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Learn about the quantization concept, energy band formation, and electron motion in materials such as metals, semiconductors, and insulators. Understand the classification of materials based on bandgap size and the range of conductivities exhibited by different materials.
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Quantization Concept plank constant Core electrons Valence electrons
Quantization Concept • The Shell Model L shell with two sub shells Nucleus s 1 K s 2 L p 2 1s22s22p2or[He]2s22p2 • The shell model of the atom in which the electrons are confined to live within certain shells and in sub shells within shells.
Quantization Concept Stable orbit has radiusr0 z z y y x x + e 1sorbital 2pxorbital v z z r o y y e x x 2pyorbital 2pzorbital (ml = 0) The planetary model of hydrogen atom in which the negatively charged electron orbits the positively charged nucleus. Orbitals
Atomic Bonding • Bonding forces in Solids • Ionic bonding (such as NaCl) • Metallic bonding (all metals) • Covalent bonding (typical Si) • Van der Waals bonding (water…) • Mixed bonding (GaAs, ZnSe…, ionic & covalent)
Energy Band Formation Allowed energy levels of an electron acted on by the Coulomb potential of an atomic nucleus. Splitting of energy states into allowed bands separated by a forbidden energy gap as the atomic spacing decreases; the electrical properties of a crystalline material correspond to specific allowed and forbidden energies associated with an atomic separation related to the lattice constant of the crystal.
Energy Band Formation • Broadening of allowed energy levelsinto allowed energy bands separated by forbidden-energy gaps as more atoms influence each electron in a solid. One-dimensional representation Two-dimensional diagram in which energy is plotted versus distance.
Energy Band Formation Energy Bandgap where ‘no’ states exist Pauli Exclusion Principle Only 2 electrons, of spin+/-1/2, can occupy the same energy state at the same point in space. As atoms are brought closer towards one another and begin to bond together, their energy levels must split into bands of discrete levels so closely spaced in energy, they can be considered a continuum of allowed energy. • Strongly bonded materials: small interatomic distances. • Thus, the strongly bonded materials can have larger energy bandgaps than do weakly bonded materials.
Energy Band Formation (Si) • Energy levels in Si as a function of inter-atomic spacing The 2N electrons in the3ssub-shell and the 2N electrons in the3psub-shell undergosp3hybridization. conduction band (empty) valence band (filled) The core levels (n=1,2) in Si are completely filled with electrons.
Energy Band Formation • Energy band diagrams. N electrons filling half of the 2N allowed states, as can occur in a metal. A completely empty band separated by an energy gapEg from a band whose 2N states are completely filled by 2N electrons, representative of an insulator.
Metals, Semiconductors, and Insulators Ef Ef Metal Semiconductor • Allowed electronic-energy-state systems for metal and semiconductors. • States marked with an “X” are filled; those unmarked are empty.
Metals, Semiconductors, and Insulators • Typical band structures of Metal Electron Energy, E Free electron Vacuum E = 0 3s Band level 2 p Band 3 p Overlapping energy bands 3 s 2 p 2 s Band 2 s Electrons 1 s 1 s SOLID ATOM • In a metal the various energy bands overlap to give a single band of energies that is only partially full of electrons. • There are states with energies up to the vacuum level where the electron is free.
Electron Motion in Energy Band Current flowing E = 0 E 0 • Electron motion in an allowed band is analogous to fluid motion in a glass tube with sealed ends; the fluid can move in a half-filled tube just as electrons can move in a metal.
Electron Motion in Energy Band E = 0 E 0 • No fluid motion can occur in a completely filled tube with sealed ends.
Energy Band Formation • Energy band diagrams. • Energy-band diagram for a semiconductor showing the lower edge of the conduction bandEc, a donor levelEd within the forbidden gap, and Fermi levelEf, an acceptor levelEa, and the top edge of the valence bandEv.
Electron Motion in Energy Band • Fluid analogy for a semiconductor • No flow can occur in either the completely filled or completely empty tube. • Fluid can move in both tubes if some of it is transferred from the filled tube to the empty one, leaving unfilled volume in the lower tube.
Metals, Semiconductors, and Insulators • Typical band structures at 0K. Insulator Semiconductor Metal
Material Classification based on Size of Bandgap • Ease of achieving thermal population of conduction band determines whether a material is an insulator, metal, or semiconductor.
Metals, Semiconductors, and Insulators • Range of conductivities exhibited by various materials. I n s u l a t o r s S e m i c o n d u c t o r s C o n d u c t o r s M a n y c e r a m i c s S u p e r c o n d u c t o r s A l u m i n a D i a m o n d I n o r g a n i c G l a s s e s M e t a l s M i c a P o l y p r o p y l e n e D e g e n e r a t e l y S o d a s i l i c a g l a s s P V D F doped Si A l l o y s B o r o s i l i c a t e P E T P u r e S n O 2 I n t r i n s i c S i A m o r p h o u s T e G r a p h i t e A g S i O N i C r I n t r i n s i c G a A s 2 A s S e 2 3 - 6 - 3 - 1 8 - 1 5 - 1 2 - 9 0 3 9 1 2 6 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 Conductivity (m)-1
Energy Band Diagram P E ( r ) r V ( x ) a a 0 x x L = a a a 2 3 x = 0 S u r f a c e S u r f a c e C r y s t a l • E-k diagram, Bloch function. PE of the electron around an isolated atom When N atoms are arranged to form the crystal then there is an overlap of individual electron PEfunctions. PE of the electron, V(x), inside the crystal is periodic with a perioda. • The electron potential energy [PE,V(x)], inside the crystal is periodic with the same periodicity as that of the crystal, a. • Far away outside the crystal, by choice, V = 0 (the electron is free and PE = 0).
Energy Band Diagram • E-k diagram, Bloch function. Schrödinger equation Periodic Potential Periodic Wave function Bloch Wavefunction • There are many Bloch wavefunction solutions to the one-dimensional crystal each identified with a particular k value, say kn which act as a kind of quantum number. • Each k (x) solution corresponds to a particular kn and represents a state with an energy Ek.
Energy Band Diagram T h e E n e r g y B a n d T h e E - k D i a g r a m D i a g r a m E k C B C o n d u c t i o n y E m p t y - - e B a n d ( C B ) k e E E c c u h u h E g E E V a l e n c e v v + + h h y O c c u p i e d B a n d ( V B ) k V B k – š / a š / a • E-k diagram of a direct bandgap semiconductor • The E-k curve consists of many discrete points with each point corresponding to a possible state, wavefunction k (x), that is allowed to exist in the crystal. • The points are so close that we normally draw the E-k relationship as a continuous curve. In the energy range Evto Ecthere are no points [k (x), solutions].
Energy Band Diagram • The energy is plotted as a function of the wave number, k, along the main crystallographic directions in the crystal. Ge Si GaAs The bottom axis describe different directions of the crystal.
Energy Band Diagram • E-k diagram E E E C B E I n d i r e c t B a n d g a p , g E C B c D i r e c t B a n d g a p P h o t o n C B E E E g E c c E P h o n o n k r v c b E E v V B v k V B V B v b k k k k k – k – – S i w i t h a r e c o m b i n a t i o n c e n t e r G a A s S i In GaAs the minimum of the CB is directly above the maximum of the VB. direct bandgap semiconductor. Recombination of an electron and a hole in Si involves a recombination center. In Si, the minimum of the CB is displaced from the maximum of the VB. indirect bandgap semiconductor
Direct and Indirect Energy Band Diagram (a) Direct transition with accompanying photon emission. (b) Indirect transition via defect level.
Energy Band • A simplified energy band diagram with the highest almost-filled band and the lowest almost-empty band. vacuum level : electron affinity conduction band edge valence band edge
Metals vs. Semiconductors • Pertinent energy levels Metal Semiconductor • Only the work function is given for the metal. • Semiconductor is described by the work functionqΦs, the electron affinityqs, and the band gap (Ec – Ev).
Metals, Semiconductors, and Insulators • Typical band structures of Semiconductor C o v a l e n t b o n d S i i o n c o r e ( + 4 e ) E l e c t r o n e n e r g y , E c E + c C o n d u c t i o n B a n d ( C B ) E m p t y o f e l e c t r o n s a t 0 K . E c B a n d g a p = E g E v V a l e n c e B a n d ( V B ) F u l l o f e l e c t r o n s a t 0 K . 0 A simplified two dimensional view of a region of the Si crystal showing covalent bonds. The energy band diagram of electrons in the Si crystal at absolute zero of temperature.
Electrons and Holes Electrons: Electrons in the conduction band that are free to move throughout the crystal. Holes: Missing electrons normally found in the valence band (or empty states in the valence band that would normally be filled). These “particles” carry electricity. Thus, we call these “carriers”
Electrons and Holes • Generation of Electrons and Holes E E l e c t r o n e n e r g y , c E + c C B E u h > E c – F r e e e g h u h o l e E g – e + H o l e h E v V B 0 Each line between Si-Si atoms is a valence electron in a bond. When a photon breaks a Si-Si bond, a free electron and a hole in the Si-Si bond is created. A photon with an energy greater then Eg can excite an electron from the VB to the CB.
Effective Mass (I) • An electron moving in respond to an applied electric field. E E within a Vacuum within a semiconductor crystal • It allow us to conceive of electrons and holes as quasi-classical particles and to employ classical particle relationships in semiconductor crystals or in most device analysis.
Carrier Movement Within the Crystal Density of States Effective Masses at 300 K Ge and GaAs have “lighter electrons” than Si which results infaster devices
Effective Mass (II) • Electrons are not free but interact with periodic potential of the lattice. • Wave-particle motion is not as same as in free space. Curvature of the band determine m*. m* is positive in CB min., negative in VB max.
Effective Mass Approximation • The motion of electrons in a crystal can be visualized and described in a quasi-classical manner. • In most instances • The electron can be thought of as a particle. • The electronic motion can be modeled using Newtonian mechanics. • The effect of crystalline forces and quantum mechanical properties are incorporated into the effective mass factor. • m* > 0 : near the bottoms of all bands • m* < 0 : near the tops of all bands • Carriers in a crystal with energies near the top or bottom of an energy band typically exhibit a constant (energy-independent) effective mass. • ` : near band edge
Impurity Doping • The need for more control over carrier concentration • Without “help” the total number of “carriers” (electrons and holes) is limited to 2ni. • For most materials, this is not that much, and leads to very high resistance and few useful applications. • We need to add carriers by modifying the crystal. • This process is known as “doping the crystal”.
Binding Energies of Impurity • Hydrogen Like Impurity Potential (Binding Energies) • Effective mass should be used to account the influence of the periodic potential of crystal. • Relative dielectric constant of the semiconductor should be used (instead of the free space permittivity). : Electronsin donor atoms : Holesin acceptor atoms Binding energies in Si: 0.03 ~ 0.06 eV Binding energies in Ge: ~ 0.01 eV
Concept of a Donor “Adding extra” Electrons Band diagram equivalent view
