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diffusion_math

# diffusion_math - 1 EE143 Notes on Dopant Diffusion N Cheung...

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1 EE143 Notes on Dopant Diffusion N. Cheung I. MATHEMATICS OF DOPING PROFILES The diffusion equation with constant D : C(x,t) t = D 2 C(x,t) x 2 has the general solution: C(x,t) = 1 4 Dt - F(x ' ) e -(x-x ' ) 2 4Dt dx ' where F(x') is the C(x,t) profile at t=0 A. Predeposition: (constant source diffusion) (1) C(x,t)=C o erfc( x 2 Dt ) (2) Dopant dose incorporated Q = 0 C( x,t ) dx = 2C o Dt * The dose incorporated depends on the square-root of predep time (3) Surface concentration C o = C (0,t) = solid solubility of dopant in Si for all times. * The surface concentration is independent of predeposition time. Use Fig. 4.6 of Jaeger for the C o values. (4) Concentration gradient C(x,t) x = -C o Dt exp [-x 2 /4Dt] B. Drive-in (limited source diffusion) Initial condition: At t=0, we approximate the initial distribution as a delta function of area Q located at x=0. This approximation applies to diffusion predeposition or ion implantation with small R p . (1) C(x,t)= Q Dt exp [ -x 2 /4Dt] for x>0 "half-gaussian" (2) Q(t) = 0 C( x,t ) dx = Q (at t=0) for all times * During drive-in, the dopant dose is conserved . (3) Surface concentration C (0,t) = Q Dt * During drive-in, the surface concentration decreases with time. (4)Concentration gradient C(x,t) x = -x 2Dt Q Dt exp [ -x 2 /4Dt ]

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2 C. Ion Implantation (1) Gaussian Approximation :C(x) = N p exp [ - ( x-R p ) 2 2 ( R p ) 2 ] "full gaussian" (2) Incorporated dose = - C(x)dx 0 C(x)dx = N p [ 2  R p ] if R p is > 3 R p (3) Peak concentration N p = 2  R p 0.4 R p (4) If R p is sufficiently deep that the implant profile is close to a full gaussian, additional drive-in step for time t will just create a gaussian with larger straggle: C(x,t) = N p [ 1 + 4Dt 2( R p ) 2 ] 1/2 exp [ - (x - R p ) 2 2( R p ) 2 + 4Dt ] [Note: For long drive-in times, the left-side of the full gaussian piles up at the surface and the above equation is not valid. We can see in the limit that 2 Dt >>R p , C(x,t) develops into a half gaussian with: C(x,t)= Dt exp [ -x 2 /4Dt] for x>0 (5) R p close to surface The exact solutions with C x = 0 at x = 0 (.i.e. no dopant loss through surface) can be constructed by adding another full gaussian placed at -R p [Method of Images]. C(x, t) = 2 ( R 2 p + 2Dt) 1/2 [e - (x - R p ) 2 2 ( R 2 p + 2Dt) + e - (x + R p ) 2 2 ( R 2 p + 2Dt) ] We can see that for (Dt) 1/2 >> R p and R p , C(x,t) e - x 2 /4Dt ( Dt) 1/2 ( the half-gaussian drive-in solution)
3 D. Common Dopant Diffusion Constants Diffusion constants of dopants in Si reported in the literature are not consistent, probably due to variability of the processing conditions. High concentration diffusion effects (discussed later) also complicates the picture. Unless specified, we will use the following concentration independent diffusion constants for calculations: D = D o exp[- E a /kT ] D o = Pre-exponent constant E a = Activation Energy in eV k = Boltzmann Constant = 8.62

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diffusion_math - 1 EE143 Notes on Dopant Diffusion N Cheung...

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