Thermodynamics of deposition flux-dependent intrinsic film stress

Vapour deposition on polycrystalline films can lead to extremely high levels of compressive stress, exceeding even the yield strength of the films. A significant part of this stress has a reversible nature: it disappears when the deposition is stopped and re-emerges on resumption. Although the debate on the underlying mechanism still continues, insertion of atoms into grain boundaries seems to be the most likely one. However, the required driving force has not been identified. To address the problem we analyse, here, the entire film system using thermodynamic arguments. We find that the observed, tremendous stress levels can be explained by the flux-induced entropic effects in the extremely dilute adatom gas on the surface. Our analysis justifies any adatom incorporation model, as it delivers the underlying thermodynamic driving force. Counterintuitively, we also show that the stress levels decrease, if the barrier(s) for adatoms to reach the grain boundaries are decreased.


Adatom Lattice Gas (ALG) model
In this model the adatoms are assumed to reside in discrete lattice sites on the surface [2]. Imagine a terrace that has site N possible number of sites, where adatoms want to rest.
If one has to put ad N adatoms on this surface, the number of different configurations can be calculated as: From this we can derive the entropy of this particular adatom gas on this particular surface: , and with the Stirling's approximation we derive: Therefore we can write the entropic term of the chemical potential of the adatom gas as: , and by defining ad site /   N N we end at:

Two Dimensional Adatom Gas (2DAG) model
In this model the adatoms are assumed to be completely delocalized and to behave as "true" particles of a 2D gas on the surface [2]. The partition function of an adatom gas containing ad N adatoms is the multiple of the partition functions of each adatom divided by ad ! N . The division by ad ! N removes the double counting of the states: The partition functions of a single adatom on a 2D surface with only in-plane (X and Y) degrees of freedom can be written as: , in which we used the quantum mechanical description of particle in a box in a 2 dimensional space. By assuming this space to be macroscopic in size and with x y   L L L, this results in: By combining Eqs. 4 and 5 the total partition function of a 2DAG becomes: The entropy of this 2DAG can be derived as: In the last step we used eff sur ad ad The entropic component of the chemical potential of 2DAG is therefore given by:

Entropic component of the chemical potential: ALG vs 2DAG
The thermodynamical derivation in the paper depends crucially on the entropic term of the chemical potential of the surface. Therefore, we plot in supplementary figure 1 ad ( / )    T S N for both the ALG as well as the 2DAG taking into account the particular values for Cu adatoms on a Cu(111) surface.

Supplementary Note 2 -Adatom density at specific sites on a terrace
Supplementary figure 2 shows a simplified cross sectional model of the film surface, in which we define the position of the first lattice row next to the ascending step edge as the origin of a terrace with width w . The adatom density at the ( 1  n ) th site away from ascending step edge ( n  ) can be derived via the general differential equation for mass conservation on the terrace: , in which d  At a certain temperature T, the adatom diffusion rate d  , the adatom evaporation rate e  , and the step growth speed s V , can be defined as following: depends on the vibration entropies of the ground and the transition states of adatoms during the hopping or desorption process [3]. It can be shown for copper at room temperature that the prefactor 0  almost equals the vibrational frequency of the adatoms within their potential well. The latter is also known as the "attempt frequency". However, it should be noticed that this equality is not valid in general [2,3].
High mobility materials have large diffusion terms and, therefore, the step speed term becomes insignificant, unless one assumes unrealistically large deposition rates or terrace widths. Similarly, materials with low vapor pressure do have a negligible adatom evaporation term. Hence, for the typical film deposition conditions of Cu, Ag, or Au at room temperature, the step speed as well as evaporation term can be safely omitted in Eq. 9 [2,4], leading to: For the steady state situation with constant deposition flux, i.e.
/ 0     n t , the second order linear differential equation, Eq.10, has the solution of the type: At the position of ascending and descending steps the adatom density would be: Fw c w c (12) To determine the constants 1 c and 2 c we make use the two following boundary conditions: mass conversation dictates that the sum of the attachment and detachment rates at the ascending as well as the descending step edges should be equal to the diffusion currents toward theses steps. The combination of attachment and detachment rates at the ascending ( 0  n ) and descending (  n w) step edge are: , where form E is the formation energy of an adatom (conceptually this is the energy difference of a surface with a step and a kink in comparison with the same surface, in which one took one atom from a kink site and placed it on the terrace), att E is the attachment barrier for adatoms to the ascending step, ES E is the Ehrlich-Schwoebel barrier for adatoms to overcome a descending step edge, and 0 s is a dimensionless correction prefactor associated with the hopping over the step. To be more precise, E kT is the ratio of the hop frequencies over step and terrace [5]. By Eqs. 13 can be simplified as: The diffusion current at any point on the terrace can be calculated as: Fn c n can be written as: The combination of Eqs. 12 and 17 delivers the following relationship,  Note that Eq. 19 produces the thermodynamically correct value for the adatom density on a terrace if the surface receives no deposition flux 0 F  : To provide insight in the adatom density variation on terraces, we calculated the density profile on Cu(111) terraces for different widths ( w ) and the presence as well as absence of an Ehrlich-Schwoebel barrier ( for most metals at room temperature), and form 0.714 eV  E [8]. Notice that two cases describe the situation of funneling at the GB's with effective terrace widths 20  w and ( 1  s ); see the main text for more explanation.
According to Gibbs-Thompson a macroscopic surface curvature change alters the attachment/detachment rates [2] and, hence, also the adatom densities n  on the terraces.
For a polycrystalline gold film, which first has been brought to its equilibrium state at 750 K 9], it has been shown that upon starting the deposition at room temperature the surface roughness/curvature increase initially before it approaches a steady state value 10]. However, the time scale of the roughness development is significantly longer than that of the reversible stress jumps. Note that the surface curvature variations involve changes in the step and kink densities and their distributions, which is expected to happen much slower than the adatom density variations (refer to the main manuscript).
Moreover, although not mentioned in 10], after the deposition has been stopped, no significant decrease of the roughness towards the equilibrium state was observed, e.g. due to the decay of 2D islands on the mounds. This is due to the rather limited activation of the surface annealing processes at room temperature [11]. For our study, we, therefore, safely omit these additional correction terms associated with surface curvature variations.

Supplementary Note 3 -Chemical potential of a grain as a function of stress
Supplementary figure 5 shows a simplified model of a grain with thickness 0 L that contains already an internal pre-stress g  . For pedagogical reason we first consider g  as a hydrostatic stress, before we (below) address the more realistic case of a film under biaxial stress. The chemical potential of (and also within) this grain can be defined as the partial derivative of its free energy with respect to the amount of atoms, N , in the grain. Adding a (thin) layer to this grain with a certain thickness t that contains t N atoms causes strain , which consequently rises the (potential) energy of the system by the amount of involved work t W . Therefore, the chemical potential of a grain can be written as: The applied work can be, in general, calculated as: , in which  is the stress in the grain that is a function of the strain  , and L is the amount of displacement of the grain boundary. Please note that ( )   is linear in the elastic regime of the material, but becomes highly non-linear once one passes this regime.
Let us start here with the simple case of the elastic regime, in which the stress is linearly related to the strain via the Young's modulus: In this case the work in Eq. 22 can be calculated as: , in which  describes the atomic volume of the material. As we assumed the film to be in the elastic regime, the pre-stress of the grain before the addition of the extra (thin) layer with thickness t is given by g 0 p r e    E , if pre  describes the pre-deformation of the grain.
By combining Eqs. 21 and 24: The generalization to the plastic regime (i.e. g  larger than the yield strength of the material) follows simply by the fact that the slope of the stress-strain curve, E , decreases to values less than the Young's modulus 0 E and, in addition, becomes strain dependent. Hence, for infinitesimal values of t, the applied work is again calculated as: , and similarly the chemical potential is derived as: Note that the / This leads to an important conclusion: the variation of the chemical potential of a grain is only a function of internal stress variation and, therefore, fully independent of the elasticity, the shape of the grain, and whether the material is in the elastic or plastic regime.
As mentioned above, Eq. 28 holds only for hydrostatic stress conditions. Let us now derive the biaxial stress case, as thin films are typically anchored to the surface in the in-plane directions while being unconfined in the out-of-plane direction. For the general case, the chemical potential is related to the stress and strain fields as [12]: