# Exploring Topological Transport in Pt2HgSe3 Nanoribbons: Insights for
Spintronic Device Integration

Rafael L. H. Freire<EMAIL_ADDRESS>Ilum School of Science,
Brazilian Center for Research in Energy and Materials (CNPEM), Campinas, SP,
Brazil.  F. Crasto de Lima<EMAIL_ADDRESS>Ilum School of Science,
Brazilian Center for Research in Energy and Materials (CNPEM), Campinas, SP,
Brazil.  Roberto H. Miwa<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal de Uberlândia, 38400-902 Uberlândia, Minas Gerais, Brazil.  A. Fazzio
<EMAIL_ADDRESS>Ilum School of Science, Brazilian Center for
Research in Energy and Materials (CNPEM), Campinas, SP, Brazil.

###### Abstract

The discovery of the quantum spin Hall effect led to the exploration of the
electronic transport for spintronic devices. Here, we theoretically
investigated the electronic conductance in large-gap realistic quantum spin
Hall system, Pt2HgSe3 nanoribbons. By an ab initio approach, we found that the
edge states present a penetration depth of about $0.9$ nm, which is much
smaller than those predicted in other 2D topological systems. Thus, suggesting
that Pt2HgSe3 allows the exploitation of topological transport properties in
narrow ribbons. Using non-equilibrium Green’s functions calculations, we have
examined the electron conductivity upon the presence of Se $\leftrightarrow$
Hg antistructure defects randomly distributed in the Pt2HgSe3 scattering
region. By considering scattering lengths up to $109$ nm, we found
localization lengths that can surpass $\mu$m sizes for narrow nanoribbons
($<9$ nm). These findings can contribute to further understanding the behavior
of topological insulators under realistic conditions and their integration
within electronic, spintronic devices.

††preprint: APS/123-QED

##  I  Introduction

Topological insulators (TIs) are an emerging class of intriguing materials
with unique electronic properties. Particularly, the wave function that
describes their electronic states spans a Hilbert space with a new topology.
The consequence is that at any interface, with an ordinary insulator, they
will present a gapless state protected by time-reversal symmetry [1].

The first indirect observation of the topological edge states was the
measurement of the quantized conductance in HgTe/CdTe quantum wells [2, 3]. On
the other hand, the direct observation of the electronic state on the surface
was possible through an angle-resolved photoemission spectroscopy (ARPES)
experiment, first in Bi_1-xSb_x [4]. Despite this, characterizing the surface
structure of such materials is not an easy task, and later combined
measurements of ARPES, scanning tunneling microscopy (STM), and scanning
tunneling spectroscopy (STS) helped to identify TI materials [5, 6].

Measurements of quantized conductance in the quantum spin Hall geometry have
been done in quantum well systems HgTe/CdTe and InAs/GaSb [2, 3, 7], on
decorated/interfaced graphene [8], and on WTe2 [9]. STM/STS of edge states
were also observed on WTe2 [10, 11], SiC/Bismuthene interface [12] and in
jacutingaite mineral (Pt2HgSe3) [13]. However, those systems are either
quantum wells, low gap decorated graphene, and/or 2D materials unstable under
oxidation [14, 15, 16]. In other words, only a few 2D materials with the
topological helical edge mode showing quantized transport have been measured.
This drives a quest to deeply understand their structural and energetic
stability, how intrinsic defects, confinement length (ribbon width), and
temperature, to cite a few, can affect the quantized transport.

Jacutingaite (Pt2HgSe3) is a naturally occurring mineral [17], stable against
oxidation [13]. It gained attention due to the large spin-orbit coupling
effect leading to a non-trivial $0.15$ eV energy gap opening [18, 19], being
the first realization of the Kane-Mele topological model [20]. While it is a
chemically stable topological insulator, the experimental synthesis of
Jacutingaite is prone to present defects and disorders. Such effects in the
electronic transport of the topological states, combined with confinement
potential, bulk density of state, and topological state penetration depth, can
change the quantized topological conductance.

In this work, we systematically study the ballistic transport in jacutingaite
nanoribbons including stoichiometric defects, by combining density functional
theory (DFT) and electronic transport calculations through non-equilibrium
Green’s functions (NEGFs). We explored different concentrations of non-
magnetic defects in a disordered geometry and the competition between
topological edge-state manifestation and confinement in narrow ribbons. Here
we showed that the localization length in the transport can surpass $\mu m$
length, that is, allowing realistic device engineering in the jacutingaite
platform despite the presence of intrinsic defects.

##  II Methodology

###  II.1 Equilibrium geometry and electronic structure

To obtain the geometric, energetic, and electronic properties of pristine and
defective systems, we performed density functional theory (DFT) calculations
as implemented in the plane wave basis Vienna Ab-Initio Simulation Package
(VASP) [21, 22]. We employed the semilocal exchange-correlation formalism
proposed by Perdew, Burke, and Ernzerhof (PBE) [23, 24]. To improve the
description of van der Waals interactions, we used the DFT-D3 pairwise
corrections proposed by Grimme [25]. The plane-wave basis energy cutoff was
set to $400$ eV for all calculations with an energy convergence parameter of
$10^{-6}$ eV, and atoms were allowed to relax until all forces were smaller
than $10^{-3}$ eV/Å. The electron-ion core interactions were described through
the projected augmented wave (PAW) potentials [26, 27]. We integrated the
Brillouin zone through a $4\times 4\times 1$ k-point mesh for relaxation. We
further included relativistic effects by considering spin-orbit coupling
corrections (SOC).

###  II.2 Electronic transport

Figure 1:  (up) Nanoribbon dimensions and illustrative device for electronic
transport measurement. (bottom) Defective building blocks indicating center
and edge antistructure defects. Gray, (purple/black), and (green/red) atoms
represent Pt, Hg, and Se atoms, respectively.

In Fig. 1, we present the setup used for the electronic transport calculation.
We have considered the Se-terminated zigzag nanoribbons (NRs) [13] where the
electronic transport takes place along the $z$-direction, and three NR widths
($W$) of $34$, $60$, and $99$ Å, parallel to the $x$-direction. We have
combined DFT and a recursive Green’s function (RGF) method [28]. The system
can be decomposed into three main parts, two semi-infinite charge reservoirs,
that are the ($i$) left and ($ii$) right electrodes, and ($iii$) the central
scattering region. Besides, within the RGF the scattering region can be
further split into several building blocks (Disorder Blocks in Fig. 1). Two
distinct building block kinds have been examined, as shown in Fig. 1(bottom),
one with the anti-structure defect located in the NR’s central region and the
other with the defect close to the NR’s edge sites. The concatenation of these
blocks along the transport-direction is random, resulting in scattering
lengths ($L$) up to $0.1$ $\mu$m. It is also important to note that the blocks
were constructed to ensure coupling only between adjacent neighbors. For the
transport calculations, we first obtain the electrodes and scattering region
Hamiltonians through SIESTA-based DFT calculations. Thus, we employed norm-
conserving Troullier-Martins pseudopotentials [29], and single
$\zeta$-polarized (SZP) basis set of numerical atomic orbitals to expand the
Kohn-Sham orbitals of the valence electrons. Our real-space mesh cutoff was
set to $300$ Ry and the Brillouin zone was integrated through a Monkhorst-Pack
$k$-points mesh $1\times 1\times 10$ ($1\times 1\times 3$) for the electrodes
(scattering) region. Our SZP basis set calculations agree with our plane-wave-
based calculations, thus being sufficient to describe the electronic structure
of the system.

##  III Results

Although the present study focuses on the topologically protected electronic
transport in Pt2HgSe3 nanoribbons, we will start our investigation by
examining key structural and electronic properties of pristine and defective
Pt2HgSe3 monolayer.

###  III.1 Pt2HgSe3 monolayer

####  III.1.1 Pristine system

Figure 2:  (a) Jacutingaite monolayer top and side views and (b) electronic
band structure with and without SOC effects. Gray, purple, and green atoms
represent Pt, Hg, and Se atoms, respectively.

The Hg atoms in Pt2HgSe3 pinpoint a honeycomb structure, purple atoms in Fig.
2(a), embedded in a PtSe matrix. At the equilibrium geometry, the electronic
band structure obtained without the inclusion of the spin-orbit coupling (SOC)
is characterized by a Dirac cone structure crossing the Fermi level at the
K-point, Fig. 2(b). Meanwhile, upon the inclusion of the SOC, we found an
energy gap at the Dirac point with the band edges composed mainly by Hg-$6s$
orbitals hybridized with Pt-$4d$ orbitals. Such behavior is typical of the
Kane-Mele model [20], and characterizes the topological phase of jacutingaite
[18, 19]. The energy gap opened by the spin-orbit coupling in the pristine
system is calculated here as $0.13$ eV. It is worth pointing out that previous
studies considering hybrid functionals (HSE) and GW calculations predicted a
higher energy gap of $0.2$ eV [19, 30] and $0.5$ eV [18]. However,
experimental STM measurements at $9$ K estimate the energy gap to be $0.08\pm
0.03$ eV [13], closer to the one obtained by using the DFT-GGA approach.

Table 1:  Intrinsic point defects’ formation energies. Defect |  $E^{f}$ ($\mathrm{eV}$)
---|---
VSe | $5.42$
VPt | $6.87$
VHg | $0.23$
$({\rm Se\leftrightarrow Hg})$ | $0.79$

####  III.1.2 Intrinsic defects

The emergence or suppression of a given topological phase as well as its
influence on electronic transport can be dictated by the presence of defects
[31, 32, 33, 34]. For instance, the trivial $\rightarrow$ non-trivial QSH
transition is mediated by the concentration of selenium vacancies (VSe) in
PtSe2 ML [33].

Here, we have considered vacancies (VX, with X = Se, Pt, and Hg), and
antistructure (Se $\leftrightarrow$ Hg) defects [35] in Pt2HgSe3 ML. The
occurrence rate of these defects can be inferred by the calculation of the
formation energy ($E^{f}$)[36, 37],

$E^{f}=E_{\rm pristine}-E_{\rm defect}-n\times E_{\rm X},$ (1)

where $E_{\rm pristine}$ and $E_{\rm defect}$ are the total energies of the
pristine and the defective Pt2HgSe3 ML; $E_{\rm X}$ is the total energy of an
isolated atom X, and $n$ is the number of missing atoms. For a single X
vacancy, VX, $n$ = 1, while for the antistructure defect, which is
stoichiometric, $n$ = 0. In the antistructure defects, we have considered
antisites, SeHg and HgSe, created adjacent to each other. Generally,
transition metals vacancy have larger defect formation energy than the
chalcogen vacancy [38], which is also verified here, as VPt formation energy
is higher than VSe. However, the formation energy of both defects is larger
compared with VHg, $0.23$ eV, followed by the (Se $\leftrightarrow$ Hg)
antistructure, 0.79 eV.

The stoichiometry of Pt2HgSe3 has been conserved in experimentally synthesized
jacutingaite samples [39, 13]. Meanwhile, naturally occurring samples exhibit
Pd doping in the Pt site while maintaining the Hg/Se ratio [17]. These
results, combined with the lower formation of the (stoichimetric)
antistructure defects, allow us to infer that (Se $\leftrightarrow$ Hg)
interchanged defect is quite likely to be present in Pt2HgSe3 [40]. In the
sequence, we will examine the effect of the local disorder induced by the
antistructure defect on the electronic transport properties, mediated by the
topologically protected edge states in Jacutingaite NRs.

###  III.2 Pt2HgSe3 nanoribbons: electronic transport

The transport geometry for the monolayer Pt2HgSe3 is presented in Fig. 1. The
leads are pristine systems, and the scattering region contains the
antistructure defects. We explored ($i$) different nanoribbons width ($W=34$,
$60$, and $99$ Å), ($ii$) different antistructure concentrations
($\lambda=1.9\times 10^{6}$, and $4.5\times 10^{6}$ defects/cm), and ($iii$)
different scattering region length ($L=15.6$, $31.2$ and $109.2$ nm).

####  III.2.1 Pristine nanoribbons

Figure 3:  Electronic band structure projected on the bulk and edge regions
(left) and conductance (right) across a Pt2HgSe3 nanoribbon with different
size width, (a) $W=34$, (b) $W=60$ and (c) $W=99$ Å. The green and gray colors
in the band structure is proportional to the edge ($|\langle{\rm
edge}|n,\vec{k}\rangle|^{2}$) and bulk ($|\langle{\rm
bulk}|n,\vec{k}\rangle|^{2}$) contribution to the system eigenstate,
respectively. For all band structures, each NR edge region for projection is
about $7.69$ Å wide (perpendicular to transport direction), while the
remaining atoms are considered as bulk projection region.

First, we analyze the conductance through a defect-free jacutingaite NR. In
such a case, the incident wave function is also an eigenstate of the
scattering region, thus leading to a suppression of scattering processes. In
Figs. 3(a)-(c), we present the electronic band structure and the conductance
results (in units of $G_{0}$ = $e^{2}/h$) for different NR widths, $W=34$,
$60$, and $99$ Å. The band structure presents the projected contribution of
the atomic orbitals from the middle of the ribbon, namely bulk-like region
$|\langle{\rm bulk}|n,\,\vec{k}\rangle|^{2}$, and the orbitals at the edge
$|\langle{\rm edge}|n,\,\vec{k}\rangle|^{2}$. We can identify two topological
edge bands close to the Fermi energy. For $W=34$, and $60$ Å, we can see a gap
opening at the Dirac crossing due to the inter-edge interaction. In contrast,
for $W=90$ Å, we find the emergence of a Dirac cone at the $\Gamma$-point.

As shown in Fig. 3(right), the energy window ($\Delta\varepsilon$) close to
the Fermi level with $G$ = $2G_{0}$, due to the edge metallic band, decreases
as the NR width increases. That is, for $W=90$ Å [Fig. 3(c)] we found
$\Delta\varepsilon$ of around $0.13$ eV, which is compatible with the
topological band gap of the single layer jacutingaite. Whereas, for $W=34$ and
$60$ Å [Figs. 3(a) and (b)], in addition to the energy gap at the Dirac point
due to the inter-edge interactions, $\Delta\varepsilon$ increases to about
$0.19$ and $0.15$ eV, which can be attributed to the confinement potential
sparsing the bulk-like bands [41].

Since the inter-edge interactions are mediated by the bulk (inner) states, the
penetration length ($\ell$) of the edge states is a key quantity in
understanding the topologically protected electronic transport along the NR’s
edge channels. Within the Kane-Mele model, the penetration depth is $\ell\sim
at\sqrt{3}/(2\lambda)$, with $a$ the lattice parameter, $t$ the nearest-
neighbor hopping, and $\lambda$ the spin-orbit strength term. Taking these
values from the jacutingaite band structure with $t/6$ the Dirac band width at
$\Gamma$-point, the SOC gap at K-point,
$E^{\text{K}}_{\text{gap}}=6\sqrt{3}\lambda$, and $a=7.5$ Å, we can estimate
$\ell=11.2$ nm. In contrast, in Ref. [18] the authors predicted a penetration
depth ($\ell=\hbar v_{F}/E_{\text{gap}}$) of about $4.7$ Å, by using an energy
gap $E_{\text{gap}}$ of $0.53$ eV, larger than that observed experimentally,
and $v_{F}$ of $3.6\times 10^{5}$ m/s. Further STM measurements, indeed,
observed a penetration length of $\sim$ 5 Å in the Se terminated zigzag
jacutingaite NRs [13].

Figure 4:  Penetration depth estimation for different NR-width. Blue curves
indicate the $yz$-planar averaged local density of states (LDOS) according to
their real-space projection as depicted for each NR-width, obeying $|\Psi_{\rm
top}(x)|^{2}=|\phi(x)|^{2}e^{-2x/\ell}$ . Orange dashed lines are the
exponential fitting to obtain the respective penetration depth $\ell$.

By projecting the edge states contribution in real space, Fig. 4, we can
explicitly calculate the penetration depth ($\ell$) by fitting the wave
function decay as $\Psi_{\rm top}(x)=\phi(x)e^{-x/\ell}$, where $\phi(x)$ is
the planar ($yz$) averaged single-particle wave-function (near the
$\Gamma$-point) of the metallic band crossing the Fermi level. We found a mean
value $\ell=0.91$ nm for $W=34$ and $60$ Å [Figs. 4(a) and (b)], and
$\ell=0.76$ nm for the widest NR, $W=99$ Å. It is worth noting that these
values of $\ell$ are lower than predicted considering the empty honeycomb
lattice (Kane-Mele model), $11.2$ nm, suggesting that the PtSe2 background’s
dielectric media improves the wave function’s screening, and thus reducing
penetration depth.

The reduced penetration depth implies that the topological electronic
transport is maintained against inter-edge scattering processes even in narrow
jacutingaite NRs. However, in the case of scattering potentials along the NRs
[42, 43], we might observe an entirely different picture. In this scenario,
the inter-edge interactions may lead to the suppression of the electronic
transport along the edge channels.

####  III.2.2 Antistructure defects in NRs

Here, we will study the effect of the scattering potentials induced by the (Se
$\leftrightarrow$ Hg) antistructure on the electronic transport properties
along the jacutingaite NRs. In Figs. 5(a) and (b), we present the total charge
rearrangement ($\Delta\rho$), defined as the total charge difference between
the defective and pristine (defect-free) systems,
$\Delta\rho=\rho_{\text{defect}}-\rho_{\text{prist}}$, induced by the
antistructure defect at the center, and at the edge sites of the NR with
$W=34$ Å. In the former, we see a charge rearrangement in diameter of $\sim
0.7$ nm, giving a scale of the antistructure defect localization. In contrast,
as depicted in Fig. 5(b), an antistructure defect at the edge site of the NR
leads to a charge rearrangement in both edges, ruled by the edge-edge coupling
present for such width. Meanwhile, by increasing the ribbon width to $W=60$ Å
[Fig. 5(c)], although both NRs present practically the same penetration
length, $\ell=0.9$ nm [Figs. 4(a) and (b)], such a charge density
rearrangement at the opposite side to the defect becomes nearly absent. That
is, as expected, the inter-edge interactions are weakened for larger values of
NR width with respect to the penetration length.

Figure 5:  Charge density difference with/without the defect presence
$\Delta\rho(\vec{r})=\rho_{\rm def}(\vec{r})-\rho_{\rm pris}(\vec{r})$, for
the $W=34$ Å nanoribbon (a) and (b) and for $W=60$ Å nanorribon in (c). In (a)
the defect is located in the center of the ribbon, while in (b) and (c) the
defect is at the ribbon edge. Blue/yellow regions indicate a decrease/increase
in the electron concentration. The isosurface value is 0.01 $e$ Å-1

Focusing on the electronic transport properties, using the simulation setup
presented in Fig. 1, we calculate the conductance, $G$, as a function of the
width, $W$, and length ($L$) of the NR. We have considered scattering lengths
ranging from $16$ up to $109$ nm with the antistructure defects randomly
distributed along the scattering region, as described in Section II-B, and
schematically depicted in Fig. 6(e).

In Figs. 6(a)-(c), we present our results of conductance at the presence of
antistructure defects ($G_{\text{defect}}$, in units of $G_{0}=e^{2}/h$) near
the Fermi level, for $W=34$, $60$, and $99$ Å and linear concentration of
defects ($\lambda$) of $4.5\times 10^{6}$ cm-1. It is noticeable that
$G_{\text{defect}}$ is smaller than the conductance of the pristine NR
($G_{\text{prist}}$) indicated by dashed lines in Fig. 6. Since the
antistructure defects are non-magnetic, the backscattering process in a single
edge is forbidden by the time-reversal symmetry, thus we can infer that the
results of $G_{\text{defect}}<G_{\text{prist}}$ are due to the partial or
total inter-edge backscattering processes within the energy window
($\Delta\varepsilon$ in Fig. 3) of the topological edge states. It is observed
that the reduction of $G_{\text{defect}}$ with respect to $G_{\text{prist}}$
becomes more pronounced, independent of the NR width, for energies below
$-0.1$ eV, i.e. $\varepsilon-\varepsilon_{\text{F}}<-0.1\,\text{eV}$,
suggesting the predominance of the bulk states in the electronic transport
along the NR. Furthermore, the following results are noteworthy:

($i$) For a given NR width and defect concentration, $\lambda$,
$G_{\text{defect}}$ decreases for higher values of $L=15.6\rightarrow 109.2$
nm, since the scattering rate increases with the length of the scattering
region. Indeed, as shown in Figs. 6(a) and (d), by reducing the concentration
of the defects, $\lambda=4.5\rightarrow 1.9\,\times 10^{6}$ cm-1, we find that
the reduction of $G_{\text{defect}}$ is mitigated.

($ii$) For a given value of $\lambda$, the reduction of $G_{\text{defect}}$ is
also mitigated by increasing the NR’s width, $W=34\rightarrow 99$ Å [Figs.
6(a) $\rightarrow$ (c)]. This can be attributed to the reduction of the inter-
edge backscattering since the ratio between the penetration length of the edge
states and the width of the NR, $\ell/W$, reduces for wider systems. For
instance, $\ell/W=0.27$ and $0.08$ for $W=34$ and $99$ Å, respectively.

The energy dependence of $G_{\text{defect}}$ and the width of the NRs [($ii$)]
reveals that the deviation of $G_{\text{defect}}$ with respect to
$G_{\text{prist}}$ is proportional to the inter-edge orbital couplings
mediated by the antistructure states. In the present study of antistructure
defects, we find that $G_{\text{defect}}\approx 2G_{0}$ for
$\varepsilon-\varepsilon_{\text{F}}$ about $-0.05$ eV, as depicted in Figs.
6(b) and (c), thus indicating that the edge-edge interaction is nearly
suppressed around $0.05$ eV below the Fermi level. Other (non-magnetic)
defects may result in different energy intervals where the inter-edge coupling
is minimized. On the other hand, in the limit of ${W}\rightarrow\infty$,
perfect conducting channels with $G_{\text{defect}}=G_{\text{prist}}$ should
be observed, independent of the scattering region length and the concentration
of the defects.

Figure 6:  Conductance as a function of nanoribbon length for defect
scattering region. (a)-(c) for linear defect concentration of
$\lambda=4.5\times 10^{6}$ cm-1 and (d) for $\lambda=1.9\times 10^{6}$ cm-1.
(a) and (d) the ribbons have width $W=34$ Å , (b) $W=60$ Å and (c) $W=99$ Å.
In panel (e) we schematically present the distribution of the defects as a
function of the ribbon length. (f) The average localization lenght $\xi$ for
different energy ranges and nanorribon with $W$ for $\lambda=4.5\times 10^{6}$
cm-1.

####  III.2.3 Localization length

Given the random distribution of the scattering potential, here induced by the
antistructure defects, at first glance our results of conductance can be
interpreted in terms of Anderson localization. In this case, we can gain
further insights by looking into the localization length $\xi$ as a function
of the energy, where $\xi$ is defined as the length in which the conductance
decays by an exponential factor [44, 45]

$G=G_{0}e^{-L/\xi},$ (2)

here, $G_{0}$ is an exponential prefactor, and $L$ is the length of scattering
region. From the last equation, we can have

$\ln(G/G_{0})=-\frac{L}{\xi}.$ (3)

Within this model, $\xi$ is inversely proportional to the density of the
scattering centers, $\rho$, namely $\xi\propto\rho^{-1}$ [45]. Thus, it is
expected that a given materials’ NRs with the same defect concentration would
present similar localization lengths. However, this is not what we found in
our systems. Instead, $\xi$ presents a NR’s width dependence due to ($i$) the
inter-edge scattering process of the topological states and ($ii$) the
distinct behavior of the electronic transport through the bulk and edge
channels, where the latter (former) is (is not) topologically protected.

The behavior of $G$ vs $L$ is presented in Fig. 6 where we estimated the
localization lengths. Given the random distribution of defects for each $L$ an
oscillation of $G/G_{0}$ for each energy is expected, that is, driven by the
defect-defect. However across energies ranges wider than the defect-defect
interaction a mean localization length is robust. In Fig. 6(f) we present
average values of $\xi$ for energy ranges with $0.05$ eV window. For energies
outside the topological edge state range $\varepsilon-\varepsilon_{\rm
F}<-0.1$ we see a localization length of $10^{-1}$ $\mu$m, that is the
localization of the system trivial bulk-like states. Within the topological
gap $0.05>\varepsilon-\varepsilon_{\rm F}>-0.1$, the localization length
increases in relation to those on the bulk-like energy range, that is, the
backscattering forbidden topological transport is dominant. Additionally,
within the topological edge states energy range there is an increase in the
localization length with respect to the ribbon width $W$ reaching up to
$10^{1}$ $\mu$m for $W=99$ Å at the Fermi energy,
$\varepsilon-\varepsilon_{\rm F}=0$. Despite for narrow ribbons ($W=34$ Å) the
non-magnetic impurity leads to edge-to-edge scattering; there exists energy
ranges that can preserve long-range transport with conductance $2G_{0}$, for
instance for $\varepsilon-\varepsilon_{\rm F}=0$ eV below the Fermi energy,
reaching localization length up to $\sim 6$ $\mu$m. Thus, regardless of
intrinsic defects, the Pt2HgSe3 platform allows exploiting the topological
states in narrow ribbons.

##  IV Conclusions

We investigated the electronic transport properties of 2D-Jacutingaite
nanoribbons including defects and disorders within the systems. We analyzed
the electronic conductance for different-width nanoribbons and different
device lengths along the transport direction. Our ab initio results indicate a
synergy between the confinement potential and manifestation of topological
edge states that increase the energy range associated with the topological
edge states in narrow ribbons. Stoichiometric defects in this system have a
spatial length of $\sim 0.7$ nm, while the edge state penetration depth, $\sim
0.9$ nm, is much lower than in other 2D topological systems. By considering
scattering lengths up to $109$ nm, we found localization length that can
surpass $\mu$m sizes for narrow nanoribbons ($<9$ nm). This allows the
exploitation of topological transport properties in narrow ribbons. In fact,
by computing the transport localization length we show that a topological-
driven transport can survive high defect densities $\sim 10^{6}$ cm-1 for
ribbons with width $W=34$ Å. Our analysis can contribute to the fundamental
understanding and design integration of TIs within spintronic devices.

## Conflicts of interest

There are no conflicts to declare.

###### Acknowledgements.

The authors acknowledge financial support from the Brazilian agencies FAPESP
(grants 20/14067-3, 19/20857-0, and 17/02317-2), CNPq, INCT-Nanocarbono, INCT-
Materials Informatics (grant 167651/2023-4), and Laboratório Nacional de
Computação Científica for computer time (project ScafMat2 and emt2D).

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# An entropic effect essential for surface entrapment of bacteria

Premkumar Leishangthem  Complex Systems Division, Beijing Computational
Science Research Center, Beijing 100193, China  Xinliang Xu
<EMAIL_ADDRESS>Complex Systems Division, Beijing Computational Science
Research Center, Beijing 100193, China  Department of Physics, Beijing Normal
University, Beijing 100875, China

###### Abstract

The entrapment of bacteria near boundary surfaces is of biological and
practical importance, yet the underlying physics is still not well understood.
We demonstrate that it is crucial to include a commonly neglected entropic
effect arising from the spatial variation of hydrodynamic interactions,
through a model that provides analytic explanation of bacterial entrapment in
two dimensionless parameters: $\alpha_{1}$ the ratio of thermal energy to
self-propulsion, and $\alpha_{2}$ an intrinsic shape factor. For $\alpha_{1}$
and $\alpha_{2}$ that match an Escherichia coli at room temperature, our model
quantitatively reproduces existing experimental observations, including two
key features that have not been previously resolved: The bacterial “nose-down”
configuration, and the anticorrelation between the pitch angle and the
wobbling angle. Furthermore, our model analytically predicts the existence of
an entrapment zone in the parameter space defined by
$\\{\alpha_{1},\alpha_{2}\\}$.

bacterial motion, fluid dynamics, soft matter

Swimming microorganisms are constantly influenced by the presence of boundary
surfaces in their natural habitat, giving rise to rich swimming behaviors
Kantsler  _et al._ (2013); Contino  _et al._ (2015). One phenomenon commonly
known as surface entrapment attracts particular interest, where the swimmer
moves along near the surface for a prolonged time Di Leonardo  _et al._
(2011). Since its first discovery in bull spermatozoa Rothschild (1963), such
entrapment is widely observed for a variety of microorganisms in both domain
Bacteria (e.g., Escherichia coli Bianchi  _et al._ (2017)) and domain
Eukaryota (e.g., Tetrahymena pyriformis Ohmura  _et al._ (2018)) in different
types of fluids Cao  _et al._ (2022), and around surfaces with different
properties Poddar  _et al._ (2020). In addition to its importance to many
biological processes such as fertilization Tung and Suarez (2021); Raveshi
_et al._ (2021), and biofilm formation Nadell  _et al._ (2016), such surface-
swimmer interaction also provides insight for the design of microfluidic
structures Dehkharghani  _et al._ (2019) as well as artificial microswimmers
Simmchen  _et al._ (2016); Liu  _et al._ (2016); Ketzetzi  _et al._ (2020) for
desired transport properties. However, the underlying mechanism is still
poorly understood.

Using E. coli as an example where experimental data are abundant, we provide a
theoretic study explaining the dynamics of flagellar bacteria near surfaces.
In an early experiment, accumulation of bacteria near a solid plane was
observed, which was attributed to an effective cell-plane attraction Frymier
_et al._ (1995). Treating one bacterium as the sum of only the leading order
singularities (e.g., the force dipole, the source dipole, etc.), the
hydrodynamic interaction involved in the far-field limit can be analytically
obtained Berke  _et al._ (2008); Spagnolie and Lauga (2012), which predicts a
seeming entrapment when extrapolated for bacteria near the wall. However, as
entrapment events are typically observed at a surface distance smaller than
bacterial cell size, such an extrapolation is not appropriate because all
terms in the multipole expansion become important Kim and Karrila (2005). At
such small surface distance, the role of hydrodynamic interactions has been
challenged by a demonstration where experimental data can be equally well
interpreted by stochastic models with or without hydrodynamic interactions Li
and Tang (2009).

Replacing the planar boundary with convex surfaces, an experiment studied the
effect of surface radius, and suggested that the near field hydrodynamic
interaction plays the major role Sipos  _et al._ (2015). In the meantime,
while stochastic effects such as rotational noise Schaar  _et al._ (2015) or
bacterial tumbling Junot  _et al._ (2022) are important for the eventual
escape of bacteria at large time scales, during the entrapment they can be
significantly suppressed Drescher  _et al._ (2011); Molaei  _et al._ (2014).
The leading role of near field hydrodynamics in surface entrapment was further
confirmed by an experiment using high resolution holographic microscopy
Bianchi  _et al._ (2017), with observation of two key features for the trapped
bacteria, i.e., the “nose-down” configuration and the anticorrelation between
the pitch angle and the wobbling angle. However, numerical simulations with
full hydrodynamic interactions are unable to reproduce these key features: for
no-slip plane the simulation results show a “nose-up” configuration Giacche
_et al._ (2010); Pimponi  _et al._ (2016), and suggest a positive correlation
between the wobbling angle and the pitch angle Eisenstecken  _et al._ (2016).
In this work we show that previous theoretic/numerical studies have neglected
a key entropic effect inherited from the drastic spatial variation of near
field hydrodynamic interactions at room temperature. When this crucial effect
is incorporated, even a simplified model of the hydrodynamic interactions is
capable in providing a quantitative explanation for the observed surface
entrapment behaviors.

Problem formulation for non-wobbling bacteria — For simplicity, we first study
a non-wobbling bacterial model which allows analytic solution. Swimming in a
fluid of viscosity $\mu$ above an infinitely large plane with no-slip boundary
at $x=0$, an E. coli bacterium is simplified as two spherical beads, a body-
bead with radius $R_{\mathrm{b}}$ and a tail-bead with radius
$R_{\mathrm{t}}$, connected by a rigid rod that separates the two centers by
$l$ (inset in Fig. 1a) Zhang  _et al._ (2021). The tail-bead is propelled by a
phantom force $\bm{F}_{\mathrm{act}}$ provided by the spinning of flagella
that is not treated explicitly. Each configuration is fully determined by the
surface distance $d$ between bacterial body and the plane, and pitch angle
$\theta$ which is positive for a “nose-down” configuration.

Since the characteristic size and speed are about 1 $\mathrm{\mu m}$ and 10
$\mathrm{\mu m/sec}$, respectively, in water the corresponding Reynolds number
is low ($10^{-5}$) so that bacterial flows are typically studied by the linear
Stokes equation. At a time resolution $\Delta t\approx 10^{-2}\mathrm{sec}$
the system is in the over-damped limit described by:

$\bm{\mathrm{\xi}}\cdot\bm{\mathrm{U}}=\bm{\mathrm{F}}^{\mathrm{P}}+\bm{\mathrm{F}}^{\mathrm{B}},$
(1)

where the resistance tensor $\bm{\mathrm{\xi}}$ for any configuration is fully
determined by hydrodynamics,
$\bm{\mathrm{U}}\equiv\left(\bm{u}_{\mathrm{b}},\bm{\omega}_{\mathrm{b}},\bm{u}_{\mathrm{t}},\bm{\omega}_{\mathrm{t}}\right)^{T}$
is the translational/rotational velocity vector with indices “$\mathrm{b}$”
and “$\mathrm{t}$” standing for body-bead and tail-bead, respectively,
$\bm{\mathrm{F}}^{\mathrm{P}}\equiv\left(\bm{F}_{\mathrm{b}},\bm{L}_{\mathrm{b}},\bm{F}_{\mathrm{t}},\bm{L}_{\mathrm{t}}\right)^{T}$
represents the nonhydrodynamic forces and $\bm{\mathrm{F}}^{\mathrm{B}}$
represents stochastic forces.

At absolute-zero temperature, $\bm{\mathrm{F}}^{\mathrm{B}}=0$. Eq. 1 then
reduces to
$\bm{\mathrm{\xi}}\cdot\bm{\mathrm{U}}=\bm{\mathrm{F}}^{\mathrm{P}}$, which
is:

$\left(\begin{array}[]{c}\bm{F}_{\mathrm{b}}\\\ \bm{L}_{\mathrm{b}}\\\
\bm{F}_{\mathrm{t}}\\\ \bm{L}_{\mathrm{t}}\\\
\end{array}\right)=\left(\begin{array}[]{cccc}\bm{\xi}_{\mathrm{bb}}^{\mathrm{FU}}&\bm{\xi}_{\mathrm{bb}}^{\mathrm{F\omega}}&\bm{\xi}_{\mathrm{bt}}^{\mathrm{FU}}&\bm{\xi}_{\mathrm{bt}}^{\mathrm{F\omega}}\\\
\bm{\xi}_{\mathrm{bb}}^{\mathrm{LU}}&\bm{\xi}_{\mathrm{bb}}^{\mathrm{L\omega}}&\bm{\xi}_{\mathrm{bt}}^{\mathrm{LU}}&\bm{\xi}_{\mathrm{bt}}^{\mathrm{L\omega}}\\\
\bm{\xi}_{\mathrm{tb}}^{\mathrm{FU}}&\bm{\xi}_{\mathrm{tb}}^{\mathrm{F\omega}}&\bm{\xi}_{\mathrm{tt}}^{\mathrm{FU}}&\bm{\xi}_{\mathrm{tt}}^{\mathrm{F\omega}}\\\
\bm{\xi}_{\mathrm{tb}}^{\mathrm{LU}}&\bm{\xi}_{\mathrm{tb}}^{\mathrm{L\omega}}&\bm{\xi}_{\mathrm{tt}}^{\mathrm{LU}}&\bm{\xi}_{\mathrm{tt}}^{\mathrm{L\omega}}\\\
\end{array}\right)\cdot\left(\begin{array}[]{c}\bm{u}_{\mathrm{b}}\\\
\bm{\omega}_{\mathrm{b}}\\\ \bm{u}_{\mathrm{t}}-\bm{u}_{0}\\\
\bm{\omega}_{\mathrm{t}}\\\ \end{array}\right)$ (2)

where
$\bm{u}_{0}\equiv(\bm{\xi}_{\mathrm{tt}}^{\mathrm{FU}})^{-1}\cdot\bm{F}_{\mathrm{act}}$.
The system is fully determined with two widely used conditions: (i) the free-
swimming condition, i.e.,
$\bm{F}_{\mathrm{b}}=-\bm{F}_{\mathrm{t}}\equiv\bm{F}_{\mathrm{eff}}$ and
$\bm{L}_{\mathrm{b}}=-\bm{L}_{\mathrm{t}}-(\bm{r}_{\mathrm{t}}-\bm{r}_{\mathrm{b}})\times\bm{F}_{\mathrm{t}}\equiv-\bm{L}_{\mathrm{eff}}+\bm{F}_{\mathrm{eff}}\times(\bm{r}_{\mathrm{b}}-\bm{r}_{\mathrm{t}})$;
(ii) the rigid body condition, i.e.,
$\bm{\omega}_{\mathrm{b}}=\bm{\omega}_{\mathrm{t}}\equiv\bm{\omega}_{0}$ and
$\bm{u}_{\mathrm{t}}=\bm{u}_{\mathrm{b}}+\bm{\omega}_{0}\times(\bm{r}_{\mathrm{t}}-\bm{r}_{\mathrm{b}})$.
Eq. 2 is then solved for $\bm{F}_{\mathrm{eff}}$, $\bm{L}_{\mathrm{eff}}$,
$\bm{\omega}_{0}$, and $\bm{u}_{\mathrm{b}}$, leading to system evolution
$\Delta\bm{\mathrm{R}}=\bm{\mathrm{U}}\Delta t$.

At finite temperature, $\bm{\mathrm{F}}^{\mathrm{B}}\neq 0$. By integrating
Eq. 1 over $\Delta t\gg\tau_{\mathrm{B}}\sim 10^{-6}\mathrm{sec}$ the Brownian
time scale but still small so that changes in configuration are not
significant, we get two additional terms in system evolution Ermak and
McCammon (1978); Grassia  _et al._ (1995); Brady and Bossis (1988):

$\Delta\bm{\mathrm{R}}=\bm{\mathrm{U}}\Delta
t+k_{\mathrm{B}}T\mathbf{\nabla}\cdot\bm{\mathrm{\xi}}^{-1}\Delta
t+\bm{\mathrm{X}}(\Delta t),$ (3)

where $\bm{\mathrm{X}}(\Delta t)$ is a random displacement characterized by a
multi-variance Gaussian distribution with $\langle\bm{\mathrm{X}}(\Delta
t)\rangle=0$ and $\langle\bm{\mathrm{X}}(\Delta t)\bm{\mathrm{X}}(\Delta
t)\rangle=2k_{\mathrm{B}}T\bm{\mathrm{\xi}}^{-1}\Delta t$.

Figure 1: (a) For a bacterium modeled as two spherical beads connected by a
rod (inset), its trajectory in real space demonstrates surface entrapment. (b)
The temporal evolutions of $d$ (black line) and $\theta$ (red line) show a
three-stage dynamics visualized by blue, yellow, and green shaded areas,
respectively.

For an entrapped E. coli, we have $d/R_{\mathrm{b}}\sim 0.1$ so small that
terms in $\bm{\mathrm{\xi}}$ become large, in agreement with experimental
observation of suppressed $\bm{\mathrm{X}}(\Delta t)$ Drescher  _et al._
(2011) that we choose to treat as negligible during entrapment. On the other
hand, while commonly neglected in previous numerical studies Giacche  _et al._
(2010); Pimponi  _et al._ (2016); Eisenstecken  _et al._ (2016), the entropic
term $k_{\mathrm{B}}T\mathbf{\nabla}\cdot\bm{\mathrm{\xi}}^{-1}$ describes the
spatial variations in diffusivity, and in the $d\to 0$ limit remains a finite
constant independent of $d$. Therefore, the evolution equation becomes
$\Delta\bm{\mathrm{R}}=\bm{\mathrm{U^{\prime}}}\Delta t$ with
$\bm{\mathrm{U^{\prime}}}\equiv\left(\bm{u^{\prime}}_{\mathrm{b}},\bm{\omega^{\prime}}_{\mathrm{b}},\bm{u^{\prime}}_{\mathrm{t}},\bm{\omega^{\prime}}_{\mathrm{t}}\right)^{T}=\bm{\mathrm{U}}+k_{\mathrm{B}}T\mathbf{\nabla}\cdot\bm{\mathrm{\xi}}^{-1}$,
where $\bm{\mathrm{U}}$ satisfies Eq. 2 and the rigid body condition changes
to
$\bm{\omega^{\prime}}_{\mathrm{b}}=\bm{\omega^{\prime}}_{\mathrm{t}}\equiv\bm{\omega}_{0}$
and
$\bm{u^{\prime}}_{\mathrm{t}}=\bm{u^{\prime}}_{\mathrm{b}}+\bm{\omega}_{0}\times(\bm{r}_{\mathrm{t}}-\bm{r}_{\mathrm{b}})$.

Numerical simulation — For a typical E. coli in water at room temperature, we
estimate that $k_{\mathrm{B}}T\approx 4\times 10^{-21}\mathrm{N\cdot m}$,
$\mu\approx 10^{-3}\mathrm{N\cdot sec/m^{2}}$, $|\bm{F}_{\mathrm{act}}|\approx
2\times 10^{-13}\mathrm{N}$, $R_{\mathrm{b}}\approx 1\mathrm{\mu m}$,
$l\approx 5\mathrm{\mu m}$, and $R_{\mathrm{t}}\approx 0.4\mathrm{\mu m}$.
Using $1\mathrm{\mu m}$ and $1\mathrm{sec}$ as the unit of length and time
respectively and setting $\mu=1$ for the unit of force, in our study such a
bacterium can be characterized by
$\\{l=5,R_{\mathrm{b}}=1,R_{\mathrm{t}}=0.4,|\bm{F}_{\mathrm{act}}|=200,k_{\mathrm{B}}T=4\\}$.
In Fig. 1a we show its dynamics moving towards the plane with initial
configuration $\\{d=20,\theta=75^{\circ}\\}$, simulated assuming only
hydrodynamic interactions without steric interactions. The key in this
simulation is getting $\bm{\xi}$ for each configuration, which is constructed
following the Stokesian dynamics simulation Brady and Bossis (1988); Swan and
Brady (2007) in a two-step procedure. Specifically, we first model the grand
mobility tensor in the far-field limit through
$\bm{M}_{\infty}=\bm{M}_{0}+\bm{\hat{M}}$. Here $\bm{M}_{0}$ is the analytic
far-field hydrodynamic interaction without plane Jeffrey and Onishi (1984),
and $\bm{\hat{M}}$ corresponds to the plane contribution which is analytically
available through the method of images Swan and Brady (2007). In the second
step, $\bm{\xi}$ is obtained through
$\bm{\xi}=\bm{M}_{\infty}^{-1}+\bm{\xi}_{\mathrm{b}}$, where
$\bm{\xi}_{\mathrm{b}}$ characterizes the lubrication between body-bead and
the plane. Once $\bm{\xi}$ is obtained, we can solve Eq. 2 for
$\bm{F}_{\mathrm{eff}}$, $\bm{L}_{\mathrm{eff}}$, $\bm{\omega}_{0}$, and
$\bm{u^{\prime}}_{\mathrm{b}}$.

As illustrated by the temporal evolution of $d$ and $\theta$ in Fig. 1b, Our
numerical results reproduce the experimentally observed three-stage dynamics
Bianchi  _et al._ (2017): the initial approach where $d$ drops very quickly
with an almost constant $\theta$; the reorientation stage where $\theta$
decreases very quickly right after $d$ falls below the size of body-bead; and
the steady swimming stage where both $\dot{d}$ and $\dot{\theta}$ gradually
decay to zero characterizing a stable entrapment. Such dynamics can also be
illustrated in the phase diagram defined by $d$ and $\theta$ (Fig. 2a), where
the steady swimming corresponds to a stable fixed point with $\theta>0$ (a
“nose-down” configuration), in agreement with previous experimental
observations. A quantitative study of the reorientation stage shows that
$\tan{\theta}$ decays exponentially for $t>t_{0}$ with $t_{0}$ the time when
$d$ falls below the size of body-bead (Fig. S1 in SI), in good agreement with
experimental observations (Fig. 3d in Bianchi  _et al._ (2017)). While this
exponential decay was attributed to steric interactions in Bianchi  _et al._
(2017), in our simulation it is obtained assuming only hydrodynamic
interactions.

Analytic solution — To understand these results, we simplify the problem with
two ideal approximations for analytic solutions. Since $l\gg R_{\mathrm{t}}$,
our first approximation assumes that the tails are not hydrodynamically
coupled with the body or the plane. Specifically, we have body-tail coupling
$\bm{\xi}_{\mathrm{bt}}=\bm{\xi}_{\mathrm{tb}}=0$, and tail self term
$\bm{\xi}_{\mathrm{tt}}$ a constant. Therefore, the only configurational
dependent term in $\bm{\mathrm{\xi}}$ is $\bm{\xi}_{\mathrm{bb}}$, which is an
analytical function of single parameter $d\textfractionsolidus R_{\mathrm{b}}$
in the $d\to 0$ limit Jeffrey and Onishi (1984). Since $d/R_{\mathrm{b}}$ is
small during entrapment, our second approximation uses this analytical
$\bm{\xi}_{\mathrm{bb}}$ function for all $d$ of interest. These two ideal
approximations lead to a simplified
$k_{\mathrm{B}}T\mathbf{\nabla}\cdot\bm{\mathrm{\xi}}^{-1}$ with only one non-
zero component: A constant translational velocity
$\bm{v}=\frac{k_{\mathrm{B}}T}{6\pi\mu R_{\mathrm{b}}^{2}}\bm{\hat{x}}$ for
bacterial body moving away from the plane (SI, Sec. A).

To obtain fixed points in the phase diagram defined by $d$ and $\theta$, we
insert $\dot{d}=\dot{\theta}=0$ to Eq. 2, which gives:

$\displaystyle(\bm{\xi}_{\mathrm{bb}}^{\mathrm{LU}})_{zy}*(\bm{u^{\prime}}_{\mathrm{b}}\cdot\bm{\hat{y}})$
$\displaystyle=$
$\displaystyle(\bm{\xi}_{\mathrm{tt}}^{\mathrm{FU}})_{yy}*(\bm{u^{\prime}}_{\mathrm{b}}\cdot\bm{\hat{y}})*l*\sin{\theta}$
(4) $\displaystyle|\bm{F}_{\mathrm{act}}|*\sin{\theta}$ $\displaystyle=$
$\displaystyle(\bm{\xi}_{\mathrm{bb}}^{\mathrm{FU}})_{xx}*(\bm{v}\cdot\bm{\hat{x}})$
(5)

Here Eq. 4 is equivalent to the fifth equation in Sipos  _et al._ (2015),
which characterizes the torque balance on the body-bead in $\bm{\hat{z}}$,
between the boundary-induced torque due to bacterial body translation
$\bm{u^{\prime}}_{\mathrm{b}}\cdot\bm{\hat{y}}$ along the plane (LHS) and the
torque arises from the friction against bacterial tail translation (RHS). Eq.
5 characterizes the force balance on the body-bead in $\bm{\hat{x}}$, between
self-propulsion (LHS) and the entropic effect we introduced (RHS). The
coefficients in Eq. 4 and Eq. 5 are available from lubrication theory as
$(\bm{\xi}_{\mathrm{bb}}^{\mathrm{LU}})_{zy}=-6\pi\mu
R_{\mathrm{b}}^{2}*\frac{2}{15}\ln{\frac{d}{R_{\mathrm{b}}}}$,
$(\bm{\xi}_{\mathrm{tt}}^{\mathrm{FU}})_{yy}=6\pi\mu R_{\mathrm{t}}$,
$(\bm{\xi}_{\mathrm{bb}}^{\mathrm{FU}})_{xx}=\frac{6\pi\mu
R_{\mathrm{b}}^{2}}{d}$.

Figure 2: (a) In the phase diagram defined by $d$ and $\theta$, bacterial
trajectory (green line) approaches to a stable fixed point (green circle). (b)
The parameter space defined by $\alpha_{1}$ and $\alpha_{2}$ can be divided
into three regions: region I (red) above the red line
$\alpha_{2}=-\frac{2}{15}\ln{\alpha_{1}}$, region II (green) between the red
line and blue line $\frac{15}{2}\alpha_{1}\alpha_{2}=e^{-1}$ with
$\alpha_{1}\leq e^{-1}$, and the region outside the above two regions. A
typical E. coli at room temperature is characterized by the yellow star in
region II. The inset in (b) shows $x\ln{x}$ as a function of $x$.

When $\bm{u^{\prime}}_{\mathrm{b}}\cdot\bm{\hat{y}}=0$, a trivial solution
$\\{\theta_{0}=\frac{\pi}{2},d_{0}=\frac{k_{\mathrm{B}}T}{|\bm{F}_{\mathrm{act}}|}\\}$
exists, where the bacterium points straightly towards the plane. For
$\bm{u^{\prime}}_{\mathrm{b}}\cdot\bm{\hat{y}}\neq 0$, Eq. 4 and Eq. 5 can be
further reduced in terms of two dimensionless parameters
$\alpha_{1}\equiv\frac{k_{\mathrm{B}}T}{|\bm{F}_{\mathrm{act}}|R_{\mathrm{b}}}$
and $\alpha_{2}\equiv\frac{R_{\mathrm{t}}l}{R_{\mathrm{b}}^{2}}$:

$\displaystyle\frac{d}{R_{\mathrm{b}}}\ln{\frac{d}{R_{\mathrm{b}}}}$
$\displaystyle=$ $\displaystyle-\frac{15\alpha_{1}\alpha_{2}}{2}$ (6)
$\displaystyle\ln{\frac{d}{R_{\mathrm{b}}}}$ $\displaystyle=$
$\displaystyle-\frac{15\alpha_{2}\sin{\theta}}{2}$ (7)

where the prefactor -15/2 arises from translation-rotation coupling
$(\bm{\xi}_{\mathrm{bb}}^{\mathrm{LU}})_{zy}$.

Dictated by Eq. 6 and Eq. 7, two curves,
$\frac{15}{2}\alpha_{1}\alpha_{2}=e^{-1}$ and
$\alpha_{2}=-\frac{2}{15}\ln{\alpha_{1}}$, become important in the parameter
space defined by $\alpha_{1}$ and $\alpha_{2}$ (the blue line and the red
line, respectively, in Fig. 2b). The first curve,
$\frac{15}{2}\alpha_{1}\alpha_{2}=e^{-1}$, arises from the fact that
$\frac{d}{R_{\mathrm{b}}}\ln{\frac{d}{R_{\mathrm{b}}}}\geq-e^{-1}$ for $d>0$
where the equality happens only at $\frac{d}{R_{\mathrm{b}}}=e^{-1}$ (Fig. 2b
inset). Thus, nontrivial fixed points ($\left\\{d_{1},\theta_{1}\right\\}$ and
$\left\\{d_{2},\theta_{2}\right\\}$) only exist when
$0\geq-\frac{15}{2}\alpha_{1}\alpha_{2}\geq-e^{-1}$, with
$0\leq\frac{d_{1}}{R_{\mathrm{b}}}\leq
e^{-1}\leq\frac{d_{2}}{R_{\mathrm{b}}}\leq 1$. The second curve,
$\alpha_{2}=-\frac{2}{15}\ln{\alpha_{1}}$, is obtained by assuming
$\sin{\theta}=1$, the largest possible value for $\sin{\theta}$. Our analysis
shows that (Sec. C in SI), for parameter choice $\\{\alpha_{1},\alpha_{2}\\}$
above the red line in Fig. 2b (region I), only the nontrivial fixed point
$\left\\{d_{2},\theta_{2}\right\\}$ with $\frac{d_{2}}{R_{\mathrm{b}}}>e^{-1}$
exists. For parameter choice $\\{\alpha_{1},\alpha_{2}\\}$ between the red
line and blue line with $\alpha_{1}\leq e^{-1}$ in Fig. 2b (region II), both
nontrivial fixed points exist. No nontrivial fixed point exists for parameter
choice $\\{\alpha_{1},\alpha_{2}\\}$ outside region I and II.

Furthermore, for various choices of $\\{\alpha_{1},\alpha_{2}\\}$ in region I
and II, we numerically studied the neighborhood of each fixed point in the
phase diagram defined by $d$ and $\theta$. Our results for all cases (SI, Sec.
C) show that $\\{d_{1},\theta_{1}\\}$ with
$\frac{d_{1}}{R_{\mathrm{b}}}<e^{-1}$ is always stable, while
$\\{d_{2},\theta_{2}\\}$ with $\frac{d_{2}}{R_{\mathrm{b}}}>e^{-1}$ is always
a saddle point. Since physically observed entrapment is associated only with a
stable fixed point which only exists for $\\{\alpha_{1},\alpha_{2}\\}$ in
region II (Fig. 2b), this region defines the entrapment zone: No physical
entrapment can be observed for $\\{\alpha_{1},\alpha_{2}\\}$ outside the zone.

For a typical E. coli at room temperature, we have $\alpha_{1}=0.02$ and
$\alpha_{2}=2$, which falls in the entrapment zone (Fig. 2b). For this set of
$\\{\alpha_{1},\alpha_{2}\\}$, we predict the stable fixed point at
$d^{*}=0.17\mathrm{\mu m}$ from Eq. 6 and then $\theta^{*}=7^{\circ}$ from Eq.
7. Using our simulation that includes hydrodynamic coupling of the tails, we
get $d^{*}=0.13\mathrm{\mu m}$ and $\theta^{*}=8^{\circ}$. Both our analytic
and simulation results are in good agreement with experimental observations
Bianchi  _et al._ (2017); Li  _et al._ (2008).

Figure 3: (a) Our model of a wobbling bacterium explicitly considers
$\bm{\omega}_{\mathrm{b}}^{0}$ and $\bm{\omega}_{\mathrm{t}}^{0}$. (b)
Trajectory of a wobbling bacterium near a plane. (c) The temporal evolutions
of $d$ (solid black), $\theta_{\mathrm{b}}$ (solid red), and
$\theta_{\mathrm{t}}$ (dash red). Blue (green) bar indicates time $t_{1}$
($t_{2}$) when $\theta_{\mathrm{b}}$ is smallest (largest) during entrapment,
with the corresponding configurations shown in the inset using actual
$\theta_{\mathrm{b}}$ and $\theta_{\mathrm{t}}$, and exaggerated $d$.

Bacterial wobbling — In previous sections, we obtained analytic predictions
for non-wobbling bacteria, where the body-tail connection is treated as rigid
without considering the self-spinning of either the bacterial body or the
flagellar bundle, i.e.,
$\bm{\omega}_{\mathrm{b}}=\bm{\omega}_{\mathrm{t}}=\bm{\omega}_{0}$. In real
world, the self-spinning of the two parts, $\bm{\omega}_{\mathrm{b}}^{0}$ and
$\bm{\omega}_{\mathrm{t}}^{0}$, respectively, are generally not collinear
(Fig. 3a). For free-swimming bacteria the nonzero angle $\gamma$ formed by the
two vectors leads to a center of mass rotation $\bm{\omega}_{0}$ generally not
aligned with its translation and therefore, bacterial wobbling. To account for
the wobbling, we generalize the non-wobbling bacterial model by explicit
consideration of $\bm{\omega}_{\mathrm{b}}^{0}$ and
$\bm{\omega}_{\mathrm{t}}^{0}$, and replace the rigid connection by a
universal joint through relation
$R_{\mathrm{b}}^{3}|\bm{\omega}_{\mathrm{b}}^{0}|\cos{\gamma}=R_{\mathrm{t}}^{3}|\bm{\omega}_{\mathrm{t}}^{0}|$
Kamdar  _et al._ (2022); Shimogonya  _et al._ (2016) where
$|\bm{\omega}_{\mathrm{t}}^{0}|=\mathrm{100Hz}$ fixed to match the experiments
Darnton  _et al._ (2007) (SI, Sec. D). The overall rotations for bacterial
body and flagellar bundle are
$\bm{\omega}_{\mathrm{b}}=\bm{\omega}_{\mathrm{b}}^{0}+\bm{\omega}_{0}$ and
$\bm{\omega}_{\mathrm{t}}=\bm{\omega}_{\mathrm{t}}^{0}+\bm{\omega}_{0}$,
respectively.

With this generalisation, we numerically study the dynamics of wobbling
bacteria near a plane, where each configuration is now determined by $d$ and
two distinct pitch angles, $\theta_{\mathrm{b}}$ and $\theta_{\mathrm{t}}$
(Fig. 3a). For illustration purposes, in Fig. 3 we highlight the self-spinning
by drawing an ellipsoid and a helix in place of the body-sphere and tail-
sphere used in our actual simulations. Our results show that bacteria can be
trapped in a clockwise circular trajectory when viewed from above (Fig. 3b).
The radii of the circles are $10\sim 200\mathrm{\mu m}$, which increase with
$d$ and $\alpha_{1}$ (Fig. S4 in SI), all of which are in agreement with
earlier experiments Lauga  _et al._ (2006); Cao  _et al._ (2022); Li  _et al._
(2008); Maeda  _et al._ (1976). While for technical reasons most experiments
focus on $\theta_{\mathrm{b}}$ with scarce data available for
$\theta_{\mathrm{t}}$, our simulation has the advantage to capture both. For a
typical bacterium ($l=5$, $R_{\mathrm{b}}=1$, $R_{\mathrm{t}}=0.4$) with
$\gamma=30^{\circ}$ at room temperature, in Fig. 3c we show our numerical
results for the temporal evolution of $d$, $\theta_{\mathrm{b}}$, and
$\theta_{\mathrm{t}}$. During the entrapment stage, all these three variables
are periodically oscillating with the same frequency determined by
$\bm{\omega}_{0}$, where $d$ and $\theta_{\mathrm{b}}$ are almost in phase
while $\theta_{\mathrm{t}}$ has nearly an opposite phase (Fig. 3c). In the
inset, we show the bacterial configurations at $t_{1}$ ($t_{2}$) with smallest
(largest) $\theta_{\mathrm{b}}$ denoted as $\theta_{\mathrm{min}}$
($\theta_{\mathrm{max}}$). The experimentally recorded average pitch angle
$\bar{\theta}$ and wobbling angle $\theta_{w}$ Bianchi  _et al._ (2017) can be
obtained through
$\bar{\theta}\equiv(\theta_{\mathrm{max}}+\theta_{\mathrm{min}})/2$ and
$\theta_{w}\equiv(\theta_{\mathrm{max}}-\theta_{\mathrm{min}})/2$. In a
similar fashion, we define
$\bar{d}\equiv(d_{\mathrm{max}}+d_{\mathrm{min}})/2$.

A variety of bacteria characterized by
$\\{l,R_{\mathrm{b}},R_{\mathrm{t}},\gamma\\}$ are then simulated at room
temperature, where for $\gamma$ we sampled $20^{\circ}$, $30^{\circ}$, and
$40^{\circ}$, and for each of the other three parameters we limit the
variations to at most $20\%$ from the corresponding value of a typical E.
coli. The scatter plot of all $\\{\bar{\theta},\theta_{w}\\}$ obtained from
the entrapment stage show an anticorrelation spreading broadly along the
$\bar{\theta}=\theta_{w}$ direction (Fig. 4a), in quantitative agreement with
previous experiment (Fig. 4 in Bianchi  _et al._ (2017)).

Figure 4: Simulation data with $\gamma=20^{\circ}$ (triangles),
$\gamma=30^{\circ}$ (crosses), and $\gamma=40^{\circ}$ (circles) yield the
scatter plot $\\{\bar{\theta},\theta_{w}\\}$ (green) in (a), normalized
$\bar{\theta}$ (red) and $\theta_{w}$ (blue) as functions of $\alpha_{2}$ in
(b). Results from Bianchi  _et al._ (2017) are shown as grey dots in (a).

Interesting results emerge when sorting out our data by $\gamma$. For each
specific $\gamma$ we observe an anticorrelation with a much narrower spread,
which is regulated by $\alpha_{2}$ in a quantitatively similar fashion.
Specifically, for each $\gamma$ we plot $\bar{\theta}$ ($\theta_{w}$) as a
function of $\alpha_{2}$, normalized by $\bar{\theta}^{\mathrm{WT}}$
($\theta_{w}^{\mathrm{WT}}$) obtained from a typical bacterium with that
particular $\gamma$. Our data from three distinct $\gamma$ collapse,
indicating existence of two master curves for
$\bar{\theta}/\bar{\theta}^{\mathrm{WT}}$ and
$\theta_{w}/\theta_{w}^{\mathrm{WT}}$ respectively (Fig. 4b).

Our observation that the anticorrelation is dictated mostly by variation in
$\alpha_{2}$ can be explained by Eq. 6 and Eq. 7, which are still valid for
wobbling bacteria if we replace $\\{d,\theta\\}$ in the equations by
$\\{\bar{d},\bar{\theta}\\}$ observed in data (Fig. S5 in SI). Since we are in
the parameter range that changes in $\ln{(\bar{d}/R_{\mathrm{b}})}$ are much
less significant than changes in $\bar{d}/R_{\mathrm{b}}$, for an estimate we
can neglect changes in $\ln{(\bar{d}/R_{\mathrm{b}})}$, which leads to
$\bar{d}/R_{\mathrm{b}}\sim\alpha_{1}\alpha_{2}$ according to Eq. 6 and
$\sin{\bar{\theta}}\approx\bar{\theta}\sim\alpha_{2}^{-1}$ independent of
$\alpha_{1}$ according to Eq. 7. This estimate correctly captures the positive
(negative) correlation between $\bar{d}/R_{\mathrm{b}}$ ($\bar{\theta}$) and
$\alpha_{2}$, in qualitative agreement with numerical fit of our data, which
gives $\bar{d}/R_{\mathrm{b}}\sim\alpha_{2}^{1.5}$ (Fig. S6a),
$\ln{(\bar{d}/R_{\mathrm{b}})}\sim\alpha_{2}^{-0.8}$ (Fig. S6b), and
$\bar{\theta}\sim\alpha_{2}^{-1.7}$ (Fig. 4b). As it is experimentally
established that wobbling can be significantly suppressed by nearby boundary
surfaces Kamdar  _et al._ (2022); Vizsnyiczai  _et al._ (2020), the positive
correlation between $\bar{d}/R_{\mathrm{b}}$ and $\alpha_{2}$ leads to
positive correlation between $\theta_{w}$ and $\alpha_{2}$ (more suppressed
wobbling at smaller $\alpha_{2}$), and thus the anticorrelation between
$\bar{\theta}$ and $\theta_{w}$.

Discussion — In this study we ignore stochastic effects
$\bm{\mathrm{X}}(\Delta t)$ and bacterial tumbling. At a cost of making
trapped bacteria incapable to escape, this simplification helps us focus on
the entrapment stage and highlight the importance of the entropic effect
$k_{\mathrm{B}}T\mathbf{\nabla}\cdot\bm{\mathrm{\xi}}^{-1}$. While surface
escape at large time scales is out of scope for current work, it is
interesting for a future study to investigate how variations in $\alpha_{1}$
and $\alpha_{2}$ influence stability of fixed points against stochastic
fluctuations, leading to parameter dependence of surface residence time.

For the entrapment stage, we show that the self propulsion for the “nose-down”
configuration needs to be balanced by a cell-plane repulsion, in sharp
contrast to the earlier notion that surface entrapment arises from an
effective attraction Frymier  _et al._ (1995). At room temperature and
$d\approx 0.1\mathrm{\mu m}$, an entropic repulsion arises naturally from the
drastic spatial variation of near field hydrodynamic interactions. By
including this entropic term, we demonstrate that hydrodynamic interactions
alone explain existing observations, with even a simplified model that
considers bacterial body (flagellar bundle) as a sphere with the size equal to
its hydrodynamic radius. More importantly, for future experiments to verify if
other mechanisms such as steric interactions or stochastic effect
$\bm{\mathrm{X}}(\Delta t)$ also contribute, our model has provided the
following predictions: (i) The entrapped configuration $\\{d,\theta\\}$ is
dictated by two dimensionless parameters
$\alpha_{1}\equiv\frac{k_{\mathrm{B}}T}{|\bm{F}_{\mathrm{act}}|R_{\mathrm{b}}}$
and $\alpha_{2}\equiv\frac{R_{\mathrm{t}}l}{R_{\mathrm{b}}^{2}}$, through two
analytic relations, i.e., Eq. 6 and Eq. 7. Extra care is needed for studies at
different temperatures, as self propulsion can triple with a slight increase
of $k_{\mathrm{B}}T$ from $20^{\circ}\mathrm{C}$ to $40^{\circ}\mathrm{C}$
Maeda  _et al._ (1976). (ii) About the flagellar bundle, the orientation
$\theta_{\mathrm{t}}$ has an opposite phase with respect to $d$ and
$\theta_{\mathrm{b}}$; and the variation in angle $\gamma$ is the main factor
underlying the broad spread of observed $\\{\bar{\theta},\theta_{w}\\}$ data
along the $\bar{\theta}=\theta_{w}$ direction. (iii) There exists an
entrapment zone within the range of $0<\alpha_{1}<e^{-1}$ and
$\alpha_{2}>2/15$.

Three implications follow from (iii). First, the existence of an upper limit
in $\alpha_{1}$ demonstrates that entrapment is a special feature for active
systems, and no entrapment is allowed for passive systems (zero activity and
thus $\alpha_{1}=\infty$) at any temperature. Second, entrapment is not
available for any active swimmers with intrinsic shape factor
$\alpha_{2}\equiv\frac{R_{\mathrm{t}}l}{R_{\mathrm{b}}^{2}}<2/15$, which
provides a guideline for controlling biological and engineering active
swimmers near surfaces. Third, while entrapment only exists at low
temperatures that satisfy
$\alpha_{1}\equiv\frac{k_{\mathrm{B}}T}{|\bm{F}_{\mathrm{act}}|R_{\mathrm{b}}}<e^{-1}$,
at the lowest temperature possible, i.e., the absolute zero temperature, our
solution becomes singular with surface distance
$\lim\limits_{k_{\mathrm{B}}T\to 0}d_{1}\to 0$, where all theoretical studies
treating the solvent as a continuum through hydrodynamics break down. This
explains why negligence of
$k_{\mathrm{B}}T\mathbf{\nabla}\cdot\bm{\mathrm{\xi}}^{-1}$ in previous
studies of hydrodynamic interactions, which is equivalent to setting the
temperature to zero, cannot reproduce the entrapment correctly. Instead, a
finite temperature is essential in achieving the physical entrapment, or in
other words, sticky bacteria are hot.

We thank X. Cheng, L. S. Luo, M. Stynes, Y. L. Wu, J. H. Yuan, and H. P. Zhang
for helpful discussions. This work is supported by NSFC No. 11974038 and No.
U2230402. We also acknowledge the computational support from the Beijing
Computational Science Research Center.

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