defects iv - rkr diffusion

7/30/2019 Defects IV - RKR Diffusion

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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)
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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

defects iv - rkr diffusion

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