Spintronic Quantum Phase Transition in a Graphene/Pb0.24Sn0.76Te Heterostructure with Giant Rashba Spin‐Orbit Coupling

Mechanical stacking of two dissimilar materials often has surprising consequences for heterostructure behavior. In particular, a 2D electron gas (2DEG) is formed in the heterostructure of the topological crystalline insulator Pb0.24Sn0.76Te and graphene due to contact of a polar with a nonpolar surface and the resulting changes in electronic structure needed to avoid polar catastrophe. The spintronic properties of this heterostructure with non‐local spin valve devices are studied. This study observes spin‐momentum locking at lower temperatures that transitions to regular spin channel transport only at ≈40 K. Hanle spin precession measurements show a spin relaxation time as high as 2.18 ns. Density functional theory calculations confirm that the spin‐momentum locking is due to a giant Rashba effect in the material and that the phase transition is a Lifshitz transition. The theoretically predicted Lifshitz transition is further evident in the phase transition‐like behavior in the Landé g‐factor and spin relaxation time.


Introduction
Of the many extraordinary promises of the two-dimensional (2D) materials revolution, perhaps none is as captivating as the idea that materials with widely varying properties can be arbitrarily stacked into heterostructures, regardless of lattice spacing or growth mechanism 1 .With a menu of thousands of 2D materials 2 , designer heterostructures are at hand with such remarkable properties as interlayer excitons in transition metal dichalcogenides 3,4 (TMDs), "magic-angle" superconductivity in graphene and TMDs 5 , and ferromagnetism from non-magnetic constituents 6 .
In many of these systems, heterostructures are more than simply the sum of their parts.Rather than each layer in a stack acting as a discrete entity, unique hybridization effects result in emergent pheonomena 7,8 .One of the most versatile and useful materials is graphene: like tofu, graphene often acquires the flavor of whatever is placed in its proximity 9,10 .
Perhaps the most compelling application of such heterostructures is in next-generation computing devices 11,12,13 .Primary among the many looming roadblocks of current computing paradigms are device bottlenecks in both energy expenditure and the physical limitations of the ubiquitous charge-based CMOS structures 14 .Spin-based "spintronic" devices offer considerably lower energy operation, higher speeds, and greater densities 15 .Because the spin diffusion length is proportional to mobility, a nearly defect-free graphene would seem like an ideal spintronic channel 16 .However, its lack of spin-orbit coupling leaves few options for the spin current control needed for device operation.Combining graphene with a high spin-orbit material such as a topological insulator 17,18 , where the spin-orbit coupling is conveyed by proximity, could be a viable solution.
One recent example of a graphene-based heterostructure with unexpected properties is the graphene/topological crystalline insulator (TCI) system graphene/Pb0.24Sn0.76Te(Gr/PST).While systemic symmetries protect the properties of all topological materials, TCIs are primarily protected by mirror symmetry 19 .Since the TCI has a significantly lower conductivity than graphene, one might expect most of the current to flow through the graphene, resulting in proximitized spin-orbit coupling, similar to other graphene heterostructures 9,10,20 .However, stacking graphene onto the PST breaks the inversion symmetry and destroys the topological state.
Due to excess charge, the structure must either change its stoichiometry or undergo a complete charge reconfiguration-a phenomena known as the polar catastrophe.The charge redistribution needed to avoid polar catastrophe results in the formation of a two-dimensional electron gas (2DEG) at the interface, analogous to the interface 2DEG famously discovered in LaAlO3/SrTiO2 and similar systems 21 .Although the topological order of the PST is destroyed in the heterostructure, electronic structure modifications as a result of electronic reconstruction due to the polar discontinuity at the interface leads to the appearance of a high-mobility 2DEG at the interface.Although this composite may not be a true topological material, it shares many properties with the parent topological PST material, such as high mobility, high spin lifetime, high spin-orbit coupling, and spin-momentum locking.
In this paper, we study the spintronic properties of the 2DEG created at the interface of graphene and PST.We fabricate and measure non-local spin valve (NLSV) devices and discover a low-temperature regime dominated by giant Rashba coupling that enables spin-momentum locking.At higher temperatures, the system transitions to a more typical NLSV device channel, attributable to a Lifshitz transition at about 40 K. Density functional theory (DFT) analysis is used to understand the spin texture and 2DEG behavior.Our DFT calculations reveal a giant Rashba spin-orbit parameter, along with the Lifshitz transition switching mechanism.We also use the Hanle effect to measure spin lifetimes as a function of temperature.We report spin lifetimes as high as 2.18 ns and spin transport persisting up to at least 500 K, properties highly desirable for spintronics applications.We also compare our measurements here with the charge transport measurements performed previously in this system.The robust spin transport and quantum phase transition in this system could be exploited for future low-power high-performance computing devices.

Non-local spin valve measurements
To measure the spintronic behavior of the Gr/PST system, we processed our heterostructures into standard NLSV devices (see methods below).Measurement of a NLSV is essential to demonstrating spin current generation and manipulation.Figure 2 is a summary of the NLSV measurements at various bias and temperature conditions.A sweeping magnetic field, applied along the easy axis of the FM contacts (Figure 2(a, inset)), switches the relative orientations of the FM magnetization at their respective coercive fields.This results in a lower (higher) measured voltage when the contacts are parallel (antiparallel).Room temperature NLSV behavior is shown in Figure 2

Non-local
Hysteresis in the NLSV measurements of a topological or high spin-orbit coupled material is a strong indication of surface state transport.Below 50 K, the NLSV exhibits a strong hysteresis on top of the magnetization switching.This is shown in Fig. 2(c) with the NLSV data at 3 K.
Again, the red (blue) curves are for positive (negative) magnetic field sweeps.The top (bottom) panel is a -0.1 µA (+0.1 µA) bias.Reversing the bias direction, and also the electrons' momentum, reverses the sign of the hysteresis.Similar hysteresis was also observed in both topological materials 22 and in 2DEGs 23 as a result of spin-momentum locking, though the origins of the effect for each channel type is different.For a topological material, there is a linear energy dispersion caused by band inversion where each massless, chiral Fermion has its momentum locked at a right angle to a spin-state that is thus protected by symmetry from backscattering.Our previous theory work on the PST/Gr heterostructure demonstrated that time reversal symmetry is broken here, resulting in the destruction of the topological state in the PST 21 .For a trivial 2DEG with parabolic energy dispersion, spin-momentum locking, a manifestation of spontaneous nonequilibrium spin polarization caused by an electric current, arises due to Rashba spin-orbit coupling (SOC) 24,25 .This is the same effect that allows a measure of control over spin relaxation using an electric field because the spin-splitting energy is proportional to the expectation value of electric field 26 .∆ ℎ decreases quickly for increasing temperature until the spin-momentum locking disappears above 40 K. ∆  also rapidly decreases with increasing temperature until equilibrating at ~150 K, where it remains up to at least 500 K (see Supplement for more details).Based on these findings, as well as the DFT calculations presented below, we posit a Lifshitz phase change that occurs around 40 K.This conclusion is further supported by previous measurements using only charge transport in this heterostructure 21 .In the Supplement, we rule out other mechanisms including a switch from ballistic-to-diffusive transport and simple thermal effects on the transport.

Band Structure Calculations
As the temperature changes, both the Fermi energy and the lattice parameter can also change slightly, driving the Lifshitz transition.Figure 3 shows the calculated band structure for a Graphene/SnTe (111) interface.From the SnTe component, our system is expected to inherit topological Dirac points, which are located at the Γand M-points 27 .Two additional Dirac points at the Γ-point are inherited from the graphene layer.This is a consequence of zone-folding, with both the K and K' points of graphene getting folded to the Γ-point in the Brillouin zone of the heterostructure.As shown in  the carbon pz orbitals.To estimate the strength of the Rashba effect in the heterostructure, the Rashba model was used, in which the SOC splits the parabolic band by shifting the carriers with opposite spin by the momentum kR.In this model, the SOC coupling parameter, αR, is defined as αR=2ER/kR, where ER is the energy difference between the crossing point and the band top or bottom at a particular kR.Near the M point, along M-Γ for the SS1 band we find that kR ≈ 0.18 Å - 1 and ER ≈ 0.13 eV.This gives αR ≈1.5 eV Å, which is a giant value for the Rashba parameter.For comparison, it is about half of the largest published value for the Rashba parameter in the literature for BiTeI. 28The average Fermi velocity is also high (1.5 x 10 5 m/s), being nearly 30% of the Fermi velocity of Bi2Se3. 29 Figure 3 The properties of the surface states SS1 and SS2 play an important role in the generation and registration of spin-polarized currents in the heterostructure.In our experiments, the NLSV device geometry is most sensitive to in-plane spins with Rashba-type spin-momentum locking.As the electronic pockets centered on K (associated with SS2) have high values of out-of-plane spin component Sz, while their in-plane spin texture is not Rashba-like in character, the current-induced spin polarization associated with SS2 is unlikely to be detected.Hence, experimentally, the role of the SS1 states is more important.The SS1 band exhibits Rashba-type spin texture, where the spin lies mostly in-plane, perpendicular to the k-vector.A peculiarity of the SS1 band is that it is flat in a wide region of k-space close to the Fermi level.As seen in Figure 3(c), the SS1 FS changes rapidly for small excursions away from EF.As a result, the FSs associated with this band can easily adjust their area, shape, and topology under the influence of external parameters such as pressure, composition, temperature, magnetic field, and so on.Abrupt modification of the topology of a Fermi surface is the hallmark of a Lifshitz transition.Therefore, we expect external stimuli such as temperature to strongly affect transport phenomena.For example, a large enhancement of thermoelectric efficiency in SnSe over a wide temperature range (10-300 K) was attributed to the pressure-induced Lifshitz transition 30 .Therefore, our calculations suggest that the system is on the verge of different Lifshitz transitions, which can easily occur under different stimuli, such as Pbdoping, Fermi level tuning, and lattice expansion (for example, through temperature modulation).
The number and complexities of FSs indicate that the experimentally observed disappearance of spin-momentum locking for temperatures above ~40 K can be ascribed to one or more effects.One such effect is simply the disappearance of some of the hole FSs, which as

Spin Properties
Another important consideration for any spintronic device is the spin polarization.In Figure 4, we plot the components of spin polarization derived from both the Rashba (black) and standard NLSV (red) channels.For a full discussion on the analysis of each of these components, see the Supplement.Comparing the Rashba polarization to the injected polarization, the Rashba component is a significant portion of the observed spin signal at low temperature before it gradually disappears.
First, we will discuss the low-temperature regime which is dominated by the Rashba effect.
Although the origin of spin-momentum locking for topological materials and 2DEGs is different, the theory to analyze the hysteresis is identical 32 .For both topological materials and 2DEGs and for both ballistic and diffusive transport, the high-and low-voltage states in the hysteresis curve are described by [32][33][34][35] where h is Planck's constant, e is the elementary charge, PFM is the polarization of the ferromagnetic detector contact (~48% for Ni80Fe20) 36 , vF is the Fermi velocity, m * is the effective mass, W is the width of the channel, pRashba is the induced spin polarization due to the Rashba effect, and mu is a unit vector along the direction of magnetization 21 .In previous studies, spontaneous, non-equilibrium spin polarization due to either the Rashba effect or topological surface state spin-momentum locking is distinguishable by having opposite signs in the vector pRashba.However, as many others have noted 22,33,37 , it is difficult to draw any conclusions about the sign of the polarization.We therefore calculate an absolute value of polarization, displayed vs temperature in Figure 4.
where ηf = Eg/kBT, Eg is the system band gap obtained from DFT calculations as a lower bound on the realistic actual band gap and kB is the Boltzmann constant.For an ideal non-degenerate system, Equation (2) reduces to the Einstein relation.The spin diffusion constants calculated from a measurement of mobility in this system at 10 K for PST, graphene, and the heterostructure are 0.172 cm 2 /s, 3.27 cm 2 /s, and 17.2 cm 2 /s, respectively.The shape of the injected polarization in Further spintronic properties of the heterostructure can be extracted from analysis of the Hanle effect, shown in Figure 5. Here, a magnetic field is applied out of plane with respect to the moments of the FM injector/detector contacts as is depicted in schematic in Figure 5(a).This causes the spins in the channel to precess at the Larmor frequency,   =     /ℏ , where g is the Landé g-factor,   is the Bohr magneton, and   is the out-of-plane magnetic field.As the magnetic field is increased, precessional dephasing results in a higher resistance, thus a magnetic where spin is injected into the graphene at x1 and t=0 and is detected at x2. S0 is the spin injection rate and vd is the electron drift velocity (equal to 0 for pure spin currents).Fits to the Hanle data provide a measure of the spin lifetime.The Landé g-factor for graphene is known to be 2. 41 However, for PST, the Landé g-factor is not accurately known 42 and is unknown for the heterostructure 2DEG because it is difficult to separate bulk and surface g-factors in the heterostructure.Therefore, we leave the Landé g-factor as a fitting parameter.Ω.In the low-temperature regime (T<50 K), the VRH model dominates the temperature-dependent conduction.A stronger thermal activation or a two-state system instead of the variable range hopping was expected in this heterostructure due to the observed temperature-dependent phase change.We attribute the behavior to variable range hopping over the relevant temperature scales; however, the data does deviate from the model at higher temperatures.
Returning to a study of the spin dynamics, Figures 5(c) and (d) display the Hanle fit parameters g and s as a function of temperature.Figure 5(c) shows two average g values; for T < 40 K, g~0.42, while for T > 40 K, g~1.7.Higher temperature fits did not converge using the lower temperature g value and vice versa.This is entirely consistent with a Lifshitz phase transition as identified above.Figure 5(d) shows that the extracted spin lifetime vs temperature also has two regimes.In regime I, the spin lifetime is higher, reaching values as large as 2.18 ns.Above 40 K, it transitions to regime II, where the spin lifetimes are markedly lower, only about 500 ps.The extracted spin lifetime vs temperature here was for one device, though behavior was similar for all devices tested (additional information is provided in the Supplement).Calculated as discussed above, the spin diffusion length   = 13.1 μm for s=2.18ns.In our previous work, we used electron-electron interactions to estimate an effective spin-orbit (SO) interaction of 4.5 meV 21 .
Using the values from the Hanle measurement here, we can also determine an effective SO interaction 46 .For the assumed D'yakonov-Perel' spin relaxation mechanism,    = h 2 / (4 Δ SO 2   ), give ΔSO = 0.206 meV.This value is an order of magnitude lower than our previous estimated interaction, which could be a result of interference between the Rashba and injection/extraction spin channels.

Conclusion
In conclusion, we demonstrated high-quality spin transport in a PST/Gr structure along with a temperature dependent Lifshitz transition.We observed spin-momentum locking in the non-local spin valve measurements due to a giant Rashba SOC, which we further calculated using DFT.Spin transport survives to at least 500 K and the devices display a long spin lifetime, long diffusion length, and high spin polarization efficiency as measured through Hanle effect measurements.The observation of a Lifshitz quantum phase transition in this heterostructure could be incorporated as the switching mechanism in a future ultra-low power computing device.Further, as results of the polar catastrophe are general for any polar/non-polar heterostructure, we predict that additional heterostructures with remarkable spintronic properties using other novel materials are imminently available and should be studied in the future.
TCI behavior, as discussed in depth elsewhere 21,47 .Graphene was grown by low pressure (5-50 mTorr) chemical vapor deposition on copper foils at 1030 C under flowing H2 and CH4 gas.After growth, the Cu substrates were etched and transferred to the PbSnTe using a wet process 48 .

Device Fabrication
To fabricate the non-local spin valve devices, we first utilized optical lithography with Shipley S1813 photoresist to pattern vias for electrical contacts on the PST followed by electron beam deposition of Ti/Au (5 nm/35nm) and lift-off in acetone.PST mesas were then defined by a second lithography step followed by ion milling in Ar/H2 plasma and cleaning in an acetone/ultrasonic bath.A subsequent O2 plasma descum removed any remaining photoresist residue.Two layers of graphene were then transferred on top of the PST mesas.One layer of graphene creates the graphene/PST heterostructure transport channel, while the second graphene layer is used in a tunnel barrier (TB) contact.Another lithography step using polymethyl methacrylate (PMMA) and deep-UV exposure followed by O2 plasma shaped the graphene into mesas.PMMA and deep-UV lithography are used here to minimize chemical contamination of the graphene.Semiconductor NLSV devices require a TB to match the conductivities between the metallic spin injection/detection contacts and the semiconductor channel 49 .We fabricated devices with a TB consisting of fluorographene/MgO.Vias for the TB contacts are defined by optical lithography using Lift Off Resist (LOR5A) and Shipley 1813, particularly designed to minimize residue.The graphene inside the vias is fluorinated by exposure to a XeF2 gas 38 , which acts to both decouple the two graphene layers and selectively transform the top layer into an insulator.
Immediately after fluorination, electron beam deposition is used to deposit MgO/Ni80Fe20 (Py)/Au (1.5 nm/30 nm/10 nm).An additional optical lithography/deposition step is performed to define Ti/Au (10 nm/40 nm) top contacts to ensure good electrical connection.Finally, the devices are fluorinated a second time to ensure that the top layer of graphene is completely insulating.
Additional characterization details can be found the Supplement.

Measurement Methods
NLSV and Hanle effect measurements are performed in a cryogen-free variable temperature cryostat set upon a rotating platform and centered between the poles of a 1 T electromagnet.From previous work on simultaneously fabricated Hall bar devices, measured

Characterization of Tunnel Barriers
In this section we characterize the tunnel barriers (TBs) for the non-local spin valve device (NLSV) in the main text.Usually, a local geometry is used to study spin-momentum locking effects [2][3][4][5] due to rapid spin relaxation and owing to rapid momentum relaxation and short spin diffusion lengths in most topological materials.To measure spin diffusion through a channel, the detection contacts should be on the order of the spin diffusion length.For topological materials, it was demonstrated that spin-momentum locking could be measured in the non-local geometry because even though the spin injected into the bulk and on the top surface relaxes quickly, the induced spin polarization on the bottom surface remains detectable.The tunnel barrier contact is inherently surface sensitive; however, the creation of the 2DEG channel may greatly increase the spin diffusion length to be greater than the PST alone.The spin diffusion constants calculated from a measurement of mobility in this system at 10 K for PbSnTe, graphene, and the heterostructure are 0.172 cm 2 /s, 3.27 cm 2 /s, and 17.2 cm 2 /s, respectively.Moreover, spin-momentum locking was previously measured in such a geometry owning to surface effects away from the diffusion through the bulk channel 6- 8 .In any case, a non-local measurement through tunnel barrier contacts is inherently surface sensitive, although in our heterostructure we cannot disregard convolution of surface and PST bulk effects with two conductive bands crossing the Fermi energy and participating in the 2DEG transport.

Additional Density Functional Theory Details
To simulate the experimental heterostructures, we used a slab-supercell approach with a vacuum gap of 25 Å along the z direction between the replica slabs.The basic (undoped) heterostructure was constructed by placing √3x√3 graphene supercell on a 1x1 cell of Teterminated (111)-SnTe thin film, the bottom of which is passivated by hydrogen (Fig. S7).The effect of Pb doping on this basic system was modeled by replacing some Sn atoms with Pb atoms.As discussed in the main text, the most important feature of the band structure in Fig. S8 is the presence of two surface bands crossing the Fermi level.These bands are labeled SS1 (green) and SS2 (orange).Both SS1 and SS2 are mainly localized on the upper graphene side and are characterized by both positive and negative effective masses, depending on k.In the vicinity of the Γ point, they hybridize with the carbon pz orbitals.The surface states SS1 and SS2 emerge because the (111)-surface of SnTe is polar.In the [111] direction, the sequence of atomic layers can be viewed as a sequence of charged layers +2, -2 +2, -2 …, if we assign to the Sn and Te ions their formal charges equal to +2 and -2.In our system, the polarity is canceled by introducing one more Te layer on the bottom, which violates the stoichiometry.This new layer leads to the formation of the surface states SS1 and SS2, which provide the needed, positive compensating charge density σext (1 hole per unit cell area) on both sides 17 .The system is metallic because together SS1 and SS2 surface bands must be approximately half-filled.The specific features of SS1 and SS2 can be understood looking at the potential averaged over (xy) plane near the Teterminated surface of SnTe (see Fig. S8(b)).The potential bends near the surface becoming more positive or less binding.This explains why the surface states SS1 and SS2, split off from the uppermost bulk valence bands.On the other hand, the potential gradient near the surface is responsible for the giant spin-splitting between SS1 and SS2 states.

Figure 1 (
a) shows an optical image of a measured Gr/PST spin valve and Figure 1(b) shows a device schematic.A charge current is applied between one outer non-magnetic reference Ti/Au contact and the adjacent inner tunnel barrier (TB)/ferromagnetic (FM) contact (injection), while monitoring the voltage across the other pair of contacts (detection).The spin injection/detection TB/FM contacts are different widths to exploit magnetic shape anisotropy.A spin-polarized charge current is injected from the FM, across the TB contact, and into the heterostructure 2DEG channel.While charge current only flows along the source-drain path, spins simultaneously diffuse in all directions.The pure spin current at the detector contact results in a spin-splitting of the chemical potential that manifests as a measurable voltage.

Figure 1 :
Figure 1: Device image and operation scheme.(a) Optical image of the Gr/PST non-local spin valve.Annotations indicate the materials used.For this device, the left and right graphene/MgO/Py/Au tunnel barrier contacts are 3 μm and 0.5 μm wide, respectively, and are separated by 1.5 μm.The channel width is 25 μm for all devices measured.(b) Device schematic of the non-local spin valve showing operation and externally applied magnetic field.As indicated, the external magnetic field is applied in-plane with the ferromagnetic contacts for non-local spin valve measurements and out-of-plane with the ferromagnetic contacts for Hanle effect measurements.
(a).The red (blue) curves are for positive (negative) magnetic field sweep directions, and the solid (dotted) lines show data a bias current of +10 µA (-10 µA), accounting for both spin extraction and injection.This is typical behavior for a NLSV with the difference in peak position coming from the difference in switching field of the injection/detection contacts.Switching peaks were observed up to 500 K (Figure2(b)), the limit of our measurement capabilities, attesting to the robustness of the heterostructure and the device.A full set of temperature-and bias-dependent NLSV measurements are presented in the Supplement.

Figure 2 :
Figure 2: Non-local spin valve device operation.Red (blue) lines are for magnetic field sweeping negative to positive (positive to negative).A constant background resistance is subtracted for all plots.(a) Non-local resistance vs magnetic field at 300 K for +/-10 μA.Inset shows a device schematic with a sweeping in-plane magnetic field.(b) Non-local resistance vs magnetic field at 500 K for +10 μA, demonstrating operation of the spin valve up to high temperature and the limits of our measurement equipment.(c) Non-local resistance vs magnetic field for +/-0.1 μA at 3 K showing a switch in the sign of the hysteresis with bias polarity.(d) Non-local resistance of the spin injection/extraction switching peak, "Non-local," and hysteretic resistance change due to Rashba spin-orbit coupling, "Rashba," as a function of temperature.The inset shows the total nonlocal resistance in the spin valve vs magnetic field at 10 K and with a -10 μA bias and defines the relevant system resistances for analysis.

Figure 2 (
Figure 2(d) summarizes of the temperature dependence of both the non-local resistance ∆  and the hysteretic resistance change due to Rashba spin-orbit coupling ∆ ℎ .The inset shows a magnetic field sweep at 10 K, with the relevant resistances defined in the annotation.

Figure 3 (
a), all the Dirac cones at the Γand M-points are gapped due to lowering of symmetry in the heterostructure.In the presence of graphene, the C3v point symmetry of SnTe is reduced to C3-symmetry, which removes the topological protection of the surface states associated with SnTe, opening the gap.Meanwhile, the SnTe film breaks the sublattice symmetry in graphene, opening gaps in the Dirac cones associated with graphene at Г.

Figure 3 .
Figure 3. (a) Calculated band structure along the high symmetry lines for the heterostructure.Shown in orange (SS1) and green (SS2) are the two surface bands crossing the Fermi level.(b) The Fermi surfaces corresponding to the band structure presented in (a), including spin texture.(c) Constant energy contours for the energies below the Fermi level corresponding to the band structure presented in (a).Electron and hole pockets are noted by e and h, respectively.
(b), we plot the spin-resolved Fermi surfaces (FSs) at the Fermi energy E=EF.The magnitude of the in-plane component of spin is represented by the length of the arrows, and the color of the arrows gives the z-components of the spins.Except for the K-centered FSs, all other FSs have a small Sz component and exhibit Rashba-type spin texture, with the spin being perpendicular to the k-vector.In contrast, the K-centered ellipses have a significant Sz component.Moreover, their in-plane spin component tends to be parallel or antiparallel to k.Because of their large areas, the electron ellipse around K and the hole star around Γ should produce the dominant contribution to the electric current induced by an applied electric field.The Supplement includes calculations for the case when some of the atoms are replaced by Pb, as is the case for our exact structure.A cross section of the band structure at various energies below the Fermi level are plotted in Figure 3(c).Additional bandstructure cross sections and those above the Fermi level are found in the Supplement.There are multiple Fermi level crossings at different k-points, resulting in many FSs.The large electron ellipse around K, denoted by e in Figure 3(c), comes from the SS2 band.There are three hole FSs, denoted by h in Figure 3(c), including a very large star centered on Γ, all related to the SS1 band.There are also two hexagonal FSs around Γ, which are derived from graphene states.
discussed have significant contribution to the spin-polarized current.Another effect is due to high anisotropy of the FSs: a measured change in resistivity, ΔRRashba, should strongly depend on the orientation of the spin-induced current relative to the heterostructure.With increasing temperature, the anisotropy of the Fermi surfaces may change in such a way that the spin transfer by the drift current becomes inefficient.A third possible effect may be related to the charge redistribution between the pz orbitals of carbon and the p orbitals of Sn and Te.At polar surfaces, the compensating charge density tends to persist and the carbon pz orbitals can be filled or emptied at the expense of Sn and Te p orbitals31 .Since the spin splitting of the graphene Dirac cone (whether n-or p-doped) leads to two Fermi surfaces with opposite spin directions at each momentum, the corresponding current-induced spin densities nearly cancel each other, resulting in the observed behavior.

Figure 4 :
Figure 4: Spin polarization efficiency.Lower bound of the temperature-dependent spin polarization efficiency due to the Rashba spin-momentum locking (red) and of the non-local spin valve (black).Experimental error bars are within the size of the depicted squares at each point.

Figure 2 (Figure 5 :
Figure 2(d) is due to a two-level measured spin relaxation time, discussed below.Comparing the Rashba polarization to the injected polarization, the Rashba component is a significant portion of the observed spin signal at low temperature before it gradually disappears.
field sweep will trace out a pseudo-Lorentzian line shape proportional to solutions to the spin diffusion equation described by the steady state spin current40

Figure 5 (
Figure 5(a) is a plot of non-local resistance as a function of applied out-of-plane field for a device at a temperature of 40 K.The pseudo-Lorentzian behavior is a signature of the Hanle effect, i.e., precessional dephasing of the spin current at the detection contact.The data are in black while the red dashed line shows the fit to Equation 3. A second-order polynomial background was subtracted from the raw Hanle data to remove trivial magnetoresistance.Figure 5(b) summarizes the change in non-local resistance (RHanle, as defined in Figure 5(a)) as a function of temperature.Although the NLSV signal persists to at least 500 K (Figure 2(b)), the Hanle signal disappears above approximately 75 K.This behavior has been observed in many previously studied spin valves 43,44 , with the NLSV signal far outlasting the Hanle signal, and has been attributed to thermal fluctuations and phonon scattering.Insight into the momentum scattering behavior is learned from temperature dependent resistivity measurements.The inset of Figure 5(b) shows the four-terminal resistivity (ρ) vs temperature measured in this spin valve device along with a fit to a model that includes both thermal activation (TA) and Mott variable range hopping (VRH), 45 σ = σΤΑ + σVRH.σTA = A exp[-Ea/2kBT] and σVRH = B exp[-(T/T0) 1/3 ].Fitting parameters are the thermal activation energy Ea= 0.765 eV, temperature constant T0=0.573K, and relative conductivities A = 0.439 Ω and B = 221 Fig.S2shows NLSV measurements ranging from 3 K up to 300 K with high-temperature data shown in Fig.S3ranging from 350 K up to 500 K. Data for Figs.S2 and S3were taken with applied currents of -10 µA and +10 µA, respectively.For measurements below ~50 K, a hysteretic resistance due to Rashba spin-orbit coupling is observed in addition to the non-local resistance switching peak (see Fig.2in the main text for more details).As temperatures approach the transition temperature (near 40 K), the two changes in resistance overlap and the switching is more difficult to extract.At 500 K (Fig.S3(d)), the highest temperature reached by our measurement equipment, the two NLSV peaks overlap, indicating that the ferromagnetic layer (Py) has softened.

Figure S2 .
Figure S2.Non-local spin valve measurements.Red (blue) lines are for magnetic field sweeps from negative to positive (positive to negative).Data are shown from 3 K up to 300 K, taken at an applied current of -10 µA.A constant background resistance was subtracted for all plots.

Figure S3 .
Figure S3.High-temperature non-local spin valve.Red (blue) lines are for magnetic field sweeps from negative to positive (positive to negative).Data are shown from 350 K up to 500 K, taken at an applied current of +10 µA.A constant background resistance was subtracted for all plots.

Figure S4 .
Figure S4.Hanle data.Hanle measurements at (a) low and (b) high temperatures taken with an applied current of -10 µA.A background subtraction was applied to remove background magnetoresistance and temperature effects.

Figure S5 .
Figure S5.Bias dependence.Bias dependence of the non-local spin valve and Hanle measurements taken at 3 K and 50 K.(a) Change in resistance peaks for the non-local spin valve measurement configuration taken at 3 K where the orange and black points are the non-local and Rashba hysteretic changes in resistance, respectively.(b) Comparison of ∆RNL at 50 K for both Hanle and NLSV device configurations.(c) Dependence of spin lifetime on bias current at 50 K.

Fig. S5
Fig.S5shows that there is no clear bias dependence of the spin valve either above(50 K)   or below (3 K) the transition temperature.At low temperatures, external noise and effects mask the Hanle signal making it impossible to fit to determine the spin lifetime and spin polarization efficiency values therefore data are shown only for negative applied currents (see Fig.S5 (a-c).)Asshown in Fig.S6, from the change in Rashba resistance and the spin lifetime, there is a slight increase in efficiency for low bias, similar to other studies.

Figure S6 .
Figure S6.Spin polarization efficiency bias dependence.(a) Rashba hysteretic spin polarization efficiency across positive and negative bias and (b) injected spin polarization efficiency across positive and negative bias at 50 K.

The
SnTe film was chosen to contain 35 atomic layers (17 Sn layers + 18 Te layers).Only the top four and bottom four layers of SnTe, as well as graphene and hydrogen layers, were allowed to relax fully, whereas the remaining 27 layers of SnTe were constrained to bulk positions to avoid large interlayer oscillations due to finite size effects.Calculations were performed as discussed in the main text and the methods.

Figure S7 .
Figure S7.Side (a) and top (b) views of the heterostructure consisting of a √3x√3 supercell of graphene on 1x1 (111)-SnTe thin film, which is terminated in Te.The lower surface of the composite is decorated with hydrogen, passivating the Te-atoms on that surface.Also shown are the x and y axes in real space and the ГM and ГK directions in the reciprocal space.

Figure S8 .
Figure S8.(a) Calculated band structure along the high symmetry lines for the heterostructure shown in Figure S1.Shown in orange and green (labeled as SS1 and SS2) are the two surface bands crossing the Fermi level.(b) The potential averaged over (xy) plane near the Te-terminated surface of SnTe, explaining the origin of the surface states SS1 and SS2 (see the corresponding text).

Figure S9 .
Figure S9.The Fermi surfaces corresponding to the band structure presented in Figure S8, without (a) and with (b) spin texture.

Figure S10 .
Figure S10.Same as in Figure S9, but for the heterostructure where the SnTe film is replaced by PbTe film.

Figure S11 .
Figure S11.Constant energy contours for the energies below the Fermi level corresponding to the band structure presented in Figure S8.

Figure S12 .
Figure S12.Same as in Figure S11, but for the energies above the Fermi level.