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Microelectronics 2

Electrical Engineering 2. Lecture 6. Microelectronics 2. Dr. Peter Ewen. (Room G08, SMC; email - pjse). MOBILITY. The MOBILITY ,  , is a measure of the ease with which an electron can move through the solid.

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Microelectronics 2

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  1. Electrical Engineering 2 Lecture 6 Microelectronics 2 Dr. Peter Ewen (Room G08, SMC; email - pjse)

  2. MOBILITY The MOBILITY,  , is a measure of the ease with which an electron can move through the solid. Electron movement is hindered by collisions, i.e. by SCATTERING. Electrons are scattered by: (A) The thermal vibrations of the atoms (i.e. PHONONS) – this is termed LATTICE SCATTERING.

  3. (B) DEFECTS (e.g. impurities) – DEFECT SCATTERING or IMPURITY SCATTERING. V: a vacancy – a place where an atom is missing. I: an interstitial – an extra atom that is residing in the space between regular atoms. D: a dislocation – here a region of the crystal has slipped relative to a neighbouring region. Im: an impurity – e.g. a donor or acceptor atom in the case of an extrinsic semiconductor V Im I D Fig. 30: Defect examples

  4. vd – drift velocity E – electric field def  = vd /E Note: • A mobility can also be defined for holes. • Units for mobility are: m2V-1s-1 • Mobility depends on temperature. For lattice scattering, mobility decreases as temperature increases because there are more thermal vibrations around at higher temperatures. For lattice scattering:l = KlT-3/2 l – indicates lattice scattering Kl – constant T – temperature in K

  5. Fig. 31 Low temperature – electron moving slowly High temperature – electron moving fast + + ionised donor atom + For impurity scattering: i = KiT3/2 For a metal:   T-1

  6. l CONDUCTION BY CHARGE DRIFT E (F = qE) Electron motion during time interval, t, in zero field. Electron motion during time interval, t, in field E. DRIFT VELOCITY: vd = l / t

  7. CONDUCTIVITY  – conductivity E – electric field J = I / A – current density J = E Ohm’s Law Metals:  = ne Semiconductors:  = nee + peh (intrinsic or extrinsic) special cases Intrinsic Semiconductors:  = eni(e + h) (n = p = ni for intrinsic) n-type:  ≈ nnee≈ NDee (nn >> pn for n-type) p-type:  ≈ ppeh≈ NAeh (pp >> np for n-type)

  8. LECTURE 6 Influence of temperature on resistivity/conductivity  Metals • Intrinsic semiconductors • Extrinsic semiconductors  Superconductivity

  9. 12. Mobility When 3 V is applied across the faces of a 2 mm thick wafer of pure Si at 300 K the electrons are found to drift a distance of 10-3 m in 5 s. Determine the electron mobility. Assuming that lattice scattering predominates over impurity scattering, what would the electron mobility be at 100 K?

  10. 2mm 12. Mobility 3V

  11. Assuming that lattice scattering predominates over impurity scattering, what would the electron mobility be at 100 K?

  12. 13. Ohm's Law in terms of current density Show that Ohm's Law can also be written as J = E where J is the current density,  the conductivity of the material, and E the electric field within the material.

  13. 13. Ohm’s Law in terms of current density l R A I V

  14. 14. Conductivity of a metal Show that the conductivity of a metal is given by  = ne where n is the (free) electron concentration and  is the electron mobility.

  15. 14. Conductivity of a metal metal wire A vdt I P Let: A be the cross sectional area of the wire n be the free electron concentration in the wire vdbe the electron drift velocity Consider a fixed point, P, along the wire and work out the amount of charge passing this point in time, t: the volume of electrons passing point P in time t will be vdt  A the number of electrons passing point P in time t will be vdt  A  n the amount of charge passing point P in time t will be vdt  A  n  e

  16. the amount, Q, of charge passing point P in time t will be vdt  A  n  e

  17. Resistance Resistance Resistance Temperature Temperature Temperature Variation of resistance with temperature for a solid Resistance Resistance Resistance Temperature Temperature Temperature

  18. For a metal:  = ne •  = -1 = 1 / ne • But for a metal • n is constant •   T-1 Hence   T Fig. 33: Resistivity vs temperature for a metal

  19. ln() Slope = –Eg/2k -ve TCR High T Low T 1/T Fig. 34(a): Conductivity vs inverse temperature for an intrinsic semiconductor Intrinsic Semiconductors ln(o)  = eni(e + h) e and h  T-3/2 ni  T3/2 exp (–Eg /2kT)   = oexp (–Eg /2kT)  ln() = ln(o) – Eg /2kT

  20. Intrinsic Semiconductors ln()  = 1/ = 1/[oexp (–Eg /2kT)]   = oexp (Eg /2kT) or ln() = ln(o) + Eg /2kT Slope = Eg/2k -ve TCR ln(o) High T Low T 1/T Fig. 34(b): Resistivity vs inverse temperature for an intrinsic semiconductor

  21. Extrinsic Semiconductors Doping density (m-3) Electron concentration ,n / m-3 ND 1019 ni -200 -100 0 100 200 300 Temperature, T / C Fig. 35: Resistivity vs temperature for n-type Si Room Temperature ln() / m Intrinsic Si ln() ~ 1/T 105 104 103 102 10 1 • Consider n-type Si: • = 1/ ≈ 1/(nnee)   T3/2 TCR -ve 1020 TCR +ve 1021 1022 1023 1024 1025 TCR zero Degenerate case For low/moderate doping   T-3/2 -200 -100 0 100 200 300 Temperature / C

  22. The table gives the resistance of four different samples at various temperatures. There is: • one metal, • one intrinsic semiconductor, • one n-type semiconductor with ND = 1025 m-3, • one p-type semiconductor with NA = 1020 m-3. • Which sample is the p-type semiconductor?

  23. Sample A is a metal because resistance  T • For a metal:  = ne •  = -1 = 1 / ne • But for a metal • n is constant •   T-1 Hence   T Fig. 33: Resistivity vs temperature for a metal

  24. Doping density (m-3) Electron concentration ,n / m-3 ND 1019 ni -200 -100 0 100 200 300 Temperature, T / C Extrinsic Semiconductors Fig. 35: Resistivity vs temperature for n-type Si Room Temperature ln() / m Intrinsic Si ln() ~ 1/T 105 104 103 102 10 1 Sample C • Consider n-type Si: • = 1/ ≈ 1/(nnee)   T3/2 Sample B 1020 1021 1022 1023 1024 1025 Sample D Degenerate case For low/moderate doping   T-3/2 -200 -100 0 100 200 300 Temperature / C

  25. SUPERCONDUCTIVITY Discovered in 1911 by H. Kamerlingh-Onnes Before 1986 Tc < 23 K 1986 Tc ~ 35 K 1987 Tc ~ 90 K 20?? Tc ~ 300 K 1994 Tc ~ 153 K Fig. 36: Loss of resistance in a superconductor at low temperatures Resistivity  Tc – critical temperature 0 Tc Temperature  0

  26. Nobel Prize-winners in the field of Superconductivity 1913 – H. Kamerlingh-Onnes, discoverer of the effect in Hg. 1972 – John Bardeen, Leon Cooper and Robert Schrieffer, theoreticians who developed the first correct theory of the effect. (This was Bardeen’s second Nobel Prize – he also got one in 1956 for co-inventing the transistor.) John Bardeen, William Shockley and Walter Brattain – inventors of the transistor, for which they were awarded the Nobel Prize for Physics in 1956.

  27. Nobel Prize-winners in the field of Superconductivity 1913 – H. Kamerlingh-Onnes, discoverer of the effect in Hg. 1972 – John Bardeen, Leon Cooper and Robert Schrieffer, theoreticians who developed the first correct theory of the effect. (This was Bardeen’s second Nobel Prize – he also got one in 1956 for co-inventing the transistor.) 1987 – Alexander Muller and Georg Bednorz, IBM Zurich, discoverers of the effect in ceramic materials at 35 K. 20?? – “For the discovery of a room temperature superconductor, this year’s Nobel Prize in Physics goes to...”

  28. Superconductivity occurs in metals, alloys, semiconductors and ceramics. Nature 444, 465-468 (23 November 2006) Tc = 0.35K

  29. Electron pairs (“Cooper pairs”) form:  each member of the pair must have equal and opposite momentum  so momentum of the pair remains unchanged by collisions, i.e. no scattering occurs Leon Cooper

  30. The Cooper Pairs take a finite time to respond to any change in electric field so for an A.C. signal the current will lag the voltage – the superconductor has an inductance L associated with it • Below Tc Cooper pairs exist alongside ordinary electrons. The superconductor can be represented by:  = 0 for D.C.  ≈ 0 for A.C. Represents the normal electrons R resistanceless wire Fig. 37 L (perfect inductor) Represents the Cooper Pairs

  31. The Meissner Effect Magnetometer 57.3 Fig. 38: Expulsion of magnetic flux for T < Tc Magnetic field Superconductor Tc

  32. APPLICATIONS OF SUPERCONDUCTIVITY • High-Field Electromagnetics Mag-lev train based on superconducting magnets. Superconducting magnet from a particle accelerator. MRI body scanner using superconducting magnets (Magnetic Resonance Imaging)

  33. Device/Chip Interconnects Faster operation (no RC time delays) Less heat generation (no Joule heating) on-chip interconnects connections between chips on pcb • Lossless Power Transmission 24KV, 2400A superconducting power cable made by Pirelli Cables

  34. SUMMARY INFLUENCE OF TEMPERATURE ONRESISTIVITY / CONDUCTIVITY The temperature dependence of the resistivity (or conductivity) of a solid is governed by the temperature dependence of the mobility and carrier concentration. METALS  = -1 = 1 / ne: n is constant and   T-1 Hence   T

  35. ln() Slope = –Eg/2k -ve TCR High T Low T 1/T INTRINSIC SEMICONDUCTORS  ln() [or ln()] is proportional to 1/T (i.e. INVERSE TEMPERATURE)  Plots of ln() [or ln()] vs. 1/T are important because they can be used to obtain Eg

  36. Doping density (m-3) 1019 EXTRINSIC SEMICONDUCTORS Room Temperature ln() / m Intrinsic Si ln() ~ 1/T The temperature dependence of the resistivity depends on the temperature range and doping concentration. 105 104 103 102 10 1   T3/2 • For light/moderate doping & temperatures below the onset of intrinsic behaviour:  T3/2 -200 -100 0 100 200 300 Temperature / C

  37. Doping density (m-3) 1019 EXTRINSIC SEMICONDUCTORS Room Temperature ln() / m Intrinsic Si ln() ~ 1/T The temperature dependence of the resistivity depends on the temperature range and doping concentration. 105 104 103 102 10 1   T3/2 • For temperatures above the onset of intrinsic behaviour the resistivity is essentially that of the intrinsic material. -200 -100 0 100 200 300 Temperature / C

  38. Doping density (m-3) 1019 EXTRINSIC SEMICONDUCTORS Room Temperature ln() / m Intrinsic Si ln() ~ 1/T The temperature dependence of the resistivity depends on the temperature range and doping concentration. 105 104 103 102 10 1   T3/2 1020 1021 • As doping is increased,  decreases and the plots move down the ln() axis. 1022 -200 -100 0 100 200 300 Temperature / C

  39. Room Temperature ln() / m Intrinsic Si ln() ~ 1/T Doping density (m-3) 105 104 103 102 10 1   T3/2 1019 1020 1021 1022 1023 1024 1025 Degenerate case -200 -100 0 100 200 300 Temperature / C EXTRINSIC SEMICONDUCTORS The temperature dependence of the resistivity depends on the temperature range and doping concentration. • For heavy doping, the temperature dependence of the mobilities for lattice and impurity scattering cancel and  is constant.

  40. Resistivity  Tc – critical temperature 0 Tc Temperature  0 SUPERCONDUCTIVITY • D.C. resistivity drops to ZERO at Tc– the CRITICAL TEMPERATURE (There is a small A.C. impedance.) • Tc's are now > liquid nitrogen temperature 77K • Superconductivity occurs in metals, alloys semiconductors and ceramics. • The MEISSNER EFFECT can be used to detect the onset of superconductivity. • Device applications - interconnects

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