Field-induced coexistence of $s_{++}$ and $s_{\pm}$ superconducting states in dirty multiband superconductors
Julien Garaud Alberto Corticelli Mihail Silaev Egor Babaev
cond-mat.supr-con





 
Department of Physics, KTH-Royal Institute of Technology, 
Stockholm, SE-10691 Sweden 

Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350, 
 
Fédération Denis Poisson FR2964, 
Université de Tours, Parc de Grandmont, 37200 Tours, France
  
Department of Physics, KTH-Royal Institute of Technology, 
Stockholm, SE-10691 Sweden 
	
Department of Physics and Nanoscience Center, 
University of Jyväskylä, P.O. Box 35 (YFL), FI-40014 
University of Jyväskylä, Finland
 
Department of Physics, KTH-Royal Institute of Technology, 
Stockholm, SE-10691 Sweden 


 

In multiband systems, such as iron-based superconductors, the superconducting 
states with locking and anti-locking of the interband phase differences, 
are usually considered as mutually exclusive. For example, a dirty two-band 
system with interband impurity scattering undergoes a sharp crossover between 
the s_± state (which favors phase anti locking) and the s_++ state 
(which favors phase locking). 

We discuss here that the situation can be much more complex in the presence 
of an external field or superconducting currents. In an external applied 
magnetic field, dirty two-band superconductors do not feature a sharp 
s_±→ s_++ crossover but rather a washed-out crossover to a finite 
region in the parameter space where both s_± and s_++ states can 
coexist for example as a lattice or a microemulsion of inclusions of different 
states.  

The current-carrying regions such as the regions near vortex cores can exhibit 
an s_± state while it is the s_++ state that is favored in the bulk.
This coexistence of both states can even be realized in the Meissner state 
at the domain's boundaries featuring Meissner currents. We demonstrate that 
there is a magnetic-field-driven crossover between the pure s_± and 
the s_++ states.



74.25.Dw,74.20.Mn,74.62.En

Field-induced coexistence of s++ 
and s+- superconducting states in dirty multiband 
superconductors
    Egor Babaev
    December 30, 2023
====================================================================================================








§ INTRODUCTION


It is widely accepted that in iron-based superconductors, the pairing between 
electrons is produced by interband electron-electron repulsion 
<cit.>. 

In such a situation, the superconducting state, which features a sign inversion 
between the two s-wave gap functions is called s_±, in contrast to the 
s_++ state, which has no sign inversion. The presence of disorder is known 
to potentially lead to a crossover from the s_± to the s_++ state 
<cit.>. In the absence of an external magnetic field, 
the crossover is sharp and has little thermodynamic features.

It was however recently demonstrated that the crossover line is accompanied 
by a non trivial transition in the core structure of vortices 
<cit.>. 

More precisely the vortices, in the vicinity of the crossover line, can acquire 
a circular nodal line around the singular point in one of the superconducting 
components <cit.>. This singular nodal line, which in 
three dimensions extends to a cylindrical nodal surface surrounding the vortex 
line, results in the formation of a peculiar “moat"-like profile in the 
subdominant component of the superconducting gap. As a result, the inner 
region of the vortex core shows a π relative phase between the gaps while 
it is zero in the outer region. This means that these moat-core vortices consist 
of an s_± phase inclusion in the vortex core, which is separated from the 
bulk s_++ phase by the nodal line. 

Here we investigate the consequences of that physics on the phase diagram of 
such dirty two-band superconductors, in an external magnetic field. In a low 
applied external field the lattices and liquids of such moat-core vortices 
represent a phase coexistence or a mircoemulsion of such s_± inclusions 
inside the bulk s_++ state. At elevated fields this results in a field-induced 
transition between the s_± and s_++ states.


We start our investigation, in Sec. <ref>, by deriving a two-band 
Ginzburg-Landau model (including the gradient terms) from the microscopic 
Usadel theory of dirty two-band superconductors and discuss the essential 
properties of the phase diagram. 

Next, Sec. <ref> is devoted to the investigation of the physical 
properties of the elementary topological excitations (the vortices) and their 
core structure in that model. There, we show that across the s_±/s_++ 
crossover line there is a structural transition in the core of topological 
excitations, and we further construct the diagram where such solutions 
occur, as a function of the system's microscopic parameters. 
In Sec. <ref> we investigate the consequences for the 
phase diagram in an applied external magnetic field, and finally our 
conclusions are presented in Sec. <ref>.




§ THEORETICAL FRAMEWORK


Within a weak-coupling approximation, two-band superconductors with a high 
concentration of impurities can be described by a system of two Usadel 
equations, coupled by interband impurity-scattering terms (see e.g.
<cit.>):
Eq:Usadel
ω_n f_i    = D_i/2 (g_i^2 f_i - f_i ∇^2 g_i) 	
			 +  Δ_i g_i									

 			       +∑_j≠iγ_ij ( g_if_j - g_jf_i) 	 .

Here ω_n = (2n+1)π T (with n∈ℤ) are the Matsubara 
frequencies. T is the temperature, D_i are the electron diffusivities, 
and γ_ij are the interband scattering rates. 

The quasiclassical propagators f_i and g_i are respectively the anomalous 
and normal Green's functions in each band, which obey the normalization 
condition |f_i|^2 + g_i^2 =1. The components of the order parameter 
Δ_j=|Δ_j|e^iθ_j are determined by the self-consistency
equations
Eq:SelfConsistency
 Δ_i =2πT  ∑_n=0^N_d 
 ∑_j λ_ij f_j (ω_n),

for the Green's functions that satisfy Eq. Eq:Usadel. Here,  
N_d=Ω_d/(2π T) is the summation cut-off at Debye frequency Ω_d. 
The diagonal elements λ_ii of the coupling matrix λ̂ 
in the self-consistency equation Eq:SelfConsistency describe the 
intraband pairing, while the interband interaction is determined by the 
off-diagonal terms λ_ij (j≠ i). The interband coupling parameters 
and impurity scattering amplitudes satisfy the symmetry relation <cit.>
Eq:Symmetry
 λ_ij= - λ_J/N_i   with  j≠i,
   and  γ_ij=ΓN_j  ,
 
where λ_J,Γ>0 and N_1,2 are the partial densities of states 
in the two-bands.  

In order to investigate the physical properties of dirty two-band superconductors 
in an external field, we consider a Ginzburg-Landau (GL) model that is derived from 
the microscopic Usadel theory of dirty superconductors. The Ginzburg-Landau 
free energy functional for multiband models is obtained as an expansion in 
several small parameters: small gaps and gradients (not to be confused with a 
symmetry-based GL expansion that uses a single small parameter τ; see also 
remark [For the question of formal validity of the multiband expansion 
see the discussion of the clean case <cit.>]). Silaev.Babaev:12
The resulting expression, including gradient terms, reads as 
<cit.>:
Eq:FreeEnergy
/_0 =   
∑_j=1^2{
 	k_jj/2|Δ_j |^2
 	+a_jj|Δ_j|^2+b_jj/2|Δ_j|^4}  
 		

      +k_12/2
 	((Δ_1)^*Δ_2+(Δ_2)^*Δ_1 )
		

      +2(a_12+c_11|Δ_1|^2+c_22|Δ_2|^2)
 	 (Δ_1^*Δ_2)
		

      +(b_12+c_12cos2θ_12)|Δ_1|^2|Δ_2|^2    
   +^2/2
		 .

Here, the complex fields Δ_j=|Δ_j| e^iθ_j represent the 
superconducting gaps in the different bands, and θ_12=θ_2-θ_1
stands for the relative phase between them. The two gaps in the different 
bands are electromagnetically coupled by the vector potential  of the 
magnetic field =∇×, through the gauge derivative 
≡+iq, and the coupling constant q parametrizes the 
penetration depth of the magnetic field. 

 
The coefficients of the Ginzburg-Landau functional a_ij, b_ij, c_ij 
and k_ij are calculated from a given set of input parameters λ_ij, 
D_i, T and Γ of the microscopic self-consistent equation. First, 
the coefficients of gradient terms are given by <cit.>
Eq:GLparameters:K
 k_ii    = 2πT N_i ∑_n=0^N_d 
  D_i (ω_n + γ_ji)^2 + γ_ijγ_ji D_j /ω^2_n(ω_n+γ_ij+γ_ji)^2 

 k_ij    = 2πT N_iγ_ij ∑_n=0^N_d 
  D_i (ω_n + γ_ji) + D_j (ω_n + γ_ij) /ω^2_n(ω_n+γ_ij+γ_ji)^2	 ,

with j≠ i. Next, the coefficients of the potential terms are
Eq:GLparameters:A
a_ii   = N_iλ_jj/det(λ̂)
-2πT ∑_n=0^N_d
(ω_n+γ_ji) N_i/ω_n(ω_n+γ_ij+γ_ji)
 , 

a_ij   =-N_iλ_ij/det(λ̂)
-2πT ∑_n=0^N_d
 γ_ij N_i/ω_n(ω_n+γ_ij+γ_ji)
 . 

The other parameters read as 
Eq:GLparameters:B
b_ii   = πT N_i∑_n=0^N_d
(ω_n+γ_ji)^4 /ω_n^3(ω_n+γ_ij+γ_ji)^4

     +
πT N_i∑_n=0^N_d
γ_ij(ω_n+γ_ji)
(ω_n^2+3ω_nγ_ji+γ_ji^2) /ω_n^3(ω_n+γ_ij+γ_ji)^4
 ,

b_ij   = -πT N_i∑_n=0^N_d
γ_ij /(ω_n+γ_ij+γ_ji)^4

 +
π   T N_i∑_n=0^N_d
γ_ij(γ_ij+γ_ji)(ω_n(γ_ij+γ_ji)
+2γ_ijγ_ji)/ω_n^3(ω_n+γ_ij+γ_ji)^4
 ,
and 
Eq:GLparameters:C
c_ii   =πT N_i 
 
   ∑_n=0^N_d
  γ_ij(ω_n+γ_ji)
(ω_n^2 +(ω_n+γ_ji)(γ_ij+γ_ji)) /ω_n^3(ω_n+γ_ij+γ_ji)^4
 ,

c_ij   =πT N_i∑_n=0^N_d
  γ_ij(ω_n+γ_ji)(ω_n+γ_ij)
(γ_ij+γ_ji) /ω_n^3(ω_n+γ_ij+γ_ji)^4
 .



In Eq. Eq:GLparameters:K, the coefficients of the gradient terms 
depend on  electronic diffusivity coefficients D_1 and D_2. The parameter 
space can be reduced by absorbing one of the electronic diffusivity coefficients 
in the gradient term. Without loss of generality, we choose D_1 to be the 
largest diffusivity coefficient (D_1>D_2). Thus, in the dimensionless units, 
the coefficients of the gradient term depend only on the ratio of diffusivities, 
or diffusivity imbalance r_d=D_2/D_1<1. We define the dimensionless 
variables by normalizing the gaps by T_c and the lengths by 
ξ_0 = √(D_1/T_c). The magnetic field by is scaled 
B_0 = T_c √(4π N_1), where N_1 is the density of states in the first 
band, and the free energy _0 =B_0^2/4π. The electromagnetic coupling 
constant is q = 2π B_0 ξ_0^2/Φ_0. In these units, the London penetration 
length λ_L is given by 
λ^-2_L=q^2(k_iiΔ_i0^2+2 k_12Δ_10Δ_20),
where Δ_i0 is the bulk value of the dimensionless gap. 






It was demonstrated in Ref. Silaev.Garaud.ea:17 that, within 
its range of applicability, the two-band Ginzburg-Landau formalism 
Eq:FreeEnergy indeed produces phase diagrams that match those of the 
microscopic theory even at temperatures substantially lower than the critical 
temperature T_c of the superconducting phase transition.

Figure <ref> shows such a phase diagram in the case of a two-band 
superconductor with nearly degenerate bands (λ_11=0.29 and 
λ_22=0.3), and intermediate repulsive interband pairing interaction 
(λ_12=λ_21=-0.05). The regions of different ground state 
relative phases clearly identify the different phases. This illustrates that the 
presence of disorder leads to a crossover from the s_± to the s_++ 
state. In the absence of an external magnetic field, the crossover is sharp as 
can be seen by the solid black line that shows the crossover between s_± 
and s_++ states, where the subdominant gap Δ_1 vanishes.

Another interesting transition between the s_++ and the s_± phases 
is possible. Indeed, as a results of the presence of impurities, a possibility 
for an additional phase appears. This extra phase, where the interband phase 
difference θ_12=θ_2-θ_1 is neither zero nor π, is termed 
the s+is state. The fact that the interband phase difference can be such that 
θ_12≠0,π follows from the existence of phase-locking terms 
∝cosθ in Eq:FreeEnergy:Interaction1 and the others that 
are ∝cos2θ in Eq:FreeEnergy:Interaction1. The competition 
between those terms is responsible for the existence of the impurity-induced 
s+is state <cit.>. The s-wave states that spontaneously 
break the time-reversal symmetry have been a subject of much interest in  
recent years; see, e.g., <cit.>.

In this work, we focus on the physics of the direct s_± to the s_++ 
impurity-induced crossover in an external magnetic field (denoted by the solid 
black line on Fig:Diagram).

Although the physics of vortices in, and in the vicinity of, the s+is phase is 
very rich, it is beyond the scope of the current work and its detailed study will 
be addressed in a subsequent work <cit.>.


Below, we discuss that in an external applied magnetic field, dirty two-band 
superconductors do not feature a sharp s_±→ s_++ crossover but there 
appears a finite region in the parameter space where both 
s_± and s_++ states can coexist in a peculiar way. 




§ STRUCTURAL TRANSITION OF VORTEX CORES

 

When going from the s_++ to s_± state, there is a transition in the 
vortex core structure: the vortices there can develop an additional circular 
nodal line around the singular point in one of the superconducting components 
<cit.>. Thus these vortices consist of an s_± phase 
inclusion in the vortex core, which is separated from the bulk s_++ phase 
by the nodal line. 

In order to investigate the properties of single-vortex solutions, the vector 
potential  and the gap functions Δ_1,2 are discretized using a 
finite-element framework <cit.>. Starting with an initial configuration 
where both components have the same phase winding (i.e., at large distances 
Δ_i∝iθ where θ is the polar angle relative to 
the vortex center), the Ginzburg-Landau free energy Eq:FreeEnergy 
is then minimized using a non-linear conjugate gradient algorithm. 
We begin by simulating the vortex solutions in zero external field. 
For that purpose, the vortices are induced only by the initial configuration 
of the phase winding. For further details on the numerical methods employed 
here, see for example the related discussion in <cit.>. 







Figure <ref> shows the numerically calculated (isolated) vortex 
solutions in the vicinity of the impurity-induced crossover, in the case of 
a two-band superconductor with nearly degenerate bands and weak repulsive 
interband pairing interaction.

Deep in the s_++ and s_± regimes (in the first and fourth column), 
the vortices have multicore structure where both components exhibit a vortex 
profile with different sizes of conventional cores, determined by the fundamental 
length scales.

The vortex profiles of the minority component Δ_1 become very different 
when approaching the crossover line. On the s_± side of the crossover, the 
minority component exhibits a strong overshoot near the core. The density overshoot 
effect, although much smaller, was also reported in the microscopic model with 
one clean and one dirty band <cit.>.







The very unconventional feature of vortices in this model of dirty two-band 
superconductors is the appearance of an additional circular nodal line of the 
minority component in addition to the usual point singularity at the center of 
the vortex <cit.>. While the bulk relative phase is zero 
(the s_++ state) far from the vortex center, due to the competition between 
gradient and potential terms, it is more favorable to achieve a θ_12=π 
relative phase (s_± state) in the vicinity of the core singularity. 
The transition between the “core" states with θ_12=π and the 
asymptotic state θ_12=0 is realized by nullifying the minority component 
at a given distance of the core, when the potential terms dominate over the 
gradient term. This can be seen in the third column of Fig:Vortices. 
Analytical explanation of such behavior was given in the previous work 
Ref. <cit.>. 

The existence of the circular nodal line results in the formation of a peculiar 
“moat"-like profile in the subdominant component of the superconducting gap.
Note that the total superconducting order parameter does not vanish there, 
rather it is only the subdominant gap (here Δ_1) that vanishes exactly at 
this nodal line. Indeed, since it separates regions with θ_12=π from 
regions with zero phase difference, it has to be exactly zero somewhere. 
This can be viewed as a real-space counterpart of the crossover on the phase diagram. 
Namely, the line of direct crossover that separates the s_++ state from the 
s_± phase is that where the subdominant gap vanishes.

 





In order to understand how generic are such solutions, we further investigate 
numerically their existence on various phase diagrams. We find that the existence 
of these kinds of “moat-core" vortices does not depend on the specific values of 
the pairing coefficients. 

By that statement we mean that, for the various different coupling matrices 
λ̂ we investigated, we have been able to identify regions of the 
(T,Γ) diagram where “moat-core" vortices do form. This can be heuristically 
addressed with the following criterion that the condition for the transition of the
vortex core structure is that the mixed-gradient energy Eq:FreeEnergy:Mixed 
dominates the Josephson energy Eq:FreeEnergy:Interaction1. This was 
discussed analytically in Ref. Garaud.Silaev.ea:17a.

The investigation of the vortex solutions, for various parameter sets, rather 
shows that the moat-core is a common feature in the vicinity of the crossover 
line. The region of their existence is shown in Fig:MoatDiagram.
It is is clearly visible that the region with moat-core vortex solutions shrinks 
close to T_c and is eventually suppressed [panel (a) and (b)]. This is to be 
expected because near critical temperature, the solutions are dominated by a 
critical mode (see corresponding analytical estimates in  <cit.>). 
For the investigated regimes, the width (in terms of impurities δΓ) 
of the regions where moat-core vortices exist increases with increasing the 
band disparity λ_ii [compare for example panels (b) and (d)]. 
By comparing panels (a), (b) and (c), it can also be seen that the width of
the moat “region” increases with the interband coupling λ_12.




§ PHASE COEXISTENCE IN EXTERNAL FIELD AND MAGNETIC-FIELD-DRIVEN CROSSOVER



In this section, we consider the effect in the presence of intervortex
interactions and its implication for states of dirty two-band superconductors 
in an external magnetic field.

First of all the existence of these moat-core vortices, where both s_± and 
s_++ phases coexist, implies that a lattice of such vortices would represent 
a special kind of phase coexistence and that, in an external field, the sharp 
crossover found in the ground state is rather washed out to a finite region 
in the parameter space. 






In order to investigate the response to an applied external magnetic 
field H=H_z e_z, perpendicular to the plane, the Gibbs 
free energy 𝒢=-· H is minimized, with 
requiring that = B_ext on the boundary (for details, 
see a related discussion in Ref. Garaud.Babaev.ea:16).


Figure <ref> demonstrates the external-magnetic-field-driven 
crossover between the s_± and s_++ states. The coupling parameters 
are those of Fig:Vortices, and the temperature and the strength of 
the interband impurity scattering place the system in the bulk s_++ state, 
close to the crossover line. The computations are thus performed for parameters 
where the single-vortex solutions have a circular nodal line similar to that 
shown in the third column in Fig:Vortices.

In zero and low external fields, the preferred phase locking in the bulk is 
θ_12=0 (the s_++ state). Upon increasing the external field, 
vortices start to enter the domain, introducing small inclusions of state 
with θ_12=π in their core. When the density of vortices becomes 
significant, the cores of the subdominant component start to overlap until the 
whole system shows the θ_12=π phase locking everywhere. Thus the 
crossover here is driven by the external magnetic field.


We find that even in a low applied field, below the first critical field, 
the Meissner state also exhibits the unusual property of coexistence of 
both s_++ and s_± states. Indeed, the s_± state can be realized 
near the  boundaries while in the  bulk, it is the s_++ state. That is, 
as can be seen from the first column of Fig:Magnetization, the 
region carrying Meissner currents shows a θ_12=π phase locking 
(thus an s_± state near boundaries), while the bulk is in the s_++ 
state (with θ_12=0). This current-carrying region with θ_12=π 
is separated from the θ_12=0 bulk by a nodal line of the subdominant 
component. That nodal line originates in the competition between phase locking 
kinetic terms favoring a θ_12=π relative phase in the current-carrying 
regions, and potential terms that favor zero phase difference. The reversal of 
the phase-locking in the current-carrying region is thus in some sense similar 
to that occurring in vortex cores in the vicinity of the crossover 
<cit.>.


As illustrated in Fig:Magnetization, the magnetization properties 
in the vicinity of the s_± /s_++ crossover are rather unusual. 
It is also interesting to consider the case of a field-cooled simulation 
close to the impurity-induced crossover. Figure <ref> 
displays a simulation of a field-cooled experiment at a constant applied 
field that starts at temperatures placing the system above the crossover 
line in the s_± phase, and that ends in the s_++ phase. This 
field-cooled path is denoted by the vertical line displayed in the phase 
diagram of Fig:Diagram.

Sufficiently far from the crossover, the relative phase corresponds to that 
of the bulk properties in zero field, i.e., the s_± phase for temperatures 
above the crossover line and s_++ below the crossover. However, unlike 
the zero field case, there is no sharp crossover between both states. 
As can be seen in the third column, after the crossover, the vortex-carrying 
regions remain mostly with the s_± phase because the cores of the subdominant 
component still overlap. When going away from the crossover, that is, when 
further decreasing the temperature, the cores in the subdominant component 
Δ_1 do not overlap anymore. Still the vortices with zero nodal line 
feature inclusion of the θ_12=π state while the intervortex spaces have 
relative phase θ_12=0. That situation of a lattice (or liquid) of 
moat-core vortices thus represents a phase  coexistence or a 
mircoemulsion of s_± inclusions inside the s_++ state. 




§ CONCLUSION



We considered a two-band superconducting system that has an impurity-driven 
s_±/s_++ crossover line. We generalize the zero-field phase diagram to 
the case where the system is subjected to external magnetic field. We report 
that it is a rather generic feature that on the s_++ side of this crossover, 
the vortex solutions are unconventional, featuring  a circular nodal line that 
segregates the s_± phase inclusions in the cores from the bulk s_++ phase. 

Further, we demonstrated that as a consequence of such vortex solutions, 
the behavior of dirty two-band superconductors in the vicinity of the crossover 
is also rich in an external field. Indeed, in contrast to the zero-field picture 
of a sharp crossover, the lattice and liquids of moat-core vortices represent 
a lattice or a “mircoemulsion" of s_± inclusions inside the s_++ state. 
Moreover, as the vortex density raises in increasing field, there is also a 
field-induced crossover from s_++ to the s_±, which can be resolved 
in local phase-sensitive probes <cit.>. 

We also pointed out that in these systems in an applied external field, the 
superconducting state near the Meissner current carrying boundary can be s_± 
while it is s_++ in the bulk. This result has direct implications for 
local probes of superconducting states, such as the one proposed in
Refs. <cit.>, and 
for Josephson junction experiments. The coexistence state will manifest itself 
in the existence of signatures of both the s_± and  s_++ states depending 
on the probe's position.

The formation of nodal lines was also found in a different phenomenological 
clean three-band model where it arises for a different reason due to frustrated 
Josephson terms <cit.> rather than originating from 
competition between mixed-gradient and potential terms. That suggests that the 
phenomenon is rather generic and applies also to other models 
featuring a crossover from the s_± to s_++ state.



The work was supported by Swedish Research Council Grants
No. 642-2013-7837, and No. VR2016-06122 and the Goran Gustafsson Foundation 
for Research in Natural Sciences and Medicine. M.S. was supported by 
the Academy of Finland (Project No. 297439).
The computations were performed on resources provided by the Swedish 
National Infrastructure for Computing (SNIC) at National Supercomputer 
Center at Linköping, Sweden.













§ GINZBURG-LANDAU COEFFICIENTS



Table <ref> shows the actual values of the coefficients 
entering the Ginzburg-Landau functional that were used for the various 
simulations throughout the paper. 

P[1]>p#1














24
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fnum [1]
 #1firstoftwo
 secondoftwo
 
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 #1firstoftwo
 secondoftwo
 





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href[1]#1@endlink
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12`$12`&12`#12`1̂2`_12`%12
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