Modulated Kubo-Toyabe functions to study fluctuated weakandPhysics
muon
Ⅲ-2. magnetism
Atomic & Solid State(Muon)
Modulated Kubo-Toyabe
functionstostudyfluctuatedweakmagnetism
andmuondiffusion at pseudogap state of underdoped La2-xSrxCuO4
diffusion
at pseudogapfunctions
state of underdoped
La2-xﬂuctuated
SrxCuO4
Modulated
Kubo-Toyabe
to study
weak

RIKEN Accel. Prog. Rep. 51 (2018)

D. Umar,and I. Watanabemagnetism and muon M.
diﬀusion
atpseudogap state of underdoped
M. D. Umar, 1, 2 and I. Watanabe 1, 2
La2−x
One of the interesting puzzles for understanding
theSrx CuO
ve been4implementing the numerical solution of Volterra
＊1,＊2

＊1,＊2

＊ ＊

＊ ＊of thepuzzles atfor theunderstanding
thevebeen implementing
numerical solution
natureOnestateof interesting
the pseudogap
cuprate-based
integralequation in theMathematica
softwareof Volterrawith the∗1,∗2∗1,∗2 inin Mathematica
nature state of superconductor
the pseudogap attheequation
softwarethehigh-temperature
is M.
theD.cuprate-based
discrepancy
trapezoidal
method,which every areaof the withpartitioned
Umar
and integral
I.
Watanabehigh-temperature
superconductor
is thetrapezoidal
which everyasareaof the partitioned
betweenneutron scatteringand muonspin discrepancy
relaxation
integration method,interval isinapproacheda trapezoid.
between neutron
scatteringof andmuon spin
relaxation
integration
is approached as a trapezoid.
experiments
the detection
spontaneous
magneticorder
Asymmetryinterval
Function
One of the ininterestingpuzzlesfor understanding
theexperimentsin theorbital
detectionof spontaneous
magnetic
order
by,forinstance,(loop)
currentor
intra-unit-cellAsymmetry Function
nature state of the pseudogap at the cuprate-based highby, forinstance,
orbital(loop) scattering
current or experimentintra-unit-cell(IUC)
magneticorder.
A neutronontemperature
superconductor
isthediscrepancy
between(IUC)1) magnetic
order.
neutron scatteringexperimenton
YBCO
has shown
the Apresenceof anorderedmagnetic
neutronscattering
and
muon
spin
relaxation
experi1)
YBCO
has shownpresenceof anordered magnetic
state
preserving
thethetranslational
symmetryof thements
the detection
spontaneoussymmetry
magneticoforderstatein preservingthe oftranslational
theunderlying
lattice, indicating
the orbital current model.
Thisby, underlying
for instance,orbital(loop)
current
orintra-unit-celllattice,indicating
theorbital
current
model.
This
result was also supported earlier by an angle-resolved(IUC)
magneticorder.
A neutron
experimentresultwas also
supported
earlierscatteringby experiment,
an angle-resolved
photoemission
spectroscopy(ARPES)
which1)
on YBCOhasshown
thepresenceof
anorderedmagphotoemission
spectroscopy (ARPES)
experiment,
confirmed
time-reversal-symmetry
breaking(TRSB) whichat theneticstate
preserving
thetranslational
symmetry
ofthe
confirmed
time-reversal-symmetry
breaking(TRSB)
atthe
pseudogap state of Bi-2212.2)2) Contrary to Neutron
pseudogap
state indicatingof Bi-2212.
Contraryto Neutron
underlying
lattice,theorbitalcurrentmodel.
scattering and ARPES results, SR has confirmed thescatteringand ARPESresults, SR
has byconfirmed
theThisresultwasalso
supported
earlieran orderedangleabsenceof TRSB
associatedwith thespontaneous
absenceof TRSB associatedwith the spontaneousorderedresolvedphotoemission
spectroscopy(ARPES)
experi3)
magnetic state in the single crystal of LSCO. 3) However,magnetic statein the single
crystal of LSCO. 4)However,ment,
conﬁrmed
time-reversal-symmetry
breakusingwhicha pulse-SRexperiment,
Watanabe et al. 4) strongly
Fig. 1. Line shapes of the muon spin relaxation functionusing
a pulse-SRexperiment,state
Watanabe
et al. 2)stronglyingindicated(TRSB)
atthepseudogapof
Bi-2212.
ConPig. 1. LineLine
shapes of ofthemuon
muonspin
spin
relaxation
function
thepresenceofofweak
magnetismatthepseudogapsummationinternal
ﬁeldsfrom function
theordered
Pig.for1. theshapes of therelaxation
indicatedthepresence
weakmagnetismatthepseudogap
trarytoNeutronscatteringand
ARPES
results,
µSR
for
the
summation
ofinternal
fieldsfrom
theorderedstate
LSCOsample.
formagnetic
the summation
fields fromthe andorderedstate ofofa internal
polycrystalline
samplenuclearstateinina apolycrystalline
polycrystalline
LSCO
sample.
5)has conﬁrmed
the absenceof TRSB
associatedwith themagnetic state
state of aa polycrystalline
polycrystallinesample
sampleandand nuclear
AAtheoreticalmagnetic
moments in of
the zero-ﬁeld condition
plottednuclear
for diﬀertheoreticalstudy
study5) onon thethe effecteffect ofof screening
screening charge
chargemoments inin the zero-fieldzero-fieldcondition
conditionplottedplotted fordifferent
different
spontaneous
magnetic
state inbeen
the single crysdensity
aorderedtomoments
ent ratiostheof the ﬁeld fromthe ordered formagnetic state
densityonon3)
amuon
muon (point(point charge)
charge) has
has been proposed
proposed toratiosofthefieldfromtheorderedmagnetic
state(ω=γB)μB)
talexplainofLSCO.
However,usingapulse-µSRexperiment,Toratiosof γthefield from the ordered magnetic state (ω=γexplainthe
thediscrepancy.
discrepancy.
To confirm
confirm thethe presence
presence ofof weakweak(ω =µ B) to the width of the dipole nuclear ﬁeldμ (∆).
4)
tothewidthof
thedipole
nuclearfield(Δ)
stronglyindicatedthepresenceof
Watanabeetal.
magnetismassociatedwiththeorbital
current
modelorIUC
to the width of the dipole nuclear field (Δ)
magnetism associated with the orbital current model or IUC
weakmagnetismatpseudogap
in abymagneticorder
totothescreeningprocess,
applying
AsymmetryFunction
Functionmagneticorderdue
duethethescreeningstateprocess,
bypolycrysapplying
AsymmetrytallineLSCO
sample.
theFormula,
the Kubo-Golden
Kubo-Golden
Formula, we
we havehave extended
extended known
known5) (muon spin relaxation functions)Kubo-Toyabe
AKubo-Toyabe
theoretical functions
study
the eﬀect
screening
chargefunctionson(muon
spin ofrelaxation
functions)
forforfour
scenarios
ofofthefieldof nuclear
density
ona muon(pointcharge)
has been proposedfour
scenarios
theinternalinternal
fielddistribution
distribution
sites
byweaktomoments
explainthediscrepancy.
To conﬁrm
the presence
momentson
onmuonmuon
sitesbroadened
broadenedby aa comparable
comparable
Theoftwosourcesmagnetism.
The scenarios
scenariosof withtwo independent
independentof magnetism.
weakmagnetismassociatedtheorbital current
includea a Gaussian
distributionmoments)
includeGaussian order
distribution(nuclearmodel
or IUC
magnetic
due to (nuclear
thescreening
probroadened
byby athe(orderedbroadeneda delta-functiondelta-function
distribution
cess,by applying
Kubo-Goldendistribution
Formula, wehave
magnetic
state Kubo-Toyabe
polycrystallinesample),magnetic
state
inin aa polycrystallinesample),
Lorentzianextendedknownfunctions(muon
spin redistribution,andVoigtian
distribution.
Another
distribution,andVoigtian
distribution.
Anotherpossibility
laxation functions) for four scenarios of the internal
ﬁeld
nuclearof
moments
belonging
Voigtiansites
forfornuclear
moments
aa Voigtian
distribution
distribution
nuclearbelonging
momentstotoon
muon
broadbroadenedbybyweak
weaksources
sources with
with aa Lorentzian
Lorentzian distribution
broadened
enedby a comparable
weak magnetism.
The scenarios
hasbeenbeendeveloped.
developed.
hasFig. 2. Line shapes of muon spin relaxation function for theof two independent sources include a Gaussian distribuPig.
2.
ofofmuonspin relaxation
function
forforSincemuon
muondiffusion
diffusionand
andthethedynamic
dynamic behavior
behavior ofof weak
weakPig.
2. LineLine shapes
shapesmuonfunction
Sincesummationof internalﬁelds spin
fromrelaxation
the orderedmagnetic
tionmagnetism(nuclearmoments)
broadenedbya
delta-function
thesummationofinternal
fieldsfromtheorderedmagnetic
inthedependenceoftemperatureanddoping
thesummationof internal fields
fromorderedmagnetic
magnetism in the dependence of temperature and dopingstateof a polycrystalline
sampleandthenuclearmoments
indistribution(orderedmagnetic
stateinpolycrystalline
stateofaa polycrystalline
sampleandnuclear
moments
in inconcentration
canalsoalsopossiblyoccur
occur inin aaa SR
SRexperiment,
state
ofpolycrystalline
sampleandnuclear
moments
concentration
canpossiblyexperiment,
thelongitudinal-ﬁeld(LF)
conditionplottedfordiﬀerentsample),
Lorentzian
distribution,and
Voigtiandistributhe
have extended
extended muon
muon spin
spin relaxation
relaxation functions
functions forthe longitudinal-fieldlongitudinal-field(LF)(LF)condition
conditionplottedplottedforfordifferent
different
wewehaveforratios ofthe
the appliedlongitudinallongitudinal ﬁeld
(ω0μB=LFγ) µto
BLFtion.dynamic
Anotherpossibility
for nuclear
moments
belongingratiosthe) tocases(modulated
Kubo-Toyabe
functions).
The0=γratios ofof the applied
applied longitudinalfield
field(ω(ω=γB
dynamic cases(modulated
Kubo-Toyabefunctions).
The0μ LF) to thethe fromﬁeld thefrom
the orderedmagneticstate(ω=γB).
µto adynamic
Voigtiancases
distribution
broadenedby weak asources
fieldorderedmagneticstate(ω=γB).
areconstructedbyapplying
strongμ B).
field from the ordered magnetic state (ω=γdynamic cases are constructed by applying a strong
μwithcollisiona Lorentzian
distribution
has internalbeen developed.
modelwithwithmodulated
field onon muonmuon
collision modelaa modulated
internal fieldongoing
research, we have been implementing the nuSincemuonasasdiﬀusion
andprocess.
the dynamic
behavior
ofsitestreatedtreatedMarkovianprocess.
The equations
equationsof theReferences
sites
a aMarkovianTheoftheReferences
mericalsolutionof Volterra
equation
modulated
Kubo-Toyabe
function for the developed
1) B. Fauque et al., Phys.
Rev. Lett.integral
96, 197001(2006). in Matheweak
magnetismin the dependence
andmodulated
Kubo-Toyabe
function offortemperature
the developed
1) B. Fauque et al., Phys. Rev. Lett. 96, 197001 (2006).
maticasoftwarewiththetrapezoidal
method,in which
scenarios
exist in the Volterra
integral equation,andthey2) A. Kamiski et al., Nature, 416, 610 (2002).
dopingconcentration
also integral
possiblyoccur inanda µSR
scenarios
exist in the canVolterra
equation,they2) A. Kamiski et al., Nature, 416, 610 (2002).
every
areaofthepartitioned
integrationinterval
is apcanbesolved
numerically.
Inourongoing
research,
weha3)G.
J.
MacDougall
etal.,
Phys.
Rev.
Lett.
101,017001(2008).
experiment,have extended
muon research,spin relaxation
can be solvedwe
numerically.
In our ongoing
we ha3) G. J. MacDougall et al., Phys. Rev. Lett. 101, 017001 (2008).
proachedasa
trapezoid.
4)
I.
Watabeetal.,
J.
Phys.
Soc.
Jpn.
73,2232(2008).
functionsfor dynamic cases (modulated Kubo-Toyabe＊14) I. Watabe et al., J. Phys. Soc. Jpn. 73, 2232 (2008).
RIKEN Nishina Center＊15) A. Shekhter et al., Phys. Rev. Lett. 101, 227004 (2008).
＊2RIKEN Nishinafunctions).
Thedynamiccases
are constructed by apDepartment
of Center
Physics,
HokkaidoUniversity5) A. Shekhter et al., Phys. Rev. Lett. 101, 227004 (2008).
＊2References
Departmentof Physics,
Hokkaido
Universityplyinga strong
collisionmodelwith a modulated inter1) B. Fauque et al., Phys. Rev. Lett. 96, 197001 (2006).
nal ﬁeld on muon sites treated as a Markovian process.
2) A. Kamiski et al., Nature (London), 416, 610 (2002).
The equations of the modulated Kubo-Toyabe function3) G. J. MacDougall et al., Phys. Rev. Lett. 101, 017001
for the developed scenarios exist in the Volterra integral(2008).
equation, and they can be solved numerically. In our4) I. Watabe et al., J. Phys. Soc. Jpn. 73, 2232 (2008).
∗1∗25) A. Shekhter et al., Phys. Rev. Lett. 101, 227004 (2008).
RIKEN Nishina Center
Department of Physics, Hokkaido University- 198 -