We present an extended hydrodynamic model describing the transport of

electrons in the axial direction of a silicon nanowire. This model has been formulated by

closing the moment system derived from the Boltzmann equation on the basis of the maximum

entropy principle of Extended Thermodynamics, coupled to the Schr¨odinger-Poisson

system. Explicit closure relations for the high-order fluxes and the production terms are

obtained without any fitting procedure, including scattering of electrons with acoustic

and non polar optical phonons. We derive, using this model, the electron mobility

Di Stefano, V.

Muscato, O.

UPCommons

Coupled quantum-classical transport in silicon nanowires

We present an extended hydrodynamic model describing the transport of
electrons in the axial direction of a silicon nanowire. This model has been formulated by
closing the moment system derived from the Boltzmann equation on the basis of the maximum
entropy principle of Extended Thermodynamics, coupled to the Schr¨odinger-Poisson
system. Explicit closure relations for the high-order fluxes and the production terms are
obtained without any fitting procedure, including scattering of electrons with acoustic
and non polar optical phonons. We derive, using this model, the electron mobility

UPCommons. Portal del coneixement obert de la UPC

Coupled  quantum-classical transport in Silicon nanowiresV International Conference on Computational Methods for Coupled Problems in Science and EngineeringCOUPLED PROBLEMS 2013S. Idelsohn, M. Papadrakakis and B. Schrefler (Eds)COUPLED QUANTUM-CLASSICAL TRANSPORT INSILICON NANOWIRESO. MUSCATO AND V. DI STEFANODipartimento di Matematica e InformaticaUniversità degli Studi di CataniaViale A. Doria 6, 95125 Catania, Italye-mail: muscato@dmi.unict.it, vdistefano@dmi.unict.itKey words: Silicon nanowires, Schrödinger-Poisson-Boltzmann system, hydrodynamicmodel, maximum entropy principleAbstract. We present an extended hydrodynamic model describing the transport ofelectrons in the axial direction of a silicon nanowire. This model has been formulated byclosing the moment system derived from the Boltzmann equation on the basis of the maxi-mum entropy principle of Extended Thermodynamics, coupled to the Schrödinger-Poissonsystem. Explicit closure relations for the high-order fluxes and the production terms areobtained without any fitting procedure, including scattering of electrons with acousticand non polar optical phonons. We derive, using this model, the electron mobility.1 INTRODUCTIONSilicon nanowires (SiNWs) are quasi one-dimensional structures in which the electronsare spatially confined in the two transversal directions and free to move in the longitudinalone. SiNW devices have been fabricated recently using lithographic techniques [1]. Theyhave attracted significant interest due to their potential to function as logic devices,thermoelectric devices, and sensors. Therefore, it is crucial to accurately model andestimate the performance of these devices.By shrinking the dimension of electronic devices, effects of quantum confinement areobserved and the wave nature of the electrons must be taken into account. The Non-Equilibrium Green Function formalism is the most advanced transport model for thesimulation of SiNW devices, but it necessitates rather intensive computational efforts sinceit requires detailed information on the propagation of the electron wave packet injectedin the device. Under reasonable hypothesis, transport in low-dimension semiconductorscan be tackled coupling quantum and semiclassical tools. In fact, the main quantumtransport phenomena in SiNW transistors at room temperature, such as the source-to-drain tunneling, and the conductance fluctuation induced by the quantum interference,11109O. Muscato and V. Di Stefanobecome significant only when the longitudinal length (called channel) is smaller than 10nm[2]. Therefore, for longer channels, semiclassical formulations based on the 1-D MultibandBoltzmann Transport Equation (MBTE) can give reliable simulation results when it issolved self-consistently with the 3-D Poisson and 2-D Schrödinger equations in order toobtain the self-consistent potential and subband energies and wavefunctions [1]. Anothersimplification comes from the use of the Effective Mass Approximation (EMA), which issupposed to be still a good solution in the confining direction in the presence of disorder,which is probably valid for semiconductor nanowires down to 5 nm in diameter, belowwhich atomistic electronic structure models need to be employed. Solving the MBTEnumerically is not an easy task, because it forms an integro-differential system in twodimensions in the phase-space and one in time, with a complicate collisional operator.The full solution of the MBTE can be obtained or by using the Monte Carlo (MC)method [3]-[10] or by using deterministic numerical solvers [11],[12],[13] at expense of hugecomputational times. Another alternative is to obtain from the MBTE hydrodynamicmodels that are a good engineering-oriented approach. This can be achieved by takingmoments of the MBTE, and by closing the obtained hierarchy of balance equations aswell as modeling the production terms (i.e. the moments on the collisional operator).2 Transport equationsIn the following we shall consider a SiNW with rectangular cross section. For a quantumwire with linear expansion in z-direction, and confined in the plane x-y, the normedelectron wave function ψ(x, y, z) can be written in the formψ(x, y, z) = χα(x, y)eikzz√Lz(1)where χα(x, y) is the wave function of the α-th subband and the term eikzz/√Lz describesan independent plane wave in z-direction confined to the normalization length, wherez ∈ [0, Lz] and kz is the wave vector number. In general the electron is subject to externalconfining potential U , such as by a discontinuity in the band gap at an interface betweentwo materials, and also to the effect of the other electrons in the system. The simplestapproximation, called Hartree approximation, is to assume that the electrons as wholeproduce an average electrostatic energy potential Vtot, and that a given electron feels theresulting total potentialVtot = U(x, y) − eΦ(x, y, z) . (2)The normed wave function satisfies the Schrödinger equation in the Effective Mass Ap-proximation, i.e. [Ec −22m∗∆ + Vtot(x, y, z)]ψ = E ψ (3)where E is the total energy, Ec the conduction band edge energy, and m∗ denotes theeffective mass of the electron in the conduction band. By inserting eq.(1) into eq.(3), in21110O. Muscato and V. Di Stefanoeach z-th cross section of the device, one obtains the following equation for the envelopefunction χαz(x, y)[−22m∗(∂2∂x2+∂2∂y2)+ U − eΦ]χαz = εαzχαz , Eαz = εαz +2k2z2m∗+ Ec (4)where εαz is the kinetic energy associated with the confinement in the x-y plane, and wehave assumed parabolic band approximation. The term Φ satisfies the Poisson equation∇ · [ǫ∇Φ(x, y, z)] = e(n − ND + NA) (5)where ND, NA are the doping profile (due to donors and acceptors) and n(x, y, z, t) is theelectron density, which depends on χαzn(x, y, z, t) =∑αρα(z, t)|χαz(x, y, t)|2 (6)where ρα is the subband linear density in the z-directionρα(z, t) =22π∫fα(z, kz, t)dkz (7)fα being the electron distribution function in the α-subband. For an assigned confiningpotential, one has to solve a coupled problem formed by eqs.(4), (5) and (6) to find εαz, χαzin each cross-section.The transport in the z-th direction is described using the MBTE [1]∂fα∂t+ vz(kz)∂fα∂z−eEz∂fα∂kz=∑α′∑ηCη[fα, fα′ ] (8)where e is the absolute value of the electron charge,  the Planck constant divided by 2π,andvz =1∂Eαz∂kz=kzm∗, Ez = −1e∂Eαz∂z(9)are respectively the electron group velocity and the electric field. In the low densityapproximation (not-degenerate case), the collisional operator writesCη[fα, fα′ ] =Lz2π∫dk′z {wη(k′, k)fα′(k′z) − wη(k, k′)fα(kz)} (10)where wη(k, k′) = wη(α, kz, α′, k′z) is the η-th scattering rate. When α = α′ we have anintra-subband scattering, otherwise we have an inter-subband scattering.Scattering mechanisms in SiNWs must comprise acoustic phonon scattering (bulk andconfined), non-polar optical phonon scattering, surface scattering, scattering with ionized31111O. Muscato and V. Di Stefanoimpurities, as well as dielectric screening [4], [5]. However in this preliminary study, forthe sake of simplicity, we shall limit ourselves to consider just scattering with optical andacoustic phonons. For the bulk acoustic phonon scattering, in the elastic equipartitionapproximation, the transition rate is given by [1]wac(k, k′) = sacGαα′δ (Eα′ − Eα) , sac =2πD2AkBTLρ�v2sLz(11)where DA is the acoustic deformation potential (9 eV), TL the lattice temperature, ρ themass density ( 2.33 gr/cm3), vs the sound speed (6960 m/sec), and Gαα′ the confinementfactorGαα′=∫|χα′(x, y)|2|χα(x, y)|2dxdy . (12)For the optical phonons we havewop(k, k′) = sop[g0 +12∓12]Gαα′δ (Eα′ − Eα ∓ �ω0) , sop =πD20ρω0Lz(13)where D0 is the optical deformation potential (11.4 108 eV/cm), �ω0 the effective opticalphonon energy (63 meV), and g0 the Bose-Einstein phonon occupation number.3 Extended Hydrodynamic modelBy multiplying the MBTE (8) by the weight functions ψA = {1, vz, εz, vzεz}, andintegrating in the kz space, one obtains the following hydrodynamic-like equations∂ρα∂t+∂(ραV α)∂z= ρα∑α′Cαα′ρ (14)∂(ραV α)∂t+2m∗∂(ραW α)∂z+em∗ραEz = ρα∑α′Cαα′V (15)∂(ραW α)∂t+∂(ραSα)∂z+ ραeEzVα = ρα∑α′Cαα′W (16)∂(ραSα)∂t+∂(ραFα)∂z+ 3em∗ραEzWα = ρα∑α′Cαα′S (17)in the unknowns (called moments)V α =2(2π)1ρα∫Rfα(z, kz, t)vzdkz (subband velocity), (18)W α =2(2π)1ρα∫Rfα(z, kz, t)εzdkz (subband energy), (19)Sα =2(2π)1ρα∫Rfα(z, kz, t)εzvzdkz (subband energy- flux) (20)41112O. Muscato and V. Di Stefanoand the higher-order flux Fα, and the production termsF α =2(2π)1ρα∫fαv2zεzdkz (21)Cαα′ρ =2(2π)1ρα∑η∫Cη[fα, fα′ ]dkz (22)Cαα′V =2(2π)1ρα∑η∫Cη[fα, fα′ ]vzdkz (23)Cαα′W =2(2π)1ρα∑η∫Cη[fα, fα′ ]εzdkz (24)Cαα′S =2(2π)1ρα∑η∫Cη[fα, fα′ ]εzvzdkz . (25)This system of PDEs is of hyperbolic type and it is not closed, i.e. there are more un-knowns than equations. The Maximum Entropy Principle leads to a systematic way forobtaining constitutive relations on the basis of the information theory [14], as alreadyproved successfully in the bulk case [15]-[19], and for quantum well structures [20], [21].Actually, in a semiconductor electrons interact with phonons describing the thermal vi-brations of the ions placed at the points of the crystal lattice. However, since we areconsidering the phonon gas as a thermal bath, one has to extremize only the electroncomponent of the entropy. We define the entropy of the electronic system asSe =∑α|χα(x, y, t)|2Sαe (26)Sαe = −2(2π)kB∫R(fα log fα − fα)dkz , (27)and, according to MEP, we estimate the fα’s as the distributions that maximize Se underthe constraints that the basic moments, which we have previously considered, are assigned.In a neighborhood of local thermal equilibrium, this distribution function writes [22]f̂α = exp(−λαkB− λαW εz) {1 − τ(λ̂αV vz + λ̂αSvzεz)}(28)where the quantities (λα, λαW , λ̂αV , λ̂αS) are known functions of the moments{ρα, V α,W α, Sα}. By using the distribution function (28) it is possible to evaluate theunknown functions appearing in the balance equations by integration. In this way thehigher-order flux term writesFα =6(W α)2m∗(29)51113O. Muscato and V. Di Stefanoas well as the production terms Cαα′ρ , Cαα′V , Cαα′W , Cαα′S have been determined. We want un-derline that this Extended Hydrodynamic model has been closed by using first principles,and it is free of any fitting parameters.4 Electron MobilityThe mobility is one of the most important parameters that determine the performanceof a field-effect transistor. At low electric field, the carrier drift velocity is proportionalto the electric field strength, and the proportionality constant is defined as the mobility.Hence a higher mobility material is likely to have higher frequency response, becausecarriers take less time to travel through the device. When the fields are sufficiently large,nonlinearities in the mobility and saturation in the drift velocity are observed. In fact, thescattering of the carriers with the lattice, the impurities, and the surface is more activefor higher fields, and the charges lose the energy gained by the electric field.Now we want to prove that our Extended Hydrodynamic model is able to predict suchbehaviours. We shall assume that the wire is surrounded by an oxide which gives rise toan infinitely deep potential barrier. In such a case, the following analytical relations forthe bottom energies and envelope functions can be used [1]εα =�2π22m∗(n2L2x+m2L2y), χα =√2Lxsin(nπLxx)√2Lysin(mπLyy)n,m ∈ N. (30)To obtain the drift velocity and the mobility, we have performed a numerical integrationof our hydrodynamic model in the stationary homogeneous case with a constant electricfield along the z direction. In this case the unknowns (ρα, V α,W α, Sα) depend on thetime only. The initial data are the equilibrium values, obtained with a global maxwellianwith lattice temperature T0 (300 K), i.e.V α(0) = 0 ,W α(0) =12kBT0 , Sα(0) = 0 (31)ρα(0) = LxLyNDexp(− εαkBT0)∑α exp(− εαkBT0) , (32)where ND is the number of donor impurities. The average drift velocity is defined as�V � =∑α ραV α∑α ρα. (33)In the figure 1 we plot the subband velocities V α (α =1,..,4) as well as the average driftvelocity versus the simulation time, for an electric field of 1000 V/cm, with Lx = Ly = 10nm. The stationary regime is reached in a few picoseconds, and the typical phenomenonof saturation is qualitatively and quantitatively well described.61114O. Muscato and V. Di Stefano0 0.2 0.4 0.6 0.8 1105106time (ps)drift velocity (cm/sec)α = 1α = 2α = 3α = 4averageFigure 1: The subband drift velocities V α and the average drift velocity (33) versus the simulation time,for an electric field of 1000 V/cm, with Lx = Ly = 10 nm.In the figure 2 we plot the average drift velocity, obtained in the stationary regime,versus the electric field from which we can evaluate the saturation velocity vs = 6.7 106cm/sec.71115O. Muscato and V. Di Stefano101 102 103 104 105103104105106107electric field (V/cm)average drift velocity  ( cm/sec)Figure 2: The average drift velocity (33), obtained in the stationary regime, versus the electric field withLx = Ly = 10 nm.The average mobility isµ =∑α ραµα∑α ρα, µα =V αEz(34)where µα is the subband mobility. In the figure 3 we plot the average mobility as functionof the electric field. We notice for low fields (≤ 1000 V/cm) the mobility is constant(i.e. µ0 = 406 cm2 V/sec) whereas, for high fields, the mobility decreases because thescattering processes become more active. In the latter figure for comparison we have alsoreported the mobility given by the Caughey-Thomas formula [23]µC = µ0[1 +(µ0Ezvs)2]−12. (35)Similar results have been obtained using MC simulations [4], but with more expensivecomputational times.81116O. Muscato and V. Di Stefano101 102 103 104 105150200250300350400450electric field (V/cm)mobility (cm2 /V Sec)hydrodynamicCaughy−ThomasFigure 3: The average mobility (34)1 (circles) versus the electric field with Lx = Ly = 10 nm, and themobility evaluated using the Caughey-Thomas formula (diamonds).5 ConclusionsAn extended hydrodynamic model for SiNWs has been formulated with the use of themaximum entropy principle, where the transport coefficients are completely determinedwithout any fitting procedure. Using this model we have evaluated the electron mobility(low and high-field), which is in agreement with MC simulation results. However, ourmodel must be improved by including other relevant scattering mechanisms such as scat-tering with impurities, surface roughness, acoustic confined phonons. Simulation of realdevice as well as the study of thermoelectric effects according to the guideline in [24]-[28]are under investigation, and they will be published in the next future.AcknowledgmentWe acknowledge the support of the Università degli Studi di Catania, MIUR PRIN 2009”Problemi Matematici delle Teorie Cinetiche e Applicazioni”, and the CINECA AwardN. HP10COORUX 2012.REFERENCES[1] Ferry, D.K. Goodnick, S.M. and Bird, J. Transport in nanostructures. CambridgeUniversity Press, (2009).91117O. Muscato and V. Di Stefano[2] Lundstrom, M. and Wang, J. Does source-to-drain tunneling limit the ultimate scalingof MOSFETs?. IEDM Tech. Dig. (2002), 707-710.[3] Ramayya, E.B. Vasileska, D. Goodnick, S.M. and Knezevic, I. Electron mobility inSilicon Nanowires. IEEE Trans. Nanotech. (2007) 6(1):113-117[4] Ramayya, E.B. Vasileska, D. Goodnick, S.M. and Knezevic, I. Electron transport insilicon nanowires: The role of acoustic phonon confinement and surface roughnessscattering. J. Appl. Physics (2008) 104:063711.[5] Ramayya, E.B. and Knezevic, I. Self-consistent Poisson-Schrödinger-Monte Carlosolver: electron mobility in silicon nanowires. J. Comput. Electron. (2010) 9:206-210.[6] Muscato, O. Monte Carlo evaluation of the transport coefficients in a n+ - n - n+silicon diode. COMPEL (2000) 19(3): 812-828.[7] Muscato, O. and Wagner, W. Time step truncation in direct simulation Monte Carlofor semiconductors. COMPEL (2005) 24(4): 1351-1366.[8] Muscato, O. Wagner, W. and Di Stefano, V. Numerical study of the systematic errorin Monte Carlo schemes for semiconductors. ESAIM: M2AN (2010) 44(5): 1049-1068.[9] Muscato, O. Wagner, W. and Di Stefano, V. Properties of the steady state distributionof electrons in semiconductors. Kinetic and Related Models (2011) 4(3): 809-829.[10] Muscato, O. Di Stefano, V. and Wagner, W. A variance-reduced electrothermalMonte Carlo method for semiconductor device simulation. Computers & Mathematicswith Applications (2013) 65(3):520-527.[11] Lenzi, M.L. Palestri, P. Gnani, E. Gnudi A., Esseni, D. Selmi, L. and Baccarani, G.Investigation of the transport properties of silicon nanowires using deterministic andMonte Carlo approaches to the solution of the Boltzmann transport equation. IEEETrans. Electr. Dev. (2008) 55(8):2086-2096[12] Ossig, G. and Schuerrer, F. Simulation of non-equilibrium electron transport in siliconquantum wires. J. Comput. Electron. (2008) 7: 367-370.[13] Majorana, A. Muscato, O. and Milazzo, C. Charge transport in 1D silicon devicesvia Monte Carlo simulation and Boltzmann-Poisson solver. COMPEL (2004) 23(2):410-425.[14] Jou, D. Casas-Vázquez, J. and Lebon, G. Extended irreversible thermodynamics.Springer-Verlag, Berlin, (2001)101118O. Muscato and V. Di Stefano[15] Muscato, O. Pidatella, R.M. and Fischetti, M.V. Monte Carlo and hydrodynamicsimulation of a one dimensional n+−n−n+ silicon diode. VLSI Design (1998) 6(1-4):247-250.[16] Muscato, O. and Di Stefano, V. Modeling heat generation in a sub-micrometricn+ − n − n+ silicon diode. J. Appl. Phys. (2008) 104(12): 124501.[17] Muscato, O. and Di Stefano, V. Hydrodynamic modeling of the electro-thermal trans-port in silicon semiconductors. J. Phys. A: Math. Theor. (2011) 44(10): 105501.[18] Muscato, O. and Di Stefano, V. An Energy Transport Model Describing Heat Genera-tion and Conduction in Silicon Semiconductors. J. Stat. Phys. (2011) 144(1): 171-197.[19] Muscato, O. and Di Stefano, V. Heat generation and transport in nanoscale semi-conductor devices via Monte Carlo and hydrodynamic simulations. COMPEL (2011)30(2): 519-537.[20] Mascali, G. and Romano, V. A non parabolic hydrodynamical subband model forsemiconductors based on the maximum entropy principle. Math. Comp. Model. (2012)55(3-4): 1003-1020.[21] Camiola, V.D. Mascali, G. and Romano, V. Numerical simulation of a double-gateMOSFET with a subband model for semiconductors based on the maximum entropyprinciple, Cont. Mech.Thermodyn. (2012) 24(4-6): 417-436.[22] Muscato, O. and Di Stefano, V. Hydrodynamic modeling of silicon quantum wires.J. Comp. Electron. (2012) 11(1): 45-55.[23] Selberherr, S. Analysis and Simulation of Semiconduct Devices. Springer, Wien, NewYork, (1984).[24] Muscato, O. and Di Stefano, V. Local equilibrium and off-equilibrium thermoelectriceffects in silicon semiconductors. J. Appl. Phys. (2011) 110(9): 093706.[25] Di Stefano, V. and Muscato, O. Seebeck Effect in Silicon Semiconductors. Acta Appl.Math. (2012) 122(1): 225-238.[26] Muscato, O. and Di Stefano, V. Electro-thermal behaviour of a sub-micron silicondiode. Semicond. Sci. Tech. (2013) 28(2): 025021.[27] Ramayya, E.B. and Knezevic, I. Ultrascaled Silicon Nanowires as Effi-cient Thermoelectric Materials, Proceed. IWCE 2009, art. no. 5091160, DOI:10.1109/IWCE.2009.5091160, (2009)[28] Aksamija, Z. and Knezevic, I. Thermoelectric properties of silicon nanostructures, J.Comput. Electron. (2010) 9: 173-179.111119

English

https://upcommons.upc.edu/bitstream/2117/192694/1/Coupled-2013-100_Coupled%20quantum-classical.pdf

Coupled quantum-classical transport in silicon nanowires

Abstract

Similar works

Full text

Available Versions

UPCommons

UPCommons. Portal del coneixement obert de la UPC