1 Semiconductors Band Formation & direct and indirect gaps 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India
2 Review of Energy Bands (1) In crystalline solids, the atoms are assembled in a periodical arrangement, in such a way as to minimize the energy of the system Example: NaCl crystal (ionic bound) In the solid, the separation between the constituting atoms is comparable to the atomic size, so the properties of the individual atoms are altered by the presence of neighbouring atoms. Kasap, S.O., Principles of electrical engineering materials and devices, McGraw-Hill, 1997
3 Energy bands Resulting from principles of quantum physics Discrete energy levels: More atoms more energy levels of electrons: There are so many of them that they form energy bands: The discrete (allowed) energy levels of an atom become energy bands in a crystal lattice 3
4 Energy Bands - Solids When atoms approach to form molecules, Pauli s exclusion principle assumes a fundamental role. When two atoms are completely isolated from each other, in a way that there s no interaction of electrons w.d.f, they can have identical electronic structures. As the space between the atoms becomes smaller, electron f.d.o superposition occurs. As stated previously, Pauli s Exclusion Principle says that two different electrons can not be described by the same quantum state; so, an unfolding of the isolated atom s discrete energy levels into new corresponding levels to the electron pair occurs.
5 In order to form a solid, many atoms are brought together. Consequently, the unfolded energy levels form, essentially, continuous energy bands. As an example, the next picture shows an imaginary Si crystal formation from isolated Si atoms. As the distance between atoms approaches the equilibrium inter-atomic separation of the Si crystal, this band unfolds into two bands separated by an energy gap, Eg.
6 Silica Diamond Semiconductors Glass Si Ge Fe Cu They have an electrical conductivity whose value is in between the metal and the insulators conductivity. Insulator semicondutores Metals -1 cm -1
7 Semiconductors Elementary Semiconductors : Si and Ge Semicondutores Composites: Binary: ZnO, GaN, SiC, InP,GaAs Ternary: AlGaAs, GaAsP, HgCdTe, Quaternarys: InGaAsP, AlInGaP. Transistors, diodes and ICs: Si e Ge LEDs: GaAs,GaN, GaP Lasers: AlGaInAs, InGaAsP, GaAs, AlGaAs Detectors: Si, InGaAsP, CdSe, InSb, HgCdTe
8 Semiconductors The semiconductor conductivity can be changed through : Temperature Optical Excitation Impurity Doping Devices based on semiconductors are fast and consume low energies; Semiconductors devices are compact and can be integrated into IC s; Semiconductors devices are cheap.
9 .:: Semiconductor, Insulators, Conductors ::. Full band Empty band All energy levels are occupied by electrons Both full and empty bands do not partake in electrical conduction. All energy levels are empty ( no electrons)
10 Electron energy.:: Semiconductor energy bands at low temperature ::. Empty conduction band At low temperatures the valance band is full, and the conduction band is empty. Recall that a full band can not conduct, and neither can an empty band. Forbidden energy gap [Eg] At low temperatures, semiconductors do not conduct, they behave like insulators. Full valance band The thermal energy of the electrons sitting at the top of the full band is much lower than that of the Eg at low temperatures.
11 Conduction Electron : Assume some kind of energy is provided to the electron (valence electron) sitting at the top of the valance band. This electron gains energy from the applied field and it would like to move into higher energy states. This electron contributes to the conductivity and this electron is called as a conduction electron. At 0 0 K, electron sits at the lowest energy levels. The valance band is the highest filled band at zero kelvin. Forbidden energy gap [Eg] Empty conduction band Full valance band
12 Semiconductor energy bands at room temperature When enough energy is supplied to the e - sitting at the top of the valance band, e - can make a transition to the bottom of the conduction band. When electron makes such a transition it leaves behind a missing electron state. This missing electron state is called as a hole. Hole behaves as a positive charge carrier. Magnitude of its charge is the same with that of the electron but with an opposite sign. +e- +e- +e- +eenergy Empty conduction band Forbidden energy gap [Eg] Full valance band
13 Doped and undoped Semicondcutors ::. Holes contribute to current in valance band (VB) as e - s are able to create current in conduction band (CB). Hole is not a free particle. It can only exist within the crystal. A hole is simply a vacant electron state. A transition results an equal number of e - in CB and holes in VB. This is an important property of intrinsic, or undoped semiconductors. For extrinsic, or doped, semiconductors this is no longer true.
14 Electron energy Bipolar (two carrier) conduction empty occupied After transition Valance Band (partly filled band) After transition, the valance band is now no longer full, it is partly filled and may conduct electric current. The conductivity is due to both electrons and holes, and this device is called a bipolar conductor or bipolar device.
15 What kind of excitation mechanism can cause an e- to make a transition from the top of the valance band (VB) to the minimum or bottom of the conduction band (CB)? Answer : Thermal energy? Electrical field? Electromagnetic radiation? To have a partly field band configuration in a semiconductor, one must use one of these excitation mechanisms. Eg Partly filled CB Partly filled VB Energy band diagram of a Semiconductor at a finite temperature.
16 1-Thermal Energy : Thermal energy = k x T = 1.38 x J/K x 300 K =25 mev Excitation rate = constant x exp(-eg / kt) Although the thermal energy at room temperature, RT, is very small, i.e. 25 mev, a few electrons can be promoted to the CB. Electrons can be promoted to the CB by means of thermal energy. This is due to the exponential increase of excitation rate with increasing temperature. Excitation rate is a strong function of temperature.
17 2- Electric field : For low fields, this mechanism doesn t promote electrons to the CB in common semiconductors such as Si and GaAs. An electric field of V/m can provide an energy of the order of 1 ev. This field is enormous. So, the use of the electric field as an excitation mechanism is not useful way to promote electrons in semicondcutors.
18 3- Electromagnetic Radiation : c E h h (6.62x10 J s) x(3x10 m / s) / ( m) E( ev ) (in m) h = 6.62 x J-s c = 3 x 10 8 m/s 1 ev=1.6x10-19 J 1.24 for Silicon Eg 1.1 ev ( m) 1.1 mn 1.1 Near infrared To promote electrons from VB to CB Silicon, the wavelength of the photons must 1.1 μm or less
19 Conduction Band e- + Valance Band photon The converse transition can also happen. An electron in CB recombines with a hole in VB and generate a photon. The energy of the photon will be in the order of Eg. If this happens in a direct bandgap semiconductor, it forms the basis of LED s and LASERS.
20 Insulators : The magnitude of the band gap determines the differences between insulators, s/c s and metals. The excitation mechanism of thermal is not a useful way to promote an electron to CB even the melting temperature is reached in an insulator. Even very high electric fields is also unable to promote electrons across the band gap in an insulator. CB (completely empty) Eg~several electron volts VB (completely full) Wide band gaps between VB and CB
21 Metals : Touching VB and CB CB VB CB VB Overlapping VB and CB These two bands looks like as if partly filled bands and it is known that partly filled bands conducts well. This is the reason why metals have high conductivity. No gap between valance band and conduction band
22 Gap Energy (ev ) Semiconductors Lattice Constant (Å)
23 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.
24 Metals, Semiconductors, and Insulators Range of conductivities exhibited by various materials. Insulators A l u m i n a M a n y c e r a m ic s Semiconductors Conductors Superconductors D ia m o n d I n o r g a n ic G la s s e s M ic a P o ly p r o p y le n e S o d a s ilic a g la s s P V D F P E T SiO 2 B o r o s ilic a te P u r e S n O 2 I n tr in s ic S i A m o rp h o u s A s S e I n tr in s ic G a A s 2 3 Metals D e g e n e r a te ly doped Si A llo y s T e G r a p h ite N ic r A g Conductivity ( m)
25 Valance band, conductance band conductance band valance band Valance band these electrons form the chemical bonds almost full Conductance bend electrons here can freely move almost empty v valance band / uppermost full c conductance band / lowermost empty 25
26 Conductors and insulators 26 forbidden gap band gap semiconductor insulator metal For Si: Wg = 1.12 ev for SiO2: Wg = 4.3 ev
27 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 gap Eg from a band whose 2N states are completely filled by 2N electrons, representative of an insulator.
28 Metals Ef Ef Metal Semiconductor Allowed electronic-energy-state systems for metal and semiconductors. States marked with an X are filled; those unmarked are empty.
29 Typical band structures of Metal Metals Electron Energy, E Free electron Vacuum level 3p 3s 2p 2s 2s Band 3s Band 2p Band E = 0 Overlappin g energy bands Electrons 1s ATOM 1s SOLID 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.
30 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.
31 Electron Motion in Energy Band E = 0 E 0 No fluid motion can occur in a completely filled tube with sealed ends.
32 Energy band diagrams. Energy Band Formation Energy-band diagram for a semiconductor showing the lower edge of the conduction band Ec, a donor level Ed within the forbidden gap, and Fermi level Ef, an acceptor level Ea, and the top edge of the valence band Ev.
33 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.
34 E-k diagram, Bloch function. Energy Band Diagram PE(r) r PE of the electron around an isolated atom V(x) 0 a a x=0 a 2a 3a Surface Crystal x=l Surface x When N atoms are arranged to form the crystal then there is an overlap of individual electron PE functions. PE of the electron, V(x), inside the crystal is periodic with a period a. 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).
35 Energy Band Diagram E-k diagram, Bloch function. 2 d 2 dx 2m 2 e E V ( x) 0 V ( x) V ( x ma) m 1,2,3... Schrödinger equation ( x ) U ( x) k k e Periodic Potential i k x 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.
36 Energy Band Diagram E-k diagram of a direct bandgap semiconductor The E-k Diagram E k The Energy Band Diagram Conduction Band (CB) E g e - E c Empty k hu E c CB e - hu Valence Band (VB) h + E v Occupied k E v h + VB k š /a š /a 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 Ev to Ec there are no points [ k (x), solutions].
37 Energy Band Diagram E-k diagram E E E Direct Bandgap k E g CB VB E c E v Photon k k VB Indirect Bandgap, E g CB E k c cb E v k vb k k E r VB CB E Phonon c E v k GaAs Si Si with a recombination center In GaAs the minimum of the CB is directly above the maximum of the VB. direct bandgap semiconductor. In Si, the minimum of the CB is displaced from the maximum of the VB. indirect bandgap semiconductor Recombination of an electron and a hole in Si involves a recombination center.
38 Direct and indirect-band gap materials : Direct-band gap semiconductors (e.g. GaAs, InP, AlGaAs) CB E For a direct-band gap material, the minimum of the conduction band and maximum of the valance band lies at the same momentum, k, values. e- + k When an electron sitting at the bottom of the CB recombines with a hole sitting at the top of the VB, there will be no change in momentum values. VB Energy is conserved by means of emitting a photon, such transitions are called as radiative transitions.
39 Indirect-band gap s/c s (e.g. Si and Ge) E For an indirect-band gap material; the minimum of the CB and maximum of the VB lie at different k-values. When an e- and hole recombine in an indirect-band gap s/c, phonons must be involved to conserve momentum. CB VB + k e - Eg Phonon Atoms vibrate about their mean position at a finite temperature.these vibrations produce vibrational waves inside the crystal. Phonons are the quanta of these vibrational waves. Phonons travel with a velocity of sound. Their wavelength is determined by the crystal lattice constant. Phonons can only exist inside the crystal.
40 The transition that involves phonons without producing photons are called nonradiative (radiationless) transitions. These transitions are observed in an indirect band gap semiconductors and result in inefficient photon producing. So in order to have efficient LED s and LASER s, one should choose materials having direct band gaps such as compound semiconductors of GaAs, AlGaAs, etc
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