Magnetic Excitations in the Spin-1 Anisotropic Heisenberg Antiferromagnetic Chain System NiCl$_2$-4SC(NH$_2$)$_2$

0.16 MB·English

by  S. A. Zvyagin

MagneticExcitations inthe Spin-1 AnisotropicHeisenberg Antiferromagnetic ChainSystem
NiCl -4SC(NH )
2 2 2
S. A. Zvyagin,1 J. Wosnitza,1 C. D. Batista,2 M. Tsukamoto,3 N. Kawashima,3
J. Krzystek,4 V. S. Zapf,5 M. Jaime,5 N. F. Oliveira, Jr.,6 and A. Paduan-Filho6
1DresdenHighMagneticFieldLaboratory(HLD),ForschungszentrumDresden-Rossendorf,01314Dresden,Germany
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
7 3Institute forSolid State Physics, University of Tokyo, Kashiwa, Chiba 227-8581, Japan
0 4NationalHighMagneticFieldLaboratory, FloridaStateUniversity, Tallahassee, FL32310, USA
0 5NationalHighMagneticFieldLaboratory,LosAlamosNationalLaboratory,MS-E536,LosAlamos,NM87545,USA
2 6Instituto de Fisica, Universidade de Sao Paulo, 05315-970 Sao Paulo, Brazil

n
a NiCl2-4SC(NH2)2 (DTN) isaquantum S = 1 chain system withstrong easy-pane anisotropy and a new
J candidatefortheBose-Einsteincondensation ofthespindegreesoffreedom. ESRstudiesofmagneticexci-
tationsinDTNinfieldsupto25Tarepresented. Basedonanalysisofthesingle-magnonexcitationmodein
0
thehigh-fieldspin-polarized phase and previous experimental results[Phys. Rev. Lett. 96, 077204 (2006)],
3
a revised set of spin-Hamiltonian parameters is obtained. Our results yield D = 8.9 K, Jc = 2.2 K, and
] Ja,b = 0.18Kfortheanisotropy,intrachain,andinterchainexchangeinteractions,respectively. Thesevalues
l areusedtocalculatetheantiferromagneticphaseboundary,magnetizationandthefrequency-fielddependence
e
- oftwo-magnonbound-stateexcitationspredictedbytheoryandobservedinDTNforthefirsttime. Excellent
r quantitativeagreementwithexperimentaldataisobtained.
t
s
.
t PACSnumbers:75.40.Gb,76.30.-v,75.10.Jm
a
m
- Antiferromagnetic (AFM) quantum spin-1 chains have ofachemicalpotential,changingthebosonpopulation.Inac-
d been the subject of intensive theoretical and experimental cordancewithmean-fieldBECtheory[12,13,14],thephase-
n
studies, fostered especially by the Haldane conjecture [1]. diagramboundaryforathree-dimensionalsystemshouldobey
o
c Due to quantumfluctuations, an isotropic spin-1 chain hasa apower-lawdependence,H−Hc1 ∼Tc3/2,asT →0.Above
[ spin-singletgroundstateseparatedfromthefirstexcitedstate theuppercriticalfield,Hc2 (definedatT = 0),thesystemis
by a gap ∆ ∼ 0.41J [2], where J is the exchange interac- inafullyspin-polarized(FSP)phase,andtheexcitationspec-
1
v tion.AsshownbyGolinellietal.[3],thepresenceofastrong trumisformedbygappedmagnons.
0 easy-planeanisotropyD cansignificantlymodifythe excita-
The compound NiCl2-4SC(NH2)2 (dichloro-tetrakis
5 tion spectrum, so that the gap size is not determined by the
thiourea-nickel(II), known as DTN), is a new candidate
7 strength of the AFM quantum fluctuations exclusively, but
1 for studying the BEC phenomenon in magnetic fields. It
dependsonthedimensionlessparameterρ = D/J. TheHal-
0 has a tetragonal crystal structure, space group I4, with two
7 danephaseispredictedtosurviveuptoρc =0.93[4],where moleculesintheunitcell[15,16]. Theanisotropy,intra-and
0 thesystemundergoesaquantumphasetransition. Forρ>ρc inter-chainexchangeparameters,D = 8.12K,J = 1.74K,
/ thegapreopens,butitsoriginisdominatedbytheanisotropy c
t andJ = 0.17K, respectively, were obtainedfroma fit of
a D, andthesystem isintheso-calledlarge-D regime. While a,b
m zero-field inelastic-neutron-scattering data [17] using gener-
theunderlyingphysicsofHaldanechainsisfairlywellunder-
alizedspin-wavetheory[18]. Itwasshownthatthe Nispins
- stood,relativelylittleisknownaboutthemagneticproperties
d are stronglycoupled along the tetragonalaxis, making DTN
(andparticularlytheelementaryexcitationspectrum)ofnon-
n asystemofweaklyinteractingS = 1chainswithsingle-ion
o HaldaneS =1AFMchainsinthelarge-Dphase.Intensethe-
anisotropy larger than the intra-chain exchange coupling.
c oreticalworkandnumerouspredictions[3,4,5,6,7,8,9,10]
Recently, it was proposed [16, 17] that the field-induced
:
v maketheexperimentalinvestigationoflarge-Dspin-1chains low-temperaturetransitionofDTNtotheAFM-orderedstate
i atopicalprobleminlow-dimensionalmagnetism.
X can be interpreted as a BEC of magnons, with values of
r Recently, weakly-coupledspin-1 chains have attracted re- Hc1 = 2.1 T and Hc2 = 12.6 T for the lower and upper
a critical fields, respectively. Although the observed overall
newed interest due to their possible relevance to the field-
picture is consistent with the BEC scenario, some questions
induced Bose-Einstein condensation (BEC) of magnons.
remain. One of the unsolved problems is a pronounced
When the field H, applied perpendicular to the easy plane,
(about 2 T) disagreement between the value for the upper
exceedsacriticalvalueHc1(definedatT =0),thegapcloses
critical field obtained experimentally and the predicted one
andthesystemundergoesatransitionintoanXY-likeAFM
phase with a finite magnetization and AFM magnon excita- (Hc2 = 10.85 T) [17]. This discrepancy is rather puzzling,
tions. If the spin Hamiltonian has axial symmetry with re- particularlyinlightoftheexcellentagreementforHc1.
specttotheappliedfield,theAFMorderingcanbedescribed InthisLetter,wereportelectronspinresonance(ESR)stud-
asBECofmagnonsbymappingthespin-1systemintoagas iesoftheelementaryexcitationspectruminDTNinmagnetic
ofsemi-hard-corebosons[11].Theappliedfieldplaystherole fieldsupto25T(∼ 2Hc2). Adistinctadvantageofthehigh-
2
tenas
2.0 2.0
H < Hc1 H > Hc2 H = JνSj·Sj+eν + [D(Sjz)2+gµBHSjz], (1)
Two-magnon bound state
nits) 1.5 1.5 Xj,ν Xj
u whereν = {a,b,c}. Acorrespondingdispersionoflow-field
rb. Ms=+1 E F magnetic excitations is shown schematically in Fig. 1 (left).
a 1.0 1.0
gy ( A Then, the frequency-field dependence of the ∆Ms = ±1
er Single-magnon state ESR transitions can be written as ωA/B = ∆ ± gµBH,
En 0.5 M=-1 0.5 where∆correspondstotheenergygapatk=0andH = 0.
s
B C These transitions were observed in our experiments and are
M=0
0.0 s 0.0 denotedinFig.2bycircles[20].Thezero-fieldsplittingmea-
-p k0 p -p 0k p sureddirectlyyields∆ = 269GHz,whichagreesquitewell
--pp withresultsofneutron-scatteringmeasurements,∆k=0 =1.1
FIG.1: (coloronline)Aschematicviewofthemagnetic-excitation meV [17]. No splitting within the excited doublet in zero
dispersion in an S = 1 Heisenberg chain with strong easy-plane field was observed, which indicates the absence of in-plane
(D > 0) anisotropy for two arbitrary fields, H < Hc1 (left) and anisotropy.Fromtheslopeofthefrequency-fielddependences
H > Hc2 (right),withamagneticfieldappliedalongtheprincipal ofthebranchesAandB,thegfactorwasdetermineddirectly,
axisz. NotethattheESRtransitionsdenotedbyA,B,C,E,andF
g =2.26[21].
occuratk=0. Two-particlecontinuapredictedforbothregionsare
notshownforsimplicity. As mentioned before, when the magnetic field exceeds
the upper critical field, Hc2, the system is in the fully spin-
polarized state (a corresponding dispersion of low-energy
magneticexcitationsisshownschematicallyinFig.1(right)).
Excitationsfromthe groundstate to the single-magnonstate
fieldapproachfordeterminingthespin-Hamiltonianparame-
have been observed in our experiments and are denoted by
ters of spin-1 AFM chains is the availability of exact theo-
squares in Fig. 2 (the resonance C). These excitations cor-
retical expressionsfor the spin-polarizedphase. Analysis of
respond to a single-spin flip from the S = −1 to S = 0
thehigh-fieldsingle-magnonbranchhasallowedustoobtain z z
stateandareuniformlydelocalizedovertheentirelatticewith
the precise value for the anisotropy parameter, D = 8.9 K.
a well-defined momentum k. The S = 0 state propagates
Usingresultsofrecentthermodynamicandneutron-scattering z
alongthelatticeasafreequasiparticlewithhoppingJ along
measurements[16, 17], we were able to refine the exchange ν
theν direction(ν = {a,b,c}),thatarisesfromthetransverse
parameters, obtaining J = 2.2 K, J = 0.18 K for intra-
c a,b
and inter-chain exchange interactions, respectively. In addi-
tion, we present magnetocaloric-effect and low-temperature
magnetization data allowing us to check the obtained set of e 683.3 GHz
parameters. These values agree well with those obtained by ncs) 294.4 GHz E
firetstiunlgtstohfeqAuFaMntupmhaMseobnoteunCdaarrlyoasnimduthlaetimonasg,naentidzantiicoenlywrieth- smitta b. uint A 216.9 GHz
producebothcriticalfields,Hc1 = 2.1TandHc2 = 12.6T. ran(ar B C T = 1.6 K
T
Furthermore,we reporton the first directand reliable obser-
vationofthe two-magnonboundstatesinspin-1AFM chain 700 E F
systemwithstrongeasy-planeanisotropy,predictedbytheory z) 600 A
H
[9]forthehigh-fieldspin-polarizedphase.Usingtheobtained G 500 C
(
parametersofthespin-Hamiltonianwewereabletocalculate cy 400
n
frequency-fielddependencesof the two-magnonbound state e 300
u
excitations.Excellentquantitativeagreementbetweenthethe- eq 200
oreticalpredictionsandexperimentisobtained. Fr 100 B
The investigation of the magnetic excitation spectra in 0
0 4 8 12 16 20 24
single-crystalline DTN samples was done using a tunable- Field (T)
frequency submillimeter-wave ESR spectrometer [19] with
FIG. 2: (color online) Top: Typical ESR transmission spectra in
externalfieldH appliedalongthetetragonalcaxis. DTN,takenatfrequenciesof216.9, 294.4, and683.3GHzatT =
1.6K,H kc. Bottom: Frequency-fielddependenceofmagneticex-
Insufficientlysmallfields, a spin-1Ni(II)system isinthe
citationsinDTN,takenattemperaturesof1.6K(themodesA,B,C,
quantum-paramagnetic (QPM) phase, having the Ms = 0 andE)and4.3K(themodeF),H kc. Symbolsdenotetheexper-
ground state. The excitation spectrum is gapped and deter- imental results, and linescorrespond toresults of calculations (see
mined by the ∆Ms = ±1 transitions. For fields H applied textfordetails).
perpendiculartotheeasyplane,theHamiltoniancanbewrit-
3
bound-stateexcitationscorrespondstoadouble-spin-fliptran-
1.4 1.2 sitionfromS = −1toS = +1. Thetransversepartofthe
z z
Heisenberg term of H mixes this state with the one that has
1.2 QPM 1.0 a pair of S = 0 states. Since the diagonal energy differ-
) z
K 1.0 ence between these two states, 2D, is much bigger than J
ure ( 0.8 FM 0.8 M and Ja (associated with hopping in the c and a directions)c,
at 0.6 /M the distance between the two Sz = 0 sites remains finite,
r
pe 0.6 sat giving rise to a two-magnon bound state. The two-magnon
m 0.4 bound states appear to be a specific feature of anisotropic
e 0.4
T spin-1 Heisenberg systems. It is worth mentioning that the
0.2 XY-AFM 0.2 two-magnonboundstates were alreadypredictedin 1970by
SilberglittandTorrance[5]forHeisenbergferromagnetswith
0.0 0.0
0 2 4 6 8 10 12 14 single-ionanisotropy. Later on, this subjectattracted a great
Filed (T)
dealofattentionduetoitspotentialrelevancetotheintrinsic
FIG. 3: (color online) Temperature-field diagram of the AFM- localizedspinmodesinanisotropicferromagnets[22]andan-
ordered phase obtained from magnetocaloric-effect measurements tiferromagnets[23].Itwassuggested[9]thatthetwo-magnon
(openedsquares), andthemagnetizationdatatakenatT = 16mK boundstatesshouldmakeadistinctcontributiontotheexcita-
(openedcircles),H kc. ResultsofthequantumMonteCarlocalcu- tionspectrumofS=1large-DAFMchainsabovetheupper
lationsofthephase-diagramboundariesandthemagnetizationusing
the set of parameters obtained as described in text are denoted by critical field Hc2 and that their effectcan be unambiguously
identifiedbyESRmeasurements.Asignatureoftwo-magnon
closedsquaresandcircles,respectively.
bound states was obtained by means of high-field ESR in
thespin-1chaincompoundNi(C H N ) Ni(CN) (knownas
2 8 2 2 4
NENC) [24]. A broad absorption was detected in the high-
partoftheHeisenberginteraction.Therearealsodiagonalen- fieldspin-polarizedphase. Basedonanalysisofthetempera-
ergygainsof−2(Jc+2Ja)duetotheIsingpart,and−Ddue ture dependenceof the ESR intensity, this feature was inter-
tothesingle-ionanisotropy. Thediagonalenergycostcomes pretedas transitionsfromthe single-magnonto two-magnon
fromtheZeemaninteractiongµBH.Then,thesingle-magnon boundstates. However,acloserlookatESRspectrainNENC
excitationdispersioncanbecalculatedexactly: uncovered a more complicated structure than predicted (in-
cluding, for instance, two non-equivalentNi sites and, most
ω(k)=gµ H −D−2(J +2J )+2(J cosk +
B c a c z likely,a finitein-planeanisotropy),whichmadethe accurate
Jacoskx+Jbcosky). (2) quantitativecomparisonwiththetheorydifficult.
Herewe reportonthe firstobservationof transitionsfrom
TheESRtransitionstakingplaceatk=0havethefrequency
thegroundstatetotwo-magnonboundstateinaspin-1AFM
ω = gµ H − D. The best fit of the ESR data denoted
C B
chain system with strong easy-plane anisotropy in the high-
by squares in Fig. 2 reveals D = 8.9 K for the anisotropy
fieldFSPphase.Thecorrespondingexcitationsaredenotedby
constant.FromtheexactexpressionforHc2,givenas[17]
trianglesinFig.2,bottom(theresonanceE,Fig.2(top)).The
1 frequency-fielddependence of the ground-state two-magnon
Hc2 = gµ (D+4 Jη), (3) bound-state excitations can be calculated exactly using the
B
η
X set of parameters obtained as described above. Results of
and using Hc2 = 12.6 T [16, 17] we obtain ηJη = corresponding calculations are shown in Fig. 2 (bottom) by
J +2J =2.557K.Thezero-fielddispersionofmagneticex- the line E. One more resonance absorption was observed at
c a
P
citations calculated using neutron-scatteringdata [17] yields higher temperatures (which indicates transitions within ex-
J /J = 0.082. Thus,inadditiontotheanisotropyconstant citedstates). ThecorrespondingdataobtainedatT = 4.3K
a c
D = 8.9 K, all three exchangeparameters, J = 2.2 K and aredenotedinFig.2(bottom)bystars. ThisESRmodecorre-
c
J =J =0.18K,canbecalculatedquiteprecisely. spondstotransitionsfromthesingle-magnontotwo-magnon
a b
Thephase boundary(obtainedfrommagnetocaloric-effect bound states (Fig. 1 (right)), which occur at k = 0. The
measurements)andthefielddependenceofthemagnetization frequency-fielddependenceofthesetransitionscanbecalcu-
atT = 16mKwerecomputedfortheobtainedsetofparam- lated,usingtheexpressionωF =ωE−ωC (whereωE andωC
eters by means of a quantum Monte Carlo simulations for a aretheexcitationfrequenciesforthemodesEandC,respec-
finite lattice of L3 sites, L = 16. Fig. 3 shows a very good tively),andisdenotedinFig.2bythelineF.Inbothcases,ex-
agreement between the calculated (closed symbols) and ex- cellentagreementwithexperimentaldatawasachieved,which
perimental(openedsymbols)data. showsthatourmodelactuallyworksfairlywelluptoT ∼J.
In addition to ordinary single-magnon states and two- Let us now discuss the low-field quantum-paramagnetic
magnon continuum, the theory [9] predicts the existence of phase. As mentioned before, in this phase the system has
two-magnon bound states (sometimes referred to as single- the M = 0 ground state and a gapped excitation spectrum
s
ionboundstates[5]).Thephysicalpictureofthetwo-magnon formedby∆M = ±1transitions. Theobtainedsetofspin-
s
4
Hamiltonian parameters can be used to calculate the lower Acknowledgments.– The authors express their sincere
critical field, employing the spin-wave theoretical approach thanks to A.K. Kolezhuk, A. Orenda´cˇova´, and S. Hill for
[18]: fruitful discussions. A portion of this work was performed
at the National High Magnetic Field Laboratory, Tallahas-
1
Hc1 = µ2−4s2µ Jη. (4) see, FL, which is supportedbyNSF CooperativeAgreement
gµBs η No. DMR-0084173,bytheStateofFlorida,andbytheDOE.
X S.A.Z. acknowledgesthe support from the NHMFL through
Theparametersµ=10.3ands2 =0.92canbeobtainedfrom the VSP No. 1382. The Monte Carlo simulation was car-
theself-consistentequations: riedoutonSGIAltrix3700Bx2attheSupercomputerCenter,
ISSP,UniversityofTokyo. A.P.FandN.F.O.,Jr. aregrateful
D =µ(1+ 1 γk) (5) forsupportfromCNPqandFAPESP(Brazil).
N ω
s k
k
X
and
s2 =2− 1 µ+s2γk, (6) [1] F.D.M.Haldane,Phys.Lett.93A,464(1983);Phys.Rev.Lett.
N ω 50,1153(1983).
s k
Xk [2] R.Botetetal.,Phys.RevB28,3914(1983);M.P.Nightingale
andH.W.J.Blote,Phys.Rev.B33,659(1986);T.SakaiandM.
whereγk = 2 νJνcoskν andωk = ωkA/B(H = 0). Itis Takahashi,Phys.Rev.B42,1090(R)(1990).
importanttomention,thatthedisagreementbetweenthecal- [3] O.Golinellietal.,Phys.Rev.B46,10854(1992).
P
culated(Hc1 = 2T)andexperimental(Hc1 = 2.1T)values [4] T.SakaiandM.Takahashi,Phys.Rev.B42,4537(1990).
isduetolimitationsofthespin-wavetheoryappliedtoquasi- [5] R.SilberglittandJ.B.Torrance,Jr.,Phys.Rev.B2,772(1970).
[6] S.T.Chiu-Tsaoetal.,Phys.Rev.B12,1819(1975).
one-dimensional spin systems. Using the spin-wave theory
[7] N. Papanicolaou and P.N. Spathis, Phys. Rev. B 52, 16001
[18] and the set of parameters obtained as described above,
(1995).
thefrequency-fielddependenceoftheESRtransitionscanbe [8] M.Orenda´cˇetal.,Phys.Rev.B52,3435(1995).
calculatedusingtheexpression[17] [9] N.Papanicolaouetal.,Phys.Rev.B56,8786(1997).
[10] A.K. Kolezhuk and H.J. Mikeska, Prog. Theor. Phys. Suppl.
ωA/B = µ2+4s2µ J ±gµ H. (7) 145,85(2002);Phys.Rev.B65,014413(2001).
η B [11] C. D. Batistaand G. Ortiz, Phys. Rev. Lett.86, 1082 (2001);
s Xη Adv.Phys.53,1(2004).
[12] I.Affleck,Phys.Rev.B43,3215(1991).
Eq.(7) predictsa valueof Hc1 = 2 T,thatis in factin rela- [13] T.GiamarchiandA.M.Tsvelik,Phys.Rev.B59,11398(1999).
tivelygoodagreementwiththeexperimentalvalueHc1 =2.1 [14] T.Nikunietal.,Phys.Rev.Lett.84,5868(2000).
T. The results of the calculations are presented by the lines [15] A.Paduan-Filhoetal.,J.Chem.Phys.74,4103(1981).
A and B in Fig. 2 (bottom) together with experimental data [16] A.Paduan-Filhoetal.,Phys.Rev.B69,020405(R)(2004).
denoted by circles. The agreement in this case is reason- [17] V.S.Zapfetal.,Phys.Rev.Lett.96,077204(2006).
[18] B.R.Cooperetal.,Phys.Rev.127,57(1962).
ablygood,consideringthatthequasi-one-dimensionalnature
[19] S.A.Zvyaginetal.,PhysicaB346-347,1(2004).
ofthesystemcausespronouncedquantumfluctuationsofthe
[20] AminorimpurityphasewasdetectedinsomeESRspectra.Al-
AFM-orderparameterinthelow-fieldQPMphase.
thoughtheoriginofthisphaseisnotclearatthemoment, we
Insummary,asystematicESRstudyofthemagneticexci- speculatethatitismostlikelyoriginatedfromasuperficiallayer
tationsinDTN,anS = 1HeisenbergAFMchainmaterialin ofDTN-crystalsattackedbytheGE-varnishsolvent,whichwas
the large-D regime, has been presented. Investigationof the usedtofixthesampleinsidethesample-holder.
high-fieldmagnonexcitationsallowedusto obtaina reliable [21] Withaccuracy better than ±0.5%. Forfurther references: the
anisotropy constant D is calculated withaccuracy better than
setofthespin-Hamiltonianparameters,whichwasemployed
±3%;theaccuracyofestimationofexchangeparametersisba-
to calculate the ESR spectrum in a broad range of magnetic
sicallydeterminedbyaccuracyofestimatingtheuppercritical
fields and frequencies. These values agree very well with fieldHc2andbyaccuracyofneutronscatteringmeasurements
the onesobtained from fitting the AFM-phase boundaryand [17],whichisbetterthan±10%.
low-temperaturemagnetizationofDTNwithresultsofquan- [22] R.F.Wallisetal.,Phys.Rev.B52,R3828(1995).
tum Monte Carlo simulations, including both critical fields. [23] R.Laietal.,Phys.Rev.B54,R12665(1996).
Theparameterswereusedtocalculatethefrequency-fieldde- [24] S.A. Zvyagin et al., Czech. J. Phys. 46, 1937 (1996); M.
Orenda´cˇetal.,Phys.Rev.B60,4170(1999).
pendence of two-magnon bound-state excitations, predicted
by theory and observedin DTN for the first time. Excellent
agreementbetweenthetheoryandexperimentwasobtained.