International Journal of Advance Engineering and Research Development APPLICATION OF MONTE CARLO METHOD FOR DEVICES SIMULATION

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1 Scietific Joural of Impact Factor (SJIF): 5.71 Iteratioal Joural of Advace Egieerig ad Research Developmet Volume 5, Issue 05, May e-issn (O): p-issn (P): APPLICATION OF MONTE CARLO METHOD FOR DEVICES SIMULATION Rajesh Kumar 1, Ajaa Kumari 2, Taru Kumar Dey 3 1 Uiversity Departmet of Electroics, B.R.A Bihar Uiversity, Muzaffarpur Departmet of Physics, N.D.College, Purea Post Graduate Departmet of Physics, L S College, Muzaffarpur Itroductio The Mote Carlo method is a umerical statistical method that uses probability to solve physical ad mathematical problems. This approach is well suited for simulatio of physical pheomea associated with the stochastic processes. I fact, MC method was applied to some problems i the eutro trasport ad the statistical physics before it was applied to the carrier trasport i semicoductor devices [1, 2]. MC simulatios o trasport properties are based o the descriptio of particle motio. The progress i plasma simulatios is helpful for the semicoductor device simulatio [3]. The MC simulatio of carrier trasport has had a great progress over the past two decades. Researchers have implemeted trasport mechaisms i the device simulatio, icludig ew scatterig processes, boudary coditios, electrostatic self-cosistecy, more comprehesive models, more efficiet simulatio algorithms, etc. The MC approach is regarded as the most importat approach for the simulatio of ultra-scale devices uder the various electric field coditios. A more accurate simulatio is the itroductio of full eergy electroic bads extracted by empirical pseudopotetial calculatios [4]. The first multi-valley MC simulatio with the parabolic bad, a sigle logitudial acoustic (LA) phoo, ad six fixed-eergy itervalley phoos was itroduced i ref. [5]. Ref. [6] cosidered the o-parabolic bad ad slightly altered phoo deformatio potetials. Novel deformatio potetials, which more closely match the available data o electro diffusio i silico, are itroduced i ref. [7] a few years later. This phoo model was the widely refereced review of the MC simulatio, ad it became the set of phoo eergies ad deformatio potetials most ofte employed i the literature over the past two decades. Scatterig with itervalley phoos are itroduced by other workers [8]. The full bad MC simulatio of silico, computed from empirical pseudopotetials, is firstly itroduced i by Tad et al., [9]. They used the simple phoo model of LA phoos, six fixed itervalley phoos as ref. [4], ad the deformatio potetials of ref. [6]. The impact ioizatio i a full bad MC simulatio with the multi-valley deformatio potetials of ref. [10] was itroduced by Sao et al. [6]. Realistic MC device simulatios usig self-cosistet full bad were first performed by Fischetti et al. [11]. They also make the distictio betwee logitudial ad trasverse acoustic (TA) phoo scatterig, usig a simple aalytic dispersio for both LA ad TA. Ref. [11] poited out the defiitio of eergy valleys i the full bad simulatio ad used two phoo potetials, i.e. the fixed-eergy optical phoo ad the LA phoo icludig dispersio. The most sophisticated MC simulatio for carrier trasport i silico was performed by ref. [12] ad ref. [13]. They employed the full phoo dispersio obtaied from a adiabatic bod-charge model ad the full bad computed from empirical pseudopotetials. The electro-phoo scatterig rates were calculated as a fuctio of wave vector ad eergy i cosistecy with the phoo dispersio ad the bad structure. Most MC simulatios foud i practice today employ full eergy bads, yet scatterig rates ad eergy exchage with the lattice are still computed with simplified phoo dispersio model. Phoo eergies ad deformatio potetials i most frequet use are those origially itroduced i ref. [7]. MC simulatio is a huge computatioal simulatio system that deals with the radom evets. Especially, the free flight time occupy a large part of the CPU time. Reductio of CPU time for this part of the simulatio is a topic issue i MC simulatio. Borsari employed the step scatterig method to improve the simulatio time [14]. Kato proposed to optimize the value of self-scatterig, ad the CPU time was further reduced sigificatly [15]. I this study, we employed the MC model which uses aalytical descriptios for both the electro bad structure ad the acoustic phoo dispersio relatioship, whe the effect of heat geeratio is icluded i the simulatio. Procedure of Mote Carlo Method Geeral processes of the MC simulatio for carriers trasport ad scatterig i semicoductors devices have bee well described [16]. This sectio provides a brief itroductio for MC algorithm. Esemble MC method used i this work preselects several tes of thousads super-particles to represet the mobile carriers iside the devices. This umber is limited by computatioal costraits, but good simulatio results ca be obtaied if the umber of super-particles ad simulatio time are larger eough. The particles are iitialized with thermal eergy distributios by expressio 3k B T/2 r, r is a radom umber uiformly distributed betwee 0 ad 1, ad with radomly orieted wave vector. Whe the simulatio is started, the particles are allowed to drift for short free flight time ( ), which is shorter tha the average time betwee collisios, the oe process of scatterig is selected. The selectio of scatterig mechaism ca be made i All rights Reserved 400

2 a way that the scatterig rate compare to the total of all scatterig rates ( ) idepedet of the carrier eergy. The free flight time ( ) of each particle ca be cosequetly determied by total scatterig rate ( ) ad a uiform radom umber (r 1 ) as [16] (1) The total scatterig rate ( ) is take to be larger tha the largest value of scatterig rate W T (E k ) to avoid a egative value of scatterig rate withi the selected eergy rage of carrier. The MC simulatios are ot suited for low-field carrier trasport, where the drift-diffusio method may be preferred. However, the MC method Figure1 Selectio of a scatterig mechaism algorithm flowchart. represets the most physically comprehesive simulatio approach for charge trasport i semicoductors. Drift Process Whe potetial eergy of carriers varies slowly as a fuctio of positio, drift process of carriers i semicoductor devices ca be treated semi-classically. Thereby, carriers ca be regarded as free particles with a effective mass. Based o the equatios of motio for carriers the chage i the wave vector durig the free flight time is obtaied by itegratig the equatio of motio with respect to time; thus [16], (2) where H is the total eergy of a carriers with a charge e give by (3) where E K is the kietic eergy of the carriers ad V(r) is the electrostatic potetial. If a electric field F is applied o a semicoductor device, Eq. (4.2) has a solutio as: (4) Scatterig Process I the scatterig process, firstly determied what scatterig mechaism by which a carrier is to be occurred, ad the idetify the carrier state after the scatterig. The selectio of a scatterig mechaism ca be made by usig fuctios E defied as [16]: All rights Reserved 401 (5)

3 which are the successive summatios of the scatterig rates ormalized with the maximum of sum of all scatterig rates. is idetical to the parameter defied by (6) ad is the total umber of scatterig mechaisms. A scatterig mechaism for carriers with eergy E k is selected by E E is geeratig a radom umber r 2 lyig betwee 0 ad 1, ad comparig r 2 to ; if the fuctios satisfied the coditio as follows: (7) -th scatterig mechaism is chose. The Pauli's exclusio priciple is ot take ito accout i Eq. (7), because the carrier occupacy i the fial states is igored. Selectio steps for scatterig are described i the flowchart show i Figure 1. Velocity Calculatio I simulatio of semicoductor device, MC simulatio is equivalet to solvig the Boltzma trasport equatio. Distributio fuctio ca be calculated by the mea velocity of carriers, ad eergy ca be calculated whe the flight time of carriers i each volume elemet of k-space is accumulated. This process demads a large amout of memory to accumulate the data i k-space. However, it is ot ecessary to do this, due to the mea values of carrier velocity ad carrier eergy ca be calculated directly by moitorig each carrier flight ad the takig a average over all flights. The istataeous carrier velocity is formed by [16] k k (8) therefore, the mea velocity of carrier durig flight time ca be formed as (9) where E k ad k are small icremets of the carrier eergy ad carrier wave vector durig flight time, respectively. Substitutig Eq. (4) ito Eq. (9), the...(10) Makig use of the mea velocity of carrier durig flight time give by Eq. (10), the mea velocity of carriers durig the total simulatio time T is give as (11) where E f is the eergy of carrier at the ed of the flight ad E i is the eergy of carrier at the start of the flight. The summatio has to be made for all free flights. Eq. (11) shows that the eergy icremet durig each free flight time. The same reasoig leads to mea eergy of carrier beig give as follows where <E> is give to a good approximatio by (12) Mote Carlo Device Simulatio MC method for device simulatio has similar procedure with the EMC simulatio. Oly a few elemets have to be added for MC device simulatio. Carriers spread i a boudary less bulk semicoductor; however, trasports of carrier are restricted by the boudary coditio. Therefore, to set up suitable boudary coditios, it is ecessary for carriers to reach the surface of the device [16]. Carriers should either be exit or eter the ohmic cotact area of the device or might reflected at the isulator surface of the device durig the simulatio. Self-cosistetly potetial ad electric field calculatio with the distributio of carriers through the solutio of the Poisso equatio with appropriate boudary coditios is other thig to be take ito accout i device simulatio. Boudary coditios applied to the carrier motio ad the Poisso equatio must be cosistet each other. Therefore, the calculatio of carrier motio with suitable boudary coditios, the self-cosistet Poisso calculatio with charge distributio, ad the treatmet of All rights Reserved 402 (13)

4 associated with the delete, exit or etrace of carrier through the surface of the device are ecessary arragemets i the Mote Carlo device simulatios. Figure 2 shows a typical flowchart of the Mote Carlo device simulatio. The geometry of the device, the material compositio, layer structure, the apply voltage, the dopig profile, ad the cotact regios are specifies i step physical system accordig to the data give by the user. Figure 2. Basic Mote Carlo algorithm flowchart. The iitial carrier distributio i real space ad k-space, ad the iitial potetial profile i the device are specifies i the subroutie iitial coditio. The umber of carrier is always varyig durig the simulatio because carriers are exit or eter i ohmic cotact regio of device. The profile of carrier desity calculated from particle distributio i subroutie charge distributio. The profile of the carrier desity obtaied is trasferred to the subroutie potetial ad electric field for the potetial ad electric field calculatio. The role of each subroutie is described i followig subsectios. Iitial Coditio At the start of the simulatio, the iitial coditio of device as the particle distributio i real space ad k-space, ad the potetial profile are specifies i the subroutie iitial coditio. The carriers are usually distributed i spatial accordace with the desity profile of the correspodig dopig cocetratio. The particles may be distributed spatially accordig to the desity profile of carriers for save computig time, which is computed i advace by a device simulatio based o the drift-diffusio method. The iitial distributio of carrier s eergy is determied by radom umbers based o the assumptio that the eergy of carrier is early at the thermal equilibrium at the start of the simulatio. Thus, the eergy of each carrier E k is calculate by [16]...(14) where k B is the Boltzma costat, T is the lattice temperature (assumed to equal the carrier temperature), r is radom umber uiformly distributed betwee 0 ad 1. The wave vector k of carrier ca be determied by the Ek-k relatio based o the eergy value give by Eq. (13). Due to the eergy of carriers is located at miimum poit of real bad structure, the simple spherical ad parabolic bad ca be assumed, the k vector is determied by the relatio...(15) For o-parabolic bads, the eergy of carrier is correct use the Ek-k relatio give by Eq. (2). The, Eq. (14) has ew form as All rights Reserved 403

5 ...(16) For ellipsoidal, parabolic bads, the compoets of k are obtaied by a similar procedure usig Eq. (15). Charge Distributio The desity profile of carrier is directly related to the particle distributio i the device. The calculatio of the desity profile of carrier is simple ad is based o coutig the umber of particles for each grid poit. The simplest earest-grid-poit method is usually employed i the MC device simulatio, i which the desity profile of carrier at a grid poit (i, j) is calculated from the total umber of particles i the cell surroudig the grid poit as show i Figure 3. Sice the particles are regarded as super-particles i the calculatio, the carrier desity is obtaied as [16]...(17) where N pp is the umber of carriers per super-particle, N(i, j) is the umber of particle i the cell (i, j), ad x y is the square of the cell for the case of the two-dimesioal device. There is ievitably statistical oise i the distributio of carrier profile sice the umber of particles employed is rather limited. The statistical oise may lead to umerical istability i some cases. Such statistical oise ca be avoided usig the cloud-i-cell method. I the cloud-i-cell method, the carrier associated with a particle is regarded as a cloud of carrier spread spatially. We report a brief descriptio of the cloud-i-cell method i follow. The fiite differece mesh is cosidered with the odes located at (x i, y j ). The costat spatial step i the x- directio ad y-directio are deote by x ad y, respectively. The, if (x, y) the poit coordiates i which oe wats to compute the desity of carrier, with x i < x < x i +1 ad y i < y < y i +1, the desity of carrier is compute i the followig way...(18) where A(i, j) = x y. The cloud-i-cell method do exist that avoid the problems of self-forces but they are ecessary whe deal with heterostructures ad the spatial step is ot regular. The cloud-i-cell method reduces the amplitude of fluctuatio i the desity profile of carrier durig the simulatio because of the spreadig of the carrier cloud. However, whe the cloud-i-cell method is applied to a device with abrupt heterojuctio, the carrier desity obtaied by the cloud-i-cell scheme may be over-estimated o oe side of the juctio ad uder-estimated o the other side. Figure 3 Cell ad grid All rights Reserved 404

6 Solutio of Poisso Equatio MC device simulatio requires potetial profiles extracted from self-cosistet solutio of the Poisso equatio. Aalytical solutio for the Poisso equatio is hardly accepted i a realistic device structure with appropriate boudary coditios, due to the potetial profile has to be determied for a large umber of charged particles. Fiite differece scheme of the Poisso equatio with oe- or two-dimesioal form is a effective method amog various umerical methods. Poisso equatio ca be solved by the followig form (19) There are two sources of charge i Eq. (19): mobile charge ad fixed charge. Mobile charges are electros ad holes, whose desities are represeted by ad p. Fixed charges are ioized door ad acceptor atoms whose desities are represeted by N D ad N A, respectively. is the permittivity of material. The subscripts (i, j) deote the (i, j)-th grid o the x-y plae. The discretizatio of the Poisso will give a algebraic system to solve if the two-dimesioal regular fiite-differece grid is applied. But this method is quite complicated to solve, because the boudary coditios are difficult to implemet i a geeric simulatio. Furthermore, this algebraic system is cosumig from the grid view of computer memory. I this sectio, o-statioary Poisso equatio will be itroduces. This equatio is easy to solve ad ca be implemeted i a geeral umerical cotext with robust ad umerical schemes. The form of the o-statioary Poisso equatio is show i the followig (20) where the variables (, p, N D, N A, ) have the same meaig as above. k S is a costat givig the right dimesios of the potetial term t. The solutio of this equatio is similar with those from classical Poisso equatios described i the precedet paragraph. Furthermore, both solutios have the same iitial potetial coditios ad the same boudary coditios. Therefore, oce Eq. (20) is umerically solved, classical Poisso equatio will be easily solved by simply gettig the solutio of the o-statioary Poisso equatio for big fial time. I the cotext of fiite differece method, same umerical scheme is obtaied applyig fiite-differece method of derivatives to the o-statioary Poisso equatio. I the fiite differece method, the value of the potetial o the grid poits ca be discretized o a equally spaced mesh as (21) where the x ad y are the spatial mesh size. The i, jis the potetial computed at time t = t i + t, i the poit (i, j). Applyig these expressios to the o-statioary Poisso equatio, oe gets the followig umerical form [17] (22) The preseted scheme is valid oly i the case of homogeeous case, but it is easy to expad it to the heterogeeous structures. Due to the fact that the iitial coditios ad the boudary coditios are icluded i the preseted scheme, it is easy to implemet this solutio i simulatio of semicoductor devices. Electric Field Calculatio It is easy to get the solutio of the electric field of device system by the solutio of the static Poisso equatio or the o-statioary Poisso equatio. The geeric defiitio of the electric field is as follows [16] (23) So, i the cotext of fiite-differece method, the solutio of electric field i the two dimesioal cells of the grid as follows:...(24) These simple expressios are used i simulatio. Although the expressios are simple, but the result values of electric field are accurate ad All rights Reserved 405

7 Coclusio Details of the applicatio of the Mote Carlo methods to device simulatio have bee described i this chapter. Calculated results o carrier trasport or performace of devices may ot precisely agree with experimetal results due to ucertaity i the kowledge of material parameters or scatterig mechaism. However, the error i the MC simulatio of semicoductor devices is acceptable for may cases. The preset approach is more tha a order of magitude faster tha the full-bad device simulatio, ad is accessible o moder computers. Mote Carlo simulatios with oparabolic bads ca be applied to the egieerig of low-voltage aoscale devices ad materials, where detailed kowledge of carrier trasport icludig the electro-phoo iteractio is required. Refereces [1] B. T. Browe, J. J. H. Miller, Numerical Aalysis of Semicoductor Devices ad Itegrated Circuits, Boole Press, Dubli [2] M. Kurata, Numerical Aalysis for Semicoductor Devices, Lexigto Press, Lexigto, Mass [3] M. S. Mock, Aalysis of Mathematical Models of Semicoductor Devices, Boole Press, Dubli [1] Yu. A. Shreider, The Mote Carlo Method, Pergamo, Oxford, 1956 [2] K. Bider, Applicatio of the Mote Carlo Method i Statistical Physics, Spriger,Berli, 1984 [3] R. W. Hockey ad J. W. Easwood, Computer Simulatio Usig Particles, Mc Graw-Hill, New York [4] J. Y. Tag ad K. Hess, Impact ioizatio of electros i silico (steady state), J.Appl. Phys. 54, o. 9, 5139 (1983). [5] C. Caali, C. Jacoboi, F. Nava, G. Ottaviai, ad A. Alberigi-Quarata, Electro drift velocity i silico, Phys. Rev. B 12, o. 4, 2265 (1975). [6] C. Jacoboi ad L. Reggiai, The Mote Carlo method for the solutio of charge trasport i semicoductors with applicatios to covalet materials, Rev. Mod. Phys. 55, o. 3, 645 (1983). [7] R. Bruetti, C. Jacoboi, F. Nava, ad L. Reggiai, Diffusio coefficiet of electros i silico, J. Appl. Phys. 52, o. 11, 6713 (1981). [8] T. Yamada, J.-R. Zhou, H. Miyata, ad D. K. Ferry, I-plae trasport properties of Si/Si1 xgex structure ad its FET performace by computer simulatio, IEEE Tras. Electro Devices 41, 1513 (1994). [9] J. Y. Tag ad K. Hess, Impact ioizatio of electros i silico (steady state), J. Appl. Phys. 54, o. 9, 5139 (1983). [10] C. Caali, C. Jacoboi, F. Nava, G. Ottaviai, ad A. Alberigi-Quarata, Electro drift velocity i silico, Phys. Rev. B 12, o. 4, 2265 (1975). [11] M. V. Fischetti ad S. E. Laux, Mote Carlo aalysis of electro trasport i small semicoductor devices icludig bad-structure ad space-charge effects, Phys. Rev. B 38, o. 14, 9721 (1988). [12] P. D. Yoder ad K. Hess, First-priciples Mote Carlo simulatio of trasport i Si, Semicod. Sci. Techol. 9, 852 (1994). [13] T. Kuikiyo, M. Takeaka, Y. Kamakura, M. Yamaji, H. Mizuo, M. Morifuji, K.Taiguchi, ad C. Hamaguchi, A Mote Carlo simulatio of aisotropic electro trasport i silico icludig full bad structure ad aisotropic impact-ioizatio model, J. Appl. Phys. 75, o. 1, 297 (1994). [14] V. Borsari ad C. Jacoboi, Mote Carlo Calculatios o Electro Trasport i CdTe, Phys. Status Solidi B 54, 649 (1972). [15] K. Kato, Hot-carrier simulatio for MOSFETs usig a high-speed Mote Carlo method, IEEE Tras. Electr. Dev., ED 35, 1344 (1988). [16] E. Pop, R. W. Dutto, ad K. E. Goodso, Aalytic bad Mote Carlo model for electro trasport i Si icludig acoustic ad optical phoo dispersio, J. Appl. Phys. 96, 4998 (2004). [17] K. Tomizawa, Numerical simulatio of submicro semicoductor devices. Artech House, All rights Reserved 406