# Current through Ballistic 2DEG Channel

• FermiFrustration
In summary, to solve this problem, you need to solve the two-dimensional Schrödinger equation with the given parameters and use the energy eigenvalues to calculate the number of available channels through the channel, which can then be used to calculate the current in the system.
FermiFrustration
Homework Statement
See attachment!
Find current through ballistic 2DEG channel assuming a parabolic potential in the channel
Relevant Equations
Schrödinger Equation, Airy's Equation, I = N (2e^2)/h V
So I am a bit uncertain what approach is best for solving this problem and how exactly I should approach it, but my strategy right now is:

1. Solve the time-independent Schrödinger Equation with the given Hamiltonian and find energy eigenvalues of system:
-Here I struggle a bit with actually solving it; if my approach is right this should be the crux of the problem
-Since the y and x-dependent parts of the Schrödinger equation are possible to separate I think it is possible to solve this as two differential equations with only one variable like:

## ( -\frac{h^2}{2m^{*}}(\frac{d^2}{dx^2} + \frac{d^2}{dy^2}) + \frac{1}{2}m^* \omega^2 y^2 - E )\Psi = 0 ##
## (-\frac{h^2}{2m^{*}}\frac{d^2}{dx^2} - E )\Psi = (\frac{h^2}{2m^{*}}\frac{d^2}{dy^2} - \frac{1}{2}m^* \omega^2 y^2)\Psi ##
## \implies ¨(-\frac{h^2}{2m^{*}}\frac{d^2}{dx^2} - E )\Psi = G ##
## \implies (\frac{h^2}{2m^{*}}\frac{d^2}{dy^2} - \frac{1}{2}m^* \omega^2 y^2)\Psi = G ##

-I am however unsure how to reassemble this into a complete solution.

2. Find energy eigenstates below the fermi energy - their number should be the number of available channels through the channel

3. Plug this new obtained N into I = N (2e^2)/h V with the given voltage from source to drainIs my approach right and how should I go about solving the differential equation?

#### Attachments

• NanoP1.png
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Your approach is close, but you need to solve the two-dimensional Schrödinger equation with the given parameters, rather than separating it into two equations. The two-dimensional version of the Schrödinger equation is:## ( -\frac{h^2}{2m^{*}}(\frac{d^2}{dx^2} + \frac{d^2}{dy^2}) + \frac{1}{2}m^* \omega^2 (x^2+y^2) - E )\Psi = 0 ##There are several methods for solving this equation, such as separation of variables, Green's function techniques, and numerical methods. Once you have solved the equation to obtain the energy eigenvalues, you can then use those values to calculate the number of available channels through the channel, which you can then plug into I = N(2e^2)/h V with the given voltage from source to drain.

## 1. What is a ballistic 2DEG channel?

A ballistic 2DEG (two-dimensional electron gas) channel is a type of electronic device that consists of a thin layer of electrons confined to a two-dimensional plane. This channel is typically created in a semiconductor material and is used to study the behavior of electrons in a controlled environment.

## 2. How is current generated in a ballistic 2DEG channel?

Current is generated in a ballistic 2DEG channel when a voltage is applied across the channel, causing the electrons to move from one end to the other. The movement of electrons creates a flow of current, which can be measured and studied to understand the behavior of electrons in this type of channel.

## 3. What factors affect the current through a ballistic 2DEG channel?

The current through a ballistic 2DEG channel is affected by several factors, including the applied voltage, the thickness of the channel, the temperature of the device, and the properties of the material used to create the channel. Additionally, any external magnetic or electric fields can also influence the current through the channel.

## 4. How is the current through a ballistic 2DEG channel measured?

The current through a ballistic 2DEG channel is typically measured using specialized equipment, such as a current probe or a Hall effect sensor. These tools are able to detect the flow of electrons through the channel and provide accurate measurements of the current at different points along the channel.

## 5. What is the significance of studying the current through a ballistic 2DEG channel?

Studying the current through a ballistic 2DEG channel can provide valuable insights into the behavior of electrons in a confined environment. This research can have practical applications in the development of new electronic devices and technologies, as well as advancing our understanding of fundamental principles in physics.

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