# Multiscale analysis of crystal defect formation in rapid solidification of
pure aluminium and aluminium-copper alloys

Tatu Pinomaa1, Matti Lindroos1, Paul Jreidini2, Matias Haapalehto1,
Kais Ammar3, Lei Wang4, Samuel Forest3, Nikolas Provatas2, and Anssi
Laukkanen1

1ICME group, VTT Technical Research Centre of Finland Ltd, Espoo, Finland,
2Department of Physics and Centre for the Physics of Materials, McGill
University, Montreal, Canada.
3MINES ParisTech, PSL University, MAT - Centre des matériaux, Evry, France.
4Federal Institute for Materials Research and Testing (BAM), Berlin, Germany

###### Abstract

Rapid solidification leads to unique microstructural features, where a less
studied topic is the formation of various crystalline defects, including high
dislocation densities, as well as gradients and splitting of the crystalline
orientation. As these defects critically affect the material’s mechanical
properties and performance features, it is important to understand the defect
formation mechanisms, and how they depend on the solidification conditions and
alloying. To illuminate the formation mechanisms of the rapid solidification
induced crystalline defects, we conduct a multiscale modeling analysis
consisting of bond-order potential based molecular dynamics (MD), phase field
crystal based amplitude expansion (PFC-AE) simulations, and sequentially
coupled phase field – crystal plasticity (PF–CP) simulations. The resulting
dislocation densities are quantified and compared to past experiments. The
atomistic approaches (MD, PFC) can be used to calibrate continuum level
crystal plasticity models, and the framework adds mechanistic insights arising
from the multiscale analysis.

##  1 Introduction

Several industrial processes, such as thermal spray coating deposition [1],
certain welding techniques [2], and metal additive manufacturing [3], operate
under rapid solidification conditions. Rapid solidification leads to
drastically altered microstructures through selection of metastable phases,
solute trapping kinetics [4, 5, 6], changes in solidification morphology [7],
and reduction of a microstructural feature size such as dendrite primary arm
spacing [7]. An overlooked, yet important, feature of these microstructures is
the formation of various types of crystalline defects [8]. These include
trapping of excess point defects [9], formation of high dislocation densities
[10, 11, 12], cavitation or microvoiding [10, 13, 14] , high microstructural
(type II-III) residual stresses [15, 16], and so-called lattice orientation
gradients [17]. These crystalline defects critically affect the material’s
mechanical properties and performance. In this work, we focus on aluminium
alloys. They are a light-weight class of materials relevant for e.g. transport
industry, where rapid solidification conditions are known to significantly
alter the mechanical properties of the material [18, 19, 16, 20]. In metal
additive manufacturing of aluminium alloys, one of the key issues is
controlling rapid solidification induced shrinkage and the associated cracking
[21, 22].

Solidification shrinkage leads to stresses in the forming solid, which induce
plastic flow accompanied by formation of dislocations [12] and e.g. stacking
faults [23]. On continuum scale, the dislocations can be classified into
geometrically necessary dislocations (GNDs) which are required to maintain
lattice continuity when orientation variations are present, as well as
statistically stored dislocations (SSDs) which occur through random trapping
processes [24, 25]. SSDs have a net-zero Burger’s vector, and thus do not
affect the lattice curvature. It should be noted that the classification to
GNDs and SSDs depends on spatial resolution. Even SSDs will cause lattice
curvature in their immediate vicinity, if the atomistic crystalline lattice is
fully resolved.

There are several experimental techniques to characterize dislocations. A view
of individual dislocations can be obtained with transmission electron
microscopy (TEM). The necessity to produce a thin foil sample is likely to
disturb the original dislocation structure, and cause projection related
inaccuracies [12]. Another technique, Electron channeling contrast imaging
(ECCI), where the latter allows for one to characterize on significantly
larger sample areas compared to TEM [26]. A commonly used technique to
estimate dislocation densities is electron back-scatter diffraction (EBSD)
imaging, where the local pixel-by-pixel crystalline orientation can be mapped
out, and the misorientation between neighboring pixels can be used to estimate
microstructure-scale lattice curvature, and through strain-gradient crystal
plasticity theories, to dislocations (GNDs) necessary to accommodate this
curvature [24]. Because the processes associated X-ray – crystals interaction
are simpler (mostly elastic) [27], the peak broadening in the resulting
diffraction patterns can be more reliably used to estimate dislocation
densities [25, 28] or e.g. lattice strains [15] in the illuminated volume at
the high intensities produced in synchrotrons. Still, e.g. instrumental
inaccuracies and sample preparation cause peak broadening, limiting the
reliability of X-ray diffraction (XRD) based techniques to isolate the
contribution of dislocation densities to the peak broadening [27].

Due to the limitations in the accuracy spatial and temporal resolution,
simulation techniques provide valuable insight into the mechanisms behind
formation of rapid solidification structures. There have been numerous
investigations into rapid solidification structures and their micromechanics
of using molecular dynamics [29, 30], phase field method [31, 32, 33], crystal
plasticity method [28, 34, 35], and rapid solidification melt pool dynamics
[36]. However, to our knowledge, formation of the aforementioned rapid
solidification induced crystalline defects have not been investigated
thoroughly with simulation approaches.

In this paper we consider pure and copper-alloyed aluminium, focusing on the
formation of rapid solidification defects on multiple length scales, from MD
to phase field crystal based amplitude expansion (PFC–AE) modeling, as well as
coupled phase field – crystal plasticity (PF–CP) simulations on the largest
scales. Crystalline defect formation mechanisms are illustrated, and we
compare the resulting average dislocation densities predicted by the
considered modeling methods (MD, PFC–AE, PF–CP), and by past experiments.

##  2 Methods

In this paper, we use a multiscale modeling framework presented in Figure 1,
consisting of molecular dynamics (MD) simulations, phase field crystal based
amplitude expansion (PFC–AE) simulations, and phase field – crystal plasticity
(PF–CP) simulations.

Figure 1: Schematic of the multiscale modeling approach for describing
crystalline defects forming in rapid solidification.

###  2.1 Molecular dynamics

Molecular dynamics simulations of rapid polycrystalline solidification are
carried out with LAMMPS [37], where we use a bond-order potential for Al-Cu
with an analytical form derived from quantum mechanics, accurate in the Al
rich end of the binary system [38]. When compared to experiments and ab-initio
simulations, this bond-order potential has an excellent description of, for
example, the pure aluminium melting temperature, as well as the point defect
characteristics and the characteristically high stacking fault energy of
aluminium. The phase diagram resulting from this bond-order potential is
presented elsewhere [39].

The free solidification conditions are initialized to contain a co-existing
solid and liquid phases as depicted in Figure 2, following Yang et al. [40],
adjusted to polycrystalline solid domain of four misoriented crystallites. For
the initialization, first a solid domain of pure aluminium FCC is generated in
a cuboid geometry as shown in Figure 2a, where the solid is split into four
misoriented grains (Figure 2b). Afterwards, randomly selected aluminium atoms
are replaced by copper atoms such that the compositions of the solid and
liquid regions are approximately equal to the solidus and liquidus values
corresponding to the equilibration temperature; the MD-based phase diagram is
presented elsewhere [39]. Then, atoms in the solid region are fixed, while the
rest of the system is melted at a relatively high temperature in an
$\mathrm{NP_{Y}AT}$ ensemble, where $Y$ is the solidification direction. In
this ensemble, the total number of particles (N) and cross-sectional area (A)
are kept constant, while the average temperature (T) and longitudinal pressure
($\mathrm{P}_{\mathrm{Y}}$) are controlled by a Nosé-Hoover thermostat-
barostat. As the four crystallites (Figure 2) contain grain boundaries with
high curvatures, we do not perform any relaxation at this point in order to
preserve the grain boundary structure.

Figure 2: a) Initial geometry for the molecular dynamics simulations, and b)
the four rectangular crystallites initialized in the solid domain.

The box dimensions are 41 nm $\times$ 41 nm $\times$ 56 nm, where the longest
dimension is the direction where solidification progresses, and a solid region
was initialized in the middle. Periodic boundary conditions are imposed in all
directions. The solid region was segmented into four domains where the lattice
orientations were rotated as indicated in Figure 2 b.

The solidification is started by setting the solute concentration in the
liquid region to a chosen value and setting the thermostat temperature to the
chosen undercooling. The solidification simulations are performed in a
$\mathrm{NP_{Y}AT}$ ensemble using a single global Nosé-Hoover thermostat. It
should be noted that the latent heat release leads to local temperature rise
above the thermostat target temperature. A detailed analysis of single crystal
Al-Cu rapid solidification kinetics, point defects, and chemical ordering is
presented in a separate article [39].

###  2.2 Phase field crystal based amplitude expansion simulations

Phase field crystal based amplitude expansion (PFC-AE) simulations were
conducted for rapid solidification of pure aluminium. In contrast to
traditional phase field methods[41, 32], PFC is capable of self-consistently
retaining lattice orientation and other atomic information while evolving
microstructure dynamically on diffusive time scales [42].

In this work, we use a PFC model based on the one proposed by Kocher et al.
[43], which accounts for both liquid and vapor disordered phases as well as an
ordered solid phase in a system consisting of a single atomic species. The
existence of a vapor phase in the model is crucial under the assumption that
extreme interface velocities in a pure material system would be capable of
trapping microvoids [10, 13, 14], in a manner analogous to solute trapping in
alloy systems [6].

The PFC model exhibits a thermodynamic phase diagram approaching that of pure
aluminum in the vicinity of the triple point [44]. The reference density was
set near the solid coexistence line at a temperature appropriate for deep
quenches. The model’s phase diagram exhibits a sharp deviation from pure
aluminum’s at densities below the critical point, due to the expanded nature
of this model’s free energy functional [42]. Further work is underway to
expand PFC approaches to more accurately span the range of densities
accessible in pure materials’ phase diagrams [45]. As the present work focuses
on solidification, good agreement with the solidus in the vicinity of the
triple point was determined to be sufficient.

Although PFC models operate in diffusive time scales, they are still limited
in spatial scale as individual atomic density peaks need to be resolved. To
achieve larger simulation system scales, the PFC model is coarse-grained into
a complex order parameter phase field model, where individual atomic peaks are
integrated out, through the use of the amplitude expansion method [46, 47].

This amplitude field model was simulated in a co-moving reference frame where
circular randomly oriented grains were initialized with an up to 10 degree
misorientation with respect to the solidification direction. Periodic
boundaries were used for system edges perpendicular to the solidifying front.
More details about rapid solidification phase field crystal simulations are
presented in another publication [48].

###  2.3 Phase field method with crystal plasticity

####  2.3.1 Phase field model for rapid solidification

On the largest scale (several micrometers), rapid solidification is simulated
with phase field method. Solute trapping kinetics according to continuous
growth model [5] is implemented through a modified anti-trapping current, as
described in [49], in a multi-order parameter model [50], and applied to thin
film rapid solidification conditions [32]. For the phase field model presented
in [32], the model parameters are summarized in Table 1. For the temperature
field, a frozen gradient approximation is imposed, representative for rapid
solidification conditions. More details about this phase field model
implementation are presented in Ref. [32] with slightly different model
parameters corresponding to a dilute Al-Cu alloy, and in Ref. [34] for 316L
stainless steel. The simulations are carried out with Z-set software [51, 52],
where the information about the order parameter and concentration fields are
forwarded to crystal plasticity simulations, whose the methodological details
are summarized in what follows.

Table 1:  Phase field model parameters for Al-4-5at%Cu alloy.

Equilibrium partition coefficient $k_{e}$ | 0.103a
---|---
Equilibrium liquidus slope $m_{l}^{e}$ | -0.579 K/at%a
Alloy concentration $c_{l}^{o}$ | 4.5 at%
Solid-liquid interface energy $\sigma_{SL}$ | 0.127 J/m2 b
Gibbs-Thomson coefficient $\Gamma$ | 1.098$\times$ 10-7 K m b
Solutal capillary length $d_{0}$ | 5.1 nm b
Interaction parameter $\omega$ | 2.0
Liquid diffusion coefficient $D_{L}$ | 3.0 $\times$10-9 m2/s
Solid diffusion coefficient $D_{s}$ | 0 m2/s
Kinetic coefficient $\beta_{0}$ | 0.04466 s/m
Capillary anisotropy strength $\epsilon_{c}$ | 0.01 c
Kinetic anisotropy strength $\epsilon_{k}$ | 0.13 d
Solute trapping velocity $V_{D}^{PF}$ | 4.9 m/s
Computational interface width $W_{0}$ | 6 nm
Trapping parameter $A$ | 0.125
Element size $dx$ | 1.0$W_{0}$

a: Estimated using Thermo-Calc’s TCAL6 database [53]
b: From molecular dynamics simulations in Ref. [39].
c: From experiments in Ref. [54]
d: From molecular dynamics simulations in Ref. [55]

####  2.3.2 Crystal plasticity model

To investigate defect formation in rapid solidification on microstructural
scales resulting from solidification shrinkage, a crystal plasticity model is
presented closely following the model by Monnet et al. [56]. We impose a small
deformation framework, where the total strain ($\underline{\epsilon}$) is
additively decomposed to elastic ($\underline{\epsilon}^{E}$), plastic
($\underline{\epsilon}^{P}$), solidification shrinkage
($\underline{\epsilon}^{S}$), and thermal ($\underline{\epsilon}^{T}$),
contributions, respectively:

$\underline{\epsilon}=\underline{\epsilon}^{E}+\underline{\epsilon}^{P}+\underline{\epsilon}^{S}+\underline{\epsilon}^{T}.$
(1)

Dislocation slip is assumed to be the dominant deformation mechanism. As these
simulations focus on the solidification, the system temperatures are close to
the alloy solidus temperature, and solid-state thermal contraction (due to the
solid’s thermal expansion effect) is insufficient to produce notable thermal
thermal strains within the simulation box.

The strain rate due to solidification shrinkage is given by

$\displaystyle\underline{\dot{\epsilon}}^{S}$
$\displaystyle=\underline{\epsilon}_{0}\frac{1}{2}\left(\dot{h}+1\right)\underline{I},$
(2)

where $\underline{\epsilon}_{0}$ is the axial dilation (shrinkage) strain
adjusted to produce approximately 7 % volumetric shrinkage from liquid to
solid, $h=h(\\{\phi_{i}\\})$ indicates the solid fraction based on the order
parameters drawn from the phase field model:

$\displaystyle h(\\{\phi_{i}\\})$
$\displaystyle=N_{orient}-1+\sum_{i=1}^{N_{orient}}\phi_{i},$ (3)

where $h\rightarrow 1$ in the solid and $h\rightarrow-1$ in the liquid, and
in-between values in the solid-liquid interface. The net elasticity tensor
($\Lambda$) is interpolated between liquid and orientation of each solid phase
as

$\displaystyle\Lambda=\frac{1-h}{2}\Lambda_{L}+\sum_{i=1}^{N_{orient}}\frac{1+\phi_{i}}{2}\Lambda_{i},$
(4)

where $\Lambda_{L}$ is the liquid elasticity tensor, and $\Lambda_{i}$ is the
elasticity tensor of a crystal corresponding to order parameter $\phi_{i}$
solid region.

Plastic strain rate is given by the sum of twelve $\\{111\\}<110>$ slip
systems in FCC aluminium.The plastic strain rate is thus defined as

$\underline{\dot{\epsilon}}^{P}=\sum_{s=1}^{N_{s}}\dot{\gamma}^{s}\underline{N}^{s}$
(5)

where $N_{s}$ is the number of slip systems, and $\underline{N}^{s}$ , and
viscoplastic flow rule yields a shear rate for slip system $s$ as

$\dot{\gamma}^{s}=\frac{1+h(\\{\phi_{i}\\})}{2}\left\langle\frac{|\tau^{s}|-\tau_{c}^{s}}{K_{\tau}}\right\rangle^{N}sign(\tau^{s}),$
(6)

where $\tau^{s}$ is the resolved shear stress and $\tau_{c}$ is the effect
slip resistance of a slip system $s$, and the solid fraction
$h(\\{\phi_{i}\\})$ was used as a prefactor in Equation 6 to ensure that
plastic deformation does not occur in the liquid. Viscous parameter $K_{\tau}$
is chosen to be equal to the solid solutions strength of the material, and
parameter $N$ defines the strain rate dependence. Critical resolved shear
stress is defined as

$\tau_{c}^{s}=\tau_{0}^{SS}+\tau_{HP}+\mu b^{s}\sqrt{\sum_{j}a_{sj}\rho^{j}},$
(7)

where $\tau_{0}^{SS}$ is the dislocation-free shear stress resistance for
large grains with solid solution strengthening effects included. The Hall-
Petch-like size effect can be described with $\tau_{HP}$ in Equation 7; for
simplicity, and due to the lack of available data, we set the size effect
strengthening to zero in the actual simulations: $\tau_{HP}=0$. However, when
the single crystal parameters were fitted with experimental stress-strain data
(fitting procedure described further below), the HP effect is included to
match the initial yield behavior of the material as follows:

$\tau_{HP}=\frac{\mu(T)}{\mu_{ref}}\frac{K_{HP}}{\sqrt{d_{grain}}},$ (8)

where $\mu(T)$ is the temperature-dependent shear modulus, $\mu_{ref}$ is the
shear modulus in room temperature, $K_{HP}$ is Hall-Petch coefficient, and
$d_{grain}$ is the effective grain size.

The last term on the right-hand side of Equation 7 leads to dislocation-
induced forest hardening, where $\mu$ is the shear modulus, $b^{s}$ is the
length of Burgers vector for slip system $s$, $a_{sj}^{\tau}$ interaction
matrix between slip systems $s$ and $j$, and $\rho^{j}$ dislocation density of
slip system $j$. Once deformation and plasticity due to solidification takes
place, dislocation density can increase rapidly.

In this paper, we assume that solid solution strengthening effect in Al-Cu due
to copper is negligible, and therefore the resistance coming from this source
was set to zero in the simulations. However, the solid solution term can have
significant effect on slip resistance in other materials and the key aspect
then is to retrieve the local concentration field from phase field
simulations, as presented elsewhere for 316L steel PF–CP simulations [34].

The evolution of statistically stored dislocation density corresponding to
slip system $s$ is governed by

$\dot{\rho}^{s}=\frac{|\dot{\gamma}^{s}|}{b^{s}}\left[\frac{1}{d_{grain}}+\frac{1}{K_{forest}}\sqrt{\sum_{j}^{forest}a_{sj}\rho_{j}}+\frac{1}{K_{coplan}}\sqrt{\sum_{j}^{coplan}a_{sj}\rho_{j}}-G_{c}\rho^{s}\right],$
(9)

where $d_{grain}$ is the mean grain size (used in polycrystal parameter
fitting), $K_{forest}$ and $K_{coplan}$ define the hardening effect from
dislocations. They may be interpreted to describe the average number of
obstacles passed by a dislocation before immobilizing. Dislocation
annihilation is controlled with a parameter $G_{c}$.

The CP model was parametrized using data in Ref. [57], where elevated
temperature polycrystalline stress-strain data of aluminium alloy in series
5083 at 673 K and a strain rate of 10 s-1. We chose this alloy as it should
have negligible strengthening effects due to precipitates, which are neglected
in this work. For the fitting procedure, we chose viscous parameter values
$K=8$ and $N=15$ in Equation 6, whereas for the PF–CP simulations presented
later, we used $K=1$ and $N=15$. For the fitting, we activated the Hall-Petch
strengthening in Equation 8 with $K_{HP}=0.15$ and $d_{grain}=180$ $\mu$m; in
the PF–CP simulations we set $\tau_{HP}=0$. We initialized a simple equiaxed
polycrystalline structure, and using the stress-strain data in Ref. [57], we
set the dislocation hardening parameters to $K_{forest}=50$ and
$K_{coplan}=150$. The used parameter values are summarized in Table 2.

Table 2: Crystal plasticity model parameters for Al-4-5at%Cu alloy.

Young’s modulus $E_{S}$ | 31.5 MPa a
---|---
Poisson’s ratio (solid) $\nu^{S}$ | 0.27 b
Shear modulus (solid) $\mu^{S}$ | $E/(2(1+\nu^{S}))$
Young’s modulus (liquid) $E^{L}$ | 0.1 GPa
Poisson’s ratio (liquid) $\nu^{L}$ | 0.49
Length of Burgers vector $b^{s}$ | 0.296 nm
Forest obstacle parameter $K_{forest}$ | 50.0
Coplanar obstacle parameter $K_{coplan}$ | 150.0
Annihilation distance $G_{c}$ | 2.96 nm
Viscous parameter for slip $K_{\tau}$ | 1.0
Viscoplastic strain rate exponent $N$ | 15.0
Solidification strain components $\epsilon^{s}_{1,2,3}$ | -0.0228 d
Interaction matrix $a_{sj}$ | See Table 1 in Ref. [56]

a: Interpolated value at 674 K for Al 6061 alloy from Ref. [58].
b: Interpolated value at 674 K for FCC 316L steel from Ref. [59].
c: Interpolated value at 674 K for Al 5083 alloy from Ref. [60].
d: Adjusted to produce a 7% volume decrease from liquid to solid.

The PF–CP simulations are performed on Z-set software [51, 52], where more
detailed simulations for a case of 316L steel are found elsewhere [34].

##  3 Results

###  3.1 Molecular dynamics (MD)

Figure 3: Atomistic simulations of polycrystalline aluminium rapid
solidification, showing the time evolution and the dislocation structure of
the solidified structures, where a) is pure aluminium (interface temperature
$\approx$ 897 K, soldification velocity $\approx$ 56 m/s), and b) is Al–1at%Cu
(interface temperature $\approx$ 898 K, solidification velocity $\approx$ 25
m/s).

In the MD simulations, a simulation box with liquid in the middle, and the
resulting rapid solidification at early time (0.1 ns) and fully solidified
structures are shown in Figure 3 for pure aluminium and Al–1at%Cu. The full
simulation geometry is shown side by side with the geometry clipped in half in
the solidification ($Y$) direction. The different phases are identified with
Ovito’s Polyhedral Template Matching (PTM) algorithm [61], where the gray
color (labeled "Other") corresponds to an unidentified coordination number,
which is predominantly liquid or amorphous phase. The grain boundaries with
normal in $X$ direction remain more or less similar throughout the
solidification, maintaining the disordered structure. In contrast, the grain
boundaries in $Z$ direction quickly form stacking faults, seen as tilted grain
boundary with HCP (red) interface layers. Stacking faults have been observed
in past Al-Cu rapid solidification MD simulations [29].

Figure 3 also shows the dislocation structures as identified with Ovito’s
Dislocation analysis tool, DXA [62]. The single layer stacking faults contain
Shockley partial dislocations, shown as wedge-shaped red layers, which
continue in $Z$ and $X$ direction due to the periodic boundary conditions. In
addition, Frank partials and perfect dislocations are identified in the
interiors of each four crystallite. Also, a small number of stair-rod and
Hirth type dislocations are identified. The resulting dislocation densities
are summarized in Table 3, and are compared to other simulation techniques and
literature values in section 4.

Rapid solidification kinetics of single crystal pure aluminium and aluminium-
copper alloy is presented in another paper [39], where we find high
interstitial and vacancy concentrations, but no dislocations. This suggests
that the generation of dislocations is intimately tied to interaction between
misoriented grains in rapid solidification of atomistically small systems. The
role of misorientations, orientation gradients, and GNDs, are further
investigated with PFC–AE simulations in what follows.

###  3.2 Phase field crystal method based amplitude expansion (PFC–AE)
simulations

In the PFC–AE simulations, we considered cases of rapid solidification and an
intermediate solidification velocity, for pure aluminium. For the rapid
solidification case, we applied a frozen temperature gradient
$G=$5\text{\times}{10}^{7}\text{\,}\mathrm{K}\text{\,}{\mathrm{m}}^{-1}$$ with
an average system temperature set to $T=$805\text{\,}\mathrm{K}$$ initially, a
significant quench below pure aluminum’s melting point of approximately
$933\text{\,}\mathrm{K}$, and the average density was chosen close to the
solid’s density; we refer to this case as a "large quench". A second case,
leading to lower solidification velocities–referred to as "intermediate
quench"—, was carried out, where a quench temperature
$T=$875\text{\,}\mathrm{K}$$ was chosen in the limit of no temperature
gradient, i.e., $G=0$. The lattice orientations in the simulated solid grains
were calculated from the atomic displacement field extracted from the complex
amplitude fields following Ref. [63].

Figure 4 (top row) shows the lattice orientations in a post-solidified
simulated sample, reconstructed from snapshots of the co-moving simulation
frame. The large quench case leads to smaller microstructural features and the
highly misoriented grains (close to orientation +10o and -10o) tend to be
dominated by grains that have an orientation closer to the solidification axis
(0o). Moreover, Figure 4a has visible orientation gradients, where it can be
seen that the orientation distribution tends towards the solidification
direction as the solidification progresses. This rapid solidification induced
formation of low angle grain boundaries has been seen earlier for melt-spun
aluminium alloys [19], as well as in additively manufactured 316L steel [17].
In contrast, the intermediate quench case leads to clearly separated cellular
tips where the orientation does not vary notably. The orientation gradients
are compared to thin film rapid solidification experiments elsewhere [48].

The orientation gradients were quantified in terms of kernel average
misorientation (KAM), shown in the middle row of Figure 4. To calculate the
KAM, the raw simulation data was first downscaled by subdividing the
simulation data into blocks, each containing 8x8 grid points, where the local
orientation was calculated as the average of each block. For the downscaled
blocks, KAM was then evaluated by summing the absolute value of local
disorientations up to fourth nearest neighbors, excluding disorientations
larger than 2 degrees following [24], as well as pixels where the density was
close to the liquid value, which occurred at high-angle grain boundaries, the
liquid domain, as well as small liquid or vapor pockets (discussed further
below). As the laplacian of the density field has a large magnitude at the
liquid or vapor interfaces, we chose an appropriate cut-off value for the
density field laplacian to mask out interfacial pixels from the KAM
evaluation. The resulting KAM distributions are shown in Figure 4 (second
raw), illuminating large KAM hot spots throughout the solidified system for
the large quench case, whereas only a few KAM hot spots are visible in the
intermediate quench case.

Maintaining strain compatibility across a polycrystalline structure with
anisotropic mechanical properties occurs by intragrain lattice curvature,
which necessitates arrays of appropriate dislocations with net non-zero
Burger’s vector. These dislocations are called geometrically necessary
dislocations (GND). Through energy minimization, these dislocations tend to
arrange themselves into cells, dividing a grain into subgrains, which can also
be called geometrically necessary boundaries [25]. These subgrains or "inner
dendrites", corresponding to small rotations caused by thermal stresses, are
visible also in FCC 316L steel in rapid solidification through additive
manufacturing [26].

Following Ref. [24], the KAM distribution ($\theta_{KAM}$) was used to
determine the GND density distribution:

$\displaystyle\rho_{GND}=2\frac{\theta_{KAM}}{r_{kernel}b},$ (10)

where $r_{kernel}$ is the kernel radius and $b$ is the Burgers vector length.
As the KAM was evaluated using fourth nearest neighbors, $r_{kernel}\approx
2a_{Al}$, where $a_{Al}\approx 0.4196$ nm is pure aluminium FCC lattice
constant near its melting point [64]. Assuming dislocations due to
$\\{111\\}\langle 110\rangle$ slip, the Burgers vector magnitude is
$b=a_{Al}/\sqrt{2}$. The resulting GND distribution is shown in Figure 4,
illustrating that the large quench case leads to subdivision into subgrains
through high KAM and GND densities, whereas for the intermediate quench case,
GND and KAM levels are low and division into subgrains is negligible. The
average GND densities are listed in Table 3 for the large and intermediate
quench cases, and compared to other simulation techniques and literature later
in section 4.

Figure 4: Phase field crystal simulations of pure Al rapid solidification,
with the kernel average misorientations (KAM), as well as the geometrically
necessary dislocation (GND) densities, where a) corresponds to large quench
(large solidification velocity), and b) to an intermediate quench and
velocity.

The slower solidification conditions (Figure 4a) result in relatively smooth
solid-liquid interfaces with few defects trapped in the solid. In contrast,
the higher driving force case (Figure 4b) displays rougher interfaces where
the interface grooves trap disordered pockets of fluid that can later become
stable defects. The trapping of defects from interface grooves, and the
resulting defect stabilization, is exemplified in Figure 5a–c (red dashed
circles). This defect formation mechanism is consistent with molecular
dynamics simulations by He et al. [65], where a notable solid-liquid interface
roughness–albeit on smaller length scales compared to our
simulations–facilitates the formation of crystalline defects, by allowing for
deep solid-liquid interface grooves to ‘fold in’ and trap non-crystalline
domains as stable defects. There is indirect experimental evidence of rapid
solidification induced microvoid formation, or cavitation, in melt-spun
aluminium alloys [14], a powder metallurgically solidified Al-Li alloy [13],
and cast high purity copper [10]. It should be noted that an alternative
formation mechanism for these experimentally observed microvoids is that trace
amounts of gases, from the processing atmosphere, can dissolve into the metal
melt and form microscopic bubbles upon solidification.

It is also interesting to analyze the healing of crystalline defects after the
defect pockets have formed. Smaller defect pockets heal into perfect crystals,
or coalesce with existing microvoids, as shown in Figure 5a–c (dashed
rectangles). The reconstructed atomic lattice is presented in Figure 5d,
illustrating two stable microvoids and smaller lattice defects.

Density profile lines were extracted in the vicinity of the solid-liquid
interface, as shown in Figure 5e, for both the large quench (blue solid line)
and intermediate quench (red solid line) cases both according to the PFC–AE
simulations. The corresponding equilibrium values are shown as dashed
horizontal lines. It can be seen that the larger quench leads to a relatively
large deviation from the material equilibrium density due to a large number of
defect pockets that form during rapid solidification. As the defect pockets
heal, the solid density starts to approach the equilibrium density. In
contrast, the intermediate quench does not form notable defect pockets, and
therefore the density remains approximately constant after solidification. The
abrupt drop in the PFC–AE simulation density profiles (red and blue solid
lines in Figure 5e) correspond to the occurence of the solid-liquid interface.
More details on the orientation gradients using this PFC-AE model are
presented elsewhere [48].

Figure 5: Phase field crystal based amplitude expansion simulations of rapid
solidification: a-c) show time evolution of the density field, illustrating
defect pocket formation, merging, and healing; d) shows a zoom-in of the
reconstructed atomic lattice, showing stabilization of lattice defects and
microvoids; e) shows typical density line profiles in the solid side close to
the liquid.

###  3.3 Phase field – crystal plasticity (PF–CP) simulations

####  3.3.1 Phase field simulations

The phase field simulations are carried out in a rectangle of size 480 $\mu$m
$\times$ 960 $\mu$m, with zero flux boundary conditions on all edges for the
order parameter and concentration fields. A frozen gradient is applied with a
pulling speed $V=$0.03 m/s, and either $G=$ 106 K/m or 2$\times$106 K/m; also,
we initialized either a single crystal three distinct orientations. The single
crystal cases are initialized as sinusoidal solid fronts, and the three
distinct crystals are initialized as circles.

The time evolution of each four case is shown in Figure 6 in terms of the
copper concentration. The single-crystal simulations approach a quasi-steady
state cellular growth mode towards the end of the system far-end boundary,
with the smaller $G$ resulting in a larger cell spacing as expected. The three
crystal simulations lead competitive growth of misoriented grains, where the
larger $G$ leads to relatively straight grain boundary trajectories as the
capillary anisotropy becomes less important at larger undercoolings [55].
Overall, the smaller $G$ leads to more heterogeneity in the solidification
microstructures, visible as stronger microsegregation. This heterogeneity is
reflected in the microstructural residual stress distribution through results
from the solidification shrinkage, analyzed further below.

Figure 6: Phase field simulations of rapid solidification for Al–4.5at%Cu with
two different thermal gradients $G$, and with either a single crystal or three
crystals initialized.

####  3.3.2 Crystal plasticity coupling

The crystal plasticity simulations were carried out by taking the order
parameter fields from the phase field simulations shown above, and coupling
their dynamics to the crystal plasticity model. Specifically, the order
parameter time evolution resulted in solidification shrinkage induced
stresses, which lead to to plastic deformation and generation of statistically
stored dislocations (SSDs). The left and bottom boundary nodes were fixed in
$X$ and $Y$ directions, respectively. The top and right boundary nodes were
allowed to move, while staying on a straight line through the multi-point
constraint technique. The initial dislocation density was set uniformly to a
small non-zero value, 107 m-2.

The evolution of concentration (from phase field simulations), von Mises
stresses, and dislocation densities are shown for the $G$ = 106 K/m case with
three crystals in Figure 7. The solidification shrinkage generates von Mises
hot spots that follow the heterogeneity in the microsegregation, with
intercellular regions having higher SSD densities.

Figure 7: Phase field – crystal plasticity coupled simulation of Al–4.5at%Cu
rapid solidification, showing the copper concentration, solidification
shrinkage induced von Mises stresses, and the densities of statistically
stored dislocations.

The concentration, von Mises stresses, and dislocation densities are shown for
all four cases are shown in Figure 8 before reaching the right system
boundary. The smaller thermal gradient leads to larger heterogeneity in the
solidification front, reflected as larger hot spots of von Mises stresses and
dislocation densities. Moreover, the single crystal simulations lead to
slightly lower dislocation densities than their multi-crystal counter-parts.
This suggests that while the generation of large dislocation densities in
rapid solidification is not inherently tied to competitive growth of
neighboring crystals, the shrinkage stresses and the dislocation density
distributions are magnified by grain-grain interactions. This stress
magnification is consistent with experimental metal additive manufacturing
(rapid solidification) of aluminium-copper alloys, where solidification
cracking is observed more commonly along grain boundaries [22].

The average dislocation densities were calculated for the dislocation
densities in Figure 8, where the last 130 nm from the right end of the system
was excluded to avoid the liquid domain ahead of the solidification front. The
average dislocation densities for each simulation case were as follows:
4.1$\times$1012 m-2 for $G$ = 106 K/m single crystal, 4.9$\times$1012 m-2 for
$G$ = 106 K/m three crystals, 3.6$\times$1012 m-2 for $G$ = 2$\times$106 K/m
single crystal, and 4.1$\times$1012 m-2 for $G$ = 2$\times$106 K/m three
crystals. These dislocation densities are summarized in Table 3 along with
other simulation cases and experimentally obtained literature-given values,
showing that the PF–CP simulations predict dislocation densities that are
roughly 10-20 times smaller than those obtained experimentally.

There are several CP model parameters that are both challenging to determine
accurately, and have a significant influence on the CP model quantitative
predictions. In particular, dislocation climb can be come relevant during
solidification, close to the alloy solidus. In this work, the effect of
dislocation climb is phenomenologically included through the visco-plastic
power-law-like strain rate rule (Equation 6), parametrized in terms of the
strain exponent $N$ and prefactor $K_{\tau}$. In our tests, decreasing the
strain rate exponent $N$ in Equation 6 from 15 to 4 did not have a significant
influence on the dislocation densities. Decreasing the dislocation hardening
parameters $K_{forest}$ and $K_{coplan}$ in Equation 9 increased the average
dislocation density levels to over 10${}^{1}4$ m-2 which would better match
the experimental results in Table 3; this parameter adjustment, however,
resulted in an unrealistically large hardening behavior. In the future,
crystal plasticity model parameters should be re-adjusted quantitatively to
elevated temperature experiments or simulations with discrete dislocation
dynamics (DDD) or MD.

Figure 8: Phase field – crystal plasticity coupled simulation of Al–4.5at%Cu rapid solidification for all four considered cases, showing the copper concentration, solidification shrinkage induced von Mises stresses, and the densities of statistically stored dislocations. Table 3: Dislocation densities in rapid solidification using various simulation methods, as well as measured in past experiments. The dislocation types are specified: geometrically necessary dislocations (GND), statistically stored dislocations (SSD), total dislocation content (Total). In MD simulations, the dislocation types were explicitly determined, where "Others" include stair-rod and Hirth type dislocations. In the past experiments listed, the measurement technique is indicated: elecron back-scatter diffraction (EBSD), X-ray diffraction (XRD), neutron diffraction (neutron diffr.), and transmission electron microscopy based line-intercept measurements (TEM line-inter.). Simulation results in this work |
---|---
|  | Perfect | Frank | Shockley | Others | Unit
MD | Pure Al | 1.9 | 1.1 | 41 | 0.42 | 1015 m-2
MD | Al–1 at%Cu | 1.7 | 1.2 | 45 | 0.39 | 1015 m-2
PFC | Pure Al, low V | 0.1 (GND) | 1015 m-2
PFC | Pure Al, high V | 1.7 (GND) | 1015 m-2
PF-CP | Al–4.5at%Cu | 0.004-0.005 (SSD) | 1015 m-2
Past experimental results |
EBSD, AM Al-Cu [66] | 0.072 (GND) | 1015 m-2
XRD, Melt spun Al 7075 [67] | 0.057 (Total) | 1015 m-2
XRD, Melt spun Al 5083 [68] | 0.14 (Total) | 1015 m-2
XRD, AM Al-Si-Mg [20] | 0.92 (Total) | 1015 m-2
Neutron diffr., AM Al-Si-Mg [16] | 0.16-0.50 (Total) | 1015 m-2
TEM line-inter., AM Al-Si-Mg [69] | 0.11-0.31 (Total) | 1015 m-2

##  4 Discussion

The MD simulations produced a relatively high density of Shockley partial
dislocations forming around stacking faults that form around some of the grain
boundaries, as shown in Figure 3. The density of intragrain dislocations
(mostly perfect and Frank type) are remarkably close to the GND densities
estimated in the rapid solidification PFC-AE simulations, as summarized in
Table 3. These dislocation densities are 10-20 times larger than the listed
experimentally measured dislocation densities [67, 68, 66, 69, 20]. As
discussed more thoroughly in the Introduction section, there are several
sources of error in the experimental evaluation of dislocation densities that
can, to some extent, explain the discrepancy between the MD and PFC–AE
simulations and the experiments.

The discrepancy can be related to the fact that some dislocations can
annihilate during cool-down, and this cool-down is not simulated in this
paper. Moreover, solidification velocities are somewhat higher in the MD and
PFC–AE simulations than in typical AM or melt spun conditions, as indicated by
the larger grain sizes in the experiments. Larger solidification velocities in
the simulations are likely to pronounce the formation of dislocations. It
should be noted that in this work, we did not perform a systematic
investigation of solidification velocity versus dislocation density; we did,
however, demonstrate a qualitative difference between "intermediate quench"
and "large quench" regimes. The "intermediate quench" case was essentially
free of intragrain GNDs with an overall lower average GND density, while the
"large quench" case had significantly higher GND densities, visible intragrain
lattice rotation, and trapped microvoid and smaller lattice defects.

Work is currently under way to match the PFC and PFC–AE models’ kinetic time
scales quantitatively to experimental conditions, which should help remedy
this issue. Finally, the secondary hard phases in Al-Si-Mg alloy dislocation
densities in Table 3, from measurements in [69, 20], are likely to have
dislocations that are generated through thermal misfit between the FCC
aluminium matrix and secondary hard phases that form during cool-down. These
secondary phase effects are not accounted for in our simulations. In contrast
to the MD and PFC–AE results, the PF–CP simulations lead to an order of
magnitude smaller dislocation densities than what in the experiments in Table
3. This discrepancy can be related to 1) the uncertainty in the high
temperature elastic and dislocation dynamics parameters used in the CP model;
2) lack of lattice curvature (orientation gradient) and GNDs in the CP model;
3) limited and indirect description of dislocation climb—a strongly thermally
activated process—through the visoplastic flow rule parameters $K_{\phi}$ and
$N$ in Equation 6, which is known to be heavily thermally activated process.

While the MD was carried out in 3D, the PFC–AE and PF–CP simulations were in
2D. Performing the PF–CP simulations in 3D will lead to more complex grain
morphologies and grain boundaries, which are likely to increase the
heterogeneity due to solidification shrinkage and thereby the dislocation
density levels. Currently, only pure aluminium was considered in the PFC
simulations. Interaction between copper segregation and defect structures
would be interesting to consider on diffusive time scales in PFC simulations,
in order to better understand the emergence of pre-precipitates in rapid
aluminium alloys. High quality mechanical properties for Al-Cu at temperatures
close to the melting point are scattered and are a significant source of
uncertainty for quantitative predictions in the PF–CP simulations. Some of
these properties, especially elastic properties, could be evaluated with MD
simulations.

##  5 Conclusion

In this paper, we presented a multiscale analysis of crystalline defect
formation in pure aluminium and dilute Al-Cu alloys using molecular dynamics
on smallest scales, phase field crystal based amplitude expansion (PFC–AE)
simulations on intermediate scales, and a coupled phase field – crystal
plasticity (PF–CP) simulation scheme on largest scales.

The mechanism of formation, merging, and healing of microvoids in rapid
solidification of pure aluminium was demonstrated with PFC–AE simulations,
being consistent with past experimental literature.

The PFC-AE and MD simulations produced dislocation densities which were
markedly close to each other, while being an order of magnitude larger than
past experimentally measured dislocation densities. In contrast, the PF–CP
framework lead to an order of magnitude smaller dislocation densities than in
past experiments, where several potential sources of error were pointed out.

The presented multiscale framework can be used to investigate the formation of
crystalline defects, as well as to perform calibration of uncertain parameters
related to phenomenological dislocation evolution equations in the crystal
plasticity models. As the submodels (MD, PFC–AE and PF–CP) underlying the
platform presented continue to evolve, it is expected that the quantitative
prediction of dislocation densities with experiments will improve
significantly.

## Funding

TP, MH, and AL wish acknowledge the support of Academy of Finland through the
HEADFORE project, Grant No. 333226, as well as CSC – IT Center for Science,
Finland, for computational resources. NP and PJ acknowledge the Natural
Sciences and Engineering Research Council of Canada (NSERC) and the Canada
Research Chairs (CRD) Program.

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††thanks: author to whom correspondence should be addressed.

# Perpendicular magnetic anisotropy in ultra-thin Cu2Sb-type (Mn-Cr)AlGe films
onto thermally oxidized silicon substrates

Takahide Kubota<EMAIL_ADDRESS>Institute for Materials
Research, Tohoku University, Sendai 980-8577, Japan  Center for Spintronics
Research Network, Tohoku University, Sendai 980-8577, Japan  Keita Ito
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
Center for Spintronics Research Network, Tohoku University, Sendai 980-8577,
Japan  Rie Y Umetsu  Institute for Materials Research, Tohoku University,
Sendai 980-8577, Japan  Center for Spintronics Research Network, Tohoku
University, Sendai 980-8577, Japan  Center for Science and Innovation in
Spintronics (Core Research Cluster), Tohoku University, Sendai 980-8577, Japan
Koki Takanashi  Institute for Materials Research, Tohoku University, Sendai
980-8577, Japan  Center for Spintronics Research Network, Tohoku University,
Sendai 980-8577, Japan  Center for Science and Innovation in Spintronics (Core
Research Cluster), Tohoku University, Sendai 980-8577, Japan

###### Abstract

Perpendicularly magnetized films showing small saturation magnetization,
$M_{\mathrm{s}}$, are essential for spin-transfer-torque writing type
magnetoresistive random access memories, STT-MRAMs. An intermetallic compound,
(Mn-Cr)AlGe of the Cu2Sb-type crystal structure was investigated, in this
study, as a material showing the low $M_{\mathrm{s}}$ ($\sim 300$ kA/m) and
high-perpendicular magnetic anisotropy, $K_{\mathrm{u}}$. The layer thickness
dependence of $K_{\mathrm{u}}$ and effects of Mg-insertion layers at top and
bottom (Mn-Cr)AlGe$|$MgO interfaces were studied in film samples fabricated
onto thermally oxidized silicon substrates to realize high-$K_{\mathrm{u}}$ in
the thickness range of a few nanometer. Optimum Mg-insertion thicknesses were
1.4 and 3.0 nm for the bottom and the top interfaces, respectively, which were
relatively thick compared to results in similar insertion effect
investigations on magnetic tunnel junctions reported in previous studies. The
cross-sectional transmission electron microscope images revealed that the Mg-
insertion layers acted as barriers to interdiffusion of Al-atoms as well as
oxidization from the MgO layers. The values of $K_{\mathrm{u}}$ were about
$7\times 10^{5}$ and $2\times 10^{5}$ J/m3 at room temperature for 5 and 3 nm-
thick (Mn-Cr)AlGe films, respectively, with the optimum Mg-insertion
thicknesses. The $K_{\mathrm{u}}$ at a few nanometer thicknesses is comparable
or higher than those reported in perpendicularly magnetized CoFeB films which
are conventionally used in MRAMs, while the $M_{\mathrm{s}}$ value is one
third or less smaller than those of the CoFeB films. The developed (Mn-Cr)AlGe
films are promising from the viewpoint of not only the magnetic properties,
but also the compatibility to the silicon process in the film fabrication.

Magnetoresistive random access memory, MRAM, is an emerging memory attracting
much interest in its non-volatility and fast operation. [1, 2, 3] For
expanding practical application of MRAMs, reduction of the critical current
density, $J_{\mathrm{c}}$, is required for the magnetization switching induced
by spin-transfer-torque, STT, in magnetic tunnel junctions, MTJs. The
$J_{\mathrm{c}}$ is proportional to saturation magnetization, $M_{\mathrm{s}}$
of the magnetization switching layer[4], thus magnetic materials showing low
$M_{\mathrm{s}}$ are important. For the STT-MRAMs with giga-bit-class memory
capacity, in addition, perpendicular magnetization is essential for small
$J_{\mathrm{c}}$ [5, 6] as well as for high-density memory cells, and the high
perpendicular magnetic anisotropy, $K_{\mathrm{u}}$, is also necessary for the
data retention. [1, 2, 3] Various materials have been studied as a class of
low $M_{\mathrm{s}}$ and high $K_{\mathrm{u}}$ system, e.g. Mn-Ga alloys
showing D022\- and L10-phases, [7, 8, 9] D022-Mn3Ge alloys, [10, 11, 12] and a
body-centered-tetragonal Mn ultra-thin film.[13] Although attractive
properties were reported for all those materials showing $M_{\mathrm{s}}\sim$
100 – 300 kA/m and $K_{\mathrm{u}}\sim$ 106 J/m3 or greater at room
temperature, a drawback is that most studies were done using single
crystalline MgO substrates which are unsuitable for practical devices. A few
studies also reported film samples onto thermally oxidized silicon substrates,
e.g. using Mn-Ga alloys, [14, 15] however, another intrinsic drawback is all
those Mn-alloys in previous studies were ferrimagnets 111Theoretically,
Mn50Ga50 with the L10 phase is a ferromagnet. However, in experiments, the
film samples were off-stoichiometric with Mn-rich composition which are
ferrimagnetic. The stoichiometric L10-Mn50Ga50 is unstable, thus there were
few experimental reports.which caused reduction of spin polarization at the
interface.[17]

As an alternative material with the Mn-alloys in previous studies, here, an
intermetallic compound (Mn-Cr)AlGe [18] is focused on. The (Mn-Cr)AlGe
compound is a ferromagnet which belongs to a family of Mn-based intermetallic
compounds showing the Cu2Sb-type crystal structure, such as MnAlGe [19, 20,
21, 18, 22] and MnGaGe. [23, 20, 24] The $M_{\mathrm{s}}$ values of the Mn-
compounds range 200 – 300 kA/m which are relatively small because of the small
magnetic moment of the ferromagnetically coupled Mn-moments showing itinerant
electronic structure. [25] The values of $K_{\mathrm{u}}$ are 7.3 $\times$ 105
and 8.1 $\times$ 105 J/m3 for (Mn0.76Cr0.28)Al0.94Ge1.02[18] and for
Mn1.0Ga1.1Ge0.9[24], respectively, at room temperature, which are much higher
than those reported for conventionally used CoFeB films showing perpendicular
magnetization. [26] These physical properties of low $M_{\mathrm{s}}$ and high
$K_{\mathrm{u}}$ are promising for the MRAM application, in addition, there is
another merit of the silicon process compatibility for those compounds, i.e.
the films with (001)-texture exhibiting perpendicular magnetization are
available on the SiO2 surfaces as well as other amorphous substrates. [27, 28,
29, 18, 22, 30] The (001)-orientation is also an essential requirement for
MTJs using (001)-textured MgO$|$CoFe(B) structure that exhibits large tunnel
magnetoresistance. [31] In addition, a recent study simulated reduction of
$J_{\mathrm{c}}$ in antiferromagnetically coupled layered structure consisting
of high- and low-$M_{\mathrm{s}}$ materials, [32] for which the Cu2Sb-type
(Mn-Cr)AlGe can be a low-$M_{\mathrm{s}}$ material layered with, e.g., CoFe(B)
showing relatively high-$M_{\mathrm{s}}$. In our previous study, the template
effect was demonstrated for the (001)-textured growth of the Cu2Sb-MnAlGe
films using (001)-textured MgO seed/capping layers, in which the MgO(001)
template promotes the (001)-texture in the MnAlGe films, however, relatively
thick dead layer around the MnAlGe$|$MgO interfaces degraded $K_{\mathrm{u}}$
for the thickness less than 10 nm. [33] The sputtered MgO layer may cause
oxidization at the interface with a metallic magnetic material, which can be
eliminated by inserting an ultra-thin metallic layer, e.g. a Mg-layer in
between. [34, 35, 36] Such an interface layer possibly improves the dead layer
problem of the interface between the Mn-compounds and MgO, thus in this study,
the effect of the Mg-interface layer was investigated for (Mn-Cr)AlGe$|$MgO
interfaces to realize high-$K_{\mathrm{u}}$ at the layer thickness of a few
nanometer.

The layered film samples were fabricated using a magnetron sputtering system
with a base vacuum pressure bellow 1 $\times$ 10-7 Pa for the deposition
chamber. All samples were fabricated onto single crystalline silicon (100)
substrates with a 650 nm-thick thermally oxidized amorphous layer on the
surface. The stacking structure of samples is as follows: Sub.$|$buffer
layers$|$MgO 1.5 nm$|$Mg $t_{\mathrm{Mg}}^{\mathrm{btm}}|$(Mn-Cr)AlGe $t|$ Mg
$t_{\mathrm{Mg}}^{\mathrm{top}}|$MgO 1.5 nm$|$cap, where
$t_{\mathrm{Mg}}^{\mathrm{btm(top)}}$ and $t$ are the thicknesses of the Mg-
insertion for the bottom (top) interface and that of the (Mn-Cr)AlGe layer,
respectively. The buffer layers consist of Ta 3 nm$|$W 0.3 nm$|$CoFeBTa 1 nm,
from bottom to top, which promotes (001)-texture of the bottom MgO layer.
222The 0.3 nm-thick W layer was not used in the series A samples, however, no
difference in magnetic properties was confirmed by comparing the samples with-
and without- the W layer, which are shown in Supplementary Material. Note that
the CoFeBTa layer is nonmagnetic in this study. The film composition of the
(Mn-Cr)AlGe layer was (Mn0.8Cr0.2)Al1.1Ge0.9 in the atomic ratio. All layers
were deposited at room temperature inside the ultra-high vacuum chamber, and a
3 nm-thick Ta was deposited on the top as a cap layer. After the capping, all
samples were annealed using a vacuum furnace. The annealing temperatures were
300, 400, and 500 °C. Magnetization curves were measured at room temperature
using vibration sample magnetometers and a superconducting quantum
interference device magnetometer. The interface properties as well as the
crystal structures were characterized using high-angle annular dark field-,
HAADF-, scanning transmission electron microscope, STEM, images. The
thicknesses of the Mg-insertion were optimized via two steps: First, the top
interface was optimized by changing $t_{\mathrm{Mg}}^{\mathrm{top}}$ from 0 to
7 nm, for which $t_{\mathrm{Mg}}^{\mathrm{btm}}$ was fixed to 0. Then, for the
second step, the bottom interface was optimized with the
$t_{\mathrm{Mg}}^{\mathrm{btm}}$ in the range of 0 to 1.4 nm, for which
$t_{\mathrm{Mg}}^{\mathrm{top}}$ was fixed to 3.0 nm. For both steps, $t$ was
fixed to 10 nm. After the two steps, three sample series were selected as
shown in Table 1 to study the $t$ dependence of $K_{\mathrm{u}}$ for the (Mn-
Cr)AlGe layer, for which $t$ was changed from 3 to 30 nm.

Figure 1: Mg-insertion thickness, $t_{\mathrm{Mg}}^{\mathrm{top}}$ or
$t_{\mathrm{Mg}}^{\mathrm{btm}}$, dependence of magnetization values, $m$, per
the sample area, $A$. The $t=10$ nm for all samples. (a) In the
$t_{\mathrm{Mg}}^{\mathrm{top}}$ dependence, $t_{\mathrm{Mg}}^{\mathrm{btm}}$
was fixed to 0, and (b) in the bottom $t_{\mathrm{Mg}}^{\mathrm{btm}}$
dependence, $t_{\mathrm{Mg}}^{\mathrm{top}}$ was fixed to 3.0 nm. The
temperatures shown in the panels are the annealing temperatures of the
samples.

The Mg-insertion dependence of magnetization, $m$, per the sample area, $A$,
is shown in figure 1.333Magnetization curves for fig. 1 appear in
Supplementary Material For the top interface, shown in fig. 1(a), the
magnetization increases with $t_{\mathrm{Mg}}^{\mathrm{top}}$ of up to the
certain thickness values depending on the annealing temperature, i.e., the
values of $t_{\mathrm{Mg}}^{\mathrm{top}}$ for the maximum $m/A$ shift towards
larger values for the higher annealing temperature, the reason for which will
be discussed in the latter part of this article. The maximum $m/A$ was about
2.9 mA for $t_{\mathrm{Mg}}^{\mathrm{top}}$ = 3 – 4 nm and the annealing
temperature of 400 °C, thus $t_{\mathrm{Mg}}^{\mathrm{top}}$ of 3.0 nm is
selected as an optimum value for the first step. The Mg-insertion for the
bottom interface, shown in fig. 1(b), exhibits moderate dependence except for
the annealing temperature of 300 °C in which $m/A$ monotonically decreases
with $t_{\mathrm{Mg}}^{\mathrm{btm}}$. For other annealing temperatures, with
increasing $t_{\mathrm{Mg}}^{\mathrm{btm}}$, $m/A$ gradually increases for 400
°C, while that exhibits gradual decrease for 500 °C. As a conclusion from the
second step, the $t_{\mathrm{Mg}}^{\mathrm{btm}}$ of 1.4 nm is selected.

Table 1:  Definition of the sample series.  Series | Mg layer thickness (nm)
---|---
| bottom | top
A | 0 | 0
B | 0 | 3.0
C | 1.4 | 3.0

Three sample series listed in table 1 are selected for the investigation on
the $t$ dependence of $K_{\mathrm{u}}$.

Figure 2:  Annealing temperature dependence of (a) saturation magnetization,
$M_{\mathrm{s}}$ and the thicknesses of dead layer, $t_{\mathrm{dead}}$
evaluated using the $t$ dependence of $m/A$ plots.

The $M_{\mathrm{s}}$ and the dead layer thickness, $t_{\mathrm{dead}}$ were
evaluated by the slope and the intercept to the horizontal axis, respectively,
in plots of $m/A$ as a function of $t$.444Magnetization curves and the plots
of $m/A$ vs $t$ used for fig. 2 are shown in Supplementary Material. The
values are summarized in figure 2. The $M_{\mathrm{s}}$ shows no dependence on
the sample series or annealing temperature within the range of error bars, and
the values are about 330 kA/m which is close to a theoretical value of 352
kA/m for (Mn0.8Cr0.2)AlGe.[18] On the other hand, $t_{\mathrm{dead}}$ shows
clear dependence on the sample series as well as the annealing temperature: On
the annealing temperature dependence, $t_{\mathrm{dead}}$ exhibit minima at
400 °C for all sample series, and among the sample series, series B and C
exhibit the minima; $t_{\mathrm{dead}}$ of about 1.2 $\pm$ 0.5 nm which is
smaller than that of 3.5 $\pm$ 0.2 nm for the series A.

Figure 3:  Magnetization curves of 5 nm-thick samples for (a) series A and (b)
C. (c) a 3 nm-thick sample for series C. The annealing temperature is 400 °C
for all samples. The marks of $\perp$ and $||$ represent applied magnetic
fields of perpendicular-to-plane and in-plane directions, respectively.

The difference among the sample series is clearly seen in magnetic properties
especially at $t$ below 10 nm. Figure 3 shows magnetization curves of 5 nm-
thick samples for series A and C, and a 3 nm-thick samples for series C. The
squareness of hysteresis curves for the perpendicular-to-plane magnetic field
direction is close to one for both 5 nm-thick samples of series A and C;
however, the magnitude of magnetization is smaller for the sample in series A
(fig. 3(a)) than that in series C (fig. 3(b)). In the series C, the
perpendicular magnetization showing a high squareness ratio is maintained down
to the $t$ of 3 nm as shown in fig. 3(c). Note that similar results were
obtained in series B for $t=$ 5 and 3 nm which are shown in Supplementary
Material.

Figure 4:  The $t$ dependence of perpendicular magnetic anisotropy energy,
$K_{\mathrm{u}}$.

The $t$ dependence of $K_{\mathrm{u}}$ was evaluated to characterize the
difference among the sample series quantitatively. The $K_{\mathrm{u}}$ is
defined using the actual layer thickness, $t-t_{\mathrm{dead}}$, with the
following equation,
$K_{\mathrm{u}}=K_{\mathrm{u}}^{\mathrm{eff}}/(t-t_{\mathrm{dead}})+(\mu_{0}/2)M_{\mathrm{s}}^{2}$,
where $K_{\mathrm{u}}^{\mathrm{eff}}$ is the effective perpendicular magnetic
anisotropy energy per unit area which was calculated from the area enclosed by
the perpendicular-to-plane and in-plane magnetization curves, e.g. shown in
fig. 3. The $K_{\mathrm{u}}$ are summarized in figure 4 as a function of $t$
for all the sample series. For series A, $K_{\mathrm{u}}$ of about $7\times
10^{5}$ J/m3 was achieved in samples down to the $t$ of 15 nm, and at the $t$
of 10 nm or thinner, $K_{\mathrm{u}}$ decreases with decreasing $t$, which is
about $1\times 10^{5}$ J/m3 at 5 nm. On the other hand for series B and C,
$K_{\mathrm{u}}$ of about $7\times 10^{5}$ J/m3 was maintained down to the $t$
of 5 nm; in addition, 3 nm-thick samples show $K_{\mathrm{u}}$ of about
$2\times 10^{5}$ J/m3. These results clearly confirm that the reduction of
$t_{\mathrm{dead}}$ efficiently improved the $K_{\mathrm{u}}$ in a few
nanometer thickness region.

Figure 5:  The cross-sectional scanning transmission electron microscope,
STEM, images of samples for (a, b) series A and (c, d) B. The annealing
temperature was 400 °C and $t=10$ nm for both samples. (a, d) The bright
field, BF, images. (b, c) high-angle annular dark field, HAADF, images. The
lines drawn inside the HAADF images are line profiles of energy dispersive
x-ray mapping for the samples. The color bars drawn on the middle describe the
possible layer components. Crystal structure of the Cu2Sb-type (Mn-Cr)AlGe is
schematically depicted in the center.

Although the Mg-insertion is found to be useful for suppressing the
$t_{\mathrm{dead}}$, a question is the relatively thick
$t_{\mathrm{Mg}}^{\mathrm{top(btm)}}$, i.e., in previous studies using MgO-
MTJs, the optimum thicknesses of Mg-insertion range 0.4 – 1 nm.[34, 35, 36] In
the present study, the optimum thicknesses are 1.4 and 3 nm for the bottom and
top (Mn-Cr)AlGe$|$MgO interfaces, respectively, which are thicker than those
in the previous studies. The cross sectional HAADF-STEM observations were
carried out to characterize the dead layer and the crystal structures more
clearly, and the images are shown in figure 5 for $t=10$ nm in series A and B.
First, concerning crystal structure of the middle region, both samples exhibit
high-(001) orientation which can be seen in the lattice images of the bright
field-, BF-, STEM images shown in figs. 5(a) and 5(d) for series A and B,
respectively. The high-orientation can be also confirmed in HAADF images shown
in figs. 5(b) and 5(c) in which periodical contrast changes are clearly
observed. The HAADF contrasts also implies the layered atomic components with
the chemical order as shown in the center of fig. 5. Second, a difference is
found for the actual layer thickness of the (Mn-Cr)AlGe layer: From the BF-
STEM images, the layer thicknesses are 8.5 $\pm$ 0.2 and 9.7 $\pm$ 0.2 nm for
series A and B, respectively. Line profiles of energy dispersive x-ray, EDX,
spectroscopy represent the difference in the upper interfaces more clearly,
which causes the difference in the layer thicknesses: For the series A sample,
Al-rich oxidized region is found between the (Mn-Cr)AlGe and the top-MgO
layer, for which the thickness is approximately 2 nm and is marked as ”Al-O”
in the color scale beside the HAADF image of fig. 5(b). On the other hand for
series B, similar Al-rich region is observed at the top interface, however,
the thickness of less than 1 nm is much smaller than that in series A.
Concerning the interface to the bottom MgO layer, the Al-O layer is also
observed in each sample series, for which the thickness of approximately 1 nm
is similar to each other. Such formation of the Al-O layer caused by
interdiffusion of Al-atoms from a metallic layer to an oxide layer has been
reported in layered samples consisting of an ultra-thin Co2FeAl film capped by
a sputtered MgO layer, in which the diffusion of Al-atoms might occur during
the deposition of the MgO upper layer onto the metallic layer containing Al.
In the present study, the formation of the Al-O layer can be qualitatively
understood with the following scenario: The (Mn-Cr)AlGe layer is amorphous in
the as-deposited state, for which the constituent elements are randomly
distributed at the surface before the deposition of the MgO layer. During the
deposition of the MgO layer, the surface of (Mn-Cr)AlGe could be exposed to
small amount of oxygen plasma which causes oxygen rich interface between the
(Mn-Cr)AlGe and the top MgO layers. Here, the formation free energy of Al-
oxide is the smallest among the constituent elements of (Mn-Cr)AlGe[40], thus
Al diffusion may occur with assistant of plasma energy during the deposition
of MgO as well as the post-annealing procedure resulting in the formation of
the Al-rich oxide region for the upper interface. For the bottom interface, Al
diffusion may also occur during the annealing process, however, the amount of
Al-diffusion is less than the top interface because the oxygen rich interface
was never formed for the bottom interface. Thus, the Mg-insertion is
considered to suppress such Al-diffusion as well as the formation of Al-O
resulting in the suppression of $t_{\mathrm{dead}}$ and the enhancement of
$K_{\mathrm{u}}$ at $t$ in a few nanometer thickness region. Based on the
discussion above, an answer can be given to the question for the different
optimum $t_{\mathrm{Mg}}^{\mathrm{top(btm)}}$ values depending on the
annealing temperatures shown in fig. 1: By considering that the diffusion
process of Al is a factor which promotes the crystallization of (Mn-Cr)AlGe,
for no Mg-insertion sample, Al atoms move inside the layer by relatively low
energy-assist, i.e. the low annealing temperature of 300 °C which is
sufficient for crystallization. However, with increasing the Mg-insertion, the
Al-diffusion was suppressed, which requires the annealing temperature of 400
°C or higher for crystallization showing large magnetization.

To conclude, the perpendicular magnetic anisotropy of the Cu2Sb-type (Mn-
Cr)AlGe films was investigated, especially focusing on a-few-nanometer-
thickness region of the (Mn-Cr)AlGe layer. The Mg-insertion layer was employed
at the (Mn-Cr)AlGe$|$MgO interfaces, for which effects on magnetic properties
and the $t_{\mathrm{dead}}$ were systematically examined. The $M_{\mathrm{s}}$
exhibited almost no dependence on the Mg-insertion thickness, while
$t_{\mathrm{dead}}$ reduced with the Mg-insertion. The cross-sectional HAADF-
STEM images presented clear contrast change for the (Mn-Cr)AlGe layer, which
represented highly oriented (001)-crystal planes as well as chemical order of
(Mn-Cr)AlGe. The STEM images also revealed role of the Mg-insertion as a
diffusion barrier to Al atoms. The $K_{\mathrm{u}}$ of about $7\times 10^{5}$
J/m3 at $t$ of 5 nm is comparable with those for thick, e.g. $t$ of 30 nm
samples, and much higher than those reported in perpendicularly magnetized
CoFeB films which are conventionally used in STT-MRAMs. Although
$K_{\mathrm{u}}$ of about $2\times 10^{5}$ J/m3 at $t$ of 3 nm is comparable
with those for the CoFeB films, the small $M_{\mathrm{s}}$ of (Mn-Cr)AlGe can
be a merit for reducing the $J_{\mathrm{c}}$. Among various materials showing
small-$M_{\mathrm{s}}$ and high-$K_{\mathrm{u}}$, another merit of (Mn-Cr)AlGe
is the silicon process compatibility: All the samples in this study were
fabricated using thermally oxidized silicon substrates with relatively thin
underlayers including the (001)-oriented MgO template.[33] The solid-phase-
epitaxy based method for crystallization is compatible with device fabrication
process for complementally metal oxide semiconductor devices as well as for
MTJs.[31] The (Mn-Cr)AlGe film showing perpendicular magnetization is also
interesting from the application point of view.

###### Acknowledgements.

TK would like to thank Dr. Yoshiaki Sonobe for fruitful discussion, Mr. Issei
Narita for technical support of film composition analysis, and the Foundation
for Promotion of Material Science and Technology of Japan for HAADF-STEM
observation and analysis. This work was partially supported by KAKENHI
(20K05296) from JSPS, and a cooperative research program (No.20G0414) of the
CRDAM-IMR, Tohoku Univ.

## Data Availability Statements

The data that support the findings of this study are available within its
supplementary material.

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  * Note [3] Magnetization curves for fig. 1 appear in Supplementary Material.
  * Note [4] Magnetization curves and the plots of $m/A$ vs $t$ used for fig. 2 are shown in Supplementary Material.
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