Resistivity of thin films

Bulk materials are primarily classified by the influence of the sample boundaries. Based on the ratio of the boundaries compared to the dimensions of the system, the influence of the boundaries is negligible. The resistivity is mainly given by the mean free path of the electrons. Due to the occurrence of a further limitation of the mean free path the resistivity increases, besides the already presented scattering opportunities.

Depending on the temperature and the purity of the crystals, the mean free path can adopt dimensions in the order of the thin film thickness and therefore further scattering mechanisms occur which are negligible in bulk. Therefore, the scattering of electrons on grain boundaries and on internal surfaces have to be taken into account. A model to describe the scattering on grain boundaries is described by Mayadas and Shatzkes [13,14],

whereas the Fuchs-Sondheimer model explains the scattering at the surface [15, 16].

2.2.1 Fuchs-Sondheimer model

In 1901, Thomson [17] had already the idea that the dimensions of a system could in-fluence the resistivity. Based on the Boltzmann transport equation, Fuchs postulated 1938 [15] a size-effect for a free-electron model which can be simplified by [18]:

• disorder of the film is independent from the film thickness

• clean and parallel surfaces of the film

• a spherical Fermi surface

• isotropic scattering events

p = 0 p = 1

d

Figure 2.3: Description of parameter pof the Fuchs-Sondheimer model, which describes the fraction of the electrons which are scattered specularly.

Fig. 2.3 illustrates the phenomenological specularity parameterp. For p= 0, a complete diffusive scattering of the electrons takes place, whereas for p = 1 the electrons were scattered according Snell’s Law, so-called specular reflection.

A way to calculate the conductivity of the film σ_f is σ_f

0, the ratio between film thickness d and the mean free path l0 adopted from a corresponding bulk system (for example density of impurities or lattice defects).

A further modification simplifies the formula since Eq. 2.3 is only numerically solvable.

For thick films (dl₀) one obtains [16]:

and for thin films (dl₀)

In summary, the Fuchs-Sondheimer theory describes the scattering of electrons on parallel and smooth surfaces with a scattering parameter p. The scattering on a rough surface cannot be explained with this model. However, inspired hereby, further works integrated the roughness into their models [12, 19–22]. Also in some models a second parameter q was introduced due to the fact that the second surfaces of a film can differ from the other one [23–25].

2.2.2 Mayadas-Shatzkes model

Mayadas and Shatzkes take another approach to describe the resistivity of thin films [13].

They considered the scattering at grain boundaries of polycrystalline metals, which occur between two crystals. The ground boundaries were treated as potential barriers, where the electron can either be transmitted or reflected. Therefore, a grain boundary reflection coefficient R_gb was introduced, whereby an increase of the resistivity for increasing R_gb was observed. In this model the differences between specular and diffusive scattering is negligible, compared to the approach of Fuchs and Sondheimer. Another length scale has to be introduced to describe the resistivity, the density of the grain boundaries, here denoted with the average grain size D.

Since Mayadas and Shatzkes developed a 1D model only, some assumptions and

sim-D

R

Figure 2.4: Sketch of the Mayadas scattering model withDthe distance between two grain boundaries andRgbthe grain boundary reflection coefficient.

plifications [26] had to be done. Only those grain boundaries were taken into account which are aligned perpendicular to the electric field. The contributions of all other grain boundaries have negligible effect on the total resistivity. In addition, the remaining perpendicular grain boundaries were described by a Gaussian distribution of planar scat-terers, with the averaged distance ¯Dand the standard deviations. Potentials of the form V_x =Sδ(x−x_n) describe these scatterers at the position x_n with S the strength of the

scatterer andδ the Delta Dirac function.

The relation between strength of the scattererS and the grain boundary reflection coef-ficient R_gb is given by

S² = ~³

2m_ev_Fk_F R_gb

1−R_gb (2.6)

with Fermi velocity v_F and Fermi wave vector k_F.

The conductivityσ_g for polycrystalline metals can be derived from the Boltzmann equa-tion. Thereby, the conductivity inside a crystalσ₀ is multiplied by the scattering rate.

σ_g =σ₀·f(α) (2.7)

l0 is the mean free path inside the crystal, according to σg. f(α) = 1−3

Since this first model of Mayadas-Shatzkes describes the scattering on grain boundaries only, they extended it for thin films according to the model of Fuchs-Sondheimer. In addition, film surfaces were also considered and lead to the expression for the film con-ductivity σf [14]:

Eq. 2.10 reveals that the specularity parameter p (Fuchs-Sondheimer) and the grain boundary reflection coefficient quantities R_gb (Mayadas-Shatzkes) of the models can be derived simultaneously. Furthermore, it is shown that p and Rgb are not independent which leads to the finding that Matthiessen’s rule is not valid for this boundary consid-eration. Nevertheless, for thick films (dl₀) Matthiessen’s rule is valid since pand R_gb are independent, so the resistivity of a thin film can be written as:

ρ_f =ρ₀+ρ_ss+ρ_gb (2.12)

The contribution of the film surface (Fuchs-Sondheimer) amounts to ρ_ss = 3

8(1−p)ρ₀l₀

d (2.13)

whereas grain boundary scattering leads to ρ_gb = 3

2ρ₀ R_gb 1−Rgb

l₀

D (2.14)

Other groups worked on simplifications and statistical models of these findings, as re-ported in Refs. [27–32].

Influence of the film thickness

Figure 2.5: Resistivity as a function of the film thickness for the Fuchs-Sondheimer (in a)) and the Shatzkes model (in b)). Two kinds of grains must be considered in the model of Mayadas-Shatzkes. The columnar grains are thickness independent, whereas the spherical grains reveal a behavior like the Fuchs-Sondheimer model. Taken from Ref. [4].

Fig. 2.5 illustrates the resistivity as a function of the film thickness for both models.

The Fuchs-Sondheimer model reveals a reciprocal behavior for the resistivity (the thinner the film thickness, the higher the resistivity), while in the Mayadas-Shatzkes model two kinds of grains have to be considered. The first kind is the columnar grain, which yields a resistivity independent of the film thickness. However, for the second kind, the spherical grains are proportional to the film thickness. Like in the model of Fuchs-Sondheimer, a reciprocal behavior of the resistivity versus the film thickness can be observed. Since for metal films the grains grow constantly with the film thickness, the thickness dependence of both models are very similar, although two types of scattering were considered.