Lorentzian lineshape, uncertainty principle and AlGaAs conduction band offset

[daffoddill]

Homework Statement

1) I try to understand the lorentzian lineshape and relate to the gaussian graph but i don't know what is the difference.

2) Uncertainty principle- if it's related to the lorentzian lineshape it will give the information about the lifetime in the well and the width of the transmission peak. This has to be applied for one dimensional two barrier of AlGaAs material to find the transmission probability.

3) For AlGaAs conduction band offset-find how the conduction band offset varies with Al concentration and why Al increases the conduction band offset.

Homework Equations

The Attempt at a Solution

 

[Rajini]

Hi,
1) In general, Lorentzian profiles are related to lifetime and and not Gaussian. Gaussian rapidly touches the x-axis and Lorentzian don't behave like that. I suggest you to plot and see the difference..(if you want i can upload two plots).
2)the FWHM of Lorentzian is equal to the lifetime (please write more information)
3)I have no idea

 

[daffoddill]

thx rajini..
yes if you can upload the two plots it will be great..because it's quit few information that i found through the internet..could you please upload if you have any other information about it?
thanks

 

[Rajini]

Hi,
Loretzian function:
$$f(x)=\frac{I_0(\Gamma/2)^2}{(x-x_0)^2+(\Gamma/2)^2}$$
$$\Gamma$$=FWHM.

Gaussian function:
$$f(x)=I_0\exp\left(\frac{-(x-x_0)^2}{2\Gamma^2}\right)$$
$$\Gamma2\sqrt{2\ln2}$$=FWHM.
For both function:
I₀=height of peak.
x₀=position of peak.
You will find 2 pictures with same height, fullwidth=1 for lorentzian and position =0. For Gaussian fullwidth=1.1774 (=$$0.5\times2\sqrt{2\ln2}$$).
Note how the Gaussian function rapidly touches the x-axis and for Lorentzian it is not!

 

[paranormal]

1) The Lorentzian lineshape is a type of spectral lineshape that is characterized by a symmetric, bell-shaped curve with a single peak. It is commonly used to describe the spectral lines of atoms and molecules. The main difference between Lorentzian and Gaussian lineshapes is the tails of the curve. Lorentzian lineshapes have longer tails compared to Gaussian lineshapes, which have shorter tails. This difference is due to the different underlying physical mechanisms that produce these lineshapes. Lorentzian lineshapes arise from a homogeneous broadening mechanism, while Gaussian lineshapes arise from an inhomogeneous broadening mechanism.

2) The uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. This principle is related to the Lorentzian lineshape in that the width of the transmission peak in the Lorentzian lineshape is directly related to the uncertainty in the energy of the particle. This uncertainty in energy is related to the lifetime of the particle in the well, as well as the width of the transmission peak.

3) The AlGaAs conduction band offset refers to the difference in energy between the conduction bands of AlGaAs and GaAs. This offset is an important factor in determining the electronic properties of AlGaAs materials. As the Al concentration increases, the conduction band offset also increases. This is because the addition of Al atoms changes the band structure of the material, resulting in a larger energy difference between the conduction bands of AlGaAs and GaAs. This change in band structure is due to the different electronegativities of Al and Ga atoms, which leads to a larger separation between their respective conduction bands.