Nb/Ni Superlattice

The electronic structure of a Nb(110)/Ni(111) superlattice is created using Questaal’s superlattice editor lmscell, and the interface relaxed with lmf, using a constrained form of molecular statics.

JMRAM devices typically consist of films of Nb, oriented on the (110) direction, followed by a Cu spacer layer, followed by Ni, with another Cu spacer layer and finally capped with Nb. Cu and Ni are oriented along (111). Ni and Cu are both fcc with room temperature lattice constants 3.524 Å and 3.615 Å, respectively. Nb on the other hand has a lattice constant of 3.300 Å. To make an A/B superlattice unit cells of the A and B regions in the plane normal to the interface must be coincident. This is a challenging requirement, in part because the lattices are not matched, but also the (110) and (111) are not compatible.

The tutorial begins by rotating Nb [110] axis and the Ni [111] axis respectively to [001]. Then superlattice vectors are identified that are nearly coincident. By applying a small shear to the Ni, the lattice vectors are made exactly coincident.

At this point superlattices can be constructed. Some are made and the site positions at the frontier layers relaxed.

A separate tutorial takes the structural information made in this tutorial to investigate Landauer-Buttiker transport through a Nb/Ni/Nb trilayer.
Note: at present that tutorial is written for Nb[110]/Ni(110)/Nb([110] trilayer

Table of Contents

• Preliminaries
• Introduction
  • Basic strategy
• 1. Reconstructed unit cells for elemental Nb and Ni
  • 1a. Structural information for elemental Nb and Ni
  • 1b. Reconstructed Nb, and Ni, coincident site lattices
    • Initial reconstruction of Nb
    • Initial reconstruction of Ni
    • Nb and Ni, coincident site lattices
• 2. Lattice structure of the Nb(m)/Ni(n)/Nb(m) superlattice
  • 2a. Nb/Ni(3n)/Nb superlattices
  • 2b. Nb/Ni(3n)+1/Nb superlattices
  • 2c. Nb/Ni(3n)+2/Nb superlattices
• 3. Relaxing the Nb(2)/Ni(3)/Nb(2) superlattice
  • 3a. Relaxing the Nb(2)/Ni(3)/Nb(2) superlattice
    • Optimizing the Nb/Ni layer spacing
    • Preparing a site file for the ASA
  • 3b. Relaxing the Nb(2)/Ni(4)/Nb(2) superlattice
  • 3c. Relaxing the Nb(2)/Ni(5)/Nb(2) superlattice
• Footnotes and references

touch init.nb; rm *.nb *.ni;
cp ~/lmf/fp/test/nbni/{init.nb,init.ni} .

blm --nfile --ctrl=ctrl --omax=-.01 nb
lmchk ctrl.nb --wspec
cp spec.nb fpspec.nb
blm --nfile --ctrl=ctrl --omax=.14 nb
lmscell --plx~m~-1,-4,0,2,-2,0,1,1,2 '--rot=(1,-1,0)-pi/2,z:pi/4,z:acos(5/sqrt(43))' nb --wsitex~fn=site2 --sort:"x3 x2"
cp site2.nb site20.nb
lmchk ctrl.nb --wspec > out.lmchk.nb
lmscell -vfile=20 ctrl.nb '--stack~rmsite@targ=11:20~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,1/sqrt(2)~wsite@fn=site10a'
lmscell -vfile=20 ctrl.nb '--stack~rmsite@targ=1:10~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,1/sqrt(2)~wsite@fn=site10b'

blm --nfile --ctrl=ctrl --omax=-.01 ni
lmchk ctrl.ni --wspec > out.lmchk.ni
cp spec.ni fpspec.ni
blm --nfile --ctrl=ctrl --omax=.14 ni
lmchk ctrl.ni --wspec > out.lmchk.ni
lmscell '--rot=z:pi/4,x:acos(sqrt(1/3)),z:acos(4/sqrt(19))' --plx~m~3,0,2,-1,0,4,-1,3,-1 ni --wsitex~fn=site2 --sort:"x3 x2"
echo 3.27871926 0.0 0.0 1.82998284 2.15665546 0.0 0.0 0.0 1.41421356 | mcx -f9f12.6 . -s6.226 | sed  's/rows 1 cols 9 real/rows 3 cols 3 real/' > pnb
echo 3.082207   0.0 0.0 1.62221421 1.9668302  0.0 0.0 0.0 1.73205081 | mcx -f9f12.6 . -s6.643 | sed 's/rows 1 cols 9 real/rows 3 cols 3 real/' > pni
cp ctrl.ni ctrl.ni.bak
echo STRUC >> ctrl.ni
echo ' DEFGRD= ' `mcx -f9f12.8 pni -i pnb -x -t -a:nr=1 s s -w:nohead . | sed 's:[0-9.]*$:1:'` >> ctrl.ni
lmchk -vfile=2 ni --wsitex~fn=site3
cp ctrl.ni.bak ctrl.ni
lmchk -vfile=2 ni | grep vol; lmchk -vfile=3 ni | grep vol
sed -i.bak 's:\(plat.*[1-9.]*\)$:\1/1.00983:' site3.ni
lmchk -vfile=2 ni | grep vol; lmchk -vfile=3 ni | grep vol
lmchk -vfile=3 ni --wsitex~scala=6.226/6.643~fn=site42
sed -i.bak 's:\(plat.*1.83\).*$:\1:' site42.ni
lmchk ctrl.ni --wspec > out.lmchk.ni
lmscell -vfile=42 ctrl.ni '--stack~rmsite@targ=15:42~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,.61~wsite@fn=site14a'
lmscell -vfile=42 ctrl.ni '--stack~rmsite@targ=1:14,29:42~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,.61~wsite@fn=site14b'
lmscell -vfile=42 ctrl.ni '--stack~rmsite@targ=1:28~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,.61~wsite@fn=site14c'
lmscell -vfile=42 ctrl.ni '--stack~assign@z=pos(3,1)~addpos@dx=0,0,-z~wsite@fn=site42a'
lmscell -vfile=42 ctrl.ni '--stack~roll@off=14~assign@z=pos(3,1)~addpos@dx=0,0,-z~shorten@quad1~wsite@fn=site42c'
lmscell -vfile=42 ctrl.ni '--stack~roll@off=-14~assign@z=pos(3,1)~addpos@dx=0,0,-z~shorten@quad1~wsite@fn=site42b'

touch spec.nb; rm *.nb *.ni
cp ../input.fp.Nb110.Ni111/{init.nb,site20.nb,spec.nb,site42.ni,spec.ni} .
cp ~/lmf/fp/test/nbni/{init.nb,site20.nb,spec.nb,site42.ni,spec.ni} .

blm --nfile --ctrl=ctrl --omax=-.01 nb
lmscell -vfile=20 ctrl.nb -vdz='1/2/sqrt(2)-.61/2' --stack~file@site42.ni@dpos=0,0,-dz~file@site20.nb~wsite@fn=sitex
cp sitex.nb sitein.dat
blm --nfile --rdsite=sitein --rdspec~fn=spec.nb~fn=spec.ni --usefiler --nosort --shorten=no --ctrl=ctrl dat
lmplan dat --pledit:q

lmchk dat --wsite~short~fn=site82 --wpos=pos0
lmchk dat --mino~dxmx=.001~xtol=.0001~nkill=10~maxit=200~11:34,49:72 --wsite~short~fn=site82 --wpos=pos
mcx pos.dat pos0.dat -- > dpos.dat

cp site82.dat sitein.nbni
blm --autobas~loc=5 --rsham~el=-.3,-1.2~rscut=1.6 --dhftol=45 --molstat~xtol=0~gtol=0.01 --nfile --rdsite --omax=-.01 --nosort --shorten=no --nit=20 --nk=4,5,2 --gmax=10 --mixb=.1 --frzwf --wsite~ib=1:10,35:48,73:82~rlx=000~long~fn=site82 nbni
cp actrl.nbni ctrl.nbni

cp ../input.fp.Nb110.Ni111/ctrl.nb ../input.fp.Nb110.Ni111/site20.nb ../input.fp.Nb110.Ni111/site82.dat ../input.fp.Nb110.Ni111/site42.ni ../input.fp.Nb110.Ni111/dpos.dat ../input.fp.Nb110.Ni111/fpspec.nb ../input.fp.Nb110.Ni111/fpspec.ni .
cp site82.dat sitein.nbni
lmscell -vfile=20 ctrl.nb -vdz0='1/2/sqrt(2)' -vdz=dz0-.61/2 -vdp3=-.00 --stack~file@site42.ni@dpos=0,0,-dz+dp3/2~file@site20.nb@dpos=0,0,dp3~setlbl@spid=Nbx@targ=11:20,nbas-19:nbas-10~setlbl@spid=Nix@targ=21:34,nbas-33:nbas-20~pad=0,0,dp3~wsite@fn=sitex
cp sitex.nb sitein.nbni
blm --rdspec~fn=fpspec.nb~fn=fpspec.ni --usefiler --rdpos=dpos.dat --autobas~loc=5 --rsham~el=-.3,-1.2~rscut=1.6 --dhftol=45 --molstat~xtol=0~gtol=0.01 --nfile --rdsite --omax=-.01 --nosort --shorten=no --nit=20 --nk=4,5,2 --gmax=10 --mixb=.2 --frzwf --wsite~ib=1:10,35:48,73:82~rlx=000~long~fn=site82 nbni
cat actrl.nbni | sed s/NFORCE=0/NFORCE=20/ > ctrl.nbni

lmf ctrl.nbni -vfile=3 -vnk1=4 -vnk2=4 -vnk3=1 -vfrzwf=t -vconv=1e-4 -vconvc=1e-3 --rpos=pos36 --rs=11,0,0 -vnit=0 --wpos=pos5
lmchk ctrl.nbni --rdpos~scale=-1~fn=dpos.dat --wpos=pos0
mcx -f3f13.7 pos5.nbni pos0.nbni  -- > dpos.2+3+2

Preliminaries

This tutorial makes extensive use of many of the Questaal packages. including lmscell, lmplan lmf, and miscellaneous utilities such as vextract. They are assumed to be in your path, or in the directory where you cloned the Questaal package (assumed here to be ~/lm).

It invokes mpix to execute MPI jobs. mpix is a proxy for mpirun, the default command that launches MPI jobs. We assume your system has 16 processors on a core, but you can use any number, or run in serial.

For drawing postscript files, this tutorial assumes you will use the Apple open. Substitute your postscript viewer of choice for open.

Introduction

This tutorial builds a Nb(110)/Ni(111) superlattice, using Questaal tools blm, lmscell and lmchk. Since Nb[110] and Ni[111] are poorly lattice matched, unit cells in the plane normal to [110] and [111] respectively must be found that are nearly coincident. Within the coincident lattice, bonds across the interface are very irregular, and there is a large attendant lattice relaxation which must be dealt with. This is done in two steps: a trial relaxation that minimizes some measure of overlaps between atomic density, and subsequently using the LDA code lmf to relax the structure by minimizing the forces.

Basic strategy

To stack unit cells of Nb and Ni on top of one another, the two lattice vectors normal the stacking plane must match (they must be coincident). This entails:

• a reconstruction of bcc Nb with 10 atoms in a plane normal to [110]. The Nb lattice is also rotated orienting the [110] axis along z.
• a reconstruction of fcc Ni with 14 atoms in a plane normal to [111]. The Ni lattice is also rotated orienting [111] axis along z.
• The Ni lattice is sheared slightly to make Nb and Ni exactly coincident in the plane normal to z. It is also strained along z to partially accommodate the volume change from the in-plane shear, according to Poisson’s ratio.

The prescription is in outline:

• Use blm to build input files for elemental Nb and Ni in reconstructed lattices.
• Use the supercell maker lmscell in combination with blm to generate different reconstructions of the elemental lattices that can be joined together.
• Use the superlattice editor   lmscell --stack   to stack reconstructed Nb and Ni lattices into superlattices.
• Use lmchk to make a rough estimate of the atomic relaxation of frontier atoms at the Nb/Ni interfaces, followed by the LDA code lmf to perform a constrained relaxation the interface, allowing atoms in the frontier planes to relax until forces on them are small.

1. Reconstructed unit cells for elemental Nb and Ni

This tuturial requires only two init files corresponding to element Nb and Ni to complete.

This tutorial concludes with a self-consistent finish

Three files are required to use lmf to make the self-consistent electronic structure for a specified lattice. All are autogenerated once supplied structural information.

• A ctrl file, which is the main input file. We will autogenerate this file using the blm utility.
• A site file with structural data. (The ctrl file can also hold structural information, but we keep it separate so only the site changes as the structure changes.) blm also makes this file.
• A basp file, which defines the basis set. The free-atom code lmfa makes this file.

We need to assemble files in several contexts:

• For the end leads, Nb
• For the central region, Ni
• Ni/Nb superlattices, built so that the interface can be relaxed in a constrained manner. The constraint is necessary for later use, when superlattices are converged into trilayer (… Nb Nb / Ni / Nb Nb …) suitable for study of transport of an active region. The semi-infinite left and right leads are required to have a periodic lattice.

1a. Structural information for elemental Nb and Ni

Copy the boxes below into files init.nb and init.ni. They contain the requisite structural information for elemental bcc Nb and fcc Ni. Here we use the 0K lattice constants.

# Init file for Nb.  Lattice constant at 0K
LATTICE
    ALAT=6.226 PLAT= -.5 .5 .5  .5 -.5 .5  .5 .5 -.5
SITE
    ATOM=Nb   X=0 0 0

and

# Init file for Ni.  Lattice constant at 0K
LATTICE
    ALAT=6.643 PLAT= .5 .5 .0  .5 .0 .5  .0 .5 .5
SPEC
    ATOM=Ni  MMOM=0,0,0.6  PZ=0,0,0
SITE
    ATOM=Ni   X=0 0 0

1b. Reconstructed Nb, and Ni, coincident site lattices

Initial reconstruction of Nb

Nb is bcc, and it will be rotated so the [110] direction points along z which will be axis of the superlattice.

Build a template ctrl and site file for bcc Nb. This file is incomplete but it sufficient for steps needed for the reconstruction.

blm --nfile --ctrl=ctrl nb

--nfile tells blm to insert the following lines into ctrl.nb :

%ifdef file | file==0
  file=   site{file}
%endif

This allows Questaal codes to read structural data from siten_.ext_{:.path}, when preprocessor variable file is assigned to n. These lines have no effect if file is not assigned, in which case Questaal codes will read from the usual file site.ext.

Next we rotate the crystal with the following (admittedly odd) construction. The total rotation consists of a sequence of three separate ones: first, a rotation of the [110] axis to the [001], then a 45 degree rotation the new z axis, finally another rotation around the same axis. --rot=(1,-1,0)-pi/2 rotates [110] to z which will be the superlattice axis. The latter rotations will be made clear when we align Ni and Nb.

lmscell --noplx '--rot=(1,-1,0)-pi/2,z:pi/4,z:acos(5/sqrt(43))' nb --wsitex~fn=site1

Next, make a superlattice of this rotated cell. It reads site information from site1.nb which lmscell generated in the previous instruction.

lmscell -vfile=1 --plx~m~-1,-4,0,2,-2,0,1,1,2 nb --wsitex~fn=site2 --sort:"x3 x2"

This superlattice is artfully constructed from a linear combination of integer multiples bcc lattice vectors. These particular multiples yield the following lattice vectors:

               plx                            plx/plat                           qlx
   3.27872   0.00000   0.00000     -1.00000  -4.00000   0.00000      0.30500  -0.25880   0.00000  ← PLAT(1)
   1.82998   2.15666   0.00000      2.00000  -2.00000   0.00000      0.00000   0.46368   0.00000  ← PLAT(2)
   0.00000   0.00000   1.41421      1.00000   1.00000   2.00000      0.00000   0.00000   0.70711  ← PLAT(3)

This superlattice consists of two planes 10 atoms each. The first plane passes through the origin, and the second through $\sqrt{2}/2$, which is the spacing between (110) planes in the bcc lattice. To confirm this, do

lmplan -vfile=2 nb --pledit~q

bcc planes are stacked in an ABAB… pattern. A and B planes are equivalent but B is translated in the lateral direction. The third plane at a height $\sqrt{2}$, coincides with the first without a lateral translation. Thus this rotation/superlattice combination puts PLAT(3) along [001] with a length $\sqrt{2}$.

Initial reconstruction of Ni

Ni is fcc, and it will be rotated so the [110] direction points along z.

Make a template ctrl and site file for fcc Ni:

blm --nfile --ctrl=ctrl ni

In the Ni case we will rotate the [111] to z using --rot=z:pi/4,x:acos(sqrt(1/3)). An additional rotation z:acos(4/sqrt(19)) is made for reasons that will soon become clear. We also need a superlattice of this rotated cell, with different superlattice vectors than the Nb case. We can accomplish both the rotation and the superlattice in one step as follows:

lmscell '--rot=z:pi/4,x:acos(sqrt(1/3)),z:acos(4/sqrt(19))' --plx~m~3,0,2,-1,0,4,1,1,1 ni --wsitex~fn=site1 --sort:"x3 x2"

Compare the first lines of site1.ni and site2.nb. The lattice vectors PLAT(1) and PLAT(2) (when scaled by their respective lattice constants) are very similar.

This particular superlattice consists of 14 atoms all in a single plane normal to the new z, which is [111] in the unrotated frame. To confirm this, do

lmplan -vfile=1 ni --pledit~q

Ni is fcc whose planes are stacked ABC. We will use as a basic unit a superlattice with three planes (42 atoms) that include all the ABC planes. As needed we will split the ABC layers into single planes.

lmscell '--rot=z:pi/4,x:acos(sqrt(1/3)),z:acos(4/sqrt(19))' --plx~m~3,0,2,-1,0,4,-1,3,-1 ni --wsitex~fn=site2 --sort:"x3 x2"

You can confirm that PLAT(1) and PLAT(2) in files site1.ni and site2.ni are the same; only the third axis now points along z.
lmplan confirms that the new cell of 42 atoms consists of 3 planes, normal to [111] in the unrotated frame.

lmplan -vfile=2 ni --pledit~q

You can confirm that sites 15:28 are uniformly translated relative to sites 1:14, and sites 29:42 are translated by twice that amount. Each plane is spaced by $1/\sqrt{3}$ in units of the lattice constant.

Nb and Ni, coincident site lattices

Note that lmscell created the superlattices using --wsitex instead of --wsite. This saves the basis vectors in the form of fractional multiples of the lattice vectors rather than Cartesian coordinates. This will be important for the shear which we need to do next, because to stack Nb and Ni into a superlattice PLAT(1) and PLAT(2) have to be exactly coincident.

From each of the files site2.nb and site2.ni , extract the lattice vectors and scale by their respective lattice constants.

echo 3.27871926 0.0 0.0 1.82998284 2.15665546 0.0 0.0 0.0 1.41421356 | mcx -f9f12.6 . -s6.226 | sed  's/rows 1 cols 9 real/rows 3 cols 3 real/' > pnb
echo 3.082207   0.0 0.0 1.62221421 1.9668302  0.0 0.0 0.0 1.73205081 | mcx -f9f12.6 . -s6.643 | sed 's/rows 1 cols 9 real/rows 3 cols 3 real/' > pni

The sed instruction is appended to render pnb and pni into the 3×3 matrices they actually are. Inspect them and you should see:

% rows 3 cols 3 real
   20.413306    0.000000    0.000000   11.393473   13.427337    0.000000    0.000000    0.000000    8.804894
% rows 3 cols 3 real
   20.475101    0.000000    0.000000   10.776369   13.065653    0.000000    0.000000    0.000000   11.506014

This shows that the superlattice vectors are nearly coincident, with the first vector in each case pointing along x. This explains why these particular superlattice vectors and rotations were chosen.

We must shear one or the other lattice to render the first two vectors PLAT(1) and PLAT(2) exactly coincident. (PLAT(3) has no such constraint and we will deal with it later.) We elect to shear the Ni lattice, since real Ni layers are disordered anyway.
Do the following:

mcx -f3f12.8 pni -i pnb -x

You should get

  0.99698194  0.00000000  0.00000000
  0.04972026  1.02768205  0.00000000
  0.00000000  0.00000000  0.76524277

This matrix tells us how much we have to shear Ni to match Nb. For now we are interested only in the 2×2 subblock since the third axis is not dictated by matching planes of the two lattices. For now let’s restrict the deformation to the (PLAT(1), PLAT(2)) in the plane normal to z.

Temporarily add these lines to ctrl.ni:

STRUC
 DEFGRD=  0.99698194  0.00000000  0.00000000  0.04972026  1.02768205  0.00000000  0.00000000  0.00000000  1

and create a new file site3.ni with the transformed structural data.

lmchk -vfile=2 ni --wsitex~fn=site3

Remove or comment out DEFGRD before continuing.

Token DEFGRD supplies a general linear transformation of the lattice: it scales both lattice and basis vectors by a 3×3 matrix. Compare lattice vectors of site2.nb and site3.ni: you should see that (PLAT(1), PLAT(2)) are coincident (remember that alat is different in the two cases)

head -1 site2.nb; head -1 site3.ni

Finally we must decide how much to scale PLAT(3). Were Ni incompressible we would know how much we should strain Ni along z: any volume change generated by DEFGRD would be compensated by a tensile strain along z. To determine the volume change induced by DEFGRD do the following (be sure you have removed or commented out the DEFGRD line first!):

lmchk -vfile=2 ni | grep vol; lmchk -vfile=3 ni | grep vol

You should find that the volume changed by this ratio:

dval 3153.75620/3078.09526 = 1.02458

As a sanity check, copy site3.ni to site3.ni.bak and scale the 9^th element of plat by 1/1.02458, or simply use sed:

sed -i.bak 's:\(plat.*[1-9.]*\)$:\1/1.02458:' site3.ni

Run lmchk again and confirm the unit cell volumes of site2.ni and site3.ni are the same. This is the required strain for the incompressible case, which corresponds to a Poisson ratio of 0.5. The opposite limit, Poisson ratio of 0, means that change in lateral area has no effect on tensile strain along z (cork is approximately like that). You can look up Poisson ratios for elements here : https://periodictable.com/Properties/A/PoissonRatio.v.html; it is 0.31 for Ni, close to 3/5.

Since Ni isn’t incompressible, this strain is tool large. This volume increase should be reduced by 2/5, which corrresponds to a strain dval 1+.02458*2/5 = 1.00983 . Replace the 9^th element of plat with dval 1.73205081/1.00983 = 1.71519. To do it on the command-line, invoke

cp site3.ni.bak site3.ni
sed -i.bak 's:\(plat.*[1-9.]*\)$:\1/1.00983:' site3.ni

Run lmchk as before and confirm the ratios of cell volumes are dval 3123.05566/3078.09526 = 1.01461 . Thus the ~2.5% strain is reduced to a ~1.5% strain.

To make the final reconstructed Ni site file, do the following

lmchk -vfile=3 ni --wsitex~scala=6.226/6.643~fn=site42

Command line argument scala causes lmchk to scale the lattice constant ALAT by a factor and the lattice and basis vectors by its reciprocal. The structure is unchanged, only the distribution between ALAT and PLAT is changed. In this context ALAT becomes the same for Ni and Nb. The name site42.ni was used simply to mark how many atoms are in the unit cell.

Note plat(3,3) is 1.83006861, which is very close to 1.83, which is a multiple of 0.61. For convenience replace plat(3,3) with 1.83, since we will need to divide this cell into three subcells corresponding to each of the three planes. Use your text editor or use sed:

sed -i.bak 's:\(plat.*1.83\).*$:\1:' site42.ni

Before closing this section, it is convenient to construct partial site files with individual Nb or Ni planes

lmscell -vfile=20 ctrl.nb '--stack~rmsite@targ=11:20~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,1/sqrt(2)~wsite@fn=site10a'
lmscell -vfile=20 ctrl.nb '--stack~rmsite@targ=1:10~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,1/sqrt(2)~wsite@fn=site10b'

lmscell -vfile=42 ctrl.ni '--stack~rmsite@targ=15:42~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,.61~wsite@fn=site14a'
lmscell -vfile=42 ctrl.ni '--stack~rmsite@targ=1:14,29:42~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,.61~wsite@fn=site14b'
lmscell -vfile=42 ctrl.ni '--stack~rmsite@targ=1:28~assign@z=pos(3,1)~addpos@dx=0,0,-z~p3@0,0,.61~wsite@fn=site14c'

2. Lattice structure of the Nb(m)/Ni(n)/Nb(m) superlattice

The Nb(m)/Ni(n)/Nb(m) superlattice (m and n refer to numbers of planes in each region) has an Nb/Ni and an Ni/Nb interface; also periodic boundary conditions apply and there is a Nb/Nb interface as well. Provided the right layer ends in a B plane and the left begins with an A plane, the Nb/Nb interface will just be a continuum of 4 bcc Nb planes along z ([110] in the unrotated lattice). Periodic boundary conditions requires that cannot be translated with respect to each other, so we impose the constraint that the left Nb lead begins with an untranslated A plane and the right B lead ends with an untranslated B plane.

The Nb/Ni and Ni/Nb interfaces are more complicated. The main difficulty is that stacking ideal Nb and Ni planes will bring some Ni-Nb bonds much too close together while leaving void regions. Atoms can be relaxed, but there are some constraints we want to follow. We would like to structure the stacks so the superlattice is built in a lego-like fashion, layer by layer. That way construction can be automatic and easily generated for any number of layers. This is greatly facilitated by restricting atomic relaxation to atoms on frontier planes. This is an approximation, but it is a reasonable one. In any case the real material will be highly non-ideal, which is very difficult to model, so there is little point in trying to allow more general relaxations. The main thing is that the frontier atoms are not pressed too close together or kept too far apart. The fact that the displacements are limited only to the interface greatly simplifies the adoption the lego-like, layer-by-layer construction, since each layer is essentially independent of the others.

To stack Ni on Nb, we must address these issues:

• What is the spacing between Nb and Ni planes?
  As a first guess we will take the average of the Nb-Nb and Ni-Ni spacing.
• Should the Ni lattice be translated relative to the Nb?
  In principle this question would be resolved by lattice relaxations. However, the interfaces are complicated and there may be many metastable configurations where forces vanish.

To handle any number of Ni layers we need to consider three distinct configurations:

1. the Nb/(A’B’C’)_n/Nb family with 3n Ni planes
2. the Nb/(A’B’C’)_nA’/Nb family with 3n+1 Ni planes
3. the Nb/(A’B’C’)_nA’B’/Nb family with 3n+2 Ni planes

Within any family the interfaces will be the same, but between families they will not, and each family must be dealt with independently of the others.

2a. Nb/Ni(3n)/Nb superlattices

We will distinguish the three kinds of lattices by denoting them as 3n, 3n+1 and 3n+2 superlattices, respectively. This section builds the simplest case, the Nb(2)/Ni(3)/Nb(2) superlattice. If the files below are not already present in your working directory set up the directory with

cp ~/lmf/fp/test/nbni/{ctrl.nb,site20.nb,spec.nb,site42.ni,spec.ni,site14[abc].ni,site10[ab].nb} .

Even without any translations or reconstructions, there are several possible stacking arrangements. Consider the following stacking possibilities: AB123AB, AB231AB, AB312AB. Here A and B refer to the two Nb planes, and 1,2,3 refer to the three Ni planes. The simplest choice: AB|123|AB turns out to be the best among the three possibilities. Build the stack with the superlattice editor lmscell starting with the reconstructed Nb lattice as a base.

lmscell -vfile=20 ctrl.nb -vdz0='1/sqrt(2)' -vdz=dz0/2-.61/2 -vdp3=-.00 --stack~file@site42.ni@dpos=0,0,-dz~file@site20.nb~wsite@fn=sitex

This makes the AB|123|AB superlattice, using the average of Nb-Nb and Ni-Ni spacings as the spacing between Nb and Ni layers. The superlattice structure is written to file sitex.nb.

Use blm to construct a temporary ctrl and site file with extension .dat. At this stage we only want a ctrl file with enough information to examine the lattice structure. blm can read structural information from a site file, a feature which we use here:

cp sitex.nb sitein.dat
blm --nfile --rdsite=sitein --rdspec~fn=spec.nb~fn=spec.ni --usefiler --nosort --shorten=no --ctrl=ctrl dat

This lattice has 20+42+20 = 82 sites. Use lmscell to see the structure of stack

lmplan dat --pledit:q

Inspect the table below this line

   ib      Class       Plane  PL       x-x.h     y-y.h     z-z.h      h       del h

delh indicates the change in projection along z compared to the preceding site. delh is always zero except for the first atom in a new plane:

    1       1:Nb          1    0      0.00000   0.00000   0.00000   0.00000   0.70711
   ...
   11      21:Nb11        2    0      2.05873   0.10783   0.00000   0.70711   0.70711
   ...
   21       2:Ni          3    0      0.00000   0.00000   0.00000   1.36566   0.65855
   ...
   35      30:Ni15        4    0      0.43389   0.05135   0.00000   1.97566   0.61000
   ...
   49      58:Ni29        5    0      0.86779   0.10270   0.00000   2.58566   0.61000
   ...
   63      41:Nb21        6    0      0.00000   0.00000   0.00000   3.24421   0.65855
   ...
   73      61:Nb31        7    0      2.05873   0.10783   0.00000   3.95132   0.70711

This confirms that planes are stacked (Nb, Nb, Ni, Ni, Ni, Nb, Nb) with Nb-Nb spacings 0.70711, Ni-Ni spacings 0.61 and Nb-Ni spacings 0.65855 (the average of 0.70711 and 0.61). Note also the very first site has a spacing 0.70711. This is the spacing between the last plane, sites 73:82, and the first plane, sites 1:10. It is $1/\sqrt{2}$, as it should be because of periodic boundary conditions.

Next we need to examine Ni-Nb bonds, and relax frontier sites to space bonds more evenly. The simplest way to do this is to minimize sphere overlaps. blm automatically determined sphere radii in the elemental compounds by enlarging the sphere radius to match the Wigner Seitz cell volume, and thus making them overlap: 14% in the bcc case, 11% in the close-packed fcc case. (blm imported these radii into ctrl.dat via --rdspec and --usefiler.) Such geometry violation is always present in ASA calculations, but in this context we want to use the sphere overlaps as a measure of how bonds are spaced.
Do the following

lmchk dat --wpos=pos0

Near the end you should see this line

 OVMIN, 1426 pairs:  fovl = 3.78727e-5   <ovlp> = 18.3%   max ovlp = 36%

The maximum overlap is large, but slightly less than what it would be if we had chosen interfaces other than AB|A’B’C’|AB, e.g. ABA|A’B’C’|AB or AB|A’B’C’|BAB.

lmchk has a facility to move atoms that minimize sphere overlaps. It does this by minimizing an objective function (e.g. the sixth power of the sum of all positive overlaps). We will use this facility to get an initial estimate of the reconstruction. As noted above, we do not want to move all sites, only the frontier ones so we can build any 3n lattice lego-style. Do the following

lmchk dat --mino~dxmx=.001~xtol=.0001~nkill=10~maxit=200~11:34,49:72 --wsite~short~fn=site82 --wpos=pos

Note that only sites 11:34,49:72 move (compare to table above).

Near the end you should find the following:

 minimized: fovl = 6.96742e-6   <ovlp> = 13.8%   max ovlp = 19.6%

The maximum overlap is reduced to 19.6% from an initial 36%. Since there is a heavy energy penalty for atoms being too close together, this is a much better starting point than the unrelaxed lattice.

lmchk wrote the unrelaxed and relaxed positions to files pos0.dat and pos.dat respectively. To see how much atoms moved, do

mcx pos.dat pos0.dat -- > dpos3.dat

Displacements are significant, on the order of 10% of the bond length. Note that if all the sites were included in the list, the maximum overlap would fall to 17.3% — not much lower than the constrained case. This provides a rough justification for constraining the relaxation to frontier atoms.

2b. Nb/Ni(3n)+1/Nb superlattices

Consider the following possibibilities stacking the Nb/Ni4/Nb superlattice. (Here A and B refer to the two Nb planes, and 1,2,3 refer to the three Ni planes). AB1231AB, AB2312AB, AB3123AB. It turns out the AB1231AB has the largest overlap (38.1%), while AB2312AB and AB3123AB both have 36% overlaps.

To build the AB3123AB stack, copy the following box into file batch3123ab

file@site14c.ni@dpos=0,0,-dz+dp3/2
file@site14a.ni@dpos=0,0,-dz+dp3/2
file@site14b.ni@dpos=0,0,-dz+dp3/2
file@site14c.ni@dpos=0,0,-dz+dp3/2
file@site20.nb@dpos=0,0,dp3

Then do

lmscell -vfile=20 ctrl.nb -vdz0='1/sqrt(2)' -vdz=dz0/2-.61/2 -vdp3=-.00 --stack~batch=batch3123ab~wsite@fn=sitex
cp sitex.nb sitein.dat
blm --nfile --rdsite=sitein --rdspec~fn=spec.nb~fn=spec.ni --usefiler --nosort --shorten=no --ctrl=ctrl dat
lmchk dat --pr20 --wpos=pos0

You should find the maximum sphere overlap is 36%. We will use the ideal AB3123AB as a starting point.

Do the crude relaxation by minimizing sphere overlaps

lmchk dat --mino~dxmx=.001~xtol=.0001~nkill=10~maxit=200~11:34,63:86 --wsite~short~fn=site82 --wpos=pos
mcx pos.dat pos0.dat -- > dpos4.dat

Near the end you should find the following:

 minimized: fovl = 6.05759e-6   <ovlp> = 13.5%   max ovlp = 19.8%

The initial maximum overlap is a bit larger than the AB123AB of Section 2a, but the minimizer nearly equalized the two.

2c. Nb/Ni(3n)+2/Nb superlattices

Use the same patterning for the Nb/Ni5/Nb superlattice as for the Nb/Ni4/Nb superlattice. To build the AB12312AB stack, copy the following box into file batch12312ab

file@site14a.ni@dpos=0,0,-dz+dp3/2
file@site14b.ni@dpos=0,0,-dz+dp3/2
file@site14c.ni@dpos=0,0,-dz+dp3/2
file@site14a.ni@dpos=0,0,-dz+dp3/2
file@site14b.ni@dpos=0,0,-dz+dp3/2
file@site20.nb@dpos=0,0,dp3

Then do

lmscell -vfile=20 ctrl.nb -vdz0='1/sqrt(2)' -vdz=dz0/2-.61/2 -vdp3=-.00 --stack~batch=batch12312ab~wsite@fn=sitex
cp sitex.nb sitein.dat
blm --nfile --rdsite=sitein --rdspec~fn=spec.nb~fn=spec.ni --usefiler --nosort --shorten=no --ctrl=ctrl dat
lmchk dat --pr20 --wpos=pos0

You should find the maximum sphere overlap is 36%. We will use the ideal AB3123AB as a starting point, as the overlaps are better than AB23123AB (37.5%) or AB31231AB (38.1%).

Minimize sphere overlaps

lmchk dat --mino~dxmx=.001~xtol=.0001~nkill=10~maxit=200~11:34,77:100 --wsite~short~fn=site82 --wpos=pos
mcx -f3f14.8 pos.dat pos0.dat -- > dpos5.dat

3. Relaxing the Nb(2)/Ni(3)/Nb(2) superlattice

If the files below are not already present in your working directory, set up this section with

cp ~/lmf/fp/test/nbni/{ctrl.nb,site20.nb,spec.nb,site42.ni,spec.ni,fpspec.nb,fpspec.ni,dpos3.dat,site14[abc].ni,site10[ab].nb,nb2ni4nb2/dpos4.dat,nb2ni4nb2/batch3123ab} .

3a. Relaxing the Nb(2)/Ni(3)/Nb(2) superlattice

The previous section supplied a suitable structure to perform a density-functional calculation of the Nb(2)/Ni(3)/Nb(2) superlattice, assuming for the Nb/Ni spacing an average of the Nb-Nb spacing and Ni-Ni spacing. It also provided a rough estimate for how sites relax around their ideal geometries (dpos3.dat).

Here we will use total energy minimization to properly relax the structure by minimizing forces. At the same time the Nb/Ni interplanar spacing should not be assumed, but calculated. We also will differentiate the frontier Nb and Ni species from their bulk counterparts. It is not crucial here but it will be when transport is calculated in the ASA.

lmscell -vfile=20 ctrl.nb -vdz0='1/sqrt(2)' -vdz=dz0/2-.61/2 -vdp3=-.00 --stack~file@site42.ni@dpos=0,0,-dz+dp3/2~file@site20.nb@dpos=0,0,dp3~setlbl@spid=Nbx@targ=11:20,nbas-19:nbas-10~setlbl@spid=Nix@targ=21:34,nbas-33:nbas-20~pad=0,0,dp3~wpos@fn=pos0~wsite@fn=sitex
cp sitex.nb sitein.nbni
blm --rdspec~fn=fpspec.nb~fn=fpspec.ni --usefiler --rdpos=dpos3.dat --autobas~loc=5 --rsham~el=-.1,-0.9~rscut=1.6 --dhftol=45 --molstat~xtol=0~gtol=0.01 --nfile --rdsite --omax=-.01 --nosort --shorten=no --nit=20 --nk=4,5,2 --gmax=10 --mixb=.2 --frzwf --wsite~ib=1:10,35:48,73:82~rlx=000~long~fn=site82 nbni
cat actrl.nbni | sed s/NFORCE=0/NFORCE=20/ > ctrl.nbni

The entire stack height can be modified by p3. This is distributed in the cell by shifting sites after the first interface by p3/2 and sites after the second by p3.

blm makes actrl.nbni and site82.nbni in a form suitable for lmf. The last command cat actrl.nbni | sed s/NFORCE=0/NFORCE=20/ > ctrl.nbni creates the ctrl we will use.

We will use this to relax the lattice via self-consistent LDA calculations. Self-consistency is not the purpose, but rather to obtain lattice displacments needed by the ASA. It will have bo be repeated for transport in the ASA, but the ASA cannot be used to relax the lattice. The bands generated here can also serve as a benchmark to validate the quality of the ASA.

It is always a good idea to confirm that sphere radii and overlaps are sensible. The input file has one radius Nb and one for Ni and two other radii for the frontier species, all of which are reasonably similar. Click on the line below for further details.

We need a starting density and basis set both supplied by lmfa:

cp site82.nbni site1.nbni
lmfa -vfile=1 ctrl.nbni
cp basp0.nbni basp.nbni
lmfa -vfile=1 ctrl.nbni

-vfile=1 tells lmfa to read site file site1.nbni, as can seen by inspecting ctrl.nbni. (We could have used -vfile=82 and read from site82.nbni, but the calculations will need to be repeated for different p3, site1.nbni will be designated as the file for p3=0.) lmfa writes the basis file to basp0.nbni and information about the atomic density to atm.nbni. It has to be run a second time once information about the local orbitals is found from the first pass (this can redistribute the weight between core and valence density).

We are ready to run lmf. Initially we want to make the system self-consistent without relaxing any atoms. Then we can see how large the forces are with our simple but rude estimate for lattice displacements, and to monitor the precision of the k point integration.

With 82 atoms, running in serial mode is not a viable option. Exactly how you run lmf will depend on the hardware available to you. lmf will parallelize over k points, but we need to know how many. Extract that information this way:

lmf ctrl.nbni -vfile=1 --quit=ham | grep BZMESH

There are 22 irreducible QP. As the calculation is a bit expensive, we will make the initial runs with a smaller k-mesh using 4×4×1 divisions in place of 4×5×2. The difference is not all that large, and we can tune the k mesh more finely near convergence, if needed.

lmf ctrl.nbni -vnk1=4 -vnk2=4 -vnk3=1 -vfile=1 --quit=ham | grep BZMESH

About parallelization: suppose you have a single node machine with 16 cores, and lmf is compiled with Intel’s MKL library (this tutorial was written using such a machine). You could do either one of these:

mpix -n 10 lmf -vbeta=.1 -vnit=30 -vnk1=4 -vnk2=4 -vnk3=1 -vfile=1 --v8 --tbeh -vrsham=2 ctrl.nbni > out
env OMP_NUM_THREADS=2 MPI_NUM_THREADS=2 | mpix -n 10 lmf -vbeta=.1 -vnit=30 -vnk1=4 -vnk2=4 -vnk3=1 -vfile=1 --v8 --tbeh -vrsham=2 ctrl.nbni > out

The latter uses 20 threads which is more than 16, but the extra cost of swapping threads is made up for making better use of the 16 available cores. If your machine has 32 cores you can do env OMP_NUM_THREADS=3 MPI_NUM_THREADS=3, and so on. You can also invoke MPI across nodes.

Caution: When this tutorial was written the --v8 option did not compute the forces correctly, so traditional algorithms are used:

mpix -n 10 lmf -vbeta=.1 -vnit=30 -vnk1=4 -vnk2=4 -vnk3=1 -vfile=1 ctrl.nbni > out

Before relaxing the structure, inspect the output file. Note in particular:

1. Do grep more= out to see the change RMS DQ in output − input charge. As the iterations proceeded it decreased to a relatively small value, less than 10⁻⁴.

   Also inspect save.nbni and observe that the Harris-Foulkes and Kohn-Sham total energies are near each other in the final iteration.

   These results are both good indications that the density is close to self-consistent.

2. The forces are not small, but they are not massive either:

   grep 'Maximum Harris force' out
   

   The maximum force should be about 67 mRy/a.u. This is a measure of the reliability of the crude relaxation by minimizing sphere overlaps.

3. The relatively crude 4×4×1 k mesh isn’t bad either. Check out the change to the band sum evaluated with Bloechl weights instead of standard tetrahedron weights:

   grep 'Bloechl correction' out
   

   1 mRy for 82 atoms is quite modest; it means that forces will not be strongly affected by a finer k mesh.

Run lmf again using the (nearly) self-consistent rst.nbni file, this time allowing frontier atoms to relax.

rm -f mixm.nbni
mpix -n 10 lmf -vfile=1 -vnk1=4 -vnk2=4 -vnk3=1 -vfrzwf=t -vconv=1e-4 -vconvc=1e-3 -vminx=5 --wpos=pos36 ctrl.nbni > out1

This uses a Broyden algorithm to relax the structure for five iterations. It writes the positions file for what would be iteration six into pos36.nbni. ‘6’ designates iteration 6, and ‘3’ designates three ML of Ni. Later relaxations of of the

Try grep RELAX out1 to see how the gradient evolved in successive relaxation steps:

 RELAX: (warning) failed to read Hessian; set to unity
 RELAX:  completed Broyden iter 1;  max shift=-0.0100  |g|=0.253
 RELAX:  completed Broyden iter 2;  max shift=0.01000  |g|=0.169
 RELAX:  completed Broyden iter 3;  max shift=-0.0100  |g|=0.129
 RELAX:  completed Broyden iter 4;  max shift=0.01000  |g|=0.0589
 RELAX:  completed Broyden iter 5;  max shift=0.00493  |g|=0.0235

|g| is the length of the 144 component force “vector”; it should drop from an initial value 0.237 to 0.0275 in the 5^th iteration.
The total energy also drops significantly as egrep -E '^[cx]' out1 shows:

c file=1 nk1=4 nk2=4 nk3=1 frzwf=1 conv=1e-4 convc=.001 minx=5 ehf=-432804.2830013 ehk=-432804.2832668
x file=1 nk1=4 nk2=4 nk3=1 frzwf=1 conv=1e-4 convc=.001 minx=5 ehf=-432804.3377062 ehk=-432804.3382388
x file=1 nk1=4 nk2=4 nk3=1 frzwf=1 conv=1e-4 convc=.001 minx=5 ehf=-432804.3724307 ehk=-432804.3727466
x file=1 nk1=4 nk2=4 nk3=1 frzwf=1 conv=1e-4 convc=.001 minx=5 ehf=-432804.4007881 ehk=-432804.4011704
x file=1 nk1=4 nk2=4 nk3=1 frzwf=1 conv=1e-4 convc=.001 minx=5 ehf=-432804.4154372 ehk=-432804.4152356

The maximum force on an individual atom can be seen from grep 'Maximum Harris force' out1

 Maximum Harris force = 67.1 mRy/au (site 64)  Max eval correction = 0.3   ← before relaxation
...
 Maximum Harris force = 26.7 mRy/au (site 70)  Max eval correction = 144.1 ← after relaxation, iteration 5

The largest forces by far are on atoms constrained not to relax (largest force is on site 47, middle Ni layer), which will not be improved by further iteration. You can confirm this by running the lmf once again at the positions for what would have been iteration 6:

rm -f mixm.nbni hssn.nbni
mpix -n 10 lmf ctrl.nbni -vfile=$file -vnk1=4 -vnk2=4 -vnk3=1 -vfrzwf=t -vconv=1e-4 -vconvc=1e-3 --rpos=pos36 --rs=11,2,1 -vnit=20

The energy drops another 2 mRy, small compared to prior changes in energy. Thus the positions in file _pos36.nbni{:.path} are nearly optimum within the constraints. You can also check the k convergence (one iteration is enough)

mpix -n 10 lmf ctrl.nbni -vfile=$file -vfrzwf=t -vconv=1e-4 -vconvc=1e-3 --rpos=pos36 --rs=12,0 -vnit=1

and you should see that the charge density and total energy are minimally affected.

As noted, frontier atoms continue to have significant forces. If need be we could reduce this force by including more layers in the relaxation. But, this complicates our ability to assemble stacks “lego” style, and since the real interface is probably rather non-ideal and almost certainly different from this one, it isn’t worth the trouble to reduce the residual forces.

Optimizing the Nb/Ni layer spacing

There is no good justification to assume that the Nb-Ni spacing is $(0,0,-\mathbf{dz/2})$. The instructions above can vary spacing via p3: spacing at each interface shrinks by p3/2.

Repeat the lmscell ... / cp ... / blm ... block of Section 3a but assign dp3 as -vdp3=-.04 and designate site2.nbni as the site file for p3=−0.04 : cp site82.nbni site2.nbni . Do not repeat the last step (cat actrl.nbni | sed s/NFORCE=0/NFORCE=20/ > ctrl.nbni), but preserve ctrl.nbni from Section 3a to minimize cancellation of errors. Sphere radii could have changed because of the different relaxation process, though they don’t in practice in this case. (If you no longer have this file, copy it from ~/lmf/fp/test/nbni/ctrl.nbni .)

Perform the same procedure as before (substituting -vfile=2 for -vfile=1) to relax the site2.nbni setup.

Make a new site file for p3=−0.08, now designating site3.nbni as the site file, and relax the site3.nbni setup; to the same for p3=−0.12, designating site4.nbni as the site file

For the four relaxations you should find the following total energies

0.00    -432804.4154372  ← site1
-.04    -432804.4578181  ← site2
-.08    -432804.4852606  ← site3
-.12    -432804.4949157  ← site4

The energy gained by allowing p3 to vary is comparable to the internal site relaxation. To estimate the minimum, copy the first two columns into a file dat and fit a second order polynomial to it:

echo q | pfit 2 dat

The fit puts the minimum at &minus.125, which is close enough to the site4 structure.

Sanity checks: the following files should be nearly identical to those in the distribution, ~/lmf/pgf/test/nbni-111 :

diff pos36.nbni ~/lmf/pgf/test/nbni-111/pos36.nbni 

Preparing a site file for the ASA

cp ~/lmf/fp/test/nbni/{ctrl.nb,site20.nb,fpspec.nb,site42.ni,fpspec.ni} .

The ASA calculation of transmission will need site information to proceed. It requires a site file without any relaxations (created below as site30.nbni), and information about the relaxed positions. This is file pos36.nbni made in the previous section.

The steps below follow the setup of Section 3a but produce site30.nbni (the unrelaxed site file corresponding to p3=−.12). ‘3’ marks three ML of Ni, and ‘0’ designates no relaxation.

lmscell -vfile=20 ctrl.nb -vdz0='1/sqrt(2)' -vdz=dz0/2-.61/2 -vdp3=-.12 --stack~file@site42.ni@dpos=0,0,-dz+dp3/2~file@site20.nb@dpos=0,0,dp3~setlbl@spid=Nbx@targ=11:20,nbas-19:nbas-10~setlbl@spid=Nix@targ=21:34,nbas-33:nbas-20~pad=0,0,dp3~wpos@fn=pos0~wsite@fn=sitex
cp sitex.nb sitein.nbni
blm --rdspec~fn=fpspec.nb~fn=fpspec.ni --usefiler --autobas~loc=5 --rsham~el=-.1,-0.9~rscut=1.6 --dhftol=45 --molstat~xtol=0~gtol=0.01 --nfile --rdsite --omax=-.01 --nosort --shorten=no --nit=20 --nk=4,5,2 --gmax=10 --mixb=.2 --frzwf --wsite~ib=1:10,35:48,73:82~rlx=000~long~fn=site30 nbni

Sanity checks: the following files should be nearly identical to those in the distribution, ~/lmf/pgf/test/nbni-111 :

xx

3b. Relaxing the Nb(2)/Ni(4)/Nb(2) superlattice

The files below are needed to set up this section

cp ~/lmf/fp/test/nbni/{ctrl.nb,ctrl.nbni,site20.nb,spec.nb,site42.ni,spec.ni,fpspec.nb,fpspec.ni,site14[abc].ni,site10[ab].nb,nb2ni4nb2/dpos4.dat,nb2ni4nb2/batch3123ab} .

We can reasonably assume p3 will be the same for the Nb(2)/Ni(4)/Nb(2) superlattice as for the Nb(2)/Ni(3)/Nb(2) superlattice. (You can confirm that this is the case by relaxing lattices for several values of p3 and minimising, as was done in the previous section.) Make a new site file for p3=−0.12, now designating site44.nbni as the site file

lmscell -vfile=20 ctrl.nb -vdz0='1/sqrt(2)' -vdz=dz0/2-.61/2 -vdp3=-.12 --stack~batch=batch3123ab~setlbl@spid=Nbx@targ=11:20,nbas-19:nbas-10~setlbl@spid=Nix@targ=21:34,nbas-33:nbas-20~pad=0,0,dp3~wpos@fn=pos0~wsite@fn=sitex
cp sitex.nb sitein.nbni
blm --rdspec~fn=fpspec.nb~fn=fpspec.ni --usefiler --rdpos=dpos4.dat --autobas~loc=5 --rsham~el=-.1,-0.9~rscut=1.6 --dhftol=45 --molstat~xtol=0~gtol=0.01 --nfile --rdsite --omax=-.01 --nosort --shorten=no --nit=20 --nk=4,5,2 --gmax=10 --mixb=.2 --frzwf --wsite~ib=1:10,35,96-9:96,35:62~rlx=000~long~fn=site82 nbni
cp site82.nbni site44.nbni

We can use the same ctrl.nbni as for the Nb(2)/Ni(3)/Nb(2) superlattice. But because the cell is becoming long, it is safer to tighten tolerances on the Ewald sums. Starting from the existing ctrl file, do

cat ctrl.nbni | awk '{ if ($3 == "rsham>1") {print "      TOL=1d-12"; print "EWALD TOL=1d-12"}; print}'  > ctrl.nbni2
cp ctrl.nbni2 ctrl.nbni

Following the relaxation procedure once again, relax the site44.nbni setup. This time write the positions file into pos46.nbni. ‘6’ designates iteration 6, and ‘4’ designates four ML of Ni.

Finally make site40.nbni (the unrelaxed site file corresponding to p3=−.12) for the ASA

blm --rdspec~fn=fpspec.nb~fn=fpspec.ni --usefiler --autobas~loc=5 --rsham~el=-.1,-0.9~rscut=1.6 --dhftol=45 --molstat~xtol=0~gtol=0.01 --nfile --rdsite --omax=-.01 --nosort --shorten=no --nit=20 --nk=4,5,2 --gmax=10 --mixb=.2 --frzwf --wsite~ib=1:10,35:48+14,82+14-9:82+14~rlx=000~long~fn=site40 nbni

3c. Relaxing the Nb(2)/Ni(5)/Nb(2) superlattice

The files below are needed to set up this section

cp ~/lmf/fp/test/nbni/{ctrl.nb,ctrl.nbni,site20.nb,spec.nb,site42.ni,spec.ni,fpspec.nb,fpspec.ni,site14[abc].ni,site10[ab].nb,nb2ni5nb2/dpos5.dat,nb2ni5nb2/batch12312ab}

Use the same p3 for the Nb(2)/Ni(5)/Nb(2) superlattice. Make a new site file for p3=−0.12, now designating site54.nbni as the site file

lmscell -vfile=20 ctrl.nb -vdz0='1/sqrt(2)' -vdz=dz0/2-.61/2 -vdp3=-.12 --stack~batch=batch12312ab~setlbl@spid=Nbx@targ=11:20,nbas-19:nbas-10~setlbl@spid=Nix@targ=21:34,nbas-33:nbas-20~pad=0,0,dp3~wpos@fn=pos0~wsite@fn=sitex
cp sitex.nb sitein.nbni
blm --rdspec~fn=fpspec.nb~fn=fpspec.ni --usefiler --rdpos=dpos5.dat --autobas~loc=5 --rsham~el=-.1,-0.9~rscut=1.6 --dhftol=45 --molstat~xtol=0~gtol=0.01 --nfile --rdsite --omax=-.01 --nosort --shorten=no --nit=20 --nk=4,5,2 --gmax=10 --mixb=.2 --frzwf --wsite~ib=1:10,35:76,110-9:110~rlx=000~long~fn=site82 nbni
cp site82.nbni site54.nbni

Use the same ctrl.nbni as for the Nb(2)/Ni(4)/Nb(2) superlattice.

cat ctrl.nbni | awk '{ if ($3 == "rsham>1") {print "      TOL=1d-12"; print "EWALD TOL=1d-12"}; print}'  > ctrl.nbni2
cp ctrl.nbni2 ctrl.nbni

Make site50.nbni (the unrelaxed site file corresponding to p3=−.12) for the ASA

blm --rdspec~fn=fpspec.nb~fn=fpspec.ni --usefiler --autobas~loc=5 --rsham~el=-.1,-0.9~rscut=1.6 --dhftol=45 --molstat~xtol=0~gtol=0.01 --nfile --rdsite --omax=-.01 --nosort --shorten=no --nit=20 --nk=4,5,2 --gmax=10 --mixb=.2 --frzwf --wsite~ib=1:10,35:48+28,82+28-9:82+28~rlx=000~long~fn=site50 nbni

Following the relaxation procedure once again, relax the site54.nbni setup.

Footnotes and references

¹ Dimitar Pashov, Swagata Acharya, Walter R. L. Lambrecht, Jerome Jackson, Kirill D. Belashchenko, Athanasios Chantis, Francois Jamet, Mark van Schilfgaarde, Questaal: a package of electronic structure methods based on the linear muffin-tin orbital technique, Comp. Phys. Comm. 249, 107065 (2020).