The Hall mobility (μH) is given by:

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  1. \(\mu_{\mathrm{H}}=\frac{\mathrm{e} \tau}{\mathrm{m}}\)
  2. μ= eτm
  3. \( \mu_{\mathrm{H}}=\frac{\mathrm{m} \tau}{\mathrm{e}}\)
  4. \( \mu_{\mathrm{H}}=\frac{\mathrm{em}}{\tau}\)

Answer (Detailed Solution Below)

Option 1 : \(\mu_{\mathrm{H}}=\frac{\mathrm{e} \tau}{\mathrm{m}}\)
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Detailed Solution

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Explanation:

Hall Mobility (μH)

Definition: Hall mobility is a parameter that quantifies the ability of charge carriers, such as electrons or holes, to move through a semiconductor material under the influence of an electric field. It is directly related to the charge carrier’s drift velocity in response to the applied electric field and is a critical parameter in characterizing the electrical properties of materials.

Formula Derivation:

Hall mobility can be expressed mathematically as:

μH=σne

Where:

  • σ is the electrical conductivity of the material.
  • n is the charge carrier concentration (number of charge carriers per unit volume).
  • e is the elementary charge of an electron (approximately 1.6×1019 C).

Alternatively, Hall mobility can also be expressed in terms of the relaxation time (τ ) and the effective mass of the charge carrier (m ):

μH=eτm

Here:

  • e is the charge of the carrier.
  • τ is the average time between two successive collisions of the charge carriers (relaxation time).
  • m is the effective mass of the charge carrier.

This equation is derived from the relationship between drift velocity (vd ), mobility (μ ), and the electric field (E ), as well as the fundamental definitions of charge carrier dynamics.

Correct Option Analysis:

The correct answer is:

Option 1: μH=eτm

This is the standard and accurate formula for Hall mobility. It directly reflects the dependence of Hall mobility on the charge of the carrier, the relaxation time, and the effective mass of the charge carrier. The higher the relaxation time or the smaller the effective mass, the greater the mobility of the charge carriers.

Important Information:

To further understand the analysis, let’s evaluate the other options:

Option 2: μH=e×τ/m

This option appears to represent the formula correctly, but the use of ambiguous formatting (e.g., missing parentheses or improper division syntax) can lead to misinterpretation. For clarity and standardization in scientific communication, it is essential to use proper mathematical formatting, such as the one provided in Option 1.

Option 3: μH=mτe

This option is incorrect because it suggests that Hall mobility is directly proportional to the product of the effective mass (m ) and the relaxation time (τ ) and inversely proportional to the charge (e ). This contradicts the actual relationship, where mobility is inversely proportional to the effective mass and directly proportional to both the charge and the relaxation time.

Option 4: μH=emτ

This option is incorrect because it implies that Hall mobility is proportional to the product of charge (e ) and effective mass (m ) and inversely proportional to the relaxation time (τ ). This is inconsistent with the correct formula, which states that mobility is proportional to charge and relaxation time but inversely proportional to the effective mass.

Conclusion:

Hall mobility is a fundamental parameter in semiconductor physics, directly influencing the performance of electronic devices. The correct formula, μH=eτm , highlights the interplay between charge, relaxation time, and effective mass in determining the mobility of charge carriers. A clear understanding of this relationship is critical for the analysis and design of materials and devices in electronics and optoelectronics.

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