Homework chapter 9

PROBLEMS

Section 9.2The pn JunctionIdeal CurrentVoltage Relationship

9.1 (a) Consider an ideal pn junction diode at T = 300 K operating in the forward-bias region. Calculate the change in diode voltage that will cause a factor of 10 increase in current. (b) Repeat part (a) for a factor of 100 incurrent.

9.3 An ideal silicon pn junction at T = 300 K is under forward bias. The minority-carrier lifetimes are n0 = 10-6 s and p0 = 10-7 s. The doping concentration in the n region is Nd = 1016 cm-3. Plot the ratio of hole current to the total current crossing the space charge region as the p-region doping concentration varies over the range 1015Na1018 cm-3. (Use a log scale for the doping concentrations.)

9.5 For a silicon pn junction at T = 300 K, assume p0 = 0.1 n0 and n = 2.4 p. The ratio of electron current crossing the depletion region to the total current is defined as the electron efficiency. Determine the expression for the electron injection efficiency as a function of (a) Nd/Na and (b) the ratio of n-type conductivity to p-type conductivity.

9.7 Consider an ideal silicon pn junction diode with the following parameters: n0 = p0 = 0.1  10-6 s, Dn = 25 cm2/s, Dp = 10 cm2/s. What must be the ratio of Na/Ndso that 95 percent of the current in the depletion region is carried by electrons?

9.9 A germanium p+n diode at T = 300 K has the following parameters: Na = 1018 cm-3, Nd = 1016 cm-3, Dp = 49 cm2/s, Dn = 100 cm2/s, p0 = n0 = 5 s, and A = 10-4 cm2. Determine the diode current for (a) a forward-bias voltage of 0.2 V and (b) a reverse-bias voltage of 0.2 V.

9.11 A silicon step junction has uniform impurity doping concentrations of Na = 5 1015 cm-3 and Nd = 1 1015 cm-3, and a cross-sectional area of A = 10-4 cm2. Let n0 = 0.4 s and p0 = 0.1 s. Consider the geometry in Figure 9.11. Calculate (a) the ideal reverse saturation current due to holes, (b) the ideal reverse saturation current due to electrons, (c) the hole concentration at xn if Va = Vbi, and (d) the electron current at x = xn + Lp for Va = Vbi.

9.13 The limit of low injection is normally defined to be when the minority carrier concentration at the edge of the space charge region in the low-doped region becomes equal to one-tenth the majority carrier concentration in this region. Determine the value of the forward-bias voltage at which the limit of low injection is reached for the diode described in (a) Problem 9.9 and (b) Problem 9.10.

9.15 Consider two ideal pn junction at T = 300 K, having exactly the same electrical and physical parameters except for the bandgap energy of the semiconductor materials. The first pn junction has a bandgap energy of 0.525 eV and a forward-bias current of 10 mA with Va = 0.255 V. For the second pn junction, “design” the bandgap energy so that a forward-bias voltage of Va = 0.32 V will produce a current of 10 A.

9.17 Assume that the mobilities, diffusion coefficients, and minority carrier lifetime parameters are independent of temperature (use the T = 300 K values). Assume that n0 = 10-6 s, p0 = 10-7 s, Nd = 5 1015 cm-3, and Na = 5 1016 cm-3. Plot the ideal reverse saturation current density from T = 200 K to T = 500 K for (a) silicon, (b) germanium, and (c) gallium arsenide ideal pn junctions. (Use a log scale for the current density.)

*9.19 A p+n silicon diode is fabricated with a narrow n region as shown in Figure 9.16, in which WnLp. Assume the boundary condition of pn = pn0 at x = xn + Wn. (a) Derive the expression for the excess hole concentration pn (x) as given by Equation (8.27). (b) Using the results of part (a), show that the current density in the diode is given by

9.21 A forward-biased silicon diode is to be used as a temperature sensor. The diode is forward biased with a constant current source and Va is measured as a function of temperature. (a) Derive an expression for Va (T) assuming that D/L for electrons and holes, and Eg are independent of temperature. (b) If the diode is biased at ID = 0.1 mA and if Is = 10-15 A at T = 300 K, plot Va versus T for 20C < T200C. (c) Repeat part (b) if ID = 1mA. (d) Determine any changes in the results of parts (a) through (c) if the change in bandgap energy with temperature is taken into account.

Section 9.3The Schottky Barrier Junction-Ideal Current-Voltage Relationship

9.23 (a) Consider a Schottky diode at T = 300 K formed with tungsten on n-type silicon. Let Nd =5 1015 cm-3 and assume a cross-sectional area of A = 5 10-4 cm2.. Let Bn = 0.68 V. Determine the forward-bias voltage required to obtain a current of 1 mA, 10 mA, and 100 mA. (b) Repeat part (a) if the temperature is increased to T = 400 .

9.25 (a) Consider a Au n-type GaAs Schottky diode with a cross-sectional area of 10-4 cm2. Let Bn = 0.86 V. Plot the forward-bias current-voltage characteristics over a voltage range of 0 VD 0.5 V. Plot the current on a log scale. (b) Repeat part (a) for an Au n-type silicon Schottky diode. Let Bn = 0.65 V. (c) What conclusions can be drawn from these results?

9.27 The reverse-saturation current densities in a pn junction diode and a Schottky diode are 5  10-12 A/cm2 and 7  10-8 A/cm2, repectively, at T = 300 K. The cross-sectional area of the pn junction diode is A = 8  10-4 cm2. Determine the vross-sectional area of the Schottky diode so that the difference in forward-bias voltages to achieve 1.2 mA is 0.265 V.

*9.29 A Schottky diode and a pn junction diode have cross-sectional areas of A = 7  10-4 cm2. The reverse-saturation current densities at T = 300 K of the Schottky diode and pn junction are 4  10-8 A/cm2 and 3  10-12 A/cm2, respectively. A forward-bias current of 0.8 mA is required in each diode. (a) Determine the forward-bias voltage required across each diode. (b) If the voltage from part (a) is maintained across each diode, determine the current in each diode if the temperature is increased to 400 K. (Take into account the temperature dependence of the reverse-saturation currents. Assume Eg = 1.12 eV for the pn junction diode and B0 = 0.82 V. for the Schottky diode.)

Section 9.4Small-Signal Model of the pn Junction

9.31 Consider a p+n silicon diode at T = 300 K. The diode is forward biased at a current of 1 mA. The hole lifetime in the n region is 10-7 s. Neglecting the depletion capacitance, calculate the diode impedance at frequencies of 10 kHz, 100 kHz, 1MHz, and 10 MHz

9.33 Consider a p+n silicon diode at T = 300 K. The slope of the diffusion capacitance versus forward-bias current is 2.5  10-6 F/A. determine the hole lifetime and the diffusion capacitance at a forward-bias current of 1 mA.

9.35 A silicon pn junction diode at T = 300 K has a cross-sectional area of 10-2 cm2. The length of the p region is 0.2 cm and the length of the n region is 0.1 cm. The doping concentrations are Nd = 1015 cm-3 and Na = 1016 cm-3. Determine (a) approximately the series resistance of the diode and (b) the current through the diode that will produce a 0.1 V drop across this series resistance.

9.37 The minimum small-signal diffusion resistance of an ideal forward-biased silicon pn junction diode at T = 300 K is to be rd = 48 . The reverse saturation current is Is = 2  10-11 A. Calculate the maximum applied forward-bias voltage that can be applied to meet this specification.

Section 9.5Generation-Recombination Currents

9.39 Consider a reverse-biased gallium arsenide pn junction at T = 300 K. Assume that a reverse-bias voltage, VR = 5 V, is applied. Assume parameter values of: Na = Nd = 1016 cm-3, Dp = 6 cm2/s, Dn = 200 cm2/s, and p0 = n0 = n = 10-8 s. Calculate the ideal reverse saturation current density and the reverse-biased generation current density. How does the relative value of these two currents compare to those of the silicon pn junction?

9.41 Consider a GaAs pn diode at T = 300 K with Na = Nd = 1017 cm-3 and with a cross-sectional area of 10-3 cm2. The minority carrier mobilities are n = 3000 cm2/V-s and p = 200 cm2/V-s. The lifetimes are p0 = n0 = n = 10-8 s. As a first approximation, assume the electron-hole generation and recombination rates are constant across the space charge region. (a) Calculate the total diode current at a reverse-bias voltage of 5 V and at forward-bias voltages of 0.3 V and 0.5 V. (b) Compare the results of part (a) to an ideal diode at the same applied voltages.

9.43 A silicon pn junction diode at T = 300 K has the following parameters: Na = Nd = 1016 cm-3, p0 = n0 = n = 5  10-7 s, Dp = 10 cm2/s, Dn = 25 cm2/s, and a cross-sectional area of 10-4 cm2. Plot the diode recombination current and the ideal diode current (on a log scale) versus forward bias voltage over the range 0.1 Va 0.6 V.

9.45 Consider, as shown in Figure P9.45, a uniformly doped silicon pn junction at T = 300 K withimpurity doping concentrations of Na = Nd = 5 1015 cm-3 and minority-carrier lifetimes of n0 = p0 = 0 = 10-7 s. A reverse-bias voltage of VR = 10 V is applied. A light source is incident only on the space charge region, producing an excess carrier generation rate of g = 4 1019 cm-3 s-1. Calculate the generation current density.

Section 9.6Junction Breakdown

9.47The critical electric field for breakdown in silicon is approximately crit = 4  105 V/cm. Determine the maximum n-type doping concentration in an abrupt p+n junction such that the breakdown voltage is 30 V.

9.49Consider an abrupt n+p GaAs junction with a p-type doping concentration of Na = 1016cm-3. Determine the breakdown voltage.

9.51 An abrupt silicon p+n junction has an n-region doping concentration of Nd =5 1015 cm-3. What must be the minimum n-region width such that avalanche breakdown occurs before the depletion region reaches an ohmic contact (punchthrough)?

9.53A diode will very often have the doping profile shown in Figure P5.28, which is known as an n+ pp+ diode. Under reverse bias, the depletion region must remain within the p region to avoid premature breakdown. Assume the p region doping is 1015 cm-3. Determine the reverse-bias voltage such that the depletion region remains within the p region and does not reach breakdown if the p region width is (a) 75 m and (b) 150 m. Form each case, state whether the maximum depletion eidth or the breakdown voltage is reached first.

Section 9.7Charge storage and Diode Transients

9.55(a) In switching a pn junction from forward to reverse bias, assume that the ratio of reverse current, IR, to forward current, IF, is 0.2. Determine the ratio of storage time to minority-carrier lifetime, ts/p0. (b) Pepeat part (a) if the ration of IR to IF is 1.0.

*9.57 Consider a diode with a junction capacitance of 18 pF at zero bias and 4.2 pF at a reverse bias voltage of VR = 10 V. The minority-carrier lifetimes are 10-7 s. The diode is switched from a forward bias with a current of 2 mA to a reverse-bias voltage of 10 V applied through a 10 k resistor. Estimate the turn-off time.

*9.59 A silicon pn junction diode at T = 300 K is to be designed to have a reverse-bias breakdown voltage of at least 50 V and to handle a forward-bias current of ID = 100 mA while still operating under low injection. The minority-carrier diffusion coefficients and lifetimes are Dn = 25 cm2/s, and n0 = p0 = 5  10-7s. The diode is to be designed for minimum cross-sectional area.