Electron-boson effects in the infrared properties of metals

26 Pages·0.48 MB·English

by  S. V. Shulga

ELECTRON-BOSON EFFECTS IN THE INFRARED PROPERTIES OF
METALS
S. V. SHULGA
Institute of Spectroscopy, RAS, Troitsk, 142190, Russia,
1 IFW-Dresden and the University of Bayreuth, Germany
0 (September1998)
0
The interpretation of optical conductivity in thenormal and superconducting states
2
isconsideredintheframeofthestandardIsotropicSinglewideBand(ISB)modelusing
n
the theory proposed by Nam (Nam S.B., Phys. Rev. 156 470-486 (1967)). The exact
a
analytical inversion ofthenormalstateNam equationsisperformed andapplied for the
J
recovery of the reliable information from the experimental data. The Allen formula is
6
derivedinthestrongcouplingapproximationandusedforthephysicallytransparentin-
1
terpretationoftheFIRabsorption. Thephenomenological ”generalised“ Drudeformula

isobtainedfromtheNamtheoryinahightemperatureapproximation. Itisshown,that
]
n the reconstruction of the shape of the spectral function α2F(ω) from the normal state
o optical data at T>T is not unique and the same data can be fitted by many spectral
c
c functions. Thisproblemisconsideredindetailfromdifferentpointsofview. Atthesame
-
r time,usingtheexactanalyticalsolution,onecangetfromthenormalstatedataauseful
p pieceofinformation,namely,thevalueofthecouplingconstant,theupperenergybound
u
oftheelectron-bosoninteractionfunction,andtheaveragedbosonfrequency. Moreover,
s
theyareevenvisuallyaccessible,iftheopticaldataarepresentedintermofthefrequency
.
at dependentoptical mass and scattering rate. The superconductingstate Nam formalism
m andrelatedsimplifiedtheoryareanalysedfrom theuserpointofview. Anoveladaptive
method of the exact numerical inversion of the superconducting state Nam equations
-
d is presented. Since this approach uses the first derivative from the experimental curve,
n thesignal/noise ratio problem is discussed in detail. Itis shown that,thefinestructure
o of the spectral function can be recovered from optical data in the case of s-wave pair-
c ing symmetry. In contrast, in the d-wave case the resulting image is approximately the
[
convolution of the input electron-boson interaction function and the Gauss distribution
1 exp(−ω2/∆2opt). Asimplifiedvisualaccessibility(VA)procedureisproposedfor”byeye“
v analyses of the superconducting state optical reflectivity of the ISB metals with s and
3 d pairing symmetry. The bosons, responsible for the superconductivity in YBa Cu O ,
2 3 7
4 exhibita”phonon-like“spectralfunction with theupperfrequencyboundless than500
2 cm−1 and theaveraged boson frequency near 300 cm−1.
1
0
1
I. INTRODUCTORY REMARKS, RESERVATIONS AND BASIC EQUATIONS
0
/
t
a This is the preliminary version of invited lecture for Nato-ASI ”Material Science, Fundamental
m • Properties and Future Electronic Applications of High-Tc Superconductors“, Albena, Bulgaria,
- September 1998, Kluwer Academic Publishers, Dortrecht, pp. 323-360 (2001)
d
n The interpretationof the normaland superconducting state Far Infrared Region (FIR) properties
o • of metals is still rather ambiguous. In this lecture I will restrictmyself to the considerationof the
c
of Isotropic Single wide Band (ISB) model based on the Nam formalism [1].
:
v
i From the optical conductivity σ(ω) one derives the so called transport spectral function α2 F(ω).
X • tr
It has the same spectral structure, i.e. the same number of peaks with similar relative positions
r (seeFig.4in[2]andFig.1in[3])asastandardα2F(ω)whichenterstheEliashbergequations. But
a
their amplitudes can be lower,and as a result the transport coupling constant λ is less than the
tr
standard λ. In this lecture I will not distinct the α2F(ω) and α2 F(ω).
tr
Iftheelectronicbandisverynarrow,theforwardscatteringcoulddominate. Inthiscasetheoptical
•
propertiesmanifeststhemselvesbythe negligiblecoupling,whilethe tunnelling spectroscopygives
reasonablefinite coupling strength. Asimilar problemariseswhen the quasiparticlesfromthe flat
or nested parts of the Fermi surface are mainly responsible for the superconductivity. Since they
1
havesmallFermivelocities,theirinfluenceinthetransportresponsefunctionwhichisproportional
to v2 is weak. Such a scenario was recently proposed for borocarbides [4].
F
In the frame of the Eliashberg theory it is impossible to elucidate the nature of the bosons which
•
are responsible for the superconductivity. In this context I will use the term boson, keeping for
short the standard electron-phonon notation α2F(ω).
Iwillrestrictmyselftotheconsiderationofthelocal(London)limitcase,thatis,tothenormalskin
•
effect scenario. The generalisation of results obtained to the nonlocal (Pippard) limit is possible
but will not be considered here due to the lack of space.
Thepoweroftheproposedmethodsisillustratedanalysingavailableexperimentaldataforasingle-
•
domainYBa2Cu3O7−dcrystal[5]. Note,thattothebestofmyknowledge,theexperimentalHTSC
superconducting state reflectivity was not been considered from the inverse problem viewpoint at
all, despite the single early attempt [6].
Theoreticalanalysisofanyphysicalquantity,asopticalconductivity,isperformedinframeofsome
•
modelwhichincludesatheoreticalformalismandsomeassumptionsaboutthebandstructure,the
anisotropy of the coupling function and so on. The considered here ISB model is the standard
default. Nevertheless, in frame of its s-wave version the question: ”What is the value of the gap?
“hasanunique answer,butford-wavecasethe correctquestionsareeither:”Whatarethe values
ofthe gaps? “ or ”What is the value of the maximumgap? “. In other words,the absolute values
and the physical meaning of the quantities are model dependent.
If the model is reach enough, one can calculate many physical properties using the same set of
•
input (material) parameters. A solution of the inverse problem means the determination of these
material parameters from the experimental data. A short list of the ISB model input parameters
is given in section IA.
As a rule, the inverse problem is ill-posed. The experimental data contain less information, than
•
we would like to know. The reasons for are the temperature induced broadening and the finite
signal to noise ratio S/N.
In this lecture we will mainly be interested in the amount of the remaining information about
•
the spectral function α2F(ω), which can be obtained from the optical data. In spectroscopy the
unit of information is a peak. A single peak needs four parameters for its description: the upper
and the low frequency bounds Ω and Ω , the position of the maximum and its amplitude.
max min
An narrow symmetric peak within the broad band needs three parameters: the amplitude, the
halfwidth, and the position. The sum of all peaks yields the upper and lower frequency bounds
Ω and Ω of the spectral band. Since the Ω of the α2F(ω) is usually equal to zero, the
max min min
remaining three band parameters of α2F(ω) can be determined even from the normal state data
[7].
A. Isotropic single band model
The standardISB model [8]is the most developedpartofthe moderntheory ofsuperconductivity. It
describesquantitativelytherenormalizationofthephysicalpropertiesofmetalsduetotheelectron-boson
interaction. TheinputquantitiesoftheISBmodelarethedensityofstatesattheFermienergyN(0),the
∗
Fermi velocity v , the impurity scattering rateγ , the Coulombpseudopotentialµ , andthe spectral
F imp
function α2F(ω)ofthe electron-bosoninteraction. The scaleoftransportandopticalpropertiesis fixed
by the plasma frequency h¯ω = 4πe2v2N(0)/3. In this section I present only the final expressions of
pl F
the Nam theory, more detailed information can be found in [1,2,9].
p
In order to calculate any physical property in the superconducting state one needs a solutions of
Eliashberg equations (EE) [11]. In my opinion, the following EE representation is most suitable for
numerical solution by iterations
2
iγ ∆˜(ω) π
Im∆˜(ω)= imp + dyα2F(ω y)
2 ω˜2(ω) ∆˜2(ω) 2 Z −
−
qω y y ∆˜(y)
coth − tanh Re (1)
(cid:20) (cid:18) 2T (cid:19)− (cid:16)2T(cid:17)(cid:21) ω˜2(y) ∆˜2(y)
−
iγ ω˜(ω) π q
Imω˜(ω)= imp + dyα2F(ω y),
2 ω˜2(ω) ∆˜(ω) 2 Z −
−
qω y y ω˜(y)
coth − tanh Re , (2)
(cid:20) (cid:18) 2T (cid:19)− (cid:16)2T(cid:17)(cid:21) ω˜2(y) ∆˜2(y)
−
q
where∆˜(ω)andω˜(ω)aretherenormalizedgapfunctionandtherenormalizedfrequencyrespectively,and
γ denotes the impurity scattering rate within the Born approximation. The real and the imaginary
imp
partsoftheEliashbergfunctions∆˜(ω)andω˜(ω)areconnectedbytheKramers-Kronigrelations. Hence,
they have the same Fourier images. This reasoning yields the fast solution procedure. The convolution
type integrals (1-2) should be calculated by the Fast Fourier Transform (FFT) algorithm. The inverse
complex Fourier transformations of the results obtained give complex values of ∆˜(ω) and ω˜(ω). If one
wouldlike touse this efficientFFT methodalsoford-wavepairingsymmetry,the processofarithmetic-
geometric mean [12] for the evaluation of the complex elliptic integrals ( see below Eqs. (67,68) is
strongly recommended. The density of states ReN(ω) and the density of pairs ReD(ω)
ω˜(ω)
N(ω)= , (3)
ω˜2(ω) ∆˜2(ω)
−
q ∆˜(ω)
D(ω)= , (4)
ω˜2(ω) ∆˜2(ω)
−
q
could be approximated by step functions. The convolution of any spectrum with the step function
sign(y2 ∆2) results in the shift by ∆ at y > 0 and by ∆ at y < 0 and its integration over
− 0 0 − 0
y. It is important, that due to the presence of singularities in N(ω) and D(ω) at ω = ∆ the more
0
′ ′ ′ ′ ′ ′
complicatedfunctions suchas dω N(ω )N(ω ω ) and dω D(ω )D(ω ω )stillcontainajump (now
− −
atω =2∆ ). Sincethe convolutionisaquitegeneralpropertyoftheGreenfunctionsapproach,onecan
0
R R
expect that the first derivatives from a spectrum (of different nature) over frequency will reproduce the
input spectral function α2F(ω) (may be with the appropriate phase distortion). When applied to the
optical conductivity, this approachgives visual accessibility (VA) procedure discussed in the sec. III.
In the normal state the set (2) is reduced to the following simple formula
γ π y ω y
Imω˜(ω)= imp + dyα2F(y) coth tanh − , (5)
2 2 2T − 2T
Z (cid:20) (cid:16) (cid:17) (cid:18) (cid:19)(cid:21)
which however looks unwieldy in comparison with the same formula presented in Matsubara formalism
ω˜(iωn) ω˜n =ω˜n−1+2πT(1+λn) (6)
≡
where
λ α2F(iν )=2 dzα2F(z)z/(z2+ν2) (7)
k ≡ k k
Z
are kernels of the spectral function, ω = πT(2k+1) and ν = 2πTk are fermion (ω ) and boson (ν )
k k k k
Matsubara energies, respectively, and ω˜ =γ /2+πT(1+λ ).
0 imp 0
The normal state ISB optical conductivity takes the following forms
ω2 tanh y+ω tanh y
σ(ω,T)= pl dy 2T − 2T (8)
8πiω ω˜(ω+y) ω˜∗(y),
Z (cid:0) (cid:1) − (cid:0) (cid:1)
3
and
ω2 n−1 1
pl
σ(iω ,T) σ = , (9)
n n
≡ 4πn ω˜k+ω˜n−k−1
k=0
X
in the real and imaginary axes techniques, respectively.
The tedious expression for the superconducting state ISB optical conductivity has the form [1,9,10]
ω2 g tanh(x/2T)
σ(ω)= pl dx rr
16πω Z ∆˜2(x) ω˜2(x)+ ∆˜2(x+ω) ω˜2(x+ω)
− −
q g∗ tanh[(x+ωq)/2T]
rr
∗
−
∆˜2(x) ω˜2(x)+ ∆˜2(x+ω) ω˜2(x+ω)
− −
(cid:18)q q (cid:19)
g tanh[(x+ω)/2T] tanh(x/2T)
ar
+ { − } ∗, (10)
∆˜2(x) ω˜2(x)+ ∆˜2(x+ω) ω˜2(x+ω)
− −
q (cid:18)q (cid:19)
where
g =1 N(x)N(x+ω) D(x)D(x+ω),
rr
− −
∗ ∗
g =1+N (x)N(x+ω)+D (x)D(x+ω) (11)
ra
are coherent factors.
II. NORMAL STATE OPTICAL CONDUCTIVITY
A. The exact solution of an inverse problem
It is amusing, that the Eq. (9) is solvable for ω˜ and Eq. (6) for λ . Hence, at least theoretically, an
k k
exact analytical solution exists. It takes three or four steps as shown below.
I) One has to perform the analytical continuation of the conductivity σ(ω) from the real energy
axis, where we are living, to the poles of the Bose distribution function (Matsubara boson energies)
iν = 2πiTk. This procedure is similar to the Kramers-Kronig analysis, which is nothing other than
k
the analytical continuation from the real frequency axis to itself. For example, if one fits the data by a
sum of Drude and Lorentz terms or by formula (33), one has simply to substitute the complex values
iν for the frequency. Note, that the non-metallic (IR active direct phonon contribution and interband
k
transition)partofthedielectricpermeabilityǫ (ω)havetobesubtractedfromthetotaloneǫ(ω)before
ph
starting of the analysis. In the genuine far infrared region one could use the real constant ǫ∞ =ǫph(0),
instead of ǫ (ω), but if the spectral range is wide, the subtraction of ǫ (ω) is not trivial.
ph ph
II) The renormalized frequencies should be calculated from (9) as follows
ω2
ω˜ = pl , (12)
0
8πσ
1
ω2
pl
ω˜ = ω˜, (13)
1 0
4πσ −
2
ω2 ω2 n−2 1
pl pl
ω˜n−1 = ω˜0,where An =σn . (14)
2πAn − −4πn ω˜k+ω˜n−k−1
k=1
X
III) InvertingEq. (6), onecouldevaluate the values ofthe spectralfunction λ =α2F(ν ) welooked
n n
for
4
ω˜ γ /2
0 imp
λ = − 1 (15)
0
πT −
ω˜n ω˜n−1
λ = − 1. (16)
n
2πT −
IV)Using,forexample,Padepolynoms[14]onehastocontinuetheelectron-bosoninteractionfunction
back to the real axis.
YBa Cu O a-axis model 2
1.00 2 3 0.9 2.0 single peak
model 2
model
single peak model
favorite polynom 1.5
0.95
ω)
R(
vity a) ωF()1.0 b)
cti0.90 α2
e
efl
r
0.5
0.85
0.0
0 500 1000 1500 0 500
ω [cm-1] ω [cm-1]
FIG. 1. Panel (a) shows the frequency dependence of the normal state reflectivity (solid line, T=100 K,
Schutzmann J. et al., Phys. Rev., B46, 512 (1992)) in comparison with the model ones. The curves, shown by
the dotted and the dotted-dashed lines, was calculated using ”model 2“ and ”single peak“ spectra (presented
on the panel (b)). The dashed line is the least square fit of YBa Cu O a-axis reflectivity by the favourite
2 3 0.9
polynom Eq. 32 (with thefixed ǫ∞ =12).
In practice, if the signal to noise ratio is not extremely large, the last step makes no sense due to the
dramaticincreaseofthe uncertaintyintheλ withelevationofn. Thepointisthatthesmallquantities
n
λ Eq. (16) and A Eq. (14) are the difference of large quantities. Note, that the uncertainty in the
n n
λ values has to be small, not in comparison with λ itself, but with the difference between λ and its
n n n
high energy asymptotic value
2λ <Ω2 >
λ˜ = 0 (17)
n <Ω2 >+ν2
n
where
∞
1
<Ω2 >= dΩ Ωα2F(Ω) (18)
λ
0 Z0
If the averageboson energy has the order of 2πT, the last condition (δλ λ˜ λ ) may be violated
n n n
≪| − |
even for n=3, since λ˜ and λ have similar values. For example, at T =100 K for the spectral function,
n n
shown by the dotted line in Fig. 1, λ˜ /λ =1.37, 1.12, 1.06, 1.03 for n=1, 2, 3, 4 respectively. Three
n n
or four λ values only form a too small parameter set to recover correctly the shape of the spectral
n
function. Actually,thepossibilitytoobtainatleastsmallamountofreliableinformation,istheadditional
advantageprovidedbythe Matsubarasolutionincomparisonwiththe realaxisone whereall valuesare
unreliable [15]. Fig. 1 shows the good agreement between the experimental normal state 1-2-3 single
crystalreflectivity data [5] and the calculatedones using for various spectralfunctions. Unfortunately,
the lack of a unique solution promoted unreasonable speculations on the mechanisms in several novel
superconducting systems in connection with proposed hand-made electron-boson interaction functions.
Inmyopinion,twelveyearsafterthediscoveryofthesuperconductivityincupratesitistimelytoreturn
to a self-consistent description of the superconductivity in this compounds.
5
The first Matsubara value of the conductivity σ(iν )( σ ) gives the value of the coupling constant
1 1
≡
λ( λ ), if the plasma frequency ω and the impurity scattering rate γ are known a priori
0 pl imp
≡
1 8π2T(1+λ ) 4πγ
0 imp
= + . (19)
σ ω2 ω2
1 pl pl
ThereflectivityR(ω)canalsobecontinuedtotheimaginaryfrequencyaxisandincludedintothesolution
of the inverse problem. Following this way, one can derive from (19) the following formula
γ ω2
imp p
λ = 1 + , (20)
0 −2πT − (2πT)2(coth2(K1/4) ǫ∞)
−
where ǫ∞ is the low-frequency value of the non-metallic (phonon and interband transition) part of
dielectric permeability, and K is the electromagnetic kernel
m
∞
2ν log 1 R(z)
m
K(ν )= dz | − |. (21)
m π z2+ν2
Z0 m
The first λ and the second λ allow ( for example ) to estimate the average boson energy and very
1 2
approximately the width of the spectral function. If there is the possibility to combine this incomplete
information about α2F(ω) with the results of other spectral measurements, for example, tunnelling or
neutron data, one can qualify the transport spectral function. The so called ”model 2“ electron-boson
interactionfunction[7]wasrecoveredassuming,thatthesolutionhastoberesemblethephonondensity
of states, measured by inelasitc neutron spectroscopy.
B. Optical mass and impurity scattering rate
In ref. [7,15] we shown, that it is useful to describe the optical conductivity in terms of the so called
“extended” Drude formula
ω2/4π ω2/4π
pl p
σ(ω,T)= = . (22)
W(ω,T) γ (ω,T) iωm∗ (ω,T)/m
opt − opt b
∗
Here the optical relaxation W(ω), the optical mass m (ω,T)/m and the optical scattering rate
opt b
γ (ω,T)asthecomplex,realandimaginarypartsofaninversenormalisedconductivityω2/4πσ(ω,T)
opt pl
have been introduced. This presentation is old and popular in spectroscopy, but the practical value of
the Eq. (22) has not been exploited much. The optical relaxation will be compared with the effective
∗
mass m (ω,T)/m and the effective scattering rate γ (ω,T), which by definition are the real and
eff b eff
imaginary parts of the renormalized frequency
∗
ωm (ω,T) iγ (ω,T)
ω˜(ω,T)= eff + eff . (23)
m 2
b
∗
When the frequency dependencies of γ (ω,T) and m (ω,T)/m are plotted, the coupling constant,
opt b
the average boson energy and the upper boundary of spectral function are visually accessible (see Fig.
2). The appropriate visual accessibility rules can be derived from the properties of the renormalized
frequencyω˜ usingtheAllenformulaEq.(26)[2]. Since suchusefulexpressionwasoriginallyobtainedin
the weakcoupling approximation(λ 1)only, below I rederiveit forthe generalcase,including strong
≪
coupling.
For any k the values of λ are positive and are of the same order of magnitude, except may be
k
λ . It means, that in the zero order the series ω˜ defined by Eq. (6) is an arithmetical progression.
0 n
Consequently, according to the Gauss rule for the arithmetical series the denominators ω˜k+ω˜n−k−1 in
(9) at a given n do not depend on k, that is, they coincide. In view of Eq. (22), Eq. (9) takes the form
Wn+1 ω˜k+ω˜n−k. (24)
≈
Hence we can expand the denominator in (9)
6
n−1
4πσ 1 1 1
n
= (25)
ωp2l ≡ Wn n k=0 Wn[1+(ω˜k+ω˜n−k−1−Wn)/Wn]
X
to the first order in power of (ω˜k +ω˜n−k−1 Wn)/Wn. After some simple algebra we arrive at the
−
sought-for Allen formula
n−1
2
W = ω˜ . (26)
n k
n
k=0
X
m* /m (ω,T=1K)
eff b
3 m* /m (ω,T=100K) b)
eff b
1 ]1500
-m
c
*m/mb2 α2F(ω) a) ng rates [ 1000 γγeff((ωω,,TT==110K0)K)
1 m*opt/mb(ω,T=1K) catteri 500 γeopfft(ω,T=1K)
m* /m (ω,T=100K) S γ (ω,T=100K)
opt b opt
0 0
0 500 1000 1500 0 500 1000 1500
ω [cm-1] energy [cm-1]
FIG.2. Thefrequencydependenciesofthenormalstateopticalandeffectivemasses(a)andscatteringrates(b)
calculatedusingthe”model2“spectralfunctionwithλ=2(showninpanel(a)forcomparison). Thequasiparticle
effective mass m∗ (ω)/m and scattering rate γ (ω) at T =1 K (solid lines, panels (a) and (b)) reproduce
eff b eff
the fine structure of the spectral function, but at T =100 K (dashed lines) this function are smeared out due
to averaging over Fermi distribution. At ω > Ω the quasiparticles become ”undressed“, m∗ (ω)/m ≈ 1
max eff b
and γ (ω) ≈ const. The optical mass m∗ (ω)/m and the scattering rate γ (ω) even at T =1 K (dotted
eff opt b opt
lines) do not reproduce the structure of the spectral function due to the energy averaging, all the more at high
temperature ones (T =100 K, dashed-dottedlines).
The real axis versions of Eq. (26) at T =0 look similar
1 ω
W(ω)= dz2ω˜(z), (27)
ω
Z0
ωm∗ (ω,T) 1 ω 2zm∗ (z,T)
opt eff
= dz , (28)
m ω m
b Z0 b
1 ω
γ (ω)= dzγ (z). (29)
opt eff
ω
Z0
and justify the following simple description of the interaction of light with the conducting subsystem
of the normal state metal. When a photon with an energy ω has been absorbed, two excited virtual
′
quasiparticles, an ”electron“ and a ”hole“ are created. If the first particle’s frequency is ω , the second
′
ones frequency should be ω ω . The exited ”electron“ and ”hole“ relax to the Fermi level according
− ′
to the quasiparticle laws (see Eqs. 5,6). Since ω can varies from 0 to ω, the optical relaxation W(ω) is
the frequency-averaged(1 ω), double (electron+hole factor 2) renormalized frequency ω˜.
ω 0 →
At T =0 the ISB effective scattering rate γ (ω) 2Imω˜(ω)
eff
R ≡
ω
γ (ω)=γ +2π dyα2F(y) (30)
eff imp
Z0
7
has a clear physical meaning. The exited quasiparticles with the energy ω can relax by the emission of
virtual bosons with the energy varying the range from 0 to ω or by the impurity scattering.
At finite temperature the Allen formula
1 tanh ω−z +tanh z
W(ω)= dz 2T 2T 2ω˜(z) (31)
ω 2
Z (cid:0) (cid:1) (cid:0) (cid:1)
and the effective scattering rate (5) have exactly the same interpretation, since the only difference
betweenthem and Eqs.(27,27,29,30)is the presence ofthe Fermi distributions insteadofstep functions
at T=0.
Figure 2 shows the frequency dependencies of the normal state optical and effective masses and
scattering rates calculated using the model spectral function [7] with λ=2. The results are plotted for
the low temperature T=1 K and for T=100 K which is close to T and has the order of Ω /2πT,
c boson
where Ω =330 cm−1 is the averaged boson frequency. Both m∗ (ω)/m and γ (ω) show the
boson eff b eff
characteristic features corresponding the peculiarities of the phonon spectrum. When the frequency ω
exceeds the upper energy bound of the boson spectrum Ω , the quasiparticle becomes ”undressed“.
max
∗
Its mass m (ω)/m 1 and effective scattering rate γ (ω) const, but of course, the absolute
eff b ≈ eff ≈
value of γ (ω ) could be huge.
eff
→∞ ∗
The optical mass m (ω)/m and scattering rate γ (ω) have no features at ω Ω due to the
opt b opt ≈ max
energy averaging(27,28,29,31). One can only calculate the approximate value of Ω as the frequency
max
∗
wherethem (ω)/m 1andγ (ω)reachthehalvesoftheirmaximummagnitudes. Fig.2shows,that
opt b− opt
this my rule can not be applied even to the calculated frequency dependence of the optical scattering
rate γ (ω), the more to the experimental curve, where the high energy behaviour is distorted by the
opt
interband transition contribution to the dielectric permeability. In contrast the frequency dependence
∗
of the optical mass m (ω)/m is suitable for the application of the simple criterion
opt b
∗ ∗
m (ω =Ω )/m 1 0.5(m (ω =0)/m 1). (32)
max b b
− ≈ −
∗
Moreoveritslowenergyvaluem (ω =0)/m 1+λ itselfcontainsausefulpieceofinformation. Fig.
opt b ≈ tr ∗
2ashows,thattheapproximatevaluesofthetransportcouplingconstantλ (m (0)/m 1)andthe
∗ tr ≈ opt b−∗
upperfrequencyboundoftheelectron-bosoncouplingfunction[m (Ω )/m 1 0.5(m (0)/m
opt max b− ≈ opt b−
1)] are visually accessible. If the spectral function is wide and its peak position (or averagedfrequency)
coincideroughlywiththeΩ /2,theopticalrelaxationW(ω)issmoothandtheelucidationofΩ at
max boson
hightemperatureisimpossible. Whenanexoticsuperconductorwiththenarrow,δ-functionlikespectral
function will be discovered, following this way one can easily construct the recipe for the evaluation of
the strength and the position of the single peak.
C. My favourite polynom
The Pade polynom analytical continuation [14] is a capricious and not completely correct procedure.
Nevertheless, if the temperature is high enough in comparison with the energy under consideration it
works splendidly. The numerical experiment has given to my great surprise the following (and to some
extent obvious) result (see Fig. 3). The good agreement between the real axis calculations and the
Matsubara ones takes place every time when the approximation using 1000 σ points gives the same
n
result as the four points Pade approximation.
This remarkable Pade polynom with only four parameters ω2/4π, A, B and C
pl
ω2/4π
pl
σ(ω)= (33)
A iω+ Bω
− ω+iC
fits well the majority of the experimental data at T T and itself is the demonstration of the result
c
≥
obtainedinsectionIIA: the fewfirstMatsubaravaluesσ areresponsibleforthe frequencydependence
k
of σ(ω).
The Pade polynom (33) is a formula of merit. At first, it has the correct analytical properties. Its
two poles are located in the lower half-plane of the complex energy. The high (ω ) and low
→ ∞
(ω 0) energy asymptotic behaviour are reasonable [23]. At second, it is ideal for the least square
→
8
fits of experimental R(ω) [24], as well as for simple analytical estimations. Having the exact solution
(see sec.IIA) we have to substitute the complex values iν for energy, perform easily the analytical
k
continuationto theimaginaryaxisandanalysethe valuesofσ . Inthewayweobtainthe interpretation
i
of A, B, C. Another possibility is a fast and easy analysis of the data in terms of the optical mass
∗
m (ω,T)/m and the optical scattering rate γ (ω,T)
opt b opt
∗
m (ω,T) BC
opt 1= (34)
m − ω2+C2
b
Bω2
γ (ω,T)=A+ . (35)
opt ω2+C2
0.0035 1.00
Reσ(ω)Matsubara 4 points 40
b
0.0030 Reσ(ω) Matsubara 4 points ω)/m 30
0.0025 RImeσσ((ωω)) rreeaall aaxxiiss 0.95 *(mopt20
0.0020 ω) b)
R(
σω) (n0.0015 a) ctivity 0.90 0 50ω 1[c0m0-11 ]50
0.0010 e
Refl0.85 Rb3C60
0.0005 fit, ω =10000cm-1
pl
0.0000
0.80
0 200 400 600 0 50 100 150
ω [cm-1] ω [cm-1]
FIG.3. Panel (a) shows thefrequency dependenciesof thereal and imaginary partsof thecalculated normal
stateopticalconductivityusingtherealaxisformalism (solidlines)andMatsubaratechnique(squares-Reσ(ω),
circles - Imσ(ω)). The analytical continuation from the imaginary axis to the real one was made by the Pade
polynom with the degree of polynomial n=4. Calculations was made for T=100 K using ”model 2“ spectral
function. Panel (b) shows the frequency dependence of the normal state reflectivity R(ω) of Rb C measured
3 60
atT=40KbyDegiorgiet al.,Phys. Rev. Lett. 69,2987(1992)(circles)incomparison withtheleastsquarefit
by the favourite polynom. Since four parameters were too much, the value of the plasma frequency ω =10000
pl
cm−1 was fixed. The obtained frequency dependenceof the optical mass is shown in theinset.
Atthird,ifthelow-frequencyreflectivityR(ω)measuredathightemperaturecannotbefittedbyEq.
(33) or the obtained value of λ contradicts the generally accepted values, it might indicate, that the
0
materialunderconsiderationisnotastandardIsotropicSinglewideBand(ISB)metalanditsproperties
havetobeanalysediftermsofanother,unconventionalmodel. Inordertoillustratetheutilityofthethis
approach,letusconsideranothersuperconductingcompoundwithrelativelyhightransitiontemperature
Rb C [16,18,17]. Since the ratio of the experimental reflectivities in the superconducting and normal
3 60
statesR /R 1 above100cm−1 [16],in frameofthe ISB modelitmeans,thatthe highenergybound
s n
of the electron≈-boson interaction function Ω < 100 cm−1. The low energy part of the normal state
max
reflectivity R(ω), measured at T=40 K, was fitted (with the adopted fixed ω =10000 cm−1) by (33)
pl
∗
(see Fig. 3b), m (ω,T)/m is shown in the inset. One can see, that the optical mass at small ω is
opt b
huge itself and gives λ > 100. This values points to heavy fermion or polaron models, rather than to
the generally accepted weak or intermediate coupling scenario. On the other hand, the analysis of the
normal state specific heat data gives an extremely small value of the coupling constant. As a result
we concluded that Rb C is not a ISB metal and its properties should be treated in terms of a more
3 60
complicated model which takes into account the self-energy and conductivity vertex corrections, the
small value of the bandwidth ∆E 100 meV, possible influence of the electron-electroncorrelation,the
≈
strong anisotropy of the coupling function and the Fermi velocity, and etc.
Andthelastbutnotleast,ifthedatacanbefittedby(33),thatisbytheformulawithfourparameters,
9
itis impossibletorecoverfromthis fourparametersthe detailedshape ofthe electron-bosoninteraction
function α2F(ω).
In conclusion, it is naturally to define the amount of the available information by the number of
poles and zeros of the function which fits well the experimental curve or the number of the determinate
Matsubara values.
D. Temperature dependence of the optical mass and relaxation rate and its two-band ersatz
ω)3000 a)
γ*(opt T=100 K b)
ω) , T=300 K 3
γes (opt2000 T=200 K ω)/mb
g rat *m(opt T=200 K
erin T=100 K ss 2
att1000 ma T=300 K
ptical sc γγ(*(ωω)) Optical
O
0 1
0 500 1000 1500 0 500 1000 1500
ω [cm-1] ω [cm-1]
4
ω=0,T)/mb3 Tc Ωω/max(m*-1)/2 12 Tc Τγ-1() [cm]ate opt34000000 γγ 2ooppπtt((λωωT==i0n,fTin)ity,T)
*(mass mopt2 0 0 2πT/Ωma2x c) scattering r2000 d)
al al 1000
c c
pti pti
O O
1 0
0 1 2 3 0 100 200 300
2πT/Ω
max T [cm-1]
FIG.4. Panels (a)and(b)showthefrequencydependenceof theopticalmass andtheopticalscattering rate
∗
forT=100K,200K,300K.Panels(c)and(d)presentthecorrespondingtemperaturedependenceofm (T)/m
opt b
and γ (T) for ω =0 and ω =∞. All curves (solid lines) was calculated using model 2 spectral function. For
opt
∗ ∗
comparison the normalised optical scattering rates γ (ω,T) = γ (ω,T)/m (ω,T)/m are plotted by dotted
opt opt b
lines in panel (a). The temperature dependence of the quantity ω(m∗−1)/2 (see text) is shown by solid line on
inset (c) together with the phenomenological line ω = 5T (dotted line). The common for γ (ω = 0,T) and
opt
γ (ω=∞,T) asymptoteγ =2πλT is shown bydotted line in panel (d).
opt
In frame of the ISB model we assume, that all quasiparticles have the same Fermi velocity, but in
real metals there is some anisotropy of v . If the impurity scattering rate γ is small in comparison
F imp
with the complete optical scattering rate γ and the electron-boson coupling is isotropic, we can use
opt
the averagevalue <v2 > and keepthe ISB model. On the other hand, if the impurity scattering,given
F
10