Spin transport in non-magnetic nano-structures induced by non-local spin injection

We review our recent achievements on optimization of spin injection from ferromagnetic into nonmagnetic metals and characterization of spin transport properties in the non-magnetic nano-structures. We have realized the efficient spin injection by solving spin resistance mismatch problem in spin diffusion process across the interface between ferromagnetic and nonmagnetic metals. We analyzed temperature dependent spin relaxation length and time in Ag within the framework of the Elliot–Yafet mechanism based on spin–orbit interaction and momentum relaxation. The spin relaxation length in a light metal Mg is found comparable to that of Ag due to its peculiar electronic band structure in which so called spin-hotspots dramatically enhance spin relaxation. Spin relaxation properties in various metals are also quantitatively discussed. We employed commonly used Hanle effect measurements to characterize spin relaxation of spin current and reexamined both theoretically and experimentally the effect of spin absorption at the interface. The affected spatial profile of chemical potential due to the longitudinal and transverse spin absorption results in the broadened Hanle curve. All the Hanle curves both in metallic and semi-conductive materials including graphene fall into the universal scaling plot. Anatomy of spin injection properties of the junction and spin transport properties in non-magnetic metal is shown in tables. & 2014 Elsevier B.V.. Published by Elsevier B.V. All rights reserved.


Introduction
Since 1988, spintronics has witnessed a variety of spin related phenomena such as giant magneto-resistance, tunneling magnetoresistance and spin transfer torque, etc., that are governed by the interaction between spins of itinerant s(p) and localized d electrons [1,2]. More importantly, spin current has also been found to play a central role in causing such spintronics phenomena to occur not only in traditional magnetic metals but also in various materials including metals, semiconductors and oxides [3,4]. The generated spin current can be used to manipulate dynamic properties of the magnetization via spin-transfer-torque and spin-orbit-torque in magnetic/nonmagnetic hetero-structures [5][6][7][8][9][10].
Pure spin currents carry only spins (spin angular momenta) unlike conventional spin-polarized currents which carry both charges and spins [4]. One of well-known methods to generate the pure spin current is non-local spin injection. When the spin-polarized current is injected from a ferromagnet (FM) into a nonmagnet (NM), spins are accumulated in the vicinity of the FM/NM interface. The accumulated spins diffuse in the NM, and thus the spin accumulation is formed in the place where no charge current is present. The non-local spin injection was first demonstrated in 1985 using micro-scale devices which consist of a 50-mm-thick Al bar with ferromagnetic junctions [11]. This experiment demonstrated a tiny spin accumulation signal of a few tens of pico-volts. The experiment was revisited in 2001 using nano-scale lateral spin valves (LSVs) [12]. This brought about an enhanced signal of about one micro-volt at room temperature, which spurred intensive research efforts in non-local LSVs for spintronic device applications.
In order to boost spintronic research excellence, the solid understanding of transport properties of the pure spin current is essential. The efficient generation of the pure spin current is indispensable. In Section 2, we describe spin injection-detection characteristics in LSVs and how to enhance the output signal of LSVs which is a direct measure of the spin accumulation in a NM nanowire. For example use of an interface MgO layer enables us to avoid spin resistance mismatch problems which hamper efficient spin injection from FM into NM. In Section 3, we study the spin transport and relaxation mechanisms of Mg as a light metal as well as Ag as a "standard" NM metal because the spin-orbit interaction plays an important role. So far, the spin relaxation mechanism in NM has mainly been studied by means of conduction electron spin resonance (CESR) [13] and the sample dimensions were limited to bulk. The intrinsic (phonon) and extrinsic (impurity, grain boundary and surface) properties of the spin relaxation in the NM nanowires are analyzed in the temperature dependence of the spin relaxation time, which results in the characteristic value of the spin-flip probability in consistent with other techniques such as CESR. In Section 4, we describe the collective precession of the pure spin current in LSVs, so-called Hanle effect [11,14]. In ballistic transport, spins can coherently rotate at a Larmor frequency proportional to the applied perpendicular magnetic field, however, in diffusive transport of the pure spin current, dephasing occurs in the collective spin precession. We have successfully formulated the Hanle effect signal of LSVs with taking into account its relaxation and anisotropic spin absorption for the transverse and longitudinal components of the spin polarization direction of the pure spin current relative to the detector magnetization-direction. We demonstrate LSV with NiFe/MgO/Ag junctions which enable to detect a highly coherent spin precession over a long-distance of several microns in a silver nanowire. The phase coherency in the spin precession is detailed and the material-independent properties of the diffusive spin current are extracted.
2. Nonlocal spin injection scheme to characterize transport properties of pure spin current

Spin injection-detection properties of lateral spin valve
The current density j s in the channel s is written with the drift term and the diffusion term characterized by the non-equilibrium electron density δn s where ϕ e and δε s are the electrical potential and the shift in the electrical chemical potential of electrons from its equilibrium value. Therefore, current density is expressed by the electro chemical potential μ s ¼eϕ e þδε s Note that j s ≡j ↑ À j ↓ is spin current density. By combining them with Eqs. (2.1)-(2.3) and the detail balanced formula N ↑ /τ ↑↓ ¼ N ↓ /τ ↓↑ where τ ss′ is the spin scattering time from spin state s to s′, one obtains [15][16][17][18][19]  ↓ is used. When the current is applied at the interface between FM and NM, the electro chemical potential depending on the spin channel is modified. The difference in the spin-dependent electro chemical potential is known as spin accumulation δμ ¼(μ ↑ -μ ↓ ). According to Eq. (2.5), δμ obeys .
(2.7) 2 sf 2 δμ λ δμ ∇ = − δμ decreases with distance, and its characteristic length is called as the spin diffusion length. The typical spin diffusion lengths range from several nanometers to several micrometers for NM and from several nanometers to tens of nanometers for FM [13]. From Eq. (2.5), the relation between the spin accumulation δμ and the spin current density is given by [20] j e 2 . (2.8) It is convenient to use the spin-resistance R N(F) ≡ρλ sf /A where A is the cross-sectional area through which the spin current flows [17,21] and ρ is the resistivity. The subscripts of N and F represent NM and FM, respectively. We consider the one dimensional spin diffusion model where spin current is injected at x¼ 0 and decays at infinity. In this case, the solution of Eq. For FM, Eq. (2.9) holds with the corrected factor of (1 À P F 2 ), i.e., I S ¼δμ/(2eR F /(1 À P F 2 )) where P ( ) / ( ) is the spin polarization of FM. It means that the spin resistance is the opposition to the passage of the spin current through a material.
In order to discuss the spin accumulation in the vicinity of the FM/NM interface, Eq. (2.7) is solved in the geometry of the junction shown in Fig. 1(a). Here, no contact resistance and no spin-flip at the interface are assumed. Also, one dimensional model is used (assuming that the spin diffusion length of FM is much shorter than the widths of FM and NM and the spin diffusion length of NM is much shorter than the width and the thickness of NM  where Θ(x) is the step function, a 1 ,b 1 ,V 1 is the constant determined by the boundary condition. a 1 and b 1 correspond the generated spin accumulation in NM and FM, respectively and V 1 is the electro chemical potential difference of the interface.  where e x and e z is the unit vector in x and z direction. From Eq. (2.9), one finds that the effective spin polarization P eff in the vicinity of the interface decreases from "bare" FM spin polarization P F due to the back-flow of the spin current. In order to obtain the electro chemical potentials for NM and FM, the coefficients a 1 , b 1 and V 1 are derived with Eq. (2.10) with using the conditions of the continuity of the electro chemical potential and the first order differential (that is, we consider the case that there is no interface resistance and the loss of spin) as  Curve is derived with material parameters shown in [22]; [28]. Fig. 1(b) and (c) show the spatial distribution of the electro chemical potential as a function of the distance from the interface.
The spin accumulation in NM is detected by using ferromagnetic electrodes which has spin-dependent electro chemical potential. Two FM electrodes in LSV work as an injector and detector as shown in Fig. 2(a). In the case of Ohmic junction, the electro chemical potential of FM and NM is continuous as shown in Fig. 2 (b), and thus the spin accumulation appears in FM attached to the NM sustaining spin accumulation. Spin relaxation takes place in FM in the length-scale of the spin diffusion length. The relaxation depends on the relative direction of the spin polarization of injected spins and the magnetization of the detection FM electrode because the density of states near the Fermi surface depends on the spin. Therefore the potential after the relaxation depends on the spin as shown in Fig. 2(b). When the magnetization configuration of injector FM and detector FM in LSV is antiparallel, the detection voltage is different. Therefore, by sweeping the magnetic field in parallel to the easy axis of the FM wires, one observes characteristic signal, spin valve signal as shown in Fig. 2(c) and (d). We note that the switching field of FM wires are need to be different to observe a clear spin valve signal. Detection voltage V is proportional to the injected current I [15], and V/I is called as spin valve signal (non-local resistance). Especially, the difference of the voltages ΔV S ≡(V P À V AP ) and its normalized value ΔR S ≡ΔV S /I are called as spin accumulation signal (voltage) and spin signal, respectively and used as performance index of the LSV.
Spin signal ΔR S can be deduced from spin diffusion equation. In similar to the previous paragraph, spatial distribution of the electro chemical potential is derived from Eq. (2.4). By taking account of the contact resistance, the boundary conditions are where G Ij s is the conductivity of the j-th interface (junction) for each spin channel s. One obtains spin signal as [15]   . From [28].
where P Fj is the spin polarization of the j-th FM, R Fj is the spinresistance of j-th FM.
( ) is the interfacial spin polarization and L is the separation between the injector and the detector. This formula includes the effect of "spin-resistance mismatch" (will be detailed in the Section 2.3). When R I c R F and R N , The principle of the detection of the spin accumulation for resistive junction (including magnetic tunneling junction) is as follows. The spin current is absorbed into the junction according to Eq. (2.13). G Ij s is different but I | | σ is same for the spin channels. Therefore the decrease of the electro chemical potentials at the junction, ( ) are different for the channels and the spin accumulation is detected via resistive junctions with a finite interfacial spin-polarization.

Sample fabrication and measurement methods
The modern device-fabrication technique opens a way to study spin transport phenomena in the sub micrometer scale ranging from 100 nm to 1 μm, which is shorter than that of the characteristic length of NM. In this section, we briefly summarize fabrication procedure of LSVs.
The samples were fabricated by using resist mask patterned by means of electron beam (e-beam) lithography. Typical procedure is as follows. Firstly, Ti(Cr)/Au electrodes and alignment marks were fabricated on Si/SiO 2 substrates by means of conventional photolithography, to reduce the time for patterning by e-beam lithography. Then the sample was patterned by e-beam lithography in various steps. For example, the FM (e.g. Py) wires were firstly prepared on the sample and subsequently NM (e.g. Cu) wires are deposited in order to bridge Py wires [12]. Ion-milling is usually performed prior to Cu evaporation to clean the surface and make the FM/NM contacts transparent in conduction. In this case, the ebeam lithography was repeated twice to prepare two wires. In the second case, the number of times to use e-beam lithography can be reduced to once by employing Shadow evaporation technique [23]. For instance, double-layer organic resist of Methyl methacrylate (MMA) and Poly methyl methacrylate (PMMA) were spun on the sample. After the e-beam lithography and the development, the resist patterns form a three dimensional structure as shown in Fig. 3. The undercut structure was created by using backscattering of e-beam on the substrate. The resist structures were examined by the scanning electron microscope SEM operated with ultra-low acceleration voltage as shown in Fig. 4. In the third example, one uses trilayer structures which consist of MMA/Ge/PMMA. In this case, the patterns of only PMMA layer was developed firstly because the developing solvent is blocked by Ge before it reached to MMA. Plasma etching is performed to transfer the pattern of MMA to the Ge mask [24]. Then the second time development and Oxygen plasma cleaning were performed to remove the MMA layer. The resist structure permits higher aspect ratio of undercut compared to the one in double layer resist. In the fourth example, one can fabricate LSV with single e-beam evaporation process without double layer resist, by using dual tilted evaporation with a high-aspect-ratio resist structure [25]. In this case, both NM and FM layers need to be evaporated with tilt, which may worsen the spin transport properties (i.e. the junction polarization, the electrical conductivity, and spin diffusion length may largely decrease.). Advantage is the acceptable range of the fabrication condition is wider than those in the second and third techniques because the bridged parts of the structures in the second and the third cases are fragile.
After the depositions, unnecessary resists and layers were removed by soaking the sample in a remover, which is called as a liftoff process. Then samples can be heat treated. The treated conditions were typically in the forming gas (the mixture of Ar with H 2 ) and the temperature at 400°C [22,26]. The sample was examined by SEM. The improper fabrication condition causes remaining resist and metal films on the sample, in which situation the edge of the sample may create high contrast in a SEM image. This situation is especially seen in the improper milling conditions when the milled layer is deposited on the side of the wires. The detail and the parameters of sample fabrication are found in Refs. [27][28][29].
The lateral spin valve signal can generally be measured electrically as follows: Electrical spin injection generates spin accumulation which diffuses to the detector junction and the spin dependent electrochemical potential can be detected by measuring the voltages of FM detector. See Fig. 2 for the principle of measurement. Electrical measurement was performed by using conventional current-bias lock-in technique or by a dc current source and nano-voltmeter. Lock-in technique enables to detect a tiny signal at a low frequency. The S/N of DC measurement is not as good as lock-in technique but enables to keep the duration of the applied current short. The DC measurement is thus suitable for high current measurements.
In order to characterize spin transport properties and the spin injection/detection properties of the junctions, one examines separation dependence of spin signals. Another characterization scheme is called as Hanle effect, i.e. the observation of collective spin precession in NM. When the magnetic field is applied perpendicular to the polarization of spin current, Larmor spin precession is induced (we will detail in Section 4). The precession angle depends on the transit-time from injector to detector. The response gives time-scale of spin transport and hence the spin relaxation time and diffusion constant for spin current can be examined. The Hanle effect is so far intensively used as the means to characterize spin transport properties [14,[30][31][32][33]. The Hanle signal can be observed both for P and AP magnetization states. Generally speaking both states should be measured because otherwise it may be difficult to exclude the AMR effect [32]. The magnetic field range required for the measurement depends on the spin relaxation time and the ratio of separation L, spin diffusion lengths λ N , and contacts resistance (which determines the strength of spin absorption effect). Generally, the metallic structure requires magnetic fields as high as 100-400 mT. Therefore one needs to align the magnetic field accurately not to have domain wall nucleation and propagation otherwise the magnetization reversals can easily take place in the injector and detector especially for Py. Also, the upper limits of applicable magnetic field were determined by the magnetization process of injector and detector [34,35]. In the case of graphene, the magnetic field is typically one order of the magnitude smaller because of the longer spin relaxation time.
Additionally, recently another type of characterization was proposed called as three-terminal Hanle effect. In this scheme, the voltage across the junction was measured with applying magnetic field perpendicular to the sample. The collective spin precession modifies the voltages across the junction [36,37]. Initially the effect was explained as it is responsible only for the spin polarization of the junction, later, the detail of the interface states are also considered responsible for this effect [38]. Later explanation is consistent with 3-terminal experiment on the metallic system where there is no modulation of the junction voltages which accounts for spin relaxation time in NM [39].

Enhanced spin accumulation signal and extended structure in LSVs
The spin-polarization P of the injected current in NM is expressed as [15,28] We note that the spin absorption of the detector is neglected for simplicity. Fig. 5(a) shows j S /j as a function of the spin resistance. The typical parameters of NM and FM is R F /R N ¼0.1, therefore the most of the spin-polarization of the current in FM relaxes at the interface. It is thus difficult to inject spins into NM, that is known as conductance mismatch problem (we call it as "spin-resistance mismatch"). Theoretically, the obstacle of the efficient spin injection can be overcome by introducing the spin-polarized interface (e.g. tunnel junction) [15,[40][41][42]. The injected spin polarization is then given by [28] ( Where R I is the interface resistance and R I ¼R I1 ¼ R I2 and P I ¼ P I1 ¼ P I2 . We note that if NM is two-dimensional system, Eqs. (2.15) and (2.16) can be used by replacing t N /ρ N in NM with the sheet conductance s N [43]. Fig. 6 shows the j S /j as a function of R I in the typical parameters of Py/MgO/Ag junctions [22]. This indicates that the back flow of spin current decreases as R I increases. Experimentally, the spin injection properties were reported with a variety of junctions such as Al/Al 2 O 3 /Co [14,44], Al/Al 2 O 3 /Co 50 Fe 50 [23,45,46], Al/Al 2 O 3 /Py [45,[47][48][49], and Co/ Al 2 O 3 /Cu [30,50], of which Al 2 O 3 layers were formed by oxidizing Al or depositing AlO x . Spin injection properties of Py/MgO/Ag [22,26,51,52] and Co 50 Fe 50 /MgO/Ag junctions [53] were also reported of which MgO is e-beam deposited. Alternative way to overcome the spin resistance mismatch is to use a FM with high spin polarization because the factor 1/(1-P F 2 ) should be taken into account for the effective spin resistance (the spin absorption was suppressed when P F becomes larger). For example, Takahashi et al. [54] reported the suppression of the spin absorption effect as well as the spin valve signal in metallic LSV using Heusler materials Co 2 Fe(Ge 0.5 Ga 0.5 ). Therefore search for high P F material is one of important research directions [55]. Table 1 summarizes the parameters of the junctions as well as the fabrication condition.
Another interesting class of material to study spin transport is graphene [33] where the spin diffusion length could be as long as 100 μm [63]. For graphene, spin resistance mismatch issue can be described in the same manner with metallic case although spin resistance of graphene is much larger than those of metals. Spin transport properties of graphene were reported in many papers but reported data still vary in a wide range. Table 1 also summarizes so far available data for junction properties. For graphene LSV, the spin injection properties were reported with Graphene (/TiO x )/AlO x /Co [33,57,58], Graphene(/TiO x )/MgO/Co [59], Graphene/MgO/Py junctions [60], and others [61,62]. We note here that some values of λ sf could be underestimated whereas some of P I could be overestimated because the spin absorption effect was not taken into account in the analyses of Hanle curves [43,64].
LSVs have various extended structure in multi-terminal geometry [65], which are not only useful to characterize spin transport properties by using middle wire, but also important to enhance spin accumulation. Dual injector LSV (DLSV) with spin-polarized interface layer enables to enhance spin accumulation compared with the single injector lateral spin valve (SLSV) consisting of two FM wires bridged by a NM wire. As shown in Fig. 7, a DLSV consists of three FM wires bridged by a NM wire. By cutting the edge of a NM wire at the injector, the undesired spin relaxation volume can be removed. The measurement geometry is shown in Fig. 7.
The formulation of the spin signal in DLSV is almost the same as that of SLSV. The electro chemical potential of the N wire in DLSV is a e a e a e x d , a 1 , a 2 and a 3 are the coefficient determining the electro chemical potential shifts due to the spin injection/absorption from FM1, FM2 and FM3, respectively, d 12 and L are the center-center separation between FM1-FM2 and FM2-FM3, respectively. The spin diffusion Eq. (2.7) with the boundary conditions for DLSV yields [35] (1 ) cos ( ) are the normalized spin-resistance of FM and the normalized interface resistance, respectively. The detected spin signal V 3 /I is calculated as It turns out that the spin accumulation is maximized when the magnetization configuration of the injectors F 1 -F 2 is antiparallel. The overall change of the spin valve signal ΔR S in the in-plane magnetic field dependence of V 3 /I is expressed as Assuming that all the electrodes and their interface are identical, Eq. (2.20) is expressed as Fig. 6. R I dependence of spin injection efficiency. From [28]. . .
The enhancement factor of DLSV compared to SLSV is defined as α increases monotonically as r I þr F increases and shows a maximum α max ≡1 þ exp( À 2d 12 /λ N )þ2exp(-d 12 /λ N ) for r I þr F ⪢ 1.
When the interface resistance R I is larger enough to prevent the spin absorption effect into FM from NM (R I ⪢ R N ), ΔR S is expressed as The spin signal is enhanced by a factor of α max compared to SLSV where the 1 þexp( À 2d 12 /λ N ) and 2exp( À d 12 /λ N ) in α max are from spin injections from FM2 and FM1, respectively. The first term of 1 represents the direct diffusive spin flow to FM3 from FM2, and the next term of exp( À2d 12 /λ N ) represents the flow to FM3 from FM2 via the reflection at the edge of the N wire near FM1. The last term of 2exp( À d 12 /λ N ) is related to spins injected from FM1. Fig. 8 shows simulated enhancement factor for various spin-resistance mismatch factor r n ≡r I þr F as a function of normalized separation d 12 /λ Ν . The experimental verification of the enhancement of spin accumulation and the effect of spin absorption in DLSV is described in Refs. [35,53].
Since LSV offers various extended structure, it is also beneficial to characterize spin Hall effect, i.e. interconversion between a longitudinal charge current and a transverse pure spin current, as a direct response of spin dependent scattering, associated with spin injection, [46,66,67]. Here, we provide the analytical expression for the LSV in middle wire geometry as illustrated in Fig. 9 [68]. Middle wire (M 2 ) is usually NM with large spin-orbit interaction (P F ¼0), but we do not fix P F ¼0 not to lose the generality. The non-local spin signal V 3 /I is expressed as In the simple case where the injector FM and detector FM wires are equivalent (F 1 ¼F 3 ), one obtains following expressions with where V with , V without are the V 3 in LSV with and without middle wire, respectively. Under the condition of ( )   8. Enhancement factor of various spin-resistance mismatch factor r n ≡r I þ r F from with spin absorption regime (r n ¼0.1) to without spin absorption regime (r n ¼ 100) as a function of normalized separation d 12 /λ Ν . .
where R M is the spin resistance of the middle wire [68].
3. Spin relaxation mechanism in nonmagnetic nanowires

Spin relaxation mechanism of conduction electrons
Spin relaxation mechanism in metals has been intensively studied for a long time not only for the fundamental interest but also for the spintronics applications. So far, three types of the spin relaxation mechanism have been proposed for conduction electron spins by Elliott-Yafet [69,70], D'yakonov-Perel' [71,72] and Bir-Aronov-Pikus [73].
In the Elliott-Yafet mechanism, the spin-orbit interaction (SOI) and momentum relaxation play key roles. Firstly, we consider the simple case where Bloch states of different spins |þ 4 and |À 4 ("up"-and "down"-) are subjected to the momentum scatterings with phonons, impurities and grain boundaries. The spin relaxation time τ sf is inversely proportional to the scattering matrix as 1/τ sf ∝ |o þ |ℋ|À 4| 2 , where ℋ is the Hamiltonian. Therefore, the spinindependent momentum scatterings alone do not induce spin relaxation. However, in real crystal, the SOI mixes the spin-"up" state and spin-"down" state. Considering small SOI, one takes perturbative approach and, original "up" state becomes the admixture of the up state and the down state of the order of λ SOI /ΔE where λ SOI is the spin-orbit splitting [69,70]. That is, the "new" spin up-state |↑4 E|þ 4 þ (λ SOI /ΔE)|À 4 has non-zero transition probability to |↑4. The spin relaxation time is proportional to the momentum relaxation time τ e and inversely proportional to the square of the spin-orbit splitting in the first order. Table 2 listed the parameters for spin relaxation in typical metals from the atomic spectrum [70]. For Li, Mg, Al, Cu, Ag and Au, the s-state is hybridized with p-state or d-state and the spin-flip probability in the concerned material depends on the hybridization condition. Maximum (λ SOI /ΔE) 2 rapidly increases with an increase of atomic number. There, the spin relaxation time is expected to be longer for light metals.
In the D'yakonov-Perel' mechanism, the SOI lifts the spin-degeneracy in the crystal lacking the inversion symmetry such as the zinc-blend semiconductor. This is equivalent to having a momentum-dependent internal magnetic field B(k) which can induce spin flips through the interaction term S B k ( )̇. The spin relaxation rate 1/τ sf is proportional to the momentum relaxation time τ e .
In the Bir-Aronov-Pikus mechanism, the spin relaxation caused by the electron-hole exchange interaction. This interaction depends on the spins of interacting electrons and act as an effective magnetic field. This mechanism is effective only for the semiconductors with a significant overlap between electron and hole wave functions [74]. Table 2 Atomic parameters to discuss spin relaxation properties in metals [70].    Fig. 9. Non-local spin valve with middle wire.

Surface spin scattering of nanowires
In LSVs, spin diffusion is limited in one dimension because the NM-wire width and thickness is much shorter than its spin diffusion length. Recently, there are several reports addressing the effect of surface on spin relaxation in LSVs but still in controversial [34,75]. The effect itself is recognized in the age of CESR [76]. A large contribution of the surface spin scattering process hampered a quantitative analysis of the spin relaxation mechanism in the NM nanowires. Therefore, the influence of a MgO capping layer on the spinflip mechanism for Ag nanowires was investigated [77]. A SEM image of fabricated LSV and the schematic diagram of the nonlocal measurement circuit are shown in Fig. 10(a). The field dependence of the spin signal for LSVs is shown in Fig. 10(b). Clear spin valve signals ΔR S were observed to be 5.06 mΩ and 2.36 mΩ for LSVs with or without MgO capping layer, respectively. ΔR S exhibits an exponential decreases with increasing L, as can be seen in Fig. 10(c), being attributable to the spin relaxation in the Ag nanowire. Here we assume a transparent interface for the Py/Ag junction of our devices, i.e. clean interfaces confirmed by TEM analyses and very low interface resistance of the Py/Ag junction [77]. Therefore, the analytical expression of ΔR S can be expressed as, where P F is the spin polarization of FM, R Ag ¼ρ Ag λ Ag /t Ag w Ag and R Py ¼ ρ Py λ Py /w Ag w Py are the spin-resistances for Ag and Py, respectively, where ρ is the resistivity, t the thickness, and w the width. The experimental data were fitted by adjusting parameters P F and λ Ag with setting the value of λ Py ¼5 nm reported by Dubois et al. [78]. denotes T-dependent surface spin-relaxation time. Fig. 11(a) shows λ Ag calculated from separation dependence of ΔR S , and the one obtained from ρ Ag and fixed spin-flip probabilities ε ph and ε imp , later of which will be detailed in the next section. Temperature variation of τ sf is also obtained by using D Ag N sf λ τ = as shown in Fig. 11(b). The calculated λ Ag and τ sf based on the E-Y mechanism for LSV with capping layer well reproduce the temperature dependence whereas those without capping layer shows the clear deviation from experimental data especially below T¼ 40 K. This means that the surface scattering is pronounced in the LSV without capping layer. We shall note here that quantitative model of T-dependent surface scattering is proposed in Ref. [34], however, cannot be directly applied on the highly conductive NM in our samples. In order to clarify the surface spin-relaxation quantitatively, further study is required. The possible origin of the reduction of surface spin relaxation is as follows. Surface spin-flip is dominated by SOI [80,81]. Therefore, the surface spin-flip probability, ε surf , is of the order of the magnitude, (αZ) 4 , where e c / 2 α = ℏ and Z is an atomic number [80,81]. Oxidization possibly forms Ag-O layer on the surface of the sample without capping layer. The MgO capping layer decreases effective Z at the surface because atomic number of Mg is much smaller than that of Ag, may result in suppression of the spin-flip scattering.

Spin-flip probabilities for phonon and impurity scatterings in Ag and Mg nanowires
According to the E-Y mechanism, the total spin relaxation time is given by  are momentum relaxation times, and ε ph and ε imp are probabilities of spin-flip scatterings. The notations "ph" and "imp" respectively correspond to phonon, and impurity (including grain boundaries and T-independent surface scattering) mediated scatterings and probabilities. For LSV with capping, we derive ε ph and ε imp by assuming the T-dependent surface spin-relaxation is suppressed to be enough small.
To compare sf where m e is the electron mass, n is the free electron density ( ¼ 5.86 Â 10 22 cm À 3 for Ag) [82], K is a constant for a given metal and Θ D is Debye temperature. Experimentally obtained phonon contribution to the resistivity ph ρ is fitted to Eq. (3.3b) with K and Θ D as fitting parameters. Then we obtain K and Θ D as 5.15 Â 10 À 6 Ω cm and 184 K, and 4.70 Â 10 À 6 Ω cm and 175 K, for LSV with and without capping, respectively. These are in good agreement with reported values [83]. Also, K is expressed by the analytical expression and K , B ph D α ρ = Θ with α B ¼4.255 [84]. Our data results in α B $ 4.2, which is in reasonable agreement. As discussed previously, since the T-dependent surface scattering can be assumed negligible in the LSV with the MgO capping layer, the constant value of τ sf at the low temperatures is considered as sf imp τ . Hence, the temperature variation of sf ph τ can be deduced from Eq. (3.2). Then, by comparing ρ with τ sf , we obtain ε ph and ε imp for the Ag nanowires in substantial agreement with reported values for the bulk: ε ph and ε imp are, 2.61 Â 10 À 3 and 4.03 Â 10 À 3 for LSV with MgO capping; ε ph ¼2.86 Â 10 À 3 and ε imp ¼2.50 Â 10 À 3 for the bulk from CESR study [85,86]. Also, Hanle effect in Section 4.2 gives ε imp ¼2.5 Â 10 À 3 for 100-nm-thick Ag. It is consistent because ε imp can be different for sample with different thickness reflecting the quality of nanowire.
Material dependence of sf ph 1 τ − is discussed by Monod and Benue [87], and more recently Fabian and Das Sarma pointed out the relation between sf ph τ and ph D ρ Θ [74]. For monovalent metals, the material dependence of sf where λ SOI is the spin-orbit splitting and ΔE is the separation to the nearest band with the same transformation properties [70]. As in Eq. (3.3b), reduced resistivity ρ/K ¼f(T/Θ D ) is material independent. Equation (3.3a) shows that the spin-lattice relaxation is also material independent after proper scaling by considering SOI. Based on E-Y mechanism, Monod and Benue estimated a magnitude of the effect of SOI. Then they found the reduced temperature dependence of 1/ sf ph τ /(λ SOI /ΔE) 2 in CESR data shows material independent B-G curve for noble and monovalent alkali metals. In Eq. (3.3), by substituting representative value of (λ SOI / ΔE) 2 into ε, spin flip rate is expressed as, ℏ is the gyromagnetic ratio, g is the g-factor and μ B is the Bohr magneton and B is constant α Β ne 2 m e /γ e . Parameters to discuss spin relaxation for various metals are summarized in Table 2 [70]. Revised Monod-Beuneu scaling [74] of C/ sf ph τ vs T / D Θ is shown in Fig. 12 for CESR  data of noble and monovalent alkali metals together with the nonlocal spin injection data for the Ag nanowire. The experimental data fall well on to the universal curve which reflects intrinsic feature of the Ag nanowire.
As can be seen in Table 2, the spin relaxation time of light metals Al and Mg is expected longer by two-three orders of the magnitude. For example, the dominated process of spin relaxation in Ag is in mediated 5p states (λ SOI /ΔE) 2 ¼0.114 whereas that for Mg is in 2p states (λ SOI /ΔE) 2 ¼1.32 Â 10 À 5 . Therefore, spin transport properties of Mg nanowires were studied. The SEM image of fabricated LSV is shown in Fig. 13(a). The field dependence of the spin valve signal for the LSV with L ¼300 nm is shown in Fig. 13(b), representing a clear spin valve behavior. ΔR S is 1.1 and 3.6 mΩ at room temperature (RT) and 10 K, respectively [88]. The interface resistance of the Py/Mg junction was measured where the current is applied between terminals 4 and 5 and the voltage is detected by using terminals 6 and 2, as shown in Fig. 13(a). The interface resistance is below the resolution ability of 1 f Ω m 2 in our measurement system, and thus we assume that the Py/Mg junctions are transparent, i.e., zero interface resistance. The amplitude of ΔR S is relatively high in the metallic LSVs with the transparent Ohmic contact such as Py/Cu, Co/Cu and Co/Al junctions [12,75,89,90] implying the high spin injection efficiency of the Py/ Mg junction. Fig. 13(c) shows the ΔR S as a function of L at RT and 10 K and for various LSVs. ΔR S decreases with increasing L due to a spin-flip scattering during the diffusive spin transport in the Mg nanowire. The experimental results are fitted to Eq. (3.1) by adjusting parameters P F and λ N . The spin diffusion length λ F of Py is fixed to the value of 5 nm from the literature [78]. The resistivity of Py is 4.7 Â 10 À 5 Ω cm at RT and 3.5 Â 10 À 5 Ω cm at 10 K, respectively. The resistivities of Mg are 1.0 Â 10 À 5 Ω cm and 4.0 Â 10 À 6 Ω cm for w Mg ¼ 170 nm at RT and 10 K, respectively. From the fitting, the values of λ N and P F are found to be 720 nm and 0.43 at 10 K, and 230 nm and 0.33 at RT, respectively. The spin diffusion length of Mg shows a similar value reported for Ag, Cu and Al [13]. Among these NM materials so far used in the LSVs, Mg is a lightest element, implying a smallest SOI. However, the obtained spin diffusion length of Mg is below micron. To discuss the origin of the comparable spin diffusion length in spite of small SOI for Mg, we should focus on the spin-flip mechanism in the NM metal. Monod reported that the spin relaxation in metals is divided into two groups: one is the monovalent alkali and noble metals, and the other is the polyvalent metals such as Al and Mg [87]. The former group shows a universal curve in the . These values are smaller than noble metals Ag and Cu and comparable to those of Al [77,93]. However, since the diffusion constant of Mg as well as Al is an order of the magnitude smaller than those of Ag and Cu, all the resulting spin diffusion lengths are comparable. For the comparison of the materials with the almost same strength of the SOI, we compare the obtained τ sf with that of Na e.g., the normalizing factor (λ SOI /ΔE) 2 is 1.32 Â 10 À 5 and 2.73 Â 10 À 5 for Mg and Na, respectively [87]. τ sf of the Mg nanowire is 14 ps at RT which is close to Θ D ¼290 K, and τ sf of Na is 22 ns at Θ D ¼ 150 K [94]. Moreover, the spin-flip probability for Mg is two order of the magnitude larger than those of Na (ε ph $ 0.067 ps / 22 ns ¼3.0 Â 10 À 6 , where the momentum relaxation time is obtained with Drude model [95]). Such a significant reduction of sf ph τ and ε ph for Mg could not be explained by the simple E-Y mechanism. This may be due to the existence of spinhot-spots pointed out by Fabian and Das Sarma [91]. Polyvalent metals such as Al and Mg have a complex Fermi surfaces and the area with enhanced spin relaxation property near the Brillouin zone boundaries, accidental degeneracy points.

Comparison of spin transport parameters in Cu, Ag, Al, Au and Mg by various measurement techniques
So far, the spin relaxation properties were characterized by means of conduction electron spin resonance (CESR) measurement, weak localization measurement, and lateral spin valve measurement. In CESR, one measures resonance spectrum of conduction spins, which gives the transverse spin relaxation time T 2 , in proportional to the half width of resonant spectrum. Longer T 2 gives rise to a sharp resonance spectrum, therefore, this scheme is beneficial to study the sample with weak spin relaxation such as alkali metals. For example, T 2 in Li, Na, and Be are reported in the temperature from 4 K to 296 K [96]. In contrast, the noble metals have relatively short T 2 and thus the T 2 is studied only in the limited range of a low temperature, e.g., CESR of Cu was reported, for the films with the thickness of tens of micrometer below and residual resistivity ratio is $ 1000 [97].
Later in 1980s, weak localization measurement was pioneered. This effect reflects the character of coherence of electron, but it also provides the property of spin relaxation. The advantage is that it can be applied for the material with short spin relaxation time if the phase coherence length is enough long. The sample form could be both thin film and wire. Recent study showed the explicit connection between the spin orbit length λ SO in weak localization and the spin diffusion length as λ SO ¼( 3 /2) sf λ . This effect can be measure only in low temperature due to the limitation of the length of the phase coherence. However, it enables to access spin relaxation properties even in the material with short λ sf (τ sf ) such as Pt [98].
Giant magneto-resistance (GMR) effect caused by spin-dependent scatterings between bottom and top electrodes is measured in FM/NM/FM multilayer structure. The spin relaxation in NM can be characterized when the thickness of the NM layer is comparable to the spin diffusion length.
Lateral spin transport provides means to access spin relaxation properties in a wide temperature range. Separation dependence of spin valve signal yields characteristic length λ sf and Hanle measurement complimentary yields τ sf . Temperature dependence of τ sf up to room temperature provides rigorous test of spin relaxation mechanism in nanowire as discussed in Section 3.3. Usually the dimensions of wires in LSV fabricated by a conventional lithography is up to 100 nm, therefore to characterize the material with short spin diffusion length, one needs to use other techniques. However, LSVs also provides it by using spin absorption technique, where the middle wire in consideration is inserted [21]. This geometry is beneficial to study particularly Spin Hall effect. In order to quantitatively analyze spin relaxation in middle wire, there needs to take care of the interfacial resistance between middle wire (Section 2.3) and also of the three dimensional distribution of spin current near the junction. Table 3 summarizes the properties of non-magnetic metals Cu, Ag, Al, Au and Mg, investigated by various measurement techniques. For Cu, Ag, and Al, large number of data were collected but for Au and Mg the number of data are limited. Transport properties such as λ sf , τ sf , D, and τ e depend on the detail of the sample. In contrast, spin-flip probability (spin relaxation ratio) ε ≡τ e /τ sf shows sample independent character if the nature of scattering under consideration is same. So far reported ε ph for Cu, Ag, and Al by means of both LSV and CESR are agreed within the accuracy of factor two with the one exception of Ref. [34]. Spin diffusion length for Cu, Ag, Al, and Mg is usually longer than the thickness of NM in GMR measurement, and therefore, GMR effect usually gives lower limit of λ sf [13]. Although one naively expects heavier metals show stronger spin relaxation, the effect of electronic structure shows strong influence on the spin relaxation properties, as discussed in Section 3.3. Spin-flip probability for non-magnetic impurity (including grain boundaries) ε imp are deviated by a factor of about hundred in each metals, implying that ε imp largely depends on the detail of the momentum scatterings.

Analytical formula of Hanle effect signal in LSVs
When the magnetic field is applied perpendicular to the spin orientation in NM nanowires, Larmor precession is induced. The collective spin precession, so-called Hanle effect, is one of the most effective methods to characterize dynamic spin transport properties as schematically shown in Fig. 14 [12,112]. This section outlines analytical expressions of the Hanle effect signal in LSVs: One is derived from a transit-time distribution model based on a diffusive transport of the pure spin current [12] and the other is derived from the Bloch-Torrey equation [113].
In the transit-time distribution model based a diffusive transport of the pure spin current, the probability P(t) of the spin reached at the detector position x ¼L after the spin is injected at the time t¼ 0 is expressed as where D N and τ sf are the diffusion constant of NM and the spin relaxation time, respectively. P(t) of a Ag nanowire calculated with the typical parameters is shown in Fig. 15(a). With an increase of the time t, spin relaxation pronounces and hence the P(t) drastically decreases. The spin valve signal of a response of the spin precession in LSVs is expressed as ℏ is the gyromagnetic ratio, μ B is the Bohr magneton, and ℏ is the Planck constant. Fig. 15   The first, second and third terms on the right-hand side come from the spin precession, spin relaxation and spin diffusion, respectively. Bloch-Torrey equation under the spin injection in the stationary state is expressed as [22,113]   a τ e is obtained with Drude model ( ¼ m e /(ne 2 ρN)). b D is converted from ρ 0 in the literature. c τ sf is obtained with D sf N sf

Evaluation of spin relaxation time
Our experimental studies are based on metallic LSVs, which have comparative advantage in designing the measurement scheme owing to clear physics of spin transport and spin relaxation mechanism as discussed in Section 3.3, good controllability of dimensions where one-dimensional transport model is applicable and comparability of junction property from low resistive transparent junctions to high resistive tunnel junctions. Fig. 16 shows Hanle signal for LSVs with Py/Ag and Py/MgO/Ag junctions, with the injector-detector separation L varied from 3.00 μm to 6.00 μm. The amplitude of the Hanle signal for the Py/Ag junctions are reasonably smaller than those for Py/MgO/Ag junctions due to the spin-resistance mismatch [22,51]: in the case of the Ohmic Py/Ag junction, the spin current in the Ag wire is absorbed into Py, which is expected from very low interface resistance R I for Py/Ag. In  n τ sf at room temperature is converted from 1/τ p ρ 0 is converted from RRR $ 1100 and one at 295 K. [111]. q τ sf is converted from half width of resonance: r ρ 0 is converted from RRR $ 1600 and one at 295 K [111].   After the long algebra, one arrives at   The boundary conditions also lead to the non-local voltage V due to the spin accumulation detected by FM2,  The effect of the magnetization process on the spin accumulation can be described by considering the vector spin polarization Pe FMi to y and z-axis and its projection as shown in Fig. 18 [47]. The experimental results are well reproduced by the present theoretical calculations using reasonable parameters listed in Table 4, as can be seen in Fig. 16. The obtained spin polarizations P F and P I agree well with our previous results [22] and values reported in Ref. [13]. The resistivity of Py was 1.75 Â 10 À 5 Ω cm. The junction resistance of Py/MgO/Ag was 20 Ω, which is enough higher compared with spin-resistance R Ag ¼ρ N λ N /A N ¼1 Ω. The interfacial resistance of Ohmic Py/Ag junctions and the spin diffusion length of Py are taken as R I A J ¼ 5 Â 10 -4 Ω (μm) 2 [13] and λ Py ¼5 nm [78], respectively from the literature. D N ¼612 719 cm 2 /s is derived from Einstein relation  1.55 Â 10 22 states/eV/cm 3 [92]. While the shape of Hanle signal is drastically modified by the junctions as in Fig. 16, the spin relaxation times for Py/Ag and Py/MgO/Ag junctions are well agreed as 40.8 76.2 ps and 40.3 77.3 ps. The spin relaxation mechanism is characterized by the spin-flip probability ε≡τ e /τ sf with respect to the momentum relaxation time τ e . For Ag, ε ¼0.10 ps/ 40 ps ¼2.5 Â 10 À 3 in this study is consistent with that (2.50 Â 10 À 3 ) deduced form conduction electron spin resonance [116]. The agreement of ε between the measurements is also reported for Al and Cu [11,117]. In addition to it, the Fourier transform of the theoretical Hanle signal agrees with the experimental P(t) not only for LSVs with Py/MgO/Ag junctions but also for Py/Ag junctions, as can be seen in Fig. 19, which complimentary supports the validity of our model. These results show that Eq. (4.2) cannot be used with the most widely used P(t) of Eq. (4.1) to analyze Hanle signal in LSVs of which R I is lower than R N due to the spin absorption effect. They may provide spurious spin relaxation times with mimicking signals or in some cases with different shapes of Hanle signals. In other words, the same spin relaxation time results in the different Hanle signal with and without spin absorption, the former of which exhibits a broader signal as shown in Fig. 16. This tendency is consistent with the reported Hanle signals in graphene based LSVs with various type of junctions, where the spin relaxation time is deduced as 448-495 ps and 84 ps for tunnel junction and transparent junction, respectively [59]. The reanalysis of data using our model provides 448-495 ps and 440 ps for tunnel junctions and transparent junctions, respectively as shown in Fig. 20 and Table 5, which allows us to separate the intrinsic and extrinsic spin flip mechanisms in graphene.
Dynamic transport properties of spin current in Ag can be also compared with Mg. Transit time distribution can be obtained by Fourier transform of Hanle signal (see Eq. (4.2) [64,118]) and Fig. 19 (c) and (d) show P(t) derived from Hanle effect in LSV with Mg nanowires. The separation L for Mg is half of that for Ag but P(t) for Mg shows longer transit-time. The transit time at which P(t) takes maximum and the velocity of spin current is approximated as t max E L/2 sf sf τ λ and v EL/t max ¼2(D N /τ sf ) 1/2 , respectively, for LSV with L ⪢ λ N . Therefore, the higher velocity in Ag can be explained by the shorter τ sf and larger D N .
Our model also enables us to determine the spin mixing conductance G ↑↓ . This will be detailed in Appendix and Table 6.   Table 4. Solid curve shows distribution including effect of spin absorption. All P(t) is normalized by P(t max ) where t max gives maximum of P(t).   Table 5 Adjusting parameters for Hanle signals for graphene based LSV with Co/Graphene junction in Ref. [59].

Towards coherent spin precession
In the diffusive pure-spin transport, the collective spin precession decoheres due to broadening of the dwell time distribution in the channel between the injector and the detector [14]. For example, the amplitude of the spin valve signal at B Z ¼0 decreases after the π rotation at B z π ¼0.16 T, as can be seen in Fig. 16(f). In order to better quantify the coherency in the collective spin precession, we define the figure of merit as the ratio R R / S S 0 Δ Δ π , where R S Δ π and R S 0 Δ are the amplitude of the spin signal right after the π rotation and that in zero field right before the rotation begins, respectively. The R R / S S 0 Δ Δ π increases with increasing L, and the experimental trend is well reproduced by Eq. (4.21) as shown in Fig. 21(b). To understand the observed trend in more detail, we employ the one-dimensional diffusion model which gives the ycomponent of net spin density at the detector , as a function of the dwell time t in the presence of B z [14]. The 〈S y 〉 vs t curves for L ¼λ Ν with B z ¼0 and B z ¼B z π are shown in Fig. 21(c). When B z ¼0, 〈S y 〉 takes a broad peak structure followed by a long exponential tail. The detected spin signal in LSVs is proportional to the 〈S y 〉 integrated over time. The distribution of S y B 0 z 〈 〉| = gets narrower as the channel length becomes longer, of which evolution is depicted in three distribution curves under B z ¼0 of Fig. 21(c)-(e). The long exponential tail observed in Fig. 21(c) diminishes in proportion to t t 1/ exp( / ) sf τ − . When B Z ¼B Z π is applied, the integrated value of 〈S y 〉 over time cancel for short L $ λ Ν (Fig. 21(c)) whereas it does not cancel for long L ⪢ λ Ν (Fig. 21(d) and (e)), indicating that the coherence of collective spin precession is well preserved for long spin transport. This trend is experimentally observed as an increase of R R / S S 0 Δ Δ π from 0.21 to 0.53 with L as shown in Fig. 21(b).
To better understand the coherence in collective spin precession, t t sf τ =˜is substituted into the distribution function at B Z ¼ 0.
We then obtain S y < > ∝    [14,46,47], (b) [119], (c) [33,57,59,120], (d) [32]. From [28]. metals, semiconductors and graphene in Fig. 22. Interestingly the relation between the coherence and the normalized separation shows a universal behavior and the experimental data are well reproduced by Eq. (4.21). We shall note here that the effective length L/λ N , not the spin relaxation time, is an important parameter to manipulate the spin precession coherently in the diffusive pure-spin transport while the spin accumulation is relaxed during the diffusive transport in the channel. Therefore, the high spin injection efficiency of the Py/MgO/Ag junction and the confinement effect in the dual-injector lateral-spin-valve structure could offer advantages for realizing giant spin accumulation as well as the coherent spin precession along a 10 μm-long Ag wire which is much longer than the spin diffusion length.

Summary
We studied spin transport properties using non-local spin injection with metallic lateral spin valves (LSVs). Fabrication method and detection scheme were overviewed. We described the spin transport of LSV in both a standard geometry consisting of two FM wires bridged by a NM wire and extended geometries consisting of three FM wires which can be used to generate enhanced spin accumulation and to characterize spin Hall effects. In particular, we discussed the spin absorption effect at the junction. It is found that the junction with the interface resistance comparable or greater than spin resistance of NM prevents the spin absorption effect. Using high spin-polarization of the junction (typically 40.10) enables efficient spin injection. The reported junction properties are summarized in Table 1 including the junction with Graphene based LSVs. We also showed that spin absorption effect hinders the expected enhancement of spin accumulation by using multi-terminal geometry whereas such structures with proper junction resistances yield the enhancement. Spin relaxation mechanism in various NM were studied and overviewed. Spin transport properties in the NM so far reported are summarized in Tables 2 and 3. We characterized intrinsic spin relaxation and extrinsic (due to impurities and surfaces) spin relaxations in Ag wire by analyzing temperature variation of the spin diffusion length. Surface spin relaxation of Ag wire are suppressed by MgO capping. Spin relaxation properties were characterized by spin-flip probabilities in each scattering event, and the intrinsic spin relaxation properties in LSVs due to phonons as well as impurities are found to be consistent with those obtained from previous conduction electron spin resonance studies. The spin relaxation time of Mg is comparable to that of Ag despite of much smaller atomic number and spin-orbit splitting, indicating that the spin transport properties of Mg wire reflect the effect of band structure that enhances spin relaxation (so-called spin hot spot). We overviewed representative characterization methods for spin transport in NM such as non-local spin injection using separation dependence and Hanle effect, giant magneto-resistance effect, conduction electron resonance, and (anti-)weak localization. We also formulated Hanle effect for non-local spin valves with various junctions including transparent and tunneling contacts to account for the effect of spin absorption. We experimentally observed different broadening in Hanle curves depending on the types of junctions either Py/Ag or Py/MgO/Ag. Fitting analyses using our formula that takes into account the spin absorption gives almost the same spin relaxation times for LSVs with and without the spin absorption as shown in Table 4. Hanle effect with transparent junction introduces transverse spin absorption which is characterized by spin mixing conductance. The obtained spin mixing conductance from Hanle measurement is consistent with that of GMR study but meaningfully smaller than that of the spin pumping as summarizes in Tables 6. Hanle curves with a wide  24 Spin pumping (F/N/F) [129] range of separation (up to L/λ N $ 7) are observed and the properties of collective spin precession are characterized. We found the coherence in the collective spin precession became longer for the longer separation. We summarized it as a universal plot of the coherence in the spin precession, applicable for materials so far reported not only metals but also semiconductive materials including graphene. This clearly indicates that the coherence in the collective spin precession is controlled by L/λ N for diffusive spin transport.
The solid understanding of spin transport properties is a basis for future spintronics researches using special experimental techniques such as high frequency measurements, electron microscopy, synchrotron radiation sources and so on, and also using wide variety of materials such as oxides, superconductor, magnetically ordered system, layered materials and so on. measurement techniques is a pressing challenge for the spintronics community engaged in spin injection physics.
See Tables A1 and A2. Spin-flip probability. Spin relaxation ratio. 14 ε F Fermi energy 2 ε imp Spin-flip probability for impurity scatterings 12 ε ph Spin-flip probability for phonon scatterings 12 ϖ surf Spin-flip probability for surface scatterings 12 f Function of B-G theory 12 ϕ e Electrical potential 2 g g -factor 13 G Ii Interface conductance of i-th junction 5 G Ij σ Interface conductance of j-th junction for spin channel σ 5 G ↑↓, k Real part of spin mixing conductance of k-th junction 18 GG sh ↑↓ Sharvin mixing conductance 23  Larmor frequency for π/2 spin precession 17 Z Atomic number 12