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1 Drude-Lorenz Free Electron Theory Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/CMP2013 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2 Drude-Lorentz Model: 1900-1904 Paul Drude (1863-1906) Hendrik Lorentz (1853-1928) Ashcroft-Mermin Chapter 1: - Based on the discovery of electrons by J. J. Thomson (1897) - Electrons as particles (Newton’s equation) - Electron “gas” (Maxwell-Boltzmann statistics) - Worked well - Explained Wiedemann-Franz law (1853) k/sT ~ 2-3 x 10-8 (W-ohm/K2) (by double mistakes) - Could not account for: - T-dependence of k alone - T-dependence of s alone - Electronic specific heat Ce (too big) - Magnetic susceptibility cm (too big) P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 1. 2. Drude’s Assumptions: 3 Drude assumed that the compensating positive charge was attached to much heavier particles, so it is immobile. In Drude model, when atoms of a metallic element are brought together to form a metal, 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. Matter consists of light negatively charged electrons which are mobile, & heavy, static, positively charged ions. 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. Neglect of the electron-ion interactions is the FREE ELECTRON APPROXIMATION. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 3. Electron-ion collisions are assumed to dominate, as these will abruptly alter the electron velocity & maintain thermal equilibrium. 4. The probability of an electron suffering a collision in a short time dt is dt/t, where 1/t the scattering rate. Electrons emerge from each collision with both the direction & magnitude of their velocity changed; the magnitude is changed due to the local temperature at the collision point. 1/t is often an adjustable parameter. Ion Mean time between collisions is t. Trajectory of mobile electron P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory • • In a single isolated atom of the metallic element has a nucleus of charge e Za as shown in Figure below. Figure represents Arrangement of atoms in a metal • where Za - is the atomic number and • e - is the magnitude of the electronic charge • [e = 1.6 X 10-19 coulomb] surrounding the nucleus, there are Za electrons of the total charge –eZa. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 5 Some of these electrons ‘Z’, are the relatively weakly bound valence electrons. The remaining (Za-Z) electrons are relatively tightly bound to the nucleus and are known as the core electrons. These isolated atoms condense to form the metallic ion, and the valence electrons are allowed to wander far away from their parent atoms. They are called `conduction electron gas’ or `conduction electron cloud’. Due to kinetic theory of gas Drude assumed, conduction electrons of mass ‘m’ move against a background of heavy immobile ions. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory The density of the electron gas is calculated as follows. A metallic element contains 6.023X1023 atoms per mole (Avogadro’s number) and ρm/A moles per m3 Here ρm is the mass density (in kg per cubic metre) and ‘A’ is the atomic mass of the element. Each atom contributes ‘Z’ electrons, the number of m Z electrons per cubic metre. N . n V A The conduction electron densities are of the order of 1028 conduction electrons for cubic metre, varying from 0.91X1028 for cesium upto 24.7X1028 for beryllium. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory These densities are typically a thousand times greater than those of a classical gas at normal temperature and pressures. Due to strong electron-electron and electron-ion electromagnetic interactions, the Drude model boldly treats the dense metallic electron gas by the methods of the kinetic theory of a neutral dilute gas. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory BASIC ASSUMPTION FOR KINETIC THEORY OF A NEUTRAL DILUTE GAS 1. In the absence of an externally applied electromagnetic fields, each electron is taken to move freely here and there and it collides with other free electrons or positive ion cores. This collision is known as elastic collision. 2. The neglect of electron–electron interaction between collisions is known as the “independent electron approximation”. 3. In the presence of externally applied electromagnetic fields, the electrons acquire some amount of energy from the field and are directed to move towards higher potential. As a result, the electrons acquire a constant velocity known as drift velocity. 4. In Drude model, due to kinetic theory of collision, that abruptly alter the velocity of an electron. Drude attributed the electrons bouncing off the impenetrable ion cores. 5. Let us assume an electron experiences a collision with a probability per unit time 1/τ . That means the probability of an electron undergoing collision in any infinitesimal time interval of length ds is just ds/τ. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory The time ‘t’ is known as the relaxation time and it is defined as the time taken by an electron between two successive collisions. That relaxation time is also called mean free time [or] collision time. Electrons are assumed to achieve thermal equilibrium with their surroundings only through collision. These collisions are assumed to maintain local thermodynamic equilibrium in a particularly simple way. Trajectory of a conduction electron P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Success of classical free electron theory It is used to verify ohm’s law. It is used to explain the electrical and thermal conductivities of metals. It is used to explain the optical properties of metals. Ductility and malleability of metals can be explained by this model. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drawbacks of classical free electron theory From the classical free electron theory the value of specific heat of metals is given by 4.5R, where ‘R’ is called the universal gas constant. But the experimental value of specific heat is nearly equal to 3R. With help of this model we can’t explain the electrical conductivity of semiconductors or insulators. The theoretical value of paramagnetic susceptibility is greater than the experimental value. Ferromagnetism cannot be explained by this theory. At low temperature, the electrical conductivity and the thermal conductivity vary in different ways. Therefore K/σT is not a constant. But in classical free electron theory, it is a constant in all temperature. The photoelectric effect, Compton effect and the black body radiation cannot be explained by the classical free electron theory. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude’s classical theory Theory by Paul Drude in 1900, only three years after the electron was discovered. Drude treated the (free) electrons as a classical ideal gas but the electrons should collide with the stationary ions, not with each other. average rms speed so at room temp. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude’s classical theory relaxation time scattering probability per unit time mean free path P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory ...this must surely be wrong.... The electrons should strongly interact with each other. Why don’t they? The electrons should strongly interact with the lattice ions. Why don’t they? Using classical statistics for the electrons cannot be right. This is easy to see: de Broglie wavelength of an electron: for RT condition for using classical statistics is some Å P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: electrical conductivity we apply an electric field. The equation of motion is integration gives and if is the average time between collisions then the average drift speed is for we get remember: P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: electrical conductivity number of electrons passing in unit time current of negatively charged electrons current density Ohm’s law and with we get P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: electrical conductivity Ohm’s law and we can define the conductivity and the resistivity and the mobility P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Validity of Ohm’s law Valid for metals. also valid for homogeneous semiconductors not valid for inhomogeneous semiconductors P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: electrical conductivity Drude’s theory gives a reasonable picture for the phenomenon of resistance. Drude’s theory gives qualitatively Ohm’s law (linear relation between electric field and current density). It also gives reasonable quantitative values for resistivity, at least at room temperature. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory The Wiedemann-Franz law k constant s Wiedemann and Franz found in 1853 that the ratio of thermal and electrical conductivity for ALL METLALS is constant at a given temperature (for room temperature and above). Later it was found by L. Lorenz that this constant is proportional to the temperature. Let’s try to reproduce the linear behaviour and to calculate L here. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory The Wiedemann Franz law Estimated thermal conductivity (from a classical ideal gas) the actual quantum mechanical result is P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Many open questions: Why does the Drude model work so relatively well when many of its assumptions seem so wrong? In particular, the electrons don’t seem to be scattered by each other. Why? How do the electrons sneak by the atoms of the lattice? Why do the electrons not seem to contribute to the heat capacity? Why is the resistance of an disordered alloy so high? 23 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 24 Density of conduction electrons in metals ~1022 – 1023 cm-3 rs – measure of electronic density rs is radius of a sphere whose volume is equal to the volume per 1/ 3 3 3 1 4rs electron V 1 rs ~ 3 N n n1/ 3 4n mean inter-electron spacing in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm) rs/a0 ~ 2 – 6 2 a0 0.529 Å – Bohr radius me 2 ● Electron densities are thousands times greater than those of a gas at normal conditions ● There are strong electron-electron and electron-ion electromagnetic interactions In spite of this the Drude theory treats the electron gas by the methods of the kinetic theory of a neutral dilute gas P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 25 t m ne2 At room temperatures resistivities of metals are typically of the order of microohm centimeters (mohm-cm) and t is typically 10-14 – 10-15 s Mean free path l=v0t v0 – the average electron speed l measures the average distance an electron travels between collisions 2 estimate for v0 at Drude’s time 1 2 mv0 3 2 k BT → v0~107 cm/s → l ~ 1 – 10 Å consistent with Drude’s view that collisions are due to electron bumping into ions At low temperatures very long mean free path can be achieved l > 1 cm ~ 108 interatomic spacings! the electrons do not simply bump off the ions! The Drude model can be applied where a precise understanding of the scattering mechanism is not required Particular cases: electric conductivity in spatially uniform static magnetic field and in spatially uniform time-dependent electric field Very disordered metals and semiconductors P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory The Drude Model - Results Main Results 1) DC Conductivity ( f e E ) ne 2t so m - mean free path l vT t (1 ~ 10) at RT v 2) Hall Coefficient ( f e( E H ) ) c 1 R nec 3) AC Conductivity ( f e E ( ) exp(i t ) ) so s ( ) 1 it 4 ne 2 2 - plasma frequency p m P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory The Drude Model - results B. Main Results 4) Thermal Conductivity ( j q k T ) 1 k v 2t cv 1 2 3 v t cv 3k B T k 3 3 2 , v , c nk B - Wiedemann-Franz law s T v 2 m 2 ne t m = 3 kB 2 ( ) T 2 e 5) Thermopower ( E QT ) Q = cv , 3ne cv 3 nk B 2 kB 2e P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude Conductivity Ohm’s “Law”: V = IR The Resistance R is a property of the conductor (e.g. a wire) which depends on its dimensions, V is a voltage drop & I is a current. In microscopic physics, it is more common to express Ohm’s “Law” in terms of a dimension-independent conductivity (or resistivity) which is intrinsic to the material the wire is made from. In this notation, Ohm’s “Law” is written E = j or j = sE (1) Here, E = the electric field, j = the current density, the resistivity & s the conductivity of the material. Consider n electrons per unit volume, all moving in the direction of the current with velocity v. The number of electrons crossing area A in time dt is nAvdt A 28 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory The charge crossing A in dt is -nevAdt, so j = -nev. (2) In the real material, we expect the electrons to be moving randomly even in zero electric field due to thermal energy. However, they will have an average, or drift velocity along the field direction. vdrift = -eEt / m (3) This comes from integrating Newton’s 2nd Law over time t. This is the velocity that must be related to j. Combining (2) & (3) gives j = (ne2t / m)E. Comparison of this with (1) gives the Drude conductivity: ne2t s m P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory (3) As is often the case for physics models, this result for σ has been obtained using some very simple assumptions, which surely cannot be correct! How can we test this result? First - does it in any way self-justify its assumptions? Does the Drude assumption of scattering from ions seem sensible? Check it by measuring s for a series of known metals, and, using sensible estimates for n, e and m, estimate t. Result - t 1014 s at room temperature. Instead of a an average scattering time t, it’s often necessary to formulate a theory of conductivity in terms of an average distance between collisions. This distance is called the mean free path between collisions. To do this, we have to also consider the average electron velocity. This should not be vdrift, which is the electron velocity in the presence of a field. Instead, it should be vrandom, the velocity associated with the intrinsic thermal energy of the electrons. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 30 Estimate vrandom by treating the electrons as a classical gas and using the well-known result: ½(m)v2random = (3/2)(kB)T (for revision see for example Halliday and Resnick, Physics (Wiley)). Result: The mean free path is l = vrandomt 110 Å Good news: this is of the order of interatomic distances. Clearly, however, you would need more to convince a skeptical world. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 31 Go after one of the most striking experimental results on metals, the Wiedemann-Franz law. Since 1853, it had been known that one of the most universal properties of metals concerns a relationship between thermal and electrical conductivity. Recall electrical result j = sE; in simple geometry j = sE = sdV/dx Thermal equivalent jq = -kdT/dx. k is the thermal conductivity. ΔV j jq -ΔT P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude’s assumption (with some experimental backup) was that k in metals is dominated by the electronic contribution. A result from elementary kinetic theory: k = (1/3)vrandomlCel (4) where Cel is the electronic specific heat per unit volume. If each electron has an energy (3/2)kBT, Cel = dEtot/VdT = (3/2) nkBT . Recalling that l = vrandomt, and dividing k by σ gives: k 1 2 kB 3 kB mvrandom 2 T s 2 e 2 e 2 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory (5) Dividing by T gives the simple result that k 3 kB sT 2 e 2 (6) This is exactly the kind of result that we always like to reach. All the parameters that might be regarded in some way as suspect have dropped out, leaving what looks like it might be a universal quantity. Experimentally, this is indeed the case. The true number is a factor of two different to the Drude result, but in his original work, a numerical error made the agreement appear to be exact! P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory So, Drude’s model appeared to be reasonably self-consistent in identifying electron-ion collisions as the main scattering mechanism, and had a triumph regarding the most universal known property of metals. As would happen today, this was enough to set it up as the main theory of metals for two decades. However, fundamental problems began to emerge: 1. It could not explain the observation of positive Hall coefficients in many metals. 2. As more became known about metals at low temperatures, it was obvious that since the conductivity increased sharply, l was far too long to be explained by simple electron-ion scattering. 3. A vital part of the thermal conductivity analysis is the use of the kinetic theory value of 3/2nkB for the electronic specific heat. Measurements gave no evidence for a contribution of this size. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory The physics of solids is deeply quantum mechanical; indeed condensed matter is arguably the best ‘laboratory’ for studying subtle quantum mechanical effects in the 21st century. Advanced general interest reading on this issue (probably more suitable some time later in the year unless you have already read quite a bit about quantum mechanics): ‘The theory of everything’, R.B. Laughlin and D. Pines, Proc. Nat. Acad. Sci. 97, 28 (2000). P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Gas of classical charged particles, electrons, moves through immobile heavy ions arranged in a lattice, vrms from equipartition theorem (which is of course derived from Boltzmann statistics) _2 1 3 me v k BT 2 2 _ vrms 2 3k BT v me Between collisions, there is a mean free path length: L = vrms τ and a mean free time τ (tau) Figure 12.11 (a) Random successive displacements of an electron in a metal without an applied electric field. 37 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory If there is an electric field E, there is also a drift speed vd (108 times smaller than vrms) but proportional to E, equal for all electrons eEt vd me Figure 12.11 (b) A combination of random displacements and displacements produced by an external electric field. The net effect of the electric field is to add together multiple displacements of length vd t opposite the field direction. For purposes of illustration, this figure greatly exaggerates the size of vd compared with vrms. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 38 neAvd dt J nevd Adt Substituting for vd ne t J E me 2 Figure The connection between current density, J, and drift velocity, vd. The charge that passes through A in time dt is the charge contained in the small parallelepiped, neAvd dt. So the correct form of Ohm’s law is predicted by the Drude model !! P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory ne t s me 40 2 2 With mean free time τ = L/vrms With vrms according to Maxwell-Boltzmann statistics 2 ne L s 3kBTme ne L s mevrms Proof of the pudding: L should be on the order of magnitude of the interatomic distances, e.g. for Cu 0.26 nm s 8.49 1022 cm 3 (6.02 1019 C )2 0.26nm 3 1.381 1023 JK 1 300K 9.109 1031 kg σCu, 300 K = 5.3 106 (Ωm)-1 compare with experimental value 59 106 (Ωm) -1, something must we wrong with the classical L and vrms P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Result of Drude theory one order of magnitude too small, so L must be much larger, this is because the electrons are not classical particles, but wavicals, don’t scatter like particles, in addition, the vrms from Boltzmann-Maxwell is one order of magnitude smaller than the vfermi following from Fermi-Dirac statistics 41 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory s 1 3k BTme 2 ne L 0.5 So ρ ~T theory for all temperatures, but ρ ~T for reasonably high T , so Drude’s theory must be wrong ! Figure 12.13 The resistivity of pure copper as a function of temperature. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 43 Phenomenological similarity conduction of electricity and conduction of heat, so free electron gas should also be the key to understanding thermal conductivity V J s x Q T K At x 1 K CV vrms L 3 k B nvrms L K 2 Ohm’s law with Voltage gradient, thermal energy conducted through area A in time interval Δt is proportional to temperature gradient Using Maxwell-Boltzmann statistics, equipartion theorem, formulae of Cv for ideal gas = 3/2 kB n Classical expression for K P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory _ vrms 2 3k BT v me Lets continue For 300 K and Cu 23 vrms 1 3 1.381 10 JK 300K 9.109 1031 kg k B nvrms L K 2 1.381 1023 JK 1 8.48 1022 cm 3 1.1681 105 ms 1 0.26nm K 2 1.381 1023 JK 1 8.48 1028 m 3 1.1681 105 ms 1 0.26 109 m K 2 Ws K 17.78 Kms -1 -1 Experimental value for Cu at (300 K) = 390 Wm K , again one order of magnitude too small, actually roughly 20 times too small P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 45 ne2 L s mevrms K With MaxwellBoltzmann This was also one order of magnitude too small, 2 e mrs 2 k B nvrms Lmevmrs k B m v 2 s 2ne L 2e _ vrms 2 3k BT v me Lorenz number classical K/σ K 3k B2 8 2 2 1.12 10 WK sT 2e 2 3 k K BT s 2e 2 Wrong only by a factor of about 2, Such an agreement is called fortuitous P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 46 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 47 replace Lfor_a_particle with Lfor_a_wavial and vrms with vfermi, s classical 2 ne L for _ a _ particle mevrms L for _ a _ wavical _ vrms 2 3k BT v me s quantum ne2 L for _ a _ wavical mev fermi mev fermis quantum ne2 v fermi 2 EF me For Cu (at 300 K), EF = 7.05 eV , Fermi energies have only small temperature dependency, frequently neglected P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 48 v fermi 2 EF me v fermi ,copper ,300 K 2 7.05 1.602 109 J 6 1 1 . 57 10 ms 9.109 1031 kg one order of magnitude larger than classical vrms for ideal gas L for _ a _ wavical mev fermis quantum ne2 L for _ a _ wavical _ cooper 9.109 1031 kg 1.57 106 ms 1 5.9 107 1m 1 8.49 1028 m 3 (1.602 1019 C )2 L for _ a _ wavical _ cooper 39nm two orders of magnitude larger than classical result for particle P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 49 s classical 2 ne L for _ a _ particle mevrms So here something two orders of two magnitude too small (L) gets divided by something one order of magnitude too small (vrms), i.e. the result for electrical conductivity must be one order of magnitude too small, which is observed !! But L for particle is quite reasonable, so replace Vrms with Vfermi and the conductivity gets one order of magnitude larger, which is close to the experimental observation, so that one keeps the Drude theory of electrical conductivity as a classical approximation for room temperature P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory . In effect, neither the high vrms of 105 m/s of the electrons derived from the equipartion theorem or the 10 times higher Fermi speed do not contribute directly to conducting a current since each electrons goes in any directions with an equal likelihood and this speeds averages out to zero charge transport in the absence of E Figure 12.11 (a) Random successive displacements of an electron in a metal without an applied electric field. (b) A combination of random displacements and displacements produced by an external electric field. The net effect of the electric field is to add together multiple displacements of length vd t opposite the field direction. For purposes of illustration, this figure greatly exaggerates the size of vd compared with vrms. 50 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 51 K classical k B nvrms Lclassical 2 Vrms was too small by one order of magnitude, Lclassical was too small by two orders of magnitude, the classical calculations should give a result 3 orders of magnitude smaller than the observation (which is of course well described by a quantum statistical treatment) So there must be something fundamentally wrong with our ideas on how to calculate K. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 52 Wait a minute, K has something to do with the heat capacity that we derived from the equipartion theorem Kclassical 1 CV _ for _ ideal _ gas vrms L for _ particle 3 We had the result earlier that the contribution of the electron gas is only about one hundredth of what one would expect from an ideal gas, Cv for ideal gas is actually two orders or magnitude larger than for a real electron gas, so that are two orders of magnitude in excess, with the product of vrms and L for particle three orders of magnitude too small, we should calculate classically thermal conductivities that are one order of magnitude too small, which is observed !!! P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 53 2 e mrs 2 k nv Lm v k m v B rms e mrs B K 2 s 2ne L 2e K 3k B2 8 2 2 1.12 10 WK sT 2e Fortunately L cancelled, but vrms gets squared, we are indeed very very very fortuitous to get the right order of magnitude for the Lorenz number from a classical treatment (one order of magnitude too small squared is about two orders of magnitude too small, but this is “compensated” by assuming that the heat capacity of the free electron gas can be treated classically which in turn results in a value that is by itself two order of magnitude too large- two “missing” orders of magnitude times two “excessive orders of magnitudes levels about out P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 54 K fermi 2 2 B kT ( )nLfor _ a _ wavical 3 mev fermi s quantum 2 ne L for _ a _ wavical mev fermi That gives for the Lorenz number in a quantum treatment K k 8 2 2.45 10 WK sT 3e 2 2 B 2 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 55 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 56 Back to the problem of the temperature dependency of resistivity Drude’s theory predicted a dependency on square root of T, but at reasonably high temperatures, the dependency seems to be linear This is due to Debye’s phonons (lattice vibrations), which are bosons and need to be treated by Bose-Einstein statistics, electrons scatter on phonons, so the more phonons, the more scattering Number of phonons proportional to Bose-Einstein distribution function n phonons 1 e / k BT 1 Which becomes for reasonably large T k BT n phonons P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory At low temperatures, there are hardly any phonons, scattering of electrons is due to impurity atoms and lattice defects, if it were not for them, there would not be any resistance to the flow of electricity at zero temperature Matthiessen’s rule, the resistivity of a metal can be written as σ = σlattice defects + σlattice vibrations P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Electric Conduction Drude’s model P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 59 Electric Conduction Drude’s model (cont’d) P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 60 Electric Conduction Drude’s model (cont’d) P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 61 Exercise 1 −8 Calculate the resistance of a coil of platinum wire with diameter 0.5 mm and length 20 m at 20 C given =11×10 m. Also determine the resistance at 1000 platinum =3.93×10 −3 C, given that for /°C. l 20 m 8 R0 (1110 m) 11 3 2 A [0.5(0.5 10 m)] To find the resistance at 1000 But 0 1 T T0 C: R l A so we have : R R0 1 T T0 Where we have assumed l and A are independent of temperature could cause an error of about 1% in the resistance change. R(1000C) (11 )[1 (3.93103 C1 )(1000 C 20 C)] = 53 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 63 Exercise 2 A 1000 W hair dryer manufactured in the USA operates on a 120 V source. Determine the resistance of the hair dryer, and the current it draws. P 1000 W I 8.33A V 120 V V IR V 120 V R 14.4 I 8.33A The hair dryer is taken to the UK where it is turned on with a 240 V source. What happens? (V ) 2 (240 V) 2 P 4000 W R 14.4 This is four times the hair dryer’s power rating – BANG and SMOKE! P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Chapter 24: Electric Current Current Definition of current A current is any motion of charge from one region to another. • Suppose a group of charges move perpendicular to surface of area A. • The current is the rate that charge flows through this area: I dQ ; dQ amount of charge that flows dt during the time interval dt Units: 1 A = 1 ampere = 1 C/s P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 64 65 Current Microscopic view of current P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 66 Current Microscopic view of current (cont’d) P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 67 Current Microscopic view of current (cont’d) • In time t the electrons move a distance x d t • There are n particles per unit volume that carry charge q • The amount of charge that passes the area A in time t is Q q(nAd t ) • The current I is defined by: dQ Q I lim nq d A t 0 dt t • The current density J is defined by: I J nq d A J nq d Current per2unit area Units: A/m Vector current density P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 68 An Overview In this chapter, we will treat conduction ‘e’ in metal as “free charges” that can be accelerated by an applied electric field, to explain the electrical and thermal conduction in a solid. Electrical conduction involves the motion of charges in a material nd under the Influence of an applied electric field. By applying Newton’s 2 law to ‘e’ motion & using a concept of “mean free time” between ‘e’ collisions with lattice vibrations, crystal defects, impurities, etc., we will derive the fundamental equations that govern electrical conduction in solids. Thermal conduction,i.e., the conduction of thermal E from higher to lower temperature regions in a metal, involves the conduction ‘e’ carrying the energy. Therefore, the relationship between the electrical conductivity and thermal conductivity will be reviewed in this textbook. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2.1 Classical theory : The DRUDE model Goal: To find out the relation between the conductivity (or resistivity) and drift velocity , and thereby its relation to mean free time and drift mobility, from the description of the current density In a conductor where ‘e’ drift in the presence of an electric field, current density is defined as the net amount of charge flowing across a unit area per unit time q J A t J : current density q : net quantity of charge flowing through an area A at Ex In this system, electrons drift with an average velocity vdx in the x-direction, called the drift velocity. (Here Ex is the electric field.) Drift velocity is defined as the average velocity of electrons in the x direction at time t, denote by vdx(t) 1 vdx [vx1 vx 2 vx 3 ... vxN ] N P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory vxi : x direction velocity of the ith electrons N : # of conduction electrons in the metal [2.1] 2.1.1 Metals and conduction by electrons Current density in the x direction can be rewritten as a function of the drift velocity q enAvdx t Jx At At J x (t ) envdx (t ) [2.2] : In time Δt, the total charge Δq crossing the area A is enAΔx, where Δx=vdxΔt and n is assumed to be the # of ‘e’ per unit volume in the conductor (n=N/V). : time dependent current density is useful since the average velocity at one time is not the same as at another time, due to the change of Ex Think of motions of a conduction ‘e’ in metals before calculating Vdx. (a) A conduction ‘e’ in the electron gas moves about randomly in a metal (with a mean speed u) being frequently and randomly scattered by thermal vibrations of the atoms. In the absence of an applied field there is no net drift in any direction. (b) Ex Ex (t ) In the presence of an applied field, Ex, there is a net drift along the x-direction. This net drift along the force of the field is superimposed on the random motion of the electron. After many scattering events the electron has been displaced by a net distance, Δx, from its initial position toward the positive terminal P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2.1.1 Metals and conduction by electrons To calculate the drift velocity vdx of the ‘e’ due to applied field Ex, we first consider the Since is the acceleration a of the ‘e’eEx [F=qE=ma], velocity vxi of the ith ‘e’ in the x direction at t. vxi in the x direction at t is given by me Let uxi be the initial velocity of ‘e’ i in the x direction just after the collision. Vxi is written as the sum of uxi and the acceleration of the ‘e’ after the collision. Here, we suppose that its last collision was at time ti; therefore, for time (t-ti), it accelerated free of collisions, as shown in Fig.2.3. uxi is velocity of ith‘e’ in the x direction after the collision However, this is only for the ith electron. We need the average velocity vdx for all such electrons along x as the following eqn. vdx 1 eEx [vx1 vx 2 vx 3 ... vxN ] (t ti ) N me (t-ti) : average free time for N electrons between collision (~ τ = mean free time or mean scattering time) Fig 2.3 Velocity gained in the x direction at time t from the electric field ( Ex) for three different electrons. There will be N electrons to consider in the metal. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2.1.1 Metals and conduction by electrons Drift mobility (vs. mean free time) : widely used electronic parameter in semiconductor device physics. Suppose that τ is the mean free time or mean scattering time. Then, for some electrons, (t-ti) will be greater than ,and for others, it will be shorter, as shown in Fig 2.3. Averaging (t-ti) for N electrons will be the same as . Thus we can substitute for (t-ti) in the previous expression to obtain t t t vdx et Ex me [2.3] Equation 2.3 shows that the drift velocity increases linearly with the applied field. The constant of proportionality a special name and symbol, called drift mobility , which is defined as md et / me vdx m d E x where m d et me has been given [2.4] [2.5] which is often called the relaxation time, is directly related to the microscopic processes that cause the scattering of the electrons in tthe, metal; that is, lattice vibration, crystal imperfections, and impurities. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2.1.1 Metals and conduction by electrons From the expression for the drift velocity vdx the current density Jx follows immediately… by substituting Equation 2.4 into 2.2, that is, J x enm d Ex Therefore, the current density is proportional to the electric field and the conductivity s enm d sEx [2.6] s term is given by [2.7] Then, let’s find out temperature dependence of conductivity (or resistivity) of a metal by considering the mean time . t t t The mean time between collisions has further significance. Its 1/ represents the mean frequency of collisions or scattering events; that is 1/ is the mean probability per unit time that the electron will be scattered. Therefore, during a small time interval probability of scattering will be . t t / t P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory t , the 2.2 Temperature dependence of resistivity To find the temperature dependence of determines the drift velocity. t s , let’s consider the temperature dependence of the mean free time , since this Fig 2.5 scattering of an electron from the thermal vibration of the atoms. The electron travels a mean distance between collisions. Since the scattering cross-sectional area is S,tin the volume Sl there must be at least one scatterer as l u t Ns Sut 1 volume t a1 t 2 1 SuNs [2.11] Ns : concentration of scattering centers When the conduction electrons are only scattered by thermal vibrations of the metal ion, then in the mobility expression refers to the mean time between scattering events by this process. m d et m t S : cross-sectional area u : mean speed a : amplitude of the vibrations e P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2.2 Temperature dependence of resistivity Lattice-scattering-limited conductivity : the resistivity of a pure metal wire increase linearly with the temperature, due to the scattering of conduction electrons by thermal vibrations of the atoms. feature of a metal (cf. semiconductors) The thermal vibrations of the atom can be considered to be simple harmonic motion, much the same way as that of a mass M attached to a spring. From the kinetic theory of matter, 1 1 Ma 2 w2 (average kinetic energy of the oscillations) kT 4 2 So a 2 T . This makes sense because raising the T increases atomic vibrations. Thus t 1 1 C or t a 2 T T Since the mean time between scattering events τ is inversely proportional to the area that scatters 2the ‘e’, a eC (to show a relation with T) results in meT mT 1 1 T 2e T AT [2.12] sT enm d e nC substituting for t in md et / me So, the resistivity of a metal m d P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2.3 MATTHIESSEN’s and NORDHEIM’s Rules. 2.3.1 Matthiessen’s rule and the temperature coefficient of resistivity : The theory of conduction that considers scattering from lattice vibrations only works well with pure metal and it fails for metallic alloys. Their resistivities are weakly T-dependent, and so, different type of scattering mechanism is required for metallic alloys. Let’s consider a metal alloy that has randomly distributed impurity atoms. We have two mean free times between collision. Strained region by impurity exerts a scattering force F = - d (PE) /dx tI t T : scattering from thermal vibration only t i : scattering from impurity only In unit time, a net probability of scattering, 1 1 t t Two different types of scattering processes involving scattering from impurities alone and thermal vibrations alone. 1 tT t 1 ti is given by [2.13] Then, since drift mobility depends on effective scattering time, effective drift mobility is given by 1 1 1 ud u L u I P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory [2.14] 2.3.1 Matthiessen’s rule and the temperature coefficient of resistivity where u L uI is the lattice-scattering-limited drift mobility, is the impurity-scattering-limited drift mobility. Since effective resistivity of the material is simply 1 1 1 enud enuL enuI 1/ enud which can be written T I [2.15] This summation rule of resistivities from different scattering mechanisms is called Matthiessen’s rule. Furthermore, in a general from, effective resistivity can be given by T R ( R : residual resistivity) : scattering E of impurities, dislocations, in ternal atom, vacancies, gain boundaries, etc Since residual resistivity shows very little T-dependence whereas ρT = AT . AT B P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory [2.17] 2.3.1 Matthiessen’s rule and the temperature coefficient of resistivity Temperature coefficient of resistivity (TCR) Eqn. 2.17 indicates that the resistivity of a metal varies with T, with A and B depending on the material. Instead of listing A and B in resistivity tables, we prefer a temperature coefficient that refers to small, normalized changes around a reference temperature. 0 1 0 T T T0 [2.18] - temp sensitivity of the resistivity of metals If the resistivity follows the behavior like in Eqn. 2.17, then Eqn. 2.18 leads to 0 1 0 (T T0 ) [2.19] where a0 is constant over a temperature range T0 to T, o & T T To P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Resistivity of various metals vs. T AT B However is only an approximation for some metals and not true for all metals. This is because the origin of the scattering may be different depending on the temperature. 100 T 2000 Inconel-825 NiCr Heating Wire 1000 10 Scattering from vibration Iron Resistivity (n m) Resistivity(n m) Tungsten Monel-400 T Tin 100 Platinum (n m) 3.5 0.1 0.01 T 3 T5 2.5 2 Copper Nickel 0.001 Silver 0.0001 T5 1.5 1 0.5 = R R 0 0 0.00001 10 100 1000 10000 10 20 40 60 80 100 T (K) Scattering from impurity 100 1000 10000 Temperature(K) Temperature (K) The resistivity of copper from lowest to highest temperatures (near The resistivity of various metals as a function of temperature above 0 melting temperature, 1358 K) on a log-log plot. Above about 100 K, °C. Tin melts at 505 K whereas nickel and iron go through a magnetic 5 to non-magnetic (Curie) transformations at about 627 K and 1043 K T, whereas at low temperatures, T and at the lowest respectively. The theoretical behavior ( ~ T) is shown for reference. temperatures approaches the residual resistivity R . The inset [Data selectively extracted from various sources including sections in shows the vs. T behavior below 100 K on a linear plot ( is too R Metals Handbook, 10th Edition, Volumes 2 and 3 (ASM, Metals small on this scale). Park, Ohio, 1991)] P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2.3.2 Solid solution and Nordheim’s rule How does the resistivity of solid solutions change with alloy composition ? In an isomorphous alloy of two metals, that is, a binary alloy that forms a solid solution (Ni-Cr alloy), we would expect Eqn 2.15 to apply, with the temperature-independent impurity contribution increasing with the concentration of solute atoms. I T I [2.15] This means that as alloy concentration increases, resistivity increases and becomes less temperature dependent as ρI, overwhelms ρT, leading to αo << 1/273. This (temperature independency) is the advantage of alloys in resistive components. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2.3.2 Solid solution and nordheim’s rule 1500 Nordheim’s rule for solid solutions: an important semiempirical Eqn. that can be used to predict the resistivity of an alloy, which relates the impurity resistivity to the atomic fraction X of solute atoms in a solid solution, as follows: Temperature (°C) How does the concentration of solute atoms affect on ρI ? US UID Q I L US LID O S 1400 LIQUID PHASE 1300 L+ 1200 1100 1000 S SOLID SOLUTION 20 0 40 100% Cu 60 80 at.% Ni 100 100% Ni (a) [2.21] C (Nordheim’s coefficient): represents effectiveness of the solute atom in increasing the resistivity. 600 Resistivity (n m) I CX (1 X ) 500 Cu-Ni Alloys 400 300 200 100 consistent 0 Nordheim rule is useful for predicting the resistivities of dilute alloys, particularly in the low-concentration region. 0 20 100% Cu 40 60 at.% Ni 80 100 100% Ni (b) (a) Phase diagram of the Cu-Ni alloy system. Above the liquidus line only the liquid phase exists. In the L + S region, the liquid (L) and solid (S) phases coexist whereas below the solidus line, only the solid phase (a %Nordheim’s rule assumes that the solid solution has the solute atoms randomly solid solution) exists. (b) The resistivity of the Cu-Ni alloy as a function distributed in the lattice, and these random distributions of impurities cause the ‘e’ to of Ni content (at.%) at room temperature. [Data extracted from Metals become scattered as they whiz around the crystal. Handbook-10th Edition, Vols 2 and 3, ASM, Metals Park, Ohio, 1991 and Constitution of Binary Alloys, M. Hansen and K. Anderko, McGraw-Hill, New York, 1958] P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 2.3.2 Solid solution and Nordheim’s rule Combination of Matthiessen and Nordheim rules leads to a general expression for ρ of the solid solution: matrix CX (1 X ) [2.22] where matrix T R is the resistivity of the matrix due to scattering from thermal vibrations and from other defect, absence of alloying elements. 160 Quenched 140 Resistivity (n m) Exception: at some concentrations of certain binary alloys, Cu and Au atoms are not randomly mixed but occupy regular sites, which decrease the resistivity. ------------ 120 100 80 Annealed 60 40 20 Cu3Au CuAu 0 0 10 20 30 40 50 60 70 80 90 100 Composition (at.% Au) Electrical resistivity vs. composition at room temperature in Cu-Au alloys. The quenched sample (dashed curve) is obtained by quenching the liquid and has the Cu and Au atoms randomly mixed. The resistivity obeys the Nordheim rule. On the other hand, when the quenched sample is annealed or the liquid slowly cooled (solid curve), certain compositions (Cu3Au and CuAu) result in an ordered crystalline structure in which Cu and Au atoms are positioned in an P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory ordered fashion in the crystal and the scattering effect is reduced. A Model for Electrical Conduction: The Drude Model Atoms Electrons A regular array of atoms surrounded by a “cloud” of free electrons Random movement under zero field P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Random movement modified by a field Drude Model F qE mea qE a me nq 2 E J nqvd t me J sE qE v f v i at v i t me nq2t s me Now take the average over all times. Then vi=0 (random movement), me 2 nq t qE qE v f vd t t me me , where t is the mean time between collisions l t vd is the mean free path P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Resistivity and Temperature Resistivity in metals is linear with temperature over a limited range 0 1 T T0 R R0 1 T T0 : temperature coefficient of resistivity 1 0 T P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory -1 (C ) Plasma: A neutral gas of heavy ions and light electrons. Metals and doped semiconductors can be treated as plasmas because they contain equal numbers of fixed positive ions and free electrons. Free electrons in this system experience no : restoring force from the medium when they interact with electromagnetic waves. driven by the electric field of a light wave. p r () 1 , (2 i) 2 Ne p 0 m0 2 where 86 1 2 . p: plasma frequency For a lightly damped system, = 0, so that 7.1 Plasma reflectivity Drude-Lorentz model: Considering the oscillations of a free electron induced by AC electric field E(t) of a light wave with polarized along the x direction: 2p r () 1 2 ~ n r , n~ is imaginary for < p, positive for for > p zero for = p, 2 d x dx m0 eE(t ) eE0e it . 2 dt at By substituting x x0e it m0 x(t ) eE (t ) . m0 (2 i) The reflectivity: ~ 1 2 n R ~ n 1 The electric displacement: D r 0 E 0 E P Therefore: Ne 2 E 0 E m0 (2 i) Ne 2 r () 1 Reflectivity of an undamped free carrier gas as a function of frequency. 2 m ( i ) P.Ravindran, PHY0750 0Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Thus optical measurements of r() are equivalent to AC conductivity measurement of s(). 7.2 Free carrier conductivity Considering the damping and the electron velocity v x , the momentum p = m0v By splitting dv m0 m0 v eE , dt dp p eE , dt t The damping time t = 1/. The shows that the electron is being accelerated by the field, but loses its momentum-14 in the -13 time t. So t is the momentum scattering time. t is typically in the range 10 —10 s, hence optical frequency must be used to obtain information about t. -it By substituting v = v0e , et 1 v (t ) E (t ) m0 1 it The current density: j Nev sE, where the AC conductivity s(): s0 s() , 1 it where s0 is the DC conductivity. Ne 2 t s0 m0 p r () 1 2 ( i) 2 into its real and imaginary components: 1 1 2 2p t 2 1 2 t 2 2p t , (1 2 t 2 ) . n, k and , the real and imaginary parts of the complex refractive index and the attenuation coefficient can 1/2be worked out. At very low frequencies that satisfy t << 1 and 2 >>1,, n k = (2 / 2 ) , thus: 1 2 1 2 2 t 2k 2( 2 / 2) . c c c Ne 2 t s0 0 2p t, m0 This gives: 2 p 2 (2s0m 0 )1/ 2 . The attenuation coefficient is proportional to the square root of the DC conductivity and the frequency. Define the skin depth : 1/ 2 2 2 s0m 0 Ne 2 is() This implies that AC field can only penetrate a short distance into a conductor such as a r () 1 1 2 metal. 0 m0 (PHY075 Condensed i) Matter Physics, 0 Spring 2013 : Drude-Lorenz P.Ravindran, Free Electron Theory 87 7.3 88 Metals 7.3.1 The Drude model The valence electrons is free. The density N is equal to the density of metal atoms multiplied by their valency; The characteristic scattering time t can be determined by the measurement of s. Experimental reflectivity of Al as a function of photon energy. The experimental data is compared to predictions of the free electron model with h =-15 15.8 eV. The dotted line is calculated with no damping. The dashed line with t = 8.010 s, which is the value deduced from the DC conductivity. All metals will become transmitting if > p ( UV transparency of metals) Free and plasma properties of some metals. The values of N are in the range 28 electron 29 density -3 10 —10 m . the very large values of N lead to plasma high electrical and thermal conductivities and plasma frequency in the UV region. The figure shows that the reflectivity of Al is over 80% up to 15 eV, and then drops off to zero at higher frequencies. From this figure, one can see that the model accounts for the general shape of the spectrum, but there are some important detials that are not explained. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 7.3.2 Interband transitions in metals Interband absorption is important in metals because the EM penetrate a short distance into the surface, and if there is a significant probability for interband absorption, the reflectivity will be reduced from the free carrier value. The interband absorption spectra of metals are determined by their complicated band structures and Fermi surfaces. Furthermore, one needs to consider transitions at frequencies in which the free carrier properties are also important. 2 1 89 Electronic configuration: [Ne]3s 3p with three valence electrons; the first Brillouin zone is completely full, and the valence electrons spread into the second, third and slightly into the fourth zones. The bands are filled up to the Fermi energy EF, and direct transitions can take place from any the states below the Fermi level to unoccupied bands directly above them on the E—k diagram. “parallel band effect” corresponding to the dip in the reflectivity at 1.5 eV originates the high density of states between the two parallel bands. Moreover, there are further transition at a whole range of photon energies greater than 1.5 eV. The density of states for these transition will be lower than at 1.5 eV because the bands are not parallel, however, the absorption rate is still significant, and accounts for the reduction of the reflectivity predicted by the Drude model.. Aluminium Copper 10 1 [Ar]3d 4s , The wide outer 4s band (1), Approximately free states 2 electron 2 Dispersion : E = h k /2m0; The narrow 3d band (10) More tightly bound Relatively dispersionless The Fermi energy lies in the middle of the 4s band above the 3d band A well-defined threshold for interband transitions from the 3d to the 4s. Band diagram of Al at the W and K points that are responsible for the reflectivity dip at 1.5 eV are labelled P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 90 7.3.2 Interband transitions in metals Copper Gold and silver In gold the interband absorption threshold occurs at a slightly higher energy than copper. In silver the interband absorption edge is around 4 e, the frequency is in the ultraviolet, and so the reflectivity remains high throughout the whole visible spectrum. The 3d electrons lie in relatively bands with very high densities of states, while the 4s are much broader with a low density of states. The Fermi energy lies in the middle of the 4s band above the 3d band. Interband transition are possible from the 3d band below EF. The lowest energy transitions are marked on the band diagram. The transition energy is 2.2 eV which corresponds to a wavelength of 560 nm. The measured reflectivity of copper. Based on the plasma frequency, one would expect near-100% reflectivity for photon energies below 10.8 eV (115nm). However, the experimental reflectivity falls off sharply above 2 eV due to the interband absorption edge. The explain why copper has a reddish colour. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 91 A model for electrical conduction (a) (b) A classical model of electrical conduction in metals: Drude model in 1900 – In the absence of an electric field, the conduction electrons move in random directions through the conductor with average speeds v ~ 106 m/s. The drift velocity of the free electrons is zero. There is no current in the conductor since there is no net flow of charge. – When an electric field is applied, in addition to the random motion, the free electrons drift slowly (vd ~ 10-4 m/s) in a direction opposite that of the electric field. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 92 Derivation of the drift velocity, vd, using Drude model Electric force on an electron: F = qE, where q = -e. Acceleration of the electron: a = F/me = qE/me. Define the following: – t = 0: the instant just after one collision has occurred; – t : the instant just before the next collision occurs; – vi: velocity of the electron at t = 0; vf: velocity of the electron at time t. Apply Newton’s 2nd law: vf = vi + at = vi + (qE/me)t. Average vf over all possible values of vi and collision time t: v v qE t qE t ; so, v qE t f i me me d me – t: average time interval between successive collisions = mean free time = relaxation time. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 93 Conductivity and resistivity in terms of microscopic quantities According to Drude model: – Conductivity s = (nq2t)/me. – Resistivity: = 1/s = me/(nq2t). – n: charge carrier density = the number of charge carriers per unit volume. – q: the charge on each carrier. For electrons, q=-e. – me: electronic mass. – t: mean free time or relaxation time According to Drude model, conductivity and resistivity do not depend on the strength of the electric field, a feature characteristic of a conductor obeying Ohm’s law. Mean free path = average distance between collisions: l v t P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 94 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 95 Sommerfeld Model: 1928 Ashcroft-Mermin Chapter 2: - Systematically recast Drude-Lorentz theory in terms of FD statistics rather than MB statistics - Wiedemann-Franz law (still) came out right - Estimated specific heat right - Difficulties remained: - Sign of Hall coefficient - Magneto-resistance - What determines the scattering time t? - What determine the density n? - Why are some elements non-metals? Arnold Sommerfeld (1868-1951) P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: why are metals shiny? Drude’s theory gives an explanation of why metals do not transmit light and rather reflect it. 96 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Some relations from basic optics wave propagation in matter plane wave complex index of refraction Maxwell relation 97 all the interesting physics in in the dielectric function! P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Free-electron dielectric function one electron in time-dependent field we write and get the dipole moment for one electron is and for a unit volume of98solid it is P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Free-electron dielectric function we use to get so the final result is is called the plasma frequency 99 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Meaning of the plasma frequency the dielectric function in the Drude model is with remember ε real and negative, no wave propagation metal is opaque ε real and positive, propagating waves metal is transparent 100 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory plasma frequency: simple interpretation values for the plasma energy 101 longitudinal collective mode of the whole electron gas P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory the Wiedemann-Franz law k constant s Wiedemann and Franz found in 1853 that the ratio of thermal and electrical conductivity for ALL METALS is constant at a given temperature (for room temperature and above). Later it was found by L. Lorenz that this constant is proportional to the 102 102 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory The Wiedemann Franz law estimated thermal conductivity (from a classical ideal gas) the actual quantum mechanical result is 103 this is 3, more or less.... P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Comparison of the Lorenz number to experimental data at 273 K metal 10-8 Watt Ω K-2 Ag 2.31 Au 2.35 Cd 2.42 Cu 2.23 Mo 2.61 Pb 2.47 Pt 2.51 Sn 2.52 W 3.04 Zn 2.31 L = 2.45 10 -8 -2 Watt Ω K 104 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Failures of the Drude model Despite of this and many other correct predictions, there are some serious problems with the Drude model. 105 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: electrical conductivity line 106 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Failures of the Drude model: he mean free path A very good example is http://www.sciencemag.org/cgi/content/full/319/5867/1226/FIG1 It seems that the electrons manage to sneak past the (close packed) atoms and by all the other electrons. How do they do this? 107 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Failures of the Drude model: electrical conductivity of an alloy • • The resistivity of an alloy should be between those of its components, or at least similar to them. It can be much higher than that of either component. 108 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Failures of the Drude model: heat capacity consider the classical energy for one mole of solid in a heat bath: each degree of freedom contributes with energy heat capacity monovalent divalent trivalent el. transl. 109 ions vib. Experimentally, one finds a value of about at room temperature, independent of the number of valence electrons (rule of Dulong P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 109 : Drude-Lorenz Free Electron Theory Many open questions: 110 Why does the Drude model work so relatively well when many of its assumptions seem so wrong? In particular, the electrons don’t seem to be scattered by each other. Why? How do the electrons sneak by the atoms of the lattice? Why do the electrons not seem to contribute to the heat capacity? Why is the resistance of an disordered alloy so high? P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 111 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Electrical properties of metals: Classical approach (Drude theory) at the end of this lecture you should understand.... Basic assumptions of the classical theory DC electrical conductivity in the Drude model Hall effect Plasma resonance / why do metals look shiny? thermal conduction / Wiedemann-Franz law Shortcomings of the Drude model: heat capacity... 112 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Some relations from basic optics wave propagation in matter plane wave complex index of refraction Maxwell relation 113 all the interesting physics in in the dielectric function! P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Free-electron dielectric function one electron in time-dependent field we write and get the dipole moment for one electron is and for a unit volume of114 solid it is P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Free-electron dielectric function we use to get so the final result is is called the plasma frequency 115 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 11 6 The free electron theory of metals The Drude theory of metals Paul Drude (1900): theory of electrical and thermal conduction in a metal application of the kinetic theory of gases to a metal, which is considered as a gas of electrons mobile negatively charged electrons are confined in a to immobile positively charged ions metal by attraction isolated atom nucleus charge eZa Z valence electrons are weakly bound to the nucleus (participate in chemical reactions) Za – Z core electrons are tightly bound to the nucleus (play much less of a role in chemical reactions) in a metal – the core electrons remain bound to the nucleus to form the metallic ion the valence electrons wander far away from their parent atoms called conduction electrons or electrons P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory in a metal Doping of Semiconductors C, Si, Ge, are valence IV , Diamond fcc structure. Valence band is full Substitute a Si (Ge) with P. One extra electron donated to conduction band N-type semiconductor P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 11 7 11 8 The basic assumptions of the Drude model 1. between collisions the interaction of a given electron with the other electrons is neglected and with the ions is neglected independent electron approximation free electron approximation 2. collisions are instantaneous events Drude considered electron scattering off the impenetrable ion cores the specific mechanism of the electron scattering is not considered below 3. an electron experiences a collision with a probability per unit time 1/τ dt/τ – probability to undergo a collision within small time dt randomly picked electron travels for a time τ before the next collision τ is known as the relaxation time, the collision time, or the mean free time τ is independent of an electron position and velocity 4. after each collision an electron emerges with a velocity that is randomly directed and with a speed appropriate to the local temperature P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 11 9 DC electrical conductivity of a metal V = RI Ohm’s low the Drude model provides an estimate for the resistance introduce characteristics of the metal which are independent on the shape of the wire E j j=I/A – the current density – the resistivity R=L/A – the resistance s = 1/ the conductivity j sE L A j env v is the average electron velocity eE v t m j sE ne2t j m E ne2t s m P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 12 0 motion under the influence of the force f(t) due to spatially uniform electric and/or magnetic fields average momentum equation of motion for the momentum per electron dp(t ) p( t ) f (t ) electron collisions introduce a frictional damping termdt for the momentum t per electron Derivation: dt dt p(t dt ) 1 p(t ) f (t )dt 0 t t scattered part fraction of electrons that does not experience scattering p(t dt ) p(t ) f (t )dt p( t ) t total loss of momentum after scattering dt O dt 2 p(t dt ) p(t ) p( t ) f (t ) O dt dt t dp(t ) p( t ) f (t ) dt t P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory average velocity p(t ) mv(t ) 1. The Drude Model A. Assumptions 1) Free Electron Model 2) Independent Electron Model 3) Relaxation Time Approximation - Equation of motion dv v m f m dt t - Elementary kinetic theory 4) Maxwell Boltzmann Statistics P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Electrical properties of metals: Classical approach (Drude theory) at the end of this lecture you should understand.... Basic assumptions of the classical theory DC electrical conductivity in the Drude model Hall effect Plasma resonance / why do metals look shiny? thermal conduction / Wiedemann-Franz law Shortcomings of the Drude model: heat capacity... 122 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude’s classical theory Theory by Paul Drude in 1900, only three years after the electron was discovered. Drude treated the (free) electrons as a classical ideal gas but the electrons should collide with the stationary ions, not with each other. average rms speed so at room temp. 123 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude’s classical theory relaxation time (average time between scattering events) mean free path 124 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory onduction electron Density n #atoms per volume calculate as #valence electrons per atom density atomic mass 125 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory ...this must surely be wrong.... The electrons should strongly interact with each other. Why don’t they? The electrons should strongly interact with the lattice ions. Why don’t they? Using classical statistics for the electrons cannot be right. This is easy to see: condition for using classical statistics is some Å de Broglie wavelength of an electron: for RT 126 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory but: In a theory which gives results like this, there must certainly be a great deal of truth. Hendrik Antoon Lorentz So what are these results? 127 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: electrical conductivity we apply an electric field. The equation of motion is integration gives and if is the average time between collisions then the average drift speed is for 128 we get remember: P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: electrical conductivity number of electrons passing in unit time current of negatively charged electrons current density Ohm’s law and with we get 129 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: electrical conductivity Ohm’s law define ivity 130 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Ohm’s law valid for metals valid for homogeneous semiconductors not valid for inhomogeneous semiconductors 131 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Electrical conductivity of materials 132 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory How to measure the conductivity / resistivity A two-point probe can be used but the contact ore wire resistance can be a problem, especially if the sample has a small P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 133 How to measure the conductivity / resistivity The problem of contact resistance can be overcome by using a four point probe. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 134 Drude theory: electrical conductivity line P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory Drude theory: electrical conductivity Drude’s theory gives a reasonable picture for the phenomenon of resistance. Drude’s theory gives qualitatively Ohm’s law (linear relation between electric field and current density). It also gives reasonable quantitative values, at least at room temperature. 136 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory To explain the high conductivities and the trend we need to have a model for both thermal and electrical conductivity, that model should be able to explain Ohm’s law, empirical for many metals and insulators, ohmic solids J = σ E current density is proportional to applied electric field U ρl R = / I for a wire R = /A Conductivity , resistivity is its reciprocal value 2 J: current density A/m σ: electrical conductivity Ω -1 -1 m , reciprocal value of electrical resistivity E: electric field V/m Also definition of σ: a single constant that does depend on the material and temperature but not on applied electric field represents connection between I and U P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory 137