1. Electrical Properties of Materials Conductivity, Bands & Bandgaps

2. Objectives To understand:  Electronic Conduction in materials  Band Structure  Conductivity l Metals l Semiconductors l Ionic conduction in ceramics  Dielectric Behavior l Polarization

3. Definitions Ohm’s Law  V = iR l V - Voltage, i - current, R -Resistance  Units l V - Volts l (or W/A (Watts/amp) or J/C (Joules/Coulomb)) l i - amps l (or C/s (Coulombs/second) R - ohms (  )

4. Definitions  Resistance Area Length i Consider current moving through a conductor with cross sectional area, A and a length, l R = V/i

5. Definitions  Conductivity,  :  Conductivity is the “ease of conduction” Ranges over 27 orders of magnitude!  = 1/  (units: (  -cm) -1 Conductivity Metals10 7 1/  cm Semiconductors 10 -6 - 10 4 1/  cm Insulators10 -10 -10 -20 1/  cm

6. Definitions  Electronic conduction: l Flow of electrons, e and electron holes, h  Ionic conduction l Flow of charged ions, Ag + Charge carriers can be electrons or ions

7. Electronic Conduction  In each atom there are discrete energy levels occupied by electrons  Arranged into: l Shells K, L, M, N l Subshells s, p, d, f

8. In Solid Materials  Each atom has a discrete set of electronic energy levels in which its electrons reside.  As atoms approach each other and bond into a solid, the Pauli exclusion principle dictates that electron energy levels must split.  Each distinct atomic state splits into a series of closely spaced electron states - called an energy band

9. Electronic Conduction Pauli Exclusion Principle - no two electrons within a system may exist in the same “state” All energy levels (occupied or not) “split” as atoms approach each other 1S 1 E A1A2 For two atomsFor many atoms

10. Banding Energy 1S 3S 3P 2P 2S 4S 3D Isolated Atom Energy 1S 3S 3P 2P 2S 4S 3D Bonded Atoms

11. Electronic Conduction Once states are split into bands, electrons fill states starting with lowest energy band. Electrical properties depend on the arrangement of the outermost filled and unfilled electron bands. “boxes of marbles analogy” Inter-atomic separation Equilibrium Separation Band Gap

12. Band Structure  Valence Band l Band which contains highest energy electron  Conduction Band l The next higher band Insulator filled empty filled empty Metal filled empty Valence Band Conduction Band Semiconductor

13. Band Structure  Fermi Energy, E f l Energy corresponding to the highest filled state  Only electrons above the Fermi level can be affected by an electric field (free electrons) EfEf E

14. Conduction in Metals- Band Model  For an electron to become free to conduct, it must be promoted into an empty available energy state  For metals, these empty states are adjacent to the filled states  Generally, energy supplied by an electric field is enough to stimulate electrons into an empty state

15. Resistivity,  in Metals  Resistivity typically increases linearly with temperature:  t =  o +  T l Phonons scatter electrons  Impurities tend to increase resistivity: l Impurities scatter electrons in metals  Plastic Deformation tends to raise resistivity l dislocations scatter electrons

16. Temperature Dependence, Metals There are three contributions to   t due to phonons (thermal)  i due to impurities  d due to deformation (not shown)  =  i +  o +  d

17. Electrical Conductivity, Metals For charge transport to occur - must have: - something the carry the charge - the ability to move  = conductivity = 1/   = ne 

18. Electrical Conductivity, Metals   = electrical conductivity  n = number of concentration of charge carriers l depends on band gap size and amount of thermal energy   = mobility l measure of resistance to electron motion - related to scattering events - (e.g. defects, atomic vibrations) “highway analogy”  = ne 

19. Temperatures Dependence, Metals  Metals,  decreases with T (  = ne  Two parameters in Ohm’s law may be T dependent: n and   Metals - number of electrons (in conduction band) does not vary with T. n = number of electrons per unit volume n  10 22 cm -3 and  10 2 -10 3 cm 2 /Vsec  10 5 -10 6 (ohm-cm) -1 All of the observed T dependence of  in metals arises from 

20. Semiconductors and Insulators  Electrons must be promoted across the energy gap to conduct  Electron must have energy: l e.g. heat or light absorptrion  If gap is very large (insulators) l no electrons get promoted low electrical conductivity, 

21. Semiconductors  For conduction to occur, electrons must be promoted across the band gap Energy is usually supplied by heat or light Note - electrons cannot reside in gap

22. Thermal Stimulation Suppose the band gap is Eg = 1.0 eV P = number of electrons promoted to conduction band

23. Stimulation of Electrons by Photons Photoconductivity Conductivity is dependent on the intensity of the incident electromagnetic radiation E = h = hc  c  m  (sec -1 ) h  E g

24. Stimulation of Electrons by Photons Provided Band Gaps: Si -1.1 eV (Infra red) Ge0.7 eV (Infra red) GaAs1.5 eV (Visible red) SiC3.0 eV (Visible blue) (If incident photons have lower energy, nothing happens when the semiconductor is exposed to light.) h  E g

25. Intrinsic Semiconductors  Intrinsic Semiconductors  Once an electron has been excited to the conduction band, a “hole” is left behind in the valence band Since neither band is now completely full or empty, both electron and hole can migrate

26. Conductivity of Intrinsic S.C.  Intrinsic semiconductor l pure material  For every electron, e, promoted to the conduction band, a hole, h, is left in the valence band (+ charge) Silicon - 1.1 eV Germanium - 0.7 eV Total conductivity  =  e +  h = ne  e + ne  h For intrinsic semiconductors: n = p &  = ne(  e +  h )

27. Extrinsic Semiconductors  Extrinsic semiconductors l impurity atoms dictate the properties  Almost all commercial semiconductors are extrinsic  Impurity concentrations of 1 atom in 10 12 is enough to make silicon extrinsic at room T!  Impurity atoms can create states that are in the bandgap.

28. Types of Extrinsic Semiconductors  In most cases, the doping of a semiconductor leads either to the creation of donor or acceptor levels p-Type semiconductors In these, the charge carriers are positive n-Type semiconductors In these, the charge carriers are negative

29. Silicon  Diamond cubic lattice  Each silicon atom has one s and 3p orbitals that hybridize into 4 sp 3 tetrahedral orbitals  Silicon atom bond to each other covalently, each sharing 4 electrons with four, tetrahedrally coordinated nearest neighbors.

30. Silicon n-type semiconductors:  Bonding model description: l Element with 5 bonding electrons. Only 4 electrons participate in bonding the extra e - can easily become a conduction electron p-type semiconductors:  Bonding model description: l Element with 3 bonding electrons. Since 4 electrons participate in bonding and only 3 are available the left over “hole” can carry charge

31.  In order to get n-type semiconductors, we must add elements which donate electrons i.e. have 5 outer electrons. l Typical donor elements which are added to Si or Ge: l Phosphorus l Arsenic l Antimony l Typical concentrations are ~ 10 -6 Doping Elements, n-Type Group V elements

32. Doping Elements, p-type  To get p-type behavior, we must add acceptor elements i.e. have 3 outer electrons. l Typical acceptor elements are: l Boron l Aluminum l Gallium l Indium Group III elements

33. Location of Impurity Energy Levels  Typically,  E ~ 1% E g  E g  E  E

34. Conductivity of Extrinsic S.C.  There are three regimes of behavior: It is possible that one or more regime will not be evident experimentally

35. n-Type Semiconductors  Band Model description: l The dopant adds a donor state in the band gap Band Gap Donor State If there are many donors n>>p (many more electrons than holes) Electrons are majority carriers “n-type” - (negative) semiconductor  =  e +  h = ne  e + ne  h  ≈ neu

36. p-Type Semiconductors  Band Model description: l The dopant adds a acceptor state in the band gap Band Gap Acceptor State If there are many acceptors p>>n (many more electrons than holes) holes are majority carriers “p-type” - (negative) semiconductor  =  e +  h = ne  e + ne  h  ≈ peu

37. III-V, IV-VI Type Semiconductors  Actually, Si and Ge are not the only usuable Semiconductors  Any two elements from groups III and Vor II and VI, as long as the average number of electrons = 4 and have sp 3 -like bonding, can act as semiconductors. l Example: Ga(III), As(V) GaAs Zn(II), Se(VI) ZnSe  Doping, of course, is accomplished by substitution, on either site, by a dopant with either extra or less electrons. In general, “metallic” dopants will substitute on the “metal” sites and “non-metallic” dopants will substitute on non-metal sites. For the case where the dopant is between the two elements in the compound, substitution can be amphoteric (i.e. on both sites)  Question: Give several p-type and n-type dopant for GaAs and ZnSe. What kind of dopant is Si in InP?

38. Ionic Conduction  Cations and Anions are Charged l In and Electric field they can migrate CURRENT! l Cations and anions migrate in opposite directions  The total conductivity: n = number of ions e = charge per ion  total =  electronic +  ionic  = ne 

39. Ionic Mobility Mobility of an Ionic species: Ionic conduction usually increases with T - number of carriers goes up and diffusion goes up

40. Dielectric Behavior  Dielectrics - electrical insulating l Dipoles exist - there is a separation of + and - charges l Dipoles interact with electric fields  Dielectric Materials - Capacitors

41. Dielectric Behavior  Capacitance Capacitance = Charge/Voltage = Coulombs/Volt = Farads If a vacuum is between plates:  o = permittivity of a vacuum C = Q/V C =  o A/l

42. If a dielectric material is between the plates: Dielectric Behavior  = dielectric medium permittivity Relative permittivity  r =  o Represents “charge storage capacity”

43. Typical Data for  Why is frequency important ? next few slides

44. What is Capacitance?  What is polarization? Polarization Vector, P Electric Dipole Moment (- to +) P = qd In the presence of an electric field, a force will tend to orient the electric dipole with the applied field The process of dipole alignment is called Polarization Surface charge density: D C/m 2   (electric field) D =  D = dielectric displacement

45. Types of Polarization  Electronic Polarization l may be induced (to some degree) in all atoms Displacement of the center of the negative electron cloud off the nucleus (only present when there is an electric field)

46. Types of Polarization  Electronic polarization

47. Types of Polarization  Ionic Polarization - Only occurs in ionic materials An applied field displaces cations in one direction and anions in another: No electric fieldElectric Field

48. Types of Polarization  Ionic Polarization

49. Types of Polarization  Orientation Polarization  Total Polarization: l P = P e + P i + P o l (electronic + ionic + orientation) Only in materials which possess permanent dipole moments No fieldElectric Field H O ++ -- example

50. Types of Polarization  Orientation (molecular) polarization

51. Frequency Dependence of the Dielectric Constant  Alternating Current (Applied voltage or electric field changes direction with time)  Dipoles try to reorient with field (This requires time)  Relaxation Frequency = 1/time to reorient

52.  Sometime dipoles can’t keep up with changing electric field: Frequency Dependence of the Dielectric Constant As frequency increases, dielectric constant decreases as orientation and ionic components go to zero

53. Barium Titanate  Perovskite Structure - excellent ionic polarization due to small Ti +4 “ball in cage”

54.  Perovskite Structure - permanent dipole created by Ti +4 offset.  Piezoelectric material l E-field dimensional change Barium Titanate

55. Superconductivity  Discontinuous reduction in resistivity  Resistivity < 13 orders of mag. Below Cu  Superconductivity lost  Above critical temperature T c, field H c and current density J c Hg Pt 4.2K Temp 

56. Meissner Effect  H < H c superconductors exclude magnetic field  Applications - Mag Lev trains

59. Ceramic Superconductors  Historical perspective l <1986 - highest T c known = 23.3K (Nb 3 Ge) l 1986 - LaBaCuO T c = 30 K l 1987 - YBa 2 Cu 3 O 7 T c = 95 K (above liquid N 2 ) l 1989 - TlBaCaCuO T c = 125 K  Holy Grail - room temperature superconducting material!!! l Computer interconnects (low current) l Transmission lines (high current)