Robustness of the Hedgehog Skyrmion

N. S. Manton111email<EMAIL_ADDRESS>ORCID: 0000-0002-2938-156X

Department of Applied Mathematics and Theoretical Physics,
University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, U.K.

###### Abstract

We investigate the radial profile function of the hedgehog Skyrmion with unit
baryon number in generic EFTs (effective field theories) of pions. The
analysis assumes chiral symmetry and ignores the pion mass term. The Skyrmion
is always smooth, because it has no point source at the origin, and terms in
the EFT with higher numbers of pion derivatives do not result in uncontrolled
large corrections or singularities there. The profile varies in quite a
limited way as the terms in the EFT change, and a universal profile function
is proposed.

Keywords: Hedgehog Skyrmion, Radial profile, Effective field theory

##  1 Introduction

In the 1960s, Skyrme introduced a nonlinear field theory of pions whose static
classical solutions are topological solitons – Skyrmions [1]. Skyrmions are
reviewed in the books [2, 3, 4]. A Skyrmion’s topological charge is identified
with its baryon number (atomic mass number, or nucleon number) $B$, and the
fundamental solution has $B=1$. This $B=1$ solution’s fields are spherically
symmetric, and because the triplet of pion fields points radially outwards, it
is often referred to as a “hedgehog”. A hedgehog configuration is completely
specified by a radial profile function $f(r)$ satisfying the boundary
conditions $f(0)=\pi$ and $f(\infty)=0$. The static field equations reduce to
a second-order ODE for $f$, which has a unique solution satisfying these
boundary conditions. The hedgehog Skyrmion, rather than a configuration with
less symmetry, is almost certainly the minimal-energy static solution in the
$B=1$ sector of the field theory. When its rotational degrees of freedom are
quantized, one finds low-energy states representing spin-half protons and
neutrons [5].

Skyrme’s original Lagrangian has three terms, a term quadratic in first
derivatives of the field (the Dirichlet term), a term quartic in first
derivatives (the Skyrme term), and a pion mass term with no derivatives. When
the pion mass term is dropped, the theory has full $SU(2)_{\rm L}\times
SU(2)_{\rm R}$ chiral symmetry, although this is spontaneously broken to
$SO(3)$ isospin symmetry by the vacuum field configuration. It is easier to
analyse the Skyrmion’s profile $f(r)$ for massless pions, where the large-$r$
tail has power rather than exponential decay, and this is what we do
throughout this paper.

The balance between terms quartic and quadratic in derivatives gives the
Skyrmion a finite size in three spatial dimensions. Furthermore, the
particular term quartic in derivatives chosen by Skyrme has the property that
in the Lagrangian, time-derivatives only appear quadratically. Subsequently,
it was realised that there is one further term, sextic in derivatives, that is
compatible with chiral symmetry and is still only quadratic in time-
derivatives [6]. This term is often included in Skyrme-type Lagrangians. It is
called the BPS-Skyrme term.

In parallel to studies of Skyrmions, chiral effective field theories (EFTs) of
pions and nucleons have been developed using perturbation theory [7, 8, 9,
10]. For a recent review and comprehensive bibliography, see [11]. In the
perturbative picture, nucleons have their own fundamental fields which act as
sources for the pion fields, and the configurations have no topological
stability. The terms in the EFT Lagrangian are not constrained to have only
two time-derivatives, and several more terms are included. Their coefficients
are fitted to data extracted (mainly) from nucleon-nucleon scattering. As the
(mass) dimensions of terms increase with the number of derivatives, the
coefficients incorporate increasing powers of the inverse of a mass cut-off.
Terms beyond a certain number of derivatives can therefore be neglected
provided one is primarily interested in physics at momenta and energies below
this cut-off scale.

From the general EFT perspective, the analysis of Skyrme’s original field
equations has been criticised on the grounds that many terms that could be
present in the Lagrangian have been ignored. Moreover, the balance between the
quartic and quadratic terms, contributing to the Skyrmion’s energy, suggests
that it is wrong to neglect terms with higher powers of derivatives, and that
these terms could substantially change the Skyrmion’s core structure. This
criticism relies on the idea that the spatial derivatives of the fields are
large in the core region, close to the nucleon sources at $r=0$. However, the
$B=1$ Skyrmion in Skyrme’s original theory is quite smooth at $r=0$ because
there is no point source. So a more reasonable expectation is that variations
in the coefficients of the EFT will make small changes to the Skyrmion
profile. This is what we aim to show here.

In detail, we will consider the most general terms that arise in a pion EFT
with chiral symmetry, and their combination into the energy functional for a
static, hedgehog $B=1$ Skyrmion. We will allow arbitrary powers of the first
radial derivative of the profile, $f^{\prime}(r)$, but exclude
$f^{\prime\prime}(r)$ and higher derivatives of $f$ from the energy density.
To avoid excessive algebra, we will mostly ignore terms that are more than
sextic in $f^{\prime}$, but show that it would be straightforward to include
them. Some of the terms we retain, quartic or sextic in $f^{\prime}$, occur in
perturbative EFT, but have previously been excluded from most analyses of
Skyrmions because they are partnered by terms in the full Lagrangian that are
quartic or sextic in time-derivatives. We do not discuss the potentially
complicated time-evolution and quantization that is associated with such time-
derivatives. Their effects are probably controllable if the time-derivatives
are small.

We will solve the second-order nonlinear ODE for $f$ up to $O(r^{3})$ for
small $r$, and up to $O\left(\frac{1}{r^{6}}\right)$ for large $r$. These
approximate solutions can be matched at some specific, adjustable radius
$r=R$. The result is a continuous profile with continuous slope. For a special
choice of the coefficient of the $O(r^{3})$ term for small $r$, generally
different from the coefficient arising in a particular EFT, $f$ has a
continuous second derivative, i.e. continuous curvature. Aside from the
spatial scale $R$, this is a “universal” profile function that can be used as
an approximate ansatz for a hedgehog Skyrmion in just about any EFT. Local
analysis does not fix $R$; it needs to be determined by minimising the total
energy in an EFT of interest. This requires some numerical integration, which
we perform in the simplest case.

In our analysis, we fix the Dirichlet and Skyrme terms to have standard,
conventional coefficients. However, physically, these coefficients should be
calibrated to fit the size and mass of a nucleon. So, although the scale $R$
varies between different Skyrme-type theories, there is freedom to calibrate
all Skyrmions to have the same leading large-$r$ behaviour. Only the detailed
shape of the core profile depends on the variant, and this shape does not vary
much. In this sense, the hedgehog Skyrmion is robust. Further numerical work
would be needed to verify this claim more completely.

Our whole approach has a lot in common with that of Marleau in ref.[12].
Marleau also solved for the hedgehog profile $f(r)$ at small $r$ and large
$r$, generating recursion relations for the coefficients of the powers of $r$
and $\frac{1}{r}$ respectively, to all orders, and wrote down a very general
profile that matches both asymptotic expansions. However, the resulting
profile still depends on three undetermined parameters, a slope at $r=0$, a
leading coefficient at large $r$, and a scale $R$. These three parameters need
to be calculated numerically, by energy minimisation. This contrasts with our
analysis, where matching is used to fix two of these parameters, at the price
of $f$ not being smooth to all orders. An important further difference is that
Marleau’s energy functional is restricted to being quadratic in $f^{\prime}$.
By contrast, we are allowing chirally-invariant terms including higher powers
of $f^{\prime}$, that arise from more general EFTs. Our classification of
chirally-invariant energy densities is essentially the same as that of
Gudnason and Nitta [13], but they did not simultaneously consider the $B=1$
hedgehog profile.

##  2 Energy Functionals

In this section, we exploit the formalism of the strain tensor of a Skyrme
field to construct general Skyrme energy functionals for a static $B=1$
hedgehog configuration with profile $f(r)$ [14].

Recall that a (static) Skyrme field $U({\bf x})$ is an $SU(2)$-valued field
that can be expressed in terms of triplet pion and singlet sigma fields as

$U({\bf x})=\sigma({{\bf x}}){\boldsymbol{1}}_{2}+i{\boldsymbol{\pi}}({{\bf
x}})\cdot{\boldsymbol{\tau}}\,,$ (1)

with ${\boldsymbol{1}}_{2}$ the unit $2\times 2$ matrix and
${\boldsymbol{\tau}}$ the Pauli matrices. The constraint

$\sigma^{2}+{\boldsymbol{\pi}}\cdot{\boldsymbol{\pi}}=1$ (2)

also needs to be satisfied (since $SU(2)$, as a manifold, is a 3-sphere), so
only the pion fields ${\boldsymbol{\pi}}$ are independent. The current
$R_{i}=\partial_{i}UU^{-1}$ plays the role of the Skyrme field gradient, and a
key quantity is the strain tensor $D_{ij}=-\frac{1}{2}{\rm Tr}(R_{i}R_{j})$,
which is a positive-definite geometrical measure of the Skyrme field’s spatial
variation. It characterises the local elastic strain of the mapping $U$ from
spatial $\hbox{\upright\hbox to0.0pt{I\hss}\kern 1.7ptR}^{3}$ to the unit
3-sphere of $SU(2)$, equipped with their standard Riemannian metrics. The
strain tensor is invariant under global chiral transformations $U({{\bf
x}})\to U_{\rm L}U({{\bf x}})U_{\rm R}^{-1}$. The eigenvalues of the strain
tensor are also invariant under local spatial rotations, and symmetric
polynomials in these eigenvalues are candidates for the Skyrme field’s energy
density.

Let us denote these eigenvalues by
$\lambda_{1}^{2},\lambda_{2}^{2},\lambda_{3}^{2}$. The Skyrme energy
functional is parity-invariant, so it involves only even, symmetric
polynomials in $\lambda_{1},\lambda_{2},\lambda_{3}$. A fundamental algebraic
basis for such polynomials is

$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}\,,$
$\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{4}+\lambda_{2}^{4}+\lambda_{3}^{4}\,,$
$\displaystyle I_{6}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{6}+\lambda_{2}^{6}+\lambda_{3}^{6}\,.$ (3)

Also useful are, alternatively,

$\displaystyle J_{2}=I_{2}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}\,,$
$\displaystyle J_{4}=\frac{1}{2}(I_{2}^{2}-I_{4})$ $\displaystyle=$
$\displaystyle\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{3}^{2}\lambda_{1}^{2}\,,$
$\displaystyle J_{6}=\frac{1}{6}(I_{2}^{3}-3I_{2}I_{4}+2I_{6})$
$\displaystyle=$
$\displaystyle\lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2}\,.$ (4)

The hedgehog ansatz for the Skyrme field is

$U_{\rm H}({{\bf x}})=\exp\left(if(r)\,\hat{\bf
x}\cdot{\boldsymbol{\tau}}\right)=\cos f(r){\boldsymbol{1}}_{2}+i\sin
f(r)\,\hat{\bf x}\cdot{\boldsymbol{\tau}}\,,$ (5)

where $\hat{\bf x}$ is the spatial, outward radial unit-vector. $f(r)$ is the
radial profile function, required to satisfy the boundary conditions
$f(0)=\pi$ and $f(\infty)=0$, so that $U_{\rm H}=-{\boldsymbol{1}}_{2}$ at the
origin and $U_{\rm H}={\boldsymbol{1}}_{2}$ (the vacuum value) at spatial
infinity. For the hedgehog ansatz, the strain tensor has eigenvalues

$\lambda_{1}^{2}=f^{\prime
2}\,,\quad\lambda_{2}^{2}=\lambda_{3}^{2}=\frac{\sin^{2}f}{r^{2}}$ (6)

(where $f$ denotes $f(r)$ from here on). The first is the strain in the radial
direction, and the second and third are the two equal strains in the angular
directions. $\sin f$ rather than $f$ occurs here because of the curvature of
the target $SU(2)$. The fundamental symmetric polynomials (3) and (4) reduce
to

$\displaystyle I_{2}$ $\displaystyle=$ $\displaystyle f^{\prime
2}+2\,\frac{\sin^{2}f}{r^{2}}\,,$ $\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle f^{\prime 4}+2\,\frac{\sin^{4}f}{r^{4}}\,,$ $\displaystyle
I_{6}$ $\displaystyle=$ $\displaystyle f^{\prime
6}+2\,\frac{\sin^{6}f}{r^{6}}\,,$ (7)

and

$\displaystyle J_{2}$ $\displaystyle=$ $\displaystyle f^{\prime
2}+2\,\frac{\sin^{2}f}{r^{2}}\,,$ $\displaystyle J_{4}$ $\displaystyle=$
$\displaystyle 2f^{\prime
2}\,\frac{\sin^{2}f}{r^{2}}+\frac{\sin^{4}f}{r^{4}}\,,$ $\displaystyle J_{6}$
$\displaystyle=$ $\displaystyle f^{\prime 2}\,\frac{\sin^{4}f}{r^{4}}\,.$ (8)

The polynomials $J_{2},J_{4},J_{6}$ are quadratic in $f^{\prime}$, so the
corresponding terms in the full Lagrangian are quadratic in time-derivatives.
It is therefore a linear combination of these polynomials that gives the
traditional energy density for a Skyrme-type theory, and Skyrme’s original
theory combines just $J_{2}$ (the Dirichlet term) and $J_{4}$ (the Skyrme
term). $J_{6}$ is the additional (BPS-Skyrme) term introduced by Jackson et
al. [6]. Here we will allow for terms with higher powers of these polynomials,
and therefore higher powers of $f^{\prime}$. It is then often more convenient
to use the simpler monomials

$K_{k,l}=f^{\prime k}\frac{\sin^{l}f}{r^{l}}\,,$ (9)

where $k$ and $l$ are even integers (not both zero). However, individually
$K_{k,l}$ is not a symmetric polynomial in the strain eigenvalues, so is not
chirally invariant. Symmetrisation is required.

Our first result concerns the allowed behaviour of the profile $f$ near $r=0$.
We show, quite generally, that $f$ has leading dependence

$f(r)=\pi-\beta r+O(r^{3})\,,$ (10)

with $\beta$ undetermined locally. This result holds provided that the energy
functional is constructed from the symmetric polynomials $I_{2},I_{4},I_{6}$,
but does not hold for arbitrary combinations of the monomials $K_{k,l}$.
Consider a general polynomial energy density of the form
$G(I_{2},I_{4},I_{6})$. For the hedgehog ansatz, this density is independent
of angles and only depends on $r$. The total energy is therefore

$E=4\pi\int_{0}^{\infty}r^{2}G(I_{2},I_{4},I_{6})\,dr\,.$ (11)

$I_{2},I_{4},I_{6}$ depend on $f^{\prime}$ and $\sin f$, as in formulae (7).
Therefore the Euler–Lagrange equation for $f$ has the form

$\displaystyle\frac{d}{dr}\left\\{r^{2}\left(2\frac{\partial G}{\partial
I_{2}}f^{\prime}+4\frac{\partial G}{\partial I_{4}}f^{\prime
3}+6\frac{\partial G}{\partial I_{6}}f^{\prime 5}\right)\right\\}$
$\displaystyle\quad-2r\left(2\frac{\partial G}{\partial I_{2}}\frac{\sin
f}{r}+4\frac{\partial G}{\partial
I_{4}}\frac{\sin^{3}f}{r^{3}}+6\frac{\partial G}{\partial
I_{6}}\frac{\sin^{5}f}{r^{5}}\right)\cos f=0\,.$ (12)

This equation is invariant under the transformations $r\to-r$ and $f\to
f+2\pi$, so it is consistent for $\pi-f(r)$ to be a series in odd powers of
$r$ for small $r$, with no term proportional to $r^{2}$. Moreover, eq.(10)
implies that $f^{\prime}=-\beta+O(r^{2})$, $\frac{\sin f}{r}=\beta+O(r^{2})$
and $\cos f=-1+O(r^{2})$, so $I_{2},I_{4},I_{6}$ and $f^{\prime}$ are
constants with $O(r^{2})$ corrections for small $r$, and have vanishing
derivatives at $r=0$. To linear order in $r$, it is therefore sufficient in
eq.(12) to differentiate $r^{2}$, obtaining $2r$; the equation is then clearly
satisfied at linear order, for any $G$, because $f^{\prime}=-\beta$,
$\frac{\sin f}{r}=\beta$ and $\cos f=-1$ at $r=0$.

All the strain eigenvalues at $r=0$ are $\beta^{2}$, so a hedgehog profile has
an isotropic strain tensor there, i.e. the Skyrme field $U$ is smooth and is
locally a conformal map from physical $\hbox{\upright\hbox to0.0pt{I\hss}\kern
1.7ptR}^{3}$ to the target $SU(2)$. Note that the symmetrisation with respect
to the strain eigenvalues is crucial for this. For example, the energy
functional

$\tilde{E}=4\pi\int_{0}^{\infty}r^{2}f^{\prime 2}\,dr$ (13)

has Euler–Lagrange equation $\frac{d}{dr}(r^{2}f^{\prime})=0$ and this is not
satisfied at $O(r)$ by $f$ of the form (10) if $\beta\neq 0$.

##  3 Standard Hedgehog Skyrmion Revisited

Here, we review the hedgehog Skyrmion profile $f(r)$ in the original Skyrme
theory with its Dirichlet and Skyrme terms, and massless pions. This is an
opportunity to develop our approach, and clarify the approximations that it
entails. The energy functional, expressed in terms of the strain eigenvalues,
is

$E=\int_{\hbox{\upright\hbox to0.0pt{I\hss}\kern
1.7ptR}^{3}}(J_{2}+J_{4})\,d^{3}x=\int_{\hbox{\upright\hbox
to0.0pt{I\hss}\kern
1.7ptR}^{3}}\left(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}+\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{3}^{2}\lambda_{1}^{2}\right)\,d^{3}x\,.$
(14)

For the hedgehog ansatz, with strain eigenvalues (6), this simplifies to

$E=4\pi\int_{0}^{\infty}\left(r^{2}f^{\prime 2}+2\sin^{2}f+2f^{\prime
2}\sin^{2}f+\frac{\sin^{4}f}{r^{2}}\right)\,dr\,.$ (15)

The resulting field equation is

$r^{2}f^{\prime\prime}+2rf^{\prime}-\sin 2f+2\sin^{2}f\,f^{\prime\prime}+\sin
2f\,f^{\prime 2}-\sin 2f\,\frac{\sin^{2}f}{r^{2}}=0\,.$ (16)

For large $r$, a consistent local solution satisfying the boundary condition
$f(\infty)=0$ is a power series in $\frac{1}{r^{2}}$. The leading terms are of
the form

$f(r)=\frac{C}{r^{2}}+\frac{D}{r^{6}}+O\left(\frac{1}{r^{8}}\right)\,,$ (17)

and substituting this into (16), one finds that $C$ is arbitrary and
$D=-\frac{1}{21}C^{3}$, so

$f(r)=\frac{C}{r^{2}}-\frac{1}{21}\frac{C^{3}}{r^{6}}+O\left(\frac{1}{r^{8}}\right)\,.$
(18)

It is the $\sin f$ factor in the angular strain eigenvalues that generates the
$O(\frac{1}{r^{6}})$ correction to the leading $\frac{C}{r^{2}}$ term. The
absence of a term proportional to $\frac{1}{r^{4}}$ is because the series for
$\sin f$ contains only odd powers of $f$.

The most useful observation is that it is only the Dirichlet term,
contributing the first three terms to eq.(16), and not the Skyrme term, that
determines the large-$r$ solution up to $O(\frac{1}{r^{6}})$. The Skyrme term
generates sources that affect the coefficient of the $O(\frac{1}{r^{8}})$
term. This is a general result that we will verify below – no terms in the
energy functional apart from the Dirichlet term affect the large-$r$ behaviour
of $f(r)$ up to $O(\frac{1}{r^{6}})$. In this sense, we have established the
long-range robustness, or universality, of the hedgehog Skyrmion, as any
$O(\frac{1}{r^{8}})$ correction to a leading $O(\frac{1}{r^{2}})$ term is
effectively a short-range correction. We will therefore, from now on, always
approximate the hedgehog profile for large $r$ by

$f(r)=\frac{C}{r^{2}}-\frac{1}{21}\frac{C^{3}}{r^{6}}\,.$ (19)

This should be a good approximation in the range $\frac{C}{r^{2}}\lesssim 2$,
i.e. $r^{2}\gtrsim\frac{1}{2}C$. The coefficient $C$ is ultimately fixed so
that $f(0)=\pi$; in this sense, $C$ does depend on all the terms in the energy
functional.

Next, we consider the form of $f(r)$ for small $r$. Taking account of the
boundary condition $f(0)=\pi$, we write $f=\pi-g$, with $g(0)=0$. Then
$f^{\prime}=-g^{\prime}$, $\sin f=\sin g$ and $\cos f=-\cos g$, so the field
equation for $g$ is

$r^{2}g^{\prime\prime}+2rg^{\prime}-\sin 2g+2\sin^{2}g\,g^{\prime\prime}+\sin
2g\,g^{\prime 2}-\sin 2g\,\frac{\sin^{2}g}{r^{2}}=0\,,$ (20)

the same as eq.(16) for $f$. A consistent solution is a series in odd powers
of $r$, conveniently written as

$g(r)=\beta r-\gamma\beta^{3}r^{3}+O(r^{5})\,.$ (21)

Substituting this into eq.(20), and working up to $O(r^{3})$ gives

$\left(20\gamma-\frac{8}{3}+40\gamma\beta^{2}-\frac{4}{3}\beta^{2}\right)\beta^{3}r^{3}=0\,.$
(22)

As anticipated from the general argument of the last section, all the terms
linear in $r$ (with coefficients both $\beta$ and $\beta^{3}$) cancel, so
there is no local constraint on $\beta$. Equation (22) requires that

$\gamma=\frac{2}{15}\left(\frac{1+\frac{1}{2}\beta^{2}}{1+2\beta^{2}}\right)$
(23)

so

$f(r)=\pi-\beta
r+\frac{2}{15}\left(\frac{1+\frac{1}{2}\beta^{2}}{1+2\beta^{2}}\right)\beta^{3}r^{3}+O(r^{5})\,.$
(24)

$\beta$ is ultimately determined by the need to satisfy the boundary condition
$f(\infty)=0$.

Even in the absence of the Skyrme term contributions, there is still a local
solution $f(r)$, with the bracketed expression in (24) replaced by $1$. This
plays a role in the bag-like Skyrmions of Gustafsson and Riska [15]. For more
general energy functionals, we shall find that $\gamma$ is always
$\frac{2}{15}$ times a ratio of polynomials in $\beta^{2}$ with leading
coefficients $1$.

##  4 Further Energy Contributions

We next investigate the modification of the hedgehog Skyrmion profile due to
further terms added to the Skyrme energy functional. Chiral symmetry requires
these additional terms to be constructed from symmetric polynomials in the
strain eigenvalues $\lambda_{1}^{2}=f^{\prime 2}$ and
$\lambda_{2}^{2}=\lambda_{3}^{2}=\frac{\sin^{2}f}{r^{2}}$.

An individual term in the energy density, before symmetrisation, has the form

$K_{m,n+p}=\lambda_{1}^{m}\lambda_{2}^{n}\lambda_{3}^{p}=f^{\prime
m}\frac{\sin^{n+p}f}{r^{n+p}}$ (25)

with $m$, $n$ and $p$ even integers (not all zero). Its contribution to the
energy is

$4\pi\int_{0}^{\infty}r^{2}\left(f^{\prime
m}\frac{\sin^{n+p}f}{r^{n+p}}\right)\,dr\,,$ (26)

leading to the contribution

$m\,\frac{d}{dr}\left(f^{\prime
m-1}\frac{\sin^{n+p}f}{r^{n+p-2}}\right)-\frac{n+p}{2}\,\sin 2f\,f^{\prime
m}\frac{\sin^{n+p-2}f}{r^{n+p-2}}$ (27)

to the field equation for $f$. The solution for small $r$ still has leading
terms

$f(r)=\pi-\beta r+\gamma\beta^{3}r^{3}+O(r^{5})\,.$ (28)

Substituting this into (27), and retaining terms up to $O(r^{3})$, gives the
contribution of $K_{m,n+p}$ to the field equation,

$\displaystyle\beta^{m+n+p-2}\Biggl{\\{}-m\left(2\beta
r-4(3m+n+p-3)\gamma\beta^{3}r^{3}-\frac{2}{3}(n+p)\beta^{3}r^{3}\right)$
$\displaystyle+(n+p)\left(\beta
r-(3m+n+p-1)\gamma\beta^{3}r^{3}-\frac{1}{6}(n+p+2)\beta^{3}r^{3}\right)\Biggr{\\}}.$
(29)

The $O(r^{5})$ part of $f$ doesn’t contribute.

The corresponding symmetrised polynomial in the strain eigenvalues is

$\displaystyle K_{m,n,p}$ $\displaystyle=$ $\displaystyle
K_{m,n+p}+K_{n,p+m}+K_{p,m+n}$ (30) $\displaystyle=$ $\displaystyle f^{\prime
m}\frac{\sin^{n+p}f}{r^{n+p}}+f^{\prime
n}\frac{\sin^{p+m}f}{r^{p+m}}+f^{\prime p}\frac{\sin^{m+n}f}{r^{m+n}}\,,$

where $m\geq n\geq p$ to avoid overcounting, and cyclic permulation of the
indices is sufficient because $\lambda_{2}^{2}=\lambda_{3}^{2}$. Gudnason and
Nitta have also worked with these polynomials [13]. Using (29), we find that
$K_{m,n,p}$ makes the contribution

$\displaystyle\Bigl{\\{}10(m^{2}+n^{2}+p^{2}-m-n-p)\gamma+mn+np+pm$
$\displaystyle\qquad\qquad-\frac{1}{3}(m^{2}+n^{2}+p^{2})-\frac{2}{3}(m+n+p)\Bigr{\\}}\beta^{m+n+p+1}r^{3}$
(31)

to the field equation at $O(r^{3})$, and as expected, the $O(r)$ terms have
cancelled due to symmetrisation.

The Dirichlet and Skyrme energy densities are, respectively

$\displaystyle K_{2,0,0}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=f^{\prime
2}+2\,\frac{\sin^{2}f}{r^{2}}\,,$ $\displaystyle K_{2,2,0}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{3}^{2}\lambda_{1}^{2}=2f^{\prime
2}\,\frac{\sin^{2}f}{r^{2}}+\frac{\sin^{4}f}{r^{4}}\,,$ (32)

and according to (31) they contribute

$\displaystyle\left(20\gamma-\frac{8}{3}\right)\beta^{3}r^{3}\,,$
$\displaystyle\left(40\gamma-\frac{4}{3}\right)\beta^{5}r^{3}$ (33)

to the field equation at $O(r^{3})$. Let us now supplement these by the
additional quartic energy density usually omitted from Skyrme-type theories,
with coefficient $\mu$,

$\mu
K_{4,0,0}=\mu(\lambda_{1}^{4}+\lambda_{2}^{4}+\lambda_{3}^{4})=\mu\left(f^{\prime
4}+2\,\frac{\sin^{4}f}{r^{4}}\right)$ (34)

which contributes

$\mu(120\gamma-8)\beta^{5}r^{3}$ (35)

to the field equation. The combination of these energy densities leads to the
modified equation for $\gamma$,

$\left(20\gamma-\frac{8}{3}+40\gamma\beta^{2}-\frac{4}{3}\beta^{2}+120\mu\gamma\beta^{2}-8\mu\beta^{2}\right)\beta^{3}r^{3}=0\,,$
(36)

whose solution is

$\gamma=\frac{2}{15}\left(\frac{1+(\frac{1}{2}+3\mu)\beta^{2}}{1+(2+6\mu)\beta^{2}}\right)\,,$
(37)

so

$f(r)=\pi-\beta
r+\frac{2}{15}\left(\frac{1+(\frac{1}{2}+3\mu)\beta^{2}}{1+(2+6\mu)\beta^{2}}\right)\beta^{3}r^{3}+O(r^{5})\,.$
(38)

$\gamma$ does not have a strong dependence on $\mu$.

The large-$r$ solution for $f$, up to $O(\frac{1}{r^{6}})$, has the unchanged
form

$f(r)=\frac{C}{r^{2}}-\frac{1}{21}\frac{C^{3}}{r^{6}}\,.$ (39)

This is because the leading modification to the field equation from quartic
terms in the energy density like $f^{\prime 4}$ and $\frac{\sin^{4}f}{r^{4}}$
is at $O(\frac{1}{r^{8}})$, just as we found for the Skyrme term, and this
only affects the solution of the equation for the profile $f$ at
$O(\frac{1}{r^{8}})$.

We see that the additional quartic energy density has little effect on the
Skyrmion profile expressed in terms of the asymptotic parameters $\beta$ and
$C$. Just the value of $\gamma$ changes somewhat. However, a larger change is
likely when a matching radius $R$ is determined by energy minimisation. The
matching of (38) and (39) can be arranged so that the profile and its first
derivative are continuous at $r=R$. $R$ will depend on $\mu$, and this will in
turn affect $\beta$ and $C$.

Similar calculations can be used to find the effect of sextic energy densities
constructed from the strain eigenvalues. The three independent symmetric
polynomials can be selected to be

$\displaystyle K_{6,0,0}=\lambda_{1}^{6}+\lambda_{2}^{6}+\lambda_{3}^{6}$
$\displaystyle=$ $\displaystyle f^{\prime 6}+2\,\frac{\sin^{6}f}{r^{6}}\,,$
$\displaystyle
K_{4,2,0}=\lambda_{1}^{4}\lambda_{2}^{2}+\lambda_{2}^{4}\lambda_{3}^{2}+\lambda_{3}^{4}\lambda_{1}^{2}$
$\displaystyle=$ $\displaystyle f^{\prime 4}\frac{\sin^{2}f}{r^{2}}+f^{\prime
2}\frac{\sin^{4}f}{r^{4}}+\frac{\sin^{6}f}{r^{6}}\,,$ $\displaystyle
K_{2,2,2}=3\lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2}$ $\displaystyle=$
$\displaystyle 3f^{\prime 2}\frac{\sin^{4}f}{r^{4}}\,,$ (40)

and of these, $K_{2,2,2}$ is the BPS-Skyrme term. Using (31) we find they
contribute additional terms to eq.(36), respectively,

$\displaystyle\left(300\gamma-16\right)\beta^{7}r^{3}\,,$
$\displaystyle\left(140\gamma-\frac{8}{3}\right)\beta^{7}r^{3}\,,$
$\displaystyle\left(60\gamma+4\right)\beta^{7}r^{3}\,.$ (41)

The numerator and denominator in the expression (37) for $\gamma$ therefore
acquire $O(\beta^{4})$ corrections whose precise form depends on the
coefficients of the sextic terms in the energy functional.

In particular, the traditional Skyrme-type theories have an energy density
combining the Dirichlet, Skyrme and BPS-Skyrme contributions. Normalising the
Dirichlet and Skyrme terms as usual, the energy density is

$K_{2,0,0}+K_{2,2,0}+\nu K_{2,2,2}\,.$ (42)

The field equation in this case has a small-$r$ solution where

$\gamma=\frac{2}{15}\left(\frac{1+\frac{1}{2}\beta^{2}-\frac{3}{2}\nu\beta^{4}}{1+2\beta^{2}+3\nu\beta^{4}}\right)\,.$
(43)

This formula shows that $\gamma$ is quite sensitive to the value of $\nu$ if
$\beta$ is $O(1)$.

We have not calculated the effect of terms of yet higher order (e.g. octic).
These have been classified up to 12th-order by Gudnason and Nitta [13].
However, such higher-order terms have higher dimensions, as they involve
higher powers of derivatives, and their coefficients are suppressed by further
inverse powers of the cut-off mass. Overall, they affect the shape of the
Skyrmion hedgehog profile mainly at small $r$. Their effect is small, and
certainly not singular at short distances, and in that sense the Skyrmion is
robust.

Mathematically, the parameter $C$ could be quite sensitive to the coefficients
of the EFT, but we still need to calibrate the length scale, i.e. convert $r$
to physical length units. The strength of the hedgehog profile’s tail, for
large $r$, is physically fixed by nucleon-nucleon forces, so this tail is
truly robust up to $O(\frac{1}{r^{6}})$.

##  5 A Universal Hedgehog Profile

The simplest hedgehog profile for small $r$ is $f(r)=\pi-\beta r$. Adding a
cubic correction $\gamma\beta^{3}r^{3}$ so that the field equation is
satisfied to $O(r^{3})$ is an improvement, but $\gamma$ depends on the energy
functional, so is not universal to all chiral EFTs. It is instead desirable to
have a universal profile that includes a cubic term, and this is proposed
below.

Before doing this, we recall the earlier, simpler version, due to Ruback and
the present author [16]. Here, we just use the leading terms for small $r$ and
large $r$, and match $f$ and $rf^{\prime}$ at some radius $R$. The profile is

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle\pi-\beta r\,,\quad r\leq
R$ $\displaystyle f(r)$ $\displaystyle=$ $\displaystyle\frac{C}{r^{2}}\,,\quad
r\geq R\,.$ (44)

If we set $y=\beta R$ and $x=\frac{C}{R^{2}}$, then the matching conditions
become

$\displaystyle\pi-y$ $\displaystyle=$ $\displaystyle x\,,$ $\displaystyle y$
$\displaystyle=$ $\displaystyle 2x\,,$ (45)

with solution $y=\frac{2}{3}\pi$ and $x=\frac{1}{3}\pi$. So the profile is

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\pi-\frac{2\pi}{3}\frac{r}{R}\,,\quad r\leq R$ $\displaystyle
f(r)$ $\displaystyle=$ $\displaystyle\frac{\pi}{3}\frac{R^{2}}{r^{2}}\,,\quad
r\geq R\,.$ (46)

$f$ and $rf^{\prime}$ are continuous at $R$, but $r^{2}f^{\prime\prime}$ has
the fairly large discontinuity $2\pi$.

$R$ is not yet determined, as a spatial rescaling does not spoil the matching
conditions; it is not surprising that two matching conditions do not uniquely
determine the three parameters $\beta$, $C$ and $R$. $R$ is determined by a
global minimisation of the energy, which requires some numerical integration
[16]. For the original Skyrme theory, the Dirichlet term contributes a
numerical multiple of $R$ to the total energy, and the Skyrme term a multiple
of $\frac{1}{R}$. The energy is minimised when these contributions are equal
(a version of Derrick’s theorem), and this fixes $R$. One finds that $R=1.26$.
The estimated energy of the Skyrmion with the approximate profile (46) is then
$E=151$, about $3\%$ greater than the energy for the true profile, calculated
by solving the field equation (16) using a shooting method.

We now discuss our universal profile $f(r)$. The first improvement is to
include the universal $O(\frac{1}{r^{6}})$ term for large $r$, as in (19). The
second is to include an $O(r^{3})$ term for small $r$, as in (24) or (38), but
for this we need to choose the coefficient $\gamma$. Our proposal is to fix
$\gamma$ so that $f$ has a continuous second derivative at the matching radius
$R$. This approach gives the same profile for any chiral EFT, although it is
less accurate in individual cases than if $\gamma$ is fixed by solving the
field equation. The proposed profile is of the form

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle\pi-\beta
r+\gamma\beta^{3}r^{3}\,,\quad r\leq R$ $\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{C}{r^{2}}-\frac{1}{21}\frac{C^{3}}{r^{6}}\,,\quad r\geq
R\,.$ (47)

We set $y=\beta R$ and $x=\frac{C}{R^{2}}$ as before. Then, matching $f$,
$rf^{\prime}$ and $r^{2}f^{\prime\prime}$ at $r=R$ leads to the constraints

$\displaystyle\pi-y+\gamma y^{3}$ $\displaystyle=$ $\displaystyle
x-\frac{1}{21}x^{3}\,,$ $\displaystyle y-3\gamma y^{3}$ $\displaystyle=$
$\displaystyle 2x-\frac{2}{7}x^{3}\,,$ $\displaystyle 3\gamma y^{3}$
$\displaystyle=$ $\displaystyle 3x-x^{3}\,.$ (48)

Elimination of $y$ and $\gamma y^{3}$ reduces these to

$\pi=5x-x^{3}\,,$ (49)

and then $y=\frac{1}{7}(9\pi-10x)$ and $\gamma y^{3}=\frac{1}{3}(\pi-2x)$. As
the local maximum value of the right-hand side of (49) is
$\frac{10}{3}\sqrt{\frac{5}{3}}\simeq 4.30$, this cubic equation for $x$ has
three real roots. The smaller of the positive roots gives the desired profile.
One finds $x=0.696$, and then $y=3.045$ and $\gamma=0.0207$. This determines
all the coefficients except the scale $R$. The explicit form of the profile is
now

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\pi-3.045\frac{r}{R}+0.583\frac{r^{3}}{R^{3}}\,,\quad r\leq R$
$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
0.696\frac{R^{2}}{r^{2}}-0.0161\frac{R^{6}}{r^{6}}\,,\quad r\geq R\,.$ (50)

For the other two roots, either $x$ or $\gamma$ is negative, giving the
profile a wrong shape.

In the original Skyrme theory, the total energy for this universal profile is
calculated to be

$E=4\pi\left(3.538R+9.615\frac{1}{R}\right)\,,$ (51)

where the first contribution is from the Dirichlet term and the second from
the Skyrme term. The minimum occurs at $R=1.649$, giving an energy $E=146.6$,
or equivalently $E=1.238\times 12\pi^{2}$ as a multiple of the topological
lower bound $12\pi^{2}$ [17]. For this value of $R$, $\beta=1.847$ and
$C=1.890$. The exact profile, found using a shooting method, has energy
$E=1.232\times 12\pi^{2}$. Although the universal profile gives an energy
accurate to $0.5\%$, $C$ is $11\%$ less than the numerically determined value
$C=2.16$. Nevertheless, this profile is closer to the exact solution than the
approximation of ref.[16].

For the Skyrme-type theory including the BPS-Skyrme sextic term with
coefficient $\nu$, $\gamma$ is given by the formula (43). If $\beta\simeq 2$
and $\nu\simeq 0.05$, then $\gamma$ is close to its value $\gamma=0.0207$ for
the universal profile. The universal profile should therefore work
particularly well for this Skyrme-type theory.

##  6 Conclusions

We have studied the asymptotic forms of the profile function $f(r)$ of a $B=1$
hedgehog Skyrmion, for small $r$ and large $r$. A complete, approximate
profile can then be found by matching these asymptotic profiles and their
first derivatives at some radius $r=R$, with $R$ determined by energy
minimisation. The analysis applies to a large range of energy functionals of
chiral EFTs, with massless pions, and not just to the original Skyrme theory
with its Dirichlet and Skyrme terms. Chiral symmetry is imposed by requiring
that the energy density is a combination of symmetric polynomials in the
strain eigenvalues of the Skyrme field.

The profile at large $r$ is insensitive to the details of the energy
functional; it is more sensitive at small $r$, but only at cubic and higher
order in $r$. In all cases, the Skyrmion is smooth at $r=0$ and has no source.
We have exploited these observations to propose a universal profile for a
hedgehog Skyrmion that is continuous and has continuous first and second
derivatives.

It is hoped that the ideas developed here could be used to demonstrate that
Skyrmions are relevant to the modelling of nucleons in a larger range of
chiral EFTs than the standard Skyrme-type theories, including in those
perturbative EFTs that have been exploited to calculate nucleon-nucleon
interactions and the excitation spectra of small nuclei.

## Acknowledgements

I’m grateful to Dionisio Bazeia and Matheus Marques for organising the
stimulating workshop “Modern Trends in Field Theory” in João Pessoa, Brazil in
March, 2024, and for the helpful comments there by Antonio de Felice. I also
thank Sven Bjarke Gudnason for comments on this paper and for checking a
calculation. This work has been partially supported by consolidated grant
ST/T000694/1 from the UK Science and Technology Facilities Council.

## References

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  * [8] H. Leutwyler, On the foundations of chiral perturbation theory, Ann. Phys. 235, 165 (1994).
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  * [10] E. Epelbaum and U.-G. Meissner, Chiral dynamics of few- and many-nucleon systems, Ann. Rev. Nucl. Part. Sci. 62, 159 (2012).
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  * [12] L. Marleau, Solving Skyrmions, arXiv:hep-ph/0403292 (2004).
  * [13] S. B. Gudnason and M. Nitta, A higher-order Skyrme model, JHEP 1709:028 (2017).
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# Surface superconductivity as the primary cause of broadening of
superconducting transition in Nb-films

A. Zeinali  V. M. Krasnov<EMAIL_ADDRESS>1Department of
Physics, Stockholm University, AlbaNova University Center, SE-10691 Stockholm,
Sweden

###### Abstract

We study the origin of broadening of superconducting transition in sputtered
Nb films. From simultaneous tunneling and transport measurements we conclude
that the upper critical field $H_{c2}$ always corresponds to the bottom of
transition $R\sim 0$, while the top $R\sim R_{n}$ occurs close to the critical
field for destruction of surface superconductivity $H_{c3}\simeq 1.7H_{c2}$.
The two-dimensional nature of superconductivity at $H>H_{c2}$ is confirmed by
cusp-like angular dependence of magnetoresistance. Our data indicates that
surface superconductivity is remarkably robust even in disordered
polycrystalline films and, surprisingly, even in perpendicular magnetic
fields.

##  I Introduction

Superconductivity occurs as a result of the second-order phase transition,
accompanied by a sudden appearance of the superconducting order parameter
below the critical temperature $T_{c}$ and the upper critical field $H_{c2}$
SaintJames . This should lead to an abrupt vanishing of resistance. However,
in reality resistive transitions are always broadened, especially in magnetic
field. This is usually ascribed to flux-flow phenomenon caused by motion of
Abrikosov vortices BardeenStephen_1965 . Broadening can also be caused by
spatial inhomogeneity (e.g. variation of $T_{c}$), or superconducting
fluctuations Varlamov_book ; Finkelstein_2012 ; Glatz_2014 , particularly for
high temperature superconductors. Finally, surface superconductivity (SSC) may
survive up to a significantly higher field $H_{c3}\simeq 1.69H_{c2}$ than bulk
superconductivity SaintJames , which can also smear the superconducting
transition. Although SSC is quite profound in polished clean superconductors
Bellau_1966 ; Park_2004 ; Casalbuoni_2005 ; Das_2008 , it is usually ignored
for disordered, polycrysstaline films because SSC is considered to be very
sensitive to the quality of the surface (e.g., surface passivation
Casalbuoni_2005  and order parameter suppression SaintJames ), surface
roughness Bellau_1966 ; Casalbuoni_2005  and surface scattering Agterberg_1996
.

The broadening is detrimental for superconducting devices such as transition-
edge sensors TES1 ; TES2  and resonators Casalbuoni_2005 . Presence of several
mechanisms makes the interpretation of broadening ambiguous. The lack of
understanding does not allow confident extraction of fundamental parameters of
superconductors, such as $H_{c2}$, because it is unclear which point at the
transition curve corresponds to $H=H_{c2}$. Arbitrary criteria, such as 10, 50
or 90 $\%$ of the normal state resistance $R_{n}$, are commonly applied which
apparently does not work for high-$T_{c}$ superconductors with very broad
transitions Zavaritsky . Therefore, clarification of the mechanism of
broadening is important both for fundamental and applied research on
superconductors.

In this work we study the origin of broadening of superconducting transitions
in sputtered Nb films. We perform simultaneous tunneling spectroscopy and
transport measurements, which allow unambiguous ascription of $H_{c2}$ to the
bottom of resistive transition $R(H_{c2})/R_{n}\sim 0$. The top of transition
corresponds to $\sim 1.7$ times higher fields, which is close to the third
critical field $H_{c3}$ for destruction of surface superconductivity. The two-
dimensional (2D) nature of SSC at $H_{c2}<H<H_{c3}$ is confirmed by
observation of cusp-like angular dependence of magnetoresistance. Thus we
conclude that surface superconductivity, rather than flux-flow, inhomogeneity
or fluctuations, is the primary cause of broadening of superconducting
transitions in magnetic field. Our data indicates that surface
superconductivity is remarkably robust even in disordered polycrystalline
films and, surprisingly, even in perpendicular magnetic fields.

Figure 1: (Color online). Resistive transitions of a 150 nm thick Nb film. (a)
Temperature dependence of the resistance in zero magnetic field. (b) and (c)
Field dependencies of resistances at $T=1.8$ K in fields (b) perpendicular and
(c) parallel to the film. Black and red circles mark the upper critical field
$H_{c2}$ and the field of the onset of resistive transition, which coincides
with the critical field of surface superconductivity $H_{on}\sim H_{c3}$.

##  II Experimental

The studied sample contains several Nb/Al-AlOx/Nb tunnel junctions with
sputtered Nb electrodes of thicknesses $d=150$ and 50 nm. Junction
characteristics in perpendicular fields were reported in Ref. MR . Due to
different thicknesses, electrodes have slightly different $T_{c}$ of 9.2 and
8.8 K. Parameters extracted from tunneling characteristics are determined by
the thinner electrode, while transport measurements are made at the thicker
electrode. This explains a minor difference in $H_{c2}$ values obtained by
those techniques. Measurements are performed in a gas-flow 4He cryostat with a
superconducting solenoid. Samples are mounted on a rotatable holder with the
alignment accuracy $\sim 0.02^{\circ}$. Details of the setup can be found
elsewhere MR .

##  III Results

In Figures 1 (a-c) we show superconducting transitions of a 150 nm thick Nb
film: (a) $R(T)$ in zero field and $R(H)$ at $T=1.8$ K for field perpendicular
(b) and parallel (c) to the film. It is seen that at zero field the $R(T)$
transition is very sharp and does not show any extended fluctuation region or
inhomogeneity. However, $R(H)$ transitions are quite broad. Interestingly,
$R(H)$ is broader when the field is parallel to the film. This confronts
interpretation of broadening in terms of vortex motion because the driving
Lorentz force is most effective in perpendicular and vanishes in parallel
field. Therefore, this broadening is not consistent with either flux-flow,
inhomogeneity or fluctuation mechanisms.

Figure 2: (Color online). Comparison of theoretically calculated (a-c) and
experimentally measured (d-f) characteristics of Nb/AlAlOx/Nb tunnel
junctions. (a) and (d) Temperature dependence of $I$-$V$ characteristics at
zero field. (b) and (e) Field dependence of $I$-$V$ characteristics for field
perpendicular to the junction/films at $T\simeq 2$ K. (c) and (f) The
corresponding differential conductances for curves from panels (b) and (e).
The field scale in (f) is normalized by the upper critical field
$H_{c2}^{\perp}=1.61$ T, which is obtained as a single scaling factor for all
the curves at different fields. Data from Ref. MR .

###  III.1 Determination of $H_{c2}$ from tunneling spectroscopy

In order to analyze surface superconductivity scenario, first of all, it is
necessary to determine bulk $H_{c2}$. For this we perform magneto-tunneling
spectroscopy. Figure 2 represents a comparison of theoretically calculated
(a-c) and experimentally measured tunneling characteristics of our
Nb/AlAlOx/Nb junction (data from Ref. MR ). Details of calculations are
described in Ref. MR ). Panels (a) and (d) show temperature dependencies of
$I$-$V$ characteristics at zero field. Panels (b) and (e) show field
dependence of $I$-$V$ characteristics for field perpendicular to the
junction/films at $T\simeq 2$ K. Panels (c) and (f) show the corresponding
differential conductances for $I$-$V$ curves from panels (b) and (e). There is
a good quantitative agreement between theoretical and experimental
characteristics. The main spectroscopic features are the sharp sum-gap peak at
$V_{p}=2\Delta/e$, where $\Delta$ is the superconducting gap and the
suppressed quasiparticle current and conductance below the sum-gap voltage.
With increasing field the peak is decreasing in height and is moving to lower
voltages. Simultaneously the sub-gap conductance is increased. All this is due
to suppression of the superconducting gap by magnetic field. The extent of
suppression depends solely on $H/H_{c2}$. Above $H_{c2}$ the superconducting
gap vanishes and the $I$-$V$ becomes linear (Ohmic). Thus, the ratio
$H/H_{c2}$ uniquely determines the shape of tunneling characteristics in
magnetic field. Therefore, as discussed in Ref. MR , the ratio $H/H_{c2}$ can
be uniquely determined from analysis of the shape of tunneling
characteristics. In Fig. 2 (f) the field is normalized by thus obtained
$H_{c2}^{\perp}=1.61$ T. We emphasize that this value is obtained as a single
fitting parameter for the whole set of d$I/$d$V(V)$ characteristics at
different $H$. This removes the uncertainty in determination of $H_{c2}$.

Fig. 3 (a) shows a set of tunneling d$I$/d$V(V)$ characteristics of a
Nb/AlOx/Nb junction at $T=1.8$ K in fields parallel to Nb films. From
comparison of Figs. 2 and 3 (a), it can be seen that the influence of magnetic
field is qualitative similar both for parallel and perpendicular field
orientations. In Ref. MR  it was shown that the peak height and the peak
voltage exhibit universal almost $T$-independent quasi-linear scaling as a
function of $H/H_{c2}(T)$. Fig. 3 (b) and (c) demonstrate such a scaling at
different temperatures for field parallel to Nb films. Dashed and dashed-
dotted lines in Fig. 3 (b) represent theoretical results from Ref. MR  for
$T=1.96$ and 4.7 K, correspondingly. The overall quality of scaling is quite
good, which allows confident extraction of $H_{c2}(T)$. Thus obtained $H_{c2}$
is unambiguous because it is deduced as a single fitting parameter for the
whole set of d$I/$d$V(V)$ characteristics at fixed $T$ for different $H$.

Figure 3: (Color online). (a) Differential conductances of a Nb/AlOx/Nb
junction (in a semi-logarithmic scale) at parallel to the films magnetic
fields and $T=1.8$ K. (b) and (c) Scaling of the sum-gap peak resistance (b)
and voltage (c) as a function of $H/H_{c2}(T)$ at different temperatures and
parallel fields. Dashed and dashed-dotted lines in (b) represent theoretical
curves at $T=1.96$ K and 4.7 K, respectively. (d) and (e) Black squares
represent upper critical fields perpendicular (d) and parallel (e) to the
films, obtained from the scaling of magneto-tunnelling characteristics. Dashed
lines represent the expected third critical field for surface
superconductivity $H_{c3}=1.69H_{c2}$. Red circles mark the top onset of the
resistive transition.(f) Angular dependence of resistance of Nb electrodes at
fields slightly below and above $\mu_{0}H_{c2}(\parallel)=2.52$ T. A cusp-like
feature at $H>H_{c2}$ indicates occurrence of the two-dimensional surface
superconductivity.

Squares in Figs. 3 (d) and (e) represent obtained $H_{c2}(T)$ dependencies for
perpendicular and parallel field orientations, respectively. Using the
relation $H_{c2}^{\perp}=\Phi_{0}/2\pi\xi^{2}$ we calculate the coherence
length $\xi_{0}\simeq 14$ nm. This small value indicates that the film is in
the dirty limit with a very short scattering length due to a disordered film
structure with nm-scale crystallites. Thus, the studied Nb film $d=150$ nm is
an order of magnitude thicker than $\xi_{0}$. This leads to an important for a
further discussion conclusion that our films are bulk three-dimensional (3D)
superconductors practically in the whole temperature range $T<T_{c}$. Red
circles in Figs. 1 (b) and (c) represent top onsets $R(H_{on})\simeq R_{n}$ of
resistive transitions. Dashed blue lines correspond to $H_{c3}=1.69H_{c2}$
expected for surface superconductivity, which is close to the onset field.
Remarkably this is true even for the perpendicular field orientation when SSC
in the uniform case is not expected SaintJames .

Figure 4: (Color online). (a) Bias dependence of $R(\Theta)$ at $T=1.8$ K and
$H=2.81$ T. (b) Deduced current-voltage characteristics at different angles
from the data in panel (a). Dashed line represents the normal state $I$-$V$.
(c) Simulated angular dependence of $R(\Theta)$ in the surface
superconductivity model, taking into account non-linearity of $I$-$V$
characteristics in the vicinity of critical current. Solid lines represent
calculated of $R(\Theta)$ at different bias. Dashed curve represents the
standard flux-flow dependence $H/H_{c3}$.

###  III.2 Angular dependence of magnetoresistance

The 2D nature of SSC should be reflected in a cusp-like angular dependence of
$H_{c3}$, given by the equation Yamafuji_1966 :

$\left[\frac{H_{c3}(\Theta)}{H_{c3}^{\parallel}}\cos\Theta\right]^{2}A(\Theta)+\left|\frac{H_{c3}(\Theta)}{H_{c3}^{\perp}}\sin\Theta\right|=1,$
(1)

where $A(\Theta)=1+(1-\sin\Theta)\tan\Theta$. It is only slightly different
from Tinkham’s 2D result with $A(\Theta)=1$.

Fig. 3 (f) shows angular-dependencies of magnetoresistance at $T=1.8$ K and at
different fields below and above $\mu_{0}H_{c2}^{\parallel}=2.52$ T. Zero
angle $\Theta=0$ corresponds to field parallel to the film. It is seen that at
2.5 T very slightly below $H_{c2}$, $R(\Theta)$ is flat at $\Theta=0$, which
is characteristic for 3D bulk Nb. However, at $H>H_{c2}^{\parallel}$ angular
dependence acquires a 2D cusp. Since the film thickness is much larger than
$\xi$, the observed 2D behavior at low $T$ may originate solely from SSC.

###  III.3 Non-linear bias dependence

The sheet surface critical current (in A/cm) is Abrikosov_1965  :

$I_{c}\simeq\frac{5}{2\sqrt{3\pi}}\frac{H_{c}}{\kappa}\left(1-\frac{H}{H_{c3}}\right)^{3/2}.$
(2)

Here $H_{c}$ is the thermodynamic critical field (in Oe) and $\kappa$ is the
Ginzburg-Landau parameter. Typically such $I_{c}$ is in the range from few to
few tens of A/m. For our films with $\kappa\gg 1$ and the width of few microns
the $I_{c}$ is in the $\mu$A range, comparable to the probe current.
Therefore, the results do depend on the bias, as illustrated in Fig. 4 (a).
This is due to strong non-linearity of current-voltage characteristics at
$I\sim I_{c}$, as demonstrated in Fig. 4 (b). In order to demonstrate how such
the non-linearity affects experimental characteristics we consider a standard
shape of $I$-$V$:

$V=R_{n}\sqrt{I^{2}-I_{c}^{2}}.$ (3)

Together with Eq. (2) it yields

$1-\left[\frac{R(\Theta)}{R_{n}}\right]^{2}=\frac{I_{c}^{2}}{I^{2}}\left[1-\frac{H}{H_{c3}(\Theta)}\right]^{\nu}.$
(4)

The exponent $\nu$ depends on the shape of the $I$-$V$ and the quality of the
surface Bellau_1966 . For the case of Eq. (3) it is $\nu=3$.

Figure 5: (Color online). (a) and (b) Excess conductance vs. magnetic field at
$T=1.8$ K plotted in (a) a double-logarithmic and (b) a semi-logarithmic
scale. Quasi-exponential decay $\Delta S(H)$ is seen. (c) The high-field part
of excess conductance. It reveals fluctuation contribution, which is positive
for parallel, but negative for perpendicular field orientation. (d) Analysis
of power-law scaling expected for surface superconductivity according to Eq.
(4). Dashed line represents a power-law with the exponent $\nu=4.8$.

In Fig. 4 (c) we show $R(\Theta)$ curves for the SSC model calculated from
Eqs. (1) and (4) for $H=1.1H_{c2}$ at different bias. Calculations are made
for $H_{c3}^{\perp}=H_{c2}$ and $H_{c3}^{\parallel}=1.69H_{c2}$ and for
$I_{c}(\Theta)=$const. For comparison we also show flux-flow type dependence
$R/R_{n}=H/H_{c3}$. It is seen that the cusp in the SSC model is much sharper,
primarily due to non-linearity of the $I$-$V$. Overall behavior is similar to
experimental data from Fig. 4 (a), even though in experiment a very sharp cusp
at $\Theta=0$ survives up to much higher current. The difference is due to an
oversimplified assumption of angular-independent $I_{c}(\Theta)=$const, used
in calculations. In reality $I_{c}(\Theta)$ has a sharp maximum at $\Theta=0$
because the Lorenz force vanishes as $\sin(\Theta)$. It is possible to get a
better fit using a more realistic $I_{c}(\Theta)$, but we don’t want to go in
to more complicated modelling because the main purpose of calculations was to
demonstrate how non-linearity of $I$-$V$’s leads to a much sharper (compared
to a simple 2D flux-flow model) cusp in $R(\Theta)$.

###  III.4 Analysis of fluctuation contribution

Finally we discuss fluctuation contribution to the broadening of resistive
transition. In Fig. 5 (a) we show normalized excess conductance $\Delta
S=(1/R-1/R_{n})R_{n}$ for the data from Figs. 1 (b) and (c) in a double-
logarithmic scale. Such graph is usually used for analysis of fluctuation
contribution to conductivity. Dashed line shows $\Delta
S\propto(H-H_{c2})^{-1}$ dependence expected for 2D fluctuations Varlamov_book
. It is seen that although there is a narrow range of fields close to $H_{c2}$
with similar behavior, the overall agreement is poor. In Fig. 5 (b) we replot
the same data in a semi-logarithmic scale. It is seen that $\Delta S$ decays
quasi-exponentially with increasing field at approximately the same rate for
both field orientations. A similar exponentially decay versus both $T$ and $H$
has been reported for high-$T_{c}$ cuprates MR ; Katterwe_2014 ; Alloul2011 ;
Luo2014 . Here we demonstrate that it is generic also for conventional
superconductors. Such behavior is not expected for fluctuations Varlamov_book
; Finkelstein_2012 ; Glatz_2014  and we argue that it is rather a signature of
SSC.

It is possible to distinguish fluctuation contribution from non-fluctuating
SSC. SSC always leads to excess conductance, but fluctuation contribution to
magnetoresistance can be both positive and negative Varlamov_book ;
Finkelstein_2012 ; Glatz_2014 . In particular, at low $T$ and at field
perpendicular to the film the density-of-state contribution to fluctuations
leads to excess resistance, rather than excess conductance Glatz_2014 . We
clearly see such a contribution in our data. In Fig. 5 (c) we show high-field
part of excess conductance in the linear scale. It is seen that in parallel
field there is always an excess conductance $\Delta S>0$, which rapidly
decreases upon approaching the surface critical field $H_{c3}\simeq
1.7H_{c2}^{\parallel}$, but never really vanishes. The remaining tail is a
signature of fluctuations that persist at any field. For perpendicular field,
$\Delta S$ at high fields becomes negative, which is consistent with
theoretical expectations for fluctuation contribution at $T\ll T_{c}$
Glatz_2014 .

In Fig. 5 (d) we check the power-law scaling suggested by Eq. (4) for SSC. It
is seen that there is a good scaling in a broad field range, although
extraction of the exponent $\nu$ is not very confident because it depends on
the chose of $H_{c3}$. The dashed line corresponds to $\nu=4.8$. Upon
approaching $H_{c3}$, deviations with opposite signs for parallel and
perpendicular field orientations appear, signalizing fluctuation
contributions. This indicates that SSC makes a dominant contribution to excess
conductivity at $H_{c2}<H\lesssim 0.8H_{c3}$, while fluctuation contribution
starts to become significant only upon weakening of SSC at $H>0.8H_{c3}$ and
takes over completely at $H>H_{c3}$.

##  IV Discussion

Our results suggest that surface superconductivity is the primary cause of
broadening of superconducting transition in magnetic field. As indicated in
Figs. 1 (b) and (c), $H=H_{c2}$ corresponds to the bottom of transition,
consistent with earlier studies Burger_1965 , and $H_{c3}$ to the top of the
resistive transition. Thus the full width of the transition is dominated by
SSC. Although SSC is well known for carefully polished single crystals
Park_2004 ; Das_2008 , it is usually considered to be insignificant for
disordered, rough or inhomogeneous superconducting films because of its
assumed fragility and sensitivity to surface conditions SaintJames ;
Casalbuoni_2005 ; Bellau_1966 ; Agterberg_1996 . Therefore, observation of a
very robust SSC in our strongly disordered polycrystalline films is rather
surprising, especially for field perpendicular to the film. In perfectly
uniform films SSC should not occur at perpendicular field orientation
SaintJames ; Abrikosov_1965 . Yet, SSC in perpendicular fields has been
directly visualized by scanning laser microscopy for similar films Werner_2013
and also reported for some layered supercondcutors LevyBertrand_2016  and
sintered polycrystalline MgB2 samples Tsindlekht_2006 . Presumably it is the
polycrystallinity of our films that allows SSC at grain boundaries even in
perpendicular fields. Thus we conclude that surface superconductivity is a
robust phenomenon that should be carefully considered in analysis of data
close to superconducting transition.

###### Acknowledgements.

The work was supported by the Swedish Research Council Grant No. 621-2014-4314
and the Swedish Foundation for International Cooperation in Research and
Higher Education Grant No. IG2013-5453.

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