Spin relaxation in fluorinated single and bilayer graphene

We present a joint experiment-theory study on the role of fluorine adatoms in spin and momentum scattering of charge carriers in dilute fluorinated graphene and bilayer graphene. The experimental spin-flip and momentum scattering rates and their dependence on the density of fluorine and carrier doping are obtained through weak localization and conductivity measurements, respectively, and suggest the role of fluorine as resonant magnetic impurities. For the estimated fluorine concentration of a few hundred ppm, the observed spin lifetimes are in the range of 1–10 ps. Theoretically, we established tight-binding electronic structures of fluorinated graphene and bilayer graphene by fitting to density-functional supercell calculations and performed a comprehensive analysis of the spin-flip and momentum scattering rates within the same devices, aiming to develop a consistent description of both scattering channels. We find that resonant scattering in graphene is very sensitive to the precise position of the resonance level, as well as to the magnitude of the exchange coupling between itinerant carriers and localized spins. The experimental data point to the presence of weak spin-flip scatterers that, at the same time, relax the electron momentum strongly, nearly preserving the electron-hole symmetry. Such scatterers would exhibit resonance energies much closer to the neutrality point than what density-functional theory predicts in the dilute limit. The inclusion of a magnetic moment on fluorine adatoms allowed us to qualitatively capture the carrier-density dependence of the experimental rates but predicts a greater (weaker) spin (momentum) relaxation rate than the measurements. We discuss possible scenarios that may be responsible for the discrepancies. Our systematic study exposes the complexities involved in accurately capturing the behavior of adatoms on graphene.

Figure 1

Density-dependent dephasing rates versus carrier density ${\tau }_{\varphi }^{-1}\left(n\right)$ on a double-log plot in fluorinated BLG devices W03 (red, $n_{\mathrm{F}}=4.4\times {10}^{12}{\mathrm{cm}}^{-2}$), W02 (blue, $n_{\mathrm{F}}=3.8\times {10}^{12}{\mathrm{cm}}^{-2}$), and W38 (green, $n_{\mathrm{F}}=2.2\times {10}^{12}{\mathrm{cm}}^{-2}$) at $T=1.7\mathrm{K}$. Square symbols are for electrons and triangles are for holes. The gray dashed line corresponds to a power law dependence of $n^{-1}$.

Figure 2

Experimental scattering rate ratio ${\tau }_{\varphi }^{-1}/{\tau }_{\mathrm{m}}^{-1}$ versus carrier density $n$ in fluorinated SLG and BLG. Solid orange triangles correspond to a SLG device (sample A) reported in Ref. [32] with $n_{\mathrm{F}}=5\times {10}^{11}{\mathrm{cm}}^{-2}$. This data was taken on the hole side. The BLG data, on the electron side, are shown in blue (W02), green (W38), and red (W03) squares. Open symbols indicate $T=1.7\mathrm{K}$, while solid symbols show the $T=0$ extrapolation ${\tau }_{\mathrm{sat}}^{-1}$. The gray dashed and dotted lines correspond to a power-law dependence of $n^{-1}$, differing by a factor of two.

Figure 3

Perturbed DOS of fluorinated (blue solid) and hydrogenated (red dashed) graphene structures: Panel (a) corresponds to SLG with impurity concentration (per carbon atom) $\eta =400\mathrm{ppm}$ and panels (b) and (c) to BLG with the d and nd adatom absorption positions, respectively, both with concentration $\eta =200\mathrm{ppm}$. The high concentration values are chosen for better visibility. The black dashed lines on the background show the unperturbed DOS. Inspecting panels (a)–(c), one observes broad (200–300 meV) resonance levels in fluorinated SLG and BLG with $E_{\mathrm{res}}^{\mathrm{SLG}}=-262\mathrm{meV}, E_{\mathrm{res},\mathrm{d}}^{\mathrm{BLG}}=-253\mathrm{meV}$, and $E_{\mathrm{res},\mathrm{d}}^{\mathrm{BLG}}=-247\mathrm{meV}$, respectively. To the contrary, hydrogen acts in all three cases as a very narrow (width below 10 meV) resonant scatterer with the corresponding resonant energies $E_{\mathrm{res}}^{\mathrm{SLG}}=16\mathrm{meV}, E_{\mathrm{res},\mathrm{d}}^{\mathrm{BLG}}=23\mathrm{meV}$, and $E_{\mathrm{res},\mathrm{nd}}^{\mathrm{BLG}}=19\mathrm{meV}$. The blue (fluorine) and red (hydrogen) vertical arrows indicate the energy positions of the resonances.

Figure 4

Spin-relaxation rate in fluorinated SLG. The experimental data (black symbols) are from sample A of Ref. [32]. The blue solid line shows the spin-relaxation rate based on the fluorine TB model with $J=0.56\mathrm{eV}, {\mathrm{\Sigma }}_{\mathrm{eh}}=64\mathrm{meV}$, and fluorine concentration $\eta =131\mathrm{ppm}$, which is based on experimental estimates. The red dashed line corresponds to the spin-relaxation rate in the alternative scenario of a strong resonant impurity represented by hydrogen, which for the same value of $\eta , J=9\mathrm{meV}$, and ${\mathrm{\Sigma }}_{\mathrm{eh}}=77\mathrm{meV}$, restores the electron-hole-symmetry. Corresponding TB parameters are provided in Appendix pp3.

Figure 5

Spin-relaxation rate in fluorinated BLG. The experimental data (black symbols) are from sample W03 (extrapolated to $T=0$). The blue solid line shows the spin-relaxation rate based on the fluorine TB model with $J=0.47\mathrm{eV}, {\mathrm{\Sigma }}_{\mathrm{eh}}=20\mathrm{meV}$, and fluorine concentration $\eta =572\mathrm{ppm}$, which is based on experimental estimates. The red dashed line corresponds to the spin-relaxation rate in the alternative scenario of a strong resonant impurity represented by hydrogen, with the same values of $\eta$ and ${\mathrm{\Sigma }}_{\mathrm{eh}}$, and $J=-40\mathrm{meV}$. Corresponding TB parameters are provided in Appendix pp3.

Figure 6

Spin to momentum relaxation rate ratio of fluorinated (a) BLG (sample W03, black symbols) and (b) SLG (sample A, black symbols). (a) The fluorine TB model (blue solid) underestimates ${\tau }_{\mathrm{m}}^{-1}$ by a factor of 40, while the alternative scenario of a strong resonant impurity, here hydrogen, (red dashed) underestimates the rate by a factor of 6. (b) The alternative scenario of a strong resonant impurity, here hydrogen, (red dashed) underestimates ${\tau }_{\mathrm{m}}^{-1}$ by a factor of 7.

Figure 7

(a) Measured $\sigma$ (${\mathrm{V}}_g$) of SLG sample A (black) and B (red) from Ref. [32]. $n_{\mathrm{F}}=5\times {10}^{11}{\mathrm{cm}}^{-2}$ in sample A and $2.2\times {10}^{12}{\mathrm{cm}}^{-2}$ in sample B. (b) $\sigma \left(n\right)$ of BLG sample W38 (olive), W02 (cyan) and W03 (magenta) from Ref. [34]. $n_{\mathrm{F}}=2.2$, 3.8, and $4.4\times {10}^{12}{\mathrm{cm}}^{-2}$, respectively. The same three samples are used in the current study.

Figure 8

(a) Sheet conductance ${\sigma }_s$ versus $B$ for BLG sample W03 measured at electron density $n=6\times {10}^{12}{\mathrm{cm}}^{-2}$. From bottom to top: $T=1.6$, 2.5, 3.5, 5.0, 7.5, 10 K. Dashed lines are fits to the WL expression in BLG of Ref. [42]. (b) Dephasing rate ${\tau }_{\varphi }^{-1}$ in sample W03 as a function of temperature at varying electron densities $n$. From bottom to top: $n=13$, 9.5, 7.5, 6.0, $5.0\times {10}^{12}{\mathrm{cm}}^{-2}$. The table presents the extracted coefficient $\alpha$ in Eq. (A2).

Figure 9

Experimental effective mass of electrons and holes in BLG using tight-binding parameters ${\gamma }_0=3.43\mathrm{eV}, {\gamma }_1=0.40\mathrm{eV}, {\gamma }_4=0.216\mathrm{eV}, {\gamma }_3=0, \mathrm{\Delta }=0.018\mathrm{eV}$ as obtained in Refs. [51, 52].

Figure 10

DFT (black dotted) and TB (blue solid) calculated electronic band structure of a $10\times 10$ supercell of fluorinated SLG graphene. The TB parameters are $\omega =5.5\mathrm{eV}$ and $\varepsilon =-2.2\mathrm{eV}$.

Figure 11

DFT (black dotted) and TB (blue solid) calculated electronic band structure of a $7\times 7$ supercell of BLG with one fluorine adatom in the dimer (left) and nondimer (right) adsorption position on the top layer. The TB parameters are ${\omega }_{\mathrm{d}}=7.0\mathrm{eV}$ and ${\varepsilon }_{\mathrm{d}}=-2.5\mathrm{eV}$ for the dimer configuration and ${\omega }_{\mathrm{nd}}=8.0\mathrm{eV}, {\varepsilon }_{\mathrm{nd}}=-3.0\mathrm{eV}$ for the nondimer configuration.

Figure 12

DFT (black dotted) and TB (blue solid) calculated electronic band structure of a $7\times 7$ supercell of BLG with one fluorine adatom in the dimer (left) and nondimer (right) adsorption position on the top layer. In addition to the TB parameters ${\omega }_{\mathrm{d}}=7.0\mathrm{eV}$ and ${\varepsilon }_{\mathrm{d}}=-2.5\mathrm{eV}$ for the dimer configuration, and ${\omega }_{\mathrm{nd}}=8.0\mathrm{eV}, {\varepsilon }_{\mathrm{nd}}=-3.0\mathrm{eV}$ for the nondimer configuration, a finite potential offset of $U=0.16\mathrm{eV}$ is assigned to the top layer of BLG in the TB calculation to account for charging effects.

Figure 13

Spin to momentum relaxation-rate ratio of fluorinated BLG. Here we consider fluorine as a charged impurity; this reduces the discrepancy between the fluorine TB model and the experimental data (black symbols, sample W03) to a factor of 15.