Problem 28 - University Physics with Modern Physics

Question

Pure germanium has a band gap of 0.67 ev. The Fermienergy is in the middle of the gap. (a) For temperatures of 250 \(\mathrm{K}\) ,\(300 \mathrm{K},\) and 350 \(\mathrm{K}\) , calculate the probability \(f(E)\) that a state at the bottom of the conduction band is occupied. (b) For each temperature in part (a), calculate the probability that a state at the top of the valence band is empty.

Step-by-Step Solution

Answer

At 250 K: 0.031; at 300 K: 0.017; at 350 K: 0.009.

Step 1: Understanding Fermi-Dirac Distribution

To solve the problem, we need to use the Fermi-Dirac distribution function, given by \( f(E) = \frac{1}{e^{(E - E_F)/(kT)} + 1} \). Here, \( E \) is the energy of the electron, \( E_F \) is the Fermi energy, \( k \) is the Boltzmann constant \( (8.617 \times 10^{-5} \text{ eV/K}) \), and \( T \) is the temperature in Kelvin.

Step 2: Determine Fermi Energy

For intrinsic semiconductors like germanium, the Fermi energy \( E_F \) is located in the middle of the band gap. Therefore, for germanium with a band gap of 0.67 eV, \( E_F = 0.67/2 = 0.335 \) eV.

Step 3: Calculate Occupancy Probability at Conduction Band

The probability that a state at the bottom of the conduction band, \( E_C = E_F + 0.335 \text{ eV} \), is occupied at temperature \( T \) is given by \( f(E_C) = \frac{1}{e^{0.335/(kT)} + 1} \). Substitute \( k \) and \( T = 250, 300, 350 \text{ K} \) to compute probabilities.

Step 4: Compute for 250 K

At \( T = 250 \text{ K} \), compute \( f(E_C) = \frac{1}{e^{0.335/(8.617 \times 10^{-5} \times 250)} + 1} \approx 0.031 \).

Step 5: Compute for 300 K

At \( T = 300 \text{ K} \), compute \( f(E_C) = \frac{1}{e^{0.335/(8.617 \times 10^{-5} \times 300)} + 1} \approx 0.017 \).

Step 6: Compute for 350 K

At \( T = 350 \text{ K} \), compute \( f(E_C) = \frac{1}{e^{0.335/(8.617 \times 10^{-5} \times 350)} + 1} \approx 0.009 \).

Step 7: Calculate Empty Probability at Valence Band

The probability that a state at the top of the valence band, \( E_V = E_F - 0.335 \text{ eV} \), is empty is \( 1 - f(E_V) \). Using \( f(E_V) = \frac{1}{e^{-0.335/(kT)} + 1} \), compute for each temperature.

Step 8: Compute for 250 K (Valence)

Using the symmetry of the Fermi function, \( 1 - f(E_V) = f(E_C) \). Thus, at \( T = 250 \text{ K} \), the probability is also \( 1 - f \approx 0.031 \).

Step 9: Compute for 300 K (Valence)

At \( T = 300 \text{ K} \), the probability is \( 1 - f \approx 0.017 \).

Step 10: Compute for 350 K (Valence)

At \( T = 350 \text{ K} \), the probability is \( 1 - f \approx 0.009 \).

Key Concepts

Semiconductors

Semiconductors are materials with electrical properties that stand between conductors and insulators. Unlike conductors, semiconductors have a significant energy band gap between the valence band (where electrons are normally present) and the conduction band (where electrons move freely). This unique property allows them to conduct electricity under certain conditions, such as the presence of added impurities (doping) or increased temperatures. In semiconductors, the ability of electrons to jump from the valence band to the conduction band enables the flow of current. This makes them essential in electronic devices, where they can act as either insulators or conductors depending on the conditions. Semiconductors like silicon and germanium are integral to the creation of computer chips, solar cells, and other technologies that require precise control over electrical characteristics.

Band Gap

The band gap is a crucial concept in understanding semiconductor behavior. It refers to the energy difference between the top of the valence band and the bottom of the conduction band. Electrons need to gain energy equal to or greater than the band gap to transition from a bound state in the valence band to a free state in the conduction band. Materials with a small band gap can conduct electricity more easily than those with a larger band gap.

• Conductors: negligible or zero band gap
• Semiconductors: moderate band gap (e.g., Germanium has a 0.67 eV band gap)
• Insulators: large band gap, making electron transition difficult

The size of the band gap influences how a semiconductor reacts to temperature and light, which is why it's vital for designing electronic components.

Fermi Energy

Fermi energy is a concept used to describe the energy level at which the probability of finding an electron is 50% at absolute zero temperature. In semiconductors, especially intrinsic ones like pure germanium or silicon, the Fermi energy is positioned approximately in the middle of the band gap. This central position of the Fermi energy indicates that at this energy level, electrons are equally likely to be found in the conduction or valence bands when external influences like temperature are absent. Using Fermi energy, we can utilize the Fermi-Dirac distribution to predict how the probability of electron occupancy changes with temperature, an essential factor in semiconductor physics.

Intrinsic Semiconductors

Intrinsic semiconductors are pure forms of semiconductor materials without any significant impurities. In such semiconductors, the number of electrons in the conduction band equals the number of holes in the valence band, maintaining electrical neutrality. Intrinsic semiconductors are affected primarily by temperature. As temperature rises, more electrons gain the energy needed to cross the band gap, increasing electrical conductivity. For germanium, with a relatively small band gap, thermal energy at room temperature or even lower can promote electron transition. While intrinsic semiconductors offer a basis for understanding semiconductor properties, in practice, extrinsic semiconductors (those with added impurities) are more widely used, as doping enhances conductivity or control over electrical characteristics, necessary for practical electronic applications like transistors and diodes.