EE43 MATHEMATICS OF DOPING PROFILES N. Cheung The diffusion equation with constant D : has the general solution: C(x,t) = C(x,t) = D 2 C(x,t) 4πDt F(x ' ) e -(x-x' ) 2 4Dt dx ' - where F(x') is the C(x,t) profile at t= A. Predeposition: (constant source diffusion) x () C(x,t)=C o erfc( 2 Dt ) (2) Dopant dose incorporated Q = C( x,t ) dx = 2C o Dt π * The dose incorporated depends on the square-root of predep time (3) Surface concentration C o = C (,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 B. Drive-in (limited source diffusion) = -C o πdt exp [-x2 /4Dt] Initial condition: At t=, we approximate the initial distribution as a delta function of area Q located at x=. This approximation applies to diffusion predeposition or ion implantation with small R p. () C(x,t)= Q πdt exp [ -x2 /4Dt] for x> "half-gaussian" (2) Q(t) = C( x,t ) dx = Q (at t=) for all times * During drive-in, the dopant dose is conserved. Q (3) Surface concentration C (,t) = πdt * During drive-in, the surface concentration decreases with time. (4)Concentration gradient C(x,t) -x = Dt Q πdt exp [ -x2 /4Dt ] C. Ion Implantation
() Gaussian Approximation :C(x) = N p exp [ - ( x-r p )2 2 ( R p ) 2 ] "full gaussian" (2) Incorporated dose φ = C(x)dx - C(x)dx = N p [ 2πδR p ] if R p is > 3δR p (3) Peak concentration N p = φ.4φ 2π R R p 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 - (x - R p ) 2 4Dt exp [ 2( R p ) 2 ]/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)= φ πdt exp [ -x2 /4Dt] for x> (5) R p close to surface The exact solutions with C = at x = (.i.e. no dopant loss x through surface) can be constructed by adding another full gaussian placed at -R p [Method of Images]. C(x, t) = (x - R p ) 2 (x + R φ )2 p 2π ( R 2 [e- 2 ( R 2 p + 2Dt) p + + e- 2 ( R 2 p + 2Dt) ] 2Dt)/2 We can see that for (Dt) /2 >> R p and R p, C(x,t) φ e - x2 /4Dt (πdt) /2 ( the half-gaussian drive-in solution) D. Common Dopant Diffusion Constants 2
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 = 8.62-5 ev /K Use Table 4. of Jaeger for the D o and E a values. Dopant D o (cm2/sec) E a (ev) B.5 3.69 P.5 3.69 As.32 3.56 E. Multiple Drive-in steps (Dt) effective = (Dt) +(Dt) 2 + (Dt) 3... will be used in the half-gaussian solution Proof: The diffusion equation can be written as : C = D(t) 2 C t Let β (t) D(t )dt D(t) = β Using C = C β β The diffusion equation becomes: C β β = β 2 C or C β = 2 C The diffusion equation becomes: C β β = β 2 C or C β = 2 C C When we compare that to a standard diffusion equation with D being time-independent: (Dt) = 2 C, we can see that replacing the (Dt) product in the standard solution by β will also satisfy the time-dependent D diffusion equation. [Note} 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
F. Two-Dimensional Drive-in Diffusion Profile (Constant D) (a) Line Diffusion Source Consider a line diffusion source with s atoms/cm at r =, the diffusion equation in cylindrical coordinates is: D 2 C r 2 + D C r r = C The 2-dimensional diffusion profile is equal to: s C(r, t) = 2πDt e-r2 /4Dt [Note: r 2 = x 2 +y 2 in rectangular co-ordinates] (b) Semi-infinite plane Diffusion Source For a half-plane of diffusion source at x=, y < (with surface concentration of Q atoms/cm 2, the diffusion profile in x-y coordinates is simply the superposition of all line sources located at (,y ) from y < : Semi-infinite source with Q atoms/cm2 diffusion mask y C(x,y,t) = K e - (x2 + (y-y ) 2 )/4Dt dy C(x,y,t) where K is a normalization factor. x Let z= y-y 2 Dt, dz= -dy 2 Dt y/2 Dt C(x,y,t) = K 2 Dt e -x2 /4Dt e -z 2 dz - = K πdt e -x2 /4Dt y [ + erf ( 2 Dt ) ] To find K, we note that for y + (far from the masking edge), C(x,y,t) one-dimensional case). So K = Q / 2πDt Q πdt e -x2 /4Dt (i.e., the Therefore C(x,y,t) = Q 2 πdt e -x2 /4Dt y [ + erf ( 2 Dt ) ] We will expect less dopants under the diffusion,mask region since there is no supply of dopant from the surface. 4
G. Interpretation of the Irvin s Curves The purpose of generating the Irvin s curves is establish the explicit relationship between the four parameters : N o (surface concentration), x j (junction depth), background concentration (N b ), and the R S (sheet resistance), if the profile is known to be either an erfc function (predepositoion) or a half-gaussian function (drive-in). Once any three parameters are known, the fourth one can be determined. The implicit relationship used to generate the curves is the dependence of the majority carrier mobility on the total dopant concentration (assumed to be universal for dopants in Si). Keep in mind there are only two parameters which will uniquely determine the erfc or halfgaussian doping profiles : N o and Dt. Given N o, N B, and x j, Dt can be solved for either of these two profiles. Since the depth dependence of the dopant concentration is known, the sheet resistance is simply an integral quantity of the (net concentration mobility ) product : R S = ρ / x j = x j q [ N(x) - N B ] µ(x) dx For IC process monitoring, x j and R S can easily be measured in the laboratory, not Dt. The Irvin s curves are arranged in such a way ( i.e., N o versus R S x j ) such that N o can be determined once R S, x j and N B are known. Alternatively, if N o can be measured ( with Hall Effect or Secondary Mass Ion Spectrometry), x j can also be inferred. 5
SOME PROPERTIES OF THE ERROR FUNCTION z erf (z) = 2 π e -y2 dy erfc (z) - erf (z) erf () = erf( ) = erf(- ) = - erf (z) 2π z for z << erfc (z) e -z2 π z for z >> d erf(z) dz = - d erfc(z) dz = 2π e-z2 z erfc(y)dy = z erfc(z) + π (-e-z2 ) erfc(z)dz = π 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 part in 7 ): erf(z) = - (a T + a 2 T 2 +a 3 T 3 +a 4 T 4 +a 5 T 5 ) e -z2 where T = +P z and P =.32759 a =.254829592 a 2 = -.284496736 a 3 =.424374 a 4 = -.4535227 a 5 =.645429... exp(-z^2). erfc(z)....2.4.6.8.2.4.6.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 z Another less-accurate but simpler approximation is erf(z) [ - e - 4z2 π ] /2. This approximation has error less than %. For calculations involving the difference of two erf functions, the % accuracy may not be adequate. 6