Non-linearNon-linear Effects in Effects in DiffusionDiffusion

on Nanoscaleon Nanoscale

Department of Solid State PhysicsDepartment of Solid State PhysicsUniversity of DebrecenUniversity of Debrecen

D.L. D.L. BekeBeke Z. Erd Z. Erdélyiélyi, I.A. Szab, I.A. Szabóó,, Cs. CserhCs. Cserháátiti

Diffusion in nanomaterialsDiffusion in nanomaterials

Two important featuresTwo important features

a)a) High numbers of grain- orHigh numbers of grain- orphase boundaries (GB or PB) and phase boundaries (GB or PB) and dislocationsdislocations- fast diffusion and solid st. reactions, segregation, etc.- fast diffusion and solid st. reactions, segregation, etc.

b) Principal problems (very shortb) Principal problems (very shortdistances and preferably no structural distances and preferably no structural defects)defects)

Principal difficultiesPrincipal difficulties::

-Short diffusion distances LShort diffusion distances L d d(continuum description fails)(continuum description fails)

- Gradient energy correctionsGradient energy corrections

- Stress effectsStress effects

Discrete (MartinDiscrete (Martin‘‘s) models) model

i-1 i i+10 1 N-1 Nx

i-1 i i+10 1 N-1 Nx

i-1 i i+10 1 N-1 Nx

i-1 i i+10 1 N-1 Nx

i-1 i i+10 1 N-1 Nx

i-1 i i+10 1 N-1 Nx

i-1 i i+10 1 N-1 Nx

i-1 i i+10 1 N-1 Nx

dcdcii/dt = -z/dt = -z

vv[c[cii(1-c(1-ci-1i-1))i,i-1i,i-1 – (1-c – (1-c

ii)c)ci-1i-1i-1,ii-1,i + +

+ c+ cii(1-c(1-c

i+1i+1))i,i+1i,i+1 – c – ci+1i+1(1-c(1-c

ii) ) i+1,ii+1,i]. ]. i,i+1i,i+1==exp(-Eexp(-Ei,i+1i,i+1/kT/kT))

„„Classical” Fick Classical” Fick I.-I.-II.II.

jji,i+1i,i+1== -D-Dii((c/c/x)/x)/ ccii//t = t = [[DDii((c/c/x)/x)/]/ ]/ xx

Ei,i +1=Eo - i + i and Ei+1,i= Eo - i - i,

V=VAB-(VAA+VBB)/2

i= [zv(ci-1+ci+1+ci+ci+2) + zl(ci+ci+1)](VAA-VBB)/2

i V.

Input parametersInput parameters: : zzvv, , zzll,,VVAAAA-V-VBBBB,,VV,T,T(Z= 2zv+zl)

-validity limit-validity limit

m= Z(Vm= Z(VAA AA –V–VBBBB)/kT)/kT

DDiihh=D(0)exp(mc)=D(0)exp(mc)

Effect of the strong concentration dependence of DEffect of the strong concentration dependence of D

0

20

40

60

80

100

30 40 50 60 70 80

depth (nm )

Ge

( %

)

heat.

as rec.

aa-S-Si/Gei/Ge

x

MoMo//VV MoMo//VV

The interface reamins sharpThe interface reamins sharp and shifts!and shifts!

T=1053 KT=1053 K

680K680K100h100h

Dissolution in ideal systemsDissolution in ideal systems: : Ni into CuNi into Cu

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

number of layers

atom

ic fr

acti

on o

f Ni

0

300

700

880

885

time

D(in Ni)<<D(in Cu)D(in Ni)<<D(in Cu)

8 Ni Cu(111)

tim

e

Linear kineticsLinear kinetics !!! !!!

Monte CarloMonte Carlo

Ni side

InterfaceCu side

Experiment: AES from the top of Ni on CuExperiment: AES from the top of Ni on Cu(111)(111)

TT=679K=679K

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10 12

time (103s)

nu

mb

er o

f at

omic

lay

ers

of N

i

0

0,2

0,4

0,6

0,8

1

0 10 20 30

number of layers

atom

ic f

ract

ion

of

Ni

01513274818171860

tim

e

Sharpening of a wide interface Sharpening of a wide interface (T=1000K, m(T=1000K, m’’=9)=9)

50 55 60 65 70

(2) 1073K +1h

(3) 1123K +1h

(4) 1173K +1h

(1) 1023K +0.5h

Inte

nsi

ty (

a.u

.)

2 (degree)

0 2 4 6 8 10 12 14 16 18 20

(0) as received (1) 1023K +0.5h (2) 1073K +1h (3) 1123K +1h (4) 1173K +1h

(4) (3)

(1) (2)

(0)

Inte

nsity

(a.

u.)

2 (degree)

Mo-VMo-V Mo-VMo-V

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

number of layers

atom

ic f

ract

ion

of N

i 0 0.0050.879 35140

Ni-C

uN

i-Cu

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

number of layers

atom

ic fr

actio

n

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

number of layers

atom

ic fr

actio

n

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

number of layers

atom

ic fr

actio

n

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

number of layers

atom

ic fr

actio

n

Growth of intermetallid layer:Growth of intermetallid layer:

i -1 i i+1

n

nd

3

33

2

22

!3

2/

!2

2/

2 x

nd

x

nd

x

ndnn

i

d

cc

x

cii

1

d

d

cc

d

cc

x

ciiii 11

2

2

i -1 i i+1

n

= =c/

3

33

2

22

!3

2/

!2

2/

2 x

nd

x

nd

x

ndnn

i

d

cc

x

cii

1

d

d

cc

d

cc

x

ciiii 11

2

2

e.g. e.g. ii= (V= (VAAAA-V-VBBBB){cZ +(z){cZ +(z

vv+Z/4)d+Z/4)d2222c/c/xx2 2 +...} = cZ(V+...} = cZ(VAAAA-V-VBBBB)+)+ii

’ ’ +...+...

  Introducing i,=exp[-(Eo-i)/kT]=ihexp[i

’/kT].

and if i/kT«1 (i.e. exp[i/kT]1+ i/kT

jji,i+1i,i+1== JJi,i+1i,i+1/q=/q= JJi,i+1i,i+1d/d/== -D-Dii((c/c/x)/x)/ + +

+ D+ Dii[2[2/f/foo’’ - d’’ - d22/24](/24](33c/c/xx33)/)/ +...+...

DDii=z=zvvdd22ii ~V~V Fick I.Fick I.

Amorphous systems?, Stress effects…Amorphous systems?, Stress effects…

-4

-3

-2

-1

0

-42 -41 -40 -39 -38 -37

ln(t)

ln(M

)

m=0.495

-10

-9

-8

-7

-6

-5

-4

-42 -40 -38 -36 -34

ln(t)

ln(M

)

m=0.73

Xo=0

'( ) (0)em cD c D

Constant concentration at the surface:Constant concentration at the surface:

Conclusions (ideal systems):Conclusions (ideal systems):

-At short distances the continuum descriptions fails At short distances the continuum descriptions fails and this strongly depends on the concentration and this strongly depends on the concentration dependence of D (non-linearity)dependence of D (non-linearity)

- Non-linearity leads to - Non-linearity leads to shift of a sharpshift of a sharp interface interface-The non-linearity leads to The non-linearity leads to

a lineara linear shift of a sharp interface shift of a sharp interfacesharpening sharpening of an originally wide interfaceof an originally wide interface

-Gradient energy corrections are important not only in Gradient energy corrections are important not only in the currents but also in the mobilitiesthe currents but also in the mobilities

PapersPapers::

CSIK, A., LANGER, G., BEKE, D.L., ERDÉLYI, Z., MENYHÁRD, CSIK, A., LANGER, G., BEKE, D.L., ERDÉLYI, Z., MENYHÁRD, M. SULYOK, A. M. SULYOK, A. Journal of Appl. Phys. Journal of Appl. Phys. 89/189/1, 804-806 (2001), 804-806 (2001)  BEKE, D.L., LANGER. G.A., CSIK, A., ERDÉLYI, Z., KIS-VARGA, BEKE, D.L., LANGER. G.A., CSIK, A., ERDÉLYI, Z., KIS-VARGA, M.,SZABÓ, I.A., PAPP, Z. M.,SZABÓ, I.A., PAPP, Z. Defect and Diff. Forum Defect and Diff. Forum 194-199194-199, 1403-1416 , 1403-1416 (2001) (2001) ERDÉLYI, Z., GIRARDEAUX., CH., BERNARDINI, J., BEKE, D.L, ERDÉLYI, Z., GIRARDEAUX., CH., BERNARDINI, J., BEKE, D.L, ROLLAND, A. ROLLAND, A. Defect and Diff. Forum Defect and Diff. Forum 194-199194-199, 1161-1166 (2001), 1161-1166 (2001)

  ERDÉLYI, Z, GIRARDEAUX, CH. TŐKEI, ZS. BEKE, D.L. ERDÉLYI, Z, GIRARDEAUX, CH. TŐKEI, ZS. BEKE, D.L. CSERHÁTI, C., ROLLAND, A. CSERHÁTI, C., ROLLAND, A. Surf. Sci., Surf. Sci., 496496/1-2, 129 (2002)/1-2, 129 (2002)

ERDÉLYI, Z, ERDÉLYI, Z, SZABSZABÓ I.A., Ó I.A., BEKE, D.L. BEKE, D.L. Phys. Lev. LettersPhys. Lev. Letters,, in printin print

Chapters:Chapters:

BERNARDINI, J, BEKE, D.L., „Diffusion in Nanomaterials” BERNARDINI, J, BEKE, D.L., „Diffusion in Nanomaterials” in „Nanocrystalline materials: Properties and Applications” in „Nanocrystalline materials: Properties and Applications” (Eds. Knauth, P., Schoonman, J.) Kluwer Academic Publ., (Eds. Knauth, P., Schoonman, J.) Kluwer Academic Publ., Boston, 2001Boston, 2001

BEKE, D.L. CSERHÁTI, C.BEKE, D.L. CSERHÁTI, C.,, ERDÉLYI, Z. ERDÉLYI, Z.,, SZABÓ, I.A. SZABÓ, I.A.,, “Segregation in Nanostructures” in „Advances in Nanophase “Segregation in Nanostructures” in „Advances in Nanophase materials and nanotechnology” Volume: „Nanoclusters” (ed. materials and nanotechnology” Volume: „Nanoclusters” (ed. H.S. Nalwa) American Scientific Publ., 2002, in printH.S. Nalwa) American Scientific Publ., 2002, in print

SIDORENKO, S., BEKE. D.L., KIKINESHI, A., “Materials SIDORENKO, S., BEKE. D.L., KIKINESHI, A., “Materials Science of Nanosctutures, Naukova Dumka, Kiyv, 2002Science of Nanosctutures, Naukova Dumka, Kiyv, 2002