Thin Solid Films 515 (2006) 1881 – 1885
www.elsevier.com/locate/tsf
Surface and grain boundary contributions in the electrical
resistivity of metallic nanofilms
Juan M. Camacho ⁎, A.I. Oliva
Centro de Investigación y de Estudios Avanzados del IPN Unidad Mérida, Depto. de Física Aplicada, A.P 73-Cordemex 97310 Mérida, Yucatán, México
Received 5 April 2005; received in revised form 11 May 2006; accepted 13 July 2006
Available online 17 August 2006
Abstract
The surface and grain boundary contributions in the electrical resistivity ρ of Au, Al and Cu films deposited by thermal evaporation with
thickness from 3 to 100 nm on glass substrates were determined. The ρ values were measured and theoretically evaluated following the Fuchs–
Sondheimer, the Mayadas–Shatzkes, and the combined models. A method to measure the grain size and its distribution from atomic force
microscopy images was implemented, finding a lognormal behavior in all cases. We obtained that surface ( p) and grain boundary reflection (R)
coefficients decrease as the film thickness increases, showing R coefficient, higher values and most important changes than p coefficient. We
concluded that high ρ values are mainly due to grain boundaries' contributions.
© 2006 Elsevier B.V. All rights reserved.
PACS: 75.50.Bk; 73.61.At; 73.63.Bd
Keywords: Atomic force microscopy; Electrical properties and measurements; Grain boundary; Metals
1. Introduction
Nanotechnology is nowadays a fertile field for materials. Lowdimensional physical properties are quiet different than for
material bulk. Electrical resistivity ( ρ) is an amazing property in
metals due to the large variations measured when thickness (t)
diminishes at nanometric scale. Different theories have been
proposed to justify this behavior. Thomson [1], proposed a free
electron gas geometrical model assuming electrons with constant
mean free path (λ) and colliding between two reflecting surfaces.
However, this model fails for t N λ and for bulk. Fuchs and
Sondheimer (FS) [2,3], proposed a better approximation by
considering the quantum effect of the free electrons by considering
a statistic distribution of the λ values and the important role of the
film surfaces. A coefficient p (from 0 to 1) is used to describe
electron scattering at the film interfaces. A recent model proposed
by Mayadas and Shatzkes (MS) [4,5], improves the previous
models by including the important effect of the grain boundaries
(GB) and mean grain size D produced during film deposition. MS
model uses a reflection coefficient R (between 0 and 1) at the GB.
⁎ Corresponding author. Tel.: +52 99 91242100; fax: +52 99 99812917.
E-mail address: [email protected] (J.M. Camacho).
0040-6090/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.tsf.2006.07.024
Actually, we know that ρ value is a complex property that depends
very strongly on the combination of the surface scattering, grain
boundary scattering [6], roughness, defects and impurities, as a
consequence of the films growth conditions. Assuming that
roughness and impurities effects can be minimized by the very
small film thickness and by the use of high purity metals, surface
and grain contributions are not effects easy to discern separately.
Al, Cu, Au and Ag and their alloys are the most important
electrical conductors for microelectronic applications. Several
experimental works has been reported trying to understand the ρ
behavior for these metals in order to evaluate the p and R
coefficients. Liu et al. [7], reported the study of Cu films with
thickness between 10 and 40 nm and found that ρ measured values
fit well with p = 0.05 and R = 0.24 of the combined model for a λ/
t = 0.3 ratio. Lim et al. [8] found, also for Cu films, p = 0 and
R = 0.32 values for the 10–500 nm thickness range. Higher values
of p = 0.6 and R = 0.5 were reported for this material but in
nanowires geometry [9]. On these and others works, authors rarely
explain why they use that p and R values and normally, they adjust
the experimental value with the theoretical models most accepted.
On the other hand, scarce efforts have been made to justify the ρ
measurements mainly by the GB effects [10]. Zhang et al. [11]
studied polycrystalline Cu films and obtained that grain boundary
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scattering is the dominant effect of the high ρ values. Cattani et al.
[10] proposed a new model to explain the ρ of Pt and Au and
concluded that grain size (GS) also plays an essential role. In this
work, we deposit Al, Au and Cu films in the 3–100 nm range on
glass substrate by thermal evaporation in order to study the
electrical resistivity. Atomic Force Microscopy (AFM) film
images were taken and analyzed to obtain the GS and its
distribution. We found that lognormal distribution fitted the GS
behavior for all t range. Results were related with the theoretical
models in order to quantify the surface and GB scattering
contributions to the ρ value; i.e., the p and R parameters. We found
that GB is the most dominant contribution of the higher ρ values
measured on this t range. Different p and R values were found as a
function of t when the GS behavior with thickness is considered.
Cu [4,8]. However, with time, different authors have been reported
different values of R for Au and Cu, being higher values for films.
2.3. Combined model
A combination of the FS and MS models has been proposed
[13] in order to include both effects. Thus, with the combined
equation and including experimental measurements, we obtain
the corresponding p and R values for very thin films. However,
the GS and its distribution need to be known as a function of
thickness. The combined model is obtained by combining Eqs.
(1) and (2), such that total ρ value can be determined by:
qtotal ¼ qFS þ qMS −q0
ð3Þ
2. Theoretical aspects
Again, different combined values of p and R have been
reported, arguing best fitting with the experimental data.
We used the FS, MS, and combined theoretical models to
justify the ρ data obtained for metals. A brief description of the
models is given:
3. Experimental procedures
2.1. FS model
Fuchs and Sondheimer (FS) [2,3] proposed a theoretical
model by considering the quantum effect of the free electrons, a
statistic distribution of the λ values in the bulk and assuming
that film surfaces play an important role in the λ value. The FS
model can be determined by:
q0
k
;
¼
qFS Up ðkÞ
Z l
1
1 3
1 1 1−e−kt
with
¼
ð1−pÞ
−
dt
Up ðkÞ k 2k 2
t 3 t 5 1−pe−kt
1
ð1Þ
where ρFS is the resistivity value due to film surfaces, ρo is the
bulk value, k = t / λo ratio being λo the mean free path of the bulk
and p is the fraction of elastically dispersed electrons by the thin
film surfaces; p = 0 gives the maximum ρ value by total surface
scattering, and p = 1 for mirror surfaces, where we obtain the
bulk ρ value. Approximations to Eq. (1) can be possible for
different k ratios.
Thin films of Al, Au and Cu were deposited on glass substrates
by thermal evaporation with thickness between 3 and 100 nm in a
vacuum chamber at 10− 3 Pa and ∼0.2 nm/s as growth rate. For
each metal, we deposit five pairs of films with different thickness
by means of a homemade gyratory substrate-holder. Film
thickness was monitored and controlled by a quartz crystal with
a Maxtek TM-400 controller during deposition. Calibration was
realized with a Dektak 8 profile-meter in order to have reliability
on the deposited thickness. Film resistivity was measured after
preparation by the four-probe technique with a Jandel head, a HP6443A power supply and a HP-3458A digital-voltmeter with high
resolution. Images of the films surface morphology were obtained
with an AFM Auto-probe CP in contact mode with a SiN
cantiliver covered with gold. Images were obtained at 1 Hz as
scan rate with 256 × 256 pixels2 resolution and only received a
flattened treatment for analysis. Images data were analyzed with
home-made software developed in Mathematica© to obtain the
grain size and its distribution. We found a lognormal distribution
of the GS in all AFM images. Experimental values of ρ were
associated with the GS through the combined model in order to
quantify the p and R contributions.
2.2. MS model
Mayadas and Shatzkes (MS) [4,5] improved the FS model by
considering the dispersion of electrons by GB, assuming that
the mean grain size (D) is the main dispersive factor. The
proposed relation is:
q0
2
1 −1
2
3
¼ 1− a þ 3a −3a ln 1 þ
;
3
a
qMS
k0 R
with a ¼
D 1−R
ð2Þ
Here, ρMS is the resistivity value due to the GB, and R is the
reflection coefficient in the GB, taking values between 0 and 1.
Initial values reported for bulk were R =0.17 for Al and R = 0.24 for
Fig. 1. Electrical resistivity vs. film thickness measured for Al, Au and Cu films.
J.M. Camacho, A.I. Oliva / Thin Solid Films 515 (2006) 1881–1885
1883
4. Results and discussion
Fig. 1 shows the ρ values measured on the Al, Au and Cu films
deposited for different thickness. For Al films we obtained the ρ
minor values between 3 and 30 nm thickness range. Contrarily, Au
and Cu films were possible to measure higher values of ρ into the
same thickness range. We attributed these lower values to the
native oxide layer formed on Al when films are exposed at
atmospheric pressure. This was demonstrated by the slow increase
of ρ detected when an Al film of 6 nm thickness was growth and
continuously monitored along 3 days; after this time, the
measurement of ρ was not possible. Thus, Al film oxidizes such
that the electrical connections between grains disappear. X-ray
diffraction analysis showed polycrystalline structure in all
deposited films. When the measured ρ data are compared with
the theoretical curve coming from the FS model, theoretical data
seems far and up to experimental data, indicating that ρ behavior in
Fig. 1 needs to include the GB contribution.
Fig. 2 shows typical surface morphology obtained from Al,
Au, and Cu for different thickness films. Differences on the GS
can be observed for each deposited film. The visualized GS from
AFM images needs to be quantifying by means of its distribution
function. For that, we developed specific software by obtaining
the numerical Laplacian of the image-matrix to enhance the graincontours by detecting the maximum values. The converted new
data-matrix, now formed by zeros (grain area) and ones (boundary
grain), permits to calculate the enclosed grain area. Thus,
assuming circular grains, we can estimate the mean grain diameter
D, its distribution and the surface density for each AFM image.
Thus, obtained D values are plotted as a function of thickness
and fitted by using a dynamical scaling law D ∝ tB behavior.
Growth exponent B fitted for Al, Au and Cu films, were 0.20 ±
0.072, 0.36 ± 0.052, and 0.75 ± 0.073 values, respectively. Fig. 3
shows this behavior for the three metallic films. The D vs. t log–
log plot shows a linear increase with thickness for the range
analyzed. Excepting for first stages of growth of Au films (minor
than 25 nm) where D was not evaluated due to the discontinuities
found on the film formation; i.e., a coalescence phenomenon
occurs by the semi-continuous films observed. However, initial
GS values for Au were estimated by assuming to follow a similar
Fig. 2. AFM topographical images (0.5 × 0.5 μm2) used for grain size
determination; (A) 15-nm thick Al (ΔZ = 9.2 nm), (B) 35-nm thick Au
(ΔZ = 10.9 nm), (C) 100-nm thick Cu (ΔZ = 8.0 nm).
Fig. 3. Grain size D vs. thickness behavior for Al, Au, and Cu films obtained
from AFM images. Different values of the growth exponent B were found for
each metal. Solid lines are curves fitted for each metallic film.
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scaling law than Al and Cu as was recently demonstrated [14].
However, different growth exponents have been reported for
different authors such that values can widely oscillate between 0.2
and 0.7 [10,15,16] for these metals. The D vs. t behavior fitted for
each metal was introduced in the combined model and plotted for
different p (from 0 to 1, and 0.1 step) and R parameters.
Histograms of the GS for the three metallic films followed in all
cases a lognormal distribution. Fig. 4 shows typical lognormal
distributions found for GS on the metallic films. Similar
distribution of the GS vs. thickness was reported before for Au
and Pt films in the 2–40 nm thickness range [12]: D increases
rapidly in the minor thickness region until become constant near
13–15 nm; after that, D increases again.
Fig. 5. Coefficient R vs. thickness obtained with the combined model for (a) Al,
(b) Au, and (c) Cu. Different R values where found for each material. Each
group of curves were calculated from p = 0 (lower) to p = 1 (upper) in the arrow
direction.
The resistivity experimental data ( ρexp) for each metallic film
were compared with the combined model (Eq. (3)), which uses
different p parameters to find the corresponding coefficient R as a
function of thickness by fitting the ρ experimental data. Fig. 5
shows three groups of data and the corresponding R values for
each studied metal. Each group shows ten curves changing from
p = 0 to p = 1 in the arrow direction. For the analyzed thickness
range, R values were found to be higher for Au and Cu than for Al
films which value tends to a R = 0.5 as minimum. Excepting for
Cu, R coefficient tends to diminish with thickness, indicating that
scattering due to GB is most important than surface scattering.
Assuming that ρexp is the sum of bulk ρo (defects, phonons,
etc.) and surface ( ρFS − ρo) and grain contributions ( ρMS − ρo);
now, we are ready to calculate the contributions of the surfaces
and the GB dispersions by means of the combined model. Given
that we know ρexp (Fig. 1) and D (determined by AFM) parameters, we assign a p value, and calculate the corresponding
R value from the following relation:
qexp ¼ qMS ðR; DÞ þ qFS ðpÞ−q0
Fig. 4. Typical lognormal grain size distribution obtained for (a) Al, (b) Au, and
(c) Cu films. Solid curves are the best fit for each distribution.
ð4Þ
These contributions are plotted in Fig. 6 for each studied metal
as a function of thickness. Each group of curves indicates the
different values of p parameter taken from 0 to 1 in the arrow
direction. Lower curves are the surface scattering contributions;
upper ones, the GB contributions. Cu films show the most
important contributions due to the GB as compared with the
surface contributions, which decreases with thickness. In all
metallic films analyzed, we also obtained that the surface contributions decreases when thickness increases and are minor than
GB contributions. Scattering surface contribution p was quantified between 0.2–0.1, meanwhile scattering grain boundary R
was between 0.9–0.5 when thickness decreased. Recently, Zhang
et al. [11], determined an R = 0.46 value for PVD copper films,
assuming that ρt vs. t plot follows a linear behavior, and they
concluded that ρ is mainly dominated by the GB. However, the R
parameter can change its value with the deposition technique and
t, mainly in the first stages of growth, as our results indicated.
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5. Conclusions
The electrical resistivity of Al, Au, and Cu films was obtained
in the 3–100 nm thickness range. The grain size, measured from
AFM images, was found to follow a lognormal distribution.
Assuming a D behavior according to the dynamical scaling law,
D ∝ t B, permits to quantify the D of each film in the thickness
range studied. The obtained D behavior was used with the ρ
experimental data and the combined model (FS + MS) to quantify
the surface and grain boundary contributions to the ρ value. We
found for the three metallic films analyzed, that the p and R
parameters change with thickness; and hence, that grain
boundaries causes the main contributions of the measured ρ
data. We obtained different and decreasing values of the R and p
parameters when the film thickness increase as compared with the
reported data, meaning that film growth process plays a key role
on the ρ during first stages, being the GB formation, the main
reason of the high ρ measured. These results are in good
agreement with the obtained in reference [16] when authors affirm
that ρ vs. t is a result of the competition between the thickness and
the roughness variation with growth time.
Maybe, by the most stable surface of the Al films due to the
native oxide layer, we found that R and p parameters determined
from the combined model, decrease since the thinnest film
prepared (3 nm). This behavior was not observed on Au and Cu
films, for this thickness region, perhaps by the no-grain formation, as observed by the AFM images. A further detailed study
of the first stage of film growth needs to be done.
Acknowledgements
This work was supported by CONACYT (México) through
project 38480-E. Authors thank the technical help given by J.E.
Corona and O. Ceh.
References
Fig. 6. Main contributions of the resistivity as determined with the combined
model and the experimental data for: (a) Al, (b) Au, and (c) Cu films. Bulk
resistivity contribution is not plotted. The p value increases in the arrow
direction.
According to the MS theory, R parameter depends on the GB
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behavior of the strength potential S values with thickness. First
approximations show that S value presents higher values for
thinnest thick, and changes abruptly when t increases, such that
at 50 nm thick, S practically reaches their minimum value.
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