With the advent of porous dielectrics, Cu drift-diffusion reliability issues in CMOS backend have only been exacerbated. In this regard, a modeling and simulation study of Cu atom/ion drift-diffusion in porous dielectrics is presented to assess the backend reliability and to explore conditions for a reliable Resistive Random Access Memory (RRAM) operation. The numerical computation, using elementary jump frequencies for a random walk in 2D and 3D, is based on an extended adjacency tensor concept. It is shown that Cu diffusion and drift transport are affected as much by the level of porosity as by the pore morphology. Allowance is made for different rates of Cu dissolution into the dielectric and for Cu absorption and transport at and on the inner walls of the pores. Most of the complex phenomena of the drift-diffusion transport in porous media can be understood in terms of local lateral and vertical gradients and the degree of their perturbation caused by the presence of pores in the transport domain. The impact of pore morphology, related to the concept of tortuosity, is discussed in terms of “channeling” and “trapping” effects. The simulations are calibrated to experimental results of porous SiCOH layers of 25 nm thickness, sandwiched between Cu and Pt(W) electrodes with experimental porosity levels of 0%, 8%, 12%, and 25%. We find that porous SICOH is more immune to Cu^{+} drift at 300 K than non-porous SICOH.

The geometric complexity of porous materials is a long-standing challenge in efficiently describing drift-diffusion transport phenomena. Porous media can be found in many diverse applications ranging from fuel cells, energy technologies, batteries, heterogeneous catalysis to geology and oil industry.^{1} In the CMOS technology, integration of nanometer-sized pores into low-k injection Interlayer Dielectric (ILD) films is one of the approaches to lower the capacitance further and by the same token the RC signal delay and thus to help sustain the continued scaling of micro-electronic devices.^{2,3} At the same time, the porosity of the dielectrics offers an additional parameter to optimize Resistive Random Access Memory (RRAM) performance. While increasing porosity of porous dielectrics achieves the desirable lowering of the dielectric constant (k), it also creates a myriad of reliability and implementation issues. On the modeling side, the drift diffusion problems have been usually addressed by using drift-diffusion continuum differential equations using the mobility concept with appropriate boundary conditions.^{4,5} However, analysis, of the interconnect samples manufactured at Intel Research Labs, shows that such an approach is grossly inadequate: (i) In non-porous low-k dielectrics such as SiCOH, the elementary diffusional jump distance λ_{o} has been estimated to be 1 nm–2 nm.^{6–8} (ii) A typical thickness of dielectric layers in the most advanced CMOS interconnect modules is about 20-25 nm. Thus, the thickness is 10–15 times the elementary displacement λ_{o}. (iii) In porous dielectrics, a typical compact pore size is a few nanometers^{9} and only slightly larger than the elementary diffusional jump length λ_{o}. (iv) Our analysis of the breakdown of the Intel samples demonstrates that the fields responsible for Cu filament formation at 300 K are around a few of 10^{6} V/cm. The mobility concept is valid when electric fields are smaller than kT/(qλ_{o}) (at 27 °C and λ_{o} = 1.5 nm is ∼2 × 10^{5} V/cm), thus significantly smaller than the fields responsible for dielectric failure. Hence, it is mandatory to evaluate ion drift-diffusion phenomena using an atomistic approach with elementary jump probabilities. Upon application of an electric field E_{el}, the energy activation E_{a} barrier of the jump frequency along the field direction is lowered, whereas against the electric field, it is enhanced.

The Metal–Insulator–Metal (MIM) structures investigated in this study were manufactured with a Cu electrode at the bottom and an island-shaped Pt or W counter-electrode at the top, with SICOH of four levels of porosity, ρ = 0%, 8%, 12%, and 25%, sandwiched in between as shown in Fig. 1(a). The steps involved in the fabrication process are described elsewhere,^{10–15} and the electrical characterization of these devices is discussed in detail in Ref. 16. The pore size distribution ranges from 2 nm to 8 nm.^{12–14} Electrical characterization has shown that at 27 °C, the critical voltage for forming Cu filaments increased with increased porosity.^{16} The conductance due to Cu diffusion was found to be highest for ρ = 0% porosity. From the data for the sample with SiCOH 25 nm thick and ρ = 0%, we can estimate the effective Cu diffusion according to the equation $Deff\u2248d22\tau o.$ For d = 25 nm and t_{0} = 4 min = 240 s, D_{eff} = 1.3 × 10^{−14} cm^{2}/s at 400 °C. It has been reported^{7} that Cu diffusion in methyl-doped SiO_{2} (a variant of SiCOH) is $D=6.1\xd710\u22129\u2009exp\u22120.71\u2009eVkT\u2009cm2/s$. This diffusion coefficient describes well also Cu diffusion for our samples. Using the equation for D_{eff}, we calibrate our atomistic simulations as follows: $\tau o=\lambda o22DoexpEakT$ using λ_{0} = 2.1 nm,^{8} D_{o} = 5 × 10^{−9} cm^{2}/s, and E_{a} = 0.71 eV, and we obtain τ_{0} = 3.2 × 10^{6} s, 13.7 s, 0.88 s, at 27 °C, 275 °C, and 400 °C, respectively. The assignment of λ_{o} and τ_{o} calibrates our random walk diffusion simulations with experiments at a given temperature.

Our method to evaluate a random walk in one dimension is based on the adjacency matrix concept. The adjacency matrix A is defined as an n × n matrix of which elements a(i, j) are 0 if the corresponding nodes are not connected with each other and 1 if they are connected. In 1-D, the probability W(n, M) that a particle, initially at node 0, arrives at node n after M jumps, which can be derived by repeated matrix multiplication assuming that only jumps to the nearest neighbors are non-vanishing [i.e., a(i, i − 1) = a(i, i + 1) = 1 otherwise zero], is identical with that of statistical random walk theory.^{17} The adjacency matrix approach can be extended^{18} to 2D(3D) and results in the 4th (6th) rank operation defined in the following equation:

The tensor element τ^{N}(i, j; n, m) after N jumps is the number of total paths between locations (i, j) and (n, m). In the case of diffusion, four elementary jump probabilities of 0.25 are assigned to the bulk nodes, three elementary jump probabilities of 0.25 to the surface nodes, and a fourth of 0.25 for stay; two jump probabilities are assigned to the corner nodes of the simulation domain plus 0.5 for staying at the node. An example of how the elementary individual jump probabilities can be assigned is shown in Fig. 1(d). The pores are approximated by squares of multiples of elementary lattice cells, to avoid nonessential modeling overhead. Here, we focus on the Cu atom transfer from the Cu electrode across the porous dielectrics to the counter-electrode as indicated in Fig. 1(a). A measure for the effective diffusivity of Cu across the porous dielectrics is a predetermined number of Cu atoms to arrive at the W/Pt-electrode assuming that Cu atoms are immobilized there. The effective diffusivity extracted from our simulations is as follows: $Deff=d22\tau oN,$ where d(=25 nm) is the thickness of the dielectric between Cu and W electrodes, *τ*_{o} is the time for one elementary diffusion jump at a given temperature, and N is the number of such jumps to transport a predetermined number of Cu atoms across the dielectric, which is a result of the simulation. The number of diffusional jumps N depends on the ambient temperature and the properties of the dielectric, i.e., on the porosity level and on the pore morphology. It should be stressed that the diffusion simulations describe the Cu transport to the Pt electrode in the initial phase of Cu nucleation on the Pt interface. The formation of the Cu filament would require a simulation of drift coupled with the Poisson equation which is beyond the scope of this work.

The jump frequency of an elementary diffusional jump over the elementary jump distance λ is given by $\upsilon =\upsilon o\u2009exp(\u2212EakT)$, where E_{a} is the activation energy, k is the Boltzmann constant, and T is the temperature in Kelvin. In the case of diffusion, the jump frequencies in the matrix material are the same in all directions. In the case of an applied electric field E, however, the effective barrier for the jump in the direction of the applied field is lowered from E_{a} to E_{a} – qλ_{0}E, and for the jump opposite to the direction of the field, it is increased to E_{a} + qλ_{0}E. Thus, we have the jump frequency in the direction of the field given by ν_{+} = $v0\u2061exp\u2212EakT\u22c5exp+q\lambda oE2kT$ and in the opposite direction $v\u2212=v0\u2009exp\u2212EakT\u22c5exp\u2212q\lambda oE2kT$, where the simplifying assumption has been made that the jump frequencies to the sites perpendicular to the electric field are unaffected by the electric field including any not local electric fields. We also assume that once the Cu atoms/ions arrive at the Pt electrode, they are immobilized due to strong whetting effects between Cu and Pt. From the elementary jump frequencies υ_{+} and υ_{_}, drift velocity, v_{drift}, can be calculated as $\upsilon drift=2\lambda v0\u2009exp\u2212EakT\u22c5sinhq\lambda oE2kT$. As described elsewhere,^{11} in our samples, the set voltage for conductive filaments lies between 1 V and 2 V. So the average electric field in our samples under set operation is 1.5 V/(25 nm) = 6.0 × 10^{5} V/cm. However, during the Cu filament formation, the electric field across the closing gap between the Cu electrode and growing filament will be significantly larger. In cases where E ≪ kT/qλ_{o}, the relation for v_{drift} can be simplified to the well-known mobility model with μ = exp(−E_{a}/kT) × qλ_{o}^{2}ν_{o}/kT = q/kT × D_{eff}. However, kT/qλ_{o} = 1.21 × 10^{5} V/cm at room temperature is much smaller than the realistically applied electric fields including those that may lead to conductive filament formation.^{11} At low temperatures and sufficiently large fields, all extant diffusive jump frequencies can be neglected relative to the field-enhanced jump frequency v_{+}. However, at elevated temperatures, such a reduction to pure drift is no longer justified. At 400 °C and E = 7.5 × 10^{5} V/cm, the field enhancement over the diffusion becomes low. Hence, at elevated temperatures, both mechanisms of diffusion and drift are bound to contribute to the Cu transport. In our characterization, the voltage is ramped from 0 to 2 V at a ramp rate from 0.2 V/s to 2.0 V/s, so it takes at most 10 s to reach 2 V. Since our samples are 25 nm thick and λ = 2.1 nm, it takes at most 11 elementary jumps at a constant field of 3 × 10^{6} V/cm to cross the dielectric for a total time of 10 × 5 s = 50 s. It would take 10 × 1.2 × 10^{4} s = 2000 min at 1 × 10^{6} V/cm. However, the assumption of an average electric field over the entire dielectric thickness does not capture the dynamics of the filament formation. As the partial filament begins to grow, the electric fields between the tip of the asperity and counter-electrode will vary between 10^{5} and 10^{7} V/cm = 2 V/(2 nm), where the Cu atom size is about 2 nm. The enhancement factor $exp+q\lambda oE2kT$ at 300 K for E = 10^{7} V/cm is immense: 7.2 × 10^{17}, meaning that at these fields the filament forming would be instantaneous, on a time scale of 10^{−13} s. We conclude that the electric field responsible for the filament formation in the final stage of the filament formation is (3-4) × 10^{6} V/cm at 300 K. This critical field is reduced at higher temperatures. Similar calculations show that the critical fields for filament formation are reduced to (1-2) × 10^{6} V/cm at 275 °C and to just below 10^{6} V/cm at 400 °C. The CMOS back-end temperature can reach 120 °C^{19} during microprocessor operation. Elevated temperatures can trigger many reliability issues including filament formation, which would result in potential shorts between the metallization lines in CMOS Back End of Line (BEOL).

In our simulations, we consider different levels of porosity and different pore morphologies. Figure 1(e) shows different levels of porosity for the same type of pore morphology. Figure 1(f) shows different pore morphologies for the same level of porosity (25%). In Fig. 1(g), it can be seen that the effective diffusivity decreases with porosity ρ as (1 − ρ), e.g., at ρ = 46%, we obtain D_{eff}/D_{plain} = 0.52 compared with 1 − ρ = 0.54. This is in good agreement with the Bruggeman relation^{1,20} given by D_{eff}(ρ, τ_{por}) = D(ρ = 0) × (1 − ρ)/τ_{por}, where τ_{por} is the tortuosity defined conceptually as τ_{por} = (effective average path length)/(shortest path between the electrodes). We note that there is no established analytical algorithm for calculating tortuosity and usually, tortuosity is extracted from diffusion simulations for the same porosity but different morphologies. In Fig. 1(h), diffusion coefficients are compared for the same ρ = const = 25% but for different morphologies. The diffusivity in the vertically oriented pores is very similar to the non-porous case S_{PLAIN}. The lowest diffusivity is observed for laterally elongated pores of the pattern S_{4}. It can be seen that the diffusivities correlate very well with the number of direct unobstructed vertical channels available to the diffusive transport; hence, the smaller the number of direct vertical channels, the smaller the diffusivity.

The strong impact of tortuosity, seen in Fig. 1(h), can be understood in terms of the roles of the vertical and lateral gradient components. The lateral and vertical gradients are visualized for the case of a virtual sample structure, shown in Fig. 1(b) consisting of a tree-shaped matrix material and in Fig. 1(c) where Cu is diffusing from the top Cu electrode through the tree-shaped matrix material defined by two rows of laterally elongated pores. In the non-porous case, there are only the vertical gradients and all horizontal gradients are zero reflecting the lateral uniformity of the diffusion. The observed retardation of diffusion in porous media can be understood as a reduction or elimination of the vertical gradients. However, the disruption of vertical gradients generates lateral gradients. The lateral gradients, in turn, allow diffusing atoms to circumnavigate the pores and thus mitigate the retardation effects due to the disrupted vertical gradients. In the case of pattern S_{1}, vertical gradients are intact along the columnar pores. Only at the top electrode, slight contribution of lateral diffusion is required to deflect diffusing atoms into the matrix material channels between the pores. As a result, the diffusivity is only slightly smaller than the non-porous sample even though the porosity is quite high, ρ = 26%. The diffusivities for pore morphologies of S_{2} and S_{3} are very close to each other and lower than S_{1} and S_{PLAIN} as they display a comparable degree of perturbation to the local vertical gradients. On these grounds, the diffusivity for S_{3} can be expected to be slightly lower than for S_{2} because the continuity of vertical gradients is more disrupted in S_{3} than in S_{2}. In the S_{2} morphology, the alignment of pores along the vertical direction leads to a rapid unobstructed transport along the vertical direction, which may be called a “channeling effect.” We find the largest retardation of the diffusivity for the S_{4} morphology. This is so since the horizontally oriented columnar pores pose a very effective barrier to diffusion. The disruption of the continuity of the vertical gradients here is severe. Only two “channels” on both sides of the domain allow for a direct Cu transport to the bottom electrode. Moreover, the matrix material between the horizontal pores constitutes traps for Cu atoms; once strayed into this region, the atoms can diffuse for a long time inside the matrix material between two elongated pores (trapping effect). The trapping effect is investigated further in simulations of the laterally elongated columnar pores of porosity between 12% and 87% shown in Fig. 1(i). A finite and infinite supply of Cu atoms at the top surface has been considered. The highest retardation is reached at an intermediate porosity of ρ = 45%. Going from ρ = 0% to ρ = 45%, the vertically open channels are not affected, but the trapping effect in the lateral matrix material between the columnar pores is increased. However, beyond ρ = 45%, trapping decreases, e.g., in the case of 87% where columnar pores have been replaced by one single large pore, thus eliminating lateral trapping spaces.

Figure 1(j) shows generic results for pure drift and pure diffusion in terms of the number of elementary time steps as a function of porosity for uniformly distributed compact pores. Two y-axes are shown: for diffusion, the y axis indicates a number of diffusion steps to collect 0.3% of surface concentration of Cu, and for drift, it indicates the number of jumps to collect 100% of Cu surface atoms (note that in the simulation surface, after N = d/λ jumps all the Cu atoms arrive at the same time, and after N − 1 jumps, not a single Cu atom would arrive in the case of drift). While the number of jumps for diffusion increases roughly linearly with porosity between ρ = 0% and ρ = 46%, the transport time for drift increases exponentially and reaches an infinite value for ρ = 46%. At this porosity level, there is not a single vertical path for Cu^{+} ions to reach the counter-electrode. Figure 1(k) shows the drift times for porosities from 0% to 47% for uniformly distributed square pores [S_{2} pattern shown in Fig. 1(e)] as a function of temperature. The dependence of the transport time as a function of temperature is similar for all porosity levels considered. It is seen that for 0% and 12% porosity, times are much smaller than for 25% and 46% porosity. This can be attributed to the presence of significantly more unobstructed vertical paths in 0% and 12% porosity cases than for 25% and 46% porosity cases. The difference between transport times for ρ = 0% and ρ = 46% decreases with increasing temperature: at 27 °C, the difference is more than two orders of magnitude, whereas at 400 °C, the difference is only factor 2-3. At elevated temperatures, the lateral jump frequencies due to diffusion help circumnavigate the pores to find new vertical channel paths along which drift accelerates the Cu transport.

At high enough temperatures and low enough fields for porosities up to 50% for uniformly distributed cubic pores, the transport time becomes independent of porosity. The pure drift transport time at 27 °C and ρ = 46% was infinite as seen in Fig. 1(j), but at elevated temperatures, the drift transport time becomes finite due to the assistance of lateral diffusion activated at higher temperatures. The drift enhancement factor for the forward jump probability in the direction of the electric field is strongly reduced with increasing temperature. The strongest reduction of the forward jump probability occurs at high fields and high temperatures. From this, one can conclude that at 300 K, the reliability of highly porous dielectrics with respect to Cu punch-through should be much better than for a non-porous material. However, at elevated temperatures, the diffusion component allows to circumnavigate the pore and the abovementioned reliability is bound to deteriorate. At 400 °C, there is basically no difference between non-porous and porous dielectrics even at ρ = 46%.

In conclusion, an algebraic atomistic method for simulating diffusion and drift phenomena at a high electric field in porous dielectrics using the elementary jump frequencies has been used to assess the reliability of porous CMOS BEOL dielectrics and the opportunities for filament formation in MIM structures. The simulations have been calibrated to experiments performed on porous SiCOH dielectrics with porosity ranging from 0% to 25%. It was found that at 27 °C and for uniformly distributed square pores the drift comes to a halt at 46% porosity. However, at elevated temperatures, the lateral diffusion component triggers an onset of the drift transport that may pose a reliability risk. In other words, Cu contamination of the interlayer dielectric for porous and non-porous dielectrics should be comparable at temperatures above 400 °C. This in agreement with the experimental findings in the literature^{4} showing that increasing porosity reduces the diffusivity providing thus a larger reliability margin in terms of Cu contamination.