1. MSE 141 Electronic Materials Semiconductors and Devices Dr. Benjamin O. Chan Associate Professor February 2012

2. Problem Set 3 • Hummel 3rd edition • Chapter 8 • 1, 4, 6, 7, 9, 13, 15, 22 • Due Feb. 16 • Final Exam • Thursday Feb. 23

3. Semiconductors • Active electrical component • Metals and their alloys are usually passive • Non-linear I-V characteristics • Diodes, transistors, lasers • Sensors, amplifiers, light source • Can manipulate conductivity • Si the most common material • Ge: first material used • Other semiconductors • GaAs, AlGaAs, GaN, InP, CdTe

4. Si Success • Ease of oxidation • Wet or dry oxidation • Natural oxide forms after a few days when exposed to air • SiO2 layer provides good insulating and masking functions • Photolithography • Masking function • Circuit Fabrication • Electrical insulation

5. Bond Modification • Hybrid states • 2 s states and 6 p states combine to form 8 sp states • 8 sp states split into 2 branches containing 1 s and 3 p states • Lower s state: 1 electron per atom • Lower p states: 3 electrons per atom • Valid for covalently bonded materials

6. Overlapping sp Levels

7. Energy Bands • Valence band contains 4N electrons • Completely filled • Conduction band can accommodate 4N electrons • But of course it is empty! • States are filled like water fills a vessel!

8. Band Structure of Si

9. Energy Gap at 0 K

10. The Energy Gap • Decreases with atomic number • Artificial diamond: C-based IC • Ge: good IR detector • Sn easily conducts at room temperature • Wavelength equivalent

11. Temperature Dependence • where Eg(0) is the energy gap at 0 K •  = 5 x 10-4 eV/K and • D = Debye temperature (Table 19.2) • Temperature needed to reach 96% of the final value for heat capacity CV • T> D : classical region • T< D : QM consideration

12. Eg vs. T • Eg decreases with increasing temperature • For Si, • D=650K, • Eg reduction = -2.4 x 10-4 eV/K

13. Intrinsic Semiconductors • Intrinsic = pure • Electrical conduction • Electrons must be excited from the valence to the conduction band • Small Problem • Thermal energy at room temperature (kBT) is only 25.8 meV. How can electrons cross Eg (1.10 eV for Si at T=300K)?

14. Ec Ev Interband Transition • Electron goes from valence to conduction band • “hole” appears in valence band • Intrinsic carrier concentration (298 K) • Si: 1.5 x 1010 cm-3 • GaAs: 1.1 x 106 cm-3 • Seatwork • Determine the concentration of intrinsic carriers relative to the number of atoms in Si and GaAs.

15. Fermi Energy • Probability of occupation= F(EF) = ½ • T = 0K • E<Ev: F(E)=1 • E>Ec: F(E)=0 • EF located in the middle of the energy gap • At higher temperatures F(E) can become less than 1 for E<Ev and F(E) can be bigger than 0 for E>Ec

16. Fermi Energy

17. Seatwork/Homework • Using the Fermi distribution, calculate the probability of occupation for E > Ec for Si and GaAs at room temperature.

18. Why should EF be in the middle of the energy gap?

19. Density of Electrons and Holes

20. Electrons and Holes • Number of electrons in the conduction band = number of holes in the valence band • Expression for Nh similar to Ne • Although me* ≠ mh*

21. Intrinsic Carrier Concentration

22. Mobility m • Conductivity is not determined by carrier concentration alone! • Drift velocity v per unit electric field E

23. Conductivity • Recall and • Substituting m expression,

24. Temperature Dependence of Mobility, Carrier Density and Conductivity

25. Extrinsic Semiconductors • Extrinsic=Impure • No such thing as pure semiconductor • Defective by nature • Doping=Intentionally Impure • Group III impurity • Missing electron=hole=p-type • Group V impurity • Extra electron=n-type

26. Group V Impurity in Si

27. Intentional Impurities • Usually in the ppm range • Group V • Donates an electron to the material • Impurity energy level just below conduction band • Group III • Accepts an electron from the material • Impurity energy level just above valence band

28. Donor and Acceptor Levels

29. Majority and Minority Carriers • Majority carrier • Usually from dopant (extrinsic carriers) • Minority carrier • Usually from interband transitions (intrinsic carriers) • Valid for reasonably low T • The picture can change at high T • Intrinsic carriers become the majority!

31. How many carriers can you put in the material? • Limited by solid solubility curve for impurity • Hume Rothery rules! • Crystal structure should be preserved • Complete solubility not required!

32. Solid Solubilities of Impurities

33. Carrier Concentration • Intrinsic • n-type • p-type

34. Carrier Concentration vs. Temperature

35. Conductivity and Temperature

36. Fermi Energy Changing with Doping and Temperature

37. Determining Carrier Concentration • Indirect measurement • Resistivity is easier to measure compared to conductivity • Correlate resistivity with carrier concentration • Direct Measurment • Hall Effect

38. sample ammeter voltmeter Resistivity Measurements

39. Problems with Resistivity Measurements • Geometrical shape required for specimen • Uniform cross-sectional area • Contact resistance • Loading effects from meter • Voltmeter: should not draw current • High impedance required • Ammeter: should not impede current flow • Zero resistance desired • Stray capacitance and inductance

40. Four Point Probe Technique(ASTM F84-84, F374-84) • Suitable for wafer geometry • Rectangular/circular/cylindrical/slab • Minimizes contact resistance and loading effects • Four probes separated equally by distance s • Outer probes: constant current supply • Inner probes: voltmeter • Sheet resistance Rs

41. Four Point Probe • Correction Factors • Geometrical in nature • Something to do with s and how big the sample dimensions are with respect to it • Computing resistivity • w=sample thickness

42. Correction Factor Limit • Rs formula valid for w « a or d • BONUS PROBLEM • In the limit as d»s, show that CF = (/ln 2) = 4.54

43. Why do you need a constant current source for the four point probe set-up? (Why is a constant voltage source not sufficient?)

44. Resistivity and Carrier Concentration

45. Empirically Generated Curves! • There are other ways to determine carrier concentration! • Non-destructive • Destructive • Rutherford Backscattering (RBS) • Secondary Ion Mass Spectrometry (SIMS) • Neutron Activation Analysis (NAA)

46. Carrier Type Determination • Is the material n-type or p-type? • Seebeck Effect • Double thermocouple • Works for • Bulk material • Films on high resistivity material • Films on opposite type substrate

47. Voltmeter/Ammeter Hot Probe (ASTM F42-77(87)) +

48. How it works… • Hot probe has greater carrier concentration • Positive terminal of meter must be on the hot probe • If p-type, electrons flow from cold to hot junction: meter records positive reading • If n-type, electrons from flow from hot to cold: meter records negative reading • If the tester leads are inverted, the deflections will be inverted! (this can be confusing!?)

49. Determining Resistivity and Carrier Type Simultaneously • Hall Effect Apparatus • Can measure concentrations down to 1012 electrons/cm3 • Sensitivity is several orders of magnitude better than any chemical analysis • Applies to bulk material • Pass a current in a material immersed in a magnetic field

50. Hall Effect • Consider a rectangular block of n-type semiconductor immersed in a magnetic field with a magnetic induction B in the z-direction • Let a current density j flow through the material in the +x direction • Flowing electrons experiencea Lorentz force deflecting them from their path