Dopant Diffusion Sources

Transcription

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

2 (d) Liquid Source. 3 Solid Solubility of Common Impurities in Si o C C (cm -3 ) 4

3 Diffusion Coefficients of Impurities in Si D D O e E A kt Cu 1-13 B,P Au 1-14 As 5 Temperature Dependence of D D E D D A, E e activation energy k Boltzman constant ev / kelvin A E A kt are tabulated. in ev 6

4 Mathematics of Diffusion C() J Fick s First Law: C J (, t ) D D : diffusion constant [ D ] cm sec (, t ) 7 C From the Continuity Equation (,t) C(,t) C(,t) + J (,t) J (,t) D C(,t) D C(,t) Diffusion Equation 8

5 Concentration independence of D If D is independent of C (i.e., D is independent of ). (, ) Ct (, ) Ct t D Concentration Independent Diffusion Equation State of of the the art art devices use use fairly high high concentrations, causing variable diffusivity and and other significant sideeffects (transient-enhanced diffusion, for for eample.) 9 A. Predeposition Diffusion Profile Boundary Conditions: (, ) C (, t) C t C solid solubility of the dopant Initial Condition: Ct (, ) Justification: Si wafers are ~5um thick, doping depths of interest are typically < several um At time, there is no diffused dopant in substrate 1

6 Diffusion under constant surface concentration C C (, t ) Dt C 1 π C erfc Dt Surface Concentration (solid solubility limit) C Dt e y dy Characteristic distance for diffusion. t 3 >t t 1 t >t 1 11 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 π e -z erfc(y)dy z erfc(z) + 1 π (1-e-z ) π e -z z for z >>1 erfc(z)dz 1 π 1

7 Practical Approimations of erf and erfc 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, this approimation 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 1+P z and P a a a a a erfc(z) ep(-z^) [1] Predeposition dose Q () t C (, t) C d Dt π t [] Concentration gradient C Co π Dt e 4Dt 14

8 B. Drive-in Profile Boundary Conditions C (, t ) C Physical meaning of C/ : No diffusion flu in/out of the Si surface.therefore, dopant dose is conserved : C() Initial Conditions : C (, t ) Co erfc ( Dt ) Predep s (Dt) 15 C(, t ) Q δ ( ) Solution of Drive-in Profile with Shallow Predeposition Approimation: ( ) Q C Dt predep π C(,t) Approimate predep profile as a delta function at C(,t) t 1 t ( ) Ct, π Q ( Dt) drive in e 4 ( ) Dt drive in 16

9 How good is the δ() approimation? Let R Dt Dt predep drive in For For reference only only C()/C Approimation over-estimates conc. here R1 Eact solution Delta function Approimation Good agreement R.5 Approimation under-estimates concentration here. 17 Summary of Predeposition + Drive-in D t t 1 D C 1 Diffusivity at Predeposition temperature Predeposition time Diffusivity at Drive-in temperature Drive-in time 1 C D t t D ( ) e π *This will be the overall diffusion profile after a shallow predeposition diffusion step, followed by a drive-in diffusion step. D t 18

10 Semilog Plots of normalized Concentration versus depth Predeposition Drive-in 19 Diffusion of Gaussian Implantation Profile Note: φ is is the implantation dose

11 Diffusion of Gaussian Implantation Profile (arbitrary Rp) The eact solutions with C at (.i.e. no dopant loss through surface) can be constructed by adding another full gaussian placed at -R p [Method of Images]. ( - R p ) φ C(, t) π ( R [e- ( R p + Dt) + e - ( + R p ) ( R p + Dt) ] p + Dt)1/ We can see that in the limit (Dt) 1/ >> R p and R p, C(,t) φe- /4Dt (πdt) 1/ (the half-gaussian drive-in solution) For For reference only only 1 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: C D(t) C t Let β (t) D(t )dt D(t) β Using C C β β The diffusion equation becomes: C β β β C or C β C

12 When we compare that to a standard diffusion equation with D being C time-independent: C, we can see that replacing the (Dt) (Dt) product in the standard solution by β will also satisfy the timedependent D diffusion equation. Eample 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. 3 Thermal Budget (Dt) (Dt effective ) i step i Temp (t) Eample Dt total of : Well drive-in well drive-in step S/D Anneal step time and S/D annealing For For a complete process flow, only only those steps with with high Dt Dt values are are important 4

13 Irvin s Curves p-type erfc n-type erfc p-type half-gaussian n-type half-gaussian Eplicit relationship between: N (surface concentration), o (junction depth), j N (background concentration), B R (sheet resistance), S Once any any three parameters are are known, the the fourth one one can can be be determined. 5 Motivation to generate the Irvin s Curves Both N B (4-point-probe), R S (4-point probe) and j (junction staining) can be conveniently measured eperimentally 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 and N B at j to determine the profile C(). (i.e., we can determine the Dt value) 3) Once the profile C() is known, the sheet resistance R S can be integrated numerically from: 4) Irvin s Curves are plots of N o versus ( R s j ) for various N B. Rs j q µ 1 ( )[ C ( ) ] N B d 6

14 Illustrating the relationship of N o, N B, j, and R S 7 -Dimensional Diffusion with constant D Eample 1 Drive-in from line source with s atoms/cm D C r + D C r r C s C(r, t) πdt e-r /4Dt Diffusion Equation in cylindrical coordinates 8

15 Two-Dimensional Drive-in Profile (cont.) Eample : Semi-Infinite Plane Source Diffusion mask y j y y j (at ) < j (at y ) equal-conc. contours j C(,y,t) Q πdt e - /4Dt y [ 1 + erf ( Dt ) ] where Q predep dose in #/cm Rule of of Thumb :: y j j ~.7-.8 jj 9