tranquada neutrons p2 - qs3 quantum science summer...

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

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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