prelimSol - ECE 315 Prelim Exam Solution Fall 2005 Name...

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ECE 315 Prelim Exam Solution Fall 2005 Name: ___________________________________ Student net ID: _______________________ For a semiconductor resistor with length of l and cross section of A , the resistance can be estimated as ( 29 p n qp qn A l A l R μ μ ρ + = = , where n and p are the electron and hole concentration, and μ n and μ p are the electron and hole mobility. Diode equations in case: W d = x n + x p bi D A si V N N q + = 1 1 2 0 ε ε , N A x p = N D x n , = 2 ln i D A bi n N N q kT V , ε 0 =8.85 × 10 -14 F/cm, ε si =11.7 is the relative dielectric constant of silicon, q =1.6 × 10 -19 coul is the elemental charge, k B the Boltzmann constant, and T the temperature (default=300K). The thermal voltage k B T/q at room temperature is 26mV. For nMOSFET with the threshold voltage V th , the drain current I D in the linear and saturation regions above threshold ( V GS > V th ) are: ( 29 ( 29 saturation 2 linear 2 2 2 th GS Dsat DS th GS ch ox D th GS Dsat DS DS DS th GS ch ox D V V V V V V C L W I V V V V V V V V C L W I - = - = - = < - - = μ μ And in the saturation region, the quasi-static circuit model can be approximated as: 1. For a semiconductor wire as shown below with a cross section of 2 μ m × 1 μ m and length of 100 μ m contains a p-n junction in the middle with N A =10 16 cm -3 and N D =10 16 cm -3 . Assume n i =10 10 cm -3 and μ n =1,000cm 2 /Vs and μ p =400cm 2 /Vs. Calculate the wire resistance in Ohm. Notice that a real number is required, instead of just formula. Show your derivation and brief justification. ( Hint: the depletion region has majority carriers much less than the doping level. ) (15 pts) g mn v GS N A =10 16 cm -3 , N D =0 N D =10 16 cm -3 , N A =0 1 μ m 1 μ m 1 μ m 100 μ m 1
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The majority carrier is the main component in drift current (or Ohm’s law resistivity), but we need to substrate out the area that is in the depletion region. m W V mV n N N q kT V i D A bi μ 43 . 0 72 . 0 10 2 10 6 . 1 10 85 . 8 7 . 11 2 72 . 0 12 60 ln 16 19 14 2 = × × × × = = × = = - - Since N A =N D , W is equally distributed on the two sides, which makes the two resistors in parallel have an effective area of 1 μ m by (1 - 0.43/2) μ m. The resistance is then: ( 29 = + × × × × × × = - - - - k R 570 400 1000 10 10 6 . 1 1 10 784 . 0 10 10 100 16 19 4 4 4 . 2. For a semi-ideal pMOSFET with its source at ground (and hence the gate and drain voltage will be negative), the threshold voltage V th is - 1.0V. (a) Draw the semi-ideal log(| I D |) vs. V GS (range - 2V to 0V) under a small constant V DS (say - 50mV). Identify the operating regions. When the transistor is off, just mark off. When the transistor is on, mark linear and saturation regions. (5 pts) (b) Repeat (a) but with V DS = - 2V on the same plot. Please remember to mark the operating
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