Fys4310 doping

1. FYS4310 DOPING Fys4310 doping DIFFUSION

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

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

4. Examples B in Si As in Si P in Si, high concentration Emitter push Zn in GaAs Diffusion program III 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 SiO2 vs Si Diffusion of metallic impurities

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

6. x x+dx Jin Jout qed J: flux C: concentration t : time Continuity equation should be well known One dimentional Derivation

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

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

9. Boundary cond. - ”drive - in” Analytic solution diffusion equation 2 don surface predep no flux Dtpre<<Dtinndr Drive in Solution

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

11. Elec field Electric field assisted diffusion - doping Flux

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

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

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

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

16. Schematic 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 Diy :diffusivity of Si by vac. w. charge y in intrinsic case

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

18. How? Assume D can be measured, From Diffusion, atomistic model consequences With measurements of D vs. n you can determine D*, D-, D= Fit to y: charge state *,-,+,= Measured cm2/s eV

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

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

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

22. Band-gap narrowing Diffusion, High doping effects Ed e-e int.act.. Donor donor interaction. High doping conc. Very low doping conc. Eg varies ni varies [V] varies.

23. Band-gap narrowing, qualitatively? Diffusion, High doping effects For 1017 ≤ N ≤ 3·1017 cm-3 .∆Egel ~ 3.5·10-8·Nd1/3 (eV) (Nd in cm-3) Van Overstraeten, R. J. and R. P. Mertens, Solid State Electron. 30, 11 (1987) 1077- 1087.

24. Cluster formation solubility ln(n) Diffusion, High doping effects Complexes/clusters Maybe precursor to segregation ln(C) Si VAs2-complex V=+2As+=VAs2 Various reactions As One model V kVl+mAs++ye=(VKAsm)(m+lk-y) As Si Complex neutral, el.neg, [V]>>[2V] Gives k=1 and m=1,2,4

25. Cluster formation solubility ln(n) Diffusion, High doping effects ln(C) V2As2-complex kVl+mAs++ye=(VKAsm)(m+lk-y) V How to find out about cluster models.? 2V-+2As+=V2As2 Én model As Si As V

26. 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. Diffusion, High doping effects surface ∆G* ∆G r* r volume bulk

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

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

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

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

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

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

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

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

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

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

37. 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) Diffusion, Measurements of diffusion profiles, D(C) So measurement of C, and n el. P is needed Example P in simple case Methods for C Methods for n SIMS, TOFSIMS Neutron activation RBS, ESCA, EDAX IR abs. Spreading Resistance Differential Hall i.e.. +strip Capacitance, C-V + strip SCM, SRM

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

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

40. 3 2 More common geometry Diffusion, Measurements of diffusion profiles 4 4 3 1 1 2 van der Peuw method

41. Hall measurements Diffusion, Measurement of electric profile Measurement-procedure Measure rs, RHS gives µH Strip a layer Repeat to d Calc individual n and µ

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

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

44. Diffusion, Measurement of diffusion profile

45. Diffusion, Measurement of diffusion profile

46. Spreading resistance measurements Diffusion, Measurement of diffusion profile needle beveling

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

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

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

50. SIMS, Diffusion, Measurement of diffusion profile