# Density functional theory study of experimentally observed Au/Ge interfaces

Olga Sikora<EMAIL_ADDRESS>Faculty of Materials Engineering and
Physics, Cracow University of Technology, Podchora̧żych 1, PL-30084 Kraków,
Poland Małgorzata Sternik  Institute of Nuclear Physics, Polish Academy of
Sciences, W. E. Radzikowskiego 152, PL-31342 Kraków, Poland Benedykt R. Jany
Marian Smoluchowski Institute of Physics, Jagiellonian University,
Prof. Stanisława Łojasiewicza 11, PL-30348 Kraków, Poland  Franciszek Krok
Marian Smoluchowski Institute of Physics, Jagiellonian University,
Prof. Stanisława Łojasiewicza 11, PL-30348 Kraków, Poland  Przemysław Piekarz
Institute of Nuclear Physics, Polish Academy of Sciences, W. E. Radzikowskiego
152, PL-31342 Kraków, Poland Andrzej M. Oleś<EMAIL_ADDRESS>Max Planck
Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart,
Germany  Institute of Theoretical Physics, Jagiellonian University, Prof.
Stanisława Łojasiewicza 11, PL-30348 Kraków, Poland

###### Abstract

In recent years, nanostructures with hexagonal polytypes of gold have been
synthesised, opening new possibilities in nanoscience and technology. As bulk
gold crystallizes in the fcc phase, surface effects can play an important role
in stabilizing hexagonal gold nanostructures. Here we investigate several
hetero-structures with Ge substrate, including the fcc and hcp phases of gold
that have been observed experimentally. We determine and discuss their
interfacial energies and optimized atomic arrangements, comparing the theory
results with available experimental data. Our calculations for the Au-
fcc(011)/Ge(001) show how the presence of defects in the interface layer can
help to stabilize the atomic pattern consistent with microscopic images. The
Au-hcp/Ge interface with the significant mismatch between two surfaces reveals
large atomic displacements, which might indicate that the (111) germanium
substrate is not responsible for the formation of the hcp phase of gold.
Finally, analyzing the electronic properties, we demonstrate that Au/Ge
systems have metallic character but covalent-like bonding states between
interfacial Ge and Au atoms are also present.

##  I Introduction

Heterophase interfaces are responsible for unique properties of many advanced
devices designed for electronics and other applications [1]. Understanding the
formation and energetics of interfaces is highly important for the nucleation
of new crystalline phases on a specific substrate [2, 3]. The structure of a
heterophase can be studied using advanced atomic-resolution experiments, such
as high-resolution electron microscopy [4], high-resolution secondary-electron
microscopy [5], scanning transmission electron microscopy [6, 7], or scanning-
tunnelling microscopy. However, experimental data provide only partial insight
into the formation of interfaces and the interpretation of microscopic images
is often ambiguous. Various simulation techniques can be used to resolve the
uncertainties and reveal mechanisms stabilizing the observed interfaces.
Deeper knowledge of surfaces and interfaces in hetero-structures can play a
crucial role in developing new methods for synthesizing such materials and in
expanding their possible applications in nanoscience and technology. As a
result of advances in computational methods and the increasing computer power,
the atomic structure and specific properties of various interfaces have been
successfully studied using first principle calculations [8, 9, 10, 11, 12,
13].

Up to very high pressure, gold crystallizes in the fcc phase [14] with the
$ABC$ repeating pattern of hexagonal planes in the (cubic) [111] direction.
Recently, nanostructures including different polytypes of gold have been
observed in various experiments and have gained interest due to their
potential applications. For example, the interface regions of polytypic gold
nanorods were found to be highly active in catalysis [15]. The hexagonal
phases seen in experiments are characterized by $ABAB$ (hcp) or $ABAC$ (dhcp)
stacking patterns. The hcp surface has been observed experimentally in
nanowires [16] and ultrathin sheets on graphene oxide [17]. Nanoribbons with
metastable dhcp structure have been also reported [18, 19] and were used to
grow the dhcp forms of several other metals. It is worth mentioning that also
stable hexagonal silver (dhcp) nanostructures have been syntesized [20, 21].

Nanostructures of hcp gold were found after growing Ge nanowires with Au as
catalyst [22], and a possible mechanism responsible for the formation of hcp
gold has been suggested, involving hexagonal AuGe $\beta$-phase present at the
intermediate stages of growth. Interestingly, while the fcc crystallites were
randomly oriented with respect to the Ge substrate, the hcp nanostructures
were typically found with (001) planes at 60-65∘ to the (111) Ge planes [22].
In a recent experiment, stable hcp nanoislands we obtained under controlled
annealing conditions on the germanium substrate [23]. After initial
crystallization of the fcc gold phase, the hcp phase grows from the eutectic
Au/Ge liquid. Again, there seems to be a preferred hcp crystal orientation
with the Au(010) plane (Au(01$\bar{1}$0) in the Miller-Bravais notation)
parallel to the Ge(111) plane.

First principles calculations of the cohesive energy and elastic constants as
well as phonon dispersion relations show stability of both hcp and dhcp
polytypes of gold [24]. Ab initio studies indicate higher stability of the fcc
phase and a tendency towards hcp $\to$ fcc phase transformation, however, the
calculated differences between the polytypes are very small [14, 25]. On the
other hand, molecular dynamics simulations of metallic nanowires show a phase
transformation from fcc to hcp below a critical diameter [26]. These findings
support possibilities of obtaining new hexagonal structures under specific
conditions where factors not dominant in bulk, e.g. surface effects, can lead
to their enhanced stability.

The purpose of this paper is to provide a quantitative decription of the
energetics of interface and to draw possible conclusions concerning the type
and formation of Au/Ge hetero-structures. Therefore, the concept of
interfacial energy and of the work of separation (adhesion) were used. We
applied first-principles methods based on density functional theory (DFT) to
study Au/Ge hetero-structures with different interfacial plane orientations.

The remaining of the paper is organized as follows. The methodology and the
calculation details are given in Sec. II A, and in Sec. II B the investigated
crystal planes and possible interfaces are introduced. Convergence tests
presented in Sec. II C are conducted for the simplest, almost strain-free Au-
fcc(001)$\parallel$Ge(001) hetero-structure. Section II D begins from
discussing the calculated surface energies for several Au and Ge planes as
well as available experimental data. In the rest of this Section two simple
Au/Ge interfaces are discussed. In Sec. III A we compare several variants of a
hetero-structure with parallel Ge(001) and Au-fcc(011) planes and their
different mutual position. To the best of our knowledge, only one of the
structures, with [110]Ge$\parallel[0\bar{1}1]$Au-fcc, has been observed [23].
We show the optimal configuration obtained from the calculation and discuss
the experimental findings. In Sec. III B we investigate the novel $hcp$
structure that is adjacent to the Ge(111) surface. From the electron
microscope image [23] one can identify the Au-$hcp$ plane parallel to Ge
substrate as $(010)$. We discuss the optimized structures and consider defects
that could stabilize the interface. In Sec. IV we analyse the electronic
properties of Au/Ge systems and the formation of Ge-Au bonds. The paper is
concluded in Sec. V. The Appendices provide surface energy details for gold
and germanium crystals (A) and the charge density distribution at the Au/Ge
interface (B).

##  II Methodology

###  II.1 Interfacial energy and the work of separation
in DFT calculations

In contrast to the bulk phase of a material, the surface atoms have an
incomplete set of neighbors and therefore unrealized bonding energy (surface
energy). When two crystalline solids bind together, the atoms or molecules
form an interface with preferred plane orientation relationship determined by
the strength of bonding, which can be quantified using the concepts of
interfacial free energy and the work of separation. Their values can be
obtained from the DFT calculations of total energies of the appropriate
systems modeling the hetero-structure as well as the bulk crystals and slabs
with a vacuum layer. The methodology details (from choosing the interface
model up to the optimization methods) differ between implementations [12, 27].
Here we focus on the approach in which two crystalline solids ($X$ and $Y$)
form a supercell with imposed periodic boundary conditions, i.e. there are two
interfaces within the supercell. The interfacial energy $\gamma_{int}$ is
defined as:

$\gamma_{int}=\frac{1}{2A_{int}}\left[E_{XY}-\left(n_{X}\varepsilon_{X}^{bulk}+n_{Y}\varepsilon_{Y}^{bulk}\right)\right],$
(1)

where $E_{XY}$ is the total energy of the XY hetero-structure,
$\varepsilon_{X(Y)}^{bulk}$ is the energy per atom in the $X(Y)$ bulk form,
$n_{X(Y)}$ is the number of $X(Y)$ atoms in the XY supercell, and $A_{int}$ is
the area of the interface.

The work of separation ($W_{sep}$) is defined as the energy required to
reversibly separate a bulk material into two semi-infinite bulks with free
surfaces. Here we calculate this energy for an $XY$ hetero-structure,
separated into two slabs, i.e., two supercells with either $X$ or $Y$ atoms
and a vacuum layer, with periodic boundary conditions imposed:

$W_{sep}=\sigma_{X}+\sigma_{Y}-\gamma_{int},$ (2)

where $\sigma_{X}$ and $\sigma_{Y}$ are the surface energies of $X$ and $Y$
phases. Their values can be obtained from the following formula:

$\sigma_{X(Y)}=\frac{1}{2A_{int}}\left(E_{X(Y)}^{slab}-n_{X(Y)}\varepsilon_{X(Y)}^{bulk}\right),$
(3)

where $E_{X}(Y)^{slab}$ is the total energy of the $X(Y)$ slab with two free
surfaces. We note that for interfaces with identical crystal plane
orientations of the slabs but different mutual positions of atoms between the
interface layers, the interfacial energy and work of separation differ by the
same value—it is sufficient to compare one of these quantities.

All calculations were based on the DFT method implemented in the plane wave
basis VASP code [28, 29]. The Perdew-Burke-Ernzerhof (PBE) functional [30]
within the generalized gradient approximation (GGA) for the exchange and
correlation energy was used. The electronic wave functions were expanded as
linear combinations of plane waves, truncated to include only plane waves with
kinetic energies below a cutoff energy $E_{\text{cut}}=350$ eV. The valence
states were optimized with the Ge($s^{2}p^{2}$) and Au($s^{1}d^{10}$) electron
configurations.

Bulk materials, surfaces and interfaces were simulated using the supercell
method with the periodic boundary conditions. Optimizations of the structural
parameters (lattice constants and atomic positions) were carried out using the
Monkhorst-Pack grid of k-points appropriate for the calculated structure: from
dense (8,8,8) grid for crystallographic cells of bulk crystal to (4,4,4) grid
for larger superlattices that model the Au-fcc(011)/Ge(001) and Au-
hcp(010)/Ge(111) hetero-structures (see below). The slabs with vacuum are
calculated using (8,8,2) or (4,4,2) k-points grids depending on the size of
the cross-section. A vacuum layer of 15 Å was selected for each structure to
eliminate interactions between two surfaces. The conditions for ending the
optimization loops for electronic and ionic degrees of freedom were defined by
the total energy difference between steps of 10-8 eV and the internal forces
of 10-2 eVÅ-1, respectively.

Figure 1: Surfaces of Au and Ge slabs investigated in our study, with
experimental interatomic distances: (a) Au-fcc(001); (b) Ge(001), dashed green
square indicates the building blocks matching the [100]Au length; (c) Au-
fcc(011); (d) Au-hcp$(010)$, or $(01\bar{1}0)$ in the Miller-Bravais notation;
(e) Ge(111). To distinguish the hcp lattice, the Au atoms are shown in dark
orange.

In all calculations, the starting configurations were prepared under
assumption that the lateral extent of the gold lattice is equal to optimized
dimensions of the germanium lattice (germanium is treated here as a
substrate). As a result, a non-zero lateral strain is created, both in the
superlattices and in (gold) bulk or slab structures. The calculated
interfacial energies capture therefore mainly the chemical bonding energy at
the interface and the energy cost of internal distortions, as the strain
coming from the lattice mismatch is already present in the bulk phases of
gold. The lattice constant perpendicular to the interface as well as the
atomic positions are fully relaxed.

###  II.2 Structural models

Under ambient pressure, germanium and gold crystallize in the diamond
($Fd\bar{3}m$) and the $fcc$ ($Fm3m$) structure, respectively. Their
experimental lattice constants are $a_{\textnormal{Ge}}=5.66$ Å for germanium
and $a_{\textnormal{Au-}fcc}=4.08$ Å for gold [31]. The hexagonal (hcp) phase
of Au formed during the specific process of annealing and cooling of gold and
germanium is characterized by the $P6_{3}/mmc$ space group with lattice
constants $a_{\textnormal{Au-}hcp}=2.90$ Å and $c_{\textnormal{Au-}hcp}=4.88$
Å [23]. These structural parameters are well reproduced in our DFT
calculations. Obtained cubic lattice constants are equal to 5.77 Å and 4.16 Å
for the Ge and Au-fcc, respectively, and the hexagonal lattice parameters are
2.93 Å and 4.89 Å for Au-hcp.

Figure 1 shows surfaces that terminate slabs which are the building blocks for
hetero-structures investigated in this study. We constructed them by looking
for matching lattice parameters and following the experimental data if
available. Moreover, different mutual positions of the slabs were considered
in order to find the best atomic arrangement.

It can be easily noticed that (001) faces of Au-fcc and Ge can form almost
strain-free hetero-structure if the gold slab as in panel (a) is joined with
the Ge slab marked with a black square in panel (b). A variant of such
superlattice was used to perform convergence tests described in the next
subsection.

Another way of exploiting this lattice match is to connect Au-fcc(011) plane
as shown in panel (c) and the same Ge(001) slab. We note that now we get a
strain-free interface only in one direction, and we need a few Au and Ge slabs
to reduce the lattice strain in the the perpendicular direction.
Alternatively, the Au-fcc(011)/Ge(001) hetero-structure can be built using a
Ge slab marked with the green square, as its side length matches the gold
lattice constant. Again, multiple cells are needed in the perpendicular
direction.

Below we discuss several variants of the Au-fcc(011)/Ge(001) superlattice in
Sec. III and compare our findings to experimental results. There, however, no
obvious way exists to build the experimentally found Au-hcp$(010)$/Ge(111)
interface [the relevant Au and Ge surfaces are shown in panels (d-e)]. As
discussed in Sec. III, we optimize a superlattice guided by the experimental
picture.

###  II.3 Convergence tests

Figure 2: Two variants of Au-fcc(001)/Ge(001) interface atomic arrangement;
the shaded surface indicates the interface Ge layer, and the second Ge layer
is visible below (crystal planes are slightly tilted to show all the atoms):
(a) variant $T_{1}$, the interface Au atoms are placed maximally away from the
Ge atoms—this structure was used for convergence tests, and (b) variant
$T_{2}$, with Au atoms directly above the interface consisting of Ge atoms.

Ideally, the thickness of the slabs should be large enough to ensure that all
the surface effects are captured within the supercell and that the two
interfaces resulting from periodic boundary conditions do not interact with
each other. To test the convergence of surface energies, interface energies
and the work of separation we considered the Ge(001) and Au-fcc(001) slabs
with different numbers of atomic layers and the epitaxial Au-fcc(001)/Ge(001)
heterojunction shown in Fig. 2 (two variants of mutual positions of the slabs
are presented). This simplest hetero-structure can be built by setting the
[110] direction of the Au crystal parallel to the [100] direction of the Ge
lattice [32], i.e. combining two fragments of the lattice planes oriented as
in Fig. 1 (a) and (b). The resulting mismatch $\epsilon$ defined as:

$\epsilon=\left(a_{\text{Au-
fcc}}\sqrt{2}-a_{\text{Ge}}\right)/a_{\text{Ge}}\times 100\%,$ (4)

is equal to only 2.0%, i.e., the hetero-structure is almost strain-free. As
only four Au atoms and two Ge atoms form a single layer of this interface, one
can easily increase the number of layers in the supercell. In variant $T_{1}$
of the considered hetero-structure, the Au atoms are maximally away from the
Ge atoms, while in variant $T_{2}$ some of the gold atoms are located on the
top of Ge sites.

Table 1: Interfacial energies ($\gamma_{int}$), work of separation values ($W_{sep}$), rumpling parameters $r_{\text{Ge}}$ ($r_{\text{Au}}$) for interfacial Ge (Au) layers, and interlayer distances at the interface ($d_{int}$) calculated for different Au/Ge hetero-structures as well as Au-fcc/Au-hcp superlattice. interface | variant |  $\gamma_{int}$ (J/m2) |  $W_{sep}$(J/m2) |  $~{}~{}~{}r_{\text{Ge}}$ (Å) |  $~{}~{}~{}r_{\text{Au}}$ (Å) |  $~{}~{}~{}d_{int}$ (Å)
---|---|---|---|---|---|---
Au-fcc(001)/Ge(001) | $T_{1}$ | 0.345 | 1.862 | 0.00 | 0.40 | 1.76
| $T_{2}$ | 0.909 | 1.298 | 0.00 | 0.13 | 2.36
Au-fcc(111)/Au-hcp(001) | – | 0.041 | 1.411 | – | 0.00 | 2.41
Au-fcc(011)/Ge(001) | $A$ | 0.590 | 1.638 | 0.08 | 0.53 | 1.55
| $B$ | 0.821 | 1.407 | 0.49 | 0.38 | 1.93
| $C$ | 0.336 | 1.892 | 0.21 | 0.61 | 1.77
| $D$ | 0.437 | 1.784 | 0.29 | 0.62 | 2.13
Au-hcp(010)/Ge(111) | – | 0.369 | 1.506 | 0.23 | 0.30 | 2.17

Our preliminary calculations indicated that variant $T_{1}$ is preferred and
for this structure we conducted convergence tests. First, separated slabs of
Ge and Au-fcc crystals terminated with (001) plane and vacuum layer of 15 Å
were used to determine the optimal slab thicknesses for surface energy
calculations. The calculated values plotted in Fig. 3 (left panel) show that
above the fifth layer the discrepancies between results are smaller than 0.03
J/m2. Large initial increase of the surface energy values seen in the data for
Ge crystal results from very small number of atoms (only six) in this slab,
and two free surfaces.

Right panels of Fig. 3 present the convergence tests performed to find the
optimal thickness of the components in the Au/Ge hetero-structure with five Ge
layers. The plot of the interfacial energy reveals much weaker dependence on
the number of Au layers. Due to lack of free space atom do not significantly
change their positions and the surface effect appear to vanish quickly with
the distance from the interface. Values of the work of separation for Au-
fcc(001)/Ge(001) hetero-junction include both the calculated surface and
interface energies, therefore again we observe differences in the results of
about 0.03 J/m2 for slabs with more than five layers of Au.

Figure 3:  Convergence tests: (left) dependence of Au-fcc(001) and Ge(001)
surface energies on the number of slab layers, (right) the interfacial energy
and work of separation for Aufcc(001)/Ge(001) hetero-junction vs. the number
of Au layers (the number of Ge layers is equal to five).

We conclude that five layers of both Au and Ge atoms in the slabs are
sufficient for our calculations. This observation is important for the
investigation of more complicated interfaces where larger numbers of layers
result in a very long computational time.

###  II.4 Results for model surfaces and interfaces

Figure 4:  Surface energies calculated for germanium (left) and fcc-gold
(right) crystal planes compared to available experimental and theory data
(values are given in Appendix A).

We consider several low-index planes of Ge, Au-fcc and Au-hcp crystals (some
of them are presented in Fig. 1). The calculated surface energies are
presented in (Fig. 4) and Appendix A. The comparison of our results with the
available theoretical and experimental data shows that different theoretical
approaches lead to discrepancies in absolute values of surface energies,
however, the relative changes between the investigated crystal planes are very
similar within each method. Experimental data for Ge crystal show the same
trend, i.e. lowering the surface energy from (001) to (111) plane, with a
slightly bigger slope.

Figure 5:  The relaxed structures of the simplest coherent interfaces. Two
variants of the Au-fcc(001)/Ge(001) hetero-structure: (a) variant $T_{1}$,
with interfacial Au atoms at hollow sites of Ge surface, and (b) variant
$T_{2}$, with interfacial Au atoms at top sites of Ge surface. Panel (c) shows
the Au-fcc(001)/Au-hcp(0001) interface with the two phases of gold marked in
yellow and dark orange, respectively. The positions of the gray atoms are
compatible with both arrangements.

The optimized atomic arrangements for our model Au-fcc(001)/Ge(001) hetero-
structures are shown in Fig. 5 (variant $T_{1}$ and $T_{2}$). The total
energies of two considered supercells are $-108.806$ eV and $-106.484$ eV for
the variant $T_{1}$ and $T_{2}$, respectively. Therefore, the realization of
the variant $T_{1}$ with interfacial Au atoms located at hollow sites of
germanium surface is more probable. The Ge-Au bond lengths obtained for this
structure, 2.50 Å, are in good agreement with the values taken experimentally
for amorphous Ge-Au alloys, 2.66 Å [33] and for thin film of Au covering
Ge(111) surface, 2.5 Å [34].

The calculated interface energy and work of separation are presented in the
first two rows of Table 1. The lower total energy of variant $T_{1}$ leads to
considerably lower surface energy (the difference is almost 0.6 J/m2). The
work of separation for variant $T_{2}$ is lower than for variant $T_{1}$ by
the same value (as the surface energies are identical). Table 1 presents also
other structural parameters useful for describing interfaces: (i) the
intralayer rumpling parameter ($r_{Ge}$ and $r_{Au}$), defined as the maximal
difference between the $z^{th}$ coordinates of atoms belonging to one layer,
(ii) the interlayer distance ($d_{int}$). The Au rumpling parameter in variant
$T_{1}$ is higher than in variant $T_{2}$ (0.4 Å vs. 0.130 Å ), as the result
of interactions between interfacial Au atoms with atoms from the next Ge
layer. The negligibly small rumpling parameter in the Ge layer reflect little
changes in the atomic positions.

We conclude discussion of model hetero-structures with results for the perfect
match fcc(111)/hcp(001) interface, shown in Fig. 5(c). It changes the $ABC$
fcc arrangement of lattice planes into the $ABAB$ pattern, and, as can be seen
in the figure, one cannot uniquely assign two planes to either of these
phases. We symmetrically divided the gray layers between the two phases and
calculated the interface energy values. As could be expected, the value of the
Au-fcc/Au-hcp interface energy is much smaller (by the order of magnitude)
than in the other interfaces. The work of separation includes also the surface
energy and is of the same order of magnitude as in other considered
interfaces.

##  III Experimentally observed hetero-structures

In this Section we present our results of ab initio calculations for the Au/Ge
interfaces observed in the experimentally grown gold nanoislands on the
germanium substrate [23]. A particularly regular fcc structure can be found on
the Ge(001) surface, with (011) gold plane orientation. When the sample is
annealed above the eutectic temperature, a hcp gold phase facing the Ge(111)
plane is also observed.

###  III.1 Au-fcc(011)/Ge(001)

As already discussed in Sec. II, there are several possibilities of building a
low-energy Au-fcc(011)/Ge(001) hetero-structure, and here we compare their
calculated interface energy values and structural parameters. First, we join
the Au-fcc(011) plane oriented as in Fig. 1(a) and the Ge(001) plane from
panel 1(b), using a cell marked with the black square as a building block for
the supercell. In such a way we can get almost strain-free hetero-structure in
the $[1\bar{1}0]$Au direction. In order to reduce strain along the
perpendicular [100]Au direction, we take four Au lattice constants and three
Ge blocks (the resulting mismatch is about 4%). To get benchmark results for
this orientation of crystal planes, we consider two different mutual positions
of the slabs presented in Figs. 6(a) and 6(b).

In variant $A$, rows of gold atoms are placed between the rows of germanium
atoms, and we can expect from previous calculations that such arrangement is
energetically preferable to interfaces in which some Au and Ge atoms are
forced to make shorter bonds. We include five Au and five Ge layers in the
supercell, and this gives altogether 70 atoms (30 Ge and 40 Au). Similarly to
the Au(001)/Ge(001) interface, the gold atoms close to the interface move
slightly from their initial positions (rumpling parameter $r_{\text{Au}}=0.53$
Å), while the stiffer germanium lattice stays almost unchanged (rumpling
parameter $r_{\text{Ge}}=0.08$ Å). In variant $B$, the shift of Au plane with
respect to Ge surface that places the column of Au atoms right over the Ge
atoms [Fig. 6(b)] leads to a significant increase in the total energy of the
system (supercell) by about 3.5 eV. Additionally, the forces induced to
preserve the Ge-Au bond length on the appropriate level lead to an increase of
the rumpling parameter of the Ge layer up to 0.49 Å. The interface energy of
variant $B$ is higher than of variant $A$ by 0.23 J/m2. These findings confirm
that structures with some Au atoms located directly above the Ge atoms tend to
be energetically expensive.

Figure 6: Variants $A$, $B$, $C$, and $D$ of the Au-fcc(011)/Ge(001) interface
(initial configurations) are shown in (a-d) panels, respectively. (a) and
(b)—Four Au slabs as shown in Fig. 1(c) are placed upon a Ge supercell
constructed from three blocks marked with a black square in Fig. 1(b). In
variant $A$ the interface Au atoms avoid Ge atoms, while in variant $B$ some
of the Au atoms are placed directly above Ge atoms. (c) and (d)—Five Au slabs
as shown in Fig. 1(c) are joined with seven Ge blocks marked by a green square
in Fig. 1(b). Variants $C$ and $D$ of this interface differ in orientation of
the Ge slabs with respect to the Au slabs.

Atomically resolved STEM-HAADF images of gold nanoislands on germanium
substrate [23] provide yet another way of matching Au-fcc(011) and Ge(001)
crystal planes. Since the distance between germanium atoms in [110] direction
closely matches the gold lattice constant (the mismatch of about 1.9%), we can
use a green square in Fig. 1(b) as a new building block for the hetero-
structure.

We note that there are two different orientations of this block with respect
to the Au-fcc(011) slab shown in Fig. 1(c). Both arrangements are consistent
with electron microscopy picture as long as the Au and Ge atoms at the
interface are not discriminated. In the perpendicular direction (with no
experimental image to guide us), we chose to join seven germanium blocks and
five gold slabs as shown in Fig. 1(c), obtaining a mismatch of about 3%.
Variant $C$ is constructed in such a way that rows of gold atoms are located
between the germanium rows (see Fig. 6(c)), and we expect this arrangement to
be energetically favored. Indeed, the interface energy obtained for a
supercell containing six layers of Ge (42 atoms) and five layers of Au (50
atoms) is equal to 0.336 J/m2, a value very close to the lowest energy
obtained earlier for the optimal Au-fcc(001)/Ge(001) structure.

Small displacements of atoms from initial positions are observed in the
interface layer ($r_{\text{Au}}=0.62$ and $r_{\text{Ge}}=0.22$ Å), and the
distance between Ge and Au interfacial planes is equal to 1.78 Å (again, very
similar to our first low energy hetero-structure). These numbers suggest that
variant $C$ is a good candidate for the observed nanostructure. However, a
detailed analysis of the scattered intensity of atomic columns [23] suggests
that the rows of Au atoms are located directly above the rows of Ge atoms.

Figure 7:  The infusion of additional Ge atoms to variant $D$ of the Au-
fcc(011)/Ge(001) hetero-structure: (a) the optimized structure of variant $D$;
(b-d) additional Ge atoms per supercell introduced to the gold layer from
which some Au atoms are removed (twice as many as the number of Ge atoms
added). The green rectangle marks the extension of the mixed Au/Ge layer. For
better visibility of the changes in the structure, Ge-Ge bonds are shown in
the left panels.

We therefore built variant $D$ of the hetero-structure, with 102 atoms (42 Ge
and 60 Au), and interface shown in Fig. 6(d). In order to match the
experimental picture, we need a different orientation of the germanium
building blocks (rotation by 90∘ with respect to variant $C$). Due to
incommensurate Au-Au and Ge-Ge distances, it is impossible to find a regular
pattern of Au-Ge bonds in the resulting interface layers (some of them might
be much shorter than in the optimal Au/Ge arrangement). After optimization, we
get similar values of rumpling parameters as in variant $C$, but the distance
between Au and Ge layers is increased by almost 0.4 Å. Because of the
different numbers of atoms in each supercell we cannot make a direct
comparison of the total energies, however, the interface energy in variant $D$
is higher by approximately 0.1 J/m2. It is still the second lowest value among
variants of the Au-fcc(011)/Ge(001) interface and the difference with respect
to the optimized variant $C$ is relatively small. The slightly higher
interface energy in variant $D$ might arise from the distortions present in
the Ge substrate visible in Fig. 7(a), and the enlarged distance between the
two phases.

In the last part of this Section, we discuss a possibility of lowering the
interfacial energy of this variant by introducing defects into the interface
layer. We investigated three supercells of variant $D$, in which some pairs of
Au atoms in the interface layer were replaced by Ge atoms, and their optimized
atomic arrangements are presented in Fig. 7(b-d). We can observe that while
one such defect, shown in panel (b), seems to disrupt the arrangement of some
atoms even further, more defects restore a remarkably regular structure with a
decreased distance between layers at the interface. The calculated interface
energy values for structures in panels (a) to (d) are: 0.437, 0.416, 0.388,
and 0.343 J/m2, respectively. The Ge atoms replacing some Au pairs gradually
improve the interface energy and the result obtained for the last modification
is very close to the interface energy of variant $C$. Figure 8 shows the
experimental image of the investigated interface together with one of our
optimized structures, shown in Fig. 7(d). We note, however, that variant $C$
as well as variant $D$ without defects would also be consistent with the
available experimental image (showing only the view as in right panels of Fig.
7), so we cannot use it to discriminate these arrangements.

Figure 8:  Experimentally observed Au-fcc(011)/Ge(001) interface: atomically
resolved STEM-HAADF image [23], and four optimized supercells as in the right
panel of Fig. 7(d), i.e. of variant $D$ with defects (six interfacial Au atoms
per a supercell are replaced by three Ge atoms). The green rectangle marks the
mixed layer.

###  III.2 Au-hcp(010)/Ge(111)

Figure 9: The Au-hcp(010)/Ge(111) interface atomic arrangements in initial
configurations (left panels) and side view of optimized configurations (right
panels): (a) the position of Au and Ge slabs chosen in such a way that the
shortest Au-Ge distance is maximal, there are four short distances per
interface (i.e., eight in the supercell) indicated by circles; (b) four Au
atoms are replaced by Ge atoms in each interface with the shortest Au-Ge
distance; (c) the positions of slabs are now chosen in a way that two Au-Ge
distances in an interface are the shortest possible ones, and the Au vacancies
are created at these positions (marked by circles).

Our last hetero-structure, including the hcp phase of gold, is the most
interesting one from the perspective of investigating new phases in nano-
structures and their possible applications. It is however also the most
challenging interface for our ab initio calculations. The mutual position of
the slabs can be partially read from electron microscopy images [23], and the
identified gold lattice plane is (010). The hcp gold crystallite grows on the
Ge(111) surface and as it can be seen from the images, the distance between
two gold columns and two closest Ge atoms at the interface is incommensurate.
The atomic arrangement is not very regular and the defects may significantly
influence the interface ordering.

A relatively low-strain interface can be formed with $4\times 3$ Au-hcp(010)
blocks connected to $3\times 2$ Ge(111) blocks as shown in Fig. 1. The Au
lattice is extended in the [100]Au-hcp direction (with a mismatch of about 3%)
and compressed in the [001]Au-hcp direction (a mismatch of 6%). The remaining
lattice strain may be a factor contributing to atom displacements away from
the hcp lattice positions in the Au part of the hetero-structure. We are
strongly limited here by computational costs required for optimizing large
supercells (our supercell contains 96 Au and 72 Ge atoms).

Figure 10:  Atomically resolved STEM-HAADF image of the Au-hcp/Ge(111)
interface [23] with the optimized supercell in which altogether four Au atoms
are replaced by the Ge atoms as shown in Fig. 9(b).

In this particular Au/Ge hetero-junction it is not possible to find a starting
configuration with similar Au-Ge distances across the interface, and
inevitably some of them are significantly shorter if compared to other
optimized structures. In the first interface used for optimization, presented
in Fig. 9(a), the mutual position of the slabs is chosen in such a way that
the shortest Au-Ge bond is maximal for a given supercell size. Atoms in four
of the Au-Ge pairs in an interface (i.e., eight pairs in the unit cell),
marked with circles, are very close to each other and can be expected to
change positions during optimization. As a consequence, DFT calculations with
full atomic displacements lead to strong changes in the Au atomic positions.
The obtained value of interface energy (0.369 J/m2) are similar to the
previously investigated variants $C$ and $D$ of Au-fcc(011)/Ge(001) hetero-
structure, however, the optimized structure is visibly distorted. We cannot
therefore confirm that the Ge(111) crystal plane stabilizes the hcp phase of
gold.

Figure 11:  The electron density of states (EDOS) for Au and Ge atoms,
averaged over all atoms in layers adjacent to the interface. The results for
the respective bulk phases are presented as filled areas. For all plots
$E_{F}=0$.

We considered also possible defects in the lattice that could lower the bond
energy of the interface and/or preserve the hcp ordering in the Au part of the
slab. In the structure shown in Fig. 9(b), we introduced Ge atoms in place of
Au atoms in the four positions at the interface where the Au-Ge distances are
minimal. While the hcp zigzags are more pronounced in the optimized supercell,
the interface energy is slightly higher (0.395 J/m2) with respect to defect-
free junction. Figure 10 shows a good agreement between the experimental
picture and the optimized structure of this variant.

Other possible defects are vacancies in the Au lattice as in Fig. 9(c). To
construct the last supercell, we first found the shift between slabs for which
the Au-Ge distance is the smallest possible (there are two such Au-Ge pairs
per interface), and removed the Au atoms. The interface energy is
significantly higher than in other considered hcp variants (0.514 J/m2),
however, the optimized lattice preserves the hexagonal arrangements. The
germanium lattice is distorted around the vacancies, but this does not
contradict the experimental picture in which some shifts of atoms can be
noticed.

Figure 12:  The charge density differences (isosurface value$=$0.0045 e/Å-3)
for variants $C$ (a) and $D$ (b) of Au-fcc(011)/Ge(001) interface compared
with the isosurface of Au-hcp(010)/Ge(111) interface (c). Yellow and cyan
regions indicate the electron accumulation and depletion, respectively.

##  IV Electronic structure

Here we discuss electronic properties for optimized structures of the defect-
free Au/Ge interfaces investigated in the previous section, i.e. variants $C$
and $D$ of the Au-fcc(011)/Ge(001) interface and the Au-hcp(010)/Ge(111)
heterostructure shown in Fig. 9(a). Figure 11 shows the electron density of
states (EDOS) for Au and Ge atoms (averaged over all atoms in layers adjacent
to the interface). These results are compared to the bulk density of states
(we note that small differences in the bulk phase between Au-fcc and Au-hcp
can be seen from the plots). First of all, electron density at the Fermi level
is finite for the Ge atoms, making the interface region metallic. This effect
is also visible for other Ge layers in our supercells with only slight
decrease in the density of states. On the other hand, the EDOS for gold atoms
remains essentially unchanged around the Fermi level with respect to the bulk
phase for all investigated interfaces. However, the big peak near -2 eV
disappears and the density is shifted towards lower energy values. This shift
can be understood as formation of bonding states with germanium electrons.

Figure 13:  (a) The isosurfaces of the charge density difference for variant
$C$ at 0.0064 e/Å-3 (yellow and blue correspond to positive and negative
values, respectively and (b) the map of charge density distribution for a
plane defined by three atoms in a red frame of panel (a), i.e., a cross-
section through the middle of two bonds. The orientation of this plane is
different from the view in panel (a).

The ability of Au atoms to form bonds with Ge substrate is correlated with the
germanium broken bonds at the interface and can be confirmed by plots of
charge density isosurfaces (see Appendix B). Even deeper insight into
formation of these bonding states can be gained by investigating the
redistribution of electrons in the interface. The changes in charge
distribution induced by the Au/Ge interface can be demonstrated by calculating
the difference between the total charge density of the interface
($\rho_{\mathrm{Au/Ge}}$) and charge densities of systems with vacuum
substituting for Au ($\rho_{\mathrm{vac/Ge}}$) or Ge
($\rho_{\mathrm{Au/vac}}$) in accordance with the formula:

$\displaystyle\Delta\rho=\rho_{\mathrm{Au/Ge}}-\rho_{\mathrm{vac/Ge/}}-\rho_{\text{Au/vac}}.$
(5)

The charge density differences calculated for variant $C$ and $D$ of the Au-
fcc(011)/Ge(001) interface and for the Au-hcp(010)/Ge(111) interface are
illustrated in Fig. 12 with yellow regions representing charge accumulation
and light blue regions indicating charge depletion. For all heterostructures,
the charge transfer from Au and Ge intefacial atoms leads to the electron
accumulation at approximately half of their distance. At presented isocharge
surface value, the enhancement of charge density between interfacal Ge atoms
and their nearest neighbors in the substrate is also observed for variant D
and Au-hcp/Ge interface.

Figure 13 shows another view of the charge transfer from Au and Ge atoms for
variant $C$ (a), and the map of charge density (b) presented for a plane
defined by three atoms in a red frame in panel (a). The density map shows a
cross-section through both Au-Ge and Ge-Ge bonds, revealing their detailed
structure. Perhaps unexpectedly, the charge density between the interface Au
and Ge atoms reaches almost the same level as for Ge-Ge covalent bonds, see
Fig. 13(b). The integrated charge density of the bonding electrons, however,
can be expected to be significantly smaller, one can also observe that Au-Ge
bond is strongly asymmetrical. Similar bonding states were already observed
for Au atoms deposited on Ge(001) surface using the X-ray photoelectron
spectroscopy spectra, supported with first-principles calculations [35].

##  V Summary and conclusions

In summary, we have used density functional theory to study the structure and
stability of phase boundaries found in gold nanoislands grown on the Ge
substrate. This is a very reliable approach which allows one: (i) to select
the most stable structures, (ii) to explain experimental results for
interfaces, and (iii) to make predictions concerning possible structural
properties. In our calculations (using the Ge substrate slab size in the
calculations for all supercells and separate slabs) the interfacial energy
values mostly capture the chemical energy of the interface. The calculations
for the Au-fcc(011)/Ge(001) interface reveal the lowest interfacial energies
for variants $C$ and $D$ of hetero-junctions consistent with available
microscopic images. Refined experimental data indicating that variant $D$
(with slightly higher interface energy) is realized, lead us to the search for
possible defects in the interface layer that might lower the energy cost of
this junction. The optimized supercell is characterized by both a low
interfacial energy and a regular atomic pattern fitting the experimental
picture. The presented approach provides helpful insight into structure and
energy costs of deformations present in several investigated variants.

We also constructed several supercells for studying the novel hcp phase of
gold. Optimization of the atomic positions in the superlattice shows
significant atomic displacements that can result from a lattice strain caused
by a limited size of the superstructure, and particularly uneven distribution
of Au-Ge distances. Again, we found that creation of defects stabilizes the
observed Au-hcp(010)/Ge(111) structure by improving the matching between the
Au layer and substrate and eliminating part of the internal stress.

Electronic structure suggests that the interface region becomes metallic but
bonding states between the interface atoms are also formed. The detailed
electron density maps show that the charge transfer is present both at the
interface and in the adjacent germanium layer. Similar electronic properties
might be found in other noble metal-semiconductor hetero-structures, and
further investigations are needed to verify the universality of our findings.

###### Acknowledgements.

The authors gratefully thank Johan Verbeeck and Nicolas Gauquelin from EMAT
University of Antwerp for atomically resolved HAADF STEM imaging of
nanoislands made for $fcc$ and $hcp$ phases of gold. The authors kindly
acknowledge support by the National Science Centre (NCN, Poland) under
Projects No.: 2017/25/B/ST3/02586 (P.P. and M.S.), 2020/04/X/ST5/00539
(B.R.J.), and 2021/43/B/ST3/02166 (A.M.O.). A.M.O. is grateful for support via
the Alexander von Humboldt Foundation Fellowship [36] (Humboldt-
Forschungspreis). Some figures in this work were rendered using Vesta software
[37]. For the purpose of Open Access, the authors have applied a CC-BY public
copyright licence to any Author Accepted Manuscript (AAM) version arising from
this submission.

##  Appendix A Surface energies

The surface energies calculated for several planes of Ge and Au-fcc crystals
are given and compared to available theoretical and experimental data in Table
2. We also present results for the hcp phase of gold, considering the
hexagonal $(001)$ plane as well as the (010) plane relevant for our
experimental interface models. It is worth noticing that the differences
between Au-fcc(111) surface and Au-hcp(001) surface begin from the third layer
only and this explains why the numerical results are almost identical for
these planes. Generally, a surface with lower energy is usually considered
more stable since it needs smaller amount of work to be done when cleaving
along this surface. Thus, the Ge(111) and Au-fcc(111) free surfaces are more
favored than others.

Table 2: Surface energies calculated for various low-index planes of Ge, Au-fcc and Au-hcp crystals. For comparison previous theoretical results and experimentally derived values are presented.  | surface energy (J/m2)
---|---
Ge-diamond |  (001) |  (011) |  (111)
this work | 1.394 | 1.241 | 1.035
theory (LDA) [38] | 1.710 | 1.510 | 1.320
exp [39] | 1.835 | 1.300 | 1.060
Au-fcc |  (001) |  (011) |  (111)
this work | 0.813 | 0.834 | 0.641
theory (GGA) [40] | 0.87 | 0.91 | 0.72
theory (LDA) [40] | 1.34 | 1.43 | 1.26
exp [41] | – | – | 1.5
Au-hcp | (001) | (010) |
this work | 0.639 | 0.840 |

##  Appendix B Charge density distribution

An emergence of the Ge-Au bonding in the modeled hetero-structures can be
illustrated using the plots of charge distribution at the Au/Ge interface. In
Fig. 14 the isocharge surfaces (isosurface of 0.051 e/Å-3) are presented. Away
from the interface, each germanium atom shares four covalent bonds with its
nearest neighbors, and gold atoms form a metallic phase. At the interface,
bonding states can also be seen between Ge and Au atoms. After increasing the
isosurface value to 0.062 e/Å-3, the bonding surface is no longer visible
between Au and Ge atoms, though it is still present between neighboring Ge
atoms (and disappears for the isosurface value of 0.07 e/Å-3).

The view from both sides of the modeled interface shows that at Au-
fcc(011)/Ge(001) interfaces each Ge atom is attached to two Au atoms, while
the interfacial Ge atoms at the Au-hcp(010)/Ge(111) interface are bonded with
one Au atom. This difference arises from the structure of germanium surfaces:
the (001) germanium crystal plane consists of atoms with two free valence
electrons, while the (111) plane consists atoms with one free valence electron
only.

Figure 14:  The isosurfaces of charge density (isosurface value equal to 0.051
e/Å-3) for variants: (a) $C$ and (b) $D$ of Au-fcc(011)/Ge(001) interface,
compared with the isosurface of Au-hcp(010)/Ge(111) interface (c).

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# Evaluation of a Motion Measurement System for PET Imaging Studies

Junxiang Wang1, Ti Wu2, Iulian Iordachita1, Peter Kazanzides2 1Dept. of
Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
(email: [jwang334, iordachita]@jhu.edu)2Dept. of Computer Science, Johns
Hopkins University, Baltimore, MD 21218, USA (email<EMAIL_ADDRESS>
###### Abstract

Positron Emission Tomography (PET) enables functional imaging of deep brain
structures, but the bulk and weight of current systems preclude their use
during many natural human activities, such as locomotion. The proposed long-
term solution is to construct a robotic system that can support an imaging
system surrounding the subject’s head, and then move the system to accommodate
natural motion. This requires a system to measure the motion of the head with
respect to the imaging ring, for use by both the robotic system and the image
reconstruction software. We report here the design and experimental evaluation
of a parallel string encoder mechanism for sensing this motion. Our
preliminary results indicate that the measurement system may achieve accuracy
within 0.5 mm, especially for small motions, with improved accuracy possible
through kinematic calibration.

##  I Introduction

Positron Emission Tomography (PET) relies on the injection of a radioactive
tracer, which is then preferentially absorbed by specific tissues (based on
the choice of tracer). The absorbed tracer emits positrons that react with
nearby electrons, creating a pair of annihilation (gamma) photons that travel
in opposite directions and are detected by the PET imaging ring. Higher
sensitivity can be achieved by placing the PET detectors as close as possible
to the subject. For brain imaging during natural activities, such as
locomotion, the ideal solution would be a wearable PET imaging ring, such as
the Helmet PET [1], except that with current technology, a PET imaging ring
with sufficient sensitivity for neuroscience research would weigh 10 kg or
more [2] (other estimates are 15-20 kg). Thus, we are instead exploring the
use of a robotic system to suspend the PET imaging ring over the subject’s
head and to actively compensate for head motion while the subject performs
activities such as walking on a treadmill, as illustrated in Fig. 1. The goal
of the robotic system is to keep the imaging ring approximately centered
around the subject’s head (coarse motion compensation), with residual motion
corrected by the image reconstruction algorithm (fine motion compensation).
This is an example of human-robot interaction, where the robot must safely
move in proximity of the human.

Figure 1: System concept: Parallel robot supports PET imaging ring around a
subject’s head, while the subject walks on a treadmill. The string encoder
system measures the position of a helmet (green), worn by the subject, with
respect to the imaging ring (blue).

Given the high safety requirement when robotically moving a 15-20 kg weight
over a human head, we plan to have a redundant measurement system consisting
of both optical and mechanical sensing of the subject’s head motion with
respect to the PET imaging ring, possibly fused with inertial sensing. This
paper focuses on the design and experimental evaluation of a mechanical
sensing system, consisting of six string encoders connected between the PET
imaging ring and a safety helmet attached to the subject’s head. Our current
plan is to use this system as the primary measurement for control of the
robotic system (coarse motion compensation) because it is robust and can
provide high-frequency measurements (on the order of 1 kHz). In contrast,
optical sensing can provide high accuracy, but provides lower-frequency
feedback (on the order of 30-60 Hz) and can occasionally fail to provide a
measurement. We have not yet determined which measurement system to use for
the fine motion compensation performed by the image reconstruction software.
According to the imaging scientists, motion measurements should be accurate to
within about 0.5 mm for this task. Thus, one of our goals is to evaluate
whether the mechanical sensing system can meet this accuracy requirement.

Motion correction is also relevant for conventional PET imaging, and several
researchers have investigated different approaches for sensing this motion.
Especially, different markerless approaches have been studied extensively in
recent years. Olesen [3, 4] describes a system that incorporates a near
infrared light emitting diode into a digital light processing projector to
result in a surface scanner, tracking the head motion by creating a 3D point
cloud on the head surface. This system is mounted directly on the PET imaging
device, and while providing high accuracy of less than 0.3 mm, adds
considerable weight to the imaging ring. This setup also requires clearance
between the patient’s head and the imaging ring for the emission and
processing of the near infrared light, which is in conflict with our desire
for a compact, highly-sensitive imaging system. Other methods rely more on
image processing, thus reducing the amount of computation required compared to
such a 3D surface imaging system. Kyme [5] and Anishchenko [6] utilized four
cameras to record the patient’s head, followed by detection of facial features
from the video, and determine the 6 degree-of-freedom (DOF) movement of the
head. However, this markerless approach could only achieve an accuracy of
about 2 mm.

Chamberland [7] uses positron emission fiducial markers to detect tumor
movement for more accurate targeting in radiotherapy. The advantage of this
approach is that motion can be directly measured in the images, which is ideal
for the fine motion correction, but is too slow to control the robot for the
coarse motion correction. Also, the fidicual marker would be visible in the
reconstructed image and could interfere with measurements of nearby brain
structures.

We previously analyzed motion capture and accelerometer data from subjects
during overground and treadmill walking, respectively, to determine the
typical range of head motion, as well as head velocity and acceleration [8].
We assumed a helmet of dimension 263 mm x 215 mm and evaluated the ability of
a simulated robot to compensate for the recorded head motion, without
colliding with an imaging ring of 300 mm diameter. Our results indicated that
this was feasible, given a robot that could compensate for typical head
motions within about 100 ms. Our goal in this work is to develop a measurement
system that will enable us to experimentally verify this assertion. Section II
presents the measurement system design and implementation, including
integration of a robot for providing ground-truth displacements. This is
followed by accuracy results in Section III and conclusions in Section IV.

##  II Methods

This section describes the design of our mechanical measurement system, using
parallel string encoders, followed by a kinematic analysis, details of the
system construction, and homing procedure (to determine initial string
lengths). We next discuss how the string lengths are adjusted to compensate
for lateral motion within their guide channels. Finally, we describe the
method for determining the transformation between the UR3 robot coordinate
system and string coordinate system.

###  II-A Mechanical design

For our mechanical measurement system, we decided to use string encoders,
attached between the imaging ring and the helmet. This creates a parallel
structure (essentially a passive parallel robot), where we wish to compute the
forward kinematics to convert the measured joint positions (string encoder
lengths) to the Cartesian pose. Given that we need at least 6 string encoders
to provide 6 DOF, there are many different ways that these can be configured.
We selected a Stewart platform structure over other commonly used cable-driven
parallel robots (CDPRs) [9] and parallel measuring structures [10], because it
is a compact and widely used mechanical structure [11] and is less susceptible
to string interference (i.e., the strings should not come into contact with
each other, the imaging ring, or the helmet).

We explored several Stewart platform designs, considering motion sensitivity
(resolution) and isotropic performance. Details of the design optimization
will published separately. The final design is based on approximating the
imaging ring as a sphere of 300 mm diameter and the helmet/head as a sphere of
250 mm diameter. We note that this is just an approximation for our initial
evaluation, as the imaging ring is expected to be cylindrical, not spherical,
and the helmet is expected to be elliptical with a major axis of about 263 mm
in diameter. We created two attachment rings – one to represent the imaging
ring and the other to represent the helmet. Due to the placement of the rings
within the respective spheres, the imaging ring diameter is 261.32 mm and the
helmet ring diameter is 235.42 mm. Figure 2 shows a CAD model of the string
encoder system generated with Creo. where the top figure indicates the
locations of the string attachment points, and the bottom figure provides a
more direct view of the attachment points on the imaging ring. The orientation
of the system is also shown in the figure, and in the nominal configuration,
the helmet coordinate axes are aligned with the base (PET imaging ring)
coordinate axes. The string attachment points are given in Table I, where
$B_{i}$ and $H_{i}$ ($i=1\dots 6$) are the base and helmet attachment points
respectively, as seen in the nominal configuration.

TABLE I: Stewart platform attachment point coordinates (units: mm) Point | X | Y | Z | Point | X | Y | Z
---|---|---|---|---|---|---|---
$B_{1}$ | 121.39 | 48.35 | 73.67 | $H_{1}$ | 96.99 | 66.70 | 42.06
$B_{2}$ | -18.82 | 129.30 | 73.67 | $H_{2}$ | 9.27 | 117.35 | 42.06
$B_{3}$ | -102.56 | 80.95 | 73.67 | $H_{3}$ | -106.26 | 50.64 | 42.06
$B_{4}$ | -102.56 | -80.95 | 73.67 | $H_{4}$ | -106.26 | -50.64 | 42.06
$B_{5}$ | -18.82 | -129.30 | 73.67 | $H_{5}$ | 9.27 | -117.35 | 42.06
$B_{6}$ | 121.39 | -48.35 | 73.67 | $H_{6}$ | 96.99 | -66.70 | 42.06
Figure 2: CAD model of string encoder system, with base (imaging ring) and
helmet attachment points labeled

###  II-B String encoder kinematics

The string encoder system is a Stewart platform and thus we use a standard
kinematics formulation, where the inverse kinematics (from Cartesian pose to
string lengths) is easily computed given the mounting points of the strings on
the base (PET imaging ring) and moving platform (helmet). Specifically, if
$B_{i}$ and $H_{i}$ ($i=1..6$) represent the coordinates of the base and
helmet attachment points, respectively, and $X$ is the Cartesian pose
(transform) of the helmet with respect to the base, the computed string
lengths, $\hat{L}_{i}$, are given by the following inverse kinematics
solution:

$\hat{L}_{i}=\left|\left|X*H_{i}-B_{i}\right|\right|$ (1)

The forward kinematics is more complex and is computed numerically, as
described in [12]. Given measured string lengths, $L_{i}$, the estimated
Cartesian pose of the helmet with respect to the base, $\hat{X}$, is computed
iteratively by:

$\hat{X}_{k+1}=\hat{X}_{k}+J(\hat{L}_{k})\left(L_{m}-\hat{L}_{k}\right)$ (2)

where $k$ is the iteration counter, $J(\hat{L}_{k})$ is the Jacobian and
$\hat{L}_{k}$ is the inverse kinematic solution, eq. (1), corresponding to
$\hat{X}_{k}$. In this equation, the Cartesian pose is represented by Euler
angles, using the intrinsic ZYX convention, i.e., with respect to the helmet
coordinate axes shown in Fig 2. We subsequently refer to the rotation angle
about the z-axis as roll, the angle about the y-axis as pitch, and the angle
about the x-axis as yaw. The iteration is terminated when the string length
error, given by $||L_{m}-\hat{L}_{k}||$ is less than a specified threshold
(0.01 mm in our case) or if the iteration counter reaches a specified limit
(50 in our implementation). We typically see convergence in 3-4 iterations.
Note that for a parallel structure, it is more practical to compute the
inverse Jacobian and then obtain the Jacobian by numerically computing the
pseudo-inverse.

###  II-C System construction

To study the motion of the helmet (head) relative to the PET imaging system,
using the above presented kinematic model, we employed an experimental setup
consisting of a string encoder measurement system, with a UR3 robot (Universal
Robots, Odense, Denmark) to move the helmet ring and a UR5 robot to support
the PET imaging ring, as shown in Fig. 3. In the experiments reported here,
the UR5 robot is not moved. In future proof-of-concept work, it will move
based on the measured head motion to attempt to keep the (mock) PET imaging
ring centered over the helmet. In the longer term, the weight of an actual PET
imaging ring exceeds the UR5 payload and therefore a custom robot, as shown in
Fig. 1, will instead be used. We interfaced to the UR3 robot via TCP, using
its real-time script interface.

The six string encoders (MPS-XXXS-200MM-P, Miran Industries, China) each have
a measurement range of 200 mm at a resolution of 60 counts/mm. The string
encoder bodies were attached on the PET imaging ring while the mobile ends
were attached to the helmet. All custom-made parts of the string encoder
measurement system were built with laser-cut acrylic plates combined with 3D
printed (Stratasys F170) ABS components. The geometry of the imaging ring,
helmet, and strings connection points were defined based on the Stewart
platform kinematics presented above.

Figure 3: Experimental setup consisting of string encoder system and UR3 and
UR5 robots

The string encoders are incremental encoders with quadrature outputs (A and B
channels) and an index pulse (I). We created a custom board to interface these
signals to an FPGA board developed for the da Vinci Research Kit (dVRK) [13].
The FPGA firmware already included a 4-channel encoder interface module, which
was modified to handle up to 8 channels. We implemented our test software on a
PC and connected to the FPGA board via a UDP socket.

###  II-D Homing procedure

Because the string encoders measure relative displacements, it is necessary to
perform a homing procedure to set the initial absolute displacement. This
homing procedure is performed by utilizing the index pulses provided by the
string encoders, which are transmitted on a different channel than the regular
quadrature pulses. Through its 200 mm range of travel, each string encoder
produces three index pulses, equally spaced apart by 4000 counts, which is
equivalent to one-third of the total measurement range. The location of the
first index pulse differs for each string encoder, and the absolute
displacement that corresponds to this first index pulse was recorded before
installing each of the string encoders.

With this information, each time the string encoder system is powered on, the
measuring ends can be brought as close to the encoder bodies as possible in
order to trigger the first index pulse, which is captured by the FPGA
firmware. By calculating the difference between these values and the known
positions, we can set the offsets to obtain the correct absolute
displacements.

###  II-E Calibration of string length adjustment

We observed that the attachment points for the string encoders are affected by
a slight deviation from the ideal model presented earlier. As demonstrated in
Fig. 4, the designed attachment point for the string is the center of its
circular guide channel on the body of the encoder, but the string in fact
makes contact with the edge of the channel instead of going through the
center. A potential solution would be to adjust the attachment point
coordinates to account for the shift. However, when the helmet platform moves,
the string also frequently shifts its position within the channel, especially
when the movement is around the center point. This would require the
attachment point coordinates to be updated during operation, which would
complicate the approach.

Figure 4: Deviation of the encoder string from the hole center

Instead, we recognized that due to the angled orientation of each string
encoder, the string almost always lies against the edge of the channel—in
fact, with our range of motion implemented below, the string never extends
from the center along the direction of the guide channel. Therefore, the
actual reading of each string encoder is less than the nominal situation where
the string passes through the center of the attachment point, and this
difference is always a fixed amount. We then attempted to identify an optimal
value for this string length offset through empirical testing, experimenting
with offset lengths ranging from -2 mm to 6 mm. For each setting, we put the
robot through a set range of motion in all 6 degrees of freedom and collected
a series of 66 data points. Evaluation of the results and determination of the
optimal string length offset are presented in the following section.

###  II-F Transformation between robot and string encoder system

To compare the displacement measured by the string encoder system with the
ground-truth displacement provided by the UR3 robot, a transformation between
the two coordinate systems is required. We accomplished this by considering
that the transformation between the PET imaging ring (base) and the robot,
${}^{B}E_{R}$, is fixed for this experiment because the UR5 robot does not
move. The string encoder system measures the transformation between the PET
imaging ring and the helmet, ${}^{B}E_{H}$, and after setting the robot tool
center point (TCP) to be coincident with the center of the helmet, the robot
interface can output the corresponding transformation between the robot and
the helmet, ${}^{H}E_{R}$. Consequently,
${}^{B}E_{R}={{}^{B}E_{H}}{{}^{H}E_{R}}$, and for a constant position of the
base, the string encoder system only needs to query the robot once to obtain
${}^{B}E_{H}$ and then compute ${}^{B}E_{R}$. Afterwards, the measurements
from the string encoder system can be converted to a 6 DOF position in the
robot coordinate system by computing ${(^{B}E_{H})^{-1}}{{}^{B}E_{R}}$, and
the full 6 DOF pose can be extracted from the resulting transformation.

##  III Results

###  III-A Experimental setup

We used a UR3 robot to provide precise motions of the helmet. We first
verified the measurement accuracy of the UR3 robot, for a $\pm$10 mm range of
motion, using a dial indicator (543-693B, Mitutoyo Corp., Japan), as shown in
Fig. 5. We limit the robot motion to this specific range since the designed
clearance between the helmet and the imaging ring is only about 15 mm, as
outlined in Section II-A. Table II summarizes the result of the verification.
The difference indicated in the third column could have been partially caused
by the rough surface on the 3D-printed part that was mounted on the UR3 end-
effector. Nevertheless, we can conclude that the UR3 accuracy is about 0.1 mm,
and hence precise enough to provide the ground-truth helmet motion in our
experiments.

Figure 5: Dial indicator for measuring robot translation accuracy TABLE II: UR3 robot accuracy evaluation, showing commanded robot displacement, measured dial indicator displacement and difference (units: mm) Robot | Dial Indicator | Difference
---|---|---
-10 | -9.880 | -0.120
-9 | -8.884 | -0.116
-8 | -7.898 | -0.102
-7 | -6.912 | -0.088
-6 | -5.911 | -0.089
-5 | -4.914 | -0.086
-4 | -3.950 | -0.050
-3 | -2.951 | -0.049
-2 | -1.946 | -0.054
-1 | -0.956 | -0.044
0 | 0.004 | -0.004
1 | 0.966 | 0.034
2 | 1.956 | 0.044
3 | 2.956 | 0.044
4 | 3.923 | 0.077
5 | 4.925 | 0.075
6 | 5.909 | 0.091
7 | 6.913 | 0.087
8 | 7.899 | 0.101
9 | 8.881 | 0.119
10 | 9.874 | 0.126

###  III-B Calibration results

For each string length offset, we obtained the vector difference $\Delta P$
between the displacement measured by the string encoder system, $P_{enc}$, and
the actual displacement produced by the UR3 robot, $P_{rob}$, at each of the
66 points. We then took the norm of these error vectors, $|\Delta P|$,
obtained the root-mean-square (RMS) as well as the maximum across all 66
points, and plotted them against the respective offset, as shown in Fig. 6.
All offsets from 2.5 mm to 5 mm share similar RMS error magnitudes of around
0.3 mm, but the 3 mm offset is shown to have the smallest maximum error
magnitude.

We also used calipers to measure the string outer diameter, which was 0.8 mm,
and the inner diameter of its guide channel, which was 2.8 mm, giving a
clearance of 1 mm. A similar phenomenon is also present on the moving end of
the encoder attached on the helmet, where the position of the string is also
shifted by about 1 mm. Additionally, we observed the difference in the encoder
readings before and after temporarily holding one string at the two designated
attachment points using tools, and noted that the string length reading
increased by 2.5 mm. Therefore, our selection of the 3 mm offset based on
error minimization is supported by physical measurement, and this value is
used in the subsequent accuracy evaluation.

Figure 6: Calibration result for string length offsets

###  III-C Accuracy evaluation

We first determined the transformation between the UR3 robot coordinate system
and the string encoder coordinate system, as described in Section II-F. Then,
the UR3 robot performed a range of motion of $\pm$10 mm or deg along each of
the axes, one axis at a time, in increments of 1 mm or deg. At each position,
the string lengths were recorded, adjusted by the 3 mm string length offset
identified above, and then the forward kinematics applied to compute the
Cartesian position. The results are shown in Table III as well as in Fig. 7.

TABLE III: Translation error, $|\Delta P|$ (mm), due to displacements (in mm or deg) from origin, $\Delta P=P_{enc}-P_{rob}$. Displacement |  $|\Delta P|$ (mm)
---|---
(mm or deg) | X | Y | Z | Roll | Pitch | Yaw
-10 | 0.340 | 0.438 | 0.176 | 0.090 | 0.569 | 0.462
-9 | 0.222 | 0.379 | 0.109 | 0.207 | 0.652 | 0.460
-8 | 0.208 | 0.301 | 0.156 | 0.178 | 0.519 | 0.397
-7 | 0.265 | 0.275 | 0.165 | 0.210 | 0.425 | 0.428
-6 | 0.116 | 0.219 | 0.227 | 0.194 | 0.433 | 0.460
-5 | 0.161 | 0.165 | 0.242 | 0.189 | 0.294 | 0.316
-4 | 0.121 | 0.146 | 0.254 | 0.162 | 0.255 | 0.258
-3 | 0.054 | 0.139 | 0.232 | 0.175 | 0.161 | 0.265
-2 | 0.129 | 0.023 | 0.108 | 0.143 | 0.164 | 0.213
-1 | 0.030 | 0.121 | 0.040 | 0.075 | 0.103 | 0.173
0 | 0.068 | 0.045 | 0.111 | 0.115 | 0.111 | 0.018
1 | 0.198 | 0.126 | 0.126 | 0.033 | 0.159 | 0.061
2 | 0.138 | 0.126 | 0.035 | 0.059 | 0.216 | 0.030
3 | 0.127 | 0.118 | 0.108 | 0.106 | 0.382 | 0.215
4 | 0.199 | 0.201 | 0.056 | 0.161 | 0.421 | 0.252
5 | 0.264 | 0.270 | 0.204 | 0.208 | 0.526 | 0.250
6 | 0.260 | 0.225 | 0.179 | 0.287 | 0.663 | 0.323
7 | 0.272 | 0.323 | 0.240 | 0.336 | 0.758 | 0.494
8 | 0.357 | 0.342 | 0.238 | 0.408 | 0.792 | 0.556
9 | 0.363 | 0.375 | 0.324 | 0.451 | 0.968 | 0.574
10 | 0.394 | 0.425 | 0.333 | 0.522 | 1.003 | 0.734
RMS | 0.223 | 0.257 | 0.193 | 0.241 | 0.529 | 0.377
Figure 7: Accuracy evaluation result of the string encoder system. Horizontal
axes represent displacements (mm or deg), and vertical axes represent the
error magnitude $|\Delta P|$ (mm)

The results indicate that we have achieved RMS errors less than 0.3 mm for
translations along all three axes, and for rotation around the z-axis, and
less than 0.6 mm for the other two rotations. For the translations, even the
maximum error magnitude within the $\pm$10 mm motion range was less than the
0.5 mm requirement for fine motion correction.

##  IV Conclusions

We developed a mechanical 6 DOF measurement system consisting of 6 parallel
string encoders, connected in a Stewart platform configuration. This system is
intended to be used to measure the motion of a helmet, worn by a human
subject, with respect to a PET imaging ring supported by a robotic system. We
performed experiments, using a robot to provide precise helmet motions, and
verified that RMS translation errors are usually less than 0.3 mm, with values
slightly above 0.5 mm for some rotations. These relatively larger errors
associated with displacements in the pitch and yaw angles might be attributed
to inaccuracy in setting the robot TCP, a process mentioned in section II-F.
Since the TCP is set to be a constant offset in the direction (Z)
perpendicular to the mounting surface of the robot end effector, an inaccuracy
in setting this offset would cause robot rotations about the tool X and Y axes
(yaw and pitch, respectively) to also cause translations that would be
measured by the string encoder system. Nonetheless, the performance described
above is sufficient for coarse motion correction, where a robot moves the PET
imaging ring to attempt to keep it centered around the head. It may be
suitable for fine motion correction, performed during image reconstruction,
where an accuracy of 0.5 mm or better is desired.

Our testing only evaluated one motion axis at a time—it is possible that
combined motions (e.g., simultaneous translation and rotation) would produce
higher errors. We have not yet identified a specification for the largest
rotation error and our testing did not measure the rotation error (we only
measured the translation error resulting from applied rotations). Considering
that the maximum radius of a human head is about 100 mm, a rotation error of
0.3 degrees would produce an error of about 0.5 mm on the skull surface.
However, for neuroscience research, targets of interest will likely be closer
to the center of the head and therefore the effect of rotation error may be
less critical. Our future work includes determination of the rotation accuracy
specification for fine motion correction. In addition, if the UR5 robot is
able to keep up with the head motion, the relative displacement of the head
with respect to the imaging ring may be much smaller than the 10 mm and 10
degree displacements tested here. In that case, we could expect higher
accuracy as our results indicate higher accuracy for displacements closer to
the center position. If necessary, we can perform a kinematic calibration of
the string encoder system using methods developed for Stewart platforms [14]
to further increase the accuracy.

Our next stage of development will be to use the UR3 robot to emulate
realistic human head motion, using the previously recorded data analyzed in
our prior work [8], and to use the UR5 robot to move the PET imaging ring
based on those measurements. Eventually, since the PET imaging ring can weigh
up to 20 kg and the UR5 has a payload of only 5 kg, the UR5 would be replaced
by a custom robot, but currently it serves as a prototype for verification of
our motion measurement and compensation system.

## Acknowledgements

Yangzhe Liu participated in early design discussions of the measurement
system.

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