Long Channel MOSFETs

Long Channel MOSFETs : DRAIN CURRENT MODEL General drain current model Simplified!!! Charge Sheet approximation: analytical expression for S/D current Conventionally V source: Grounded Vds, Vg : > 0V Vbs: < 0 (initially assumed ‘0’ P-type: Na f(x, y): band bending, or intrinsic potential w.r.t. bulk intrinsic potential V(y): electron quasi-Fermi potential at y w.r.t. Fermi potential of the n+ source

Long Channel MOSFETs : DRAIN CURRENT MODEL No bias between Source and Substrate  ff(source) = ff (bulk) Channel to substrate diode (n+/p-) Quasi Fermi potential stays essentially constant across the depletion region V(y) does not change with x  no current flow:  V(y=L) =Vds Inversion charge density as a function of Quasi Fermi Potential, V(y) and  Under surface inversion 

Long Channel MOSFETs : DRAIN CURRENT MODEL Gradual-Channel Approximation 1-D MOSFET Model: Gradual channel approximation (GCA) Assuming dE/dy << dE/dx, valid for channel region except beyong the pinch-off Hole current, generation & recombination current are negligible Current continuity equation is to electron current in the y-direction Ids is constant along y V(y) ~ quasi Fermi potential, playing the role of fn Ids(y) at the position, y Qi(y) at the position, (y, z)

Long Channel MOSFETs : DRAIN CURRENT MODEL Gradual-Channel Approximation PAO & SAH’s Double Integral PAO & SAH’s Double Integral

Long Channel MOSFETs : MOSFET I-V PAO & SAH’s Double Integral Boundary Conditions 1. 2. Charge Sheet Approximation : for analytical solution For an analytical solution Inversion layer thickness is zero (like a sheet of charge) No potential drop or band bending across the inversion layer Depletion approximation is applied to the bulk depletion region

Long Channel MOSFETs : MOSFET I-V Since I-V using charge sheet model For a given Vg, Ids first increases linearly with the Vds  linear (tiode) region Then, gradually levels off to a saturated value (saturation region)

Long Channel MOSFETs : MOSFET I-V Linear (Triode) Region If Vds is small, from power of Vds, keep only 1st order of Vds Vt is the gate voltage when the surface potential or band bending reaches 2yB and the Si charge is the bulk depletion charge for that potential. Typically 0.6 ~ 0.9V If Vg < Vt : little current flow  MOSFET subthreshold region

Long Channel MOSFETs : MOSFET I-V Linear region  MOSFETs act like a R:  Modulated by Vg Vt can be determined from Ids-Vg plot at low Vds Extrapolated intercept of the linear portion of the Ids(Vg ) plot with the Vg axis at low Vds This value is slightly higher than “2yB” Vt due to the inversion layer capacitance and other effects Ids(Vg ) is not linear near Vt  CSA is not valid in this regime

Long Channel MOSFETs : MOSFET I-V Characteristics in the Saturation Region For a large Vds , 2nd term in power series of Vds, m: body effect coefficient, typically 1.1<m<1.4 As Vds increase, Ids followed a parabolic curve until a maximum or saturation value is reached when Vds=Vdsat=Vt-Vg Valid only Vds<Vdsat Beyond Vds=Vdsat(Vds>Vdsat)

Long Channel MOSFETs : MOSFET I-V Characteristics in the Saturation Region Valid only Vds<Vdsat Beyond Vds=Vdsat(Vds>Vdsat)

Long Channel MOSFETs : MOSFET I-V Onset of pinch-off and current saturation Drain current saturation can be understood from the inversion charge density. When V<2yB Shaded area If Vds is small (linear region) Qi(drain end) ~ Qi (source end) (slightly lower) As Vds↑, Ids↑ and Qi(drain end)↓ to zero when Vds = Vdsat = (Vg-Vt)/m Ids is at its maximum Surface c hannel vanishes at the drain end Pinch-off

Long Channel MOSFETs : MOSFET I-V Onset of pinch-off and current saturation As Vds↑(over Vdsat), pinch-off point moves toward source  Ids remains essentially the same because for Vds>Vdsat, V at pinch-off point remains at Vdsat.  L changes to L’ (L’<L)  channel length modulation Drain current: Integrating from 0 to y after multiplying by dy Substituting Ids from

Long Channel MOSFETs : MOSFET I-V Onset of pinch-off and current saturation V(y) and –Qi/mCox=(Vg-Vt)/m-V(y) for several Vds At low Vds  V(y) ↑ LINEARLY As Vds ↑  Qi(L) ↓ due to electron quasi-EF ↓  dV/dy ↑ to maintain current continuity Vds =Vdsat=(Vg-Vt)/m  Qi(L) =0 V(y)/dy=∞ The Ey changes more rapidly than Ex, GCA breaks down Beyond Pinch-Off (Vds>2yB) Carriers not confined to the surface channel 2-D Poisson;s equation for carrier injection from pinch-off into the drain depletion region Not power series in Vds General form from Qi=0 and V=Vdsat(dIds/dVds=0) from Vds (several X >2yB), slight change ≈5% in Idsat compared to CSA 

Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS On a linear scale Id approach ‘0’ immediately below Vt. On a log scale, Id remains at nonnegligible levels for several tenths of a volt below Vg. (yB<yS<2yB) The inversion charge density does not drop to zero abruptly Exponential dependence on ys or Vg. Important in low-voltage, low power application

Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS Drift and Diffusion Components of Drain Current Strong inversion region , 2yB<yS  Drift current dominates Subthreshold conduction , yB<yS<2yB  Diffusion current dominates Both Diffusion and Drift included Ratio varies along the channel At low Vds, diffusion and drift components CAN BE separated If qV.kT<<1: only the 1st order terms of V kept Qi(V) replaced by its 0th order value Qi(V=0) Current continuity  V must vary linearly from S to D Total current dV/dy  the drift current E-field or dyS/dy (or dyS/dV(y) since constant dV/dy)  The drift fraction is given by the change of yS(band bending) w.r.t. quasi-Fermi potential, dyS/dV If V 0

Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS Drift and Diffusion Components of Drain Current Voltage drop across the oxide given by the last term of Weak inversion, yB<yS<2yB Numerator << 1 DIFFUSION component dominates Strong inversion, yS>2yB dyS/dV≈1 DRIFT current dominattes

Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS Subthreshold Current Expression By Gauss’s Law In weak inversion, << Expanding into a power series: 0th order=-Qd, 1st order term = -Qi Surface potential yS is related to the Vg through Since the inversion charge density is small, yS can be considered as a function of Vg only, independent of V. The electric field along the channel direction is small. The drift current is negligible. Substituting Qi into

Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS Subthreshold Current Expression yS can be expressed in terms of Vg yS only slightly deviates from the threshold value, 2yB.  | yS - 2yB | << 2yB and expand the square-root term around yS = 2yB :