Fundamentals and Applications of Ternary Diffusion. Proceedings of the International Symposium on Fundamentals and Applications of Ternary Diffusion, Hamilton, Ontario, Canada, August 27–28, 1990

204 Pages·1990·8.976 MB·English

by  G. R. Purdy (Eds.)

Titles of Related Interest—
Ashby ENGINEERING MATERIALS 1
Ashby ENGINEERING MATERIALS 2
Brook IMPACT OF NON-DESTRUCTIVE TESTING
Koppel AUTOMATION IN MINING, MINERAL AND METAL PROCESSING 1989
R٧hle METAL-CERAMIC INTERFACES
1iiya METAL MATRIX COMPOSTTES
Other CIM Proceedings Published by Pergamon
Bergman FERROUS AND NON-FERROUS ALLOY PROCESSES
Blckert REDUCTION AND CASTING OF ALUMINUM
Bouchard PRODUCTION. RERNING, FABRICAnON AND RECYCUNG OF LIGHT
METALS
ChalMey TAILING AND EFFLUENT MANAGEMENT
Closset PRODUCTION AND ELECTROLYSIS OF LIGHT METALS
Dobby PROCESSING OF COMPLEX ORES
Embury HIGH TEMPERATURE OXIDATION AND SULPHIDAHON PROCESSES
Jaeck PRIMARY AND SECONDARY LEAD PROCESSING
Jonas DIRECT ROLUNG AND HOT CHARGING OF STRAND CAST BILLETS
Kachaniwaky IMPACT OF OXYSEN ON THE PRODUCTIVITY OF NON-FERROUS
METALLURGICAL PROCESSES
Lalt F WEINBERG INTERNATIONAL SYMPOSIUM ON SOLIDIFICATION
PROCESSING
Maemillan QUALITY AND PROCESS CONTROL IN REDUCTION AND CASTING
OF ALUMINUM AND OTHER LIGHT METALS
Mostaghaci PROCESSING OF CERAMIC AND METAL MATRIX COMPOSTTES
Plumpton PRODUCTION AND PROCESSING OF ΠΝΕ PARnCLES
Rigaud ADVANCES IN REFRACTORIES FOR THE METALLURGICAL INDUSTRIES
RucMIe ACCELERATED COOUNG OF ROLLED STEEL
Salter GOLD MEIΔLLURGY
Thompson COMPUTER SOFTWARE IN CHEMICAL AND EXTRACTIVE
METALLURGY
IWIgge-Molecey MATERIALS HANDLING IN PYROMETALLURGY
IWIgge-Mdecey PROCESS GAS HANDLING AND CLEANING
lyson FRACTURE MECHANICS
Wilkinson ADVANCED STRUCTURAL MATERIALS
Related Journals
(Frae MmpI* coptot avallabto upon raqueat)
ACTA METALLURGICA
CANADIAN METALLURGICAL QUARTERLY
MATERIALS RESEARCH BULLETIN
MINERALS ENGINEERING
SCRIPTA METALLURGICA
PROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM ON
FUNDAMENTALS AND APPLICATIONS OF TERNARY DIFFUSION
HAMILTON, ONTARIO, CANADA, AUGUST 27-28, 1990
Fundamentals and
Applications of
Ternary Diffusion
Editor
G.R. Purdy
Department of Materials Science and Engineering
McMaster University
Hamilton, Ontario, Canada
Symposium organized by The Metallurgical Society of CIM
29th ANNUAL CONFERENCE OF METALLURGISTS OF CIM
29e CONFERENCE ANNUELLE DES MÉTALLURGISTES DE L'ICM
Férgamon Press
Member of Maxwell Macmillan Pfergamon Publishing Corporation
New York Oxford Beijing Frankfurt Sao Paulo Sydney Tbkyo Toronto
Pergamon Press Offices:
U.S.A. Pergamon Press, Inc., Maxwell House, Fairvlew Park,
Elmsford, New York 10523, U.S.A.
U.K. Pergamon Press pic, Headlngton Hill Hall,
Oxford 0X3 OBW, England
PEOPLE'S REPUBLIC Pergamon Press, 0909 China World Tower, No. 1 Jian
OF CHINA Quo Men Wai Avenue, Beijing 1000004, People's Republic
of China
FEDERAL REPUBLIC Pergamon Press GmbH, Hammerweg 6,
OF GERMANY D-6242 Kronberg, Federal Republic of Germany
BRAZIL Pergamon Editora Ltda. R٥a Ega de Queiros, 346
CEP 04011, Paraνso, Sao Paulo, Brazil
AUSTRALIA Pergamon Press Australia Pty Ltd., RO. Box 544,
Potts Point, NSW 2011, Australia
JAPAN Pergamon Press, 8th Floor, Matsuoka Central Building,
1-7-1 Nishishinjuku, Shinjuku-ku, Tokyo 160, Japan
CANADA Pergamon Press Canada Ltd., Suite 271,
253 College Street, Toronto, Ontario M5T1R5 Canada
Copyright © 1990 Pergamon Press, Inc.
All rights reserved. No part of this publication may
be reproduced In a retrieval system or transmitted In
any form or by any means: electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission In writing from the
publishers.
Library of Congress Cataloging in Publication Data
ISBN 0-08-040412-X
Printing: 1 2 3 4 5 6 7 89 Year: 0 1 2 3 4 5 6 7 89
Printed In ttie United States of America
The paper used in this publication meets the minimum require­
ments of American National Standard for Information Sciences-
Permanence of Paper for Printed Library Materials, ANSI Ζ 39.48-
1984
Foreword
In 1955, as a junior professor at McGill University and a summer employee of Alimiinium Labora­
tories in Kingston, Ontario, Jack Kirkaldy encountered the problems of diffusion and phase transfor­
mations in multicomponent alloys and as a physidst-cum-metallurgist decided to undertake a systematic
ejqjerimental and theoretical study of the phenomena with a view to establishing a technological problan-
solving capability. Upon joining McMaster University in 1957, he established a large and productive
research group, focussing on aluminum and iron-carbon base solid alloys, but not excluding the full
range of solid and liquid material manifestations. His synthesis, with D.J. Yoimg, appeared in 1987
as the definitive volume on multicomponent effects. Diffusion in the Condensed State. The present
volume may be regarded as an appendix and update on that synthetic contribution.
Taking a lead from Onsager's irreversible thermodynamic foundations, Kirkaldy, in 1958, recognized
the importance of matrix methods in generalizing binary theory to temary and higher order systems.
Elements of this intelligence are to be found in the new contributions of Miller, Morral et al and
"Kirkaldy et al, in this volume. Jack recognized early on that temary cross effects would often be weak,
as for example in dilute solutions, so that pseudo-binary constmctions would be adequate for problem-
solving. While establishing appropriate theoretical criteria, he also sought to discover material or ex­
perimental conditions where such cross-effects would be aihanced. Papers in the present volume co-
authored with Young, Okongwu, Zou and Buchmayr expand this particular knowledge to a range
of new material systems.
There is a broad trend toward interaction enhancement at concentrations rich enough to produce phase
transformations. Kirkaldy and co-workers have accordingly extensively studied the multicomponent-
multiphase phenomena in both stable and unstable interface configurations. This work in Fe<:-X alloys
has had an important technological impact through the successful marketing of Jack's software pack­
age pertaining to quality control and optimization in the heat treatment of alloy steels. The papers
of Dayananda and Liu, Jan et al,, van Loo et al. Stringer, Brechet, Balasubramanian, Zou, Larson
and Buchmayr bear directly on this problem area and exploit Jack's earlier contributions.
In 1963-66, Lane and Kirkaldy demonstrated that the Kirkendall Effect associated with a vacancy diffii­
sion mechanism in solids produces a temary diffusion cross-effect through the drift velocity which
can be comparable in magnitude to purely thermodynamic effects. In his present short contribution,
he makes an explicit comparison of magnitudes and demonstrates that vacancy correlation effects on
diffiision profiles are usually unimportant in close-packed materials.
In early studies of unstable temary transformation interfaces. Jack recognized a close analogy with
instabilities generated in cellular solidification of binary alloys in a temperature gradient. This has led
him into a deep study of pattern formation based within the variational principles of irreversible ther­
modynamics and formal logic. While excluded from this volume, the elements are to be found in Chapter
13 of the above-mentioned 1987 volume and the Pergamon symposium volume "Advances in Phase
Transformations", which was arranged to celebrate Jack's 60th birthday. With this volume, presented
upon his retirement, we gratefully acknowledge Jack Kirkaldy's many contributions to, and intellec­
tual leadership in, üie field of temary diffusion over more than three decades.
Gary Purdy
Hamilton, Ontario
August 1990
Impurity diffusion in a doubly-doped elemental
semiconductor with pair formation
D.J. Young
School of Materials Science and Engineering, University of New South Wales,
Kensington, N.S.W. 2033, Australia
R.C. Dorward
Kaiser Chemical and Aluminum Corp., Pleasanton, California, U,S.A,
J.S. Kirkaldy
Institute for Materials Research, McMaster University,
Hamilton, Ontario, Canada, L8S 4M1
Abstract
The diffusion of substitutionally dissolved donors and acceptors has been
described in terms of point defect movement and an irreversible thermodynamic
treatment has been used to identify the interactions among the different diffusing
species. The principal diffusive interactions for a donor species have been shown to be
countercurrent donor-vacancy movement, countercurrent donor-solvent movement,
countercurrent donor-acceptor movement and co-current donor-acceptor movement.
The relative magnitude of kinetic coefficients can be calculated specifically from an
absolute rate theory approach. This paper deals specifically with the common case of
countercurrent donor-acceptor movement subject to the complication of donor-acceptor
pair formation. This is incorporated into the kinetics using a linear expansion of the
logarithm of the activity coefficient. A representative diffusion profile is given
showing significant attractive segregation of the donor on a gradient of acceptor
despite the extreme diluteness of the reactants.
Introduction
Diffusion against a concentration gradient in a ternary or higher order couple
can occur for a variety of reasons including thermodynamic interactions, Onsager
correlation cross-effects, vacancy winds and inhomogeneities, electrostatic neutraliza­
tion interactions in ionic solids (1) and electron-solute interactions in semiconductors
(2). This paper deals with the exceptionally strong effect of an electrostatic pair
reaction (1-3) as represented by combination of donor and acceptor dopants in a semi­
conductor (e.g. Si). At the very dilute concentrations involved thermodynamic
solution and Onsager effects (1) are by contrast of no observational consequence.
An elemental semiconductor such as silicon will in general contain a variety of
defect types, including impurities. In describing the diffusion of tJhose impurities, it is
necessary to take into account their chemical interactions with all other species in the
solid. An appropriate methodology for calculating these effects is provided by irrever­
sible thermodynamics, which provides a phenomenological relationship between
fiuxes and driving forces and interrelates the different species. For any system of
Schottky and (or) Frenkel defects we adopt the notation of Kroger and Vink (4)
wherein the lattice species are represented by the symbol SV. Here the subscript
represents the normal occupancy in a perfect crystal of the site in question, and the
principal symbol represents the species actually occupying the site. The superscript
represents the charge of the species relative to normal site occupancy with a prime
denoting a negative, a dot positive and a cross, zero charge.
When defects diffuse under isothermal, field-free conditions, they must on the
average move in groups which conserve both electrical charge and sites. These groups
FUNDAMENTALS AND APPLICATIONS OF TERNARY DIFFUSION
satisfy the definition of "relative building units" formulated by Schottky (5) and by
Kroger et al. (6) in their description of point defect equilibria. The usefulness of this
concept in treating diffusion in ionic solids has since been demonstrated (7, 8).
Kirkaldy and Young (1) have adapted that treatment to the case of an elemental
semiconductor containing multiple vacancy charge states, as well as impurities. They
have also discussed procedures for introducing the diffusion kinetics via absolute rate
theory, and this will be expanded in the present context of diffusing building units.
Finally, the effect of donor-acceptor pair formation on the observed diffusion
behaviour is analyzed. Although the treatment is quite general, the specific case of
silicon is used as an example.
The Relative Building Units For A Doubly-Doped Semiconductor
This section summarizes the results of the irreversible thermodynamic entropy
source method for identifying the building units Ui (1). In this method, the linear
dependencies contained in the set of fluxes appearing in the entropy source term are
eliminated using the flux constraints due to site and charge conservation.
For silicon containing a donor, D, an acceptor. A, and vacancies which can be
positively or negatively charged, as well as neutral, there exist 18 distinguishable
pairs of species fluxes not involving the donor. Elimination of these flux pairs reveals
the building units containing D, in serial order (i = 1-13):
Dsi-Vsi + 2«} {Dsi-Vsi + 2h- }
ÍDsi-Asi + 2e'} {Ds,-A3^-2h- }
Only 13 of 18 distinguishable units result because all six units involving V reduce to
{Dg¿-Vg¿}. It should be noted that these units reflect the simplest forms of
correlation, as any linear combination of the units would obviously satisfy all
thermodynamic and conservative requirements.
The relative building units involving D g. necessarily reflect the flux
constraints which via the entropy source expression impose the requirements that
movement of D be balanced by an opposite movement of another lattice species, and
that any net charge flow thereby arising be neutralised. Thus four principal
correlations are seen to exist; countercurrent donor-vacancy movement, counter-
current donor-solvent movement, countercurrent donor-acceptor movement and
cocurrent donor acceptor movement.
The relative importance of these various possible correlations will be
determined by the probabilities of finding the required species in adjacent positions
within the solid (i.e., of locating the required relative building unit). Thus the most
likely of the various possible donor-vacancy exchange mechanisms will be determined
by the vacancy ionization equilibria. All correlations involving the simultaneous
movement of two constituents require the presence of a vacancy. If the vacancy is
explicit in the corresponding relative building unit, then net vacancy transport occurs.
If the unit contains no vacancy, then the vacancies function only as intermediaries. It
follows then that donor-solvent countercurrent diffusion can occur with and without
net vacancy movement. The favoured mechanism will depend on the relative
FUNDAMENTALS AND APPLICATIONS OF TERNARY DIFFUSION
abundance of V'^. and Vg¿. Cocurrent donor-acceptor diffusion can also occur both
with and without'a vacancy ñux. Given that the probability of finding a Si'^si species
approaches unity, the unit {Dg, +Ag^ —2Siy adjacent to an additional Vsi is more
probable than {D c: +A «.-2V*g.} alone. The zero vacancy flux situation therefore
seems more probable. Countercurrent donor-acceptor diffusion cannot involve a net
vacancy flux but must involve an electronic current. The relative importance of
cocurrent and countercurrent donor-vacancy correlations will depend on kinetic
factors and on the relative availability of neutral vacancies and free carriers.
Applying the same procedures to an analysis of the diffusion of acceptors A' it is
found that the relative building units U'. involved are in line serial order (i = 1-13)
ÍAsi-Vg,-2e}{Ag^-Vg^+2h- }
{^Si - Si^i " e'} {A' - Si^^ + h- } {Ag^ - 2Si^^ + VgJ
(2)
ÍAsi-Dg^-2e'} {Ag,-Dg^+2h- }
K^'^Si -2V^iHAg, + Dg^-2Si^^}
The mechanism by which D can move is via the relative building units listed in
Eq. 1. A linear combination of the fluxes arising therefrom represents a complete and
exhaustive accounting for diffusion of D , viz.,
13
JD = - Σ ^i^níu.} (3)
i = l
where q(Ui) is the electrochemical potential of building unit Ui of standard form
η. =μ. + ζ. Ρψ (4)
where μι is the chemical potential, zi is the number of electronic charges with due
regard to sign, F is the Faraday and ψ is the electrochemical potential. As usual ψ
vanishes in the transformations. Substituting for q(Ui) from Eq. 1 one obtains after
rearrangement
JD = - Σ 'i ^^D - «12 + ·ΐ3 - \ o- \l> ^^·A + «7 + 'β + ^^9 + 21 ,3) Vp^. (5)
i = l
and the cross-terms in the diffusion matrix are seen to be explicitly identified.
Similarly it is found for diffusion of A'
,
13
<f A = - + ^3 - \o - "^D - Σ ^ + + ^8 + + 2X^3) Vpg. (6)
i=l
Further simplification results if the solid solution is Henrian, for then in terms
of concentration mj in moles/unit volume
νμ. = RT Vm./m. (7)
and, for low vacancy concentrations,
FUhTOAMENTALS AND APPLICATONS OF TERNARY DIFFUSION
(8)
Substitution of (7) and (8) into (5) and (6) followed by rearrangement yields
J.
RT I mj, " D
[LAD f^AA I^ASil (9)
RT D •"A '
where the Ly are defined as the coefficients in Eqns. (5) and (6).
To proceed further, the as-yet unidentified coefficients €[ and λ{ must be
evaluated. To this end we must turn to a kinetic description of the process. We may
anticipate, however, that because the units UIQ to U13 of Eq. 1 involve precisely the
same lattice and free carrier species as the analogous units U'lo to U'ls of Eq. 2, the
kinetics of the species interchange processes will be the same, and hence
li = Ai , i = 10,11,12,13 (10)
In consequence of this eventuality, the synunetry expected of the L-matrix is arrived
at.
Defect Diffusion Kinetics
The diffusion of point defects can be calculated by the methods of absolute rate
theory (9) when proper account is taken of local electrostatic field effects (10). The
method has been presented elsewhere (11). For the interchange of lattices species i
and vacancies between a^acent sites separated by a distance d
J = - V. K. a. a V{n. - η } (11)
RT IV IV I V ^ h ^v'
Here Ν is the volume concentration of lattice sites, viv is a frequency term, and Kiv is
the equilibrium constant relating the activity of the transition-state complex to those
of the reactants (ai and ay). This expression is used to evaluate the rates of the several
pair-wise interchanges which constitute the movement of a complete relative building
unit.
Applying Eqn. (11) to the diffusion of Uio, for example, we find
Nd^
(12)
RT
(13)
RT
In addition, it is possible to write for the overall process from initial to final state
JiO = - vK^aj^a^a^afvWD ) - η(Α ) + 2n(e )} (15)
The quantity vKc is evaluated from the quasi-steady-state condition
FUNDAMENTALS AND APPLICATONS OF TERNARY DIFFUSION
J=J<»> = jib)^j(c)
and the combined activity product corresponding to the four reactions of Eqs. 12-14
(16)
It is then found that
a^xV{n(D)-ri(A') + 2n(e')} (17)
where new concentration variables are defined by
(18)
^A = ^AW^AV^' "^e^W^
The coefficient ho is thus evaluated. Values found in this way for all the li, λι are
listed in the Appendix. It is seen here explicitly that
i= 1,2...9
(19)
1. = λ. , i = 10,...13
and the L-matrix is symmetric.
The relative importance of the different diffusion mechanisms depends upon the
concentrations of the individual species involved which in turn depend on the
equilibrium constants for the formation reactions (1, 12). Even though all relative
building units, when they move, represent diffusion without a net current, several of
them contain a net charge, e.^., {Dgi—Vg.}. Movement of these units obviously
involves two equal and opposite electrical currents. However, the existence of
sufficient units to provide a diffusive flux necessarily implies a space charge unless
local compensating, oppositely charged defects are present. This, in turn, implies that
the probability of finding the required species in adjacent positions within the solid
(i.e. of locating the required relative building unit) is low. In other words, because
thev possess mobile free carriers, semiconductors cannot support large space charges
and tnerefore the concentration of uncompensated building units is necessarily low.
Accordingly, diffusion contributions from the following units are taken to be
negligible:
^1' U,,, U,,,
U,, U,. U,, U,,, U,,
The contributions of the remaining units are now used to evaluate the coefficients in
the diffusion equations (9). It is convenient to recast these as
RT ^DD - ^DSi ^DA - ^DSi
(20)
:ÍA
RT ^AD " ^ASi ^AA - ^ASi
'"si "Si
Noting from definitions (18) that for Henrian behaviour
Vm./m. = Vm./m. (21)
11 11
and substituting for the Lij from term-by-term comparison with Eqns. (5) and (6) then
if only charge-neutral relative building units are considered, we obtain (dropping the
primes for clarity)
FUNDAMENTALS AND APPLICATONS OF TERNARY DIFFUSION
(22)
^A= -I^AD^"^D-I^AA^"^A
2m 2m,
DP
m^ + mj^a^x m^ + m^^ m^, + nijj
D, 2m^
DA
+ 2
Nd"
D 2m. mj^m^mj^ (23)
AD
+ 2 a
Nd m^ + m^ mg.mj^(mj^ + m^a^x)
D 2m. 2m^
AA V'
Nd'2 mj^ + m^a^ m^+ m^, + m^
Note that the m's in Eqs. 22 as defined incorporate factors of form vK.
Consequences of Defect Pairing
Point defects may be able to interact strongly with one another (1-3,13) forming
complexes which are relatively inunobile within tJie host lattice. An example of this
additional kind of local equilibriiun is pair formation between two oppositely charged
donors and acceptors in a semiconductor represented by the chemical equation
D -H A' = DA (24)
for which we may write the mass law, assuming ideal or Henrian solution behaviour
(25)
^DA^^D- ^A' - ^
In this case the concentrations of species actually free to move, XD and XA', differ
from the analytically measured concentrations of these components, Xi and X2, i.e.
(26)
^D - ^1 ~ ^DA ^A' " ^2 ~ ^DA
Since diffusion experiments on solids invariably rely on analytical measurements of
total component concentration, the diffusion equations must be couched in these
terms. It is necessary, therefore, to describe the thermodynamics of the solution in
terms of total component concentrations.
Whilst the free species D and A', if dilute, will be Henrian, it is most
improbable tiiat the solute components Xi and X2 can be so regarded. This is an
immediate consequence of the fact that Xi, X2 are not direct measures of XD and XA'
because of the formation of pairs. Consequently activities referred to the pure solutes
are expected to deviate markedly from their values in the absence of pair formation.
One particular solution model is now proposed and its consequences for the diffusion
equations explored.
If the two solutes have negligible solubility for the solvent and each other, then
the phase boundaries of the solution may be defined, at constant temperature and
constant activity lines (Fig. 1).
YjX^ = l and ¥3X2=1 (27)