1. Lecture V METALS dr hab. Ewa Popko

2. Measured resistivities range over more than 30 orders of magnitude Material Resistivity (Ωm) (295K) Resistivity (Ωm) (4K) 10 -12 “Pure” Metals Copper 10 -5 Semi- Conductors Ge (pure) 5  10 2 10 12 InsulatorsDiamond10 14 Polytetrafluoroethylene (P.T.F.E) 10 20 10 14 10 20 Potassium 2  10 -6 10 -10 Metals and insulators

3. Metals, insulators & semiconductors? At low temperatures all materials are insulators or metals. Semiconductors: resistivity decreases rapidly with increasing temperature. Semiconductors have resistivities intermediate between metals and insulators at room temperature. Pure metals: resistivity increases rapidly with increasing temperature. 10 20 - 10 10 - 10 0 - 10 -10 - Resistivity (Ωm) 1002003000 Temperature (K) Diamond Germanium Copper

4. Core and Valence Electrons Simple picture. Metal have CORE electrons that are bound to the nuclei, and VALENCE electrons that can move through the metal. Most metals are formed from atoms with partially filled atomic orbitals. e.g. Na, and Cu which have the electronic structure Na 1s 2 2s 2 2p 6 3s 1 Cu 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 1 Insulators are formed from atoms with closed (totally filled) shells e.g. Solid inert gases He 1s 2 Ne 1s 2 2s 2 2p 6 Or form close shells by covalent bonding i.e. Diamond Note orbital filling in Cu does not follow normal rule

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

6. Bound States in atoms Electrons in isolated atoms occupy discrete allowed energy levels E 0, E 1, E 2 etc.. The potential energy of an electron a distance r from a positively charge nucleus of charge q is V(r) E2E1E0E2E1E0 r 0 Increasing Binding Energy

7. Bound and “free” states in solids V(r) E2E1E0E2E1E0 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. r 0 0 + ++++ R Nuclear positions V(r) lower in solid (work function). Naive picture: lowest binding energy states can become free to move throughout crystal V(r) Solid

8. Energy Levels and Bands + E +++ + position Electron level similar to that of an isolated atom Band of allowed energy states. 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

9. 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’. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ++ + + + ++ + +++ + + + ++ + + ++++++ Na + ions: Nucleus plus 10 core electrons 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. (iii) 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. Ohms law and electron drift V = E/L = IR(Volts) Resistance R =  L/A(Ohms) Resistivity  = AR/L(Ohm m) E = V/L =  =  j(Volts m -1 ) Conductivity  (low magnetic field) j =  E(Amps m -2 ) I = dQ/dt(Coulomb s -1 ) Area A dx vdvd L Area A Electric field E Force on electron F Drift velocity v d Current density j = I/A n free electrons per m 3 with charge –e ( e = +1.6x10 -19 Coulombs ) Force on electrons F = -eE results in a constant electron drift velocity, v d. 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/   is called the ‘scattering time’, or ‘momentum relaxation time’. 3/  is independent of the initial electron momentum & energy.

14. Momentum relaxation Consider electrons, of mass m e, moving with a drift velocity v d 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  p  is the characteristic momentum or drift velocity relaxation time. p(t = 0) = m e v d (t = 0) dp/p(t) = - dt/  p  dp/dt = -p(t)/  p p(t) = p(0)exp(-t/  p  ) t/  p If, in a particular conductor, the average time between scattering events is  s and it average takes 3 scattering event to randomise the momentum. Then the momentum relaxation time is  p  3  s.

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 = -nev d = -nep/m e = (ne 2  p /m e ) E So the conductivity is  = j/E = ne 2  p /m e The electron mobility,  is defined as the drift velocity per unit applied electric field  = v d / E = e  p /m e (units m 2 V -1 s -1 )

16. The Hall Effect An electric field E x causes a current j x to flow. E x, j x EyEy BzBz v d = v x The Hall coefficient is R H = E y /j x B z = -1/ne The Hall resistivity is    = E y /j x = -B/ne j x = -nev x so E y = -j x B z /ne Therefore E y = +v x B z F = -e (E + v  B). In equilibrium j y = 0 so F y = -e (E y - v x B z ) = 0 A magnetic field B z 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 E y. j For a general v x. v x +ve or -ve

17. The Hall Effect The Hall coefficient R H = E y /j x B z = -1/ne The Hall angle is given by tan  = E y /E x =  H  For many metals R H 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. j=j x EyEy BzBz v d = v x ExEx E

18. Sign of Hall Effect E x, j x EyEy BzBz vdvd EyEy BzBz vdvd Hall Effect for free particles with charge +e ( “holes” ) Hall Effect for free particles with charge -e ( electrons ) E y = +v x B z = - v d B z j x = -nev x = ne v d E y = -j x B z /ne R H = E y /j x B z = -1/ne E y = +v x B z = v d B z j x = nev x = ne v d E y = j x B z /ne R H = E y /j x B z = 1/ne

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

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

22. k- space Free Classical Electrons states Defined by position (x,y,z) and momentum (p x, p y, p z ) Electron state defined by a point in k-space x z y pxpx pypy pzpz Free Quantum Electrons states Uniquely determined by the wavevector, k. Or equivalently by (p x, p y, p z ) = (  k x,  k y,  k z ). Equal probability of electron being anywhere in conductor. kxkx kyky kzkz

23. 3D analog of energy levels for a particle in a box!! Eigenstates & energies The allowed values of n x, n y, and n z, 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 n x, n y, and n z. The radius n rs of the sphere :

25. 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: Density of states

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

27. 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. f(E) Fermi-Dirac function for T=0. For T=0 all states are occupied up to an energy E F, called the Fermi energy, and all states above E F are empty. The probability of occupation of a particular state of energy E is given by the Fermi-Dirac distribution function, f(E).

28. The Fermi Energy Calculated E F 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 The number of occupied states per unit volume in the energy range E to E+dE is n(E)dE n(E) at T = 0

29. Free Electron Fermi Surface Metals have a Fermi energy, E F. Free electrons so E F =  2 k F 2 /2m At T=0 All the free electron states within a Fermi sphere in k-space are filled up to a Fermi wavevector,k F. The Fermi wavelength  k F The surface of this sphere is called the Fermi surface. On the Fermi surface the electrons have a Fermi velocity v F = hk F /m e. The Fermi Temperature,T F, is the temperature at which k B T F = E F. When the electron are not free a Fermi surface still exists but it is not generally a sphere.

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

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 = (k B T/2) x n x (number of degrees of freedom = 3) So the specific heat at constant volume C V = dU/dT= 3/2nk B. At room temperature the lattice specific heat, 3nk B ( n harmonic oscillators with 6 degrees of freedom). In most metals, at room temperature, C V is very close to 3nk B. The absence of a measureable contribution to C V 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 m 3 in a metal can be written as E = E o (T=0) +  E(T). Where E o (T=0) is the value at T=0. n(E)dE Each of these increases its energy by ~k B T The number of electrons that increase their energy is ~n(k B T/E F )where n is the number of electrons per m 3 At a temperature T only those electrons within ~ k B T of E F have a greater energy that they had at T=0.

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

34. Low Temperature 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. Metal  calc. (JK -1 mol -1 )  expt (JK -1 mol -1 ) Cu 5  10 -7 7  10 -7 Pb 1.5  10 -6 3  10 -6 where  and  are constants Predicted electronic specific heat

35. Dynamics of free quantum electrons Classical free electrons F = -e (E + v  B) = dp/dt and p =m e v. Quantum free electrons the eigenfunctions are ψ(r) = V -1/2 exp[i(k.r-  t) ] 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 /dt(Ehrenfest’s Theorem) for E =  2 k 2 /2m e 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 Densityj = -nev d Conductivity  = j/E = ne 2  p /m e Mobility  = v d / E = e  p /m e Hall coefficientR H = E y /j x B z = -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 = ΛvC v /3where v is the root mean square electron speed, Λ is the electron mean free path & C v is the electron heat capacity per m 3 For T<< T F we can set v = v F and Λ = v F  p  and C v = (    2) nk B (T/T F ) So K =  2 nk 2 B  p T/3mnow  = ne 2  p /m Therefore K /  T = (   / 3)(k B /e) 2 = 2.45 x 10 -8 W  K -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 -8 W  K -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. Solid state N~10 23 atoms/cm 3 2 atoms 6 atoms Energy band theory

41. Metal – energy band theory

42. Insulator -energy band theory

43. diamond

44. semiconductors

45. Intrinsic conductivity ln(  ) 1/T

46. ln(  ) Extrinsic conductivity – n – type semiconductor

47. Extrinsic conductivity – p – type semiconductor

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

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