Fincher-Burke spin excitations and ω/T scaling in insulating La1.95Sr0.05CuO4
Wei Bao,1 Y. Chen,2, 3 K. Yamada,4 A. T. Savici,5 P. L. Russo,6 J. E. Lorenzo,7 and J.-H. Chung2, 3
Condensed Matter and Thermal Physics, Los Alamos National Laboratory, Los Alamos, NM 87545
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899
Dept. of Materials Science and Engineering, University of Maryland, College Park, MD 20742
Institute of Materials Research, Tohoku University, Sendai 980-8577, Japan
Brookhaven National Laboratory, Upton, NY 11973
TRIUMF, Vancouver, BC V6T 2A3, Canada
Institut Néel, CNRS, BP 166X, F-38043, Grenoble, France
(Dated: November 27, 2018)
Insulating La1.95Sr0.05CuO4 shares with superconducting cuprates the same Fincher-Burke spin
excitations, which usually are observed in itinerant antiferromagnets. The local spectral function
satisfies ω/T scaling above ∼16 K for this incommensurate insulating cuprate. Together with
previous results in commensurate insulating and incommensurate superconducting cuprates, these
results further support the general scaling prediction for square-lattice quantum spin S = 1/2
systems. The width of incommensurate peaks in La1.95Sr0.05CuO4 scales to a similar finite value
as at optimal doping, strongly suggesting that they are similarly distant from a quantum critical
point. They might both be limited to a finite correlation length by partial spin-glass freezing.
It is well known that the (π, π) peak of the Néel anti-
ferromagnetic order of the parent compounds is replaced
by a quartet of incommensurate peaks when cuprates are
sufficiently doped to have become superconducting1. The
very recent excitement is that spin excitations measured
in superconducting La2−xSrxCuO4 (LSCO) (x = 0.10
and 0.16)2, YBa2Cu3O6.6 (YBCO)
3, as well as in the
“stripe-ordered” La1.875Ba0.125CuO4 with a suppressed
4 share a common spectral feature: broad excitation
continua, originating from the incommensurate quartet,
disperse towards the (π, π) point with increasing en-
ergy at the rate of the spin-wave velocity of the par-
ent compounds. The spectrum is distinct from the
spin-waves in two important ways: 1) the excitations
are not resolution-limited; 2) there are no the outward
branches. Such a spin excitation spectrum previously
has been observed in itinerant spin-density-wave anti-
ferromagnets such as elemental metal Cr5 and strongly
correlated metal V2−yO3
6, and the single-lobed disper-
sive continuum is referred as the Fincher-Burke mode. A
self-consistent theory has been developed by Moriya and
others to describe the mode7, and quantitative agree-
ment has been achieved for three-dimensional itinerant
antiferromagnets6. Itinerant theories have also been de-
veloped to account for the Fincher-Burke-like modes in
superconducting cuprates8.
Sandwiched between the parent antiferromagnetic in-
sulator and the high-TC superconductor in the phase dia-
gram of LSCO is a distinct doping regime from x = 0.02
to 0.0559. Cuprates in the regime are insulators with-
out the long-range Néel order, and a spin-glass transi-
tion occurs at Tf . 10 K
10,11. This doping range is
often referred to as the spin-glass regime, although the
spin-glass phase extends to both lower and higher dop-
ings in the Néel and superconducting states. Magnetic
correlations were extensively investigated and regarded
as being commensurate, as in the parent compound12,13.
However, with improved single-crystal samples, mag-
netic correlations show a novel incommensurate dou-
blet, which also differs distinctly from the quartet in the
superconductors9,14. In this paper, we report that spin
excitations of La1.95Sr0.05CuO4 in the insulating spin-
glass regime are also composed of the Fincher-Burke
modes, originating from the incommensurate doublet,
with a velocity the same as in superconducting LSCO.
Although cuprates in the spin-glass regime are insula-
tors, they are not the usual band insulators, and part
of the Fermi surface may have survived15,16. It would
be interesting to investigate whether the Fincher-Burke
modes reported here can be accounted for by extending
theories for similar spin excitations of superconducting
cuprates.
Meanwhile, for quasi-two-dimensional (2D) spin
S=1/2 cuprates, the temperature range where spin fluc-
tuations are investigated is within T << J/kB ∼
1000 K17, where classical statistical mechanics has to be
replaced by quantum statistical mechanics18. The gen-
eral scaling argument of 2D quantum statistical systems
leads to a prediction for samples which are not exactly
at a quantum critical point (QCP) that for T ≪ J/kB
but above a low-temperature limit TX , the energy scale
for 2D spin fluctuations at long wavelengths is kBT
18,19.
Hence, for TX < T ≪ J/kB, the spin excitation spectrum
follows the ω/T scaling19, which is not a robust feature
of the Moriya theory7. For T < TX , a constant energy
gap is predicted19. We test these predictions against the
distinct incommensurate spin excitations from the dou-
blet in La1.95Sr0.05CuO4. We also compare our results
with previous investigations of incommensurate spin fluc-
tuations from the quartet in superconducting cuprate20
and commensurate ones at the (π, π) point in insulating
cuprate21. Our data support the scaling above TX but no
gap is observed below TX . Surprisingly, scaling analysis
of the q and ω dependent spin excitation spectra indi-
cates that La1.95Sr0.05CuO4 and optimally doped LSCO
are similarly distant from the QCP.
http://arxiv.org/abs/0704.1161v3
A single piece of La1.95Sr0.05CuO4 crystal of 5.2 g
was used in this work. It was grown using a traveling-
solvent floating-zone method as described previously9,14.
We use the orthorhombic Cmca unit cell (a = 5.338Å,
b = 13.16Å, c = 5.404Å at 1.5 K) to describe the q-
space for measurements at NIST using the cold neutron
triple-axis spectrometer SPINS. The sample temperature
was controlled by a pumped He4 cryostat. The horizon-
tal collimations before and after the sample were both
80′, and a Be filter cooled by liquid nitrogen was used
after the sample to reduce higher order neutrons pass-
ing through the pyrolytic graphite (002) used for both
monochromator and analyzer. The intensity of magnetic
neutron scattering was counted against a flux monitor
placed before the sample in a fixed Ef = 5 meV configu-
ration and normalized to yield S(q, ω) in absolute units.
Such a cold neutron spectrometer readily resolves the in-
commensurate doublet near (100), i.e., the (π, π) point
(see Fig. 1), while it is difficult to resolve the doublet us-
ing a thermal neutron triple-axis spectrometer due to its
coarser resolution. Hence, we will present only the cold
neutron scattering results here.
We first present the nominal elastic signal at various
temperatures. Fig. 1(a) shows constant-energy ~ω=0
scans through the incommensurate doublet from 1.2 to 80
K. The sharp peak at (100) is due to higher-order diffrac-
tion of (200). Its width indicates the instrument resolu-
tion. The magnetic doublets at qIC=(1,0,±0.058(2)) are
obviously broader than the resolution. The deconvoluted
peak width yields the in-plane correlation length for the
nominally elastic spin correlations, ξ� = 34(2)Å at 1.2
K, about 9 nearest-neighbor Cu spacings. With increas-
ing temperature, the doublet monotonically decreases in
intensity without appreciable change in either the peak
width or position. Above ∼ 20 K, the doublet disappears,
consistent with previous studies9.
At finite energies, however, the temperature depen-
dence of the doublet is entirely different from that at
~ω = 0. Fig. 1(b) shows constant-energy ~ω=0.5 meV
scans measured in the same temperature range. Instead
of monotonically decreasing, the intensity first increases,
reaches a maximum between 15 and 20 K, and then de-
creases with further rising temperature. The intensity at
the peak shoulder, e.g., at q = (1, 0, 0.2), in Fig. 1(a)-(b)
measures a temperature-independent background, which
has been subtracted in Fig. 1(c). Note that at 1.2 K,
S(q, ω) at 0.5 meV is more than one order of magni-
tude weaker than at ~ω = 0. This reflects the fact
that the energy spectrum of S(q, ω) shows a prominent
sharp “central peak” at ω = 0 at low temperatures, see
Fig. 1(c). The “central peak” is energy-resolution-limited
at the SPINS spectrometer, with the full-width-at-half-
maximum (FWHM) of 0.3 meV. However, the nominal
elastic signal from La1.95Sr0.05CuO4 is not truly static
at T > Tg ≈ 5 K as determined by our µSR measure-
ments, which has an energy resolution of ∼10−6 meV.
Details of the µSR study will be published elsewhere.
Similar “central peak” phenomenon has been reported
FIG. 1: (color online) The generalized spin correlation func-
tion S(q, ω) as a function of q along the c-axis measured at
(a) ~ω = 0 and (b) 0.5 meV, respectively, at various temper-
atures. (c) S(q, ω) as a function of energy at q = (1, 0, 0.05)
and at 1.2 and 20 K, respectively. The “central peak” at 1.2
K is also shown on a 1/3 scale with open circles. Background
has been subtracted in (c).
for Li-doped La2CuO4 and YBa2Cu3O6+x, and the very
slow spin dynamics is attributed to a partial spin-glass
transition22,23.
What is the energy dependence of the doublet?
Fig. 2(a) shows scans at various energies at 20 K. Below
3 meV, the two incommensurate peaks are clearly distin-
guishable. As energy increases, the doublet merges into
a flat-top peak. The scans in Fig. 2(a) can be fitted using
two gaussians of the same width. The peak positions are
shown as the black circles in Fig. 2(b). The dispersion
is consistent with the inner branches of spin-waves (solid
lines)17. The same dispersion rate has been reported for
superconducting La2−xSrxCuO4 (x = 0.10 and 0.16)
The shaded area in Fig. 2(b) covers the FWHM, which
grows slowly with energy from 0.089(3) Å−1 at 0.5 meV.
FIG. 2: (color online) (a) Constant-~ω scans at 20 K. The
solid lines are two equal-width gaussians. The horizontal bars
indicate instrument resolution (FWHM). (b) The double peak
positions. The shaded area represents the FWHM. The spec-
tral shape strongly resembles that for La1.84Sr0.16CuO4
2 and
V1.97O3
6. The lines denote the spin-wave branches with a
velocity of 850 meVÅ of La2CuO4
17. The FWHM/2 mea-
sured at those energies listed in (a) is shown as a function of
temperature in (c) and reduced temperature in (d).
The width is comparable to that for the x = 0.16 sample.
Because of the smaller doublet separation, the merge of
the peaks occurs at ∼4 meV, much lower than at ∼40
meV for La1.84Sr0.16CuO4
2. Thus, despite very differ-
ent ground states and spatial spin correlations, insulating
and superconducting LSCO share the common Fincher-
Burke spectral shape.
Now we turn to examination of scaling behavior of
spin excitations. Historically, it was done in the spin-
glass regime through the local spin correlation function
S(ω) =
dqS(q, ω) using cuprate samples showing com-
mensurate spin correlations12,13. It is re-examined here
with the very different incommensurate spin correlations
of our improved sample. The imaginary part of the local
dynamic magnetic susceptibility relates to S(ω) by the
fluctuation-dissipation theorem
χ′′(ω) = π(1 − e−~ω/kBT )S(ω). (1)
Fig. 3 shows χ′′(ω) from 1.2 to 80 K, which is well de-
scribed by the Debye relaxor model
χ′′(ω) =
χ0(~ω/Γ)
1 + (~ω/Γ)2
, (2)
where χ0 is the local staggered static magnetic suscep-
tibility, and Γ the spin relaxation constant. In previous
studies12,13, the local χ′′(ω) was modeled by
χ′′(ω) = I(ω) arctan[a1(~ω/kBT ) + a2(~ω/kBT )
2]. (3)
The arctan function is stipulated by the marginal Fermi
liquid theory12,24, but I(ω) has no determined analytic
form13. Hence, we opt for the well-known Debye relaxor
FIG. 3: (color online) The local dynamic magnetic suscep-
tibility χ′′(ω) measured at various temperatures. The solid
lines are the least-square fit to Eq. (2). The relaxation con-
stant Γ is shown on the ~ω-T plane as a function of T , and
departs from the dashed line Γ = 0.73kBT below TX ≈ 16 K.
model, Eq. (2), to fit our data. The Debye relaxor model
has also successfully described measured χ′′(ω) of insu-
lating La2Cu0.94Li0.06O4, which has commensurate mag-
netic correlations21.
On the base plane of Fig. 3, the spin relaxation con-
stant Γ obtained from the least-squares fit is shown as
a function of temperature. The good instrument reso-
lution, 0.3 meV (FWHM), has a negligible effect during
fitting. One interesting result is that Γ = 0.73(2)kBT
for temperatures above TX ≈ 16 K. Hence, when χ
′′(ω),
normalized by its maximum χ0/2 at ~ω = Γ, is plot-
ted as a function of ~ω/kBT , Eq. (2) dictates that all
data collected above TX collapse onto a single universal
curve y = 2/[1 + (x/0.73)2] and Fig. 4(a) bears this out.
The result is commonly referred to as the ω/T scaling,
and Γ/kBT = O(1) is a hallmark of quantum magnetic
theory18,19. For T < TX , Fig. 3 shows that Γ departs
from the proportionality to temperature. Consequently,
the low temperature data would not follow the scaling
curve, as demonstrated by Fig. 4(b). Note that the spec-
tral function Eq. (2) does not become gapped below TX ,
contrary to non-random quantum theory19, but can be
explained by dopant scattering25,26.
The ω/T scaling and its departure below TX shown
in Fig. 4 for La1.95Sr0.05CuO4 bears a striking similar-
ity to what reported for La2Cu0.94Li0.06O4
21. The two
cuprates have similar hole concentration and develop
spin-glass at similar Tg. However, they differ in several
important ways: i) The dopants are out of the CuO2
plane in the Sr compound, but directly replace Cu2+
in the Li compound. ii) The former becomes a super-
conductor with additional 0.5% more holes, but the lat-
ter always remains an insulator. iii) Magnetic correla-
tions are incommensurate in the former, but commen-
surate in the latter. iv) The in-plane correlation length
ξ� ≃ 34(2)Å for the glassy spin component in the former,
but ξ� ≫ 274Å in the latter, and the κ(ω, T ) shown in
Fig. 2(c) is more than double that in the latter22. v)
FIG. 4: (color online) The normalized local dynamic magnetic
susceptibility χ′′(ω)2/χ0 as a function of ~ω/kBT . (a) The
ω/T scaling is followed for data taken above TX ≃ 16 K. The
solid curve is the scaling function y = 2/[1 + (x/0.73)2]. (b)
Data taken below TX does not follow the scaling function.
Γ/kBT = 0.73 for the former, and 0.18 for the latter
In spite of these differences, Γ saturates at Γ0 ∼ 1 meV
and χ′′(ω) becomes essentially T -independent (see Fig. 3)
for both cuprates below TX . As a consequence, Eq. (1)
requires a reduced S(ω) at low energies when the temper-
ature decreases below TX , as observed in Fig. 1(b) and
(c), in sharp contrast to a magnet at the QCP.
Scaling of spin excitations has also been examined near
the optimal doping for La1.86Sr0.14CuO4
20. The mate-
rial is concluded to be near a QCP, namely, TX→0, with
some caveat27. The χ′′P /ω in [
20], equaling to χ0/Γ of
Eq. (2), would saturate below TX , but TC = 35 K sets
the upper limit for measurable TX in La1.86Sr0.14CuO4.
Hence it cannot be determined whether La1.86Sr0.14CuO4
or La1.95Sr0.05CuO4 is closer to a QCP with a lower TX .
Another method to assess the distance from the QCP is
to examine the width of constant-~ω scans, see Fig. 2(c).
Adapting the ansatz in [20], the κ(ω, T ) plotted as a func-
tion of a reduced temperature better collapses our data
in Fig. 2(d), and the solid line is
κ(ω, T )2 = κ20 + (kBT/c)
2[1 + (~ω/0.73kBT )
2], (4)
where κ0 = 0.044(1)Å
−1 and c = 4(1) × 102 meVÅ. At
the QCP, κ0 is expected to be zero, and its value for
La1.95Sr0.05CuO4 is comparable to κ0 in the supercon-
ducting state, but narrower than κ0 at 40 K in the normal
state for La1.84Sr0.16CuO4
2. Hence, La1.95Sr0.05CuO4
and the optimally doped LSCO with a short 1/κ0, about
6 Cu-Cu spacings, seem equally distant from the QCP.
In conclusion, the Fincher-Burke modes, the broad and
dispersive spin excitations of itinerant antiferromagnets,
are observed in the spin-glass regime of La2−xSrxCuO4.
Theoretical understanding of similar excitation modes in
superconducting cuprates now has an added task in the
insulators. Befitting to the generality of its theoretical
argument, the ω/T scaling is shown to be valid for a new
type of cuprates above TX . Spin excitations below TX
remain gapless contrary to the prediction of non-random
quantum theory. The spin-glass transition at finite dop-
ing introduces an extra component of slow spin fluctu-
ations. It would be interesting to explore whether the
glassy state22,23,28, limiting the correlation length of the
rest of spins, is responsible for the equal distance from
the QCP for La1.95Sr0.05CuO4 and La1.84Sr0.16CuO4.
Work at LANL is supported by U.S. DOE, and SPINS
partially by NSF under Agreement No. DMR-0454672.
1 J. M. Tranquada, in Handbook of High-Temperature Super-
conductivity, edited by J. R. Schrieffer (Springer, Berlin,
2007), p. 257.
2 N. B. Christensen, D. F. McMorrow, H. M. Rønnow,
B. Lake, S. M. Hayden, G. Aeppli, T. G. Perring,
M. Mangkorntong, M. Nohara, and H. Takagi, Phys. Rev.
Lett. 93, 147002 (2004).
3 S. M. Hayden, H. A. Mook, P. Dai, T. G. Perring, and
F. Doğan, Nature (London) 429, 531 (2004).
4 J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D.
Gu, G. Xu, M. Fujita, and K. Yamada, Nature (London)
429, 534 (2004).
5 C. R. Fincher, G. Shirani, and S. A. Werner, Phys. Rev.
B 24, 1312 (1981).
6 W. Bao, C. Broholm, S. A. Carter, T. F. Rosenbaum,
G. Aeppli, S. F. Trevino, P. Metcalf, J. M. Honig, and
J. Spalek, Phys. Rev. Lett. 71, 766 (1993); W. Bao, C. Bro-
holm, J. M. Honig, P. Metcalf, and S. F. Trevino, Phys.
Rev. B 54, R3726 (1996).
7 T. Moriya, Spin Fluctuations in Itinerant Electron Mag-
netism (Springer-Verlag and Berlin, 1985).
8 A. Abanov and A. V. Chubukov, Phys. Rev. Lett. 83,
1652 (1999); U. Chatterjee et al., Phys. Rev. B 75, 172504
(2007).
9 M. Fujita, K. Yamada, H. Hiraka, P. M. Gehring, S. H.
Lee, S. Wakimoto, and G. Shirane, Phys. Rev. B 65, 64505
(2002).
10 B. J. Sternlieb, G. M. Luke, Y. J. Uemura, T. M. Rise-
man, J. H. Brewer, P. M. Gehring, K. Yamada, Y. Hidaka,
T. Murakami, T. R. Thurston, et al., Phys. Rev. B 41,
8866 (1990).
11 F. C. Chou, F. Borsa, J. H. Cho, D. C. Johnston, A. Las-
cialfari, D. R. Torgeson, and J. Ziolo, Phys. Rev. Lett. 71,
2323 (1993).
12 B. Keimer, N. Belk, R. J. Birgeneau, A. Cassanho, C. Y.
Chen, M. Greven, M. A. Kastner, A. Aharony, Y. Endoh,
R. W. Erwin, et al., Phys. Rev. B 46, 14034 (1992).
13 M. Matsuda, R. J. Birgeneau, Y. Endoh, Y. Hidaka, M. A.
Kastner, K. Nakajima, G. Shirane, T. R. Thurston, and
K. Yamada, J. Phys. Soc. Japn 62, 1702 (1993).
14 S. Wakimoto, G. Shirane, Y. Endoh, K. Hirota, S. Ueki,
K. Yamada, R. J. Birgeneau, M. A. Kastner, Y. S. Lee,
P. M. Gehring, et al., Phys. Rev. B 60, R769 (1999).
15 M. L. Tacon, A. Sacuto, A. Georges, G. Kotliar, Y. Gallais,
D. Colson, and A. Forget, Nature Physics 2, 537 (2006).
16 T. Yoshida, X. J. Zhou, T. Sasagawa, W. L. Yang, P. V.
Bogdanov, A. Lanzara, Z. Hussain, T. Mizokawa, A. Fuji-
mori, H. Eisaki, et al., Phys. Rev. Lett. 91, 027001 (2003).
17 S. M. Hayden, G. Aeppli, R. Osborn, A. D. Taylor, T. G.
Perring, S.-W. Cheong, and Z. Fisk, Phys. Rev. Lett. 67,
3622 (1991).
18 D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod.
Phys. 77, 579 (2005).
19 S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
20 G. Aeppli, T. E. Mason, S. M. Hayden, H. A. Mook, and
J. Kulda, Science 278, 1432 (1997).
21 W. Bao, Y. Chen, Y. Qiu, and J. L. Sarrao, Phys. Rev.
Lett. 91, 127005 (2003).
22 Y. Chen, W. Bao, Y. Qiu, J. E. Lorenzo, J. L. Sarrao,
D. L. Ho, and M. Y. Lin, Phys. Rev. B 72, 184401 (2005).
23 C. Stock, W. J. L. Buyers, Z. Yamani, C. L. Broholm, J.-
H. Chung, Z. Tun, R. Liang, D. Bonn, W. N. Hardy, and
R. J. Birgeneau, Phys. Rev. B 73, 100504(R) (2006).
24 C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abra-
hams, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 1996
(1989).
25 S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B
51, 14874 (1995).
26 Y. L. Liu and Z. B. Su, Phys. Lett. A 200, 393 (1995).
27 S. Chakravarty, Science 278, 1412 (1997).
28 C. Panagopoulos, A. P. Petrovic, Hillier, J. L. Tallon, C. A.
Scott, and A. P. Rainford, Phys. Rev. B 69, 144510 (2004).

  Insulating La1.95Sr0.05CuO4 shares with superconducting cuprates the same
Fincher-Burke-like spin excitations, which usually are observed in itinerant
antiferromagnets. The local spectral function satisfies omega/T scaling above
~16 K for this incommensurate insulating cuprate. Together with previous
results in commensurate insulating and incommensurate superconducting cuprates,
these results further support the general scaling prediction for square-lattice
quantum spin S=1/2 systems. The width of incommensurate peaks in
La1.95Sr0.05CuO4 scales to a similar finite value as at optimal doping,
strongly suggesting that they are similarly distant from a quantum critical
point. They might both be limited to a finite correlation length by the partial
spin-glass freezing.

Fincher-Burke spin excitations and ω/T scaling in insulating La1.95Sr0.05CuO4
Wei Bao,1 Y. Chen,2, 3 K. Yamada,4 A. T. Savici,5 P. L. Russo,6 J. E. Lorenzo,7 and J.-H. Chung2, 3
Condensed Matter and Thermal Physics, Los Alamos National Laboratory, Los Alamos, NM 87545
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899
Dept. of Materials Science and Engineering, University of Maryland, College Park, MD 20742
Institute of Materials Research, Tohoku University, Sendai 980-8577, Japan
Brookhaven National Laboratory, Upton, NY 11973
TRIUMF, Vancouver, BC V6T 2A3, Canada
Institut Néel, CNRS, BP 166X, F-38043, Grenoble, France
(Dated: November 27, 2018)
Insulating La1.95Sr0.05CuO4 shares with superconducting cuprates the same Fincher-Burke spin
excitations, which usually are observed in itinerant antiferromagnets. The local spectral function
satisfies ω/T scaling above ∼16 K for this incommensurate insulating cuprate. Together with
previous results in commensurate insulating and incommensurate superconducting cuprates, these
results further support the general scaling prediction for square-lattice quantum spin S = 1/2
systems. The width of incommensurate peaks in La1.95Sr0.05CuO4 scales to a similar finite value
as at optimal doping, strongly suggesting that they are similarly distant from a quantum critical
point. They might both be limited to a finite correlation length by partial spin-glass freezing.
It is well known that the (π, π) peak of the Néel anti-
ferromagnetic order of the parent compounds is replaced
by a quartet of incommensurate peaks when cuprates are
sufficiently doped to have become superconducting1. The
very recent excitement is that spin excitations measured
in superconducting La2−xSrxCuO4 (LSCO) (x = 0.10
and 0.16)2, YBa2Cu3O6.6 (YBCO)
3, as well as in the
“stripe-ordered” La1.875Ba0.125CuO4 with a suppressed
4 share a common spectral feature: broad excitation
continua, originating from the incommensurate quartet,
disperse towards the (π, π) point with increasing en-
ergy at the rate of the spin-wave velocity of the par-
ent compounds. The spectrum is distinct from the
spin-waves in two important ways: 1) the excitations
are not resolution-limited; 2) there are no the outward
branches. Such a spin excitation spectrum previously
has been observed in itinerant spin-density-wave anti-
ferromagnets such as elemental metal Cr5 and strongly
correlated metal V2−yO3
6, and the single-lobed disper-
sive continuum is referred as the Fincher-Burke mode. A
self-consistent theory has been developed by Moriya and
others to describe the mode7, and quantitative agree-
ment has been achieved for three-dimensional itinerant
antiferromagnets6. Itinerant theories have also been de-
veloped to account for the Fincher-Burke-like modes in
superconducting cuprates8.
Sandwiched between the parent antiferromagnetic in-
sulator and the high-TC superconductor in the phase dia-
gram of LSCO is a distinct doping regime from x = 0.02
to 0.0559. Cuprates in the regime are insulators with-
out the long-range Néel order, and a spin-glass transi-
tion occurs at Tf . 10 K
10,11. This doping range is
often referred to as the spin-glass regime, although the
spin-glass phase extends to both lower and higher dop-
ings in the Néel and superconducting states. Magnetic
correlations were extensively investigated and regarded
as being commensurate, as in the parent compound12,13.
However, with improved single-crystal samples, mag-
netic correlations show a novel incommensurate dou-
blet, which also differs distinctly from the quartet in the
superconductors9,14. In this paper, we report that spin
excitations of La1.95Sr0.05CuO4 in the insulating spin-
glass regime are also composed of the Fincher-Burke
modes, originating from the incommensurate doublet,
with a velocity the same as in superconducting LSCO.
Although cuprates in the spin-glass regime are insula-
tors, they are not the usual band insulators, and part
of the Fermi surface may have survived15,16. It would
be interesting to investigate whether the Fincher-Burke
modes reported here can be accounted for by extending
theories for similar spin excitations of superconducting
cuprates.
Meanwhile, for quasi-two-dimensional (2D) spin
S=1/2 cuprates, the temperature range where spin fluc-
tuations are investigated is within T << J/kB ∼
1000 K17, where classical statistical mechanics has to be
replaced by quantum statistical mechanics18. The gen-
eral scaling argument of 2D quantum statistical systems
leads to a prediction for samples which are not exactly
at a quantum critical point (QCP) that for T ≪ J/kB
but above a low-temperature limit TX , the energy scale
for 2D spin fluctuations at long wavelengths is kBT
18,19.
Hence, for TX < T ≪ J/kB, the spin excitation spectrum
follows the ω/T scaling19, which is not a robust feature
of the Moriya theory7. For T < TX , a constant energy
gap is predicted19. We test these predictions against the
distinct incommensurate spin excitations from the dou-
blet in La1.95Sr0.05CuO4. We also compare our results
with previous investigations of incommensurate spin fluc-
tuations from the quartet in superconducting cuprate20
and commensurate ones at the (π, π) point in insulating
cuprate21. Our data support the scaling above TX but no
gap is observed below TX . Surprisingly, scaling analysis
of the q and ω dependent spin excitation spectra indi-
cates that La1.95Sr0.05CuO4 and optimally doped LSCO
are similarly distant from the QCP.
http://arxiv.org/abs/0704.1161v3
A single piece of La1.95Sr0.05CuO4 crystal of 5.2 g
was used in this work. It was grown using a traveling-
solvent floating-zone method as described previously9,14.
We use the orthorhombic Cmca unit cell (a = 5.338Å,
b = 13.16Å, c = 5.404Å at 1.5 K) to describe the q-
space for measurements at NIST using the cold neutron
triple-axis spectrometer SPINS. The sample temperature
was controlled by a pumped He4 cryostat. The horizon-
tal collimations before and after the sample were both
80′, and a Be filter cooled by liquid nitrogen was used
after the sample to reduce higher order neutrons pass-
ing through the pyrolytic graphite (002) used for both
monochromator and analyzer. The intensity of magnetic
neutron scattering was counted against a flux monitor
placed before the sample in a fixed Ef = 5 meV configu-
ration and normalized to yield S(q, ω) in absolute units.
Such a cold neutron spectrometer readily resolves the in-
commensurate doublet near (100), i.e., the (π, π) point
(see Fig. 1), while it is difficult to resolve the doublet us-
ing a thermal neutron triple-axis spectrometer due to its
coarser resolution. Hence, we will present only the cold
neutron scattering results here.
We first present the nominal elastic signal at various
temperatures. Fig. 1(a) shows constant-energy ~ω=0
scans through the incommensurate doublet from 1.2 to 80
K. The sharp peak at (100) is due to higher-order diffrac-
tion of (200). Its width indicates the instrument resolu-
tion. The magnetic doublets at qIC=(1,0,±0.058(2)) are
obviously broader than the resolution. The deconvoluted
peak width yields the in-plane correlation length for the
nominally elastic spin correlations, ξ� = 34(2)Å at 1.2
K, about 9 nearest-neighbor Cu spacings. With increas-
ing temperature, the doublet monotonically decreases in
intensity without appreciable change in either the peak
width or position. Above ∼ 20 K, the doublet disappears,
consistent with previous studies9.
At finite energies, however, the temperature depen-
dence of the doublet is entirely different from that at
~ω = 0. Fig. 1(b) shows constant-energy ~ω=0.5 meV
scans measured in the same temperature range. Instead
of monotonically decreasing, the intensity first increases,
reaches a maximum between 15 and 20 K, and then de-
creases with further rising temperature. The intensity at
the peak shoulder, e.g., at q = (1, 0, 0.2), in Fig. 1(a)-(b)
measures a temperature-independent background, which
has been subtracted in Fig. 1(c). Note that at 1.2 K,
S(q, ω) at 0.5 meV is more than one order of magni-
tude weaker than at ~ω = 0. This reflects the fact
that the energy spectrum of S(q, ω) shows a prominent
sharp “central peak” at ω = 0 at low temperatures, see
Fig. 1(c). The “central peak” is energy-resolution-limited
at the SPINS spectrometer, with the full-width-at-half-
maximum (FWHM) of 0.3 meV. However, the nominal
elastic signal from La1.95Sr0.05CuO4 is not truly static
at T > Tg ≈ 5 K as determined by our µSR measure-
ments, which has an energy resolution of ∼10−6 meV.
Details of the µSR study will be published elsewhere.
Similar “central peak” phenomenon has been reported
FIG. 1: (color online) The generalized spin correlation func-
tion S(q, ω) as a function of q along the c-axis measured at
(a) ~ω = 0 and (b) 0.5 meV, respectively, at various temper-
atures. (c) S(q, ω) as a function of energy at q = (1, 0, 0.05)
and at 1.2 and 20 K, respectively. The “central peak” at 1.2
K is also shown on a 1/3 scale with open circles. Background
has been subtracted in (c).
for Li-doped La2CuO4 and YBa2Cu3O6+x, and the very
slow spin dynamics is attributed to a partial spin-glass
transition22,23.
What is the energy dependence of the doublet?
Fig. 2(a) shows scans at various energies at 20 K. Below
3 meV, the two incommensurate peaks are clearly distin-
guishable. As energy increases, the doublet merges into
a flat-top peak. The scans in Fig. 2(a) can be fitted using
two gaussians of the same width. The peak positions are
shown as the black circles in Fig. 2(b). The dispersion
is consistent with the inner branches of spin-waves (solid
lines)17. The same dispersion rate has been reported for
superconducting La2−xSrxCuO4 (x = 0.10 and 0.16)
The shaded area in Fig. 2(b) covers the FWHM, which
grows slowly with energy from 0.089(3) Å−1 at 0.5 meV.
FIG. 2: (color online) (a) Constant-~ω scans at 20 K. The
solid lines are two equal-width gaussians. The horizontal bars
indicate instrument resolution (FWHM). (b) The double peak
positions. The shaded area represents the FWHM. The spec-
tral shape strongly resembles that for La1.84Sr0.16CuO4
2 and
V1.97O3
6. The lines denote the spin-wave branches with a
velocity of 850 meVÅ of La2CuO4
17. The FWHM/2 mea-
sured at those energies listed in (a) is shown as a function of
temperature in (c) and reduced temperature in (d).
The width is comparable to that for the x = 0.16 sample.
Because of the smaller doublet separation, the merge of
the peaks occurs at ∼4 meV, much lower than at ∼40
meV for La1.84Sr0.16CuO4
2. Thus, despite very differ-
ent ground states and spatial spin correlations, insulating
and superconducting LSCO share the common Fincher-
Burke spectral shape.
Now we turn to examination of scaling behavior of
spin excitations. Historically, it was done in the spin-
glass regime through the local spin correlation function
S(ω) =
dqS(q, ω) using cuprate samples showing com-
mensurate spin correlations12,13. It is re-examined here
with the very different incommensurate spin correlations
of our improved sample. The imaginary part of the local
dynamic magnetic susceptibility relates to S(ω) by the
fluctuation-dissipation theorem
χ′′(ω) = π(1 − e−~ω/kBT )S(ω). (1)
Fig. 3 shows χ′′(ω) from 1.2 to 80 K, which is well de-
scribed by the Debye relaxor model
χ′′(ω) =
χ0(~ω/Γ)
1 + (~ω/Γ)2
, (2)
where χ0 is the local staggered static magnetic suscep-
tibility, and Γ the spin relaxation constant. In previous
studies12,13, the local χ′′(ω) was modeled by
χ′′(ω) = I(ω) arctan[a1(~ω/kBT ) + a2(~ω/kBT )
2]. (3)
The arctan function is stipulated by the marginal Fermi
liquid theory12,24, but I(ω) has no determined analytic
form13. Hence, we opt for the well-known Debye relaxor
FIG. 3: (color online) The local dynamic magnetic suscep-
tibility χ′′(ω) measured at various temperatures. The solid
lines are the least-square fit to Eq. (2). The relaxation con-
stant Γ is shown on the ~ω-T plane as a function of T , and
departs from the dashed line Γ = 0.73kBT below TX ≈ 16 K.
model, Eq. (2), to fit our data. The Debye relaxor model
has also successfully described measured χ′′(ω) of insu-
lating La2Cu0.94Li0.06O4, which has commensurate mag-
netic correlations21.
On the base plane of Fig. 3, the spin relaxation con-
stant Γ obtained from the least-squares fit is shown as
a function of temperature. The good instrument reso-
lution, 0.3 meV (FWHM), has a negligible effect during
fitting. One interesting result is that Γ = 0.73(2)kBT
for temperatures above TX ≈ 16 K. Hence, when χ
′′(ω),
normalized by its maximum χ0/2 at ~ω = Γ, is plot-
ted as a function of ~ω/kBT , Eq. (2) dictates that all
data collected above TX collapse onto a single universal
curve y = 2/[1 + (x/0.73)2] and Fig. 4(a) bears this out.
The result is commonly referred to as the ω/T scaling,
and Γ/kBT = O(1) is a hallmark of quantum magnetic
theory18,19. For T < TX , Fig. 3 shows that Γ departs
from the proportionality to temperature. Consequently,
the low temperature data would not follow the scaling
curve, as demonstrated by Fig. 4(b). Note that the spec-
tral function Eq. (2) does not become gapped below TX ,
contrary to non-random quantum theory19, but can be
explained by dopant scattering25,26.
The ω/T scaling and its departure below TX shown
in Fig. 4 for La1.95Sr0.05CuO4 bears a striking similar-
ity to what reported for La2Cu0.94Li0.06O4
21. The two
cuprates have similar hole concentration and develop
spin-glass at similar Tg. However, they differ in several
important ways: i) The dopants are out of the CuO2
plane in the Sr compound, but directly replace Cu2+
in the Li compound. ii) The former becomes a super-
conductor with additional 0.5% more holes, but the lat-
ter always remains an insulator. iii) Magnetic correla-
tions are incommensurate in the former, but commen-
surate in the latter. iv) The in-plane correlation length
ξ� ≃ 34(2)Å for the glassy spin component in the former,
but ξ� ≫ 274Å in the latter, and the κ(ω, T ) shown in
Fig. 2(c) is more than double that in the latter22. v)
FIG. 4: (color online) The normalized local dynamic magnetic
susceptibility χ′′(ω)2/χ0 as a function of ~ω/kBT . (a) The
ω/T scaling is followed for data taken above TX ≃ 16 K. The
solid curve is the scaling function y = 2/[1 + (x/0.73)2]. (b)
Data taken below TX does not follow the scaling function.
Γ/kBT = 0.73 for the former, and 0.18 for the latter
In spite of these differences, Γ saturates at Γ0 ∼ 1 meV
and χ′′(ω) becomes essentially T -independent (see Fig. 3)
for both cuprates below TX . As a consequence, Eq. (1)
requires a reduced S(ω) at low energies when the temper-
ature decreases below TX , as observed in Fig. 1(b) and
(c), in sharp contrast to a magnet at the QCP.
Scaling of spin excitations has also been examined near
the optimal doping for La1.86Sr0.14CuO4
20. The mate-
rial is concluded to be near a QCP, namely, TX→0, with
some caveat27. The χ′′P /ω in [
20], equaling to χ0/Γ of
Eq. (2), would saturate below TX , but TC = 35 K sets
the upper limit for measurable TX in La1.86Sr0.14CuO4.
Hence it cannot be determined whether La1.86Sr0.14CuO4
or La1.95Sr0.05CuO4 is closer to a QCP with a lower TX .
Another method to assess the distance from the QCP is
to examine the width of constant-~ω scans, see Fig. 2(c).
Adapting the ansatz in [20], the κ(ω, T ) plotted as a func-
tion of a reduced temperature better collapses our data
in Fig. 2(d), and the solid line is
κ(ω, T )2 = κ20 + (kBT/c)
2[1 + (~ω/0.73kBT )
2], (4)
where κ0 = 0.044(1)Å
−1 and c = 4(1) × 102 meVÅ. At
the QCP, κ0 is expected to be zero, and its value for
La1.95Sr0.05CuO4 is comparable to κ0 in the supercon-
ducting state, but narrower than κ0 at 40 K in the normal
state for La1.84Sr0.16CuO4
2. Hence, La1.95Sr0.05CuO4
and the optimally doped LSCO with a short 1/κ0, about
6 Cu-Cu spacings, seem equally distant from the QCP.
In conclusion, the Fincher-Burke modes, the broad and
dispersive spin excitations of itinerant antiferromagnets,
are observed in the spin-glass regime of La2−xSrxCuO4.
Theoretical understanding of similar excitation modes in
superconducting cuprates now has an added task in the
insulators. Befitting to the generality of its theoretical
argument, the ω/T scaling is shown to be valid for a new
type of cuprates above TX . Spin excitations below TX
remain gapless contrary to the prediction of non-random
quantum theory. The spin-glass transition at finite dop-
ing introduces an extra component of slow spin fluctu-
ations. It would be interesting to explore whether the
glassy state22,23,28, limiting the correlation length of the
rest of spins, is responsible for the equal distance from
the QCP for La1.95Sr0.05CuO4 and La1.84Sr0.16CuO4.
Work at LANL is supported by U.S. DOE, and SPINS
partially by NSF under Agreement No. DMR-0454672.
1 J. M. Tranquada, in Handbook of High-Temperature Super-
conductivity, edited by J. R. Schrieffer (Springer, Berlin,
2007), p. 257.
2 N. B. Christensen, D. F. McMorrow, H. M. Rønnow,
B. Lake, S. M. Hayden, G. Aeppli, T. G. Perring,
M. Mangkorntong, M. Nohara, and H. Takagi, Phys. Rev.
Lett. 93, 147002 (2004).
3 S. M. Hayden, H. A. Mook, P. Dai, T. G. Perring, and
F. Doğan, Nature (London) 429, 531 (2004).
4 J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D.
Gu, G. Xu, M. Fujita, and K. Yamada, Nature (London)
429, 534 (2004).
5 C. R. Fincher, G. Shirani, and S. A. Werner, Phys. Rev.
B 24, 1312 (1981).
6 W. Bao, C. Broholm, S. A. Carter, T. F. Rosenbaum,
G. Aeppli, S. F. Trevino, P. Metcalf, J. M. Honig, and
J. Spalek, Phys. Rev. Lett. 71, 766 (1993); W. Bao, C. Bro-
holm, J. M. Honig, P. Metcalf, and S. F. Trevino, Phys.
Rev. B 54, R3726 (1996).
7 T. Moriya, Spin Fluctuations in Itinerant Electron Mag-
netism (Springer-Verlag and Berlin, 1985).
8 A. Abanov and A. V. Chubukov, Phys. Rev. Lett. 83,
1652 (1999); U. Chatterjee et al., Phys. Rev. B 75, 172504
(2007).
9 M. Fujita, K. Yamada, H. Hiraka, P. M. Gehring, S. H.
Lee, S. Wakimoto, and G. Shirane, Phys. Rev. B 65, 64505
(2002).
10 B. J. Sternlieb, G. M. Luke, Y. J. Uemura, T. M. Rise-
man, J. H. Brewer, P. M. Gehring, K. Yamada, Y. Hidaka,
T. Murakami, T. R. Thurston, et al., Phys. Rev. B 41,
8866 (1990).
11 F. C. Chou, F. Borsa, J. H. Cho, D. C. Johnston, A. Las-
cialfari, D. R. Torgeson, and J. Ziolo, Phys. Rev. Lett. 71,
2323 (1993).
12 B. Keimer, N. Belk, R. J. Birgeneau, A. Cassanho, C. Y.
Chen, M. Greven, M. A. Kastner, A. Aharony, Y. Endoh,
R. W. Erwin, et al., Phys. Rev. B 46, 14034 (1992).
13 M. Matsuda, R. J. Birgeneau, Y. Endoh, Y. Hidaka, M. A.
Kastner, K. Nakajima, G. Shirane, T. R. Thurston, and
K. Yamada, J. Phys. Soc. Japn 62, 1702 (1993).
14 S. Wakimoto, G. Shirane, Y. Endoh, K. Hirota, S. Ueki,
K. Yamada, R. J. Birgeneau, M. A. Kastner, Y. S. Lee,
P. M. Gehring, et al., Phys. Rev. B 60, R769 (1999).
15 M. L. Tacon, A. Sacuto, A. Georges, G. Kotliar, Y. Gallais,
D. Colson, and A. Forget, Nature Physics 2, 537 (2006).
16 T. Yoshida, X. J. Zhou, T. Sasagawa, W. L. Yang, P. V.
Bogdanov, A. Lanzara, Z. Hussain, T. Mizokawa, A. Fuji-
mori, H. Eisaki, et al., Phys. Rev. Lett. 91, 027001 (2003).
17 S. M. Hayden, G. Aeppli, R. Osborn, A. D. Taylor, T. G.
Perring, S.-W. Cheong, and Z. Fisk, Phys. Rev. Lett. 67,
3622 (1991).
18 D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod.
Phys. 77, 579 (2005).
19 S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
20 G. Aeppli, T. E. Mason, S. M. Hayden, H. A. Mook, and
J. Kulda, Science 278, 1432 (1997).
21 W. Bao, Y. Chen, Y. Qiu, and J. L. Sarrao, Phys. Rev.
Lett. 91, 127005 (2003).
22 Y. Chen, W. Bao, Y. Qiu, J. E. Lorenzo, J. L. Sarrao,
D. L. Ho, and M. Y. Lin, Phys. Rev. B 72, 184401 (2005).
23 C. Stock, W. J. L. Buyers, Z. Yamani, C. L. Broholm, J.-
H. Chung, Z. Tun, R. Liang, D. Bonn, W. N. Hardy, and
R. J. Birgeneau, Phys. Rev. B 73, 100504(R) (2006).
24 C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abra-
hams, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 1996
(1989).
25 S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B
51, 14874 (1995).
26 Y. L. Liu and Z. B. Su, Phys. Lett. A 200, 393 (1995).
27 S. Chakravarty, Science 278, 1412 (1997).
28 C. Panagopoulos, A. P. Petrovic, Hillier, J. L. Tallon, C. A.
Scott, and A. P. Rainford, Phys. Rev. B 69, 144510 (2004).
