Introduction
With hundreds of millions of computer hard drives sold every year, magnetism is currently, by far, the main repository of information storage. This dominance will only increase with the expected proliferation of data centers for “cloud” access over the Internet. It is the electron “spin,” the elementary nanomagnet, that carries this information. Beyond storage, spin is foreseen as the foundation for a new paradigm for information processing toward low-power-consumption nonvolatile “green” electronics. This is the aim of spintronics.
However, despite intense research, a simple device such as the spin transistor proposed in 1990Reference Datta and Das1 has remained elusive. Whereas it was soon realized that fundamental constraints on the physics of spin transport would make this concept very difficult to achieve with conventional semiconductors such as GaAs or siliconReference Schmidt, Ferrand, Molenkamp, Filip and van Wees2 (indeed, electrical injection of a spin current directly into silicon was demonstrated only recentlyReference Dash, Sharma, Patel, de Jong and Jansen3), a suitable material was still sought.
Recently, because of its fascinating electronic properties, graphene has become the focus of expectations for producing breakthroughs in many areas of nanoelectronics.Reference Castro Neto, Guinea, Peres, Novoselov and Geim4–Reference Jozsa, van Wees, Tsymbal and Zutic8 For spintronics, graphene’s obvious attraction is mainly the long spin lifetime expected from the small spin–orbit coupling of carbon atoms and the absence of nuclear spins for the main isotope. The combination of this expected long spin lifetime with a high electron velocity, related to the linear dispersion relation of electrons in graphene, underlies the potential of graphene for spintronics. The ability to transport spin information efficiently over practical distances could further enable complex spintronic devices, such as the reconfigurable logic gate integrating both memory and logic proposed by Dery et al.,Reference Dery, Wu, Ciftcioglu, Huang, Song, Kawakami, Shi, Krivorotov, Zutic and Sham9 and eventually open the way to spin information processing.
The spin diffusion distances observed in graphene are very long, in the 100-μm range, much longer than those in conventional metals and semiconductors. This is a unique advantage for several concepts of spintronic devices, particularly for complex architectures in which information is coded by pure spin currents and processed by series of logic gates acting on their spin polarization.
Indeed, although a suitable platform for such devices remains to be identified, initial steps in this direction have already been taken, such as non-charge-based “beyond-CMOS” memory and logic devices highlighted in the Emerging Research Devices chapter of the International Technology Roadmap for Semiconductors (ITRS).10 Among several other spintronic devices, so-called “all-spin-logic” circuits based on the transport and processing of information coded by spin currents have been proposed.Reference Behin-Aein, Datta, Salahuddin and Datta11
Research on a suitable platform for spin transport started decades ago and focused first on conventional inorganic semiconductors and metals. However, these materials have shown limited spin signals and/or spin diffusion lengths that are typically in the range of only a few tenths of a micrometer at room temperature. Graphene, in contrast, appears to be potentially well adapted for the transport of spin information over relatively long distances in the 100-μm range with limited spin losses. Some organic molecules or carbon nanotubes could also be expected to provide this performance, but graphene is more convenient for practical devices.
Although the injection and detection processes still need to be improved and the relaxation mechanisms need to be understood, it has been demonstrated that spin-polarized currents in graphene can give rise to large electrical signals. In this article, we describe these experiments and the theoretical framework that enables their interpretation and optimization. In particular, matching the resistance of the tunneling contacts to spin relaxation in graphene is key to achieving efficient spin transport.
Early results on exfoliated graphene
The first experiment on spin transport in graphene was reported in 2006, for a graphene flake connected to NiFe electrodes.Reference Hill, Geim, Novoselov, Schedin and Blake12 The device was a lateral spin valve (LSV), in which a spin-polarized current injected from one ferromagnetic electrode travels through the graphene before being detected by a second electrode. In analogy to optical systems, the ferromagnetic electrodes act as a polarizer/analyzer set. The current flowing through the device depends on the persistence of the spin polarization and whether the electrodes are magnetically aligned in a parallel or antiparallel configuration.
This measurement was rapidly followed by several other spin-transport measurements on single-layer graphene (SLG) and multilayer graphene (MLG).Reference Tombros, Jozsa, Popinciuc, Jonkman and van Wees13–Reference Goto, Kanda, Sato, Tanaka, Ootuka, Odaka, Miyazaki, Tsukagoshi and Aoyagi16 Mainly, two configurations are reported for the measurements: one “local” and the other “nonlocal”. The local configuration is a simple two-terminal device acting as an LSV. In the nonlocal configuration, four terminals are used, in a geometry slightly different from that used in conventional four-point measurements. Specifically, as seen in Figure 1, the current path is separated from the voltage measurement zone. The nonlocal configuration was originally developed to extract low signalsReference Jedema, Heersche, Filip, Baselmans and van Wees18 in semiconductors and metals, where the non-spin-aligned current would overwhelm the signal in local LSV measurements.
(a) A four-terminal LSV device based on a single-layer exfoliated graphene flake showing the graphene shape. (b) Nonlocal measurement geometry of a graphene spin-valve device. The current I is injected between electrodes 3 and 4, and a voltage V is detected between electrodes 1 and 2. (c) A nonlocal spin signal appears as a difference in the nonlinear resistance R nl = V/I in different parallel and antiparallel configurations of the magnetic electrodes as the magnetic field H is swept. These measurements were made at room temperature in a 600-nm-wide device with a 3-μm interelectrode spacing between central electrodes 2 and 3. (d) Schematic representation of the spin-dependent chemical potentials μ↑ and μ↓ induced by injecting a spin-polarized current from a ferromagnetic electrode into the graphene lateral channel of the device in (b). The difference μ↑ – μ↓is the spin accumulation giving rise to the measured spin signal. The arrows indicate the magnetization directions of the electrodes. The green (red) dots correspond to the chemical potential being probed with the magnetization pointing downward (upward). Adapted from Reference Reference Popinciuc, Jozsa, Zomer, Tombros, Veligura, Jonkman and van Wees17.
As an example of this nonlocal technique, Figure 1a shows the SLG device studied by Popinciuc et al.Reference Popinciuc, Jozsa, Zomer, Tombros, Veligura, Jonkman and van Wees17 Injecting a spin-polarized current at electrode 2 (Figure 1b) creates an out-of-equilibrium spin population in the graphene layer.
The difference between the electrochemical potentials of the spin-up and spin-down carriers, Δμ = (μ↑ – μ↓), is called the “spin accumulation” [μ↑ (μ↓) = –eV + E F↑ (E F↓) at 0 K] (see Figure 1d). This polarization diffuses and is measured below electrodes 3 and 4 away from the electrical current (Figure 1b). The spatial spread of the spin polarization in a material is characterized by the spin diffusion length, l sf, which is related to the spin–lattice relaxation time, τsf (spin lifetime), in that material by l sf = (Dτsf)1/2, where D is the diffusion coefficient.
An example of a room-temperature nonlocal spin signal associated with the measurement of the difference in spin accumulation amplitude between these contacts is shown in Figure 1c. The amplitude of the signal depends not only on the length but also on the mean contact resistance, R b, of the tunnel barrier between the graphene and the electrodes. A larger resistance prevents spin escape into the electrodes and preserves a larger spin polarization. The measurement is usually analyzed according to a one-dimensional model based on the drift–diffusion equations (see the later section “Analysis of experimental results on graphene” for details). In the notation used in this article, the nonlocal spin signal, ΔR nl, can be expressed asReference Popinciuc, Jozsa, Zomer, Tombros, Veligura, Jonkman and van Wees17
$$\Delta R_{{\rm{nl}}} = \pm {{2{\rm{\gamma }}^2 {\rm{\rho }}_{{\rm{sq}}} l_{{\rm{sf}}} } \over w}{{\left( {{{wR_{\rm{b}} } \mathord{\left/ {\vphantom {{wR_{\rm{b}} } {{\rm{\rho }}_{{\rm{sq}}} l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {{\rm{\rho }}_{{\rm{sq}}} l_{{\rm{sf}}} }}} \right)^2 \exp \left( {{{ - L} \mathord{\left/ {\vphantom {{ - L} {l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {l_{{\rm{sf}}} }}} \right)} \over {\left( {1 + {{wR_{\rm{b}} } \mathord{\left/ {\vphantom {{wR_{\rm{b}} } {{\rm{\rho }}_{{\rm{sq}}} l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {{\rm{\rho }}_{{\rm{sq}}} l_{{\rm{sf}}} }}} \right)^2 - \exp \left( {{{ - 2L} \mathord{\left/ {\vphantom {{ - 2L} {l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {l_{{\rm{sf}}} }}} \right)}}$$where γ is the injection polarization, ρsq is the square (sheet) resistance of the material (graphene in this case), l sf is its spin diffusion length, w is its width, and L is the distance between injection and detection.
This expression predicts an exponential decay, exp(–L/l sf), of the spin signal as a function of the device length, except when l sf is very long, leading to slower 1/L decay. Based on the experimentally observed exponential variation, a spin diffusion length of 1.6 μm was derived for the graphene sample.Reference Popinciuc, Jozsa, Zomer, Tombros, Veligura, Jonkman and van Wees17
The spin-transport parameters can also be extracted using the Hanle effect, in which a magnetic field is applied in a direction perpendicular to the spin accumulation and causes precession and dephasing of the spins in the lateral channel (see Figure 2a). This eventually leads to an oscillating decay of the spin signal, as illustrated by the experimental results of Yang et al.Reference Yang, Balakrishnan, Volmer, Avsar, Jaiswal, Samm, Ali, Pachoud, Zeng, Popenciuc, Guntherodt, Beschoten and Ozyilmaz20 in Figure 2b–c. When the alignment between the applied field and the spin polarization suppresses the precession effects, the spin signal is restored. Fitting of the Hanle curves with solutions of Bloch equations leads to the determination of the diffusion constant D, spin lifetime τsf, and spin diffusion length l sf. For their graphene samples, Popinciuc et al. found τsf and l sf values of up to 0.2 ns and 2.2 μm, respectively, at room temperature. Values in the same range were also later found by the Kawakami group.Reference Han, McCreary, Pi, Wang, Li, Wen, Chen and Kawakami19, Reference Han, Pi, McCreary, Li, Wong, Swartz and Kawakami21–Reference Han, Chen, Wang, McCreary, Wen, Swartz, Shi and Kawakami23 Indeed, using the nonlocal techniques, they found τsf ≈ 1 ns at 4 K and τsf ≈ 0.3 ns at room temperature. Figure 3 shows the local and nonlocal signals at 4 K and room temperature, illustrating the opposite signs of the two types of signal.
Hanle effect: (a) Application of an external magnetic field perpendicular to the electrode’s magnetization (i.e., perpendicular to the channel plane) forces the precession of the polarization of the spin current. Top: Spin current is conserved in the absence of a perpendicular field. Bottom: The applied field is just strong enough to induce one-half precession during channel transport. (b–c) Room-temperature Hanle curves recorded by Han et al.Reference Han, McCreary, Pi, Wang, Li, Wen, Chen and Kawakami19 (on nonlocal open devices as shown in Figure 5a) in both the parallel and antiparallel states of electrode magnetization. Oscillating and decaying device resistances as a function of the applied field are observed. The spin lifetime τsfand diffusion constants were extracted from fits of the data. The results in (b) reflect direct ferromagnet/graphene contact, whereas in (c), the contact resistance is increased by the insertion of a tunnel barrier. An increase in spin lifetime from 84 ps to 448 ps was obtained when tunnel barriers were used, showing the importance of spin escape to the electrodes (see description in text).
(a) Schematic illustration of the measurement configurations for LSVs, contrasting the nonlocal configuration (top), in which the current I is injected between two neighboring electrodes and the voltage V = IR nl is measured at two other electrodes, and the local configuration (bottom), in which the voltage V = IR is measured between the same electrodes in which the current is injected. (b) Typical signals reported for local measurements at 4 K, with observed ΔR values limited to the 100-Ω range. Reproduced with permission from Reference Reference Han, McCreary, Pi, Wang, Li, Wen, Chen and Kawakami19. ©2012, Elsevier. (c) Similar results obtained at room temperature. Reproduced with permission from Reference Reference Popinciuc, Jozsa, Zomer, Tombros, Veligura, Jonkman and van Wees17. ©2007, Nature Publishing Group. Note that signals that are clear in the nonlocal configuration become noisier in the local configuration for the same sample.
Recent results on large-scale graphene
Whereas most of the spin-transport measurements discussed so far were made on exfoliated graphene on SiO2, recent publications have also reported experiments performed on graphene grown by chemical vapor deposition (CVD) on copper foils and transferred onto SiO2.Reference Avsar, Yang, Bae, Balakrishnan, Volmer, Jaiswal, Yi, Ali, Guntherodt, Hong, Beschoten and Ozyilmaz24 These experiments showed spin-transport properties similar to those of exfoliated graphene and introduced the interesting possibility of large-scale production of spin-transport devices, because large areas of CVD graphene can be fabricated easily.
Another very interesting alternative for large-scale integration is epitaxially grown graphene on silicon carbide (SiC). In addition to a large size, epitaxial graphene (EG) samples also show very high mobilityReference Sprinkle, Siegel, Hu, Hicks, Tejeda, Taleb-Ibrahimi, Le Fèvre, Bertran, Vizzini, Enriquez, Chiang, Soukiassian, Berger, de Heer, Lanzara and Conrad25 (see also the articles in this issue by Ruan et al. and Nyakiti at al.). However, graphene layers grown on the silicon-terminated face (Si face) of SiC show different structures and properties from those grown on the carbon-terminated face (C face), and this is also true for spin transport.
For the Si face, which allows easier control of layer growth, Maassen et al.Reference Maassen, van den Berg, Ijbema, Fromm, Seyller, Yakimova and van Wees26 performed nonlocal spin transport measurements on multilayer epitaxial graphene (MEG). They found an average mobility of ~1900 cm2 V−1 s−1 and τsf values on par with those of the best exfoliated samples (up to 2.3 ns at low temperature) but with surprisingly small diffusion constants and l sfvalues in comparison.
For the C face, Dlubak et al.Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 explored spin transport on MEG (~10 layers) (see Figure 4). Although the number of layers is more difficult to control, this type of graphene is composed of uncoupled monolayer graphene sheets (it is not simply a thin graphite layer), leading to better transport properties. They found very high mobilities of ~17,000 cm2 V−1 s−1. In these samples, cobalt/alumina tunnel junctions of very large resistance, in the megaohm range, were used as injectors and detectors.Reference Dlubak, Seneor, Anane, Barraud, Deranlot, Deneuve, Servet, Mattana, Petroff and Fert28
LSV device presented in Reference Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27: (a) Plan-view scanning electron microscope image of a two-terminal local LSV. The width of the epitaxial graphene (EG) channel (gray) on SiC (blue) is 10 μm, and the distance between the two Al2O3/Co electrodes (red) is L = 2 μm. (b) Optical image of the entire structure, including contact pads. (c) Large local ΔR spin signals measured at 4 K. Note that the arrows indicate the sweep directions. Reproduced with permission from Reference Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27. ©2012, Nature Publishing Group.
Local magnetoresistance (MR) curves obtained with MEGReference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 are shown in Figure 4c with local spin signals (ΔR) in the megaohm range. These observed spin signals, much larger than the resistances of the cobalt electrodes and graphene channel, are the largest spin signals ever observed with graphene. In Figure 5b, the variation ΔR/R ∝ 1/LR b identifies the regime expected for a very large tunnel-barrier resistance compared to the spin resistance of the graphene channel (defined as 
$R_{{\rm{ch}}}^{\rm{S}} = {{{\rm{\rho }}_{{\rm{sq}}} l_{{\rm{sf}}} } \mathord{\left/ {\vphantom {{{\rm{\rho }}_{{\rm{sq}}} l_{{\rm{sf}}} } w}} \right. \kern-\nulldelimiterspace} w}$) and the electrode resistances and a very large l sf value compared to L. The corresponding physics depends on the ratio between the electron dwell time (also called transit time) which is proportional to LR b and τsf. The impressive spin signals of these devices can be explained only by l sfin the 100-μm range and above. Such spin diffusion lengths, much longer than those reported previously, are probably related to the very high mobility and high quality of C-face SiC epitaxial graphene.
(a) Expected evolution of ΔR/R, normalized to the maximum achievable value γReference Schmidt, Ferrand, Molenkamp, Filip and van Wees2/(1 – γ2), as a function of barrier-to-channel spin resistance ratio, 
${{R_{\rm{b}} } \mathord{\left/ {\vphantom {{R_{\rm{b}} } {R_{{\rm{ch}}}^{\rm{S}} }}} \right. \kern-\nulldelimiterspace} {R_{{\rm{ch}}}^{\rm{S}} }}$, as derived from Equation 2 in the LC case. Two scenarios are given: L = l sf/125 (blue), which corresponds to the range of data in Reference Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 for which the maximum γ2/(1 – γ2) can be achieved, and L = l sf/2 (purple), which corresponds to an extrapolation to much larger devices (for the same other parameters) for which the maximum γ2/(1 – γ2) is no longer achievable. The colored oval represents the experimental data range from Reference Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27. An example is given of how the efficiency of a representative device drops (dashed arrow) if its length is changed from l sf = 125L (blue circle) to l sf = 2L (purple circle) with all other parameters remaining constant. The solid horizontal arrow shows how one needs to tune R b precisely to recover a high signal efficiency. The width of the high-efficiency window (the range of resistance) scales with l sf/L. (b) Data from Table 1 of Reference Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27, shown together with plots of Equation 4 expressed as a function of the product of R b and L for l sfvalues of 4 μm (average literature value), 50 μm, 150 μm, and 250 μm.
Spin diffusion lengths l sf in the channel of epitaxial graphene lateral spin valves (LSVs) obtained by fitting Equation 3 to experimental resultsReference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 with only l sfas a free parameter.
All other parameters (except γ) extracted from sample measurements. The calculations were performed with γ = 0.32, the maximum value of γ foundReference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 for the spin polarization of tunneling from cobalt through alumina in similar devices, and provide only a lower bound of l sf.
L, device length; R P, device resistance in the parallel magnetic configuration of the cobalt electrodes; R b, interface resistance as defined in the text; ΔR, spin signal (i.e., the absolute change in device resistance between the parallel and antiparallel magnetic configurations); MR = ΔR/R P, magnetoresistance.
Spin relaxation in graphene
Considering spin relaxation mechanisms, it was observed by the van Wees groupReference Jozsa, Maassen, Popinciuc, Zomer, Veligura, Jonkman and van Wees29 that l sf increased linearly with the diffusion constant (proportional to τp, the momentum scattering time). This led the authors to suggest that, for their samples at least, the mechanism of spin relaxation was of the Elliot–Yafet (EY) type.Reference Elliott30 This mechanism predicts that, for each carrier momentum scattering event, there is a small probability (related to spin–orbit coupling) of spin flip and hence spin information loss. Thus, τsf is expected to increase with τp.
Another interesting result was more recently obtained for MLG by the same group.Reference Maassen, Dejene, Guimaraes, Jozsa and van Wees31 They found that τsf increased with the number of layers. In the framework of the EY mechanism,Reference Elliott30 this increase in τsfcan be attributed to better screening of the external scattering potentials, as reported for suspended graphene.Reference Bolotin, Sikes, Jiang, Klima, Fudenberg, Hone, Kim and Stormer32
In agreement with the conclusions of the van Wees group,Reference Jozsa, Maassen, Popinciuc, Zomer, Veligura, Jonkman and van Wees29 the Kawakami group found a decrease in τsf as τp decreased, suggesting the dominance of EY30 spin relaxation. However, interestingly, in experiments on bilayer graphene (BLG), they found the opposite behavior: a τsf value of up to 6.2 ns at low temperature and close to 1 ns at room temperature.Reference Han and Kawakami22 (Concurrently, a similar τsf value of 2 ns at room temperature was also found by Yang et al.;Reference Han, McCreary, Pi, Wang, Li, Wen, Chen and Kawakami19 see Figure 2b.) Compared to the experiments on SLG by the same group, surprisingly, this inverse dependence on τp suggests the dominance of the Dyakonov–Perel (DP)Reference Dyakonov and Perel33 mechanism, which relates spin flips to the accumulation of lattice-induced precession of the spin between scattering events. Thus, increased τp leads to increased τsf. Yang et al. also foundReference Han, Chen, Wang, McCreary, Wen, Swartz, Shi and Kawakami23 that organic-ligand-bound gold nanoparticles, although introducing faster momentum scattering by localized charges, had no effect on τsf.
More recently, the interpretation of the observed variation of τsf with τp and its relationship to the two mechanisms has been investigated. In most experiments, the variation of τp (which is proportional to μ, the mobility) is controlled through the variation of the charge density n through a gate, for which τp ∝ 1/n. However, it was shownReference Ochoa, Castro-Neto and Guinea34 that, away from the Dirac point, for the EY mechanism, τsf ∝ 1/τp, and the EY mechanism could lead to DP-like behavior. In addition, it was suggested that the DP mechanism could in some cases lead to EY-like behavior.Reference Zhang and Wu35
As discussed in the following section, however, the value of τsf observed in experiments corresponds to the fastest relaxation pathway in the devices. This pathway might not necessarily be in the graphene channel, as relaxation can occur through spin escape to the electrodes, where relaxation is much faster. For example, it was shown that at least a threefold increase in τsfcould be obtained by replacing pinhole contacts with tunnel contacts, which better isolate the channel from the electrodes.Reference Han, McCreary, Pi, Wang, Li, Wen, Chen and Kawakami19
With contradictory results for τsf for different types of samples, the mechanism of spin relaxation in graphene is not yet clear, and it appears that no straightforward distinction can be made between the Elliot–Yafet and Dyakonov–Perel mechanisms with the available experimental data. One direction for future work is the study (both theoretical and experimental) of EG samples for which the longest τsf (a few hundred nanoseconds) and l sf(a few hundred micrometers) were found.Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27
Theoretical model and device physics
We now present a theoretical picture of spin transport in a graphene LSV and show how it can be applied to the experimental determination of τsf or l sf.
General discussion
The analysis of spin transport developed for metal or semiconductor LSVs has to be adapted to describe similar experiments with graphene (or carbon nanotubes). Because the spin relaxation in graphene is considerably slower than that in metals, a strong relaxation-rate mismatch occurs between the graphene channel and the ferromagnetic electrodes. Hence, to prevent the escape and relaxation of spin accumulation into the electrodes (also called backflowReference Popinciuc, Jozsa, Zomer, Tombros, Veligura, Jonkman and van Wees17, Reference Fukuma, Wang, Idzuchi, Takahashi, Maekawa and Otani36) and to obtain a large spin signal, it is necessary to isolate the lateral channel from the electrodes by interface resistances (usually tunnel barriers). However, if the interface resistances are too large, the electron dwell time in the lateral channel becomes longer than τsf, and the spin signal drops. In the next section, we show that large spin signals occur only within a narrow window of interface resistances.
The spin accumulation, and thus the spin signal, can be preserved even more by considerably reducing the volume available for spin relaxation, that is, by working in geometries where the distance L between the current and voltage contacts is much shorter than l sf. Spreading and relaxation of spin accumulation outside the active region can be prevented by working with devices with a confined geometry (see Figure 6).
Schematic representations of the different types of LSVs on a graphene (or other) channel: (a) nonlocal detection and open (nonconfined) channel (NLO), (b) local detection and open channel (LO), (c) local detection and confined channel (LC), and (d) nonlocal detection and confined channel (NLC). In all cases, L is the distance between injection electrode and the detection electrode.
The propagation of spin currents in lateral devices can generally be described within the framework of the drift–diffusion equations first introduced by van Son et al.Reference Jaffrès, George and Fert37 and Johnson and SilsbeeReference van Son, van Kempen and Wyder38 and then extended by Valet and FertReference Johnson and Silsbee39 for the interpretation of giant magnetoresistance, in which the current flow is perpendicular to the layers. Solution of the drift–diffusion equationsReference Valet and Fert40 gives the typical exponentially decaying electrochemical potential profiles μ↑ and μ↓ shown in Figure 1 and, thus, Δμ = (μ↑ – μ↓). The spin signal is expressed as ΔV (or ΔR = ΔV/I), the difference in voltage (or resistance) between the parallel and antiparallel magnetic configurations, which scales with Δμ/e (or Δμ/eI). The amplitude of Δμ is controlled by the balance between the injected spin current (proportional to the current I) and the relaxation of the spin accumulation in the whole device, including the channel and, importantly, the magnetic electrodes.
Indeed, the typical situation of a much higher relaxation rate in the electrodes (cobalt, iron, etc.) than in the channel (here, graphene) impacts the device physics. We define the spin resistance of the two-dimensional, nonmagnetic channel as 
$R_{{\rm{ch}}}^{\rm{S}} = {{{\rm{\rho }}_{{\rm{sq}}} l_{{\rm{sf}}} } \mathord{\left/ {\vphantom {{{\rm{\rho }}_{{\rm{sq}}} l_{{\rm{sf}}} } w}} \right. \kern-\nulldelimiterspace} w}$. The corresponding spin resistance of the ferromagnetic electrodes is 
$R_{\rm{F}}^{\rm{S}} = {{{\rm{\rho }}_{\rm{F}} l_{{\rm{sf}}}^{\rm{F}} } \mathord{\left/ {\vphantom {{{\rm{\rho }}_{\rm{F}} l_{{\rm{sf}}}^{\rm{F}} } A}} \right. \kern-\nulldelimiterspace} A}$, where 
${\rm{\rho }}_{\rm{F}} $ and 
$l_{{\rm{sf}}}^{\rm{F}} $ are the resistivity and spin diffusion length in the electrode, respectively, and A is the cross section of the current flow path. We note that the condition 
$R_{\rm{F}}^{\rm{S}} \ll R_{{\rm{ch}}}^{\rm{S}} $ almost always applies and hence is assumed in the following discussion. The mean contact resistance R b and barrier spin polarization coefficient γ characterize the spin contact resistances R ± = 2(1 ± γ)R b.
In a simple physical description, the spin accumulation, Δμ, and the spin signal, ΔR = ΔV/I ≈ Δμ/eI, are controlled by γ, the spin asymmetry coefficient of the interface, and by the ratio 
${{R_{\rm{b}} } \mathord{\left/ {\vphantom {{R_{\rm{b}} } {R_{{\rm{ch}}}^{\rm{S}} }}} \right. \kern-\nulldelimiterspace} {R_{{\rm{ch}}}^{\rm{S}} }}$. This ratio fixes the proportion between the two main relaxation paths: (1) spin escape/backflow to the electrodes, where the spin quickly relaxes, leading to an overall relaxation rate proportional to 1/R b (blue zone in both plots of Figure 7), and (2) intrinsic spin relaxation in the lateral channel, which is proportional to 
${1 \mathord{\left/ {\vphantom {1 {R_{{\rm{ch}}}^{\rm{S}} }}} \right. \kern-\nulldelimiterspace} {R_{{\rm{ch}}}^{\rm{S}} }}$ for the open channels in Figure 6a–b or to 
$\left( {{L \mathord{\left/ {\vphantom {L {l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {l_{{\rm{sf}}} }}} \right)R_{{\rm{ch}}}^{\rm{S}} $ for the confined channels in Figure 6c–d (red zone in both plots of Figure 7).
Expected variations of (a) the total spin signal, �R, and (b) the relative spin signal, MR = ΔR/R P, as functions of the ratio between the interface resistance, R b, and the channel spin resistance, 
$R_{ch}^S $, for the LSV configurations in Figure 6a–c. The calculation was performed using Equation 2 with L = l sf/5. The symbols indicate—schematically, not quantitatively—the experimental results on carbon nanotubes (crosses) and graphene (triangles) discussed in the text. The curves for the NLC LSV of Figure 5 are similar to the LC curves and are not shown. The blue and red colors indicate the zones where the behavior is dominated by spin relaxation by escape to the electrodes and relaxation within the channel, respectively.
The nonlocal open (NLO), local open (LO), and local confined (LC) (nonlocal confined [NLC] is similar to LC) curves in Figure 7 show what is expected for the variation of the spin signals of the devices in Figure 6 in the aforementioned limit 
$R_{\rm{F}}^{\rm{S}} \ll R_{{\rm{ch}}}^{\rm{S}} $. The curves of ΔR = ΔV/I (Figure 7a) and 
${{\Delta R} \mathord{\left/ {\vphantom {{\Delta R} {R^{\rm{P}} }}} \right. \kern-\nulldelimiterspace} {R^{\rm{P}} }} = {{\Delta V} \mathord{\left/ {\vphantom {{\Delta V} {V_{{\rm{bias}}}^{\rm{P}} }}} \right. \kern-\nulldelimiterspace} {V_{{\rm{bias}}}^{\rm{P}} }}$ (Figure 7b, where R P is the resistance between the source and drain in the parallel magnetic configuration) as functions of 
${{R_{\rm{b}} } \mathord{\left/ {\vphantom {{R_{\rm{b}} } {R_{{\rm{ch}}}^{\rm{S}} }}} \right. \kern-\nulldelimiterspace} {R_{{\rm{ch}}}^{\rm{S}} }}$ are shown for L = l sf/5.
To explain this behavior, we first consider the simplest geometry, LC, shown in Figure 6c, consisting of two contacts and confined spin accumulation (within L). The expressions for ΔR and ΔR/R P (in the limit 
$R_{\rm{F}}^{\rm{S}} \ll R_{{\rm{ch}}}^{\rm{S}} $) areReference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27, Reference Valet and Fert40
$$\Delta R = {{4{\rm{\gamma }}^2 R_{\rm{b}} } \over {2\cosh \left( {{L \over {l_{{\rm{sf}}} }}} \right) + \left( {{{R_{\rm{b}} } \over {R_{{\rm{ch}}}^{\rm{S}} }} + {{R_{{\rm{ch}}}^{\rm{S}} } \over {R_{\rm{b}} }}} \right)\sinh \left( {{L \over {l_{{\rm{sf}}} }}} \right)}}$$and
$${{\Delta R} \over {R^{\rm{P}} }} = {{{\rm{\gamma }}^2 } \over {1 - {\rm{\gamma }}^2 }}{2 \over {2\cosh \left( {{L \over {l_{{\rm{sf}}} }}} \right) + \left( {{{R_{\rm{b}} } \over {R_{{\rm{ch}}}^{\rm{S}} }} + {{R_{{\rm{ch}}}^{\rm{S}} } \over {R_{\rm{b}} }}} \right)\sinh \left( {{L \over {l_{{\rm{sf}}} }}} \right)}}$$ These equations give a steplike curve for ΔR, as seen in Figure 7a (curve LC), and a bell-like curve for ΔR/R P, as seen in Figure 7b (curve LC), as functions of 
$\log \left( {{{R_{\rm{b}} } \mathord{\left/ {\vphantom {{R_{\rm{b}} } {R_{{\rm{ch}}}^{\rm{S}} }}} \right. \kern-\nulldelimiterspace} {R_{{\rm{ch}}}^{\rm{S}} }}} \right)$. Three zones can be defined: 
$R_{\rm{b}} \ll \left( {{L \mathord{\left/ {\vphantom {L {l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {l_{{\rm{sf}}} }}} \right)R_{{\rm{ch}}}^{\rm{S}} $, 
$\left( {{L \mathord{\left/ {\vphantom {L {l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {l_{{\rm{sf}}} }}} \right)R_{{\rm{ch}}}^{\rm{S}} < R_{\rm{b}} < \left( {{{l_{{\rm{sf}}} } \mathord{\left/ {\vphantom {{l_{{\rm{sf}}} } L}} \right. \kern-\nulldelimiterspace} L}} \right)R_{{\rm{ch}}}^{\rm{S}} $, and 
$R_{\rm{b}} > \left( {{{l_{{\rm{sf}}} } \mathord{\left/ {\vphantom {{l_{{\rm{sf}}} } L}} \right. \kern-\nulldelimiterspace} L}} \right)R_{{\rm{ch}}}^{\rm{S}} $.
On the left (tinted blue), for 
$R_{\rm{b}} \ll \left( {{L \mathord{\left/ {\vphantom {L {l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {l_{{\rm{sf}}} }}} \right)R_{{\rm{ch}}}^{\rm{S}} $ [which is the conventional channel resistance, i.e., 
$$R_{{\rm{ch}}} = \left( {{L \mathord{\left/ {\vphantom {L {l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {l_{{\rm{sf}}} }}} \right)R_{{\rm{ch}}}^{\rm{S}} $$], the interface resistance, R b, is too low, corresponding to the reported “impedance-mismatch” regime. The physical interpretation is that, at low R b, the relaxation occurs mainly through spin escape. In this case, the spin signal is very low, 
$\Delta R \approx {{2\gamma ^2 R_{\rm{b}} ^2 } \mathord{\left/$, and decreases with decreasing R b. Because the device resistance is dominated by the channel resistance, R P ≈ R ch, one obtains 
${{\Delta R} \mathord{\left/$.
We next focus on the zone corresponding to the beginning of the steepest slope of the steplike curve of ΔR (Figure 7a) and the maximum of the bell-like curve of ΔR/R (Figure 7b). This is the regime in which 
$\left( {{L \mathord{\left/$. The physical interpretation of ΔR at this point is that the increase of the interface resistance R b progressively reduces the spin escape to the electrodes as 1/R b, thus increasing the spin accumulation and ΔR in proportion to R b with �R = 2γ2R b. With both ΔR and the device resistance, 
$R^{\rm{P}} \approx 2\left( {1 - \gamma ^2 } \right)R_{\rm{b}} $ (which holds for 
$R_{\rm{b}} \gg R_{{\rm{ch}}} $), increasing in proportion to R b, 
${{\Delta R} \mathord{\left/$ becomes practically constant. This corresponds to the peak of ΔR/R P in the LC curve in Figure 7b.
The data for carbon nanotubes in Reference Reference Hueso, Pruneda, Ferrari, Burnell, Valdés-Herrera, Simons, Littlewood, Artacho, Fert and Mathur41, represented by crosses in Figure 7b, are approximately in this regime because ΔR approximately follows the increase in the interface resistance (R P increasing from 33 MΩ to 150 MΩ), whereas the values of ΔR/R P remain in a narrow range between 58% and 72%. It is worth noting that, as explained earlier, in this regime, the spin relaxation mainly occurs through spin escape to the high-relaxation-rate electrodes. Hence, for example, Hanle experiments performed in this range mainly determine τsflinked to spin escape to the electrodes and not relaxation in the channel.
We finally focus on the third zone for the curves. This corresponds to the right of the step of the ΔR curve and the right of the ΔR/R bell curve (crossover and red tint in Figure 7). For 
$R_{\rm{b}} > \left( {{{l_{{\rm{sf}}} } \mathord{\left/ {\vphantom {{l_{{\rm{sf}}} } L}} \right. \kern-\nulldelimiterspace} L}} \right)R_{{\rm{ch}}}^{\rm{S}} $, the competition between the two relaxation mechanisms is inverted. The limiting factor for spin accumulation is no longer spin relaxation by spin escape (∝1/R b) but rather spin relaxation in the length L of the channel [
$ \propto \left( {{{l_{{\rm{sf}}} } \mathord{\left/$]. Then, as seen in Figure 7a, ΔR progressively saturates at the value of 
$\Delta R = {{4\gamma ^2 R_{\rm{b}} } \mathord{\left/$. In the limit 
$l_{{\rm{sf}}} \gg L$ of the figure, this leads to 
$\Delta R = {{4\gamma ^2 R_{{\rm{ch}}}^{\rm{S}} l_{{\rm{sf}}} } \mathord{\left/$. Whereas ΔR saturates, 
$R^{\rm{P}} \approx 2R_{\rm{b}} (1 - \gamma ^2 )$ still increases in proportion to R b, so ΔR/R P decreases as 1/R b, as shown in Figure 7b (also see Equation 4 in the next section). In the variation as 1/LR b, the proportionality to 1/L is related to the confinement of the spin relaxation within L. It is only after this saturation of ΔR that the Hanle effect becomes directly related to the spin relaxation inside the channel.
The experimental data on graphene reported in Reference Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27, characterized by a decrease in ΔR/R P as 1/R b (see Equation 4), are distributed between the second and third zones, as represented by the triangle symbols in Figure 7.
We have presented the most interesting case of 
$l_{{\rm{sf}}} \gg L$. However, this highly favorable case does not correspond to many earlier experiments, where l sf was found to be on the order of L. As L increases and approaches l sf, the range 
$\left( {{L \mathord{\left/$ of the ideal regime (top of the bell curve) shrinks. In other words, the window of large MR in Figure 7 progressively narrows. The cosh and sinh functions in Equations 2 and 3 can no longer be approximated as 1 and L/l sf, respectively. This restores the less favorable exponential variations of ΔR as a function of L/l sf that are usually reportedReference Takahashi and Maekawa42 in place of the linear dependences on L, reflecting the balance between the spin relaxation in the channel L and the spin escape of Equation 4, for example.
We now turn to the open (unconfined) configurations in Figure 6a–b (curves NLO and LO in Figure 7), in which the spin accumulation/relaxation is not confined between the electrodes but spreads a distance l sf on both sides of the device. In this case, the channel relaxation rate is no longer driven by 
$\left( {{L \mathord{\left/$ (the ratio L/l sf arises from the confinement over approximately L) but rather exhibits the higher rate of 
${1 \mathord{\left/ {\vphantom {1 {R_{{\rm{ch}}}^{\rm{S}} }}} \right. \kern-\nulldelimiterspace} {R_{{\rm{ch}}}^{\rm{S}} }}$, as generally 
${L \mathord{\left/ {\vphantom {L {l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {l_{{\rm{sf}}} }} \ll 1$ or, at most, 
${L \mathord{\left/ {\vphantom {L {l_{{\rm{sf}}} }}} \right. \kern-\nulldelimiterspace} {l_{{\rm{sf}}} }} \approx 1$. Hence, as one can see for curves LO and NLO in Figure 7, the crossover to the regime controlled by relaxation inside the channel occurs earlier as a function of increasing R b at about 
${{R_{\rm{b}} } \mathord{\left/$. Accordingly, the level of the saturation plateau now scales with 
$R_{{\rm{ch}}}^{\rm{S}} $ instead of 
$\left( {{{l_{{\rm{sf}}} } \mathord{\left/$. We found saturation at 
$\Delta R = {\rm{\gamma }}^2 R_{{\rm{ch}}}^{\rm{S}} $ for curve NLO (configuration in Figure 6a) and 
$\Delta R = 2{\rm{\gamma }}^2 R_{{\rm{ch}}}^{\rm{S}} $ for curve LC (configuration in Figure 6c), in agreement with the standard expressions of Takahashi and MaekawaReference Takahashi and Maekawa42 (or the relevant limit of Equation 1). This is well below the saturation level 
$\Delta R \approx {{4{\rm{\gamma }}^2 R_{{\rm{ch}}}^{\rm{S}} l_{{\rm{sf}}} } \mathord{\left/$ of curve LC (configuration in Figure 6c) for a confined geometry. It is then not surprising to find that the previously reported ΔR values were much smaller than the ΔR values expected from a confined geometry (curve LC), in the device-favorable situation of l sf ≫ L. This is also true for the nonlocal confined (NLC) case in Figure 6d. For example, when the distance between the outer contacts is 3L, the spin signal ΔR (not represented in Figure 7) still saturates to 
$\Delta R \approx {{{\rm{\gamma }}^2 R_{{\rm{ch}}}^{\rm{S}} l_{{\rm{sf}}} } \mathord{\left/$, well above the saturation level of the nonconfined LSVs,Reference Jaffrès, George and Fert37 again showing the amplification factor of l sf/L due to the confinement. An example of similar amplification by confinement can be found in Reference Reference Laczkowski, Vila, Nguyen, Marty, Attane, Jaffres, George and Fert43 for a metallic LSV.
As we demonstrate next, the regime of the LC curve in Figure 7 can be clearly identified in experimental results on epitaxial graphene. Other results, on structures of types NLO and LO in Figure 6a–b, correspond to the behavior illustrated for LO in Figure 7.
Analysis of experimental results on graphene
In this section, we analyze the experimental results obtained by Dlubak et al.Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 on high-mobility MLG grown epitaxially on C-face SiC. A schematic of the LC-type device, with a graphene channel in contact with two cobalt electrodes through alumina tunnel barriers, is shown in Figure 4. The main parameters of the devices analyzed, with channel lengths of 0.8 μm or 2 μm and tunnel resistances R b varying between 3 MΩ and 75 MΩ, are listed in Table I, together with the corresponding experimental results.
As indicated previously, the variation of the spin signals in Table I (represented in Figure 5b) is characteristic of the crossover from a regime in which the relaxation is dominated by the interface barrier to one in which it is dominated by the channel. Here, ΔR is expected to progressively saturate, and ΔR/R is expected to drop to zero as 1/LR b in the limit of large R b, as shown by Equation 4 in the next paragraph. This variation is consistent with the experimental results shown in Figure 5. The data from Table I are also represented by the triangle symbols in Figure 7a–b.
The experimental results of Table I and Figure 5 were analyzed in Reference Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 on the basis of the general expression in Equation 3. To obtain better insight into the parameters involved in the analysis, we rewrite the asymptotic limit of this equation at large R bas
$${{\Delta R} \over {R^{\rm{P}} }} = {{2{\rm{\gamma }}^2 } \over {1 - {\rm{\gamma }}^2 }}{{l_{{\rm{sf}}} R_{{\rm{ch}}}^{\rm{S}} } \over {LR_{\rm{b}} }} = {{2{\rm{\gamma }}^2 } \over {1 - {\rm{\gamma }}^2 }}l_{{\rm{sf}}} {{{\rm{\rho }}_{{\rm{sq}}} } \over {wLR_{\rm{b}} }},$$where the graphene square resistance is ρsq ≈ 1 kΩ (from independent measurements on the same graphene layer) and w is the width of the channel. The two unknown parameters are γ and l sf, with ΔR/R P being an increasing function of both. Dlubak et al.Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 assumed the largest value found in previous experiments on cobalt/alumina junctions, namely, γ = 0.32, to derive a lower bound for l sf. The best fits between Equation 3 and the experimental results for the different samples were found for the spin diffusion lengths listed in Table I, all in the 100-μm range or slightly above. Additionally, relative to the maximum magnetoresistance of MR = γReference Schmidt, Ferrand, Molenkamp, Filip and van Wees2/(1 – γ2) expected for a symmetric double tunnel junction without any spin relaxation in the intervening conducting material, the MR values of samples with smaller values of R bcorrespond to an efficiency close to 80%.
It should be emphasized that the l sf values listed in Table I are only lower bounds. Smaller values of the spin injection parameter γ, such as those reported in many experiments with graphene, or use of standard expressions that are valid for nonconfined geometriesReference Takahashi and Maekawa42 would lead to greater l sf values. Nevertheless, it is also important to rule out potential spurious effects. First, the resistances of the graphene channel and electrodes are in the kiloohm and ohm ranges, respectively; hence, they are several orders of magnitude smaller than the experimentally observed ΔR values. Second, some contact effect, such as an anisotropic MR effect of the tunnel resistances, would be expected to increase with the tunnel resistance, in contradiction to the observations.
One might also ask whether spin signals could be observed in much longer devices, say, 50 or 100 μm. As explained in the earlier section “Analysis of experimental results on graphene” (also see Figure 7), large values of ΔR/R can be observed only in the window 
$\left( {{L \mathord{\left/$. When L increases and tends progressively to l sf, this window shrinks considerably. For moderate values of L such as L = l sf/10, the maximum MR value can still be obtained but at the cost of precise tuning of the interface resistance, which is not always possible experimentally. For L above this range, the maximum value of MR, γ2/(1 – γ2), is no longer achievable even with very precise tuning of the barrier. This is illustrated in Figure 5a, where one can see that, for devices similar to those described in Reference Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 but with L ≈ l sf/2, a large MR (although not as large as the maximum value) can be obtained only with highly tunable tunnel resistances that are several orders of magnitude smaller than that of the experiment.
We finally discuss the origin of the much greater l sf value found for MEG in Table I, through comparisons with other experiments. One cannot exclude the possibility that the smaller l sf values in some other experiments could be due to lower-quality tunnel barriers. Another origin of the discrepancy could be incorrect interpretation of Hanle measurements when spin relaxation is mainly due to spin escape (backflow) and not to intrinsic spin relaxation in graphene. However, the main difference arises from the superior properties of MEG grown on C-face SiC compared to graphene obtained by chemical vapor deposition or exfoliation. One of the advantages of this type of graphene comes from the screening of substrate-induced scattering in the top layers of the MLG. This is also the probable origin of the very high mobility achieved without treatment or suspension of the graphene. The second point is that graphene multilayers are flatter and have less corrugation than monolayers. This should reduce ripple-induced spin–orbit coupling and its contribution to spin relaxation and might also explain the longer τsfvalues found in samples with increasing numbers of layers.
Actually, the 100-μm range for the samples in Reference Reference Dlubak, Martin, Deranlot, Servet, Xavier, Mattana, Sprinkle, Berger, de Heer, Petroff, Anane, Seneor and Fert27 should not be the upper limit of l sf values in graphene. Improvements in the quality of epitaxial graphene, substrates with smaller interactions, and suspended graphene can probably all lead to greater l sfvalues and further increase the potential of graphene for large-scale spintronic devices, such as spin-only logic circuits.
Conclusions
Graphene has been experimentally identified as a pertinent medium for the transport of spin information over macroscopic distances with limited losses, thus enabling further work on more complex spin architectures (see Figure 8). The next step in developing graphene for spintronics applications is a thrilling challenge and concerns gate manipulation of the spin information transported in the graphene channel. However, a pertinent implementation has yet to be identified. Several theoretical predictions have been formulated, and experimental testing of these predictions has begun. A few interesting proposals are interfaces with ferromagnetic materials or impurities,Reference Jayasekera, Kong, Kim and Nardelli50, Reference Krompiewski51 interfaces with large spin–orbit or Rashba interactions,Reference Ochoa, Castro-Neto and Guinea34, Reference Castro Neto and Guinea52, Reference Varykhalov, Sanchez-Barriga, Shikin, Vescovo, Rybkin, Marchenko and Rader53 and also structural tuning of magnetic properties (using nanoribbon edges, nanomeshes, etc.).Reference Wunsch, Stauber, Sols and Guinea54–Reference Yazyev57 Other interesting types of control of electronic properties are also offered by the Dirac character of the electronic states and their associated pseudospin.Reference Abanin, Morozov, Ponomarenko, Gorbachev, Mayorov, Katsnelson, Watanabe, Taniguchi, Novoselov, Levitov and Geim58–Reference San-Jose, Prada, McCann and Schomerus60
Comparison of device performance (ΔR) versus channel properties (l sf) for different spin-transport media (metals, semiconductors, carbon allotropes), as reported in the literature in the case of simple local or nonlocal lateral-spin-transport devices. The dark gray arrow indicates the increasing l sf requirement for large signals, from spin transistors to more complex spin circuits. Appelbaum et al. also reported interesting results on silicon, with spin-information propagation over distances in the 100-μm range.Reference Appelbaum, Huang and Monsma48 However, those devices used multiterminal hot-electron structures that allowed the transmission of only a very small fraction (<10−4) of the injected current, giving spin signals not comparable with those reported here.Reference Jansen49
Experimentalists will now have to test the implementation of several possible concepts of graphene-based spin logic gates, and this will certainly keep them busy for some time. Nevertheless, the prospects are highly promising for green spin-based low-power memory and logic devices, as well as a global graphene-based electronics platform, fitting a “More than Moore” scenario in the short term and augmenting classic scaling of silicon complementary metal-oxide semiconductor (CMOS), as well as a long-term beyond-CMOS vision.
Acknowledgments
We thank R. Mattana, F. Petroff, J.-M. George, J. Peiro, C. Berger, W.A. de Heer, and M. Sprinkle for helpful discussions. We acknowledge financial support from the EU FP7 work program under Grant GRAFOL. P.S. also thanks the Institut Universitaire de France for a junior fellowship.
