# MOS Device Equation (second order effects)

MOSFET Current Voltage Characteristics Consider the cross-sectional view of an n-channel MOSFET operating in linear mode (picture below) VGS > VT0 VS =0 n+ Source VDS Channel x=0 x y p-type substrate + - n+ Drain x=L Depletion Region VB =0 We assume the threshold voltage is constant along the channel.

The channel voltage Vc has boundary conditions: Vc at x=0=VS=0 and Vc at x=L=VDS The channel is inverted from the source end to the drain end. Other voltages of interest are: VGSVT0 and VGD=VGSVDSVT0 MOSFET Voltage Characteristics The channel current (drain current ID) is caused by electrons in the channel region traveling from source to drain under the influence of the lateral electric field. If the total mobile electron charge in the surface inversion layer is assigned the vaiable QI(x), we can thus express this charge as a function of the gateto-source voltage VGS and the channel voltage Vc(x) QI(x)=-Cox[VGS-Vc(x)-VT0] The thickness of the inversion layer tapers along the channel from the source towards the drain because the influence of Vgate-tochannel decreases from source to drain. If we consider a small incremental resistance dR for a differential segment of the channel assuming constant electron mobility n at the

surface we have: dR dx W n QI ( x ) MOSFET Voltage Current Characteristic The variable W represents the channel width. The electron surface mobility n depends on the doping concentration of the channel region. We further assume that the channel current density is uniform across the segment where we are measuring the incremental resistance. ID flows between the source and drain. Applying Ohms law for this segment yields the voltage drop along the incremental segment dx: dV I dR I dx D c D W n QI ( x ) The above equation can now be integrated along the channel from x=0 to x=L using the boundary conditions for Vc

We get: VDS I D L W n C ox VGS Vc VT 0 dVc 0 MOSFET Voltage Current Characteristics Assuming that the channel voltage Vc is the only variable that depends on position x, the drain current is determined to be: C W I D n ox 2VGS VT 0 VDS VDS2 2 L This equation shows the dependence of the drain current on the process parameters such as oxide capacitance, carrier mobility, and bulk to source voltage. The drain current ID also depends on the devices channel length and width. MOSFET Voltage Current Characteristics The equations: I ds

0 when V gs Vt 2 V I V gs Vt V ds for ds ds 2 0 V V gs Vt equvalently VGS VT 0 ; VGD VGS V DS VT 0 ds 2 V gs Vt I for 0 V gs Vt V ds ds 2 represent a simple view of the MOS transistor DC Voltage current equations. There are models that better calculate the MOS transistors operation with accuracy.

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