Influence of Lifshitz transitions and correlation effects on the scattering rates of the charge carriers in iron-based superconductors

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by  J. Fink

epl draft
Influence of Lifshitz transitions and correlation effects on the scat-
tering rates of the charge carriers in iron-based superconductors.
J. Fink1
1 Leibniz Institute for Solid State and Materials Research Dresden, Helmholtzstr. 20, D-01069 Dresden, Germany
6
1
0
2

n PACS 74.70.Xa–pnictides
a PACS 74.20.Pq–electronic structure calculations
J PACS 74.20.Pq–Strongly correlated electron systems

5 PACS 74.40.Kb–Quantum critical phenomena
2
Abstract – Minimum model calculations on the co-action of hole vanishing Lifshitz transitions

] and correlation effects in ferropnictides are presented. The calculations predict non-Fermi-liquid
l
e behaviour and huge mass enhancements of the charge carriers at the Fermi level. The findings
- are compared with recent ARPES experiments and with measurements of transport and ther-
r
t mal properties of ferropnictides. The results from the calculation can be also applied to other
s
. unconventional superconductors and question the traditional view of quantum critical points.
t
a
m
-
d
n
o Introduction . – Badmetalsandunconventionalsu- Besides transport and thermal properties, angle-
c perconductivity are both one of the most active research resolvedphotoemissionspectroscopy(ARPES) [9]isaver-
[
fields in solid state physics. Extensive research efforts satile method to study the electronic structure of solids.

1 have revealed that these phenomena appear near partic- Intheunconventionalsuperconductingferropnictides [10]
v ular points, termed quantum critical points (QCP), in near optimal doping, ARPES has detected changes of the
6 the phase diagram temperature vs. a control parameter. topologyofFermisurfaces(Lifshitztransitions [11]ofthe
1
These control parameters can be pressure, chemical pres- pocket vanishing type) for the electron doped systems
5
sure, doping, or magnetic field. Near the QCP, the nor- [12–19] in which the hole pockets disappear and in hole
6
0 mal state properties are characterized by non-Fermi liq- doped systems [20] in which the electron pockets disap-
. uid behaviour, dubbed bad metal. At low temperatures pear. In addition, ARPES has detected non-Fermi-liquid
1
0 very often unconventional superconductivity is observed. (linear in energy) scattering rates Γ of the charge carriers
6 SincetheQCPappearsattheendofanantiferromagnetic independent of the control parameter [21–23]. ARPES
1 range,awidespreadviewofthesephenomenaisrelatedto is especially valuable since it can detect the momentum
:
v a scenario in which at low temperatures the charge car- dependence and the orbital dependence of the scattering
i riers are strongly coupled to antiferromagnetic quantum rates in multi band systems. The scattering rates and the
X
fluctuations [1–3]. These fluctuations could be the glue related imaginary part of the self-energy (cid:61)Σ [24] charac-
r
a mediating unconventional superconductivity. They could terizethemany-bodypropertiesofquasiparticlesinsolids.
alsoaccountforthestrangenormalstatenon-Fermi-liquid In this article we first describe the appearance of a
behaviour as visible in transport and thermal properties linear-in-energy dependence of Γ in systems close to a
of ferropnictides. The former have manifested themselves pocket vanishing Lifshitz transition. Next we discuss the
e.g. by a linear temperature dependence of the resistivity linear in energy scattering rates in correlated materials.
which was observed in various highly correlated systems Then we describe the fact that the non-Fermi-liquid be-
such as heavy fermion systems [4], cuprates [5], and iron haviour near a Lifshitz transition is enhanced by correla-
based superconductors [6,7]. In this context we mention tion effects. On the basis of this understanding and the
the phenomenological model of a marginal Fermi-liquid, experimentalARPESresultsweconcludethataco-action
sometimes also called singular liquids, which was devel- of the non-Fermi liquid scattering rates with a Lifshitz
oped to understand Raman experiments and transport transition of the pocket vanishing type, i.e., a crossing of
properties of doped cuprates [8]. a flat band through the Fermi level, leads to a huge mass
p-1
J. Fink
Inthefollowing,wedonotevaluatethesusceptibilitiesby
(a) (d) (g)
Et taking into account the full band structure. To simplify
EF k EF k EF matters, a one band model is used (see fig 1(a)). For the
ferropnictidesatwobandmodel((seefig1(d-f))wouldbe
E E E a better approximation. The sum over the momentum is
(b) (e) (h) neglectedandξ(E(cid:48),(cid:15)k,(cid:15)k+Q,T)isapproximatedbyacon-
Et stant density of states D(0) at the Fermi level multiplied
EF k EF k EF by the Fermi function NF(E,T) = 1/(1+exp(E/kBT))
(k is the Boltzman constant):
B
E E E
(j) ξ ∝D(0)N (E,T). (2)
(c) (f) Et F
EF k EF k EF χ(E,(cid:15)k,(cid:15)k−Q,T)isapproximatedbytheconvolutionof
the occupied part of the band with the unoccupied part
E E E oftheband, theFermiedgesofwhichbeingbroadenedby
N (E,T) and 1−N (E,T), respectively:
F F
Figure 1: (Color online) Electron-electron Auger-like scatter- (cid:90)
ing process. Top row: initial state with a photo hole. Middle χ(E,T)∝ dE(cid:48)
(3)
row: Auger-like transitions. Lower row: final state with a re-
laxed photo hole and an electron-hole excitation. (a-c) For a ×D(E(cid:48))NF(E(cid:48),T)D(E−E(cid:48))(1−NF(E−E(cid:48),T)
single conduction band. (d-f) Same as (a-c) but for a band
For the occupied part of the band D(E(cid:48)) = D(0) is used
structurewithholeandelectronpocketstoillustrateelectron-
while for the unoccupied part D(E(cid:48)) = D(0) for E(cid:48) > E
electron scattering in ferropnictides. (g-j) Same as (a-c) but t
inadifferentpresentationandforapartiallyfilledconduction and D(E(cid:48)) = 0 for E(cid:48) < Et is used (Et is negative, see
band close to a Lifshitz transition. In this case the top of the fig.1(g-i)). BymovingEttozerowecansimulateapocket
conduction band at E is close to the Fermi level. vanishing type Lifshitz transition.
t
First we discuss the scattering rate for zero tempera-
ture and a wide conduction band (E (cid:28) 0). In this case
enhancement. Thisscenarioprovidesanalternativemodel t
ξ is constant up to the Fermi energy and the convolu-
tounderstandthestrangenormalstatepropertiesandpos-
tion of the two Fermi edges leads to χ which is linear in
sibly also unconventional superconductivity in correlated
energy. Performing the integration leads to quadratic in-
matter.
crease of the scattering rate as a function of energy. This
Scattering rates near a Lifshitz transition. – In is observed in ARPES of normal metals, such as Mo [26]
this section the scattering rate of the charge carriers Γ where the scattering rate for electron-electron interaction
ee
due to electro- electron interaction is described in a mini- Γ is proportional to E2. This also leads to a quadratic
ee
mummodel. Inafirstapproximationthescatteringrateis temperature dependence of the resistivity in normal met-
determined by an Auger process in the conduction band, als at low temperatures where phonons can no more be
i.e., a relaxation of an excited photo hole to lower ener- excited.
gies relative to the Fermi level. The received energy is For the case near a Lifshitz transition (E → 0) the
t
transfered to an electron-hole excitation (see fig 1(a-c)) unoccupied part is a δ function at the Fermi level and
[24]. Following this description the scattering rate can be therefore χ is constant in energy. Performing the integra-
calculated by the relation [24,25] tion yields a non-Fermi-liquid linear-in-energy scattering
rate.
(cid:90)E In fig. 2(a) we show the exponent of the energy de-
(cid:88)
Γee(E,T)∝U2 dE(cid:48) (1) pendence of the scattering rate as a function of Et and
Q 0 the binding energy. Near the Lifshitz transition (Et = 0)
×ξ(E(cid:48),(cid:15) ,(cid:15) ,T)χ(E(cid:48),(cid:15) ,(cid:15) ,T). a non-Fermi liquid behaviour is obtained for all ener-
k k+Q k k−Q
gies. With increasing distance to the Lifshitz transi-
Q is the momentum transfer for the excitations, T is the tion (more negative E values) the scattering rate trans-
t
temperature, E the energy relative to the Fermi energy, forms to a Fermi liquid behaviour with a quadratic en-
and (cid:15) is the bare particle dispersion. For the Coulomb ergy dependence. A similar calculation was performed
k
interaction between two holes, a local approximation is for the temperature dependence instead of the energy de-
used, i.e., it is described by the on-site Coulomb poten- pendence (see fig. 2(b)). Near the Lifshitz transition a
tial U. ξ(E(cid:48),(cid:15) ,(cid:15) ,T) describes the probability for the linear-in-temperature non-Fermi-liquid behaviour is ob-
k k+Q
hole relaxation in the occupied part of the band while served. WithincreasingdistancetotheLifshitztransition
χ(E(cid:48),(cid:15) ,(cid:15) ,T)describesthesusceptibilityforelectron- aquadratictemperatureFermi-liquidbehaviourisprevail-
k k−Q
holeexcitationsbetweentheoccupiedandtheunoccupied ing at low temperatures. We emphasize that in these cal-
part of the band (e.g. the Lindhardt susceptibility [24]). culationsforanuncorrelatedsystem,non-Fermi-liquidbe-
p-2
Lifshitz transitions and correlation effects
N(k) (a)
(a)
(a)
k k
N(k) (b) F
- - - -
tt k k
F
(b)
Figure 3: (Color online) Momentum density N(k) together
with an Auger-like scattering process in the conduction band.
(a) For a weakly correlated Fermi liquid. (b) For a highly
correlated metal.
quasiparticles, is small and a large fraction (proportional
to 1−Z) is due to incoherent particles. When we add a
coherentquasiparticlewithenergyE tothesystem(e.g. a
photohole)thephasespaceforthedecayofthequasipar-
- - - -
ticle is again proportional to E. However, in this case the
t relaxation energy can be transferred also to the incoher-
ent charge carriers. These can be excited not only within
Figure 2: (Color online) (a) False colour presentation of the a narrow energy range E below the Fermi level but the
exponent of the energy dependence of the scattering rate as a electron-holeexcitationsoftheincoherentchargecarriers,
functionofthetopoftheconductionband(Et)andtheenergy. which are no more restricted by the Pauli principle, can
(b) The exponent of the energy dependence of the scattering now occur in the entire range of the band width W. Thus
rate as a function of the top of the conduction band (E ) and
t inthiscasethetotalphasespacefortherelaxationprocess
the thermal energy k T.
B isproportionaltoEW whichforE (cid:28)W isproportionalto
E. ThisleadstothemarginalFermiliquidmodelinwhich
haviouroccursonlyveryclosetotheLifshitztransitionfor one assumes that the weight of the coherent particles is
E >|E |. close to zero, i.e., Z (cid:28) 1. Thus different from a normal
t
Fermiliquid,theinteractionofthecoherentquasiparticles
Scattering rates in correlated systems. – In this with the incoherent particles yield strong low-energy re-
section we try to understand the difference of the energy laxationprocesses. Inthisviewitisdifficulttorationalize
dependence of the scattering rate between a normal and a Fermi liquid behaviour of a strongly correlated single
a highly correlated metal. In fig. 3(a) the momentum band metal since the total relaxation process is a sum of
density of a weakly correlated Fermi liquid together with two components, one which proportional to ZE2 and a
the relaxation of an added photo hole due to an electron- second one which is proportional (1−Z)E.
hole excitation, already discussed in the previous section,
is presented. The phase space for the hole relaxation is Scattering rates of a correlated system near a
proportionaltotheenergyE. BecauseofthePauliprinci- Lifshitz transition. – In this Section the co-action of
ple electrons can only excited with an energy E below the a Lifshitz transition and correlation effects is discussed
Fermi level. Thus the total phase space for the scattering usingtheARPESresultsoftheelectronicstructureofthe
process is proportional to E2 [24], already demonstrated holepocketinferropnictides [23]. Intheseexperimentson
in the previous section. electrondopedcompounds, alinear-in-energydependence
In the case of strong correlation effects the hight of the ofΓwasobservedinalargerangeofthecontrolparameter.
stepofthemomentumdensityatk whichisproportional It is assume that the experimental results for Γ can be
F
to the renormalization factor Z is reduced and large frac- extrapolatedtolowenergies(wellbelowatypicalARPES
tions are shifted from regions below k to regions above energy resolution of about 5 meV). From Γ the imaginary
F
k (see fig.3(b)) [24]. In the case of very strong corre- partoftheself-energy(cid:61)ΣiscalculatedbytherelationΓ=
F
lation effects, Z which determines the weight of coherent Z(cid:61)Σ [24]. Athighenergiesweassumethataboveacutoff
p-3
J. Fink
in fig.4 by red lines. In fig.4 we also present the effec-
tive masses m∗/m (blue lines), derived from the second
0
10 derivativeoftherenormalizeddispersion. Infig.4(b),(d),
(a) (b)
and (f) m∗/m is presented as a function of the momen-
5 m0 0
m*/ tum k. Similar data have been presented already in Ref.
0 [23]. Here in fig.4 (a), (c), and (e) m∗/m0 values as a
function of energy are added.
Inthecasewherethetopofthebareparticleholepocket
just touches the Fermi level, i.e., when at the Fermi level
thebareparticledispersionisflat(fig.4(d)),whichoccurs
attheLifshitztransition,(cid:60)Σisveryclosetothebarepar-
10
(c) (d) ticle band and therefore at the Fermi level the renormal-
5 m0 ized band is very flat. This corresponds to a very high
m*/ effective mass m∗/m ≈ 8 at k = 0 (fig.4(d)) and at the
0
0
Fermi level (fig.4(c)). Moving away from k =0 or E =0
the effective mass is strongly reduced to a value between
1 and 2.
Whenweshiftthetopoftheholepocket50meVabove
the Fermi level (fig.4(a) and (b)), which in real systems
10 correspondstoachangingofthecontrolparameterbydop-
(e) (f)
ing,bychemicalpressure,orbypressure,thedispersionof
5 m0
MFL E = 0 m*/ therenormalizedbandisalsoreduced. However,theslope
0 0 of this band at the Fermi level is strongly reduced when
compared with the case where the bare band touches the
Fermi level. This reduction of the Fermi velocity is con-
nected with the effective masses at E which is reduced
F
to a value of about 2.
Finally,whenweshiftthetopoftheholepocket50meV
below the Fermi level (fig.4(e) and (f)), the band is still
renormalized but the effective mass at k =0 is reduced to
a value of about 4.
Figure 4: (Color online) Calculation of the renormalized dis-
persion(redline)andtheeffectivemassm∗/m (bluelines)as Summarizingtheresultsofthecalculationspresentedin
0
afunctionofenergy(leftpanels)andasafunctionofmomen- fig.4, at the Lifshitz transition and only at this transition
tum (right panels) using a bare particle parabolic dispersion one obtains a correlation induced particularly enhanced
(black line) and a (cid:60)Σ derived from ARPES experiments on flattening of the renormalized band which is related to an
ferropnictides. (a)and(b)ForEt =−50meV,(c)and(d)For almost diverging effective mass at the Fermi level and at
Et =0, (e) and (f) For Et =50 meV. the momentum where the Lifshitz transition occurs. The
correlation effects also lead to a pinning of the top of the
hole pocket at the Fermi level.
energy E , corresponding to the band width, (cid:61)Σ remains
c
In fig.5 we present similar calculations but with a ther-
constant. Thus the complex self-energy Σ corresponds to
mal energy of 0.03 eV corresponding to a temperature of
that proposed in the marginal Fermi liquid model [8]:
348K.InthecaseofaLifhitztransition(fig.5(c)and(d))
1 E π theeffectivemassattheFermilevelisreducedformabout
Σ (E)= λ Eln c +i λ x. (4)
MF 2 MF x 2 MF 8 to 4. However away from the Lifshitz transition, when
we shift the top of the hole pocket to -50 meV (fig.5(a)
x=max(|E|,k T) and k T is the thermal energy. λ
B B MF and (b)) the effective mass at the Fermi level is enhanced
is a dimensionless coupling constant. In the calculations
from about 2 to 4 due to a broadening of the maximum
presented in fig.4 zero thermal energy and typical values
above the Fermi level.
derivedfromARPESexperimentsontheinnerholepocket
Finally a calculation of the effective mass as a function
of electron doped ferropnictides [23] are used: λ =1.4
MF of the thermal energy k T and the shift of the top of
B
and E = 1.5 eV. The parabolic bare particle dispersion
c the hole pocket is presented in fig.6. An enhancement of
(black lines in fig.4) was taken from a fit to bands of the
the effective mass near the Lifshitz transition and at low
innerholepocketofferropnictidesderivedfromDFTband
temperatures is realized.
structure calculations. Using the above given parameters,
(cid:60)Σ was calculated with the help of eq. (4) (green lines in Discussion . – The present calculations offer a new
fig.4). Thedifferencebetweenthebareparticledispersion explanation of the non-Fermi-liquid behaviour of corre-
and (cid:60)Σ yields the renormalised dispersions [24] depicted lated systems near a QCP which is different from the tra-
p-4
Lifshitz transitions and correlation effects
10
(a) (b)
5 m0
m*/
0
10 t
(c) (d)
5 m0
m*/
Figure 6: (Color online) Calculation of the effective mass as
0
afunctionofthermalenergyk T andashiftofthetopofthe
B
hole band (E ).
t
by London penetration depth [29,30] and de Haas-van
10 Alphen effect [31] measurements have detected a huge
(e) (f)
5 m0 massenhancementattheQCPwhichwasinterpretedbya
m*/ coupling to antiferromagnetic quantum fluctuations. The
0 present calculations provide an alternative explanation of
the mass enhancement in terms of a co-action of corre-
lation effects and a Lifshitz transition (see fig. 6). Close
to the QCP the resistivity data on BaFe As P [7,32]
2 2−x x
show a linear temperature dependence over a large range,
but moving away from the QCP a Fermi liquid behaviour
is detected at low temperatures. It is difficult to inter-
pret these results on the basis of the present calculations
since the conductivity depends in a non-trivial way not
Figure5: (Coloronline)Thesamecalculationasinfig.4,but
only on the scattering rate but also on the effective mass
with the addition of a finite thermal energy of 0.03 eV.
and the number of charge carriers. The present calcula-
tions reveal that all three parameter are changing near a
Lifshitz transition. According to previous work [33] the
ditional view related to a coupling of the charge carriers
conductivity should be not renormalized by many-body
to quantum spin fluctuations. In the present scenario the
interactions. PossiblytheARPESresultscouldberelated
strange normal state properties follow from a co-action of
to “hot” sections of the Fermi surface mediating antifer-
a Lifshitz transition and strong local electron-electron in-
romagnetismandsuperconductivitywhile“cold”sections,
teraction which lead to a large number if incoherent and
not detected in the ARPES experiments, could be related
a small number of coherent charge carriers. At the end
tothe normalstate conductivity. Possiblyalso theFermi-
of the 3d-row, correlation effects are caused by the Mott-
liquidbehaviouroftheferropnictidesdetectedintheopti-
Hubbard on-site Coulomb interaction while Hund’s-rule
cal conductivity [34] could be related to the “cold” spots
exchange interaction is more important in the middle of
. In this context one should mention that the conduc-
the row. That local on-site correlation effects are dom-
tivity of electron doped ferropnictides is predominately
inant in the scattering rates related to electron-electron
determined by electronic states of the electron pockets
interactionisalsosupportedbytheoreticalcalculationsin
[35], which may be less correlated because there is no en-
the framework of DFT combined with dynamical mean-
hancement by the proximity to a Lifshitz transition. The
field theory (DMFT). Interestingly, non-Fermi-liquid-like
thermoelectric power is connected to the derivative of the
self-energies have been reported [27,28] which could be
densityofelectronicstatesattheFermilevel. Thusthedi-
fittedbysub-linearpowerlawsoverarangeofelevateden-
vergent thermopower detected in Ba(Fe Co ) As near
ergies. Theincoherent-metalbehaviorhasbeenattributed 1−x x 2 2
optimal doping [36] can be readily explained by the sce-
to an interplay of Hubbard and Hund’s-rule couplings in
nario described in this contribution: a coaction of a Lif-
a multi-band system.
shitztransitionandcorrelationeffects. Thepresentcalcu-
In the following we compare the present calculations lations of the mass enhancement at the Lifshitz transition
with normal state transport and thermal properties of canprobablyalsoexplainthethermalexpansiondataand
ferropnictides. Several investigations on BaFe As P theassociatedGru¨neisenparameterofBa(Fe Co ) As
2 2−x x 1−x x 2 2
p-5
J. Fink
[37]. Furthermore,alsothethermaldataonthesupercon- [9] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod.
ducting heat capacity jump in BaFe As P [7] can be Phys. 75, 473 (2003).
2 2−x x
explained within this model. We also mention that the [10] D. C. Johnston, Adv. Phys. 59, 803 (2010).
enhancedrelaxationrate1/T T neartheQCPdetectedin [11] I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960).
1
[12] W. Malaeb et al., J. Phys. Soc. Jpn. 78, 123706 (2009).
NMRexperimentsonBa(Fe Co ) As canberelatedto
1−x x 2 2 [13] C. He et al., Phys. Rev. Lett. 105, 117002 (2010).
aLifshitztransition [38]. Moreover,thedatafortheeffec-
[14] S.Thirupathaiahetal., Phys.Rev.B81,104512(2010).
tive mass as a function of the control parameter, i.e., the
[15] S.Thirupathaiahetal., Phys.Rev.B84,014531(2011).
band shift, resemble very much data on the temperature
[16] Z.-H. Liu et al., Phys. Rev. B 84, 064519 (2011).
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[24] G. D. Mahan, Many-Particle Physics, Kluwer Aca-
[40].
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Phys. Rev. Lett. 83, 2085 (1999).
aflatbandrelativetotheFermilevel. Fromthesecalcula-
[27] P. Werner et al., Nat Phys 8, 331 (2012).
tionswederiveamodelinwhichthestrangenormalstate
[28] Z. P. Yin, K. Haule, and G. Kotliar, Phys. Rev. B 86,
transport and thermal properties near optimal substitu-
195141 (2012).
tion/dopingcanbeexplainedbyaco-actionofamarginal
[29] K. Hashimoto et al., Science 336, 1554 (2012).
Fermiliquidself-energywithavanHovesingularityatthe
[30] P. Walmsley et al., Phys. Rev. Lett. 110, 257002 (2013).
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theMigdal’stheoremisviolatedintheunconventionalsu- [34] A. Tytarenko, Y. Huang, A. de Visser, S. Johnston, and
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[35] F. Rullier-Albenque, Comptes Rendus Physique 17, 164
the BCS-BE crossover. The results can be generalised to
(2016).
other unconventional superconductors and possibly are a
[36] S. Arsenijevic et al., Phys. Rev. B 87, 224508 (2013).
recipe for future search of new high-T superconductors.
c [37] C. Meingast et al., Phys. Rev. Lett. 108, 177004 (2012).
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∗∗∗
[39] A. B. Migdal, Soviet Physics JETP−USSR 34, 996
(1958).
This work was financially supported by the German [40] S. N. Bose, Z. phys 26, 178 (1924).
Research Foundation the DFG through the priority pro-
gram SPP1458. Valuable discussions with S. Borisenko,
B. Bu¨chner, P. Fulde, D. van der Marel, and C. Monney
are acknowledged.
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