5.1 p-n junction formation and built-in potential

P-n junctions are the backbone of semiconductor devices. When p-type and n-type materials meet, charge carriers flow between them, creating a depletion region and built-in potential.

Understanding p-n junction formation is key to grasping how diodes, transistors, and solar cells work. We'll explore the physics behind carrier diffusion, space charge regions, and energy barriers that make these devices tick.

Formation of p-n junctions

• P-n junctions are fundamental building blocks of semiconductor devices $pn$ junctions form when p-type and n-type semiconductors are brought into contact, allowing charge carriers to flow between the two regions
• The formation of p-n junctions is crucial for the operation of diodes, transistors, and solar cells understanding the principles behind p-n junction formation is essential for designing and optimizing semiconductor devices

Bringing p-type and n-type semiconductors into contact

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• P-type semiconductors are doped with acceptor impurities $acceptors$ create holes as majority carriers and have a Fermi level closer to the valence band
• N-type semiconductors are doped with donor impurities $donors$ provide extra electrons as majority carriers and have a Fermi level closer to the conduction band
• When p-type and n-type semiconductors are brought into contact, a p-n junction is formed at the interface between the two regions $interface$ allows charge carriers to move across the junction

Diffusion of majority carriers

• Due to the concentration gradient of majority carriers across the p-n junction, holes from the p-type region and electrons from the n-type region diffuse across the junction
• Holes diffuse from the p-type region into the n-type region, leaving behind negatively charged acceptor ions in the p-type region near the junction
• Electrons diffuse from the n-type region into the p-type region, leaving behind positively charged donor ions in the n-type region near the junction
• The diffusion of majority carriers continues until an equilibrium condition is reached

Drift current and electric field

• As majority carriers diffuse across the p-n junction, they create a space charge region near the junction consisting of positively charged donor ions on the n-type side and negatively charged acceptor ions on the p-type side
• The space charge region gives rise to an electric field directed from the n-type region to the p-type region, opposing the diffusion of majority carriers
• The electric field causes a drift current of minority carriers $electrons$ in the p-type region and holes in the n-type region) in the opposite direction of the diffusion current
• The drift current balances the diffusion current, resulting in a net current of zero at equilibrium

Equilibrium condition in p-n junctions

• At equilibrium, the diffusion current and drift current in the p-n junction are equal and opposite, resulting in a net current of zero across the junction
• The equilibrium condition is characterized by a constant Fermi level throughout the p-n junction, as the Fermi levels of the p-type and n-type regions align
• The alignment of Fermi levels results in a built-in potential across the junction, which is the potential difference between the p-type and n-type regions at equilibrium
• The built-in potential creates an energy barrier that prevents further diffusion of majority carriers across the junction at equilibrium

Depletion region in p-n junctions

• The depletion region, also known as the space charge region, is a critical component of p-n junctions it plays a significant role in the operation of semiconductor devices and determines their electrical characteristics
• Understanding the properties of the depletion region, such as its width and the factors affecting it, is essential for analyzing and designing p-n junction-based devices

Depletion of majority carriers

• As majority carriers $holes$ in the p-type region and electrons in the n-type region) diffuse across the p-n junction, they leave behind charged ions near the junction
• The region near the junction becomes depleted of majority carriers, forming the depletion region or space charge region
• The depletion region extends into both the p-type and n-type regions, with the p-type side containing negatively charged acceptor ions and the n-type side containing positively charged donor ions

Space charge region

• The space charge region is formed by the positively charged donor ions on the n-type side and the negatively charged acceptor ions on the p-type side of the depletion region
• The space charge creates an electric field directed from the n-type region to the p-type region, which opposes the diffusion of majority carriers
• The electric field in the space charge region is responsible for the drift current of minority carriers, which balances the diffusion current at equilibrium

Depletion region width

• The width of the depletion region depends on the doping concentrations of the p-type and n-type regions and the applied voltage across the junction
• In an abrupt p-n junction, the depletion region width $W$ can be calculated using the following equation: $W = \sqrt{\frac{2\varepsilon}{q}\left(\frac{N_A + N_D}{N_AN_D}\right)(V_{bi} - V_a)}$

where $\varepsilon$ is the permittivity of the semiconductor, $q$ is the elementary charge, $N_A$ and $N_D$ are the acceptor and donor doping concentrations, $V_{bi}$ is the built-in potential, and $V_a$ is the applied voltage.

Factors affecting depletion width

• Doping concentrations: Higher doping concentrations lead to a narrower depletion region, while lower doping concentrations result in a wider depletion region
• Applied voltage: Applying a reverse bias voltage increases the depletion region width, while applying a forward bias voltage decreases the depletion region width
• Temperature: Increasing temperature can slightly increase the depletion region width due to increased thermal generation of charge carriers
• Semiconductor material properties: The permittivity and bandgap of the semiconductor material affect the depletion region width

Built-in potential in p-n junctions

• The built-in potential is a fundamental property of p-n junctions that arises from the alignment of Fermi levels and the difference in work functions between the p-type and n-type regions
• Understanding the origin and calculation of the built-in potential is crucial for analyzing the behavior of p-n junctions and their applications in semiconductor devices

Origin of built-in potential

• When a p-n junction is formed, the Fermi levels of the p-type and n-type regions must align at equilibrium to maintain a constant Fermi level throughout the junction
• The alignment of Fermi levels results in a potential difference across the junction, known as the built-in potential $V_{bi}$
• The built-in potential creates an energy barrier that prevents further diffusion of majority carriers across the junction at equilibrium

Fermi level alignment

• In a p-type semiconductor, the Fermi level is closer to the valence band, while in an n-type semiconductor, the Fermi level is closer to the conduction band
• When the p-type and n-type semiconductors are brought into contact, electrons flow from the n-type region to the p-type region, and holes flow from the p-type region to the n-type region until the Fermi levels align
• The alignment of Fermi levels establishes a constant Fermi level throughout the p-n junction at equilibrium

Calculation of built-in potential

• The built-in potential $V_{bi}$ can be calculated using the following equation: $V_{bi} = \frac{kT}{q}\ln\left(\frac{N_AN_D}{n_i^2}\right)$

where $k$ is the Boltzmann constant, $T$ is the absolute temperature, $q$ is the elementary charge, $N_A$ and $N_D$ are the acceptor and donor doping concentrations, and $n_i$ is the intrinsic carrier concentration of the semiconductor.

• The built-in potential depends on the doping concentrations and the intrinsic carrier concentration of the semiconductor material

Dependence on doping concentrations

• The built-in potential increases with increasing doping concentrations in the p-type and n-type regions
• Higher doping concentrations lead to a larger difference in the Fermi levels of the p-type and n-type regions before junction formation, resulting in a higher built-in potential
• The logarithmic dependence of the built-in potential on the doping concentrations implies that the built-in potential is less sensitive to changes in doping levels compared to the depletion region width

Energy band diagram of p-n junctions

• Energy band diagrams are essential tools for visualizing and understanding the behavior of p-n junctions they illustrate the energy levels of the conduction and valence bands, the Fermi level, and the built-in potential
• Analyzing energy band diagrams helps in understanding carrier transport, energy barriers, and the effects of applied bias on p-n junctions

Band bending in equilibrium

• When a p-n junction is formed, the alignment of Fermi levels results in a bending of the conduction and valence bands near the junction
• The conduction and valence bands bend upward on the n-type side and downward on the p-type side, creating a potential barrier known as the built-in potential
• The band bending in equilibrium is a consequence of the space charge region and the electric field created by the ionized donors and acceptors

Electron and hole energy barriers

• The band bending in a p-n junction creates energy barriers for electrons and holes attempting to cross the junction
• For electrons, the energy barrier is the difference between the conduction band edge on the p-type side and the Fermi level
• For holes, the energy barrier is the difference between the Fermi level and the valence band edge on the n-type side
• These energy barriers prevent the flow of majority carriers across the junction at equilibrium

Relation between built-in potential and energy barriers

• The built-in potential $V_{bi}$ is directly related to the energy barriers for electrons and holes in a p-n junction
• The electron energy barrier $\phi_n$ and the hole energy barrier $\phi_p$ are given by: $\phi_n = qV_{bi} - (E_F - E_v)_p$ $\phi_p = (E_c - E_F)_n - qV_{bi}$

where $E_F$ is the Fermi level, $E_c$ is the conduction band edge, $E_v$ is the valence band edge, and the subscripts $n$ and $p$ denote the n-type and p-type regions, respectively.

• The sum of the electron and hole energy barriers is equal to the bandgap energy $E_g$ of the semiconductor: $\phi_n + \phi_p = E_g$

Quasi-Fermi levels under bias

• When a p-n junction is subjected to an applied bias, the Fermi level is no longer constant throughout the junction

• Under bias, the Fermi level splits into two quasi-Fermi levels: one for electrons $E_{Fn}$ and one for holes $E_{Fp}$

• The quasi-Fermi levels describe the population of electrons and holes in the conduction and valence bands under non-equilibrium conditions

• The separation between the quasi-Fermi levels at the junction is equal to the applied voltage $V_a$: $qV_a = E_{Fn} - E_{Fp}$

• The quasi-Fermi levels help in understanding the carrier transport and current flow in p-n junctions under applied bias

Charge neutrality in p-n junctions

• Charge neutrality is a fundamental principle in p-n junctions that ensures the overall charge balance in the device
• Understanding charge neutrality and the charge distribution in the depletion region is essential for analyzing the electrical properties of p-n junctions

Charge distribution in depletion region

• In the depletion region of a p-n junction, there is a non-uniform distribution of charge due to the presence of ionized donors and acceptors
• The p-type side of the depletion region contains negatively charged acceptor ions, while the n-type side contains positively charged donor ions
• The charge distribution in the depletion region creates an electric field that opposes the diffusion of majority carriers

Ionized donors and acceptors

• In the depletion region, the majority carriers $holes$ in the p-type region and electrons in the n-type region) are swept away by the electric field, leaving behind ionized donors and acceptors
• The ionized donors are positively charged, having donated an electron to the conduction band, while the ionized acceptors are negatively charged, having accepted an electron from the valence band
• The concentration of ionized donors $N_D^+$ and acceptors $N_A^-$ in the depletion region depends on the doping concentrations and the width of the depletion region

Charge neutrality condition

• The charge neutrality condition in a p-n junction states that the total charge on the p-type side of the depletion region must be equal and opposite to the total charge on the n-type side
• Mathematically, the charge neutrality condition can be expressed as: $\int_{-x_p}^{0} \rho_p(x)dx = -\int_{0}^{x_n} \rho_n(x)dx$

where $\rho_p(x)$ and $\rho_n(x)$ are the charge densities in the p-type and n-type regions, respectively, and $x_p$ and $x_n$ are the depletion region widths on the p-type and n-type sides.

• The charge neutrality condition ensures that the net charge in the depletion region is zero, maintaining the overall charge balance in the p-n junction

Poisson's equation in depletion region

• Poisson's equation relates the electric field and potential distribution in the depletion region to the charge density
• In one dimension, Poisson's equation for the depletion region can be written as: $\frac{d^2\phi}{dx^2} = -\frac{\rho(x)}{\varepsilon}$

where $\phi$ is the electrostatic potential, $\rho(x)$ is the charge density, and $\varepsilon$ is the permittivity of the semiconductor.

• By solving Poisson's equation with appropriate boundary conditions and the charge neutrality condition, the electric field and potential distribution in the depletion region can be determined
• The solution of Poisson's equation provides valuable insights into the behavior of p-n junctions and helps in calculating important parameters such as the depletion region width and the built-in potential

Capacitance of p-n junctions

• P-n junctions exhibit capacitive behavior due to the presence of the depletion region and the variation of the depletion width with applied voltage
• Understanding the capacitance of p-n junctions is important for analyzing their dynamic behavior and designing applications such as varactor diodes and voltage-controlled oscillators

Depletion capacitance

• The depletion region in a p-n junction acts as a parallel-plate capacitor, with the ionized donors and acceptors forming the plates and the depletion region width serving as the dielectric thickness
• The depletion capacitance $C_D$ is given by: $C_D = \frac{\varepsilon A}{W}$

where $\varepsilon$ is the permittivity of the semiconductor, $A$ is the cross-sectional area of the junction, and $W$ is the depletion region width.

• The depletion capacitance is inversely proportional to the depletion region width, which varies with the applied voltage

Variation with applied voltage

• The depletion region width and, consequently, the depletion capacitance, vary with the applied voltage across the p-n junction
• Under reverse bias, the depletion region width increases, leading to a decrease in the depletion capacitance
• Under forward bias, the depletion region width decreases, resulting in an increase in the depletion capacitance
• The voltage dependence of the depletion capacitance is described by the following equation: $C_D = \frac{C_0}{\sqrt{1 - \frac{V_a}{V_{bi}}}}$

where $C_0$ is the depletion capacitance at zero bias, $V_a$ is the applied voltage, and $V_{bi}$ is the built-in potential.

Diffusion capacitance

• In addition to the depletion capacitance, p-n junctions also exhibit diffusion capacitance, which arises from the diffusion of minority carriers across the junction
• Diffusion capacitance is significant under forward bias conditions when there is a large flow of minority carriers
• The diffusion capacitance $C_{diff}$ is proportional to the forward bias current $I_F$ and the minority carrier lifetime $\tau$: $C_{diff} = \frac{\tau I_F}{kT/q}$

where $k$ is the Boltzmann constant, $T$ is the absolute temperature, and $q$ is the elementary charge.

• Diffusion capacitance is usually negligible under reverse bias conditions

Measuring capacitance-voltage characteristics

• The capacitance-voltage $C-V$ characteristics of p-n junctions provide valuable information about the doping profile, built-in potential, and other device parameters
• C-V measurements are performed by applying a small AC voltage signal superimposed on a DC bias voltage and measuring the resulting capacitance
• The C-V characteristics can be analyzed using techniques such as the Mott-Schottky plot, which plots $1/C^2$ versus the applied voltage
• From the C-V characteristics, important parameters such as the doping concentrations, built-in potential,