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Chapter 8: Problem 1

(A) Consider an ideal pn junction diode at \(T=300 \mathrm{~K}\) 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 increase in current.

Short Answer

Expert verified
(a) 60 mV; (b) 120 mV.

Step by step solution

01

Understand the relationship between current and voltage

For a pn junction diode, the current (\( I \)) flowing through it in the forward-bias region can be described by the diode equation:\[ I = I_0 \left( e^{\frac{V}{nV_T}} - 1 \right) \]where \( I_0 \) is the reverse saturation current, \( V \) is the forward voltage across the diode, \( n \) is the ideality factor (usually close to 1 for silicon diodes), and \( V_T \) is the thermal voltage. At \( T=300 \; K \), \( V_T \approx 26 \; mV \).
02

Derive the change in voltage for a factor of 10 increase in current

Write the equation for the current increase by a factor of 10:\[ 10I = I_0 \left( e^{\frac{V_2}{nV_T}} - 1 \right) \]Divide by the original current expression:\[ 10 = \frac{e^{\frac{V_2}{nV_T}} - 1}{e^{\frac{V_1}{nV_T}} - 1} \]Now assume \( e^{\frac{V_1}{nV_T}} \gg 1 \) and \( e^{\frac{V_2}{nV_T}} \gg 1 \), thus the equation simplifies to:\[ 10 \approx e^{\frac{(V_2 - V_1)}{nV_T}} \]Solving for \( V_2 - V_1 \), we get:\[ \Delta V = nV_T \ln(10) \]For \( n = 1 \), \( \Delta V \approx 60 \; mV \) as \( V_T \approx 26 \; mV \).
03

Calculate voltage change for a factor of 100 increase in current

Repeating the process for a 100-fold increase,\[ 100 = \frac{e^{\frac{V_3}{nV_T}} - 1}{e^{\frac{V_1}{nV_T}} - 1} \]Assuming the same condition \( e^{\frac{V_1}{nV_T}} \gg 1 \) and \( e^{\frac{V_3}{nV_T}} \gg 1 \),:\[ 100 \approx e^{\frac{(V_3 - V_1)}{nV_T}} \]Solving for \( V_3 - V_1 \), we get:\[ \Delta V = nV_T \ln(100) \]Thus, \( \Delta V \approx 120 \; mV \) for \( n = 1 \), since \( V_T \approx 26 \; mV \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diode Equation
The diode equation is a fundamental formula used to describe the behavior of diodes, particularly pn junction diodes, in the forward-bias region. This equation helps to quantify the relationship between the current flowing through the diode and the voltage applied across it. The equation is given by:\[ I = I_0 \left( e^{\frac{V}{nV_T}} - 1 \right) \]Here:
  • \( I \) represents the diode current.
  • \( I_0 \) is the reverse saturation current, a small constant current.
  • \( V \) is the forward voltage across the diode.
  • \( n \) is the ideality factor, usually close to 1 for silicon diodes.
  • \( V_T \) is the thermal voltage.
This equation highlights how exponentially the current increases with an increase in forward voltage. It's crucial for understanding how diodes conduct electricity when connected in the forward direction and is foundational in circuit design.
Thermal Voltage
Thermal voltage, denoted as \( V_T \), is an essential parameter in evaluating the performance of pn junction diodes. At a given temperature \( T \), thermal voltage is defined by the relationship:\[ V_T = \frac{kT}{q} \]Where:
  • \( k \) is the Boltzmann constant, approximately \(1.38 \times 10^{-23} \, J/K\).
  • \( T \) is the absolute temperature in Kelvin.
  • \( q \) is the charge of an electron, about \(1.6 \times 10^{-19} \, C\).
At room temperature, approximately \( 300 \, K \), the thermal voltage \( V_T \) rounds to around \( 26 \, mV \). This seemingly small voltage plays a significant part in the diode equation, helping to describe how temperature impacts the current-voltage characteristics of the diode. Understanding \( V_T \) is vital since it indicates how sensitive the diode鈥檚 behavior is to changes in temperature.
Forward-Bias Region
A pn junction diode is said to be in the forward-bias region when the positive side of the external voltage is connected to the p-type material, and the negative side is connected to the n-type material. This configuration reduces the barrier potential, allowing a significant current to flow through the diode. In the forward-bias region:
  • The applied voltage decreases the width of the depletion region at the junction.
  • It allows charge carriers (holes and electrons) to recombine at the junction, resulting in current flow.
  • The diode conducts current with minimal resistance, unlike in the reverse-bias condition, where current is practically negligible.
The effect of forward biasing is evident through the exponential rise in current with an increase in the forward voltage, described by the diode equation. This is the typical mode for diodes when used in applications like rectifiers, where converting AC to DC is required.
Current-Voltage Relationship
The current-voltage (I-V) relationship in a pn junction diode provides insight into how the diode operates at different voltages in both the forward and reverse-bias regions. In the forward-bias condition, a small increase in voltage results in an exponential increase in current as explained by the diode equation.Key points about the I-V relationship include:
  • In the forward-bias region, the current increases swiftly as the applied forward voltage surpasses the built-in potential barrier, causing a low-resistance path.
  • For reverse bias, the diode current is very minimal, close to the reverse saturation current \( I_0 \), unless breakdown occurs.
  • The slope of the I-V curve in the forward-bias region is steep, displaying high sensitivity to small changes in voltage due to the exponential nature of the equation \( I = I_0 \left( e^{\frac{V}{nV_T}} - 1 \right) \).
Understanding this relationship is crucial for designing efficient semiconductor devices and predicting their behavior under different electrical conditions.

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Most popular questions from this chapter

Sketch the energy-band diagram of an abrupt pn junction under zero bias in which the p region is degenerately doped and \(E_{C}=E_{F}\) in the \(\mathrm{n}\) region. Sketch the forward- and reverse-biased current-voltage characteristics. This diode is sometimes called a backward diode. Why?

A silicon pn junction has impurity doping concentrations of \(N_{d}=2 \times 10^{15} \mathrm{~cm}^{-3}\) and \(N_{a}=8 \times 10^{15} \mathrm{~cm}^{-3} .\) Determine the minority carrier concentrations at the edges of the space charge region for \((a) V_{a}=0.45 \mathrm{~V},(b) V_{a}=0.55 \mathrm{~V}\), and (c) \(V_{a}=-0.55 \mathrm{~V}\)

A forward-biased silicon diode is to be used as a temperature sensor. The diode is forward biased with a constant current source and \(V_{a}\) is measured as a function of temperature. ( \(a\) ) Derive an expression for \(V_{a}(T)\) assuming that \(D / L\) for electrons and holes, and \(E_{g}\) are independent of temperature. \((b)\) If the diode is biased at \(I_{D}=0.1 \mathrm{~mA}\) and if \(I_{s}=10^{-15} \mathrm{~A}\) at \(T=300 \mathrm{~K}\), plot \(V_{a}\) versus \(T\) for \(20^{\circ} \mathrm{C}

Consider a silicon pn junction diode with an applied reverse-biased voltage of \(V_{R}=\) 5V. The doping concentrations are \(N_{a}=N_{d}=4 \times 10^{16} \mathrm{~cm}^{-3}\) and the cross-sectional area is \(A=10^{-4} \mathrm{~cm}^{2}\). Assume minority carrier lifetimes of \(\tau_{0}=\tau_{n 0}=\tau_{p 0}=10^{-7} \mathrm{~s}\). Calculate the ( \(a\) ) ideal reverse- saturation current, (b) reverse-biased generation current, and ( \(c\) ) the ratio of the generation current to ideal saturation current.

We want to consider the effect of a series resistance on the forward-bias voltage required to achieve a particular diode current. \((a)\) Assume the reverse-saturation current in a diode is \(I_{x}=10^{-10} \mathrm{~A}\) at \(T=300 \mathrm{~K}\). The resistivity of the \(\mathrm{n}\) region is \(0.2 \Omega-\mathrm{cm}\) and the resistivity of the \(\mathrm{p}\) region is \(0.1 \Omega-\mathrm{cm}\). Assume the length of each neutral region is \(10^{-2} \mathrm{~cm}\) and the cross-sectional area is \(2 \times 10^{-5} \mathrm{~cm}^{2}\). Determine the required applied voltage to achieve a current of \((i) 1 \mathrm{~mA}\) and \((i i) 10 \mathrm{~mA}\). (b) Repeat part ( \(a\) ) neglecting the series resistance.
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