Table Of ContentPolar charge and orbital order in 2H-TaS
2
Jasper van Wezel
Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
Itwasrecentlydiscoveredthatinspiteofthescalarnatureofitsorderparameter,thechargeorder
in 1T-TiSe can be chiral. This is made possible by the emergence of orbital order in conjunction
2
withthechargedensitymodulations. Hereweshowthatacloselyrelatedchargeandorbitalordered
statearisesin2H-TaS . Inbothmaterials,themicroscopicmechanismdrivingthetransitionisbased
2
ontheinteractionbetweenthreedifferentlypolarizeddisplacementwaves. Therelativephaseshifts
between these waves lead both to the formation of orbital order and to the breakdown of inversion
symmetry. Incontrastto1T-TiSe however,thepresenceofamirrorplaneinthelatticeof2H-TaS
2 2
prevents the distorted structure in this material from being chiral, and a polar charge and orbital
orderedstatearisesinstead. Itisstressedthatbulkexperimentsareindispensableindifferentiating
2
between the chiral and polar phases.
1
0
2 PACSnumbers: 71.45.Lr,11.30.Rd,71.30.+h
n
a
INTRODUCTION
J
0
2 The emergence of charge density wave (CDW) order
in a material implies modulations in both its electronic
] density and, because of the non-zero electron-phonon
l
e coupling, the equilibrium positions of its atomic lattice.
-
r While the electronic density modulations already break
t
s thetranslationalsymmetryoftheunderlyinglattice,and
t. profoundly affect both the transport and equilibrium
a
properties of the material [1, 2], the ionic displacements
m
may in certain materials conspire to break additional
-
symmetries, and give rise to supplemental ordering. The
d
n breakdownofrotationalsymmetryindichalcogenidesfor
o example, can give rise to a structural transition within
c the charge ordered phase [3], while the broken inversion
[
symmetryinSnTeallowstheappearanceofferroelectric- FIG. 1: A comparison between the lattice structures of 2H-
2 ity in conjunction with the CDW in that material [4]. TaS and1T-TiSe . Left: asingleunitcellof2H-TaS . The
2 2 2
v trigonal prismatic coordination of the Ta atoms is invariant
Only recently, it was discovered that it is possible for
5 underreflectionintheindicatedmirrorplane. Thissymmetry
the atomic displacements accompanying charge order to
3 is not broken by the formation of charge order. Right: two
0 break the inversion symmetry of the atomic lattice in a unit cells of 1T-TiSe . Notice that, owing to the octahedral
2
2 chiral way [5, 6]. How such a helical configuration of coordination of the Ti atoms, the mirror plane is absent in
1. charge density can be formed is not a priori obvious, this structure.
1 because CDW materials, unlike spin density waves, have
1 a scalar order parameter which cannot straightforwardly
1 be made chiral. This apparent paradox was resolved by a state with chiral electronic structure could affect both
:
v interpretingtheobservedchiralCDWphasein1T-TiSe the spin structure of its Cooper pairs and the symmetry
2
i astheresultofsimultaneousorbitalandchargeorder[7]. ofthegap,althoughnosucheffectswereobservedinthis
X
Since the prerequisites for the formation of such a chiral case [9].
r
a charge and orbital ordered phase are rather generic, it Herewepointoutthatinspiteoftheapparentlychiral
may be expected that a chiral CDW can be identified in structureobservedinSTMexperiments, thelatticesym-
a broad class of charge ordered materials [8]. metry of 2H-TaS does in fact not allow the formation
2
Basedonananalysisofscanningtunnelingmicroscope of a helical state. It is shown that as in 1T-TiSe , the
2
(STM) images of its surface, it was recently suggested charge order in this compound arises from a microscopic
that the CDW in 2H-TaS closely resembles the charge mechanism which involves orbital ordering as well as a
2
order seen in 1T-TiSe2, and in particular that it is also modulation of the charge density. Unlike the 1T struc-
chiral [9]. This suggestion is especially interesting be- ture however, the lattice of the 2H material has a mirror
cause at low temperatures, the charge density modula- symmetry,whichpreventsitsCDWfrombecomingchiral
tions in 2H-TaS are known to coexist with a supercon- (see Fig. 1). Using a Landau order parameter analysis
2
ductingphase. Theemergenceofsuperconductivityfrom we show how instead a polar orbital and charge ordered
2
FIG. 2: The electronic structure of 2H-TaS . (a) The electronic band structure close to the Fermi energy (adapted from 11).
2
Only the lowest lying band derived from Ta-d electrons, and none of the S-derived bands, cross the Fermi level. (b) The
orbitalcharacteroftheelectronsinthelowestTa-dband. Therealspaceorbitalshapeofelectronswithagivenmomentumis
shownfortheelectronicstatesathighsymmetrypointsinthefirstBrillouinzone. Atthezonecentre,theelectronsarealigned
perpendicular to the Ta planes, while at the zone edges the orbitals lie within that plane. The dotted line gives a schematic
indication of the position of the Fermi surface in the first Brillouin zone. (c) The Fermi surface sheets obtained from the
modelofnon-interactingone-dimensionalchains. TheprojectionofthefirstBrillouinzonetothea∗-b∗-planeshowsflatsheets
of Fermi surfaces extending along the c∗ direction as straight dashed lines. Each pair of sheets corresponds to a particular
orientation of Ta-d orbitals. Two pairs of sheets can be simultaneously connected by a single nesting vector, as shown for Q(cid:126) .
1
state arises in 2H-TaS . This polar state looks identi- at the Y and M points in Fig. 2b they are aligned with
2
cal to the chiral state in 1T-TiSe if only the topmost the crystallographic a and −a−b axes respectively. All
2
atomic layers are considered, as observed in the recent of these orbital wave functions can be written as differ-
STM experiments. Bulk measurements which may be ent linear combinations of the d , d and d orbitals,
z2 x2 y2
employed to confirm the presence of a polar charge and defined with respect to the c-axis.
orbital ordered state are proposed.
The Fermi surface of 2H-TaS lies approximately two
2
thirdsofthewaybetweenthecentreoftheBrillouinzone
and its edges. As a first order approximation, we can
ONE-DIMENSIONAL CHAINS thus describe the electrons at the Fermi surface using
the three in-plane d -like orbitals which become exact
z2
at the zone boundaries. These orbitals overlap strongly
The hexagonally ordered sheets of Tantalum atoms
along one-dimensional chains in the crystal lattice, while
in the quasi two-dimensional, layered transition metal
theexchangeintegralswithneighboringchainsaresmall.
dichalcogenide 2H-TaS are sandwiched between sheets
2
Basedonthisobservation,ithasbeensuggestedthatthe
ofSulphuratoms. ThelocalcoordinationoftheTaatoms
charge density wave formation in this material may be
is trigonal prismatic, as indicated in Fig. 1, resulting in
understood as the result of the ‘hidden nesting’ of non-
a crystal field splitting of the Ta-d orbitals [10]. It is
interacting,one-dimensionalchainsofTaorbitals[11],in
known from band structure calculations that only the
close analogy to the mechanism underlying for example
lowest Ta-d band, and none of the Se-p bands, cross the
the formation of the charge ordered structure in elemen-
Fermi energy, as shown in Fig. 2a [11–13]. It may thus
tal Tellurium [6, 14].
be expected that the CDW in 2H-TaS results predomi-
2
nantly from the hybridization of its metal atoms [11]. The four different d -like orbitals are not mutually
z2
The orbital character of the d-bands can be obtained orthogonal, and form an overcomplete basis for the de-
from a tight binding fit to the band structure. At the scription of the lowest d-band. The approximate de-
centre of the Brillouin zone, the electrons in the band scription of 2H-TaS as a system of non-interacting one-
2
closest to the Fermi energy are purely of the d type, dimensionalchainscanthereforenotbetenablethrough-
z2
with the z direction oriented along the crystallographic out the first Brillouin zone. In particular, we know
c-axis [11, 12]. Going towards the edges of the Brillouin thattheinteractionsbetweenin-planeorbitals,whichare
zone, the character of the electrons changes, until pre- strictly zero at the points X, Y, and M, must become
cisely at the edge (the X-point in Fig. 2b), the orbital strong at the Brillouin zone centre. We will nevertheless
is again of the d type, but now with its the quantiza- neglecttheseinteractionshere,becausethetightbinding
z2
tionaxisalongthecrystallographicb-axis. Goingaround fit of the band structure shows that at the zone centre,
the edge of the Brillouin zone, the shape of the orbital the lowest band lies far above the Fermi energy. As long
remains the same, but its orientation is rotated so that as we only consider low energy processes which occur
3
close to the Fermi level, the description in terms of non- The order parameter consists of the modulation α((cid:126)r) of
interactingchainswillbeagoodapproximation. Wethus theaveragechargedensity,whichcanbewrittenasasum
consider the Hamiltonian of three complex components ψj = ψ0ei(Q(cid:126)j·(cid:126)r+ϕj). The
Hˆ =2t(cid:88)(cid:110)cos((cid:126)k·(cid:126)a)aˆ†kaˆk+cos((cid:126)k·(cid:126)b)ˆb†kˆbk aamndpltithuedpesroopfagaallticoonmvpeocnteonrstsQ(cid:126)ajrearaessuthmeedCDtoWbeneesqtiunagl
k vectors. The phases ϕ are to be determined from the
(cid:111) j
+ cos((cid:126)k·d(cid:126))dˆ†dˆ . (1) minimization of the free energy, which is given by [19]:
k k
(cid:90)
Here (cid:126)a, (cid:126)b and d(cid:126) = −(cid:126)a−(cid:126)b are unit cell vectors of the F = d(cid:126)x(cid:8) aα2+bα3+cα4+dα6
2H crystal structure (see Figs. 1 and 2b), and the three
differentfermionicoperatorscreateelectronsinthethree + f(cid:2)|ψ1ψ2|2+|ψ2ψ3|2+|ψ3ψ1|2(cid:3)(cid:9). (3)
different in-plane d -like states.
z2
The powers of α represent the terms allowed by symme-
The three one-dimensional chains share one electron
try in the expansion of both the electronic Coulomb en-
per unit cell, resulting in three sets of flat Fermi surface
ergy and the elastic energy cost of deforming the lattice.
sheets, as indicated in Fig. 2c. The perfect nesting of
The cross terms in the last line signify the competition
these sheets gives rise to three concurrent CDW insta-
between different CDW components over the available
bilities. Choosingthepropagationvectorsoftheseinsta-
Fermi surface.
bilities so that each charge density wave simultaneously
TotakeintoaccountalsotheUmklappprocessesasso-
modulates the electron distributions of two different sets
ciated with the presence of a discrete lattice, we expand
of electrons, allows the system to maximize its energy
the coefficients of the expression for the free energy such
gain [15]. We thus recover the three experimentally ob-
servedCDWvectorsQ =a∗/3, Q =(−a∗+b∗)/3, and that they reflect the lattice symmetry [19].
1 2
Q =−b∗/3 [16–18].
3The atomic displacement waves accompanying the a=a0+a1(cid:88)eiG(cid:126)i·(cid:126)r+γa1(cid:88)eiG(cid:126)i·((cid:126)r+R(cid:126)i(cid:48))+.. (4)
charge order will share its propagation vectors, but their i i,i(cid:48)
polarizations depend on the anisotropy of the electron-
Here G(cid:126) are the shortest reciprocal lattice vectors and
latticecouplingasdeterminedbytheorbitalstructure[7]. i
R(cid:126) connect a Tantalum site to its Sulphur neighbors in
Since each species of in-plane orbital is arranged along i
the same unit cell. Because a arises from Umklapp ef-
one-dimensional chains with negligible electronic overlap 1
fects, it is proportional to the electron-phonon coupling.
between them, their individual displacements can only
The factor γ reflects the difference in strength of the
be directed along the chains. Each nesting vector con-
electron-phonon coupling on Ta and S sites. From the
nectstwotypesofFermisurfacesheets,belongingtotwo
tight-bindingfittothebandstructure,weknowthatthe
differently oriented orbitals. The overall displacement
distortions in 2H-TaS result from the hybridization of
imposed by the wave propagating along a given nesting 2
Ta-dorbitals,withtheSulphuratomsplayingonlyasec-
vector is thus the linear superposition of the displace-
ondary role. We therefore assume γ to be small, and set
ments induced in the two orbital sectors affected by it
it to zero from here on.
[7, 14]. Combining all CDW components, we then find
The free energy of Eq. (3) can be evaluated directly,
the total displacement pattern
and its phase dependent part is given by
(cid:126)u((cid:126)r)∝[d(cid:126)−(cid:126)a]sin(Q(cid:126) ·(cid:126)r)+[(cid:126)a−(cid:126)b]sin(Q(cid:126) ·(cid:126)r)
1 2
3
+[(cid:126)b−d(cid:126)]sin(Q(cid:126)3·(cid:126)r). (2) F = 2ψ03b0cos(ϕ1+ϕ2+ϕ3)
1 (cid:88)
Noticethatthepresentanalysisdoesnotuniquelydeter- + ψ3b cos(3ϕ )
4 0 1 j
mine the relative signs of the polarization components. j
WiththesignsasinEq. (2)however,wepreciselyrepro- 3 (cid:88)
+ ψ4c cos(2ϕ −ϕ −ϕ )
ducethelatticestructurethathadbeensuggestedforthe 2 0 1 j j+1 j−1
CDW phase of 2H-TaS before the possibility of a chiral j
2
phase became apparent [16–18]. + 5ψ6d (cid:88)cos(3ϕ −3ϕ ), (5)
8 0 0 j j+1
j
RELATIVE PHASES where we took into account terms up to sixth order.
The coefficients b and c arise from the coupling of the
1 1
To describe the microscopic interactions between the electronic charge modulation to the ionic lattice. To
electronic density modulations and the lattice which de- favor bonding between Ta atoms, b should be chosen
1
termine the relative phases of the displacement wave positive, and c negative. The minimum energy solu-
1
components, we construct a Ginzburg-Landau theory. tion up to fourth order is then the normal CDW with
4
ϕ = ϕ = ϕ = π/3, which corresponds to the non-
1 2 3
chiral structure mentioned before. Including a finite,
non-zerovalueford however,changesthesolutionqual-
0
itatively. Thephasesofthedifferentchargedensitywave
components can no longer all be equal, and the corre-
sponding atomic displacements break the inversion sym-
metry of the lattice. In the extreme case with d (cid:29) c ,
0 1
the stable solution obeys ϕ =π/3, ϕ =ϕ ±2π/9 and
1 2 1
ϕ =ϕ ±4π/9, while for intermediate values of d , the
3 1 0
phase differences interpolate smoothly between the two
extreme solutions.
Eventhoughd enterstheexpansionofthefreeenergy
0
only at sixth order, it arises directly from the electronic
Coulomb interactions, while the fourth-order coefficient
c correspondstoanUmklappeffect,andisproportional
1
to the electron-phonon coupling on the Ta sites. The ra-
tio d /c is therefore not in general negligibly small, and
0 1
the interplay between Coulomb interaction and electron-
latticecouplingmaygiverisetotheformationofnon-zero
phase differences. The pattern observed in recent STM
FIG.3: Thenon-polarandpolarchargedensitywaves. Top:
experiments on 2H-TaS , which was suggested to be chi-
2 The extrema of the three components of the charge density
ral [9], corresponds to such a regime, in which the sixth modulation α = (cid:80) ψ cos(Q(cid:126) ·(cid:126)r +ϕ ), in the absence of
order term does not dominate the expression of Eq. (5), j 0 j j
any relative phase differences, are drawn as dashed lines in
but also cannot be neglected. the planar projection on the left. Centres of inversion, in-
dicated by triangles, exists at the points in the plane where
threeextremameet. Thepositionofoneoftheinversioncen-
THE POLAR CDW tres within the three-dimensional atomic structure is shown
schematically on the right, along with two horizontal mirror
planes. Bottom: Fornon-zerorelativephasedifferences,the
Theeffectsoftheabsenceorpresenceofrelativephase
extrema of two charge density wave components are shifted.
differences between the charge density wave components Theresultingstructurenolongerhasanycentreofinversion.
areschematicallyindicatedinFig. 3. Intheformercase, The horizontal mirror planes indicated in the three dimen-
theextremaofallthreecomponentsofthechargedensity sionalstructurehowever,areunaffectedbytherelativephase
wave conicide on or in between Ta sites. The resulting shifts. The atomic lattice in this state thus breaks inversion
symmetry,butretainsitsmirrorsymmetries,sothatitispo-
atomic displacements do not break the inversion sym-
lar rather than chiral.
metryoperationscenteredatthesamepositions,andthe
distortedlatticestructureconcurswiththepreviouslyre-
ported3×3superstructure[16–18]. Iftherelativephases
tice structure is not chiral. The modulations do however
become non-zero however, the extrema of two compo-
break the inversion symmetry of the lattice, so that the
nents start to shift. The extreme situation with d dom-
0
final lattice structure is found to be polar.
inating Eq. (5), is shown in the bottom of Fig. 3. In
practice, this extreme situation will not be reached, and The polar and the non-polar phases shown in Fig. 3
the actual shifts of the charge density wave extrema in both have 3×3 superstructures defined by the same set
2H-TaS will be smaller. Regardless of the size of the ofCDWpropagationvectors,anddifferonlyintheirval-
2
relativephaseshiftshowever,theinversioncentresofthe ues for the relative phases between the charge density
original lattice will no longer coincide with the extrema wavecomponents. Bothphasesthereforeagreewithpre-
of all three charge density wave components, and the in- viouslyreportedneutronscatteringandX-raydiffraction
version symmetry of the lattice is broken. measurements of the electronic superstructure [16–18].
The lattice structure of the 2H polytype, as shown in The atomic displacements of only the top two atomic
Figs. 1 and 3, consists of two TaS sandwich layers per layers do not show the presence of the mirror plane or
2
unit cell, and the top and bottom Sulphur atoms within the absence of chirality, so that their Fourier transform
asinglesandwichareaffectedbythechargedensitymod- will look identical in character to the patterns observed
ulations in the same way. In contrast to 1T-TiSe , the in a chiral state. The presence of relative phase differ-
2
central Ta atoms therefore always define a mirror plane, encesleadstoadisplacementoftheextremaoftwoCDW
even in the presence of non-zero relative phases. Since it componentsawayfromthecentralTaatoms,thusgiving
is not possible to define left or right handed patterns in rise to the observed variations in electronic density on
the presence of this mirror symmetry, the distorted lat- these atoms. The similarity of the recent STM images
5
of the surface of 2H-TaS to those obtained on chiral
2
1T-TiSe clearly indicates that both materials harbor a
2
three-component charge density wave, and that in both
materials the phase differences between the components
canbenon-zero[5,7,9]. Whereasinthecaseof1T-TiSe
2
thosephasedifferencesimplyachirallatticestructure,in
2H-TaS the displacements form a polar pattern.
2
Experimental verification of the proposed polar lat-
tice structure requires bulk probes. The reduction of the
space group of 2H-TaS from P6 /mmc (no. 194) to Pm
2 3
(no. 6)asthepolarphaseisenteredshouldbeobservable
in for example X-ray diffraction experiments. Indirect
evidence of the absence of chirality may also be gained
FIG.4: Theorbitalorderinthepolarphaseof2H-TaS . Only
from optical reflectometry. The polarization dependence 2
thoseTa-dorbitalswhoseoccupationvariesmostsignificantly
of reflected light was shown to be two-fold symmetric
as the system goes from the uniform to the polar phase, are
in the chiral phase of 1T-TiSe , in contrast to the three-
2 indicated in the planar projection of the atomic lattice. Red
foldsymmetryexpectedforanon-chiralthree-component orbitals identify a decrease in occupation, while blue orbitals
charge density wave [5]. representanincrease. Asinfig. 3,thedashedlinescorrespond
Finally,likein1T-TiSe ,theshiftingoftheCDWcom- totheextremaofthethreechargedensitywavecomponents.
2
Noticethattheorbitalorderdirectlyevidencesthebreakdown
ponents due to the non-zero phase differences also au-
of both rotational symmetry and inversion symmetry in the
tomatically implies the formation of orbital order in the
polarphase,butdoesnotaffecttheplanarmirrorsymmetry.
system. Thepolarizationsofthedisplacementwavecom-
ponentsinEq. (2)eachoriginateinacombinationoftwo
specific orbital sectors. The relative differences between
the amplitudes of the three charge density wave com- ofamirrorplaneinitslattice,whichcannotbedestroyed
ponents on a given Ta site therefore also correspond to by the phase shifts of the three CDW components. Bulk
relativedifferencesbetweentheelectronicoccupationsof experiments are indispensable in identifying whether a
the three in-plane Ta-d orbitals. In the polar phase, the given material is chiral or polar, because surface probes
variations in orbital occupation are schematically shown arenotsensitivetotheexistenceofthismirrorsymmetry.
inFig. 4,inwhichonlytheorbitalsmostsignificantlyaf-
In spite of the different character of their final atomic
fectedbythechargemodulationsaredrawn. Noticethat
configurations,thechargedensitywavesin1T-TiSe and
2
thevariationsinoccupationbetweendifferentsitesdefine
2H-TaS arecloselyrelated. Bothofthemarisefromsim-
2
an orbital density wave, rather than an arrangements of
ilar ‘hidden’ one-dimensional chains of orbitals, which
completely filled and empty orbitals. The presence of
give rise to differently polarized components of a dis-
this type of orbital order in 2H-TaS is a direct conse-
2 placement wave. The central role played by the elec-
quence of the presence of the relative phase differences
tronicorbitalsindefiningthesepolarizations,isreflected
which define the polar charge density wave.
in the fact that the formation of both this type of polar
andofchiralchargeorderarenecessarilyaccompaniedby
the emergence of orbital order. Finally, the phase shifts
CONCLUSIONS
which give rise to the breakdown of inversion symmetry
in both compounds arise from a balance between similar
In summary, we have shown that the lattice structure
terms in the Landau expansion of their free energies.
in the charge ordered phase of 2H-TaS may be under-
2
These similarities, and the generic nature of the in-
stoodasresultingfromtheinteractionbetweenthreedif-
teractionsunderlyingthem,stronglysuggestthatorbital
ferently polarized displacement waves. The longitudinal
order and the loss of inversion symmetry may be general
polarizations of these waves are due to the relative ori-
characteristicsofamuchbroadergroupofmaterialswith
entations of the electronic orbitals involved in the CDW
multiple charge density waves.
formation. As in the similar compound 1T-TiSe , the
2
interplay between the local Coulomb interaction and the
electron-latticecouplinggivesrisetothepresenceofnon-
zerorelativephasedifferencesbetweenthecomponentsof
thechargedensitymodulation,whichbreaktheinversion
Acknowledgements
symmetryoftheoverallstructure. Unlikethecaseof1T-
TiSe however, the distortions in the lattice of 2H-TaS
2 2
yieldapolar,ratherthanachiral,phase. Theemergence ThisresearchwassupportedbytheUSDOE,Officeof
of chirality in this material is forestalled by the presence Science, under Contract No. DE-AC02-06CH11357.
6
(2011).
[10] R.Huisman,R.deJonge,etal.,J.SolidStateChem.3,
56 (1971).
[1] R.Peierls,QuantumTheoryofSolids (OxfordUniversity [11] M. H. Whangbo and E. Canadell, J. Am. Chem. Soc.
Press, Oxford, 1955). 114, 9587 (1992).
[2] G. Gru¨ner, Rev. Mod. Phys. 60, 1129 (1988). [12] L. F. Mattheiss, Phys. Rev. B 8, 3719 (1973).
[3] P. B. Littlewood and T. Rice, Phys. Rev. Lett. 48, 27 [13] J. E. Inglesfield, J. Phys. C 13, 17 (2000).
(1982). [14] H. Fukutome, Prog. Theor. Phys. 71, 1 (1984).
[4] P.B.Littlewood,CRCCrit.Rev.SolidStateMater.Sci. [15] M.Whangbo,E.Canadell,P.Foury,andJ.Pouget,Sci-
11, 229 (1984). ence 252, 96 (1991).
[5] J.Ishioka,Y.H.Liu,etal.,Phys.Rev.Lett.105,176401 [16] J.A.Wilson,F.J.DiSalvo,andS.Mahajan,Adv.Phys.
(2010). 24, 117 (1975).
[6] J.vanWezelandP.B.Littlewood,Physics3,87(2010). [17] D. Moncton, J. Axe, and F. DiSalvo, Phys. Rev. B 16,
[7] J. van Wezel, Europhys. Lett. 96, 67011 (2011). 801 (1977).
[8] J.vanWezel,ECRYS-11ConferenceProceedings (2011). [18] R. Brouwer and F. Jellinek, Physica B 99, 51 (1980).
doi:10.1016/j.physb.2012.01.028 [19] W. McMillan, Phys. Rev. B 12, 1187 (1975).
[9] I.Guillamon,H.Suderow,etal.,NewJ.Phys.13,103020
