Keywords

1 Introduction

The fabrication of nanocomposite nanowires has recently received much attention because of the interesting magnetic, superconducting and magnetotransport properties and their application in the development of high-density magnetic recording media and novel magnetic sensors [1, 2]. These studies were triggered by the discovery of giant magnetoresistance (GMR) effects [3] in certain metallic multilayers in which the interlayer exchange coupling results in antiferromagnetic alignment of the adjacent magnetic layers [4]. Specifically, NiCo [5], Co/Cu, NiFe/Cu or CoNiCu/Cu [6, 7] nanowires show anisotropic magnetoresistance (AMR) or current perpendicular to the planes giant magnetoresistance (CPP-GMR). This type of nanowires can be grown at low cost by electrodeposition from aqueous solution into nanochannel templates [8].

While designing such nanoscale devices, it is necessary to know their structural and thermo-mechanical properties since the mechanical strength of nanowires plays an important role in maintaining the structural integrity of the devices. For feature sizes of several atomic layers, surface stress may induce a large intrinsic compressive stress in the interior of nanowires, and their structure becomes instable. Thus, under some conditions, the nanowires may transform from one orientation to another spontaneously [9] or, even initiate phase transformations [10]. Molecular dynamics (MD) simulations have been performed extensively and reported in published literature for composite nanowire property characterizations including phase transformation behavior at nanoscale, mechanisms of dislocation formation, plastic region and breaking under stress.

In this work, the stress–strain relations of Co/Cu multilayered nanowires under extension and compression and different conditions by means of MD simulations are obtained. The elastic modulus and initial yield stress are mainly analyzed. The knowledge of these mechanical properties will be helpful to the design, manufacture and manipulation of nanodevices. Effects of composition, size, strain rate and temperature on the extension properties are discussed in detail as a function of the crystallographic orientation. Influence of internal structure and interface on the mechanical behavior of nanostructures is discussed. The nanostructural deformation for tensile and compressive stresses is presented together with an identification of dislocations.

2 Computational Method

The interaction among atoms is essentially determined by the atomic potential. In our case, the used potential for the Co–Cu system was elaborated by Levanov et al. [11] in the second-moment tight-binding approximation (TB-SMA) [12]. This potential fits experimentally lattice constants, cohesive energies and elastic constants for Cu (face-centered cubic, fcc) and Co (hexagonal close-packed, hcp) and takes into account ab initio properties of small Co clusters on Cu surfaces. In this model, the cohesive energy E i of an atom i is:

$$ \begin{gathered} E_{i} = \sum\limits_{j( \ne i)}^{{}} {\frac{1}{2}V^{\alpha \beta } (r_{ij} )} + F(\rho_{i} ) = \sum\limits_{j( \ne i)}^{{}} {\frac{1}{2}V^{\alpha \beta } (r_{ij} )} - (\rho_{i} )^{1/2} \hfill \\ \quad \rho_{i} = \sum\limits_{j( \ne i)}^{{}} {\Upphi^{\alpha \beta } (r_{ij} )} \hfill \\ \end{gathered} $$
(1)

where α and β refer to the chemical nature of the atoms i and j, respectively; and r ij is the distance between them. The first term, repulsive term, is described by a pairwise potential energy of the modified Born-Mayer type. The second term, attractive term, contains the many-body interaction that models the band energy.

From this magnitude, the stress tensor σ kl can be obtained as the derivative of the total energy per unit volume [13]. Using a TB-SMA potential, this tensor can be expressed as:

$$ \begin{gathered} \sigma^{kl} = \frac{1}{V}\sum \limits_{i} \frac{{\partial E_{i} }}{{\partial \varepsilon^{kl} }} = \frac{1}{V}\mathop \sum \limits_{i} \left( { - m_{i} v_{i}^{k} v_{i}^{l} + \frac{1}{2}\mathop \sum \limits_{{j\left( { \ne i} \right)}} \left( {h_{ij} } \right)\frac{{r_{ij}^{k} r_{ij}^{l} }}{{r_{ij} }}} \right) \hfill \\ h_{ij} = \frac{{\partial V^{\alpha \beta } }}{{\partial r_{ij} }}\left( {r_{ij} } \right) + \left( {\frac{\partial F}{\partial \rho_i}\left( {\rho_i} \right) + \frac{\partial F}{\partial \rho_j}\left( {\rho j} \right)} \right)\frac{{\partial \Upphi^{\alpha \beta } }}{{\partial r_{ij} }}\left( {r_{ij} } \right) \hfill \\ \end{gathered} $$
(2)

where ε kl is the strain-tensor Cartesian component kl (k,l ∈ x,y,z), v k is the speed component k, and V the system volume. Each element of the summation divided by the effective atomic volume is the local stress at the atom site i. The main part of this work is carried out on a so-called reference nanowire. This system is a set of seven epitaxial ultrathin layers composed of two fcc oriented Co and Cu sublayers of about 10 nm thick each one stacked up alternately to build a Co/Cu multilayered nanowire. For this thickness, Co/Cu nanowires exhibit a large giant magnetoresistance (GMR) of 20–30% [14]. The presence of Co/Cu interfaces and the lattice mismatch favors an fcc phase for Co [14]. A circular cross-sectional shape of 5.11 nm in diameter has been considered. This size was selected since it can be built and investigated experimentally [15], and even though the simulation time is long, it is still reasonable. The thickness ratio between Co and Cu sublayers and the transverse and longitudinal dimensions have been varied.

Prior to uniaxial loading, the nanowires were relaxed and thermally equilibrated at 300 K using a Nosé–Hoover thermostat [16] while keeping periodic boundary conditions in the main nanowire direction. Then, boundary conditions were removed and the system was again relaxed at that temperature by using a constant pressure algorithm for 30 ps [17]. This procedure causes the pre-stress to be set to zero in the lattice. Finally, the uniaxial tensile and compressive loading process of nanowires was simulated at the temperature of 300 K and a strain rate of 109 s−1. Nevertheless, other strain rates and temperatures have also been considered in the present work. The Nosé–Hoover thermostat gives rise to modest temperature fluctuations, which leads to correct canonical averages of system properties. The uniaxial loading is reached by applying velocity to the atoms of both ends. This velocity varies linearly from zero to the maximum value, thus creating a ramp velocity profile. Such a ramp velocity is used to avoid the emission of shock waves.

To provide information on the atomic displacement in dislocations, we calculate the slip vector [18] and the maximum magnitude of slip vector, defined respectively for an atom i as:

$$ \vec{s}_{i} = - \frac{1}{{n_{s} }}\sum\limits_{j( \ne i)}^{n} {\left( {\vec{r}_{ij} - \vec{R}_{ij} } \right)} \;;\quad \quad s_{i,\max } = \mathop {\max }\limits_{{j = 1,..,n_{s} }} \left| {\vec{r}_{ij} - \vec{R}_{ij} } \right| $$
(3)

where r ij and R ij are the vector differences of atoms i and j at the current and reference positions, respectively; n is the number of nearest neighbors to atom i; and n s is the number of slipped neighbors. The reference positions are associated with the state of zero mechanical stress. The slip vector coincides with the Burgers vector for the slip of adjacent atomic planes, where the atom lies on one of those planes. For any other inhomogeneous deformation, the slip vector will provide quantitative information about the deformation. The maximum magnitude of slip vector has also been calculated to assess the margin of error in predictions. The internal relocation of atoms has been studied by the common-neighbor analysis (CNA) [19] and the lattice unit vectors (crystallographic orientation) in each grain [20].

3 Results and Discussion

In a similar way to that for pure metals [21], the stress curve as a function of the engineering strain for a Co/Cu composite nanowire in tensile loading at normal and low strain rates (<5·109 s−1) increases up to an absolute maximum in the elastic region. In Fig. 1, stress–strain curves at different strain rates and orientation are shown. The 〈100〉 orientation exhibits a very linear stress–strain relationship [22]. The initial yield stress (or strain) is defined as the stress (or strain) at the elastic limit that, for these values of strain rate, coincides with the absolute maximum. After reaching the peak value, the stress goes down dramatically due to the occurrence of dislocations. Then, the plastic zone is developed. In this region, the curve is saw-shaped, and at high strain rates it softly decreases. Plasticity comes from the emission and movement of different dislocation systems. The elastic zone can end in a relative maximum at rates of the order of 5·1010 s−1, especially for 〈110〉 and 〈100〉 directions. Then, a strain hardening region is developed and it is necessary to distinguish between yield and ultimate (absolute maximum) stresses.

Fig. 1
figure 1

Stress–strain curves for a reference nanowire at different strain rate and crystallographic orientations and 300 K

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The stress curve is linear only at low strain in the elastic region. Wu [22] indicated that the nonlinearity may come from the inhomogeneity of nanowire cross-section, especially important in multilayers due to the lattice mismatch. The Young’s or elastic modulus is determined from the slope by applying a linear fit at low strain. The stress required for homogeneous dislocation nucleation is highly dependent on the crystallographic orientation and the uniaxial loading conditions [23]; thus, certain orientations require a higher stress in tension, e.g., 〈111〉 nanowires. Besides, these nanowires are the hardest one (larger elastic modulus). In Fig. 2, the Young’s modulus (rough, in some cases) and ultimate stress as a function of the strain rate are indicated. 146, 209, 76 GPa are the elastic moduli; and 3.76, 8.10, 5.74 GPa, the yield stresses used for the normalization in 〈110〉, 〈111〉 and 〈100〉 directions, respectively. The elastic modulus and yield stress are approximately independent of strain rate, except at high strain rates (>5·109 s−1). The effect of strain rate on the mechanical behavior is similar for the three crystallographic orientations, except for 〈110〉 nanowires. In this case, the increase in elastic modulus with increasing strain rate is due to the nonlinearity of the stress curve. If the nanowire diameter decreases, this distinguishing behavior does not exist.

Fig. 2
figure 2

Normalized Young’s modulus and ultimate stress as a function of the strain rate for the reference nanowire with different crystallographic orientations and 300 K

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Size effects on the elastic behavior have also been analyzed. A parameter as the volume-to-surface-area (vol/sa) ratio (normalized by the vol/sa ratio of the Cu unit cell) is introduced. The dependence of Young’s modulus and initial yield stress on this parameter is represented in Fig. 3 for the tensile regime. The smallest nanowire had 1.50 nm maximum wide (5-Co/Cu-layer long), and the largest one, 5.11 nm (reference nanowire). Below a ratio of about 10, both mechanical magnitudes diverge from the bulk value, increasing in the case of 〈110〉 orientation and decreasing for the other two directions. This behavior is in accordance with previous works on pure Cu, Ag and Au nanowires [24–26]. The surface effect plays an important role on the mechanical properties, since, in fact, the different ‘core–shell’ elasticity [27] and the nonlinear elastic response of the nanowire core induced by the surface stress [24] are determining factors of the elastic modulus.

Fig. 3
figure 3

Normalized Young’s modulus and initial yield stress as a function of the normalized vol/sa ratio for nanowires with different crystallographic orientations (at 300 K and 109 s−1)

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Two details deserve to be mentioned in the break-up layer of a nanowire of 2.24 nm in diameter (ratio about 6.5): For 〈100〉 nanowires, grains undergo a crystallographic reorientation, thus it is possible to find mainly 〈111〉-oriented and also 〈110〉-oriented grains. A similar reorientation process towards 〈110〉 direction is observed in an fcc gold 〈100〉 nanowire [28]. Besides, after the break-up the number of hcp atoms is larger for 〈100〉 nanowires (2.5 times) than the number of fcc atoms unlike 〈111〉 and 〈110〉 directions (the former has 3.2 times more fcc than hcp atoms, and the later 4.5). These hcp atoms are preferably in the Co sublayer for 〈100〉 direction (there are two times more hcp Co than Cu atoms), and in the Cu sublayer for 〈111〉 (two times more hcp Cu than Co atoms) and 〈110〉 (there are only hcp Cu atoms) directions. The fcc atoms are preferably in the Cu sublayer for 〈100〉 nanowires (two times more fcc Cu than Co atoms) and in the Co sublayer for 〈111〉 and 〈110〉 nanowires (two times more fcc Co than Cu atoms in both cases). Therefore, in the break-up zone Co recovers an hcp structure more easily for 〈100〉 nanowires, and in addition Cu is more stable in its fcc structure for 〈111〉 and 〈110〉 nanowires.

In Fig. 4, the Young’s modulus and initial yield stress at different temperatures for the reference nanowire are represented. Both magnitudes decrease approximately linearly with increasing temperature [29]. The deviation from the horizontal line is larger in the case of 〈100〉 direction, and therefore, temperature has more influence on the mechanical behavior of this orientation. The drop of the elastic modulus with increasing temperature is associated with the significantly weakened bond forces due to the larger thermal vibrations of atoms [30]. Nevertheless, temperature has no substantial effect on the deformation mechanism [30]. The dislocation nucleation takes place more easily at higher temperatures. Theoretical models of dislocation emission from a crack tip show a definitive relationship between activation energy and temperature [31]. This effect leads to decrease the yield stress. Absolute values of yield stress in these three directions at different temperature are slightly smaller than other published values on pure Cu nanowires due probably to the different simulation potential [32].

Fig. 4
figure 4

Normalized Young’s modulus and initial yield stress as a function of temperature for the reference nanowire with different crystallographic orientations at 109 s−1

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The elastic modulus of Co/Cu nanowires decreases with increasing thickness of Cu sublayer in a similar way to that obtained in other bimetallic multilayers [33]. In fact, both magnitudes are much larger for Co nanowires than for Cu nanowires. In Fig. 5, the Young’s modulus and yield stress are represented as a function of the thickness ratio between Co and Cu sublayers. While the Young’s modulus decreases gradually, yield stress falls steeply from pure Co value to the pure Cu value for any Cu sublayer thickness. In fact, lattice distortion takes place firstly and fundamentally in Cu sublayers, even for small thicknesses of these [33]. A reason is that for this potential the stacking-fault energy of Cu is 19 mJm-2, and of Co is 25 mJm−2.

Fig. 5
figure 5

Normalized Young’s modulus and initial yield stress as a function of the thickness ratio between Co and Cu sublayers for the reference nanowire with different crystallographic orientation (at 300 K and 109 s−1). For pure Co, thickness ratio is 0; and for Cu, 1. Corresponding values of the yield strain (%) are indicated numerically from small to large thickness

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The lattice misfit at the interface may also have influence on the mechanical properties. Cu has a lattice parameter of 3.615 Å and fcc Co of 3.544 Å. Therefore, Cu sublayers should experience an additional tensile stress, and Co sublayers should experience a compressive stress (Poisson effect). Figure 6 shows the normalized Young’s modulus and yield stress as a function of the number of interfaces for a Co/Cu nanowire with the same length. The elastic modulus and yield stress increase with increasing number of interfaces, especially for 〈100〉 and 〈110〉 directions. Diameter of these nanowires is 3.5 times lower than that necessary to create a misfit dislocation. Therefore, in this case internal stresses influence only moderately the mechanical properties. Additional studies on this subject are necessaries, especially for different diameters and/or large misfit [33, 34].

Fig. 6
figure 6

Normalized Young’s modulus and initial yield stress as a function of the number of interfaces for nanowires with different crystallographic orientations and the same length (at 300 K and 109 s−1)

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The magnitude of Burgers vector of a {111}〈112〉 partial dislocation in Cu is 1.48 Å, and of a {111}〈110〉 full dislocation is 2.56 Å. Therefore, from the distribution of magnitude of slip vector, it is possible to know the existence of stacking faults and full dislocations. The components of slip vector allow to know the specific partial dislocation nucleated on a {111} plane. In Fig. 7, an image of three nanowires in tension and their corresponding distributions of magnitude and maximum magnitude of slip vector are shown. Yielding of nanowires occurs through the nucleation from edges and propagation across the nanowires of {111}〈112〉 partial dislocations (see in Fig. 7 the top nanowire). Dislocation nucleation in a nanowire seems to be related to the effective critical resolved shear stress (RSS) caused by the external forces in the leading slip direction on the slip plane [35]. According to the distributions of magnitude and maximum magnitude of slip vector in Fig. 7, 〈110〉 nanowires slip via {111}〈112〉 partial dislocation nucleation and propagation. Unlike, in 〈111〉 nanowires, both {111}〈112〉 and {111}〈110〉 slips occur, the latter occurs through two successive {111}〈112〉 slips on the same {111} plane. Finally, in 〈100〉 nanowires, the magnitude of slip vector is not always an indicative factor of the type of generated dislocations since other defects mask slips, given the small space in which the yielding process takes place [36]. Due to this, the maximum magnitude of slip vector is used. In this case, both {111}〈112〉 and {111}〈110〉 slips occur, although the frequency of appearance of the first type is larger. Au〈100〉 nanowires show the same behavior, however Au 〈111〉 nanowires, in tensile yielding, only nucleate {111}〈112〉 partial dislocations [37].

Fig. 7
figure 7

Projection of three nanowires of 2.24 nm in diameter (at 300 K and 109 s−1). The vertical axis (nanowire axis) corresponds to: 〈110〉 , 〈111〉 and 〈100〉 directions (from top to bottom), and the horizontal one, to 〈00-1〉 , 〈11-2〉 and 〈001〉 directions, respectively. Colors are assigned according to the values of s i,max : black (s i,max  < 0.7), grey (orange) (0.7 < s i,max  < 2), light grey (2 < s i,max  < 3) and light black (blue) (s i,max  > 3). Beside this representation, the distributions of magnitude s and maximum magnitude s max of slip vector are shown

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Other interesting aspect is that the Young’s modulus and yield stress are larger in compression than in tension, except for 〈100〉 direction. This asymmetry may be primarily due to effects of surface stresses and the different slip systems active during tensile and compressive yielding [35]. In Fig. 8 both magnitudes are represented as a function of the thickness ratio. The uniaxial Schmidt factor for leading partial slip is larger under compression than under tension for 〈100〉 nanowires unlike for the other two directions; and thus, for these nanowires, it is easier to yield under compression [26, 35]. Differences between tension and compression for both magnitudes are larger for 〈110〉 direction, especially in the case of the elastic modulus. Besides, differences for this magnitude are independent of the thickness ratio. Unlike, the gap for the yield stress between both loading processes is larger in general whether the nanowire is made only of Co.

Fig. 8
figure 8

Young’s modulus and initial yield stress as a function of the thickness ratio between Co and Cu sublayers for the reference nanowire with different crystallographic orientation and tensile and compressive stresses (at 300 K and 109 s−1)

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According to the distributions obtained from the vector slip, nanowires show differences in the slip mechanism for 〈110〉 and 〈100〉 directions (not for 〈111〉 direction). In compression, the former system slip via both {111}〈112〉 and {111}〈110〉 dislocations, and the latter only through {111}〈112〉 dislocations. It is well known that for bulk single crystal the plastic deformation mode depends on the relationship between the Schmidt factor of the leading partial dislocation and that of the trailing partial dislocation. Full dislocation slipping prefers to be activated if the Schmidt factor of the trailing partial dislocation is larger than that of the leading partial dislocation [26, 38]. In our case, the only case in that this rule is not verified is for the 〈111〉 direction in tension in a similar way to what happens in other systems like Au [26].

4 Summary and Conclusions

The mechanical deformation properties of Co/Cu multilayered nanowires (5 nm in diameter ×  140 nm long) have been studied by MD under uniaxial tensile and compressive stresses at different crystallographic orientations (〈110〉, 〈111〉, 〈100〉). The elastic modulus and initial yield stress are approximately independent of strain rate, except at high strain rates. The effect of strain rate on the mechanical behavior is similar for the three crystallographic orientations, except the increase in elastic modulus with increasing strain rate for 〈110〉 nanowires due to the nonlinearity of the stress curve.

Below a normalized vol/sa ratio of about 10, elastic modulus and initial yield stress for tensile strength diverge from the bulk value, increasing in the case of 〈110〉 orientation and decreasing for the other two directions. In the break-up Co/Cu layer for 〈100〉 nanowires, grains undergo a crystallographic reorientation towards 〈111〉 and 〈110〉 directions. Besides, in this layer after the break-up the number of hcp atoms is larger than the number of fcc atoms only for 〈100〉 nanowires. These hcp atoms are preferably in the Co sublayer unlike the other two directions. The fcc atoms are preferably in the Cu sublayer for 〈100〉 nanowires unlike the other two directions. Young’s modulus and yield stress decrease approximately linearly with increasing temperature. Temperature has more influence on the mechanical behavior of 〈100〉 orientation.

〈110〉 nanowires slip via {111}〈112〉 partial dislocation nucleation and propagation. Unlike, in 〈111〉 nanowires, both {111}〈112〉 and {111}〈110〉 slips occur, the latter occurs through two successive {111}〈112〉 slips on the same {111} plane. In 〈100〉 nanowires, both {111}〈112〉 and {111}〈110〉 slips occur, although the frequency of appearance of the first type is larger. Young’s modulus and yield stress are larger in compression than in tension, except for 〈100〉 direction. The uniaxial Schmidt factor for leading partial slip, in addition to effects of surface stresses and the different active slip systems explain this circumstance. Differences between tension and compression for both magnitudes are larger for 〈110〉 direction, and in the case of elastic modulus they are independent of the thickness ratio. Tension and compression show differences in the slip mechanism for 〈110〉 and 〈100〉 directions. In compression, the former system slips via both {111}〈112〉 and {111}〈110〉 dislocations; and the latter, only through {111}〈112〉 dislocations. The relationship between the Schmidt factor of the leading and the trailing partial dislocations can explain this effect for these two directions, but not for 〈111〉 direction that does not show differences in the slip mechanisms.