Derivation of width of depletion layer in the pn-junction

In summary, the derivation for the width of the depletion region in "SEMICONDUCTOR DEVICE FUNDAMENTALS" by Robert F. Pierret is given by the equations x_N = \sqrt{\frac{N_A^2}{N_D N_A (N_A+N_D)}} and W = \sqrt{\frac{(N_A+N_D)^2}{N_D N_A}}. This may be confusing as it is different from the expected outcome W = x_N + x_P = \sqrt{\frac{N_D^2+N_A^2}{N_A\cdot N_D\cdot(N_A+N_D)}}. However, this result has been found in various sources on the internet. The symbols used in the equations are defined
  • #1
tcsv018
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Hello,

I read a derivation for the width of the depletion region [itex]W[/itex] in "SEMICONDUCTOR DEVICE FUNDAMENTALS" by Robert F. Pierret in which at one point it says:
http://imageshack.com/i/ipbgKsK9p
Here again for better readability:

[itex]x_N = \sqrt{\frac{2 K_S \epsilon_0}{q}\frac{N_A}{N_D\cdot(N_A+N_D)}V_{bi}}[/itex]
[itex]x_P = \sqrt{\frac{2 K_S \epsilon_0}{q}\frac{N_D}{N_A\cdot(N_A+N_D)}V_{bi}}[/itex]
[itex]W = x_N + x_P = \sqrt{\frac{2 K_S \epsilon_0}{q}\frac{N_D+N_A}{N_A\cdot N_D}V_{bi}}[/itex]

Which is confusing to me as I would expect the same containing:
[itex]W = x_N + x_P = \sqrt{\frac{2 K_S \epsilon_0}{q}\frac{N_D^2+N_A^2}{N_A\cdot N_D\cdot(N_A+N_D)}V_{bi}}[/itex]This same outcome though is found on various places in the internet.

Does anyone know what I am missing?

Kind regards,

Name

Symbols:
[itex]x_N[/itex] length of n-doped region
[itex]x_P[/itex] length of p-doped region
[itex]N_A[/itex]density of acceptors
[itex]N_D[/itex]density of donors
[itex]V_{bi}[/itex] builtin potential

All the others are known constants
 
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  • #2
##\sqrt{x}+\sqrt{y} \neq \sqrt{x+y}##

Ignoring the common prefactors,
$$x_N = \sqrt{\frac{N_A^2}{N_D N_A (N_A+N_D)}}$$
$$x_N = \sqrt{\frac{N_A^2}{N_D N_A (N_A+N_D)}}$$
$$W = \sqrt{\frac{1}{N_D N_A (N_A+N_D)}} \left(\sqrt{N_A^2} + \sqrt{N_D^2}\right) = \sqrt{\frac{1}{N_D N_A (N_A+N_D)}} \left(N_A + N_D\right) \\= \sqrt{\frac{(N_A+N_D)^2}{N_D N_A (N_A+N_D)}} = \sqrt{\frac{N_A+N_D}{N_D N_A}}$$
 
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  • #3
Thanks,
Shame on me
 

Related to Derivation of width of depletion layer in the pn-junction

1. What is a pn-junction?

A pn-junction is a type of semiconductor device that consists of a p-type semiconductor (with positively charged carriers) and an n-type semiconductor (with negatively charged carriers) joined together. This results in a depletion layer or barrier between them, creating a one-way flow of electrons.

2. How is the width of the depletion layer in a pn-junction derived?

The width of the depletion layer can be derived using the formula W = (2εrε0Vbi / (qNa))1/2, where W is the width, εr is the relative permittivity, ε0 is the permittivity of free space, Vbi is the built-in potential, q is the elementary charge, and Na is the acceptor doping concentration.

3. What factors affect the width of the depletion layer?

The width of the depletion layer is affected by the doping concentrations of the p and n regions, the built-in potential, and the permittivity of the materials. It is also influenced by external factors such as voltage applied to the junction and temperature.

4. Why is the width of the depletion layer important in pn-junctions?

The width of the depletion layer plays a crucial role in the functioning of pn-junctions. It determines the barrier for electron flow and influences the device's electrical properties, such as the forward and reverse bias behavior and the junction capacitance.

5. How is the derivation of the width of the depletion layer used in practical applications?

The derivation of the width of the depletion layer is used in the design and analysis of various semiconductor devices, such as diodes, transistors, and solar cells. It helps in understanding the behavior of pn-junctions under different conditions, allowing for the optimization of device performance.

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