Page 1 of 51
`
`SAMSUNG ET AL. EXHIBIT 1085
`Samsung et al. v. Elm 3DS Innovations, LLC
`IPR2016-00387
`
`

`

`aterias cienoe
`of
`
`Thin Fi ms
`
`Ii———————£
`
`Milton Ohring
`Stevens Institute of Technology
`Department of Materials Science and Engineering
`Hoboken, New Jersey
`
`~riu‘9 hawk
`
`befaubs “H ~.
`
`\[t/omuz waj
`312x 389'’
`Crzp/1/(em
`
`zt4?r¥o<:z,
`3%?-~90}.
`
`ACADEMIC PRESS, INC.
`Harcourt Brace Jovanovich, Publishers
`
`Boston San Diego I New York
`London Sydney Tokyo Toronto
`
`Page 2 of 51
`
`

`

`This book is printed on acid—free paper.
`
`Copyright © 1992 by Academic Press, Inc.
`All rights reserved.
`No part of this publication may be reproduced or
`transmitted in any form or by any means, electronic
`or mechanical, including photocopy, recording, or
`any information storage and retrieval system, without
`permission in writing from the publisher.
`
`Designed by Elizabeth E. Tustian
`
`ACADEMIC PRESS, INC.
`1250 Sixth Avenue, San Diego, CA 92101
`
`United Kingdom Edition published by
`ACADEMIC PRESS LIMITED
`
`24-28 Oval Road, London NW1 7DX
`
`Library of Congress Cata1oging—in—Publication Data
`
`Ohring, Milton, date.
`The materials science of thin films / Milton Ohring.
`p.
`cm.
`Includes bibliographical references and index.
`ISBN 0-12-524990-X (Alk. paper)
`1. Thin films.
`I. Title.
`TA418.9.T45047
`1991
`620’.44—dc20
`
`Printed in the United States of America
`91929394
`987654321
`
`Page 3 of 51
`
`

`

`"-
`Chapter 8FT...
`
`Enterdiffusion and
`
`Reactions in Thin Films
`
`8.1 .
`
`INTRODUCTION
`
`There is hardly an area related to thin-film formation, properties, and perfor-
`mance that is uninfluenced by mass-transport phenomena. This is especially
`true of microelectronic applications, where very small lateral as well as depth
`dimensions of device features and film structures are involved. When these
`
`characteristic dimensions (d) become comparable in magnitude to atomic
`diffusion lengths, then compositional changes can be expected. New phases
`such as precipitates or layered compounds may form from ensuing reactions,
`altering the initial film integrity. This, in turn, frequently leads to instabilities
`in the functioning of components and devices that are manifested by such
`effects as decrease in conductivity as well as short— or even open—circuiting of
`conductors, lack of adhesion, and generation of stress. The time it takes for
`such effects to evolve can be roughly gauged by noting that the diffusion length
`is given by ~ 2 x/D7 , where D and t are the appropriate diffusivity and time,
`respectively. Therefore t = d2 /4D. As we shall see, D values in films are
`relatively high even at low temperatures, so small film dimensions serve to
`make these characteristic times uncomfortably short. Such problems frequently
`surface when neighboring combinations of materials are chemically reactive.
`
`355
`
`Page 4 of 51
`
`

`

`356
`
`lnterdiffusicm and Reactions in Thin Films
`
`For example, consider the the pitfalls involved in designing a Cu—Ni film
`couple as part of the Contact structure for solar cells (Ref. 1). Readily available
`high-temperature data in bulk metals extrapolated to 300 °C yield a value of
`3.8 X lO‘24 cml/sec for the diffusion coefficient of Cu in Ni. For a 10003
`thick Ni film, the interdiffusion time is thus predicted to be (10‘5)2 /4(3.8 X
`10”“) sec, or over 200,000 years! Experiment, however, revealed that these
`metals intermixed in less than an hour. When colored metal films are involved,
`as they are here, the eye can frequently detect the evidence of interdiffusion
`through color or reflectivity changes. The high density of defects, e.g., grain
`boundaries and vacancies, causes deposited films to behave differently from
`bulk metals, and it is a purpose of this chapter to quantitatively define the
`distinctions. Indeed, a far more realistic estimate of the Cu-Ni reaction time
`can be made by utilizing the simple concepts developed in Section 8.2. Other
`examples will be cited involving interdiffusion effects between and among
`various metal film layer combinations employed in Si chip packaging applica-
`tions. Practical problems associated with making both stable contacts to
`semiconductor surfaces and reliable interconnections between devices have
`
`been responsible for generating the bulk of the mass-transport-related concerns
`and studies in thin films. For this reason,
`issues related to these extremely
`important subjects will be discussed at length.
`While interdiffusion phenomena are driven by chemical concentration gradi-
`ents, other mass-transport effects take place even in homogeneous films. These
`rely on other driving forces such as electric fields,
`thermal gradients, and
`stress fields, which give rise to respective electromigration, thermomigration,
`and creep effects that can similarly threaten film integrity. The Nernst—-Ein-
`stein equation provides an estimate of the characteristic times required for such
`transport effects to occur. Consider a narrow film stripe that is as wide as it is
`thick. If it can be assumed that the volume of film affected is ~ d3 and the
`
`mass flows through a cross—sectional area dz, then the appropriate velocity is
`d/ t. By utilizing Eq. 1-35, we conclude that tz RTd/DF. Large driving
`forces (F), which sometimes exist in films, can conspire with both small d and
`high D values to reduce the time to an undesirably short period. As circuit
`dimensions continue to shrink in the drive toward higher packing densities and
`faster operating speeds, diffusion lengths will decrease and the surface—area—
`to-volume ratio will increase. Despite these tendencies, processing tempera~
`tures and heat generated during operation are not being proportionately re-
`duced. Therefore,
`interdiffusion problems are projected to persist and even
`worsen in the future.
`
`In addition to what may be termed reliability concerns, there are beneficial
`mass~transport effects that are relied on during processing heat treatments in
`
`Page 5 of 51
`
`

`

`3.2. Fundamentals of Diffusion
`
`357
`
`films. Aspects of both of these broad applications will be discussed in this
`chapter in a fundamental way within the context of the following subjects:
`
`8.2. Fundamentals of Diffusion
`
`8.3. Interdiffusion in Metal Alloy Films
`8.4. Electromigration in Thin Films
`8.5. Metal—Serniconductor Reactions
`8.6. Silicides and Diffusion Barriers
`
`8.7. Diffusion During Film Growth
`
`Before proceeding, the reader may find the survey of diffusion phenomena
`given in Chapter 1 useful and wish to review it.
`
`8.2. FUNDAMENTALS or DIFFUSION
`
`8.2.1. Comparative Diffusion Mechanisms
`Diffusion mechanisms attempt
`to describe the details of atomic migration
`associated with mass transport through materials. The resulting atom move-
`ments reflect the marginal properties of materials in that only a very small
`fraction of the total number of lattice sites, namely, those that are unoccupied,
`interstitial, or on surfaces, is involved. An illustration of the vacancy mecha-
`nism for diffusion was given on p. 36. Similarly, the lattice diffusivity DL, in
`terms of previously defined quantities, can be written as
`
`~
`
`DL = Doexp ~ EL/RT,
`
`(8-1)
`
`where EL is the energy for atomic diffusion through the lattice on a per-mole
`basis. In polycrystalline thin films the very fine grain size means that a larger
`proportion of atom~defect combinations is associated with grain boundaries,
`dislocations, surfaces, and interfaces, relative to lattice sites, than is the case in
`bulk solids. Less tightly bound atoms at these nonlattice sites are expected to
`attract different point—defect populations and be more mobile than lattice
`atoms. Although the detailed environment may be complex and even varied,
`the time—averaged atomic transport
`is characterized by the same type of
`Boltzmann behavior expressed by Eq. 8-l. Most importantly, the activation
`energies for grain-boundary, dislocation, and surface diffusion are expected to
`be smaller than EL , leading to higher diffusivities. Therefore, such hetero-
`geneities and defects serve as diffusion paths that short-circuit the lattice.
`In order to appreciate the consequences of allowing a number of uncoupled
`transport mechanisms to freely compete, we consider the highly idealized
`
`Page 6 of 51
`
`

`

`Interdiffusion and Reactions in Thin Film
`
`Figure 8-1. Highly idealized polycrystalline film containing square grains, grain ,
`boundaries, and dislocations.
`
`polycrystalline film matrix in Fig. 8-1. Grain—boundary slabs of width 6 serve l
`as short—circuit diffusion paths even though they may only be 5-10 3 wide.
`e
`They separate square—shaped grains of side 1. Within the grains are dissociated ‘
`dislocations oriented normal to the film surface. They thread the latter with a -
`density pa, per cm2, and diffusion is assumed to occur through the dislocation
`core whose cross—sectional area is A d. Parallel transport processes normal to t
`the film plane are assumed to occur for each mechanism. Under these .
`conditions, the number of atoms (fti) that flow per unit time is essentially equal
`to the product of the appropriate diffusivity (D,-), concentration gradient V
`( dc / dx) ,, and transport area involved. Therefore,
`
`Lattice:
`Grain Boundary:
`Dislocation:
`
`.
`
`dx L
`
`dx 1,
`
`,
`r'zL = DL12(£)
`,
`rib = <5Dbl(£i£)
`rid = AdDdlzpd ( 3-) ,
`
`dc
`X L!
`
`where L, b and d refer to lattice, grain-boundary, and dislocation quantities.
`The importance of short—circuit mass flow relative to lattice diffusion can be
`quantitatively understood in the case of face—centered cubic metals where data
`for the individual mechanisms are available. A convenient summary of result—
`ing diffusion parameters is given by (Ref. 2)
`Lattice:
`DL ~ 0.5 exp — l7.OTM/Tcmz/sec,
`Grain
`
`(8-3a) .i
`
`Boundary :
`
`5D,, at 1.5 X lO”3exp — 8.9TM/Tcm3/sec,
`
`(8-3b)
`
`Dislocation: AdDd z 5.3 X l0’15exp —— l2.5TM/T cm4/sec.
`
`(8—3c)
`
`Page 7 of 51
`
`

`

`8.2. Fundamentals of Diffusion
`
`These approximate expressions represent average data for a variety of FCC
`[I metals normalized to the reduced temperature T/ TM, where TM is the melting
`point. As an example, the activation energy for lattice self—diffusion in Au is
`p, easily estimated through comparison of Eqs. 8-] and 8-3a, which gives
`'1 EL /RT = l7.0TM/T. Therefore, EL = l7.ORTM or (l7.0)(1.99 cal/mole-
`} K)(l336 K) = 45,200 cal /mole. As a first approximation,
`the preceding
`equations can be assumed to be valid for both self- and dilute impurity
`Z diffusion. Generalized Arrhenius plots for D,_ as a function of TM/ T have
`I already been introduced in Fig. 5-6 for metal, semiconductor, and alkali halide
`I matrices.
`
`Regimes of dominant diffusion behavior, normalized to the same concentra-
`
`tion gradient, can be mapped as a function of I and p d by equating the various
`iz, in Eq. 8-2. The equations of the boundary lines separating the operative
`transport mechanisms are thus
`
`l_ D1.
`1
`50b’
`
`D1.
`p :
`" AdDd’
`
`and
`
`1
`
`AdDdPd
`5D,,
`
`'
`
`These are plotted as ln 1/] versus pd in Fig. 8-2 at four levels of T/ TM,
`
`IIIIIIIIIIII
`
`[_T/TM=o.5,.-.. _-~,
`1
`E
`
`I iIIIIIIII
`
`E_|
`T/TM=O.3 ,._-_- 7
`I
`
`|'l|1|"T‘i| ._._.._...__...._
`
`U’
`
`024-53iOl202468|O|2
`
`Log ;%(cm'3)
`Log %(o'n'3)
`Figure 8-2. Regimes of dominant diffusion mechanism in FCC metal films as a
`function of temperature. (Reprinted with permission from Elsevier Sequoia, S.A. from
`R. W. Balluffi and J. M. Blakely, Thin Solid Films 25, 363, 1975).
`
`Page 8 of 51
`
`

`

`360
`
`lnterdiffusion and Reactions in Thin Films 1'
`
`employing Eqs. 8~3a, b, c. The broken rectangles represent the range of thin
`film values for l and ,0d that occur in practice. For typical metal films with a V
`grain size of 1 pm or less, grain-boundary diffusion dominates at all practical
`temperatures. Similarly, for dislocation—free epitaxial films where 1/! = 0,
`lattice diffiision dominates. Transport at these extremes is intuitively obvious.
`Where the film structure is such that combinations of mechanisms are opera-
`tive, different admixtures will occur as a function of temperature. Generally,
`lower temperatures will favor grain boundary and dislocation short—circuiting 7
`relative to lattice diffusion.
`
`V
`
`Surface diffusion is another transport mechanism of relevance to thin films 7
`because of the large ratio of the number of surface—to-bulk atoms. As noted in
`Chapter 5,
`this mechanism plays an important role in film nucleation and
`
`T (‘‘C)
`43001200 M00 1000 900
`
`T (‘’C)
`1200 M00 4000 900
`
`48V
`29V
`39V
`
`56V
`
`E3\
`N
`
`E3Q
`
`, I-2
`
`E9u
`
`.u.
`‘<30
`
`29c
`
`o
`:3
`LI.
`Eo
`
`0.7
`
`0.8 0.85
`
`0.65 0.7
`
`I000/T (K-1)
`1000/T u<")
`Figure 8-3. Diffusion coefficients of various elements in Si and GaAs as a function
`of temperature. (Reprinted with permission from John Wiley and Sons, from S. M.
`Sze, Semiconductor Devices: Physics and Technology, Copyright © 1985, John
`Wiley and Sons).
`
`Page 9 of 51
`
`

`

`G 8.2. Fundamentals of Diffusion
`
`361
`
`growth processes. Reduced parameters describing measured surface transport
`in FCC metals have been suggested (Ref. 3); e. g.,
`
`D3 z 0.0l4exp —-
`
`6.54T
`T M cmz/sec
`
`T
`for -74 > 1.3.
`
`It is well known, however, that surface diffusion varies strongly with ambient
`,1 conditions, surface crystallography, and the nature and composition of surface
`and substrate atoms.
`Systematics similar to those depicted in Fig. 8-2 also govern diffusion
`G behavior in ionic solids and semiconductors where grain boundaries and
`:5. dislocations are known to act as short-circuit paths. However, complex space-
`charge effects in ionic solids make a clear separation of lattice and grain-
`boundary diffusion difficult in these materials. In semiconductors a great deal
`of impurity diffusion data exists, and these are used in designing and analyzing
`doping treatments for devices. This is a specialized field, and complex
`i modeling (Ref. 4) is required to accurately describe diffusion profiles. Due to
`.
`the importance of Si and GaAs films, preferred lattice dopant diffusion data are
`1 presented in Fig. 8—3 (Ref. 5). Some very recent data on diffusion of noble
`metals Au, Ag, and Cu in amorphous Si films interestingly reveals that the
`activation energy for diffusion in the disordered matrix is very similar to
`i values obtained for lattice diffusion in crystalline Si. (Ref. 6).
`
`L 8.2.2. Grain-Boundary Diffusion
`
`T Of all the mass-transport mechanisms in films, grain—boundary (GB) diffusion
`has probably received the greatest attention. This is a consequence of the rather
`small grain size and high density of boundaries in deposited films. Rapid
`p diffusion within individual GBs coupled with their great profusion make them
`the pathways through which the major amount of mass is transported. Low
`diffusional activation energies foster low-temperature transport, creating seri-
`ous reliability problems whose origins can frequently be traced to GB involve-
`ment. This has motivated the modeling of both GB diffusion and phenomena
`related to film degradation processes.
`The first treatment of GB diffusion appeared nearly 40 years ago. The
`Fisher model (Ref. 7) of GB diffusion considers transport within a semi-in-
`finite bicrystal film initially free of diffusant, as shown in Fig. 8-4. A diffusant
`whose concentration C0 is permanently maintained at plane y = 0 diffuses
`into the GB and the two adjoining grains. At low temperatures in typical
`polycrystalline films, it is easily shown that there is far more transport down
`
`Page 10 of 51
`
`

`

`lnterdiffusion and Reactions in Thin Films
`
`/CO ///
`
`c i
`
`Figure 8-4. Representation of diffusional penetration down a grain boundary (y
`direction) with simultaneous lateral diffusion into adjoining grains (x direction).
`
`the GB than there is into the matrix of the grains. The ratio of these two fluxes
`can be estimated through the use of Eqs. 8-2 and 8-3 for FCC metals; i.e.,
`
`6D,,
`izb
`r'zL_lDL_
`
`3 x10'8
`1
`
`exp
`
`8.1 TM
`T
`
`Assuming I: 10*“ cm and T/T = 1/3, we have rib/fl, = 1.1 X 107. For
`this reason, we may envision transport to consist primarily of a deep rapid
`penetration down the GB from which diffusant subsequently diffuses laterally
`into the adjoining grains, building up the concentration level there. This is
`shown schematically in Fig. 8-4 and described mathematically by
`1/2
`
`,
`
`(8—4)
`
`—-—-——€——)
`CL<’“”’> = C0“? ‘ (513 m
`
`b
`
`2t/D
`
`x
`
`y-erfc
`
`7257
`
`L
`
`where C,_( x, y, t) is the diffusant concentration at any position and time.
`The Fisher analysis of the complex, coupled GB-lattice diffusion process
`yields simplified decoupled solutions——an exponential diffusant profile in the
`GB and an error function profile within adjoining grains. Experimental verifi-
`cation of Eq. 8-4 is accomplished by measurement of the integrated concentra— -
`tion 5 within incremental slices A y thick (e.g., by sputtering) normal to the
`y = 0 surface; i.e.,
`
`T
`
`E ’ I C,r,(X, J’. t) dxAy = const e‘(2\/13;/‘5Db\/T7)‘/2’.
`'—oo
`
`(8-5)
`
`Page 11 of 51
`
`

`

`8.2. Fundamentals of Diffusion
`
`The last equation suggests that a plot of ln 5 versus y is linear. Therefore, the
`‘ useful result
`
`5D
`
`1.0
`1’ ” w;
`
`dy
`
`dln5 ‘Y
`the value of DL in the same
`in order to obtain 6Db,
`I emerges. However,
`; system must be independently known. This poses no problem usually, since
`A. lattice diffusivity data are relatively plentiful. Exact, but far more complicated,
`integral solutions, that are free of the simplifications of the Fisher analysis,
`7f have been obtained by Whipple (Ref. 8) and Suzuoka (Ref. 9). A conclusion,
`, based on these analyses, that has been extensively used is
`
`50, = —O.66
`
`dln5 ”5’3(4DL)‘/2
`dys/5
`t
`
`(8-7)
`
`_
`
`f;
`l
`
`Apart from overriding questions of correctness, the difference between Eqs.
`8-6 and 8-7 is that ln 5 is plotted versus y in the former and versus y‘’/5 in
`the latter. Frequently, however, the experimental concentration profiles are not
`sufficiently precise to distinguish between these two spatial dependencies. It
`:7 does not matter that actual films are not composed of bicrystals, but rather
`polycrystals with GBs of varying type and orientation; the general character of
`the solutions is preserved despite the geometric complexity. A schematic
`representation of equiconcentration profiles in a polycrystalline film containing
`an array of parallel GBs is shown in Fig. 8-5. At elevated temperatures the
`extensive amount of lattice diffusion masks the penetration through GBs. At
`the lowest temperatures, virtually all of the diffusant is partitioned to GBs. In
`F“ between,
`the admixture of diffusion mechanisms results in an initial rapid
`F“ penetration down the short-circuit network, which slows down as atoms leak
`Q
`into the lattice. The behaviors indicated in Fig. 8-5 represent the so~called A-,
`. B—, and C-type kinetics (Ref. 10). Polycrystalline film diffusion phenomena
`; have been studied in the B to C» range for the most part. Excellent reviews of
`
`t\\\\\\\\\\\\\\‘_
`
`7
`
`_..&m\\\\\\\\\\\
`
`.«’
`
`4
`i
`
`A KINETICS
`
`B KINETICS
`
`C KINETICS
`
`F.
`
`Schematic representation of type A (highest-temperature), B, and C
`— Figure 8-5.
`V
`(lowest—temperan1re) diffusion kinetics. (From Ref. 10).
`
`Page 12 of 51
`
`

`

`364
`
`Interdiffusion and Reactions in Thin Films
`
`the mathematical theories of GB diffusion including discussions of transport in
`these different temperature regimes, and applications to thin-film data are
`available (Refs. 11, 12). The best general source of this information is the
`volume Thin Films——Inz‘erdiffusion and Reactions, edited by Poate, Tu, and L
`Mayer (Ref. 12) which also serves as an authoritative reference for much of
`the material discussed in this chapter. This book also contains a wealth of
`experimental mass transport data in thin—fllm systems.
`The experimental measurements of the penetration of radioactive ‘95Au into
`epitaxial (Fig. 8-6a) and polycrystalline (Fig. 8—6b) Au films provide a test of
`the above theories. They also importantly illustrate how the spectrum of
`diffusion behavior can be decomposed into the individual component mecha-
`nisms through judicious choice of film temperature and grain size. These data —
`
`were obtained by incrementally sputter-sectioning the film, collecting the ;
`removed material in each section, and then counting its activity level. Very low p
`
`- 352°c,2_:e x :04 sec
`~ 325°C, 4.39 x :0“ sec
`A 247.5°C, L82 x I06 sec
`o 295.4°c, 6.82 no“ see
`o 27s.o°c, 3.456 x :05 sec
`
`3[
`
`2zD
`>-
`0:
`
`<c
`
`:Ca
`
`n
`on
`
`S>
`
`-CZl
`
`—-
`o<1
`
`2EQL
`
`UCL(1')
`Ln
`93
`<1
`
`IO
`
`I5
`
`50
`45
`40
`35
`30
`25
`20
`PENETRATlON DISTANCE (1o'5 cm)
`
`55
`
`60
`
`65
`
`70
`
`Figure 8-6a. Diffusional penetration profiles of 195Au in (001) epitiaxial Au films at ,:
`indicated temperatures and times. Lattice and dislocation diffusion dominate. (From
`Ref. 13).
`
`Page 13 of 51
`
`

`

`8.2. Fundamentals of Diffusion
`
`PENETRATION DISTANCE (Io“‘cm)
`I
`I
`'
`’
`0.2
`0.4
`Q6
`0.8
`I._o, o.o QOIS
`I -7
`|
`>
`
`0
`
`Y‘
`
`
`
`Au'95sI=EcIFIcACTIVITY(ARBITRARYUNITS)
`
`
`
`
`
`2
`
`BLANK RuN\j
`V°“”’Wmvvvv~V-v-wqifiiiggMIN.)
`itI
`.imm,..,-.,\a\afi°c (I MIN.)
`\_
`I-x.
`§
`x
`~;—n-law?‘ <I37°c (I MIN.)
`‘\
`**** *ee«x-4.- _
`«is I-—as-*-,.‘*_*_*_/4-3Z:C (10 MIN.)
`AA%
`in
`’
`._
`‘°*~=~M..‘M_A[I27°c (Io MIN.)
`A‘n_A_A._
`
`°«o~u-lo.o.n.o.°-o_°_o_°J°<_‘Jg.T°C (IO MIN)
`‘f°°‘<><>o<>o00O0o_[l 17°C (10 MIN.)
`o~o_.<,_o___o_
`
`:
`
`II«lid
`
`II’t'ni'il1Iin-Ii
`
`I(i]IUJ.|i_.LJ.iuiI
`
`i
`o.o o.oI
`I.
`l.4
`1.2
`o 0.2 0.4 0.6 0.8 L0
`TRACER PENETRATION DISTANCI-26/5
`(lO'5cm6/5)
`
`Figure 8-6b. Diffusional penetration profiles of 195Au in polycrystalline films at
`indicated temperatures and times. Only GB diffusion is evident.
`(Reprinted with
`V
`if permission from Elsevier Sequoia, S.A., From D. Gupta and K. W. Asai, Thin Solid
`'. Films 22, 121, 1974).
`
`; concentration levels can be detected because radiation—counting equipment is
`i quite sensitive and highly selective. This makes it possible to measure shallow
`profiles and detect penetration at very low temperatures. The epitaxial film
`i data display Gaussian—type lattice diffusion for the first 1000 to 1500 K,
`* followed by a transition to apparent dislocation short—circuit transport beyond
`this depth. Rather than high~angle boundaries, these films contained a density
`II of some 101° to 10" dissociated dislocations per crnz. On the other hand,
`pi extensive low—temperature GB penetration is evident in fig. 8-6b without much
`lattice diffusion. The large differences in diffusional penetration between these
`two sets of data, which are consistent with the systematics’ illustrated in Fig.
`8-2, should be noted. For epitaxial Au a mixture of lattice and dislocation
`diffusion is expected for pd -~ 101°/cm2 at temperatures of ~ O.4TM. Only
`GB diffusion is expected, however, at temperatures of ~ 0.3TM for a grain
`size of 5 X 10’5 cm, and this is precisely what was observed.
`
`Page 14 of 51
`
`

`

`366
`
`lnterdiifusion and Reactions in Thin Films
`
`8.2.3. Diffusion in Miscible and Compound-Forming Systems
`
`It is helpful to initiate the discussion on diffu~
`8.2.3.1. Miscible Systems.
`sion in miscible systems by excluding the complicating effects of grain
`boundaries. Bulk materials contain large enough grains so that the influence of
`GBs is frequently minimal. For thin films a couple where both layers are single
`crystals (e.g., a heteroepitaxial system) must be imagined. Under such condi-
`tions, the well-established macroscopic diffusion analyses hold. Upon interdif—
`fusion in miscible systems, there is no crystallographic change, for this would
`imply new phases. Rather, each composition will be accessed at some point or
`depth within the film as a continuous range of solid solutions is formed. When
`the intrinsic atomic diffusivities are equal,
`i.e., DA = DB,
`the profile is
`symmetric and Eq. 1-27a governs the resultant diffusion. On the other hand, it
`is more common that DA =# DB, so A and B atoms actually migrate with
`unequal velocities because they exchange with vacancies at different rates.
`As an example of a miscible system, consider the much-studied Au—Pd
`polycrystalline thin-film couple in Fig. 8-7 (Ref. 15). Both AES sputter
`sectioning and RBS methods were employed to obtain the indicated profiles,
`whose apparent symmetry probably reflects the lack of a strong diffusivity
`dependence on concentration.
`It
`is very tempting to analyze these data by
`fitting them to an error-function~type solution. Effective diffusivity values
`could be obtained, but they would tend to have limited applicability because of
`the heterogeneous character of the film matrix. It must not be forgotten that
`
`O
`
`if
`
`é
`
`CO0
`
`
`
`PdCONCENTRATION(ATOMIC%)
`
`-.
`O
`
`. BACK SCATTERING
`
`A
`§§ AES
`
`.90 MRS '
`200 HRS
`
`V
`
`o
`
`I
`
`_
`
`v
`u
`
`V
`'
`\ '
`u
`o
`'
`
`_
`
`I
`
`_
`
`r
`I
`1400 16.00
`
`1200
`
`1800 2000
`
`.,
`DISTANCE FROM SURFACE (A)
`Figure 8-7. Palladium concentration profiles in a Au—Pd thin-film diffusion couple 7,
`measured by RBS and AES techniques.
`(Reprinted with permission from Elsevier
`Sequoia, S.A., from P. M. Hall, J. M. Morabito and J. M. Poate, Thin Solid Films
`33, 107, 1976).
`
`:
`
`Page 15 of 51
`
`

`

`8.2. Fundamentals of Diffusion
`
`367
`
`the
`the dominant mechanism in this couple. Therefore,
`G GB diffusion is
`appropriate GB analysis is required in order to extract fundamental transport
`parameters. With this approach, it was found that a defect-enhanced admixture
`i of GB and lattice diffusion was probably responsible for the large changes in
`the overall composition of the original films. Diffusional activation energies
`obtained for this film system are typically 0.4 times that for bulk diffusion, in
`accord with the systematics for GB diffusion.
`
`8.2.3.2. Compound-Forming Systems. Many of the interesting binary
`combinations employed in thin-film technology react
`to form compounds.
`Since the usual configuration is a planar composite structure composed of
`elemental films on a flat substrate,
`layered compound growth occurs. The
`concentration—position profile in such systems is schematically indicated in
`Fig. 88. Each of the terminal phases is assumed to be in equilibrium with the
`intermediate compound. The compound shown is stable over a narrow rather
`than broad concentration range. Both types of compound stoichiometries are
`observed to form. With time,
`the compound thickens as it consumes the oz
`phase at one interface and the B phase at the other interface.
`It is instructive to begin with the simplified analysis of the kinetics of
`compound growth based on Fig. 8-8. Only the 1 phase (compound) interface
`in equilibrium with at is considered. Since A atoms lost from 0: are incorpo-
`rated into 7, the shaded areas shown are equal. With respect to the interface
`
`yCOMPOUND
`i<-~Lo~—-—-> X7
`
`Figure 8-8. Depiction of intermediate compound formation in an A~B diffusion
`couple. Reaction temperature is dotted in on phase diagram.
`
`Page 16 of 51
`
`

`

`368
`
`lnterdiflusion and Reactions in Thin Films
`
`moving with velocity V, the following mass fluxes of A must be considered:
`
`flux into interface = COCV — D4
`
`dc, )
`
`dx mt-
`
`flux away from interface = Cy V.
`
`These fluxes remain balanced for all times, so by equating them we have
`
`dX
`V: _ =
`dt
`
`D dCa a’
`oz(
`/26)’
`C, — C7
`
`(H)
`
`where X is the compound layer thickness. From the simple geometric
`construction shown, dCa / dx can be approximated by (CA — Ca)/LO. There-
`fore as growth proceeds, L0 increases while V decreases. If the shaded area
`within the 01 phase can be approximated by (1 /2) L0(CA — Ca), and this is set
`equal to C,/X,
`then L0 = ZCYX/(CA —— Ca). Substituting for dC<, /dx in
`Eq. 8-8 leads to
`
`(CA — C11)2
`Do:
`dX
`dz‘ " 2X (c,,— C7)C '
`
`'
`
`Upon integrating, the 01-7 portion of the compound layer thickness is obtained
`as
`
`[D,,(c,, — C,,)2]'/2:1/2
`X= .
`“Cot ' C'y)C*ri
`/
`
`(3-10)
`
`A similar expression holds for the B—'y interface, and both solutions can be
`added together to yield the final compound layer thickness X7; i.e.,
`
`X7 = const :1/2.
`
`(8-11)
`
`The important feature to note is that parabolic growth kinetics is predicted.
`Thermally activated growth is also anticipated, but with an effective activation
`energy dependent on an admixture of diffusion parameters from both or and ;3
`phases.
`
`Among the important and extensively studied thin-film compound—forming
`systems
`are Al~Au (used in interconnection/contact metallurgy)
`and
`metal—silicon (used as contacts to Si and SiO2); they will be treated later in the
`chapter. Parabolic growth kinetics is almost always observed in these systems.
`When diffusion is sufficiently rapid; however, growth may be limited by the
`speed of interfacial reaction. Linear kinetics varying simply as t then ensue,
`
`Page 17 of 51
`
`

`

`8.2. Fundamentals of Diffusion
`
`369
`
`times. For longer times linear growth gives way to
`but only for short
`diffusion-controlled parabolic growth.
`
`8.2.4. The Kirkendall Effect
`
`The Kirkendall effect has served to illuminate a number of issues concerning
`solid—state diffusion. One of its great successes is the unambiguous identifica—
`tion of vacancy motion as the operative atomic transport mechanism during
`interdiffusion in binary alloy systems. The Kirkendall experiment requires a
`diffusion couple with small inert markers located within the diffusion zone
`between the two involved migrating atomic species. An illustration of what
`happens to the marker during thin-film silicide formation is shown in Fig. 8-9.
`Assuming that metal (M) atoms exchange sites more readily with vacancies
`than do Si atoms, more M than Si atoms will sweep past the marker. In effect,
`more of the lattice will move toward the left! To avoid lattice stress or void
`
`generation, the marker responds by shifting as a whole toward the right. ‘The
`reverse is true if Si is the dominant migrating specie. Such marker motion has
`indeed been observed in an elegant experiment (Ref. 16) employing RBS
`methods to analyze the reaction between a thin Ni film and a Si wafer.
`Implanted Xe, which served as the inert maker, moved toward the surface of
`
`Si DIFFUSION
`
`V/
`
`W M
`
`ARKERM...»
`METAL DIFFUSION
`
`Schematic of Kirkendall marker motion during silicide formation. (Re~
`Figure 8-9.
`printed with permission from John Wiley and Sons, from J. M. Poate, K. N. Tu and J.
`W. Mayer, eds., Thin Films: Irzterdiffusion and Reactions, Copyright © 1978, John
`Wiley and Sons).
`
`Page 18 of 51
`
`

`

`370
`
`lnterdiffusion and Reactions in Thin Films
`
`the couple during formation of Ni2Si. The interpretation, therefore, is that Ni
`is the dominant diffusing specie.
`
`8.2.5. Diffusion Size Effects (Ref. 17)
`
`A linear theory of diffusion has been utilized to describe the various transport
`effects we have considered to this point and, except for this section, will be
`assumed for the remainder of the thin—f1lm applications in this book. The
`macroscopic Fick diffusion equations defined by Eqs. 1-21 and 1-24 suffice as
`an operating definition of what is meant by linear diffusion theory. There are,
`however, nonlinear diffusion effects that may arise in thin—f1lm structures when T
`relatively large composition changes occur over very small distances (e.g.,
`superlattices). To understand nonlinear effects, we reconsider atomic diffusion
`between neighboring planes in the presence of a free—energy gradient driving
`force. As a convenient starting point, two pertinent equations (1-33 and 1-35)
`describing this motion are reproduced here:
`
`GD ‘ AG
`rN = 21: exp — fislmfi,
`
`v = DF/RT.
`
`(3-12)
`
`(8-13)
`
`An expansion of sinh(AG / R T) yields
`
`RT
`
`3! RT
`
`5! RT
`
`(8-14)
`
`7
`
`sinh
`
`RT
`
`AG _ AG +i(AG)3+1(AG)5
`Under conditions where AG / RT < l, the higher-order terms are small com— I
`pared with the first, which is the source of the linear effects expressed by the
`Nernst—Einstein equation. Linear behavior is common because the lattice i
`cannot normally support large energy gradients.
`Now consider nonideal concentrated alloys where the free energies per atom
`or chemical potentials, p.,-, at nearby planes 1 and 2 are defined by (Eq. 1~9)
`
`/1.1 = 14° + kTlna, = u° + kT1n'y1C1,
`
`(8~l5)
`M2 = ,5’ + kT1n dz = ,l‘’ + /(Tin 7202.
`Here [L = G /NA, NA is Avogadro’s number, the activity a is defined by the
`product of the activity coefficient 7 and concentration C, and it” is the .7;
`chemical potential of the specie in the standard state. The force on an atom (f)
`if
`is defined by the negative spatial derivative of [LI
`/*‘2—i"'l
`
`f=_—‘z
`dx
`
`9
`
`N,a0
`
`Page 19 of 51
`
`

`

`8.2. Fundamentals of Diffusion
`
`I
`
`and the force on a mole of atoms is
`
`If F is also defined as 2 AG / a0 (Eq. 1-35), then
`
`F = NA f.
`
`_
`
`,
`
`846
`
`1“('Y2C2 /7101)
`“OF
`AG
`—-—=
`)
`(
`2N,
`2RT
`RT
`where N, is the number of lattice spacings (an) included between planes 1 and
`2. The ratio 'y2 C2 /71C, typically ranges from 10 to l03. In conventional thin
`films, N, > 100, so AG/RT is small compared with unity. Only the first
`‘
`term in the expansion of sinh(AG/RT) need be retained, which, as noted
`.
`.. earlier, defines diffusion in the linear range.
`Imagine now what happens when N, is about 5 to 10 so that film dimensions
`T’ of only 10-20 A are involved. At the highest values of 1/2C2 /'y,C', ,
`the
`1 quantity AG / RT is approximately unity. The higher-order terms in Eq. 8-14
`: can no longer