# Growth of 2D topological material Bi on InSb(111)B with fractal surface
structures

Yoshiyuki Ohtsubo<EMAIL_ADDRESS>National Institutes for Quantum Science and
Technology, Sendai 980-8579, Japan  Graduate School of Frontier Biosciences,
Osaka University, Suita 565-0871, Japan  Department of Physics, Graduate
School of Science, Osaka University, Toyonaka 560-0043, Japan  Takuto Nakamura
Graduate School of Frontier Biosciences, Osaka University, Suita 565-0871,
Japan  Department of Physics, Graduate School of Science, Osaka University,
Toyonaka 560-0043, Japan  Shin-ichi Kimura<EMAIL_ADDRESS>Graduate School of Frontier Biosciences, Osaka University, Suita 565-0871,
Japan  Department of Physics, Graduate School of Science, Osaka University,
Toyonaka 560-0043, Japan  Institute for Molecular Science, Okazaki 444-8585,
Japan

###### Abstract

Bismuth (Bi) atomic layers are known as 2D topological materials with variety
of the electronic structures and topological orders depending on the number of
stacking layers. Recently, it is reported that few layers of Bi grown on
semiconductor substrate InSb(111)B exhibit the Sierpiński-triangle (ST)
fractal patterns on the surface. In this work, we have grown Bi layers on
InSb(111)B and traced the evolution of the atomic and electronic structures of
Bi. The surface atomic structures and growth modes were monitored by using
reflective high-energy electron diffraction and core-level photoelectron
spectroscopy. It is suggested that the single layer of the ST-phase Bi grows
on InSb(111)B and the following Bi deposition causes layer-by-layer growth up
to nominally 4 atomic layers. Diffuse band dispersion and quantum well states
observed by angle-resolved photoelectron spectroscopy are consistent with the
small surface domains and variation of the thickness even during the layer-by-
layer growth region. The further Bi evaporation changes the growth mode to the
3D island formation. The unveiled growth behavior of Bi on InSb(111)B would
provide a new interesting playground to study 2D topological electronic
structure of quasi-periodic 2D atomic layers.

##  I Introduction

Bismuth (Bi) is one of the most widely used elements as a component of
topological materials [1] such as the 3D topological insulators (TIs) Bi2Se3
[2, 3] and Bi1-xSbx [4], because it is the heaviest non-radioactive element
with a strong spin-orbit interaction. Not only the 3D crystals, 2D atomic
layers of Bi(111) with buckled honeycomb lattice is also expected as a 2D TI
with fertile topological phases depending on its thickness [5, 6]. Moreover,
it is also expected that some external perturbations, such as lattice
distortion, external fields, or point defects, could modify and control the
topological phase and electronic structure of the 2D Bi layers [7, 8, 9].

From experimental aspect, the atomic and electronic structure and its topology
of the 2D Bi(111) layers have been studied on various substrates. The Bi
atomic layers down to 6 atomic layers are known to grow on semiconductor
Si(111) and Ge(111), while the thinner Bi(111) layers are not obtained because
of the structural phase transition to phosphorene-like Bi(110) layers [10, 11,
12, 13]. Note that another unit, bi-atomic layer (BL), is used in some
references. On a TI with close in-plane lattice constant, Bi2Te3, Bi(111)
layers grow from 1 BL [14, 15, 16, 17]. However, the 2D electronic structure
of Bi(111) coexists with topological surface states of the substrate, Bi2Te3
[14]. Although the single-layer Bi (Bismuthene) on SiC is free from substrate
electronic structure around the Fermi level ($E_{\rm F}$), its atomic
structure is strongly influenced by the substrate lattice, resulting in the
flat Bi layer in contrast to the buckled ones in the free-standing case [18].
Therefore, to fabricate the few atomic layers (typically 1-4) of free-standing
2D Bi(111) was an unresolved problem of 2D material growth, in order to
explore its topological electronic structures.

Recently, epitaxial growth from 4 layers of Bi(111) was reported on
InAs(111)A; the (111) surfaces of zinc-blende structures are often called as A
(B) face with group-III (V) element termination [19]. Although the detailed
evolution of Bi 2D electronic structures are not reported yet, the zinc-blende
(111) surfaces are spotlighted as promising substrate for growth of few layers
of 2D Bi(111) thanks to their semiconducting electronic structures and
triangular lattice with close values of the in-plane lattice constants to
Bi(111); 4.53 Å, 4.28 Å and 4.58 Å for Bi(111), InAs(111) and InSb(111),
respectively. Moreover, the other attractive result is reported in this year
that Bi on InSb(111)B grows with Sierpiński-triangle (ST)-type fractal surface
patterns by Scanning Tunneling Microscopy (STM) and theoretical model
calculations [20]. ST is a self-similar fractal with a fractional Hausdorff
dimension [21]. This exotic surface structure provides an attractive
playground to study the topological electronic structures in 2D quasicrystals.
For further studies of this exotic surface structure, experimental exploration
about the growth modes and surface electronic structures of Bi layers on
InSb(111)B is awaited.

In this work, we have traced the evolution of the atomic and electronic
structures of the Bi(111) 2D layers grown on InSb(111)B substrates by using
reflective high-energy electron diffraction (RHEED) and photoelectron
spectroscopy (PES). It is suggested that the single layer of the ST-phase Bi
grows on InSb(111)B and the following Bi deposition causes layer-by-layer
growth up to nominally 4 atomic layers. Afterward, the growth behavior of Bi
changes to the 3D mode with small, isolated Bi(111) islands on InSb(111)B
substrates, suggested from diffused angle-resolved PES (ARPES) band
dispersion.These results support the previous observation of the fractal ST-
phase Bi and provide wide-range picture of Bi 2D layer growth on lattice-
matched InSb(111)B.

##  II Experimental Methods

The InSb(111)B substrates were cleaned by repeated cycles of Ar-ion sputtering
(with acceleration energy of 0.5 keV) and annealing at 540 K, resulting in In-
rich (3$\times$3) surface periodicity. Then, homo-epitaxial growth of InSb
layers by co-evaporating In and Sb in Sb-rich condition with the substrate
temperature of 550 K changed the substrate surface periodicity to Sb-rich
(2$\times$2) as shown in the RHEED pattern (Fig. 1 (a)) [22]. Bi was
evaporated from an Alumina crucible to the InSb(111)B-(2$\times$2) substrates
at room temperature. During the Bi evaporation, intensities of the RHEED
diffraction rods were monitored in-situ.

Core-level PES and ARPES measurements were performed at BL5U of UVSOR-III with
linearly polarized photons with the energies ranging from 40 to 120 eV. The
energy resolutions and positions of the Fermi level ($E_{\rm F}$) were
calibrated by the Fermi edge of Ta foils touching the sample. The energy
resolution was evaluated to be $\sim$30 meV. All the measurements were
performed at room temperature with the photon spot size of $\sim$50 $\mu$m on
the sample.

##  III Results and Discussion

Figures 1 (a, b) show the RHEED patterns before and after the Bi growth on
InSb(111)B. As shown in Fig. 1 (b), the Bi layer after growth shows low-
background and intense (1$\times$1) patterns indicating high quality of the
obtained Bi film. During the Bi growth, the surface fractional rods derived
from the (2$\times$2) superlattice of the Sb-terminated InSb(111)B substrate
gradually disappeared within 5 min. We found no distinct change of the in-
plane lattice constant reflected to the integer-order rods before and after
the growth. It would be because the difference of the in-plane lattice
constant of 2D Bi(111) (4.53 Å) and InSb(111) (4.58 Å) is too small to be
detected from the current RHEED diffraction rod width.

Figure 1 (c) is the intensity variations of the RHEED diffraction rods during
the Bi growth. One can find some significant peaks indicated by triangle
markers, which would be derived from completion of each Bi atomic layer. The
time interval between the peaks is 3.5$\pm$0.5 min. in most cases. The time
between the start of the evaporation and the first peak is 1.5-2.0 min.,
different from the others. It would be because the first peak is from the
wetting layer consists not only of Bi but also of surface Sb and In atoms. The
major difference between the (0 0) and (2 0) rods is the second peak. The
second peak of (0 0) is the most intense and its peak time ($\sim$ 5 min.) is
different from that of (2 0) ($\sim$6 min.). The ST phase would not contribute
to the in-plane diffraction rods other than (0 0), where all the information
of the ordered surface, both in-plane and out-of-plane ones, contributes. It
is because the ST patterns do not obey the in-plane uniform-interval
periodicity due to the self-similar fractal patterning. Therefore, the intense
peak appearing only at (0 0) rod would be derived from the ST phase. After
$\sim$15 min. with 5 peaks (4 layers and the wetting layer, while the 5th peak
is not so significant as the former ones), the oscillation of the RHEED
intensity disappears. It suggests that the growth mode of Bi changes and the
layer-by-layer growth becomes no longer valid after this period, while the in-
plane periodicity of Bi(111) remains as shown in Fig. 1 (b).

Figure 2 (a) is the core-level PES spectra of Bi/InSb(111)B during the Bi
growth. The 4$d$ peaks of In and Sb decrease together with the increase of Bi
5$d$ ones. This behavior is consistent with the growth of Bi layers capping
the InSb substrate. As indicated by the vertical dashed lines, Sb 4$d$ has
tails in higher kinetic energy region at the clean substrate (0 min.), while
In 4$d$ appears as a simple spin-orbital doublet. It would be due to the
surface termination layer of Sb of the Sb-rich InSb(111)B-(2$\times$2)
surface. This tail structure of Sb 4$d$ disappears with the Bi evaporation and
the peak energy shifts to higher Kinetic energies. At 6 min., both In and Sb
4$d$ levels are weak but simple doublets without tails. This change would come
from the formation of the Bi layers together with breaking the Sb-(2$\times$2)
termination layers of the substrate. As shown in Fig. 2 (b), the peak energy
of the Bi 5$d$ levels also changes from the initial period (1.5 min.) to the
thicker region (6.0 min. or higher). This behavior is consistent with the
RHEED oscillation suggesting the wetting-layer formation, where the
surrounding elements of Bi is different from the upper layers; only the Bi
atoms in the wetting layer would touch the substrate InSb. The peak positions
of the Bi 5$d$ in the 6 min. and 45 min. spectra are nearly the same,
suggesting that the surrounding conditions of the Bi atoms above the wetting
layer are almost identical in whole range of Bi growth.

Figure 2 (c) is the evolution of the In and Sb core-level areas normalized by
corresponding Bi areas. The core-level areas are obtained from the PES spectra
by subtracting linear backgrounds estimated around each peak; for example, 55
to 61 eV for In 4$d$. Both In and Sb peaks decay linearly in the logarithmic
plot, indicating the uniform exponential decay of the core-level area.
Moreover, the slope of the linear decay changes in 12-15 min. for both In and
Sb peaks, as guided by the fat lines in Fig. 2 (c). It suggests that the
growth mode of Bi changes in this region. Combined with the disappearance of
the RHEED oscillation around 15 min., it would be a change from the layer-by-
layer to 3D island-type growth modes. The latter mode has the smaller slope
than that of the former mode. This behavior can be explained by the assumed
change of the growth mode, because the 3D island growth could remain some area
with thin Bi thicknesses.

Figure 3 is the ARPES intensity plots of the Bi layers on InSb(111)B, measured
along $\bar{\Gamma}-\bar{\rm M}$ with linearly polarized photons ($h\nu$ = 40
eV). As guided by the white dashed lines in Figs. 3 (a, b), the hole-like
bands (light and heavy holes of zinc-blende-type semiconductors) observed
before Bi evaporation disappears soon after starting the Bi evaporation. At
1.5 min. corresponding to the the wetting layer formation, no significant
bands were observed. It suggests that the wetting layer of Bi on InSb(111)B
has no well-defined 2D atomic and electronic structures. Then, at 6.0 min.,
where the ST-phase would appear, the new bands S1 and S2 dispersing upwards to
$\bar{\Gamma}$ and downwards to $\bar{\rm M}$, respectively, appear. However,
these bands, S1 and S2, remain in the similar energy ranges in thicker
regions, as shown in Figs. 3 (d, e), far above the disappearance of ST
structures [20]. Therefore, these bands would be from the 2D Bi(111) layers,
rather than from the fractal ST surface structures. We found no other new band
structures around $E_{\rm F}$ in 4.5-6.0 min. samples. It would be because of
the same reason why the formation of the ST phase only appears in the RHEED (0
0) diffraction rod; the in-plane periods of the ST phase are not uniform and
thus the atomic and electronic structures from the ST phase cannot be observed
as the 2D periodic diffraction nor bands.

The binding energy of the bottom of S2 decrease together with the increasing
thickness of Bi, as shown in Figs. 3 (c-f). In the thick region (Figs. 3 (e,
f)), the second feature S2’ lying in deeper binding energies than S2 appears.
These can be understood by assuming S2 and S2’ as the quantum-well (QW) states
of Bi ultra-thin layers appearing around $\bar{\rm M}$ [11]. Since the number
of QW bands increases together with the number of the layers, one QW band, S2
in this case, should move to uniform directions during the growth so that the
larger numbers of QW bands could be filled in the corresponding bulk-band
projection. This trend is consistent with the assumed layer-by-layer growth
mode of the 2D Bi layers on InSb(111)B.

However, some features of the ARPES plots cannot be understood
straightforwardly by the layer-by-layer growth mode. At first, the observed
band dispersion of the thick sample shown in Fig. 3 (f) is different from the
bulk-like Bi(111) layers. For example, the electron pocket forming the
hexagonal Fermi contour [11, 13] is not observed in this work. Secondly, the
QW state appears even in the thick sample (Fig. 3(f)) where the layer-by-layer
growth mode should be already broken. These features could be explained by
supposing that the surface atomic structures of the obtained 2D Bi layers have
small commensurate areas. The earlier work reported that even after the
disappearing of the fractal ST phase, the surface atomic structure of
Bi/InSb(111)B contains many domain walls and the commensurate 2D atomic
structure is smaller than 10 atomic units [20]. This would prevent the
formation of the well-defined 2D surface bands as observed on the other
substrates [11, 14, 17, 13]. This hypothesis could also explain the diffused
bands in whole ARPES data, comparing to the other substrates. With such in-
plane disorder and variation of the heights of the 3D islands, the QW states
from each island would be averaged as an uniform background and only the
common 2D layers, which would have the maximum height among the islands in
each growth period, would provide the diffused QW states to the APRES
intensity plots, as Fig. 3 (f).

As shown above, the averaged 2D electronic structure of the Bi layers on
InSb(111)B is diffused and it would be difficult to find out unique electronic
characteristics from there. However, at the same time, it encourages the
electronic structure exploration with the better spatial resolution. For
example, the electron microscopy technique together with ARPES [23] or ARPES
with nano-focused photons [24] would be useful to obtain the electronic
structures localized in self-similar triangle edges and corners on the ST Bi
layers. Since the majority area of the Bi layers lacks the well-defined 2D
periodicity, there would be full of the 1D edges with different
characteristics (lengths, corner angles, and so on). Therefore, the Bi layers
on InSb(111)B would be a fertile playground for the next-generation high-
spatial resolution techniques to study the low-dimensional (0D or 1D)
topological electronic structures.

##  IV Summary

In this work, we have studied the evolution of the atomic and electronic
structures of the Bi(111) layers grown on InSb(111)B substrates by using RHEED
and PES. It is suggested that the single layer of the ST-phase Bi grows on
InSb(111)B and the following Bi evaporation causes layer-by-layer growth up to
nominally 4 atomic layers. Afterward, the growth behavior of Bi changes to the
3D mode with small, isolated Bi(111) islands on InSb(111)B substrates
suggested from diffused ARPES band dispersion. These results support the
previous observation of the fractal ST-phase Bi and provide wide-range picture
of Bi 2D layer growth on lattice-matched InSb(111)B, encouraging the following
researches with high spatial resolution to unveil the low-dimensional
electronic structures of the Bi fractal triangles.

##  V Acknowledgement

We acknowledge J. K. Modak, S. Ideta and K. Tanaka for their technical support
during the experiments. The core-level PES and ARPES measurements were
performed under UVSOR proposals 19-563 and 20-765. This work is supported by
JSPS KAKENHI (Grants Nos. JP20K03859, JP19H01830, and JP20H04453).

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Figure 1:  (a, b) RHEED patterns of (a) the InSb(111)B-(2$\times$2) substrate
and (b) Bi grown on InSb(111)B with the evaporation time of 45 min. Electron
energy is 12.5 keV. Rectangles in (b) represent the area where the RHEED
intensities were monitored. (c) RHEED intensity oscillations during Bi growth
on InSb(111)B-(2$\times$2). Triangle markers indicate the peak positions of
each diffraction rod.  Figure 2:  (a) Core-level PES spectra upon Bi
evaporation on InSb(111)B. Vertical dashed lines are guides to the eye for
peak positions of In 4$d$ and Sb 4$d$ levels. (b) Close-up spectra of Bi 5$d$
levels from the same data as (a). Each spectra is normalized by the maximum
values in the kinetic energy range shown here (46 to 54.5 eV). (c) Evolution
of In and Sb core-level areas normalized by corresponding Bi core-level areas.
Figure 3:  ARPES intensity plots obtained from Bi layers grown on InSb(111)B
along $\bar{\Gamma}-\bar{\rm M}$ (parallel to [11$\bar{2}$]). The inset in (a)
depicts the $k_{//[11\bar{2}]}$ region observed. Dashed lines guide the
observed band dispersion.


11institutetext: Center for Cosmology and Astrophysics, Alikhanian National
Laboratory and Yerevan State University, Yerevan, Armenia

# On the role of $\Lambda$ on accretion disks

Sh. Khlghatyan

(Received: date / Revised version: date)

###### Abstract

The dynamics of accretion disk is considered taking into account the Lense-
Thirring precession in the presence of cosmological constant $\Lambda$ of
Schwarzschild – de Sitter metric. The nodal and apsidal frequencies are
obtained and the role of $\Lambda$ is revealed in their properties, including
the consequences for the Bardeen-Petterson effect.

###### pacs:

PACS-04.50.KdModified theories of gravity

##  1 Introduction

The accretion disks are known to be crucial structures which exist both in
stellar scale in the vicinity of individual stars, and especially, in centers
of galaxies, essentially determining the observational properties of those
objects LB ; SS ; LBP ; GO ; Bal . Among the effects influencing the structure
and evolution of accretion disks, General Relativity (GR) predictions i.e.
Lense-Thirring, Bardeen-Petterson effects BP  have been considered both for
stellar and galactic nuclei scale accretion processes, see 1  and references
therein.

At the same time, the observational indications for the dark matter and dark
energy have triggered the study of variety of modified gravity models. Among
the considered modified gravity models is the one involving the cosmological
constant in the weak-field limit of GR G ; GS1 ; GS2 . $\Lambda$ acts as
second gravity constant along with the Newtonian constant $G$, as one of the
fundamental constants of Universe GS3 . Then the resulting metric is that of
Schwarzschild – de Sitter R ; GS1

$g_{tt}=1-\frac{2GM}{c^{2}r}-\frac{\Lambda r^{2}}{3},\quad
g_{rr}=\left(1-\frac{2GM}{c^{2}r}-\frac{\Lambda r^{2}}{3}\right)^{-1}.$ (1)

For the current the numerical value $\Lambda=1.11\times 10^{-52}m^{-2}$ Pl  it
is predicted that with an increase in accuracy and statistics in observational
data, the observations of gravitational lensing, groups and clusters of
galaxies, the flow of galaxies in the vicinity of the Local Group, properties
of cosmic voids GK  can provide informative tests to detect the effects
associated with the cosmological constant GS6 . The influence of $\Lambda$ on
various effects and the possibility to obtain constraints associated with
$\Lambda$-term is considered in L ; Con1 ; GKS ; SK ; GEM . Particularly, it
was shown that the $\Lambda$-term is able to provide a solution for the Hubble
tension problem GS4 .

In this paper we study the role of the metric Eq.(1) and of the $\Lambda$-term
in the structure and evolution of accretion disks, particularly, regarding the
Lense-Thirring precession and its consequences. We obtain expressions for both
nodal and apsidal frequencies with the $\Lambda$ term and reveal the
consequences for the Bardeen-Petterson effect BP .

##  2 Basic equations

The main equations describing the structure of accretion disks i.e. of mass,
angular momentum and energy balance are SS ; 4

$\displaystyle\frac{\partial}{\partial t}(2\pi
r\Sigma)+\frac{\partial\dot{M}}{\partial r}=0,$
$\displaystyle\frac{\partial}{\partial t}(2\pi r\Sigma
j)+\frac{\partial\dot{J}}{\partial r}=0,$
$\displaystyle\frac{\partial}{\partial t}(2\pi r\Sigma
e)+\frac{\partial\dot{E}}{\partial r}+4\pi rF=0,$

where $\Sigma$ is the surface mass density, $\dot{M}$ is the rate of
accretion, $j$ is the angular momentum per unit mass, $\dot{J}$ is the rate of
angular momentum flow, $e$ is the total mechanical energy per unit mass,
$\dot{E}$ is the rate at which energy flows across a cylinder with a radius
$r$ and $F$ is the rate of energy which is radiated locally from the disk
surface. These quantities are related to the model-dependent parameters of the
disk and to the following basic quantities as follows

$v=\sqrt{r\frac{d\Phi}{dr}},\quad\Omega=\sqrt{\frac{1}{r}\frac{d\Phi}{dr}},\quad
j=\sqrt{r^{3}\frac{d\Phi}{dr}},$ (2)

where $v$ is the rotational velocity, $\Omega$ is the angular velocity and
$\Phi$ is the gravitational potential. Moreover, in order to consider the
relativistic effects it is convenient to use the so-called GR potential which
is defined as 4

$\Phi_{GR}=-\frac{GM}{r-r_{g}}$ (3)

where $r_{g}=\frac{2GM}{c^{2}}$ is the gravitational radius.

##  3 $\Lambda$ corrections

For the metric Eq.(1) one can see that $\Lambda$ enters in the weak field
limit of GR and at the same time influences in the relativistic precession (or
Einstein’s precession) Br  and the Lense-Thirring effect. In the weak-field
limit the gravitational potential is written as

$\Phi=-\frac{GM}{r}-\frac{\Lambda c^{2}r^{2}}{6}.$ (4)

On the other hand, for relativistic precession we will get

$\Omega_{GR}=\frac{2GJ}{c^{2}\tilde{a}^{3}(1-\tilde{e}^{2})^{3/2}}+\frac{\Lambda
J}{3M},$ (5)

where $\tilde{a}$ and $\tilde{e}$ are the semi-major axis and the eccentricity
of the orbit respectively. Then, the Lense-Thirring precession changes to

$\Omega_{LT}=\frac{2GJ}{c^{2}r^{3}}+\frac{\Lambda J}{3M},$ (6)

where $J$ is the angular momentum of the central body with mass $M$.

According to the Eqs.(4-6) the contribution of the $\Lambda$-term increases at
large distances from the object. Namely, the distance beyond which the
$\Lambda$-term becomes dominant is

$r^{3}_{cr}=\frac{6GM}{\Lambda c^{2}}.$ (7)

This critical distance $r_{cr}$ is tabulated in Table 1 for certain typical
objects.

Table 1:  Critical distance for different objects
---
Central Object |  Mass (Kg) |  Radius (m)
Earth |  5.97 $\times 10^{24}$ |  4.92 $\times 10^{16}$
Sun |  1.98 $\times 10^{30}=M_{\odot}$ |  4.30 $\times 10^{18}$
M87∗ |  6.35 $\times 10^{9}M_{\odot}$ |  4.06 $\times 10^{21}$

Considering Eq.(4) and the form of the GR potential in Eq.(3), the main
parameters of accretion disk can be written as

$\displaystyle
v=\frac{1}{\sqrt{3}}\sqrt{\frac{3GM}{r}\left(1+\frac{2r_{g}}{r}\right)-\Lambda
c^{2}r^{2}}$ (8)
$\displaystyle\Omega=\sqrt{\frac{GM}{r}\left(1+\frac{2r_{g}}{r}\right)-\frac{\Lambda
c^{2}}{3}}$ (9) $\displaystyle
j=\sqrt{GMr\left(1+\frac{2r_{g}}{r}\right)-\frac{\Lambda c^{2}r^{4}}{3}}.$
(10)

##  4 Nodal and apsidal frequencies

Using the post-Newtonian approximation and taking into account the corrections
for the momentum, the equation for the disk takes the following form 1

$\frac{d\mathbf{v}}{dt}=-\frac{1}{\rho}\nabla
P+\mathbf{v}\times\mathbf{h}-\nabla\Phi+\mathbf{S}_{visc},$ (12)

where $S_{visc}$ is the viscous force per unit mass, $\rho$ is the density,
$P$ is the pressure and $\mathbf{v}\times\mathbf{h}$ is the gravitomagnetic
force according to gravito-electromagnetic (GEM) formalism GEM1 ; GEM . In
order to consider the relativistic effects the GR potential in Eq.(3) is used
in the above equation for the gravitational potential $\Phi$.

The Lense-Thirring precession causes the plane of an orbit inclined to the
$(x,y)$ plane to precess around the angular momentum $(z)$ axis. This
precession is taken into account by adding the second term in Eq.(12).

The nodal precession frequency that arises due to the inclusion of the
gravitomagnetic force term in the equation of motion is defined by the
derivative of the effective potential 1 ; 2

$\Omega^{2}=\frac{\partial^{2}\Phi_{eff}}{\partial z^{2}}\Big{|}_{z=0},$ (13)

where the effective potential includes the contribution of the gravitomagnetic
force and is given as 3

$\Phi_{eff}=\Phi(r)+\Phi^{\prime},\quad\mathbf{v}\times\mathbf{h}=-\nabla\Phi^{\prime}.$
(14)

##  5 $\Lambda$-modification

Considering the corrections given in the section 3, we can obtain the modified
nodal and apsidal frequencies. In our case, we are interested in the second
and third terms in the Eq.(12), which are modified due to the presence of the
$\Lambda$ term.

Then, $\mathbf{h}$ will take the following form

$\mathbf{h}=\left(\frac{2}{r^{3}}+\frac{\Lambda}{3M}\right)\mathbf{S}-\frac{6(\mathbf{S}\mathbf{r})}{r^{5}}\mathbf{r}$
(15)

and the gravitational potential is modified as

$\Phi=-\frac{GM}{r}\left(1+\frac{r_{g}}{r}\right)-\frac{\Lambda
c^{2}r^{2}}{6}.$ (16)

Taking into account the modified expressions we find the effective potential

$\begin{split}\Phi_{eff}=&\frac{4S}{5r^{5/2}}\sqrt{GM}\,\,{}_{2}F_{1}\left(-\frac{5}{6},-\frac{1}{2};\frac{1}{6};\frac{\Lambda
c^{2}}{3GM}r^{3}\right)-\frac{2r}{3}\sqrt{\frac{G}{M}}S\Lambda\,\,{}_{2}F_{1}\left(-\frac{1}{2},\frac{1}{6};\frac{7}{6};\frac{\Lambda
c^{2}}{3GM}r^{3}\right)\\\
&-\frac{3Sz^{2}}{r^{4}}\sqrt{\frac{GM}{r}-\frac{\Lambda
c^{2}r^{2}}{3}}-\frac{GM}{r}\left(1+\frac{r_{g}}{r}\right)+\frac{\Lambda
c^{2}r^{2}}{6},\end{split}$ (17)

where ${}_{2}F_{1}$ is the hypergeometric function.

Note, that the cosmological constant $\Lambda$ enters in two places and
modifies two phenomena. First, it enters in the potential of the gravitational
field $\Phi$ and second, it influences the pure relativistic Lense-Thirring
precession frequency. Furthermore, as can be seen from the Eq.(15), the
additional frequency with the $\Lambda$ term does not depend on the distance
or other parameters of the disk.

When $\Lambda\to 0$ the Eq.(17) is reduced to the ordinary Newtonian
potential. The difference between the potentials $\Delta\Phi$ due to the
presence of $\Lambda$-terms for a black hole of M87* type is shown in Figure
1. One can see that while this difference is negligible for galactic nuclei
scales, the presence of $\Lambda$-terms becomes considerable at larger
distances.

Figure 1: The difference between effective potentials for M87*.

Using the expression for the effective potential and the equation for the
nodal frequency $\Omega$ we find the $\Lambda$-modified expression for the
nodal precession. In addition, in order to obtain a dimensionless frequency,
we normalize the so-called post-Keplerian frequency by taking the $\Lambda$
term into account

$\Omega^{2}=-\frac{6S}{r^{4}\omega_{p-K,\Lambda}^{2}}\sqrt{\frac{GM}{r}-\frac{\Lambda
c^{2}r^{2}}{3}}.$ (18)

The post-Keplerian frequency is obtained from the following equation

$g_{tt,r}\left(\frac{dt}{d\phi}\right)^{2}+2g_{t\phi,r}\left(\frac{dt}{d\phi}\right)+g_{\phi\phi,r}=0,$
(19)

where $g_{tt},g_{t\phi},g_{\phi\phi}$ are the components of the Kerr-de Sitter
metric. Accordingly, considering the definition
$\omega_{p-K,\Lambda}=\frac{d\phi}{dt}$ we have

$\omega_{p-K,\Lambda}^{2}=\frac{MG}{r^{3}}-\frac{\Lambda
c^{2}}{3}-\frac{4a}{c}\left(\frac{MG}{r^{3}}-\frac{\Lambda
c^{2}}{3}\right)^{3/2}.$ (20)

Finally, using the nodal frequency according to the Eq.(18) we find the
apsidal frequency

$\kappa^{2}=4\Omega^{2}-\frac{9}{r^{5}}\frac{\frac{2\Lambda
c^{2}r^{3}}{3}-3GM}{\sqrt{\frac{GM}{r}-\frac{\Lambda c^{2}r^{2}}{3}}}.$ (21)

##  6 Bardeen-Petterson effect

The influence of the Lense-Thirring precession on the dynamics of the
accretion disk leads to Bardeen-Petterson effect when the nodal angle changes
and as a result, a spiral-like patterns have to appear in the disk BP . The
rotation angle of the line of nodes is given by

$\alpha=\frac{r\Omega_{LT}}{v_{r}},$ (22)

where $v_{r}$ is the velocity of the accretion disk that flows into the black
hole and consequently it spends time $t=r/v_{r}$ at radius $r$. For the
Shakura-Sunyaev accretion disk SS  the velocity $v_{r}$ is

$v_{r}=\frac{a}{t_{d}}$ (23)

and $t_{d}$ is the diffusion time defined as

$t_{d}=\frac{a}{\nu}.$ (24)

In the above equation $a$ is the semi-major axis and the diffusion coefficient
$\nu$ is related to the rotation angle of nodes $\alpha$, the Keplerian
frequency $\omega_{K}$ and to the scale height $H$ as

$\nu=\alpha\omega_{k}H^{2}.$ (25)

Thus we can state that the presence of the $\Lambda$ contributes to the values
$\Omega_{LT}$ by the Eq.(6), $\omega_{p-K,\Lambda}$ by Eq.(20) and to the
scale height $H$. Taking into account all these corrections, we get the
following expression for

$\alpha^{2}=\Omega_{LT}\frac{ar}{\omega_{p-K,\Lambda}}\left(\frac{\mu}{kT}\right)^{2}\left(\frac{GM}{a^{2}}-\frac{\Lambda
c^{2}}{3}a\right)^{2},$ (26)

where $k$ is the Boltzmann constant, $\mu$ is the mass of the gas particles
and the temperature $T$ is obtained by the condition of equilibrium between
the heating rate due to viscosity.

##  7 Conclusion

We studied the dynamics of the accretion disk to reveal the contribution of
the $\Lambda$-term in the equations of modified GR. We showed that the
$\Lambda$-term enters in both relativistic and non-relativistic equations and
contributes in two different phenomena accordingly. For the weak-field limit
$\Lambda$ enters in the gravitational potential $\Phi$, while for relativistic
effects one has the modification due to $\Lambda$ both in Einstein and Lense-
Thirring precessions. We obtained the critical distance at which the
contribution of $\Lambda$ becomes dominant and by considering the presence of
$\Lambda$ in the above mentioned effects we calculated the nodal and apsidal
frequencies by solving the relevant equations analytically. For typical
astrophysical objects including for accretion disks in galactic centers the
real contribution of the $\Lambda$ is negligible. This issue, however, can
have sense for consideration in the context of galactic nuclei and galactic
larger structure evolutionary interrelations. One can recall that, initially
the gravitational lensing was also considered as non-observable effect.
Finally, we have analyzed the Bardeen-Petterson effect and have shown that due
to the presence of the $\Lambda$ in $\Omega_{LT}$ the post-Keplerian frequency
$\omega_{p-K,\Lambda}$ and the key parameter, i.e. the rotation angle of the
line of nodes are modified.

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