# Second degree DE for pn-junction carrier concentration

• rire1979
In summary, the expression d2DPn(x)/dx2 can be calculated as DPn(x)/Lh2, where DPn(x) represents the excess minority carriers (holes) concentration in the n-type part of the pn junction. The roots to the characteristic equation are +/- 1/Lh, with the solution being DPn(x) = Ae-1/Lh + Be1/Lh. The solution for DPn(x) is actually DPn(0)e-x/Lh, with initial conditions of DPn(0) at x = 0 and DPn(Lh) = 0 at x = Lh. However, the process to arrive at this solution is unclear and assistance is needed.
rire1979
Upon some calculation I arrive to the expression:

d2DPn(x)/dx2 = DPn(x)/Lh2

Where:

DPn(x) = Pn(x) + Pno - excess minority carriers (holes) concentration in the n-type part of the pn junction.

Now the roots to the characteristic equation are +/- 1/Lh where Lh is the length of the diffusion.

Therefore the solution looks like:

DPn(x) = Ae-1/Lh + Be1/Lh

I know for a fact the solution is DPn(x) = DPn(0)e-x/Lh

The initial conditions would be that:
@ x = 0 we have hole concentration DPn(0)
@ x = Lh we have DPn(Lh) = 0

But I have no idea how to arrive at the solution in bold. I'm missing something and I was thinking you could help.

Thank you.

Try writing down the characteristic equation again and verifying the roots.

Lord Crc said:
Try writing down the characteristic equation again and verifying the roots.

r2 - 1/Lh2 = 0

r = +/- 1/Lh as I've mentioned.

I'm sorry, bit late here... misread that and thought the char. equation was r^2 - 1/Lh^2 r = 0, which, from my quick glance, could get you where you wanted. My bad.

## 1. What is a pn-junction?

A pn-junction is a type of interface between two semiconductor materials, one with a surplus of positive charge carriers (p-type) and the other with a surplus of negative charge carriers (n-type). This junction allows for the flow of electrons and holes, making it a key component in electronic devices.

## 2. What is carrier concentration?

Carrier concentration refers to the number of charge carriers (electrons or holes) present in a material. It is typically measured in units of carriers per cubic centimeter (cm-3). In the case of a pn-junction, the carrier concentration can vary depending on the type of semiconductor material and the presence of impurities.

## 3. How is second degree DE used in pn-junction carrier concentration?

Second degree differential equations (DE) are used to describe the behavior of carrier concentration in a pn-junction. This is because the rate of change of carrier concentration is directly related to the electric field and the mobility of carriers in the semiconductor material.

## 4. What factors affect carrier concentration in a pn-junction?

The carrier concentration in a pn-junction is affected by several factors, including the doping concentration of the semiconductor materials, the temperature, and the applied voltage. Additionally, the presence of impurities and defects in the material can also impact carrier concentration.

## 5. How is carrier concentration in a pn-junction measured?

Carrier concentration in a pn-junction can be measured using various experimental techniques, such as Hall effect measurements, capacitance-voltage measurements, and current-voltage measurements. These techniques involve applying an external voltage or magnetic field and measuring the resulting change in carrier concentration.

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