J Comput Electron DOI 10.1007/s10825-010-0319-6

Implementation of the Wigner-Boltzmann transport equation within particle Monte Carlo simulation Damien Querlioz · Jérôme Saint-Martin · Philippe Dollfus

© Springer Science+Business Media LLC 2010

Abstract In this paper, we detail the main numerical issues of the Monte Carlo method developed to solve the WignerBoltzmann transport equation and simulate the quantum transport in semiconductor nanodevices. In particular, we focus on the boundary conditions regarding the injection of particles and the limits of integration for the calculation of the Wigner potential which are of crucial importance for the physical correctness of simulation results. Through typical examples we show that this model is able to treat correctly purely quantum coherent and semi-classical transport situations as well. It is finally shown that to investigate devices operating in mixed quantum/semi-classical regimes and to analyze the transition between both regimes, this approach takes advantage of its full compatibility with Boltzmann algorithm. Keywords Wigner function · Quantum transport · Monte Carlo simulation

1 Introduction Quantum simulations are recommended when device dimension reduces to the deca-nanometer scale, i.e. when device operation and performance may rely on coherent quantum effects. In spite of numerical difficulties in including scattering, the most widely used quantum simulation method is the non-equilibrium Green’s function formalism (NEGF) [1–3]. During the last decade this technique has been developed and used to study many kinds of devices, D. Querlioz · J. Saint-Martin · P. Dollfus () Institut d’Electronique Fondamentale, CNRS, Univ. Paris-Sud, UMR 8622, Bâtiment 220, 91405 Orsay, France e-mail: [email protected]

in particular to nano-MOSFETs [4] and to nanowire and nanotube-based devices [5–11]. Another option is to use the Wigner’s formalism of quantum transport which is based on a function defined in the phase space as a Fourier transform of the density operator [12–14]. In the classical limit this function reduces to the classical Boltzmann distribution function. The dynamical equation of the Wigner function, i.e. the Wigner transport equation, is very similar to the Boltzmann counterpart, except in the influence of the potential whose rapid space variations generate quantum effects. The strong analogy between Wigner and Boltzmann formalisms makes it possible to adapt the standard Monte Carlo technique to solve the Boltzmann transport equation by just considering the Wigner function as an ensemble of pseudo-particles. Under some approximations on the treatment of scattering (weak and fast scattering limit) the collision operator of the Wigner transport equation is similar to that of the Boltzmann equation [15, 16], which gives rise to the so-called WignerBoltzmann transport equation (WBTE) [17]. Wigner Monte Carlo simulation can thus reconcile semi-classical and quantum simulations [18]. It gives access also to time simulation of realistic devices with possible coupling of quantum and semi-classical descriptions of transport [16]. Beyond the direct numerical solution of the WignerBoltzmann transport equation [19–24], a possible particle Monte Carlo method consists in a stochastic interpretation of the electron-potential interaction through a scattering mechanism which results in the generation of “positive” and “negative” particles [17, 25, 26]. Here we focus on the “affinity” Monte Carlo technique that has been used in the simulation of resonant tunneling diodes and silicon nanoscaled transistors [27–29]. In this technique, each particle is weighted by a parameter called affinity [30] which can take negative values and evolves continuously along the

J Comput Electron

particle trajectory. The carrier affinities carry all the quantum information on the system. In this article we describe the main numerical issues of the affinity method and we show that it is able to treat correctly purely quantum coherent and semi-classical transport situations as well. It is thus shown to be very well suited to bridge these two extreme regimes of transport, in particular though scattering-induced decoherence effects.

2 Model and numerical issues Let us consider a one-dimensional transport problem. For a statistical ensemble of particles described by a density operator ρ the Wigner function is defined in the phase-space (x, k) as a differential Fourier transform of the density matrix ρ(x, x  ), also called Weyl-Wigner transform, i.e.     −1 fw (x, k) = (2π) d x  e−i k x ρ x + x  /2, x − x  /2 . (1) In the Wigner-Boltzmann formulation, the dynamical equation of the Wigner function writes as a quantum counterpart of the Boltzmann equation, i.e.  ∂fw ∂fw + k = Qfw + Cfw , ∂t m ∂x where



Qfw (x, k) =

    dk  Vw x, k − k  fw x, k 

(2)

(3)

is the quantum evolution term including the non-local effect of the potential through the Wigner potential defined by

Vw (x, k) =

   1 dx  exp −i k x  i2π      x x −V x − , × V x+ 2 2

and Cfw (x, k, t) =



(4)

    d k si k , k fw x, k  , t

i

  − si k, k fw (x, k, t)

(5)

is the Boltzmann collision term including scattering effects in the weak and fast scattering approximation [15, 17, 31]. The scattering probabilities per unit of time si (k, k ) are calculated for each scattering process i in the 3D reciprocal space within the first order perturbation theory. It should be noted that in some cases it is relevant to separate the potential V (x) may be separated into a slowly varying part Vslow (x) and a rapidly varying part Vrapid (x), or in other

words into a classical component Vcl (x) and a quantum mechanical component Vqm (x), i.e. [31] V (x) = Vslow (x) + Vrapid (x) = Vcl (x) + Vqm (x).

(6)

Actually, these two terms can be treated separately in the transport equation. The semi-classical component yields a Boltzmann-like evolution of the WF via the classical force term, while only the quantum mechanical component is included in the Wigner potential (4) [31]. The transport equation (2) becomes  ∂fw 1 ∂ Vcl ∂fw ∂fw + k − = Qfw + Cfw . ∂t m ∂x  ∂x ∂k

(7)

This property makes it possible to switch easily from semiclassical to quantum descriptions of transport or to mix them within the same device by using the same universal code. 2.1 The continuous affinity method: principles In this approach, initially suggested by Shiffren et al. [30, 32], the Wigner function is represented as a sum of Dirac excitations of the form     δ x − xj (t) δ k − kj (t) Aj (t). (8) fw (x, k, t) = j

In contrast to classical particles, these excitations are weighted by an amplitude Aj , called affinity, which evolves continuously under the action of the quantum evolution term of the Wigner-Boltzmann equation (2) which describes the non-local effect of the potential. Since the Wigner function can take negative values in the presence of quantum transport effects, the affinity may be negative too. Consistently with the Heisenberg inequalities, such excitations of negative weight cannot represent physical particles and will be called pseudo-particles. They should be considered as mathematical objects useful to solve the Wigner transport equation. They are taken into account in the reconstruction of the Wigner function and in the computation of all physical averages. Compared to the semi-classical Monte Carlo algorithm, one of the main changes consists of adding, at each time step, the update of the Wigner function and of the particle affinities. In a mesh of the phase-space M(x, k) the quantum evolution term (3) induces the change of the affinity of particles in the mesh according to j ∈M(x,k)

dAj = Qfw (x, k) dt

(9)

which means that at each time step the affinity of all pseudoparticles in a mesh of the phase-space is updated according to the value of Qfw in this mesh. The non-local effect of

J Comput Electron

the potential is thus fully applied to the affinity evolution, in contrast to the semi-classical case where the local effect of the potential gradient induces the change of wave vector. The simple idea on which is based this quantum simulation method now appears clearly. Along its trajectory a pseudo-particle scatter as a classical particle, and during a free flight, if the potential has been separated in slowly and rapidly varying parts (6), the coordinates of the j -th particle obey, in the effective mass approximation, ⎧ d  ⎪ ⎪ ⎨ x j = kj , dt m (10) ⎪ ⎪ ⎩ d kj = − 1 ∂Vcl (x) . dt  ∂x If the whole potential is included in the quantum evolution term, i.e. Vcl = 0, the wave vector of each particle is constant during a free-flight and can take a new value only after scattering. At the end of a time step the Poisson equation is solved, the Wigner potential and the quantum evolution term Qfw are updated and the particle affinities are recalculated according to (9). One should keep in mind that using this technique the potential may have two effects. If one part of the potential is treated as a slowly varying potential Vslow it induces an evolution of the pseudo-particle wave vectors as in semi-classical simulation. The rest of the potential Vrapid induces the evolution of affinities at each time step. In the semi-classical limit, i.e. if the full potential is treated as a slowly varying quantity V = Vslow , the quantum evolution term Qfw is zero and the particle affinity is constant. The method turns out to be equivalent to the semiclassical EMC. It should be noted that the strong similarity and even compatibility of this technique with the conventional MC solution of the BTE is one of its highest advantage, which will be illustrated later. 2.2 Pseudo-particle injection and boundary conditions First of all, it should be reminded that in semi-classical device simulation with Ohmic contacts, the only condition of particle injection is the neutrality of real-space meshes adjacent to the Ohmic contacts. After each time step, if particles are missing in some “Ohmic” meshes with respect to the charge neutrality, the appropriate number of carriers (of affinity equal to 1) is injected in these meshes to recover the neutrality. Assuming these “Ohmic” regions to be in thermal equilibrium, an equilibrium distribution is used to select randomly their wave vector components. In this way the consistence between the distribution of potential and the average number and the distribution of particles in the device is reached. Obviously, this condition of particle injection should still be used in Wigner simulation of Ohmic contacts

if the transport in the contact region is assumed to be essentially semi-classical. However, it is not enough to ensure the conservation of total affinity and charge within an algorithm in which the particle affinity evolves continuously. Indeed, one of the most important difficulties in this MC method lies in the fact that even a particle with zero affinity may gain finite affinity through the quantum evolution term Qfw according to (9). It means that if there is no particle in a particular region of the phase space where the affinity should evolve, a significant error may occur with possible non-conservation of charge since the contribution of each particle to the total charge in the device is weighted by its affinity. This problem is very important for device simulation and should be fixed by implementing an appropriate algorithm to inject particles of convenient affinity. The correct approach consists in filling the phase-space with pseudo-particles of zero affinity as follows. After each time step the quantum evolution term Qfw (x, k) is calculated in the full phase-space. If in a mesh M(x, k) of the phase-space, even inside the device, the quantity |Qfw (x, k)| is finite, a pseudo-particle of zero affinity is injected in the mesh [27]. In summary, it is necessary to combine the “semi-classical injection” of particles of affinity equal to 1 at Ohmic contacts to guarantee the electrical neutrality near the contacts and the “quantum injection” of pseudo-particles of 0 affinity in all regions of the phasespace where particles are missing and where Qfw takes significant values. Another fundamental problem lies in the choice of the limits of integration for the calculation of the Wigner potential (4). There are two approaches depending on whether the contacts are assumed to be coherent or non-coherent. In the former case the integration is cut at a maximum size from the contact corresponding to the “coherence length” beyond which no quantum effect may occur [17], which raises the question of the relevant choice of the coherence length in the contact. In the latter case, the integration should be limited to positions x  such that both x + x  /2 and x − x  /2 belong to the device [33]. This approach is used in the model we have developed. However, we have checked that in the devices considered in the next chapters all limits of integration larger than that corresponding to the hypothesis of decoherent contacts yield the same results. This insensitivity is certainly due to the fact that in these cases (RTD, MOSFET, . . .) contact regions, or access regions, have a semi-classical behaviour dominated by scattering. 2.3 Problems of discretization The numerical treatment of physical equations always lies on a discretization scheme whose choice may be crucial to converge to accurate and stable results. This choice is discussed here regarding the affinity evolution and Wigner potential evaluation.

J Comput Electron

2.3.1 Affinity evolution To describe the time evolution of pseudo-particle affinities, a very stable discretization scheme is required. Indeed, we observed that due to the noise inherent in the technique, the MC simulation acts as a stiff problem. In our model, an implicit Backward Euler scheme was finally implemented, i.e. 1 Ai (t + dt) − Ai (t) = dt × Qfw (x, k, t + dt), N

(11)

where N is here the number of pseudo-particles in the mesh M(x, k) of the phase space. This backward Euler scheme is implicit. It may be implemented by matrix inversion of the quantum evolution operator Q, or by using a predictor/corrector technique of high order, at least 4th order. The two techniques give the same results but the predictor/corrector one is faster. All higher precision schemes were proved to be detrimental to the simulation stability and required longer simulation time to obtain good average quantities. In particular Cayley’s scheme, known to be the best technique for the evaluation of the time derivative in the deterministic solution of the WBTE, leads to instable results in MC simulation. 2.3.2 Artificial periodicity of the Wigner potential Let us remind ourselves about the full expression of the onedimensional Wigner potential which writes     L x 1  −i k x  Vrapid x + dx e Vw (x, k) = i2π −L 2   x , (12) − Vrapid x − 2 where the value of L may depend on x, as discussed above. The Wigner potential can be discretized by introducing a grid space dx, i.e., Vw (x, k) =

1 dx π

L/dx j =1

   dx sin(j kdx) Vrapid x + j 2

  dx . − Vrapid x − j 2

(13)

This expression (13) is periodic in k-space, with a period of 2π/dx. Of course this periodicity is not physical and disappears for dx → 0. Hence, when implementing a Wigner transport model, one should pay attention to the fact that the maximum value of k must be always smaller than π/dx. In general this condition can be quite easily satisfied using reasonable values of dx. It should be also noted that since (11) is very close to a discrete Fourier transform, the k-space meshing must be a

priori chosen in accordance with the real-space meshing, i.e. dk = π/L. The literature even suggests that this condition is mandatory for a direct solution of the WTE [34]. Unfortunately, it is incompatible with the contact model in which L depends on the position x. However, we have observed within the MC technique that – If L is uniform throughout the device, all mesh sizes satisfying the condition dk < π/L give the same result. – If L is non-uniform, any meshing satisfying the xdependent condition dk < π/L(x) gives the same result. It confirms the numerical robustness of the Wigner Monte Carlo method and offers some freedom in the choice of realspace and k-space meshing for the correct evaluation of the Wigner potential.

3 Validation and applications 3.1 Validation of the quantum mechanical treatment: interaction of a wave packet with a tunnelling barrier We verify here that the Wigner MC method described above is able to reproduce quantum mechanical effects accurately. To test the method, we consider the typical situation previously suggested in [30]. It consists in simulating the interaction of a ballistic Gaussian wave packet with a tunnel barrier. The Gaussian wave packet is thus in the form   (x − x0 )2 exp[ik0 x]. ψ(x) = N exp − 2σ 2

(14)

This packet corresponds to the Wigner function  

(x − x0 )2 exp −(k − k0 )2 σ 2 . (15) fw (x, k) = N  exp − 2 σ Such a simple situation without collision term can be solved easily from the Schrödinger equation. In Fig. 1 an excellent agreement is shown between our particle approach of the WTE and the solution of the Schrödinger equation. In particular, the oscillations of the wave-function are very well reproduced. This shows that the method can accurately model the existence of space correlations and interferences. However, it should be noted that some particle noise inherent in the method remains visible. 3.2 Validation of the semi-classical treatment: N+ /N/N+ diode In our MC simulation technique the potential felt by electrons may be treated two different ways with a priori two very different effects. It may induce the evolution of the

J Comput Electron Fig. 1 Time evolution of a wave packet interacting with a square tunnel barrier. Comparison between (a), (b), (c) the direct solution of the Schrödinger equation (left side) and (d), (e), (f) the Wigner Monte Carlo simulation (right side)

electron wave vector according to (10) if it is treated semiclassically, or it may be considered to act only on the electron affinity through (9) if it is treated quantumly. These two treatments seem mathematically very different. It is interesting and relevant to check that they coincide in semi-classical situation to prove that the quantum model is able to describe it properly. A simple GaAs N+ /N/N+ structure is considered, with + N regions doped to 1018 cm−3 and the 50 nm-long central region doped to 1016 cm−3 . The conduction band structure is limited to the  valley with an effective mass of 0.06m0 . The scattering mechanisms included in the collision operator are interaction with acoustic phonons (in the elastic approximation), polar optical phonons and ionized impurities at room temperature. The transport simulation is selfconsistently coupled with the 1D Poisson equation. It is performed using both the semi-classical EMC algorithm and the quantum EMC model. In the latter case the Poisson potential is fully included in the quantum evolution term, i.e.

V = Vrapid , which means that the electron wave vector is constant during free flights and may change only when the electron is scattered by a phonon or an impurity. After each time step the Wigner potential associated with the Poisson potential is recalculated and the affinity of particles are updated according to (9). The current flowing through the device may be calculated either by counting all affinities entering and leaving the device through the Ohmic contacts, or by averaging the electron velocities and using the Ramo-Shockley theorem I=

e Ai vi , L

(16)

i

where L is here the device length. These two methods give the same results but the Ramo-Shockley approach is less noisy and converges more rapidly to the steady-state solution. Thus it was used to obtain all results presented here. For this structure with a 50 nm-long central region, no quan-

J Comput Electron Fig. 2 (a) Current-voltage characteristics obtained for a 50 nm/50 nm/50 nm N+ /N/N+ structure using semi-classical (V = Vslow ) and quantum (V = Vrapid ) Wigner Monte Carlo simulation; (b) Conduction band profiles at 0.2 V bias voltage for both types of simulation

tum effect is expected to occur. Accordingly, the I –V characteristics plotted in Fig. 2a are identical for semi-classical and quantum treatment of the potential. The results are superimposed on three orders of magnitude of current and voltage. The conduction band profiles plotted in Fig. 2b for V = 0.2 V are also identical. These results clearly show that in semi-classical situation, i.e. for slowly varying potential, the semi-classical evolution of wave vectors and the quantum evolution of affinities are fully equivalent. It is an important validation step for the Wigner MC technique. 3.3 Transition between quantum and semi-classical transport To illustrate how the Wigner-Boltzmann MC simulator can bridge the quantum and the semi-classical regimes of transport, we consider here two typical examples of nanodevices: the Double-Gate (DG) MOSFET and the resonant tunnelling diode (RTD). We first consider the multi-subband simulation in the mode-space approximation [29, 35] of ultra-scaled end-ofroadmap double gate MOSFETs using both semiclassical and quantum Monte Carlo simulations. We also compare the results with that obtained using ballistic quantum simulation based on the non-equilibrium Green’s function (NEGF) formalism. The gate length LG , equivalent gate oxide thickness EOT and silicon channel thickness TSi are 6 nm, 1 nm and 3 nm, respectively. The source and drain access regions are 15 nm long and doped to 1020 cm−3 . All simulations approaches include the Pauli principle. To exacerbate the quantum effects, in particular the source-drain tunneling [28] we plot in Fig. 3a the transfer characteristics obtained at low temperature (T = 77 K). In subthreshold regime Wigner and ballistic-NEGF simulations give the same results, which shows again that (i) scattering is actually negligible in this bias range [28] and (ii) the Wigner formalism describes very well the transport regime dominated by coherent tunneling. Indeed, since it does not include this effect, Boltzmann simulation gives currents several orders of magnitude lower than quantum simulations. In contrast, above threshold the

ballistic NEGF approach strongly overestimates the current, while Wigner and Boltzmann currents are very close. In some cases, in particular with smaller doping density in the source and drain access, the Wigner on-current may even be smaller than the Boltzmann one as a consequence of quantum reflection on the sharp potential drop at the drain-end of the channel [28]. Here, with degenerate source and drain access, the barrier controlling the injection in the channel is still significant at VGS = VDD , which results in a nonnegligible source-drain tunneling current, even in on-state. In this case, quantum reflection and tunneling compensate each other. The ballistic NEGF approach strongly overestimates the on-current. At high gate voltage, scattering has actually an important influence, especially in highly-doped source and drain access, both at low VDS (Ohmic regime) and at high VDS (saturation regime) [29]. Additionally, in such device scattering has been shown to cause the emergence of semi-classical transport, i.e. of the localization of electrons [36]. The simulated RTD consists of a 5 nm-thick GaAs 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 0.06m0 . The transport simulation is self-consistently coupled with 1D Poisson’s equation. The I –V characteristics obtained from our model including scattering (squares) is compared in Fig. 3b with the ballistic simulation (circles) and with a simulation where scattering rates have been artificially multiplied by five (diamonds). It is thus essential to consider them properly for room-temperature simulation of RTDs. It is interesting to analyze the possible influence of scattering-induced decoherence on this behavior. The coherence and the phononinduced decoherence in the RTD is actually clearly visualized in Figs. 4a–c which represent the density matrix at V = 0.3 V in the same scattering situations as in Fig. 3b. The

J Comput Electron Fig. 3 I –V characteristics of (a) the DG-MOSFET at T = 77 K using three types of mode-space simulation and (b) the RTD at room temperature using Wigner MC simulation, with scattering artificially deactivated (circles), with standard scattering rates (squares) and with scattering rates artificially multiplied by 5 (diamonds)

Fig. 4 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

elements of the density matrix (DM) were extracted from the Wigner-Boltzmann simulation by inverse Fourier transform of the Wigner function. 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, 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 (or semi-classical) tunneling situation. This phonon induced transition tends to suppress the resonant tunnelling peak shown on Fig. 3b 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.

boundary conditions regarding the injection of particles and the limits of integration for the calculation of the Wigner potential are of crucial importance for the physical correctness of simulation results. Some important questions related to the discretization were also addressed, in particular on the time evolution of particle affinity and on the risk of artificial periodicity of the Wigner potential. Then some typical examples were considered to validate this Monte Carlo Wigner-Boltzmann approach in the two extreme situations of fully coherent and full-diffusive transport, which shows the universality of the model able to bridge coherent and semi-classical transport. Its full compatibility with Boltzmann algorithm makes it suitable for (i) the investigation of devices operating in 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. Acknowledgements This work has been supported in part by the European Community through Network of Excellence NANOSIL (ICT-216171).

References 4 Conclusion This paper has detailed the main numerical issues of a state-of-the-art Monte Carlo technique developed to solve the Wigner quantum transport equation in nanodevices. The

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Implementation of the Wigner-Boltzmann transport ...

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