* The authors contributed equally to this paper.

# Vibrational properties of LiNbO3 and LiTaO3 under uniaxial stress

Ekta Singh<EMAIL_ADDRESS>Institut für Angewandte Physik,
Technische Universität Dresden, 01062 Dresden, Germany  Mike N. Pionteck  :
<EMAIL_ADDRESS>Institut für Theoretische Physik and
Center for Materials Research (LaMa), Justus-Liebig-Universität Gießen, 35392
Gießen, Germany  Sven Reitzig  Institut für Angewandte Physik, Technische
Universität Dresden, 01062 Dresden, Germany  Michael Lange  Institut für
Angewandte Physik, Technische Universität Dresden, 01062 Dresden, Germany
Michael Rüsing  Institut für Angewandte Physik, Technische Universität
Dresden, 01062 Dresden, Germany  Lukas M. Eng  Institut für Angewandte Physik,
Technische Universität Dresden, 01062 Dresden, Germany  ct.qmat: Dresden-
Würzburg Cluster of Excellence—EXC 2147, TU Dresden, 01062 Dresden, Germany
Simone Sanna  Institut für Theoretische Physik and Center for Materials
Research (LaMa), Justus-Liebig-Universität Gießen, 35392 Gießen, Germany

###### Abstract

Structural strain severely impacts material properties such as the linear and
non-linear optical response. Moreover, strain plays a key role, e.g., in the
physics of ferroelectrics and in particular of their domain walls.
$\upmu$-Raman spectroscopy is a well-suited technique for the investigation of
such strain effects, as it allows to measure the lattice dynamics locally.
However, quantifying and reconstructing strain fields from Raman maps requires
knowledge on the strain dependence of phonon frequencies. In this work, we
have analyzed both theoretically and experimentally the phonon frequencies in
the widely used ferroelectrics lithium niobate and lithium tantalate as a
function of uniaxial strain via density functional theory and $\upmu$-Raman
spectroscopy. Overall, we find a good agreement between our _ab initio_ models
and the experimental data performed with a stress cell. The majority of
phonons show an increase in frequency under compressive strain, while the
opposite is observed for tensile strains. Moreover, for E-type phonons, we
observe the lifting of degeneracy already at moderate strain fields (i.e. at
$\pm$0.2\text{\,}\%$$) along the x and y directions. This work hence allows
for the systematic analysis of 3D strains in modern-type bulk and thin-film
devices assembled from lithium niobate and tantalate.

Lithium niobate, lithium tantalate, strain, stress, ferroelectric, domain
walls, Raman spectroscopy, DFT

##  I Introduction

Lithium niobate (LN) and Lithium tantalate (LT) are cornerstone materials in
nonlinear optics and optoelectronics, with established applications ranging
from second-harmonic generation [1, 2], over optical ring resonators, to
holographic memory devices [3], to name just a few. These devices are based on
bulk single crystal platforms and have been constantly optimized over the last
decades via meticulous analyses of crystal compositional aspects [4, 5, 6],
degradation effects [7], defect distribution, and the precise imaging of
domain structures [8, 9, 10]. Single crystalline LN (or LT) thin films on
insulating substrates have recently opened up a completely new scenario. This
new platform allows for much higher integrability into existing nanoelectronic
and nanophotonic structures [11, 12] and outperforms bulk-based devices due to
the submicron-scale confinement of optical modes [13, 14, 15].

Nevertheless, the transition towards thin-film LN (TFLN) devices emphasizes
the importance of crystal properties that so far have played a minor role in
LN bulk investigations and device fabrication. Among these, we count the
increased influence of mechanical stress ($\sigma$) fields [16, 17], e.g., due
to the bonding process of the LN thin film onto the substrate, or occurring as
a result of the poling procedure or waveguide fabrication, which is paramount
for the reliable and reproducible production of highly efficient nonlinear
optical devices. Accumulated stress does not only increase the chance of
critical failure (e.g. due to delamination) [16, 17], but also influences
optical key properties such as refractive indices [18, 19] and thus endangers
the optical responses of a periodically-poled domain grid [20].

Among established imaging techniques applied for the analysis of LN platforms,
Raman spectroscopy can provide critical information about internal mechanical
stress. To discriminate the influence of stress fields from other factors,
e.g. electrical fields [21] or compositional heterogeneity [22], and to
quantify occurring stress markers, it is essential to gain fundamental
knowledge on the relation of phonon properties and mechanical stress fields.
To date, the only investigations in this regard were conducted via hydrostatic
pressure-cells [23, 24]. However, such experiments only investigate the
influence of isotropically applied stress. In reality, effects like stress
fields in the vicinity of LN DWs, or mechanical stress in LN thin films, are
not isotropic [25, 26]. Under these environments, the directionality of stress
largely changes the materials properties. Therefore, it is necessary to
theoretically predict and experimentally verify the influence of uniaxial
stress on phonon properties. This then provides a fundamental reference for
future investigations of mechanical stress fields, in analogy to the reference
work of Stone _et al._ [21] for incident electrical fields.

In this work, we firstly provide the theoretical prediction on the evolution
of the phonon properties in LN and LT crystals upon uniaxial compression along
all three crystal axes via density functional theory (DFT) in Sec. II.1.
Secondly, we combine a piezo-driven uniaxial stress cell with a $\mu$-Raman
spectroscopy setup to enable the experimental study of stress on the LN
crystals. Experimentally, we compare stoichiometric and 5$\%$ MgO-doped LN
samples compressed along the x,y, and z axes [see Sec. II.2]. Sec. III.3 then
provides a discussion of the calculated and measured data.

With this work, we thus provide a basis for the quantitative, local stress
analysis in LN and LT platforms via $\mu$-Raman spectroscopy and give a
reference for the deeper understanding and further optimization of LN- and LT-
based devices.

##  II Methodology

###  II.1 Computational Approach

The DFT calculations are performed within the generalized gradient
approximation (GGA) [27] in the formulation of Perdew, Burke, and Ernzerhof
(PBE) [28] as implemented in the Vienna Ab Initio Simulation Package (VASP)
[29]. Thereby, projector augmented wave (PAW) [30] potentials with projectors
up to $l=3$ for Nb and Ta, and $l=2$ for Li and O have been used, which proved
to be accurate enough as demonstrated in a previous work [31]. The electronic
wave functions are expanded into a plane-wave basis set up to kinetic energy
of $475\text{\,}\mathrm{e}\mathrm{V}$. In order to model uniaxial stress, the
lattice constants of the rhombohedral unit cell were adjusted according to the
expression

$\boldsymbol{a}_{n}(\mathrm{\epsilon})=\left(\boldsymbol{I}+\mathrm{\epsilon}\right)\boldsymbol{a}_{n}(0)\mathrm{,}$
(1)

where $\mathrm{\epsilon}$, $\boldsymbol{I}$, $\boldsymbol{a}_{n}(0)$, and
$\boldsymbol{a}_{n}(\mathrm{\epsilon})$ denote the strain tensor, the identity
matrix, and the nth unstrained and strained lattice vector, respectively. The
strain tensor was then calculated using the elastic compliance tensor $S$ from
Ref. [32],

$\mathrm{\epsilon}=S\mathrm{\sigma}\mathrm{,}$ (2)

and the boundary conditions for uniaxial stress. This means that all
components of the stress tensor $\mathrm{\sigma}$ are set to zero, except for
the component in the direction of which uniaxial stress is to be exerted. For
instance, if uniaxial stress is to be calculated along the x direction,
$\mathrm{\sigma}_{1}\neq
0=\mathrm{\sigma}_{2}=\mathrm{\sigma}_{3}=\mathrm{\sigma}_{4}$ (Voigt
notation) is set as boundary condition. Due to constraints of the unit cell
sizes, only stoichiometric LN and LT is calculated. However, as later
demonstrated by experiments by comparing 5% MgO-doped and stoichiometric LN,
it is shown that the results for 5% MgO-doped and stoichiometric LN are
comparable, therefore this assumption is justified, to only calculate
stoichiometric LN and LT.

The calculations were performed in two ranges for LN and LT: First, the
effects of strain, i.e. compressive as well as tensile strain, on the phonon
frequencies have been calculated in fine increments up to values reachable in
the experiments. Second, calculations for higher values of strains, but more
coarse increments, were performed. For LN and LT this procedure has been
performed for lower strains up to $0.10\text{\,}\%$, as well as for higher
values up to $2.4\text{\,}\%$. Here, increments of $0.02\text{\,}\%$ and
$0.4\text{\,}\%$ have been chosen, respectively. The Hellmann-Feynman forces
are minimized under a threshold value of
$0.005\text{\,}\mathrm{e}\mathrm{V}\mathrm{/}\text{\AA}$ by relaxation of the
atomic positions. For our calculations of unstrained structures we have used
commonly accepted experimental lattice constants
($a_{\text{R}}^{\text{LN}}=$5.494\text{\,}\text{\AA}$$,
$\alpha^{\text{LN}}=$55.867\text{\,}{}^{\circ}$$ and
$a_{\text{R}}^{\text{LT}}=$5.474\text{\,}\text{\AA}$$,
$\alpha^{\text{LT}}=$56.171\text{\,}{}^{\circ}$$) [31].

Table 1: Description of the samples used for the experimental stress measurement analysis. sample name | description | size | stress axis
---|---|---|---
sLN-x | z-cut stoichiometric LN | y $\times$ z: 580 $\times$ 113 $\upmu$m2 | x
sLN-y | z-cut stoichiometric LN | x $\times$ z: 580 $\times$ 113 $\upmu$m2 | y
5Mg-LN-x | z-cut 5$\%$-MgO doped LN | y $\times$ z: 500 $\times$ 140 $\upmu$m2 | x
5Mg-LN-y | z-cut 5$\%$-MgO doped LN | x $\times$ z: 400 $\times$ 100 $\upmu$m2 | y
5Mg-LN-z | x-cut 5$\%$-MgO doped LN | y $\times$ x: 400 $\times$ 100 $\upmu$m2 | z

The $\Gamma$-centered phonon frequencies and eigenvectors are derived by the
frozen-phonon method [33] without symmetry constraints. For the calculation of
the Hessian matrix, atomic displacements of $0.015\text{\,}\text{\AA}$ in each
Cartesian direction are considered. Since our approach does not take into
account the long-range electric fields accompanying the longitudinal-optical
(LO) phonons, our calculations are restricted to transversal-optical (TO)
phonons, only. A $8\times 8\times 8$ k-point mesh is used to sample the first
Brillouin-zone corresponding to the orthorhombic unit cell, which yields 192
irreducible points.

###  II.2 Experimental Procedure

In order to complement the above theoretical investigations, we measured the
frequency shift of the phonon modes in LN by combining $\upmu$-Raman
spectroscopy with a uniaxial stress cell [34]. All measurements were performed
using a LabRAM HR spectroscope from Horiba in 180∘ back-scattering geometry.
In this setup, a HeNe laser at $632.8\text{\,}\mathrm{nm}$ wavelength was
focused onto the samples using a 100X and 0.9 NA objective. For each data
point, two acquisitions were collected for $10\text{\,}\mathrm{s}$ each, while
using a diffraction grating of 1800 grid lines per millimeter for spectral
analysis. For these settings, the corresponding spectral resolution is
$0.56\text{\,}\mathrm{c}\mathrm{m}^{-1}$, which was determined as part of the
calibration using a He-Ne gas discharge lamp. Here, the plasma line at a
wavelength of 671.7 nm has a natural line width of 0.025 nm
($0.56\text{\,}\mathrm{c}\mathrm{m}^{-1}$), which corresponds to the spectral
resolution i.e. the standard deviation, which only means how well the two
overlapping peaks can be resolved. On the other hand, with the help of the
standard error obtained from peak fitting, each peak center can be pinpointed
with a 100-1000 times more precision, and this is the quantity we are
interested in.

All the samples used in this work were firmly attached to the uniaxial stress
cell using black stycast epoxy resin [35] such that the strain is fully
transmitted to the sample through the epoxy resin. The force applied to the
sample was directly measured through a force sensor attached to the back of
the device. This force was later converted to stress using the relation

$\sigma=\dfrac{F}{A},$ (3)

where $A$ denotes the cross-section of the samples provided in table 1, and
$F$ is the force measured by the force sensor. In order to compensate for any
charge-induced effects due to the piezoelectric effect, the top and bottom
faces of the sample surfaces were electrically grounded using
$10\text{\,}\mathrm{nm}$-thick chromium electrodes. All the measurements were
always performed across the chromium electrodes, which at the given thickness
are still sufficiently transparent and do not affect our measurements, as also
shown in the past for in situ second-harmonic microscopy studies [36].

A total of five samples were measured [see Tab. 1]. First, stoichiometric LN,
being close to the ideal crystals considered for the theoretical calculations,
is measured and used to compare theory with experiment. Second, 5$\%$ MgO-
doped LN samples are chosen, because they are widely used among the
ferroelectrics and integrated optics community due to their low
photorefraction and highly conductive domain walls [12, 37, 38, 39]. Comparing
the results on the stoichiometric and MgO-doped crystals allows us to see
whether or not there are any stoichiometry-specific changes in the responses,
which would also not be covered by the DFT calculations.

####  II.2.1 Data extraction method

The Raman spectra of single-crystalline LN and LT are well understood and have
been investigated thoroughly before [31, 40, 41, 42, 43]. The optical phonons
of LN and LT consist of four A1, five A2, and nine doubly-degenerated E phonon
branches. Here, A1 and E are optical branches and are both Raman and Infrared
active. In contrast, the A2 branch is optically inactive. Furthermore,
depending on the scattering geometry these branches either appear as TO or LO
modes. Conveniently LN and LT peaks are labeled by type of symmetry (A1 or E),
followed by the specific type (TO or LO) and consecutive number counting from
low to high frequency. For example, the phonon E(TO9) labels the ninth, i.e.
highest frequency, E-type TO phonon. The assignment of peaks in this work has
been performed according to the most recent accepted assignment by Margueron
_et al._ , Sanna _et al._ and Rüsing _et al._ [41, 42, 31].

Figure 1(a), (b) and (c) shows example Raman spectra of the 5Mg-LN-x, 5Mg-
LN-y, and 5Mg-LN-z samples, respectively. These spectra are measured in
different scattering geometries represented by the Porto notation
$k_{i}(e_{i},e_{s})k_{s}$, where $k_{i}$ and $k_{s}$ represent the direction
of incident and scattered light in crystal coordinates, and $e_{i}$ and
$e_{s}$ denote the direction of polarization of both incoming and Raman
scattered beams, respectively [41]. Here, each plot contains 3 spectra, where
spectrum A belongs to the unstressed ($0\text{\,}\mathrm{MPa}$) state of the
sample while spectrum B describes a compressed state of the sample. The third
is the difference spectrum of the former two. To eliminate the intensity
influences in the difference spectrum and to solely highlight the effect of
the frequency shift due to applied stress, all the spectra were first
normalized individually by subtracting the noise background (if applicable)
and divided by the intensity of the highest peak [10]. From the different
spectra of Fig. 1(b), it can be clearly noticed that the frequency of the
E(TO1) phonon mode in $\mathrm{\overline{Z}}$(XY)Z geometry increased under
compression, while the frequency for A1(TO1) in $\mathrm{\overline{X}}$(ZZ)X
geometry decreased. Apart from these large differences, further changes are
difficult to notice. So, to quantify frequency shifts as a function of stress,
the fitting of the spectra by Lorentzian functions is performed. In this way,
the frequencies at the maxima of the phonon peaks were determined for
different stress values and plotted against applied stress, such as shown in
Fig. 2. Via statistical analysis of the phonon shifts, we are able to deduce
the experimental phonon-stress-shift coefficients with a precision of 0.01
cm-1/$\%$.

Additionally, it is well-known [44, 45] that the spectra of congruent or MgO-
doped LN show broader and less resolvable peaks as compared to the
stoichiometric LN. Here, various peaks overlap with each other, that then
leads to a more challenging peak assignment. For example, the pairs E(TO3)-
A1(LO1), A1(LO2)-E(TO4), and A1(LO3)-E(TO7) overlap with each other in the
$\mathrm{\overline{Z}}$(XX)Z and $\mathrm{\overline{Z}}$(YY)Z geometry of MgO-
doped LN. Therefore, in Sec. III.1, to avoid any misleading conclusions, for
these peaks the Raman shifts with the stress have not been fitted [see Table 2
and 4].

Furthermore, to find an optimum location on the sample, a series of
optimization measurements [see SI for further information] are performed. As a
result, all the measurements were performed in the center of the sample and 10
$\upmu$m below the sample surface.

Figure 1: (a) $\mathrm{\overline{Z}}$(XX)Z Raman spectra of unstressed (0 MPa)
and x compressed (-85.7 MPa) 5$\%$ MgO-doped LN (5Mg-LN-x). (b)
$\mathrm{\overline{Z}}$(XY)Z Raman spectra of unstressed (0 MPa) and y
compressed (-140.7 MPa) 5$\%$ MgO-doped LN (5Mg-LN-y). (c)
$\mathrm{\overline{X}}$(ZZ)X Raman spectra of unstressed (0 MPa) and z
compressed (-175 MPa) z-axis of 5$\%$ MgO-doped LN (5Mg-LN-z). The difference
curve in all (a)-(c) graphs represents the difference spectra between the
stressed and unstressed states of the sample.

##  III Results

Figure 2:  Measured response of the E(TO1) phonon mode (a)-(b) in the samples
sLN compressed along the x and y axes, respectively and (c)-(d) in the sample
5Mg-LN compressed along the x and the y axes, respectively. The stoichiometric
and 5$\%$ MgO-doped LN behave similarly.

###  III.1 Experimental Results

For the sake of simplicity, we only show the pressure dependency for selected
phonon modes; the complete results are shown in the Supplement information
(SI).

Fig. 2 shows the response of the E(TO1) phonon mode under x and y compression
for both the stoichiometric and 5$\%$-MgO doped LN samples. The three curves
in a single plot represent the three different scattering geometries given in
Porto’s notation. Since the change of the frequency ($\Delta\omega$) with
respect to the unstressed state is the quantity of interest, we have plotted
the frequencies with respect to the $0\text{\,}\mathrm{MPa}$ value as a
function of stress. Hence, we also account for any offset due to sample
mounting or bending effects as discussed in the SI.

The coloured region around the fitted lines show the one-sigma confidence
interval, calculated from linear fitting which included the standard error of
individual peaks. The E(TO1) mode shows a linear positive slope for the
compression in all cases. In the case of x compression for both sLN-x and 5Mg-
LN-x samples, the slope is larger in $\mathrm{\overline{Z}(YY)Z}$ measurement
geometry. On the other hand, when the same samples are compressed along the y
crystallographic axis, the slope is larger for $\mathrm{\overline{Z}(XY)Z}$
and $\mathrm{\overline{Z}(XX)Z}$ scattering geometries. We have observed this
behavior for almost all the phonon modes [see SI]. The value of different
slopes in different measurement geometry can be interpreted as the lifting of
the two-fold degeneracy of the E-modes predicted by theory as discussed below.

When comparing 5Mg-LN-x/-y and sLN-x/-y, the slopes are similar as shown in
Fig. 2. The origin of the small deviations could be due to the presence of
defects in doped-LN. This argument is consistent with the work done by
Tejerina _et al._ , where they show that doping increases the phonon shift
under hydrostatic pressure ($\Delta\omega$/$\Delta\sigma$) [46], especially
for the A1(TO2), A1(TO3), and E(TO4) phonon modes. The comparison for other
peaks is also summarized in Table 2.

As will later be shown by theory, the phonon frequency should decrease under
tension with a slightly different magnitude. That is why the phonon response
of sample sLN-x is also measured experimentally in
$\mathrm{\overline{Z}(YY)Z}$ scattering geometry under tension. For tensile
strain, the opposite response is observed as expected: the frequency decreased
on applying tension to the sample. For example, phonon mode E(TO6) shows the
largest response as for compression with a slope of $\mathrm{-8.15}$ $\pm$
2.88 cm-1/GPa. The plot for E(TO6) and the slopes for all peaks are provided
in SI.

Table 2:  Experimentally determined Raman shift changes $\Delta\omega$ as a function of stress $\sigma$ for x compression. The slopes were obtained after linearly fitting the experimental changes in frequency upon x compression. An asterisk labels silent modes [41]. In the case of 5$\%$ MgO-doped LN, many peaks overlap with each other [see Fig. 1(a)-(b)], therefore extracting the peak frequency with an acceptable error was not possible. This is represented by the ’-’ sign. | sLN-x (cm-1/GPa) | 5Mg-LN-x (cm-1/GPa)
---|---|---
Phonon modes |  $\mathrm{\overline{Z}}$(XX)Z |  $\mathrm{\overline{Z}}$(XY)Z |  $\mathrm{\overline{Z}}$(YY)Z |  $\mathrm{\overline{Z}}$(XX)Z |  $\mathrm{\overline{Z}}$(XY)Z |  $\mathrm{\overline{Z}}$(YY)Z
$\mathrm{A_{1}(LO_{1})}$ | 2.13 $\pm$ 0.52 | * | 2.06 $\pm$ 0.58 | - | * | -
$\mathrm{A_{1}(LO_{2})}$ | 3.29 $\pm$ 0.17 | * | 3.66 $\pm$ 0.17 | - | * | -
$\mathrm{A_{1}(LO_{3})}$ | 0.39 $\pm$ 0.94 | * | 0.29 $\pm$ 0.71 | - | * | -
$\mathrm{A_{1}(LO_{4})}$ | 1.96 $\pm$ 0.21 | * | 2.32 $\pm$ 0.19 | 3.07 $\pm$ 0.83 | * | 3.79 $\pm$ 0.80
$\mathrm{E(TO_{1})}$ | 0.70 $\pm$ 0.26 | 0.63 $\pm$ 0.26 | 1.98 $\pm$ 0.27 | 0.71 $\pm$ 0.35 | 1.05 $\pm$ 0.22 | 2.68 $\pm$ 0.29
$\mathrm{E(TO_{2})}$ | 0.34 $\pm$ 0.17 | 0.26 $\pm$ 0.25 | 1.68 $\pm$ 0.16 | -0.51 $\pm$ 0.45 | -0.63 $\pm$ 0.52 | 1.52 $\pm$ 0.3
$\mathrm{E(TO_{3})}$ | 1.28 $\pm$ 0.44 | 1.39 $\pm$ 0.65 | 1.67 $\pm$ 0.42 | - | - | -
$\mathrm{E(TO_{4})}$ | 2.12 $\pm$ 0.35 | 1.56 $\pm$ 0.59 | 3.02 $\pm$ 0.4 | - | - | -
$\mathrm{E(TO_{5/6})}$ | 6.36 $\pm$ 0.98 | 6.25 $\pm$ 1.5 | 8.17 $\pm$ 0.87 | 5.87 $\pm$ 4.23 | 3.77 $\pm$ 3.69 | 9.38 $\pm$ 2.16
$\mathrm{E(TO_{7})}$ | 0.44 $\pm$ 0.37 | 0.86 $\pm$ 1.51 | 1.24 $\pm$ 0.22 | - | - | -
$\mathrm{E(TO_{8})}$ | -0.47 $\pm$ 0.30 | 0.37 $\pm$ 0.36 | 3.03 $\pm$ 0.27 | -0.29 $\pm$ 0.8 | 1.66 $\pm$ 0.69 | 4.32 $\pm$ 0.67

###  III.2 Strain Simulation

First, to assess the quality of the models in this work, the calculated phonon
frequencies of unstrained stoichiometric LN and LT are compared with previous
works [31] (all data appear in the SI). Compared to our experiment, the
largest deviation from the experimental values for LN and LT occurs for the
A1(TO4) mode with $20\text{\,}\mathrm{cm}^{-1}$ and
$22\text{\,}\mathrm{cm}^{-1}$, respectively. The mean deviation from
experimental values for LN and LT is about $8.7\text{\,}\mathrm{cm}^{-1}$ and
$8.4\text{\,}\mathrm{cm}^{-1}$. Thus, compared with the mean deviation of the
theoretical results of Ref. [31] which was about
$10.7\text{\,}\mathrm{c}\mathrm{m}^{-1}$ and
$10.8\text{\,}\mathrm{c}\mathrm{m}^{-1}$ for LN and LT, there is a close
agreement with both the theoretical and experimental values. In addition, the
typical deviations are also consistent with literature values, such as Refs.
[41, 47].

In order to demonstrate the dependence of the phonon frequencies predicted by
our calculations, the E(TO4) mode of LN was chosen [see Fig. 3]. Figures 3(a)
and (b) show the results of the calculations with lower and higher strains,
respectively. The frequency shift of the E(TO4) mode has an approximately
linear dependence on the applied compression. Such a linear dependence is
observed in all of our calculations for each mode for both the compressive and
tensile strain in stoichiometric LN and LT. This is consistent with the
results of LN and LT under hydrostatic pressure by Mendes-Filho _et al._ [23].
In Fig. 3(a) the frequency increases up to $6\text{\,}\mathrm{cm}^{-1}$ and in
(b) up to $0.3\text{\,}\mathrm{cm}^{-1}$.

Figure 3: DFT calculated relative frequency shifts of the E(TO4) mode of
stoichiometric LN as a function of strain in z-direction in (a) steps of
$0.4\text{\,}\%$ and (b) steps of $0.02\text{\,}\%$. The linear fit and the
standard deviation of both are plotted and their parameters displayed. The
slopes for strain are defined as
$\mathrm{(\Delta\omega)}/(\mathrm{|\Delta\epsilon|)}$ and displayed with the
fitting error.

Our calculations predict for stoichiometric LN and LT that the degeneracy of
the E modes is lifted under strain applied along the x and y direction due to
the reduction of the threefold symmetry. This results in splitting into two
branches, which is consistent with previous studies on various 2D materials,
that reported a mode splitting under strain due to the reduction of symmetry
[48, 49]. Consequently, for strains applied along the x and y direction, all E
modes are considered separately. For example, the E(TO9) mode for compression
in x and y directions has a significant splitting [see Fig. 4(a)]. In
contrast, it is noticeable that the E(TO9) mode remains degenerate for z
compression since the considered strains in the z direction of both LN and LT
do not lead to any reduced symmetry. This is the case for all E modes of LN
and LT under strain.

Figure 4: The DFT calculated relative frequency shifts of (a) E(TO9) (first
row), (b) E(TO5) (second row) and (c) A1(TO2) (third row) modes of
stoichiometric LN as a function of strain in steps of $0.4\text{\,}\%$ along
the x, y and z direction. The slopes for strain are defined as
$\mathrm{(\Delta\omega)}/(\mathrm{|\Delta\epsilon|)}$ and displayed with the
fitting error.

Since the slopes, for strains in the x and y directions of LN and LT, are very
similar [see SI] due to the symmetry of the crystals, Table 3 shows only the
slopes for strain in the x direction. With the exception of the A1(TO1) and
A1(TO4) modes, compressive strain leads to an increase in frequency, while
tensile strain leads to a decrease in frequencies, which is consistent with
results from previous research on other materials [50, 51]. Comparing the
slopes under the compressive and tensile strain of LN, we find that they
hardly differ for the individual modes. This behavior is not surprising,
however, it is not universal, as shown for a 2D material by Pak _et al._ [50].
Although under both compressive and tensile strain the E(TO5/6) modes have the
largest frequency shifts, the E(TO6) mode has a larger slope than the E(TO5)
mode under compression. However, this relation turns around under tensile
strain as can be seen in Table 3. This applies to strain in the x and y
direction for both LN and LT (see SI).

Table 3: Calculated slopes of transversal A1 and E modes of stoichiometric LN under compressive and tensile strain, and stoichiometric LT under compressive strain in the x-direction at higher strains in $\mathrm{cm}^{-1}$/%. The slopes for strain are defined as $\mathrm{(\Delta\omega)}/(\mathrm{|\Delta\epsilon|)}$. Phonon | LN comp. | LN tens. | LT comp.
---|---|---|---
mode | $\mathrm{\Delta\omega}\mathrm{[cm^{-1}/\%]}$ | $\mathrm{\Delta\omega}\mathrm{[cm^{-1}/\%]}$ | $\mathrm{\Delta\omega}\mathrm{[cm^{-1}/\%]}$
$\mathrm{A_{1}(TO_{1})}$ | 1.31 | -2.87 | 3.16
$\mathrm{A_{1}(TO_{2})}$ | 3.64 | -4.78 | 4.51
$\mathrm{A_{1}(TO_{3})}$ | 5.18 | -7.08 | 4.84
$\mathrm{A_{1}(TO_{4})}$ | 3.95 | -3.79 | 3.93
$\mathrm{E(TO_{1})}$ | 0.68/3.26 | -1.56/-3.37 | 1.32/1.45
$\mathrm{E(TO_{2})}$ | 1.52/0.32 | -3.69/0.04 | 2.68/0.91
$\mathrm{E(TO_{3})}$ | 1.09/2.24 | -4.40/-1.56 | 2.62/2.45
$\mathrm{E(TO_{4})}$ | 2.19/1.98 | -3.42/-1.65 | 2.73/2.40
$\mathrm{E(TO_{5})}$ | 6.58/6.09 | -9.57/-13.57 | 7.39/8.31
$\mathrm{E(TO_{6})}$ | 12.84/12.39 | -4.93/-3.76 | 14.82/13.58
$\mathrm{E(TO_{7})}$ | 4.18/2.72 | 1.01/-1.52 | 0.83/2.02
$\mathrm{E(TO_{8})}$ | 3.63/0.91 | -0.67/-4.51 | 4.65/1.54
$\mathrm{E(TO_{9})}$ | 1.36/4.94 | 0.70/-5.12 | 4.00/3.45

In addition, the results of LT under compression are listed in Table 3 for the
same modes. In direct comparison with the results of LN, they are very similar
along both directions, as well as magnitude. This is in agreement with Mendes-
Filho _et al._ [23] and expected since LN and LT are isostructural.
Nevertheless, there are some small deviations. In particular, the degeneracy
lifting is attenuated compared to LN. For instance, the upper branch of the
E(TO1) mode has a notable lower slope than that of LN. Also, the E(TO9) mode
exhibits a significantly attenuated degeneracy lift for LT under compression
than for LN. These deviations can be explained with the help of the
eigenvectors. In Fig. 5 the eigenvectors of A1(TO1-4) and E(TO1-9) are shown
in the rhombohedral unit cell [41, 31]. Such modes with high vibrational
contributions from Nb and Ta ions, like the E(TO1) mode, lead to a slightly
different vibrational behavior [31]. The modes that have in turn low
contributions of Nb and Ta ions with respect to their eigenvectors, are very
similar under compression of LN and LT. Since in the E(TO9) mode, the
displacement of O is more involved than that of Nb/Ta, the reason for the
difference between LN and LT may be due to the nature of Nb/Ta-O bond, similar
to observations in Refs. [31, 41]. The mean Nb-O distance
($2.02\text{\,}\text{\AA}$) is larger than that of Ta-O
($2.00\text{\,}\text{\AA}$). Stronger bonding with the oxygen atoms, and thus
larger deformation of the oxygen octahedron, might be the reason for the
difference between LN and LT concerning the E(TO9) mode. Also, slight
differences in E(TO7) and E(TO8), which are pure distortions of the oxygen
octahedron, might be related to the different Nb/Ta-O bonds.

Figure 5: The eigenvectors of the Raman active vibrations of unstrained
stoichiometric LN and LT in the rhombohedral unit cell of the transversal
phonon modes with A1 and E symmetry. Thereby light grey is denoted as Li, dark
grey as Nb/Ta, and red as O ion, while the green arrows represent the
displacement direction.

The E(TO5/6) and A1(TO2) modes are characteristic for LN and LT under strain,
since these have particularly high slopes, as shown in Table 3 and Fig. 4(b)
and (c). This is also in agreement with Ref. [23]. The E(TO5/6) are especially
sensitive to strain in the x and y directions. This can be explained by a
simple analysis of their eigenvectors. In Fig. 5 it can be seen that these two
modes, in contrast to all other modes, are characterized by particularly large
displacements of the lithium ions parallel to the x-y-plane. Hence, these
modes are most affected when the distances between the ions parallel to the
x-y-plane are shortened. The A1(TO2) mode has among the largest slope for
strain along the z direction. This can be reasonably explained in terms of the
eigenvectors as well. As seen in Fig. 5, the ions of the A1(TO2) mode vibrate
in z direction. When compression in the z direction shortens the bond length
between the ions parallel to their displacement, the frequency is expected to
increase. As the eigenvectors of the A1(TO2) mode point most in the z
direction, it consequently has the largest slope compared to all other modes
when compressed in the z-direction. All further data and results on LN and LT
under compressive and tensile strain are provided in the SI.

Figure 6: Comparison of the experimental and theoretical frequency shift (at
lower strains): (a)-(b) E(TO6) phonon mode under compression along the x and y
axes of stoichiometric LN. The shaded region around all the curves is the
confidence interval of the linear fit.

###  III.3 Discussion: Comparing theory and experiment

As mentioned in the introduction section, the calculations are performed in
units of strain and the experiments are performed in units of stress. In order
to compare both results we have converted the experimental values of stress
into values of strain using the equation given below:

$\sigma=E\epsilon\mathrm{,}$ (4)

where $E$ is Young’s Modulus of a sample [52]. The calculated value of Young’s
Moduli of LN along the x, y, and z axes are $173.6\text{\,}\mathrm{GPa}$,
$173.07\text{\,}\mathrm{GPa}$, and $201.0\text{\,}\mathrm{GPa}$, respectively
[see calculations in SI]. For every measured stress, we calculated the
respective strain value and plotted the corresponding frequency shift along
with the theoretical data as a function of strain [see Fig. 6]. Since the
stoichiometric samples are the closest system to the stoichiometric system
considered for theoretical calculations, they are compared with theoretical
data in Fig. 6.

For x compression, theoretically, the E(TO6) mode shows lifting of the
degeneracy in accordance with the results of Sec. III.2. For a more accurate
comparison, the slopes for higher strains for y and z compression are listed
in Table 4. The theoretical results on the degeneracy lifting offer a new
interpretation of the experimental results. In particular, comparing the data
for the E(TO7) mode with compression in the x direction, it can be seen that
the data for the $\mathrm{\overline{Z}}$(YY)Z resembles the upper branch, and
its slope of the theoretical data, and the data for the
$\mathrm{\overline{Z}}$(XY)Z and $\mathrm{\overline{Z}}$(XX)Z geometry is
similar to the theoretical data and slope of the lower branch. For y
compression, the same can be observed, with the $\mathrm{\overline{Z}}$(XY)Z
and $\mathrm{\overline{Z}}$(XX)Z geometries having the higher slope,
respectively, and the $\mathrm{\overline{Z}}$(YY)Z having the lower slope,
i.e. the slopes flip for y compression. Consequently, it can be assumed that
with the help of different scattering geometries the lifting of the degeneracy
by x and y compression can be reproduced experimentally.

Based on this interpretation, it can be concluded that the experimental and
theoretical data fit very well. For a more accurate comparison, the slopes for
higher strains for y and z compression are listed in Table 4.

Interpreting the experimental data as lifting the degenerate E modes on x and
y compression can also help characterize compressed LN. From the data, it can
be implied that one has x compressed LN when the $\mathrm{\overline{Z}}$(YY)Z
geometry measures a higher slope than for the $\mathrm{\overline{Z}}$(XY)Z and
$\mathrm{\overline{Z}}$(XX)Z geometries. If, on the other hand, one has y
compressed LN, this arguement just reverts. In contrast, if no splitting E
modes are observed in different scattering geometries, it can be concluded
that the LN under investigation must be a z-compressed sample. Furthermore,
the experimental comparison in Fig. 6 and Table 4, confirms the prediction
that the E(TO6) mode is the mode with highest slope for x and y compression.
In comparison with Sec. III.1 and III.2, the slopes for x compression are very
similar to those for y compression.

Table 4:  Comparison between the theoretical response at higher strains with the experimental response of the samples sLN-y and 5Mg-LN-z. Asterisks label modes that are Raman silent in the given scattering geometry. | sLN-y (cm-1/%) | 5Mg-LN-z (cm-1/%)
---|---|---
Modes |  $\mathrm{\overline{Z}}$(XX)Z |  $\mathrm{\overline{Z}}$(XY)Z |  $\mathrm{\overline{Z}}$(YY)Z | Theory |  $\mathrm{\overline{X}}$(ZZ)X |  $\mathrm{\overline{X}}$(YZ)X |  $\mathrm{\overline{X}}$(YY)X | Theory
$\mathrm{A_{1}(LO_{1})}$ | 8.01 $\pm$ 1.41 | * | 6.31 $\pm$ 0.68 | * | * | * | * | *
$\mathrm{A_{1}(LO_{2})}$ | 7.95 $\pm$ 0.35 | * | 6.18 $\pm$ 0.68 | * | * | * | * | *
$\mathrm{A_{1}(LO_{3})}$ | 4.24 $\pm$ 1.94 | * | 2.82 $\pm$ 0.68 | * | * | * | * | *
$\mathrm{A_{1}(LO_{4})}$ | 6.75 $\pm$ 0.33 | * | 5.12 $\pm$ 0.68 | * | * | * | * | *
$\mathrm{A_{1}(TO_{1})}$ | * | * | * | 1.15 | -4.97 $\pm$ 0.93 | * | -5.10 $\pm$ 5.63 | -2.97
$\mathrm{A_{1}(TO_{2})}$ | * | * | * | 3.83 | 7.49 $\pm$ 1.08 | * | 5.98 $\pm$ 4.73 | 4.44
$\mathrm{A_{1}(TO_{3})}$ | * | * | * | 4.47 | 4.54 $\pm$ 1.24 | * | -0.19 $\pm$ 1.4 | 0.65
$\mathrm{A_{1}(TO_{4})}$ | * | * | * | 3.87 | -4.94 $\pm$ 0.65 | * | -6.94 $\pm$ 0.60 | -4.91
$\mathrm{E(TO_{1})}$ | 6.27 $\pm$ 0.63 | 5.97 $\pm$ 0.63 | 4.41 $\pm$ 0.60 | 3.25/0.83 | * | 0.40 $\pm$ 0.44 | -0.15 $\pm$ 0.54 | -0.61
$\mathrm{E(TO_{2})}$ | 5.48 $\pm$ 0.35 | 4.84 $\pm$ 0.35 | 1.56 $\pm$ 0.34 | -0.70/2.36 | * | 0.61 $\pm$ 0.53 | 1.08 $\pm$ 1.51 | -2.55
$\mathrm{E(TO_{3})}$ | 7.56 $\pm$ 1.04 | 5.81 $\pm$ 1.04 | 4.68 $\pm$ 1.07 | 1.34/2.21 | * | 1.46 $\pm$ 1.75 | 1.44 $\pm$ 7.44 | 0.50
$\mathrm{E(TO_{4})}$ | 4.5 $\pm$ 0.69 | 4.44 $\pm$ 0.69 | 2.42$\pm$ 0.68 | 2.03/2.12 | * | 5.43 $\pm$ 0.92 | -3.84 $\pm$ 11.74 | 2.40
$\mathrm{E(TO_{5})}$ | * | * | * | 6.73/6.55 | * | * | * | -0.68
$\mathrm{E(TO_{6})}$ | 12.9 $\pm$ 1.85 | 11.13 $\pm$ 1.85 | 9.16 $\pm$ 2.02 | 13.00/11.86 | * | 0.72 $\pm$ 2.31 | 1.90 $\pm$ 1.9 | -0.94
$\mathrm{E(TO_{7})}$ | 3.41 $\pm$ 0.68 | 1.64 $\pm$ 0.68 | 1.23 $\pm$ 0.82 | 4.12/2.01 | * | 7.21 $\pm$ 2.64 | 7.69 $\pm$ 0.59 | 4.75
$\mathrm{E(TO_{8})}$ | 8.56 $\pm$ 0.48 | 6.72 $\pm$ 0.48 | 3.27 $\pm$ 0.70 | 3.75/1.00 | * | 6.73 $\pm$ 0.62 | 6.20 $\pm$ 5.94 | 3.35
$\mathrm{E(TO_{9})}$ | - | - | - | 5.41/0.91 | - | - | -7.95 $\pm$ 13.41 | 0.90

As concluded from the experimental results in Sec. III.1, the 5$\%$ MgO-doped
LN responds equally well as the stoichiometric LN, upon compression. We
compare the 5Mg-LN-z sample with theoretical data of stoichiometric LN for z
compression. As theoretically predicted and confirmed by the experiment, both
E(TO7) and A1(TO4) modes show a linear dependence on z compression [see
Fig.7(a)-(b)]. For the E(TO7) mode, no degeneracy lifting occurs for z
compression, which can also be seen in the experimental data. This
investigation provides another rule for characterizing compressed LN.

Figure 7: Comparison of theoretical stoichiometric and experimental 5Mg-LN-z
sample when compressing along the z-direction, for frequency shifts of (a)
E(TO7) and (b) A1(TO4). E(TO7) shows a positive slope while A1(TO4) shows a
negative slope for z compression.

As described in Sec. III.2 the frequencies of most modes are linearly
increasing with increasing compression. For a few modes, this is not the case,
e.g., for the A1(TO4) modes. For this mode, a linear decrease in frequency
with increasing compression is theoretically predicted. This is consistent
with experimental measurements, as shown in Fig. 7(b). This unusual behavior
can also be explained with the help of the eigenvectors. In Fig. 5 it can be
seen that the oxygen ions move towards the positively charged niobium ions and
then away from them. If z compression reduces the distance between the
cations, then as the oxygen ions displace away from niobium, the restoring
force towards niobium may be weakened by the shortened distance to the lithium
ions, which also exert the Coulomb force on the negatively-charged oxygen
ions. Thus, the frequency for z compression might be reduced. In a similar
way, this explanation can also be applied to A1(TO1) modes. All other modes
for comparison for z compression are shown in Table 4.

##  IV Conclusions

Experimental and theoretical phonon frequencies as a function of uniaxial
strain have been evaluated and compared for LN and LT. This investigation has
shown that all phonon modes are affected by strain. Both theoretical and
experimental results are consistent with the roughly linear behavior of the
modes and can be reasonably explained by an analysis of the calculated
eigenvectors. The linearity of the slopes confirms the previous works under
hydrostatic pressure and under external electric fields [21, 23].

Accordingly, the obtained slopes provide reference values for the peak shifts.
Furthermore, with the help of the slopes, it is possible to characterize
compressed LN crystals and, hence, the vicinity of domain walls and wave
guides in LN. Our experimental and theoretical investigations show that x and
y compressed LN has particularly high peak shifts for the E(TO5/6) modes and z
compressed LN for the A1(TO2), E(TO7) and E(TO8) modes [23].

Furthermore, calculations have shown that the degeneracy of the E modes is
lifted for x and y compression upon symmetry reduction. The splitting has also
been observed in our experiments when measured under different scattering
geometries. Based on our theoretical and experimental investigations three
rules can be formulated to characterize the direction of uniaxial compression
of a LN sample using different scattering geometries. This only concerns the E
modes. The sample is x compressed if higher frequency shifts are observed for
the $\mathrm{\overline{Z}}$(YY)Z geometry than for the
$\mathrm{\overline{Z}}$(XY)Z and $\mathrm{\overline{Z}}$(XX)Z geometries when
applying the same compressive stress. For y compression, the opposite is the
case. In contrast, when no differences are observed between the frequency
shift of the E modes for various scattering geometries, it can be concluded
that the sample under investigation must be z compressed.

Furthermore, the calculation has shown that the behavior of LT under strain is
not significantly different from that of LN. Experimentally, it has
additionally revealed that the frequency shifts under compression of
stoichiometric and 5% MgO-doped LN are comparable. As DWs in LN and LT can be
represented in the first approximation as strained lattices, our investigation
might be helpful to understand the Raman signal of DWs. Further studies of
strained samples, concerning e.g., their linear and non-linear optical
response might further help to characterize ferroelectric DWs.

###### Acknowledgements.

We thank Dr Clifford W. Hicks for the help in designing the stress cell used
in the measurements. The authors gratefully acknowledge the financial support
by the Deutsche Forschungsgemeinschaft (DFG) through joint DFG-ANR project
TOPELEC (No. EN-434/41-1 and No. ANR-18-CE92-0052), FOR5044 (ID: 426703838;
http:
www.For5044.de), as well as the Würzburg-Dresden Cluster of Excellence
”ct.qmat” (EXC 2147). Calculations for this research were conducted on the
Lichtenberg high-performance computer of the TU Darmstadt and at the
Höchstleistungrechenzentrum Stuttgart (HLRS). The authors furthermore
acknowledge the computational re- sources provided by the HPC Core Facility
and the HRZ of the Justus-Liebig-Universität Gießen. Furthermore, we thank D.
Bieberstein and T. Gemming from IFW Dresden for assistance with dicing of the
wafers and support from the International Max Planck Research School for
Chemistry and Physics of Quantum Materials.

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in 1,…,17 See pages \x, of supplements.pdf


# Elementary Lift and Single Element Coextension of A Binary Gammoid

###### Abstract.

Elementary lift is a generalized splitting operation. Splitting of a binary
gammoid need not be a gammoid. This paper finds forbidden minors for a binary
gammoid whose elementary lift is a binary gammoid. We also find forbidden
minors for a binary gammoid whose single element coextension is a binary
gammoid.

Shital Dilip Solanki1, Ganesh Mundhe2 and S. B. Dhotre3
_1\. Ajeenkya DY Patil University, Pune-411047, Maharashtra, India
2\. Army Institute of Technology, Pune-411015, Maharashtra, India
3\. Department of Mathematics, Savitribai Phule Pune University,
Pune - 411007, Maharashtra, India_
E-mail: 1\<EMAIL_ADDRESS>2<EMAIL_ADDRESS>3.
<EMAIL_ADDRESS>

Mathematics Subject Classification (2010): 05B35,05C83, 05C50
Keywords:Binary Matroid, Splitting, Binary Gammoid, Minor, Quotient,
Coextension, Lift.

##  1\. Introduction

Initially, the splitting operation in graphs is introduced by Fleischner [6]
as follows.
Let $a=(v_{1},v)$ and $b=(v_{2},v)$ be two arcs incident at node $v$. Then
remove arcs $a$, $b$ and adding new node $v^{\prime}$ and arcs
$a=(v_{1},v^{\prime})$ and $b=(v_{2},v^{\prime})$. This operation is called
splitting using two arcs. This operation is extended to binary matroid by
Raghunathan et al. [9] as a splitting of binary matroid using two element set.
Later, this operation is generalized as a splitting of matroid using n-element
set as follows, by Shikare et al. [8].

######  Definition 1.1.

[8] Let a binary matroid $B$ with $H\subseteq E(B)$ and let $A$ be the matrix
representing the binary matroid $B$. Obtain a new matrix $A_{H}$ by adding a
new row at the bottom of the matrix $A$ with entries 0 everywhere except 1 at
the columns representing elements of $H$. Then the vector matroid of $A_{H}$
is called splitting matroid $B_{H}$, and the operation is called splitting
using a set.

Later, for a binary matroid, Azadi [1] defined the element splitting operation
as follows.

######  Definition 1.2.

[1] Let a binary matroid is $B$ and $H\subseteq E(B)$. Let the matrix $A$
represent binary matroid $B$. Obtain a matrix $A_{H}^{\prime}$ by adding a row
at the bottom of the matrix $A$ with entries $0$ everywhere except entries $1$
in the columns corresponding to the elements in $H$ and adding one column with
entries zero except in the last row where the entry is $1$. Then the vector
matroid corresponding to the matrix $A_{H}^{\prime}$ is the element splitting
matroid $B_{H}^{\prime}$, and the operation is known as the element splitting
operation.

Later, for binary matroid, Azanchilar [2] defined the es-splitting operation
as follows.

######  Definition 1.3.

Let the matrix $A$ represents a binary matroid $B$ and $H\subseteq E(B)$ with
$e\in H$. Let $N$ be a matrix obtained from $A$ by adding a row labeled
$\gamma$, similar to a row labeled by $e$. Let $M$ be the vector matroid of
the matrix $N$, then the element splitting matroid $M_{H}^{\prime}$ of $M$ is
called es-splitting matroid of $B$, and $B_{H}^{e}$ denotes it, and the
operation is called es-splitting operation.

For undefined concepts, we refer to Oxley [10].
Let $M$ and $Q$ be matroids and let matroid $N$ be such that for some subset
$X$ of $E(N)$, $M=N\backslash X$ and $Q=N/X$. Then $Q$ is called the quotient
of $M$, and $M$ is called the lift of $Q$. If $X$ is a singleton set, then $Q$
is called the elementary quotient of $M$ and $M$ is called the elementary lift
of $Q$. From the definition of splitting and coextension, it is observed that
the elementary lift of $Q$ is the splitting matroid $Q_{T}$ for some $T$. From
the definition of element splitting operation, it is clear that the single
element coextension of the given matroid $B$ is the element splitting matroid
$B_{T}^{\prime}$ for some $T\subseteq E(B)$.

It was observed that the elementary lift or splitting operation and single
element coextension or element splitting operation of binary matroid do not
preserve the properties like graphicness, cographicness, contentedness etc.
Borse [11] characterized graphic matroid, which gives graphic matroid under
splitting operation using two-element set. Similarly, graphic matroid is
characterized by Mundhe [5] whose elementary lift is graphic.
Thus, in the same way, it is very interesting to characterize the class of
binary gammoid whose elementary lift and single element coextension is
gammoid.

Characterization of a binary gammoid is given as follows.

######  Theorem 1.1.

[10] For a matroid $M$ the following are equivalent.
(i) $M$ is a binary gammoid.
(ii)$M$ is a graphic gammoid.
(iii) $M$ is a regular gammoid.
(iv) $M$ has no minor isomorphic to $U_{2,4}$ or $M(K_{4})$.

Notation: Let $\mathcal{G}_{k}$ denote the class of a binary gammoid which do
not yield a binary gammoid after splitting using the k-element set.

Borse [3] characterized a binary gammoid whose splitting and element splitting
is a binary gammoid as follows.

######  Theorem 1.2.

[3] Let a binary gammoid be $M$, then $M\in\mathcal{G}_{2}$ if and only if $M$
contain $M(G_{1})$ minor, where $G_{1}$ is shown in Figure 1.

$x$$y$$G_{1}$ Figure 1. Minimal minor of the class $\mathcal{G}_{2}$.

######  Theorem 1.3.

[3] Let $M$ be a binary gammoid, then element splitting matroid
$M_{H}^{\prime}$ is a binary gammoid, for any $H\subseteq E(M)$ with $|H|=2$
if and only if $M$ does not contain $M(G_{6})$ minor, where $G_{6}$ is as
shown in Figure 2.

$G_{6}$ Figure 2. Forbidden minor for element splitting of a binary gammoid.

This paper aims to characterize a binary gammoid whose splitting and element
splitting is a binary gammoid. We prove the main theorems as stated below.

######  Theorem 1.4.

Let a binary gammoid be $M$ and $k\geq 2$ such that $M\in\mathcal{G}_{k}$.
Then $M$ contain a minor $P$, for which one of the following holds.
(i) $P$ is isomorphic to the one element extension of some minimal minor of
the class $\mathcal{G}_{k-1}$.
(ii) $P=M(Q_{i})$ or extension of $M(Q_{i})$ not more than k elements, where
the graph $Q_{i}$ is shown in Figure 5, for $i=2,3,4$.

######  Theorem 1.5.

Let a binary gammoid be $M$ then $M\notin\mathcal{G}_{3}$ if and only if $M$
do not contain $M(G_{i})$ minor, where the graph $G_{i}$ is shown in Figure 3,
for $i=2,3,4$.

$G_{2}$$G_{3}$$G_{4}$$x$$y$$z$$x$$y$$z$$x$$y$$z$ Figure 3. Minimal minor of
the class $\mathcal{G}_{3}$

######  Theorem 1.6.

Let a binary gammoid be $M$, then element splitting matroid $M_{H}^{\prime}$
is a binary gammoid for any $H\subseteq E(M)$, with $|H|\geq 3$ if and only if
$M$ do not contain minor $M(G_{6})$, where $G_{6}$ is as shown in Figure-2.

This paper also characterizes a binary gammoid whose es-splitting is a binary
gammoid. We proved the theorem stated as below.

######  Theorem 1.7.

Let $M$ be a binary gammoid then es-splitting matroid $M_{T}^{e}$ is a binary
gammoid, for $T\subseteq E(M)$, with $|T|\geq 2$ and $e\in T$ if and only if
$M$ do not contain a minor $M(G_{7})$, where $G_{7}$ is as shown in Figure 4.

$G_{7}$$x$$y$ Figure 4. Forbidden minor for es-splitting of a binary gammoid

##  2\. Preliminary Results

For $k\geq 2$, $\mathcal{G}_{k}$ denote a collection of a binary gammoid whose
splitting is not a binary gammoid using a set with k elements. In this paper,
we aim to characterize minimal minors that belong to the class
$\mathcal{G}_{k}$, for $k\geq 2$.

######  Lemma 2.1.

Let $M$ be a binary gammoid such that $M_{H}$ has $M(K_{4})$ minor for some
$H\subseteq E(M)$ with $|H|\geq 2$. Then $M$ contains a minor $P$ with
$H\subseteq E(P)$, such that $P$ satisfies one of the following conditions.

  1. (i).

$P_{H}\cong M(K_{4})$;

  2. (ii).

$P_{H}/H^{\prime}\cong M(K_{4})$, for some $H^{\prime}\subseteq H$;

  3. (iii).

$P$ is a one element extension of some minimal minor in the class
$\mathcal{G}_{k-1}$, for $k\geq 2$.

###### Proof.

Let a binary gammoid $M$ be such that $M_{H}$ contains $M(K_{4})$ minor. Thus
there exists subsets $H_{1}$ and $H_{2}$ of $E(M)$ such that $M_{H}\backslash
H_{1}/H_{2}\cong M(K_{4})$. Let $H_{i}^{\prime}=H\cap H_{i}$ and
$H_{i}^{\prime\prime}=H_{i}-H_{i}^{\prime}$, for $i=1,2$. Then
$M_{H}\backslash H_{1}^{\prime\prime}/H_{2}^{\prime\prime}\cong(M\backslash
H_{1}^{\prime\prime}/H_{2}^{\prime\prime})_{H}$ as each $H_{i}^{\prime\prime}$
is disjoint from $H$. Let $P=M\backslash
H_{1}^{\prime\prime}/H_{2}^{\prime\prime}$. Then $P$ is a minor of $M$
containing $H$. Consider $P_{H}\backslash
H_{1}^{\prime}/H_{2}^{\prime}\cong(M\backslash
H_{1}^{\prime\prime}/H_{2}^{\prime\prime})_{H}\backslash
H_{1}^{\prime}/H_{2}^{\prime}\cong M_{H}\backslash
H_{1}^{\prime\prime}/H_{2}^{\prime\prime}\backslash
H_{1}^{\prime}/H_{2}^{\prime}\cong M_{H}\backslash H_{1}^{\prime}\cup
H_{1}^{\prime\prime}/H_{2}^{\prime}\cup H_{2}^{\prime\prime}\cong
M_{H}\backslash H_{1}/H_{2}\cong M(K_{4})$. That is $P_{H}\backslash
H_{1}^{\prime}/H_{2}^{\prime}\cong M(K_{4})$.
If $H_{1}^{\prime}=H_{2}^{\prime}=\emptyset$ Then (i) holds.
If $H_{1}^{\prime}=\emptyset$ and $H_{2}^{\prime}\neq\emptyset$ Then (ii)
holds.
If $H_{1}^{\prime}\neq\emptyset$. $H_{1}^{\prime}\subseteq H$ then
$|H_{1}^{\prime}|\leq k$. Now, if $|H_{1}^{\prime}|=k$, then
$H_{1}^{\prime}=H$ and $H_{2}^{\prime}=\emptyset$ it follows that
$M(K_{4})\cong P_{H}\backslash H_{1}^{\prime}\cong P_{H}\backslash H\cong
P\backslash H$. Thus, a binary gammoid $M$ contains $M(K_{4})$ minor, a
contradiction. Hence $|H_{1}^{\prime}|\neq k$. If $0<|H_{1}^{\prime}|<k$. Let
$h\in H_{1}^{\prime}$, $T=H-{h}$ and $T^{\prime}=H_{1}^{\prime}-{h}$. Assume
that $N=P\backslash h$ then $N$ is a minor of $P$ and hence it is a minor of a
binary gammoid $M$, thus $N$ is a binary gammoid and $|T|=k-1$.
$P_{H}\backslash h\cong(P\backslash h)_{(H-{h})}\cong N_{T}$. As
$P_{H}\backslash H_{1}^{\prime}/H_{2}^{\prime}\cong M(K_{4})$ implies that
$N_{T}\backslash T^{\prime}/H_{2}^{\prime}\cong M(K_{4})$, thus $N$ is the
minimal minor in the class $\mathcal{G}_{k-1}$. As $N=P\backslash h$, $P$ is
the extension of $N$ by one element. ∎

######  Lemma 2.2.

Let a binary gammoid $P$ be as stated in Lemma 2.1. There is a binary gammoid
$Q$ with $a\in E(Q)$, such that $Q\backslash a\cong M(K_{4})$ and $P\cong Q/a$
or $P$ is coextension of $Q/a$ by not more than k elements.

###### Proof.

Let $H\subseteq E(P)$ and $P$ be a binary gammoid such that $P_{H}\cong
M(K_{4})$ or $P_{H}/H^{\prime}\cong M(K_{4})$ for some subset $H^{\prime}$ of
$H$. From the definition of splitting and element splitting operation, we can
easily verify the relations, $P_{H}^{\prime}\backslash a\cong P_{H}$ and
$P_{H}^{\prime}/a\cong P$ where $P_{H}^{\prime}$ is element splitting of the
binary gammoid $P$ using a set $H$.
Case-(i). If $P_{H}\cong M(K_{4})$, We assume that $Q=P_{H}^{\prime}$ where
$E(Q)=E(P)\cup a$. Then $Q/a\cong P_{H}^{\prime}/a\cong P$ and
$P_{H}^{\prime}\backslash a\cong Q\backslash a\cong P_{H}\cong M(K_{4})$. Thus
$P\cong Q/a$
Case-(ii). If $P_{H}/H^{\prime}\cong M(K_{4})$ then we assume that
$Q=P_{H}^{\prime}/H^{\prime}$ thus $Q\backslash a\cong
P_{H}^{\prime}/H^{\prime}\backslash a\cong P_{H}^{\prime}\backslash
a/H^{\prime}\cong P_{H}/H^{\prime}\cong M(K_{4})$. Hence $Q/a$ is quotient of
$M(K_{4})$. Also $Q/a=P_{H}^{\prime}/H^{\prime}/a\cong
P_{H}^{\prime}/a/H^{\prime}\cong P/H^{\prime}$. Hence $P$ is coextension of
$Q/a$ by $H^{\prime}$ and $|H^{\prime}|\leq k$. Thus we say that $P$ is
coextension of $Q/a$ by not more than k elements. ∎

For a binary matroid $M$, if $M\cong Q\backslash a$ for some matroid $Q$ with
$a\in E(Q)$, then $Q/a$ is called quotient of $M$ and $M$ is called elementary
lift of $Q/a$. Thus to find minimal minor of the class $\mathcal{G}_{k}$, we
need to find minimal minor $P$ of a binary gammoid $M$, such that $P_{H}\cong
M(K_{4})$ or $P_{H}/H^{\prime}\cong M(K_{4})$ for some subset $H^{\prime}$ of
$H$, where $H\subseteq(M)$. Thus by Lemma 2.2, $P\cong Q/a$ or $P$ is
extension of $Q/a$ not more than k elements. Thus we need to find $Q/a$, that
is quotient of $M(K_{4})$. Hence, in the following lemma, we find graphic
quotient of $M(K_{4})$.

######  Lemma 2.3.

Every graphic quotient of $M(K_{4})$ is isomorphic to one of the matroid
$M(Q_{i})$ where $Q_{i}$ is shown in Figure 5, for $i=1,2,3,4$.

$Q_{1}$$Q_{2}$$Q_{3}$$Q_{4}$ Figure 5. Quotient for $K_{4}$

###### Proof.

Let $N$ be binary matroid with $x\in E(N)$ such that $N\backslash x\cong
M(K_{4})$ and $N/x$ be a binary gammoid and hence graphic by Lemma 1.1. Then
for some connected graph $G$, $N/x\cong M(G)$. If ${x}$ is cocircuit or
circuit of $N$ then $N/x\cong N\backslash x\cong M(K_{4})$. Thus $G\cong
Q_{1}$.

If ${a}$ is not a circuit or a cocircuit of $N$. Then $N\backslash x$ contains
4 nodes and 6 arcs. Then $N$ contains 4 nodes and 7 arcs. Thus $r(N/x)=2$ and
$E(N/x)=6$. Thus graph $G$ contains 3 nodes and 6 arcs. $G$ cannot be simple
graph as there is no simple graph on 3 nodes and 6 arcs.
If $G$ contains more than one loop, then $M(G)\cong N/x$ contains more than
one loop, then $N\backslash x\cong M(K_{4})$ will contain a loop or 2-circuit,
a contradiction. Hence $G$ does not contains more than one loop. Also, If $G$
contains more than two arcs having the same end nodes. Then $M(G)\cong N/x$
contains more than two parallel elements; thus $N\backslash x\cong M(K_{4})$
will contain 2-circuit, a contradiction. Thus $G$ contains not more than two
parallel arcs having the same end nodes. Hence there are two cases.
Case-(i). If $G$ contains one loop, then $G$ can be obtained by adding a loop
to a graph with 3 nodes and 5 arcs. By Harary [4] (page 226), there is only
one graph on 3 nodes and 5 arcs. Hence $G\cong Q_{2}$ and $G\cong Q_{3}$.
Case-(ii). Suppose $G$ does not contain a loop. Then $G$ contains three pairs
of parallel arcs. By Harary [4] (page 226), there is only one graph on 3 nodes
and 6 arcs. Hence $G\cong Q_{4}$. ∎

##  3\. Elementary Lifts of Binary Gammoid

In this section, we characterize a binary gammoid whose elementary lift is a
binary gammoid. We state the theorem again.

######  Theorem 3.1.

Let a binary gammoid be $M$ and $k\geq 2$ such that $M\in\mathcal{G}_{k}$.
Then $M$ contain a minor $P$ for which one of the following holds.
(i) $P$ is isomorphic to the single element extension of some minimal minor of
the class $\mathcal{G}_{k-1}$.
(ii) $P=M(Q_{i})$ or extension of $M(Q_{i})$ by at most k elements, where the
graph $Q_{i}$ is shown in Figure 5, for $i=2,3,4$.

###### Proof.

Let a binary gammoid $M$ be such that $M\in\mathcal{G}_{k}$. Thus, by Lemma
1.1, $M_{H}$ contain a minor $M(K_{4})$ for some subset $H$ of $E(M)$ with
$|H|\geq 2$. Then by Lemma 2.1, $M$ contains a minor $P$ containing $H$ such
that one of the following holds.
(i). $P_{H}\cong M(K_{4})$ or $P_{H}/H^{\prime}\cong M(K_{4})$ for some
$H^{\prime}\subseteq H$;
(ii). $P$ is a one element extension of some minimal minor of the class
$\mathcal{G}_{k-1}$.
Condition (i) follows immediately from Lemma 2.1. Now, if $P_{H}\cong
M(K_{4})$ or $P_{H}/H^{\prime}\cong M(K_{4})$ for some $H^{\prime}\subseteq H$
then by Lemma 2.2 there is a minor $Q$ with $a\in E(Q)$ such that $Q\backslash
a\cong M(K_{4})$ and $P=Q/a$ or extension of $Q/a$ by at most k elements.
By Lemma 2.3, it is proved that, when $Q\backslash a\cong M(K_{4})$ then
$Q/a\cong M(Q_{i})$, where the graph $Q_{i}$ is as shown in Figure 5, for
i=1,2,3,4. If $P=M(Q_{1})$ and $P$ is a minor of a binary gammoid, a
contradiction. Hence, either $P=M(Q_{i})$ or P is an extension of $M(Q_{i})$
not more than k elements, for i=2,3,4. ∎

######  Corollary 3.2.

Let $M$ be a binary gammoid and $H\subseteq E(M)$, with $|H|\geq 2$. Then
$M_{H}$ is a binary gammoid if and only if $M$ does not contain a minor
$M(Q_{i})$, for $i=2,3,4$.

###### Proof.

If $M$ contain minor $M(Q_{i})$, then easy to prove that $M_{H}$ is not a
binary gammoid,for $i=2,3,4$.
Conversely, let $M$ be such that it does not contain $M(Q_{i})$ minor, for
$i=2,3,4$. On contrary, assume that $M_{H}$ is not a binary gammoid. Then
$M\in\mathcal{G}_{k}$ for some $k$. By Theorem 3.1, $M$ contains $M(Q_{i})$
minor, for $i=2,3,4$, a contradiction. Hence $M_{H}$ is a binary gammoid. ∎

In this section, we give alternate proof to the theorem given by Borse [3]. We
state the theorem again.

######  Theorem 3.3.

[3] Let a binary gammoid be $M$ then $M\in\mathcal{G}_{2}$ if and only if $M$
contain $M(G_{1})$ minor, where graph $G_{1}$ is shown in the Figure 1.

###### Proof.

Let a binary gammoid $M$ contain minor $M(G_{1})$. Then it is easy to prove
that $M_{x,y}$ is not a binary gammoid, where $x,y$ are shown in Figure 1.

Conversely, let $M$ do not contain minor isomorphic to $M(G_{1})$, then we
prove that $M\notin\mathcal{G}_{2}$. Suppose, $M\in\mathcal{G}_{2}$ , that is
$M_{T}$ is not a binary gammoid for some $T=\\{x,y\\}\subseteq E(M)$, then by
Lemma 1.1, $M_{T}$ contains minor $M(K_{4})$. Then by Lemma 2.1, there is
minor $P$ which is isomorphic to the one element extension of some minimal
minor in $\mathcal{G}_{k-1}$ or $P=M(Q_{i})$ or $P$ is an extension of
$M(Q_{i})$ by one or two elements, where the graph $Q_{i}$ is as shown in
Figure 5, for $i=2,3,4$. It is observed that the class $\mathcal{G}_{1}$ is
empty. Thus $P\cong M(Q_{i})$ or extension of $M(Q_{i})$ not more than k
elements, for $i=2,3,4$. By Theorem 1.1, $P$ is a graphic gammoid thus $P\cong
M(G)$. Let $G$ contains edge cut of two cardinality, say $\\{a,b\\}$. Then it
is cocircuit of $P$ and $\\{a,b\\}$ contains cocircuit of $Q_{i}\backslash a$
for some $i$, which is contradiction. Hence $G$ does not contain edge cut
containing two elements.
Case-(i) If $P\cong M(Q_{1})$, a contradiction, as $P$ is a minor of a binary
gammoid. Hence we discard $Q_{1}$. Case-(ii) If $P\cong M(Q_{2})$ then
$M(Q_{2})_{T}$ will contain a loop or a pair of parallel arcs for any two-
element set $T$. Hence $M(Q_{2})_{T}\ncong M(K_{4})$, thus graph $G$ is
obtained by taking coextension of $Q_{2}$ by one or two elements such that it
does not contain edge cut of two cardinalities. Extensions of $Q_{2}$ are
given in the Figure 6. Note that, $a_{1}\cong G_{1}$. Hence we discard
$a_{1}$. splitting $a_{2}$ and $a_{3}$ using two elements does not give
$M(K_{4})$; hence we take coextensions of $a_{2}$ and $a_{3}$ by one element
such that it does not contain a cut set of two cardinalities.

$a_{1}$$a_{2}$$a_{3}$ Figure 6. Coextensions of $Q_{2}$ by one element

Coextensions of $a_{1}$ are the graphs $b_{1}$ and $b_{2}$ and coextension of
$a_{3}$ is $b_{3}$, where $b_{1},b_{2},b_{3}$ are given in Figure 7.
Coextensions $b_{1}$, $b_{2}$ and $b_{3}$ contains minor $G_{1}$, hence we
discard $b_{1},b_{2},b_{3}$ and hence $Q_{2}$.

$b_{1}$$b_{2}$$b_{3}$ Figure 7. Coextensions of $a_{1}$ and $a_{2}$

Case(iii) If $P\cong M(Q_{3})$ or $P\cong M(Q_{4})$ then using similar
argument made above we can discard $Q_{3}$ and $Q_{4}$.

Thus, from all the cases discuss above, we conclude that $M_{T}$ is a binary
gammoid for any two element set $T\subseteq E(M)$, that is
$M\notin\mathcal{G}_{2}$. ∎

We now find the minimal minors for the class $\mathcal{G}_{3}$. We state the
theorem again.

######  Theorem 3.4.

Let a binary gammoid be $M$ then $M\in\mathcal{G}_{3}$ if and only if $M$
contain $M(G_{i})$ minor, where Figure 3, shows graph $G_{i}$, for $i=2,3,4$.

###### Proof.

Let a binary gammoid $M$ contains minor $M(G_{i})$, where $G_{i}$ is shown in
Figure 3, for $i=2,3,4$. Then we prove that $M\in\mathcal{G}_{3}$.
Let $H=\\{x,y,z\\}$ be as shown in graph $G_{2}$ in Figure 3 and $A_{2}$ be
the matrix representing the matroid $M(G_{2})$ as given below.
$A_{2}=\left[\begin{array}[]{cccccc}x&y&z&&&\\\ 1&0&0&1&0&1\\\
0&1&0&0&1&1\end{array}\right].$ Then
$(A_{2})_{H}=\left[\begin{array}[]{cccccc}x&y&z&&&\\\ 1&0&0&1&0&1\\\
0&1&0&0&1&1\\\ 1&1&1&0&0&0\end{array}\right].$

It is observed that vector matroid $M(G_{2})_{H}$ of the matrix $(A_{2})_{H}$
is isomorphic to $M(K_{4})$. Thus by Theorem 1.1, $M(G_{2})_{H}$ is not a
binary gammoid. Similarly, we prove that, if $M$ contain minors $M(G_{3})$ and
$M(G_{4})$ then $M_{H}$ is not a binary gammoid. Hence $M\in\mathcal{G}_{3}$.

Conversely, if $M$ does not contain any of $M(G_{i})$ minor, for $i=2,3,4$.
Then we prove that $M_{H}$ is a binary gammoid for any $H\subseteq E(M)$, with
$|H|=3$. Suppose, $M_{H}$ is not a binary gammoid for some subset $H$ of
$E(M)$. Then $M_{H}$ has a minor $M(K_{4})$. Thus by lemma 2.1, there exists a
minor $P$ of $M$, such that $P_{H}\cong M(K_{4})$ or $P_{H}/H^{\prime}\cong
M(K_{4})$ for some subset $H^{\prime}$ of $H$. By Lemma 2.2, there is a binary
gammoid $N$ with $a\in E(N)$, such that $N\backslash a\cong M(K_{4})$ and
$P\cong N/a$ or $P$ is extension of $N/a$ not more than three elements. By
Lemma 2.3, $N/a\cong M(Q_{i})$ for $i=1,2,3,4$. We discard the case of
$M(Q_{1})$ as it is not a binary gammoid. Note that, for $i=2,3,4$,
$M(Q_{i})\cong M(G_{i})$, that is $M$ contains minor $M(G_{i})$ for $i=2,3,4$,
a contradiction. Hence $M_{H}$ is a binary gammoid for any set $H$ containing
three elements. Hence proof.

∎

##  4\. Single Element Coextension of Gammoid

From the definition of element splitting for a binary matroid $M$, it is clear
that the single element coextension of $M$ is the element splitting matroid
$M_{H}^{\prime}$ for some $H\subseteq E(M)$. We have characterized a binary
gammoid whose single element coextension is a binary gammoid. We state the
theorem again.

######  Theorem 4.1.

Let $M$ be a binary gammoid then element splitting matroid $M_{H}^{\prime}$ is
a binary gammoid, for $H\subseteq E(M)$, with $|H|\geq 3$ if and only if $M$
do not contain $M(G_{6})$ minor, where $G_{6}$ is as shown in Figure 8.

$G_{6}$$x$$y$$z$ Figure 8. Forbidden minor for single element coextension of a
binary gammoid

###### Proof.

Let $M$ be a binary gammoid such that it contains a minor $M(G_{6})$, then it
is straightforward to prove that $M_{H}^{\prime}$ is not a binary gammoid for
$H=\\{x,y,z\\}$, where $x,y,z$ are shown in Figure 8.
Conversely, let $M$ do not contain $M(G_{6})$ minor then we need to prove that
element splitting $M_{H}^{\prime}$ is a binary gammoid. Suppose that
$M_{H}^{\prime}$ is not a binary gammoid, then by Theorem 1.1,
$M_{H}^{\prime}$ contains a minor $M(K_{4})$. Thus for some $H_{1}$ and
$H_{2}$ in $E(M)$, $M_{H}^{\prime}\backslash H_{1}/H_{2}\cong M(K_{4})$. There
are following cases.
Case-(i). If $a\notin H_{1}\cup H_{2}$ then $M_{H}^{\prime}\backslash
H_{1}/H_{2}/a\cong M(K_{4})/a$ thus we have $M_{H}^{\prime}/a\backslash
H_{1}/H_{2}\cong M(G_{6})$, but, $M_{H}^{\prime}/a=M$ which gives that
$M\backslash H_{1}/H_{2}\cong M(G_{6})$, which is a contradiction.
Case-(ii). If $a\in H_{1}$ then, $M_{H}^{\prime}\backslash a\backslash
H_{1}-{a}/H_{2}\cong M(K_{4})$, but we have $M_{H}^{\prime}\backslash
a=M_{H}$. Thus we say that $M(K_{4})$ is a minor of splitting matroid $M_{H}$.
Then by Corollary 3.2, $M$ contains minor $M(Q_{i})$, for $i=2,3,4$ and each
$M(Q_{i})$ contains a minor $M(G_{6})$, which is a contradiction.
Case-(iii). If $a\in H_{2}$ then $M_{H}^{\prime}/a\backslash
H_{1}/H_{2}-{a}\cong M\backslash H_{1}/H_{2}-{a}\cong M(K_{4})$, a
contradiction. Thus from above cases we say that $M_{H}^{\prime}$ is a binary
gammoid. ∎

##  5\. Es-splitting of a Binary Gammoid

In this section, we characterize a binary gammoid whose es-splitting is a
binary gammoid. We state the theorem below.

######  Theorem 5.1.

Let $M$ be a binary gammoid then $M_{H}^{e}$ is a binary gammoid, for
$H\subseteq E(M)$, with $|H|\geq 2$ and $e\in H$ if and only if $M$ do not
contain a minor $M(G_{7})$, where $G_{7}$ is as shown in Figure 9.

$G_{7}$$x$$y$ Figure 9. Forbidden minor for es-splitting of a binary gammoid

###### Proof.

Let $M$ be a binary gammoid such that it contain $M(G_{7})$ minor then it is
straightforward to prove that $M_{H}^{e}$ is not a gammoid, for $H=\\{x,y\\}$
and $e\in H$.

Conversely, let the binary gammoid $M$ not contain $M(G_{7})$ minor. Let $G$
be a graph corresponding to $M$, and $F$ be the graph obtained from $G$ by
adding one parallel edge. Suppose that $M_{H}^{e}$ is not a binary gammoid.
From the definition of es-splitting operation, it is clear that
$M(F)_{H}^{\prime}=M_{H}^{e}$ then $M(F)_{H}^{\prime}$ is also not a binary
gammoid. Thus, by Theorem 4.1, $F$ contain minor $M(G_{6})$. Hence $M$
contains minor $M(G_{7})$, a contradiction. Hence $M_{H}^{e}$ is a binary
gammoid. ∎

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