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Chapter 4: Problem 52

Consider GaAs at \(T=300 \mathrm{~K}\) with \(N_{d}=0 .(\) a) Plot the position of the Fermi energy level with respect to the intrinsic Fermi energy level as a function of the acceptor impurity concentration over the range of \(10^{14} \leq N_{a} \leq 10^{17} \mathrm{~cm}^{-3} \cdot(b)\) Plot the position of the Fermi energy level with respect to the valence-band energy over the same acceptor impurity concentration as given in part \((a)\).

Short Answer

Expert verified
The Fermi energy shifts down with increasing acceptor concentration.

Step by step solution

01

Energy Levels in Semiconductors

Understand the definitions: The Fermi energy level ( E_f) is the chemical potential for electrons, the intrinsic Fermi energy ( E_{i)) is the Fermi level when no doping is present, and the valence-band energy ( E_v) represents the top of the valence band in a semiconductor. GaAs is assumed to be an intrinsic semiconductor with only acceptor impurities (no donor impurities, N_d = 0).
02

Fundamental Equations

For an intrinsic semiconductor, the shift in Fermi energy level due to doping can be calculated using:\[E_f - E_i = -kT \, ext{ln} \left(\frac{N_a}{n_i}\right)\]where \(k\) is the Boltzmann constant, \(T\) is the temperature, \(N_a\) is the acceptor concentration, and \(n_i\) is the intrinsic carrier concentration.
03

Calculate the Intrinsic Carrier Concentration\(n_i\)

For GaAs at \(300\, \text{K}\), the intrinsic carrier concentration \(n_i\) is approximately \(2.1 \times 10^{6}\, \text{cm}^{-3}\).
04

Plot \(E_f - E_i\) with Varying \(N_a\)

Using the equation from Step 2, calculate the shift \(E_f - E_i\) for \(N_a\) values ranging from \(10^{14}\) to \(10^{17}\, \text{cm}^{-3}\). Use values of \(k = 8.6173 \times 10^{-5}\, \text{eV/K}\) and \(T = 300 \, \text{K}\). Plot the results with \(N_a\) on the x-axis and \(E_f - E_i\) on the y-axis.
05

Calculate \(E_f - E_v\)

The Fermi energy level can also be expressed in terms of the valence band:\[E_f - E_v = -kT \, \text{ln} \left(\frac{N_a}{N_v}\right)\]where \(N_v\) is the effective density of states in the valence band, which for GaAs at \(300\, \text{K}\) is approximately \(1.04 \times 10^{19} \, \text{cm}^{-3}\).
06

Plot \(E_f - E_v\) with Varying \(N_a\)

Using the equation from Step 5, calculate the shift \(E_f - E_v\) for the same range of \(N_a\). Plot the results with \(N_a\) on the x-axis and \(E_f - E_v\) on the y-axis.

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Key Concepts

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

Intrinsic Semiconductor
An intrinsic semiconductor is a pure semiconductor without any significant impurity atoms present. In an intrinsic semiconductor, such as Gallium Arsenide (GaAs), the number of electrons in the conduction band equals the number of holes in the valence band. This balance is due to thermal generation when the semiconductor is kept at thermal equilibrium.
Intrinsic semiconductors exhibit a unique set of properties:
  • They serve as the idealized version of a semiconductor.
  • Contain an equal concentration of electrons and holes.
  • Their electrical properties are determined by the material itself and not any impurities.
  • At room temperature, these materials have a certain intrinsic carrier concentration, denoted as \(n_i\).
For GaAs at room temperature \( (300 \, \text{K}) \), this intrinsic carrier concentration \(n_i\) is approximately \(2.1 \times 10^6 \, \text{cm}^{-3}\). This value is crucial for analyzing how the material behaves when doped with impurities and affects the position of the Fermi energy level.
Acceptor Impurity Concentration
In semiconductors, doping involves introducing impurities to modify their electrical properties. Acceptor impurities in a semiconductor accept electrons from the material, creating holes. These are typically atoms that have one less valence electron than the semiconductor material.
Acceptor doping transforms a normally intrinsic semiconductor into a p-type semiconductor, characterized by a predominance of holes as charge carriers:
  • An increase in acceptor impurity concentration \(N_a\) leads to more holes in the material.
  • The Fermi energy level shifts closer to the valence band as \(N_a\) increases.
  • For GaAs, it鈥檚 essential to plot and understand the behavior of the Fermi level across a range from \(10^{14}\) to \(10^{17} \, \text{cm}^{-3}\).
The shift of the Fermi level \((E_f - E_i)\) is computed using the formula: \[E_f - E_i = -kT \, \text{ln} \left(\frac{N_a}{n_i}\right)\]where \(k\) is the Boltzmann constant and T is the temperature. This negative relationship indicates that as acceptor concentration increases, the Fermi level moves closer to the valence band.
Valence-Band Energy
The valence-band energy \(E_v\) is a critical reference point in the energy band structure of semiconductors. It signifies the highest energy level that can be occupied by holes. In doped semiconductors, understanding the relationship between the Fermi energy level \(E_f\) and the valence band is key to predicting electrical behavior.
For analyzing how doping affects this relationship, one must calculate \(E_f - E_v\), which signifies how the Fermi level moves in relation to the valence band:
  • The equation used for this analysis is \[E_f - E_v = -kT \, \text{ln} \left(\frac{N_a}{N_v}\right)\]where \(N_v\) represents the effective density of states in the valence band.
  • In GaAs at \(300 \, \text{K}\), \(N_v\) approximates \(1.04 \times 10^{19} \, \text{cm}^{-3}\).
  • As acceptor levels increase, \(E_f\) moves closer to \(E_v\), demonstrating a more significant presence of holes.
Understanding this relationship helps predict how different concentrations of acceptor impurities will affect the semiconductor's conductivity, which is fundamental in designing electronic devices.

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Most popular questions from this chapter

Assume that silicon, germanium, and gallium arsenide each have dopant concentrastions of \(N_{d}=1 \times 10^{13} \mathrm{~cm}^{-3}\) and \(N_{a}=2.5 \times 10^{13} \mathrm{~cm}^{-3}\) at \(T=300 \mathrm{~K}\). For each of the three materials: \((a)\) Is this material n type or \(\mathrm{p}\) type? (b) Calculate \(n_{0}\) and \(p_{0}\)

(a) The carrier effective masses in a semiconductor are \(m_{n}^{*}=1.21 m_{0}\) and \(m_{p}^{*}=0.70 m_{0}\). Determine the position of the intrinsic Fermi level with respect to the center of the bandgap at \(T=300 \mathrm{~K} .(b)\) Repeat part \((a)\) if \(m_{n}^{*}=0.080 m_{0}\) and \(m_{p}^{*}=0.75 m_{0}\).

(a) The maximum intrinsic carrier concentration in a silicon device must be limited to \(5 \times 10^{11} \mathrm{~cm}^{-3} .\) Assume \(E_{g}=1.12 \mathrm{eV}\). Determine the maximum temperature allowed for the device. ( \(b\) ) Repeat part ( \(a\) ) if the maximum intrinsic carrier concentration is limited to \(5 \times 10^{12} \mathrm{~cm}^{-3}\).

A particular semiconductor material is doped at \(N_{d}=2 \times 10^{14} \mathrm{~cm}^{-3}\) and \(N_{a}=1.2 \times\) \(10^{14} \mathrm{~cm}^{-3}\). The thermal equilibrium electron concentration is found to be \(n_{0}=1.1 \times\) \(10^{14} \mathrm{~cm}^{-3} .\) Assuming complete ionization, determine the intrinsic carrier concentration and the thermal equilibrium hole concentration.

A new semiconductor material is to be "designed." The semiconductor is to be \(\mathrm{p}\) type and doped with \(5 \times 10^{15} \mathrm{~cm}^{-3}\) acceptor atoms. Assume complete ionization and assume \(N_{d}=0 .\) The effective density of states functions are \(N_{c}=1.2 \times 10^{19} \mathrm{~cm}^{-3}\) and \(N_{\mathrm{x}}=1.8 \times 10^{19} \mathrm{~cm}^{-3}\) at \(T=300 \mathrm{~K}\) and vary as \(T^{2} .\) A special semiconductor device fabricated with this material requires that the hole concentration be no greater than \(5.08 \times 10^{15} \mathrm{~cm}^{-3}\) at \(T=350 \mathrm{~K}\). What is the minimum bandgap energy required in this new material?
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