Diffusione nei solidi: teoria e applicazioni a materiali avanzati

Transcription

1 Università Ca Foscari Venezia Diffusione nei solidi: teoria e applicazioni a materiali avanzati Prof. Elti Cattaruzza Department of Molecular Sciences and Nanosystems, Università Ca Foscari Venezia, via Torino 155/b, Venezia-Mestre, Italy elti.cattaruzza@unive.it

2 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

3 n enlightening example Copper-nickel pair before a thermal treatment at high temperature Copper (nickel) atoms are present only in the left (right) region tomic concentration as a function of the position inside the sample

4 ...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 tomic concentration as a function of the position inside the sample

5 What is diffusion? DIFFUSION : 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

6 Imperfections in crystals Point defects Line defects (dislocations) Surfaces Grain boundaries The most important imperfections for the diffusion are: point defects

7 Point defects a) a missing atom within a crystal b) two (three, four,...) missing atoms c) a missing ion-pair of opposite charge d) an extra atom within a crystal structure e) a displaced ion from the lattice

8 Thermal atomic vibrations For T>0 K, atoms have kinetic energy (i.e. vibrations) toms have different energies (i.e. distribution) The fraction _ n/n tot of atoms with energy greater than E (for E»E) is: n N tot e E kt k is the Boltzmann s constant (k= J/K) Units of E: Joules Prob. T 1 T 1 < T 2 < T 3 T 2 E n/n tot T 3 kinetic energy

9 ... 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 bonds and jump to new locations, thus originating vacancies... DIFFUSION n N tot Me E kt The minimum energy required is called activation energy Q (or E) (units of Q or E can be J/atom, ev/atom, J/mole, or calories/mole)

10 Diffusion mechanisms n 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 substitutional atom leaves its normal lattice site and enters an interstitial position toms move by a simple exchange mechanism or by a ring mechanism a), b) are the main ones

11 ctivation energy for diffusion low activation energy Q diffusion favoured

12 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

13 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/m 2 sec

14 Fick s first law (rate of diffusion) The flux J of atoms is proportional to the concentration gradient J = D dc dx Fick s first law J = atoms flux (atoms/m 2 sec) D = diffusivity (m 2 /sec) C = concentration (atoms/m 3 ) Stationary diffusion: J(x,t)= J(x)

15 factors influencing the diffusion coefficient (diffusivity) D The diffusivity D depends on: 1) the nature of the solute atoms Small atoms have a higher diffusivity than large ones toms can be neutral or charged 2) the nature of the solid structure toms diffuse more readily in weakly bonded solids toms diffuse more readily in low packed solids 3) the temperature Higher temperatures provide higher diffusivities D (an rrhenius-type equation) )= D e ( E, T 0 E kt D 0 is the proportional constant k= J/K E is the activation energy per atom (Joules)

16 temperature and the diffusion coefficient D E kt D( E, T)= D0e D Q, T)= D e ( 0 Q RT D 0 is the proportional constant R is the gas constant (R=8.314 J/mole K, or cal/mole K) Q is the activation energy (J/mole, or cal/mole) The logarithm of the diffusivity D: ln D = ln D0 E kt Data of D are usually plotted in rrhenius-type plots ln D = ln D0 Q RT

17 rrhenius-type plot of diffusivity D ln D = ln D0 E kt ln D = ln D0 Q RT

18 Fick s 2 nd 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: J x C = D x x The equation of continuity: J x C = t

19 still Fick s second law Being, the final differential equation is: = x C D x t C Fick s second law 2 2 x C D t C =...and if the diffusivity D does not depend on composition: t C x J = (...for the ideal dilute case Nernst-Einstein have shown that D=ũkT) ũ = mobility of diffusing species = average velocity/unit force

20 solving the Fick s second law The solution of the Fick s 2 nd law depends on the boundary conditions! practical case: the initial concentration of the diffusing atoms in the material is uniform (C 0 ); the concentration value of the diffusing atoms at the surface of the material is a constant (C s ). t = 0, C=C 0 for any positive x t > 0, C=C s for x=0 and C=C 0 for x= Solution: C t) = C ( C C x ( s s 0 ) erf 2 x Dt (C x (t) is the concentration at depth x, at time t)

21 Error function and composition profile z 2 2 erf ( z) = y e dy, z = π 2 0 x Dt In non steady-state diffusion, the concentration depth profile of diffusing species changes with time C x (t) The thickness of the region interested by the diffusion increases with (Dt) 1/2 x

22 a practical case: the carburizing of steels thermal treatment of a solid in controlled atmosphere exhibits the described boundary conditions! Carburization of steel: thermal treatment of steel at high temperature in a carbon-rich atmosphere (often with CH 4 ) (the induced formation of iron carbide on the surface increases the hardness of the material)

23 The ion-exchange (an example of diffusion in silicate glasses) Silicate (e.g. SiO 2 -based) glass structure crystal glass glass with modifiers Crystalline silica: long-range structural order Vitreous silica: limited structural order (max. 3 nm) Silicate glass: SiO 2 with other oxides

24 the ion exchange in silicate glasses E bond 1 ev (for alkaline ions)...easy diffusion! Process parameters: 1. bath composition 2. bath temperature 3. exchange time n alternative method: Field-assisted solid-state ion exchange (FSSIE) *. Quaranta, E. Cattaruzza, F. Gonella, Modelling the ion exchange process in glass: Phenomenological approaches and perspectives, Mat. Sci. Eng. B 149 (2008)

25 (not conventional) ion exchange in silicate glasses RF magnetron sputtering isolation metal contact oven Metallic film nm thick glass substrate deposited film metal contact u metallic film 200 nm thick (ohmic contact) + E Silicate glass slide + M + O et Na + i µ t + pparatus configuration PRMETER RNGE (temperature) T 100 to 500 C (electric field) E 10 to 500 V/mm (process time) t minutes; hours

26 the ion exchange in silicate glasses Ion diffusion in glass Hypothesis transport of charge entirely due to metallic ions two ionic diffusing species ( and B) with the same charge vacancy (V) concentration nearly constant (equilibrium) (local) thermodynamic equilibrium diffusion by site changes -B, -V, B-V Diffusion towards x direction J J B = = L T L T B µ x µ x L T L T B BB µ B x µ B x J + J B + JV = 0 L: phenomenological parameters µ: chemical potential T: (absolute) temperature

27 the ion exchange in silicate glasses Nernst-Planck equation(s) J (lnγ ) c ( x, t) = D 1 + ( x, t) u E( x, t) c( x, t) (ln F) x + D: diffusion coefficient γ: activity coefficient F: ion atomic fraction u: ion mobility c: concentration E: electric (local) field L LB D = k c cb q L L u = + kt c c B q: ion charge u =? Dq kt No Nernst-Einstein: u = Dq ktf f: correlation coefficient

28 the ion exchange in silicate glasses lternative (equivalent) description B (or V): two equilibrium level system with an energy barrier, D e D kt T )=, ( 0 Kinetic reaction at the surface *, B*= liquid phase (salt);, B =solid phase (glass) * * B B + = + ), ( ), ( ), ( ) (ln ) (ln 1 ), ( ), ( ), ( ), ( ) (ln ) (ln 1 ), ( t x c t x E u t x x c F D t x J t x c t x E u t x x c F D t x J B B B B B B B + + = + + = γ γ then diffusion

29 the ion exchange in silicate glasses Charge balanced E=E(F, γ, D) x c F F D F D D D t x J B B B γ + + = ) (ln ) (ln 1 ), ( q =q B =q Final equation to solve: Hypotheses usually done: c = constant at the surface =c S (good good) c = 0 in the glass at the beginning (good good) D does not depend on x (STRONG STRONG) Solution: = = Dt x erfc c Dt x erf c t x c s s ), ( x c D t x J ~ ), ( = = x c D x t c ~

30 the ion exchange in silicate glasses By solving the diffusion equation: Refractive index and g depth profile g + Na + ion exchange in SLG Good control of the doping profile graded-index waveguide g + Na + ion exchange in SLG

31 the ion exchange in silicate glasses Soda-lime glass (SLG) wt% at. % (approx.) SLG SLG g + Na SiO Si 15.2 Na 2 O 10.2 Na 1.1 K 2 O 0.5 K 6.5 CaO 2.4 Ca 5.1 MgO 2.6 Mg 1.8 l 2 O l FSSIE Borosilicate glass (BSG) wt% at. % (approx.) (0.7 others) 59.6 O Cu + Na + SLG BSG 69.6 SiO Si 8.4 Na 2 O 5.5 Na 8.4 K 2 O 3.7 K 9.9 B 2 O B 2.5 BaO 0.3 Ba (1.2 others) 61.0 O

32 the FSSIE in silicate glasses T=400 C, E=500 V/mm (t=120 min) Ion yield (counts/s) SLG matrix O Ca Si u K Na Co 2+ u 3+ Er 3+ Ion yield (counts/s) Depth (nm) T=500 C, E=200 V/mm O Ca Si Er K Na Depth (nm)

33 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)

34 two-step MNCGs synthesis by diffusion processes (1 st example) 1. Metal-alkali ion exchange 2. Thermal treatment in controlled atmosphere (H 2, O 2, r, ) Ion exchange: a silicate glass is immersed in a molten salt bath containing the doping atoms; they replace alkali ions of the glass by diffusion the activation energy for the diffusion of ions is usually high!

35 the first step (ion exchange) waveguide Silicate glass refractive index: n glass Metal-doped silicate glass refractive index : n doped-glass n 1 θ i θ r n doped-glass > n glass Snell s law: n 1 sinθ i = n 2 sinθ t n 2 (n 2 > n 1 ) θ t n air Waveguide! n doped-glass n glass (n doped-glass >n air,n glass )

36 two-step MNCGs synthesis by diffusion processes (1 st example) glass containing Na (and K, Ca, Mg,...) ions: soda-lime glass Na + Cu + ion exchange in CuSO 4 :Na 2 SO 4 bath at T= 545 C for 10 minutes 1 st step thermal treatment in r H 2 (4%) at T= 160 C for 5 hours 2 nd step

37 two-step MNCGs synthesis by diffusion processes (1 st example) 1 st step copper diffuses inside the glass both as Cu + and Cu ++ ions depletion of Na, accumulation of Cu 2 nd step Hydrogen diffusion inside the doped glass induces precipitation and aggregation of copper atoms

38 two-step MNCGs synthesis by diffusion processes (2 nd example) 1. Metal ion implantation 2. Thermal treatment in controlled atmosphere (H 2, O 2, r, ) Ion implantation: 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)

39 two-step MNCGs synthesis by diffusion processes (2 nd example) glass (SiO 2, silicate glass, ) u + ion implantation at E=190 kev, F= ions/cm 2, j 2 µ/cm 2 1 st step thermal treatment in air at T= 900 C for 1 hour 2 nd step

40 two-step MNCGs synthesis by diffusion processes (2 nd example) 1 st step gold implanted atoms are dispersed inside the glass surface 2 nd step Inside the doped region, the O 2 diffusion induces the gold atoms to move and aggregate to form metal particles in the nm range of size

41 two-step MNCGs synthesis by diffusion processes (3 rd, 4 th examples ) glass (SiO 2, silicate glass*, ) Na + g + ion exchange* OR g + ion implantation 1 st step laser irradiation (λ=527 nm, ε=0.5 J/cm 2 ) OR ion irradiation (1 MeV Xe +, ions/cm 2 ) 2 nd step

42 two-step MNCGs synthesis by diffusion processes (3 rd, 4 th examples ) 1 st step 2 nd step Na + g + ion exchange 2 nd step laser irradiation (λ=527 nm, ε=0.5 J/cm 2 ) ion irradiation (1 MeV Xe +, ions/cm 2 )

43 Ion exchanged silicate glasses for solar cells covering: down-shifting properties Conversion efficiency of the solar cells WFER THIN FILM about 8% of power falls in the near-uv region

44 scientific frame Modification of the incoming light spectrum! Silicate glasses doped with transition elements such as g and Cu are known to be luminescent materials downshifting ion exchange Main parameters: 1) bath composition 2) atm. composition 3) T exc and t exc B + substrate B ) and also subsequent thermal treatments, if needed

45 glasses and baths Float-glass? Sn 2+ contamination! in form of nanoparticles{ One-side contamination Sn 2+ in the first few microns Reducing agent: Sn 2+ Sn 4+ Salt bath { CuSO 4 :Na 2 SO 4 (46:64),T=550 C, t=20 min gno 3 :NaNO 3 (1:99),T=320 C, t=60 min (annealed) Too much Cu 2+, Cu 0 Too much g 0 change of (Cu) BTH (g) IMMERSION CuCl:ZnCl (11:89) 2 Floated (F), not Dipped (D)

46 optical absorption COPPER g precipitation favoured by Sn presence CuCl:ZnCl 2 (11:89) (as-exchanged) gno 3 :NaNO 3 (1:99) T=320 C, t=60 min (annealed) SILVER mainly Cu +

47 (g) photoluminescence Band centered around 600 nm: presence of charged few-atoms aggregates Band centered at nm: interaction between g + -g + pairs the downshifting takes place controlling the silver aggregation

48 (Cu) PL and XNES PL and PLE XNES Band centered around 500 nm: high value of the Cu + /Cu 2+ ratio

49 P-against-V cell tests Power (mw) Gas cell output power Cu-doped glass (350 C-3h) Cu-doped glass (350 C-20min) g-doped glass (floating) Current (m) Power (mw) I MP Fill Factor = V MP I SC V OC V MP Voltage (mv) Voltage (mv) 6.2 Reference pure glass Voltage (mv) an encouraging result*! * E. Cattaruzza et al., Sol. Energy Mater. Sol. Cells 130 (2014)

50 silver again How to maximize the formation of (charged) few-atoms aggregates avoiding g nanoparticles? Formation Mechanism of Silver Nanoparticles Stabilized in Glassy Matrices nne Simo et al., J. m. Chem. Soc.134 (2012) a threshold T ann exists, for any given glass!

51 new experiments Element tomic % ION EXCHNGE (T=320 C, t=60 min) gno 3 :NaNO 3 (1:99) gno 3 :NaNO 3 (0.1:99.9) Tin-free soda-lime glass composition 1 % 0.1 % as-exc 1h 4h 16h 1h 4h 16h 1h 4h 16h 380 C 410 C 440 C NNELING (in air) 380 C T = 410 C 440 C t = { { 1 h 4 h 16 h

52 1 mol% g-doped glass PL intensity (arb. units) Ref. glass as-exc 380 C-1h C-4h 380 C-16h 410 C-1h 410 C-4h C-16h 440 C-1h 440 C-4h C-16h λexc = 350 nm BEST 440 C 410 C longer times 380 C very long time Optical density (arb. units) Wavelength (nm) small silver clusters (d < 1 nm) Ref. glass as-exc 380 C-1h 380 C-4h 380 C-16h 410 C-1h 410 C-4h 410 C-16h 440 C-1h 440 C-4h 440 C-16h Wavelength (nm) 410 C-16h should be the best sample WORST 440 C longer times 410 C medium time

53 P-against-V cell test Power (W) Si cell Ref. glass as-exc 380 C-1h 380 C-4h 380 C-16h 410 C-1h 410 C-4h 410 C-16h 440 C-1h 440 C-4h 440 C-16h Power (W) Voltage (V) Voltage (V) yield still lower than with the pure glass best sample: 410 C-16h

54 looking for the best conditions 3D contour plot of nm integrated PL intensity (λ exc =350 nm, 1% exc. samples) 3D Contour Plot of Integrale ( nm), emissione, 350 nm, 1% against t (h) and T ( C) Integrale ( nm), emissione, 350 nm, 1% = Spline V3 1V4 2V4 Most promising annealing conditions T ( C) 410 1V2 2V 2V1 410 C T 420 C V1 2V2 2V t (h) > 9E9 < 9E9 < 8E9 < 7E9 < 6E9 < 5E9 < 4E9 < 3E9 < 2E9 10h t 12h 3D Contour Plot of Integrale ( nm), emissione, 350 nm, 1% against t (h) and T ( C) Integrale ( nm), emissione, 350 nm, 1% = Negative Exponential Smoothing 450 3D Contour Plot of Integrale ( nm), emissione, 350 nm, 1% against t (h) and T ( C) Integrale ( nm), emissione, 350 nm, 1% = Distance Weighted Least Squares V3 1V4 2V V3 1V4 2V T ( C) 410 1V2 2V 2V1 T ( C) 410 1V2 2V 2V V1 2V2 2V t (h) > 9E9 < 9E9 < 8E9 < 7E9 < 6E9 < 5E9 < 4E9 < 3E9 < 2E V1 2V2 2V t (h) > 1E10 < 1E10 < 8E9 < 6E9 < 4E9 < 2E9

55 PL quantum yield bsolute fluorescence quantum yield (number of emitted photons / number of absorbed photons) λ exc =350 nm * E. Cattaruzza et al., Ceramics International 41 (2015)