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FREE ELECTRON THEORY ARC TOPICS TO BE COVERED Classical free electron theory and its limitations Deduction of Ohm’s law, Statement of Weidemann-Franz law Quantum theory of free electrons, Fermi level Density of states, Fermi-Dirac distribution Thermionic emission, Richardson equation Chief Characteristics of Metals • Metal possesses high electrical and thermal conductivity • Metals obey Ohm’s law • Conductivity of metals decreases with rise of temperature • Metals obey Wiedemann-Franz law • Near absolute zero, certain metals exhibit superconductivity Classical Free Electron Theory • Initially stated by Drude in 1900 He was working prior to the development of quantum mechanics, so he began with a classical model. (1863-1906) CONCEPT • In Drude model, the valence electrons from each atom become detached and wander freely through the metal, while the metallic ions remain intact and play the role of the immobile positive particles. • Each electron behaves as a perfect gas molecule • Each electron is free to move through the entire volume of the metal • System of free electrons in a metal = free electron gas Free Electron Model Schematic model of metallic crystal, such as Na, Li, K, etc. Equilibrium positions of the atomic cores are positioned on the crystal lattice + + + + + and surrounded by a sea of conduction electrons. + + + + + + + + + + + + + + + + + + + + Drude’s Assumptions 1. Matter consists of light negatively charged electrons which are mobile & heavy, positively charged ions. 2. The only interactions are electron-ion collisions, which take place in a very short time t. • The neglect of the electron-electron interactions is THE INDEPENDENT ELECTRON APPROXIMATION. • The neglect of the electron-ion interactions is THE FREE ELECTRON APPROXIMATION 3. Electron-ion collisions are assumed to dominate. These will abruptly alter the electron velocity & maintain thermal equilibrium. 4. The mean time between collisions is 𝜏 5. Electrons emerge from each collision with their velocity changed. •Till the application of an external electric field, the electrons move about in a random manner Drift Velocity In presence of applied electric field, electrons move in a specific direction. This directional motion of the free electrons is called DRIFT. Average velocity gained during this drift motion is called DRIFT VELOCITY. Steady state drift velocity produced for unit electric field is called MOBILITY (μ) Fig. Ref. Google Relaxation Time (𝜏) When the applied electric field is switched off, the electrons again undergo collision. The electron gas resumes its equilibrium condition. Such a process which leads to the establishment of equilibrium in a system from which it was previously disturbed is called the relaxation process. The time taken for this process is RELAXATION TIME. < 𝑣𝑥 > = < 𝑣𝑥 >0 𝑒 −𝑡 τ Mean free path (λ) It is the average distance travelled by the conduction electron between successive collisions with the lattice ions. Mean collision time (𝜏𝑪 ) The average time taken by an electron between two successive collisions of an electron with lattice points during its motion. (averaging is done over a large number of collisions) Drift Velocity Expression An electric field is applied. The equation of motion of free electron of mass 𝑚𝑒 is Integrating, we get If is the average time between collisions then the average drift velocity is 𝒆𝑬𝝉 𝒗𝒅 = − 𝒎 10 Ohm’s Law • Basic law concerning the flow of electricity. • Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. • Constant of proportionality, resistance, is introduced • In mathematical terms, V = I x R where V is voltage, I is current, and R is resistance • When an electric field, E is applied to a conductor, an electric current begins to flow, and the current density by Ohm’s law is J E • Materials that obey Ohm’s law are said to be ohmic E Ohm’s Law Experimental observation: V I V RI L I A I L LJ A V J L E J If J = current density for electric field E, then J E , where σ = conductivity Amount of charge passing per unit time = -𝑛𝑒𝑣𝑑 𝐴 = I So, current density 𝐼 −𝑛𝑒𝑣𝑑 𝐴 J= = = −𝑛𝑒𝑣𝑑 ……… (1) 𝐴 Distance covered =𝑙 = 𝑣𝑑 𝑡 𝐴 We know, 𝒆𝑬𝝉 𝒗𝒅 = − 𝒎 ………… (2) S𝑜, 𝑣𝑜𝑙𝑢𝑚𝑒 𝑐𝑜𝑣𝑒𝑟𝑒𝑑 𝑖𝑛 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒 = 𝑣𝑑 𝐴 From (1) and (2), 𝑛𝑒 2 𝞽 𝐸 𝐽= 𝑚 σ This is the form of Ohm’s Law in terms of free electrons. Wiedemann-Franz Law Wiedemann and Franz law states that the ratio of thermal and electrical conductivity of all metals is constant at a given temperature (for room temperature and above). Thermal conductivity Electrical Conductivity 𝐾 σ = constant Later it was found by L. Lorenz that this constant is proportional to the absolute temperature 𝐾 3 𝑘𝐵2 = 𝑇 2 σ 2 𝑒 L = Lorentz Number Drawbacks of Classical Free Electron Theory • Specific Heat: Classical free electron theory all valence electrons in a metal can absorb thermal energy. So, molar electronic specific heat 1.5 times R, where R = universal gas constant. This is about 100 times greater than experimentally predicted values. • Mean free path: Experimental value of λ is much greater than the theoretical value • Temperature (T) Dependence of Electrical Conductivity(σ) Classical free electron theory σ is inversely proportional to 𝑇 experiments σ is inversely proportional to 𝑇 Drawbacks (continued..) • Wiedemann- Franz law: At low temperatures, K/σT is not a constant. But in classical free electron theory, it is a constant at all temperatures. • Paramagnetism of Metals: Theoretical value of paramagnetic susceptibility is greater than the experimental value. Experimental fact that paramagnetism of metals is nearly independent of temperature could not be explained Salient features of Quantum Free Electron Theory Proposed by Sommerfeld in 1928 • Electrons obey the laws of quantum mechanics • Energy levels of electrons are quantized • Electrons occupy energy orbitals according to Pauli’s exclusion principle • Distribution of electrons in different energy levels are according to FermiDirac statistics • Retained concept of free electrons moving in a uniform potential but prevented them from escaping the crystal by very high potential barriers at the surfaces FERMI LEVEL : Highest energy level occupied by electrons at Absolute zero. All the energy states upto Fermi level are OCCUPIED and all energy levels above Fermi level are VACANT. FERMI ENERGY: Energy corresponding to Fermi Level. Constant for a particular system Probability of an electron occupying a particular energy level ‘E’ is given by Fermi-Dirac Distribution f (E) 1 e ( E E F ) / k BT 1 On increasing the temperature, electrons get excited to higher energy level. Distribution of electrons in different energy levels gets determined by Fermi-Dirac Distribution function. At T = 0 K and for E < E F , f(E) = 1 for E > EF , f(E) = 0 For lower energies, f tends to 1. For higher energies, f tends to 0. Fermi Distribution Function at Different Temperatures For temperatures greater than zero, Fermi function plot begins to fall close to E F 𝟏 and at E = EF , f(E) = 𝟐 FERMI VELOCITY = velocity associated with Fermi Energy i.e. velocity of electrons occupying Fermi Level FOR SODIUM, 𝐸𝐹 = 3.2 𝑒𝑉 1 𝑚𝑣𝐹2 2 𝑣𝐹 = = 𝐸𝐹 = 3.2 X 1.6 X 10−19 J 2𝐸𝐹 𝑀 So, 𝑣𝐹 = 1.1 𝑋 106 𝑚/𝑠 FERMI VELOCITY FERMI TEMPERATURE = Temperature associated with Fermi energy 𝑘𝐵 𝑇𝐹 = 𝐸𝐹 So, 𝑇𝐹 = 37100 𝐾 Ref: Eisberg, R. and Resnick, R. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. New York: Wiley, 1985. FERMI TEMPERATURE DENSITY OF ENERGY STATES • In a macroscopically small energy interval, there are many discrete energy levels. • Difference between neighbouring energy levels is as small as 10−6 eV CONCEPT OF DENSITY OF ENERGY STATES simplifies OUR CALCULATION. It helps in finding the number of energy states with a specific energy. Density of states (DOS) of a system describes the number of available states in a unit volume per unit energy range. In a system, if N(E) = number of electrons with energy E, g(E) = number of energy states with energy E, f(E) = probability of an electron to occupy energy state E, then N(E) dE = g(E)dE f(E) We consider a free electron of mass ‘m’ trapped inside a cubical metal block of side length, ‘a’. According to quantum mechanics, energy of the free electron, 𝐸= 𝑛2 ℎ 2 ----- (1) where h = Planck’s constant 2 8𝑚𝑎 where 𝑛2 = 𝑛𝑥2 + 𝑛𝑦2 + 𝑛𝑧2 𝑛𝑥 , 𝑛𝑦 , 𝑛𝑧 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑧𝑒𝑟𝑜 Let us consider a space of points represented by coordinate system 𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 along the three mutually perpendicular directions. Let each point with integer values of the coordinates represent an energy state. Let n be the radius vector from origin (0,0,0) to a point represented by (𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 ). So, 𝑛2 = 𝑛𝑥2 + 𝑛𝑦2 + 𝑛𝑧2 -------- (2) All points on the surface of the sphere of radius ‘n’ will have the same energy. As per quantum condition, values of 𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 are restricted to be positive. Only in one octant of the sphere, each point corresponds to only positive values of 𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 . SPACE OF POINTS If n = radius of sphere whose octant encloses all the points upto an energy ‘E’, then Number of allowed energy values upto an energy E = number of points in the octant of sphere of radius ‘n’ We consider another sphere of radius n+dn whose octant encloses all points upto an energy ‘E+dE’, then Number of allowed energy values upto an energy E + dE = number of points in the octant of sphere of radius ‘n + dn’ So, number of allowed energy states in energy range dE = number of points in the space between two octant shells of radii n and n+dn = (volume of space between two octant shells of radii n and n+dn) X (number of points / unit volume) 1 = 4 π 𝑛2 𝑑𝑛 𝑋 1 8 1 2 = π 𝑛2 𝑑𝑛 ---------------- (3) Since 𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 are all integers, A unit volume of plot consists of Just one point If g(E) = number of energy states per unit energy range, then number of energy states in the energy interval dE = g(E)dE π So, g(E) dE = 2𝐸 8𝑚𝑎 From (1), 𝑛2 = ℎ2 𝑛= 2 𝑛 𝑛 𝑑𝑛 ------------ (4) ---------- (5) 8𝑚𝑎2 𝐸 ℎ2 1 2 ---------- (6) Differentiating (5) , we get 2 𝑛 𝑑𝑛 = So, 𝑛 𝑑𝑛 = 8𝑚𝑎2 𝑑𝐸 ℎ2 2 1 8𝑚𝑎 2 ℎ2 dE ----------- (7) Using (4), (6) and (7), we get π 𝑔 𝐸 𝑑𝐸 = 4 3 8𝑚𝑎2 2 ℎ2 1 𝐸2 𝑑𝐸 ---------- (8) Each energy value is applicable to two energy states, one for an electron with spin-up, and the other for an electron with spin down (Pauli’s exclusion principle). So, the number of allowed energy states in the energy interval dE 𝑔 𝐸 𝑑𝐸 = = π 2𝑋 4 3 8𝑚𝑎2 2 ℎ2 3 8𝑚𝑎2 2 π 2 ℎ2 1 2 𝐸 𝑑𝐸 1 2 𝐸 𝑑𝐸 Hence, the number of energy states present in unit volume having energy values lying between E and E + dE (DOS) is given by = π 8𝑚 2 ℎ2 3 2 1 2 𝐸 𝑑𝐸 [𝑎3 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑚𝑒𝑡𝑎𝑙 𝑏𝑙𝑜𝑐𝑘 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 𝑎𝑟𝑒 𝑝𝑟𝑒𝑠𝑒𝑛𝑡] Density of energy states for a free electron gas General Expression for 𝑁 𝐸 𝑑𝐸 = = π 2 3 8𝑚𝑎2 2 ℎ2 1 2 𝐸 𝑑𝐸 𝑓(𝐸) TASK: Find out the number of electrons present per unit volume of a cubical metal block at absolute zero temperature Thermionic Emission The emission of electrons from a metal under the effect of thermal energy is called THERMIONIC EMISSION. Emitted electrons are called THERMIONS. Electrons are free to move inside the metal Electrons cannot come out of the metal surface on its own as high potential barrier is present at the surface but when the temperature of the metal is sufficiently high, electrons gain sufficient energy to overcome the barrier and ESCAPE from the metal surface Free electron theory assumes that the potential within the metal is constant. The minimum energy to be supplied to the electron for its emission from the metal is termed as WORK FUNCTION (Ф) of the metal Richardson’s Equation If W = minimum energy of the electron for its emission from the surface, E F= Fermi energy of the metal, then, Ф = W – EF = work function of the metal. No. of energy states / unit volume in energy range E to E + dE, π 8𝑚 D(E)dE= 2 ℎ2 3 2 1 2 𝐸 𝑑𝐸 We know, 𝐸 = 𝑝2 2𝑚, So, density of energy states per unit volume in momentum range p to p+dp, 8π𝑝2 𝑑𝑝 2 2 𝐷 𝑝 𝑑𝑝 = = 4π𝑝 𝑑𝑝 ℎ3 ℎ3 ----------- (1) We construct a plot in ‘momentum space’ such that each point represents a particular combination of momenta components 𝑝𝑥 , 𝑝𝑦, 𝑝𝑧 of an electron along x-, y- and z-directions. So, 𝑝2 = 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 Volume element in momentum space, 𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑍 = 4π𝑝2 𝑑𝑝 -------- (2) From (1) and (2), the density of states in momentum space, 𝐷(𝑝𝑥 𝑝𝑦 𝑝𝑧 )𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 = 2 𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 ℎ3 --------- (3) Hence, no. of electrons/unit volume having momenta in the range 𝑝𝑥 and 𝑝𝑥 + 𝑑𝑝𝑥 , 𝑝𝑦 and 𝑝𝑦 + 𝑑𝑝𝑦 , 𝑝𝑧 and 𝑝𝑧 + 𝑑𝑝𝑍 , 𝑛(𝑝𝑥 𝑝𝑦 𝑝𝑧 )𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 = ℎ23 𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 −−−−− − 𝐸 − 𝐸𝐹 𝑒 𝑘𝑇 +1 (4) Now, we consider the metal plate to be in Y-Z plane. Electrons will be emitted in a direction perpendicular to Y-Z plane i.e. along x-axis. Only those electrons will be emitted whose energy, E > W. 𝐸 − 𝐸𝐹 ≫ kT. So, 1 in denominator can be neglected. We know, 𝐸 = 𝑝2 2𝑚 2 𝐸𝐹 𝑛 𝑥 = 3 𝑒 𝑘𝑇 ℎ 2 𝐸𝐹 𝑛 𝑥 = 3 𝑒 𝑘𝑇 ℎ ∞ ∞ ∞ 𝑒 −(𝑝 𝑝𝑥0 ∞ 2 2𝑚𝑘𝑇 2 2 )𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 −∞ −∞ ∞ ∞ 2 𝑒 −(𝑝𝑥 + 𝑝𝑦 + 𝑝𝑧 ) 𝑝𝑥0 −∞ −∞ 2𝑚𝑘𝑇 )𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 2 𝐸𝐹 𝑛 𝑥 = 3 𝑒 𝑘𝑇 ℎ ∞ ∞ ∞ 2 𝑒 −𝑝𝑥 2𝑚𝑘𝑇 𝑑𝑝𝑥 𝑝𝑥0 𝑒 2 + 𝑝2 ) 2𝑚𝑘𝑇 −(𝑝𝑦 𝑧 𝑑𝑝𝑦 𝑑𝑝𝑧 −∞ −∞ Standard Integral Form ∞ 2 𝑒 −𝛼𝑥 𝑑𝑥 = −∞ 4𝜋𝑚𝑘𝑇 𝐸𝐹 𝑛 𝑥 = 𝑒 𝑘𝑇 3 ℎ ∞ 𝑒 −𝑝𝑥2 2𝑚𝑘𝑇 𝑑𝑝𝑥 𝑝𝑥0 So, current density, 𝑝𝑥 𝐽 = 𝑛 𝑥 𝑒𝑣𝑥 = 𝑛 𝑥 𝑒 𝑚 = = 𝑒 4𝜋𝑚𝑘𝑇 𝐸 /𝑘𝑇 ∞ −𝑝2 2𝑚𝑘𝑇 𝑥 𝑒 𝐹 𝑒 𝑝𝑥 𝑑𝑝𝑥 3 𝑝 𝑚 ℎ 𝑥0 𝑒 4𝜋𝑚𝑘𝑇 𝐸 /𝑘𝑇 −𝑝2 2𝑚𝑘𝑇 𝐹 𝑥 𝑒 [𝑒 (2𝑝𝑥 𝑚 ℎ3 2𝑚𝑘𝑇) 𝑝𝑥 ] ∞ 𝑝𝑥0 𝜋 𝛼 J = 𝑒 4𝜋𝑚𝑘𝑇 𝐸 /𝑘𝑇 −𝑝2 2𝑚𝑘𝑇 𝐹 𝑒 . 𝑒 𝑥0 (𝑚𝑘𝑇) 3 𝑚 ℎ J= 4𝜋𝑚𝑘 2 𝑇 2 𝐸 /𝑘𝑇 −𝑊 𝑘𝑇 𝑒 𝑒 𝐹 .𝑒 3 ℎ 2 [since 𝑊 = 𝑝𝑥0 2𝑚] J= 4𝜋𝑚𝑘 2 𝑇 2 𝐸 /𝑘𝑇 −(𝐸 + Ф)/𝑘𝑇 𝐹 𝐹 𝑒 𝑒 . 𝑒 ℎ3 [since 𝑊 = 𝐸𝐹 + Ф] J= 4𝜋𝑚𝑒𝑘 2 ℎ3 𝑇 2 𝑒 − Ф/𝑘𝑇 𝐽 = 𝐴 𝑇 2 𝑒 −Ф 𝑘𝑇