Observation of Field-Induced Transverse Néel Ordering in the Spin Gap System TlCuCl$_3$

5 Pages·0.19 MB·English

by  H. Tanaka

typeset using JPSJ.sty <ver.1.0b>
Observation of Field-Induced Transverse N´eel Ordering
in the Spin Gap System TlCuCl
3
Hidekazu Tanaka∗, Akira Oosawa, Tetsuya Kato1 Hidehiro Uekusa2, Yuji Ohashi2,
1 Kazuhisa Kakurai3 and Andreas Hoser4
0
0 Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo152-8551
2 1Faculty of Education, Chiba University, Inage-ku, Chiba 263-8522
2Department of Chemistry, TokyoInstitute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551
n 3Neutron Scattering Laboratory, Institute for Solid State Physics, The University of Tokyo, Tokai, Ibaraki
a
319-1106
J
4Hahn-Meitner-Insitut, GlienickerStraße 100, D–14109 Berlin, Germany
9
1 (Received )

l] Neutron elastic scattering experiments have been performed on the spin gap system TlCuCl3
e in magnetic fields parallel to the b-axis. The magnetic Bragg peaks which indicate the field-
tr- iTndautceQd=N´e(ehl,o0r,dl)erwinigthwoedred olbisnertvheedaf∗or−mc∗agpnleatnice.fieTldhehsipghinersttrhuacntutrheeingatphefioelrddeHregd≈pha5s.5e
s was determined. The temperature and field dependence of the Bragg peak intensities and the
.
t phaseboundaryobtainedwerediscussed inconnection witharecenttheorywhichdescribesthe
a
field-induced N´eel ordering as a Bose-Einstein condensation of magnons.
m
-
d KEYWORDS: TlCuCl3,spingap,field-inducedmagneticordering,spinstructure,neutronelasticscattering,Bose-
Einsteincondensation ofmagnons
n
o
c
[

The singletgroundstate withtheexcitationgap(spin meV on the planar dimer Cu Cl in the double chain.
2 2 6
v gap) is a notable realization of the macroscopic quan- The neighboring dimers couple magnetically along the
6 tum effect in quantum spin systems. When a mag- chain and in the (1,0,−2) plane.
7 netic field is applied in the spin gap system, the gap Our previous magnetic measurements revealed that
2 ∆ is suppressed and closes completely at the gap field TlCuCl undergoes 3D magnetic ordering in magnetic
3
1
H = ∆/gµ . For H > H the system can undergo fields higher than the gap field H ≈5.5 T.9) The mag-
0 g B g g
magnetic ordering due to three-dimensional (3D) inter- netization exhibits a cusplike minimum at the ordering
1
0 actions. Such field-induced magnetic ordering was stud- temperature TN. The phase boundary on the tempera-
/ ied first for Cu(NO ) · 5H O.1) However,the magnetic turevsfielddiagramisindependentofthefielddirection
t 3 2 2 2
a properties near H = H have not been investigated be- whennormalizedbytheg-factor,andcanberepresented
g
m
cause of the very low ordering temperature, the maxi- by the power law
- mum of which is 0.18 K. Recently, the study of field-
d (g/2)[H (T)−H ]∝Tφ , (1)
inducedmagneticorderinghasbeenrevived,becausethe N g
n
o spingaphasbeenfoundinmanyquantumspinsystems. with φ = 2.2, where H (T) is the transition field at
N
c Field-induced 3D ordering has been observed in several temperature T. These features cannot be explained by
v: quasi-one-dimensionalspin gap systems.2–6) the mean-field theory.13,14)
i This paper is concerned with field-induced magnetic Nikuniet al.15) arguedthatfield-inducedmagneticor-
X
ar ostrrduecrtiunrgein(spTalcCeugCrlo3u.pTPh2is1/cco)m.7p)oTulnCduhCal3s caomntoaninosclpinlaic- dcoenridnegnisnatTiolnCu(CBlE3Cca)nobfeexrecpitreedsetnrtiepdleatssa(mBaogsne-oEnisn)s.teIinn
nar dimers of Cu2Cl6, in which Cu2+ ions have spin-21. the magnon BEC theory, the external field H and the
These dimers are stacked on top of one another to form total magnetization M are related to the chemical po-
infinite double chains parallel to the crystallographic a- tentialµandthe number ofmagnonsN,respectively,as
axis. These double chainsarelocatedatthe cornersand µ = gµ (H −H ) and M = gµ N. The observed tem-
B g B
center of the unit cell in the b−c plane, and are sep- perature dependence of the magnetization and the field
arated by Tl+ ions. The magnetic ground state is the dependence of the ordering temperature9) were qualita-
spin singlet with the excitation gap ∆/kB ≈ 7.5 K.8,9) tively well described by the magnon BEC theory based
The magnetic excitations in TlCuCl3 were investigated on the Hartree-Fock (HF) approximation.15)
by Oosawa et al.,10) who found that the lowest excita- IfthemagnonsundergoBECatorderingvectorQ for
0
tion occurs at Q= (0,0,1) and its equivalent reciprocal H >H ,thenthetransversespincomponentshavelong-
g
points, as observed in KCuCl3.11,12) The origin of the rangeorder,whichischaracterizedbythesamewavevec-
gap is the strong antiferromagnetic interaction J =5.26 tor Q . When the magnetization is small, i.e., magnons
0
are dilute, the transverse magnetization m⊥ per site is
∗E-mail: [email protected].
2 HidekazuTanakaet al.
Table I. AtomiccoordinatesofTlCuCl3.
x y z 3000
Tl 0.7780 0.1700 0.5529 TlCuCl3
Cu 0.2338 0.0486 0.1554
2500 H // b
Cl(1) 0.2656 0.1941 0.2597
Cl(2) 0.6745 −0.0063 0.3177 k]
Cl(3) −0.1817 0.0966 −0.0353 n 48 2000 Q=(1 0 -3)
o
m
s/ H=12T T=1.9K
unt 1500
o
expressed by y [c
m⊥ =gµBpnc/2 , (2) ensit 1000
nt
I
wheren isthecondensatedensity. Thusthecondensate
c 500
density nc is proportional to the magnetic Bragg peak H=0T
intensity.
In order to confirm the transverse spin ordering cor- 0
58.0 58.5 59.0 59.5 60.0 60.5 61.0
responding to the off-diagonal long-range order of the
BEC,andtoinvestigatethetemperatureandfielddepen- 2θ (deg.)
denceofthetransversemagnetizationm⊥ corresponding
tothecondensatedensityn ,weperformedneutronelas-
c
tic scattering experiments on TlCuCl3. Fig. 1. θ−2θ scans for (1,0,−3) reflection measured at H = 0
The details of sample preparation were reported in and12Tat1.9K.
reference 9. Since the precise structural analysis for
TlCuCl hasnotbeenreported,weperformedthesingle-
3
crystal X-ray diffraction. The lattice parameters at
room temperature were determined as a = 3.9815˚A, Table II. Observedandcalculated magneticBraggpeakintensi-
b=14.1440˚A, c=8.8904˚A and β =96.32◦. The atomic ties at T = 1.9 K and H = 12 T for Hkb. The intensities are
normalizedtothe(0,0,1)M reflection. Risthereliabilityfactor
coordinates obtained are shown in Table I.
givenbyR=Ph,k,l|Ical−Iobs|/Ph,k,lIobs.
Neutron scattering experiments were carried out at
the E1 spectrometer of the BER II Research Reactor (h,k,l) Iobs Ical
oftheHahn-MeitnerInstitute withthe verticalfieldcry- (0,0,1)M 1 1
omagnetVM1. Theincidentneutronenergywasfixedat (0,0,3)M 0 0.014
Ei =13.9 meV, and the horizontal collimation sequence (0,0,5)M 0.063 0.107
was chosen as 40’-80’-40’. A single crystal of ∼ 0.4 cm3 (1,0,1)M 0.031 0.009
wasused. The sample wasmountedinthe cryostatwith (1,0,−1)M 0.074 0.127
(1,0,3)M 0.056 0.024
its(0,1,0)cleavageplaneparalleltothescatteringplane, (1,0,−3)M 0.398 0.380
∗ ∗
so that the reflections in the a −c plane were investi- (2,0,1)M 0.012 0.013
gated. The samplewascooledto 0.05K usingadilution (2,0,−1)M 0.128 0.102
refrigerator. Anexternalmagneticfieldofupto12Twas R 0.12
applied along the b-axis. Lattice parameters a = 3.899
˚A, c = 8.750 ˚A and β = 95.55◦ were used at helium
temperatures.
Magnetic Bragg reflections were observed for H > ever,veryweaknuclearpeakswerealsoobservedforodd
H ≈ 5.5 T, which could be indexed by Q = (h,0,l) l. To refine the magnetic structure, we used the atomic
g
withoddl. Figure1showstheθ−2θ scansfor(1,0,−3) coordinates shown in Table I and the nuclear scattering
reflection measured at H = 0 and 12 T at 1.9 K. These lengths bTl = 0.878, bCu = 0.772 and bCl = 0.958 with
reciprocal lattice points are equivalent to those for the the unit of 10−12 cm.16) The magnetic form factors of
lowest magnetic excitation at zero field.10) This result Cu2+ were taken from reference 17. The extinction ef-
indicates that the magnetic unit cell is the same as the fectwasevaluatedbycomparingobservedandcalculated
chemical one. Ferromagnetic Bragg reflections due to intensities for various nuclear Bragg reflections.
theinducedmomentcouldnotbedetectedinthepresent Figure 2 shows the spin structure determined at 1.9 K
measurements. The integrated intensities of nine mag- and 12 T. The directions of spins are in the a−c plane
netic reflections were measuredat 1.9 K under anexter- whichis perpendicular to the applied field. Spins on the
nalfield of12T. The results aresummarizedinTable II same dimers represented by thick lines in Fig. 2 are an-
together with the calculated intensities. The intensities tiparallel. Spinsarearrangedinparallelalongaleginthe
◦
ofnuclearBraggreflectionswerealsomeasuredtodeter- double chain, and make an angle of 39 with the a-axis.
mine the magnitude of the magnetic moment per site. The spins on the same legs in the double chains located
The nuclear peaks were observed at Q = (h,0,l) with at the corner and the center of the unit cell in the b−c
even l, as expected from the space group P2 /c. How- plane are antiparallel. Although scans in the reciprocal
1
Field-InducedTransverseN´eelOrderinginTlCuCl3 3
a
12
H=12.0T
TlCuCl
3
k] 10 6
0
8
2
n H // b
mo 8 10.0T m
s/ ⊥
unt Q=(1 0 -3) 4 (12
o 0
3ensity [10c 64 7.0T78..20TT TN B/Cu)µ-222+
k Int 6.6T6.8T 2
a
Pe 2 6.4T
6.2T
c
6.0T
5.8T
0 0
0 2 4 6 8 10
T (K)
Fig. 2. SpinstructureintheorderedphaseofTlCuCl3. Theex-
ternalfieldisappliedalongtheb-axis. Thedoublechainlocated
Fig. 3. Temperaturedependence oftheBraggpeakintensityfor
atthecornerandthecenterofthechemicalunitcellintheb−c
(1,0,−3) reflection in various magnetic fields. The square of
planearerepresentedbysolidanddashedlines,respectively.
thetransversemagnetization persite m2 isshownontheright
⊥
abscissa.
∗
plane including the b -axis were notperformed, the spin
structure shown in Fig. 2 is uniquely determined within surement,18) the smearing of the peak intensity around
thepresentmeasurements. Thusthetransversemagnetic T (H) is attributed not to the intrinsic smearing of the
N
ordering predicted by the theory13–15) was confirmed. ordering temperature due to additional effects such as
Comparing magnetic peak intensities with those of the staggered inclination of the principal axis of the g-
nuclear reflections, we evaluate the magnitude of the tensorandthe Dzyaloshinsky-Moriyainteraction,butto
transverse magnetization at 1.9 K and 12 T as m⊥ = the diffuse scattering. The amount of diffuse scattering
gµBhS⊥i = 0.26(2) µB. Since the N´eel temperature aroundTN(H)seemstobeindependentofpeakintensity
for H = 12 T is TN = 7.2 K, the magnitude of m⊥ at T =0.
at 1.9 K must be very close to that at zero tempera- Figure4showsthefielddependence ofthe Bragginten-
ture (see Fig. 2). For T → 0, the condensate density sityfor(1,0,−3)reflectionmeasuredatvarioustempera-
nc approximates the density of the magnon n, which is tures. Onthe rightabscissa,weshowm2⊥. Theintensity
given by n = 2m/gµB with the magnetization m per remainsalmostzerouptothetransitionfieldHN(T),and
site. The value of n at 12 T and 1.8 K is obtained as then increasesrapidly. It is clear that field-induced N´eel
n ≈ 0.028 with m ≈ 0.03 µB and g = 2.06.8,9) Sub- ordering takes place at HN(T). At 0.2 K, the intensity
stituting nc ≈ n ≈ 0.028 into eq. (2), m⊥ ≈ 0.24 µB. (m2⊥) is almost proportional to H −Hg just above Hg,
Thisvalueagreeswithm⊥ =0.26(2)µB obtainedbythe and is slightly concave at higher fields. With increasing
present measurement. temperature, the transition field H (T) increases, and
N
Figure 3 shows the temperature dependence of the the slope just above H (T) increases.
N
magnetic peak intensity for (1,0,−3) measured in var- The magnon BEC theory based on the HF approxi-
ious magnetic fields. On the right abscissa, we show mation15) predicts that the magnon density n and the
m2⊥ which corresponds to the condensate density nc = condensate density nc are both concave functions of
2(m⊥/gµB)2. The N´eel ordering can be detected at the µ = gµB(H −Hg) for T→0.15) The deviation from the
temperatures indicated by the thick arrows. With in- linearfunctionofµisduetothenoncondensatemagnons,
creasing external field, the ordering temperature TN(H) and is proportional to µ3/2. The field dependence of
and the intensity of the magnetic Bragg peak increase. the Bragg intensity for sufficiently low temperatures is
The anomaly around TN(H) becomes smeared as the explainable by the magnon BEC theory. However, the
magnetic field approaches Hg, which leads to the dif- magnetization for H > Hg is the slightly convex func-
ficulty in determining the N´eel temperature from the tionofH,8) asobservedinlow-dimensionalspinsystems.
temperature dependence of the Bragg peak intensity for This behavior disagrees with the magnon BEC theory.
H ≤ 6.6 T. Since the sharp anomaly is observed at We summarize the phase transition temperature
the transition field HN(T) in the field dependence of TN(H) and field HN(T) in Fig.5, in which the data ob-
the Bragg intensity (see Fig. 4), and the sharp phase tainedfromthepreviousmagneticmeasurementsarealso
transition was detected through the specific heat mea- plotted. Because the transition points determined from
4 HidekazuTanakaet al.
12 14
TlCuCl
3 TlCuCl H // b
k] 10 6 3
80 T=200mK 3.5K 12
n 2 H // b
mo 8 1.9K m TN (Magnetization)
unts/ Q=(1 0 -3) 5.0K 4 ⊥ (12 10 HTNN ((NMeaugtnroenti)zation)
3y [10co 6 2.7K 6.0K B0/Cµ-22 H (T) 8 HN (Neutron) φ=2.0(1)
nsit 4 u2+
e )
nt 4.2K 2
ak I 6
e 2
P
0 0 4
0 2 4 6 8 10 12 14 0 2 4 6 8
H (T) T (K)
Fig. 4. Magnetic field dependence of the Bragg peak intensity Fig. 5. ThephaseboundaryinTlCuCl3forHkbdeterminedfrom
for (1,0,−3) reflection at various temperatures. The square of the results of the temperature variation(closed rectangles) and
the transversemagnetization persite m2 isshownontheright fieldvariation(closedcircles)oftheBraggpeakintensity. Open
⊥
abscissa. circles and rectangles denote the transition points determined
fromthepreviousmagnetizationmeasurements. Thedashedline
isaguide forthe eyes. The solidlinedenotes the fitby eq. (1)
thepresentneutronscatteringexperimentsandthemag- with(g/2)Hc(0)=5.7(1)Tandφ=2.0(1).
neticmeasurementsliealmostonthesameline,they are
consistent with each other. This supports the predic-
tion of the magnon BEC theory that the magnetization
exhibits a cusplike minimum at the transition tempera- 1) K. M. Diederix, J. P. Groen, L. S. J. M. Henkens, T. O.
ture.15) Klaassenand N.J. Poulis: PhysicaB 94(1978) 9 andrefer-
ences therein.
Thephaseboundarycanbeexpressedbythepowerlaw
2) Z. Honda, H. Asakawa and K. Katsumata: Phys. Rev. Lett.
of eq. (1). In the previous magnetic measurements, the 81(1998)2566.
lowesttemperaturewas1.8K.Thelow-temperaturedata 3) P.R.Hammar,D.H.Reich,C.BroholmandF.Trouw: Phys.
contribute greatly to the determination of the exponent Rev.B57(1998)7846.
φ. Therefore,addingthepresentdatadownto0.05K,we 4) G.Chaboussant, P.A.Crowell,L.P.L´evy, O.Piovesana, A.
MadouriandD.Mailly: Phys.Rev.B55(1997)3046.
reevaluate the value of φ. We fit eq. (1) to the data for
5) M. Hagiwara, H. Aruga Katori, U. Schollwo¨ck and H. -J.
T < 4 K, which is lower than half the gap temperature Mikeska: Phys.Rev.B62(2000)1051.
∆/kB = 7.5 K. The best fit is obtained with Hg = 5.54 6) H. Manaka, I. Yamada, Z. Honda, H. Aruga Katori and K.
T and φ = 2.0(1), which is slightly smaller than the Katsumata: J.Phys.Soc.Jpn.67(1998)3913.
previous value of φ = 2.2. The exponent obtained, φ = 7) K.Takatsu,W.ShiramuraandH.Tanaka: J.Phys.Soc.Jpn.
66(1997)1611.
2.0(1),issomewhatlargerthanthe φ=1.5predictedby
8) W. Shiramura, K. Takatsu, H. Tanaka, M. Takahashi, K.
the magnon BEC theory. The difference between these
Kamishima, H. Mitamura and T. Goto: J. Phys. Soc. Jpn.
values may be attributed to the fluctuation effect which 66(1997)1900.
is disregardedin the HF approximation. 9) A. Oosawa, M. Ishii and H. Tanaka: J. Phys. : Condens.
Inconclusion,wehavepresentedtheresultsofneutron Matter11(1999) 265.
10) A. Oosawa, T. Kato, H. Tanaka, K. Nakajima, K. Nishi
elasticscatteringexperimentsperformedonthespingap
and K. Kakurai: J. Phys. Soc. Jpn. Suppl. in press, cond-
systemTlCuCl inhighmagneticfields alongtheb-axis.
3 mat/0004108.
The field-induced transverse N´eel ordering was clearly 11) T.Kato,K.Takatsu,H.Tanaka,W.Shiramura,M.Mori,K.
observed for H > H ≈ 5.5 T. The phase boundary NakajimaandK.Kakurai: J.Phys.Soc.Jpn.67(1998)752.
g
can be described by the power law as predicted by the 12) N.Cavadini,W.Henggler,A.Furrer,H.-U.Gu¨del,K.Kr¨amer
andH.Mutka: Eur.Phys.J.B7(1999) 519.
magnon BEC theory. However, the exponent obtained,
13) M. Tachiki and T. Yamada: J. Phys. Soc. Jpn. 28 (1970)
φ = 2.0, is somewhat larger than the predicted value of
1413.
φ=1.5. 14) M. Tachiki and T. Yamada: Suppl. Progr. Theor. Phys. No.
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Rev.Lett.84(2000)5868.
K.expresstheirgratitudetothe Hahn-MeitnerInstitute
16) P.J.Sears: International Tables for Crystallography, vol. C,
for their kind hospitality during the experiment.
ed.A.J.C.Wilson(Kluwer,Dordrecht,1992)p.383.
Field-InducedTransverseN´eelOrderinginTlCuCl3 5
17) P.J.Brown: InternationalTablesforCrystallography,vol.C,
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