Semiconductor surface engineering

[LUFER]

Homework Statement

What is the relationship between the optical cross-section and the physical cross-section of a particle?

Relevant Equations

σ sc= (S(spread))/(I (incident))

Regarding surface engineering, specifically the reflection and scattering of a semiconductor material, I have the following question in response to the problem below.

On a surface with reflectance R = 0.35, a power of 5 x 10⁻⁵ W/µm² is applied. Knowing that the total power scattered by the particle is 5 x 10⁻² W, determine the optical section. What is the relationship between the optical section and the physical section of the particle?

Scattering Section or Optical Section = σ sc
Physical or geometric section = σ geom
Scattering efficiency factor = Q sc

Substrate reflectivity (R): 0.35.
Incident intensity (I): 5*10E-5 w/(μm^2)
Total scattered power (S or Psc): 5*10E-2 W
Determination of the Optical Section σsc = ?


σ sc= Q sc* σ geom

S: Total power of scattered light (measured in Watts).
I: Intensity of incident light (measured in Watts/μm²).

σ sc= (S(spread))/(I (incident)) or σ sc= (P/ I )

σ sc=((5 ×10^(-2) ) W) / (5 ×10^(-5) W/(μm^2 )) = 10^3 μm^2


What I understood:

The irradiance that effectively interacts with the particle (Ief) is the result of modulating the incident power by the reflectance capacity of the substrate. Mathematically, the presence of the substrate defines the light available for scattering.
Surface reflectance (R) = 0.35
Incident Irradiance (I inc) = 5 *E-5 W/(µm^2 )

I ef= R * I inc
I ef= 0,35 * (5 * 10^(-5) )
I ef= 1,75 *10^(-5) W/(µm^2 )


σ opt=P esp/I ef

σ opt=((5 * 10^(-2) ))/((1.75 * 10^(-5) ) ) = 2857 µm^2




I'm honestly not sure if the concepts I used in the equations are correct. I would appreciate it if someone could correct me and, most importantly, provide me with bibliographic sources to help me delve deeper into the subject.

 

[Steve4Physics]

Hi @LUFER. Your post is difficult (for me at least) to follow. Here are some comments...

In the original question the phrase, "a power of 5 x 10⁻⁵ W/µm²" is wrong. The quantity 5 x 10⁻⁵ W/µm² is an intensity, not a power.

The wording refers to "a surface" and to "the particle". This is unclear/ambiguous. Which is it? There are two very different interpretations.

In the original question you are asked "What is the relationship between the optical section and the physical section of the particle?". This is not clear terminology. Are you being asked to find the ratio of the scattering cross-section to the physical cross-section for a (spherical) particle?

Another problem is that you have used many different variable names and there is duplication. Here is a list of the variables you have used (as best as I can do them quickly):

R Reflectance
R Substrate reflectivity
$σ_{sc}$ Scattering Section or Optical Section
$σ_{geom}$ Physical or geometric section
$Q_{sc}$ Scattering efficiency factor
$I$ Incident intensity
$S$ Total scattered power
$S$ spread
$P_{sc}$ Total scattered power
$P$ Unclear – possibly the same as $P_{sc}$
$S$ Total power of scattered light
$I_{inc}$ Incident irradiance
$σ_{opt}$ not defined
$P_{esp}$ not defined
$I_{ef}$ the irradiance that effectively interacts with the particle
To be honest, that’s all a bit of a mess - there are unused terms and duplication.

You need to uniquely define necessary variables.

Give any formulae which relate variables. (Each variable should appear in at least one formula or you do not need it.)

Show subscripts correctly. E.g. use LaTeX. A link to the LaTeX Guide is at the bottom-left of the edit-window. Or click this link.

Sorting out the above problems will help you to get useful replies.

 

[LUFER]

Hi @LUFER. Your post is difficult (for me at least) to follow. Here are some comments...

In the original question the phrase, "a power of 5 x 10⁻⁵ W/µm²" is wrong. The quantity 5 x 10⁻⁵ W/µm² is an intensity, not a power.

The wording refers to "a surface" and to "the particle". This is unclear/ambiguous. Which is it? There are two very different interpretations.

In the original question you are asked "What is the relationship between the optical section and the physical section of the particle?". This is not clear terminology. Are you being asked to find the ratio of the scattering cross-section to the physical cross-section for a (spherical) particle?

Another problem is that you have used many different variable names and there is duplication. Here is a list of the variables you have used (as best as I can do them quickly):

R Reflectance
R Substrate reflectivity
$σ_{sc}$ Scattering Section or Optical Section
$σ_{geom}$ Physical or geometric section
$Q_{sc}$ Scattering efficiency factor
$I$ Incident intensity
$S$ Total scattered power
$S$ spread
$P_{sc}$ Total scattered power
$P$ Unclear – possibly the same as $P_{sc}$
$S$ Total power of scattered light
$I_{inc}$ Incident irradiance
$σ_{opt}$ not defined
$P_{esp}$ not defined
$I_{ef}$ the irradiance that effectively interacts with the particle
To be honest, that’s all a bit of a mess - there are unused terms and duplication.

You need to uniquely define necessary variables.

Give any formulae which relate variables. (Each variable should appear in at least one formula or you do not need it.)

Show subscripts correctly. E.g. use LaTeX. A link to the LaTeX Guide is at the bottom-left of the edit-window. Or click this link.

Sorting out the above problems will help you to get useful replies.

I apologize, I will make the requests you asked for and edit the response to make the changes more understandable.


Light Scattering in Reflective Substrates
Fundamental Concept
Light scattering is the physical phenomenon in which light, upon encountering an obstacle (such as a contaminant particle), is deflected from its original straight path in various directions. In Surface Engineering, this concept is essential for detecting and measuring the size of particles that can cause short circuits in high-precision devices (such as silicon wafers and MOS devices).

Definition of Terms
To avoid duplication, we will use the following standardization:
• $\sigma_{sc}$ (Optical Section or Scattering Section): This is the "light signature" of the particle. It is defined by the ratio between the total scattered power and the intensity of the incident or effective light. Its purpose is to allow the detection of the contaminant.

• $\sigma_{geom}$ (Physical or Geometric Section): This is the area of the actual cross-section (physical size) of the particle.

• $Q_{sc}$ (Scattering Efficiency): The factor that relates the optical cross-section to the physical cross-section.

________________________________________
Problem Data
• $R$ (Substrate Reflectivity) = 0.35
• $I_{inc}$ (Incident Intensity/Irradiance) = 5 \cdot 10^{-5} W/μm²
• $P_{esp}$ (Total Scattered Power) = 5 \cdot 10^{-2} W
________________________________________
Solution: Determination of the Optical Cross-Section ($\sigma_{sc}$)
Below, we present the two calculation approaches based on the interpretation of the interaction of light with the substrate.

Approach 1: Standard Method (Adapted TIS)
In this scenario, the optical cross-section is calculated by directly relating the scattered power to the intensity of the incident light source.

$$\sigma_{sc} = \frac{P_{esp}}{I_{inc}}$$
Substituting the values:
$$\sigma_{sc} = \frac{5 \cdot 10^{-2}}{5 \cdot 10^{-5}}$$
Final Answer 1:
$$\sigma_{sc} = 10^{3} \text{ μm}^2 = 1000 \text{ μm}^2$$
Approach 2: Considering the Effective Irradiance on the Reflecting Surface
In reality, the particle is not isolated in space. The presence of a substrate with reflectance ($R = 0.35$) alters the energy density that the particle effectively "sees". The effective irradiance ($I_{ef}$) modulates the light available for scattering.

1. Calculation of Effective Irradiance ($I_{ef}$):
$$I_{ef} = R \cdot I_{inc}$$
$$I_{ef} = 0.35 \cdot (5 \cdot 10^{-5})$$
$$I_{ef} = 1.75 \cdot 10^{-5} \text{ W/μm}^2$$
2. Determination of Optical Section ($\sigma_{sc}$):
$$\sigma_{sc} = \frac{P_{esp}}{I_{ef}}$$
$$\sigma_{sc} = \frac{5 \cdot 10^{-2}}{1.75 \cdot 10^{-5}}$$
Final Answer 2:
$$\sigma_{sc} \approx 2857 \text{ μm}^2$$

 

[Steve4Physics]

I apologize

No need to apologise!

I’m not familiar with. Surface Engineering so can't help much. I'll add a few questions/ comments and then hopefully leave it to someone who knows what they're talking about.

It might help if you said what text book you are using and/or where the question comes from.

(This appears to be a rather specialist topic, so maybe a Mentor will move it.)

Light Scattering in Reflective Substrates
Fundamental Concept
Light scattering is the physical phenomenon in which light, upon encountering an obstacle (such as a contaminant particle), is deflected from its original straight path in various directions. In Surface Engineering, this concept is essential for detecting and measuring the size of particles that can cause short circuits in high-precision devices (such as silicon wafers and MOS devices).

It is not clear to me what system we are talking about. It could be:

1) light being scaterred inside a transparent medium (e.g. light passing through a medium containing impurity particles);

2) light scaterring (diffuse reflection) from the surface of a material, where surface imperfections are modelleled as scattering particles;

3) something else.

A diagram clearly showing where reflection and scattering occur would help.

Problem Data
• $R$ (Substrate Reflectivity) = 0.35

The meaning isn't clear. What is the boundary at which reflectivity is being measured?

Approach 1: Standard Method (Adapted TIS)

In this scenario, the optical cross-section is calculated by directly relating the scattered power to the intensity of the incident light source.
$\sigma_{sc} = \frac{P_{esp}}{I_{inc}}$

Substituting the values:
$\sigma_{sc} = \frac{5 \cdot 10^{-2}}{5 \cdot 10^{-5}}$

Final Answer 1:

$\sigma_{sc} = 10^{3} \text{ μm}^2 = 1000 \text{ μm}^2$

This doesn't use the given value of reflectivity! Is that acceptable?

Approach 2: Considering the Effective Irradiance on the Reflecting Surface

I've no idea what 'Effective radiance' is in this context, so can't comment.

Good luck!