MOSFET Models. The basic MOSFET model consist of: We will calculate dc current I D for different applied voltages.

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1 MOSFET Model The baic MOSFET model conit of: junction capacitance CBS and CB between ource (S) to body (B) and drain to B, repectively. overlap capacitance CGO and CGSO due to gate (G) to S and G to overlap, repectively. S to B and to B diode. We will calculate dc current for different applied voltage. 4March04 HO #18: ELEN 51 MOSFET Model Saha #1

2 MOSFET C Model Let u apply a large potential at the gate of an MOS capacitor to caue inverion, i.e. G > th. The conductance, 1/R of the inverion layer i: x n L G x P W g x 1 W q n ( x) dx n R L µ (1) 0 Where, n (x) e denity in the inverion layer x depth of the inverion layer µ n e mobility in the inverion layer 4March04 HO #18: ELEN 51 MOSFET Model Saha #

3 nverion Layer Conductance The inverion layer mobility i, alo, called the urface mobility and i 1/ bulk mobility. Now, the inverion layer charge per unit area i given by: x Q q n ( x) dx 0 Auming µ n contant, we get from (1) and (): g W µ Q L n ( e ) (3) and the inverion layer reitance of an elemental length dy i: dy dr (4) W µ nq Eq. (4) for MOS capacitor can be directly ued to derive drain current for MOSFET under different biaing condition. 4March04 HO #18: ELEN 51 MOSFET Model Saha #3 ()

4 Level1 MOSFET Model Let u conider the the following MOSFET tructure. The gate bia G provide the control of urface carrier denitie. f G > th, inverion layer exit. a conducting channel exit S and current will flow. S G n+ n+ P th i determined by the propertie of the tructure and i given by the equation we derived for MOS capacitor. That i, Q f q N A K ε o(φ F) th φ MS + φ F + (5) Co Co For G < th, the tructure conit of two diode back to back and only leakage current flow ( o of PN junction); i.e., ~ 0. 4March04 HO #18: ELEN 51 MOSFET Model Saha #4

5 Characteritic + G + x y n+ n+ Q (y) Q B (y) P nverion layer epletion region Note: The depletion region i wider around the drain becaue of the applied drain voltage. The potential along the channel varie from at y L to 0 at y 0 between the drain and ource. The channel charge Q and the bulk charge Q B will in general be f(y) becaue of the influence of, i.e. potential varie along the channel length. 4March04 HO #18: ELEN 51 MOSFET Model Saha #5

6 Characteritic Uing (4), the voltage drop acro an elemental length dy of MOSFET channel i: d Now, at any point in ilicon the induced charge due to G i: Again, dr Wµ n dy Q (y) Q (y) Q (y) + Q B (y) (7) G FB Q /C OX + φ (8) Note that FB include φ MS and Q f. Combining (7) and (8), we get: Q (y) { G FB φ (y)}c OX Q B (y) (9) Since the urface i inverted, φ φ F plu any revere bia that exit between the channel and ubtrate (due to or Sub ). (6) 4March04 HO #18: ELEN 51 MOSFET Model Saha #6

7 Characteritic φ (y) (y) + φ F (10) Alo, we know from MOS capacitor analyi that Note: Q B ( y) qn qk ( y) ε N [ ( y) + φ A we move from S, (y) due to R drop in the channel. x d max a we move toward the drain Q B a we move toward the drain. A x d Subtituting (10) and (11) into (9), we have: max o A F ] (11) Q ( y) { + G FB qk ε ( y) φ o N A F } C [ ( y) + φ OX F ] (1) 4March04 HO #18: ELEN 51 MOSFET Model Saha #7

8 Characteritic Subtituting (1) in (6), we get: dy Wµ n( { G FB ( y) φ F } COX qk εon A[ ( y) + φ F ])d f we aume Q B (y) contant, i.e. neglect the influence of channel voltage on Q B, then we can write: dy Wµ Wµ n n C C OX OX qk ε on A[ ( y) + φ G FB ( y) φ F COX [ ( y) ]d G th n Eq. (13) th include the effect of φ F, φ MS, and Q B (y0). y 0 y L ntegrating within the limit: to 0 W µ ncox [ G th ] L ] d (13) 4March04 HO #18: ELEN 51 MOSFET Model Saha #8 F (14)

9 Characteritic: Linear Region Eq. (14) i ued in Level 1 MOSFET SPCE model to calculate. f we define: κ µ n C OX proce tranconductance β µ n C OX (W/L) κ(w/l) gain factor of the device β[ G th ] f < 0.1, we can implify (15) to: β ( G th ) (15) (16) Eq. (16) how that current varie linearly with. Thi region of MOSFET operation i called the linear region of operation. From (16), the effective reitance between the ource and drain i: R ch β ( ) G 1 th (17) 4March04 HO #18: ELEN 51 MOSFET Model Saha #9

10 Characteritic plot for different G from Eq. (15) how for higher. While the meaured aturate at higher. Thi dicrepancy i due to the breakdown of gradual channel approximation near the drain end of the channel at high. Eq. (15) i valid only a long a an inverion layer exit all the way from S. The maximum value of v. plot occur at G th SAT drain aturation voltage (18) For > SAT, a channel will not exit all the way to the drain. 4March04 HO #18: ELEN 51 MOSFET Model Saha #10

11 Characteritic: Saturation What happen for > SAT? e travelling in the inverion layer are injected into the depletion region. There the high εfield pull e into the drain. Further increae of do not change (to a firt order). contant for > SAT. The boundary between the linear and the aturation region i decribed by: G th SAT (18) n+ nverion layer end (pinch off) + G l n+ + > SAT G th 6 Linear Saturation Region Region 5 P pinch of epletion region length March04 HO #18: ELEN 51 MOSFET Model Saha #11

12 Characteritic: Saturation Subtituting (18) in (16), we can how that the aturation drain current i given by: W ( ) SAT µ ncox G th (19) L quare law theory of MOSFET device. At > SAT, channel i pinchedoff and the effective channel length i given by: L eff L l d L(1 l d /L) W ( L l ) ( ) Typically, l d << L, then from (0): d SAT µ ncox G th (0) l d 1 L 4March04 HO #18: ELEN 51 MOSFET Model Saha #1 l + d SAT 1 (1) L

13 Characteritic: Saturation Since l d f( ), we can write: l L d λ () Here, λ f( ) channel length modulation model parameter. SAT ( + λ ) 1 (3) From (3), 0 at 1/ λ. λ can be extracted from the intercept of v. plot at 0 a hown in Fig. G5 > G4 > G3 > G > G1 G5 G4 G3 G G1 1/λ 0 4March04 HO #18: ELEN 51 MOSFET Model Saha #13

14 Level 1 MOSFET Model: Summary Current Eq: 0 G th (cutoff region) β[ G th ] β ( G th) ( 1+ λ). G th, SAT (linear region) G th, SAT (aturation region) (4) Model Parameter: TO threhold voltage at B 0 KP proce tranconductance GAMMA body factor LAMBA channel length modulation factor PH φ F bulk Fermipotential. 4March04 HO #18: ELEN 51 MOSFET Model Saha #14

15 Aumption: Level 1 MOSFET Model: Summary gradual channel approximation (GCA) i valid majority carrier current i negligibly (uch a hole current for nmosfet i neglected) recombination and generation are negligible current flow in the ydirection (along the length of the channel) only carrier mobility µ in the inverion layer i contant in the y direction current flow i due to drift only (no diffuion current) bulk charge Q B i contant at any point in the ydirection. The accuracy of Level 1 model i poor even for long device. 4March04 HO #18: ELEN 51 MOSFET Model Saha #15

16 4March04 HO #18: ELEN 51 MOSFET Model Saha #16 Level Model: BulkCharge Model n reality, Q B varie along the channel form ource at y 0 to drain at y L becaue of applied bia. Then from (11) with back bia B we have: Uing (6) in (6) and integrating form y 0 to y L we get: Eq (7) i the current ued by SPCE Level MOSFET model. ) ( ] ) ( [ ) ( ) ( max y C y N qk y x qn y Q B F OX B F A o d A B φ γ φ ε (5) ( ) ) ( ) ( ) ( y y C y Q B F F FB G OX + + φ γ φ (6) ( ) ( ) }] { 3 ) [( 3 3 B F B F F FB G OX n L C W φ φ γ φ µ (7)

17 Level 1 v. Level Characteritic 6 4 Eqn. (14) G 5 Eqn. (7) G We ee that for a typical device the approximate Eq. (14) work well at low but predict higher current at large. Thi i becaue ome of the gate charge upport Q B at higher which i ignored by (14). Both Eqn. (14) and (7) are valid only a long a an inverion layer exit all the way from S. The expreion for aturation drain voltage for Level model i given by: SAT γ γ G FB φ F + γ G FB + B + 4 (8) 4March04 HO #18: ELEN 51 MOSFET Model Saha #17

18 Level MOSFET Model Current Conideration of Q B variation along the channel offer more accurate modeling in the linear and aturation region. The current given by (7) i accurate, however, complex. i implified uing a new model parameter: 0.5 δ φ + f 0 β[ β α G α ] ( ) ( 1+ λ ). G F B α 1+ δγ, th then th (9) (30) G th (cutoff region) G th, SAT (linear region) G th, SAT (aturation region) (31) 4March04 HO #18: ELEN 51 MOSFET Model Saha #18

19 Subthrehold Region Model S th GS Simple long channel device equation aume: S 0 for GS < th. n reality, S 0 and varie exponentially with GS in a manner imilar to a bipolar tranitor. n order to develop a theory of ubthrehold conduction, let u conider MOS band diagram with applied ource (S) and gate (G) bia meaured with repect to the ubtrate (Sub), that i: S Sub G Sub. φ GSub GS Gate SiO psi φ SSub φ F φ F SSub E c E i E F E v 4March04 HO #18: ELEN 51 MOSFET Model Saha #19

20 Subthrehold Region Model n the ubthrehold (weak inverion) region, we know the following: 1 A E F i pulled above E i, the number of minority carrier e at the urface increae exponentially with E F E i. When φ φ F, trong inverion (n urf p bulk ) i achieved and we obtain th. 3 For φ < φ F, the dominant charge preent near the urface are ionized acceptor atom, i.e. n urf << N A. Thu, there i no εfield laterally along the urface ince Poion equation i the ame everywhere. Thu, any current flow mut be due to diffuion only. φ 4March04 HO #18: ELEN 51 MOSFET Model Saha #0 φ F E c E i E F E v

21 Subthrehold Region Model dn Aq (3) dx 4 The e gradient along the channel (dn/dx) mut be contant in order to maintain contant current. Aq[{n(0) n(l)}/l] n+ n+... (33) X Now, we can now ue carrier tatitic to calculate n(0) and n(l). Referring band diagram on page 0: n i q ( φ φ ) ( φ φ ) kt Where φ i the urface potential with repect to the ubtrate. Alo, note that φ contant along the channel [ε(x) 0]. 4March04 HO #18: ELEN 51 MOSFET Model Saha #1 P So few e that ε x 0 q SSub F kt ( 0) nie (34) n( L) n e Sub F N A (35)

22 Subthrehold Region Model Since the charge in the ubtrate i aumed uniform (N A ), then from Poion equation: d φ qn A de (36) dx ε dx φ varie parabolically and E varie linearly with ditance. Since the e concentration fall off a e qφ/kt away from the urface, eentially, all of the minority carrier e are contained in a region in which the potential drop by kt/q. The depth of the inverion layer, X inv φ/e. where φ kt/q, and E i the εfield at the urface. Again, from Gau Law: ε E Q ε qn φ (37) ε kt q kt B X inv (38) ε q qn qn Aφ Aφ ε 4March04 HO #18: ELEN 51 MOSFET Model Saha # A

23 Subthrehold Region Model Uing (34), (35), and (38) in (33), we get: q W q ni e L Here A WX inv. Uing Sub S + S Sub, ( φ φ ) q( φ φ ) SSub kt F e Sub kt F kt q ε qn φ A (39) ( φ φ ) q SSub F W ε kt ktni e 1 e L qnaφ q kt To make ue of thi equation, we need to know how φ varie with the externally applied potential G. qε N φ 4March04 HO #18: ELEN 51 MOSFET Model Saha #3 S (40) A GSub FB + φ + (41) Co

24 Subthrehold Region Model Generally, it i more common to ue the ource potential a the reference o that: G Sub GS + SSub φ ψ + S Sub q( ψ φf ) W ε kt ktni e 1 e L qn Aφ qε N A ψ + GS FB + ψ + Co The depletion layer capacitance i: Therefore, from (43): d dψ GS 1 qε N A C o ( ψ + ) SSub ( ) C C SSub C o q kt n S qε ( ψ + ) N A SSub (4) (43) (44) (45) 4March04 HO #18: ELEN 51 MOSFET Model Saha #4

25 Subthrehold Region Model n order to eliminate ψ from (4), we expand GS in a erie about the point ψ 1.5φ F (weak inverion correpond to φ F ψ φ F ). Where * and i obtained from (43). GS GS ψ 1. 5φ F ( ψ 1. φ ) GS GS + n ψ φ F (46) F Combining (46) with (4) to eliminate ψ in the exponential and uing (44) to eliminate the quare root of ψ in (4), we obtain: * q( GS GS ) qφ F qs W kt nkt kt kt µ Cni e 1 e (47) L q 4March04 HO #18: ELEN 51 MOSFET Model Saha #5

26 Subthrehold Region Model Thu, the ubthrehold current i given by: * q( ) GS GS qφ F q S W kt nkt kt kt µ Cni e 1 e (47) L q From (47) we note that: depend on S only for mall S, i.e. S 3kT/q, ince exp[ q S /kt) 0 for larger S. depend exponentially on GS but with an ideality factor n > 1. Thu, the lope i poorer than a BJT but approache to that of a BJT in the limit n 1. N A and S Sub enter through C. 4March04 HO #18: ELEN 51 MOSFET Model Saha #6

27 S (A) 1.E04 1.E05 1.E06 1.E07 1.E08 1.E09 1.E10 1.E11 1.E1 Subthrehold Slope (Sfactor) n order to change by one decade, we get from (47): q nkt S GS 60 ln10 S m decade kt q 1 + C C GS () o ln10( n) th kt q ln C C kt C Slope ln10 1 q + C m 60 ( n) decade 4March04 HO #18: ELEN 51 MOSFET Model Saha #7 o (@ room T)... (48) o (@ room T)

28 Subthrehold Model Final Note n weak inverion or ubthrehold region, MOS device have exponential characteritic but are le efficient than BJT becaue n > 1. Subthrehold lope S doe not cale and i contant. Therefore, th can not be caled a required by the ideal caling law. S affect th a well a ubthrehold current. n order to optimize S, the deirable parameter are: thin oxide low N A high Sub. 4March04 HO #18: ELEN 51 MOSFET Model Saha #8

29 Level /3 MOSFET Model Current Uing ubthrehold region model, for G < on, we can how: on e q ( ) G on nkt (49) Here on G at the point of interection between the trong and weak inverion while G on Thu, the complete longchannel C MOSFET model i given by: ( ) q G on exp nkt α β[ G th β G th α on ] ( ) ( 1+ λ ). G on (cutoff region) G th, SAT (linear region) G th, SAT (aturation region) (50) 4March04 HO #18: ELEN 51 MOSFET Model Saha #9

30 nverion Layer Mobility n our derivation of, we have conidered µ contant which i not true under applied G and. A the vertical field E x and lateral field E y increae with increaing G and repectively, carrier undergo increaed cattering. µ f(e x, E y ) For implicity of calculation an effective mobility defined a the average mobility of carrier i ued in MOSFET: µ eff X inv 0 µ ( x, X inv 0 y) n( x, y) dx n( x, y) dx (51) 4March04 HO #18: ELEN 51 MOSFET Model Saha #30

31 Mobility egradation due to G n reality, µ i ignificantly reduced by large vertical e field. The vertical efield, pull the inverion layer e toward the urface cauing more urface cattering and interaction of e with oxide charge (Q f, N it ) cauing coulomb cattering. Effective mobility v. effective normal efield how: at high field: univeral behavior independent of doping denity. at low field: dependence on 1) doping denity and ) interface charge. univeral behavior 4March04 HO #18: ELEN 51 MOSFET Model Saha #31

32 Mobility egradation due to G Mobility veru effective normal electric field curve i modeled uing three component: coulomb cattering (due to ionized impurity) phonon cattering almot contant urface roughne cattering (at Si/SiO interface). At low normal field: le inverion charge denity ionized impurity cattering dominate and µ eff f(n A ). At high normal field: higher inverion charge denity cloe to the interface urface roughne cattering dominate. 4March04 HO #18: ELEN 51 MOSFET Model Saha #3

33 Mobility egradation due to G The mobility correction ha everal flavor depending on the level of MOSFET model. n SPCE MOSFET Level 3 and 4 ue a imple form uch a: µ eff 1+ θ µ ( ) Where the model parameter are: G th θ mobility degradation parameter due to G. µ 0 low field mobility. 0 (5) The effective mobility due to G i called lowfield urface mobility. 4March04 HO #18: ELEN 51 MOSFET Model Saha #33

34 Mobility egradation due to An additional complication arie becaue of the high lateral e field E(y). We aumed, v d µe. However, for e in ilicon v d aturate near E ~ 10 4 /cm. Average efield for hort channel device > 10 4 /cm. Therefore, mall geometry MOS device will operate with: v d v at 10 7 cm/ec. Since µ contant, we need to account for high lateral field effect in the expreion for derived from imple theory. 4March04 HO #18: ELEN 51 MOSFET Model Saha #34

35 Mobility egradation due to The velocity aturation of inverion carrier due to increaed lateral field caue: current aturation ooner than predicted by SAT G th lower SAT than the predicted value by imple theory. The drift velocity due to high field effect i given by: v µ [( ) β ] 1 E E β y 1+ y C E Where E C v at /µ critical lateral field for velocity aturation. For implicity of numerical olution, β 1 i ued for modeling. µ Ey v 1 + µ E v (53) y at 4March04 HO #18: ELEN 51 MOSFET Model Saha #35

36 Modeling Mobility egradation The effective mobility due to combine effect of G and i given by: µ eff µ θ ( G th ) + θbb + θc (54) Where θ C 1/LE C L channel length of MOSFET. Thu, µ eff i modeled by the parameter et: {(µ 0, θ, θ b, and v at (E C )}. The model parameter are obtained by curve fitting the experimental data to the model equation. 4March04 HO #18: ELEN 51 MOSFET Model Saha #36

37 Modeling SAT at High efield Aume a piecewie linear model, i.e., v d aturate abruptly at E E c : v d µ eff E, E 1+ E v at, c ( E E ) ( E E ) c c (55) Where E lateral efield. E c critical efield at which carrier are velocity aturated, i.e. v v at. Now, the current at any point x in the channel i given by: (x) WC o [ G th (x)]v(x) (56) Where (x) potential difference between the drain and channel at x. v(x) carrier velocity at x. 4March04 HO #18: ELEN 51 MOSFET Model Saha #37 v v at µ o E c E

38 Modeling SAT at High efield Subtituting (55) in (56) we get: d( x) E( x) Wµ C [ x ] dx eff o G th ( ) E ntegrating (57) from x 0 to x L and (x) 0 to (x), we get in the linear region ( S SAT ): 1 Wµ effco G th S S (58) + S L 1 EcL Let u define, SAT the drain aturation voltage due to v at. i.e. at E E c. Uing thi condition in (57) we get in the aturation region ( S SAT ): W µ eff C o ( ) G th SAT E c c (57) (59) 4March04 HO #18: ELEN 51 MOSFET Model Saha #38

39 Modeling SAT at High efield given by (58) and (59) mut be S SAT. Thu, Equating (58) and (59) we can olve for SAT : SAT EC L E L + C ( G th) ( ) G th G th G 1+ E L (60) From (60), if L i large, SAT ( G th ) ued in the imple theory. Eq (60) i ued to calculate drain current for > SAT in BSM3 and BSM4 MOSFET model. C th 4March04 HO #18: ELEN 51 MOSFET Model Saha #39

40 HotCarrier Effect For S > SAT, lateral efield increae beyond E c and reache a maximum value at the drain end. From peudod analyi it can be hown that maximum efiled i: E (61) c S SAT E m + l p E E m Where, l p depend on: gate oxide thickne t ox E c S L X junction depth x j l 1. 73t x p (6) ox j f ( S SAT ) /l p >> E c, then: E m l S SAT (63) p 4March04 HO #18: ELEN 51 MOSFET Model Saha #40

41 HotCarrier Effect Channel electron traveling through high electric field near the drain end can: n+ Source Gate g hot e hole ub n+ rain become highly energetic, i.e. hot caue impact ionization and generate e and hole hole go into the ubtrate creating ubtrate current, ub. Some channel e have enough energy to overcome the SiO Si energy barrier generating gate current, g. The maximum efield, E m near the drain ha the greatet control of hot carrier effect. 4March04 HO #18: ELEN 51 MOSFET Model Saha #41

42 Subtrate Current Modeling mpact ionization coefficient, α A i exp( B i /E). Then the ubtrate current due to impact ionization i: ub αdx o l A e Where x o tart of impact ionization region with E E c. x l p length of impact ionization region with E E m. Changing the limit of integration to E in (19) and after integration we get: A i B i ub l pem exp (65) Bi Em A E m, ub. l p p o i B i E ( x) dx (64) 4March04 HO #18: ELEN 51 MOSFET Model Saha #4