# Density functional study of twisted graphene
$L1_{0}$-FePd heterogeneous interface

Mitsuharu Uemoto<EMAIL_ADDRESS>Department of Electrical and
Electronic Engineering, Graduate School of Engineering, Kobe University, 1-1
Rokkodai-cho, Nada-ku, Kobe 651-8501, Japan  Hayato Adachi  Department of
Electrical and Electronic Engineering, Graduate School of Engineering, Kobe
University, 1-1 Rokkodai-cho, Nada-ku, Kobe 651-8501, Japan  Hiroshi Naganuma
Center for Innovative Integrated Electronics Systems (CIES), Tohoku
University, 468-1 Aramaki Aza Aoba, Aoba, Sendai, Miyagi, 980-8572, Japan
Center for Spintronics Integrated Systems (CSIS), Tohoku University, 2-2-1
Katahira Aoba, Sendai, Miyagi 980-8577 Japan  Center for Spintronics Research
Network (CSRN), Tohoku University, 2-1-1 Katahira, Aoba, Sendai, Miyagi
980-8577 Japan  Graduate School of Engineering, Tohoku University,  6-6-05,
Aoba, Aoba-ku, Sendai, Miyagi, 980-8579, Japan Tomoya Ono  Department of
Electrical and Electronic Engineering, Graduate School of Engineering, Kobe
University, 1-1 Rokkodai-cho, Nada-ku, Kobe 651-8501, Japan

###### Abstract

Graphene on $L1_{0}$-FePd(001), which has been experimentally studied in
recent years, is a heterogeneous interface with a significant lattice symmetry
mismatch between the honeycomb structure of graphene and tetragonal alloy
surface. In this work, we report on the density functional study of its
atomic-scale configurations, electronic and magnetic properties, and
adsorption mechanism, which have not been well understood in previous
experimental studies. We propose various atomic-scale models, including simple
nontwisted and low-strain twisted interfaces, and analyze their energetical
stability by performing structural optimizations using the van der Waals
interactions of both DFT-D2 and optB86b-vdW functionals. The binding energy of
the most stable structure reached $E_{\mathrm{B}}=-0.22$ eV/atom for DFT-D2
($E_{\mathrm{B}}=-0.19$ eV/atom for optB86b-vdW). The calculated FePd-graphene
spacing distance was approximately 2 Å, which successfully reproduced the
experimental value. We also find out characteristic behaviors: the modulation
of $\pi$-bands, the suppression of the site-dependence of adsorption energy,
and the rise of  moiré-like  corrugated buckling. In addition, our atomic
structure is expected to help build low-cost computational models for
investigating the physical properties of $L1_{0}$ alloys/two-dimensional
interfaces.

Spintronics, First-principles, FePd

††preprint: APS/123-QED

##  I  Introduction

Iron-palladium [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] is a binary ordered
alloy with a tetragonal $L1_{0}$ structure that has attracted much attention
as a material for spintronic applications [1, 2] because of its high
perpendicular magnetic anisotropy (PMA) of $K_{u}\sim
10^{7}~{}\mathrm{erg}/\mathrm{cm}^{3}$[3, 4, 5], and low Gilbert damping of
thin films ($\alpha\sim 10^{-2}$) [8, 9]. These properties are desirable for
high-density magnetic random access memory (MRAM)[13] and magnetic tunneling
junction (MTJ)[2] devices. MgO is usually utilized as a barrier layer for
FePd-based junctions [14, 1]; however, the lattice mismatch approaches
$10~{}\%$ which poses a serious problem for a smooth interface and obtaining a
high tunnel magnetoresistance (TMR) ratio. Graphene is considered a potential
replacement for MgO layers. A recent experimental study successfully realized
the integration of graphene layers deposited on the $(001)$surface of FePd
(FePd$(001)$/Gr) using chemical vapor deposition (CVD) [6, 12]. It has been
reported that graphene protects Fe atoms on the surface from oxidation and
provides stable crystallinity, atomic thickness, and flatness control without
degrading the perpendicular magnetic properties. Graphene grown via CVD is
expected to have an energetically stable orientation when formed on tetragonal
FePd epitaxial films. In contrast to the MgO barrier and FePd interface, van
der Waals force bonding can be expected to form a flat interface between the
graphene and FePd layers. However, the actual atomic configuration of the C
atoms on the surface is not yet well understood.

Graphene on metal surfaces has been intensively studied both experimentally
and theoretically, particularly for lattice-matched fcc-metals (e.g.,
Ni(111)/Gr, Cu(111)/Gr, Al(111)/Gr, etc.) [15, 16, 17, 18, 19, 20, 21]. Their
$(111)$-surface has a threefold symmetry, which is in good agreement with the
honeycomb structure of graphene. C atoms are placed on characteristic
”adsorption sites” relative to metallic atoms [15]. The bonding mechanism of
metal-graphene interfaces can be divided into two regimes: physisorption and
chemisorption [16]. In the former case, graphene is weakly attracted to
metallic atoms via van der Waals (vdW) interactions. In the latter case, a
strong hybridization between the graphene $p\pi$-state and metal $d$-state
modifies the electronic band structure and shortens the graphene-metal spacing
distance.  For example , the typical physisorbed and chemisorbed spacings are
estimated as $\sim 3~{}\mathrm{\AA}$ and $\sim 2$ Å, respectively [16]. In
addition, interfaces with a small lattice constant mismatch (e.g.,
Ir(111)/Gr, Pt(111)/Gr, etc.) exhibit vertical buckling of the graphene,
reflecting their  moiré  superstructures[21, 22, 23], where the spacing varies
continuously between $2\sim 3$ Å.

FePd(001)/Gr exhibited a significant lattice symmetry mismatch. For the
tetragonal $L1_{0}$ structure, Fe (or Pd) atoms on the $(001)$ face are
arranged in a square lattice, which is distinct from that of graphene. The
atomic configuration at the interface and the bonding mechanism are not as
obvious as those for the fcc-metal surfaces; therefore, theoretical and
computational studies are desired.

This study investigates atomic-scale configurations, electronic and magnetic
properties, and adsorption mechanisms using density functional theory
(DFT)-based calculations. We propose a few designs of the interface structure:
simple non-twisted models (simple models) and low-strain twisted models
(twisted models). The former is superior in terms of computational costs, and
the latter is physically suitable. We performed structural optimization of
various interface models and analyzed their binding and strain energies. Their
influence on the geometrical parameters of graphene (bond length and angle)
reveals the reduction of graphene strain energy to obtain a stable structure,
and using exhaustive exploration, we find a few possible twisted models. The
binding energy of the most stable structure reaches -0.19 eV/atom (-18
kJ/mol), which is of the same order as that of Ni(111)/Gr. The structural and
magnetic properties of our model agree with recent experiments [6, 12]. In
addition, we also find out characteristic behaviors: the modulation of
$\pi$-bands, the suppression of the site-dependence of adsorption energy, and
the rise of  moiré -like corrugated buckling.

The remainder of this paper is organized as follows: in Sec. II, we present
the interface structure models and the computational conditions. In Sec. III,
we report the optimized energies for each structure and discuss their
electronic structures. Finally, in Sec. IV, we present the conclusion.

##  II  Method

Figure 1:  Conceptual illustration of simple non-twisted model. (a) Bulk
$L1_{0}$-FePd crystal structure and square Fe lattice on the $(001)$ plane.
(b) Bulk-graphene structure. The periodic units of both structures are marked
as gray-colored areas. (c) $S_{nm}$ interface structure consisting of $n$-FePd
units and $m$-graphene periodic units ($n=5,m=3$).

In this study, we propose two atomic-scale structural models of the
FePd(001)/Gr interface: a simple non-twisted model (simple model) and a low-
strain twisted model (twisted model). The ”simple model” is designed to reduce
the number of atoms per supercell, which can significantly reduce the
computational time. However, there is a possibility of a large tensile strain
due to the deformation of graphene. The ”twisted model” is a more generalized
interface that accounts for twist angles. Although it enables a decrease in
strain by using adequate supercells, a large computational effort is usually
required.

###  II.1 Interface models

####  II.1.1 Simple non-twisted interface model (Simple model)

Figure 1(a) shows the bulk FePd unit cell with $L1_{0}$-ordered structure,
which contains single Fe and Pd atoms. The lattice parameters are
$a_{\mathrm{FePd}}=2.67~{}\mathrm{\AA}$ and
$c_{\mathrm{FePd}}=3.70~{}\mathrm{\AA}$, which were determined from the
structural optimization under equilibrium conditions, in good agreement with
previous experimental and theoretical studies [7, 10, 11]. Figure 1(b)
illustrates the honeycomb lattice structure of graphene; the gray rectangular
area is a single unit cell containing four C atoms in each cell. The in-plane
interatomic distance is $a_{\mathrm{Gr}}=1.42~{}\mathrm{\AA}$. Owing to the
difference in lattice symmetry between FePd and graphene, the atomic interface
structures were not obvious.

Figure 2:  Initial structure of $S_{nm}$-type FePd(001)/Gr interface model.
There are seven atomic layers of FePd, a single graphene layer, and an
$8.0~{}\mathrm{\AA}$ vacuum region. The Fe-C interlayer distance is assumed to
be $2~{}\mathrm{\AA}$ in the initial condition.

As seen Fig. 1(c), the armchair (AC) and zigzag (ZZ) directions of the
graphene layer are parallel to the $[100]$ and $[010]$ directions of FePd,
respectively. This structure is constructed based on the following guiding
principles:

  1. 1.

Owing to the attractive interaction between Fe and C atoms, the arrangement in
which these atoms are considered stable.

  2. 2.

The honeycomb structure of graphene on the surface should not have significant
distortions.

The period of the graphene ZZ direction is
$\sqrt{3}a_{\mathrm{Gr}}=2.46~{}\mathrm{\AA}$. However, because the period in
the AC direction, $3a_{\mathrm{Gr}}=4.26~{}\mathrm{\AA}$, has a significant
mismatch with $a_{\mathrm{FePd}}$, a long-period supercell structure is
required in this direction. We refer to the interface structure consisting of
$n$-period FePd units and $m$-period graphene units as $S_{n,m}$ [see Fig.
1(c)]. This design can be characterized by a pair of integers $n$ and $m$.

To evaluate the stability of the $S_{n,m}$ interface structure, we considered
slab supercells as shown in Fig. 2, which consists of seven atomic layers of
FePd with a graphene layer terminating on the top side of the slab. The bottom
two layers were fixed to simulate the bulk properties during structural
optimization. We also used a vacuum layer of approximately
$8.0~{}\mathrm{\AA}$, which was sufficiently large to truncate the
interactions under periodic boundary conditions. The initial spacing between
the top Fe layer and graphene ($d_{\mathrm{FeC}}$) is determined to be
$2.0~{}\mathrm{\AA}$. The diameter of the $S_{n,m}$ supercell is
$2.67n~{}\mathrm{\AA}\times 2.67~{}\mathrm{\AA}\times 21.1~{}\mathrm{\AA}$,
and the numbers of Fe, Pd, and C atoms are $4n$, $3n$, and $4m$, respectively.

####  II.1.2 Low-strain twisted interface model (Twisted model)

Figure 3:  Illustration of low-strain twisted interface model represented as
$T_{n_{1},n_{2},m_{11},m_{12},m_{21},m_{22}}$; as an example,
$T_{4,8,-5,2,-1,7}$ structure is depicted. (a) $n_{1}\times n_{2}$ square
lattice of top-most Fe layer. (b) twisted graphene layer. The green arrows and
gray area represent the primitive unit vectors and cell, respectively. The
blue arrows ($\vec{s}_{1}$ and $\vec{s}_{2}$) are supercell lattice vectors,
and the red arrows are the AC and ZZ axes.

In the twisted model shown in Fig. 3, we considered the interface structure
with  the twist angle between FePd and graphene. The initial arrangement of C
atoms was generated by linear transformations involving rotation and resizing
onto the original graphene lattice.

Here, we consider the C atom coordinates of freestanding graphene
$\vec{r}^{\mathrm{(c)}}$ as follows:

$\displaystyle\vec{r}^{\mathrm{(c)}}_{k}=$ $\displaystyle
k_{1}\vec{a}^{\mathrm{(Gr)}}_{1}+k_{2}\vec{a}^{\mathrm{(Gr)}}_{2}+\vec{\tau}^{\mathrm{(Gr)}}_{k_{3}}$
(1)

where $k\equiv(k_{1},k_{2},k_{3})$ are integers, $\vec{a}^{\mathrm{(Gr)}}_{1}$
and $\vec{a}^{\mathrm{(Gr)}}_{2}$ are primitive unit vectors of
$(3a_{\mathrm{Gr}}/2,\pm\sqrt{3}a_{\mathrm{Gr}}/2)$ (green arrows in Fig.
3(b)), and $\vec{\tau}^{\mathrm{(Gr)}}_{1}$ and
$\vec{\tau}^{\mathrm{(Gr)}}_{2}$ are the sublattice vectors of $(0,0)$ and
$(\vec{a}^{\mathrm{(Gr)}}_{1}+\vec{a}^{\mathrm{(Gr)}}_{2})/3$, respectively.
In the presence of a twist angle, the transformed C atom coordinates
$\vec{r}^{\mathrm{(C)}\prime}_{k}$ are given by

$\displaystyle\vec{r}^{\mathrm{(C)}\prime}_{k}=$
$\displaystyle\hat{\mathcal{A}}\vec{r}^{\mathrm{(C)}}_{k}$ (2)

with $2\times 2$ transform matrix, $\hat{\mathcal{A}}$ (Affine map). As seen
in Fig. 3(a), we assume a supercell including $n_{1}\times n_{2}$ square Fe
lattice; the unit vectors are represented as
$\vec{s}_{1}=n_{1}\vec{a}^{\mathrm{FePd}}_{1}$ and
$\vec{s}_{2}=n_{2}\vec{a}^{\mathrm{FePd}}_{2}$ (blue arrows) with
$\vec{a}^{\mathrm{FePd}}_{1}=(a_{\mathrm{FePd}},0)$ and
$\vec{a}^{\mathrm{FePd}}_{2}=(0,a_{\mathrm{FePd}})$ (green arrows in Fig. 3).
Because the Fe and C layers have the same periodicity, the following
conditions are required:

$\displaystyle\left\\{\begin{aligned}
\vec{s}_{1}=&m_{11}\hat{\mathcal{A}}\vec{a}^{\mathrm{(Gr)}}_{1}+m_{12}\hat{\mathcal{A}}\vec{a}^{\mathrm{(Gr)}}_{2}\\\
\vec{s}_{2}=&m_{21}\hat{\mathcal{A}}\vec{a}^{\mathrm{(Gr)}}_{1}+m_{22}\hat{\mathcal{A}}\vec{a}^{\mathrm{(Gr)}}_{2}\end{aligned}\right.\;,$
(3)

where $m_{11}$, $m_{12}$, $m_{21}$., and $m_{22}$ are the integers. By solving
the linear system of equations, we obtain

$\displaystyle\left\\{\begin{aligned}
\hat{\mathcal{A}}\vec{a}^{\mathrm{(Gr)}}_{1}=&\mu_{11}n_{1}\vec{a}^{\mathrm{Fe}}_{1}+\mu_{12}n_{2}\vec{a}^{\mathrm{Fe}}_{2}\\\
\hat{\mathcal{A}}\vec{a}^{\mathrm{(Gr)}}_{2}=&\mu_{21}n_{1}\vec{a}^{\mathrm{Fe}}_{1}+\mu_{22}n_{2}\vec{a}^{\mathrm{Fe}}_{2}\end{aligned}\right.$
(4)

with

$\displaystyle\begin{pmatrix}\mu_{11}&\mu_{12}\\\
\mu_{21}&\mu_{22}\end{pmatrix}=$ $\displaystyle\begin{pmatrix}m_{11}&m_{12}\\\
m_{21}&m_{22}\end{pmatrix}^{-1}\;.$ (5)

Therefore, we have

$\displaystyle\hat{\mathcal{A}}=$
$\displaystyle\left(\mu_{11}n_{1}\vec{a}^{\mathrm{Fe}}_{1}+\mu_{12}n_{2}\vec{a}^{\mathrm{Fe}}_{2}\right)\left(\vec{b}^{\mathrm{(Gr)}}_{1}\right)^{\mathrm{T}}$
$\displaystyle+\left(\mu_{21}n_{1}\vec{a}^{\mathrm{Fe}}_{1}+\mu_{22}n_{2}\vec{a}^{\mathrm{Fe}}_{2}\right)\left(\vec{b}^{\mathrm{(Gr)}}_{2}\right)^{\mathrm{T}}$
(6)

where $\vec{b}^{\mathrm{(Gr)}}$ are reciprocal vectors of
$\vec{a}^{\mathrm{(Gr)}}$. The above formalism provides a universal
description of the twisted interface, which can be characterized by six
integers: ${n_{1},n_{2},m_{11},m_{12},m_{21},m_{22}}$. Therefore, we represent
the structure as $T_{n_{1},n_{2},m_{11},m_{12},m_{21},m_{22}}$; as an example,
Fig. 3(a)–(b) shows $T_{4,8,-5,2,-1,7}$ model. Similar to Fig. 2, slab
supercell structures containing the twisted graphene layer on seven atomic
layers of FePd were employed in the calculations.

###  II.2 Computation

For computation, we performed density functional theory (DFT) calculations
using the Vienna ab initio simulation package (VASP) 6.2, which uses the
projector augmented wave (PAW) method [24] and a plane-wave basis set. We also
employed the generalized gradient approximation (GGA) exchange-correlation
functional of the Perdew–Burke–Ernzerhof (PBE) [25] with Grimme’s DFT-D2
method for vdW interactions [26]. Besides, for comparison, we also use
optB86b-vdW functional [27, 28]. We used a plane-wave basis cutoff energy of
$400~{}\mathrm{eV}$. The first Brillouin zone was sampled using Gamma-centered
$k$-point grids.  The magnetization is treated by spin-collinear calculations.
In the case of the simple model $S_{n,m}$, we used $n_{k}\times 9\times 1$
grids, where $n_{k}$ is the smallest integer satisfying $n_{k}\times n\geq 9$.
For the twisted model, we used $n_{k_{1}}\times n_{k_{2}}\times 1$ with
$n_{k_{i}}\times n_{i}\geq 9$.

##  III  Result and Discussion

Here, we consider a few different ”simple model” interface structures
($S_{3,2}$, $S_{4,2}$, $S_{4,3}$, $S_{5,3}$, $S_{5,4}$, and $S_{7,4}$). To
determine their stability, we introduce the binding energy $E_{\mathrm{B}}$
using the following formula:

$\displaystyle E_{\mathrm{B}}(n,m)=$
$\displaystyle\frac{E_{\mathrm{FePd/Gr}}(n,m)-E_{\mathrm{FePd}}(n)-E_{\mathrm{Gr}}(m)}{n_{\mathrm{C}}(m)}$
(7)

with the number of C atoms $n_{\mathrm{C}}(m)=4m$, where
$E_{\mathrm{FePd/Gr}}(n,m)$ is the total energy of the relaxed FePd(001)/Gr
slab structure and $E_{\mathrm{FePd}}(n)$ and $E_{\mathrm{Gr}}(m)$ are that of
the pristine FePd slab and the pristine free-standing graphene in the
equilibrium conditions, respectively. Additionally, we calculated the strain
energy $E_{\mathrm{S}}$ as follows:

$\displaystyle E_{\mathrm{S}}(n,m)=$
$\displaystyle\frac{\tilde{E}_{\mathrm{Gr}}(n,m)-E_{\mathrm{Gr}}(m)}{n_{\mathrm{C}}(m)}\;,$
(8)

where $\tilde{E}_{\mathrm{Gr}}(n,m)$ is the energy of deformed graphene with
the same C atom coordinates as $E_{\mathrm{FePd/Gr}}(n,m)$.

Table 1:  Geometrical parameters of the initial atomic configuration of the simple models ($S_{n,m}$) and the calculated energies of the relaxed structures: the mean absolute error (MAE) of the C atoms’ bond length $a_{\mathrm{C}}$ and angle $\theta_{\mathrm{C}}$ from the equilibrium free-standing graphene ($1.42~{}\text{\AA}$ and $120^{\circ}$) are indicated as MAE $a_{\mathrm{C}}/a_{\mathrm{Gr}}$ and MAE $\theta_{\mathrm{C}}$. The binding energies $E_{\mathrm{B}}$, strain energies $E_{\mathrm{S}}$, and spacing distances $d_{\mathrm{FeC}}$ are calculated by both of DFT-D2 and optB86b-vdW (the values in parentheses) functionals.  | Model | $n/m$ | $\mathrm{MAE}~{}a_{\mathrm{C}}/a_{\mathrm{Gr}}$ | $\mathrm{MAE}~{}\theta_{\mathrm{C}}$ |  $E_{\mathrm{B}}$ (eV/atom) |  $E_{\mathrm{S}}$ (eV/atom) |  $d_{\mathrm{FeC}}$ (Å)
---|---|---|---|---|---|---|---
a | $S_{3,2}(\sim S_{6,4})$ | $1.50$ | $15~{}\%$ | $6^{\circ}$ | $-0.02~{}(+0.01)$ | $+0.25~{}(+0.25)$ | 2.04 (2.04)
b | $S_{4,2}(\sim S_{2,1},S_{6,3})$ | $2.00$ | $28~{}\%$ | $6^{\circ}$ | $+0.24~{}(+0.25)$ | $+1.47~{}(+1.47)$ | N/A
c | $S_{4,3}$ | $1.33$ | $16~{}\%$ | $10^{\circ}$ | $+0.27~{}(+0.29)$ | $+0.68~{}(+0.67)$ | N/A
d | $S_{5,3}$ | $1.67$ | $16~{}\%$ | $2^{\circ}$ | $-0.11~{}(-0.09)$ | $+0.25~{}(+0.25)$ | $1.98~{}(1.99)$
e | $S_{5,4}$ | $1.25$ | $17~{}\%$ | $12^{\circ}$ | $+0.38~{}(\color[rgb]{0,0,1}+0.27)$ | $+0.63~{}(+0.63)$ | N/A
f | $S_{7,4}$ | $1.75$ | $19~{}\%$ | $0.4^{\circ}$ | $+0.00~{}(+0.03)$ | $+0.44~{}(+0.44)$ | 1.96 (1.96)

Table 1 shows $E_{\mathrm{B}}$ and $E_{\mathrm{S}}$ values of $S_{n,m}$
models, and for comparison, we tested both the DFT-D2 and optB86b-vdW
functionals. In the DFT-D2 results, most of the structures have a positive
$E_{\mathrm{B}}$ value, which is energetically unstable, and only two
structures have a negative $E_{\mathrm{B}}$. $S_{5,3}$ is the most stable,
with $E_{\mathrm{B}}=-0.11$ eV/atom. In the optB86-vdW results, $S_{5,3}$ is
also stable, with $E_{\mathrm{B}}=-0.09$ eV/atom. $S_{5,3}$ is the structure
with the smallest $E_{\mathrm{S}}=0.25$ eV/atom, and the magnitude of
$E_{\mathrm{S}}$ is comparably large to $E_{\mathrm{B}}$. We expect that
strain in graphene is essential for the stability of these simple models.

Figure 4:  Estimated modulation of the bond length and bond angle of the
graphene layer in the $S_{nm}$ structure. (a) Definition of bond length
$a^{(1)}_{\mathrm{C}}$ , $a^{(2)}_{\mathrm{C}}$, and bond angle
$\theta^{(1)}_{\mathrm{C}}$ , $\theta^{(2)}_{\mathrm{C}}$. (b)
$a^{(1)}_{\mathrm{C}}$ and $a^{(2)}_{\mathrm{C}}$ as functions of the $n/m$
ratio. (c) $\theta^{(1)}_{\mathrm{C}}$ and $\theta^{(2)}_{\mathrm{C}}$ as
functions of the $n/m$ ratio.

Moreover, $d_{\mathrm{FeC}}$ in Table 1 represents a distance between the
topmost Fe and C layer. For some of the energetically unstable
($E_{\mathrm{B}}\gg 0$) interface models, C atoms are not adsorbed on metal
surfaces, or not possible to construct a flat atomic surface. For example, in
the case of $S_{5,4}$, the vertical distances from the Fe surface to the
nearest / farthest C atom are widely distributed between $1.5\sim 4.2$ Å. [See
S1 in the supplementary material] Therefore, for a few unstable structures,
$d_{\mathrm{FeC}}$ values in Table 1 are indicated as N/A.

We then consider the bond length and bond angle distributions in the initial
state of the optimization. Here, we consider the initial coordinates of C
atoms in $S_{n,m}$. The bond lengths are represented by two lengths
$a_{\mathrm{C}}^{(1)}$ and $a_{\mathrm{C}}^{(2)}$, as shown in Fig. 4(a);
$a_{\mathrm{C}}^{(1)}$ and $a_{\mathrm{C}}^{(2)}$ can be analytically
expressed as functions of the ratio $n/m$:

$\displaystyle a_{\mathrm{C}}^{(1)}=$
$\displaystyle\frac{1}{3}\left(\frac{n}{m}\right)a_{\mathrm{FePd}}$ (9)
$\displaystyle a_{\mathrm{C}}^{(2)}=$
$\displaystyle\sqrt{\frac{1}{4}+\frac{1}{36}\left(\frac{n}{m}\right)^{2}}a_{\mathrm{FePd}}\;.$
(10)

In Fig. 4(b), we plotted the $n/m$ dependence of $a_{\mathrm{C}}^{(1)}$ and
$a_{\mathrm{C}}^{(2)}$. As the equilibrium bond length of graphene is
approximately $a_{\mathrm{Gr}}=1.42~{}\mathrm{\AA}$,  to reduce  the strain,
the value of $n/m$ should be approximately $\approx 1.5$. Using a similar
procedure, the bond angles $\theta_{\mathrm{C}}^{(1)}$ and
$\theta_{\mathrm{C}}^{(2)}$ shown in Fig. 4(a), can be calculated using the
following equation:

$\displaystyle\theta_{\mathrm{C}}^{(1)}=$
$\displaystyle\cos^{-1}\left[-\frac{(n/m)}{6\sqrt{1/4+(1/36)(n/m)^{2}}}\right]$
(11) $\displaystyle\theta_{\mathrm{C}}^{(2)}=$ $\displaystyle
2\cos^{-1}\left[\frac{(n/m)}{6\sqrt{1/4+(1/36)(n/m)^{2}}}\right]\;.$ (12)

Figure3(c) shows the n/m dependence of $\theta_{\mathrm{C}}^{(1)}$ and
$\theta_{\mathrm{C}}^{(2)}$; the angle approaches $120^{\circ}$ at
approximately $n/m\approx 1.7$. This result indicates that the distortion of
the graphene lattice is minimized in the n/m range of $1.5\sim 1.7$, which is
consistent with Table 1. This trend can be attributed to the strain in the
honeycomb structure of graphene.

Figure 5:  Potential energy surface for lateral shifts of the graphene layer
on FePd(001). (a) Schematic illustration. $\Delta{x}$ and $\Delta{y}$
represent the displacements of the $\mathrm{C}_{1}$ atom from the top of a
single Fe site. (b) Dependence of the binding energy difference $\Delta{E}$ on
$(\Delta{x},\Delta{y})$.  ($S_{5,3}$-type model is employed in this
calculation)

Next, we calculated the potential energy surface (PES) profile: adsorption-
energy dependency on the lateral shift of the graphene layer with respect to
that of the topmost Fe layer. Figure 5(a) shows a schematic illustration.
$\Delta{x}$ and $\Delta{y}$ represent the magnitudes of the displacements of
$\mathrm{C}_{1}$ atom from Fe atom. Structural optimizations were performed by
fixing the lateral position of the $\mathrm{C}_{1}$ atom and obtaining the
binding energy as a function of displacement
$E_{\mathrm{B}}(n,m;\Delta{x},\Delta{y})$.  We use $S_{5,3}$-type interface
model.  Figure 5(b) plots $\Delta{E}(\Delta{x},\Delta{y})$ defined as:

$\displaystyle\Delta{E}(\Delta{x},\Delta{y})=$ $\displaystyle
E_{\mathrm{B}}(n,m;\Delta{x},\Delta{y})-E_{\mathrm{B}}(n,m;0,0)\;.$ (13)

Equivalent points appear periodically in the PES profile because of the
translational symmetry of the system. An energetically stable minimum appears
in the vicinity of $(\Delta{x},\Delta{y})=(0,0)$, which corresponds to the
case when a single C atom is at the top of the Fe site, and $\Delta{E}$
becomes unstable when every C atom is equally distant from the Fe atoms. This
sub-$a_{\mathrm{Fe}}$ scale change is a result of the attractive short-range
interaction between Fe and C atoms, which reflects chemisorption. In contrast,
$\Delta{E}$ did not exceed $0.006~{}\mathrm{eV/atom}$, which is less than 5%
of the binding energy. This 5 % change is below the typical value obtained for
a chemisorbed interface such as Ni(111)/Gr [16]. This behavior was similar to
that of the physisorbed case;  it will be also discussed in the last section
of this paper.

Figure 6:  Charge density profile of the $S_{5,3}$-type FePd(001)/Gr
interface. $\mathrm{C}_{1}$ and $\mathrm{Fe}_{1}$ represent the closest pair,
and $\mathrm{C}_{2}$ and $\mathrm{Fe}_{2}$ are examples of sites with no close
C or Fe atoms.

Figure 6 shows the electron-density profile of the $S_{5,3}$ structure. A
high-density area is found between labeled $\mathrm{C}_{1}$ and the
neighboring C atom, which represents a C-C $\mathrm{sp}^{2}\sigma$ bond
orbital. Although most $\mathrm{C}-\mathrm{Fe}$ bonds are not seen as
$\mathrm{C}-\mathrm{C}$, there is an area of slightly higher density between
$\mathrm{C}_{1}$ and $\mathrm{Fe}_{1}$ (displayed by the broken circle), which
is one of the closest Fe-C pairs. In addition, the average Fe-C binding
distance is approximately $2~{}\mathrm{\AA}$, which is a typical value for
chemisorbed metal/Gr interfaces [16, 29].

Table 2:  Geometrical parameters and energies of the twisted FePd(001)/Gr models. The size of supercell $n_{1}\times n_{2}$, and the twisted angle $\theta^{\mathrm{(AC)}}_{\mathrm{twist}}$ and $\theta^{\mathrm{(ZZ)}}_{\mathrm{twist}}$ are explained in Fig. 3. $n_{\mathrm{C}}$ is the number of C atoms in the supercell. similar to the table 1, $E_{\mathrm{B}}$, $E_{\mathrm{S}}$, and $d_{\mathrm{FeC}}$ represents the binding energy, strain energy and the spacing distance calculated by using DFT-D2 and optB86b-vdW (the values in parentheses) functionals.  | Model | $n_{1}\times n_{2}$ | $n_{\mathrm{C}}$ | $\theta_{\mathrm{twist}}^{\mathrm{(AC)}},\theta_{\mathrm{twist}}^{\mathrm{(ZZ)}}$ | $\mathrm{MAE}~{}a_{\mathrm{C}}/a_{\mathrm{Gr}}$ | $\mathrm{MAE}~{}\theta_{\mathrm{C}}$ |  $E_{\mathrm{B}}$ (eV/atom) |  $E_{\mathrm{S}}$ (eV/atom) |  $d_{\mathrm{FeC}}$ (Å)
---|---|---|---|---|---|---|---|---|---
g | $T_{4,6,-5,2,-1,7}$ | $4\times 6$ | $66$ |  $6.5^{\circ}$, $7.5^{\circ}$ | $0.5~{}\%$ | $0.5^{\circ}$ | $-0.20~{}(-0.16)$ | $0.02~{}(0.02)$ | $2.11~{}(2.10)$
h | $T_{4,7,-5,2,-1,8}$ | $4\times 7$ | $76$ |  $6.8^{\circ}$, $6.4^{\circ}$ | $0.3~{}\%$ | $0.3^{\circ}$ | $-0.21~{}(-0.18)$ | $0.01~{}(0.01)$ | $2.08~{}(2.08)$
i | $T_{4,8,-5,2,-1,8}$ | $4\times 8$ | $86$ |  $6.9^{\circ}$, $5.7^{\circ}$ | $0.5~{}\%$ | $0.8^{\circ}$ | $-0.22~{}(-0.19)$ | $0.02~{}(0.02)$ | $2.07~{}(2.07)$
Figure 7:  Radial distribution function of the C-Fe bonds for the stable
structures of simple model (a) and twisted models (b). The typical distances
for chemisorption ($\sim 2~{}\mathrm{\AA}$) and physisorption ($\sim
3~{}\mathrm{\AA}$) are labeled for convenience.  Figure 8:  Height profile of
the Gr layer of optimized simple model $S_{5,3}$ (a) and the twisted model
$T_{4,8,-5,2,-1,8}$ (b). The black points represent lateral ($xy$-) position
of C atoms, and the colormaps show their vertical ($z$-) position displacement
from the average level.  The white lines indicates the periodicity of the
supercells.  Figure 9:  Spin-polarized  density of states (DoS) and partial
DoS (pDoS) profiles: (a) DoS for the pristeine FePd (b) pDoS of topmost Fe(3d)
for the pristeine FePd (c) pDoS of C(s) and C(p) for free-standing graphene.
(d-f) DoS and pDoS for FePd(001)/Gr interface of the simple $S_{5,3}$ model
[illustrated in Fig. 1(c)]. (g-i) FePd(001)/Gr interface of the twisted
$T_{4,8,-5,2,-1,8}$ model [illustrated in Fig. 3].

Next, we consider low-strain twisted interface models. The twisted model
$T_{n_{1},n_{2},m_{11},m_{12},m_{21},m_{22}}$ has six individual parameters
and numerous structures. We exhaustively searched for every possible set of
$n_{1}$, $n_{2}$, $m_{11}$, $m_{12}$, $m_{21}$, and $m_{22}$ with the
restriction that supercell size $n_{1}\times n_{2}$ is smaller than $9\times
9$. From the analysis of $S_{nm}$ model, we assumed that a screening condition
that reduces the strain in the initial structure is essential for stability.
In Table 2, we selected three structures with minimal MAE
$a_{\mathrm{C}}/a_{\mathrm{Gr}}$: $T_{4,6,-5,2,-1,7}$, $T_{4,7,-5,2,-1,8}$ and
$T_{4,8,-5,2,-1,8}$ models, for convenience, which are also represented as g,
h, and i, respectively. Model i, illustrated in Fig. 3, is the most stable
structure among the three cases. $E_{B}$ was $-0.22$ eV/atom ($\approx-21$
kJ/mol) in DFT-D2 and $-0.19$ ($\approx-18$ kJ/mol) in optB86b-vdW. Because
this value is of the same order as that of the well-known chemisorbed
Ni(111)/Gr ($\approx-12$ kJ/mol) [16], the adsorption state is considered very
stable. The above twisted models can dramatically decrease strain; in model g,
h and i, MAE $a_{\mathrm{C}}/a_{\mathrm{Gr}}$ is lesser than 0.5 %, and
$E_{\mathrm{S}}$ is $0.01\sim 0.02$ eV/atom. Although
$\mathrm{MAE}~{}a_{\mathrm{C}}/a_{\mathrm{Gr}}$ and
$\mathrm{MAE}~{}\theta_{\mathrm{C}}$ became larger in the order of model
$\text{g}<\text{h}<\text{i}$, model i is the most stable amongst the three. In
such a low-strain model, the Fe-C interaction is more essential than MAEs.

Next, we considered the radial distribution function of C-Fe bonds, which is
defined as follows:

$\displaystyle d(r)$
$\displaystyle=\frac{1}{n_{\mathrm{c}}}\sum_{k}^{\mathrm{(Fe)}}\sum_{k^{\prime}}^{\mathrm{(C)}}\delta\left(r-\|\vec{r}^{\mathrm{(Fe)}}_{k}-\vec{r}^{\mathrm{(C)}}_{k^{\prime}}\|\right)\;,$
(14)

where $\vec{r}^{\mathrm{(Fe)}}_{k}$ and $\vec{r}^{\mathrm{(C)}}_{k^{\prime}}$
represent the coordinates of $k$ and $k^{\prime}$-th topmost Fe and C atoms,
respectively. For computational convenience, we used a Gaussian of
$\sigma=0.1~{}\mathrm{\AA}$ as the $\delta$-function above. Figure 7(a) shows
$d(r)$ of the stable $S_{5,3}$ model (d in Table. 1). $d(r)$ has a broad band
from $2~{}\mathrm{\AA}$ to $3~{}\mathrm{\AA}$. Generally, the spacing of the
typical chemisorbed fcc-metal/Gr interface is approximately $2~{}\mathrm{\AA}$
and that of the physisorbed interface is approximately $3~{}\mathrm{\AA}$. In
the case of FePd(001)/Gr, the averaging of the interatomic interactions
between C and Fe resulted in an intermediate behavior between chemisorption
and physisorption. Similar behavior was observed in the twisted model [see
Fig. 7(b)]. d(r) around $r=2$ Å  becomes  slightly larger in the order of the
model $\text{g}<\text{h}<\text{i}$, representing the number of Fe-C pairs at a
typical chemisorption distance. This trend is consistent with $E_{\mathrm{B}}$
in Table 2; in the low-strain case, Fe-C inter-atomic interaction is also
essential.

As seen in Fig. 6, the height of the C atoms is not flat but rather buckled
vertically. Generally, in the typical fcc-metal/Gr case, buckled graphene is
often observed at the interface with a  moiré  structure by small lattice
constant mismatches, such as Ir(111)/Gr and Pt(111)/Gr. Their buckling
amplitudes are usually more than $1~{}\text{\AA}$ [21] and the  moiré  pattern
because of the long-period (more than a few nanometers) hexagonal lattice
[22]. For FePd(001)/Gr, the colormaps in Fig. 8 shows the height profile of
the graphene layer in the optimized structures. Surprisingly, well-defined
vertical corrugations of approximately $0.4\sim 0.5~{}\text{\AA}$ were
observed. Unlike the  moiré  pattern, the model in our present study has a
stripe-like pattern instead of a hexagonal pattern owing to the lattice
symmetry mismatch. Some C atoms, which are close to the top Fe sites, feel a
relatively stronger attractive interaction than any other C atoms. This
inhomogeneity results in buckling, but their amplitudes are largely suppressed
by averaging the interatomic interactions. The C atom at the origin in Fig. 8,
which is situated on top of the Fe sites, is in the valley region of the
corrugation.

Figure 9 presents the density of states (DoS) and partial DoS (pDoS) profiles.
The plots (a)–(c) represent DoS and pDoS for pristine FePd and graphene.
Figure 9 (d)–(f) and Figure 9 (g)–(i) represent the simple and twisted
interface models, respectively. In Fig. 9 (b), the pDoS of Fe(d) exhibits
well-defined exchange splitting representing ferromagnetic order, and no
significant difference between Fig. 9 (b), (e), and (h). Because this behavior
does not change, as shown in Fig. 9(e) and (h), the ferromagnetic properties
are not degraded by the presence of graphene, which is consistent with recent
experimental measurements [6]. As shown in Fig. 9(c), the free-standing
graphene has a characteristic $\mathrm{p}\pi$ band edge, the so-called  Dirac
core , where pDoS becomes zero at $E_{\mathrm{F}}$. For FePd(001)/Gr, as shown
in Fig. 9(f) and (i), graphene has a hybrid state of $\mathrm{p}\pi$ and spin-
polarized d-bands around $E_{\mathrm{F}}$. This orbital hybridization
originates from the chemical bonds between C and Fe, which leads to the
induced magnetization of graphene, which is often observed in the graphene-
ferromagnet interfaces [30, 19].

Figure 10:  Local magnetic moment on C atom (a) and topmost Fe atom (b) as a
function of distance to nearest Fe (or C) atom. Both of the simple $S_{5,3}$
model (blue point) and twisted $T_{4,8,-5,2,-1,8}$ model (orange triangle) are
also plotted. The blue line indicates that on the Fe atom in bulk FePd.

Figure 10(a) shows the local magnetic moment of C atoms as a function of the
distance to the nearest Fe atoms. The simple model (blue points) and twisted
model (orange triangles) indicate the same distribution. The induced magnetic
moment $\mu$ of C atoms far from Fe atoms is positive (the majority spin),
which is the same polarization component of Fe(3d); this behavior results from
charge transfer from the substrate to graphene. In contrast, when the C atom
nears the Fe atom, the magnetic moment gradually changes to the minority spin
side. This is owing to the presence of minority spin d-states near $E_{F}$
[see Fig. 9(b)], where stronger hybridization increases the minority spin pDoS
of $\mathrm{p}\pi$ band at $E_{\mathrm{F}}$. The total magnetic moments of the
graphene layer, because the contribution from each C atom is averaged,
approaches a small value $|\mu|<0.02\mu_{\mathrm{B}}$, which is below that of
typical graphene on ferromagnetic metals  for example  Ni(111)/Gr of
$\mu=0.05\mu_{\mathrm{B}}\sim 0.10\mu_{\mathrm{B}}$ [30] . The local magnetic
moment of the topmost Fe atoms is plotted in Fig. 10(b). The decrease in the
distance to the nearest C atom led to a decrease in the magnetic moment from
that of a Fe atom in bulk FePd ($\mu\approx
2.87\color[rgb]{0,0,1}\mu_{\mathrm{B}}$). Similar behavior has been
theoretically predicted for Ni(111)/Gr [19].

Figure 11:  Comparison of the computational results and observation by the
electron microscopy. (a) STEM image of trilayer graphene on FePd interface.
Detailed STEM images can be also seen in Ref [12] (b) Estimated interlayer
distances of single-layer, AB-stacked bilayer, and ABA- and ABC-stacked
trilayer graphene on FePd.  The functionals of DFT-D2 and optB86b-vdW (the
values in parentheses) are employed in this calculation.

Finally, figure 11(a) shows the cross-sectional scanning tunneling electron
microscopy (STEM) images of a sample in which three layers of graphene are
formed on an epitaxial FePd film. Three complementary detection modes are
presented: BF, ABF, and HAADF. From the BF and ABF-STEM images, it can be
confirmed that the interatomic distance between the first layer of graphene
and FePd is approximately $1.9~{}\text{\AA}$, whereas the upper layers of
graphene are approximately $4~{}\text{\AA}$ thick. The HADDF-STEM image shows
that Fe and Pd are alternately aligned in the out-of-plane direction,
indicating a highly ordered $L1_{0}$ structure. The fact that the top surface
layer is Fe is in agreement with the model shown in Fig. 2. In Fig. 11(b), we
show the optimized interatomic distances of single-layer, bilayer, and
trilayer graphene on a FePd slab.  The distance between the lower graphene and
Fe layers ($d_{\mathrm{FeC}}$) is close to that of the experimentally observed
value. The interlayer distances of the graphite-graphite layer
($d_{\mathrm{CC1}}$ and $d_{\mathrm{CC2}}$) is about $3.3~{}\text{\AA}$
(details are given in the supplementary material S2-S3) has discrepancy with
experimental results.

##  IV  Summary

This study investigated the atomic-scale structures, electronic and magnetic
properties, and adsorption mechanism of the FePd(001)/Gr heterogeneous
interface. We proposed various atomic-scale models, including simple
nontwisted and low-strain twisted interfaces. The ”simple model” is designed
to consist of a small number of atoms, thereby saving computational costs. It
provides an energetically (meta)stable structure; however, it has
unrealistically large strain energy. The later ”twisted model” requires large
supercells containing many atoms, but the strain can be greatly reduced. We
explored possible interface structures that are smaller than $9\times 9$
supercell structures with lower strain (small MAE $a_{\mathrm{C}}$). The
optimized structure had a binding energy of $E_{\mathrm{B}}=-0.22$ eV/atom in
DFT-D2 ($E_{\mathrm{B}}=-0.19$ eV/atom in optB86b-vdW), which is comparable in
order of magnitude to that of other well-understood metal/Gr interfaces. The
stability is governed by the strain of graphene and– C-Fe interatomic
attraction. Additionally, more stable structures are likely to exist in larger
supercells (more than $10\times 10$). However, the present structure contains
more than 300 atoms and larger supercell models are computationally difficult.
Further, because the strain energy $E_{S}$ in the present structure is already
sufficiently small ($E_{S}\leq 0.02$ eV/atom), no significant $E_{B}$
improvements are expected even in the larger supercells.

The potential energy surface (PES) profiles shown in Fig. 5 shows that the
change in $E_{\mathrm{B}}$ is small ($\leq 5~{}\%$) for atomic-scale lateral
shifts of graphene, which corresponds to the robustness of the absorption. The
flatness of PES also indicates suppression of the site-dependence of
$E_{\mathrm{B}}$. Typical lattice-matched interfaces (e.g., Ni(111)/Gr), owing
to the fixed relative positions between the C and metal atoms, have sharp
minimum points in the PES, resulting in characteristic adsorption sites. For
FePd(001)/Gr with a lattice symmetry mismatch, the relative positions of each
Fe-C were uniformly distributed. As shown by the radial distribution function
in Fig. 7, the interatomic attraction is averaged over the various
configurations, which leads to a lack of site dependency in the PES. Besides,
the coexistence of chemisorption and physisorption distances results in
intermediate bonding mechanisms.

For interfaces with small lattice constant mismatches (for example,
Ir(111)/Gr), the  moiré  structure and site-dependent attracting interaction
originate from vertical buckling. Such behavior in the symmetry-mismatched
interface is non-trivial; however, in the case of FePd(001)Gr, our results
predict the existence of  moiré -like buckling, as shown in Fig. 8. Their
amplitudes ($\leq 0.5~{}\text{\AA}$) are much smaller than those of the small-
lattice-constant mismatched interface; this is considered to be the result of
averaging the inter-atomic attraction. The DoS in Fig. 9 indicates that the
macroscopic electronic and magnetic properties of FePd were not significantly
changed by the presence of graphene, which is also in good agreement with the
experimental results [6]. We expect FePd(001)/Gr to exhibit desirable
properties for MTJ applications. Because the graphene coverage is sufficiently
stable to protect Fe from oxidization without significantly degrading the
macroscopic magnetic properties, the FePd(001)/Gr interface can avoid large
lattice mismatches such as those exhibited by the FePd(001)/MgO interface.

In addition, comparing the DoS of the ”simple model” and ”twisted model” in
Fig. 9, it is remarkable that the choice of the interface structure does not
make a significant difference, especially near $E_{\mathrm{F}}$. In a typical
lattice-matched interface, the choice of adsorption site strongly influences
their electronic properties. However, in lattice symmetry-mismatched
interfaces, site dependency is suppressed by the uniform averaging of Fe-C
interactions. This finding justifies the use of the simple model as a
reasonable substitute for realistic twisted models in calculations of physical
properties, which are mainly dominated by electronic and magnetic structures.
(The Simple models are, due to the large strain, inaccuracies for calculation
related to structural energies and forces such as molecular dynamics and
phonon properties.)  In the future, we expect that the simple models will
become a useful testbed to reduce the computational costs of time-consuming
calculations  that is , electron transport properties [31, 32] and
magnetocrystalline anisotropies [33] of FePd(001)/Gr and any other $L1_{0}$
alloy-based interfaces.

## SUPPLEMENTARY MATERIAL

See the supplementary material for optimized of atomic positions of the simple
models (S1), and detailed computational results of bilayer (S2) and trilayer
(S3) graphene on FePd(001) surface.

###### Acknowledgements.

This work is supported by MEXT as ”Program for Promoting Researches on the
Supercomputer Fugaku” (Quantum-Theory-Based Multiscale Simulations Toward the
development of next-generation energy-saving semiconductor devices,
JPMXP1020200205) and used the computational resources of super computer Fugaku
provided by the RIKEN Center for Computational Science (Project ID:
hp200122/hp150273). T.O. acknowledge the financial support from JSPS KAKENHI
(grant number:JP16H03865). M.U., H.N., and T.O. acknowledges the JSPS Core-to-
Core Program (grant number:JPJSCCA20160005). This work is performed with the
approval of the Photon Factory Program Advisory Committee (H.N.) (No.
2019S2-003). The visualization of the atomic structures in this study is
performed using VESTA software [34] . The computation is performed by the
facilities of the Supercomputer Center, the Institute for Solid State Physics,
the University of Tokyo, and also performed by Oakforest-PACS at JCAHPC
through the Multidisciplinary Cooperative Research Program in CCS, the
University of Tsukuba.

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# Compositional Zero-Shot Learning for Attribute-
Based Object Reference in Human-Robot Interaction

Peng Gao1∗, Ahmed Jaafar1∗, Brian Reily2, Christopher Reardon3, and Hao Zhang1
1University of Massachusetts Amherst, 2DEVCOM Army Research Laboratory,
3University of Denver
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
*Authors contributed equally to this paper

###### Abstract

Language-enabled robots have been widely studied over the past years to enable
natural human-robot interaction and teaming in various real-world
applications. Language-enabled robots must be able to comprehend referring
expressions to identify a particular object from visual perception using a set
of referring attributes extracted from natural language. However, visual
observations of an object may not be available when it is referred to, and the
number of objects and attributes may also be unbounded in open worlds. To
address the challenges, we implement an attribute-based compositional zero-
shot learning method that uses a list of attributes to perform referring
expression comprehension in open worlds. We evaluate the approach on two
datasets including the MIT-States and the Clothing 16K. The preliminary
experimental results show that our implemented approach allows a robot to
correctly identify the objects referred to by human commands.

> Keywords: Object Reference, Zero-Shot Learning, Human-Robot Interaction

##  1 Introduction

Figure 1: A motivating scenario of object reference for human-robot
interaction.

Natural language-enabled robots have recently attracted considerable attention
to enable intuitive, efficient, and transparent human-robot interaction and
teaming [1, 2, 3], which has a wide variety of real-world applications
throughout society, such as in elderly care, hospital assistance, education,
inspection, and search and rescue [4, 5, 6]. Object reference, defined as the
capability of identifying a particular object that a human teammate refers to,
is essential for intelligent robots to appropriately communicate with humans
[7]. In language-based communication, a language-enabled robot must comprehend
a referring expression in order to recognize and localize a particular object
from its visual perception using a set of referring attributes extracted from
natural language by the human teammate.

While robot perception has shown promising performance for recognizing object
categories, they are insufficient for referring expression comprehension to
represent and identify object instances from language and vision. First, an
object instance can be referred to through language before it is observed, and
visual data may not be available for the object at the time it is referenced,
meaning that the robot may have to visually identify an unfamiliar object that
has not previously observed. Second, most object recognition and detection
techniques focus on identifying the same categories of objects but are
generally unable to discriminate between object instances, which are more
typically referred to. Third, in an open world, even given a bounded number of
object categories, the number of object instances and the number of object
attributes can be unbounded. To enable referring expression comprehension and
address the above challenges, we implement an attribute-based compositional
zero-shot learning (CZSL) approach that composes a list of seen attributes and
object labels to identify unseen object-attribute pairs from a robot’s
perception data in an open world.

##  2 Related Work

In this study, our primary focus is on compositional zero-shot learning (CZSL)
[8]. CZSL involves working with a training dataset that comprises various
combinations of states and objects. The primary objective in CZSL is to
recognize and identify previously unseen combinations of these attributes and
objects during testing. We generally divide the existing methods into two
groups, including pure visual-based methods and language-prior methods.

For the first group of methods, several approaches have been proposed. Some
methods tackle CZSL by separately learning classifiers for objects and states
and then combining these classifiers to construct the final recognition model,
such as using an SVM classifier trained for known combinations, and class
weights for new combinations are inferred using a Bayesian framework [9].
LabelEmbed [10] introduces a transformation network built on top of pre-
trained state and object classifiers. Another work [11] suggests encoding
objects as vectors and states as linear operators that transform these
vectors. Similarly, a recent work [12] enforces symmetrical representations of
objects based on their state transformations. For the second group of methods,
the mainstream is to minimize the loss between visual and linguistic learning
space for unseen attribute-object pairs [13]. Some recent works use graph
structure to leverage information transfer between seen to unseen pairs using
graph convolutional networks [14] or graph attention networks [15] along with
modular networks.

In this paper, we implement an attribute-based compositional zero-shot
learning (CZSL) approach that composes a list of seen attributes and object
labels to identify unseen object-attribute pairs from the robot’s perception
data. The approach allows robots to disentangle attributes and object
appearances from observed compositions and predict unseen or unknown
compositions in open-world scenarios.

##  3 Approach

###  3.1 Problem Formulation

We implement an approach for object reference in the context of human-robot
interaction. In this scenario, we assume a shared environment containing a
human and an assistant robot, such as a living room or a workspace. The
primary objective is for the assistant robot to accurately identify the object
to which the human is referring, based on the natural language instructions
provided.

Specifically, let’s consider a typical human referring expression like “black
pen”. This expression can be readily parsed by existing natural language
models, such as BERT [16], into its attributes and class label. We denote the
attribute “black” as $w_{attr}$ and the class label “pen” as $w_{obj}$. Based
upon the parsed keywords $w_{attr},w_{obj}$, the robot then undertakes the
task of identifying the referred object in the shared environment, drawing
upon its own observation sequence, which we denote as
$\mathcal{{L}}=\\{\mathcal{{I}}_{1},\dots,\mathcal{{I}}_{n}\\}$. Formally, we
formulate the referring expression comprehension as follows:

$s_{i}=cos\left(\phi(\mathcal{{I}}_{i}),\psi\left(w_{attr},w_{obj}\right)\right)=\frac{\phi(\mathcal{{I}}_{i})\psi(w_{attr},w_{obj})}{||\phi(\mathcal{{I}}_{i})||\;||\psi(w_{attr},w_{obj})||}$
(1)

where $\phi$ denotes the network to encode visual features of
$\mathcal{{I}}_{i}\in\mathcal{{L}}$, which is constructed based on a ResNet
[17] or a Vision-Transformer [18] followed by an Average Pooling operation,
then the final visual feature is computed through a linear layer. $\psi$
denotes the network to encode word features of $w_{attr},w_{obj}$, which is
based on GLoVe [19] followed by a multi-layer perception (MLP). $cos$ denotes
the cosine function that computes the similarity. $s_{i}$ denotes the
similarity between the query text command and the robot observation. Given the
similarity, the object reference is formulated as a classification problem,
which applies the SoftMax function to the similarity score and predicts which
class of attribute-object pairs the observation belongs to, thus achieving
object reference.

###  3.2 Attribute-Based CZSL

To make the classification generalizable to unseen attribute-object pairs,
based on the existing work [20, 21], we use an attribute-based compositional
zero-shot learning method to disentangle attributes of objects for unseen
object-attribute pair prediction. Formally, the approach introduces two images
$\mathcal{{I}}_{attr}$ and $\mathcal{{I}}_{obj}$ into the learning process.
$\mathcal{{I}}_{attr}$ has the same attribute as $\mathcal{{I}}$ but with a
different object. $\mathcal{{I}}_{obj}$ has the same object as $\mathcal{{I}}$
but with different attributes. For example, if $\mathcal{{I}}$ is a black pen,
then $\mathcal{{I}}_{attr}$ can be a black fork with the same attribute
“black” as $\mathcal{{I}}$ but with a different object class “fork”, denoted
as $w_{non-obj}$. $\mathcal{{I}}_{obj}$ can be a sliver pen with the same
object class as $\mathcal{{I}}$ but with a different attribute “sliver”,
denoted as $w_{non-attr}$. Our goal is to make a robot not only recognize the
black pen that is seen in training but also recognize the silver fork by
composing the disentangled attributes and object classes.

Formally, first the correlation between $\mathcal{{I}},\mathcal{{I}}_{attr}$
and between $\mathcal{{I}},\mathcal{{I}}_{obj}$ needs to be computed to
disentangle the correlated attribute and object class. The correlation matrix
is defined as:

$\mathbf{C}^{attr}=\frac{\phi(\mathcal{{I}}_{i})\phi(\mathcal{{I}}_{attr})}{||\phi(\mathcal{{I}}_{i})||_{2}\;||\phi(\mathcal{{I}}_{attr})||_{2}}\quad
and\quad\mathbf{C}^{obj}=\frac{\phi(\mathcal{{I}}_{i})\phi(\mathcal{{I}}_{obj})}{||\phi(\mathcal{{I}}_{i})||_{2}\;||\phi(\mathcal{{I}}_{obj})||_{2}}$
(2)

where $\phi$ denotes the visual feature encoder based on a ResNet or a Vision
Transformer. $\mathbf{C}^{attr}$ denotes the correlation matrix between
$\mathcal{{I}}$ and $\mathcal{{I}}_{attr}$ on the attribute $w_{attr}$, and
$\mathbf{C}^{obj}$ denotes the correlation matrix between $\mathcal{{I}}$ and
$\mathcal{{I}}_{obj}$ on the object class $w_{obj}$. Given the correlation
matrix, row-wise and column-wise SoftMax operations are performed on it to get
the mask for different attributes and object classes, which is defined as

$\mathbf{A}_{i}=\frac{e^{\mathbf{C}^{attr}_{i,:}}}{\sum_{i=1}^{d}e^{s_{i,j}}}\quad
and\quad\mathbf{A}^{attr}_{j}=\frac{e^{\mathbf{C}^{attr}_{:,j}}}{\sum_{i=1}^{d}e^{s_{i,j}}}\quad
and\quad\mathbf{A}^{obj}_{j}=\frac{e^{\mathbf{C}^{obj}_{:,j}}}{\sum_{i=1}^{d}e^{s_{i,j}}}$
(3)

where $d$ denotes the dimensions of the feature channels , $\mathbf{A}_{i}$
and $\mathbf{A}^{attr}_{j}$ denotes the correlation masks between
$\mathcal{{I}}$ and $\mathcal{{I}}_{attr}$ in the feature space.
$\mathbf{A}^{obj}_{j}$ denotes the correlated region of $\mathcal{{I}}_{obj}$
on the object class in the feature space. To further disentangle the unseen
attributes and object class for generalizable prediction, the following based
on the negative correlation matrix is computed:

$\mathbf{A}^{non-
obj}_{j}=\frac{e^{-\mathbf{C}^{attr}_{:,j}}}{\sum_{i=1}^{d}e^{s_{i,j}}}\quad
and\quad\mathbf{A}^{non-
attr}_{j}=\frac{e^{-\mathbf{C}^{obj}_{:,j}}}{\sum_{i=1}^{d}e^{s_{i,j}}}$ (4)

where $\mathbf{A}^{non-obj}_{j}$ denotes the non-correlation mask between
$\mathcal{{I}}$ and $\mathcal{{I}}_{attr}$ about the object class $w_{non-
obj}$. $\mathbf{A}^{non-attr}_{j}$ denotes the non-correlation mask between
$\mathcal{{I}}$ and $\mathcal{{I}}_{obj}$ about the attribute $w_{non-attr}$.
To get the final masks, the elements in these masks are added up along their
feature channels, which is defined as $m_{j}=\sum_{i=1}^{d}\mathbf{A}_{ij}$,
where $\mathbf{m}=\\{m_{j}\\}^{l}$ and $l=49$ denotes the feature length in
this paper. Similarly, $\mathbf{m}^{attr}$, $\mathbf{m}^{obj}$,
$\mathbf{m}^{non-attr}$ and $\mathbf{m}^{non-obj}$ can be obtained based on
$\mathbf{A}^{attr},\mathbf{A}^{obj},\mathbf{A}^{non-attr},\mathbf{A}^{non-
obj}$ respectively. Given the feature masks, the disentangled feature is
computed as follows:

$\mathbf{v}_{attr}=\mathbf{m}\cdot\phi(\mathcal{{I}}_{attr})+\mathbf{m}^{attr}\cdot\phi(\mathcal{{I}}),$
(5)
$\mathbf{v}_{obj}=\mathbf{m}\cdot\phi(\mathcal{{I}}_{obj})+s\mathbf{m}^{obj}\cdot\phi(\mathcal{{I}})$
(6) $\mathbf{v}_{non-attr}=\mathbf{m}^{non-
attr}\cdot\phi(\mathcal{{I}}_{attr}),$ (7) $\mathbf{v}_{non-
obj}=\mathbf{m}_{non-obj}\cdot\phi(\mathcal{{I}}_{obj})$ (8)

where $\mathbf{v}_{obj},\mathbf{v}_{attr},\mathbf{v}_{non-
obj},\mathbf{v}_{non-attr}$ denote the disentangled features of
$w_{obj},w_{attr},w_{non-obj},w_{non-attr}$ respectively. Given the
disentangled features, classification is performed given Eq. (1) by replacing
$\phi(\mathcal{{I}})$ with the disentangled features to predict the object-
attribute pair given human text command. Cross entropy loss is used to train
the network, in which the loss is minimized between all the visual features
defined in Eqs. (5-8) and their associated word embedding features.

##  4 Preliminary Experimental Results

Table 1: Quantitative Results using the MIT-States Dataset based on the AUC ($\%$) metric. | Seen | Unseen | Object | Attr
---|---|---|---|---
Val $@1$ | 0.3368 | 0.2965 | 0.3683 | 0.3190
Val $@2$ | 0.4501 | 0.4382 | 0.4800 | 0.4531
Val $@3$ | 0.5206 | 0.5195 | 0.5508 | 0.5320
Test $@1$ | 0.316 | 0.2585 | 0.3336 | 0.2844
Test $@2$ | 0.4252 | 0.3811 | 0.4442 | 0.4035
Test $@3$ | 0.4987 | 0.4577 | 0.5160 | 0.4793

We use two existing datasets to evaluate the approach: MIT-States [22] and
Clothing16K [23]. The MIT-States dataset contains object classes such as fish,
rooms, etc. It also has attributes, such as mossy, dirty, etc. Clothing16K
contains clothes, such as suits and pants, with different attributes, such as
pink and black. In the training stage, all of the objects and attributes are
seen. In the testing stage, the compositions of seen and unseen
objects/attributes are used to evaluate the generalization of the approach. In
addition, various scenarios are evaluated including seen object-attribute
pairs (Seen), unseen object-attribute pairs (Unseen), unseen objects with seen
attributes (Object), and seen objects with unseen attributes (Attr).

Area Under the Curve (AUC) is utilized to evaluate the accuracy of text-to-
image retrieval. For each scenario, the top K ($@k$) candidates are retrieved
as the results.

Table 1 presents the quantitative results using the MIT-States dataset. From
these results, we observe that the approach can successfully retrieve the
correct images given the human text commands. The performance of unseen pairs
retrieval is slightly worse than the seen pairs, which indicates the
generalization capability of the approach for object references in human-robot
interaction.

(a) Evaluation on Clothing16K Dataset

(b) Evaluation on Real Robot

Figure 2: Experiment setups. (a) Retrieval of unseen object-attribute pairs
given text query command. (b) Case studies on object reference in human-robot
interaction.

We further implement a method based on the recent work [21], it obtains the
quantitative results using the Clothing16K dataset, as shown in Figure 2(a).
Given the query text command, the approach can correctly retrieve the unseen
object-attribute pairs. We deploy our implemented work on a physical robot,
Triton, to conduct a case study. In this study, the Triton moves around its
environment to identify the unseen attribute-object pair given via a human
textual command. As shown in Figure 2(b), the human command is “blue shirt”
which is unseen in the training stage. Eventually, the robot can correctly
identify the “blue shirt” and move forward to it, given that some of the
attribute-objects it was trained on are “red shirt” and “blue shorts”.

##  5 Conclusion

In this paper, we implement a new attribute-based compositional zero-shot
learning (CZSL) approach to enable referring expression comprehension for
human-robot interaction. The approach composes a list of unknown attributes
and object labels to identify unseen object-attribute pairs from the robot’s
perception data in an open world. Our preliminary experimental results have
validated the effectiveness of the approach.

Our current implementation has some limitations. In the future, we will
further study the following aspects, including 1) integrating natural language
processing with the vision network to perform human-robot interaction tasks,
2) adding robot search and navigation modules to complete the loop, 3) doing a
deeper analysis with more sophisticated robots in real-world experiments.

#### Acknowledgments

This research was partially supported by the ONR grant N00014-21-1-2418, the
ARL A2I2 Program W911NF-23-2-0005, and the NSF CAREER Award IIS-2308492.

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