Critical test for Altshuler-Aronov theory: Evolution of the density of
  states singularity in double perovskite Sr$_2$FeMoO$_6$ with controlled
  disorder
Critical test for Altshuler-Aronov theory: Evolution of the density of states
singularity in double perovskite Sr2FeMoO6 with controlled disorder
M. Kobayashi,1 K. Tanaka,2 A. Fujimori,1 Sugata Ray,3, 4 and D. D. Sarma4, 5
Department of Physics and Department of Complexity Science and Engineering,
University of Tokyo, Kashiwa, Chiba 277-8561, Japan
Department of Applied Physics and Stanford Synchrotron Radiation Laboratory,
Stanford University, Stanford, California 94305, USA
Materials and Structures Laboratory, Tokyo Institute of Technology, Midori, Yokohama 226-8503, Japan
Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India
Centra for Advanced Materials, Indian Association for the Cultivation of Science, Kolkata 700032, India
(Dated: October 31, 2018)
With high-resolution photoemission spectroscopy measurements, the density of states (DOS) near
the Fermi level (EF) of double perovskite Sr2FeMoO6 having different degrees of Fe/Mo antisite
disorder has been investigated with varying temperature. The DOS near EF showed a systematic
depletion with increasing degree of disorder, and recovered with increasing temperature. Altshuler-
Aronov (AA) theory of disordered metals well explains the dependences of the experimental results.
Scaling analysis of the spectra provides experimental indication for the functional form of the AA
DOS singularity.
PACS numbers: 71.20.-b, 71.23.-k, 71.27.+a, 79.60.-i
Disordered electronic systems, which have random po-
tentials deviating from an ideal crystal, have been inves-
tigated from both fundamental and application points of
view [1]. Ever since the finding of filling-control metal-
insulator transitions (MIT) in transition-metal oxides
known as strongly correlated system, disorder has at-
tracted even more attention because not only electron-
electron interaction but also disorder are supposed to
play fundamentally important roles in the MIT. Altshuler
and Aronov [2] studied the effect of electron-electron in-
teraction in a disordered metallic medium, and predicted
that the density of states (DOS) near the Fermi level
(EF) shows a singularity of |E −EF|1/2 and the DOS at
EF increases with increasing temperature in proportion
T . The theory has been applied to the low tempera-
ture conductivity of disordered metals such as disordered
Au and Ag films [3], amorphous alloy Ge1−xAux [4], and
transition metal chalcogenide Ni(S,Se)2 [5].
In a previous work, Sarma et al. [6] have reported
photoemission (PES) measurements on B-site disordered
perovskites LaNi1−xMxO3 (M=Mn and Fe), which show
MIT as a function of x, and shown that the disor-
der affects the DOS near EF in such a way that had
been theoretically predicted by Altshuler and Aronov
[2]. In a similar B-site substituted transition-metal ox-
ide SrRu1−xTixO3, which demonstrates MIT at x ∼ 0.3
(SrRuO3 is metallic), the depletion of the DOS near EF
has shown an unusual |E − EF|1.2 dependence in both
metallic and insulating phases [7]. In addition, although
it is believed that a disorder-induced insulator shows
a soft Coulomb gap characterized by a (E − EF)2 de-
pendence of the DOS near EF [8, 9], the unexpected
|E − EF|3/2 dependence of the DOS near EF related
to charge density wave has been observed in insulating
BaIrO3 [10]. It is considered that fine structure in the
DOS in the vicinity of EF is sensitive to both the degrees
of disorder and electron correlation, and therefore experi-
mental confirmation of a basic theory for disordered elec-
tronic system such as the Altshuler-Aronov (AA) theory
is necessary for understanding of the DOS singularity.
While AA theory makes specific predictions about both
E and T dependences, photoelectron spectroscopy has
been used only to probe the E dependence with abso-
lutely no reference to the T dependence. Therefore, de-
tailed high-resolution temperature-dependent PES mea-
surements are also highly desired to verify the AA theory.
The present paper reports on high-resolution PES
experiments on the B-site ordered double perovskite
Sr2FeMoO6 (SFMO), where we have controlled the de-
gree of Fe/Mo antisite disorder (AD) in the sample prepa-
ration procedure. Through detailed analysis for the tem-
perature and degree of disorder dependences of the PES
spectra near EF, the results provide experimental confir-
mation of the AA theory of disordered metals. SFMO
has been investigated intensively due to the theoreti-
cal prediction of half-metallic nature and the observa-
tion of large magnetoresistance under low magnetic fields
at room temperature [11]. In this system, there are
characteristic defects known as Fe/Mo AD at the B-
site, which remarkably affects the physical properties of
SFMO [12, 13, 14, 15, 16]. By controlling the degree of
AD, one can investigate disorder effects in the metal with-
out changing the chemical composition and other condi-
tions.
Polycrystalline SFMO samples having different degrees
of Fe/Mo AD prepared as follows; First, the samples with
the highest degree of AD 45% were prepared. Then they
were annealed at 1173 K, 1673 K, and 1523 K for a pe-
riod of 5 hours under 2% H2/Ar to obtain the degrees of
AD 40%, 25%, and 10%, respectively (SFMO having AD
50% is the same as ordinary perovskite SrFe0.5Mo0.5O3).
Details of the sample preparation are given in Ref. [12]
http://arxiv.org/abs/0704.1688v2
2.0 1.5 1.0 0.5 0
Binding Energy (eV)
0.4 0.2 0.0
1.5 1.0 0.5 0.0
Binding Energy (eV)
0.5 0.2
Fe eg↑ Fe t2g↓-Mo t2g↓
AD 40%
T = 10 K
 fractured
 scraped
 AD 40%
 AD 10%
 background
Fe t2g↓-
 Mo t2g↓
Fe eg↑
FIG. 1: Valence-band photoemission spectra of Sr2FeMoO6
taken at 10 K with He-I radiation. (a) Comparison between
the spectra taken from the fractured and scraped surfaces.
The spectra have been normalized to the area from 0 eV to
0.8 eV (Fe t2g↓-Mo t2g↓ states). (b) Comparison between
different degrees of antisite disorder (AD) 10% and 40%. The
spectra have been normalized to the area from 0.8 eV to 2
eV (Fe eg↑ states). Top: Background subtraction. The insets
show enlarged plots near EF.
and [13]. Using x-ray diffraction, the degree of disorder
was quantified from the intensity of a supercell-reflection
peak. Scanning electron microscopy in conjunction with
energy dispersive x-ray analysis revealed no change in
composition during the annealing. Transport measure-
ments were performed on the AD 10% and 40% sam-
ples by a standard four prove technique using a Physi-
cal Property Measurement System (Quantum Design Co.
Ltd). PES spectra were recorded using a spectrometer
equipped with a monochromatized He resonance lamp
(hν = 21.2 eV), where photoelectrons were collected with
a Gammadata Scienta SES-100 hemispherical analyzer in
the angle integrated mode. The total resolution of the
spectrometer was ∼10 meV, and the base pressure was
1.0×10−8 Pa. Clean surfaces were obtained by repeated
scraping in situ with a diamond file. The position of EF
was determined by measuring PES spectra of evaporated
gold which was electrically in contact with the sample.
The line shapes of the PES spectra obtained were al-
most the same as those in a previous report on single
crystalline SFMO taken with synchrotron radiation [17].
In order to examine the influence of surface treatment,
we measured valence-band spectra taken from fractured
and scraped surfaces. In the binding-energy (EB) range
from 2 eV to 10 eV, there were appreciable differences
such as the sharpness of structures in the O 2p and the
Fe t2g↑ bands (not shown), the background intensity, and
to some extent the Fe eg↑ band. On the other hand,
within ∼ 1 eV of EF, i.e., in the Fe t2g↓ +Mo t2g↓ conduc-
tion bands, the line shapes of the fractured and scraped
samples were similar to each other as shown in Fig. 1(a).
0.3 0.2 0.1 0
Binding Energy (eV)
0.01 0.00 -0.01
0.3 0.2 0.1 0.0 -0.1
Binding Energy (eV)
0.01 0.00 -0.01
AD 10%
AD 25%
AD 40%
AD 45%
T= 10 K
T= 100 K
T= 200 K
T= 300 K
(a) T=10 K (b)
AD 10%
Au    8 K
150 K
300 K
FIG. 2: Photoemission spectra of Sr2FeMoO6 near the Fermi
level. The spectra have been normalized to the area from
EB = 0.3 eV to 0.6 eV. (a) Degree of Fe/Mo AD dependence
at 10 K. (b) Temperature dependence of the AD 10% sample.
As a reference, Au spectra are also shown. The insets show
an enlarged plot in the vicinity of EF.
Since in previous PES studies it has been reported that
LDA+U calculation well explains the valence-band spec-
tra taken from the fractured surface [17, 18], we consider
that the different surface treatments have not affected
the spectra near EF which reflect the bulk properties.
In contrast, the spectra are intensively influenced by AD
and temperature as we shall see below.
Figure 1(b) shows valence-band spectra for different
degrees of disorder, i.e., AD 40% and 10%. Although
the peak due to the localized Fe eg↑ states was nearly
identical between AD 10% and AD 40%, there was a
clear difference in the Fe t2g↓-Mo t2g↓ conduction band
between the two spectra. This result suggests that the
disorder influences the DOS near EF. Figure 2 shows the
temperature and degree of disorder dependences of the
spectra near EF normalized to the area in the region from
EB = 0.3 eV to 0.6 eV, in which the spectra were iden-
tical and independent of the degree of AD as shown in
Fig. 1(b). For a fixed temperature, the intensity of the
spectra near EF was depleted with degree of disorder.
This behavior is consistent with the previous report on
a disordered metal system LaNi1−xMnxO3 [6]. Indeed,
temperature dependent resistivity measurements on the
AD 40% sample showed a minimum around 40 K, i.e., the
resistivity increased with decreasing temperature below
40 K, while on the AD 10% one there was no minimum
till the lowest measured temperature (20 K). The obser-
vations are consistent with previous reports of transport
measurements on SFMO having various degrees of dis-
order [16, 19] and on SFMO with high Fe/Mo ordering
[13, 20]. For a fixed degree of disorder, the intensity at
EF increased with temperature as shown in Fig. 2(b).
3456789
23456789
√X  (=√|²|/kBT )
AD 45%   10K, 100K,
200K, 300K
AD 40%   10K, 100K,
200K, 300K
AD 25%   10K, 100K,
200K, 300K
AD 10%   10K, 100K,
200K, 300K
j(X) = [X
+(1.07)
T=300 K
100 K
FIG. 3: Scaling analysis for the spectral depletion. Scaled
spectra as a function of
X (X = |ǫ|/kBT ) are plotted on
a bilogarithmic scale. An analytical form of ϕ = [X2 +
(1.07)4]1/4 is also plotted.
This behavior indicates that SFMO differs from normal
metals such as Au in which PES spectra at various tem-
peratures have temperature-independent intensity at EF
and intersect at EF irrespective of temperature as shown
in Fig 2(b), representing the simple Fermi-Dirac distri-
bution function.
Now, we analyze the temperature and degree of disor-
der dependences of the spectra based on the theory for
disordered metal suggested by Altshuler and Aronov [2].
The theory predicted that electron-electron interaction
accompanied by impurity scattering leads to an anomaly
in the DOS around EF and the resulting singular part of
the DOS δD is given by
δD(ǫ)
(~D)3/2
ϕ (|ǫ|/kBT ) ,
ϕ (X) =
X (X ≫ 1)
1.07 +O(X2) (X ≪ 1)
where D0 is the original DOS at EF, ǫ is the energy mea-
sured from EF, H is the inverse Debye radius, and D is
the diffusion coefficient due to impurity scattering.
In order to see whether experimental spectra satisfy
Eq. (1), we have made the following scaling analysis.
According to Eq. (1), the PES spectra I(ǫ, T ) near EF
should be proportional to D′0+ δD(ǫ), where D′0 is a con-
stant and D′0 < D0. Therefore, I(ǫ, T ) can be parame-
terized as
I(ǫ, T ) = A+G
kBT ϕ
, (2)
0.3 0.2 0.1 0 -0.1
Binding Energy (eV)
3002001000
Temperature (K)
403020100
Disorder (%)
a + g [²
+(1.07)
 Fitted
AD 10%
AD 25%
AD 40%
AD 45%
AD 45%
(b) (c)
300 K
200 K
100 K
FIG. 4: Results of fitting of the photoemission spectra of
Sr2FeMoO6 to the analytical curves of Altshuler-Aronov the-
ory. (a) Fitted spectra for the AD 45% sample at various
temperatures. (b) Degree of disorder dependence of the val-
ues of γ/α. G/A is also plotted. The dotted line is a guide
to the eye. (c) Temperature dependence of the DOS at EF.
Solid curves demonstrate fitted curves using Eq. (4) with pa-
rameters deduced from the spectral line-shape fitting of panel
where A and G are only dependent on the degree of dis-
order. (I − A)/G
kBT plotted against X = |ǫ|/kBT
should fall onto the same curve if Eq. (1) is valid, and
can be used to evaluate the functional form of ϕ if A and
G are chosen to satisfy the condition that ϕ(0) → 1.07
as ǫ/kBT → 0. In Fig. 3, the results are plotted on a log-
arithmic scale, where the PES spectra have been divided
by the Fermi-Dirac function convoluted with the experi-
mental resolution estimated from the Au spectra. Notice
that all the low energy part of the spectra fell onto the
same curve, which we attribute to the scaling function
ϕ(X). Deviation from the scaling function ϕ(X) occurs
at high energies where the original DOS starts to deviate
from the constant one. We find that ϕ(X) approaches√
X for X > 1, corresponding to AA theory. The results
ensure the validness of analysis based on AA theory for
the depletion of PES spectra near EF.
In order to analyze the spectra using Eq. (1), we pro-
pose an analytical form of ϕ(X) = [X2+(1.07)4]1/4 inter-
polating both the limits of large and small X of Eq. (1).
This form is shown to accurately reproduce the experi-
mentally scaling function ϕ(X) deduced above as shown
in Fig. 3. Therefore, we employ a model function
{α+ γ [ǫ2 + (1.07)4(kBT )2]1/4} f(ǫ), (3)
where f(ǫ) is the Fermi-Dirac function, α and γ are fit-
ting parameters. γ/α depends only on degree of disorder
and are independent of a way of intensity normalization.
Figure 4(a) shows fitted results for the spectra of the
AD 45% sample at various temperatures. The fitting
function given by Eq. (3) well reproduced the spectra
near EF, where the fitted ranges were chosen to the valid
range of the scaling function ϕ(X) as shown in Fig. 3 [21].
Figure 4(b) shows the values of the coefficients which rep-
resent the strength of the DOS singularity as a function
of disorder. The γ/α value is independent of temperature
and approximately linearly increases with degree of disor-
der as shown in Fig. 4(b), indicating that the DOS singu-
larity near EF is enhanced with increasing degree of dis-
order as predicted by AA theory. The constant G will be
relative to the value of γ. Actually, the G/A value shows
the same dependence as γ/α [Fig. 4(b)]. Equation (3) as
a function of X without f , i.e., α + γ[X2 + (1.07)4]1/4,
well reproduced the line shape of the depletion as shown
in Fig. 3, where the parameters were chosen α = 0 and
γ = 1 to correspond with the analytical ϕ(X) [22]. The
result demonstrates validness of our assumption for the
functional form of ϕ given by Eq. (3).
Equation (1) indicates that the singular contribution
to the DOS δD/D0 at EF is proportional to
T . In or-
der to study the temperature dependence of the DOS at
EF, comparison was made between the experimental and
theoretical δD/D0 at EF. The PES intensity (or DOS)
at EF is plotted as a function of temperature in Fig. 4(c).
For a fixed temperature, the DOS at EF increased with
decreasing degree of disorder. For a fixed degree of dis-
order, the DOS at EF increased with increasing temper-
ature. According to Eq. (1), temperature dependence of
the DOS at EF is expressed as
α+ 1.07 γ
kBT , (4)
corresponding Eq. (3) at ǫ = 0. Using the values of γ/α
obtained by fitting to the PES spectra, Eq. (4) well re-
produces the temperature-dependence of the DOS at EF
as shown in Fig. 4(c). The result is consistent with theo-
retically predicted temperature dependence of δD/D0 at
EF. It follows from the arguments described above that
AA theory applies to not only the disorder dependent
depletion near EF but also the temperature dependent
DOS at EF.
As mentioned above, AA theory treats the scattering
processes on general ground and does not depend on the
functional form of the potential of the scattering center.
This theory can be applied to the case that the mean free
path of itinerant electrons is larger than or comparable
with its wave length.
In conclusion, we have performed high-resolution pho-
toemission experiments on polycrystalline Sr2FeMoO6
samples having different degrees of Fe/Mo antisite dis-
order. The photoemission spectra near the Fermi level
depended on degree of the Fe/Mo antisite disorder as
well as on temperature. The Altshuler-Aronov theory on
disordered metal well explained both the dependences.
Scaling analysis for the spectral depletion clarifies the
functional form of the density of states singularity near
the Fermi level. We believe that the findings will provide
an indicator for degrees of disorder and electron-electron
correlation, and promote spectral analysis for the index
of the density of states depletion near the Fermi level.
The present results point to a need for taking into ac-
count both electron-electron interaction and disorder ef-
fects for an understanding of the electronic structure of
metallic correlated electron system.
The authors thank H. Yagi and M. Hashimoto for help
in experiments. This work was supported by a Grant-
in-Aid for Scientific Research in Priority Area “Inven-
tion of Anomalous Quantum Materials” (16076208) from
MEXT, Japan. D.D.S. thanks DST and BRNS for fund-
ing this research. S.R. thanks JSPS postdoctoral fellow-
ship for foreign researchers. M.K. acknowledges support
from the Japan Society for the Promotion of Science for
Young Scientists.
[1] P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57,
287 (1985).
[2] B. L. Altshuler and A. G. Aronov, Solid State Commun.
30, 115 (1979).
[3] S. Schmitz and S. Ewert, Solid State Commun. 74, 1067
(1990).
[4] W. L. McMillan and J. Mochel, Phys. Rev. Lett. 46, 556
(1981).
[5] A. Husmann, D. S. Jin, Y. V. Zastavker, T. F. Rosen-
baum, X. Yao, and J. M. Honig, Science 274, 1874
(1996).
[6] D. D. Sarma, A. Chainani, S. R. Krishnakumar, E.
Vescovo, C. Carbone, W. Eberhardt, O. Rader, C. Jung,
C. Hellwing, W. Gudat, H. Srikanth, and A. K. Ray-
chaudhuri, Phys. Rev. Lett. 80, 4004 (1998).
[7] J. Kim, J.-Y. Kim, B.-G. Park, and S.-J. Oh, Phys. Rev.
B 73, 235109 (2006).
[8] A. F. Efros and B. I. Shklovskii, J. Phys. C 8, L49 (1975).
[9] J. G. Massey and M. Lee, Phys. Rev. Lett. 75, 4266
(1995).
[10] K. Maiti, R. S. Singh, V. R. R. Medicherla, S. Rayaprol,
and E. V. Sampathkumaran, Phys. Rev. Lett. 95, 016404
(2005).
[11] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura,
and Y. Tokura, Nature 395, 677 (1998).
[12] D. D. Sarma, E. V. Sampathkumaran, S. Ray, R. Natara-
jan, S. Majumdar, A. Kimar, G. Nalini, and T. N. G.
Row, Solid State Commun. 114, 465 (2000).
[13] D. D. Sarma, Sugata Ray, K. Tanaka, and A. Fujimori,
PRL in press.
[14] J. Navarro, J. Nogués, J. S. Muñoz, and J. Fontcuberta,
Phys. Rev. B 67, 174416 (2003).
[15] B. J. Park, H. Han, J. Kim, Y. J. Kim, C. S. Kim, and
B. W. Lee, J. Magn. Magn. Mater. 272-276, 1851 (2004).
[16] Y. H. Huang, M. Karppinen, H. Yamauchi, and J. B.
Goodenough, Phys. Rev. B 73, 104408 (2006).
[17] T. Saitoh, M. Nakatake, A. Kakizaki, H. Nakajima, O.
Morimoto, S. Xu, Y. Moritomo, N. Hamada, and Y.
Aiura, Phys. Rev. B 66, 035112 (2002).
[18] J.-S. Kang, J. H. Kim, A. Sekiyama, S. Kasai, S. Suga,
S. W. Han, K. H. Kim, T. Muro, Y. Saitoh, C. Hwang,
C. G. Olson, B. J. Park, B. W. Lee, J. H. Shin, J. H.
Park, and B. I. Min, Phys. Rev. B 66, 113105 (2002).
[19] Y.-H. Huang, H. Yamauchi, and M. Karppinen, Phys.
Rev. B 74, 174418 (2006).
[20] Y. Tomioka, T. Okuda, Y. Okimoto, R. Kumai, K.-I.
Kobayashi, and Y. Tokura, Phys. Rev. B 61, 422 (2000).
[21] The starting points are about 76 meV, 109 meV, 129
meV, and 145 meV for 10 K, 100 K, 200 K, and 300 K,
respectively.
[22] The analytical form of ϕ(X) can be expressed as more
general formula: ϕn(X) = [X
2n + (1.07)4n]1/4n, which
also satisfies both the limits ofX. Using this function, the
scaling function has been fitted well as n = 1.09 ± 0.14.
This result emphasizes the validness of the analytical
ϕ(X) because the exponent n is nearly 1.
