# Multi-band $s_{++}$ superconductivity in $\textrm{V}_{3}\textrm{Si}$
determined
from the response to a controlled disorder

Kyuil Cho  Ames Laboratory, Ames, IA 50011, USA  Department of Physics &
Astronomy, Iowa State University, Ames, IA 50011, USA  M. Kończykowski
Laboratoire des Solides Irradiés, CNRS UMR 7642 & CEA-DSM-IRAMIS, École
Polytechnique, F-91128 Palaiseau cedex, France  S. Ghimire  Ames Laboratory,
Ames, IA 50011, USA  Department of Physics & Astronomy, Iowa State University,
Ames, IA 50011, USA  M. A. Tanatar  Ames Laboratory, Ames, IA 50011, USA
Department of Physics & Astronomy, Iowa State University, Ames, IA 50011, USA
Lin-Lin Wang  Ames Laboratory, Ames, IA 50011, USA  V. G. Kogan  Ames
Laboratory, Ames, IA 50011, USA  R. Prozorov<EMAIL_ADDRESS>Ames
Laboratory, Ames, IA 50011, USA  Department of Physics & Astronomy, Iowa State
University, Ames, IA 50011, USA

(December 7, 2021)

###### Abstract

The London penetration depth, $\lambda(T)$, was measured in a single crystal
V3Si. The superfluid density obtained from this measurement shows a distinct
signature of two almost decoupled superconducting gaps. This alone is
insufficient to distinguish between $s_{\pm}$ and $s_{++}$ pairing states, but
it can be achieved by studying the effect of a controlled non-magnetic
disorder on the superconducting transition temperature, $T_{c}$. For this
purpose, the same $\text{V}_{3}\text{Si}$ crystal was sequentially irradiated
by 2.5 MeV electrons three times, repeating the measurement between the
irradiation runs. A total dose of 10 C/cm2 ($6.24\times 10^{19}$
electrons/$\textrm{cm}^{2}$) was accumulated, for which $T_{c}$ has changed
from 16.4 K in a pristine state to 14.7 K (9.3 $\%$). This substantial
suppression is impossible for a single isotropic gap, but also it is not large
enough for a sign-changing $s_{\pm}$ pairing state. Our electronic band-
structure calculations show how five bands crossing the Fermi energy can be
naturally grouped to support two effective gaps, not dissimilar from the iron
pnictides physics. We analyze the results using two-gap models for both,
$\lambda(T)$ and $T_{c}$, which describe the data very well. Thus, the
experimental results and theoretical analysis provide strong support for an
$s_{++}$ superconductivity with two unequal gaps,
$\Delta_{1}\left(0\right)\approx 2.53\;\textrm{meV}$ and
$\Delta_{2}\left(0\right)\approx 1.42\;\textrm{meV}$, and a very weak inter-
band coupling in $\text{V}_{3}\text{Si}$ superconductor.

##  I Introduction

At the time of its discovery in 1953 [1], a cubic (A15 structure)
$\text{V}_{3}\text{Si}$ compound had the highest superconducting transition
temperature, around 17 K. Despite showing a clear exponential attenuation of
all thermodynamic quantities upon cooling towards $T=0$, which signaled a
fully gapped Fermi surface, most of spectroscopic [2, 3], transport [4, 5] and
thermodynamic measurements [3, 6, 7, 8] showed unconventional behavior or at
least some unusual features. Such behavior can be associated with a peculiar
electronic band-structure showing Van Hove singularities in the density of
states (DOS) close to the Fermi level [9, 10, 3, 11]. While this certainly
plays an important role, now we know that a multi-gap superconductivity is
needed as well to understand the measurements. Here we focus on a multi-band,
multi-gap nature of superconductivity in this fascinating material.

While $\textrm{MgB}_{2}$ [12] is commonly accepted as the first confirmed two-
gap superconductor [13, 14, 15], the multi-band superconductivity was studied
much earlier, albeit only theoretically. Soon after the development of the
microscopic model of superconductivity [16] the possibility of “overlapping
bands” was studied [17, 18, 19], eventually leading to a general description
of multi-band superconductivity [20, 21, 22, 23], in particular the effects of
disorder [23]. Nevertheless, before $\textrm{MgB}_{2}$, there was no attempt
to interpret the unusual properties of $\text{V}_{3}\text{Si}$ through the
prism of multi-band effects. The observations of the unconventional London
penetration depth [2, 3], anisotropic upper critical field, $H_{c2}$ [6], an
unexpectedly large decrease of $T_{c}$ with nonmagnetic disorder, either after
neutron irradiation [24, 5] or naturally present in real material [25], a
large $T_{c}/T_{F}\sim 0.01$ ratio ($T_{F}$ is Fermi temperature) [3] and a
variety of vortex lattice configurations [26] all pointed to an unconventional
behavior of a confirmed $s-$wave superconductor. Surely, modern re-
interpretation of many of these results is consistent with, if not fully
explained, by multi-gap superconductivity. Therefore, retrospectively,
$\textrm{V}_{3}\textrm{Si}$ is a much earlier than $\textrm{MgB}_{2}$, “the
first” two-band superconductor.

Experimental observation of a two-gap superconducting state relies on a
substantial decoupling between the two bands and a substantial difference
between them either in terms of dimensionality, electronic properties, pairing
mechanism and/or scattering rates [7, 27, 28, 29]. In terms of more recent
measurements, when multi-band superconductivity became widely accepted and
discussed, circa 2001, while some reports support single-gap conventional
$s-$wave BCS superconductivity in $\textrm{V}_{3}\textrm{Si}$ [28], many more
experimental and theoretical studies point to two distinct energy gaps in this
material [30, 7, 31, 32]. There is a complication, though. Perhaps due to a
variation of stoichiometry, atomic disorder or extremely strain-sensitive
structure of Van Hove singularities in the vicinity of the Fermi level,
$\textrm{V}_{3}\textrm{Si}$ samples show a spread of behaviors, especially in
the properties related to a two-gap superconductivity [33, 4, 10, 24, 30, 8,
11]. Furthermore, establishing a multi-band nature from thermodynamic
measurements is necessary but insufficient for the microscopic understanding
of superconductivity, because the order parameter enters thermodynamic
quantities in the even powers, therefore the gaps of the same or opposite
signs on different bands contribute similarly [7, 27, 34, 35]. In this
situation, phase-sensitive experiments are needed, but it is often difficult
to implement experimentally [36, 22, 37]. While in high$-T_{c}$ cuprates
direct measurements that depend on the phase variation along the Fermi surface
have provided a definitive proof of a sign-changing $d-$wave order parameter
[36], in multi-band iron-based superconductors, a similar simple arrangement
in real space is not possible, and more complicated approaches are needed
[37]. The interpretations of more complex phase-sensitive experiments, such as
quasiparticles interference, are not straightforward either [38, 39].

Scattering off non-magnetic impurities is a phase-sensitive method, albeit
indirect, that was successfully used in iron-based superconductors to probe
the sign-changing multi-band $s_{\pm}$ order parameter [40, 41, 42]. As we
discuss in Section III.E, the suppression of $T_{c}$ formally depends on the
Fermi surface averaging of the order parameter in the first power,
$\left\langle\Delta\left(\mathbf{k}\right)\right\rangle_{FS}$, which is sign-
sensitive. For example, isotropic $s-$wave angular part averages to 1, but
$d-$wave averages to 0. Of course, more than one measurement is needed for
different levels of scattering in the system. In our approach, simultaneous
measurements of normal state resistivity (to characterize the amount of
introduced disorder), the superconducting transition temperature, $T_{c}$
(phase-sensitive measurement), and low-temperature variation of the London
penetration depth, $\lambda\left(T\right)$ (to estimate the anisotropy of the
order parameter amplitude), provide enough information to make that
conclusion. Here we show that this scheme can be applied to prove the
existence of two distinct gaps of the same sign, or $s_{++}$ order parameter,
in the title material, $\textrm{V}_{3}\textrm{Si}$. The utility of such an
approach was extended significantly by recent theoretical analysis of the
impurity scattering in superconductors with non-trivial multi-band structure
[43, 44]. For example, it is possible to have a singlet unconventional pairing
with a sign-changing superconducting order parameter, yet fully gapped Fermi
surface, similar to what we uncover here. However, the electronic band-
structure should support such an unconventional scenario in a first place. We
note that nodeless unconventional superconductivity has been studied in a
context of triplet pairing, such as $p-$wave, which shows a variety of nodal
and nodeless behaviors depending on the material and experimental conditions,
for example, in some heavy-fermion superconductors [45, 46]. However, most
superconductors have singlet pairing states and the studies of the effects of
a controlled disorder is a powerful tool to study unconventional and exotic
states, including multi-gap superconductivity [42]. For example, a similar
combination of London penetration depth measurements of electron-irradiated
samples was used to study disorder-driven transitions of the superconducting
gap [47], the interplay of ferromagnetism and superconductivity [48], proving
fully-gapped superconductivity in a heavy-fermion superconductor [49], and
following the doping evolution of the order parameter [50]. If we include
other types of irradiation, many studies employed neutrons and protons to
induce non-magnetic disorder. Such disorder was used to induce a two-gap to a
single-gap crossover in MgB2 [51], trace the evolution of $s_{\pm}$ symmetry
in iron-pnictides [52] and studying its cross-over to $s_{++}$ state [41], or
significantly suppress the superfluid density [53]. Evidently, a controlled
disorder in conjunction with thermodynamic measurements is a well-established
approach to tune and probe the superconducting state.

##  II Experimental

Our $\textrm{V}_{3}\textrm{Si}$ crystals with $T_{c}\approx 16.4$ K were cut
out of a “master boule” single crystal studied previously, for example in
Refs.[54, 55] and references therein. The resistivity above $T_{c}$ of
pristine samples was in range of $5-10\>\mathrm{\mu\Omega}\cdot\text{cm}$,
consistent with the previous reports [25, 55, 54]. The samples were of sub-mm
size. In particular, the crystal used in electron irradiation study was
$0.73\times 0.62\times 0.2\;\textrm{mm}^{3}$.

The variation of the in-plane London penetration depth, $\Delta\lambda(T)$,
was measured using a self-oscillating tunnel-diode resonator (TDR) technique
[34, 56, 27]. The TDR circuit resonates approximately at 14 MHz, and the
frequency shift is measured with a precision better than one part per billion
(ppb). Its inductor coil generates ac magnetic field,
$H_{ac}<20~{}\text{mOe}$, so that the sample is always in the Meissner state
at temperatures of interest. Details of the technique and its principles are
given in Ref.[57, 58, 59] and the detailed calibration procedure is described
in Refs.[60, 58]. The sample was mounted on a 1 mm diameter sapphire rod and
inserted into a 2 mm diameter inductor coil. The coil and the sample were
mounted in a vacuum inside a ${}^{3}\textrm{He}$ cryostat. The TDR circuit was
actively stabilized at 5 K, and the sample was controlled from 0.4 K and up by
independent LakeShore controllers. It is straightforward to show that the
change of the resonant frequency when a sample is inserted into the coil is
proportional to the sample magnetic susceptibility as long as the change of
the total inductance is small and one can expand, $\Delta
f/f_{0}\approx\varDelta L/2L_{0}$ where $2\pi f_{0}=1/\sqrt{CL_{0}}$ with sub-
index $"0"$ referring to an empty resonator. The coefficient of
proportionality that includes the demagnetization correction is measured
directly by pulling the sample out of the resonator at the base temperature
[60].

Figure 1: (a) $\text{V}_{3}\text{Si}$ unit cell with two formula units, $Z=2$.
(b) Brillouin zone (BZ) and Fermi surfaces (FS) of the five different bands
crossing the Fermi level ($E_{F}$) for $\text{V}_{3}\text{Si}$. The band
numbers correspond to the numbers in the text. (The small FS of Band 5 is
circled for clarity.) (c) Energy band dispersion along the high symmetry
directions of BZ and (d) Partial density of states as function of energy for
the five bands crossing $E_{F}$.

The low-temperature 2.5 MeV electron irradiation was performed at the SIRIUS
Pelletron facility of the Laboratoire des Solides Irradiés (LSI) at the École
Polytechnique in Palaiseau, France. The acquired irradiation dose is
conveniently expressed in C/cm2 and measured directly as a total charge
accumulated behind the sample by a Faraday cage. Therefore, 1
C/cm${}^{2}\approx 6.24\times 10^{18}$ electrons/cm2. In the experiment, the
London penetration depth was measured, then the sample was irradiated, and the
cycle repeated. The irradiation was carried out with the sample immersed in
liquid hydrogen at about 20 K. Low-temperature irradiation is needed to slow
down recombination and migration of defects. Upon warming up to room
temperature, a quasi-equilibrium population of atomic vacancies remains due to
a substantial difference in the migration barriers between vacancies and
interstitials. An example of such incremental irradiation/measurement sequence
showing the resistivity change measured in-situ, as well as the annealing
after warming up, is given elsewhere [61]. In the present case, the sample was
dispatched between the lab and the irradiation facility for the measurements
and irradiation, and then the sequence was repeated until the sample had
accumulated a substantial dose of 10 C/cm2 $\approx 6.24\times 10^{19}$
electrons/cm2. Further information on the physics of electron irradiation can
be found elsewhere [62, 63].

Density functional theory (DFT) with Perdew-Burke-Ernzerhof (PBE) exchange-
correlation functional [64] has been used to calculate the band structure of
$\textrm{V}_{3}\textrm{Si}$ at the experimental lattice constant of
$a=4.741\>\text{Å}\;$ [65]. The DFT calculations have been done in VASP [66]
using projected augmented wave method and a plane-wave basis set with a
kinetic energy cutoff of 246 eV. The charge density is converged on a
($8\times 8\times 8$) Monkhorst-Pack $k-$point mesh, including the
$\Gamma-$point. For the Fermi surface (FS) calculations, a much denser
($30\times 30\times 30$) $k-$point mesh is used. The Fermi velocity for each
band has been calculated by the derivative of the DFT band dispersion, i.e.,
group velocity, and then averaged over the Fermi surface of each band in the
Brillouin zone (BZ), the same method as employed previously [67, 68].

##  III Results and Discussion

###  III.1 Electronic band-structure

Table 1:  Electronic band structure parameters relevant to the $\gamma-$model fitting. The bands are naturally grouped in two effective bands $I$ and $II$. The calculated parameter $\gamma=\left(n_{1}v_{1}^{2}+n_{2}v_{2}^{2}\right)/\sum_{i=1}^{5}n_{i}v_{i}^{2}=n_{I}v_{I}^{2}/\left(n_{I}v_{I}^{2}+n_{II}v_{II}^{2}\right)=0.109$, to be compared with the experimental best fit, $\gamma=0.175$. The effective quantities remapped on two effective bands are shown in the last two columns. band | $v_{F}^{2}\times 10^{15}$ | DOS | Two | $v_{F}^{2}\times 10^{15}$ | DOS
---|---|---|---|---|---
$\left(\textrm{cm}/\textrm{s}\right)^{2}$ | states/eV/cell | bands | $\left(\textrm{cm}/\textrm{s}\right)^{2}$ | st./eV/cell
1 | $2.22$ | $1.22$ | $I$ | 3.69 | 1.74
2 | $4.48$ | $2.26$
3 | $7.11$ | $6.11$ | $II$ | 7.70 | 6.80
4 | $8.18$ | $7.48$
5 | $0.00315$ | $0.11$ | $\times$ |  |

$\textrm{V}_{3}\textrm{Si}$ has a primitive cubic crystal structure in space
group 223 ($\text{Pm}\overline{3}\text{m}$) with V sitting at 6c and Si at 2a
positions as shown in Fig.1(a). The band structure of
$\textrm{V}_{3}\textrm{Si}$, Fig.1(c) has flat pieces along $\Gamma$-X,
$\Gamma$-M and $\Gamma$-R directions, which is similar to
$\text{Nb}_{3}\text{Sn}$, another A15 superconductor with the same group of
elements. There are five bands crossing the $E_{F}$ as highlighted in
different colors in Fig.1(c). The corresponding partial densities of state
(DOS) of these bands are plotted in Fig.1(d) and summarized in Table 1. Among
them, bands 1 and 2 are hole bands with states gathering along the M-R
direction, the edges of the cubic Brillouin zone (BZ) (see Fig.1(b)((1) and
(2)). On the other hand, band 5 has a very small electron pocket around the R
point, and the contribution to DOS is negligibly small. In contrast, bands 3
and 4 are dominant in DOS at $E_{F}$, which corresponds to most of the flat
band contributions around the $\Gamma$ point as seen in Fig.1(c). The three-
dimensional (3D) Fermi surfaces (FS) in Fig.1(b)((3) and (4)) show complex FS
for both bands 3 and 4, which have multiple sheets at $E_{F}$. Analysis of
Fig.1(b) suggests that the five bands can be naturally grouped in two
effective ones. Specifically, bands 1 and 2 are well separated in energy from
bands 3 and 4 at intermediate $\mathbf{k}-$values inside the BZ, making the
inter-band transitions improbable. Furthermore, bands 1 and 2 are much closer
in energy, and this is also true for bands 3 and 4, but at different
$\mathbf{k}$. This suggests grouping bands 1 and 2 into an effective band $I$,
bands 3 and 4 into another effective band $II$, and discarding negligible-DOS
band 5. Electronic parameters of all five bands are reported in Table.1. The
“effective” parameters of two effective bands, $I$ and $II$, are given in the
last two columns. The multi-band average for band $I$ is
$v_{F,I}^{2}=(n_{1}v_{F,1}^{2}+n_{2}v_{F,2}^{2})/(n_{1}+n_{2})$, and similar
for the effective band $II$. As we explain in the two-band $\gamma-$model
Section III.C, the relative contribution of each band to the superfluid
density, $\rho_{s}=\gamma\rho_{I}+\left(1-\gamma\right)\rho_{II},$is given by
the parameter,
$\gamma=n_{I}v_{I}^{2}/\left(n_{I}v_{I}^{2}+n_{II}v_{II}^{2}\right)$ (hence,
$\gamma-$model). As shown in Table 1, we estimate $\gamma=0.109$, which is
quite close to the experimental $\gamma=0.109$, discussed in the next Section
III.C.

The high DOS at $E_{F}$ in $\textrm{V}_{3}\textrm{Si}$ indicates electronic
instability, consistent with literature reports [11, 10, 3]. Although one way
to reduce such instability is to promote an exchange splitting, giving a
magnetic solution at the DFT level, experimentally $\textrm{V}_{3}\textrm{Si}$
is not magnetic. Another way to lift the electronic instability is through the
electron-phonon coupling. Similar band structure with flat bands in
$\text{Nb}_{3}\text{Sn}$ is susceptible to lattice distortion by a phonon mode
[69], indicating a strong electron-phonon coupling in such compounds, hence
obvious connection to superconductivity. Thus, the non-magnetic electronic
band structure of $\textrm{V}_{3}\textrm{Si}$ provides important microscopic
details for superconductivity models, such as DOS at $E_{F}$ and Fermi
velocity, which have been used successfully for MgB2, the first proven two-
band superconductor. In fact, we calculated the electronic band structure of
$\text{MgB}_{2}$ as a benchmark to compare with with the original
$\gamma-$model [7, 29] and one of the first DFT calculations of a two-gap
system [67], and obtained similar results.

###  III.2 London penetration depth and superfluid density

Figure 2: Temperature dependent London penetration depth measured in a V3Si
single crystal in pristine state and after three doses of electron
irradiation. The inset zooms at the low temperature region showing a clear
signature of a second gap developing at around 10 K.

We begin by examining the superfluid density obtained from the measured London
penetration depth. Figure 2 shows temperature dependent variation of London
penetration depth,
$\varDelta\lambda\equiv\lambda\left(T\right)-\lambda\left(T_{min}\right)$,
with the increasing dose of electron irradiation. Considering the exponential
low-temperature behavior, we can safely assume that
$\lambda\left(T_{min}=0.4\>\textrm{K}\right)\approx\lambda\left(0\right)$ and
then the normalized superfluid density is calculated as
$\rho_{s}=\left(\varDelta\lambda\left(0\right)/\varDelta\lambda\left(T\right)\right)^{2}=\left(1+\varDelta\lambda/\lambda\left(0\right)\right)^{-2}$.
The inset in Fig.2 zooms at the low-temperature region. There is a clear,
almost knee-like feature in $\varDelta\lambda\left(T\right)$ around 10 K,
which we now know is expected for a two-gap superconductor with different and
weakly-coupled gaps [27]. Similar features were reported in high quality
$\textrm{V}_{3}\textrm{Si}$ crystals before and not surprisingly was
interpreted as a definitive evidence of a two-gap superconductivity [7, 30, 7,
30]. The temperature of this kink-like feature is suppressed upon irradiation
approximately at the same rate as $T_{c}$, signaling that both gaps change at
a similar rate. Furthermore, in Fig.2, normal state values above $T_{c}$ are
determined by the skin depth. They increase upon irradiation due to the
increase of residual resistivity following the Matthiessen’s rule [70].
Importantly, the superconducting transition temperature, $T_{c}$, is
monotonically and noticeably suppressed from 16.4 K to 14.7 K (9.3$\%$). The
upper cut-off at $T_{c}$ is determined by the normal-metal skin depth, which
allows us to estimate the resistivity in a contact-less way using
$\rho=\mu_{0}\pi f\delta^{2}$ where $\delta\left(T_{c}\leftarrow
T\right)\approx 2\lambda\left(T\rightarrow T_{c}\right)$ and
$\lambda\left(T\right)=\varDelta\lambda\left(T\right)+\lambda\left(0\right)$,
where $\lambda(0)=130\;\textrm{nm}$ from Ref.[3, 4]. The extracted resistivity
values are 8.6, 11, 13.6, and 15.4 $\mu\Omega$cm for 0, 2.1, 5.7 and 10 C/cm2
electron irradiation doses, respectively. These values appear to be quite
comparable with the literature [4, 5, 25]. We note that in a large body of
work on $\mathrm{V}_{3}\mathrm{Si}$, a spread of $\lambda(0)$ values ranging
from 83 nm to 230 nm can be found. They were obtained using different methods,
and in samples of different forms (crystal vs. polycrystalline) and purity
[71, 72, 73]. The value we use is within the statistical maximum of the
current literature values. Importantly, our results and conclusions are
independent of the particular value of $\lambda(0)$.

Figure 3: (a) Symbols show the superfluid density in pristine sample
calculated from the data shown in Fig.2. Blue and green solid lines show
labeled partial superfluid densities, $\rho_{1}$ and $\rho_{2}$, obtained in
the least squares fitting. The thick orange line behind the data shows an
excellent agreement of the data with the fitted total superfluid density,
$\rho_{s}=\gamma\rho_{1}+\left(1-\gamma\right)\rho_{2}$. Best fit parameters
are: $\nu_{11}=0.700$, $\nu_{22}=0.578$, $\nu_{12}=0.005$, and $\gamma=0.175$.
(b) Best fit solutions of the self-consistency gap equations, Eq.2. The $T=0$
gap ratios are, $\varDelta_{1}/T_{c}=1.787$ and $\varDelta_{2}/T_{c}=1.005$.

Figure 3 (a) shows the normalized superfluid density of a
$\textrm{V}_{3}\textrm{Si}$ crystal in pristine state. A similar curve for a
different crystal, cut from the same master boule, showing the same two-gap
structure, was published in our earlier paper where a self-consistent
$\gamma-$model based on Eilenberger formalism was introduced [7]. In the
original $\gamma-$model, two isotropic $s-$wave gaps are obtained from the
solution of the self-consistency equation, and then all thermodynamic
quantities, including the superfluid density, can be calculated. The model was
further generalized to include anisotropic or even nodal gaps [74]. Here, it
is sufficient to consider the original isotropic approach.

###  III.3 The isotropic $\gamma-$model

The $\gamma-$model considers two bands with Fermi velocities, $v_{i}$, and the
partial densities of states, $n_{i}=N_{i}\left(0\right)/N\left(0\right)$,
where $N\left(0\right)$ is the total density of states at Fermi level, so that
$n_{1}+n_{2}=1$. The dimensionless effective interaction constants are defined
as $\nu_{ik}=N(0)V_{ik}$, where $V_{ik}$ is the electron-electron interaction
matrix. Note that in the original paper we used $\lambda$ for the interaction
matrix. To avoid confusion with the London penetration depth, here we use
$\nu_{ik}$. Also note that this definition differs from that used in the
literature, $g_{ik}=n_{k}\nu_{ik}$. Our notation has an advantage of being
symmetric, $\nu_{ik}=\nu_{ki}$. Therefore, for two bands, we have three
coefficients of the interaction matrix, two in-band, $\nu_{11}$ and
$\nu_{22}$, and an inter-band coupling, $\nu_{12}$. In the analysis, we
perform a least-squares fit of the experimental superfluid density shown in 3
(a) in Matlab. If all normal-state parameters of a material are known,
$\nu_{ik}$ are the three fitting parameters. They are reduced to two free
parameters by the equation for $T_{c}\left(\nu_{ik}\right)$,

$1.7638k_{B}T_{c}=2\hbar\omega_{D}\exp(-1/\widetilde{\nu})$ (1)

where we assume conventional electron-phonon mechanism of superconductivity
with $\omega_{D}$ being the Debye frequency. In general, if energy of bosonic
pairing “glue” is known, it should be substituted instead of $\hbar\omega_{D}$
in Eq.1. The pre-factor comes from the weak-coupling approximation used in the
$\gamma-$model. The effective interaction constant,
$\widetilde{\nu}\left(\nu_{ik}\right)$, is obtained from the solution of
algebraic equations containing all coefficients, $\nu_{ik}$, see Section II.A
of Ref. [7]. To fit the superfluid density, first the self-consistent gap
equation is solved at each temperature. Introducing dimensionless quantities,
$\delta_{i}=(\Delta_{i}/T)/(2\pi t)$, where $t=T/T_{c}$, the gaps equations
are given by

$\displaystyle\delta_{i}$
$\displaystyle=\sum_{k=1,2}n_{k}\nu_{ik}\delta_{k}\left(\widetilde{\nu}^{-1}-\ln
t-A_{k}\right),$ $\displaystyle A_{k}$
$\displaystyle=\sum_{n=0}^{\infty}\left[\left(n+1/2\right)^{-1}-\left(\delta_{k}^{2}+\left(n+1/2\right)^{2}\right)^{-1/2}\right]$
(2)

Note that we often set Boltzmann constant, $k_{B}=1$, where it is obvious, and
use it explicitly to emphasize the numerical values or proper dimensions,
e.g., Eq.1. For a given set of the coupling constants, $\nu_{ik}$, and partial
densities of states, $n_{i}$, this system can be solved numerically for
$\delta_{i}\left(t\right)$ and therefore provide the energy gaps,
$\Delta_{i}\left(t\right)=2\pi T\delta_{i}(t)$. This is a crucial step missing
in the so-called $\alpha-$model description of the two-band superconductivity
[13]. While it was useful early on to explain experimental signatures of two-
gap superconductivity in $\textrm{MgB}_{2}$, the fitting parameters of
$\alpha-$model have little physical meaning. A follow-up study used two-gap
functions pre-calculated from the microscopic theory and showed an excellent
agreement between experimental and theoretical superfluid density,
$\rho_{s}\left(t\right)$ [75]. Indeed, the $s-$wave $\textrm{MgB}_{2}$ for
which all normal-state parameters are known is a perfect demonstration of the
$\gamma-$model where different quantities are calculated from $\nu_{ik}$
obtained from the fit of $\rho_{s}\left(t\right)$ [7, 29].

Figure 3(b) shows two gaps calculated self-consistently from Eq.2. The
individual gap ratios are, $\Delta_{1}/T_{c}=1.787$ and
$\Delta_{2}/T_{c}=1.005$. This should be compared with the results of
microwave surface impedance measurements where similar apparent two-gap
behavior was observed in the superfluid density and the values of
$\Delta_{1}/T_{c}=1.8$ and $\Delta_{2}/T_{c}=0.95$, quite close to ours, were
derived [30]. In the absolute units we obtain,
$\Delta_{1}\left(0\right)\approx 2.53\;\textrm{meV}$ and
$\Delta_{2}\left(0\right)\approx 1.42\;\textrm{meV}$. After the gaps are
calculated, the total superfluid density,
$\rho_{s}=\gamma\rho_{1}+\left(1-\gamma\right)\rho_{2}$ can be evaluated and
fitted to the experimental data. The partial contributions to the superfluid
density are given by [7],

$\displaystyle\rho_{i}$
$\displaystyle=\delta_{i}^{2}\sum_{n=0}^{\infty}\left[\delta_{i}^{2}+(n+1/2)^{2}\right]^{-3/2},$
$\displaystyle\gamma$
$\displaystyle=\frac{n_{1}v_{1}^{2}}{n_{1}v_{1}^{2}+n_{2}v_{2}^{2}}$ (3)

where $v_{i}$ are the Fermi velocities (not to be confused with Greek
$\nu_{i}$ of the interaction matrix). Analyzing Fig.1, we group bands 1 and 2
into one effective band $I$, and bands 3 and 4 into another band $II$, and we
can safely neglect band 5. (Here we use Roman numerals $I$ and $II$ to index
these “effective” bands). For the first effective band, we find
$\gamma=\left(n_{1}v_{1}^{2}+n_{2}v_{2}^{2}\right)/\sum_{i=1}^{5}n_{i}v_{i}^{2}=0.109$.
If we included band 5, it’d make the difference only in 6th decimal digit.
Using $\gamma$ as another fitting parameter, the best fit of this model to the
data gave $\nu_{11}=0.700$ (fixed by $T_{c}$, Eq.1), $\nu_{22}=0.578$,
$\nu_{12}=0.005$, and $\gamma=0.175$, with the effective
$\widetilde{\nu}=0.350$ (see Eq.1). Remarkably, the best-fit value of $\gamma$
is quite close to the estimate from the electronic band-structure
calculations, see Table 1 where we find $\gamma=0.109$, . This gives
confidence in the model and shows its applicability to describe the
superconductivity in $\textrm{V}_{3}\textrm{Si}$. Naturally, overall, smaller
partial density of states on band $I$, somewhat counter-intuitively, leads to
a larger gap, which is the property of the self-consistent two-band model [7,
35]. We note that the possible uncertainty in the experimental value of
$\lambda\left(0\right)$ leads to some uncertainty in the fitting parameters,
but not large enough to alter the general conclusion of the relative
amplitudes of the obtained interaction matrix.

Figure 4: Knock-out defects creation cross-sections for vanadium and silicon
ions in $\textrm{V}_{3}\textrm{Si}$ as function of electron energy assuming
the displacement energy threshold, $E_{d}=25\>\mathrm{eV}$. At the operating
energy of 2.5 MeV, the total cross-section is $\sigma=72$ barn, which leads to
the estimate of $4.4\times 10^{-4}$ displacements-per-atom (dpa) per 1 C/cm2
of the irradiation.

###  III.4 Effect of electron irradiation

In the last three decades, many studies involving particle irradiation were
performed on various conventional and unconventional superconductors and there
is a vast literature on this topic [76, 77, 78, 42] Due to the differences in
the rest mass and irradiation temperature, the number and the morphology of
the created defects varies significantly between different projectile
particles. It appears that MeV electrons, thanks to a small rest mass,
transfer just enough energy upon collision with ions, of the order of tens of
eV, to produce well-defined point-like scattering centers [48]. Much larger
energy transfer, for example from protons, produce many secondary collisions
and less localized damage. A more detailed discussion of electron irradiation
and created defects in solids can be found elsewhere [62, 63].

Figure 4 shows the ion-type-resolved cross-sections of the defects creation
calculated using SECTE (“Sections Efficaces Calcul Transport d’Électrons”)
software, developed at École Polytechnique (Palaiseau, France) by members of
the “Laboratoire des Solides Irradiés”, specifically for the interpretation of
MeV-range electron irradiation using their Pelletron-type linear accelerator,
SIRIUS [79]. Basically, this is a computer-assisted atomic-weights-averaged
interpolation of the ion knock-out cross-sections tabulated by O. S. Oen [80].
In the absence of microscopic calculations, we used the commonly assumed value
of the ion displacement energy upon a head-on collision, $E_{d}=25$ eV. The
partial cross-sections are very similar, and we expect a roughly equal number
of defects on vanadium and silicon sites. At the operational energy of 2.5
MeV, the total cross-section is estimated as $\sigma=72$ barn, which means
that roughly 1.8 defects are produced per 1000 unit formulas per 1 C/cm2. This
is a small number of defects that cannot change the electronic structure in
any appreciable way, and a significant reduction of $T_{c}$ observed in our
experiments must have a different explanation, such as its pair-breaking
nature.

First, let us examine the effect of electron irradiation on superfluid
density. While we do not know how much $\lambda\left(0\right)$ changes, we
attempted to adjust its value to scale all curves onto a pristine one. As
shown in Fig.5 this worked rather well with a small increase of
$\lambda\left(0\right)$ values shown in the legend. This indicates that
scattering does not alter the gap values themselves and, due to very small
inter-band coupling, has practically no effect on the total superfluid
density. Each isotropic band follows the Anderson theorem [81] and the change
in $T_{c}$ comes mostly from the inter-band scattering between order
parameters of different magnitude. We note that the relative change of
$\lambda\left(0\right)$ can be estimated from Tinkham’s widely-used approach
[82] that gives for moderate scattering,
$\lambda\approx\lambda_{clean}\sqrt{1-\xi_{0}/\ell}$ , where $\xi_{0}\approx
60$ nm is the BCS coherence length, and $\ell\approx 30$ nm is the electronic
mean free path in the pristine state. Both numbers are estimated for V3Si from
$T_{c}$, Fermi velocity and resistivity, see Table 1 and Fig.2. As shown in
Fig.2, at the maximum irradiation dose, the resistivity doubles. Therefore, we
expect the increase of $\lambda\left(0\right)$ by a factor of about 1.3, which
is not large and does not alter our conclusions, especially considering an
apparent scaling shown in Fig.5 .

Figure 5: Evolution of superfluid density ($\rho_{s}$) upon irradiation. For
the pristine case, we used $\lambda(0)=130$ nm from Ref. [3]. The curves
representing irradiated state were calculated with the penetration depth
$\lambda(0)$ values shown in the legend. They were chosen to collapse the
curves on the pristine one.

###  III.5 Suppression of $T_{c}$ by disorder in a two-band superconductor

While it is clear that the superfluid density shows a convincing two-distinct-
gaps features implying small inter-band coupling, this still leaves an
unanswered important question of the relative sign of the order parameter on
each band. This is because superfluid density, as well as any other
thermodynamic quantity includes even powers of the gap function, so that an
$s_{\pm}$ state cannot be distinguished from an $s_{++}$ state if the gaps are
the same, see, for example, Eq.3. The suppression of $T_{c}$ on the other hand
is very sensitive to the overall anisotropy of the order parameter, including
a generalized view when two bands are considered side by side along the common
path on the entire Fermi surface [83]. This situation can be analyzed
employing a very useful ansatz that temperature and angular parts of the order
parameter can be separated,
$\Delta\left(T,\mathbf{k}_{F}\right)=\Psi\left(T\right)\Omega\left(\mathbf{k}_{F}\right)$,
where $\mathbf{k}_{F}$ is Fermi wave vector and the angular part obeys the
normalization condition for the Fermi surface average,
$\left\langle\Omega^{2}\right\rangle_{FS}=1$ [83, 84]. For example, for a
single band $s-$wave, $\Omega=1$ and for a $d-$wave,
$\Omega=\sqrt{2}\cos\left(2\varphi\right)$. For a two-gap superconductor Kogan
introduced two gaps each described by its own angular part $\Omega_{i}$ [83].
In this case, the normalization reads,

$\left\langle\Omega^{2}\right\rangle=n_{1}\left\langle\Omega_{1}^{2}\right\rangle+n_{2}\left\langle\Omega_{2}^{2}\right\rangle=1$
(4)

In the case of an anisotropic gap, even non-magnetic (no spin-flip) scatterers
suppress superconducting transition temperature, $T_{c}$. With spin-flip
scattering both channels reduce $T_{c}$. Openov gives a generalized Abrikosov-
Gor’kov [85] type expression where gap anisotropy is explicitly taken into
account [86, 87]. We note that a more general theory of the $T_{c}$
suppression by disorder scattering, extended to topologically non-trivial
superconductors, is discussed elsewhere [44, 43]. Here it suffice to consider
the $\Omega-$ approach, which gives,

$\ln
t_{c}=\psi\left(\frac{g+g_{m}}{2t_{c}}+\frac{1}{2}\right)-\psi\left(\frac{1}{2}\right)\\\
-\left<\Omega\right>^{2}\left[\psi\left(\frac{g+g_{m}}{2t_{c}}+\frac{1}{2}\right)-\psi\left(\frac{g_{m}}{t_{c}}+\frac{1}{2}\right)\right]$
(5)

where $t_{c}=T_{c}/T_{c0}$ with $T_{c0}$ being the transition temperature in a
pristine state, and $\psi$ is the digamma function. Dimensionless magnetic and
non-magnetic scattering rates are given by

$g_{(m)}=\frac{\hbar}{2\pi k_{B}T_{c0}}\frac{1}{\tau_{(m)}}$ (6)

where $\tau$ and $\tau_{m}$ are non-magnetic and magnetic (spin flip)
scattering times, respectively. (Note that original Abrikosov-Gor’kov theory
uses a different definition of the scattering rate, $\rho=\hbar/\left(\pi
k_{B}T_{c}\tau\right)$, with the actual (suppressed) $T_{c}$). The effect of
gap anisotropy can be immediately seen from Eq.5 \- it contains $\Omega$ in
the first power. For a single-band $s-$wave,
$\left\langle\Omega\right\rangle=1$ and we obtain $t_{c}=1,$ recovering the
Anderson theorem [81]. For a $d-$wave, $\left\langle\Omega\right\rangle=0$ and
we obtain an expression where both magnetic and non-magnetic impurities
suppress $T_{c}$. It is interesting to note that the critical value for the
complete $T_{c}$ suppression of an $s-$wave order parameter by magnetic
impurities, $g_{m}=0.14$, is exactly half of the value for a $d-$wave order
parameter suppression by non-magnetic impurities, $g=0.28$.

We can now use Eq.5 with the two-gaps $\Omega-$approach, Eq.4 and that
$\left\langle\Omega\right\rangle=n_{1}\left\langle\Omega_{1}\right\rangle+n_{2}\left\langle\Omega_{2}\right\rangle$.
Specifically, we consider two isotropic gaps described by constant values
$\Omega_{1}$ and $\Omega_{1}$. In other words, two bands are represented in a
generalized single angular coordinate, - band 1 from $0$ to $2\pi$, and band 2
from $2\pi$ to $4\pi$, each with its own density of states $N_{i}$, so that
the partial densities of states are $n_{i}=N_{i}/\left(N_{1}+N_{2}\right)$.
Introducing the gap ratio, $r=\Omega_{2}/\Omega_{1}$ and the ratio of the
partial densities of state, $n=n_{2}/n_{1}$=N${}_{1}/N_{2}$, we obtain for the
total average in this two-gap model:

$\left\langle\Omega\right\rangle^{2}=\frac{(nr+1)^{2}}{(n+1)\left(nr^{2}+1\right)}$
(7)

Without magnetic scattering ($g_{m}=0$), the transition temperature of a two-
band superconductor is,

$\ln
t_{c}=\psi\left(\frac{g}{2t_{c}}+\frac{1}{2}\right)-\psi\left(\frac{1}{2}\right)\\\
-\frac{(nr+1)^{2}}{(n+1)\left(nr^{2}+1\right)}\left[\psi\left(\frac{g}{2t_{c}}+\frac{1}{2}\right)-\psi\left(\frac{1}{2}\right)\right]$
(8)

It is important to emphasize that the superfluid density as a function of
(reduced) temperature and $T_{c}$ are two independent measurements, which
makes the analysis better defined and constrained.

Figure 6: Suppression of $T_{c}$ upon different types of particle irradiation.
Normalized $T_{c}$ suppression upon electron irradiation (current study) is
compared with two different previous studies by proton and neutron
irradiations. It is clearly shown that the electron irradiation is most
effective in suppressing $T_{c}$.

To compare the experimentally observed decrease of $T_{c}$ with our model, we
need a proper parameter characterizing the scattering rate. The problem is
that different sources of disorder produce somewhat different effects. Figure
6 compares the relative change of the transition temperature, $\Delta
t_{c}\equiv\left(T_{c}-T_{c0}\right)/T_{c0}$, in $\text{V}_{3}\text{Si}$ per
$1\>\mu\Omega\cdot\textrm{cm}$ of resistivity increase caused by electron
irradiation in the current study with two previous studies where defects were
induced by proton [88] and neutron [5] irradiation. The rates of the relative
change are, $d\Delta t_{c}/d\rho=-0.013$
$\left(\mu\Omega\cdot\textrm{cm}\right)^{-1}$ (electron irradiation), -0.008
$\left(\mu\Omega\cdot\textrm{cm}\right)^{-1}$ (neutron), and -0.006
$\left(\mu\Omega\cdot\textrm{cm}\right)^{-1}$ (proton). Due to their small
rest mass and matching range of the energy transfer (1-100 eV), electrons
produce the most efficient point-like defects and have the largest suppression
rate. A similar trend is observed in other materials, for example, well-
studied iron-based superconductors [42, 48]. On the other hand, the observed
rates are not too different, roughly 0.01
$\left(\mu\Omega\cdot\textrm{cm}\right)^{-1}$, and we can put it in a
perspective by comparing with other superconductors. For that, we need to
calculate the dimensionless scattering rate, Eq.6. In our case of measured
$\lambda\left(T\right)$ and $\rho\left(T\right)$ the simplest estimate of the
scattering time, $\tau$, is via the London and Drude electrodynamics,
$\tau\left(T_{c}\right)=\mu_{0}\lambda_{clean}^{2}\left(0\right)/\rho\left(T_{c}\right)$.
Note that clean-limit value, $\lambda_{clean}\left(0\right)$, needed for the
density of states in the normal metal, enters this estimate, whereas (normal
metal) scattering time comes from resistivity. This approach is well justified
in isotropic $s-$wave superconductors and $s_{++}$ compounds assuming that the
gap smearing caused by the modest amounts of non-magnetic disorder is much
smaller than the gap amplitudes.

In our case, we can use Tinkham’s widely-used approach [82] that gives for
moderate scattering, $\lambda\approx\lambda_{clean}\sqrt{1-\xi_{0}/\ell}$ ,
where $\xi_{0}\approx 60$ nm is the BCS coherence length, and $\ell\approx 30$
nm is the electronic mean free path in the pristine state. Both numbers are
estimated for V3Si from $T_{c}$, Fermi velocity and resistivity, see Table 1
and Fig.2. As shown in Fig.2, at the maximum irradiation dose, the resistivity
doubles. Therefore, we expect the increase of $\lambda\left(0\right)$ by a
factor of about 1.3. This is an insignificant change to alter the main
features reported here - the exponential attenuation at low temperatures and a
higher temperature kink signaling of two barely-coupled gaps of different
magnitude. This is further confirmed by the apparent scaling of the superfluid
density for all doses of electron irradiation, Fig.5.

The experimental dimensionless scattering rate can be estimated as,

$g\approx\frac{\hbar}{2\pi
k_{B}\mu_{0}}\frac{\rho(T_{c})}{T_{c0}\lambda_{clean}(0)^{2}}$ (9)

Note that we measure resistivity change with respect to the pristine sample to
subtract inelastic scattering, but this also removes background impurity
scattering in samples before irradiation. Fortunately, judging by very low
pinning, this correction is negligible [26]. Also, note that this
dimensionless rate contains unmodified $T_{c0}$, which is different from the
original Abrikosov-Gor’kov definition [85].

Figure 7: Normalized change of superconducting transition temperature, $\Delta
t_{c}\equiv\left(T_{c}-T_{c0}\right)/T_{c0}$ as function of the dimensionless
scattering rate, $g$. (a) comparison of $\text{V}_{3}\text{Si}$ single crystal
with known nodeless and nodal $s_{\pm}$ superconductors shown in the legend.
The theoretical curves from Eq.8 are shown by lines. Clearly, all sign-
changing $s_{\pm}$ superconductors show suppression rate larger than in
$\text{V}_{3}\text{Si}$. (b) similar comparison with another $s_{++}$ two-band
superconductor, $\text{NbSe}_{2}$ [89] and unconventional Dirac semi-metal
compound, $\text{PdTe}_{2}$ [43, 90]. As soon as CDW is suppressed,
$\text{NbSe}_{2}$ shows a similar suppression rate as $\text{V}_{3}\text{Si}$.

Figure 7 shows the normalized change of superconducting transition
temperature, $\Delta t_{c}\equiv\left(T_{c}-T_{c0}\right)/T_{c0}$, as function
of the dimensionless scattering rate, $g$. Figure 7(a) compares
$\text{V}_{3}\text{Si}$ single crystal with known nodeless and nodal $s_{\pm}$
superconductors, isovalently substituted
$\text{Ba}(\text{Fe}_{0.76}\text{Ru}_{0.24})_{2}\text{As}_{2}$ and
$\text{BaFe}_{2}(\text{As}_{0.7}\text{P}_{0.3})_{2}$, and hole- doped
underdoped $\text{Ba}_{0.81}\text{K}_{0.19}\text{Fe}_{2}\text{As}_{2}$ and
optimally-doped $\text{Ba}_{0.66}\text{K}_{0.34}\text{Fe}_{2}\text{As}_{2}$
(BaK122) [42]. The theoretical curves from Eq.8 are shown by different lines.
Clearly, all sign-changing $s_{\pm}$ superconductors show suppression rate
significantly higher than in two-gap $\text{V}_{3}\text{Si}$, which is
consistent with $s_{++}$ theoretical curve for gap ratio, $r=+0.4$, while the
same gap ratio, but of opposite signs, $r=-0.4$, is close to BaK122 data.
Interestingly, a nodal multi-band $s_{\pm}$ superconductor,
$\text{BaFe}_{2}(\text{As}_{0.7}\text{P}_{0.3})_{2}$, shows even greater rate
of $T_{c}$ suppression, most likely because in this case the inter-band and
in-band scattering channels are both pair-breaking. Figure 7(b) compares
$\text{V}_{3}\text{Si}$ with another $s_{++}$ two-band superconductor,
$\text{NbSe}_{2}$ [89], and unconventional Dirac semi-metal compound,
$\text{PdTe}_{2}$ [43, 90]. In $\text{NbSe}_{2}$, the situation is complicated
by the charge-density wave (CDW), whose competition with superconductivity
(SC) leads to the initial increase of $T_{c}$. However, as soon as CDW is
destroyed by disorder, further suppression of $T_{c}$ is quite similar to our
subject compound, $\text{V}_{3}\text{Si}$ [89]. The second compound,
unconventional $\text{PdTe}_{2}$ shows the rate of suppression quite similar
to $\text{V}_{3}\text{Si}$. Moreover, it also has a fully-gapped Fermi surface
leading to exponential attenuation of the penetration depth. However,
peculiarities of the electronic band structure of $\text{PdTe}_{2}$ support
unconventional pairing mechanism [43, 90], whereas $\text{V}_{3}\text{Si}$
does not have such topological features and is consistent with BCS-type two-
gap superconductivity. This is a good example showing that measurements alone
cannot answer objectively. They must be supported by theoretical analysis.

##  IV Conclusions

We used controlled point-like disorder induced by 2.5 MeV electron irradiation
at different doses to study superconducting order parameter in a
$\text{V}_{3}\text{Si}$ single crystal. Simultaneous measurements of London
penetration depth and superconducting transition temperature, $T_{c}$, set
stringent experimental boundaries on possible superconducting states.
Specifically, we observe: (1) exponentially attenuated low-temperature
behavior of $\lambda\left(T\right)$ (which means a fully gapped Fermi
surface); (2) a kink at higher reduced temperatures (signaling of two barely-
coupled gaps); (3) a significant shift of $T_{c}$ (gaps of different
amplitude). The discussed analysis is applicable for any choice of
$\lambda\left(0\right)$. Using a two-band analysis for both quantities,
$\rho_{s}\left(T\right)$ and $\Delta T_{c}$, we conclude that $s_{++}$ pairing
with two barely-coupled gaps of different amplitudes,
$\Delta_{1}\left(0\right)\approx 2.53\;\textrm{meV}$ and
$\Delta_{1}\left(0\right)\approx 1.42\;\textrm{meV}$, provide an excellent fit
and overall self-consistent description of the experiment. This makes
$\text{V}_{3}\text{Si}$ the earliest (superconductivity discovered in 1953)
proven $s_{++}$ superconductor, preceding $\text{MgB}_{2}$ (superconductivity
discovered in 2001) by half a century.

###### Acknowledgements.

We thank David Christen for providing excellent single crystals well-
characterized in his earlier papers. This work was supported by the US
Department of Energy (DOE), Office of Science, Basic Energy Sciences,
Materials Science and Engineering Division. Ames Laboratory is operated for
the US DOE by Iowa State University under contract DE-AC02-07CH11358. The
authors acknowledge support from the EMIR&A French network (FR CNRS 3618) on
the “SIRIUS” platform under proposal # 18-5155. We thank the whole SIRIUS
team, O. Cavani, B. Boizot, V. Metayer, and J. Losco, for operating electron
irradiation facility.

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