compute sna/atom command — LAMMPS documentation

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# compute sna/atom command

# compute snad/atom command

# compute snav/atom command

# compute snap command

# compute sna/grid command

# compute sna/grid/local command

## Syntax

    compute ID group-ID sna/atom rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
    compute ID group-ID snad/atom rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
    compute ID group-ID snav/atom rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
    compute ID group-ID snap rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
    compute ID group-ID snap rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
    compute ID group-ID sna/grid nx ny nz rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
    compute ID group-ID sna/grid/local nx ny nz rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...

- ID, group-ID are documented in compute command

- sna/atom = style name of this compute command

- rcutfac = scale factor applied to all cutoff radii (positive real)

- rfac0 = parameter in distance to angle conversion (0 < rcutfac < 1)

- twojmax = band limit for bispectrum components (non-negative integer)

- R_1, R_2,… = list of cutoff radii, one for each type (distance units)

- w_1, w_2,… = list of neighbor weights, one for each type

- nx, ny, nz = number of grid points in x, y, and z directions (positive integer)

- zero or more keyword/value pairs may be appended

- keyword = rmin0 or switchflag or bzeroflag or quadraticflag or chem or bnormflag or wselfallflag or bikflag or switchinnerflag or sinner or dinner or dgradflag

      rmin0 value = parameter in distance to angle conversion (distance units)
      switchflag value = 0 or 1
         0 = do not use switching function
         1 = use switching function
      bzeroflag value = 0 or 1
         0 = do not subtract B0
         1 = subtract B0
      quadraticflag value = 0 or 1
         0 = do not generate quadratic terms
         1 = generate quadratic terms
      chem values = nelements elementlist
         nelements = number of SNAP elements
         elementlist = ntypes integers in range [0, nelements)
      bnormflag value = 0 or 1
         0 = do not normalize
         1 = normalize bispectrum components
      wselfallflag value = 0 or 1
         0 = self-contribution only for element of central atom
         1 = self-contribution for all elements
      switchinnerflag value = 0 or 1
         0 = do not use inner switching function
         1 = use inner switching function
      sinner values = sinnerlist
         sinnerlist = ntypes values of Sinner (distance units)
      dinner values = dinnerlist
         dinnerlist = ntypes values of Dinner (distance units)
      bikflag value = 0 or 1 (only implemented for compute snap)
         0 = descriptors are summed over atoms of each type
         1 = descriptors are listed separately for each atom
      dgradflag value = 0 or 1 (only implemented for compute snap)
         0 = descriptor gradients are summed over atoms of each type
         1 = descriptor gradients are listed separately for each atom pair

## Examples

    compute b all sna/atom 1.4 0.99363 6 2.0 2.4 0.75 1.0 rmin0 0.0
    compute db all sna/atom 1.4 0.95 6 2.0 1.0
    compute vb all sna/atom 1.4 0.95 6 2.0 1.0
    compute snap all snap 1.4 0.95 6 2.0 1.0
    compute snap all snap 1.0 0.99363 6 3.81 3.83 1.0 0.93 chem 2 0 1
    compute snap all snap 1.0 0.99363 6 3.81 3.83 1.0 0.93 switchinnerflag 1 sinner 1.35 1.6 dinner 0.25 0.3
    compute bgrid all sna/grid/local 200 200 200 1.4 0.95 6 2.0 1.0

## Description

Define a computation that calculates a set of quantities related to the bispectrum components of the atoms in a group. These computes are used primarily for calculating the dependence of energy, force, and stress components on the linear coefficients in the snap pair_style, which is useful when training a SNAP potential to match target data.

Bispectrum components of an atom are order parameters characterizing the radial and angular distribution of neighbor atoms. The detailed mathematical definition is given in the paper by Thompson et al. (Thompson)

The position of a neighbor atom i’ relative to a central atom i is a point within the 3D ball of radius \(R_{ii'}\) = rcutfac \((R_i + R_i')\)

Bartok et al. (Bartok), proposed mapping this 3D ball onto the 3-sphere, the surface of the unit ball in a four-dimensional space. The radial distance r within R_ii’ is mapped on to a third polar angle \(\theta_0\) defined by,

\[\theta_0 = {\sf rfac0} \frac{r-r_{min0}}{R_{ii'}-r_{min0}} \pi\]

In this way, all possible neighbor positions are mapped on to a subset of the 3-sphere. Points south of the latitude \(\theta_0\) = rfac0 \(\pi\) are excluded.

The natural basis for functions on the 3-sphere is formed by the representatives of SU(2), the matrices \(U^j_{m,m'}(\theta, \phi, \theta_0)\). These functions are better known as \(D^j_{m,m'}\), the elements of the Wigner D-matrices (Meremianin, Varshalovich, Mason) The density of neighbors on the 3-sphere can be written as a sum of Dirac-delta functions, one for each neighbor, weighted by species and radial distance. Expanding this density function as a generalized Fourier series in the basis functions, we can write each Fourier coefficient as

\[u^j_{m,m'} = U^j_{m,m'}(0,0,0) + \sum_{r_{ii'} < R_{ii'}}{f_c(r_{ii'}) w_{\mu_{i'}} U^j_{m,m'}(\theta_0,\theta,\phi)}\]

The \(w_{\mu_{i'}}\) neighbor weights are dimensionless numbers that depend on \(\mu_{i'}\), the SNAP element of atom i’, while the central atom is arbitrarily assigned a unit weight. The function \(f_c(r)\) ensures that the contribution of each neighbor atom goes smoothly to zero at \(R_{ii'}\):

\[\begin{split}f_c(r) = & \frac{1}{2}(\cos(\pi \frac{r-r_{min0}}{R_{ii'}-r_{min0}}) + 1), r \leq R_{ii'} \\ = & 0, r > R_{ii'}\end{split}\]

The expansion coefficients \(u^j_{m,m'}\) are complex-valued and they are not directly useful as descriptors, because they are not invariant under rotation of the polar coordinate frame. However, the following scalar triple products of expansion coefficients can be shown to be real-valued and invariant under rotation (Bartok).

\[\begin{split}B_{j_1,j_2,j} = \sum_{m_1,m'_1=-j_1}^{j_1}\sum_{m_2,m'_2=-j_2}^{j_2}\sum_{m,m'=-j}^{j} (u^j_{m,m'})^* H {\scriptscriptstyle \begin{array}{l} {j} {m} {m'} \\ {j_1} {m_1} {m'_1} \\ {j_2} {m_2} {m'_2} \end{array}} u^{j_1}_{m_1,m'_1} u^{j_2}_{m_2,m'_2}\end{split}\]

The constants \(H^{jmm'}_{j_1 m_1 m_{1'},j_2 m_ 2m_{2'}}\) are coupling coefficients, analogous to Clebsch-Gordan coefficients for rotations on the 2-sphere. These invariants are the components of the bispectrum and these are the quantities calculated by the compute sna/atom. They characterize the strength of density correlations at three points on the 3-sphere. The j2=0 subset form the power spectrum, which characterizes the correlations of two points. The lowest-order components describe the coarsest features of the density function, while higher-order components reflect finer detail. Each bispectrum component contains terms that depend on the positions of up to 4 atoms (3 neighbors and the central atom).

Compute snad/atom calculates the derivative of the bispectrum components summed separately for each LAMMPS atom type:

\[-\sum_{i' \in I} \frac{\partial {B^{i'}_{j_1,j_2,j} }}{\partial {\bf r}_i}\]

The sum is over all atoms i’ of atom type I. For each atom i, this compute evaluates the above expression for each direction, each atom type, and each bispectrum component. See section below on output for a detailed explanation.

Compute snav/atom calculates the virial contribution due to the derivatives:

\[-{\bf r}_i \otimes \sum_{i' \in I} \frac{\partial {B^{i'}_{j_1,j_2,j}}}{\partial {\bf r}_i}\]

Again, the sum is over all atoms i’ of atom type I. For each atom i, this compute evaluates the above expression for each of the six virial components, each atom type, and each bispectrum component. See section below on output for a detailed explanation.

Compute snap calculates a global array containing information related to all three of the above per-atom computes sna/atom, snad/atom, and snav/atom. The first row of the array contains the summation of sna/atom over all atoms, but broken out by type. The last six rows of the array contain the summation of snav/atom over all atoms, broken out by type. In between these are 3*N rows containing the same values computed by snad/atom (these are already summed over all atoms and broken out by type). The element in the last column of each row contains the potential energy, force, or stress, according to the row. These quantities correspond to the user-specified reference potential that must be subtracted from the target data when fitting SNAP. The potential energy calculation uses the built in compute thermo_pe. The stress calculation uses a compute called snap_press that is automatically created behind the scenes, according to the following command:

    compute snap_press all pressure NULL virial

See section below on output for a detailed explanation of the data layout in the global array.

New in version 3Aug2022.

The compute sna/grid and sna/grid/local commands calculate bispectrum components for a regular grid of points. These are calculated from the local density of nearby atoms i’ around each grid point, as if there was a central atom i at the grid point. This is useful for characterizing fine-scale structure in a configuration of atoms, and it is used in the MALA package to build machine-learning surrogates for finite-temperature Kohn-Sham density functional theory (Ellis et al.) Neighbor atoms not in the group do not contribute to the bispectrum components of the grid points. The distance cutoff \(R_{ii'}\) assumes that i has the same type as the neighbor atom i’.

Compute sna/grid calculates a global array containing bispectrum components for a regular grid of points. The grid is aligned with the current box dimensions, with the first point at the box origin, and forming a regular 3d array with nx, ny, and nz points in the x, y, and z directions. For triclinic boxes, the array is congruent with the periodic lattice vectors a, b, and c. The array contains one row for each of the \(nx \times ny \times nz\) grid points, looping over the index for ix fastest, then iy, and iz slowest. Each row of the array contains the x, y, and z coordinates of the grid point, followed by the bispectrum components. See section below on output for a detailed explanation of the data layout in the global array.

Compute sna/grid/local calculates bispectrum components of a regular grid of points similarly to compute sna/grid described above. However, because the array is local, it contains only rows for grid points that are local to the processor subdomain. The global grid of \(nx \times ny \times nz\) points is still laid out in space the same as for sna/grid, but grid points are strictly partitioned, so that every grid point appears in one and only one local array. The array contains one row for each of the local grid points, looping over the global index ix fastest, then iy, and iz slowest. Each row of the array contains the global indexes ix, iy, and iz first, followed by the x, y, and z coordinates of the grid point, followed by the bispectrum components. See section below on output for a detailed explanation of the data layout in the global array.

The value of all bispectrum components will be zero for atoms not in the group. Neighbor atoms not in the group do not contribute to the bispectrum of atoms in the group.

The neighbor list needed to compute this quantity is constructed each time the calculation is performed (i.e. each time a snapshot of atoms is dumped). Thus it can be inefficient to compute/dump this quantity too frequently.

The argument rcutfac is a scale factor that controls the ratio of atomic radius to radial cutoff distance.

The argument rfac0 and the optional keyword rmin0 define the linear mapping from radial distance to polar angle \(theta_0\) on the 3-sphere, given above.

The argument twojmax defines which bispectrum components are generated. See section below on output for a detailed explanation of the number of bispectrum components and the ordered in which they are listed.

The keyword switchflag can be used to turn off the switching function \(f_c(r)\).

The keyword bzeroflag determines whether or not B0, the bispectrum components of an atom with no neighbors, are subtracted from the calculated bispectrum components. This optional keyword normally only affects compute sna/atom. However, when quadraticflag is on, it also affects snad/atom and snav/atom.

The keyword quadraticflag determines whether or not the quadratic combinations of bispectrum quantities are generated. These are formed by taking the outer product of the vector of bispectrum components with itself. See section below on output for a detailed explanation of the number of quadratic terms and the ordered in which they are listed.

The keyword chem activates the explicit multi-element variant of the SNAP bispectrum components. The argument nelements specifies the number of SNAP elements that will be handled. This is followed by elementlist, a list of integers of length ntypes, with values in the range [0, nelements ), which maps each LAMMPS type to one of the SNAP elements. Note that multiple LAMMPS types can be mapped to the same element, and some elements may be mapped by no LAMMPS type. However, in typical use cases (training SNAP potentials) the mapping from LAMMPS types to elements is one-to-one.

The explicit multi-element variant invoked by the chem keyword partitions the density of neighbors into partial densities for each chemical element. This is described in detail in the paper by Cusentino et al. The bispectrum components are indexed on ordered triplets of elements:

\[\begin{split}B_{j_1,j_2,j}^{\kappa\lambda\mu} = \sum_{m_1,m'_1=-j_1}^{j_1}\sum_{m_2,m'_2=-j_2}^{j_2}\sum_{m,m'=-j}^{j} (u^{\mu}_{j,m,m'})^* H {\scriptscriptstyle \begin{array}{l} {j} {m} {m'} \\ {j_1} {m_1} {m'_1} \\ {j_2} {m_2} {m'_2} \end{array}} u^{\kappa}_{j_1,m_1,m'_1} u^{\lambda}_{j_2,m_2,m'_2}\end{split}\]

where \(u^{\mu}_{j,m,m'}\) is an expansion coefficient for the partial density of neighbors of element \(\mu\)

\[u^{\mu}_{j,m,m'} = w^{self}_{\mu_{i}\mu} U^{j,m,m'}(0,0,0) + \sum_{r_{ii'} < R_{ii'}}{\delta_{\mu\mu_{i'}}f_c(r_{ii'}) w_{\mu_{i'}} U^{j,m,m'}(\theta_0,\theta,\phi)}\]

where \(w^{self}_{\mu_{i}\mu}\) is the self-contribution, which is either 1 or 0 (see keyword wselfallflag below), \(\delta_{\mu\mu_{i'}}\) indicates that the sum is only over neighbor atoms of element \(\mu\), and all other quantities are the same as those appearing in the original equation for \(u^j_{m,m'}\) given above.

The keyword wselfallflag defines the rule used for the self-contribution. If wselfallflag is on, then \(w^{self}_{\mu_{i}\mu}\) = 1. If it is off then \(w^{self}_{\mu_{i}\mu}\) = 0, except in the case of \({\mu_{i}=\mu}\), when \(w^{self}_{\mu_{i}\mu}\) = 1. When the chem keyword is not used, this keyword has no effect.

The keyword bnormflag determines whether or not the bispectrum component \(B_{j_1,j_2,j}\) is divided by a factor of \(2j+1\). This normalization simplifies force calculations because of the following symmetry relation

\[\frac{B_{j_1,j_2,j}}{2j+1} = \frac{B_{j,j_2,j_1}}{2j_1+1} = \frac{B_{j_1,j,j_2}}{2j_2+1}\]

This option is typically used in conjunction with the chem keyword, and LAMMPS will generate a warning if both chem and bnormflag are not both set or not both unset.

The keyword switchinnerflag with value 1 activates an additional radial switching function similar to \(f_c(r)\) above, but acting to switch off smoothly contributions from neighbor atoms at short separation distances. This is useful when SNAP is used in combination with a simple repulsive potential. For a neighbor atom at distance \(r\), its contribution is scaled by a multiplicative factor \(f_{inner}(r)\) defined as follows:

\[\begin{split} = & 0, r \leq S_{inner} - D_{inner} \\ f_{inner}(r) = & \frac{1}{2}(1 - \cos(\frac{\pi}{2} (1 + \frac{r-S_{inner}}{D_{inner}})), S_{inner} - D_{inner} < r \leq S_{inner} + D_{inner} \\ = & 1, r > S_{inner} + D_{inner}\end{split}\]

where the switching region is centered at \(S_{inner}\) and it extends a distance \(D_{inner}\) to the left and to the right of this. With this option, additional keywords sinner and dinner must be used, each followed by ntypes values for \(S_{inner}\) and \(D_{inner}\), respectively. When the central atom and the neighbor atom have different types, the values of \(S_{inner}\) and \(D_{inner}\) are the arithmetic means of the values for both types.

The keywords bikflag and dgradflag are only used by compute snap. The keyword bikflag determines whether or not to list the descriptors of each atom separately, or sum them together and list in a single row. If bikflag is set to 0 then a single bispectrum row is used, which contains the per-atom bispectrum descriptors \(B_{i,k}\) summed over all atoms i to produce \(B_k\). If bikflag is set to 1 this is replaced by a separate per-atom bispectrum row for each atom. In this case, the entries in the final column for these rows are set to zero.

The keyword dgradflag determines whether to sum atom gradients or list them separately. If dgradflag is set to 0, the bispectrum descriptor gradients w.r.t. atom j are summed over all atoms i’ of type I (similar to snad/atom above). If dgradflag is set to 1, gradients are listed separately for each pair of atoms. Each row corresponds to a single term \(\frac{\partial {B_{i,k} }}{\partial {r}^a_j}\) where \({r}^a_j\) is the a-th position coordinate of the atom with global index j. This also changes the number of columns to be equal to the number of bispectrum components, with 3 additional columns representing the indices \(i\), \(j\), and \(a\), as explained more in the Output info section below. The option dgradflag=1 requires that bikflag=1.

Note

Using dgradflag = 1 produces a global array with \(N + 3N^2 + 1\) rows which becomes expensive for systems with more than 1000 atoms.

Note

If you have a bonded system, then the settings of special_bonds command can remove pairwise interactions between atoms in the same bond, angle, or dihedral. This is the default setting for the special_bonds command, and means those pairwise interactions do not appear in the neighbor list. Because this fix uses the neighbor list, it also means those pairs will not be included in the calculation. One way to get around this, is to write a dump file, and use the rerun command to compute the bispectrum components for snapshots in the dump file. The rerun script can use a special_bonds command that includes all pairs in the neighbor list.

## Output info

Compute sna/atom calculates a per-atom array, each column corresponding to a particular bispectrum component. The total number of columns and the identity of the bispectrum component contained in each column depend of the value of twojmax, as described by the following piece of python code:

    for j1 in range(0,twojmax+1):
        for j2 in range(0,j1+1):
            for j in range(j1-j2,min(twojmax,j1+j2)+1,2):
                if (j>=j1): print j1/2.,j2/2.,j/2.

There are \(m(m+1)/2\) descriptors with last index j, where m = \(\lfloor j \rfloor + 1\). Hence, for even twojmax = 2(m-1), \(K = m(m+1)(2m+1)/6\), the m-th pyramidal number, and for odd twojmax = 2 m-1, \(K = m(m+1)(m+2)/3\), twice the m-th tetrahedral number.

Note

the diagonal keyword allowing other possible choices for the number of bispectrum components was removed in 2019, since all potentials use the value of 3, corresponding to the above set of bispectrum components.

Compute snad/atom evaluates a per-atom array. The columns are arranged into ntypes blocks, listed in order of atom type I. Each block contains three sub-blocks corresponding to the x, y, and z components of the atom position. Each of these sub-blocks contains K columns for the K bispectrum components, the same as for compute sna/atom

Compute snav/atom evaluates a per-atom array. The columns are arranged into ntypes blocks, listed in order of atom type I. Each block contains six sub-blocks corresponding to the xx, yy, zz, yz, xz, and xy components of the virial tensor in Voigt notation. Each of these sub-blocks contains K columns for the K bispectrum components, the same as for compute sna/atom

Compute snap evaluates a global array. The columns are arranged into ntypes blocks, listed in order of atom type I. Each block contains one column for each bispectrum component, the same as for compute sna/atom. A final column contains the corresponding energy, force component on an atom, or virial stress component. The rows of the array appear in the following order:

- 1 row: sna/atom quantities summed for all atoms of type I

- 3*N rows: snad/atom quantities, with derivatives w.r.t. x, y, and z coordinate of atom i appearing in consecutive rows. The atoms are sorted based on atom ID.

- 6 rows: snav/atom quantities summed for all atoms of type I

For example, if K =30 and ntypes=1, the number of columns in the per-atom arrays generated by sna/atom, snad/atom, and snav/atom are 30, 90, and 180, respectively. With quadratic value=1, the numbers of columns are 930, 2790, and 5580, respectively. The number of columns in the global array generated by snap are 31, and 931, respectively, while the number of rows is 1+3*N+6, where N is the total number of atoms.

Compute sna/grid evaluates a global array. The array contains one row for each of the \(nx \times ny \times nz\) grid points, looping over the index for ix fastest, then iy, and iz slowest. Each row of the array contains the x, y, and z coordinates of the grid point, followed by the bispectrum components.

Compute sna/grid/local evaluates a local array. The array contains one row for each of the local grid points, looping over the global index ix fastest, then iy, and iz slowest. Each row of the array contains the global indexes ix, iy, and iz first, followed by the x, y, and z coordinates of the grid point, followed by the bispectrum components.

If the quadratic keyword value is set to 1, then additional columns are generated, corresponding to the products of all distinct pairs of bispectrum components. If the number of bispectrum components is K, then the number of distinct pairs is K(K+1)/2. For compute sna/atom these columns are appended to existing K columns. The ordering of quadratic terms is upper-triangular, (1,1),(1,2)…(1,K),(2,1)…(K-1,K-1),(K-1,K),(K,K). For computes snad/atom and snav/atom each set of K(K+1)/2 additional columns is inserted directly after each of sub-block of linear terms i.e. linear and quadratic terms are contiguous. So the nesting order from inside to outside is bispectrum component, linear then quadratic, vector/tensor component, type.

If the chem keyword is used, then the data is arranged into \(N_{elem}^3\) sub-blocks, each sub-block corresponding to a particular chemical labeling \(\kappa\lambda\mu\) with the last label changing fastest. Each sub-block contains K bispectrum components. For the purposes of handling contributions to force, virial, and quadratic combinations, these \(N_{elem}^3\) sub-blocks are treated as a single block of \(K N_{elem}^3\) columns.

If the bik keyword is set to 1, the structure of the snap array is expanded. The first \(N\) rows of the snap array correspond to \(B_{i,k}\) instead of a single row summed over atoms \(i\). In this case, the entries in the final column for these rows are set to zero. Also, each row contains only non-zero entries for the columns corresponding to the type of that atom. This is not true in the case of dgradflag keyword = 1 (see below).

If the dgradflag keyword is set to 1, this changes the structure of the global array completely. Here the snad/atom quantities are replaced with rows corresponding to descriptor gradient components on single atoms:

\[\frac{\partial {B_{i,k} }}{\partial {r}^a_j}\]

where \({r}^a_j\) is the a-th position coordinate of the atom with global index j. The rows are organized in chunks, where each chunk corresponds to an atom with global index \(j\). The rows in an atom \(j\) chunk correspond to atoms with global index \(i\). The total number of rows for these descriptor gradients is therefore \(3N^2\). The number of columns is equal to the number of bispectrum components, plus 3 additional left-most columns representing the global atom indices \(i\), \(j\), and Cartesian direction \(a\) (0, 1, 2, for x, y, z). The first 3 columns of the first \(N\) rows belong to the reference potential force components. The remaining K columns contain the \(B_{i,k}\) per-atom descriptors corresponding to the non-zero entries obtained when bikflag = 1. The first column of the last row, after the first \(N + 3N^2\) rows, contains the reference potential energy. The virial components are not used with this option. The total number of rows is therefore \(N + 3N^2 + 1\) and the number of columns is \(K + 3\).

These values can be accessed by any command that uses per-atom values from a compute as input. See the Howto output doc page for an overview of LAMMPS output options. To see how this command can be used within a Python workflow to train SNAP potentials, see the examples in FitSNAP.

## Restrictions

These computes are part of the ML-SNAP package. They are only enabled if LAMMPS was built with that package. See the Build package page for more info.

## Related commands

pair_style snap

## Default

The optional keyword defaults are rmin0 = 0, switchflag = 1, bzeroflag = 1, quadraticflag = 0, bnormflag = 0, wselfallflag = 0, switchinnerflag = 0,

(Thompson) Thompson, Swiler, Trott, Foiles, Tucker, J Comp Phys, 285, 316, (2015).

(Bartok) Bartok, Payne, Risi, Csanyi, Phys Rev Lett, 104, 136403 (2010).

(Meremianin) Meremianin, J. Phys. A, 39, 3099 (2006).

(Varshalovich) Varshalovich, Moskalev, Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore (1987).

(Mason) J. K. Mason, Acta Cryst A65, 259 (2009).

(Cusentino) Cusentino, Wood, Thompson, J Phys Chem A, 124, 5456, (2020)

(Ellis) Ellis, Fiedler, Popoola, Modine, Stephens, Thompson, Cangi, Rajamanickam, Phys Rev B, 104, 035120, (2021)