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04 Semiconductor in equilibrium

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  • 1. Chapter 4 The Semiconductor in Equilibrium 23/09/08 1Department of Microelectronics and Computer Engineering
  • 2. Topics • Thermal-equilibrium concentration of electron and holes • Intrinsic carrier concentration • Intrinsic Fermi-level position • Dopant atoms and energy levels • Extrinsic carrier concentration and temperature dependence • Ionization energy of dopant atoms in silicon • Fermi level in extrinsic semiconductor • Degenerate semiconductors • Fermi level in two systems in contact with each other and at thermal equilibrium. 23/09/08 2 Department of Microelectronics and Computer Engineering
  • 3. Why? • Current is determined by flow rate and density of charge carriers. • The density of electron and holes are related to the density of states function and the Fermi distribution (or probability) function. 23/09/08 3Department of Microelectronics and Computer Engineering
  • 4. Conduction band Valence band 23/09/08 4Department of Microelectronics and Computer Engineering
  • 5. Density of states * Density of quantum states per unit volume at energy EFor conduction band For valence band 23/09/08 5Department of Microelectronics and Computer Engineering
  • 6. Fermi-Dirac distribution (or probability) function The probability that a quantum state at an energy E will be occupied by an electron The ratio between filled and total quantum states at any energy E 23/09/08 6Department of Microelectronics and Computer Engineering
  • 7. Fermi-Dirac distribution (or probability) function 23/09/08 7Department of Microelectronics and Computer Engineering
  • 8. Distribution of electron and holesNumber of electrons at E n (E ) = g c (E ) fF (E )(in conduction band) Fermi-Dirac Density of states at E probability functionNumber of holes at E p ( E ) = g v ( E ) [1 − f F ( E ) ](in valence band) 23/09/08 8Department of Microelectronics and Computer Engineering
  • 9. conduction band valence band 23/09/08 9Department of Microelectronics and Computer Engineering
  • 10. Electron concentration ∫ T o p o f c o n d u c tio n b a n dn0 = D e n s ity o f s ta te s ∗ P ro b a b ility fu n c tio n d E B o tto m o f c o n d u c tio n b a n d ∞ EC The equation is valid for both intrinsic and extrinsic semiconductors 23/09/08 10 Department of Microelectronics and Computer Engineering
  • 11. Condition: E - EF >> kT Boltzman approximation 23/09/08 September/Oktober 2004 11Department of Microelectronics and Computer Engineering
  • 12. Comparison of Fermi-Dirac probability function and Maxwell-Boltzmann approximation 3kT Maxwell-Boltzmann approximation and Fermi-Dirac function are within 5% of each other when E-EF ≥3KT 23/09/08 12 Department of Microelectronics and Computer Engineering
  • 13. n0 Gamma function: 1 π 2 23/09/08 13Department of Microelectronics and Computer Engineering
  • 14. Nc= effective density of states function in the conduction band The equation is valid for both intrinsic and extrinsic semiconductors 23/09/08 14Department of Microelectronics and Computer Engineering
  • 15. Hole concentration 23/09/08 15Department of Microelectronics and Computer Engineering
  • 16. Nv= effective density of states function in the valence band 23/09/08 16Department of Microelectronics and Computer Engineering
  • 17. Effective density of states function and effective mass values Nc(cm-3) Nv(cm-3) mn*/m0 mp*/m0 Si 2.8 x 1019 1.04 x 1019 1.08 0.56 Gallium Arsenide 4.7 x 1017 7.0 x 1018 0.067 0.48 Germanium 1.04 x 1019 6.0 x 1018 0.55 0.37 23/09/08 17Department of Microelectronics and Computer Engineering
  • 18. Intrinsic semiconductor• Intrinsic electron concentration = Intrinsic hole concentration ni = pi Intrinsic carrier concentration Why? • charge carriers due to thermal excitation • thermally generated electrons and holes always created in pairs. 23/09/08 18 Department of Microelectronics and Computer Engineering
  • 19. INTRINSIC Semiconductor Intrinsic Fermi level Intrinsic carrier concentration 23/09/08 19Department of Microelectronics and Computer Engineering
  • 20. Commonly accepted values of ni at T=300K Semiconductor Ni Silicon 1.5 x 1010 cm-3 Gallium Arsenide 1.8 x 106 cm-3 Germanium 2.4 x 1013 cm-3 23/09/08 20Department of Microelectronics and Computer Engineering
  • 21. 1 Eg log10 (ni ) = log10 ( N c N v ) − log10 e 2 2kT Plot of log10(ni) vs. 1/T is straight line. Slope is negative. From the slope Eg can be calculated 23/09/08 21Department of Microelectronics and Computer Engineering
  • 22. Application of the intrinsic semiconductors • High Electron Mobility Transistor • High resistivity substrate for RF circuits • amorphous-Si Solar Cells Structure of solar cell 23/09/08 22Department of Microelectronics and Computer Engineering
  • 23. Where is the intrinsic Fermi level? Conduction band Valence band EFi (Intrinsic Fermi level): EF at which electron and hole concentration becomes equal 23/09/08 23Department of Microelectronics and Computer Engineering
  • 24. Electron concentration Hole concentrationEven in intrinsic semiconductor, Fermi level is not exactly at centre betweenconduction and valence bands. 23/09/08 24Department of Microelectronics and Computer Engineering
  • 25. 23/09/08 25Department of Microelectronics and Computer Engineering
  • 26. Intrinsic silicon lattice 23/09/08 26Department of Microelectronics and Computer Engineering
  • 27. Acceptor and Donor Impurities: • In Si four electrons in the valence shell participate in bonding. • atom with more than 4 valence electrons  donor impurity • less than 4  acceptor impurity. 23/09/08 27 Department of Microelectronics and Computer Engineering
  • 28. Donor Impurity: Sb Silicon lattice doped with donor impurity • At very low temperature, the donor (excess) electron is still bound to the impurity atom. • However, the donor electron is loosely bound to the impurity atom and can become free with small amount of thermal energy. Impurity atom is then ionized and positively charged. 23/09/08 28 Department of Microelectronics and Computer Engineering
  • 29. Donor electron energy level: • little energy required to move donor electrons from donor states to conduction band. • positively charged donor ions are fixed but donor electrons in the conduction band can move through the crystal. 23/09/08 29 Department of Microelectronics and Computer Engineering
  • 30. Acceptor Impurity:• One covalent bond is incomplete for Si.• With little thermal energy, a valence electron can break from another covalent bond andcan occupy this position, thus creating a hole at the location of the broken covalent bond.• The acceptor impurity is then ionized and negatively charged. 23/09/08 30 Department of Microelectronics and Computer Engineering
  • 31. Acceptor Energy Level: • little energy required to move valence electrons to acceptor levels. • negatively charged acceptor ions are fixed but holes in the valence band can move through the crystal. 23/09/08 31Department of Microelectronics and Computer Engineering
  • 32. Electron concentration vs. temperature in n-type semiconductor Electron concentration vs. temperature showing partial ionization, extrinsic and intrinsic regions. 23/09/08 32 Department of Microelectronics and Computer Engineering
  • 33. Electron concentration vs. temperature in extrinsic semiconductor• At low temperatures, donor impurities are partially ionized. Astemperature increases the percentage of ionized donor impurities alsoincreases• Once all donor impurities are ionized, there is no increase in carrierconcentration. Even though intrinsic carrier concentration continues toincrease, it is still small compared to extrinsic concentration.•At high temperatures, intrinsic carrier concentration dominates andelectron concentration continues to increase again. 23/09/08 33 Department of Microelectronics and Computer Engineering
  • 34. Ionization energy: The ionization energy is the energy necessary to remove an electron from the neutral atom. In case of donor atoms, the ionization energy is the energy necessary to elevate an electron from the donor level to conduction band. 23/09/08 34Department of Microelectronics and Computer Engineering
  • 35. Ionization energy:• In the next few slides, we will calculate the approximate ionizationenergy for donor atoms.•We use Bohr atomic model for these calculations. For hydrogenatom, Bohr model and quantum mechanics give similar results.•Donor impurity atom can be visualized as one donor electronorbiting the positively charged donor ion. This condition is similar tothat in a hydrogen atom.•However we have to consider permittivity of silicon instead ofpermittivity of free space. 23/09/08 35Department of Microelectronics and Computer Engineering
  • 36. Angular Momentum Quantization• Bohr proposed that circumference of electron orbit = integer number of wavelengths  Then angular momentum, angular momentum of electron is quantized. 23/09/08 36 Department of Microelectronics and Computer Engineering
  • 37. Ionization energy calculation: Coulomb attraction force Centripetal force Angular momentum quantization 23/09/08 37 Department of Microelectronics and Computer Engineering
  • 38. As defined in Chapter 2, Bohr radius = orbiting electron radius = Bohr radiusFor silicon, m* For n=1,ε r = 11.7 and = 0.26 m0 23/09/08 38Department of Microelectronics and Computer Engineering
  • 39. • r1/a0=45 or r1=23.9A0 • This radius ~ 4 lattice constants of Si. • Each unit cell contains 8 silicon atoms. • Donor electron thus loosely bound to the donor atom. • We will next find the approximate ionization energy. 23/09/08 39Department of Microelectronics and Computer Engineering
  • 40. Total energy Kinetic energy Potential energy ( refer slide 37) 23/09/08 40Department of Microelectronics and Computer Engineering
  • 41. • Ionisation energy of Hydrogen in lowest energy state = -13.6eV • For Si, it is -25.8meV << band gap. • Calculations using Bohr model give only the order of magnitude of the ionisation energy. Actual values differ. Impurity ionization energies in gallium arsenide Impurity Ionization energy (eV) Donors Impurity ionization energies in Silicon and Germanium Selenium 0.0059 Impurity Ionization energy (eV) Tellurium 0.0058 Si Ge Silicon 0.0058 Donors Germanium 0.0061Phosphorus 0.045 0.012Arsenic 0.05 0.0127 Acceptors Beryllium 0.028 Acceptors Zinc 0.0307 Cadmium 0.0347Boron 0.045 0.0104 Silicon 0.0345 23/09/08 41Aluminum 0.06 0.0102 Germanium 0.0404 Department of Microelectronics and Computer Engineering
  • 42. EXTRINSIC Semiconductor Majority carrier Minority carrier electrons electrons n-TYPE p-TYPE Majority carrier holes Minority carrier holes 23/09/08 42Department of Microelectronics and Computer Engineering
  • 43. The equation is valid for both intrinsic and extrinsic semiconductors 23/09/08 43Department of Microelectronics and Computer Engineering
  • 44. Another form (relation between E F and E Fi) Intrinsic carrier concentration n0p0=ni2 23/09/08 44Department of Microelectronics and Computer Engineering
  • 45. Where is the Fermi level? 23/09/08 45Department of Microelectronics and Computer Engineering
  • 46. Condition for theBoltzmann approximation EC-EF > 3KT 23/09/08 46Department of Microelectronics and Computer Engineering
  • 47. If the impurity concentration is very high.... Fermi level will be very close to conduction band or valence band. No Boltzmann approximation Use The Fermi-Dirac Integral 23/09/08 47Department of Microelectronics and Computer Engineering
  • 48. Boltzman approximation Only if EC-EF > 3KT 23/09/08 48Department of Microelectronics and Computer Engineering
  • 49. Fermi-Dirac Integral If ηF>1, then EF>EC 23/09/08 49Department of Microelectronics and Computer Engineering
  • 50. Degenerate Semiconductors 23/09/08 50Department of Microelectronics and Computer Engineering
  • 51. If the impurity atoms are very close each other... • Donor electrons interact with each other • The single discrete donor energy will split into a band • The band may overlap the conduction band • If the concentration exceed Nc, EF lies within the conduction band 23/09/08 51Department of Microelectronics and Computer Engineering
  • 52. Degenerated Semiconductor Nd > Nc Na > Nv Fermi level in the conduction band: Metallic conduction 23/09/08 52Department of Microelectronics and Computer Engineering
  • 53. Statistics of donors and acceptors 23/09/08 53Department of Microelectronics and Computer Engineering
  • 54. How many electrons still in the donor levels compared to the total number of electrons? depends on the temperature and the Fermi level.... 23/09/08 54Department of Microelectronics and Computer Engineering
  • 55. Probability function for donor & acceptor levelsDensity of electronsoccupying donor states Density of donor atoms This same as the Fermi-Dirac probability function except the pre-exponential coefficient of ½. Concentration of ionized donors 23/09/08 55 Department of Microelectronics and Computer Engineering
  • 56. similar for holes: g=degeneration factor; 4 for GaAs and Si acceptor levels 23/09/08 56Department of Microelectronics and Computer Engineering
  • 57. Low temperature Moderate temperature High temperature 23/09/08 57Department of Microelectronics and Computer Engineering
  • 58. Moderate temperature Ed-EF>>kTIf Ed-EF >> kT, then even Ec-EF >> KT Then, 23/09/08 58Department of Microelectronics and Computer Engineering
  • 59. Fraction of electrons still in the donor statesWith Phosphorus doping of Nd=1016cm-3, atT=300K, nd/(nd+n0)=0.41% Almost complete ionization at Room Temp! 23/09/08 59Department of Microelectronics and Computer Engineering
  • 60. Extremely low temperature (T=0K) nd=Nd Freeze-out EF > Ed 23/09/08 60Department of Microelectronics and Computer Engineering
  • 61. High temperature (because of thermally generated electrons) (because of thermally generated holes) • At very high temperature behavior is just like the intrinsic semiconductor 23/09/08 61Department of Microelectronics and Computer Engineering
  • 62. Complete ionizationFreeze-out 23/09/08 62Department of Microelectronics and Computer Engineering
  • 63. Compensated semiconductor• Both donor and acceptor impurities in the same region• If Nd > Na  n-type compensated semiconductor• If Nd < Na  p-type compensated semiconductor• If Nd = Na  completely compensated (will behave like intrinsic material)• Practical semiconductor is always compensated semiconductor. Eg. Substrate is predoped usually p-type. All other dopings are done on top of this. 23/09/08 63Department of Microelectronics and Computer Engineering
  • 64. Charge neutrality : Ionized acceptors Ionized donors 23/09/08 64Department of Microelectronics and Computer Engineering
  • 65. Recall Using the relation n0 is not simply Nd 23/09/08 65Department of Microelectronics and Computer Engineering
  • 66. Similarly in p-type semiconductor, 2 Na − Nd  Na − Nd  2 p0 = +   + ni 2  2 Minority carrier concentration 2 2 n ni p0 = i n0 = n0 po (n-type material) (p-type material) 23/09/08 66Department of Microelectronics and Computer Engineering
  • 67. 23/09/08 67Department of Microelectronics and Computer Engineering
  • 68. Where is the Fermi level of an extrinsic semiconductor? N-type: Nd>>ni then n0≈Nd 23/09/08 68 Department of Microelectronics and Computer Engineering
  • 69. Where is the Fermi level of a p-type extrinsic semiconductor? P-type: Na>>ni then p0≈Na  Nv  E F − Ev = kT ln  N   a 23/09/08 69 Department of Microelectronics and Computer Engineering
  • 70. (a) (b) Position of Fermi level for an (a) n-type and (b) p-type semicondcutor. 23/09/08 70Department of Microelectronics and Computer Engineering
  • 71. Different expression for the n-type... Another expression for the p-type  p0 E − E = k T ln    ni  Fi F 23/09/08 71Department of Microelectronics and Computer Engineering
  • 72. Variation of EF with doping concentration: 23/09/08 72Department of Microelectronics and Computer Engineering
  • 73. Variation of EF with temperature T  n0 E F − E Fi = kT ln   n   i  p0E − E = k T ln    ni  Fi F• At higher temperatures, thesemiconductor becomes moreintrinsic. ni increases and Fermi levelmoves towards mid-gap Variation of Fermi level with temperature for•At T=0, Fermi level is above Ed in n- different doping concentrationstype and below Ea in p-typesemiconductor 23/09/08 73 Department of Microelectronics and Computer Engineering
  • 74. EF must be equal when different systems are in contact and in thermodynamic equilibriumConsider a material A, with Fermi level material B with Fermi level EFB.EFA. Bands below EFA are full and aboveare empty. 23/09/08 74 Department of Microelectronics and Computer Engineering
  • 75. EF must be equal when different systems are in contact and in thermodynamic equilibrium • When A and B are brought in contact, electrons will flow from A into lower energy states of B, until thermal equilibrium is reached. • Thermal equilibrium  EF same in A & B 23/09/08 75Department of Microelectronics and Computer Engineering
  • 76. Summary  ( Ec − EF ) Holds for both • Electron concentration n0 = N c exp  −  intrinsic as well  kT  as extrinsic  ( E − Ev ) • Hole concentration p0 = N v exp  F  semiconductor  kT • Intrinsic carrier concentration :  ( Ec − Ev )   Eg  ni 2 = n0 p0 N c = v exp  N − N c Nv exp  = −  kT   kT • In intrinsic semiconductor, Fermi level is close to but not exactly in the centre between conduction and valence bands. 3  m p*  EFi − Emidgap = kT ln  *  4  m   n  23/09/08 76 Department of Microelectronics and Computer Engineering
  • 77. Summary• In extrinsic semiconductor, Fermi level is close to conduction band (n-type) or valence band (p-type)• Position of Fermi level in extrinsic semiconductor  n0  EF − EFi kT ln  =   ni • In compensated n-type semiconductor electron concentration is given by 2 N d − Na  Nd Na − 2 n0 =  +2 ni  + 2  • When two different systems are in contact and in thermal equilibrium, EF must be the same in both systems. 23/09/08 77 Department of Microelectronics and Computer Engineering

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