# ELEC 3908, Physical Electronics, Lecture 23. The MOSFET Square Law Model

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1 ELEC 3908, Physical Electronics, Lecture 23 The MOSFET Square Law Model

2 Lecture Outline As with the diode and bipolar, have looked at basic structure of the MOSFET and now turn to derivation of a current model in terms of potentials and physical quantities Because of the structure of the device, the derivation is more complex than for the diode or BJT the fine details of the math are not as important as the overall approach Derivation will be for triode operation, with saturation modeling considered in lecture 24 End result will be the familiar square law model Page 23-2

3 Co-Ordinate System For MOSFET derivations, the following co-ordinate system will be used x is the distance into the substrate, x = 0 is the substrate surface y is the distance along the channel from source to drain, y = 0 is the source end of the channel, the drain end is y = L z is the distance along the channel parallel to the source and drain, z = 0 is the same point as x = y = 0, the other side is z = W Page 23-3

4 Model Assumptions Uniformity in z: when the MOSFET channel is wide (in z) compared to its length (in y), it is reasonable to assume that the total effect of current flow can be determined by analysing the device at a single point in z then scaling the result by the width W Page 23-4

5 Model Assumptions (con t) Steady state operation: All time derivatives are zero (leading to a static or dc model). n-channel device: The derivation will be for a device constructed in a p-well. The same results apply with the appropriate bias polarity changes. Drift Conduction: Any diffusion component is ignored Non-saturated carrier velocity: Electric fields will be assumed to be low enough so that v E Zero subthreshold current: For V GS < V T, I D = 0 Uniformly doped substrate Gradual channel approximation (GCA): The lateral E (due to V DS ) is assumed to be small compared to the transverse E (due to V GS ). This allows a one-dimensional charge balance between the gate and the substrate Page 23-5

6 Triode Model - Channel J Integration If carrier transport is only due to drift, the current density will be J ( x, y) = qn( x, y) μ ( x, y) E ( x, y) n n y This expression can be integrated over x from the surface (x = 0) to a point deep enough into the substrate to enclose the channel (x = x C ), and over z from one side of the channel (z = 0) to the other (z = W) W x C C J ( x, y) dxdz = qn( x, y) μ ( x, y) E ( x, y) dxdz 0 0 n 0 0 W x n y Page 23-6

7 Triode Model - Channel Integral Simplification W x C C J( x, ydxdz ) = qnx (, y) μ ( x, y) E ( x, ydxdz ) 0 0 n 0 0 The left hand side of this equation is simply the terminal current I D, since at steady state the total channel current must be constant along the channel On the right hand side, the integral wrt z can be performed to give a factor W (since the device is uniform in z by assumption) The result of these simplifications is x C W This equation must be further simplified to arrive at a solution x I = Wq n( x, y) μ ( x, y) E ( x, y) dx D n y 0 n y Page 23-7

8 Triode Model - Channel Integration Approximations The value of mobility is being integrated over the channel, and is not constant due to the changing E, however it is normally approximated (badly) as being constant at an effective surface mobility value In the previous expression, the channel electric field E y is integrated over the channel depth x C - it is a fair approximation to assume that E y is not a function of x, that is that E y is constant over the channel With these simplifications, the result is x C I D = W μ ne y( y) q n( x, y) dx 0 Page 23-8

9 Triode Model - Mobile Charge Term The remaining integral term is now x C q n( x, y) dx This represents the total electron charge per unit gate area at the point y, so the following definition can be made This quantity is termed the mobile charge, since it is distinct from fixed charge arising due to substrate acceptors in the depletion region With this definition, the current expression can be written 0 x C Q \$ m ( y) = q n( x, y) dx 0 I = μ WE ( y) Qˆ ( y) D n y m Page 23-9

10 Triode Model - Integration over y This expression can now be integrated over y from one end of the channel to the other to give L I dy = μ W ( y) Qˆ ( y) dy E L D n y m 0 0 Since I D is a constant along the channel, the left hand side is simply I D L In order to evaluate the right hand side, the variable of integration is changed from the channel position y to the surface potential ψ S Page 23-10

11 Triode Model - Surface Potential The surface potential ψ S is the potential at the substrate surface, i.e. the point x = 0 The electric field in the y direction can be written using Helmholtz s theorem as dψ S E y ( y) = dy With the above substitution and some rearrangement, the current expression becomes I W L Q y d S = μ \$ ψ ( ) dy dy D n m 0 L Page 23-11

12 Triode Model - Change of Integration Variable The variable of integration can now be changed from y, position along the channel, to ψ S, surface potential At the source end y = 0, the ψ S boundary condition is ψ S,S At the drain end y = L, the ψ S boundary condition is ψ S,D These boundary conditions can be found in terms of the bulk potential and the applied biases on each junction ψ ψ = 2φ + V = 2φ + V SS, B SB SD, B DB The integral expression can therefore be written I ψ S, D B DB W W = μ Q\$ d = Q d L ( ψ ) ψ μ \$ L ( ψ ) ψ D n m S S n m S S ψ 2φ + V S, S 2φ B + V SB Page 23-12

13 Triode Model - Charge Balance and Gate Charge To determine the mobile charge in terms of surface potential, the gradual channel approximation is used to write a charge balance between the gate and substrate ( ) Q \$ = Q \$ = Q \$ + Q \$ G S m dep The gate charge is found by the potential across the oxide capacitance and the oxide capacitance itself as Q\$ = V V ψ C\$ ( ) G GB FB S ox Page 23-13

14 Triode Model - Square Law Depletion Charge In the square law model, a severe approximation is made as to the nature of the depletion charge under the surface The positional dependence is ignored, and the source value is applied along the channel ( ψ ) = 2 ε ( 2φ + ) \$Q q N V dep S Si A B SB This results in a simple expression for I D, but sacrifices a great deal of accuracy Page 23-14

15 Triode Model - Final Mobile Charge Expression The mobile charge is therefore found as ( ψ ) = ( ψ ) ( ψ ) = ( V V ψ ) C\$ + 2qε N ( 2φ + V ) Q\$ Q\$ Q\$ m S G S dep S GB FB S ox Si A B SB This expression can then be substituted into the integral equation for I D to give I 2φ + V B DB W = μ Q\$ d L ( ψ ) ψ D n m S S 2φ + V B SB C W 2φ B+ VDB = μ \$ n ox ( VGB VFB S) L ψ 2φ B+ VSB ( φ + ) 2qε N 2 V C\$ Si A B SB ox dψ S Page 23-15

16 Threshold Modulation Once the integration is performed, the result can be written compactly with a generalized definition of the threshold voltage V ( 2 2 ) V = V + γ φ + V φ T To B SB B ( φ ) 2qεSiNA 2 B 2qε = V + 2φ + γ = C\$ C\$ To FB B ox These equations model the dependence of V T on source substrate bias, an effect called threshold modulation V To is the zero bias threshold voltage, and γ is the threshold modulation parameter Si ox N A Page 23-16

17 Example 23.1: γ Calculation Calculate the threshold modulation parameter γ for a MOSFET with a substrate doping of 2x10 16 /cm 3, and an oxide thickness of 20 nm. Page 23-17

18 Example 23.1: Solution By direct substitution, using the value of oxide capacitance found previously γ= 2 q = 047. V Page 23-18

19 Triode Model - Final Current Expression With some simplification and the generalized definition of threshold voltage, the drain current expression for the square law model in the triode region becomes I C W V DS D = μ \$ 2 n ox ( VGS VT) VDS V V, V V, L 2 GS T DS DS sat Since this equation is based on the mobile charge expression below, it is only valid until the point of pinchoff, when the mobile charge goes to zero at the drain end ( ψ ) = ( ψ ) ( ψ ) = ( V V ψ ) C\$ + 2qε N ( 2φ + V ) Q\$ Q\$ Q\$ m S G S dep S GB FB S ox Si A B SB Page 23-19

20 Square Law V DS,sat The triode region expression is valid until the mobile charge at the drain end goes to zero This happens at a drain end surface potential of 2φ B +V DB,sat which can be written 2φ B + V DS,sat +V SB The pinchoff condition can therefore be written ( ψ ) = 0 = ( 2φ, ) Q\$ V V V V C\$ m S GB FB B DS sat SB ox ( φ ) + 2qε N 2 + V Si A B SB This expression can be solved for V DS,sat to give ( φ + ) 2qεSiNA 2 B VSB VDS, sat = VGB VSB VFB 2φ B C\$ = V ox 3 GS = V T (generalized) = V V GS T Page 23-20

21 Other V DS,sat Models Important to realize that V DS,sat = V GS -V T is the pinchoff voltage for the square law model only - it arises in this form because of the model used for the depletion charge ( ψ ) = 0 = ( 2φ, ) Q\$ V V V V C\$ m S GB FB B DS sat SB ox ( φ ) + 2qεSiN A B + VSB from constant \$ term A model which uses a more accurate depletion charge expression, or which makes different assumptions, would give a different expression for V DS,sat Q dep Page 23-21

22 Saturation Region Model For the simple square law model, the current is assumed to be constant after pinchoff at the pinchoff value I C W V D = μ \$ n ox ( VGS VT ) VDS, sat L = μ n ( V V ) 2 DS, sat C W 2 \$ GS T ox V V V V L, 2 2 GS T DS DS, sat Page 23-22

23 Example 23.2: Square Law Current Calculation Calculate the drain current flowing in the MOSFET whose layout (1.0 μm/div) is shown below at the potentials shown. Assume 0.1 μm of lateral diffusion under the gate. Use the gate oxide thickness and substrate doping of the previous examples. Use an effective surface mobility of 530 cm 2 /Vsec. Page 23-23

24 Example 23.2: Solution From the layout, the channel width W is 8 μm, and the channel length L is 2 μm less the lateral diffusion of the source and drain under the gate (0.1 μm each), 1.8 μm. Page 23-24

25 Example 23.2: Solution (con t) From the individual terminal potentials, the potentials required for the square law model are VSB = VS VB VGS = VG VS = 1 0= 1 = 3 1= 2 V = V V = 4 1= 3 DS D S Since V SB 0, the threshold voltage must be recalculated as ( ) V T = = V Page 23-25

26 Example 23.2: Solution (con t) The saturation drain source voltage is therefore VDS, sat = VGS VT = = 157. V Since V DS is larger than V DS,sat, the device is in the saturation region. The current is therefore I D = ( ) = A Page 23-26

27 Lecture Summary The simple square law model for MOSFET drain current was determined by Writing an expression for current density by drift, and integrating the current density through a thin slice of the channel (x and z) Performing a final integration along L, which involves an important simplification of the depletion charge dependence on channel position The resulting model shows that for non zero V SB, threshold modulation will occur The end of the triode region of operation is the saturation voltage V DS,sat, which is V GS -V T for the square law model, but is different for other models with different assumptions about the charge Page 23-27

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