Field induced nucleation in nano-structures
FField induced nucleation in nano-structures
V. G. Karpov ∗ and D. Niraula , † Department of Physics and Astronomy, University of Toledo, Toledo, OH 43606, USA
We predict the probability of field induced nucleation (FIN) of conductive filaments across thenano-thin dielectric layers in memory and switching devices. The novelty of our analysis is thatit deals with a dielectric layer of thickness below the critical nucleation length. We show howthe latter constraint can make FIN a truly threshold phenomenon possible only for voltage (not thefield) exceeding a certain critical value that does not depend on the dielectric thickness. Our analysispredicts the possibility of threshold switching without memory under certain thickness dependentvoltages. In parallel, the thermal runaway mechanism of electronic switching is described analyticallyleading to results consistent with the earlier published numerical modeling. Our predictions offerthe possibility of experimental verifications deciding between FIN and thermal runaway switching.
I. INTRODUCTION
The presence of conductive filaments (CFs) is criti-cally important for functionality of phase change mem-ory (PCM), resistive random access memory (RRAM) ,and threshold switches (TS) . Also, CFs are responsiblefor the breakdown phenomena in gate dielectrics.While the important role of CFs is commonly recog-nized, their underlying physics is not sufficiently under-stood. The two types of mechanisms of CF formationhave been proposed in the literature: (1) the direct elec-tric field induced nucleation (FIN) of a new phase in theform of conductive filamentary pathway characterized byits critical length and nucleation barrier, and (2) theelectronic filament precursor raising the local tempera-ture enough to trigger a phase transformation or remainas such for the case of TS. It is worth recalling here that CFs in the above men-tioned devices are initially created by the electro-formingprocess that requires a certain forming voltage ∼ − H ∼ −
20 nm, and the insulatinggap width h gap ∼ − The voltages required tocreate and bridge the insulating gaps in CFs are typicallyseveral times lower than the forming voltages.
Here we consider a constrained FIN where the volumeavailable for nucleation is limited. That condition canbe important for the modern nanometer devices whereconductive bridges form through narrow, h gap ∼ − h c (cid:29) h gap ,say, h c ∼
10 nm, as illustrated in Fig. 1. Also, itcan be relevant for the electroforming processes if thedielectric layer thickness is small enough, < ∼
10 nm.The paper is organized as follows. In Sec. II we ap-ply the standard field induced nucleation theory to thenarrow ( h c (cid:29) h gap ) gap case. Sec. III will introduce adifferent FIN scenario driven by the free energy originat-ing from the effective capacitor formed by the CF tip and the opposite electrode. Sec. IV discusses the interplaybetween the two FIN scenarios. The thermal runawayelectronic mechanism of switching is analyzed in Sec. V.Sec. VI contains final conclusions. II. THE STANDARD FIN SCENARIO
We recall that FIN is a process where a small metallicneedle-shaped embryo nucleates in a non-conductive hostdue to the polarization energy gain in a strong externalelectric field E . Similar to other nucleation processes,FIN is characterized by the critical nucleation length h c and nucleation barrier W representing respectivelythe length above which the embryo grows spontaneously,and its corresponding energy. Along the standard lines,the latter quantities are defined without any spacial con-straints.As demonstrated earlier, nucleation in TS and non-volatile memory is described similarly; for the sake ofspecificity, here, we consider the case of TS. Assuminga high aspect ratio nucleus, h (cid:29) r its free energy in a FIG. 1: Left: critical length ( h c ) conductive embryo formedin a uniform electric field within a gap of height H signifi-cantly exceeding h c . That condition corresponding to that ofthe standard FIN theory can relate to a pristine (before ‘form-ing’) device structure. Right: a conductive filament with aninsulating gap of thickness h gap in a ‘formed’ device. A smallconductive embryo will nucleate inside the gap of h gap < h c ,i. e. beyond the limits of applicability of the standard FIN.Dashed arrows represent the electric field. a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r FIG. 2: The free energies of a conductive embryo vs. itsheight in the electric fields of three different strengths, cor-responding (from right to left) to the unstable, metastable,and thermodynamically stable embryos. The initial state ofthe system corresponds to h ≈ ∼ kT ). W is the nucleation barrier height, W ms is the barrier determining the lifetime of a metastableembryo. The inset diagram shows the transition to unstablestate upon the field removal. uniform field of strength E is presented as, F ( h, r ) = 2 πrhγ − αh εE . (1)where γ is the surface tension. The second term in Eq.(1) represents the electrostatic energy gain and is writtenin a truncated form where α stands for a combinationof numerical and logarithmic multipliers. For example, α = {
12 ln[(2 h/r ) − } − for the ellipsoidal CF, and α = { ln[(2 h/r ) − } − for the cylindrical CF, assuming h (cid:29) r . In what follows we neglect the logarithmicdependence of α treating it as a constant smaller thanone, say, α ∼ . r = h = 0) and the newphase state where F decreases with h and r . It was shownthat such a barrier is provided by the absolute minimumof F ( h, r ) located at the boundary region of the minimumallowed value of r (in sub-nanometer range) denoted hereas r min . The latter is determined by the conditions ofelectric and mechanical integrity for CF. For r = r min , the free energy is a minimum at h c = (cid:114) πr min γ εα E (2)corresponding to the barrier height W ( E ) = F [ h c ( E ) , r min ] = 2 / ( πr min γ ) / / ( αε ) / E (3)as illustrated in Fig. 2.For the future references, it is convenient to representthe nucleation barrier through the electric potential U = EH , W ( U ) = W U U (4)with W = 2 πγr min H, and U = (cid:114) πr min γ αε . (5)Because the nucleation time τ = τ exp( W/kT ) where τ = const, it follows that FIN takes place under anyvoltage U over the time that exponentially decreases with U as τ ∝ exp[( W /kT )( U /U )] . (6)This type of dependence was experimentallyobserved. Considering FIN in a narrow gap, the weak field region,
E < E ms [explicitly given in Eq. 7)] can be determinedby the condition h c > h gap . In that region, the free en-ergy within the gap is increasing with h making embryosunstable. On the other hand, under higher fields, E > E s [also, explicitly given in Eq. 7)], the barrier is at shorterdistance than h c , and yet, the energy F ( h gap , r min ) ispositive. Therefore, for the field strengths in the inter-val, E ms ≡ (cid:115) πr min γ εαh < E < (cid:115) πr min γεαh ≡ E s , (7)a metastable embryo within a gap ( h c < h gap ) can beformed. Shown in Fig. 2 the barrier determining thatembryo lifetime is given by W ms = W ( E ) − F ( h gap ) (8)with W ( E ) and F ( h ) from Eqs. (3) and (1). Themetastable embryo lifetime is given by τ ms = τ exp( W ms /kT ) (9)with W ms = 0 when E = E ms .The upper characteristic field E s in Eq. (7) is deter-mined by the condition F ( h gap ) = 0, i.e. the embryo re-mains stable as long as the field E is applied. When thefield is removed, the second term in Eq. (1) disappears,and the embryo free energy linear in h makes it decay asillustrated in Fig. 2 relevant for TS functionality.It follows from the above that while the nucleation ofunconstrained CF is possible for the field of any strength(with exponentially field dependent nucleation times),the confined CFs require the field strength E > E ms andthe corresponding voltages U > U ms = E ms h gap . Simi-larly, stable CFs can nucleate when U > U s = E s h gap . Itfollows from Eq. (7) that both U s and U ms are indepen-dent of h gap and U s = √ U ms . Therefore, as opposed tothe case of unconstrained CF, the confinement seems tomake nucleation a truly threshold voltage phenomenon.Interestingly, the latter expressions for U s and U ms predict numerical values consistent with the data in theorder of magnitude, U s = (cid:114) πγr min εα = (cid:18) / (cid:19) U ∼ . − . (10)Here and in what follows we use the numerical valueslisted in Table I.Also, we note that the transition from the regime ofconstant field to constant voltage with dielectric thinningbelow 10 nm was observed in PCM devices, although itsoriginal explanation was different. It was demonstratedthat in 30 nm PCM structures switching is dominated bythe field strength rather than voltage. TABLE I: Some parameters related to FINParameter H , nm a r min , nm b ε c γ , dyn/cm d α e κ , cm /s f Value 3 0.3 25 10-100 0.1 0.1 a Following published estimates. b We use r min discussed in the early work on FIN. c We use the dielectric permittivity of HfO . d Because the values of interfacial energies in materials undergoingFIN are not available, we use the ballpark of typical values for avariety of other systems. e See the discussion after Eq. (1) f Thermal diffusivity estimated or HfO based RRAM devices. III. ANOTHER SCENARIO OF FIN
The preceding section analysis tacitly assumed the po-larization energy of a metal needle remaining cubic inlength regardless of its closeness to opposite electrode.That assumption remains valid through almost the en-tire gap, since the electric field is strongly different fromthe uniform only in a small region ∼ r /h (cid:28) h gap aroundthe tip. However, there is another gap related effect signifi-cantly contributing to the free energy: capacitive inter-action between the CF tip and its opposite electrode.That interaction can be thought of as due to a flat platecapacitor formed by the tip’s end face and the oppositeelectrode, leading to the energy contribution, F C = − CU ≈ − εr U h gap (11)where C is the above mentioned capacitance approxi-mated with that of the flat plate capacitor, U is the volt-age between the electrodes. Note that assuming othergeometrical shapes of the tip lead to slightly differentresults, where, for example, r is replaced with theelectrode size in Eq. (11) for the case of hyperboloid ofrevolution. A comment is in order to explain the negative sign of F C in Eq. (11). In the process of changing the gap ca-pacitance ∆ C with its width h gap , we assume that the FIG. 3: The free energy of CF in a narrow gap between twoelectrodes. Symbols represent the results of the COMSOLnumerical modeling for 0.5 nm radius CF between the twocoaxial circular metal electrodes of 10 nm radius each. Thesolid line is a fit by Eq. (12). The dashed line is the free energycorresponding to the same surface tension and polarization,but without the capacitive interaction. system remains at constant voltage, so the capacitor en-ergy changes by U ∆ C/
2. However, simultaneously thecharge ∆ Q = U ∆ C passes through the power source,which requires the energy − U ∆ Q = − U ∆ C making thetotal energy change equal − U ∆ C/ F ( h, r ) = 2 πγrh − αh ε ( U/H ) − εr U H − h ) . (12)A major feature added here is a sharp energy decrease forrelatively small gap widths h gap (cid:28) H due to interactionbetween the CF tip and opposite electrode. We haveverified the above heuristic free energy form of Eq. (12)with COMSOL modeling for a variety of CF dimensions;one example is presented in Fig. 3 for a cylinder shapedCF of radius r = 0 . ∼
10 %of relative error.The free energy of Eq. (12) has a stationary pointat certain h and r that can be determined analytically.Here we skip the corresponding cumbersome equationsnoting that, for that point, D ≡ ∂ F/∂h ∂ F/∂r − ( ∂ F/∂h∂r ) > ∂ F/∂h < ∂ F/∂r <
0, whichidentifies it as a maximum.The case of a maximum stationary point in free en-ergy landscape is similar to that of the standard fieldinduced transformation scenario described in the preced-ing section: the phase transformation barrier is deter-mined by the minimum acceptable radius r min . Its re-lated gap width is given by dF ( h, r min ) /dh = 0 with F from Eq. (12). Because the polarization term [second onthe right hand side of Eq. (12)] is relatively unimportantfor H − h (cid:28) H , we get, H − h = H U U γ where U γ ≡ (cid:115) πγH εr min . (13)The corresponding barrier height is given by, W = W (1 − U/U γ ) , (14)leading to the nucleation time exponentially dependenton bias. We conclude that the above introduced capac-itive interaction acts as a sort of clutch decreasing CFenergy in a narrow interval of its heights close to the op-posite electrode. The corresponding nucleation barrierdecreases with voltage linearly unlike the standard FINdependence in Eq. (6).Note that the parameters from Table I, yield a nu-merical estimate of the characteristic voltage U γ ∼ h gap < ∼ τ T ∼ h /κ < ∼ . κ is the thermal diffusivity ensuring local quasi-equilibrium. The dissipation will increase the quasi-equilibrium temperature, thus accelerating the nucle-ation. On the other hand, the concomitant electric short-ing, would decrease the electrostatic energy of the en-tire device towards its final value corresponding to thefully formed CF. That decrease aggravates the free en-ergy falloff again helping nucleation.The quantitative description of the above mentionedtransient temperature increase and shorting effect fall be-yond the present scope. Therefore, we should admit adegree of uncertainty making the proposed FIN scenarioquestionable for small enough voltages. If that scenariodoes not work, the structures with nano-gaps can workonly as TS and not memory when their applied biasesare below U ms . The viability of ∼ IV. INTERPLAY OF TWO FIN SCENARIOS
The interplay between the standard and here devel-oped FIN models is described by two relations, U γ U s = √ α Hr min (15)and W new W stand = (cid:18) / (cid:19) U (1 − U/U γ ) U s (16) FIG. 4:
Left : A sketch of the FIN barrier voltage dependence,linear in low voltage region and hyperbolic for higher voltages.
Right : The filament free energies with (solid line) and without(dashed line) the tip electrode interaction [the last term in Eq.(12)] taken into account. where the nucleation barriers W stand and W new are givenrespectively by Eqs. (4) and (14). Comparing the bar-rier shapes in Figs. 3 and 2 we then conclude that thestandard FIN scenario dominates (i. e. W stand < W new )when U > U c ≡ / (3) / U s . That is the same range ofvoltages where the standard FIN theory predicts the volt-age dependence of τ in Eq. (6). We recall, in addition,that the transition between the two FIN scenarios hasa threshold nature at voltage U ms = E ms h gap [see Eq.(7)]. That is slightly above U c , since, based on their def-initions, U c = (2 / U ms . Such a behavior is illustratedin Fig. 4 (Left).It should be noted in addition that the criterion ofnucleation in the standard FIN scenario depends on thepost-nucleation CF evolution and conditions determiningthe desired lifetime of the initially metastable CF createdunder voltage above U ms . In general, the sufficient volt-age is determined by the condition that the nucleated CFsurvives the desired time as specified in Eq. (9). How-ever, the results presented in Sec. III show that the tip-electrode interaction can transform the metastable CFinto a stable one as illustrated in Fig. 4 (Right).The above consideration predicts that, given all otherfactors, the switching voltage should decrease when thegap H decreases. Indeed, using τ = τ exp( W/kT ) alongwith Eq. (14), the voltage capable of triggering nucle-ation over time τ becomes, U ( τ ) = (cid:20) H − kT πγr min ln (cid:18) ττ (cid:19)(cid:21) (cid:114) πγεr min , (17)It increases with H for any given exposure time. Thelatter prediction is consistent with the data showinghow the switching voltage in a formed structure is by afactor 2-4 lower than the forming voltage. V. ELECTRONIC SWITCHING
A number of recent publications theoreticallystudied purely switching (without any structural trans-formations) that might pertain to threshold switches op-erating as selector devices in modern solid state memoryarrays. Their underlying thermal runaway scenario uti-lizes a non-linear current voltage characteristic where theconductivity is thermally and field activated, σ = σ exp( − w/kT ) with w = w − δw ( E ) . (18)Here δw ( E ) is the field induced decrease in the activationenergy of conductivity. A local lateral variation of cur-rent will then generate excessive local heat and temper-ature additionally increasing the current density at thatlocation, etc.; hence, thermal electron instability evolv-ing into a narrow filament carrying high electric current.The formation of such a filament is identified as switch-ing.Because the underlying modeling remains numer-ical, here, we give a simple analytical treatment al-lowing to examine the corresponding parameter ranges.We start with the standard 3D heat transfer equation c∂T /∂t = χ ∇ T + σE = 0 where the thermal capaci-tance ( c ), thermal conductivity ( χ ), and electric conduc-tivity taking their respective values in the semiconductorand metal electrode materials forming a structure withaxial symmetry shown in Fig. 5.Given the ambient temperature T at the exter-nal surfaces and the Neumann boundary conditions atthe electrode-semiconductor interfaces, it was shown that averaging the semiconductor temperature along thetransversal direction (between the electrodes) reduces theheat transport equation to the form c s χ s ∂T∂t = ∇ T + β ( T − T ) + σE χ s , β ≡ χ e χ s HL . (19)Here T depends on the radial coordinate r , indexes s and e denote respectively the semiconductor and electrode FIG. 5: A sketch of a semiconductor layer between two metalelectrodes of thickness L each. The bottom view shows a hotspot caused by the runaway instability. FIG. 6: A sketch of the graphical solution of Eq. (21) wherethe straight line and the curve represent its left- and right-hand sides. materials. β is a reciprocal thermal length characteriz-ing the decay of radial nonuniformities. Note that thesecond and third terms on the right-hand-side representsrespectively heat dissipation and evolution.Eq. (19) along with Eq. (18) allows the standard lin-ear stability analysis constituting a natural approach tostudying run-away phenomena. Introducing the averagelateral temperature T and substituting δT ≡ T − T = (cid:88) q T q exp( iω q t − iqr ) ( (cid:28) T ) , (20)yields in the linear approximation,( T − T ) β χ s σ E = exp (cid:18) − wkT (cid:19) (21)and c s χ s iω q = − q − β + σ E χ s wkT exp (cid:18) − wkT (cid:19) . (22)The instability takes place for positive real values of iω .For the modes with q below a certain wave number q ,the latter criterion reduces to the form, λ ≡ T − T T wkT β β + q ≥ λ cr = 1 . (23)While derived differently, the inequality in Eq. (23) issimilar to the classical criterion λ ≥ λ cr ≈ .
88 for1D thermal instability with q = 0 and β = H/ T l ) corresponding to the uniform current flow, high( T h ) for the hot spot, and intermediate ( T unst ), which isunstable as seen from the relationship between the heatevolution and dissipation [cf. the note after Eq. (19)].That conclusion agrees with the general phenomenolog-ical analysis of thermal instabilities. Furthermore, thecondition for critical field (straight line tangential to theexponential curve in Fig. 6) self-consistently coincideswith the criterion in Eq. (23) when q = 0.The above simplistic analysis provides approximatenumerical results given the same parameters as in thepublished numerical modeling. For example, assum-ing values E = 10 V/m, σ = 10 S/m, χ s = 0 . − K − , H = 10 nm, L = 30 nm, χ e /χ s = 170, R = 60 nm, yields β ≈ m − , and w/kT =ln { σ E / [ β χ s ( T − T )] } ≈
7. Here we have rather ar-bitrarily approximated ( T − T ) = T under the loga-rithm, which has no significant effect when it appearswith a multiplier that is by many orders of magnitudebigger than ( T − T ). Substituting into Eq. (23) gives7( T − T ) /T = 1, i. e. the switching temperature T ≈ Furthermore, the estimated T ≈
340 K and w/kT = 7yield w ≈ .
21 eV, which must be interpreted as thebarrier for the above used E = 10 V/m. Any addi-tional assumptions about that barrier field dependenceare not necessary as long as w remains an adjustableparameter, which was the case for all the preceding nu-merical modeling carried out under the assumptionsof δw ( E ) following the Pool-Frenkel law. We would liketo emphasize here that the thermal runaway switchingwill take place for any model of thermally activated con-ductivity with or without field or voltage dependent bar-rier. In particular, fitting the data with Pool-Frenkel lawbased modeling does not appear indicative of that law.Of course, the switching field will depend on that barrierassumptions when the value of zero field barrier w ispostulated.Some additional observations are as follows.(i) Using the available data, Eq. (23) shows thatswitching takes place when the barrier is still signifi-cant, w/kT (cid:29)
1. With that in mind, one can approx-imate the required barrier suppression vs. gap width, δw ( E ) = const + kT ln H , which dependence may be tooweak to resolve experimentally.(ii) Unlike FIN, the runaway model per se does not pre-dict any delay time between voltage pulse and switch-ing. The observed delay was attributed in that modelto the thermalization time or left without explicitinterpretation. As long as the measured delay time isexponential in voltage, the only relevant mechanism isthe conductivity activation where τ ∝ exp[ − δw ( E ) /T ];hence, the prediction of delay time vs. voltage beingreciprocal of the voltage dependent conductivity, is openfor experimental verifications. Note that FIN predictionsfor τ ( U ) are quite different as specified in Eqs. (6) and(14). (iii) The above analysis will describe the finite area ef-fects when we set q = 2 π/R where R is the device ra-dius taking into account, along the general lines, thatthe strongly oscillating terms with q > π/R in theexpansion of Eq. (20) are immaterial. With that inmind, Eqs. (21) and (23) predict that switching requirestemperature increase ( T − T ) that are by the factor of1 + 4 π ( β R ) − greater than that for infinitely large de-vices. Since ( T − T ) ∝ E , we conclude that the switch-ing fields and potentials scale as (cid:112) π ( β R ) − withdevice size, which is at least qualitatively consistent withthe results of numerical modeling. That scaling be-comes practically important for nano-sized devices. Nosuch scaling is predicted by FIN where switching fieldremains independent of device size.The above (ii) and (iii) can be used to experimentallydecide between the switching mechanisms of FIN andthermal runaway. A general concern about the latterarises for the case of submicron and especially nanome-ter thick amorphous structures with transversal conduc-tion varying by many orders of magnitude between dif-ferent spots, which strongly affect the thermal runawaymechanism. VI. CONCLUSIONS
We have shown that,(1) FIN in narrow (shorter than the critical nucleationlength) gaps between two metal electrodes exhibits newtrends governed by certain characteristic voltages in theparctically interesting range.(2) There exists the characteristic voltage U ms indepen-dent of the gap width, below which the nucleation barrierdecreases with voltage linearly, while it is reciprocal ofvoltage above U ms , and the transition is of thresholdnature independent of the gap width.(3) For some materials, nucleation of CF can result infunctionality of only TS without memory.(4) Our analytical consideration of thermal runawayswitching provides results consistent with the earlierpublished numerical modeling and leads to severalpredictions offering experimental verifications decidingbetween the mechanisms of FIN and thermal runaway.This work was supported in part by the SemiconductorResearch Corporation (SRC) under Contract No. 2016-LM-2654. We are grateful to I. V. Karpov and R. Kotlyarfor useful discussions. ∗ Electronic address: [email protected] † Electronic address: [email protected] Phase Change Memory: Device Physics, Reliability andApplications , Edited by Andrea Redaelli (Editor), Springer2017. Resistive Switching: From Fundamentals of Nanoionic Re-dox Processes to Memristive Device Applications , Editedby Daniele Ielmini and Rainer Waser, Wiley-VCH, (2016). Z. Wang, M. Rao, R. Midya, S. Joshi, H. Jiang, P. Lin, W.Song, S. Asapu, Ye Zhuo, C. Li, H. Wu, Q. Xia, J. J. Yang,
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