Stress and strain in polycrystalline thin films

Abstract

Polycrystalline thin films on substrates usually are in a “stressed” state. In the present paper, recent results on stress in polycrystalline films will be reviewed.

For Cr and CrN films, it has been shown that the stress is not uniform over the thickness of the film. High tensile stresses are observed near the substrate-film interface. Lower tensile stresses are observed further away from the interface. Moreover, it has been shown that the tensile stress is generated at the grain boundaries. In the case for which the deposition of the film is accompanied by an ion bombardment, a compressive stress is generated. The tensile and compressive stresses in these films are independent and additive. This description, however, does not hold for all high melting point films, notably not for TiN.

For c-BN films, the challenge is to form these metastable films and at the same time control the compressive stress within an acceptable range. Here some promising results have been obtained.

For low melting point films, some new deposition experiments have been performed and new theories have been formulated. Here the situation is still less clear than with the high melting point films.

Introduction

When today one searches for “stress AND (thin films OR coatings)” on the web of science, one gets over ten thousand hits. This huge amount of papers demonstrates the interest in stresses in thin films and coatings. These extensive search results do not include a reference to the pioneering work by Stoney, who in his 1909 paper derived the archetype of the equation that now goes by his name [1]. Some work on evaporated iron films was already done in the 1950s by Finegan and Hoffman [2] but the field really took off in the 1970s. In 1970, Francois d'Heurle suggested that the compressive stresses measured in sputter deposited Mo films might be due to a “shot-peening” action of the depositing atoms. This effect is now generally accepted and goes by the name “ion-peening” (Ref. [3]).

In 1968, Klokholm and Berry proposed constrained shrinkage as an explanation for the measured tensile stresses in evaporated metal films [4]. In 1972, Doljack and Hoffman published their paper on tensile stress in evaporated Ni films [5]. In that paper, they argue that tensile stress is generated at the grain boundaries. For Cr- and CrN-films, we now have evidence that the Doljack and Hoffman explanation is appropriate, see section “Stress in zone T films”. For WC–W multilayers, constrained shrinkage may very well be the appropriate description in view of the very high apparent interfacial stresses (see section “Stress in multilayers”).

In 1978, Abermann published the first in-situ stress measurements on a growing film [6]. These measurements, made during evaporation of Ag, show a complicated compressive–tensile–compressive–tensile (CTCT) behavior in the development of stress with thickness. In Fig. 1 a result by Abermann et al. is reproduced. This result, today, still lacks a satisfactory explanation.

Various aspects of the field have been reviewed by Hoffman in 1976, Doerner and Nix in 1988, Windischmann in 1992, Koch in 1994, and Spaepen in 2000 (Refs. [7], [8], [9], [10], [11]). In the present review, I shall limit myself to polycrystalline films that have no epitaxial relation to the underlying substrate.

Films or coatings on a substrate are usually in a stressed state. The film “wants” to be smaller or larger than the substrate allows it to be, hence the film is in tensile stress (film “wants” to shrink) or compressive stress (film “wants” to expand). The unit of stress is Pascal [Pa]. Stresses in thin films typically range from minus a few GPa to plus a few GPa. Tensile stresses are designated by a positive sign, compressive stresses by a negative sign.

The trivial reason for a film to be in a stressed state is deposition at a different (higher) temperature than the temperature at which the stress is measured. Due to a difference in thermal expansion coefficients between the film and substrate, the film is in a stressed state. This dependency in differential form is [8]:

$$\frac{d{\sigma }_{\text{f}}}{dT}=({\alpha }_{\text{s}}-{\alpha }_{\text{f}})\left(\frac{E_{\text{f}}}{1-v_{\text{f}}}\right)$$

with σ_f the stress in the film, T the temperature, α_s the coefficient of linear thermal expansion of the substrate, α_f the coefficient of linear thermal expansion of the film, and E_f / (1 − ν_f) the biaxial modulus of the film. In addition to this thermal stress, an intrinsic stress may also be observed. The present review is entirely concerned with this intrinsic stress. The stresses observed during the thermal cycling of thin films on silicon wafers are not considered in this paper. These stresses are the combined effect of Eq. (1) and temperature dependent stress relaxation mechanisms. A starting point for the study of stress during thermal cycling is the review by Doerner and Nix [8].

Large stresses have to be avoided for many applications. Large tensile stresses may lead to cracking of the film. Large compressive stresses may lead to buckling [12], [13], [14]. Conventional wisdom is to strive for a compressive stress of a few hundred MPa. This can be understood based upon results from the widely applied “scratch test” for protective coatings. In this test, a diamond stylus is pushed into the coated material with a prescribed force and subsequently moved laterally [15]. In the wake of the moving indenter, a tensile stress occurs leading to film cracking. In the case for which the film was under compressive stress prior to the “scratch testing”, a higher load can be applied before the resulting stress becomes tensile and the film fails [16].

We have knowledge of stress in thin films in two ways: (1) by directly measuring the crystal lattice strain in the film using X-ray diffraction [17] and (2) by measuring the elastic deformation of the substrate, usually a single crystal silicon wafer. The wafer becomes curved due to the stress in the film [18]. Since we know the elastic properties of the wafer, we can calculate the stress in the film. Mechanical equilibrium requires the net force (F) and bending moment (M) equilibrium to vanish on any film/substrate cross section [15]. Thus

$$F=\int \sigma \left(z\right)dA=0$$

and

$$M=\int z\sigma \left(z\right)dA=0$$

with σ(z) the stress at height z in the film, z the moment lever arm, and A the sectional area through the film and substrate.

In this review, I will treat stress measurement by wafer curvature, both conceptual as well as practical. I shall touch only slightly on X-ray strain measurements, since an excellent review on that topic has been published recently [17]. The actual stress measurements, that are discussed, are ordered according to the observed microstructure, either zone T or zone II, (Refs. [19], [20], [21]). A special topic, with its own section is stress relaxation in c-BN.