Chapter Two. Energy Bands and Effective Mass
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1 Chapter Two Energy Bands and Effective Mass
2 Energy Bands Formation At Low Temperature At Room Temperature Valence Band Insulators Metals Effective Mass Energy-Momentum Diagrams Direct and Indirect Semiconduction
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5 Band Theory of Solids Electrons surrounding an atoms nucleus have certain well defined energies. These can be calculated using Quantum Mechanic Theory. Pauli s Exclusion Principle requires that each electron in an system of atoms (solid) occupy a unique energy level. The number of resulting energy levels will be the same as the number of atoms being brought together to make the solid a very large number. How many atoms are in 1 cm 3 of Silicon? Atomic Mass of Si = 28.1 kg/mole 6.023x10 26 = 28.1x x1022 atoms per gram A cm3 of Si weighs 2.3 g, and thus will contain 4.93x10 22 atoms. Again a very large number of atoms, each with unique energy levels.
6 Energy Band Formation in Solids The energy band is on the order of 1eV. The spread between discrete levels will be ~1eV/number of atoms, or about ev. Solving E=kT tells us that these levels will be indiscernible above K Thus we call it a Band.
7 A simple but powerful model.
8 A simple but powerful model.
9 This is a simple but powerful model. At 0K there are no electrons of holes to carry charge (provide electrical current). Free electrons and/or holes can be induced via temperature, light, or doping the material. Very high electric field only seldom serves to induce conduction.
10 Thermal energy excites some valence band electrons to jump the energy gap into the conduction band. Results in empty energy states in the valence bands (holes) and some occupied energy levels in the conduction band (electrons)
11 E = kt at 300K = 25meV The Si energy gap is 1.1 ev. How can any electrons overcome the energy gap at room temperature? The Excitation Rate is a strong exponential function of temperature ev 1.1 Excitation Rate Silicon = (constant)e kt
12 Boltzmann s Constant = X 10-5 ev K -1 E = 8.617x = ev E = 8.617x = ev e =3x10 19 e =5.75x10 10 Doubling the temperature increased the exponential a billion times. a STRONG temperature dependence.
13 Light - Yes Photon Energy (ev) = 1.24/photon wavelength (10-6 m). From E = 1.24/1.1 = 1.12 ev we deduce that wavelengths of 1.1 microns (near infrared) or longer (visible light) will promote electrons to the Silicon conduction band. The opposite, exciting photons by recombining electrons from the conduction band with holes in the valence band, is the basis of LEDs and semiconductor lasers but require direct semiconductors. E-Field Typically No The electron mean free path in a solid is around 10 nm. This means that an E-field of 10 8 (100 million) V/m is needed to excite the electron to 1 ev. Most devices must operate at much lower electric fields.
14 Much Wider Energy Gap Thermally induced carriers very unlikely due to wide energy gap. Many will remain insulators until they melt. Some can be doped to create high temperature devices (diamond and SiC)
15 Semiconductors and insulators differ only by the magnitude of the energy gap. Bands Touch or Overlap Plenty of available allowed energy levels for conduction. Can be viewed as a single partially filled band. Conduction exclusively by electrons as far as we are concerned. Some divalent metals like Beryllium exhibit hole conduction, but that is beyond our field of interest.
16 Electrons traveling through a solid crystal behave differently than electrons traveling in a vacuum. Acceleration due to e-field differs in a crystal than in free space. Speed and acceleration also depend on the crystalline direction of travel. Changing the electron mass from that of the actual (vacuum) mass allows use of free-electron-type equations. This changed mass is called Effective Mass.
17 Helps us understand energy-momentum diagrams In order to characterize photon producers To understand the working of the Gunn diode, the simplest of all semiconductor devices. To understand Fermi-level positioning To grapple with negative mass for a hole. Provides additional insight to forbidden and allowed energy levels.
18 Electrons (particles) have wave-like properties. Diffraction (Transmission Electron Microscopy) De Broglie wavelength, λ Bragg Diffraction Condition nλ = 2dsin θ Or nλ = 2d when beam is normal to atomic plane
19 A touch of Quantum Mechanics More on the De Broglie Wavelength, λ, and the Bragg Condition (diffraction) Planck s Constant which leads us to Energy-Momentum Diagrams
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