Anisotropic Spin Relaxation in Graphene

Spin relaxation in graphene is investigated in electrical graphene spin valve devices in the nonlocal geometry. Ferromagnetic electrodes with in-plane magnetizations inject spins parallel to the graphene layer. They are subject to Hanle spin precession under a magnetic field $B$ applied perpendicular to the graphene layer. Fields above 1.5 T force the magnetization direction of the ferromagnetic contacts to align to the field, allowing injection of spins perpendicular to the graphene plane. A comparison of the spin signals at $B=0$ and $B=2\mathrm{T}$ shows a 20% decrease in spin relaxation time for spins perpendicular to the graphene layer compared to spins parallel to the layer. We analyze the results in terms of the different strengths of the spin-orbit effective fields in the in-plane and out-of-plane directions and discuss the role of the Elliott-Yafet and Dyakonov-Perel mechanisms for spin relaxation.

Figure 1

Spin transport in graphene. (a) A SEM picture of a single layer of graphene contacted by 6 cobalt electrodes (sample A). Spins traveling a distance of $5\mu \mathrm{m}$, from cobalt electrode $F2$ to $F3$, are probed using the nonlocal geometry. The voltage circuit ($F3$-graphene-$F6$) is completely separated from the current circuit ($F1$-graphene-$F2$). (b) Hanle type spin precession experiment, the magnetization of the spin injector $F2$ is set antiparallel to the magnetization of spin detector $F3$. Spins are injected parallel to the graphene plane. (c) Application of a strong external magnetic field ($\sim 1.4\mathrm{T}$) perpendicular to the graphene layer results in injector and detector magnetizations aligned parallel to the external magnetic field. Spins are injected perpendicular to the graphene plane. (d) Hanle spin precession in case of parallel ($\uparrow \uparrow$, black curve) and antiparallel ($\uparrow \downarrow$, gray curve) magnetizations.

Figure 2

Anisotropic spin relaxation at high electron density $n$ (a) $n=2.1\times {10}^{12}{\mathrm{cm}}^{-2}$. Initially, spins are injected parallel to the graphene plane having the magnetization of the spin injector set parallel (gray line) or antiparallel (black line) to the detector (at a distance $L=5\mu \mathrm{m}$). From the fits (dashed line) we extract the diffusion constant $D$ and the relaxation time $T_{\Vert }$. A magnetic field of $\sim 1.4\mathrm{T}$ is needed to align the magnetization of the cobalt electrodes out of their easy magnetization axis, in this case the spins are injected perpendicular to the graphene layer having a spin relaxation time $T_{\perp }$. $T_{\perp }$ is 19% smaller to $T_{\Vert }$. The small decrease in the nonlocal signal found between 1.5 and 2 T is attributed to a background due to orbital magnetoresistance effects. The same experiment has been performed on sample $B$ for (b) $L=4\mu \mathrm{m}$ $n=3.5\times {10}^{12}{\mathrm{cm}}^{-2}$), (c) $L=2\mu \mathrm{m}$ ($n=2.8\times {10}^{12}{\mathrm{cm}}^{-2}$) and (d) $L=500\mathrm{nm}$ $n=3.5\times {10}^{12}{\mathrm{cm}}^{-2}$).

Figure 3

Anisotropic spin relaxation at the Dirac point for $L=500\mathrm{nm}$ (a) Gate voltage dependence of the graphene resistance between electrodes $F_2$ and $F_3$ [Fig. 1a]. The charge neutrality point is found at $Vg\approx -75\mathrm{V}$. (b) The nonlocal resistance for spins injected parallel to the graphene layer (black line). This is defined as the difference in the signal obtained for injector and detector magnetizations set parallel ($R_{NL,\uparrow \uparrow }$) and the signal for injector and detector set to antiparallel ($R_{NL,\uparrow \downarrow }$). Our model (gray line) which takes into account the finite contact resistance gives a qualitatively good fit to the data for $V_g<30\mathrm{V}$. (c) Hanle spin precession at the Dirac point (see Fig. 2 and text).

Figure 4

Dependence of the spin signal as function of electrode spacing $L$ for spins injected parallel to the graphene plane. (a) The gate voltage $V_g$ is set in such a way that for $L=0.5$, 2 and $4\mu \mathrm{m}$ we have the same value of $R=2.2\times {10}^{-6}\mathrm{m}$ ($n\simeq 5.0\times {10}^{12}{\mathrm{cm}}^{-2}$, $D=0.04{\mathrm{m}}^2{\mathrm{s}}^{-1}$). We obtain a relaxation length of $\lambda =1.5\pm 0.2\mu \mathrm{m}$ which is similar to the length extracted from the spin precession measurements (see Fig. 2). Model calculations for $\lambda =0.7\mu \mathrm{m}$ and $\lambda =5\mu \mathrm{m}$ are shown for comparison. (b) Moving towards the Dirac point using a gate voltage in such a way that we decrease the $R$ value to $1.0\times {10}^{-6}\mathrm{m}$ ($n\simeq 2\times {10}^{12}{\mathrm{cm}}^{-2}$, $D=0.03{\mathrm{m}}^2{\mathrm{s}}^{-1}$) has a strong influence in $\lambda$, as it decreases to $1\mu \mathrm{m}$ and (c) to $0.8\mu \mathrm{m}$ (for $R=5.5\times {10}^{-7}\mathrm{m}$, $n\simeq 1.0\times {10}^{12}{\mathrm{cm}}^{-2}$, $D=0.02{\mathrm{m}}^2{\mathrm{s}}^{-1}$).