Open Porosity

If Ps is trapped in pores that are big enough and interconnected with each other, it may be able to diffuse over long distances within the interconnected porous network. ( Ps with a room temperature thermal velocity of 8 X 10⁶ cm/sec makes some 1-2 million collisions per lifetime in pores roughly 5-10nm in diameter and therefore could diffuse several microns/lifetime) The travel distance of Ps can be greater than the porous film thickness ( usually less than 1 micron in low-k films). As a result, Ps can easily diffuse out of the film and into the surrounding vacuum, as depicted in the upper panel of Figure 1. The observable effect on the Ps lifetime is that most of the Ps annihilates with the vacuum lifetime of 142 ns, a telltale indicator that the pores in the film are interconnected.

An examples of lifetime spectrum acquired with a porous silica film is shown in Figure 2. There is a prompt peak characterized by positron and Ps lifetimes less than 0.5 ns (typical of annihilation in nonporous, bulk samples) and a long lifetime component indicative of copious (35 - 40%) Ps formation in the pores. The fitted value of t for this long component in the as-received film is 140 ns, suspiciously indistinguishable from the vacuum value. It indicates that the pores are so highly interconnected that mobile Ps is able to diffuse out of the film and into the surrounding vacuum system.

It is necessary to deposit a thin capping layer on top of the film to effectively keep the Ps corralled in the porous film, as shown in the lower panel of Figure1. We capped the porous silica film with 80 nm of Al (other materials, such as silica, TEOS, silicon nitride can be used as well). Since Ps samples the entire interconnected pore network, a single lifetime, t, is fitted to be 41 ns (Figure 2). Using our rectangular extension to the Tao-Eldrup model, the lifetime can be converted into a pore size, or more specifically, since the geometry of the pores is not known and may be somewhat ill-defined for open interconnected porosity, a mean free path, l(=4V/S, V: volume, S: surface area). The mean free path corresponding to 41 ns Ps lifetime is 2.4 nm.

Figure 1. Ps behavior in open pores. A capping layer is required to get the correct Ps lifetime in the interconncted pores.

Figure 2. Ps lifetime distribution in the porous MSSQ film

Summary:

Ps can readily diffuse into vacuum through the open channels and a high intensity of Ps decaying in vacuum can be observed and used as an indicator of pore interconnectivity.
A capping layer is required to keep Ps from escaping into the vacuum, and a single Ps lifetime will then be fitted that corresponds to the average Ps mean free path or pore size throughout the film thickness.

References:

D. W. Gidley, W. E. Frieze, T. L. Dull, A. F. Yee, E. T. Ryan and H. M. Ho, Positronium annihilation in mesoporous thin films, Physical Review B, 60, 8, R5157 (1999)
D. W. Gidley, T. L. Dull, W. E. Frieze, J. N. Sun and A. F. Yee, Probing Pore Characteristics in Low-K Thin Films Using Positronium Annihilation Lifetime Spectroscopy, Materials Research Society Symposium Proceeding, 612, D4.3.1 (2000)