Table Of ContentUnified character of correlation effects in unconventional Pu-based
superconductors and δ-Pu
A. B. Shick,1,2 J. Kolorenc,2 J. Rusz,3,2 P. M. Oppeneer,3
A. I. Lichtenstein,4 M. I. Katsnelson,5 and R. Caciuffo1
1European Commission, Joint Research Centre,
Institute for Transuranium Elements, Postfach 2340, D-76125 Karlsruhe, Germany
2Institute of Physics, ASCR, Na Slovance 2, CZ-18221 Prague, Czech Republic
3Dept. of Physics and Astronomy, Uppsala University, Box 516, S-75120 Uppsala, Sweden
4University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany
3 5Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
1 (Dated: January 17, 2013)
0
2 Electronicstructurecalculations combiningthelocal-density approximationwithanexact diago-
nalization oftheAnderson impuritymodelshowan intermediate5f5-5f6-valencegroundstate and
n delocalization of the 5f5 multiplet of the Pu atom 5f-shell in PuCoIn5, PuCoGa5, and δ-Pu. The
a
5f-localmagneticmomentiscompensatedbyamomentformedinthesurroundingcloudofconduc-
J
tion electrons. For PuCoGa5 and δ-Pu the compensation is complete and the Anderson impurity
6
ground state is a singlet. For PuCoIn5 the compensation is partial and the Pu ground state is
1
magnetic. We suggest that the unconventional d-wave superconductivity is likely mediated by the
5f-states antiferromagnetic fluctuations in PuCoIn5, and by valence fluctuationsin PuCoGa5.
]
l
e PACSnumbers: 74.70.Tx,74.45.+c,74.20.Mn,74.20.Pq
-
r
t
s
Providinga consistentdescriptionof correlationef- ments of the gap amplitude and its temperature de-
.
t
a fects in the electronic structure of elementalactinides pendence.
m and their compounds is a complex problem due to Recently, superconductivity has been discovered
- the interplay between the localized and the itinerant also in PuCoIn [11], with T = 2.5 K. The exper-
d 5 c
nature of the 5f electrons. It is commonly accepted imental studies of this compound were immediately
n
o that 5f-electrons in light actinides form rather broad followed by conventional density functional theory
c conductionbands whereasfor the heavy actinides the (DFT) calculations in the local-density generalized-
[ 5f statesareatomic-like. Johansson[1]describedthis gradient approximation (LDA/GGA) [12, 13]. Keep-
1 situationasa“Motttransitioninthe5f-electronsub- ing in mind a well known failure of DFT in the case
v system”taking place betweenPuandAm whenmov- of δ-Pu [6], it can be expected that LDA/GGA does
3 ing along the Periodic Table. Katsnelson et al. [2] not provide an accurate description of the electronic
1 linked the broadening of the 5f band to the “atomic structure for this strongly correlated material. A few
7
collapse” characterizing the transformation from the static mean-field correlated band theory calculations
3
. high-temperature expanded and the low-temperature were also performed [12, 14], making use of different
1 compressed phases of Pu. flavorsoftheLDA/GGAplusCoulomb’sU (LDA+U)
0
3 A quantitative description of the Mott transition method. While being an improvement over the con-
1 in actinides [3] was obtained by the dynamical mean- ventional band theory, the LDA(GGA)+U falls short
: field theory (DMFT) [4] more than 20 years after the indescribingtheitinerant-to-localizedcrossoverofthe
v
i concept was formulated. Further DMFT studies sug- 5f manifold in δ-Pu [5] and PuCoGa5 [10].
X
gestedanintermediate-valencenatureofthePu-atom Here, we report electronic structure calculations of
r 5f shell [5] and provided justification for the experi- PuCoIn , PuCoGa and δ-Pu performed by combin-
a 5 5
mentally proved absence of magnetism in δ-Pu [6]. ing LDA with the exact diagonalization (ED) [15] of
a discretized single-impurity Anderson model [16]. In
The intermediate-valence and nonmagnetic char-
thisapproach,thebandstructureobtainedbytherel-
acter of the 5f shell can play an important role
ativistic version of the full-potential linearized aug-
in stabilizing the superconducting state exhibited by
mented plane wave method (FP-LAPW) [17] is con-
PuCoGa below a critical temperature T of 18.5
5 c
sistently extended to account for the full structure of
K. [7–9]. The unconventional character of supercon-
the 5f-orbital atomic multiplets and their hybridiza-
ductivity in this compound is now generally accepted
tion with the conduction bands [18].
but the microscopic mechanism responsible for elec-
tron pairing remains unknown. The d-wave symme- The starting point of our approach is the multi-
try of the superconducting gap in PuCoGa has been band Hubbard Hamiltonian [19] H = H0 + Hint.
5
provenbypoint-contactspectroscopyexperiments[10] H0 = H0 c† c , where i,j label lattice
i,j,γ iγ1,jγ2 iγ1 jγ2
that also provided the first spectroscopic measure- sites anPd γ = (lmσ) mark spinorbitals {φ }, is the
γ
2
one-particle Hamiltonian found from ab initio elec- isusedtoconstructaneffectiveLDA+U potentialV ,
U
tronicstructurecalculationsofaperiodiccrystal;Hint which is inserted into Kohn–Sham-like equations:
is the on-site Coulomb interaction [19] describing the
f-electroncorrelation. Weassumethatelectroninter- [−∇2+VLDA(r)+VU +ξ(l·s)]Φbk(r)=ǫbkΦbk(r).
actionsinthe s, p,anddshellsarewellapproximated (3)
in DFT.
These equations are iteratively solved until self-
The effects of the interaction Hamiltonian Hint
on the electronic structure are described by a k- consistencyoverthechargedensityisreached. Ineach
iteration,anewGreen’sfunctionG (z)[whichcor-
independentone-particleselfenergyΣ(z),wherez isa LDA
responds to G(z) from Eq.(2) with the self-energy Σ
(complex) energy. The selfenergy is constructed with
set to zero] , and a new value of the 5f-shell occupa-
the aid of an auxiliary impurity model describing the
tionare obtainedfromthe solutionof Eq.(3). Subse-
completeseven-orbital5f shell. Thismulti-orbitalim-
quently, a new self-energy Σ(z) corresponding to the
purity model includes the full spherically symmetric
updated 5f-shell occupation is constructed. Finally,
Coulomb interaction, the spin-orbit coupling (SOC),
the next iteration is started by evaluating the new
and the crystal field (CF). The corresponding Hamil-
local Green’s function, Eq.(2).
tonian can be written as [16]
Himp = [ǫk]σmmσ′′b†kmσbkm′σ′ + ǫffm†σfmσ
kXmm′ Xmσ
σσ′
+ ξl·s+∆CF σmmσ′′fm†σfm′σ′
mmX′σσ′(cid:2) (cid:3)
+ [Vk]σmmσ′′fm†σbkm′σ′ +h.c. (1)
kXmm′(cid:16) (cid:17)
σσ′
1
+ 2 Umm′m′′m′′′fm†σfm†′σ′fm′′′σ′fm′′σ,
mmX′m′′
m′′′σσ′
wheref† createsanelectroninthe5f shellandb†
mσ mσ
createsanelectroninthe“bath”thatconsistsofthose FIG. 1. (Color online) The Pu atom LDA j = 5/2, 7/2
host-band states that hybridize with the impurity 5f projected DOS, and LDA hybridization function ∆(ǫ) =
shell. Theenergypositionǫf oftheimpuritylevel,and −π1ImTr[G−1(ǫ+iδ)]. TheinsetshowsthePuCoIn5crys-
tal structure.
the bath energies ǫk are measured from the chemical
potential µ. The parameter ξ specifies the strength
In order to determine the bath parameters Vk
of the SOC and ∆ is the crystal-field potential at
CF and ǫk, we assume that the LDA represents the
the impurity. The parameter matrices Vk describe
non-interacting model. We then associate the LDA
the hybridization between the 5f states and the bath
Green’s function G (z) with the Hamiltonian of
orbitals at energy ǫk. LDA
Eq. (1) when the coefficients of the Coulomb interac-
The band Lanczos method [15] is employed to find
tionmatrixaresettozero(Umm′m′′m′′′ =0). Thehy-
the lowest-lying eigenstates of the many-body Hamil-
bridizationfunction ∆(ǫ) is thenestimated as ∆(ǫ)=
tonianHimp and to calculate the one-particle Green’s −1 ImTr[G−1 (ǫ+iδ)]. The curveobtained for ∆(ǫ)
function [G (z)]σ σ′ in the subspace of the f or- π LDA
imp mm′ is shown in Fig. 1, together with the j = 5/2,7/2-
bitals at low temperature (k T = 1/500 eV). The
B projected LDA densities of the f-states. The results
self-energy [Σ(z)]σ σ′ is then obtained from the in-
mm′ also show that the hybridization matrix is, to a good
verse of the Green’s-function matrix [G ].
imp approximation,diagonalin the {j,j } representation.
z
Once the self-energy is known, the local Green’s
Thus,weassumethe firstandfourthterms inthe im-
function G(z) for the electrons in the solid,
puritymodel,Eq.(1),tobediagonalin{j,j },sothat
z
we only need to specify one bath state (six orbitals)
[G(z)] = 1 d3k z+µ−H (k)−Σ(z) −1 , withǫk=1 andVk=1 , andanotherbathstate (eight
γ1γ2 V Z LDA γ1γ2 j=5/2 j=5/2
BZ BZ (cid:2) (cid:3) orbitals) with ǫk=1 and Vk=1 . Assuming that the
(2) j=7/2 j=7/2
most important hybridization is the one occurring in
is calculated in a single-site approximation as given
the vicinity of E , the numerical values of the bath
in [18]. Then, with the aid of the local Green’s F
parameters Vk=1 are found from the relation [20]
function G(z), we evaluate the occupation matrix 5/2,7/2
n = −1 Im EF dz[G(z)] . The matrix n |Vj|2δ(ǫj − ǫ) = −∆(ǫ)/N integrated over the
γ1γ2 π −∞ γ1γ2 γ1γ2 k k k f
R P
3
energyinterval, E −0.5 eV ≤ǫ≤E +0.5 eV, with In the case of PuCoGa , on the other hand, the
F F 5
N = 6 for j = 5/2 and N = 8 for j = 7/2. The hybridized ground state of the impurity is a non-
f f
bath-state energies ǫk=1 shown in Table I are ad- magnetic singlet with all angular moments of the 5f-
5/2,7/2
justed to approximately reproduce the LDA 5f-state bathclusterequaltozero(S =L=J =0). Itconsists
occupations n5f/2 and n7f/2. ofhnfi=5.30f statesandhnbathi=8.70bathstates.
In a pictorial way, we can imagine that the magnetic
moment of the 5f shell (for which we get S = 2.18,
f
TABLE I. 5f-states occupations n5/2 and n7/2, and bath Lf = 4.05, Jf = 2.43) is completely compensated by
f f
state parameters ǫ1 (eV), V1 (eV) for Pu-atom themomentcarriedbytheelectronsintheconduction
5/2,7/2 5/2,7/2
in PuCoIn5, PuCoGa5, and δ-Pu from LDA calculations. band. Asthevalueofthe5f magneticmomentfluctu-
atesintime,becauseoftheintermediate-valenceelec-
Material n5/2 n7/2 ǫ5/2 V5/2 ǫ7/2 V7/2
f f 1 1 1 1 tronic configuration, this compensation must be un-
PuCoIn5 4.78 0.39 0.36 0.21 -0.25 0.25 derstood as dynamical in nature. The same situation
PuCoGa5 4.38 0.76 0.25 0.29 -0.07 0.34
is realized in δ-Pu (S =2.11, L =4.21, J =2.62),
δ-Pu 4.16 0.85 0.33 0.27 -0.01 0.36 f f f
whose groundstate is found to be a nonmagnetic sin-
glet with hn i=5.21 and hn i=8.79.
f bath
In the calculations we used an in-house implemen- The 5f-orbital density of states (DOS) obtained
tation [21, 22] of the FP-LAPW method that in- from Eq. (2) for the three investigated compounds
cludes both scalar-relativisticand spin-orbit coupling is shown in Fig. 2. Below the Fermi energy E the
F
effects. The calculations were carried out assuming DOS exhibits the three-peak structure typical for Pu
a paramagnetic state with crystal structure param- and for a number of its compounds, and its shape is
eters for PuCoIn5, PuCoGa5, and δ-Pu taken from in good agreement with experimental photoemission
Refs. [11, 23, 24], respectively. The Slater integrals spectra. It can be noticed that the multiplets for the
were chosen as F0 = 4.0 eV, and F2 = 7.76 eV, F4 = atomic f6 configuration (f6 → f5 transition, lying
5.05 eV, and F6 = 3.07 eV [25]. They corresponds closer to EF) are better resolvedthan for the f5 part
to commonly accepted values for Coulomb’s U = 4.0 of the spectrum (f5 →f4 transition).
eV and exchange J = 0.64 eV. The SOC param- Comparison with previous LDA+Hubbard-I (HIA)
eters ξ = 0.28 eV for PuCoIn5 and PuCoGa5 and calculation for δ-Pu [18], and PuCoGa5 [26] shows
0.29 eV for δ-Pu were determined from LDA calcula- that the three-peak manifold lying above 2 eV bind-
tions. CFeffectswerefoundtobenegligibleand∆CF ing energy has a slight upright shift towards EF. At
was set to zero. For the double-counting term enter- binding energies around 4 eV, the LDA+HIA peaks
ing the definition of the LDA+U potential, VU, we are substantially modified, and in the LDA+ED cal-
have adopted the fully-localized (or atomic-like)limit culations they are spread over a ∼ 3 eV energy in-
(FLL)Vdc =U(nf−1/2)−J(nf−1)/2. Furthermore, terval. These changes in the DOS are induced by the
wesettheradiioftheatomicspheresto3.1a.u.(Pu), hybridizationandsuggestpartialdelocalizationofthe
2.3 a.u. (Co), 2.3 a.u. (Ga), and 2.5 a.u. (In). The f5 multiplet. This is a situation suggested first by
parameter RPu×Kmax =10.54 determined the basis Hanzawa[27] in intermediate-valence rare-earthcom-
set size, and the Brillouin zone (BZ) sampling was pounds such as SmS or SmB , where fluctuations oc-
6
performed with 1152 k points. The self-consistent cur between two atomic-like 4f configurations. Here,
procedure defined by Eqs. (1)–(3) was repeated un- the5f statesremainlocalizedforthef6 configuration
til the convergence of the 5f-manifold occupation nf but become itinerant for the f5 one.
was better than 0.01. As the many-body resonances lying closer to the
We are now ready to discuss the solutionof Eq.(1). Fermienergyareproducedbyf6 →f5multiplettran-
For PuCoIn5, the ground state of the cluster formed sitions,they arein a wayanalogsto the Racah peaks,
by the 5f shell and the bath is given by a superposi- specific transitions between Racah multiplets [28] of
tion of a magnetic sextet (23%) and a non-magnetic fn and fn±1. On the other hand, these structures
singlet (77%), with occupation numbers hnfi = 5.40 determine the metallic character of the investigated
in the f shell and hnbathi = 8.40 in the bath states. materialsthatcanthereforebeconsideredasarealiza-
This ground state is not a singlet and carries a non- tionofaRacahmetal,situatedbetweenthetwolimit-
zeromagneticmoment. Forthe5f shellalone,theex- ing cases represented by fully localized intermediate-
pectationvaluesofthe spin(Sf),orbital(Lf)andto- valence rare-earth compounds and metallic systems
tal(Jf)angularmomentscanbecalculatedashXˆf2i= (e.g.,nickel)withanon-integernumberofdelectrons.
Xf(Xf + 1) (Xf = Sf,Lf,Jf), giving Sf = 2.27, Both PuCoGa5 and δ-Pu display a temperature-
Lf = 3.90, and Jf = 2.09. The individual compo- independent magnetic susceptibility at low tempera-
nents ofthe moments vanish, hSˆzi=hLˆzi=0,unless tures [6, 29]. Analogous to the intermediate-valence
f f
thesymmetryisbrokenbyanexternalmagneticfield. rare-earth compounds [30], the magnetic susceptibil-
4
6 at E , and Z is a quasiparticle weight, Z =
j=5/2 a) F
j=7/2 (Tr[N(EF)(1− dΣd(ǫǫ))|ǫ=EF]/Tr[N(EF)])−1. We get
V) Tfc = 72 meV (∼ 850 K) for PuCoGa5 and Tfc = 63
1/e4 meV(∼750K)forδ-Pu. SinceTfc ishigh, χremains
es ( constant for T ≪Tfc, as observed experimentally for
at PuCoGa and δ-Pu. The situation is different in the
St 5
of case of PuCoIn5 where the ground state of the impu-
y rity is not a pure singlet due to weakerhybridization.
sit2
n Consequently,thetemperaturedependence ofχisex-
e
D
pected to be more pronounced.
The electronic specific-heat coefficient can be esti-
0 mated as γ = π2k2 Tr[N(E )(1− dΣ(ω))| ]. For
−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 3 B F dω ω=0
Energy (eV) δ-Pu,weget≈44mJK−2 mol−1,inverygoodagree-
ment with experimental data. For PuCoGa , we get
6 5
j=5/2 b) ≈43mJK−2 mol−1 whichissmallerthantheexperi-
j=7/2 mentalvalue of80–100mJK−2 mol−1. ForPuCoIn ,
5
V) the estimated γ value of ≈ 52 mJ K−2 mol−1 is even
e
1/4 furtherawayfromtheexperimentalvalueof≈180mJ
es ( K−2 mol−1. Inthiscase,itisdifficulttoobtainanac-
at
St curate value for γ due to the sharp DOS peak in the
of vicinity of E (see Fig. 2). When taken right at the
y F
sit2 DOS peak position, the γ value of 95 mJ K−2 mol−1
n
e isobtained. Also,notethatapossibleenhancementof
D
γ due to the electron-phonon interaction is not taken
into account.
0
−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6
Energy (eV)
6
j=5/2 c)
j=7/2
V)
e
1/4
s (
e
at
St
of
y
sit2
n
e
D
0
−8 −6 −4 −2 0 2 4 6
Energy (eV)
FIG. 2. (Color online) f-electron density of states (DOS,
j = 5/2, 7/2 projected) for the Pu atom in PuCoIn5 (a),
PuCoGa5 (b) and δ-Pu (c).
ityisanticipatedtobehaveasχ∼1/(T+T ),where
fc
the temperature T describes fluctuations between
fc
the 5f and conduction band electron states. T cor-
fc FIG. 3. (Color online)(Top) The band structure with f-
respondsindeedtothebroadeningofthequasiparticle
weightfatbandsforPuCoIn5,and(bottom)theFermisur-
resonancenearE duetovalencefluctuations[31]. As
F face of PuCoGa5 and PuCoIn5 obtained from LDA+ED
the ground state of the impurity is a singlet, we esti- calculations. The shade of colors encodes the size of the
mate T using arenormalizedperturbationtheoryof energy gradient.
fc
the Anderson model [16], T = −π2Z[∆(E )/N ],
fc 4 F f
where [∆(E )/N ] is the hybridization per orbital Figure 3 shows the band structure and the corre-
F f
5
sponding Fermi Surface (FS) for PuCoIn , calculated
5
from the solutions of Eq. (3), which represents an ex-
tendedLDA+U static-mean-fieldbandstructurewith
[1] B. Johansson, Phys.Rev.B 11, 2740 (1975).
the5f-statesoccupationmatrixobtainedfromthelo-
[2] M. I. Katsnelson, I. V. Solovyev, and A. V. Trefilov,
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Fig. 3 shows also the FS for PuCoGa (Fig. S2 of
5 [3] S.Y.Savrasov,G.Kotliar,andE.Abrahams,Nature
Ref. [10]). Close similarities in the band structure of 410, 793 (2001).
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513 (2007).
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5
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financial support from Czech Republic Grants No. 108, 017001 (2012).
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and from DFG Grant No. 436 TSE 113/53/0-1. (2007).
