# Hall anomaly by vacancies in pinned lattice of vortices: A quantitative
analysis on the thin-film data of BSCCO

Ruonan Guo  Shanghai Center for Quantitative Life Sciences & Physics
Department, Shanghai University, Shanghai 200444, China  Shanghai Key
Laboratory of High Temperature Superconductors, Shanghai University, Shanghai
200444, China  Yong-Cong Chen<EMAIL_ADDRESS>Shanghai Center for
Quantitative Life Sciences & Physics Department, Shanghai University, Shanghai
200444, China  Ping Ao<EMAIL_ADDRESS>Shanghai Center for Quantitative
Life Sciences & Physics Department, Shanghai University, Shanghai 200444,
China

###### Abstract

Hall anomaly, as appears in the mixed-state Hall resistivity of type-II
superconductors, has had numerous theories but yet a consensus on its origin.
In this work, we conducted a quantitative analysis of the magnetotransport
measurements on BSCCO thin films by Zhao et al. [Phys. Rev. Lett. 122, 247001
(2019)] and validate a previously proposed vacancy mechanism [cf. J. Phys.
Condens. Matter. 10, L677 (1998)] with many-body vortex correlations for the
phenomenon. The model attributes the Hall anomaly to the motion of vacancies
in pinned fragments of vortex lattice. Its validity is first examined by an
exploration on the vortex states near the Kosterlitz-Thouless transition on
the vortex crystal. Comparisons are then carried out between the measured
activation energies with the calculated creation energy of the vortex-anti-
vortex pair and the vacancy energy on the flux-line lattice, with no
adjustable parameter. Our analysis elucidates the theoretical basis and
prerequisites of the vacancy model. In particular, the vacancy activation
energies are an order of magnitude smaller than that of a sole vortex line.
The proposed mechanism may provide a macro-theoretical framework for other
studies.

Keywords: High-temperature superconductor; BSCCO thin film; Hall anomaly;
Vacancy and flux-line lattice; Vortex many-body effect; Activation energy;
Kosterlitz-Thouless transition

##  I Introduction

The Hall anomaly in the mixed-state Hall resistivity of superconductors, i.e.
the sign reversal of the Hall resistivity below the superconducting transition
temperature and in the presence of flux-line vortices, was discovered as early
as in the 1950s Niessen and Staas (1965); Reed  _et al._ (1965). Much
attention has since been set on its physical origin. Prior to the discovery,
Onsager Onsager (1949) had established the framework of the vortex theory
based on fluid dynamics in 1949. A modern version of his work, the equation
for a $j^{th}$ vortex of unit length in a superconductor takes the same
Langevin equation as a charged particle in the presence of a magnetic field Ao
(1998),

$m\ddot{\mathbf{r}}_{j}=q\left(\frac{n_{s}}{2}\right)h\left(\mathbf{v}_{s,t}-\dot{\mathbf{r}}_{j}\right)\times\hat{\mathbf{z}}-\eta\dot{\mathbf{r}}_{j}+\mathbf{F}_{p}+\mathbf{f},$
(1)

where the overhead of dots stands for time derivatives. The unit length vortex
at $\mathbf{r_{j}}$ has an effective mass $m$, subject to a pinning force
$\mathbf{F}_{p}$, a fluctuating force $\mathbf{f}$, viscosity $\eta$, and
moves in a background of a superfluid with total velocity $\mathbf{v}_{s,t}$
(which includes contributions from all other vortices). For the parameters in
Eq. (1), $q=\pm 1$ indicates the vorticity (under the usual right-hand rules),
$h$ is the Planck constant, $n_{s}$ the superfluid carrier density, and
$\mathbf{\hat{z}}$ the unit vector in the direction of the magnetic field. The
term with the velocity $\dot{\mathbf{r}}_{j}$ at the right-hand side is also
known as the Magnus force. Note that CGS units are assumed throughout this
work, in line with the majority of work in the literature.

With Eq. (1), two idealized pictures can be drawn. Take e.g. $q=+1$, Fig. 1(a)
shows that in the absence of the pinning force $\mathbf{F}_{p}$ and the
frictional force $(\eta=0;\mathbf{F}_{p}=0)$, the vortex velocity
$\dot{\mathbf{r}}_{j}$ matches that of superfluid in both direction and
magnitude. In other words, for an average charge $e$ in the vortex, the Magnus
Force (due to the electric field $\mathbf{E}$ generated by moving vortex),
i.e. $e\mathbf{E}$ cancels the Lorentz force
$(e/c)\mathbf{v}_{s}\times\mathbf{H}$, leading to the same Hall effect as in a
normal metal. One has no reason to expect a sign reversal on the Hall
resistance $R_{xy}$ below the superconducting transition temperature $T_{c}$.
Fig. 1(b) presents an alternative scenario of an extreme situation: The vortex
is firmly trapped by a strong pinning force $\mathbf{F}_{p}$ such that there
is no Magnus force. Moreover, the pinning force is opposite to the Lorentz
force and there will be no change of sign on the Hall resistance $R_{xy}$
either. These scenarios raise an apparent paradox between conceptual reasoning
and experimental observation.

|
---|---
(a) | (b)
Figure 1: Analysis of vortex dynamics in two ideal situations. In the picture
$\mathbf{E}$ is the electric field generated by moving vortex and $\mathbf{H}$
is the applied magnetic field. (a) With no pinning force $\mathbf{F}_{p}=0$
and no frictional force $(\eta=0;\mathbf{f}=0)$, the vortex moves at the same
velocity as the superfluid $\dot{\mathbf{r}}_{j}=\mathbf{v}_{s}$. (b) The
vortex is firmly trapped by strong pinning force $\mathbf{F}_{p}$ such that
$\dot{\mathbf{r}}_{j}=0$.

To address the abnormal sign reversal, several groups Nozières and Vinen
(1966); Hagen  _et al._ (1990, 1993); Wang and Ting (1991) have proposed a
phenomenological model, which adds an adjustable parameter $\alpha$ directly
in front of the Magnus force. This modifies the vortex dynamic equation to

$m\ddot{\mathbf{r}}_{j}=q\left(\frac{n_{s}}{2}\right)h(\mathbf{v}_{s,t}-\alpha\dot{\mathbf{r}}_{j})\times\hat{\mathbf{z}}-\eta\dot{\mathbf{r}}_{j}+\mathbf{F}_{p}+\mathbf{f}.$
(2)

In a steady-state where the vortex is subject to the friction force only, the
Hall resistivity $\rho_{xy}(0)$ and the longitudinal resistivity
$\rho_{xx}(0)$ can be obtained respectively

$\displaystyle\rho_{xx}(0)$ $\displaystyle=$ $\displaystyle-\frac{\eta
qhH}{ec\left(\alpha^{2}q^{2}{n_{s}}^{2}h^{2}+4\eta^{2}\right)},$ (3)
$\displaystyle\rho_{xy}(0)$ $\displaystyle=$ $\displaystyle\frac{\alpha
n_{s}q^{2}h^{2}H}{2ec\left(\alpha^{2}q^{2}{n_{s}}^{2}h^{2}+4\eta^{2}\right)}.$
(4)

This allows the sign reversal of the Hall resistivity $\rho_{xy}(0)$ when
$\alpha<0$, though it is unclear how $\alpha$ can be so adjusted. Furthermore,
there should also be resistivity coming from the normal component of electrons
at finite temperatures. The latter follows a parallel connection to the
superconducting circuit so that we have

$\displaystyle\rho_{xx}(T)$ $\displaystyle=$
$\displaystyle-\frac{\eta\eta^{\prime}qhH}{e\left[c\eta^{\prime}\alpha^{2}q^{2}n_{s}^{2}(T)h^{2}-eh\eta
qH(n_{s}(0)-n_{s}(T))+4c\eta^{\prime}\eta^{2}\right]},$ (5)
$\displaystyle\rho_{xy}(T)$ $\displaystyle=$ $\displaystyle\frac{\alpha
Hn_{s}(T)q^{2}h^{2}}{ce\left[2\alpha^{2}q^{2}n_{s}^{2}(T)h^{2}+\alpha
n_{s}(T)q^{2}h^{2}(n_{s}(0)-n_{s}(T))+8\eta^{2}\right]}.$ (6)

Here $n_{s}(T)$ is the density of superfluid electron at temperature $T$ and
the density of normal electron can be written as $n_{s}(0)-n_{s}(T)$. The
change in the sign of $\alpha$ has to overcome the normal-state electron
contribution to achieve the sign reversal on $\rho_{xy}(T)$. The details are
presented in Part A of the Supplemental Material (SM) to this worksup .

The adjustable parameter $\alpha$ itself has not had a consistent origin in
the approach. Hall and Vinen Hall and Vinen (1956a, b) in 1956 assumed a big
drag force of the same as the Magnus force. Nozieres and Vinen Nozières and
Vinen (1966), Hagen et al. Hagen  _et al._ (1990, 1993) further developed a
theoretical explanation for the existence of such a force. In contrast, Wang
and Ting Wang and Ting (1991) argued that the parameter $\alpha$ should be
zero. They based their theory on the well-known normal-core model for flux
lines in extreme type-II superconductors and correctly took into account the
backflow current due to pinning forces, which concluded in no Magnus force. As
a result, there can be a small $\alpha$ with either sign, leading to a signing
inversion of the Hall coefficient under certain circumstances. However, Ao and
Thouless Ao and Thouless (1993) in 1993 found that $\alpha\equiv 1$ at zero
temperature. And three years later, Thouless, Ao and Niu Thouless  _et al._
(1996) extended the result to low but finite temperature. Additionally, from
the Ginzburg-Landau theory, another phenomenological model was proposed by Xu
et al. Xu  _et al._ (2002) in 2002 for the Hall anomaly in High-temperature
superconductors. And in 2004, Ghenim et al. Ghenim  _et al._ (2004) also
suggested a model for conventional superconductors.

Despite that the Hall anomaly attracted numerous theoretical studies, a lack
of advancement in experimental techniques has hindered a consensus on its
origin. But recently, there has been a steady improvement in the situation
over the last two decades. In 1996, Zhu et al. Zhu  _et al._ (1997) designed a
mechanical experiment to directly measure the total transverse force on moving
vortices in a type II superconductor for the first time, and their result is
consistent with the Ao-Thouless theory Ao and Thouless (1993). In 2001, Zhu
and Nyeanchi Zhu and Nyeanchi (2001) demonstrated that certain features of the
Kosterlitz-Thouless transition of a vortex lattice are preserved near the
superconducting transition temperature. In 2019, Chen’s team Yu  _et al._
(2019) developed a fabrication process which can produce intrinsic monolayer
crystals of BSCCO. In 2021, Richter et al. Richter  _et al._ (2021) measured
the resistivity, Hall effect, and anisotropic superconducting coherence
lengths in HBCCO thin films with morphological variations. Excellent
measurement of Hall effects in an atomically thin high-temperature
superconductor by a Harvard group Zhao  _et al._ (2019) in 2019 drastically
extended the region which displays the Hall sign reversal.

In this work, we present an in-depth analysis of the activation energies
deduced from the experimental data in Zhao  _et al._ (2019). It is then
compared to the predicted activation energy of independent vortices and the
energy of vortex many-body correlation under flux-line lattices. In the next
section, we examine whether the abnormal Hall effect on the BSCCO film meets
the pre-requisites of the vacancy model by analyzing the state of the vortices
near the KT transition temperature. In section III, we first review the
fundamentals of our methodology, namely the pinning and dynamics of vacancies
in a vortex lattice. The core concept presented follows throughout the entire
section. The experimental data are extracted and compared with the theoretical
calculations of activation energies of carriers in the BSCCO film, under
varying magnetic fields. The excellent agreement between them elucidates the
conformation of the Hall anomaly to the vacancy model. Some concluding remarks
and possible connections to other works and future directions are discussed in
the final section.

##  II Vortex states around the Kosterlitz-Thouless transition

To validate that the Hall anomaly complies with the pre-requisites of the
vacancy model, it is crucial to clarify the states of vortex crystal near the
Kosterlitz-Thouless (KT) transition. In this section, we will first review
some basics of the vortex lattice state near the KT transition which melts the
crystal. Then we discuss the nature of the KT transition from the
thermodynamic perspective and estimate the theoretical value of the transition
temperature $T_{KT}$ of the BSCCO thin film. Finally, the question of whether
the concept and presence of vacancies still apply above $T_{KT}$ is addressed.

###  II.1 KT transition of molten crystals

The melting transition of most solid materials at the present has not been
well understood as there is a lack of theories explaining the transition on
the microscopic scale. Furthermore, the mechanism of melting depends on the
interaction details between the constituents forming a crystal lattice. In
particular various defects which reduce the translational order of the crystal
play a major role. It should be noted that in two dimensions, only edge
dislocations and not the screw ones are important in the melting transition.
The core energy of the dislocations upon which these effects can form must be
sufficiently low for their spontaneous appearance. The specific analysis of
the dislocation formation energy in a 2D BSCCO film will be presented in the
subsequent section.

These dislocations in question are topological point defects, which implies
that a single one cannot be created isolated by an affine transformation
without cutting the hexagonal crystal up to infinity (i.e. up to its borders).
Hence they must be created in pairs with antiparallel Burgers vectors. When a
large number of dislocations were e.g. thermally excited, the discrete
translational order of the crystal would be destroyed. Simultaneously, the
shear modulus and Young’s modulus would disappear, signaling the starting of
the molten transition from a solid to a fluid phase.

However, it is possible that the orientational order is not yet fully
destroyed (as indicated by lattice lines in one direction) and one finds -
very similar to liquid crystals - a fluid phase with typically a six-folded
direction field. This so-called hexatic phase still has an orientational
stiffness. Such an anisotropic fluid phase can appear when the dislocations
dissociate into isolated five-folded and seven-folded disclinations Gasser
_et al._ (2010). This two-step melting phenomenon is described within the so-
called Kosterlitz-Thouless-Halperin-Nelson-Young-theory (KTHNY theory), based
on two separate transitions of the Kosterlitz-Thouless-type. In 2010, Urs
Gasser et al. Gasser  _et al._ (2010) presented the first conclusive evidence
for the existence of the hexatic phase and two continuous phase transitions in
2D melts in a colloidal model system with repulsive magnetic dipole-dipole
interaction.

###  II.2 Thermodynamics of KT transition

The topological phase transition in a 2D superfluid was predicted by
Berezinskii Berezinskii (1970) and Kosterlitz and Thouless Kosterlitz and
Thouless (1973) and elaborated by Halperin and Nelson Nelson and Halperin
(1979); Nelson (1978). A simple thermodynamic argument allows us to understand
the intrinsic quality of the KT transition Blatter  _et al._ (1994). The
Helmholtz free energy is given by the difference between the energy $E$ and
the entropy of a dislocation $S$ multiplied by the temperature $T$,

$F=E-TS.$ (7)

The energy $E$ is given by Eq. (15), which is contributed by a dislocation
pair over a large distance. For convenience, we rewrite it as

$E=\frac{d\varepsilon_{0}}{2\sqrt{3}\pi}\ln\left(\frac{L}{a}\right).$ (8)

The entropy can be estimated from the number of places the dislocation can be
positioned, namely on each of the $\sim L^{2}$ plaquette of the lattice, i.e.,

$S=k_{B}\ln\left(\frac{L^{2}}{a^{2}}\right).$ (9)

Accordingly, the free energy is given by

$F=\left[d\frac{\varepsilon_{0}}{2\sqrt{3}\pi}-2k_{B}T\right]\ln\left(\frac{L}{a}\right).$
(10)

Evidently, there exists a temperature above which a vast number of
dislocations are preferred. Such a transition is of topological nature and is
referred to as the KT transition. The transition temperature $T_{KT}$ could be
expressed as

$T_{KT}=\frac{1}{4\sqrt{3}\pi
k_{B}}\left(\frac{\Phi_{0}}{4\pi\lambda}\right)^{2}d\cong 57.3K.$ (11)

Ideally at $T<T_{KT}$, thermally excited dislocations form pairs of close
compact, namely a dislocation pair. But they are spontaneously separated when
$T>T_{KT}$. The KT transition temperature $T_{KT}$ measured by a Harvard group
is 60 K Zhao  _et al._ (2019), which validates the effective thickness and the
London penetration depth chosen below for our analysis.

###  II.3 The presence of vacancies above $T_{KT}$

Let us for the moment assume a vortex solid-liquid phase transition near (the
first) $T_{KT}$, around 60 K. Then the Arrhenius behavior of longitudinal
resistance places the Hall sign reversal within the thermally activated flux
flow regime above the vortex lattice melting temperature Ao (2020). An urgent
question is whether the Hall anomaly mainly appears in the vortex-liquid
regime. Scilicet fades away as the vortex liquid freezes into a solid-state
crystal.

However, there is no evidence for such a solid-liquid phase transition in the
BSCCO film. As the transition would be a first-order phase transition in which
observation of latent heat was to be expected, the resistance of the BSCCO
film should measure an abrupt change at the transition. More crucially
Kosterlitz-Thouless transition is classified as a topological phase
transition, the third type of phase transition. Experientially, all curves
intercepted by the isotherm of $T_{KT}=60$ K are smooth, with no sign of jump
around the node.

According to the generic nature of phase transitions in two-dimensional
materials Gammel  _et al._ (1988a); Safar  _et al._ (1993); Gammel  _et al._
(1988b), we can reasonably deduct the phase diagram as shown in Fig. 2. In the
illustration, $T_{KT}$ is the lowest with a second melting temperature $T_{m}$
for the solid-liquid phase transition below the superconducting transition
temperature $T_{c}$. For $T_{KT}<T<T_{m}$, there should exist local fragments
of vortex lattice, and in each of them long-range order out to be preserved.
This sets the prerequisites for the existence of vacancies and the
applicability of the vacancy model proposed by Ao et. al. Ao (1998).

Figure 2: Schematic phase diagram of the superconducting gap vs. temperature.
The solid curve represents the relationship between the order parameter and
temperature of the type-II superconductor under a constant external magnetic
field. When $T_{KT}<T<T_{m}$, the vortex lattice possesses quasi-long order.
Three insets show the states of the vortex lattice at diverse temperature
intervals.

##  III Hall anomaly by vacancies of the vortex lattice

In this section, we intend to quantitatively demonstrate that the Hall anomaly
is in full concert with the theoretical basis of the vacancy hypothesis.
Namely, it is due to the movement of vacancies, a direct result of many-body
vortex interaction and the origin of the Hall anomaly. First, the dynamics of
vacancies in a vortex lattice are discussed in detail. Then some results are
applied to fit the experimental data by the Harvard group Zhao  _et al._
(2019) with no adjustable parameters. The vacancy activation/formation energy
under a diverse set of magnetic fields is found to be the same order of
magnitude as theoretical predictions. In particular, both theoretical and
experimental results show that the energy of an independent vortex line or a
vortex-anti-vortex pair is an order of magnitude larger than the vacancy
energy.

###  III.1 Properties of vacancies in a pinned vortex lattice

In 1993, Thouless and Ao Ao and Thouless (1993) proved that the existence of
Magnus force is a universal essence of superconductor vortex line. In 1998, Ao
Ao (1998) further established a set of processes leading to the Hall effect as
a result of moving vacancies in a background of pinned vortex lattice(s). It
manifests in particular that neither a modification on the vortex equation nor
an assumption of two types of carriers is necessary. One only needs to study
the vortex dynamics equation proposed by Niu, Ao and Thouless Niu  _et al._
(1994) in 1994. To recite the theory, we turn to a crucial quantitative result
regarding the motion of vacancies in a pinned vortex lattice used in the
subsequent analysis, namely the vacancy formation energy in a flux line
lattice.

First we look at the energy scale of dislocations in the lattice. In a type-II
superconductor with mixed states, the many-body correlation between the
vortices and the pinning forces usually cannot be ignored. On a two-
dimensional flux line lattice (FLL) of a thin film of thickness $d$,
spontaneous nucleation of a pair of edge dislocations costs an energy Brandt
(1995); Blatter  _et al._ (1994)

$e_{d}(r)\cong
da^{2}\left(\frac{c_{66}}{4\pi}\right)\ln\left(\frac{r}{a}\right),$ (12)

where $a=(2\Phi_{0}/\sqrt{3}B)$ is the lattice spacing, $r\gg a$ is the
distance between two dislocations, and $c_{66}$ is the shear modulus of the
FLL, cf. Kosterlitz (2016). For uniform distortions the elastic moduli Brandt
(1995) of a triangular FLL reads

$c_{66}\approx\left(\frac{B\phi_{0}}{16\pi\lambda^{2}\mu_{0}}\right)\left(1-\frac{1}{2\kappa^{2}}\right)\left(1-b^{2}\right)\left(1-0.58b+0.29b^{2}\right).$
(13)

Here $\lambda^{2}=(m^{\ast}c^{2}/8\pi\rho_{s}e^{2})$ is the London penetration
depth (with the effective mass $m^{\ast}$ and superfluid density $\rho_{s}$ of
the underlying carriers of charge $2e$), $\Phi_{0}=(hc/2\left|e\right|)$ is
the flux quantum of a Cooper pair, $\kappa$ is the GL parameter and
$b=H/H_{c2}$ (with the applied magnetic field strength $H$ the upper critical
field $H_{c2}$ of the superconductor). The 2-d FLL is then a uniaxial elastic
medium similar to that of an isotropic and bulk superconductor. In the limit
of large $\kappa$ and relatively small magnetic field, we have

$c_{66}\approx\left(\frac{B\Phi_{0}}{16\pi\lambda^{2}\mu_{0}}\right).$ (14)

Using Eqs. (12)-(14) we can re-express the energy for a dislocation pair as

$e_{d}(r)=\frac{d\,\varepsilon_{0}}{2\sqrt{3}\pi}\ln\left(\frac{r}{a}\right).$
(15)

Here the major variable
$\varepsilon_{0}\equiv\left(\Phi_{0}/4\pi\lambda\right)^{2}$ defines vortex
creation energy per unit-length, $(d\,\varepsilon_{0})$ then sets the scale
for both the vortex-vortex and strong pinning interactionsAo (1998). The
energy scale $(\varepsilon_{0}/2\sqrt{3}\pi)$ for the dislocation pair is
about ten times smaller than $\varepsilon_{0}$, it is energetically favourable
to have close-distance dislocation pairs in the lattice as carriers of
transverse current.

Thus at temperature $k_{B}T\ll(d\,\varepsilon_{0})$ we can ignore the
contribution to the current from the vortices hopping out of pinning as well
as thermal activation of vortex-antivortex pairs. This is because the entire
vortex lattice, formed via inter-vortex interactions should be effectively
pinned down. Instead, we should look into the vacancies and interstitials
which can be viewed as the smallest dislocation pairs Friedel (2013). The
vacancy formation energy $\varepsilon_{v}$ per unit length can be estimated Ao
(1998) by setting $r\sim 2a$ in $e_{d}(r)$ together with an extra factor

$\varepsilon_{v}\sim\frac{\varepsilon_{0}\ln
2}{2\sqrt{3}\pi}\left(\frac{a}{\xi}\right).$ (16)

Note that in Ao’s initial valuations, the effect of the magnetic field on the
energy barrier in the activation process has been ignored. It is anticipated
that such an effect should relate to the ratio of the lattice constant as a
function of a magnetic field to the coherence length $\xi$ in some way. In
this work, we attempt to take the effect into account with the simplest
multiplication factor $(a/\xi)$.

###  III.2 Vacancy activation energy under magnetic field

We can now analyze the experimental data from the SM of Zhao  _et al._ (2019)
with the Arrhenius empirical formula whose validity in solid-state kinetics
has been illustrated in e.g. Mossel  _et al._ (2000). Taking that the dominant
source of resistance $R$ is the thermal activation of the dissipative
vacancies in the superconductor film, we have

$\;\;R=A\exp\left(E_{a}/k_{B}T\right)\;\Rightarrow\;\\\
\log_{10}R(T)=\log_{10}A-(\log_{10}e)[E_{a}(T)/k_{B}T],\;\;\;\;$ (17)

where $A$ is a pre-factor for the exponential term, and $T$ is the temperature
with $k_{B}$ the usual Boltzmann constant. Both $A$ and the activation energy
$E_{a}$ can be themselves temperature-dependent. Now in the GL analysis, the
penetration depth $\lambda(T)$ is given by

$\lambda(T)=\lambda(0)/[1-(T/T_{c})]^{1/2}.$ (18)

The length scale is greatly affected by temperature, while its dependence on
the magnetic field is negligible Pippard (1950). Substituting $\lambda(T)$
into the vacancy formation energy Eq. (16), we get

$\displaystyle E_{a}(T)$ $\displaystyle=$ $\displaystyle\frac{d\ln
2}{2\sqrt{3}\pi}\left(\frac{a}{\xi}\right)\left(\frac{\Phi_{0}}{4\pi\lambda(T)}\right)^{2}.$
(19)

Observe that the fitting requires the magnetic field $B$ to be in the middle
of the lower critical magnetic field $H_{c1}$ and the upper critical magnetic
field $H_{c2}$. Although for thicker films the thickness in Eq. (17) depends
on various parameters such as magnetic field, pinning, temperature and
anisotropy. The case here is relatively simple. We take $d$ to be $50\%$ of
the physical thickness of the thin film, which is based on the observation
that the ratio roughly corresponds to the “superconductive” portion of the
material (along the $c$-axis).

To proceed further, we set $\lambda(0)=2690$ ÅProzorov  _et al._ (2000) for
BSCCO-2212, which can be compared to other values from i) reversible
magnetization, $\lambda\approx 2100$ ÅKogan  _et al._ (1993); ii) uSR,
$\lambda\approx 1800$ ÅLee  _et al._ (1993); and iii) lower critical field
measurements, $\lambda\approx 2700$ ÅNideröst  _et al._ (1998). The
superconducting transition temperature $T_{c}=89$ K and $\log_{10}A\approx 2$
are read off from the experimental figure Zhao  _et al._ (2019) for the
BSCCO-2212 film. Other experimental parameters include the effective film
thickness $=50\%$ of a 3-layer film $d=1.5UC=2\times 1.5\times 15.35\times
10^{-8}$ cm (The half height a unit cell in Bi-2212 is 15.35 Å Dou  _et al._
(2018)), the GL parameter $\kappa=86$ Stintzing and Zwerger (1997), the
coherence length $\xi=\lambda(T)/\kappa$, together with the flux quantum of
Cooper pair $\Phi_{0}=hc/2\left|e\right|=2.07\times 10^{-7}$ G$\cdot$ cm2 and
the Boltzmann constant $k_{B}=1.38\times 10^{-16}$ erg/K. All quantities are
in CGS units. We then apply Eq. (17) to Fig 7 of SM in Zhao  _et al._ (2019).
The result that requires no extra fitting parameter is shown in Fig. 3. The
theoretical values of the average energy of vacancy formation under diverse
magnetic fields are presented in Table 1. Both display excellent agreement
between theory and experiment, with no adjustable parameters.

Figure 3: An Arrhenius plot of the resistance vs. temperature with the solid
lines and symbols from the experimental data Zhao  _et al._ (2019) and the
open symbols and dash lines from Eq. (17). In these dashed lines, there is an
external magnetic field, the activation energy is governed by the vacancy
motion, which is determined by Eq. (19). In the absence of an external
magnetic field, the activation energy is caused by the independent vortex
motion, which depends on Eq. (21).

###  III.3 Vortex-anti-vortex pairs in the absence of a magnetic field

In the zero-field case, the activation $E_{a}(T)$ is in the order of an
independent vortex energy $\varepsilon_{in}$, which reads Blatter  _et al._
(1994)

$\varepsilon_{in}=d\left(\frac{\Phi_{0}}{4\pi\lambda(T)}\right)^{2}\ln\kappa.$
(20)

For a vortex-anti-vortex pair, the distant current contribution cancels out,
hence the item $\ln\kappa$ should drop out. Therefore, for an impact pair, the
creation energy can be taken as

$E_{a}(T)=2d\,\varepsilon_{0}.$ (21)

Substituting the above formula along with parameters taken for the other
estimates to Eq. (17), we can compare the result to Fig 7 of SM in Zhao  _et
al._ (2019). The outcome with no further variable again is shown in Fig. 3,
with the calculated average formation energy of vortex-anti-vortex pair in
Table 1. Both the field-dependent and zero-field predictions are in excellent
agreement with experimental measurements. They are further discussed below.

B/T | 0 | 0.5 | 1 | 3 | 5
---|---|---|---|---|---
E/K(Experimental value) | 1273 | 368.3 | 293.0 | 218.5 | 188.7
E/K(Theoretical value) | 2414 | 412.5 | 330.8 | 205.6 | 172.9
Table 1: The energy of the 3UC BSCCO film on various magnetic fields. The
experimental value is the activation energy of the film, which is gotten by
Eq. (17). The theoretical values of the film at $B=0$ T and $B\neq 0$ T are
determined by Eq. (21) and Eq. (19), respectively.

###  III.4 A comparison of vacancy and vortex-pair activation energies

Eq. (17) states that the slopes of the curves in Fig 7 of SM in Zhao  _et al._
(2019) match to $E_{a}/k_{B}$. Their average values are tabulated in Table 1.
Observe that the activation energy at zero field $B=0$ is an order of
magnitude larger than that at $B>0$, which represents one of the Primary
outcomes of the present paper. For $B=0$ there are no vacancies and the
activation energy is at the same order of magnitude as the independent vortex
energy.

On the other hand, the prime contribution to the activation energy comes from
vacancies for $B\neq 0$, the influence of the independent vortex is very small
and can be ignored beyond the likely melting temperature $T_{m}$, cf. Fig. 2.
Keeping the same temperature, the films under a magnetic field have much
larger resistance than they have under a zero field, cf. Fig 7 of SM in Zhao
_et al._ (2019). The corresponding activation energies at $B\neq 0$ T are all
in the same order of magnitude as the vacancy formation energies in Table 1.

By quantitative comparisons between their experimental and the theoretical
values, we conclude that the Hall anomaly is a consequence of many-body vortex
interactions, which further conforms to the theoretical analysis of the
vacancy model. In particular, according to the above analysis and Table 1, the
vortex-anti-vortex energy is an order of magnitude larger than the vacancy
energy. The latter constitutes one of our primary arguments.

##  IV Discussions

The current microscopic understandings of the Hall anomaly in the 3UC BSCCO
crystal may be classified into two opposite physical models, which are based
on disparate theoretical approaches and give contradictory interpretations.
The Harvard group Ao (1998) fabricated a few unit-cell (UC) thick BSCCO and
detected the reversal depicted by vortex dynamics in this system. It laid the
foundation to differentiate diverse models experimentally. The group Zhao  _et
al._ (2019) has made a theoretical explanation based on the individual vortex
dynamic model Feigel’Man  _et al._ (1995). Inversely, Ao proposed a multi-body
correlation model Ao (1998) in which vacancy movements on pinned vortex
lattice are the primary mechanism for the phenomenon.

In this paper we elaborate, via quantitative analysis, on the three main
aspects of the Ao’s model. Firstly, the precondition and theoretical basis of
the Hall anomaly warrant the vacancy model in a pinned vortex lattice or
fragments of the lattice. Secondly, both theory and experiment reveal that the
energy of the vortex-anti-vortex pair is about an order of magnitude higher
than the vacancy energy. Last but not least, distinct theoretical models of
the Hall anomaly can be quantitatively distinguished via experiment.
Particularly, our theoretical predictions have no adjustable parameters. As
discussed in Part E of SM sup , several predictions of the vacancy model
explain the Hall anomaly of BSCCO in the mixed state better than some other
models.

We also look forward to providing a macro-theoretical framework for related
topics in future works. For example, Yang et al. Yang  _et al._ (2022)
observed linear-in-temperature and linear-in-magnetic field resistance on
nano-patterned YBCO film arrays over extended temperature and magnetic field.
Meanwhile, the low-field magneto-resistance oscillates with a period dictated
by the superconducting flux quantum. It is possible that the unexpected
signatures may be explained by a pinned vortex lattice model in a similar
consideration. In particular, the plasticity of a vortex lattice pinned by the
periodic nano holes can lead to diminishing of super concurrent as a result of
free-energy minimisation. Further exploration will be presented elsewhere.

###### Acknowledgements.

One of us (NG) is grateful to Prof. Chuanbing Cai for useful discussions on
the research. This work was supported in part by the National Natural Science
Foundation of China No. 16Z103060007 (PA).

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