Defect Phenomena: Diffusion

• structures are getting smaller in all• structures are getting smaller in all technological developments (e.g. nano-design)

• especially in semiconductor h l d itechnology: device structures are

even smaller than diffusion length of dopants (45 nm technology of CPUs)

• diffusion starts to become a problem when device is exposed to high temperatures e.g. after ion implantation (annealing atimplantation (annealing at 900…1000°C for 30 s)

• diffusion can strongly be influenced by co-implantation of electrically ineffective ions (e g carbon coineffective ions (e.g. carbon co-implantation in B:SiC)

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Atomic change of sites & diffusion

• diffusion in solids = material transport in lattice as a result of atomic change of sites

• for a single atom: random path• 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 plastic

deformation and ion implantationdeformation and ion implantation- sintering- layer growth at surfaces, …

• 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 observedobserved.

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Most simple mechanism: jump of a vacancy

• jump of a vacancy (Leerstelle) = movements of an atom to the vacancy sitey

• jump rate νv= reciprocal mean duration of stay of a vacancy at a given lattice site

• the jump rate follows an Arrhenius law:

Z … number of next neighbors

v 0 vexp( / )mZ E kTν = ⋅Γ ⋅ −

gΓ0 … jump attempt frequencyEv

m ... migration energy k … Boltzmann constant

• diffusion constant for diffusing impurity in

• typical frequency Γ0 ≈ 1012 s-1

• thus temperature T1 where 1 step per d i b d • diffusion constant for diffusing impurity in

a diluted alloy:second is observed:

• for metals with Tmelt=1300 K: already at 1 v/ K 380 / eVmT E= ⋅

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D d Z= Γ

3Z … coordination numberd … distance to next neighbor

melt y300K vacancy mechanism of diffusion works well

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Vacancy model of diffusion

• self-diffusion in metals and alloys, in many ionic crystals, and in ceramic materials often occurs i h ivia vacancy mechanism

• atomic fraction of vacancies in thermal equilibrium

F Fexp( )exp( ) vacancy formation entropy and enthalpyS HC S H

• typical values of Cv are 10-4 … 10-3 near the melting point

F Fv F Fexp( )exp( ) , ... vacancy formation entropy and enthalpy C S H

kT kT= − −

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divacancy mechanism of diffusion

• at very high temperatures (near melting point) number of divacancies becomes id bl lconsiderably 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 more complicated

• structure of lattice and size of atoms is obviously• structure of lattice and size of atoms is obviously important for jump

• difference: self- and impurity diffusion• interstitial diffusion is often activated already at

very low temperatures, i.e.

• migration energy extremely low in Au and Nb:v im mE E>

• migration energy extremely low - in Au and Nb: self-interstitials move below 1K !

• self-interstitial annealing after low-temperature electron irradiation of Cu:

tyR

esis

tivit

resi

dual

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Interstitial diffusion• when interstitials exist in a large concentration: interstitial diffusionwhen 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)

• not important for self diffusion in dense

ring vacancy interstitialmechanism

• not important for self-diffusion in dense metallic lattice (but vacancy mechanism)

• self-interstitials in metals have a much larger formation enthalpy compared to vacanciespy p

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Hydrogen diffusion in metals

• hydrogen is very small: rapid diffusion• technological application: storage of

h d i t l f i f l llhydrogen in metals for use in fuel cells (e.g. in Ti)

• permeation of hydrogen through Pd membrane: method for purificationmembrane: method for purification

• isotopic effects are found: DH>DD>DT

• deviation of DH below RT from A h i l l i d b

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Arrhenius low was explained by quantum effects (tunneling)

Interstitial diffusion in metals

• C, N, and O often dissolve interstitially in metals (e.g. in Nb)( g )

• comparison with Nb self-diffusion shows orders of magnitude difference

• interstitial diffusivity near melting pointinterstitial 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• D differs by 20 orders of magnitude• slope is determined by migration enthalpy (Wanderungsenthalpie)slope is determined by migration enthalpy (Wanderungsenthalpie)

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Dissociative interstitial-substitutional exchange mechanism:Frank-Turnbull mechanismFrank Turnbull mechanism

• atom starts from regular lattice site, moves to interstitial position, and diffuses as interstitial relatively fast (B in Si)interstitial relatively fast (B in Si)

• vacancy is required; diffusion ends at the vacancy sitei sB V = B+

11• also called: dissociative mechanism• example: fast diffusion of Cu in Ge

Interstitial-substitutional exchange mechanism without vacancies:Kick-out mechanismKick 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

i i iA +B = B A A self-interstitials+• 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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• example: rapid diffusion of Au, Pt, and Zn in Si; also several dopants in Si

Abnormal fast diffusion in Si

abnormal fast diff si it in Si is d e• 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 ( ideo)1. direct interstitial mechanism (video)2. vacancy mechanism (video)3. Frank-Thurnbull mechanism (video)4. Kick-out mechanism (video)( )

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Macroscopic description: Fick’s laws

• 1. Fick’s law (A. Fick 1855) describes diffusion (“current” I of atoms) along a concentration gradient dn/dx:g d e d /d :

2 Fi k’ l (diff i ti ) d ib i

d D ... diffusion coefficientd

nI Dx

= −

• 2. Fick’s law (diffusion equation) describes in addition time dependence

2n nD∂ ∂=

• is second order, linear partial differential equation• solution requires starting and boundary conditions

2Dt x∂ ∂

• D is measured in cm2/s• typical values:

• gases (normal conditions): 10-5 10-4 m2/s• gases (normal conditions): 10 5 … 10 4 m2/s• liquids (RT): 10-9 m2/s• solids: 10-9 … 10-24 m2/s

15• example: Au self-diffusion at RT: D =10-24 m2/s• this means about 10-10 m/day: 1 atomic distance

Special solutions of the diffusion equation

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)s p e (se e us g sou ce)• concentration after time t is then described as

2

( , ) exp( )4

M xc x tDtDt

= −

• the quantity is the typical diffusion lengthDt

4DtDtπ

The error function solution• if at t = 0 the concentration of diffusing species is c(x,0) = 0 and if for t > 0 the

t ti t 0 i t i d t b (0 t) t th l ti f th diff iconcentration at x = 0 maintained to be c (0, t) = cs = const., the solution of the diffusion equation is:

( , ) where 1sxc x t c erfc erfc z erf z⎛ ⎞

= ⋅ = −⎜ ⎟⎝ ⎠

• 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)

( )2s f f f

Dt⎜ ⎟⎝ ⎠

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Experimental determination of diffusion coefficient

Tracer method• radioisotopic tracer atoms are deposited at surface of solid by e.g. electro-deposition• isothermal diffusion is performed for a given time tso e d us o s pe o ed o g ve e t• often quartz ampoules are used (T<1600°C)• sample is then divided in small slices either mechanically, chemically or by sputtering

techniquesh i ll f diff i l th f > 10 D>10 11 2/• mechanically: for diffusion length of > 10 µm; D>10-11 cm2/s

• sputtering of surface: for small diffusion length (at low temperatures) 2nm … 10µm• for the range D = 10-21 … 10-12 cm2/s

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Experimental determination of diffusion coefficient

• example: diffusion of Fe in SiFe3Si

• from such figures the diffusion constant can be determined with an accuracydetermined with an accuracy of a few percent

• stable isotopes can be used as well, when high-resolution SIMS i dSIMS is used

• this technique is more difficult

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Diffusion studies using MBE δ-layers and SIMS

• sophisticated method: growth of layer structure of material of yinterest including δ-layers of diffusing impurity

• example: B diffusion in Si with and without Si implantationand 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 presentis suppressed when C is present at high concentration

• Si self-interstitials are strongly suppressed due to presence of C

• B diffusion is impeded (diffuses via kick-out mechanism)

• diffusion profiles were analyzed numerically; different diffusion

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numerically; different diffusion mechanism can be separated

Rene Scholz, Ph.D. Thesis, Halle 1999

Diffusion is strongly temperature-dependent

• description of temperature behavior can often be described by an Arrhenius yrelation

0 exp( ) B

QD Dk T

= −

... activation enthalphy of diffusionBk T

Q

• the pre-exponential factor D0 can be written as:

' exp( )SD D Δ=

00

'

exp( )

... diffusion entrophy geometry factors jump frequency

B

D Dk

SD

=

Δ

• the so-called Arrhenius plot of diffusion shows log (diffusivity) = f (1/T); when

0... geometry factors, jump frequencyD

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shows log (diffusivity) = f (1/T); when Q is temperature independent, a straight line with slope –Q kB

-1 is found

Diffusion depends on lattice structure 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

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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

weldingwelding

copperbrass

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Zn diffusion in GaP

• 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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Literature

• “Diffusion in Condensed Matter”, J. Kärger, P. Heitjans, R. HaberlandtFriedr. Vieweg & Sohn Verlagsgesell. mbH Braunschweig 1998

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