Lecture - 16 : P-N Junction

PN Junction in Equilibrium

Introduction to PN Junction

• The lecture introduces the concept of a PN junction, which consists of p-type and n-type semiconductor materials.

• It explains that the interface between these two types is known as the metallurgical junction, which can be created using epitaxial processes like molecular beam epitaxy (MBE) and metal-organic chemical vapor deposition (MOCVD).

Fabrication Techniques

• Discusses different fabrication techniques for junctions, including diffusion and implantation methods.

• Diffusion involves introducing impurities (e.g., arsenic for n-type and boron for p-type), resulting in graded profiles rather than abrupt changes.

Band Diagrams and Charge Carriers

• The band diagram illustrates the energy levels in p-type and n-type semiconductors, highlighting how Fermi levels align at equilibrium.

• In p-type semiconductors, holes are mobile charge carriers while immobile charges consist of positive ions; conversely, n-type semiconductors have electrons as mobile carriers.

Charge Neutrality and Concentration Changes

• At equilibrium, Fermi energy aligns at the interface leading to band bending; this requires adjustments in conduction bands and valence bands.

• The total charge neutrality condition is discussed: it includes contributions from holes, electrons, acceptor ions, and donor ions.

Diffusion Process

• Highlights that only mobile particles (holes and electrons) can move across the junction due to concentration gradients.

Understanding Charge Movement in Semiconductors

Charge Dynamics and Current Flow

• The movement of charge carriers (holes and electrons) is influenced by their respective diffusion coefficients, with holes moving in the positive X direction while current flows negatively due to electron movement.

• In semiconductors, only mobile charges (electrons and holes) move; fixed donor ions do not. This results in a non-zero charge density near interfaces when these carriers migrate.

Drift and Diffusion Currents

• Two primary mechanisms for current flow are identified: drift current, which depends on electric fields, and diffusion current, driven by concentration gradients.

• A one-dimensional model illustrates that half of the charge carriers will move left while half will move right, leading to net carrier movement based on concentration differences.

Mathematical Representation of Carrier Movement

• Using Taylor series expansion allows for calculating net carrier movement across a small section of semiconductor material based on local concentrations.

• The net flow of carriers can be expressed as Δn over ΔT, linking it to thermal velocity and establishing a relationship between flow rate and carrier density.

Current Density Calculation

• The expression for current density (J) incorporates charge (Q), thermal velocity, area, and the gradient of carrier concentration (d n/d x), indicating J's proportionality to this gradient.

• The diffusion coefficient emerges from the relationship between J and d n/d x; it reflects how both electrons and holes diffuse from high to low concentration areas.

Equilibrium Conditions in Semiconductor Regions

• At equilibrium without external bias, the Fermi level remains constant. This implies zero net current flow despite local variations in charge distribution near interfaces.

• Depletion regions form where mobile carriers are absent due to migration effects; remaining fixed charges create an imbalance leading to localized electric fields within these regions.

Interaction Between Electric Fields and Carrier Movement

• The presence of positive and negative charges establishes an electric field directed from positive to negative regions. This field influences carrier motion oppositely compared to diffusion forces.

Equilibrium in P-N Junctions

Understanding Electron and Hole Currents

• The equilibrium state in a p-n junction occurs when electron drift equals electron diffusion, and hole drift equals hole diffusion.

• The electric field direction is negative, indicating it points in the minus X direction. This is crucial for understanding the behavior of charge carriers.

Fermi Level Alignment

• In equilibrium, the Fermi levels of both sides align; on the P side, it is below the intrinsic level by kT log(n_a/n_i) .

• The energy bands bend due to this alignment, with differences between intrinsic energy levels contributing to potential differences across the junction.

Built-in Potential Calculation

• The built-in potential ( V_bi ) can be expressed as V_bi = kT/q log(n_a n_d/n_i^2) , where kT/q represents thermal voltage.

• At room temperature, thermal voltage is approximately 26 mV. This value plays a significant role in determining junction characteristics.

Depletion Region Approximation

• A step junction profile assumes an abrupt transition; however, real profiles may exhibit curvature. This approximation simplifies calculations for further analysis.

• Charge neutrality requires that total positive and negative charges balance within the depletion region defined by widths x_p (P side) and x_n (N side).

Electric Field Calculation Using Gauss's Law

• The electric field can be derived from Gauss's law: nabla cdot E = rho/epsilon . Here, Epsilon_0 K denotes dielectric permittivity.

Understanding Depletion Width and Built-in Potential in Semiconductor Junctions

Overview of Energy and Potential in P-N Junctions

• The discussion begins with the concept of depletion width, relating it to the Fermi level in a P-type semiconductor. The energy levels are described as being influenced by the potential, where energy is represented as -Q times V.

• It is explained that since -Q represents the charge on an electron, the corresponding potential V will be lower on the side with higher energy.

• The relationship between electric field direction and potential is highlighted; moving against the field results in an increase in potential.

Calculating Built-in Potential

• The formula for electric field E is introduced: E = -dV/dx. This leads to understanding that potential can be derived from integrating this electric field over distance.

• The built-in potential is calculated as half of the product of maximum electric field (E_max) and depletion width (W_d), emphasizing that this area under a triangular curve represents potential.

Relationship Between Doping Concentration and Depletion Width

• A key insight reveals that depletion width inversely correlates with doping concentration; higher concentrations lead to smaller depletion widths.

• Formulas for calculating x_n (depletion width on N-side) and x_p (depletion width on P-side) are provided, showing how they depend on doping concentrations.

One-Sided Junction Example

• An example of a one-sided junction illustrates scenarios where one side (P or N type) is heavily doped compared to the other. This affects Fermi level positioning relative to band edges.

• In heavily doped regions, such as P+, Fermi levels approach valence band edges, while normal doping conditions apply to N-side calculations based on donor concentration.

Implications for Built-in Potential Calculation

• The built-in potential becomes primarily dependent on donor concentration at the N-side due to fixed conditions at P-side.