Lecture V METALS dr hab. Ewa Popko

1. Lecture V METALS dr hab. Ewa Popko

2. Resistivity (Ωm) (295K) Resistivity (Ωm) (4K) Material 10-5 Potassium 10-12 “Pure”Metals 10-10 2  10-6 Copper Semi-Conductors Ge (pure) 5  102 1012 1014 Insulators Diamond 1014 Polytetrafluoroethylene (P.T.F.E) 1020 1020 Metals and insulators Measured resistivities range over more than 30 orders of magnitude

3. Metals, insulators & semiconductors? 1020- At low temperatures all materials are insulators or metals. Diamond 1010- Resistivity (Ωm) Germanium Pure metals: resistivity increases rapidly with increasing temperature. 100 - Copper 10-10- 0 100 200 300 Temperature (K) Semiconductors: resistivity decreases rapidly with increasing temperature. Semiconductors have resistivities intermediate between metals and insulators at room temperature.

4. Note orbital filling in Cu does not follow normal rule Core and Valence Electrons Most metals are formed from atoms with partially filled atomic orbitals. e.g. Na, and Cu which have the electronic structure Na 1s2 2s2 2p63s1 Cu 1s2 2s2 2p6 3s23p63d104s1 Insulators are formed from atoms with closed (totally filled) shells e.g. Solid inert gases He 1s2 Ne 1s2 2s2 2p6 Or form close shells by covalent bonding i.e. Diamond Simple picture. Metal have CORE electrons that are bound to the nuclei, and VALENCE electrons that can move through the metal.

5. e- Attract Attract Repel Repel Na+ Na+ Attract Attract e- Metallic bond Atoms in group IA-IIB let electrons to roam in a crystal. Free electrons glue the crystal Additional binding due to interaction of partially filled d – electron shells takes place in transitional metals: IIIB - VIIIB

6. 0 V(r) E2 E1 E0 Increasing Binding Energy r Bound States in atoms Electrons in isolated atoms occupy discrete allowed energy levels E0, E1, E2 etc. . The potential energy of an electron a distance r from a positively charge nucleus of charge q is

7. + + + + + Nuclear positions R Bound and “free” states in solids The 1D potential energy of an electron due to an array of nuclei of charge q separated by a distance R is Where n = 0, +/-1, +/-2 etc. This is shown as the black line in the figure. 0 V(r) E2 E1 E0 V(r) Solid V(r) lower in solid (work function). Naive picture: lowest binding energy states can become free to move throughout crystal r 0

8. + + + + + Electron level similar to that of an isolated atom Energy Levels and Bands In solids the electron states of tightly bound (high binding energy) electrons are very similar to those of the isolated atoms. Lower binding electron states become bands of allowed states. We will find that only partial filled band conduct Band of allowed energy states. E position

9. Na+ ions: Nucleus plus 10 core electrons + + + + + + + + + + + + + + + + + + + + + + + + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Why are metals good conductors? Consider a metallic Sodium crystal to comprise of a lattice of Na+ ions, containing the 10 electrons which occupy the 1s, 2s and 2p shells, while the 3s valence electrons move throughout the crystal. The valence electrons form a very dense ‘electron gas’. We might expect the negatively charged electrons to interact very strongly with the lattice of positive ions and with each other. In fact the valence electrons interact weakly with each other & electrons in a perfect lattice are not scattered by the positive ions.

10. Electrons in metals P. Drude: 1900 kinetic gas theory of electrons, classical Maxwell-Boltzmann distribution independent electrons free electrons scattering from ion cores (relaxation time approx.) A. Sommerfeld: 1928 Fermi-Dirac statistics F. Bloch’s theorem: 1928 Bloch electrons L.D. Landau: 1957 Interacting electrons (Fermi liquid theory)

11. Free classical electrons:Assumptions • We will first consider a gas of free classical electrons subject to external electric and magnetic fields. Expressions obtained will be useful when considering real conductors • (i) FREE ELECTRONS: The valence electrons are not affected by the electron-ion interaction. That is their dynamical behaviour is as if they are not acted on by any forces internal to the conductor. • (ii) NON-INTERACTING ELECTRONS: The valence electrons from a `gas' of non-interacting electrons. They behave as INDEPENDENT ELECTRONS; they do not show any `collective' behaviour. • ELECTRONS ARE CLASSICAL PARTICLES: distinguishable, p~exp(-E/kT) • (iv) ELECTRONS ARE SCATTERED BY DEFECTS IN THE LATTICE: ‘Collisions’ with defects limit the electrical conductivity. This is considered in the relaxation time approximation.

12. L Area A Electric field E Force on electron F Drift velocity vd Current density j = I/A dx Area A vd n free electrons per m3 with charge –e ( e = +1.6x10-19 Coulombs ) Ohms law and electron drift V = E/L = IR (Volts) Resistance R = rL/A (Ohms) Resistivity r = AR/L (Ohm m) E = V/L = rI/A = rj (Volts m-1) Conductivity s = 1/r (low magnetic field) j = sE (Amps m-2) I = dQ/dt (Coulomb s-1) Force on electrons F = -eE results in a constant electron drift velocity, vd. Charge in volume element dQ = -enAdx

13. Relaxation time approximation At equilibrium, in the presence of an electric field, electrons in a conductor move with a constant drift velocity since scattering produces an effective frictional force. Assumptions of the relaxation time approximation : 1/ Electrons undergo collisions. Each collision randomises the electron momentum i.e. The electron momentum after scattering is independent of the momentum before scattering. 2/ Probability of a collision occurring in a time interval dt is dt/t. t is called the ‘scattering time’, or ‘momentum relaxation time’. 3/ t is independent of the initial electron momentum & energy.

14. t/tp Momentum relaxation Consider electrons, of mass me, moving with a drift velocity vd due to an electric field E which is switch off at t=0. At t=0 the average electron momentum is In a time interval dt the fractional change in the average electron momentum due to collisions is integrating from t=0 to t then gives tpis the characteristic momentum or drift velocity relaxation time. p(t = 0) = mevd(t = 0) dp/p(t) = - dt/tp ; dp/dt = -p(t)/tp p(t) = p(0)exp(-t/tp) If, in a particular conductor, the average time between scattering eventsis tsand it average takes 3 scattering event to randomise the momentum. Then the momentum relaxation time istp = 3ts.

15. Electrical Conductivity In the absence of collisions, the average momentum of free electrons subject to an electric field E would be given by The rate of change of the momentum due to collisions is At equilibrium Now j = -nevd= -nep/me = (ne2tp /me) E So the conductivity is s = j/E = ne2tp /me The electron mobility, m, is defined as the drift velocity per unit applied electric field m = vd / E =etp /me (units m2V-1s-1)

16. Ex, jx Ey vd = vx Bz For a general vx. vx+ve or -ve The Hall Effect An electric field Ex causes a current jx to flow. • A magnetic field Bz produces a Lorentz force in the y-direction on the electrons. Electrons accumulate on one face and positive charge on the other producing a field Ey . F = -e (E + v  B). In equilibrium jy = 0 so Fy = -e (Ey - vxBz) = 0 Therefore Ey = +vxBz jx = -nevx so Ey = -jxBz/ne The Hall resistivity isrH= Ey/jx = -B/ne j The Hall coefficient is RH = Ey/jxBz = -1/ne

17. Bz E The Hall Effect j=jx vd = vx Ey The Hall coefficient RH = Ey/jxBz = -1/ne The Hall angle is given by tanf = Ey/Ex =rH/r For many metals RH is quiet well described by this expression which is useful for obtaining the electron density, in some cases. However, the value of n obtained differs from the number of valence electrons in most cases and in some cases the Hall coefficient of ordinary metals, like Pb and Zn, is positive seeming to indicate conduction by positive particles! This is totally inexplicable within the free electron model. Ex

18. Ex, jx Ey vd Ex, jx Ey vd Ey = +vxBz = - vd Bz jx = -nevx = ne vd Ey = -jxBz/ne RH = Ey/jxBz = -1/ne Ey = +vxBz = vd Bz jx = nevx = ne vd Ey = jxBz/ne RH = Ey/jxBz = 1/ne Bz Bz Sign of Hall Effect Hall Effect for free particles with charge +e ( “holes” ) Hall Effect for free particles with charge -e ( electrons )

19. The (Quantum)Free Electron model: Assumptions (i) FREE ELECTRONS: The valence electrons are not affected by the electron-ion interaction. That is their dynamical behaviour is as if they are not acted on by any forces internal to the conductor. (ii) NON-INTERACTING ELECTRONS: The valence electron from a `gas' of non-interacting electrons. That is they behave as INDEPENDENT ELECTRONS that do not show any `collective' behaviour. (iii) ELECTRONS ARE FERMIONS: The electrons obey Fermi-Dirac statistics. (iv) ‘Collisions’ with imperfections in the lattice limit the electrical conductivity. This is considered in the relaxation time approximation.

20. U(r) U(r) Free electron approximation Neglect periodic potential & scattering (Pauli) Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)

21. U(r) Ek |k| Eigenstates & energies

22. y py px x z pz Free Quantum Electrons states Uniquely determined by the wavevector, k. Or equivalently by (px, py, pz) = (kx, ky, kz). Equal probability of electron being anywhere in conductor. ky kx kz k- space Free Classical Electrons states Defined by position (x,y,z) and momentum (px, py, pz) Electron state defined by a point in k-space

23. Eigenstates & energies 3D analog of energy levels for a particle in a box!! The allowed values of nx, ny, and nz, are positive integers for the electron states in the free electron gas model.

24. Density of states The number of states that have energies in a given range dE is called the density of states g(E). Let us think of a 3D space with coordinates nx, ny, and nz. The radius nrs of the sphere :

25. Density of states Each point with integer coordinates represents one quantum state. Thus the total umber of points with integer coordinates inside the sphere equals the volume of the sphere: and for integer numbers are positive –only 1/8 the total volume: Including spin, the number of allowed electron states is equal to:

26. Density of states (DOS) g(E) E

27. f(E) Fermi-Dirac function for T=0. Fermi-Dirac distribution function The Density of States tells us what states are available. We now wish to know the occupancy of these states. Electrons obey the Pauli exclusion principle. So we may only have two electrons (one spin-up and one spin-down) in any energy state. The probability of occupation of a particular state of energy E is given by the Fermi-Dirac distribution function, f(E). For T=0 all states are occupied up to an energy EF, called the Fermi energy, and all states above EF are empty.

28. The Fermi Energy The number of occupied states per unit volume in the energy range E to E+dE is Calculated EF for free electrons by equating the sum over all occupied states at T=0 to the total number of valence electrons per unit volume, n i.e. i.e. This gives n(E)dE n(E) at T = 0

29. Free Electron Fermi Surface Metals have a Fermi energy, EF. Free electrons so EF = 2kF2/2m At T=0 All the free electron states within a Fermi sphere in k-space are filled up to a Fermi wavevector,kF. The Fermi wavelength l = 2p/kF The surface of this sphere is called the Fermi surface. On the Fermi surface the electrons have a Fermi velocity vF = hkF/me. The Fermi Temperature,TF, is the temperature at which kBTF = EF. When the electron are not free a Fermi surface still exists but it is not generally a sphere.

30. The effects of temperature At a temperature T the probability that a state is occupied is given by the Fermi-Dirac function Fermi-Dirac function for a Fermi temperature TF =50,000K, about right for Copper. where μ is the chemical potential. For kBT << EF μ is almost exactly equal to EF. n(E)dE The finite temperature only changes the occupation of available electron states in a range ~kBT about EF.

31. Electronic specific heat capacity Consider a monovalent metal i.e. one in which the number of free electrons is equal to the number of atoms. If the conducting electrons behaved as a gas of classical particles the electron internal energy at a temperature T would be U = (kBT/2) x n x (number of degrees of freedom = 3) So the specific heat at constant volume CV = dU/dT= 3/2nkB. At room temperature the lattice specific heat, 3nkB ( n harmonic oscillators with 6 degrees of freedom). In most metals, at room temperature, CV is very close to 3nkB. The absence of a measureable contribution to CV was historically the major objection to the free classical electron model. If electrons are free to carry current why are they not free to absorb heat energy? The answer is that they are Fermions.

32. Electronic Specific heat The total energy of the electrons per m3 in a metal can be written as E = Eo(T=0) + DE(T). Where Eo(T=0)is the value at T=0. n(E)dE At a temperature T only those electrons within ~ kBT of EF have a greater energy that they had at T=0. The number of electrons that increase their energy is ~n(kBT/EF) where n is the number of electrons per m3 Each of these increases its energy by ~kBT

33. The electronic specific heat is therefore A full calculation gives (Kittel p151-155) For a typical metals this is ˜ 1% of the value for a classical gas of electrons. E.g. Copper (kT/EF) ~ 300/50,000 = 0.6% At room temperature the phonon contribution dominates.

34. Metal gcalc. (JK-1 mol-1) gexpt(JK-1mol-1) Cu 5  10-7 7  10-7 Pb 1.5  10-6 3  10-6 Low Temperature Specific Heat Predicted electronic specific heat At low temperature one finds that The first term is due to the electrons and the second to phonons. The linear T dependent term is observed for virtually all metals. However the magnitude of γ can be very different from the free electron value. where g and a are constants

35. Dynamics of free quantum electrons Classical free electrons F = -e (E + v  B) = dp/dt and p =mev. Quantum free electrons the eigenfunctions are ψ(r) = V-1/2 exp[i(k.r-wt) ] The wavefunction extends throughout the conductor. Can construct localise wavefunction i.e. a wave packets The velocity of the wave packet is the group velocity of the waves The expectation value of the momentum of the wave packet responds to a force according to F = d<p>/dt (Ehrenfest’s Theorem) for E = 2k2/2me Free quantum electrons have free electron dynamics

36. Conductivity & Hall effect Free quantum electrons have free electron dynamics Free electron expressions for the Conductivity, Drift velocity, Mobility, & Hall effect are correct for quantum electrons. Current Density j = -nevd Conductivity s = j/E = ne2tp /me Mobility m = vd / E = etp /me Hall coefficient RH = Ey/jxBz = -1/ne

37. Electronic Thermal Conductivity Treat conduction electrons as a gas. From kinetic theory the thermal conductivity, K, of a gas is given by K = ΛvCv/3 where v is the root mean square electron speed, Λ is the electron mean free path & Cv is the electron heat capacity per m3 For T<< TF we can set v = vF and Λ = vFtpand Cv = (p2/2) nkB (T/TF) So K = p2nk2BtpT/3m now s = ne2tp/m Therefore K /sT = (p2/3)(kB/e)2 = 2.45 x 10-8WWK-2 (Lorentz number) The above result is called the Wiedemann-Franz law Measured values of the Lorentz number at 300K are Cu 2.23, In 2.49, Pb 2.47, Au 2.35 x 10-8WWK-2 (very good agreement)

38. Free electron model:Successes • Introduces useful idea of a momentum relaxation time. • Give the correct temperature dependent of the electronic specific heat • Good agreement with the observed Wiedemann-Franz Law for many metals • Observed magnitudes of the electronic specific heat and Hall coefficients are similar to the predicted values in many metals • Indicates that electrons are much more like free electrons than one might imagine.

39. Free electron model:Failures • Electronic specific heats are very different from the free electron predictions in some metals • Hall coefficients can have the wrong sign (as if current is carried by positive particles ?!) indicating that the electron dynamics can be far from free. • Masses obtained from cyclotron resonance are often very different from free electron mass and often observe multiple absorptions (masses). More than one type of electron ?! • Does not address the central problem of why some materials are insulators and other metals.

40. Energy band theory Solid state N~1023 atoms/cm3 2 atoms 6 atoms

41. Metal – energy band theory

42. Insulator -energy band theory

43. diamond

44. semiconductors

45. Intrinsic conductivity ln(s) 1/T

46. Extrinsic conductivity – n – type semiconductor ln(s) 1/T

47. Extrinsic conductivity – p – type semiconductor

48. Conductivity vs temperature ln(s) 1/T

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