1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.012 MICROELECTRONIC DEVICES AND CIRCUITS Problem Set No. 2 Issued: September 16, 2009 Due: September 23, 2009 Reading Assignments: Lecture 3 (9/17/09) - Chap. 4 (4.2, 4.3), Chap. 6 (begin) Lecture 4 (9/22/09) - Chap. 6 (all), Chap. 7 (7.1, 7.2) Lecture 5 (9/24/09) - Chap. 5 (5.1) Problem 1 - We say that we can approximate the minority carrier lifetime in Si as beinginfinite even though it is really several hundred milliseconds, because this is a longtime compared to the time it takes a hole or an electron to drift or diffuse across a device to a contact or junction. To verify this statement consider the following two questions: a) With drift, the time it takes a carrier to go (transit) a distance d is: τtr,Drift = d/µE. i) How long does it take an electron (µe = 1600 cm2/V-s) to drift 10 µm in a field of 10 V/cm? ii) Repeat a) i) for a hole (µh = 600 cm2/V-s). iii) Compare your answers in a) i) and a) ii) to a lifetime of 100 ms. Is approximating this lifetime as “infinite” in this situation reasonable? b) With diffusion, the time it takes a carrier to transit a distance d is: τtr,Diff = d2/2D. i) How long does it take an electron (De = 40 cm2/s) to diffuse 10 µm in a concentration gradient of 1015 cm -3 per micron? ii) Repeat b) i) for a hole (Dh = 15 cm2/s). iii) Compare your answers in a) i) and a) ii) to a lifetime of 100 ms. Is approximatingthis lifetime as “infinite” in this situation reasonable? . Problem 2 - Do Problem 3.9, Parts a thru d, in the course textbook. Problem 3 - Two short problems: i) Do Problem 4.3 in the course textbook. ii) Do Problem 4.2 in the course textbook. Problem 4 - The n-type silicon sample illustrated on the top of the next page is 10 microns(µm) long and has metal ohmic contacts, A and B, on either end. The net donor concentration is 1 x 1015 cm -3; the electron mobility, µe, is 1600 cm2/V-s; and the hole mobility, µh, is 600 cm2/V-s. The electrostatic potential of the metal relative to intrinsic silicon is 0.2V, and the intrinsic carrier concentration at room temperature is 1010 cm -3. Use the 60 mV rule to calculate the electrostatic potential, i.e. use (kT/q) ln 10 = 0.06 V.2 n-type (1 x 10 cm )ABOhmicOhmic05.0x [µm]15-3-5.0a) What are the thermal equilibrium (i.e., no light, vAB = 0 V) hole and electron concentrations and electrostatic potential, φn, in this silicon? b) Sketch the electrostatic potential, φ(x), with vAB = 0 V, going from the metal on theleft, through the silicon, and into the metal on the right. Dimension your sketch; label any significant features. c) Assume now that vAB = 0.5 V. Sketch the electrostatic potential, φ(x), with vAB = 0.5 V, going from the metal on the left, through the silicon, and into the metal on theright (where you should assume the value of ø is unchanged, i.e. 0.2 V). Dimension your sketch and label any significant features, including φ(0). e) When vAB = 0.5 V, what are the electron and hole drift current densities, Jedr and Jhdr ,respectively, at x = 0 µm Next consider a sample similar to our original sample, except that it is doped p-type with a net acceptor concentration of 1 x 1015 cm -3 in the region from x = 0 µm to x = +5 µm as shown below. Note: This is a diode with the p-side to the right. ABOhmicOhmic0+5.0x [µm]p-type (1 x 10 cm )15-3-5.0n-type (1 x 10 cm )15-3f) Sketch the electrostatic potential, φ(x), going from the metal on the left, through the silicon, and into the metal on the right, with vAB = 0 V. Dimension your sketch and label any significant features, including the value of φ(0). g) The diode is now reverse biased by vAB = 0.5 V. Sketch the electrostatic potential, φ(x), now, going from the metal on the left, through the silicon, and into the metal on the right. Keep φ in the metal to the right of x = 5 µm at the value it had in Part f.Dimension your sketch and label any significant features, including the value of φ(0).MIT OpenCourseWarehttp://ocw.mit.edu 6.012 Microelectronic Devices and Circuits Fall 2009 For information about citing these materials or our Terms of Use, visit:
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