Ultralong 100 ns Spin Relaxation Time in Graphite at Room Temperature

Graphite has been intensively studied, yet its electron spins dynamics remains an unresolved problem even 70 years after the first experiments. The central quantities, the longitudinal ($T_1$) and transverse ($T_2$) relaxation times were postulated to be equal, mirroring standard metals, but $T_1$ has never been measured for graphite. Here, based on a detailed band structure calculation including spin-orbit coupling, we predict an unexpected behavior of the relaxation times. We find, based on saturation ESR measurements, that $T_1$ is markedly different from $T_2$. Spins injected with perpendicular polarization with respect to the graphene plane have an extraordinarily long lifetime of $100$ ns at room temperature. This is ten times more than in the best graphene samples. The spin diffusion length across graphite planes is thus expected to be ultralong, on the scale of $\sim 70~\mu$m, suggesting that thin films of graphite -- or multilayer AB graphene stacks -- can be excellent platforms for spintronics applications compatible with 2D van der Waals technologies. Finally, we provide a qualitative account of the observed spin relaxation based on the anisotropic spin admixture of the Bloch states in graphite obtained from density functional theory calculations.

Graphite has been intensively studied, yet its electron spins dynamics remains an unresolved problem even 70 years after the first experiments. The central quantities, the longitudinal (T 1 ) and transverse (T 2 ) relaxation times were postulated to be equal, mirroring standard metals, but T 1 has never been measured for graphite. Here, based on a detailed band structure calculation including spin-orbit coupling, we predict an unexpected behavior of the relaxation times. We find, based on saturation ESR measurements, that T 1 is markedly different from T 2 . Spins injected with perpendicular polarization with respect to the graphene plane have an extraordinarily long lifetime of 100 ns at room temperature. This is ten times more than in the best graphene samples. The spin diffusion length across graphite planes is thus expected to be ultralong, on the scale of ∼ 70 µm, suggesting that thin films of graphite -or multilayer AB graphene stacks -can be excellent platforms for spintronics applications compatible with 2D van der Waals technologies. Finally, we provide a qualitative account of the observed spin relaxation based on the anisotropic spin admixture of the Bloch states in graphite obtained from density functional theory calculations.
Spintronic devices require materials with a suitably long spin-relaxation time, τ s . Carbon nanomaterials, such as graphite intercalated compounds 1 , graphene 2 , fullerenes 3 , and carbon nanotubes 4 , have been considered [5][6][7] for spintronics [8][9][10] , as small spin-orbit coupling (SOC) systems with low concentration of magnetic 13 C nuclei which contribute to a long τ s . However, experimental data and the theory of spin-relaxation in carbon based materials face critical open questions. Chiefly, the absolute value of τ s in graphene is debated with values ranging from 100 ps to 12 ns [11][12][13][14][15][16] , and theoretical investigations suggest an extrinsic origin of the measured short τ s values 17 .
A remarkably simple example of an anisotropic carbonbased material is graphite, which, while being one of the most extensively studied crystalline materials, still holds several puzzles. Specifically, the spin-relaxation, its anisotropy, and the g-factor are not yet understood in graphite, and this represents a 70-year-old challenge. As early as 1953, the first spin spectroscopic study of graphite 42,43 used conduction electron spin resonance (CESR). The CESR linewidth, ∆B, yields directly the spin-decoherence time: T 2 = (γ∆B) −1 , where γ/2π ≈ 28 GHz/T is the electron gyromagnetic ratio (which is related to the g-factor as |γ| = gµ B / ). Magnetic resonance is characterized by two distinct relaxation times, T 1 and T 2 , which denote the relaxation of the components parallel and perpendicular to the external magnetic field, respectively 44 . In zero magnetic field, T 1 = T 2 = τ s holds and the latter parameter is measured in spin-injected transport studies.
The ESR linewidth and g-factor have a peculiar anisotropy in graphite. ∆B is about twice as large for a magnetic field perpendicular to the graphene layers (denoted as ∆B ⊥ ) as compared to when the magnetic field is in the (a, b) plane (denoted as ∆B ). The respective g-factors are also strongly anisotropic: g ⊥ is considerably shifted with respect to the free-electron value of g 0 = 2.0023, and is temperaturedependent, whereas g is barely shifted and is temperature independent 45,46 .
Although several explanations have been proposed [45][46][47][48] , no consistent picture has emerged yet for these anomalous findings in graphite. Nevertheless, understanding the spinrelaxation mechanism would be important for the advancement of spin-relaxation theory in general, but especially for spintronics applications of mono-or few-layer graphene.
We unravel the anomalous spin relaxation in graphite by studying the details of the spin-orbit coupling, its dependence over the Fermi surface, and its anisotropy. We find that the SOC is strongly anisotropic in graphite due to the symmetry 47 : the (pseudospin) spin-orbit field is oriented along z. This results in a significant anisotropy of the electron spin dynamics, spin-relaxation time, and a giant g-factor anisotropy. We demonstrate that the ESR data is compatible with the SOC anisotropy scenario. Remarkably, the theory predicts an ultralong spin-relaxation time for spins that are polarized perpendicular to the graphite (a, b) plane. Indeed, saturated ESR experiments reveal a spin-relaxation time longer than 100 ns at room temperature for this geometry. These findings qualify graphite thin film as a strong candidate for spintronics technology.

I. THEORETICAL PREDICTIONS
We performed first-principles calculations of the electronic structure of graphite in the presence of spin-orbit coupling. Figure 1. shows the band dispersions near the K point of the Brillouin zone (BZ). We limit our qualitative considerations to the shown energy scale, which corresponds to the quasiparticle energy smearing up to room temperature. The full calculated band structure along high symmetry lines is presented in the Supporting Information (SI).
In Elliott-Yafet's theory of spin relaxation 49,50 , which is dominant in centrosymmetric metals such as graphite, spinrelaxation is due to the mixing of the otherwise pure spin up/down states 9,51-55 . For a selected spin quantization axis, the two degenerate Bloch states can be characterized as up, |↑ , or down, |↓ , in the absence of SOC. Due to spin-orbit coupling, the spin up/down states have a (typically small) admixture of the Pauli spin down/up spinors: Here | ↑ , and | ↓ are the Bloch states in the presence of the SOC. The spin-flip probability is proportional to b 2 k . Thanks to our high precision calculation of the dispersion relation, we could determine b 2 k near the Fermi level in Fig.  1. For the electron spin quantized along the c-axis, the spin admixture probability is rather small, less than 10 −6 , without a significant momentum dependence. However, b 2 k exhibits peaks at the band crossings (in fact, those are anticrossings as SOC opens a gap of 24 µeV such as in graphene 56 ) when the spin quantization axis is in the plane. Then the spin admixture probabilities are orders of magnitude higher than elsewhere in the Brillouin zone. In fact, we find here so-called spin hotspots 51 , similar to what happens in monolayer graphene at the Dirac point 57 . Fig. 1. reveals that the dominant contribution to the spin admixture comes from the momenta along K → Γ.
While calculating Fermi-level averages of b 2 k is beyond the scope of the present work (mainly due to the tiny Fermi surface of graphite and the presence of the spin hot-spots), our calculation suggests that the spin-relaxation rate in an in-plane magnetic field is expected to be at least an order of magnitude faster than in a magnetic field parallel to the c-axis.
To relay on the spin-relaxation rate to the ESR line, we evoke the theory of Yafet 50 developed for the case of an anisotropic SOC. For graphite, the major features could be nicely followed in Fig. 2. He argued that both the T 1 and T 2 relaxation times are caused by fluctuating magnetic fields δB = (δB a , δB b , δB c ) due to the SOC. Fluctuating fields along a given direction give rise to spin relaxation of a spin component perpendicular to them. (A more recent and rigorous derivation of the ESR relaxation times is found in Ref. 58). Following Yafet 50 , we introduce δB 2 ⊥ and δB 2 for the squared magnitude of the fluctuating fields along the crystalline c axis, and in the (a, b) plane, respectively. The results for the corresponding spin-relaxation rates are: This is illustrated in Fig. 2. E.g., for B ⊥ , spins precess around the c axis and T 1 is caused entirely by δB (Eq. (3)), however T 2 is caused by both δB and δB ⊥ fluctuating fields as these SOC fields are perpendicular to spin component which is in the plane (Eq. (4)). For the B orientation, the spins precess in the a − c plane thus T 1 is caused by both the δB and δB ⊥ which also yields T 2,⊥ = T 1, (Eqs. (4) and (5)). Yafet discussed the case of the extreme uniaxial anisotropy, i.e., when δB 2 = 0 while δB 2 ⊥ is finite, which is in fact, due to symmetry considerations, the theoretical prediction for graphene (Ref. 59). We suppose that this is the case for graphite, as well, which we check experimentally. It is interesting to notice, that somewhat counterintuitively, the extreme anisotropy corresponds to just a factor of 2 anisotropy in the ESR linewidth as (T 2 ) −1 ⊥ / (T 2 ) −1 = ∆B ⊥ /∆B = 2 (see Eqs. (4) and (6)).

II. RESULTS AND DISCUSSION
A. Anisotropy of the ESR linewidth and g-factor Fig. 3 shows the temperature dependence of the ESR linewidth for the two major magnetic field orientations in highly-oriented pyrolytic graphite (HOPG) samples. Comparing the data with previous measurements 43,46,60-62 , we find good agreement for the 50 − 300 K temperature range. Due to the finite penetration depth of microwaves, the ESR lineshape is asymmetric. This is taken into account by fitting the ESR spectra to the so-called Dysonian lines 63 that are well approximated by a mixture of absorption and dispersion components of a Lorentzian curve 64 . Additional data including typical ESR spectra, the g-factor, the temperature-dependent ESR intensity, angular dependence of the ESR spectra, and linewidth are provided in the SI.
Above 50 K, the linewidth increases with decreasing temperature for both orientations. Linewidth data for B ⊥ are not shown below 30 K where the significant line broadening and the relatively weak signal intensity prevent a reliable analysis. Annealing at 300 • C in a dynamic vacuum affects the ESR linewidth only below 50 K. This is plausible since due to its semi-metallic character and the small Fermi surface even a few ppm of paramagnetic impurities can dominate the signal at low temperatures due to their Curie spin susceptibility 65 . Higher annealing temperatures (up to 600 • C) did not influence further the linewidth.
From a calibrated measurement of the spin-susceptibility, we obtained that the Curie contribution to the ESR signal is due to 5(1) ppm of S = 1/2 spins. We show in Fig. S2 that the susceptibility contribution from the localized and delocalized spins is equal at about 15 K. Given that any influence of the localized species on the g-factor and the linewidth is weighted with the spin-susceptibility 65 , we conclude that above 50 − 100 K (especially at the technologically important room temperature region), the observed ESR linewidth is the intrinsic property of graphite. Although the present work focuses on the anisotropy of the ESR linewidth, we also mention that the conventional Elliott-Yafet's theory predicts an opposite temperature dependence, which may be related to the near band degeneracy around the graphite Fermi surface. Its full description requires additional work though.
The temperature-dependent g-factor data (see Fig. S5 of SI) confirms the earlier experimental data 46,[60][61][62] and the theoretical prediction. The measured factor of 2 anisotropy in the ESR linewidth of graphite (see the bottom image of Fig.  3) matches exactly the theoretical prediction of a vanishing SOC for spins polarized in the (a, b) plane and a finite SOC for spins polarized along the c-axis. Strictly speaking, the linewidth anisotropy ratio is 2 only between 100 and 300 K, while it deviates somewhat upwards below 100 K and downwards above 300 K, which calls for more detailed modeling.

B. The ultralong spin-relaxation time, T1
An important consequence of the absence of spin-orbit fields in the graphene planes is the predicted ultralong spinlattice relaxation time, T 1,⊥ , according to Eq. (3), whereas the other three relaxation times remain finite, below 50 ns. To experimentally verify this prediction, we performed saturation ESR measurements (see Fig. S7 of SI). Although the transversal spin-relaxation time, T 2 is accessible directly from the linewidth, T 1 does not directly affect the lineshape in conventional ESR studies. In principle, both relaxation times can be measured with a spin-echo technique but it is limited to relaxation times when both T 1 and T 2 are longer than ∼ 1 µs, which is not the case herein. However, saturation ESR experiments 44 yield relaxation times down to ∼ 10 ns. The method is based on monitoring the variation of the ESR signal intensity and the linewidth as a function of the irradiating microwave power. A characteristic drop in the signal intensity, accompanied by a line broadening allows determining T 1 . Further technical details are given in the SI together with the raw saturation ESR data.
The T 1 results for both orientations of the magnetic field are shown in Fig. 4. The data is given only above 300 K for B ⊥ and above 370 K for B as below these temperatures the shortening of the respective T 2 prevents the observation of the saturation effect. Nevertheless, having in mind the spintronics application, this is the relevant temperature range. Red triangles and the dashed line in Fig. 4 confirm that the T 1, = T 2,⊥ prediction from Eqs. (4) and (5) is indeed satisfied, which provides further proof that the extreme SOC anisotropy is realized in graphite.
The most striking experimental observation is the presence of T 1,⊥ relaxation times beyond 100 ns and the approximate factor 10 anisotropy of T 1 . Although it is observed in an ESR experiment, i.e., in the presence of an external magnetic field, this result can be readily extended to the case of zero magnetic fields: it implies that τ s,⊥ in graphite would be as long as 100 ns when electrons are injected with a spin perpendicular to the graphene planes and it would be about a factor 10 times shorter when the spins lie in the graphene planes. This is also depicted in Fig. 4. The giant anisotropy may find a number of applications, e.g., it could be exploited to control the spinrelaxation times in spintronic devices. It is known 1 that electrons travel mainly in the graphene planes in graphite and interlayer transport is diffusion limited and it leads to a conduction anisotropy σ /σ ⊥ beyond 10 3 − 10 4 . This allows to approximate the spin diffusion in graphite while considering that electrons reside on a given graphene layer. The large value of τ s,⊥ leads to an ultra-long spin-diffusion length in graphite: δ s = vF √ 2 √ τ τ s (the factor 2 appears from the two-dimensional diffusion equation). It gives δ s ≈ 70 µm with typical values of v F = 10 6 m/s and τ = 10 −13 s. This is already a macroscopic length scale, which could bring spintronic devices closer to reality.

III. SUMMARY
In conclusion, we unraveled the anomalous dynamics of itinerant electron spins in graphite. We first studied the details of the band-structure-related spin-orbit coupling which pre-dicted a hitherto hidden, symmetry-related extreme anisotropy of the spin-orbit coupling. This anisotropy in fact explains the known anomalous anisotropic properties of graphite: a strongly anisotropic g-factor and the 1 : 2 anisotropy of the ESR linewidth. We recognized that the latter should in principle be accompanied by a giant anisotropy of the spin-lattice relaxation time. This prediction was examined with saturation ESR studies and we observe ultralong (T 1 in excess of 100 ns) spin-relaxation times for spins aligned perpendicular to the graphene planes. When extrapolated to the zero-field limit, this predicts similarly long spin-relaxation times in spintransport studies and a macroscopic spin-diffusion length.

a. Experiment
We studied high-quality HOPG (highlyoriented pyrolitic graphite) from Structure Probe Inc. (SPI Grade I) using electron spin resonance (ESR). The HOPG had a mosaicity of 0.4 • ± 0.1 • . We studied X-band (0.33 T, 9.4 GHz) in a commercial ESR spectrometer (Bruker Elexsys E500) in the 4 − 673 K temperature range. The spectrometer is equipped with a goniometer which allows reliable and reproducible sample rotations. We used HOPG disk samples of 3 mm diameter and 70 µm thickness.
The samples were sealed in quartz ampules under 20 mbar He for the ESR measurement. Annealing of some of the samples was performed in a high dynamic vacuum in a furnace up to 300 − 600 • C. The goals in this study are the accurate measurement of the ESR linewidth, and temperature-dependent ESR intensity (penetration effects for a bulky sample prevent the determination of the absolute spin-susceptibility). We also monitored the g-factor as a function of temperature to enable comparison with existing literature data. We employed a low magnetic field modulation to avoid signal distortion, we also used a low (150 µW) microwave power to avoid saturation of the normal data and we monitored the change in the cavity resonance frequency. We quote the widths of derivative Lorentzian curves (or half width-half maximum data, HWHM) fitted to the experimental data which is related to the peak-to-peak linewidth as: ∆B HWHM = ∆B pp / √ 3. Saturated ESR experiments were performed in a commercial microwave cavity (Bruker ER 4122 SHQ, Super High Q Resonator) with an unloaded Q 0 = 7500 and with the sample Q L = 5500. This cavity produces AC magnetic fields of where p is the microwave power in Watts and we used microwave powers up to 0.2 W.
b. Theory The electronic band structure of graphite in the presence of spin-orbit coupling is determined by the full potential linearized augmented plane waves (LAPW) method, based on density functional theory implemented in Wien2k 66 . For exchange-correlation effects, the generalized gradient approximation was utilized 67 . In our three-dimensional calculation the graphene sheets of lattice constant a = 1.42 √ 3Å are separated by the distance of c = 3.35Å. Integration in the reciprocal space was performed by the modified Blöchl tetrahedron scheme, taking the mesh of 33 × 33 k-points in the irreducible Brillouin zone (BZ) wedge. As the plane-wave cut-off, we took 9.87Å −1 . The 1s core states were treated fully relativistically by solving the Dirac equation, while spinorbit coupling for the valence electrons was treated within the muffin-tin radius of 1.34 a.u. by the second variational method 68 .
The algorithm and the code used to calculate the b 2 k values were proven to be correct for various other materials previously 69 . These include WS 2 70 , WSe 2 , MoSe 2 71 , various heterostructures 72,73 , etc. We are confident that it provides adequate results for the case of graphite as well. However, averaging the spin admixture, b 2 , over the tiny Fermi-surface pockets is, at the moment, not feasible. The main reason is that the admixture varies strongly (over several orders of magnitude) close to the band degeneracies (spin hot spots), requiring very dense sampling of the Fermi surface. At the moment, this is beyond the computational capabilities of our available computing infrastructure.

DATA AVAILABILITY
The data needed to evaluate and reproduce the conclusions are present in the paper and the Supporting Information (Online Content). Additional data related to this paper are available from the corresponding author upon request.

ONLINE CONTENT
Online content includes supporting information.