Depletion Region and Built-in Potential in p-n Junctions

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Hello! Let's dive into the next topic in our "Semiconductor Physics and Transport" module.

In our last lesson, we established how to control the concentration of electrons and holes in a semiconductor through doping, creating P-type and N-type materials. We derived the expressions for carrier concentrations ( $n$ , $p$ ) and established the crucial law of mass action, $n p = n_{i}^{2}$ . This sets the stage for today's topic.

The logical next question is: what happens when we bring these two types of materials together? The interface between a P-type and an N-type semiconductor forms a p-n junction, arguably the most important structure in all of electronics. It is the fundamental building block for diodes, solar cells, LEDs, and transistors.

Our goal for this lesson is to analyze the formation of the depletion region and the built-in potential in a p-n junction by solving the Poisson equation. We will first build a qualitative physical picture of the junction at equilibrium and then develop a quantitative model to calculate its key properties.

1. The P-N Junction at Equilibrium: A Qualitative Picture

Imagine we have a block of P-type semiconductor and a block of N-type semiconductor, separate from each other. As we saw last time, the Fermi level $E_{F}$ in the P-type material is close to the valence band, while in the N-type material, it's close to the conduction band.

What happens when we bring them into intimate contact? The system must reach thermal equilibrium. A fundamental requirement of thermal equilibrium for a system of electrons is that the Fermi level must be constant everywhere. To achieve this, electrons and holes will move, forcing the energy bands to bend in the vicinity of the junction.

This process is driven by two competing mechanisms:

Diffusion: There's a massive concentration gradient for both carrier types at the junction. Holes are abundant on the P-side and scarce on the N-side, so they diffuse from P to N. Similarly, electrons diffuse from N to P. This movement of charge constitutes a diffusion current.
Drift: As majority carriers diffuse across the junction, they leave behind the fixed, ionized dopant atoms. On the P-side, we are left with negatively charged acceptor ions ( $N_{a}^{-}$ ). On the N-side, we are left with positively charged donor ions ( $N_{d}^{+}$ ). This creates a region near the junction that is depleted of mobile carriers, known as the depletion region or space-charge region.

This layer of fixed positive and negative charges establishes an internal electric field, $E$ , pointing from the N-side to the P-side. This field exerts a force on any remaining mobile carriers, pushing electrons towards the N-side and holes towards the P-side. This motion constitutes a drift current that opposes the diffusion current.

Equilibrium is reached when, for each carrier type, the drift current exactly balances the diffusion current. The net current flow is zero, but a dynamic balance of opposing currents persists.

To visualize this process, please watch the following video. It provides an excellent animation of the carrier dynamics and the resulting energy band diagram.

Electronic Devices: pn junction under equilibrium - depletion width, build-in potential

This video from the channel techgurukula provides a clear, step-by-step qualitative explanation of how a p-n junction reaches equilibrium. It covers the initial diffusion of carriers, the formation of the space-charge region, and the establishment of the internal electric field.

Watch the segments from 5:32 to 10:32. Focus on how the diffusion of electrons and holes uncovers fixed dopant ions, how this creates an electric field, and how that leads to the bending of the energy bands at equilibrium.

The result of this band bending is a built-in potential barrier, $V_{bi}$ , which prevents the vast majority of majority carriers from diffusing across the junction.

Electrostatic Variables for the Equilibrium p-n Junction — This figure summarizes the key electrostatic variables at an equilibrium p-n junction. (a) The energy band diagram shows the band bending and the built-in potential $qV_{bi}$. (b) The electrostatic potential $V(x)$ shows the potential barrier. (c) The electric field $\mathcal{E}(x)$ is confined to the depletion region. (d) The charge density $\rho(x)$ shows the fixed positive and negative charges of the ionized dopants.

2. Quantitative Analysis: Solving the Poisson Equation

To calculate the built-in potential and the width of the depletion region, we need a quantitative model. The governing equation is the 1D Poisson's equation, which you'll recall from electrostatics relates the electric potential to the charge density:

\frac{d ^{2} V}{d x ^{2}} = - \frac{ρ ( x )}{ϵ}

where $ρ (x)$ is the net charge density and $ϵ = ϵ_{r} ϵ_{0}$ is the dielectric permittivity of the semiconductor.

Solving this exactly is complex because the mobile carrier concentrations $n (x)$ and $p (x)$ are themselves complicated functions of the potential $V (x)$ . We can make excellent progress by adopting the depletion approximation.

The Depletion Approximation

This approximation simplifies the charge profile dramatically:

Inside the depletion region ( $- x_{p} < x < x_{n}$ ), we assume the mobile carrier concentrations are negligible ( $n \approx 0, p \approx 0$ ). The charge density is therefore due only to the fixed ionized dopants:
- $ρ (x) = - q N_{a}$ on the P-side ( $- x_{p} < x < 0$ )
- $ρ (x) = + q N_{d}$ on the N-side ( $0 < x < x_{n}$ )
Outside the depletion region, we assume the material is perfectly charge-neutral.

With this step-function charge profile, we can solve the Poisson equation by direct integration.

Step 1: Deriving the Electric Field $E (x)$

The electric field is related to the potential by $E = - d V / d x$ . Therefore, the Poisson equation can be written as $\frac{d E}{d x} = \frac{ρ ( x )}{ϵ}$ . Integrating this with respect to $x$ gives the electric field.
The result is a triangular field profile that peaks at the metallurgical junction ( $x = 0$ ) and goes to zero at the edges of the depletion region ( $x = - x_{p}$ and $x = x_{n}$ ).

Step 2: Deriving the Potential $V (x)$

Integrating the electric field ( $V (x) = - \int E (x) d x$ ) gives the potential profile. The potential varies quadratically with position inside the depletion region. The total potential difference across the junction is the built-in potential, $V_{bi} = V (x_{n}) - V (- x_{p})$ .

The following resource provides a detailed, step-by-step mathematical derivation of these quantities.

PN and Metal–Semiconductor Junctions

The textbook chapter 'PN and Metal–Semiconductor Junctions' by Chenming Hu presents the standard, formal derivation of the p-n junction properties. It uses the depletion approximation to solve the Poisson equation rigorously.

Please read section 4.2, 'Depletion-Layer Model' (pages 94-96). Follow the derivation starting from the charge density profile in Fig. 4-6c, through the integration to find the electric field (Eq. 4.2.3 and 4.2.4), and the second integration to find the potential (Eq. 4.2.6 and 4.2.7).

3. Key Parameters: Built-in Potential and Depletion Width

The derivation from the Poisson equation yields expressions for $V_{bi}$ and the total depletion width $W = x_{n} + x_{p}$ in terms of the doping concentrations.

Built-in Potential ( $V_{bi}$ )

While $V_{bi}$ emerges from solving the Poisson equation, it can also be found more directly from thermodynamic arguments. The requirement of a constant Fermi level means that the total band bending must be $q V_{bi}$ . This leads to a fundamental relationship connecting the built-in potential to the doping concentrations and the intrinsic carrier concentration:

V_{bi} = \frac{k _{B} T}{q} ln (\frac{N _{a} N _{d}}{n _{i}^{2}})

This equation is extremely important. It tells us that the potential barrier is determined by the doping levels on both sides of the junction. Higher doping leads to a larger built-in potential.

Depletion Width ( $W$ )

The depletion region as a whole must be charge neutral, which means the total negative charge on the P-side must equal the total positive charge on the N-side: $N_{a} x_{p} = N_{d} x_{n}$ . This implies that the depletion region extends further into the more lightly doped side.

By combining this charge neutrality condition with the solution to the Poisson equation, we arrive at the expression for the total depletion width:

W = x_{n} + x_{p} = \frac{2 ϵ}{q} (\frac{1}{N _{a}} + \frac{1}{N _{d}}) V_{bi}

This equation shows that the width of the depletion region increases with the built-in potential but decreases with higher doping levels.

The same textbook chapter you just read derives these results and provides a helpful example.

PN and Metal–Semiconductor Junctions

Let's revisit the textbook chapter to consolidate these final, crucial equations and see them in action.

First, review section 4.1.2 (page 93) for the derivation of the built-in potential (Eq. 4.1.2). Then, read section 4.2.2 (page 96) for the derivation of the depletion-layer width (Eq. 4.2.8). Finally, work through Example 4-1 on page 97 to see how these formulas are applied to a practical case.

Test your understanding!

Consider a silicon P+N junction where the P-side is much more heavily doped than the N-side ( $N_{a} ≫ N_{d}$ ). How do the expressions for $V_{bi}$ and $W$ simplify? Where does most of the depletion region lie?

Show answer

Built-in Potential ( $V_{bi}$ ): The expression $V_{bi} = \frac{k _{B} T}{q} ln (\frac{N _{a} N _{d}}{n _{i}^{2}})$ does not simplify significantly, but since $ln (N_{a})$ changes slowly, $V_{bi}$ is primarily determined by the lighter doping, $N_{d}$ .
Depletion Width ( $W$ ): In the term $(\frac{1}{N _{a}} + \frac{1}{N _{d}})$ , since $N_{a} ≫ N_{d}$ , we have $\frac{1}{N _{a}} ≪ \frac{1}{N _{d}}$ . Therefore, the expression for the width simplifies to:
$W \approx \frac{2 ϵ V _{bi}}{q N _{d}}$
The width is almost entirely determined by the doping on the lightly doped side.
Location: From the charge neutrality condition $N_{a} x_{p} = N_{d} x_{n}$ , we get $\frac{x _{n}}{x _{p}} = \frac{N _{a}}{N _{d}} ≫ 1$ . This means $x_{n} ≫ x_{p}$ . The depletion region extends almost entirely into the lightly doped N-side, and we can approximate $W \approx x_{n}$ .

For the computationally inclined: Simulating a P-N Junction

Given your interest in simulations, you might find it rewarding to numerically solve the coupled drift-diffusion and Poisson equations that we've discussed. The Python package oedes is designed for 1D semiconductor device simulation. The documentation includes a tutorial that walks you through building a p-n junction model from scratch.

This is an excellent way to see how the theoretical equations are implemented in practice. You could try to set up a simulation with the parameters from Example 4-1 in the textbook and see if your numerical solution for the equilibrium potential profile matches the analytical result.

oedes Documentation

The oedes documentation provides a tutorial for simulating a p-n junction.

This is an optional activity. If you're interested, you can follow the 'Tutorial: PN junction' (Chapter 4, pages 7-11). Pay close attention to how the PoissonEquation is defined and how the pn_doping profile is implemented. Sections 5.3 and 5.17 provide the underlying theory for the Poisson equation used in the package.

Conclusion

Today, we have dissected the p-n junction, the heart of modern electronics. By analyzing the interplay of carrier diffusion and drift, we developed a physical model for the junction at equilibrium.

Key Takeaways:

When P-type and N-type semiconductors are joined, carrier diffusion creates a depletion region of fixed ionized dopants near the interface.
This space charge gives rise to a built-in electric field and a corresponding built-in potential ( $V_{bi}$ ), which opposes further diffusion.
The system reaches equilibrium when the drift and diffusion currents for each carrier type exactly balance.
Using the depletion approximation, we can solve the Poisson equation to find the electric field (triangular profile) and electrostatic potential (parabolic profile) across the junction.
The built-in potential and depletion width are determined by material properties and doping concentrations:
- $V_{bi} = \frac{k _{B} T}{q} ln (\frac{N _{a} N _{d}}{n _{i}^{2}})$
- $W = \frac{2 ϵ}{q} (\frac{1}{N _{a}} + \frac{1}{N _{d}}) V_{bi}$

Preview of Next Lesson:
We have now analyzed the equilibrium state of the most important semiconductor interface. The next few lessons will shift focus from interfaces back to bulk transport phenomena. We will introduce a more powerful and general framework for describing how charge carriers move under the influence of external forces: the Boltzmann transport equation. Applying it in the relaxation time approximation will allow us to derive expressions for fundamental transport coefficients like electrical and thermal conductivity, which are essential for understanding how a device actually functions when it's part of a circuit.