مطالعات تجربی و شبیه سازی مقاومت در فیلم مس در مقیاس نانو

The effect of film thickness on the resistivity of thin, evaporated copper films (approximately 10–150 nm thick) was determined from sheet resistance, film thickness, and mean grain-size measurements by using four-point probe, profilometer, and electron backscatter diffraction (EBSD) and X-ray diffraction (XRD) methods, respectively. The resistivity of these films increased with decreasing film thickness in a manner that agreed well with the dependence given by a versatile simulation program, published earlier, using the measured values for the mean grain size and fitting parameters for surface and grain boundary scattering. Measurements of the change in sheet resistance with temperature of these films and the known change in resistivity with temperature for pure, bulk copper were used to calculate the thickness of these films electrically by using Matthiessen’s rule (this is often referred to as an “electrical thickness”). These values agreed to within 3 nm of those obtained physically with the profilometer. Hence, Matthiessen’s rule can continue to be used to measure the thickness of a copper film and, by inference, the cross-sectional area of a copper line for dimensions well below the mean free path of electrons in copper at room temperature (39 nm).

The increase in the effective resistivity of copper interconnects as physical dimensions approach the bulk mean free path of electrons (approximately 39 nm at 25 °C [1]) is a serious concern because of the impact it has on reducing circuit speed. A model and a highly versatile simulation program, published earlier [2] and [3], were used to examine and predict size effects on the resistivity of thin metal films and lines [2] and [3]. The experimental effort reported here is intended to characterize the resistivity behavior of actual films and compare the results with the predictions of the previously introduced simulation program [3]. This program uses three input parameters. One is the experimentally determined mean grain size for a given film thickness. The other two are fitting parameters that characterize the scattering of electrons from the surfaces and from the grain boundaries within the film. To a good approximation, metals such as aluminum and copper obey Matthiessen’s rule [4], which states that the resistivity of a metal is equal to the sum of the temperature dependent resistivity of the pure, bulk form of the metal plus a temperature independent residual resistivity. This means that the change in resistivity with temperature of the metal is not affected by impurities or other sources of scattering that contribute to the increase in resistivity above that of the pure, bulk form of the metal. Hence, (dρ/dT)pure,bulk = dρ/dT of the metal as long as the metal obeys Matthiessen’s rule. This is a powerful result because, with the value for (dρ/dT)pure,bulk, available in the literature [4], the thickness of a metal film and the cross-sectional area of a metal line can be calculated, respectively, from sheet resistance measurements of the film and from line resistance measurements made at two temperatures, respectively (the thickness of the films measured with this method will be regarded as “electrical thickness” throughout the paper). For films, t = (dρ/dT)pure,bulk/(dRS/dT), where RS is the sheet resistance and t is the film thickness. It is therefore important to determine if and how Matthiessen’s rule is impacted by size effects. To obtain experimentally the dependence of copper resistivity on film thickness (size effect), copper films of varying thicknesses were evaporated on Pyrex and silicon wafers, and the sheet resistance and physical thickness of the films were measured. To look for a size effect on Matthiessen’s rule, the change in sheet resistance with temperature, dRS/dT, was measured for each film thickness. The value for (dρ/dT)pure,bulk used in this paper is 0.0067 μΩ cm/°C [4].

The size effect on the resistivity of evaporated copper films, ranging in thickness from 9 nm to 167 nm, was determined experimentally from measurements of the electrical sheet resistance and the physical thickness. To evaluate a previously published simulation program [2] for studying the size effect in metals, such as copper, the mean grain size was measured for each film thickness, as it is a key input parameter to the program. Good agreement was obtained between the experimental results and the simulated variation of resistivity with film thickness as judged by how well the resistivity versus film thickness curves compared (see Fig. 4). The simulation program is available to others by using the flowcharts and the program code that have been provided [3]. The Electron Back Scatter Diffraction (EBSD) and the X-ray Diffraction (XRD) methods were used, in combination, to determine the mean grain size of the grains in the plane of the film. While the EBSD method is designed to provide such a measure of grain size, which is needed by the simulation program, useful images of the grains could be obtained only for the two thickest films. The XRD method provided data for every film thickness, but it was of the mean thickness of the grains in each film. By using the assumption that the mean grain size in the plane of the film (GSxy) is directly proportional to the mean grain thickness (GSz) for a given film thickness, the mean grain size in the plane of the film was obtained for all films by using the value for GSxy/GSz obtained from the thickest film. That such a relationship between the size and thickness of the grains may exist, as was indicated from our observations, could be useful to other researchers in seeing fruitful avenues for further nucleation studies. Measurements of the change in sheet resistance with temperature for each film thickness were used to calculate the electrical thickness of the films using Matthiessen’s rule. The level of agreement between the physical (Dektak) and the electrical measurements of film thickness over the entire range of film thicknesses showed that Matthiessen’s rule can be used to measure the thickness of copper films as thin as 20 nm. For thinner films, the simulation program predicts significant underestimates of actual film thickness. This prediction could not be evaluated because the differences in the thicknesses noted were less than the uncertainty in the precision of the physical thickness measurements (3–4 nm). Future work may involve using narrow, patterned features rather than the films used in the present work. Each approach involves tradeoffs: the evaporated film may exhibit different conduction characteristics from a patterned feature; resistivity results from a patterned feature may be dominated by non-uniform cross-section and sidewalls making determination of a unique modeling solution challenging [22]. Recent work by a team including one of the authors of this paper describes a novel fabrication method to produce patterned copper features with extremely uniform sidewalls [23] and [24].