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**Unformatted text preview: **Solutions to Chapter 4 Exercises 4.1. Under the scaling transformation, W W / , L L / , t ox t ox / , V ds V ds / , V g V g / , and V t V t / , Eq. (3.23) becomes I C W L V V V I ds eff ox g t ds ds - = ( ) / / ; and Eq. (3.28) becomes I C W L m V V I ds eff ox g t ds - = ( ) / / 1 2 2 . Note that both m = 1 + 3 t ox / W dm and eff which is a function of E eff given by Eq. (3.53) are nearly invariant under constant field scaling. 4.2. Under the same scaling rules as above, Eq. (3.40) becomes I C W L m kT q e ds eff ox q V V mkT g t - - ( ) / / ( ) ( )/ 1 2 . The exp(- qV ds / kT ) term has been neglected since typically V ds >> kT / q . In subthreshold, V g < V t and exp[ q ( V g- V t )/ mkT ] > exp[ q ( V g- V t )/ mkT ] (note that > 1), therefore, the subthreshold current increases with scaling faster than I ds . If the temperature also scales down by the same factor, i.e., T T / , then I ds I ds / , same as the drift current in Exercise 4.1. 4.3. Since the factor eff ( V g- V t )/( mv sat L ) is invariant under the scaling transformation, W W / , L L / , t ox t ox / , V ds V ds / , V g V g / , and V t V t / , the square-root expression and therefore the fraction in Eq. (3.79) is unchanged after scaling. The saturation current I dsat of Eq. (3.79) then scales the same way as the fully saturation-velocity limited current, Eq. (3.81), i.e., I C W v V V I dsat ox sat g t dsat - ( ) . SOLUTION 1 4.4. Under the generalized scaling rules, W W / , L L / , t ox t ox / , V g ( / ) V g , and V t ( / ) V t , Eq. (3.79) transforms as I C Wv V V V V mv L V V mv L dsat ox sat g t eff g t sat eff g t sat - +-- +- + ( ) ( ) / ( ) ( ) / ( ) 1 2 1 1 2 1 . In the limit of eff ( V g- V t )/( mv sat L ) >> 1 (short channel or velocity saturation), the fractional factor in the above equation equals 1 independent of , so I dsat ( / ) I dsat . On the other hand, if eff ( V g- V t )/( mv sat L ) << 1 (long channel), the fractional factor can be approximated by eff ( V g - V t )/(2 mv sat L ), and one has I dsat ( 2 / ) I dsat . In general, the scaling behavior is between the two limits. 4.5. (a) ) ( 2 s dm ox a B fb t x W C qN V V- + + = 2 4 s a B si dm x qN W + = Eliminate N a from the above eqs, dm B fb t ox B si s W V V C x--- = ) 2 ( 4 V fb = - E g /2 q B = -1.06 V, so x s = 17 nm....

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