# Fluctuation Spectroscopy in Granular Superconductors with Application to
Boron-doped Nanocrystalline Diamond

D. T. S. Perkins<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom  G.
M. Klemencic  School of Physics and Astronomy, Cardiff University, Queen’s
Building, The Parade, Cardiff, CF24 3AA, United Kingdom  J. M. Fellows  School
of Physics, H. H. Wills Physics Laboratory, University of Bristol, Tyndall
Avenue, Bristol, BS8 1TL, United Kingdom  R. A. Smith  School of Physics and
Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United
Kingdom

###### Abstract

We perform a detailed calculation of the various contributions to the
fluctuation conductivity of a granular metal close to its superconducting
transition. We find three distinct regions of power law behavior in reduced
temperature, $\eta=(T-T_{c})/T_{c}$, with crossovers at $\Gamma/T_{c}$ and
$E_{Th}/T_{c}$, where $\Gamma$ is the electron tunneling rate, and $E_{Th}$ is
the Thouless energy of a grain. The calculation includes both intergrain and
intragrain degrees of freedom. This complete theory of the fluctuation region
in granular superconductors is then compared to experimental results from
boron-doped nanocrystalline diamond, using the assumption of a constant phase
breaking rate, $\tau_{\phi}^{-1}$. We find a semi-quantitative agreement
between the theoretical and experimental results only in the case of large
phase breaking. We argue that there may be a novel phase breaking mechanism in
granular metals worthy of further experimental and theoretical investigation.

##  I Introduction

Electronic transport in granular materials has been studied both theoretically
and experimentally in great depth since the 1960s, with earlier works
focussing heavily on the behavior near the metal insulator transition [1, 2].
In the late 1990s and early 2000s, theoretical developments allowed for
calculation of transport properties of granular materials in the metallic
regime; see [3] for a review. In these works a granular diagrammatic theory
was used to determine the effects of weak localization [4, 5], electron-
electron interactions [6, 7, 8], and superconducting fluctuations in the
presence of magnetic fields [9, 10, 11] on the electrical conductivity of a
granular metal. This approach only considered the intergrain degrees of
freedom (DOFs), and generally led to the same temperature dependences seen in
homogeneous materials, albeit with a change to the relevant physical
parameters, such as the effective diffusion constant.

In contrast, Lerner et. al. [12] considered superconducting fluctuations in
granular metals from the perspective of the intragrain DOFs only. They
predicted two crossovers in the temperature dependence of the fluctuation
conductivity, $\sigma_{fl}$, as one moves away from the superconducting
transition temperature, $T_{c}$. They assumed that, for a typical granular
metal, $\delta\ll\Gamma\lesssim E_{Th}\lesssim T_{c}$, where $\delta$ is the
mean level spacing, $\Gamma$ is the tunneling rate of electrons between
grains, $E_{Th}=\mathcal{D}_{0}/a^{2}$ is the Thouless energy of a single
grain, $\mathcal{D}_{0}$ is the intragrain diffusion coefficient, and $a$ is
the typical grain size. This assumption means that the system is in the
metallic limit with the dimensionless tunneling conductance
$g_{T}=\Gamma/\delta\gg 1$. The granular nature of the system is ensured since
$g_{g}=E_{Th}/\delta\gtrsim g_{T}$, where $g_{g}$ is the dimensionless
conductance of an isolated grain. Lerner et. al. [12] predicted the first of
these crossovers to occur as the reduced temperature, $\eta=(T-T_{c})/T_{c}$,
approached $\Gamma/T_{c}$, and the second to happen at $E_{Th}/T_{c}$.

Two recent papers by Klemencic et. al. [13, 14] characterised the granular
transport properties in films of boron-doped nanocrystalline diamond (BNCD),
where $T_{c}\lesssim 4$K. In [13] they measured the corrections to the
electrical conductivity due to superconducting fluctuations and observed two
crossovers in the behavior of the fluctuation conductivity. As $T$ increased,
the power law behavior, $\sigma_{fl}\sim\eta^{\alpha}$, changed from
$\alpha=-1/2$ (close-to-$T_{c}$ region), to $\alpha=-3$ (intermediate region),
and back to $\alpha=-1/2$ (far-from-$T_{c}$ region). This matched Lerner et.
al.’s prediction for the close-to-$T_{c}$ and intermediate regions, but not
the far-from-$T_{c}$ region, where they predicted a power law with
$\alpha=-2$.

In this paper we combine the approaches of Beloborodov et. al. [3] and Lerner
et. al. [12] to consider both internal (intragrain) and external (intergrain)
DOFs simultaneously. This enables us to consistently treat the various
temperature regimes: the external DOFs are needed to consider the close-
to-$T_{c}$ to intermediate crossover; the internal DOFs are needed to consider
the intermediate to far-from-$T_{c}$ crossover. We first tackle the problem
analytically for a general granular metal, and obtain power laws for the
different contributions to $\sigma_{fl}$ in each region. We then use the
material parameters obtained by Klemencic et. al. to numerically evaluate the
predictions in BNCD for comparison to experiment.

The rest of the paper is structured as follows: Section II outlines and
extends the granular diagrammatic formalism to include both internal and
external DOFs. The three regions of behavior in $\sigma_{fl}$ emerge
naturally, with simple power law relations appearing deep inside each region
due to the changing energy scale set by the pole of the pair propagator.

In Section III we perform the main diagrammatic calculation, which includes
both internal and external DOFs. We analytically consider the limiting power
law behavior of the Aslamazov-Larkin (AL), Maki-Thompson (MT), and density of
states (DOS) contributions in each region, and summarise these in table 1. In
Section IV we discuss the different power laws produced by each contribution
in each region, and their relative sizes. This allows us to make experimental
predictions for superconducting fluctuations in granular metals. In Section V
we compare our theory to the experimental measurements of fluctuation
conductivity in BNCD by Klemencic et. al. [13]. We find that the inclusion of
phase breaking is necessary to obtain semi-quantitative agreement with
experiment.

##  II Diagrammatic Theory for Granular Systems

We assume a very general form for the Hamiltonian of a granular system with
superconducting correlations,

$\begin{split}H&=\sum_{i}\sum_{\sigma}\sum_{\mathbf{k}}\xi_{\mathbf{k}}c_{i\sigma\mathbf{k}}^{\dagger}c_{i\sigma\mathbf{k}}+\sum_{i}\sum_{\sigma}\sum_{\mathbf{k},\mathbf{q}}U_{i}(\mathbf{q})c_{i\sigma\mathbf{k}+\mathbf{q}}^{\dagger}c_{i\sigma\mathbf{k}}+\sum_{i,j}\sum_{\sigma}\sum_{\mathbf{k},\mathbf{p}}t_{ij}^{\mathbf{k}\mathbf{p}}c_{i\sigma\mathbf{k}}^{\dagger}c_{j\sigma\mathbf{p}}\\\
&\quad+\frac{1}{2}\sum_{i,j}\sum_{\sigma,\sigma^{\prime}}\sum_{\mathbf{k},\mathbf{p}}V_{ij}c_{i\sigma\mathbf{k}}^{\dagger}c_{j\sigma^{\prime}\mathbf{p}}^{\dagger}c_{j\sigma^{\prime}\mathbf{p}}c_{i\sigma\mathbf{k}}-\lambda\sum_{i}\sum_{\begin{subarray}{c}\mathbf{k},\mathbf{p},\\\
\mathbf{q}\end{subarray}}c_{i\uparrow\mathbf{k}}^{\dagger}c_{i\downarrow\mathbf{q}-\mathbf{k}}^{\dagger}c_{i\downarrow\mathbf{q}-\mathbf{p}}c_{i\uparrow\mathbf{p}}.\end{split}$
(1)

The first term is the single grain free-electron Hamiltonian, where
$\xi_{\mathbf{k}}=k^{2}/(2m)-\mu$ is the electron energy relative to the Fermi
surface; the second term describes the random scattering from impurities in a
single grain; the third term accounts for tunneling between grains with the
matrix element $t_{ij}^{\mathbf{k}\mathbf{p}}$ being associated to tunneling
from state $\mathbf{p}$ in the $j$th grain to state $\mathbf{k}$ in the $i$th
grain. The last two terms describe electron-electron interactions. The fourth
term is the Coulomb repulsion, $V_{ij}$, where we neglect dependence upon
intragranular momenta, $q\sim a^{-1}$, since these are much larger than the
intergranular momenta, $Q\sim L^{-1}$, where $L=\mathcal{N}^{1/3}a$ is the
typical system length, and $\mathcal{N}$ is the number of grains. Finally, the
fifth term is the standard s-wave BCS interaction. We may assume that the BCS
interaction acts within a grain, so that our system forms a Josephson junction
array for $T<T_{c}$.

Figure 1: Dyson equation for the granular electron Green’s function.

In this paper we focus on the corrections to the electrical conductivity due
to superconducting fluctuations. Our methodology differs from the previous
granular diagrammatic literature [9, 10, 6, 7, 4, 15, 3, 11, 5], in that we
work in granular real space as opposed to lattice momentum space. Only after
we have summed over the internal DOFs do we transform to the common picture of
lattice momentum space. These ideas closely follow the approach of [16], where
standard rules of diagrammatics [17] are used with a few additional rules to
include granularity into the problem. These new rules are as follows:

  1. 1.

Each electron line receives a grain index, denoted by a Latin character (e.g.
the $j$th grain). All internal grain labels are then summed over.

  2. 2.

Each tunneling vertex, represented by a crossed circle, only allows for
nearest neighbor hopping. The tunneling matrix elements are Gaussian
distributed [3], analogous to the disorder present within a grain, according
to

$\begin{split}\langle t_{ij}^{\mathbf{k}\mathbf{p}}\rangle&=0,\\\ \langle
t_{ij}^{\mathbf{k}\mathbf{p}}t_{lm}^{\mathbf{k}^{\prime}\mathbf{p}^{\prime}}\rangle&=\begin{cases}t^{2}(\delta_{im}\delta_{jl}+\delta_{il}\delta_{jm})\delta_{\mathbf{k}+\mathbf{k}^{\prime}=\mathbf{p}+\mathbf{p}^{\prime}},\\\
0,\qquad\text{otherwise.}\end{cases}\end{split}$ (2)

Each pair of correlated tunneling events then carries a factor of
$t^{2}a^{d}$, analogous to the homogeneous disorder factor $(2\pi
N(0)\tau_{0})^{-1}$, and conserves intragranular momentum.

  3. 3.

Each current vertex carries a factor of
$t_{ij}^{\mathbf{k}\mathbf{p}}ea/\sqrt{\mathcal{N}}$.

The averaged tunneling matrix elements can be related to $\Gamma$ via Fermi’s
golden rule,

$\Gamma=2\pi N(0)a^{d}t^{2},$ (3)

where $N(0)$ is the single spin density of states per unit volume at the Fermi
surface, and $d$ is the dimensionality of the system.

The electron Green’s function for granular systems maintains the same form as
in the homogeneous case,

$G_{i}(\mathbf{k},i\varepsilon)=\frac{1}{i\varepsilon-\xi_{\mathbf{k}}+\frac{i}{2\tau}\text{sgn}(\varepsilon)}.$
(4)

However, $\tau^{-1}$ now contains scattering rates due to both impurities
within a grain, $\tau_{0}^{-1}$, and tunneling back and forth between nearest
neighbor grains,

$\frac{1}{\tau}=\frac{1}{\tau_{0}}+z\Gamma,$ (5)

where $z$ is the coordination number of the lattice. This is represented by
the Dyson series shown in fig. 1, where the thin solid lines denote the free
electron Green’s functions within a single grain, whilst thick solid lines
represent electron Green’s functions including impurity scattering and
tunneling. The dashed lines between pairs of impurity scattering events and
pairs of tunneling events denote their correlation due to averaging over the
Gaussian distributions.

We now define the granular diffuson for the system,
$\Gamma_{ph}(\mathbf{Q},\mathbf{q},i\varepsilon+i\omega,i\varepsilon)$, via
the series shown in fig. 2. Under the assumption that an electron scatters
several times within a grain before tunneling, $l\ll a$ ($l$ is the elastic
mean free path) – or equivalently $\tau_{0}^{-1}\gg z\Gamma$ and hence
$\tau^{-1}\gg z\Gamma$ – we find the granular diffuson to have the form

$\begin{split}\Gamma_{ph}(\mathbf{Q},\mathbf{q},i\varepsilon+i\omega,i\varepsilon)=\frac{1}{2\pi
N(0)\tau^{2}}\frac{\Theta(-\varepsilon(\varepsilon+\omega))}{\mathcal{D}_{0}q^{2}+|\omega|+\Gamma\lambda_{\mathbf{Q}}}.\end{split}$
(6)

In the above, $\lambda_{\mathbf{Q}}=z(1-\gamma_{\mathbf{Q}})$, where
$\gamma_{\mathbf{Q}}=z^{-1}\sum_{\alpha}e^{i\mathbf{Q}\cdot\mathbf{a}_{\alpha}}$
is the structure factor, the $\mathbf{a}_{\alpha}$ are the lattice vectors
connecting nearest neighbor centres, and the sum is over nearest neighbors.
This provides an alternative derivation of the same granular diffuson obtained
by Beloborodov et. al. [3], in which the internal scattering and external
tunneling events are treated on an equal footing. The details are given in the
appendix.

The cooperon,
$\Gamma_{pp}(\mathbf{Q},\mathbf{q},i\varepsilon+i\omega,i\varepsilon)$, is
also obtained in a similar manner, and has the same form as eq. 6 with an
additional phase breaking rate, $\tau_{\phi}^{-1}$, in the denominator

$\Gamma_{pp}(\mathbf{Q},\mathbf{q},i\varepsilon+i\omega,i\varepsilon)=\frac{1}{2\pi
N(0)\tau^{2}}\frac{\Theta(-\varepsilon(\varepsilon+\omega))}{\mathcal{D}_{0}q^{2}+|\omega|+\Gamma\lambda_{\mathbf{Q}}+\tau_{\phi}^{-1}}.$
(7)

In general we will use capital letters to denote the external momenta,
$\mathbf{Q}$, and lowercase letters to denote the internal momenta,
$\mathbf{q}$.

Figure 2: Dyson equation for the granular diffuson.

From the granular cooperon we now obtain the granular pair propagator,

$L(\mathbf{Q},\mathbf{q},i\omega)=-\frac{1}{N(0)}\left[\ln\left(\frac{T}{T_{c,0}}\right)+\psi\left(\frac{1}{2}+\frac{\mathcal{D}_{0}q^{2}+|\omega|+\Gamma\lambda_{\mathbf{Q}}+\tau_{\phi}^{-1}}{4\pi
T}\right)-\psi\left(\frac{1}{2}\right)\right]^{-1},$ (8)

where $\psi(x)$ is the digamma function, $T_{c,0}$ is the bare transition
temperature, and $T_{c}$ is the observed transition temperature with
suppression due to phase breaking. The latter are related via

$\ln\left(\frac{T_{c}}{T_{c,0}}\right)+\psi\left(\frac{1}{2}+\frac{1}{4\pi
T_{c}\tau_{\phi,c}}\right)-\psi\left(\frac{1}{2}\right)=0,$ (9)

where $\tau_{\phi,c}^{-1}$ is the phase breaking rate at $T_{c}$.

In general, following Lerner at. al. [12], we will assume that $\Gamma\ll
E_{Th}\lesssim T_{c}$, as well as considering the specific sub-case where
$E_{Th}\ll T_{c}$. In the small $\Gamma$ limit we may expand the digamma
function to obtain,

$\begin{split}L(\mathbf{Q},\mathbf{q},i\omega)=-\frac{1}{N(0)}\Bigg{[}\ln\left(\frac{T}{T_{c}}\right)&+\psi\left(\frac{1}{2}+\frac{\mathcal{D}_{0}q^{2}+\tau_{\phi}^{-1}}{4\pi
T}\right)-\psi\left(\frac{1}{2}+\frac{1}{4\pi T_{c}\tau_{\phi,c}}\right)\\\
&\qquad\qquad\quad+\psi^{\prime}\left(\frac{1}{2}+\frac{\mathcal{D}_{0}q^{2}+\tau_{\phi}^{-1}}{4\pi
T}\right)\frac{|\omega|+\Gamma\lambda_{\mathbf{Q}}}{4\pi
T}\Bigg{]}^{-1}.\end{split}$ (10)

We write this in a more convenient notation as

$L(\mathbf{Q},\mathbf{q},i\omega)=-\frac{1}{N(0)}\left[\epsilon(\mathbf{q})+\alpha_{1}(\mathbf{q})\frac{|\omega|+\Gamma\lambda_{\mathbf{Q}}}{4\pi
T}\right]^{-1},$ (11)

where

$\alpha_{n}(\mathbf{q})=\psi^{(n)}\left(\frac{1}{2}+\frac{\mathcal{D}_{0}q^{2}+\tau_{\phi}^{-1}}{4\pi
T}\right),$ (12a) and
$\epsilon(\mathbf{q})=\ln\left(\frac{T}{T_{c}}\right)+\alpha_{0}(\mathbf{q})-\alpha_{0,c}(\mathbf{0}).$
(12b)

Here we have used the subscript $c$ on $\alpha_{0,c}(\mathbf{0})$ to denote
that it is evaluated at the transition temperature. It is worth noting that
$\epsilon(\mathbf{0})\simeq\eta=(T-T_{c})/T_{c}$ when $\eta\ll 1$ and the
phase breaking rate is small. This is true in all regions of behavior, except
the extreme part of the far-from-$T_{c}$ region where $T$ approaches $2T_{c}$.

Moving onto the sub-case, $E_{Th}\ll T_{c}$, we may further expand the
denominator of the pair propagator to yield

$L(\mathbf{Q},\mathbf{q},i\omega)=-\frac{1}{N(0)}\Bigg{[}\epsilon(\mathbf{0})+\alpha_{1}(\mathbf{q})\frac{|\omega|+\mathcal{D}_{0}q^{2}+\Gamma\lambda_{\mathbf{Q}}}{4\pi
T}\Bigg{]}^{-1}.$ (13)

As in previous literature [3], we assume the grains form a cubic lattice of
side length $a$, so that

$\lambda_{\mathbf{Q}}=2\sum_{\alpha=1}^{d}\left[1-\cos(Q_{\alpha}a)\right].$
(14)

Considering small $Q$, we see that $\Gamma\lambda_{\mathbf{Q}}\simeq\Gamma
a^{2}Q^{2}$, allowing us to identify $\mathcal{D}_{T}=\Gamma a^{2}$ as the
granular diffusion coefficient.

Figure 3: Leading order contributions to the electrical conductivity due to
superconducting fluctuations in granular systems. (a) and (c) are density of
states (DOS) diagrams; (b) and (d) are Maki-Thompson (MT) diagrams; (e) is the
Aslamazov-Larkin (AL) diagram. Diagrams (c) and (d) cancel exactly in the DC
conductivity limit, and so are not considered.

Finally, we note that the internal momenta are quantized according to the
Neumann boundary condition corresponding to zero current normal to a grain’s
surface,

$\hat{\mathbf{n}}\cdot\nabla\varphi=0,$ (15)

where $\varphi$ is the electron wavefunction. We shall assume, without loss of
generality, that the grains are cubic with side length $a$, so that the
internal momenta are quantized as

$\mathbf{q}=\frac{\pi}{a}(n_{x},n_{y},n_{z}),$ (16)

where $n_{i}=0,1,2,...$. Clearly the above is for a three dimensional system,
but the $d$ dimensional equivalent is trivial to deduce.

Since we have quantized our internal momenta according to eq. 16, we may
analyze the singular nature of the pair propagator to understand the existence
of three regions of behavior as one approaches $T_{c}$ from above. We begin by
identifying the physical energy scales in $L(\mathbf{Q},\mathbf{q},i\omega)$
as $\mathcal{D}_{0}q^{2}$ and $\Gamma\lambda_{\mathbf{Q}}$, which are
associated with the internal and external DOFs respectively. These appear as
the dimensionless combinations $\mathcal{D}_{0}q^{2}/T$ and
$\Gamma\lambda_{\mathbf{Q}}/T$. The smallest non-zero value of
$\mathcal{D}_{0}q^{2}$ is $E_{Th}$, so the dimensionless internal energy scale
is $E_{Th}/T_{c}$. Similarly, $\lambda_{\mathbf{Q}}$ is, at most, of order
unity, so $\Gamma/T_{c}$ is the dimensionless external energy scale.

Looking at the most singular contributions that arise from the pair
propagator, we see that when $\epsilon(\mathbf{0})\simeq\eta\ll\Gamma/T_{c}$,
any non-zero $\mathbf{q}$ gives a significantly less singular function.
However, we may consider small non-zero $\mathbf{Q}$ which give rise to small
changes in the denominator of eq. 13, that are equally singular to the zero
lattice momentum piece. We may therefore treat $\mathbf{Q}$ as being
continuous in this region. It follows that only the external DOFs are
physically relevant here, and so the system appears to be $d$ dimensional.
This is the close-to-$T_{c}$ region.

Next consider $\Gamma/T_{c}\ll\eta\ll E_{Th}/T_{c}$, where again any non-zero
$\mathbf{q}$ leads to less singular contributions, and hence the internal DOFs
again play no role here. However, no choice of $\mathbf{Q}$ will generate a
notable change in the propagator, so that the external DOFs are effectively
unseen in this region. Therefore, neither the internal or external DOFs are
physically relevant in this regime, and so the system appears to be quasi-zero
dimensional. This is the intermediate region.

Finally consider $E_{Th}\ll\eta\leq 1$. As in the intermediate case, the
external DOFs are effectively unseen. However, non-zero $\mathbf{q}$ can give
equally singular contributions to the zero momentum term. The internal momenta
can then be treated as continuous, leading to the system appearing to be $d$
dimensional again. This is the far-from-$T_{c}$ region.

The above gives a concrete definition of the dimensional crossovers described
in the introduction and the works of Lerner et. al. [12] and Klemencic et. al.
[13]. Now that we have established the key ideas behind the granular
diagrammatic method, we proceed to calculate the corrections to the electrical
conductivity due to superconducting fluctuations.

##  III General fluctuation corrections

By analogy to the homogeneous system, the complete set of diagrams describing
the leading order fluctuation corrections to conductivity are shown in fig. 3
[18]. As we are calculating the electromagnetic response function,
$K_{\alpha\beta}(i\Omega)=\Omega\sigma_{\alpha\beta}(i\Omega)$, where $\Omega$
is a bosonic Matsubara frequency, the diagrams should cancel when $\Omega=0$.
We find that this is the case working in the granular diffusive limit,
$\mathcal{D}_{T}Q^{2}\ll z\Gamma$, which is equivalent to $Q\ll a^{-1}$. In
the following we therefore replace $\Gamma\lambda_{\mathbf{Q}}$ with
$\mathcal{D}_{T}Q^{2}$.

The new features arising in the granular diagrams of fig. 3 compared to the
homogeneous diagrams are the correlated fourth order tunneling events shown in
figs. 3c, 3d, and 3e. These are the leading order contributions at
$\mathcal{O}(t^{4})$ after averaging over the tunneling matrix elements. Other
choices of event placement or correlation pairing either contribute nothing or
generate higher order corrections.

We now focus on the explicit calculation of the density of states (DOS),
Aslamazov-Larkin (AL), and Maki-Thompson (MT) diagrams, in that order. We
assume that the system is in the granular metallic limit, $\delta\ll\Gamma\ll
E_{Th}\ll T_{c}$, so that $1\ll g_{T}\ll g_{g}$. We first consider the DOS
contribution in detail, constructing the electromagnetic response function in
lattice real space, transforming to lattice momentum space, and treating both
the internal and external DOFs simultaneously. This yields the most general
form of the DOS contribution, after which we begin to consider the three
limiting regimes that occur naturally in the pair propagator:
$\eta\ll\Gamma/T_{c}$, $\Gamma/T_{c}\ll\eta\ll E_{Th}/T_{c}$, and
$E_{Th}/T_{c}\ll\eta\ll 1$. We then summarise the expected temperature
dependence of the DOS correction to the electrical conductivity deep inside
each temperature region.

We then move on to the AL and MT diagrams in turn, presenting fewer
mathematical details here, as the same ideas and methods are used as in the
DOS calculation. In each case we first present the most general form of the
response function with both sets of DOFs, before considering the different
limiting behaviors. We finally summarise the temperature dependences of all
corrections in each region. A complete set of our results is presented in
table 1 at the end of this section.

###  III.1 DOS corrections

To the diagram in fig. 3a we associate the electromagnetic response function

$\begin{split}K_{\alpha\alpha}(i\Omega)=-\frac{4e^{2}a^{2}t^{2}T^{2}}{a^{2d}\mathcal{N}}&\sum_{\begin{subarray}{c}i,l,\\\
m\end{subarray}}\sum_{\varepsilon,\omega}\sum_{\begin{subarray}{c}\mathbf{k},\mathbf{p},\\\
\mathbf{q}\end{subarray}}\bigg{\\{}G_{i}(\mathbf{k},i\varepsilon+i\Omega)^{2}G_{i}(\mathbf{q}-\mathbf{k},i\omega-i\varepsilon-i\Omega)\\\
&\times[G_{i+\alpha}(\mathbf{p},i\varepsilon)+G_{i-\alpha}(\mathbf{p},i\varepsilon)]L_{lm}(\mathbf{q},i\omega)\\\
&\times
C_{il}(\mathbf{q},i\varepsilon+i\Omega,i\omega-i\varepsilon-i\Omega)C_{im}(\mathbf{q},i\varepsilon+i\Omega,i\omega-i\varepsilon-i\Omega)\bigg{\\}},\end{split}$
(17)

where $\varepsilon$ is a fermionic Matsubara frequency, whilst $\Omega$ and
$\omega$ are bosonic Matsubara frequencies. The functions
$C_{ij}(\mathbf{q},i\varepsilon+i\omega,i\varepsilon)=2\pi
N(0)\tau\,\Gamma_{pp,ij}(\mathbf{q},i\varepsilon+i\omega,i\varepsilon)$ are
the closed cooperons at either end of the pair propagator. The labels $i$ and
$i\pm\alpha$ belong to the electron propagators, whilst $l$ and $m$ arise from
the closed cooperons allowing for tunneling to a new grain, from which the
pair propagator begins or ends. The associated electron lines are not seen
explicitly in the diagrams, however, as their Green’s functions have already
been summed over in deriving the closed cooperon.

These cooperons lead to two possible sign choices for the frequencies
$\varepsilon$ and $\omega-\varepsilon-\Omega$. We first focus on the case
where $\varepsilon+\Omega>0$, $\omega-\varepsilon-\Omega<0$, and
$\varepsilon<0$. Before obtaining the conductivity from $K(i\Omega)$, we need
to move to the lattice momentum picture. We start by performing the
intragranular momentum sums of the electron Green’s functions at the Fermi
surface, leaving the sum over the small intragranular momenta, $\mathbf{q}$.
We then introduce the granular spatial Fourier transforms of the remaining
functions via,

$L_{ij}(\mathbf{q},i\omega)=\frac{1}{\mathcal{N}}\sum_{\mathbf{Q}}L(\mathbf{Q},\mathbf{q},i\omega)e^{i\mathbf{Q}\cdot\mathbf{R}_{ij}},$
(18)

where $\mathbf{R}_{ij}=\mathbf{R}_{i}-\mathbf{R}_{j}$ is the relative position
vector connecting the centres of grains $i$ and $j$, located at
$\mathbf{R}_{i}$ and $\mathbf{R}_{j}$ respectively.

Now we perform analytic continuation of the Matsubara frequency $i\Omega$ to
real frequency $\Omega+i\delta$, where $\delta$ is a positive infinitesimal,
to find the retarded electromagnetic response function,
$K^{R}_{\alpha\beta}(\Omega)$. The conductivity tensor is then found using

$K^{R}_{\alpha\beta}(\Omega)=-i\Omega\sigma_{\alpha\beta}(\Omega).$ (19)

Given that the $\mathcal{O}(\Omega^{0})$ terms cancel, we expand
$K^{R}_{\alpha\beta}(\Omega)$ to $\mathcal{O}(\Omega)$ to find the DC
conductivity corrections. We further simplify the problem by only retaining
the $\omega=0$ piece of the Matsubara sum to neglect dynamical effects, as in
the homogeneous calculation. Doing this leads to

$\sigma_{DOS}^{(1)}=\frac{N(0)\Gamma
a^{2}e^{2}}{4\pi^{2}Ta^{d}\mathcal{N}}\sum_{\mathbf{Q}}\sum_{\mathbf{q}}\psi^{\prime\prime}\left(\frac{1}{2}+\frac{\mathcal{D}_{0}q^{2}+\Gamma\lambda_{\mathbf{Q}}+\tau_{\phi}^{-1}}{4\pi
T}\right)L(\mathbf{Q},\mathbf{q},0).$ (20)

Given that $\Gamma\ll T_{c}$, we may neglect the $\Gamma\lambda_{\mathbf{Q}}$
term appearing in the digamma derivative, $\psi^{\prime\prime}(x)$.

We may replace the $\mathbf{Q}$ sum by an integral, use the granular diffusive
limit to replace all occurrences of $\Gamma\lambda_{\mathbf{Q}}$ with $\Gamma
a^{2}Q^{2}$ in the integrand, and take the upper limit to be $1/a$. At this
point we focus on the $d=3$ case in order to compare to the experimental
results of [13]. We are thus left with

$\sigma_{DOS}^{(1)}=\frac{\Gamma
e^{2}}{8\pi^{4}Ta}\sum_{\mathbf{q}}\int_{0}^{1}dQ\frac{\alpha_{2}(\mathbf{q})Q^{2}}{\epsilon(\mathbf{q})+\frac{\alpha_{1}(\mathbf{q})\Gamma}{4\pi
T}Q^{2}}.$ (21)

We note that the alternative sign choice, $\varepsilon+\Omega<0$,
$\omega-\varepsilon-\Omega>0$, and $\varepsilon<0$, produces an identical
contribution, and so we find the correction due to the DOS to be twice that of
eq. 21,

$\sigma_{DOS}=\frac{e^{2}}{\pi^{3}a}\sum_{\mathbf{q}}\frac{\alpha_{2}(\mathbf{q})}{\alpha_{1}(\mathbf{q})}\left[1-\sqrt{\frac{4\pi
T\epsilon(\mathbf{q})}{\Gamma\alpha_{1}(\mathbf{q})}}\arctan\left(\sqrt{\frac{\Gamma\alpha_{1}(\mathbf{q})}{4\pi
T\epsilon(\mathbf{q})}}\right)\right].$ (22)

This is as much progress as can be made with exact analytics. We now consider
the different temperature regions by taking the appropriate limits of eq. 22.

In the close-to-$T_{c}$ region, $\eta\ll\Gamma/T_{c}$, only the
$\mathbf{q}=\mathbf{0}$ component gives any significant contribution. All non-
zero internal momentum contributions are less singular than the zero momentum
piece due to the quantized nature of $\mathbf{q}$ and the fact that the
Thouless energy is much larger than the tunneling rate. We therefore reproduce
the expected behavior analogous to homogeneous case,

$\sigma_{DOS}=\frac{e^{2}}{\pi^{3}a}\frac{\alpha_{2}(\mathbf{0})}{\alpha_{1}(\mathbf{0})}\left[1-\sqrt{\frac{4\pi
T\eta}{\Gamma\alpha_{1}(\mathbf{0})}}\arctan\left(\sqrt{\frac{\Gamma\alpha_{1}(\mathbf{0})}{4\pi
T\eta}}\right)\right],$ (23)

which is approximately constant for $\eta\ll\Gamma/T_{c}$ [19]. In this region
the characteristic size of a fluctuating Cooper pair is much larger than the
typical size of a grain. The granular system thus appears to be a homogeneous
disordered medium, with the tunneling events acting as the source of disorder.

To obtain the correction in the intermediate region, $\Gamma/T_{c}\ll\eta\ll
E_{Th}/T_{c}$, we again note that only the $\mathbf{q}=\mathbf{0}$ term gives
any significant contribution, so that eq. 23 is still valid. The argument of
the $\arctan$ is now small, so we may expand this to leading order to obtain

$\sigma_{DOS}=\frac{\alpha_{2}(\mathbf{0})}{12\pi^{4}}\frac{\Gamma}{T}\frac{e^{2}}{a}\frac{1}{\eta}.$
(24)

In the far-from-$T_{c}$ region, $E_{Th}/T_{c}\ll\eta\ll 1$, we need to
consider the $\mathbf{q}\neq\mathbf{0}$ terms. Since $\eta\gg E_{Th}/T_{c}$,
the summand will be slowly varying for small $\mathbf{q}$ values, and so we
may approximate the sum by an integral with an appropriate upper cut-off at
$q=q_{c}$. A natural cut-off arises from the diffusive limit, so it appears
that $\mathcal{D}_{0}q_{c}^{2}=\tau^{-1}$. However, we need to remember that
the digamma derivatives decay quickly and so become small when
$\mathcal{D}_{0}q^{2}\geq 4\pi T$, and so we define the cut-off via
$\mathcal{D}_{0}q_{c}^{2}=4\pi T$. As the most singular behavior occurs for
$\mathcal{D}_{0}q^{2}\ll 4\pi T$, we expand the digamma functions and their
derivatives so that

$\alpha_{n}(\mathbf{q})\simeq\alpha_{n}(\mathbf{0})+\frac{\mathcal{D}_{0}q^{2}}{4\pi
T}\alpha_{n+1}(\mathbf{0}),$ (25a)
$\epsilon(\mathbf{q})\simeq\ln\left(\frac{T}{T_{c}}\right)+\alpha_{0}(\mathbf{0})-\alpha_{0,c}(\mathbf{0})+\frac{\alpha_{1}(\mathbf{0})}{4\pi
T}\mathcal{D}_{0}q^{2}.$ (25b)

Before performing the $\mathbf{q}$ integral, we are able to expand the
$\arctan$ again, noting that $\epsilon(\mathbf{q})\gg\Gamma/T_{c}$ in this
case. We therefore arrive at the integral,

$\sigma_{DOS}=\frac{\alpha_{2}(\mathbf{0})\Gamma
a^{2}e^{2}}{24\pi^{6}T}\int_{0}^{q_{c}}dq\frac{q^{2}}{\eta+\frac{\alpha_{1}(\mathbf{0})}{4\pi
T}\mathcal{D}_{0}q^{2}},$ (26)

where $q_{c}=\sqrt{4\pi T/E_{Th}}$, and we noted that $\ln(T/T_{c})\simeq\eta$
when $\eta\ll 1$. This leads to the result

$\sigma_{DOS}=\frac{\alpha_{2}(\mathbf{0})}{3\alpha_{1}(\mathbf{0})}\frac{e^{2}}{a}\sqrt{\frac{\Gamma^{2}T}{\pi^{9}E_{Th}^{3}}}\left[1-\sqrt{\frac{\eta}{\alpha_{1}(\mathbf{0})}}\arctan\left(\sqrt{\frac{\alpha_{1}(\mathbf{0})}{\eta}}\right)\right].$
(27)

This correction has the same form as the close-to-$T_{c}$ region, albeit with
a few changes to the constants appearing alongside $\eta$. We see that for
$\eta\ll 1$, the DOS contribution will again be approximately constant. This
behavior can be attributed to the fluctuating Cooper pairs being much smaller
than the typical grain size, and hence the system again appears homogeneous.

We have therefore shown that there are three temperature regions with
different power law relations between $\sigma_{DOS}$ and $\eta$,

$\sigma_{DOS}\sim-\frac{e^{2}}{a}\begin{cases}const.\,,\quad&\eta\ll\frac{\Gamma}{T_{c}}\\\\[5.0pt]
\frac{\Gamma}{T}\eta^{-1}\,,\quad&\frac{\Gamma}{T_{c}}\ll\eta\ll\frac{E_{Th}}{T_{c}}\\\\[5.0pt]
const.\times\sqrt{\frac{\Gamma^{2}T}{E_{Th}^{3}}}\,,\quad&\frac{E_{Th}}{T_{c}}\ll\eta\ll
1.\end{cases}$ (28)

###  III.2 Aslamazov-Larkin corrections

The Aslamazov-Larkin diagram of fig. 3e has the electromagnetic response
function,

$K_{\alpha\beta}(i\Omega)=-\frac{4T}{a^{d}\mathcal{N}}\sum_{\begin{subarray}{c}l,m,\\\
n,s\end{subarray}}\sum_{\mathbf{q}}\sum_{\omega}\bigg{[}\widetilde{B}_{\alpha,ls}(\mathbf{q},i\omega)\widetilde{B}_{\beta,nm}(\mathbf{q},i\omega)L_{ml}(\mathbf{q},i\omega+i\Omega)L_{sn}(\mathbf{q},i\omega)\bigg{]},$
(29)

where the functions $\widetilde{B}_{\alpha,ls}(\mathbf{q},i\omega)$ represent
the triangular blocks either side of the diagram. These may be written
explicitly as

$\begin{split}\widetilde{B}_{\alpha,ls}(\mathbf{q},i\omega,&i\Omega)=\frac{Teat^{2}}{a^{d}}\sum_{i}\sum_{\varepsilon}\sum_{\mathbf{k},\mathbf{p}}\Big{[}G(\mathbf{k},i\varepsilon+i\Omega)G(\mathbf{q}-\mathbf{k},i\omega-i\varepsilon)G(\mathbf{q}-\mathbf{p},i\omega-i\varepsilon)\\\
&\times
G(\mathbf{p},i\varepsilon)C_{li}(\mathbf{q},i\varepsilon,i\omega-\varepsilon)\Big{\\{}C_{si+\alpha}(\mathbf{q},i\varepsilon,i\omega-i\varepsilon)-C_{si-\alpha}(\mathbf{q},i\varepsilon,i\omega-i\varepsilon)\Big{\\}}\Big{]}.\end{split}$
(30)

Noting that only the pair propagator and cooperons have explicit dependence on
the grain indices, we may move easily to the lattice momentum space picture,
so that

$K_{\alpha\alpha}(i\Omega)=-\frac{4T}{a^{d}\mathcal{N}}\sum_{\mathbf{Q}}\sum_{\mathbf{q}}\sum_{\omega}\bigg{[}B_{\alpha}(\mathbf{Q},\mathbf{q},i\omega)^{2}L(\mathbf{Q},\mathbf{q},i\omega+i\Omega)L(\mathbf{Q},\mathbf{q},i\omega)\bigg{]},$
(31)

where

$\begin{split}B_{\alpha}(\mathbf{Q},\mathbf{q},i\omega,i\Omega)=\frac{2Teat^{2}}{a^{d}}&\sin(Q_{\alpha}a)\sum_{\varepsilon}\sum_{\mathbf{k},\mathbf{p}}\bigg{[}G(\mathbf{k},i\varepsilon+i\Omega)G(\mathbf{q}-\mathbf{k},i\omega-i\varepsilon)\\\
&\qquad\times
G(\mathbf{p},i\varepsilon)G(\mathbf{q}-\mathbf{p},i\omega-i\varepsilon)C(\mathbf{Q},\mathbf{q},i\varepsilon,i\omega-i\varepsilon)^{2}\bigg{]}.\end{split}$
(32)

As in the homogeneous case [18, 20, *AL_Phys_Lett], we may take
$\Omega=\omega=0$ inside the blocks to isolate the most singular behavior of
the pair propagators. Computing the momentum and frequency sums within the
granular diffusive limit yields,

$B_{\alpha}(\mathbf{Q},\mathbf{q})=\frac{ea^{2}\Gamma N(0)}{2\pi
T}\psi^{\prime}\left(\frac{1}{2}+\frac{\mathcal{D}_{0}q^{2}+\tau_{\phi}^{-1}}{4\pi
T}\right)Q_{\alpha}.$ (33)

We analytically continue the $\omega$ sum in the usual manner [18], and take
the $\mathcal{O}(\Omega)$ term to arrive at,

$\sigma_{AL}=\sum_{\mathbf{Q}}\sum_{\mathbf{q}}\frac{B_{\alpha}(\mathbf{Q},\mathbf{q})^{2}}{\pi
TNa^{d}}\int_{-\infty}^{+\infty}dz\frac{\text{Im}[L^{R}(\mathbf{Q},\mathbf{q},z)]^{2}}{\sinh^{2}\left(\frac{z}{2T}\right)},$
(34)

where $L^{R}(\mathbf{Q},\mathbf{q},z)$ is the retarded form of the
analytically continued pair propagator. By approximating $\sinh[z/(2T)]\simeq
z/(2T)$, we perform the $z$ integration, and replacing the $\mathbf{Q}$ sum by
an integral, we find

$\begin{split}\sigma_{AL}=\frac{Te^{2}}{6\pi^{2}\Gamma
a}\sum_{\mathbf{q}}\Bigg{[}3\sqrt{\frac{\alpha_{1}(\mathbf{q})\Gamma}{4\pi
T\epsilon(\mathbf{q})}}&\arctan\left(\sqrt{\frac{\alpha_{1}(\mathbf{q})\Gamma}{4\pi
T\epsilon(\mathbf{q})}}\right)\\\
&\qquad\qquad-\frac{\alpha_{1}(\mathbf{q})\Gamma[5\alpha_{1}(\mathbf{q})\Gamma+12\pi
T\epsilon(\mathbf{q})]}{\left(\alpha_{1}(\mathbf{q})\Gamma+4\pi
T\epsilon(\mathbf{q})\right)^{2}}\Bigg{]}.\end{split}$ (35)

At this point we consider the behavior of $\sigma_{AL}$ in the different
temperature regions.

In the close-to-$T_{c}$ region, $\eta\ll\Gamma/T_{c}$, we take the
$\mathbf{q}=\mathbf{0}$ term of eq. 35 to obtain

$\sigma_{AL}=\frac{1}{8}\sqrt{\frac{\alpha_{1}(\mathbf{0})T}{\pi^{3}\Gamma}}\frac{e^{2}}{a}\frac{1}{\eta^{1/2}},$
(36)

and hence we recover the standard $d=3$ result.

In the intermediate region, $\Gamma/T_{c}\ll\eta\ll E_{Th}/T_{c}$, we again
consider just the $\mathbf{q}=\mathbf{0}$ component, but expand in terms of
$\alpha_{1}(\mathbf{0})\Gamma/(4\pi T\eta)$ to obtain the leading order
correction. This results in

$\sigma_{AL}=\frac{\pi}{1920}\frac{\Gamma^{2}}{T^{2}}\frac{e^{2}}{a}\frac{1}{\eta^{3}},$
(37)

and we see that the AL term has the quasi-zero dimensional behavior,
$\sigma_{AL}\sim\eta^{-3}$, reproducing the result of [12]. This is due to the
fluctuating Cooper pairs being of a comparable size to the typical grain.

Finally, for the far-from-$T_{c}$ region, $E_{Th}/T_{c}\ll\eta\ll 1$, we find

$\sigma_{AL}=\frac{1}{960}\sqrt{\frac{\alpha_{1}(\mathbf{0})^{3}\Gamma^{4}}{\pi^{9}TE_{Th}^{3}}}\frac{e^{2}}{a}\frac{1}{\eta^{3/2}}.$
(38)

This is to be expected given the lack of a $q^{2}$ factor in the integrand due
to the current vertices.

In review, we see that the AL term also has three distinct regions, each with
different dependences upon the reduced temperature,

$\sigma_{AL}\sim\frac{e^{2}}{a}\begin{cases}\sqrt{\frac{T}{\Gamma}}\,\eta^{-1/2}\,,\quad&\eta\ll\frac{\Gamma}{T_{c}}\\\\[5.0pt]
\frac{\Gamma^{2}}{T^{2}}\,\eta^{-3}\,,\quad&\frac{\Gamma}{T_{c}}\ll\eta\ll\frac{E_{Th}}{T_{c}}\\\\[5.0pt]
\sqrt{\frac{\Gamma^{4}}{TE_{Th}^{3}}}\,\eta^{-3/2}\,,\quad&\frac{E_{Th}}{T_{c}}\ll\eta\ll
1.\end{cases}$ (39)

###  III.3 Maki-Thompson corrections

Here we are interested in the diagram shown in fig. 3b. The linear response
function written in lattice momentum space for this diagram is

$\begin{split}K_{\alpha\alpha}(i\Omega)=\frac{4e^{2}a^{2}t^{2}T^{2}}{a^{3d}\mathcal{N}}\sum_{\mathbf{Q}}&\sum_{\begin{subarray}{c}\mathbf{k},\mathbf{p},\\\
\mathbf{q}\end{subarray}}\sum_{\varepsilon,\omega}\Bigg{[}\cos(Q_{\alpha}a)G(\mathbf{k},i\varepsilon+i\Omega)G(\mathbf{q}-\mathbf{k},i\omega-i\varepsilon-i\Omega)\\\
&\times
G(\mathbf{p},i\varepsilon)G(\mathbf{q}-\mathbf{p},i\omega-\varepsilon)L(\mathbf{Q},\mathbf{q},i\omega)\\\
&\times
C(\mathbf{Q},\mathbf{q},i\varepsilon+i\Omega,i\omega-i\varepsilon-i\Omega)C(\mathbf{Q},\mathbf{q},i\varepsilon,i\omega-i\varepsilon)\Bigg{]}.\end{split}$
(40)

Within the granular diffusive limit $\cos(Q_{\alpha}a)$ can be approximated by
unity.

There are two types of contribution to the MT term according to the different
sign choices of the Matusbara frequencies of the electron Green’s functions.
The three possible configurations, starting from the bottom left Green’s
function and moving clockwise in fig. 3b, are $++--$, $--++$, and $-+-+$. The
$++--$ and $--++$ terms together produce a result, $\sigma_{MT}^{(reg1)}$,
that is identical to $\sigma_{DOS}/2$; the $-+-+$ term produces a
contribution, $\sigma_{MT}^{(reg2)}$ (identical to $\sigma_{DOS}/2$ in the
absence of phase breaking and when $E_{Th}\ll T_{c}$, but differs otherwise),
and a more singular (anomalous) piece, $\sigma_{MT}^{(an)}$.

Focusing on the $\omega=0$ component, the second regular (non-anomalous) part
can be written as,

$\sigma_{MT}^{(reg2)}=\frac{N(0)\Gamma}{\pi\mathcal{N}}\frac{e^{2}a^{2}}{a^{d}}\sum_{\mathbf{Q}}\sum_{\mathbf{q}}\frac{\alpha_{2}(\mathbf{q})\Gamma\lambda_{\mathbf{Q}}L(\mathbf{Q},\mathbf{q},0)}{\mathcal{D}_{0}q^{2}+\Gamma\lambda_{\mathbf{Q}}+\tau_{\phi}^{-1}},$
(41)

where we have not yet assumed anything about the size of $E_{Th}$. Considering
$d=3$, we perform the $\mathbf{Q}$ integral to yield,

$\begin{split}\sigma_{MT}^{(reg2)}&=S_{\phi}+\frac{1}{2\pi^{3}}\frac{e^{2}}{a}\sum_{\mathbf{q}}\frac{\alpha_{2}(\mathbf{q})}{\alpha_{1}(\mathbf{q})}\\\
&\\!\\!\\!\\!\times\Bigg{[}1-\frac{\alpha_{1}(\mathbf{q})\Gamma\tau_{\phi}}{4\pi
T\tau_{\phi}\epsilon(\mathbf{q})-\alpha_{1}(\mathbf{q})(1+\mathcal{D}_{0}q^{2}\tau_{\phi})}\left(\frac{4\pi
T\epsilon(\mathbf{q})}{\alpha_{1}(\mathbf{q})\Gamma}\right)^{3/2}\arctan\left(\sqrt{\frac{\alpha_{1}(\mathbf{q})\Gamma}{4\pi
T\epsilon(\mathbf{q})}}\right)\Bigg{]},\end{split}$ (42)

where

$\begin{split}S_{\phi}=\frac{1}{2\pi^{3}}\frac{e^{2}}{a}\sum_{\mathbf{q}}\Bigg{[}&\frac{\alpha_{2}(\mathbf{q})\Gamma\tau_{\phi}}{4\pi
T\tau_{\phi}\epsilon(\mathbf{q})-\alpha_{1}(\mathbf{q})(1+\mathcal{D}_{0}q^{2}\tau_{\phi})}\\\
&\qquad\qquad\qquad\times\left(\frac{1+\mathcal{D}_{0}q^{2}\tau_{\phi}}{\Gamma\tau_{\phi}}\right)^{3/2}\arctan\left(\sqrt{\frac{\Gamma\tau_{\phi}}{1+\mathcal{D}_{0}q^{2}\tau_{\phi}}}\right)\Bigg{]}.\end{split}$
(43)

In the absence of phase breaking, $S_{\phi}=0$ and eq. 42 collapses to half of
eq. 22 when $E_{Th}\ll T_{c}$, as expected. The major new feature we see in
the granular $\sigma_{MT}^{(reg2)}$ contribution is the presence of
$S_{\phi}$.

The close-to-$T_{c}$ behavior is given by the $\mathbf{q}=\mathbf{0}$ piece of
eq. 42. To find the intermediate behavior, we expand the $\arctan$ containing
$\eta$ in eq. 42 to third order in its argument, and then take
$\mathbf{q}=\mathbf{0}$,

$\sigma_{MT}^{(reg2)}=\frac{1}{2\pi^{3}}\frac{\alpha_{2}(\mathbf{0})}{\alpha_{1}(\mathbf{0})}\frac{e^{2}}{a}\left[1+\frac{\alpha_{1}(\mathbf{0})\Gamma\tau_{\phi}}{4\pi
T\tau_{\phi}\eta-\alpha_{1}(\mathbf{0})}\left(\frac{1}{3}-\frac{4\pi
T\eta}{\alpha_{1}(\mathbf{0})\Gamma}+\frac{\arctan(\sqrt{\Gamma\tau_{\phi}})}{(\Gamma\tau_{\phi})^{3/2}}\right)\right].$
(44)

Finally, the far-from-$T_{c}$ behavior is found by performing the $\mathbf{q}$
sum numerically in the case that we are unable to perform an expansion in
$\mathcal{D}_{0}q^{2}$. If, however, $E_{Th}$ is sufficiently small to allow
this expansion, instead of taking the $\mathbf{q}=\mathbf{0}$ piece to obtain
the intermediate result, we may expand $\epsilon(\mathbf{q})$ to first order
in $\mathcal{D}_{0}q^{2}$, whilst setting $\mathbf{q}=\mathbf{0}$ in
$\alpha_{n}(\mathbf{q})$, and replace the $\mathbf{q}$ sum by an integral.
This integral has the usual cut-off, and so yields,

$\begin{split}\sigma_{MT}^{(reg2)}=S_{\phi}+\frac{1}{12\pi^{5}}\frac{\alpha_{2}(\mathbf{0})}{\alpha_{1}(\mathbf{0})}\left(\frac{4\pi
T}{E_{Th}}\right)^{3/2}&\frac{1}{4\pi
T\tau_{\phi}\eta-\alpha_{1}(\mathbf{0})}\\\
&\times\left\\{\Gamma\tau_{\phi}\eta\left[1-\sqrt{\eta}\arctan\left(\frac{1}{\sqrt{\eta}}\right)\right]-\alpha_{1}(\mathbf{0})\right\\},\end{split}$
(45)

where $S_{\phi}$ will have to be handled numerically.

It is clear that $\sigma_{MT}^{(reg2)}$ will only be singular in the
intermediate region, similar to the DOS contribution. In the close-to-$T_{c}$
and far-from-$T_{c}$ regions, the behavior is approximately constant for small
phase breaking rates. For larger phase breaking rates, the temperature
dependence of $\tau_{\phi}^{-1}$ may become important.

Moving onto the anomalous contribution, we consider only the $\omega=0$
component to yield

$\sigma_{MT}^{(an)}=\frac{N(0)\Gamma}{\pi\mathcal{N}}\frac{e^{2}a^{2}}{a^{d}}\sum_{\mathbf{Q}}\sum_{\mathbf{q}}\frac{\alpha_{1}(\mathbf{q})L(\mathbf{Q},\mathbf{q},0)}{\mathcal{D}_{0}q^{2}+\Gamma\lambda_{\mathbf{Q}}+\tau_{\phi}^{-1}}.$
(46)

Taking $d=3$, we perform the $\mathbf{Q}$ integration to give

$\begin{split}\sigma_{MT}^{(an)}&=\frac{2T\tau_{\phi}}{\pi^{2}}\frac{e^{2}}{a}\sum_{\mathbf{q}}\frac{\alpha_{1}(\mathbf{q})}{\alpha_{1}(\mathbf{q})(\mathcal{D}_{0}q^{2}\tau_{\phi}+1)-4\pi
T\tau_{\phi}\epsilon(\mathbf{q})}\\\
&\\!\\!\\!\\!\times\Bigg{[}\sqrt{\frac{\mathcal{D}_{0}q^{2}+\tau_{\phi}^{-1}}{\Gamma}}\arctan\left(\sqrt{\frac{\Gamma}{\mathcal{D}_{0}q^{2}+\tau_{\phi}^{-1}}}\right)-\sqrt{\frac{4\pi
T\epsilon(\mathbf{q})}{\alpha_{1}(\mathbf{q})\Gamma}}\arctan\left(\sqrt{\frac{\alpha_{1}(\mathbf{q})\Gamma}{4\pi
T\epsilon(\mathbf{q})}}\right)\Bigg{]}.\end{split}$ (47)

For simplicity we initially set $\tau_{\phi}^{-1}=0$, and return to the
discussion of a non-zero value later.

In the close-to-$T_{c}$ region ($\eta\ll\Gamma/T_{c}$), we recover the
analogous $d=3$ homogeneous result

$\sigma_{MT}^{(an)}=\sqrt{\frac{T}{2\pi\Gamma}}\frac{e^{2}}{a}\frac{1}{\eta^{1/2}}.$
(48)

In the intermediate region ($\Gamma/T_{c}\ll\eta\ll E_{Th}/T_{c}$) we find,

$\sigma_{MT}^{(an)}=\frac{1}{4\pi}\frac{e^{2}}{a}\frac{1}{\eta},$ (49)

which has the same power law form as the DOS term, but with a prefactor of
order unity as opposed to $\Gamma/T_{c}$. Finally, in the far-from-$T_{c}$
region ($E_{Th}/T_{c}\ll\eta\ll 1$), we obtain

$\sigma_{MT}^{(an)}=\sqrt{\frac{\Gamma^{2}T}{8\pi
E_{Th}^{3}}}\frac{e^{2}}{a}\frac{1}{\eta^{1/2}}.$ (50)

In summary, we see that the Maki-Thompson correction has the following
dependence upon the reduced temperature,

$\sigma_{MT}^{(an)}\sim\frac{e^{2}}{a}\begin{cases}\sqrt{\frac{T}{\Gamma}}\,\eta^{-1/2}\,,\quad&\eta\ll\frac{\Gamma}{T_{c}}\\\\[5.0pt]
\eta^{-1}\,,\quad&\frac{\Gamma}{T_{c}}\ll\eta\ll\frac{E_{Th}}{T_{c}}\\\\[5.0pt]
\sqrt{\frac{\Gamma^{2}T}{E_{Th}^{3}}}\,\eta^{-1/2}\,,\quad&\frac{E_{Th}}{T_{c}}\ll\eta\ll
1.\end{cases}$ (51)

The complete set of the DOS, AL, and MT corrections, and their temperature
dependence in the three regions of behavior are summarised in table 1.

Table 1: Summary of the regional dependences of the DOS, AL, and MT corrections upon $\eta$ and the energy scales $\Gamma$, $E_{Th}$, and $T_{c}$, when $\tau_{\phi}^{-1}=0\,$K. Diagram | $\eta\ll\frac{\Gamma}{T_{c}}$ | $\frac{\Gamma}{T_{c}}\ll\eta\ll\frac{E_{Th}}{T_{c}}$ | $\frac{E_{Th}}{T_{c}}\ll\eta\ll 1$
---|---|---|---
$\sigma_{DOS}$ | $1$ | $\frac{\Gamma}{T}\,\eta^{-1}$ | $\sqrt{\frac{\Gamma^{2}T}{E_{Th}^{3}}}$
$\sigma_{AL}$ | $\sqrt{\frac{\Gamma}{T}}\,\eta^{-1/2}$ | $\frac{\Gamma^{2}}{T^{2}}\,\eta^{-3}$ | $\frac{\Gamma^{2}}{\sqrt{TE_{Th}^{3}}}\,\eta^{-3/2}$
$\sigma_{MT}^{(an)}$ | $\sqrt{\frac{\Gamma}{T}}\,\eta^{-1/2}$ | $\eta^{-1}$ | $\sqrt{\frac{\Gamma^{2}T}{E_{Th}^{3}}}\,\eta^{-1/2}$

Having demonstrated the existence of three distinct temperature regions in a
granular system, we will now discuss the relative magnitude and temperature
dependence of the various contributions, to determine which power law
behaviors we expect to see experimentally.

Figure 4: Fluctuation conductivity contributions with no phase breaking. The
black vertical lines show where the power law crossovers were seen in
Klemencic et. al.’s experiment [13]. The dashed lines act as guides for power
law behavior. (a) Total fluctuation conductivity; (b) DOS plus regular MT
contribution; (c) AL contribution; (d) anomalous MT contribution.

##  IV Theoretical discussion

Unlike the AL term, the MT and DOS terms have the same power law behavior in
both their close-to-$T_{c}$ and far-from-$T_{c}$ regions. This is due to the
MT and DOS terms being $\mathcal{O}(t^{2})$, whilst AL is
$\mathcal{O}(t^{4})$. The $\mathcal{O}(t^{4})$ behavior results in the
generation of a $\sin(Q_{\alpha}a)$ term, and hence an additional factor of
$Q$, within each block of the AL diagram, as in the homogeneous case. However,
no additional factors of $q$ are generated, so the internal and external DOFs
act differently. In contrast, the MT and DOS terms do not gain additional
factors of either internal or external momenta. It follows that the $Q$ and
$q$ integrals are equivalent in close-to-$T_{c}$ and far-from-$T_{c}$ regions,
leading to the same power law behavior.

Examining the magnitude of the diagrams, we see that the AL correction cannot
dominate the anomalous MT correction in the absence of a significant phase
breaking rate. In the close-to-$T_{c}$ region,
$\sigma_{AL}\sim\sigma_{MT}^{(an)}$, whereas
$\sigma_{AL}\sim(\Gamma/T)^{2}\eta^{-3}$ and $\sigma_{MT}^{(an)}\sim\eta^{-1}$
in the intermediate region. Despite the AL term being more singular in $\eta$,
its prefactor is much smaller since $\Gamma\ll T_{c}\lesssim T$. Simple
comparison of these terms shows that the $\eta^{-3}$ behavior of the AL
correction can only dominate over the anomalous MT term when
$\eta\lesssim\Gamma/T_{c}$, which is clearly not in the region where we expect
these power laws to exist. Thus, if $\tau_{\phi}^{-1}=0$, then all behavior
seen in the intermediate regime would be due to the anomalous MT contribution.

The DOS term is of order $\Gamma/T_{c}$, so the AL contribution may be able to
dominate over the DOS correction within part of the intermediate regime. The
region of AL dominance would depend on the relative size of
$\sqrt{\Gamma/T_{c}}$ and $E_{Th}/T_{c}$. In the far-from-$T_{c}$ region, the
AL term will only dominate over the DOS when
$\eta\lesssim(\Gamma/T_{c})^{2/3}$, which may not occur within this region,
again depending upon the size of $(\Gamma/T_{c})^{2/3}$ compared to
$E_{Th}/T_{c}$.

Looking at the total fluctuation conductivity, in the absence of phase
breaking mechanisms, we therefore expect the anomalous MT and AL terms to
dominate in the close-to-$T_{c}$ region, whilst the anomalous MT and DOS
contributions will dominate in the intermediate and far-from-$T_{c}$ regions.
The resulting fluctuation conductivity would hence be

$\sigma_{fl}\sim\frac{e^{2}}{a}\begin{cases}\sqrt{\frac{T}{\Gamma}}\frac{1}{\eta^{1/2}}\,,\quad&\eta\ll\frac{\Gamma}{T_{c}}\\\\[5.0pt]
\frac{1}{\eta}\,,\quad&\frac{\Gamma}{T_{c}}\ll\eta\ll\frac{E_{Th}}{T_{c}}\\\\[5.0pt]
\sqrt{\frac{\Gamma^{2}T}{E_{Th}^{3}}}\frac{1}{\eta^{1/2}}\,,\quad&\frac{E_{Th}}{T_{c}}\ll\eta\ll
1.\end{cases}$ (52)

Including a phase breaking mechanism, $\tau_{\phi}^{-1}\neq 0$, suppresses the
anomalous MT term. If the suppression is large enough, then the anomalous MT
and DOS terms may cancel almost perfectly, as they have the same intermediate
behavior and are opposite in sign. In this case, the $\eta^{-3}$ power law of
the AL term may be able to dominate in the intermediate region and therefore
be observable. The exact temperature dependence of the phase breaking
mechanisms present in granular systems is currently not known, and is beyond
the scope of this paper. We therefore cannot rule out the possibility of a
novel phase breaking mechanism that alters the temperature dependence of the
anomalous MT term, such that it now produces an $\eta^{-3}$ power law in the
intermediate region.

Figure 5: Experimental measurements of the fluctuation conductance from the
work of Klemencic et. al. [13] with power law fittings in each region. These
measurements were taken on a 329 nm thick BNCD film. The shaded areas show the
approximate regions for the two crossovers.

##  V Comparison to experiment

In what follows, we make use of eqs. 22, 35, and 47 with the parameters
$T_{c}$, $E_{Th}$, $\Gamma$, $a$, and $\delta$ being taken from the works of
Klemencic et. al. [13, 14], and proceed to compute these numerically to make
comparison to experimental observation. We take $T_{c}=3.8\,$K, $E_{Th}=1\,$K
($\mathcal{D}_{0}=13.1\text{ cm s}^{-1}$), $\Gamma=2.62\times 10^{-2}\,$K,
$a=10^{-7}\,$m, and $\delta=5.6\times 10^{-3}\,$K (this corresponds to a
carrier concentration of $n=10^{27}\text{ m}^{-3}$). With these values we
cannot approximate the $\mathbf{q}$ sum as an integral at any point, and thus
the sum must be performed explicitly. Due to our assumption of the diffusive
limit, $\mathcal{D}_{0}q^{2}\ll\tau_{0}^{-1}$, we need only include the
$\mathbf{q}=\mathbf{0}$ term. We therefore do not include the internal DOFs in
our following analysis.

In the absence of phase breaking mechanisms, we plot the theoretically
expected fluctuation conductivity in fig. 4a. Comparing this to the
experimental data of Klemencic et. al. [13] in fig. 5 we see that the
anomalous MT and AL contributions dominate in the close-to-$T_{c}$ region,
producing the expected $\eta^{-1/2}$ behavior. In the intermediate region, the
anomalous MT term overtakes the AL correction leading to an $\eta^{-1}$ power
law. Finally, in the far-from-$T_{c}$ region, the anomalous MT term generates
an $\eta^{-1/2}$ dependence, whilst dominating the AL contribution. Note that
this far-from-$T_{c}$ behavior does not originate from the internal DOFs.

Figure 6: Fluctuation corrections with various constant phase breaking rates,
$\tau_{\phi}^{-1}$. The dashed black line acts as a guide for $\eta^{-3}$
behavior. (a) shows the total fluctuation conductivity using parameters based
upon Klemencic et. al.’s experiment [13]. (b) shows the total fluctuation
conductivity using a custom set of parameters.

This prediction matches the experimental data shown in fig. 5 in the close-
to-$T_{c}$ and far-from-$T_{c}$ regions. However, the data shows a clear
$\eta^{-3}$ power law in the intermediate region, which can be attributed to
the AL term. This implies that the anomalous MT contribution must experience
some form of suppression to allow for the AL power law to dominate.

Let us consider a simple mechanism of suppression in the form of a constant
phase breaking rate. In real systems, $\tau_{\phi}^{-1}$ will also have a
temperature dependence, but that is beyond the scope of this paper. Phase
breaking is often neglected in 3D systems, and only used as a momentum cut-off
in the 1D and 2D cases to avoid infrared divergence issues. In reality,
however, phase breaking processes exist in all dimensionalities and so we
should not neglect them. In fig. 6a we consider the effect of a range of
constant phase breaking rates upon the total fluctuation conductivity. For
large phase breaking rates, this simple mechanism suppresses the anomalous MT
term sufficiently that it almost cancels the DOS correction in the close-
to-$T_{c}$ and intermediate regions. Moving further away from the transition,
the anomalous MT term begins to deviate from the DOS behavior, and starts to
dominate in the far-from-$T_{c}$ region. This constant $\tau_{\phi}^{-1}$ is a
simple and not entirely physical mechanism, and leads to a minimum in the
fluctuation conductivity when $\tau_{\phi}^{-1}\sim T$.

For comparison, let us consider a different set of physical parameters to try
and shift the theoretical close-to-$T_{c}$ to intermediate crossover to higher
values of $\eta$. In fig. 6b we choose $\Gamma=0.1\,$K, and $E_{Th}=3\,$K.
Here we can see that the close-to-$T_{c}$ to intermediate crossover in
$\sigma_{fl}$ occurs at a value of $\eta$ nearer to that seen in experiment.
The $\eta^{-3}$ power law still requires a large phase breaking rate to
appear, and persists over a smaller range of $\eta$. We again observe a
minimum occurring in the conductivity for large $\tau_{\phi}^{-1}$.

The minimum in $\sigma_{fl}$ that appears for large $\tau_{\phi}^{-1}$ is
extremely small, as is the magnitude of the fluctuation data in the far-
from-$T_{c}$ region. As a result, the shape of this data is very sensitive to
the fitting applied in the high temperature region ($T\gg T_{c}$). A small
change in the fitting parameters for the high temperature data can lead to a
significant change in the far-from-$T_{c}$ fluctuation data. Therefore,
obtaining the exact $\eta$ dependence of the far-from-$T_{c}$ region is not
trivial.

In order to truly compare the theoretical and experimental results, one would
need a model of the temperature dependence of $\tau_{\phi}$, which has not
been fully addressed in the literature. In particular, one would need to
determine how the granularity of a material influences $\tau_{\phi}$ in the
fluctuation region. We are left with two possibilities: either the anomalous
MT term is suppressed such that the AL term dominates in the intermediate
region, or the phase breaking rate has a novel temperature dependence that
gives the anomalous MT correction an $\eta^{-3}$ power law. In either case, we
can attribute the close-to-$T_{c}$ behavior to the AL and anomalous MT terms,
and the far-from-$T_{c}$ behavior to the anomalous MT term. To fully
understand the intermediate region, further study of the phase breaking rate
in granular systems is required both theoretically and experimentally.

##  VI Conclusions

In this paper we have demonstrated that in a metallic superconducting granular
system close to and above $T_{c}$, three regions of behavior exist within the
fluctuation conductivity. This was shown by including both internal and
external DOFs into the theoretical analysis, to reflect the character of a
Cooper pair’s varying size compared to the typical grain size. This can be
understood in terms of the existence of two coherence lengths in the granular
system,

$\xi_{g}=\sqrt{\frac{\pi\mathcal{D}_{0}}{8T_{c}\eta}},\qquad\xi_{T}=\sqrt{\frac{\pi\mathcal{D}_{T}}{8T_{c}\eta}},$
(53)

where $\xi_{g}$ is the intragrain coherence length, and $\xi_{T}$ is the
intergrain coherence length. These can be obtained from the prefactors of
$q^{2}$ and $Q^{2}$, respectively, in the pair propagator of eq. 13. The
close-to-$T_{c}$ region occurs when $\xi_{T}\gtrsim a$, the intermediate
regime occurs when $\xi_{T}\lesssim a\lesssim\xi_{g}$, and the far-
from-$T_{c}$ region occurs when $\xi_{g}\lesssim a$.

In order for the $\eta^{-3}$ power law, which is seen experimentally, to be
observable, we found that the inclusion of a significant phase breaking rate
was required, $\tau_{\phi}^{-1}\sim T_{c}$ – such values have been seen
experimentally [22]. This is necessary to suppress the anomalous MT term, such
that it cancels almost perfectly with the DOS contribution, thus allowing the
AL behavior to dominate. As the phase breaking rate is increased, the
observable $\eta^{-3}$ region becomes larger, but eventually an
uncharacteristic minimum develops in the fluctuation conductivity. Looking at
larger values of $\tau_{\phi}^{-1}$ leads to the DOS term dominating, and
hence a negative $\sigma_{fl}$. We note that the far-from-$T_{c}$ fluctuation
conductivity extracted in experiment is sensitive to the high temperature
fitting, and so a minimum may actually be present in the data. The assumption
of a constant phase breaking rate is not realistic, and a detailed
understanding of the temperature dependence of $\tau_{\phi}^{-1}$ may be
necessary for better fitting of theory to experiment.

###### Acknowledgements.

The author D. T. S. Perkins would like to thank Manjinder Kainth and Rose
Davies for useful discussions and feedback regarding the contents of this
paper and its presentation. D. T. S. Perkins acknowledges funding from the UK
Engineering and Physical Sciences Rsearch Council (EPSRC). G. M. Klemencic
wishes to acknowledge financial support by the European Research Council under
the EU Consolidator Grant ‘SUPERNEMS’ (647471) and the UK EPSRC under the
grant ‘A Diamond Bridge to Phase Slip Physics’ (EP/V048457/1).

*

##  Appendix A Derivation of the Granular Diffuson

Here we provide an alternative derivation of the granular diffuson to that of
Beloborodov et. al. [3], with the inclusion of external DOFs. From the diagram
in fig. 2 the first two terms form the first piece of the series, and may be
written respectively as

$\begin{split}S^{(1)}_{ji}&=\frac{1}{2\pi N(0)\tau_{0}}\delta_{ij},\\\
S^{(2)}_{ji}&=a^{d}t^{2}\delta_{\langle ji\rangle},\end{split}$ (54)

where

$\delta_{\langle ji\rangle}=\begin{cases}1,\qquad i,\,j\text{ nearest
neighbors}\\\ 0,\qquad\text{otherwise.}\end{cases}$ (55)

We first compute the diffuson self-energy

$\Pi_{ml}=\frac{\delta_{ml}}{a^{d}}\sum_{\mathbf{k}}G(\mathbf{k}+\mathbf{q},i\varepsilon+i\omega)G(\mathbf{k},i\varepsilon),$
(56)

which is calculated exactly as in the homogeneous case to yield

$\Pi_{ml}(\mathbf{q},i\omega)=2\pi
N(0)\tau(1-\omega\tau-\mathcal{D}_{0}q^{2}\tau)\delta_{ml}.$ (57)

The Dyson equation for the diffuson can then be written as,

$\Gamma_{ph,ji}(\mathbf{q},i\omega)=S^{(1)}_{ji}+S^{(2)}_{ji}+\sum_{l,m}\Big{[}S^{(1)}_{li}+S^{(2)}_{li}\Big{]}\Pi_{ml}(\mathbf{q},i\omega)\Gamma_{ph,jm}(\mathbf{q},i\omega).$
(58)

To solve this we move to lattice momentum space via the transform in eq. 18,
to obtain

$\Gamma_{ph}(\mathbf{Q},\mathbf{q},i\omega)=\Big{[}\Big{\\{}S^{(1)}(\mathbf{Q})+S^{(2)}(\mathbf{Q})\Big{\\}}^{-1}-\Pi(\mathbf{Q},\mathbf{q},i\omega)\Big{]}^{-1}.$
(59)

We next rewrite $S^{(1)}_{ji}$ and $S^{(2)}_{ji}$ as

$S^{(1)}(\mathbf{Q})+S^{(2)}(\mathbf{Q})=\frac{1}{2\pi
N(0)\tau_{0}}+a^{d}t^{2}\sum_{\alpha}e^{iQ_{\alpha}a}=\frac{1}{2\pi
N(0)\tau}\left[1+z\Gamma\tau(\gamma_{\mathbf{Q}}-1)\right],$ (60)

which may be substituted into eq. 59.

Finally, we assume that an electron scatters several times within a grain
before tunneling to a neighboring grain, so that $z\Gamma\tau_{0}\ll 1$ and
hence $z\Gamma\tau\ll 1$. We may now expand the reciprocal of eq. 60 to
leading order in $z\Gamma\tau(\gamma_{\mathbf{Q}}-1)$, and substitute the
result into eq. 59. This yields the diffuson given in eq. 6.

## References

  * Abeles  _et al._ [1975] B. Abeles, P. Sheng, M. D. Coutts, and Y. Arie, Advances in Physics 24, 407 (1975).
  * Adkins [1995] C. J. Adkins, Hopping conductivity in granular metals revisited, in  _Metal-Insulator Transitions Revisited_ , edited by P. P. Edwards and C. N. R. Rao (Taylor and Francis, 1995).
  * Beloborodov  _et al._ [2007] I. S. Beloborodov, A. V. Lopatin, V. M. Vinokur, and K. B. Efetov, Rev. Mod. Phys. 79, 469 (2007).
  * Beloborodov  _et al._ [2004] I. S. Beloborodov, A. V. Lopatin, and V. M. Vinokur, Phys. Rev. B 70, 205120 (2004).
  * Biagini  _et al._ [2005] C. Biagini, T. Caneva, V. Tognetti, and A. A. Varlamov, Phys. Rev. B 72, 041102(R) (2005).
  * Beloborodov  _et al._ [2001] I. S. Beloborodov, K. B. Efetov, A. Altland, and F. W. J. Hekking, Phys. Rev. B 63, 115109 (2001).
  * Beloborodov  _et al._ [2003] I. S. Beloborodov, K. B. Efetov, A. V. Lopatin, and V. M. Vinokur, Phys. Rev. Lett. 91, 246801 (2003).
  * Efetov and Tschersich [2003] K. B. Efetov and A. Tschersich, Phys. Rev. B 67, 174205 (2003).
  * Beloborodov and Efetov [1999] I. S. Beloborodov and K. B. Efetov, Phys. Rev. Lett. 82, 3332 (1999).
  * Beloborodov  _et al._ [2000] I. S. Beloborodov, K. B. Efetov, and A. I. Larkin, Phys. Rev. B 61, 9145 (2000).
  * Skrzynski  _et al._ [2002] B. S. Skrzynski, I. S. Beloborodov, and K. B. Efetov, Phys. Rev. B 65, 094516 (2002).
  * Lerner  _et al._ [2008] I. V. Lerner, A. A. Varlamov, and V. M. Vinokur, Phys. Rev. Lett. 100, 117003 (2008).
  * Klemencic  _et al._ [2017] G. M. Klemencic, J. M. Fellows, J. M. Werrell, S. Mandal, S. R. Giblin, R. A. Smith, and O. A. Williams, Phys. Rev. Materials 1, 044801 (2017).
  * Klemencic  _et al._ [2019] G. M. Klemencic, J. M. Fellows, J. M. Werrell, S. Mandal, S. R. Giblin, R. A. Smith, and O. A. Williams, Scientific Reports 9, 4578 (2019).
  * Beloborodov  _et al._ [2005] I. S. Beloborodov, A. V. Lopatin, and V. M. Vinokur, Phys. Rev. B 72, 125121 (2005).
  * Perkins and Smith [2020] D. T. S. Perkins and R. A. Smith, Ann. Phys. 418, 168170 (2020).
  * Abrikosov  _et al._ [1975] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,  _Methods Of Quantum Field Theory In Statistical Physics_ (Dover Publications, 1975).
  * Larkin and Varlamov [2005] A. I. Larkin and A. A. Varlamov,  _Theory of Fluctuations in Superconductors_ (Oxford University Press, 2005).
  * [19] In eq. 23, $\eta$ should really be $\epsilon(\mathbf{0})$ – the replacement of $\epsilon(\mathbf{0})$ by $\eta$ is only valid when the phase breaking rate is small. We have written $\eta$ in this equation as we are initially interested in deriving the power law behaviors in the reduced temperature, in the absence of phase breaking. The same is true in eqs. 24, 36, 37, and 44.
  * Aslamazov and Larkin [1968a] L. G. Aslamazov and A. I. Larkin, Sov. Phys. Solid State 10, 875 (1968a).
  * Aslamazov and Larkin [1968b] L. G. Aslamazov and A. I. Larkin, Phys. Lett. 26, 238 (1968b).
  * Meiners-Hagen and Gey [2001] K. Meiners-Hagen and W. Gey, Phys. Rev. B 63, 052507 (2001).