# Pure Odd Frequency Superconductivity at the Cores of Proximity Vortices

Mohammad Alidoust  Department of Physics, Faculty of Sciences, University of
Isfahan, Hezar Jerib Avenue, Isfahan 81746-73441, Iran  Alexander Zyuzin
Department of Theoretical Physics, The Royal Institute of Technology,
Stockholm, SE-10691 Sweden  A. F. Ioffe Physical - Technical Institute, 194021
St. Petersburg, Russia  Klaus Halterman  Michelson Lab, Physics Division,
Naval Air Warfare Center, China Lake, California 93555, USA

###### Abstract

After more than a decade, direct observation of the odd frequency triplet
pairing state in superconducting hybrid structures remains elusive. We propose
an experimentally feasible setup that can unambiguously reveal the zero energy
peak due to proximity-induced equal spin superconducting triplet correlations.
We theoretically investigate a two dimensional Josephson junction in the
diffusive regime. The nanostructure consists of a normal metal sandwiched
between two ferromagnetic layers with spiral magnetization patterns. By
applying an external magnetic field perpendicular to the junction plane,
vortices nucleate in the normal metal. The calculated energy and spatially
resolved density of states, along with the pair potential, reveal that
remarkably, only triplet Cooper pairs survive in the vortex cores. These
isolated odd frequency triplet correlations result in well defined zero energy
peaks in the local density of states that can be identified through tunneling
spectroscopy experiments. Moreover, the diffusive regime considered here rules
out the possibility of Andreev bound states in the vortex core as contributors
to the zero energy peaks.

###### pacs:

72.10.-d, 72.10.Bg, 73.63.-b, 73.25.+i, 74.78.-w

Introduction. In analogy to the pairing mechanism in 3He, spin triplet Cooper
pairing was predicted to coexist with spin singlet correlations in hybrid
structures consisting of $s$-wave superconductors (S) and inhomogeneous
ferromagnets (IFMs)Buzdin2005 ; first ; Bakurskiy1 ; Keizer2006 ;
Halterman2007 ; Bobkova ; Khaydukov ; Moor ; singh ; Eschrig2015 ; longrg ;
jason_natphys . For these types of systems, spin triplet correlations with
nonzero ($\pm 1$) projections along a given spin quantization axis can result
in long range proximity effects Buzdin2005 ; first . It was argued that traces
of the triplet pairing state could be revealed in measurements of the critical
supercurrent Buzdin2005 ; first ; kup_rmp ; crnt_1 ; crnt_2 ; crnt_hoz ;
longrg ; Eschrig2015 ; Khaydukov ; Keizer2006 ; Baker  and local density of
states (LDOS) buz_zep ; golubov ; klaus_zep ; golu . In the former case, the
critical supercurrent should show a slow damping behavior as a function of
spin singlet depairing factors (such as the thickness of a uniform magnetic
layer), while in the later case, the LDOS should exhibit a peak at zero
energy. Unfortunately, an unambiguous and direct observation of the spin
triplet pairing state in F/S hybrid platforms remains elusive due to the
difficulty in isolating the triplet pairs entirely, even when a half-metallic
ferromagnet is incorporated klaus_zep ; bernard ; km_hf ; singh ; buzhf ;
Eschrig2015 . Thus, it is preferable to find a practical way to manipulate the
pair correlations so that the singlet and triplet components occupy separate
regions of space. In contrast to current approaches buzhf ; crnt_2 ; golu ;
crnt_1 ; golubov ; Buzdin2005 ; first ; klaus_zep ; bernard ; km_hf ; singh ;
Eschrig2015 ; longrg ; jason_natphys ; golu , controlling the pair
correlations in this way can be achieved by applying a magnetic field to the
F/S structure longrg , inducing proximity vortices with normal state cores.
This may consequently create a favorable situation where the singlet and
triplet pair correlations can be fully separated at the vortex cores.

The first experimental observation of nonmagnetic proximity induced vortices
recently occurred in two dimensional normal metal (N) Josephson junctions
vrtx_exp . It was observed that applying an external magnetic field
perpendicular to a wide SNS Josephson junction causes nucleation of a vortex
lattice in the normal metal parallel to the SN interfaces. The number of
induced vortices depends on the intensity of the externally applied magnetic
field. The proximity-induced vortices in two dimensional Josephson structures
was first discussed theoretically in connection with the Fraunhofer and
anomalous critical supercurrent responses in Josephson junctions with both
normal metal berg_fr ; mj_nf  and ferromagnetic elements longrg . This concept
was also recently extended to disordered surface states of topological
insulators and Dirac materials in the quasiclassical regime ZAD_3DTI .

In this paper, we study the diffusive S-${\rm Ho}$/N/${\rm Ho}$-S Josephson
junction structure shown in Fig. 1 as a system for fully isolating the odd
frequency spin-1 superconducting triplet correlations. The existence of the
triplet pairs is directly revealed in the form of DOS signatures. The role of
the Holmium (${\rm Ho}$) layers is that of a spin-1 triplet pairing source,
while the superconducting phase gradient across the junction drives the
triplet pairs into the N region. By taking advantage of the fact that an
external magnetic field applied perpendicularly to the junction plane induces
vortices in the N region, while expelling the spin singlet pairs from the
vortex centers (creating a normal core), we demonstrate that spin-1 triplet
correlations occupy the normal core region, as revealed through peaks in the
zero energy DOS. We support our findings by a spin parameterization technique
to the Green function of system that allows for fully identifying the behavior
of each individual pair correlation longrg . Since the N layer is a diffusive
metal with numerous strong scattering sources, the superconducting coherence
length is much larger than the mean free path, and therefore bound states
cannot form at the centers of the vortices, ruling out Andreev bound states as
contributors to the zero energy peak (ZEP). Consequently, the spin-1 triplet
channel is highly dominant within the vortex core, causing the ZEP in the DOS.

Results and Discussions. It is now firmly established that the electronic
properties of a diffusive hybrid superconducting structure can be described by
the Usadel equation within the quasiclassical framework Buzdin2005 ; first ;
kup_rmp . The Usadel equation in the normal region reads usadel :

$\displaystyle
D\hat{{\bm{\nabla}}}(\hat{G}\hat{{\bm{\nabla}}}\hat{G})+i[\varepsilon\hat{\rho}_{z},\hat{G}]=0,~{}~{}\hat{G}(\varepsilon,\mathbf{R})=\left(\begin{array}[]{cc}G^{A}&G^{K}\\\
0&G^{R}\end{array}\right),~{}~{}~{}~{}~{}$ (3)

where $D$ represents the diffusion constant in the N and S regions and
$\varepsilon$ is the quasiparticle energy measured from the Fermi level. We
normalize all lengths by the superconducting coherence length,
$\xi_{S}=\sqrt{\hbar D/|\Delta_{0}|}$, energies by the superconducting gap at
zero temperature, $|\Delta_{0}|$, and adopt natural units where
$\hbar=k_{B}=1$. The Green function $\hat{G}(\varepsilon,\mathbf{R})$ is
composed of the advanced, $G^{A}(\varepsilon,\mathbf{R})$, retarded,
$G^{R}(\varepsilon,\mathbf{R})$, and Keldysh, $G^{K}(\varepsilon,\mathbf{R})$,
propagators, which carry the complete physical information of the system
considered. In the presence of an external magnetic field, ${\bf
H}=(0,0,H_{z})$, directed perpendicular to the junction plane, the derivatives
can be replaced by their covariants, i.e.
$\hat{{\bm{\nabla}}}={\bm{\nabla}}-[ie{\bm{A}}\hat{\rho}_{z},...].$ Here
${\bm{A}}$ is the vector potential associated with the external field
${\bm{H}}$. In equilibrium, as considered throughout the paper, the advanced
and Keldysh propagators can be expressed via the Retarded Green function. In
this case, one can show that
$G^{A}(\varepsilon,\mathbf{R})=-\\{\hat{\rho}_{z}G^{R}(\varepsilon,\mathbf{R})\hat{\rho}_{z}\\}^{\dagger}$
and
$G^{K}(\varepsilon,\mathbf{R})=\\{G^{R}(\varepsilon,\mathbf{R})-G^{A}(\varepsilon,\mathbf{R})\\}\tanh(\varepsilon
k_{B}T/2)$, where $k_{B}$ is the Boltzmann constant, and the system
temperature is denoted by $T$. Therefore, it suffices to focus on the retarded
Green function, and then eventually construct the total propagator using the
simple relations above. One useful limit for F/S structures is the so-called
low proximity limit. This limit permits linearization of the Green function,
yielding a linear set of differential equations that are in general coupled
ZAD_3DTI ; longrg . Although highly useful transport characteristics can be
captured in this limit, the full proximity regime allows for the study of
energy-resolved and spatially-resolved DOS, and other relevant physical
quantities. Hence, we first employ the full proximity limit, resulting in a
complex set of nonlinear coupled differential equations longrg  and then
compliment our findings with a spin parameterization technique in the low
proximity limit.

In establishing a numerically stable algorithm in the full proximity limit, we
use the so-called Riccati parametrization riccati , where it is convenient to
introduce two correlated functions $\gamma$ and $\tilde{\gamma}$, which are in
effect unknown 2$\times$2 matrices. In this parameterization scheme, the
retarded Green function takes the following form:

$\displaystyle
G^{R}(\varepsilon,\mathbf{R})=\left(\begin{array}[]{cc}(1-\gamma\tilde{\gamma})\Gamma&2\gamma\tilde{\Gamma}\\\
2\tilde{\gamma}\Gamma&(\tilde{\gamma}\gamma-1)\tilde{\Gamma}\end{array}\right),$
(6)

in which $\Gamma=(1+\gamma\tilde{\gamma})^{-1}$ and
$\tilde{\Gamma}=(1+\tilde{\gamma}\gamma)^{-1}$. Substituting the Riccati
parameterized Green function into the Usadel equation, Eq. (3), and
considering the external magnetic field, we arrive at the following equations
for $\gamma$ and $\tilde{\gamma}$ in the N region of Fig. 1:

$\displaystyle\partial_{k,k^{\prime}}^{2}\gamma-2(\partial_{k,k^{\prime}}\gamma)\tilde{\gamma}\Gamma\partial_{k,k^{\prime}}\gamma-(2eH_{z}k^{\prime})^{2}\left\\{2\Gamma-1\right\\}\gamma$
$\displaystyle-4ieH_{z}k^{\prime}\left\\{\partial_{k}\gamma-(\partial_{k}\gamma)\tilde{\Gamma}-\Gamma\partial_{k}\gamma\right\\}=-2i\frac{\varepsilon}{D}\gamma,$
(7a)
$\displaystyle\partial_{k,k^{\prime}}^{2}\tilde{\gamma}-2(\partial_{k,k^{\prime}}\tilde{\gamma})\gamma\tilde{\Gamma}\partial_{k,k^{\prime}}\tilde{\gamma}-(2eH_{z}k^{\prime})^{2}\left\\{2\tilde{\Gamma}-1\right\\}\tilde{\gamma}$
$\displaystyle+4ieH_{z}k^{\prime}\left\\{\partial_{k}\tilde{\gamma}-(\partial_{k}\tilde{\gamma})\Gamma-\tilde{\Gamma}\partial_{k}\tilde{\gamma}\right\\}=-2i\frac{\varepsilon}{D}\tilde{\gamma}.$
(7b)

For compactness, we have defined $k\equiv x$ and $k^{\prime}\equiv y$ for the
spatial coordinates so that
$\partial_{k,k^{\prime}}\equiv\partial_{x}+\partial_{y}$. We have also
employed the Coulomb gauge, so that ${\bm{\nabla}}\cdot{\bm{A}}=0$.

Figure 1:  (Color online) Schematic of the S-Ho/N/Ho-S junction subject to a
perpendicular external magnetic field ${\bf H}$. The junction plane resides in
the $z=0$ plane and the N/Ho interfaces are located at $x=\pm L/2$. The
junction has a length and width of $L$ and $W$, respectively. The two
helimagnets (Holmium type) with internal fields ${\bf h}$ (see text) are
attached to the diffusive normal metal (N) solely for producing spin-1 triplet
pair correlations. The perpendicular external magnetic field induces proximity
vortices in the N region depicted schematically.

We consider a realistic situation where the junctions are well described by a
tunneling process boundary_c . The appropriate boundary conditions for this
regime are the Kupriyanov-Zaitsev boundary conditions boundary_c :

$\displaystyle\zeta\hat{G}\mathbf{n}\cdot\hat{{\bm{\nabla}}}\hat{G}=[\hat{G},\hat{G}_{\text{S}}],~{}~{}G^{R}_{\text{S}}=\left(\begin{array}[]{cc}{\cal
C}&{\cal S}e^{+i\varphi}\\\ {\cal S}e^{-i\varphi}&-{\cal
C}\end{array}\right),~{}~{}$ (10)

where $\zeta$ is the ratio of the barrier resistance to the resistivity of the
normal layer, and the components of the retarded superconducting bulk solution
boundary_c , can be expressed by ${\cal C}\equiv\cosh\theta\sigma_{0}$ and
${\cal S}\equiv i\sinh\theta\sigma_{y}$, in which
$\theta=\text{atanh}(\Delta/\varepsilon)$. The superconducting phase is
denoted by $\varphi$ and the unit vector normal to the interfaces is denoted
by $\mathbf{n}$. Inserting the Riccati parameterized Green function into the
boundary conditions, Eq. (10), we find the following first order differential
equations at $x=\mp L/2$ with $\varphi=\pm\phi/2$:

$\displaystyle\partial_{k}\gamma+2ieH_{z}k^{\prime}\gamma=\pm(2\frac{{\cal
C}}{{\cal S}}+\gamma e^{\mp i\phi/2}-\frac{e^{\pm
i\phi/2}}{\gamma})\frac{{\cal S}\gamma}{\zeta},~{}~{}~{}$ (11a)
$\displaystyle\partial_{k}\tilde{\gamma}-2ieH_{z}k^{\prime}\tilde{\gamma}=\pm(2\frac{{\cal
C}}{{\cal S}}+\tilde{\gamma}e^{\pm i\phi/2}-\frac{e^{\mp
i\phi/2}}{\tilde{\gamma}})\frac{{\cal S}\tilde{\gamma}}{\zeta}.$ (11b)

Next, to generate spin-1 triplet correlations and have them occupy the N
region, several practical ways can be considered Eschrig2015 . For example,
the triplets can be generated in a SF/N/FS type junction with the aid of
uniform noncollinear magnets, or texturized magnets Eschrig2015 ; longrg .
Another option would be the use of spin-active interfaces in the form of
magnetic insulators or materials with strong spin-orbit coupling in the
presence of a Zeeman field Eschrig2015 . Nonetheless, we emphasize that there
are a number of ways to generate spin-1 triplet correlations that would yield
essentially the same results presented here. Therefore, to simplify the setup
and proposed experiment, we consider the structure sketched in Fig. 1, with
the assumption that the spin-1 triplet correlations have been induced in the N
region by the Holmium (Ho) layers which can be described by additional terms
$(\mathbf{h}\cdot\mathbf{\sigma})\gamma-\gamma(\mathbf{h}\cdot\mathbf{\sigma}^{\ast})$,
and
$\tilde{\gamma}(\mathbf{h}\cdot\mathbf{\sigma})-(\mathbf{h}\cdot\mathbf{\sigma}^{\ast})\tilde{\gamma}$
in the Usadel equation Eq. (7a) and (7b), respectively. One can also solve the
Usadel equation (3) without the kinetic part and derive new solutions to the
bulk superconductors in the presence of an inhomogeneous magnetization, the
effect of which is to renormalize $\hat{G}_{\text{S}}$ in Eq. (10), including
the ${\cal C}$ and ${\cal S}$ terms. The Holmium-like magnetization pattern is
coordinate dependent and can be described by
$\mathbf{h}=h_{0}(\cos\psi,\sin\psi\sin\beta x/a,\sin\psi\cos\beta x/a),$ in
which $\psi$ and $\beta$ are the apex and azimuthal angles of the cone that
constitutes the spiral pattern (see Fig. 1), and $a$ is the atomic interlayer
distance volvik . Here we take the widely used values, $\psi=4\pi/9$ and
$\beta=\pi/6$ volvik .

To determine the signatures of various proximity induced superconducting
correlations, we calculate the singlet pair potential $U_{pair}$:

$\displaystyle U_{pair}(\mathbf{R})=-\frac{{\cal
N}_{0}\lambda}{8}\text{Tr}\left\\{\frac{\rho_{x}-i\rho_{y}}{2}\tau_{z}\int
d\varepsilon G^{K}(\varepsilon,\mathbf{R})\right\\},~{}~{}~{}$ (12)

and the local density of states:

$\displaystyle{\cal N}(\varepsilon,\mathbf{R})=\frac{{\cal
N}_{0}}{2}\text{Re}\left[\text{Tr}\\{\hat{G}(\varepsilon,\mathbf{R})\\}\right],$
(13)

where ${\cal N}_{0}$ is the density of states per spin at the Fermi level and
$\lambda$ is the pairing interaction constant.

Figure 2:  (Color online) ($a$) Color map of normalized local density of
states as a function of quasiparticles’ energy $\varepsilon$ and location
along the junction width $y$ at $x=0$ inside the diffusive normal metal N when
the outer Ho layers are off i.e. $h_{0}=0$. ($b$) Corresponding spatial map of
normalized singlet pair potential. Figure 3:  (Color online) ($a$) Local
density of states as a function of quasiparticles’ energy $\varepsilon$ at the
vortex core, $x=y=0$ (shown in Fig. 2) for four values of internal field in
the Ho layers, $h_{0}=0.0,2.0\Delta,4.0\Delta,$ and $6.0\Delta$. ($b$)
Corresponding singlet pair potential, $U_{pair}$, along the junction length,
$x$, in the same location as the vortex core i.e. $y=0$. ($a$) and ($b$)
panels are obtained within the full proximity limit and the other parameters
are identical to those of Fig. 2. ($c$)-($e$) show the modulus of singlet
$|\text{S}(\varepsilon)|$, spin-1 $|\text{T}_{y}(\varepsilon)|$, and spin-0
$|\text{T}_{z}(\varepsilon)|$ triplets against $\varepsilon$ within the low
proximity limit. The solid lines show the correlations at the vortex core
while the dashed lines correspond to a representative location outside of the
vortex core: $x=0.35L,y=0$.

To begin, we present the local DOS and pair potential in Fig. 2 for the case
$h_{0}=0$, i.e. the Ho layers in Fig. 1 have been replaced with normal metals,
and consequently no triplet correlations exist. To further simplify our
analysis, we consider a sufficiently wide junction, $W\gg L$, and set the
external magnetic field so that only a single magnetic flux quantum $\Phi_{0}$
passes through the N region. We also assume representative values of
$\zeta=4$, the system temperature set at $T=0.05T_{c}$ (with critical
temperature $T_{c}$), and a superconducting phase difference $\phi=\pi$. This
choice of $\phi$ only shifts the vortex core to $x=y=0$ ZAD_3DTI ; longrg
without affecting the final outcome. Due to the single magnetic quantum flux
in N, a single proximity vortex is induced berg_fr . As seen in panel ($b$),
the pair potential vanishes at $x=y=0$, coinciding with the normal core of the
vortex. To shed more light on the influence of proximity effects on the vortex
behavior, we have also calculated the corresponding LDOS shown in panel ($a$)
as a function of the quasiparticle energy, $\varepsilon$, and location along
the junction width, $y$ (at $x=0$). It is apparent that the LDOS at $x=y=0$ is
equal to unity which clearly demonstrates that no singlet superconducting
correlation exist in the vicinity of $x=y=0$, where the singlet pair potential
is zero. Note that $U_{pair}$ only involves the spin singlet component of the
Green function even in the presence of an external magnetic field. This can be
clearly seen in the low proximity regime where the contributions from the
singlet and triplet channels can be decomposed longrg . Panel ($a$) shows that
the LDOS becomes reduced at locations away from $x=y=0$. This can be
understood by noting that the singlet pair correlations are responsible for
inducing a minigap in the hybrid structure. This is reflected in the behavior
of the pair potential [panel ($b$)] which shows that $U_{pair}$ increases as
one moves away from the normal core of vortex ($x=y=0$).

Panel ($a$) of Fig. 3 exhibits the LDOS at the center of vortex ($x=y=0$) vs
the quasiparticle energy $\varepsilon$, and at differing values of the
exchange field: $h_{0}=0,2\Delta,4\Delta$, and $6\Delta$. Panel ($b$)
illustrates the corresponding singlet pair potential along the junction length
in the $x$ direction (at $y=0$). To have absolute comparisons, the parameters
are kept the same as those used in Fig. 2. The normalized DOS for $h_{0}=0$ is
equal to unity, corresponding to the normal phase at the vortex core as
discussed in relation to Fig. 2. Switching $h_{0}$ to nonzero values
immediately induces a peak at zero energy and its amplitude increases with
stronger, more inhomogeneous internal fields $\mathbf{h}$. This follows from
the fact that stronger IFMs can more effectively convert singlet correlations
into triplet ones. Although not shown here, our results demonstrated a
disappearance of the ZEP when $\mathbf{h}$ is uniform and collinear. To
clearly determine the type of superconducting correlations responsible for the
ZEPs, one can simply consider the singlet pair potential shown in panel ($b$).
It is apparent that $U_{pair}$ completely vanishes at $x=0$ where the
associated LDOS is calculated in panel ($a$). Therefore, the results clearly
demonstrate that the only nonvanishing pair correlations at $x=y=0$ are the
spin-1 triplet pairs, and therefore are responsible for the induction of the
ZEPs. When the opacity of the interfaces is large enough, e.g. $\zeta=10-20$,
the normal and anomalous Green functions can be approximated by $|G|\sim 1$
and $|F|\ll 1$ and one can expand the Green function around the bulk solution.
This regime allows for the spin parameterization of the Green function via
${F}({\bm{R}},\varepsilon)=i[\text{S}({\bm{R}},\varepsilon)+{\bm{\tau}}\cdot{\textbf{T}}({\bm{R}},\varepsilon)]\tau_{y},$
as exhaustively described in Ref. longrg, . The Green function can then be
decomposed into its singlet $\text{S}({\bm{R}},\varepsilon)$, spin-1
$\text{T}_{y}({\bm{R}},\varepsilon)$, and spin-0
$\text{T}_{z}({\bm{R}},\varepsilon)$ triplet components. Panels ($c$)-($e$)
illustrate the behaviour of these correlations at the vortex core ($x=y=0$)
and at $x=0.35L,y=0$, outside of the core. It is clearly seen that the largest
nonvanishing component within low energies at the center of the vortex is the
odd frequency equal spin component $\text{T}_{y}({\bm{R}},\varepsilon)$.
Although we have focused on precisely the vortex core in our calculations, the
spin singlet component should be practically enough suppressed (and triplets
dominate) within a circle with a radius of the magnetic penetration depth
around the vortex core to experimentally reveal the predicted signatures
above.

It is known that vortices in clean superconductors can host bound states that
are separated in energy by an amount $\sim\Delta^{2}/\varepsilon_{F}$
vrx_theo1 . These bound states yield a peak in the LDOS of a vortex core at
the Fermi level vrx_theo1  that was first observed in Ref. vrx_bs_exp,  and
then followed up by numerous theoretical and experimental works vrx_bs_theo ;
vrx_theo3 ; vrx_exp1 ; vrx_exp2 ; vrx_exp3 ; vrx4 ; vrx1 ; vrx2 ; vrx3 . These
low-energy states reflect relevant details of the bulk gap structure of the
superconducting state balat . It is important to emphasize that the vortices
discussed here are in the diffusive limit where the quasiparticles move in
random directions after each collision with the scattering sources and
$\xi_{S}$ is much larger than the mean free path, thus excluding the existence
of Andreev bound states at the vortex cores. To achieve optimal DOS
signatures, the STM tip should be placed near the vortex core, where the odd
frequency triplet correlations are revealed through an enhancement of the zero
energy quasiparticle states in close vicinity of the tip. Finally, it is worth
mentioning that in light of the specific system parameters used for producing
spin-1 triplet correlations, the thickness of the ${\rm Ho}$ layers and the
actual magnetization patterns can play important roles in the singlet to
triplet conversion process longrg ; crnt_1 ; crnt_2 . In this work, the two
${\rm Ho}$ layers are considered identical and of thickness 10nm longrg ;
crnt_1 ; crnt_2 .

Conclusions. To summarize, motivated by recent experimental progress related
to the proximity induced vortices vrtx_exp , we have proposed an
experimentally accessible platform that utilizes the cores of proximity
vortices to isolate the equal spin triplet pairings Buzdin2005 ; first . We
showed that a proximity-induced vortex can be generated in the normal layer of
a two dimensional diffusive S-Ho/N/Ho-S junction by applying an external
magnetic field to the junction plane, with the Holmium (Ho) layers serving as
sources of spin-1 triplet correlations. We then demonstrated that one can
directly probe the equal spin triplet pairings via a tunneling spectroscopy
experiment at the normal core of the vortex.

###### Acknowledgements.

M.A. would like to thank G. Sewell for his valuable instructions in the
numerical parts of this work. A.Z. was financially supported by the Swedish
Research Council Grant No. 642-2013-7837. K.H. is supported in part by ONR and
a grant of HPC resources from the DOD HPCMP.

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value in the row corresponding to each data-points true feature value. This
shows the overall impact of the location feature, and allows one to view this
impact separately for each value it attains. (A): $h=5$, (B) $h=30$, (B)
$h=60$, (B) $h=180$

It is first important to note that we only have one year of data, so the
season variable here could actually capture some long term systematic change
in response time or incident severity. However, we are not told to expect this
from industry experts. When we consider incidents in the winter, we see the
output is mixed at a 5 minute horizon (Figure 14A), increased at a horizon of
30 minutes (Figure 14B) where as it is decreased at the later horizons. The
effect of spring appears to be systematically smaller in size than the other
seasons.

##  Appendix J AUROC Validation

Whilst we have looked at C-index and Brier score, for different prediction
time, prediction horizon pairs as was done in Lee et al. (2020), we can
equally look at the area under the receiver operator curve as a performance
metric, as was done in Jarrett et al. (2020). We show results for this, at
particular horizons, averaged over all input windows, in Table 11, computed
using the methods discussed in Pölsterl (2020) and Lambert and Chevret (2016).

Table 11: AUROC values for the dynamic models. Larger values indicate a better model. We see similar conclusions are present here that were made when inspecting the time-varying C-index values. Model | Prediction Horizon (minutes) | Mean Over
---|---|---
15 | 30 | 45 | 60 | 120 | 180 | 240 | Horizons
Cox | 0.693 | 0.716 | 0.655 | 0.615 | 0.558 | 0.557 | 0.540 | 0.619
RSF | 0.805 | 0.745 | 0.692 | 0.653 | 0.582 | 0.585 | 0.572 | 0.662
SW (LN) | 0.857 | 0.798 | 0.759 | 0.714 | 0.622 | 0.580 | 0.552 | 0.697
SW (NP) | 0.862 | 0.808 | 0.771 | 0.731 | 0.642 | 0.602 | 0.568 | 0.712
SW (Kernel) | 0.878 | 0.818 | 0.780 | 0.735 | 0.640 | 0.594 | 0.558 | 0.715
SW (Raw) | 0.877 | 0.809 | 0.764 | 0.712 | 0.613 | 0.571 | 0.544 | 0.698

##  Appendix K MAPE as a Function of Time

We have looked at the error at different percentiles into an incident because
of the national criteria set by Highways England when considering incident
duration modelling. However, we note that some other papers consider error at
different minutes into an incident. These are clearly two different methods of
aggregation. The first averages across different events, taking predictions
from potentially very different time points, but that represent some fraction
of the entire incident. The second averages across the same prediction time
for every incident, but disregards the fact that this might be a very short
time relative to the total duration of some events, and a very long time
relative to others. We present results for the second averaging here to give
clear comparison and show our results align generally with the expected
behaviour of other dynamic prediction works. Absolute percentage error (APE),
computed as a function of minutes into an incident for some of the models
considered is given in Figure 15 for events longer than 60 minutes. This
behaviour is generally consistent with that seen in Figure 2 in Li et al.
(2015).

Figure 15: APE (both mean and median shown) for some dynamic models as a
function of the minutes into an incident. We show 30 minutes onwards, which is
the point where only incident related information is being fed into the
sliding window model, and the landmarking models are no longer fit with minor
events, as these have already ended.

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