Electronic conduction in semiconductors

Prof. Stanislav S. Fedotov, Prof. Dmitry Aksyonov

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Materials Chemistry

Center for Energy Science and Technology

October 8^th, 2025

Main application: active microelectronic devices, such as transistors and microchips

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Semiconductors

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Silicon ingot

• A semiconductor material has an electrical conductivity value falling between that of a conductor, such as metallic copper, and an insulator
• Its conducting properties may be altered in useful ways by introducing impurities ("doping") into the crystal structure
• A semiconductor is a material with a band gap

Germanium rods

GaAs

Electronic structure of semiconductor: presence of a band gap

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Two types of carriers: holes and electron

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No field

Field applied: e-h appear

Field applied: e, h migrate

Direct vs indirect band gaps

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How to measure?

Measuring band gap: optical adsorption

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Is silicon direct or non-direct?

Band structure of silicon

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What about other elements?

Semiconductors or metalloids

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Si, Ge or

III-V

or

II-VI

Four electrons and formation of sp³ bonds

sp3 hybridization

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Bonding/antibonding and covalency

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Band gap formation in Ge, Si, and C (diamond)

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• In C, Si, and Ge, when the sp³ hybrid bond is formed, each of the four atoms involved forms two orbitals: one bonding and one antibonding.
• The four electrons fill the bonding orbitals, leaving the antibonding orbitals empty.

Cohesive energy vs band gap of semiconductors

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Since the bandgap energy increases as the lattice parameter decreases, one might expect that bandgap energy would increase with pressure in semiconductors or not? Homework task.

E_C = 5.47 + 1.78E_g

Typical semiconductors and band gaps

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Since the lattice parameter

increases with temperature due to thermal expansion, the bandgap energy decreases with temperature (only one reason).

from Kittel

Why for PbS it is increasing?

Intrinsic electrical conductivity

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intrinsic carrier concentration:

e is elementary charge

n is concentration of carriers

μ is mobility of carriers

Fermi energy and Fermi level and carrier concentration

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• For intrinsic semiconductors the Fermi-level is between valence and conduction bands
• Due to Fermi-Dirac statistics at finite temperature only small amount electrons will be excited to the conduction band and the same amount of holes will be in valence band depending on band gap n ~ exp(E_g/2kT)

Carrier concentration of semiconductors is a function of band gap and temperature

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n ~ exp(-E_g/2kT)

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Mobility and effective mass

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Charge carrier mobility

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• Mobilities are usually larger for smaller band gaps
• Mobilities of holes are usually smaller

Extrinsic conductivity, n-type

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IP = me⁴/(4πε₀)²ℏ² - first ionization energy of a hydrogen atom, for P ~ 0.05 eV, kT is 0.026 eV at room T

Extrinsic conductivity, p-type

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Ionization energies (eV)

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Fermi-level in doped semiconductors

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Fermi level as a function of T in doped materials

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As the temperature is increased, the n_i and p_i increases, while the number of donors (or acceptors) remains constant (assuming they are all ionized), which drives E_F back toward Eⁱ_F

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Intrinsic vs extrinsic temperature dependence

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Impact of impurity concentration on mobility

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Impact of T on mobility electrons and holes

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At low concentration less than 10²⁴ the mobility is decreasing due to thermal scattering

For n>10²⁴ T has almost no impact on mobility

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Conductivity vs T in doped semiconductor

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Conductivity increases exponentially due to increased number of ionized impurities

σ ~ exp(E_g/2kT)

exponential growth

due to intrinsic conductivity

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The plot can be used to determine E_g

Decrease of mobility due to thermal scattering

100K

25K

The Hall effect to determine charge concentration and mobility

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R_H is Hall coefficient, I_x is current, B_z is field

For metals:

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The p-n rectifying junction (diode)

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Forward bias

Reverse bias

No bias

The current–voltage characteristics of a p–n junction

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Rectifying of AC current using a pn junction

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Bipolar (junction) transistor

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• Amplifying device
• Switching device

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h_FE is gain (10-100)

Bipolar transistor as a switch

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Field effect transistor (FET)

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FET operates by using an electric field to control the flow of current between the source and drain terminals through a semiconductor channel. The voltage applied to the gate terminal modulates the conductivity of the channel, allowing the FET to act as a switch or amplifier.

Summary on semiconductors

• Semiconductors are characterized with the band gap
• The conductivity increases with T
• Two types of intrinsic carriers are always present: electrons and holes
• The extrinsic carriers can be introduced by doping which allows to manipulate conductivity and type of carriers - highly important for electronics

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Task 1

Calculate the electrical conductivity of intrinsic silicon at 150 C.

n_i = 4x10¹⁹ m^-3, μ_e = 0.06 m²/V-s and μ_h=0.022 m²/V-s

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Task 2

To high-purity silicon is added 10²³ m^-3 arsenic atoms.

(a) Is this material n-type or p-type?

(b) Calculate the room-temperature electrical conductivity of this material.

(c) Compute the conductivity at 100C.

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Task 3

An extrinsic p-type silicon material is desired having a room-temperature conductivity of 50 (Ohm-m)^-1. Specify an acceptor impurity type that may be used as well as its concentration in atom percent to yield these electrical characteristics.

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Individual studies:

• Reading
• W. D. Callister, Jr. Materials Science and Engineering An Introduction
• R. Hummel, Electronic Properties of Materials
• A. Sutton, Electronic Structure of Materials
• C. Kittel, Introduction to Solid state physics

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α = ΔR/R_room/ΔT

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ΔR = α R_roomΔT=

0.004*5*30 = 0.6 Ohm