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SEMICONDUCTOR PHYSICS

Lecturer Information

• Arvin P. Dela Roca, MSc, REE RME, ECT, CPM

Band Theory of Solids

• Energy Levels

  • Every atom has its own energy level.

  • Atoms apart do not interact; when close together, energy levels split into two.

• Band Theory

  • Explains the formation of energy bands within solids.

Energy Level Splitting in Solids

• Energy Band Structure

  • (a) Energy band structure reflects actual spacing between atoms in a solid.

  • (b) Energy level splitting occurs as a function of distance.

  • (c) Isolated atoms possess discrete energy levels.

Energy Bands in Semiconductors

• Energy Level Distribution

  • 2N Levels leading to 4N States.

  • 4N Levels resulting in 8N States.

• Filling of Levels

  • Empty Levels and Filled Levels distributed between Valence Energy Band and Conduction Band.

  • Band gap separates filled and empty states.

• Silicon Crystal Energy Level Configuration

Valence and Conduction Bands

• Valence Band

  • Defined as the band formed by a series of energy levels containing valence electrons. It can be fully or partially filled.

• Conduction Band

  • Formed by free electron energy values that have broken covalent bonds. It is the next permitted energy band and may be empty or partially filled.

Forbidden Energy Gap

• Definition

  • The gap between the valence band and conduction band is referred to as the Forbidden Energy Gap or Band Gap.

• Formation

  • It comprises non-permitted energy levels above the valence band and below the conduction band.

• Notation

  • Energy gap denoted as $E_g$ and is the energy required to excite an electron from the valence band to the conduction band, measured in eV.

Classification of Solids Based on Energy Band Diagram

• Insulators

  • Materials that do not conduct current, with a large energy gap greater than 5 eV.

• Semiconductors

  • Materials exhibiting conductivity between insulators and conductors, characterized by an energy gap on the order of 1 eV.

• Conductors

  • Materials that conduct current, where the valence band overlaps with the conduction band indicating zero energy gap.

Types of Semiconductors

• Intrinsic Semiconductor

  • Chemically pure semiconductor.

• Extrinsic Semiconductor

  • Semiconductor doped with impurities.

• Doping

  • The process of introducing impurities into an intrinsic semiconductor, modifying its electrical properties.

Semiconductor Types

• N-type Semiconductor

  • Doped with pentavalent impurities (e.g., P, As) that donate free electrons (Donor impurities) creating energy levels close to the conduction band, denoted by E_D.

• P-type Semiconductor

  • Doped with trivalent impurities (e.g., Al, B) that accept free electrons (Acceptor impurities) creating discrete energy levels close to the valence band, denoted by E_A.

Concept of Holes

• Definition of Holes

  • A hole is defined as the absence of an electron, interpreted as a deficiency of negative charge, and treated as a positive charge carrier.

• Properties

  • Holes are mobile as electrons move from adjacent sites, resulting in a shift of holes in the opposite direction. Analogy: Holes resemble bubbles in a liquid.

Fermi Level & Fermi Energy

• For Conductors

  • The highest occupied energy level at 0 K is termed the Fermi Level, with corresponding energy termed the Fermi Energy.

• For Semiconductors

  • The Fermi Level serves as a reference energy level, correlating to the center of gravity of conduction electrons and valence holes.

Fermi-Dirac Distribution Function

• Function Definition

  • $f(E) = rac{1}{1 + ext{exp}\big( rac{E - E_F}{kT}\big)}$ where:

    • $f(E)$ = probability that energy level with energy $E$ is occupied at temperature $T$ (in Kelvin).

    • $E_F$ = Fermi energy; $k$ = Boltzmann's constant.

Effect of Temperature on Fermi Function

• Temperature Variation

  • Graph represents the change in $f(E)$ as a function of temperature, with a crossover occurring at point C where $f(E) = 1/2$.

Fermi Level in Intrinsic Semiconductors

• At 0 K

  • Number of electrons in conduction band equals the number of holes in valence band: $n_e = n_h = n_i$; Fermi Level positioned in the middle of the band gap.

• At Non-zero Temperature

  • The Fermi Level shifts based on effective mass:
    $E_F = E_i + rac{3}{4}kT ext{ln}\bigg( rac{m_h + m_e}{2}\bigg)$

  • Where conditions are: m_h > m_e.

Fermi Level in P-type Semiconductor

• Charge Distribution

  • In P-type, n_h > n_e thus shifting Fermi Level towards the valence band.

Fermi Level in N-type Semiconductor

• Charge Distribution

  • In N-type, n_e > n_h resulting in the Fermi Level shifting towards the conduction band.

Variance of Fermi Level with Impurity Concentration in N-type Semiconductors

• Figures

  • (a) Low impurity concentration.

  • (b) Moderate impurity concentration.

  • (c) High impurity concentration.

Drift Current

• Definition

  • Drift Current arises from the motion of charge carriers influenced by an applied electric field.

• Mathematical Representation

  • $J_e^{(drift)} = n_e imes e imes <br>u_e imes E$ for electrons and fragment:

  • $J_h^{(drift)} = p imes e imes <br>u_h imes E$ for holes, where $n_e$ and $n_h$ are the mobilities of electrons and holes.

Diffusion Current

• Definition

  • Diffusion current results from the directional movement of charge carriers due to a concentration gradient.

• Mathematical Representation

  • $J_e^{(diffusion)} = -eD_e rac{dn}{dx}$ for electrons and $J_h^{(diffusion)} = -eD_h rac{dp}{dx}$ for holes, where $D_e$ and $D_h$ are diffusion coefficients for electrons and holes, respectively.

Conductivity of Semiconductors

• Intrinsic Semiconductor

  • Conductivity formula: $ext{σ}_i = n_i e ( ext{μ}_e + ext{μ}_h)$

• Extrinsic Semiconductor

  • N-type: $ext{σ}_n = n imes e imes ext{μ}_e = N_D imes e imes ext{μ}_e$

  • P-type: $ext{σ}_p = p imes e imes ext{μ}_h = N_A imes e imes ext{μ}_h$ where $N_D$ and $N_A$ are donor and acceptor impurity concentrations, respectively.

P-N Junction Diode

• Diode Characteristics

  • Functions as a current valve allowing unidirectional current flow.

  • Conductive exclusively in forward bias; ideally non-conductive in reverse bias, utilized in rectification.

  • Exhibits nonlinear voltage-current characteristics.

Formation of P-N Junction Diode

• Charge Carrier Distribution

  • P-region: holes as majority; electrons as minority charge carriers.

  • N-region: electrons as majority; holes as minority charge carriers.

  • P-N junction is the interface separating these two regions.

Physical Structure and Symbol of Diode

• Components

  • Anode and Cathode connections for p-type and n-type semicondictors constructing the physical structure of the diode.

• Circuit Symbol

Biasing of P-N Junction Diode

• Forward Bias

  • Positive battery terminal to the p-side, negative to the n-side permits current flow.

• Reverse Bias

  • Positive battery terminal to n-side, negative to p-side prevents current flow.

Energy Band Diagram of P-N Junction Diode

• At Equilibrium

  • Energy levels of electrons increase, while holes decrease in energy.

• In Forward Bias

  • Shift in energy levels that favors current flow.

• In Reverse Bias

  • Energy of holes and electrons positioned to suppress conduction.

V-I Characteristics of a P-N Junction Diode

• Graphical Representation

  • Voltage-current characteristics of the diode illustrating operational cutoff points:

  • V_F < V_R, $V_F = V_O$, and properties related to cut-in voltage.

Pictorial History of Transistors

• Transistor Development Timeline

  • Highlighting significant years in the evolution of transistors from inception to modern advancements.

Transistor Basics

• Definition

  • A transistor can be conceptualized as a device allowing current transit from low resistance to high resistance paths, also known as a Bipolar Junction Transistor.

Modes of Operation of Transistor

• Key Configurations

  • Common base mode.

  • Common emitter mode.

  • Common collector mode.

• Operation Conditions

  • For efficient transistor action, the emitter-base junction must be forward biased and collector connected in reverse bias.

Energy Band Diagram of a Transistor

• Unbiased Mode

  • Energy states illustrated for both electrons and holes in the n-p-n transistor configuration.

N-P-N Transistor in Common Base Mode

• Charge Flow Representation

  • Illustrative representation revealing charge dynamics through conduction paths and recombination paths.

Hall Effect

• Definition

  • Hall voltage, $V_H$, refers to the potential difference generated perpendicular to both current and magnetic field during the placement of a current-carrying conductor in a magnetic field, an effect discovered by Edwin H. Hall in 1897.

• Mathematical Formulation

  • Described by the Lorentz force: $F = -e imes v imes B$.

Application of Hall Effect

• Practical Uses

  • Determining semiconductor type and concentration of charge carriers (p or n).

  • Assessing the mobility of charge carriers and measuring magnetic fields.