8.2.5 Effect of Thermal
`Oxidation on the
`Diffusion Coefficient*
`
`8.2 ATOMIC DIFFUSION MECHANISMS
`
`383
`
`The thermal oxidation of silicon produces a higher-than-equilibrium
`number of silicon interstitials that will in turn increase the diffusivity
`of any atom that has an interstitialcy component. Conversely, if an
`atom diffuses solely by vacancies, extra interstitials will depress the
`number of vacancies present and decrease the diffusivity. Boron,
`phosphorus, and arsenic all show enhancements, while antimony
`shows a retardation. The enhancement effect is seen in both wet and
`dry oxidations and is more pronounced at lower oxidation/diffusion
`temperatures. Early work reported a retarding effect above 1 I 50°C-
`1200°C, which corresponds approximately to the temperature where
`oxidation-induced stacking faults begin to shrink, but more recent
`data merely show the effect disappearing at high temperature.
`The number of interstitials generated at the interface will
`depend on the oxidation rate, which decreases with time (thick-
`ness). Plus, some of the interstitials diffuse into the oxide and
`recombine with incoming oxygen, while the rest diffuse into the
`silicon and eventually recombine. If the additional interstitials
`affect only the interstitial component of D, then, from Eq. 8.10,
`D = D*(I + fi/Vi /NT), and the average diffusivity <D> over a
`diffusion/oxidation time t is given by
`
`<D> = f i Dt 0*(1 + 4)1(N, — Npdt
`N'y
`0
`
`8.22
`
`N, is the concentration of interstitials where diffusion is occurring,
`and NI is their equilibrium concentration. Various complex expres-
`sions for the interstitial supersaturation have been derived (16-20).
`It is predicted that for low temperatures and dry oxygen, the super-
`saturation starts at zero and increases with time. At longer times, it
`is experimentally deduced to be given by
`
`N, — =
`
`l"
`dt
`
`8.23
`
`where p is in the 0.2-0.4 range and K is a constant. Since dxIdt
`decreases as the oxide layer thickness increases, the supersaturation
`decreases with time in this region. The supersaturation also de-
`creases with depth into the wafer. A simple analysis gives
`
`N, —
`
`— et"-
`
`8.24
`
`where L is a characteristic length, measured experimentally to be
`30 p.m at 1000°C and 25 p.m at 1100°C (21). Fig. 8.6 shows the mag-
`nitude of the enhancement of D for boron, arsenic, and phosphorus.
`
`'See references 11-20.
`
`Elm Exhibit 2159, Page 400
`
`

`
`384
`
`CHAPTER 8 IMPURITY DIFFUSION
`
`10-' '
`
`---12
`to
`
`a \
`
`---Dry oxygen
`inert
`
`---Dry oxygen
`Inert
`
`10-13 'a.
`
`\ Phosphorus
`
`1044
`
`-
`
`Arsenic
`
`\\
`
`Hy's
`
`N
`N
`(00)
`
`(100)
`(100)
`
`( 100)
`
`Boron
`
`N (100)
`(III)
`(100)
`
`Diffusion coefficient (cm2/s)
`
`FIGURE 8.6
`
`Effect of oxidation during dif-
`fusion on diffusion coefficient.
`(Source: D.A. Antoniadis et al..
`App!. Phys. Len. 33. p. 1030,
`1978. and D.A. Antoniadis et al..
`J. Electrochem. Soc. 125. p. 813,
`1978. Reprinted by permission of
`the publisher. The Electrochemi-
`cal Society. Inc.)
`
`10-16
`0.65
`
`1 0.7
`
`0.8
`0.75
`1000/ T(K)
`
`0.85
`
`0.65
`
`01.7
`
`0.75
`0.8
`1000/ T(K)
`
`0.85
`
`0.9
`
`8.2.6 Effect of Diffusing
`Direction (Orientation)
`
`If the crystal in which diffusion is occurring is anisotropic. the value
`of D will depend on direction. In the general case, Eq. 8.4 becomes
`
`./i = D, aNax;
`
`8.25
`
`from which it is seen that the diffusion coefficient is a second-rank
`tensor. In cubic crystals, examples of which are Si and GaAs, prop-
`erties that are second-rank tensors are independent of crystallo-
`graphic orientation (22) (which is the reason that Eq. 8.5 is identical
`to Eq. 8.7). However, recalling that, depending on the diffusion
`mechanism, a deviation of the density of vacancies and/or intersti-
`tials may affect D, it can be surmised that a nonisotropic flow of
`either of them could cause a nonisotropic effective D. It is. in fact,
`experimentally observed that in silicon, orientation-dependent dif-
`fusions are often encountered (23-26) during simultaneous thermal
`oxidation and diffusion. For diffusion in an inert atmosphere, there
`is no evidence of anisotropic diffusion coefficients (14, 27). This ori-
`entation-dependent phenomenon can be explained by the different
`rates at which nonequilibrium numbers of interstitials are generated
`by oxidation on different crystallographic surfaces. The effect be-
`comes greater as the temperature is lowered, and its magnitude at
`any temperature is in the order of (100) > (110) > (111). Impurities
`showing the effect will be the same ones showing oxidation-en-
`hanced diffusion (OED), and the magnitude of the effect can be
`judged by the curves of Fig. 8.6.
`
`Elm Exhibit 2159, Page 401
`
`

`
`8.2.7 Effect of
`Heavy Doping
`
`8.2 ATOMIC DIFFUSION MECHANISMS
`
`385
`
`As was shown in the discussion on vacancies, the density of charged
`point defects can change with doping concentration when the level
`approaches ni. Most silicon dopants have a diffusion component
`depending on charged vacancies and will thus show a doping-level
`dependency. In addition, the field enhancement discussed in the pre-
`vious section (and which becomes noticeable only when N > ni)
`must also be included where appropriate. in the case of boron, and
`perhaps the other acceptors as well, diffusion is apparently by a
`neutral complex—for example, B- V. . Arsenic and antimony dif-
`fuse by V° and V- ; phosphorus, by V° and V= ; and boron and gal-
`lium, by V° and V. . Thus, the expressions for D* are as follows
`(28):
`
`As, Sb
`
`D = g(D" + D* -1
`ni
`
`P
`
`D = g D*" -4- D* = (1)2]
`/I;
`
`B, Ga, Al
`
`D = (D*" + D* 4-11 )
`
`
`ni
`
`When N > ni, the neutral vacancy contribution to the total D
`generally becomes small. If, at the same time, the diffusing profile
`is such that an appreciable electric field occurs and g (from Eq. 8.21)
`approaches 2, then, if D* is the diffusivity in intrinsic material, for
`donors,
`As, Sb D 2D*
`
`P
`
`D
`
`2D* (-11 -
`
`For acceptors, field-aided diffusion appears to be negligible, so
`
`B, Ga, Al
`
`D
`
`D*
`
`ni
`
`In intermediate doping ranges and at some temperatures, contribu-
`tions from charge states other than those just listed may also become
`important. Fig. 8.7 is a curve of ni versus temperature for silicon
`and can be used to tell when the doping concentrations are in the
`region where these effects may occur.
`When the concentration of dopants in silicon approaches the
`1021 atoms/cc range, precipitates, complexes, and atomic misfit
`strain begin to play a role in diffusion. For arsenic, complexes begin
`to form at slightly above 1020 atoms/cc, have a reduced diffusivity,
`
`Elm Exhibit 2159, Page 402
`
`

`
`386
`
`CHAPTER 8 IMPURITY DIFFUSION
`
`FIGURE 8.7
`
`- -20
`0
`1
`
`—
`
`versus temperature for sili-
`con. (Source: From data in F.J.
`Morin and J.P. Maim. Phvs. Rev.
`96, p. 28. 1955.)
`
`I0'
`
`io' 7
`
`700 800 900 1000 1100 1200 1300
`Temperature (°C)
`
`and are not fully ionized. The result is that DA, versus concentration
`has a maximum as shown in Fig. 8.8 (29). This figure also shows the
`increase in diffusivity with concentration up to the point where com-
`plexes form. In the case of phosphorus, the strain effect (see next
`section) becomes important for concentrations above about 4 x le
`atoms/cc.
`
`From Eq. 8.6, the simple expression for D is of the form D =
`. When the doping level is below ni over the whole temper-
`D„e
`ature range being covered and when there are no oxidation enhance-
`ment effects—that is, in the intrinsic diffusivity range—Eq. 8.6 is
`appropriate. E will be different for each element, and in cases where
`differing diffusing conditions can favor a particular charge state for
`the point defect involved, different E values for each state are re-
`quired. Diffusion data are sometimes presented in graphic form, in
`which case D versus lagives a straight line if Eq. 8.6 is appropriate.
`When the data give a straight line, an alternative is to present D„ and
`E in tabular form, as is done in Table 8.2 for silicon. Unfortunately,
`most gallium arsenide diffusants are not well enough behaved to be
`depicted in this manner. However, the table still lists various n- and
`p-dopants.
`
`8.2.8 Effect of
`Temperature
`
`8.2.9 Effect of
`Stress/Strain
`
`Lattice strain will change the bandgap (30, 31) and thus can affect
`the distribution of vacancies and the diffusivity (28, 31). It has also
`been suggested that, in the case of GaAs, it affects the jump fre-
`
`Elm Exhibit 2159, Page 403
`
`

`
`1000° C
`
`FIGURE 8.8
`
`Kr"
`
`Kr"
`
`Diffusion coefficient (cm2/s)
`
`Effect of concentration on ar-
`senic diffusivity. (The peak po-
`sition at intermediate tempera-
`tures follows the line.)
`(Source: Adapted from R.B. Fair.
`Semiconductor Silicon/8l. p. 963.
`Electrochemical Society. Pen-
`nington, N.J., 1981.)
`
`850'C
`
`10-16
`1019
`
`I
`
`III tilt!!
`10"
`Arsenic concentration (atoms/cc)
`
`1021
`
`TABLE 8.2
`
`Diffusion Coefficients
`
`Silicon
`Do ((wets)
`
`E (eV)
`
`Donors
`Phosphorus
`V°
`V .-
`V''
`Arsenic
`V°
`V
`Antimony
`V°
`V
`Acceptors
`Boron
`V°
`V'
`Aluminum
`V°
`V'
`Gallium
`V'
`V'
`Self-Interstitial
`
`3.85
`4.44
`44.2
`
`0.066
`12
`
`0.214
`15
`
`0.037
`0.41
`
`1.385
`2480
`
`0.374
`28.5
`
`Source: From references 28 and 29.
`
`3.66
`4.0
`4.37
`
`3.44
`4.05
`
`3.65
`4.08
`
`3.46
`3.46
`
`3.41
`4.20
`
`3.39
`3.92
`
`Gallium Arsenide
`E (eV)
`
`Do (mills)
`
`0.0185
`3000
`
`2.6
`4.16
`
`Donors
`Sulfur
`Selenium
`Tellurium
`
`Acceptors
`Beryllium
`
`7.3 x 10'
`
`1.2
`
`Magnesium
`
`0.026
`
`Zinc
`
`Cadmium
`Mercury
`
`Gallium
`Arsenic
`
`0.1
`0.7
`
`3.2
`5.6
`
`387
`
`Elm Exhibit 2159, Page 404
`
`

`
`388
`
`CHAPTER 8 IMPURITY DIFFUSION
`
`FIGURE 8.9
`
`Diffusion interactions.
`
`Emitter
`diffusion
`
`Base
`
`Base
`
`J
`
`Distance
`
`(a) Emitter push
`
`(b) Emitter pull
`
`(c) Emitter dip
`
`8.2.10 Emitter Push,
`Pull, and Dip
`
`quency directly (32). Strain can arise from atomic misfit of diffused
`or implanted atoms (see section 8.7) or from layers such as SiO2 or
`silicon nitride deposited on the surface. In particular, the stress
`around windows cut in oxide and nitride produces noticeable
`changes in D values (32, 33).
`
`Simple theory predicts that each impurity will diffuse quite indepen-
`dently of others that may be present. There are, however, at least
`three phenomena—emitter push, pull, and dip—where this is not
`true (10, 34). These interactions are shown in Fig. 8.9. The earliest
`one reported (in 1959) was emitter push, which caused a gallium
`base to move deeper under a diffused phosphorus emitter (35). The
`effect was also present with boron base diffusions and indeed, in
`some cases, made it difficult to obtain the desired narrow bipolar
`base widths. Sometime later, when gallium bases and arsenic emit-
`ters were used, the reverse situation, emitter pull, was reported.
`Emitter push and pull can be explained in terms of the concentration
`of vacancies associated with the emitter diffusion (3, 31). The other
`interactive effect shown in Fig. 8.9 is a dip in the concentration of
`base impurity that is sometimes seen at or near the intersection of
`the emitter profile with the base profile. In this case, internal field
`retardation appears to cause the dip (34, 36).
`
`8.3
`SOLUTIONS TO THE
`DIFFUSION
`EQUATIONS
`
`Impurity diffusion mathematics parallels that for carriers, except
`that the diffusion coefficient has more causes of variability in the
`region of interest. Combining Fick's first law with the continuity
`equation and assuming that D is independent of concentration give
`Fick's second law:
`
`Elm Exhibit 2159, Page 405
`
`

`
`8.3 SOLUTIONS TO THE DIFFUSION EQUATIONS
`
`389
`
`aN
`at
`
`a'N
`a2
`
`where t is the diffusion time. In three dimensions,
`
`aN
`at
`
`= DV2N
`
`8.26
`
`8.27
`
`When D is a function of the concentration N, Eq. 8.26 becomes
`
`aN
`at
`
`a(DaN/ax) aD aN
`ax ax
`ax
`
`+
`
` 2N
`a
`D
`ax-
`
`8.28
`
`In regions where the diffusion coefficients are concentration depen-
`dent, closed-form analytical solutions for the boundary conditions
`of interest are usually not available. In those cases, numerical inte-
`gration and computer modeling are used to obtain solutions. Large
`programs are available that are designed to run on mainframe com-
`puters—for example, SUPREM, the Stanford University process
`engineering model for diffusion and oxidation. Nevertheless, it is
`still very instructive to first look at solutions of Eq. 8.26 for various
`boundary conditions applicable to semiconductor processing. To a
`first approximation, such solutions are valid. However, of more im-
`portance, the use of the simple closed-form solutions in studying a
`process does not screen the engineer from the physical processes as
`the massive computer modeling programs do.
`Eq. 8.26 may be solved by separation of the variables (37)—that
`is, by considering that
`
`N(x,t) = X(x)Y(t)
`
`8.29
`
`where X is a function only of x and Y is a function only of the time
`t. The general solution to Eq. 8.29 is
`
`N(x,t) = L
`
`
`
`A [A(X) cos Xx + B(X) sin Xx]e - '"idX
`
`8.30
`
`Solutions with boundary conditions appropriate for semicon-
`ductor processing are given in the following sections. Where multi-
`ple impurities are present, it is assumed that each impurity species
`diffuses independently of all others. Thus, separate, independent so-
`lutions to the diffusion equation can be obtained for each impurity.
`The total impurity concentration N'(x,t) can be determined by sum-
`ming the individual distributions. The net impurity concentration is
`given by 12,N, —
`
`Elm Exhibit 2159, Page 406
`
`

`
`390
`
`CHAPTER 8 IMPURITY DIFFUSION
`
`8.3.1 Diffusion from
`Infinite Source on
`Surface
`
`This condition is one of the more common conditions used and gives
`the profiles that were shown in Figs. 8.2 and 8.3. The impurity con-
`centration at the surface is set by forming a layer of a doping source
`on the surface. The layer either is thick enough initially or else is
`continually replenished so that the concentration No is maintained
`at the solid solubility limit of the impurity in the semiconductor dur-
`ing the entire diffusion time. Assuming that the semiconductor is
`infinitely thick, the solution is as follows (37):
`
`N(x,t) = N,[1 — (4--) J e-z3dz
`v
`o
`
`8.31
`
`t. The integral is a converging infinite series re-
`where z = x/2
`ferred to as the error function, or erf(z), and occurs in the solution
`of many diffusion problems. Toward the end of this chapter, some
`properties of the error function and a short table of values are given
`(see Table 8.12). Using the erf abbreviation, Eq. 8.31 can be written
`as
`
`N(x,t) = No I — erf (
`x )]
`
`2V T5i
`
`8.32
`
`The expression 1 — erf(z) is often referred to as the complementary
`error function erfc(z). If there is a background concentration N, of
`the same species at the beginning of diffusion, Eq. 8.32 becomes
`
`N(x,t) = N, + (No — N
`
`x
`— erf
`(2N/Lii)
`
`8.33
`
`If there is a background concentration N, of a different species, then
`they act independently, and
`
`N(x,t) = N,
`
`N„[I — erf(
`
`
`
`
`2VT)i)]
`
`8.34
`
`If the species are of opposite type, a junction will occur when
`
`No[ 1 — erf(
`
`)] = N,
`
`2\15t
`
`The value of x where this occurs is given by
`
`erf-'
`
`o
`
`which can be rewritten as
`
`"r. = AVi
`
`8.35
`
`8.36
`
`8.37
`
`Elm Exhibit 2159, Page 407
`
`

`
`8.3 SOLUTIONS TO THE DIFFUSION EQUATIONS
`
`391
`
`FIGURE 8.10
`
`Position of junction after diffu-
`sion times in ratio of 1:4:9. (x;
`is linear with Vi regardless of
`NoIN, ratio.)
`
`where A is a constant given by A = 2[erf-1(1 — Ni/N0)]\/D. Thus,
`the junction depth x; increases as the square root of the diffusion
`time, as illustrated in Fig. 8.10 where the diffusion times are in the
`ratio of 1:4:9. A linear plot of x; versus Nfi provides assurance that
`D is remaining constant over the range of concentrations covered.
`Then, a best-fit curve of several values of x, and t can be used to
`determine a particular .v.; and t that can be substituted into Eq. 8.36
`to determine D.
`
`EXAMPLE 0 If No is 5 x 10° atoms/cc and a diffusion is made into an opposite-
`type background of 1016 atoms/cc, how long will it take to diffuse
`to the point where the junction is I p.m deep if the temperature is
`such that D = 10-'1 cmVs?
`Substituting into Eq. 8.36 (cm and s are a consistent set of
`units) gives
`
`10-4 cm = (2 x 10-7 cm/ Vi) x V1 erf-1(1 — 1016/5
`x 10°)5 x 102Vi =
`erf -'(0.998)
`
`From Table 8.12, erf(0.998) = 2.18. Thus, t = 52,900 s, or 14.7
`hours.
`
`Because No is set by the solubility limit of the dopant being
`used, at a given temperature, only one Na is available per dopant
`per semiconductor. The use of a diffusion with the concentration set
`
`Elm Exhibit 2159, Page 408
`
`

`
`392
`
`CHAPTER 8 IMPURITY DIFFUSION
`
`8.3.2 Diffusion from
`Limited Source
`on Surface
`
`by solid solubility gives good control, but the surface concentration
`N, may be much higher than desired. To provide greater flexibility
`in the choice of N0, a two-step diffusion process comprised of a
`short diffusion from an infinite source followed by its removal and a
`continued diffusion from the thin layer of impurities already diffused
`in is often used. The total number of impurities S available is set by
`the first diffusion and is given by
`
`S = I N(x)dx
`
`8.38
`
`=
`
`N„[I — erf(
` ) dr = Nrrr —2N(rii
`205t
`
`The behavior of the second diffusion is described in the next
`section.
`
`At t = 0, a•fixed number S/cm2 of impurities is on the surface. N is
`given by (37)
`
`N(x,t) —
`V7rDt
`
`e-x:14th
`
`8.39
`
`In this case, the distribution is Gaussian' and not erf. The surface
`concentration N(0,t) continually diminishes with time as shown in
`Fig. 8.11 and is given by
`
`N(0,t) —
`
`8.40
`
`Thus, in the limited-source case, the surface concentration de-
`creases linearly with
`The junction depth, unlike that of the pre-
`vious case, varies in a more complex manner:
`
`--014DE
`e
`
`=
`
`Nn
`S
`
`rrDt
`
`Or
`
`X. = 14D/ logl
`
`
`N8 1 BN,FrrDt)
`
`11/2
`
`8.4I a
`
`8.4 1 b
`
`Eq. 8.39 was derived based on S being located at x = 0, but if
`S actually is in a layer from a previous short diffusion of the type
`described by Eq. 8.32, and if the Dt product of the second diffusion
`
`KWunctions of the form V = e
`
`are referred to as Gaussian.
`
`Elm Exhibit 2159, Page 409
`
`

`
`8.3 SOLUTIONS TO THE DIFFUSION EQUATIONS
`
`393
`
`1.0
`
`0.5
`
`Normalized concentration
`
`FIGURE 8.11
`
`Diffusion from limited source
`plotted on linear scale for dif-
`fusion times in ratio of 1:4:9.
`(Note that S decreases as
`square root of time.)
`
`is as much as four or five times that of the first one, the impurities
`from the first can still be considered as all lying at x = 0. S for the
`second diffusion will be the amount of impurities introduced during
`the first diffusion. Substituting the value for S from Eq. 8.38 into
`Eq. 8.39 gives
`
`N(x:t ,t' ) =
`
`(2N (,)(Dt I/2 _ v.2/4/Ye
`e .
`DY
`
`and the surface concentration is
`
`N„(t,t') =
`
`2N
`'Tr
`
`/ Dt
`D' t'
`
`o\
`
`8.42
`
`8.43
`
`where the prime values indicate values for the second diffusion.
`When the second Dt product is not large compared to the first, as is
`likely during short, low-temperature diffusions, the exact solution
`instead of the approximations of Eqs. 8.42 and 8.43 can be used (38,
`39):
`
`N(x,t,t') =
`
`2
`V IT s. Ts
`
`,
`erf(ourt)cint
`
`8.44
`
`Elm Exhibit 2159, Page 410
`
`

`
`394
`
`CHAPTER 8 IMPURITY DIFFUSION
`
`where
`
`DI
`
`=
`
`13 — 4(Dt Die)
`
`8.3.3 Diffusion from
`Interior Limited Source
`
`(Representative values for the integral are tabulated in Table 8.13.)
`The surface concentration is given by
`
`2
`Nsur = IV() —
`
`tan I
`
`Di
`Ur'
`
`8.45
`
`Since tan ' x = x — x3/3 + .0/5 — . . . , the difference between the
`exact solution for M and that given by Eq. 8.43 is only a few percent
`when D'I' > 5Dt.
`Ion implantation is an alternative to the first diffusion. The ions
`implanted can be counted very accurately, and they are quite close
`to the surface (see Chapter 9). They will not have the atoms all lo-
`cated on one plane, however, and the boundary conditions are much
`more like those discussed in the next section.
`
`This condition is like the case just described, except that the sheet
`of impurity atoms is located at x = X, where X„ is in the interior of
`the semiconductor so that the source is depleted by diffusion in both
`directions. In this case (37), assuming that X„ is far removed from a
`free surface,
`
`N(x,t) —
`
` e
`2\77r .5/
`
`xni!bim
`
`8.46
`
`Fig. 8.12 shows two profiles with their respective diffusion times
`differing by a factor of 4. This exponential profile has the same
`shape as the Gaussian or normal distribution often encountered in
`probability theory. For that application, it is written as
`
`Ni
`(TV-2-71r
`
`e
`
`-- x0) 2,2u2
`
`8.47
`
`where N, is the number of observations and o• is the standard
`deviation." A comparison of Eqs. 8.46 and 8.47 shows that N1
`
`"68.3% of the area under the curve is contained in the portion of the curve between
`hr and x — X„ — hr.
`— X„
`
`Elm Exhibit 2159, Page 411
`
`

`
`8.3 SOLUTIONS TO THE DIFFUSION EQUATIONS
`
`395
`
`0.1 -
`
`0.01
`
`0.001
`
`0.0001
`
`on (atoms/cc)
`
`Normalized concentra
`
`FIGURE 8.12
`
`Profile for diffusion from
`source S of width = 0 located
`at x = X,.
`
`EXAMPLE
`
`3i. This correlation is useful when
`corresponds to S and a =
`considering subsequent diffusions after an ion implant since the ini-
`tial ion implant profile is often described by a total flux (1)(S), a range
`(X0), and a standard deviation SR,, (a or VT/Ti).
`
`q Suppose that an ion implant is characterized by a total dose 4, of
`10" atoms/cm', a range of 0.2 p.m, and a standard deviation of 0.1
`Ian. What will the profile look like after a 20 minute heat cycle at
`a temperature where D = 10 '' cm'/s? Neglect the fact that be-
`cause of the shallowness of the implant, there might be a surface
`boundary effect.
`4) is the total number of impurities and hence corresponds to
`S. The scatter during implant is equivalent to a first diffusion from
`an initial sheet source. S/2„ = cr = 0.1 urn = VITIt = 10 cm.
`Solving for Dr gives 5 x 10 - " cm'. The subsequent diffusion has
`a Dr product of 10- " cm:7s x 1800 s = 1.8 x 10 "cm', which is
`much less than the equivalent DI of the implant. Hence, it would
`be expected to change the profile very little. However, as will be
`discussed in section 8.3.13, in this case the final distribution can
`be calculated by substituting a (DIL = ED,r, + D2i, = 5 x 10 "
`+ 1.8 x 10 " = 6.8 x 10 " cm' for the DI of Eq. 8.46.
`
`q
`
`8.3.4 Diffusion from
`Layer of Finite
`Thickness
`
`This condition is a case similar to the one just described, except that
`initially a rectangular rather than a sheet source distribution is lo-
`cated internally to the body of the semiconductor. As is shown in
`Fig. 8.13a, if the initial thickness 2C is much greater than VFW, it is
`
`Elm Exhibit 2159, Page 412
`
`

`
`396
`
`CHAPTER 8 IMPURITY DIFFUSION
`
`FIGURE 8.13
`Profile for diffusion from inte-
`rior layer of width = 2t lo-
`cated at A- = X„.
`
`2Q
`
`0.001
`
`xo
`
`(a)
`
`(b)
`
`best treated as the two separate step distributions to be described in
`the next section. Of more interest is the case where Viit is greater
`than C, where the peak distribution N0 decreases with increasing
`diffusion time.
`Application for this set of boundary conditions can occur if the
`dopant redistribution of multiple layers of molecular beam epitaxy
`after heat treatments is being considered. It is less applicable to con-
`ventional epitaxial processing since autodoping will usually over-
`shadow the diffusion.
`To simplify the algebra, consider that a new origin is chosen
`with its 0 at the X0 of Fig. 8.13a. If the width of the doped layer is
`2e, N(x,t) is given by (37)
`
`N(x,t) =
`
`+
`N
`
`erf
`+ erf(e
`
`2 2V5i 105i
`
`8.48
`
`where 2e is the width of the layer. As diffusion progresses, impuri-
`ties will diffuse out in both directions as shown in Fig. 8.13b. The
`peak concentration N0 will decrease as erf(C/2\15i) and thus will
`change imperceptibly until e/2-V—Dt becomes less than about 2.
`
`Elm Exhibit 2159, Page 413
`
`

`
`8.3.5 Diffusion from
`Concentration Step
`
`8.3 SOLUTIONS TO THE DIFFUSION EQUATIONS
`
`397
`
`The initial boundary conditions for this case, illustrated in Fig. 8.14
`as the abrupt steps of t = 0, approximate the conditions just after a
`conventional epitaxial deposition and very closely match those after
`molecular beam epitaxy (MBE). If there are a series of thin MBE
`layers, the previous case may be more appropriate. This case also
`provides an approximation of a grown junction crystal distribution.
`However, grown junction technology has now been obsolete for 20
`years. Diffusion from each side of the step can be considered to
`proceed independent y as described by Eqs. 8.49 and 8.50 (37):
`
`N,
`[1 — erf(
`N(x,t) = —
`22
`
`)]
`
`N(x,t) =
`
`N,
`2
`
`+ erf(
`
`x
`2//Wt
`
`8.49
`
`8.50
`
`N, is the doping level for —x and t = 0; N,, for +x and t = 0. D,
`is the diffusion coefficient for the N, species; D,, for the N, species.
`Eqs. 8.49 and 8.50 may be added together to give a single expression
`covering diffusion from both sides of the step:
`
`N(x
`
`
`N ,
`,t) —
`2 [
`
`,
`N
`x
`1 — erf
`
`
`M ( 2 V, )1 + 2 11 + erf(2Viit)
`
`8.51
`
`t=
`
`10-
`
`10-2
`
`10-3
`
`Normalized impurity concentration
`
`I0-6
`
`x= 0
`
`FIGURE 8.14
`
`Profile for diffusion occurring
`at a concentration step. (In this
`example. N, and N2 have the
`same Di product.)
`
`Elm Exhibit 2159, Page 414
`
`

`
`398
`
`CHAPTER 8 IMPURITY DIFFUSION
`
`Normalized impurity concentration
`
`FIGURE 8.15
`
`Dip in net concentration occur-
`ring near original high/low
`boundary because of a more
`rapid diffusant in the high-re-
`sistivity layer.
`
`0
`
`Fig. 8.14, along with the initial conditions, also shows a profile for
`each separate diffusion (Eqs. 8.49 and 8.50) and their sum (Eq. 8.51)
`for the case of the same impurity on each side of the step (D,t =
`D,t). When the two Dt's are not equal, substantially different distri-
`butions can sometimes occur. For example, if N1 >> N, and D,>
`D,, it is possible to get a dip in the concentration curve as shown in
`Fig. 8.15 (40). This dip might occur, for example, if an epi substrate
`were doped with antimony and overgrown with an arsenic layer. The
`maximum possible dip is a factor of 2 and occurs before the N, dif-
`fusion has perceptibly moved into the N, region. If a heavily doped
`n-substrate partially compensated by having some spurious p-dop-
`ant present is overgrown with a high-resistivity n-layer, then if N, <
`N1, and if 0,„ << < D2p, a p-layer could occur in the epitaxial layer as
`shown in Fig. 8.16 (41). This happens if low-resistivity antimony
`substrates become contaminated with boron. The resulting p-layer
`is often referred to as a "phantom" p-layer. although it is indeed
`real. If N, and N, are of opposite type, the junction will move into
`the more lightly doped region as diffusion proceeds, regardless of
`the relative values of D, and D,.
`
`This case, as shown by Fig. 8.17, describes diffusion from a heavily
`doped substrate into an epitaxial layer being grown at a velocity v
`in cm/s. However, this case is ordinarily of little importance both
`because autodoping (see Chapter 7) will usually overshadow diffu-
`sion and because growth is generally much more rapid than diffu-
`
`8.3.6 Diffusion from
`Concentration Step into
`Moving Layer
`
`Elm Exhibit 2159, Page 415
`
`

`
`8.3 SOLUTIONS TO THE DIFFUSION EQUATIONS
`
`399
`
`Phantom p-layer
`
`N n
`
`p
`
`10'
`
`ty concentration
`
`8
`
`FIGURE 8.16
`
`Origin of phantom p-layer that
`occurs in n-epi layers because
`of n-substrate contamination
`by a faster diffusing p-dopant.
`
`io-'
`
`= 0
`
`sion. In the event that the epitaxial growth temperature is reduced
`to the point where autodoping is negligible, diffusion will be reduced
`to the point that the solution for diffusion into an infinite thickness
`(Eq. 8.50) can be used. Even at higher temperatures, the use of Eq.
`8.50 causes little error in most cases.
`To account for the moving boundary, instead of Eq. 8.26, the
`equation to be solved is as follows (42, 43):
`
`a2N SIN
`<)N
`= -7- + v—
`ar.
`it t
`ax.
`
`8.52
`
`where v is the epitaxial growth velocity. In the solutions to follow,
`v is assumed to remain constant. One solution represents diffusion
`from an infinitely thick substrate into the growing layer and is gen-
`erally the one of interest. In the event that the "substrate" is a thin
`epitaxial layer just grown or a thin high-concentration layer diffused
`into a lightly doped wafer, then it is not infinite in extent, and its
`concentration will decrease with growth (diffusion) time. The other
`represents diffusion from the growing layer into the substrate. For
`the case of out-diffusion from an infinite substrate, the boundary
`conditions are as follows:
`
`N(— x,0) = N1
`N( —004) = N1
`aN
`J= — D —
`
`= (K + v)N at x,
`
`FIGURE 8.17
`
`Geometry for diffusion during
`epitaxial growth.
`
`Substrate
`
`Epi
`layer
`
`Uniform
`velocity v
`
`0
`
`X,
`
`Elm Exhibit 2159, Page 416
`
`

`
`400
`
`CHAPTER 8 IMPURITY DIFFUSION
`
`where K is a rate constant describing the loss of dopant at the epi—
`ambient interface and .x; is the location of the interface. The solution
`is given by
`
`x
`N(x,t) 1
`rf
`N I — 2 e c 2-V5t
`+ v)
`(K
`2K
`
`( 2e(viv)(0- x erfc
`
`
`v1
`x \
`2
`V-5 )
`
`8.53
`
`2K + v)ei(K+riamiu; of -xi
`(
`
`2K
`(2(K + v)t — xl
`x erfc
`2Aibt
`
`EXAMPLE
`
`When the rate constant K approaches infinity (no impediment at the
`surface), Eq. 8.53 will predict a smaller N for a given x than will Eq.
`8.50. When K goes to zero (no loss at the surface), Eq. 8.53 will
`predict a higher value for N at a given x than will Eq. 8.50. For
`v'tID greater than about 10, Eqs. 8.53 and 8.50 will very closely
`match, regardless of the K value. The reason for the insensitivity to
`K in this case is that the layer appears infinitely thick and no impur-
`ity gets to the boundary. Little data exist on K values, but data in
`reference 43 suggest values in the range of 2 x 10 'cm/s. In the
`event that the substrate is not infinitely thick—for example, for one
`epitaxial layer deposited on top of another or on top of a diffused
`layer—then the solution is more related to diffusion from the finite
`thickness sources discussed earlier. (See reference 42 for further
`details.)
`
`q For the case of a 1200°C silicon epitaxial deposition, calculate the
`v'tID values for a layer 0.05 p.m thick if the deposition rate is 0.5
`p.m/minute and the substrate is doped with antimony.
`The D value for antimony, found from Fig. 8.26, which ap-
`pears in a later section, is about 10 13 cm'/s. The time t to grow
`0.05 p.m at 0.5 p.m/minute is 0.1 minute, or 6 s. The rate of 0.5 iti.m
`per minute equals 8.3 x 10 -7 cm/s. Thus, v2t1D =.= 42. As the film
`gets thicker, the expression gets larger. It also gets larger as v in-
`creases and as D decreases. D, as will be seen later, decreases
`rapidly as the temperature decreases. Typical silicon epitaxial de-
`position conditions are 1100°C at 1µm/minute where v'tID will be
`much larger than 42. Thus, in most epitaxial depositions, 1,2tID>>
`10, and Eq. 8.50 is applicable for thicknesses greater than a very
`small fraction of a micron.
`
`The other half of the diffusion, that from the moving layer into
`the initial material, is considerably simpler since there is no out-
`
`Elm Exhibit 2159, Page 417
`
`

`
`8.3 SOLUTIONS TO THE DIFFUSION EQUATIONS
`
`401
`
`diffusion problem. When the initial material is infinitely thick, the
`boundary conditions are as follows:
`
`N( - x,0) = 0
`N(x„,t) = N,
`N( CO, = 0
`
`and when applied to Eq. 8.52 give
`
`N(x,t) =
`
`2
`
`+ erf(
`
`
`
`2V-5
`
`(2v1 -
`+ e(r/DN'I-x)erfc
`2030/
`
`8.54
`
`When v2tID is large, this equation reduces to the nonmoving bound-
`ary solution of Eq. 8.50.
`
`8.3.7 Out-Diffusion with
`Rate Limiting at Surface
`
`This set of conditions would describe the motion of impurities in an
`initially uniformly doped semiconductor heated to a high tempera-
`ture. The impurity flux J, across the semiconductor-ambient inter-
`face at x = 0 is given by
`
`J, = K(N,, -
`
`where K is the same rate constant discussed in the preceding case,
`N, is the concentration of impurity in the semiconductor at the sur-
`face, and Ne is the equilibrium concentration of impurity in the am-
`bient adjacent to the semiconductor. Note that if Ne > N„ there will
`be in-diffusion rather than a loss of impurities by out-diffusion. If,
`as is common, Ne is assumed to be much less than N, then J, =
`- KN,., and, by setting the growth velocity 1, equal to zero, Eq. 8.53
`can be used to calculate the impurity profile following out-diffusion
`(43). Thus,
`
`N(x,t)
`
`( - x + emu:, - oeifc(2Kt
`- err
`2Vit
`
`
`2V5t
`
`8.55
`
`For the limit of K = 0, nothing will escape; for the other limit of K
`very large so that there is no surface barrier, Eq.