Lina Henao

Karim Mazeau

The Molecular Basis of the Adsorption of Bacterial Exopolysaccharides on Montmorillonite Mineral Surface

Keywords: molecular modeling, clay, polysaccharide

Rhizospheric exopolysaccharides (EPS) spontaneously aggregate mineral particles. Molecular dynamics simulations with Cerius2 and Material Studio programs have been performed to study the adsorption of chemical groups, monosaccharides and oligosaccharides, onto the basal surface of Na-montmorillonite. The estimated enthalpies of adsorption of chemical groups nicely reproduced the expected values. Mono-and oligosaccharides have a preferred geometry of adsorption. Monosaccharides maintained their shape upon adsorption whereas oligosaccharides flattened more or less on the surface, depending on their flexibility. The conformational adaptability was shown to be the leading factor that determined the interaction strength between the EPS and the mineral surface.

Introduction

Organic/inorganic hybrid composites are materials that comprise nanometer-size mineral particles dispersed in a polymer matrix. Compared to pure matrix, these materials exhibit enhanced properties : an increase of the mechanical and thermal performances, and a barrier effect to gas diffusion [START_REF] Ray | Polymer/layered silicate nanocomposites: a review from preparation to processing[END_REF][START_REF] Alexandre | Polymer-layered silicate nanocomposites: preparation, properties and uses of a new class of materials[END_REF]. Most of the matrices considered so far are synthetic polymers; preparing such materials is difficult because of the low compatibility between hydrophobic polymers and the hydrophilic mineral. To overcome this difficulty, the cations located in the interlayer space of the clays are exchanged by tensioactive molecules (principally alkyl ammoniums). The inter layer space thus gain a hydrophobic character and become compatible with polymers.

Such hybrid complexes do exist in nature; rhizospheric microorganisms, i.e. living in the vicinity of plant roots, produce exo-polysaccharides (EPS); these EPS possess two key properties: they aggregate mineral particles [START_REF] Dorioz | The role of roots, fungi and bacteria on clay particle organization. An experimental approach[END_REF] and they can retain a large amount of water [START_REF] Chenu | Clay-or sand-polysaccharide associations as models for the interface between microorganisms and soil: water related properties and microstructure[END_REF].

The benefit for bacteria is obvious as they are protected by organo-mineral aggregates against [START_REF] Chenu | Clay-or sand-polysaccharide associations as models for the interface between microorganisms and soil: water related properties and microstructure[END_REF][START_REF] Roberson | Relationship between desiccation and exopolysaccharide production in a soil Pseudomonas sp[END_REF][START_REF] Alami | Rhizosphere soil aggregation and plant growth promotion of sunflowers by an exopolysaccharide-producing Rhizobium sp. strain isolated from sunflower roots[END_REF][START_REF] Ashraf | Inoculating wheat seedlings with exopolysaccharide-producing bacteria restricts sodium uptake and stimulates plant growth under salt stress[END_REF][START_REF] Hartel | Role of extracellular polysaccharide production and clays in the desiccation tolerance of cowpea bradyrhizobia[END_REF][START_REF] Chenu | Organic matter influence on clay wettability and soil aggregate stability[END_REF][START_REF] Cosentino | Aggregate stability and microbial community dynamics under drying-wetting cycles in a silt loam soil[END_REF]. There is also a benefit for the soil, as the presence of aggregates is correlated with soil fertility [START_REF] Chenu | Polymer bridging: A mechanism of clay and soil structure stabilization by polysaccharides[END_REF]. Polysaccharides have indeed the additional properties of being biocompatible and biodegradable, two supplementary advantages exploited in an emerging class of biomaterials [START_REF] Simon | Adsorption of cellulose derivatives onto montmorillonite: a SEC-MALLS study of molar masses influence[END_REF][START_REF] Park | Green\" Nanocomposites from Cellulose Acetate Bioplastic and Clay: Effect of Eco-Friendly Triethyl Citrate Plasticizer[END_REF][START_REF] Park | Effect of Sequential Mixing and Compounding Conditions on Cellulose Acetate/Layered Silicate Nanocomposites[END_REF][START_REF] Wang | Biopolymer chitosan/montmorillonite nanocomposites: Preparation and characterization[END_REF][START_REF] Darder | Biopolymer-Clay Nanocomposites Based on Chitosan Intercalated in Montmorillonite[END_REF][START_REF] Darder | Chitosan-clay nanocomposites: application as electrochemical sensors[END_REF].

The macroscopic properties reflect the associations occurring at the molecular scale between polysaccharides, mineral particles and water. The surface of montmorillonite for example, bears negative charges; one may intuitively think that the most favorable organic molecules for adsorption onto such mineral are poly-cations. The situation is apparently more complicated as many rhizospheric EPS are shown to interact with minerals [START_REF] Alami | Rhizosphere soil aggregation and plant growth promotion of sunflowers by an exopolysaccharide-producing Rhizobium sp. strain isolated from sunflower roots[END_REF][START_REF] Labille | Destabilization of montmorillonite suspensions by Ca2+ and succinoglycan[END_REF][START_REF] Amellal | Colonization of wheat roots by an exopolysaccharide-producing Pantoea agglomerans strain and its effect on rhizosphere soil aggregation[END_REF][START_REF] Chenu | Mechanical strength of clay minerals as influenced by an adsorbed polysaccharide[END_REF][START_REF] Chenu | Influence of a fungal polysaccharide, scleroglucan, on clay microstructures[END_REF][START_REF] Dontsova | Anionic polysaccharide sorption by clay minerals[END_REF][START_REF] He | Partial deflocculation of mutual flocs of allophane and halloysite by xanthan and chitosan and relevance to particle arrangement in the flocs[END_REF][START_REF] Rao | Influence of starch polysaccharide on the remolded properties of two Indian clay samples[END_REF]; they are either neutral or anionic (with one or two charges per repeat unit, generally carboxylic acids).

The role of the fine structure of the polysaccharide on its adsorption mechanism onto mineral surfaces remains largely unclear. However, EPS offer a unique opportunity to establish structure-property relationships; their experimental responses to flocculate a colloidal suspension of clays are effectively structure dependant [25 , 26] and EPS possess a wide variety of structures: they can be linear or branched. When present, the ionisable group is located either on the backbone or on the side chains (see figure 1). To complement the experimental efforts, our objective was thus to reveal, by molecular modeling, the structural factors (if any) of microbial polysaccharides that are responsible for their adsorption on mineral surfaces. Montmorillonite is the chosen mineral; it is widespread on earth, its crystal structure is reported [START_REF] Tsipursky | The distribution of octahedral cations in the 2:1 layers of dioctahedral smectites studied by oblique-texture electron diffraction[END_REF] and it is widely studied by molecular modeling (in particular in interactions with organic species) [START_REF] Capkova | Structure of montmorillonite cointercalated with stearic acid and octadecylamine: Modeling, diffraction, IR spectroscopy[END_REF][START_REF] Gaudel-Siri | Molecular Dynamics Study of e-Caprolactone Intercalated in Wyoming Sodium Montmorillonite[END_REF][START_REF] Capkova | Effect of surface and interlayer structure on the fluorescence of rhodamine B-montmorillonite: modeling and experiment[END_REF][START_REF] Fermeglia | Computer simulation of nylon-6/organoclay nanocomposites: prediction of the binding energy[END_REF][START_REF] Capkova | Modeling of aniline-vermiculite and tetramethylammonium-vermiculite; test of force fields[END_REF]. Only the basal surface of montmorillonite was considered in this study as it is statistically the most abundant surface.

Material and Methods.

In this study, we have used the modeling softwares Cerius 2 and Material studio (accelrys inc.) running on silicon graphics workstations at the Centre d'Expérimentation et de Calcul Intensif, CECIC, Grenoble, France.

Construction of the initial structures.

Mineral surface.

Montmorillonite is a smectite type mineral clay, it is a hydrophilic mineral that consists of a nanometer-thick layers formed by sandwiching an aluminum octahedron sheet between two silicon tetrahedron sheets. The initial unit cell was built from relevant crystallographic coordinates of a pyrophillite crystal, published by Tsipursky and Drits [START_REF] Tsipursky | The distribution of octahedral cations in the 2:1 layers of dioctahedral smectites studied by oblique-texture electron diffraction[END_REF], using the Crystal to a surface area of 5.4 nm 2 . This surface was used to study the adsorption of small molecules (chemical functions and monosaccharides). A larger surface was also built to model the adsorption of the largest structures (oligomers). Cell parameters of the large surface model were 72.8 Å x 62.1 Å, (area of 45 nm 2 ). The clay mineral is then minimized, equilibrated by molecular dynamics (NPT at 300 K and 1 atm) and then optimized again. To generate a mineral surface convenient for the simulation, the lattice constant c of the cell was extended to 100 Å. Surfaces were then hydrated by a monolayer of water molecules.

→3)-β-D-Glcp-(1→ 4)-2deoxy-β-D-GlcpA-(1→ 4)-β-D-Glcp-(1→ 4)-α-L-Rhap-(1→ 6 ↑ 1 α-D-Glcp-(6 ←1)-β-D-Glcp RMDP17 →3)-β-D-Glcp-(1→ 4)-2deoxy-β-D-GlcpA-(1→ 4)-β-D-Glcp-(1→ 4)-α-L-Rhap-(1→ 6 ↑ 1 α-D-Glcp-(6 ←1)-β-D-Glcp RMDP17 → 4)-β-D-GlcpA-(1→ 4)-β-D-Glcp-(1→ 4)-β-D-Glcp-(1→ 6)-α-D-Glcp-(1→ 4)-α-D-Galp-(1→ 6 ↑ 1 β-D-Galp-(1→ 3)-β-D-Galp / \ 4 6 \ / C / \ H 3 C COOH YAS34 → 4)-β-D-GlcpA-(1→ 4)-β-D-Glcp-(1→ 4)-β-D-Glcp-(1→ 6)-α-D-Glcp-(1→ 4)-α-D-Galp-(1→ 6 ↑ 1 β-D-Galp-(1→ 3)-β-D-Galp / \ 4 6 \ / C / \ H 3 C COOH YAS34 Succinoglycan COCH 3 ↓ 6 → 4)-β-D-Glcp-(1→ 4)-β-D-Glcp-(1→ 3)-β-D-Galp-(1→ 4)-β-D-Glcp-(1→ 6 ↑ 1 β-D-Glcp-(1→ 3)-β-D-Glcp-(1→3)-β-D-Glcp-(1→6)-β-D-Glcp / \ 6 4 6 ↑ \ / COCH 2 CH 2 COOH C / \ H 3 C COOH Succinoglycan COCH 3 ↓ 6 → 4)-β-D-Glcp-(1→ 4)-β-D-Glcp-(1→ 3)-β-D-Galp-(1→ 4)-β-D-Glcp-(1→ 6 ↑ 1 β-D-Glcp-(1→ 3)-β-D-Glcp-(1→3)-β-D-Glcp-(1→6)-β-D-Glcp / \ 6 4 6 ↑ \ / COCH 2 CH 2 COOH C / \ H 3 C COOH

Organic chemicals.

We studied the adsorption of chemical groups, monosaccharides and oligosaccharides; they are all related to the EPS indicated in Figure 1. The chemical groups considered are : methane (CH 4 ), ethane (CH 3 CH 3 ), methanol (CH 3 OH), ethanol (CH 3 CH 2 OH), 2-propanol (CH 3 CH 2 OHCH 3 ), methanoic acid (HCOOH), ethanoic acid (CH 3 COOH), methanal (HCHO), ethanal (CH 3 CHO), dimethyl ketone (CH 3 (CO)CH 3 ), methoxymethane (CH 3 OCH 3 ), methyl ethanoate (CH 3 (CO)OCH 3 ), sodium acetate (CH 3 COO -Na + ). The twenty monosaccharides 2; the two hydroxyl groups involved in glycosidic bonds have been systematically changed to OMe groups.

→3)-β-D-Glcp-(1→ 4)-β-D-GlcpA-(1→ 4)-β-D-Glcp-(1→ 4)-α-L-Rhap-(1→ 6 ↑ 1 α-D-Glcp-(6 ←1)-β-D-Glcp Rhamsan →3)-β-D-Glcp-(1→ 4)-β-D-GlcpA-(1→ 4)-β-D-Glcp-(1→ 4)-α-L-Rhap-(1→ 6 ↑ 1 α-D-Glcp-(6 ←1)-β-D-Glcp Rhamsan Xanthan → 4)-β-D-Glcp-(1→ 4)-β-D-Glcp-(1→ 3 ↑ 1 β-D-Manp-(1→4)-β-D-GlcpA-(1→2)-α-D-Manp / \ 6 4 6 ↑ \ / COCH 3 C / \ H 3 C COOH Xanthan → 4)-β-D-Glcp-(1→ 4)-β-D-Glcp-(1→ 3 ↑ 1 β-D-Manp-(1→4)-β-D-GlcpA-(1→2)-α-D-Manp / \ 6 4 6 ↑ \ / COCH 3 C / \ H 3 C COOH → 4)-β-D-ManpA-(1→ 4)-β-D-Glcp-(1→ 4)-β-D-Galp-(1→ 3 ↑ 1 β-D-Glcp KYGT207 → 4)-β-D-ManpA-(1→ 4)-β-D-Glcp-(1→ 4)-β-D-Galp-(1→ 3 ↑ 1 β-D-Glcp KYGT207 β-Kdop 2 ↓ 3 → 3)β-D-Glcp-(1→ 4)-β-D-Glcp-(1→ 4)-α-L-6dTalp-(1→ 2 ↑ COCH 3 MWAP71 β-Kdop 2 ↓ 3 → 3)β-D-Glcp-(1→ 4)-β-D-Glcp-(1→ 4)-α-L-6dTalp-(1→ 2 ↑ COCH 3 MWAP71 → 6)α-D-Glcp-(1→
The oligomers considered in this part correspond to a unique repeat unit of selected EPS synthetized by rhizospheric microbes: dextran, MWAP71, RMDP17, Rhamsan, xanthan, YAS34 and succinoglycan; their primary structures are reported in Figure 1. A degree of polymerisation of 6 was chosen for the dextran to obtain a molar mass comparable to the other oligomers (see Table 4). Polysaccharides are usually highly polydisperses; dextran for example represents a family of homopolysaccharides that feature a substantial number of consecutive α-(1→6) linkages in their main chain, usually more than 50% of the total linkages. Dextrans also possess variable amount of side chains constituted by mono and/or oligosaccharides linked to the main chain by (1→3) and (1→2) linkages. It is obviously not possible to consider all the possible chemical structures for a given polysaccharide so that each model correspond only an idealized structure; consequently, the dextran modelled in our study is only represented by its dominant structural feature (see Figure 1). In addition, some of the polysaccharides are known to form multiple helices in solution (xanthan for example); only the simple chain was considered in our study.

Chemical groups, monosaccharides and repeat units of EPS were constructed from standard geometries of bond lengths and bond angles, thanks to the sketcher module. The initial conformation of the pyran ring of monosaccharides is 4 C 1 for Glc, Man, Gal and Kdo residues, or 1 C 4 for Rha and Tal. In the case of oligosaccharides, values of the conformational parameters describing the relative orientation of two consecutive monomers (torsion angles Φ, Ψ and ω) were derived from the conformational analysis of model disaccharides [33 , 34].

Torsion angle values correspond to the lowest energy minimum and/or auxiliary minima of the (Φ, Ψ) potential energy surfaces. Explicit hydrogen atoms were used in all model systems.

Each structure was then relaxed to minimize energy and avoid atom overlaps.

Modelling the adsorption.

An organic molecule is then inserted in the simulation box above the mineral surface in a random orientation. Configurational sampling was then performed using a combination of energy minimization and molecular dynamics at elevated temperatures, typically 400 to 600 K. A typical dynamic simulation lasts 500 ps for the shortest ones, up to 2 ns for the longest ones. Once adsorbed, a 20Å slab of amorphous water molecules was added to the system. The resulting hydrated model was first minimized, equilibrated for 10 ps and minimized again

Computational details.

The Universal force field was used, UFF [START_REF] Rappe | UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations[END_REF]. This choice resulted from a compromise between good accuracy and availability of the parameters for all atom types present in the molecular models. This force field was successfully used in independent studies of organoclay species [29, 36, 37: Pospisil, 2004 #54]. Furthermore, preliminary tests reveal that this force field correctly reproduces the particular conformational properties of carbohydrates: puckering of the pyran rings and the exo-anomeric effect. The charge equilibration method was used to calculate charges for each atom [START_REF] Rappe | Charge equilibration for molecular dynamics simulations[END_REF]. Long-range interactions were treated by the Ewald summation technique [START_REF] Ding | Atomic level simulations on a million particles: the cell-multipole method for Coulomb and London nonbond interactions[END_REF]. The simple point charge model (SPC) was used for the water molecules.

The minimization uses the conjugate gradient procedure with the convergence criterion of the root-mean-square of the atomic derivatives of 0.05 kcal mol -1 Å -1 .

Molecular dynamic calculations were based on the canonical NVT ensemble (constant number of particles, volume, and temperature). The equations of motion were solved using the standard Verlet algorithm [START_REF] Verlet | Computer \"experiments\" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules[END_REF], with a time step of 1 fs. The system is coupled to a bath at the desired temperature using Nose's algorithm [START_REF] Evans | The Nose-Hoover thermostat[END_REF]. During the production MD simulations, the positions of the mineral surface atoms were fixed, but all remaining system components (counter ion, organic molecule, water) were allowed to move accordingly.

Properties.

Enthalpies of adsorption.

Ten structures (snapshots) were randomly selected and minimized from the molecular dynamic trajectory. The total potential energy of a microstate may be written as:

E tot = E clay + E ligand + E inter
where the first three terms represent the energy of the total system, the energy of montmorillonite alone, and the energy of the ligand molecule, and consist of both valence and nonbonded components. The last term is the interaction energies between the clay and the ligand (consisting of nonbonded terms only). The enthalpy of adsorption is taken as the negative of the interaction energies of the 10 selected structures: Surface area in contact.

The estimation of the molecular surface areas was performed with the Connolly dot algorithm, [START_REF] Connolly | Computation of molecular volume[END_REF] with a probe radius of 1.4 Å.

Hydrogen bonds.

The method traditionally used to detect a hydrogen bond is geometric: two oxygen atoms are considered hydrogen bonded if the distance between the hydrogen of the donor and the oxygen acceptor is lower than 2.5 Å and if the angle between the oxygen donor, the hydrogen donor and the oxygen acceptor larger than 90°.

Results and discussion.

A hierarchical approach was chosen to reveal the adsorption behavior of polysaccharides onto the basal surface of the clay mineral. Adsorption of chemical groups was examined first, followed by adsorption of monosaccharides. Finally the adsorption of oligosaccharides was considered. Idealized chemical structures of the EPS are given in Figure 1; it was experimentally shown that all of them do adsorb on mineral surfaces. Predicted quantities are the energies associated with the binding and also the geometry of the complexes.

During the dynamics trajectory, the total energy decreases when the saccharide adsorbs onto the surface of the clay mineral, suggesting that the whole modeled system reaches a more stable state. Consequently, the positive values of ∆H ads (or the negative values of E int ) indicate that the adsorption is a favorable process. During the dynamics simulation, the chemical groups explore several adsorption sites of similar energy. Table 1 gives their average interaction energies with the mineral surface. The counter ion has no noticeable effect on the interaction of hydrophobic groups (Methyl and Ethyl) with the clay surface. In contrast, it strongly participates to the interaction with polar groups. Alcohol and carboxyl groups form Hydrogen Bonds with the oxygen atoms of the mineral surface; electrostatic interactions (ionic like) also occur, between the oxygen atoms of the organic molecule and the counter ion.

The interaction of very small molecular fragments of saccharides with the mineral surface can be considered unperturbed by structural effects, the interaction is thus optimal and the estimated values of the enthalpies of adsorption give the orders of magnitude of the energies involved upon adsorption. The interaction of the different functions on the mineral surface is globally weak; the energy of a hydrogen bond O-H…O and a hydrophobic interaction are typically in the range 4 to 5 kcal/mol and 1 to 3 kcal/mol, respectively. Our results are in agreement with these values, suggesting that the force field correctly reproduces the basic interaction energies. Dynamical behavior of mono-and oligosaccharides differ from that of functional groups.

They touched the mineral surface and explored several orientations prior to converging to their preferred adsorption geometry. This suggests that the potential energy surface has many energy minima. The energy decreases gradually with time during the course of the simulation and only the final parts of the trajectories were considered (when the total energy is stabilized to its minimal value), in order to estimate the interaction energies. On average, only 30 to 40% of the total accessible surface of monosacharides is in direct contact with the mineral surface; this value is remarkably unchanged for the oligomeric fragments.

Monosaccharides.

All the monomers considered in this study are indicated in Table 2 with the structural and energy details of their adsorption on the mineral surface.

The adsorption enthalpy strongly depends on several factors, including the number of pendant groups, their nature, position and orientation with respect to the pyran ring. Figure 2 compares the preferred geometry of interaction of two glucoses 1,6 dimethylated (α and β).

Inverting the configuration on the anomeric carbon atom changed the adsorption features of the monosaccharide on the mineral surface.

Here again, the counter ion favors the adsorption of the monosaccharides. The average interaction energy was -30 kcal/mol in the absence of counter ions; it reached -106 kcal/mol in their presence. Inspection of the models reveals the strong similarity of the binding of monosaccharides with that of the functional groups: hydrogen bonds and ionic interactions.

Hydrophobic interactions seem to play a particular role. Methyl groups were in direct contact with the surface and the monosaccharide-surface interaction was stronger when more Omethoxy groups were present (see Table 2). The group contribution method clearly overestimated the interaction energies. For example, group method predicts an energy of interaction of about -200 kcal/mol for a simple disubstituted Glucose molecule. The energies really involved with the adsorption of such molecules are much lower than this expected value, they ranged between -104 kcal/mol and -120 kcal/mol. This is not surprising as the group contribution assumes an optimal interaction whereas, in reality, the adsorption is far from perfect. In addition, structural effects are neglected in the group method; it cannot discriminate between different stereo isomers.

The changes in the conformational energies (∆E) were minimal for the monosaccharides.

They are made of a pyran ring which adopts a dominant chair conformation, except for very rare cases. The chair geometry remained when the monosaccharide is adsorbed, by contrast, side chains showed flexibility. Given the results of the previous section, simulations on the oligomeric structures of carbohydrates were systematically performed in the presence of the counter ions. Their energies of interaction with the clay surface are given in Table 3, the geometrical aspects of the adsorption are given in Table 4.

∆H Inspecting the molecular models shows that the monomers in the adsorbed conformational state do not systematically adopt the unperturbed orientation revealed in the preceding section. To illustrate this, Figure 3 gives the adsorbed fragment of dextran ; among its six α-Glc residues ; four of them adsorb in an identical geometry already observed in the study of free monosaccharides ; by contrast, the remaining two α-Glc behave differently. Chemical Hydrogen atoms and water molecules are omitted for clarity.

The presence of the surface systematically induced an increase of the internal energy of the EPS fragments. The conformation was thus less stable when adsorbed than when isolated; the difference reaches 20% in the case of RMDP17 and succinoglycan. The presence of the surface also induced an increase of the total accessible surface of the organic molecule. The The RMDP17 oligosaccharide flattened on the surface, all the monomer units were in direct contact with the surface (Figure 4). On the other hand the pendent monomer Kdo (which bears the carboxylic group) of MWAP71 was not in contact with the clay. RMSD compares coordinates of the conformation when adsorbed with the free conformation; it indicates the extent of modification of the shape of the sugar with adsorption. RMSD strongly vary from one EPS to the other; it reached 17% for succinoglycan and it was minimal for the MWAP71 and YAS34. change its shape upon interaction with the mineral surface. In other words, the more flexible the structure, the stronger the interaction. It is thus obvious that the deformation ability (to optimize the interaction) of linear polymers is larger than that of branched polymers for which the side chains induce a steric hindrance that restrict the flexibility of the whole molecule.

Accordingly, the fragment of MWAP71 (Figure 4) is rigid; its conformational adaptation is restricted to 3 glycosidic bonds and the Kdo pendant group is not in interaction with the surface. In contrast the large flexibility of RMDP17 (5 glycosidic bonds) allows all its monomer units being in contact with the surface.

Comparison with the experiments.

The accuracy of the models could be assessed by comparing the predicted data to the experimental data. Our results revealed the strong participation of the counter ion in the stability of the complex. They considerably increased the interaction energy between the EPS segment and the mineral surface. The crucial role of electrostatic (ionic) interactions through counter ions is also revealed by many experiments [START_REF] Dontsova | Anionic polysaccharide sorption by clay minerals[END_REF][START_REF] Fuller | Flocculation and coagulation of Ca-and Mg-saturated montmorillonite in the presence of a neutral polysaccharide[END_REF][START_REF] Kanaani | Biofilm interactions in water reuse systems: adsorption of polysaccharide to kaolin[END_REF][START_REF] Gu | The interaction of polysaccharides with Silver Hill illite[END_REF]. Complexes were also stabilized by hydrogen bonds and hydrophobic interactions. Experimental data show that adsorption of ethyl(hydroxyethyl)cellulose on talc is destabilized by urea, which is an hydrogen bond breaker [START_REF] Simon | Adsorption of cellulose derivatives onto montmorillonite: a SEC-MALLS study of molar masses influence[END_REF]. Finally, the importance of the hydrophobic interactions is counter intuitive; it is however consistent with NMR results which indicate that aliphatic chains are in direct contact with montmorillonite [START_REF] Simpson | The application of 1H high-resolution magic-angle spinning NMR for the study of clay-organic associations in natural and synthetic complexes[END_REF]. Our models also showed that the oligosaccharides undergo conformational changes between the solution states and the adsorbed state. Quartz crystal microbalance with dissipation monitoring (QCM-D) experiments reveal that the adsorbed conformation of dextran on alumina differs from that in solution [START_REF] Kwon | Model Bacterial Extracellular Polysaccharide Adsorption onto Silica and Alumina: Quartz Crystal Microbalance with Dissipation Monitoring of Dextran Adsorption[END_REF].

Experiments are also performed to establish the structure-property relationships. The ability of several EPS to flocculate a colloidal suspension of clay particles is compared [START_REF] Labille | Flocculation of colloidal clay by bacterial polysaccharides: effect of macromolecule charge and structure[END_REF]. it is well recognized that such characteristics strongly impact on polymer adsorption on surfaces. In addition, modeling is performed on idealized chemical structures of the EPS and the analysis of the results was based only on the enthalpies of adsorption. The free energy of adsorption would be the choice parameter but the entropic contribution is hardly accessible.

The entropy depends on two major contributions: de-solvation of the surfaces, which is a favorable process, and the conformational restrictions of the polymer upon adsorption; which gives non-favorable entropy.

Conclusion.

Atomic level molecular simulations were applied to investigate the interaction features of model molecules mimicking bacterial exopolysaccharides (in their neutral forms) with the basal surface of Na-montmorillonite surface. The study has been performed using Cerius2 and material studio modeling programs.

It was found that hydrogen bonds, electrostatic and hydrophobic interactions are present in the model systems and provide a significant effect on the adsorption strength of EPS fragments on the mineral surface. The adsorption of monosaccharides was dependant on the nature of the monomer and its side chains. The six-membered ring stayed in its preferred geometry when adsorbed. In contrast, oligosaccharides changed their shape upon adsorption. It was found that the enthalpies of adsorption linearly increase with increasing the interacting surface area, suggesting that the key factor determining the adsorption is the ability of the EPS to unfold in order to maximize its surface in contact. The most flexible structures were the most favorable candidates for adsorption. The present modelling was able to predict a complex physical adsorption phenomenon in agreement with the experimental results.

  The Molecular Basis of the Adsorption of Bacterial Exopolysaccharides on Montmorillonite Mineral Surface.

  changes and hydric stress

  The experimental lattice is monoclinic, with space group C2/m, and the lattice parameters: a=5.20, b=9.20, c=10.13 and α=90º, β=99º, γ=90º. The unit cell is duplicated along the two directions parallel to the sheet planes. Then, we performed substitution of some Al 3+ by Mg 2+ in the octahedral layer and substitution of some Si +4 atoms by Al3+ in the tetrahedral layer in order to obtain the desired final chemical composition of the Wyoming Na-montmorillonite : Na 0.75 (Si 7.75 Al 0.25 )(Al 3.5 Mg 0.5 )O 20 (OH) 4 . This structure has a cation exchange capacity (CEC) of 1.01meq/g; the CEC is defined as the amount of the exchangeable cations retained by the clay to neutralize the negative charge (milliequivalent of positive charge per gram of clay). Periodic boundary conditions are applied in all the three dimensions. Typical a, b dimensions of the periodic cell are 17.8 Å x 30.4 Å, corresponding

Figure 1 .

 1 Figure 1. Chemical structures of the polysaccharides considered in this study.

Dextran → 6

 6

  analysis. Calculations have been repeated three times on several test cases, results were consistent.

7 ∆H

 7 ads = -<E inter > We were interested by the energy change of the carbohydrate structure when it goes out of the solution to the adsorbed state : ∆E. Dynamic trajectories of the carbohydrate structures were thus performed, in the absence of the mineral surface.∆E = E ads -E Free where E ads and E Free are the energies of the adsorbed and isolated (in the absence of the mineral surface) conformations, respectively.

Figure 2 .

 2 Figure 2. Equilibrium adsorbed geometry of β-glucose (top) and α-glucose (bottom) methylated at positions 1 and 6. Left side views, right front views. Hydrogen atoms are omitted for clarity.

S

  totcompl : solvent accessible surface of the adsorbed conformation, S Free solvent accessible surface of the free conformation, S interact area in interaction. Molecular masses of the different fragments of EPS are also indicated (Mw).

Figure 3 .

 3 Figure 3. Equilibrium adsorbed geometry of the dextran fragment. Hydrogen atoms and water molecules are omitted for clarity.

  constraints that limit the number of accessible orientations of the constituent monosaccharides with respect to the mineral surface. However, the potential energy surface contains a huge amount of energy minima; monomer units that are part of a polymer simply explore secondary potential wells.

Figure 4 .

 4 Figure 4. Equilibrium adsorbed geometry of the MWAP71 (top) and the RMDP17 (bottom).

Figures 3

 3 Figures3 and 4show the repeat units of MWAP71 and RMDP17 adsorbed on the surface.

Figure 5 .

 5 Figure 5. Energy of adsorption as a function of the interacting area.

Figure 5

 5 Figure5gives the correlation between the estimated enthalpies of adsorption and the effective interacting area. The linear relationship suggests that the real area in interaction is the critical factor on which depends the adsorption of the tested EPS. Structural factors play a role in the deviation with respect to the master line.

  Experiments measure the average size of the flocs which indirectly reflects the efficiency of the EPS to bridge different mineral particles. Such measurements thus give an indication of the affinity of the EPS with the mineral surface. An interaction efficiency coefficient (IEC) was defined in order to allow the comparison between the experimental and modeling approaches. The IEC corresponded to the enthalpy of adsorption, normalized by the effective interaction area. Experimental floc size in the presence of succinoglycan (335 µm) is much larger than that of xanthan (158 µm); even though the molar mass of the xanthan sample is twice the mass of succinoglycan. In qualitative agreement with the experiments, IEC of succinoglycan (1027 cal/molÅ 2 ) was estimated larger than that of xanthan (882 cal/molÅ 2 ).

  flocculation ability. Estimated IEC reproduced the experimental tendency: they were estimated at 1017 and 1152 cal/molÅ 2 for dextran and YAS34, respectively.Quantitative comparison between experimental data and the present modeling results is not possible because of both experimental difficulties and assumptions in modeling. Molar masses of the different polysaccharides studied experimentally differ by almost a factor of 20;

Table 1 .

 1 Energies (kcal/mol) and interacting areas (Å 2 ) of several chemical moieties adsorbed onto the mineral surface. E

			Without counter-ion			With counter-ion
	Group	E c	E vw	E tot	S int	E c	E vw	E tot	S int
	CH 4	0	-6	-6	18	-1	-6	-7	18
	CH 3 OH	-3	-7	-10	22	-28	-3	-31	24
	HCHO	-5	-6	-11	16	-21	-4	-25	23
	HCOOH	-1	-7	-8	22	-17	-5	-22	23
	CH 3 CH 3	0	-9	-9	25	-4	-8	-12	25
	CH 3 CH 2 OH	-2	-11	-13	27	-30	-5	-35	28
	CH 3 CHO	-5	-10	-15	22	-23	-8	-31	28
	CH 3 COOH	-1	-11	-12	30	-20	-8	-28	30
	CH 3 OCH 3	0	-11	-11	34	-31	-6	-37	32
	CH 3 CHOHCH 3	-2	-13	-15	30	-29	-8	-37	30
	CH 3 (CO)CH 3	-2	-13	-15	35	-25	-10	-35	34
	CH 3 (CO)OCH 3	-2	-14	-16	36	-29	-12	-41	41

tot : total energy, E c and E vw are its coulomb and Van der Waals components. Standard deviations on energies ranged between 1 to 3 kcal/mol and those on surfaces ranged between 1 to 5 Å 2 .

Table 2 .

 2 Energetic and geometric characteristics of selected monomers of carbohydrates

	10
	Å 2 ). Standard deviations on energies ranged between 2 to 8 kcal/mol and those on surfaces
	ranged between 1 to 14 Å 2 .
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

adsorbed onto the clay mineral surface in the presence or absence of counter-ion. Energy of interaction (E int in kcal/mol); difference in energy between the free state and the adsorbed state (∆E in kcal/mol). Total solvent exposed surface (S tot in Å 2 ), surface in interaction (S int in

Table 3 .

 3 Selected characteristics of the repeat units of EPS. ∆H Ads : enthalpy of adsorption of the oligomer on the mineral surface, E adsorb : average energy of the adsorbed conformation, E Free : average energy of the equilibrium conformation in the absence of the surface, RMSD root mean square deviation per atom between the free and the adsorbed conformations.

		Ads	E adsorb	E Free	RMSD
	dextran	351	725	646	0,0234
	KYGT207	240	486	442	0,0304
	MWAP71	151	421	394	0,0145
	rhamsan	333	686	615	0,0362
	RMDP17	316	603	509	0,0465
	succinoglycan	564	950	785	0,0245
	xanthan	268	592	541	0,0230
	YAS34	445	958	841	0,0153
	Energies are given in kcal/mol.
		S totcompl	S Free	S interact	Mw
	dextran	806	743	345	1018
	KYGT207	550	531	239	708
	MWAP71	599	595	202	778
	rhamsan	796	737	340	1016
	RMDP17	802	745	314	1000
	succinoglycan	1244	1071	549	1554
	xanthan	773	735	304	982
	YAS34	942	967	386	1264

Table 4 .

 4 Geometric characteristics (in Å 2 ) of the adsorption of the repeat units of EPS.

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Mingjun Yang
email: mjyang@nano.ku.dk.

P Mark Rodger

John H Harding

S L S Stipp

Molecular dynamics simulations of peptides on calcite surface

Keywords: molecular simulation, peptides, biomineralization, calcite, surface molecular simulation, peptide, biomineralization, calcite, surface mineralisation

A series of Molecular Dynamics (MD) simulations has been carried out to investigate the interaction between peptides and a calcite ( ) 4 1 10 surface in water. A 16-amino acid and a 17-amino acid peptide have been built and three different configurations for each peptide are used as starting configurations. The dynamic behaviour of these peptides has been investigated by calculating their radii of gyration and distribution of dihedral angles. For comparison, the simulations of peptides in vacuum and water have also been carried out. The simulations indicate that these peptides generally have strong interactions with the calcite surface and the peptides changed their configuration to favour this interfacial interaction. Continuum electrostatic calculations based on the Poisson-Boltzmann equation (PBE) have also confirmed strong electrostatic interactions between peptides and the calcite surface. The results suggest that peptides can control calcite crystallisation and that strong electrostatic interactions between peptides and calcite surfaces dominate the interfacial interactions.

Knowledge of the interactions between proteins and calcite surfaces is a key to understanding biomineralization. Progress in experiment has greatly improved our understanding of the role of peptides and proteins in the mechanism of biomineralization and shows that the structure of these biomolecules is very important in controlling crystal growth. A study on mollusk shell proteins AP7 and AP24

indicates that some structural features of these proteins will greatly influence their kinetic behavior in mineralization [1]. Again, the control of crystal growth by proteins is demonstrated in the formation of eggshell. Recently, ovocleidin-17 has been identified as the major protein of calcified eggshell, and the three-dimensional structure of this protein is believed to closely relate to its functionality in the biomineralization process [2]. Ajikumar and colleagues have used charged peptides to mimic the function of eggshell proteins and the results show that these peptides were able to facilitate the nucleation and growth of polycrystalline calcium carbonate [3].

Metzler and colleagues used X-ray absorption near edge spectroscopy (XANES) to study the electronic structure of crystalline calcium carbonate and peptides at the surface, and hence the mutual effects of calcite on peptides and peptides on calcite during biomineralization [4]. A study on mollusk shell proteins AP7 and AP24

indicates that some structural features have an important influence in their kinetic behavior during mineralization, which suggests protein structure plays a very important role in controlling biomineralization [1].

However, current understanding of the interaction between biomolecules and minerals during biomineralisation is far from complete. Here, computer simulations can be an Deleted: [2] Deleted: demonstrated

Deleted: interaction

Deleted: mineralisation Deleted: [3] Deleted: mineralisation

Deleted: [4] Deleted: Experimental evidence has greatly contributed to our understanding about the biomineralisation in the presence of peptides and proteins, and shows that the structures of these biomolecules plays a key role in controlling crystal growth , but complete description of the mechanism is lacking. to simulate the interaction between polysaccharides and calcite, and facilitate the simulations of biomolecules and inorganic surfaces [7]. Peptides are highly charged molecules so the electrostatic interactions between peptides and mineral surfaces dominate the interfacial interactions.

The Poisson-Boltzmann equation (PBE) has been proposed as an effective continuum model to evaluate the electrostatic properties [8]. This model uses implicit solvent and can determine the contribution of electrostatic to the forces and energies of a molecular system.

In this study, we used MD simulations to simulate the dynamical behaviour of In the modeling, we put water and a peptide molecule on a calcite surface as shown in Fig. 3. Because all the peptides under study were negatively charged with -4e, two calcium ions were added to each peptide in order to neutralize the system. The temperature was 300 K and a three-dimensional periodic boundary was used. All the MD simulations were carried out using the DL_POLY 2.18 code [12] with a set of force fields designed for use at bio-inorganic interfaces [6]. Full details about the force fields between the organic molecule and the mineral are given in that reference.

The interaction potentials used for CaCO 3 were those derived by Pavese et al. [13] for modeling a range of properties of calcite and aragonite crystals. The potentials for peptides were obtained from the ff03 force field in AMBER; the flexible TIP3P potential was used for water [14].

The coordinates of both peptides and the calcite surface obtained from the final configuration of MD simulations were used as input for PBE calculations. After taking away all the water molecules, PBE calculations were performed with the APBS program [8]. In the calculation, the solute intramolecular interactions were computed by the usual molecular mechanics mothods, while the solute-solvent and solventsolvent interactions were computed by a mean-field approximation through the use of PB electrostatic theory. The dielectric constant for water was set to 80.0, and 8.0 for the calcite surface and peptides.

Peptides in vacuum

The MD simulations of peptides in vacuum were carried out at 300 K for 2 ns. From Fig. 4 we can see that after the initial equilibration period, the configurations of the peptides became stable, as characterized by their radii of gyration. The potential energy underwent variance during the simulations. The results show that the potential energy of a peptide in vacuum is closely linked to its specific configuration.

Peptides in water

The simulations of peptides in water were carried out at 300 K for 2 ns. The potential energy and gyration radius for the peptide molecule were examined for each simulation, and results are shown in Fig. 5. The solvation of peptides by water molecules can greatly stabilize peptide molecules. There is no obvious difference in potential energy between the peptides. Their larger gyration radius in water indicates that they tend to maintain an extended structure such as PI (B) and PII (B). The difference in dynamic behaviour of peptides with and without water molecules, as seen from the gyration radius calculations, implies that the presence of water molecules has great influence on the functionality of peptides during biomineralization.

Peptides on calcite surfaces in water

Formatted: English U.K. These simulations were used to simulate adsorption of peptides on the calcite surface.

Deleted

To make sure that the adsorption of peptides onto the calcite surface is that for a low energy (if possible the equilibrium) configuration, the following procedure was used to build the system. First, the peptide was put about 4 Å away from the surface and the MD simulation ran for 2.5 ns, to allow adsorption to be complete. Second, water molecules were added to the system and a series of stepwise MD simulations were used to relax the system. After the system has equilibrated, the potential energy and the radius of gyration of the peptide could be obtained from production simulations of 2 ns, and results are shown in Fig. 6. The results suggest that although there are differences in the gyration radius, the variability of potential energy among the peptides is insignificant. The adsorption energy could be obtained by comparing the potential energy of the system before and after adsorption using the method described by Yang et al. [7]. The results are shown in Table 1. Low negative adsorption energy means strong adsorption between the peptide and the surface.

The peptides with the beta-turn configuration have the strongest adsorption

Changes in dihedral angles of the peptides

The gyration radius of a peptide can not display all the information about a peptide configuration. Therefore, the dihedral angles of the peptides have been calculated to indicate the change in configuration with more detail. For each residue unit, three dihedral angles can be defined as in Figure 7.

The distribution of each type of dihedral angle could be obtained from the trajectory of the MD simulations. For example, the distributions of all dihedral angles for PI(A)

during the former simulations are shown in Fig. 8.

During the simulations, the dihedral angle "omega" hardly changed from around 180°, which agrees with the expected chemical structure. The figures clearly show that the change of peptide configuration can be reflected by the change in the dihedral angles "phi" and "psi". For simulation of PI (A) in water, the distribution of the dihedral angle "phi" has a major peak around 290°, and a small peak around 230°, while the distribution of the dihedral angle "psi" has a major peak around 340°, and a small peak around 230°. The distribution of these dihedral angles shows that peptides can keep their configuration in water without significant changes. From simulations of peptides on calcite surface, we can see some new peaks and slight shifts of previous peaks. The "phi" type of dihedral angle has two new distribution peaks around 80° and 170°, while "psi" has peaks around 70° and 160°. The difference in distribution of dihedral angles shows the alteration of configuration of peptides after they interact with the calcite surface. From the isosurfaces, we can see that the dominating electrostatic interaction is a key factor in peptide interactions with calcite. Two types of strong binding can be identified: 1) interactions between carbonate groups of the calcite surface and amino groups of the peptides; and 2) interactions between calcium ions in calcite and carboxyl groups of peptides. ... [1] ... [4] ... [2] ... [5] ... [8] ... [6] ... [9] ... [7] ... [10] ... [13] ... [11] ... [14] ... [12] .

Figure 2

 2 Figure 2 was built with Materials Studio 4.0 [11].

Formatted

  Configurations with coiled structure (smaller gyration radius) are energetically favourable since molecules with this kind of configuration are able to interact with themselves so their potential energy is lowered by these intramolecular interactions. PI(A ) and PII(B) have the largest gyration radius in all of the PI and PII configurations during the simulations, and as a result, their potential energies are the largest. The change of peptide configuration as indicated from the gyration radius will induces a change of potential energy of the peptide. For example, the gyration radius of PII(A) decreased sharply (about 10%) at t=1300 ps and accordingly, its potential energy increased from around -2100 to -2400 kJ/mol.

(

  adsorption energies: -204.4 kJ/mol for PI (C), and -136.8 for PII (C)); those of alphahelix configuration have the weakest adsorption, and those of extended configuration have adsorption energies between those of the beta-turn and alpha-helix configurations. The dependence of adsorption energy on peptide configurations indicates that the configuration of peptides plays a key role in biomineralization controlled by peptides or proteins. The results agree well with the experiments, which showed that peptides with specific configurations can facilitate calcite crystal growth [

3 . 5

 35 Electrostatic potential isosurfacesFrom the MD simulations, we are able to investigate the molecular system in atomistic detail. It is still instructive to plot the potential isosurfaces of peptide and calcite with the PBE calculations. The electrostatic potential isosurfaces represent points of electrostatic potential within a volume of space. The results from PBE calculations with the APBS program are shown in Figure9.

Formatted

  were used to investigate the interaction between peptides and this calcite surface. The results indicate that the specific configuration of peptides plays an important role in their adsorption. The calculated adsorption energies show that beta-turn configuration peptides have the strongest interaction with the calcite surface, extended configurations have less strong interactions, and alpha-helix configurations have the least interaction. From the continuum electrostatic calculations with PBE, two types of strong interaction have been identified: 1) interactions between carbonate groups of the calcite surface and amino groups of the peptides; and 2) interactions between the calcite surface and carboxyl groups of peptides. This helps to explain the biological control in biomineralization.

Formatted

Figure 2 Figure 3

 23 Figure 2 Calcite (10.4) surface 78x34mm (300 x 300 DPI)

  Arg Arg Glu Glu Trp Trp Asp Asp Arg Arg Glu Glu Trp TrpAsp Asp) and a 17-amino acid peptide (Arg Arg Glu Glu Trp Trp Asp Asp Pro Arg Arg Glu Glu Trp Trp Asp Asp) were built with the Amber 9 program[9]. These two

	peptides on calcite. The adsorption of peptides on calcite ( ) 4 1 10	has been calculated
	and electrostatic interactions between the peptides and calcite surface have been
	examined with PBE continuum electrostatic calculations.
	2 2 2 2. . . . Methods		Deleted: to study the
			Deleted: mineralisation
	A 16-amino acid peptide (		Deleted: of calcite
	F o r		Deleted: mineralisation
	P
	e		Field Code Changed
	e r		Formatted: Font: Italic Field Code Changed Deleted: mineralisation
	R		Field Code Changed
	e
	v i e
	w	Field Code Changed Deleted:
		O n l	Deleted: is able to
		y

peptides have been reported to facilitate the nucleation, growth and aggregation of calcite crystal

[3]

. A peptide can have a variety of configurations; each configuration may have a different influence on calcite crystal growth. Therefore, for each peptide simulated, we generated three configurations, alpha-helix, extended and beta-turn.

Table 1

 1 Adsorption energies for peptides adsorbed on the calcite ( )

	.. [15]
	... [16]
	... [17]
	... [18]
	... [19]
	... [20]
	... [21]
	... [22]
	... [23]
	... [24]
	... [25]

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Acknowledgements M.Y. and J.H.H. acknowledge funding from EPSRC under Grant No GR/S80103/01 and M.Y. and S.L.S.S. acknowledge funding from Maersk Olie og Gas AS. Computer resources were provided by the Danish Centre for Scientific Computing (DCSC) and the MOTT facility (EPSRC Grant GR/S84415/01), run by the STFC e-Science Centre.

We also thank Colin L. Freeman (University of Sheffield, UK) for helpful discussions.


S Martinie

D Munteanu

G Le Carval

J L Autran

MARTINIE Mr Sébastien
email: sebastien.martinie@cea.fr

Analytical modelling and performance analysis of Double-Gate MOSFET-based circuit including ballistic/quasi-ballistic effects

Keywords: Double-Gate MOSFET, ballistic/quasi-ballistic transport, compact model, ring oscillator Double-Gate MOSFET, ballistic/quasi-ballistic transport, compact model, Ring Oscillator

Oscillation frequency (Hz) Charge capacitance (pF) Ballistic transport Analytical model Numerical model t si =5 nm t ox = 1.2nm L c =25 nm

We want to thank the reviewers for their comments and very pertinent suggestions. The manuscript has been revised according to their recommendations as shown in the following.

1) Reviewer's comment point 1:

Response: We agree with the reviewer comment, there is typographic error in equation ( 1): Considering the corresponding signs of in (1-R)/(1+R), we obtain:

                  + - + -       + - - = - - T k qV T k qV th T GS ox D DS DS e R R e R R v V V C W I . . 1 1 1 1 . 1 1 .
). .( .

(1)

2) Reviewer's comment point 2:

Response: Figure 6.a illustrates the impact of the channel length on the oscillation frequency. For the smallest length the oscillation frequency increases due to SCE/DIBL. But for long channel case the transport (ballistic or quasi-ballistic) is predominant, thus oscillation frequency decrease in quasi-ballistic case because this depends on the channel length and is constants for ballistic transport as the I on . But, effectively we suspect that the increase of the oscillation frequency in the ballistic case is not realistic due to numerical noise in our code. In fact this is not really important for the objective of our study, which is dedicated to the simulation of short length.

3) Reviewer's comment point 3:

Response: The figure 9 has been corrected to include the structural parameter of the DG MOSFET:

Introduction

As the MOSFET continues to shrink rapidly, emerging physical phenomena, such as ballistic transport, have to be considered in the modelling and simulation of ultra-scaled devices. Future Double-Gate MOSFETs (DGMOS Fig. 1), designed with channel lengths in the decananometer scale, are expected to be more ballistic or quasi-ballistic than diffusive. At this level of miniaturisation is essential to directly evaluate the impact of ballistic and quasi-ballistic transport at circuit level through simulation of several circuit demonstrators. The implementation of compact models in Verilog-A (Smartspice [1]) environment offers the opportunity to describe as accurately as possible the physics of transport and to analyze its impact on various circuit elements.

Several analytical models based on the Drift-Diffusion formalism demonstrate that it is possible to introduce the diffusive transport in compact modelling. However, when the channel length approaches the value of mean free path, the mobility definition can no more strictly explain the electronic transport in the device [2]. In this case we use the flux theory and the main parameter of this approach is the backscattering coefficient, which expresses the ballistic and the quasi-ballistic transport. Some well-known works performed by Lundstrom et al [3][4][5] demonstrate the usefulness of the flux theory in qualitatively describing quasi-ballistic transport in compact modelling.

More recently several works [6][7][8] proposed some accurate solution to describe transport for nanoscale DG MOSFETs including quantum confinement or the effect of tunneling through the source-drain potential barrier on devices characteristic. But few works [9][10] exposed the influence of ballistic/quasi-ballistic transport on the operation of circuit demonstrator. In this work we demonstrate the feasibility of a simulation study of ballistic/quasiballistic transport at circuit level and we show the impact of this advanced transport on the switch of CMOS inverter and the oscillation frequency of ring oscillator. This paper is organized as follows: the section I explains the model physics and the corresponding assumptions. In part II, we explain our device simulation in ballistic and quasi-ballistic case. The part III highlights the qualitative connection between physics of quasiballistic transport and its impact on circuit performance. Finally, we compare our model with numerical simulation in terms of CMOS switch and oscillation frequency.

DG MOSFET model

A. Ballistic and quasi-ballistic transport

The proposed analytical model (implemented in Verilog-A environment) is based on the well-known work of Natori [10] and Lundstrom [3]. This formula describes the ballistic and quasi-ballistic current:

                  + - + -       + - - = - - T k qV T k qV th T GS ox D DS DS e R R e R R v V V C W I . . 1 1 1 1 . 1 1 . ). .( . ( 1
)
where W is the gate width, C ox is the gate oxide capacitance, V GS is the gate to source voltage, V DS is the drain to source voltage, V T is the threshold voltage, k is the Boltzmann constant, q is the electron charge and T is the lattice temperature.

In addition, Lundstrom et al. developed physical compact models describing the device operation in quasi-ballistic regimes using the backscattering coefficient R [4]:

λ + = kT kT L L R ; DS c kT V q T k L L . . . = (2)
where λ is the mean free path and L kT is the distance over which the channel potential drops by kT/q compared to the peak value of the source to channel barrier. Physically, L kT represents the critical distance over which scattering events modify the current; the limit and the demonstration of the backscattering coefficient are exposed [12].

We consider here the Dynamical mean Free Path which is a "local free path of ballistic carriers" [8]. This characteristic length represents the average distance to be crossed before the next scattering event. This approach considers that each carrier is ballistic as long as any event does not disturb its trajectory. The dynamic mean free path (dfp) connects the ballistic velocity and all interactions (coulomb and acoustic/optical phonon interaction) experienced by carriers crossing the channel. The scattering process with impurities (τ imp ) and phonon interactions (τ ph ) are calculated as in [8] and the value of dfp used here is 27 nm in the intrinsic silicon channel. In practice, dfp replaces λ to describe quasi-ballistic transport:

tot bal v dfp τ . = ;          + = = + = - - - DS bal bal bal ph imp tot V q T k m v . . . 2 3 * . 2 1 1 1 ε ε τ τ τ (3)
where m* is the mass in direction of transport, v bal the ballistic velocity, τ tot the total scattering rate and ε bal the carrier energy.

B. Short channel effects

To obtain an accurate model and describe all electrostatic effects, we have also introduced Short Channel Effect and Drain Induced Barrier Lowering (SCE/DIBL) using the Suzuki's model [13] where η=ψ s (x)-V GS , ψ s (x) is the surface potential, ε Si and ε ox are, respectively, the silicon and the oxide permittivity. The solution of equation (5.a) is:

            +       - = λ λ η λ η η c D c S L sh x sh x L sh x . . ) ( (6)
where the boundary conditions are η S =ψ S (0)-V GS at the source and η D =ψ S (L c )-V GS at the drain. ψ S (0) is the built-in potential equal to (k.T/q).ln(N D .N A /ni²), N D and N A are, respectively, the doping level in the source/drain regions and in the channel.

The expression of the threshold voltage shift (∆V T ) due to SCE/DIBL depends on the value of the minimum potential, given by:

λ η η ψ . 4 min min . . . 2 ) ( c L D S GS S e V x - + = (7)         - = S D c L x η η λ ln . 2 2 min ( 8
)
∆V T is then obtained from the following equation [13]:

λ η η . 4 . . . 2 c L D S T e V - = ∆ (9)
After some algebraic manipulation, we find the analytical expression of ∆V T . Thus, V T in equation ( 1) is modified by ∆V T , as follows:

T th T V V V ∆ - = (9)
where V th and ∆V T are the long channel threshold voltage and its variation due to SCE/DIBL, respectively. Finally, the above-threshold regime is linked to the subthreshold regime using an interpolation function based on the subthreshold swing S parameter, also defined by Suzuki in [13]:

1 min min ) ( ). 10 ln( . . -         = GS S dV x d q T k S ψ (10)
This assures the perfect continuity of our model between on-state regime and off-state regime.

Simulation RESULTS

A. Device simulations

After implementation in Verilog-A environment, the model has been used to simulate the DGMOS structure schematically presented in Figure 1. The source and drain regions are heavily doped (1×10 20 cm -3 ) and an intrinsic thin silicon channel is considered. The channel length varies from 10 nm to 200 nm; a gate oxide of 1.2 nm thick and a midgap metal gate are also considered.

It is well-known that the ballistic current is independent of channel length [11] except when SCE or DIBL appears. In order to clearly confirm this point, simulations have been performed for several length (20, 25, 30, 40, 50, 100 and 200 nm) and considering two types of transport (quasi-ballistic and ballistic; Fig. 2). Note that for the ballistic case, the mean free path value has been chosen to be extremely large compared to the channel length. In contrast to the ballistic case, the quasi-ballistic transport has the same behaviour as that of diffusive transport and the form of the output characteristics depends on L c .

Figure 3 shows the drain current versus the gate voltage characteristics for the DGnMOS and DGpMOS simulated devices at V DS =0.7V. In this approach, we suppose that the transport description (for ballistic and quasi-ballistic case) for holes is identical to that of electrons, with uniquely changing the thermal velocity value in non- degenerate conditions (Fig. 1b et 1c, [14]). As expected, the ballistic and quasi-ballistic current shows a perfect continuity between the above and the subthreshold regime (as illustrated on Fig. 3a). Finally, figure 3b shows that the DGnMOS and DGpMOS have the same behaviour in terms of transport, with different current levels due to the different values of thermal velocity.

B. Circuit simulations

In addition to the simulation of single device operation, we have simulated different circuit elements such as CMOS inverters and ring oscillators (Fig. 1b and1c) to show the impact of ballistic/quasi-ballistic transport at circuit level.

The output voltage (V out ) of the CMOS inverter switches more sharply from the "1" state to the "0" state in the ballistic case than in quasi-ballistic transport (Fig. 4). The switch of the CMOS inverter depends on the limit between linear and saturation region, which controls the switch between transistors. When SCE/DIBL occur, the transition between linear and saturate regime is modified, and the switch from the "1" state to the "0" state is less sharp. In the quasi-ballistic case, the abruptness of the CMOS characteristic is strongly deteriorated. In conclusion, these results prove that the ballistic transport improves the switch and the static performances of the CMOS inverter.

Figure 5 shows the oscillation frequency as a function of the charge capacitance for two channel lengths: 100 and 30 nm. As expected the oscillation frequency is reduced when the charge capacitance increases, due to variation of the propagation time through the inverters. We can also note the strong influence of short channel effects that increase the current value and reduce the difference between the oscillation frequencies in quasiballistic transport compared with ballistic transport. Moreover, we thoroughly studied the influence of parasitic elements and phenomena on circuit performances. Figure 6.a illustrates the impact of two charge capacitances on the oscillation frequency versus the channel length. -Firstly, the benefit of ballistic transport (versus quasi-ballistic) on the switch of the CMOS inverters is hidden (inset (a) and (b) of Fig. 4).

-Secondly, the I on current strongly increased (Fig. 6.b).

Consequently the oscillation frequency in ballistic and quasi-ballistic case is less influenced by the value of the dynamic mean free path.

These results show that the oscillation frequency is directly influenced by the type of transport (ballistic versus quasi-ballistic) which changes strongly the static and the transient performances. However, parasitic phenomena such as interconnect capacitances or SCE/DIBL are essential parameters in the analysis of circuit performances even when the intrinsic behaviour of transistors is dominated by ballistic transport.

Comparison with numerical Simulation

A. Numerical simulation of ballistic transport using the quasi-ballistic mobility concept

In a previous work we enhanced the pioneering approach of quasi-ballistic mobility proposed by Rhew et al [15] and we introduced it into TCAD simulator, in order to numerically simulate the ballistic/quasi-ballistic transport using a TCAD tool. This approach, extensively presented in [9], is used here to validate our analytical model. In the following, we shortly remind the basic equations of this approach, based on the flux theory or McKelvey's flux method [2], [15].

) ( ) ( ) ( . . x S x D x D r r dx d F F F ← → - = (11)
) ( ) ( ) ( . . x S x D x S r r dx d F F F ← → - = ( 12
)
where r is the scattering probabilities in the presence of an electric field and the symbols "→" and "←" of scattering probabilities represent the carrier's velocity components that are parallel or anti-parallel, respectively, to the electric field direction. Adding equations ( 11) and ( 12) and considering F=F D -F S and n=(F D +F S )/v th (where v th is the thermal velocity) we obtain:

( ) 0 = - = dx d dx d S D F F F (13) dx dn r r v n v r r r r th th . . . ← → ← → → ← + +         + - = F ( 14
)
We use here the expression of scattering probabilities proposed in [15] (with the assumption of non-degenerate gas) for the negative charge carrier:

1 - ← = mfp r ; T k x E q mfp r . ) ( . 1 - = - → ( 15
)
where k is the Boltzmann constant, T is the lattice temperature, q is the electron charge, mfp is the mean free path and E is the electric field in the direction of transport. By analogy to the classical drift-diffusion approach, we obtain the Einstein relation; we define then a new mobility-like parameter, µ qb , called quasi-ballistic mobility [7]:

) ( . . 1 . 2 x E q T k mfp v th qb + = µ (16)
This quasi-ballistic mobility is introduced in TCAD software [9] (ATLAS SILVACO [16]). This approach represents an enhanced Drift-Diffusion-like approach to include quasi-ballistic and ballistic effects in the TCAD simulator. The quasi-ballistic mobility model is implemented using an explicit C-interpreter function describing the relation between the mobility and the parallel electric field (equation ( 16)). All other quantities (such as the electrostatic potential, electric field, carrier concentration, current densities ...) are computed as usual in TCAD simulation, by solving the Poisson equation coupled to drift-diffusion transport equation. The details and the validation of this approach are explained in [9] and illustrated with simulations of small circuits based on DGMOS technology.

B. Comparison between numerical and analytical simulations

In order to validate the analytical model, we have compared our results with numerical simulation. Figure 8 compares the switch of the CMOS inverter in ballistic case for different channel lengths and illustrates the difference between our analytical model and numerical simulation.

For the long channel case (L c =100 nm), the ballistic switch for numerical and analytical simulation perfectly matches. But for short channel (L c =20 nm) a small error is found in the comparison with numerical data. This is due to the fact that for L c =20 nm we approach the validity limit of our analytical model (which is L c =2.t Si in the description of ∆V T ). Finally, the numerical simulations confirm the previous remark (exposed in paragraph 3) that short channel effects deteriorate the switch of the CMOS inverter.

Figure 9 compares the analytical and numerical oscillation frequency (for t Si = 5 nm, t ox = 1.2 nm, L c =25 nm) in the ballistic case for different charge capacitances. We note that an identical behaviour is obtained for the numerical and the analytical data, which confirms the result of paragraph 3. Although, a small error exists (error attributed to the I on value, which is not exactly the same in analytical and numerical case), our analytical model shows a good agreement with the numerical simulation.

Conclusion

In this work, a compact model for DGMOS taking into account ballistic and quasiballistic transport has been proposed and implemented in Verilog-A environment. Short channel effects and an interpolation function to link the above and the subthreshold The dynamical mean free path definition was used to describe scattering processes with impurities and phonons. Finally, the model has been used to simulate two different small-circuits (CMOS inverter and ring oscillators) and to show the significant impact of ballistic/quasi-ballistic transport on the switch of CMOS inverter and the oscillation frequency of ring oscillator. Our simulation results prove that the ballistic transport improves the switch and the static performances of the CMOS inverter, and increases the oscillation frequency of ring oscillators. Finally, we have compared this model with numerical simulation in ballistic case; this comparison also validates our conclusions on the influence of short channel effects and ballistic transport on the operation of circuit elements.

This work also demonstrates the feasibility of a simulation study of ballistic/quasiballistic transport at circuit level and highlights the direct relation between the type of transport and static or transient performances of small-circuits.

Acknowledgements

Introduction

As the MOSFET continues to shrink rapidly, emerging physical phenomena, such as ballistic transport, have to be considered in the modelling and simulation of ultra-scaled devices. Future Double-Gate MOSFETs (DGMOS Fig. 1), designed with channel lengths in the decananometer scale, are expected to be more ballistic or quasi-ballistic than diffusive. At this level of miniaturisation is essential to directly evaluate the impact of ballistic and quasi-ballistic transport at circuit level through simulation of several circuit demonstrators. The implementation of compact models in Verilog-A (Smartspice [1]) environment offers the opportunity to describe as accurately as possible the physics of transport and to analyze its impact on various circuit elements.

Several analytical models based on the Drift-Diffusion formalism demonstrate that it is possible to introduce the diffusive transport in compact modelling. However, when the channel length approaches the value of mean free path, the mobility definition can no more strictly explain the electronic transport in the device [2]. In this case we use the flux theory and the main parameter of this approach is the backscattering coefficient, which expresses the ballistic and the quasi-ballistic transport. Some well-known works performed by Lundstrom et al [3][4][5] demonstrate the usefulness of the flux theory in qualitatively describing quasi-ballistic transport in compact modelling.

More recently several works [6][7][8] proposed some accurate solution to describe transport In this work we demonstrate the feasibility of a simulation study of ballistic/quasiballistic transport at circuit level and we show the impact of this advanced transport on the switch of CMOS inverter and the oscillation frequency of ring oscillator. This paper is organized as follows: the section I explains the model physics and the corresponding assumptions. In part II, we explain our device simulation in ballistic and quasi-ballistic case. The part III highlights the qualitative connection between physics of quasiballistic transport and its impact on circuit performance. Finally, we compare our model with numerical simulation in terms of CMOS switch and oscillation frequency.

DG MOSFET model

A. Ballistic and quasi-ballistic transport

The proposed analytical model (implemented in Verilog-A environment) is based on the well-known work of Natori [10] and Lundstrom [3]. This formula describes the ballistic and quasi-ballistic current:

                  + - + -       + - - = - - T k qV T k qV th T GS ox D DS DS e R R e R R v V V C W I . . 1 1 1 1 . 1 1 . ). .( . ( 1
)
where W is the gate width, C ox is the gate oxide capacitance, V GS is the gate to source voltage, V DS is the drain to source voltage, V T is the threshold voltage, k is the Boltzmann constant, q is the electron charge and T is the lattice temperature.

In addition, Lundstrom et al. developed physical compact models describing the device operation in quasi-ballistic regimes using the backscattering coefficient R [4]:

λ + = kT kT L L R ; DS c kT V q T k L L . . . = (2)
where λ is the mean free path and L kT is the distance over which the channel potential drops by kT/q compared to the peak value of the source to channel barrier. Physically, L kT represents the critical distance over which scattering events modify the current; the limit and the demonstration of the backscattering coefficient are exposed [12].

We consider here the Dynamical mean Free Path which is a "local free path of ballistic carriers" [8]. This characteristic length represents the average distance to be crossed before the next scattering event. This approach considers that each carrier is ballistic as long as any event does not disturb its trajectory. The dynamic mean free path (dfp) connects the ballistic velocity and all interactions (coulomb and acoustic/optical phonon interaction) experienced by carriers crossing the channel. The scattering process with impurities (τ imp ) and phonon interactions (τ ph ) are calculated as in [8] and the value of dfp used here is 27 nm in the intrinsic silicon channel. In practice, dfp replaces λ to describe quasi-ballistic transport:

tot bal v dfp τ . = ;          + = = + = - - - DS bal bal bal ph imp tot V q T k m v . . . 2 3 * . 2 1 1 1 ε ε τ τ τ (3)
where m* is the mass in direction of transport, v bal the ballistic velocity, τ tot the total scattering rate and ε bal the carrier energy.

B. Short channel effects

To obtain an accurate model and describe all electrostatic effects, we have also introduced Short Channel Effect and Drain Induced Barrier Lowering (SCE/DIBL) using the Suzuki's model [13] where η=ψ s (x)-V GS , ψ s (x) is the surface potential, ε Si and ε ox are, respectively, the silicon and the oxide permittivity. The solution of equation (5.a) is:

            +       - = λ λ η λ η η c D c S L sh x sh x L sh x . . ) ( ( 6
)
where the boundary conditions are η S =ψ S (0)-V GS at the source and η D =ψ S (L c )-V GS at the drain. ψ S (0) is the built-in potential equal to (k.T/q).ln(N D .N A /ni²), N D and N A are, respectively, the doping level in the source/drain regions and in the channel.

The expression of the threshold voltage shift (∆V T ) due to SCE/DIBL depends on the value of the minimum potential, given by:

λ η η ψ . 4 min min . . . 2 ) ( c L D S GS S e V x - + = (7)         - = S D c L x η η λ ln . 2 2 min ( 8
)
∆V T is then obtained from the following equation [13]:

λ η η . 4 . . . 2 c L D S T e V - = ∆ (9)
After some algebraic manipulation, we find the analytical expression of ∆V T . Thus, V T in equation ( 1) is modified by ∆V T , as follows:

T th T V V V ∆ - = (9)
where V th and ∆V T are the long channel threshold voltage and its variation due to SCE/DIBL, respectively. Finally, the above-threshold regime is linked to the subthreshold regime using an interpolation function based on the subthreshold swing S parameter, also defined by Suzuki in [13]:

1 min min ) ( ). 10 ln( . . -         = GS S dV x d q T k S ψ (10)
This assures the perfect continuity of our model between on-state regime and off-state regime.

Simulation RESULTS

A. Device simulations

After implementation in Verilog-A environment, the model has been used to simulate the DGMOS structure schematically presented in Figure 1. The source and drain regions are heavily doped (1×10 20 cm -3 ) and an intrinsic thin silicon channel is considered. The channel length varies from 10 nm to 200 nm; a gate oxide of 1.2 nm thick and a midgap metal gate are also considered.

It is well-known that the ballistic current is independent of channel length [11] except when SCE or DIBL appears. In order to clearly confirm this point, simulations have been performed for several length (20, 25, 30, 40, 50, 100 and 200 nm) and considering two types of transport (quasi-ballistic and ballistic; Fig. 2). Note that for the ballistic case, the mean free path value has been chosen to be extremely large compared to the channel length. In contrast to the ballistic case, the quasi-ballistic transport has the same behaviour as that of diffusive transport and the form of the output characteristics depends on L c .

Figure 3 shows the drain current versus the gate voltage characteristics for the DGnMOS and DGpMOS simulated devices at V DS =0.7V. In this approach, we suppose that the transport description (for ballistic and quasi-ballistic case) for holes is identical to that of electrons, with uniquely changing the thermal velocity value in non- degenerate conditions (Fig. 1b et 1c, [14]). As expected, the ballistic and quasi-ballistic current shows a perfect continuity between the above and the subthreshold regime (as illustrated on Fig. 3a). Finally, figure 3b shows that the DGnMOS and DGpMOS have the same behaviour in terms of transport, with different current levels due to the different values of thermal velocity.

B. Circuit simulations

In addition to the simulation of single device operation, we have simulated different circuit elements such as CMOS inverters and ring oscillators (Fig. 1b and1c) to show the impact of ballistic/quasi-ballistic transport at circuit level.

The output voltage (V out ) of the CMOS inverter switches more sharply from the "1" state to the "0" state in the ballistic case than in quasi-ballistic transport (Fig. 4). The switch of the CMOS inverter depends on the limit between linear and saturation region, which controls the switch between transistors. When SCE/DIBL occur, the transition between linear and saturate regime is modified, and the switch from the "1" state to the "0" state is less sharp. In the quasi-ballistic case, the abruptness of the CMOS characteristic is strongly deteriorated. In conclusion, these results prove that the ballistic transport improves the switch and the static performances of the CMOS inverter.

Figure 5 shows the oscillation frequency as a function of the charge capacitance for two channel lengths: 100 and 30 nm. As expected the oscillation frequency is reduced when the charge capacitance increases, due to variation of the propagation time through the inverters. We can also note the strong influence of short channel effects that increase the current value and reduce the difference between the oscillation frequencies in quasiballistic transport compared with ballistic transport. Moreover, we thoroughly studied the influence of parasitic elements and phenomena on circuit performances. Figure 6.a illustrates the impact of two charge capacitances on the oscillation frequency versus the channel length. -Firstly, the benefit of ballistic transport (versus quasi-ballistic) on the switch of the CMOS inverters is hidden (inset (a) and (b) of Fig. 4).

-Secondly, the I on current strongly increased (Fig. 6.b).

Consequently the oscillation frequency in ballistic and quasi-ballistic case is less influenced by the value of the dynamic mean free path.

These results show that the oscillation frequency is directly influenced by the type of transport (ballistic versus quasi-ballistic) which changes strongly the static and the transient performances. However, parasitic phenomena such as interconnect capacitances or SCE/DIBL are essential parameters in the analysis of circuit performances even when the intrinsic behaviour of transistors is dominated by ballistic transport.

Comparison with numerical Simulation

A. Numerical simulation of ballistic transport using the quasi-ballistic mobility concept

In a previous work we enhanced the pioneering approach of quasi-ballistic mobility proposed by Rhew et al [15] and we introduced it into TCAD simulator, in order to numerically simulate the ballistic/quasi-ballistic transport using a TCAD tool. This approach, extensively presented in [9], is used here to validate our analytical model. In the following, we shortly remind the basic equations of this approach, based on the flux theory or McKelvey's flux method [2], [15].

) ( ) ( ) ( . . x S x D x S r r dx d F F F ← → - = ( (11)
) 12
where r is the scattering probabilities in the presence of an electric field and the symbols "→" and "←" of scattering probabilities represent the carrier's velocity components that are parallel or anti-parallel, respectively, to the electric field direction. Adding equations ( 11) and ( 12) and considering F=F D -F S and n=(F D +F S )/v th (where v th is the thermal velocity) we obtain:

( ) 0 = - = dx d dx d S D F F F (13) dx dn r r v n v r r r r th th . . . ← → ← → → ← + +         + - = F ( 14
)
We use here the expression of scattering probabilities proposed in [15] (with the assumption of non-degenerate gas) for the negative charge carrier:

1 - ← = mfp r ; T k x E q mfp r . ) ( . 1 - = - → ( 15
)
where k is the Boltzmann constant, T is the lattice temperature, q is the electron charge, mfp is the mean free path and E is the electric field in the direction of transport. By analogy to the classical drift-diffusion approach, we obtain the Einstein relation; we define then a new mobility-like parameter, µ qb , called quasi-ballistic mobility [7]:

) ( . . 1 . 2 x E q T k mfp v th qb + = µ (16)
This quasi-ballistic mobility is introduced in TCAD software [9] (ATLAS SILVACO [16]). This approach represents an enhanced Drift-Diffusion-like approach to include quasi-ballistic and ballistic effects in the TCAD simulator. The quasi-ballistic mobility model is implemented using an explicit C-interpreter function describing the relation between the mobility and the parallel electric field (equation ( 16)). All other quantities (such as the electrostatic potential, electric field, carrier concentration, current densities ...) are computed as usual in TCAD simulation, by solving the Poisson equation coupled to drift-diffusion transport equation. The details and the validation of this approach are explained in [9] and illustrated with simulations of small circuits based on DGMOS technology.

B. Comparison between numerical and analytical simulations

In order to validate the analytical model, we have compared our results with numerical simulation. Figure 8 compares the switch of the CMOS inverter in ballistic case for different channel lengths and illustrates the difference between our analytical model and numerical simulation.

For the long channel case (L c =100 nm), the ballistic switch for numerical and analytical simulation perfectly matches. But for short channel (L c =20 nm) a small error is found in the comparison with numerical data. This is due to the fact that for L c =20 nm we approach the validity limit of our analytical model (which is L c =2.t Si in the description of ∆V T ). Finally, the numerical simulations confirm the previous remark (exposed in paragraph 3) that short channel effects deteriorate the switch of the CMOS inverter.

Figure 9 compares the analytical and numerical oscillation frequency (for t Si = 5 nm, t ox = 1.2 nm, L c =25 nm) in the ballistic case for different charge capacitances. We note that an identical behaviour is obtained for the numerical and the analytical data, which confirms the result of paragraph 3. Although, a small error exists (error attributed to the I on value, which is not exactly the same in analytical and numerical case), our analytical model shows a good agreement with the numerical simulation.

Conclusion

In this work, a compact model for DGMOS taking into account ballistic and quasiballistic transport has been proposed and implemented in Verilog-A environment. Short channel effects and an interpolation function to link the above and the subthreshold The dynamical mean free path definition was used to describe scattering processes with impurities and phonons. Finally, the model has been used to simulate two different small-circuits (CMOS inverter and ring oscillators) and to show the significant impact of ballistic/quasi-ballistic transport on the switch of CMOS inverter and the oscillation frequency of ring oscillator. Our simulation results prove that the ballistic transport improves the switch and the static performances of the CMOS inverter, and increases the oscillation frequency of ring oscillators. Finally, we have compared this model with numerical simulation in ballistic case; this comparison also validates our conclusions on the influence of short channel effects and ballistic transport on the operation of circuit elements.

This work also demonstrates the feasibility of a simulation study of ballistic/quasiballistic transport at circuit level and highlights the direct relation between the type of transport and static or transient performances of small-circuits.

Acknowledgements

  for V T . The starting point of this model is the Poisson equation, where the depleted and the electron charges are neglected: a parabolic dependence in the y-direction:

Figure 6 .

 6 b shows the I on current and V T length. This last figure demonstrates the strong influence of SCE/DIBL effect on the transient performance. The explanation is that when SCE/DIBL effects occur:

Figure 7

 7 Figure 7 shows the schematics of the flux evolution in the channel (in the direction of

  have been included to obtain a complete description of current characteristics.

  This work was supported by the French Ministry of Research (ANR-05-NANO-002 project "MODERN"). The authors would like to acknowledge O. Rozeau, M. Reyboz, V. Barral, T. Poiroux, M-A. Jaud, O Bonno and S. Barraud from CEA-LETI MINATEC (Grenoble, France) for helpful discussions.

Figure 1 .

 1 Figure captions

Figure 2 .

 2 Figure 2. Drain current versus V DS for L c =200, 100, 50, 40, 30, 25 and 20 nm and

Figure 3 .

 3 Figure 3. Drain current ((a) log and (b) lin scale) versus V GS for L c =100, 30 and 20 nm.

Figure 4 .

 4 Figure 4. V out versus V in in a CMOS inverter. [Inset (a) and (b): transfer curve in

Figure 5 .Figure 6 .Figure 7 .

 567 Figure 5. Oscillation frequency versus the charge capacitance for Lc=100 and 30nm

Figure 8 .

 8 Figure 8. V out versus V in in a CMOS inverter for t Si = 5 nm and t ox = 1.2 nm.

Figure 9 .

 9 Figure 9. Oscillation frequency versus charge capacitance for t Si = 5 nm, t ox = 1.2 nm,

Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 8 .Figure 9 .

 2345689 Figure 1. Martinie et al.

  for nanoscale DG MOSFETs including quantum confinement or the effect of tunneling through the source-drain potential barrier on devices characteristic. But few works[9- 10] exposed the influence of ballistic/quasi-ballistic transport on the operation of circuit demonstrator.

  for V T . The starting point of this model is the Poisson equation, where the depleted and the electron charges are neglected: a parabolic dependence in the y-direction:

Figure 6 .

 6 b shows the I on current and V T length. This last figure demonstrates the strong influence of SCE/DIBL effect on the transient performance. The explanation is that when SCE/DIBL effects occur:

Figure 7

 7 Figure 7 shows the schematics of the flux evolution in the channel (in the direction of

  included to obtain a complete description of current characteristics.

  This work was supported by the French Ministry of Research (ANR-05-NANO-002 project "MODERN"). The authors would like to acknowledge O. Rozeau, M. Reyboz, V. Barral, T. Poiroux, M-A. Jaud, O Bonno and S. Barraud from CEA-LETI MINATEC (Grenoble, France) for helpful discussions.

Figure 1 .

 1 Figure captions

Figure 2 .

 2 Figure 2. Drain current versus V DS for L c =200, 100, 50, 40, 30, 25 and 20 nm and

Figure 3 .

 3 Figure 3. Drain current ((a) log and (b) lin scale) versus V GS for L c =100, 30 and 20 nm.

Figure 4 .

 4 Figure 4. V out versus V in in a CMOS inverter. [Inset (a) and (b): transfer curve in

Figure 5 .Figure 6 .Figure 7 .

 567 Figure 5. Oscillation frequency versus the charge capacitance for Lc=100 and 30nm

Figure 8 .

 8 Figure 8. V out versus V in in a CMOS inverter for t Si = 5 nm and t ox = 1.2 nm.

Figure 9 .

 9 Figure 9. Oscillation frequency versus charge capacitance for t Si = 5 nm, t ox = 1.2 nm,

Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 23456 Figure 1. Martinie et al.

Figure 8 .Figure 9 .

 89 Figure 7. Martinie et al.

  2,3 1 CEA-LETI MINATEC, 17 rue des Martyrs, 38054 Grenoble, Cedex 9, France.

		ABSTRACT
	In this paper we present a compact model of Double-Gate MOSFET architecture
	including ballistic and quasi-ballistic transport down to 20 nm channel length. In
	F o r addition, this original model takes into account short channel effects (SCE/DIBL) by a simple analytical approach. The quasi-ballistic transport description is based on Lundstrom's backscattering coefficient given by the so-called flux method. We also F o r
		P P
		e e
		e r e r
		R R
		e e
		v i e v i e
		w w
	Corresponding author: Ring Oscillator.	O n l O n l
	Mr. Sébastien MARTINIE	y y
	CEA-LETI MINATEC,
	17 rue des Martyrs,
	38054 Grenoble, Cedex 9, France
	Phone: (33) 4 38 78 18 40
	Fax:	(33) 4 38 78 51 40
	E-mail: sebastien.martinie@cea.fr

2 IM2NP-CNRS, UMR CNRS 6242, Bât. IRPHE, 49 rue Joliot Curie, BP 146, 13384 Marseille Cedex 13, France. 3 University Institute of France (IUF), 75000 Paris, France. *include an original description of scattering of processes by introducing the "dynamical mean free path" formalism. Moreover, we implemented our model in a Verilog-A environment, and applied it to the simulation of circuit elements such as CMOS inverters and Ring Oscillators to analyze the impact of ballistic/quasi-ballistic transport on circuit performances. Finally, in order to validate our work, we confronted this model with numerical simulation of CMOS and Ring Oscillator in ballistic case. Keywords: Double-Gate MOSFET, ballistic/quasi-ballistic transport, compact model,

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