Neutron Scattering with Examples from Cuprate Superconductors Part 2 John Tranquada Quantum Science Summer School Cornell University June 22, 2018 center for emergent superconductivity
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Neutron Scatteringwith Examples from Cuprate Superconductors
Part 2
John Tranquada
Quantum Science Summer School Cornell University
June 22, 2018
center for emergentsuperconductivity
Coherent magnetic scattering
Form factor in cuprates
Sr2CuO3
hybridized orbital
ionic 3dx2-y2 orbital
form factor
Walters et al.,Nat. Phys. (2009)
Differential cross section with magnetic terms
neutron spin state
neutron spin operator x 2
nuclear spin operatormagnetic interaction vector
Magnetic scattering
Dynamical structure factor
Instantaneous correlations
Sum rule
Polarized-beam scattering
• The component of S⊥ that is perpendicular to the incident neutron polarization flips the neutron spin
• If we spin-polarize the incident neutron beam, and analyze the polarization of the scattered beam, we can separate “spin-flip” and “non-spin-flip” scattering
• Polarization analysis is expensive in terms of intensity
• I’m not going to discuss this
Magnetic diffraction
Collinear spins
Average over domains
Antiferromagnetic order doubles unit cell
Real space Reciprocal space
Crystalstructure
Magneticstructure
Coordinates mustrotate by 45º todescribe orthorhombic cell
Q = (h, k, l)
in units of (2π/a,2π/b,2π/c)
Octahedral tilt pattern in LTO issame as AF order; however, structure factor for tilted octahedra at Q = (1/2,1/2,L) is zero if L = 0.
Tilt and spin orders
(a)
(b)
Sche
mat
ic r
epre
sent
atio
n of
the
tilt
patte
rn w
ithin
one
pla
ne o
f C
uO^
octa
hedr
a in
the
LT
O a
nd L
TT
pha
ses.
In
the
LT
T p
hase
the
pat
tern
is
rota
ted
by 9
0° (f
ourf
old
glid
e) in
the
next
laye
r to
esta
blis
h te
trag
onal
sym
met
ry.
Tem
pera
tuje
386
Sche
mat
ic re
pres
enta
tion
of th
e be
havi
or o
f the
sof
t-X-p
oint
pho
non
mod
es i
n th
e lo
w-t
empe
ratu
re p
hase
s. w
^(T^
= w
jCJJ)
= 0
due
to
vani
shin
g ha
rmon
ic e
nerg
y te
rms.
a),(
7",)
= 0
due
to v
anis
hing
qu
artic
ani
sotr
opy
ener
gy.
such
mat
eria
ls. S
ince
the
dom
ain
conf
igur
atio
ns o
f the
LT
O a
nd L
TT s
truct
ures
are
qui
te d
iffer
ent,
it w
ould
be
inte
rest
ing
to c
ompa
re th
e SC
tran
spor
t pro
perti
es o
f
LTT
and
LTO
mat
eria
ls w
hich
are
nea
riy id
entic
al in
ot
her
resp
ects.
(Of c
ours
e, th
is p
resu
ppos
es th
e ex
isten
ce
of S
C in
LTO
mod
ifica
tions
, in
acco
rdan
ce w
ith a
ccep
ted
com
mon
bel
ief.)
2.
As
poin
ted
out b
y Th
io e
t al.
[18]
, the
wea
k bu
cklin
g of
th
e C
uOj p
lane
s as
soci
ated
with
the
LT p
hase
s cau
ses
a ca
ntin
g of
the
antif
erro
mag
netic
(A
F) sp
in s
truct
ure.
B
ecau
se o
f fru
strat
ion
effe
cts,
even
sm
all c
antin
g ca
n gr
eatly
influ
ence
the
mag
netic
inte
rpla
nar c
oupl
ing.
A
lthou
gh d
opin
g de
stro
ys A
F lo
ng-ra
nge
orde
r, A
F co
rrel
atio
ns a
re st
ill si
gnifi
cant
at d
opin
g le
vels
nec
essa
ry
for
SC, a
nd m
ay m
edia
te S
C sp
in-p
airin
g. If
this
is s
o,
both
the
inte
rpla
nar
SC a
nd A
F co
upH
ng a
re d
irect
ly
influ
ence
d by
the
spin
-can
ting
and
stru
ctur
al b
uckl
ing
in
the
CuO
pla
nes.
Thi
s su^
ests
that
the
SC tr
ansf
orm
atio
n te
mpe
ratu
res
of th
e LT
O a
nd L
TT m
odifi
catio
ns m
ay
diffe
r sig
nific
antly
.
In th
is c
onne
ctio
n it
is n
otew
orth
y th
at fo
r a
sequ
ence
of
La2_
^Ba
Cu04
_j, s
ampl
es p
repa
red
iden
tical
ly to
thos
e of
this
st
udy,
Moo
denb
augh
et a
l. [1
0] fi
nd
that
with
incr
easi
ng x
th
e SC
tran
sfor
mat
ion
firs
t inc
reas
es to
abo
ut 2
0 K
for
X =
0.0
8, th
en b
road
ens
and
decr
ease
s to
aboi
it 5
K n
ear
X =
0.1
2, in
crea
ses a
gain
to n
ear
25 K
at x
= 0
.15,
and
then
dr
ops o
ff ag
ain.
The
low
erin
g of
T^
at in
term
edia
te d
opin
g ro
ughl
y co
rrel
ates
with
the
occu
rren
ce o
f a m
ajor
fra
ctio
n of
LT
T p
hase
pre
sent
at l
ow te
mpe
ratu
res
[13]
. A r
elat
ed
J. D
. A
XE
ET
AL
. IB
M S
. R
ES.
DE
VE
LO
P.
VO
L.
33
NO
. 3
MA
Y
IM9
erties are also important as mentioned above. Stimulated bythose findings, we aimed to clarify the static magnetic prop-erties in the spin-glass phase of La1.98Sr0.02CuO4, in whichthe 3D antiferromagnetic long-range ordering just disap-pears. The most interesting issue is how the long-range 3Dantiferromagnetic ordering disappears and the spin-glass be-havior appears with hole doping.In this study we find that below !40 K quasielastic mag-
netic peaks develop at the positions where magnetic Braggpeaks exist in La2CuO4. This means that spin correlationsdevelop perpendicular to the CuO2 plane in addition to par-allel to it, suggesting that the 2D spin fluctuations, whichexist at high temperatures, at least in part freeze with freez-ing temperature enhanced due to the interplanar interaction.Our most important findings are that the spin correlations inthe CuO2 plane are anisotropic at low temperatures and thatthe spin directions in the spin clusters differs from that inpure La2CuO4. The spin cluster dimension along the aorthoaxis (!160 Å ) is !6 times longer than that along the borthoaxis (!25 Å ) at 1.6 K. From these results, it is concludedthat the spin-glass state can be described as a random freez-ing of quasi-3D spin clusters with anisotropic spin correla-tions.The format of this paper is as follows: The scattering
geometry used in this study is summarized in Sec. II. Experi-mental details are described in Sec. III. The experimentalresults of the elastic neutron measurements are presented andthe spin structure of the short-range ordered state is summa-rized in Sec. IV. The experimental results on the inelasticneutron measurements are presented in Sec. V. In Sec. VI wediscuss the magnetic properties of the spin-glass state.
II. SCATTERING GEOMETRY
Figure 1"a# shows the scattering geometry in the (H ,H ,0)scattering plane in the high-temperature tetragonal phase.La1.98Sr0.02CuO4 undergoes a structural phase transition fromtetragonal (I4/mmm) to orthorhombic (Bmab) structure at485 K. Figure 1"b# shows the scattering geometry in the(H ,K ,0) scattering plane in the low-temperature orthorhom-bic phase. The structure is slightly distorted with the aorthoand bortho axes almost along the diagonal directions of thea tetra and b tetra axes. Ideally, the experiments should be per-formed with a single-domain crystal. However, four domainsare expected to exist in real crystals since a twin structure isenergetically stable. For small rectangular samples it is pos-sible to prepare a single-domain crystal by heating thesample above the transition temperature and then cooling itdown while applying pressure along the aortho or bortho axis.However, the heat treatment is not efficient for large crystalsso that preparation of a single-domain sample is extremelydifficult in practice. Consequently in this experiment, wewere forced to use a four-domain crystal.The scattering geometry for the four-domain crystal is
shown in Fig. 1"c#. (H ,K ,L)ortho and (K ,H ,L)ortho nuclearBragg peaks are observed at nearby positions. As a result,four peaks are observed around (1,1,0)tetra and three peaksare observed around (2,0,0)tetra in the (H ,K ,0) scatteringplane. Figure 2"a# shows an elastic scan $scan A as indicatedin Fig. 1"c#% at (2,0,0)tetra . The two side peaks at (2,0,0)tetraoriginate from two of the domains and the central peak,
which is factor of !2 larger than each side peak, originatesfrom the other two domains. From these results, we estimatethat the four domains are equally distributed. Due to the twinstructure, the (H ,0,L)ortho and the (0,K ,L)ortho scatteringplanes are superposed upon each other as shown in Fig. 1"d#.Scan B in Fig. 1"d# corresponds to scan C in Fig. 1"c#. Sincethe vertical resolution is quite broad, the (2,0,0)ortho$(0,2,0)ortho% peak shown in Fig. 2"b# actually originatesfrom two separate (2,0,0)ortho $(0,2,0)ortho% peaks from twodomains above and below the scattering plane.There are two ways to express Miller indices; the high-
temperature tetragonal phase notation (H ,K ,L) tetra and thelow-temperature orthorhombic phase notation (H ,K ,L)ortho .Since all of the results shown in this paper are observed in
FIG. 1. Diagrams of the reciprocal lattice in the (H ,K ,0) scat-tering zone in the tetragonal phase "a# and the orthorhombic phase"b# for a single-domain crystal. The lower figures show diagrams ofthe reciprocal lattice in the (H ,K ,0) scattering zone "c# and in the(H ,0,L)ortho scattering zone "d# for a crystal with four domains.
FIG. 2. The results of elastic neutron scans A and B as shown inFigs. 1"c# and 1"d#, respectively.
PRB 61 4327FREEZING OF ANISOTROPIC SPIN CLUSTERS IN . . .
erties are also important as mentioned above. Stimulated bythose findings, we aimed to clarify the static magnetic prop-erties in the spin-glass phase of La1.98Sr0.02CuO4, in whichthe 3D antiferromagnetic long-range ordering just disap-pears. The most interesting issue is how the long-range 3Dantiferromagnetic ordering disappears and the spin-glass be-havior appears with hole doping.In this study we find that below !40 K quasielastic mag-
netic peaks develop at the positions where magnetic Braggpeaks exist in La2CuO4. This means that spin correlationsdevelop perpendicular to the CuO2 plane in addition to par-allel to it, suggesting that the 2D spin fluctuations, whichexist at high temperatures, at least in part freeze with freez-ing temperature enhanced due to the interplanar interaction.Our most important findings are that the spin correlations inthe CuO2 plane are anisotropic at low temperatures and thatthe spin directions in the spin clusters differs from that inpure La2CuO4. The spin cluster dimension along the aorthoaxis (!160 Å ) is !6 times longer than that along the borthoaxis (!25 Å ) at 1.6 K. From these results, it is concludedthat the spin-glass state can be described as a random freez-ing of quasi-3D spin clusters with anisotropic spin correla-tions.The format of this paper is as follows: The scattering
geometry used in this study is summarized in Sec. II. Experi-mental details are described in Sec. III. The experimentalresults of the elastic neutron measurements are presented andthe spin structure of the short-range ordered state is summa-rized in Sec. IV. The experimental results on the inelasticneutron measurements are presented in Sec. V. In Sec. VI wediscuss the magnetic properties of the spin-glass state.
II. SCATTERING GEOMETRY
Figure 1"a# shows the scattering geometry in the (H ,H ,0)scattering plane in the high-temperature tetragonal phase.La1.98Sr0.02CuO4 undergoes a structural phase transition fromtetragonal (I4/mmm) to orthorhombic (Bmab) structure at485 K. Figure 1"b# shows the scattering geometry in the(H ,K ,0) scattering plane in the low-temperature orthorhom-bic phase. The structure is slightly distorted with the aorthoand bortho axes almost along the diagonal directions of thea tetra and b tetra axes. Ideally, the experiments should be per-formed with a single-domain crystal. However, four domainsare expected to exist in real crystals since a twin structure isenergetically stable. For small rectangular samples it is pos-sible to prepare a single-domain crystal by heating thesample above the transition temperature and then cooling itdown while applying pressure along the aortho or bortho axis.However, the heat treatment is not efficient for large crystalsso that preparation of a single-domain sample is extremelydifficult in practice. Consequently in this experiment, wewere forced to use a four-domain crystal.The scattering geometry for the four-domain crystal is
shown in Fig. 1"c#. (H ,K ,L)ortho and (K ,H ,L)ortho nuclearBragg peaks are observed at nearby positions. As a result,four peaks are observed around (1,1,0)tetra and three peaksare observed around (2,0,0)tetra in the (H ,K ,0) scatteringplane. Figure 2"a# shows an elastic scan $scan A as indicatedin Fig. 1"c#% at (2,0,0)tetra . The two side peaks at (2,0,0)tetraoriginate from two of the domains and the central peak,
which is factor of !2 larger than each side peak, originatesfrom the other two domains. From these results, we estimatethat the four domains are equally distributed. Due to the twinstructure, the (H ,0,L)ortho and the (0,K ,L)ortho scatteringplanes are superposed upon each other as shown in Fig. 1"d#.Scan B in Fig. 1"d# corresponds to scan C in Fig. 1"c#. Sincethe vertical resolution is quite broad, the (2,0,0)ortho$(0,2,0)ortho% peak shown in Fig. 2"b# actually originatesfrom two separate (2,0,0)ortho $(0,2,0)ortho% peaks from twodomains above and below the scattering plane.There are two ways to express Miller indices; the high-
temperature tetragonal phase notation (H ,K ,L) tetra and thelow-temperature orthorhombic phase notation (H ,K ,L)ortho .Since all of the results shown in this paper are observed in
FIG. 1. Diagrams of the reciprocal lattice in the (H ,K ,0) scat-tering zone in the tetragonal phase "a# and the orthorhombic phase"b# for a single-domain crystal. The lower figures show diagrams ofthe reciprocal lattice in the (H ,K ,0) scattering zone "c# and in the(H ,0,L)ortho scattering zone "d# for a crystal with four domains.
FIG. 2. The results of elastic neutron scans A and B as shown inFigs. 1"c# and 1"d#, respectively.
PRB 61 4327FREEZING OF ANISOTROPIC SPIN CLUSTERS IN . . .
Octahedral tilts Magnetic moments
Same unit cell within one plane for both, but correlation with second plane in unit cell is different.
First experiments: neutron powder diffraction
Yang et al., JPSJ 56, 2283 (1987)
La2CuO4
Main panel: Ei = 14.7 meV
Insert (a): Ei = 5.1 meV, T = 150 K
Indexing based on Cmca space group
High resolution required to distinguish possible orthorhombic reflections.
First report mistakenly interpreted LTO (021) superlattice peak as due to AF order.
Ordered moment
spin wave theory: <S> = 0.303g ≈ 2.2
m ≈ 0.67 μB
arXiv:cond-mat/0512115
Elastic vs. inelastic weight
2D antiferromagnet with S=1/2
Total scattering: <S2> = S(S+1) = 3/4
Elastic weight: <S>2 = (0.303)2 = 0.09
Inelastic weight: <S2> - <S>2 = 0.66 = 0.88 <S2>
Most of the spin scattering is inelastic!This violates the premise of perturbative
spin-wave theory.
Antiferromagnetic spin waves
Assume ordered spins along z
Linear dispersion spin-wave velocity
coordination number(4 for square lattice)
Spin waves in La2CuO4
Headings et al., PRL (2010)
Exchange parameters
Headings et al., PRL (2010)
Coldea et al., PRL (2001)
Large J requires intermediate coupling
J = 4t2/ U
Suppose U = 8t
then J = t/2
J = 143 meV gives t = 0.3 eV
KCuF3: S = 1/2 spin chain system
T = 300 K
T = 6 K T = 50 K
T = 150 K
Lake et al., Nat. Mat. (2005)
Spinons and the 2-spinon spectrum
Zaliznyak, Nat. Mat. (2005)
S = 1/2, two-leg spin ladder
Jrung = 1.09 meV
Jleg / Jrung = 0.29
ground state: singlets on rungs
excited state: triplet can disperse along ladder
Savici et al., PRB (2009)
Magnetic critical scattering
Neutron scattering on single crystal La2CuO4 Shirane et al., PRL 59, 1613 (1987)
ElasticBraggintensity
Energy-integrated2D intensity
Rods of scattering from 2D spin correlations
Cu spins maintain 2D correlations to high T
S(q2D) ~ 1 / [(q2D)2 + ξ-2]
ξ = spin-spin correlation length
ξ-1 ~ exp(-αJ/T)
J = 135 meV ~ 1500 K
Theory:
Chakravarty, Halperin,+Nelson,PRB 39, 2344 (1989)
Hasenfratz+Niedermayer,Phys. Lett. B 268, 231 (1991)
Expt: Birgeneau et al., JPCS 56, 1913 (1995)
Single CuO2 layer should order AF at T=0
Chakravarty et al., PRL (1988)
ξ(T) is consistent withRenormalized Classical behavior and not Quantum Disordered
1/8 problem LTT
LTO
Moodenbaugh et al., PRB (1988)
Axe et al., PRL (1989)
Stripes and superlattice peaks
Spin and charge stripe order
1/2
1/2h
(0 2 0)
B
A(0 1 0)
SDW
CDW
0
k
(1 0 0)
(b) CDW 3.6K
75Kscan-B
1.70 1.75 1.800
800
1000
1200
Counts
/10m
in
k (r.l.u.)
(a) SDW 3.6Kscan-A
0.3 0.4 0.5 0.6 0.70
200
400
600
1400
Counts
/2m
in
La1.875Ba0.125CuO4, =0meV
Fujita et al., PRB (2004)
Charge order: stripes, not checkerboardIn
tens
ity -
Back
grou
nd
Qz (r.l.u.)
Hard (100-keV) X-ray diffraction study of La1.48Nd0.4Sr0.12CuO4 shows that there are
4 CuO2 layers per supercell
Only makes sense in terms ofstripe order!
Zimmermann et al.,Europhys. Lett. (1998)
Constant-energy slices through magnetic scattering
Stripe-ordered La1.875Ba0.125CuO4
T = 12 K
Tc < 6 K
JMT et al., Nature (2004)
La1.875Ba0.125CuO4
χ´´
Fujita et al., PRB (2004)
Spectral weight and dispersion
Universal magnetic spectrum
JMT et al., Nature (2004)Vignolle et al., Nat. Phys. (2007)Hayden et al., Nature (2004)Stock et al., Phys. Rev. B (2005), (2010)Xu et al., Nat. Phys. (2009)
0 10 20 30 40 50 60
Temperature (K)
1x10-5
1x10-4
1x10-3
1x10-2
1x10-1
1
-(M
-M0
)/H
ab
1x10-4
1x10-3
1x10-2
1x10-1
1
Resis
tivity (m
Ohm
-cm
)
!ab
Resis
tivity (m"
cm
)Tco
TsoTBKTTc T3D T1**T2**
!c # 10-3
H = 2 Oe
ZFC
H ! ab
H ! c
Td2
$4% [ &
$ &
(60 K
) ]
LBCO x=1/83D SC 2D SC Tco
Stripe order:
compatible with 2D SCat 40 K
frustrates interlayer Josephson coupling
Q. Li et al., PRL (2007)JMT et al., PRB (2008)
Intertwined Orders
Antiferromagnetism
Spin Stripes
Uniform d-wave Superconductor
Pair Density Wave
Intertwinedantiferromagnetism
andsuperconductivity
2D SC and Pair-Density-Wave Superconductor
CDW SDW
PDWFrustration of interlayer coupling: Himeda et al., PRL (2002) Berg et al., PRL (2007)
Intertwined superconductivity and antiferromagnetism
P.A. Lee, PRX (2014)
Intertwined orders(a)
(c)
(d)
(b)
Fradkin et al., RMP (2015)
AF order
SDW+CDW
d-wave SC
Pair density wave+SDW+CDW
Spallation Neutron Source
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HYSPEC
HYbrid SPECtrometer BL14B at the SNS (ORNL) Time-of-flight with area detector Polarization analysis
Ovi GarleaAndrei Savici
Barry Winn
Travis Williams
Melissa Graves-Brook
NIST Center for Neutron Research
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