Stacking fault energies of face-centered cubic concentrated solid

solution alloys

Shijun Zhao1,*, G. Malcolm Stocks1,*, Yanwen Zhang1,2

1
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831,

USA
2
Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA

Email: zhaos@ornl.gov and stocksgm@ornl.gov.

Keywords: concentrated solid solution alloys, stacking fault energy, first-principles calculations,
vibrational entropy

1
Abstract

We report the stacking fault energy (SFE) for a series of face-centered cubic (fcc) equiatomic

concentrated solid solution alloys (CSAs) derived as subsystems from the NiCoFeCrMn and NiCoFeCrPd

high entropy alloys based on ab initio calculations. At low temperatures, these CSAs display very low

even negative SFEs, indicating that hexagonal close-pack (hcp) is more energy favorable than fcc

structure. The temperature dependence of SFE for some CSAs is studied. With increasing temperature, a

hcp-to-fcc transition is revealed for those CSAs with negative SFEs, which can be attributed to the role of

intrinsic vibrational entropy. The analysis of the vibrational modes suggests that the vibrational entropy

arises from the high frequency states in the hcp structure that originate from local vibrational mode. Our

results underscore the importance of vibrational entropy in determining the temperature dependence of

SFE for CSAs.

2
1. Introduction

Single-phase concentrated solid solution alloys (CSAs) with two or more multiple principal elements

situated in a simple face-centered cubic (fcc) lattice have received great interest owing to their exceptional

mechanical properties [1–3] and significantly improved radiation resistance [4–8] compared to

conventional alloys. Most notably, several Ni-based fcc CSAs including high entropy alloys (HEAs) such

as NiCoCr [9] and NiCoFeCrMn [10] exhibit remarkable ultimate tensile strength, uniform elongation

and fracture toughness at cryogenic temperatures. Experimental observations demonstrate that the high

strength of the above CSAs is closely related to a transition in the deformation mechanism from

conventional dislocation glide to nano-twining as the temperature is decreased [1–3]. However, the

detailed information on the deformation and dislocation properties is still rather limited in this unusual

class of alloys. The atomic-level stiffness and lattice instability may have a critical impact on local

minimum states in the energy landscape that controls energy dissipation and defect evolution in a non-

equilibrium radiation condition. Understanding the stacking fault energies (SFEs) in CSAs may pave the

way for designing materials intrinsically radiation resistant [8].

It is well established that the deformation mechanism and dislocation behavior in materials depend largely

on their SFEs [11,12]. SFE represents the energy associated with interrupting the normal stacking

sequence of a crystal plane, which significantly affects the mobility of defects, defect clusters and

dislocations, and therefore influences the defect evolution and material performance in extreme

environments (temperature, pressure, irradiation, etc). It is commonly assumed that deformation twinning

is favored in low SFE materials and dislocation slip dominates in high SFE materials. To fully interpret

the lattice stiffness, dislocation mobility and brittle-ductile transition, the generalized stacking fault

energy (GSFE) curve (γ surface) is often utilized, which involves both intrinsic (stable) and unstable

stacking-fault energies. The γ surface gives the interplanar potential energy landscape associated with the

sliding between two adjacent planes in a slip system. It also provides the restoring force described in the

famous Peierls-Nabarro model for dislocations [13,14]. In fcc structures, {111}<110> shear deformation

3
is the major operative slip system [15]. As a result, the γ surface for this slip system can be used to

understand a vast number of phenomena related to dislocation loop formation and growth, dislocation

movement, plastic deformation, crystal growth and phase transitions under both thermal and irradiation

conditions. Indeed, previous atomistic simulations have shown that the γ surface in a fcc lattice within the

{111} plane gliding along the <112> direction provided sufficient fundamental information to properly

classify the deformation modes [11,12].

The properties of SFE and dislocation properties in several CSAs have been studied previously [16–20].

Specifically, the calculation based on the ab initio derived elastic constants and the stacking fault

probability measurements using X-ray diffraction analysis shows that NiFe, NiFeCr, NiFeCrCo, and

NiFeCrCoMn CSAs have rather low SFEs [16]. A low SFE for NiCoFeCrMn is also consistent with

experimental observations regarding dislocation dissociation and deformation twins [17,18]. Interestingly,

SFEs in different CSAs are found to have only a weak dependence on the number of elements present in

the alloy system, indicating the importance of both the types of alloying elements and their concentrations.

Using the coherent potential approximation (CPA) method based on a supercell consisting of 9 fcc [111]

layers, a large positive temperature dependence of the SFE in NiFeCrCoMn has been predicted [19],

which is in line with the deformation mechanism transition observed experimentally. Whilst these studies

shed considerable light on the properties of SFEs in CSAs, the correlation between the effect of chemical

disorder and the magnitude of SFE in CSAs is still not clear. Studies of the γ surface of CSAs, which are

of great significance in understanding the dislocation and ductility of these alloys, have potential to reveal

the relationship between the number and types of alloying elements and the mechanical response or

radiation response of CSAs, and therefore their possible applications in nuclear technologies [8].

In this work, we study the SFEs and γ surfaces for a series of Ni-based CSAs based on first-principles

calculations using both supercell methods and the axial interaction model (AIM) [21,22]. In the supercell

method, the whole γ surface is obtained by sliding the upper half of the cell with respect to the lower half.

4
By mapping the stacking sequence into a one-dimensional Ising model [21,22], the SFE can also be

determined within the AIM method from the energies of structures having different stacking sequences.

In this work, three structures, namely fcc, hexagonal close-pack (hcp) and double hexagonal close packed

(dhcp) are considered. Including fcc and hcp corresponds to keeping the interaction terms up to the

second neighbor planes (AIM1), while the dhcp structure is required to account for interactions from the

third neighbor planes (AIM2). We show that some CSAs exhibit low, even negative SFE at low

temperature that suggests hcp is more stable than fcc. However, calculation of the temperature

dependence of SFE for some CSAs reveals a hcp-to-fcc transition that is driven by the vibrational entropy.

These results may help to understand the lattice stability and dislocation behaviors in CSAs.

2. Methods

2.1 First-principles calculations

Ab initio total-energy calculations were based on density-functional theory as implemented in the Vienna

ab initio simulation package (VASP) [23]. A gradient corrected functional in the Perdew-Burke-

Ernzerhof (PBE) form was used to describe the exchange and correlation interactions [24]. Electron-ion

interactions were treated within the projector-augmented-wave PAW method [25]. Standard PAW

pseudopotentials distributed with VASP were adopted. The energy cutoff for the plane-wave basis set was

set to be 270 eV unless indicated. The energy convergence was set to be 10-6 eV. All calculations were

performed with spin-polarization to account for the magnetic properties of considered alloys.

2.2 Special quasirandom structures

The chemical disorder of CSAs was modeled using special quasirandom structures (SQS) developed to

predict self-averaging quantities of alloys using finite size supercells [26]. In this work, the SQS structure
𝑚
was constructed by optimization of the Warren-Cowley short range order (SRO) parameters 𝛼𝑖𝑗 [27,28]

used to describe the chemical ordering around an atomic species as defined by:

5
 𝑚
𝑚 𝑝𝑖𝑗
𝛼𝑖𝑗 =1− 𝑐𝑗
. (1)

𝑚
Here 𝑝𝑖𝑗 is the probability of finding atomic species j around an atom of type i in m-th neighboring shell

and 𝑐𝑗 is the atomic concentration of type j. For a totally random solution, all SRO parameters should be

zero. A departure from zero indicates the extent to which atom-atom correlations exist. Positive values

show a tendency toward clustering (the predominance of i-i and j-j pairs) while negative values suggest a

tendency toward ordering (the predominance of i-j pairs).

The optimization of SRO was achieved by swapping element species with a Monte Carlo algorithm. In

this process, the SRO was firstly calculated for a randomly populated supercell of the desired composition.

A cost function f was defined based on the assigned weight (wk) and the input current SRO parameter

𝑖𝑛𝑝𝑢𝑡 𝑚
(𝛼𝑖𝑗 and 𝛼𝑖𝑗 ) at each shell:

f   w
m i, j
m |  ijinput   ijm | , (2)

where the summation is over all the considered shells and all the short-range order parameters in each

shell. For the choices of weights, we ensure the highest weight is given to the first-nearest-neighbor shell

since this shell is the most important. The optimization of the cost function proceeded as follows: firstly

an event was defined regarding whether to attempt a random exchange between a pair of different atoms.

The decision whether to accept or to reject the exchange was made according to the standard Metropolis

scheme. Specifically, if the change of Δf =f(new)-f(initial) was negative, then the exchange was accepted.

If the change was positive, then the exchange was accepted with the probability

∆𝑓
𝑝 = exp [− ], (3)
𝑘𝐵 𝑇

where kB is the Boltzmann constant and T is the temperature at current step. By simulated annealing from

a high temperature (ranging from 5000 to 50000 K), the SRO was optimized and the atomic

configurations evolved to a totally random structure. For all the structures used in the calculations, the

SROs were optimized to those corresponding to a total random alloy for at least the first three neighbor

6
shells. In this way, the closest random structure at a given supercell size is generated. Multiple structures

corresponding to the same alloy composition are generated through different annealing temperatures.

2.3 Stacking fault energy calculation

Two approaches were used to calculate SFE. In the first approach, a parameterized model is obtained by

mapping the stacking sequence onto a one dimensional axial next-nearest neighbor Ising (ANNNI) model

[21,22]. In this model, the ab initio total energy is expressed as a sum of coupling energies between

individual planes that is then truncated to a finite number of neighboring layers, under the assumption that

the fault interactions are short ranged. The SFE can then be deduced from the calculated coupling

constants. This approach can be generalized to the AIM model by including higher order terms in the

expansions. In the second approach, a supercell is used to represent the disordered state and a stacking

fault is then introduced by sliding one half of the crystal (supercell) with respect to the other half with an

appropriate glide vector. In this method, a slab configuration is usually adopted. Note that a periodic

supercell including two stacking faults can also be used to calculate SFE, but it would require a

significant large size to avoid interactions between adjacent stacking faults. While the intrinsic and

extrinsic stacking fault energies as well as twin boundary energies can be obtained at the same time using

the parameters obtained from the AIM model, the supercell approach is indispensable in order to calculate

the energy variation during the sliding process, i.e. γ surface.

In the AIM model [21,22], three structures were used to calculate the coupling constant, namely, fcc, hcp

and dhcp structures. In this way, the energies of the intrinsic stacking fault (ISF) can be expressed in

terms of the energy of these three structures:

(𝐸ℎ𝑐𝑝 +2𝐸𝑑ℎ𝑐𝑝 −3𝐸𝑓𝑐𝑐 )

𝛾𝐼𝑆𝐹 = 𝐴
, (4)

where Edhcp, Ehcp and Efcc are the total energy per atom of the dhcp, hcp and fcc phase, respectively. In

practice, these energies were calculated from SQS supercells with 144 atoms. As a first-order

7
approximation, the stacking fault energy can be given by the energy difference between hcp and fcc

structures: γISF ~ γESF ~ 2(Ehcp-Efcc)/A. These two methods are denoted as AIM1 and AIM2, respectively.

Both the cell parameters and internal coordinates were fully optimized in the calculations. Therefore, a

higher energy cutoff of 350 eV was used since the change of cell volume may induce errors in the Pulay

stress. To minimize the effect of configuration dependent energies, the energies from three independent

SQS supercells were calculated and the results were averaged.

In the supercell approach, a SQS consisting of 9 [1̅1̅1] planes containing 108 atoms was constructed by

considering three-dimensional periodic boundary conditions. After the configuration was obtained, a

vacuum region of more than 10 Å was added and a stacking fault was inserted by rigidly shifting the

upper four [1̅1̅1] layers in the [112] or [1̅10] directions. This process was repeated for every possible

position of the stacking fault and the stacking sequence was manually changed in order to ensure that the

stacking fault was always located in the middle of the supercell. For example, the sequence of perfect

stacking A1B1C1A2B2C2A3B3C3 was rearranged to C2A3B3C3A1B1C1A2B2 in order to calculate the GSFE

curve between the A1B1 planes. In this way, 9 GSFE curves were calculated and subsequently averaged to

obtain a single, configurationally averaged GSFE curve. The standard deviation for the averaged SFE

ranges from 11-55 mJ/m2, depending on the alloy complexity. The following numerical details are

adopted in the calculations. The Brillouin zone (BZ) was sampled using Gamma-point-based 4×4×2 mesh.

The Methfessel-Paxton smearing method with 0.1 eV smearing width was used for integration over the

BZ [29]. For each rigid shift, the atomic positions were relaxed along the close-packed direction by

minimizing the Hellmann-Feynman forces on each atom to less than 10-2 eV/Å, with the cell parameters

fixed. The SFE was calculated from:

1
𝛾𝐼𝑆𝐹 = 𝐴 (𝐸𝐼𝑆𝐹 − 𝐸0 ), (5)

where EISF and E0 are the energies of configurations with and without the ISF respectively and A is the

ISF area. For each slip system studied, the stacking-fault vectors were varied from 0.0b to 1.0b in steps of

8
0.2b, where b was the corresponding Burgers vector. Further calculation details and some discussions

regarding the precision of the calculations are provided in the Appendix.

Available experimental [30] and theoretical [4,30] evidence suggests that the magnetic ground states of

CSAs are rather complicated. Experimentally, some of the CSAs studied (e.g. NiFe, NiCo and NiCoFe)

exhibit robust ferromagnetism, some (e.g. NiCoCr) show magnetically order only at low temperature

(<100K), whilst others (e.g. NiFeCoCrMn) show no signs of magnetic ordering at any temperature.

Nevertheless, for all of the systems studied, spin-polarized DFT admit a collinear spin polarized state that

is lower in energy than the non-magnetic (non-spin polarized) solution (Appendix A.3). As a result, all

calculations were performed spin polarized in order to account for the energy associated with moment

formation. The lattice parameter for the ferromagnetic state is found to be much closer to experimental

value than the non-spin polarized solution, a strong indication that the energetics is much better described

by spin-polarized calculations. This is a well-known consequence of tendency of non-spin polarized DFT

calculations to underestimate the equilibrium atomic volume of metallic magnets and the volume

expansion that accompanies moment formation in spin-polarized calculations.

2.4 Free energy calculation

In order to estimate the SFE at finite temperature based on DFT, the AIM1 model was used. In particular,

the Helmholtz free energy of the hcp and fcc structure at volume V and temperature T was calculated

using:

F(V,T)=E0(V)+Fele(V,T)+Fvib(V,T)+Fmag(T)+Fconfig(T)+PV, (6)

where E0(V) is the total energy of the structure at 0K, Fele, Fvib, Fmag and Fconfig are the thermal electronic,

vibrational, magnetic and configurational free energy, respectively. A standard approach to the

calculation of Fele and Fvib was employed [31]. In particular, the phonon free energy was calculated using

the supercell method with the force constants obtained from density functional perturbation theory, as

implemented in VASP code. The code PHONOPY [32] was then used to calculate the phonon properties

9
within the harmonic approximation. Specifically, a 72-atom SQS supercell was employed for both the fcc

and hcp structures. A 4×4×2 K-point mesh was used to perform reciprocal space integrals with an energy

cutoff of 350 eV in force calculations. We have carefully checked the phonon dispersions and no negative

frequencies are observed.

Currently, there is no direct method to calculate Fmag within DFT. To investigate the importance of Fmag,

we have resorted to the widely used semi-empirical method [33] as implemented in the ATAT code [34].

In this method, the required input is the Curie temperature and the mean atomic magnetic moment. We

have calculated the Fmag for those CSAs with known Curie temperature and the results show that Fmag is

rather small in all cases. For example, the calculated maximal Fmag is only -0.1 meV/atom for NiCoFeCr

based on a Curie temperature of 120 K and average magnetic moment of 0.65 μB/atom. This value is an

order of magnitude less than the contribution from lattice vibrations. As a result, this term is not taken

into consideration throughout this work. It should be noted that the small Fmag may come from the

shortcoming of this method as indicated in a previous CPA study that Fmag also contributes to the phase

stability of fcc NiCoFeCrMn [35].

Once all of the above contributions to the free energy have been determined, within the AIMD1 model,

the temperature dependence of the SFE can be estimated from:

FISF(T)=2(Fhcp(T)-Ffcc(T))/A=2(ΔE0+ΔFele+ΔFph)/A, (7)

where ΔE0, ΔFele and ΔFph are the differences in zero-temperature energy, electronic free energy and

phonon free energy between the hcp and fcc structures, respectively. Here, the PV term is omitted as the

ambient pressure is set to 0 GPa and the volume change between hcp and fcc is negligible. The

configurational free energy Fconfig(T)=Econfig-TSconfig(T) can be obtained from mean field approximation

with Econfig being the energy of an idealized alloy system having a lattice parameter corresponding to the

temperature investigated and Sconfig is approximated by the ideal entropy of mixing Sconfig=−kB Σci lnci

where ci was the mole fraction of species i. Since Sconfig is equal for both fcc and hcp structure (the same

10
concentration in both structures), this term has no effect on phase stability and consequently is ignored

[36].

3. Results

3.1 Stacking fault energies of pure Ni

The γ surface for pure Ni is studied first. In the GSFE curve for {1̅1̅1}<112> shear deformation, the first

energy maximum determines the unstable stacking fault energy (γUSF), which represents the energy barrier

[112]
for the generation of a stacking fault. It occurs around 𝑏𝑝 /2 , where the Burgers vector is

[112] [112]
𝑏𝑝 =(1/6)[112]a0 with a length of 𝑎0 /√6. In the calculations, the barrier occurs at 0.56𝑏𝑝 . The first

[112]
energy minimum is located at 𝑏𝑝 which defines the intrinsic stable stacking fault energy (γ ISF). The

[112]
position of 𝑏𝑝 corresponds to the intrinsic stacking fault structure, where a full dislocation dissociates

into a pair of Shockley partials. The GSFE curve corresponding to {1̅1̅1}<[1̅10]> shear deformation

shows a symmetric behavior with a single peak at the center of the deformation. The present calculation

̅ 10] direction is higher than that along [112] direction. The calculated
suggests that the γUSF along the [1

γISF and γUSF for Ni are 133 and 281 mJ/m2 respectively, in agreement with previous theoretical and

experimental results [37]. From the AIM method, the γISF is determined to be 127 mJ/m2 in line with a

previous study [38]. A detailed comparison with existing results is provided in Table 1. These results

suggest that the simulation setup (slab or alias shear) has little influence on the calculated γISF in pure Ni.

Moreover, the calculation of γISF in Ni is found to be in better agreement with experiment when based on

the generalized gradient approximation (GGA) [24] than when based on the local density approximation

(LDA) [39].

11
Table I Calculated stable stacking fault energy (γISF) and unstable stacking fault energy (γUSF) in pure Ni.

Different methods are denoted by exchange-correlation functional (LDA [39], PBE [24] and PW91[40])

and simulation setups.

γISF (mJ/m2) γUSF (mJ/m2) Refs.

133 281 Present, supercell
127 Present, AIM
182 LDA, slab [41]
110 273 PW91, slab [42]
121 230 PBE, slab [43]
127.2 PBE, slab [38]
130.35 PBE, slab [44]
142 863 PBE, slab [45]
131-137 305 PW91, simple alias shear [19]
127-136 263 PW91, pure alias shear [19]
120-130 Expt.[37]

3.2 The γ surfaces of concentrated solid solution alloys

̅ 10] directions for the full series of CSAs that were

Fig. 1 presents the GSFE curves along the [112] and [1

considered. All CSAs exhibit low γISF compared to pure Ni. Most notably, negative γISF are observed for

NiCo, NiCoCr, NiCoFeCr and NiCoFeCrMn. The implication of the latter results for the stability of a

stacking fault as well as the relative stabilities between fcc and hcp structures will be discussed in the

following section.

The GSFE curve in the [112] direction indicates that CSAs with negative γISF also have larger γUSF than

other CSAs, which suggests a low ratio of γISF/γUSF. Since γUSF determines the energy barrier for the

nucleation of trailing dislocations and γISF measures the stability of a stacking fault, the low ratio γISF/γUSF

indicates a high probability of twinning [46]. For the {1̅1̅1}<[1̅10]> slip system, it is observed that a

[1̅10] dislocation dissociates into two [1̅10]/2 paritial dislocations that further dissociate into two

Shockley partial dislocations. This process can be represented by: [1̅10]/2 → [2̅11]/6 + [1̅21̅]/6, which

12
 ̅ 10] direction, the GSFE curve is symmetric in pure Ni. For
is typical of a fcc lattice [47]. Along the [1

CSAs, the random arrangement of different elements leads to site-dependent energies, which generates an

asymmetric γ surface along this direction. However, the deviation is rather limited and the fcc structure of

these CSAs is well preserved.

3.3 Stacking fault energy of concentrated solid solution alloys

The SFE from both supercell and AIM methods are summarized in Fig. 2 for all CSAs studied. These is

an overall consistency between the results from these two methods. In particular, the CSAs NiCo, NiCoCr,

NiCoFeCr and NiCoFeCrMn all have negative SFEs in both supercell and AIM approaches. For NiCo, a

previous calculation gave values between 0-32 mJ/m2 [38]. The negative SFE about -10 mJ/m2 found in

this work indicates that the fcc lattice of NiCo becomes unstable with respect to the formation of an

intrinsic stacking fault or hcp phase at 0K. The calculations based on the AIM model indeed show that the

hcp phase of NiCo has a lower energy than fcc phase by around 2.1 meV/atom. This results in a negative

SFE about -12 mJ/m2, in good agreement with the average result from the supercell method. Therefore,

based on ab initio calculations, the fcc phase is not the true stable phase for NiCo at 0 K. This is

inconsistent with the Ni-Co phase diagram where that the fcc-hcp transition does not occur until the

concentration of Co exceeds 65% [48]. A similar inconsistency was also observed in other alloy systems,

e.g. Cu-Al [49]. The SFE for NiFe is 105 mJ/m2, in good agreement with previous result of 100 mJ/m2

calculated by combing the experimental data of stacking fault probability and DFT calculations of elastic

constants [16]. However, our results for NiCoFeCr and NiCoFeCrMn are negative at 0 K compared to

their results. For NiCoFeCrMn, our calculated stability of hcp relative to fcc is in accordance with

previous work [35]. As to be discussed later, this inconsistency between the established phase diagram

and ground state DFT calculations at 0 K can be resolved by including the effects of temperature. In

particular, the differences in vibrational (phonon) entropy between the fcc and hcp phases that originate

from the disorder induced difference in the interlayer spacing and local lattice distortions between the two

phases.

13
The similar results from AIM1 and AIM2 shown in Fig. 2 suggest that the range of interatomic

interaction in these CSAs is restricted to the first two neighbor shells. The error bars represent the

fluctuation induced by the random arrangement of different elements. Since interatomic interactions are

short-ranged in these CSAs, the SFE depends on the details of the specific stacking fault plane. Taking

NiCo as an example, because Co favors the hcp structure [50], the more Co in the fault plane, the lower

the SFE becomes. Consequently, the SFE in CSAs exhibits a distribution that significantly influences

local dislocation behaviors as observed experimentally [18] and theoretically [20]. The results in Fig. 2

are averaged over all the planes from SQS supercell, which corresponds to configurational averaging,

thereby reflecting the overall fault energies. It is worth noting that these results calculated from SQS cells

assume homogeneous alloy structures. If short range order is present, the inhomogeneity should be taken

into account in order to analyze the fault energy at specific local chemical composition [18].

The impact of elemental coupling on the SFE can be clarified by studying the different constituent

elements in these CSAs. The comparison among binaries suggests that the coupling between Ni and Co

has the largest effect on reducing the SFE. The results for ternaries reveal that Cr decreases the SFE

significantly compared to Fe. For quaternaries, Cr tends to decrease the SFE more than Mn while Mn is

more effective in reducing the SFE than Pd in quinaries. It has long been considered that the d-electrons

play an important role in determining SFE in transition metals and their alloys [51]. Thus, to better

understand the underlying physical mechanism, the orbital and spin resolved local density of states (DOS)

of d electrons are analyzed (Fig. 3) for the alloy pairs NiCoCr/NiCoFe, NiCoFeCr/NiCoFeMn and

NiCoFeCrMn/NiCoFeCrPd. In each pair, the CSAs have SFE of opposite sign, but only a single species is

substituted. Within Fig. 3, the upper panels ((a), (b), and (c)) correspond to the CSAs having negative

SFE while the lower panels correspond to CSAs with positive SFE. The total DOS of d states at Fermi

level N(Ef) calculated by summation of the spin-up and spin-down channels is also presented in each

panel. For these transition metal systems, the DOS are dominated by d electrons, an indication of the

14
short-range interactions between atoms. When comparing these pairs of systems it is usefully to focus on

the local DOS of d states of the substituted species within the [111] plane. Comparing the NiCoCr and

NiCoFe systems, the relevant Cr and Fe DOS presented in Fig. 3(a) and (b) show only minor changes as

the structure transforms from perfect fcc stacking (black) to the stacking fault configurations (red). In

addition, the occupancy is barely affected, suggesting that the hybridization between states changes little

during the glide process. The most striking difference between Cr and Fe containing systems is in the

distribution of electronic states around Fermi level. As indicated in Fig. 3, the N(Ef) is higher for Cr than

for Fe. For low magnetic moment Cr-sites (MCr ~ ‒0.14 μB) the spin splitting between majority and

minority d-bands is small with the result that the Fermi energy lies in the high DOS region for both spin

channels, see Fig. 3(a). For high moment Fe-sites (MCr ~ 2.62 μB) the large spin splitting results in the

Fermi energy being above the high DOS majority-spin d-band and a diminished N(Ef). Therefore, the

decrease of SFE in NiCoCr relative to NiCoFe correlates closely with the N(Ef). This is in agreement with

the argument that SFE in transition metals decreases with increasing electron state density at the Fermi

energy [52]. Comparing the DOS on the substituted Cr and Mn atoms in the NiCoFeCr/NiCoFeMn and

the substituted Mn and Pd atoms in the NiCoFeCr/NiCoFeMn pair leads to the same conclusion. This is

true for all CSAs analyzed, supporting the past conjecture that the relative magnitude of SFE can be

deduced from the electronic structure.

3.4 Temperature dependence of the stacking fault energies

To study the temperature dependence of SFE, the AIM1 model is employed to calculate the free energy

difference between fcc and hcp phases. The temperature dependence of the contributions from the

electrons (ΔFele) and from the phonons (ΔFph), together with their combined effect on the SFEs is

provided in Fig. 4 for pure Ni and a subset of the CSAs studied. For pure Ni, ΔFele is much larger than

ΔFph and both decrease with increasing temperature. As a result, the SFE of pure Ni decreases with

increasing temperature. This is in agreement with previous results using the pure alias shear method [19].

15
The decrease of SFE is almost linear with respect to temperature above 300K and the coefficient of dγ/dT

is around -0.04 mJ/m2/K.

As can be seen in Fig. 4, the temperature dependence of ΔFele, ΔFph and corresponding SFE of the CSAs is

significantly different from that of Ni. For NiCo and NiCoCr, both ΔFele and ΔFph increase with

increasing temperature. In addition, the contribution from ΔFph is larger than that from ΔFele. As a result,

the SFE for both NiCo and NiCoCr increases with temperature. For NiCoFe, the dominant ΔFph decreases

with increasing temperature, which decreases SFE at high temperatures. Note that because of the strong

jagged EDOS of NiCoCr around the Fermi level, the calculated ΔFele is relatively scattered.

The above observations can be traced back to the different electronic and phonon dispersion properties of

the various CSAs studied. For pure Ni, the temperature dependent SFE is dominated by the variation of

ΔFele. Inspection of the energy and entropy terms included in ΔFele reveals that electron-hole entropy

dominates the variation of ΔFele. This contribution mainly arises from the electronic DOS (EDOS) near EF.

As a result, it is the differences in the DOS of hcp and fcc Ni near EF that dominates the temperature

dependence of the SFE in pure Ni. As shown in Fig. 5, the DOS of fcc Ni increases slowly in the spin-

down channel whereas that of hcp Ni exhibits a sharp peak located slightly above EF. The enlarged DOS

is given in the inset. The increase of temperature induces a shift of EF to higher energies, which results in

higher contribution to the electron-hole entropy from hcp Ni. Therefore, ΔFele becomes the dominant

factor in determining the stability of hcp and fcc phase for pure Ni. The phonon density of states (PDOS)

only makes a minor contribution in pure Ni because the difference between the PDOS of fcc and hcp

phase is small. For NiCo, the difference between EDOS of hcp and fcc NiCo is extremely small near EF.

For the majority channel EF lies in the low DOS sp-state dominated part for both hcp and fcc. While EF

does fall in the high DOS d-band region for both hcp and fcc, the DOS of both structures are very similar.

This is due to the alloying effect that leads to the smearing of the band structure, as found in previous

Korringa-Kohn-Rostoker coherent-potential-approximation (KKR-CPA) calculations of the Bloch

16
spectral functions [4]. These calculations found that disorder smearing of the electronic band structure is

more pronounced in the spin-down channel, which makes the corresponding EDOS flat, in contrast to the

sharply peaked EDOS of pure Ni. Consequently, ΔFele is small for NiCo. On the other hand, ΔFph

becomes dominant. Consideration of the PDOS in Fig. 5 shows that the main difference between hcp and

fcc NiCo lies in the high-frequency region, where the PDOS of hcp phase extends to higher-frequency

compared to the fcc phase. In the harmonic approximation, the difference of vibrational entropy for two

phases α and β at high temperatures (above the Debye temperature) can be obtained from [53]:

𝛽−𝛼 ∞
∆𝑆𝑣𝑖𝑏 = 3𝑘𝐵 ∫0 (𝑔𝛼 (𝜀) − 𝑔𝛽 (𝜀)) ln(𝜀) 𝑑𝜀. (8)

Therefore, the entropy difference from lattice vibrations is proportional to the logarithmic moment of the

difference between the PDOS. To illustrate this, the integrand (𝑔ℎ𝑐𝑝 (𝜀) − 𝑔 𝑓𝑐𝑐 (𝜀)) ln(𝜀) is displayed in

Fig. 5. Clearly, the large contribution to ΔFph that is shown in Fig. 4 indeed arises from the high-

frequency region.

The above arguments also apply to other CSAs. Fig. 5 shows that both spin channels of NiCoCr have

more smeared out EDOS than those of pure Ni (both spin channels) or NiCoFe (the majority spin

channel). The corresponding PDOS shows that the hcp NiCoCr has more extended high-frequency states

than fcc phase. On the other hand, the PDOS of NiCoFe are similar to pure Ni for both hcp and fcc phase.

As a result, the temperature dependence of ΔFele for NiCoFe is similar to that of NiCo (increases with

temperature), while the dependence of ΔFph is similar to that of pure Ni (decrease with temperature). The

final results suggest that the SFE of NiCoCr increases with temperature while that of NiCoFe decreases.

For all CSAs, the vibrational entropy is the dominant factor in determining the temperature dependence of

the stability of fcc with respect to hcp. The importance of vibrational entropy has been previously noted in

CPA studies of NiCoFeCrMn [35]. The additional high-frequency states in the hcp phase lead to an

increased ΔFph with increasing temperature. By analyzing the vibrational modes, it is found that the high-

17
frequency states in hcp originate from local vibrational mode ascribed to the local interaction among

atoms. These interactions depend sensitively on the local environment. A detailed study of the structural

parameters reveals that the fcc and hcp structures of these CSAs exhibit different features. The c/a ratios

calculated for different structures as well as the determined dγ/dT are summarized in Table II. For hcp

structures, the ideal c/a value is 1.633. Using the same definition, the c/a for fcc structures are also

calculated. Table II shows that the c/a for fcc CSAs exhibits a small deviation from the ideal value due to

atomic displacement fluctuations. The fact that all CSAs considered crystallize in fcc structure is clearly

in line with the small variations of c/a for these alloys. However, the deviation from the ideal c/a for hcp

depends on alloy composition. The CSAs with positive dγ/dT show a lower c/a ratio for hcp compared to

fcc phase, while those with negative dγ/dT have a higher c/a ratio for hcp phase. It should be noted that

the volume change from fcc to hcp phase is rather small. Therefore, the high frequency states in hcp phase

are largely from the compressed interlayer interactions. In these close-packed structures, the atomic

interaction is much stronger along the close-packed direction. The decreased c/a ratio suggests that the

interlayer bonds become stiffer and therefore contribute to the high frequency vibrations [36]. As a result,

the vibrational entropy decreases for the hcp phase and the fcc phase is stabilized. The deviation of c/a

should be attributed to the different elemental couplings as these elements exhibit different c/a in hcp

phase [54].

Table II Temperature dependent coefficients of SFE (dγ/dT) and c/a values in fcc and hcp phases, which

are the ratios between the distance of two close-packed layers and the distance between neighboring

atoms in a close packed layer. For each structure type, c/a results corresponding to the average from three

independent SQS supercells.

dγ/dT (mJ/m2/K) c/a (fcc) c/a (hcp)

NiCo 0.04 1.634 1.623
NiFe -0.02 1.629 1.642
NiCoFe -0.01 1.631 1.634
NiCoCr 0.09 1.636 1.612

18
 NiCoFeMn 0.03 1.634 1.620
NiCoFeCr 0.07 1.636 1.615
NiCoFeCrMn 0.11 1.631 1.606
NiCoFeCrPd 0.05 1.629 1.615

Dislocation-mediated slip and deformation twinning are two important competing mechanisms for plastic

deformation and energy dissipation in fcc materials. It is generally accepted that the deformation

mechanism has an intimate relationship with SFE. Deformation twinning dominates in low SFE alloys

while dislocation glide usually happens in high SFE metals. The different SFEs and their temperature

dependence suggest that these CSAs exhibit different deformation mechanisms at different temperatures.

For CSAs with negative SFE and positive dγ/dT, such as NiCoCr, it is to be expected that dislocation-slip

is dominant at high temperature and twining increases with decreasing temperatures; an expectation that

is consistent with experimental observations to date [1,9,55].

5. Conclusion

We have investigated the SFEs for a series of CSAs using first-principles calculations. The temperature

dependence of SFE for some representative CSAs is studied. In general, these CSAs studied exhibit low

even negative SFEs, depending on the alloying elements. The analysis of negative SFE found for some

CSAs suggests the importance of vibrational entropy in stabilizing their fcc structures. We further show

that the bond stiffness difference between the hcp and fcc phases is responsible for the observed hcp-to-

fcc transition as temperature increases. As the value of c/a can be deduced from the alloy structures, these

results suggest that the temperature dependence of SFEs can be inferred directly by measuring the detail

structure of the CSAs. The present study also indicates that elemental coupling determines the magnitude

of SFE as well as its temperature dependence. While the SFE is related to the electronic properties of the

constituent elements, its dependence on temperature as well as the stability of fcc structure is governed by

phonon properties.

19
Acknowledgment

This work was supported as part of the Energy Dissipation to Defect Evolution (EDDE), an Energy

Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy

Sciences.

Appendix

In this appendix, we discuss the calculation details and analyze possible errors introduced by the

calculation.

A.1 Simulation setup

The simulation setup is schematically illustrated in Fig. 6 for both AIM and supercell models employed in

this work. For AIM calculations, both cell vectors and internal coordinates are fully optimized. In this

way, we ensure that the minimal energy of the system is achieved after relaxation, though the cell vectors

are slightly deformed. The angular deviations from the orthogonal axes are very small for both fcc and

hcp structures (<0.1°). The optimized structures are used to determine the value of c/a. For supercell

calculations, a vacuum layer is introduced as illustrated in Fig. 6(b).

A.2 The choice of supercell in stacking fault energy calculations

In the supercell method, a slab configuration including 9 [111] layers was used. The effect of layer

number on the calculated SFE was tested based on empirical potentials. The results show that increasing

layer number from 9 to 200 leads to a SFE difference of less than 1 mJ/m2 for pure Ni, which suggests

that 9 [111] layers are sufficient to obtain accurate SFEs. Similar conclusions were also reported

previously [43].

20
The number of atoms used in supercell and AIM calculations was 108 and 144, respectively. As a result,

the exact equiatomic concentration of a given CSA is not always preserved, especially for quinaries.

However, the results based on these two methods are consistent, indicating the concentration effect is

small. The agreement is also an indication of the convergence of the results with respect to the supercell

size. We have checked the convergence of AIM approach by comparing the results from AIM1

calculations using a 72-atom supercell and a 144-atom supercell. The results are -4.33 mJ/m2 and -5.38

mJ/m2 for NiCoFeCrPd, which suggests that the 72-atom supercell is rather well converged. However,

applying the AIM2 model requires that the same number of atoms is used for all required structures. For

our simulation setup, there are 12 atoms in each layer, which leads to a minimum of 144 atoms in order to

make sure the number of atoms is equal in fcc (4 ABC layers), hcp (6 AB layers) and dhcp (3 ABAB

layers) structures. Therefore, the 144-atom supercells are used throughout.

The SFE calculated by supercell methods is averaged from all the possible planes in the SQS cell.

Different SQS supercells generated by different annealing temperature were tested to check the

convergence of the SFE with respect to the specific cell realization. The averaged results from nine planes

for NiCo calculated from three independent SQS supercells are -10.25, -14.47 and -14.21 mJ/m2,

indicating that the SFE presented in this work are fairly independent of the specific supercell employed.

Note that the SFE difference of around 4 mJ/m2 corresponds to an energy difference about 0.7 meV/atom.

This difference is a reflection of the configurational variation of different SQSs.

The energy required for the AIM model is calculated by three independent SQS cells for each structure

(fcc, hcp and dhcp). It is found that the standard deviation is rather small, which suggests a good

convergence of the energy with respect to the SQS cell. For example, the standard deviation for NiCo is

0.33 meV/atom, 0.24 meV/atom and 0.58 meV/atom for three independent fcc, hcp and dhcp calculations.

In principle, more SQSs can be used to improve the statistics. Because of the computational cost, only

three SQS structures are considered in this work.

21
The AIM1 model is used as the basis for calculating the temperature dependence of the free energy. Since

it is shown that the contribution from vibrational entropy is vital to the variation of ΔFph, the accurate

determination of the PDOS is extremely important. For random solid solutions, the disorder poses a great

challenge to theoretical modeling. For electrons in compositionally disordered alloys, ab initio KKR-CPA

(and related) methods [4] provide a well-developed first-principles theory to describe the effect of

disorder on the configurationally averaged electronic structure; at least within a mean field approximation.

However, for phonons in compositionally disordered alloys, the methods and corresponding codes are

much less well developed. The current state of the art is embodied in the itinerant coherent-potential-

approximation (iCPA) [56]. While some codes exist that implement the iCPA and can, therefore, treat

both site diagonal (mass) and off-diagonal (force constant) disorder, they are currently limited to binary

alloys. As a result, in this study, the calculation of the configurationally averaged phonon DOS is

addressed using SQS structures. To investigate the sensitivity of PDOS on SQS configurations, two

independent calculations with different SQS structure were performed. The calculated results for NiCo

are shown in Fig. 7, where it can be seen that the PDOS from two independent SQS structure are very

similar, exhibiting the same set of major features and overall distributions.

A.3 Influence of magnetic moments on total energy calculations

Within DFT, all the CSAs studied in this work admit a magnetic (spin polarized) solution. As an

illustrative of the influence of magnetism, the energy-volume curves corresponding to the magnetic and

nonmagnetic fcc and hcp phase of NiCoFeCrMn are shown in Fig. 8. Clearly, the magnetic structures

have lower energy than the nonmagnetic ones. Notably, the hcp structure has lower energy than fcc

structure in both magnetic and nonmagnetic cases, indicating hcp is more stable than fcc at 0 K. For the

full set of binary through quinary CSAs considered here, the nature of the magnetic ground state is found

to depend on the specific alloying elements in a rather simple way. For alloys containing Ni, Fe or Co the

single-site (local) moments associated with each species align parallel to one another; which is indicative

22
of the tendency of these elements to couple ferromagnetically with themselves and each other. On the

other hand, for alloys that also contain either Cr or Mn (or both), the single site moments corresponding

to these elements mostly align antiparallel to those of Fe, Co and Ni; which is indicative of the tendency

of these elements to couple antiferromagnetically to themselves and each other. These results are in good

agreement with those calculated by some of the authors of this paper and by others using ab initio

implementations (KKR-CPA and EMTO-CPA) of the CPA method for the direct calculation of the

configurationally averaged properties of solid solution alloys [4,30,35].

For all alloys, the total energies of magnetic and nonmagnetic structures have been compared by directly

optimizing both their cell vectors and internal coordinates. It turns out that the magnetic structure always

exhibits lower energy than the nonmagnetic structure. Therefore, the results from magnetic calculations

are presented in this study.

A.4 Influence of lattice parameters on stacking fault energy results

The lattice parameters (a0) of CSAs were firstly determined by calculating the energy-volume relation for

each alloys and then fitting with the Murnaghan equation of state [57]. The results shown in Table III

reveal that the majority of the calculated lattice parameters are slightly smaller than those measured

experimentally; the exception being NiPd for which the calculated lattice parameter is larger. In fact,

different experimental results are reported for this alloy. The general underestimation can be understood

in part by the thermal expansion effect. However, there are also differences that result from the particular

exchange-correlation functional (PBE) used in this work [58]. Since PBE functional is generally better

than LDA in describing the magnetism of metals in most cases [59], we used PBE all through this work.

This can also be justified by the results presented in Table I. The model for the stacking fault calculations

was constructed based on these lattice parameters.

23
We have examined the influence of underestimation of lattice constant on the calculated SFE. In this case,

the ternary NiCoFe is used to study the effect of lattice expansion by increasing the lattice constant used

in the supercell approach. The results show that SFE of NiCoFe changes only 4 mJ/m2 when the lattice

constant increases by 1.7%. Similar results are also obtained for NiCoCr, where only a slight change of

SFE is obtained when changing the lattice constant. The limited change of SFE suggests that the

underestimation of lattice constants is not responsible for the observed relative stability between hcp and

fcc phase.

Table III lattice parameters of considered CP-SCAs. The experimental results are adopted from Ref. [55]

unless indicated.

a0 (Å) Exp. Error

Ni 3.5148 3.5238 -0.26%
NiFe 3.5788 3.5826 -0.11%
NiCo 3.5251 3.5345 -0.27%
NiPd 3.7653 3.7074, 3.745a 1.56%, 0.41%
NiCoCr 3.5323 3.5590 -0.75%
NiCoFe 3.5593 3.5690 -0.27%
NiCoFeMn 3.5539 3.5919 -1.06%
NiCoFeCr 3.5397 3.5715 -0.89%
NiCoFeCrMn 3.5406 3.5991 -1.63%
NiCoFeCrPd 3.6721 3.6730 -0.02%
a
Value at 20°C[60].

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