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Defect Phenomena: Diffusion
structures are getting smaller in alltechnological developments (e.g.nano-design)
especially in semiconductortechnology: device structures areeven smaller than diffusion lengthof dopants (45 nm technology ofCPUs)
diffusion starts to become a problemwhen device is exposed to hightemperatures e.g. after ionimplantation (annealing atC for 30 s)
diffusion can strongly be influencedby co-implantation of electricallyineffective ions (e.g. carbon co-implantation in B:SiC)
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Atomicchange of sites & diffusion
diffusion in solids = material transport in
lattice as a result of atomic change of sites
for a single atom: random path
diffusion always important for processes at
elevated temperatures, such as:
- ordering and disordering processes in
alloys (formation of precipitation)
- doping of semiconductors
- defect annealing after plasticdeformation and ion implantation
- sintering
- OD\HUJURZWKDWVXUIDFHV
proved: diffusion is realized by jumps of
interstitials or vacancies/divacancies
simultaneous change of a ring of atoms
needs too high energy. It has never been
observed.
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Most simple mechanism: jump of a vacancy
jump of a vacancy (Leerstelle) =
movements of an atom to the vacancy site jump rate Qv= reciprocal mean duration of
stay of a vacancy at a given lattice site
the jump rate follows an Arrhenius law:
=QXPEHURIQH[WQHLJKERUV
*0 MXPSDWWHPSWIUHTXHQF\
Evm ... migration energy
N%ROW]PDQQFRQVWDQW
v 0 vexp( / )mZ E kTQ *
typical frequency *0
|1012 s-1
thus temperature T1 where 1 step per
second is observed:
for metals with Tmelt=1300 K: already at
300K vacancy mechanism of diffusion
works well
1 v/ K 380 / eVmT E
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Vacancy model of dif fusion
self-diffusion in metals and alloys, in many ionic crystals, and in ceramic materials often occurs
via vacancy mechanism
atomic fraction of vacancies in thermal equilibrium
typical values ofCv in metals are 10-4 -3 near the melting point (not in semiconductors)
F Fv F Fexp( )exp( ) , ... vacancy formation entropy and enthalpy
S HC S H
kT kT
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divacancy mechanism of diffusion
at very high temperatures (near melting point) number of divacancies becomesconsiderably large
vacancy mechanism of diffusion is accompanied by divacancy mechanism
however vacancy mechanism dominates below 2/3 Tm
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Interstitial diffusion interstitial (Zwischengitteratom) diffusion is morecomplicated
structure of lattice and size of atoms is obviously
important for jump
difference: self- and impurity diffusion
interstitial diffusion is often activated already atvery low temperatures, i.e.
migration energy extremely low - in Au and Nb:
self-interstitials move below 1K !
self-interstitial annealing after low-temperature
electron irradiation of Cu:
v i!!m mE E
residualRes
istivity
dR
dT
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Interstitial diffusion
when interstitials exist in a large concentration: interstitial diffusion
especially important when small atoms diffuse: e.g. hydrogen in metals
but also self-diffusion (e.g. in Si, since diamond lattice is relatively open)
ring vacancy interstitial
mechanism
not important for self-diffusion in dense
metallic lattice (there: vacancy mechanism)
self-interstitials in metals have a much larger
formation enthalpy compared to vacancies
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Hydrogen diffusion in metals
hydrogen is very small: rapid diffusion
technological application: storage of
hydrogen in metals for use in fuel cells
(e.g. in Ti)
permeation of hydrogen through Pdmembrane: method for purification
isotopic effects are found: DH>DD>DT
deviation of DHbelow RT from
Arrhenius low was explained by
quantum effects (tunneling)
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Interstitial diffusion in metals
C, N, and O often dissolve
interstitially in metals (e.g. in
Nb)
comparison with Nb self-
diffusion shows orders of
magnitude difference
interstitial diffusivity near
melting point may be as high
as in liquids
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Interstitial diffusion in metals
diffusivity of interstitially dissolved atoms can be very different
Ddiffers by 20 orders of magnitude
slope is determined by migration enthalpy (Wanderungsenthalpie)
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Dissociative interstitial-substitutional exchange mechanism:Frank-Turnbull mechanism
atom starts from regular lattice site, moves to interstitial position, and diffuses as
interstitial relatively fast (B in Si)
vacancy is required; diffusion ends at the vacancy site
also called: dissociative mechanism
example: fast diffusion of Cu in Ge
i sB V =B
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Interstitial-substitutional exchange mechanism without vacancies:K ick-out mechanism
impurity atom B starts from interstitial site, diffuses there and kicks out an atom
at regular lattice site, which by itself starts interstitial diffusion
diffusion of B ends at a regular lattice site, but can start there again, after being
kicked out again
example: rapid diffusion of Au, Pt, and Zn in Si; also several dopants in Si
A i A i iA +B =B A A ... self-interstitials
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Abnormal fastdiffusion in Si
abnormal fast diffusivity in Si is due
to interstitial-substitutional exchange
mechanism (kick-out mechanism)
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Summary of diffusion mechanisms
1. direct interstitial mechanism (video)
2. vacancy mechanism (video)
3. Frank-Thurnbull mechanism (video)
4. Kick-out mechanism (video)
http://localhost/var/www/apps/conversion/tmp/scratch_9/diff_i.avihttp://localhost/var/www/apps/conversion/tmp/scratch_9/diff_v.avihttp://localhost/var/www/apps/conversion/tmp/scratch_9/diff_ft.avihttp://localhost/var/www/apps/conversion/tmp/scratch_9/diff_ko.avihttp://localhost/var/www/apps/conversion/tmp/scratch_9/diff_ko.avihttp://localhost/var/www/apps/conversion/tmp/scratch_9/diff_ft.avihttp://localhost/var/www/apps/conversion/tmp/scratch_9/diff_v.avihttp://localhost/var/www/apps/conversion/tmp/scratch_9/diff_i.avi7/30/2019 Defects IV - RKR Diffusion
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0DFURVFRSLFGHVFULSWLRQ)LFNVODZV
)LFNVODZ$)LFNGHVFULEHVGLIIXVLRQ
FXUUHQWIof atoms) along a concentration
derivation/gradient dn/dx:
)LFNVODZGLIIXVLRQHTXDWLRQGHVFULEHVLQ
addition time dependence
is second order, linear partial differential equation
solution requires starting and boundary conditions
dD ... diffusion coefficient
d
nI D
x
2
2n nDt x
w ww w
Dis measured in m2/s (often in cm2/s)
typical values:
gases (normal conditions): 10-5 -4 m2/s
liquids (RT): 10-9 m2/s
solids: 10-9 -24 m2/s
example: Au self-diffusion at RT: D =10-24 m2/s
this means about 10-10 m/day: 1 atomic distance
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description of temperature behavior can
often be described by an Arrheniusrelation
the pre-exponential factorD0 can be
written as:
the so-called Arrhenius plot of diffusion
shows log (diffusivity) = f (1/T); when
Q is temperature independent, a straight
line with slopeQ kB-1 is found
Diffusion isstrongly temperature-dependent
0 exp( )
... activation enthalphy of diffusion
B
QD D
k T
Q
0
0
'
0
'
exp( )
... diffusion entrophy... geometry factors, jump frequency
B
SD D
k
SD
'
'
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Thin-film solution
thin layer of diffusing species (amount M per unit area) is located at x=0 of a semi-infinite
sample (self-exhausting source) concentration after time t is then described as
the quantity is the typical diffusion lengthDt
Spec ial solutions of the diffusion equation
2
( , ) exp( )4
M xc x t
DtDt
S
Theerror function solution
if at t = 0 the concentration of diffusing species is c(x,0) = 0 and if fort > 0 the
concentration at x= 0 maintained to be c(0, t) = cs = const., the solution of the diffusion
equation is:
these conditions describe the in-diffusion of a diffusor into semi-infinite solid with a non-
volatile (non-exhausting) source (e.g. diffusor from gas phase)
( , ) where 12
s
xc x t c erfc erfc z erf z
Dt
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Trace r method
Only method for self-diffusion, but works in general
radioisotopic tracer atoms are deposited at surface of solid by e.g. electro-deposition isothermal diffusion is performed for a given time t
often quartz ampoules are used (T 10 m; D>10-11 cm2/s
VSXWWHULQJRIVXUIDFHIRUVPDOOGLIIXVLRQOHQJWKDWORZWHPSHUDWXUHVQPP
for the range D= 10-21 -12 cm2/s
Experimental determination of diffusion coefficient
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example: diffusion of Fe inFe3Si
from such figures the
diffusion constant can be
determined with an accuracy
of a few percent
stable isotopes can be used as
well, when high-resolutionSIMS is used
this technique is more
difficult
Experimental determination of diffusion coefficient
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sophisticated method: growth of
layer structure of material ofinterest including G-layers of
diffusing impurity
example: B diffusion in Si with
and without Si implantation
(upper panel)
after implantation: strong
enhancement of diffusion due to
implantation-induced defects
lower panel: enhancement of
diffusion by implantation defects
is suppressed when C is present
at high concentration
Si self-interstitials are stronglysuppressed due to presence of C
B diffusion is impeded (diffuses
via kick-out mechanism)
diffusion profiles were analyzed
numerically by MC methods
different diffusion mechanism
can be separated this way
Diffusion studies using MBE G-layers and SIMS
Rene Scholz, Ph.D. Thesis, Halle 1999
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Typical schematic of a
dynamical SIMS instrument. High energy ions are
supplied by an ion gun (1 or
2) and
focused on to the target
sample (3), which ionizes
and sputters some atoms off
the surface.
These secondary ions are
then collected by ion lenses
(5) and filtered according to
atomic mass (6), then
projected onto an electron
multiplier (7, top), Faradaycup (7, bottom), or CCD
screen (8).
SI MS: Secondary Ion Mass Spectroscopy
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Table-top SI MS System
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Diffusion depends on latticestructure and defect density
diffusivity is much higher along grain boundaries and dislocations
diffusion also depends on crystal lattice structure, i.e. the phase of an alloy (fcc and bcc Fe)
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The Kirkendall effect
when two metals A and B are in direct contact, A atoms diffuse into B, and vise versa
diffusion may be different, so at one side vacancy clusters are formed, the other material swells
welding
copperbrass
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Zn diffusion in G aP
Zn diffusion in GaP (also in GaAs) creates a large number of monovacancies
in contradiction to all existing diffusion models
further research required to fully understand diffusion Positron annihilation result
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L iterature
'LIIXVLRQLQ&RQGHQVHG0DWWHU-Krger, P. Heitjans, R. Haberlandt
Friedr. Vieweg & Sohn Verlagsgesell. mbH Braunschweig 1998
Analytical solutions of diffusion equation
7KHMathematics of'LIIXVLRQ-RKQCrank, Oxford University Press; 2. Ed. (1979, Reprint
2004.
'LIIXVLRQ- 0HWKRGHQGHU0HVVXQJXQG$XVZHUWXQJ:-RVW9HUODJYRQ'U'LHWULFKSteinkopff, Darmstadt 1957