Consistent two-lifetime model for spectral functions of superconductors

1.3 MB·

by  František Herman

Consistent two-lifetime model for spectral functions of superconductors
Frantiˇsek Herman and Richard Hlubina
Department of Experimental Physics, Comenius University, Mlynska´ Dolina F2, 842 48 Bratislava, Slovakia
Recently it has been found that models with at least two lifetimes have to be considered when
analyzing the angle resolved photoemission data in the nodal region of the cuprates [T. Kondo et
al.,Nat. Commun. 6,7699(2015)]. Inthispaperwecomparetwosuchmodels. Firstweshowthat
thephenomenologicalmodelusedbyKondoetal. violatesthesumrulefortheoccupationnumber.
Next we consider the recently proposed model of the so-called Dynes superconductors, wherein the
twolifetimesmeasurethestrengthsofpair-conservingandpair-breakingprocesses. Wedemonstrate
7 that the model of the Dynes superconductors is fully consistent with known exact results and we
1 study in detail the resulting spectral functions. Finally, we show that the spectral functions in the
0 nodal region of the cuprates can be fitted well by the model of the Dynes superconductors.
2
PACSnumbers: PACS
n
a
J
I. INTRODUCTION Gor’kov propagator violates the sum rule for the occu-
6
pation number and therefore the model used in Ref. 8
1
Recent experimental progress in angle resolved pho- should be discarded.
] toemission spectroscopy (ARPES)1 enables not only to Our second goal is to demonstrate that, neverthe-
n determinethepositionoffeaturesintheelectronspectral less, a fully consistent two-lifetime phenomenology for
o function,butalsotostudymoresubtleissuessuchasthe superconductors does exist. To this end we consider
c
- spectral lineshapes. At least for the conventional low- the recently proposed model of the so-called Dynes
r temperaturesuperconductors,theredoesexistatheoret- superconductors,9 wherein the two-lifetime phenomenol-
p
u icaltechniquewhichcanaddresssuchissues-namelythe ogyresultsasaconsequenceoftakingintoaccountboth,
s Eliashberg theory,2 which allows for strong coupling be- thepair-conservingandthepair-breakingscatteringpro-
t. tween the electrons and bosonic collective modes. How- cesses. In Section 3 we present detailed predictions for
a ever, it is not obvious whether this type of theory is ap- the spectral functions of the Dynes superconductors and
m
plicabletothecuprates. Moreover,evenconventionalsu- we explicitly demonstrate the applicability of this ap-
- perconductors may possess complicated phonon spectra proachtothelow-temperatureARPESdatainthenodal
d
and/or they can exhibit substantial elastic scattering,3 region of the cuprates. Furthermore, in the Appen-
n
o bothofwhichcomplicatetheEliashberganalysis. Forall dices we show that the model of the Dynes supercon-
c thesereasons,itisdesirabletohaveasimpleandgeneric ductors is fully consistent with known exact results, and
[ theory which does take finite quasiparticle lifetimes in we present explicit formulas for the momentum distribu-
1 superconductors into account. tionfunctions10 withintheEliashbergtheory. Finally, in
Section 4 we present our conclusions.
v LetusstartbynotingthatspectralfunctionsofaBCS
0 superconductor in presence of elastic pair-conserving
3 scattering can be found even in textbooks.2,4 However,
4 II. THE MODEL USED BY KONDO ET AL.
it is well known that the density of states (or the so-
4
calledtomographicdensityofstatesincaseofanisotropic
0 superconductors5)impliedbysuchspectralfunctionsex- Followingprevioustheoreticalsuggestions,11,12 theau-
.
1 hibits a full spectral gap consistent with the Anderson thors of Ref. 8 fit their high resolution ARPES data
0 theorem. On the other hand, experimentally, the gap in the nodal region of optimally doped and overdoped
7
isquiteoftenonlypartial3,5 andthetunnelingdensityof Bi2212 samples to spectral functions derived from the
1
statesisbetterdescribedbythephenomenologicalDynes phenomenological self energy
:
v formula.6,7 This means then that, in order to take the 2
i ∆
X non-trivial density of states into account, also a second Σ(k,ω)=−iΓ + (1)
r typeofscatteringprocesses-whicharenotsubjecttothe 1 ω+εk+iΓ0
a Anderson theorem - have to be considered. Two-lifetime and they interpret the scattering rates Γ and Γ as the
1 0
phenomenology of precisely this type has in fact been single-particleandpairscatteringrates,respectively. We
appliedquiterecently,8 withtheaimtoparameterizethe shall comment on these identifications later.
high-resolution ARPES data in the nodal region of the In order to demonstrate the physical meaning of the
cuprates. phenomenological self energy Eq. (1), let us first note
Thegoalsofthispaperaretwofold. First, inSection2 that it implies that the electron Green function in the
wedemonstratethatthemodelusedinRef.8canbecast superconducting state can be written in the form
into a form consistent with the generalized Eliashberg
(ω+iγ)+(ε +iγ(cid:48))
theory, and that it exhibits several attractive features. G(k,ω)= k , (2)
However, we also show that the resulting 2×2 Nambu- (ω+iγ)2−(ε +iγ(cid:48))2−∆2
k
2
where we have introduced γ = (Γ + Γ )/2 and γ(cid:48) =
0 1 0.3 ε =-5Δ
(Γ −Γ )/2. According to Ref. 8, throughout the super- k
con0duct1ing phase γ(cid:48) < 0. With increasing temperature Γ0=0
|γ(cid:48)|decreasesanditvanishesatthecriticaltemperature. Γ1=Δ
ω)0.2
The main observation of this Section is that Eq. (2) k,
(
shouldformtheupperleftcomponentofthegeneral2×2 A11 2Γ1/Δ
Nambu-Gor’kov Green function for a superconductor: 0.1
ωZ(k,ω)τ +[ε +χ(k,ω)]τ +φ(k,ω)τ
Gˆ(k,ω)= 0 k 3 1,
[ωZ(k,ω)]2−[εk+χ(k,ω)]2−φ(k,ω)2 -010 -5 0 5 10
(3)
ω/Δ
where τ are the Pauli matrices, Z(k,ω) is the wave-
i
function renormalization, and φ(k,ω) is the anomalous FIG. 1: Spectral function A (k,ω) of an electron inside the
11
self-energy. Thefunctionχ(k,ω)describestherenormal- Fermi sea according to the model (2). The widths of the
ization of the single-particle spectrum and it vanishes in electron-like branch at ω ≈ −E and of the hole-like branch
k
aparticle-holesymmetrictheory;thatiswhyitisusually atω≈Ek,whereEk isthequasiparticleenergyEq.(10),are
neglected. For the sake of completeness, let us mention essentially determined by Γ1 and Γ0, respectively.
that the matrix τ does not enter Eq. (3), because we
2
work in a gauge with a real order parameter.
in agreement with the experimental findings of Ref. 5.
Since in the high-frequency limit the functions χ, φ,
Thesquareroothastobetakensothatitsimaginarypart
and ω(Z −1) should stay at most constant, Eq. (2) in
is positive and we keep this convention throughout this
this limit can be uniquely interpreted in terms of the
paper. InEq.(7)N denotesthenormal-statedensityof
general expression Eq. (3) with 0
states. Note that, although the particle-hole symmetry
Z(ω)=1+iγ/ω, χ(ω)=iγ(cid:48), φ(ω)=∆, (4) is broken due to χ(cid:54)=0, N(ω) is an even function of ω.
The broken particle-hole symmetry is clearly visible
where we have chosen a real anomalous self-energy φ. already in the normal state with ∆=0, in which case
OnechecksreadilythatEq.(4)reproducesEq.(2)forall
1 1
frequencies. The finite value of χ is unusual but seems G (k,ω)= , G (k,ω)= .
11 ω−ε +iΓ 22 ω+ε +iΓ
to be attractive, since the cuprates, being doped Mott k 1 k 0
(8)
insulators, might be expected to break the particle-hole
These results show that Γ and Γ should not be inter-
symmetry. Note also that all three functions Z, χ and φ 0 1
preted as pair and single-particle scattering rates as has
do not depend on k, which is the standard behavior.
beendoneinRef.8,butratherasthescatteringratesfor
It is well known that the 2×2 formalism leads to a
the holes and for the electrons, respectively.
redundant description, and therefore the Green function
Unfortunately, Eqs. (8) turn out to be mutually in-
has to satisfy additional constraints. These constraints
consistent, but in a quite subtle way. In order to show
are most clearly visible in the Matsubara formalism,
this,letusintroducethespectralfunctionsA (k,ω)with
thereforeletusreformulateEq.(3)ontheimaginaryaxis, ii
i=1,2, corresponding to the Green functions G (k,ω).
allowing explicitly only for frequency-dependent func- ii
Applying standard procedures, the following exact sum
tions Z =Z(iω ), χ =χ(iω ), and φ =φ(iω ):
n n n n n n rules can be established,
(cid:0) (cid:1)
Gˆ(k,ωn)=−iω(cid:0)nωZnZτ0(cid:1)+2+ε(cid:0)kε++χnχτ(cid:1)32++φφn2τ1. (5) (cid:90) ∞ 1+deω−ω/TA11(k,ω) = (cid:104)ck↑c†k↑(cid:105)=1−nk↑,
n n k n n −∞
(cid:90) ∞ dω
Due to the redundancy of the 2×2 formalism, singlet A (k,ω) = (cid:104)c† c (cid:105)=n .
1+e−ω/T 22 −k↓ −k↓ −k↓
superconductorshavetoexhibitthefollowingsymmetry: −∞
Butinasingletsuperconductorwehaven =n ,and
G (k,ω )=−G (k,−ω ). (6) k↑ −k↓
22 n 11 n therefore the following exact relation should hold:
NotethatthefunctionsEq.(4)readasZn =1+iγ/|ωn|, (cid:90) ∞ dω
χn = iγ(cid:48), and φn = ∆ on the imaginary axis. It is easy 1+e−ω/T [A11(k,ω)+A22(k,ω)]=1. (9)
to see that when these expressions are plugged in into −∞
Eq. (5), the Green function does satisfy Eq. (6). Making use of Eqs. (8) at T = 0, the integrals on the
Anadditionalattractivefeatureofthephenomenology left-hand side can be taken easily and the results are
Eq. (4) is that it leads (also for finite values of γ(cid:48)) to the
Dynes formula for the tunneling density of states, (cid:90) ∞dωA (k,ω) = 1 + 1 arctan(cid:18)εk(cid:19),
11 2 π Γ
(cid:34) (cid:35) 0 1
ω+iγ (cid:90) ∞ 1 1 (cid:18)ε (cid:19)
N(ω)=N0Re (cid:112)(ω+iγ)2−∆¯2 , (7) dωA22(k,ω) = 2 − π arctan Γk .
0 0
3
0.6 Γn=Δ/2 Γn=Δ Γn=2Δ
Γ/Γn=0.
0.5
Γ/Γn=0.25
ω)0.4
k,0.3 Γ/Γn=0.5
A(11 2Γn/Δ Γ/Γn=1.
0.2
0.1 2Γn/Δ
0.0 2Γn/Δ
-5 -3 -1 1 3 5 -5 -3 -1 1 3 5 -5 -3 -1 1 3 5
ω/Δ ω/Δ ω/Δ
FIG. 2: Spectral functions A (k,ω) of the Dynes superconductor for an electron inside the Fermi sea with ε = −3∆. The
11 k
total scattering rate Γ =Γ+Γ increases from the left to the right panel. The curves in each panel differ by the strength of
n s
the pair-breaking scattering rate Γ, while Γ is kept fixed. The color coding is the same in all panels.
n
It can be seen readily that, if Γ (cid:54)= Γ , the sum rule (9) bytheGreenfunctionEq.(11)isdescribedbytheDynes
0 1
is violated by these results. formula Eq. (7) with γ = Γ, and that is why supercon-
One might have the impression that the particle-hole ductors described by Eq. (11) have been called Dynes
asymmetry which causes the sum rule violation is an ar- superconductors in Ref. 9. In this work we shall keep
tifact of our generalization of the Green function (2) to this term.
thematrixform(3). Thatthisisnotthecasecanbeseen It is worth pointing out that in absence of pair-
by plotting the spectral function directly for Eq. (2), see breaking processes, Eq. (11) reproduces the textbook re-
Fig. 1, which clearly shows that the electron- and hole- sultsforpair-conservingscattering,seee.g. Refs.2,4. On
like branches exhibit different scattering rates. the other hand, in the opposite limit Γ = 0 when only
s
We conclude that the phenomenology (4) is internally pair-breaking processes are present, Eq. (11) coincides
consistent only if Γ = Γ = γ, in which case γ(cid:48) = 0. with the phenomenology (4) in the physically consistent
0 1
But then the Green function (3) has a simple two-pole case with Γ = Γ = Γ. Moreover, in the normal state
0 1
structure with poles at ω =±E −iγ where with ∆=0 the Green function Eq. (11) becomes diago-
k
nal and its matrix elements are
(cid:113)
E = ε2 +∆2, (10)
k k 1 1
G (k,ω)= , G (k,ω)= ,
implying that the spectral function is a sum of two 11 ω−ε +iΓ 22 ω+ε +iΓ
k n k n
Lorentzians. However, the authors of Ref. 8 stress that (13)
the experimentally observed lineshapes are asymmetric. where Γ = Γ+Γ is the total scattering rate, which
n s
This means then that the phenomenology (4) is not ap- involves both, pair-breaking as well as pair-conserving
plicable to the nodal spectral functions of the cuprates. scatteringprocesses. NotethatEq.(13)doesnotexhibit
the pathologies implied by Eq. (8).
One checks readily that the Green function Eq. (11)
III. DYNES SUPERCONDUCTORS is analytic in the upper half-plane of complex frequen-
cies, as required by causality, and that Gˆ(k,ω) ∝ τ /|ω|
0
Very recently, a consistent two-lifetime phenomenol- for |ω| → ∞. In Appendix B, we present an explicit
ogy for superconductors has been derived within the co- proof that Eq. (11) satisfies the well-known sum rules
herent potential approximation, assuming a Lorentzian for the zero-order moments of the electron spectral func-
distribution of pair-breaking fields and an arbitrary dis- tion, in particular also the sum rule Eq. (9). Moreover,
tribution of pair-conserving disorder.9 If we denote the inAppendixDweprovethattheelectronandholespec-
pair-breaking and pair-conserving scattering rates as Γ tralfunctionsarepositive-definite,asrequiredbygeneral
and Γ , respectively, then the result of Ref. 9 for the considerations.
s
Nambu-Gor’kov Green function of the disordered super- In view of these observations, we believe that Eq. (11)
conductor can be written as represents the simplest internally consistent Green func-
(1+iΓ /Ω)(cid:2)(ω+iΓ)τ +∆¯τ (cid:3)+ε τ tion for a superconductor with simultaneously present
Gˆ(k,ω)= s 0 1 k 3, (11) pair-breaking and pair-conserving scattering processes.
(Ω+iΓs)2−ε2k This generic BCS-like Green function is parameterized
by three energy scales: scattering rates Γ and Γ , as well
where s
as by the gap parameter ∆. In what follows we present
(cid:113)
Ω(ω)= (ω+iΓ)2−∆¯2. (12) a detailed analysis of its spectral properties.
Spectral functions of the Dynes superconductor for an
SomeusefulpropertiesofthefunctionΩ(ω)aredescribed electron with momentum k fixed to lie inside the Fermi
in Appendix A. The tunneling density of states implied sea are shown in Fig. 2. The BCS quasiparticle peaks at
4
0.8 Γn=Δ/2 Γn=Δ Γn=2Δ Γ/Γn=0.
)0.6 Γ/Γn=0.25
ω
k,0.4 Γ/Γn=0.5
A(11 Γ/Γn=1.
0.2
0.0
-5 -3 -1 1 3 5 -5 -3 -1 1 3 5 -5 -3 -1 1 3 5
ω/Δ ω/Δ ω/Δ
FIG. 3: Spectral functions A (k,ω) of the Dynes superconductor for an electron directly at the Fermi surface, ε = 0. The
11 k
total scattering rate Γ =Γ+Γ increases from the left to the right panel. The curves in each panel differ by the strength of
n s
the pair-breaking scattering rate Γ, while Γ is kept fixed. The color coding is the same in all panels.
n
ω ≈±E areseentobebroadenedbythetotalscattering Ref. 8. We have determined the fitting parameters by
k
rate Γ , irrespective of the ratio between pair-breaking the standard least-squares technique in the interval from
n
and pair-conserving scattering processes. The relative -100 meV to 100 meV; their values are ∆ = 25 meV,
importance of the two types of processes becomes im- Γ = 3.7 meV, Γ = 16 meV, and Λ = 103 meV for
s
portant only in the vicinity of the chemical potential. the fit using Eq. (11). On the other hand, we have
For Γ = 0 a full spectral gap appears for |ω| < ∆, in found ∆ = 27 meV, Γ = 0 meV, Γ = 12 meV, and
0 1
agreement with the Anderson theorem, and additional Λ=87 meV for the fit using Eq. (1).
peaks appear in the spectral function at ω =±∆. After
switching on a finite pair breaking rate Γ (cid:54)= 0, the spec-
Bothfitsfindroughlythesamevalueofthegap∆and
tral gap starts to fill in, and at the same time the peaks
of the background parameter Λ, but the scattering rates
at ω =±∆ get smeared away. Finally, when Γ=Γ and
n turnouttobequitedifferent. Inviewofthelatterobser-
the pair-conserving processes disappear completely, the
vation it seems to be worthwhile to repeat the analysis
spectral function is given by a sum of two Lorentzians
of Ref. 8, but with the ansatz Eq. (11) for the electron
centered at ω =±E .
k Green function. It remains to be seen whether this type
SpectralfunctionsforanelectrondirectlyattheFermi
ofanalysiscanbeappliedalsoattemperaturesaboveT ,
c
surface, ε = 0, are somewhat different and they are
k andwhatistheresultingtemperaturedependenceofthe
showninFig.3. Thedifferenceiscausedbythefactthat
scattering rates Γ and Γ .
s
the quasiparticle energies ±E in this case coincide with
k
±∆. Thereforeonlytwopeaksarepresentinthespectral
function, in contrast to the general case with four peaks. The small value of the pair-breaking rate Γ with re-
However, the rest of the phenomenology can be simply spect to the large pair-conserving rate Γs implied by
related to the case ε (cid:54)= 0: the high-energy form of the Fig.4isconsistentwiththeobservationthattheconcept
k
spectral functions is controlled exclusively by the total ofthetomographicdensityofstatesisusefulintheanal-
scattering rate Γ , whereas finite pair-breaking fills in ysis of the ARPES data.5,13 Since in an anisotropic su-
n
the spectral gap and smears the peaks at ω =±∆. perconductor large-angle scattering is pair-breaking, the
smallness of Γ implies that the dominant scattering pro-
In Appendix C we complement the discussion of elec-
cesses (at least in the nodal region and at low tempera-
tronspectralfunctionsbystudyingtheso-calledmomen-
tum distribution functions.10 In addition to presenting tures) have to be of the forward-scattering type.
explicit formulas valid for any Eliashberg-type super-
conductorwithonlyfrequency-dependentfunctionsZ(ω) The importance of forward-scattering processes in the
and φ(ω), we also show that making use of the momen- nodalregionhasbeenconfirmedrecentlyalsobyananal-
tum distribution functions, one can determine the total ysis of the momentum distribution curves.14 As for the
scattering rate Γn of a Dynes superconductor in an al- microscopic origin of the forward scattering, it has been
ternative way. argued that it can be caused by elastic scattering on dis-
In Fig. 4 we demonstrate that Eq. (11) can fit the ex- orderlocatedoutsidetheCuO planes.15 Otherexplana-
2
perimentally observed symmetrized spectral functions in tions include scattering on (quasi) static long-range fluc-
the nodal region of the cuprates with at least compara- tuations, perhaps due to competing order, or scattering
ble quality as Eq. (1). The number of fitting parameters on low-energy long-wavelength emergent gauge fields.16
is the same for both fits: two scattering rates, the gap Thesedifferentscenariacanbedistinguishedbydifferent
∆, and the energy scale Λ which determines the phe- dependence on temperature and/or Fermi-surface loca-
nomenological background |ω|/Λ2. This type of back- tionandfurtherexperimentalworkisneededtodiscrim-
ground description has been used in all fits presented in inate between them.
5
2.5 inAppendixC,thescatteringrateΓ canbedetermined
n
from the width of the momentum distribution functions,
2.0 and it enters also the analysis of optical conductivity.17
Obviously, description of superconductors making use
)1.5 ofEq.(11) canbequantitativelycorrect onlyatenergies
ω
k, smaller than the typical boson energies of the studied
(111.0 system. At higher energies, application of a full-fledged
A
Eliashberg-type theory2 - but extended so as to allow
0.5 for processes leading to Eq. (11) at low energies - is un-
avoidable. For completeness, in Appendix C we have de-
0.0 scribed a procedure which, starting from the assumption
-100 -50 0 50 100
of only frequency-dependent Eliashberg functions Z(ω)
ω(meV) and ∆(ω), allows for their complete determination from
ARPESdatabycombiningtwoapproaches: themomen-
FIG. 4: Experimentally observed low-temperature sym- tum distribution technique and the tomographic density
metrized spectral functions at the Fermi level reported in of states.
Ref. 8 for optimally doped Bi2212 at angle φ = 24◦. Also
Finally, in Fig. 4 we have demonstrated that the low-
shown are least-square fits in the region from -100 meV to
temperature ARPES data in the nodal region of the
+100meVaroundtheFermilevelwhichmakeuseofEq.(11)
cuprates can be fitted well using Eq. (11). Our results
(red solid line) and of Eq. (1) (blue dashed line). The values
confirmpreviousclaimsabouttheimportanceofforward-
of the fitting parameters are shown in the main text.
scattering processes in this region, but identification of
their physical origin will require further detailed angle-
and temperature-dependent studies.
IV. CONCLUSIONS
Tosummarize,wehaveshownthatthephenomenolog-
Appendix A: Properties of the function Ω(ω)
ical self-energy Eq. (1), which has been proposed theo-
retically in Refs. 11,12 and applied recently in Ref. 8, is
internally consistent only in the case when Γ = Γ ; in Let us decompose the function Ω(ω) defined by
0 1
this case the electron spectral function in the supercon- Eq.(12)intoitsrealandimaginaryparts, Ω=Ω1+iΩ2.
ducting state is a sum of two Lorentzians. One finds readily that Ω1,2 should satisfy the relations
The simplest consistent genuine two-lifetime Green
function of a superconductor is given by Eq. (11). This Ω1Ω2 =ωΓ, Ω21−Ω22 =ν2, (A1)
modeldependsontwoscatteringrates: thepair-breaking
scattering rate Γ and the pair-conserving scattering rate whereν2 =ω2−∆2−Γ2. Oursignconventionleadsthen
Γs. TheGreenfunctionEq.(11)impliesthatthedensity to the following explicit expressions for Ω1,2:
of states is described by the Dynes formula Eq. (7) with
γ = Γ and the electron spectral functions exhibit more (cid:114)
(cid:104)(cid:112) (cid:105)
structure than might be expected naively, see Figs. 2,3. Ω (ω) = sgn(ω) ν4+4ω2Γ2+ν2 /2,
1
The Green function Eq. (11) is analytic in the upper
(cid:114)
half-plane,ithasthecorrectlarge-frequencyasymptotics, (cid:104)(cid:112) (cid:105)
Ω (ω) = ν4+4ω2Γ2−ν2 /2.
2
itsdiagonalspectralfunctionsarepositive-definite,andit
satisfiestheexactsumrulesEq.(9)andEq.(B3). More-
over,inthethreelimitingcasesofeitherΓ=0,orΓs =0, Note that Ω1(ω) is an odd function of ω, while Ω2(ω) is
or ∆ = 0, it reduces to the well-known results. There- positive definite and even. A straightforward calculation
fore, although Eq. (11) has been originally derived only shows that for ω >0 the following inequalities are valid:
for a special distribution of pair-breaking fields within
the coherent potential approximation, we believe that it Ω ≤ω, Ω ≥Γ. (A2)
1 2
represents a generic two-lifetime Green function of a su-
perconductor. These inequalities will be used in Appendix D.
Our results provide a (in principle) straightforward ItisworthpointingoutthatthefunctionΩ (ω)charac-
1
recipe for extracting the scattering rates Γ and Γ = terizingtheDynessuperconductorisinprincipledirectly
n
Γ+Γ from experimental data: the pair-breaking scat- measurable in low-temperature tunneling experiments.
s
tering rate Γ is best determined from the tunneling (or, In fact, it is well known that in such experiments the
inanisotropicsuperconductors,tomographic5)densityof derivative of the current-voltage characteristics, dI/dV,
states, whereas the total scattering rate Γ may be ex- is proportional to the tunneling density of states N(ω)
n
tracted from the widths of the quasiparticle peaks in with ω = eV. But, since N(ω) ∝ dΩ /dω, the function
1
spectral functions, see Fig. 2. Alternatively, as shown Ω (ω)isproportionaltothemeasuredfunctionI =I(V).
1
6
4 Using the oddness of the function ϕ(ω) and of its
εk=0,Γ=0.1Δ,Γs=Δ asymptotic values, one finds readily that Eq. (B2) im-
plies the matrix equation
2
(cid:90) ∞
dωAˆ(k,ω)=τ , (B3)
0
−∞
0
inperfectagreementwiththewell-knownexactsumrule
for the zero-order moment of the spectral function.
-2 ln|F(ω)/Δ2| Next we prove that Eq. (B2) satisfies the exact sum
rule Eq. (9). Since TrAˆ(k,ω)=−π−1∂ϕ/∂ω, we have to
φ(ω)
prove that
-4
-4 -2 0 2 4 (cid:90) ∞ dω ∂ϕ
=−π.
ω/Δ 1+e−ω/T ∂ω
−∞
Bycalculatingtheintegralontheleft-handsidebyparts,
FIG. 5: Real and imaginary parts of the function H(ω).
our task reduces to proving the equality
(cid:90) ∞ dω
Appendix B: Sum rules for the Dynes ϕ(ω)=0.
4T cosh2(ω/2T)
superconductors −∞
But,sinceϕ(ω)isodd,thislastequalityistriviallysatis-
In this Appendix we prove that Eq. (11) satisfies the fied. Thus we have proven that the Dynes superconduc-
sumrulesforthezero-ordermomentsoftheelectronspec- tors satisfy Eq. (9).
tral function. To this end, let us introduce an auxiliary For the sake of completeness, let us note that the full
complex function F(ω) of the real frequency ω, matrix form of the sum rule Eq. (9) reads
F(ω)=ε2k−[Ω(ω)+iΓs]2. (cid:90) ∞ dω Aˆ(k,ω)=(cid:18)1−nk bk (cid:19), (B4)
−∞ 1+e−ω/T bk nk
NotethatthefunctionF(ω)dependsalsoonthemomen-
tumk,butforthesakeofsimplicitythisdependencewill where n = n = n , b = (cid:104)c c (cid:105) = (cid:104)c† c† (cid:105),
k k↑ −k↓ k k↑ −k↓ −k↓ k↑
not be displayed explicitly. and the thermodynamic expectation values n and b
k k
Let us furthermore define the function are given by
1 (cid:90) ∞ dω ∂ϕ ω
H(ω)=lnF(ω)=ln|F(ω)|+iϕ(ω). n = − tanh ,
k 2 2π ∂ε 2T
0 k
In the second equality we have represented the complex (cid:90) ∞ dω ∂ϕ ω
b = tanh .
functionF(ω)=|F(ω)|exp{iϕ(ω)}intermsofitsampli- k 2π ∂∆ 2T
tude |F(ω)| and phase ϕ(ω) constrained to the interval 0
(−π,π). A plot of the real and imaginary parts of the It should be pointed out that sum rules for higher-
function H(ω) is shown in Fig. 5. Note that the phase order moments of the spectral function which generalize
ϕ(ω) is an odd function of frequency and its asymptotic Eqs. (B3,B4) can be also derived, but their right-hand
values are ϕ(±∞)=∓π. sides depend on the Hamiltonian of the problem. Such
Making use of the function H(ω), the Nambu-Gor’kov sum rules therefore do not provide useful checks of the
Green function (11) can be written in the following ele- phenomenological Green function Eq. (11).
gant form:
(cid:20) (cid:21)
1 ∂H ∂H ∂H Appendix C: Momentum distribution functions in
Gˆ(k,ω)= τ − τ − τ . (B1)
2 ∂ω 0 ∂∆ 1 ∂ε 3 the Eliashberg theory
k
From here follows the following explicit expression for Wehavealreadynotedthat,withintheEliashbergthe-
the Nambu-Gor’kov spectral function, defined as usual ory, the functions Z(ω) and φ(ω) usually depend only
by Aˆ(k,ω)=−π−1ImGˆ(k,ω): on frequency ω and are independent of the momentum
k. Quite some time ago, it has been pointed out that
(cid:20) (cid:21)
Aˆ(k,ω)= 1 −∂ϕτ + ∂ϕτ + ∂ϕτ . (B2) in such cases it is useful to study the spectral function
2π ∂ω 0 ∂∆ 1 ∂εk 3 A11(k,ω) for fixed frequency ω as a function of the bare
electron energy ε , the so-called momentum distribution
k
Equation (B2) forms the starting point of our discussion function.10 Tosimplifytheformulas,inthisAppendixwe
of the sum rules. will replace A (k,ω) by A(ε,ω).
11
7
resolve all three terms in Eq. (C2), together with their
4 Γ=0.1Δ,Γs=Δ
relative weights. But at sufficiently large frequencies we
should expect that Γ(cid:101) (cid:28) |Ω(cid:101)|, see Fig. 6. In this case the
2 γ/Δ Γ/Δ following approximate equality is valid
0 4πΩ(cid:101)2
δ (ε−Ω(cid:101))δ (ε+Ω(cid:101))≈δ (ε−Ω(cid:101))+δ (ε+Ω(cid:101)),
Γ(cid:101) Γ(cid:101) Γ(cid:101) Γ(cid:101) Γ(cid:101)
-2
whichmeansthattheproductoftwoLorentzianscannot
-4 ω/Δ Ω/Δ be distinguished from their sum. Inserting this equality
intoEq.(C2),onefindsreadilythatthespectralfunction
-4 -2 0 2 4
A(ε,ω) is given by a sum of only two Lorentzians,
ω/Δ
(cid:20) (cid:21) (cid:20) (cid:21)
1 ω 1 ω
FIG. 6: Functions ω(cid:101)(ω), γ(cid:101)(ω), Ω(cid:101)(ω), and Γ(cid:101)(ω) for a Dynes A(ε,ω)≈ (cid:101) +1 δ (ε−Ω(cid:101))+ (cid:101) −1 δ (ε+Ω(cid:101)).
superconductor. 2 Ω(cid:101) Γ(cid:101) 2 Ω(cid:101) Γ(cid:101)
But if this is the case, then from fits to the momentum
Instead of the two complex functions Z(ω) and φ(ω),
let us introduce the following four real functions of fre- distribution function one can determine only Ω(cid:101), Γ(cid:101), and
ω, but not γ. In other words, we do not have access to
quency ω(cid:101)(ω), γ(cid:101)(ω), Ω(cid:101)(ω), and Γ(cid:101)(ω): t(cid:101)hepairingf(cid:101)unctionφ2(ω)=(ω(cid:101)+iγ(cid:101))2−(Ω(cid:101)+iΓ(cid:101))2 inthis
ωZ = ω+iγ, frequency limit.
(cid:101) (cid:101)
(cid:112)
(ωZ)2−φ2 = Ω(cid:101) +iΓ(cid:101). (C1) Thereisyetanotherreasonwhyfitstothemomentum
distributionfunctioncanprovidereliableestimatesofthe
Toillustratetheirsymmetriesandtypicalform, inFig.6 Eliashberg parameters only for |ω| (cid:46) ∆: namely, this
we plot the functions ω(cid:101)(ω), γ(cid:101)(ω), Ω(cid:101)(ω), and Γ(cid:101)(ω) for a technique requires that both ratios, γ(cid:101)/Γ(cid:101) and ω(cid:101)/Ω(cid:101), are
Dynes superconductor. sufficiently different from 1, so that the weights of the
After a tedious but straightforward calculation the second and third terms in Eq. (C2) can be determined
spectral function of a general Eliashberg superconductor precisely. But Fig. 6 clearly shows that this criterion is
can be written as satisfied only for |ω|(cid:46)∆.
(cid:20) (cid:21) (cid:20) (cid:21)
1 γ 1 γ DoesthismeanthattheEliashbergproblemoffinding
A(ε,ω) = (cid:101) +1 δ (ε−Ω(cid:101))+ (cid:101) −1 δ (ε+Ω(cid:101))
2 Γ(cid:101) Γ(cid:101) 2 Γ(cid:101) Γ(cid:101) the functions Z(ω) and ∆(ω) can not be solved in the
frequency range ∆(cid:46)|ω|? The answer is no: it has been
+ 1(cid:20)ω(cid:101) − γ(cid:101)(cid:21)4πΩ(cid:101)2δ (ε−Ω(cid:101))δ (ε+Ω(cid:101)), (C2) pointed out recently18,19 that, by applying the powerful
2 Ω(cid:101) Γ(cid:101) Γ(cid:101) Γ(cid:101) Γ(cid:101) inversion technique developed in Ref. 20, it is possible
to extract the complex gap function ∆(ω) = φ(ω)/Z(ω)
where we have introduced the notation
from the measured tomographic density of states. When
1 Γ(cid:101) this knowledge is combined with the momentum distri-
δ (x)=
Γ(cid:101) πx2+Γ(cid:101)2 bution technique - which allows for a relatively straight-
forward determination of Ω(cid:101) and Γ(cid:101) in the limit ∆ (cid:46) |ω|
for a Lorentzian with width Γ(cid:101). According to Eq. (C2), with one Lorentzian only - making use of the expression
the spectral function A(ε,ω), when viewed as a function
of energy ε at fixed frequency ω, consists of three terms.
Ω(cid:101) +iΓ(cid:101)
The first two terms are Lorentzians, whereas the third Z(ω)=
(cid:112)
term is a product of two Lorentzians. ω2−∆2(ω)
Whenthemeasuredmomentumdistributionfunctions
are fitted by Eq. (C2), Ω(cid:101) can be determined from the one can determine also the second Eliashberg function
positions of the Lorentzians and Γ(cid:101) is given by their Z(ω), thereby solving the Eliashberg problem.18
widths. Finally, from the relative weights of the three
Finally, let us note that the momentum distribution
terms in Eq. (C2) one can determine the ratios γ(cid:101)/Γ(cid:101) and functions can be useful also in the special case of the
ω(cid:101)/Ω(cid:101). Withallfourfunctionsω(cid:101)(ω),γ(cid:101)(ω),Ω(cid:101)(ω),andΓ(cid:101)(ω) Dynes superconductors described by Eq. (11). In fact,
known, one obtains full information about the supercon- sinceinthefrequencyrange∆(cid:46)|ω|thewidthoftheob-
ducting state. This idea has been used in an impressive servable Lorentzian in the momentum distribution func-
set of recent papers, see Ref.18 and references therein. tionofaDynessuperconductorisΓ(cid:101) ≈Γn,thisgivesusan
One should note, however, that in order to determine independentprocedureformeasuringthetotalscattering
all four parameters ω(cid:101), γ(cid:101), Ω(cid:101), and Γ(cid:101), it is necessary to rate Γn =Γ+Γs.
8
Appendix D: Proof of the inequalities A (k,ω)≥0 InviewofthesymmetriesillustratedbyFig.6,onechecks
ii
for the Dynes superconductors easily that it is sufficient to prove that these inequalities
hold for ω >0.
InthisAppendixwewillprovethatthediagonalspec- So far, our discussion was valid for any Eliashberg su-
tralfunctionsA (k,ω)oftheDynessuperconductorsare perconductor. NowwespecializetotheDynessupercon-
ii
positive-definite, as required by general considerations. ductors. Making use of Eqs. (C1),(11) one finds easily
To this end, let us first note that the diagonal compo- that in this case the quantities ω(cid:101), Ω(cid:101), Γ(cid:101), and γ(cid:101) can be
nents of the Nambu-Gor’kov Green function within the written in terms of the functions Ω and Ω introduced
1 2
Eliashberg theory read as in Appendix A as
ω+iγ±ε
Gii(k,ω)= (Ω(cid:101)(cid:101)+iΓ(cid:101)(cid:101))2−kε2. Ω(cid:101) = Ω1,
k Γ(cid:101) = Ω2+Γs,
Since Aii(k,ω) = −π−1ImGii(k,ω), from here it follows ω = ω+Γ ωΩ2−ΓΩ1,
that the requirement A (k,ω)≥0 is equivalent to (cid:101) s Ω2+Ω2
ii 1 2
ωΩ +ΓΩ
2Ω(cid:101)Γ(cid:101)ω(cid:101)−γ(cid:101)(Ω(cid:101)2−Γ(cid:101)2)≥−γ(cid:101)ε2k∓2Ω(cid:101)Γ(cid:101)εk, γ(cid:101) = Γ+Γs Ω21+Ω22.
1 2
which has to hold for all ε and ω. Maximizing the ex-
k
Let us note in passing that these expressions justify the
pression on the right-hand side with respect to ε , this
k
results plotted in Fig. 6.
requirement can be rewritten as
Next we plug the expressions for ω(cid:101), Ω(cid:101), Γ(cid:101), and γ(cid:101)
γ(cid:101)2Γ(cid:101)2 +ω(cid:101)Ω(cid:101) 2γ(cid:101)Γ(cid:101) ≥Ω(cid:101)2, iEnqtos. (EAq1s.) a(Dnd1)o.f thIfe oinneequmalaitkieess Eusqes.o(fA2th),eafetqeurasliotmiees
γ(cid:101)2+Γ(cid:101)2 γ(cid:101)2+Γ(cid:101)2 straightforward algebra one can check that the inequal-
ities Eqs. (D1) are satisfied. This completes the proof
whichhastobevalidforallfrequenciesω. Sincethefirst
that, for the Dynes superconductors, the inequalities
termontheleft-handsideisobviouslypositive,itfollows
A (k,ω)≥0 are valid.
that it is sufficient to show that ii
(cid:32) (cid:33)
ω(cid:101) ≥ 1 γ(cid:101) + Γ(cid:101) .
Ω(cid:101) 2 Γ(cid:101) γ(cid:101) Acknowledgments
In order to prove this latter inequality, we will prove the
This work was supported by the Slovak Research and
following two simpler inequalities:
Development Agency under contracts No. APVV-0605-
14 and No. APVV-15-0496, and by the Agency VEGA
ω(cid:101) ≥ γ(cid:101), ω(cid:101) ≥ Γ(cid:101). (D1) under contract No. 1/0904/15.
Ω(cid:101) Γ(cid:101) Ω(cid:101) γ(cid:101)
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