T Garandel

R Arras

X Marie

P Renucci

L Calmels

Electronic structure of the Co(0001)/MoS2 interface, and its possible use for electrical spin injection in a single MoS2 layer

Keywords: 72.25.Dc, 72.25.Hg, 72.25.Mk, 73.20.At, 73.63.Rt, 75.70.Ak, 75.70.Cn

The ability to perform efficient electrical spin injection from ferromagnetic metals into twodimensional semiconductor crystals based on transition metal dichalcogenide monolayers is a prerequisite for spintronic and valleytronic devices using these materials. Here, the hcp Co(0001)/MoS2 interface electronic structure is investigated by first-principles calculations based on the density functional theory. In the lowest energy configuration of the hybrid system after optimization of the atomic coordinates, we show that interface sulfur atoms are covalently bound to one, two or three cobalt atoms. A decrease of the Co atom spin magnetic moment is observed at the interface, together with a small magnetization of S atoms. Mo atoms also hold small magnetic moments which can take positive as well as negative values. The charge transfers due to covalent bonding between S and Co atoms at the interface have been calculated for majority and minority spin electrons and the connections between these interface charge transfers and the induced magnetic properties of the MoS2 layer are discussed. Band structure and density of states of the hybrid system are calculated for minority and majority spin electrons, taking into account spin-orbit coupling. We demonstrate that MoS2 bound to the Co contact becomes metallic due to hybridization between Co d and S p orbitals. For this metallic phase of MoS2, a spin polarization at the Fermi level of 16 % in absolute value is calculated, that could allow spin injection into the semiconducting MoS2 monolayer channel.

Finally, the symmetry of the majority and minority spin electron wave functions at the Fermi level in the Co-bound metallic phase of MoS2 and the orientation of the border between the metallic and semiconducting phases of MoS2 are investigated, and their impact on spin injection into the MoS2 channel is discussed.

I.INTRODUCTION

Triggered by the success of graphene, the field of two-dimensional (2D) semiconductor (SC) crystals based on transition metal dichalcogenide (TMDC) monolayers encounters a spectacular development [1]. This new class of exciting materials presents several original characteristics. Their direct band gap [2] allows investigations by optical techniques, and the strong spin-orbit coupling combined with the lack of inversion symmetry result in nonequivalent valleys in the reciprocal space, that can be selectively addressed by circularly polarized light due to optical selection rules [3]. In a sense, the valley index could constitute a novel degree of freedom to carry and process information ("valleytronics" [4]). The exploration of the spin and valley degrees of freedom by all-optical experiments have been carried in the past years [5,6,7]. However, electrical spin injection yet remains elusive in these systems [8].

The ability to inject spin polarized currents would pave the way for new spintronic and spinoptronic devices [9,10,11], where one can imagine to take benefit from optical and spin properties of these materials. It could also benefit to future valleytronic devices, where electrical generation and control of the valley index is required. Due to the unique correlation between the spin and valley indices of charge carriers, this point is directly linked to electrical injection of spin polarized (and energy selected) carriers in TMDC. The problem of electrical injection into 2D semiconductors like MoS 2, MoSe2, WSe2, or WS2 is thus a new challenge in this field, after the recent exploration of electrical spin injection into tridimensional (3D) semiconductors such as GaAs [9,10,11], Si [12], Ge [13], and in 2D Graphene [14]. The first experimental proof of electrical spin injection in 2D semiconductors has just been reported very recently: Ye et al. have shown that spin polarized holes can be injected in WSe2 from the ferromagnetic semiconductor GaMnAs that acts as a spin aligner [15]. Unfortunately, this ferromagnetic material has a Curie temperature near 200K, far below room temperature. An alternative way consists in using ferromagnetic metals (FM) like Cobalt, Iron or Nickel, with Curie temperatures well above room temperature. In the context of the electrical spin injection into 3D semiconductors in the diffusive regime, it is well established that one has to overcome the impedance mismatch [16] between the FM and the SC layers: this is usually performed by inserting an oxide layer or a tunnel barrier [17]. The electrical spin injection from a ferromagnetic electrode into a single TMDC layer should, however, be totally different from the case of spin injection in a 3D semiconductor and the problem has to be reconsidered. In particular, the semiconductor nature of MoS 2 just below the metallic contact is questionable because covalent bonds could be formed between the metal and the TMDC monolayer, as discussed by Allain et al. [8]. Modification of the magnetic properties of MoS2 below the contact, in particular a spin-polarization of the electron states near the Fermi level and induced magnetic moments, could also occur due to the direct contact between MoS2 and the magnetic layer. First-principles calculations of the electronic structure based on the density functional theory (DFT) constitute a tool of choice to understand the bonding mechanisms between the FM and the TMDC layers and to give a clear description of the spin-polarized electron states at their interface.

Most of the first-principles studies on interfaces between metals and a single TMDC layer have focused on non-magnetic metals like Ir, Pd, Ru, In, Ti, Au, Mo or W [18,19]. Only few studies have concerned interfaces between Fe, Co or Ni and a TMDC monolayer: Dolui et al. have studied the Giant Magnetoresistance of Fe/MoS2/Fe magnetic tunnel junctions and demonstrated that MoS2 becomes conductive when the MoS2 spacer only contains one or two layers, due to the strong interaction between Fe and S atoms; this behavior could, however, be due to the fact that Fe is present on both sides of the MoS2 layer, and the case of a single Fe/MoS2 interface has not been investigated there [20]. Considering an interface between a single Co atomic layer and a MoS2 sheet, Chen et al. showed that the electronic structure of these two monolayers is drastically modified by a strong interface binding [21]; their results are however strongly influenced by the extreme thinness of the Co layer (which is so thin that it even becomes half-metallic), while Co electrodes in real Co/MoS2-based devices would certainly be thicker and contain several Co atomic layers. Leong et al. and Co/MoS2 interfaces [23]. Their study of the Fe(111)/MoS2 interface is based on a supercell in which a 3x3 slab of Fe(111) and a 4x4 cell of MoS2 are stacked together. For Co/MoS2, they used a supercell in which a MoS2 monolayer is sandwiched between face-centered cubic (fcc) Co layers and an atomic structure which should not correspond to that of a real Co/MoS2 interface, not only because Co actually crystallizes in the hexagonal compact (hcp) structure, but also because their multilayer is based on 4x4 Co(111) atomic layers superimposed on a 3x3 MoS2 single layer: considering the experimental values of the lattice parameters (0.2507 nm for hcp Co and 0.312 nm for MoS2), this interface corresponds to a relatively high lattice mismatch of 6.9%. Moreover, the MoS2 layer would only be bound to Co on one of its two sides, in a realistic MoS2/Co contact. For all these reasons, the genuine atomic structure of the Co/MoS2 interface may probably be different from that used by these authors.

In the present paper, we considered supercells built from 5x5 Co(0001) atomic layers with the hcp stacking and a 4x4 MoS2 single layer. These stacking would correspond to the very small lattice mismatch of 0.4% and would be more realistic for calculating the physical properties of the Co/MoS2 interface between a Co magnetic electrode and a single MoS2 layer.

After a brief description of the first-principles methods that we have used, we will first describe the atomic structure of the Co(0001)/MoS2 interface, before giving details on the electronic states, magnetic moments, and charge transfers at the interface. We finally discuss the physical properties of the Co(0001)/MoS2 interface in the perspective of spin injection in the 2D semiconductor MoS2, and give insights on the possible utilization of the low strained Co/MoS2 contact for in plane spin transport in lateral channels [17].

II.FIRST-PRINCIPLES METHODS

The ground state energy, the charge and the spin densities of all the supercells have been calculated self-consistently using the full-potential augmented plane waves + local orbitals (APW+lo) method implemented in the code WIEN2k [START_REF] Blaha | WIEN2k, an augmented plane wave+local orbitals program for calculating crystal properties[END_REF]. The Kohn-Sham equation has been solved in the framework of the density functional theory (DFT), using the parametrization proposed by Perdew, Burke and Ernzerhof for the exchange and correlation potential that was treated within the generalized gradient approximation (GGA) [START_REF] Perdew | [END_REF]. In all our supercells, we used atomic sphere radii of 1.8, 1.8, and 2.0 atomic units (a. u.), respectively for Co, S and Mo atoms. The largest wave vector Kmax used for expanding the Kohn-Sham wave functions in the interstitial area between atomic spheres is given by the dimensionless parameter RminKmax=6.0, where Rmin is the smallest atomic sphere radius of the supercell; this corresponds to an energy cut-off of 151 eV. The irreducible wedge of the Co/MoS2 supercell two-dimensional Brillouin zone was sampled with a k-mesh of typically 24 different k-vectors, generated with a special k-grid used to perform Brillouin zone integrations with the modified tetrahedron integration method.

To model the Co/MoS2 interface, we used rather big symmetric supercells consisting in a Co slab with hcp stacking and a thickness of five 5x5 (0001) monolayers (MLs), covered on each of its two sides by a 4x4 MoS2 single layer, followed by vacuum. The thickness of the Co layer is sufficient to recover most of the electronic structure of bulk hcp Co at the center of the slab (shape of the density of states curves, value of the spin magnetic moment) and the thickness of the vacuum separation between periodically adjacent MoS2 layers is above 1 nm, large enough to avoid interaction effects between them. We have chosen to use a periodic slab with a single MoS2 layer on both sides of Co to get also identical surfaces on both sides of the Co slab and vacuum separation, which avoids artefacts such as charge transfer between surfaces across the Co layer. This kind of huge supercells contain 125 Co, 64 S and 32 Mo atoms and up the 36 non-equivalent atoms, depending on the relative positions of S and Co atoms at the interface. This high number of atoms is a necessary requirement to treat the problem of the realistic low-strained single MoS2 contact that can be found, for example, at the source or the drain of a spin-field effect transistor (FET) based on a MoS2 channel. The lattice parameter that we chose for the Co(5x5)/MoS2(4x4) unit cell corresponds to four times the lattice parameter calculated for MoS2 (0.319 nm).

The atomic structure of all the different MoS2/Co(5MLs)/MoS2 supercells that we have considered has been obtained by minimizing the forces acting on the different atoms. The minimum energy is achieved when all the atoms have reached their equilibrium position in the supercell. London dispersion forces are not included in the calculation.

Once the atomic structure has been calculated, we can choose to include spin-orbit coupling effects at each self-consistent loop, to check whether these effects modify the electronic structure of the Co(0001)/MoS2 interface. This is performed within the second-order perturbation theory, for which we chose to include unoccupied electron states up to the maximum energy of 48 eV above the Fermi level, the magnetization being oriented along the Co(0001) axis.

III. ATOMIC STRUCTURE OF THE Co(0001)/MoS2 INTERFACE

We considered three different supercells, which correspond to three different manners of superimposing the 4x4 unit cell of 1H-MoS2 (shown on Figure 1a) on the 5x5 unit cell of Co(0001) (Fig. 1b). The first supercell (further labelled Supercell1) corresponds to the case where one of the interface S atoms of the 4x4 MoS2 cell (for instance, the S atom at the corners of the 4x4 MoS2 cell shown in Fig. 1a) is located just above one of the fcc hollow atomic sites of the Co(0001) surface. The second supercell (Supercell2) corresponds to the case where one of the interface S atoms is on a top atomic site (i. e. just above an atom of the surface Co ML). We finally considered the last supercell (Supercell3), for which one of the interface S atoms is above one of the hcp hollow sites of the Co(0001) surface (i. e. just above an atom of the subsurface Co ML). These three supercells are represented in Figure 2. The atomic structure of these three supercells has been calculated self-consistently without including spinorbit coupling. After all the atoms have reached their equilibrium position, we observed that the ground state energy is the lowest for Supercell1, and respectively 0.0827 eV and 0.813 eV higher for Supercell2 and Supercell3. These energy differences correspond to the whole supercells, which all contain 2 MoS 2/Co interfaces with 16 MoS2 formula units in the 4x4 MoS2 cell. Consequently, the difference per MoS2 formula unit, between the ground state energies of the different supercells is only of 2.6 meV between Supercell2 and Supercell1, and of 25.4 meV between Supercell3 and Supercell1. From now, we will only consider the physical properties of Supercell1, which corresponds to the lowest energy interface structure. remaining S and Co interface atoms. These interatomic distances are very close to those which have been measured for bulk cobalt sulfides (0.232 nm for CoS2 with the pyrite structure [26]).

Each of the two

This clearly shows that bonding between the MoS2 single layer and the Co(0001) surface has a covalent nature and is not due to the Van der Walls interaction. Each of the interface Co atoms are bound to a single S atom, except the four ones marked by a star in Fig. 3.

The atomic layers present a small warping near the Co/MoS2 interface, due to the different numbers of chemical bounds formed by the non-equivalent interface S atoms. Based on the averaged values of the z-coordinates (the z-axis being perpendicular to the interface) calculated for the atoms in the different atomic layers, we can estimate the average distance between successive atomic layers: we obtain an average distance of 0.205 nm between the interface Co and interface S atomic layers, and average distances of 0.160 and 0.154 nm between the Mo layer and the interface S and external S layers, respectively. These later distances can be compared to the distance of 0.157 nm calculated between the S and Mo atomic layers of an isolated MoS2 sheet. The undulation of the successive atomic layers (difference between the highest and lowest z-coordinates) is respectively of 0.011 nm, 0.029 nm, 0.026 nm and 0.023 nm for the interface Co, interface S, Mo and external S atomic layers.

IV. ELECTRONIC STRUCTURE OF THE Co(0001)/MoS2 INTERFACE

The majority and minority spin band structures of the Co/MoS2 slab are shown in Figures 4b and4c. They contain a huge number of bands, some of them corresponding to cobalt bands and others to MoS2 bands, folded on themselves, with important band gap opening at the center and the edges of the two-dimensional Brillouin zone. The shaded areas which appear in these band structures correspond to the Co d-band continuum (below -0.5 eV for majority spin and across the Fermi level for minority spin electrons). Additional bands, which do not correspond to the folded bands of Co or MoS2 also appear in this figure; this is for instance the case between -0.5 eV and 0.5 eV for majority spin electrons. These new bands correspond to interface Bloch states involving covalently bound Co and S atoms. They strongly modify the physical properties of MoS2, giving a metallic behavior to this layer, induced by interface covalent bonding. Some of these interface bands have a non-negligible dispersion near the Fermi level (electrons in the metallic phase of MoS2 at the Co/MoS2 interface will not have a high effective mass).

Each energy band in Figs. 4b and4c is drawn with small circles, the radius of which is proportional to the contribution of the MoS2 layer to the corresponding Bloch states. The band of the Co/MoS2 slab corresponding to the bottom of the conduction band of the MoS2 single layer appears near 0.32 eV above the Fermi level (it is indicated with a blue arrow on Fiq.4b). This band is more clearly visible for minority spin (Fig. 4b) than for majority spin electrons for which it is located in the continuum of Co d-bands (Fig. 4c). The difference between the energy of this band and the Fermi level corresponds to the height of the Schottky barrier [19], the value of which can be estimated at 32 . 0 = B φ eV for the Co/MoS2 interface. Our calculation of the Schottky barrier height is performed at the DFT level. As discussed by Zhong et al. [27], this seems to give results in a better agreement with experimental ones than those computed with the GW method, probably due to the fact that many-electron effects are greatly depressed by the charge transfer at the MoS2/metal interface, which significantly screens electron-electron interaction [27].

Figure 5 shows the density of states (DOS) of the MoS2 layer at the Co(0001)/MoS2 interface. This DOS curve is continuous between -8 eV and energies well above the Fermi level, confirming the metallic character of the MoS2 layer when it is covalently bound to the cobalt surface. This holds for majority as well as for minority spin electrons. Despite the strong modification of the electronic structure of MoS2, important DOS peaks can be identified in Fig. 5 as belonging to the valence and conduction bands of the isolated MoS2 layer: The DOS curves of the isolated MoS2 sheet has also been represented in Fig. 5, where it has been shifted to coincide to the main DOS peaks of MoS2 at the Co/MoS2 interface. The bottom of the conduction band in the shifted DOS curve of the isolated MoS2 layer is near 0.32 eV above the Fermi level, which confirms the value of the Schottky barrier height estimated from the band structure of the supercell. This estimation of the Schottky barrier height is larger than the values measured experimentally (between 60 meV in [28] and 121 meV [29]). These measurements are however performed on exfoliated MoS2 flakes. In these cases, the flakes are exposed to air before Co is deposited. It means that impureties and molecules can be trapped at the interface, resulting in localized states within the gap that can modify the pinning of the fermi level, and thus the Schottky barrier height. It would be interesting to compare our result with a full epitaxial Co/MoS2 hybrid system elaborated in ultra-high vacuum, when it will be available.

Note that our estimation of the Schottky barrier height of the Co/MoS2 interface is comparable to values found in previous studies for similar interfaces involving another metal, like the Ti/MoS2 system [19].

The majority and minority spin densities of states give access to the spin polarization S P near the Fermi level in the MoS2 layer. This important quantity, defined as the difference between the majority and the minority spin densities of states divided by their sum, clearly when Mo and S atoms keep the same positions as in Supercell1 and Co atoms have all been removed. The majority spin space-dependent charge transfer can finally be obtained from

( ) ( ) ( ) ( ) { } r r r r 2 , , MoS Co n n n n ↑ ↑ ↑ ↑ + - = ∆
. The minority spin charge transfer is similarly given by 7 shows a top and a side view (the direction of observation corresponds to the blue arrow in Fig. 3) of the calculated three-dimensional majority and minority spin charge transfers interface is due to an excess of minority spin electrons, concomitant with a lack of majority spin electrons on interface Co atoms. Similarly, the magnetic moment of Mo atoms is due to an excess of majority spin and a lack of minority spin electrons for the Mo atoms which have a positive spin magnetic moment (right hand side of Figs. 7b and7d), and mostly to an excess of minority spin electrons for Mo atoms which have a negative magnetic moment (left hand side of Fig. 7d).

( ) ( ) ( ) ( ) { } r r r r 2 , , MoS Co n n n n ↓ ↓ ↓ ↓ + - = ∆ . Figure

VII. PERSPECTIVES: TOWARDS ELECTRICAL SPIN INJECTION IN MoS2

In the context of spin injection into bulk semiconductor materials, it has been established that the so-called conductivity mismatch [16] between the ferromagnetic metal injector and the semiconductor constitutes a fundamental obstacle for efficient spin injection at the ferromagnetic metal/semiconductor interface in the diffusive regime. To circumvent this problem, it has been shown that a thin tunnel barrier must be introduced between FM and SC, inducing an effective spin dependent resistance [17]. A possible tunnel barrier that can be exploited is the natural Schottky barrier that occurs between the two materials. As spin injection in tunnel regime is desirable, a careful engineering of the doping in the semiconductor close to the interface is required in order to make the Schottky barrier thin enough to behave as a tunnel barrier. It has been realized successfully in Fe/GaAs system [10] where GaAs was gradually n-doped up to 10 19 cm -3 at the interface, resulting in a very efficient electrical spin injection from Fe into GaAs in tunnel regime.

In the context of TMDCs, the general problem of electrical contacts on TMDCs for transport in a two-dimensionnal channel is a challenging task (tackled for example by phase engineering techniques [31,32], as well as using contacts based on heterostructures involving graphene [22]. Concerning spin injection with Co/MoS2 (or other FM/TMDC [33]) interfaces, one has to consider the Schottky barrier between the metallic phase of MoS2 just below the contact, labeled hereafter (MoS2)*, and the semiconductor MoS2 channel out of the contact.

As in GaAs, one could imagine to strongly increase the doping in MoS2 close to the (MoS2)*/MoS2 one-dimensional border, see Figure 8. Even if, up to now, spatially controlled chemical doping of a single layer TMDC is still a challenge, such an in-plane localized doping has been already obtained with the help of additional gate electrodes developed successfully for in-plane P-I-N junctions in TMDC Light Emitting Diodes [34]. Considering the electrical spin injection tunnel process at this (MoS2)*/MoS2 (n-doped) one-dimensional interface, one has to compare, for majority and minority spin, the compatibility of the symmetries of the electron wave functions in (MoS2)* at the Fermi level, with the ones in the conduction band of the MoS2 single layer semiconductor channel, in order to estimate the efficiency of the tunneling process.

The conduction electron wavefunctions in the isolated MoS2 channel exhibit a strong Mo dz 2 character (as well as a smaller Mo s character), and a S (px+py) character. The DOS at the Fermi level calculated for (MoS2)* is qualitatively different from the DOS at the bottom of the conduction band of MoS2. Due to the strong hybridization between Co and S atomic orbitals at the Co/MoS2 interface, Bloch electron states at the Fermi level result from a non-trivial combination of atomic orbitals that involve Mo-dz 2 , S-(px+py) and also other orbitals, see Table 1. However, the contribution of Mo-dz 2 and S-(px+py) atomic orbitals is not negligible and still represents 33% and 37% of the total DOS of (MoS2)* at the Fermi level, respectively for majority and minority spin electrons. The symmetry of the majority and minority spin electron states at the Fermi level in (MoS2)* is thus partly compatible with the symmetry of Bloch states in the conduction band in the MoS2 channel.

The efficiency of electrical injection in the MoS2 channel in the tunnel regime should also depend on the direction of the one dimensional border between (MoS2)* and MoS2. We know that the Bloch vector of electrons in the conduction band of the MoS2 channel, after tunneling through the Schottky barrier, corresponds to one of the K-valleys. We also know, using the simple model of a free electron with mass m and energy E propagating in a plane towards a straight one-dimensional potential step where the potential jumps from 0 to

E U >

, that the penetration depth that characterizes the exponential decay of the wave function after the step is given by

2 1 2 // 2 2 2 2 -                   + - m k E U m h h ; //
k is the component of the electron two- dimensional Bloch vector parallel to the straight border, and

TABLE CAPTIONS:

Table 1: Total and partial (for the most important atomic orbitals) Mo and S majority and minority spin density of states at the Fermi level for the Co/MoS2 interface. All the results are given in the same arbitrary unit.

FIGURE CAPTIONS

  have studied the Ni(111)/MoS2 interface, but they mostly focus their study on the consequences of the insertion of a graphene layer between Ni and MoS2, than on a detailed description of the Ni(111)/MoS2 interface [22]. Finally, Yin et al. have recently calculated the electronic structure of the Fe/MoS2

  Co/MoS2 interfaces in the supercell involves 25 Co and 16 S atoms. One of these 16 interface S atoms (further labelled S3) is covalently bound to 3 different but equivalent interface Co atoms (labelled Co3); 3 of the 16 interface S atoms (labelled S2) are each bound to 2 different but equivalent Co atoms (Co2), and all the 12 remaining interface S atoms are bound to a single Co atom. The position of all these atoms of the Co/MoS2 interface is shown in Figure 3. The distance between interface S and Co atoms is of 0.236 nm between S3 and Co3, 0.234 nm between S2 and Co2, and of 0.221 or 0.222 nm between the 12

.

  Red and green areas respectively correspond to a local excess and a local lack of electrons. As expected, we see on this figure that charge transfers occur between interface Co and S atoms along the Co-S covalent bonds. This figure shows that the reduction of the Co magnetic moment at the

.

  It follows that the straight (MoS2)*/MoS2 border should be perpendicular to the (ΓK) direction of the two-dimensional Brillouin zone, in order to get a spin-polarized current with maximum intensity reaching a K-valley in the MoS2 was granted access to the HPC resources of CALMIP supercomputing center under the allocation p1446 (2014-2016).

Figure 1 :

 1 Figure 1: Atomic structure (top views) of (a): the 4x4 unit cell of MoS2, and (b): the 5x5 unit cell of a 5MLs Co(0001) slab.

Figure 2 :

 2 Figure 2: Side views of the atomic structure of the Co/MoS2 supercells corresponding to (a): Supercell1 (b): Supercell2, and (c): Supercell3.

Figure 3 :

 3 Figure 3: Top view of the atomic structure of the Co(0001)/MoS2 interface for Supercell1. The atoms Co2 (dark blue spheres), Co3 (medium blue), S2 (red) and S3 (orange) discussed in section III are indicated. The Co atoms marked by a star are those which are not directly bound to a S atom.

Figure 4 :

 4 Figure 4: Band structure of (a): the 4x4 MoS2 single layer supercell, (b): the Co/MoS2 slab for majority spin, (c): the Co/MoS2 slab for minority spin, (d): the 5x5 Co(0001) slab for majority spin and (e): the 5x5 Co(0001) slab for minority spin. The radius of the red circles in panels (b) and (c) is proportional to the MoS2 contribution to the electron states. The blue arrow in panel (a) indicates the minimum of the conduction band in MoS2; its corresponding position in the band structure of the Co/MoS2 slab is also shown with a blue arrow in panel (b), where it corresponds to the height of the Schottky barrier.

Figure 5 :

 5 Figure 5: Contribution of a MoS2 monolayer to the density of states of Supercell1 (dark curves).The density of states of an isolated MoS2 layer is also represented after an energy shift (red curves). The upper and lower parts of the figure respectively correspond to majority and minority spin electrons.

Figure 6 :

 6 Figure 6: Spin polarization in the MoS2 layer, calculated without (red curve) and with (dark curve) spin-orbit coupling.

Figure 7 :

 7 Figure 7: (a): Top view and (b): side view of the majority spin charge transfer

Figure 8 :

 8 Figure 8: Sketch representing (a): the bottom view (below the Co contact) and (b): the side view of the physical area involving the Co(0001) contact, (MoS2)*, the Schottky barrier (MoS2)*/ MoS2 obtained by suitable doping and the MoS2 channel. The one-dimensional border which should be designed to lower the exponential decay of wave functions in the Schottky barrier is indicated by red a dashed line in (a), together with the (ΓK) direction of MoS2 (red arrow). The corresponding Schottky barrier profile is represented in panel (c).

Figure 3 Figure 4 Figure 5

 345 Figure 3

Figure 6

 6 Figure 6

ACKNOWLEDGEMENTS

The authors acknowledge the Université Fédérale de Toulouse-Midi-Pyrénées and Région Midi-Pyrénées for the PhD grant SEISMES as well as the grant NEXT n° ANR-10-LABX-0037 in the framework of the « Programme des Investissements d'Avenir". This work

indicates if a spin-polarized current can be injected through the Co/MoS2 interface to a MoS2 channel. DOS curves shown in Fig. 5 The estimation of the spin polarization at the Fermi level for Co/MoS2 interface is of importance for spin injection. As pointed out by Mazin [30], an accurate determination of the electric current spin polarization would however require complementary DFT-based calculations (including the transmittance of the whole complex structure and the effects of the bias voltage).

V. SPIN MAGNETIC MOMENTS AT THE Co(0001)/MoS2 INTERFACE

The spin magnetic moment of Co atoms is on average 8% lower at the Co(0001)/MoS2 interface than in bulk hcp Co (1.69 µB), with values that depend on the kind of interface S atom to which they are covalently bound: The interface Co atoms that show the highest spin magnetic moment (1.66 µB and 1.63 µB) are the 4 atoms which are not bound to S atoms, followed by Co3 (1.62 µB) and Co2 atoms (1.57 µB). All the other interface Co atoms have a spin-magnetic moment between 1.48 and 1.50 µB: the lowering of the interface Co atom magnetic moment is more important when Co atoms are more strongly bound to S atoms, with a shorter Co-S bond length. All the interface S atoms have a small spin magnetic moment with the same sign as the Co atom magnetic moments, with values between 0.012 and 0.016 µB.

The spin magnetic moment of S atoms in the external S layer is even smaller (0.003 to 0.004 µB). The spin magnetic moment of Mo atoms (between -0.029 and -0.024 µB) is antiferromagnetically coupled to the Co and S magnetic moments, except when these Mo atoms are bound to one or to two S2 atoms (in this case, Mo spin magnetic moments respectively take the positive values 0.008 and 0.050 µB).

VI. CHARGE TRANSFER AT THE Co(0001)/MoS2 INTERFACE

To calculate the charge transfer between atoms induced by covalent bonding at the Co(0001)/MoS2 interface, we proceeded as follows: first, we calculated the majority spin electron density ( )

VIII. CONCLUSIONS

In this paper we have investigated the electronic structure of a single low-strained hcp Co(0001)/MoS2 interface, using first-principles calculations based on the functional density theory, in order to estimate the potentiality of a cobalt injector for electrical spin injection into a MoS2 monolayer, in view of spintronic devices as spin-FET. We first described the lowest energy atomic structure and show that interface S atoms are covalently bound to one, two or three interface Co atoms. A lower spin magnetic moment is observed for Co atoms at the interface, together with a small magnetization of S atoms. Mo atoms also hold small magnetic moments which can takes positive as well as negative values. The induced magnetic moments have been interpreted in terms of majority and minority spin charge transfers at the interface.

Band structures and density of states curves have been calculated for minority and majority spin electrons in the hybrid system, taking into account spin-orbit coupling. We demonstrate that MoS2 just below the cobalt contact becomes metallic due to hybridization with Co d orbitals.

TABLES:

FIGURES:


Sudhanya Chatterjee

Olivier Commowick

Onur Afacan

Simon K Warfield

Christian Barillot

MULTI-COMPARTMENT MODEL OF BRAIN TISSUES FROM T2 RELAXOMETRY MRI USING GAMMA DISTRIBUTION

Keywords: T 2 relaxometry, microstructure, brain

The brain microstructure, especially myelinated axons and free fluids, may provide useful insight into brain neurodegenerative diseases such as multiple sclerosis (MS). These may be distinguished based on their transverse relaxation times which can be measured using T 2 relaxometry MRI. However, due to physical limitations on achievable resolution, each voxel contains a combination of these tissues, rendering the estimation complex. We present a novel multi-compartment T 2 (MCT2) estimation based on variable projection, applicable to any MCT2 microstructure model. We derive this estimation for a three-gamma distribution model. We validate our framework on synthetic data and illustrate its potential on healthy volunteer and MS patient data.

INTRODUCTION

MRI voxels of the human brain are heterogeneous in terms of tissue types due to the limited imaging resolution and physical constraints. Each voxel in the white matter (WM) contains a large number of myelinated and non-myelinated axons, glial cells and extracellular fluids [START_REF] Tomasch | Size, distribution, and number of fibres in the human corpus callosum[END_REF][START_REF] Lancaster | Three-pool model of white matter[END_REF]. For example, every square millimeter of the corpus callosum in a human brain has more than 100,000 fibers (myelinated and non-myelinated) of varying diameters [START_REF] Tomasch | Size, distribution, and number of fibres in the human corpus callosum[END_REF]. These tissues can be distinguished based on their T 2 relaxation times. Myelin being a tightly wrapped structure has a very short T 2 relaxation time of 10 milliseconds (ms) [START_REF] Lancaster | Three-pool model of white matter[END_REF]. The estimated T 2 relaxation time of the myelinated axons is 40ms [START_REF] Lancaster | Three-pool model of white matter[END_REF]. The ventricles and tissue injury regions contain free fluids which have a high T 2 relaxation time (>1000ms). The T 2 relaxation values between those of myelin and myelinated axons and the free fluids correspond to the glial cells and extracellular tissues [START_REF] Lancaster | Three-pool model of white matter[END_REF]. An ability to obtain the condition of these tissues can help us gain better insights into the onset and progress of neurodegenerative diseases such as multiple sclerosis (MS). Myelin water fraction (MWF) has been computed from T 2 relaxometry images using a variety of approaches [START_REF] Mackay | Magnetic resonance of myelin water: An in vivo marker for myelin[END_REF][START_REF] Akhondi-Asl | Fast myelin water fraction estimation using 2D multislice CPMG[END_REF]. Most of these methods primarily focus on the MWF estimation. However, MWF alone might not be able to convey the en-tire information since it is a relative measurement. For example, in MS patients a decrease in MWF in a WM lesion might be caused by myelin loss or fluid accumulation due to tissue injury or both. Hence for relative measurements like water fraction (WF), all the WF maps should be observed simultaneously for a complete understanding of the tissue condition. Here we propose an estimation framework to obtain brain microstructure information using a multi-compartment tissue model from T 2 relaxometry MRI data. The T 2 space is modeled as a weighted mixture of three continuous probability density functions (PDF), each representing the tissues with short, medium and high T 2 relaxation times. We estimate the PDF parameters using variable projection (VARPRO) approach [START_REF] Golub | Separable nonlinear least squares: the variable projection method and its applications[END_REF]. We derive this generic estimation framework for gamma PDFs. We validate the proposed method using synthetic data against known ground truth. We then illustrated it on a healthy subject and MS patient.

METHOD AND MATERIALS

Theory

Signal model The T 2 space is modeled as a weighted mixture of three PDFs, representing each of the three T 2 relaxometry compartments: short-, medium-and high-T 2 . Thus the voxel signal at the i-th echo time (t i ) is given as:

s (t i ) = M 0 3 j=1 w j ∞ 0 f j (T 2 ; p j ) EPG (T 2 , T E, i, B 1 ) dT 2 (1)
Each compartment is described by a chosen PDF, f j (T 2 ; p j ), where p j = {p j1 , . . . , p jn } ∈ R n are the PDF parameters. In Eq. (1), w j is the weight of the j-th distribution with j w j = 1. ∆T E, B 1 and M 0 are the echo spacing, field inhomogeneity and magnetization constant respectively. Imperfect rephasing of the nuclear spins after application of refocusing pulses leads to the generation of stimulated echoes [START_REF] Hennig | Calculation of Flip Angles for Echo Trains with Predefined Amplitudes with the Extended Phase Graph (EPG)-Algorithm: Principles and Applications to Hyperecho and TRAPS Sequences[END_REF]. Hence the T 2 decay is not purely exponential. The stimulated echoes are thus obtained using the EPG algorithm [START_REF] Prasloski | Applications of stimulated echo correction to multicomponent T2 analysis[END_REF]. EPG(•) is the stimulated echo computed at t i = i ∆TE where i = {1, . . . , m} and m is the number of echoes.

Optimization M 0 and w j can be combined into a single term α j ∈ R + without any loss of generality. In that case, the weight w j corresponding to each compartment is obtained as w j = α j / i α i . In the most general case, the least squares minimization problem is thus formulated as:

α, p, B1 = arg min α,p,B1 m i=1   y i - 3 j=1 α j λ j (t i ; p, B 1 )   2 = arg min α,p,B1 Y -Λ (p, B 1 ) α 2 2 (2)
where Y ∈ R m is the observed signal and m is the number of echoes;

α ∈ R + 3 ; Λ ∈ R m×3 ; p = {p 1 , p 2 , p 3 } ∈ R k ,
where k = 3n. In Eq. ( 2), each element of Λ, Λ ij =λ (t i ; p j , B 1 ), is computed as:

Λ ij = ∞ 0 f j (T 2 ; p j ) EP G (T 2 , T E, i, B 1 ) dT 2 (3)
Due to the EPG formulation, there is no closed form derivative solution for the optimization of B 1 [START_REF] Prasloski | Applications of stimulated echo correction to multicomponent T2 analysis[END_REF]. Hence, we opt for an alternate optimization scheme where we iterate between optimization of {p, α} with a fixed value of B 1 and optimization for B 1 using the obtained {p, α} values. The terms Λ (p, B 1 ) and α in Eq. ( 2) are linearly separable. Hence we can use the VARPRO approach to solve for {p, α} [START_REF] Golub | Separable nonlinear least squares: the variable projection method and its applications[END_REF]. The unknown α is substituted by Λ (p)

+ Y, where Λ (p) + is the Moore-Penrose generalized inverse of Λ (p). The VARPRO cost function is computed as:

arg min p I -Λ (p) Λ (p) + Y 2 2 (4)
where, I -Λ (p) Λ (p) + is the projector on the orthogonal complement of the column space of Λ (p). Since p ∈ R k , the Jacobian matrix J ∈ R k×m and its columns are computed as shown in [START_REF] Golub | Separable nonlinear least squares: the variable projection method and its applications[END_REF]. To compute the elements of J, we need to obtain ∂Λ/∂p ji , ∀i, j [START_REF] Golub | Separable nonlinear least squares: the variable projection method and its applications[END_REF]. After solving Eq. ( 4) for p, the values of α are obtained as Λ (p) + Y. The optimization for {α, p} and B 1 is performed alternatively until convergence. B 1 is optimized using a gradient free optimizer (BOBYQA), as it does not have any closed form solution [START_REF] Prasloski | Applications of stimulated echo correction to multicomponent T2 analysis[END_REF]. Multi-compartment model using gamma PDF The previous estimation framework is generic as it does not depend on the chosen PDF. We choose here to use gamma PDF for f j (•) for j = {1, 2 ,3} since their non-negativity and skewed nature are well suited to describe the compartments used to model the T 2 space. The mean T 2 values of myelin, myelinated axons, inter-and extra-cellular and free fluids in the brain are well studied in the literature [START_REF] Lancaster | Three-pool model of white matter[END_REF][START_REF] Mackay | Magnetic resonance of myelin water: An in vivo marker for myelin[END_REF]. Hence we parameterized each f j in terms of its mean (µ j ) and variance (v j ) rather than the usual shape and scale parameter representation (refer Eq. ( 5)). Using this parametric form of the gamma PDF makes the choice of optimization bounds convenient.

f (T 2 ; µ j , v j ) = T (µ 2 j /vj )-1 2 Γ µ 2 j /v j (v j /µ j ) µ 2 j v j exp -T 2 v j /µ j (5)
Hence we have, There are almost no echoes available which correspond to the high-T 2 compartment. The robustness and accuracy of the implementations to simultaneously estimate the weights and all the PDF parameters has been found to be not reliable [START_REF] Kj Layton | Modelling and estimation of multicomponent T 2 distributions[END_REF].

p = {µ s , v s , µ m , v m , µ h , v h }
Hence we choose to estimate only the mean of the gamma PDF corresponding to the medium-T 2 compartment.

Using the VARPRO approach we thus estimate four parameters of the signal model: mean of the medium-T 2 gamma PDF (µ m ) and the three weights corresponding to each compartment. Hence only ∂Λ/∂µ m is required for computing the Jacobian matrix and is obtained as:

∂Λ ∂µ m = ∞ 0 f (T 2 ; µ m , v m ) µ m v m 2 log T 2 µ m v m - (6)
2Ψ µ 2 m v m + 1 - T 2 v m EP G (T 2 , T E, i, B 1 ) dT 2
where Ψ(•) is the digamma function. The remaining gamma PDF parameter values are pre-selected for the three compartments based on histology findings reported in the literature [START_REF] Mackay | Magnetic resonance of myelin water: An in vivo marker for myelin[END_REF] and are set as {µ s , µ h } = {30, 2000} ms and {v s , v m , v h } = {50, 100, 6400} ms 2 . We assume a reasonable bound on µ m of 100-125ms for its optimization. The minimization problem in Eq. ( 4) is solved for µ m using the analytically obtained derivative in Eq. ( 6) with a gradient based optimizer [START_REF] Svanberg | A class of globally convergent optimization methods based on conservative convex separable approximations[END_REF]. The short-T 2 compartment here indicates the condition of myelin and myelinated axons [START_REF] Lancaster | Three-pool model of white matter[END_REF]. The medium-T 2 compartment's WF conveys information on the condition of axons, glial cells and extracellular fluids [START_REF] Lancaster | Three-pool model of white matter[END_REF]. The condition of free fluids (such as in ventricles and fluid accumulation due to tissue injuries) is indicated by the high-T 2 WF values.

Experiments

Synthetic data. MS patient data. The method was finally tested on T 2 relaxometry MRI data of a MS patient. The observations from our estimation maps were compared with the pathological findings on MS lesion reported in the literature [START_REF] Lassmann | Heterogeneity of multiple sclerosis pathogenesis: Implications for diagnosis and therapy[END_REF][START_REF] Cr Guttmann | The evolution of multiple sclerosis lesions on serial MR[END_REF]. We observed whether the estimation maps obtained from our method were able to provide insight into MS lesion which corroborate with the pathological findings. The acquisition details are the same as for the healthy volunteer data. The ventricles and other regions with free fluids have a higher µ m compared to the normal appearing white matter (NAWM) tissues. This is relevant as free fluids have a higher T 2 value compared to the relatively tightly bound tissues present in NAWM. MS patient data. Two lesions are present in the MR image of the MS patient shown in Fig. 3 that are marked with red and blue arrows. We observe absence of short-T 2 WF in both lesions. The lesions and their neighboring tissues have higher w m values than NAWM tissues. The w h map shows fluid accumulation in lesion-2 but not in lesion-1 . The estimated medium-T 2 gamma PDF map shows a higher PDF mean for both lesions compared to the NAWM. In lesion-1 the estimated µ m increases (with respect to the neighboring NAWM) as it approaches the core of the lesion, but is less than the µ m estimated at the ventricles where there is free fluid.

RESULTS

Synthetic

DISCUSSION

Our method was successfully validated against synthetic data with known ground truth for all SNR values (refer Fig. 1).

High w s values in the genu of the CC of healthy volunteer data (refer Fig. 2) is due to the high myelin and myelinated fibers density in this region compared to any other part of the brain [START_REF] Tomasch | Size, distribution, and number of fibres in the human corpus callosum[END_REF]. In the estimation maps of the MS patient (refer Fig. 3), the absence of short-T 2 WF in the lesions and in its neighboring regions can be explained by demyelination of the nerve fibers caused by MS [10][3]. Demyelination at the onset of MS is followed by macrophage intervention leading to an increased cellular activity in the MS lesion regions [START_REF] Lassmann | Heterogeneity of multiple sclerosis pathogenesis: Implications for diagnosis and therapy[END_REF] whose T 2 relaxation time is greater than myelin and myelinated axons but less than those of free fluids [START_REF] Lancaster | Three-pool model of white matter[END_REF]. This phenomenon might explain the high w m values in the lesions and in neighboring regions. Demyelination is followed by progressive axonal damage and fluid accumulation (due to tissue injuries) in MS lesions [START_REF] Lassmann | Heterogeneity of multiple sclerosis pathogenesis: Implications for diagnosis and therapy[END_REF]. The extent of axonal damage and fluid accumulation in the lesions can provide useful information regarding the lesion state and its response to a treatment. Lesion-2 has fluid accumulation unlike lesion-1 , possibly indicating that the two lesions are in different stages. The continuous axonal damage in the MS lesions [10][11] explains the higher µ m in the lesion regions compared to the neighboring NAWM as a higher µ m value indicates tissues with less tightly bound water. The increment in µ m values in lesion-1 as we approach the lesion core from the lesion boundary might also indicate a reduction in axon density. This is in accordance with the pathology of MS lesion evolution [10][11].

CONCLUSION

We proposed a generic estimation method to obtain estimates of tissue microstructure in brain by modeling the T 2 spectrum as a weighted mixture of three gamma PDFs. The maps estimated from our method can be effective in understanding the heterogeneity of lesions [START_REF] Lassmann | Heterogeneity of multiple sclerosis pathogenesis: Implications for diagnosis and therapy[END_REF] in MS patients and be used as potential biomarkers to have information on the MS lesion growth stage. As a part of the future study we intend to validate the observations by applying the proposed estimation framework on more healthy controls and MS patient datasets.

  data. The results of the synthetic data simulation are shown in Fig 1. It shows that with increasing SNR the weights estimation gets more accurate. The bars around the mean value are the 95% confidence intervals (CI) which is obtained as 1.96 times the standard deviation of the estimates. The CIs of the estimation improve with with increasing SNR for all three WFs. The ground truth lies in the CI of the mean estimated weights for all three compartments.

Fig. 1 .

 1 Fig. 1. Mean of estimated weights with a 95% confidence interval for 100 signal averages for the synthetic data study. Healthy volunteer data. The estimation maps for the healthy volunteer are shown in Fig 2. The genu of the corpus callosum (CC) has higher w s values compared to any other region

Fig. 2 .

 2 Fig. 2. Estimation maps for a healthy volunteer.

Fig. 3 .

 3 Fig. 3. Estimation maps for MS patient data. Lesion-1 and lesion-2 are marked with red and blue arrows respectively.

  where, (•) s , (•) m and (•) h are the PDF parameters describing the short-, medium-and high-T 2 compartments respectively. Due to practical limitations such as feasible acquisition time, coil heating and specific absorption rate (SAR) guidelines, T 2 relaxometry MRI sequences have limitations on the shortest echo time and number of echoes per acquisition. The high-T 2 compartment aims at capturing of free fluids in the brain, and hence has a T 2 relaxation time larger than 1 second[START_REF] Mackay | Magnetic resonance of myelin water: An in vivo marker for myelin[END_REF]. A standard T 2 spin echo multi-contrast sequence has the shortest T 2 acquisition (first echo) at around 8 -10ms and has 20 -40 acquired echoes. Hence for the short-T 2 compartment, we usually have a very limited (around 3-4) number of echoes.

  The proposed method was first validated against synthetic data generated following a known ground truth, composed of three gamma PDFs with parameters, {µ s , µ m , µ h } = {25, 120, 1900} and {v s , v m , v h } = {40, 90, 6000}. The weights chosen for each compartment were, {w s , w m , w h } = {0.2, 0.7, 0.1}. The B 1 , M 0 and T 1 values considered for this simulation were 1.3, 950 and 1000 respectively. The experiments are carried out for SNR values ranging from 5 to 100 in steps of 5. We simulated T 2 relaxometry data with the following parameters: first echo (TE 0 ) is at 9ms; TE= 9ms; 32 echoes, 100 signal averages.

	Healthy volunteer data. The method was tested on T 2
	relaxometry data acquired on a healthy volunteer (male,
	age: 26) with the following acquisition details: Siemens 3T
	MRI scanner; 2D multislice CPMG sequence; 32 echoes;
	TE 0 = 9ms; echo spacing of 9ms; TR = 3000ms; single
	slice acquired; slice thickness of 4mm; in plane resolution of
	1.04mm × 1.04mm; matrix size of 192 × 192.