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nMOSFET Schematic Four structural masks: Field, Gate, Contact, Metal. Reverse doping polarities for pMOSFET in N-well. nMOSFET Schematic polysilicon gate Source terminal: Ground potential. Gate voltage: Vg Drain voltage: Vds Substrate bias voltage: −Vbs gate oxide Vg Vds z n+ source depletion region 0 y L n+ drain x inversion channel p-type substrate -Vbs W ¾ψ(x,y): Band bending at any point (x,y). ¾V(y): Quasi-Fermi potential along the channel. ¾V(y=0) = 0, V(y=L) = Vds. Drain Current Model Electron concentration: ni 2 q ( ψ −V ) kT e n( x , y) = Na Electric field: 2 2kTN a dψ E ( x, y ) = = ε si dx 2 − qψ / kT qψ ni 2 − qV / kT qψ / kT qψ (e + − 1 + 2 e − 1) − e kT kT Na Condition for surface inversion: ψ ( 0 , y ) = V ( y) + 2ψ B Maximum depletion layer width at inversion: Wdm ( y ) = 2ε si [V ( y ) + 2ψ B ] qN a Gradual Channel Approximation Assumes that vertical field is stronger than lateral field in the channel region, thus 2-D Poisson’s eq. can be solved in terms of 1-D vertical slices. Current density eq. (both drift and diffusion): dV ( y) Jn ( x, y) = −qµ n n( x, y) dy Integrate in x- and z-directions, dV dV I ds ( y ) = − µ eff W Qi ( y ) = − µ eff W Qi (V ) dy dy xi where Qi ( y ) = − q ∫ n( x , y )dx is the inversion charge/area. 0 Current continuity requires Ids independent of y, integration with respect to y from 0 to L yields W Vds I ds = µeff ∫ ( −Qi (V )) dV L 0 Pao-Sah’s Double Integral Change variable from (x,y) to (ψ,V), ni 2 q (ψ −V )/ kT n( x, y) = n(ψ , V ) = e Na 2 q (ψ −V ) / kT ψB ψ s ( n / N )e dx i a Qi (V ) = − q ∫ n(ψ , V ) dψ = − q ∫ dψ ψs ψ B E (ψ , V ) dψ Substituting into the current expression, 2 W Vds ψ s (ni / N a )e q (ψ −V ) / kT I ds = qµ eff dψ dV ∫ψ B ∫ 0 L E (ψ , V ) where ψs(V) is solved by the gate voltage eq. for a vertical slice of the MOSFET: Qs V g = V fb + ψ s − = V fb + ψ s + Cox 2ε si kTN a qψ s ni 2 q (ψ s −V ) / kT + e Cox Na2 kT 1/ 2 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 ψs = 2ψB + V(y). Depletion charge: Total charge: Inv. charge: Qd = −qN a Wdm = − 2ε si qN a (2ψ B + V ) Qs = −Cox (Vg − V fb − ψ s ) = −Cox (Vg − V fb − 2ψ B − V ) Qi = Qs − Qd = −Cox (V g − V fb − 2ψ B − V ) + 2ε si qNa ( 2ψ B + V ) W Vds Substituting in I ds = µeff ∫0 ( −Qi (V )) dV and integrate: L 2 2ε si qN a W Vds 3/ 2 3/ 2 I ds = µeff Cox Vg − V fb − 2ψ B − [(2ψ B + Vds ) − (2ψ B ) ] Vds − L 2 3Cox Linear Region I-V Characteristics For Vds << Vg , 4ε si qN aψ B W I ds = µ eff Cox Vg − V fb − 2ψ B − L Cox 4ε si qNaψ B Cox is the MOSFET threshold voltage. 1E+0 0.8 1E-2 0.6 1E-4 0.4 1E-6 0.2 1E-8 1E-10 0 0.5 1 1.5 2 Gate Voltage, Vg (V) V on≈V t 2.5 3 0 Linear Ids (arbitrary scale) Log(Ids ) (arbitrary scale) where Vt = V fb + 2ψ B + W Vds = µ eff Cox (Vg − Vt )Vds L Saturation Region I-V Characteristics Keeping the 2nd order terms in Vds: I ds = µ eff Cox where m = 1 + ε si qN a / 4ψ B Cox = 1+ Cdm 3t = 1 + ox Cox Wdm W L m 2 ( V − V ) V − Vds t ds g 2 is the body-effect coefficient. (Vdsat, Idsat) I ds = I dsat 2 W (Vg − Vt ) = µ eff Cox 2m L when Vds = Vdsat = (Vg − Vt)/m. Drain Current Vg4 Vg3 Vg2 Vg1 Typically, m ≈ 1.2. Drain Voltage Pinch-off Condition From inversion charge density point of view, Q i (V ) = −C ox (V g − Vt − mV ) W while I ds = µeff L V ds ∫ (−Q (V ))dV i 0 −Qi (V ) At Vds = Vdsat = (Vg − Vt)/m, Qi = 0 and Ids = max. Cox (Vg − Vt ) ∝ Ids 0 Source Vdsat Vds Drain = Vg − Vt m V Pinch-off from Potential Point of View V ( y) = Vg − Vt m 2 V − Vt y Vg − Vt − g − 2 L m m At the pinch-off point, dV/dy → ∞ V (y ) V g −V t m ⇒ Gradual channel approximation breaks down. L′ Vds |Qi|/mCox V(y) 0 y 2 + V ds L Vds 0 Source Current is injected into the bulk depletion region. Vds L Drain y Beyond Pinch-off Subthreshold Region Vds Saturation region Subthreshold region Linear region Vt 1E+0 Vg − Vt m Drain Current (arbitrary units) Vds = Vg Low Drain Bias 1E-1 1E-2 Diffusion Component 1E-3 1E-4 1E-5 1E-6 Drift Component 1E-7 1E-8 0 0.5 1 Gate Voltage (V) 1.5 2 Subthreshold Currents qψ s ni 2 q (ψ s −V ) / kT − Qs = ε si E s = 2ε si kTN a + 2e kT N a 1/ 2 Power series expansion: 1st term Qd, 2nd term Qi, 2 ε si qN a kT ni q (ψ s −V )/ kT − Qi = e 2ψ s q N a ⇒ W I ds = µ eff L ε si qN a kT 2ψ s q 2 2 ni qψ s / kT − qV ds / kT e e 1 − N a ( 2 ) kT q (Vg −Vt )/ mkT W ( 1 − e − qVds / kT ) or, I ds = µ eff Cox (m − 1) e L q Inverse subthreshold slope: −1 d (log I ds ) mkT kT Cdm = 2.3 1 + S = = 2.3 dV q q C g ox Body Effect: Dependence of Threshold Voltage on Substrate Bias If Vbs ≠ 0, I ds = µeff Cox W L 2 2ε si qN a Vds − V − V − 2 ψ − V ( 2ψ B + Vbs + Vds )3/ 2 − ( 2ψ B + Vbs )3/ 2 g ds fb B 2 3Cox [ ⇓ ε si qN a / 2(2ψ B + Vbs ) dVt = dVbs Cox tox=200 Å Threshold Voltage, Vt (V) 2ε si qNa ( 2ψ B + Vbs ) Cox ] 1.8 ⇓ Vt = V fb + 2ψ B + 1.6 Na=1016 cm−3 Vfb=0 1.4 1.2 Na=3×1015 cm−3 1 0.8 0.6 0 2 4 6 Substrate Bias Voltage, Vbs (V) 8 10 Dependence of Threshold Voltage on Temperature For n+ poly gated nMOSFET, Vfb = − (Eg /2q) − ψB ⇒ 4ε si qNaψ B Vt = − + ψB + Cox 2q Eg ε si qNa / ψ B 1 dEg dVt =− + 1 + 2q dT dT Cox ⇒ dψ 1 dE g dψ B =− + ( 2 m − 1) B 2q dT dT dT dVt k N c N v 3 m − 1 dE g = − (2m − 1) ln + + dT q N a 2 q dT From Table 2.1, dEg/dT ≈ −2.7×10−4 eV/K and (NcNv)1/2 ≈ 2.4×1019 cm−3. For Na ∼ 1016 cm−3 and m ∼ 1.1, dVt/dT is typically −1 mV/K. MOSFET Channel Mobility µ eff ∫ = xi µ n n( x )dx 0 ∫ xi 0 n( x )dx It was empirically found that when µeff is plotted against an effective normal field Eeff, there exists a “universal relationship” independent of the substrate bias, doping concentration, and gate oxide thickness (Sabnis and Clemens, 1979). 1 1 Here Eeff = Qd + Qi ε si 2 Since Qd = 4ε si qN a ψ B = Cox (Vt − V fb − 2ψ B ) and |Qi| ≈ Cox(Vg − Vt), ⇒ For n+ Eeff = Vt − V fb − 2ψ B 3tox poly gated nMOSFET, + Vg − Vt 6tox Eeff = Vt + 0.2 Vg − Vt + 3tox 6tox 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, µ eff ≈ 32500 × E −1/ 3 High field region (> 1 MV/cm): Limited by surface roughness scattering (less temp. dependence). Temperature Dependence of MOSFET Current P-channel MOSFET Mobility Eeff = 1 1 Q Qi + d 3 ε si In general, pMOSFET mobility does not exhibit as “universal” behavior as nMOSFET. Electron and Hole Mobilities vs. Field 1000 -0.3 tox=35 Å tox=70 Å Eeff 100 2 (cm /V-s) 300 µeff 500 200 Electron -0.3 Eeff -2 Eeff Hole -1 50 30 0.1 1 µm CMOS 0.2 0.3 0.1 µm CMOS 0.5 1 Eeff (MV/cm) Eeff 2 3 Intrinsic MOSFET Capacitance −1 1 1 C = WL + Subthreshold region: g ≈ WLCd Cox Cd Linear region: Cg = WLCox Saturation region: y Qi ( y ) = −Cox (Vg − Vt ) 1 − L ⇒ 2 Cg = WLCox 3 V (y ) V g −V t m L′ |Qi|/mCox V(y) 0 Vds 0 Source Vds L Drain y Inversion Layer Capacitance Gate In the charge-sheet model, Ci = ∞ and Qi = Cox(Vg − Vt). Vg Cox −Qs (inversion) C d Q d Q i (low freq.) p-type substrate n+ channel C i In reality, inversion layer has a finite thickness and finite capacitance. d ( − Qi ) C ox Ci 1 = ≈ C ox 1 − dV g C ox + Ci + C d 1 + Ci / C ox Inversion Layer Capacitance 1st order approximation, Ci ≈ |Qi|/(2kT/q) and |Qi| ≈ Cox(Vg − Vt), therefore, Ci/Cox = (Vg − Vt)/(2kT/q). Inversion Charge Density, Qi (µC/cm2) 2 kT q (Vg − Vt ) − Qi = Cox (Vg − Vt ) − ln1 + 2 q kT 0.8 Na=5×1016 cm−3 Note: Linearly extrapolated threshold voltage is typically (2-4)kT/q higher than the threshold voltage Vt at ψs(inv.) = 2ψB. tox=100 Å 0.6 Vfb=0 Cox(Vg−Vt) 0.4 0.2 0 Vt 0 0.5 1 1.5 2 2.5 Gate Voltage, Vg (V) 3 3.5 Short-Channel Effect If L ↓, I ds = I dsat 2 W (Vg − Vt ) = µ eff Cox ↑ 2m L source drain Threshold voltage becomes sensitive to channel length and drain bias. 1E-3 1 Log scale 1E-4 0.8 1E-5 0.6 Slope ~q/kT 1E-6 1E-7 1E-8 Vt 0 0.2 0.4 0.6 0.8 Gate Voltage (V) 0.4 Linear scale 1 0.2 0 1.2 Source-Drain Current (mA/µm) gate-controlled barrier Source-Drain Current (A/µm) 2 And Cg = WLCox ↓. But …… 3 Short-Channel Vt Roll-off Drain-Induced Barrier Lowering 1E+1 Drain current (A/cm) 1E+0 Vds= ]3.0 V L=0.2 µm ]50 mV 1E-1 1E-2 1E-3 1E-4 L=2.0 µm 1E-5 1E-6 1E-7 1E-8 -0.5 L=0.35 µm 0 0.5 Gate voltage (V) tox=100 Å Na=3×1016 cm−3 1 1.5 Lateral Field Penetration 2-D Poisson’s Eq.: qN ∂E x ∂E y ρ + = =− a ∂ x ∂ y ε si ε si εsi∂Ex/∂x: gate controlled depletion charge. εsi∂Ey/∂y: S/D controlled depletion charge. Note that the characteristic length of exponential decay is independent of channel length. MOSFET and Scale Geometry Length 2-D AnalysisGeometry in a Simplified MOSFET L Gate n+ poly Source n+ t ox Na Drain Wd n+ Substrate A 2-D boundary-value problem with Poisson’s equation Gate n+ poly -t ox Source 0H A n+ B Wd C Na D Drain y GL F E n+ x Substrate To eliminate the boundary condition at the Si/oxide interface, the oxide region is replaced by an equivalent Si region (εsi/εox)tox ≈ 3tox thick. Subthreshold region: In AFGH (oxide), ∂ 2ψ ∂ 2ψ + =0 ∂ x 2 ∂ y2 In ABEF (silicon), ∂ 2ψ ∂ 2ψ qN a + = ∂ x 2 ∂ y2 ε si Boundary conditions: ψ ( −3tox , y ) = Vg − V fb ψ ( x ,0 ) = ψ bi ψ ( x , L ) = ψ bi + Vds ψ (Wd , y ) = 0 along GH, along AB, along EF, along CD. General approach to a 2-D boundary value problem Gate Source n+ poly -t ox 0H A n+ Wd B C Na D Drain y GL F E n+ x Substrate Let: ψ ( x , y ) = v ( x , y ) + uL ( x , y ) + uR ( x , y ) + uB ( x , y ) v(x,y) is a solution to the inhomogeneous equation and satisfies the top boundary condition. uL, uR, uB are solutions to the homogeneous equation such that ψ(x,y) satisfies the other B.C.’s. Gate Source For satisfying the Boundary conditions: nπ ( L − y ) sinh W + 3t ∞ nπ ( x + 3tox ) * ox d uL ( x , y ) = ∑ bn sin Wd + 3tox nπ L n =1 sinh Wd + 3tox nπy sinh W + 3t ∞ nπ ( x + 3tox ) * ox d uR ( x, y ) = ∑ cn sin nπL Wd + 3tox n =1 sinh Wd + 3tox nπ ( x + 3tox ) sinh ∞ L * sin nπy uB ( x, y ) = ∑ d n nπ (Wd + 3tox ) L n =1 sinh L n+ poly -t ox 0H A n+ Wd B C Na D Drain y GL F E n+ x Substrate Note that for u=sin(kx), d2u/dx2=-k2u; And that for u=sinh(ky), d2u/dy2=k2u. MOSFET Scale Length Short-channel threshold roll-off (V) 24t ox ∆Vt = Wdm ψ bi (ψ bi + Vds ) e 1 1.2 1.4 To keep short-channel effect under control, Lmin should be kept larger than about 2λ. 1.5 0.4 0.2 0 Wdm + 3t ox λ ≡ Wdm + (εsi/εox)tox 1.3 0.6 πL / 2 Define scale length, m= 0.8 − 0 1 2 Lmin / (Wdm + 3tox ) 3 4 m = ∆Vg /∆ψs = 1 + 3tox /Wdm Depletion Width Scaling 0 Wdm = 4ε si kT ln( N a / ni ) q2 N a Maximum Depletion Width (µm) 10 1 0.1 0.01 1.0E+14 1.0E+15 1.0E+16 1.0E+17 1.0E+18 Substrate Doping Concentration (cm-3) 1.0E+19 Generalized Scale Length εi Source Gate ψ1 εsi ψ2 In the one-region model, the eigenvalues are: ti Wd kn = Drain For two regions, assume eigenvalues: Body L kn = π λn D. Frank et al., EDL 10/98 ∞ π ( L − y) π ( x + t i ) sin u L1 ( x, y ) = ∑ bn1 sinh λ λ n =1 n n π ( L − y ) π ( x − Wd ) sin u L 2 ( x, y ) = ∑ bn 2 sinh λ λ n =1 n n ∞ nπ Wd + 3t ox B.C. at x = 0: uL1 = uL2 (duL1/dy = duL2/dy) and εiduL1/dx = εsiduL2/dx ⇒ εsi tan(π ti /λn) + εi tan(π Wd /λn) = 0 Generalized Scale Length Lowest eigenvalue: εsi tan(π ti /λ1) + εi tan(π Wd /λ1) = 0 Normalized Gate Insulator Thickness, ti /λ1 1 ∆ψSCE∝exp(−πL/2λ1) Lmin ~ 2λ1 ε i /ε si= 0.8 3 1 1/3 (SiO2) Note that: λ1>Wd, and λ1>ti 0.6 10 0.4 λ1=2Wd =2ti is always a point of symmetry regardless of εi, εsi. 0.2 0 0 0.2 0.4 0.6 0.8 Normalized Si Depletion Depth, Wd /λ1 1 If εi=εsi, λ1=Wd + ti MOSFET Body Effect Depletion boundary Gate Oxide Silicon Potential ∆ψ change s ∆Vg 3tox Wdm (εsi /εox =3) m = ∆Vg /∆ψs = 1 + 3tox /Wdm MOSFET Design Space with SiO2 Gate Oxide Thickness (nm) 10 λ=20 nm 8 SiO2 To obtain a good subthreshold slope, the body-effect coefficient, 15 nm 10 nm 6 5 nm m = ∆Vg /∆ψs = 1 + 3tox /Wdm 4 Wd =10tox 2 0 m=1.3 0 5 10 15 Depletion Depth in Si (nm) In the intercept region, λ = Wd + 3tox is a good approximation. should be kept close to unity. 20 Lmin ~1.5λ ~ 20t ox High-k Gate Insulator Normalized Gate Insulator Thickness, ti /λ 1 High-k gate insulator is an active area of Si research because it may replace SiO2 thereby circumventing the tunneling problem. ε i /ε si= 0.8 3 1 1/3 (SiO2) 0.6 10 0.4 0.2 0 0 0.2 0.4 0.6 Normalized Si Depletion Depth, 0.8 Wd /λ 1 But λ ~ Wd + (εsi/εi)ti is valid only when ti << λ. In general, requires ti < λ/2, regardless of εi. Velocity Saturation Because of velocity saturation, the saturation of drain current in a shortchannel device occurs at a much lower voltage than Vdsat = (Vg − Vt)/m for long channel devices. This causes the saturation current, Idsat, to deviate from the ∝ (Vg − Vt)2 behavior and from the 1/L dependence. Velocity-Field Relationship v= µ eff E E n 1 + Ec 1/ n At low fields, v = µeffE : Ohm’s law. As E → ∞, v = vsat = µeffEc. Critical Field: Ec = vsat µ eff It is commonly believed that: n = 2 for electrons, n = 1 for holes. vsat is independent of µeff (vertical field), but Ec depends on µeff. Only the n = 1 case can be solved analytically. Analytical Solution for n=1 I ds = −WQi v = −WQi (V ) µ eff ( dV / dy ) 1 + ( µ eff / v sat )( dV / dy ) Current continuity requires that Ids be a constant, independent of y. µeff I ds dV I = − µ WQ ( V ) + ⇒ ds i eff vsat dy Multiplying dy on both sides and integrating from y = 0 to L and from V = 0 to Vds, one solves for Ids: V ds I ds = − µeff (W / L )∫ Qi (V )dV 0 1 + ( µeff Vds / vsat L ) Charge-sheet model: Therefore, I ds = Qi (V ) = −Cox (Vg − Vt − mV ) [ µ eff Cox (W / L ) (Vg − Vt )Vds − ( m / 2 )Vds 2 1 + ( µ eff Vds / vsat L ) ] Saturation Drain Voltage and Current The saturation voltage, Vdsat, can be found by solving dIds/dVds = 0: And the saturation current is: Drain saturation current (mA/µm) 0.6 Vdsat = 2(Vg − Vt ) / m 1 + 1 + 2 µeff (Vg − Vt ) / ( mvsat L ) I dsat = CoxWvsat (Vg − Vt ) 1 + 2µeff (Vg − Vt ) / ( mvsat L ) − 1 1 + 2µeff (Vg − Vt ) / ( mvsat L ) + 1 tox=100Å 0.5 0.4 L=2.5 µm L=0 0.3 L=0.5 µm (Dashed: long-ch. model, solid: velocity sat. model) 0.2 0.1 L=10 µm 0 0 1 2 3 Gate Overdrive, Vg−Vt , (V) 4 5 Velocity-Saturation-Limited Current At the drain end of the channel when Vds = Vdsat, Qi ( y = L ) = −Cox (Vg − Vt − mVdsat ) and Idsat = −WvsatQi(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 dsat = CoxWvsat (Vg − Vt ) When (Vg − Vt) << mvsatL/2µeff, 2 W (Vg − Vt ) I dsat = µ effCox L 2m Long channel limit. 1 + 2µeff (Vg − Vt ) / ( mvsat L ) − 1 1 + 2µeff (Vg − Vt ) / ( mvsat L ) + 1 In the limit of L → 0, I dsat = CoxWvsat (Vg − Vt ) Velocity saturation limited current. 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. Velocity Overshoot Monte-Carlo simulation: ¾ Even in a 30 nm device, nFET/pFET velocity and therefore current ratio is still ≈ 2 because of the difference in effective masses. ¾ The velocity at the source does not greatly exceed 107 cm/s; therefore, current does not greatly exceed I dsat = CoxWvsat (Vg − Vt ) Distribution Function Fermi-Dirac distribution under equilibrium: f ( E) = 1 1+ e ( E − E f )/ kT The standard semi-classical transport theory is based on the Boltzmann transport equation (BTE): eE ∂f + v ⋅ ∇r f + ⋅ ∇ 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 ) ]} h ∂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). Velocities at the Source and at the Drain Ids = WQi v Ids = WCox (Vg - Vt )vs Source E0 Ids = WCox (Vg - Vt - Vdsat)vd 1 Drain r Inversion charge density at the source is given by Cox(Vg-Vt). Inversion charge density at the drain is much lower because of the drain bias. Current continuity is maintained consistent with band bending. Scattering theory At high drain bias, T’=0, Source E0 I ds = TI + Let r=ns-/ns+, the backscattering coefficient. 1 Drain r Ref. Lundstrom, EDL, p.361, 1997 Then 1− r I ds / W = Cox (Vg − Vt )v T r + 1 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 ds / W = Cox (Vg − Vt )vT Carrier Thermal Injection Velocity For 2-D nondegenerate carriers, ∞ EN ( E ) f ( E ) dE ∫ ⟨E⟩ = = kT ∫ N ( E ) f ( E )dE 0 ∞ 0 so vrms = 2 kT m For uni-directional injection, vT = ∫ ∞ 0 ∫ 2 v x exp(−mv x / 2kT )dv x ∞ 0 2 exp(−mv x / 2kT )dv x = 2kT πm Injection Velocity in the Degenerate Case At 0 K, all states below the Fermi energy are filled. In 2-D, define a Fermi circle with velocity vF. ∫∫ v dv dv x ⟨ vx ⟩ = x v x >0 ∫∫ dv dv x y y 4 = vF 3π v x >0 Since Cox (Vg − Vt ) m 1 1 2 N ( E ) EF = 2 mvF = ns = πh 2 q 2 vT = 4h 2Cox (Vg − Vt ) 3m qπ I-V Curves of a Ballistic MOSFET Natori, JAP, p. 4879, 1994. I ds / W = 4h 2Cox Cox (Vg − Vt )3 / 2 (T=0 K) 3m qπ Independent of L!