Department of Electrical and Computer Engineering, Cornell University ECE 3150: Microelectronics Spring 018 Homework 3 Due on Feb. 15, 018 by 7:00 PM Suggested Readings: a) Lecture notes Important Note: 1) MAKE SURE THAT YOU INDICATE THE UNITS ASSOCIATED WITH YOUR NUMERICAL ANSWERS. OTHERWISE NO POINTS WILL BE AWARDED. ) Lab 1 is scheduled for the week of Feb. 1 3) Unless noted otherwise, always assume room temperature. 4) START EARLY ON THIS HOMEWORK SET Problem 3.1: (PN Junctions warm up) Consider two PN diodes, A and B, as depicted below. Suppose the junction area for each diode is 1.0 square-m. a) At D 0 value (in microns)? V Volts, which diode has the wider total depletion region width xno x po and what is its b) At V D 0 Volts, in which diode the magnitude of the maximum electric field in the depletion region is the largest and what is its value (in V/cm)? c) Under a reverse bias V D 0, which diode will breakdown first (i.e. at a smaller magnitude of the negative bias). 1
Problem 3.: (PN Junction Capacitances) Consider a PN junction diode, as depicted below. Suppose the junction area of the diode is 100.0 square- m. Unfortunately, the N- and P- dopings of the PN-diode are unknown. However, it is known that N-side is much more heavily doped than the P-side. An undergraduate student comes up with a technique to figure out the dopings. He measures the depletion region capacitance of the PN diode in reverse bias, at different values of the reverse bias. He then plots the inverse square of the capacitance (i.e. 1 C j ) as a function of the reverse bias voltage and obtains the following curve: 6 x 10 10 1/C j (1/Farads ) 8 6 4 0-4 -3 - -1 0 1 V (Volts) D a) From the data shown above, figure out the built in voltage B. b) From the information given, the data shown above, and your answer in part (a) above, figure out the dopings, N a and N d, on both sides of the junction. Enamored by his experimental success, the undergraduate student decides to extend the negative voltage range of his data and obtains the following plot.
7 x 10 1/C j (1/Farads ) 1.5 1 0.5 You can see that for values of V D less than -5 Volts, the data changes slope (the magnitude of the slope increases by a factor of two). c) Can you explain why the slope changes in the data shown above? What information can you obtain about the doping from the change in the slope in the data shown above? Explain your answers. Hint: This problem will test your skills with electrostatics. The quantity of interest here is: 1 C j VD Imagine what will happen if the doping were to change slightly as a function of position (i.e. if N a or both were a function of position: as sa N d x or x N a ). The capacitance j W, where W is the width of the depletion region. So in order to find the value of W 0-10 -5 0 V D (Volts) N d or C can always be written 1 C j, you need to find VD W V D and relate it to the doping right at the edges of the depletion region. Lesson: Capacitance measurement (or capacitance spectroscopy) is a very useful tool to characterize the doping in semiconductor devices. Problem 3.3: (PN Diodes in forward bias) Consider the following PN diode structure. Area = A 3
Wn Wp m 16 3 Nd 10 1/ cm p n n 930 cm /V s p 310 cm /V s 4 A 10 cm If V D 0.476 V, then: a) What is the TOTAL hole density (in #/cm 3 ) at x W n xn (at the metal contact)? b) What is the TOTAL hole density (in #/cm 3 ) at x xn (at the edge of the depletion region)? c) What is the total positive charge Q P (in Coulombs) due to the excess holes that got injected into the N- side from the P-side in steady state? d) What is the hole current density 0 J p at x 0 (in Amps/cm )? e) If the total negative charge Q N (in Coulombs) due to the excess electrons, which got injected into the P-side from the N-side, in steady state is one-half (in magnitude) of the total positive charge (in Coulombs) due to the excess holes that got injected into the N-side from the P-side in steady state (and found in part (c) above) then what is the doping N a on the P-side of the diode? f) Using your results in parts (c) and (e), find the total diffusion capacitance C d (in Farads) of the PN diode at the bias point? The diffusion capacitance is defined as: dqn QP Cd dvd g) What is the ratio of the electron current density J n 0 at x 0 to the hole current density J p 0 at x 0? Which one is larger and why? h) Plot (with proper units and labels) the electron and the hole current densities in the entire device (pay attention to the sign as well as the magnitude). i) Find the diffusion component Jp diff x of the hole current density x Wp xp x xp on the P-side of the junction (in Amps/cm )? J p in the range j) Using your answer in part (i), find the drift component Jp drift x of the hole current density x the range Wp xp x xp on the P-side of the junction (in Amps/cm )? k) The drift component Jp drift x of the hole current density x on the P-side of the junction can be written as, J p in J p in the range Wp xp x xp 4
x ppo p xe x qhppoex drift Jp qh ' Using your answer in part (j), Find the electric field in the quasi-neutral P-side (units: V/cm)? Of course, you will find that the field is non-zero. But it should be small. So quasi-neutral regions do have non-zero fields after all they are just small. l) Find the diode current (in Amps)? m) Suppose you needed to redesign the PN diode such that the electron current contribution to the total diode current is 5 times larger than the hole current contribution to the total diode current. How will you choose the doping N a of the P-side (in #/cm 3 ) to achieve this objective? 5