# Neural network variational Monte Carlo for positronic chemistry

Gino Cassella1<EMAIL_ADDRESS>W.M.C. Foulkes1 David Pfau2,1 James
S. Spencer2 1Dept. of Physics, Imperial College London, London SW7 2AZ, United
Kingdom  2DeepMind, London N1C 4DJ, United Kingdom

###### Abstract

Gamma rays produced by positron annihilation are used as a sensitive probe of
matter at atomic length scales. Technologies for manipulating positrons and
studying their interactions with ordinary matter are rapidly progressing. This
motivates the development of accurate _ab initio_ methods for modelling
positronic interactions with molecular matter. Here, we apply the recently
developed Fermionic neural network (FermiNet) wavefunction ansatz to the
problem of finding the ground-state properties of mixed positron-electron
systems. We find that FermiNet produces highly accurate, in some cases state-
of-the-art, ground-state energies across a range of atoms and small molecules
with a wide range of qualitatively distinct positron binding characteristics.
We highlight the capabilities of our method by calculating the positron
binding energy of the challenging non-polar Benzene molecule.

††preprint: AIP/123-QED

Since the existence of the positron – the positively charged anti-particle of
the electron – was first postulated by Dirac [1], many predictions have been
made concerning the formation of bound states between positrons and ordinary
matter. The formation of bound electron-positron states results in greatly
enhanced annihilation rates. This process has recently been successfully
exploited in positron annihilation spectroscopy experiments [2, 3, 4, 5, 6, 7,
8, 9], where the energy-dependent annihilation rate enables sensitive
measurements of the positron binding energy [10].

Enhanced positron annihilation _in materio_ has valuable applications in
medical physics [11], materials science [12, 13], and astrophysics [14].
Experimental apparatus for trapping large numbers of positrons continues to
grow in sophistication, offering a glimpse into a world of exotic antimatter
chemistry [15]. These advances motivate the development of a strong _ab
initio_ description of positronic bound states to accelerate the continued
development of new antimatter-based technology.

This problem has been addressed using many standard computational chemistry
tools. Positron binding to atoms and molecules has been studied using
wavefunction expansions in explicitly correlated Gausians [16, 17, 18, 19]
(ECG), configuration interaction (CI) methods [20, 21, 22, 23], and quantum
Monte Carlo (QMC) methods [24, 25, 26, 27, 28, 29, 30, 31]. The annihilation
rate and lifetime of positrons in solids, particularly in the presence of
defects, has been studied using density functional theory [32, 33] and quantum
Monte Carlo methods [34, 35].

Despite the intense theoretical interest, describing the positronic
wavefunction remains challenging for several reasons. Due to the repulsive
potential between nuclei and positrons, expanding the positronic wavefunction
in a basis often requires including diffuse basis functions and basis
functions with large angular momenta. In many cases, it is not appropriate to
consider the positronic wavefunction to be centred on the nuclei of a
molecule, introducing additional difficulties in choosing appropriately
centred basis functions. These difficulties have historically resulted in very
slow convergence of CI calculations and limited accuracy for QMC methods which
utilise wavefunction ansatze based on Hartree-Fock orbitals.

The most successful description of positron binding to molecules comes from a
recent work that develops a many-body theory of positron binding to molecules
and produces positron binding energies in close agreement with experimental
measurements [36]. In their work, Hofierka et al. highlight the shortcomings
of QMC methods applied to positron binding – particularly, the lack of
calculations for large non-polar molecules, which constitute the majority of
experimentally studied systems. Here, we address this shortcoming.

We propose a new approach to calculating the ground state properties of
molecular positronic bound states, based on recently developed neural network
wavefunction ansatze for QMC [37]. The Fermionic neural network (FermiNet)
models the many-body wavefunction without referencing a set of basis
functions. This conveniently sidesteps a number of the aforementioned
difficulties in describing positronic wavefunctions. We extend FermiNet to
represent the positronic component of the wavefunction on an even footing with
the electronic component. With a minimal alteration to the neural network
architecture, we obtain a flexible and accurate ansatz for mixed electron-
positron wavefunctions. We present results for a range of systems with
qualitatively distinct mechanisms for positron binding and obtain state-of-
the-art accuracy for the ground-state energy in these systems. Finally, we
show that our method obtains the positron binding energy for Benzene in close
agreement with the experimental value and the many-body theory of Hofierka et
al [36].

## Results

We benchmark the accuracy of FermiNet for a series of well-studied positronic
systems. These are presented here in (approximate) order of increasing
complexity. Unless otherwise specified, all results presented herein were
obtained using the network architecture and training protocol detailed by
Spencer [38]. We pre-train the electron orbitals to the Hartree-Fock solution
of the bare molecule and do not pre-train the positron orbitals. Errors in
energy expectation values are evaluated using a reblocking approach [39] to
account for sequential correlations in the Metropolis-Hastings sampling.

_Positronium hydride –_ The positronium (Ps) atom (consisting of a bound
electron and positron) and hydrogen form a stable molecule. Here we work
within the Born-Oppenhimer approximation, neglecting the proton’s motion. Near
exact ECG calculations are available for this system [16], yielding a ground-
state energy of $E_{0}=-0.7891794$ Hartrees. We obtain a ground state energy
of $E_{0}=-0.789144(3)$ Hartrees.

_Sodium and magnesium atoms –_ The first ionisation energy of sodium, 0.1886
Hartrees, is smaller than the binding energy of the Ps atom, 0.25 Hartrees.
The positronic sodium atom is then more accurately described as a bound
complex of a positronium atom and a sodium cation. The binding energy of this
complex is calculated as
$\epsilon=E\left([\text{Na}^{+},\text{Ps}]\right)-E(\text{Na}^{+})-0.25$. We
fail to predict binding without variance matching, obtaining $\epsilon=-0.37$
milliHartrees. Utilizing a variance matching procedure, we obtain
$\epsilon=0.32(15)$ milliHartrees, predicting binding in agreement with
previous FCSVM ($\epsilon_{\text{FCSVM}}=0.4$ milliHartrees) and CI
($\epsilon_{\text{CI}}=0.21$ milliHartrees) calculations.

We obtain a positron affinity of $0.01618(9)$ Hartrees for the magnesium atom.
This agrees with previous FCSVM ($0.015612$ Hartrees) and CI ($0.01615$
Hartrees) results.

| LiH | BeO
---|---|---
Method |  Bare |  Positronic |  Binding energy |  Bare |  Positronic |  Binding energy
FermiNet-VMC |  -8.07051(1) |  -8.10775(1) |  0.03723(2) |  -89.90572(4) |  -89.93082(3) |  0.02510(7)
SJ-VMC |  -8.06307(3) [25] |  -8.08034(4) [25] |  0.01727(7) |  -89.3173(25) [30] |  -89.3365(13) [30] |  0.0192(38)
SJ-FN-DMC |  -8.0704(1) [25] |  -8.10718(11) [25] |  0.03678(2) |  -89.7854(13) [30] |  -89.8134(12) [30] |  0.02800(25)
CISD |  -8.03830 [20] |  -8.05530 [20] |  0.017 |  - |  - |  -
MRD-CI |  -8.06827 [21] |  -8.09764 [21] |  0.02937 |  -89.759352 [23] |  -89.773133 [23] |  0.013781
ECG |  -8.07054 [18] |  -8.10747 [18] |  0.03693 |  - |  - |  -
GW |  - |  - |  0.039(1)*[36] |  - |  - |  -
* Theoretical (not statistical) uncertainty estimated from the comparison between different levels of theory as described by Hofierka _et al._ , caption of Table I
Table 1: Ground state energy of LiH and BeO, and their positronic complexes,
obtained via various computational methods at equilibrium bond-length.
Statistical errors are omitted where they are smaller than the reported
precision or otherwise omitted in the referenced source. The lowest
variational energy in each column is bolded.

_Lithium hydride and beryllium oxide –_ We have calculated the ground-state
energy of LiH and its positronic complex for a range of interatomic
separations (see Data Tables in Supplementary Material). Fitting these
potential energy surfaces using Nesterov’s algorithm, implemented in the
MOLCAS package, we obtain equilibrium bond distances of 3.0196 Bohr, with
ground-state energy -8.07050(1) Hartrees for LiH, and 3.371 Bohr with a ground
state energy of -8.10774(1) Hartrees for [LiH, e+]. These deviate very
slightly from the widely accepted literature values of 3.015 Bohr (LiH) and
3.348 Bohr ([LiH, e+]). We have calculated ground-state energies at the
canonical separations for comparison with previous results, shown in Table 1.
Our calculations are not only in excellent agreement with previous work but
are seen to yield the most accurate variational result for [LiH, e+] reported
to date.

We have calculated the ground-state wavefunction of BeO over a range of
interatomic separations – from below the equilibrium separation to
dissociation. We fit the potential energy surfaces using MOLCAS and obtain an
equilibrium bond distance of 2.515 Bohr with a ground-state energy of
-89.90572(4) Hartrees for BeO, and 2.530 Bohr with a ground-state energy of
-89.93082(3) Hartrees for [BeO, e+]. We compare our results against previous
calculations in Table 1 and find that FermiNet yields the most accurate
variational result for [BeO, e+].

The electronic ground state of BeO transitions from the spin-singlet
configuration at the equilibrium interatomic separation to the spin-triplet
configuration at the dissociative limit. We enforce the appropriate ground-
state spin configuration by choosing $S_{z}=1$ ( as FermiNet is a spin-
assigned wavefunction) at wide interatomic separations. The resulting
potential energy surfaces are plotted in Fig. 1. At $\sim 4$ Bohr, the
electronic ground-state transitions between the spin-singlet and spin-triplet,
causing a sharp change in the ground-state dipole moment and the resulting
positron binding energy with the molecular ground-state, which vanishes almost
completely. At $\sim 6$ Bohr, we see a smooth transition between two
qualitatively distinct binding modes between the molecule and the positron –
binding to the molecular dipole field below this separation and binding
exclusively to the ‘lone’ beryllium atom beyond this separation. This is
readily seen by visualizing the ground-state positron density on either side
of the maximum, as shown in Fig. 1 for the triplet state, and corroborated by
the dipole moment falling below the critical threshold for binding ($\sim
1.625$ Debye) at the transition.

Figure 1: (a) Ground-state positronic density, projected into the molecular
plane, of a positron attached to a beryllium oxide molecule over a range of
interatomic distances, accumulated via MCMC sampling. Image scale has been
normalized by bondlength. The scale bar in the bottom left of each panel
indicates the relative size of one Bohr radius, $a_{0}$. (b) Energy (of the
bare molecule and positronic complex), dipole moment (of the bare molecule),
and positron affinity of beryllium oxide over a range of interatomic
distances. Green (black) markers in the dipole moment and positron affinity
plots indicate values accumulated for the electronic singlet (triplet)
projected wavefunction. Solid lines and dots in the dipole moment plot
indicate the corresponding values obtained by Buenker et al. [23]. Horizontal
red lines in the dipole moment and positron affinity plots indicate the mean-
field critical value for positron binding to the molecular dipole field and
the positron affinity of a lone beryllium atom, respectively. Figure 2:
Orthographic projections of the ground-state one-particle density for
positronic (a) dilithium and (b) benzene molecules. Left/right columns show
the electronic/positronic density, $\rho_{e^{-}}$/$\rho_{e^{+}}$. In the
dilithium molecule, the positron is seen to be strongly localized to a torus
wrapping the covalent bond. In the benzene molecule, the positronic orbital is
extremely diffuse, resulting in a much noisier Monte Carlo estimate of the
one-particle density. The positronic orbital resembles a $p$-orbital
‘sandwiching’ the aromatic ring. The scale bar in the bottom left of each
subfigure indicates the relative size of one Bohr radius, $a_{0}$.

_Dilithium –_ For the dilithium molecule, we obtain a vertical positron
affinity of 66.75(2) milliHartrees at an equilibrium bond length of 5.015
Bohr. The molecular binding energy of Li2 from our calculations is $\sim 38$
milliHartrees, and the positron affinity of a lone lithium atom is known from
the literature to be $\sim 2$ milliHartrees, meaning this system is very
stable against the
$[\text{Li}_{2},\text{e\textsuperscript{+}}]\to[\text{Li},\text{e\textsuperscript{+}}]+\text{Li}$
dissociation channel. The one-particle density of the positronic ground-state
of dilithium is shown in Fig. 2(a).

_Benzene –_ The experimental positron binding energy of Benzene is 5.51
milliHartrees [10]. Utilizing our variance-matching procedure (described in
detail for Benzene in the Supplementary Material), we obtain a finite positron
binding energy of 4.1(3) milliHartrees. This falls in very close agreement
with the binding energy obtained by Hofierka et al. of $\sim$4.26
milliHartrees [36]. The one-particle density of the positronic ground-state of
benzene is shown in Fig. 2(b).

## Discussion

The selection of systems presented here spans a broad range of positron
binding phenomena: positronium formation, binding with an induced atomic
dipole moment, binding with a static molecular dipole moment, and binding due
to correlations with covalent bonding electrons in molecules. In all cases
where benchmarks are available, FermiNet VMC produces excellent and, in some
cases, state-of-the-art results for the positron affinity. From the density
plots presented, it is clear that this performance is consistent between
wavefunctions with very different qualitative characteristics. An identical
ansatz is used for every system studied – we have not employed any fine-tuning
or system-specific treatments. An important aspect of our approach is that,
due to being a basis-set-free method, we do not provide any information about
the nature of the positron binding in the input to the calculation (e.g. via
the placement of basis set functions). Rather, the location and nature of the
positron binding emerges naturally during optimization of the wavefunction.
The high level of accuracy achieved across various systems shows that FermiNet
offers a flexible and accurate ansatz for mixed electron-positron
wavefunctions.

To date, quantum Monte Carlo calculations of positron binding have focused on
small, polar molecules. As pointed out by Hofierka et al. [36], accurate
quantum Monte Carlo results for large non-polar organics, which comprise the
majority of experimentally relevant systems, are lacking. We believe that the
present work amends this gap. Our results for positronium hydride, sodium and
magnesium atoms, and small diatomic molecules demonstrate that our approach
can achieve state-of-the-art accuracy compared with previous work. Further,
our results for the non-polar dilithium and benzene molecules demonstrate that
this accuracy is retained when describing modes of positron binding governed
entirely by strong electron-positron correlation effects. The results in Fig.
2 offer an intuitive understanding of the binding mechanism between non-polar
molecules and positrons: correlation-dominated binding is facilitated by the
presence of a centre of increased electronic density away from the atomic
nuclei of a molecule. In dilithium, this is the covalent bond, and in benzene,
this is the increased electronic density in the centre of the molecule from
the delocalisation of the $\pi$-bonds in the ring.

Our result for the positron binding energy of benzene fall within ‘chemical
accuracy’ ($\sim$ 1.6 milliHartrees) of the experimental value, but this level
of accuracy is insufficient for other species of experimental interest. Many
chemical species possess positron binding energies below chemical accuracy,
and obtaining this level of accuracy represents a significant challenge to
computaitonal chemistry methodsis a signi. We believe that the necessary
improvement in accuracy, and obviation of our need to utilise variance
matching, can be achieved by adopting the recently introduced PsiFormer
architecture [40], an improvement upon the FermiNet architecture utilising
Transformer networks [41]. The PsiFormer obtains more accurate total energies
for the ground-state energy of the bare benzene molecule than FermiNet.
Investigating the accuracy of the PsiFormer applied to positron binding energy
calculations is a promising avenue for future work.

As with previous work utilizing FermiNet, computational scaling remains an
issue. Calculations involving $\gtrsim 50$ particles are too expensive for
presently available computational resources, prohibiting the application of
the method to large molecules. One approach to calculating the ground-state
wavefunction of large molecules is utilizing pseudopotentials to remove core
electrons from the calculations. The positronic component of the wavefunction
is often small near the atomic nuclei, so we would expect the correlation
effect between the core electrons and the positron to be small.
Pseudopotentials have been successfully employed in previous QMC calculations
of positron lifetimes in solids [35]. As the architecture employed herein only
extends the original FermiNet architecture to treat different particle types
separately, we expect that future advances in the computational efficiency of
FermiNet and related neural network wavefunctions will be able to be readily
combined with our approach to model positron binding.

## Conclusion

We have shown that the FermiNet ansatz for VMC calculations can be extended to
include positrons naturally, treating positrons on an equal footing to
electrons. Because this ansatz does not depend on a basis set, our treatment
sidesteps traditional methods’ issues in selecting and converging an
appropriate basis set for describing positronic wavefunctions. Our method
produces highly accurate results for several molecules with various binding
mechanisms for positrons without any system-specific tuning. We expect that
the simplicity of this method will lend itself to many challenging
applications beyond those presented here, e.g. calculations involving multiple
positrons. With additional computational effort, this method can provide
accurate predictions for positron annihilation experiments.

## Methods

We find the ground-state wavefunction and corresponding ground-state energy of
the many-body Coulomb Hamiltonian in the Born-Oppenheimer clamped nuclei
approximation,

$\mathcal{H}=-\frac{1}{2}\sum_{i}\nabla_{i}^{2}+\sum_{i,j}\frac{q_{i}Z_{j}}{|\mathbf{r}_{i}-\mathbf{R}_{j}|}+\sum_{i>j}\frac{q_{i}q_{j}}{|\mathbf{r}_{i}-\mathbf{r}_{j}|}+\sum_{i>j}\frac{Z_{i}Z_{j}}{|\mathbf{R}_{i}-\mathbf{R}_{j}|},$
(1)

where $q_{i},\mathbf{r}_{i}$ are the particle charges and positions, and
$Z_{i},\mathbf{R}_{i}$ are the charges and positions of fixed nuclei. Here,
and throughout, we utilise Hartree atomic units: $\hbar=e=m_{e}=1$. For the
mixed electron-positron systems considered here, $q_{i}=\pm 1$. We solve for
the many-body ground-state wavefunction using the variational Monte Carlo
(VMC) method: [42] a many-body wavefunction $\Psi_{\theta}$, parameterized by
$\theta$, is continuously updated via a gradient descent procedure to minimize
the energy expectation value,

$\langle
E\rangle_{\theta}=\frac{\int\Psi_{\theta}^{*}(\bm{r})\mathcal{H}\Psi_{\theta}(\bm{r})d\bm{r}}{\int\Psi_{\theta}^{*}(\bm{r})\Psi_{\theta}(\bm{r})d\bm{r}},$
(2)

where $\bm{r}=(\bm{r}_{1},\ldots,\bm{r}_{N})$. This integral, and its gradient
with respect to $\theta$, are evaluated via Monte Carlo integration.

The FermiNet represents the many-body wavefunction as a sum of block-diagonal
determinants containing many-particle orbitals which depend upon the
coordinates of all particles in a permutation equivariant manner. This is
written

$\Psi(\bm{r})=\sum_{k}^{n_{\text{det}}}\prod_{\chi}\text{det}\left[\psi^{k\chi}_{i}(\bm{r}^{\chi}_{j};\\{\bm{r}^{\chi}_{/j}\\};\\{\bm{r}^{/\chi}_{j}\\})\right],$
(3)

where the set $\\{\bm{r}_{/j}\\}$ includes all particle coordinates except
$\bm{r}_{j}$, and $\chi=(\sigma,q)$ labels species of particles which are
distinguished by their spin $\sigma\in(\uparrow,\downarrow)$ and charge
$q\in(+,-)$. Here we have made a slight abuse of notation for the sake of
brevity: permutation invariance for the set $\\{\mathbf{r}_{j}^{/\chi}\\}$ is
only maintained between particles of the same species. We emphasise that these
are not the dense determinants discussed in recent works extending FermiNet
[43], with the exception of the Benzene calculation for which dense
determinants were used for the electronic component of the wavefunction. The
many-particle orbitals $\psi_{i}$ are represented by a deep neural network
[37] (architecture described in the Supplementary Material). Multiplicative
coefficients are omitted from the sum as they are trivially absorbed into the
orbitals. Gradient descent is performed via the Kronecker-factored approximate
curvature (KFAC) algorithm, an approximation of natural gradient descent [44]
which scales well to large neural networks [45]. Natural gradient descent is
closely related to the stochastic reconfiguration method, well-known in the
quantum Monte Carlo literature [46]. The present work introduces two
alterations to the original FermiNet architecture. Firstly, we have included
positronic orbitals as additional species, i.e. additional blocks in the
determinant. Secondly, we utilise distinct weights in the neural network
layers for every species (unlike the original FermiNet, where spin up and down
electronic orbitals shared weights).

A single determinant of the form in Eq. 3 can represent any fermionic many-
body wavefunction [47]. In practice, the argument for the universality of
FermiNet determinants depends upon the representation of discontinuous
functions which cannot be constructed using realistic neural networks. Despite
this, FermiNet-VMC calculations obtain state-of-the-art accuracy in ground
state energy calculations for a range of molecules and solids [37, 38, 48, 49,
50, 43].

We only consider calculations involving a single positron in the present work.
We have discussed the treatment of the positronic spin coordinate only to
demonstrate how our technique may be extended to calculations involving many
positrons, as such systems have recently attracted theoretical interest.

Ground-state wavefunctions for bare and positronic molecules are not
guaranteed to be similarly converged after an equal number of gradient descent
steps. This introduces uncontrolled error in calculating positron binding
energies via VMC. Previous work has shown that FermiNet-VMC calculations yield
ground-state energies within chemical accuracy ($\sim 1.5$ milliHartrees) of
exact results for many small molecules [37, 38]. With this level of accuracy,
the uncontrolled error will be negligible for molecules with a large positron
binding energy. However, for molecules with very small binding energies, or
large molecules for which the uncontrolled error may become large compared to
the positron binding energy, there is no guarantee that an accurate estimate
of the positron binding energy will be obtained by comparing ground-state
calculations of different quality. In these cases, we employ the variance
matching technique described by Entwistle _et al._ [51], addressed in the
Supplementary Material.

###### Acknowledgements.

This work was undertaken with funding from the UK Engineering and Physical
Sciences Research Council (EP/T51780X/1) (GC). Calculations were carried out
with resources provided by the Baskerville Accelerated Compute Facility
through a UK Research and Innovation Access to HPC grant, and we acknowledge
PRACE for awarding us access to JUWELS at GCS@FZJ, Germany. Via his membership
of the UK’s HEC Materials Chemistry Consortium, which is funded by EPSRC
(EP/R029431), Foulkes used the UK Materials and Molecular Modelling Hub for
computational resources, MMM Hub, which is partially funded by EPSRC
(EP/T022213).

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## Appendices

### Variance matching

Figure 3: Top: Binding energy obtained using bare molecular ground-state
energies obtained from checkpoints taken after varying training iterations.
Bottom: Local energy variance difference between bare molecular checkpoints
and positronic molecule at the end of training.

Variance matching was utilised in the main text to calculate the positron
binding energy for the sodium atom and the benzene molecule. Here we will
describe the procedure for obtaining benzene’s positron binding energy. We
note that all variances are evaluated by freezing the wavefunction parameters
and taking 10k samples of the local energy by MCMC sampling. The variance in
the local energy is calculated using a reblocking procedure to account for
sequential correlations in the MCMC sampling [39].

After a fixed number of optimisation steps (i.e. at the end of training), we
calculate the variance of the local energy for the positronic ground-state
wavefunction. We then calculate the variance of the local energy for the bare
molecular ground-state wavefunction at a series of regularly spaced
checkpoints during training – approximately every 50k iterations from 150k to
300k. We refine the variance matching by calculating the variance between the
two checkpoints where the difference crosses zero. We additionally calculate
the variance at several other randomly chosen points in the range of interest
to validate our implicit assumption that the variance is approximately
monotonic. The closest variance match is obtained after 175k training
iterations of the bare molecule. The binding energy and variance difference as
a function of the bare molecule training iterations is shown in Fig. 3. The
obtained variance difference as a function of training iteration is not
completely monotonic due to the stochastic optimisation procedure.

### Fermionic neural networks

The many-body orbitals $\psi_{i}^{k\chi}$, which enter into FermiNet
determinants (Eq. 3), are represented by a deep neural network architecture.
Electron-nuclear and electron-electron coordinates are input to the network.
The vector separations and their norms are used as inputs: the norm provides
the necessary discontinuity to allow the – otherwise smooth – network to
satisfy the Kato cusp conditions [52]. We write these inputs as

$\displaystyle\mathbf{h}_{i}^{1\chi}=\left(\mathbf{r}_{i}^{\chi}-\mathbf{r}_{I},\left\lVert\mathbf{r}_{i}^{\chi}-\mathbf{r}_{I}\right\rVert\quad\forall\quad
I\right)$ (4)
$\displaystyle\mathbf{h}_{ij}^{1\chi\xi}=\left(\mathbf{r}_{i}^{\chi}-\mathbf{r}_{j}^{\xi},\left\lVert\mathbf{r}_{i}^{\chi}-\mathbf{r}_{j}^{\xi}\right\rVert\right),$
(5)

with capitalized subscripts referring to atomic co-ordinates, $\chi$ and $\xi$
labelling fermionic species as in the main text, and
$\left\lVert.\right\rVert$ the Euclidean norm.

Consecutive transformations update these input features,

$\displaystyle\mathbf{h}^{l+1\chi}_{i}=\tanh\left(\underline{\underline{\mathbf{V}}}^{l}\mathbf{j}^{l\chi}_{i}+\mathbf{b}^{l}\right)+\mathbf{h}^{l\chi}_{i}$
(6)
$\displaystyle\mathbf{h}^{l+1\chi\xi}_{ij}=\tanh\left(\underline{\underline{\mathbf{W}}}^{l}\mathbf{h}^{l\chi\xi}_{ij}+\mathbf{c}^{l}\right)+\mathbf{h}^{l\chi\xi}_{ij}.$
(7)

where

$\mathbf{j}_{i}^{\ell\chi}=\text{concatenate}\left(\mathbf{h}_{i}^{\ell\chi},\mathbf{H}_{i}^{\ell\chi}\right)$
(8)

and

$\mathbf{H}^{\ell\chi}_{i}=\text{concatenate}_{\xi}\left(\frac{1}{n^{\xi}}\sum_{j=1}^{n^{\xi}}\mathbf{h}_{j}^{\ell\xi},\frac{1}{n^{\xi}}\sum_{j=1}^{n^{\xi}}\mathbf{h}_{ij}^{\ell\chi\xi}\right).$
(9)

The first transformation (one subscripted index) is referred to as the one-
electron stream, and the second (two subscripted indices) is the two-electron
stream. These are the ‘layers’ of the network. The linear dimension of the
weight matrices $\underline{\underline{\mathbf{V}}}^{l}$ and
$\underline{\underline{\mathbf{W}}}^{l}$ determine the ‘width’ of the layers.
These widths, and the total number of layers $L$, are the hyperparameters
which determine the network architecture.

The outputs from the $L$th transformation are subject to a final, spin-
dependent, linear transformation and multiplied by (in open boundary
conditions) an exponentially decaying envelope [38], which enforces the decay
of the wave function as $\mathbf{r}_{i}\to\infty$,

$\displaystyle\psi_{i}^{k\chi}(\mathbf{r}_{j},\\{\mathbf{r}_{/j}\\})=\phi_{i}^{k\chi}(\mathbf{r}_{j},\\{\mathbf{r}_{/j}\\})f^{k\chi}_{i}(\mathbf{r}_{j})$
(10) where
$\displaystyle\phi_{i}^{k\chi}(\mathbf{r}_{j},\\{\mathbf{r}_{/j}\\})=(\mathbf{w}_{i}^{k\chi}\cdot\mathbf{h}^{L\chi}_{j}+g_{i}^{k\chi})$
(11) and $\displaystyle
f_{i}^{k\chi}(\mathbf{r}_{j})=\left[\sum_{m}\pi_{im}^{k\chi}\text{exp}\left(-\sigma_{im}^{k\chi}|(\mathbf{r}_{j}^{\chi}-\mathbf{R}_{m})|\right)\right].$
(12)

The functions $\psi_{i}^{k\chi}$ are used as the inputs to the determinants,
Eq. 3.

The determinant expansion in Eq. 3 does not require multiplicative
coefficients as they can be trivially absorbed into the determinants [38].
Hutter showed that a single determinant of such many-body functions could
represent any antisymmetric function [47]. However, the proof depends upon the
construction of discontinuous functions that cannot be represented in practice
by a finite network of a reasonable size.

The set of parameters,

$\theta=\\{\underline{\underline{\mathbf{V}}}^{l},\underline{\underline{\mathbf{W}}}^{l},\mathbf{w}_{i}^{k\chi},\mathbf{b}^{l},\mathbf{c}^{l},g_{i}^{k\chi},\pi_{im}^{k\chi},\sigma_{im}^{k\chi}\\},$
(13)

are all learnable. The electronic orbitals, $\chi=(\sigma,-)$, are pretrained
to the Hartree-Fock orbitals of the bare molecule, even for positronic
calculations, according to the procedure outlined in [37]. For a more
extensive description of the FermiNet architecture, see Pfau et al. [37].

### Experimental setup

Kind | Parameter | Value
---|---|---
Optim | Batch size | 4096
Optim | Training iterations | 3e5
Optim | Pretraining iterations | 1e4
Optim | Learning rate | $(1e4+t)^{-1}$
Optim | Local energy clipping | 5.0
KFAC | Momentum | 0
KFAC | Covariance moving average decay | 0.95
KFAC | Norm constraint | 1e-3
KFAC | Damping | 1e-3
MCMC | Steps between parameter updates | 10
Table 2: FermiNet hyperparameters for all experiments in the paper. For the
Benzene calculations, the decay of the learning rate was disabled (i.e. the
value was fixed at 1e-4 throughout training).

Four A100 GPUs were used for all calculations presented, except the Benzene
molecule, for which sixteen A100 GPUs were used. Single-precision floating
point numbers were used in all calculations. The FermiNet was implemented
using the JAX Python library [53], extending a development version of the
FermiNet [54]. Optimization used a JAX implementation of the Kronecker-
factored approximate curvature (KFAC) gradient descent algorithm [38, 45, 55].

Monte Carlo moves are accepted/rejected on a per-species basis, including spin
species. All particles of a given species are moved simultaneously within each
Monte Carlo step. Each species has an independent Monte Carlo step proposal
size, and this step size is adapted during training to maintain a $\sim 50\%$
acceptance rate.

We found the convergence of the Benzene calculations was accelerated when the
decay of the learning rate (see Table 2) was disabled. We found that following
this aggressive training regime with an additional 2000 training iterations
with the appropriately decayed learning rate greatly improved the energies
obtained during inference. Understanding the training dynamics of the FermiNet
continues to be an open question of great interest.

## Data Tables

### Lithium hydride

| Energy [Hartrees]
---|---
Bondlength [$a_{0}$] |  Bare |  $\sigma_{\text{Bare}}$ |  Positronic |  $\sigma_{\text{Positronic}}$
2.815 |  -8.069021 |  0.000004 |  -8.100914 |  0.000004
2.882 |  -8.069882 |  0.000005 |  -8.102727 |  0.000009
2.948 |  -8.070345 |  0.000003 |  -8.104151 |  0.000010
3.015 |  -8.070507 |  0.000004 |  -8.105283 |  0.000004
3.082 |  -8.070371 |  0.000007 |  -8.106226 |  0.000007
3.148 |  -8.069958 |  0.000003 |  -8.106885 |  0.000005
3.215 |  -8.069331 |  0.000004 |  -8.107324 |  0.000003
3.281 |  -8.068203 |  0.000278 |  -8.107583 |  0.000003
3.348 |  -8.067207 |  0.000287 |  -8.107737 |  0.000004
3.415 |  -8.066352 |  0.000007 |  -8.107712 |  0.000011
3.481 |  -8.065065 |  0.000005 |  -8.107542 |  0.000004
3.548 |  -8.063663 |  0.000005 |  -8.107234 |  0.000005
3.615 |  -8.062156 |  0.000004 |  -8.106949 |  0.000006
3.682 |  -8.060571 |  0.000004 |  -8.106551 |  0.000008
3.749 |  -8.058894 |  0.000008 |  -8.105952 |  0.000010
3.816 |  -8.057171 |  0.000005 |  -8.105373 |  0.000132
3.883 |  -8.055407 |  0.000007 |  -8.104972 |  0.000006
3.950 |  -8.053575 |  0.000007 |  -8.104321 |  0.000010
4.017 |  -8.051748 |  0.000008 |  -8.103593 |  0.000004
4.084 |  -8.049849 |  0.000008 |  -8.102936 |  0.000005
4.151 |  -8.047965 |  0.000005 |  -8.102246 |  0.000010

### Beryllium oxide

| Energy [Hartrees]
---|---
Bondlength [$a_{0}$] |  Bare |  $\sigma_{\text{Bare}}$ |  Positronic |  $\sigma_{\text{Positronic}}$
| Singlet
2.115 |  -89.845004 |  0.000024 |  -89.865185 |  0.000027
2.215 |  -89.875412 |  0.000024 |  -89.896963 |  0.000019
2.315 |  -89.893706 |  0.000014 |  -89.916586 |  0.000019
2.415 |  -89.903115 |  0.000042 |  -89.927073 |  0.000033
2.515 |  -89.905721 |  0.000014 |  -89.930761 |  0.000029
2.615 |  -89.903574 |  0.000075 |  -89.929260 |  0.000088
2.715 |  -89.897711 |  0.000070 |  -89.924101 |  0.000020
2.815 |  -89.889610 |  0.000022 |  -89.915860 |  0.000043
2.915 |  -89.879387 |  0.000073 |  -89.906657 |  0.000029
3.015 |  -89.869008 |  0.000073 |  -89.895173 |  0.000032
3.115 |  -89.857212 |  0.000018 |  -89.884496 |  0.000039
3.215 |  -89.846269 |  0.000021 |  -89.872785 |  0.000118
3.315 |  -89.838491 |  0.000082 |  -89.860431 |  0.000095
3.415 |  -89.830544 |  0.000024 |  -89.845536 |  0.000027
3.515 |  -89.820883 |  0.000027 |  -89.834588 |  0.000032
3.615 |  -89.812532 |  0.000032 |  -89.824085 |  0.000029
3.715 |  -89.805062 |  0.000019 |  -89.812757 |  0.000077
3.815 |  -89.798934 |  0.000097 |  -89.801516 |  0.000026
3.915 |  -89.786483 |  0.000030 |  -89.792682 |  0.000042
4.015 |  -89.794044 |  0.000051 |  -89.794320 |  0.000043
| Triplet
3.515 |  -89.824472 |  0.000018 |  -89.822049 |  0.000031
4.015 |  -89.784895 |  0.000023 |  -89.788294 |  0.000031
4.515 |  -89.754943 |  0.000024 |  -89.763941 |  0.000031
5.015 |  -89.738802 |  0.000023 |  -89.745154 |  0.000032
5.515 |  -89.732889 |  0.000022 |  -89.736450 |  0.000033
6.015 |  -89.731046 |  0.000022 |  -89.732024 |  0.000033
6.515 |  -89.730538 |  0.000023 |  -89.730718 |  0.000034
7.015 |  -89.730307 |  0.000022 |  -89.731238 |  0.000031
7.515 |  -89.730150 |  0.000021 |  -89.731904 |  0.000030
8.015 |  -89.730040 |  0.000021 |  -89.731992 |  0.000036
8.515 |  -89.730035 |  0.000025 |  -89.732091 |  0.000031
9.015 |  -89.729969 |  0.000022 |  -89.732011 |  0.000030

### Benzene

System |  $\langle E\rangle$ [Hartrees] |  $\sigma_{E}^{2}$
---|---|---
Positronic @ 300k iter. |  -232.223293 |  0.000847
Bare @ 134k iter. |  -232.216986 |  0.001202
Bare @ 138k iter. |  -232.217466 |  0.001170
Bare @ 150k iter. |  -232.217369 |  0.001040
Bare @ 176k iter. |  -232.219174 |  0.000871
Bare @ 200k iter. |  -232.221368 |  0.000639
Bare @ 226k iter. |  -232.222084 |  0.000747
Bare @ 240k iter. |  -232.222453 |  0.000568
Bare @ 260k iter. |  -232.223825 |  0.000680
Bare @ 276k iter. |  -232.223786 |  0.000576
Bare @ 300k iter. |  -232.225837 |  0.000668


# Electron correlation mediated site-selective charge compensation in
polar/non-polar heterointerface

Nandana Bhattacharya  Department of Physics, Indian Institute of Science,
Bengaluru 560012, India  Arpita Sen  Theoretical Sciences Unit, Jawaharlal
Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560 064,
India  Jianwei Zhang  Key Laboratory of Polar Materials and Devices, East
China Normal University Shanghai, Shanghai 200241, China  Ranjan Kumar Patel
Department of Physics, Indian Institute of Science, Bengaluru 560012, India
Siddharth Kumar  Department of Physics, Indian Institute of Science, Bengaluru
560012, India  Prithwijit Mandal  Department of Physics, Indian Institute of
Science, Bengaluru 560012, India  Shashank Kumar Ojha  Department of Physics,
Indian Institute of Science, Bengaluru 560012, India  Jyotirmay Maity
Department of Physics, Indian Institute of Science, Bengaluru 560012, India
Zhan Zhang  Advanced Photon Source, Argonne National Laboratory, Lemont,
Illinois 60439, USA  Hua Zhou  Advanced Photon Source, Argonne National
Laboratory, Lemont, Illinois 60439, USA  Fanny Rodolakis  Advanced Photon
Source, Argonne National Laboratory, Lemont, Illinois 60439, USA  Padraic
Shafer  Advanced Light Source, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Christoph Klewe  Advanced Light Source,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA  John
William Freeland  Advanced Photon Source, Argonne National Laboratory, Lemont,
Illinois 60439, USA  Zhenzhong Yang  Key Laboratory of Polar Materials and
Devices, East China Normal University Shanghai, Shanghai 200241, China  Umesh
Waghmare  Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced
Scientific Research, Jakkur P.O., Bangalore 560 064, India  Srimanta Middey
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science,
Bengaluru 560012, India

###### Abstract

One of the boundary conditions of the classical electromagnetic theory demands
continuous electric potential across any boundary, which may not be naturally
satisfied in atomically engineered heterostructures. Such polarity mismatch in
oxide heterointerfaces is compensated through some
(electronic/chemical/structural) reconstructions, leading to a myriad of
emergent phenomena. The question we are posing is whether conventional
semiconductor band bending framework is sufficient to comprehend compensation
mechanisms in oxide heterostructures since, unlike semiconductors, complex
oxides host strong electron correlations whose effects are indispensable. To
address this, we investigate the interface between a prototypical insulating
double perovskite Nd2NiMnO6 and a wide-bandgap insulator SrTiO3. This
polar/non-polar interface offers a similar scenario as the famous
LaAlO3/SrTiO3 system but with an exception - two transition metal sites with
two individual correlated energy scales. By combining several experimental
techniques and density functional theory, we establish a site-selective charge
compensation process that occurs explicitly at the Mn site of the film,
leaving the Ni sites inert. This surprising selectivity, which cannot be
accounted by existing polar compensation mechanisms, is directly attributed to
the TM cations’ relative correlation energy scales. This discovery presents
that site-specific charge compensation can be a designer’s tool for tailoring
emergent phenomena in oxide heterostructures.

Understanding the consequences of electrostatic boundary conditions at the
interface between two dissimilar materials has been a long-standing and ever-
explorative question in condensed matter physics [1]. Semiconductor
heterostructures became a starting ground to study the effect of polarity
mismatch, also termed as polar catastrophe, wherein Ge/GaAs heterostructure
came to be studied in great detail since the 1970s [2]. Although both charge
and atomic rearrangements were theoretically demonstrated as feasible ways to
compensate the diverging potential [2, 3], no electronic reconstruction was
observed experimentally [4, 5], which can be attributed to the lack of
multiple valence states of elements present in a typical semiconductor
heterostructure [6]. From this aspect, complex oxide heterostructures are
fundamentally different as transition metal (TM) ions can easily access
multiple valence states and have strong interplay amongst spin, charge,
lattice, and orbital degrees of freedom [7, 8, 9]. In recent years, the impact
of polarity mismatch in oxide heterostructures has become a research theme in
itself, wherein a plethora of emergent phenomena such as two-dimensional
electron gas [10, 11], superconductivity [12], ferromagnetism [13, 14, 15],
superparamagnetism [16], etc. have been demonstrated. Unlike the case in a
semiconductor, the potential divergence can be avoided here through either
electronic/chemical reconstruction, resulting in a change in cationic valency
[17, 18, 19, 20, 21, 22, 23] or structural effect such as interfacial
intermixing [24, 25], intermediate layer formation [26], etc.

Another intricate factor inherent to TM oxides, specifically 3 _d_ TMOs, is
that they are strongly correlated electron systems and are traditionally
discussed by characteristic energy scales like the on-site coulomb repulsion
energy ($U$), charge transfer energy ($\Delta$) and the hopping parameter
($t$) [27]. To examine whether these energies decide polar compensation
process or a simple semiconductor-like band bending scenario is sufficient, we
focus on the thin films of double perovskite (DP) oxide (A2 _BB’_ O6) [28]
where _B_ , _B’_ are the TM elements with their individual correlated energy
scales. These materials themselves are interesting as they exhibit high-
temperature ferromagnetism [29], room temperature magnetoresistance [30],
multiglass behavior [31], insulating ferromagnetism [32, 33], etc. with lots
of potential for technological applications.

In this work, we have investigated thin films of Nd2NiMnO6 (NNMO), grown on
single crystalline NdGaO3 (NGO) (110)or [or represents orthorhombic coordinate
system] and SrTiO3 (STO) (001) substrates by pulsed laser deposition (PLD)
technique. Bulk NNMO is a monoclinic ferromagnetic insulator with a Curie
temperature of 200 K [34]. Contrary to the NNMO film on NGO where both film
and substrate consist of alternating +1 and -1 charged plane (hence no
interfacial polarity mismatch), the NNMO/STO heterostructure encompasses
interfacial polarity mismatch, similar to the well-known LaAlO3/SrTiO3
(LAO/STO) heterostructure [10]. We have explored the effect of polar
catastrophe using element-sensitive X-ray absorption spectroscopy (XAS) and
scanning transmission electron microscopy (STEM) measurements along with first
principles calculations. Our findings establish that polar compensation
happens through the formation of Mn3+ within the film side near the interface,
while the Ni site remains unperturbed. The additional electron comes due to
surface oxygen vacancies. This site-selective charge compensation emanates
from the relative difference between $U$ and $\Delta$ of Ni and Mn,
demonstrating a direct connection between polarity compensation and electronic
structure parameters in complex oxide heterostructures.

Figure 1: Structural characterizations Reciprocal space mapping (RSM) in (a)
_H-L_ plane (b) _K-L_ plane for a 20 uc film on STO around (113)pc Bragg
reflection. (c) HAADF-STEM image of a 20 uc film on STO. EELS element map of
(e) Ti, (f) Nd, (g) Mn, (h) Ni and (d) the corresponding area of HAADF-STEM
image. Crystallographic unit cells of NNMO and STO have been also shown on the
left of the image (d).

Sample fabrication and characterization: NNMO films of several film
thicknesses [2, 3, 5, 10, and 20 uc, uc= unit cell in pseudocubic
($\mathrm{pc}$) notation] were grown on TiO2 terminated STO (001) substrates
in layer-by-layer fashion using a PLD system, connected with a high-pressure
reflection high energy electron diffraction (RHEED) setup. For comparison
purposes, 5, 10, and 20 uc NNMO films were also grown on GaO2 terminated NGO
(110)or [(001)pc] substrates. The epitaxial strain is calculated to be +0.4%
and +1.5% for NNMO on NGO, and STO, respectively. The details of sample growth
conditions and characterizations using RHEED, X-ray diffraction (XRD), and
X-ray reflectivity (XRR) have been discussed in the Methods section. Fig. 1(a)
and (b) show reciprocal space map (RSM) for the 20 uc NNMO film on STO in
$H-L$ and $K-L$ plane around the (113)pc Bragg reflection of the substrate,
recorded with synchrotron X-ray. The observation of the same $H$ and $K$ value
for the film and substrate confirms that the film has the same in-plane
lattice constants as the underlying STO. Using this RSM image, the out-of-
plane lattice constant is found to be 3.803 Å, which is also very similar to
the value found from a (00$L$) crystal truncation rod measurement. RSM
measurements for films grown on NGO also confirm the proper epitaxial relation
between the film and substrate.

The epitaxial growth has been further confirmed by a cross-sectional scanning
transmission electron microscopy (STEM) experiment on a 20 uc NNMO film on STO
(experimental details are in the Methods section). As depicted in Fig.
1(c)-(d), high-angle annular dark field (HAADF) image along the zone axis
[100] confirmed coherent epitaxial growth of the film on STO, devoid of any
defect or dislocation. Atomically resolved electron energy loss spectroscopy
(EELS) mapping [Fig. 1(e)-(h)] has found that the extent of cationic
intermixing across the interface is limited to 1 uc on the film side, which is
also consistent with the film/substrate interface roughness obtained from XRR
fitting. Since such minuscule intermixing is very likely to be present in all
films due to the use of same growth condition, it can not be the source of
systematic changes in electronic structures with the film thickness, which are
shown in the latter part of the paper. Another notable observation is that we
do not find any NiO precipitation, which was reported to be the polar
compensation mechanism for a thick ($\sim$ 40 nm) La2NiMnO6 film [35].

Figure 2:  X-ray absorption spectroscopy (XAS) on Ti $L_{3,2}$ and Ni
$L_{3,2}$ edges (a) Schematic of element sensitive XAS experiment performed at
grazing incidence (X-ray beam incident at 15∘ to the sample plane). Total
electron yield (TEY) and total fluorescence yield (TFY) detectors record the
surface-sensitive and bulk-sensitive spectra, respectively. All XAS
experiments have been performed at room temperature and TEY data have been
shown here. (b) Ti _L_ 3,2 XAS data recorded for 2, 3 and 5 uc films on STO,
compared with standard Ti4+ spectra of STO. The positions of all four peaks do
not change with the film thickness. (c) Ni _L_ 3,2 XAS data for 2, 3, 5, 10
and 20 uc films on STO and 10 uc film on NGO performed in TEY mode. Reference
for Ni2+ spectrum of Ca2NiOsO6 has been adapted from Ref. [36].

Probing charge compensation on substrate side: The smaller band gap of NNMO
($\sim$1-1.5 eV) [34, 37] compared to STO (3.25 eV) should result in a
straddled band alignment, similar to the well-studied LaMnO3/SrTiO3 (LMO/STO)
heterostructure [19, 16, 38, 39, 40, 41]. While in such cases, the electronic
reconstruction should occur within the film side of the interface [19, 38],
there are also reports of charge compensation within the STO substrates for
LMO/STO case [39, 40]. Therefore, it is crucial to probe the electronic
structures of both film and substrate as well for NNMO/STO heterostructures.
For checking the valence state of the TM sites, we have performed element-
sensitive soft X-ray absorption spectroscopy (XAS) experiments for all samples
grown on STO and NGO substrates. For a 3$d$ transition metal, the XAS with
soft X-ray probes 2$p^{6}$3$d^{n}\rightarrow$ 2$p^{5}$3$d^{n+1}$ transition
and the spectral line shape is very sensitive to the valency, spin character
of the initial state and crystal field environment of the system [42]. All XAS
data, discussed in the main text, were recorded in surface-sensitive total
electron yield (TEY) with a grazing incidence geometry (see Fig. 2(a) for
experimental arrangement).

The origin of the metallic interface for LAO/STO heterostructure is attributed
very often to the polar catastrophe-driven electronic reconstruction
mechanism, resulting in Ti3+ in STO near the heterointerface [17]. The
presence of oxygen vacancy in STO can also result in metallic behavior due to
electron doping on Ti-sites [43]. We start with our attempt to find out any
presence of Ti3+ in NNMO/STO heterostructures. As shown in Fig. 2(b), spectral
features of Ti _L_ 3,2 edges are very similar to that of Ti4+ of bulk
stoichiometric STO. On further analyzing the energy separation between the two
main peaks of each of the _L_ 2 and _L_ 3 edges, which are known to detect
even small amount of Ti3+ ions [40], we do not find any signature of electron
doping on Ti. This is further corroborated by our electrical transport
measurements using wire-bonded contacts, which do not find any measurable
interfacial conductivity, similar to LaMnO3 /SrTiO3 [19], LaCrO3 /SrTiO3
heterostructures [24]. The presence of any bulk oxygen vacancy within the
substrates can also be discarded as all NNMO films were annealed in-situ for
30 minutes under a 500 Torr ultrapure oxygen atmosphere immediately after the
deposition.

Probing charge compensation on film side: In the absence of any charge
compensation in STO substrate, we proceed to probe toward the film side. In
bulk NNMO, Ni has +2 oxidation state, irrespective of any change in the degree
of Ni-Mn site-disorder [34, 44]. In Fig. 2(c), we show Ni $L_{3,2}$ edge XAS
spectra of several films on STO, one 10 uc film on NGO along with a Ni2+
standard: double perovskite Ca2NiOsO6 [36]. Admittedly, Ni remains in Ni2+
state for all films on both substrates, implying that the polar mismatch does
not affect Ni ions of NNMO.

Figure 3:  Demonstration of charge compensation on Mn sites. (a) Mn _L_ 3,2
edge XAS data recorded at room temperature for 5, 10 and 20 uc films on NGO
(upper panel) and 2, 3, 5, 10 and 20 uc films on STO (lower panel) in TEY
mode. References Mn2+, Mn3+ and Mn4+ are obtained from Mn _L_ 3,2 spectra of
MnO, LaMnO3 and LiNi0.5Mn0.5O2, respectively adapted from Ref. [44]. Dotted
lines, corresponding to the dominant feature of $L_{3}$ edge for Mn2+, Mn3+
and Mn4+ have been shown as a guide to the eye. (b) Layer resolved out-of-
plane lattice constant for the 20 uc NNMO film on STO, evaluated using the
HAADF-STEM image. (c) Schematic structure of 2 uc NNMO/ 3.5 uc STO / 2 uc NNMO
used in DFT calculation. Magnetic moments of the Ni and Mn cations are also
marked. (d) The in-plane average (oscillating cyan line) and macroscopic
average (magenta line) of the electrostatic potential obtained from DFT for 2
uc NNMO/ 3.5 uc STO / 2 uc NNMO slab. The macroscopic average has been
performed over a length scale of 3.91 Å.

In sharp contrast, we find a systematic change in Mn XAS spectra as shown in
Fig. 3(a). For easy comparison, Mn XAS spectra for the standard of Mn2+, Mn3+
and Mn4+ (adapted from Ref. 44), have also been shown in Fig. 3(a). The
spectra for 20 uc and 10 uc films on NGO substrate (no interfacial polar
mismatch) confirm +4 oxidation state of Mn, similar to bulk NNMO [34].
However, there is a small feature of Mn2+ around 639.4 eV, which becomes much
more prominent for the 5 uc film on NGO. On the contrary, there is an
additional change in Mn XAS for the films on STO substrate. Apart from the
Mn2+ feature which is also found here with a similar trend with the film
thickness, an additional feature for Mn3+ states is observed, which increases
with the lowering of film thickness. These Mn3+ signatures become comparable
to Mn4+ for the 5 uc film and even become the dominant contributor for the 3
uc film. The 2 uc film on STO shows primarily features of Mn2+ and Mn3+. We
have also checked Mn XAS of 20 uc NNMO film on NGO and STO in bulk-sensitive
total fluorescence (TFY) mode. We do not find any signature of Mn2+ feature
there, signifying it is present near the surface (surface contribution is
masked by bulk Mn4+ contribution in TFY signal). It may be noted that surface
oxygen vacancies leading to Mn2+ ions is very common for manganite thin films
[45, 38] and it has also been designated to compensate the polarity for
LMO/STO heterostructures [38]. However, such surface Mn2+ cannot compensate
for polar catastrophe for NNMO as it is present in films on both STO and NGO
substrates. Rather, the compensation is happening through the formation of
Mn3+ as it is exclusively present only for films on STO. It is very important
to note that the TEY mode probes $\sim$ 4-5 nm length scale from the surface
and the contribution of the deeper layers decreases exponentially from the top
surface [42]. Thus, the decrease of Mn3+ feature with the increase in NNMO
film thickness indicates that the Mn valence is non-uniform along the out-of-
plane film direction and there are few layers of the film with a prevailing
contribution of Mn3+ ion.

Identifying the regions with dominant contribution of Mn3+ and Mn2+ : The
ionic radii of Mn2+, Mn3+ and Mn4+ ions in octahedral environment with high
spin configuration are 0.83 Å, 0.645 Å, and 0.53 Å [46]. Thus, the layer-
resolved out-of-plane lattice constant ($c_{\mathrm{pc}}$) should vary
according to the Mn oxidation state if they are in different layers. Fig. 3(b)
shows the variation of the layer resolved $c_{\mathrm{pc}}$, evaluated using
the HAADF-STEM image, demonstrating film has three types of layers. The
$c_{\mathrm{pc}}$ of the intermediate layers (4-14 uc from the interface) is
very similar to the value obtained from X-ray diffraction measurement (Fig.
1(a), (b)). The higher value of $c_{\mathrm{pc}}$ for few outermost layers
establishes Mn2+ are indeed located near the film surface. Most importantly,
the $c_{\mathrm{pc}}$ value for the first three unit cells of the NNMO film is
slightly higher, compared to the intermediate layers, implying that 2-3 uc
near the interface regions are enriched with Mn3+. Thus, our comprehensive
results and detailed analysis from XAS and STEM experiments confirm that site-
selective electron doping occurs for interfacial Mn states to compensate for
the polar catastrophe in NNMO films on STO. In the forthcoming section, we
will discuss in great details regarding the proposed mechanism that deals with
this site-selective electron doping in Mn.

Density functional theory: To determine implications of polar catastrophe to
the electronic structure, particularly at the interface between the NNNO film
and SrTiO3 substrate, we present analysis based on first-principles DFT
calculations of bulk STO, bulk NNMO and a symmetric heterostructured slab
consisting of 2 uc of NNMO/3.5 uc of STO/2 uc of NNMO Fig. 3(c) (details are
provided in the Methods section). Our estimates of the band gap of bulk STO
and NNMO are 3.44 eV, and 1.5 eV respectively, which is comparable within the
typical DFT errors with the experimental values reported previously [10, 34].
Planar and macroscopic averaged electrostatic potentials of the 2 uc NNMO/3.5
uc STO/ 2 uc NNMO slab (shown in Fig. 3(d)) reveal deviation at the interface
from the bulk, reflecting on the mismatch in polarity. The internal
electrostatic field obtained as the slope of macroscopic averaged potential is
$\sim$ 0.15 V/A, which is comparable to the value reported for the LMO/STO
interface [19]. In the self-consistent ground state of the relaxed structure
of NNMO:STO:NNMO slab, magnetic moments of Ni ions located at the surface and
of those at the film/substrate interface are quite similar. In sharp contrast,
Mn ion located at the surface layer has more electrons than the Mn located at
the interfacial layer, which is consistent with the Mn2+ observed at the
surface from our XAS (Fig. 3(a)) and STEM (Fig. 3(b)) results. Moreover, the
calculated electronic structure reveals that (a) the bands near the Fermi
energy and of relatively narrow energy width are localized on oxygen atoms at
the surface, which correlate with stabilization of oxygen vacancies in the
surface layer seen experimentally, and (b) electronic structure projected onto
interfacial atoms exhibits a gap, in agreement with experiment.

Figure 4:  Mechanism of site-selective charge compensation Schematic depicting
atomic terminations of (a) GaO2 terminated NNMO on NGO heterostructure: type-
III interface with no polar catastrophe. A linear potential with a negative
slope has been subtracted to get the final profile[1]. (b) TiO2 terminated
NNMO on STO heterostructure: type-II interface exhibiting polar catastrophe.
Corresponding electric field (E) and potential (V) have been plotted following
electrostatic boundary conditions. The surface layers are depicted considering
surface oxygen vacancies, as found from XAS experiments. (c) Proposed charge
compensation mechanism for NNMO on STO demonstrating transfer of 1/2 e- per
unit cell to the interface (yellow highlighted region) alleviating polar
catastrophe. (d) Energy band representation of upper and lower Hubbard bands
(UHB and LHB) of Ni2+ and Mn4+ cations in bulk NNMO (left) and of Ni2+ and
Mn3+ cations post site-selective charge compensation (right). Relative
positions of energy bands have been drawn using values of _U_ and $\Delta$
adapted from Ref. [47]. (e) O _K_ edge X-ray absorption spectroscopy of 2, 3,
5, 10 and 20 uc films on STO at room temperature compared with a reference
spectra of 5 uc $\gamma$ Al2O3 (GAO) film on STO. The reference spectra for
NiO adapted from Ref. [48]. The pink shaded line represents Mn
$e_{g}^{\uparrow}$-O 2$p$ and Mn $t_{2g}^{\downarrow}$-O 2$p$, the yellow
shaded line represents Ti $t_{2g}$ \- O 2$p$, and the blue shaded line
represents Ni UHB - O 2$p$ and Ti $e_{g}$ \- O 2$p$ contributions to the
spectra and are meant as guide to the eye.

##  I Discussion

While numerous oxide interfaces with interfacial polar mismatch have been
investigated over the last twenty years, the novelty of our work lies in the
specific participation of the Mn site and the inertness of the Ni site in the
compensation process. Based on our results of extensive XAS and STEM
experiments, the following charge compensation [schematically depicted in Fig.
4(a)-(c)] can be considered as a means to alleviate polar catastrophe. The
diverging potential for NNMO/STO heterostructure can be compensated by the
transfer of 1/2 electron per uc to the interfacial Mn resulting in the
formation of Mn3+ cations [22]. It can be easily understood that such electron
transfer would be energetically unfavorable if the top layer had Mn4+ akin to
bulk NNMO to start with as Mn does not have access to +5 oxidation state.
However, some of the surface Mn2+ ions can easily donate electrons and
compensate for the polarity mismatch for the NNMO/STO interface. Surprisingly,
Ni2+, which is also capable of donating electrons to avoid polar catastrophe
[49], remains silent in the present case.

While the potential profile discussed above, based on the ideas initially
proposed for semiconductor heterostructures, can account for the polarity
compensation in the present case, it fails to explain why only the Mn site
takes part in the compensation mechanism. Generally, one would expect electron
transfer to occur in any of the transition metal cations present since the
potential slope is independent of the nature of the cation. In order to
explain such site selectivity, we need to invoke the physics of strong
electron correlation for 3$d$ TMOs. In oxygen environment, Ni2+ has a typical
value of $U\sim$ 6.5 eV and $\Delta\sim$ 4 eV [see page 102 of Ref. 47]. The
corresponding parameters for Mn4+ are $\sim$ 5 eV and, 1 eV, respectively. In
Fig. 4(d), we have drawn the upper Hubbard band (UHB) and lower Hubbard band
(LHB) for both Ni2+ and Mn4+ with a common O 2$p$ band. The Mn UHB is at an
energetically lower value than the Ni UHB, making it a preferred host to an
electron addition. The resultant energy level diagram after the addition of an
electron to the Mn is also depicted in Fig. 4(d), highlighting the significant
overlap between UHBs of Ni2+ and Mn3+ [$U\sim$ 5 eV and $\Delta\sim$ 4 eV for
Mn3+, Ref. 47]. Thus, the observed site-selective charge compensation is
related to the intrinsic energy scales of correlated physics.

The above modification in unoccupied electronic states is further verified by
O $K$-edge XAS measurements. From a pure ionic picture, the dipole transition
1$s\rightarrow$2$p$ is forbidden for O2- due to fully filled O 2$p$ shell.
However, due to hybridization between TM $d$ and O 2$p$ orbitals in real
systems, a sizable spectral weight of O 2$p$ is introduced into the unoccupied
density of states. This makes O $K$-edge XAS a useful method to find out
relative energy positions of cation states above the Fermi level [50]. O
$K$-edge XAS spectra for NNMO films on STO substrates have been shown in Fig.
4(e). To identify the features, related to Ti and Ni states, XAS spectra for a
5 uc $\gamma$-Al2O3/SrTiO3 film [51] and bulk NiO [48] have also been plotted.
The feature of Ti $t_{2g}$ \- O 2$p$ hybridized states of STO substrates are
strongly present in the spectra of 5 uc film along with the films with lower
thicknesses, and is weakly visible for the 10 uc NNMO film. The feature
related to Ni UHB states appears around 532 eV for all NNMO films [48]. The
most notable observation is the evolution of the feature around 528 eV with
the NNMO film thickness, which is contributed by the Mn $e_{g}^{\uparrow}$-O
2$p$ and Mn $t_{2g}^{\downarrow}$-O 2$p$ states [52]. The gradual suppression
of this 528 eV peak with the lowering of film thickness establishes the
movement of Mn UHB to higher energy, as depicted in Fig. 4(d), due to the
charge compensation on Mn sites near the NNMO/STO interfacial regions. This is
further supported by our observation that the 528 eV feature remains unaltered
with the lowering of NNMO film thickness, grown on NGO substrate.

## Conclusion

To summarize, the present study unveils a surprising site-selective charge
compensation in Nd2NiMnO6 thin films on SrTiO3 (001) substrates via electron
transfer from surface to the interfacial Mn states resulting in Mn3+ cations
at the interface. Despite being uniformly distributed as Mn, the Ni sites
remain unaffected by the process. By using O _K_ edge XAS, we further
demonstrate that this site-selectivity is related to the relative energy
position of the upper Hubbard band of Mn4+ with respect to the Ni2+. Thus, our
results demonstrate the crucial role and inevitability of strongly correlated
electron physics in understanding polar compensation for oxide
heterostructures.

More than thousands of double perovskites have been synthesized in the
laboratory [28] so far, and we believe such a site-selective charge
compensation mechanism will also be pertinent for other insulating double
perovskites. Epitaxial stabilization of double perovskites on (111)-oriented
perovskite substrates will be another interesting direction to explore as the
power of geometrical lattice engineering in realizing novel quantum many-body
phases has been demonstrated for simple perovskites [53, 54, 55]. Double
perovskites with the desired properties are also being predicted in recent
years using machine learning and artificial intelligence [56, 57], and our
findings may add another dimension for designing/predicting new double
perovskite heterostructures within such an approach.

## Methods

Experimental details. Polycrystalline NNMO powder was synthesized using solid-
state synthesis [34] and pelletized to make a target for pulsed laser
deposition (PLD) growth. TiO2 terminated STO and GaO2-terminated NGO
substrates were obtained through thermal annealing of as-received substrates
from Shinkosha, Japan [58, 59]. A KrF excimer laser operating at $\lambda$ =
248 nm with a fluence of 2 J/cm2 and a repetition rate of 2 Hz was used for
the deposition. The growth conditions were maintained the same for films on
both STO and NGO substrate, with 150 mTorr dynamic oxygen pressure during the
deposition at 750∘C and post-annealing of the films at 500 Torr of oxygen
pressure at the same temperature. The layer-by-layer growth with unit cell
precision was confirmed by the observation of intensity oscillation of the
specular spot of RHEED pattern. The observation of streaks in the RHEED image
of the film, taken after cooling to the room temperature, testifies to the
smooth surface morphology of the film. Details for GAO/STO sample preparation
and characterization can be found in Ref. [51]. XRD and XRR patterns were
recorded using a lab-based Rigaku smartlab X-ray diffractometer. Reciprocal
space mapping of several films was performed at the 33 ID-D beamline of the
Advanced Photon Source (APS), USA with an incident photon energy of 15.5 keV.
A spherical aberration-corrected Grand ARM300F operated at 300 kV used for
performing STEM and EELS map experiments. XAS measurements were performed in
TEY mode at a grazing incidence of 15∘ at the BL 4.0.2 beamline at Advance
Light Source, USA. XAS spectra for a few films were also recorded in TFY mode
at the 29-ID-D beamline of APS.

Calculation details. Our first-principles calculations are based on density
functional theory (DFT) using the SCAN meta-GGA exchange-correlation
functional implemented in the VASP code [60, 61, 62]. For bulk NNMO, in order
to incorporate the correlation effects of $f$ electrons of Nd, $d$ electrons
of Ni and Mn, we used Hubbard $U$ parameters for the mentioned elements as 6
eV, 3.5 eV and 3.5 eV, respectively whereas for STO, the value of $U$ was
taken to be 8.5 eV for both Sr and Ti. We have chosen these $U$ parameters as
reported by Chen et al [19]. Spin-polarized DFT was used for all calculations.
Projector augmented wave (PAW) potentials were used here, which contain a
contribution from kinetic energy density of the core electrons [63]. A kinetic
energy cutoff of 600 eV was used to truncate the plane-wave basis for
representing wave functions. The Brillouin zone integration was sampled using
uniform meshes of $3\times 3\times 3$ _k_ -vectors for estimating optimized
lattice parameters of bulk materials. Self-consistent numerical solution of
the Kohn-Sham equation was obtained iteratively to converge total energy
within 10-5 eV. We relaxed the structures to minimize the energy until the
Hellman-Feynman forces on each atom was less than 0.002 eV/Å.

With the optimized lattice parameters obtained for bulk materials, we
simulated the NNMO/ STO /NNMO heterostructure. The unit cell of cubic bulk STO
was doubled in $x-y$ plane (lattice parameters $a$ and $b$) first and then
made thrice the thickness of it along lattice parameter $c$. The top and
bottom of the STO slabs were terminated with extra TiO2 layers along (100)
direction. These TiO2 layers were interfaced with (110)or surface of NNMO with
thickness of 2 uc at top and bottom of the heterostructure. The
heterostructure was simulated in a manner such that the top and bottom atomic
layers of NNMO are placed with $\sigma_{h}$ symmetry (as a mirror image of
each other) keeping STO in the middle of the heterostructure. We further
relaxed the heterostructure. The Brillouin zone integration was sampled over
uniform meshes of $3\times 3\times 1$. The vacuum layer was taken about 23 Å
along $c$ vector to separate between two successive slabs in the simulation
cell.

## Acknowledgement

The authors acknowledge XRD and wire bonding facilities of the Department of
Physics, IISc Bangalore. SM acknowledges funding support from a SERB Core
Research grant (Grant No. CRG/2022/001906). SM would also like to acknowledge
the seed funding from Quantum Research Park (QuRP), funded by Karnataka
Innovation and Technology Society (KITS), K-Tech, Government of Karnataka. NB
acknowledges funding from the Prime Minister’s Research Fellowship (PMRF),
MoE, Government of India. JM acknowledges UGC, India for fellowship. This
research used resources of the Advanced Photon Source, a U.S. Department of
Energy Office of Science User Facility operated by Argonne National Laboratory
under Contract No. DE-AC02-06CH11357. This research used resources of the
Advanced Light Source, which is a Department of Energy Office of Science User
Facility under Contract No. DE-AC02-05CH11231. ZY acknowledges the National
Natural Science Foundation of China (Grant No. 52202235).

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