Diffusione nei solidi: teoria e applicazioni a materiali ... · Università Ca’ Foscari Venezia Diffusione nei solidi: teoria e applicazioni a materiali avanzati Prof. Elti Cattaruzza

Università Ca’ Foscari Venezia

Diffusione nei solidi: teoria e applicazioni a materiali avanzati

Prof. Elti CattaruzzaDepartment of Molecular Sciences and Nanosystems, Università Ca’ Foscari Venezia,via Torino 155/b, 30172 Venezia-Mestre, Italy

[email protected]

Why to study diffusion?

– The movement of atoms is favoured by thermal energy

– Such a movement can be described by equations

– Suitable modifications of materials can be proposed and realized

– Material properties can be improved

An enlightening example

Copper-nickel pair before a thermal treatment at high temperature

Copper (nickel) atoms are present only in the left (right) region

Atomic concentration as a function of the position inside the sample

←

←

←

...after a thermal treatment below melting temperature

In the central region a Cu-Ni alloy has formed

Copper and nickel atoms migrated in the opposite region

Atomic concentration as a function of the position inside the sample

←

←

←

What is “diffusion”?

DIFFUSIONDIFFUSION :: the movement of atoms or molecules in a material

To understand the diffusion in solids, we need to know something about:

• imperfections in solids (in crystals)

• thermal atomic vibrations

Imperfections in crystals

– Point defects

– Line defects (dislocations)

– Surfaces

– Grain boundaries

The most important imperfections for the diffusion are: point defectspoint defects

Point defects

a) a missing atom within a crystalb) two (three, four, ...) missing atomsc) a missing ion-pair of opposite charge

d) an extra atom within a crystal structuree) a displaced ion from the lattice

Thermal atomic vibrations

– For T>0 K, atoms have kinetic energy (i.e. vibrations)

– Atoms have different energies (i.e. distribution)

– The fraction n/Ntot of atoms with energy greater than E (for E»E) is:

_

eNkTE

tot

n −∝

k is the Boltzmann’s constant (k=1.38×10-23J/K)Units of E: Joules

E

kinetic energy

T1

T2

T3

T1< T2 < T3Prob.

n/Ntot

... an important consequence of the thermal energy

– There is a (small) fraction of atoms having very large energies:if this energy is large enough, some atoms can break their bondsand jump to new locations, thus originating vacancies...

The minimum energy required is called activation energy Q (or E)

““DIFFUSIONDIFFUSION””

(units of Q or E can be J/atom, eV/atom, J/mole, or calories/mole)

MeNkTE

tot

n −∝

Diffusion mechanisms

An atom leaves its lattice site to fill a nearby vacancy (thus creating a new vacancy)←

When a small interstitial atom is present, the atoms moves from one interstitial site to anoth←

A substitutional atom leaves its normal lattice site and enters an interstitial position←

Atoms move by a simple exchange mechanism or by a ring mechanism←

a), b) are the main ones

Activation energy for diffusion

low activation energy Q

diffusion favoured

⇓

…a more realistic sketch of the first example shown (Cu-Ni pair)

Point defects are very important for diffusion, allowing the atoms movement

Temperature is very important for diffusion, increasing the movement of atoms and the creation of new vacancies

The diffusion equations

Diffusion depends on time. How can we define the atoms flux during diffusion?

flux, J: number of atoms passing per unit time through a plane of unit area (⊥ to the diffusion direction)

Units of J: atoms/m2 sec

Fick’s first law (rate of diffusion)

The flux J of atoms is proportional to the concentration gradient

J = atoms flux (atoms/m2•sec)D = diffusivity (m2/sec)C = concentration (atoms/m3)

Stationary diffusion: J(x,t)= J(x)

dxdCDJ −=

Fick’s first law

…factors influencing the diffusion coefficient (diffusivity) D

The diffusivity D depends on:1) the nature of the solute atoms

2) the nature of the solid structure

3) the temperature

Small atoms have a higher diffusivity than large onesAtoms can be neutral or charged

Atoms diffuse more readily in weakly bonded solidsAtoms diffuse more readily in low packed solids

Higher temperatures provide higher diffusivities

D0 is the proportional constant k=1.38×10-23J/KE is the activation energy per atom (Joules)eDD kT

E

TE −=),( 0

(an Arrhenius-type equation)

…temperature and the diffusion coefficient D

eDD kTE

TE −=),( 0 eDD RTQ

TQ −=),( 0

D0 is the proportional constant R is the gas constant (R=8.314 J/mole•K, or 1.986 cal/mole•K)Q is the activation energy (J/mole, or cal/mole)

The logarithm of the diffusivity D:

kTEDD −= 0lnln

RTQDD −= 0lnln

Data of D are usually plotted in Arrhenius-type plots

Arrhenius-type plot of diffusivity D

kTEDD −= 0lnln

RTQDD −= 0lnln

Fick’s 2nd law (composition profile)

In most cases, the diffusion is a dynamic process!

C(x) =C(x,t) (the concentration C of diffusing atoms changes with time, so does the flux J)

From the Fick’s first law:

⎟⎠⎞

⎜⎝⎛−=

xCD

xxJ

∂∂

∂∂

∂∂

The equation of continuity:

tC

xJ

∂∂

∂∂

−=

…still Fick’s second law

Being , the final differential equation is:

⎟⎠⎞

⎜⎝⎛=

xCD

xtC

∂∂

∂∂

∂∂

Fick’s second law2

2

xCD

tC

∂∂

∂∂

=

...and if the diffusivity D does not depend on composition:

tC

xJ

∂∂

∂∂

−=

(...for the ideal dilute case Nernst-Einstein have shown that D=ũkT)

ũ = mobility of diffusing species = average velocity/unit force

…solving the Fick’s second law

The solution of the Fick’s 2nd law depends on the boundary conditions!

A practical case:

the concentration value of the diffusing atoms at the surface of the material is a constant (Cs).

Solution:Solution:

⎟⎠

⎞⎜⎝

⎛−−=DtxerfCCCtC ssx 2

)()( 0

(Cx(t) is the concentration at depth x, at time t)

the initial concentration of the diffusing atoms in the material is uniform (C0);

t > 0, C=Cs for x=0 and C=C0 for x=∞

t = 0, C=C0 for any positive x

Cx(t)

x

Error function and composition profile

Dtxzdyyzerf

ze 2

,2)(0

2=−= ∫π

In non steady-state diffusion, the concentration depth profile of diffusing species changes with time

The thickness of the region interested by the diffusion increases with (Dt)1/2

…a practical case: the carburizing of steels

A thermal treatment of a solid in controlled atmosphere exhibits the describedboundary conditions!

Carburization of steel:Carburization of steel: thermal treatment of steel at high temperature in a carbon-rich atmosphere (often with CH4 )

(the induced formation of iron carbide on the surface increases the hardness of the material)

The ionThe ion--exchange exchange (an example of diffusion in silicate glasses)(an example of diffusion in silicate glasses)

Silicate (e.g. SiO2-based) glass structure

Crystalline silica: long-range structural orderVitreous silica: limited structural order (max. 3 nm)Silicate glass: SiO2 with other oxides

crystal glassglass with modifiers

…the ion exchange in silicate glassesEbond ≈ 1 eV (for alkaline ions) ...easy diffusion!

Process parameters:Process parameters:

1. bath composition2. bath temperature3. exchange time

An alternative method:Field-assisted solid-stateion exchange (FASSIE)

* A. Quaranta, E. Cattaruzza, F. Gonella, “Modelling the ion exchange process in glass: Phenomenological approaches and perspectives”, Mat. Sci. Eng. B 149 (2008) 133–139

… (not conventional) ion exchange in silicate glasses

metal contact

metal contact

glass substrate

isolation

+ –

µAit

oven

deposited film

RF magnetron sputtering

Metallic film 50-200 nm thick

Au metallic film 200 nm thick

(ohmic contact)

PARAMETER RANGE

(temperature) T 100 to 500 °C

(electric field) E 10 to 500 V/mm

(process time) t minutes; hours

Apparatus configurationSilicate glass slide

−−

Met+

Na+

O−

+

E

+

…the ion exchange in silicate glasses

Ion diffusion in glassIon diffusion in glass• transport of charge entirely due to metallic ions• two ionic diffusing species (A and B) with the same charge• vacancy (V) concentration nearly constant (equilibrium)• (local) thermodynamic equilibrium

• diffusion by site changes A-B, A-V, B-V

Hypothesis

0=++ VBA JJJ

Diffusion towards “x” direction

xTL

xTLJ

xTL

xTLJ

BBBABAB

BABAAAA

∂∂µ

∂∂µ

∂∂µ

∂∂µ

−−=

−−=

L: phenomenological parametersµ: chemical potentialT: (absolute) temperature


Nernst-Planck equation(s)

),(),(),()(ln)(ln1),(

txctxEutxx

cF

DtxJ AAA

A

AAA +⎟⎟

⎠

⎞⎜⎜⎝

⎛+−=

∂∂

∂γ∂

D: diffusion coefficientγ: activity coefficientF: ion atomic fractionu: ion mobilityc: concentration

E: electric (local) field⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

A

AB

A

AAA

B

AB

A

AAA

cL

cL

kTqu

cL

cLkD

q: ion charge

kTqDu A

A =? No NernstNo Nernst--Einstein:Einstein:

kTfqDu A

A =

f: correlation coefficient


Alternative (equivalent) description

A–B (or A–V): two equilibrium level system with an energy barrier, ∆

eDD kTT∆

−=∆ ),( 0

Kinetic reaction at the surface…A*, B*= liquid phase (salt); A, B =solid phase (glass) ** BABA +=+

),(),(),()(ln)(ln1),(

),(),(),()(ln)(ln1),(

txctxEutxxc

FDtxJ

txctxEutxxc

FDtxJ

BBB

B

BBB

AAA

A

AAA

+⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

+⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

∂∂

∂γ∂

∂∂

∂γ∂…then diffusion


Charge balanced → E=E(FA, γA, D)

xc

FFDFDDDtxJ A

A

A

BBAA

BAA ∂

∂∂

γ∂⎟⎟⎠

⎞⎜⎜⎝

⎛+

+−=

)(ln)(ln1),(

qA=qB=q

Final equation to solve:

Hypotheses usually done: • cA = constant at the surface =cS (goodgood)• cA= 0 in the glass at the beginning (goodgood)• D does not depend on x (STRONGSTRONG)˜

Solution: ⎟⎠

⎞⎜⎝

⎛=⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−=Dtxerfcc

Dtxerfctxc ssA 22

1),(

xcDtxJ A

A

~),(∂∂

−=

⎟⎠⎞

⎜⎝⎛=

xcD

xtc AA

~∂∂

∂∂

∂∂


Ag+– Na+ ion exchange in SLG

By solving the diffusion equation:

Good control of the doping profile

Ag+– Na+ ion exchange in SLG

Refractive index and Ag depth profile

“graded-index”waveguide


Soda-lime glass (SLG)

wt% at. % (approx.)

69.6 SiO2 24.1 Si

15.2 Na2O 10.2 Na

1.1 K2O 0.5 K

6.5 CaO 2.4 Ca

5.1 MgO 2.6 Mg

1.8 Al2O3 0.6 Al

(0.7 others) 59.6 O

Borosilicate glass (BSG)

wt% at. % (approx.)

69.6 SiO2 23.7 Si

8.4 Na2O 5.5 Na

8.4 K2O 3.7 K

9.9 B2O3 5.8 B

2.5 BaO 0.3 Ba

(1.2 others) 61.0 O

Ag+– Na+

Cu+– Na+

SLG BSG

SLG SLG

FASSIE

…the FASSIE in silicate glasses

10-1

100

101

102

103

104

105

0 200 400 600

Ion

yiel

d (c

ount

s/s)

Depth (nm)

OCaSi

AuK

Na

T=400 °C, E=500 V/mm (t=120 min)

SLG matrix

Co2+ Au3+

Er3+

10-1

100

101

102

103

104

105

0 100 200 300 400 500 600 700

Ion

yiel

d (c

ount

s/s)

Depth (nm)

OCaSi

ErK

Na

T=500 °C, E=200 V/mm

…MNCGs (metal nanocluster composite glasses) synthesis

MNs embedded in glass enhance its optical third-order susceptibility (Kerr effect)...

intensity-dependent refractive index(useful for photonics applications)

↵

…two-step MNCGs synthesis by diffusion processes (1st example)

a silicate glass is immersed in a molten salt bath containing the doping atoms; they replace alkali ions of the glass by diffusion

Ion exchange:

the activation energy for the diffusion of ions is usually high!

1. Metal-alkali ion exchange2. Thermal treatment in controlled atmosphere (H2, O2, Ar, …)

…the first step (ion exchange) →waveguide

Silicate glass refractive index: nglassMetal-doped silicate glass refractive index : ndoped-glass

ndoped-glass > nglass

θi θr

θt

n1

n2(n2 > n1)

Snell’s law:n1 sinθi = n2 sinθt

Waveguide! ndoped-glass

nair

nglass (ndoped-glass > nair , nglass)

glass containing Na (and K, Ca, Mg, ...) ions:soda-lime glass

Na+–Cu+ ion exchange in CuSO4:Na2SO4bath at T= 545 ˚C for 10 minutes

thermal treatment in Ar–H2(4%) at T= 160 ˚C for 5 hours

←

←

←

1st step

2nd step


depletion of Na,accumulation of Cu

copper diffuses inside the glass both as Cu+ and Cu++ ions

Hydrogen diffusion inside the doped glass induces precipitation and aggregation of copper atoms

1st step

2nd step


1. Metal ion implantation2. Thermal treatment in controlled atmosphere (H2, O2, Ar, …)

a silicate glass is bombarded with metal ions accelerated by a potential difference of several kV; they enter the glass and dope it (atoms can also diffuse during the implantation by RED)

Ion implantation:

…two-step MNCGs synthesis by diffusion processes (2nd example)


glass (SiO2, silicate glass, …)

Au+ ion implantation at E=190 keV, F=3×1016 ions/cm2, j≤2 µA/cm2

thermal treatment in air at T= 900 ˚C for 1 hour

←

←

←

1st step

2nd step

gold implanted atoms are dispersed inside the glass

Inside the doped region, the O2diffusion induces the gold atoms to move and aggregate to form metal particles in the nm range of size

1st step

2nd step

surface


…two-step MNCGs synthesis by diffusion processes (3rd, 4th

examples…)

glass (SiO2, silicate glass*, …)

laser irradiation (λ=527 nm, ε=0.5 J/cm2)OR

ion irradiation (1 MeV Xe+, 3×1016 ions/cm2)

←

←

←

1st step

2nd step

Na+–Ag+ ion exchange*OR

Ag+ ion implantation

ion irradiation (1 MeV Xe+, 3×1016 ions/cm2) laser irradiation (λ=527 nm, ε=0.5 J/cm2)

Na+–Ag+ ion exchange

1st step

2nd step

…two-step MNCGs synthesis by diffusion processes (3rd, 4th

examples…)

2nd step

Ion exchanged silicate glasses for solar cells

covering: down-shifting properties

Conversion efficiency of the solar cells

WAFER THIN FILM

about 8% of power falls in the

near-UV region

downshiftingSilicate glasses doped with transition elements such as Ag and Cu are known to be

luminescent materials

Modification of the incoming light spectrum!

… scientific frame…

ion exchange substrate

A+A+

B+B+Main parameters:

1) bath composition2) atm. composition3) Texc and texc

4) … and also subsequent thermal treatments, if needed…

in form of nanoparticles{

… glasses and baths…

Float-glass?SnSn2+2+ contamination!contamination!

• One-side contamination• Sn2+ in the first few microns• Reducing agent: Sn2+ → Sn4+

Salt bathSalt bathCuSO4:Na2SO4 (46:64),T=550°C, t=20 min

AgNO3:NaNO3 (1:99),T=320°C, t=60 min (annealed)

Too muchCu2+, Cu0

Too muchAg0

… change of(Cu) BATH CuCl:ZnClCuCl:ZnCl22 (11:89)

(Ag) IMMERSION Floated (F)Floated (F), not Dipped (D)

{

… optical absorption…

AgNO3:NaNO3 (1:99)T=320°C, t=60 min

(annealed)

CuCl:ZnCl2 (11:89)(as-exchanged)

Ag precipitation favoured by Sn presence

… mainly Cu+SILVERSILVER

COPPERCOPPER

… (Ag) photoluminescence…

Band centered at 450-500 nm: interaction between Ag+ -Ag+ pairs

Band centered around 600 nm: presence of charged few-atoms aggregates

• the downshifting takes place…• …controlling the silver aggregation

… (Cu) PL and XANES…

Band centered around 500 nm: high value of the Cu+/Cu2+ ratio

PL and PLE XANES

… P-against-V cell tests…

GaAs celloutput power

6.2

6.3

6.4

6.5

6.6

6.7

660 680 700 720 740 760

Cu-doped glass (350°C-3h)

Cu-doped glass(350°C-20min)

Ag-doped glass(floating)

Reference pure glass

Pow

er (m

W)

Voltage (mV)

0

2

4

6

8

10

0 200 400 600 800 1000

Cur

rent

(mA

)

Voltage (mV)

Fill Factor = 0.77

ISC

VOC

VMP

IMP

0

1

2

3

4

5

6

7

0 200 400 600 800 1000

Pow

er (m

W)

Voltage (mV)

VMP

…an encouraging result*!

* E. Cattaruzza et al., Sol. Energy Mater. Sol. Cells 130 (2014) 272–280

… silver again…

How to maximize the formation of (charged) few-atoms aggregates

avoiding Ag nanoparticles?Formation Mechanism of Silver Nanoparticles Stabilized in Glassy Matrices

Anne Simo et al., J. Am. Chem. Soc.134 (2012) 18824−18833

…… a threshold Ta threshold Tannann exists, for any given glass!exists, for any given glass!

… new experiments…

Tin-free soda-lime glass composition

Element Atomic %ION EXCHANGEION EXCHANGE

(T=320°C, t=60 min)

AgNO3:NaNO3 (1:99)AgNO3:NaNO3 (0.1:99.9)

ANNEALING (in air)ANNEALING (in air)

380°CT = 410°C

440°C

1 ht = 4 h

16 h

{{

1 %

0.1 %

380°C 410°C 440°C

1h 4h 16h 1h 4h 16h 1h 4h 16has-exc

… 1 mol% Ag-doped glass…

0.4

0.8

1.2

1.6

2Ref. glassas-exc380°C-1h380°C-4h380°C-16h410°C-1h410°C-4h410°C-16h440°C-1h440°C-4h440°C-16h

300 350 400 450 500 550 600 650

Opt

ical

den

sity

(arb

. uni

ts)

Wavelength (nm)

small silver clusters(d < 1 nm)

0

1 107

2 107

3 107

4 107

5 107Ref. glassas-exc380°C-1h380°C-4h380°C-16h410°C-1h410°C-4h410°C-16h440°C-1h440°C-4h440°C-16h

400 450 500 550 600 650 700

PL in

tens

ity (a

rb. u

nits

)

Wavelength (nm)

λexc = 350 nm BEST440°C410°C longer times380°C very long time

WORST

440°C longer times410°C medium time

410°C-16h should be

the best sample

… P-against-V cell test…

0.034

0.035

0.036

0.037

0.038Ref. glassas-exc380°C-1h380°C-4h380°C-16h410°C-1h410°C-4h410°C-16h440°C-1h440°C-4h440°C-16h

0.36 0.38 0.4 0.42 0.44

Pow

er (W

)

Voltage (V)

Si cell

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.1 0.2 0.3 0.4 0.5 0.6

Pow

er (W

)

Voltage (V)

• …yield still lower than with the pure glass• best sample: 410°C-16h

3D Contour Plot of Integrale (400-700 nm), emissione, 350 nm, 1% against t (h) andT (°C)

Integrale (400-700 nm), emissione, 350 nm, 1% = Spline

> 9E9 < 9E9 < 8E9 < 7E9 < 6E9 < 5E9 < 4E9 < 3E9 < 2E9

1VA1

1VA2

1VA3 1VA4

2VA 2VA1

2VA2 2VA3

2VA4

0 2 4 6 8 10 12 14 16 18

t (h)

370

380

390

400

410

420

430

440

450T

(°C

)

1VA1

1VA2

1VA3 1VA4

2VA 2VA1

2VA2 2VA3

2VA4

… looking for the best conditions…


Integrale (400-700 nm), emissione, 350 nm, 1% = Negative Exponential Smoothing

> 9E9 < 9E9 < 8E9 < 7E9 < 6E9 < 5E9 < 4E9 < 3E9 < 2E9

1VA1

1VA2

1VA3 1VA4

2VA 2VA1

2VA2 2VA3

2VA4

0 2 4 6 8 10 12 14 16 18

t (h)

370

380

390

400

410

420

430

440

450

T (°

C)

1VA1

1VA2

1VA3 1VA4

2VA 2VA1

2VA2 2VA3

2VA4


Integrale (400-700 nm), emissione, 350 nm, 1% = Distance Weighted Least Squares

> 1E10 < 1E10 < 8E9 < 6E9 < 4E9 < 2E9

1VA1

1VA2

1VA3 1VA4

2VA 2VA1

2VA2 2VA3

2VA4

0 2 4 6 8 10 12 14 16 18

t (h)

370

380

390

400

410

420

430

440

450

T (°

C)

1VA1

1VA2

1VA3 1VA4

2VA 2VA1

2VA2 2VA3

2VA4

3D contour plot of 400-700 nm integrated PL intensity (λexc=350 nm, 1% exc. samples)

Most promisingannealing conditions

410°C ≤ T ≤ 420°C

10h ≤ t ≤ 12h

… PL quantum yield…

Absolute fluorescence quantum yield (number of emitted photons / number of absorbed photons)

λexc=350 nm

* E. Cattaruzza et al., Ceramics International 41 (2015) 7221–7226

Documents

Diffusione nei solidi: teoria e applicazioni a materiali ... · Università Ca’ Foscari Venezia Diffusione nei solidi: teoria e applicazioni a materiali avanzati Prof. Elti Cattaruzza