Mimicing the Kane-Mele type spin orbit interaction by spin-flexual phonon coupling in graphene devices

On the efforts of enhancing the spin orbit interaction (SOI) of graphene for seeking the dissipationless quantum spin Hall devices, unique Kane-Mele type SOI and high mobility samples are desired. However, common external decoration often introduces extrinsic Rashba-type SOI and simultaneous impurity scattering. Here we show, by the EDTA-Dy molecule decorating, the Kane-Mele type SOI is mimicked with even improved carrier mobility. It is evidenced by the suppressed weak localization at equal carrier densities and simultaneous Elliot-Yafet spin relaxation. The extracted spin scattering time is monotonically dependent on the carrier elastic scattering time, where the Elliot-Yafet plot gives the interaction strength of 3.3 meV. Improved quantum Hall plateaus can be even seen after the external operation. This is attributed to the spin-flexural phonon coupling induced by the enhanced graphene ripples, as revealed by the in-plane magnetotransport measurement.

However, in order to achieve the QSH, the SOI type has to be the Kane-Mele (KM) desired type with the Hamiltonian of ~τzzsz, because the common Rashba SOI (~⃗ • ( ⃗ × ⃗)) may break the QSH transport by mixing different branches of spin channels [16]. This is reasonable since the KM type SOI is originated from the symmetry of the graphene honeycomb lattice. Furthermore, the above external decoration may simultaneously suppress the mobility of graphene and subsequently destroy rather fragile QSH states.
Here we show that the KM type SOI can be mimicked by a spin-flexual phonon coupling picture, which is implemented by decorating the graphene devices using Na[Dy(EDTA)(H2O)3]·5H2O (EDTA-Dy) molecules. The Elliot-Yafet (EY) plot of spin scattering time and carrier elastic scattering time gives the SOI strength of 3.3 meV. The graphene devices even present improved half-integer quantum Hall transport after decorated.

Experimental
The graphene devices (figure 1(a)) were fabricated on a single crystal grain of chemical vapor deposition grown graphene sheet, which were transferred onto the Si substrate with a 300-nm-thick SiO2 layer. Standard e-beam lithography and e-beam evaporation was used to make the Hall bar devices. The EDTA-Dy complex is prepared according the improved methods [29] (see Supporting Materials, Note I).
The influence of EDTA-Dy coating on graphene was investigated. Transport measurements were done in Cryomagnetics' C-Mag system with the low-frequency lock-in technique. Pristine graphene was measured in the system and then it was processed by a little drop of EDTA-Dy solution. The processed graphene was kept in fuming cupboard for several minutes to dry off. EDTA-Dy decorated graphene was then recooled and measured in the system. A vector magnet is used in the in-plane field measurement.  Interestingly, the performances of the graphene devices are improved after the EDTA-Dy decoration, which is also seen in graphene grafted with Pt-porphyrins [30].

Improved transport after the decoration of EDTA-Dy
The carrier mobility unexpectedly changes from 1919 cm 2 V -1 s -1 to 3226 cm 2 V -1 s -1 after decoration. And the quantum Hall effect (QHE) is observed at 12 T and 2 K, as shown in figure 1(d), with the Hall conductivity xy goes quantized and the longitudinal resistivity ρxx approaches zero. The 4(n + 1/2) plateau series is the characteristic half-integer QHE of monolayer graphene [31,32].

Suppressed WL and selectively-enhanced KM-type SOI
The SOI alters the interference in a pair of time-reversal loops, which causes the phase shift and quantum modification to the low-field magnetoresistance (MR) [33,34]. For graphene, E. McCann's weak localization (WL) theory describes the low-field WL correction with a KM-type SOI to the magnetoconductivity (MC) as below [35,36]  The SOI strength increases with rising temperature both in pristine and decorated graphene as shown in figure 2(d). It is believed that the SOI (ΔD0) which generates the Dirac cone gap will not change with temperature. We therefore extrapolate the data along the linear fitting and expect ΔD0 from the intercepts at zero temperature. It is found that ΔD0 is quite small for pristine graphene, while it achieves 3.3 meV after the EDTA-Dy decoration. Such ΔD0 has been large enough to accommodate the QSH states. The temperature dependent SOI is attributed to the electronic scattering by the impurity centers [38,39].

Decoration induced ripples revealed by in-plane magnetic field
As stated above, both the suppressed WL and successful EY plot reveals the selectively enhanced KM-type SOI while decorating the graphene by EDTA-Dy. Such decoration should induce more scattering, but the present devices even show improved quantum transport, which forms the main question of this work. We propose the ripples caused by the stretching EDTA-Dy coating might contribute to the pseudo magnetic field (gauge field), which prefers the KM-type SOI in graphene.
Such ripples are revealed by the atomic force microscopy (AFM) as shown in figure   3(a). The root-mean-square (RMS) roughness of pristine graphene is 0.66 nm. After EDTA-Dy decoration, the RMS roughness increases to 1.64 nm. Similar ripples have been described in PTSA coated graphene [40]. It can be understood by the interfacial stretching due to the different temperature dependent shrinking of graphene and the organic film. Such ripples even detach some part of graphene from the substrates and suppress the charge scattering from the substrates [41,42]. To where θ is the angle between the current flow and B||, as shown in the inset of figure   3(b). Z is RMS height and R is correlation length. The graphene before and after EDTA-Dy decoration is measured with θ ≈ 10˚ and 80˚, which gives Z 2 /R ≈ 0.14 nm and 0.67 nm respectively by fitting Eq. (2).
Further measurements have to be carried out to obtain the RMS (Z). Parallel-field -generated magnetic flux through the ripples causes orbital effect, due to which the phase-coherent WL can be suppressed. For this, we design the experiment to measure the WL features driven by a perpendicular field while applying a series of fixed parallel fields.
The WL features driven by a perpendicular field can be formulated as below [44] where −1 = 4 ⊥ /ℏ . In figure 3(c), the WL scattering rates are extracted by Folk et al [43], with Z 2 R = 1.07 nm 3 .

Spin-flexural phonon coupling mode in rippled graphene
Here we propose that the EY spin relaxation can be interpreted by the flexural phonon mode in rippled graphene. This is partially based on the previous study on all symmetry-adapted spin-phonon couplings [45], where the phonons vibration This finally leads to a EY-type spin relaxation time that satisfy accounting for the results in figure 2.
To further study the temperature dependent spin lifetime, we resort to a second quantization form of the phonon field. In this form, the spin-phonon interaction is reduced to where , , ′ = ℏ 2 √ ′ , ω is the phonon dispersion, is the mass density of the Carbon atoms, and is a four-dimensional electron spinor in sublattice and spin space. Evaluating the spin-flip scattering probability and using the Fermi's golden sum rule, the spin relaxation rate can be obtained. For higher temperature region with > ℏ / , which is the experimental case [49], we arrive at where A and B are the second order and first order T dependent coefficient respectively, while C is a constant representing the zero temperature residue contribution [45]. As an estimation, neglecting the dispersion of phonon and replacing the integral by an averaged frequency ω ̅, then A and B can be found to satisfy = 64g 2 k B 2 k F 3 πℏ 2 v F ρ 2 ω ̅ 4 and = 64g 2 k B k F 3 πℏv F ρ 2 ω ̅ 3 . For experimental values, we take ρ = 7.6 ×10 -7 kg•m -2 and ω ̅ = 10 meV. In addition, g is related to the strength of KM-like SOC. From figure 2(d), g can be set to be g ~ 2 eV•Å -2 . Then the parameters can be estimated to be A ~ 2.0×10 -6 K -2 ps -1 and B ~ 2.3×10 -4 K -1 ps -1 .
In figure 4, we fit the experimental

Conclusion
We can selectively enhance the KM-type SOI in graphene by the EDTA-Dy decoration, as evidenced by the suppressed WL and simultaneous EY spin relaxation.
The SOI strength of 3.3 meV is achieved. The QHE can be seen with a 4(n + 1/2) series of plateaus even after decoration. We believe the interfacial stretching and local ripple configuration may achieve the simultaneous external SOI control and transport