotropy. diffuses along the spin channel toward the detector electrode F2. The Within this framework, recent experiments are reviewed on single- magnitude of the local spin accumulation at the F2 position is quan- layer and bilayer graphene as well as on graphene-based het- tified by measuring the voltage drop V nl between F2 and M2.23 An erostructures. Finally, future experiments to further advance the external magnetic field B applied along the easy-axis of the ferromag- understanding of spin related phenomena in these systems are nets controls the relative orientation of F1 and F2. A typical nonlocal discussed. measurement, shown in Fig. 1(b), presents sharp transitions that FIG. 1. (a) Schematics of a graphene lateral spin device with spin-sensitive electrodes acting as the spin injector (F1) and spin detector (F2). (b) Typical nonlocal spin resistance Rnl as a function of the in-plane magnetic field B applied along the easy-axis of F1 and F2. The sharp transitions from positive to negative Rnl mark the switching from the parallel to antiparallel configurations between F1 and F2, as illustrated by the arrows. (c) Spin precession measurements for the parallel and antiparallel alignment of F1 and F2 magnetizations (symbols). Solid lines correspond to the fitting to Eq. (1) that allows the extraction of the spin dependent parameters. For the shown example, τs = 0.21 ns, Ds = 0.012 m2 s−1 , and λs = 1.58 μm. ∥ ∥ APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 7, 120701-2 © Author(s) 2019 APL Materials RESEARCH UPDATE scitation.org/journal/apm indicate the change from the parallel to antiparallel configurations shape anisotropy. Therefore, spins are injected in the plane of the of the F1 and F2 magnetizations. The difference in V nl between the channel. The first reported method to determine the spin lifetime parallel and the antiparallel configurations, ΔVnl = Vnl↑↑ − Vnl↑↓ , is anisotropy in graphene uses a large B– (>1 T) to tilt the mag- the nonlocal spin signal, which is commonly expressed in terms of netization of the F electrodes along the field direction in order resistance units as ΔRnl = ΔV nl /I.23 to enable the injection of spins perpendicular to the graphene plane. The spin lifetime anisotropy ζ is then extracted by com- B. Spin precession experiments paring the spin signals when the injected spins are oriented in plane (B– = 0) and perpendicular to it (B– > 1 T). The disadvan- Information about the spin relaxation length and the spin life- tage of extracting ζ with this method arises from the large mag- time is typically obtained by means of (Hanle) spin precession exper- netoresistive effects that are present in graphene, which limits its iments. Here, a magnetic field B– is applied perpendicular to the range of validity to sufficiently large carrier concentrations n.36 substrate causing an in-plane spin precession as spins diffuse from Therefore, alternative experimental approaches have been devel- F1 to F2.21,33 The shape of a typical Hanle spin precession curve in oped, which produce out-of-plane spin populations by spin preces- the high contact resistance limit is given by the expression sion and require much lower magnetic fields. Raes et al. discussed Pi Pd ∞ 1 L2 two such approaches under an oblique magnetic field34 or under Rnl (B) = ± 2 ∫ √ exp[− ] a magnetic field applied in the graphene plane and perpendicu- e NA 0 4πDs t 4Ds t lar to the F metal electrodes.35 They focused on the oblique case × cos(ωL t) exp(−t/τs∥ )dt, (1) because the large rotation of the F electrodes magnetizations for in- where the + (−) sign accounts for the parallel (antiparallel) config- plane fields generally complicates the data analysis and increases uration of the injector and detector, and Pi and Pd , e, N, and A are the uncertainty in the determination of ζ.35 Shortly afterwards, the spin polarization of the injector and the detector, the electron Ghiasi et al.37 and Benítez et al.38 implemented anisotropy mea- charge, the density of states at the Fermi energy, and the contact area, surements with in-plane magnetic fields in graphene modified by ∥ proximity of a TMDC, where the anisotropy is so large that it respectively. Ds and τs stand √ for the spin diffusion constant and the can still be readily visualized and quantified. In Subsections III A ∥ ∥ spin lifetime while λs = Ds τs is the spin relaxation length. The and III B, the characteristics of these methods are discussed in Larmor frequency is given by ωL = γc B– , with γc = gμB /h,̵ where g is detail. the gyromagnetic factor, μB the Bohr magneton, and h is the reduced ̵ Planck constant. The spin projection over the detector is captured in A. Oblique spin precession L2 the term cos(ωL t), whereas √4πD 1 t exp[− 4D st ] is the probability that 1. Homogeneous spin channel s ∥ the injected spins reach the detector at a time t and exp(−t/τs ) rep- A schematic of the oblique spin precession method is shown resents the effect of spin relaxation. Spin precession measurements in Fig. 2(a). The magnetic field B, characterized by the angle β, for the parallel and antiparallel configurations are shown in Fig. 1(c). is applied in a plane that contains the F metal electrode long axis The spin signal is maximum at B = 0, and its absolute value decreases and is perpendicular to the substrate. The spin density ⃗s under with increasing B and vanishes at large magnetic fields due to spin an oblique magnetic field satisfies the one-dimensional steady state ∥ ∥ dephasing; τs , Ds , and λs are extracted by fitting the Hanle curve to Bloch equation, Eq. (1). Until 2016, most experiments used the configuration described ⃗ − τs−1 ⋅ ⃗s = 0, Ds ∇2⃗s − γc ⃗s × B (2) above to determine spin relaxation properties, with the magnetic field applied perpendicular to the substrate. In this situation, spins where Ds is the spin diffusion constant and τs−1 is a 3 × 3 tensor with ∥ ∥ ∥ precess in the plane of the spin channel, and, therefore, τs and λs entries τs and τs⊥ . are actually the spin lifetime and spin relaxation length for spins The simplest approach to solve Eq. (2) is to transform the coor- oriented in-plane (thus, the ∥ symbol). As discussed in Sec. III, dinates (ex , ey , ez ) into the rotated Cartesian axis system character- Raes et al. demonstrated that the application of oblique magnetic ∥ ⊥ ized by unit vectors (ex , eB , eB ). In the new coordinate system, the fields34 or, alternatively, in-plane magnetic fields perpendicular to applied magnetic field B ⃗ = (0, B, 0) can only induce precession to the the magnetizations of F1 and F235 enables the determination of the ∥ ⊥ spin lifetime τs⊥ and spin relaxation lengths λ⊥s for spins oriented spin population ⃗s = (sx , sB , sB ) perpendicular to it. The solution of out-of-plane. The simplicity of this experimental approach com- Eq. (2) for a homogeneous spin channel is described in detail in Ref. bined with its lack of artifacts has led to an increasing number of the- 35. Refined models include contact-induced spin relaxation due to oretical and experimental studies to investigate the spin relaxation the finite contact resistance39 and inhomogeneous spin lifetimes in anisotropy and gather information on spin relaxation mechanisms the graphene channel.37,38 and SOC. The solution of Eq. (2) for different values of ζ with B ori- ented at β = 45○ is shown in Fig. 3(a). It is observed that at B ≡ Bd ∼ 70 mT, the spin component perpendicular to B is fully III. DETERMINATION OF THE SPIN-LIFETIME dephased. For B > Bd , the precessional motion is completely sup- ANISOTROPY pressed, and thus, Rnl saturates to the remanent nonprecessional The magnetization of the F electrodes on a typical device spin component that lies along the magnetic field direction. In this situation, Rnl at the F detector is given by Rnl = Rnl (ζ, β) cos2 (β∗ ), β [Fig. 1(a)] is oriented in-plane along their long axis due to magnetic APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 7, 120701-3 © Author(s) 2019 APL Materials RESEARCH UPDATE scitation.org/journal/apm FIG. 2. Determination of the spin relaxation anisotropy ratio ζ by means of spin precession. (a) Schematics of the oblique spin precession method. The magnetic field B is applied at an angle β in a plane that contains the easy-axis of the ferromagnetic injector (F1) and that is perpendicu- lar to the substrate. (b) Schematics of the out-of-plane spin precession method. The magnetic field B is applied in plane and perpendicular to the easy-axis of the ferromagnetic injector (F1). (c) Illustration of a graphene-based lateral spin device with a nonhomogeneous spin channel. Regions I, II, and IV indicate the pristine graphene channel while region III is covered with a TMDC. As a spin diffuses from F1 to F2 under an oblique (a) or in-plane (b) magnetic field, it undergoes precession. As it approaches the TMDC, it gath- ers an out-of-plane component. Inset in (c) represents the graphene electronic band structure close to the Dirac point for graphene/TMDC. The colors and arrows represent the out-of-plane spin sub-bands. εk and Kx,y account for the energy band and the wave vectors respectively. where β∗ = β − γ(β, B) takes into consideration the tilting angle √ L √ 1 γ(β, B) of the magnetization of the F electrodes that results from Rnl (ζ, β) = f (ζ, β) exp[− ∥ ( − 1)]R0nl . (3) f (ζ, β) the applied magnetic field, as calculated below. The factor cos2 (β∗ ) λs accounts for the projection of the injected spins along the magnetic −1 field direction and the subsequent projection along the easy-axis of In this expression, f (ζ, β) = (cos2 (β) + 1ζ sin2 (β)) and R0nl is the the F detector. The term Rnl (ζ, β) is given by34 value of the nonlocal resistance at B = 0. For the isotropic case, it is FIG. 3. Spin precession line shapes Rnl (B, ζ) obtained numerically using Eq. (2) in an anisotropic medium. (a) Rnl (B, ζ) for the oblique spin precession method at fixed β = 45○ for anisotropy ratios ζ = 0.2, 0.5, 1.5, and 2. The dashed line corresponds to the isotropic case (ζ = 1). As ζ increases, the asymp- totic value of the spin signal increases. (b) Numerically extracted spin preces- sion line shapes Rnl (B, ζ) for the out- of-plane spin precession method for anisotropy ratios ζ = 0.2, 0.5, 1.5, 2, and 2.5. The dashed line corresponds to the isotropic case (ζ = 1). As ζ decreases, the position of the minima shifts to higher magnetic fields. Rnl (B, ζ) for magnetic fields applied out of plane (B– ) and in plane (B∥ ) for the isotropic (c) and anisotropic (d) cases. The perfect over- lap between the two curves in (c) is a consequence of the isotropic spin relax- ation, while in (d), the difference between Rnl for B– and B∥ allows us to easily identify the anisotropy nature of the spin transport. All the curves are normalized to their value at B = 0 and are obtained for L = 7 μm, Ds = 0.02 m2 s−1 . The in-plane spin lifetime is constant in all ∥ curves, τs = 0.5 ns, and only τs⊥ is changed. APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 7, 120701-4 © Author(s) 2019 APL Materials RESEARCH UPDATE scitation.org/journal/apm straightforward to verify that Rnl (ζ = 1, β) is independent of β and minimum in Rnl corresponds to an average spin rotation angle of its value corresponds to Rnl (ζ = 1, β) = R0nl . ±π. When B increases, spins start to acquire an out-of-plane com- The tilt angle γ of the F electrodes magnetizations can be calcu- ponent, which relaxes slower than the in plane one when ζ > 1. lated by considering the Stoner-Wohlfarth model, which provides a Then, for an average spin rotation of ±π and ζ ≫ 1, Rnl can be good approximation for a coherent magnetization rotation at low B. larger than its value at B = 0 (when spins are always in plane and The relationship between γ and β is given by the expression40 the spin lifetime is short). In contrast, for ζ < 1, the out-of-plane spin component relaxes faster than the in-plane one, implying that B sin(β) the minimum in Rnl is less pronounced and can eventually disappear γ = arcsin[ ], (4) Bs + B cos(β) for ζ ≪ 1. In addition, Fig. 3(b) shows that the position of the minima, where Bs is the saturation magnetic field. In the specific case of β = 90○ , γ = arcsin(B/Bs ). The functional dependence of γ which occurs for a collective π spin precession, changes with ζ. A spin contributes a precession angle ϕ in a diffusion time t with a with B is followed closely by experimental results on anisotropic probability P(t) such that t = ϕ/ωL . Here, P(t) is determined by the magnetoresistance and Hanle spin precession experiments, from product of the diffusion-time distribution function and the probabil- which Bs is readily obtained.34 ity that the spin has not flipped during t. The latter is proportional to A fundamental design parameter to correctly determine ζ using exp(−t/τ s ), resulting in a suppressed probability at long t or, equiv- Eq. (3) is the spin channel length L. In particular, a minimum L alently, at large ϕ. Such a suppression is more significant for short is required to observe complete dephasing of the spin component spin lifetimes, which implies that the collective spin precession angle perpendicular to the magnetic field at sufficiently low B, typically for a given B will increase with longer spin lifetimes. This explains Bd ≲ 0.15 T, thus avoiding magnetoresistance effects. Consider- why the minima in Fig. 3(b) develop at lower B as ζ increases.35 Fig- ing no contact effects in the spin relaxation process, the contribu- ure 3(c) compares Rnl when B is applied in-plane (B∥ , out-of-plane tion to the spin signal stemming from the spin component per- spin precession) and out-of-plane (B– , in-plane spin precession) for pendicular to the magnetic field R⊥nl can be written as R⊥nl (B) ∝ ∥ the isotropic case ζ = 1, where no differences are observed between exp(−L((B γc /2Ds )1/2 − 1/(Ds τs )1/2 )).39 Defining a threshold value the two configurations. These results are in stark contrast to those sat −η for the saturation such that Rnl = 10 √ Rnl (B = 0), the condition when ζ ≠ 1. In this case, Rnl clearly depends on the magnetic field ⊥ sat 2Ds Rnl (Bd ) ≤ Rnl establishes that L ≥ Bd γc η ln 10. Assuming high- orientation, as shown in Fig. 3(d). For out-of-plane spin precession, quality devices with λs = 30 μm, Ds = 0.075 m2 s−1 , and a saturation Rnl lies above (below) the in-plane spin precession line shape when threshold 10−η = 10−3 ,13,39 the minimum channel length is L ≈ 16 μm ζ > 1 (ζ < 1). Comparing the in-plane and out-of-plane to achieve full dephasing at Bd = 0.15 T. spin precession curves is a straightforward way to check the isotropic/anisotropic character of the system. 2. Inhomogeneous spin channel In the case of a nonuniform spin channel, Eqs. (1) and (3) are IV. SPIN RELAXATION ANISOTROPY EXPERIMENTS no longer valid to obtain the spin related parameters and ζ. This is IN LATERAL SPIN DEVICES particularly relevant for heterostructures comprising graphene with Having introduced the fundamental tools to determine the spin other materials covering partially the spin channel [see Fig. 2(c)]. lifetime anisotropy, this section first overviews recent implementa- Here, the diffusive process is governed by Eq. (2) with boundary tions in single- and bilayer graphene lateral spin devices. Then, it conditions, x = 0 : ⃗sI = ⃗sII ; ∂x⃗sI − ∂x⃗sII = ⃗s0 introduces graphene-based heterostructures in which the graphene SOC is enhanced by proximity of a high spin-orbit material, in par- x = l : ⃗sII = ⃗sIII ; ∂x⃗sII = ∂x⃗sIII ticular, in van der Waals heterostructures formed by graphene and (5) x = w : ⃗sIII = ⃗sIV ; ∂x⃗sIII = ∂x⃗sIV TMDCs. x → ±∞ : ⃗s = 0, where I, II, and IV denote the pristine graphene and III the het- A. Spin relaxation anisotropy in single-layer graphene erostructure region. Spins are injected in the y direction ⃗s0 = The first experimental efforts to extract ζ in graphene lateral (0, s0 , 0), where s0 is a constant that depends on the injected elec- spin devices required the application of out-of-plane magnetic fields, trical current and the contact polarization. Spin absorption effects as described in Sec III. The magnitude of the spin signal was found are not taken into account. to be smaller for out-of-plane spin injection than for in-plane spin ∥ injection. Tombros et al.42 reported ζ ∼ 0.8 with τs = 60 ps for B. Out-of-plane spin precession graphene on SiO2 . A similar approach was later used by Guimarães A schematic of the out-of-plane spin precession method is et al.36 in graphene fully encapsulated with hBN with enhanced spin shown in Fig. 2(b). Here, the magnetic field B is applied in plane ∥ relaxation times τs = 2 ns and higher carrier mobilities. Double and in a direction perpendicular to the easy-axis of the F metal encapsulation with hBN enables the control of the electric field E and electrodes. The precession occurs in a plane perpendicular to the the carrier density n independently by applying top- and back-gate ∥ substrate containing the easy axis, thus probing both τs⊥ and τs . The voltages. The values of ζ were reported to be electric-field tunable, solution of Eq. (2) for this configuration is shown in Fig. 3(b). For with ζ ∼ 0.75 for E = 0 V/nm to ζ ∼ 0.65 for E = 0.7 V/nm. ζ ≫ 1, Rnl strongly deviates from the isotropic case ζ = 1; therefore, a However, the determination of τs⊥ with perpendicularly mag- large anisotropy can be easily identified.37,38,41 The magnitude of the netized electrodes requires the use of intense magnetic fields, APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 7, 120701-5 © Author(s) 2019 APL Materials RESEARCH UPDATE scitation.org/journal/apm ∥ which represents a major drawback stemming from magnetoresis- are in excellent agreement with ζ ∼ 1, and thus, τs = τs⊥ which is tive effects. The graphene layer resistance and associated diffusion independent of n. A more recent experiment in low-mobility CVD constant are magnetic-field dependent, reducing the range of car- graphene on SiO2 by Ringer et al.44 combines the oblique and out-of- rier densities in which the method can be applied. These effects are plane spin precession experiments to extract ζ, yielding ζ ∼ 0.9–0.7, more pronounced close to the charge neutrality point (CNP) and in ∥ with τs between 1.1 ns and 0.7 ns. Nevertheless, measurements as a samples with large carrier mobilities;43 in such a scenario, they have function of n and temperature in highest quality graphene devices to be taken into consideration even at moderate applied magnetic and a careful analysis of the F magnetization with B in plane are still fields. Figure 4(a) shows the squared resistance Rsq of graphene as a necessary. function of B– for different n (back-gate voltages V g ). As expected, Initially, models based on the Elliot-Yafet (EY) and Dyakonov- magnetoresistance is largest nearby the CNP, with variations in Rsq Perel (DP) mechanisms were adopted to explain spin relaxation in in excess to 35% at B ≈ 1 T. For B below ∼0.2 T, the variations graphene. In the EY mechanism, the spin relaxation arises from spin are less than 3%, independently of V g [see the inset in Fig. 4(a)]. flips between scattering events leading to a linear scaling between the Figure 4(b) shows spin precession measurements with the magnetic ∥ momentum relaxation time τ p and τs . In the DP mechanism, spins field applied out-of-plane. It shows that Rnl increases for B > 0.5 T precess around an effective SOF between scattering events leading ∥ −1 because of the out-of-plane rotation of the electrodes’ magnetiza- tion. As B increases, a saturation of Rnl is expected when the rotation to a scaling τp ∝ τs .6 While the EY and DP mechanisms have is completed. However, further increase in B shows that Rnl presents successfully explained spin relaxation in metals and semiconduc- a monotonous decrease after reaching a maximum value at B ∼ 1.5 T. tors, experiments in graphene attempting to discriminate between This behavior is observed even far from the CNP and can be partially them have yielded contradictory results; reports searching for scal- ∥ explained by a suppressed spin injection and detection efficiency due ing laws between τ p and τs suggested spin relaxation consistent with to the increased graphene resistance. Following the analysis reported EY, DP, or a combination of both.45–48 Nevertheless, several con- in Refs. 36 and 42, for a signal decrease of 45% and 70%, as displayed clusions can be drawn from the spin lifetime anisotropy results. For in Fig. 4(b), ζ = 0.55 and ζ = 0.3, respectively. This result differs from DP mechanisms with Rashba SOFs, ζ = 0.5 is expected, while for EY the actual ζ ∼ 1 extracted with the oblique spin precession method driven by phonon scattering, ζ ≈ 0;6 therefore, they cannot explain in the same device at low magnetic fields [Fig. 5(b)]. ζ = 1 on their own. The presence of resonant scattering by local mag- Figure 5 shows typical spin precession measurements for a netic moments, arising from adatoms or vacancies, could explain the representative set of β values for graphene onto two different sub- experimental results. Calculations considering sp3 bonded hydro- ∥ strates, SiO2 34 and hBN. In both cases, τs is between 0.2 ns and gen adatoms can account for the observed spin lifetimes even for 0.4 ns. When applying the oblique precession method (Sec. III A), very small hydrogen concentration (∼1 ppm).15,16,18 The paramag- the precessional motion is suppressed at Bd ∼ 0.1 T for all β and netic nature of the local magnetic moments results in a random spin the remanent nonlocal spin signal is independent of B. The rema- orientation after each scattering event, so ζ = 1. Additionally, the nent signal derives from the nonprecessing spin component that lies common features observed in the case of single-layer graphene inde- β along the B direction, whose magnitude Rnl depends on β as expected pendently of the type of substrate (Fig. 5) suggest that the substrate from Eq. (3). Figures 5(b) and 5(d) show Rnl above Bd [marked by plays a minor role in making the spin relaxation isotropic, at least β for graphene with moderate mobility. a vertical line in Figs. 5(a) and 5(c)] normalized to 1, Rnl /R0nl , as a 2 ∗ function of cos (β ). From Eq. (3), for the isotropic case, i.e., ζ = 1, Rnl /R0nl = cos2 (β∗ ) so Rnl /R0nl as a function of cos2 (β∗ ) results in a β β B. Spin relaxation anisotropy in bilayer graphene β straight line with slope equal to one. For the anisotropic case, Rnl /R0nl Besides high mobilities at room temperature, gate-tunable car- lies above the straight line (ζ > 1) or below it (ζ < 1), as extracted rier density, and parabolic bands near the K and K ′ points, bilayer from Eq. (3). For both SiO2 and hBN substrates, experimental results graphene offers the possibility to tune the bandgap with a transverse FIG. 4. (a) Graphene squared resistance Rsq for different back-gates V g as a func- tion of the applied out-of-plane magnetic field B– . (Inset) Normalized Rsq at low values of B– . The dashed line marks the limit of B– where the oblique spin pre- cession measurements are typically car- ried out. (b) Spin precession measure- ments at high magnetic fields for different V g . The blue arrows mark the signal dif- ference between B = 0, B– = 1.5 T, and B– = 3.5 T. Reproduced with permission from Raes et al., Nat. Commun. 7, 11444 (2016). Copyright 2016 Springer Nature. APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 7, 120701-6 © Author(s) 2019 APL Materials RESEARCH UPDATE scitation.org/journal/apm FIG. 5. Room temperature spin preces- sion measurements acquired for the par- allel magnetization configuration of the spin injector and detector for oblique magnetic fields B applied at different angles β (from 90○ to 30○ ) in single- layer graphene on SiO2 (a) and hBN (c). The carrier densities are n = −2.3 × 1012 cm−2 and n = 1.2 × 1012 cm−2 , respectively. Rnl normalized by Rnl,0 vs cos2 β∗ for the SiO2 (b) and hBN (d). The solid lines in (b) and (d) represent Rnl for the indicated value of ζ calculated with Eq. (3), and the black straight line corresponds to the isotropic case, ζ = 1. [(a) and (b)] Reproduced with permission from Raes et al., Nat. Commun. 7, 11444 (2016). Copyright 2016 Springer Nature. electric field.49,50 First principles calculations show that the intrin- demonstrated that ζ can be large and tunable at low temperatures sic SOC around the K and K ′ is about 12 μeV and points out of (75 K). By controlling the carrier concentration n with a back-gate the bilayer graphene plane, similar to single-layer graphene.49 The voltage applied to the insulating substrate, ζ was found to vary space inversion symmetry breaking, caused by the substrate or an from 3.5 at n = 6 × 1011 cm−2 to about 8 at the CNP [Fig. 6(a)]. external electric field, results in an extrinsic Bychkov-Rashba SOC Because ζ is larger close to the CNP, the driven spin relaxation pointing in-plane that removes the spin and valley degeneracy. The anisotropy is attributed to the intrinsic out-of-plane spin orbit fields Bychkov-Rashba SOC increases with the carrier density n, whereas that induce a significant spin-valley coupling.51 Similar results in the intrinsic SOC decreases with n. Thus, spin relaxation driven by dual-gate devices were independently observed by Xu et al.41 The SOC can be inferred from ζ. use of a global back-gate and a local top-gate allowed them to con- Spin relaxation anisotropy measurements using the oblique trol the carrier density and the transverse electric field. A value of spin precession method have been recently reported in bilayer ζ = 12 is extracted at low temperatures (100 K) near the CNP under graphene encapsulated with hBN.41,51 Leutenantsmeyer et al.51 a transverse electric field E = −0.5 V/nm [Fig. 6(b)]. In addition, Xu FIG. 6. Spin anisotropy experiments in bilayer graphene. (a) Rnl normalized by Rnl,0 vs cos2 β close to the CNP of fully encapsulated bilayer graphene.51 (b) Spin anisotropic behavior in the presence of a transverse electric field E = −0.5 V/nm and close to the charge neutrality point (CNP) in dual-gate devices.41 (a) Reproduced with permis- sion from Leutenantsmeyer et al., Phys. Rev. Lett. 121, 127702 (2018). Copyright 2018 American Physical Soci- ety, (b) reproduced with permission from Xu et al., Phys. Rev. Lett. 121, 127703 (2018). Copyright 2018 American Physical Society. APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 7, 120701-7 © Author(s) 2019 APL Materials RESEARCH UPDATE scitation.org/journal/apm et al. observed isotropic behavior for n ≥ 4 × 1012 cm−2 . Temper- C. Spin relaxation anisotropy in single-layer ature dependent measurements show a decrease in ζ, resulting in graphene/ TMDC heterostructures isotropic behavior at room temperature. Besides the highly tunable Emerging graphene-based spintronics applications, such as ∥ value of the spin lifetime anisotropy, τs is about 10 ns, two orders of spin-logic devices, require tunability of a large SOC via electric magnitude larger than those observed in van der Waals heterostruc- fields. Although not accessible in pristine graphene, this can be tures with similar values of ζ (Sec. IV C), which can be ascribed to achieved by stacking graphene with high spin-orbit materials. Lay- the larger SOC in the latter. ered semiconducting TMDCs, which comprise heavy transition FIG. 7. Spin anisotropy experiments in lateral spin devices comprising single-layer graphene and a transition metal dichalcogenide (TMDC). [(a), (b), (e), and (f)] The out- of-plane spin precession method with an in-plane magnetic field (Bx ). [(a) and (e)] Spin precession in homogeneous single-layer graphene reference devices. [(b) and (f)] Spin precession in nonhomogeneous devices with a single-layer graphene channel partially covered with WS2 (b) at room temperature and MoSe2 (f) at 75 K. (c) Room temperature spin precession curves obtained by the oblique precession method at the indicated β in a graphene/WS2 device. The precession curves correspond to the parallel magnetization configuration of the spin injector and detector. Full dephasing of the precessing component is observed at B > 0.12 T. (d) Rnl normalized by R0nl vs cos2 (β∗ ). The data represented by full symbols are extracted from (c) at B = 0.16 T. The solid blue line represents the modeled response for ζ = 10. The data represented by open symbols correspond to the reference device with ζ = 1 in (a). [(a)–(d)] Reproduced with permission from Benítez et al., Nat. Phys. 14, 303 (2018). Copyright 2018 Springer Nature. [(e) and (f)] Reproduced with permission from Ghiasi et al., Nano Lett. 17, 7528 (2017). Copyright 2017 American Chemical Society. APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 7, 120701-8 © Author(s) 2019 APL Materials RESEARCH UPDATE scitation.org/journal/apm metal atoms (such as Mo or W) and chalcogen atoms (such as S, The strongly anisotropic behavior is manifested in Rnl normalized to Se, or Te), are of special relevance. First principles calculations have Rnl (B = 0) vs cos2 (β∗ ) [Fig. 7(d)]. Results similar to those in Figs. 7(a) shown a proximity-induced enhancement of the SOC in graphene and 7(b) have been independently observed in graphene/MoSe2 by TMDCs in the millielectron-volt range.52 When graphene is in ∥ at T = 75 K [Figs. 7(e) and 7(f)].37 The reported values for τs proximity with a TMDC, it preserves its linear Dirac band structure ⊥ and τs are 3–3.5 ps and 30–40 ps, respectively, with ζ ∼ 10, for and a bandgap opens due to broken pseudospin symmetry, while both graphene/MoSe2 and graphene/WS2 . The experimental val- ∥ the proximity-induced SOC and broken space inversion symmetry ues of τs and τs⊥ are well reproduced by numerical simulations remove the spin degeneracy and lead to a strong spin-valley cou- and microscopic theories of spin relaxation anisotropy that con- pling53 [see schematics in the inset of Fig. 2(c)]. About the K and sider a strong intervalley scattering in the graphene/TMDC het- K ′ points, spins have an out of plane component, alternating from erostructure.59,60 Intervalley scattering arises from short-range elas- up to down as the energy of the bands increases, and a winding in- tic scattering centers, e.g., structural defects in graphene or vacancies plane component, as a consequence of the Rashba-type SOC. The in the TMDC. Due to the strong spin-valley coupling present in out-of-plane spins and in-plane texture reverse the sign between the graphene/TMDC heterostructures, the out-plane spin relaxation fol- K and K ′ points.52 Such spin texture has a direct impact on the spin lows the Dyakonov-Perel mechanism with the out-plane spin relax- ∥ dynamics, leading to distinct spin lifetimes τs and τs⊥ . ation rate controlled by the momentum scattering time (τ p ), i.e., Weak (anti)localization measurements have been commonly 1/τs⊥ ∝ τp , while the in-plane spin relaxation is dominated by the used to evidence SOC enhancement in graphene/TMDC.54–57 In ∥ intervalley scattering time (τiv ), i.e., 1/τs ∝ τiv . With strong inter- such experiments, spin-orbit scattering times are extracted from valley scattering, τiv ≪ τ p , the spin lifetime anisotropy ζ extracted quantum corrections to the magnetoconductivity, the most notable ∥ from numerical simulations lies between 20 and 200, with τs ∼ 1 ps signature being a decrease in the spin lifetime to the picosecond ⊥ and τs ∼ 20–100 ps, which agrees well with the experimental results. range. However, it is not straightforward to distinguish between the influence of symmetric and antisymmetric spin-orbit scattering times, which are related to the in-plane and out-of-plane spin relax- V. CONCLUSIONS AND PERSPECTIVES ation times.54,58 In addition, weak (anti)location measurements are This Research Update describes the state-of-the-art techniques restricted to low temperatures (typically bellow 10 K), which makes to determine the spin lifetime anisotropy in lateral spin devices. the method limited in comparison to spin precession experiments The so-called oblique and out-of-plane spin precession methods (see Sec. III). are considered the most reliable. Their development has moti- Spin precession experiments on graphene/TMDC heterostruc- vated a growing interest in evaluating anisotropic spin dynamics, tures use a lateral spin device geometry schematically shown in by modeling and experiments, as a means of gathering information Fig. 2(c). It consists of a nonhomogeneous spin channel in which about graphene intrinsic SOC and about proximity-induced SOC in single-layer graphene is partially covered with a TMDC. Spins are van der Waals heterostructures. The discussed experiments repre- injected through F1 and diffuse along the graphene channel toward sent the starting point of this lively research area that aims at inves- F2. Due to the proximity-induced SOC in the graphene/TMDC tigating new physical phenomena and novel spintronics applications region, the in-plane and out-of-plane spins have different associated using 2D materials. lifetimes. Future experiments will aim at identifying intrinsic spin relax- Experiments carried out by Ghiasi et al.37 in graphene/MoSe2 ation in graphene, without the presence of dominant magnetic scat- heterostructures at low temperatures (75 K) and by Benítez et al.38 tering centers, and at achieving full control of the spin dynamics in graphene/WS2 (and MoS2 ) heterostructures at room temperature by manipulating the SOC with external electric fields. Spin relax- have demonstrated strongly anisotropic spin transport behavior. ation anisotropy must be studied in ultraclean devices with large Figures 7(a) and 7(b) summarize the features of the spin precession mobilities.13 Encapsulating graphene with a top hBN dielectric could response in the out-of-plane precession configuration [Fig. 2(b)] in protect graphene from chemicals and aggressive device fabrica- single-layer graphene and when WS2 is placed over the graphene tion processes, which can introduce adatoms or impurities acting spin channel.38 For isotropic spin relaxation in graphene [Fig. 7(a)], as magnetic scattering centers.61 Such extrinsic contributions can Rnl has its highest value at B = 0 and starts decreasing with B due to actively mask or alter the intrinsic spin-relaxation processes. On the spin precession and dephasing. At sufficiently large B, the magneti- other hand, recent first-principles calculations predict a giant spin zation of the F electrodes rotates and eventually saturates along the relaxation anisotropy in graphene on hBN depending on the stack- field direction. As injected spins are parallel to B, they do not precess ing configuration and external electric fields,62 which has not been any longer and Rnl recovers its value at B = 0. This is in stark contrast implemented in the experiments so far. with the features observed in the graphene/TMDC heterostructure The case of graphene-based heterostructures is of special rele- [Fig. 7(b)]. As B increases, Rnl becomes much larger than its value at vance because they can play a key role in the development of spin B = 0. The maximum magnitude of Rnl is reached when the diffus- logics and novel devices based on coupled spin-valley dynamics. ing spins rotate ∼π/2 at the TMDC location and point roughly out Spin lifetime anisotropy experiments are consistent with proximity- of plane. The subsequent decrease in Rnl is due to spin dephasing induced SOC and intervalley scattering as the origin of the spin and the rotation of the magnetization of the F contacts. The anoma- relaxation. However, the experimentally extracted values are some- lous enhancement of Rnl is a clear indication of anisotropic spin what smaller than those obtained by numerical simulations. In ∥ relaxation, with λ⊥s > λs and thus ζ ≫ 1. Figure 7(c) further shows this regard, high interface quality of graphene with other high Rnl for different β using the oblique spin precession configuration. spin-orbit layered materials is essential to ultimately control the APL Mater. 7, 120701 (2019); doi: 10.1063/1.5124894 7, 120701-9 © Author(s) 2019 APL Materials RESEARCH UPDATE scitation.org/journal/apm magnitude of the induced SOC by the action of external electric Schönenberger, P. Seneor, C. Stampfer, S. O. Valenzuela, X. Waintal, and fields. B. van Wees, 2D Mater. 2, 030202 (2015). 18 Finally, the investigation of spin relaxation anisotropy is inher- D. Kochan, S. Irmer, M. Gmitra, and J. Fabian, Phys. Rev. Lett. 115, 196601 ently related to spin-charge conversion (SCC), as both phenom- (2015). 19 D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. Lett. 103, 146801 ena derive from SOC characteristics.63 In recent years, there has (2009). been a growing interest in observing proximity-induced SCC in 20 S. 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