Introduction

The tantalizing prospect of graphene spintronics was initiated by Tombros and coworkers1, who first reported long spin diffusion length in large area graphene. The small spin-orbit coupling (SOC) in carbon, plus the absence of a hyperfine interaction, suggested unprecedented spin lifetimes (τs) 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 effects1,8,9,10,11,12,13,14,15, the measured τs are orders of magnitude shorter, even for high-mobility samples. Extrinsic sources of SOC, including adatoms16,17,18,19 or lattice deformations20,21, have been proposed to explain this discrepancy. Moreover, the nature of the dominant spin relaxation mechanism in graphene is elusive and debated. The conventional Dyakonov-Perel (DP)22 and Elliot-Yafet (EY)23 mechanisms, usually describing semiconductors and disordered metals, remain inconclusive in graphene because neither effect can convincingly reproduce the observed scaling between τs and the momentum relaxation time τp8,11. Although generalizations of both mechanisms have been proposed, they do not allow an unambiguous interpretation of experiments6,20,21,24,25.

It should be noted that the achieved room-temperature spin lifetime in graphene is already long enough for the exploration of spin-dependent phenomena such as the spin Hall effect26,27,28, or to harness proximity effects as induced for instance by magnetic oxides29 or semiconducting tungsten disulphide30. However, a comprehensive picture of the spin dynamics of massless Dirac fermions in the presence of weak spin-orbit coupling fields is of paramount importance for further exploitation and manipulation of the spin, pseudospin and valley degrees of freedom7,31,32,33.

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 (electron-hole puddles)34. For graphene on a SiO2 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 in-plane Rashba SOC field. For the case of a hexagonal boron nitride (hBN) substrate, the role of electron-hole puddles is reduced to an effective energy broadening and the spin lifetime is limited by pure dephasing35,36. These situations, however, share a common fingerprint - an M-shape energy dependence of τs that is minimal at the Dirac point. Taken together, our results provide deeper insight into the fundamentals of spin lifetimes in graphene dominated by electron-hole puddles.

Results

Disorder and Transport time

Electron-hole puddles are real-space fluctuations of the chemical potential, induced by the underlying substrate, which locally shift the Dirac point37,38,39. 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 coworkers37, electron-hole puddles can be modeled as a random distribution of long range scatterers, , where ξ = 10 and 30 nm denote the effective puddle ranges for SiO2 and hBN substrates, respectively38,40, and is randomly chosen within [−Δ, Δ]. Based on experimental data, typical impurity densities are ni = 1012 cm−2 (Ni/Ntot = 0.04%, the percentage of impurity sites) for SiO2 and ni = 1011 cm−2 %) for hBN substrates38,41. 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 SiO2 substrates, respectively. From such information, we can tune Δ to obtain suitable disorder profiles for the onsite energy of the π-orbital. Figure 1 (main frame) shows the calculated onsite energy distribution corresponding to hBN and SiO2 substrates, where we set Δ = 50 meV for SiO2 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 (SiO2 case).

Figure 1
figure 1

Onsite energy distribution of the carbon atoms in the graphene sample, which mimics the chemical potential induced by hBN (green) and SiO2 (black) substrates together with their Gaussian fitting lines. Inset: Real space vizualization of the energy landscape for a graphene sample with 0.04% Gaussian impurities (SiO2 case). Absolute values are pictured.

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To fully characterize the role of electron-hole puddles, we evaluate the transport time τp using a real-space order-N approach, which computes the diffusion coefficient D(E, t). We extract τp 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 moderate43), and from this we obtain τp(E) for hBN and SiO2 substrates using the scaling law37.

where I1(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 τp are shown in Fig. 2(a) for both substrates. For SiO2, τp is on the order of a few ps, while for hBN τp is more than two orders of magnitude larger. The spin precession time used in our calculations, , is shown for comparison.

Figure 2
figure 2

(a) Transport times for graphene on SiO2 and hBN substrates (solid black and red curves, respectively). The dashed line shows the spin precession time. (b) Time-dependent spin polarization for out-of-plane (solid red line) and in-plane (solid black line) spin injection for the SiO2 substrate, plus the fits to the exponential damping (dashed lines). The blue curves show the same information for the hBN substrate with out-of-plane injection.

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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 oxide46. 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 hBN-encapsulated 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 electron-hole puddles

We now analyze the spin dynamics for puddles corresponding to the SiO2 and hBN substrates. The blue curve in Fig. 2(b) shows the time-dependent spin polarization for the hBN substrate at the Dirac point for an initial out-of-plane 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 TΩ and the spin relaxation time τs can be evaluated.

Figure 2(b) also shows for the SiO2 substrate with initial spin polarization in-plane (α = ||) and out-of-plane (α = ⊥). In contrast to the hBN case, for which exhibits significant precession, the disorder strength of electron-hole puddles for SiO2 is sufficient to interrupt spin precession. As a result, the polarization for SiO2 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 τs for the SiO2 case when rotating the initial spin polarization (out-of-plane vs. in-plane), and when varying the impurity concentration (0.04%, 0.08%, and 0.16%). Figure 3 shows the extracted τs for the out-of-plane (a) and in-plane (b) cases. The energy dependence of τs exhibits an M-shape increasing from a minimum at the Dirac point, with a saturation and downturn of τs for E ≥ 200 meV. The values of τs range from 50 to 400 ps depending on the initial polarization and impurity density. We observe an increase of τs with ni, which shows that a larger scattering strength reduces spin precession and dephasing, resulting in a longer spin lifetime, as described by the so-called motional narrowing effect48. 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 limit2,3. These results are fully consistent with the DP spin relaxation mechanism20,21,48.

Figure 3
figure 3

Spin lifetimes for out-of-plane (a) and in-plane (b) spin injection for SiO2 substrate at impurity densities of 0.04% (black solid curves), 0.08% (red dashed curves), and 0.16% (blue dotted curves). (c) Spin lifetime with out-of-plane spin injection for the hBN substrate at impurity densities of 0.004% (black curve) and 0.016% (red curve).

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Figure 3(c) shows for the hBN substrate ( and 0.016%) where a similar M-shape is observed. While is similar to near the Dirac point, it is much larger at higher energies, reaching nearly 1 ns (for λR = 37.4 μeV). A striking difference is that the scaling of τs with ni is opposite to that of the SiO2 case, with an increase in puddle density resulting in a decrease in τs, 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 SiO2 substrates

Figure 4 provides a global view of our results, where we plot τs vs. 1/τp for the SiO2 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), τs decreases strongly with decreasing τp. However, with increasing defect density (SiO2 substrate) this trend reverses and τs scales almost linearly with 1/τp, 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 SiO2 scaled inversely with the mobility, while devices on hBN appear to show the opposite trend.

Figure 4: Low-energy spin lifetimes versus 1/τp (for initial out-of-plane spin polarization).
figure 4

Squares (circles) are for graphene on hBN (SiO2) substrate. Closed (open) symbols are for spin relaxation at the Dirac point (at E = −200 meV). The blue dashed line shows the spin lifetime assuming only energy broadening (top axis). Inset: spin lifetime in absence of puddles computed using the TB model in real space (red circles) or k-space (blue solid line), and the low-energy model in k-space (green dashed line), with η = 13.5 meV.

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While the SiO2 results of Fig. 4 show DP behavior, the nature of the spin relaxation for weak electron-hole puddles is less clear. The fact that τs and τp decrease together suggests the EY mechanism, but we find τs ≤ τp near the Dirac point and τs ≪ τp at higher energies. This contrasts with the usual picture of EY relaxation, where charge carriers flip their spin when scattering off impurities, giving τs = τp/α, where α ≪ 1 is the spin flip probability6. Instead, this situation matches that described in ref. 48; when τp > TΩ, 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 τs with decreasing τp. We verify this by removing the real-space disorder (setting Δ = 0) and modeling the electron-hole puddles with an effective Lorentzian energy broadening η*. The results are shown in Fig. 4 (main frame, blue dashed line), where we plot τs vs. η* at E = −200 meV (top axis). For small η*, the scaling matches well with the real-space 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 τs saturates at very large η*. There, the scaling of τs vs. η* clearly fails to replicate the DP behavior seen in the real-space simulations, since the effective broadening model does not induce the momentum scattering necessary for motional narrowing48.

Next we explain the origin of the M-shaped energy dependence of τs. At low energies, the spin dynamics are dominated by strong spin-pseudospin coupling36, which yields fast dephasing and a minimum of τs at the Dirac point, in agreement with experimental data. At higher energies, the origin of the downturn of τs depends on the substrate. For the case of SiO2 substrate it is driven by the conventional DP mechanism, where . For the case of hBN, the downturn of τs can be explained by comparing the spin dynamics in the TB model (Eq. (2) in Methods) with the low-energy 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 real-space approach and with a standard k-space approach and give identical τs (inset of Fig. 4, red circles and blue solid line), indicating the equivalence of the real- and k-space 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 low-energy Hamiltonian (see Methods for and green dashed line in Fig. 4 inset for results) clearly cannot capture the saturation and downturn of τs(E), i.e. its full M-shape. To qualitatively reproduce the M-shape of τs(E), the first-order 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 spin-orbit interaction36.

In addition to their different energy dependence, the TB and low-energy models also yield very different spin lifetimes. A value of τs = 10 ns is obtained at the Dirac point for the low-energy model, which is two orders of magnitude larger than τs from the TB Hamiltonian, indicating a strong spin dephasing induced by the high-order k-terms. Interestingly, by studying the changes of τs(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 λR = 5 μeV (electric field of 1 V/nm4), a spin lifetime of  ns is deduced at the Dirac point, whereas at higher energies τs 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 electron-hole puddles. For hBN substrates, spin relaxation is dominated by dephasing arising from an effective energy broadening induced by the puddles, and τs scales with τp. In contrast, for SiO2 substrates dephasing is limited by motional narrowing, leading to a DP regime with . Remarkably, both regimes exhibit similar values of τs at the Dirac point and a similar M-shape 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 Elliot-Yafet or Dyakonov-Perel mechanisms11.

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 time-dependence of the spin polarization we observe that the spin precession frequency is non-uniform 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 τp and τs 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 τs and τp would scale oppositely with respect to electron density if 6. Similarly, for our results the scaling of τp and τs 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 τs and τs 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 Elliot-Yafet 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 spin-orbit coupling, λR, 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 electric-field dependent λR should be at play. This might explain why, especially for the hBN substrate, the simulations show a larger variation of τs in energy than the gate voltage dependent spin lifetimes reported in experiments13,14.

Finally, in a recent experiment by Guimarães and coworkers, external magnetic and electric fields were used to investigate the spin lifetime anisotropy in hBN-encapsulated 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 out-of-plane fields49 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 electron-hole puddles on SiO2 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 in-plane 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 electron-hole puddles

The tight-binding (TB) Hamiltonian for describing spin dynamics in graphene is given by

where γ0 is the nearest-neighbor π-orbital hopping, VI is the intrinsic SOC, and VR is the Rashba SOC. In the low-energy limit, this Hamiltonian is often approximated by a continuum model describing massless Dirac fermions in a single Dirac cone, , where vF is the Fermi velocity, is the momentum, are the spin (pseudospin) Pauli matrices, , and . The value λI = 12 μ eV is commonly used for the intrinsic SOC of graphene4 while the Rashba SOC is electric field-dependent. Here, we let λR = 37.4 μeV, taken from an extended sp-band TB model for graphene under an electric field of a few V/nm4,5. Higher-order SOC terms in the continuum model beyond allow an extension to higher energy50. We note that the single cone approximation can be inappropriate in case of strong valley mixing.

Spin dynamics methodology

The time-dependent spin polarization of propagating wavepackets is computed through36

where are the Pauli spin matrices and is the spectral measure operator. The wavepacket dynamics are obtained by solving the time-dependent Schrödinger equation42, starting from a state which may have either out-of-plane (z-direction) or in-plane spin polarization. An energy broadening η is introduced for expanding through a continued fraction expansion of the Green’s function42, and mimics an effective disorder. This method has been used to investigate spin relaxation in gold-decorated graphene36. Here, we focus on the expectation value of the spin z-component and the spin x-component .

Additional Information

How to cite this article: Tuan, D. V. et al. Spin dynamics and relaxation in graphene dictated by electron-hole puddles. Sci. Rep. 6, 21046; doi: 10.1038/srep21046 (2016).