The abilities to inject and detect spin carriers are fundamental for research on transport and manipulation of spin information1, 2. Pure electronic spin currents have been recently studied in nanoscale electronic devices using a non-local lateral geometry, both in metallic systems 3 and in semiconductors4. To unlock the full potential of spintronics we must understand the interactions of spin with other degrees of freedom. Such interactions have been explored recently, for example, by using spin Hall5, 6, 7 or spin thermoelectric effects6, 8, 9. Here we present the detection of non-local spin signals using non-magnetic detectors, through an as-yet-unexplored nonlinear interaction between spin and charge. In analogy to the Seebeck effect10, where a heat current generates a charge potential, we demonstrate that a spin current in a paramagnet leads to a charge potential, if the conductivity is energy dependent. We use graphene11 as a model system to study this effect, as recently proposed12. The physical concept demonstrated here is generally valid, opening new possibilities for spintronics.
At a glance
Previous reports on detection of spin signals using non-magnetic contacts have made use of spin–orbit interaction through the (inverse) spin Hall effect5, 6. Recently, large non-local signals in graphene have been attributed to an effect with similar phenomenology, given by the difference in Hall resistance between two (spin) channels induced by an applied perpendicular magnetic field7. Both effects produce a charge potential transversal to the direction of the spin current and are valid in the linear regime. In the present work we deal with a different concept based on a nonlinear interaction between spin and charge which results in charge potentials longitudinal to the spin current12. This effect is solely based on the energy dependence of the conductivity σ(ε), not requiring spin–orbit interaction or external magnetic fields.
To explain the concept of detection of spin signals used here it is useful to make an analogy with thermoelectrics. As shown in Fig. 1a, a temperature gradient sets up a heat current. Under open-circuit conditions, this results in a built-up voltage V =−S(T2−T1), with S the Seebeck coefficient of the conducting system. For the case of diffusive spin transport13 (Fig. 1b) the electrochemical potential of each spin channel can be described as μ±=μavg±Δμ, with Δμ the spin accumulation (created by electrical spin injection) in the conductor, which decays with a characteristic spin-relaxation length λ. The gradient in spin accumulation sets up a spin current, which results, for ferromagnetic or ferrimagnetic materials with a spin polarization of the conductivity13 β, in a built-up voltage V =−(β/e)(Δμ2−Δμ1). While S is a general property of conductors, β is in general zero for paramagnetic materials. Therefore, pure spin currents are not expected to generate charge voltages in a paramagnet such as graphene. The latter is not true if we consider spin transport away from the Fermi level. When a sizable Δμ is present, each spin channel experiences a different conductivity even in a paramagnet, as long as the conductivity is energy dependent. So we consider a spin polarization of the conductivity induced by Δμ, which can be approximated as12 β=−Δμσ−1∂σ/∂ε.
To complete the analogy, we define α=σ−1∂σ/∂ε. The previous expressions for V are only valid when the coefficients S and β are independent of the driving forces T and Δμ, respectively. In reality, the Seebeck coefficient is given by the Mott formula10, 14 S=−L0eαT, with L0=(π2/3)(kB2/e2)the Lorentz number. In the limit T≈T2−T1, the Seebeck voltage depends quadratically on the driving force as V ∝L0eα(T)2. Similarly, for spin transport in a paramagnet, the induced spin polarization mentioned above (β=−αΔμ) also results in a quadratic dependence on the driving force as V ∝(α/e)(Δμ)2. Owing to the common factor α the effect described here has similar behaviour to the Seebeck voltage, showing opposite polarity for electron and hole regimes. Furthermore, because Δμ is (to lowest order) linear in the injection current3, 15, the result is a second-order signal V ∝I2.
For our proof of concept we use graphene, which, apart from being a two-dimensional platform for relativistic quantum mechanics11, has proved to be an excellent system for spin transport, where large values of Δμ≈1 meV can be obtained15. The sample is shown in Fig. 2a. It consists of a lateral graphene field-effect transistor covered with a thin aluminium oxide barrier that yields high-resistance contacts for efficient spin transport16 and electrostatic gating through the Si/SiO2 substrate, as previously reported15, 17. As well as using magnetic Co contacts for electrical spin injection and detection (Fig. 2b), we also include Ti/Au contacts. These non-magnetic contacts are used to electrically detect spins in the configuration shown in Fig. 2c. There are two key aspects to such a measurement configuration. First, the use of non-magnetic detectors simplifies the analysis of the non-local signal, because for the case of using magnetic detectors both linear and nonlinear signals are expected owing to direct detection of Δμ (refs 12, 18). Second, the use of two magnetic contacts as source and drain for charge current enables us to measure the non-local signal for both parallel V Pa and antiparallel V Ap alignment of their magnetizations and thereby to focus on the difference between the two states ΔV. This way we can exclude background signals that do not depend on Δμ and can be present in non-local measurements18. We use a lock-in technique to determine the linear V1 and second-order V2 components of the resulting root-mean-square signal. From them we extract the non-local resistances Ricontributing to the total signal V=R1I+R2I2. All measurements are at room temperature, unless otherwise noted.
We start by characterizing spin transport in the linear regime. The results in Fig. 3 show a non-local spin-valve effect, demonstrating spin transport between contacts 2 and 3, for a centre-to-centre separation of L=1.0 μm and width of graphene w≈1.1 μm. The obtained spin resistance ΔR1≈4 Ω is nearly constant versus the gate voltage Vg applied to the substrate. Vgcontrols the charge-carrier density ng in graphene as ng=γ(Vg−VD), with VD the condition for charge neutrality (Dirac point) and γ=7.2×1014 m−2 V−1. On the other hand, the graphene square resistance Rsq depends on Vg, changing by a factor of five. The observed ΔR1 versus Rsq behaviour can be understood by the standard relation15 ΔR1=(P2Rsqλ/w)exp(−L/λ), with P the spin polarization of the magnetic contacts, as being due to the charge-carrier-density dependence of λ with a minimum at the Dirac point. The latter is given by the behaviour of the spin-diffusion constant D and spin-relaxation time τ in graphene19, 20, 21 (Supplementary Section SA). Furthermore, the previous relation is valid only for contact resistances RcRsqλ/w, where the contacts do not affect the spin transport16, 17, 22. We take into account both considerations in our modelling below, by parameterizing λ (Fig. 3d) and by including the finite resistance of the contacts used for spin injection and detection.
Next, we demonstrate nonlinear detection of spins by using non-magnetic contacts. In Fig. 4b is shown a clear spin-valve effect in the second-order component V2 of the non-local signal. The transitions in V2 occur at the switching fields of the magnetic contacts used for current injection. We observe at zero gate voltage that V2Ap>V2Pa, consistent with the presence of a larger Δμ for the antiparallel magnetic configuration15 and a positive sign of the parameter α for electron transport12. Therefore, we expect that the sign of the nonlinear spin resistance ΔR2 should follow that of α and change sign when going from transport in the electron (Vg>VD) to the hole (Vg<VD) regime. The latter is confirmed by observing that ΔR2<0 for gate voltages VgVD. We remark that the measured spin-valve signal cannot be explained by spurious detection of potentials in the current-carrying part of the sample, owing to the absence of spin-valve signal in the first-order response (Fig. 4a).
The gate-voltage dependence of the nonlinear spin resistance is presented in Fig. 4c. The ΔR2 versus Vg curve shows a maximum of ≈5 k Ω A−1 for electron transport. We did not observe a clear sign change when crossing the Dirac point, whereas for hole transport there was a minimum value of only ≈−2 kΩ A−1. To understand this electron–hole asymmetry we looked into the charge-transport properties of the detector circuit, between contacts 4 and 5. The Dirac curve in Fig. 5a shows that, while there is a reasonable symmetry for Vg close to VD=−9 V, this is not the case for larger Vg, as shown by the kink visible at Vg=−55 V. Such kinks in the Dirac curve have been described as arising owing to electron doping from metal contacts with a thin oxide layer that prevents charge-density pinning23. Our contacts are deposited onto a thin oxide barrier. Therefore, we interpret the Dirac curve of the detector circuit as being composed of two contributions, the graphene under (and next to) the contacts and that away from the contacts (curves 1 and 2 in Fig. 5a, Supplementary Section SB).
Having described both spin and charge transport in the linear regime, we now construct a minimal one-dimensional model that enables quantitative comparison with experiment. As mentioned above, we model Δμ by considering the induced conductivity spin polarization β, finite-resistance contacts and gate-voltage dependence of λ. We use a fixed P=9% for the magnetic contacts (extracted in the regime Rc≥5Rsqλ/w) and a width profile for graphene as extracted from atomic force microscopy. Furthermore, we describe the Dirac curve for each graphene region using the approximate relation24 σ=νe(ng2+ni2)1/2, with ν the carrier mobility and ni a background carrier density due to the presence of electron–hole puddles and thermally generated carriers. We then extract the parameter α for each region12 (Fig. 5b) in a similar way to the extraction of the Seebeck coefficient of graphene from the Dirac curve 14, 24.
The model, schematically shown in Fig. 5c, has the extension of the graphene region 1 beyond the contact edge as the only free parameter. Scanning photocurrent work25 has shown that the doping in graphene decays gradually from the contact edge, extending up to a distance of ≈0.3 μm. For simplicity we consider a constant doping up to a distance of 0.15 μm. The modelled ΔR2-versus- Vg curve (solid line in Fig. 4c) successfully reproduces both the trend and the magnitude of the data. The agreement imparts certainty to our interpretation of the measured ΔR2 signal as arising owing to the nonlinear interaction between spin and charge.
The magnitude of ΔR2 is only slightly limited by the finite resistance of our contacts (Rc≈7 kΩ). Assuming infinite contact resistance, we predict only up to a twofold increase in ΔR2. On the other hand, the use of high-quality tunnel contacts22 with P=30% would yield a tenfold increase. Moreover, as , similar to the Seebeck coefficient14, decreasing ni by two orders of magnitude by using a boron nitride substrate26 would yield a further tenfold increase. Therefore, the herewith demonstrated effect is a real candidate for spin detection. It can be regarded as a step in the logical progression from linear interactions between spin and charge towards interactions between spin and heat, as studied in the field of spin caloritronics27.
Graphene is obtained from highly oriented pyrolytic graphite by mechanical exfoliation and deposited on a highly n-doped Si substrate covered with a thermal oxide layer 300 nm thick. The Si substrate is used as electrostatic gate. Graphene is first covered by 0.8 nm Al followed by natural oxidation to obtain a thin aluminium oxide layer. Electron-beam lithography, deposition by evaporation and lift-off were carried out twice, first for the Ti/Au contacts and then for Co. We observed in previous samples a lower resistance of the Ti/Au contacts, which prevented us from using them for spin detection16. This problem was possibly related to the required baking of poly(methyl methacrylate) for the second electron-beam lithography step, which may cause diffusion of Ti/Au through the oxide barrier. To solve it we increased the thickness of the oxide barrier only for the Ti/Au contacts by deposition and oxidation of an extra 0.3 nm of Al. This yielded non-invasive Au contacts with similarly high resistances to those of the Co contacts.
Characterization took place at room temperature and at 77 K (Supplementary Section SA) in a cryostat with a base pressure ≈1×10−7 mbar. The sample was first annealed under vacuum at 130 °C for ≈24 h for removal of physisorbed water, resulting in low hysteresis in the Dirac curve (ΔVD<10 V). This hysteresis leads to the increased uncertainty of ΔV2 around Vg=−20 V seen in Fig. 4c. Measurements were made using a 1–5 μA a.c. current source and recording simultaneously the first- and second-harmonic responses using two lock-in systems. All current and voltage signals reported are root-mean-square values. Therefore, the resistances were extracted as R1=V1/I and . Excitation frequency was kept at ≤3 Hz to prevent signals due to capacitive coupling, as determined by frequency scans. Contribution from higher harmonics was found to be negligible.
We developed a one-dimensional finite-element code using the program MATLAB to find numerical solutions for the two-channel spin diffusion equations13
with the inclusion of an element-specific conductivity spin polarization12 β, as described in the main text. Element length was kept ≤10 nm. I± and Δμ were set to be continuous across element boundaries. We consider point contacts located at the centre of the fabricated electrodes, which could either inject a spin current PI or detect the electrochemical potential μavg+PΔμ. Spin relaxation under contacts with finite resistance (4.9, 9.2, 7.3 and <1 kΩ for contacts 2, 3, 4 and 5, respectively) was implemented as in ref. 17, with the extra consideration of contact spin polarization P. We use a phenomenological description of each Dirac curve by the relation σ=νe(ng2+ni2)1/2, which describes the data using a constant mobility ν by including the effect of electron–hole puddles through a background carrier density ni (ref. 24). For each Dirac curve the parameter α=σ−1∂σ/∂ε is extracted using ∂σ/∂ε=(∂σ/∂ng)(∂ng/∂ε), in a manner similar12 to that used in the literature for the Seebeck coefficient14, 24. The extracted parameters for the curve Dirac 2 (graphene between contacts) were ν=3,900 cm2 V−1 s−1 and ni=3.5×1015 m−2, consistent with previous experiments on SiO2 substrates24, whereas for Dirac 1 we obtained ν=800 cm2 V−1 s−1 and ni=2×1016 m−2. To obtain ΔR2 we calculated the difference in potential V=−μavg/e between the Au detectors, for both parallel and antiparallel configurations and d.c. currents I=±5 μA. The result was then fitted with ΔV =ΔR1I+ΔR2I2. The odd contribution |ΔR1|<3 mΩ was found to be negligible.
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We would like to thank J. G. Holstein, B. Wolfs and M. de Roosz for technical assistance. This work was financed by the Zernike Institute for Advanced Materials and by European Union Seventh Framework Programme Information and Communication Technologies grant no 257159 MACALO.
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