PCS224-Lecture09-Rebello-Student

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Course Information

  • PCS 224 - Solid State Physics

  • Semester: Fall 2024

  • Lecture 09

  • Topic: Doping in Semiconductors and Intro to Band Structure

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Covered Topics

  • Electrostatics (Lec 01 & 02)

    • Electric Force and Electric Field

    • Charge Distribution (i.e. parallel plate)

    • Electric Potential Energy and Potential Difference

    • Capacitance

  • Photoelectric Effect (Lec 03, Lec 04)

  • Introduction to Semiconductors (Lec 04, Lec 05)

    • Intrinsic vs. Extrinsic Semiconductors

    • Holes as Charge Carriers

    • Fundamental Equation of Semiconductors: n0 = ni^2

  • Current and Conductivity (Lec 06)

    • Current, Conductivity, Resistance, Drift Speed, Current Density, Hall Effect

  • Bohr’s Model of the Atom (Lec 08)

    • Overview of previously covered material

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Lecture Goals

  • Review Bohr’s Model of the Atom

  • Topic Focus: Doping in Semiconductors

    • Understanding how semiconductors are engineered (p-type and n-type)

    • Introduction of new terminology: dielectric constant, donors, acceptors, compensated semiconductors

    • Conceptual understanding of the usefulness of doping in engineering

    • Reference: Neamen 4.2

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Naturally Occurring Semiconductors

  • Silicon (Si) and Germanium (Ge) are intrinsic semiconductors.

    • Intrinsic Semiconductors: Naturally occurring, defined by their pure state without impurities.

    • Extrinsic Semiconductors: Engineered by adding impurities to modify electrical properties.

    • Valence Electrons: Crucial for forming covalent bonds in semiconductors.

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Electron and Hole Concentration

  • Definitions:

    • n0: Free electron concentration (number density) in thermal equilibrium.

    • p0: Hole concentration (number density) in thermal equilibrium.

  • Why are two variables necessary?

    • Expectation is that number of electrons does not equal number of holes due to doping and thermal motion.

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Intrinsic Concentration

  • For intrinsic semiconductors: n0 = p0 = ni (where ni is the intrinsic carrier concentration).

  • Table 1.2: Intrinsic Carrier Concentrations @ 300 K

    • Silicon: n_i = 1.02 x 10^10 cm^-3

    • Gallium Arsenide (GaAs): n_i = 2.1 x 10^6 cm^-3

    • Germanium (Ge): n_i = 2.4 x 10^13 cm^-3

  • Key Takeaway: ni is a property of the material, affects its carrier dynamics when doped.

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What is Doping?

  • Doping Definition:

    • Adding impurities (different types of atoms) into semiconductor.

  • Donors vs. Acceptors:

    • Donors: Atoms that provide extra free electrons (denoted as N_D).

    • Acceptors: Atoms that create holes (denoted as N_A).

  • Impact of impurities: Enhance electrical properties by increasing charge carriers (electrons or holes).

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N-Type and P-Type Materials

  • N-Type Material:

    • More electrons (n0 > p0), electrons are majority charge carriers.

  • P-Type Material:

    • More holes (p0 > n0), holes are majority charge carriers.

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N-Type Doping

  • Replacement of Silicon atom with a Group V atom (e.g., Phosphorus).

  • Phosphorus atoms have five valence electrons, four are utilized in covalent bonding.

  • Leads to extra free electron and increases n0 in semiconductor.

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N-Type Semiconductor Formation

  • Phosphorus impurity: Substitutes a Silicon atom, leading to excess electrons.

  • For N-type semiconductors, approximately: n0 = N_D

    • Conceptual analogy: Similar to adding acid to water, resulting in a large concentration of hydrogen ions.

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Lecture 08 Recap

  • Formulas and principles reviewed:

    • Binding energy and potential energy relations.

    • Key parameters from the Bohr model were discussed (energy levels, radii).

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Recap: Orbital Radii

  • Energy level formulas and implications of orbital distances relative to atomic structures were reinforced.

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Binding Energy of Donor Electron

  • Leftover electron interacts with the positively charged donor (e.g., from Phosphorous).

  • Ionization energy similar to that of hydrogen.

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Dielectrics

  • The polarization of molecules occurs when an external electric field is applied, resulting in a change in dielectric behavior.

  • The dielectric constant expresses how much the electric field is reduced in the material.

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Dielectric Properties of Semiconductors

  • Property Table for Silicon, Gallium Arsenide, and Germanium at T=300 K:

    • Silicon (Si):

      • Atoms: 5.0 x 10^22 cm^-3

      • Dielectric Constant: 11.7

      • Bandgap Energy: 1.12 eV

  • Influence of dielectric constants expressed in effective density of states.

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Continue: Binding Energy of Donor Electron

  • Impact of dielectric constant on binding energies was emphasized.

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Summary of Donor (N-Type)

  • Binding energy for donor electrons lower than intrinsic energy levels.

    • Key Aspect: Donor electrons are easily freed at room temperature.

    • Concentration denoted as N_D; typically assume all donors are ionized.

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Example Question for Donor Concentration

  • Problem illustrating the adjustments in donor levels to achieve specific electrical properties in silicon.

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P-Type Doping

  • Replacement of Silicon with a Group III atom (e.g., Boron).

  • Boron lacks a full complement of valence electrons, resulting in the formation of holes.

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Extrinsic P-Type Semiconductor

  • Boron substitutes Si atom, creating holes that can lead to better charge carrier mobility.

    • Understanding of mobility in discussing how electrons move to fill these holes.

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Summary of Acceptor Behavior

  • Acceptor states become easily filled at room temperature, leading to increased hole concentration.

    • Denoted concentration typically as N_A in calculations.

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Properties of Silicon and Other Semiconductors

  • Reevaluation of conductors, semiconductors, and insulators based on intrinsic properties.

    • Include relevant properties as required for semiconductor analysis.

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Example Question and Conceptual Questions

  • Problem regarding calculations for determining donor concentrations and their impact on resistance.

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Continuing Example Questions

  • Further exploration of acceptor states influence on resistance calculations.

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Electroneytrality Equation

  • Concept of electric neutrality within a semiconductor emphasizing the balance of charge carriers.

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Compensated Semiconductors

  • Simulation of extrinsic doping creating complex scenarios with balance across n-type and p-type within a single semiconductor.

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Violations of Electroneytrality

  • Impact of disruptions leading to electric current and tendency toward reestablishment of neutrality.

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Overview of Doping Effects

  • Comprehensive review showing how donor and acceptor impurities affect conductivity and charge carrier concentrations under normal and doped conditions.

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Lecture Goals Moving Forward

  • Transitioning to band structure concepts and understanding their implications in conductivity and insulator behavior from an energy level perspective.

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Energy Levels within Solids

  • Discussion about effective energy levels as dictated by quantum mechanics in multi-atom configurations related to conduction and valence bands.

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Quantum Physics Overview

  • Differentiation between classical and quantum mechanics as it pertains to electron behavior in solids.

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Wave Functions in Quantum Mechanics

  • Defining particles by their spatial probability distribution for energy states in solids.

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Energy Levels in Hydrogen Model

  • Review of energy transitions in hydrogen and its quantum energy state variations with emission of light.

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Electron Orbitals and Quantum States

  • Review of how electrons are organized within shells, subshells, and their corresponding states.

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Electron Orbital Characteristics

  • Describing differences in electron likelihoods across various subshells in terms of spatial probability.

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Chemists’ Notation for Electron Configuration

  • Describing orbital configurations and maximum capacity limits.

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Energy Levels in Multi-Electron Atoms

  • Illustrating the complexity arising from multiple electron interactions affecting energy states.

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Pauli Exclusion Principle

  • Principle emphasizing that no two electrons can occupy the same quantum state, establishing energy state occupancy rules.

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Filling Energy Levels

  • Emphasizing energy preferences and how electrons fill from lowest to highest available states.

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Multiple Atoms and Energy Levels

  • Dependency of atomic interactions on electron energy distributions across removed atom geometries.

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Energy Levels for Multiple Atomic Configurations

  • How energy levels evolve as atoms approach, emphasizing complexity due to interactions.

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Level Splitting and Formation of Energy Bands

  • Energy band formation due to atom proximity encouraging energy line redistributions and band structures forming.

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Energy Band Formation

  • Resulting continuum of energy states blending processes, becoming bands due to substantial atom aggregation.

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Solid-State Energy Bands

  • Understanding defined conduction bands and valence bands within semiconductors, inclusive of forbidden energy gaps.

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Energy Band Components

  • Characteristics of energy levels and their implications in charge density and stability.

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Energy Level (Band) Diagrams

  • Mechanisms for representing electron occupation states across energy levels, emphasizing the absence of states within gaps.

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Conduction and Valence Bands Described

  • Distinction between filled and unfilled bands—fundamental to understanding conduction mechanisms.

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Conductors vs Insulators

  • Explicitly differentiating materials characterized by their conduction capabilities under ground state conditions.

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Semiconductor Behavior at Low Temperature

  • Insight into semiconductors acting as insulators at low temperatures, with potential for conduction evolving as temperatures rise.

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Semiconductor Temperature Responses

  • Understanding thermal excitation mechanisms contributing to semiconductor transitions between states at varying temperatures.

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Inquiry on Material Classification

  • Identify nature of materials based on given data, categorizing them appropriately as conductors, insulators, or semiconductors.

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Current Flow Generation

  • Detailed exploration of conditions necessary for current flow in semiconductors at T=0.

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Conditions for Current Flow

  • Transaction requirements for electrons to move and overcome energy barriers for conduction.

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Challenges in Non-Metal Conductivity

  • Assessing difficulties faced by intrinsic semiconductors in facilitating current flow compared to conductive materials.

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Hall Effect Inquiry

  • Understanding conceptually the dynamics of electron and hole interactions in semiconductor behaviors.

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Energy of a Photon Overview

  • Understanding how light energy facilitates electron excitation in semiconductors across energy bands.

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Photon Energy for Excitation

  • Estimating relevant wavelengths necessary for electron excitation across defined band gaps in materials.

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Homework Problem - Photon Energy

  • Assessing photon energy numbers in relevance to conducting semiconductor energy transitions.

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Doped Semiconductors at Absolute Zero

  • Understanding energy states unique to n-type and p-type semiconductors when at absolute zero temperature conditions.

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Above Absolute Zero Conditions

  • Evaluating the charge carrier behaviors under typical operational temperatures with complete ionization considerations.

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Energy Band Diagrams for Doped Semiconductors

  • Visualization of energy states and their implications for conduction and carrier density in doped materials.

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Role of Free Electrons

  • Description of how free electrons (n-carriers) migrate and conduct under applied fields in n-type semiconductors.

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Holes as Current Carriers

  • Understanding positional movements and implications of holes and their behaviors under electric fields as p-carriers.

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Questions on Intrinsic Semiconductor Behavior

  • Analyzing conceptual questions about intrinsic charge dynamics in semiconductors.

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Summary of the Energy Level Diagram

  • Consolidating knowledge on drawing and interpreting energy band diagrams relevant for conduction types.

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Progression of Semiconductor Behavior with Temperature

  • Advanced exploration of temperature impacts on charge carrier distribution within semiconductors.

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Elaborating on Solid Energy Bands

  • Recap of definitions and roles of energy bands and gaps within solid-state physics conventions.

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Fermi Energy and Its Role

  • Clarifying significance of Fermi energy as it relates to electron states occupancy at absolute zero.

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Fermi Energy Position in Bands

  • Positioning of Fermi energy within band structures dictates a material’s conductive properties.

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Fermi Energy Across Material Types

  • Overview of Fermi level differences in semiconductors versus metals and insulators.

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Vacancy in States vs. Probability

  • Understanding the determination factors surrounding occupied states and their relation to the Fermi energy.

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Additional Resources and References

  • Recommended links for extended reading on semiconductor devices and energy band theory.

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Key Attributes to Retain

  • Importance of comprehending semiconductor engineering fundamentals.

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Advanced Understanding Goals

  • Deepening comprehension of energy bands and their governing physics in current flow dynamics.

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Questions and Student Engagement

  • Encouragement for comprehending all discussed topics and preparation for future discussions on Band Structure and Fermi Energy.

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