Properties of Error Function erf(z) And Complementary Error Function erfc(z)

Transcription

1 Properties of Error Function erf(z) And Complementary Error Function erfc(z) z erf (z) π e -y dy erfc (z) 1 - erf (z) erf () erf( ) 1 erf(- ) - 1 erf (z) d erf(z) dz π z for z <<1 erfc (z) 1 - d erfc(z) dz d erf(z) dz - 4 π z e -z z erfc(y)dy z erfc(z) + 1 π e -z π (1-e-z ) π e -z z for z >>1 erfc(z)dz 1 π 1

2 The value of erf(z) can be found in mathematical tables, as build-in functions in calculators and spread sheets. If you have a programmable calculator, you may find the following approximation useful (it is accurate to 1 part in 1 7 ): erf(z) 1 - (a 1 T + a T +a 3 T 3 +a 4 T 4 +a 5 T 5 ) e -z 1 where T and P P z a a a a a exp(-z^).1 erfc(z) z

3 Dopant Diffusion (1) Predeposition dopant gas * dose control SiO Si SiO Doped Si region () Drive-in * profile control (junction depth; concentration) SiO SiO Si SiO Turn off dopant gas or seal surface with oxide Note: Dopant predeposition by diffusion can also be replaced a shallow implantation step 3

4 Dopant Diffusion Sources (a) Gas Source: AsH 3, PH 3, B H 6 (b) Solid Source BN Si BN Si SiO (c) Spin-on-glass SiO +dopant oxide 4

5 (d) Liquid Source. 5

6 Diffusion Mechanisms (a) Interstitial (b) Substitutional 6

7 Diffusion Mechanisms : ( c) Interstitialcy 7

8 Diffusion Mechanisms : (d) Kick-Out, (e) Frank Turnbull 8

9 Mathematics of Diffusion C(x) J Fick s First Law: ( ) C x, J ( x, t ) D x D : diffusion constant [ D ] cm sec x t 9

10 Using the Continuity Equation C ( x, t ) + J ( x, t ) t C t J x x D C x Diffusion Equation 1

11 If D is independent of C (i.e., C is independent of x). ( ) C ( x t ) C x, t, D t x Concentration Independent Diffusion Equation 11

12 Temperature Dependence of D D E D A D, E e activation energy k Boltzman A E A kt 5 constant ev / kelvin are tabulated. in ev 1

13 Diffusion Coefficients of Impurities in Si D D O e E A kt 13

14 A. Predeposition Diffusion Profile Boundary Conditions: C x, t C solid solubility of the dopant ( ) C x, t ( ) Initial Condition: C x, t ( ) 14

15 Solid Solubility of Common Impurities in Si 15

16 C C ( x, t ) Dt C 1 π C erfc x Dt e y dy Characteristic distance for diffusion. Surface Concentration (solid solubility limit) x Dt C t 3 >t x t 1 t >t 1 x 16

17 [1] Predeposition dose Q ( t ) C ( x, t ) C dx Dt π t [] Conc. gradient C x π Co Dt e x 4 Dt 17

18 B. Drive-in Profile Boundary C ( x, t ) Conditions : C X x C(x) Initial C ( x, t ) Conditions Co : erfc x ( Dt ) x Predep s (Dt) t 18

19 Shallow Predep Approximation: C(x) C ( x, t ) Q δ ( x) At t ( ) Q C Dt predep π x C(x) t 1 t Solution of Drive-in Profile : C x, t ( ) x π Q Dt ( ) drive in e x 4 ( ) Dt drive in 19

20 How good is the δ(x) approximation? R Dt Dt predep drive in C(x)/C Approximation over-estimates conc. here R1 Exact solution Delta function Approximation Good agreement R.5 Approximation under-estimates conc here x

21 1 Summary of Predep + Drive-in ( ) t D x e t D t D C x C t D t D π Diffusivity at Predep temperature Predep time Diffusivity at Drive-in temperature Drive-in time

22 Semilog Plots of normalized Concentration versus depth Predep Drive-in

23 Diffusion of Gaussian Implantation Profile Note: φ is the implantation dose 3

24 Diffusion of Gaussian Implantation Profile (arbitrary R p ) The exact solutions with C at x (.i.e. no dopant loss through surface) x can be constructed by adding another full gaussian placed at -R p [Method of Images]. φ C(x, t) π ( R [ep + Dt)1/ (x - R p ) (x + R p ) ( R p + Dt) + e- ( R p + Dt) ] We can see that in the limit (Dt) 1/ >> R p and R p, C(x,t) φe- x /4Dt (πdt) 1/ (the half-gaussian drive-in solution) 4

25 The Thermal Budget Dopants will redistribute when subjected to various thermal cycles of IC processing steps. If the diffusion constants at each step are independent of dopant concentration, the diffusion equation can be written as : t Let β (t) D(t) β t Using C t D(t )dt C β β t C t D(t) C x The diffusion equation becomes: C β β t β t C x or C β C x 5

26 When we compare that to a standard diffusion equation with D being timeindependent: C C, we can see that replacing the (Dt) product in (Dt) x the standard solution by β will also satisfy the time-dependent D diffusion equation. Example Consider a series of high-temperature processing cycles at { temperature T1, time duration t1},{ temperature T, time duration t}, etc. The corresponding diffusion constants will be D1, D,.... Then, β D1t1+Dt +... (Dt)effective ** The sum of Dt products is sometimes referred to as the thermal budget of the process. For small dimension IC devices, dopant redistribution has to be minimized and we need low thermal budget processes. 6

27 T(t) Thermal ( Dt) effective Budget i ( Dt) i Examples: Well drive-in and S/D annealing steps well drive-in step S/D Anneal step time * For a complete process flow, only those steps with high Dt values are important 7

28 Irvin s Curves * 4 sets of curves See Jaeger text p-type Erfc n-type Erfc p-type half-gaussian n-type half-gaussian Establish the explicit relationship between: N (surface concentration), o x (junction depth), j N (background concentration), B R (sheet resistance), S Once any three parameters are known, the fourth one can be determined. 8

29 Motivation to generate the Irvin s Curves Both N B (4-point-probe), R S (4-point probe) and x j (junction staining) can be conveniently measured experimentally but not N o (requires secondary ion mass spectrometry). However, these four parameters are related. Approach 1) The dopant profile (erfc or half-gaussian )can be uniquely determined if one knows the concentration values at two depth positions. ) We will use the concentration values N o at x and N B at xx j to determine the profile C(x). (i.e., we can determine the Dt value) 3) Once the profile C(x) is known, the sheet resistance R S can be integrated numerically from: Rs x j q µ ( x )[ C ( x ) ] N B dx 4) The Irvin s Curves are plots of N o versus ( R s x j ) for various N B. 1 9

30 Figure illustrating the relationship of N o,n B,x j, and R S 3