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Estimation of spin-diffusion length from the magnitude of spin current absorption

We demonstrate the method to calculate the spatial distributions of the spin current and accumulation in the multi-terminal ferromagnetic/nonmagnetic hybrid structure using an approximate electro-transmission line. The analyses based on the obtained equation yield the results in good agreement with the experimental ones. This implies that the method allows us to determine the spin diffusion length of additionally connected electrically floating wire from the reduction of the spin signal.


I. INTRODUCTION
Spin-dependent electron transport phenomena in nanostructured ferromagnet(F)/normal metal(N) hybrid systems show intriguing characteristics in association with the spin transfer and accumulation. The devices based on spin transport phenomena have great advantages over the conventional electronic devices because of additional spin functionalities. 1 Therefore, understanding responsible physics on the spin diffusion process becomes essential to realize spin-electronic devices. Especially, electrical spin injection from F into N, semiconductor or superconductor is an important technique for developing such devices. 2 When a spin-polarized current flows across the junction between F and N, the spin-splitting in the chemical potential is induced in the N layer because of the sudden change in spin dependent electrical conductivity. 3 This leads to the spin accumulation in the vicinity of the F/N interface. The injected spin can be flipped due to spin-orbit interaction, magnon, phonon scattering and etc. The length scale over which the traveling electron spin memorizes the initial direction is known as the spin diffusion length, an important measure to realize the efficient spin injection.
First electrical spin injection into N was demonstrated by Johnson and Silsbee using a single-crystalline Al bulk bar. 4 They found that the Al bar has a long spin diffusion length of a few hundred microns. However, the obtained output spin signal was quite small in the range of nano ohms due to large sample dimensions. Current perpendicular to plane (CPP) giant magnetoresistance measurements in magnetic multilayers of alternating F and N layers enable us to perform more detailed study on the spin-diffusion process in the N layer because the relevant scale of the spin-diffusion length can be controlled by the spacer thickness. 5 This technique is suitable for the N layer with magnetic or even non-magnetic impurities which reduce the spin diffusion length dramatically. 6 However, for Ns in which long spin diffusion lengths are expected, the precise estimation of the spin diffusion length is unattainable because of difficulty in preparing CPP device with the thickness of the N spacer as thick as the spin diffusion length. On the contrary, planar mesoscopic spin-valve device is suitable to study the spin dependent electron transport of the material which has long spin diffusion length. Recently, Jedema et al. succeeded in detecting the clear spin signal in the lateral structure even at room temperature by using nonlocal spin-valve (NLSV) measurements similar to the Johonson's potentio-metric method. 7 In order to detect clear spin signals in the NLSV measurements, the spacing between the F injector and detector should be shorter than the spin-diffusion length of the material. This means that, for the Ns with spin diffusion length of sub hundred nanometers, the NLSV technique cannot be applied because of the technical limit of device fabrication. We have experimentally demonstrated that in such lateral structures an additional floating probe which does not carry the charge current, affects the spin transport when the probe is located within the spin diffusion length from the spin injector. 8 In this article, we further extend the experiment for quantitative estimation of the spin-diffusion length using the spin absorption effect. (i.e. spin sink)

II. BASIC EQUATIONS
The continuous charge current I C in the F subsection provides the spin current I S = α F I C , with α F of the spin polarization in the F subsection. When the spin current flows across the F/N junction, the spin splitting in the chemical potential is induced in both F and N subsections, as shown in Fig. 1(a). Here, we devide the spin splitting chemical potential into two components. One is the voltage drop due to the charge current with the ohmic resistance in each layer shown in Fig. 1(b). In this case, the distribution of the chemical potential is obtained by considering the conventional electrical circuit consisting of a series jucntion with two different resistances. A constant spin current I S is induced by the continuous charge current in the F subsection. However, no spin current exists in the N subsection.
The other one is the spin dependent the chemical potential without the contribution of ohmic-resistance voltage drop as shown in Fig. 1(c), where spin currents are generated both in the N and F layers. Since the sorce of the induced spin currents I S is originated from the charge current of the F layer, the relation I S = I SF + I SN is satisfied at the interface. Here I SF and I SN are respectively the spin currents in the F and N subsections induced by the spin splitting of the chemical potential. Thus, the F/N junction with the charge current I C acts as a spin-current source with the magnitude of α F I C .
We consider the spin-splitting component of the chemical potential in which there is no charge current. The essential features for the spin-current and spin-accumulation distributions in the diffusive regime are described by the spin-dependent Boltzman equation. 9 In general, from the one-dimensional diffusion equation, the induced spin-splitting voltage ∆V S = ∆µ/e = (µ ↑ − µ ↓ )/e, where µ ↑ and µ ↓ are the chemical potential of the up-and down-spins, respectively, is given by Here λ is the spin diffusion length. In the N layer, using the spin-dependent ohmic law I ↑,↓ = (−σ N S/2)∂V ↑,↓ /∂x, the spin current I SN = I ↑ − I ↓ is calculated as where, S N , σ N and λ N are the cross sectional area, the conductivity and the spin diffusion length of the N subsection, respectively. The spin resistance R SN is a measure of the difficulty for spin mixing and is defined as 2λ N /(σ N S N ). The situation described by Eqs. 1 and 2 is equivalent to the electrical transmission line of the characteristic impedance R SN with the attenuation constant 1/λ N . 10 We can also obtain the similar relation for the F layer. In the F subsection, the spin current I SF is given by where, S F and σ F are the cross sectional area and the conductivity in the F subsection, respectively. Since there is no charge current, the relation α F ∂∆V S /∂x = −(∂V ↑ /∂x + ∂V ↓ /∂x) is satisfied. Therefore, the spin current I SF in the F layer is given by where, λ F is the spin diffusion length in the F subsection. Thus, in the F subsection, R SF is given by ( Eqs. 2 and 5 mean that the characteristic spin resistance over the spin-diffusion length is comparable to the characteristic impedance for the spin current and spin-splitting voltage. Using these relations, we can simplify the calculation of the spatial distributions of spin current and accumulation in complex multi-terminal F/N hybrid structures consisting of different characteristic spin resistances. We will show that the characteristic spin resistance R S is an important measure to design the spin-dependent-transport property of the system. First, we consider a cascaded transmission line of two different characteristic spin resistances as shown in Fig. 2(a). Here, the first and second subsections have the characteristic resistance R S1 with the spin-diffusion length λ 1 and the characteristic resistance R S2 with the spin diffusion length λ 2 , respectively. The length of the first subsection is d and the second subsection extends to infinity. The basic equations ∆V This relation and the continuities of I S and ∆V S at should note that the situations where V 1− > 0 and V 1− < 0 respectively correspond to the spin-current absorption into the second line -and the reflection from it due to the resistance mismatch at the boundary. We then define the transmission coefficient T as the ratio of the spin splitting voltage V S1 at the junction to the voltage V S0 at the input terminal. The transmission coefficient T ≡ ∆V S1 /∆V S0 is given by where, Q is the ratio R S2 /R S1 . Defining the input spin resistance as ∆V S0 /I S0 where I S0 is the spin current at the input terminal, We obtain the input spin resistance in the series connection R S−Series as The input spin resistance R S−Parallel in a parallel connection shown in Fig. 2(b) is also calculated as Above basic equations can be applied for calculating the characteristics in multi-terminal spin-transport systems.
In the above calculation, we assume a transparent interface between the different subsections with zero resistance. This assumption is justified only in limited materials. However, in this article, we treat Cu/Py and Cu/Au interfaces. These interfaces are known to have small resistance compared with other interfaces such as Cu/Pd and Cu/Pt. 5,6,11 Therefore, we neglected the interface resistance.

A. Single junction
We now consider a single F/N cross junction with the charge current I C flowing across the F/N interface as shown in Fig. 2(c). As mentioned above, the charge current across the junction plays a role of the spin current source with the magnitude of α F I C . The induced spin current diffuses into each subsection according to the magnitude of characteristic spin resistance. This can be described as the parallel spin resistance circuit consisting of two F and two N subsections shown in Fig. 2(c). We can now calculate the induced spin-splitting voltage V S0 at the interface as .
The spin splitting decays exponentially over the spin diffusion length in each subsection. Here, R Sd is the spin resistance from the detector junction and is given by From Eq. 7, R Si is deduced as The total spin resistance of the device R S is given as the parallel sum of R N , two R F and R Si : Also using Eq. 6, the transmission coefficient of the device T 2 corresponding to the ratio of the spin-splitting voltage between the injector and detector F/N interfaces can be calculated as The induced spin-splitting voltage at the injector junction is given by I S R S . Then, the induced spin-splitting voltage ∆V S2 at the detector junction can be calculated by ∆V S2 = T 2 I S R S . Since the NLSV signal ∆V N2 is given by α F ∆V S2 12 , we obtain We experimentally demonstrate the precise estimation of the spin diffusion lengths of F and N strips and the spin polarization of F strips using Eq. 14. The lateral spin valves for this study consist of double Py/Cu junctions. Figure 3 Cu strips are respectively 26.8 µΩcm and 2.08 µΩcm at room temperature. We vary the distance between the injector and detector junctions from 270 nm to 700 nm and measure the NLSV signal at room temperature as a function of the distance. In order to compare quantitatively Eq. 14 with the experimental spin signal, the cross sections S Cu and S Py have to be known to obtain the spin resistance of each strip. We thus have to carefully estimate the cross sections. From previous reports, λ Cu is of the order of several hundred nanometer and λ Py is a few nanometers. 7,13,14,15 Since the spin-current flows along the Cu strip over a few hundred of nanometers the area S Cu for estimating R Cu should be given by the cross section of the Cu strip (100 nm × 80nm). On the other hand, in the Py strip, the cross section is not appropriate for the area S Py for R SPy because the spin current diminishes in the vicinity of the junction. Therefore, for the Py strip S Py should be the junction-area 100 nm × 100 nm. 16 A NLSV signal at a distance of 270 nm as a function of the external magnetic field parallel to the Py strip is shown in the inset of Fig. 3(b). We observe the clear spin-accumulation signal at room temperature. Figure 3(b) shows the NLSV signal as a function of the distance d. The obtained signal decreases monotonically with increasing the distance d due to the spin relaxation. This result is fitted to Eq. 14. As shown in the Fig. 3(b), the fitted curve is in good agreement with the experimental results. From the fitting parameter, we obtain the spin-diffusion length of the Cu strip and that of the Py strip as 500 nm and 3 nm, respectively. And also, the polarization α Py of Py strip is determined as 0.25 at room temperature. Here, λ Cu = 500 nm is quite long compared with other reported values. 7, 14 We notice that the resistivity of our Cu is rather smaller than that of the reported one. This supports the long spin diffusion length of our Cu strip. The λ Py = 3 nm and α Py = 0.25 are reasonable although the values are slightly smaller than other group's values. This may be due to rather higher resistivity of our Py than others. Using these values, we can calculate the characteristic spin resistance of each strip as R SCu = 2.60 Ω and R SPy = 0.34 Ω.

C. Triple junction
We extend the above formalism to a triple junction in Fig. 2(e). In this case, the output spin-splitting voltage induced between the F2 and N strip by the current injection from F1 is influenced by the M strip which absorbs the induced spin current. The middle strip has the spin resistance of R SM with the spin diffusion length λ M . The magnitude of the spin-current and spin-accumulation can be calculated by using the method explained above . From the calculation, the position of the middle wire was found not to affects the spin signal. Therefore, for simplicity we assume that the middle wire is located in the center between F1 and F2 strips which have the same spin resistances of R SF . When the current is injected from F1, the NLSV voltage ∆V N3 at junction of F2/N is given by  Fig. 4 . This is because a small amount of the spin current is absorbed into the middle Cu strip.  When the spin resistance of the additional strip is much smaller than that of the N strip, the spin currents tend to flow into the additional strip to be relaxed. We also emphasize that, using the absorption effect, the spin-diffusion length of the additional strip can be estimated quantitatively from the reduction of the spin signal in the NLSV measurement.
The situation of the present experiment may be directly compared with the spin pumping effect observed in magnetic multilayers. 18