ABSTRACT. Siddharth Potbhare Master of Science, 2005

Transcription

1 ABSTRACT Ttle of Thess: CHARACTERIZATION OF 4H-SC MOSFETs USING FIRST PRINCIPLES COULOMB SCATTERING MOBILITY MODELING AND DEVICE SIMULATION Sddharth Potbhare Master of Scece 005 Thess drected by: Professor Nel Goldsma Deartmet of Electrcal ad Comuter Egeerg Detaled aalyss of a 4H-SC MOSFET has bee carred out by umercally solvg the steady state semcoductor Drft-Dffuso equatos. Moblty models for bul hoo scatterg surface hoo scatterg surface roughess scatterg Coulomb scatterg by terface tras ad oxde charges ad hgh feld effects have bee develoed ad mlemeted. A frst rcles Coulomb scatterg moblty model has bee develoed secfcally to model the hyscs of the verso layer 4H-SC MOSFETs. The Coulomb scatterg model taes to accout scatterg of moble charges by occued terface tras ad fxed oxde charges dstrbuto of moble charges the verso layer ad screeg. Smulated IV curves have bee comared to exermetal data. Desty of states for the terface tras have bee extracted ad seem to be agreemet wth exermetal measuremets. Smulatos dcate that occued terface tras 4H-SC MOSFETs are resosble for moblty degradato low currets ad hgh threshold voltages. Ther effect dmshes at hgh temeratures due to reducto tra occuacy ad at hgh gate voltages due to creased screeg. At hgh gate voltages surface roughess scatterg lays the maor role moblty degradato 4H-SC MOSFETs.

2 CHARACTERIZATION OF 4H-SC MOSFETs USING FIRST PRINCIPLES COULOMB SCATTERING MOBILITY MODELING AND DEVICE SIMULATION by Sddharth Potbhare Thess submtted to the Faculty of the Graduate School of the Uversty of Marylad College Par artal fulfllmet of the requremets for the degree of Master of Scece 005 Advsory Commttee: Professor Nel Goldsma Char Professor Pamela Abshre Professor Mart C. Pecerar

3 DEDICATION To my arets

4 ACKNOWLEDGEMENTS Frst ad foremost I would le to tha my advsor Dr. Goldsma for all the advce ad suort he has gve me ever sce I oed hs grou. I tha hm for ths wllgess to lste to whatever I had to say ad hs atet gudace over the years whch has eabled me to do ths wor. I would le to tha Dr. Abshre ad Dr. Pecerar for tag tme off ther busy schedules ad readg my thess ad servg o my thess commttee. I would le to say tha you to Dr. Gary Pegto for hs hel ad valuable uts throughout ths wor. I would le to tha Mr. Avars Lels ad hs grou at ARL for rovdg me wth exermetal data that was so ecessary for ths wor. I would le to tha my arets ad my famly for ther uvaryg belef me. Tha you to my lab-mates A Amrt Bo Datta Gary Latse ad Zeye for lots ad lots of fu-flled dscussos o Slco Carbde ad lfe geeral. I would also le to tha all my freds ad roommates here at College Par for mag ths lace a home away from home.

5 TABLE OF CONTENTS Lst of Fgures....v Lst of Tables... x. Itroducto..... Motvato Outle Prevous wor SC devce smulato My Aroach...6. Drft Dffuso Modelg ad Numercal Aalyss Drft Dffuso Model Posso s Equato Curret Equatos Cotuty Equatos Steady State Drft Dffuso Model Geerato ad Recombato Numercal Methods for Drft-Dffuso Smulato of SC MOSFETs Boudary Codtos Fte Dfferece Dscretzato of the Semcoductor Equatos Numercal Methods D Mesh... 3 v

6 3. Coulomb Scatterg Moblty Model Need for a Robust Coulomb Scatterg Moblty Model for 4H-SC MOSFETs Dervg the quas D Coulomb Scatterg Rate Equato Deth Deedet Coulomb Scatterg Moblty Equato Physcs of the Coulomb Scatterg Moblty Model Comlete Moblty Model ad Smulato Techque for 4H-SC MOSFET Comlete Moblty Model Bul Moblty Surface Phoo Moblty Surface Roughess Moblty Coulomb Scatterg Moblty Hgh Feld Moblty Total Moblty Smulato Techque ad Alcato to 4H-SC MOSFETs Iterface Tra Model Fxed Oxde Charge Desty H-SC MOSFET Parameter Extracto Scheme Smulato Results ad Iterretatos Room Temerature Fts to Exermetal Data Room Temerature Physcs of 4H-SC MOSFETs Iterface Tras 4H-SC MOSFETs Surface Roughess... 8 v

7 5..3. Total Low Feld Moblty Electro Cocetrato ad Curret Desty Comarg Coulomb Moblty ad Surface Roughess Moblty Effect of Screeg Devce Performace Predctos based o Smulatos Reducto of the Occued Iterface Tra Desty Reducto Surface Roughess Combed Reducto of Iterface Tra Desty ad Surface Roughess Future Wor...09 v

8 LIST OF FIGURES Fgure.. 4H-SC MOSFET devce structure... 7 Fgure.. Wor fucto dfferece betwee -tye olyslco gate ad -tye elayer 4H-SC MOSFET... Fgure.3. Flowchart for solvg the drft-dffuso semcoductor equato system Fgure 3. Comlex oles ( ad ) of the term sde the tegral whch gves the quas-d matrx elemet H D (Equato 3.) Fgure 3.. The quas-d scatterg wavevector show as dfferece betwee the fal ad the tal wavevectors Fgure 4.. Iterface tra desty of states for 4H-SC MOSFET. D t edge cm - ev - D t md cm - ev - ad t Fgure 4.. Ferm dstrbuto fucto ad the desty of occued accetor-tye terface tras at a low gate voltage of -V... 6 Fgure 4.3. Ferm dstrbuto fucto ad the desty of occued accetor-tye terface tras at a gate voltage of V Fgure 4.4. Occuato of accetor-tye terface tras by electros at two dfferet temeratures Fgure 4.5. Measured threshold voltage ad threshold voltage calculated by excludg the effect of fxed charge desty ad occued terface tra desty at dfferet temeratures Fgure 4.6. Plot of the dfferece betwee fxed oxde charge desty ad the occued terface tra desty as a fucto of temerature obtaed from threshold voltage measuremets Fgure 5.. Dra curret versus gate-source voltage o a logarthmc scale for the 00µm 00µm 4H-SC MOSFET at room temerature ad a dra-source voltage of 0.5V v

9 Fgure 5.. Dra curret versus gate-source voltage o a lear scale for a 00µm 00µm 4H- SC MOSFET at room temerature ad a dra-source voltage of 0.5V Fgure 5.3. Dra curret versus dra-source voltage for a 00µm 00µm 4H-SC MOSFET at room temerature Fgure 5.4. Iterface tra desty of states rofle for a 4H-SC MOSFET Fgure 5.5. Comarso of the verso charge desty (N v ) ad the occued terface tra desty (N t ) at dfferet gate voltages at room temerature Fgure 5.6. Coulomb scatterg moblty ( µ C ) lotted as a fucto of deth sde the semcoductor at dfferet gate voltages Fgure 5.7. Surface roughess moblty ( µ SR ) lotted as a fucto of deth sde the semcoductor... 8 Fgure 5.8. Fgure showg the comarso betwee the varous low feld moblty comoets at a gate-source voltage of 6V ad at room temerature Fgure 5.9. Total low feld moblty ( µ LF ) as a fucto of deth at the ceter of the chael. Gate-source voltages rage from -V to 8V Fgure 5.0. Varato of electro cocetrato wth deth at the ceter of the chael show for dfferet gate-source voltages ad at room temerature Fgure 5.. Curret desty varato wth deth for a 4H-SC MOSFET at room temerature.. 89 Fgure 5.. Average deth of the verso layer (Z avg ) lotted as a fucto of gate-source voltage at room temerature Fgure 5.3. Comarso betwee Coulomb scatterg moblty ( µ C ) ad surface roughess moblty ( µ SR ) at a gate-source voltage of V Fgure 5.4. Comarso betwee Coulomb scatterg moblty ( µ C ) ad surface roughess moblty ( µ SR ) at a gate-source voltage of V v

10 Fgure 5.5. Low feld dra curret versus gate-source voltage characterstcs for the two cases of screeed ad uscreeed Coulomb scatterg moblty models Fgure 5.6. Coulomb scatterg moblty lotted as a fucto of deth sde the semcoductor for the case of uscreeed Coulomb scatterg Fgure 5.7. Comarso of the verso charge desty (N v ) ad the occued terface tra desty (N t ) at dfferet gate voltages for two dfferet terface tra desty of states rofle Fgure 5.8. Coulomb scatterg moblty ( µ C ) lotted as a fucto of deth sde the semcoductor at dfferet gate voltages for the case of reduced terface tra destes Fgure 5.9. Curret desty varato wth deth at room temerature for a 4H-SC MOSFET wth reduced desty of states of the terface tras Fgure 5.0. Dra curret versus gate-source voltage for a 4H-SC MOSET at room temerature ad dra-source voltage of 0.5V... 0 Fgure 5.. Dra curret versus gate-source voltage curve for a 4H-SC MOSET at room temerature ad a dra-source voltage of 0.5V Fgure 5.. Comarso of the mrovemet currets due to reducto surface roughess ad the reducto terface tra desty of states at a gate-source voltage of 4V. A Fgure 5.3. Curret desty varato wth deth at the ceter of the chael a 4H-SC MOSFET after reducto the desty of terface states at the bad edge ad a reducto the surface roughess by a factor of 0 each Fgure 5.4. Dra curret versus gate-source voltage at room temerature wth the surface roughess ad the terface tra desty of states reduced by a factor of 0 each x

11 LIST OF TABLES Table.. Imortat hyscal roertes of S ad olytyes of SC... Table 4.. Parameter values used for calculatg the bul moblty a 4H-SC MOSFET Table 4.. Parameter values for calculatg surface hoo moblty 4H-SC MOSFETs x

12 CHAPTER. Itroducto Slco Carbde s fast becomg a materal of recog for maufacturg of hgh temerature hgh ower tegrated crcuts. I addto to a large badga hgh thermal coductvty ad a very hgh breadow feld the ma characterstc that sets SC aart from other wde badga materals s ts ablty to grow a atural oxde. Le slco t s ossble to grow slco doxde o a slco carbde substrate. Ths characterstc of SC maes t easy to maufacture SC MOSFETs ad other devces whch have bee tradtoally made usg slco. Through the years sce the frst tegrated crcut slco based devces have bee able to meet the eed for smaller faster cheaer more relable ad more dverse electroc crcuts. But all slco based ICs have faced lmtatos terms of oeratg temeratures (<50 o C) ad usable ower. SC based devces ad tegrated crcuts wll wor at much hgher temeratures ad much hgher ower as comared to S based devces. Table. shows a comarso of the materal roertes of SC ad S. Owg to ts larger badga ad sueror thermal coductvty t should be ossble to use SC based devces at temeratures u to 500 o C. Also because of the large badga ad huge breadow electrc feld SC devces ca oerate at voltages excess of several hudred volts. A large badga eables very low leaage currets duced by the buld-u of free charges ad thermal ruaway resultg from mact ozato rocess. Devces are

13 exected to have fast swtchg seeds ad low eergy loss eve at hgh temeratures. SC ca drve the mcroelectroc revoluto orgated by S to the hgh temerature hgh ower realm. Table.. Imortat hyscal roertes of S ad olytyes of SC S 3C-SC 6H-SC 4H-SC Badga (ev) Bul Electro Moblty (cm /Vs) Saturato Velocty (0 7 cm/s) Breadow Feld (0 6 V/cm) 3 3 Thermal Coductvty (W/K cm) Statc Delectrc Costat Curretly the bggest challege develomet of SC devces s low surface moblty at the SC-SO terface. The reaso for ths low surface moblty has bee show to be extremely large destes of occued terface tras at the SC-SO terface. The earlest S devces had hgh terface tra destes at the S-SO terface. By rocessg refemets over the decades very hgh qualty SO ca ow be grow o S substrates elmatg almost all terface tras. Hece devce smulators for S devces have lttle eed for models for the effects of terface tras o devce erformace. I order to further develo SC MOS techology t s meratve to uderstad deth the causes for moblty degradato ad the effect of varous scatterg mechasms artcularly surface roughess scatterg surface hoo scatterg ad Coulomb scatterg of verso layer moble charges by fxed oxde charges ad occued terface tras. Usg a advaced drft dffuso MOSFET devce smulator ad by develog ew models for terface tras ad Coulombc scatterg of moble charges I have tred to extract ad solate causes of devce degradato the varous regos of devce oerato.

14 .. Motvato The motvato of ths thess s to study the devce erformace of 4H-SC MOSFETs usg drft dffuso devce smulato ad to address the ssues causg erformace degradato SC MOSFETs. For ths a methodology has bee develoed to model the terface tras occurrg at the SC-SO terface ther occuacy ad ther effects scatterg of verso layer moble carrers. Ths has bee acheved by develog a comrehesve Coulomb scatterg moblty model secfcally for scatterg by occued terface tras ad by fxed oxde charges SC MOSFETs. The model has bee develoed from basc frst rcles hyscs of scatterg of moble charges by statoary charged scatterg ceters. I have tred to mold the scatterg hyscs equatos the form of a Coulomb Moblty whch ca be drectly cororated the drft dffuso smulator. Usg ths ew moblty model t s ow ossble to searate out ad deedetly study the effect of the hghly dese terface tras SC MOSFETs ad dscuss whether they or some other mechasms are resosble for the low surface moblty of these devces... Outle The thess teds to address detal the ssues related to umercal smulato of SC devces ad dscuss the moblty models that have bee mlemeted the drft dffuso smulator to model the behavor of 4H SC MOSFETs. I the frst chater I descrbe the curret ad ast wor that has bee doe SC devce smulato. I have tred to comle a reasoably comlete lst of devce modelg that has bee carred out for SC devces. It s followed by a short descrto of my aroach to address the ssues of SC devce modelg ad smulato. 3

15 I Chater I descrbe the drft-dffuso model for semcoductor devces ad the umercal smulato scheme for a 4H-SC MOSFET. The chater descrbes the dscretzato of the semcoductor equatos steady state the meshg scheme used to extract maxmum hyscs of the verso layer the umercal methods used to solve the couled semcoductor equatos ad the varous hyscal quattes that ca be extracted from the drft dffuso model. The ew Coulomb scatterg moblty model develoed for 4H SC MOSFETs s descrbed Chater 3. I tal detal about the hyscs of Coulomb scatterg semcoductors ad how t s very mortat for SC devces. I descrbe the methodology of dervg a Coulomb Moblty that s easy to cororate a MOSFET drft dffuso smulator from basc hyscs of scatterg semcoductors. I Chater 4 other moblty models reresetg varous other scatterg mechasms have bee descrbed. I have cluded the scatterg due to bul hoos surface hoos surface roughess ad hgh lateral felds addto to the occued terface tra scatterg the devce smulator. I also descrbe detal the modelg of terface tra states SC ad a method for extractg the fxed oxde charge desty from temerature deedet threshold voltage data ths chater. Fally the ffth chater I descrbe the smulato results for 4H SC MOSFETs. I dscuss the varous devce erformace related results that I have extracted by comarg smulated IV curves to exermetally measured data ths chater. 4

16 .3. Prevous wor SC devce smulato Zeg et al. [] have show a method of extractg eergy deedet terface tra desty for 6H ad 4H-SC MOSFETs the subthreshold rego of oerato. Scozze et al. [] have show the detrmetal effects of terface-traed charge o the SC MOSFET characterstcs. Arold et al. [3] have show that terface tras SC are resosble for decrease trascoductace lower moble verso charge desty ad low drft moblty of verso layer electros. Sas et al. [4] have show that severe trag of electros at the SC- SO terface 4H-SC devces causes a reducto the umber of free electros the verso layer ad also a dro the moblty. Correlato betwee chael moblty ad terface tras SC MOSFETs has also bee descrbed by Suzu et al. [5][6] All these observatos have bee made o the bass of exermets or comact modelg of SC devces. Physcs based smulatos of dee submcro 4H-SC MOSFETs have bee reseted by Dubarc et al. [7] but the authors have assumed that there are o terface charges at the SC-SO terface. Rosche et al. [8] have roosed electro moblty models for 4H 6H ad 3C SC whch descrbe the deedece of the electro moblty o dog cocetrato temerature ad electrc feld. They too have ot show ay effect of terface tras ther moblty models. Nlsso et al. [9][0] have descrbed Mote Carlo ad drft dffuso smulatos of 4H ad 6H- SC feld effect trasstors. But they have oly cosdered acoustc hoo scatterg olar ad o-olar otcal hoo scatterg ad ozed murty scatterg. Mcevcus et al. [] have carred out Mote Carlo smulatos for SC but he has focused o hoo scatterg ad ozed murty scatterg. Roldá et al. [] clude the effect of a et terface charge ther wor. However ther terface tra model s ot eergy deedet. But exermetal measuremets of terface state desty of states for SC have show that the desty of states s eergy deedet. Vathulya et al. [3] have used a eergy deedet terface state desty of states model for 4H 5

17 ad 6H-SC MOSFETs ad have extracted the verso layer moblty by fttg the curret equatos to exermetal data. They have used comact modelg ad umercal methods to solve sets of equatos smultaeously to extract a average moblty. Powell et al. [4][5] have descrbed detal varous moblty models for 6H SC MOSFETs. Ther moblty model cororates the effect of scatterg due to occued terface tras. But they do ot show how scatterg vares wth deth sde the verso layer. Ther method for dealg wth screeg of the scatterg charge ceters dee verso s emrcal..4. My Aroach My am s to address the ssue of the eed for a roer comrehesve moblty model for SC MOSFETs that ca accurately model the hyscs of the verso layer SC MOSFETs ad s smle eough to be cluded a drft dffuso smulator. I addto to the usual scatterg mechasms the moblty model should be able to descrbe the effect of scatterg of verso layer moble charges by the occued terface tras a the SC-SO terface fxed oxde charges dstrbuted at the terface ad the oxde. I the rocess I wll be buldg uo the wor doe by others ad develo a sohstcated drft dffuso devce smulator for SC MOSFETs. To solve ths roblem I started at the basc hyscs of Coulombc scatterg of charges. I develoed a quas D Coulomb scatterg model whch descrbed how verso layer electros are scattered by statoary charges located at the semcoductor-oxde terface ad/or some dstace sde the oxde tself. Usg ths scatterg model I was able to wrte a equato for the Coulomb moblty whch was deedet o the desty of occued tras desty of fxed oxde charges dstace betwee the scatterg charges ad verso layer moble charges screeg of 6

18 the scatterg charges by the moble charges ad temerature. The Coulomb moblty equato was smle eough to be cluded the drft dffuso modelg scheme. I order to verfy the accuracy of my model ad also to udge the mortace of varous scatterg mechasms a 4H-SC MOSFET I comared my smulatos to exermetal data. To mae cofdet redctos about the future erformace caabltes of SC MOSFETs I frst use the exermetal data to calbrate my smulator. Oce I have determed the roer materal roertes for my model ad have the comarsos of varous erformace degradg mechasms I ca redct the erformace of future SC MOSFETs. My smulatos also show that the terface tras lyg at dfferet eergy levels sde the SC badga have effects at dfferet gate voltage levels. The terface state tra desty rofle extracted from smulatos ad comarsos wth measured IV data seems to agree wth values measured by exermets. 7

19 CHAPTER. Drft Dffuso Modelg ad Numercal Aalyss I ths chater I wll reset the drft-dffuso model whch serves as the bass for the umercal smulato of a SC MOSFET devce. I beg by revewg the drft-dffuso equatos ad how they are used for the secfc case of smulatg a MOSFET devce. I wll also reset the dscretzato scheme for these equatos ad the umercal methods that have bee used to solve them... Drft Dffuso Model The drft dffuso equatos serve as the basc buldg blocs for semcoductor devce modelg ad ca be derved drectly from Maxwell s equatos for electromagetc radato ad the Boltzma trasort equato of etc theory. They are essece equatos derved from the Boltzma trasort equato by dog certa aroxmatos. The drft dffuso equatos cosst of the Posso s equato the equatos for electro ad hole currets ad the curret cotuty equatos for electros ad holes. I ths secto I descrbe these equatos bref ad the elaborate a lttle o the umercal methodology that s mlemeted to solve them for a SC MOSFET devce. 8

20 ... Posso s Equato The frst of the drft-dffuso equatos s the Posso s equato. It relates the electrostatc otetal () to the et charge desty () sde a semcoductor. r r ( ) (.) s the et charge desty ad s the delectrc ermttvty of the materal whch the charge s reset. Isde the semcoductor the charge artcle cocetrato cossts of egatvely charged electro cocetratos () ostvely charged hole cocetratos () ad the ozed doat cocetratos (D). The doats are further searated to ostvely charged doors (N D ) ad the egatvely charged accetors (N - A ). Substtutg these values for the et charge desty the Posso equato ca be rewrtte as: where q s the et charge o a sgle electro ad r r (.) ( ) q ( D) D (.3) N D N A The electro () ad hole () cocetratos are wrtte terms of the electro ad hole quas-ferm levels ( ad ). ex (.4) VT ex (.5) VT where s the trsc carrer cocetrato ad V T s the thermal voltage. 9

21 ... Curret Equatos There are two heomea causg curret flow a semcoductor: () Drft ad () Dffuso. Presece of a electrc feld a semcoductor wll cause the free electros ad holes to drft alog the feld les. Ths method roduces a curret whch s ow as the drft curret. If there s cocetrato gradet of the electro or hole cocetratos the electros ad holes wll flow from the hgher cocetrato rego to ther lower cocetrato rego geeratg a curret. Ths curret s termed as the dffuso curret. The cotrbuto of carrer trasort due to drft ca be formulated as: r r (.6) J drft qv r r (.7) J drft qv where v r ad v r are the drft veloctes of electros ad holes resectvely due to a aled electrc feld. These veloctes ca be exressed terms of the mobltes ( µ ad µ ) ad the aled feld ( E r ). r r v µ E (.8) r r v µ E (.9) The dffuso comoet of carrer trasort s due to radom moto ad gradets the moble charge cocetrato ad s descrbed by the followg: r ( ) J dff q D r ( ) J dff q D (.0) (.) D ad D are the electro ad hole dffuso coeffcets resectvely. They ca be related to the electro ad hole mobltes by the Este relatos: 0

22 D D BT (. q µ BT (.3) q µ Here B s the Boltzma s costat ad T s the temerature. Combg both drft ad dffuso comoets the total exresso for the electro ad hole currets sde a semcoductor s gve as: r J r J r J r J drft drft r J r J dff dff r r qµ E q (.4) ( D ) r r qµ E q (.5) ( D ) Wrtg the electrc feld E as a gradet of the electrostatc otetal E r r the curret equatos ca be rewrtte as: r J r J r r qµ q (.6) ( D ) r r qµ q (.7) ( D )..3. Cotuty Equatos The cotuty equatos are based o the coservato of moble charge. They relate the chage moble charge cocetrato tme to the gradet of the curret desty ad the rates of geerato ad recombato of carrers. For electros the cotuty equato s wrtte as t r r J q R G (.8) The cotuty equato for holes s gve by t r r J q R G (.9)

23 where G ad G are the electro ad hole geerato rates R ad R are the electro ad hole recombato rates ad r J r ad r J r are the et flux of electros ad holes ad out of the secfc volume. The curret cotuty equatos smly state that the total curret flow to or out of a volume of sace s equal to the tme varyg charge desty wth that volume lus ay addtos due to geerato or recombato that may occur...4. Steady State Drft Dffuso Model I steady state there s o et chage electro ad hole cocetrato over tme. Hece the cotuty equatos for electros ad holes ca be equated to zero. t r r J q R G 0 (.0) t Substtutg the formulae for r r J q J r ad J r R G 0 (.) from the curret equatos the above equatos we have r r r r [ ( D )] R G 0 µ (.) r r [ ( D )] R G 0 µ (.3) Equatos.0 ad. alog wth the Posso s equato (.) form the steady state drft dffuso system of equatos. These equatos are to be solved for the electrostatc otetal () the electro cocetrato () ad the hole cocetrato () sde the devce. It s aaret from the above equatos that the moblty geerato ad recombato of electros ad holes lay mortat roles the hyscs of carrer trasort a devce. I wll be

24 descrbg the moblty model used for 4H-SC MOSFET devces detal the ext two chaters. I wll brefly touch uo the geerato-recombato mechasms ext...5. Geerato ad Recombato For SC MOSFET smulato two tyes of recombato mechasms have bee modeled. The recombato occurrg due to tra ceters (Shocley-Read-Hall) ad due to drect artcle recombato (Auger). The geerato of artcles due to mact ozato s also cluded. Shocley-Read-Hall (SRH) Recombato The cature ad emsso of holes ad electros by tras that resde the md-bad eergy zoe s modeled usg the well ow Shocley-Read-Hall (SRH) mechasm of recombato. The SRH recombato rate s gve by R SRH ( ) ( ) (.4) where ad are the morty carrer lfetmes of electros ad holes resectvely. The morty carrer lfetmes for SC are order of a few aosecods. They are tycally aroud orders of magtude less tha those for S. For a -doed SC slab whe there s a curret flowg through the devce the electro cocetrato s much hgher tha the hole cocetrato( >> ). If the carrer lfetmes of electros ad holes are tae to be equal ( ) wrtte as the the SRH recombato rate ca be 3

25 R SRH (.5) At room temerature the trsc carrer cocetrato of 4H-SC s aroud 0-8 cm -3. For a slab of SC doed wth -tye murty of the order of 0 8 door atoms er cubc cetmeters at room temerature the hole cocetrato s gog to be aroud (0-8 ) /0 8 whch s aroxmately 0-30 cm -3. Hece t s easy to see that the SRH recombato rate SC s gog to be very small (aroud 0-0 cm -3 /s). Auger Recombato SC s a drect badga semcoductor. Hece the robablty of a drect bad-to-bad recombato by trasfer of eergy to aother carrer s very small. Hece ths recombato mechasm ow as the Auger recombato s rare SC devces. I the drect recombato rocess a free electro the coducto bad combes wth a free hole the valece bad ad the et mometum of the two artcle system s carred off by a thrd free artcle whch ca be a electro or a hole. The Auger recombato rate s gve by R Auger ( )( C C ) (.6) where C ad C are the coeffcets reresetg teractos whch the remag carrer s a electro ad a hole resectvely. Imact Iozato Geerato Geerato of artcles occurs whe a artcle wth hgh eergy colldes wth a boded artcle resultg oe addtoal free electro ad oe addtoal free hole. Ths tye of geerato s referred to as mact ozato geerato. I SC t s see that the free artcles do ot atta very hgh eergy due to the very hgh badga. Hece the mact ozato 4

26 geerato rate s very small. The mact ozato geerato rate s modeled as roortoal to the carrer curret desty [6]. G II r r ( J J ) (.7) q where b ex r (.8) E G II s the et mact ozato geerato rate a () s the er ut legth geerato coeffcet for electros (holes) ad b () s the electrc feld at whch mact ozato geerato becomes sgfcat. 5

27 .. Numercal Methods for Drft-Dffuso Smulato of SC MOSFETs I ths secto I wll descrbe the umercal methods used for solvg the drft dffuso equatos for the secfc case of a SC MOSFET devce. I order to solve the couled artal dfferetal equatos comrsg the drft dffuso system of equatos a roer methodology has to be followed. The equatos are solved for D structure whch s a secto cut arallel to the chael of the MOSFET. These equatos are solved for the electrostatc otetal () the electro cocetrato () ad the hole cocetrato () at dscreet mesh ots sde the devce the source dra bul ad the oxde regos. Arorate boudary codtos for ad based o the aled voltages at the varous regos are bult to the system. I wll descrbe the boudary codtos the meshg scheme the dscretzato of the semcoductor equatos ad the umercal methods used for solvg the dscretzed system of equatos.... Boudary Codtos The MOSFET devce structure s show fgure.. It s dvded to the source the dra the bul the oxde ad the terface regos where the semcoductor equatos are to be solved. The boudary regos are the regos where exteral voltage s aled or a artfcal boudary s created. 6

28 Fgure.. 4H-SC MOSFET devce structure 7

29 Ohmc Cotact The source bul ad dra cotacts show fgure. are modeled as Ohmc cotacts. That s there s o resstace of the cotact tself. Hece all the voltage aled at these cotacts s trasferred to the semcoductor below. The boudary codto for the electrostatc otetal s therefore gve by for Ohmc cotacts o -tye materal ad (.9) C VC (.30) C VC for Ohmc cotact o -tye materal. Here ad are the bult otetals for -tye ad -tye semcoductors thermal equlbrum. For a -tye materal wth a dog of N the bult otetal s smly D N D VT l (.3) ad for a -tye semcoductor doed of N the bult otetal s A N l A VT (.3) where s the trsc carrer cocetrato at temerature T ad V T s the thermal voltage. As there s o ower loss the Ohmc cotact ad the carrers are at thermal equlbrum we ca say that there exsts charge eutralty the cotact volume. Hece the total charge desty s equal to zero. where D N D N. A ( ) 0 q D (.33) 8

30 Ohmc cotact ad Sce we are at thermal equlbrum Smlarly for a -tye semcoductor rego at the Ohmc cotact ad. Hece for a -tye semcoductor at the (.34) D D 4 (.35) (.36) D D 4 (.37) Gate Cotact There are o moble charge carrers sde the gate oxde. Hece oly the Posso s equato s solved sde the oxde wth the charge desty tae as zero. The semcoductor equatos are ot solved for ad sde the oxde; hece o boudary codtos are eeded for ad at the gate cotact. The boudary codto for the electrostatc otetal o the gate cotact s defed as VGB (.38) G V G V G s the aled gate voltage ad VGB s the bult- gate voltage. It s equal to the metal-semcoductor wor fucto dfferece betwee the gate metal ad the semcoductor elayer of the 4H-SC MOSFET. The 4H-SC MOSFET used for measuremets has a -tye olyslco gate doed at 0 0 cm -3 ad a -tye e-layer doed at cm -3. Hece as 9

31 show Fgure. the wor fucto whch s the dfferece betwee the Ferm levels of the olyslco gate ad the 4H-SC e-layer ca be wrtte as E S F S Eg S S E S g 4.06eV S B ND BT l S oly (.39) E 4H F SC 4H 4H SC SC 6.97eV E E 4H g 4H g SC SC 4H B SC N BT l 4H e A SC (.40) S 4H SC ( E E ).9V VGB MS q F F (.4) Here S 4.05eV 4H-SC S 3.95eV E g.ev E 4H-SC g 3.6eV N oly D 0 0 cm -3 e N A cm -3 S. 0 0 cm -3 4H-SC cm -3 T 300 o K q C ad B J o K - 0

32 S 4.05eV 4 SC 3.95eV S E f S B S 0.5 E g MS 4H-SC 0.5 E g 4H-SC B 4H-SC E f -tye olyslco SO -tye 4H-SC Fgure.. Wor fucto dfferece betwee -tye olyslco gate ad -tye e-layer 4H- SC MOSFET.

33 Artfcal Boudares Artfcal boudares cosst of all boudares whch the devce structure ceases to exst for smulato uroses but realty ths boudary may ot exst o the devce hyscally. The artfcal boudares are laced far eough away from the carrer trasort actvty where the chage electrostatc otetal electro cocetrato ad hole cocetrato across the boudary s eglgble. Hece at the artfcal boudares we have the codtos N N N 0 (.4) where N s the dervatve tae the drecto ormal to the artfcal boudary.... Fte Dfferece Dscretzato of the Semcoductor Equatos I order to solve the system of couled dfferetal equatos comrsg the drft dffuso model each equato must be dscretzed sace. I ths secto I reset the dscretzato scheme for the Posso s equato ad the curret cotuty equatos. The Posso s equato s solved sde the semcoductor ad the oxde whereas the curret cotuty equatos are solved oly sde the semcoductor. At the semcoductor-oxde terface the Gauss s law s mlemeted order to solve for the electrostatc otetal. Each equato s dscretzed two dmesos usg the fte dfferece method where each osto (x y) the devce s maed to a mesh ot ( ). The osto of x at the th mesh le s desgated by the otato x ; lewse the osto of y at the th mesh le s desgated by the otato y. If t s eeded addtoal ots ca be defed as lyg betwee two cosecutve mesh ots. These ots are desgated by ± or ±. The dstace betwee two mesh ots are desgated by the varables h ad.

34 h x x (.43) y y (.44) Other varables wll be defed for the uroses of smlfyg the wrtg of the dscretzed equatos. Posso s Equato Posso s equato gves a aalytcal reresetato of the relatosh betwee electrostatc otetal () ad the et charge dstrbuto. Semcoductor-Isulator Iterface At the semcoductor-oxde terface t s assumed that there are o free electros ad holes ad that the dfferece the electrc dslacemet vectors the sulator ad the semcoductor s equal to the effectve surface charge. r r a ˆ (.45) surf ( D Ds ) Qsurf where D r s the electrc dslacemet vector ad Qsurf s the effectve surface charge desty at the semcoductor-oxde terface ad â surf s a ut vector the drecto of the semcoductor-oxde terface. Ths equato ca be rewrtte form of Gauss s law by wrtg the electrc felds at the semcoductor-oxde terface. E E Q (.46) ox ox s s surf Here the electrc felds are the felds eredcular to the terface. Wrtg them usg the electrostatc otetals we have ox s Qsurf (.47) y y ox s 3

35 4 Wrtg the dscretzed forms of the frst order dervatves we have the equato for the electrostatc otetal at the terface. 0 surf ox s ox s Q F ox ox ox ox ox ox ox ox (.48) where ox reresets the mesh-le whch defes the terface. Q surf s the et effectve surface charge at the mesh-ot ( ox ). It s the sum of the fxed oxde charge ad the terface traed charge at that mesh-ot. Isde the Oxde There s o charge reset sde the oxde. Hece the Posso s equato wll loo le a smle Lalaca. 0 (.49) Fte dfferece dscretzato of the above equato has the followg form. 0 h h h h F (.50) where ( ) ( ) ( ) ( ) h h h h h h h h (.5-54) Isde the Semcoductor Rewrtg the Posso s equato sde the semcoductor T T s D V V q ex ex (.55) Fte dfferece dscretzato of the above equato has the followg form.

36 5 0 ex ex T T s D V V q h h h h F (.56) Steady State Electro Curret Cotuty Equatos The steady state electro ad hole curret cotuty equatos are solved at all meshots sde the semcoductor. They are dscretzed by the Scharfetter-Gummel scheme. The temerature term the equato s modeled as a local temerature T. If the drft dffuso equatos are couled wth the heat flow equato the we would be able to extract the heat characterstcs of the devce. But I have ot used the heat flow equato my smulatos so the local temerature s effect costat ad s equal to the oeratg temerature of the devce. The Scharfetter-Gummel dscretzato of the steady state electro curret cotuty equato s gve as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 " " # $ % % & ' h h V G R V V h V h V F T T T T T µ ( µ ( µ ( µ ( µ ( µ ( µ ( µ ( (.57) s the electro cocetrato at mesh-ot ( ). ( ) ) ( s the Beroull fucto defed as

37 6 ( ) ( ) ex ) ) ) ( (.58) ad T T T T V V V V (.59-5) The moblty terms the above equato are wrtte as x x µ µ µ ± ± ad y y µ µ µ ± ± (.63 64) Here x µ ad y µ stad for the x-drecto ad the y-drecto electro moblty resectvely at mesh-ot ( ) Steady State Hole Curret Cotuty Equatos The hole curret cotuty equato ca be dscretzed the same way as the electro curret cotuty equato. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 " " # $ % % & ' h h V G R V V h V h V F T T T T T µ ( µ ( µ ( µ ( µ ( µ ( µ ( µ ( (.65) The moblty terms the above equato are wrtte as x x µ µ µ ± ± ad y y µ µ µ ± ± (.66 67)

38 Here µ ad x µ stad for the x-drecto ad the y-drecto hole moblty resectvely at y mesh-ot ( )..3. Numercal Methods There are two umercal methods that are used to solve the set of dscretzed equatos show the secto above. The frst method s a teratve Gummel Bloc method whch starts wth a tal guess for ad at all mesh-ots sde the devce ad solves the three equatos cosecutvely to get to a soluto. Ths soluto acts le the tal guess for the ext solver whch s the Newto solver. The Newto solver solves for all three varables ad at all mesh-ots smultaeously usg the Gauss-Newto Algorthm. Gummel Bloc Method The Posso electro curret cotuty ad the hole curret cotuty equatos are solved for the electrostatc otetal () electro quas-ferm level ( ) ad the hole quas-ferm level ( ) resectvely oe after the other usg a teratve solver. Frst the dscretzed Posso s equato s solved for at all ots the mesh usg a teratve method. I ths case ad are et costat. Next usg the ew calculated over the mesh the dscretzed electro curret cotuty equato s solved for over the mesh. I ths case ad are ot allowed to chage. Fally usg the ew ad values the hole curret cotuty equato s solved at all mesh ots ad the ew s obtaed. The dscretzed fte-dfferece equatos are solved usg the teratve Gauss-Sedel method. Here as a examle I show how the soluto for s obtaed teratvely. 7

39 The dscretzed Posso s equato s arraged the form show below f ( ) 0 (.68) where deotes the curret terato. The solver begs wth a tal guess for at all mesh ots the devce. Accordg to Newto s method for solvg olear equatos we ca wrte f ( f ) ( ) * f 0 (.69) Hece we have ad * f ( f ) (.70) * (.7) Here electrostatc otetal at the mesh ot ( ). The teratos cotue tll the error * falls below a rescrbed error crtero. Oce covergece s reached for the electrostatc otetal a smlar method s used to solve for the electro quas-ferm level ( ) ad the for the hole quas-ferm level ( ). Oce we have covergece for all three we swtch to the faster Newto s method. Newto Method Ule the Bloc Gummel Method the drft-dffuso equatos rema couled ths method. So ad are comuted smultaeously. Ths s accomlshed by defg a Jacoba matrx ad solvg for the chages the three varables betwee teratos. Ths rocess s rereseted as matrx equato as 8

40 9 " " " # $ % % % & ' " " " " # $ % % % % & ' * * * " " " " " # $ % % % % % & ' F F F F F F F F F F F F (.7) where the vectors r r r sgfy that the matrx oerato s erformed for all o-boudary mesh ots of the varables ad. Oce the above matrx equato s solved the ew values for ad are obtaed as * * ad * ( ) F F ad F are as show equatos ad.65. The Jacoba matrx s made u of the dervatves of these fuctos wth resect to ad. The Newto solver s allowed to terate tll t reaches the secfed covergece codto for all three varables. The the curret s calculated sde the chael at the dra cotact at the source cotact ad at the substrate cotact. If the curret s cotuous across the devce the the smulato s roer ad all relevat data s stored varous fles. A smle flowchart for the smulator s show Fgure.3.

41 Set u the devce dmesos materal roertes temerature bas voltages dog rofle etc. Dscretzato of the semcoductor equatos Ital Guess for ad Iteratve Gummel Bloc Method. Solve for Coverged? N Y Newto s Method for better accuracy N Coverged? Y Curret Cotuty? N Y Extract electro ad hole cocetratos moblty curret desty IV characterstcs etc. Fgure.3. Flowchart for solvg the drft-dffuso semcoductor equato system 30

42 ..4. D Mesh A D o-uform mesh has bee created for dscretzg the drft-dffuso equatos for the MOSFET structure. The mesh s very fe ear the dra ad the source uctos where there s rad chage otetal ad charge cocetrato. Whereas ear the ceter of the devce the mesh s coarse as there s ot much varato these hyscal quattes there. I order to cature the hyscs of the verso layer the mesh s et very fe ear the terface. The mesh sacg s et as low as Å ear the terface. Ths has eabled me to extract detaled hyscs of the verso layer. For examle moblty varatos as a fucto of deth ear the terface or the curret desty varato as a fucto of deth a 4H-SC MOSFET. The mesh s carefully crafted so that the electrostatc otetal does ot vary by more tha the thermal voltage wth adacet mesh ots. 3

43 CHAPTER 3 3. Coulomb Scatterg Moblty Model I ths chater I descrbe ew Coulomb scatterg moblty model detal. I frst tal about the eed for a robust moblty model for Coulombc scatterg of verso layer electros due to the traed charges at the terface ad the fxed charges the oxde. The I descrbe the dervato of the Coulomb scatterg rate ad the Coulomb moblty from basc frst rcles hyscs. At the ed I reset my observatos o ths moblty model ad how well t reflects the varous hyscal heomea occurrg the verso layer of a 4H-SC MOSFET. 3.. Need for a Robust Coulomb Scatterg Moblty Model for 4H-SC MOSFETs Coulomb scatterg of verso layer moble charges taes lace due to the resece of occued terface tras ad fxed oxde charges at the SC-SO terface. The Coulomb otetal due to these traed charges ad oxde charges decreases wth dstace. Hece moble charges located close the terface exerece more scatterg as comared to the oes located away from the terface. Hece a Coulomb scatterg moblty model for the moble charges eeds to have a deedece o the dstace betwee the moble charge ad the scatterg charge ceter. Further 3

44 the Coulomb otetal s screeed by the moble charges. Hece the Coulomb moblty should also clude the effect due to screeg. As the desty of traed charge s very hgh for SC MOSFETs Coulomb scatterg s a mortat heomeo lmtg the coductvty moblty the verso layer. All these factors romted me to develo a quas D Coulomb scatterg moblty model whch taes to accout the effect of scatterg due to terface traed charges ad fxed oxde charges dstrbuto of the fxed charges sde the oxde dstrbuto of moble charges sde the verso layer ad screeg of the scatterg charges by verso layer moble charges. The Coulomb moblty s also temerature deedet because the desty of occued tras decreases wth temerature ad the effect of screeg reduces wth temerature. Moblty models for Coulomb scatterg of electros by murtes the semcoductor have bee descrbed lterature [6][8]. The murtes are dstrbuted the verso layer ad so the Coulomb scatterg s three dmesoal ature. Ths model s ot alcable to 4H SC MOSFETs because all the scatterg charges are located at the terface or some dstace sde the oxde whle all moble charges are located below the terface. Broos-Herrg model descrbed Coulomb scatterg for a 3D electro gas. D scatterg by a Coulomb otetal has bee descrbed by Ado et al. [9]. The authors have assumed the verso layer to be a sheet of charge ad a average moblty s calculated. Sah et al. [0] have descrbed D Coulomb scatterg of electros by surface oxde charges. They have cluded the effect of electros beg dstrbuted the verso layer ad also of dstrbuto of fxed charges sde the oxde []. But ther moblty model does ot have a exlct z-deedece that ca be cororated a D devce smulator. They assume varous dstrbuto fuctos for verso layer electros ad scatterg charges ad calculate a average moblty based o that. Gamz et al. [] have descrbed a very comlex D Coulomb scatterg model whch s ot sutable for cluso a drft dffuso devce smulator. My moblty model does ot requre ay secfc dstrbuto fucto for electros the verso layer or ay dstrbuto fucto for fxed charges sde 33

45 the oxde. It drectly uses fxed charge destes at secfc deths sde the oxde ad the calculated electro cocetrato at varous deths sde the semcoductor to calculate the scatterg rate. Hece t s easly cororated a D devce smulator. 3.. Dervg the quas D Coulomb Scatterg Rate Equato The followg assumtos are made about the dstrbuto of the charged murtes causg Coulomb scatterg of moble carrers. The charged murtes are of two tyes fxed oxde charges ad terface traed charges. All these charges are located at the terface or sde the oxde. The ostos of the scatterg charge ceters at the terface ad sde the oxde are gve by z whle the ostos (deth) of the moble carrers sde the semcoductor are gve as z. z z 0 at the terface. I wll be dervg the scatterg model for electros as the moble carrers. Hece from ow o electros are tae as the moble charges the verso layer. We start wth a 3D screeed Coulomb otetal [8]: e V r ( ) r e 4 Here s the average ermttvty gve as [9] qscr (3.) Ad the screeg wave vector qsc ox SC (3.) s gve as the verse of the Debye legth for semcoductors q e N v sc (3.3) SC Z av BTe 34

46 Here N v s the average D verso charge desty at ay ot alog the chael of the MOSFET. I calculate ths D verso charge desty by tegratg the 3D electro cocetrato over the deth sde the semcoductor. I obta the electro cocetrato at each mesh ot sde the devce by solvg the semcoductor equatos descrbed Chater. Hece the D verso charge desty s calculated by tegratg these dscreet electro cocetrato values umercally. N v z z 0 ( z) dz (3.4) Also Z avg s the average deth of the verso layer at ay ot alog the chael of the MOSFET. It s evaluated by tegratg the roduct of the deth ad the 3D electro cocetrato ad dvdg t by the verso charge desty. Z avg z z z z 0 ( z) z dz 0 (3.5) ( z) dz I order to obta the scatterg rate I eed to formulate the erturbato factor (the screeed Coulomb otetal) to the Ferm Golde Rule. The Ferm Golde Rule gves the trasto rate / ) for a scatterg mechasm as ( r r. - ( E r r ) / r r H 0 E. - h - (3.6) Where r -. case as E r ad E r - rereset the eergy of the moble charge at the tal ad fal states r ad The matrx elemet Ferm s Golde rule ca be wrtte for the 3D Coulomb scatterg r r r r r r r r q r 3 D 3- D 3D H 3 D e V ( ) e V ( ) e d (3.7) r 35

47 where r r r ' q3d 3D 3D. Ths loos le a 3D Fourer trasform. Substtutg for ( r ) V r from equato (3.) we ca wrte the tegrato usg shercal co-ordates as Usg chage of varable t cos e q r q e r dr d e r r sc 3 Dr cos H3D sd (3.8) 4 H 3D e qscr q3 Drt re dr e r 0 t dt (3.9) H 3D q3 Dr e q e e scr re dr q Dr r 0 3 H e r 0 e s ( q r) sc 3D 3D q r dr q r 3 D (3.0) (3.) H qscr e ' e % & qsc q [ q s( q r) q cos( q r) ] 3D sc 3D 3D 3D 3D $ " # 0 (3.) O alyg the lmts we get H 3D e (3.3) q q sc 3D Ths s the matrx elemet for the case of 3D Coulombc scatterg. I am gog to treat the Coulomb Scatterg of verso layer charges as a quas D scatterg heomeo. As all the scatterg charges are located at the terface or sde the oxde ad the moble carrers beg scattered are located at dfferet deths the verso layer the Coulomb scatterg otetal see by the moble charges deeds uo the dstace betwee them ad the scatterg charges. Hece the verso charges located at dfferet dstaces from the terface are scattered at dfferet rates. For wrtg the scatterg equatos for a quas D scatterg heomeo I eed to fd the quas D scatterg matrx elemet H D. I assume that the scatterg taes lace the X-Y lae ad s dfferet for charges located at dfferet ostos the z drecto. 36

48 37 Hece by tag the Iverse Fourer Trasform of the 3D matrx elemet alog the q z drecto I wll get the matrx elemet for a quas D scatterg heomeo. I frst slt the 3D scatterg wave vector to a D comoet ad a z-comoet. The 3D scatterg wave vector s wrtte as 3 z D D q q q (3.4) Tag z q r the drecto of z I ca wrte the D matrx elemet by tag the Iverse Fourer Trasform as z z q D D dq e H < H z 3 (3.5) z z q sc z D D dq e q q q e < H z (3.6) z sc D z sc D z z q D dq q q q q q q e < e H z (3.7) I solve ths comlex tegral usg the Resdue method by cosderg the comlex cotours c ad c show Fgure 3.. The comlex tegral ca be solved as Re z D z z q q q z D z z q D q q q e s e H (3.8) ( ) ex z D z D D q q z q q e H (3.9) sc D sc D D q q e e H z q q (3.0) Ths s the z-deedet quas-d scatterg matrx elemet that wll be used to calculate the Coulomb scatterg rate.

49 Fgure 3. Comlex oles ( ad ) of the term sde the tegral whch gves the quas-d matrx elemet H D (Equato 3.). c ad c are the comlex cotours that are used to evaluate the comlex tegral by the resdue method. 38

50 Usg the matrx elemet of equato (3.0) ad the Ferm Golde Rule gve equato (3.7) the quas D trasto rate s gve by / r r. - H h D 0 4 D sc e e ( E r E r ) 0 ( E r E r ) - h 4 q q z q D q sc - (3.) Here r r ad - are D wave-vectors reresetg the mometum of the moble carrer before ad after the scatterg evet. before ad after the scatterg evet. E ad Er - r rereset the eerges of the moble carrer The scatterg charges are assumed to be dstrbuted at the terface ad sde the oxde wth a D desty gve by N D (z ) at deth z sde the oxde. ( 0) 8Nt N f z 0 N D ( z ) 7 (3.) 6N f ( z ) z < 0 where N s the terface tra desty at the terface ad N ( ) t f z s the fxed oxde desty at a dstace z sde the oxde. Hece the above trasto rate equato ca be rewrtte as D D sc ( z z ) sc ( E r r ) 4 e e / r r N ( ) E. - D z 0 h q q q q - (3.3) The ext ste s to evaluate the relaxato tme from the trasto rate equato. Coulomb scatterg s elastc ature ad usg the Bor aroxmato the D trasort relaxato tme s gve as [9] 39

51 ( z z E) - - cos 4 d ( ) d / r r. - (3.4) where s the relaxato tme for the scatterg of a moble charge located at a deth z sde the semcoductor havg the fal eergy E caused by Coulombc teracto wth a scatterg charge located at a deth z sde the oxde. s the scatterg agle ad r - s the fal state of the moble charge after the scatterg evet. Substtutg the trasto rate from the Ferm Golde rule I get the Coulomb scatterg rate equato as ( z z E) q D qsc ( z z ) ( z ) e 4 e N D 4 h d s 0 ( E r E r - - ) d (3.5) q D q sc To get a exresso for the Coulomb moblty from the above scatterg rate some algebra s volved. As a frst ste the tegral over the fal wave vector r - s reformulated terms of the D scatterg wave vector q r D. 40

52 Fgure 3.. The quas-d scatterg wavevector show as dfferece betwee the fal ad the tal wavevectors. Coulomb scatterg s elastc. Hece the magtudes of the fal ad tal wavevectors rema the same (electro/hole eergy does ot chage). > s the scatterg agle. 4

53 As show Fgure 3. the D scatterg wave vector ca be wrtte as r r r q D - : q D -s Here q D reresets the magtude of the vector q r D whle s the scatterg agle. (3.6) Wrtg / I ca rewrte equato (3.5) as ( z z E) ( ) sc s 0 ( E E ) r r d ( z ) ex 4- s q ( z z ) 4 e N D -d - h s q sc - (3.7) I the substtute the fal wave vector by the eergy ( E ) r by wrtg - Hece I ca wrte h - E r (3.8) - m * The scatterg rate equato the becomes e * m d - de r (3.9) - h E * r ex - s q ( z z ) sc 4 * N D ( z ) m h 0 ( E E ) de r r r * - - h E 0 h 0 8m E r r - s q sc r - 8m h E s d (3.30) Relacg E r by E ad solvg the above tegral for de gves the followg eergy deedet - scatterg rate equato. 4

54 43 ( ) ( ) ( ) d q E m z z q E m m z N e E z z sc sc D " " # $ % % & ' 0 * * 3 * 4 s s 8 s 8 ex h h h (3.3) By a smle algebrac maulato the above tegral ca be rewrtte as ( ) ( ) ( ) d E m q q z z q E m E m m z N e E z z sc sc sc D 0 * * * 3 * 4 s 8 s 8 ex 8 h h h h (3.3) The tegrato the above term s ot solvable aalytcally. I rewrte the above equato as ( ) ( ) ( ) E z z F E z N e E z z D 6 4 h (3.33) where F(z z E) s the eergy deedet form factor gve by ( ) ( ) d z z q E m q E m q E z z F sc sc sc 0 * * s 8 ex s 8 h h (3.34)

55 3.3. Deth Deedet Coulomb Scatterg Moblty Equato The scatterg rate equato gve (3.33) s eergy deedet. The average scatterg rate s calculated usg by aroxmatg the average eergy as E B Te because I have a quas D scatterg mechasm. Usg ths aroxmato I ca wrte the average scatterg rate as ( z ) 4 e N D 6 h T B e F ( z z T ) e (3.35) where T e s the electro (or hole) temerature ad F s the form factor s gve as F q ' * 8m T 0 8m BTe " " $ % s q & h # sc h sc B e ( z z Te ) ex% s qsc ( z z ) d * (3.36) The Coulomb moblty for a carrer at osto z sde the semcoductor due to scatterg by Coulombc teracto wth scatterg charges located at z sde the oxde s the gve by µ C ( z z T ) e m e * * m e N 6 3 D h B ( z ) T e F ( z z T ) e (3.37) The total Coulomb moblty for a moble carrer at a deth z ca be obtaed by addg u the scatterg rates due to scatterg charges located at dfferet deths sde the oxde usg Mathesse s rule as µ C < ( z T ) µ ( z z T ) e z C e (3.38) 44