Key Semiconductor Properties

Why This Matters

Semiconductors are the foundation of nearly every electronic device you'll encounter—from the transistor in your phone to the solar panels on rooftops. In condensed matter physics, you're being tested on your understanding of how band structure governs electronic behavior, why doping transforms material properties, and what physical mechanisms enable devices to function. These concepts connect directly to quantum mechanics, statistical mechanics, and electromagnetism, making semiconductors a perfect testing ground for integrated problem-solving.

Don't just memorize definitions—know why the Fermi level shifts with doping, how carrier mobility affects device performance, and what distinguishes direct from indirect band gap materials. Exam questions often ask you to connect microscopic properties (like effective mass) to macroscopic behavior (like conductivity), so focus on the chain of cause and effect. Master these relationships, and you'll handle both conceptual questions and quantitative problems with confidence.

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Band Structure Fundamentals

The electronic behavior of any semiconductor begins with its band structure—the arrangement of allowed and forbidden energy states that electrons can occupy. The band gap determines whether a material conducts, insulates, or behaves as a semiconductor.

Band Structure and Energy Gaps

• Band gaps separate valence and conduction bands—electrons must gain energy equal to or greater than $E_g$ to conduct electricity
• Semiconductor band gaps typically range from $0.1$ to $3$ eV, small enough for thermal or optical excitation but large enough to control conductivity
• Optical and electrical properties both depend on $E_g$: it sets the absorption edge for photons and determines intrinsic carrier concentration at any temperature

Direct and Indirect Band Gaps

• Direct band gap materials allow electrons to transition between bands without momentum change—enabling efficient light emission in LEDs and laser diodes
• Indirect band gap semiconductors (like silicon) require phonon assistance for transitions, meaning a lattice vibration must supply the momentum difference
• Device implications are significant—direct gap materials dominate optoelectronics, while indirect gap materials work well for electronics where light emission isn't needed

Effective Mass of Carriers

• Effective mass ($m^*$) describes how carriers accelerate under applied fields—it's determined by the curvature of the band structure, $m^* = \hbar^2 \left( \frac{d^2E}{dk^2} \right)^{-1}$
• Lighter effective mass means higher mobility and faster devices, since carriers respond more readily to electric fields
• Electrons and holes have different effective masses in the same material, affecting transport properties and requiring separate treatment in calculations

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Carrier Statistics and the Fermi Level

Understanding where electrons are likely to be found—and how that distribution changes—is central to predicting semiconductor behavior. The Fermi level acts as the chemical potential for electrons, governing carrier populations in both bands.

Fermi Level and Its Temperature Dependence

• The Fermi level ($E_F$) marks the energy where electron occupation probability equals 50%, described by the Fermi-Dirac distribution
• In intrinsic semiconductors, $E_F$ sits near mid-gap; doping shifts it toward the conduction band (n-type) or valence band (p-type)
• Temperature increases push $E_F$ toward mid-gap in extrinsic semiconductors as intrinsic carriers begin to dominate—this is the intrinsic transition

Intrinsic and Extrinsic Semiconductors

• Intrinsic semiconductors are undoped, with carrier concentration determined solely by thermal excitation across the band gap: $n_i = \sqrt{N_c N_v} \exp\left(-\frac{E_g}{2k_BT}\right)$
• Extrinsic semiconductors contain intentional impurities that dramatically increase carrier concentration at room temperature
• The transition from extrinsic to intrinsic behavior occurs at high temperatures when thermally generated carriers exceed dopant-provided carriers

Doping and Carrier Concentration

• n-type doping introduces donor atoms (extra valence electrons) that contribute electrons to the conduction band
• p-type doping uses acceptor atoms (fewer valence electrons) that create holes in the valence band
• Carrier concentration can be tuned over many orders of magnitude—from $10^{14}$ to $10^{20}$ cm$^{-3}$—giving precise control over conductivity

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Transport Properties

How carriers move through a semiconductor determines device speed and efficiency. Transport depends on both how many carriers exist and how freely they can move.

Electron and Hole Mobility

• Mobility ($\mu$) quantifies carrier drift velocity per unit electric field: $\mu = \frac{v_d}{E} = \frac{q\tau}{m^*}$, where $\tau$ is the scattering time
• Scattering mechanisms include lattice vibrations (phonons), ionized impurities, and defects—each dominates in different temperature and doping regimes
• Higher mobility enables faster switching in transistors and better performance in high-frequency applications

Electrical Conductivity

• Conductivity combines carrier concentration and mobility: $\sigma = q(n\mu_e + p\mu_h)$, capturing contributions from both electrons and holes
• Temperature affects conductivity through competing mechanisms—carrier concentration increases with $T$, but mobility typically decreases due to phonon scattering
• Doping level determines whether a semiconductor behaves more like an insulator (lightly doped) or approaches metallic conductivity (heavily doped)

Hall Effect

• The Hall effect produces a transverse voltage when current flows through a semiconductor in a magnetic field—$V_H = \frac{IB}{nqt}$
• Hall measurements reveal both carrier concentration and carrier type (the sign of $V_H$ indicates n-type vs. p-type)
• The Hall coefficient $R_H = \frac{1}{nq}$ is a fundamental experimental tool for characterizing semiconductor materials

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Generation, Recombination, and Temperature Effects

Carriers are constantly being created and destroyed in semiconductors. The balance between generation and recombination determines steady-state carrier populations and device behavior.

Recombination and Generation Processes

• Generation creates electron-hole pairs through thermal excitation, photon absorption, or impact ionization
• Recombination annihilates pairs via radiative (photon emission), Auger (energy to third carrier), or Shockley-Read-Hall (trap-assisted) mechanisms
• Carrier lifetime ($\tau$) measures average time before recombination—critical for solar cells, where longer lifetimes mean more collected carriers

Temperature Dependence of Semiconductor Properties

• Intrinsic carrier concentration rises exponentially with temperature: $n_i \propto T^{3/2} \exp\left(-\frac{E_g}{2k_BT}\right)$
• Mobility decreases at high temperatures due to increased phonon scattering, typically as $\mu \propto T^{-3/2}$ for acoustic phonon scattering
• Device operation limits exist because excessive temperature causes loss of extrinsic behavior and increased leakage currents

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Junction Physics

When differently doped regions meet, the resulting interfaces enable nearly all semiconductor devices. The p-n junction is the building block of diodes, transistors, and solar cells.

p-n Junctions and Depletion Regions

• The depletion region forms at the junction as electrons and holes diffuse across and recombine, leaving behind ionized dopants that create a built-in electric field
• Built-in potential ($V_{bi}$) opposes further diffusion and equals $V_{bi} = \frac{k_BT}{q}\ln\left(\frac{N_AN_D}{n_i^2}\right)$
• Forward bias reduces the barrier and allows current flow; reverse bias widens the depletion region and blocks current—this asymmetry creates diode behavior

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Optical Properties

Semiconductors interact with light in ways that enable lasers, LEDs, photodetectors, and solar cells. The band gap determines which photon energies can be absorbed or emitted.

Optical Properties and Absorption

• Photons with energy $h\nu \geq E_g$ can excite electrons across the band gap, creating electron-hole pairs
• The absorption coefficient ($\alpha$) describes how quickly light intensity decays in the material—direct gap materials have sharp absorption edges
• Below-gap transparency means semiconductors are transparent to photons with $h\nu < E_g$, which is why silicon appears opaque to visible light but transparent to infrared

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Advanced Structures

Modern devices exploit engineered materials and nanoscale effects to achieve properties impossible in bulk semiconductors. Quantum mechanics becomes essential when dimensions approach the de Broglie wavelength.

Quantum Confinement Effects in Nanostructures

• Quantum confinement occurs when carrier motion is restricted to dimensions comparable to the de Broglie wavelength—creating discrete energy levels instead of continuous bands
• Quantum dots confine carriers in all three dimensions, producing atom-like spectra with size-tunable emission wavelengths
• Enhanced optical properties include sharper emission peaks and higher oscillator strengths, enabling applications in displays and biological imaging

Semiconductor Alloys and Heterostructures

• Alloys (like $\text{Al}_x\text{Ga}_{1-x}\text{As}$) allow continuous tuning of band gap by varying composition $x$
• Heterostructures stack different semiconductors to create band offsets that confine carriers or guide light
• Applications include quantum well lasers, high-electron-mobility transistors (HEMTs), and multi-junction solar cells with record efficiencies

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Quick Reference Table

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Self-Check Questions

1. Both mobility and carrier concentration affect conductivity—if you double the doping level but mobility drops by half due to impurity scattering, what happens to conductivity?

2. Compare how temperature affects intrinsic carrier concentration versus carrier mobility. At what temperature regime does each effect dominate conductivity behavior?

3. Why is silicon (indirect band gap) used for solar cells but not for LEDs, while GaAs (direct band gap) is preferred for light-emitting devices?

4. A Hall effect measurement gives a negative Hall coefficient. What does this tell you about the semiconductor, and how would the Fermi level position differ from a sample with a positive Hall coefficient?

5. Explain why quantum dots can emit different colors of light depending on their size, connecting your answer to the concepts of band gap and quantum confinement.