Applied Physics Consolidated Notes PDF

UNIT 1 - QUANTUM MECHANICS INTRODUCTION: Quantum mechanics is a physical science dealing with the behaviour of matter and energy on the scale of atoms and subatomic particles or waves. The term "quantum mechanics" was first coined by Max Born in 1924. The acceptance by the general physics communi...

UNIT 1 - QUANTUM MECHANICS INTRODUCTION: Quantum mechanics is a physical science dealing with the behaviour of matter and energy on the scale of atoms and subatomic particles or waves. The term "quantum mechanics" was first coined by Max Born in 1924. The acceptance by the general physics community of quantum mechanics is due to its accurate prediction of the physical behaviour of systems, including systems where Newtonian mechanics fails. DUAL NATURE OF LIGHT: There are some phenomena such as interference, diffraction and polarization which can be explained by considering light as wave only. On the other hand phenomenon such as photoelectric effect and Compton Effect can be explained by considering light as a particle only. When we visualize light as a wave, we need to forget its particle aspect completely and vice versa. This type of behavior of light as a wave as well as particle is known as dual nature of light. Einstein’s theory of photoelectric effect: When a photon of energy hυ is incident on the surface of the metal, a part of energy Φ is used in liberating the electron from the metal. This energy is known as the work function of the metal. The rest of energy is given to the electron so that is acquires kinetic energy ½ mv2. Thus a photon of energy hυ is completely absorbed by the emitter. Energy of photon = Energy needed to liberate the electron + Maximum K.E of the liberated electron hυ = Φ + KEmax hυ = Φ + ½ mv2max The above equation is called Einstein’s photoelectric equation. This equation can explain all the features of the photoelectric effect. Compton Effect When a beam of high frequency radiation (x-ray or gamma- ray) is scattered by the loosely bound electrons present in the scatterer, there are also radiations of longer wavelength along with original wavelength in the scattered radiation. This phenomenon is known as Compton Effect. When a photon of energy hν collides with the electron, some of the energy is given to this electron. Due to this energy, the electron gains kinetic energy and photon loses energy. Hence scattered photon will have lower energy hν ’ that is longer wavelength than the incident one. (λ’ – λ) = h/mc [1-cosΦ] where h/mc = λC = Compton wavelength = 0.02424Å Page 1 of 93 De Broglie hypothesis: Louis De Broglie a French Physicist put forward his bold ideas like this “Since nature loves symmetry, if the radiation behaves as a particle under certain circumstances and waves under other circumstances, then one can even expect that entities which ordinarily behave as particles also exhibit properties attributable to waves under appropriate circumstance and those types of waves are termed as matter waves. All matter can exhibit wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave. The concept that matter behaves like a wave was proposed by Louis de Broglie in 1924. It is also referred to as the de Broglie hypothesis of matter waves. On the other hand de Broglie hypothesis is the combination of wave nature and particle nature. If ‘ E ’ is the energy of a photon of radiation and the same energy can be written for a wave as follows E = mc2 ---(1) (particle nature) and E = hν = hc/λ ---(2) (wave nature) Comparing eqns (1) & (2) we get mc2 = hc/λ or λ = h/mc = h/p λ = h/p ; where λ = De Broglie wavelength Particles of the matter also exhibit wavelike properties and those waves are known as matter waves. Expression for de Broglie wavelength of an accelerated electron De Broglie wavelength for a matter wave is given by λ = h/p ; where λ = De Broglie wavelength -------------(1) From eqn. (1) we find that, if the particles like electrons are accelerated to various velocities, we can produce waves of various wavelengths. Thus higher the electron velocity, smaller will be the de-Broglie wavelength. If velocity v is given to an electron by accelerating it through a potential difference V, then the work done on the electron is eV. This work done is converted to kinetic energy of electron. Hence, we can write ½ mv2 = eV mv = (2meV)1/2 -------------(2) But eqn.(1) can be written as λ = h/mv -------------------(3) Substituting eqn.(2) in eqn.(3) we get λ = h/(2meV)1/2 Page 2 of 93 PROPERTIES OF MATTER WAVES: 1. The wavelength of a matter wave is inversely related to its particles momentum 2. Matter wave can be reflected, refracted, diffracted and undergo interference 3. The position and momentum of the material particles cannot be determined accurately and simultaneously. 4. The amplitude of the matter waves at a particular region and time depends on the probability of finding the particle at the same region and time. Wave packet: Two or more waves of slightly different wavelengths alternately interfere and reinforce so that an infinite succession of groups of waves or wave packets are produced. The velocity of the individual wave in a wave packet is called phase velocity of the wave and is represented by Vp. + = + Vg Vp Phase, Group and particle velocities: According to de Broglie each particle of matter (like electron, proton, neutron etc) is associated with a de Broglie wave; this de Broglie wave may be regarded as a wave packet, consisting of a group of waves. A number of frequencies mixed so that the resultant wave has a beginning and an end forms the group. Each of the component waves propagates with a definite velocity called wave velocity or phase velocity. Expression for Phase velocity: A wave can be represented by Y= A sin (ωt – kx) ---------- (1) Where k = ω/v = wave number (rad/m) ; ω = Angular frequency (rad/s) When a particle moves around a circle ν times/s, sweeps out 2πν rad/s In eqn.(1) the term (ωt – kx) gives the phase of the oscillating mass (ωt – kx) = constant for a periodic wave d (ωt – kx) /dt = 0 or ω – k(dx/dt) = 0 or dx/dt =ω/k Page 3 of 93  vp = ω/k When a wave packet or group consists of a number of component waves each traveling with slightly different velocity, the wave packet (group) travels with a velocity different from the velocities of component waves of the group; this velocity is called Group velocity. Expression for Group velocity: A wave group can be mathematically represented by the superposition of individual waves of different wavelengths. The interference between these individual waves results in the variation of amplitude that defines the shape of the group. If all the waves that constitute a group travel with the same velocity, the group will also travel with the same velocity. If however the wave velocity is dependent on the wavelength the group, velocity will be different from the velocity of the individual waves. The simplest wave group is one in which two continuous waves are superimposed. Let the two waves be represented by y1 = a cos (ω1t – k1x) and y2 = a cos (ω2t – k2x) The resultant y = y1 + y2 = a cos (ω1t – k1x) + a cos (ω2t – k2x)   -    k - k      2   k1  k 2   y  2a cos  1 2 t -  1 2  x cos  1 t -   x  2   2    2   2    1  2   k  k2  Let     and  1 k  2   2    -    k - k   y  2a cos  1 2 t -  1 2  x  cos( t - kx)  2   2   This equation represents a wave of angular frequency ω and wave number k whose amplitude is modulated by a wave of angular frequency (ω1 – ω2)/2 and wave number (k1 – k2)/2 and has a maximum value of 2a. The effect of this modulation is to produce a succession of wave groups as shown below: The velocity with which this envelope moves, i.e., the velocity of the maximum amplitude of the group is   2  given by vg  1  k1  k2 k If a group contains a number of frequency components in an infinitely small frequency interval (for Δk →0), then the above expression may be written as Page 4 of 93  d vg  This is the expression for group velocity dk Two or more waves of slightly different wavelengths alternately interfere and reinforce so that an infinite succession of groups of waves or wave packets are produced. The de Broglie wave group associated with a particle travels with a velocity equal to the particle velocity. Relation between group velocity (vg) and phase velocity(vp) We know that  d vp  ------ (1) and vg  ------ (2) k dk    k vp d dv  vg  (k v p )  k p  v p dk dk dv  dv  d   v g  v p  k p  v p  k  p   dk  d  dk   2     dv p  2  vp         2  d   dv  v g  v p -   p   d  Relation between the particle velocity of a matter wave and is its group velocity d vg  - - - - - - - - (1) dk E where   2  2    h   2  d    dE - - - - - - - -(2)  h  Also, we know that 2  p  2  k   2   p   h   h   2  dk    dp - - - - - - - - - (3)  h  d dE   - - - - - - - - - - - - - - - -(4) dk dp From eqn.(1) and (4) one can write dE vg  dp Exp ression for kinetic energy can be written as p2 E  2m p dE  dp m dE p   v p a rticle        (5) dp m comp aring eqns.(1), (4) & (5) we get v group  vparticle Page 5 of 93 Expression for de Broglie wavelength using group velocity We know that the expression for group velocity is given by d vg  - - - - - - - - (1) dk where   2  d  2d 2 1 k  dk  2d    d  vg  1 d   Also, we know that vgroup  v particle d  v particle       ( 2) 1 d   We can write T otal energy  potential energy  kinetic energy 1 i.e E  mv 2  V - - - - (3) also one can write E  h - - - - - (4) 2 Equating eqn.(3) & (4) 1 h  mv 2  V - - - - - - - -(5) 2 T aking derivative on both sides of eqn.(5) we get h d  mv dv (If we treat V  constant then dV  0)  mv   d    dv - - - - - - - - - (6)  h  Substituting eqn.(6) in eqn.(2) we get 1 m d    dv   h  T aking integration on both sides we get 1 mv   const.  h h   mv Page 6 of 93 Relation between phase velocity, particle velocity and velocity of light Since de Broglie wave is associated with a moving particle therefore, it is very much essential to know that if both the particle and wave associated with them travel with the same velocity or with different velocity. vp = ω/k = 2πυ/(2π/λ) = λν = (h/mv)(mc2/h) ∴ vp = c2/v As the velocity of material particle is always less than the velocity of light c, it means that the propagation velocity of de Broglie wave is always greater than c. Thus it seems that both the particle & de Broglie wave associated with the particle do not travel together with the same velocity & the wave would leave the particle behind. However, these difficulties can be ruled by considering that a moving material particle is equivalent to a wave packet rather than a single wave. Principle of complimentarity : The experiment of Davisson & Germer demonstrated the diffraction of electron beams. The wave nature of electrons can also be demonstrated by interference with a double slit. But it is an extremely difficult task to prepare a suitable double slit that can transmit an electron beam. But the experiment was done by Jönson in 1961. He passed a 50,000eV beam of electron through a double slit. The pattern obtained by him was very similar to the interference pattern obtained by Young with visible light. In an experiment of the above type it is rather tempting to try to find out through which slit an electron has passed. If we design a suitable device for detecting the passage of an electron through one of the slits, the interference pattern is found to vanish. If the electron is to behave like a classical particle, it has to pass through one of the two slits. On the other hand, if it is a wave, it can pass through both the slits! When we try observing the passage of electron through one of the slits, we are examining its particle aspect. However, when we observe the interference pattern we are investigating the wave aspect of electron. At a given moment and under given circumstances the electron will behave either as a particle or as a wave but not as both. In other words, the particle and wave nature of a physical entity cannot be observed simultaneously. Heisenberg’s Uncertainty principle. Physical quantities like position, momentum, time, energy etc. can be measured accurately in macroscopic systems (i.e. classical mechanics). However, in the case of microscopic systems, the measurement of physical quantities for particles like electrons, protons, neutrons, photons etc are not accurate. If the measurement of one is certain and that of other will be uncertain. Page 7 of 93  A wave packet that represents and symbolizes all about the particle and moves with a group velocity describes a de Broglie wave. According to Bohr’s probability interpretation, the particle may be found anywhere within the wave-packet. This implies that the position of the particle is uncertain within the limits of the wave packet. As the wave packet has a velocity spread, there is an uncertainty about the momentum of the particle. Thus according to uncertainty principle states that the position and the momentum of a particle in an atomic system cannot be determined simultaneously and accurately. If Δx is the uncertainty associated with the position of a particle and Δpx the uncertainty associated with its momentum, then the product of these uncertainties will always be equal or greater than h/4π. That is Δx Δpx ≥ h/4π Different forms of uncertainty principle ΔE Δt ≥ h/4π Δω Δθ ≥ h/4π Applications Heisenberg’s Uncertainty principle (Nonexistence of electron in the nucleus) The radius ‘r’ of the nucleus of any atom is of the order of 10-14m so that if an electron is confined in the nucleus, the uncertainty in its position will be of the order of 2r = ∆x (say) i.e diameter of the nucleus But according to HUP ∆x ∆p ≥ h/4π (∆p = uncertainty in momentum) ∆x ~ 2x10-14m Therefore, ∆p = h/(4π ∆x) = 6.625E-34 / (4π x 2x10-14) = 2.63 x 10-21 kg-m/s Taking ∆p ~ p we can calculate energy using the formula E2 = c2(p2 + mo2c2) = (3x108)2x [(2.63 x 10-21)2 + (9.1x10-31)2x (3x108)2] = 7.932x10-13J = 4957745 eV ~ 5 MeV However, the experimental investigations on beta decay reveal that the kinetic energies of electrons must be equal to 4MeV. Since there is a disagreement between theoretical and experimental energy values we can conclude that electrons cannot be found inside the nucleus. a) Wave function (ψ): Water waves ------------ height of water surfaces varies Light waves -------------- electric & magnetic fields vary Matter waves -------------- wave function (ψ) Ψ is related to the probability of finding the particle. Max Born put these ideas forward for the first time. Page 8 of 93  The wave function ψ indicates the state of the particle. However it has no direct physical significance. There is a simple reason why ψ cannot be interpreted in terms of an experiment. The probability that something be in a certain place at a given time must lie between 0 & 1 i.e. the object is definitely not there and the object is definitely there respectively.  An intermediate probability, say 0.2, means that there is a 20% chance of finding the object. However, the amplitude of a wave can be negative as well as positive and a negative probability -0.2 is meaningless. Hence ψ by itself cannot be an observable quantity.  Because of this the square of the absolute value of the wave function ψ is considered and is known as probability density denoted by | ψ |2  The probability of experimentally finding the body described by the wave function ψ at the point x, y, z at the time t is proportional to the value of | ψ |2. Small value of |ψ|2 ----- Less possibility of presence As long as |ψ|2 is not actually zero somewhere however, there is a definite chance, however small, of detecting it there. Max Born first made this interpretation in 1926. If we know the momentum of a particle, we can find the wavelength of the associated matter wave by using the equation λ = h /mν. We have now to realize how we can describe the amplitude of a matter wave. That is we have to find out just what is waving. A particle of mass ‘m’ traveling in the increasing x- direction with no force acting on it is called a free particle. According to Schrodinger the wave function ψ(x,t) for a free particle moving in the positive x direction is given by ψ(x,t) = ψo ei(kx – ωt), here ψo = amplitude and ψ(x,t) = complex b) Probability density : If ψ is a complex no. then its complex conjugate is obtained by replacing i by –i, ψ alone don’t have any meaning but only ψψ* gives the probability of finding the particle. In quantum mechanics we cannot assert where exactly a particle is. We cannot say where it is likely to be P (x) = ψψ* = [ψo ei(kx – ωt)] [ψo e-i(kx – ωt)] = |ψo|2  Large value of |ψ|2 ----- Strong possibility of presence of particle  Small value of |ψ|2 ----- Less possibility of presence of particle Page 9 of 93 c) Normalization of wave function: The probability of finding the particle between any two coordinates x1 & x2 is determined by summing the probabilities in each interval dx. Therefore there exists a particle between x1 & x2 in any interval dx. This situation can be mathematically represented by x2   ( x) 2 dx = 1 x1 If a particle exists anywhere in a region of space within a small volume element dv, then the normalized condition can be represented as   dV  1 2 - Time independent one dimensional Schrodinger wave equation : A wave eqn. for a debroglie wave is given by   A ei(kx - t) ------------- (1) Differentiating twice eqn.(1) with respect to t we get  - 2 A ei(kx - t)   2 t 2  2  - 2 -------------- (2) t 2 Displacement ‘y’ of a wave is given by y = A sin[ωt-kx]-------(A)  y 2 Differentiating twice w.r.t ‘t’ we get 2  - 2 y ----------(B) t 2y   2 2 1 Similarly differentiating twice w.r.t ‘x’ we get  - k 2 y    y     2 y ----(C) x 2 v v 2y  1  2y Comparing eqns (B) & (C) we get   ------ (3) x 2  v 2  t 2 By analogy eqn. for a traveling de Broglie wave is given by  2  1   2   ------ (4) x 2  v 2  t 2  2   2 Comparing eqns. (2) & (4)  -    ; where ω = 2πν and v = νλ x 2 v  2  4 2  1  1   2  -  2  or  -  2  2 --------(5) x 2   2  4   x For a particle of mas ‘m’ moving with a velocity ‘v’ Kinetic energy = ½ mv2 = m2v2/2m = p2/2m But p = h/λ Page 10 of 93 Therefore, KE = [1/ λ2] [h2/2m] Substituting for 1/ λ2 from eqn (5) we get  1   h 2   2  h 2   1   2 KE = -  2    2  -  2    2  4    2m  x  8 m     x Total energy is given by  h 2   1   2 E = PE + KE = V -  2    2  8 m     x  2  - 82 m   (E - V)   x 2  h  2  2   8 2m   (E - V)   0 x 2  h 2  Properties of wave function. Ψ should satisfy the law of conservation of energy i.e Total energy = PE + KE be consistent with de Broglie hypothesis i.e λ = h/p be single valued ( because probability is unique) be continuous be finite be linear so that de Broglie waves have the important superposition property Eigen value & Eigen function: A wave function Ψ, which satisfies all the properties is said to be Eigen function (Eigen = proper)  An operator O is a mathematical operator (differentiation, integration, addition, multiplication, division etc.) which may be applied on a function Ψ(x), which changes the function to another function Ф(x). This can be represented as  O (x)  (x)  If a function is Eigen function, then by result of operation with an operator O , we get the same function as  O (x)   (x) Ψ(x) = eigen function, λ = eigen value, Ô = operator and Ψ(x) = operand d2 d2 Eg.  (sin 2 x )  4(sin2x) , Here Ô =  ; λ = 4 ; ψ(x) = sin2x dx 2 dx 2 Page 11 of 93 Energy Eigen values and Eigen function for a particle trapped in a potential well of infinite height A particle moving freely in one-dimensional “box” of length ‘L’ trapped completely within the box is imagined to be as a particle in a potential well of infinite depth. Initial conditions V(x) = 0 ; 0 < x < L V(x) = ∞ ; x < 0, x > L If the walls of the box are perfectly rigid, the particle must always be in the box and the probability for finding it elsewhere must be zero. Thus outside the box we have Ψ (x) = 0 ; x < 0, x > L Schrodinger wave equation is  2  82 m  2 ( E  V )  0 --------------- (1) x 2 h Inside the well V = 0, thus equation (1) becomes  2  8 2 m  2 E  0 ---------------- (2) x 2 h 8 2 m Let 2 E  k2 ----------------- (3) h Substituting eqn.(3) in eqn.(2) we get  2  k2   0 ----------------- (4) x 2 The solution for above differential eqn. can be written as Ψ (x) = A sin kx + B coskx ---------------- (5) Let us solve equation (5) outside the boundaries Case I : For x ≤ 0, Ψ = 0 Therefore, Ψ(0) = A sin0 + B cos0 => B = 0 ------------------- (6) Page 12 of 93 Case II : For x ≥ L, Ψ = 0 Therefore, Ψ(L) = A sin kL => A sin kL = 0 => Either A = 0 or sin kL = 0 A ≠ 0 because Ψ is finite inside the box Therefore, sin kL = 0 => k L = n π => k = n π / L ------------------ (7) Where n = 1,2,3………… Thus the solution to the Schrodinger equation for a particle trapped in a linear region of length ‘L’ is a series of standing de Broglie waves. Only certain values of k are permitted and thus only certain values of E may occur. Thus the energy is quantized. Substituting eqn.(6) in eqn.(3) we get 8 2 m  n  2 E   or  L 2 h 2 2 n h En  ------------------- (8) where n = 1,2,3………. 2 8mL Equation (8) is the expression for energy Eigen values for a particle trapped in a potential well of infinite depth. However, the particle must be present somewhere inside the well, thus L  dx  1 2 0 Substituting eqn.(6) & eqn.(7) in eqn.(5) we get Ψ (x) = A sin[nπ / L]x  Lx dx  1 L 1 A 2 sin 2 n However, we know that sin 2   (1  cos 2) 0 2 Page 13 of 93  1 L    L 1  A   dx -  cos 2n x dx   1 2 L 2 0 20  L A 2  L  L   2n    x  0 -   sin  x   1 2   2n   L   0 A2   L    L-  sin 2nL  1 But sin 2nπ = 0 2   2n   Therefore, A2L 2 1 or A  2 L Hence, we can write the wave function as 2  n  ( x )  sin   x L L Energy Eigen values for a free particle A particle moving in any region of space without the influence of force is called as a free particle. We know that Schrodinger wave equation can be written as  2  82 m  2 (E  V)  0 --------------- (1) x 2 h Let us treat V = 0, thus equation (1) becomes  2  8 2 m  2 E  0 ---------------- (2) x 2 h 8 2 m Let 2 E  k2 ----------------- (3) h Substituting eqn.(3) in eqn.(2) we get  2  k2   0 ----------------- (4) x 2 The solution for above differential eqn. can be written as Ψ (x) = A sin kx + B cos kx ---------------- (5) Page 14 of 93 It is not possible to apply the boundary condition and solve the eqn. (5). Because Ψ is finite everywhere in the space. Hence energy eigen value for a free particle can be written as Therefore for a free particle, the energy Eigen values are not quantized and is equal to the kinetic energy of the particle itself. Page 15 of 93  UNIT 2 - Electrical & Thermal properties of materials Postulates of Classical Free Electron Theory (CFET) or Drude-Lorentz theory Electrical conductivity in Metals: Drude-Lorentz theory Large number of atoms combines to form a metal; the boundaries of the neighbouring atoms slightly overlap on each other. Due to such an overlapping, though the core electron remain unaffected, the valence electron find continuity from atom to atom and thus can move easily throughout the body of the metal. The free movement of electrons means that none of them belongs to any atom in particular, but each of them belongs to the metal to which they are confined to. Thus, each such electron is named a free electron. The electrons in the closed shells are called core electrons and those in the outer incomplete shell are called valence electrons. The core electrons are strongly attracted by comparatively immobile nucleus (+vely charged metallic ions). The valence electrons in the constituent atoms are free electrons. In 1900, Drude assumed that the electrons in a metal are free to move and form ‘electron gas’.Lorentz predicted that the kinetic theory of gases could be applied to the free electron gas. When the atoms are brought closer to form metal, the valence electrons get detached and move freely through the metal. Hence, they are called free or conduction electrons. The concentration of the free electrons is ~ 1028/m3. Assumptions of classical free electron theory: 1. A metal is imagined as a structure of 3-dimensional array of ions in between which, there are freely moving valence electrons confined to the body of the material. Such freely moving electrons cause electrical conduction under an applied field and hence referred to as conduction electrons. 2. The free electrons are treated as equivalent to gas molecules and thus they are assumed to obey the laws of kinetic theory of gases. In the absence of the field, the energy associated with each electron at a temp.T is given by (3/2) kT, where k is Boltzmann constant. It is related to the KE through the relation (3/2)kT = (1/2)m(vth)2, where vth = thermal velocity. 3. The electric potential due to the ionic cores is taken to be essentially constant throughout the body of the metal and the effect of repulsion between the electrons is considered insignificant. 4. The electric current in a metal due to an applied field is a consequence of the drift velocity in a direction opposite to the direction of the field. Page 16 of 93 What do you understand by Drift velocity? Drift velocity: The disconnection of a valence electron from the parent atom results in a virtual loss of one negative charge for that atom. Consequently, the electrical neutrality of the atom is lost and it becomes an ion. The structure formulation due to the array of such fixed ions in 3-dimensions is called lattice. “The nucleus of an atom together with the electrons in the inner shells is called the ionic core” A free e- while moving across the metal, knocks against the lattice corners. Its direction of motion will be continuously changing. The random motion of the free e- will be retained in the metal even after the application of an electric field. As the e-s have –ve charge, the net motion or drift of the e-s will be in a direction opposite to that of the applied electric field. The velocity of this overall motion of the e-s is called drift velocity. In the absence of an electric field, the free electrons in a metal will be moving at random in all directions and will be at thermal equilibrium. Then by the kinetic theory of gases ½ mv2 = 3/2kT The force acting on an electron under the application of field E will Ee. The resulting acceleration will be Ee/m. The drift velocity is small compared to the random velocity v. Further, the drift velocity is not retained after a collision with an atom because of the relatively large mass of the atom. Hence, just after a collision, the drift velocity is zero. If the mean free path is λ then the time that elapses before the next collision takes place is λ/v. Hence the drift velocity acquired just before the next collision takes place is Drift velocity = (acceleration)x(time constant) = (Ee/m)x(λ/v) Define the term and Mean free path, Mean collision time and Relaxation time. The average distance traversed by the free electron between two successive collisions with the lattice corners is called Mean free path (λ). The average time interval between two successive collisions of an electron with the lattice corners is called Mean collision time (τ). Relaxation time: In the absence of an external electric field, the free electrons in a metal will be moving at random in all directions. Hence, the average velocity vav in any particular direction will be zero. When an external electric field is applied, the electrons will have a net average velocity vnav in a direction opposite to the direction of the applied field. If the external field is turned off, the average velocity reduces exponentially from the value vnav to zero. Vav = 0 (in the absence of the field) Vav = Vnav (in the presence of the field due to drift velocity) If the field is turned off suddenly, the average velocity Vav reduces exponentially to zero from Vnav Page 17 of 93  vav  vnav e  (t  r ) ; where  r  Re laxation time , t  time counted when the field is turned off at t   r vav  1 vnav  e When the external electric field is removed then the time required for the average velocity of the conduction electrons in a metal to be reduced to (1/e) times its initial value at the time of removal of the field is called Relaxation time. Expression for electrical conductivity in metals: Under the influence of electric field E on a conductor, the electrons having charge e will get a force F = eE ----------- (1) If m is the mass of the electron, then from Newton’s II law of motion F = ma = m(dv/dt) ------------ (2) Comparing eqns. (1) & (2) eE = m(dv/dt) → dv = (Ee/m)dt By taking integration on both sides we get v = (Ee/m)t ------------ (3) Let t = τ = collision time (average) Since by definition, the collision time applies to an average value, the corresponding velocity in eqn.(3) also becomes the average velocity vav. Therefore, vav = (Ee/m) τ --------------(4) We know that Current density (J) α Applied field (E) I I A J=σE ; where σ = electrical conductivity Therefore, σ = J/E -----------(5) L But J = I/A current per unit cross sectional area σ AE = I ------------- (6) Conductor cross section Let a current carrying conductor of length L & area of cross section A is having ‘n’ number of e-s. Page 18 of 93 Then we can write total charge q = (nAL) e ; t = L/vav ; I = q/t Now the average velocity of electrons is given by vav = distance/time vav = distance for a unit time Therefore, volume = vav x area & current (I) = [(nAL)e]/[L/vav] = nA vav e ------------(7) Substituting the values of ‘I’ and ‘vav’ from eqns. (4) & (6) we get σ AE = nA[(Ee/m) τ]e σ = [ne2/m] τ ------------- (8) Explain the failures of classical free electron theory Molar specific heat at constant volume (CV) Specific heat capacity is the measure of heat energy required to increase the temperature of a substance by one degree Kelvin (when the unit quantity is mole then the term molar specific heat capacity is used). Molar specific heat of a gas is CV = (3/2)R but Specific heat of a metal by its conduction e- is CV = 10-4 RT This deviation in the results of Cv is not explained by the classical theory. Page 19 of 93 Temperature dependence of σ We know that 1 3 mv 2th  kT 2 2 3kT v th  m v th  T However, mean collision time τ is inversely proportional to vth. Therefore, 1 1    v th T ne 2 But   m     1   T 1 But experiment ally it is observed that   T Dependence of electrical conductivity on concentration of electrons We have σ = [ne2/m] τ ⟹ σ α n According to this dependence σ will be high for the metals having large electron concentration. But some of the metals having less electron concentration is found to have higher σ and vice versa. These experimental observations were not explained by classical free electron theory. Page 20 of 93 Write a short note on Fermi-Dirac statistics Fermi-Dirac statistics: Particles of half integral spin (1/2, 3/2……..) which obey Pauli’s exclusion principle such as electrons or nucleons are known as fermions and their distribution function is the Fermi-Dirac distribution. Fermi- Dirac distribution function is given by 1 f FD (E)  e (E -E F ) / kT 1 General observation is that all of the distribution functions fall to zero at large values of E When E >>kT, the occupation probability is very small fFD never becomes larger than one just as we expect for particles that obey the Pauli’s exclusion principle. At T = 0, all energy levels up to EF are occupied (fFD = 1.0 ) and all energy levels above EF are empty (fFD = 0). As T increases, some levels above EF are partially occupied (fFD > 0), while some levels below EF are partially empty (fFD < 1). At higher temperature, the “Spread out” of fFD becomes more. When E = EF and T ~ 0K fFD = 0.5. Thus an alternative definition of EF is “ the point at which the occupation probability is exactly 0.5 or 50%. EF is almost constant for most materials. For semi-conductors, the density of conduction electrons can change significantly with temperature and thus EF in these materials is temperature dependent. f(E) is the Fermi factor which gives the probability of occupation of a given energy state for a material in thermal equilibrium. Probability of occupation for E < EF at T = 0 1 1 1 f FD (E)  -   1 e 1  1  0 1    1 e  f FD (E)  1 for E  E F Energy levels are certainly occupied and E  E F applies to all the energy levels below E F Probability of occupation for E > EF at T = 0 1 1 1 f FD (E)     0 e 1  1  f FD (E)  0 for E  E F Energy levels are not occupied at E  E F When E = EF and T ~ 0K fFD = 0.5. Thus an alternative definition of EF is “ the point at which the occupation probability is exactly 0.5 or 50%. Page 21 of 93 Define Fermi energy (EF) and Fermi velocity (VF) Fermi Energy (EF): The energy of the electron in the highest occupied state is known as Fermi energy of the metal. Fermi velocity (VF): The energy of the electrons which are at the Fermi level is EF. The velocity of the electrons which occupy Fermi level is known as Fermi velocity EF = ½ mvF2 or vF = (2EF/m)1/2 Failures of classical free electron theory Molar specific heat at constant volume (CV) Specific heat capacity is the measure of heat energy required to increase the temperature of a substance by one degree Kelvin (when the unit quantity is mole then the term molar specific heat capacity is used). Molar specific heat of a gas is CV = (3/2)R but Specific heat of a metal by its conduction e- is CV = 10-4 RT This deviation in the results of Cv is not explained by the classical theory. Temperature dependence of σ We know that 1 3 mv 2th  kT 2 2 3kT v th  m v th  T However, mean collision time τ is inversely proportional to vth Therefore, 1 1    v th T ne 2 But   m     1   T 1 But experiment ally it is observed that   T Page 22 of 93 Dependence of electrical conductivity on concentration of electrons We have σ = [ne2/m] τ ⟹ σ α n According to this dependence σ will be high for the metals having large electron concentration. But some of the metals having less electron concentration is found to have higher σ and vice versa. These experimental observations were not explained by classical free electron theory. Assumptions of quantum free electron theory (QFET): Concept of free electron: The total energy of the electron is contributed by 1. Potential energy of the electron which depends on its distance from the proton and 2. Kinetic energy due to its motion round the proton. When an electron absorbs some energy given o it, it moves to a larger distance from proton and finds itself in a larger orbit, due to which there will be an increase in the total energy. All the energy levels except the ground level are known as excited energy levels or excited states. For instance, the electron can never be found in an orbit in which its energy is of any value between -1.51eV & -3.40eV. This kind of restriction on energy values is called the quantization of energy. However, it must be noted that the quantization of energy occurs because of the reason that the electron is under the influence of field due to proton. A free electron can of course, have continuous energy values. Assumptions of Quantum theory of free electron: Two assumptions of the classical free electron theory are retained in the quantum theory. They are 1. The electrons travel under a constant potential. However, their movement is confined to the boundaries of the metal. 2. The attraction between the e-s and the ions at the lattice corners as well as the repulsion between e-s are ignored. There are two more assumptions which are peculiar to the quantum theory. They are 3. The energy levels of the conduction electrons are quantized 4. The distribution of electrons among the various permitted energy levels is subject to Pauli’s exclusion principle Page 23 of 93 Merits of QFET  Only the electrons whose energies differ from EF by kT can absorb heat and that there are 2kT/ EF such electrons. These electrons are the one which are very close to the Fermi level. But free electron theory assumes that all the valence electrons in a metal can absorb thermal energy. For one kmol of a metal, there will be NA free electrons. The total energy of the electron is given by U = 3/2 NAkT When heat is supplied to the material, the free electrons also absorb part of that heat ∴ CV = dU/dT = 3/2 NAk = 3/2 R = 3/2 x 6.023x1026x1.38x10-23 = 12.5 kJ/mol/K This calculated value of molar specific heat at constant volume is found to be hundred times greater than experimentally predicted value.  In a high quality metal, when the atoms vibrate, the lattice is no longer ideal and presents an effective cross sectional area of πr2 for scattering, where r is the amplitude of vibration. The electron mean free path λ is inversely proportional to the scattering cross section λ = 1/ πr2 But the energy of the vibrating atom E~ r2 ~kT ∴ λ ~ 1/T but σ α λ ⟹ λ α 1/T or σ α 1/T  If we compare the cases of Cu & Al, the value of ‘n’ for Al is 2.13 times higher than that of Cu. But the value of the ratio (λ /vF) for Cu is 3.73 times higher than that of Al. Hence not only σ α λ but also σ α (λ /vF). Density of States The Fermi function does not by itself gives us the number of electrons, which have certain energy. It gives us only the probability of occupation of an energy state by a single electron. Since even at the highest energy the difference between neighboring energy levels is as small as 10-6 eV, we can say that in a macroscopically small energy interval dE, there are still many discrete energy levels. To know the actual number of electrons with a given energy, one must know the number of states in the system, which has the energy under consideration. Then by multiplying the number of states by the probability of occupation, we get the actual number of electrons. Page 24 of 93 Evaluation of density of states for the electrons in a 3-dimensional solid of unit volume We have the equation for the allowed energy for a particle in one-dimensional potential well is n2 h 2 E ------- (1) 8mL2 For 3-dimensional well (n2x  n2y  n2z )h 2 E ----------- (2) where n2 = n2x + n2y + n2z 8mL2 nz nz dn n n ny ny nx nx The number of available states within a sphere of radius ‘n’ is given by 1/8[4/3πn3]. The factor 1/8 accounts for the fact that only +ve integers are allowable & thus only one octant of the sphere is available. Again the no. of states within a sphere of radius (n+dn) is therefore 1/8[4/3π(n+dn)3]. Thus the no. of energy states having energy values between E & E+dE is Z(E)dE = 1/8[4/3π(n+dn)3] - 1/8[4/3πn3] = π/6 (3n2dn) = π/2 (n2dn) ------------ (3) From eqn.(1) we can write Page 25 of 93  1  2  2 1 8mL2 E  8mL  n2   n   E 2 h2  h2    Differentiating w.r.t E we get  2  8mL  1  8mL2  2n dn    dE  dn    dE  h2  2n  h 2      Eqn.(4) in eqn.(3) leads to  2   2 π 2  1   8mL   π   8mL  Z(E) dE  n     dE   n   dE 2  2n   h 2   4   h2      1  2  2  8mL2   π  8mL    12 Z(E) dE      E dE  4  h 2   h2      3  2  2  π  8mL  1 Z(E) dE     E 2 dE  4  h 2    If should be now remembered that the Pauli’s exclusion principle permits two electrons in each state, so that the no. of energy levels actually available are 3  2 2 1  2π  8mL  Z(E) dE      E 2 dE   4  h 2    The actual no. of electrons N(E) dE in a given energy interval dE will now be obtained by multiplying the no. of energy states in the interval by Fermi distribution function N(E) dE = Z(E) F(E) dE 3  π  8m  2 1 2 dE N(E) dE    E  2  2  h   1  exp[(E  E F ) / kT] At T=0K when E kBT 1 𝐸−𝐸𝑓 𝑓(𝐸) = 𝐸−𝐸𝑓 = exp [− ( 𝑘 )] ----------------(4) exp( ) 𝐵𝑇 𝑘𝑏 𝑇 Hence eqn(1) becomes 𝐸 +𝜑 4𝜋 3⁄ 1⁄ 𝐸−𝐸𝑓 𝑛 = ∫𝐸 𝐶 ( 3 ) (2𝑚𝑒∗ ) 2 (𝐸 − 𝐸𝐶 ) 2 exp [− ( )] 𝑑𝐸 𝐶 ℎ 𝑘𝐵 𝑇 Here E1/2 has been replaced by (E-Ec)1/2 since Ec is lower energy level in conduction band and Φ is the work function of the metal. The combination of Ec + Φ turns out to be huge and we can replace it by ∞. 4𝜋 3⁄ ∞ 1⁄ 𝐸−𝐸𝑓 𝑛 = (ℎ3 ) (2𝑚𝑒∗ ) 2 ∫𝐸 (𝐸 − 𝐸𝐶 ) 2 exp [− ( 𝑘 )] 𝑑𝐸----------------(5) 𝐶 𝐵𝑇 Evaluating the above integral by taking E = Ec +x and dE = dx and limit of the integration as 0 to ∞ (assuming the lowest energy almost to be equal to zero). ∞ 4𝜋 3 1 𝐸𝑓 − 𝐸𝑐 − 𝑥 𝑛 = ( 3 ) (2𝑚𝑒∗ ) ⁄2 ∫ (𝑥) ⁄2 exp ( ) 𝑑𝑥 ℎ 𝑘𝐵 𝑇 0 ∞ 4𝜋 3 1 𝐸𝑓 − 𝐸𝑐 −𝑥 𝑛 = ( 3 ) (2𝑚𝑒∗ ) ⁄2 ∫ (𝑥) ⁄2 exp ( ) exp ( ) 𝑑𝑥 ℎ 𝑘𝐵 𝑇 𝑘𝐵 𝑇 0 ∞ 4𝜋 3 𝐸𝑓 − 𝐸𝑐 1 −𝑥 𝑛 = ( 3 ) (2𝑚𝑒∗ ) ⁄2 exp ( ) ∫ (𝑥) ⁄2 exp ( ) 𝑑𝑥 ℎ 𝑘𝐵 𝑇 𝑘𝐵 𝑇 0 𝑥 𝑑𝑥 Let 𝑦 = 𝑡ℎ𝑒𝑛 𝑑𝑦 = 𝑘𝐵 𝑇 𝑘𝐵 𝑇 3⁄ 𝐸𝑓 −𝐸𝑐 3 1⁄ 4𝜋 ∞ 𝑛 = (ℎ3 ) (2𝑚𝑒∗ ) 2 exp ( ) (𝑘𝐵 𝑇)2 ∫0 (𝑦) 2 exp (−𝑦) 𝑑𝑦 ----------------(6) 𝑘𝐵 𝑇 1 ∞ 1⁄ 𝜋 ⁄2 Using gamma integral function we can write ∫0 (𝑦) 2 exp (−𝑦) 𝑑𝑦 = 2 Page 54 of 93  1 4𝜋 3⁄ 𝐸𝑓 − 𝐸𝑐 3 𝜋 ⁄2 ∗ 𝑛 = ( 3 ) (2𝑚𝑒 ) 2 exp ( ) (𝑘𝐵 𝑇)2 ( ) ℎ 𝑘𝐵 𝑇 2 𝟑⁄ 𝟐𝝅𝒎∗𝒆 𝒌𝑩 𝑻 𝟐 𝑬𝒇 −𝑬𝒄 𝑬𝒇 −𝑬𝒄 𝒏 = 𝟐( ) 𝐞𝐱𝐩 ( ) = 𝑵𝒄 𝐞𝐱𝐩 ( ) ----------------(7) 𝒉𝟐 𝒌𝑩 𝑻 𝒌𝑩 𝑻 𝟑⁄ 𝟐𝝅𝒎∗𝒆 𝒌𝑩 𝑻 𝟐 Where 𝑵𝒄 = 𝟐 ( ) and equation (7) represents concentration of electrons in conduction band 𝒉𝟐 Expression for concentration of holes in valence band: The concentration of holes in valence band can be calculated as follows 𝑛 = ∫ 𝑁(𝐸)[1 − 𝑓(𝐸)] 𝑑𝐸 ----------------(1) By substituting the values for N(E) & f(E) and solving the integral as we did for the concentration of electrons in conduction band we can arrive at the following equation 𝟑⁄ 𝟐𝝅𝒎∗𝒉 𝒌𝑩 𝑻 𝟐 𝑬𝒗 −𝑬𝒇 𝑬𝒗 −𝑬𝒇 𝒏 = 𝟐( ) 𝐞𝐱𝐩 ( ) = 𝑵𝒗 𝐞𝐱𝐩 ( ) ----------------(2) 𝒉𝟐 𝒌𝑩 𝑻 𝒌𝑩 𝑻 𝟑⁄ 𝟐𝝅𝒎∗𝒉 𝒌𝑩 𝑻 𝟐 Where 𝑵𝒗 = 𝟐 ( ) 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 ℎ𝑜𝑙𝑒𝑠 𝑖𝑛 𝑣𝑎𝑙𝑒𝑛𝑐𝑒 𝑏𝑎𝑛𝑑 𝒉𝟐 and equation (2) represents concentration of holes in valence band Page 55 of 93 Expression for intrinsic carrier concentration: In an intrinsic semiconductor, the concentration of holes and electrons are equal, i.e. n = p. The product of hole and electron concentrations for a given material is constant at a given temperature. If impurity is added to increase n, there will be corresponding decrease in p so that the product n x p remains constant and is sometimes called law of mass action. Therefore in an intrinsic semiconductor n x p = ni x pi = ni2 𝟑⁄ 𝟑⁄ 𝟐𝝅𝒎∗𝒆 𝒌𝑩 𝑻 𝟐 𝑬𝒇 − 𝑬𝒄 𝟐𝝅𝒎∗𝒉 𝒌𝑩 𝑻 𝟐 𝑬𝒗 − 𝑬𝒇 𝑛𝑖2 = [𝟐 ( ) 𝐞𝐱𝐩 ( )] [𝟐 ( ) 𝐞𝐱𝐩 ( )] 𝒉𝟐 𝒌𝑩 𝑻 𝒉𝟐 𝒌𝑩 𝑻 𝟐𝝅𝒌𝑩 𝑻 𝟑 𝟑 𝑬𝒄 − 𝑬𝒗 𝑛𝑖2 = 4 ( 𝟐 ) (𝒎∗𝒉 𝒎∗𝒆 ) ⁄𝟐 𝐞𝐱𝐩 [− ( )] 𝒉 𝒌𝑩 𝑻 Substituting (Ec - Ev) = Eg and the values of constants we get 𝟑 𝟐𝝅 𝒙 𝟏. 𝟑𝟖 𝒙 𝟏𝟎−𝟐𝟑 𝟑 𝑬𝒈 𝑛𝑖2 = 4( −𝟑𝟒 𝟐 ) (𝒎∗𝒉 𝒎∗𝒆 ) ⁄𝟐 𝑻𝟑 𝐞𝐱𝐩 [− ( )] (𝟔. 𝟔𝟑𝟒 𝒙 𝟏𝟎 ) 𝒌𝑩 𝑻 𝟑 𝟑⁄ 𝟐𝝅 𝒙 𝟏. 𝟑𝟖 𝒙 𝟏𝟎−𝟐𝟑 𝒙 𝟗. 𝟏 𝒙 𝟏𝟎−𝟑𝟏 𝒎∗𝒉 𝒎∗𝒆 𝟐 𝑬𝒈 𝑛𝑖2 = 4( ) ( ) 𝑻𝟑 𝐞𝐱𝐩 [− ( )] (𝟔. 𝟔𝟑𝟒 𝒙 𝟏𝟎−𝟑𝟒 )𝟐 𝒎𝟐 𝒌𝑩 𝑻 𝟑⁄ 𝒎∗𝒉 𝒎∗𝒆 𝟐 𝑬𝒈 𝑛𝑖2 = 2.322 𝑥 10 43 ( ) 𝑻𝟑 𝐞𝐱𝐩 [− ( )] 𝒎𝟐 𝒌𝑩 𝑻 𝟑⁄ 𝟑⁄ 𝟐𝝅 𝒎 𝒌𝑩 𝑻 𝟐 𝒎∗𝒉 𝒎∗𝒆 𝟒 𝑬𝒈 𝑛𝑖 = 2 ( ) ( ) 𝐞𝐱𝐩 [− ( )] 𝒉𝟐 𝒎𝟐 𝟐𝒌𝑩 𝑻 𝟑⁄ 𝑬𝒈 𝒏𝒊 = 𝑨𝒐 𝑻 𝟐 𝒆𝒙𝒑 [− (𝟐𝒌 𝑻)] 𝑩 1 ∗ ∗ 𝟑⁄𝟒 21 𝒎𝒉 𝒎𝒆 Where 𝐴𝑜 = 𝐴 ⁄2 = 4.819 𝑥 10 ( 𝟐 ) 𝒎 Page 56 of 93 Fermi level in an intrinsic semiconductor: In an intrinsic semiconductor, the concentrations of electrons and holes are equal. Therefore we can write 𝑬𝒇 − 𝑬𝒄 𝑬𝒗 − 𝑬𝒇 𝑵𝒄 𝐞𝐱𝐩 ( ) = 𝑵𝒗 𝐞𝐱𝐩 ( ) 𝒌𝑩 𝑻 𝒌𝑩 𝑻 𝑵𝒄 𝑬𝒇 − 𝑬𝒄 𝑬𝒗 − 𝑬𝒇 = 𝐞𝐱𝐩 [− ( )] 𝐞𝐱𝐩 ( ) 𝑵𝒗 𝒌𝑩 𝑻 𝒌𝑩 𝑻 𝑵𝒄 𝑬𝒗 + 𝑬𝒄 − 𝟐𝑬𝒇 = 𝐞𝐱𝐩 [( )] 𝑵𝒗 𝒌𝑩 𝑻 𝑵𝒄 𝑬𝒗 + 𝑬𝒄 − 𝟐𝑬𝒇 𝒍𝒏 [ ]= 𝑵𝒗 𝒌𝑩 𝑻 𝑵𝒄 𝑬𝒗 + 𝑬𝒄 − 𝟐𝑬𝒇 = 𝒌𝑩 𝑻 𝒍𝒏 [ ] 𝑵𝒗 𝑬𝒗 + 𝑬𝒄 𝒌𝑩 𝑻 𝑵𝒄 𝑬𝒇 = − 𝒍𝒏 [ ] 𝟐 𝟐 𝑵𝒗 If the effective masses of the holes and electrons are equal, then 𝑬𝒗 + 𝑬𝒄 𝑬𝒇 = 𝟐 Hence it is clear from the above equation that for an intrinsic semiconductor, the Fermi level lies at the middle of the energy gap, when the effective masses of the holes and electrons are equal. Page 57 of 93 Expression for conductivity of semiconductors: The conductivity of a metal can be written as 𝜎 = 𝑛𝑒𝜇 -------------------------- (1) Where n = concentration of electrons ; 𝜇 = mobility of electrons In a semiconductor, both electrons & holes are charge carriers. Therefor eqn (1) can be written as 𝜎 = 𝑛𝑒𝜇𝑒 + 𝑝𝑒𝜇ℎ -------------------------- (2) Where n & p= concentrations of electrons & holes ; 𝜇𝑒 & 𝜇ℎ = mobility of electrons & holes For an intrinsic semiconductor, the concentrations of electrons & holes are equal, hence eqn. (2) can be written as 𝜎 = 𝑛𝑖 𝑒(𝜇𝑒 + 𝜇ℎ ) -------------------------- (3) In a p-type semiconductor, since p >> n eqn. (2) can be written as 𝜎 = 𝑝𝑒𝜇ℎ -------------------------- (3) In an n-type semiconductor, since n >> p eqn. (2) can be written as 𝜎 = 𝑛𝑒𝜇𝑒 -------------------------- (4) Substituting the value for ni in eqn. (3) we get 3⁄ ∗ ∗ 3⁄ 2𝜋 𝑚 𝑘𝐵 𝑇 2 𝑚ℎ 𝑚𝑒 4 𝐸𝑔 𝜎 = 2( ) ( ) exp [− (2𝑘 𝑇)] 𝑒(𝜇𝑒 + 𝜇ℎ ) ----------------(5) ℎ2 𝑚2 𝐵 Eqn. (5) can be written as 𝐸𝑔 𝜎 = 𝐴 exp [− (2𝑘 𝑇)] ---------------- (6) 𝐵 3⁄ ∗ ∗ 3⁄ 2𝜋 𝑚 𝑘𝐵 𝑇 2 𝑚ℎ 𝑚𝑒 4 Where 𝐴 = 2 ( ) ( ) 𝑒(𝜇𝑒 + 𝜇ℎ ) ------ ℎ2 𝑚2 ---------- (7) Hence eqn. (6) can be rewritten as 1 𝐸𝑔 𝑙 = 𝐴 exp [− (2𝑘 𝑇)] = ---------------- (8) 𝜌 𝐵 𝑅𝑎 𝑙 𝐸𝑔 𝐸𝑔 𝑅 = 𝐴𝑎 exp [𝑘 𝑇] = 𝐶 exp [2𝑘 𝑇] ---------------- (9) 𝐵 𝐵 Here C = l/Aa, where a = area of cross section ; l = length of the specimen. Taking natural log on both sides of eqn. (9) we get ln R = ln C + Eg/2kBT ---------------- (10) Eqn.(10) is similar to equation of straight line y = mx + c, where y = ln R, x = 1/T, m = Eg/2kB and c = ln C. If a plot is drawn between ln R verses 1/T. The value of Eg can be determined from the slope of the straight line. Hence Eg = 2kB x slope. Page 58 of 93 Hall Effect : In a conductor, the flow of electric current is the movement of charges due to the presence of an electric field. If a magnetic field is applied in a direction perpendicular to the direction of motion of the charges, the moving charges accumulate such that opposite charges lie on opposite faces of the conductor. This distribution of charges produces a potential difference across the material that opposes the migration of further charge. This creates a steady electrical potential as long as the charges are flowing in the material and the magnetic field is on. This is the Hall effect. Consider a rectangular slab of a semiconductor material in which a current I is flowing in the positive x-direction. Let the semiconducting material be of n-type, which means that the charge carriers are electrons. Let a magnetic field B be applied along the z- direction as shown in the figure. Under the influence of the magnetic field, the electrons experience the Lorentz force FL given by FL = -Bev ------------ (1) Applying the Flemming’s left hand rule, we see that the force is exerted on the electrons in the negative y-direction. The electrons are therefore deflected downwards. As a result, the density of the electrons increases in the lower end of the material, due to which its bottom edge becomes negatively charged. On the other hand, the loss of electrons from the upper end causes the top edge of the metrical to become positively charged. Hence a potential VH, called the Hall voltage appears between the upper and lower surfaces of the semiconductor material which establishes an electric field EH called the Hall field across the conductor in the negative y-direction. The field EH, exerts an upward force FH on the electrons given by FH = -eEH ------------ (2) Now as the deflection of electrons continues in the downward direction due to the Lorentz force FL, it also contributes to the growth of Hall field. As a result , the force FH which acts on the electron in the upward direction also increases. These two opposing forces reach equilibrium at which stage, FL = FH Using eqns. (1) & (2), above equation becomes -Bev = -eEH or EH = Bv ------------ (3) If d is the distance between the upper and lower surfaces of the slab, then, EH = VH/d or VH = EH d = Bvd ------------ (4) Let w be the thickness of the material in the z-direction Its area of cross section normal to the direction of I is = wd. The current density J = I/wd ------------ (5) But, we know that J= nev = ρv ------------ (6) Therefore, ρv = I/wd or v = I/wdρ ------------ (7) Comparing equations (4) & (7) we get VH = BI/new Hall coefficient (RH) : Page 59 of 93 For a given semiconductor, the Hall field EH depends upon the current density J and the applied field B. EH α JB EH = RH JB, where RH is called the Hall Coefficient Therefore, RH = EH /JB = Bv/ nevB = 1/ne RH = 1/ne Note: 1. Positive value of RH indicates that the charge carriers are holes. 2. Negative value of RH indicates that the charge carriers are electrons. Application of Hall Effect 1. Hall Effect is used to find whether a semiconductor is N-type or P-type. 2. Hall Effect is used to find carrier concentration. 3. Hall Effect is used to calculate the mobility of charge carriers (free electrons and holes). 4. Hall Effect is used to measure conductivity. 5. Hall Effect is used to measure a.c. power and the strength of magnetic field. 6. Hall Effect is used in an instrument called Hall Effect multiplier which gives the output proportional to the product of two input signals. ***** End ***** Page 60 of 93  UNIT 4 - LASERS AND OPTICAL FIBERS Introduction on LASER  Light Amplification by Stimulated Emission of Radiation is LASER. A laser is a generator of highly monochromatic, coherent, intense light. The production of laser is the consequence of interaction of radiation with matter, which occurs under appropriate conditions leading to the transition of the system from one energy level to another.  The transition of an atom or molecule from one energy level to another occurs in a jump & is called quantum transitions. Quantum transition may be induced by the various causes. In particular, they can occur when the atoms interact with optical radiation.  The system of energy levels of atoms governs the behavior of electrons performing the transition in atoms.  Unlike other sources of light, the laser produces radiation with highly regular light field, outstanding in its coherence, monochromaticity and directivity. Interaction of radiation with matter Interactions of radiation with matter can take place through three possible ways. They are (1) Induced Absorption: Let the energy of the lower level be E1 and that of the upper level be E2. Assume that the atom is in lower level and a photon of energy hυ = E2- E1 travels nearer the atom. The atom can absorb this photon and rise from the level E1 to E2 thus making a transition upon the absorption of a quantum of light. This is called induced or stimulated absorption and can be represented as atom + photon = atom* E2 E2 E1 + hυ = E2 E1 E1 (2) Spontaneous Emission: When an atom is in the level E2 makes transition to E1 by spontaneously emitting a photon of energy E2 – E1 without any stimulus, then the process is called Spontaneous emission. In this case, atom* = atom + photon E2 hυ = E2 – E1 E1 Due to spontaneous emission, the photons are emitted in all possible directions. There is no phase relationship between the photons. Thus, the emitted light is incoherent. The number of spontaneous transitions taking place in a system depends only on the number of atoms N2 in the excited state E2. (3) Stimulated Emission: Page 61 of 93 The emission of photons by an atom under the influence of a passing photon of just right frequency due to which the atom makes a transition from a higher energy state to a lower energy state is called stimulated emission. A photon of energy E2 – E1 induces the excited atom to make downward transition releasing the energy in the form of photon. Thus, the interaction of a photon with an excited atom triggers the exited atom to drop to the lower energy state emitting a photon. The phenomenon of forced emission of photons is known as stimulated emission a photon emitted during stimulated emission has the same energy as the incident photon. It is emitted in the same direction and as the same phase as the incident photon thus the photons are coherent. If these two coherent photons then interact with two more excited state atoms, four coherent photons are produced and so on. Therefore the stimulated process leads to photo amplification. The process is represented by Atom* + Photon = Atom + 2 photons E2 hυ hυ E1 The stimulated transition depends on both the energy density of incident radiation and the number of atoms N2 available in the excited state E2 for de excitation. Conditions, requisites & properties of laser (i) Conditions for Laser Presence of metastable states: An atom can remain in the excited state for a limited time of about 10- 8 s. However, there exist such excited states in which the lifetime is greater than 10-8s. These states are called as metastable states. Population inversion: For lasing action to occur there should be more number of atoms in the higher state than the lower state. Over a period of time atomic density in the higher level decreases. The process of achieving back higher atomic density in the higher level is known as population inversion. (ii) Requisites of a Laser system Energy Source (Excitation Source) : Which will rise the system to an excited state. Active medium: Which when excited achieves population inversion. It may be a solid, Liquid or gas. Optical Cavity: Consisting of two mirrors facing each other. The active medium is enclosed in this cavity. One of the mirrors is 100% reflective and the other is partially transparent to let the some of the radiation to pass through. The optical cavity is made use of to make the stimulated emission possible in more number of atoms in the active medium. This naturally increases the intensity of the laser beam. Page 62 of 93 Einstein’s coefficients and expression for energy density of photons at a given frequency and temperature. Einstein’s coefficients are transition probabilities, which tell us about the extent of stimulated & spontaneous emission and induced absorption. Let A21 = Spontaneous emission transition probability / unit time E(υ) = Energy density of the atomic field E1 & E2 = Two energy levels occupied by N1 & N2 atoms B12 E(υ) = Absorption transition probability/unit time B21 E(υ) = stimulated emission transition probability/unit time The no. of atoms that fall from level E2 to E1 / unit time = [A21 + B21E(υ)]N2 The no. of atoms that raise from level E1 to E2 / unit time = B12E(υ)N1 Therefore net rate of change of atoms in the level E2 / unit time is [dN2/dt] = {B12E(υ)N1 - [A21 + B21E(υ)]N2} ------------------------ (1) Under equilibrium conditions, the net rate of change of atoms in any level must be zero. So [dN2/dt] = 0 ------------------------(2) Comparing eqns.(1) & (2) B12E(υ)N1 = [A21 + B21E(υ)]N2 N2/N1 = {B12E(υ) / [A21 + B21E(υ)]} ------------------------(3) According to Boltzmann, the atomic population at different energy levels at a given temperature T is given by N2/N1 = exp[-hυ/kBT] ------------------------(4) Comparing eqns. (3) & (4) B12E(υ) / [A21 + B21E(υ)] = exp[-hυ/kBT] [B12 exp(hυ/kBT) - B21]E(υ) = A21 ( where kB = 1.38x10-23 J/K ) E(υ) = (A21/ B12) / [exp(hυ/kBT) – (B21/ B12)] ------------------------(5) This is the expression for the energy density of the photons at a given frequency & temperature Planck’s formula for the energy density of radiation at a given temperature T is given by E(υ) = (8πhυ3/c3) / [exp(hυ/kBT) – 1] ------------------------(6) Comparing eqns. (5) & (6) A21/ B12 = 8πhυ3/c3 and B21/ B12 = 1 Note: (1) eqn. (5) can be written as [A21/ B12 E(υ)] = [exp(hυ/kBT) – 1] If hυ > kT then A21/ B12 E(υ) is +ve and large. Then spontaneous emission is much more probable than stimulated emission. (2) If hυ < kT then A21/ B12 E(υ) is +ve and small. Then stimulated emission may become Important (3) The ratio A21/ B12 α υ3. This shows that the probability of spontaneous emission increases rapidly with the energy difference between two states Page 63 of 93 Construction and working of He-Ne laser Helium – Neon gas laser: Construction: The active lasing species in this laser is It consists of fused quartz tube, the ends of which are sealed the neon atoms. Neon atoms are excited with a transparent material and are called windows. These by the transfer of energy from excited windows are inclined at an angle known as Brewster’s angle He atoms which in turn are excited by in order to get polarized laser out put. The tube contains two collisions with the electrons. Thus, electrodes at either side. A power supply (D.C) is connected population inversion involves a) Electron impact (of He) or across the electrodes. Two external plain mirrors are mounted collision of first kind on either side of the tube. One of them is fully silvered so that b) Excitation transfer (of Ne from it is 100% reflective but the other is partially transparent. A He) or collision f second kind mixture containing He at a pressure of 1mm of Hg and Ne at a pressure of 0.1mm of Hg and in the ratio 10:1 is filled in the tube. This mixture acts as an active medium. Adjustment for reflector Perfect reflector Partial reflector CH CH Mixture of He & Ne gas Excitatio n Source Working: When the DC discharge voltage is applied across the electrodes, fast moving electrons are produced. These electrons collide with more abundant helium atoms in the mixture. As a result of this collision, the helium atoms are excited from the ground level to the higher metastable states. This kind of collision is called collision of first kind and can be represented as He + e-  He* Since the excited energy levels of He atoms are nearly same as that of Ne atoms, the Ne atoms in the ground stage are able to receive the energy due to the collision with excited He atoms. This kind of collision is called collision of second kind, and can be represented as He*+Ne  Ne*+He As a result of collision of second kind, the energy levels 3S and 2S of Neon are well populated. Hence population inversion builds up between the energy levels 3S and 3P, 3S and 2P, 2S and 2P. The laser transition takes place continuously between the levels 3S and 3P, yielding a laser light of wavelength 3.39 micrometer, 3S and 2P transition yielding a laser light of wavelength 6328 Å. Transition between 2S and 2P yields a laser light of wavelength 1.15 micrometer, hence Helium Neon gas laser is a continuous laser. Out of many laser transitions which are possible in the active medium, a laser beam of desired wavelength can be trapped by adjusting the distance between the mirrors while the other radiations can be suppressed. On the other hand a methane chamber can be used to absorb IR radiations. Hence the laser cavity is called resonant cavity for that radiation. Page 64 of 93 Construction and working of semiconductor laser A semiconductor laser is a specially fabricated pn junction device (both the p and n regions are highly doped) which emits coherent light when it is forward biased. It is made from Gallium Arsenide (GaAs). They are of very small size (0.1 mm long), efficient, portable and operate at low power. These are widely used in Optical fiber communications, in CD players, CD-ROM Drives, optical reading, laser printing etc. p and n regions are made from same semiconductor material (GaAs). A p type region is formed on the n type by doping zinc atoms. The diode chip is about 500 micrometer long and 100 micrometer wide and thick. The top and bottom faces have metal contacts to pass the current. The front and rear faces are polished to constitute the resonator. Active medium: The active medium in GaAs is GaAS. But it is also commonly said that depletion region is the active medium in semiconductor laser. The thickness of the depletion layer is usually very small (0.1 μm). Pumping Source: Forward biasing is used as pumping source. The p-n junction is made forward biased that is p side is connected to positive terminal of the battery and n side to negative. Under the influence of forward biased electric field, conduction electrons will be injected from n side into junction area, while holes will enter will enter the junction from the p side. Thus, there will again be recombination of holes and electrons in depletion region and thus depletion region becomes thinner. If Efc is the quasi conduction Fermi level and if Efv is the quasi Fermi valence level then the condition for lasing action is Efc - Efv > Eg. Optical resonator system: The two faces of semiconductor which are perpendicular to junction plane make a resonant cavity. The top and bottom faces of diode, which are parallel to junction plane are metallised so as to make external connections. The front and back faces are roughned to suppress the oscillations in unwanted direction. Achievement of population inversion: When p-n junction diode is forward biased, then there will be injection of electrons into the conduction band along n-side and production of more holes in valence band along p-side of the junction. Thus, there will be more number of electrons in conduction band Page 65 of 93 Applications of laser  Measurement of Pollutants in Atmosphere: Atmospheric optics uses lasers for the remote probing of the atmosphere, including the measurement to traces of pollutant gases, temperature, water vapour concentration, sometimes at ranges greater than 8 to 15 km away. The pollutants in the atmosphere include carbon monoxide, nitrous oxide, sulphur dioxide, Freon, ethylene, vinyl chloride (the cancer causing agent from plastic industries) and other matter. The laser technique consists of a laser source, a transreceiver optical system, a signal processing electronic unit and retro reflector. The principle involves projecting a laser beam, through the atmosphere. The area where the pollutants are to be measured. The system employs a pulsed laser as a source of light energy. The measurement is based on the spectral absorption of the laser beam. Since different gases in atmosphere absorb laser energy at different wavelengths, the amount of absorption by each wavelength indicates the amount of pollutant in the atmosphere. Light that is back scattered by the congestion of matter is detected by a photo detector. The reflected laser beam undergoes attenuation due to spectral absorption by the pollutants in the atmosphere. The distance of the scattering matter is calculated from the time the laser pulse takes to go to the matter and return back to the system. The attenuated beam received at the photo detector and the beam energy is integrated and compared with reference laser energy source, the difference in the energy called error signal is analyzed and converted into a direct readout by the computer. The reading indicates the concentration and the distribution of pollutants in different vertical sections.  Laser welding The spot to be welded is focused by a laser beam. Due to the generation of heat, the material melts over a tiny area on which the laser beam is focused. The impurities in the material float up on the surface of the material and upon cooling the material becomes homogeneous solid structure which makes the stronger joint.  Laser Drilling The spot to be drilled is focused by a laser beam. Drilling is done with high power pulsed lasers. The light pulses will be of the order of 10-4s to 10-3s duration. The principle of laser drilling is mainly to heat the metal to its boiling point and vapourise it or remove it by high pressure vapour. When high power pulsed lasers are projected onto the metal spot, a vapour keyhole surrounded with molten metal is formed. With continued heating by laser beam, the metal vapourises and atoms ionize. The metal vapour then interacts with light beam and thereby the electrons get accelerated by electromagnetic radiation. This further heats the metal vapour, increasing the number of ions, forming plasma. Plasma absorbs laser light and emits blackbody radiation. Since the plasma interactions in the keyhole generate detonation waves, combined with the high power pressure, the molten metal gets ejected from the keyhole, thereby forming drilling operation.  Laser Cutting Lasers are found to be very effective in cutting different types of material. The material to be removed in laser cutting can be accomplished by melting and blowing out the molten metal. For blowing out the molten metal, a very high velocity gas jet of some inert gas is used. This process is known as Gas assisted laser beam machining. Page 66 of 93 Holography : The light waves reflected by an object are characterized by their amplitude and phase. In conventional photography, the photographic plate records only the intensity and not the phase. Hence, it loses the 3D nature of the object. An ordinary photograph represents a 2D recording of a 3D scene. This inherent limitation in conventional photography is eliminated in holography. In holography, the photographic plate records the intensity and phase distribution of the electromagnetic wave scattered from an object. The principle of holography involves two steps: (i) Recording of the image. (ii) Reconstruction of the image. (i) Recording of the image: During the recording process, the scattered waves from the object superimpose on the reference wave. These two waves interfere in the plane of the recording medium and produce interference pattern which are characteristic of the object. The recording medium records the intensity distribution in the interference pattern which has also recorded the phase of the electromagnetic wave scattered from the object. The recorded wave is called the hologram. The hologram has little resemblance to the object and it has in it a coded form of the object wave. (ii) Reconstruction of the image: In this process, the hologram is illuminated with a reference beam whose position is the same as that of the recording. This beam is called reconstruction beam. When the hologram is illuminated by the reconstruction beam, two waves are produced. One wave appears to diverge from the object and provides the virtual image of the object. The second wave converges to a second image which is real and thus can be recorded on a screen or photographed. Applications:  Used for character recognition i.e. to identify finger prints, postal addresses, etc.  Used in data storage devices. To produce gratings. Page 67 of 93 Principle of ray propagation in an optical fiber : Optical fibers are wires of either glass of plastics and are used in modern communication systems. The optical fiber consists of a central core, medium of higher refractive index n1, and an outer cladding of slightly lesser refractive index n2. The Optical fiber is protected from the external corrosive atmosphere and handling with an outer polyurethane jacket. Light propagates in a optical fiber by the phenomena of Total Internal reflection at the core cladding interface. If the angle of incidence at the core cladding interface is greater than the critical angle of incidence then the ray of light would be totally reflected back into the optical fiber’s core. The ray propagates through the fiber through reflections at the core cladding interfaces. The angle of incidence θo at the core end face has to be less than a critical value called as the angle of acceptance. The cylindrical symmetry of the optical fiber corresponds to a conical shape for the rays with an acceptance angle θo. Hence, it also called as the acceptance cone half angle. Condition for ray propagation in an optical fiber : n1> n2 ic n1= R I of core Өc Өa n2= R I of cladding A ray of light incident at an angle of incidence θo on the end face of the fiber undergoes refraction. The angle of refraction being θ1. Applying Snell’s law at the end face of the core we have n0 sin  0  n1 sin 1 n0 sin 1  Sin 0  (1) n1 The ray travels in the core and undergoes refraction at the core cladding interface. The angle of incidence at the interface is θ2 (equal to 90- θ1). If this angle is equal to the critical angle for Total internal refraction then the ray travels along the core cladding interface and the angle of refraction is 900. Applying Snell’s law at the core cladding interface n1 sin(90  1 )  n2 sin 90 Page 68 of 93  n2  cos 1  ----- ( 2) n1 From equations 1 and 2 we have 2 n0  n2  n12  n22 Sin  0  sin 1  (1  cos 1 )  2 1  n    n1  1  n1 n12  n22 sin  0   (3) n0 Thus a ray of light has to be incident at an angle less than or equal to the acceptance angle as given by equation 3 for the ray to propagate through the fiber. If the optical fiber is placed in air then sin  0  n12  n22 This is also referred to as the Numerical Aperture (NA) of the optical fiber and determines the maximum angle of incidence for a given combination of core and cladding refractive indices. Numerical Aperture  sin  0  n12  n22 The Numerical aperture of an optical fiber measures the ability of the fiber to gather light incident on it. It also is an index of the fiber’s capability to carry multiple modes of communication. Types of optical fibers and modes of propagation : Optical fibers are classified according to their ability to propagate an optical signal effectively through the fiber and the number of modes supported by the fiber. In practice three types of fibers are in use  Single mode step index fiber  Multi mode step index fiber  Graded Index multimode fibers. Single mode fiber In a single mode fiber only a single mode of propagation is allowed. In these types of fibers the diameter of the core is about 8μm to 10μm and a cladding diameter o