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1 nmosfet Schematic Four structural masks: Field, Gate, Contact, Metal. Reverse doping polarities for pmosfet in N-well.

2 nmosfet Schematic 0 y L n + source n + drain depletion region polysilicon gate x z V g p-type substrate -V bs gate oxide inversion channel V ds W Source terminal: Ground potential. Gate voltage: V g Drain voltage: V ds Substrate bias voltage: V bs ψ(x,y): Band bending at any point (x,y). V(y): Quasi-Fermi potential along the channel. V(y0) 0, V(yL) V ds.

3 Drain Current Model Electron concentration: Electric field: E ( x, y) dψ dx ktn ε si a e qψ / kt + nxy (, ) qψ n 1 + kt N i a e n N i a qv / kt e ( e q( ψ V) kt qψ / kt qψ 1) kt Condition for surface inversion: Maximum depletion layer width at inversion: W ψ ( 0, y) V( y) + ψ B dm ( y) si [ V( y) + ψ ] ε qn a B

4 Gradual Channel Approximation Assumes that vertical field is stronger than lateral field in the channel region, thus -D Poisson s eq. can be solved in terms of 1-D vertical slices. Current density eq. (both drift and diffusion): J x y q n x y dv ( y ) n(, ) µ n (, ) dy Integrate in x- and z-directions, I y W dv dy Q y W dv dy Q V ds( ) µ eff i( ) µ eff i( ) x i where Qi ( y ) q n ( x, y ) dx 0 is the inversion charge/area. Current continuity requires I ds independent of y, integration with respect to y from 0 to L yields W V ds Ids µ eff ( Qi V ) dv L ( ) 0

5 Pao-Sah s Double Integral Change variable from (x,y) to (ψ,v), ni nxy (, ) n( ψ, V) N e Q ( V ) q i ψ ψ s B dx n( ψ, V ) dψ q dψ Substituting into the current expression, I ds qµ eff W L a ψ ψ s B q( ψ V)/ kt ( n i / Na ) e E( ψ, V ) q( ψ V ) / kt q( ψ V ) / kt Vds ψ s ( n i / Na ) e dψ 0 ψ B E( ψ, V ) dψ dv where ψ s (V) is solved by the gate voltage eq. for a vertical slice of the MOSFET: V Q ε ktn s si a qψ s ni V V C C kt N e ψ s + ψ fb + ψs + + g fb s ox ox a q( V)/ kt 1 /

6 Charge Sheet Approximation Assumes that all the inversion charges are located at the silicon surface like a sheet of charge and that there is no potential drop across the inversion layer. After the onset of inversion, the surface potential is pinned at ψ s ψ B + V(y). Depletion charge: Total charge: Inv. charge: Q qn W ε qn ( ψ + V) d a dm si a B Q C ( V V ψ ) C ( V V ψ V ) s ox g fb s ox g fb B Q Q Q C ( V V ψ V) + ε qn ( ψ + V) i s d ox g fb B si a B W V ds Substituting in Ids µ eff ( ) and integrate: L Qi ( V) dv 0 I C W V qn ds ε ds eff ox Vg Vfb B V si a µ ψ ds + L 3Cox [ ψ B V 3 / ds ψ 3 / ( ) ( B) ]

7 Linear Region I-V Characteristics For V ds << V g, I C W L V V 4εsiqN aψ B ds µ eff ox g fb ψ B Cox 4ε qn ψ si a B where Vt Vfb+ ψ B+ is the MOSFET threshold voltage. C ox V µ C W ( L V V ) V ds eff ox g t ds 1E Log( ) (arbitrary scale) Ids 1E- 1E-4 1E-6 1E Linear (arbitrary scale) Ids 1E Gate Voltage, V g (V) V on V t

8 Saturation Region I-V Characteristics Keeping the nd order terms in V ds : I C W L V V V m ds µ eff ox g t ds V ds ( ) εsiqna / 4ψ B Cdm tox where m is the body-effect coefficient. C ox C ox W dm (V dsat, I dsat ) V g4 I I C W ( Vg Vt) ds dsat µ eff ox L m when V ds V dsat (V g V t )/m. Drain Current V g V g3 V g1 Typically, m 1.. Drain Voltage

9 Pinch-off Condition From inversion charge density point of view, while Q V i ( ) C ( V V) ox g t Q ( V ) C ( V V mv ) I i ox g t W V ds µ L ( ) 0 ds eff i ( Q V ) dv At V ds V dsat (V g V t )/m, Q i 0 and I ds max. I ds V dsat 0 V ds V Source Drain Vg Vt m

10 Pinch-off from Potential Point of View V( y) V g Vt g Vt g Vt m V m y V L m Vds + y L V ds V (y ) At the pinch-off point, dv/dy V g V t m Q i /mc ox V ds L Gradual channel approximation breaks down. V(y) 0 0 Source V ds L Drain y Current is injected into the bulk depletion region.

11 Beyond Pinch-off

12 Subthreshold Region V ds Subthreshold region Saturation region Linear region V ds Vg Vt m Drain Current (arbitrary units) 1E+0 1E-1 1E- 1E-3 1E-4 1E-5 1E-6 1E-7 Low Drain Bias Drift Component Diffusion Component V t V g 1E Gate Voltage (V)

13 Q Subthreshold Currents s ε E si s 1/ qψ s ni q( ψ s V )/ kt ε siktn a + e kt Na Power series expansion: 1st term Q d, nd term Q i, Q i ε siqn ψ s a kt q n N i a e q( ψ V )/ kt s I ds µ eff W L ε siqn ψ s a kt q n N i a e ds ( 1 e ) qψ / kt qv / kt s or, I C W L m kt ds eff ox e µ ( 1) 1 q ( e ) qv ( V)/ mkt qv / kt g t ds Inverse subthreshold slope: S d(log Ids ) dv g 1 mkt kt q q 1 C C dm ox

14 Body Effect: Dependence of Threshold Voltage on Substrate Bias If V bs 0, I C W V qn ds ε ds eff ox Vg Vfb B V si a µ ψ ds L 3Cox V V + ψ + t fb B dv dv t bs ε qn εsiqna( ψb + Vbs ) C ox / ( ψ + V ) C si a B bs ox Threshold Voltage, (V) Vt t ox 00 Å V fb 0 3 / 3 / [( ψ B Vbs Vds) ( ψb Vbs) ] Substrate Bias Voltage, (V) V bs N a cm 3 N a cm 3

15 Dependence of Threshold Voltage on Temperature For n + poly gated nmosfet, V fb (E g /q) ψ B V t Eg + ψb + q 4ε qn ψ si a B C ox dv dt 1 de + 1+ q dt t g si a B ε qn / ψ dψ de B 1 g dψb + ( m 1) Cox dt q dt dt dv dt k ( m 1) ln q N N t c v N a m q 1 de dt g From Table.1, de g /dt ev/k and (N c N v ) 1/ cm 3. For N a cm 3 and m 1.1, dv t /dt is typically 1 mv/k.

16 MOSFET Channel Mobility x i µ nnxdx ( ) 0 µ eff xi nxdx ( ) 0 It was empirically found that when µ eff is plotted against an effective normal field E eff, there exists a universal relationship independent of the substrate bias, doping concentration, and gate oxide thickness (Sabnis and Clemens, 1979). 1 Here 1 Eeff Qd + Qi ε si Since Q 4ε qn ψ C ( V V ψ ) and Q i C ox (V g V t ), d si a B ox t fb B E eff V For n + poly gated nmosfet, t ox V fb ψ B Vg Vt + 3t 6t E eff ox Vt + 0. Vg V + 3t 6t ox ox t

17 N-channel MOSFET Mobility Low field region (low electron density): Limited by impurity or Coulomb scattering (screened at high electron densities). Intermediate field region: Limited by phonon scattering, 1/ E µ eff High field region (> 1 MV/cm): Limited by surface roughness scattering (less temp. dependence).

18 Temperature Dependence of MOSFET Current

19 P-channel MOSFET Mobility E eff 1 1 Qd + Qi ε 3 si In general, pmosfet mobility does not exhibit as universal behavior as nmosfet.

20 Electron and Hole Mobilities vs. Field (cm /V-s) µ eff E eff Electron -0.3 E eff Hole t ox35 Å t 70 Å ox - E eff 50 1 µm CMOS 0.1 µm CMOS -1 E eff E eff (MV/cm)

21 Intrinsic MOSFET Capacitance Subthreshold region: Linear region: C g Cg WL + WLC C ox C d WLC ox d Saturation region: Q( y) C ( V V ) 1 i ox g t C g WLC 3 ox y L V (y ) V g V t m Q i /mc ox V ds L V(y) 0 0 Source V ds L Drain y

22 Inversion Layer Capacitance Gate V g In the charge-sheet model, C i and Q i C ox (V g V t ). Q s C ox (inversion) C d Q d Q i C i In reality, inversion layer has a finite thickness and finite capacitance. p-type substrate (low freq.) n + channel d( Q ) dv g i C C ox C + C + C ox i d i Cox C / i C ox

23 Inversion Layer Capacitance 1st order approximation, C i Q i /(kt/q) and Q i C ox (V g V t ), therefore, C i /C ox (V g V t )/(kt/q). Inversion Charge Density, (µc/cm ) Q i N a cm 3 t ox 100 Å V fb 0 kt Qi Cox ( Vg Vt) ln 1 + q V t C ox (V g V t ) Gate Voltage, (V) V g qv ( g Vt) kt Note: Linearly extrapolated threshold voltage is typically (-4)kT/q higher than the threshold voltage V t at ψ s (inv.) ψ B.

24 Short-Channel Effect If L, I I C W ( Vg Vt) ds dsat µ eff ox L m And Cg WLCox. But 3 source gate-controlled barrier drain Threshold voltage becomes sensitive to channel length and drain bias. Source-Drain Current (A/µm) 1E-3 1E-4 1E-5 1E-6 1E-7 Log scale Slope ~q/kt V t Linear scale 1E Gate Voltage (V) Source-Drain Current (ma/µm)

25 Short-Channel V t Roll-off

26

27 Drain-Induced Barrier Lowering Drain current (A/cm) 1E+1 1E+0 1E-1 1E- 1E-3 1E-4 1E-5 L0. µm L.0 µm V ds ]3.0 V ]50 mv 1E-6 1E-7 L0.35 µm t ox 100 Å N a cm 3 1E Gate voltage (V)

28 Lateral Field Penetration -D Poisson s Eq.: E x y + E x y ρ qn ε ε ε si E x / x: gate controlled depletion charge. ε si E y / y: S/D controlled depletion charge. si si a Note that the characteristic length of exponential decay is independent of channel length.

29 MOSFET Geometry and Scale Length -D Analysis in a Simplified MOSFET Geometry L Gate Source n+poly Drain n+ tox N a n+ W d Substrate A -D boundary-value problem with Poisson s equation

30 Gate Source -t ox n+poly Drain 0 H G L y A F n + N a n+ W B C D E d Subthreshold region: In AFGH (oxide), ψ x ψ + y 0 x Substrate In ABEF (silicon), ψ x ψ + y qn a ε si To eliminate the boundary condition at the Si/oxide interface, the oxide region is replaced by an equivalent Si region (ε si /ε ox )t ox 3t ox thick. Boundary conditions: ψ ( 3t, y) V V ψ ( x, 0 ) ψ bi ψ ( xl, ) ψ bi + Vds ψ ( Wd, y) 0 ox g fb along GH, along AB, along EF, along CD.

31 General approach to a -D boundary value problem Source Gate -t ox n+poly Drain 0 H G L y A F n + N a n+ W B C D E d x Substrate Let: ψ ( x, y) vxy (, ) + u ( x, y) + u ( x, y) + u ( x, y) L R B v(x,y) is a solution to the inhomogeneous equation and satisfies the top boundary condition. u L, u R, u B are solutions to the homogeneous equation such that ψ(x,y) satisfies the other B.C. s.

32 Gate For satisfying the Boundary conditions: Source -t ox n+poly Drain 0 H G L y A F n + N a n+ W B C D E d u ( x, y) b L n 1 * n nπ ( L y) sinh Wd + 3t ox nπ ( x+ 3tox ) sin n L Wd + t π ox sinh 3 Wd + 3t ox x Substrate u ( x, y) c R n 1 * n u ( x, y) d B n 1 nπy sinh Wd + 3t nπl sinh Wd + 3t * n ox ox nπ ( x+ 3tox ) sin Wd + 3tox nπ ( x+ 3tox ) sinh L nπy sin nπ ( Wd + 3tox) L sinh L Note that for usin(kx), d u/dx -k u; And that for usinh(ky), d u/dy k u.

33 MOSFET Scale Length V t 4t ox ψbi ( ψbi + Vds ) e W dm πl/ W + 3t dm ox Short-channel threshold roll-off (V) L /( W + 3 min dm t ) ox m Define scale length, λ W dm + (ε si /ε ox )t ox To keep short-channel effect under control, L min should be kept larger than about λ. m V / ψ 1 + 3t /W g s ox dm

34 Depletion Width Scaling W 0 dm 4ε sikt ln( N a / ni ) qn a 10 Maximum Depletion Width (µm) E E E E E E+19 Substrate Doping Concentration (cm-3)

35 Generalized Scale Length Source Gate ε i ψ 1 ε si ψ Body L t i W d Drain D. Frank et al., EDL 10/98 In the one-region model, the eigenvalues are: k n W d nπ + 3t ox For two regions, assume eigenvalues: k n π λ n u u L1 L ( x, y) ( x, y) n 1 n 1 b b n1 n π ( L sinh λn π ( L sinh λn y) π ( x + t sin λn y) π ( x W sin λn i ) d ) B.C. at x 0: u L1 u L (du L1 /dy du L /dy) and ε i du L1 /dx ε si du L /dx ε si tan(π t i /λ n ) + ε i tan(π W d /λ n ) 0

36 Generalized Scale Length Lowest eigenvalue: ε si tan(π t i /λ 1 ) + ε i tan(π W d /λ 1 ) 0 t i /λ 1 Normalized Gate Insulator Thickness, ε /ε 1 i si 1/3 (SiO ) Normalized Si Depletion Depth, W d /λ 1 ψ SCE exp( πl/λ 1 ) L min ~ λ 1 Note that: λ 1 >W d, and λ 1 >t i λ 1 W d t i is always a point of symmetry regardless of ε i, ε si. If ε i ε si, λ 1 W d + t i

37 MOSFET Body Effect Gate Oxide Silicon Depletion boundary ψ s Potential change V g 3t ox W dm (ε si /ε ox 3) m V / ψ 1 + 3t /W g s ox dm

38 MOSFET Design Space with SiO Gate Oxide Thickness (nm) nm 10 nm 15 nm λ0 nm SiO W d 10tox m1.3 To obtain a good subthreshold slope, the body-effect coefficient, m V / ψ 1 + 3t /W g s ox dm should be kept close to unity Depletion Depth in Si (nm) In the intercept region, λ W d + 3t ox is a good approximation. L min ~1.5λ ~ 0tox

39 High-k Gate Insulator t /λ i Normalized Gate Insulator Thickness, ε /ε 1 i si 1/3 (SiO ) Normalized Si Depletion Depth, W /λ d High-k gate insulator is an active area of Si research because it may replace SiO thereby circumventing the tunneling problem. But λ ~ W d + (ε si /ε i )t i is valid only when t i << λ. In general, requires t i < λ/, regardless of ε i.

40 Velocity Saturation Because of velocity saturation, the saturation of drain current in a shortchannel device occurs at a much lower voltage than V dsat (V g V t )/m for long channel devices. This causes the saturation current, I dsat, to deviate from the (V g V t ) behavior and from the 1/L dependence.

41 Velocity-Field Relationship eff v n 1/ n E 1 + E At low fields, v µ eff E : Ohm s law. µ As E, v v sat µ eff E c. Critical Field: E c E It is commonly believed that: c v µ n for electrons, n 1 for holes. sat eff v sat is independent of µ eff (vertical field), but E c depends on µ eff. Only the n 1 case can be solved analytically.

42 Analytical Solution for n1 µ eff ( dv / dy) Ids WQv i WQi( V) 1+ ( µ / v )( dv / dy) Current continuity requires that I ds be a constant, independent of y. µ eff Ids dv Ids µ effwqi ( V) + v sat dy Multiplying dy on both sides and integrating from y 0 to L and from V 0 to V ds, one solves for I ds : Charge-sheet model: Therefore, I ds I ds eff sat ds eff W L Qi V dv µ ( / ) ( ) 0 1+ ( µ V / v L) µ V eff ds sat Q ( V ) C ( V V mv ) i ox g t [ ] C ( W / L) ( V V ) V ( m/ ) V eff ox g t ds ds 1+ ( µ V / v L) eff ds sat

43 Saturation Drain Voltage and Current The saturation voltage, V dsat, can be found by solving di ds /dv ds 0: V dsat ( Vg Vt)/ m µ ( V V )/( mv L) eff g t sat And the saturation current is: Drain saturation current (ma/µm) I C Wv ( V V ) dsat ox sat g t 0.6 t ox 100Å L0 L.5 µm 0.3 L0.5 µm L10 µm Gate Overdrive, V g V t, (V) 1+ µ eff ( Vg Vt )/( mvsatl) 1 1+ µ ( V V )/( mv L) + 1 eff g t sat (Dashed: long-ch. model, solid: velocity sat. model)

44 Velocity-Saturation-Limited Current At the drain end of the channel when V ds V dsat, Q( y L) C ( V V mv ) i ox g t dsat and I dsat Wv sat Q i (y L), i.e., carriers move at the saturation velocity. This implies that dv/dy at the drain. Therefore, the gradual channel approximation breaks down and the carriers are no longer confined to the surface channel. I C Wv ( V V ) dsat ox sat g t When (V g V t ) << mv sat L/µ eff, I C W ( Vg Vt) dsat µ eff ox L m Long channel limit. 1+ µ ( V V )/( mv L) 1 eff g t sat 1+ µ ( V V )/( mv L) + 1 eff g t sat In the limit of L 0, I C Wv ( V V ) dsat ox sat g t Velocity saturation limited current.

45 Velocity Overshoot: Monte Carlo Simulation Velocity saturation is derived from the drift and diffusion model which assumes that carriers are always in thermal equilibrium with the silicon lattice. But if the MOSFET is only a few mean free path (~10 nm) long, carriers do not travel enough distance to establish equilibrium velocity overshoot, i.e., carrier velocity at the high field region near the drain can exceed the saturation velocity.

46 Velocity Overshoot Monte-Carlo simulation: Even in a 30 nm device, nfet/pfet velocity and therefore current ratio is still because of the difference in effective masses. The velocity at the source does not greatly exceed 10 7 cm/s; therefore, current does not greatly exceed I C Wv ( V V ) dsat ox sat g t

47 Distribution Function Fermi-Dirac distribution under equilibrium: 1 f ( E) ( f)/ 1 + kt e E E The standard semi-classical transport theory is based on the Boltzmann transport equation (BTE): f t + v r f + ee h k f { S( k, k) f ( r, k, t) [ 1 f ( r, k, t) ] S( k, k ) f ( r, k, t) [ 1 f ( r, k, t) ]} k where r is the position, k is the momentum, f(r,k,t) is the distribution function, v is the group velocity, E is the electric field, S(k,k ) is the transition probability between the momentum states k and k. The summation on the right hand side is the collison term, which accounts for all the scattering events. The terms on the left hand side indicate, respectively, the dependence of the distribution function on time, space (explicitly related to velocity), and momentum (explicitly related to electric field).

48 Velocities at the Source and at the Drain I WC (V - V )v ds ox g t s I ds WQ iv Source E 0 I ds WC ox (V g - V t - V dsat)vd 1 Drain r Inversion charge density at the source is given by C ox (V g -V t ). Inversion charge density at the drain is much lower because of the drain bias. Current continuity is maintained consistent with band bending.

49 Scattering theory Source E 0 At high drain bias, T 0, I ds TI Let rn s- /n s+, the backscattering coefficient. + 1 Drain r Ref. Lundstrom, EDL, p.361, 1997 Then I / W C ( V V ) v ds ox g t T 1 1+ r r Note that r depends on the low-field mobility near the source. In the ballistic limit, no collisions in the channel, i.e., r 0, and I / W C ( V V ) v ds ox g t T

50 Carrier Thermal Injection Velocity For -D nondegenerate carriers, EN ( E) f ( E) de 0 E N ( E) f ( E) de so 0 kt v rms m kt For uni-directional injection, v T v x 0 0 exp( mv exp( mv x x / kt ) dv / kt ) dv x x kt πm

51 Injection Velocity in the Degenerate Case At 0 K, all states below the Fermi energy are filled. In -D, define a Fermi circle with velocity v F. Since m πh v x 1 mv 1 N( E) E F F v x > 0 v x n v > 0 s x dv dv x C x dv dv ox y ( V y g q V ) 4 v 3π t F v T 4 Cox( Vg Vt ) h 3m qπ

52 I-V Curves of a Ballistic MOSFET Natori, JAP, p. 4879, I ds 4h Cox / W Cox( Vg Vt 3m qπ Independent of L! 3/ ) (T0 K)

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