Ye Peng

Mainak Mookherjee

Andreas Hermann

Geeth Manthilake

David Mainprice

R D Nance

Anomalous elasticity of talc at high pressures: Implications for subduction systems

Keywords: Talc, Elasticity, Seismic anisotropy, Hydrous minerals, Subduction zone Handling

Talc is a layered hydrous silicate mineral that plays a vital role in transporting water into Earth's interior and helps to explain many anomalous geophysical observations in subduction zone settings. In this study, we explored the structure, equation of state, and elasticity of both triclinic and monoclinic talc under high pressures up to 18 GPa using first principles simulations based on density functional theory corrected for dispersive forces. Our results indicate that principal components of the full elastic constant tensor C 11 and C 22 , shear components C 66 , and several off-diagonal components show anomalous pressure dependence. This non-monotonic pressure dependence of elastic constant components is likely related to the structural changes and is often manifested in a polytypic transition from a low-pressure polytype talc-I to a highpressure polytype talc-II. The polytypic transition of talc occurs at pressures within its thermodynamic stability. However, the bulk and shear elastic moduli show no anomalous softening. Our study also shows that talc has low velocity, extremely high anisotropy, and anomalously high V P /V S ratio, thus making it a potential candidate mineral phase that could readily explain unusually high V P /V S ratio and large shear wave splitting delays as observed from seismological studies in many subduction systems.

Introduction

Water plays a vital role in the solid Earth by influencing melting processes, i.e., lowering solidus temperature, facilitating mantle convection by affecting rheological properties of rocks, and affecting geophysical observations such as seismic velocities and geoelectric fields. As plates subduct, bending-related faults often lead to hydration of the oceanic lithosphere, which in turn stabilizes a suite of hydrous minerals [START_REF] Ranero | Bending-related faulting and mantle serpentinization at the Middle America trench[END_REF]. Talc, Mg 3 Si 4 O 10 (OH) 2 , is one such hydrous mineral and contains ~4.7 wt.% water locked up in its crystal structure as hydroxyl ions. Talc and other hydrous minerals play an important role in transporting water into the Earth's interior. These hydrous minerals have limited thermal stabilities. Talc has maximum thermal stability of ~800-820 ºC at ~3 GPa, i.e., 95 km. Due to its negative Clapeyron slope (dP/dT <0) at pressures greater than 3 GPa, the thermal stability of talc is reduced to 710 ºC at 5 GPa, i.e., 155 km (Pawley and Wood, 1995). Upon reaching adequate depths, hydrous minerals dehydrate and release aqueous fluids. The upward migration of these aqueous fluids further stabilizes hydrous minerals such as talc and serpentine in the relatively cooler regions of the overlying mantle wedge [START_REF] Evans | Stability of chrysotile and antigorite in the serpentine multisystem[END_REF]. The dehydration of these hydrous minerals is often manifested as the double seismic zones [START_REF] Brudzinski | Global prevalence of double Benioff zones[END_REF][START_REF] Dorbath | Double seismic zone of the Nazca plate in northern Chile: High-resolution velocity structure, petrological implications, and thermomechanical modeling[END_REF][START_REF] Rondenay | Seismic imaging of subduction zone metamorphism[END_REF].

Effective transfer of water through the subduction zone is achieved by the transfer of water from progressively denser hydrous mineral phases. In other words, if the geothermal gradient overlaps with the stability field of hydrous minerals, water is likely to be effectively sequestered within the hydrous minerals and dragged along with the subducting plate. Serpentine is a hydrous mineral that is thermodynamically stable up to a depth of ~200 km in a relatively cold subduction zone. As serpentine dehydrates, it often transfers water to a denser hydrous mineral phase, phase A, which is stable at a deeper depth. Often the thermodynamic stability fields of serpentine and phase A do not fully overlap, i.e., the geothermal gradient passes through a region where no hydrous mineral is stable, creating a thermodynamic choke point in the effective transport of water into the deep Earth. Experimental studies have shown that at such a thermodynamic choke point, talc may intercalate water in its interlayer and stabilize the 10-Å phase that is crucial in transporting water deep down to dense hydrous mineral phases (Pawley and Wood, 1995;Pawley et al., 1995;[START_REF] Ohtani | Water transport into the deep mantle and formation of a hydrous transition zone[END_REF][START_REF] Gleason | Pressuretemperature studies of talc plus water using X-ray diffraction[END_REF][START_REF] Rashchenko | In situ spectroscopic study of water intercalation into talc: New features of 10 Å phase formation[END_REF].

Talc is a layered hydrous silicate mineral with a crystal structure that consists of an octahedral (O) layer sandwiched by two tetrahedral (T) layers (TOT), i.e., three MgO 6 octahedral units are sandwiched between two [Si 2 O 5 ] 2-units such that their apical oxygen atoms are pointing towards each other. The two hydroxyls (OH) -attached to the MgO 6 octahedra point within the di-trigonal rings formed by the tetrahedral units and are perpendicular to the (001) plane. The adjacent TOT layers are held by weak van der Waals (vdW) forces. Due to the layered crystal structure, talc is one of the softest minerals with a Mohs hardness scale of 1 [START_REF] Wenk | Minerals: Their Constitution and Origin[END_REF] and has a very low coefficient of friction ~0.16-0.23 at room temperature [START_REF] Escartín | Relationships between the microstructural evolution and the rheology of talc at elevated pressures and temperatures[END_REF][START_REF] Moore | Talc friction in the temperature range 25-400 ℃: Relevance for fault-zone weakening[END_REF]. Talc is often resistant to slip when the rate of sliding is faster [START_REF] Moore | Talc friction in the temperature range 25-400 ℃: Relevance for fault-zone weakening[END_REF]. These characteristics make talc an ideal lubricant to allow slow and stable sliding/creeping along the subduction interface, inhibiting the elastic energy build-up and preventing fast sliding and unstable slip related to large earthquakes [START_REF] Wibberley | Talc at fault[END_REF].

Talc is a common mineral and is often found in metamorphosed or hydrothermally altered lithologies. Because of its widespread occurrence, stability, and unique physical properties, the presence of talc is often invoked to explain anomalous geophysical observations. For example, it is suggested that talc might explain observations along many oceanic faults that occur in slow and ultra-slow spreading midocean ridges [START_REF] Schroeder | Strain localization on an oceanic detachment fault system, Atlantis Massif, 30 N, Mid-Atlantic Ridge[END_REF][START_REF] Boschi | Occurrence and significance of serpentinite-hosted, talc-and amphibole-rich fault rocks in modern oceanic settings and ophiolite complexes: An overview[END_REF] and stable fast creeping sections of the San Andreas fault [START_REF] Moore | Talc-bearing serpentinite and the creeping section of the San Andreas fault[END_REF][START_REF] Wibberley | Talc at fault[END_REF]. It is proposed that a talc-rich mélange zone forms in the slab-mantle interface and contributes to the low-velocity zone observed at or above the slab surface [START_REF] Bebout | Tectonic and metasomatic mixing in a high-T, subduction-zone mélange-insights into the geochemical evolution of the slab-mantle interface[END_REF][START_REF] Spandler | The importance of talc and chlorite "hybrid" rocks for volatile recycling through subduction zones; evidence from the high-pressure subduction mélange of New Caledonia[END_REF][START_REF] Marschall | Arc magmas sourced from mélange diapirs in subduction zones[END_REF]. The stable-sliding behavior of talc leads to aseismicity and therefore controls the maximum depth of earthquakes in subduction zones [START_REF] Peacock | Hydrous minerals in the mantle wedge and the maximum depth of subduction thrust earthquakes[END_REF]. It is also suggested that talc may promote slow-slip earthquakes, i.e., an earthquake-like event that releases comparable energies with a much slower rate, five-order slower, than those of ordinary great earthquakes [START_REF] Kawasaki | Silent earthquakes occurring in a stable-unstable transition zone and implications for earthquake prediction[END_REF][START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF]). In the central Mexican subduction zone, a thin layer at the top of the slab, which is characterized by ultra-slow velocity, low viscosity, high P-to-S velocity ratio (V P /V S ), and free of earthquakes, has also been linked to the presence of talc [START_REF] Kim | Generation of talc in the mantle wedge and its role in subduction dynamics in central Mexico[END_REF]. In the flat subduction system of central Mexico, talc is also suggested to play an important role in the dynamic process of slab-flattening [START_REF] Manea | Subduction zone evolution and low viscosity wedges and channels[END_REF][START_REF] Kim | Generation of talc in the mantle wedge and its role in subduction dynamics in central Mexico[END_REF].

Despite the importance of talc in subduction zone settings, little is known about its elasticity. Owing to the difficulty in finding good quality single crystals, experimental data on full elastic constants are currently lacking. Attempts have been made to estimate the shear modulus from ultrasonic compressional sound wave velocity [START_REF] Bailey | Experimental determination of elastic properties of talc to 800°C, 0.5 GPa; calculations of the effect on hydrated peridotite, and implications for cold subduction zones[END_REF]; however, so far, no known Brillouin scattering result exists for talc. In contrast to the full elastic constant tensor, a significant amount of experimental studies has provided constraints on the bulk modulus from the studies of the equation of state [START_REF] Pawley | Equation of state measurements of chlorite, pyrophyllite, and talc[END_REF][START_REF] Gleason | Pressuretemperature studies of talc plus water using X-ray diffraction[END_REF][START_REF] Gatta | On the crystal structure and compressional behavior of talc: a mineral of interest in petrology and material science[END_REF].

So far, the equation of state and elasticity of talc have been explored by at least three prior first principles simulation studies [START_REF] Stixrude | Talc under tension and compression: Spinodal instability, elasticity, and structure[END_REF][START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF][START_REF] Ulian | The compressional behaviour and the mechanical properties of talc [Mg 3 Si 4 O 10 (OH) 2 ]: a density functional theory investigation[END_REF]. The equation of state study using first principles explored the high pressure evolution of the crystal structure and linear compressibility of triclinic talc but did not explore the full elastic constant tensor [START_REF] Stixrude | Talc under tension and compression: Spinodal instability, elasticity, and structure[END_REF]. This was followed by a study that determined the full elastic constant tensor and elastic anisotropy of both monoclinic and triclinic talc up to 12 GPa [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF]. However, since talc is a layered hydrous silicate, the vdW interactions often play a vital role, and relevant correction is needed to better describe its structure and elastic properties [START_REF] Weck | Relationship between crystal structure and thermo-mechanical properties of kaolinite clay: beyond standard density functional theory[END_REF]. In addition, the predicted pressure dependence of the elastic constants and elastic moduli for the triclinic and monoclinic polymorphs were quite different, and the pressure dependence also indicated a significant scattering due to poorly converged energetics.

More recent work on talc explored the structure and elastic properties using first principles simulation integrated with dispersive forces contribution [START_REF] Ulian | The compressional behaviour and the mechanical properties of talc [Mg 3 Si 4 O 10 (OH) 2 ]: a density functional theory investigation[END_REF].

However, the full elastic constant results were limited to ambient pressure. It is well known that pressure significantly influences the elasticity of layered hydrous minerals.

Hence, in this study, we explored the structure, equation of state, and elasticity of both triclinic and monoclinic talc using first principles simulations incorporating dispersive interaction corrections at high pressures up to 18 GPa.

Methods

In nature, talc (Mg 3 Si 4 O 10 (OH) 2 ) is reported to crystallize in either monoclinic space group symmetry (C2/c) with two TOT layers (Z= 4) [START_REF] Gruner | The crystal structures of talc and pyrophyllite[END_REF] or triclinic space group symmetry (C ) [START_REF] Rayner | The crystal structure of talc[END_REF][START_REF] Perdikatsis | Strukturverfeinerung am talk Mg 3 [(OH) 2 Si 4 O 10[END_REF] with one TOT unit (Z=2). Hence, in this study, we explored both triclinic and monoclinic polymorphs. We explored the high-pressure behavior of crystal structure, equation of state, and elasticity of talc using static density functional theory as implemented in the Vienna ab-initio simulation package (VASP) [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF][START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF][START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF]Kresse and Furthmüller, 1996a, b;[START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF]. First principles simulations based on density functional theory are crucial in providing valuable insights into the energetics, structure, and elasticity of hydrous minerals [START_REF] Tsuchiya | First-principles study of hydrogen bond symmetrization of phase D under high pressure[END_REF][START_REF] Tsuchiya | A first-principles calculation of the elastic and vibrational anomalies of lizardite under pressure[END_REF]Tsuchiya, 2009, 2011;[START_REF] Chantel | The elasticity of lawsonite at high pressure and the origin of low velocity layers in subduction zones[END_REF][START_REF] Ulian | The compressional behaviour and the mechanical properties of talc [Mg 3 Si 4 O 10 (OH) 2 ]: a density functional theory investigation[END_REF]Mookherjee et al., 2015[START_REF] Mookherjee | Crystal structure, equation of state, and elasticity of hydrous aluminosilicate phase, topaz-OH (Al 2 SiO 4 (OH) 2 ) at high pressures[END_REF][START_REF] Mookherjee | Anomalous elastic behavior of phase Egg, AlSiO 3 (OH), at high pressures[END_REF][START_REF] Mookherjee | Elasticity of superhydrous phase, B, Mg 10 Si 3 O 14 (OH) 4[END_REF][START_REF] Peng | Elasticity of phase-Pi (Al 3 Si 2 O 7 (OH) 3 )-A hydrous aluminosilicate phase[END_REF][START_REF] Yang | Elasticity of superhydrous phase B at the mantle temperatures and pressures: Implications for 800 km discontinuity and water flow into the lower mantle[END_REF]Ulian andValdrè, 2017, 2019;[START_REF] Peng | Thermoelasticity of tremolite amphibole: Geophysical implications[END_REF][START_REF] Lee | Seismic anisotropy in subduction zones: Evaluating the role of chloritoid[END_REF][START_REF] Saha | Assessing the presence of volatile-bearing mineral phases in cratonic mantle as a possible cause of mid-lithospheric discontinuities[END_REF]. We used generalized gradient approximation (GGA) [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] and the highly accurate projector augmented wave method (PAW) implemented in VASP [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF].

It is well known that dispersive interaction corrections often provide better prediction of the structure and properties of layered minerals [START_REF] Ulian | The compressional behaviour and the mechanical properties of talc [Mg 3 Si 4 O 10 (OH) 2 ]: a density functional theory investigation[END_REF][START_REF] Weck | Relationship between crystal structure and thermo-mechanical properties of kaolinite clay: beyond standard density functional theory[END_REF]. Hence, to account for the long-range dispersive interactions, we computed the equation of state and elasticity of talc with optB88 vdW density functional as implemented in VASP [START_REF] Dion | Van der Waals density functional for general geometries[END_REF][START_REF] Klimeš | Chemical accuracy for the van der Waals density functional[END_REF][START_REF] Klimeš | Van der Waals density functionals applied to solids[END_REF][START_REF] Román-Pérez | Efficient implementation of a van der Waals density functional: application to double-wall carbon nanotubes[END_REF].

We performed a series of convergence tests on talc by varying the cutoff energy and k-points. Unlike the previous study where a low cutoff energy E cut of 400 eV was used [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF], we found that a cutoff energy E cut = 900 eV was required to describe the ground state energy of talc adequately (Supplementary Data Table 1).

Similarly, we found that it needed a k-point mesh of 6 3 3 Monkhorst-Pack grid with 27 irreducible k-points [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF]) (Supplementary Data Table 2).

We determined the elastic constants by straining the lattice parameters by ±1%, as outlined in previous studies [START_REF] Mookherjee | Structure and elasticity of serpentine at highpressure[END_REF][START_REF] Mookherjee | Trench parallel anisotropy and large delay times: Elasticity and anisotropy of antigorite at high pressures[END_REF][START_REF] Chheda | Structure and elasticity of phlogopite under compression: Geophysical implications[END_REF][START_REF] Peng | Elasticity of phase-Pi (Al 3 Si 2 O 7 (OH) 3 )-A hydrous aluminosilicate phase[END_REF][START_REF] Peng | Thermoelasticity of tremolite amphibole: Geophysical implications[END_REF]. Next, we computed the compressional and shear wave velocities V P (n) and V S (n) as a function of propagation direction n [START_REF] Mainprice | A FORTRAN program to calculate seismic anisotropy from the lattice preferred orientation of minerals[END_REF]. The P-wave anisotropy (AV P ) is defined as

AV P = (V Pmax -V Pmin )/[(V Pmax +V Pmin )/2]×100%.
In an anisotropic media, the shear wave splits into two waves. We denoted the fastest and slowest wave as V S1 and V S2 , respectively. We calculated the shear wave polarization anisotropy (AV S ) along each direction by AV S = (V S1 -V S2 )/[(V S1 +V S2 )/2]×100%.

Results

Equation of state and crystal structure

The energy-volume and pressure-volume behavior of talc can be well described by the 4 th -order Birth-Murnaghan equation of state (EoS) [START_REF] Murnaghan | A theory of elasticity[END_REF][START_REF] Birch | Finite strain isotherm and velocities for single-crystal and polycrystalline NaCl at high pressures and 300 K[END_REF][START_REF] Katsura | A simple derivation of the Birch-Murnaghan Equations of State (EOSs) and comparison with EOSs derived from other definitions of finite strain[END_REF] (Fig. 1). To better compare with prior EoS studies, we also fitted our data with Murnaghan EoS and 3 rd -order Birth-Murnaghan EoS (Table 1).

Our predicted zero-pressure volume for triclinic and monoclinic talc is V 0 tri = 461.8±0.1 Å 3 and V 0 mon = 923.44±0.08 Å 3 , respectively (Table 1). Experimental studies reported that the zero-pressure volume of triclinic talc is 453.6±0.4 Å 3 [START_REF] Gleason | Pressuretemperature studies of talc plus water using X-ray diffraction[END_REF] and 454.7±0.9 Å 3 [START_REF] Gatta | On the crystal structure and compressional behavior of talc: a mineral of interest in petrology and material science[END_REF], while that of monoclinic talc is 912±2 Å 3 [START_REF] Pawley | Equation of state measurements of chlorite, pyrophyllite, and talc[END_REF] and 915±8 Å 3 [START_REF] Gleason | Pressuretemperature studies of talc plus water using X-ray diffraction[END_REF]. Our results are ~1%-2% larger than the experimental results. We noted that the conventional GGA tends to underestimate the binding energy and thus overestimate the volume, while the conventional local density approximation (LDA) is the opposite. But the prediction with the vdW correction showed a much better agreement with the experimental results (Supplementary Data Fig. 1). The 4 th -order Birth-Murnaghan EoS fitting also yields the zero-pressure bulk modulus K 0 tri = 38.5±3 GPa and the pressure derivative of the bulk modulus K' 0 tri = 13.2±0.1 and K'' 0 tri = -4.3±0.1 GPa -1 for triclinic talc, and K 0 mon = 40.0±0.2 GPa and K' 0 mon = 12.2±0.3 and K'' 0 mon = -3.6±0.3 GPa -1 for monoclinic talc (Table 1). Our analysis indicates that different types of EoS fitting affect the fitting results significantly. In addition, there is also a strong correlation between K 0 and K' 0 which is known for other minerals. Such a correlation means that the energy-volume data can be fitted equally well by varying the value of K 0 and K' 0 . This must be considered when comparing a set of EoS parameters with previous results [START_REF] Angel | Equations of state[END_REF].

The lattice parameters of talc decrease with pressures for both triclinic and monoclinic structures (Fig. 2). However, the change in the c-axis is much more considerable compared with the a-and b-axis. We used f l -F l relations to determine the linear incompressibility along each axis, where f l is linear finite strain and F l is the linear normalized pressure (the subscript "l" represents each axis) [START_REF] Davies | Effective elastic moduli under hydrostatic stress-I. quasi-harmonic theory[END_REF][START_REF] Weaver | Application of finite strain theory to non-cubic crystals[END_REF][START_REF] Meade | Static compression of Ca(OH) 2 at room temperature: observations of amorphization and equation of state measurements to 10.7 GPa[END_REF][START_REF] Wentzcovitch | Crystal chemistry of forsterite: A first-principles study[END_REF][START_REF] Chheda | Structure and elasticity of phlogopite under compression: Geophysical implications[END_REF].

Both triclinic and monoclinic talc show the relation of K a ~Kb >>K c , indicating a large anisotropy in the linear compressibility (Supplementary Data Table 3, Fig. 2). Upon compression, the volume and average bond distance of each polyhedral unit decrease (Supplementary Data Fig. 2). The hydroxyl bond distance slightly increases upon compression, and beyond 1.4 GPa, it decreases monotonously with pressure. The tetrahedral rotation angle = (∑ 6

i=1 | i -120°|)/12 captures the degree of distortion in the six-membered tetrahedral rings, where  i is the angle among three adjacent basal oxygen atoms. Our results show that the tetrahedral rotation angle increases upon compression consistent with prior experimental studies [START_REF] Perdikatsis | Strukturverfeinerung am talk Mg 3 [(OH) 2 Si 4 O 10[END_REF][START_REF] Gatta | On the crystal structure and compressional behavior of talc: a mineral of interest in petrology and material science[END_REF] (Supplementary Data Fig. 2).

Elasticity

To determine the elasticity of talc, we considered both triclinic and monoclinic polymorphs. The talc crystal with triclinic symmetry has 21 independent elastic constants, while the talc crystal with monoclinic symmetry has 13 independent ones [START_REF] Nye | Physical Properties of Crystals: Their Representation by Tensors and Matrices[END_REF]. In the pressure range explored in this study, the principal components of elastic constant exhibit the following relation of C 11 ~C22 >C 33 (Figs. 3 and4, Tables 2 and3).

Upon compression, the principal components C 11 and C 22 first soften up to pressures of 3 GPa. Upon further compression beyond 3 GPa, C 11 and C 22 stiffen. In contrast to C 11 and C 22 , the C 33 increases monotonously. We also noted that C 12 , C 15 , C 25 , C 36 , and C 45 of triclinic talc and C 12 of monoclinic talc show initial softening till ~3 GPa, following which they show normal pressure dependence. While C 66 and C 14 of triclinic talc and C 66 of monoclinic talc all exhibit opposite pressure dependence, i.e., they first stiffen up to 3 GPa and then soften upon compression (Figs. 3 and4). The transition occurs within the thermodynamic stability of talc and is observed in both triclinic and monoclinic polymorphs. It is a rather unique behavior and is likely related to the reported polytypic transition between the low-pressure polytype talc-I to the high-pressure polytype talc-II [START_REF] Scott | High-pressure infrared spectra of talc and lawsonite[END_REF][START_REF] Gleason | Pressuretemperature studies of talc plus water using X-ray diffraction[END_REF]. This transition is related to the shearing between the weakly bonded adjacent TOT layers along the a-b plane, i.e., the change of the stacking angle of TOT layers. The shearing is likely to facilitate the water intercalation into talc and thus the formation of the 10-Å phase [START_REF] Gleason | Pressuretemperature studies of talc plus water using X-ray diffraction[END_REF].

Owing to such shearing between the adjacent TOT layers, C ij s with stresses acting along the a-and b-directions, i.e., subscripts of 1, 2, and 6, tend to be more sensitive and hence

show anomalous pressure dependence.

The polytypic transition is often characterized by pressure-induced structural adjustments, resulting in iso-symmetric changes in the stacking sequences of the tetrahedral and octahedral layers. In addition to talc, similar transitions have been observed in several layered hydrous silicates relevant to subduction zones, including kaolinite [START_REF] Welch | Pressure-induced transformations in kaolinite[END_REF][START_REF] Basu | Intercalation of water in kaolinite (Al 2 Si 2 O 5 (OH) 4 ) at subduction zone conditions: Insights from Raman Spectroscopy[END_REF], chlorite [START_REF] Welch | Novel high-pressure behavior in chlorite: A synchrotron XRD study of clinochlore to 27 GPa[END_REF], and serpentine [START_REF] Nestola | Antigorite equation of state and anomalous softening at 6 GPa: an in situ single-crystal X-ray diffraction study[END_REF]. Such transitions are often associated with the pressure-induced anomalies of elastic constants. For instance, prior first principles predictions of serpentine polymorphs-lizardite [START_REF] Mookherjee | Structure and elasticity of serpentine at highpressure[END_REF][START_REF] Tsuchiya | A first-principles calculation of the elastic and vibrational anomalies of lizardite under pressure[END_REF] and antigorite [START_REF] Mookherjee | Trench parallel anisotropy and large delay times: Elasticity and anisotropy of antigorite at high pressures[END_REF], and chlorite [START_REF] Mookherjee | Unusually large shear wave anisotropy for chlorite in subduction zone settings[END_REF] reported anomalous elasticity under compression.

Experimental studies on the pressure dependence of elastic constants are often rare, except for one Brillouin scattering study of antigorite, which also revealed the pressureinduced anomaly [START_REF] Bezacier | High-pressure elasticity of serpentine and seismic properties of the hydrated mantle wedge[END_REF]. Hence, such transitions are often manifested in the changes in the pressure-volume relations and high-pressure vibrational spectroscopy, as documented in serpentine polymorphs antigorite [START_REF] Nestola | Antigorite equation of state and anomalous softening at 6 GPa: an in situ single-crystal X-ray diffraction study[END_REF] and lizardite [START_REF] Auzende | High-pressure behaviour of serpentine minerals: a Raman spectroscopic study[END_REF], chlorite [START_REF] Welch | Novel high-pressure behavior in chlorite: A synchrotron XRD study of clinochlore to 27 GPa[END_REF], kaolinite [START_REF] Welch | Pressure-induced transformations in kaolinite[END_REF][START_REF] Basu | Intercalation of water in kaolinite (Al 2 Si 2 O 5 (OH) 4 ) at subduction zone conditions: Insights from Raman Spectroscopy[END_REF], and dickite [START_REF] Johnston | Novel pressure-induced phase transformations in hydrous layered materials[END_REF][START_REF] Dera | Pressure-controlled polytypism in hydrous layered materials[END_REF].

At pressures beyond the anomalous behavior, the rate of stiffening of C 11 and C 22 , i.e., the pressure dependence dC ii /dP, where i= 1 or 2, is significantly lower than that of C 33 , i.e., dC 33 /dP. Similar behavior has been often observed in other layered hydrous silicates, including serpentine polymorph lizardite [START_REF] Mookherjee | Structure and elasticity of serpentine at highpressure[END_REF][START_REF] Tsuchiya | A first-principles calculation of the elastic and vibrational anomalies of lizardite under pressure[END_REF], antigorite [START_REF] Mookherjee | Trench parallel anisotropy and large delay times: Elasticity and anisotropy of antigorite at high pressures[END_REF][START_REF] Bezacier | High-pressure elasticity of serpentine and seismic properties of the hydrated mantle wedge[END_REF], chlorite [START_REF] Mookherjee | Unusually large shear wave anisotropy for chlorite in subduction zone settings[END_REF], and phlogopite [START_REF] Chheda | Structure and elasticity of phlogopite under compression: Geophysical implications[END_REF]. We noticed that the pressure dependence of C 13 and C 23 is much larger than that of C 12 , and similarly, the pressure dependence of C 44 and C 55 is much larger than that of C 66 for the entire pressure range explored in this study.

The comparison of elastic constants between our study and prior estimates showed significant differences in C ij s (Figs. 3 and4) [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF]). These differences are likely due to both the dispersive interaction correction and the high cutoff energy (900 eV) used in this study. In contrast, the previous study did not use any correction for the dispersive force and used much lower cutoff energy (400 eV). Owing to the low cutoff energy and very likely non-converged results, prior results exhibited scattered data of elasticity versus pressure [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF]. However, there is a qualitative agreement between the prior study and our work in the talc-I to talc-II transition, and both capture the change in pressure dependence of elastic constants.

We determined the Hill averages of bulk and shear moduli based on Voigt and Reuss bounds (Fig. 5). The zero-pressure bulk and shear moduli are in good agreement with previous experimental studies [START_REF] Babeyko | Calculation of elastic properties in lower part of the Kola borehole from bulk chemical compositions of core samples[END_REF][START_REF] Bailey | Experimental determination of elastic properties of talc to 800°C, 0.5 GPa; calculations of the effect on hydrated peridotite, and implications for cold subduction zones[END_REF].

Our results show that the elastic moduli of talc increase with pressure. In addition, the results of triclinic and monoclinic talc crystal match very well, indicating that the symmetry has almost no effect on the elastic properties of talc. At 0 GPa, the bulk and shear moduli of triclinic talc crystal are 58.0 and 26.9 GPa, respectively (Table 2). While the bulk and shear moduli of monoclinic talc crystal are 58.9 and 29.6 GPa, respectively (Table 3). The predicted P-and S-wave velocity of talc also increase with pressure.

Similarly, the symmetry has almost no effect on the velocity of talc. At 0 GPa, the V P and V S of triclinic talc are respectively 6.0 and 3.2 km/s at a density of 2.70 g/cm 3 . For monoclinic talc, the V P and V S are 6.0 and 3.3 km/s with a density of 2.72 g/cm 3 .

Prior simulation results show a scattering in the pressure dependence [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF]. In addition, the shear modulus and velocities predicted in earlier studies show significant divergence between the different symmetries [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF] and are significantly higher than our current results [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF][START_REF] Ulian | The compressional behaviour and the mechanical properties of talc [Mg 3 Si 4 O 10 (OH) 2 ]: a density functional theory investigation[END_REF]. (Fig. 5). Again, the scattering in the pressure dependence of the elasticity data of the prior study is likely due to the lower cutoff energy and non-converged results.

Discussion

Anisotropy

It is well known that talc is an extremely anisotropic mineral, as evident from its structure and resulting elastic properties. It is also known to develop a strong crystallographic preferred orientation along interfaces of thrust faults and between subduction slab and mantle wedge [START_REF] Lee | Lattice preferred orientation of talc and implications for seismic anisotropy in subduction zones[END_REF][START_REF] Nagaya | Crystallographic preferred orientation of talc determined by an improved EBSD procedure for sheet silicates: Implications for anisotropy at the slab-mantle interface due to Si-metasomatism[END_REF]. Thus, we determined the seismic wave velocities V P , V S1 , and V S2 along different propagation directions (Supplementary Data Figs. 345678). At ambient conditions, the single-crystal anisotropy of talc is AV P tri ~68%, AV S tri ~150%, AV P mon ~69%, and AV S mon ~141% (Fig. 6). Upon compression, both AV P and AV S decrease significantly. When the pressure increases to ~6 GPa, the anisotropy is almost converged with AV P tri ~29%, AV S tri ~43%, AV P mon ~28%, and AV S mon ~39%. Compared with other layered hydrous silicates, including chlorite [START_REF] Mookherjee | Unusually large shear wave anisotropy for chlorite in subduction zone settings[END_REF] and antigorite serpentine [START_REF] Mookherjee | Trench parallel anisotropy and large delay times: Elasticity and anisotropy of antigorite at high pressures[END_REF], talc has the largest anisotropy at low pressures.

Especially, AV S is almost three times larger than that of chlorite and antigorite. However, previous work significantly underestimated AV S [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF]. Again, it is owing to the use of low cutoff energy and poorly converged results. The stereographic projection of the P-wave, V P , shows that the propagation is slow perpendicular to the basal plane, i.e., along the c-axis or [001] direction. The stereographic projection of AV S , i.e., the difference between V S1 and V S2 , shows that AV S is relatively large along the caxis. The crystal symmetry has little or no effect on anisotropy (Fig. 6).

For an anisotropic mineral such as talc, V P /V S ratio can be further examined as V P /V S1 and V P /V S2 and their variations along pressures and propagation directions (Supplementary Data Fig. 9). It is noted that the different crystal symmetry has almost no effect on V P /V S ratio. Along the c-axis, i.e., [001] direction, the V P /V S1 of both triclinic and monoclinic talc ranges between 1.4-1.5 over the entire explored pressure range and is relatively insensitive to compression. In contrast, the V P /V S2 of talc is quite large and is ~5-7 at zero pressure. Upon compression, V P /V S2 of triclinic talc decreases to ~3.0 at 1.5

GPa and ~2.1 at 4 GPa, while V P /V S2 of monoclinic talc decreases to ~3.4 at 1.5 GPa and ~2.1 at 4 GPa. Beyond 4 GPa, the V P /V S2 is largely insensitive to pressures. Moreover, for the randomly oriented talc aggregate, the bulk V P /V S of both triclinic and monoclinic talc ranges between 1.7-1.9 that is in-between the V P /V S1 and V P /V S2 . Additionally, the stereographic projection shows that the ratios vary a lot with propagation directions, especially V P /V S2 (Supplementary Data Fig. 9).

Poisson's ratio

When materials are compressed along a certain axis, they tend to expand in the direction perpendicular to the compression axis, and Poisson's ratio quantitatively captures this phenomenon. It is defined as the negative ratio of the transverse deformation to the axial deformation. It is expected that softer material will expand more with the unit compression force, resulting in a larger Poisson's ratio.

We estimated the Poisson's ratio of triclinic talc along various extension directions (Supplementary Method; Fig. 7). At low pressures, i.e., below 4 GPa, the Poisson's ratio is negative for many directions. This means that if talc is stretched along the axial direction, the transverse directions will also expand. This is a special class of materials known as "auxetic materials" [START_REF] Yang | Review on auxetic materials[END_REF]. Talc exhibits auxetic behavior at low pressures, i.e., P< ~4 GPa. At ~1.4 GPa, i.e., the depth of ~45 km, the

Poisson's ratio is as low as -0.62. It gradually transitions to a normal behavior at high pressures. For instance, at P≥ 5.6 GPa, talc exhibits the normal positive Poisson's ratio in all directions (Fig. 7). We noted that Poisson's ratio is still very anisotropic at higher pressures. The Poisson's ratio varies greatly with different directions, and even in a certain direction, the maximum and minimum directions yield very different Poisson's ratios.

Implication

In the central Mexican subduction zone, a series of seismological studies based on the Meso-America Subduction Experiment (MASE) have revealed the presence of an ultra-slow velocity layer between the subducting Cocos plate and the overriding North American plate (Fig. 8). This layer is characterized by a small thickness of ~2-8 km, lowvelocity, low-viscosity, and low-strength (i.e., aseismic) [START_REF] Pérez-Campos | Horizontal subduction and truncation of the Cocos Plate beneath central Mexico[END_REF][START_REF] Song | Subducting slab ultra-slow velocity layer coincident with silent earthquakes in southern Mexico[END_REF][START_REF] Kim | Geometry and seismic properties of the subducting Cocos plate in central Mexico[END_REF][START_REF] Kim | Generation of talc in the mantle wedge and its role in subduction dynamics in central Mexico[END_REF][START_REF] Manea | Flat-slab thermal structure and evolution beneath central Mexico[END_REF][START_REF] Dougherty | Seismic structure in central Mexico: Implications for fragmentation of the subducted Cocos plate[END_REF]. In addition to the existence of an ultra-slow velocity layer, the subduction zone is This thin, low-velocity, and weak layer is interpreted as being made up of either hydrated mineralogy (talc and chlorite) or high pore fluid pressures [START_REF] Audet | Seismic evidence for overpressured subducted oceanic crust and megathrust fault sealing[END_REF][START_REF] Song | Anisotropic uppermost mantle in young subducted slab underplating Central Mexico[END_REF].

Prior studies have also found that this flat segment of the slab is characterized by (a) a constant depth of ~45 km; (b) a low shear velocity of 2.4-3.4 km/s, i.e., even slower than the topmost of the slab by 1.3-1.5 km/s; (c) a high V P /V S ratio of 2.050.25 based on a V P of ~5.54 km/s; and (d) a slab surface temperature of 500-800 C [START_REF] Kim | Geometry and seismic properties of the subducting Cocos plate in central Mexico[END_REF].

Considering all these constraints, the pore fluid pressure is unlikely since the fluid is likely to be lost by this depth [START_REF] Manea | Flat-slab thermal structure and evolution beneath central Mexico[END_REF]. In contrast, the hydrated mineral such as talc is likely to form by the reaction of serpentines and slab-derived silica-rich fluids along the slab-wedge interface, which lies in the thermodynamic stability field of talc (~800C at 1-2 GPa) [START_REF] Bose | Experimental and theoretical studies of the stabilities of talc, antigorite and phase A at high pressures with applications to subduction processes[END_REF]Pawley and Wood, 1995). In addition, geophysical observations on MASE did show that fluids release due to slab dehydration and then hydrate the upper plate. For example, a ~13 km thick continental root is expected below the 2.2 km elevation of the Trans-Mexican Volcanic Belt based on isostatic equilibrium, but it is missing. Instead, low-density minerals induced by hydration are suggested to compensate for the negative buoyancy [START_REF] Kim | Geometry and seismic properties of the subducting Cocos plate in central Mexico[END_REF]. Also, several regions of non-volcanic tremor have been reported above the flat slab suggesting lasting hydration and weakening of the overlying continental crust (Fig. 8) [START_REF] Manea | Flat-slab thermal structure and evolution beneath central Mexico[END_REF]. Thus talc is a potential candidate for explaining the ultra-slow velocity layer in central Mexico [START_REF] Kim | Generation of talc in the mantle wedge and its role in subduction dynamics in central Mexico[END_REF].

However, apart from being thermodynamically stable, talc also needs to fulfill the crucial constraints of low shear velocity and high V P /V S ratio observed in this subduction zone setting. We evaluated the potentiality of talc to explain the ultra-slow velocity layer using our predicted elasticity and anisotropy. We compiled the reported V P /V S vs. V S in the central Mexican subduction zone [START_REF] Kim | Geometry and seismic properties of the subducting Cocos plate in central Mexico[END_REF]. We compared the geophysical compilation with that of relevant hydrous minerals, including lizardite, antigorite, phlogopite, and chlorite [START_REF] Mookherjee | Structure and elasticity of serpentine at highpressure[END_REF][START_REF] Bezacier | Elasticity of antigorite, seismic detection of serpentinites, and anisotropy in subduction zones[END_REF][START_REF] Chheda | Structure and elasticity of phlogopite under compression: Geophysical implications[END_REF][START_REF] Manthilake | The anomalous seismic behavior of aqueous fluids released during dehydration of chlorite in subduction zones[END_REF] (Fig. 8). At ~1.5 GPa, i.e., ~45 km depth, the V P /V S1 ratio of triclinic talc varies between 1.2 and 1.8 and V P /V S2 varies between 1.2 and 6.3

with the bulk V P /V S ~1.7. At similar conditions, the V P /V S1 of monoclinic talc varies between 1.1 and 1.8 and V P /V S2 varies between 1.2 and 5.6 with the bulk V P /V S ~1.8

(Supplementary Data Fig. 9). The bulk V P /V S assumes an inherently homogenous nature, i.e., random orientation, of the minerals/rocks, while the V P /V S1 and V P /V S2 are estimated along specific propagation directions. However, it is likely for talc to develop a strong crystallographic preferred orientation along interfaces between subduction slab and mantle wedge [START_REF] Lee | Lattice preferred orientation of talc and implications for seismic anisotropy in subduction zones[END_REF][START_REF] Nagaya | Crystallographic preferred orientation of talc determined by an improved EBSD procedure for sheet silicates: Implications for anisotropy at the slab-mantle interface due to Si-metasomatism[END_REF]. The range of V P /V S for talc due to different propagation directions can explain the V P /V S ratio as high as 2.05±0.25 observed in the ultra-slow velocity layer of the central Mexican subduction zone.

Seismologically, the strength and geometry of anisotropy are generally reflected by the delay time between the fast and slow shear waves and the polarization direction of the fast shear wave, respectively [START_REF] Long | Shear wave splitting from local events beneath the Ryukyu arc: Trench-parallel anisotropy in the mantle wedge[END_REF][START_REF] Long | The subduction zone flow field from seismic anisotropy: A global view[END_REF][START_REF] Faccenda | Fault-induced seismic anisotropy by hydration in subducting oceanic plates[END_REF]. Many subduction zones, including Ryukyu and Tonga, are often characterized by a large delay time of >1 s [START_REF] Long | The subduction zone flow field from seismic anisotropy: A global view[END_REF][START_REF] Katayama | Trench-parallel anisotropy produced by serpentine deformation in the hydrated mantle wedge[END_REF]. The delay time is proportional to the ray path length and the strength of anisotropy along the ray path. Although olivine is the dominant mantle mineral, the elastic anisotropy of olivine is ~10%, and significant thickness of mantle with preferred orientated olivine is required to explain a large delay time. In contrast, the extremely anisotropic mineral such as talc could easily explain the large delay time within a thin layer. To further illustrate the degree of anisotropy associated with talc, we took the central Mexican subduction zone as an example. If we were to assume a talc layer of ~4±1 km thick [START_REF] Song | Subducting slab ultra-slow velocity layer coincident with silent earthquakes in southern Mexico[END_REF][START_REF] Kim | Geometry and seismic properties of the subducting Cocos plate in central Mexico[END_REF][START_REF] Kim | Generation of talc in the mantle wedge and its role in subduction dynamics in central Mexico[END_REF][START_REF] Manea | Flat-slab thermal structure and evolution beneath central Mexico[END_REF], the estimated delay time is likely to be ~0.03±0.01 s to 2.1±0.5 s depending on the incidence angle based on the single-crystal anisotropy of triclinic talc. The largest delay time is observed when the ray path is parallel to the (001) plane of talc, i.e., along a-b plane of triclinic talc with the azimuth angle of 150º (Fig. 6). Similarly, the predicted delay time for monoclinic talc is ~0.04±0.01 s to 1.8±0.4 s. The largest delay time is observed when the ray path is parallel to the (001) plane and along the b-axis of monoclinic talc (Fig. 6).

However, since talc is extremely anisotropic, the predicted delay time along the c-axis is still very high, with triclinic and monoclinic talc being ~1.8±0.4 s and 1.5±0.4 s, respectively. Clearly, a relatively thin layer of extremely anisotropic minerals such as talc could easily explain the large delay time >1 s, assuming a perfect crystal alignment.

However, our understanding of the elasticity of talc in subduction zone settings is likely to be enhanced by further investigations on the slip system and how non-hydrostatic stresses are likely to affect the crystallographic preferred orientation of talc bearing lithology.

Conclusions

In summary, we explored the structure, equation of state, and elasticity of both triclinic and monoclinic talc under high pressures up to 18 GPa using first principles simulations based on density functional theory corrected for dispersive forces. Owing to the dispersive correction and high cutoff energy of 900 eV, our results are significantly improved compared to the previous high-pressure elasticity study [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF]. The prior estimation was widely used to explain the geophysical observations or build geophysical models relevant to subduction zone settings. However, our results

indicate that the prior estimation greatly overestimated the shear moduli and shear wave velocity and significantly underestimated the shear wave anisotropy. Thus, it is required to re-evaluate the velocity models and/or interpretations using the previous data set.

Our results also show that some components of the full elastic constant tensor, including C 11 , C 22 , and C 66 , exhibit anomalous pressure dependence. The non-monotonic pressure dependence of elastic constant components is very likely related to the structural changes, which usually manifests in a polytypic transition from a low-pressure polytype talc-I to a high-pressure polytype talc-II. This polytypic transition occurs at ~3 GPa, which is within the thermodynamic stability field of talc. The transition could be a structural precursor of the 10-Å phase, a hydrous mineral that is thermodynamically stable at a deeper depth and is crucial for transporting water into the deep Earth.

Our study also shows that talc bearing hydrated lithology could readily account for anomalously low velocity, extremely high anisotropy, and anomalously high V P /V S ratio observed in subduction zone settings. In addition, talc bearing lithologies could also explain a large delay time in shear wave splitting as observed from seismological studies in many subduction systems. Mineral abbreviation: tlc-tri-triclinic talc (this study; [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF]; tlc-mon: monoclinic talc (this study; [START_REF] Mainprice | Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure[END_REF]; atg-antigorite serpentine [START_REF] Mookherjee | Trench parallel anisotropy and large delay times: Elasticity and anisotropy of antigorite at high pressures[END_REF]; and chl-chlorite [START_REF] Mookherjee | Unusually large shear wave anisotropy for chlorite in subduction zone settings[END_REF].

(b) Stereographic projections down the c-axis of V P and AV S of both triclinic and monoclinic talc at ~1.3-1.4 GPa, which corresponds to a depth of ~45 km. The black square symbols denote the maximum values, and the white circle symbols denote the minimum values. The solid black lines mark the fixed azimuth angle for Fig. 8c, i.e., 150° for triclinic talc and 90° for monoclinic talc. [START_REF] Mookherjee | Structure and elasticity of serpentine at highpressure[END_REF], antigorite serpentine (atg) [START_REF] Bezacier | Elasticity of antigorite, seismic detection of serpentinites, and anisotropy in subduction zones[END_REF], phlogopite (phl) [START_REF] Chheda | Structure and elasticity of phlogopite under compression: Geophysical implications[END_REF], and chlorite (chl) [START_REF] Manthilake | The anomalous seismic behavior of aqueous fluids released during dehydration of chlorite in subduction zones[END_REF]. The gray and open diamonds represent the data of the upper and lower oceanic crustal (OC) layer of the subducting slab of the central

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. : SCND: single-crystal neutron diffraction; SCXRD: single-crystal X-ray diffraction; SEDXRPD: synchrotron energydispersive; XRPD: X-ray powder diffraction; LDA: local density approximation; GGA: generalized gradient approximation; PAW: projector augmented wave; B3LYP-D*: a hybrid exchange functional with the dispersive force correction; vdW: van der Waals correction. d : Results from this study have been fitted by Murnaghan fitting, 3 rd -order Birch-Murnaghan fitting, and 4 th -order Birch-Murnaghan fitting. The  2 of triclinic and monoclinic talc are 1.1E-2 and 3.9E-4 using Murnaghan fitting, 5.0E-3 and 2.2E-4 using3 rd -order Birch-Murnaghan fitting, 8.6E-5 and 4.3E-6 using 4 th -order Birch-Murnaghan fitting, respectively.

Table 2. Elastic constants for triclinic talc. Note: C ij,0 is the C ij at zero pressure in the unit of GPa, C ij ' is the pressure derivative of C ij which is unitless, and C ij '' is the second order pressure derivative of C ij in the unit of GPa -1 . Note: C ij,0 is the C ij at zero pressure in the unit of GPa, C ij ' is the pressure derivative of C ij which is unitless, and C ij '' is the second order pressure derivative of C ij in the unit of GPa -1 .

Research Highlights

 Talc is one of the most elastically anisotropic mineral in subduction zone settings.



Talc-bearing lithology used to explain ultra-slow velocity, high V P /V S ratio, and large delay time



  also unique because-(a) it has a shallow-flat-steep subduction configuration, and (b) the subducting slab is just located below the Moho and thus is almost devoid of overlying mantle wedge and directly overlain by the continental crust of North American plate.

Figure 1 :

 1 Figure 1: Comparison between experimental results and our predicted pressure vs. unit-

Figure 2 :

 2 Figure 2: (a-c) Lattice parameters (a) a-, (b) b-, (c) c-axis of talc as a function of

Figure 3 :

 3 Figure 3: Elastic constants C ij of triclinic talc as a function of pressure. Triclinic talc has

Figure 4 :

 4 Figure 4: Elastic constants C ij of monoclinic talc as a function of pressure. Monoclinic

Figure 5 :

 5 Figure 5: (a) Voigt-Reuss-Hill averaging bulk (K) and shear (G) moduli of talc as

Figure 6 :

 6 Figure 6: (a) Compressional and shear wave anisotropy, AV P and AV S , as a function of

Figure 7 :

 7 Figure 7: Poisson's ratio of triclinic talc along different directions at (a) low and (b) high

Figure 8 :

 8 Figure 8: (a) Schematic diagram of the central Mexican subduction zone with the

  triclinic; mon: monoclinic. b : M: Murnaghan fitting; III-BM: 3 rd -order Birch-Murnaghan fitting; IV-BM: 4 th -order Birch-Murnaghan fitting.

  c

Talc

Table 1 .

 1 Equation of state parameters for talc compared with previous studies. Number in the parenthesis is the error in the last digit.

		Symmetry a	Pressure range (GPa)	E 0 (eV)	V 0 (Å 3 )	K 0 (GPa)	K'	K'' (GPa -1 )	EoS fitting b	Methods c
	Gatta et al., 2013	tri	0-15.5		454.7( 9)	57(3)	4.9( 5)		M	SCND+SC XRD
		tri	0-15.5		454.7( 10)	56(3)	5.4( 7)		III-BM	SCND+SC XRD
	Pawley et al., 2002	mon	0-6.2		912(2) 41(4)	6(2)		M	SEDXRPD
	Pawley et al., 1995	mon	0-6.05		902.7( 2)	41.6( 9)	6.5( 4)		M	XRPD
	Gleason et al., 2008	tri	0-10.0		453.6( 4)	30(17 )	18(2 7)		III-BM	XRPD
		mon	0-10.4		915(8) 44(2)	18(1 )		III-BM	XRPD
	Stixrude et al., 2002	tri/primitiv e cell	-1.10-26.07		217.5( 1)	37.8( 4)	13.6 (2)	-4.0	IV-BM	LDA static
		tri/primitiv e cell	-1.10-26.07		221.8( 2)	29(5)	17	-8.6	IV-BM LDA 300 K
	Mainprice et al., 2008	tri	0-11.60		452.0	32.1	17.4	-9.1	IV-BM	GGA PAW
		mon	0-12.52		913.2	34.3	11.5	-3.0	IV-BM	GGA PAW
	Ulian et al., 2014	tri	-2.2-10.0		450.3 4	56.25 5.66		III-BM	B3LYP-D*
	this study d	tri	-2.4-18.1	-231.76( 2)	466.5( 8)	33(2)	8.3( 6)		M	GGA PAW+vdW
		mon	-1.4-10.8	-463.51 8(5)	924.4( 5)	43(1)	7.2( 3)		M	GGA PAW+vdW
	this study d	tri	-2.4-18.1	-231.75( 1)	464.7( 6)	34(1)	10.7 (7)		III-BM	GGA PAW+vdW
		mon	-1.4-10.8	-463.51 4(4)	924.2( 4)	41.8( 9)	8.4( 4)		III-BM	GGA PAW+vdW
	this study d	tri	-2.4-18.1	-231.75 5(2)	461.8( 1)	38.5( 3)	13.2 (1)	-4.3(1 )	IV-BM	GGA PAW+vdW
		mon	-1.4-10.8	-463.51 0(1)	923.4 4(8)	40.0( 2)	12.2 (3)

Table 3 .

 3 Elastic constants for monoclinic talc.

	V	P	C11	C12	C13 C15	C22	C23 C25	C33	C35 C44 C46 C55	C66	KH	GH
	(Å 3 )								(GPa)
	940	-0.7	221. 7	77. 7	7.8 1.7	226. 8	0.4 1.5 43.7 0.7 1.0 0.4	22. 1	73. 0	54.5	23. 2
	920	0.2	215. 8	68. 5	12. 1	3.2	218. 1	2.1 2.6 63.4 1.1 2.9 0.9	30. 3	73. 8	60.4	30. 3
	900	1.3	212. 0	64. 2	19. 3	4.8	215. 7	7.0 3.5 83.2 1.7 6.9 1.5	38. 8	74. 4	66.8	37. 5
	880	2.5	207. 1	61. 6	25. 1	6.5	216. 5	12. 6	4.2	102. 1	2.5	11. 0	2.4	47. 5	74. 8	73.9	44. 6
													5.1	79. 3	72. 8	110. 8	67. 8
									finite strain fit
	924. 4	Cij0	217. 3	69. 5	12. 4	2.9	218. 5	1.1 2.4 59.1 1.1 1.8 0.7	28. 4	73. 7	58.9	29. 6
		Cij'	-4.9	-4.6	4.9 1.7 -1.9	4.3 0.9 20.2 0.7 4.9 0.8 8.8 0.8	6.2	6.2
		Cij''	1.4	1.8	-0.0	-0.4	1.2	0.3	-0.2	-3.3	-0.1	-0.1	-0.1	-1.4	-0.4	-0.3	-0.5

  Pressure dependence of the components of the elastic constant tensor exhibits

	anomalous behavior X3 0.0785 0.0132 0.5600 0.0063 0.1366 0.0119 0.5825 0.0070 0.0325 0.6059 -0.0122 0.1428 0.5912 -0.0105 0.8842 0.9458 -0.2239 0.8329 -0.3296 0.2523 0.0496 0.8760 -0.3491 0.9474 -0.2170 0.8788 -0.0615 0.8315 0.0177 0.8410 0.0098 0.9377 0.9098 -0.0871 1.0038 -0.2615 -0.3851 0.2635 0.0991 0.9356 -0.3733 1.0002 -0.2494 0.9087 -0.0796 0.8437 0.0117 0.8369 0.0149 -0.0677 1.0078 -0.2235 1.2266 -0.5246 1.1244 -0.5850 0.1531 0.2590 1.2120 -0.6211 1.2844 -0.5615 1.0315 -0.2398 0.7995 0.0187 0.7966 0.0197 0.3265 0.3696 0.0433 0.3458 0.1081 0.3278 1.1913 -0.6237 0.2671 0.2127 0.2897 0.1824 0.1313	0.4448 0.0357 0.1759 0.1453 0.0474 0.3233 0.0827 0.4492 0.0431 0.3390 0.0462 0.4901 0.4847 0.0300 0.4622 0.0894 0.2831 0.2379 0.2024 0.4069 0.0602 0.2021 0.2852 0.2873 0.1803	X3 0.0925 0.2606 0.1284 0.4997 0.1159 0.1651 0.1287 0.1509 0.1005 0.4915 0.0782 0.2574 0.2820 0.1153 0.5050 0.0694 0.4987 0.5038 0.1076 0.1128 0.4604 0.2923 0.1206 0.5031 0.1176 0.1141 0.1851 0.5075 0.0803 0.3513 0.0660 0.4686 0.0339 0.4653 0.4828 0.0354 0.3827 0.0472 0.2089 0.1710 0.3794 0.2613 0.3923 0.0888 0.2122 0.1056 0.2656 0.0444 0.0776 0.3978 0.2312 0.3925 0.1738 0.2756 0.1620 0.2178
	X1	0.0393	0.2987	0.0355	0.3021 0.0330	0.2981 0.0715	0.3227 0.0965	X2	X1	0.1616	0.2867 0.1440	0.1478 0.2871	0.1688	0.2857	0.2874 0.1903	X2

Acknowledgments

This work is supported by the US National Science Foundation grant EAR 1763215 and EAR 1753125. YP and MM acknowledge computing resources from XSEDE facilities (GEO170003) and the High-Performance Computing, Research Computing Center, Florida State University. AH acknowledges computing resources from the UK's National Supercomputer Service through the UK Car-Parrinello Consortium (EPSRC Grant No. EP/P022561/1) and project ID d56 "Planetary Interiors". GM acknowledges funding from the INSU-CNRS and the French Government Laboratory of Excellence initiative n°ANR-10-LABX-0006, the Région Auvergne, and the European Regional Development Fund (ClerVolc contribution number 530).

Supplementary data to this article can be found online at XXXX

Mexican subduction zone [START_REF] Kim | Geometry and seismic properties of the subducting Cocos plate in central Mexico[END_REF]. (c) Delay time/ thickness of the anisotropic layer (V S /V S1 V S2 ) as a function of incidence angle of both triclinic and monoclinic talc at ~1.3-1.4 GPa, which corresponds to a depth of ~45 km. The profile of the data is marked by the solid black lines in Fig. 6, i.e., the azimuth angle is 150° for triclinic talc and 90° for monoclinic talc. The incidence angle is between the ray path and c-axis, i.e., 0° indicates the ray path is along the c-axis while 90° indicates the ray path is parallel to a-b plane. Supplementary Data Table 1. The total energy as a function of increasing cut-off energy (Ecutoff) for triclinic talc using GGA with van der Waals correction. The K-points remain unchanged.

Supplementary Data Table 2. The total energy as a function of increasing K-points for triclinic talc using GGA with van der Waals correction. The cut-off energy of 900 eV remains unchanged. Supplementary Data Table 3. Lattice parameters and linear compressibility for talc.

Supplementary Data Table 4. The elastic constants as a function of cut-off energy (Ecutoff) for triclinic talc using GGA.

Supplementary Data Figures

Supplementary Data Figure 1. Comparison of total energy using GGA with van der Waals correction (GGA + vdW) and conventional GGA for triclinic talc. The vertical bars indicate the zero-pressure unit-cell volume: red-GGA + vdW (this study); blueconventional GGA (this study); black-conventional LDA [START_REF] Stixrude | Talc under tension and compression: Spinodal instability, elasticity, and structure[END_REF]; and greenexperiments [START_REF] Gatta | On the crystal structure and compressional behavior of talc: a mineral of interest in petrology and material science[END_REF][START_REF] Gleason | Pressuretemperature studies of talc plus water using X-ray diffraction[END_REF].

Supplementary Data

Declaration of competing interest


M Orlita
email: milan.orlita@lncmi.cnrs.fr

C Faugeras

R Grill

A Wysmolek

W Strupinski

C Berger

W A De Heer

G Martinez

M Potemski

Carrier scattering from dynamical magneto-conductivity in quasi-neutral epitaxial graphene

Keywords: numbers: 71.70.Di, 76.40.+b, 78.30.-j, 81.05.Uw

The energy-dependence of the electronic scattering time is probed by Landau level spectroscopy in quasi neutral multilayer epitaxial graphene. From the broadening of overlapping Landau levels we find that the scattering rate 1/τ increases linearly with energy ǫ. This implies a surprising property of the Landau level spectrum in graphene -the number of resolved Landau levels remains constant with the applied magnetic field. Insights are given about possible scattering mechanisms and carrier mobilities in the graphene system investigated.

Understanding carrier scattering in graphene [1] is one of the fundamental but still unclear issues in the physics of electronic properties of this noble material. Initial, model prediction for graphene [2,3] done for short-and long-range scatterers independent of carrier density assumed the relaxation rate 1/τ that is directly proportional to carrier energy ǫ. This is a simple consequence of the linear density of states with energy, D(ǫ) ∼ |ǫ|, in graphene which displays the Dirac band structure with a constant Fermi velocity v F . The 1/τ ∼|ǫ| relation implies that the low temperature (Boltzmann) conductivity σ ∼ v 2 F τ (E F )D(E F ) of graphene is insensitive to changes of carrier concentration n and/or position of the Fermi energy E F ∼ |n| [2,3]. Surprisingly, experiments on gated graphene flakes show different behavior, the conductivity varies linearly with carrier density: σ(n) ∼ |n| [4,5]. The simple τ (ǫ) ∼ |ǫ| -1 rule may not apply if dominant scatterers are of the specific resonant type or ripples type [6,7], as well as for charge impurities [8][9][10][11]. Nevertheless, the unexpected, strong dependence of σ(n) on carrier concentration remains puzzling [6,12] and it is not fully clarified whether this behavior is intrinsic to graphene or has a well-defined extrinsic origin.

In this Letter we present a positive test of the 1/τ ∼ |ǫ| rule, which we have performed on quasi-neutral layers of epitaxial graphene [13][14][15][16]. To probe this graphene system with a fixed Fermi energy we profit of the dynamical magneto-conductivity method and follow the energy dependence of the spectral broadening Γ of inter Landau level (LL) transitions measured in our infrared transmission experiments. Notably, the optical conductivity response of graphene involves interband excitations across the Dirac point and allows us to to determine the parameter Γ over a broad energy range. We have found that in the limit of strongly overlapping LLs, which corresponds to the quasi-classical limit in graphene and in which the simple relation Γ = /τ (ǫ) is justified, this spectral broadening becomes Γ = α • |ǫ|. This linear dependence on energy is valid at all magnetic fields and for all transition indexes (α = 0.026 in the sample investigated). This experimentally determined scattering rate 1/τ (ǫ) ∼ |ǫ| dependence rules out specific extrinsic scattering mechanisms, such as resonant scatterers or charge impurities, in the epitaxial graphene layers which are protected from the environment. A straightforward but intriguing consequence of our finding is that the number of resolved LLs is constant, independent of the magnetic field. Possible consequences of the 1/τ ∼ |ǫ| relation on the carrier mobility in extrinsically unperturbed graphene are discussed.

The studied sample is a standard multilayer epitaxial graphene (MEG) grown on the C-terminated surface of silicon carbide (4H-SiC[000 1]) [13] with intentional thickness of 50 layers. A significant part of layers displays an electronic band structure as that of an isolated graphene monolayer, which results from the characteristic rotational stacking of these graphene sheets [17]. This is confirmed by micro-Raman spectra measured on our sample, which show a single-component 2D band [18,19], with some Bernal-stacked residuals on selected locations. The sheets studied in this experiment are quasineutral [20], only several layers close to the interface and on the surface of MEG become significantly doped (up to 10 13 cm -2 ) [21].

To measure the infrared transmittance, the sample was exposed to the radiation of a globar, which was analyzed by a Fourier transform spectrometer and delivered to the sample via light-pipe optics. The transmitted light was detected by a composite bolometer which operated at T = 2 K and which was placed directly below the sample. Measurements were done in a Faraday configuration, using a superconducting magnet. The sample is fully opaque in the energy range 85-120 meV and also displays a relatively weak transmission around 200 meV. Both absorption bands are related to phonon-related absorption in the SiC substrate. The spectra presented here are either conventional magneto-transmission spectra normalized by the zero field transmission T (ω, B)/T (ω, 0) or differential spectra T (ω, B + ∆B)/T (ω, B -∆B). The differential technique is more precise and helps to correct for possible magnetic field induced changes in the response of the bolometer.

A representative magneto-transmission spectrum at B = 4 T (normalized to the response at B = 0) of our MEG sample is shown in Fig. 1(a),(b). This spectrum is composed of a series of absorption lines which, accordingly to previous studies [14,20,[22][23][24], can be easily identified as due to optically active transitions between Landau levels L ±n of massless Dirac fermions, with the characteristic energies: E ±n = ±v F 2e B|n|, n = 0, 1, 2 . . . These are the interband transitions: L -(n+1) →L n and L -n →L n+1 , which are reflected in the spectra of our quasi-neutral graphene sheets. From the characteristic field dependence of these transitions,

ω = v F √ 2e B( √ n + √ n + 1), we find v F = (1.025 ± 0.005) × 10 6 m.s -1
, in agreement with previous estimations [14,20]. The L -1(0) →L 0(1) transition disappeared from spectra (due to occupation effect at LL filling factor ν ≈ 6) at fields of 100 mT which implies density n ≈ 10 10 cm -2 in the studied layers.

For a more quantitative analysis of the measured magneto-transmission T (ω, B), we first note that the absorption of light by our graphene sheets is relatively weak and therefore:

T (ω, B) = 1 -C • Re[σ xx (ω, B)]/(ǫ 0 c) (1)
where the proportionality factor C = N eff • 1 2 (κ 2 +3)/(κ 2 + 1) accounts for the effective number N eff of graphene sheets in our sample including the effect of possible partial coverage of the SiC surface, and for multi-reflections of light from the SiC substrate with the refractive index κ = 2.6 [14]. Re[σ xx (ω, B)] is the real part of the dynamical longitudinal conductivity of a single graphene sheet expressed in Kubo-Greenwood formalism [25]:

Re[σ xx (ω, B)] = e 3 2 B ω r,s M r,s (f s -f r )δ Γ (E r -E s -ω), (2)
where 0 ≤ f r ≤ 1 stands for the occupation factor of the r th LL and δ Γ (E) for a Lorentzian of width Γ. M r,s = v 2 F γδ |r|,|s|±1 are velocity operator matrix elements in which γ = 2 if r = 0 or s = 0 and otherwise γ = 1. Equation ( 2) is a simplified version of a more general Kubo formula (see e.g. Refs. 26 and 27) which includes initial and final spectral functions. Our simplified formula can be strictly derived for energy-independent and purely imaginary self-energies of the initial and final LLs, iΓ s and iΓ r , and in the limit of relatively narrow levels/transitions [25], i.e., Γ r,s ≪ ω and Γ r,s ≪ |E r,s |. On the other hand, since we deal with interband excitations only, the condition Γ r,s ≪ |E r,s -E r+1,s+1 | does not have to be fulfilled and adjacent LLs thus may overlap. With these approximations, the broadening Γ in Eq. 2 corresponds to Γ = Γ r + Γ s , and in a view of remarkable electron-hole symmetry of bands in graphene, also corresponds to Γ ≈ 2Γ r,s . At a given magnetic field, the broadening Γ can be viewed as a function of the resonance energy/frequency Γ = Γ( ω s→r ), where

L -1(-2) -> L 2(1) L 0(-1) -> L 1(0) L -n(-n-1) -> L (n+1)n 0 n = 1
ω s→r = |E r | + |E s | ≈ 2|E r,s |.
In further, we also accept the possibility that Γ = Γ( ω) is a continues function of the energy/frequency. As a matter of fact, the conductivity (2) is practically identical if Γ( ω r→s ) is fixed for each transition or Γ( ω) is a slowly varying function over the interval ω r→s ± Γ( ω r→s ). This latter condition as well as Γ r,s ≪ ω and Γ r,s ≪ |E r,s | are a posteriori fulfilled in our experimental data.

Since we probe here quasi-neutral graphene layers at low temperatures, we assume that the Fermi energy is pinned to the 0 th Landau level and therefore f s = 1 and f r = 0 in the case of our interband (r, s = 0), L s →L r transitions. As for a pair of degenerate transitions, L -1 →L 0 and L 0 →L 1 , which involve the 0 th LL, we can, without loss of generality, set f 1 = 0 and f 0 = f -1 = 1 for any magnetic field, since the sum of intensities of L -1 →L 0 and L 0 →L 1 transitions is independent of de facto partial population of the L 0 level (|ν| < 2) [28]. The phenomenologically introduced broadening parameter Γ is the main focus of this work. In general, this parameter depends on the magnetic field, on the transition index, or, as discussed above on frequency ω. Notably, the physical interpretation of Γ is different in the case of well-overlapped or well-resolved LLs.

In the limit of well separated Landau levels, Γ = Γ r +Γ s may correspond to a single-particle quantum level broadening Γ s of the density of states (DOS) [25]. This regime has been theoretically explored in the case of longrange and short-range disorder, especially for scattering centers insensitive to carrier concentration [2]. For shortrange scatterers, Γ s was found to be independent of the LL index but to increase as √ B. In the more complex case of long-range scattering, Γ s may also vary as √ B but the broadening of the 0 th LL is found to be significantly (by a factor √ 2) larger than that of any other levels. Charged impurities, if present, lead to a narrowing of LLs. The width of the LL is then predicted to oscillate with the filling factor and shows an overall 1/B field dependence, Γ ∼ 1/D(E F ), instead of the √ B field dependence [26,27]. As concluded by the authors of Ref. 26, this excludes charge impurities as the dominant source of scattering in MEG, in which √ B-type broadening is characteristic of inter-LL transitions [20].

In the limit of densely spaced and therefore strongly overlapping LLs, attained in graphene at sufficiently high energies -see Fig. 1(d) for illustration -the broadening parameter Γ can be interpreted as the inverse of a transport relaxation time, Γ = /τ within a factor of 2. This interpretation can be justified by the classical Drude approach to describe the carrier motion in a magnetic field [29,[START_REF] Kono | Methods in Materials Research[END_REF], in which the magnetic field effects are apparent when the cyclotron frequency (ω c = v 2 F eB/ǫ for graphene) is comparable to the carrier relaxation rate τ -1 . The transport relaxation time is the only parameter which accounts for broadening effects in the dynamical conductivity response of graphene at zero magnetic field [3] and very likely in the magneto-dynamical conductivity response in the classical limit when the density of states is weakly modulated by the magnetic field [START_REF] Pyatkovskiy | [END_REF]. Notably, our assumption Γ = /τ is favored by the electron-hole symmetry in graphene and, therefore, in neutral sheets likely identical relaxation rates for electrons and holes at energies +ǫ and -ǫ, respectively. The transport relaxation time becomes comparable with the single-particle relaxation time if short-range scattering dominates [32], which is, as shown later on, consistent with our data and which thus facilitates our interpretation.

To compare with experiment, we re-consider the (relative) magneto-transmission spectrum measured at B = 4 T in Fig. 1. To model this spectrum, we assume that σ xx (ω, B = 0) = e 2 /4 . This is also the numerical result from Eq. 2 in the limit of strongly overlapping levels. Then following Eqs. 1 and 2 we fit T (ω, B)/T (ω, 0) with two adjustable parameters/functions C and Γ(ω). In the first and simplest approach, we set Γ to be constant, independent of the transition energy. As shown in Fig 1(a), we then fail to reproduce the high energy part of the spectrum, but can well reproduce the two lowest, bestseparated transitions. We find C = 1.1 and Γ = 3.9 meV by best fitting the shape of the L -1(0) →L 0(1) transition.

However, most of the spectrum consists of overlapping transitions (in fact with the exception of the two lowestenergy transitions) that can be very well reproduced by assuming Γ = α • ω. The best values are α = 0.026 and C = 1.3 from the fit shown in Fig. 1(b). This latter value implies a surprisingly low effective number of layers, N eff ≈ 2, in the present sample. We related this to a par- FIG. 2: (color online) Differential transmission spectra T (B + ∆B)/T (B -∆B). The lowest two spectra have been taken for ∆B = 0.05 T, the other with ∆B = 0.125 T. For clarity, the subsequent curves have been shifted by a factor of 0.045. Experimental curves are accompanied by theoretical fits, assuming the broadening parameter Γ = 0.026 ω. Let us note that the condition B/∆B > 2n must be fulfilled to prevent significant overlap of adjacent transitions that involve n th LL. The dashed line shows the simulated curve for differential transmission at B = 10 T and ∆B = 0.125 T for a constant broadening Γ = 8 meV, which has been set to reproduce the L -1(-2) →L 2(1) line.

tial coverage of the SiC surface by graphene sheets and to the partial AB-stacking, e.g. into bilayers recently identified as a minor component in MEG [33,34]. Perhaps another reason could be an increased dielectric screening inside MEG [35], which can reduce the total absorption significantly. Remarkably, the Γ = 0.026 ω rule is universal, in the sense that it applies to spectra measured at any magnetic field within the range of the present measurements. This is illustrated in Fig. 2 with simulations of differential spectra measured in the region of overlapping n ≥ 1 transitions, at five distinct magnetic fields. The calculated traces, which accurately fit the data, are obtained from Eqs. 1 and 2 with a common set of parameters (α = 0.026, C = 1.3). Here we note that despite a clear overlap between transitions, each of them is still well-defined in a sense that the above discussed assumption of narrow absorption lines and of slowly varying Γ is indeed well satisfied, Γ = 0.026 ω ≪ ω.

To sum up, the broadening parameter Γ seems to be constant for transitions between well-separated levels, but tends to increases linearly with energy when levels start to overlap significantly. This behavior remains unchanged up to the highest resolved transitions, i.e., those which originate from practically merging LLs. This is shown in Fig. 1(d), where the DOS has been plotted for the experimentally derived parameters. As discussed above, in the regime of strongly overlapping levels, the broadening of electronic states is governed dominantly by the zero-field relaxation rate and the observed linear in energy increase of Γ indicates a 1/τ ∼ |ǫ| dependence. Interestingly, the same conclusion can be drawn using a simple semi-classical argument: the number of resolved transitions in our spectra is roughly magnetic field independent, i.e., the onset of LL quantization remains always pinned to a certain level (n 0 ∼10, see Fig. 2). The energy of this onset moves to higher energies as |ǫ| ∼ √ B and the cyclotron frequency corresponding to this onset thus follows a linear in energy dependence ω c = v 2 F eB/|ǫ| ∼ |ǫ|. If we employ the classical condition for this onset, ω c τ ∼ 1, we conclude 1/τ ∼ |ǫ|.

Having established the /τ = α|ǫ| dependence, let us crudely assume that it is not additionally affected by changes in the carrier concentration. The Fermi energy independent conductivity of "our graphene" is then σ = 2 α • e 2 h (σ ≈ 80e 2 /h for α = 0.026) and we can speculate on a low-temperature carrier mobility, µ = ev 2 F τ /E F , in such a model system at various charge densities. As an order of magnitude, we obtain µ ≈ 10 6 cm 2 /(V.s) for n = 10 10 cm -2 , which is consistent with our recent measurements on quasi-neutral graphene [20]. The mobility falls down to µ ≈ 10 4 cm 2 /(V.s) for the carrier density as high as n = 10 12 cm -2 , what reflects fairly well typical mobilities measured for charged, single-layer epitaxial graphene on C-face of SiC or interface layer of MEG [21,36]. Our hypothetical graphene with n = 10 13 cm -2 would have a mobility of µ ≈ 10 3 cm 2 /(V.s), in striking agreement with typical values measured for high concentration graphene on Si-face of SiC [37].

The 1/τ (ǫ) dependence reported here is expected for graphene and points towards conventional scattering mechanisms in quasi-neutral epitaxial graphene. Apart from charge impurities, excluded already earlier in MEG due to √ B-broadening of inter-LL transitions [26], we can also for certain exclude the dominant role of resonant scatterers, which produce the mid-gap states and therefore modify the predicted 1/τ ∼ |ǫ| relation. Those resonant scatters are by definition extrinsic, and maybe important for environment exposed graphene, but are naturally expected to be absent in our protected layers.

To complete our discussion of scattering mechanisms in MEG, we focus on the two lowest energy transitions of Fig. 1(a),(b). These are the two best separated transitions in our spectra, apparently related to nearly nonoverlapping LLs (|n| = 0, 1 and 2) and therefore, their behavior can be tentatively analyzed in reference to theory in Ref. 2. Figure 1(c) shows that the width of both transitions scales roughly as √ B. Moreover, the width of the n = 0 transitions is roughly the same if not even smaller than the width of the n = 1 transition. According to the theoretical analysis of the dynamical magnetoconductivity in the regime of well-separated Landau levels, this observation points towards short-rather than long-range scattering mechanism (both insensitive to carrier energy). Another possibility is the electron-electron scattering which also provides a scattering rate that increases linearly with energy [38].

Concluding, we have investigated the dynamical magneto-conductivity of quasi-neutral epitaxial graphene layers and studied the carrier scattering in these layers. We find that the relaxation rate is linear with the carrier energy. This dependence has been initially expected for graphene, nevertheless, not clearly observed yet. Its interesting implication is that the number of resolved Landau levels is fixed, independent of the magnetic field. The dominant scattering mechanism in the extrinsically unperturbed layers of epitaxial graphene is conventional, very likely due to short-range potentials or due to electron-electron interactions.

1 LFIG. 1 :

 11 Photon energy (meV) Relative transmission

We acknowledge funding received from EuroMag-NETII under the EU contract 228043, from the Keck Foundation, the NSF-MRSEC grant 0820382 from the Partner University Fund at the Embassy of France. This work has been further supported by the projects RTRA DISPOGRAPH, PICS-4340 & 395/N-PICS-FR/2009/0, MSM0021620834 and GACR P204/10/1020, as well as by grants 670/N-ESF-EPI/2010/0, 671/N-ESF-EPI/2010/0 and GRA/10/E006 (EPIGRAT).