Ferroelectric/ferroelastic domain_wall_builder creates domain walls and interface structure for the perovskites and ceramics. The code reads the structure in the vasp format, and then creates domain walls with user-defined sizes and angles.
There are three main scripts dwbuilder.py, dbuilder.py and hibuilder.py which build domain wall, single domain structures and experimentally observed atomic interface strutures with known orientation relationships (ORs), respectively. For the domain wall and interfaces, you only need to run dwbuilder.py. These scripts can be executed from the command line. I also have included examples of ferroelectric domain walls.
The package requires:
- numpy - http://www.numpy.org/
- metplotlib - https://matplotlib.org/
- ASE - https://wiki.fysik.dtu.dk/ase/
- Pymatgen - https://pymatgen.org/
For installation simply clone or download the code in your terminal and in the main directory of the package type:
> pip install .
This will copy the modules to your active python site-packages, thereby making them importable in any python script and will put the scrpits in the python bin, thereby making them executable in the shell.
After installation, you can start the software by running the following command in your terminal:
> dwbuilder
To execute the code, one should run the dwbuilder command from the command line. Upon running, the user will be prompted to input the name of the input structure file, as well as the domain wall angle and size (expressed in number of unit cells). This script builds the domain wall structure based on the space group of the input structure with the assumed polarization directions as indicated below.
R3c (Polarization direction (1-11))
R3m (Polarization direction (001))
P4mm (Polarization direction (001))
p6_3cm (Polarization direction (-1-10))
Pnma (Polarization direction (110))
Pmc2_1 (user-defined)
Amm2 (user-defined)
However, if you are unsure about the polarization dirction, you can run polarization.py to determine the polarization direction. This script uses the point charge model and you need to have both optimized CONTCAR and high symmtry structure POSCAR files to determine the polarization.
Example of using dwbuilder.py
ex:
> dwbuilder
***************************************************
* *
* Welcome to DWBuilder *
* *
***************************************************
* Hey, you must know what you are doing. *
* Otherwise, you might get wrong results. *
***************************************************
* Core Developer: M.Z.Khalid *
* Main Contributors: S.M.Selbach *
* Email: [email protected] *
***************************************************
Select a script to run:
1: dwbuilder.py 6: supercell.py
2: dbuilder.py 7: vasp2cif.py
3: hibuilder.py 8: bondanalysis.py
4: slab.py 9: rotationmatrix.py
5: polarization.py
Enter the number of the script to run: 1
Enter the input file name (with extension): BiFeO3_R3c.vasp
Space group number: 161
International symbol: R3c
Lattice type: rhombohedral
Select the domain wall angle:
R180 - R180
R71 - R71
R109 - R109
ALL - ALL
ALL
Enter the domain wall size (in number of unit cells): 3
Enter the size of the supercell along the a direction: 1
Enter the size of the supercell along the b direction: 1
Enter the size of the supercell along the c direction: 1
Enter the cutoff distance in angstroms to remove close atoms: 0
Lattice strain for R109 DW:
strain along a (%): 0.00
strain along b (%): 0.00
strain along c (%): 0.00
Lattice strain for R71 DW:
strain along a (%): 0.00
strain along b (%): 0.00
strain along c (%): 0.00
Lattice strain for R180 DW:
strain along a (%): 0.00
strain along b (%): 0.00
strain along c (%): 0.00
Domain wall structures and supercells created successfully! Log file written to LOGFILE.txt
Once you have developed the desired domain wall structure, you can visualize in vesta or ase to further refine domain wall artifacts.
Example of using hibuilder.py
In this example, I have reproduced an orientation relationship reported in the following article CMS.
> dwbuilder
***************************************************
* *
* Welcome to DWBuilder *
* *
***************************************************
* Hey, you must know what you are doing. *
* Otherwise, you might get wrong results. *
***************************************************
* Core Developer: M.Z.Khalid *
* Main Contributors: S.M.Selbach *
* Email: [email protected] *
***************************************************
Select a script to run:
1: dwbuilder.py 6: supercell.py
2: dbuilder.py 7: vasp2cif.py
3: hibuilder.py 8: bondanalysis.py
4: slab.py 9: rotationmatrix.py
5: polarization.py
Enter the number of the script to run: 3
Enter the bulk phase 1file name (with extension): Fe_unitcell.vasp
Enter the bulk phase 2 file name (with extension): Fe2Al5.vasp
Bulk phase 1 Space group number: 229
Bulk phase 1 International symbol: lm-3m
Bulk phase 1 Lattice type: cubic
Bulk phase 2 Space group number: 63
Bulk phase 2 International symbol: Cmcm
Bulk phase 2 Lattice type: orthrhombic
Enter three comma-separated values for bulk 1 lattice direction a: 1,0,1
Enter three comma-separated values for bulk 1 lattice direction b: 1.5,1.5,-1.5
Enter three comma-separated values for bulk 1 lattice direction c: -2,4,2
Enter three comma-separated values for bulk 2 lattice direction a: 0,0,-1
Enter three comma-separated values for bulk 2 lattice direction b: 1,0,0
Enter three comma-separated values for bulk 2 lattice direction c: 0,-2,0
Enter the stacking direction 0 for a, 1 for b and 2 for c): 2
strain along a (%): 1.1378
strain along b (%): -0.5492
strain along c (%): 8.2378
Angular strain along a (radians): 0.0
Angular strain along b (radians): 0.0
Angular strain along c (radians): 0.0
Warning: This code could generate interface/domain wall artifacts (i.e., Oxygen atoms, duplicate atoms etc,) at the interface, thus it requires manual adjustment.
hibuilder.py generates an initial configuration that needs additional refinement along a, b and c-directions. To determine the optimal interface distance along the a, b, and c axes, you can either perform static DFT or forcefield simulations. Additionally, you may need to explore various interfacial configurations to identify the thermodynamically stable structure.
The above code generates the following interface structure (the periodic image) visualized by VESTA:
It is also important to know that all the domain wall and interface structures built in this package follow the orthogonality condition, which means lattice vectors a,b and c are mutually perpendicular (i.e., orthognal) to each other and fulfill these conditions:
a x b = c
a x c = b
b x c = a
- A note on the microscopic degrees of freedom:
To find the global minimum of an interface structure, it is essential to calculate the optimal distance between the different bulk phases at the interface. You also need to identify the termination site that corresponds to the structure with the lowest energy, as well as consider the homogeneous interface structures of the periodic images.
To achieve these goals, I often use first-principles methods such as density functional theory (DFT), specifically with the Vienna Ab-initio Simulation Package VASP at 0 Kelvin. This computational approach helps determine the initial structures for further first-principles calculations.
Ferroelectric materials with a rhombohedral space group of R3m exhibit three types of ferroelectric domain walls (FDWs): R71, R109, and R180. Due to neutrality and mechanical compatibility, these FDWs develop along specific planes. For the R180 FDW, the {1-10} plane is used, while for R71 and R109, the FDWs are determined by the sum of the two polarization vectors.
- R71 FDW: Lies parallel to the diagonal
(1-10)plane in the primitive cell. - R180 FDW: Developed along the
{1-10}plane. - R109 FDW: The R109 FDW lies along the
(100)plane, and FDWs are simply developed by stacking the primitive cell along this direction.
To analyze the R71 and R180 FDWs, the pseudocubic rhombohedral unit cell is transformed into a 10-atom cell by rotating it 45° around the z-axis relative to the parent cubic unit cell. The transformation matrix applied is given as:
[ 1 0 1 ]
T = [ 1 0 -1 ]
[ 0 1 0 ]
In the script the above transformation matrix is applied in lattice vectors form as given below:
a=[1, 1, 0], b=[0, 0, 1], c=[1, -1, 0]
R71
The R71 domain walls are developed by stacking along the normal direction to the [1-10] plane, which aligns with the a-axis in the rotated coordinate system. Additionally, the mirror symmtry was developed by tranforming [1-10] to [-110] and from [001] to [00-1].
For the R71 FDWs, the orientation relationships for both domains (D1 and D2) are:
[ 1 0 1 ] [ 1 0 -1 ]
D1 = [ 1 0 -1 ] D2 = [ 1 0 1 ]
[ 0 1 0 ] [ 0 -1 0 ]
R180
Similar to the R71 domain wall, the R180 {1-10} wall is also parallel to the diagonal plane in the primitive cell. For the R180 FDW:
- A supercell is made by stacking the rotated 10-atom unit cell.
- The final FDWs are developed by stacking along the
[1-10]plane.
[ 1 0 1 ] [ -1 0 1 ]
D1 = [ 1 0 -1 ] D2 = [ -1 0 -1 ]
[ 0 1 0 ] [ 0 -1 0 ]
The transformation ensures that the polarization vectors in the two domains are oriented 180 degrees apart, achieving the necessary mirror symmetry for the R180 domain wall.
R109
The R109 FDW lies along the (100) plane. For this FDW:
- The domain walls are developed by stacking the primitive cell along the
(100)direction.
This straightforward stacking ensures that the polarization vectors align as required for the R109 domain wall configuration.
[ 1 0 0 ] [ 1 0 0 ]
D1 = [ 0 1 0 ] D2 = [ 0 -1 0 ]
[ 0 1 1 ] [ 0 0 -1 ]
Reference:
Taherinejad M, Vanderbilt D, Marton P, Stepkova V, Hlinka J. Bloch-type domain walls in rhombohedral BaTiO3. Physical Review B. 2012 Oct 22;86(15):155138.
Similar to the R3m, the R3c structure also possesses three FDWs: R71, R109, and R180. The R3c domain walls are developed by converting the reference R3c structure into a pseudocubic cell. This is done by cutting along the (1-11) plane, which is also the polarization axis.
The transformation vectors used to convert the R3c structure into a pseudocubic cell are given by the transformation matrix Y:
[ 1 -1 1 ]
T = [ 1 1 -1 ]
[-1 1 1 ]
The R71 , R109 , and R180 domain walls are found to adopt (001), (110), and (-110) pseudocubic domain wall planes, respectively. The transformation matrix to develop respective domains and domain walls are given as:
[ 1 0 1 ] [-1 0 1 ]
R71_D1 = [-1 0 1 ] R71_D2 = [ 1 0 1 ]
[ 0 1 0 ] [ 0 -1 0 ]
[ 0 1 0 ] [ 0 -1 0 ]
R109_D1 = [ 1 0 0 ] R109_D2 = [-1 0 0 ]
[ 0 0 1 ] [ 0 0 1 ]
[ 1 0 -1 ] [-1 0 -1 ]
R180_D1 = [ 1 0 1 ] R180_D2 = [-1 0 1 ]
[ 0 -1 0 ] [ 0 1 0 ]
The R109 domain walls are formed by stacking the D1 and D2 domains along the [001], while R71 and R180 are stacked along the [110] and [-110] directions, respectively.
Reference:
Wang Y, Nelson C, Melville A, Winchester B, Shang S, Liu ZK, Schlom DG, Pan X, Chen LQ. BiFeO 3 domain wall energies and structures: a combined experimental and density functional theory+ U study. Physical review letters. 2013 Jun 24;110(26):267601.
In the P4mm space group, with the polarization direction along the [001] direction, there are two types of domain walls allowed and developed in dwbuilder:
-
T180 Domain Walls (T180):
- Orientation: These domain walls are on the
(100)planes. - Polarization: The polarization vectors in neighboring domains are parallel but oriented in opposite directions.
- Orientation: These domain walls are on the
-
T90 Domain Walls:
- *Orientation: These domain walls are on the
(101)planes. - *Polarization: The polarization vectors on either side of the domain wall are almost perpendicular to each other.
- *Orientation: These domain walls are on the
The transformation matrix used for both domain walls are given as:
[ 1 0 0 ] [ -1 0 0 ]
T180_D1 = [ 0 1 0 ] T180_D2 = [ 0 1 0 ]
[ 0 0 1 ] [ 0 0 -1 ]
[ 0 -1 1 ] [ 0 -1 1 ]
T90_D1 = [ 1 0 0 ] T90_D2 = [ -1 0 0 ]
[ 0 1 1 ] [ 0 -1 1 ]
The T180 and T90 FDWs are develed by stacking along the [010] and [101] direction.
Reference:
Meyer B, Vanderbilt D. Ab initio study of ferroelectric domain walls in PbTiO 3. Physical Review B. 2002 Mar 1;65(10):104111.
The hexagonal manganites RMnO3 ( R= Sc, Y, Dy-Lu) belongs to the P6_3cm hexagonal space group. This space group is non-centrosymmetric, thus lacking inversion symmetry, which is a necessary condition for ferroelectricity. The polar axis for this group is along the [001] direction.
The polarization within hexagonal manganites originates from the structural tilt trimerization of the MnO5 polyhedra. Trimerization is characterized by two angles:
- Tilt Angle (θ):
- This is the angle between the
[001]direction and the Mn-apical O bond. - It gives the magnitude of the tilting and can be regarded as the order parameter.
- This is the angle between the
- Azimuthal Angle (φ):
- This angle describes the orientation of the tilting.
- During the phase transition, (φ) adopts one of six values separated by 60°, corresponding to trimerization around three possible origins with either in- or out-orientation of the tilting.
Based on the stacking site, there exist two kinds of domain walls within hexagonal manganites:
-
Neutral Domain Walls (NDW):
- Formation: NDWs are developed by stacking the positive and negative domains perpendicular to the
[001]axis. - Polarization: This stacking creates net zero polarization at the domain walls due to the cancellation of the positive and negative polarization of both domains.
- Formation: NDWs are developed by stacking the positive and negative domains perpendicular to the
-
Charged Domain Walls (CDW):
- Formation: CDWs are formed by stacking two different domains along the
[001]direction. - Polarization: This results in a net polarization at the domain walls, leading to charged domain walls.
- Formation: CDWs are formed by stacking two different domains along the
In dwbuilder, we have developed two scripts, ndw.py and cdw.py, to create these domain walls while maintaining hexagonal symmetry. To run these scripts, users need to provide the input domain structures for both domains.
ndw.py: This script is used to develop neutral domain walls (NDW).cdw.py: This script is used to develop charged domain walls (CDW).
By providing the necessary input structures, these scripts will generate the respective domain walls within the hexagonal manganites.
Note: You do not explicitly need to run these scripts directly. DWBuilder automatically detects the space group and, based on the space group, prompts users to provide details of the input structure and parameters.
Reference:
Kumagai Y, Spaldin NA. Structural domain walls in polar hexagonal manganites. Nature communications. 2013 Feb 26;4(1):1540.
The (Pnma) space group is non-polar, yet some materials within this group exhibit ferroelastic domain walls. DWBuilder generates domain walls for materials that typically crystallize in a non-polar (Pnma) structure under ambient conditions. This structure features antiferrodistortive (AFD) distortions, involving the tilting and rotation of the oxygen octahedra, which also lead to spontaneous ferroelastic strain.
- Parent Cubic Phase: In its parent cubic phase, the material exhibits a secondary ferroelectric (FE) instability.
- Polar Displacement: This instability arises from the polar displacement of ions (e.g., titanium ions within the oxygen octahedra), leading to ferroelectric polarization.
DWBuilder generates twin ferroelastic domain walls between the [110] and [1-10] domains. These walls suppress the primary AFD order parameter, activating the secondary FE instability at the twin walls. Consequently, these twin domain walls possess their own dipole moments within a ferroelastic paraelectric matrix, contributing to the overall properties of the material.
The transformation matrix used for this space group is given as:
[ 1 0 1 ] [ -1 0 1 ]
D1 = [ 1 0 -1 ] D2 = [ 1 0 1 ]
[ 0 1 0 ] [ 0 1 0 ]
The domain walls lie along the [110] direction.
References:
Barone, Paolo, Domenico Di Sante, and Silvia Picozzi. "Improper origin of polar displacements at CaTiO3 and CaMnO3 twin walls." Physical Review B 89.14 (2014): 144104.
Goncalves-Ferreira, L., Redfern, S.A.T., Artacho, E., Salje, E.K.H. "Ferrielectric twin walls in CaTiO3." Physical Review Letters, 101(9), 097602 (2008).
The Pmc2_1 and Amm2 space groups belong to the orthorhombic crystal system. Materials within these space groups can exhibit O90 and O180 (O for orthorhombic) domain walls. DWBuilder generates these domain walls by applying a rotation matrix around the c-axis.
-
User Input:
- The program initially prompts the user to enter the input structure in VASP format.
- The user is then asked to select the polar axis.
-
Rotation Matrix Application:
- Based on the user selection,
DWBuilderapplies the rotation matrix to develop the domain walls. - The rotation matrix is applied around the c-axis.
- Based on the user selection,
[ cos(γ) -sin(γ) 1 ]
Rc(γ) = [ sin(γ) cos(γ) -1 ]
[ 0 0 0 ]
- Polar Axis Adjustment:
DWBuilderensures that the polar axis does not lie along the c-axis before applying the rotation matrix.- If the user selects the a-axis as the polar axis, the structure is kept as is.
- If the b-axis or c-axis is selected as the polar axis, the structure is rotated by applying a transformation matrix as given below, aligning the a-axis as the polar axis.
- After aligning the a-axis as the polar axis,
DWBuilderapplies the rotation matrix around the c-axis by applying(Rc(γ)).
[ 1 0 0 ] [ 0 0 1 ] [ 0 1 0 ]
Ra = [ 0 1 0 ] Rb = [ 1 0 0 ] Rc = [ 0 0 1 ]
[ 0 0 1 ] [ 0 1 0 ] [ 1 0 0 ]
- O120 Head-to-Head/Tail-to-Tail and Head-to-Tail domain walls:
For O120 domain walls, DWBuilder builds two variants: Head-to-Head/Tail-to-Tail (O120_HH_TT) and Head-to-Tail (O120_HT) domain walls. The O120_HH_TT domain wall is developed by stacking the rotated domains parallel to each other, resulting in polarization meeting head-to-head on one end and tail-to-tail on the other side due to the periodic boundary condition. In contrast, the O120_HT domain wall is developed by stacking the rotated domains perpendicular to each other, ensuring a head-to-tail configuration.
DWBuilder develops domain wall supercells by generating and stacking two different domains, with the termination of the domains depending on the input unit cell termination. For example, PbTiO3 in the P4mm space group can have TiO- or PbO-terminated domain wall structures. To develop both variants, the user needs to define the input unit cell structure such that in one variant, the Pb atom is in the center, leading to PbO-terminated domain walls, while in the other, the Ti atom is in the center, resulting in TiO-terminated domain walls (See example). Additionally, for complex systems, DWBuilder includes a function to add one or two layers by defining the domain size in fractional form (e.g., 2.5), providing precise control over the domain wall structure and thickness.
-
Khalid MZ, Williamson BAD, Eggested K, Selbach SM. Hydrogen Doping at Proper/Improper Ferroelectric Domain Walls. To be submitted. 2024.
If you use DWBuilder, please cite the paper!
If you have any questions, find an issue you can get in contact with me.
MIT.