# Depletion width of linearly doped PN-junction

• Mr_Allod
In summary, the conversation discusses deriving expressions for electric field and potential, setting them equal to each other to ensure continuity, and the difficulty in finding an expression for the depletion region width. The solution to this problem involves substituting the biasing voltage into the equation for electric field and solving a cubic equation to find the values for ##x_n## and ##x_p##.
Mr_Allod
Homework Statement
Derive expressions for the electric field distribution and potential for a PN Junction with linear doping gradient then use this to calculate the depletion width as a function of the bias voltage ##V_bi##
Relevant Equations
##x<0## (P-Region) doping concentration: ##N_A = -Sx##
##x>0## (N-Region) the doping concentration is: ##N_D = Kx##
Electric Field in P-Region: ##E_P=\frac {qS}{2\epsilon}(x^2 - x_p^2)##
Electric Field in N-Region: ##E_N=\frac {qK}{2\epsilon}(x^2 - x_n^2)##
Potential in P-Region: ##V = V_p + \frac {qM_A}{\epsilon} \left( \frac {x_p^2x}{2} - \frac {x^3}{6} + \frac {x_p^3}{3} \right)##
Potential in N-Region: ##V = V_n + \frac {qM_A}{\epsilon} \left( \frac {x_n^2x}{2} - \frac {x^3}{6} - \frac {x_p^3}{3} \right)##
Continuity at x = 0:
$$Sx_p^2 = Kx_n^2$$
$$V_{bi} = V_n - V_p = \frac {q}{3\epsilon} \left( Sx_p^3 + Kx_n^3\right)$$
Hello there, I have derived the expressions for electric field and potential to be the ones above, then for continuity at ##x = 0## I set the electric fields and potentials to be equal to yield the expressions:
$$Sx_p^2 = Kx_n^2$$
$$V_{bi} = V_n - V_p = \frac {q}{3\epsilon} \left( Sx_p^3 + Kx_n^3\right)$$

I am now stuck on how to find an expression for the depletion region width, which in our class we have defined as ##|x_p|+x_n##, where ##-x_p## and ##x_n## are the edges of the depletion layer in the P and N regions respectively. I just can't seem to find a way to isolate them without ending up with one as a function of the other, which is not terribly useful so I would really appreciate some help with this.

The solution to this problem is to substitute the equation for the biasing voltage into the equation for electric field:$$\frac{q}{3\epsilon} \left( Sx_p^3 + Kx_n^3\right) = Sx_p^2 - Kx_n^2$$Rearranging this equation gives us:$$Kx_n^3 + (K-S)x_n^2 - \frac{q}{3\epsilon} = 0$$This is a cubic equation in terms of ##x_n##, which can be solved using the Cardano formula. Once we have the solution for ##x_n##, we can then solve for ##x_p## by substituting it into the equation for electric field and solving for ##x_p##.

## 1. What is the depletion width of a linearly doped PN-junction?

The depletion width of a linearly doped PN-junction is the region around the interface between the p-type and n-type semiconductors where the majority carriers (electrons in the n-type and holes in the p-type) are depleted, creating a region with no free charge carriers.

## 2. How is the depletion width of a linearly doped PN-junction calculated?

The depletion width can be calculated using the following equation: W = sqrt((2 * ε_r * ε_0 * (V_bi + V_applied)) / (q * N_a * N_d)), where W is the depletion width, ε_r is the relative permittivity of the semiconductor, ε_0 is the permittivity of free space, V_bi is the built-in potential, V_applied is the applied voltage, q is the elementary charge, N_a is the acceptor doping concentration, and N_d is the donor doping concentration.

## 3. How does the depletion width of a linearly doped PN-junction affect the device's performance?

The depletion width plays a crucial role in the operation of a PN-junction device. It controls the width of the depletion region, which in turn affects the device's capacitance, resistance, and current flow. A larger depletion width can result in a higher breakdown voltage and lower leakage current, while a smaller depletion width can lead to a faster response time.

## 4. What factors can affect the depletion width of a linearly doped PN-junction?

The depletion width is primarily influenced by the doping concentrations of the p-type and n-type semiconductors, the applied voltage, and the material properties of the semiconductor. Temperature and impurities can also have an impact on the depletion width.

## 5. How can the depletion width of a linearly doped PN-junction be controlled?

The depletion width can be controlled by adjusting the doping concentrations of the p-type and n-type semiconductors and the applied voltage. It can also be modified by using different semiconductor materials with varying properties. Additionally, the depletion width can be altered by introducing impurities or using techniques such as ion implantation or diffusion.

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