NUMERICAL SIMULATION OF ION TRANSPORT DURING ANODIC BONDING

Transcription

1 NUMERICAL SIMULATION OF ION TRANSPORT DURING ANODIC BONDING Maurco Fabbr José Roberto Sbraga Senna Laboratóro Assocado de Sensores e Materas - LAS Insttuto Naconal de Pesqusas Espacas CP 515, São José dos Campos SP Brazl Abstract. We descrbe a stable numerc dscretzaton procedure of the onc non-lnear transport equatons that governs the depleton layer dynamcs durng electrostatc bondng, n the absence of carrer dffuson. The numercal scheme allows for ndependent caton and anon moblty models, and can be accommodated to handle more complcated effects, such as the oxde layer buld-up at the metal-glass nterface and the nterdependence between sodum and oxygen transport. For Pyrex-on-slcon (s-g) bondng, results from our smulatons show that the correct global transent behavor s acheved when the oxygen moblty n the glass s around one-ffth of the value for the sodum ons; under those condtons, the depleton-layer charge profles are rather concentrated around the depleton front (away from the s-g nterface), yeldng an essentally neutral depleton zone almost everywhere. Consderaton of an addtonal potental drop due to oxde electrochemcal deposton at the bondng nterface does not change qualtatvely the model predctons. Keywords: anodc bondng, electrostatc bondng, electrc depleton layer, movng boundary problems

2 1. INTRODUCTION Anodc bondng s an mportant usual technque n mcrofabrcaton for metal-glass sealng, and was ntroduced by Walls and Pomeranz n Typcally, a metal-nsulator juncton s made by clean mechancal contact between a sodum contanng glass plate and a metal or semconductor. The juncton s then heated around 3 o C and a hgh constant voltage ( 1V) s appled between the metal (anode) and the glass (cathode) a metal contact beng provded to the free glass surface. At those moderated hgh temperatures, sodum catons Na + from the dssocaton of Na 2 O nsde the glass mgrate towards the cathode. Snce the dssocated oxygen anons have a small moblty (and the temperature s stll too low for any sgnfcant electronc conducton), a depleton layer s formed n the glass, n the vcnty of the glass-metal nterface. The hgh electrostatc feld brngs the surfaces nto ntmate contact, and metal oxde s then formed at the nterface (t s beleved that the source of oxygen atoms s the natural humdty present n the glass). That provdes a clear and homogeneous weldng whch conforms to large contact areas and allow bondng between surfaces that are not perfectly planar - for that reason t has been ntensvely used for cavty sealng and devce encapsulaton. In practce, usually a large bondng surface (around 1cm 2 ) s frst assembled, submtted to the bondng process and then cut nto fnal unts. The behavor of the transent electrc current durng the bondng process provdes ndrect nformaton about the electrc transport nsde the glass and the dynamcs of the depleton layer buld-up. It s clearly observed dstnct short-term and long-range regmes, whch have been modeled ether by equvalent-crcut models (Anthony, 1983; Albaugh, 1991) or by mcroscopc mean-feld ohmc transport (Carlson et al., 1972; Ros et al., 2). In the mproved model by Ros et al. (2), a crtcal feld value s ntroduced, above whch oxygen transport s allowed through the glass, thus provdng a threshold for dstnct short and longterm behavor. Nevertheless, all of those models so far have faled to account for the global observed electrcal transent, as they predct ether a too-short ntal regme or an ncorrect long-range behavor. The correct pcture of charge transfer nsde the depleton layer requres a detaled charge-compensaton mechansm that has not been adequately taken nto account by earler works, whch amed at models wth analytc closed solutons. In ths work, we descrbe a numercal scheme that allows for ndependent sodum and oxygen moblty models, and consders the nterdependence between sodum and oxygen transport. Results then show that the correct global transent behavor seems to be acheved when the oxygen anons moblty s enough to yeld an essentally neutral depleton zone almost everywhere, wth a rather concentrated charge profle around the depleton front (away from the glass-metal nterface). Oxygen moblty s thus lkely to set up anon transport nto the glass at very early stages of the bondng process. Nevertheless, mportant detaled dscrepances stll reman between the numercal smulatons and the known expermental behavor of the electrc transent. We thus nvestgate the ssue further by takng nto account the nsulatng oxde layer buld-up at the slcon-glass nterface. 2. ELECTROSTATIC MODEL FOR THE DEPLETION-LAYER BUILD-UP We denote the normalzed sodum (caton) and oxygen (anon) charge denstes nsde the glass by ρ + and ρ -, respectvely, such that ρ +,- 1 (ρ +,- = n +,- /n +, where n + s the ntal sodum content n the glass). Intally (tme t=) the glass s unformly neutral, ρ + = ρ -, the net charge ρ = ρ + - ρ - s zero and the electrc feld E = E s unform throughout. Then sodum catons Na +, from the dssocaton of Na 2 O nsde the glass, begn to be dragged by

3 the electrc feld, whle oxygen anons are stll mmoble due to ther low moblty. If we neglect carrer dffuson, then the Na + charge content nsde the glass s a stepwse profle, whch travels towards the cathode wth an nstantaneous speed v(t). Referrng to Fg. (1), denotng by J + the normalzed charge flux of the sodum catons, the depleton layer front Γ s dsplaced by an amount dx durng the tme nterval dt such that dx = J + dt (1) Note that the glass-metal nterface at x= s not a source of sodum ons, and therefore J + = for x Γ. The electrc feld buld-up n the depleton layer follows Gauss-Posson law: E = ρ/ ε x (2), where ε s the delectrc constant of the depleton layer medum. Outsde the depleton regon, the electrc feld s unform. Assumng that the entre potental drop occurs wthn the glass thckness L, the electrc feld profle must satsfy L V = Edx (3), where V s the appled voltage. The local charge flux relates to the local electrc feld value and local charge denstes accordng to a consttutve relatonshp J = J(E,ρ) that s usually wrtten n the form J + = µ + ρ + f(e) (4), where µ s the on moblty. In the absence of oxygen transport, Eqs. (1) to (4) determne the tme evoluton of the depleton layer, Γ(t). The smple case of ohmc behavor for Na + transport, J + = µ + ρ + E, yelds the same soluton that Albaugh (1991) found through a varable-capactance equvalent crcut model. Note that, as the depleton layer front advances, the local electrc feld ncreases at the glass-metal nterface and decreases n the layer front; therefore, the speed of advance v(t) decreases n tme. Now f one consders the actual sodum content of commercal glasses sutable for anodc bondng, t can be shown that sodum drft nsde the glass cannot proceed for long tmes wthout charge compensaton. Wth Na + transport alone, the electrc feld around the glassmetal nterface would quckly reach values well above the glass delectrc strength (Ros et al., 2). Incluson of anon transport s done by a contnuty equaton for the oxygen charge densty and by a correspondng consttutve relatonshp ρ J = (5) t x J - = µ - ρ - f(e) (6)

4 Qualtatvely, as the electrc feld values n the depleton layer get hgher, the counter-flux of oxygen anons towards the glass-metal nterface becomes apprecable. We then expect that an essentally charge-neutral regon n the depleton layer develops wth tme, whch lmts the value of the electrc feld at the glass-metal nterface. Fg. (1) shows typcal profles for charge denstes and electrc feld that are expected durng the depleton layer buld-up. METAL GLASS 1 ρ + 1 x ρ _ ρ = ρ + ρ_ -1 E I E E Γ Γ Fgure 1 Typcal charge profles and electrc feld durng development of the depleton layer, wth both sodum (+) and oxygen(-) on transport. 3. NUMERICAL SCHEME The numercal strategy takes nto account that (1) oxygen transport nsde the glass s neglgble outsde the depleton layer, where the electrc fled s unform and steady decreasng n tme; (2) the depleton layer front speed decreases n tme, and (3) a stable numercal scheme s needed for the tme ntegraton of Eq. (5). Neglectng the oxygen transport outsde the depleton layer s also justfed from the pont of vew that, for x>γ, even n the presence of sodum drft, oxygen atoms reman essentally bounded to Na 2 O onc pars; wth that n mnd, we actually mpose a null-flux of O -- when the local Na + charge densty s unperturbed. Fnte-dfference dscretzaton s done usng fxed tme-steps t. Eqs. (1) to (6) s then treated as a movng boundary problem, whch s ntegrated accordng to the followng algorthm: 1. from the current value of the electrc feld at Γ, obtan the local flux of the Na + catons from the consttutve relaton, Eq. (4); 2. calculate the dsplacement x of the depleton-layer front durng the tme-nterval t, accordng to Eq. (1);

5 3. from the current values of the electrc feld nsde the depleton-layer regon and the anon charge densty profle ρ - (x), obtan the local flux of the oxygen anons at the current tme, usng the consttutve relaton, Eq. (6); 4. obtan the charge densty profle ρ - (x) at tme t+ t by a Crank-Ncholson sem-mplct ntegraton of the contnuty equaton, Eq. (5); 5. from the net charge feld ρ = ρ + - ρ -, the electrc feld at tme t+ t s found by ntegraton of the Posson Eq. (2) subjected to the normalzaton of the total potental, Eq. (3). Intal condtons (when t= and Γ=) are E = E = V /L, ρ + = ρ - =1. (unform felds). Note that, as tme goes on, step 2 establshes a varable space mesh over the doman x Γ. 3.1 Detals of the varable-mesh fnte-dfference Cranck-Ncholson scheme In the followng, A and A stands for the value of the feld A at mesh poston at tmes (t) and (t+ t), respectvely, whle ρ and ρ are values at (t) and (t+ t) of the charge densty value over the mesh nterval (, +1). In ths secton, ρ and J refer to the anon charge densty and flux; we omt the charge label " - " for clarty. Dscretzaton of the contnuty equaton for the oxygen anons s done nsertng the consttutve relaton Eq. (6) nto Eq. (5) and adoptng a sem-mplct approxmaton for the charge densty: J ρ + ρ = µ f (E ) (7) 2 Electrc feld and local moblty values are treated explctly n tme. Values of the electrc feld and charge fluxes are specfed at mesh nodes, whle charge denstes are gven stepwse at the mesh ntervals. We number the mesh nodes as =1(1)(M+1), such that x 1 = and x M+1 =Γ. For the charge densty at the nodes, we adopt a smple lnear nterpolaton: ρ 1 + ρ ρ = (8) 2, wth boundary condtons ρ 1 = ρ1 and ρ M =1.. In that manner, we arrve at a smple trdagonal system of algebrac equatons for =1(1)M, to be solved at each tme-step. 3.2 Space mesh renormalzaton As the mesh sze M ncreases and the current mesh nterval x at Γ decreases n tme, a renormalzaton of the spatal mesh s done perodcally after a fxed number of tme-steps, thus preservng CPU tme by avodng a too-hgh spatal resoluton far from the depletonlayer advancng front (where the charge feld s lkely to be nearly unform). The strategy s to merge two adjacent mesh ntervals when the charge feld gradent s smaller than a gven threshold. Ths s done numercally by comparng the weghted numercal dsperson between two adjacent nterval values of the charge densty wth a gven mnmum. ρ,

6 4. NUMERICAL RESULTS Values for the observed external electrc current are proportonal to the sodum catons flux, whch s evaluated from Eq. (4) at the nterface poston Γ. In the smple lnear (ohmc) model, J + has the same behavor of the local electrc feld at Γ. Numerc results are shown n Fg. (2), usng reasonable known estmates for the physcal constants, and the same expermental settngs used by Albaugh, 1991 (anodc bondng of Slcon to Pyrex glass). Those are lsted n Table 1. Table 1. Dmensonal values of parameters employed n the numerc smulatons Parameter Symbol Value Expermental settngs Appled voltage V 1V Glass thckness L 3,2mm Physcal propertes Delectrc constant ε 7ε Permttvty of free space ε F/m Sodum charge content n C/m 3 Ionc resstvty of Na + r Ω.m Characterstc values of length D and tme τ for the depleton zone dynamcs can be defned as 2εV D = (9a) n + 2εrL τ = (9b) D, and they are D = m ; τ = 2.6s for the settngs of Table J/J µ_ = 3x Tme [s] Fgure 2 Decay of the normalzed electrc current n tme, as a functon of the oxygen anon moblty µ. Open crcles are expermental ponts measured by Albaugh (1991). Sold curves are the results of the numerc smulatons. The moblty of catons s held fxed at the value µ + = 1.

7 The numerc results shown n Fg. (2) for µ = was obtaned by Albaugh (1991) wth an equvalent crcut model. There s a far general agreement between our model and the observed behavor of the electrc current decay, both for short and long tme regmes, as far as the oxygen moblty n the glass reaches 2% of the sodum moblty. In that case, most of the depleton layer s almost neutral and unform, the net charge profle resemblng a delta functon around Γ (Fg. 3). 6 ( E/ E ) electrc feld dstance from the nterface ( x/ d ) Fgure 3 Electrc feld and net charge profle n he depleton zone for µ =.2, at tme 21.4τ (44.1s). The nset s an enlargement of the net charge profle around the depleton layer front Γ. Values of the computed depleton layer wdth can be seen n Fg. (4). Depleton lengths are dramatcally ncreased for hgh anon moblty, and that s n accordance wth the expermental results of Walls (197), that reported depleton wdths of the order 5 D Na + depleton wdth [n d unts] 1 3x1-5.3 µ_ = Tme [mn] Fgure 4 The computed depleton layer advance n tme, as a functon of the oxygen anon moblty µ.

8 The electrc feld at the slcon-glass nterface attans lower values, more compatble wth the delectrc strength of pure slca ( 1 7 V/cm), as the anon moblty ncreases. Ths s shown n Fg. (5). Electrc feld at s-g nterface [E/E ] µ_ = 1 5 3x E r Tme [s] Fgure 5 Numercal results for the electrc feld at the slcon-glass (s-g) nterface durng the anodc transent, as a functon of the oxygen anon moblty µ. E r s the rupture feld n pure slca. 5. EFFECTS OF THE OXIDE LAYER GROWTH Deposton of oxygen at the anode results n a contnuous growth of an nsulatng oxde layer at the metal-glass nterface, of thckness x (t). In the numerc smulatons, that can be taken nto account by consderng an unform electrc feld n the layer regon, -x x, and modfyng Eq. (3) as L = V Edx (1) x The detaled growth dynamcs of the oxde layer should consder that oxygen atoms comng from the glass accumulate at the glass-metal nterface and only reach metal atoms through dffuson across the layer x. As a frst smplfed model, we here consder that oxygen atoms are nstantaneously ncorporated nto the metal matrx to form an unform oxde layer. In that case, the ncrease dx of the oxde layer durng the tme nterval dt can be found from the anon flux J - at the metal-glass nterface (x=) and the value of the oxygen charge densty n the oxde, n ox : dt dx = J (,t) (11) n ox

9 Incluson of Eqs. (1) and (11) n the numerc model s straghtforward. Calculatons were done for a slcon-glass nterface. Each molecule of slcon oxde, SO 2, carres two oxygen anons, whch correspond to four elementary charges. From the known molecular densty SO2 n = 2, molecules/m 3 (Ruska, 1987), we thus have n ox = C/m 3. On the assumpton that, n the glass, the only source of oxygen atoms s the dssocaton of Na 2 O, usng the known data for the sodum content n Pyrex glass (Albaugh, 1991) gves a rough estmate n 18 n ox + 6 (12) Fg. (6) shows results of the numerc smulatons wth our smplfed homogeneous oxde layer growth. Unfortunately, t seems that consderaton of the potental drop through the oxde layer does not mprove the qualtatve agreement between the model and the expermental data. 1. J/J.8 wthout oxd layer.6.4 observed µ_ =.2 µ_ = Tme [s] Fgure 6 Numercal estmate of the nfluence of the oxde layer growth at the slconglass nterface on the measured electrc current transent. That s to be compared to Fg. (2). The numbers that label the curves are the values for the normalzed oxygen charge densty n the oxde, Eq. (12). 6. CONCLUSIONS AND COMMENTS The numerc smulatons pont towards the mportance of anon transport n the glass even at very early tmes durng electrostatc slcon-glass (s-g) bondng process. Hgh oxygen moblty n the glass lmts the electrc feld to moderated values at the s-g nterface and provded for large depleton layer wdths. Those fndngs are n accordance to the known expermental data, as far as electrc transents are concerned.

10 Expermental data from Albaugh (1991) seems to ndcate that a large electrc current s sustaned for long tmes (an order of magntude greater than the characterstc depleton tme) at ntal stages of the process. Our model, wth the current assumptons cannot explan that. Incluson of an addtonal potental drop due to oxde growth at the s-g nterface does not mprove qualtatvely the numerc predctons. Further expermental data under controlled condtons are thus desrable, and are beng devsed at our facltes at INPE. Computatons were done n a 2GHz Asus-Athlon PC machne wth the gcc compler. Stablty of the adopted Crank-Ncholson sem-mplct scheme requres smaller values for the tme-step t as the anon moblty µ ncreases. As a typcal example, ntegraton for µ =.2 (τ=2.6) durng the frst 9 seconds of the depleton layer growth requred a maxmum tme-step of τ and takes nearly an hour and a half of CPU tme. The model here descrbed s sutable to encompass a whole class of moblty and crossonc transport models. Improvements can be made both n the fxed-tme ntegraton strategy and mesh renormalzaton, towards a truly adaptve algorthm. In the present form, excessve CPU tme s requred as the rato of anon to caton mobltes approach 1.. REFERENCES Anthony, T.R., Anodc bondng of mperfect surfaces. J. Appl. Phys. 54, pp Albaugh, K.B., Electrode Phenomena durng Anodc Bondng of Slcon to Sodum Boroslcate Glass J. Electrochem. Soc. 138, pp Carlson, D.E.; Hang, K.W. and Stockdale, F., Electrode Polarzaton n Alkalcontanng Glasses J. Am. Cer. Soc. 55, pp.337. Ros, A.N.; Gracas, A.C.; Maa, I.A. and Senna, J.R., 2. Modelng the Anodc Current n Slcon-Glass Bondng. Rev. Bras. Aplc. Vacuo 19, pp Ruska, W.S., Mcroelectronc Processng. McGraw-Hll. Walls, G. and Pomeranz, D.I., Feld Asssted Glass-Metal Sealng J. Appl. Phys. 4, pp Walls, G., 197. Drect-Current Polarzaton durng Feld-Asssted Glass-Metal Sealng. J. Am. Cer. Soc. 53, pp.563.