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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