Abstract
The understanding of spin dynamics and relaxation mechanisms in clean graphene, and the upper time and length scales on which spin devices can operate, are prerequisites to realizing graphenebased spintronic technologies. Here we theoretically reveal the nature of fundamental spin relaxation mechanisms in clean graphene on different substrates with Rashba spinorbit fields as low as a few tens of μeV. Spin lifetimes ranging from 50 picoseconds up to several nanoseconds are found to be dictated by substrateinduced electronhole characteristics. A crossover in the spin relaxation mechanism from a DyakonovPerel type for SiO_{2} substrates to a broadeninginduced dephasing for hBN substrates is described. The energy dependence of spin lifetimes, their ratio for spins pointing outofplane and inplane, and the scaling with disorder provide a global picture about spin dynamics and relaxation in ultraclean graphene in the presence of electronhole puddles.
Introduction
The tantalizing prospect of graphene spintronics was initiated by Tombros and coworkers^{1}, who first reported long spin diffusion length in large area graphene. The small spinorbit coupling (SOC) in carbon, plus the absence of a hyperfine interaction, suggested unprecedented spin lifetimes (τ_{s}) at room temperature (from μs to ms)^{2,3,4,5,6,7}.
However, despite significant progress in improving graphene quality, resolving contact issues, and reducing substrate effects^{1,8,9,10,11,12,13,14,15}, the measured τ_{s} are orders of magnitude shorter, even for highmobility samples. Extrinsic sources of SOC, including adatoms^{16,17,18,19} or lattice deformations^{20,21}, have been proposed to explain this discrepancy. Moreover, the nature of the dominant spin relaxation mechanism in graphene is elusive and debated. The conventional DyakonovPerel (DP)^{22} and ElliotYafet (EY)^{23} mechanisms, usually describing semiconductors and disordered metals, remain inconclusive in graphene because neither effect can convincingly reproduce the observed scaling between τ_{s} and the momentum relaxation time τ_{p}^{8,11}. Although generalizations of both mechanisms have been proposed, they do not allow an unambiguous interpretation of experiments^{6,20,21,24,25}.
It should be noted that the achieved roomtemperature spin lifetime in graphene is already long enough for the exploration of spindependent phenomena such as the spin Hall effect^{26,27,28}, or to harness proximity effects as induced for instance by magnetic oxides^{29} or semiconducting tungsten disulphide^{30}. However, a comprehensive picture of the spin dynamics of massless Dirac fermions in the presence of weak spinorbit coupling fields is of paramount importance for further exploitation and manipulation of the spin, pseudospin and valley degrees of freedom^{7,31,32,33}.
In this study, we show numerically that a weak uniform Rashba SOC (tens of μeV), induced by an electric field or the substrate, yields spin lifetimes from 50 ps up to several nanoseconds. The dominant spin relaxation mechanism is shown to be dictated by long range potential fluctuations (electronhole puddles)^{34}. For graphene on a SiO_{2} substrate, such disorder is strong enough to interrupt the spin precession driven by the uniform Rashba field, resulting in motional narrowing and the DP mechanism. We also find the ratio , demonstrating the anisotropy of the inplane Rashba SOC field. For the case of a hexagonal boron nitride (hBN) substrate, the role of electronhole puddles is reduced to an effective energy broadening and the spin lifetime is limited by pure dephasing^{35,36}. These situations, however, share a common fingerprint  an Mshape energy dependence of τ_{s} that is minimal at the Dirac point. Taken together, our results provide deeper insight into the fundamentals of spin lifetimes in graphene dominated by electronhole puddles.
Results
Disorder and Transport time
Electronhole puddles are realspace fluctuations of the chemical potential, induced by the underlying substrate, which locally shift the Dirac point^{37,38,39}. Since measured transport properties usually result from an average around the charge neutrality point, it is generally difficult to access the physics at the Dirac point. As shown by Adam and coworkers^{37}, electronhole puddles can be modeled as a random distribution of long range scatterers, , where ξ = 10 and 30 nm denote the effective puddle ranges for SiO_{2} and hBN substrates, respectively^{38,40}, and is randomly chosen within [−Δ, Δ]. Based on experimental data, typical impurity densities are n_{i} = 10^{12} cm^{−2} (N_{i}/N_{tot} = 0.04%, the percentage of impurity sites) for SiO_{2} and n_{i} = 10^{11} cm^{−2} %) for hBN substrates^{38,41}. In addition, the onsite energy profiles were found to obey a Gaussian distribution, with standard deviations of σ = 5.5 and 56 meV for hBN and SiO_{2} substrates, respectively. From such information, we can tune Δ to obtain suitable disorder profiles for the onsite energy of the πorbital. Figure 1 (main frame) shows the calculated onsite energy distribution corresponding to hBN and SiO_{2} substrates, where we set Δ = 50 meV for SiO_{2} and Δ = 5 meV for hBN in order to match the experimental onsite energy profiles. The inset of Fig. 1 illustrates an energy landscape for a sample with 0.04% Gaussian impurities (SiO_{2} case).
To fully characterize the role of electronhole puddles, we evaluate the transport time τ_{p} using a realspace orderN approach, which computes the diffusion coefficient D(E, t). We extract τ_{p} from the saturation of D(E, t) since ^{42}. For numerical convenience, the calculations are first made using a larger value Δ = 0.27 eV (for which intervalley scattering remains moderate^{43}), and from this we obtain τ_{p}(E) for hBN and SiO_{2} substrates using the scaling law^{37}.
where I_{1}(x) is the modified Bessel function of the first kind, is a dimensionless parameter dictating the strength of the Gaussian potential, and the carrier density n*(E) is modified from the pristine graphene density n(E) by ^{37,44,45}. The computed τ_{p} are shown in Fig. 2(a) for both substrates. For SiO_{2}, τ_{p} is on the order of a few ps, while for hBN τ_{p} is more than two orders of magnitude larger. The spin precession time used in our calculations, , is shown for comparison.
We observe that the obtained values are consistent with experimental estimates. Monteverde and coworkers found a similar energy dependence (as in our Fig. 2(a)) of room temperature transport times for monolayer graphene on silicon oxide^{46}. Their experimental data range from 50 fs to 100 fs, whereas our numerical results predict values close to Dirac point of about 400 fs. This difference is likely due to temperature effects, additional adsorbed impurities or other structural defects which are not considered in our simulations. Similarly, the values we obtained for the case of hBN substrates are consistent with current best measurements of hBNencapsulated graphene, which report long mean free paths up to 30 μm and mobilities up to ^{47}. Our numerical results for the transport time in graphene on hBN are close to 100 ps at the Dirac point (which gives 100 microns for the mean free path), and therefore differ by less than one order of magnitude with respect to the most recent experimental data.
Spin dynamics and lifetimes in the presence of electronhole puddles
We now analyze the spin dynamics for puddles corresponding to the SiO_{2} and hBN substrates. The blue curve in Fig. 2(b) shows the timedependent spin polarization for the hBN substrate at the Dirac point for an initial outofplane polarization, (see Methods). The polarization exhibits oscillations with period ps, corresponding to the spin precession induced by the Rashba field. Simultaneously, the polarization decays in time, and by fitting , both T_{Ω} and the spin relaxation time τ_{s} can be evaluated.
Figure 2(b) also shows for the SiO_{2} substrate with initial spin polarization inplane (α = ) and outofplane (α = ⊥). In contrast to the hBN case, for which exhibits significant precession, the disorder strength of electronhole puddles for SiO_{2} is sufficient to interrupt spin precession. As a result, the polarization for SiO_{2} is better fit with . The absence of precession for compared to is consistent with the ratio between transport time and precession frequency, since whereas .
To scrutinize the origin of the dominant relaxation mechanism, we first examine the spin lifetimes τ_{s} for the SiO_{2} case when rotating the initial spin polarization (outofplane vs. inplane), and when varying the impurity concentration (0.04%, 0.08%, and 0.16%). Figure 3 shows the extracted τ_{s} for the outofplane (a) and inplane (b) cases. The energy dependence of τ_{s} exhibits an Mshape increasing from a minimum at the Dirac point, with a saturation and downturn of τ_{s} for E ≥ 200 meV. The values of τ_{s} range from 50 to 400 ps depending on the initial polarization and impurity density. We observe an increase of τ_{s} with n_{i}, which shows that a larger scattering strength reduces spin precession and dephasing, resulting in a longer spin lifetime, as described by the socalled motional narrowing effect^{48}. Additionally, the ratio (not shown) changes from 0.3 to 0.45 when is varied from 0.04 to 0.16%. Such behavior is expected when enhanced scattering drives more randomization of the direction of the Rashba SOC field, which ultimately yields in the strong disorder limit^{2,3}. These results are fully consistent with the DP spin relaxation mechanism^{20,21,48}.
Figure 3(c) shows for the hBN substrate ( and 0.016%) where a similar Mshape is observed. While is similar to near the Dirac point, it is much larger at higher energies, reaching nearly 1 ns (for λ_{R} = 37.4 μeV). A striking difference is that the scaling of τ_{s} with n_{i} is opposite to that of the SiO_{2} case, with an increase in puddle density resulting in a decrease in τ_{s}, which indicates a different physical origin. For hBN, this behavior is reminiscent of the EY mechanism, but we will argue below that its origin is different.
Crossover in spin relaxation behavior for hBN and SiO_{2} substrates
Figure 4 provides a global view of our results, where we plot τ_{s} vs. 1/τ_{p} for the SiO_{2} and hBN substrates (black and red symbols respectively) at the Dirac point and at E = −200 meV (closed and open symbols respectively). For low defect densities (hBN substrate), τ_{s} decreases strongly with decreasing τ_{p}. However, with increasing defect density (SiO_{2} substrate) this trend reverses and τ_{s} scales almost linearly with 1/τ_{p}, according to the DP relationship . At E = −200 meV, ν = 1, fitting the usual DP theory. At the Dirac point, the scaling is somewhat weaker, with ν = 1/4. These results are reminiscent of those summarized in Fig. 5(a) of Drogeler et al.^{13}, where spin lifetimes of graphene devices on SiO_{2} scaled inversely with the mobility, while devices on hBN appear to show the opposite trend.
While the SiO_{2} results of Fig. 4 show DP behavior, the nature of the spin relaxation for weak electronhole puddles is less clear. The fact that τ_{s} and τ_{p} decrease together suggests the EY mechanism, but we find τ_{s} ≤ τ_{p} near the Dirac point and τ_{s} ≪ τ_{p} at higher energies. This contrasts with the usual picture of EY relaxation, where charge carriers flip their spin when scattering off impurities, giving τ_{s} = τ_{p}/α, where α ≪ 1 is the spin flip probability^{6}. Instead, this situation matches that described in ref. 48; when τ_{p} > T_{Ω}, the spin precesses freely until phase information is lost during a collision, in analogy to the collisional broadening of optical spectroscopy. More collisions result in a greater loss of phase, reducing τ_{s} with decreasing τ_{p}. We verify this by removing the realspace disorder (setting Δ = 0) and modeling the electronhole puddles with an effective Lorentzian energy broadening η^{*}. The results are shown in Fig. 4 (main frame, blue dashed line), where we plot τ_{s} vs. η^{*} at E = −200 meV (top axis). For small η^{*}, the scaling matches well with the realspace simulations of hBN, indicating that the puddles can be represented as a uniform energy broadening (see supplementary material). Larger values of η^{*} lead to stronger mixing of different spin dynamics and τ_{s} saturates at very large η^{*}. There, the scaling of τ_{s} vs. η^{*} clearly fails to replicate the DP behavior seen in the realspace simulations, since the effective broadening model does not induce the momentum scattering necessary for motional narrowing^{48}.
Next we explain the origin of the Mshaped energy dependence of τ_{s}. At low energies, the spin dynamics are dominated by strong spinpseudospin coupling^{36}, which yields fast dephasing and a minimum of τ_{s} at the Dirac point, in agreement with experimental data. At higher energies, the origin of the downturn of τ_{s} depends on the substrate. For the case of SiO_{2} substrate it is driven by the conventional DP mechanism, where . For the case of hBN, the downturn of τ_{s} can be explained by comparing the spin dynamics in the TB model (Eq. (2) in Methods) with the lowenergy model in the absence of puddles (Δ = 0). In this regime , and spin dephasing and relaxation are driven by a combination of energy broadening and a nonuniform spin precession frequency. For the TB model, spin dynamics are calculated with the realspace approach and with a standard kspace approach and give identical τ_{s} (inset of Fig. 4, red circles and blue solid line), indicating the equivalence of the real and kspace approaches in the clean limit when accounting for the full TB Hamiltonian. We observe that while for all models, the spin lifetime shows a minimum at the Dirac point, spin transport simulations with the widely used lowenergy Hamiltonian (see Methods for and green dashed line in Fig. 4 inset for results) clearly cannot capture the saturation and downturn of τ_{s}(E), i.e. its full Mshape. To qualitatively reproduce the Mshape of τ_{s}(E), the firstorder term of the Rashba Hamiltonian, , needs to be included in . This term introduces stronger dephasing at higher energy, driven by the anisotropy of the Rashba spinorbit interaction^{36}.
In addition to their different energy dependence, the TB and lowenergy models also yield very different spin lifetimes. A value of τ_{s} = 10 ns is obtained at the Dirac point for the lowenergy model, which is two orders of magnitude larger than τ_{s} from the TB Hamiltonian, indicating a strong spin dephasing induced by the highorder kterms. Interestingly, by studying the changes of τ_{s}(E) with respect to the Rashba SOC strength, we observe the scaling behavior , meaning the spin relaxes after a finite number of precession periods β ( close to the Dirac point), see Supplementary Material. This suggests that dephasing is the limiting factor of spin lifetimes in the ultraclean case. We finally note that by taking λ_{R} = 5 μeV (electric field of 1 V/nm^{4}), a spin lifetime of ns is deduced at the Dirac point, whereas at higher energies τ_{s} could reach about 10 ns.
Discussion
Our results show a clear transition between two different regimes of spin relaxation, mediated solely by the scattering strength of the electronhole puddles. For hBN substrates, spin relaxation is dominated by dephasing arising from an effective energy broadening induced by the puddles, and τ_{s} scales with τ_{p}. In contrast, for SiO_{2} substrates dephasing is limited by motional narrowing, leading to a DP regime with . Remarkably, both regimes exhibit similar values of τ_{s} at the Dirac point and a similar Mshape energy dependence (Fig. 3), making it a signature of spin relaxation in graphene for all puddle strengths. The crossover between both mechanisms occurs when , which might have been realized in some experiments. This could explain some conflicting interpretations of experimental data in terms of either ElliotYafet or DyakonovPerel mechanisms^{11}.
We note the large discrepancy between our conclusions and the former theoretical work by C. Ertler et al.^{3}. Indeed, the conclusions of ref. 3 (the spin lifetime maximum at the Dirac point and reaching values in the millisecond range) are fully inconsistent with the main experimental features, which are a minimum of the spin lifetime at the Dirac point and an increase for higher energy, and with spin lifetimes on the order of hundreds of ps to a few nanoseconds. The fundamental difference of the model used in C. Ertler et al.^{3} and our present study turns out to be essential. In their study, the spin precession frequency was assumed to be uniform in energy, while our approach is a fully quantum study of spin dynamics without any approximation. As a result, from our analysis of the timedependence of the spin polarization we observe that the spin precession frequency is nonuniform in energy, which is one essential aspect explaining a faster decay of spin lifetime close to the Dirac point.
Our findings suggest alternative options for determining the spin relaxation mechanism in graphene from experimental measurements. Indeed, the typical approach, to examine how τ_{p} and τ_{s} scale with electron density and to assign either the EY or DP mechanism accordingly, is not always appropriate. For example, the EY mechanism in graphene is given by , such that τ_{s} and τ_{p} would scale oppositely with respect to electron density if ^{6}. Similarly, for our results the scaling of τ_{p} and τ_{s} with energy suggest an EY mechanism near the Dirac point and a DP mechanism at higher energies, but Figs. 3 and 4 indicate a richer behavior. Therefore, to determine the spin relaxation mechanism it would be more appropriate to study how τ_{s} and τ_{s} scale with defect density or mobility at each value of the electron density. We stress that the decay of the spin lifetime with increasing impurity density (for the hBN substrate) is reminiscent of the conventional ElliotYafet mechanism, but is actually a totally different mechanism, being driven by dephasing effects in a ballistic regime.
It should be noted that our simulations are performed using a constant Rashba spinorbit coupling, λ_{R}, which is attributed to substrate effects (mirror symmetry breaking and interface interaction). In the experiments, by applying large electrostatic coupling to reach higher charge densities, an additional electricfield dependent λ_{R} should be at play. This might explain why, especially for the hBN substrate, the simulations show a larger variation of τ_{s} in energy than the gate voltage dependent spin lifetimes reported in experiments^{13,14}.
Finally, in a recent experiment by Guimarães and coworkers, external magnetic and electric fields were used to investigate the spin lifetime anisotropy in hBNencapsulated graphene was found to range between 0.6 to 0.75 by varying the electric field. The origin of such values and their variation or possible connection to outofplane fields^{49} remains to be understood. Indeed, this anisotropy factor provides important information for understanding the microscopic origin of spin relaxation. In our simulations, the DP mechanism dominates for sufficiently strong disorder (such as electronhole puddles on SiO_{2} substrates). However the case of the ultraclean hBN substrate is more complex. Here, the transport time becomes larger than the spin precession frequency, making the DP mechanism inefficient. As discussed in the Supplementary Material, for inplane spin injection, additional effects are needed to yield spin relaxation, such as an external perpendicular magnetic field (as in Hanle spin precession measurements). More experimental and theoretical work remains to be done to fully determine the various mechanisms at play and the spin lifetime anisotropy in the limit of ultraclean graphene devices.
Model of homogeneous SOC and electronhole puddles
The tightbinding (TB) Hamiltonian for describing spin dynamics in graphene is given by
where γ_{0} is the nearestneighbor πorbital hopping, V_{I} is the intrinsic SOC, and V_{R} is the Rashba SOC. In the lowenergy limit, this Hamiltonian is often approximated by a continuum model describing massless Dirac fermions in a single Dirac cone, , where v_{F} is the Fermi velocity, is the momentum, are the spin (pseudospin) Pauli matrices, , and . The value λ_{I} = 12 μ eV is commonly used for the intrinsic SOC of graphene^{4} while the Rashba SOC is electric fielddependent. Here, we let λ_{R} = 37.4 μeV, taken from an extended spband TB model for graphene under an electric field of a few V/nm^{4,5}. Higherorder SOC terms in the continuum model beyond allow an extension to higher energy^{50}. We note that the single cone approximation can be inappropriate in case of strong valley mixing.
Spin dynamics methodology
The timedependent spin polarization of propagating wavepackets is computed through^{36}
where are the Pauli spin matrices and is the spectral measure operator. The wavepacket dynamics are obtained by solving the timedependent Schrödinger equation^{42}, starting from a state which may have either outofplane (zdirection) or inplane spin polarization. An energy broadening η is introduced for expanding through a continued fraction expansion of the Green’s function^{42}, and mimics an effective disorder. This method has been used to investigate spin relaxation in golddecorated graphene^{36}. Here, we focus on the expectation value of the spin zcomponent and the spin xcomponent .
Additional Information
How to cite this article: Tuan, D. V. et al. Spin dynamics and relaxation in graphene dictated by electronhole puddles. Sci. Rep. 6, 21046; doi: 10.1038/srep21046 (2016).
References
 1
Tombros, N., Jozsa, C., Popinciuc, M., Jonkman, H. & Van Wees, B. Electronic spin transport and spin precession in single graphene layers at room temperature. Nature 448, 571 (2007).
 2
HuertasHernando, D., Guinea, F. & Brataas, A. Spinorbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps. Phys. Rev. B 74, 155426 (2006).
 3
Ertler, C., Konschuh, S., Gmitra, M. & Fabian, J. Electron spin relaxation in graphene: The role of the substrate. Phys. Rev. B 80, 041405 (2009).
 4
Gmitra, M., Konschuh, S., Ertler, C., AmbroschDraxl, C. & Fabian, J. Bandstructure topologies of graphene: Spinorbit coupling effects from first principles. Phys. Rev. B 80, 235431 (2009).
 5
Ast, C. R. & Gierz, I. spband tightbinding model for the bychkovrashba effect in a twodimensional electron system including nearestneighbor contributions from an electric field. Phys. Rev. B 86, 085105 (2012).
 6
Ochoa, H., Castro Neto, A. H. & Guinea, F. Elliotyafet mechanism in graphene. Phys. Rev. Lett. 108, 206808 (2012).
 7
Han, W., Kawakami, R. K., Gmitra, M. & Fabian, J. Graphene spintronics. Nature Nanotechnology 9, 794807 (2014).
 8
Pi, K. et al. Manipulation of spin transport in graphene by surface chemical doping. Phys. Rev. Lett. 104, 187201 (2010).
 9
Yang, T.Y. et al. Observation of long spinrelaxation times in bilayer graphene at room temperature. Phys. Rev. Lett. 107, 047206 (2011).
 10
Avsar, A. et al. Toward wafer scale fabrication of graphene based spin valve devices. Nano Letters 11, 2363–2368 (2011).
 11
Zomer, P. J., Guimarães, M. H. D., Tombros, N. & van Wees, B. J. Longdistance spin transport in highmobility graphene on hexagonal boron nitride. Phys. Rev. B 86, 161416 (2012).
 12
Dlubak, B. et al. Highly efficient spin transport in epitaxial graphene on sic. Nature Physics 8, 557–561 (2012).
 13
Drögeler, M. et al. Nanosecond spin lifetimes in single and fewlayer graphene hbn heterostructures at room temperature. Nano Letters 14, 6050–6055 (2014).
 14
Guimarães, M. H. D. et al. Controlling spin relaxation in hexagonal bnencapsulated graphene with a transverse electric field. Phys. Rev. Lett. 113, 086602 (2014).
 15
Venkata Kamalakar, M., Groenveld, C., Dankert, A. & Dash, S. P. Long distance spin communication in chemical vapour deposited graphene. Nature Communinations 6, 6766 (2015).
 16
Castro Neto, A. H. & Guinea, F. Impurityinduced spinorbit coupling in graphene. Phys. Rev. Lett. 103, 026804 (2009).
 17
Fedorov, D. V. et al. Impact of electronimpurity scattering on the spin relaxation time in graphene: A firstprinciples study. Phys. Rev. Lett. 110, 156602 (2013).
 18
Wojtaszek, M., VeraMarun, I. J., Maassen, T. & van Wees, B. J. Enhancement of spin relaxation time in hydrogenated graphene spinvalve devices. Phys. Rev. B 87, 081402 (2013).
 19
Kochan, D., Gmitra, M. & Fabian, J. Spin relaxation mechanism in graphene: Resonant scattering by magnetic impurities. Phys. Rev. Lett. 112, 116602 (2014).
 20
HuertasHernando, D., Guinea, F. & Brataas, A. Spinorbitmediated spin relaxation in graphene. Phys. Rev. Lett. 103, 146801 (2009).
 21
Zhang, P. & Wu, M. W. Electron spin relaxation in graphene with random rashba field: comparison of the dyakonov perel and elliott yafetlike mechanisms. New Journal of Physics 14, 033015.
 22
Dyakonov, M. I. & Perel, V. I. Spin relaxation of conduction electrons in noncentrosymmetric semiconductors. Soviet Physics Solid State 13, 3023–3026 (1972).
 23
Yafet, Y. Solid State Physics (F. Seitz and D. Turnbull, 1963).
 24
Dora, B., Muranyi, F. & Simon, F. Electron spin dynamics and electron spin resonance in graphene. Europhysics Letters 92, 17002.
 25
Roche, S. & Valenzuela, S. O. Graphene spintronics: puzzling controversies and challenges for spin manipulation. Journal of Physics D: Applied Physics 47, 094011.
 26
Balakrishnan, J., Koon, G. K. W., Jaiswal, M., Castro Neto, A. H. & Ozyilmaz, B. Colossal enhancement of spinorbit coupling in weakly hydrogenated graphene. Nature Physics 9, 284287 (2013).
 27
Balakrishnan, J. et al. Giant spin hall effect in graphene grown by chemical vapor deposition. Nature Communications 5, 4748 (2014).
 28
Wang, Z. et al. Strong interfaceinduced spinorbit interaction in graphene on ws2. Nature Communications 6, 8339 (2015).
 29
Wang, Z., Tang, C., Sachs, R., Barlas, Y. & Shi, J. Proximityinduced ferromagnetism in graphene revealed by the anomalous hall effect. Phys. Rev. Lett. 114, 016603 (2015).
 30
Avsar, A. et al. Spinorbit proximity effect in graphene. Nature Communications 5, 4875 (2014).
 31
Roche, S. et al. Graphene spintronics: the european flagship perspective. 2D Materials 2, 030202 (2015).
 32
Son, Y.W., Cohen, L. S. G. & Marvin, L. Halfmetallic graphene nanoribbons. Nature 444, 347–349 (2006).
 33
Pesin, D. & MacDonald, A. H. Spintronics and pseudospintronics in graphene and topological insulators. Nature Materials 11, 409416 (2012).
 34
Adam, S., Brouwer, P. W. & Das Sarma, S. Crossover from quantum to boltzmann transport in graphene. Phys. Rev. B 79, 201404 (2009).
 35
Rashba, E. I. Graphene with structureinduced spinorbit coupling: Spinpolarized states, spin zero modes, and quantum hall effect. Phys. Rev. B 79, 161409 (2009).
 36
Van Tuan, D., Ortmann, F., Soriano, D., Valenzuela, S. & Roche, S. Pseudospindriven spin relaxation mechanism in graphene. Nature Physics 10, 857 (2014).
 37
Adam, S. et al. Mechanism for puddle formation in graphene. Phys. Rev. B 84, 235421 (2011).
 38
Martin, J. et al. Observation of electron hole puddles in graphene using a scanning singleelectron transistor. Nature Physics 4, 144–148 (2008).
 39
Deshpande, A., Bao, W., Miao, F., Lau, C. N. & LeRoy, B. J. Spatially resolved spectroscopy of monolayer graphene on SiO2 . Phys. Rev. B 79, 205411 (2009).
 40
Zhang, Y., Brar, V., Girit, C., Zettl, V. & Crommie, M. Origin of spatial charge inhomogeneity in graphene. Nature Physics 5, 722 (2009).
 41
Xue, J. et al. Scanning tunnelling microscopy and spectroscopy of ultraflat graphene on hexagonal boron nitride. Nature Materials 10, 282–285 (2011).
 42
Roche, S. Quantum transport by means of O(n) realspace methods. Phys. Rev. B 59, 2284–2291 (1999).
 43
Ortmann, F., Cresti, A., Montambaux, G. & Roche, S. Magnetoresistance in disordered graphene: The role of pseudospin and dimensionality effects unraveled. EPL (Europhysics Letters) 94, 47006.
 44
Rycerz, A., Tworzydo, J. & Beenakker, C. W. J. Anomalously large conductance fluctuations in weakly disordered graphene. EPL (Europhysics Letters) 79, 57003.
 45
Kłos, J. W. & Zozoulenko, I. V. Effect of short and longrange scattering on the conductivity of graphene: Boltzmann approach vs tightbinding calculations. Phys. Rev. B 82, 081414 (2010).
 46
Monteverde, M. et al. Transport and elastic scattering times as probes of the nature of impurity scattering in singlelayer and bilayer graphene. Phys. Rev. Lett. 104, 126801 (2010).
 47
Banszerus, L. et al. Ultrahighmobility graphene devices from chemical vapor deposition on reusable copper. Science Advances 1 (2015).
 48
Žutić, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323–410 (2004).
 49
Tombros, N. et al. Anisotropic spin relaxation in graphene. Phys. Rev. Lett. 101, 046601 (2008).
 50
Rakyta, P., Kormányos, A. & Cserti, J. Trigonal warping and anisotropic band splitting in monolayer graphene due to rashba spinorbit coupling. Phys. Rev. B 82, 113405 (2010).
Acknowledgements
This work has received funding from the European Union Seventh Framework Programme under grant agreement 604391 Graphene Flagship. S.R. acknowledges the Spanish Ministry of Economy and Competitiveness for funding (MAT201233911), the Secretaria de Universidades e Investigacion del Departamento de Economia y Conocimiento de la Generalidad de Cataluña and the Severo Ochoa Program (MINECO SEV20130295). F.O. would like to acknowledge the Deutsche Forschungsgemeinschaft (grant OR 349/11). Inspiring discussions with Sergio O. Valenzuela, Shaffique Adam, and Jaroslav Fabian are deeply acknowledged.
Author information
Affiliations
Contributions
S.R. directed the project. The elaboration of the electronic model and spin transport simulations were performed by D.V.T., D.S. and A.W.C.; D.V.T., F.O., A.W.C. and S.R. carried out analyses and interpretation. S.R., A.W.C., F.O. and D.V.T. wrote the main manuscript text.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Van Tuan, D., Ortmann, F., Cummings, A. et al. Spin dynamics and relaxation in graphene dictated by electronhole puddles. Sci Rep 6, 21046 (2016). https://doi.org/10.1038/srep21046
Received:
Accepted:
Published:
Further reading

Tailoring the performance of electrochemical biosensors based on carbon nanomaterials via aryldiazonium electrografting
Bioelectrochemistry (2021)

LightEnhanced Spin Diffusion in Hybrid Perovskite Thin Films and Single Crystals
ACS Applied Materials & Interfaces (2020)

Induced spin polarization in graphene via interactions with halogen doped MoS2 and MoSe2 monolayers by DFT calculations
Nanoscale (2020)

Linear scaling quantum transport methodologies
Physics Reports (2020)

Adiabatic and nonadiabatic quantum charge and spin pumping in zigzag and armchair graphene nanoribbons
Journal of Applied Physics (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.