Monte Carlo simulation of nanoelectronic devices

J Comput Electron (2009) 8: 174–191DOI 10.1007/s10825-009-0295-x

Monte Carlo simulation of nanoelectronic devices

F. Gamiz · A. Godoy · L. Donetti · C. Sampedro ·J.B. Roldan · F. Ruiz · I. Tienda · N. Rodriguez ·F. Jimenez-Molinos

Published online: 15 October 2009© Springer Science+Business Media LLC 2009

Abstract The Monte Carlo simulation method is used toanalyze the behavior of electron and hole mobility in dif-ferent nanoelectronic devices including double gate transis-tors and FinFETs. The impact of technological parameterson carrier mobility is broadly discussed, and its behaviorphysically explained. Our main goal is to show how mobil-ity in multiple gate devices compares to that in single gatedevices and to study different approaches to improve the per-formance of these devices. Simulations of ultrashort channeldevices taking into account quantum effects are also shown.

Keywords Numerical simulation · Monte Carlo method ·Nanoelectronics · Low-field mobility · Quantum-welldevices · Quantum-wire devices

1 Introduction

The modeling and simulation of semiconductor devices haveplayed a central role in the semiconductor miniaturizationprocess. As the semiconductor industry has developed newfabrication techniques and device concepts during the last45 years to obey Moore’s Law [1], the simulation commu-nity has been improving in parallel the computational mod-els in order to predict and optimize the performance of nextgeneration devices. These efforts have allowed an importantreduction in design time and cost. As a consequence, there is

F. Gamiz (�) · A. Godoy · L. Donetti · C. Sampedro ·J.B. Roldan · F. Ruiz · I. Tienda · N. Rodriguez ·F. Jimenez-MolinosDepartamento de Electronica y Tecnologia de Computadores,Universidad de Granada, 18071 Granada, Spaine-mail: [email protected]

a wide spectrum of available approaches to describe the be-havior of a considered technology. Different solutions fromclassical to full quantum models can be considered depend-ing on the needed accuracy, the computational resources andthe available time to perform the simulations [2–6]. Nu-merical simulations of semiconductor devices pursue differ-ent goals. Among many others, we could cite the followingones:

1. We can use simulation to predict the behavior of a partic-ular realistic device, for example, calculating the currentobtained in their terminals when a given bias is appliedto them. This is useful for improving the device perfor-mance by changing both the technological structure ofthe device or the polarization range, or to understand thedependencies and limiting physical mechanisms in thedevice/circuit performance (e.g. effects of noise, limitson frequency/gain, trap effects, effects of geometry, etc.)

2. We can also use simulation to study the transport prop-erties of charge carriers (mobility, diffusion coefficient,velocity overshoot) in a particular material at differentconditions of temperature, quantization, strain, crystallo-graphic orientation, and many others.

3. Simulation is also useful to understand and explain thephysics underneath the experimental measurements on aparticular device. In this context simulation can be usedto develop and improve device compact models whichare essential in the design process of new integrated cir-cuits.

Simulation is not a tool or method specific of the semi-conductor community. Many other disciplines use simula-tion to foresee the behavior of their target systems. In thecase of device simulation, the system under study is an en-semble of many electrically charged particles that interactwith externally applied electric and magnetic fields and with

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J Comput Electron (2009) 8: 174–191 175

each other. Particles are distributed both in geometrical andmomentum space, following the classical particle distribu-tion function:

f = f (r,k, t) (1)

which gives the probability of finding a particle with mo-mentum between k and k + dk and in the region r to r + drat time t . The evolution of f (r,k, t) is governed by theBoltzmann Transport Equation, BTE [7]:

∂f (r,k, t)

∂t+ v · ∇rf + 1

�

∂p∂t

· ∇kf = ∂f

∂t

∣∣∣∣coll

. (2)

The distribution function allows us to calculate ensemble av-erages over momentum and space, and for example, we canevaluate the carrier density and the density of current in thedevice:

n (r, t) = 1

V

∑

k

f (r,k, t) , (3)

J (r, t) = e

V

∑

k

v (k) f (r,k, t) . (4)

So that, simulation aims to solve the Boltzmann transportequation to evaluate f (r,k, t) in a given semiconductor de-vice.

Let’s have a closer look to Eq. 2. The first term of the lefthand side of the BTE represents a change in the distribu-tion function,f (r,k, t), due to particles added or subtractedthrough scattering, by trapping or detrapping or by gener-ation or recombination. v is the group velocity of carriers,given by

v = 1

�∇kE (k) (5)

where E (k), the dispersion relationship, takes into accountthe interaction of the carriers with the periodic potential ofthe crystal. The third term of the LHS of Eq. 2 is propor-tional to the acceleration driven by the Lorentz force becauseof the electric, F, and magnetic, B, fields:

∂p∂t

= e (F + v × B) . (6)

Finally, the right hand side of the BTE, represents thechanges in f because of scattering, mainly, phonon scatter-ing, interface roughness scattering, and Coulomb scattering.The scattering mechanisms take into account the deviationsof a semiconductor system from a perfect crystal:

∂f

∂t

∣∣∣∣coll

=∑

k′

[

f (r,k′, t)W(k′,k) − f (r,k, t)W(k,k′)]

(7)

where W(k,k′) is the transition rate per particle from k tok′. In summary, to calculate f (r,k, t) we need to know(i) the bandstructure of the semiconductor device, E(k),(ii) the local forces which affect the carriers, and (iii) thescattering rates for the scattering mechanisms involved inthe carrier transport. The solution of the BTE for a real-istic device is not an easy task. Difficulties arise from thecomplexity of the band structure, the lack of accurate mod-els for describing the different scattering mechanisms, andnon-homogeneous device structures, including non-uniformdoping. Moreover, the electric field in Eq. 6 is determinedby the electrostatic potential, which must be obtained bysolving the Poisson equation in a self-consistent way. Dif-ferent approaches have been used to solve the BTE in real-istic devices. Among the most common approaches are theself-consistent solution of Poisson and Drift Diffusion equa-tions [2, 4], or the Monte Carlo method [3, 8, 9]. The MonteCarlo method is a stochastic method used to solve the BTEwith no assumptions about the distribution functions underthe semiclassical approximation, i.e.:

1. Carriers are considered point particles with a well definedposition and momentum

2. Effective mass approximation is used to include thequantum effects of the periodic potential of the lattice

3. The particle motion is described as a set of free-flightsinterrupted by scattering events

4. Carrier trajectory and velocity are calculated using clas-sical mechanics whereas the scattering probabilities arecalculated according to quantum mechanical rules.

In this paper we have summarized our main simulationresults on different semiconductor devices obtained by us-ing the Monte Carlo method. The continuing scaling ofCMOS technology requires significant innovations in differ-ent fields, from short channel effect suppression to carriertransport enhancement [10–12]: (i) Firstly, multi-gate de-vices exhibit a scaling advantage due to better gate controlof the channel [13]. However, this in itself is not enough.(ii) The second key to the further improvement of CMOStechnology is the enhancement of carrier mobility in the de-vice channel [10]. In recent years, much research activity hasbeen focused on this task, with the use of specific dopingprofiles, the growth of low-doped epitaxial layers on highdoped substrates [14], and even the use of silicon-relatedmaterials instead of silicon. In relation to the latter proposal,a significant step was taken with the introduction of strainedsilicon to build the MOSFET channel [15]. We shall ana-lyze the behavior of electron and hole mobility in differentmultigate structures comprising double gate transistors andFinFETs.

This paper is organized as follows. In Sect. 2 we shall re-view the behavior of electron mobility in double gate transis-tors. The effect of technological parameters on carrier mo-

176 J Comput Electron (2009) 8: 174–191

bility is broadly analyzed, and its behavior physically ex-plained. Our main goal is to show how mobility in multi-ple gate devices compares to that in single gate devices andto study different approaches for further improvement. InSect. 3, we shall consider hole transport. CMOS technologyneeds from both n-channel and p-channel devices. The be-havior of hole mobility in multigate devices is of course ofgreat importance [16]. However hole mobility has been the-oretically much less studied than electron mobility becauseof the complexity of the valence band, and large researchefforts are still required. In Sects. 2 and 3, we shall focuson the evaluation of low-field mobility, i.e., carriers are con-sidered in stationary and homogeneous conditions under theeffect of a constant drift electric field. Therefore, a singleparticle Monte Carlo simulation is used; however, in Sect. 4we will study the behavior of a realistic ultrashort-channeldevice, and therefore, the BTE has to be self-consistentlysolved with the Poisson and Schroedinger equations. Differ-ent approaches will be reviewed.

2 Electron transport in multigate MOSFETs

A planar Dual-Gate-Silicon-On-Insulator (DGSOI) struc-ture consists, basically, of a silicon slab sandwiched betweentwo oxide layers. A metal or a polysilicon film contacts eachoxide (Fig. 1). Each of these films acts as a gate electrode(front and back gate), which can generate an inversion re-gion near the Si–SiO2 interfaces if an appropriate bias isapplied. Thus, we would have two MOSFETs sharing thesubstrate, source and drain.

The outstanding feature of these structures lies in the con-cept of volume inversion, introduced by Balestra et al. [17]:if the Si film is thicker than the sum of the depletion regionsinduced by the two gates, no interaction is produced betweenthe two inversion layers. The operation of this device is sim-ilar to that of two conventional MOSFETs connected in par-allel (as in Fig. 2(b)). However, if the Si thickness is reduced,the whole silicon film is depleted and an important interac-tion occurs between the two potential wells. In such con-ditions the inversion layer is formed not only at the top and

Fig. 1 Schematic of a double-gate transistor

bottom of the silicon slab (i.e., near the two silicon-oxide in-terfaces) but throughout the entire silicon film thickness. Itis then said that the device operates in volume inversion, i.e.,carriers are no longer confined at one interface, but distrib-uted throughout the entire silicon volume (Fig. 2(a)). Thiseffect has also a strong impact on the transport properties ofcarriers.

2.1 Subband modulation effect in DGSOI MOSFETs

The extension of the electrons in the direction perpendicu-lar to the Si–SiO2 interfaces is limited by the silicon thick-ness which is comparable to the De Broglie wavelength ofthe carriers. This means that we have to self-consistentlysolve the Poisson and Schroedinger equations to evaluatethe electron distribution. From the self-consistent solutionof the Poisson and Schroedinger equations, the followingfacts have been obtained.

(a) An important effect caused by the reduction of thesilicon film thickness is the subband modulation effect.This is related to the splitting of the degeneracy of theSi conduction-band minima. As a consequence of the sizequantization in the (100) silicon inversion layer, the de-generacy in the Si conduction-band minima breaks and theelectrons are distributed into two ladders of subbands: oneladder rises from the two valleys showing the longitudi-

Fig. 2 Electron distribution and potential well for two DGSOIdevices with different silicon-layer thicknesses and two values ofthe inversion charge concentration: Ninv = 1 × 1012 cm−2, andNinv = 8 × 1012 cm−2


Fig. 3 Relative population of subbands in each ladder as a function ofthe silicon layer thickness in a DGSOI transistor

nal mass, ml , in the direction perpendicular to the inter-face (E0,E1, . . .) and the other one from the four equiva-lent valleys showing the transverse mass, mt , in the samedirection (E′

0, E′1, . . .). The conduction effective mass for

electrons in non-primed subbands, mc, is equal to mt , whilethe conduction effective mass for electrons in primed sub-bands, m′

c , is equal to (2mtml)/(mt + ml), which impliesthat m′

c > mc. Therefore, the higher the population of non-primed subbands, the lower the average conduction effec-tive mass, and therefore the higher the mobility. As a con-sequence, in order to improve the electron transport proper-ties in silicon inversion layers, one should try to reduce thepopulation of primed subbands, thus decreasing the electronconduction effective mass and at the same time increasingelectron mobility. To do so, the subband energy differencebetween non-primed and primed subbands, i.e. E′

0 − E0,should be increased. This can be achieved in ultrathin SOI-inversion layers by reducing the silicon film thickness. Theself-consistent solution of the Poisson and Schroedingerequations in the structure described above demonstrates thatthe separation between the energy levels of the two subbandladders depends on the thickness of the silicon layer. Animportant consequence of this fact is that the population ofnon-primed subbands increases at the expense of the primedsubband population as the silicon layer thickness is reduced.This can be observed in Fig. 3 where the relative populationof the non-primed and primed subbands is shown versus thesilicon layer thickness. According to the discussion above,a reduction in the conduction effective mass occurs as Tw isreduced. This reduction of the conduction effective mass ofelectrons as the silicon layer thickness decreases can clearlybe observed in Fig. 4, for different inversion charge concen-trations.

This redistribution of the inversion electrons as the sili-con layer shrinks also produces a reduction in the intervalleyscattering rate between nonequivalent valleys (f scattering)due to the greater separation of the energy levels related toprimed subbands with respect to the non-primed ones.

Fig. 4 Average conduction effective mass versus the silicon thicknessfor different values of the inversion charge concentration in a DGSOItransistor

(b) Another important effect that appears in SOI-inversionlayers as the silicon layer thickness is reduced is an increasein the phonon-scattering rate. We have observed that in thecase of silicon films thinner than 20 nm, the film (Tw) isthinner than the width of the inversion charge distributionin bulk inversion layers for the same inversion charge con-centration. This means that the electron confinement in ul-trathin SOI inversion layers is greater than in bulk-inversionlayers. That is, the uncertainty concerning the location ofthe electrons in the direction perpendicular to the interfaceis lower in SOI samples than in bulk samples. In accor-dance with the uncertainty principle, there is a wider dis-tribution of the electron’s momentum perpendicular to theinterface. In other words, due to size quantization, the elec-tron’s interface-directed momentum does not have a singlevalue (as in 3-D electrons), but a distribution of likely val-ues that expands as the silicon layer thickness is reduced.Taking into account the momentum conservation principle,there are more bulk phonons available that can assist in tran-sitions between electron states, and therefore an increase inthe phonon-scattering rate is expected. As a consequence,for the same inversion charge concentration, the phononscattering rate is greater in thinner films than in thicker ones(since the confinement is greater), and therefore a mobilityreduction can be expected.

Thus, for the same inversion charge concentration, thephonon scattering rate is greater in thinner films than inthicker ones (since the confinement is greater), and thereforewe expect a reduction in mobility. Numerically, this effect isreflected in the following form factor:

Iμν =∫

∣∣ψμ(z)

∣∣2 |ψν(z)|2 dz (8)

which multiplies the phonon scattering rates [18] whereψν(z) is the envelope of the electron wavefunction in thedirection perpendicular to the interface in the ν-th subband.When confinement is greater (the overlap integral of enve-lope wavefunctions is also larger) the phonon scattering rateincreases. We have just seen that the energy-level distribu-tion (and therefore the wavefunctions) of electrons in thinner


Fig. 5 Evolution of form factor for the ground subband as a functionof silicon thickness for a DGSOI inversion layer

DGSOI inversion layers is quite different from the energy-level distribution in thicker films and, therefore, it is also dif-ferent from those found in Single-Gate SOI (SGSOI) inver-sion layers. The above form factor plays an important rolein the transport properties of electrons in inversion layerssince it is the dominant factor determining the phonon scat-tering rate. Figure 5 shows the form factor for the groundsubband as a function of silicon thickness for two differentvalues of the transverse effective field. For thinner samples,the form factor is very large, due to the geometrical con-finement of electrons in a very narrow space. As the sili-con slab thickness increases, the form factor is quickly re-duced, until a minimum is reached in the region between5–15 nm. Then, it increases to approach, for thicker samples,(Tw > 20 nm), the value presented in bulk inversion layers.As can be seen in the figure, in the range Tw = 5–15 nm theform factor for DGSOI is lower than the one correspond-ing to bulk inversion layers (Tw → ∞). Consequently, inthe range (Tw = 25 nm → 5 nm, for EEFF = 1 × 105 V/cm;and Tw = 15 nm → 5 nm, for EEFF = 5 × 105 V/cm) thephonon scattering rate in DGSOI inversion layer decreases,instead of increasing as expected. This is an important result,a direct consequence of the volume inversion effect.

2.2 Mobility calculation

Figure 6 shows the electron mobility versus silicon layerthickness for two different electric field values.

Basically, our Monte Carlo results reproduce, qualita-tively, the results obtained by Shoji et al. [19] for DGSOIinversion layers using the relaxation time approximation. Assilicon thickness is reduced, the phonon-limited mobility in-creases gradually to a maximum around Tw = 10 nm (forEEFF = 5 × 105 V/cm), decreases in the Tw = 5–10 nmrange to values below those of this parameter in conven-tional bulk MOSFETs, rises rapidly to another maximum inthe vicinity of Tw = 3 nm and finally falls. For higher elec-tric fields, the maximum mobility is smaller and is shiftedto lower Si thicknesses. Figure 6 reveals the existence of

Fig. 6 Evolution of the phonon-limited mobility with silicon slabthickness for a DGSOI MOSFET

three regions with different behavior in the DGSOI phononlimited-mobility:

i. The first region corresponds to thick silicon slabs. InDGSOI inversion layers the two channels are suffi-ciently separated from each other and no interaction ap-pears between them. This situation corresponds to twoconventional inversion layers in parallel, separated bya large potential barrier. The behavior of electrons ineach of these inversion layers is the same as that ob-served in a bulk silicon inversion layer. As the siliconthickness is reduced, the interaction of the two inver-sion layers causes the electrons to occupy the entire sili-con volume. This is the beginning of the second region,which strongly depends on the value of the transverseeffective field, since for high electric fields a potentialbarrier, which obstructs the mutual influence of the twochannels, is formed in the middle of the silicon slab.

ii. In the second region, the electron mobility in DGSOIinversion layers is up to 20% larger than the mobilityin bulk inversion layers. The limits of this region andthe values of the mobility depend on the electric fieldconsidered. This is the region in which volume inver-sion occurs. In this region of silicon thickness, both sub-band energy levels and wavefunctions vary significantlyas a consequence of the two channels interacting. It isfor this reason that the form factor which multiplies thephonon scattering rate decreases [20] compared to itsvalue in conventional bulk MOSFETs in the same trans-verse effective field (see Fig. 5). This happens down to acertain value of silicon thickness. For lower thicknesses,although the electrons are distributed throughout the en-tire silicon layer, their confinement is greater (due to thegeometrical confinement), and therefore, the form fac-tor and the phonon scattering rate increase, as shown inFig. 5. This marks the beginning of the third region.


iii. In the third and last region (Tw < 4 nm), the mobilityfor DGSOI falls abruptly. In this zone, mobility is lim-ited by the thickness of the silicon slab; that is to say,limitations on mobility are imposed by the geometri-cal confinement of the carriers. The limits of this re-gion do not depend on the transverse electric field. Infact, electron mobility in this region is hardly modifiedby the transverse electric field. As can be appreciatedin the mobility curves of Fig. 6, electron mobility in-creases abruptly in the range 4 nm → 3 nm. This isa little surprising since, as stated above, the form fac-tor and the phonon scattering rate were expected to in-crease. However, the sharp increase in electron mobilitycan be understood by taking into account the evolutionof the phonon scattering rate and the relative popula-tion of electrons in primed and non-primed subbandsas shown in Fig. 3. In Fig. 5, it is shown that fromTw = 4 nm to Tw = 3 nm, the form factor (and there-fore the phonon scattering rate) increases by about 20%(as predicted in the discussion above). However, Fig. 3shows that the relative population of the non-primed lad-der (where electrons have a lower conduction effectivemass) also increases by more than 30%, reaching almost90% of the total. Thus we have two diverging trends,and in the case Tw = 4 nm → 3 nm, the reduction in theconduction effective mass dominates the increase in thephonon scattering rate. As a consequence, mobility in-creases. When a smaller silicon slab is considered, theincrease in the phonon scattering rate is greater than thereduction in the conduction effective mass, and thereforeelectron mobility falls abruptly.

Up to now, phonon confinement has been neglected whenmodeling the interaction of electrons with phonons; the bulkphonon model was believed to provide a good enough ap-proximation for the calculation of electron transport prop-erties. However, this assumption may be questioned whenthe silicon layer in the devices being considered is only afew atomic layers thick and evidence of confined phononhas been obtained experimentally [21]. If bulk phonons areconsidered, the electron mobility displays a complex be-havior as a function of silicon layer thickness, as shown inFig. 6. To check whether the results obtained with the bulkmodel are still valid when phonon confinement is taken intoaccount, we introduced [22, 23] a confined phonon modelfor ultrathin SOI devices. We used this model for the studyof DGSOI devices, computing the electron-phonon scatter-ing rates and the electron mobility for several device struc-tures in order to show the influence of phonon confinementand its dependence on the device geometry. The confinedphonon model assumes simplified boundary conditions (ei-ther rigid or free) at the external surfaces of the two sil-icon dioxide layers: we have analyzed the difference be-tween the computed mobility for the two cases, and dis-cuss those cases where the results are (almost) independent

Fig. 7 Electron mobility in a DGSOI inversion layer as a functionof the silicon layer thickness, calculated assuming a confined phononmodel with different boundary conditions

of the boundary conditions imposed. A detailed explana-tion on the derivation of the model for confined acousticphonons can be found elsewhere [24]. We considered thethree-layer SiO2/Si/SiO2 structure shown in Fig. 1. In eachlayer, acoustic phonons are modeled as elastic waves in anisotropic medium.

We have considered two types of boundary conditions: (i)in the case of rigid boundary conditions the external surfaceof the SiO2 layers are considered rigid and fixed; (ii) in thecase of free boundary conditions, the external surfaces arefree to vibrate and unconstrained [25]. Once phonon statesare calculated, the interaction Hamiltonian can be computed.Its matrix elements give us the transition probability be-tween two states, according to the Fermi golden rule [8]. Theelectron scattering rates are then obtained summing overphonon absorption or emission, over phonon branches, andover the final electron state. The scattering rates are com-puted assuming an elastic approximation, that is, the elec-tron energy does not change in a scattering event with aphonon. Indeed, the phonon energy is typically negligiblecompared to the electron energy (at room temperature) andit can be observed that this assumption does not significantlyaffect the scattering rate [22]. The electron mobility is thenshown in Fig. 7. Mobility is reduced when phonon confine-ment is taken into account for both sets of boundary con-ditions. This is because SiO2 is softer then Si, i.e., soundvelocity in silicon dioxide is lower than in silicon. In anycase, Fig. 7 shows that there is a region of silicon thicknesseswhere electron mobility is higher in DGSOI inversion layersthan in bulk MOSFETs because of volume inversion effect.So, in addition to a better electrostatic control of the chan-nel, and therefore, a better control of short channel effects,


DGSOI devices also provide a better low-field mobility (atleast, for a range of silicon thickness).

3 Hole transport in ultrathin DGSOI transistors

The analysis of holes transport properties in electronic de-vices requires simulation tools different from those used forelectrons, because of the characteristics of the valence band.The strong non-parabolicity and anisotropy of the energydispersion of heavy-hole, light-hole and split-off bands donot allow the ellipsoidal approximation and the relativelysimple approach to non-parabolicity used in the descriptionof conduction band valleys (see Fig. 8).

The hole dispersion relations in a confining potentialcan be obtained with good accuracy in the framework ofthe k · p model. However, solving the k · p equation self-consistently with the Poisson equation is very demandingfrom a computational point of view. For this reason a tri-angular well was assumed in Ref. [26], while the potential

Fig. 8 Hole dispersion relationships in a p-channel MOSFET:(a) iso-energy lines for first subband and (b) dispersion of first 20 sub-bands along [100] direction

profile obtained with an effective mass approach was consid-ered in Ref. [27]; only recently self-consistent calculationshave been achieved employing a limited number of pointsthanks to the use of an efficient interpolation [28] or us-ing a trigonometric basis [29]. Alternatively, an hybrid self-consistent calculation has been proposed in Ref. [30], wherethe Poisson equation is coupled to the k ·p Schroedinger-likeequation only at k‖ = 0, using the density of states (DOS)obtained with a full solution of the k · p equation in a trian-gular well. In all these cases the carrier mobility was thenobtained using the Kubo-Greenwood formula. In Ref. [31]an alternative approach was used: hole dispersion was com-puted using non-local pseudopotential calculations, and thenthe transport properties were studied using a Monte Carlosimulator. In Ref. [32] the authors employed a fitted effec-tive mass in order to obtain the energy levels correspondingto k‖ = 0, and then fitted the energy dispersion relation inthe k‖ plane for the three types of holes (obtained with thek · p method for a triangular well) using an analytic expres-sion.

Our approach consists in a fully self-consistent proce-dure: Poisson equation and k · p equation are repeatedlysolved until self-consistence is obtained. The computationalcost is kept reasonably low by a proper discretization of thek‖ plane which allows a good accuracy with a relativelysmall number of points.

3.1 Self-consistent k · p

The k · p model is an approximate method used to computesemiconductor bandstructures [33–35]. Valence and con-duction bands can be studied, with maxima and/or minimaaround k = 0 or other values of wave-vector k. Many bandscan be taken into account; in the case of silicon, at least sixbands have to be considered, due to the small value of thespin-orbit splitting from light hole and heavy hole bands. Inthis way a 6 × 6 Hamiltonian is obtained for a bulk semi-conductor with a corresponding wave equation

H(k)ψk = E(k)ψk. (9)

Here, H(k) = Hkp(k) + Hso + Hstrain is the sum of thek · p Hamiltonian, the spin-orbit Hamiltonian and the strainBir-Pikus Hamiltonian whose expression can be found inRef. [36], and ψk is a six-component vector. The eigenval-ues of Eq. 9 provide us with the energy dispersion of thedifferent hole bands of bulk semiconductors.

The k · p theory has also been extended to heterostruc-tures. If translation invariance is only lost along one dimen-sion (along the z axis, to fix the notation), the correspondingcomponent of wave-vector kz is substituted with the usualderivative term −i ∂

∂z: k = (k‖, kz) → (k‖,−i ∂

∂z). At the in-

terfaces between different materials, operator ordering in theelements of Hkp(k) becomes relevant, because kz now does


not commute with material constants. Proper boundary con-ditions at the interfaces can be obtained corresponding to anon-symmetric operator ordering [37, 38].

In our solver, we discretize the resulting k ·p Hamiltonian[

H(

k‖,−i∂

∂z

)

+ V (z)I]

ψk‖ = E(k‖)ψk‖ (10)

(where I is the 6×6 identity matrix) on a non-uniform meshof Nz points in the z confinement direction. As mentionedbefore, heterointerfaces and arbitrary multilayer structurescan be considered: bulk MOSFET devices, Silicon On In-sulator (SOI) devices, different channel materials (Si, Ge)and different gate oxides (including multi-layer high-k gatestacks). The k ·p equation is solved not only in the semicon-ductor channel, but also in the dielectric layer (or layers):the finiteness of the barrier height at the interfaces betweenchannel and dielectric layers is taken into account and a pen-etration of the wavefunction in the oxide region is obtained.The appropriate rotations on the k space are performed if thesubstrate orientation is different from (001). Strain is takeninto account through the Bir-Pikus Hamiltonian; this enablesus to consider the effects of biaxial stress or uniaxial stress inan arbitrary direction. In SOI devices the k · p Hamiltonianis discretized in the whole semiconductor layer and the sur-rounding dielectric layers; on the other hand, for bulk MOS-FET devices, the very large semiconductor region cannotbe considered as a whole because the resulting 6Nz × 6Nz

matrix would be impractical from a computational point ofview, so that k · p equations are only solved in the channelregion. However, this is not a problem because the carri-ers are confined in the inversion region in the channel, nearthe interface with the gate dielectric, and the carrier densityrapidly decreases in the semiconductor.

The energy and wavefunction of the first Nsub subbandsare computed as eigenvalues and eigenvectors, respectively,of the matrix corresponding to the discretized k · p Hamil-tonian. Notice that, unlike the electron case where we candecouple confinement (Schroedinger equation) and trans-port, the discretized k · p Hamiltonian has a non-trivial de-pendence on the wave-vector component in the transportplane (k‖). Therefore, such a matrix has to be computed anddiagonalized several times, using k‖ = 0 and an orthogo-nal mesh of Nφ × Nk points in the k‖ = (k,φ) plane. Theφ-mesh is uniform in the irreducible range [0,2π/Nsimm],where Nsimm depends on crystal symmetry: it depends onsubstrate orientation and, in case of strained layers, on thetype and direction of strain. The values φn = n�φ forn = 0, . . . ,Nφ with �φ = 2π/(NsimmNφ) are used. In con-trast, a non-uniform mesh in the k direction is employed:the region near the origin of k‖ plane is the most populatedone in each subband and strong non-linearities are present.Therefore we consider a smaller step size close to k = 0 inorder to achieve an higher accuracy in such region while

Fig. 9 Example of a mesh for the k plane

limiting the total number of points. We take km = m2�k

for m = 1, . . . ,Nk with �k = kmax/N2k . To obtain a trian-

gulation of the k‖ plane, an extra point (k′m,φ′

n) = ((m +0.5)2�k, (n + 0.5)�φ) is considered inside every trapezoidof the polar grid as can be seen in Fig. 9.

For each subband, the energy is computed at the gridpoints mentioned and the dispersion relation Eν(k‖) of theν-th subband is then linearly interpolated in an unambigu-ous way in each triangle. The function Eν(k‖) is thereforerepresented as a piecewise linear function and can be usedfor the calculation of the DOS (and in the Monte Carlo sim-ulator, as reported in the following section). The DOS of theν-th subband is defined as

ρν(E) = 1

(2π)2

∫

dk‖δ(E − Eν(k‖)) (11)

where δ(x) is the Dirac delta function. In our case, the inte-gral is reduced to the sum of the contribution of all relevanttriangles; in each one, the constant-energy curve is a linesegment and the gradient is constant so that the integrationis trivial. The subband energies Eν and the wavefunctionsψν

0 (z) (obtained at k‖ = 0) are then used, together with thedensity of states, to obtain the “quantum component” of thecharge distribution. Indeed we compute separately a “quan-tum” and a “classical” component of the charge distributionby considering an energy limit Elim taken at the energy ofthe (Nsub + 1)-th subband at k‖ = 0. The density of states ofthe Nsub considered subbands is computed only up to Elim

while the parabolic bulk dispersion is used for larger ener-gies. This method is employed to obtain a smoother transi-tion in the charge density at the edge of the region consid-ered in the k · p discretization, and to limit the number ofsubbands that need to be considered in weak inversion.

The k · p solver described is coupled to a Poisson solver;a fast convergence is obtained using a predictor-correctorscheme [39]. In each step of the (k · p)—Poisson loop, thediagonalization of the k · p Hamiltonian at different meshpoints in the k‖ plane is independent; therefore the pro-gram can be parallelized in an efficient way and the com-putational time can be substantially reduced using multipleprocessors. Employing a computational cluster composed of


8 bi-processor nodes with 2.40 GHz CPU, the time neededfor the self-consistent calculation at one polarization valuevaries between a few minutes and one hour, depending onthe number of points needed for the discretization of the z

axis and the k‖ plane and the number of subbands consid-ered. In this way, the confining electrostatic potential, carrierdensity and the subband dispersion relations can be self-consistently computed for Si and Ge inversion layers, SOIdevices, arbitrarily oriented and strained substrates.

3.2 Mobility computation

The effective mobility for holes in inversion layers is com-puted using a one-particle Monte Carlo simulator. As in thecase of electrons, the motion of a carrier is simulated ina semi-classical framework as a sequence of flights inter-rupted by scattering events: carrier dynamics follows clas-sical laws while scattering rates are calculated using quan-tum mechanics using carrier wave-functions. The main dif-ference with respect the electron case derives from the factthat carrier dispersion relationships are not known analyt-ically, but are obtained using the self-consistent k · p andPoisson solver and, moreover, these are different for eachsubband.

Since the dispersion relationships are not analytical, wehave to employ a simplex Monte Carlo technique [40]; inthis case the k‖ space is two-dimensional, therefore the sim-plexes are triangles [27]. The basis of a simplex Monte Carlois the linearization of the energy of each subband as a func-tion of k‖ in every triangle, as already explained in Sect. 3.1.

The rates for the optical and acoustic phonon scatteringare computed in the isotropic approximation [26]:

1

τμ,νop

= D2op

4πρω0

(

N + 1

2± 1

2

)

×∫

d2kf δ(Eν(kf ) − Eμ(ki ) ± �ω0)Sμ,ν(ki ,kf )

(12)

and

1

τμ,νac

= D2ackBT

2π�ρv2l

∫

d2kf δ(Eν(kf ) − Eμ(ki ))Sμ,ν(ki ,kf )

(13)

where Dop and Dac are the optical and acoustic phonon cou-pling constants, ρ is density of channel material, �ω0 the op-tical phonon energy, vl the longitudinal sound velocity, kB

the Boltzmann constant, the Dirac delta function accountsfor energy conservation and Sμ,ν is the overlap integral, de-fined as:

Sμ,ν(ki ,kf ) =∫

dz

∣∣∣ψ

μ†ki

(z) · ψνkf

(z)

∣∣∣

2(14)

and ψμ

ki(z) represents the wavefunction in the μ-th subband

evaluated at k‖ = ki . However, the usual procedure is to em-ploy only the wavefunctions ψν

0 (z) obtained at k‖ = 0 in thecomputation of the scattering rate [26]. In this way, the termSμ,ν(0,0) can be taken out of the integrals defining the scat-tering rates. A direct consequence of this approximation isthat acoustic and optical phonon scattering rates do not de-pend on the direction of the particle momentum, but onlyon the particle energy. The remaining integrals in Eqs. 12and 13 are proportional to the DOS and are evaluated as ex-plained in Sect. 3.1.

Substituting the wavefunction ψνk (z) with ψν

0 (z) is gen-erally considered a good approximation; however, this re-sults in a substantial underestimation of the overlap inte-gral between each pair of subbands, which are degener-ate at k‖ = 0 (and differ only for the “spin” state). In-deed, in this case, Sμ,ν(0,0) is smaller (by orders of mag-nitude) than the typical values of Sμ,ν(ki ,kf ), which in-stead then have the same order of magnitude as the corre-sponding self-overlapping factors Sμ,μ(0,0) and Sν,ν(0,0)

(which are essentially equal to each other). Therefore wecorrect Sμ,ν(0,0) to make it the average of Sμ,μ(0,0) andSν,ν(0,0) (and essentially equal to each of them) for eachpair of subbands (see Ref. [41] for details).

Another scattering mechanism we consider in the MonteCarlo simulator is that due to surface roughness (SR). Itsrate is computed as:

1

τμ,νSR

= �2SR�2

SR

2�

∫

d2kf

|Mμ,νsr (kf − ki )|2

1 + �2SR(kf − ki )2/2

× δ(Eν(kf ) − Eμ(ki )) (15)

where Mμsr is the matrix element for SR scattering. For bulk

MOSFETs it is computed as in Ref. [42], while in the caseof SOI devices the model of Ref. [43] has to be adapted. In-deed, when two interfaces are present not only the roughnessof both surfaces produces carrier scattering, but the presenceof the both must be taken into account in the calculation ofthe scattering potential. This step is done by computing theself-consistent potential in the “original” device with nomi-nal silicon thickness, and in a “modified” device. In such de-vice the channel thickness is increased by an amount equalto �SR, that is the quadratic mean of interface fluctuations(which can be different for the two interfaces). Therefore,the calculation of perturbation potential requires the self-consistent solution of k · p and Poisson equation in two dif-ferent structures. However, we can obtain very accurate re-sults taking the “original” system as a starting point, with-out the need to solve the entire k · p problem again [44]. Todo so, we assume that the dispersion relations of the holesubbands only change with respect to the original unper-turbed device because of a rigid shift in energy. Denotingwith E

unpi (k‖) the dispersion of the i-th subband in the orig-

inal devices, we compute the energy levels Ei only at k‖ = 0


and obtain the new dispersion relation Ei(k‖) by shifting the“original” one by the difference obtained at k‖ = 0, that is:

Ei(k‖) = Eunpi (k‖) + (

Ei(0) − Eunpi (0)

)

. (16)

Once the perturbation potential is obtained, the matrix el-ements are computed as for n-channel SOI devices. Then,dielectric screening is taken into account as in Ref. [26]; inthis step, the differences between bulk and SOI devices areonly due to the different expression of the Green’s functions.Turning to Eq. 15, the delta function due to energy conser-vation converts the integral on the two-dimensional k‖ spaceinto line integrals along iso-energy contours. In this case theenergy is linearly interpolated inside each triangle of the k‖mesh, and the integration has to be performed over line seg-ments contained in the relevant triangles, in a similar way tothat carried out in the DOS computation. The resulting scat-tering rates depend on the initial moment ki and not just onthe initial energy; therefore the rates are computed at the be-ginning of the simulation as a function of the energy and theangle φ of ki in the k‖ plane and are saved in large look-uptables.

After the scattering rates have been tabulated, the car-rier dynamics is simulated, as is usual in the Monte Carlotechnique, as a sequence of flights interrupted by scatteringevents. According to the piecewise linear approximation ofthe hole dispersion in our approach, the carrier speed, beingproportional to the energy gradient, is constant in each trian-gle. The carrier motion during a flight between two scatter-ing events is therefore simulated with a constant speed whileits wavevector k‖ stays inside a triangle. Therefore, the mainissue in the simulation of the carrier motion is to determinethe time needed for the hole wavevector to exit the currenttriangle in the k‖-space. To do this we need to compute theintersection of the k‖-trajectory with the sides of the trian-gle, excluding the side from which the trajectory starts. Theflight then continues inside other triangles at different speedsuntil a scattering event takes place.

Alternatively, hole mobility can also be computed em-ploying Kubo-Greenwood formula [26]. Although this im-ply a linearization of Boltzmann transport equation, thisapproach also has some advantages with respect to MonteCarlo simulations. First of all, Monte Carlo approach re-quires larger simulation times to obtain acceptably small sta-tistical errors for the mobility. Another advantage is that itallows considering a single scattering mechanism at a time,to evaluate the separate importance of each mechanism [44].On the contrary, Monte Carlo calculations need at least oneinelastic scattering mechanism (such as optical phonons), toallow for dissipating the energy carriers gain because of thedrift electric field.

The Kubo-Greenwood formula for holes reads

μxx = e

4�2π2kBTps

∑

ν

∫

dk‖δ(E − Eν(k‖))(

∂Eν

∂kx

)2

× τ (ν)x (k‖)f0

(

Eν(k‖)) [

1 − f0(

Eν(k‖))]

. (17)

Here, mobility μxx along x direction is computed, ps =∑

ν pν and pν is the population of subband ν, τ(ν)x (k‖) is

the relaxation time for the x component of momentum insubband ν, and f0 is the Fermi-Dirac distribution f0(E) ={1 + exp[(E − EF )/(kBT )]}−1. When isotropic scatteringmechanisms are considered, such as acoustic and opticalphonon, the relaxation times coincides with the inverse scat-tering rate. Surface roughness scattering, however, is notisotropic, so that the corresponding momentum relaxationtimes must be computed as

1

τμ,νSR

= �2SR�2

SR

2�

∫

d2kf

(

1 − v(ν)x (kf )

v(μ)x (ki )

)

× |Mμ,νsr (kf − ki )|2

1 + �2SR(kf − ki )2/2

δ(Eν(kf ) − Eμ(ki )).

(18)

3.3 Results

First of all we used the self-consistent k · p and Poissonsolver to compute the valence band properties of conven-tional bulk p-channel Si MOSFETs. Monte Carlo simula-tions were employed to obtain low-field mobility, in verygood agreement [41] with universal mobility curve fromTakagi et al. [45], as can be seen in Fig. 10.

Arbitrary uniaxial and biaxial stress can also be takeninto account in the valence band and transport simulators.In particular we considered two different kind of devices:one with biaxially strained silicon channel grown on relaxedSi1−xGex substrates and the other with uniaxial strain in thechannel direction. In the former case we considered the Silayer to be sufficiently thick that the only influence of theSiGe layer consists in the induced strain; the used scatter-ing mechanisms and parameters are the same as in the un-

Fig. 10 Comparison of simulated mobility with experimental univer-sal mobility


Fig. 11 Hole mobility in devices with strained silicon channels

Fig. 12 Hole mobility in devices with different surface and channelorientations

strained case. As expected, a substantial hole mobility im-provement is found with an applied uniaxial stress while asmaller increase is obtained for Si layers grown on substratewith a Ge fraction larger than 10%. The obtained mobilityimprovement with respect to unstrained devices show a goodagreement with experimental data [41] (Fig. 11).

Substrate orientations different from the usual (001) andarbitrary channel directions can also be considered.

As we can see in Fig. 12, for (001) substrate mobilitydoes not depend on channel direction while (111) substratedevices show larger mobility, also independently of channeldirection. On the contrary (011) substrate shows a stronganisotropy and mobility is much larger for [011̄] channelthan [100]; the mobility of both cases is, however, largerthan (001) and (011) substrate orientations [41].

Double gate SOI structures with different silicon layerthicknesses have also been studied. We found that a rangeof silicon thicknesses exists, in which the mobility is higherthan that computed for conventional bulk devices or thickerSOI layers [44] (see Fig. 13). The extension and position of

Fig. 13 Hole mobility in DG SOI devices as a function of silicon layerthickness

such range depends on the considered value of the effectivefield, and on surface roughness parameters. This phenom-enon is related to volume inversion and can produce mobil-ity improvements larger than 25% over bulk values.

4 Device simulation. Multisubband ensemble MonteCarlo method

Up to now, we have considered the calculation of carriertransport properties in a boundless semiconductor struc-ture, and therefore, we used a single particle Monte Carlomethod. However when we need to simulate the whole be-havior of a realistic device, the single particle Monte Carloapproach is not appropriate because: (i) the motion of par-ticles are spatially restricted in the device region, so weneed to set up suitable boundary conditions for the parti-cles, and (ii) the Boltzmann transport equation must be self-consistently solved with the Poisson and Schroedinger equa-tions. Different solutions coming from classical to full quan-tum models can be considered depending on the needed ac-curacy, the computational resources and the available timeto perform device simulations. The most commonly usedapproach until the 90’s based on the coupled solution ofPoisson and Drift Diffusion equations [2], cannot be usedanymore in a direct way since confinement effects are ofspecial importance in State-of-the-Art (SoA) devices. Atthe opposite end of the spectrum, full quantum simulatorsbased on numerical solutions of the Schroedinger equationor the Non-Equilibrium Green’s Functions theory (NEGF)have also been developed [46]. In a quantum model, thetransport of charged particles is treated coherently accord-ing to a quantum wave equation. In the simplest case of asingle-particle Hamiltonian, carriers are considered as non-interacting waves described by the Schroedinger equation.The introduction of scattering in the simulations involves


a very high computational cost and for this reason, onlysimplified models can be used in practical quantum simu-lations [47].

Between these extreme approaches, Ensemble MonteCarlo (EMC) simulators are widely used since they presentseveral advantages compared to full quantum approxima-tions. A reduced computational cost, the possibility of con-sidering a wide variety of scattering mechanisms and highaccuracy for devices with silicon thicknesses as low as a fewnanometers [48] are some of the advantages of such simula-tors. To include quantum effects in EMC codes, two are themain solutions proposed in the bibliography:

1. The addition of a correction term to the electrostatic po-tential, i.e. quantum correction, to mimic the carrier con-centration profile obtained when the Schroedinger equa-tion is solved in the structure. Different models havebeen developed following this philosophy, giving a goodaccuracy-computational cost ratio. The most commonlyused approaches include the Density Gradient [49], theEffective Potential [50], or the Multi-Valley EffectiveConduction Band-Edge method (MV-ECBE) [51].

2. The coupling of the Boltzmann Transport Equation(BTE) solved by the Monte Carlo method in the transportplane with the Schroedinger equation in the confinementdirection evaluated in different slices of the considereddevice. This method, called Multi Subband EnsembleMonte Carlo approach (MSB-EMC) [52–55] provideswhat is to date the most detailed description of carriertransport in the device, since the scattering rates are ob-tained from a quantum solution.

In the following sections, the different EMC approachesto the quantum problem will be presented and compared.

4.1 Quantum correction methods

The easiest way to include quantum effects in semi-classicalsimulators such as EMC codes is by adding a correctionterm to the electrostatic potential obtained from the solutionof Poisson’s equation. These approaches are widely used inMC codes since the early 2000s due to their computationalefficiency (close to their semiclassical counterpart) and goodaccuracy. However, the practical implementation needs, ingeneral, a previous calibration to obtain the set of fitting pa-rameters to obtain accurate results from the transport pointof view.

4.1.1 The effective potential method

The effective potential model [50] obtains the corrected po-tential from the convolution of the electrostatic potential,V

(

x′), and a Gaussian function which represents the “ef-fective size” of the particle. The one dimensional form of

the correction is given by

Veff (x) = 1√2πa0

∫

V(

x′) exp

(

−(

x − x′)2

2a20

)

dx′ (19)

where a0 represents the spreading of the wave-packet. Themaximum of the carrier distribution is then shifted fromthe oxide interface, reproducing the total inversion charge.However, the charge profile results incorrect when is com-pared to the 1D Schroedinger solution [49]. Therefore, themethod is not appropriate when magnitudes that depend onoverlapping integrals involving envelope functions extractedfrom carrier distributions need to be calculated. This is thecase of surface roughness models or inter-valley scatteringrates when size quantization is taken into account.

4.1.2 The density gradient approach

A better approximation obtained from the Wigner poten-tial approximation is the Density Gradient (DG) [56]. Thismethod, which is one of the first quantum corrections im-plemented on a simulation code, produces very good resultsespecially in drift diffusion simulations. The correction termis given by the following expression

Vq = 2bn

∇2(√

n)

√n

(20)

where bn = �2/4rm∗

n, � is the reduced Planck’s constant,m∗

n represents the electron effective mass and r is a parame-ter whose value varies from 1 for pure states (low tempera-tures or very strong confinement) to 3 for mixed states (hightemperatures or weak confinement). The last two parametersare used to fit the carrier profile to the one obtained from thesolution of the 1D Schroedinger equation.

Figure 14 shows a comparison of the electron concentra-tion profile obtained from the solution of the Schroedingerequation, the Density Gradient approach and the effectivepotential for a MOS structure [49]. As can be observed, al-though both the approximated techniques fit the inversioncharge, in the case of the Density Gradient, it is also possi-ble to reproduce the inversion charge profile.

The drawback of DG approach is the difficult implemen-tation of the model in EMC codes. The main reason is thedependence of the driving force with the third derivative ofthe carrier concentration which is a very noisy magnitudein EMC simulators. As a consequence, it is a hard task toobtain convergence and the corrections to the field shouldbe calculated from different magnitudes trying to keep theaccuracy of DG simulations.

4.1.3 The multi valley effective conduction band-edge(MV-ECBE) method

In MC simulations, it is useful to express the quantum cor-rection in terms of the electrostatic potential, which is a


Fig. 14 Electron concentration profile obtained from the solution ofthe Schroedinger equation, the Density Gradient approach and the ef-fective potential for a MOS structure. As can be observed, the DensityGradient fairly reproduces the inversion charge profile [49]

smooth magnitude compared to the carrier density. The Ef-fective Conduction Band-Edge method (ECBE) was devel-oped to obtain the benefits of the Density Gradient in MCsimulations [57].

Starting from the Density Gradient, Eq. 20, and assum-ing an exponential relation between the electron concentra-tion and the effective potential, n ∝ exp(

qV ∗kBT

), the correctedpotential V ∗ can be written as

V ∗ = V +Vq = V + q�2

4rm∗nkBT

(

∇2V ∗ + q

2kBT

(∇V ∗)2)

.

(21)

The equation is solved in a self-consistent way with Pois-son and the BTE using the boundary conditions proposedby [58]. Depending on the device geometry and bias point,the population of the different valleys may change. As aconsequence, there is a variation in the confinement effec-tive mass when only one valley is considered which has tobe taken into account to reproduce the charge profile ob-tained from the solution of the 1D Poisson-Schroedingersystem. The practical consequence is the use of the effec-tive mass as a fitting parameter. To avoid this issue, an im-proved version of the ECBE was developed to take into ac-count the effect of different valleys and the orientation ef-fects on the device performance. This new approach is calledMulti-Valley ECBE (MV-ECBE) [51] and the main differ-ences with the standard version is the existence of differentcorrection terms depending on the valley occupied by theconsidered carrier. Orientation effects are taken into accountby considering the effective mass as a tensor. Following thecalculations in [51], the correction term for the nth valley

can be written as follows:

V ∗j V + �

2

4qrVT

⎧

⎨

⎩∇ ·

⎡

⎣

(←→1

m

)

j

· ∇V ∗j

⎤

⎦

+ 1

2VT

⎡

⎣∇V ∗j ·

(←→1

m

)

j

· ∇V ∗j

⎤

⎦

⎫

⎬

⎭(22)

which is the ECBE equation for a multi-valley system andarbitrary confinement directions. It is easy to demonstratethat Eq. 22 reduces to the standard density gradient theory,Eq. 21, when only one valley and a diagonal effective masstensor with the same value for all the elements are assumed.

The drawback of the MV-ECBE approach is the necessityof an a-priori calculated valley population. This fact limitsthe application of the model to drift diffusion simulations.However, the redistribution of the carriers among the dif-ferent valleys can be obtained in MC simulations thanks tothe intervalley scattering. In quantum-corrected MC calcu-lations (QC-EMC), the electron gas is not considered as 3Dbut pseudo-2D. In this way, the position of the carriers in theconfinement direction is known but a form factor is includedin the scattering rate:

Fij (x) = LD

∫

TSi

| i(x, z)|2| j(x, z)|2dz (23)

where i and j stand for the initial and final valley in the scat-tering process LD is the Debye length, x and z the transportand confinement directions respectively and

| i(x, z)|2 = ni(x, z)∫

TSini(x, z)dz

(24)

is the envelope of the electron wave function in the pseudo2D gas approach. The scattering rates are then weighted byFij (x). In this way, size quantization is taken into accountand the valley population is calculated self-consistently. Sur-face roughness scattering is implemented using a pseudo-2Dgas version of the model proposed in [42], where an expo-nential spectrum model is assumed to represent the rough-ness at the silicon-oxide interface and the matrix elementsdepend on the overlapping integral between the perturbationpotential and, once again, the envelope function of the cor-responding valley.

4.2 The multi-subband approach

As mentioned before, a second option to face the prob-lem of quantum confinement in EMC simulators is the useof the Multi-Subband method (MSB-EMC). Based on themode-space approach of quantum transport [59], the quan-tum problem is considered as decoupled in the transportplane. Therefore, the Schroedinger equation is only solved


Fig. 15 DGSOI structure used for MSB-EMC simulations. 1DSchroedinger equation is solved for each grid point in the transport di-rection whereas Boltzmann Transport equation is solved by the MonteCarlo method in the transport plane

in the confinement direction. This model is especially use-ful in the study of standard nano-transistors. However, it isnot suitable for the study of some structures where coher-ence phenomena are of special interest. From the simulationpoint of view, our simulated device is considered as a stackof slices perpendicular to the channel direction (Fig. 15).The 1D Schroedinger equation is then solved for each sliceand valley self-consistently with the 2D Poisson equation.The evolution of the eigen-energies, Ei,ν(x), and the wavefunctions, ξi,ν(x, z) are obtained in this way along the trans-port axis, x, for the i-th valley and the ν-th subband. Con-cerning the transport, the BTE is solved by the MC methodin the transport plane (Fig. 15).

Non parabolicity effects are included in the equations de-scribing both transport and confinement following [52]. Inopposition to standard EMC codes, the driving field under-gone by a simulated super-particle is not obtained from thegradient of the electrostatic potential, including or not thequantum correction term. According to the space-mode ap-proach, the drift field is calculated from the spatial derivativeof Ei,ν(x), obtaining a different driving force for each sub-band. This can be inferred from Fig. 16 where the differentevolution of the three first energy levels along the transportchannel is represented for a 4 nm channel thick DGSOI.

Concerning the calculation of scattering rates, a 2D elec-tron gas is assumed since a full quantum description of theproperties of the carriers is used for the confinement direc-tion. Acoustic and intervalley phonon scattering rate calcu-lations are detailed in [52] whereas surface roughness scat-tering is described in [42]. All the models consider nonpar-abolic and ellipsoidal bands. As the quantum well used tosolve the Schroedinger equation is different for each consid-ered slice, the obtained eigen-energies and wave functionsare also different. Thus, it is necessary to calculate a scat-tering table for each grid point, i.e. slice, and update themfor every new solution of the quantum problem performed

Fig. 16 Subband profile along the transport direction corresponding tothe three first energy levels for a 10 nm length and 4 nm thick DGSOIdevice

to keep the self-consistency of the calculations. The sub-band population, Ni,ν(x), is calculated by resampling thesuper-particles belonging to a given subband and slice usingthe cloud-in-cell method [60]. The electron concentration,n(x, z), is obtained adding all the densities of probability|ξi,ν(x, z)|2 weighted by the corresponding subband popu-lation:

n(x, z) =∑

i,ν

Ni,ν(x)∣∣ξi,ν(x, z)

∣∣2. (25)

The electrostatic potential is updated by solving the 2D Pois-son equation using the previous n(x, z) as input. This ap-proach is especially appropriate for the study of 1D confine-ment in nanoscale devices. The drawback (compared withsemi-classical MC codes) is an important increase of thecomputational effort. This issue can be partially overcomethanks to a high efficient parallel implementation of the codeand the possibility of grid computing.

4.3 EMC simulation of nanodevices

As commented before, MSB-EMC simulators represent themost accurate approach for the study of carrier transportnanodevices. Semiclassical EMC and the MV-EMC can beused with long channel or partially depleted SOI devicesrespectively. However, as channel length and thickness arereduced, it is necessary an accurate description of subbandprofiles in order to take into account ultra-thin film effectssuch as thickness fluctuations or phonon confinement [22].Figure 17 shows the electron concentration profile of a 4nm thick DGSOI with midgap metal gate and 1 V appliedto each gate for the different simulation approaches. It isshown how, from the electrostatic point of view, the MV-ECBE method still presents acceptable results whereas thesemiclassical approach is not valid for such kind of devices.

However, the study of the output characteristics for a 10nm channel length device with the previous confinement


Fig. 17 Electron concentration profile for a 4 nm thick DGSOI deviceobtained from Classical EMC, MV-ECBE and MSB-EMC. It can beobserved how MV-ECBE method still reproduces the profile obtainedfrom quantum calculations

structure shows an overestimation of the drain current whensemiclassical or MV-ECBE codes are used, Fig. 18. In thesemiclassical case the higher current is due to the overesti-mation of the inversion charge, a small effect of confinementand the surface roughness scattering model based on diffu-sive reflections. In the case of MV-ECBE approach, sinceonly one subband per valley is considered, inter-valley andsurface roughness scattering are under-estimated. Therefore,it is necessary to use the Multi-Subband approach to cor-rectly catch the behavior of such ultra-thin devices.

Device orientation impact on carrier transport is an im-portant issue to be studied in ultrashort transistors. This ef-fect is of special relevance with the introduction of multigatevertical devices, i.e. FinFETs. The confinement and channelorientations can be changed by simply rotating the devicewhile a standard (100) wafer is kept as substrate [61]. Themain advantage is the possibility of including different ori-entations in a single layout in order to maximize both elec-tron and holes mobilities [62–64] without using more exoticand expensive substrates as the obtained with the HybridOrientation Technology (HOT) [65].

A set of MSB-EMC simulations have been carried out onthe previously described device to establish the impact ofconfinement and transport orientations. ID vs VDS curvesfor the main orientation directions are depicted in Fig. 19when 1 V is applied to both gates. The highest performanceis obtained for the (001)/〈110〉 and the (110)/〈001〉 devices.Since both transistors present similar inversion charge val-ues, a direct extrapolation from velocity/mobility results tooutput characteristics can be assumed [64].

To explain this behavior, several factors have to be takeninto account. The first one is the variation of the transport ef-fective mass with the channel orientation. In this way, driftvelocity profiles change for each case as can be observed inFig. 20. Higher velocities are observed for orientations with

Fig. 18 Drain current comparison for different EMC approaches.Semiclassical and MV-ECBE codes clearly over-estimate the value ob-tained with the MSB-EMC method

Fig. 19 Drain current for different channel orientations. Highest draincurrent is obtained for (100) 〈110〉. Transport mass anisotropy is shownfor (110) results. Note the important differences for the 〈1–10〉 and the〈001〉 oriented channels

Fig. 20 Drift velocity profile for the considered channel orientationswith VDS = 300 mV. The highest drift velocity is obtained for the (100)〈110〉 device. Drain current differences for (110) devices comes fromtransport mass anisotropy which produces important drift velocity vari-ations for the 〈1–10〉 and the 〈001〉 oriented channels


small transport mass, i.e. (001)/〈110〉 and (110)/〈001〉. Thechannel orientation dependence can be clearly observed forthe (110) devices. The device faced to 〈001〉 shows a ve-locity profile close to the (001)/〈110〉 with the best perfor-mance. However, a 90 rotation of the channel, i.e. 〈11̄0〉, de-grades the drift velocity leading to the poorest performance.In the case of (111) and (001) orientations, there is no de-pendence of the performance with the channel orientationsince valley ellipsoids are symmetrically placed. The scat-tering also plays a role that can not be neglected. The transittime along the channel defined as

τt =∫

LCh

1

v‖(x)dx (26)

increases for smaller drift velocities. This means that carri-ers spend more time in the channel and besides the similarscattering rates for the considered energies, each carrier suf-fer a bigger amount of scattering events. As a consequence,the drift velocity is reduced more than expected. This ex-plains how the difference among the velocities is increasedas the carriers approach to the end of the channel (Fig. 20).The increase in the number of scattering events in the chan-nel implies a reduction in the intrinsic ballisticity, Bint de-fined as the fraction of carriers injected at the source thatreach the drain contact suffering zero scattering events in thechannel region [66]. As a consequence it is observed a bet-ter ballistic behavior in the case of the channel orientationscorresponding to higher performance.

4.4 Transient simulations

Quantum corrected EMC simulators can also be used tostudy the transient behavior of the devices under consider-ation. The main limitation is that the results must conformwith the quasi-stationary condition, i.e. the switch time mustbe much greater than the self- consistency time step. An in-teresting application of transient simulations is the study ofthe performance of non-conventional devices such as realspace transfer (RST) based devices. This concept introducedby Gribnikov [67] and Hess [68] is based on the spatial re-distribution of carriers, which implies a modulation of mo-bility. Different structures have been proposed in the liter-ature, based on physical phenomena like hot electron ef-fects or tunneling throughout a potential barrier ([69, 70])with applications in negative differential resistance (NDR)devices [71] and ultra-high speed switches based on the ve-locity modulation (VM) effect [72].

We have used the MV-ECBE simulator to study veloc-ity modulation transistors (VMT) based on DGSOI devices.In this case, two channels with different transport propertiesare created in the proximity of each interface. The most ap-propriate way to obtain an intentionally degraded mobility

Fig. 21 Contour plots of the electron concentration during a VMTswitch. It can be observed how the charge is transferred from one in-terface to the other. The total inversion charge in the device is keptconstant during the transition

in one of the channels is to create a bad quality Si–SiO2 in-terface in order to maximize the effects of surface roughnessscattering [73]. The current modulation is obtained by trans-ferring the charge carriers from one channel to the other,considering the OFF state when the electrons are in the lowmobility channel and the ON state when the current flowsthrough the non-degraded channel. To keep the sheet charge


density constant during the operation, an asymmetric biasshould be applied to each gate. This means that the switchtime is not limited by the transit time from source to drainbut by the time necessary to transfer the carriers from oneadjacent channel to the other. Figure 21 shows differentsnapshots of the carrier distribution in the device when thecharge is transferred from one interface to the other in a 10nm thick DGSOI. These devices are expected to be oper-ated in the Tera-Herz range with ION/IOFF ratios near to10 which are the recommended values for the use in HighFrequency and High Performance MSI, LSI circuits.

Acknowledgements This work was supported by the Spanish Gov-ernment under projects TEC2008-06758-C02-01 and FIS2008-05805,and by Junta de Andalucia under projects TIC1899 and TIC3580.Financial support from EU EUROSOI+ Thematic Network (FP7-CA-216373) and EU NANOSIL Network of Excellence (FP7-NOE-216171) is also acknowledged.

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Monte Carlo simulation of nanoelectronic devices

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