At threshold, the concentration of electrons at the surface equal to the concentration of holes in the bulk:If Vds <VT (or in subthreshold), diffusion current dominates.VgsVdsSGDL0Vgs’CdVoxVgsVds0LVgsVdsSGDL0SaturationOhmic ModeVgsVdsSGDL0SaturationOhmic ModeThreshold Voltage for an MOS Junction At threshold, the concentration of electrons at the surface equal to the concentration of holes in the bulk: 0psurfacepn = At threshold, Bsψψ2=. It is assumed that all the charge in the semiconductor is due to the depletion region. BToxBToxTHdEVVψψ22,,+=+= BTsoxSEdψεε2,+= Boundary: oxoxssEEεε= BoxDCQψ2+= By Gauss’s Law DssQE =ε With dCoxoxε=: Oxide Capacitance per unit Area BoxDTHCQVψ2+= For a real MOSFET, FBTHTHVVV −=′ oxDoxssmsTHCQCQV ±±−=ψϕ2 (+ for p-type, - for n-type) where ssQis the charge trapped at the oxide/semiconductor interface.EI EF EV EV Bqψ Bqψ BSqqψψ2=MOSFET (Metal Oxide Semiconductor Field Effect Transistor) If Vds <VT (or in subthreshold), diffusion current dominates. [ ( )/ ]0(0)sq x kTpn neψ= × 0( ) [ ( (0) ) / ] (0)p s dsn L n Exp q V kT nψ= − << TdsnndiffusiondsdriftdsVVLLnnqDEnqJJ≅−=)()0(,,µ ox D,ox(0) C +C, =Cgssds ds diffusionqVqJ J Exp Exp wherekT kTϕηη≈∝ ∝ Vgs’ Cox Cd Vox sψ Gate Semiconductor ( ( ))() ()ox gsoxs gsox D ox DC V VxCx VxCC CCψ−′= =++ Vgs Vds S G D p n+ n+ L 0 W dOhmic Mode (Triode): Source and Drain have ohmic-like contacts to the channel. ))(()( xVVVCxQTHgsoxn−−= With dCoxoxε= 0)0( =V and dsVLV =)( Resistance of a length dx: ()ndx dxdRA qn x Aρµ= = ()qn x A=Charge per unit length in x direction=)(xWQn WxQdxdRnn)(µ= dRIdVD= ∫∫=−−LdVdsTHgsoxndxIdVxVVVWC00))((µ −−=2)(2dsdsTHgsoxndVVVVCLWIµ Vgs Vds p n+ n+ L 0 x dx dOnset of Saturation (Pinch-Off) ))(()( xVVVCxqnTHgsox−−=)()(dsTHgsoxVVVCLqn −−= For ,ds gs TH ds satVVV V≥− =, () 0nL ≈: the channel is pinched-off near the drain Ideally, further increase of dsVdoesn’t increase the source-drain current ,00( ( ))Vds satLn ox gs TH dW C V V V x dV I dxµ−− =∫∫ 22,,()22d n ox gs TH n ox ds sat d satWWI C V V CV ILLµµ= −≅ ≡ dI dsV gsV 2dsV∝ Saturation Mode Ohmic Mode 1gsV 2gsV 3gsV 4gsV THgsVV −1 Vgs Vds S G D p n+ n+ L 0 x dx d L’Beyond Pinch-off : Channel Length Modulation After Saturation, i.e., Vds > Vgs – VTH Ideally: 22,()22d n ox gs TH n ox dssat d satWWI C V V CV ILLµµ= −≅ ≡ Realistically: 22,'()22d n ox gs TH n ox ds satWWI C V V CVLLµµ= −≅′, or 2the channel length modulation parameter( ) (1 ) 2 d n ox gs TH dsWI CV V VLµλλ= −+ dI dsV gsV 2dsV∝ Saturation Mode Ohmic Mode 1gsV 2gsV 3gsV 4gsV THgsVV −1 Vgs Vds S G D p n+ n+ L 0 x dx d L’ ∆LSmall-Signal Model 2( )(1 )1( ) ( ) & dm n ox gs TH dsgsdo n ox gs TH d ods oIWg CV V VVLIWg CV V I rVL gµλµ λλ∂== −+∂∂== −≈ =∂May 4, 2011, Intel, Tri-gate