Effect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study

Before exploring how the stacking faults influence the MAE of bulk Co, we would like to gain an idea of their formation energy. Within the SKKR-ASA scheme, the LSDA total energy can be cast into contributions related to individual atomic cells, E_i, comprising the kinetic energy, the intracell Hartree energy and the exchange–correlation energy, and into the two-cell Madelung (or intercell Hartree) energy, E_Mad [14]. For a simple bulk metal, like hcp Co, E_Mad is, in practice, negligible, while in the presence of stacking faults it gives a non-negligible contribution due to charge redistributions. However, from our self-consistent calculations we found that E_Mad is in the order of 0.1–0.2 meV per stacking fault. Since the typical formation energy of stacking faults is about two orders of magnitude larger than this value, in the following we consider only the layer-resolved (cell-like) contributions to the total energy. In order to check these contributions for artefacts of the appending of the bulk potential, we compared E_i for i =− 10 (layer with effective potential from a self-consistent stacking fault calculation) to E_i for i = 10 (layer with appended bulk potential), since due to the mirror symmetry, these two contributions should be identical. Reassuringly enough, they agreed to within a relative error of 10⁻⁹, which is well within the intrinsic and numerical error of our computational method.

The layer-resolved contributions to the total energy across the systems containing the stacking faults I₁,I₂, E and T₂ are shown in figure 1. Herein we observe the expected mirror symmetry and that the layer-resolved total energy contributions approach the bulk total energy, E_Co =− 37 839.459 eV, towards the edges of each system. From this figure it is obvious that the deviation of E_i from E_Co is significant up to about 15 layers away from the centre of the stacking fault.

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The stacking fault formation energy is defined as the difference in the total energy caused by the presence of the stacking fault. In order to ensure we include enough atomic layers on either side of the stacking fault, we consider the cumulative sums

$$\begin{eqnarray} &&\Delta {E}_{({\mathrm{I}}_{1},\mathrm{E},{\mathrm{T}}_{2})}(N)=\sum _{-N}^{N}{E}_{i}-(2 N+1){E}_{\mathrm{Co}}, \end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray} &&\Delta {E}_{{\mathrm{I}}_{2}}(N)=\sum _{-N+\frac{1}{2}}^{N-\frac{1}{2}}{E}_{i}-2 N{E}_{\mathrm{Co}}. \end{eqnarray} \tag{ 2 }$$

The formation energy of a given stacking fault $X={\mathrm{I}}_{1},{\mathrm{I}}_{2},\mathrm{E},{\mathrm{T}}_{2},{E}_{\mathrm{form}}^{(X)}$, is then defined as

$$\begin{eqnarray} &&{E}_{\mathrm{form}}^{(X)}=\mathop{\lim }\limits _{N\rightarrow \infty }\left (\Delta {E}_{X}(N)\right ). \end{eqnarray} \tag{ 3 }$$

The calculated values of ΔE_X(N) are shown in figure 2. Quite obviously, for all types of stacking faults, nearly 30 atomic layers (i.e., 15 layers on either side of the stacking fault) are required in order to obtain well-converged stacking fault formation energies. The fact that the effect of the stacking fault is relatively long-ranged could have significant impact on nano-sized systems as the formation energy and, consequently, the likelihood of occurrence of a stacking fault could be different depending on its location in relation to, e.g., other imperfections as well as surfaces or interfaces in the sample. We obtain the following formation energies, with a possible error of ∼0.1–0.2 meV due to the Madelung energy not being included:

$$\begin{eqnarray*} &&{E}_{\mathrm{form}}^{({\mathrm{I}}_{1})}\approx 1 6~\mathrm{meV}~\approx 4 0~\mathrm{mJ}~{\mathrm{m}}^{-2} \end{eqnarray*}$$

$$\begin{eqnarray*} &&{E}_{\mathrm{form}}^{({\mathrm{I}}_{2})}\approx 4 8~\mathrm{meV}~\approx 1 2 2~\mathrm{mJ}~{\mathrm{m}}^{-2} \end{eqnarray*}$$

$$\begin{eqnarray*} &&{E}_{\mathrm{form}}^{(\mathrm{E})}\approx 6 2~\mathrm{meV}~\approx 1 6 0~\mathrm{mJ}~{\mathrm{m}}^{-2} \end{eqnarray*}$$

$$\begin{eqnarray*} &&{E}_{\mathrm{form}}^{({\mathrm{T}}_{2})}\approx 3 9~\mathrm{meV}~\approx 1 0 0~\mathrm{mJ}~{\mathrm{m}}^{-2}. \end{eqnarray*}$$

As expected, all stacking faults incur a positive change in the total energy. Of the four types of stacking faults considered here, the intrinsic stacking fault I₁ has the lowest formation energy and the stacking fault E exhibits the highest one. While there is no available experiment in literature, the overall results agree well with e.g. [8]: the extrinsic stacking fault formation energy for the close-packed fcc metals in this study is generally significantly larger than that of the twin fault. Moreover, our calculated values for the hcp Co deformation stacking fault I₂ and the hcp Co extrinsic fault E are close to those reported by Crampin et al for Ni (which is next to Co in the periodic table) [8]: 180 (125) mJ m⁻² for the intrinsic deformation stacking fault and 150 mJ m⁻² for the extrinsic fault.

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Because it is calculated directly from the band energy, the MAE can naturally be resolved into layer-dependent contributions, D_i, see [19]. These layer-resolved contributions are depicted in figure 3 for the different types of stacking faults. Note that the mirror symmetry is well reproduced in the layer-resolved MAE contributions for all stacking faults. Moreover, the MAE approaches the bulk MAE, K_Co = 84.4 μeV, towards the edges of all four systems. For stacking faults of type I₁,I₂ and T₂, the MAE contributions become negative at the centre of the fault, favouring thus an in-plane easy axis in these layers. This could indicate that these types of stacking faults may act as pinning sites. For the type E stacking fault, the layer-resolved MAE contributions near the centre are also reduced, retaining, however, very small positive values.

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Furthermore, we note that all stacking faults induce long-ranged oscillations in the MAE. For layers of about |i| > 15, the four stacking faults exhibit very similar trends in the layer-resolved MAE contributions. In other words, at about 15 layers away from the stacking fault, the presence of a stacking fault still influences the MAE, while the particular type of the stacking fault is less significant. This will, however, obviously depend on the size of the sample.

The long-ranged oscillations in the MAE could cause significant finite-size effects in the experimental determination of the MAE of nano-sized samples. We therefore consider the following cumulative sums,

$$\begin{eqnarray} &&{K}_{({\mathrm{I}}_{1},\mathrm{E},{\mathrm{T}}_{2})}(N)=\sum _{-N}^{N}{D}_{i}-(2 N+1){K}_{\mathrm{Co}}, \end{eqnarray} \tag{ 4 }$$

and

$$\begin{eqnarray} &&{K}_{{\mathrm{I}}_{2}}(N)=\sum _{-N+\frac{1}{2}}^{N-\frac{1}{2}}{D}_{i}-2 N{K}_{\mathrm{Co}}, \end{eqnarray} \tag{ 5 }$$

where the MAE of the stacking fault systems of finite width is related to the MAE of hcp Co.

Figure 4 shows K_X(N) for the four different stacking faults as a function of N. Surprisingly, for N ≥ 5 the I₂ type stacking fault appears to increase the MAE, i.e., to strengthen the easy axis (0001) (positive effect). As seen from figure 3, this is due to the positive MAE contributions induced by the stacking fault on neighbouring layers. These apparently outweigh the strongly negative MAE contributions induced in the centre of stacking fault type I₂. This is an unexpected result as stacking faults are typically reported to lower the MAE (see e.g. [4]). It should be noted, however, that, of the stacking faults studied here, type I₂ has the next highest formation energy and it is therefore less likely to occur in an equilibrated sample. For stacking faults of types I₁, E and T₂, the overall change in the MAE with respect to hcp Co is negative (negative effect). As noted earlier, in the vicinity of these stacking faults, the easy direction is rotated normal to the (0001) axis. This is consistent with the reduction in the total MAE observed experimentally by Sokalski et al in [4].

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It is quite a remarkable feature that, as seen from figure 4, the layer-resolved MAE contributions do not settle until at about approximately 35 layers on either side of the stacking fault. This long-ranged behaviour could give rise to significant finite-size effects in nano-sized samples. Moreover, this might have consequences for theoretical investigations into the formation and effects of stacking faults on magnetic properties. Typically, in Monte Carlo simulations of stacking faults, interaction between stacking faults is kept to around three neighbouring planes [4]. In light of our results, this appears to be an uncertain assumption.

Experimentally, the presence of stacking faults is normally quantified in terms of the stacking fault density, which is partly a measure of how close the stacking faults are located. As the simplest assumption, the change in the MAE due to the presence of a number stacking faults in a sample is approximated by the sum of the changes in the MAE due to each individual stacking fault. If this were the case, the effect of an isolated stacking fault on the MAE could quite straightforwardly be transformed into the change in MAE as a function of the stacking fault density. However, the long-ranged oscillations in the MAE caused by the presence of each stacking fault indicate that the situation is far more complex.

In particular, we considered two stacking faults of type I₁, separated by three atomic layers. In other words, the system exhibits the composite stacking fault:

$$\begin{eqnarray*} &&\ldots ~\mathrm{A}~\mathrm{B}~\mathrm{A}~\mathbf{B}~\mathrm{C}~\mathrm{B}~\mathrm{C}~\mathbf{B}~\mathrm{A}~\mathrm{B}~\mathrm{A}\ldots . \end{eqnarray*}$$

Note that by removing one of the two C–B pairs of atomic layers, the twin-like stacking fault T₂ is obtained. We have chosen three atomic layers between the centres of the two stacking faults, since in dynamical models it is often used as the distance beyond which the interaction between stacking faults is neglected (see e.g. [4]). Moreover, we deal with a pair of I₁ type stacking faults because this type of stacking fault has the lowest formation energy and is, therefore, expected to occur more commonly than the other three types of stacking faults.

The difference between the layer-resolved MAE contributions and the MAE of bulk hcp Co,

$$\begin{eqnarray} &&\Delta {D}_{i}^{({\mathrm{I}}_{1}{\mathrm{I}}_{1})}={D}_{i}^{({\mathrm{I}}_{1}{\mathrm{I}}_{1})}-{K}_{\mathrm{ Co}}, \end{eqnarray} \tag{ 6 }$$

is shown in figure 5 for the composite stacking fault. As a comparison, we also show the average deviations in the layer-resolved MAE contributions from the bulk MAE of two independent type I₁ stacking faults,

$$\begin{eqnarray} &&\Delta {D}_{i}^{({\mathrm{I}}_{1}+{\mathrm{I}}_{1})}=\frac{1}{2}\left ({D}_{i+2}^{({\mathrm{I}}_{1})}+{D}_{i-2}^{({\mathrm{I}}_{1})}\right )-{K}_{\mathrm{ Co}}. \end{eqnarray} \tag{ 7 }$$

If $\Delta {D}_{i}^{({\mathrm{I}}_{1}{\mathrm{I}}_{1})}$ and $\Delta {D}_{i}^{({\mathrm{I}}_{1}+{\mathrm{I}}_{1})}$ were equal for each atomic layer i, the change of the MAE due to the presence of the composite stacking fault would be exactly twice that of a single I₁ stacking fault. However, as shown in figure 5, $\Delta {D}_{i}^{({\mathrm{I}}_{1}{\mathrm{I}}_{1})}$ and $\Delta {D}_{i}^{({\mathrm{I}}_{1}+{\mathrm{I}}_{1})}$ deviate significantly, particularly in the layers |i| ≤ 2, i.e., in the layers between the two stacking faults. Beyond |i| > 3, the magnitudes of the MAE contributions are similar for $\Delta {D}_{i}^{({\mathrm{I}}_{1}{\mathrm{I}}_{1})}$ and $\Delta {D}_{i}^{({\mathrm{I}}_{1}+{\mathrm{I}}_{1})}$, but with a phase shift of approximately one layer.

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Similarly to the case of single stacking faults, we calculate the cumulative sum of the MAE contributions of the composite stacking fault,

$$\begin{eqnarray} &&{K}_{{\mathrm{I}}_{1}{\mathrm{I}}_{1}}(N)=\sum _{i=-N}^{N}{D}_{i}^{({\mathrm{I}}_{1}{\mathrm{I}}_{1})}-(2 N+1){K}_{\mathrm{ Co}}, \end{eqnarray} \tag{ 8 }$$

and plot it in figure 6. Apparently, K_I1I1(N) converges to approximately −1.18 meV for large N, which is almost three times the change of the MAE of the single type I₁ stacking fault (∼−0.40 meV, see figure 4). Also shown in figure 6 is the difference K_I1I1(N) − 2K_I1(N), which appears to settle at approximately −0.38 meV. In other words, the two stacking faults interact to yield a stronger negative effect on the total MAE as compared to two isolated type I₁ stacking faults. This appears to be mainly due to MAE contributions from the atomic layers located in between the two type I₁ stacking faults. This could have significant consequences for predicting the resulting MAE in dynamical models used to explain experimental data. To draw any definite conclusions, a systematic study of the stacking fault types and separations would be required. We expect that such a study would be computationally extremely intensive as interlayers (or supercells) of up to approximately 160 atomic layers would be required in order to reach the limit in which the two stacking faults are far enough apart not to interact.

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