- T. Yu, L. Zhang, J. Shen, Y. Fu, Y. Fu,Dalton Trans. 43 ,
13115 – 13121 (2014). - F. Zhanget al.,Chem 7 , 774–785 (2021).
- S. Silver, S. Xun, H. Li, J.-L. Brédas, A. Kahn,Adv. Energy Mater.
10 , 1903900 (2020). - W. Zhaoet al.,J. Mater. Chem. A Mater. Energy Sustain. 8 ,
9919 – 9926 (2020). - A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, G. E. Scuseria,J.
Chem. Phys. 125 , 224106 (2006). - S. Grimme, J. Antony, S. Ehrlich, H. Krieg,J. Chem. Phys. 132 ,
154104 (2010). - F. Zhanget al.,J. Am. Chem. Soc. 141 , 5972–5979 (2019).
- D. L. McGottet al.,Joule 5 , 1057–1073 (2021).
- H. Zhuet al.,J. Am. Chem. Soc. 143 , 3231–3237 (2021).
- S. P. Dunfieldet al.,Adv. Energy Mater. 10 , 1904054 (2020).
- Y. Houet al.,Science 367 , 1135–1140 (2020).
- M. V. Khenkinet al.,Nat. Energy 5 , 35–49 (2020).
ACKNOWLEDGMENTS
Funding:The work was partially supported by the US Department
of Energy under contract DE-AC36-08GO28308 with Alliance for
Sustainable Energy, the manager and operator of the National
Renewable Energy Laboratory. The authors acknowledge the
support on 2D structure design, first-principle calculations,
synthesis of PDAI 2 and DMePDAI 2 , single-crystal synthesis and
analysis, and optoelectronic characterizations (such as TRPL
and TRMC) from the Center for Hybrid Organic-Inorganic
Semiconductors for Energy (CHOISE), an Energy Frontier Research
Center funded by the Office of Basic Energy Sciences, Office of
Science within the US Department of Energy. The authors also
acknowledge the support on device fabrication and
characterization and general thin-film perovskite characterizations
from the De-Risking Halide Perovskite Solar Cells program of the
National Center for Photovoltaics, and the support on SnO 2 ETL
synthesis along with the corresponding device fabrication and
characterization from DE-FOA-0002064 and award DE-
EE0008790, funded by the US Department of Energy, Office of
Energy Efficiency and Renewable Energy, Solar Energy
Technologies Office. Portions of this research were carried out at
the Stanford Synchrotron Radiation Lightsource, SLAC National
Accelerator Laboratory, supported by the US Department of
Energy, Office of Science, Office of Basic Energy Sciences under
contract DE-AC02-76SF00515. L.D.H. and E.L.R. acknowledge
funding support on UPS characterization and analysis from the
Office of Naval Research under award N00014-20-1-2440. X.Z. and
Y.-L.L. acknowledge support on SCLC characterization and analysis
from the National Science Foundation, under grant CMMI-1824674,
and funding from the Princeton Center for Complex Materials, a
National Science Foundation (NSF)–MRSEC program (DMR-
1420541). The DFT calculations were performed by using
computational resources sponsored by the US Department of
Energy’s Office of Energy Efficiency and Renewable Energy and
located at the National Renewable Energy Laboratory and
resources of the National Energy Research Scientific Computing
Center (NERSC), a US Department of Energy Office of Science
User Facility located at Lawrence Berkeley National Laboratory,
operated under contract DE-AC02-05CH11231. The views
expressed in the article do not necessarily represent the views of
the US Department of Energy or the US government.Author
contributions:K.Z. and F.Z. designed the experiment. F.Z. and
S.Y.P. carried out the experimental study on device fabrication and
characterizations. C.Y. conducted DFT calculations, with help
from X.W., under the supervision of Y.Y.; H.L. synthesized PDAI 2 ,
DMePDAI 2 , and the corresponding single crystals. S.P. tested and
analyzed the structures of single crystals. B.W.L. performed the
TRMC and analyzed the TRMC data and some corresponding single
crystals data. C.X. performed the AFM, CAFM, and KPFM tests. S.P.D.
conducted the XPS and analyzed the data, with the guidance from
G.T. and J.J.B.; S.U., L.T.S., and K.H.S. performed the GIWAX and
analyzed the data, with help from L.E.M.; X.Z. performed the
SCLC measurement and analysis, under the supervision of Y.-L.L.;
L.D.H. conducted UPS and analyzed the data, with the guidance
from E.L.R.; X.C. performed the TRPL and analyzed the data,
under the supervision of M.C.B.; F.Z. performed SEM and XRD
measurements. J.J.B performed supplemental XRD measurements.
F.Z., Y.Y., B.W.L, and K.Z. wrote the first draft of the paper. All
authors discussed the results and contributed to the revisions of
the manuscript. K.Z. supervised the project.Competing interests:
F.Z. and K.Z. are inventors on a provisional patent (US patent
application number 63/197,652) related to the subject matter of
this manuscript.Data and materials availability:All data needed
to evaluate the conclusions in the paper are present in the paper
or the supplementary materials. The accession numbers for the
crystal structure cif. files reported in this paper are CCDC 2048509
([PDAPbI 4 ] 15 • [PDAI 2 ]), CCDC 2048508 (BDAPbI 4 ), and CCDC
2048510 (DMePDAPbI 4 -1), which are archived at the Cambridge
Crystallographic Data Centre.SUPPLEMENTARY MATERIALS
science.org/doi/10.1126/science.abj2637
Materials and MethodsFigs. S1 to S30
Tables S1 to S7
References ( 34 – 44 )30 April 2021; resubmitted 20 August 2021
Accepted 12 November 2021
Published online 25 November 2021
10.1126/science.abj2637HEAVY FERMIONSEvidence for a delocalization quantum phase
transition without symmetry breaking in CeCoIn 5
Nikola Maksimovic1,2*, Daniel H. Eilbott1,2, Tessa Cookmeyer1,2, Fanghui Wan1,2, Jan Rusz^3 ,
Vikram Nagarajan1,2, Shannon C. Haley1,2, Eran Maniv1,2, Amanda Gong1,2, Stefano Faubel1,2,
Ian M. Hayes1,2, Ali Bangura^4 , John Singleton^5 , Johanna C. Palmstrom^5 , Laurel Winter^5 ,
Ross McDonald^5 , Sooyoung Jang1,2, Ping Ai^2 , Yi Lin^2 , Samuel Ciocys1,2, Jacob Gobbo1,2,
Yochai Werman1,2, Peter M. Oppeneer^3 , Ehud Altman1,2, Alessandra Lanzara1,2,, James G. Analytis1,2*The study of quantum phase transitions that are not clearly associated with broken symmetry is
a major effort in condensed matter physics, particularly in regard to the problem of high-temperature
superconductivity, for which such transitions are thought to underlie the mechanism of superconductivity
itself. Here we argue that the putative quantum critical point in the prototypical unconventional
superconductor CeCoIn 5 is characterized by the delocalization of electrons in a transition that
connects two Fermi surfaces of different volumes, with no apparent broken symmetry. Drawing
on established theory of f-electron metals, we discuss an interpretation for such a transition that
involves the fractionalization of spin and charge, a model that effectively describes the anomalous
transport behavior we measured for the Hall effect.C
eCoIn 5 is an f-electron metal with
notable similarities to high-temperature
superconducting copper oxides, for ex-
ample, in crystal structure, transport
properties, and unconventional super-
conductivity ( 1 – 10 ). Both CeCoIn 5 and this
category of copper oxides also exhibit signa-
tures of a quantum phase transition (QPT), a
phase transition induced by a nonthermal
parameter, underlying the superconducting
state. However, in many unconventional super-
conductors, it is unclear whether the underly-
ing QPT can be understood in the conventional
sense as a process of separating phases with
different symmetries. Unconventional types of
QPTs, such as non–symmetry breaking ( 11 ) or
weakly symmetry breaking ( 12 , 13 ) QPTs, have
therefore become a subject of intense study. In
this work, we provide evidence that CeCoIn 5
is proximate to a QPT in which the density of
itinerant electrons changes, apparently with-
out the breaking of symmetry. Establishedtheory of f-electron metals provides a means
to interpret such a transition.
At the microscopic level, f-electron metals
such as CeCoIn 5 are described by a Kondo
lattice model. Each Ce atom hosts a single f-level
valence electron, which contributes a localized
spin-½ moment. These local moments coexist
with a sea of itinerant conduction electrons.
In the conventional metallic ground state of
the Kondo lattice, the f-electrons appear to
become an integral part of the itinerant metal.
In particular, they join the conduction elec-
trons, contributing their full share to the total
Fermi volume as prescribed by Luttinger’s
theorem ( 14 ). This phenomenon occurs through
the formation of Kondo singlet correlations
between the local f moments and the conduc-
tion electrons, which effectively hybridize the
f level with the conduction bands.
A long-standing challenge has been to
characterize a QPT in which the f electrons
recover their localized character and withdraw
from the itinerant Fermi volume. Superficially,
the remaining Fermi volume without f elec-
trons is in apparent violation of Luttinger’s
theorem. The loss of Fermi volume when f
electrons localize is therefore convention-
ally accompanied by a transition to a spin-
density wave state, whereby Luttinger’s theorem
is recovered in the appropriately folded Brillouin
zone associated with translational symmetry
breaking ( 15 – 19 ). In this paper, we present76 7 JANUARY 2022•VOL 375 ISSUE 6576 science.orgSCIENCE
(^1) Department of Physics, University of California, Berkeley,
Berkeley, CA 94720, USA.^2 Materials Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA.^3 Department of Physics and Astronomy, Uppsala
University, Box 516, S-75120 Uppsala, Sweden.^4 National
High Magnetic Field Laboratory, Tallahassee, FL 32310, USA.
(^5) National High Magnetic Field Laboratory, Los Alamos, NM
97545, USA.
*Corresponding author. Email: [email protected]
(N.M.); [email protected] (J.G.A.)
RESEARCH | REPORTS