Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 2 Part 8 pps

25.1.1 Diffusion Limited by the Flow of Intrinsic Defects Hybrid diffusion controlled by the slow flow products or transport products of self-interstitials and/or vacancies emerges, if the

k −1 =

C eq s

These ratios are constants, which depend on temperature

Simplifications of Eqs (25.3) to (25.6), which are usually justified underconditions of diffusion experiments [4, 5], concern local equilibrium betweenthe various species For local equilibrium between the partners of Eqs (25.1)

or (25.2), the laws of mass action can be written as

Cs

C i C V =

C eq s

generality Particular solutions C X (x, t) are determined by the pertinent

ini-tial and boundary conditions Often, numerical methods of computationalphysics must be used to derive solutions In the following, we consider firstthe two limiting cases of practical importance already given in Chap 24.Then, we report an example for an intermediate case, which requires numer-ical treatment

25.1.1 Diffusion Limited by the Flow of Intrinsic Defects

Hybrid diffusion controlled by the slow flow products (or transport products)

of self-interstitials and/or vacancies emerges, if the relationship

C I eq DI + C V eq DV  C eq

is fulfilled This condition can be realised in material that is completely orvirtually free of inner sinks and sources of point defects Dislocation-free Siwafers are commercially available3 Such wafers are the base materials for thefabrication of microelectronic devices Ge and GaAs materials are available atleast with very low values of the dislocation density If dislocations (and otherextended defects) are absent, the sink and source terms in Eqs (25.5) and

(25.6) can be omitted, i.e K V = K I = 0 Then, the interstitial-substitutional

3 This situation is very different from metals Carefully grown metal single crystalsstill have dislocation densities of about 1010m−2

exchange reactions and the pertinent establishment of the A iequilibrium arethe fastest processes and Eq (25.3) can be replaced by

C i ≈ C eq

This approximation may be approximately true either for in-diffusion in a nite region beneath the surface of thick crystals or throughout the entirewidth of a thin wafer, provided that the diffusion time is not too short.Taking into account the exchange reactions via local equilibria, Eqs (25.10)and (25.11), equations (25.4), (25.5), and (25.6) can be simplified to

We recognise in Eq (25.15) the diffusivity for defect-controlled diffusion of

hybrid solutes given in Eq (24.10) D I+V ef f contains a self-interstitial

compo-nent, D I ef f (kick-out mechanism), and a vacancy component, D V ef f ciative mechanism) For in-diffusion in dislocation-free material, the kick-outterm leads to a supersaturation of self-interstitials and the dissociative term

(disso-to an undersaturation of vacancies Let us point out some further interestingfeatures of the combined diffusion described by Eq (25.14):

– The defect-limited diffusion mode of hybrid solutes is closely related toself-diffusion of the host (see Chap 23) The tracer self-diffusion coefficient

D ∗ = f I C eq

I DI + f V C V eq DV (25.16)equals the sum of the flow products of self-interstitials and vacancies mod-

ified by their correlation factors f I and f V, respectively This equationprovides a valuable key to deduce these contributions from diffusion ex-periments of hybrid solutes (see Chap 23)

4 The assumptions made to derive Eq (25.15) are:p

C V eq C I eq C s and C V eq

C eq

s These are not severe restrictions, because the equilibrium concentrations ofpoint defects in semiconductors are very small compared to the substitutionalsolubility of hybrid diffusers As explicitly shown in the case of kick-out diffusion

by G¨osele et al.[2], the use of the local equilibrium condition Eq (25.11) also

imposes the restriction C s ≥pC I eq C s eq Therefore, Eq (25.15) can be used withvirtually no further loss of generality

– If both mass action laws for the dissociative reaction, Eq (25.10), and thekick-out reaction, Eq (25.11), are fulfilled

CI CV = C V eq C I eq (25.17)holds automatically This means that the equilibrium between vacanciesand self-interstitials can be established via the reactions of Eqs (25.1) and(25.2) even if the direct reaction of Frenkel pair annihilation and creation

is hampered, for example, by high activation barriers for spontaneousFrenkel-pair recombination or formation

– The self-interstitial component D I

ef f of Eq (25.15) depends strongly on

the actual local concentration C s (x, t), whereas the vacancy component

D V ef f is independent of concentration The concentration dependence of

D I

ef f provides an important key for the identification of kick-out diffusionfrom experimental determined concentration profiles (see below).– Under practical diffusion conditions, usually one of the two terms on theright-hand side of Eq (25.15) predominates Such positive identificationhas been made, for example, for Cu in Ge and Co in Nb (dissociativediffusion) and for Au, Pt, and Zn in Si (kick-out diffusion) Some keyresults that led to these conclusions are presented in Sects 25.2 and 25.3

25.1.2 Diffusion Limited by the Flow of Interstitial Solutes

Hybrid diffusion controlled by the low flow products of foreign interstitialsemerges if the inequality

C i eq Di  C eq

I DI + C V eq DV (25.19)

is fulfilled In this case, Eqs (25.3) and (25.4) together with the local rium conditions yield for the substitutional concentration a normal diffusionequation with an effective diffusion coefficient

This effective diffusivity is independent of the actual concentration, C s The

approximation made on the right-hand side, C i eq  C eq

s , is readily fulfilledfor most hybrid diffusers We recognise that Eq (25.20) is the interstitial-controlled diffusivity of hybrid solutes given in Eq (24.11) This diffusionmode can be expected under the following conditions:

1 The interstitial flow product of hybrid atoms is smaller than the combinedpoint defect flow products This implies that

C V ≈ C eq

and C I ≈ C eq

This condition is virtually maintained during in-diffusion by the large flowproduct of intrinsic defects as compared to that of foreign interstitials.This holds even for dislocation-free substrates.

2 Equation (25.20) can also hold for crystals with high dislocation ties, regardless of the validity of Eq (25.19) This is because the presence

densi-of sinks (sources) for self-interstitials (vacancies) shortcircuits the flows

of intrinsic point defects, whereas the foreign interstitials have to cover

a long distance from the surface of the crystal Under such conditions,the equilibrium concentration of intrinsic point defects can be establishedalmost instantaneously everywhere Then point defect equilibrium is vir-

tually maintained during in-diffusion, although the fast penetration of A i with the subsequent changeover to A s tends to create a supersaturation(undersaturation) of self-interstitials (vacancies)

Let us consider as a simple example, diffusion of a hybrid solute limited

by its interstitial flow product5 During in-diffusion of foreign atoms into

a thick sample from an inexhaustible source at the surface x = 0, the surface concentration C s (0) is kept at C eq

s Then, an erfc-type diffusion profile typical

of a concentration-independent diffusion coefficient is expected (see Chap 3and [6, 7]):

25.1.3 Numerical Analysis of an Intermediate Case

So far, the basic concepts of interstitial-substitutional diffusion have beenelucidated with the aid of limiting cases In practice, however, one is oftenconfronted with more complex ‘intermediate’ cases Then, numerical simu-lations can provide a more complete picture Figure 25.1 shows the result

of computer simulations within the kick-out model [4] using the soft warepackage ZOMBIE [8] The numbers chosen for the numerical simulation arerepresentative for in-diffusion of Zn in Si at 1380◦C for 280 s The in-diffusionprofiles of Fig 25.1 (a) represent distributions of substitutional atoms, C s (x),

in dislocation-free material for various magnitudes of the interstitial flow

product, C i eq D i, relative to a fixed value of the self-interstitial flow product,

5 In case of a highly dislocated sample, we further assume that dislocations act

as sources or sinks of intrinsic defects only Trapping of foreign atoms is notconsidered

Fig 25.1a,b Computer simulation of in-diffusion into a thick sample via the

kick-out mechanism according to Stolwijk [4] The left part (a) shows the tration of substitutional foreign atoms The right part (b) shows the associated

concen-self-interstitial supersaturations The values of the ratio α j ≡ C eq

self-interstitials, C i eq Di, according to Eq (25.19) This profile is of the erfc-typeand the self-interstitial concentration is practically at its equilibrium value.The upper profiles in Figs 25.1 (a) and (b) visualise solutions of Eq (25.14),

with D I+V ef f ≡ D I

ef f generating a convex near-surface shape for C s (x) due

to the factor (C eq

s /Cs)2and a significant self-interstitital supersaturation At

greater depths, however, the C s (x) profile changes its shape to concave, which relates to the violation of C i ≈ C eq

i adopted in the derivation of Eq (25.14)

Figures 25.1 (a) and (b) also exhibit intermediate cases, where C i eq Di and

C I eq DI are not too much different

25.2 Kick-out Mechanism

25.2.1 Basic Equations and two Solutions

Kick-out diffusion involves the interchange of the diffusing hybrid atoms tween substitutional and interstitial sites with the aid of self-interstitials If

be-we follow G¨osele et al.[2] and assume local thermal equilibrium, this type

of diffusion simplifies to the following set of equations:

C i eq

As discussed in the previous section, the a priori interstitial-limited

dif-fusion as well as diffusion in highly dislocated material imply that

self-interstitial equilibrium is virtually maintained, i.e that C I ≈ C eq

I holds.Then, Eqs (25.24) and (25.25) yield for the substitutional concentration

a normal diffusion equation with the diffusion coefficient given by Eq (25.20)

The lowest C s (x) profile in Fig 25.1 provides an example of such

a case

The self-interstitial controlled limit emerges for C I eq D I  C eq

i D i For

negligible sink and source density, the K I-term in Eq (25.23) can be

omit-ted Then, the kick-out reaction and the establishment of the A i equilibriumare the fastest processes Using the local mass-action law Eq (25.25) and

The effective diffusion coefficient D I

ef f is the self-interstitial limited diffusivity

of Eq (25.15) In the usual case of in-diffusion, the kick-out reaction leads to

a supersaturation of self-interstitials Then, the incorporation of hybrid eign atoms on substitutional sites is limited by the outflow of self-interstitials,

for-C I eq DI The concentration dependence of D I

ef f gives rise to distinct features

of the kick-out diffusion For example, its 1/C2-dependence speeds up sion at low concentrations and leads to convex-shaped diffusion profiles Inwhat follows, we mention two particular solutions of Eq (25.26), which play

diffu-a rˆole in experiments discussed in the next section

1 Diffusion into a Dislocation-free Thick Specimen: In-diffusion of

hybrid atoms from the surface x = 0 of a thick specimen is described

by Eq (25.26) with the boundary condition C s (x = 0, t) = C eq

s and the

initial condition C (x, t = 0) = C0 For a dislocation-free crystal Seeger

and coworkers [9, 10] derived an analytical solution in parametric

form The approximate solution for C s0 C eq

s is

eq s

crys-2 Diffusion into a Dislocation-Free Wafer: Next, we consider diffusion

into a wafer of thickness d and take advantage of the wafer symmetry by choosing the wafer center at x = 0 and the wafer surfaces at x = ±d/2.

For a dislocation-free wafer a solution of Eq (25.26) has been derived for

the boundary condition C s (x = ±d/2, t) = ∞ Such a solution is useful

for not too long diffusion times and in the vicinity of the wafer center.G¨osele et al[2] have shown that for C s0= 0 solutions of the form

±



πC I eq C s eq X(0) erf



ln[X(x)/X(0)] = x (25.31)with

Fig 25.2 In-diffusion of Au into a thick dislocation-free Si specimen according to

Stolwijk et al [13] The solid line represents a fit of the kick-out mechanism The dashed line is an attempt to fit a complementary error function predicted for

a concentration-independent diffusivity

25.2.2 Examples of Kick-Out Diffusion

Au and Pt Diffusion in Silicon: Diffusion of Au in Si has been

stud-ied by several authors For a list of references see [10] For example, Wilcox

et al.[11, 12] report that the diffusion profiles in thick, dislocation-free ples differ considerably from the erfc-type shape Accurate measurements byStolwijk et al.[13], using neutron-activation analysis and grinder section-ing for depth-profiling, confirmed the non-erfc nature of Au profiles in Si

sam-A comparison between an experimental sam-Au-profile and the predictions of thekick-out mechanism, Eq (25.27), and an erfc-profile is shown in Fig 25.2.The kick-out model permits a successful fit

Stolwijk et al [13] also investigated the diffusion of Au into thin,dislocation-free Si wafers and found U-shaped diffusion profiles displayed

in Fig 25.3 Similar observations were reported by Hill et al [15], whoused the spreading-resistance technique for depth profiling (see Chap 16).U-shaped profiles are in qualitative agreement with both the kick-out andthe dissociative mechanism A distinction between the two mechanisms ispossible via a quantitative analysis of the profile shape Figure 25.4 shows

a comparison of an experimental profile from Fig 25.3 with the relationship

Fig 25.3 In-diffusion of Au into dislocation-free Si wafers for various annealing

times at 1273 K according to [13] The Au concentration C Auis given in units of the

solubility C Au eq ; the penetration depth x as a fraction of the wafer thickness d For one penetration profile of the two 4.27 h anneals d ≈ 300 µm, otherwise d ≈ 500 µm

erf ln

&

Cs

C m s

as the spreading-resistance technique by Hauber et al [16] The resultssuggest that diffusion of Pt in Si is also dominated by the kick-out mechanism

Zn Diffusion in Si: Diffusion of Zn in Si provides a well-studied example

of interstitial-substitutional exchange diffusion dominated by the kick-outmechanism Both cases – in-diffusion limited by the defect flow product or

by the flow product of interstitial Zn – have been observed in the work ofBracht et al.[17] Figure 25.5 shows concentration profiles of Zn s mea-sured on dislocation-free and on highly dislocated Si Both wafers were si-multaneously exposed to Zn vapour from an elemental Zn source in a closedquartz ampoule This source yields at both wafer surfaces a concentration of

C Zn eq

s ≈ 2.5 × 1016 cm−3, which corresponds to the solubility limit of Zn in

Si in equilibrium with the vapour phase from a pure Zn source Also shown

is a Zns profile in dislocation-free Si obtained at the same temperature butusing a 0.1 molar solution of Zn in HCl as diffusion source, providing a much

Fig 25.4 Comparison of Au diffusion into a dislocation-free Si wafer (1.03 h at

1273 K) with the prediction of the kick-out mechanism (solid line) [13] Circles:

x < 0; squares x > 0

lower Zn vapour pressure Then, the boundary concentration observed at

both surfaces of the wafer is C Zn eq

s ≈ 4.5 × 1014cm−3, which is a factor of 55

smaller than the solubility limit reached in equilibrium with an elemental Znsource Figure 25.5 demonstrates that Zn diffusion in Si depends sensitively

on the defect structure as well as on the prevailing ambient conditions The

diffusion profile in dislocation-free Si (crosses) is convex, whereas the

pro-file in the highly dislocated wafer, apart from the central region, is concave

(squares) Also the profile obtained for lower Zn concentrations is convex (circles) The kick-out mechanism yields a consistent description of all three

profiles in Fig 25.5, as illustrated by the solid lines

In addition, Bracht et al [17] studied the time evolution of Zn ration in dislocation-free and in highly dislocated Si for various temperaturesapplying a specially designed short-time diffusion method Typical examples

incorpo-are displayed in Fig 25.6 The solid lines in the top part show the result of

successful fitting of all experimental profiles based on the kick-out mechanism

The solid lines in the bottom part represent complementary error functions.

1 Dislocation-free Si and high boundary concentration: In dislocation-free

material no sinks and sources for self-interstitials exist High boundaryconcentrations of Zn maintain

C i eq D i  C eq

representing self-interstitial limited flow The supply of Znifrom the face occurs more rapidly than the decay of self-interstitial supersatura-

sur-Fig 25.5 In-diffusion of substitutional Zn measured on dislocation-free (crosses)

and highly dislocated (squares) Si samples; Zn was diffused simultaneously at

1115◦C for 2880 s into both samples, using metallic Zn as vapour source The lower

profile (circles) represents in-diffusion into a dislocation-free wafer also at 1115 ◦Cusing a Zn solution in HCl as diffusion source After Bracht et al [17]

tion by out-diffusion to the surface For long diffusion times the

Zns profiles in a semilogarithmic representation are distinctly U-shaped

like the profile (crosses) in Fig 25.5 and the Au-profiles in Fig 25.3.

These features reflect that Zns and Aus incorporation is controlled bythe out-diffusion of self-interstitials

2 Dislocation-free Si and low boundary concentration: During Zn-diffusion the Dirichlet surface condition, C i (0, t) = C i eq, is maintained by a con-stant Zn vapour pressure in the diffusion ampoule This vapour pressurealso determines the Zns equilibrium concentration C eq

s , which is

propor-tional to C i eq , whereas the properties D i, C I eq and D I are not influenced

by the ambient conditions Therefore, at sufficiently low boundary centration the relationship

con-C i eq D i  C eq

is fulfilled As a consequence, the kick-out reaction is not able to establish

self-interstitial supersaturation and C I ≈ C eq

I holds during the diffusionprocess Then, the incorporation of Zn is limited by the flow product of Zninterstitials with a diffusivity given by Eq (25.20) Under these conditions

diffusion profiles are described by complementary error functions (25.22),like the lower profile in Fig 25.5.

3 Highly dislocated Si: In Si with a high dislocation density a

supersatura-tion of self-interstitials can decay by annihilasupersatura-tion at these defects The

profiles in the bottom part of Fig 25.6 were obtained on plastically

de-formed Si with a dislocation density of about 1012m−2 This corresponds

Fig 25.6 Diffusion profiles of substitutional Zn (Zns) at 1021◦C for various

diffu-sion times according to Bracht et al [17] Top: Dislocation-free Si wafers; solid

lines show calculated profiles based on the kick-out model using one set of

param-eters C eq

s and C I eq D I Bottom: Highly dislocated Si; solid lines show fitting with

complementary error functions