J Comput Electron (2009) 8: 324–335 DOI 10.1007/s10825-009-0281-3

Wigner-Boltzmann Monte Carlo approach to nanodevice simulation: from quantum to semiclassical transport Damien Querlioz · Huu-Nha Nguyen · Jérôme Saint-Martin · Arnaud Bournel · Sylvie Galdin-Retailleau · Philippe Dollfus

Published online: 19 August 2009 © Springer Science+Business Media LLC 2009

Abstract In this paper, we review and extend our recent works based on the Monte Carlo method to solve the Wigner-Boltzmann transport equation and model semiconductor nanodevices. After presenting the different possible approaches to quantum mechanical modelling, the formalism and the theoretical framework are described together with the particle Monte Carlo implementation using a technique fully compatible with semiclassical simulation. Examples are given to highlight the importance of considering both quantum and scattering effects in nanodevices operating at room temperature, such as resonant tunnelling diode (RTD), double-gate MOSFET and carbon nanotube FET. Quantum and semiclassical approaches are compared for transistor simulation. Finally, the phonon-induced electron decoherence in RTD and MOSFET is examined through the analysis of the density matrix elements computed from the Wigner function. This formalism is shown to be relevant for the quantitative analysis of devices operating in mixed quantum/semiclassical regime and to understand the transition between both regimes or between coherent and sequential tunnelling processes. Keywords Wigner function · Quantum transport · Monte Carlo simulation · Decoherence

D. Querlioz · H.-N. Nguyen · J. Saint-Martin · A. Bournel · S. Galdin-Retailleau · P. Dollfus () Institut d’Electronique Fondamentale, CNRS, Univ. Paris-Sud, UMR 8622, 91405 Orsay, France e-mail: [email protected] Present address: D. Querlioz Stanford University, Clark Center W1.3, 318 Campus Dr. W, Stanford, CA 94305, USA

1 Introduction For about one decade, the emergence of nanoelectronics has led to a remarkable renewal in the community of device engineering and computational electronics. During the previous decade, the day where the validity of semiclassical models should be seriously questioned was expected with perplexity. Fischetti wrote in 1996 [1] We remain anxious to see when (and if) more rigorous (quantum) transport formalisms will become necessary to explain the behavior of charge carriers at dimensions even smaller than those that we employ today. Now, actually, in nano-objects emerging from bottom-up nanotechnology or even in ultra-scaled top-down nanotransistors, a new physics including many quantum features appears and cannot be properly captured by the conventional models of device physics. To include quantum effects in device simulation several approaches have been developed simultaneously. A first idea was to incorporate quantum corrections into a semiclassical description of transport, through the concepts of density gradient [2, 3] or effective potential [4]. Initially based on a rough Gaussian description of wave packets, the latter has been improved by considering either a quantum force formulation based on the Wigner formalism [5], a direct solution of Schrödinger’s equation [6], or a Pearson distribution for the wave packets [7]. These techniques are able to mimic some first order quantum effects but cannot describe properly advanced effects as resonant tunneling. A second approach consisted in transferring quantum transport models developed for mesoscopic physics in the 90s, as the recursive technique to compute either the wave

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function [8] or the Keldysh/Kadanoff/Baym Green’s functions [9, 10]. It was initially expected that such approaches may be able to replace the semiclassical ones rapidly to be universally applied to any nanoelectronic device. Since pioneering works on 2D transistor simulation [11, 12] many efforts have been done to improve the numerical techniques and to include atomistic description and scattering effects with different levels of approximation [13–22]. However, these models do not reach yet the same degree of maturity as semiclassical transport simulators in terms of robustness and versatility and very few works have investigated the transistor operation in realistic situation as in [23]. The simplest way to model the statistics of a quantum system consists in using the concept of density matrix (DM) and the associated Liouville equation. When expressing the DM in the reciprocal space this formalism may model the electron-phonon interaction accurately including collisional broadening and retardation and intra-collisional field effect [24–26]. However, it does not allow the study of real spacedependent problems. DM-based device simulation is possible using the Pauli master equation that takes into account only the diagonal elements of the DM [27, 28]. However, in spite of recent improvements [29, 30] the modeling of terminal contacts in an open system is difficult within this formulation which is thought to be valid only for devices smaller than the electron dephasing length [28]. Alternatively, an option is to use the Wigner function that is defined in the phase space as a Fourier transform of the density matrix. In the classical limit this function reduces to the classical distribution function. The dynamical equation of the WF, i.e. the Wigner transport equation (WTE), is very similar to the Boltzmann counterpart, except in the influence of the potential whose rapid space variations generate quantum effects. The WF is a standard tool in atomic physics [31] and in quantum optics [32, 33]. Quite early, it has been used in electron device simulation [34–37] in spite of numerical difficulties inherent in the discretization scheme and the boundary conditions [38]. More recently, a renewed interest in this formalism has arisen from the development of particle Monte Carlo (MC) techniques [39–43] and from improved numerical techniques [44–46]. The strong analogy between Wigner and Boltzmann formalisms makes it possible indeed to adapt the standard MC technique to solve the Boltzmann transport equation (BTE) by just considering the WF as an ensemble of pseudo-particles. Under some approximations leading to the Wigner-Boltzmann formulation, scattering effects may be included easily by using the same collision operator as in the BTE [41]. It gives access to time simulation of realistic devices with possible coupling of quantum and semiclassical descriptions of transport. This approach is still limited to 1D transport problems but the possibility to compute 2D Wigner functions within a Monte Carlo algorithm has been suggested recently [47].

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The work presented in the present paper is based on the self-consistent MC solution of the Wigner-Boltzmann equation. The formalism and the algorithm are detailed in Sect. 2. The model is then applied in Sect. 3 to some typical nanodevices as the resonant tunnelling diode (RTD), the doublegate Metal-oxide-semiconductor field-effect transistor (DGMOSFET) and the carbon nanotube field-effect transistor (CNTFET). We specially emphasize the role of scattering in such devices operating at room temperature and we compare quantum and semiclassical simulations in terms of output characteristics and phase-space representation of both Wigner and Boltzmann functions. The occurrence of quantum decoherence in such devices of size smaller than the electron wave length and mean free path is becoming an important subject of experimental and theoretical research. The decoherence induced by contact coupling in ballistic nanostructures has been discussed [48]. By solving the time-dependent Schrödinger equation the entanglement between two electrons has been analyzed as a function of time [49]. The phonon-induced decoherence in a bulk semiconductor has been investigated from the time-evolution of the generalized Wigner’s function of the electron-phonon system [50]. By means of Wigner-Boltzmann Monte Carlo simulation, we report here in Sect. 4 on the electron decoherence induced by the coupling to the phonon bath in GaAs RTD and in silicon DGMOSFET operating at room temperature.

2 Wigner-Boltzmann formalism and particle Monte Carlo solution 2.1 Wigner function and transport equation For a statistical ensemble of particles described by a density operator ρ the Wigner formalism of quantum transport is based on the Wigner function fw defined in the phasespace (r, k) as a differential Fourier transform of the density matrix ρ(r, r ), i.e. [51]  1 fw (r, k) = dr exp(−ik · r )ρ(r + r /2, r − r /2), (2π)d (1) where d is the real-space dimension of the transport problem. Although it is not positive definite, this Wigner function (WF) has similar properties to that of a distribution function via its relation with all relevant physical quantities of the system as electron density, energy, current [51, 52]. The dynamical equation of the of the WF in a potential U (r) is the Wigner transport equation (WTE) which writes ∂fw + v · ∇r fw = Qfw + Cfw , ∂t

(2)

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where C is the collision operator and Qfw is the quantum evolution term resulting from the non-local effect of the potential, defined by:  (3) Qfw (r, k) = dk Vw (r, k − k )fw (r, k ), from the Wigner potential Vw given by  1 dr exp(−ik · r ) Vw (r, k) = i(2π)d      r r −U r− . × U r+ 2 2

(4)

An alternative form for the quantum term Qfw leads to an expansion in powers of  and high order derivatives of the potential energy [51]. For slowly varying potential the first order approximation of this form reduces to 1 Qfw (r, k) = − ∇r U (r) · ∇k fw (r, k), 

(5)

which is nothing but the effect of the classical force on the distribution function in the Boltzmann transport equation (BTE). Thus, the BTE can be seen as the classical approximation of the WTE if the same collision operator C is used in both cases, i.e. if quantum collision effects are neglected. One thus immediately understands the potential advantage of this formalism to study the transport in nanodevices operating in between the ballistic and the diffusive limits and in particular to investigate the transition between the well known semiclassical regime and the less understood quantum regime.

A rigorous description of electron-phonon scattering in the Wigner formalism requires the use of the generalized Wigner function of the full electron-phonon system. For a single phonon mode q of energy ω0 and occupation number n, the Wigner transport equation then reads [39]   k ∂ ∂  + − Q + iω0 (n − n ) fw (r, k, n, n , t) ∂t m ∂r (6)

where the electron-phonon coupling is described by the collision term Cfw (r, k, n, n , t)     √ q = F (q) eiqr n + 1fw r, k − , n + 1, n , t 2   √ q   − n fw r, k + , n, n − 1, t 2

(7)

Here, F (q) is the electron-phonon matrix element. Practically, for device simulation it is convenient and most probably necessary to make the following approximations [52, 53]: (i) a weak coupling approximation, which makes it possible to consider that only the consecutive phonon numbers are coupled through the generalized WF, (ii) the assumption of fast scattering, i.e. faster than other relevant phenomena involved in the time dependence of the WF, which makes the intracollisional term of the WTE vanishing, (iii) the fact that the phonon mode remains coupled to a thermostat, which makes it possible to separate the generalized WF into a phonon part and the reduced WF fw (r, k, t) of the electron system. Then by tracing the resulting WTE over the phonon numbers and by generalizing to the case of many phonon modes, one obtains the WTE including phonon scattering as   ∂ k ∂ + − Q fw (r, k, t) ∂t m ∂r 2πF 2 (q) = q

× {δ[ω0 − E(k) + E(k − q)] × [nfw (r, k − q, t) − (n + 1)fw (r, k, t)] + δ[ω0 + E(k) − E(k + q)]

2.2 Collision operator—Wigner-Boltzmann equation

= Cfw (r, k, n, n , t),

   √ q + e−iqr − nfw r, k + , n − 1, n , t 2   √ q   . + n + 1fw r, k − , n, n + 1, t 2

× [(n + 1)fw (r, k + q, t) − nfw (r, k, t)]},

(8)

where the right hand side is exactly the collision term traditionally used in the BTE. In this approach the quantum dynamics of electrons is modelled accurately in the effective mass framework. The main approximation is in the treatment of scattering since finite collision time effects (like the collisional broadening and retardation or the intra collisional field effects), are neglected. A similar derivation can be done for the electron-ionized impurity scattering, which leads again to the same collision term as in the BTE [54]. Consider for instance the shortrange Coulomb potential created by an ionized impurity e2 exp(−β|r − ri |)/4πε|r − ri |, where ε is the semiconductor permittivity and β is the screening factor in the static screening approximation. The demonstration starts with the derivation of the Wigner potential associated with this Coulomb potential, from which a quantum evolution term is derived. After some tedious but straightforward calculations, considering a large number of dopants in density ND

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and the fast collision approximation, the electron-impurity collision term finally writes [54]  2  1 e 4 ND  Cfw (r, k) = dk 4π 2 ε 2 (k − k )2 + β 2 × δ[E(k) − E(k )][fw (r, k ) − fw (r, k)], (9) which is the same as that conventionally used in the BTE. On can conclude that, under some conditions, the collision term widely used in the BTE can be also used in the WTE. It is a strong result and one of the main advantages of the Wigner function with a view of device modelling. All the knowledge acquired in the past in the treatment of scattering in semiclassical transport may be still reused for quantum transport in the Wigner formalism. It makes it possible to study new problems such as the scattering-induced decoherence and the transition from quantum to semiclassical transport regimes. The Wigner transport equation including the Boltzmann collision term is usually called the WignerBoltzmann transport equation (WBTE) [41]. The collision term thus derives from the transition probabilities per unit of time of each scattering process calculated in the first order perturbation theory of the Fermi golden rule [55]. 2.3 Particle Monte Carlo solution of WBTE In what follows, we will consider only 1D quantum transport problems, which means that particles are either free to move in the two other directions, or so well confined that the mode-space approximation may be considered for simplicity. The Wigner function is thus defines in a 2D phase space with x and k coordinates. To develop a particle approach to solving the Wigner transport equation, we consider the Wigner function fw as a sum of Dirac excitations fi (x, k, t) = Ai (t)δ[x − xi (t)]δ[k − ki (t)], localized in both real and reciprocal spaces. Such excitations or pseudo-particle have a real-space coordinate xi , a reciprocal space coordinate ki and a magnitude Ai which is called affinity as in Ref. [40]. This latter parameter is not necessary in semiclassical transport since the Boltzmann distribution function is always positive but it is required to reconstruct the Wigner function that can locally assume negative values. It contains the information on the quantum state of the system. Consistently with the Heisenberg inequalities, such excitations, that we will call pseudo-particles, do not represent physical particles [51]. They are mathematical tools for the solution of the WBTE. The advantage of this formulation is that the evolution of these excitations follows simple equations. To consider the quantum evolution term Qfw of the WBTE within particle Monte Carlo (MC) simulation, Nedjalkov observed that this term is similar to a scattering term [41, 52]. It can be thus formally treated this way by introducing an additional “quantum scattering” rate into the Monte

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Carlo scattering process. When selected, this quantum scattering generates pseudo-particles of positive and negative integer affinity. However, this method is naturally instable since it leads to an exponential growth of the absolute values of the affinities of particles. An algorithm making use of particle multiplication and recombination has been proposed to obtain convergence [41] and self-consistence with Poisson’s equation has been reported using this technique [56]. In our approach the quantum term Qfw is considered as in [40] where it induces the continuous evolution of the pseudo-particle affinity. Pseudo-particles behave and scatter as classical particles, except that the potential does no longer influence the wave vector but only the affinity through the quantum evolution term Qfw (x, k). The wave vector can only change after a scattering event. Inside a mesh M(x, k) of the phase space the particle coordinates obey the following motion equations during a free flight ⎧d 1 ⎪ ⎨ dt xi =  ∇k E(k), d (10) dt ki = 0, ⎪ ⎩ d i∈M dt Ai = Qfw (x, k). The affinity can take negative values, in accordance with the fact that the Wigner function may be negative too. Concretely, during a Poisson time step, the phase-space coordinates of pseudo-particles evolve like in the conventional MC method. The affinity evolution equation is applied at the end of each time step. For solving this time evolution an implicit backward Euler scheme has been implemented using a predictor/corrector technique of 4th order. A critical point of this technique is the particle injection in the device. In Boltzmann simulation, thermal particles are only injected at Ohmic contacts to ensure the neutrality and quasi-equilibrium conditions in the cells adjacent to the contacts. In the case of a quantum simulation, it should be noted that if there is no particle in a given phase-space region it is not possible to make the affinity in this region evolving according the quantum evolution term (10), which may lead to strong errors related to the violation of the conservation of charge. It is thus necessary not only to still inject particles of affinity equal to 1 to get charge neutrality at contacts but also to inject particles with zero-affinity in the device to properly compute the Wigner function in the full phase space. Each mesh M(x, k) of the phase space must always contain at least one pseudo-particle. To fulfill this requirement, we showed that it is necessary to additionally inject zero-affinity particles in all empty mesh where the quantum evolution term Qfw (x, k) is not null [42]. This condition has to be paid by a quite large number of simulated particles, i.e. typically between 300,000 and 700,000 according to the type of device and to the applied bias. Using such a procedure, Boltzmann and Wigner algorithms are fully compatible. Both of them can be applied

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Fig. 1 Schematic of the simulated resonant tunneling diode. The GaAlAs barriers and the GaAs quantum well are 3 nm- and 5 nm-thick, respectively

in the same device depending on whether we consider the classical access regions or the active regions where quantum effects take place.

Fig. 2 I –V characteristics for a RTD obtained from Wigner simulation, with scattering artificially deactivated (empty circles), with standard scattering (squares), and scattering rates artificially multiplied by 5 (diamonds)

3 Application to nanodevices The Wigner/Boltzmann Monte Carlo code is here applied to the simulation of some typical nanodevices where quantum effects are likely to take place with a possible influence of scattering at room temperature, i.e. (i) the resonant tunnelling diode (RTD) whose operation is governed by the coherent tunnelling process, (ii) the ultra-small double-gate MOSFET and (iii) the carbon nanotube transistor (CNTFET). These two types of field-effect transistors may operate in quasi-ballistic regime which makes it possible for electron wave function to behave coherently, at least partially, over the active region. The expected quantum effects influencing the I –V characteristics in theses devices are the direct source-drain tunnelling through the gate-controlled potential barrier and the quantum reflexions on the steep potential gradient at the drain-end of the channel. For both types of transistors comparison between quantum (Wigner) and semiclassical (Boltzmann) simulations are presented below to analyze these effects. 3.1 Resonant tunnelling diode (RTD) We study here the typical GaAs/GaAlAs RTD schematized in Fig. 1 consisting of a 5 nm-thick quantum well sandwiched between two AlGaAs barriers 0.3 eV high and 3 nm wide. The quantum well, the barriers, and 9.5 nm-thick buffer regions surrounding the barriers are slightly doped to 1016 cm−3 . The 50 nm-long access regions are doped to 1018 cm−3 . The temperature is 300 K. The scattering mechanisms considered are those due to polar optical phonons, acoustic phonons and ionized impurities, in a single band with effective mass of 0.06 m0 . The transport simulation is self-consistently coupled with 1D Poisson’s equation. On Fig. 2, the I –V characteristics obtained from our model including scattering (squares) is compared with the ballistic simulation (circles) and with a simulation where

Fig. 3 Detail of the conduction band of the RTD obtained by Wigner simulation, at peak (V = 0.3 V, red line, squares) and valley (V = 0.475 V, green dashed line) voltages from simulation with scattering, and at the peak bias (V = 0.3 V, blue line, circles) from ballistic simulation

scattering rates have been artificially multiplied by five (diamonds). The ballistic curve has been compared with the result of a well-established ballistic NEGF simulation (not shown here) [42, 57]. An excellent agreement was found, which suggests that our Wigner-Boltzmann Monte Carlo approach correctly handles the quantum transport effects including the resonance on a quasi-bound state. It is also clearly seen here that scattering effects dramatically reduce the peak-to-valley ratio. It is thus essential to consider them properly for room-temperature simulation of RTDs. The conduction band profiles plotted in Fig. 3 highlight the importance of the self-consistence for RTD simulation, as previously shown by other authors [58–60]. In particular, when scattering is included a potential drop appears in the emitter region while the conduction band is flat in the ballistic case. This potential drop may induce an energy spreading of electrons, which modifies the resonant condition at V = 0.3 V for electrons reaching the double barrier and contributes to the suppression of current peak at the resonance. The additional effect of scattering-induced decoherence will be analyzed in Sect. 4.

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Fig. 4 Electron density in a RTD, obtained by Wigner simulation, at peak (V = 0.3 V, solid line) and valley (V = 0.475 V, dashed line) voltages from simulation with scattering

Fig. 5 Schematic cross-section of the simulated Double Gate MOSFET structure. Highly doped Source and Drain access regions are 15 nm long

As shown in Fig. 4a peak of electron density appears in the quantum well under resonant condition (V = 0.3 V), which is in accordance with the peak observed on the Wigner function [57]. In off-resonance bias V = 0.475 V, this peak strongly suppresses and an electron accumulation is formed in front of the double-barrier. Finally, it should be pointed out that the simulated peak-to-valley ratios have been shown to be in good agreement with that experimentally obtained for similar structures at 300 K and 77 K [57]. 3.2 Double-gate metal-oxide-semiconductor field-effect transistor (DG-MOSFET) We consider here the multi-subband simulation of ultrascaled end-of-roadmap double gate MOSFETs using both semiclassical [61–63] and quantum [43, 64–66] Monte Carlo simulations. A schematic cross-section of the simulated structure is presented in Fig. 5. The gate length LG , equivalent gate oxide thickness EOT and silicon channel thickness TSi are 6 nm, 0.5 nm and 3 nm, respectively. The highly-doped source and drain access regions are 15 nm long. This multi-subband approach is based on the mode-space approximation which decouples the gate-to-gate z direction

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Fig. 6 Logarithmic and linear plots of the silicon MOSFET drain current ID as a function of gate voltage VGS , obtained using Wigner (circles), Boltzmann (squares) and ballistic Green’s function (diamonds) simulations, at high VDS (0.7 V)

and the xy plane parallel to interfaces. Assuming the potential V to be y-independent, the formation of uncoupled subbands resulting from reduced channel thickness may be simply deduced from the effective 1D Schrödinger’s equation. According to the real-space meshing used for solving Poisson’s equation, this 1D Schrödinger’s equation has to be solved self-consistently at each position xi in the channel to determine the envelope functions ξn (x, z) and the subband profile En (x) to be used as potential energy for the particle transport and the Wigner potential calculation (4) along the source-to-drain axis in the n-th subband [63, 64]. To treat the 2D electron gas, the MC procedure makes use here of intra- and inter-subband scattering rates calculated according to the envelope functions ξn (xi , z) whose dependence on time and position generates an additional difficulty. In contrast to the case of standard MC, it is no longer possible to store the scattering rates in a look-up table prior to the simulation. They have to be regularly updated throughout the simulation, which contributes to weigh down the simulation time. Phonon and impurity scattering rates are derived as in Ref. [67] and Si/SiO2 roughness scattering rates as in Ref. [61]. Figure 5 shows the drain current versus gate voltage at room temperature and VDS = 0.7 V. Semiclassical Boltzmann and quantum Wigner Monte Carlo results are compared with results of ballistic Green’s function calculation [68] to emphasize the impact of scattering. All three models use the same effective mass approximation of the conduction band structure via ellipsoidal valleys with four subbands. In the subthreshold regime (low VGS ) the results based on the Wigner’s function are very close to the ballistic Green’s function ones. However, the currents are significantly higher than that obtained using semiclassical simulation. This shows that scattering has a weak impact in this regime and that the subthreshold current is strongly enhanced by direct source-to-drain tunneling through the high potential barrier in the channel. Above the threshold (high VGS ), the behavior is very different. The ballistic approach

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Fig. 9 Schematic cross-section of the simulated CNTFET with Ohmic contacts. The diameter of the (19,0) CNT is dt = 1.49 nm Fig. 7 Silicon MOSFET drain current ID as a function of drain voltage VDS , at high VGS (0.7 V), obtained using Wigner (circles) and Boltzmann (squares) models

exhibit the same form with two high value regions in the phase space which correspond to electrons in the source and drain regions. Besides, a long mouse tail-like curve is seen in the channel which describes the flow of ballistic carriers from S to D. However, while the distribution function remains positive-definite, there exist small domains around the tail in Fig. 8(b) where the WF oscillates and assumes locally negative values. This illustrates the presence of quantum reflections and emphasizes that the Wigner function cannot be interpreted as a probability function in the presence of quantum transport. 3.3 Carbon nanotube field-effect transistor (CNTFET)

Fig. 8 Cartography of (a) semi-classical distribution function (Boltzmann simulation) and (b) Wigner function of the first sub-band of a silicon MOSFET, obtained at VGS = 0.45 V and VDS = 0.7 V

strongly overestimates the current due to the efficiency of scattering and the Wigner current become similar to the current resulting from semiclassical simulation. It is remarkable that the Wigner current is actually smaller than the semiclassical current at high VGS . In this bias regime the source-drain tunneling is negligible and the quantum reflection effect due to the steep potential drop at the drain-end of the channel causes this current reduction [43]. This effect manifests itself on the ID –VDS characteristics plotted in Fig. 7. At low VDS , electrons behave as semiclassical particles and the same drain current is obtained from Boltzmann and Wigner simulations. When increasing VDS quantum particles are partially reflected by the voltage drop at the drainend and the terminal drain current becomes slightly smaller than for semiclassical particles which are fully transmitted to the drain. Figures 8(a) and 8(b) compare the Wigner function fw at high VGS with the distribution function from the semiclassical calculation at the same bias. Generally, both functions

As last example of device simulation, we now consider the CNTFET. Indeed, carbon nanotubes show unprecedented ballistic transport ability and have become key materials to envision the future of beyond-CMOS nanoelectronics [69]. Starting from a code initially developed for semiclassical simulation [70, 71], we have introduced the WignerBoltzmann Monte Carlo algorithm in the simulation of coaxially-gated CNTFET with Ohmic source and drain contacts. This version of the simulator couples the 1D transport equation, in either quantum or semiclassical formulation, with the 2D Poisson’s equation for the cylindrical device symmetry. The device parameters used for the simulation are as follows (see also schematic cross-section of Fig. 9): the gate length is 25 nm with an equivalent gate oxide thickness of 0.4 nm, the source and drain access regions are 30 nmlong with an N-type doping of 0.34 nm−1 . A semiconducting zigzag nanotube (19,0) is considered with a bandgap of 0.55 eV. The two first subbands are taken into account with effective masses of 0.048 m0 and 0.129 m0 , respectively. Phonon scattering is included using the model described in [72]. In Fig. 10 we plot the cartography in the phase space of the Boltzmann function fb (from semiclassical calculation) and the Wigner function fw (from quantum calculation) at VGS = 0.2 V, VDS = 0.4 V. The two functions look very different, much more different than in the case of 6 nm-long DG-MOSFET (Fig. 8). Due to quantum capacitance control

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(a)

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Fig. 11 ID –VGS characteristics obtained from Boltzmann (squares) and Wigner (circles) simulations of a CNTFET for the gate length of 25 nm at VDS = 0.4 V

drain tunnel is always negligible and the main possible quantum effect is the quantum reflection occurring at the drainend of the channel, as previously observed in 6 nm-long DGMOSFET, which do not generate a strong reduction of drain current. The reduction reaches 55% at VGS = 0.1 V and only 9% at VGS = 0.3 V. 4 Decoherence and quantum/semiclassical transition (b) Fig. 10 Cartography of (a) Boltzmann and (b) Wigner functions from semi-classical and Wigner MC simulation, respectively, for a CNTFET at VGS = 0.2 V and VDS = 0.4 V. The gate is located from x = 30 to 55 nm

the potential is nearly flat in the channel and the sourcedrain bias voltage essentially takes place at the drain-end of the channel [70]. In the semiclassical case, the carriers are then abruptly accelerated by the strong resulting electric field at the drain-end of the channel. In contrast, in the quantum case, the acceleration seems much slower as if the particles feel the electric field in advance, which is consistent with the idea of delocalized electrons with finite extension of their wave function and submitted to non-local effects of the potential. Moreover, the positive-negative oscillations of the Wigner function at the drain-end of the gate are the signature of a strongly coherent transport with typical quantum effect as tunnelling and reflexion. The quantum coherence between incident and reflected electrons also appears in the oscillations about k = 0 along the source-drain axis. Such oscillations are not observed in DG-MOSFET operating at room temperature (Fig. 8b) but have been evidenced at a lower temperature of 77 K due to reduced phonon scattering [66]. In spite of these very strong differences observed at microscopic level, it is remarkable that the two types of simulation do not give drastically different terminal currents, as shown in Fig. 11. Indeed, for L = 25 nm, the direct source-

The knowledge of the Wigner function gives access to the electron density matrix by inverse Fourier transform of the WF defined in (1). The diagonal terms of the DM ρ(x, x) simply provide the electron density in the device, while the off-diagonal terms ρ(x, x  ) informs about the spatial coherence and the delocalization of the electrons [53, 73]. We recently analyzed the competition between coherence and decoherence for different transport situations in typical semiconductor structures where electrons interact with a phonon bath. Here we focus the analysis on devices studied in Sect. 3, i.e. the RTD and the DG-MOSFET. According to its strength, the electron-phonon scattering is shown to induce the transition from coherent to sequential transport in RTD and the transition between quantum and semiclassical regime in MOSFET. 4.1 Decoherence in GaAs/GaAlAs RTD We already observed in Fig. 2 that scattering effects dramatically reduces the peak-to-valley ratio in the simulated RTD. We now analyze the possible influence of scattering-induced decoherence on this behavior. The coherence and the phonon-induced decoherence in the RTD is actually clearly visualized in Figs. 12(a–c) which represent the density matrix at V = 0.3 V in the same scattering situations as in Fig. 2. In the ballistic case a strong coherence is observed between electrons in the quantum well and in the emitter region. The amplitude of off-diagonal elements is even significant between electrons in collector and emitter regions,

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Fig. 12 Density Matrix modulus of the RTD operating at the peak voltage with (a) no scattering, (b) standard scattering rates, and (c) scattering rates multiplied by five

Fig. 13 Density Matrix modulus of the MOSFET in the ON-state (VGS = VDS = 0.7 V) for (a) ballistic channel, (b) standard scattering rates in the channel, and (c) scattering rates in the channel multiplied by 5. The gated part of the channel extends from x = 14 nm to x = 28 nm

which is a clear indication of a coherent transport regime. When including standard scattering rates the off-diagonal elements strongly reduce. When phonon scattering rates are artificially multiplied by 5, the off-diagonal elements of the DM vanish, i.e. the coherence between electrons on left and right side almost disappears. The process of double barrier tunneling is thus no longer fully resonant. Electrons can be seen as entering and leaving the quasi-bound state in distinct processes, with the possibility of energy exchange with the phonons. This illustrates the well-known coherent vs. sequential tunneling situation. This phonon induced transition tends to suppress the resonant tunnelling peak shown on Fig. 2 while the valley current increases to such a point that the negative differential conductance effect almost disappears. The structure tends to behave as two incoherent tunnelling resistances connected in series. 4.2 Emergence of semiclassical behavior in DG-MOSFET We now analyze the coherence in a double-gate MOSFET. The device studied here is similar to that simulated in Sect. 3 but differs in the gate length LG = 14 nm, in the equivalent oxide thickness EOT = 1 nm and in the doping of the source/drain access regions ND = 1020 cm−3 . These regions are 15 nm-long. Figure 13 shows the obtained density matrices in different cases for a gate voltage VGS and a drain voltage VDS both equal to 0.7 V (ON-state). In Fig. 13(a), scatterings have been artificially deactivated inside the channel (ballistic channel). In Fig. 13(b), standard scattering rates are used in the whole device. In Fig. 13(c), the scattering rates inside the channel have been artificially multiplied by five. To help

Fig. 14 Cut of the density matrix modulus |ρ(xC , x)|, where xC = 22 nm is the middle of the channel, in the same scattering rate situations as in Fig. 13

interpretation, a cut of the modulus of the density matrix |ρ(xC , x)| is plotted in Fig. 14, with xC = 22 nm the centre of the channel. The diagonal elements ρ(x, x), i.e. the density along the channel, is nearly independent of the strength of scattering rates, which is clearly quantified in Fig. 14 for ρ(xC , xC ) in the centre of the channel. In the ballistic case (Fig. 13(a)), a strong coherence is observed in the channel: the diagonal and off-diagonal terms of the DM have similar values. It suggests that electrons are strongly delocalized along the channel. If realistic scattering are included the off-diagonal terms of the density matrix reduces slightly (Fig. 13(b)) though a significant coherence is still observed. If the scattering rates are increased by a factor of 5 (Fig. 13(c)), the off-diagonal terms are strongly reduced and electrons become really localized in the channel, as also shown in Fig. 14 in the centre of the channel.

J Comput Electron (2009) 8: 324–335

333

In the semiclassical backscattering theory in nanoMOSFET, efficient backscattering mechanisms occur in a short (localized) region (generally called the kT layer) near the top of the source/channel injection barrier [74]. Though it has been already shown from both semiclassical and quantum simulations that scattering in the second half of the channel still plays an important role in the device operation of ultra-short MOSFET due to electrostatic effects [13, 75], it is meaningful to evaluate this model in the case of strongly delocalized electrons by direct comparison between quantum and semiclassical simulations as function of the scattering strength. The 14 nm gate length transistor is studied in this order and three types of simulation have been performed: channel with standard scatterings (sca), ballistic channel (bal), and channel where scattering have been artificially switched off only in the second half of the channel (sca-bal channel). To quantify the role of scattering in the second half of the channel and the influence of electron delocalization the following parameter R has been defined R = (Isca-bal − Isca ) / (Ibal − Isca )

(11)

According to the backscattering theory, the current in the “sca-bal” situation should be close to the current for the standard device (sca), hence the parameter R should be small. However, in a semiclassical (BTE) self-consistent simulation R is equal to 50%. As previously reported [13, 75], the scattering in the second part of the channel has an indirect influence on the kT layer by modifying the electron density and thus the electrostatic behavior. To hide this effect, we focus on non self-consistent simulations. The potential profile has been extracted from standard channel and frozen in all devices. For the BTE case R is reduced to 19%. This suggests that in the semiclassical approach, backscattering is indeed mostly caused by scatterings in the beginning of the channel. However in the case of Wigner simulation, R reduces to 30%: the end of the channel seems to play a more significant role due to the electron delocalization in this 14 nm-long channel. The same simulations were repeated for different strengths of scattering. Scattering rates in the channel were artificially multiplied by an enhancement factor α of 0.5, 2, 3.5 and 5. The resulting R evolution is plotted in Fig. 15, for both semiclassical and Wigner simulations. In semiclassical transport, R remains between 17 and 19%, and thus seems to be a kind of universal value. In contrast, in Wigner simulation R reaches 40% for scattering rates multiplied by a factor 0.5, which means that both parts of the channel thus play nearly a comparable role on the backscattering. When increasing the scattering rates, R decreases to the same value as for the semiclassical situation. This is consistent with the observation of the density matrices (cf. Fig. 13(c)) which suggested that the semiclassical behavior, i.e. the localization of electrons, emerges when increasing scatterings.

Fig. 15 Parameter R defined in the body text, as a function of the scattering rate multiplication factor in the channel, from Wigner (circles) and Boltzmann (squares) simulations. A multiplication factor of 1 corresponds to the standard scattering rates in the channel

5 Conclusion This paper has reviewed a state-of-the-art Monte Carlo technique developed to solve the Wigner quantum transport equation in nanodevices, and some results obtained for RTD, silicon MOSFET and CNTFET, including an analysis of scattering-induced electron decoherence. We may draw the following conclusions. The Wigner/Boltzmann formalism is shown to be an excellent tool for studying quantum electron transport in semiconductor systems. Its advantages lie in the fact that it is defined in a quantum phase space and it is straightforwardly connected to both the density matrix and the semiclassical Boltzmann formalism. It can then easily include scattering mechanisms within an accurate time-description of quantum effects related to potential variations though possible extensions to 2D or 3D quantum problems and to realistic band structures still remain to be demonstrated. Beyond the description of quantum effects, the full compatibility of this quantum Monte Carlo method with Boltzmann algorithm makes it suitable for (i) the investigation of devices operating in the mixed quantum/semiclassical regimes, (ii) the quantitative study of electron decoherence via extraction of the density matrix and (iii) the analysis of the transition between fully ballistic quantum transport and semiclassical diffusive transport. In particular, further analyses using this approach should allow us to grasp why the semiclassical approximation still apparently holds much beyond its theoretical limit of validity in spite of quasi ballistic and partially coherent effects. Acknowledgements This work was supported in part by the European Community through Network of Excellence NANOSIL (ICT216171) and Integrated Project PULLNANO (IST-4-026828) and in part by the Agence National de la Recherche through projects MODERN (ANR-05-NANO-002) and ACCENT (ANR-06-PNANO-069). The authors would like to thank Do Van Nam for sharing his NEGF code of DG-MOS simulation and Hugues Cazin d’Honincthun for many discussions on CNTFET simulation.

334

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