NUMERICAL SIMUALTION OF NANOSCALE

Transcription

1 NUMERICAL SIMUALTION OF NANOSCALE SEMICONUCTOR EVICES Rolad STENZEL, Wilfried KLIX eartmet of Electrical Egieerig Uiversity of Alied Scieces resde Friedrich-List-Platz 1, -169 resde Ja HÖNTSCHEL AM Saxoy LLC & Co. KG Wilschdorfer Ladstraße 11, -119 resde Germay Abstract evice modelig of ovel semicoductor devices requires adated hysical models which iclude quatum mechaical effects. The quatum hydrodyamic as well as the quatum drift diffusio model offers effective ossibilities for the simulatio of aoscale devices, articularly if tuelig rocesses aear. The models ca be imlemeted effectively i covetioal device simulatio systems. Keywords Naoelectroics, quatum devices, device simulatio, hetero structures 1. Itroductio Numerical device simulatio is a imortat rocedure for the desig ad otimizatio of ovel semicoductor devices. Advatages are the calculatio of the electrical behaviour before the fabricatio rocess, the calculatio ad visualizatio of ier-electroic values, which are ot measurable, the diagosis/fault-detectio i the techological rocess ad cosequetly the savig of costs. The advaced miiaturizatio i the semicoductor techology requires a ugrade of covetioal simulatio models because several quatum mechaical effects like tuelig through otetial barriers or article accumulatio i quatum wells aear i these aometer scaled structures. The quatum hydrodyamic (QH models, which are based o a quatum fluid dyamic model, offers exadig ossibilities for the uderstadig as well as the desig of such quatum sized semicoductor devices. The advatage of this model is its macroscoic character, because a descritio without kowledge of quatum mechaical details like iitial wave fuctio is obtaied. The derivatio of the full three-dimesioal QH trasort model, based o the Wiger-oltzma equatio with a shifted momet exasio, delivers the same coservatio law as the classical coversatio law, but the eergy desity ad the eergy balace equatio as well as the stress tesor have additioal quatum terms [1]. The cosequece of these additioal quatum terms is the extesio of the classical hydrodyamic model for the semicoductor device simulatio by exressios i the trasort ad i the eergy balace equatios, which describe a iteral quatum otetial i the trasort equatio as well as a quatum heat flux i the eergy balace equatio. These additioal terms i the classical hydrodyamic model allow to describe a cotiuous electro ad hole distributio i a semicoductor device, accumulatios of carriers i otetial wells ad resoat tuelig of carriers, resectively. Microscoic/macroscoic models, which are based o the self-cosistet 1-dimesioal or -dimesioal solutio of Schrödiger ad Poisso equatio, deedig o the cofiemet of electros, deliver a cotiuity descritio of the electro desity distributio from the iitial wave fuctio as well, but they do ot iclude tuelig rocesses ad furthermore the solutio is very timecosumig []. O the other had a quatum hydrodyamic descritio takes ito accout quatum mechaical effects i all directios of a comlex structure at accetable comutatio times. I this aer the basic model equatios are rereseted together with the umerical solutio method. Thereafter results of umerical device simulatios are demostrated.. Simulatio model The basic equatios of the QH model are the Poisso equatio ( ( ϕ = q( + N N ε (1 s, : hole ad electro desity ϕ : electrostatic otetial N, N : ioized door ad accetor desity A ε S : ermittivity of semicoductor q: elemetary charge the cotiuity equatios (idex : holes, idex : electros J = q R G + ( t A

2 Numerical simualtio of aoscale semicoductor devices J = q R G + ( t J, J : curret desities R, G: recombiatio ad geeratio rate the trasort equatios J J = = qµ k k µ qµ µ ( ϕ λ Θ q( ( T ( ϕ + λ + Θ + q( ( T µ, µ : carrier mobilities λ, λ : quatum correctio otetials Θ, Θ : bad arameters, : diffusio coefficiets T, T : carrier temeratures k : oltzma costat the eergy balace equatios S = J k T S = J k T E k ( T T τ w τw T TL E k τ L k t ( T 1 ( R G qλ ( G R 1 q ( λ 1 ( ( R G + qλ ( G R + q ( λ w τ w S, S : eergy flux desities E: electric field stregth T L : lattice temerature τ w, τ w : eergy relaxatio times the eergy flux desity equatios S t k ( T t 1 5 k ( T + T J λ J t = κ + (8 q 5 k S = κ ( T T J + λ J (9 q κ, κ : thermal coductivities ad the equatios for the quatum correctio otetial + (4 (5 (6 (7 Further aroaches are ecessary for the carrier mobilities, the geeratio ad recombiatio rates, the diffusio coefficiets ad the eergy relaxatio times, which are almost material deedetly. Equatios (1 to (11 are solved self-cosistetly for the variables (ϕ,,, T, T, λ, λ. If equatios (1 ad (11 are eglected, that meas for λ, = λ =, the QH model ca be reduced to the covetioal hydrodyamic (H model. If the carrier temeratures are set to the lattice temerature ad equatios (6 to (9 are eglected, the quatum drift diffusio (Q model ad additioally for λ, = λ = the covetioal drift diffusio ( model ca be obtaied. Additioally to iclude self-heatig effects the heat flow equatio TL ( λ S TL + H = ρ c (1 t H: ower desity (H = J E λ S : semicoductor thermal coductivity ρ: desity c: effective heat caacity ca be solved together with the other equatios for calculatio of the lattice temerature distributio. The solutio of the equatios are achieved by a successive algorithm (so called Gummel algorithm as rereseted i Fig. 1 (without heat flow equatio. For the solutio of the artial differetial equatios a box method is used. The resultig o-liear equatio systems are solved by the NEWTON-method ad the corresodig liear equatio systems by recoditioed gradiet methods. All models are imlemeted full threedimesioal i the rogram system SIMA []. Start solutio Poisso equatio Quatum correctio otetial equatios Cotiuity ad trasort equatios γ h λ = (1 6 q m Eergy balace ad flux desity equatios γ h λ = (11 6 m q m, m : carrier effective masses h : reduced Placks costat γ, γ : quatum correctio coefficiet. Fig. 1 Covergece YES Successive solutio algorithm NO

3 I (ma/mm Numerical simualtio of aoscale semicoductor devices. Results Resoat tuelig devices show arts of egative differetial resistace i their characteristics ad offer ew ossibilities for RF- ad logic alicatios. The double barrier is the basic structure, the so-called resoat tuelig diode (RT. The device structure of the RT, which was used for the simulatio, is rereseted i Fig. together with the doig desities. This structure was exerimetal ivestigated i [4]. Fig. shows the calculated curret-voltage characteristic by the QHmodel. Additioally the results of a simulatio by a trasfer matrix method (TM-method ad measuremet data [4] are iserted. The calculated characteristics agree quatitatively with the exerimetal values ad show a egative differetial resistace. shows the electro desity at V GS = -1V, V S =. Straightly uder the gate a otable electro ehacemet ca be detected due to the tuelig rocess. I Fig. 4 Electro desity of the MOSFET V GaAs 1 cm GaAs 1 cm GaAs 1 cm GaAs 1 cm m 5 m Al.4Ga.6As 5 m arrier GaAs 5 m Well Al.4Ga.6As 5 m arrier 5 m 5 m This results i a corresodig gate leakage curret which ca be see i the trasfer characteristics (Fig. 5 articularly i the cut-off regio. The covetioal model caot simulate this effect x x1 - V S =1V Fig. Structure of the RT x1-1. x x1-5 TOX=1.5m TOX=m Q-Model Fig. 7 J (x1 A/m QH-model measuremet data TM-method 1 V (V Curret-voltage characteristics of the RT I ovel MOSFET devices gate oxides thickess are reduced to a few aometers. I this case gate tuelig currets occur. For the iclusio of these effects withi the device simulatio quatum models are essetially. Fig. 4 Fig x x V GS (V -Model MOSFET trasfer characteristics (TOX: oxide thickess Quatum wires are oe-dimesioal electro gases which aear i quatum hetero structures as curret chaels or as itercoectios. Fig. 6 shows a simle ossibility to create a quatum wire. The differet bad arameters of GaAs ad Al. Ga.7 As result i a coductio bad discotiuity ad therefore electros are accumulated at the GaAs-side of the iterface. The calculated electro desity at thermodyamic equilibrium is rereseted i Fig. 7. The electro desity for differet edge legths a is deicted i Fig. 8. The simulatio shows that oly at small edges a oe-dimesioal electro gas occurs, otherwise the electro gases are slit-off ito differet arts.

4 Numerical simualtio of aoscale semicoductor devices y a x GaAs Al Ga As..7 a 18-1 cm -doig to study the ifluece of these effects o carrier distributios ad o static ad dyamic device characteristics of AlGaN/GaN-HFETs. Comarisos with exerimetal results demostrate the imortace of the cosideratio of olarizatio charges i the desig ad aalysis of itride-based HFET structures. Fig. 9 shows the device structure, used for the simulatio [5]. The effects of sotaeous ad iezoelectric olarizatio are icluded by a ositive iterlayer charge σ = cm - at the heterojuctio iterface. / 5 % /! /, Fig. 6 Quatum wire structure with rectagular cross sectio 5 /,! / = K E@ # # / = K E@ # / = # N &?!! # / = K E@ - (cm # / = K E@.4.45 x (µm y (µm.47 Fig. 9 GaN/AlGaN-heterojuctio FET Fig. 7 Electro desity distributio of the quatum wire - (cm a=6m a=1m a=16m γ =6.6 The calculated outut characteristics of the Al.5 Ga.75 N/GaN-HFET are rereseted i Fig. 1. I Fig. 11 the trasfer characteristic ad the corresodig trascoductace are lotted. The comarisos with exerimetal results show a good agreemet. 15 V (to=1v, V =-1V GS GS x (µm I (ma/mm 1 Fig. 8 Electro desity for differet edge legths a GaN-based heterojuctio field-effect trasistors (HFETs offers ovel caabilities for high-seed, highower ad high-temerature alicatios. The large iterest comes from the uique material roerties of the grou-iii-itrides. The resece of iteral strai may lead to the geeratio of large iezoelectric olarizatio fields. The sotaeous ad iezoelectric olarizatio causes additioal sheet charges at the heterojuctio iterfaces of the structures. Numerical simulatios have bee carried out 1 Fig. 1 Simulated ad exerimetal outut characteristics

5 Numerical simualtio of aoscale semicoductor devices I (ma/mm 15 1 V =8V S 1 g (ms/mm m Refereces [1] GARNER,C.L.: The quatum hydrodyamic model for semicoductor devices, SIAM J. Al. Math., vol. 54, o, , Ar [] PIGORSCH,C.-KLIX,W.-STENZEL, R.: Quatum wire slittig i aostructures, Microelectroics Egieerig, 4-44,. 5-, [] KLIX,W.: Numerische Simulatio elektroischer auelemete, er adere Verlag, 4. Fig V (V GS Trasfer characteristics ad trascoductace The RF-gais maximum stable gai ad maximum available gai (MSG/MAG as well as the curret gai (h 1 at differet gate legths are rereseted i Fig. 1. The cut-off frequecies are raged betwee f max = 59 GHz - 8 GHz ad f t = GHz - 7 GHz for L G =.4 µm -.1 µm, resectively. Gai (d 5 1 Fig. 1 MSG/MAG V S=V, V GS=-4.5V L (to=.1µm, L =.1µm G h f (GHz RF-gais at differet gate legths 4. Summary ifferet model levels for the simulatio of ovel semicoductor devices have bee rereseted. With the quatum hydrodyamic model it is ossible to take ito accout quatum mechaical effects i all directios of a aoscale device. The models are imlemeted i our simulatio system SIMA which is caable to calculate differet semicoductor devices ad device materials. The examles demostrate the fuctioality of the models. G [4] HENRIKS,A.M.P.J.-MAGNUS,W.-VAN E ROER,T.G.: Accurate modelig of the accumulatio regio of a double barrier resoat tuelig diode. Solid-State Electroics, 9,. 7-71, [5] STENZEL,R.-HÖNTSCHEL,J.-KLIX,W.-WIESZT,A.-EHTAH, R.-LEIER,H.: RF-behavior otimizatio of GaN-based HFETs by couled electrical-thermal simulatio. Proc. of 1th Worksho o Physical Simulatio of Semicoductor evices, Ilkley, UK,. Author's otes... Rolad STENZEL - bor i received the il.-ig. degree i electrical egieerig, the r.-ig. degree ad the habilitatio degree from the resde Uiversity of Techology i 1979, 198 ad 1989, resectively. From 1979 to 1989 he was with the Istitute of Electro evices ad Systems of the resde Uiversity of Techology. From 1989 to 199 he was with the GaAs-ivisio at the Istitute of Electro Physics of the Germa Academy of Scieces erli. From 199 to 199 he was a rofessor of theoretical electrical egieerig at the Uiversity of Trasortatio, resde ad sice 199 he has bee with the Uiversity of Alied Scieces resde as a rofessor for theoretical electrical egieerig ad semicoductor devices. His currets research iterests iclude umerical simulatio of HFETs ad HTs as well as ovel quatum devices ad MOSFETs. Wilfried KLIX - bor i received the il.-ig. degree i electrical egieerig, the r.-ig. degree ad the habilitatio degree from the resde Uiversity of Techology i 1979, 1987 ad, resectively. From 1979 to he was with the Istitute of Electro evices ad Systems of the resde Uiversity of Techology. Sice Setember he has bee a rofessor of theoretical electrical egieerig ad otoelectroics at the Uiversity of Alied Scieces resde. His currets research iterests iclude umerical methods for the simulatio of semicoductor devices as well as umerical simulatio of MOSFETs ad otoelectroic devices. Ja HÖNTSCHEL - bor i received the il.-ig. degree i electrical egieerig from the Uiversity of Alied Scieces resde i. From to 4 he worked for the Ph.. thesis at the Uiversity of Alied Scieces resde, Germay. Sice 4 he has bee a device egieer at AM resde, Germay. His currets research iterests icludig desig of SOI-MOSFET devices.