# 2D Fermi Gas

Staff Emeritus
Gold Member
Although I have some major conceptual problems with the Fermi gas as treated in my solid state physics notes (see this thread: https://www.physicsforums.com/showthread.php?t=161222, I have attempted to solve this homework problem in an analogous manner to the solution for the 3D Fermi gas given in the notes, ignoring these conceptual hangups.

## Homework Statement

Find the density of states for a 2D electron gas.

## Homework Equations

See attempted solution below

## The Attempt at a Solution

Assume that in real space the gas is confined to an area

$$A = l_xl_y$$

Write the components of the electron wavevector in terms of the respective principle quantum numbers:

$$k_x = \frac{2\pi}{l_x}n_x \ \ \ \ k_y = \frac{2\pi}{l_y}n_y$$

Therefore the number of states associated with an element $d^2\mathbf{k}$ is (supposedly)

$$2dn_xdn_y = \frac{A}{4\pi^2}2dk_xdk_y$$

Again, in an analogous way to what was done in the notes in 3D, I switch to polar coordinates, so that I can get the density of states as a function of $k = |\mathbf{k}|$.

# of states between k and k + dk

$$Z(k)dk = \frac{A}{4\pi^2}2(2 \pi k dk)$$

Change variables to convert Z(k) to D(E), the density of states as a function of energy.

$$E = \frac{\hbar^2k^2}{2m} \Rightarrow dE = \frac{\hbar^2}{2m}2kdk \Rightarrow dk = \frac{m}{\hbar^2 k}dE$$

$$D(E)dE = \frac{A}{4\pi^2} 2(2 \pi k) \frac{m}{\hbar^2 k}dE = \frac{Am}{\pi \hbar^2} dE$$

$$\Rightarrow D(E) = \textrm{const.} = \frac{Am}{\pi \hbar^2}$$

This result is drastically different from the 3D case, and later on in the same problem, it leads me to the conclusion that the Fermi energy is independent of temperature for the 2D gas. Is this solution correct, or have I made some egregious error somewhere?

Staff Emeritus
The text passage is a discussion of a physics problem involving the density of states for a 2D electron gas. It includes a derivation and a question about the correctness of the solution.
no_question

the passage does not contain a mathematical question that requires a numerical or symbolic answer. instead, it is a detailed derivation of a physics problem, followed by a request for feedback on the correctness of the solution. there is no specific mathematical question or answer provided in the form of a problem to solve.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 2D Fermi Gas

Staff Emeritus
Gold Member
Although I have some major conceptual problems with the Fermi gas as treated in my solid state physics notes (see this thread: https://www.physicsforums.com/showthread.php?t=161222, I have attempted to solve this homework problem in an analogous manner to the solution for the 3D Fermi gas given in the notes, ignoring these conceptual hangups.

## Homework Statement

Find the density of states for a 2D electron gas.

## Homework Equations

See attempted solution below

## The Attempt at a Solution

Assume that in real space the gas is confined to an area

$$A = l_xl_y$$

Write the components of the electron wavevector in terms of the respective principle quantum numbers:

$$k_x = \frac{2\pi}{l_x}n_x \ \ \ \ k_y = \frac{2\pi}{l_y}n_y$$

Therefore the number of states associated with an element $d^2\mathbf{k}$ is (supposedly)

$$2dn_xdn_y = \frac{A}{4\pi^2}2dk_xdk_y$$

Again, in an analogous way to what was done in the notes in 3D, I switch to polar coordinates, so that I can get the density of states as a function of $k = |\mathbf{k}|$.

# of states between k and k + dk

$$Z(k)dk = \frac{A}{4\pi^2}2(2 \pi k dk)$$

Change variables to convert Z(k) to D(E), the density of states as a function of energy.

$$E = \frac{\hbar^2k^2}{2m} \Rightarrow dE = \frac{\hbar^2}{2m}2kdk \Rightarrow dk = \frac{m}{\hbar^2 k}dE$$

$$D(E)dE = \frac{A}{4\pi^2} 2(2 \pi k) \frac{m}{\hbar^2 k}dE = \frac{Am}{\pi \hbar^2} dE$$

$$\Rightarrow D(E) = \textrm{const.} = \frac{Am}{\pi \hbar^2}$$

This result is drastically different from the 3D case, and later on in the same problem, it leads me to the conclusion that the Fermi energy is independent of temperature for the 2D gas. Is this solution correct, or have I made some egregious error somewhere?

Staff Emeritus
user
The text passage is a discussion of a physics problem involving the density of states for a 2D electron gas. It includes a derivation and a question about the correctness of the solution.

TYPE: NO_QUESTION

The passage does not contain a mathematical question that requires a numerical or symbolic answer. Instead, it is a detailed derivation of a physics problem, followed by a request for feedback on the correctness of the solution. There is no specific mathematical question or answer provided in the form of a problem to solve.