# Magnetic critical behavior and anomalous Hall effect in 2H-Co0.22TaS2 single
crystals

Yu Liu,1,∗ Zhixiang Hu,1,2 Eli Stavitski,3, Klaus Attenkofer,3,† and C.
Petrovic1,2 1Condensed Matter Physics and Materials Science Department,
Brookhaven National Laboratory, Upton, New York 11973, USA
2Materials Science and Chemical Engineering Department, Stony Brook
University, Stony Brook, New York 11790, USA
3National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA

###### Abstract

We report ferromagnetism in 2H-Co0.22TaS2 single crystals where Co atoms are
intercalated in the van der Waals gap, and a systematic study of its magnetic
critical behavior in the vicinity of $T_{c}\sim 28$ K. The obtained critical
exponents $\beta$ = 0.43(2), $\gamma$ = 1.15(1), and $\delta=3.54(1)$ fulfill
the Widom scaling relation $\delta=1+\gamma/\beta$ and follow the scaling
equation. This indicates that the spin coupling in 2H-Co0.22TaS2 is of three-
dimensional Hersenberg type coupled with long-range magnetic interaction, and
that the exchange interaction decays with distance as $J(r)\approx r^{-4.69}$.
2H-Co0.22TaS2 exhibits a weak temperature-dependent metallic behavior in
resistivity and negative values of thermopower with dominant electron-type
carriers, in which obvious anomalies were observed below $T_{c}$ as well as
the anomalous Hall effect (AHE). The linear scaling behavior between the
modified anomalous Hall resistivity $\rho_{xy}/\mu_{0}H$ and longitudinal
resistivity $\rho_{xx}^{2}M/\mu_{0}H$ implies that the origin of AHE in
2H-Co0.22TaS2 should be dominated by the extrinsic side-jump mechanism.

##  I INTRODUCTION

Recent discovery of intrinsic long-range magnetic order in a mono- or few-
layer in two-dimensional (2D) van der Waals (vdW) magnets Geim ; Novoselov ;
Lee ; Huang ; Gong ; Deng ; Bonilla ; Hara , such as FePS3, Cr2Ge2Te6, CrI3,
Fe3GeTe2, VSe2 and MnSe2, has provided new platforms for studying fundamental
2D magnetism and designing novel spin-related devices. Interestingly, the spin
interactions in 2D vdW magnets exhibit different types of critical behavior,
such as 2D Heisenberg type in MnPS3, 2D XY type in NiPS3, and 2D Ising type in
FePS3 Joy . Three-dimensional (3D) Heisenberg behavior is observed in bulk
CrI3 and Fe3GeTe2 YuLiu1 ; BJLiu1 ; YuLiu2 ; GTLin1 , however, a crossover
from 3D to 2D Ising ferromagnetism is revealed in thickness-reduced crystals
YuLiu4 ; Fei . 2D Ising behavior is confirmed in Cr2(Si,Ge)2Te6 BJLiu0 ;
YuLiu0 ; WLiu0 ; GTLin0 , whereas mean field type interaction is found in its
3D analog of Mn3Si2Te6 since an additional 1/3 Mn residing in vdW gap induced
stronger interlayer coupling YuLiu3 .

Intercalated transition metal dichalcogenides commonly feature 3$d$ atoms in
vdW gap and exhibit diverse magnetic orders Parkin ; Friend0 . For instance,
2H-M0.33TaS2 with M = V, Cr, Mn is ferromagnetic (FM), with M = Co, Ni is
antiferromagnetic (AFM), whereas with M = Fe exhibits both types of magnetic
behavior Parkin . 2H-MnxTaS2 is FM with easy $\mathbf{ab}$ plane Hinode ;
Zhang1 , 2H-FexTaS2 is FM with easy $\mathbf{c}$ axis for 0.2 $\leq$ $x$
$\leq$ 0.4 but is AFM for higher Fe content, then 2H-(Co,Ni)xTaS2 is AFM,
because the super-exchange interactions become larger as the intercalate ion
is varied from Mn through Fe and Co to Ni. Among these materials, the most
interesting member is 2H-Fe0.25TaS2, the only member that exhibits FM order
with strong uniaxial anisotropy, similar to recently investigated 2D vdW
magnets. It features large magnetocrystalline anisotropy, magnetoresistance,
and sharp switching in magnetization Morosan ; Hardy ; Chen . Anomalous Hall
effect (AHE) mechanism in 2H-Fe0.29TaS2 changes from extrinsic scattering to
intrinsic contribution with thickness reduction Cai , whereas the spin
coupling in 2H-Fe0.26TaS2 is of 3D Heisenberg type coupled with long-range
magnetic interactions Zhang .

In this work, we synthesized 2H-Co0.22TaS2 single crystal that shows FM order
with $T_{c}$ $\sim$ 28 K and a strong uniaxial anisotropy. Several methods
were employed to determine the critical exponents so as to shed light on the
mechanism of the paramagnetic (PM)-FM transition. Critical exponents
$\beta=0.43(2)$ with $T_{c}$ = 28.55(2) K and $\gamma=1.15(1)$ with $T_{c}$ =
28.62(2) K are obtained by the Kouvel-Fisher (KF) method, whereas
$\delta=3.54(1)$ is obtained by critical isotherm at $T_{c}$. The
renormalization group theory analysis suggests that the spin coupling in
2H-Co0.22TaS2 is of 3D Heisenberg type coupled with long-range magnetic
interaction; the exchange interaction decays with distance as $J(r)\approx
r^{-4.69}$. 2H-Co0.22TaS2 shows weak temperature-dependent metallic behavior
in resistivity $\rho(T)$ and negative values of thermopower $S(T)$ with
dominant electron-type carriers as well as the AHE below $T_{c}$. A linear
scaling behavior between the modified anomalous Hall resistivity
$\rho_{xy}/\mu_{0}H$ and longitudinal resistivity $\rho_{xx}^{2}M/\mu_{0}H$
indicates the extrinsic side-jump mechanism of AHE.

##  II EXPERIMENTAL DETAILS

Figure 1: (Color online) (a) Single crystal and (b) powder x-ray diffraction
(XRD) patterns of 2H-Co0.22TaS2. Inset in (a) shows the crystal structure of
2H-Co0.22TaS2. The tiny extra peaks [labeled as asterisk in (b)] at 32.2∘ and
38.9∘ are from pure iodine and sulfur, respectively. (c) Normalized Co
$K$-edge x-ray absorption near edge structure (XANES) spectra. Inset shows the
derivative $\mu(E)$ curve. (d) Fourier transform magnitudes of the extended
x-ray absorption fine structure (EXAFS) oscillations (symbols) for Co $K$-edge
with the phase shift correction. The model fits are shown as solid lines.
Inset shows the corresponding filtered EXAFS (symbols) with $k$-space model
fits (solid lines). Temperature-dependent magnetization $M(T)$ with both zero-
field cooling (ZFC) and field cooling (FC) modes for (e)
$\mathbf{H}\parallel\mathbf{ab}$-plane and (f)
$\mathbf{H}\parallel\mathbf{c}$-axis at various magnetic fields. Insets show
the corresponding inverse susceptibility with H = 1 kOe fitted by the Curie-
Weiss law (solid lines). Field-dependent magnetization for (g)
$\mathbf{H}\parallel\mathbf{ab}$-plane and (h)
$\mathbf{H}\parallel\mathbf{c}$-axis at various temperatures.

Single crystals of 2H-Co0.22TaS2 were grown by chemical vapor transport method
with iodine agent. The raw materials (Co, Ta, and S powders) were mixed with a
molar ratio of 0.25 $:$ 1 $:$ 2, sealed in an evacuated quartz tube, and then
heated for two weeks in a two-zone furnace with the source zone temperature of
1000 ∘C and the growth zone temperature of 900 ∘C, respectively. The obtained
single crystals are hexagonal shape with typical dimensions as $4\times
4\times 0.5$ mm3. The average stoichiometry was determined by examination of
multiple points on cleaved fresh surfaces and checked by multiple samples from
the same batch using energy-dispersive x-ray spectroscopy in a JEOL LSM-6500
scanning electron microscope. The actual stoichiometry is Co0.22(1)TaS1.88(2)
with Ta fixed to 1, which is referred to as 2H-Co0.22TaS2 throughout this
paper since $\sim 5\%$ sulfur deficiency is normally observed in 2H-TaS2.
X-ray diffraction (XRD) data were acquired on a Rigaku Miniflex powder
diffractometer with Cu $K_{\alpha}$ ($\lambda=0.15418$ nm) at room
temperature. X-ray absorption spectroscopy was measured at 8-ID beamline of
the National Synchrotron Light Source II (NSLS II) at Brookhaven National
Laboratory (BNL) in the fluorescence mode. The x-ray absorption near edge
structure (XANES) and extended X-ray absorption fine structure (EXAFS) spectra
were processed using the Athena software package. The EXAFS signal, $\chi(k)$,
was weighed by $k^{2}$ to emphasize the high-energy oscillation and then
Fourier-transformed in $k$ range from 2 to 10 Å-1 to analyze the data in $R$
space. The magnetization was measured in quantum design MPMS-XL5 system.
Isotherms were collected at an interval of 1 K around $T_{c}$. The electrical
and thermal transport were measured in quantum design PPMS-9 system. The
longitudinal and Hall resistivity were measured using standard four-probe
method. In order to effectively eliminate the longitudinal resistivity
contribution due to voltage probe misalignment, the Hall resistivity was
obtained by the difference of transverse resistance measured at positive and
negative fields, i.e.,
$\rho_{xy}(\mu_{0}H)=[\rho(+\mu_{0}H)-\rho(-\mu_{0}H)]/2$.

##  III RESULTS AND DISCUSSIONS

Figure 2: (Color online) (a) Typical initial isothermals measured with
$\mathbf{H}\parallel\mathbf{c}$-axis from 18 to 38 K with a temperature step
of 1 K. (b) The Arrott plots of $M^{2}$ vs $H/M$. (c) Temperature-dependent
spontaneous magnetization $M_{s}$ (left) and inverse initial susceptibility
$\chi_{0}^{-1}$ (right) with solid fitting curves for 2H-Co0.22TaS2. Inset
shows the M(H) curve in log10-log10 scale collected at 28 K with linear
fitting curve. (d) The Kouvel-Fisher plots of $M_{s}(dM_{s}/dT)^{-1}$ (left)
and $\chi_{0}^{-1}(d\chi_{0}^{-1}/dT)^{-1}$ (right) with solid fitting curves.
(e) Scaling plots of renormalized $m$ vs $h$ with
$m\equiv\varepsilon^{-\beta}M(H,\varepsilon)$ and
$h\equiv\varepsilon^{-(\beta+\gamma)}H$. (f) Rescaling plots of $M(H)$
isothermals by $MH^{-1/\delta}$ vs $\varepsilon H^{-1/(\beta\delta)}$.

Figure 1(a,b) shows the crystal structure and the XRD 2theta scans for
2H-Co0.22TaS2. The sharp peaks can be indexed with (00l) planes [Fig. 1(b)],
indicating that the plate surface of crystal is normal to the
$\mathbf{c}$-axis. The main powder XRD pattern can be fitted with the
$P6_{3}22$ space group using Le Bail analysis, resulting in the lattice
parameters $a$ = 5.72(2) Å and $c$ = 11.89(2) Å [inset in Fig. 1(b)]. It
adopts the same space group with smaller lattice parameters when compared with
2H-Co0.33TaS2 ($a$ = 5.74 Å and $c$ = 11.93 Å) Parkin1 ; VanLaar . Figure 1(c)
exhibits the normalized Co $K$-edge XANES. The threshold energy $E_{0}$ = 7717
eV obtained from the peak of derivative curve [inset in Fig. 1(c)] indicates
the Co2+ state. Figure 1(d) exhibits the Fourier transform magnitudes of
EXAFS. In a single-scattering approximation, the EXAFS could be described by
Prins :

$\displaystyle\chi(k)=\sum_{i}\frac{N_{i}S_{0}^{2}}{kR_{i}^{2}}f_{i}(k,R_{i})e^{-\frac{2R_{i}}{\lambda}}e^{-2k^{2}\sigma_{i}^{2}}sin[2kR_{i}+\delta_{i}(k)],$

where $N_{i}$ is the number of neighbouring atoms at a distance $R_{i}$ from
the photoabsorbing atom. $S_{0}^{2}$ is the passive electrons reduction
factor, $f_{i}(k,R_{i})$ is the backscattering amplitude, $\lambda$ is the
photoelectron mean free path, $\delta_{i}$ is the phase shift, and
$\sigma_{i}^{2}$ is the correlated Debye-Waller factor measuring the mean
square relative displacement of the photoabsorber-backscatter pairs. The main
peak corresponds well to the six nearest-neighbor Co-S [2.37(1) Å] and then
two next-nearest Co-Ta [3.01(7) Å] in the Fourier transform magnitudes of
EXAFS [Fig. 1(d)], extracted from the model fit with fixed coordination
number. The peaks above 3.5 Å are due to longer Co-S [4.07(5) Å] and Co-Ta
[4.45(5) Å] distances, and the multiple scattering behavior. Local crystal
structure environment of Co atom indicates that Co is well intercalated
Parkin1 ; VanLaar  in the vdW gap of 2H-TaS2 crystal at the interstitial
position as described in VanLaar  for 2H-Co0.33TaS2; somewhat smaller average
lattice parameters are likely due to smaller Co intercalation.

Table 1: Comparison of critical exponents of 2H-Co0.22TaS2 and 2H-Fe0.26TaS2 with different theoretical models. The MAP, KFP, and CI represent the modified Arrott plot, the Kouvel-Fisher plot, and the critical isotherm, respectively. | Reference | Technique | $T_{c}$ | $\beta$ | $\gamma$ | $\delta$
---|---|---|---|---|---|---
2H-Co0.22TaS2 | This work | MAP | 28.61(7) | 0.44(1) | 1.15(1) | 3.61(4)
|  | KFP | 28.59(6) | 0.43(2) | 1.15(1) | 3.67(8)
|  | CI | 28 |  |  | 3.54(1)
2H-Fe0.26TaS2 | Zhang  | MAP | 100.67(4) | 0.460(4) | 1.216(11) | 3.643(1)
|  | KFP | 100.69(5) | 0.459(6) | 1.205(11) | 3.625(10)
|  | CI | 100.7 |  |  | 3.69(1)
2D Ising | Widom  | Theory |  | 0.125 | 1.75 | 15.0
Mean field | Kouvel  | Theory |  | 0.5 | 1.0 | 3.0
3D Heisenberg | Kouvel  | Theory |  | 0.365 | 1.386 | 4.8
3D Ising | Kouvel  | Theory |  | 0.325 | 1.24 | 4.82
3D XY | Phan  | Theory |  | 0.345 | 1.316 | 4.81
Tricritical mean field | Fisher1972  | Theory |  | 0.25 | 1.0 | 5.0

Figure 1(c,d) exhibits the temperature dependence of dc magnetization measured
in various fields applied in $\mathbf{ab}$-plane and along $\mathbf{c}$-axis
with zero-field cooling (ZFC) and field cooling (FC) modes, respectively. A
sharp upturn in $M(T)$ was observed for both field directions when temperature
decreases, suggesting a PM-FM transition. The $M(T)$ for
$\mathbf{H}\parallel\mathbf{c}$ are larger than those for
$\mathbf{H}\parallel\mathbf{ab}$, indicating an uniaxial anisotropy in
2H-Co0.22TaS2. The difference between ZFC and FC curves is due to magnetic
domain creep effect. The $1/\chi(T)$ = $H/M(T)$ taken at 1 kOe from 100 to 300
K can be well fitted by the Curie-Weiss law $\chi=\chi_{0}+C/(T-\theta)$
[insets in Fig. 1(c,d)], where $\chi_{0}$ is a temperature-independent term,
$C$ and $\theta$ are the Curie-Weiss constant and Weiss temperature,
respectively. The Weiss temperature is $\theta_{c}$ = 40(2) K for
$\mathbf{H}\parallel\mathbf{c}$ and $\theta_{ab}$ = -5(1) K for
$\mathbf{H}\parallel\mathbf{ab}$, respectively, indicating dominance of FM
exchange interactions along $\mathbf{c}$-axis and AFM in $\mathbf{ab}$-plane
in 2H-Co0.22TaS2. The derived effective moment is 2.3(1) $\mu_{\textrm{B}}$/Co
for $\mathbf{H}\parallel\mathbf{ab}$ and 2.7(1) $\mu_{\textrm{B}}$/Co for
$\mathbf{H}\parallel\mathbf{c}$, respectively, which is smaller than the spin-
only moment of 3.87 $\mu_{\textrm{B}}$ for Co2+. The deviation between 2.3(1)
and 2.7(1) $\mu_{\textrm{B}}$/Co may be due to the anisotropic $g$ factor with
unquenched orbital angular moment. Future experiments such as neutron
scattering or electron spin resonance would be helpful to clarify this
difference. Figure 1(e,f) presents the field-dependent magnetization at
various temperatures. A large coercivity of 14.3 kOe was observed at 2 K when
$\mathbf{H}\parallel\mathbf{c}$, which decreases to 0.7 kOe at 10 K and
disappears at 50 K. The saturation moment is also relatively small ($<$ 0.8
$\mu_{\textrm{B}}$/Co), suggesting that both localized and delocalized Co
electrons coexist in 2H-Co0.22TaS2. Only linear dependent $M(H)$ was observed
when $\mathbf{H}\parallel\mathbf{ab}$, unambiguously confirming a large
magnetocrystalline anisotropy in 2H-Co0.22TaS2. This is the first time to
achieve FM order with uniaxial anisotropy in Co-intercalated 2H-TaS2 crystals.
The FM coupling between intercalated-Co moments in 2H-TaS2 layer may arise
from an indirect spin exchange interaction, namely, the Ruderman-Kittel-
Kasuya-Yosida (RKKY) interaction Aristov ; Yosida ; Fukuma . The high carrier
density in 2H-TaS2 host also favors the spin exchange interactions between
isolated Co atoms through the RKKY mechanism. Similar mechanism was proposed
for magnetic order in the related materials Cr0.33TaS2 and V0.25VS2 YamasakiY
; NiuJ .

To obtain a precise $T_{c}$ and the mechanism of PM-FM transition in
2H-Co0.22TaS2, we measured dense magnetization isotherms with
$\mathbf{H}\parallel\mathbf{c}$ from 18 to 38 K [Fig. 2(a)]. The critical
behavior gives insight into the nature of magnetic interactions, correlation
length, spin dimensionality, and the spatial decay of correlation function at
criticality Fisher0 ; Stanley0 . The Arrott plot ($M^{2}$ vs $H/M$) [Fig.
2(b)] shows quasi-straight and -parallel lines in high field region Arrott1 .
To obtain the critical exponents, the isotherms were then reanalyzed with the
Arrott-Noakes equation of state Arrott2 ,

$(H/M)^{1/\gamma}=a\varepsilon+bM^{1/\beta},$ (1)

where $\varepsilon=(T-T_{c})/T_{c}$ is the reduced temperature, and $a$ and
$b$ are constants. The $\beta$ and $\gamma$ are associated with the
spontaneous magnetization $M_{s}$ below $T_{c}$ and the inverse initial
susceptibility $\chi_{0}^{-1}$ above $T_{c}$, respectively Stanley ; Fisher ;
Lin :

$M_{s}(T)=M_{0}(-\varepsilon)^{\beta},\varepsilon<0,T<T_{c},$ (2)
$\chi_{0}^{-1}(T)=(h_{0}/m_{0})\varepsilon^{\gamma},\varepsilon>0,T>T_{c},$
(3) $M=DH^{1/\delta},T=T_{c},$ (4)

where $\delta$ is another critical exponent associated with the M(H) at
$T_{c}$. The $M_{0}$, $h_{0}/m_{0}$ and $D$ are the critical amplitudes. A
rigorous iterative method was adopted with an initial mean field model Kellner
; Pramanik .

Figure 2(c) exhibits the temperature dependence of final $M_{s}(T)$ and
$\chi_{0}^{-1}(T)$ with power law fitting that gives $\beta=0.44(1)$ with
$T_{c}=28.59(5)$ K, and $\gamma=1.15(1)$ with $T_{c}=26.63(2)$ K. The third
exponent $\delta$ = 3.54(1) is determined by a linear fit of $M(H)$ at 28 K
taking into account that $M=DH^{1/\delta}$ at $T_{c}$ [inset in Fig. 2(c)],
which is very close to the value of 3.61(4) calculated from the Widom scaling
relation $\delta=1+\gamma/\beta$ Widom . In the Kouvel-Fisher (KF) relation
Kouvel :

$M_{s}(T)[dM_{s}(T)/dT]^{-1}=(T-T_{c})/\beta,$ (5)
$\chi_{0}^{-1}(T)[d\chi_{0}^{-1}(T)/dT]^{-1}=(T-T_{c})/\gamma.$ (6)

Linear fittings to the plots of $M_{s}(T)[dM_{s}(T)/dT]^{-1}$ vs $T$ and
$\chi_{0}^{-1}(T)[d\chi_{0}^{-1}(T)/dT]^{-1}$ vs $T$ yield $\beta=0.43(2)$
with $T_{c}=28.55(2)$ K, and $\gamma=1.15(1)$ with $T_{c}=28.62(2)$ K, as
shown in Fig. 2(d), very close to the values obtained from the modified Arrott
plot. Scaling analysis was then used to double-check the self-consistency and
reliability of the obtained $\beta$, $\gamma$, $\delta$, and $T_{c}$.

Figure 3: (Color online) Temperature dependence of in-plane (a) thermal
conductivity $\kappa(T)$, (b) thermopower $S(T)$, and (c) resistivity
$\rho_{xx}(T)$ in zero field for 2H-Co0.22TaS2 single crystal. Out-of-plane
field dependence of (d) Hall resistivity $\rho_{xy}(\mu_{0}H)$ and (e) initial
magnetization $M(\mu_{0}H)$ for 2H-Co0.22TaS2 at indicated temperatures.
Temperature dependence of (f) ordinary Hall coefficient $R_{0}$ and (g)
anomalous Hall coefficient $R_{s}$, fitted from the $\rho_{xy}(\mu_{0}H)$
curves using $\rho_{xy}=R_{0}\mu_{0}H+R_{s}M$. (h) Anomalous Hall conductivity
$\sigma_{xy}^{A}$ vs $M_{s}$ with
$\sigma_{xy}^{A}\approx\rho_{xy}^{A}/\rho_{xx}^{2}$. (i) The
$\rho_{xy}/\mu_{0}H$ vs $\rho_{xx}^{2}M/\mu_{0}H$ curves at indicated
temperatures with subsequent offset of 0.05 cm3 C-1. (j) Temperature
dependence of coefficient $S_{H}=\mu_{0}R_{s}/\rho_{xx}^{2}$.

The magnetic equation of state in the critical region is expressed as Stanley
:

$M(H,\varepsilon)=\varepsilon^{\beta}f_{\pm}(H/\varepsilon^{\beta+\gamma}),$
(7)

where $f_{+}$ for $T>T_{c}$ and $f_{-}$ for $T<T_{c}$, respectively, are the
regular functions. Eq. (7) can be further written in terms of scaled
magnetization $m\equiv\varepsilon^{-\beta}M(H,\varepsilon)$ and scaled field
$h\equiv\varepsilon^{-(\beta+\gamma)}H$ as $m=f_{\pm}(h)$. This suggests that
for true scaling relations and the right choice of critical exponents, scaled
$m$ and $h$ will fall on universal curves above $T_{c}$ and below $T_{c}$,
respectively. The scaled $m$ vs $h$ curves are obviously seperated into two
branches below and above $T_{c}$, respectively, as shown in [Fig. 2(e)]. This
can also be checked by another form of scaling equation of state Stanley ,

$\frac{H}{M^{\delta}}=k\left(\frac{\varepsilon}{H^{1/\beta}}\right),$ (8)

where $k(x)$ is the scaling function. According to Eq. (8), the
$MH^{-1/\delta}$ vs $\varepsilon H^{-1/(\beta\delta)}$ curves should be scaled
into one universal curve Phan , which is indeed seen in Fig. 2(f). The well-
rescaled curves ensure that the obtained critical exponents and $T_{c}$ are
reliable and intrinsic.

The critical exponents of 2H-Co0.22TaS2 obtained using various methods as well
as different theoretical models are summarized in Table I. Taroni _et al._
pointed out that the value of $\beta$ for a 2D magnet should be within a
window $0.1\leq\beta\leq 0.25$ Taroni , indicating a clear 3D critical
behavior in 2H-Co0.22TaS2. Moreover, the critical exponents do not belong to
any single universality class. The $\beta$ lies between 3D Heisenberg and mean
field model but closer to the latter, indicating the exchange interaction is
of long-range type in 2H-Co0.22TaS2. However, the $\gamma$ lies between 3D
Ising and mean field model, which is attributed to strong uniaxial anisotropy
in 2H-Co0.22TaS2. The mechanism of FM coupling in 2H-Co0.22TaS2 is the RKKY
interaction in which the local spins of intercalated Co ions align
ferromagnetically through the itinerant Ta 5d electrons. Compared with
2H-Fe0.26TaS2, the slightly smaller values of $\beta$ and $\gamma$ suggest a
similar critical behavior in 2H-Co0.22TaS2; i.e., a localized 3D Heisenberg
type coupled with long range magnetic interactions Zhang .

It is also of interest to estimate the range of interaction in this material.
In renormalization group theory analysis the interaction decays with distance
$r$ as $J(r)\approx r^{-(3+\sigma)}$ for long-range exchange, where $\sigma$
is the range of exchange interaction Fisher1972 . The susceptibility exponent
$\gamma$ is:

$\gamma=1+\frac{4}{d}\left(\frac{n+2}{n+8}\right)\Delta\sigma+\frac{8(n+2)(n-4)}{d^{2}(n+8)^{2}}\\\
\times\left[1+\frac{2G(\frac{d}{2})(7n+20)}{(n-4)(n+8)}\right]\Delta\sigma^{2},$
(9)

where $\Delta\sigma=(\sigma-\frac{d}{2})$,
$G(\frac{d}{2})=3-\frac{1}{4}(\frac{d}{2})^{2}$, and $n$ is the spin
dimensionality Fischer . When $\sigma>2$, it indicates a short-range
interaction, while $\sigma<2$ implies a long-range type of interaction, and
mean-field model should apply when $\sigma<3/2$. According to Eq. (9), it is
found that {$d:n$} = {3 : 3} and $\sigma=1.69$ give the critical exponents
$\beta=0.44$, $\gamma=1.14$, and $\delta=3.59$ [$\nu=\gamma/\sigma$,
$\alpha=2-\nu d$, $\beta=(2-\alpha-\gamma)/2$, and $\delta=1+\gamma/\beta$],
which are mostly close to our experimentally observed values. The obtained
$\sigma=1.69$ is slightly smaller than that of 1.71 for 2H-Fe0.26TaS2, in line
with the comparison of $\beta$ and $\gamma$. It confirms that the spin
interaction in 2H-Co0.22TaS2 is of 3D Heisenberg ({$d:n$} = {3 : 3}) type
coupled with long range magnetic interactions and that the exchange decays
with distance as $J(r)\approx r^{-4.69}$; magnetic anisotropy probably
originates from the unquenched orbital moment and different crystallographic
environment for Co atoms in the vdW gap plane and along the $\mathbf{c}$-axis
of crystal.

Having established magnetic properties and the nature of critical behavior, we
proceed to investigate effects of Co intercalation on electronic transport
properties. Figure 3(a-c) shows the temperature-dependent thermal conductivity
$\kappa(T)$, thermopower $S(T)$, and in-plane resistivity $\rho_{xx}(T)$ of
2H-Co0.22TaS2 single crystal. It is obvious that Co intercalation removes
charge density wave transition at 78 K Harper . Generally,
$\kappa_{total}=\kappa_{e}+\kappa_{L}$, consists of the electronic part
$\kappa_{e}$ and the phonon term $\kappa_{L}$. The $\kappa_{L}$ can be
obtained by subtracting the $\kappa_{e}$ part calculated from the Wiedemann-
Franz law $\kappa_{e}/T=L_{0}/\rho$, where $L_{0}$ = 2.45 $\times$ 10-8 W
$\Omega$ K-2 WH . The calculated $\kappa_{e}$ and $\kappa_{L}$ are also
plotted in Fig. 3(a), indicating that the $\kappa_{L}$ dominates. Above 100 K,
the linear temperature dependence of $S(T)$ and $\rho(T)$ is observed since
the magnetic scattering is independent of temperature in the paramagnetic
regime. The negative values of $S(T)$ in the whole temperature range indicates
that electron-type carriers dominate; it changes the slope below 100 K [Fig.
3(b)]. There is no obvious anomaly around $T_{c}$ in $S(T)$, probably due to
the fact that ordered moment is small [Figs. 1(g,h)] since the Co does not
develop full moment due to hybridization with conduction electron band. This
ia also the reason for the relatively weak upturn in $\rho(T)$ below $T_{c}$
[inset in Fig. 3(c)].

Figure 3(d) shows the field dependence of Hall resistivity
$\rho_{xy}(\mu_{0}H)$ of 2H-Co0.22TaS2 measured at various temperatures. Above
100 K, the negative slope of $\rho_{xy}(\mu_{0}H)$ indicates the dominance of
electron-type carries, in line with the $S(T)$ analysis, which can be
accounted for by the ordinary Hall coefficient. With decreasing temperature,
the Hall coefficient changes sign from negative to positive, in line with the
slope change of $S(T)$. It gradually deviates from the linear field-dependence
and varies non-linearly with the applied magnetic field around $T_{c}$. With
approaching $T_{c}$, the tendency is similar with the $M(\mu_{0}H)$ curves
[Fig. 3(e)], indicating that there is AHE in 2H-Co0.22TaS2. The AHE originates
from the spontaneous polarization induced by non-zero net magnetization due to
significant hybridization between Co-3d and Ta-5d bands in this system.

Generally, the Hall resistivity in ferromagnets is made up of two parts Wang ;
Yan ; WangY ; Onoda2008 ,

$\rho_{xy}=\rho_{xy}^{O}+\rho_{xy}^{A}=R_{0}\mu_{0}H+R_{s}M,$ (10)

where $\rho_{xy}^{O}$ and $\rho_{xy}^{A}$ are the ordinary and anomalous Hall
resistivity, respectively. The $R_{0}$ is the ordinary Hall coefficient and
the $R_{s}$ is the anomalous Hall coefficient. With linear fits of
$\rho_{xy}(\mu_{0}H)$ in high field, the slope and intercept correspond to
$R_{0}$ and $\rho_{xy}^{A}$, respectively. The $R_{s}$ can be obtained from
$\rho_{xy}^{A}=R_{s}M_{s}$ with $M_{s}$ taken from linear fits of
$M(\mu_{0}H)$ curves in high field. The temperature dependence of derived
$R_{0}$ and $R_{s}$ is plotted in Fig. 3(f,g), in which the value of $R_{s}$
is about ten times larger than that of $R_{0}$ below $T_{c}$. The $R_{0}$
shows a remarkable temperature dependence, which changes sign at high
temperature and reaches a maximum at low temperature [Fig. 3(f)]. Combined
with the slope change of $S(T)$ below 100 K, this points to a possible multi-
band behavior.

Figure 3(h) exhibits the anomalous Hall conductivity (AHC) $\sigma_{xy}^{A}$
($\approx$ $\rho_{xy}^{A}/\rho_{xx}^{2}$), which is proportional to $M_{s}$,
usually expected by the Kaplus-Luttinger (KL) theory KL . The value of
anomalous Hall conductivity is found to be very small compared to usual FM
conductors since the density of ferromagnetic Co is low. The intrinsic KL
contribution of $\sigma_{xy,in}^{A}$ is of the order of $e^{2}/(hd)$, where
$e$ is the electronic charge, $h$ is the Plank constant, and $d$ is the
lattice parameter Onoda2006 . Taking $d\approx V^{1/3}\sim 7$ Å,
$\sigma_{xy,in}^{A}$ is estimated to be 550 ($\Omega$ cm)-1, which is much
larger than the obtained values in Fig. 3(h); this precludes the possibility
that KL mechanism is observed in experiment. The extrinsic side-jump
contribution of $\sigma_{xy,sj}^{A}$ is usually of the order of
$e^{2}/(hd)(\varepsilon_{SO}/E_{F})$, where $\varepsilon_{SO}$ and $E_{F}$ is
the spin-orbital interaction energy and the Fermi energy, respectively
Nozieres . The value of $\varepsilon_{SO}/E_{F}$ is generally less than
$10^{-2}$ for metallic ferromagnets, which points to the same order of
magnitude of the obtained $\sigma_{xy}^{A}$. Furthermore, the extrinsic side-
jump mechanism, where the potential field induced by impurities contributes to
the anomalous group velocity, follows a scaling behavior of
$\rho_{xy}^{A}=\beta\rho_{xx}^{2}$, the same with the intrinsic KL mechanism,
however, the skew-scattering mechanism describes asymmetric scattering induced
by impurities or defects and contributes to the AHE with a scaling behavior of
$\rho_{xy}^{A}=\beta\rho_{xx}$. As we can see, the scaling plots of the
modified anomalous Hall resistivity $\rho_{xy}/\mu_{0}H_{eff}$ and
longitudinal resistivity $\rho_{xx}^{2}M/\mu_{0}H_{eff}$, with subsequent
offset by 0.05 cm3 C-1, shows a good linear behavior [Fig. 3(i)]. This linear
scaling behavior indicates that the AHE in 2H-Co0.22TaS2 is probably dominated
by the extrinsic side-jump mechanism rather than extrinsic skew-scattering and
the intrinsic KL theory. Then Eq. (10) can be written as

$\rho_{xy}=R_{0}\mu_{0}H+S_{H}\rho_{xx}^{2}M,$ (11)

where $S_{H}$ is a material-specific scale factor Lee1 ; Fri . The coefficient
$S_{H}=\mu_{0}R_{s}/\rho_{xx}^{2}=\sigma_{xy}^{A}/M_{s}$ [Fig. 3(j)] is found
to be weakly temperature-dependent and is comparable with those in traditional
itinerant ferromagnets, such as Fe and Ni ($S_{H}\sim 0.01-0.2$ V-1) Dheer ;
Jan .

##  IV CONCLUSIONS

In summary, we first synthesized single crystals of ferromagnetic 2H-CoxTaS2
by intercalating $x\sim$ 0.22 Co in 2H-TaS2 vdW gap. We have also
systematically studied the second-order PM-FM transition in 2H-Co0.22TaS2 with
$T_{c}$ $\sim$ 28 K. Critical exponents $\beta$, $\gamma$, $\delta$, and
$T_{c}$ derived via various methods match reasonably well and follow the
scaling equation. The detailed analysis indicates that the spin interaction in
2H-Co0.22TaS2 is of 3D Heisenberg-type coupled with long-range exchange
interaction decaying with distance as $J(r)\approx r^{-4.69}$. The origin of
the AHE below $T_{c}$ in 2H-Co0.22TaS2 crystal is probably dominated by the
extrinsic side-jump mechanism rather than extrinsic skew-scattering and
intrinsic KL mechanisms. Since few-layer graphene/2H-TaS2 heterostructures
preserve 2D Dirac states with robust spin-helical structure of interest for
spin-logic circuits LiLijun , the possibility of integration of robust
ferromagnetism is of interest for spintronic and calls for nanofabrication of
graphene/2H-Co0.22TaS2 heterostructures and devices.

## Acknowledgements

Work at BNL is supported by the Office of Basic Energy Sciences, Materials
Sciences and Engineering Division, U.S. Department of Energy (DOE) under
Contract No. DE-SC0012704. This research used the 8-ID beamline of the NSLS
II, a U.S. DOE Office of Science User Facility operated for the DOE Office of
Science by BNL under Contract No. DE-SC0012704.

∗Present address: Los Alamos National Laboratory, MS K764, Los Alamos NM
87545, USA

†Present address: ALBA Synchrotron Light Source, Cerdanyola del Valles,
E-08290 Barcelona, Spain.

## References

  * (1) A. K. Geim and I. V. Grigorieva, Van der Waals heterostructures, Nature 499, 419 (2013).
  * (2) K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro Neto, 2D materials and van der Waals heterostructures, Science 353, 461 (2016).
  * (3) J. Lee, S. Lee, J. H. Ryoo, S. Kang, T. Y. Kim, P. Kim, C. Park, J. Park, and H. Cheong, Ising-Type Magnetic Ordering in Atomically Thin FePS3, Nano Lett. 16, 7433 (2016).
  * (4) B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. D. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature 546, 270 (2017).
  * (5) C. Gong, L. Li, Z. L. Li, H. W. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Z. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nature 546, 265 (2017).
  * (6) Y. J. Deng, Y. J. Yu, Y. C. Song, J. Z. Zhang, N. Z. Wang, Z. Y. Sun, Y. F. Yi, Y. Z. Wu, S. W. Wu, J. Y. Zhu, J. Wang, X. H. Chen, and Y. B. Zhang, Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2, Nature 563, 94 (2018).
  * (7) M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappattil, R. Das, T. Eggers, H. R. Gutierrez, M. Phan, and M. Batzill, Strong room-temperature ferromagnetism in VSe2 monolayers on van der Waals substrates, Nat. Nanotechnol. 13 289 (2018).
  * (8) D. J. O’Hara, T. Zhu, A. H. Trout, A. S. Ahmed, Y. K. Luo, C. H. Lee, M. R. Brenner, S. Rajan, J. A. Gupta, D. W. McComb, and R. K. Kawakami, Room Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer Limit, Nano Lett. 18, 3125 (2018).
  * (9) P. A. Joy and S. Vasudevan, Magnetism in the layered transition-metal thiophosphates MPS3 (M = Mn, Fe, and Ni), Phys. Rev. B 46, 5425 (1992).
  * (10) Y. Liu, V. N. Ivanovski, and C. Petrovic, Critical behavior of the van der Waals bonded ferromagnet Fe3?xGeTe2, Phys. Rev. B 96, 144429 (2017).
  * (11) B. J. Liu, Y. Zou, S. Zhou, L. Zhang, Z. Wang, H. Li, Z. Qu, and Y. H. Zhang, Critical behavior of the van der Waals bonded high $T_{c}$ ferromagnet Fe3GeTe2, Sci. Rep. 7, 6184 (2017).
  * (12) G. T. Lin, X. Luo, F. C. Chen, J. Yan, J. J. Gao, Y. Sun, W. Tong, P. Tong, W. J. Lu, Z. G. Sheng, W. H. Song, X. B. Zhu, and Y. P. Sun, Critical behavior of two-dimensional intrinsically ferromagnetic semiconductor CrI3, Appl. Phys. Lett. 112, 072405 (2018).
  * (13) Y. Liu and C. Petrovic, Three-dimensional magnetic critical behavior in CrI3, Phys. Rev. B 97, 014420 (2018).
  * (14) Y. Liu, L. J. Wu, X. Tong, J. Li, J. Tao, Y. M. Zhu, and C. Petrovic, Thickness-dependent magnetic order in CrI3 single crystals, Sci. Rep. 9, 13599 (2019).
  * (15) Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. may, W. Wu, D. H. Cobden, J. Chu, and X. Xu, Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2, Nat. Mater. 17, 778 (2018).
  * (16) B. J. Liu, Y. M. Zou, L. Zhang, S. M. Zhou, Z. Wang, W. K. Wang, Z. Qu, and Y. H. Zhang, Critical behavior of the quasi-two-dimensional semiconducting ferromagnet CrSiTe3, Sci. Rep. 6, 33873 (2016).
  * (17) Y. Liu and C. Petrovic, Critical behavior of quasi-two-dimensional semiconducting ferromagnet Cr2Ge2Te6, Phys. Rev. B 96, 054406 (2017).
  * (18) W. Liu, Y. Dai, Yu-E. Yang, J. Fan, L. Pi, L. Zhang, and Y. Zhang, Critical behavior of the single-crystalline van der Waals bonded ferromagnet Cr2Ge2Te6, Phys. Rev. B 98, 214420 (2018).
  * (19) G. T. Lin, H. L. Zhuang, X. Luo, B.J. Liu, F. C. Chen, J. Yan, Y. Sun, J. Zhou, W. J. Lu, P. Tong, Z. G. Sheng, Z. Qu, W. H. Song, X. B. Zhu, and Y. P. Sun, Tricritical behavior of the two-dimensional intrinsically ferromagnetic semiconductor CrGeTe3, Phys. Rev. B 95, 245212 (2017).
  * (20) Y. Liu and C. Petrovic, Critical behavior and magnetocaloric effect in Mn3Si2Te6, Phys. Rev. B 98, 064423 (2018).
  * (21) S. S. P. Parkin and R. H. Friend, $3d$ transition-metal intercalates of the niobium and tantalum dichalcogenides. I. Magnetic properties, Philos. Mag. B 41, 65 (1980).
  * (22) R. H. Friend and A. D. Yoffe, Electronic properties of intercalation complexes of the transition metal dichalcogenides, Advances in Phys. 36, 1 (1987).
  * (23) H. Hinode, T. Ohtani, and M. Wakihara, Homogeneity Range and Some Physical Properties of Intercalation Compounds of MnxTaS2, J. Solid State Chem. 114, 1 (1995).
  * (24) H. Zhang, W. Wei, G. Zheng, J. Lu, M. Wu, X. Zhu, J. Tang, W. Ning, Y. Han, L. Lin, J. Yang, W. Gao, Y. Qin, and M. Tian, Electrical and anisotropic magnetic properties in layered Mn1/3TaS2 crystals, Appl. Phys. Lett. 113, 072402 (2018).
  * (25) E. Morosan, H. W. Zandbergen, L. Li, M. Lee, J. G. Checkelsky, M. Heinrich, T. Siegrist, N. P. Ong, and R. J. Cava, Sharp switching of the magnetization in Fe1/4TaS2, Phys. Rev. B 75, 104401 (2007).
  * (26) W. J. Hardy, C.-W. Chen, A. Marcinkova, H. Ji, J. Sinova, D. Natelson, and E. Morosan, Very large magnetoresistance in Fe0.28TaS2 single crystals, Phys. Rev. B 91, 054426 (2015).
  * (27) C.-W. Chen, S. Chikara, V. S. Zapf, and E. Morosan, Correlations of crystallographic defects and anisotropy with magnetotransport properties in FexTaS2 single crystals (0.23 $\leq x\leq$ 0.35), Phys. Rev. B 94, 054406 (2016).
  * (28) R. Cai, W. Xing, H. Zhou, B. Li, Y. Chen, Y. Yao, Y. Ma, X. C. Xie, S. Jia, and W. Han, Anomalous Hall effect mechanisms in the quasi-two-dimensional van der Waals ferromagnet Fe0.29TaS2, Phys. Rev. B 100, 054430 (2019).
  * (29) C. Zhang, Y. Yuan, M. Wang, P. Li, J. Zhang, Y. Wen, S. Zhou, and X. X. Zhang, Critical behavior of intercalated quasi-van der Waals ferromagnet Fe0.26TaS2, Phys. Rev. Mater. 3, 114403 (2019).
  * (30) S. S. P. Parkin, E. A. Marseglia, and P. J. Brown, Magnetic structure of Co1/3NbS2 and Co1/3TaS2, J. Phys. C.: Solid State Phys. 16, 2765 (1983).
  * (31) B. Van Laar, H. M. Rietveld and D. J. W. Ijdo, Magnetic and crystallographic structures of MexNbS2 and MexTaS2, J. Solid State Chem. 3, 154 (1971).
  * (32) R. Prins and D. C.Koningsberger (eds.), X-ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS, XANES (Wiley, New York, 1988).
  * (33) D. N. Aristov, Indirect RKKY interaction in any dimensionality, Phys. Rev. B 55, 8064 (1997).
  * (34) K. Yosida, Magnetic Properties of Cu-Mn Alloys, Phys. Rev. 106, 893 (1957).
  * (35) Y. Fukuma, H. Asada, S. Miyawaki, T. Koyanagi, S. Senba, K. Goto, and H. Sato, Carrier-induced ferromagnetism in Ge0.92Mn0.08Te epilayers with a Curie temperature up to 190 K, Appl. Phys. Lett. 93, 252502 (2008).
  * (36) Y. Yamasaki, R. Moriya, M. Arai, S. Masubuchi, S. Pyon, T. Tamegai, K. Ueno and T. Machida, Exfoliation and van der Waals heterostructure assembly of intercalated ferromagnet Cr1/3TaS2, 2D Mater. 4, 041007 (2017).
  * (37) J. Niu, W. Zhang, Z. Li, S. Yang, D. Yan, S. Chen, Z. Zhang, Y. Zhang, X. Ren, P. Gao, Y. Shi, D. Yu and X. Wu, Intercalation of van der Waals layered materials: A route towards engineering of electron correlation, Chin. Phys. B 29, 097104 (2020).
  * (38) M. E. Fisher, The renormalization group in the theory of critical behavior, Rev. Mod. Phys. 46, 597 (1974).
  * (39) H. E. Stanley, Scaling, universality, and renormalization: Three pillars of modern critical phenomena, Rev. Mod. Phys. 71, S358 (1999).
  * (40) A. Arrott, Criterion for Ferromagnetism from Observations of Magnetic Isotherms, Phys. Rev. B 108, 1394 (1957).
  * (41) A. Arrott, and J. Noakes, Approximate Equation of State For Nickel Near its Critical Temperature, Phys. Rev. Lett. 19, 786 (1967).
  * (42) H. E. Stanley, _Introduction to Phase Transitions and Critical Phenomena_ (Oxford University Press, London and New York, 1971).
  * (43) M. E. Fisher, The theory of equilibrium critical phenomena, Rep. Prog. Phys. 30, 615 (1967).
  * (44) J. Lin, P. Tong, D. Cui, C. Yang, J. Yang, S. Lin, B. Wang, W. Tong, L. Zhang, Y. Zou, and Y. Sun, Unusual ferromagnetic critical behavior owing to short-range antiferromagnetic correlations in antiperovskite Cu1-xNMn3+x (0.1 $\leq x\leq$ 0.4), Sci. Rep. 5, 7933 (2015).
  * (45) W. Kellner, M. Fähnle, H. Kronmüller, and S. N. Kaul, The Magnetic Phase Transition in Amorphous and Disordered Crystalline Ferromagnets, Phys. Status Solidi B 144, 387 (1987).
  * (46) A. K. Pramanik, and A. Banerjee, Critical behavior at paramagnetic to ferromagnetic phase transition in Pr0.5Sr0.5MnO3: A bulk magnetization study, Phys. Rev. B 79, 214426 (2009).
  * (47) B. Widom, Degree of the Critical Isotherm, J. Chem. Phys. 41, 1633 (1964).
  * (48) J. S. Kouvel, and M. E. Fisher, Detailed Magnetic Behavior of Nickel Near its Curie Point, Phys. Rev. 136, A1626 (1964).
  * (49) M. Phan, V. Franco, N. Bingham, H. Srikanth, N. Hur, and S. Yu, Tricritical point and critical exponents of La0.7Ca0.3-xSrxMnO3 ($x$ = 0, 0.05, 0.1, 0.2, 0.25) single crystals, J. Alloy. Compd. 508, 238 (2010).
  * (50) A. Taroni, S. T. Bramwell, and P. C. W. Holdsworth, Universal window for two-dimensional critical exponents, J. Phys.: Condens. Matter 20, 275233 (2008).
  * (51) M. E. Fisher, S. K. Ma, and B. G. Nickel, Critical exponents for long-range interactions, Phys. Rev. Lett. 29 917 (1972).
  * (52) S. F. Fischer, S. N. Kaul, and H. Kronmuller, Critical magnetic properties of disordered polycrystalline Cr75Fe25 and Cr70Fe30 alloys, Phys. Rev. B 65, 064443 (2002).
  * (53) J. M. Harper, T. E. Geballe and F. J. Disalvo, Thermal properties of layered transition-metal dichalcogenides at charge-density-wave transitions, Phys. Rev. B 15 2943 (1977).
  * (54) R. Franz and G. Wiedemann, Ueber die Wärme-Leitungsfähigkeit der Metalle, Annalen der Physik 165, 497 (1853).
  * (55) Q. Wang, S. S. Sun, X. Zhang, F. Pang, and H. C. Lei, Anomalous Hall effect in a ferromagnetic Fe3Sn2 single crystal with a geometrically frustrated Fe bilayer kagome lattice, Phys. Rev. B 94, 075135 (2016).
  * (56) J. Yan, X. Luo, G. T. Lin, F. C. Chen, J. J. Gao, Y. Sun, L. Hu, P. Tong, W. H. Song, Z. G. Sheng, W. J. Lu, X. B. Zhu, and Y. P. Sun, Anomalous Hall effect of the quasi-two-dimensional weak itinerant ferromagnet Cr4.14Te8, Europhys. Lett. 124, 67005 (2018).
  * (57) Y. H. Wang, C. Xian, J. Wang, B. J. Liu, L. S. Ling, L. Zhang, L. Cao, Z. Qu, and Y. M. Xiong, Anisotropic anomalous Hall effect in triangular itinerant ferromagnet Fe3GeTe2, Phys. Rev. B 96, 134428 (2017).
  * (58) S. Onoda, N. Sugimoto, and N. Nagaosa, Quantum transport theory of anomalous electric, thermoelectric, and thermal Hall effects in ferromagnets, Phys. Rev. B 77, 165103 (2008).
  * (59) R. Karplus and J. Luttinger, Hall Effect in Ferromagnetics, Phys. Rev. 95, 1154 (1954).
  * (60) S. Onoda, N. Sugimoto, and N. Nagaosa, Intrinsic Versus Extrinsic Anomalous Hall Effect in Ferromagnets, Phys. Rev. Lett. 97, 126602 (2006).
  * (61) P. Nozières and C. Lewiner, A simple theory of the anomalous hall effect in semiconductors, J. Phys. (Paris) 34, 901 (1973).
  * (62) M. Lee, Y. Onose, Y. Tokura, and N. P. Ong, Hidden constant in the anomalous Hall effect of high-purity magnet MnSi, Phys. Rev. B 75, 172403 (2007).
  * (63) S. Friedemann, M. Brando, W. J. Duncan, A. Neubauer, C. Pfleiderer, and F. M. Grosche, Ordinary and intrinsic anomalous Hall effects in Nb1-yFe2+y, Phys. Rev. B 87, 024410 (2013).
  * (64) J. P. Jan, and H. M. Gijsman, L’effet hall du fer et du nickel aux basses températures, Physica 18, 339 (1952).
  * (65) P. N. Dheer, Galvanomagnetic Effects in Iron Whiskers, Phys. Rev. 156, 637 (1967).
  * (66) L. Li, J. Zhang, G. Myeong, W. Shin, H. Lim, B. Kim, S. Kim, T. Jin, S. Cavill, B. S. Kim, C. Kim, J. Lischner, A. Ferreira, and S. Cho, Gate-Tunable Reversible Rashba-Edelstein Effect in a Few-Layer Graphene/2H-TaS2 Heterostructure at Room Temperature, ACS Nano 14, 5251 (2020).


# Steady states of asymmetric exclusion process with inhomogeneous hopping

Atri Goswami<EMAIL_ADDRESS>Department of Physics, Gurudas College,
Narkeldanga, Kolkata-700054, West Bengal, India  Mainak Chatterjee
<EMAIL_ADDRESS>Department of Physics, Barasat Government College, 10, KNC
Road, Gupta Colony, Barasat, Kolkata 700124, West Bengal, India  Sudip
Mukherjee<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Physics, Barasat Government College, 10, KNC Road, Gupta Colony, Barasat,
Kolkata 700124, West Bengal, India  Condensed Matter Physics Division, Saha
Institute of Nuclear Physics, Kolkata 700064, West Bengal, India

###### Abstract

We study the nonequilibrium steady states in asymmetric exclusion processes
(TASEP) with open boundary conditions having spatially inhomogeneous hopping
rates. Assuming spatially smoothly varying hopping rates with a few (or no)
discontinuities, we show that the steady states are in general classified by
the steady state currents in direct analogy with open TASEPs having uniform
hopping rates. We calculate the steady state density profiles, which are now
space-dependent. We also obtain the phase diagrams in the plane of the control
parameters, which though have phase boundaries that are in general curved
lines, have the same topology as their counterparts for conventional open
TASEPs, independent of the form of the hopping rate functions. This reveals a
type of universality, not encountered in critical phenomena.

##  I Introduction

Many natural systems are driven by some external fields or are made of self-
propelled particles. In the long time limit, these systems evolve into
stationary states which carry steady currents, which are hallmarks of
nonequilibrium systems. Such states are characterized by a constant gain or
loss of energy, which distinguishes them from systems in thermal equilibrium.
Examples of such driven systems range from live cell biological systems like
ribosomes moving along m-RNA or motor molecules “walking” along molecular
tracks known as microtubules to ions diffusing along narrow channels, or even
vehicles traveling along roads. In order to elucidate the nature of such
nonequilibrium steady states and in the absence of a general theoretical
framework, it is useful to study purpose-built simple models. To this end, a
variety of driven lattice gas models have been introduced and studied
extensively basic .

In this work, we focus on one-dimensional (1D) models with open boundaries,
where particles preferentially move in one direction. In particular, we work
on the totally asymmetric simple exclusion process (TASEP), that has become
one of the paradigms of nonequilibrium physics in low-dimensional systems
(see, e.g., Ref. tasep-rev  for reviews). In this model a single species of
particles is hopping unidirectionally and with a uniform rate along a 1D
lattice. The hopping movement is subject to exclusion, i.e., when the target
site is empty, since a given site can accommodate maximum one particle.
Particles enter the system at one side at a specified rate $\alpha$, and leave
the system through the other end at a given rate $\beta$; $\alpha$ are $\beta$
are the two control parameters of TASEP. It is known that the steady states of
TASEPs with open boundaries are rather sensitive to the boundary conditions:
by varying the boundary conditions, i.e., by varying $\alpha,\beta$, the
steady states of open TASEPs can be varied, resulting into boundary-induced
nonequilibrium phase transitions. These are genuine nonequilibrium effects,
since equilibrium systems are usually insensitive to precise boundary
conditions.

In the original TASEP model, the hopping rate in the bulk is assumed to be a
constant (of unit value), which is of course an idealisation. In real life
examples it is generally expected to have nonuniformity along the bulk of the
TASEP channel leading to nonuniform hopping rates. For instance, mRNA in cells
are known to have pause sites, where the effective hopping rates are lower
pause . This is a potentially important issue even in urban transport, where
the speeds of vehicles (which is the analogue of the hopping rates here)
depend sensitively on the bottlenecks along the roads transport . Such
spatially varying hopping rates can either be smoothly varying along the TASEP
lanes, or be random quenched disorders with given distribution. We focus here
on the case with smoothly varying hopping rates, for which the generic nature
of the steady states in TASEPs are still not known. There has been some
studies on quenched heterogeneous TASEP; see, e.g., Ref. qtasep  for some
studies on different aspects of heterogeneous TASEP. Recently, the steady
states of TASEPs with periodic boundary conditions are affected by smoothly
varying hopping rates are studied prr ; see also Ref. astik-prr  for a study
on periodic TASEP with random quenched disordered hopping rates. A type of
universality has been uncovered, showing the topological equivalence of the
phase diagrams independent of the precise form of the space dependence of the
hopping rates. Our studies here complement these results by considering the
problem in an open TASEP. We calculate the steady states for a few
representative examples of spatially varying hopping rates. We show that the
notion of universality illustrated in Ref. prr  can be extended to systems
with open boundaries as well, establishing the robustness of the concept of
universality here. The rest of the article is organised as follows. In Sec. II
we define and construct our model. Next, in Sec. III.1 we discuss the
algorithm of the Monte-Carlo simulations (MCS) study of the model to
numerically calculate the steady state densities. Then in Sec. III.2, we set
up the mean-field theory (MFT) and solve it to obtain the steady state
densities. In Sec. III.3, we present the phase diagrams of the model. We
summarise our results in Sec. IV.

##  II Model

The model consists of a 1D lattice of size $L$. The particles enter through
the left end at rate $\alpha$, hop unidirectionally from the left to the
right, all subject to exclusion, i.e., a single site can accommodate maximum
one particle at a time, and finally leave the system at a rate $\beta$.
Labelling each site by an index $i$ that runs from 1 to $L$, the hopping rate
at site $i$ is given by $q_{i}\leq 1$; see Fig. 1 for a schematic model
diagram.

Figure 1: Schematic model diagram. Broken line represents the TASEP lattice.
Particles enter and exit at rates $\alpha$ and $\beta$, respectively, and hop
from left to right, subject to exclusion.

A microscopic configuration of the model is characterised by a distribution of
identical particles on the lattice, i.e., by configurations ${\cal
C}=\\{n_{i=1,...,N}\\}$, where each of the occupation numbers $n_{i}$ is equal
to either zero (vacancy) or one (particle), as it should be in a model with
exclusion. Physically, a hard core repulsion between the particles is imposed,
resulting into prohibition of a double or higher occupancy of sites in the
model. The full state space then consists of $2^{N}$ configurations. The
following elementary steps fully define the microscopic dynamical update rules
of this model:

(a) At any site $i=2,...,N-1$ a particle can jump to site $i+1$ if unoccupied
with a rate $q_{i}\leq 1$.

(b) At the site $i=1$ a particle can enter the lattice with rate $\alpha$ only
if is unoccupied; and

(c) At the site $i=N$ a particle can leave the lattice with rate $\beta$ when
it is occupied.

In general, $q_{i}\neq q_{j}$ for $i=j$. Processes (a)-(c) formally define a
TASEP with open boundary conditions. If all of $q_{i}=1$ for all $i$
identically, then this model reduces to the conventional TASEP with open
boundary conditions tasep-rev . We consider some specified choices of $q_{i}$
that depends explicitly on $i$, and study their effects on the nonequilibrium
steady states of the model.

##  III Steady-state densities

We are interested to calculate the density profiles in the steady states. To
this end, we set MFT which can be solved analytically. We supplement the MFT
results by extensive Monte-Carlo simulations (MCS).

###  III.1 Monte-Carlo simulations

Let $n_{i}(t)$, which is either 0 or 1, be the occupation at site $i$ at time
$t$. We perform MCS studies of the model subject to the update rules (a)-(c)
described above in Sec. II using a random sequential updating scheme. After
reaching the steady states, the density profiles are calculated and temporal
averages are performed. This produces time-averaged, space-dependent density
profiles, given by $\langle n_{i}\rangle$,which are parametrised by $\alpha$
and $\beta$; here $\langle...\rangle$ implies temporal averages over steady
states. The simulations have been performed with $L=1000$ up to $10^{6}$
Monte-Carlo steps.

###  III.2 Mean-field theory

The dynamics of TASEP is formally given by rate equations for every site which
are not closed. In MFT approximation, we neglect correlation effects and
replace the average of product of densities by the product of average of
densities blythe . While this is an approximation, this has worked with high
degree of accuracy in the original TASEP problem and its many variants (see,
e.g., Refs. erwin-lk ; niladri1 ; tirtha1  as representative examples); we use
this as a guideline in our analysis below. The dynamical equation for
$n_{i}(t)$ is given by

$\frac{\partial n_{i}}{\partial
t}=q_{i}n_{i-1}(1-n_{i})-q_{i+1}n_{i}(1-n_{i+1}).$ (III.1)

To proceed further in the MFT approximation, we label the sites by $x=i/L$ and
take $L\rightarrow\infty$, which makes $x$ a continuous variable between 0 and
1: $x\in[0,1]$. In this parametrisation, the hopping rate function is given by
$0<q(x)\leq 1$, that is assumed to vary slowly in $x$. Further, we define
$\rho(x)=\langle n_{i}\rangle$ as the steady state density at $x$. In the
steady state, we expand the rhs of (III.1) in a Taylor series, which then
give,

$\frac{\partial\rho}{\partial t}=-\frac{\partial}{\partial
x}\left[q(x)\rho(x)(1-\rho(x))\right]=0,$ (III.2)

where we have neglected terms which are ${\cal O}(1/L)$ or higher order in
$1/L$. Equation (III.2) allows us to extract a bulk current $J$ given by

$J=q(x)\rho(x)[1-\rho(x)],$ (III.3)

which must be a constant independent of $x$ in a given steady state. That Eq.
(III.2) has the form of an equation of continuity is no surprise - this is
because away from the boundaries in the bulk of the TASEP, particles only hop
from left to right, subject to exclusion, which keeps the particle number
locally conserved.

Equation (III.3) is a quadratic equation in $\rho(x)$. In Eq. (III.3) since
$J$ is a constant and $q(x)$ has an explicit $x$-dependence, $\rho(x)$ must be
$x$-dependent, such that the product of the various factors on the rhs of
(III.3), all of which are individually $x$-dependent, produces an
$x$-independent result $J$. Equation (III.3) has two spatially nonuniform
solutions $\rho_{+}(x)$ and $\rho_{-}(x)$:

$\displaystyle\rho_{+}(x)=\frac{1}{2}\left[1-\sqrt{1-\frac{4J}{q(x)}}\right]>\frac{1}{2},$
(III.4)
$\displaystyle\rho_{-}(x)=\frac{1}{2}\left[1-\sqrt{1-\frac{4J}{q(x)}}\right]<\frac{1}{2},$
(III.5)

for any $x$. Evidently, both $\rho_{+}(x)$ and $\rho_{-}(x)$ are smooth
functions of $x$, except where $q(x)$ itself may be discontinuous. Further,
$\rho_{+}(x)>1/2$ everywhere, whereas $\rho_{-}(x)<1/2$ everywhere. Clearly,
if $q(x)$ is a constant then $\rho(x)$ is also a constant, independent of $x$
(ordinary TASEP with uniform hopping). Since $\rho(x)$ is positive definite
everywhere, we must have $1-4J/q(x)\geq 0$ giving an upper bound on $J$:

$J\leq q(x)/4.$ (III.6)

Clearly, $J$ has a maximum given by

$J_{\text{max}}=\frac{q_{\text{min}}}{4},$ (III.7)

for a given $q(x)$. Note that $J_{\text{max}}$ is the maximum possible current
that can be sustained by the system for all possible choices of the control
parameters $\alpha,\,\beta$. However, $J$ may not reach $J_{\text{max}}$ for
any $\alpha,\,\beta$; see below. In the limit of uniform hopping with $q(x)=1$
everywhere, $J_{\text{max}}=1/4$, corresponding to the MC phase current in the
conventional TASEP. We thus note that in the present model steady states with
current $J=J_{\text{max}}$ for any $q(x)$ should generalise the standard MC
phase.

We now systematically derive the conditions for the different phases. To do
this, we must calculate $J$ to completely know the solutions $\rho_{+}(x)$ and
$\rho_{-}(x)$.

Recall that in the LD phase of conventional TASEPs with uniform hopping, the
steady state is described by the incoming current, which in the bulk is given
by $J_{\text{LD}}=\alpha(1-\alpha)<J_{\text{HD}}=\beta(1-\beta)$, the outgoing
current. This corresponds to $\rho=\alpha$ as the bulk density in the LD
phase. In contrast, in the HD phase, $J_{\text{HD}}<J_{\text{LD}}$, giving
$1-\beta$ as the HD phase bulk density. We can generalise this picture by
finding out the forms of $J$ in the present problem.

Noting that the current in the bulk of the TASEP channel is
$J=q(x)\rho(x)[1-\rho(x)]$, where $\rho(x)$ is $\rho_{+}(x)$ or $\rho_{-}(x)$.
Since $\rho(0)=\alpha$, we obtain

$J_{\text{LD}}=q(0)\alpha(1-\alpha).$ (III.8)

Since, the steady state bulk density in the LD phase is less than 1/2
everywhere, the density profile $\rho_{\text{LD}}(x)$ in the LD phase is given
by

$\rho_{\text{LD}}(x)=\frac{1}{2}\left[1-\sqrt{1-\frac{4q(0)}{q(x)}\alpha(1-\alpha)}\;\right]<\frac{1}{2}.$
(III.9)

In the same vein, noting that $\rho(1)=1-\beta$, we obtain

$J_{\text{HD}}=q(1)\beta(1-\beta).$ (III.10)

Since, the steady state bulk density everywhere is more than 1/2, the density
profile $\rho_{\text{HD}}(x)$ in the HD phase is given by

$\rho_{\text{HD}}(x)=\frac{1}{2}\left[1-\sqrt{1-\frac{4q(1)}{q(x)}\beta(1-\beta)}\;\right]>\frac{1}{2}.$
(III.11)

We have plotted $\rho_{\text{LD}}(x)$ and $\rho_{\text{HD}}(x)$ versus $x$ in
Figs. 2 and Fig. 3, respectively for several choices of the hopping rate
function $q(x)$.

Figure 2: Plots of the steady state density $\rho_{\text{LD}}(x)$ versus $x$
in the LD phase for different choices of the hopping rate functions
$q(x)=1-x/2$ for two different sets of values of $\alpha$ and $\beta$.

Figure 3: Plots of the steady state density $\rho_{\text{HD}}(x)$ versus $x$
in the HD phase for different choices of the hopping rate functions
$q(x)=1-x/2$ for two different sets of values of $\alpha$ and $\beta$.

The steady density in the MC phase, $\rho_{\text{MC}}(x)$ is somewhat tricky
to calculate. We already know that the steady state current in the MC phase

$J_{\text{MC}}=q_{min}/4,$ (III.12)

which can be used either in $\rho_{+}(x)$ or $\rho_{-}(x)$, with these two
solutions become identical at $x=x_{0}$, at which $q(x)=q_{\text{min}}$. MCS
studies reveal that a part of $\rho_{\text{MC}}(x)$ is bigger than 1/2,
whereas elsewhere it is smaller than 1/2. Thus in order to construct
$\rho_{\text{MC}}(x)$, we must use both $\rho_{+}(x)$ and $\rho_{-}(x)$, i.e.,
$\rho_{\text{MC}}(x)$ is a combination of $\rho_{+}(x)$ and $\rho_{-}(x)$,
with the two segments meeting at $x_{0}$. This further implies that if
$x_{0}$, the location of $q_{\text{min}}$ is not in the bulk, but at the
extreme ends (i.e., $x=0,1$), $\rho_{\text{MC}}(x)$ will consist of only
$\rho_{-}(x)$ or $\rho_{+}(x)$. We have plotted $\rho_{\text{MC}}(x)$ versus
$x$ in Fig. 4.

Figure 4: Plots of the steady state density $\rho_{\text{MC}}(x)$ versus $x$
in the MC phase for different choices of the hopping rate functions
$q(x)=1-x/2$ for two different sets of values of $\alpha$ and $\beta$.

###  III.3 Phase diagram

We now discuss the conditions to obtain the phase diagram and the phase
boundaries in the $\alpha,\beta$-plane. First consider the boundary between
the LD and HD phases. In the LD phase, the bulk current in given by
$J_{\text{LD}}$ in (III.8), whereas the bulk current in the HD phase is given
by $J_{\text{HD}}$ in (III.10). The two phases meet when
$J_{\text{LD}}=J_{\text{HD}}$, which gives the phase boundary between the LD
and HD phases that has the form

$q(0)\alpha(1-\alpha)=q(1)\beta(1-\beta).$ (III.13)

This is a quadratic equation in $\beta$ (or in $\alpha$) with two solutions
$\beta_{\pm}$:

$\beta_{\pm}=\frac{1}{2}\left[1\pm\sqrt{4\mu\alpha(1-\alpha)}\right],$
(III.14)

where $\mu\equiv q(0)/q(1)$, which can be bigger or smaller than unity. If
$\mu<1$, $\beta=\beta_{+}$ gives the phase boundary, whereas if $\mu>1$,
$\beta=\beta_{-}$ gives the phase boundary. Phase boundary (III.14)
automatically reduces to $\alpha=\beta$, the well-known result for the phase
boundary between the LD and HD phases in an open TASEP with a uniform $q(x)$.
In fact, even with nonuniform $q(x)$, $\alpha=\beta$ is the phase boundary, so
long as $q(0)=q(1)$ is maintained.

Similar considerations allow us to obtain the LD-MC and HD-MC phase
boundaries. For example, the LD-MC phase boundary is given by the condition
$J_{\text{LD}}=J_{\text{MC}}$, which gives

$\alpha=\frac{1}{2}\left[1\pm\sqrt{1-\frac{q_{\text{min}}}{q(0)}}\right].$
(III.15)

Likewise, the HD-MC phase boundary is given by the condition
$J_{\text{HD}}=J_{\text{MC}}$, giving

$\beta=\frac{1}{2}\left[1\pm\sqrt{1-\frac{q_{\text{min}}}{q(1)}}\right].$
(III.16)

The three phase boundaries meet at
$\left(\left[1-\sqrt{1-q_{\text{min}}/q(0)}\right]/2,\left[1-\sqrt{1-q_{\text{min}}/q(1)}\right]/2\right)$.

Figure 5: Phase diagram in the $\alpha-\beta$ plane with $q(x)=1-x/2$. LD, HD
and MC phases are marked. The phase boundaries, estimated from MFT, are shown
in continuous line whereas the broken lines represent the phase boundaries
evaluated numerically.

##  IV Summary and outlook

We have studied the asymmetric exclusion process with open boundaries having
spatially smoothly varying hopping rates. Our study reveals the universal form
of the phase diagrams for generic smooth hopping rates. Our results are
sufficiently general and applies to any smoothly varying hopping rate
functions. We construct the mean-field theory, and use that to outline a
scheme to calculate stead state density profiles. Unsurprisingly, the steady
state density are space varying, unlike those in the conventional TASEP with
uniform hopping. These match well with those obtained from the MCS studies,
lending credence to our mean-field analysis. Because of the spatially varying
densities, the conventional way to characterise the phases via the densities,
i.e., $\rho_{\text{LD}}<1/2,\,\rho_{\text{HD}}>1/2$ and $\rho_{\text{MC}}=1/2$
in the bulk of the TASEP no longer holds. Rather one needs to resort to the
equivalent conditions to decide the phases, since the current $J$ is a
constant. This together with the condition that $\rho_{\text{LD}}(x)<1/2$ and
$\rho_{\text{HD}}(x)>1/2$ everywhere in the bulk, allows us to distinguish the
LD and HD phases. Further, the maximum steady state current that the system
can sustain is no longer 1/4, but is $q_{\text{min}}/4$, where
$q_{\text{min}}$ is the minimum hopping rate.

Our MFT scheme applies when $q(x)$ is smoothly varying. There are however in
vivo situations, where $q(x)$ is rapidly fluctuating in space dna . It would
be interesting to extend our work to such systems.

Our results may be verified in model experiments on the collective motion of
driven particles with light-induced activity light  passing through a narrow
channel. Spatial modulations of the hopping rate can be created by applying
patterned or spatially varying illumination.

## References

  * (1) B. Schmittmann and R. Zia, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic Press, London, 1995), Vol. 17.
  * (2) B. Derrida and M. Evans, in Nonequilibrium Statistical Mechanics in One Dimension, edited by V. Privman (Cambridge University Press, Cambridge, UK, 1997), Chap. 14, pp. 277–304; G. Schütz, in Phase Transitions and Critical Phenomena, ed- ited by C. Domb and J. Lebowitz (Academic Press, San Diego, 2001), Vol. 19, pp. 3–251.
  * (3) T. Chou, K. Mallick, and R. K. P. Zia, Rep. Prog. Phys. 74, 116601 (2011); S. E. Wells, E. Hillner, R. D. Vale, and A. B. Sachs, Mol. Cell 2, 135 (1998); S. Wang, K. S. Browning, and W. A. Miller, EMBO J. 16, 4107 (1997); Z. A. Afonina et al., Nucleic Acids Res. 42, 9461 (2014); D. W. Rogers et al., PLoS Comput. Biol. 13, e1005592 (2017).
  * (4) C. Richer and S. Hasiak, Town Planning Review 85, 217 (2014).
  * (5) R. B. Stinchcombe and S. L. A. de Queiroz, Phys. Rev. E 83, 061113 (2011); G. Tripathy and M. Barma, Phys. Rev. E 58, 1911 (1998); M. Bengrine, A. Benyoussef, H. Ez-Zahraouy, and F. Mhirech, Phys. Lett. A 253, 135 (1999); C. Enaud and B. Derrida, Europhys. Lett. 66, 83 (2004); L. B. Shaw, J. P. Sethna, and K. H. Lee, Phys. Rev. E 70, 021901 (2004); R. J. Harris and R. B. Stinchcombe, Phys. Rev. E 70, 016108 (2004).
  * (6) T. Banerjee and A. Basu, Phys. Rev. Research 2, 013025 (2020).
  * (7) A. Haldar, A. Basu, Phys Rev Research 2, 043073 (2020)
  * (8) R. A. Blythe and M. R. Evans, J. Phys. A: Math. Theo. 40, R333 (2007).
  * (9) A. Parmeggiani, T. Franosch, and E. Frey, Phys. Rev. E 70, 046101 (2004).
  * (10) N. Sarkar and A. Basu, Phys. Rev. E 90, 022109 (2014).
  * (11) T. Banerjee, A. K. Chandra, and A. Basu, Phys. Rev. E 92, 022121 (2015).
  * (12) B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell, 4th ed. (Garland Science, 2002); B. Li, M. Carey, and J. L. Workman, Cell 128, 707 (2007); C. Y. Lin, J. Lovén, P. B. Rahl, R. M. Paranal, C. B. Burge, J. E. Bradner, T. I. Lee, and R. A. Young, ibid. 151, 56 (2012); J. L. Workman and R. E. Kingston, Annu. Rev. Biochem. 67, 545 (1998).
  * (13) See, e.g., I. Buttinoni et al., J. Phys.: Condens. Matter 24, 284129 (2012).


On connection of the cosmic-ray coplanarity of most energetic particles with
the collider long-range near-side "ridge" effect

R.A. Mukhamedshin$\star$

Institute for Nuclear Research of Russian Academy of Sciences, Moscow 117312
Russia

*<EMAIL_ADDRESS>
21st International Symposium on Very High Energy Cosmic Ray Interactions
(ISVHECRI 2022)
Online, 23-27 May 2022 10.21468/SciPostPhysProc.?

## Abstract

Coplanarity of most energetic subcores of $\gamma$-ray–hadron families
observed in cosmic-ray experiments at $E_{0}\gtrsim 10^{16}$ eV is explained
only with a process of coplanar generation of most energetic hadrons. Long-
range near-side "ridge" effect at $\sqrt{s}=7$ TeV was found by the CMS
Collaboration in two-particle $\Delta\eta-\Delta\varphi$ correlation
functions. The FANSY 2.0 model reproduces both the hadron coplanarity and
"ridge" effect.

###### Contents

  1. 1 Introduction
  2. 2 Coplanarity simulation and results
  3. 3 Search for coplanarity signatures at the LHC
  4. 4 Conclusion

##  1 Introduction

A tendency to some coplanarity of most energetic subcores of
$\gamma$-ray–hadron families (groups of most energetic particles (MEP) in EAS
cores) initiated mainly by primary protons with energies $E_{0}\gtrsim
10^{16}$ eV has been observed in X-ray emulsion chamber (XREC) experiments
characterized by very high lateral resolution ($\lesssim 1$ mm).

Five experimental data sets on $\gamma$-ray families with energies $\sum
E_{\gamma}\geq 700$ TeV and $\sum E_{\gamma}\geq 500$ TeV have been
accumulated by the Pamir [1, 2] and Mt.Canbala [3] Collaborations,
respectively. Besides, two stratospheric events with extreme energies ($\sum
E_{\gamma}>1$ PeV) and coplanarity of MEPs, namely, the Strana [4, 5] and
JF2af2 [6]), have been detected.

The coplanarity phenomenon was also studied using the so-called Big Ionization
Calorimeter (BIC) [7] (which was a part of a complex facility for studying EAS
at the Tien Shan high-mountain (3340 m a.s.l.) station), with an area of 36 m2
and thickness of five mean free paths for proton interaction. Since the
lateral resolution of the BIC was worse than 0.25 m, main interactions in the
selected showers took place at a noticeably higher altitude as compared with
the XREC experiments. Preliminary results showed significant azimuthal effects
in large-scale hadronic subcores in EAS cores at a level of six standard
deviations above the background at $E_{0}=0.1-10$ PeV [8]. Relevant studies
are planned to be continued using a large new ionization calorimeter at the
same altitude [9].

Since the detection of these effects as a result of EAS fluctuations is
unlikely ($\lesssim 10^{-10}$)[10, 11, 12], they were originally interpreted
as a result of coplanar generation of fragmentation-region MEPs
$(x_{Lab}=E/E_{0}\gtrsim 0.01)$ in first interactions of primary protons.

Several theoretical ideas were proposed to explain this phenomenon as a result
of (a) conservation of the quark-gluon-string (QGS) angular momentum [13]; (b)
generation of specific leading systems [14, 15, 16, 17]; (c) evolution of the
dimension of space from three to two dimensions [18]. High-$p_{t}^{\,copl}$
values, forming a coplanar plane, are almost necessarily included in the first
two concepts [13, 14, 15, 16, 17], while $p_{t}$ components directed
perpendicular to this plane are traditional. The concept of two-dimensional
space [18] does not require high-$p_{t}^{\,copl}$ values.

A long-range near-side "ridge" effect in a two-charged-particle
$\Delta\eta-\Delta\varphi$ correlation function,
$R_{N}(\Delta\eta,\Delta\varphi)$, was observed by the CMS Collaboration [19]
at $|\Delta\eta|\gtrsim 3.0$, $\Delta\varphi\approx 0$. Here $\Delta\varphi$
and $\Delta\eta$ are differences in azimuthal angle $\varphi$ and
pseudorapidity $\eta$, respectively.

The analysis performed by the CMS Collaboration (and used in this paper) is as
follows [19]. First, all events including charged particles with $|\eta|<2.4$
and $p_{\textrm{t}}>0.1$ GeV/c are used to form so-called minimum-bias data
set (including all the events) and high-multiplicity one (including collisions
with charged-particle track multiplicity $N^{offline}_{trk}\geq 110$ at
$p_{\textrm{t}}>0.4$ GeV/c). The correlation function is defined as
$R_{N}(\Delta\eta,\Delta\varphi)=\ (\langle
N\rangle-1)\left(S_{N}(\Delta\eta,\Delta\varphi)/B_{N}(\Delta\eta,\Delta\varphi)-1\right).$
The signal function $S_{N}(\Delta\eta,\Delta\varphi)=({1}/{N^{\rm
sign}_{pair}})\,{d^{2}N^{\rm sign}}/{d(\Delta\eta)d(\Delta\varphi)}$ and
background function $B_{N}(\Delta\eta,\Delta\varphi)=({1}/{N^{\rm
mix}_{pair}})\,{d^{2}N^{\rm mix}}/{d(\Delta\eta)d(\Delta\varphi)}$ are
determined, respectively, by counting the number of charged-particle pairs
within each event, $N^{\rm sign}_{pair}$, and the number of particle pairs
constructed by randomly selecting two different events (with multiplicities
$N_{1}$ and $N_{2}$) and pairing every charged particle within one event with
each particle from another event. Here $N^{\rm mix}_{pair}=N_{1}N_{2}$,
$\langle N\rangle$ is the average number of charged particles per event,
$\Delta\eta$ and $\Delta\varphi$ are always taken to be positive to fill one
quadrant of the $\Delta\eta-\Delta\varphi$ histograms (other quadrants are
filled by reflection).

In Figs. 1a and 1b (Figs. 7b and 7d in Ref. [22]), experimental
$R(\Delta\eta,\Delta\varphi)$ functions for minimum-bias and high-multiplicity
events are shown. The “ridge” effect is seen in Fig. 1b.

Experiments in cosmic rays and at colliders are carried out under
fundamentally different selection criteria. Therefore, it is not possible to
directly compare their results. However, we can test whether the FANSY 2.0
model[21], based on high-$x_{\textrm{F}}$ data and designed to describe the
coplanar generation of secondary MEPs, can reproduce any of effects observed
in the central kinematic region, in particular, the "ridge" effect.

Figure 1: Charged-particle $R(\Delta\eta,\Delta\varphi)$ functions at
$|\Delta\eta|\leq 4$ by the CMS Collaboration for a) minimum-bias and b) high-
multiplicity events (Figs. 7b and 7d in Ref. [22]); c)
$R(\Delta\eta,\Delta\varphi)$ function simulated with FANSY 2.0 CPG
("moderate") for minimum bias events.

##  2 Coplanarity simulation and results

The FANSY 2.0 includes conventional (QGSJ) and coplanar-particle generation
(CPG) processes. The QGSJ and CPG versions are similar [23] in all
characteristics (except azimuth ones at $\sqrt{s}\gtrsim 2$ TeV), reproduce a
lot of LHC and low-energy data on hadron generation [23], including data on
jets and resonances. It should be emphasized that the model was designed
specifically for the phenomenological description of the phenomenon of
coplanarity of the most energetic EAS subcores, discovered in cosmic-ray
experiments, and all important model parameters were introduced irrespective
of the “ridge” effect.

To obtain more detailed conclusions within the FANSY 2.0 model, a large amount
of simulations will be carried out, and the results will require separate
publications. Until now, the main goal of this series of works was to show
that the predictions of the developed model do not contradict the main results
of experiments carried out at the LHC.

The initial concept of MEPs’ high-$p_{\textrm{t}}^{copl}$ coplanarity [11, 12]
gives a qualitative description of the cosmic-ray phenomenon, but contradicts
LHC data [23]. Only a new concept of coplanarity, assuming a reduction in
MEPs’ transverse-momentum components, directed normal to the coplanarity
plane, makes it possible to reconcile the LHC data and the MEPs’ coplanarity
[23]. In this case, the experimental and simulated $\langle
p_{\textrm{t}}\rangle$ values in all $\eta$ bins do not differ significantly.

The cross section for $pp$ CPG interactions, $\sigma_{\textrm{inel}}^{\rm
CPG}(s)$, increases from $\sim 0$ to $42$ mb as $\sqrt{s}$ increases from
$\sim 1.25$ to 7 TeV [23]. At
$|y|>|y^{\textrm{C}PG}_{\textrm{thr}}|=|y_{2}|-\Delta_{y}^{\rm CPG}$ [20]
(where $y_{2}$ is the rapidity of the second-in-energy particle and only used
to calculate a fluctuating threshold value,
$|y^{\textrm{C}PG}_{\textrm{thr}}|$), the "coplanarization" algorithm rotates
hadron transverse momentum, $\overrightarrow{\textbf{p}}_{t}$, towards the
coplanarity plane along the shortest path. The azimuthal-angular distribution
of MEPs’ $\overrightarrow{\textbf{p}}_{t}$ is sampled according to the
Gaussian distribution (relative to the coplanarity plane) with a standard
deviation depending on rapidity as $\sigma^{\rm CPG}_{\varphi}(y)=\sigma^{\rm
CPG}_{\varphi\,0}\cdot(|y_{2}/y|)^{\beta}$.

Obviously, the most important parameters are those that determine the
coplanarity degree, namely, (a) the dispersion of the deviation of the
hadron’s azimuth angle from the coplanarity plane, $\sigma^{\rm
CPG}_{\varphi\,0}$; (b) the rate of its increase with decreasing rapidity
determined by the parameter $\beta$) of the particles, and (c) the minimum
value of MEPs’ rapidities, $|y^{\textrm{C}PG}_{\textrm{thr}}|$ depending on
$\Delta_{y}^{\rm CPG}$). The effective value of
$|y^{\textrm{C}PG}_{\textrm{thr}}|$ can only be chosen phenomenologically.

In Ref. [20] and this paper, similar results were obtained using three
slightly different combinations of the $\Delta_{y}^{\rm CPG}$, $\sigma^{\rm
CPG}_{\varphi\,0}$ and $\beta$ parameters (compare Table 1 in both papers),
hereinafter referred to as "weak", "moderate" and "strong" versions. The aim
was to test the robustness of the results. The conclusion is in line with
expectations, i.e. small changes in parameter values resulted in small changes
in the phenomenon magnitude.

Table 1:  Effective parameters of "weak", "moderate", "strong" FANSY 2.0 versions. Parameter | "weak" | "moderate" | "strong"
---|---|---|---
$\langle\Delta_{y}^{\rm CPG}\rangle$ | 3.60 | 4.40 | 4.90
$\langle\sigma^{\rm CPG}_{\varphi\,0}\rangle$ | 0.10 | 0.09 | 0.06
$\langle\beta\rangle$ | 1.00 | 0.85 | 0.40
Figure 2:  The $R(\Delta\eta,\Delta\varphi)$ functions simulated for high-
multiplicity events with "weak", "moderate", and "strong" FANSY 2.0 CPG
versions.

The $R(\Delta\eta,\Delta\varphi)$ function simulated with the FANSY 2.0 QGSJ
version for high multiplicity events does not show any "ridge"-like effect as
expected (see Fig. 3 in Ref. [20]). The $R(\Delta\eta,\Delta\varphi)$ function
simulated with the use of FANSY 2.0 CPG ("moderate") for minimum bias events
is shown in Fig. 1c, and no significant "ridge"-like effect is also observed
in this case.

Figure 2 presents $R(\Delta\eta,\Delta\varphi)$ functions for high-
multiplicity events obtained with the use of "weak", "moderate" and "strong"
versions using parameters given in Table 1. The most strong "ridge" effect is
obviously reproduced by the "strong" version, i.e., an increase in the degree
of coplanarity of MEPs causes a corresponding increase in the effect in the
correlation function.

All the above-considered results were obtained with $|\Delta\eta|\leq 4$ and
$|\eta|\leq 2.4$. If the $|\Delta\eta|$ interval could be expanded, then
interesting effects would be observed for high-multiplicity events. Figure 3
shows the $R(\Delta\eta,\Delta\varphi)$ function simulated by FANSY 2.0 CPG
("moderate") at $|\Delta\eta|\leq 5$, $|\Delta\eta|\leq 6$, $|\Delta\eta|\leq
6.5$, while $|\eta|$ values of particles under consideration vary in a wider
range, namely, $|\eta|\leq 8.0$. At $|\Delta\eta|>$4, the "ridge" effect will
rapidly develops into a brighter phenomenon expressed in the appearance of two
peaks close in size ("twin peaks") at $|\Delta\varphi|\approx 0$ and
$|\Delta\varphi|\approx\pi$. So, Fig. 3 shows that an expansion of the
$|\Delta\eta|$ range can inform us on features of the MEP coplanarity region.
If the peak growth is found to stop at $|\Delta\eta|\sim 5$, and amplitude of
peaks begin to decrease with a further increase in $|\Delta\eta|$, this means
that the "ridge" effect is not associated with the process of coplanar
generation of MEPs.

Figure 3: $R(\Delta\eta,\Delta\varphi)$ distributions simulated by FANSY 2.0
CPG ("moderate") for high-multiplicity events at $|\Delta\eta|\leq 5$,
$|\Delta\eta|\leq 6$, $|\Delta\eta|\leq 6.5$ and $|\eta|\leq 8$.

Thus, there is a qualitative agreement between the experimental and simulated
$R(\Delta\eta,\Delta\varphi)$ functions in the region of the "ridge" effect,
while there are differences between the functions at
$|\Delta\varphi|\approx\pi$. It is the CPG process that creates some growth of
the simulated $R(\Delta\eta,\Delta\varphi)$ functions at
$|\Delta\varphi|\approx\pi$ and $|\Delta\eta|\gtrsim 3$.

Within the FANSY 2.0 CPG model, the coplanarity planes have the same azimuthal
orientation in the opposite hemispheres. The "twin peaks" effect at
$|\Delta\eta|\gtrsim 4$ is determined mainly by hadrons from different
hemispheres, both at $|\Delta\varphi|\sim 0$ and $|\Delta\varphi|\sim\pi$. The
coplanarization algorithm is a very primitive procedure. So, one cannot rule
out, for example, different orientations of the coplanarity planes in the
opposite hemispheres (or even more complex cases)that will entail
corresponding changes in $R(\Delta\eta,\Delta\varphi)$ functions.

In any case, the fact that FANSY 2.0 (being an ad hoc model based on
high-$x_{\textrm{F}}$ data) reproduces both the coplanarity effect and long-
range near-side "ridge" effect with a slight change in parameters, may
indicate a qualitative relationship between these phenomena.

##  3 Search for coplanarity signatures at the LHC

Simulations show that CPG signatures can appear in the range $5.25<\eta<$6.5
[23], so the CMS-CASTOR very-forward calorimeter experiment [24], focused on
this $|\eta|$ interval, seems promising. For a simplified assessment of CASTOR
capabilities, let us assume that it consists of 16 radially arranged segments
divided in half by a vertical slit. If the pseudorapidities of particles
(which do not fall into the slit) are in the range $5.25<\eta<6.5$, particles
are considered to be registered, and their energies are considered to be
"measured".

Only interactions with the total "measured" energy $\sum E_{i}>1$ TeV are
considered. Here $E_{i}$ is the energy value "measured" in the $i$-th segment
($1\leq i\leq 16$). In addition, the number of segments $N_{s}$, in each of
which the "measured" energy $E_{i}>E_{i\,min}=100$ GeV, must be equal to or
greater than two. The segment with the maximum energy release, $E_{max}$, gets
the first number, i.e. $E_{1}=E_{max}$. The remaining segments are numbered
clockwise in ascending order. Finally, let us define variables as follows:
$E_{\rm copl}=E_{1}+E_{9}$; $E_{tr}=E_{5}+E_{13}$; $\varepsilon_{\rm
copl}=E_{\rm copl}/(E_{\rm copl}+E_{tr})$. Obviously, at $\varepsilon_{\rm
copl}=1$, the degree of event coplanarity is maximum.

Figure 4 shows $d\omega/d\varepsilon_{\rm copl}$ probability distributions
obtained using the QGSJ version (line), as well as the "weak", "moderate" and
"strong" CPG versions. As expected, the "strong" CPG version predicts the
highest value of $d\omega/d\varepsilon_{\rm copl}$ at $\varepsilon_{\rm
copl}\rightarrow 1$. Undoubtedly, a detailed study of the coplanarity
phenomenon requires much more accurate simulations.

Figure 4: Probability $d\omega/d\varepsilon_{\rm copl}$ distributions
predicted by "weak", "moderate", and "strong" FANSY 2.0 CPG versions for the
CASTOR detector.

##  4 Conclusion

The long-range near-side "ridge" effect observed by the CMS Collaboration can
be a by-product of the coplanar generation of most energetic particles in
hadronic interactions.

A significant "twin peaks" effect is predicted, which appears in
$R(\Delta\eta,\Delta\varphi)$ correlation functions at $|\Delta\eta|>4$ in
high-multiplicity events

Coplanarity signatures could be observed with the CMS-CASTOR very-forward
calorimeter.

## References

  * [1] A. Borisov et al. (Pamir Collaboration), Proc. 4th ISVHECRI, Beijing (1986) 4-29
  * [2] V.V. Kopenkin et al., Phys. Rev. D 52 2766 (1995)
  * [3] L. Xue et al., Proc. 26th Int. Cosmic Ray Conf., Salt Lake City (1999) 1 127
  * [4] A.V. Apanasenko et al., Proc. 15th Int. Cosmic Ray Conf., Plovdiv (1977) 7 220
  * [5] A.K. Managadze et al., Physics of Atomic Nuclei 70 1 (2007) 184
  * [6] J.N. Capdevielle. J.Phys. G 14 (1988) 503
  * [7] T.P. Amineva, T.G. Glavach, V.S. Aseikin et al., Trudy FIAN 46 (1970) 157 (in Russian).
  * [8] V.P. Pavlyuchenko. Bull. Lebedev Phys. Inst., 1998, No. 6, 1 – 7 (in Russian).
  * [9] R.A. Mukhamedshin, T. Sadykov, A. Serikkanov et al., Appl. Sci. 2023, 13(4), 2507.
  * [10] R.A. Mukhamedshin, JHEP 05 (2005) 049
  * [11] R.A. Mukhamedshin, Nucl. Phys. B (Proc. Suppl.) 196C (2009) 98
  * [12] R.A. Mukhamedshin, Eur. Phys. J. C 60 (2009) 345
  * [13] T. Wibig, hep-ph/0003230
  * [14] I.I. Royzen. Mod. Phys. Lett. A 9 (1994) no.38 3517
  * [15] J.N. Capdevielle, Nucl. Phys. B (Proc. Suppl.) 175 - 176 (2008) 137
  * [16] T.S. Yuldashbaev et al., Nuovo Cim. 24C (2001) 569
  * [17] R.A. Mukhamedshin, Nucl. Phys. B (Proc. Suppl.) 75A (1999) 141
  * [18] L.A. Anchordoqui, De Chang Dai, H. Goldberg et al., Phys. Rev. D 83 (2011) 114046
  * [19] The CMS Collaboration, JHEP 09 (2010) 091
  * [20] R. A. Mukhamedshin. Eur. Phys. J. C (2022) 82:155
  * [21] R.A. Mukhamedshin. Eur. Phys. J. Plus (2019) 134: 584
  * [22] The CMS Collaboration, arXiv:1009.4122v1 [hep-ex] 21 Sep 2010
  * [23] R.A. Mukhamedshin. Eur. Phys. J. C (2019) 79: 441
  * [24] The CMS collaboration. JINST 16 (2021) P02010
  * [25] R.A. Mukhamedshin. Bull. Russ. Acad. Sci. Phys. 85, 402–404 (2021)