MOSFET DC Models. In this set of notes we will. summarize MOSFET V th model discussed earlier. obtain BSIM MOSFET V th model

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1 n thi et of note we will MOSFET C Model ummarize MOSFET V th model dicued earlier obtain BSM MOSFET V th model decribe V th model parameter ued in BSM develop piece-wie compact MOSFET S model: baic equation BSM equation decribe S model parameter ued in BSM develop ubtrate current ub model to characterize device degradation due to high field ect. HO #17: ELEN MOS C Model Page 1

2 BSM V th Model We know that for uniformly doped ubtrate, N ub, the long channel threhold voltage i: V th TH 0 where V TH 0 V γ V th 2qε + γ V Si ox N ( φ V φ ub BS 0 BS body ect coicient A unified expreion for V th ued to model the nonuniform vertical channel doping profile i: ( V BS K VBS V th V TH 0 + K1 φ 2 where K 1 and K 2 model the vertically non-uniform doping ect on V th. (1 φ (2 HO #17: ELEN MOS C Model Page 2

3 BSM V th Model We derived V th model due to non-uniform vertical and lateral doping profile a: ( N LX V th V TH K φ VBS φ K VBS K φ L where V TH0 i a model parameter extracted from the meaure S v. V GS data at V BS 0 K 1 and K 2 model the ect of non-uniform vertical channel doping profile on V th and are fitting parameter extracted from the meaured data N LX model the non-uniform lateral profile on V th and i a fitting parameter extracted from the meaured data. (3 HO #17: ELEN MOS C Model Page 3

4 Modeling SCE due to BL By olving Poion Eq along the channel, V th hift due to BL can be hown a: V th θ th (L[2(V bi - φ + V S ] (4 V bi built-in voltage of the S/ junction given by: V bi kt N ln N CH 2 q ni S where N S ource-drain doping concentration N CH channel doping concentration (5 Eq (4 how that V th depend linearly with V S. And,V th decreae a V S increae due to BL. HO #17: ELEN MOS C Model Page 4

5 Now, θ ( L e where l l t th t i a Modeling SCE due to BL ε T Si ε L l OX ox + 2e X dep L l 2 t t (15 characteritic length given by : η with X dep ( φ V η fitting parameter account for the approximation ued to obtain l t. n order to account for non-uniform vertical channel doping concentration, (15 i modified o that from (13: V thd + ( e VT 0 ( e SUB L 2l VT 1 t 0 L 2l t + 2e + 2e SUB L l VT 1 t 0 L l t 2ε Si HO #17: ELEN MOS C Model Page 5 qn ( V φ bi CH BS ( ETA0 + ETABV BS V S (16

6 Narrow Width Effect - An Empirical Model V th hift due to narrow width ect can be hown a: TOX VthW φ (17 W To model both narrow width and revere narrow width ect, the model i expreed in term of fitting parameter: K 3, K 3B, and W 0 to get: V thw ( K K 3 + 3B BS TOX W + W where W' ective channel width. V ntroducing SCE in narrow device, we add to (18: V thwl VT 0W ( e VT 1W W 2l L tw 0 + 2e φ VT 1W W l L tw (18 ( V φ bi (19 HO #17: ELEN MOS C Model Page 6

7 V th V + K 1 e TH 0 + K 1 Complete V th Model ( φ V φ N b LX 1+ 1 φ + ( K 3 + K 3B L 1 K V 2 b V 2 3 b T W L L L L VT1 VT1 2lt lt 0 VT 0 ( + 2e bi VT 0W ( e VT 1W W 2l L tw OX φ SUB SUB 2lt lt ( V φ ( e + 2e ( ETA0 + ETABV BS V S + 2e 4 VT 1W W l L tw ( V φ Small L, W ect 1 vertical non-uniform channel doping ect 2 lateral non-uniform channel doping ect 3 narrow width ect 4 SCE due to BL bi (30 HO #17: ELEN MOS C Model Page 7

8 V th Model Parameter T OX T OXM X J N CH N SUB V TH0 V FB K 1 K 2 K 3 gate oxide thickne nominal T OX at which parameter are extracted junction depth channel doping concentration ubtrate doping concentration threhold V b 0 for large L Flat band voltage firt-order body ect coicient econd-order body ect coicient narrow width coicient K 3B body ect coicient of K 3 HO #17: ELEN MOS C Model Page 8

9 V th Model Parameter W 0 N LX VT0W VT1W VT2W VT0 VT1 VT2 V BM narrow width parameter lateral non-uniform doping coicient firt coicient of narrow width ect on V th at mall L econd coicient of narrow width ect on V th at mall L body-bia coicient of narrow width ect on V th at mall L firt coicient of SCE on V th econd coicient of SCE on V th body-bia coicient of SCE on V th maximum applied body bia in V th calculation HO #17: ELEN MOS C Model Page 9

10 Piece-Wie S Model: nverion Layer Conductance Let u apply a large potential at the gate of an MOS capacitor to caue inverion, i.e. V G > V th. The conductance, 1/R the inverion layer i: of x n L V G x P W g x 1 W q n n ( x dx 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 HO #17: ELEN MOS C Model Page 10

11 nverion Layer Conductance The inverion layer mobility, referred to a the urface mobility, µ 1/2 of bulk mobility. Now, the inverion layer charge per unit area i given by: Q x q n 0 ( x dx Auming µ contant, we get from (1 and (2: g W L µ Q ( e The inverion layer reitance of an elemental length dy i: dr dy W µ Q Eq. (4 can be ued to derive drain current for MOSFET under appropriate biaing condition. (2 (3 (4 HO #17: ELEN MOS C Model Page 11

12 Strong nverion Region Let u conider the following MOSFET tructure. The gate bia V G provide the control of urface carrier denitie. S G n+ n+ P f V G > V th, an inverion layer exit. a conducting channel exit from S and current will flow. V th i determined by the propertie of the tructure and i given by: Q f 2q N SUBε Si (2φ F φ + 2φ + (5 V th MS C ox F For V G < V th, the tructure conit of two diode back to back and only leakage current flow ( o of PN junction; i.e., ~ 0. C ox HO #17: ELEN MOS C Model Page 12

13 Strong nverion Region +V G +V 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 V. The potential along the channel varie from V 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 V, i.e. potential varie along the L only Gradual channel approximation. y 0 y L HO #17: ELEN MOS C Model Page 13

14 rain Current Model From (4, the voltage drop acro an elemental length dy in the MOSFET channel i: dy dv dr (6 Wµ Q (y n Now, at any point in ilicon the induced charge due to V G i: Q (y Q (y + Q B (y (7 Again, V G V FB Q /C OX + φ (8 Here V FB include φ MS and Q f. Combining (7 and (8: Q (y {V G V FB φ (y}c OX Q B (y (9 Since the urface i inverted, φ 2φ B ( 2φ F plu any revere bia between the channel and ubtrate (due to V or V BS. φ (y V(y + 2φ B (10 HO #17: ELEN MOS C Model Page 14

15 rain Current Model Alo, we know from MOS capacitor analyi that the bulk or depletion charge at any point y in the channel i: Q b ( y qn SUB X dep ( y 2qε SiN SUB[ V ( y + 2φ B ] Note that a we move from S, V(y due to R drop in the channel. X dep a we move toward the drain and Q b a we move toward the drain. (11 Subtituting (10 and (11 into (9, we get the expreion for inverion charge: Q ( y { V + G V FB 2qε N Si V ( y 2φ } C SUB B OX [ V ( y + 2φ ] B (12 HO #17: ELEN MOS C Model Page 15

16 Baic rain Current Model Subtituting (12 in (6, we get: dy Wµ ({ V V V ( y 2φ } C 2qε N [ V ( y + 2φ ]dv G FB f we aume Q b (y contant, i.e. neglect the influence of channel voltage on Q b, then: dy Wµ C Wµ C ox ox n (13 V th include the ect of φ B, φ MS, and Q b (y 0. B ox 2qε SiN SUB[ V ( y + 2φ B ] VG VFB V ( y 2φ B dv C OX [ V V V ( y ]dv (13 G y 0 ntegrating within the limit: V 0 W V µ Cox[ VG Vth ] V L 2 th to SUB HO #17: ELEN MOS C Model Page 16 Si y L V V B (14

17 Baic rain Current Model: Linear Region Eq. (14 i the Level 1 MOSFET S model. f we define, κ µ C ox proce tranconductance β µ C ox (W/L κ(w/l gain factor of the device V β[ VG Vth ] V 2 f V < 0.1 V, we can implify (15 to: β ( V G V th V (15 (16 (16 how that current varie linearly with V. Thi i defined a the linear region of operation. From (16, the ective reitance between the ource and drain i: R ch V β ( V V G 1 th (17 HO #17: ELEN MOS C Model Page 17

18 Baic rain Current Model: Linear Region V. V from (15 with different V G how for higher V. While the meaured aturate at higher V. Thi dicrepancy i due to the breakdown of gradual channel approximation near the drain end of the channel at high V. Eq. (15 i valid only a long a an inverion layer exit all the way from S. The maximum value of v. V plot occur at: V V G V th V SAT drain aturation voltage (18 For V > V SAT, a channel will not exit all the way to the drain. HO #17: ELEN MOS C Model Page 18

19 Baic rain Current Model: Saturation HO #17: ELEN MOS C Model Page 19

20 Baic rain Current Model: Saturation When V > V SAT, e- travelling in the inverion layer are injected into the pinch-off region near the drain-end. n+ nverion layer end (pinch off +V G l n+ +V > V SAT P pinch of epletion region length The high ε-field in the pinch-off region pull e- into the drain. further increae of V do not change (to a firt order. contant for V > V SAT The boundary between the linear and the aturation region i decribed by: V G V th V SAT. V G - V th 6 Linear Saturation Region Region V HO #17: ELEN MOS C Model Page 20

21 Baic rain Current Model: Saturation Subtituting (18 in (16, we can how that the aturation drain current i given by: W ( 2 SAT µ ncox VG Vth (19 2L quare law theory of MOSFET device. At V > V SAT, channel i pinched-off and the ective channel length i given by: L L l d L(1 -l d /L W 2 SAT µ ncox ( VG Vth ( L l (20 2 l d d 1 L l Typically, l d << L, then from (20: + d SAT 1 (21 L HO #17: ELEN MOS C Model Page 21

22 Baic rain Current Model: Saturation Since l d depend on (V - V SAT, we can write: ld λ ( V VSAT (22 L Here, λ channel length modulation (CLM parameter. SAT ( + ( V V λ (23 1 SAT From (23, 0 at (V V SAT 1/λ. λ can be extracted from the V -intercept of v. V plot at 0 a hown in Fig. 1 λ V A CLM Early voltage due to CLM V G5 > V G4 > V G3 > V G2 > V G1 V G5 V G4 V G3 V G2 V G1 1/λ 0 V HO #17: ELEN MOS C Model Page 22

23 Baic rain Current Model: Summary Current Eq: 0 V G V th (cut-off region V β[ VG Vth ] V 2 β 2 ( VG Vth ( 1+ λv. 2 V G V th, V V SAT (linear region V G V th, V V SAT (aturation region (24 Model Parameter: VTO threhold voltage at V B 0 KP proce tranconductance GAMMA body factor LAMBA channel length modulation factor PH 2 φ B bulk Fermi-potential. HO #17: ELEN MOS C Model Page 23

24 Baic rain Current Model: Summary Aumption: 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 y-direction (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 y-direction. The accuracy of The baic model i poor even for long channel device. HO #17: ELEN MOS C Model Page 24

25 Bulk Charge Effect n reality, Q b varie along the channel form the ource at y 0 to drain at y L becaue of the applied bia V. Then from (11 with back bia V BS we have: Q b Q ( y qn C ox SUB γ X dep 2φ + V B ( y BS 2qε N + V ( y Si [ V ( y + 2φ + V For the implicity of computation, we implify (25 by Taylor erie expanion and by neglecting higher order term, we get: 1 V ( y Qb ( y Coxγ ( 2φ B + VBS ( 2φ B + VBS SUB ( V V 2 V ( y γ 2φ + V V ( HO #17: ELEN MOS C Model Page 25 B BS ] (25 ( y C φ y (26 ox G FB B B BS + C ox γ ( 2φ + V B BS ( 2φ + V B BS V ( y C ox γ [ 2φ + V + δ. V ( y ] B BS (27

26 where δ 2φ F V B Bulk Charge Effect From (26 and (27, Q ( y C C C C C where α ox ox ox ox ox ( V V V ( y 2φ G ( VG VFB V ( y 2φ B γ 2φ F + VBS + V ( y ( V ( 2 { G VFB V y φb γ 2φ F + VBS + δ. V ( y } ( V V 2φ γ 2φ + V ( 1+ δγ V ( y G ( V V αv ( y G FB FB th bulk charge factor 1+ We will ue (28 to develop advanced F B + Q B b 2φ S ( y BS 0.5γ + B V BS model. (28 (29 HO #17: ELEN MOS C Model Page 26

27 MOSFET S Model with Bulk Charge Effect Q b variation along the channel offer more accurate modeling in the linear and aturation region. The implified MOSFET drain current model uing the bulk-charge factor α a a fitting parameter i given by: 0 β[ V β 2α Where V G V αv 2 ] V 2 ( V V ( 1+ λv. G SAT th α 1+ th ( V V / α G 2φ + th 0.5γ B V BS V G V th V G V th, V V SAT V G V th, V V SAT (cut-off region (linear region (aturation region (30 (31 (29 HO #17: ELEN MOS C Model Page 27

28 High-Field Effect in S Surface mobility µ contant aumed in MOSFET current expreion i not true under high V G and V. A the vertical field E x and lateral field E y increae with increaing V G and V, repectively, carrier uffer increaed cattering. µ f(e x, E y For the implicity of calculation an ective mobility defined a the average mobility of carrier i ued: µ X inv X inv 0 µ ( x, y n( x, y dx 0 n( x, y dx (32 W L µ V S 0 Q dv (33 HO #17: ELEN MOS C Model Page 28

29 Effect of High Vertical Electric Field n reality, µ i highly reduced by large vertical e-field. The vertical e-field pull the inverion layer e- toward the +V urface cauing: G V 1 more urface cattering 2 coulomb cattering due to interaction of e- with oxide charge (Q f, N it. Since e-field varie vertically through the inverion layer, the average field in the inverion layer i: E (E x1 + E x2 /2 (34 where E x1 vertical e-field at the Si-SiO 2 interface E x2 vertical e-field at the channel-depletion layer interface. From Gau Law: E x1 E x2 Q inv /K ε o and E x2 Q B /K ε o (35 From (34 & (35 we can how: E [(0.5Q inv + Q B ]/K ε o (36 S n+ y E x x P n+ HO #17: ELEN MOS C Model Page 29

30 Univeral Mobility Behavior n general, E [(Q B + ηq inv ]/K ε o where for <100> Si η 1/2 for electron η 1/3 for hole. Meaured µ v. E plot at low V how: a univeral behavior independent of doping concentration at high vertical field. dependence on: 1 doping concentration and 2 interface charge at low vertical field. univeral behavior HO #17: ELEN MOS C Model Page 30

31 Univeral Mobility Behavior The experimentally oberved mobility behavior i due to the relative contribution of different cattering mechanim et by the trength of vertical e-field: 1 Coulomb cattering by ionized impuritie and oxide charge. 2 Phonon cattering 3 Surface roughne cattering at the Si-SiO 2 interface. HO #17: ELEN MOS C Model Page 31

32 Univeral Mobility Behavior Surface roughne cattering a carrier confinement cloe to the interface at high vertical e-field. µ a E. The experimentally oberved univeral behavior occur becaue phonon cattering i weakly dependent on vertical e- field. eviation from the univeral behavior occur in heavily doped ubtrate at low field due to: dominant ionized impurity cattering at low inverion charge denitie. µ f(n ch. coulomb cattering by ionized impuritie in the depletion region oxide charge. Phonon cattering ha the tronget temperature dependence on µ. HO #17: ELEN MOS C Model Page 32

33 Effective Mobility due to High V G Subtituting for Q b from V th Eqn and Q in (36, we get: E VGS + V 6T OX th The dependence of µ on V G i decribed by an empirical relation: µ 0 µ ν E 1 + E0 (38 Where µ o the maximum extracted value of low-field mobility at a given doping concentration low-field urface mobility. ν 0.25 for electron and ν 0.15 for hole. E c 2.7x10 4 V/cm critical e-field above which µ o. (37 HO #17: ELEN MOS C Model Page 33

34 Mobility egradation due to V G By Taylor erie expanion of (38 and introducing V BS dependence, the vertical field mobility degradation model can be hown a: µ 0 µ 2 (39 V + 1 GS + Vth + + VGS Vth U a U b TOX TOX where U a and U b are the model parameter extracted from -V G characteritic. The V BS dependence i included by model parameter U c : µ µ 0 V + 1 ( GS + Vth VGS Vth U a U cvbs U b TOX TOX 2 (40 HO #17: ELEN MOS C Model Page 34

35 Effect of High Lateral Electric Field Additional complication arie becaue of the high lateral e-field. We ee that for e- in ilicon v d aturate near E ~ 10 4 V/cm and v d µe doe not hold. Since average e-field for hort channel device > 10 4 V/cm. mall geometry MOSFET device will operate at v d v at 10 7 cm/ec. Since µ contant, we mut account for high lateral e-field ect in the expreion for derived from imple theory. HO #17: ELEN MOS C Model Page 35

36 Mobility egradation due to V The velocity aturation of inverion carrier due to increaed lateral field E y caue: current aturation ooner than predicted by V SAT V G V th lower SAT than predicted by imple theory. The drift velocity due to high field ect i given by: µ Ey v d, for E > E β 1 β c 1+ E E ( [ ] y C Where E C v at /µ critical lateral field for velocity aturation β 2 for electron; β 1 for hole. For implicity of numerical olution, β 1 i ued. µ Ey vd 1+ µ E v y at (41 HO #17: ELEN MOS C Model Page 36

37 Mobility egradation due to V G and V The ective mobility due to the combine ect of V G and V i given by: µ Where θ C 1/LE C µ θ L channel length of MOSFET. Thu, µ i modeled by the parameter et: {(µ 0, θ, θ b, v at (E C }. ( VG Vth + θbvb + θcv (42 the model parameter are obtained by curve fitting the experimental data to the model equation. HO #17: ELEN MOS C Model Page 37

38 S Model in Strong nverion Aume a piecewie linear model, i.e., v d aturate abruptly at E E c : µ E vd, ( E Ec E 1+ Ec v E E (43 at, ( c v v at µ o E c E Where E lateral e-field and E c critical e-field at which carrier are velocity aturated, i.e. v v at. Now, the current at any point y in the channel i given by: Where (y WC ox [V G V th αv(y]v(y (44 V(y potential difference between the drain and channel at y. v(y carrier velocity at any point y in the channel. HO #17: ELEN MOS C Model Page 38

39 S Model in Strong nverion Subtituting (43 in (44 we get: dv ( y E( y Wµ C [ V V V y ] dy ox G th α ( E ntegrating (45 from y 0 to y L with correponding V(y 0 to V(y V, we get in the linear region (V S V SAT current: c (45 W µ C o 1 VG Vth α V 2 V L + S 1 EcL S V S (46 HO #17: ELEN MOS C Model Page 39

40 S Model in Strong nverion Let u define, V SAT the drain aturation voltage due to v at, i.e. at E E c. Uing thi condition in (45, we get in the aturation region (V S V SAT : ( V V V Wµ Cox G th α 2 SAT ncluding channel length modulation (CLM: E c (47 Wµ CoxEc 1 2 V S SAT ( V + G Vth αvsat VACLM where V ACLM channel length modulation parameter V (48 HO #17: ELEN MOS C Model Page 40

41 S Model in Strong nverion Since given by (46 and (47 mut be V S V SAT, therefore, equating (46 (47: V SAT E α E C C ( G Vth + ( V V L V L G th (49 Note that from (43, v at µ E c /2. Thu, the drain current model in trong inverion: Wµ Co 1 VG Vth αvs VS,for VGS > Vth, VS < V V S 2 L 1 + EcL SAT V V WCoxvat > V A S SAT ( VG Vth αvsat 1+,for VGS > Vth, VS VSAT (50 HO #17: ELEN MOS C Model Page 41

42 3. Sub-threhold Region Model S V th V GS Simple long channel device equation aume: S 0 for V GS < V th. n reality, S 0 and varie exponentially with V GS in a manner imilar to a bipolar tranitor. n order to develop a theory of ub-threhold conduction, let u conider MOS band diagram with applied ource (S and gate (G bia meaured with repect to the ubtrate (Sub, that i: V G Sub V GS φ φ V S Sub φ F φ B V S Sub E c E i E F E v V SSub V GSub. Gate SiO 2 p-si HO #17: ELEN MOS C Model Page 42

43 Sub-threhold Region Model n the ub-threhold (weak inverion region, we know: 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 2 when φ 2φ B, trong inverion (n urf p bulk i achieved and we obtain V th. φ φ B E c E i E F E v 3 For φ < 2φ F, the dominant charge preent near the urface are ionized acceptor atom, i.e. n urf << N A. Thu, there i no ε-field laterally along the urface ince Poion equation i the ame everywhere. Thu, any current flow mut be due to diffuion only. HO #17: ELEN MOS C Model Page 43

44 Sub-threhold Region Model dn Aq (51 dy n n+ 4 The e- gradient along the channel (dn/dy mut be contant in order to P maintain contant current. y... Aq[{n(0 n(l}/l] (52 So few e- that ε x 0 Now, we can now ue carrier tatitic to calculate n(0 and n(l. Referring band diagram on page 41: n n ( φ V φ q SSub B kt ( 0 nie (53 q( φ VSub φb kt ( L nie (54 Where φ i the urface potential with repect to the ubtrate. Alo, note that φ contant along the channel [ε(y 0]. N A HO #17: ELEN MOS C Model Page 44

45 Sub-threhold Region Model Since the charge in the ubtrate i aumed uniform (N ub, then from Poion equation: 2 d φ qn A de 2 (55 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 ε-field at the urface. Again, from Gau Law: ε E Q 2ε qn φ (56 X inv ε kt q 2ε qn SUB φ kt q b ε 2qN SUB φ SUB (57 HO #17: ELEN MOS C Model Page 45

46 Sub-threhold Region Model Uing (53, (54 and (57 in (52, we get: q W q ni e L Here A WX inv. ( φ V φ q( φ V φ SSub Uing V Sub V S + V SSub, kt F e Sub kt F kt q ε 2qN φ A (58 ( φ V φ q SSub F W ε kt ktni e 1 e L 2qNSUBφ qv kt S (59 To make ue of thi equation, we need to know how φ varie with the externally applied potential V G. V GSub V FB + φ + 2qε N C ox SUB φ (60 HO #17: ELEN MOS C Model Page 46

47 Sub-threhold Region Model Generally, it i more common to ue the ource potential a the reference o that: V GSub V GS + V SSub φ ψ + V SSub q( ψ φf W ε kt ktni e 1 e L 2qNSUBφ V GS V FB + ψ The depletion layer capacitance i: Therefore, from (62: + 2qε N SUB C ( ψ + V ox SSub C qv kt S qε N 2 ( ψ + V SUB SSub (61 (62 (63 dvgs dψ 1 qε N A C 2 ox ( ψ + V SSub C C ox n (64 HO #17: ELEN MOS C Model Page 47

48 Sub-threhold Region Model n order to eliminate ψ from (61 we expand V GS in a erie about the point ψ 1.5φ F (weak inverion correpond to φ F ψ 2φ F. * Where V and i obtained from (62. GS V GS ψ 1. 5φ F ( ψ 1. φ VGS VGS + n ψ φ F (64 F Combining (64 with (61 to eliminate ψ in the exponential and uing (63 to eliminate the quare root of ψ in (61, we obtain: * 2 q( V GS VGS qφf qvs W kt nkt 2kT kt µ Cni e 1 e (65 L q HO #17: ELEN MOS C Model Page 48

49 Sub-threhold Region Model Thu, the ub-threhold current i given by: * 2 q( V GS VGS qφf qvs W kt nkt 2kT kt µ Cni e 1 e (66 L q From (47 we note that: depend on V S only for mall V S, i.e. V S 3kT/q, ince exp[ qv S /kt 0 for larger V S. depend exponentially on V 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 SUB and V SSub enter through C. HO #17: ELEN MOS C Model Page 49

50 Sub-threhold Slope (S-factor n order to change by one decade, we get from (47: S (A 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 q V nkt S GS 60 ln10 S mv decade kt q C 1 + C V GS (V o ln10( n V th kt q C ln C o (@ room T... (67 kt C Slope ln10 q 1 + C mv 60 ( n decade o (@ room T HO #17: ELEN MOS C Model Page 50

51 Sub-threhold Model - Final Note n weak inverion or ubthrehold region, MOS device have exponential characteritic but are le icient than BJT becaue n > 1. Subthrehold lope S doe not cale and i contant. Therefore, V th can not be caled a required by the ideal caling law. V S affect V th a well a ubthrehold current. n order to optimize S, the deirable parameter are: thin oxide low N A high V Sub. HO #17: ELEN MOS C Model Page 51

52 Sub-threhold Region Model The ub-threhold current Eq. ued in BSM model i: ( qvs q VG Vth Voff S on 1 exp( exp VGS < V nkt nkt, for where V off offet voltage i a model parameter. (68 Thu, the piece-wie drain current model for different region of MOSFET operation: Wµ C qv kt 1 q V ( V V S G th off on 1 exp( exp, for VGS < ox VG Vth αvs VS, for VGS > Vth, V S 2 + nkt SAT HO #17: ELEN MOS C Model Page 52 V S < V L 1 EcL WC oxvat V V V A > S SAT ( VG Vth αvsat 1+,for VGS > Vth, VS VSAT th V th

53 MOS Threhold Voltage, V th Extraction V th i obtained by linear extrapolation from the maximum lope to d 0 of d - V g plot. Vd We know, d β Vg Vth Vd 2 efine, V th V d 0 and V d 50 mv Vd d β Vg Vth Vd 0 2 Vd Vth Vg 2 V 50 mv V 2 HO #17: ELEN MOS C Model Page 53 V th V g d V G

54 Subtrate Bia ependence of V th Vd 0.05V Tox 1.5nm HO #17: ELEN MOS C Model Page 54

55 Subtrate Bia ependence of V th Subtrate bia dependence of V th ubtrate i given by: for uniformly doped γ i obtained from: ( 2φ ± V φ V th V th0 ± γ 2 where γ th 2q N th C ub ox K F Sub F ε o body factor ( V V 0 V. 2φ + V 2φ F BS F Vth V th0 where lope γ y intercept γ 2φ γ γ 2φ F 2φ F + V BS HO #17: ELEN MOS C Model Page 55

56 Subtrate Bia ependence of V th Subtrate bia dependence of V th for non-uniformly doped ubtrate i given by the plot: th where ( V V 0 V. 2φ + V 2φ th γ 1 (k 1 body ect due to channel concentration γ 2 (k 2 body ect due to ubtrate concentration. V th i given by: F BS F Vth V th0 γ 2φ F γ 1 ( k 1 γ 2 ( k 2 2φ F + V BS ( ( φ V 2φ + γ 2φ V φ V th V th0 + γ F Sub F 2 F Sub F HO #17: ELEN MOS C Model Page 56

57 rain nduced Barrier Lowering (BL BL i defined a the hift in V th due to V d, epecially, in hort channel device. BL i defined a: 1 V th V th (V d-low V th (V d V dd 2 V V th d V th ( V d low ( V dd V V th ( V d d low V dd BL i calculated from: log( d V. V g plot at V d-low 50 mv V d V dd. Example: From Figure we get, V th ( V 80 mv HO #17: ELEN MOS C Model Page 57

58 Sub-threhold Slope (S S i the invere of log( d V. V g plot at V d-low and i given by: S 2.3 d dv g ( log( d To extract S: extract: d1 d (V th d2 2 dec. below d1 Calculate lope S 1/lope. HO #17: ELEN MOS C Model Page 58

59 on and off on and off can be extracted from d V g plot at V d V dd. on dat at off d at V d V g V dd V d V dd V g 0 Example: From Fig., on 440 µa/µm off 3 na/µm HO #17: ELEN MOS C Model Page 59

60 Home Work 7: ue June 2, For a ilicon MOSFET, conidering bulk charge ect in current calculate (a VTO, (b LAMBA, (c GAMMA, and (d BETA from the meaured data hown in table. VGS (V VS (V VBS (V (ma f E [0.5Q inv + Q B ]/ε i, where Q inv and Q B are the inverion charge and bulk/depletion charge under the gate, repectively and ε i i the dielectric contant of ilicon. The dependence of urface mobility, µ on proce parameter uch a T OX, N ub etc. and terminal voltage i lumped in E. Aume V GS > V th and mall V S : (a Show that E (V GS + V th /6T OX. (b f the ective mobility i modeled by: µ µ 0 /[1 + E /E 0 ] η, where µ µ V GS 0 and E 0 and η are parameter determined from the meaured data. Ue the expreion for E in part (a to how that: µ 0 µ 2 V 1 GS + Vth + + VGS Vth + U a U b TOX TOX Where U a and U b are the model parameter that are determined experimentally. HO #17: ELEN MOS C Model Page 60