A p-n junction includes p -type silicon, with donor atom density ND, adjacent to n -type silicon

Name: Problem 50, Chapter 42: University Physics with Modern Physics
Uploaded: 2021-11-28T11:35:52-08:00
Duration: 9 min 58 s
Channel: Keshav Singh

Question

A $p-n$ junction includes $p$ -type silicon, with donor atom density $N_{\mathrm{D}},$ adjacent to $n$ -type silicon, with acceptor atom density $N_{\mathrm{A}}$. Near the junction, free electrons from the $n$ side have diffused into the $p$ side while free holes from the $p$ side have diffused into the $n$ side, leaving a "depletion region" of width $W$ where there are no free charge carriers (Fig. $\mathbf{P} \mathbf{4 2 . 5 0}$ ). The width of the depletion region on the $n$ side is $a_{n}$, and the width of the depletion region on the $p$ side is $a_{p} .$ Each depleted region has a constant charge density with magnitude equal to the corresponding donor atom density. Assume there is no net charge outside the depleted regions. Define an $x$ -axis pointing from the $p$ side toward the $n$ side with the origin at the center of the junction. An electric field $\vec{E}=-E \hat{\imath}$ has developed in the depletion region, stabilizing the diffusion. (a) Use Gauss's law to determine the electric field on the $p$ side of the junction, for $-a_{p} \leq x \leq 0 .$ (Hint: Use a cylindrical Gaussian surface parallel to the $x$ -axis, with the left end outside the depletion region (where $\overrightarrow{\boldsymbol{E}}=0$ ) and the right side inside the region.) Note that the dielectric constant of silicon is $K=11.7$. (b) Determine the electric field on the $n$ side of the junction, for $0 \leq x \leq a_{n}$ (c) Use continuity at $x=0$ to determine a relationship between the depths $a_{p}$ and $a_{n}$ and in terms of $N_{\mathrm{A}}$ and $N_{\mathrm{D}}$. (d) The electric potential $V(x)$ may be obtained by integrating the electric field, as shown in Eq. $(23.18) .$ Determine $V(x)$ in the region $-a_{p} \leq x \leq 0$ using the convention that $V(x)=0$ for $x<-a_{p}$. (e) Similarly, determine $V(x)$ in the region $0 \leq x \leq a_{n}$. (f) What is the "barrier potential" $V_{b}=V\left(a_{n}\right)-V\left(-a_{p}\right) ?(\mathrm{~g})$ The $p$ side is doped with boron atoms with density $N_{\mathrm{A}}=1.00 \times 10^{16} \mathrm{~cm}^{-3} .$ The $n$ side is doped with arsenic atoms with density $N_{\mathrm{D}}=5.00 \times 10^{16} \mathrm{~cm}^{-3} .$ The $n$ side depletion depth is $a_{n}=55.0 \mathrm{nm} .$ What is the $p$ side depletion depth $a_{p} ?$ (h) What is the peak magnitude of the electric field, at $x=0 ?$ (i) What is the value of the barrier potential $V_{\mathrm{b}}$ ?

   A $p-n$ junction includes $p$ -type silicon, with donor atom density $N_{\mathrm{D}},$ adjacent to $n$ -type silicon, with acceptor atom density $N_{\mathrm{A}}$. Near the junction, free electrons from the $n$ side have diffused into the $p$ side while free holes from the $p$ side have diffused into the $n$ side, leaving a "depletion region" of width $W$ where there are no free charge carriers (Fig. $\mathbf{P} \mathbf{4 2 . 5 0}$ ). The width of the depletion region on the $n$ side is $a_{n}$, and the width of the depletion region on the $p$ side is $a_{p} .$ Each depleted region has a constant charge density with magnitude equal to the corresponding donor atom density. Assume there is no net charge outside the depleted regions. Define an $x$ -axis pointing from the $p$ side toward the $n$ side with the origin at the center of the junction. An electric field $\vec{E}=-E \hat{\imath}$ has developed in the depletion region, stabilizing the diffusion. (a) Use Gauss's law to determine the electric field on the $p$ side of the junction, for $-a_{p} \leq x \leq 0 .$ (Hint: Use a cylindrical Gaussian surface parallel to the $x$ -axis, with the left end outside the depletion region (where $\overrightarrow{\boldsymbol{E}}=0$ ) and the right side inside the region.) Note that the dielectric constant of silicon is $K=11.7$.
(b) Determine the electric field on the $n$ side of the junction, for $0 \leq x \leq a_{n}$ (c) Use continuity at $x=0$ to determine a relationship between the depths $a_{p}$ and $a_{n}$ and in terms of $N_{\mathrm{A}}$ and $N_{\mathrm{D}}$. (d) The electric potential $V(x)$ may be obtained by integrating the electric field, as shown in Eq. $(23.18) .$ Determine $V(x)$ in the region $-a_{p} \leq x \leq 0$ using the convention that $V(x)=0$ for $x<-a_{p}$. (e) Similarly, determine $V(x)$ in the region $0 \leq x \leq a_{n}$. (f) What is the "barrier potential" $V_{b}=V\left(a_{n}\right)-V\left(-a_{p}\right) ?(\mathrm{~g})$ The $p$ side is doped with boron atoms with density $N_{\mathrm{A}}=1.00 \times 10^{16} \mathrm{~cm}^{-3} .$ The $n$ side is doped with arsenic atoms with density $N_{\mathrm{D}}=5.00 \times 10^{16} \mathrm{~cm}^{-3} .$ The $n$ side depletion depth is $a_{n}=55.0 \mathrm{nm} .$ What is the $p$ side depletion depth $a_{p} ?$ (h) What is the peak magnitude of the electric field, at $x=0 ?$ (i) What is the value of the barrier potential $V_{\mathrm{b}}$ ?

University Physics with Modern Physics

Roger A. Freedman,… 15th Edition

Chapter 42, Problem 50

Instant Answer

Solved by Expert Keshav Singh

11/28/2021

Step 1

We use a cylindrical Gaussian surface parallel to the x-axis, with the left end outside the depletion region (where $\vec{E}=0$) and the right side inside the region. Gauss's law gives us $\int \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0}$, where $\vec{E}$ is Show more…

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A $p-n$ junction includes $p$ -type silicon, with donor atom density $N_{\mathrm{D}},$ adjacent to $n$ -type silicon, with acceptor atom density $N_{\mathrm{A}}$. Near the junction, free electrons from the $n$ side have diffused into the $p$ side while free holes from the $p$ side have diffused into the $n$ side, leaving a "depletion region" of width $W$ where there are no free charge carriers (Fig. $\mathbf{P} \mathbf{4 2 . 5 0}$ ). The width of the depletion region on the $n$ side is $a_{n}$, and the width of the depletion region on the $p$ side is $a_{p} .$ Each depleted region has a constant charge density with magnitude equal to the corresponding donor atom density. Assume there is no net charge outside the depleted regions. Define an $x$ -axis pointing from the $p$ side toward the $n$ side with the origin at the center of the junction. An electric field $\vec{E}=-E \hat{\imath}$ has developed in the depletion region, stabilizing the diffusion. (a) Use Gauss's law to determine the electric field on the $p$ side of the junction, for $-a_{p} \leq x \leq 0 .$ (Hint: Use a cylindrical Gaussian surface parallel to the $x$ -axis, with the left end outside the depletion region (where $\overrightarrow{\boldsymbol{E}}=0$ ) and the right side inside the region.) Note that the dielectric constant of silicon is $K=11.7$. (b) Determine the electric field on the $n$ side of the junction, for $0 \leq x \leq a_{n}$ (c) Use continuity at $x=0$ to determine a relationship between the depths $a_{p}$ and $a_{n}$ and in terms of $N_{\mathrm{A}}$ and $N_{\mathrm{D}}$. (d) The electric potential $V(x)$ may be obtained by integrating the electric field, as shown in Eq. $(23.18) .$ Determine $V(x)$ in the region $-a_{p} \leq x \leq 0$ using the convention that $V(x)=0$ for $x<-a_{p}$. (e) Similarly, determine $V(x)$ in the region $0 \leq x \leq a_{n}$. (f) What is the "barrier potential" $V_{b}=V\left(a_{n}\right)-V\left(-a_{p}\right) ?(\mathrm{~g})$ The $p$ side is doped with boron atoms with density $N_{\mathrm{A}}=1.00 \times 10^{16} \mathrm{~cm}^{-3} .$ The $n$ side is doped with arsenic atoms with density $N_{\mathrm{D}}=5.00 \times 10^{16} \mathrm{~cm}^{-3} .$ The $n$ side depletion depth is $a_{n}=55.0 \mathrm{nm} .$ What is the $p$ side depletion depth $a_{p} ?$ (h) What is the peak magnitude of the electric field, at $x=0 ?$ (i) What is the value of the barrier potential $V_{\mathrm{b}}$ ?

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Key Concepts

Semiconductor Dielectric Constant

The dielectric constant of a semiconductor, such as silicon with a relative constant of 11.7, influences the electric field and potential calculations by modifying the effective permittivity of the material. It plays a critical role in accurately applying Gauss's law in the depletion region and in understanding the electrostatic behavior within semiconductor devices.

Electric Potential and its Relation to the Electric Field

The electric potential in the depletion region is determined by integrating the electric field. The potential profile, often referred to as the barrier potential, is crucial for understanding the charge carrier dynamics and the built-in voltage in the junction. This integration provides the voltage drop across both sides of the junction, which is central to device operation.

Electric Field and its Spatial Distribution

In a PN junction, the electric field is established within the depletion region to counteract further diffusion of carriers and maintain equilibrium. The field has different expressions on the p-side and n-side based on the uniform charge densities. Continuity of the electric field at the junction and the appropriate boundary conditions are essential for accurately determining its variation across the depleted regions.

Charge Neutrality and Depletion Width Relationship

The widths of the depletion regions on the p-side and n-side are interrelated through the condition of overall charge neutrality. This means that the negative charge due to acceptor ions on the p-side must balance the positive charge due to donor ions on the n-side, leading to the relationship a_p * N_A = a_n * N_D. This concept is fundamental in determining the spatial extent of the depletion region and the electric field distribution in the junction.

Depletion Region

The depletion region is the area near the PN junction where mobile charge carriers have diffused away, leaving behind ionized dopant atoms. This region is characterized by a fixed space charge due to the exposed donor and acceptor ions, and it plays a key role in establishing the internal electric field and barrier potential across the junction.

PN Junction

A PN junction is formed by joining p-type and n-type semiconductors, which are doped with acceptor and donor atoms respectively. At the junction, charge carriers (holes and electrons) diffuse across the interface, leading to recombination and the formation of a built-in potential. This structure is fundamental in semiconductor devices like diodes and transistors and underlies concepts of rectification and charge carrier transport.

Gauss's Law in Semiconductor Analysis

Gauss's law is used to relate the electric flux through a closed surface with the charge enclosed by that surface. In the context of a PN junction, it is applied to the depletion region (which has a uniform charge density due to ionized dopants) to derive expressions for the electric field. The presence of a dielectric material, such as silicon with its specific dielectric constant, is also taken into account in this analysis.

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