Giant Spin Lifetime Anisotropy in Graphene Induced by Proximity Effects

TL;DR

This paper investigates how proximity effects from transition metal dichalcogenides induce giant spin lifetime anisotropy in graphene, revealing potential for advanced spintronic device design.

Contribution

It provides first-principles based models quantifying spin lifetime anisotropy in graphene with TMDCs, highlighting the role of spin-valley locking and disorder.

Findings

Anisotropy can reach hundreds with strong intervalley scattering

Anisotropy reduces to 1/2 with weak disorder

Spin-valley locking mediates the anisotropy

Abstract

We report on fundamental aspects of spin dynamics in graphene interfaced with transition metal dichalcogenides (TMDCs). By using realistic models derived from first principles we compute the spin lifetime anisotropy, defined as the ratio of lifetimes for spins pointing out of the graphene plane to those pointing in the plane. In the presence of strong intervalley scattering the anisotropy can reach unprecedented values of tens to hundreds, while it reduces to 1/2 for weak disorder. This behavior is mediated by spin-valley locking, which is strong in TMDCs and is imprinted onto graphene. Such giant spin transport anisotropy, driven by proximity effects, provides an exciting paradigm for designing novel spin device functionalities.

Equations

$$H_0=\hslash v_F(\kappa {\sigma }_x{k}_x+{\sigma }_y{k}_y),$$

$$H_{\Delta }=\Delta {\sigma }_z,$$

$$H_I^{A/B}=\frac{1}{2}[{\lambda }_I^A({\sigma }_z+{\sigma }_0)+{\lambda }_I^B({\sigma }_z-{\sigma }_0)]\kappa s_z,$$

$$H_{PIA}^{A/B}=\frac{a}{2}[{\lambda }_{PIA}^A({\sigma }_z+{\sigma }_0)+{\lambda }_{PIA}^B({\sigma }_z-{\sigma }_0)](k_x{s}_y-k_y{s}_x),$$

$$H_R={\lambda }_R(\kappa {\sigma }_x{s}_y-{\sigma }_y{s}_x).$$

$$H_I={\lambda }_I\kappa {\sigma }_z{s}_z,$$

$$H_{VZ}={\lambda }_{VZ}\kappa s_z,$$

$$H_{PIA}=a{\lambda }_{PIA}{\sigma }_z(k_x{s}_y-k_y{s}_x),$$

$$H_{{\Delta }_{PIA}}=a{\Delta }_{PIA}(k_x{s}_y-k_y{s}_x),$$

$$H=H_0+\frac{1}{2}\hslash \vec{\omega }(t)\cdot \vec{s},$$

$$\hslash {\omega }_x=-2(ak{\Delta }_{PIA}\pm {\lambda }_R)\sin \theta ,$$

$$\hslash {\omega }_y=2(ak{\Delta }_{PIA}\pm {\lambda }_R)\cos \theta ,$$

$$\hslash {\omega }_z=2\kappa {\lambda }_{VZ},$$

$$\overline{{\omega }_{\alpha }(t){\omega }_{\beta }(t^{\prime })}={\delta }_{\alpha \beta }\overline{{\omega }_{\alpha }^2}{\mathrm{e}}^{-|t-t^{\prime }|/{\tau }_{c,\alpha }},$$

$${\tau }_{s,x}^{-1}=\overline{{\omega }_z^2}{\tau }_{iv}+\overline{{\omega }_y^2}{\tau }_p,$$

$${\tau }_{s,y}^{-1}=\overline{{\omega }_z^2}{\tau }_{iv}+\overline{{\omega }_x^2}{\tau }_p,$$

$${\tau }_{s,z}^{-1}=(\overline{{\omega }_x^2}+\overline{{\omega }_y^2}){\tau }_p.$$

$$\frac{{\tau }_{s,\perp }}{{\tau }_{s,\parallel }}={\left(\frac{{\lambda }_{VZ}}{{\lambda }_R}\right)}^2\left(\frac{{\tau }_{iv}}{{\tau }_p}\right)+1/2.$$

$${\tau }_{s,x}^{-1}=\overline{{\omega }_y^2}{\tau }_p^{\ast },$$

$${\tau }_{s,y}^{-1}=\overline{{\omega }_x^2}{\tau }_p^{\ast },$$

$${\tau }_{s,z}^{-1}=(\overline{{\omega }_x^2}+\overline{{\omega }_y^2}){\tau }_p^{\ast },$$

$$\frac{{\tau }_{s,\alpha }^K}{{\tau }_{s,\alpha }^{K^{\prime }}}=\frac{(g{\mu }_B{B}_z+2{\lambda }_{VZ}{)}^2{\tau }_p^2+1}{(g{\mu }_B{B}_z-2{\lambda }_{VZ}{)}^2{\tau }_p^2+1},$$

Full text

Introduction. Following the discovery of graphene in 2004 Novoselov et al. (2004), a host of other two-dimensional (2D) materials have been synthesized and studied, each demonstrating unique properties and showing promise for technological applications Bhimanapati et al. (2015). Currently, there is a great deal of interest in layered heterostructures of these materials Geim and Grigorieva (2013); Novoselov et al. (2016), where the combined system might be engineered for specific applications Ferrari et al. (2015) or might enable the exploration of new phenomena Li et al. (2016); Costanzo et al. (2016). In the field of spintronics, graphene has exceptional charge transport properties but weak spin-orbit coupling (SOC) on the order of 10 $\mu$eV Gmitra et al. (2009), which makes it ideal for long-distance spin transport Banszerus et al. (2016); Drögeler et al. (2016); Roche et al. (2015) but ineffective for generating or manipulating spin currents. To advance towards spin manipulation, recent work has focused on heterostructures of graphene and magnetic insulators Yang et al. (2013); Wang et al. (2015a); Dushenko et al. (2016); Hallal et al. (2017); Leutenantsmeyer et al. (2017) or strong SOC materials such as transition metal dichalcogenides (TMDCs) and topological insulators Jin and Jhi (2013); Kaloni et al. (2014); Gmitra and Fabian (2015). The SOC induced in graphene by a TMDC could enable phenomena such as topological edge states Gmitra et al. (2016) or the spin Hall effect Avsar et al. (2014); Garcia et al. (2017); Torres et al. (2017).

To this end, a variety of recent experiments have explored spin transport in graphene/TMDC heterostructures Avsar et al. (2014); Wang et al. (2015b); Yang et al. (2016); Wang et al. (2016); Omar and van Wees (2017); Yan et al. (2016); Dankert and Dash (2017). Magnetotransport measurements revealed that graphene in contact with WS2 exhibits a large weak antilocalization (WAL) peak, revealing a strong SOC induced by proximity effects Wang et al. (2015b); Yang et al. (2016); Wang et al. (2016); Yang et al. (2017). Fits to the magnetoconductance yielded spin lifetimes $\tau_{s}\approx 5$ ps, which is two to three orders of magnitude lower than graphene on traditional substrates Drögeler et al. (2016); Kamalakar et al. (2015). It was later asserted that after the removal of a temperature-independent background, $\tau_{s}$ becomes at most only a few hundred femtoseconds Wang et al. (2016). Nonlocal Hanle measurements, meanwhile, have revealed spin lifetimes up to a few tens of picoseconds Omar and van Wees (2017); Yan et al. (2016); Dankert and Dash (2017) that can be tuned by a back gate Yan et al. (2016); Dankert and Dash (2017). Finally, charge transport measurements on a Hall bar demonstrated a large nonlocal signal that was related to the spin Hall effect Avsar et al. (2014). Fits to experimental measurements have estimated the induced SOC in graphene to be 10-20 meV Avsar et al. (2014); Wang et al. (2016), while most density functional theory (DFT) and tight-binding (TB) calculations find values closer to 1 meV Kaloni et al. (2014); Wang et al. (2015b); Yang et al. (2016); Gmitra and Fabian (2015); Gmitra et al. (2016); Alsharari et al. (2016). While these studies have demonstrated that TMDCs induce strong SOC in graphene, the estimated values of $\tau_{s}$ vary by three orders of magnitude and nothing is yet known about the mechanisms governing spin dynamics and relaxation in these systems.

In this Letter, we employ dissipative quantum spin dynamics arguments, and quantum mechanical numerical simulations, to elucidate the nature of spin relaxation in graphene/TMDC heterostructures. We find that spin relaxation follows the D’yakonov-Perel’ (DP) mechanism, with $\tau_{s}=1-100$ ps for realistic momentum relaxation rates and Fermi energies. Remarkably, the spin lifetime anisotropy, defined as the ratio of lifetimes for spins pointing out of the graphene plane to those pointing in the plane, can reach unprecedented values of tens to hundreds in the presence of intervalley scattering. This behavior is mediated by spin-valley locking induced in graphene by the TMDC, which ties the in-plane spin lifetime to the intervalley scattering time. In the absence of valley mixing this ratio reduces to 1/2, typical of systems dominated by Rashba SOC Fabian et al. (2007). A giant spin lifetime anisotropy thus represents a qualitatively new regime of spin relaxation not typically seen in 2D systems, and its measurement Raes et al. (2016, 2017) should be an experimental probe of systems with strong spin-valley coupling, which includes both graphene/TMDC heterostructures and TMDCs themselves. Furthermore, systems with giant spin lifetime anisotropy could serve as an exciting new platform for the manipulation of spin and the implementation of new spintronic devices.

Dissipative spin dynamics model. To clarify the nature of spin relaxation in graphene/TMDC systems, we follow the approach in Fabian et al. (2007), which describes spin dynamics in a randomly fluctuating magnetic field. The low-energy ($E_{F}<300$ meV) Hamiltonian of graphene on a TMDC substrate is given by $H=H_{0}+H_{\Delta}+H_{I}^{A/B}+H_{R}+H_{PIA}^{A/B}$, where Gmitra et al. (2016)

[TABLE]

In Eq. (1), $v_{F}$ is the Fermi velocity, $\kappa=1(-1)$ for the K (K*′) valley, $\sigma_{i}$ ($s_{i}$) are the sublattice (spin) Pauli matrices, $k_{i}$ are the wave vector components relative to K or K′*, and $a=0.246$ nm is the graphene lattice constant. $H_{0}$ represents the graphene Dirac cone, and $H_{\Delta}$ is a staggered sublattice potential induced by the TMDC. $H_{I}^{A/B}$ and $H_{PIA}^{A/B}$ are the intrinsic and the pseudospin inversion asymmetry (PIA) SOC, respectively, the latter of which is permitted by broken z/-z symmetry in graphene Kochan et al. (2017). Due to the broken sublattice symmetry, these terms can have different strengths and signs on the A and B sublattices ($\lambda_{I}^{A/B}$ and $\lambda_{PIA}^{A/B}$). Finally, $H_{R}$ is the Rashba SOC induced by a perpendicular electric field Kane and Mele (2005); Gmitra et al. (2009).

While Eq. (1) is useful for TB calculations, analytically it is more transparent to combine the sublattice-dependent terms, giving $H=H_{0}+H_{\Delta}+H_{I}+H_{VZ}+H_{R}+H_{PIA}+H_{\Delta_{PIA}}$ with

[TABLE]

where $\lambda_{I}=(\lambda_{I}^{A}+\lambda_{I}^{B})/2$, $\lambda_{VZ}=(\lambda_{I}^{A}-\lambda_{I}^{B})/2$, $\lambda_{PIA}=(\lambda_{PIA}^{A}+\lambda_{PIA}^{B})/2$, and $\Delta_{PIA}=(\lambda_{PIA}^{A}-\lambda_{PIA}^{B})/2$. In this form, $H_{I}$ is the usual intrinsic SOC in graphene, which opens a topological gap 2$\lambda_{I}$ at the Dirac point Kane and Mele (2005). $H_{VZ}$ is a valley Zeeman term, which locks valley to spin and polarizes the bands out of the graphene plane with opposite orientation in the K and K*′* valleys. $H_{PIA}$ renormalizes the Fermi velocity, while $H_{\Delta_{PIA}}$ leads to a $k$-linear splitting of the bands, as in traditional 2D electron gases with Rashba SOC Bychov and Rashba (1984). Except for the PIA terms, this Hamiltonian is the same as that considered in previous works Wang et al. (2015b); Yang et al. (2016); Wang et al. (2016); Alsharari et al. (2016).

The next step is to derive the effective spin-orbit field arising from the SOC terms. This is done by rewriting Eq. (2) in the basis of the eigenstates of $H_{0}$ and projecting onto the conduction and valence bands. At Fermi energies away from the Dirac point ($E_{F}\gg 1$ meV), this gives

[TABLE]

where $k$ is the wave vector magnitude, $\theta$ is the direction of $k$ with respect to $k_{x}$, and $\vec{\omega}$ is the spin precession frequency of the effective spin-orbit field. The in-plane components of $\vec{\omega}$ give a Rashba-like spin texture, where $+(-)$ is for the conduction (valence) band. Strong PIA SOC thus leads to electron-hole asymmetry, as will be seen for graphene on WS2. The out-of-plane component of $\vec{\omega}$ is determined by $\lambda_{VZ}$ and changes sign between valleys. The overall texture of the effective spin-orbit field is depicted in Fig. 1.

Owing to momentum scattering, each component of $\vec{\omega}$ will fluctuate in time. A simple model for the correlation of the fluctuating spin-orbit field is Fabian et al. (2007)

[TABLE]

where the correlation time of fluctuation $\tau_{c,\alpha}$ depends on the component of $\vec{\omega}$. The in-plane components $\omega_{x/y}$ depend only on $\theta$, and thus $\tau_{c,x}=\tau_{c,y}=\tau_{p}$, the momentum relaxation time. Meanwhile, the out-of-plane component $\omega_{z}$ depends only on the valley index, giving $\tau_{c,z}=\tau_{iv}$, the intervalley scattering time. Assuming that $\tau_{c,\alpha}\omega_{\alpha}\ll 1$, applying Eqs. (3) and (4) to the equation of motion for the density matrix Fabian et al. (2007) yields the final expressions for the spin relaxation rates

[TABLE]

In Eq. (5), the out-of-plane spin relaxation follows the usual DP relation, $\tau_{s,\perp}^{-1}\equiv\tau_{s,z}^{-1}=[2(ak\Delta_{PIA}\pm\lambda_{R})/\hbar]^{2}\tau_{p}$, with the Rashba SOC augmented by the PIA term. However, the in-plane relaxation includes contributions from both the intervalley and the overall momentum scattering, and is given by $\tau_{s,\parallel}^{-1}\equiv\tau_{s,x}^{-1}=\tau_{s,y}^{-1}=(2\lambda_{VZ}/\hbar)^{2}\tau_{iv}+\tau_{s,z}^{-1}/2$. The nature of the spin relaxation, with $\tau_{s,\parallel}$ determined by $\tau_{iv}$ and $\tau_{s,\perp}$ by $\tau_{p}$, is shown schematically in Fig. 1. Ignoring the PIA term, the spin lifetime anisotropy is

[TABLE]

Equation (6) is the main result of this work, and indicates that a giant spin lifetime anisotropy, with the in-plane spins relaxing much faster than the out-of-plane spins, should be a defining characteristic of systems with strong spin-valley locking. Using DFT values of $\lambda_{VZ}=1.2$ meV and $\lambda_{R}=0.56$ meV for graphene on WSe2 Gmitra et al. (2016), and assuming relatively strong intervalley scattering ($\tau_{iv}\sim 5\tau_{p}$), we obtain a spin lifetime anisotropy of $\sim$20. This represents a qualitatively different regime of spin relaxation than the usual case of 2D Rashba systems, where without valley Zeeman SOC the anisotropy is 1/2, with the in-plane spins relaxing more slowly than the out-of-plane spins.

Equation (5) assumes strong intervalley scattering, $\tau_{iv}\omega_{z}\ll 1$, such that fast fluctuation of $\omega_{z}$ results in motional narrowing of the in-plane spin precession and an inverse dependence of $\tau_{s,\parallel}$ on $\tau_{iv}$. In contrast, when $\tau_{iv}\rightarrow\infty$, electrons experience a constant out-of-plane spin-orbit field and only the in-plane components fluctuate with time. In this limit, the procedure above yields

[TABLE]

where $\tau_{p}^{*}=\tau_{p}/(\omega_{z}^{2}\tau_{p}^{2}+1)$. Without intervalley scattering the spin lifetime anistropy thus collapses to $1/2$, as found in Rashba systems Fabian et al. (2007). Interestingly, in this regime an external perpendicular magnetic field $B_{z}$ can induce an imbalance in the spin population of each valley by enhancing (canceling) the spin-orbit field at K (K*′*). The ratio of spin lifetimes in each valley thus becomes

[TABLE]

where $g$ is the electron g-factor and $\mu_{B}$ is the Bohr magneton. For graphene on WSe2 with $\tau_{p}=100$ fs, the difference in $\tau_{s}$ can reach 10% for $B_{z}\approx 4$ T. Although this difference is too modest to achieve a complete valley-spin imbalance, it should be considered when observing spin relaxation in these structures in a magnetic field.

Numerical simulations. To verify the above results, we perform numerical simulations of spin relaxation. The graphene/TMDC system is described by the TB form of Eq. (1), to which we add a disorder term $H_{dis}=\sum\nolimits_{i,s}V_{dis}(\vec{r}_{i})\,c_{is}^{\dagger}c_{is}$, where $c_{is}^{\dagger}(c_{is})$ is the creation (annihilation) operator at site $i$ with spin $s$, and $V_{dis}(\vec{r}_{i})$ is the potential at site $i$. We assume the disorder consists of Gaussian-shaped electron-hole puddles Adam et al. (2011), such that $V_{dis}(\vec{r}_{i})=\sum\nolimits_{j=1}^{N}\epsilon_{j}\mathrm{exp}(-|\vec{r}_{i}-\vec{r}_{j}|^{2}/2\xi^{2})$, with the strength $\epsilon_{j}$ of each scatterer randomly chosen within $[-\epsilon,\epsilon]$, and with a uniform width $\xi=\sqrt{3}a$. In the dilute limit, $\tau_{p}$ and $\tau_{iv}$ are inversely proportional to the number of scatterers $N$, while $\epsilon$ controls their relative magnitude, with larger $\epsilon$ giving stronger intervalley scattering Zhang et al. (2009); Ortmann et al. (2011).

To calculate charge and spin transport, we employ a real-space wavepacket propagation method that allows for efficient simulation of large-scale disordered graphene systems Roche (1999); Cummings et al. (2014); Van Tuan et al. (2016). For charge transport we use the mean-square spreading of the wavepacket $\langle X^{2}(E,t)\rangle$ to calculate the diffusion coefficient $D(E,t)=\partial\langle X^{2}(E,t)\rangle/\partial t$, which in turn gives the momentum relaxation time $\tau_{p}(E)=\max D(E,t)/2v_{F}^{2}$. We simultaneously calculate the expectation value of the spin of the wavepacket $\vec{s}(E,t)$, from which the spin lifetime is evaluated by fitting to $\exp(-t/\tau_{s,\alpha})$ or $\exp(-t/\tau_{s,\alpha})\cos(\omega_{z}t)$, as appropriate. The density of charge scatterers is characterized as a percentage of the number of carbon atoms, $n=N/N_{C}\times 100\%$. We consider a 500 nm $\times$ 500 nm system with 9.2 million carbon atoms, and TB parameters are taken from Table I of Ref. Gmitra et al. (2016).

Figures 2(a) and (b) show $\vec{s}$ and $\tau_{p}$ for disorder profiles corresponding to strong and weak intervalley scattering, respectively. In the former we set $n=0.1\%$ and $\epsilon=2.8$ eV, and in the latter $n=1\%$ and $\epsilon=0.5$ eV. The $\tau_{p}$ for these two cases are shown in the insets, with values typical of those found experimentally Yang et al. (2016); Wang et al. (2015b, 2016). The different energy dependence of $\tau_{p}$, with a minimum or maximum at the Dirac point, is indicative of the contribution of intervalley scattering Roche et al. (2012). In Fig. 2(a), where intervalley scattering is strong, the in-plane component of $\vec{s}$ decays much more quickly than the out-of-plane component, and spin precession is suppressed. Meanwhile, in Fig. 2(b) the in-plane spin precesses about the effective spin-orbit field with frequency $\omega_{z}=2\lambda_{VZ}/\hbar$, and relaxes more slowly than the out-of-plane spin. This behavior is consistent with Eqs. (5)-(7).

Figure 3 shows the numerical spin lifetimes in the case of strong intervalley scattering for graphene on (a) WSe2 and (b) WS2. The solid lines, for $n=0.1\%$, indicate a giant anisotropy with $\tau_{s,\perp}=20-200$ ps and $\tau_{s,\parallel}\approx 1$ ps. There is also a significant electron-hole asymmetry in $\tau_{s,\perp}$ for graphene on WS2, arising from the larger PIA SOC in this system; $\lambda_{R}=0.56$ meV and $\Delta_{PIA}=75$ $\mu$eV for WSe2, while $\lambda_{R}=0.36$ meV and $\Delta_{PIA}=1.4$ meV for WS2 Gmitra et al. (2016). The open circles are the values of $\tau_{s}$ estimated from Eq. (5), showing good agreement between the numerical simulations and the spin dynamics model. To fit $\tau_{s,\parallel}$ we assumed $\tau_{iv}=5\tau_{p}$; although our calculations do not permit an exact determination of $\tau_{iv}$, this ratio is consistent with prior numerical results Zhang et al. (2009). As shown by the dashed lines, increasing the disorder density to $n=1\%$ scales $\tau_{s}$ by a factor of 10, confirming the inverse relationship between $\tau_{s}$ and $\tau_{p,iv}$.

The numerical spin lifetimes in the absence of intervalley scattering are shown in Fig. 4, where $\tau_{s,\parallel}$ is now larger than $\tau_{s,\perp}$. The agreement with the predictions of Eq. (7), shown as the open circles, is very convincing. However, we note that the agreement worsens at low energies, as the effective spin-orbit field in Eq. (3) is only valid for energies away from the Dirac point. The insets of Fig. 4 show the numerical values of the spin lifetime anistropy. As predicted by the semiclassical theory, the anisotropy is giant in the case of strong intervalley scattering, and collapses toward 1/2 otherwise.

Summary and conclusions. Using realistic quantum spin dynamics modeling and numerical simulations, we have presented a unified picture of the spin relaxation in graphene on TMDCs. We predict a giant spin relaxation anisotropy, which emerges in graphene due to proximity effects but should exist in any system with strong spin-valley locking, including TMDCs themselves. In the absence of spin-valley locking or intervalley scattering the anisotropy falls to 1/2, as expected for Rashba systems. This large variation indicates a qualitatively new regime of spin relaxation in graphene and other 2D materials.

It should be noted that the theory presented here is applicable when spin relaxation is dominated by SOC, but other spin relaxation mechanisms can take over when the SOC is small. This appears to be the case for graphene on SiO2, where measurements yielded no anisotropy, i.e., $\tau_{s,\perp}=\tau_{s,\parallel}$ Raes et al. (2016, 2017). In these systems the SOC is small and spin relaxation is likely dominated by paramagnetic impurities Kochan et al. (2014); Soriano et al. (2015). Meanwhile, very recent measurements have confirmed our prediction of giant spin lifetime anisotropy in graphene/TMDC heterostructures, with an anisotropy of $\sim$11 (40) for graphene on MoSe2 (WSe2) at a temperature of 75 K Ghiasi et al. . Another recent measurement found an anisotropy of $\sim$10 in graphene on WS2 at room temperature Benítez et al. , suggesting that temperature-dependent effects driven by electron-electron or electron-phonon scattering should have a weak impact.

These results also have important implications for the WAL analysis of magnetotransport in graphene/TMDC heterostructures. Previous analyses have concluded that the spin relaxation is dominated by Rashba SOC Yang et al. (2016, 2017), which is seemingly at odds with the presence of giant spin lifetime anisotropy. By reanalyzing the magnetoconductance measurements of Ref. Yang et al. (2017), and introducing valley Zeeman SOC into the analysis, the experimental results can be shown to be consistent with our theory 111See Supplemental Material for an analysis of the magnetoconductance measurements of Ref. Yang et al. (2017). This Supplemental Material also includes Refs. McCann and Fal’ko (2012); Ochoa et al. (2012).

On the more applied side, the giant spin lifetime anisotropy in graphene/TMDC heterostructures might be utilized for practical purposes in spin logic devices Dery et al. (2012); Wen et al. (2016) or in relation with opto-valleytronic spin injection in graphene/TMDC spin valves Luo et al. (2017); Avsar et al. . One possible application would be the design of a linear spin polarizer, where the in-plane components of an incoming spin current would be filtered out, leaving only the net out-of-plane polarization.