Table Of ContentNon-Fermi-liquid scattering rates and anomalous band dispersion in ferropnictides
J.Fink1, A.Charnukha1, E.D.L.Rienks2, Z.H. Liu1, S. Thirupathaiah2, I. Avigo3, F.Roth4, H.S.Jeevan5,
P.Gegenwart5, M.Roslova1,6, I.Morozov1,6, S.Wurmehl1,7, U.Bovensiepen3, S.Borisenko1, M. Vojta8, B.Büchner1,7
1Leibniz Institute for Solid State and Materials Research Dresden, Helmholtzstr. 20, D-01069 Dresden, Germany
2Helmholtz-Zentrum Berlin, Albert-Einstein-Strasse 15, D-12489 Berlin, Germany
3Fakultät für Physik, Universität Duisburg-Essen, Lotharstr. 1, D-47075 Duisburg, Germany
4Center for Free-Electron Laser Science / DESY, Notkestrasse 85, D-22607 Hamburg, Germany
5Institut für Physik, Universität Augsburg, Universitätstr.1,D-86135 Augsburg, Germany
6Department of Chemistry, Lomonosov Moscow State University, 119991 Moscow, Russia
5 7Institut für Festkörperphysik, Technische Universität Dresden, D-01062 Dresden, Germany
1 8 Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
0
2 (Dated: March 25, 2015)
r
Angle-resolved photoemission spectroscopy (ARPES) is used to study the band dispersion and
a
M thequasiparticlescatteringratesintwoferropnictidessystems. OurARPESresultsshowlinear-in-
energydependentscatteringrateswhichareconstantinawiderangeofcontrolparameterandwhich
depend on the orbital character of the bands. We demonstrate that the linear energy dependence
4
2 gives rise to weakly dispersing band with a strong mass enhancement when the band maximum
crosses the chemical potential. In the superconducting phase the related small effective Fermi
] energy favors a Bardeen-Cooper-Schrieffer (BCS)[1]-Bose-Einstein (BE)[2] crossover state.
n
o PACSnumbers: 74.25.Jb,74.72.Ek,79.60.âĹŠi
c
-
r
p Introduction. Unconventional/high temperature su-
u perconductivity is observed in heavy fermion systems,
s
cuprates, molecular crystals, and ferropnictides close to
. M
at a point in the phase diagram where, as a function of a Iy
II
m controlparametersuchaspressure,chemicalpressure,or 6
III
- doping, the antiferromagnetic order is suppressed[3, 4].
d Awidespreadviewisthatatthispoint, whichiscalleda 7 5
on quantum critical point (QCP), strong antiferromagnetic 21G GGG 348 Mx
fluctuations are a candidate for the glue mediating su-
c
d
[ perconductivity and that these fluctuations would also yz
d
2 account for the normal state non-Fermi-liquid behavior xz
which is expressed, e.g. in a linear temperature depen- d
v xy
5 dence of the resistivity or to large mass enhancement of
3 the charge carriers. In the ferropnictides[5] the strange
1 normalstatepropertieshavebeenobserved,e.g. intrans-
2
port[6] and thermal properties[7]. Theoretically a QCP
0
in ferronpnictides has been predicted in Refs.8 and 9.
.
1 ARPESisaversatilemethodtoobtaininformationon Figure1. (Coloronline)SchematicFermisurfacesatkz =0
0 of ferropnictides formed by three hole pockets at Γ and elec-
5 the energy (ω) and momentum (k) dependent real and tron pockets at M. The orbital character which was adopted
1 the imaginary part of the self-energy function Σ(ω,k)
from Ref.[18] is indicated by different colors.
: whicharerelatedtothemassenhancementandthescat-
v
i tering rate, respectively[10][11].
X
In this contribution we present ARPES results on electron pockets. In the wide range of the investigated
r
a the band dispersion and the scattering rate as a control parameter we realize a linear-in-energy depen-
function of the control parameter of two ferropnic- dence of the scattering rates independent of the control
tide[5] systems: the chemically pressurized "122" com- parameter and no enhancement at the supposed QCP.
pounds BaFe (As P ) and EuFe (As P ) , and Furthermoreweseethatatoptimal"doping"weobserve
2 1 x x 2 2 1 x x 2
theelectrondoped"−111"compoundsNaFe− Co Asand a crossing of the top of the hole band through the Fermi
1 x x
−
NaFe Rh As. TherearenumerousARPESstudieson level. On the basis of these results we propose the fol-
1 x x
−
these systems published in the literature[12–17]. In the lowing scenario: a co-action between a highly correlated
present study we have measured along three directions electronliquidwithalinear-in-energyscatteringrateand
in the Brillouin zone, shown in Fig.1, to reach the high acrossingofathetop(bottom)ofahole(electron)pocket
symmetry points on the three hole pockets and on the throughthechemicalpotentialcausesananomalousband
2
energy range of 5 ω 120 meV, which can be de-
(a) (b) scribed by Γ = α≤+ βω≤. While α is determined by
elastic scattering, β is determined by inelastic scatter-
ing processes. β is much larger for the inner hole pocket
compared to the value of the middle hole pocket. Simi-
(c)
lar results have been obtained previously for the system
Ba(Fe Co ) As [21]. In Fig.2(d) we present a com-
1 x x 2 2
pilation−of all available data of β for BaFe (As P )
(d) 2 1 x x 2
and EuFe (As P ) taken at k =0. −
2 1 x x 2 z
(e) −
InFig.2(e)weshowthetopofthemiddleandtheinner
hole pocket as derived from a parabolic fit to the mea-
sureddispersion. Intheunsubstitutedsamplesasplitting
between xz and the yz states of the order of 50 meV is
observed in the tetragonal paramagnetic phase which is
probably caused by nematic fluctuations[22, 23]. As ex-
pectedfromotherstudies[23]thesplittingdecreasesand
Figure2. (Coloronline)ARPESdataBaFe2(As1 xPx)2(cir-
cles)andEuFe2(As1 xPx)2 (squares)(a)Energy−distribution disappears near optimal "doping". While we observe a
map along cut I nea−r Z of BaFe2(As1 xPx)2 x = 0.27. The decrease of the Fermi wave vector at Γ with increasing
redlinesdepictdispersionsderivedfro−mfits. (b)Momentum "doping" concentration, at Z the Fermi wave vector in-
distribution curve taken from the data in (a) at the binding creases. Thus we observe a Lifshitz transition of a Fermi
energy of 80 meV. (c) Scattering rates Γ as a function of en-
cylinder at low P concentrations to an ellipsoid around
ergy for the inner and the middle hole pocket together with
the Z point at higher P concentrations. This Lifshitz
a linear fit (black). (d) Compilation of all β values for vari-
transitionhasbeendetectedinseveralARPESstudiesof
ous high symmetry points for kz = 0 as a function of the P
concentration. (e) Top of the inner (green) and the middle 122systems[13,24]. InthepresentARPESstudy,within
(blue) hole pocket as a function of the P concentration. The error bars, we have not found any major differences in
color code corresponds to that of the high symmetry points the electronic structure between BaFe (As P ) and
2 1 x x 2
in Fig.1. The dashed lines are guides to the eye. EuFe (As P ) . −
2 1 x x 2
−
The 111 compounds are ideal for ARPES experiments
since they do not form a polar surface after cleaving and
dispersionattheFermilevelwhichleadstoastrongmass therefore, one is certain to find bulk properties[25]. On
enhancement in the normal state and to a small effec- the other hand, they have the disadvantage that close to
tive Fermi energy favoring a Bardeen-Cooper-Schrieffer theFermilevelthemiddleholepockethybridizeswiththe
(BCS)[1]-Bose-Einstein(BE)[2]crossoverstateinthesu- inner hole pocket. As a consequence, reliable results on
perconducting phase. thescatteringratescanbeonlyderivedathigherbinding
Experimental. Single crystals were grown using the energies. InFig.3(a)weshowanenergydistributionmap
self-flux technique[19, 20]. ARPES measurements were of the optimally doped NaFe Rh As x = 0.027 close
1 x x
conductedatthe12-and13-ARPESendstationsattached to the Γ point along cut I. All−three hole pockets can be
to the beamline UE112 PGM 2 at BESSY with energy clearly resolved. In Fig.3(b) we depict a cut at constant
and angle resolutions between 4 and 10 meV and 0.2 ◦, EB =40 meV together with a fit by six Lorentzians.
respectively. If not otherwise stated the measuring tem- FromthosefitswederiveΓasafunctionoftheenergy
perature at the 12- and 13-ARPES endstations were 30 aspresentedinFig.3(c). Againforallthreeholepockets,
and 0.9 K, respectively. For the unsubstituted samples Γ increases linearly with increasing energy in the range
temperature above the Néel temperatures were used to where there is no band overlapping. For the inner, mid-
keep the samples in the paramagnetic state. dle,andouterholepocketswederiveβ valuesof1.0, 0,
≈
Results. In Fig.2(a) we show an energy distribu- and 0.69, respectively. Comparing the superconducting
tion map along cut I (Fig.1) for the optimally "doped" gaps ∆ for the similar compound LiFeAs [26, 27] with
BaFe (As P ) x = 0.27 measured close to the Z = the present values of Γ shows that the strength of the
2 1 x x 2
(0,0,π/c) −point, where c is the lattice constant. The scatteringrateisdirectlyrelatedtothesizeofthesuper-
two inner hole pockets can be clearly resolved. Due to conducting gap. In Fig.3(d) we present a compilation of
matrix element effects the outer hole pocket is more dif- allβ valuesforNaFe Co AsnearΓasafunctionofCo
1 x x
−
ficult to see in this system. In Fig.2(b) we show a cut concentration. As in the case of the 122 compounds, the
at constant binding energy E = 80 meV together with β valuesareratherindependentofthecontrolparameter
B
a fit to four Lorentzians. From those fits the life-time and no enhancement at optimal doping (x = 0.027) is
broadening Γ as a function of the energy is derived as observed. Finally, inFig.3(e)thetopoftheholepockets
presented in Fig.2(c). For both hole pockets, we find a as a function of the Co concentration is depicted. As in
non-Fermi-liquid linear-in-energy dependence in a large the case of the 122 compounds, the crossing of the top
3
(b)
(a)
(b)
(a)
e f
(c)
c
(d) (c) (d) (e)
(e)
Figure3. (Coloronline)ARPESdataof111compounds. (a) Figure 4. (Color online) (a) Energy distribution map along
Exn=erg0y.0d2i7s.triTbhuetiornedmlainpeaslocnorgrecsuptoInndeatroΓdoisfpNerasFioen1−sxdRehrixvAeds cfiutt(rIednelainre)ΓaonfdBtahFeeb2a(Ares1p−axrtPicxl)e2dxisp=er0si.o2n4 (tbolgaectkhelirnew)i.th(ba)
from fits. (b) Momentum distribution curve taken from the EnergydistributionmapalongcutInearΓofNaFe1 xRhxAs
−
datain(a)atEB =40meVtogetherwithafit. (c)Scattering x=0.027togetherwiththefitteddispersionfortheouterhole
rates Γ as a function of energy for the three hole pockets pocket(redline)andthebareparticledispersion(blackline).
together with a linear fit (black). (d) Compilation of all β (c) Bare particle dispersion (black line) with the top of the
values of NaFe1 xCoxAs for various high symmetry points hole pocket at the Fermi level together with calculations of
−
for kz = 0 as a function of the Co concentration. (e) Top ΣMF (greenline)andtherenormalizeddispersion(redline).
<
of the inner , the middle , and the outer hole pocket as a Inadditionweshowtheeffectivemassm∗/m0 (blueline)and
functionoftheCoconcentration. Thecolorcodecorresponds the m∗/m0 = 1 line (thin dashed blue line). (d) The same
tothatofthehighsymmetrypointsinFig.1. Thehorizontal as (c) but with the top of the hole pocket 50 meV above the
dashed lines are guides to the eye. Fermi level. (e) The same as (c) but with the top of the hole
pocket 50 meV below the Fermi level.
of the inner hole pocket, having the strongest scattering
rate, occurs very close to optimal doping. behavior has been seen only for hole but not for electron
For all four compounds we have obtained similar re- doping, appears inconsistent with our results.
sults at the other high-symmetry points: the β values In the highly correlated systems inelastic scattering
are not dependent on the control parameter and there is processes are determined by intra (having the same or-
no k dependence[11]. bital character) and inter-band (having different orbital
z
Discussion. Our experimental results can be com- character) transitions. The difference in the scattering
pared with those from theoretical calculations in the rates between the hole pockets have been predicted by
framework of density functional theory (DFT) combined theoreticalcalculations[18,30]asthescatteringbyintra-
with dynamical mean-field theory (DMFT). Interest- bandtransitionsshouldbelargerthanthatbyinter-band
ingly, non-Fermi-liquid-like self-energies have been re- transitions.
ported[28, 29] which could be fitted by sub-linear power From the scattering rates we have calculated Σ
= ∝
laws over a range of elevated energies. This incoherent- Γ[10, 11]. Making the assumption that Σ is also linear
=
metal behavior has been attributed to an interplay of in energy at low energies and constant above a ultravio-
Hubbard and Hund’s-rule couplings in a multi-band sys- let cutoff energy ωc , also used in the phenomenological
tem: thereisasizableregimewherecarriersarestrongly marginal Fermi liquid theory[31], one can define a cou-
scattered by slowly fluctuating unquenched spins. Im- pling constant λMF by the relation =ΣMF = π2λMFu,
portantly, this source of scattering is essentially local in where u = max(ω ,kT) and kT is the thermal energy.
| |
space, implying that it is weakly influenced by longer- The comparison with experimental data near point 1 for
range spin correlations and can thus persist over an ex- BaFe2(As1 xPx)2 yields a value of λMF = 1.5 which is
tended part of the phase diagram. We believe that slightly lar−ger than values derived for cuprates[32]. Fur-
our observation of a strong, weakly doping-dependent, thervaluesforλMF arepresentedintheSupplement[11].
and approximately energy-linear single-particle scatter- Since Σ and Σ are connected by the Kramers-
< =
ing rate is consistent with originating from local corre- Kronig relation, we also use a non-Fermi-liquid Σ
<
lation physics as the one described by DMFT. We point which can be derived from a numerical calculation or
out, however, that the doping dependence of the scatter- from the analytic expression Σ = 1λ ωlnωc [31].
< MF 2 MF u
ing rate as reported in Ref.[28], where non-Fermi-liquid In Fig.4(a) we show an energy distribution map of
4
BaFe (As P ) x = 0.24 measured near Γ along cut the hole pocket having there a constant density of states
2 1 x x 2
I using a v−ertical photon polarization. This dispersion (DOS) typical of a two-dimensional electronic structure
could be fitted combining a parabolic bare particle dis- plays an important role for these correlated systems. It
persion derived from DFT calculations[33] and Σ . canexplainthedivergentmassenhancementinthetrans-
MF
<
Choosing a cut-of energy ω = 1.5 eV we obtain a port properties observed in BaFe (As P ) near opti-
c 2 1 x x 2
λ = 1.4 which is close to the value derived from the mal"doping"[6]andthestrangebehav−iorofthethermal
MF
scattering rate. Analogous data for NaFe Rh As are properties of Ba(Fe Co ) As [7].
1 x x 1 x x 2 2
shown in Fig.4(b). Here we have fitted th−e dispersion −
of the outer band (red line) using Σ . A coupling There are several ARPES publications on iron-based
MF
constant λ =1.2 is obtained. In<Fig.4(a) and (b) we superconductors which show clear indications of weakly
MF
dispersing bands at the Fermi level, which cannot be
obtain a dispersion which at the Fermi level is strongly
explained on the basis of DFT calculations. In a re-
reduced compared to the bare particle dispersion. We
cent ARPES study of ferrochalcogenides an apparently
emphasize that the weak dispersion near the Fermi level
in NaFe Rh As is not caused by a coherence peak in non-dispersive peak at the chemical potential has been
1 x x
− detected which is not in agreement with DFT calcula-
the superconducting state since the measurement pre-
tions[34]. Moreover, in the same system "shallow pock-
sented in Fig.4(b) was performed in the normal state at
T =26 K. ets" have been detected which could lead to a BCS-
BE crossover in the superconducting phase[35]. These
Using Σ , we can now calculate the renormalized
MF
< anomalousbanddispersionscanbewelldescribedbythe
dispersion close to the Fermi level as a function of the
scenario presented here.
band position. We used the parameters typical of the
inner hole pocket of BaFe (As P ) : λ = 1.5,
2 1 x x 2 MF Different from the iron-based superconductors, in the
ω = 1.5 eV, and a bare particl−e dispersion from DFT
c cuprates there is already in the bare particle band struc-
calculations on BaFe As [33]. In Fig.4(c) we show such
2 2 ture a peaked DOS due to a saddle point in the band
calculations for a bare particle hole pocket touching the
structure. There is a longstanding discussion on the im-
chemical potential , for Σ, and for the renormalized
< portance of this van Hove singularity, which crosses the
band. Due to the fact that Σ scales to zero logarith-
< Fermi level near optimal doping[36, 37] and on a break-
mically and is very close to the bare particle dispersion,
down of the Migdal theorem[38] which was the basis
there is a strong renormalization close to the Fermi level
of the standard BCS theory of superconductivity. This
which leads to a weakly dispersive band. The extension
breakdownrequirestheinclusionofnon-adiabaticeffects
of this singularity increases with increasing λ and in-
MF andthegeneralizationoftheEliashbergequations[39]in-
creasingm . Themassenhancementisenergydependent
0 terpolating between the BCS theory and the BE con-
and diverges at zero energy. In Fig.4(d) we depict an
densation[40]. In an ARPES study on YBa Cu O
2 3 7 δ
analogous calculation but with the top of the bare hole −
an "extended" van Hove singularity has been found[41]
pocket shifted 50 meV above the Fermi level. At the Γ
which nicely can be explained by the scenario described
point the calculated dispersion shows a renormalization
above: a co-action between a highly correlated electron
by a factor of four. One immediately realizes that the
liquidbehaviorandaweaklydispersingband(thesaddle
extension of the weakly renormalized dispersion at the
point) moving through the Fermi level.
Fermilevelisstronglyreduced. Moreover,thedivergence
oftheeffectivemasscurveisshiftedbyE 2.5meVinto Summary. We have measured the scattering rates
≈
theunoccupiedstates. BelowandattheFermilevelthere of two ferropnictide systems at various high symmetry
is only a moderate mass enhancement for the occupied points. In a wide range of the control parameter we ob-
electronic states. We emphasize that in this case, where serve constant linear-in-energy scattering rates. In par-
thetopoftheholepocketisnotattheFermilevel, there ticular, we see no enhancement of the scattering rates
is no divergent mass enhancement. This means that the closetotheexpectedQCP.Furthermoreatoptimal"dop-
strong low-energy scattering rates of a moderately cor- ing" the top of the hole pockets having the largest scat-
related system alone cannot describe the strange normal tering rates crosses the Fermi level. The strange nor-
state properties at a specific value of the control param- mal state transport and thermal properties near optimal
eter. In Fig.4(e) we present analogous calculations for a "doping"andprobablyalsotheunconventionalsupercon-
bareparticleholepocketshiftedbelow50meVbelowthe ductivity can be related to a proposed scenario in which
Fermi level. At Γ the bare particle dispersion is renor- a co-action of the real part of a non- Fermi liquid self-
malized by a factor 5 and the mass enhancement is energy with an edge in the DOS at the Fermi level leads
≈
strongly reduced. Analogous calculations, in which Σ to a weakly dispersing band at the chemical potential
=
is extrapolated below 5 meV by a Fermi liquid behavior and to strongly enhanced effective masses. The results
( Σ ω2) yield also a large mass enhancement at the canbegeneralizedtootherunconventionalsuperconduc-
= ∝
point where the hole pocket crosses the Fermi level[11]. tors and possibly are a recipe for future search of new
Thesecalculationsshowthatthecrossingofthetopof high-T superconductors.
c
5
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Supplementary material for: Non-Fermi-liquid scattering rates and anomalous band
dispersion in ferropnictides
J.Fink1, A.Charnukha1, E.D.L.Rienks2, Z.H. Liu1, S. Thirupathaiah2, I. Avigo3, F.Roth4, H.S.Jeevan5,
P.Gegenwart5, M.Roslova1,6, I.Morozov1,6, S.Wurmehl1,7, U.Bovensiepen3, S.Borisenko1, M. Vojta8, B.Büchner1,7
1Leibniz Institute for Solid State and Materials Research Dresden, Helmholtzstr. 20, D-01069 Dresden, Germany
2Helmholtz-Zentrum Berlin, Albert-Einstein-Strasse 15, D-12489 Berlin, Germany
3Fakultät für Physik, Universität Duisburg-Essen, Lotharstr. 1, D-47075 Duisburg, Germany
4Center for Free-Electron Laser Science / DESY, Notkestrasse 85, D-22607 Hamburg, Germany
5Institut für Physik, Universität Augsburg, Universitätstr.1,D-86135 Augsburg, Germany
5 6Department of Chemistry, Lomonosov Moscow State University, 119991 Moscow, Russia
1 7Institut für Festkörperphysik, Technische Universität Dresden, D-01062 Dresden, Germany
0 8 Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
2
r (Dated: March 25, 2015)
a
M In this Supplement we provide additional information on the scattering rate of the quasiparti-
cles in the "122" compounds BaFe2(As1 xPx)2 and EuFe2(As1 xPx)2 and the "111" compounds
4 NaFe1 xCoxAs and NaFe1 xRhxAs near−the high-symmetry poi−nts in the Brillouin zone. Further-
− −
2 more we present additional calculations on the band dispersion and the effective mass in a model
wherethescatteringratesatlowenergyobeyaFermiliquidbehaviorandathigherenergyamarginal
] Fermi liquid behavior.
n
o
PACSnumbers: 74.25.Jb,74.72.Ek,79.60.âĹŠi
c
-
r
p Spectral function. ARPES measures the energy (ω) one has to solve equation (2) numerically to derive the
u
and momentum (k) dependent spectral function A(ω,k) dispersion.
s
. multipliedbyatransitionmatrixelement[1]. Theformer From the measured width W at constant energy one
t
a is given by can determine the life time broadening or the scattering
m
rateofthequasi-particlesΓ=Wv∗wherev∗istherenor-
- 1 Σ(ω,k) malized velocity. The imaginary part of the self-energy
d A(ω,k) = = ,(1)
n −π[ω−(cid:15)k−<Σ(ω,k)]2+[=Σ(ω,k)]2 is then given by the relation =Σ = Γvvkk∗ = Γmm∗0, where
o vk is the bare velocity, m0 is the bare band mass, and
where (cid:15) is the bare particle energy, Σ(ω,k) is the
c k < m∗ istherenormalizedmass[1]. Inthepresentstudym0
[ real part and Σ(ω,k) is the imaginary part of the self-
= wastakenfromDFTcalculationsofthe"undoped"com-
energy. For small Σ(ω,k) the maxima of the spectral
2 = pounds[3, 4] while m∗ values were determined from a fit
v function(thedispersion)aredeterminedbytheequation of the measured dispersion.
5
Additional data of scattering rates. In Fig.1(a) we
3 ω (cid:15) Σ(ω,k)=0. (2)
− k−< show an energy distribution map of of the inner hole
1
2 In the case when Σ(ω,k) = λω as in a Fermi liq- pocket near Γ of undoped EuFe2As2. In Fig.2(b) we
0 < − showamomentumdistributioncurve(MDC)takenfrom
uid close to the chemical potential and when Σ(ω,k)
1. is independent of the momentum, the dispersio<n follows the data in (a) at a binding energy EB = 100 meV to-
0 gether with a fit to two Lorentzians. From the width
ω = (cid:15) /(1+λ), i.e., with increasing coupling constant
5 k of the Lorentzians and from the bare particle velocity
λ the maximum of the spectral function is shifted to
1 taken from density functional theory (DFT) calculations
: lower binding energy which means the effective mass is
v we obtain the scattering rates Γ as a function of binding
enhanced. In a marginal Fermi liquid, a concept origi-
i energy. Afitclearlyshowsthatalsointheundopedpnic-
X nally devised to phenomenologically describe the normal
tides the scattering rate of the quasi-particles is linear in
r state of the cuprate superconductors, one assumes that
a energy over a large energy range.
the scattering rates at low energy are much higher com-
In Fig.2 we show analogous data for the electron
pared to a normal Fermi liquid due to strong correlation
pocketatA=(π,0,π)forNaFe Co Asx=0.057. Also
effects and/or nesting. The self-energy is given by[2] a c 1−x x
fortheelectronpocketinthisoverdopedpnictidealiner-
1 ω π in-energy scattering rate is observed in a wide energy
c
Σ= λMFωln i λMFu, (3) range. We remark that the evaluation of the scattering
2 u − 2
ratesoftheelectronpocketsisonlyfeasibleforoverdoped
where u=max(ω ,kT), kT is the thermal energy, λ compounds, since for underdoped systems the bottom of
MF
| |
is a coupling constant, and ω is an ultraviolet cutoff the electron pockets is very close to the Fermi level.
c
energy. In this case Σ is no longer linear in energy and In Fig.3 we show analogous data for the inner hole
<
2
(a)
(a)
(b)
(b)
(c)
(c)
Figure 1. (Color online) ARPES data of EuFe2As2 recorded
at a temperature T = 200 K with vertically polarized pho-
tonshavinganenergyhν=96eV.(a)Energydistributionmap Figure 3. (Color online) ARPES data of overdoped
alongcutI(seeFig.1ofthemainpaper)nearΓ. Theredlines NaFe1 xCoxAs x =0.057 recorded at T =1K with horizon-
−
tally polarized photons having an energy of hν=41 eV. (a)
correspond to dispersions derived from fits. (b) Momentum
Energy distribution map along cut I (see Fig.1 of the main
distributioncurvetakenfromthedatain(a)atabindingen-
paper) near Γ. The red lines depict dispersions derived from
ergy of 100 meV together with a fit to two Lorentzians. (c)
Scattering rate Γ at point 1 (see Fig.1 of the main paper) as fits. (b) Momentum distribution curve taken from the data
in(a)atabindingenergyof60meV.(c)ScatteringrateΓat
afunctionofenergy(green)fortheinnerholepockettogether
point1(seeFig.1ofthemainpaper)asafunction ofenergy
with a linear fit (black).
(green) for the inner hole pocket together with a linear fit
(black).
(a)
(b)
(a) (b)
(c)
Figure 4. (Color online) kz-dependence of the slopes β
of the scattering rates of quasi-particles in ferropnictides
for cut I (see Fig.1 of the main paper) and kz = 0. (a)
BaFe2(As1 xPx)2 x = 0.27, inner hole pocket (green) and
Figure 2. (Color online) ARPES data of overdoped middle hol−e pocket (blue) (b) NaFe1 xRhxAs x=0.027, mid-
−
NaFe1 xCoxAs x=0.057 recorded at T =1 K with vertically dleholepocket(blueandouterholepocket(red). Thedashed
polariz−edphotonshavinganenergyofhν=58eV.(a)Energy lines are guides to the eye.
distribution map along cut III (see Fig.1 of the main paper)
nearA(π,0,π)whereaandcarelatticeconstants. Thered
a c
linesdepictdispersionsderivedfromfits. (b)Momentumdis- β =0.62.
tributioncurvetakenfromthedatain(a)atabindingenergy
In Fig.4 we show the k dependence of the scattering
of20meVtogetherwithafit. (c)ScatteringratesΓatpoint z
5 (see Fig.1 of the main paper) and kz = πc as a function rparteessenotfdqautaasi-opfaorptitcilmesalilnyfderorpoepdniBctaiFdees(.AIsn FigP.4)(a)xw=e
of energy for the hole pocket (yellow-green) together with a 2 1 x x 2
linear fit (black). 0.27oftheinnerandthemiddleholepocket.−InFig.4(b)
wedepictanalogousdataofthemiddleandtheouterhole
pockets in NaFe Rh As x=0.027. Although there are
1 x x
−
pocket of overdoped NaFe Co As x =0.057. Using severalreportsclaimingachangeoftheorbitalcharacter
1 x x
horizontallypolarizedphoton−sthematrixelementislarge oftheholepocketsasafunctionofkz [5–7],inbothcases,
only for the inner hole pocket. Also in this case a linear- within error bars, there is no change of the scattering
in-energyscatteringrateisderived,asshowninFig.3(c). rates as a function of kz.
Fromthefitweobtainfortheslopeofthescatteringrate In Fig.5(a) we present a compilation of the slopes
3
Table I. Slopes of the scattering rates β of the investigated
ferropnictides BaFe2(As1 xPx)2 (BFAP), EuFe2(As1 xPx)2
(EFAP), NaFe1 xCoxAs− (NFCA), and NaFe1 x−RhxAs
(a) (b) − −
(NFRA)nearoptimal"doping"togetherwithcalculatedcou-
pling constants λMF .
Point BFAP EFAP NFCA NFRA
β λ β λ β λ β λ
MF MF MF MF
1 0.75 1.5 0.81 1.6 1.0 6.3 0.58 2.7
2 0.22 0.96 0 0 0.36 2.3
≈ ≈
7 0.69 5.1 0.46 3.4
3 0.92 1.83 0.80 8.1
4 0.22 0.73 0.11 0.70
Figure 5. (Color online) (a) Compilation of β values of the 8 0.14 0.55 0.51 3.7
holepocketsofNaFe1 xCoxAsasafunctionofCoconcentra- 5 0.68 0.44 0.57 0.84 0.61
−
tion. (a)AtthreehighsymmetrypointsalongcutI(seeFig.1 6 0.87
ofthemainpaper)nearZ =(0,0,π). (b)Atthreesymmetry
c
points along cut II along the diagonal of the Brillouin zone
(seeFig.1ofthemainpaper). Thecolorcodecorrespondsto
that used for the high-symmetry points in Fig.1 in the main
paper.
of the scattering rates β of the hole pockets near Z =
(0,0,π) of NaFe Co As as a function of the Co con-
c 1−x x
centration. As in the case of k =0 (see Fig. 3(d) of the
z
mainpaper)withinerrorbarsweseenovariationinx. In
Fig.5(b) we show an analogous compilation of β related
toacutIIalongthediagonaloftheBrillouinzoneforthe
hole pockets of NaFe Co As as a function of the Co
1 x x
−
concentration. Again, within error bars no dependence
of x is realized. The fact that the average value of β at
point 1 is larger than that at point 3 possibly indicates
an anisotropic scattering rate for the inner hole pocket.
Figure 6. (Color online) Calculation of the renormalized
Thisanisotropywasalsodiscussedintermsofanisotropic
dispersion (red line) using the bare particle dispersion from
nestingconditionsandanisotropicorbitalcharacter[8]as
DFT calculations (black line) and the real part of the self-
depicted in Fig.1 in the main paper.
energy(greenline)describedinthetext. Inadditionweshow
In Table I we compile the β values at the high- the effective mass m∗ (blue line) and the m∗ = 1 line (thin
m0 m0
symmetry points at kz = 0 for the investigated four fer- dashed blue line).
ropnictide systems near optimal "doping" together with
values for λ calculated from the β values using the
MF
effective bare masses m0 from DFT calculations[3, 4]. below ω = 5 meV (the lower energy border in our
0
The coupling constants for the 111 systems are consid- ARPES results) the linear ARPES Σ by a quadratic
erably higher than those for the 122 compounds. This =
energy dependence (Fermi liquid behavior) . The result
result is at present unclear and more systematic studies
for the case where the top of the bare particle band
are necessary to clarify this point.
touches the Fermi level is presented in Fig.6. The result
From this compilation one realizes large differences of is rather independent on the strength of the coupling
the scattering rates between the three hole pockets in constantoftheFermiliquidpart. Comparingthepresent
BaFe2(As1 xPx)2andareductionofthedifferenceswhen calculation with that presented in Fig.4(c) of the main
going to N−aFe1 xCoxAs and finally to NaFe1 xRhxAs . paper there is still a weakly dispersing renormalized
− −
This evolution can be explained by the strength of the band at the Fermi level but the effective mass at the Γ
scattering potential of the substituents, being weak for point is now reduced to m∗/m0 7. The strong mass
≈
the isovalent replacement of As by P, which is outside of enhancement now appears at ω since at this energy we
0
the Fe layers, being stronger when Fe is replaced by Co, have a discontinuity in Σ. Analogous calculations for
=
which occurs inside the Fe layer, and being strongest for the hole pocket shifted by 50 meV above the Fermi level
replacement of Fe by the heavy element Rh. (not shown) yield that when the top of the bare particle
Calculations for a Fermi-liquid/non-Fermi-liquid hole pocket is not at the Fermi level, the renormalized
model. Herewepresentcalculationsinwhichweextend dispersion is very close to that presented in Fig.4(d)
4
of the main paper and the mass renormalization at the
chemical potential is reduced to 4
≈
These calculations show that in both cases, the lin-
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ear or the quadratic extension at low energies of the Phys. 75, 473 (2003).
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=
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[6] S.Thirupathaiahetal., Phys.Rev.B81,104512(2010).
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