State-of-the-art semi-classical Monte Carlo method for carrier transport in nanoscale

transistors P. Palestri, E. Caruso, F. Driussi, D. Esseni, D. Lizzit, P. Osgnach, S. Venica, L. Selmi

DIEG, University of Udine, via delle scienze 206, 33100, Udine, Italy palestri@uniud.it

Abstract - We review the Monte Carlo method to model semi-classical carrier transport in advanced semiconductor devices. We report examples of the use of the Multi-Subband Monte Carlo method to simulate MOSFETs with III-V compound semiconductor channel. Monte Carlo transport modeling of graphene-based transistors is also addressed.

I. INTRODUCTION The models implemented in commercial TCAD tools

for semiconductor devices are derived from the Boltzmann Transport Equation (BTE) under assumptions that make them inaccurate when applied to nanoscale devices (LG<100 nm), or to the study of hot carriers.

Exact numerical solutions of the BTE via the Monte Carlo (MC) method have been used since the 60s and the 70s to analyze uniform transport in semiconductors [1]. Monte Carlo simulators describe the carrier motion as a sequence of “free-flights” (i.e. collision-less motion according to Newton’s law) interrupted by scattering events, which abruptly change the particle’s momentum. The simulation is statistical in the sense that the duration of the free-flight, the selection of the scattering mechanism and of the state-after-scattering are randomly chosen according to the respective probability density functions.

During the 80s, MC models have become increasingly complex by including full-band descriptions of the E(k) relation and achieving the capability to describe realistic 2D devices [2],[3]. During the 90s, full-band MC simulators evolved by including comprehensive sets of scattering mechanisms and by improving the numerical efficiency. Mutual agreement between MC simulators developed by different groups was consolidated [4] and a wide range of successful applications to the study of I-V, RF and reliability properties of scaled MOSFETs [5][6], BJTs [7] and non-volatile-memories [8] was reported.

Quantization effects have become more and more relevant in scaled MOSFETs, and quantum corrections have been proposed for semi-classical MC simulators starting from the year 2000 [9]-[12]. These corrections describe the electrostatic effects due to the displacement of the inversion charge from the channel/dielectric interface, and sometimes also for the valley splitting [13]. However, other important aspects related to the quantized nature of the carrier gas (e.g. scattering rates modulated by size- and bias-induced quantization, strong degeneracy of the carrier gas) are not captured by these corrections.

These limitations led to the development of the Multi-Subband Monte Carlo (MSMC) approach.

This paper reviews the main features of the MSMC method and reports sample applications to the study of MOSFETs with III-V channel materials and comparison with full-quantum transport simulations. The use of the MC method to study graphene-based devices is reported too. We thus demonstrate that the MSMC method is a very powerful and useful technique to assess the potential of new, technology-boosted, MOSFET architectures.

II. THE MULTI -SUBBAND MONTE CARLO APPROACH

The MSMC methodology has been initially proposed in [14]. Since then, MSMC simulators have been developed by various groups [15]-[20]. A typical simulation flowchart is reported in Figure 1.

Figure 1 Top: Flowchart of a MSMC simulator; Bottom: sketch of a double-gate MOSFET with indication of the slices normal to the transport direction where the 1D quantization problem is solved.

The simulation proceeds as follows [21]. Starting from an initial guess of the 2D potential profile, the device is divided in sections along the channel (x) direction. In each section, the 1D Schrödinger equation (SE) is solved for all relevant valleys. The discrete energies levels in each section are used to build subband profiles along the

MIPRO 2015/MEET 5

channel; the derivative of the subband energy along x provides the driving force to move particles in the MC simulation module. The solution of the SE also provides the wave-functions that are used to compute the scattering matrix elements for intra- and inter-subband transitions and, at a later stage, the charge distribution in the direction normal to transport.

Scattering rates and subband profiles feed the MC transport module that solves the multi-subband BTE [21]; the particle state contains only the position along the channel and the 2D wave-vector in the transport plane (normal to quantization). In other words, the MC module computes, for each subband and at each position, the occupation probability of the k-states in the transport plane. During the MC step, scattering events are rejected based on the availability of the final state [22], in order to enforce Pauli’s principle that is of paramount importance to obtain accurate Fermi-Dirac occupation in highly doped regions at equilibrium and to account for the strong degeneration of the inversion channel. At the end of the MC step, the electron concentration profile is available. Then, the solution of the 2D Poisson equation provides a new estimate of the potential. The procedure is iterated until convergence is reached, i.e. until the drain current update among iterations is less than a few %.

In this paper, we show results using the MSMC tool developed at the University of Udine over the last 10 years. The simulator has been initially developed for silicon MOSFETs (bulk or SOI) [20]. Local phonon, Coulomb and surface roughness scatterings are commonly regarded as the dominant collision mechanisms in these devices. Only ' valleys are relevant, and their weak non-parabolicity can be accounted for in the calculation as a correction of the in-plane dispersion relation, resorting to the conventional Effective Mass Approximation (EMA) when solving the 1D Schrödinger equation. Screening of the scattering events has been implemented following either the scalar or the matrix dielectric function approach [23]. Extension of the model to sSi [24] and Ge [25] required a more flexible treatment of multiple sets of valleys (CB minima) with arbitrary energy shift and inter-valley phonons. Alloy scattering was then added to enable the simulation of SiGe channels [26]. Furthermore, specific scattering models were added to simulate MOSFETs with high-k dielectrics [27]; in particular, remote polar phonon scattering (i.e. the scattering potential induced in the channel by the polar phonons in the dielectric) and remote Coulomb scattering (scattering induced by fixed charges in the gate stack). Models for metallic source and drain were also developed as described in [28]. Among possibly relevant but still missing scattering mechanisms, it is worth to mention that so far we do not include long-range Coulomb interactions [29][30] whose effect on the current of short MOSFETs is still debated [31].

The multi-valley energy relation and the wide variety of scattering mechanisms allow extending the use of the MSMC model to III-V MOSFETs. Due to the highly defective interfaces of III-V semiconductors, a model for interface states has been recently implemented [32]. Generation of e-h pairs by band-to-band tunneling have been added to enable the simulation of tunnel-FETs [33].

A MSMC for pMOSFETs has been developed too [34][35].

The rich portfolio of physical effects and scattering mechanisms increases the computational burden. In spite of this, competitive execution times and remarkably good numerical efficiency was achieved thanks to optimization and parallelization of the code [36].

III. APPLICATIONS OF MSMC TRANSPORT MODELLING TO III-V CHANNEL MOSFETS

As mentioned in the previous section, the MSMC method was initially developed for group IV semiconductors. Its extension to III-V compounds required additional models for polar phonon scattering [21] and interface states [32] and a quantization model taking into account the strong non-parabolicity of the * valley [37] and the layered structure of the channel.

The non-parabolic quantization model of ref. [37] not only corrects the subband energies, but also modifies the in-plane E(k) relation. As a result (see Figure 2), when the well thickness decreases, the effective mass (i.e. the inverse curvature of the E(k) relation) of the lowest subband increases. This result is consistent with the accurate quantization models based on DFT, tight-binding or k�p calculations [38].

Figure 2: Effective mass (calculated from the inverse curvature of the in-plane E(k) of the lowest subband) vs. well thickness for different III-V compounds. The model of Ref.[37] has been used to account for the non-parabolicity of the * valley.

Figure 3: Comparison between simulated and measured [39] velocity-

field curves in bulk In0.53Ga0.47As.

One of the advantages of the MC method is that it handles equally well long and short channel devices. The former are important for calibration purposes. As an example, Figure 3 compares the simulated and experimental velocity-field curves for bulk In0.53Ga0.47As

6 MIPRO 2015/MEET

(the bulk structure is simulated in the MSMC as a thick well considering a large number of subbands), whereas Figure 4 compares MSMC results and measured mobility in In0.53Ga0.47As inversion layers. The results in Figure 3 validate the bulk band structure and the phonon and alloy scattering models, whereas Figure 4 was used to calibrate the parameters of the surface roughness model. We employ here the “standard” linear model for surface roughness [43]. A more rigorous non-linear model has been developed [44][45] and is being implemented in the MSMC.

Figure 4: Comparison between simulated and measured [40][41] low-field mobility in bulk In0.53Ga0.47As MOSFETs.

After calibration on long channel devices, the MSMC has been used to simulate the current drive of short channel III-V nMOSFETs [46],[42]. An example of transfer characteristics is reported in Figure 5 for a device with LG=14nm, designed according to the ITRS roadmap for HP devices. These calculations account for all the relevant scattering mechanisms, namely: local and remote polar phonons, surface roughness, alloy scattering, intra- and inter-valley non-polar acoustic and optical phonons. Interface states have not been considered in these simulations. The on-current of the In0.53Ga0.47As device is larger than in silicon but comparable to the one of highly strained (2GPa) silicon. The subthreshold swing is also improved in In0.53Ga0.47As as discussed in details in [46].

The analysis of the internal variables in Figure 6 helps us understand the differences between In0.53Ga0.47As and sSi channel. The profile of the first subband energy along the channel (right axis) shows that, due to the low density-of-states, the electron gas in the source and drain regions is strongly degenerate in In0.53Ga0.47As. The screening length of a highly degenerate electron gas is much longer than in a weakly degenerate one (as is the case of sSi, where the subband energy in the source and drain is close to the Fermi levels). As a result, the potential induced by the gate bias penetrates much deeper inside the source and drain in the In0.53Ga0.47As device than in the sSi one. The “effective length” of the In0.53Ga0.47As device is thus larger than in the sSi one [46] and this explains the better SS.

Figure 5 Top: simulated trans-characteristics for a MOSFET with LG=14nm considering different channel materials. TOX=4nm. The volumetric gate thickness is TGATE=4nm. TW=5nm. The device is

sketched in the lower panel. The source and drain junctions are abrupt and perfectly aligned to the gate edge.

Figure 6: Internal quantities (subband profile, inversion density and average velocity) for the device of Figure 5 considering sSi and InGaAs

channels. The energy reference (E=0) is the Fermi level in the source contact.

MIPRO 2015/MEET 7

The inversion density profile (Nsh(x), Figure 6, top graph) is, as expected, lower in In0.53Ga0.47As than in sSi, due to the lower density of states. Conversely, the average velocity profiles along the channel (bottom graph) show that Vx is larger in In0.53Ga0.47As than in sSi due to the lower effective mass in the transport direction. In particular, the velocity at the injection point (maximum of the potential barrier, indicated by the vertical arrows in Figure 6) is larger by a factor of about three, and this compensates the lower inversion charge, so that the current is almost the same in the two devices. The digital switching time can be estimated from the static simulations [46] and it is 0.327ps for the In0.53Ga0.47As device vs. 0.362ps for the sSi one. This means that the charge modulation in the S/D regions due to the low DoS almost compensates the larger injection velocity of In0.53Ga0.47As.

IV. SEMI-CLASSICAL MC VS. FULL-QUANTUM MODELS

The MSMC describes quantization normal to the transport direction in a rigorous way. On the other hand, transport from source to drain is semi-classical, i.e. electrons are treated as point particles obeying Newton’s law. This means that phenomena such as source-to-drain tunneling, that is relevant in the sub-threshold region of short channel devices are not accounted for. To overcome this limitation, following the effective potential approach already used for quantum corrections in semi-classical MC [9], we have proposed the subband smoothing procedure sketched in Figure 7 [47]: convolution between the subband profile and a Gaussian (envelope) function mimics the distributed nature of the electron wave-packet. In other words, the electron “feels” an average of the driving force in its surroundings, but it is still treated as a point particle in the MC transport core. The Gaussian shape is assigned to the electron when solving the Poisson equation [47].

Figure 7 Sketch of the subband smoothing procedure applied in the

MSMC. At low VDS the smoothening leads to more penetration of the electrons from the S/D into the channel. At VDS>0, the top of the potential barrier at the injection point is lowered: the enhanced

thermionic emission mimics the source-to-drain tunneling.

�

Figure 8 Comparison between MSMC and NEGF [48] simulations for Si MOSFETs with different gate length. Double-gate devices with 3nm

film thickness and 0.9nm EOT.

The only parameter of this model is the standard deviation of the Gaussian function. Figure 8 compares the MSMC results with and without subband smoothening with NEGF results obtained by using “nanomos” [48]. Silicon MOSFETs are considered. A single value of 1.1nm for the standard deviation of the Gaussian function allows the MSMC to reproduce the NEGF results for devices down to 5nm gate length. Applicability of this approach to III-V channel MOSFETs is under investigation.

V. MONTE CARLO MODELING OF GRAPHENE DEVICES Due to the large mobility and its monoatomic

thickness [49], graphene has been widely investigated as material for electron devices, either as a replacement of the silicon channel in FETs or as part of alternative device structures able to exploit the graphene unique features.

Although quantum-mechanical effects are very relevant in graphene due to its gap-less nature and to the long coherence length, semi-classical transport and MC methods have been used as well to predict device performance, as shown in the following two examples.

A. Graphene FETs (GFETs) The MC method has been used by several authors

studied dissipative transport in GFETs [50]-[54]. When considering graphene foils, band-to-band-tunneling (BTBT) must be taken into account [52],[54] in order to describe the strong ambipolarity of the devices. We have shown in [54] that a simple local BTBT model is adequate to reproduce more accurate NEGF calculations.

As explained above, where we showed results for III-V MOSFETs, one of the main advantages of the MC approach is that the same model can be used for long channels (for calibration purpose) as well as for short channels (to predict the performance of ultimately scaled devices). In this spirit, Figure 9 compares MC simulations and experiments for mobility in graphene nano-ribbons. Here the dominant scattering mechanisms are edge roughness and remote polar phonons from the insulator [53]. A more extensive comparison for different

8 MIPRO 2015/MEET

charge sheet densities and temperature can be found in [53].

Note that MC simulation of graphene foils does not require the MSMC approach, since there is no quantization-induced subband splitting. On the other hand, it also differs from a conventional 3D semi-classical MC since the k-space is 2D. Nano-ribbons, instead, may demand the description of quantization in the ribbon width [55]. In Figure 9, however, we use a 2D model with a ribbon width dependent energy dispersion relationship [51].

Figure 9: Simulated mobility as a function of the graphene ribbon width (W). Experiments are reported for comparison [56][57][58]. The

simulated sheet charge density is low (1011cm-2)

Having verified the correctness of the band structure model and of the scattering mechanisms in long channel devices, we have then run MC simulations of short channel devices. Sample results are illustrated in Figure 10 for a 50 nm long graphene FET. BTBT is responsible for the non-saturating output characteristic (circles): in fact, the simulation without BTBT (triangles) exhibits well behaved current saturation at large VDS. The inclusion of scattering (filled circles) significantly reduces the current compared to ballistic transport.

Figure 10 Simulated output characteristic of a graphene FET with 50nm channel. The top dielectric is 5nm of Al2O3, with the substrate consists

of 300nm of SiO2.

B. Graphene Base Transistor (GBT) Ambipolarity strongly limits the performance of

graphene FETs both in the digital (large off-current) and in the analog (high output conductance) domains [59]-[61]. For this reason, new device concepts exploiting graphene have been proposed. A relevant example is the Graphene Base Transistor (GBT) [62]-[64], sketched in Figure 11. A graphene base controls the tunnel injection of electrons from the (metal of silicon) emitter to an emitter/base insulator (EBI). The carriers then go through the graphene base, enter the conduction band of the base/collector insulator (BCI) and are eventually collected by the metal “collector”.

EMITTER

BCI

COLLECTO

R

0tEBI tBCI

CB

xinj

EBI

x

GRAPHENE BASEElectron

COLLECTOR

EMITTEREBI

BCI

Figure 11: Conduction band diagram of the GBT. Electrons are injected in the EBI conduction band at xinj. The inset shows the GBT structure.

Modeling the GBT mainly involves calculating tunneling currents [65]-[67] since this is the current limiting mechanism. Nevertheless, the MC approach has been useful to describe dissipative transport in the conduction band of the EBI and BCI layers and the energy distribution of the carriers. As an example, Figure 12 reports the average velocity (top) and the concentration profile (bottom) in a template GBT. The model considers electron scattering with the phonons of the dielectric (polar optical and non-polar acoustic) [68], and a non-parabolic description of the conduction band. We see that the velocity and concentration profiles in the BCI are essentially constant. This information was used to derive simplified electrical models to simulate the GBT under high injection conditions [67].

The MC simulation also provides the fraction of the electrons entering the BCI conduction band that are back-scattered into the base. These electrons are most likely captured by the graphene layer and contribute to the base current. Consequently, back-scattering in the BCI sets an upper limit to the common base current gain DF=IC/IE (see values in the legend of Figure 12). According to our results, the simulated DF is much higher than the experiments [63] (mainly because the assumption that the graphene base is fully transparent to electrons arriving from the EBI is not verified), but, nevertheless, small enough to pose a serious limitation to the device performance.

MIPRO 2015/MEET 9

Figure 12 Average velocity vx (a) and electron concentration nx (b)

along the GBT for a few values of VCB. The concentration is low since the injected current is a few A/m2.

VI. CONLUSIONS In this paper, we have reviewed the Monte Carlo

method for carrier transport in electron devices. Applications to MOSFETs with III-V channels and to graphene devices have been reported. This demonstrates that MC transport models are versatile tools, well suited to help in the assessment of the performance of modern nanoscale devices, and to support researchers in the evaluation of new materials and device architectures.

Compared to the conventional drift-diffusion model used in commercial TCAD, modern MSMC transport techniques provide a much more rigorous description of far-from-equilibrium transport and of carrier quantization. Compared to full-quantum transport methods, they can handle larger devices, allowing for an easier calibration with experimental data, and a more efficient and complete treatment of scattering. However, comparison between MC and full-quantum transport simulators is of paramount importance to verify and calibrate the many model ingredients; e.g., the quantization model with non-parabolic corrections used for III-V MOSFETs [38], the subband smoothening procedure [47] and the BTBT model for graphene [54].

ACKNOWLEDGMENT The research leading to these results has received

funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under Grant Agreement III–VMOS Project No. 619326, Grant Agreement E2Switch project No. 619509, Grant Agreement GRADE project No. 317839 and by the Italian MIUR, through the Futuro in Ricerca project RBFR10XQZ8.

REFERENCES [1] C. Jacoboni and L. Reggiani, “The Monte Carlo method for the

solution of charge transport in semiconductors with applications to covalent materials,” Rev. Mod. Phys.,vol. 55, pp. 645–705, 1983.

[2] M.V. Fischetti and S.E. Laux, “Monte Carlo analysis of electron transport in in small semiconductor devices including band-structure and space-charge effects,” Phys. Rev.B, vol. 38, pp. 9721–9745, 1988.

[3] F.Venturi, R.K.Smith, E.Sangiorgi, M.R.Pinto, and B.Ricc`o, “A General Purpose Device Simulator Coupling Poisson and Monte

Carlo Transport with Applications to Deep Submicron MOSFETs,” IEEE Trans. on Computer-Aided Design, vol. CAD-8, p. 360, 1989.

[4] A. Abramo, et al.,“A comparison of numerical solutions of the Boltzmann transport equation for high-energy electron transport in silicon,” IEEE Transactions on Electron Devices, vol.41, p. 1646, 1994.

[5] M.V. Fischetti and S.E. Laux, “Monte Carlo simulation of transport in technologically significant semiconductors of the Diamond and Zinc-Blende structures - Part II: submicrometer MOSFET’s,” IEEE Trans. on Electron Devices, vol. ED-38, p. 650, 1991.

[6] P. Palestri, D. Esseni, S. Eminente, C. Fiegna, E. Sangiorgi, and L. Selmi, “Understanding Quasi-Ballistic Transport in nano-MOSFETs. Part I: Scattering in the Channel and in the Drain,” IEEE Trans. on Electron Devices, vol. 52, no. 12, pp. 2727–2735, 2005

[7] E. F. Crabbè, J. M. C. Stork, G. Baccarani, M. V. Fischetti, S. E. Laux., “The impact of nonequilibrium transport on breakdown and transit time in bipolar transistors,” in IEDM Tech. Digest, p. 463, 1990

[8] A. Ghetti, L. Selmi, and R. Bez, “Low Voltage Hot Electrons and Soft-Programming Lifetime Prediction in Non-Volatile Memory Cells,” IEEE Trans. on Electron Devices, vol. 46, no. 4, p. 696, 1999

[9] D.K. Ferry, R. Akis, and D. Vasileska, “Quantum Effects in MOSFETs: Use of an Effective Potential in 3D Monte Carlo Simulation of Ultra-Short Channel Devices,” in IEDM Tech. Digest, pp. 287–290, 2000.

[10] H. Tsuchiya, B. Fischer, and K. Hess, “A Full-BAnd Monte Carlo Model for Silicon Nanoscale Devices with a Quantum Mechanical Correction of the Potential,” in IEDM Tech. Digest, pp. 283–286, 2000.

[11] B. Winstead and U. Ravaioli, “A Quantum Correction Based on Schr¨odinger Equation Applied to Monte Carlo Device Simulation,” IEEE Trans. on Electron Devices, vol. 50, pp. 440–446, 2003.

[12] P. Palestri, S. Eminente, D. Esseni, C. Fiegna, E. Sangiorgi, and L. Selmi, “An improved semiclassical Monte-Carlo approach for nano-scale MOSFET simulation,” Solid State Electronics, vol. 49, no. 5, pp. 727–732, 2005.

[13] C. Sampedro-Matarin, F. Gamiz, A. Godoy, and F.J. Gracia Ruiz, “The Multivalley Effective Conduction Band-Edge Method for Monte Carlo Simulation of Nanoscale Structures,” IEEE Trans. on Electron Devices, vol. 53, pp. 2703–2710, 2006.

[14] M.V. Fischetti and S.E. Laux, “Monte Carlo study of electron transport in silicon inversion layers,” Physical Review B, vol. 48, pp. 2244–2274, 1993.

[15] C. Jungemann, A. Edmunds, and W.L. Engl, “Simulation of Linear and Nonlinear Electron Transport in Homogeneous Silicon Inversion Layers,” Solid State Electronics, vol. 36, no. 11, pp. 1529–1540, 1993.

[16] F. Gamiz, J.A. Lopez-Villanueva, J.B. Roldan, J.E. Carceller, and P. Cartujo, “Monte Carlo Simulation of Electron Transoprt Properties in Extremely Thin SOI MOSFET’s,” IEEE Trans. on Electron Devices, vol. 45, no. 5, pp. 1122–1126, 1998.

[17] S.C. Williams, K.W. Kim, and W.C. Holton, “Ensemble Monte Carlo study of channel quantization in a 25nm n-MOSFET,” IEEE Trans. on Electron Devices, vol. 47, no. 10, pp. 1864–1872, 2000.

[18] T. Ezaki, P. Werner, and M. Hane, “Self-Consistent Quantum mechanical Monte Carlo MOSFET device simulation,” Journal of Computational Electronics, vol. 2, no. 2–4, pp. 97–103, 2003.

[19] J. Saint-Martin, A. Bournel, F. Monsef, C. Chassat, and P. Dollfus, “Multi sub-band Monte Carlo simulation of an ultra-thin double gate MOSFET with 2D electron gas ,” Semicond. Sci. Technol., vol. 21, pp. L29–L31, 2006.

[20] L. Lucci, P. Palestri, D. Esseni, L. Bergagnini and L. Selmi, “Multisubband Monte Carlo Study of Transport, Quantization, and Electron-Gas Degeneration in Ultrathin SOI n-MOSFETs”, IEEE Trans. on Electron Devices, Vol. 54, no. 5, pp. 1156-1164, 2007.

[21] D. Esseni, P. Palestri, L. Selmi, “Nanoscale MOS transistors: Semi-classical Transport and Applications”, Cambridge, 2011.

10 MIPRO 2015/MEET

[22] P. Lugli and D. K. Ferry, “Degeneracy in the ensemble Monte Carlo method for highfield transport in semiconductors,” IEEE Trans. on Electron Devices, vol. 32, no. 11, pp. 2431–2437, 1985

[23] P. Toniutti, D. Esseni, and P. Palestri. "Failure of the scalar dielectric function approach for the screening modeling in double-gate SOI MOSFETs and in FinFETs." IEEE Trans. on Electron Devices, vol. 57, n.11, pp. 3074-3083, 2010.

[24] D. Esseni, F. Conzatti, M. De Michielis, N. Serra, P. Palestri and L. Selmi, “Semi-classical transport modelling of CMOS transistors with arbitrary crystal orientations and strain engineering”, Journal of Computational Electronics, vol. 8, n..3-4, pp. 209–224, 2009.

[25] F. Conzatti, P. Toniutti, D. Esseni, P. Palestri and L. Selmi, “Simulation study of the on-current improvements in Ge and sGe versus Si and sSi nano-MOSFETs”, in IEDM tech Digest, pp. 363-366, 2010.

[26] D. Lizzit, P. Palestri, D. Esseni, A. Revelant, and L. Selmi, “Analysis of the Performance of n-Type FinFETs With Strained SiGe Channel”, IEEE Trans. on Electron Devices, vol.60, no.6, pp. 1884-1891, 2013.

[27] Toniutti, P.; Palestri, P.; Esseni, D.; Driussi, F.; De Michielis, M.; Selmi, L., "On the origin of the mobility reduction in n- and p-metal–oxide–semiconductor field effect transistors with hafnium-based/metal gate stacks," Journal of Applied Physics , vol.112, no.3, pp.034502,034502-12, 2012.

[28] V. Gudmundsson, P. Palestri, P.-E. Hellstrom, L. Selmi, M. Ostling, “Simulation of low Schottky barrier MOSFETs using an improved Multi-subband Monte Carlo model.” Solid State Electonics, vol. 79, p.172-178, 2013.

[29] M.V. Fischetti and S.E. Laux, “ Long-range Coulomb interactions in small Si devices. Part I: Performance and reliability,” Journal of Applied Physics, vol. 89, no. 2, pp. 1205–1231, 2001

[30] K. Nakanishi, T. Uechi, and N. Sano, “Self-Consistent Monte Carlo Device Simulations under Nano-Scale Device Structures: Role of Coulomb interaction, Degeneracy, and Boundary Condition,” in IEDM Tech. Digest, pp. 79–82, 2009.

[31] F. Monsieur, D. Rideau, V. Quenette et al., “The importance of the spacer region to explain short channel mobility collapse in28nm Bulk and FDSOI technologies”, in ESSDERC 2014, p.254

[32] P. Osgnach, E. Caruso, D. Lizzit, P. Palestri, D. Esseni, L. Selmi, “The impact of interface states on the mobility and drive current of InGaAs semiconductor n-MOSFETs”, to appear on Solid-State Electronics. DOI 10.1016/j.sse.2014.12.011

[33] A. Revelan, P. Palestri, P. Osgnach, L. Selmi, “Calibrated multi-subband Monte Carlo modeling of tunnel-FETs in silicon and III–V channel materials”, Solid-State Electronics, v. 88, p.54, 2013.

[34] F. Conzatti, M. De Michielis, D. Esseni, P. Palestri, ''Drain current improvements in uniaxially strained p-MOSFETs: A Multi-Subband Monte Carlo study'', Solid State Electronics, vol.53, pp.706-721, 2009.

[35] M. De Michielis, D. Esseni, P. Palestri, L. Selmi, ''Semiclassical Modeling of Quasi-Ballistic Hole Transport in Nanoscale pMOSFETs Based on a Multi-Subband Monte Carlo Approach'', IEEE Transactions on Electron Devices, vol.56, n.9, pp.2081-2091, 2009.

[36] P. Osgnach, A. Relevant, D. Lizzit, P. Palestri, D. Esseni and L. Selmi, “Toward computationally efficient Multi-Subband Monte Carlo simulations of nanoscale MOSFETs”, SISPAD, pp. 176-179, 2013.

[37] S. Jin, M. Fischetti, and T.-W. Tang, “Modeling of electron mobility in gated silicon nanowires at room temperature: Surface roughness scattering, dielectric screening, and band nonparabolicity,” Journal of Applied. Physics., vol. 102, no. 8, p. 083715, 2007.

[38] E. Caruso, G. Zerveas, G.Baccarani, L.Czornomaz, N.Daix, D.Esseni, E.Gnani, A.Gnudi, R.Grassi, M.Luisier, T. Markussen, P.Palestri, A.Schenk, L.Selmi, M.Sousa, K. Stokbro, M. Visciarelli, “Modeling approaches for band-structure calculation in III-V FET quantum wells”, Proc. of EUSOSOI-ULIS, pp. 101-104, 2015.

[39] Balynas V, Krotkus A, Stalnionis A, Gorelionok A T, Shmidt N M and Tellefsen J A, "Time-resolved, hot-electron conductivity measurement using an electro-optic sampling technique", Applied Physics A, vol. 51, n. 4, pp 357-360, 1990.

[40] C. Hinkle, A. Sonnet, R. Chapman, and E. M. Vogel, “Extraction of the effective mobility of In0.53Ga0.47As MOSFETs,” IEEE Electron Device Lett., vol. 30, no. 4, pp. 316–318, Apr. 2009.

[41] Y. Xuan, Y. Wu, T. Shen, T. Yang, and P. Ye, “High performance submicron inversion-type enhancement-mode InGaAs MOSFETs with ALD Al2O3 , HfO2 and HfAlO as gate dielectrics,” in IEDM Tech. Digest, pp. 637–640, 2007.

[42] E. Caruso, D. Lizzit, P. Osgnach, D. Esseni, P. Palestri and L. Selmi, “Simulation Analysis of III-V n-MOSFETs: Channel Materials, Fermi Level Pinning and Biaxial Strain”, in IEDM Tech. Digest, pp. 7.6.1– 7.6.4, 2014.

[43] D. Esseni, “On the Modeling of Surface Roughness Limited Mobility in SOI MOSFETs and its Correlation to the Transistor Effective Field ,” IEEE Trans. on Electron Devices, vol. 51, no. 3, pp. 394–401, 2004

[44] D. Lizzit, D. Esseni, P.Palestri, and L. Selmi, “Surface roughness limited mobility modeling in ultra-thin SOI and quantum well III-V MOSFETs”, in IEDM Tech. Digest, pp. 5.2.1–5.2.4., 2013.

[45] D. Lizzit, D. Esseni, P.Palestri, and L. Selmi,” A new formulation for surface roughness limited mobility in bulk and ultra-thin-body metal–oxide–semiconductor transistors”, Journal of Applied Physics, vol. 16, pp. 223702-1, 223702-13, 2014.

[46] D. Lizzit, D. Esseni, P.Palestri, Patrik Osgnach, and L. Selmi,” Performance Benchmarking and Effective Channel Length for Nanoscale InAs, In0.53Ga0.47As, and sSi n-MOSFETs”, IEEE Trans. On Electron Devices, Vol. 61, no. 6, pp. 2027-2034, 2014.

[47] P. Palestri, L. Lucci, S. Dei Tos, D. Esseni, and L. Selmi. “An improved empirical approach to introduce quantization effects in the transport direction in multi-subband Monte Carlo simulations”. Semiconductor Science Technology, vol. 25,.n.5, p. 055011, 2010.

[48] Z. Ren, R. Venugopal, S. Goasguen, S. Datta, and M.S. Lundstrom, “nanomos 2.5: A two-dimensional simulator for quantum transport in double-gate mosfets,” IEEE Trans. on Electron Devices, vol. 50, no. 9, pp. 1914–1925, 2003.

[49] K.S. Novoselo, A.K.Geim,S.V. Morozov,D. Jiang, M.I. Katsnelso, I.V. Grigorieva, et al. “Two-dimensional gas of massless Dirac fermions in graphene”. Nature, v.438(7065) pp. 197–200, 2005.

[50] V. Perebeinos and P. Avouris, “Inelastic scattering and current saturation in graphene,” Physical Review B, vol. 81, p. 195442, 2010.

[51] M. Bresciani, P. Palestri, D. Esseni, and L. Selmi, “Simple and efficient modeling of the E-k relationship and low-field mobility in graphene nano-ribbons,” Solid-State Electronics, vol. 54, pp. 1015–1021, 2010.

[52] J. David, L. Register, and S. Banerjee, “Semiclassical Monte Carlo analysis of graphene fets,” IEEE Trans. On Electron Devices, vol. 59, no. 4, pp. 976–982, 2012.

[53] M. Bresciani, P. Palestri, D. Esseni, L. Selmi, B. Szafranek, and D. Neumaier, “Interpretation of graphene mobility data by means of a semiclassical Monte Carlo transport model,” Solid-State Electronics, vol. 89, pp. 161–166, Nov. 2013

[54] A. Paussa, G. Fiori, P. Palestri, M. Geromel, D. Esseni, G. Iannaccone, L. Selmi, “Simulation of the Performance of Graphene FETs With a Semiclassical Model, Including Band-to-Band Tunneling”, IEEE Trans. on Electron Devices, Vol. 61, no. 5, pp. 1567- 1574, 2014

[55] A. Betti, G. Fiori, G. Iannaccone, Y. Mao, “Physical insights on graphene nanoribbon mobility through atomistic simulations.”, in IEDM Tech Digest, pp. 897–900, 2009

[56] X. Wang, Y. Ouyang, X. Li, H. Wang, J. Gu, H. Dai, “Room-temperature all semiconducting sub-10-nm graphene nanoribbon field-effect transistors.” Phys. Rev. Lett., v.100(20), p. 206803, 2008.

[57] L. Jiao, X. Wang, G. Diankov, H. Wang, H. Dai, “Facile synthesis of high-quality graphene nanoribbons.”, Nature Nanotechnology., v. 5(5), pp.321–5, 2010.

[58] M. Lemme, T. Echtermeyer, M. Baus, B. Szafranek, J. Bolten, M. Schmidt, et al. “Mobility in graphene double gate field effect transistors”, Solid-State Electronics, vol. 52(4), pp. 514–8, 2008.

[59] B. Obradovic, R. Kotlyar, F. Heinz, P. Matagne, T. Rakshit, M.D. Giles, M.A. Stettler, “Analysis of graphene nanoribbons as a

MIPRO 2015/MEET 11

channel material for field-effect transistors”, Applied Physics Letters, vol. 88, n.14, p. 142102, 2006

[60] I. Meric, M. Y. Han, A. F. Young, B. Ozyilmaz, P. Kim, and K. L. Shepard, “Current saturation in zero-bandgap, top-gated graphene field-effect transistors,” Nature Nanotechnology, vol. 3, no. 11, pp. 654–659, Nov. 2008.

[61] I. Imperiale, S. Bonsignore, A. Gnudi, E. Gnani, S. Reggiani, and G. Baccarani, “Computational study of graphene nanoribbon FETs for RF applications,” in IEDM Tech Digest, pp. 732–735, 2010

[62] W. Mehr, J. Dabrowski, J. C. Scheytt, G. Lippert, Y.-H. Xie, M. C. Lemme, M. Ostling, G. Lupina, “Vertical Graphene Base Transistor”, IEEE Electron Device Letters, vol. 33, no. 5, pp. 691–693, 2012.

[63] S. Vaziri, G. Lupina, C. Henkel, A. D. Smith, M. Ostling, J. Dabrowski, G. Lippert, W. Mehr, and M. C. Lemme, “A Graphene–Based Hot Electron Transistor”, Nano Letters, vol. 13, no. 6, pp. 2370–2375, 2013.

[64] C. Zeng, E. B. Song, M. Wang, S. Lee, C. M. Torres, J. Tang, B. H. Weiller, and K. L. Wang, “Vertical Graphene–Base Hot–

Electron Transistor”, Nano Letters, vol. 13, no. 4, pp. 1435–1439, 2013.

[65] F. Driussi, P. Palestri, and L. Selmi, “Modeling, simulation and design of the vertical graphene base transistor,” Microelectron. Eng., vol. 109, pp. 338–341, Sep. 2013.

[66] V. Di Lecce, R. Grassi, A. Gnudi, E. Gnani, S. Reggiani and G. Baccarani, “DC and Small–Signal Numerical Simulation of Graphene Base Transistor for Terahertz Operation”, Proceedings of ESSDERC, pp. 314–317, 2013.

[67] S. Venica, F. Driussi, P. Palestri, D. Esseni, S. Vaziri, L. Selmi, “Simulation of DC and RF Performance of the Graphene Base Transistor”, IEEE Trans. On Electron Devices, Vol. 61, no. 7, pp. 2570- 2576, 2014.

[68] M. V. Fischetti, D. J. DiMaria, S. D. Brorson, T. N. Theis, J. R. Kirtley, “Theory of High-Field Electron Transport in Silicon Dioxide”, Physical Review B, vol. 31, pp.8124–8142, 1985.

12 MIPRO 2015/MEET