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Lecture1 Introduction High Speed Semiconductor Devices Where are they used? At Microwave/Millimeter Wave/RF/Optical frequencies For COMMUNICATION -Point to Point -Broadcasting SIGNAL PROCESSING INTERCONNECTION NETWORKING Examples: RADAR, Satellite Communication, Mobile Communication, Metropolitan and Wide Area Network Using high speed devices Oscillators Amplifiers Mixers/Modulations/Demodulation For Amplifiers/Oscillators/Filters/Mixers/Modulators MESFETs, BJTs, HEMTs, Klystrons, Magnetrons, IMPATT Diodes, and Gunn diodes. Detector/Mixer Schottky diodes, PIN diodes, Varactor diodes. Now what makes a device high speed device? Or Why do we need special device considerations at high frequency? Why cannot we use a BJT from a basic electronics lab, use it to make an amplifier at 10 GHz? Why a special BJT? Well let us look at a simple circuit Fig.1.1 Fig.1.2 no charge on C for t < 0 V in = unit step at t = 0 RC small steady state is reached more quickly We can go for frequency domain analysis to understand this in another way Fig.1.3 Cut off frequency As RC decreases, high speed. increases and circuit becomes capable of handing high frequency – So high speed/frequency Same idea in devices also low RC must reduce RC in devices. There is a different delay mechanism know as transit time delay. We assume in normal circuits that as soon as the voltage (Electric Field) is applied, current flows immediately in the external circuit. Nature is, however not so kind. The carriers which are influened by the electric field has to start from one terminal of the device and reach the other terminal to show any observable current in the external circuit. Due to finite velocity of the carriers, there is a delay, called the transit time delay as will be explained in the following section. Resistivity and cut off frequency Think about a piece of semiconductor carrying current I . Fig.1.4 A = area of cross-section L = length = resistivity We know I = JA, J is the current density. v = drift velocity, = mobility for (n) electrons and (p) holes, respectively and q the electronic charge. Hence, Therefore, cut off frequency associated with this device is: So to increase we must increase q and are fundamental constants for Si. If So we must increase mobility and therefore drift velocity . Increase doping. However, increase in doping reduces mobility & velocity. Therefore optimization needed. We need to study the basics of solid state physics to understand about mobility, carrier concentration, etc. Lecture2 To study this – we will review Simple quantum mechanics Energy band theory Density of states for carriers Recombination and generation of carriers Carrier transport Quantum Theory Started in 1901 when Max Planck explained blackbody radiation. Solid objects emit radiation when heated. Black body radiation curves showing peak wavelengths at various temperatures:- Fig.1.5 For more insight, play with Java Applets of Black body radiation from the following links: 1. http://www.lon-capa.org/~mmp/applist/blackbody/ black.htm 2. http://webphsics.davidson.edu/alummi/MiLee/java/_mjl.htm Many attempts were made to explain the Black Body radiation pattern i.e, RaylieghJeans, Wiens. Fig.1.6 Finally Planck solved it. Vibrating atoms in material could only radiate or absorb energy in discrete packets. For a given atomic oscillator vibration at a frequency Hz. Planck postulated that the energy of the oscillator was restricted to quantized values. and is the circular frequency h = Planck's constant = One quantum of energy velocity of light in free space. , where = Wavelength (m) and ‘c' is the Bohr's theory of atoms To explain why atoms when energized (heat, electric discharge) emit discrete spectral lines , i = integer as shown in Fig.1.7 Fig.1.7 Bohr said that electrons in an atom move in well defined orbits, each orbit has a fixed energy level and angular momentum. Hydrogen atoms, atomic No. = 1, and has only one electron Angular momentum of this atom = electron rest mass, = linear electron velocity = radius of the orbit for a given n Now at equilibrium, centrifugal force = electric force for n th orbit Total energy Light energy emitted by a Hydrogen atom when heated or excited discrete in nature and equal to as shown in Fig .1.8 Fig.1.8 Quantum mechanics To extend Bohr theory to He, Li, we need Quantum Mechanics which requires Wave Particle Duality of De Broglie Total energy , where m is the mass of a photon. Therefore, photon mass = Photon momentum and . Therefore an argument could be made that an electron with a mass also has a wavelength , and can then and momentum be represented by wave, where for this particle momentum p can be associated with a wave of wavelength , where particle momentum Schrödinger and Heisenberg electrons, etc. quantum mechanics to describe small particles like Postulates of Quantum Mechanics • There exist a wave function variables and t is the time. From this in system, is complex. • The of a particle where x, y and z are the space one can find the dynamic behaviour of a particle for a given system is determined by Schrödinger equation: , where m is the mass of the particle and the potential energy operator for the system. • and is must be finite, continuous and single-valued for all x, y, z and t. • probability that the particle is in a spatial volume element dV. So, , integration over all space. • A mathematical operator can be associated with each dynamic system variable i.e. position, momentum, velocity, etc. The expectation value, when this operator operates on As examples, the space variables x, y, z are given as The momentum are given as is given as where Similarly Energy E is derived with an operator Therefore the expected value of energy from postulate 5 is If energy = E 0 (constant) or or Schrödinger equation. This is just like time harmonic case for EM field where the time dependence is Therefore, now Schrödinger equation can be written as or where . This is known as time independent Schrödinger equation and is much easier to solve. A few simple problems will be looked into the next lecture such as: • Free Particle. • Particle in a potential well. • Infinite well. • Finite well. Lecture3 Solution of Schrödinger Equation Free Particle A particle (electron), alone in the universe. Particle Alone mass m and a fixed total energy E. no force on the particle. constant potential energy everywhere U(x, y, z) = constant, say zero Let universe = one Dimensional only x variation Time independent Schrödinger equation which can be written as or , a one dimensional differential equation.Where or E has a parabolic dependence on k. General solution for the simple equation can be written as unknown constants. where Therefore, total wave function (space and time dependence) Compare this with a time-harmonic electromagnetic wave in free-space where k = constant of propagation = are Hence wave function of free particle consists of a traveling wave. If particle moves in+x then and Where Normalizing, = constant for all values of x. Thus the probability of finding the particle in any dx is equal/same If we take the universe to be infinite Probability = 0 If we take the universe to be finite but large no such difficulty Now momentum operator = Therefore, expected value of momentum Same as classical. Therefore, DeBroglie relationship Fig.2.1 Particle in an infinite potential well Particle of mass m and fixed total energy E confined to a relatively small segment of one dimensional space between x = 0 and x = a. In terms of the potential energy we can view that the particle is trapped in an infinitely deep one dimensional potential well with constant for 0 < x < a Fig.2.2 Now time independent Schrödinger equation is Boundary conditions Solving which can also be represented as A + exp (+jkx) + A - exp (jkx) Since , as vanishes for all x, so discrete values. Therefore, for modes or eigen functions , Energy So particle can have only a few discrete energy states/levels Fig.2.3 Wave function standing wave particle is bouncing back and forth between the walls of the potential well. average value of momentum at any x is 0. So to calculate momentum we have to isolate forward wave going in +x or backward wave going in –x. For +x wave, on the continuous and -x going wave parabola of a free particle. Discrete points lie Fig.2.4 The integer n is called the quantum number Value of from the equation that Lecture4 Finite Potential well Fig.2.5 U(x) = 0 for 0 < x < a U(x) = U 0 otherwise We assume a particle with energy For 0 < x < a, For x <0, For x > 0, confined to the potential well The three sets of solution are .....(1) Now we set up the boundary conditions. Far from the potential well, wave functions must go to zero as the probability of finding the particle away from potential region = 0 Therefore, must be continuous at well boundary must be continuous at well boundaries x = 0 & x = a From boundary conditions And we get four simultaneous equations elimination and we obtain if identically. Therefore, or This is a transcendental equation We define a dimensionless quantity is constant of the system. and from definition of k & Eigen value equation becomes . Fig. 2.6 Let then if intersect at one points intersect at 2 points intersect at 3 points So for, there are m points. In the limit (E finite). . Finite potential In finite potential wells, we talked about Eigen value equation (1) for a given system constant We have a discrete no. of solutions where LHS or RHS of (1) intersect. Fig.2.7 Wave Function of Finite Potential Wells For each intersection a value of particular and outside the well of potential One can clearly see from this that there is a finite probability of existence outside in the classically forbidden region (classically a particle with an energy cannot exist outside the well). If the potential well is slightly modified as in the figure below. Fig.2.8 The wave function will be nonzero in region C also. Thus the particle will have a finite probability of existing or coming in region C past the potential barrier B In C the particle can appear as a free particle. This quantum mechanical phenomenon of passing through a barrier is known as tunneling. The wave function is different inside and outside. Therefore there exists a finite probability of reflection at the well walls, called quantum mechanical reflection at the well walls. In analogy to optics it may be looked at as two partially reflecting mirrors, where an infinite potential well could be visualized as two 100% reflecting mirrors. It is used in Tunnel diodes and operation of many other solid-state devices. One of the usage of this phenomenon for high frequency decive is called Resonant Tunnelling Diode (RTD), which would be used as an oscillator and even as an amplifier. Lecture5 LINEAR HARMONIC OSCILLATOR A harmonic oscillator is a particle which is bound to an equilibrium position by a force which is proportional to the displacement from that position. Thus we have, (1) where is the spring constant. The potential is expressed as, (2) The linear harmonic oscillator can then be visualized on a mass connected to a spring of spring constant on shown in Fig. 2.8 Fig.2.8 The time independent Schrödinger equation is given by, or, To solve equation (3), we consider a dimension less quantity, (4) and (5) using (5) and (4) (6) For large values of y we can neglect we get equation (6) on, (7) Equation (7) is satisfied approximately by the solution, (8) Substituting equation (8) in (7) we get, (9) This indicates that equation (8) satisfied equation (7) approximately and hence we consider the exact solution as, (10) Putting the value of from (10) in (6) (11) The trick next, is to linearize the above equation. Equation (11) can be solved by using the power series method. Let the trail solution be, (12) (13) (14) (15) Putting equation (13), (14) and (15) in (11), This equation must hold for all values of of , and therefore the coefficient of each power must vanish separately. This gives in the recursion relation between and a n , (16) It seems that knowing (16), and can be calculated by using equation Thus we can write equation (12) as, (17) If in the equation (16) . But since should be zero for some value of the index n, then is a multiple of so on, all the succeeding coefficients which are related to by the recursion relation (16) would vanish, and one or the other bracketed series in equation (17) would terminate to become a polynomial of degree n. This occurs, when, or, (n = 0, 1, 2 ..) (18) Energy Quantization : We have obtained the condition when the wave function is acceptable as (n = 0, 1, 2 . . .) again was (19) The variation of the energy levels is shown in the Fig.2.10 Fig.2.10 Reference: 1. R.F. Pierret, Advanced Semiconductor Fundamentals, ( Volume VI in Modular Series on Solid devices), Addison Wesley Publishing House Reading , MA , 1989. 2. Solid State and Semiconductor Physics, 2 nd Edn. J.P.Mckelvey, 1966, Harper and Row, New Y 3. Introduction to Solid State Physics, 7 th Edition by C. Kittel (John Wiley & Sons, 1996). Lecture6 Energy Band Theory Semiconductors Si, Ge, GaAs, InP, CdTe…. - all are crystalline solids. This is a 3D regular arrangement of atoms called the lattice. Some elemental semiconductors such as Si & Ge have only one kind of material at the lattice sites, where as GaAs, InP,…are compound semiconductors and are composed of two interpenetrating 3D lattices. phycomp.technion.ac.il/~sshaharr/nonbravias.html www.chem.lsu.edu/.../MERLO/flattice/20zns.html There may be defects in the lattice and each atom in a lattice vibrates. http://simple-semiconductors.com/ Defects and vibrations are second order phenomenon which would be discussed separately. If we have a charged ion at x = 0 and an electron outside the attractive force between the ion and the electron is The PE of the and PE =0 at Fig. 3.1 If we consider two ions separated by ‘a', the potential profile in on shown in Fig.3.2 Fig.3.2 To get a feel for the electron energies in a crystal semiconductor let us consider an electron in a 1D lattice of atoms. The simplest lattice structure is a 1 D lattice. A regular array of atoms placed periodically. Fig.3.3 At each lattice site (x = 0, a, 2a, ….) there exists an ion with some net charge Bloch Theorem Relates value of wave function within any unit cell of a periodic potential to an equivalent point in any other unit cell. Concentrate on the behavior of any unit cell in the whole array. For 1D system the Bloch theorem says that if (1) if U(x) is the periodic such that U(x + a) = U(x) then (2) Equivalently, where = wave function for an unit cell. and Boundary conditions are imposed at end points of the periodic potential. Now, the wave number k in periodic potential set has several properties as: • It can be shown that and only two distinct values of k exist for each and every allowed values of E i.e. . • For a given E, values of k differing by a multiple of give rise to one and same wave function solution. As k is periodic or multiple-valued with a range we take range . Usually . • If the periodic potential is assumed to be in extent, running from to then there are restrictions on k. k can take a continumm of values. k must be real otherwise exp(jkx) or thus will blow up at either or . • In dealing with crystals of finite extent, information about the boundary conditions on crystal surface may be lacking. To avoid this we assume a periodic boundary condition assuming the lattice is a ring of N atoms. Fig. 3.4 Therefore, or So for the 1D lattice of ions the periodic set of finite potential wells, for large array x may be assumed to go from . Thus for a finite crystal k can only assume a set of discrete values, but as N is large therefore one has closely spaced discrete values. Lecture7 A simplified picture of the periodic potential is given in Fig. 3.5 Fig. 3.5 It is quite similar to the finite potential analysis periodicity = a + b , where k is continuous We will first consider the case: 0 < E < U 0 Inside well wave function = Outside well wave function = Schrödinger equation for 0 < x < a: Schrödinger equation for -b < x < 0: The general solutions to equations (1) and (2) are: Now applying the continuity condition on wave function and its derivative at x = 0 Fig. 3.6 Fig. 3.6 continuity requirement Note that x = -b is the same boundary as that of x = a. Now wave function and its derivative must observe Bloch Theorem With periodicity requirements. Similarly, Applying Boundary and Periodicity conditions Eliminating and by using first two equations in last two equations 2 eqns in 2 unknown constants. For non trivial values of formed from coefficient should be equal to zero. , the determinant Lecture8 Eigen value equation We get the following eigen value equation by simplifying the determinant: Now, say: (system constant) (normalized energy) For Putting these values in our eigen value equation: Now consider the 1D crystal lattice. Fig 3.3 does not necessarily mean that the electron is outside the crystal lattice. Therefore, we can also consider the case of here. For Putting these values in the eigen value equation: Now we assumed infinitely long lattice in K-P model. Therefore, k can take any value. RHS of eigen value eqn. LHS The plot of with value can thus take any value between +1 and -1. The between +1 and -1 gives allowed solutions for as a function of . is shown in Fig 3.7 Fig 3.7 Note that the allowed values of are shown by the shaded regions where Therefore there are allowed bands of i,e. allowed bands of E(energy bands). . Lecture9 Energy in Brilloun Zone representation. We learned about energy bands E or values for which has a solution and between these bands where it is not possible to find a value of E or are called forbidden gaps. Discrete number of such bands are separated by band gaps. Now for If we plot the allowed values of energy as a function of k, we obtain the E-k diagram for the one dimensional lattice. Brillouin zone When we plot the expanded E-k diagram of Periodic potential perturbation we notice the dissimilarity with the free space E-K diagram (given by the dotted line), Free Particle Solution How in particular can the periodic potential solution with an adjustable k approach the free particle solution with a fixed k in the limit where E > >U 0 ? In this regard it must be remembered that the wave function for an electron in a crystal is the product of two and Q(x) where Q(x) is the wave function in the unit cell. Q(x) is also a function of k. Increasing or decreasing k by in such a way that the product of modifies both approach the free particle. The k-value associated with given energy band is called a Brillouin Zone. 1 st Brillouin Zone 2 nd Brillouin Zone and Q(x) One way of drawing is to k between in basically eigenvalue equation you notice that increasing or decreasing range. In the by no effect on the allowed electron energy of E(k) is periodic with a period of has . Fig. 3.8 Fig. 3.8 Therefore all the electron energies can be represented within , by changing the k values by , where n is an integer. This representation of the electron is called the reduced Brillouin zone representation as shown in Fig. 3.9 where the bands of energies are identified. As the number of electrons in the system increases the bands starts to be filled up from the lowest available energies. Normally most of the bands are completely full of electrons as allowed by Pauli's Exclusion Principle. At low temperature, there could be a band completely empty. The one below it is usually completely full, called the valence band. None of these electrons can now conduct electricity. If now a condition arises that some of the electrons from the completely filled band can be excited into the completely empty band, then current can be conducted by the electrons in the empty band called the conduction band. Also according to the Bloch theorem there are two and only two k values associated with each allowed energy, one for the electron moving in the +ve direction and the other for the electron moving in the –ve direction. Also note that at & k = 0. This is a property of all E-k plots. Fig. 3.9 Lecture10 Bloch parameter k:- For free particles k = wave number = expected value of momentum For a particle bound to a periodic potential or crystal , = crystal momentum. is not the actual momentum but the momentum related to the constant of motion which incorporates the crystal interaction. This crystal momentum parameter k is also periodic with a period of . The E-K diagram is therefore the Energy versus crystal momentum characteristics of an electron in the crystal. Energy Band Solution Energy band solution indicates only the allowed energy and momentum states but not about time evolution of electron's position etc.given E, k gives possible values of position of finding the electron with a certain probability i.e, position is uncertain. Heisenberg Uncertainty Principle. If E is known exactly, uncertainty in t is infinity, we can not find anything about the electron's position. Therefore for a particle motion we need wave packets grouped about a peak energy. constant E wave function Probability of finding the represented particle in a given region of space = 1 for some specified time. Center of mass of a particle moving with a velocity wave packet also a mass – QM idea – classical idea. Packet of traveling wave with center frequency and center wave number k then describes the particle motion represented by this group. group velocity, E, k gives the center values of energy and crystal momentum. Lecture11 Effect of External Force External force F acting on the wave packet would be any force other than the crystalline force associated with periodic potential. External force may arise from dopants or external electric field. Force F acting over a short distance dx energy increases by work on the wave packet we know Rate of change of momentum = = Mass x Acceleration = therefore, Effective mass At k = 0, curvature of (b) band is more than the curvature of (a), i.e. Fig.4.1 wave packet Fig.4.1 Heavy mass slower movement, larger transit time. Mobility of a carrier curvature of seen in Fig.4.2 Fig.4.2 Now consider the band segment of the Kronig-Penny model as shown in Fig. 4.3 Fig. 4.3 near the bottom or minimum of bands. near the top part of each band. For In response to applied force the particle/electron will accelerate in a direction opposite to than expected from purely classical calculation. In most cases we do not know the exact equation of the E-k diagram (difficult even for the Kronig-Penny model) but it been found that top or bottom of the band edge/Brillouin zone edge of the E-k relation is approximately parabolic as shown in Fig. 4.3 where A is constant. Therefore, band. constant at the edge of the Brillouin zone/top or bottom of an energy Therefore, band. constant near the edge of the Brillouin zone/top or bottom of an energy Current Flow If N atoms are there in 1 dimensional crystal then the distinct k values in each band = N spaced apart by For the sake of discussion we assume each atom gives 2 electrons to the crystal a total of 2N free electron in the crystal. 2N electrons will be distributed among the available energy states. At temperature bands). Fig. 4.4 2N electrons will fit the 2N states in the lowest 2 bands (valence Fig. 4.4 At room temperature sufficient thermal energy is there and a few electrons from the top of the 2 nd band will move up to the bottom of the 3 rd band as shown in Fig. 4.5. Fig. 4.5. If a voltage applied to the crystal a current will flow through the crystal. 4 th band – no electron at room temperature – no current as totally empty bands do not contribute to the charge transport process. Concept of Holes Now consider the 1st band where N states are filled by N electrons at room temperature. With external voltage, electrons will move with velocity symmetric about . For every electron with a given another electron with the same Thus first band But the band is direction, there will be in –x direction. no current, So totally filled energy band do not contribute to the charge transport process. Thus first band on current, So totally filled energy band do not contribute to the charge transport process. Asymmetry applied electric field. For nearly empty third band current where L length of 1D crystal and For the nearly filled second band current This summation is more difficult to evaluate as we have very large number of states filled. Now if the band was completely filled, Therefore, we can write This current is the same as if a positively charge particle is placed on the empty states and the remaining states are unoccupied. Overall motion of electrons in nearly filled energy band can be described by the empty electronic states provided that the effective mass of the empty states is taken to be negative of , i.e. Now in this example empty energy states are at the top of the bands effective mass there. Therefore, Empty energy states at the top of a band +ve charge with +ve Normally the electron or holes stay mostly at the edge of a band (holes in top of Valence band and electrons in the bottom of conduction band) where parabolic band approximation can be applied. So in semiconductors electrons and holes generally have constant effective mass and the classical treatment/behaviour is a good approximation. Valence Band 1. Maxima occurs at zone center at k = 0 2. Valence band = three sub bands Two bands are degenerate (have same allowed energy) at k = 0. Third band max at reduced energy k =0. 1. At k = 0, the shape and curvature are orientation independent. Conduction Band 1. Somewhat similar, but minimum where electrons will gather material to material. 2. Ge C Band minima at zone boundary (8 such min) varies from 3. Si C Band minima at from the zone center. 4. GaAs C Band min at k = 0. Direct bandgap good for light emission conservation of momentum transition from V Band to C Band. 5. 3D structure is tensor. Band gap Energy It is the range from Valence Band maximum to Conduction Band minimum. At Si GaAs A decrease in T results in a contraction of crystal lattice stronger atomic bonds increase in Band gap energy is a good model E G 0) Ge 0.7437 eV 235 Si 1.170 eV 636 GaAs 1.519 eV 204 Lecture 12 Mobility Mobility is the measure of ease of carrier motion within a semiconductor crystal. Motion is impeded by collision which results in decreased mobility. Current Density holes electrons where E is the applied electric field and is the mobility. q the electronic charge and n & p are the respective carrier concentrations. For Si with at k, and Where GaAs with at k and Lattice Vibrations Consider the ID lattice Fig.5.1 Fig.5.1 Hooke's law is obeyed exactly If one atom is displaced by x from equilibrium position Potential energy of the atom is a function of distance from neighboring atoms Displaced position potential energy is changed by (1) potential energy when distance from neighbour =x Expanding term in Taylor series about a and keying only 1st significant potential function of a harmonic oscillator But this assumption that only one atom oscillates since atoms are bond to each other we expect that vibration of one atom in lattice will set other atom to vibration let displacement of and atom from equilibrium Fig.5.2 Fig.5.2 l = integer force acting on l th atom, a distance from adjacent atoms and for two adjustment atoms putting it to themselves. Force F on l th atom 2 nd law to atom …............(i) Then (i) become wave equation (ii) Complex oscillation possible let regular frequency = from equation (i). where q = wave no. proper Sign should be chosen to make Fig.5.3 Fig.5.3 Classical single harmonic oscillator there is a single frequency Lattice vibration are characterized by a continum of frequencies with a limiting maximum value But repetition is there for or So Brillouin zone concept is coming out. Now We consider a Crystal of finite member of atoms N Boundary condition for the displacement of lattice atom at the end of crystal may be taken periodic boundary condition to be no displacement. Periodic boundary condition The lattice vibration we talked about applicable to material like in which all atoms identical and one atom/unit cell. Material with 2 atoms/unit cell or 2 types of atoms/unit cell=Semiconductors. Double periodicity Fig.5.4 Fig.5.4 Two equations describes the vibrations (2 atom 2 period case) Right hand side is wavelike Oscillation Frequency Frequency = and wave vector Then Assume time harmonic solution. A&B constants Substituting 2 equations 2 unknowns, or For nontrivial A & B Dispersion relation between Solving for where, Two values of for any wave number q. Fig.5.5 Solving for When constants. The ratio of amplitude for lower branch. Vibrations of neighboring atoms are in phase Similar to those when an acoustic wave propagation in the crystal, optical mode lower frequency. 3-D crystal: 2 distinct modes of transverse vibration also along with optical or acoustic longitudinal mode Actual vibration If we associate a vibration mode of wave number q quantum mechanical wave function Schrödinger equ. For a harmonic oscillator each mode of vibration different energy state with energy give by particles Picture of lattice vibrations Each mode of vibration number of particles each with energy State of vibration changes by creation or annihilation of such particles q = Pseudo momentum because of periodicity Thus the vibration of the lattice with a wave vector collection of particles called phonons which associated by the energy and pseudo momentum . At a particular temp. Lattice atoms execute random vibrations and phonons of various energies & characteristics would be present. Hamiltonian of harmonic oscillator Hamiltonian of lattice system is and normalized is Kronig penney model with stationary lattice atoms is not entirely accurate as lattice vibration Hamiltonian of vibrating lattice + electrons individual electrons and lattice atoms The eigen value equation is eigenvalues are modified energy instant due to changing position of lattice atoms is the periodic component of Block function out of perturbed lattice. Lecture 14 Lattice Scattering for Mobility The position of the conduction band and valance bond extrema in a semiconductor depends on the Lattice spacing. These spacing. changes during vibration and there is perturbation in the and . Fig.5.6 Fig.5.7 Effect of lattice vibration is small for bound electrons with lower energy. But the effect is more on conduction electrons which are loosely bound to atom/nucleus. Acoustic Vibration Compression & expansion in lattice crystal. In compressed position the Energy band is and altered, So that forbidden gap is increased. If expanded forbidden band gap width decreases. Potential Energy out of electron-electrons interaction is neglected as it is small under adiabatic condition. The incident wave Electron may be reflected at step or it may be transmitted. Fig.5.7 Fig.5.7 Reflected perfectly elastic collision Transmitted over the barrier But lattice wave causing moving with acoustic velocity Doppler shift which make reflected momentum different from incident momentum – can be neglected assuming electron moves with high velocity. Transmitted looses energy. is continuous at We also have is continuous or is continuous b or Deformation Potential and Mean force path with step height to be small is related to strain as = deformation potential constant = shift of conduction per unit dilational strain. Since thermal energy is the major come of vibration maximum pressure coming volume change constant. . If compressibility In a distance probability of scattering or reflection = mean free path = linear dimension of volume If an electron is in state after a perturb potential Probability of transition from the probability that it will be in state after time t is applied per unit time is Transition prob. is max if . In collision of energy is conserved. A real change of electron state from to may occur if energy associated with lattice vibration may change by creation or absorption of a phonon. The jump – 1st approximation strain is volume Shift of conduction band edge per unit strain Acoustic phonon scattering Strain is produced only be longitudinal vibration shear associated with transverse vibrations does not affect energy eigenvalues. Passage of longitudinal classic wave/phonon = coherent regions of compression/extension which are of the order of in linear extent l = length of disturbance. Volume of this linear extent is subject to a dilatorily stress producing a max pressure and volume change stored Strain energy = Source of strain energy is thermal then stored strain energy Or Compressibility Reflection coefficient In a distance Where probability of scattering is = mean free path In going a linear dimension the probability of reflection Assume is independent of velocity mean free time between scattering/collisions. From Boltzman's Theory of free particle in a gas. above Shockley has shown from a full Quantum Mechanical treatment that where C 11 = clastic constant or young's modulus for longitudinal extension <110> The mobility in Due to lattice Scattering Valence band deformation potential constant. Conduction band deformation potential constant Much more rigorous analysis is possible with perturbation theory and 3 possible models • Deformable lattice ion model the atom remains the same • Rigid ion model change distribution gets distorted but the position of ion if gets displaced the potential gets displaced. • Deformation potential model displaced. potential membrane gets distorted if the ion gets Lecture15 Scattering in semiconductor 1. 2. 3. 4. 5. Phonon or lattice vibration scattering Ionized (dopant) impurity scattering Scattering by neutral impurity atoms and defects Carrier-Carrier scattering Piezo electric scattering Of these, (1) and (2) dominates. For most high speed devices where large carrier concentrations area not involved. Where the mobility due to each scattering mechanism written separately, the overall mobility is For Phonon or lattice vibration scattering, C 11 = average longitudinal elastic constant of semiconductor, of the carrier, = the effectives mass E ds = displacement of edge of the band per unit dilation of lattice, and T = the absolute temperature. For lon scattering N 1 = ionized dopant or impurity density = permittivity Therefore overall mobility is only for GaAs Mobility parameters Specifically for Si, N = dopant concentration i.e at room temperature are determined experimentally Parameters for Si: 1358 461 1352 459 1345 458 1298 448 1248 437 986 378 801 331 Parameters for GaAs: 0.4 1100 7100 0.542 0.8 200 8000 0.551 0.9 100 8100 0.594 Temperature Dependence Again from experiment we find all have a temperature dependence of the form For Si we have: Electrons Holes N ref (cm -3 ) 2.4 92 54.3 -0.57 1268 406.9 -2.33(electrons),-2.23(holes) 0.91 0.88 0.146 GaAs behavior slightly different High field effect We know that drift velocity But linear proportionality is valid only at low temperatures. For E field intensity Nonlinear behavior is seen. and E are no longer directly proportional. Now geometry is often small (submicron) so even with 1 V across So mobility concept may fail in such high field zones. For very high E field saturates. dimension, For Si at 300 K, for both electrons and holes. Model, For GaAs initially decreases with E after a critical field is 2 slowly increases, In a region of high velocity 10 3 v/cm and then very which can be considered as saturated . However, in GaAs for holes the velocity saturates and Independent of E field. The variation of the drift velocity with electric field for holes and electrons are shown in Fig 5.8 and 5.9 Fig 5.8 Fig 5.9 Intervalley Electron Scattering GaAs conduction band minimum is at Secondary minimum of conduction band is at L (<111>) L valley is sparsely populated at room temperature. With E field, valley electrons gain energy between scattering events. If valley electron gain energy intervalley transfer becomes possible and the population at L valley becomes enhanced at the expense of valley. At center of the valley . At L valley . Effective mass increases in L valley mobility and Drift velocity decreases in L valley as Lower valley mobility population ; upper valley mobility total electron . Arrange is strong function of E. is strong function of E. for Lecture16 Carrier Density Density of states Between the energies E 1 and E 2 no. of allowed states available to electron/holes in the cited energy range per unit volume of the crystal. From Band theory difficult A good approximation can be made from band edges, the regions of bands normally populated by carriers. Fig.6.1 For electrons near the bottom of C Band the band forms a pseudo-potential well. The bottom lies at E c and termination of the band at the crystal surface forms the walls of the well. The energy of electrons is relatively small compared with surface barriers. One can think being in a 3 Dimensional box. The density of states at the band edges density of states available to a particle of mass m* in a box with dimensions of the potential box Schrödinger equation Consider a particle of mass a m and total energy E. Size of box U(x, y, z) = Constant everywhere = 0 Time independent Schrödinger equation, (1) (2) Solve using separation of variables Put in equation 1. (3) Since k is a constant such that Thus for x = 0, x = a Thus Similarly (no. of modes in a wave resonator) Thus Such that where area integers. Thus a few discrete energy values are allowed inside. Allowed solutions/energy/levels If abc is large, small increments in to large no. of states allowed to . k space and draw. Fig.6.2 Fig.6.2 k space vector end points of all such vectors be dots. k space unit cell of volume = Thus contains one allowed solution. This is not complete. Now for = (-1, 1, 1). . . . . six other possible states for which E is same. If one counts all points on the k space it is thus necessary to divide by eight to obtain the number of independent solutions. Thus But electrons have 2 spin states Thus Now number of states with k value between arbitrary k and k+dk = Now Going from k to E space No. of electron energy states with energy between E and E+dE = Density of energy states with energy between E and E+dE Now for actual CB or VB densities of state near band edges m If E c = min CB energy and = average effective mass ( or Ge m is complicated. = max VB energy effective mass Bias for GaAs or compound semiconductors) for Si Fermi Dirac statistics - Assumption not all allowed states are filled - Electrons are indistinguishable. - Each state one electron (Pauli exclusion principle) - Total no. of electrons = fixed = constant, total energy Electrons are viewed as indistinguishable “balls” which are placed in allowed state “boxes”. Each box one single ball. Total energy of system is fixed. Balls are grouped in rows energy level no. of boxes in each energy level energy. Fermi function E F = Fermi energy k = Boltzman's constant = For closely spaced levels Continuous variable E Fig 6.3 no. of states of allowed electronic states at a given Fig 6.3 f(E) occupancy factor for electrons at energy E. g(E) density of states at energy E 1-f(E) occupancy factor for holes at energy E Distribution of Electron The distribution of electrons in conduction band No. of electrons is CB with energy between E and + dE (E > E C ) The distribution of holes in the valence band = The total carrier concentration in a band: conduction, The total no. of holes in valence band, If for CB and for VB [(Fermi Dirac integral of order 1/2)] Where, N c = effective density of conduction band states, N v = effective density of valence states. At room temperature (300k) we have: Semiconductor Ge Si GaAs Properties of function gamma function. here with a max error of Thus for From Fig 6 (d) is closely approximated by exp Thus Fermi level lies the band gap more than 3kT from either band edge said to be nondegenerate. Fig.6.5 Semiconductor is Lecture17 Maxwell-Boltzman approximation Simplified form Maxwell-Boltzman distribution (energy distribution at high temp for molecules in a low density gas.) Intrinsic semiconductor Thus we can write Also we know Where E G = Bandgap energy. Fig 6.6 shows vs T. Carrier Concentration variation with Energy It is the plot of allowed electron energy states as a function of position along a direction Fig. 7.1 Carrier distribution vs. E for electrons can be graphed by multiplying g(E) and f(E). Carrier distribution vs holes can be graphed by multiplying This is shown in Fig. 7.2 and (1-f(E)). Fig. 7.2 When we have an electric field energy band diagram, bands bend with x. potential energy Therefore, Kinetic energy is (E- Ec) in conduction band and Change Neutrality Equation Maxwell's equation gives semiconductor dielectric constant. in valence band Lecture18 Extrinsic Doping : Imperities could be incorporated into the crystal which could either contribute extre electrons or could accept electrons.In the former,afer the contribution of the electron/s the impurity itself becomes positively charged(as it was neutral to start with)and into concentration is called ND+.In the latter case the impurity after accepting electron/s becomes negatively charged and its concentration is called NA - Assuming uniform doping under equilibrium , Donors produce +ve ions, Acceptors produce –ve ions. if , this is the charge neutrality relation. Fig.6.6 Relationship for and Now let N D = no. of donor atoms and = no. of donor atoms ionized is the ratio of no. of empty states to total no. of states in donor energy g D = 2(standard value) Similarly, g A = 2(standard value) In the above expressions, g D and g A are the degeneracy factors From charge neutrality condition. We have solving this equation given N V , N c , N A , E c , E v , T, E D , E A F etc Free-out/Extrinsic T Suppose N D > > N A and N D > > n: electron concentration > > hole concentration. Also . we can find E where (a computable constant at a given T) Therefore, Solving this quadratic equation, , (+ve root chosen as ) or, Similar result can be obtained for acceptor doped material. is typically much grater than N D , p doped Si, T = 300 k n = 0.9996 , So 99.96% of P atoms are ionized at room temperature At T = 77 K i.e Liquid N 2 temperature, Special case of High Temperature When a semiconductor is kept at a high temperature, most dopants area ionized But we know that Therefore, and donor doped acceptor doped Position of Ei 1) Exact position of E i Intrinsic semiconductor . In this case, n = p, and E F = E i E i in Si is 0.0073 eV below midgap. GaAs is 0.0403 eV above midgap, at 300 k. 2) Freeze out/Extrinsic T. this is useful for low temperature calculations. 3) Extrinsic Therfore if the doping is a function of position, the bands will bend as shown in Fig. 7.2 Lecture19 Generation Recombination process in semiconductors When a semiconductor is perturbed from equilibrium state – carrier numbers are modified. Recombination generator process – order restoring mechanism – carrier excess or deficit inside the semiconductor stabilized or removed. Perturbation optical excitation, electron bombardment current injection from a contact Drift and diffusion currents are also there along with process of recombination Band to Band Recombination Direct Thermal recombination Fig. 8.1 Fig. 8.1 It is Direct annihilation of CB electron and a VB hole. An electron falls from an allowed CB state to hole in VB. The process of recombination is usually radiative with the production of a photon. Band to Band Generation Thermal or photon energy is absorbed Fig. 8.2 Fig. 8.2 Band to Band Recombination Photons are almost mass less very small momentum Therefore, photon assisted transition vertical on E-k plot In GaAs direct bandgap semiconductor, there is little change in momentum is needed for recombination process to proceed. Conservation of energy and momentum is simply met by release of photon. Fig. 8.3 Fig. 8.3 In an indirect bandgap semiconductor there is a change of crystal momentum associated with a recombination process. The emission of a photon will conserve energy but not momentum. For Band to Band recombination in an indirect bandgap semiconductor to proceed a photon must be emitted or absorbed - lattice vibration. Fig. 8.4 Fig. 8.4 Band to Band process in indirect semiconductors is complicated. This means that a diminished rate band to –band recombination in indirect semiconductors. In indirect semiconductors R-G, center recombination dominates. All info on band to band radiative recombination optical absorption coefficient (Direct BG) Fig. 8.5 . Fig. 8.5 Photon electron hole. generation rate Planck's radiantion law Radiative recombination rate Thermal equilibrium Therefore, The excess carrier Radiative recombination rate Let and and rate at which electrons and holes disappear. has units of sec-1 and so we can define as carriers lifetime (minority carrier) Radiative recombination lifetime n type semiconductor Donor . At high level injection . Ge Si GaAs Gap at 300K. R-G Centers/Traps Deep Energy level Impurity Atoms Crystal defects allowed energy levels in the midgap region. deep level states from CB and hole from VB comes to the RG center and gets annihilated. It can also be the considered on capture of having one electron from CB to deep state and then the same electron can jump to VB canceling a RG recombination non radioactive Heat/lattice vibration produced Fig. 8.6 Fig. 8.6 Fig. 8.7 Fig. 8.7 Dominant mechanism in G- R in semiconductors. rate of change of electron concentration due to RG recombination + generation rate of change of holes No of R-G traps/cm 3 filled with No of R-G traps/cm 3 empty. Total no of traps or RG centers per cm 3 . A mechanism in trap recombination or generation. Electron capture no. of electrons to be captured empty trap states. more captured n Constant. The probability that the recombination-generation center is occupied by an electron -ve sign to indicate that electron capture acts to reduce the no. of electrons in CB units of cm 3 /sec. = thermal velocity capture cross section. Lecture20 Diffusion and Continuity Equations Diffusion is a process, in which a particle tend to spread out a redistribute as a result of random thermal motion migrating from regions of high particle concentration to low particle concentration to produce uniform distribution. Electrons and holes are charged so when they diffuse diffusion current. Derivation of Diffusion current Assumptions • One dimensional only • All carriers move with the same velocity (in practice a distribution of velocity) • The distance moved by carriers between collisions is a fixed length L. (L is actually mean distance moved by carrier between collisions). Randomness of thermal motion equal no. of particles moving in +x and –x Fig.9.1 Fig.9.1 Derivation Within and section equal outflow of particles per second from any interior section to neighbouring sections on the right and left. But because of concentration gradient the no. of particles moving from right to left second is greater than no. of particles moving from left to right. Fig.9.2 Fig.9.2 Of the holes in a volume LA on either side of x = 0 Will move in proper direction so as to cross x = 0 plane = holes moving in +x which cross x=0 plane in time = holes moving in +x which cross x = 0 plane in time But = net no of +x directed holes with cross x = 0 plane in time Net current cross x=0 plane. We define Generalizing are constants in cm 2 /sec Einstein relation Fermi level inside a material is not dependent on position. Under equilibrium, if there is no current. non zero electric field is established inside a nonuniformly doped semiconductor under equilibrium. Under equilibrium drift and diffusion currents balance. or general form of Einstein relationship. When Then Similarly Continuity Equations Current in semiconductor Under ac and transit condition we need to add displacement current also. Now in general we need to include carrier generation and recombination also. There will be a change in carrier concentration within a given small region if there is an imbalance between total carrier currents in and out of the region. Recombination Generation For holes Continuity equations. Lecture21 Diodes We use varactor diodes, GUNN, Tunnel, IMPATT, Schottky (metal+semiconductor) and PIN diodes in Microwave Apllications such as mixers (heterodyning) (RF-IF conversion) as in Fig 10.1 Fig 10.1 Detectors as in Fig 10.2 Fig 10.2 Switches Lecture22 P-N junction Diodes Abrupt p-n junction Fig.10.3 Fig.10.3 Built-in potential= At equilibrium Thus Thus minority carrier in n side minority carrier in p side Thermal equilibrium E field in neutral regions=0 Thus Total-ve charge in P side-total + ve charge in n side Fig.11.4 Fig.11.4 Poissons Equation In n side . Integrating for for Maxwell's field at x=0(junction) Also we obtain W=Total depletion region width Eliminating from previous 3 equations. More accurate expression for depletion region width If one side is heavily doped the depletion region is in weakly doped place) Depletion –layer capacitance per unit area , Incremental increase in charge per unit area for a voltage increase of Inside depletion layer assuming the following • Boltzman relation is an approximation • Abrupt Depletion Layer • Low injection (injected minority carrier < majority carrier) • No generation current inside depletion layer where and When voltage is applied the minority carrier densities on both sides are changed and are imrefs or quasi Fermi level for electrons and holes (E Fn and E Fp ) under nonequilibrium condition as applied voltage/bias or optical field as shown in Fig. 10.4 …………….. Forward bias Reversed bias Now = Similarly Electron and hole current densities is proportional to gradient of quasi Fermi level. Now applied voltage across the junction. Now at the boundary of depletion layer at p side i.e. at We have from (A) & (B) Where is the equilibrium electron density on the p side. Similarly Where is the equilibrium hole density on the p side. n side Continuity equation in steady state Let recombination rate Then in 1-D we obtain (steady state) for n side Charge neutrality holds approximately ultiplying first by we get and second equation by and taking Einstein relation as =lifetime as type in n- Low injection assumption Neutral region Now Boundary condition (injection) assuming large structure exponential decay of electron and hole current components in the Depletion region as shown in Fig. 10.5 Fig. 10.5 Diode Currents Where At Similarly we obtain at the p side …..Total current …..Total current as shown in Fig 10.6 Fig 10.6 Abrupt junction with P + doping voltage drop in p + is neglected V = Constant for one sided abrupt junctions. For Ge ideal equation is valid. For GaAs and Si only qualitative agreement as shown in Fig 10.7 Fig 10.7 This variation is due to:• surface effects. • generation and recombination. • tunneling of carriers between states in the bandgap. • High injection conditions. • Series resistance effect. Under Reverse Bias the generation current, for a generation rate of G per unit volume is Therefore, one sided diode (N A >>N D ) the reverse bias current J R is dominated by the diffusion and Shockley equation is followed. For Si and GaAs, dominant. may be small and the generation term may be comparable or Under Forward bias the major R-G process in the depletion region is a capture process. The Recombination current density is given by where is the effective recombination lifetime. for and Lecture23 Diffusion Capacitance Deflection layer capacitance-reverse junction when forward biased another contribution from rearrangement of minority carrier density-Diffusion capacitance Applied voltage Current Forward Bias Small signal amplitude Small signal as component of hole density in n For . Similar expression for electron density in p side. Now for holes n in n side we know that continuity equation is S Thus Or Non Time varying case So here effect of frequency dependence Small signal current admittance due to diffusion only low frequency low frequency and are function of bias frequency. will dominate at high frequencies. so deflection layer capacitance Transient response of Diode The diode will not respond to the reverse voltage until excess minority carriers in neutral n and p regions have been withdrawn. Model of diode If we apply reverse bias suddenly then the diode passes a reverse current higher than reverse saturation current for some t. The current then falls as the stored minority carriers are withdrawn, eventually reaching . One can use charge storage model to understand this transient behaviour. Consider a is p only. junction-depletion region on the n side then the essential minority carrier Fig 10.7 depicts the hole distribution in neutral n-region. Fig.10.7 Multiplying continuity equation by qAdx and integrating it over the entire neutral n region from x n to w 1 , we get which can be written as where is the excess minority carrier stored in a n region at any time. is hole current at x = w 1 and is related to the transit time and neutral n region of width w 1 A time constant is defined by the relation, is related to the passage time through the neutral n-region of width W n , so we can write where the switching trajectory is shown in Fig. 10.12 Fig.10.12 Now in storage phase almost constant so the equation has the solution where at t = 0 , hence is a constant and is determined by the condition that . Now we assume a triangular hole distribution where the charge remains constant in the nregion at , as depicted in Fig. 10.13 Fig.10.13 and where is the time till the diode remains forward biased Thus The stored charge is equal to the area of the triangle with base , so writing and solving for we get Solving More accurate derivation of involves solution of time-dependent continuity equation with appropriate boundary conditions and is given as Lecture 24 Varactor Diode Structure The word varactor comes from variable reactor, it is a device whose reactance can be varied in a controlled manner with a bias voltage. The symbol for this diode is It is used widely in amplifier and harmonic generator, few examples are -mixing -detection - variable voltage tuning Let the doping distribution be The abrupt doing profile is achieved by epitaxy or ion-implantation, where m=0 In the varactor diode one side is heavily doped and on the other side the doping and the impurity concentration decreases with distance as shown in Fig 11.1[http://www.vias.org/feee/varactor_diodes.html] Fig 11.1 [http://www.vias.org/feee/varactor_diodes.html] where m is a number < 0 like -5/3, -3/2,-1 etc Poission's equation in the n-side is given by Applying the boundary conditions and (applied voltage) the depletion width, we get the capacitance as Where A is the area of junction and Therefore the slope of this variation is given by The variation of C vs. V is shown in Fig 11.2 (built-in voltage), where W is Fig 11.2 If then (Hyperabrupt junction), when then (abrupt junction)and when m=0 then S=1/2(graded junction). This is depicted in Fig 11.2 The larger the value of S, larger is the variation of with biasing. The equivalent circuit of varactor diode is shown in Fig 11.3 Fig 11.3 Where, is the junction capacitance, is the series resistance, is the parallel equivalent resistance of the generation-recombination current, diffusion current, and the surface leakage current. Both voltage. decreases with reverse bias voltage and increases with reverse bias The Quality factor Q of the varactor The maximum Quality Factor therefore is The variation of Q with frequency is shown in Fig 11.4 Fig 11.4 Therefore major considerations for a varactor diodes are (i) Capacitance, (ii) Voltage, (iii) Variation of capacitance with voltage, (iv) Maximum working voltage, (v) Leakage current and at high frequencies for a good varactor diode the equivalent circuit can be represented with lumped elements as Varactor application Varactor voltage , where i(t) is the current flowing, where, This is difficult to solve and therefore the analysis is done in the frequency domain where a set of coupled non-linear algebraic equation are solved. In this case and where m and n are harmonic numbers and P m,n is the average power flowing into nonlinear harmonics n An important application of pumped varactors is the parametric amplifier. When the varactor is pumped at a frequency F p and a signal is introduced as F c then at F s the varactor behaves as an impedance with a negative real parts. The negative part can be used for amplification. The series resistance limits the frequencies F p and F s and introduces noise. For losses reactances the power is Therefore the Manley-Rowe frequency-power formula for the case of lossless reactance are The output voltage is If the parametric amplifier is designed such that only power can flow at input frequency and output frequency is available at nF p , frequency dividers, rational fraction generators. Hence , P 1 corresponds to power at f p . Parametric small signal amplifiers and frequency converters If the RF signal at frequency is small compared to pump signal at frequency power exchanged at the side band frequencies and for , the is negligible. The corresponding Manley-Rowe equation is Under this condition the optimum gain is where, is the modulation ratio, f c is the dynamic cut-off frequency given by , R s being the series resistance and S max -S min is the elastance swing Lecture 25 PIN Diode In PIN diode the i region is sandwiched between the p and n region as shown in Fig 11.5 Fig 11.5 The i-region is either a high resistivity layer. PIN diodes are fabricated by -epitaxial process -Diffusion of p and n in high resistability substrate -ion drift method The concentration, charge density and electric field profiles are shown in Fig 11.6 Fig 11.6 PIN diodes are used widely in microwave wave circuits such as microwave switch with constant depletion layer and high power. The switching speed Where W is the total depletion region width and region. is the saturation velocity across i In addition the PIN diode can be used as -variable attenuator by varying device resistance that change approximately likely with forward current -Modulate signals up to GHz range. -Photo detection of internal modulated light in reverse bias. Under Riverse Bias the junction capacitance is and the series resistance is Where is the i-region resistance and is the contact resistance. The reverse bias current is Where is the ambipolar life time. The I-V characteristics is shown in Fig 11.7 Fig 11.7 PIN application PIN diodes finds application in switching circuits. The diode admittance(Y r ) in the reverse bias state and impedance (Z f ) in the forward bias can be expressed as and where, is the diode cut-off frequency and is the reverse-bias series resonance frequency Beam-lead PIN diodes are usually used in such circuits Lecture26 When a metal-semiconductor junction is formed such that the carries see a barrier to flow from one terminal to the other,it as called a schottky barries,as shown in 12.6. When the Schottky diode is forward biased (negative with respect to metal) by a voltage the barrier for electrons in Semiconductor decreases from electrons flows from Semiconductor to metal to . More increases greatly as shown in Fig 12.6 Fig 12.6 But remains unchanged because no voltage drop across metal and unchanged. Reverse bias drops across semiconductor increasing the barrier remains to where V R is negative now, decreases more but remains almost unchanged. Small reverse current flows from Semiconductor to metal as shown in Fig 12.7 Fig 12.7 Now suppose we have a semiconductor. with as shown in Fig 12.4 Fig 12.4 No depletion layer is formed in Semiconductor, no barrier exists in semiconductor or in metal. Metal +n-type is rectifying for Opposite is true for p-type Fig12.5 and non rectifying rectifying contact otherwise ohmic as shown in Fig12.5 For rectifying contact Maximum field occurs at The space charge per unit area Or throughout Fig10.5 ps . Thus if is constant Fig10.5 Schottky effect lowering of the barrier When an electron is at a distance of x from metal surface +ve charge induced on metal surface. The force of attraction between electron and induced +ve charge.=force that will exist between electron and (positive charge at –x) image charge. Attractive force (image force) transfer from . The work done on electron in the course of its to x Potential energy of an electron at a distance x from the metal surface. This energy must be added to barrier energy potential energy of electron This is shown in Fig 12.8 . to obtain total Fig 12.8 Putting So in effect the barrier height varies with field Current Transport in Metal-semiconductor Diodes The Metal-Semiconductor diodes is a majority current device. The mechanisms of current transport in M-S diodes are • Transport of electrons from semiconductor over the potential barrier into the metal (dominating factor for moderately doped semiconductor with at room temperature). operated • Quantum mechanical tunnelling of electrons through the barrier (important for heavily semiconductor and responsible for most of ohmic contacts. • Electron hole recombination in their depletion region. • Electron hole recombination in the neutral semiconductor region. Transport over potential barrier For high mobility materials thermionic emission theory For low mobility materials diffusion theory processes. thermionic emission theory Actually a combination of two • Barrier ht. • Thermal equilibrium • Net current flow does not effect this equilibrium 2 currents flux one from Metal to Semiconductor, other from semiconductor to metal, shape of the barrier profile is not important current flow depends only on . current density concentration of electrons with energies min energy needed to thermionically emit into metal. Carrier velocity in x direction electron density in energy range E and E+dE density of states occupancy function We assume that all energy in conduction band=Kinetic energy Then is the number of electrons /unit volume that have speed between v and v + dv over all direction as shown in Fig 12.9 Fig 12.9 Rectangular coordinates from radial v is the minimum velocity required in x direction to go over the potential barrier Again, Effective Richardson Const. Free electrons Richardson const. For GaAs, isotropic in the lowest minimum of conductive band. rest mass. Multi vally semiconductors are direction cosines of the normal to the emitting plane relative to the principal axes ellipsoid For are the components of the effective mass tensor. transverse mass longitudinal mass Barrier height for electrons moving from metal to semiconductor remains the same, unaffected by a voltage. It must be therefore equal and opposite to the current V=0 at room temperature Thus Thus Total current density for a = majority carriers. Diffusion theory (Schottky) for current It is for the current from semiconductor to the metal again • Barrier • Effect of electron collision within depletion region is included • Carrier conductance at x=0 and x=w are unaffected by current flow • Impurity concentration of semiconductor is non nondegenerate Current in depletion region depends on local field and concentration gradient with Steady state, current density is independent of x Thus Integrating both sides with an integrating factor Boundary condition are Putting these Boundary conditions Similar to thermoinic expression- drift depends on Temperature T Diffusion explanation is dominant if n semiconductor is heavily doped. Tunneling current When semiconductor is highly doped then depletion region is very narrow and electrons can tunnel through it. Lecture27 MESFET (Metal Semiconductor Field Effect Transistor) MESFETs are Mostly made with GaAs Analog and digital applications. contenders for GaAs IC 10 5 transistor mark -Communication technology satellite and fibre optic -Cell phone - Oscillator and 168 GHz for amplifier New materials for MESFET SiC and GaN wide band gap semiconductor -higher breakdown voltage 100kV -higher thermal conductivity -GaN has higher electron velocity than GaAs Also SiGe are used MESFETs are suitable for compound semiconductor where oxide making is difficult. Silicon dioxide material are easy to obtain in Si, and therefore mostly MOSFETs are used in Si/SiGe.n-channel is always better as faster electron transport than holes in most materials. Gate channel Metal-Semiconductor/ Schottky diode Normally ON MESFETs (Depletion Mode) with zero gate (–ve threshold) voltageis used. Normally OFF MESFETs (Enhancement Mode) have positive threshold voltage on the gate. Ion implantation and electrode metallization are needed for the fabrication of MESFETs on a Semi-insulating GaAs substrate. Simpler technology than MOSFETs or HEMTs. Basic FET operation Normally on type From the simplified diagram shown in Fig 13.1 we have Fig 13.1 Resistance of channel L=length Z=width and Channel acts like a constant resistor Now if increases-voltage distribution over channel changes as V(x) and the with of the depletion region becomes Therefore, when source is grounded Along the length of the channel the Resistance changes and the overall resistance of the channel now i However as V D increases, after a certain voltage, I D saturates at Where V P is called the pinch-off voltage as shown in Fig 13.2 Fig 13.2 When the additional voltage appears across the depletion region, so the depletion region widens, this is shown in Fig 13.3 . Fig 13.3 For W=a, , Lecture28 Drain Current In deriving the current components the following assumptions are made 1. Constant mobility assumed 2. Uniformly doped channel Considering Fig 13.4 Fig 13.4 we get Electron drift velocity Put and W(x) expression together and integrate from source to drain for For small drain voltage Expand in a taylor series and retian 1 st two terms This represents the 1-V characteristics of linear region given more negative i.e, Finally as is made more and decreases. , transistor to turn off. Turn off voltage or threshold voltage. Transconductance for small in linear region. in saturation region. Drain current in saturation region. Independent of Short channel MESFETs depends on E. In the above equation the value of V G should be placed with sign. Field Dependant Mobility. V s = electron saturation velocity Integrating from source to drain The I-V characteristics are shown in Fig 13.6 Fig. 2 Fig 13.6 Reduction in due to field dependent mobility. is same as before. 1. Drain saturation voltage 2. is reduced, Saturated velocity model The variation of velocity with electric field are shown in Fig 13.7 and 13.8 Fig 13.7 Fig.13.8 i ndependent of In practice a combination of and Better explaination of experimental results of High Frequency Performance: Smalll signal input current and output current is used. characterstics. So cut-off frequency In the saturated velocity model as shown in Fig 13.8 Fig 13.8 Transconductance is And cut-off frequency is Lecture 29 Semiconductor Heterojunction . For faster transit time, RC should be decreased. This is used in making HBTs/HEMTs. Two semiconductors with two different bandgaps can be grown one on top of the other or a material can be grown with variable band as shown in Fig 14.1 For a constant bandgap semiconductor electrons and holes move in opposite direction with the application of the electric field. In a heterostructure as shown both can move same direction !! Fig 14.1 Material that has a higher Band Gap is denoted by and that having lower Band Gap is denoted by n or p as usual. So we can have pN, nP or p+ N heterojunction. Few applications of heterostructures are HBT, diode laser, LED, Photodetector, Quantum well devices, solar cells. The materials that are grown together must be latticed matched. Few lattice matched compound semiconductors are and and and on GaAs. is basically solid solution (alloy) of Al, As and GaAs with no interface traps We use molecular beam Epitaxy (MBE) or metal organic chemical vapour deposition (MOCVD) to growth these structures. Liquid phase epitaxy is also used. Calibration is done with Reflection high energy electron diffraction (RHEED) For Bandgap minimum remains at keeping the alloy as direct bandgap and for Bandgap minimum occurs at X making the material indirect bandgap as shown in Fig 14.2 Fig 14.2 The energy diagram of the direct and indirect bandgap materials are shown in Fig 14.3 Fig 14.3 Now consider an abrupt p-n junction p type GaAs and N type AlGaAs First we draw their energy band diagram side by side as shown in Fig 14.4 then the two energy band diagrams are brought in contact, keeping the E F same on both sides as shown in Fig 14.5 Electron flows from N side to and a depletion region is formed. Fig 14.4 side and holes from p side to N side Fig 14.4 Fig.14.5 Fig.14.5 Hence at thermal equilibrium Fermi level lines up on both sides. Hence we get We note that the two sides has two different permittivity. The Poisson Equation are Now theE field is zero at and Thus and The potential profile is obtained by integrating potential is given by The built-in potential on the p-side is and assuming , the And the built-in potential on the N-side is The electric flux density is continuous across the jiunction, hence Also , (the charge conservation relationship) Considering all these we can show that for applied Also So, Current transport Junction diode and M-S diode too more complex. minority + majority current flow also. Now think of an junction or Now think of an junction Fig. 14.6 Fig. 14.6 Effective electron mass of Where Effective hole mass is and Where, and The Bandgap is given by for and and The dielectric constant is The electron affinity is for and The conduction band density of states is related to Where and The valence band density of states is where for by Lecture 30 is the forward transit time which is equal to the average time an electron spends in the Base and is Where is the physical base width, diffusion constant. is the Fudge factor and is the electron In practice we can make Ist order model of BJT (CE-forward active) Now we can define a frequency short circuit current gain is Where is finite base current And emitted base current So we have the collector current as being the low frequency current gain The first order model is given in Fig 15.4 at which Fig 15.4 Current gain is modeled as In case of electrons diffusing across base, The transit time is given by And the cut-off frequency is given by where is the time required to change the base- junction resistance. Space charge transit time junction. by charging up the capacitances through is time required to drift through depletion region in BC being the Base-Collector depletion region width. Collector charging time Where and are emitter and collector resistances. A more detailed BJT model is the Gummel-Poon model, shown in Fig 15.5, in this model we have The Early effect is depicted in Fig 15.5 Fig 15.5 Lecture 31 BJT: Current Model Now we go back to the current model of BJT where we have stated that electrons injected from the emitter will diffuse across the Base. One aim in making BJT is always to reduce the base width to reduce this unwanted base current. Good designs tries to make electron transfer from emitter to collector to be maximum and I b (due to the hole back injection) to be minimum. Electron-Hole diffusion Electron diffusing across the base and collected in the collector gives rise to the collector current I c given by where is the emitter area, thickness, is the minority diffusion coefficient, is the intrinsic concentration in base and N base is the base doping level. The base current due to hole back injection is now the ratio of desired to undesired current component is The ratio is the Base is roughly of the order of 1 and can be controlled by doping. To make , the base doping should be lower than the emitter doping. This works for most transistors, however for high speed operation the base resistance R B and the junction capacitance C BE are required to be low. R B decreases with increase in N base and C BE decreases with decreasing N emit . Therefore is difficult to achieve. So high speed(f T ) and high ( ratio) cannot be both achieved simultaneously and an optimization is needed. A recourse to this is to keep N emit low and N base moderate, but increase the electron injection efficiency by using a hot electron injection and field assisted base transport as done in HBT. Lecture 32 HBT : Heterojunction Biplar Transister In normal homo junction BJT we have In HBT the emitter is of wider band gap as shown in Fig 16.1 Fig.16.1 The electron can be easily injected from the emitter to the base but holes from p to Nemitter see a much larger energy barrier and hence the current I b due to hole back injection is reduced. Thus the base region can be made highly doped which reduces the base resistance R B . This is one of the advantage of HBT over BJT. In abrurpt HBT, we have Instead of using abrupt HBT we can use graded HBT made of the materials and Here the aluminium concentration is graded and hence called Graded Band Gap heterostructure. In a Graded Band Gap Heterojunction the barrier for the hole can be made larger then electron barrier by and we have Thus can be made even large than abrupt Heterojunction. Here we consider an example For As/GaAs graded hetero junction With no Heterojunction And with abrupt Heterojunction (useless device) is a good material for Heterojunction Other good Hetero junction are The HBT structure is shown in Fig 16.2 Fig.16.2 Components of the Base currents The various components of the Base currents are shown in Fig 16.3, they are Fig 16.3 • Back injection of holes • Extrinsic base surface recombination current. • Base contact surface recombination current • Bulk recombination current in Base layer • Depletion region recombination current in B-E depletion region. Bulk recombination current in the base region (I Bbulk ) is the dominant base current and the current gain is given by Where, is the minority electron recombination lifetime in Base and carrier transit time in Base. Now is the minority when the Base is too thin. can be decreased by an E field in base arising due to the linearly grading the bandgap in the Base region. A linearly graded Si/SiGe HBT is shown in Fig 16.4 Fig 16.4 The Poisson Equation is given by Base-Collector junction where the Base doping is greater than the Collector Doping the depletion region is mostly inside the Collector, where the Electric field is high and electrons travel mostly with saturation velocity. The electron carrier concentration The electron carrier concentration inside the collector is given by is the Collector current density. Thus the Electric field in Base-Collector depletion region is given by . When is small , where N C is the Collector doping concentration As increases slope becomes more negative as shown in Fig 16.5 Fig 16.5 While the current density increases and the area under the field profile should remain constant the depletion region thickness would continue to increase until it reaches x= X c as shown in Fig 16.6 Fig 16.6 As the current density increases to a level such that x=X c , the net charge inside the junction becomes zero and the field profile is constant as shown in Fig 16.7 Fig 16.7 When J c increases further such that x>X c the net charge inside the junction becomes negative, the electric field takes negative values as shown in Fig 16.8 Fig 16.8 When there is no more field to prevent holes from spilling, base pushout or Kirk effect occurs as shown in Fig 16.9 The current gain decreases as the transit time associated with the thickened base layer increases Fig 16.9 Emitter-Collector transit time Emitter-Collector transit time is Where is the time required to change the base potential by charging the capacitances (B-E junction capacitance) and the differential Base-Emitter junction resistance. (B-C junction capacitance) through is the base transit time. is the transit time through B-C depletion region and is the Collector charging time Hence the cut-off frequency is Lecture33 HFET (hetrojunction FET) OR HEMT –high electron mobility transistor or (two dimensional electron gas(2DEG))or MODFET A potential is formed at junction of two dissimilar semiconductors(AlGaAs/GaAs), wher is higher than the occupation levels of the electrons in the conduction band. The electrons accumulated in this potential well and form a sheet of electron similar to the inversion layer in a MOS structure. The thickness of this sheet is 10nm, smaller than the De-broglie wavelength of the electron in that material. This sheet of free electrons behaves like free atoms in a gas and hence it is called electron gas.. In this structure has been observed. This structure is similar to FET with 2DEG(Two Dimentional Electron Gas )as channel. Application of a bias voltage to gate modulates charges in the 2DEG and thus channel conducts current, this is similar to FET which is faster than MESFET operation. The formation of 2DEG is shown in Fig 17.1 Fig.17.1 QUANTUM WELL : PICTURE It can be seen that if the temperature is high(seldom encountered) or the applied field(V DS ) is high, the electrons are excited to high energies and may escape from the triangular well. It may also be scattered into conduction band of the barrier(spatial transfer) and the carriers would be lost from the channel. To avoid this a quantum well may be introduced at the interface. Actually in this case we work at several possibilities of hetrostructures. Charge transfer occurs leading to a conducting channel within single quantum well or multiple quantum well. The energies of the electrons are quantized in the step like density of states. Therefore many carriers can be put in the channel with a narrower dispersion of energies. Electron wave inside well is This is Kane model. Where x is the growth direction and k is the transverse electron wave vector. block wave form and is the envelope wave function which is the solution of Where is the effective mass, energy of carriers. is the potential and is the confinement Boundary conditions are should be continuous at the interfaces. The triangular potential well is shown in Fig 17.1 The triangular quantum well Potential is linear for and at x = 0 The Schrödinger equation is The boundary condition is The above equation has two independent solutions. One solution that is nonsingular at function is is AIRY function (Ai), so the resulting wave as shown in Fig 17.2 Fig.17.2 The quantized energy levels are Where, is the nth zero of Ai(x), hence So The basic idea of HFET or HEMT also known as MODFET- Modulation Doped Field Effect Transistor is that at equilibrium charge transfer occurs at the heterojunction to equalize the Fermi level on both side. Doping the N side gives wide base. Electrons are transferred to the GaAs side until an equilibrium is reached, this occurs because electron transfer raises the Fermi level on the GaAs side due to filling of the conduction band by electrons and also raises the electrostatic potential of the interface region because of the more numerous ionizer donors in the AlGaAs side. This charge transfer effect makes possible an old dream of semiconductor technologist, ie getting conducting electrons in a high mobility, High purity semiconductor without having to introduce mobility limiting donor impurities. The various charge transfer mechanisms in heterojunctions are 1. Electric charges and field near the interface determine the energy band bendings in the barrier and in conducting channel. 2. The quantum calculation of the electron energy levels in the channel determines the confined conduction band levels 3. The thermodynamic equilibrium conduction determine the density of transferred electrons. Assuming that before the charge transfer the potential is flat band. After charge transfer of electrons the electric field in the potential well created can be taken as constant to first order and is given by Gauss' Electrostatic potential is given by Hence the Schrödinger equation for the electron envelope wave function is The energy level in infinite triangular potential well for the ground state is Where and is usually determined experimentally. As charge transfer increases, potential created by transferred electrons also increases, leading to the lowering of the bottom of the Conduction . A 2DEG is formed when the Conduction band goes below the Fermi level, hence we get and the energy in channel is given by In AlGaAs Fermi energy level is pushed downwards by electrostatic potential up at the interface, where Where W is the width of depletion region( Where ) in AlGaAs is the donor on AlGaAs Calculating energies from the bothom of Conduction Band, we get and built Where is the donor binding energy in AlGaAs. The donor concentration is equal to i.e, the number of electron transferred. Hence we have From the above equation N S can be calculated if the other parameters are known. The Fermi level is determined empirically by the model given by In practice undoped AlGaAs spacer layer of thickness is used to separate Donor atom from channel electrons (2DEG) to prevent coloumb interactions resulting in an increased mobility is decreased as shown in Fig 17.2 The charge in the conduction band is Where We have calculated the 2DEG charge density N S and it can be related to the gate voltage by with Where , Where is the 2DEG capacitance per unit area as given by and is usually determined experimentally. As charge transfer increases, potential created by transferred electrons also increases, leading to the lowering of the bottom of the Conduction . A 2DEG is formed when the Conduction band goes below the Fermi level, hence we get and the energy in channel is given by In AlGaAs Fermi energy level is pushed downwards by electrostatic potential up at the interface, where Where W is the width of depletion region ( ) in AlGaAs built Where is the donor on AlGaAs Calculating energies from the bothom of Conduction Band, we get Where usually is the distance of the centroid of 2DEG from x = 0 as shown in Fig 17.3 and is Fig 17.3 Again the threshold voltage or pinch off voltage is given by Where is the Schottky barrier height on the donor layer as shown in Fig 17.4 Fig 17.4 At room temp also modulates the bound carrier density in Donor layer and free electron in Donor layer. For a simplest model independent of is so large that all electrons in 2DEG channel move with as shown in Fig 17.5 and Fig 17.5 Where is the electron density per surface area and is given by width and is the gate length. The transconductance is So we have , z is the gate Simplest model. With we have and with The voltage in the channel we have and the current is given by Hence current is given by Where is the position of entrance to channel on the source side. The saturation current is the current for which field on the drain side at reaches Where, just is is the source resistance For we get a linear behavior for a short highly conductive channel. The I-V characteristic is shown in Fig 17.6 Fig 17.6 for we get