APPLIED PHYSICS LETTERS 96, 163501 共2010兲
Shunting path formation in thin film structures M. Nardone, M. Simon,a兲 and V. G. Karpov Department of Physics and Astronomy, University of Toledo, Toledo, Ohio 43606, USA
共Received 25 February 2010; accepted 13 March 2010; published online 19 April 2010兲 We present a model for shunt formation in thin films containing small volume fractions of conductive components, below the critical volume fraction of percolation theory. We show that in this regime shunting is due to almost rectilinear conductive paths, which is beyond the percolation theory framework. The criteria of rectilinear paths shunting versus the percolation cluster scenario are established. The time and temperature dependence of shunting statistics is predicted with possible applications in phase change memory and thin oxides. © 2010 American Institute of Physics. 关doi:10.1063/1.3378813兴 Shunting 共i.e., local loss of transversal resistance兲 is omnipresent in thin film technology causing reliability issues in many applications. It is caused by conductive paths through a resistive film between two opposing electrodes. These paths are thought of as chains of electrically connected conductive particles created in a resistive host, in the course of device operations.1–6 Our primary example here is chalcogenide phase change memory 共PCM兲.7,8 It operates by applying appropriate voltage pulses to a chalcogenide that reversibly switches between its high resistive amorphous phase and low resistive crystalline phase. The former tends to spontaneously crystallize creating nuclei that can form conductive paths.1–3 Thin oxide films in metal oxide semiconductor field-effect transistors 共MOSFETs兲 represent another important example. In the course of operations, they accumulate point defects. A pathway consisting of such defects can be shunting between the gate and semiconductor material.5,6 Our third example refers to photovoltaics where shunting remains a major degradation mode.9,10 The usual approach to shunting employs the percolation concept1,2,6 modeling the device as a lattice with spacing 2R where R is the effective radius of a conductive particle 关Fig. 1共a兲兴. The conductive particles randomly occupy a fraction v of the lattice sites forming clusters, some of which can shunt through the film. For PCM, a typical value is R ⬃ 3 nm corresponding to the nucleation radius for the crystalline phase. For thin oxides, there is some debate as to the defect size5 but it is generally agreed upon to lie between 0.5–1.5 nm. The practically interesting film thicknesses are in the range from several nanometers to ⬃100 nm. The percolation theory 共see, e.g., Ref. 11兲 predicts shunting regardless of film thickness when v exceeds its threshold value vc⬁ ⬇ 0.3, above which the infinite percolation cluster is formed. However, the infinite percolation cluster is not exactly relevant here because its correlation radius11 Lc ⬃ 2R兩v − vc⬁兩− with ⬇ 0.9 will exceed the practical film thickness L when v is close to vc⬁. In the domain of Lc Ⰷ L, the percolation cluster backbone will play almost no role against the background of multiple finite clusters connecting the electrodes as illustrated in Fig. 1共b兲. The concept of percolation theory becomes applicable when v ⬎ vc⬁ + 共2R / L兲1/ ⬅ vc. a兲
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Here, we concentrate on the range of v ⬍ vc which is important because it determines the statistics of early failures 共before the percolation cluster is formed兲.3,4 We show that linear chains nearly perpendicular to the electrodes 关short arrows in Fig. 1共b兲兴, which are not part of the percolation cluster, are responsible for shunting in that range. We denote such chains as rectilinear pathways 共shunts兲. Prior work6 described the pre-threshold range by means of computer simulations. Our analytical approach here is more explicit explaining the nature of deviations from the percolation theory and predicting early failure statistics. To better introduce the concept, we start with a numerical experiment involving three-dimensional percolation on a simple cubic lattice. We considered a system of 10 by 10 by L lattice points where L is the device thickness in lattice units 2R 共corresponding to the particle diameter兲. The sites of the lattice were randomly populated with conducting particles up until a given volume fraction v. When the nearestneighbor of an occupied site is also occupied, they are considered to be connected and part of the same cluster. The condition for shunting is that there exists a cluster which spans the device 共i.e., shunting occurs when a cluster starts at the base of the system and ends at the opposing side a distance L away兲. For each parameter setting, 1000 trials were run. For each trial, it was determined whether a shunt forms and, if so, the number N of particles in the spanning cluster. Figure 2 reveals a maximum in the probabilistic distribution of N observed for not too small v ⬎ 0.1. As shown in Fig. 3共b兲, the most probable number of particles N0 in a shunting pathway was found to be linear in the film thickness 共L兲 implying that these pathways are almost linear and perpendicular to the
FIG. 1. 共Color online兲 共a兲 A fragment of the shunting path and 共b兲 the topology of the infinite percolation cluster with its conductive backbone. The thick dashed line represents the case of a very thin structure. Note multiple short rectilinear shunts that would not be effective for a thicker structure.
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FIG. 4. 共Color online兲 Two scenarios of shunting: 共a兲 percolation; and 共b兲 through a nearly rectilinear path.
FIG. 2. Numerically modeled statistics of the number of particles N in shunting paths for a 10⫻ 10⫻ 5 cube with v = 0.2. Spanning cluster formation occurred 143 times in 1000 trials. The curve represents the Poisson distribution.
electrodes 共for comparison, random walk pathways would have N0 ⬀ L2兲. For extremely small v ⬍ 0.1, we mostly observed exactly linear pathways of length L. Also, our modeling results in Fig. 3共a兲 showed the linear area dependence of the shunting probability. The above findings can be attributed to shunting by nearly rectilinear chains of particles 关Fig. 4共b兲兴 with a relatively insignificant degree of winding compared to the percolation cluster topology. The maximum in Fig. 2 can be understood as arising from two competing tendencies: 共1兲 larger clusters are more likely to be able to form a shunt and 共2兲 the probability of finding an N-particle cluster decreases with cluster size N. Next we give analytical description of such nearly rectilinear chains and their statistics. For a film of area A Ⰷ R2, the probability of forming a chain of N particles shunting through thickness L is given by p共N , L兲 = pN共L兲pv共N兲. Here pN共L兲 is the conditional probability of an N-particle chain to have projection of length L onto axis z perpendicular to the electrode; pv共N兲 is the probability to form an N-particle chain in a system where randomly distributed particles occupy volume fraction v. The total probability of shunting through the film is, P共L兲 =
A 共2R兲2
冕
⬁
pN共L兲pv共N兲dN,
共say, in the case of very large areas兲, it is to be interpreted as the number of shunting pathways formed. Chains of randomly dispersed conductive particles have the topology of a random walk with step size 2R. The probability distribution for such walks in a half-space to end at the point 共x , y , z兲 is given by,12 关9z/共32N2R4兲兴exp关− 3共x2 + y 2 + z2兲/8NR2兴. For large enough electrodes, integrating over x and y from −⬁ to ⬁ gives the normalized probability for the walk to end at a slab of thickness 2R a perpendicular distance L away from the starting electrode, pN共L兲 =
pv共N兲 =
where the multiplier A / 共2R兲 gives the number of possible shunting pathways starting at one electrode and the lower limit of integration is the minimum number of particles required to for shunting. If the P共L兲 in Eq. 共1兲 is larger than 1
FIG. 3. 共Color online兲 共a兲 Modeling results for the area dependence of the shunting probability. 共b兲 Numerical simulation of the typical number of sites in the shunt 共N0兲 in 10⫻ 10⫻ L unit systems for various volume fractions v. Note the strong increase in probability for v = 0.3 reflecting the beginning of percolation cluster formation.
共2兲
¯ N exp共− N ¯ 兲 exp共− ⌫N兲 N ⬇ 冑2N , N!
共3兲
where ⌫ ⬅ −ln v + v − 1. In the composite probability p共N , L兲 = pN共L兲pv共N兲, the two multipliers exhibit opposite trends favoring smaller and larger N’s, respectively. As a result, it is a maximum at a particular number of particles N0 =
2
冊
Another multiplier pv共N兲 in the integrand of Eq. 共1兲 is given by the Poisson distribution with the average number of par¯ = Nv in the volume of an N-particle chain. Using ticles N Stirling’s approximation, one gets
共1兲
L/2R
冉
3L 3L2 exp − . 2NR 8NR2
L 2R
冑
3 , 2⌫
共4兲
obtained by a straightforward optimization; this explains the results of numerical modeling in Fig. 2. Equation 共4兲 is only valid for ⌫ ⬍ 3 / 2 and its corresponding v ⬎ v0 ⬇ 0.09, since N0 must always be greater than the minimum number of particles L / 2R spanning the system. For lower volume fractions, one has to use N0 = L / 2R for the most likely number of particles in a shunting chain. For volume fractions in the range v0 ⬍ v ⬍ vc, the most probable number of particles in the connecting chain is between L / 2R ⬍ N0 ⬍ 1.5L / 2R. Hence, shunting is due to nearly rectilinear paths, in agreement with the results of the above numerical modeling. Depending on where the peak of the composite probability lies with respect to the lower limit of integration, different approximations to the integral may be used. For v ⬍ 0.1 the integral is best approximated by the integrand evaluated at N = L / 2R, while for 0.1⬍ v ⬍ 0.3 the method of steepest descent may be used which gives
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冦
Appl. Phys. Lett. 96, 163501 共2010兲
Nardone, Simon, and Karpov
3A 4R2
冑
冋 冉 冊册 冉 冑 冊
R L 3 exp − +⌫ L 2R 2
A 冑3 L 2 exp − 8R R
3⌫ 2
共 v ⬍ v 0兲
冧
共v0 ⬍ v ⬍ vc兲. 共5兲
Setting P共L兲 = 1 gives the thickness for which shunting will occur with certainty for a particular area and volume fraction. By then setting v = vc we define the critical thickness, Lc = R
冑
冉 冊
A冑3 2 ln , 3⌫共vc兲 8R2
共6兲
such that systems with thickness L ⬍ Lc are typically shunted by rectilinear paths, while the percolation cluster scenario dominates for L ⬎ Lc. Similarly, a critical area can be defined such that for a given thickness, systems with A ⬎ Ac are typically shunted by rectilinear paths, while the percolation cluster scenario dominates for A ⬍ Ac, Ac =
8R2
冑3
冋冑 册
exp
L R
3⌫共vc兲 . 2
共7兲
As a numerical example, for a PCM cell with R ⬃ 3 nm and A ⬃ 2500 nm2, we have Lc ⬃ 14 nm. By setting rather arbitrarily L ⬃ 45 nm 共within the range of current PCM thicknesses兲, we find Ac ⬃ 19 m2. For technology applications, it is important that Eq. 共5兲 can describe the temporal dependence of the shunting probability through the time dependent volume fraction v共t兲. For example, if conductive particles nucleate at a fixed rate, then the volume fraction will be linear with time, v = at. From Eqs. 共3兲 and 共5兲, when v0 ⬍ at ⬍ vc we then have,
冋 冑
P共L兲 ⬀ exp −
L R
册
3 共− 1 − ln at兲 . 2
共8兲
As another example, consider a disordered system such as chalcogenide glass, where local nucleation barriers are uniformly distributed on some fixed energy interval ⌬W. This leads to an exponential dispersion in nucleation times and a volume fraction which is logarithmic13 in time, v = 共kT / ⌬W兲ln共t / 兲, where corresponds to the shortest nucleation time in the system. For v0 ⬍ v ⬍ vc, this yields
冋 冑冉
P共L兲 ⬀ exp −
L R
3 ⌬W − 1 + ln 2 kT ln t/
冊册
.
共9兲
The temporal dependencies of Eqs. 共8兲 and 共9兲 are plotted in Fig. 5 in the Weibull coordinates commonly used in reliability literature. The plots were generated for a system with dimensions L / 2R = 15 and A / 共2R兲2 = 225 using the experimentally determined nucleation rate from Ref. 14. For the case of random barriers, we used the induction times from Ref. 14 and supplemented it with the barrier dispersion ⌬W = 0.6 eV assumed rather arbitrarily to reflect the condition ⌬W Ⰷ kT. An observation of practical interest here is that the predictions of the random barrier model are closely approximated by Weibull law 共straight lines兲 in an exponen-
FIG. 5. 共Color online兲 Weibull plots of failure statistics for the cases of fixed 共䊐兲 and random 共䊊兲 nucleation barriers for various temperatures. Straight lines show linear approximation 共Weibull law兲. Here log represents the base 10 logarithm.
tially broad time interval; this observation remains in both the domains of v0 ⬍ v ⬍ vc and v ⬍ v0. In conclusion, when the volume fraction is below its critical value of v ⬃ 0.3, the topology of conductive crystalline clusters in PCM changes from that of the percolation cluster to almost rectilinear paths. Our model predicts the probability of such paths, which can determine the statistics of earlier failures in data retention. In a broader sense, our model shows that shunting by rectilinear paths can dominate in thin-film structures of practically interesting thickness. The predicted failure statistics are closely approximated by the Weibull distribution in an exponentially broad time interval. The authors acknowledge the Intel grant supporting our research on phase change memory. Useful discussions with I. V. Karpov are greatly appreciated. 1
U. Russo, D. Ielmini, and A. L. Lacaita, IEEE Int. Reliab. Phys. Symp. Proc 2007, 547. 2 U. Russo, D. Ielmini, A. Redaelli, and A. L. Lacaita, IEEE Trans. Electron Devices 53, 3032 共2006兲. 3 B. Gleixner, A. Pirovano, J. Sarkar, F. Ottogalli, E. Tortorelli, M. Tosi, and R. Bez, IEEE Int. Reliab. Phys. Symp. Proc 2007, 542. 4 D. Mantegazza, D. Ielmini, A. Pirovano, and A. L. Lacaita, IEEE Electron Device Lett. 28, 865 共2007兲. 5 S. Lombardo, J. H. Stathis, B. P. Linder, K. L. Pey, F. Palumbo, and C. H. Tung, J. Appl. Phys. 98, 121301 共2005兲. 6 J. H. Stathis, J. Appl. Phys. 86, 5757 共1999兲. 7 R. Bez, Tech. Dig. - Int. Electron Devices Meet. 2009, 89. 8 D. Kau, S. Tang, and I. Karpov, Tech. Dig. - Int. Electron Devices Meet. 2009, 617. 9 S. H. Demtsu, D. S. Albin, J. W. Pankow, and A. Davies, Sol. Energy Mater. Sol. Cells 90, 2934 共2006兲. 10 T. J. McMahon, T. J. Berniard, and D. S. Albin, J. Appl. Phys. 97, 054503 共2005兲. 11 A. L. Efros and B. I. Shklovskii, Electronic Properties of Doped Semiconductors 共Springer, Berlin, 1979兲. 12 M. Slutsky, Am. J. Phys. 73, 308 共2005兲. 13 V. G. Karpov, Y. A. Kryukov, I. V. Karpov, and M. Mitra, J. Appl. Phys. 104, 054507 共2008兲; I. V. Karpov, M. Mitra, G. Spadini, U. Kau, Y. A. Kryukov, and V. G. Karpov, ibid. 102, 124503 共2007兲. 14 J. A. Kalb, C. Y. Wen, and F. Spaepen, J. Appl. Phys. 98, 054902 共2005兲.
