Numerical Solution of Hydrodynamic Semiconductor Device Equations Employing a Stabilized Adaptive Computational Technique

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1 Numerical Solutio of Hydrodyamic Semicoductor Device Equatios Emloyig a Stabilied Adative Comutatioal Techique YIMING LI * ad CHUAN-SHENG WANG a Natioal Nao Device Laboratories, ad Microelectroics ad Iformatio Systems Research Ceter Natioal Chiao Tug Uiversity a Deartmet of Mathematics, Natioal Tsig Hua Uiversity * Corresodig address: 11 Ta-Hsueh Rd., Hsichu city, Hsichu 3 TAIWAN Abstract: - I this aer, we geeralie our roosed earlier comutig method [1-11] to solve hydrodyamic semicoductor device equatios. For submicro MOSFET devices, we simulate their temerature distributio by solvig carrier eergy balace equatio with adative comutatioal techique. This robust method based o: (1 the fiite volume (FV discertiatio scheme; (2 the mootoe iterative (MI algorithm; (3 a osteriori error estimatio; ad (4 the 1-irregular mesh refiemet; is successfully develoed ad imlemeted. Numerical results ot oly have a good agreemet with hysical heomea but also demostrate that our methodology has good comutatioal efficiecy. Covergece roerty for arbitrary iitial guesses for the begiig of simulatio is also reorted to show the robustess of the method. Key-Words: - Hydrodyamic Equatios, Semicoductor Device Simulatio, MOSFET, Adative Comutatio, Error estimatio, Mootoe Iterative Techique, Carrier Temerature 1. Itroductio For microelectroics, the rogress of semicoductor fabricatio techology for the advaced metal oxide semicoductor field effect trasistor (MOSFET has bee of great iterests [1-2] i recet years. The device chael legths are so small that olocal effects ad source/drai egieerig become more imortat for the device characteristics ad erformace. Desite sigificat advaces over the last decade, TCAD (techology comuter-aided desig must rogress eve more dramatically durig the 2s if simulatio is to live u the high exectatios of the user commuity. The TCAD aroach [2] rovides a direct alterative to study the itrisic ad extrisic electrical behavior for these MOSFET structures at a very fudametal hysical level. I recet years, esecially for the ultra-small MOSFET device, it has become a very imortat tool i the develomet of ew devices ad fabricatio techologies [2-15]. The aim of our works is to develo a efficiet ad itelliget hysical-based TCAD tools for the day-to-day micro ad ao advace CMOS desig. To study the effects metioed above for ultra-short chael MOSFET devices, a eergy balace equatio should be solved umerically for carrier temerature distributio [1-13]. Simulated results ca be alied for the further otimal desig ad characteriatio. To study the variatio of carriers temerature ad their hysical mechaism iside a submicro MOSFET device, a hydrodyamic model icludig at least the eergy balace equatio i a device simulatio should be solved, ad this aroach has bee received may otices. Various simulatio aroaches have bee roosed for the umerical solutio of this equatio efficietly [3-15]. The raid variatio of electro temerature withi very small regios leads to a sigificat difficulty i semicoductor device simulatio. This electro temerature model roblem is so-called a sigular oliear boudary value roblem. It has the umerical stability ad the covergece roblems whe solvig the system of oliear algebraic equatios arisig from the discretiatio of such eergy balace equatio. The well kow Scharfetter-Gummel-Tag (SGT scheme [13-15] for solvig this roblem holds oly for very fie mesh artificially. However, it goes without sayig that the arragemet of the fie mesh to fit the SGT assumtio is surely a difficult task. I this aer, we further geeralie our roosed earlier comutig algorithm for the itelliget

2 umerical solutio of semicoductor device eergy balace equatio usig ot oly mootoe iterative algorithm but also adative mesh refiemet rule. First of all, we solve a set of drift diffusio (DD equatios with our develoed device simulator earlier [3-9]. The comuted hysical quatities, such as electro cocetratio, electrostatic otetial, ad electro curret are used as the iut data for carrier temerature simulatio. Accordig to our methodology [3-11], the eergy balace equatio is trasformed ito aother artial differetial equatio (PDE with havig a self-adjoit form ad the discretied usig adative fiite volume method with 1-irregular ustructured mesh. The adative FV discretied eergy balace equatio leads to a system of oliear algebraic equatios with a diagoal domiate roerty. We solve the oliear system by alyig a mootoe iterative method directly. With the develoed carrier temerature simulator, we study basic hysical eergy variatio mechaism for a submicro NMOSFET uder high bias coditio. The umerical results ad covergece roerty of the algorithm are also reorted i this work. This aer is orgaied as follows. I Sec. 2, we briefly state eergy balace equatios associated hysical models. I Sec. 3, we reset the overall adative comutatioal rocedure for the carrier temerature calculatio. I Sec. 4, simulatio results for a submicro N-MOSFET device are reseted to demostrate the robustess ad efficiecy of the method. Sec. 5 draws the coclusio ad suggests the future work. the electro ad hole cotiuity equatios. The Eqs. (4 ad (5 are electros ad holes curret equatios, resectively. The ukow φ = φ(x,y to be solved is the electrostatic otetial; ad are electros ad holes cocetratios. The fuctio D = ( N + D N A, i Eq. (1, is the secified ioied et doig rofile, ad R = R(, is the recombiatio rate for electros 19 ad holes [1]. The quatity q = C is the elemetary charge; ε s = 11.9ε is silico + ermittivity. The N, ad N are ioied door ad accetor imurities, ad D A 14 ε = F / cm is the ermittivity i vacuum. The D, D, µ, ad µ are electro ad hole diffusio coefficiets ad mobility fuctios, resectively [1]. This DD model cotais Eqs. (1 (5, ad was solved successfully i our earlier works [3-9]. y + x Gate Oxide - Chael Drai + 2. A Hydrodyamic Model We state a hydrodyamic (HD model that icludes a drift diffusio (DD model ad eergy balace equatios for electros ad holes. The DD model was derived from the Maxwell s equatio as well as charge coservatio law. q φ = ( + D, εs (1 1 R(,, q (2 1 J q = R(,, (3 J = qµ φ + qd, (4 J = qµ φ qd. (5 Eq. (1 is so-called the Poisso equatio. The Eqs. (2 ad (3 derived from the charge coservatio law are Substrate Fig. 1. A two-dimesioal cross sectio domai for a submicro N-MOSFET device. The HD model cotais eergy balace equatios for electros ad holes. It will be solved for studyig the carrier temerature variatio. ( ω ω S = J E, (6 τ ( T w w ( ω ω S = J E. (7 τ ( T J J = ω Q, q q (8 J J S = ω + kbt + Q, q + q (9 S + k BT +

3 where the otatios as well as the hysical models are followed our covetioal symbols ad ca be foud i [1-11]. I Sec. 3, we aly the adative FV ad MI methods to solve the above HD equatios. With those comuted hysical quatities from the DD equatios, the eergy balace equatios are the solved to obtai the carrier temerature distributio over the device structure. As show i Fig. 1, the HD equatios (1-(9 are subject to mixed tye boudary coditios i a two-dimesioal simulatio domai. O the left ad right sides, the homogeeous Neuma tye boudary coditio is cosidered. O the, Gate, Drai ad Substrate cotacts, the Dirichlet tye boudary coditio is alied [1,3-15]. The roosed adative solutio rocedure for the umerical solutio of these equatios will be discussed i Sec. 3. Solve drift diffusio model with adative fiite volume ad mootoe iterative methods Comute hysical quatities for carrier temerature simulatio Solve carrier eergy balace equatios with fiite volume ad mootoe iterative methods Comute carrier temerature distributios Temerature coverged? Error > TOL? No Post-rocessig No Yes Mesh refiemet Fig. 2. A overall simulatio rocedure for a HD semicoductor device simulatio. 3. Adative Simulatio Techique As show i Fig. 2, with the comuted results from the DD model i advaced, we solve eergy balace equatios (6-(9, subsequetly. To calculated electrostatic otetial, electric field, carrier cocetratios, ad curret flows, Eqs. (1-(5, the drift diffusio equatios are solved with the Gummel s decouled algorithm [3-9,14-15] ad mootoe iterative method [3-9]. Here we briefly state the flowchart ad umerical methods for the solutios of drift diffusio model which ca be foud i our earlier works [3-9]. The well-kow Gummel s decouled method is that the device equatios are solved sequetially For the umerical solutio of semicoductor device DD model, the 1 Poisso s equatio is solved for ( g φ + give the revious states u (g ad v (g. The electro curret ( g φ cotiuity equatio is solved for u (g+1 give ad v (g. The hole curret cotiuity equatio is solved for v (g+1 ( g give φ ad u (g. The suerscrit idex g deotes the Gummel s iteratio loos. Each the decouled PDE is solved adatively [3-9]. Decouled equatio is discretied with the FV method. The corresodig oliear system is solved with the MI algorithm. Whe a coverged solutio is comuted, we erform error estimatio of the results for all elemets. If the solutio does ot satisfy the secified stoig criterio (TOL, we will ru the mesh refiemet ad reeat the comutatio. The results obtaied from the DD model are used as the iut data for the umerical solutios of the carrier balace equatios (6-(9. The covetioal algorithm for the umerical solutio of electros ad holes eergy balace equatio i semicoductor device simulatio cosists of: (i alyig the FV (or so-called the fiite box method to discretie the equatios; (ii usig Scharfetter-Gummel-Tag scheme, a exoetial fittig liked algorithm, [1-15] for the electro ad hole eergy balace equatios to costruct a system of oliear algebraic equatios; (iii solvig the oliear system with Newto s iteratio method; ad (iv reeatig the ste (iii util the solutio coverged. Our simulatio algorithm, as show i Fig. 2, ot oly relaces Newto s iteratio by the MI method but also alies the adative comutig method [3-9] for a osteriori error estimatio ad automatic mesh refiemet. The system of oliear algebraic equatios arisig from adative FV discretiatio o a 1-irregular ustructured mesh, as show i Fig. 3, forms to a oliear system (1, where the matrix A ca be roved that it is still a M-matrix. A = -F(, (1 where is the ukow vector, F is the oliear vector form, ad A is the corresodig matrix, resectively. Based o our revious observatios

4 [1-11] the oliear roerty of the right had side of the eqatios (6-(9 is the mootoe fuctios i its ukow, we have a similar result cocerig the well-osed roblem for the adatively discretied eergy balace equatios. Furthermore, the mootoe iterative scheme for the corresodig discretied oliear system (1 is of the followig form [3-11]: ( m+ 1 ( m ( m ( m ( D + λ I Z = ( L + U F( Z + λiz, (13 where is the ukow vector, F is the oliear vector form, ad D, L, U, ad I are diagoal, lower triagular, uer triagular, ad idetity matrices, resectively. The mootoe iterative arameter λ is determied ode-by-ode deedig o the device structure, doig cocetratio, bias coditio, ad oliear roerty of each decouled equatio. Dash lie forms the cotrol volume with resect to ode (x i (x i +1 (x i+1/2 +1/2 : Irregular oit : Itegral ath for the cotrol volume Fig tyes 1-irregulr mesh structure. For a certai tye of the 1-irregular mesh structures, as show i Fig. 4, the oliear system i comoet-wise form ad the etries of the matrix A is as followed. j 1 j 1 j + 1 j + 1 i 1, j i 1, j j j i + 1 / 2, j + 1 / 2 i + 1 / 2, j + 1 / 2 i + 1, j = F i + 1, j ( j, (11 I Eq. (8, it ca be verified that all coefficiets are oegative ad satisfy the coditios ξ ξ j i + 1, j i 1, j j + 1 j 1 ξ, (12 i + 1 / 2, j + 1 / 2 for all discretiatio idex (j i the device domai. For other cases, we have the similar results ad roerties; therefore we ca rove the followig result immediately. Theorem 1 The oliear system (1 arisig from above adative fiite volume discretiatio scheme has at most a solutio. + (x i-1 (x i (x i+1 Dash lie idicates the ath of lie itegral (x i -1 Fig. 4. Oe of the 13 tyes of the 1-irregular mesh structure. The MI method alied here for semicoductor device temerature simulatio [3-11] is ot oly ready for aralleliatio but also a global method (i.e., it does ot require a sufficietly accurate iitial guess to begi with the solutios. Oce the aroximated solutios are comuted we do a error estimatio ad the the mesh refiemet followig the rule of the 1-irregular ustructured mesh [3-9]. 4 Simulatio Results ad Discussio I this sectio, to test the robustess of the method we reset here the comuted electro temerature distributio for a.25 µm N-MOSFET device at high bias coditio. Similarly, holes temerature ca be calculated with the roosed method. The required electro cocetratio, electrostatic otetial, electric field, ad electro curret desity are comuted directly from the DD model i advace. This tested device has a Gaussia distributio doig

5 rofile. The gate oxide thickess equals 7 m; the ratio of device width ad chael W/Leff equals 4/.25 ad VBS = V. The rofiles have their maximum values 12/cm3 i both the source ad dra where the substrate doig is of 116/cm3. T (k 4 3 Drai Gate.8 2 Substrate Drai Gate Fig. 7. Comuted electro temerature of the N-MOSFET with the iitial mesh Substrate Fig. 5. Iitial mesh for the adative comutatio. Fig. 5 shows the iitial mesh alies to simulate the N-MOSFET device temerature distributio. It cotais 256 elemets. The alied voltage for this device is VDS = VGS = 2.V, resectively. The Fig. 6 shows a fial refied mesh for the umerical solutios of the carrier eergy balace equatios. I our calculatio exeriece, it takes about 7 refiemets to satisfy the secified stoig criterio. This refied mesh cotais about 48 elemets i the simulatio domai. With the error estimatio, the ustructured mesh is geerated automatically. Fig. 7 resets the comuted electro temerature with the iitial mesh. The result is a rather rough data. After the error estimatio ad the 1-irregular mesh refiemet; we fid, as show i Fig. 8, the result is excellet ad has its hysical meaig that the temerature attais its maximum ear the chael ad the drai side. Comared with the covetioal structure mesh, our adative mesh ad solutio techique have their robustess ad imrove the quality of the solutio sigificatly. T (k Drai Gate Fig. 8. Electros temerature with the fial refied mesh. This tested.25µm N-MOSFET devices is with VDS = VGS =2.V Fig. 6. A refied mesh for the adative comutatio. With the develoed HD simulator, we have calculated the electro temerature for the covetioal N-MOSFET structure ad it is about 49 k at VDS = VGS = 2.V. The high temerature is due to the high electric field at the drai side ad ca be further imroved with the LDD drai egieerig [1,1-11]. As show i Fig. 9, we reset the global covergece roerty of the adative FV ad MI

6 algorithms for the umerical solutio of eergy balace equatios. It cofirms that the MI method for device simulatio has its robustess. Maximum orm error of T T (x,y = 3k for all (x,y T (x,y = 6k for all (x,y T (x,y = 9k for all (x,y Number of iteratios Fig. 9. Covergece behavior for ukow T i the mootoe iteratio loo. 5 Coclusio I this aer we have successfully geeralied our roosed earlier simulatio techique to calculate electro temerature by solvig a HD model adatively. This aroach relied o the MI method, a osteriori error estimatio ad 1-irregular mesh refiemet techique has bee develoed ad imlemeted. Numerical results for a submicro N-MOSFET show the robustess ad efficiecy of the method. I the future work, we la to exted this adative comutig method for advaced quatum trasort equatios simulatio. Ackowledgemet This work was suorted i art by the Natioal Sciece Coucil of Taiwa uder Cotract No. NSC M Refereces: [1] S. M. Se, Physics of Semicoductor Devices, 2d Ed., Wiley-Itersciece, New York, [2] R. Dutto, et al., Persectives o Techology ad Techology - Drive CAD., IEEE Tras. CAD, 19, 2, 2, [3] Y. L et al., Mootoe Iterative Method ad Adative Fiite Volume Method for Parallel Numerical Simulatio of Submicro MOSFET Devices, i Toics i Alied ad Theoretical Mathematics ad Comuter Sciece Ed. by V. V. Kluev, et al., WSEAS Press, Dec. 21, [4] Y. L et al., A Domai Partitio Aroach to Parallel Adative Simulatio of Dyamic Threshold Voltage MOSFET, Abst. Book Coferece o Comutatioal Physics 21, Aache, Se. 21,. O38. [5] Y. L et al., Mootoe Iterative Method for Parallel Numerical Solutio of 3D Semicoductor Poisso Equatio, i Advaces i Scietific Comutig, Comutatioal Itelligece ad Alicatios Ed. by N. Mastorakis, et al., WSES Press, July 21, [6] Y. L et al., Parallel Dyamic Load Balacig for Semicoductor Device Simulatios o a Liux Cluster, Tech. Proc th Iter. Cof. Modelig ad Simulatio of Microsystem, South Carolia, March 21, [7] Y. L A Mootoe Iterative Method for Semicoductor Device Drift Diffusio Equatios, WSEAS TRANSACTIONS ON SYSTEMS, 1, 1, 22, [8] Y. L et al., A ew arallel adative fiite volume method for the umerical simulatio of semicoductor devices, Comut. Phys. Comm., 142, 1-3, 21, [9] Y. L et al., A Novel Aroach for the Two-Dimesioal Simulatio of Submicro MOSFET's Usig Mootoe Iterative Method, IEEE Proc. Iter. Sym. VLSI-TSA, Taie Jue 1999, [1] Y. L A Novel Aroach to Carrier Temerature Calculatio for Semicoductor Device Simulatio usig Mootoe Iterative Method. Part I: Numerical Algorithm, Proc. 3 rd WSEAS Sym. Math. Methods Comut. Tech. Electrical Eg. (MMACTEE 21, Athes, Dec. 21, [11] Y. L A Novel Aroach to Carrier Temerature Calculatio for Semicoductor Device Simulatio usig Mootoe Iterative Method. Part II: Numerical Results, Proc. 3 rd WSEAS Sym. Math. Methods Comut. Tech. Electrical Eg. (MMACTEE 21, Athes, Dec. 21, [12] K. Blotekjaer, ''Trasort Equatios for Electros i Two-Valley Semicoductors,'' IEEE Tras. Elec. Dev., 17, 197, [13] T.-W. Tag, ''Extesio of the Scharfetter- Gummel Algorithm to the Eergy Balace Equatio,'' IEEE Tras. Elec. Dev., 31, 12, 1984, [14] J. W. Jerome, Aalysis of Charge Trasort: A Mathematical Study of Semicoductor Devices, Sriger-Verlag, New York, [15] M. Ieog, et al., Ifluece of Hydrodyamic Models o the Predictio of Submicrometer Device Characteristics, IEEE Tras. Eelc. Dev., 44, , 1997.