PHYSICAL REVIEW B 87, 165141 (2013)

Strongly hybridized electronic structure of YbAl2 : An angle-resolved photoemission study
M. Matsunami,1,2,* T. Hajiri,1 H. Miyazaki,1,† M. Kosaka,3 and S. Kimura1,2
1

2

UVSOR Facility, Institute for Molecular Science, Okazaki 444-8585, Japan
School of Physical Sciences, The Graduate University for Advanced Studies (SOKENDAI), Okazaki 444-8585, Japan
3
Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan(Received 26 October 2012; published 30 April 2013)
We report the electronic structure of a prototypical valence fluctuation system, YbAl2 , using angle-resolved
photoemission spectroscopy. The observed band dispersions and Fermi surfaces are well described in termsof band structure calculations based on local density approximation. Strong hybridization between the
conduction and 4f bands is identified on the basis of the periodic Anderson model. The evaluated small mass
enhancement factor and the high Kondo temperature qualitatively agree with those obtained from thermodynamic
measurements. Such findings suggest that the strong hybridization suppresses band renormalization and is
responsible for the valence fluctuations in YbAl2 .
DOI: 10.1103/PhysRevB.87.165141

PACS number(s): 71.27.+a, 71.20.Eh, 75.30.Mb, 79.60.−i

I. INTRODUCTION

Heavy-fermion or valence fluctuation systems are characterized by the strong correlations of f electrons, resulting
in an enhanced effective mass of quasiparticles due to theband renormalization effect.1 Consequently, the band structure
and Fermi surface (FS) topology are modified from thosepredicted in band structure calculations based on local density
approximation (LDA). To estimate the band renormalization
effect, a direct comparison of the experimental band structure
as well as the FS with the calculations is necessary. An
angle-resolved photoemission spectroscopy (ARPES), which
can probe both band structure and FS, is the most suitable tool
for this purpose.
Another important concept in heavy-fermion systems is ahybridization effect between conduction (c) and f electrons.
The c-f hybridization-derived electronic structure has beenverified by ARPES studies.2–13 However, most of those studies
have been performed on the basis of the surface Brillouinzone (BZ)2–6 or by using the 4d-4f resonant process7–9 in
which kz (the momentum normal to surface) dependence on
the electronic structure is neglected owing to its quasi-twodimensionality. In general, ARPES along the bulk BZ withwell-defined kz should be ideal for the three-dimensional
materials. Furthermore, ARPES data acquired at the highsymmetry point in the bulk BZ can be closely comparedwith the band structure calculations. In the case of Ce-based
compounds, however, resonant photoemission is intrinsicallyrequired to directly probe the f -derived feature in the bonding
band of the c-f hybridized bands because of the small
f -electron occupation number (nf  1). On the other hand, inYb-based compounds, the large occupation number (nf  13)
enables us to probe the dispersions of not only the bonding
band but also the antibonding band without any resonant
process, and hence the photon-energy (hν) tuning of kz into thehigh-symmetry point is available. The Kondo resonance peakalso exists on the occupied side, in contrast to Ce compounds,
and hence is accessible with ARPES.
Here, we focus on YbAl2 , which is known as one of the
prototypical valence fluctuation systems. In a recent hard x-ray
photoemission spectroscopy study, the mean valence of Ybions was estimated as + 2.2 below 300 K.14 Correspondingly,
an extremely high Kondo temperature (TK ) exceeding 2000 K
1098-0121/2013/87(16)/165141(5)has been suggested by the magnetic susceptibility15 or the
inelastic neutron scattering.16 The mass enhancement factor
is particularly small among heavy-fermion systems known so
far,17 implying the small renormalization effect. Thus YbAl2
can be a suitable system to investigate the applicable limit of
LDA calculations in relation with the fluctuating valence forthe heavy-fermion systems.
In this paper, we report the results of ARPES performed on
YbAl2 . The band structure and FSs obtained are comparable
to those of the LDA calculation. The c-f hybridized band
dispersions near the Fermi level (EF ) are clearly observed andanalyzed on the basis of a periodic Anderson model (PAM)
without interaction. The estimated mass enhancement factor
and TK are consistent with the other experiments. These results
suggest that the strong c-f hybridization suppresses the band
renormalization and hence the electron correlation effect ofYb 4f states does not work significantly in YbAl2 , even thoughrare-earth compounds are generally believed to have a strong
correlation effect.
II. METHODSSingle crystals of YbAl2 were grown by the lithium flux
method.18 The ARPES experiment was carried out at theundulator beamline BL7U of UVSOR-II in the Institutefor Molecular Science, with hν = 16–29 eV and using ahemispherical electron analyzer (MBS A-1, MB Scientific).19
The total energy resolution and the measurement temperature
were set to 20 meV and 12 K, respectively. The vacuum wasmaintained below 5×10−9 Pa during the experiment. The crystal orientation was determined by back-reflection Laue x-ray
diffraction prior to the ARPES measurements. Clean sample
surfaces for ARPES measurements were obtained by cleavingin situ at 12 K along the (111) plane. The LDA calculation
was performed by the full-potential linearized-augmentedplane-wave plus local-orbital (LAPW + lo) method including
spin-orbit coupling implemented in the WEIN2k code.20
III. RESULTS

Figure 1(a) shows the BZ of YbAl2 in which kz corresponds
to the (111) direction of fcc BZ. The FS in the kx -kz plane is
obtained by the hν-dependent ARPES, as shown in Fig. 1(b).

165141-1©2013 American Physical Society

MATSUNAMI, HAJIRI, MIYAZAKI, KOSAKA, AND KIMURA
3.5

L

W

-1

kz

(a)

kz (Å )

(b)

W

L
X
kx

3.0

K
2.5
2.0
-1.0

U

K

ky

0.5

-0.5

0.0
M’

0.5

20 eV
16 eV
1.0

(c)

-1

ky (Å )

h = 29 eV

M’

K

0.0

h = 20 eV

-0.5
-1.0

-0.5

0.0
0.5
-1
kx (Å )

1.0

FIG. 1. (Color online) (a) BZ of YbAl2 , in which kz corresponds
to the (111) direction of the fcc BZ. In the kx -ky plane, the midpoint of
the zone edge (K) is referred to as M  point, which is also equivalent
to the midpoint between L and U points. (b) and (c) FS mappingin the kx -kz and kx -ky planes, respectively. The FS is obtained by aplot of the integrated intensity within EF ± 50 meV in the ARPES
spectra. The color scale is shown at the right of each map. The FSs
obtained by the LDA calculation are overlapped in (b) and (c).

The inner potential is determined to be 13.7 eV from thesymmetry of the FS-image contrast. Figure 1(c) shows the“in-plane” FS in the kx -ky plane, measured at hν = 20 eV.
The FSs obtained by the LDA calculation are also plottedin Figs. 1(b) and 1(c). Both inner and outer FSs around the point, as predicted in the LDA calculation, are observed inboth the kx -kz and kx -ky planes. A further comparison between
the experimental results and calculations is given below on thebasis of the band structure.
Figures 2(a) and 2(b) show an intensity plot of the
ARPES spectra taken along the -K direction and theirangle-integrated spectrum, respectively, measured at hν =

0.0

(a)

(b)

Binding Energy (eV)

4f7/2
0.5

1.0

Bulk

1.5

Fit

SS
4f7/2
S

4f5/2

YbAl2

PHYSICAL REVIEW B 87, 165141 (2013)20 eV. Both the widely dispersive c bands and the nearly
dispersionless f bands are clearly visible in the ARPES image.
Regarding the f bands, which are of the Yb2+ component
corresponding to the 4f 13 final state, the main features at thebinding energy (EB ) of 0.17 and 1.49 eV are attributed tothe spin-orbit split components 4f7/2 and 4f5/2 , respectively,in the bulk. The 4f7/2 peak was fitted by a Lorentzian
multiplied by a Fermi-Dirac distribution (FDD) and an integral
background, as shown by the thin solid line in Fig. 2(b). The
peak position and width obtained were 0.171 and 0.155 eV,
respectively, which are consistent with those in the hard andsoft x-ray photoemission spectra.14 Additional fine structures
can be seen at EB ∼ 1.09 and 1.25 eV, which might be derived
from the subsurface and surface 4f7/2 states, respectively.21
Such nonbulk components have little impact on the bulk 4f7/2
band, which is discussed in more detail in the following.
Figures 3(a)–3(d) show the series of ARPES spectra and the
ARPES intensity plot near EF along the -K direction and the
direction from  to the midpoint of the zone edge (K), hereafter, referred to as M  point. The band dispersions obtained by
the LDA calculation are also plotted in Figs. 3(b) and 3(d). The
observed Yb 4f7/2 band around EB = 0.17 eV shows good
agreement of energy with the calculation without any shift.
The degree of coincidence is much greater than that of theother nearly divalent Yb system YbCu2 Ge2 .13 To emphasize
the c-band dispersions, each energy distribution curve (EDC)
in the ARPES image was normalized to the angle-integrated
spectrum. Alternatively, the same results can be achieved bynormalization for each momentum distribution curve (MDC)
area. The ARPES spectra and image obtained, shown in
Figs. 3(e)–3(h), can be regarded as “effective off-resonance”
ARPES spectra and image, representing hybridized-band
dispersions. It should be noted that, unlike purely divalentsystems,22 the existence of these hybridized bands is responsible for the valence fluctuations in YbAl2 . Their overall feature
is well described in terms of the LDA calculation. For the -K
direction, among the antibonding bands located in the lower
EB region compared with the 4f7/2 band, the innermost band
matches very well. On the other hand, the experimental outerband has a slightly weaker dispersion. This band might not be
separated from the more outer band, which does not cross EF
as shown in the LDA calculation. A similar slight deviationfrom the LDA calculation is recognized in the bonding bands
shown in Fig. 3(f). For the -M  direction, the experimental
inner and outer bands are highly mixed. The “mixed” band islocated on the inside of the calculated one and has almost thesame Fermi vector (kF ) as the inner band in the -K direction.
This finding suggests that the inner FS is more circular thanthat of the LDA calculation, as expected from Fig. 1(c).

h = 20 eV
T = 12 K

2.0
-0.5

0.0
-1
kx (Å )

0.5

IV. ANALYSIS AND DISCUSSION

Intensity(arb. units)

FIG. 2. (Color online) (a) Intensity plot of the ARPES spectraof YbAl2 taken along the -K direction. The color scale is shownat the bottom left. (b) Angle-integrated spectrum obtained from (a).
The 4f7/2 peak is fitted by a Lorentzian multiplied by an FDD and an
integral background. “S” and “SS” denote the surface and subsurface
states, respectively.

In order to quantify the impact of the c-f hybridization
effect apart from the LDA calculation, the observed band
dispersions were analyzed in terms of PAM. In the noninteracting case with Uff = 0, the band dispersion of PAM
is given by the form of hybridized bands asεk + Ef,k ± (εk − Ef,k )2 + 4Vk2
,
(1)
Ek± =
2

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STRONGLY HYBRIDIZED ELECTRONIC STRUCTURE OF . . .

PHYSICAL REVIEW B 87, 165141 (2013)

0.0

(e)

(a)

(b)

0.2

0.6
0.4
0.2
0.0
Binding Energy (eV)

0.6
0.4
0.2
0.0
Binding Energy (eV)

(b)

(f)

0.0

PAM
Mixed

PAM
Inner
Outer
0.4
+kx
–kx

0.6

+ky
–ky

0.16

4f7/2

Integrated

0.2
0.18

0.4

-0.5

0.0
-1
kx (Å )

0.5

-0.5

0.0
-1
kx (Å )

Intensity (arb. units)

(g)

0.6
0.4
0.2
0.0
Binding Energy (eV)

0.6
0.4
0.2
0.0
Binding Energy (eV)

(d)

(h)

0.0

0.6
0.5

-0.5

0.0
-1
ky (Å )

0.8 0

0.2

0.4
-1
|ky| (Å )m∗∂ 2 εk  ∂ 2 Ek+
=
,
mb
∂k 2
∂k 2

0.4

0.0
-1
ky (Å )

(d)

0.4
0.6
-1
|kx| (Å )0.6

The mass enhancement is simply related to the dispersion of
Ek+ and εk at each kF by the relation

0.2

-0.5

0.2

Integrated

FIG. 4. (Color online) (a) and (b) Analysis of band dispersions
along the -K and -M  directions, respectively, by c-f hybridized
bands based on PAM. The dotted lines indicate the c bands andf states before hybridization. The experimental data points wereobtained by a Lorentzian fitting of MDC or EDC. (c) and (d)Expansion of dispersion for the 4f7/2 band. The angle-integrated
data were obtained by the fitting of the angle-integrated spectra.

0.5-M’ direction
(c)

4f7/2

(c)
0

0.6

Binding Energy (eV)

M'

K

(a)

Binding Energy (eV)Binding Energy (eV)Intensity (arb. units)-K direction

0.5

FIG. 3. (Color online) (a) and (c) Series of ARPES spectra near
EF along the -K and -M  directions, respectively. (b) and (d)ARPES intensity plots of (a) and (c), respectively. (e)–(h) The samedata as in (a)–(d), normalized to the angle-integrated spectrum. Theband dispersions obtained by the LDA calculation are overlapped in(b), (d), (f), and (h).

where εk , Ef,k , and Vk are the c-band, f -band, and hybridization energies, respectively.1 As a simplification to fit
the experimental data using this formula, a parabolic electronband was assumed for εk and the k dependence of Ef,k andVk was neglected. The fitting results are shown in Figs. 4(a)and 4(b) in which the experimental data points were obtainedby a Lorentzian fitting of MDC or EDC and are plotted as
the absolute value of kx or ky . The band dispersions arewell reproduced by this simple model with the parameterssummarized in Table I. From the value of Ef , TK is evaluatedas 2540, 2030, and 2400 K for the inner, outer, and mixed
bands, respectively. In this case, the lowest TK is meaningful
and consistent with that obtained by other experiments.15,16(2)

where mb is the unhybridized band mass. For each band, themass enhancement factor m∗ /mb is evaluated as listed inTable I. Since the electronic specific heat coefficient γ of
YbAl2 has been reported as ∼10–17 mJ/K2 mol,18,23 andthat of LuAl2 as a reference is 5.7 mJ/K2 mol,24 the ratio,
γ (YbAl2 )/γ (LuAl2 ) = 1.8–3, is comparable to the mass
enhancement obtained in our ARPES study (m∗ /mb = 1.42–
2.79). It is quite reasonable that the outer band, with weak
dispersion compared with the LDA calculation, has a heavier
mass and a weaker hybridization.
The mass enhancement observed in YbAl2 can be describedin terms of a c-f hybridization effect. On the other hand,
the renormalization effect due to the electron correlation Uff
cannot contribute significantly. Even if the parameters obtainedby this PAM analysis include a renormalization effect insome manner such as Ẽf and Ṽ , the impact would be verysmall, as expected from the degree of mass enhancement.
More importantly, the coincidence of the 4f7/2 band position
with the LDA calculation strongly suggests an extremely
small renormalization. It is also consistent with the small
TABLE I. Parameters Ef and V used to fit the band dispersions in
Figs. 4(a) and 4(b) via Eq. (1), and the evaluated mass enhancement
factor m∗ /mb .
DirectionBandEf (eV)V (eV)m∗ /mb

-K
-K
-M 

Inner
Outer
Mixed

0.219
0.175
0.207

0.272
0.186
0.263

1.42
2.79
1.51

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MATSUNAMI, HAJIRI, MIYAZAKI, KOSAKA, AND KIMURAPHYSICAL REVIEW B 87, 165141 (2013)

renormalization factor suggested in the optical conductivityof YbAl2 .25 In the overall valence band, the localized f partat EB = 5–12 eV corresponding to the 4f 12 final state, which
is separated by Uff from the unoccupied 4f 14 final state, hasa very weak intensity as evidenced by the Yb valence.14,21
Therefore PAM analysis under the condition of Uff = 0 canbe useful for YbAl2 .
Finally, we consider the 4f7/2 peak in connection with a
Kondo resonance peak, not merely a shallow core level. As
shown in Figs. 4(c) and 4(d), this peak shows a small but
clear dispersion.26 Such behavior, which is also observed forthe strongly correlated YbRh2 Si2 ,4–6 is consistent with thetheoretical predictions on the basis of PAM.27 As a possible
point of view, therefore, this finding can be regarded as
an evidence for the dispersion of a resonance peak in thecoherent Kondo ground state of YbAl2 . On the other hand,theoretically, the “Kondo” resonance is defined only in the“Kondo lattice” regime (near Yb3+ state), not in the valencefluctuation (mixed valence) regime such as YbAl2 .31 The
relevant issues have been discussed on the basis of several
photoemission studies of YbAl3 , which has larger Yb valence
than that of YbAl2 .29,30 Indeed, the identification of the Kondo
effect, which represents a typical many-body phenomenon,1
may seem to be unreasonable for YbAl2 exhibiting an extremely small renormalization effect. In Yb-based compounds,
however, the valence fluctuation state with the energy scale of
TK can appear over a broad range of Yb valence between +2
and +3, based on the fact that the spin-orbit splitting of Yb 4f
electrons is generally larger than the 4f bandwidth.28 This
contrasts with the Ce case, in which the valence fluctuationis restricted near +3 since the Ce 4f spin-orbit splitting issmaller than the 4f bandwidth. In the case of YbAl2 , the Yb

valence is nearly divalent +2.2 as a consequence of the strong
c-f hybridization.14 With decreasing hybridization strength
toward Yb3+ , the resonance peak will move closer to EF andbecome sharper, providing a well-defined Kondo resonancepeak.31,32 Such an evolution of the Kondo resonance may
bring about the drastically modified FS topology and thehighly renormalized band structure compared with the LDA
calculation. In contrast, at the Yb2+ limit, the resonance peakturns into the shallow core level of the 4f7/2 states and does
not contribute to the FS.22 As a result, for YbAl2 , the coherent
Kondo state at much lower temperature than TK coexists
sufficiently well with the nearly unrenormalized band structure
to match with the LDA calculation.

*

8

†

matunami@ims.ac.jp
Present address: Center for Fostering Young and Innovative Researchers, Nagoya Institute of Technology, Nagoya 466-8555, Japan.
1
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V. CONCLUSION

In conclusion, we have performed ARPES of a prototypicalvalence fluctuation system, YbAl2 . The band dispersions
obtained around the  point, particularly for the energy
position of the Yb 4f7/2 band, are well described by the
LDA calculation without any modification. This may indicatethat the electron correlation of Yb 4f states does not work
significantly. A strongly c-f hybridized nature is identified
on the basis of noninteracting PAM. These results suggest awell-balanced coexistence between the coherent Kondo state
and the nearly unrenormalized band structure in YbAl2 .
ACKNOWLEDGMENTS

We would like to thank T. Ito for fruitful discussions. This
work was performed by the Use-of-UVSOR Facility Programof the Institute for Molecular Science (2011).

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Generally, the Kondo temperature TK is assumed as 3Tmax , where
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165141-4

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PHYSICAL REVIEW B 87, 165141 (2013)
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The 4f7/2 state has eightfold degeneracy. Our findings indicatethat the two or three orbitals among them are involved in the
hybridization and the residual orbitals remain nearly unhybridized
and dispersionless.
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