Orbital Dependent Electronic Masses in Ce Heavy Fermion Materials studied via Gutzwiller Density Functional Theory

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by  Ruanchen Dong

Orbital Dependent Electronic Masses in Ce Heavy Fermion Materials studied via
Gutzwiller Density Functional Theory
Ruanchen Dong,1 Xiangang Wan,2 Xi Dai,3 and Sergey Y. Savrasov1
1Department of Physics, University of California Davis, Davis, California 95616, USA
2National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
3Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China
(Dated: March 14, 2014)
Aseries ofCerium basedheavy fermionmaterials isstudiedusing acombinationof localdensity
functionaltheoryandmany–bodyGutzwillerapproximation. Computedorbialdependentelectronic
mass enhancements parameters are compared with available data extracted from measured values
4
of Sommerfeld coefficient. The Gutzwiller density functional theory is shown to remarkably follow
1
the trends across a variety of Ce compounds, and to give important insights on orbital selective
0
2 mass renormalizations that allow better understanding of wide spread of data.

r PACSnumbers:
a
M
I. INTRODUCTION electrons. Thisexpressionsuggeststwomaineffectsthat
2
may occur when correlations are brought into consider-
1
ation on top of a band theory such as LDA: first the
Heavy fermion materials pose one of the greatest
crystal field correction to a local f–electron level is con-
] challenges in condensed matter physics. Their low–
l trolled by Σ (0) and, second, the actual band narrow-
e temperature linear specific heat coefficient can be up to α
- 1000 times larger than the value expected from the free– ingiscontrolledbyaquasiparticleresidueparameterzα.
r
Both effects would affect our comparisons of γ in Table
t electrontheory;theirmagneticmomentscanbescreened
s 1.
. by the Kondo effect and their electrical resistivity is fre-
at quently divergent but sometimes superconductivity can Recently, advanced many body approaches based on
m emerge at low temperatures1. combinationsofDensityFunctionalandDynamicalMean
FieldTheory(DMFT)havebeenimplemented26tostudy
- Theoretical calculations based on density functional
d theory (DFT) in its popular Local Density Approxima- heavy fermion systems27,28 where self–consistent solu-
n tion(LDA)2 failtoreproducestronglyrenormalizedelec- tions of either Anderson or Kondo impurity problems
o have been done using most accurate Continues Time
tronicmassesinheavyfermionmaterialsduetoimproper
c Quantum Monte Carlo method29,30. While DMFT deals
[ treatment of many–body correlation effects. Consider,
for example, a well–known class Cerium class of heavy with full frequency dependent self–energy and is a lot
3 more computationally demanding than traditional LDA,
fermion materials, such as two famous phases (α and γ)
v of Cerium itself3–5, so called Ce–115s systems: CeXIn it taught us an important lesson on the so called orbital
2 5
(X=Co,Rh,Ir)6–8 and numerous Ce–122 compounds9–21: selectivityintheMotttransitionproblem,i.e. whencrys-
1
tal field dependent self–energies can reduce effective de-
9 CeX2Si2 (X=Mn,Fe,Co,Ni,Cu,Ru,Rh,Pd,Ag). A Table 1
generacy of the impurity. This affects the proximity of
5 gives evaluated via LDA densities of states specific heat
. coefficients γ for these materials as compared to experi- thequasiparticleresidueztobecomeequalzerowhenthe
1
ratio between Hubbard U and bandwidth W changes.
0 ments, where in many cases a factor of 2–20 error exists
4 in underestimating γ while in some cases, such, e.g., as A recently introduced Gutzwiller Density Functional
1 CeCo Si , an overestimation occurs. It is the purpose of Theory (GDFT) and the so called LDA+G method22–25
2 2
: this work to show that a better treatment of electronic is a simplified variational approach that relies on the
v
i correlationsviaarecentlyintroducedGutzwillerDensity Gutzwiller approximation initially introduced to study
X Functional Theory22–25 can correct most of these errors itinerant ferromagnetism of the one–band Hubbard
r and uncover which exactly orbitals become heavy as is model31, and later extended to multi–band systems32,33.
a also illustrated in Table I. In this method, the atomic configurations of correlated
orbitals are treated by adjusting their weights using a
Thephysicsoftheelectronicmassenhancementiscon-
variational procedure. This leads to renormalized en-
trolledbyalowfrequencybehaviorofthelocalelectronic
ergy bands and mass enhancements for the electrons.
self–energy which can be encoded in a simple Teilor like
The approximation was extensively studied in the limit
form
of infinite dimensions34,35, and was shown to be equiv-
Σ (ω)=Σ (0)+(z−1−1)ω+... (1) alent to a slave–boson mean–field theory36 for both
α α α
single–band and multiband models37–39. In LDA+G,
where we assume, for simplicity, the existence of some the Gutzwiller type trial wave function Pˆ|0(cid:105) is adopted
crystalfieldrepresentation|α(cid:105)thatdiagonalizestheself– with |0(cid:105) and Pˆ being the LDA ground state and the
energy matrix in a spin–orbital space of the localized f Gutzwiller projector respectively. After further applica-
2
tionoftheGutzwillerapproximation,aneffectiveHamil- A. Gutzwiller Approximation
tonian describing the dynamics of quasiparticles was ob-
tained as Heff = PˆHLDAPˆ, which contains two impor- We first illustrate the method using a general multi–
tant features discussed above: the modification of the orbital Hubbard model. The Hamiltonian is:
crystal/spin–orbital fields and the quasiparticle weight
(cid:88) (cid:88)(cid:88)
zα. Thus, the method exactly casts the effect encoded H =H0+Hint = tαijβc†iαcjβ + Uiαβnˆiαnˆiβ
into the low–frequency behavior of Σ(ω), Eq. (1). ij,αβ i α(cid:54)=β
(2)
The benchmark of the LDA+G scheme was demon- where α = 1,...,2N is the spin–orbital index of the lo-
strated in Ref. 22. For a non–magnetic correlated metal calizedorbital,N isthenumberoforbitals,e.g. 7forthe
SrVO , it produced narrower bands and larger effec- f–orbital. The first term is a tight–binding Hamiltonian
3
tive masses than those found in standard LDA. Also the which can be extracted from the LDA calculation. The
method was able to get a photoemission peak missed in second term is the on–site interaction which has been
theLDAcalculation. Theseimprovementsareveryclose restricted to the density–density interaction.
to the experimental results. Later, in Ref. 40 a complex In the atomic limit, for the localized orbital there
phase diagram of Na CoO was correctly reproduced. are 2N different states which can be either occupied or
x 3
Recently, this method has been successfully applied to empty. Therefore there is total 22N configurations |Γ(cid:105).
FeAs–based superconductors25,41 and to Ce metal42,43. All these configurations form a complete basis and the
density–density interaction is diagonal in this configura-
In this work we address the physics of Cerium heavy tionspace. Itisobviousthattheseconfigurationsshould
fermion materials via the use of the LDA+G approach. not be equally weighted. In the Gutzwiller method, we
A great spread in the extracted values of mass enhance- adjust the weight of each configuration. Therefore it is
mentsdatashowninTableItogetherwithsomeunphysi- convenienttoconstructprojectionoperatorsthatproject
calvaluesofz >1promptsusthatinmanyclassesofreal onto a specific configuration Γ at site i:
compounds both the orbital selectivity encoded via the
mˆ =|iΓ(cid:105)(cid:104)iΓ| (3)
shifts Σ (0) on top of the LDA as well as the quasiparti- iΓ
α
cleresiduesplayanimportantroleandhavetobetreated
When the interaction term is absent, the ground state
on the same footing. It therefore represents a stringiest
is the Hatree uncorrelated wave function (HWF) |Ψ (cid:105)
0
test of a many body electronic structure method such
whichisaSlaterdeterminantofthesingle–particlestates.
as LDA+G to heavy fermion materials. In particular,
When the interaction is switched on, this wave function
in its recent application to elemental Cerium43, it has
is no longer a good approximation. In the Gutzwiller
been shown that the spin–orbital splitting of the f–level
method, we project the wave function to a Gutzwiller
isrenormalizedbycorrelationsandpushesenergiesofthe
wave function (GWF) |Ψ (cid:105) by adjusting the weight of
G
J=7/2manifolduprelativetoJ=5/2states.Thisresulted
each configuration through variational parameters λ
iΓ
in lowering the degeneracy from 14 to 6 and in a greater (0(cid:54)λ (cid:54)1):
iΓ
mass enhancement of J=5/2 manifold as compared to a
non–spin orbit coupled calculation. |Ψ (cid:105)=Pˆ|Ψ (cid:105)=(cid:89)Pˆ|Ψ (cid:105) (4)
G 0 i 0
i
The paper is organized as follows. In Sec. II, the
LDA+G method is described. The results for several where
typical families of heavy fermion materials are presented Pˆ =(cid:88)λ mˆ (5)
in Sec. III. Finally, Sec. IV is the conclusion. i iΓ iΓ
Γ
Notice that when all λ = 1, the GWF is going back
iΓ
to HWF. At the same time, setting λ = 0 removes
iΓ
configuration Γ at site i. Therefore, perfectly localized
atomic state of site i is described by all λ = 0 except
iΓ
for one, and in this way, the Gutzwiller wave function
captures both the itinerant and localized behavior of the
II. THE LDA+G METHOD system.
ItisadifficulttasktoevaluateGWF.However,within
the Gutzwiller method, we can map any operator Oˆ act-
The Gutzwiller Density Functional Theory and ing on the GWF to a corresponding effective AˆG which
the LDA+G approximation have been described acts on the HWF:
previously22–25. Here we merely summarize the equa-
(cid:104)Ψ |Aˆ|Ψ (cid:105)=(cid:104)Ψ |Pˆ†AˆPˆ|Ψ (cid:105)=(cid:104)Ψ |AˆG|Ψ (cid:105) (6)
tions of the method that we implement using a linear G G 0 0 0 0
muffin–tin orbital formalism that includes both the full
where
potential terms and relatvistic spin–orbit coupling oper-
ator variationally44 . AˆG =Pˆ†AˆPˆ (7)
3
TABLEI:ComparisonbetweencalculatedusingLDAdensityofstatesandexperimentallyextractedspecificheatcoefficientsγ
andtheextractedquasiparticleresiduesz =γ /γ foranumberofCeriumbasedheavyfermioncompoundsconsidered
exp LDA exp
inthiswork. Thelastcolumnsshowthepredictionsofγ andz usingtheLDA+GmethodwiththevaluesofU=4and5eVas
well as the reference to a specific orbital degeneracy of the j =5/2 manifold that exibits strongest enhancement.
N (0) γ γ Ref. z γ γ z z Orbital
LDA LDA exp exp LDA+G LDA+G LDA+G LDA+G
St./(Ry·cell) mJ/(mol K2) mJ/(mol K2) to γ (U=4 eV) (U=5 eV) (U=4 eV) (U=5 eV) j =5/2
exp
α-Ce 36 6.2 13 3 0.48 0.55 0.33 Γ ,Γ
7 8
γ-Ce 49 8.5 0.14 0.11 Γ ,Γ
7 8
115s:
CeCoIn 150 26.0 290 6 0.09 70 104 0.20 0.13 Γ ,2×Γ
5 6 7
CeRhIn 156 27.2 420 7 0.065 75 120 0.19 0.12 Γ ,2×Γ
5 6 7
CeIrIn 165 28.6 720 8 0.04 79 210 0.14 0.07 Γ ,2×Γ
5 6 7
122s:
CeMn Si 184 31.9 47 9 0.68 52 84 0.51 0.43 Γ ,2×Γ
2 2 6 7
CeFe Si 85 14.7 22 10 0.67 21 24 0.53 0.47 Γ ,2×Γ
2 2 6 7
CeCo Si 110 19.0 10 11 1.9? 35 43 0.47 0.36 Γ ,2×Γ
2 2 6 7
CeNi Si 115 19.9 33 12 0.60 27 29 0.47 0.43 Γ ,2×Γ
2 2 6 7
CeCu Si 64 11.1 1000 13–15 0.01 430 ∞ 0.10 0 Γ
2 2 6
CeRu Si 103 17.8 350 16–18 0.05 70 190 0.33 0.13 2×Γ
2 2 7
CeRh Si 106 18.3 130 19 0.14 35 120 0.36 0.14 2×Γ
2 2 7
CePd Si 100 17.3 65-110 20,21 0.15-0.26 170 ∞ 0.045 0 Γ
2 2 7
CeAg Si 205 35.5 140 430 0.10 0.06 Γ
2 2 6
Specifically, when the operator Aˆ is a single–particle residues: 0 (cid:54) z (cid:54) 1. These are determined by the
iα
operator, e.g. Aˆ = (cid:80) Aαβc† c where Aαβ = configuration weights:
ij,αβ ij iα jβ ij
(cid:104)iα|Aˆ|jβ(cid:105), the Gutzwiller effective operator can be writ-
√
ten as: z =(cid:88) miΓmiΓ(cid:48)|(cid:104)iΓ(cid:48)|c†iα|iΓ(cid:105)| (9)
iα (cid:112)
AˆG = (cid:88) √z Aαβ√z c† c +(cid:88)Aαα(1−z )c† c ΓΓ(cid:48) niα(1−niα)
iα ij jβ iα jβ ii iα iα iα
ij,αβ i,α
(8) where m = (cid:104)Ψ |mˆ |Ψ (cid:105) and n are the occupation
iΓ G iΓ G iα
where z are the orbital–dependent quasiparticle numbers for the orbitals.
iσ
B. Combination with LDA
Similar to the idea of the LDA+U or LDA+DMFT methods, we add the interaction term on top of the LDA
calculation. The Hamiltonian is given by:
H =H +H −H (10)
LDA int DC
where H is the LDA Hamiltonian, which casts the same form as H in Eq.(2), H is the on–site interaction
LDA 0 int
term for the set of correlated orbitals, such as f–orbitals of heavy–fermion materials considered in this work. Since
the LDA calculation has already included the Coulomb interaction in some averaged level, we need to subtract the
double–counting term H from LDA. Various forms of H will be discussed later.
DC DC
The Kohn–Sham approach uses a non–interacting system as a reference which keeps the same density as the
interacting one. However, now our ground state is the GWF instead of the HWF. Therefore, we need to transform
the Hamiltonian into the effective one in the |Ψ (cid:105) basis. Since the Hamiltonian H is a single–particle operator,
0 LDA
following Ref. 24 we obtain
HG = (cid:104)Ψ |H |Ψ (cid:105)=
LDA G LDA G
(cid:32) (cid:33)  
(cid:88)√ (cid:88) (cid:88)√ (cid:88)
zα|φαi(cid:105)(cid:104)φαi|+1− |φαi(cid:105)(cid:104)φαi| HLDA zβ|φβj(cid:105)(cid:104)φβj|+1− |φβj(cid:105)(cid:104)φβj|+
αi αi βj βj
(cid:88)
(1−z )|φ (cid:105)(cid:104)φ |H |φ (cid:105)(cid:104)φ |
α αi αi LDA αi αi
αi
4
where |φ (cid:105)represents a complete basis set of the correlated orbitals, and where we omit site index i from the quasi-
αi
particle residues z due to lattice periodicity.
a
The interaction term acting on the GWF produces:
(cid:88)
HG =(cid:104)Ψ |H |Ψ (cid:105)= E m (11)
int G int G Γ Γ
iΓ
The expectation value of the total Hamiltonian gives us the total energy as a functional of the density ρ and
confugurational weights m :
Γ
(cid:88)
E(ρ,{m })=(cid:104)Ψ |HG |Ψ (cid:105)+ E m −E (12)
Γ 0 LDA 0 Γ Γ DC
Γ
AminimizationsimilartoLDAisnowperformed. RepresentingdensityintermsoftheKohn–Shamstates,produces
the equations for the quasiparticles:
(cid:32) (cid:20) (cid:21) (cid:33)
∂E(ρ,{mΓ}) = HG +(cid:88) ∂E ∂zα − ∂EDC |φ (cid:105)(cid:104)φ | |ψ (cid:105)=(cid:15) |ψ (cid:105) (13)
∂(cid:104)ψ LDA ∂z ∂n ∂n α α nk nk nk
nk α α α
α
∂E(ρ) =(cid:88) ∂E ∂zα +E =0 (14)
∂m ∂z ∂m Γ
Γ α Γ
α
Recalling the self–energy linear expansion, Eq.(1), we see from Eq.(13) that the effective Hamiltonian to be diagonal-
ized casts the following form
(cid:88)
HG =HG + (Σ (0)−V )z |φ (cid:105)(cid:104)φ | (15)
LDA α DC,α α α α
α
where Σ (0) and V are directily associated with various total energy derivatives appeared in (13).
α DC,α
C. Gutzwiller Projected Hamiltonian
It is convinient to represent all matrices in the space of the Bloch eigenvalues (cid:15) and wave functions |kj(cid:105) that are
kj
obtained from the LDA calculation. The Gutzwiller hamiltonian to be diagonalized is given by
(cid:32) (cid:33)
(cid:88) (cid:88)√ (cid:88)
(cid:104)kj(cid:48)|HG|kj(cid:105) = (cid:15) z (cid:104)kj(cid:48)|φ (cid:105)(cid:104)φ |kj(cid:48)(cid:48)(cid:105)+δ − (cid:104)kj(cid:48)|φ (cid:105)(cid:104)φ |kj(cid:48)(cid:48)(cid:105) × (16)
k(cid:48)(cid:48)j(cid:48)(cid:48) α α α j(cid:48)j(cid:48)(cid:48) α α
j(cid:48)(cid:48) α α
 
(cid:88)√ (cid:88) (cid:88)
 zb(cid:104)kj(cid:48)(cid:48)|φβ(cid:105)(cid:104)φβ|kj(cid:105)+δj(cid:48)(cid:48)j − (cid:104)kj(cid:48)(cid:48)|φβ(cid:105)(cid:104)φβ|kj(cid:105)+ (1−zα)(cid:104)kj(cid:48)|φα(cid:105)(cid:15)α(cid:104)φα|kj(cid:105)+
β β α
(cid:88)
(Σ (0)−V )z (cid:104)kj(cid:48)|φ (cid:105)(cid:104)φ |kj(cid:105)
α DC,α α α α
α
where a subset of correlated orbitals |φ (cid:105) is introduced. Their levels are given by
α
(cid:88)
(cid:15) = (cid:15) (cid:104)φ |k(cid:48)(cid:48)j(cid:48)(cid:48)(cid:105)(cid:104)k(cid:48)(cid:48)j(cid:48)(cid:48)|φ (cid:105) (17)
α k(cid:48)(cid:48)j(cid:48)(cid:48) α α
k(cid:48)(cid:48)j(cid:48)(cid:48)
The quasiparticle residues z and the level shifts Σ (0) are obtained using the Gutzwiller procedure24. The double
α α
counting potential V corrects for the fact that the LDA already accounts for some of the correlation effects in a
DC,a
mean field manner. The eigenvalue problem
(cid:88)
((cid:104)kj(cid:48)|HG|kj(cid:105)−δ E )Bkn =0 (18)
j(cid:48)j kn j
j
produces renormalized energy bands E and wave functions (cid:80) Bkn|kj(cid:105) of the quasiparticles.
kn j j
5
D. Charge Density
In order to find a new density, we calculate a Gutzwiller density matrix operator in the LDA representation
(cid:32) (cid:33)
(cid:88) (cid:88)√ (cid:88)
ρGk = f z (cid:104)kj(cid:48)|φ (cid:105)(cid:104)φ |k(cid:48)(cid:48)(cid:105)+(cid:104)kj(cid:48)|kj(cid:48)(cid:48)(cid:105)− (cid:104)kj(cid:48)|φ (cid:105)(cid:104)φ |kj(cid:48)(cid:48)(cid:105) × (19)
j(cid:48)j kj(cid:48)(cid:48) α α α α α
j(cid:48)(cid:48) α α
 
(cid:88)√ (cid:88) (cid:88)
 zβ(cid:104)kj(cid:48)(cid:48)|φβ(cid:105)(cid:104)φβ|kj(cid:105)+(cid:104)kj(cid:48)(cid:48)|kj(cid:105)− (cid:104)kj(cid:48)(cid:48)|φβ(cid:105)(cid:104)φβ|kj(cid:105)+ (cid:104)kj(cid:48)|φα(cid:105)(1−zα)ρα(cid:104)φa|kj(cid:105)
β β α
where
(cid:88)
ρ = f (cid:104)kj|φ (cid:105)(cid:104)φ |kj(cid:105) (20)
α kj α α
kj
Diagonalizing it produces new occupation numbers n
kλ
(cid:88)
(ρGk−δ n )Ckλ =0 (21)
j(cid:48)j j(cid:48)j kλ j
j
so that the density of quasiparticles in real space is given by
  
(cid:88) (cid:88) (cid:88)
ρG(r)= nkλ Cjk(cid:48)λ|kj(cid:48)(cid:105) Cjkλ∗(cid:104)kj| (22)
kλ j(cid:48) j
E. Incompleteness of Basis F. Double Counting Potential
AsoneseesfromEq.(16)theactualself–energycorrec-
To see the importance of the issue, let us examine a
tion used in the LDA+G calculation is Σ (0)−V .
shift of the LDA eigenstates (cid:15) by arbitrary value x. α DC,α
kj Frequently,asocalledLDA+Uversion46ofdoublecount-
We obtain the new Gutzwiller Hamiltonian as the old
ing potential V is used, that for our case is just an
one plus the correction DC,α
orbital–independent energy shift given by
(cid:104)kj(cid:48)|H˜G|kj(cid:105)=(cid:104)kj(cid:48)|HG|kj(cid:105)+xo (k) (23) VLDA+U =U(n −1/2), (25)
j(cid:48)j DC f
where n is the average number of f electrons which at
f
where oj(cid:48)j(k) is a matrix that can be proved to be equal Ce f–shell is close to unity. As one sees, this correction
to δj(cid:48)j only under the assumption that the LDA wave has just an overall level shift by U/2 and does not mod-
functions form a mathematically complete basis set, i.e. ify the Gutzwiller extracted spin–orbit and crystal fields
encoded in the α dependence of Σ (0). Unfortunately,
α
(cid:88) it is not exactly clear whether the overall level shift of
(cid:104)φ |kj(cid:48)(cid:48)(cid:105)(cid:104)kj(cid:48)(cid:48)|φ (cid:105)=δ (24)
α β αβ Ce f electrons has a physical effect, since the standard
j(cid:48)(cid:48) LDA+U double counting was introduced in connection
to the Hartree Fock value of the self–energy which is the
Unfortunately,modernelectronicstructuremethodsdeal value at infinite frequency, Σ(∞). It therefore may not
withfinitebasissets,andthelastrelationshipisonlyap- be suited for correcting low energy physics of the heavy
proximately satisfied. As a result, different choices of fermion systems.
energy zero for the LDA eigenvalues (cid:15) may lead to As a result, in this work we adopt a different strategy
kj
slightly different output, although in our application to in order to elucidate the physics of orbital selectivity in
well–localized Cerium 4f electrons, this introduces only Ce heavy fermion compounds: our calculations are first
minor noise in our calculated results. In the following, performed without Σa(0) correction assuming that the
we always assume that the LDA eigenvalues are mea- double counting potential
sured with respect to the Fermi energy which is the only
physically relevant energy in this problem. V(1) =Σ (0). (26)
DC,α α
6
It has an important justification that the LDA calcu- Here(cid:60)Σistherealpartand(cid:61)Σistheimaginarypartof
lated Fermi surfaces do not acquire any modifications as the electronic self–energy. Under the assumption of the
was the past evidence for some heavy fermion uranium locality of self–energy, the quasiparticle residues
compounds47. Second, we introduce the crystal field av-
eraged double counting (cid:18) ∂(cid:60)Σ(ω)(cid:19)−1
z = 1− (30)
∂ω
N
V(2) = 1 (cid:88)Σ (0) (27)
DC N α can be extracted by comparing ARPES spectra against
α
calculated LDA energy bands.
which keeps the average position of the f–level intact The dHvA effect is another powerful experimental
but allows for its crystal field modifications found self– technique which measures the properties of the Fermi
consistently via the LDA+G procedure. Since both surface properties under applied magnetic field45. LDA
V(1) and V(2) rely on Gutzwiller extracted Σ (0), calculation can identify each cyclotron orbit seen by
DC,α DC,α α the dHvA experiment and find corresponding effective
whichitselfisobtainedfromtheLDA+Gfunctionalmin-
masses. However, complexity in shapes of 3D Fermi sur-
imizationprocedure,theentiremethodisstillvariational
facesinrealsystemsalsomakesthismethodnotperfect.
andallowsanaccurateestimateofthetotalenergy. Com-
paring calculations with two types of double counting, It is remarkable that the LDA+G calculation returns
important conclusions can be drawn on which exactly the orbital dependent quasiparticle residues zα directly.
orbitals of a given heavy fermion system play a major In Table I, we are quoting these mass enhancement data
role in its electronic mass enhancement. and not the ones obtained via our calculated LDA+G
densities of states.
III. RESULTS AND DISCUSSION
A. α-Ce and γ-Ce
Weareinterestedincalculatingthemassenhancement
parametersofCeriumfelectronswhichinasimplesingle We first discuss our calculations for Cerium metal
bandtheorywouldbegivenbytheratioofm∗/m . A which is famous for its iso–structural phase transition
LDA
common approach to extract this data is to compare the from its α to γ phase that is accompanied by 15% of its
values of the Sommerfeld coefficient γ evaluated using volume expansion, and has attracted great attention in
the LDA density of states at the Fermi level N (0) the past4,5,48–51 and current43 literature. It is remark-
LDA
able that our LDA+G calculation can correct most of
π2 the error in predicting the volume of the α phase for
γ = k N (0) (28)
LDA 3 B LDA both types of the double countings that we explore in
this work. Moreover, as Fig. 1 illustrates, it clearly
withthemeasuredelectronicspecificheatwhich,accord-
shows a double well type of behavior of the energy vs
ing to the Fermi liquid theory, behaves at low tempera-
volume (with smaller/larger minima corresponding to
tures as γT. However, some care should be taken when
α/γ phases) when using the crystal field averaged dou-
adopting this procedure. First, densities of states of
ble counting V(2) and the Hubbard U’s within the range
realsystemsincludemulibandfeaturesandcontributions DC
between3.5and5.5eV.Thisisinaccordwiththeprevi-
from both heavy and light electrons. Second, LDA den-
ouspreviousLDA+DMFTstudiesforCerium49. Similar
sities of states assume some crystal field effects which
behaviorhasbeenalsofoundwhenstudyingα→δ tran-
should in general be supplemented by many body cor-
sition in metallic Plutonium52.
rections encoded in Σ (0). Therefore not only band nar-
a
The specific heat measurement of α–Ce gives its Som-
rowing but also level shifts are expected to occur in real
merfeld coefficient γ ∼ 13 mJ/(mol·K2).3. The optical
life on top of LDA. Third, many of the materials dis-
spectroscopy experiment53 estimates the effective mass
cussed in our work undergo either antiferromagnetic or
to be ∼ 6m in α–Ce and to be ∼ 20m in γ–Ce, indi-
superconducting transition before reaching T → 0 limit. e e
catingtheitinerant/localizedfeaturesoftheα/γ phases.
We quote the data for γ in Table I using the data for
exp
However, the estimated optical effective masses are fre-
their lowest temperature paramagnetic phases.
quency dependent.
There are several other ways to access this informa-
Fig. 2 shows the dependence of the quasiparticle
tion that we are going to use in this work. Optical
residuesasafunctionoftheHubbardUforvolumescor-
spectroscopy experiments can provide access to effec-
responding to α– and γ–Ce where the left/right plots
tive masses but those are frequency dependent. Angle–
resolved photoemission spectroscopy (ARPES) experi- represent our calculations with V(1)/V(2) type of double
DC DC
ments measure directly one–electron spectral functions countings. It is clear when crystal/spin–orbital correc-
tions are not taken into account (left plot), the effective
|(cid:61)Σ(k,ω)| masses for various Cerium orbitals are very similar in
A(k,ω)= (29)
(ω−(cid:60)Σ(k,ω))2+(cid:61)Σ2(k,ω) values and are not very strongly enhanced even for large
7
FIG.2: (coloronline)Calculateddependenceofquasiparticle
residues as a function of Hubbard U in α and γ phases of
Cerium metal using the LDA+G approach.
B. Ce-115s
FIG.1: (coloronline)Calculatedtotalenergyvsvolumeusing
the LDA+G method with several values of Hubbard U.
We next discuss our applications to the so–called 115
series of Cerium heavy fermion compounds CeMIn (M
5
= Co, Rh, Ir). They have a tetragonal HoCoGa –type
5
values of U. (We use relativistic cubic harmonics repre- structure which results in additional splitting of all Γ
8
sentation where j = 5/2 level is split onto Γ7, Γ8 and quadruplets into Γ6 and Γ7 doublets. Despite having
j = 7/2 onto Γ6,Γ7,Γ8 states.) The situation changes similar structure and almost identical LDA electronic
dramatically when we account for the local self–energy structures, these systems show very different proper-
correction (right plot): a Coulomb renormalized spin– ties which has attracted a great interest. CeCoIn is a
5
orbit splitting pushes Γ7,Γ8 states of j = 5/2 manifold heavy fermion superconductor with critical temperature
down and Γ6,Γ7,Γ8 states of j =7/2 manifold up. This Tc = 2.3 K, highest known in Ce–based systems6 and
resultsinredistributingtheoccupanciesofthef–electrons with the Sommerfeld coefficient γ ∼= 290 mJ/(mol·K2)
which now reside mainly at j = 5/2 level. Thus, the ef- measured just above T . CeIrIn is also a superconduc-
c 5
fectivedegeneracyis6insteadof14andthequasiparticle torwithT =0.4K8 withγ=720mJ/(mol·K2)aboveT
c c
residues become much more sensitive to the values of U. thatisnearlytemperatureindependent. CeRhIn ,onthe
5
Actual comparison of the quasiparticle residues with otherhand,isanantiferromagnetwithN´eeltemperature
experiment needs an accurate estimate of the Hubbard T =3.8 K but becomes a superconductor with T = 2.1
N c
U for Cerium f–electrons which is typically around 5 eV, KaboveacriticalpressureP ∼16kbar7. FromtheC/T
c
although, if the Hunds rule J˜1eV (important for the data, there is a peak at T =3.8 K, indicating the onset
N
virtual f2 state) is taken into account, the effective inter- of magnetic ordering. In order to find the electronic spe-
actionisreducedalittlebittoU−J.Duetoagreatsen- cific heat, one needs to use isostructural, nonmagnetic
sitivity of quasiparticle residues to this parameter, and LaRhIn to subtract the lattice contribution to C. How-
5
for the purposes of our work, we simply show in Table I ever, it is difficult to define precisely the electronic spe-
the range of z’s that one can obtain for the values of U cificheataboveT duetothepeakedstructure. Asimple
N
intherangebetween4and5eV.Forα–Cewefindthem entropy–balance construction gives a Sommerfeld coeffi-
between 0.55 and 0.33 which are close to their experi- cient γ (cid:62)420 mJ/(mol·K2) for T >T .
N
mental estimates. This is also consistent with the result Those different properties are considered as the result
oftheLDA+DMFTcalculation54. Whilethereisnospe- of the localized vs itinerant nature of the 4f electrons.
cific heat data for the γ phase, optical measurements53 ThedHvAmeasurementsforCeCoIn haveshowntheef-
5
showafactorof3–4enhancedmassesascomparedtothe fectivecyclotronmasseswithintherangefrom9to20m
0
ones of the α phase, and our reduced values of z(cid:48)s are in which is consistent with the specific heat data55,56. The
accord with this trend. LDA calculation with a model of itinerant f electrons
8
shows a reasonable agreement with the dHvA data57
while complete localization of the f electrons is needed
togettheagreementwiththeangle–resolvedphotoemis-
sion spectroscopy(ARPES) data58. The dHvA experi-
ment has been also performed for the antiferromagnetic
state of CeRhIn 59,60. Although earlier LDA calculation
5
with the itinerant model shows some agreement with
the experimental data60, the localized nature of the f
electrons was confirmed by the dHvA measurements in
Ce La RhIn 61 and by comparing the Fermi surfaces
x 1−x 5
between CeRhIn and LaRhIn 62. With the application
5 5
ofpressure,thedramaticchangeoftheFermisurfacewas
observedindicatingthechangefromthelocalizedantifer-
romagnetic state to the itinerant heavy fermion state63.
ForCeIrIn ,thereissomeexperimentalcontroversy. The
5
effective cyclone mass m∗ is observed in the range from
c
6.3to45m indicatingalargeenhancement64. LDAwith
e
itinerant f electrons57,64 explains well geometry and the
volume of the Fermi surface but the band masses are
FIG.3: (coloronline)Calculateddependenceofquasiparticle
muchsmallerofthecyclotronmasses. Thephotoemission
residues as a function of Hubbard U in CeCoIn using the
5
spectrum is well described by the LDA+DMFT calcula- LDA+G approach.
tion27 where the degree of itineracy in CeIrIn is though
5
to be even larger than in CeCoIn 65. Also, this method
5
shows the calculated effective masses to be of the same
germanium. Afteritsfirstdiscoveryofsuperconductivity
order with the experimental one66. On the other hand, with a transition temperature T (cid:39)0.5 K in CeCu Si 13,
the ARPES study67,68 shows that CeIrIn and CeRhIn c 2 2
5 5 the interest in this family awakens, especially due to the
have nearly localized 4f electrons.
interplay between antiferromagnetic and superconduct-
The results of our paramagnetic LDA+G calculations ingorders. HerewefocusonthesubclassCeM Si (M =
2 2
for all three 115 compounds are presented in Table I. Mn, Fe, Co, Ni, Cu, Ru, Rh, Pd, Ag) where all members
Similar to our calculation for Cerium, we find that the have body–centered tetragonal ThCr Si –type structure
2 2
calculatedeffectivemassesareonlymoderatelyenhanced with space group I4/mmm.
(z˜0.3−0.5)ifwedonotaccountforthecrystalfield/spin
For M = 3d series, no magnetic order is found except
orbit corrections to the self–energy on top of the LDA.
for CeMn Si where the Mn local moments order below
2 2
When we perform self–consistent calculation including
379 K70. For CeCu Si , the electronic specific heat co-
2 2
the level shifts, much smaller values of the quasiparticle efficient γ (cid:39)1000 mJ/(mol·K2) is the largest one among
residues can be reached. Fig. 3 illustrates this behav-
this family. On the other hand, CeFe Si , CeCo Si and
2 2 2 2
ior for CeCoIn . The situation here is similar to Cerium
5 CeNi Si areweakparamagnetswithrelativelysmallval-
2 2
wherethespin–orbitcouplinggetsrenormalizedbycorre-
uesofγvaluewhichisshowninTableI.Thesethreecom-
lationsmakingtheeffectivedegeneracyofthef–electrons
pounds are also known as valence fluctuation systems.
equal6. ActualvaluesofthequasiparticleresiduesinTa-
For M = 4d series, first, CeRu Si is known as a
bleIaregivenforUequal4and5eV:weseethatthees- 2 2
archetypal Kondo lattice compound: it is a paramag-
timated z is the largest for CeCoIn while the f electrons
5 net with a relative large γ (cid:39) 350 mJ/(mol·K2). The
aremorelocalizedinCeRhIn andCeIrIn . Theresidual
5 5 other three are antiferromagnets at low temperatures:
discrepancies can be attributed to the above discussed
CeRh Si has the highest ordering temperature 36–39
uncertainties seen in experiments and also to the intrin- 2 2
K71,72 while CePd Si 21,71 and CeAg Si 71,73 order an-
sic error connected to the Gutzwiller procedure as our 2 2 2 2
tiferromagnetically at 8.5-10 K an 8-10 K, respectively.
prior studies of the performance of this method against
With the application of pressure, superconductivity was
QuantumMonteCarloapproachhaveshowna30%type
found in CeRh Si 74 and in CePd Si 75. CeRu Si is
of error69. 2 2 2 2 2 2
uniqueinthissubclasssincethesuperconductivityisnot
observed down to a few mK. This makes it best tar-
get material for studying its heavy fermion state. In-
C. Ce-122s terestingly, here a metamagnetic transition was found
and extensively studied76. The cyclotron effective mass
We finally discuss our applications to Ce 122 types m ∼ 120m is observed in the dHvA experiment in-
c e
of systems. By itself, RM X is an enormous class of dicating the renormalized heavy fermion state77. The
2 2
ternary intermetallic compounds with over hundreds of electronic structure calculation using LDA with itiner-
members, where R is a rare–earth element, M denotes a ant f–electron model qualitatively explains the dHvA
transitionmetal(3d. 4d,or5d)andX iseithersiliconor data78–80 The metamagnetic transition was also stud-
9
of U above 4 eV. In fact, it is beginning to reach almost
zerovalueswhenUapproaches5eV.Itisalsoknownex-
perimentally that this system shows enormous mass en-
hancementjustbeforeitgoesintosuperconductingstate.
The 122 compounds with 4d elements (Ru, Rh, Pd,
Ag) exhibit strongly renormalized quasiparticle masses.
Our LDA+G calculations listed in Table I correctly fol-
low this trend where we see a strong reduction of z’s as
compared to our calculations with 3d elements. We also
find another interesting effect: with increasing U the ef-
fective degeneracy of the f electrons acquires a reduction
not only due to the Coulomb assisted renormalization of
spin–orbit splitting but also renormalization of the crys-
tal fields: Fig. 4 shows this behavior for Ru-122 where
we see that the values of z become different for various
crystal field levels of the j = 5/2 manifold: one Γ and
6
two Γ doublets. We find that a similar effect occurs in
7
FIG. 4: (color online) Calculated dependence of quasiparti-
all other 4d types of Ce 122s and the last column of Ta-
cle residues as a function of Hubbard U in CeMn Si and
2 2 ble I lists our results showing which particular orbitals
CeRu Si using the LDA+G approach.
2 2 exhibit the strongest mass enhancement.
ied by dHvA experiments81–83 demonstrating that the
f–electron character is changed from itinerant to local- IV. CONCLUSION
izedacrossthemetamagnetictransition. Allmembersof
thisfamilyofcompoundswereputintotheDoniach(T
N In conclusion, using a recently proposed LDA+G ap-
vs J ) phase diagram84. Later, using the LDA+DMFT
K proach we have studied quasiparticle mass renormaliza-
scheme, the phase diagram was renewed28.
tions in several classes of Ce heavy fermion compounds.
Results of our applications to Ce 122s are presented
We find that the calculation gives correct trends across
in Table 1. They assume a paramagnetic heavy fermion
various systems as compared to the measured Sommer-
state for all systems and the experimental γ(cid:48)s are ex-
feld coefficient and reproduces the order of magnitude of
tracted from the specific heat data measured above
the experimental value. We also uncover an interesting
the temperatures of antiferromagnetic/superconducting
orbital dependency of the quasiparticle residues for each
transition. Forthe122systemswith3delementssuchas
studiedcompoundwhichprovidesanimportantphysical
Mn,Fe,Co,Ni, the LDA+G procedure returns only mod-
insightonhowcorrelationsaffecttheeffectivedegeneracy
eratelyenhancedelectronmasseswhichwefindinagree-
of Cerium f–electrons placed in various crystallographic
ment with the experiment. An example of the depen-
environments.
dence of z vs U is shown on Fig. 4 for Mn based 122,
α
where the Coulomb renormalizing spin–orbit splitting is
essential to reduce the orbital degeneracy from 14 to 6 Acknowledgements
when using the crystal field averaged double counting,
V(2). This is similar to our findings in Ce and Ce–115 TheauthorsaregratefultoY.–F.Yangforusefulcom-
DC
systems. Our calculation for Cu based 122 shows that ments. ThisworkwassupportedbyUSDOENuclearEn-
its quasiparticle residues are very sensitive to the values ergy University Program under Contract No. 00088708.
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