1. Fys4310 doping DOPING

2. Diffusion program I Ficks laws Intrinsic diffusion - extrinsic diffusion Solutions of Fick’s laws ’predeposition’, ’drive-in’ electrical field enhancement Concentration dependant diffusion ( As) phenomenological atomistic Si-Self diffusion Vacancy diffusion Charged vacancy model Impurity diffusion Vacancy -dopant interactions

3. Diffusion program II Numerical solutions of Ficks laws Computer packages Math & practical experimental./ref.reading Measurements of diffusion SIMS, Radio tracer, RBS Differential Hall, SRM, CV, stain etch High concentration effects Pairs, quartets, defects Materials science Bolzman Matano method Diffusion equipment, furnaces, boilers, sources

4. Diffusion program III B in Si As in Si P in Si, high concentration Emitter push Zn in GaAs Examples etc. These points/topics are woven into other headings High doping effects Solubility, Metastable doping Dopant interactions Defects introduced in diffusion doping, Stresses Defining doping areas, masking, Diffusion in SiO 2 vs Si Diffusion of metallic impurities

5. Diffusion phenomenological Ficks laws Ficks 1st lov Defines D phonomenologic Continuity equation Ficks 2nd law J: flux

6. Continuity equation should be well known Derivation One dimentional J: flux C: concentration t : time xx+dx J out J in

7. Diffusion in general microscopically Random walk Brownian motion Microscopic lattice modelling Comparison gives D Phenomenologically Ficks 1. og 2. a: jump distance v:trial rate Z: geometry E a = E m +E v : migr.+vac 6: std for cubic

8. Analytic solution diffusion equation 1 Boundary cond. - ”predeposition” from gas or solid source C(z,0) =0 C(0,t)=C s C(∞,t)=0 Solution Total amount

9. Analytic solution diffusion equation 2 Boundary cond. - ”drive - in”  on surface Solution no flux predep Drive in Dt pre <<Dt inndr

10. Curve shape for idealized case INTRINSIC Diffusion Fig. 3-7 a b, Cambell Pre deposition Drive in

11. Electric field assisted diffusion - doping Elec field Flux

12. Atomic diffusion -self diffusion Usually requires little energy exhange vacancy- diffusion 1 2 1 2 Interstitial diffusion V V Interstitials can knock out regular atoms - interstitialcy

13. Atomic diffusion Interstitials can move from on interstitial position to the next- Interstitial diff. Usually requires little energy substitutional interstitial Si Int O, Fe, Cu, Ni, Zn Sub P, B, As, Al, Ga, Sb, Ge Substitutionals may move in various ways exchange vacancy- diffusion 1 2 1 2 V V

14. Atomic diffusion Fig 3.6 Fig 3.5 Various diffusion mechanisms for doping atoms B and P may diffuse this way -also depends whether have Si(I) or V Frank-Turnbull ’kick out’ ’Interstitialcy’

15. Atomic diffusion Assume D Si can be measured Si self diffusion - guess vacancy diffusion Can be measured by labeling the Si atoms by radioactive Si* P conc.. D Si dependant on the doping density Why? Finds:

16. Atomic diffusion Vacancy diffusion Vacancy concentration depends on doping concentration Probability for jump via vacancy depends on vacancy concentration i.e. Diffusivity D depends on doping concentration Schematic D i y :diffusivity of Si by vac. w. charge y in intrinsic case

17. atomistic diffusion consequences For dopant atoms n=n(C)n: electron concentration p=p(C)C:doping concentration i.e. D=D(C) and C=C(x) ie EXTRINSIC DIFFUSJON Diffusion equationMust be solved numerically

18. Diffusion, atomistic model consequences Assume can measure DHow? From Measure D vs. n the we can determine D*, D - and D = Fit to y: charge *,-,+,=

19. Diffusion, atomistic model consequences Assume D can be measured, How? From With measurements of D vs. n you can determine D*, D -, D = Fit toy: charge state *,-,+,= Measured cm 2 /seV

20. Diffusion, typical profiles Fig 3.8 B high conc. B acceptor B - V - Fig 3.9 As high conc. As donor, As + As + V -, As + V + n>>n i : h=2,

21. Diffusion, High doping effects Cluster formation Dislocations Band gap shrinkage Segregation

22. Diffusion, High doping effects Dislocations Stress-no disloc. f.example B doping Disadvantage for stress in membranes for MEMS, ie for pressure sensors Stress in heavy B doping of Si tried compensated by adding Ge Elastic energy released by creating a dislocation

23. Diffusion, High doping effects Band-gap narrowing Very low doping conc. High doping conc. e-e int.act..Donor donor interaction. EdEd E g varies n i varies [V] varies.

24. Diffusion, High doping effects Band-gap narrowing, qualitatively? Van Overstraeten, R. J. and R. P. Mertens, Solid State Electron. 30, 11 (1987) 1077- 1087. For 10 17 ≤ N ≤ 3·10 17 cm -3.∆E g el ~ 3.5·10 -8 ·N d 1/3 (eV) (N d in cm -3 )

25. Diffusion, High doping effects Cluster formation Various reactions ln(C) ln(n) solubility VAs 2 -complexSi V As V = +2As + =VAs 2 kV l +mAs + +ye=(V K As m ) (m+lk-y) Complex neutral, el.neg, [V]>>[2V] Gives k=1 and m=1,2,4 Complexes/clusters Maybe precursor to segregation One model

26. Diffusion, High doping effects Cluster formation How to find out about cluster models.? ln(C) ln(n) solubility V 2 As 2 -complex 2V - +2As + =V 2 As 2 kV l +mAs + +ye=(V K As m ) (m+lk-y) Én model As Si V V

27. Diffusion, High doping effects Segregation Requires nucleation of new phase, i.e. the concentration must correspond to super saturation before precipitation occurs, The density of precipitates depends upon nucleation rate and upon diffusion. It means measurements of solubility can require patience particularly at low temperatures. r*r*r ∆G∆G surface volume bulk ∆G*

28. Diffusion, Numerical calculation considerations Finite differencing Solving the diffusion equations (by simulations) General methods Finite element Monte Carlo Spectral Variational

29. Diffusion, numeric ’finite difference’ Representations of time and space differentials example. diff.eq. Combining a and b giver representation FTCS Forward Time Centered Space Forward Euler differentiation, Accuracy to 1ste order ∆t NB Unstable* FTCS explicit so U j n+1 can be calculated for each j from known points Stability achieved byStability just as much art as science

30. Diffusion, numeric ’finite difference’ Leapfrog accuracy 2nd order in ∆t FTCS: Often stability means many time steps Here F: Flux C: Flux (explicitly) Stab. criteria: Diffusion equation in simple case

31. FTCS: =Crank-Nicholson method (explicit) (implicit) Backward time Combining the two (explicit) (implicit)Crank-Nicholson Allows large time-steps Diffusion, numeric ’finite difference’

32. Diffusion equation w. Crank -Nicholsen collect left and right ie

33. Diffusion equation w Crank -Nicholsen A and B tridiagonal i.e. Boundary cond, j=1 i.e. surface R=1 reflection R=0 trapping R=0.5 segregation coeff j=2 ie.

34. Diffusion equation w Crank -Nicholsen unknown = known If D=D(C) i.e. we have a nonlinear set of equations Can readily be solved when D only a function of x and t but not of C.

35. Diffusion, numeric examples schematic Implanted profile Vacancy creation on surface Vacancy annihilation in ion-damaged region Ge i Si Implanted profile Vacancy creation on surface Vacancy annihilation in ion-damaged region Surface segregation As i Si

36. Diffusion, numeric general unknown known If D=D(C) ie we have a nonlinear system of equations But assumeas 1st approx. then calculate C n+1, and put in A for next it. vector matrixInitial conditions This equation is solved for example by Newton’s method in the general case

37. Diffusion, numeric SOFTWARE PACKAGES SUPREM III (1D), SUPREM IV(2D) [Stanford] TSUPREM4, SSUPREM4[Silvaco], ATHENA[Silvaco] ICECREAM, DIOS, STORM Floops, Foods, ACES, ’curve-fitting' models / physical models 1D, 2D, 3D All processing stages/ all thermal/ just diffusion Integrated, -masks, device, system-simulation (icecream silvaco) Required for ULSI

38. Diffusion, Measurements of diffusion profiles, D(C) SIMS, TOFSIMS Neutron activation RBS, ESCA, EDAX IR abs. So measurement of C, and n el. P is needed In many situations we have C=C(x), We can find D(C) by measuring C(x) at different T Can find D*, D +, D -, D = from the dependency D(n) Example P in simple case Methods for CMethods for n Spreading Resistance Differential Hall i.e.. +strip Capacitance, C-V + strip SCM, SRM

39. Diffusion, Measurement of diffusion, process surveillance sphere Top view

40. Diffusion, Measurement of diffusion, process surveillance ’Four-point probe’ Measures so-called sheet resistance Can find resistivity if the depth distribution of n and µ is found Mostly for process surveillance-reproducibility tests

41. Diffusion, Measurements of diffusion profiles van der Peuw method 2 1 4 3 More common geometry 1 2 3 4

42. Diffusion, Measurement of electric profile Hall measurements Measurement-procedure Measure  s, R HS gives µ H Strip a layer Repeat to d Calc individual n and µ

43. Diffusjon, Measurement of electric profile Schottky p-n junction electrolyte

44. Diffusion, Measurement of diffusion profile Scanning Probe Microscopy SSRM SCM

45. Diffusion, Measurement of diffusion profile

47. Spreading resistance measurements beveling needle

48. Diffusion, Measurement of diffusion profile Scanning capacitance microscopy Electric

49. Diffusion, Measurement of diffusion profile Scanning capacitance microscopy Elektriske

50. Diffusion, Measurement of diffusion profile Scanning capacitance microscopy Elektriske

51. Diffusion, Measurement of diffusion profile SIMS,

52. Physical Electronics University of Oslo Equipment at MRL-UiO: Installed April-Aug. 2004 principle SIMS Cameca 6F UiO SIMS

53. Characteristics of SIMS: SIMS karakteristiske trekk Sensitivity: All elements can be detected Sensitivity depends on element. in Si: B, P 10 14 cm -3 Accuracy: Through standards ( not absolute spectro-scopy/-metry) Good reproducibility Matrix-effects: Ionization coeff. depends on surface, oxidation, sputter w. oxygen yields good reproducibility Interface-effects Depth resolution: Limited by sputter process, 2 nm - 50 nm ion mixing, segregation, run away erosion Well suited for measurements of diffusion profiles in high tech 1/2 conductors:

54. Diffusion, Measurement of diffusion profile RBS Absolute measurement Depth resolution 10-30 nm Sensitivity: Matrix effects. Heavy in light matrix 10 18 cm -3

55. Detection limits

56. Analysis depths

57. Diffusion, doping sources

58. Bolzman Matano method Point: Extract D(C) from C(x) Introduce a new coordinate system x’ by C=C(x’,t) is a function only of i.e. C=C( ) We may write Ficks 2 law by LHS RHS Setting LHS=RHS

59. Bolzman Matano method Here we have the same differential on both sides, so we integrate Integration between C=0 and C=C 1 and assume C 1 is one particular conc. we wish to investigate Slight rewriting: I Put in to [I ] and choose a particular experimentally measured time for diffusion t 1 : A B 1. x’ origo from B 2. dC/dx’ from curve 3. Int(x’,C=C 1..C 0 ) 1+2+3 gives D from A Method

60. Diffusjon, example P P donor, P + P + V -, P + V + ? observations Vacancies are created at the surface Conc of vac depends on n at surface V = + P + -> VP - E C -E F depends on depth when dC/dx <0 VP- breaks up -> exess vac’ Conc. of vacancy controlled by surface and not by local doping Fair -Tsai model

61. Diffusion, ”Emitter-push” Many models Fair-Tsai diff. Model for P + Band gap narrowing Describes results adequately

62. Diffusion, Example: Zn i GaAs Zn acceptor in GaAs, Substitutes for Ga V Ga +, V Ga *..Zn i +, Zn s - Confinement of waveguides, etc..