Table Of ContentStrong Pressure Dependent Electron-Phonon Coupling in FeSe
Subhasish Mandal,1 R. E. Cohen,2,3 and K. Haule4
1Geophysical Laboratory, Carnegie Institution of Washington, Washington D.C. 20015, USA
2Geophysical Laboratory, Carnegie Institution of Washington, Washington D.C. 20015, USA
3Department of Earth Sciences, University College London,
Gower Street, WC1E 6BT, London, United Kingdom
4Department of Physics, Rutgers University, Piscataway, New Jersey 08854, USA
WehavecomputedthecorrelatedelectronicstructureofFeSeanditsdependenceontheA1g modeversus
compression. Usingtheself-consistentdensityfunctionaltheory-dynamicalmeanfieldtheory(DFT-DMFT)
with continuous time quantum Monte Carlo (CTQMC), we find that there is greatly enhanced coupling
4
1 between some correlated electron states and the A1g lattice distortion. Superconductivity in FeSe shows a
0 verystrongsensitivitytopressure,withanincreaseinTc ofalmostafactorof5withinafewGPa,followed
2 by a drop, despite monotonic pressure dependence of almost all electronic properties. We find that the
maximum A1g deformation potential behaves similar to the experimental Tc. In contrast, the maximum
n
deformationpotentialinDFTforthismodeincreasesmonotonicallywithincreasingpressure.
a
J
PACSnumbers: 74.70.Xa,74.25.Jb,75.10.Lp
5
1
Sofarthereisnopredictivetheoryforsuperconductiv- to cuprates[23–25], strong coupling phonons have also
]
ity in the cuprate and iron-superconductors, hence these been proposed to play a role in both sets of materials
n
o superconductorscanbeclassifiedasnon-conventionalsu- [3, 26–30]. The study of strong electron-phonon cou-
c perconductors, since there is a well-developed, predic- pling in correlated solids is in its infancy due to ex-
-
r tive theory for electron-phonon superconductors whose treme computational complexity. Only in the non-self-
p normal state is well-represented by conventional density consistent Hubbard I and LDA+U approximations has
u
functional theory (DFT) [1–3]. The unconventional su- it proved tractable so far [31]. The role of lattice vi-
s
. perconductors are very sensitive to applied pressure, so brations in the mechanism of SC in unconventional su-
t
a pressure provides a control to test theories and develop perconductors is still controversial. The observation of
m a better understanding [4–6]. Here we study supercon- strong electron-phonon and spin-phonon coupling, both
- ductivity under applied pressure in pure FeSe; unlike incuprates[32–35]andironsuperconductors[36–41]indi-
d
cuprates, superconductivity in FeSe arises without dop- catesthattheelectron-phononcoupling(EPC)mayplay
n
ing. FeSeisanidealsystemtostudytheelectronpairing animportantroleintheunconventionalsuperconductors,
o
c mechanism due to the simplicity of its crystal and elec- at least to explain the observed Fe-isotope effect[42], the
[ tronic structure. It shows a very strong enhancement of anomalous temperature dependence of the local Fe-As
T uponapplicationofmodestpressurewithdramaticin- displacement[43], gap anisotropy, and the correlation of
1 c
v creaseofTc from8Kto∼37K[4,7,8]andthendecreases Tc withtheFe-anionheight[44]. Theseobservationsalso
2 upon further application of pressure. Why does T in- suggestpolaronand/orbipolarondrivensuperconductiv-
c
8 crease with pressure and then decrease for rather small ity in this material [45–50].
7
latticecompression? ThisquestionwasaddressedinRef.
3 Here we examine the effects of pressure on the elec-
8, where it was found that applied pressure (P) inten-
. tronic structure and electron-phonon interaction (EPI)
1
sified antiferromagnetic spin fluctuations (SF). However,
0 in FeSe using DMFT in combination with the DFT as
this did not explain the decrease in T with further com-
4 c implemented in [51].
1 pression.
Electronic and magnetic properties of FeSe are very
:
v Thediscoveryoftheironsuperconductorsshowedthat sensitivetotheposition(z )oftheseleniumlayerswith
Se
i high T is not specific to the cuprates, and suggests
X c respect to the iron layers[15, 44, 52–56]. The magnetic
a wider field of potential high T materials [9]. Al-
c orderingisfoundtobestronglyaffectedbythechalcogen
r
a thoughDFTgivesmanypropertiesreasonablyaccurately height[52]. An accurate estimation of z is essential to
Se
forbothcuprates[10–12]andFe-superconductors[13–15],
studytheelectronicstructureofFeSe. Weoptimizedz
Se
there is also significant indication of the importance of
for P=0, 3.4, 7.2, and 11 GPa and notice a monotonic
correlations and fluctuating local moments beyond DFT
increasewithP.Thereportedexperimentalvaluesofz
Se
specially for the Fe-superconductors which are paramag-
at ambient pressure are consistently estimated to z =
Se
netic metals in room temperature, and Dynamical Mean
0.267[57]. Our DFT+DMFT computations give value of
Field Theory (DMFT) has proved to be a good approxi-
z =0.27atambientpressure(Fig. S2inSupplementary
Se
mation [16–22].
Information),whereasLDAandspin-polarizedGGAgive
While most studies suggest a spin fluctuation cou- 0.234[13] and 0.26 respectively. The inclusion of local
pling mechanism for SC in Fe-superconductors similarly spin fluctuations by DMFT is hence crucial to describe
2
(d)
0GPa
0.2
(a)
0GPa
V)
0.1
X M
y
(e0.0
e2
e1
g
er-‐0.1
n
E
-‐0.2
h1
Γ X
M
Γ Z
R
A
Γ Γ h2
X
0.2
(b)
3.4
GPa
h3
V)
0.1
y
(e0.0
(e)
3.4GPa
erg-‐0.1
X M
En e2
e1
-‐0.2
Γ X
M
Γ Z
R
A
Γ
0.2
(c)
11
GPa
V)
0.1
Γ h2
X
e
y
(0.0
h3
g
Ener-‐0.1
(f)
11GPa
0.5
Equilibrium (DFT)
-‐0.2
X M 0.4
A Distortion(DFT)
1g
Γ X
M
Γ Z
R
A
Γ e2
e1
0.3
Equilibrium (DMFT)
(g)
A Distortion(DMFT)
1g
0.2
)
0.1
δE~0.04 eV (DFT)
V
Γ X (e 0.0
h3
rgy-0.1
δE~0.12 eV(DMFT)
e
n
E-0.2
-0.3
FIG. 1. (Color online) Pressure evolution of DFT-DMFT
-0.4
spectralfunction(leftpanel,a-c)andtheFermisurfaceonthe
kz=0 plane (right panel, d-f). The Fermi surface is colored 0.5M
Γ
Z
R
in red, green, and blue according to its dominating orbital
character of xy, xz, and yz respectively. (g) Quasiparticle
weight as a function of pressure for the Fe-d orbitals. FIG. 2. (Color online). (a)Maximum and Fermi surface
average of deformation potentials (D) for the A distor-
1g
tion computed in DFT-DMFT and DFT as a function of
pressure indicates the presence of strong EPC in FeSe; in-
the structural properties in the disordered paramagnetic set shows deformation potential as a function of E at
F
state,andhassimilareffectonz asthepresenceoflong P=0. (b)Demonstration of huge local electron-lattice cou-
Se
ranged order in standard DFT. plingfortheA1g distortioninourDFT-DMFTcomputations
at P=0GPa for a selective part of the BZ; red and blue lines
We compute the DFT-DMFT spectral function
represent GGA bands. The common Fermi energy is consid-
(A(ω,k)) and the Fermi surface (FS) for the optimized
ered for the equilibrium position and denoted by the single
value of zSe at different pressures and summarized them horizontal line for both DFT and DFT-DMFT methods.
in the Fig. 1. A dramatic change in A(ω,k) is noticed
when the pressure is increased from 0 to 3.4 GPa. There
are three DFT-DMFT bands crossing the Fermi energy while the inner hole pocket h2 shrinks with increasing
(E )fromΓtoX-pointatP=0,whilethereareonlytwo pressure. Theinnerholepocketh1,whichshowsmostk
F z
bands crossing E at P=3.4 GPa. The strong 3D band dependence, vanishes above P=3.4 GPa.
F
whichcrossesE fromΓtoZ atP=0,doesnotcrossE It is important to know how the electron correlation
F F
above3.4GPa. Further, weconsidertheorbitalresolved changes with increasing pressure. In order to inves-
FS at k =0 plane as a function of pressure. From Fig. tigate the degree of correlation, we calculated Z =
z A
1(d-f) we notice that the electron pockets (e1 and e2) (1 − δΣ)−1 . In a Fermi-liquid it is the quasiparticle
δω ω=0
at the M point do not change much with increasing the weight, which is unity for non-interacting system, and is
pressurewhereastheholepockets(h1,h2,andh3)atthe much smaller than unity for strongly correlated system.
Γpointchangessignificantly. Thenumberofholepocket We have calculated Z for all the Fe-d orbitals and plot-
A
reducesfromthreetoonebyincreasingthepressure. The ted them as a function of P in Fig. 1(g). Though the
outer hole pocket h3 are mainly of xy character, while d ,d orbital becomes less correlated with the in-
z2 x2−y2
the inner hole pockets h1 and h2 consists of xz and yz crease of P, the t orbitals (d , d , and d ) remain
2g xz yz xy
character. With increasing pressure three hole pockets correlatedandinparticularthed orbital,whichcarries
xy
behave differently. The outer hole pocket h3 expands, more magnetic moment, becomes even more correlated
3
with increasing pressure. The predicted Z of 0.41 for remainsalmostunchangedafterP=3.4GPa;whereasthe
A
d orbital at P=0 GPa and at T=300 K differs from maximum D within DFT for this mode is insensitive to
xy
0.287 calculated at T = 116 K in Ref.[21] due to the increasing pressure except at P=11 GPa, where the ex-
use of slightly different lattice parameters and z . By perimental T is found to decrease. At P=11 GPa, the
Se c
comparing results with Ref. [21], we found that among sensitivebandcrossestheE fortheGGA,whichleadsto
F
other t orbitals, d shows the strongest temperature ahighDm forGGA.Forpressureabove3.4GPa, weno-
2g xy
dependence in Z at P=0GPa. tice that the maximum D in DMFT is from the electron
A
Calculated electron density of states, optical conduc- pocket centered at M-point. Dm also strongly depends
tivity both along ab-plane and c-axis shows a monotonic on the E . The inset of Fig. 2(a) shows the behavior of
F
behavior as a function of compression (details in Supple- maximum and average value of D as a function of E at
F
mentary). We now turn our discussion to the electron- P=0 calculated with DFT-DMFT. So the movement of
phonon coupling and polaron formation in FeSe. The E duetodefectorpressurecansignificantlychangethe
F
coupling between electronic states and atomic displace- FS topology and hence the D. The momentum resolved
ments (the electron-phonon matrix element) and hence spectral function A(ω,k) is shown in Fig. 2(b) for both
λisdirectlyrelatedtotheshiftintheenergyeigenvalues equilibriumpositionandA distortionbetweenthehigh
1g
at E . We calculate this shift by calculating δE (details symmetric points, where the most sensitive band crosses
F
in Supplementary Information ). We estimate the av- the E . The solid red and blue lines represent corre-
F
erage as well as the maximum value of the deformation sponding GGA bands for equilibrium position and A
1g
potential (D = δE) upon Se atom displacement (δQ) in distortion respectively. From the Fig. 2(b), we notice
δQ
z of a particular phonon mode (A distortion) for the that at P=0, the shift in energy (δE) over the atomic
Se 1g
entire FS. In Fig. 2(a) we have plotted the deforma- displacement of 0.0276 ˚A is ∼ 0.12 eV in DFT-DMFT
tion potential for both DFT and DFT-DMFT methods. and∼0.04eVinGGA,respectively. Soasreflectedfrom
The equation of states (pressure points on the axis) are Fig. 2,theD isaboutthreetimeshigherinDFT-DMFT
obtained from the experiment[58]. First we notice that forthisparticularregionoftheBZ.Ifwenoticecarefully
the average D increases in DFT-DMFT over that ob- in Fig. 2(b), the shift of the bands due to A1g distortion
tained in DFT for all pressure. At P=0 the average D2 is very non-uniform in DFT-DMFT; a strong deforma-
increases ∼1.5 times in DFT-DMFT; D is 0.84 eV/˚A in tion potential is noticed only in Γ to Z region while for
DFT-DMFTwhile0.69eV/˚AinGGA.Asimilarincrease the other part of the BZ, deformation potential is found
wasfoundbyBoerietal. [28]whenmagneticsoftnesswas to be small. This leads to a strong non-uniform EPC at
includedintheircalculations. AtP=3.4GPatheaverage P=0, which is reflected in Fig. 2(a) where maximum D
D2 increases ∼ 2.25 times in DFT-DMFT. However this isfoundtobeaboutthreetimeshigherthantheaverage.
isstillnotsufficienttoobtain37Kbutnotsmallenough We found this similar non-uniform EPC for P=1.4, 2.6,
to ignore as suggested earlier by Boeri et al.[28]. More and -2.0 GPa.
interesting is the pressure dependence of the maximum WeestimateλusingD(SupplementaryInformationfor
D. At ambient pressure we notice that the maximum details). Whiletheaverageλisstillsmall,themaximum
deformation potential (Dm) in DFT-DMFT is about 3 λ in DFT-DMFT is found to be 0.98 at P=0. At P=2.6
times higher than that obtained in GGA. One band is GPa, themaximumλreaches1.159. Wefoundthatonly
very sensitive to this deformation; it crosses the E for certain electronic states have very strong λ while the av-
F
P=0, 1.4, 2.6 and -2.0 GPa ( indicated by h1 in the Fig. erage λ is not strong enough to explain 37K. So a con-
1d for P=0). This sensitive pocket gives a very high ventionalelectron-phononmechanismseemsunlikely. On
value of Dm for this pressure range within DFT-DMFT the other hand, this also indicates that local EPC can
method, where the experimental T is also observed to be important and one can use polaron model, where a
c
be high. Dm obtained from the GGA is found to be single electron can strongly couple with the lattice and
very small at P=0. We found that the D obtained from form polarons. Formation of polaron has been exper-
DFT-DMFTisdifferentfordifferentpartsoftheFS(de- imentally found in both Fe-superconductors[41, 60, 61]
tailsintheSupplementaryInformation). Forexampleat and cuprates[47]. The anomalous temperature depen-
P=0, h1 gives Dm of ∼ 4.4 eV/ ˚A, while e1 gives only dence of the local Fe-As displacement, observed in Ref.
∼ 1.5 eV/˚A with DMFT method. The largest contribu- [43]indicatesthatlocalratherthanglobalelectron-lattice
tion in the enhanced deformation potential is from the interactionispresentinFe-basedsuperconductorsandas
hole pocket (h1), located in the Γ-region. A nonuniform suggested in Ref. [46], polaron formation is responsible
and anisotropic nature of the electron phonon coupling for the observed anomalies[43]. Though the formation of
has also seen in the cuprates, where the average EPC polaron depends on a lot of factors, like the band-filling,
wasfoundtobeoneorderofmagnitudesmallerthanthe temperature, EPC strength, phonon frequency etc, our
maximum [59]. This is consistent with our results. results suggest to use a polaron model. We consider
Withincreasingpressure,maximumvalueofthedefor- the electronic state corresponding to maximum λ (∼ 1)
mationpotentialwithinDFT-DMFTthendecreasesand forms a polaron, which is a quasiparticle consisting of
4
electron and the surrounding lattice distortion. Then laronic states or Cooper pairs [29, 41, 47, 48]. For high
the polaronic binding energy (E ) will linearly depend T cuprates, Mu¨ller et al. showed that the normal state
p c
[62, 63] on maximum λ and hence on square of the max- polaron can form Cooper pairs and can be responsible
imum deformation potential. Taking the polaronic band for superconductivity[47]. Such polaronic states may in-
into account, Alexandrov and Mott [62] described that volve spin as well as charge fluctuations [29], leading to
T exponentially depends on function of E . Under hy- a problem that still requires significant development for
c p
drostatic pressure, we found that electronic properties a predictive theory. Even if the giant correlated EPI we
change monotonically while only |Dm|2 (and hence E ) predict is not directly responsible for superconductivity,
p
initially grows (to 3.4 GPa) and then drops, similarly to it should help explain some experimental observations,
experimentalT . ThisindicatingthatastronglocalEPC such as significant and unusual isotope effects [30].
c
plays an important role in Fe-based superconductors. The high T superconductors so far are all rather bad
c
It is important to mention that Tc was found to in- metals in the normal state, have low densities of state
crease rapidly for the low pressure range (0-3 GPa) and at E , are ionic metals containing transition metal ions,
F
can reach up to 27K at 1.48 GPa[64]. The disagreement andarequasi-two-dimensionalandhighlyanisotropic,so
in the pressure dependence of experimental Tc and our itseemsanycompetitivetheoryshouldexplainwhy. Spin
DFT-DMFTcalculationofmaximumDcanbeduetothe fluctuations may play a major role, but it seems that is
presence of the mixed phase in low temperature crystal notenoughtoexplainallofthesecommonfeatures. The
structure in experiment while our calculations are based factthattheyareanisotropicbadmetalswithlowdensi-
on room temperature tetragonal (PbO-type) structure. tiesofstatesmaybebecausethisdecreasesthescreening
The behavior of the DFT-DMFT deformation poten- oftheattractiveinteraction,partlyfrompoorlyscreened
tial with pressure hints that superconductivity in FeSe Coulomb fluctuations from phonons or polarons. Corre-
may have partially phonon or polaron origin and local lations may enhance the EPI as we found here, and in
EPI plays a very important role in superconductivity in addition lead to lower entropy from fluctuations among
theunconventionalsuperconductors. Analysisofthecon- multiplets in the normal state leading to higher T ’s.
c
tributions of each many-body state reveals that charge We thank I. I. Mazin, I. I. Naumov, M. Ahart, P,
fluctuations due to correlations and charge transfer from Zhang,X.ChenandV.Struzhkinforhelpfuldiscussions.
Fe to Se are coupled to the A1g mode. ThisresearchwassupportedaspartofEFree, anEnergy
Our computations predict that applied pressure sig- Frontier Research Center funded by the US Department
nificantly changes the FS around Γ-point. We show the of Energy Office of Science, Office of Basic Energy Sci-
Fermi surface average of the deformation potential is en- encesunderAwardde-sc0001057. K.Hacknowledgesthe
hanced up to 50 % in DFT-DMFT when compared with supports from NSF DMR 0746395. Computations were
standardDFT,andisstillnothighenoughtogivehighTc performed at NERSC supercomputing facility.
inFeSe. Thisreflectsandconfirmsthatasimpleelectron-
phonon coupling mechanism seems unlikely as relevant
from many experimental findings. Calculated electronic
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