Magnetism in Quasi-One-Dimensional A$_2$Cr$_3$As$_3$ (A=K,Rb) superconductors

4.4 MB·English

by  Xianxin Wu

MagnetisminQuasi-One-DimensionalA Cr As (A=K,Rb)superconductors
2 3 3
Xianxin Wu,1 Congcong Le,1 Jing Yuan,1 Heng Fan,1,2 and Jiangping Hu1,3,2,∗
1 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
2Collaborative Innovation Center of Quantum Matter, Beijing, China
3Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA
(Dated:April22,2015)
Wepredictthattherecentlydiscoveredquasi-onedimensionalsuperconductors, A Cr As (A=K,Rb), pos-
2 3 3
sessstrongfrustratedmagneticfluctuationsandarenearbyanovelin-outco-planarmagneticgroundstate.The
frustratedmagnetismisverysensitivetoc-axislatticeconstantandcanthusbesuppressedbyincreasingpres-
5
sure. Ourresultsqualitativelyexplainstrongnon-Fermiliquidbehaviorsobservedinthenormalstateofthe
1
superconductorsastheintertwiningbetweenthemagnetismandsuperconductivitycancreatealargequantum
0
criticalregioninquasi-onedimensionalsystemsandalsosuggestthatthematerialssharesimilarphasediagrams
2
andsuperconductingmechanismwithotherunconventionalsuperconductors,suchascupratesandiron-based

r superconductors.
p
A
PACSnumbers:74.70.-b,74.25.Ha,74.20.Pq,74.20.Rp

1
2
Oneofmajorchallengesincondensedmatterphysicsisto where the superconductivity emerges in a vicinity to a novel

understandtheroleofelectron-electroncorrelationinuncon- magnetically ordered state. We predict that the materials are
]
n ventionalsuperconductors. Theeffectofelectron-electronin- characterized by strong frustrated magnetic fluctuations and
o teractionbecomesmoreimportantasthedimensionofasys- are nearby a novel in-out co-planar(IOP) magnetic ground
c
tem is lowered. Indeed, many unconventional superconduc- state. Themagnetismcanbedescribedbyaminimumeffec-
-
r tors discovered in the past are quasi-two dimensional(Q2D) tivemodelwiththreemagneticexchangeparameters:theanti-
p
electron systems. The superconductivity in these unconven- ferromagneticJ andJ(cid:48) betweentwonearest-neighbor(NN)
u 1 1
tional superconductors appears in a vicinity to a magneti- CratomsandtheferromagneticJ betweentwonextNNCr
s 2
. cally ordered state. Magnetic fluctuations which are caused atoms along c-axis. The frustrated magnetism is very sensi-
t
a byelectron-electroninteractionhavebeenwidelyconsidered tivetoc-axislatticeconstantandcanthusbesuppressedbyin-
m
to be responsible for superconductivity and many non-Fermi creasingpressure.Theresultssuggestthatthematerialshosta
- liquidbehaviorsinnormalstates. typicalphasediagramsimilartothoseoftheQ2Dunconven-
d
While there are many representatives of Q2D unconven- tional superconductors, such as cuprates[10] and iron-based
n
o tional superconductors, it has been difficult to find one superconductors[11]. Thenewmaterialscanbeidealsystems
c in quasi-one dimensional(Q1D) systems even if the effect tostudytheintimaterelationsbetweenmagnetismandsuper-
[ of the electron-electron correlation is expected to be en- conductivity since a Q1D model can be solved theoretically

hanced further. The Q1D superconductors discovered pre- withhighcontrollability.
2
v viously, including Bechgaard salts[1, 2] , Tl2Mo6Se6[3] and It is known that calculations based on the Density Func-
2 Li0.9Mo6O17[4–7], are not attributed to 3d-orbital electrons tional Theory (DFT) can successfully identify the possible
1 whichcanexhibtstrongelectron-electroninteraction. magnetic ground state in some correlated electron systems.
4
Very recently, two novel Q1D materials K Cr As [8] and Typical examples are the iron-based superconductors where
0 2 3 3
Rb Cr As [9] have been synthesized and found to be super- theoretical calculations consistently agree well with experi-
0 2 3 3
conductingbelowthetransitiontemperature6.1Kand4.8K mental measurements[12–18] even if the magnetic origin is
.
1 respectively. ThestructureofA Cr As (A=K,Rb)ischarac- stilldebatable[19].Here,wedeploysimilarDFTcalculations.
2 3 3
0
terizedbyone-dimensional(Cr As )chains(Fig.1(a)),which OurDFTcalculationsemploytheprojectoraugmentedwave
5 3 3
contain Cr distorted octahedral clusters. The alkali metal (PAW)methodencodedinViennaabinitiosimulationpack-
1 6
: ionsareintercalatedbetweenthe(Cr3As3)chains. Bothnew age(VASP)[20–22], andgeneralized-gradientapproximation
v
materials show strong non-fermi liquid behaviors in normal (GGA)[23] for the exchange correlation functional is used.
i
X states, as well as unconventional superconducting properties Throughout this work, the cutoff energy of 450 eV is taken
in superconducting (SC) states. Moreover, just like cuprates forexpandingthewavefunctionsintoplane-wavebasis. The
r
a and iron-based superconductors, the electronic physics in number of these k points are 6×6×13 for K Cr As and
2 3 3
these new materials are likely attributed to 3d-oribtals of Cr Rb Cr As [24]. The GGA plus on-site repulsion U method
2 3 3
atoms. Thereforethematerialmayexhibitstrongmagnetism (GGA+U) in the formulation of Dudarev et al.[25] is em-
andelectron-electroninteraction. ployed to describe the electron correlation effect associated
Inthispaper,weshowthatthenewmaterialscanbeacrit- withtheCr3dstatesbyaneffectiveparameterUeff. Theval-
ical representative of Q1D unconventional superconductors ues of U=2.3 eV and J=0.96 eV on Cr are adopted for our
GGA+U calculations[26]. We relax the lattice constants and
internalatomicpositions,wheretheplanewavecutoffenergy
is600eV.Forcesareminimizedtolessthan0.01eV/A˚ inthe
∗Electronicaddress:[email protected] structuralrelaxation.
2
the studies of iron based superconductors[17, 27, 28] and it
mayberelatedtothestrongspinfluctuation,whichisbeyond
DFTcalculation.Weadopttheexperimentalparametersinthe
followingcalculationunlessotherwisespecified.
The band structure, the density of states(DOS) and Fermi
surfaces for K Cr As have been calculated in Ref.29. Our
2 3 3
results as shown in Appendix A are similar to their calcula-
tions. Ingeneral,the3dstatesofCrarelocatedfrom-2.5eV
to 2.0 eV and the states near the Fermi level are mainly at-
z tributedtoCrd ,d andd orbitals. TheAs4pstates
z2 xy x2−y2
mainlylie1.0eVbelowtheFermilevelandhybridizestrongly
x y withtheCr3dstates. Alongk direction,thebandsarevery
z
dispersiveduetothestrongcouplingbetweendoribialsalong
J1 z direction. The bands attributed to Cr2 dxy and dx2−y2 or-
bitalsareveryclosetohalf-filling. Astheseparationbetween
, chains is large, the inplane band dispersion is very small but
J
1 J not negligible. However, we want to point out an important
2
featurethatisignoredinRef.29. Cr1ionshavemoredelec-
tronsthanCr2ions,whichisconsistentwithanalysisfromthe
crystalstructure. Wewill showlaterthat thisdifferencealso
resultsindifferentmagneticmomentsatCr1andCr2sitesin
magneticstates.
FIG.1: SchematicviewofthestructureofA Cr As (a). (b)and
2 3 2 ThebandstructureofRb Cr As ,shownalsoinAppendix
(c)showtheall-inandin-outnoncollinearmagneticstates. Theex- 2 3 3
A,sharesmanycommonfeatureswiththatofK Cr As ,but
changecouplingparametersJ , J(cid:48) andJ aredefinedin(d). The 2 3 3
1 1 2 they also differ in some details. As the radius of Rb is big-
orangeandpurplespheresrepresentCr1andCr2.
ger than K, Cr1As layer can get more electrons and d and
xy
d bandsofCr1arefullyoccupied. Meanwhile,thed
x2−y2 xy
TABLE I: Experimental and optimized structural parameters of anddx2−y2 bandsofCr2aremuchlessoccupied. Itleadsto
K Cr As using GGA in the paramagnetic phase. Deviations be- great difference between the 3D Fermi surfaces of the two
2 3 3
tweentheoptimizedandexperimentalvaluesaregiveninparenthe- materials. Another difference is that there is an additional
sesin%. electron Fermi surface near A(0,0,π) point in Rb Cr As ,
2 3 3
whichisattributedtod andd orbitalsofCr2. Finally
GGA EXP xy x2−y2
a(A˚) 10.113(+1.3) 9.983 thedxz anddyz orbitalsaremuchclosetotheFermilevelin
c(A˚) 4.147(-1.98) 4.230 Rb2Cr3As3.ThecalculatedN(EF)inK2Cr3As3(Rb2Cr3As3)
Cr1-Cr1/Cr2-Cr2(A˚) 2.498;2.588 2.615;2.691 is 8.76(9.13) eV−1/f.u.. The calculated Pauli susceptibility
Cr1-As1/Cr2-As2(A˚) 2.513;2.494 2.51;2.49 and specific heat coefficient are χ0 = 2.83(2.95) × 10−4
Cr1-As2/Cr2-As1(A˚) 2.522;2.506 2.516;2.506 emu/molandγ =20.7(21.5)mJ/(K2 mol). Thecalculatedγ
α1/α2(◦) 59.6;62.51 62.8;65.4 isaboutonlyonethirdoftheexperimentalvalueinK2Cr3As3,
β1/β2(◦) 60.8;60.87 62.8;62.9 suggestingstrongcorrelationinthesesystems.
γ /γ (◦) 110.6;111.6 114.4;115.2
1 2 Now we focus on the magnetic properties. We consider
fourpossiblecollinearmagneticstates,theparamagneticstate
(PM), the ferromagnetic (FM) state, the interlayer antiferro-
Duetotheasymmetricdistributionofalkalimetalionsand magnetic(AFM)stateandtheup-up-down-down(↑↑↓↓)mag-
the absence of inversion symmetry, there are two kinds of netic state. Due to strong magnetic frustration, we also con-
Cr ions in K Cr As (Rb Cr As ): Cr1 and Cr2. The Cr1 sidertwoadditionalco-planarantiferromagneticstates: all-in
2 3 3 2 3 3
ions are surrounded by six inplane A ions and the inplane magneticstate(Fig.1(b))andin-outmagneticstate(Fig.1(c)).
bondlengthis2.615(2.57)A˚. TheCr2ionsaresurroundedby Weperformcalculationswithspinorbitalcoupling(SOC)and
three inplane alkali ions and the inplane bondlength is a bit the calculated magnetic moments and the total relative ener-
longer,2.69(3.16)A˚.FromthenumberofAionssurrounding giesofabovemagneticstatesaresummarizedinTable.II.The
Cr ions, it is expected that the Cr1As layer will obtain more IOPstateisthegroundstateinK Cr As (Rb Cr As ),with
2 2 2 2 2 2
electronsthantheCr2Aslayer. a large energy gain of 39(762) meV/cell relative to the PM
The optimized and experimental structural parameters of state. The magnetic moments are 0.90(1.75) and 0.94(2.34)
K Cr As aresummarizedinTableI.Wefindthatthelattice µ on Cr1 and Cr2 sites, respectively. The initial FM state
2 3 3 B
constantsarecomparablewithexperimentalvaluesbuttheCr- converges to a PM or AFM state, indicating that the inter-
Crbondlengthsareunderestimatedby0.1A˚. Thus,theCr1- layermagneticcouplingJ(cid:48) asindicatedinFig.1(d)isantifer-
1
As1-Cr1(α )/Cr2-As2-Cr2(α ), Cr1-As1-Cr2(β )/Cr2-As2- romagneticandrelativelystrong.IntheAFMstate,theenergy
1 2 1
Cr1(β ) and Cr1-As2-Cr1(γ )/Cr2-As1-Cr2(γ ) angles are gainis25(486)meV/cellandthethreemagneticmomentsat
2 1 2
also underestimated. Similar cases have been also noted in Cr1 or Cr2 site are different. It manifests that the intralayer
3
magneticcouplingsamongthreeCratomsJ1 isAFM.Thus, 1.5 1.5
strong magnetic frustration exists in A Cr As . Our results
2 3 3
aredifferentfromthoseinRef.[29]whichmissedtoidentify 1 1
total
theIOPasthetruemagneticgroundstateintheircalculations
Toconfirmourresults,wehaveperformedall-electroncalcu- 0.5 0.5 Cr dx2-y2 dxy
lfaotriocnoslliannedarfisntdatethsaatrreerlaattihveerecnloesregiteostahnodseminagTnaebtliecImI.oments y (eV) 0 0 Cr dCxzr ddyzz2
TheIOPstategainsenergyrapidlyandbecomesveryrobust g
er
ifweincludeadditionalU, theonsiteelectron-electoncorre- En-0.5 -0.5
lation. The results from GGA+U calculations are given in
-1 -1
Table.III.TheIOPstateinK Cr As (Rb Cr As )hasalarge
2 2 2 2 2 2
energy gain of 392(466) meV/cell relative to the AFM state.
-1.5 -1.5
Themagneticmomentsaregreatlyenhancedto2.49(2.76)and
2.56(2.92)µ onCr1andCr2sitesaswell. Itisalsoimpor-
B -2 -2
tanttonotethatthemagnetismisalwaysfoundtobestronger Γ K M Γ A H L A 0 10 20 30 40 50
in Rb Cr As than in K Cr As . In the presence of U, as
2 3 3 2 3 3
thecalculationcanbeconvergedfordifferentmagneticstates, FIG.2: BandstructureandDOSforK2Cr3As3 inin-outco-planar
we can extract the magnetic exchange parameters from their (IOP)magneticstate.
energydifferences[30]. IntheGGA+Ucalculations,thelocal
magneticmomentsonCrareveryclosetoeachother. Thus,
TABLE II: The total energies for different magnetic states of
we can can extract the magnetic exchange parameters from
A Cr As . They are given relative to the total energy of the PM
theirenergydifferenceswithintheHeisenbergModel.Theen- 2 3 3
stateandtheunitismeV/cell. Themagneticmomentsaregivenin
ergiescontributedbymagneticinteractionsinthesefivemag-
µ .
B
neticstatespercellarewrittenas,
A Cr As relativeenergy(meV/cell) M M
2 3 3 Cr1 Cr2
E /S2 = 6J +12J(cid:48) +6J , PM 0 0 0
FM 1 1 2
E /S2 = 6J −12J(cid:48) +6J , FM toPM
AFM 1 1 2 A=K AFM -25 0.43 0.53
E↑↑↓↓/S2 = 6J1−6J2, ↑↑↓↓ toPM
E /S2 = −3J +6J(cid:48) +6J , all-in -3(toin-out) 0.06 0.35
all−in 1 1 2
in-out -39 0.90 0.94
E /S2 = −3J −6J(cid:48) +6J . (1)
in−out 1 1 2 PM 0 0 0
FM -589(toAFM) 0.40 2.29
Using the relative energies and the above equations, we can
A=Rb AFM -486 1.26 2.35
obtainfourequations. Butthereareonlythreeexchangepa-
↑↑↓↓ -525 0.01 1.83
rameters. For this overdetermined system of equations, we
all-in -594(toin-out) 0.34 2.15
can obtain these parameters by using a least-squares tech- in-out -762 1.75 2.34
nique. The estimated exchange couplings are J = 0.090
1
eV/S2,J(cid:48) =0.060eV/S2,J =−0.010eV/S2forK Cr As
1 2 2 3 3
and J = 0.075 eV/S2, J(cid:48) = 0.045 eV/S2, J = −0.020
1 1 2 to the increase of pressure, the magnetic moments decrease,
eV/S2 for Rb Cr As in the GGA+U calculations. The NN
2 3 3 as well as the energy gains. When c = 4.06 A˚, the energy
exchange couplings J and J(cid:48) are strongly AFM. However,
1 1 gain reaches zero, indicating the vanish of IOP magnetic or-
thenextNNexchangecouplingJ isferromagnetic. TheIOP
2 ders. We also directly calculate the pressure effect using the
statesavesenergyfromalltheseeffectiveexchangecouplings.
GGA+Ucalculation.Consistentresultsareobtainedasshown
Fig.2(a) and (b) show the calculated band structure and
inFig.3(b). Theseresultsallowustoconjectureaphasedia-
DOS in the IOP magnetic state of K Cr As . The material
2 3 3 gramofthesenewsuperconductorssimilartotheoneshared
remainsmetallicinthepresenceofthestaticmagneticorder.
bymanyQ2Dunconventionalsuperconductorsassketchedin
ThebandsneartheFermilevelsplitduetoSOCandmagnetic
Fig.4(a). Thenon-Fermiliquidbehaviorsobservedinexperi-
order but the orbital characters have little changes compared
mentscanbenaturallyexplainedasaresultofstrongmagnetic
withthoseinPMstate. Thissplittingdependsonorbitalchar-
fluctuationsinthecriticalregion.
acters of the bands: it is larger for d ,d and d , d
xy x2−y2 xz yz We can construct a minimum effective Hamiltonian to de-
bandsthanford bands.
z2 scribethemagnetismforaQ1DchaininA Cr As withthe
As the size of Rb atoms is much larger than the size 2 3 3
above mentioned magnetic exchange couplings. The Hamil-
of K atoms, the stronger magnetism in Rb Cr As than in
2 3 3 tonianis
K Cr As suggeststhatapplyingpressurecansuppressmag-
2 3 3
netic fluctuations. To simulate the effect of pressure, we in- (cid:88) (cid:88)
H = J S ·S +J(cid:48) S ·S
vestigate the magnetism as a function of the lattice constant M 1 iα iβ 1 iα jβ
c. Fig.3(a) shows the calculated magnetic moments and rel- i,αβ (cid:104)ij(cid:105),αβ
(cid:88)
ativeenergiesoftheIOPmagneticstate(relativetoPMstate) + J S ·S , (2)
2 iα jβ
as a function of c. As the decrease of c, which is equivalent
(cid:104)(cid:104)ij(cid:105)(cid:105),αβ
4
(a) (b)
TABLE III: The total energies for different magnetic states of T
A Cr As withGGA+Ucalculations. Theyaregivenrelativetothe
2 3 3 Spin Fluctuation
totalenergyoftheAFMstateandtheunitismeV/cell.Themagnetic
momentsaregiveninµ .
B
IOP
A Cr As relativeenergy(meV/cell) M M
2 3 3 Cr1 Cr2 K
FM +1495 2.47 2.58
Rb
AFM 0 2.48 2.41 SC
A=K ↑↑↓↓ +854 1.50 2.44 P
all-in +311 2.35 2.43
in-out -392 2.49 2.56
FM +1045 2.09 2.84 FIG. 4: (a) The conjectured phase diagram with pressure of
AFM 0 2.70 2.97 A Cr As . (b) The calculated spin-wave dispersion of A Cr As
2 3 3 2 3 3
A=Rb ↑↑↓↓ +792 2.72 2.90 alongk direction,withJ(cid:48)=0.7J andJ =-0.15J .
z 1 1 2 1
all-in +184 2.90 2.96
in-out -466 2.76 2.92
isAFMbetweentwoNNCratomsthathasaCr-As-Crangle
∼ 72◦ andFMbetweentwonextNNCratomsthathasaCr-
(a) (b) As-Crangle∼123◦.Thesetwomagneticexchangecouplings
inCrAsresembleJ (J(cid:48))andJ . Itisinterestingtonotethat
1 1 2
theexchangecouplingsbetweentwoCratomsareagainstthe
simpleGoodenough-Kanamori-Anderson(GKA)ruleswhich
would suggest opposite signs for J (J(cid:48)) and J . Such a vi-
1 1 2
olation suggests that multi-magnetic mechanisms may work
togetherinCrAsbasedstructuresandJ mayincludesignifi-
2
cantcontributionfromthedouble-exchangemechanismsince
theaveragevalenceofCratomsinA Cr As is2.3.
2 3 3
WehaveignoredtheDzyaloshinskii-Moriya(DM)termthat
canbeinducedbyspinorbitalcouplingsintheabsenceinver-
sion center between two Cr atoms. In CrAs, the DM term
is important in determining the incommensurate wavevector
inthedoublehelimagneticstructure. However,inA Cr As ,
FIG.3: Therelativeenergiesandmagneticmomentsasfunctionof 2 3 3
lattice constant c (a) and pressure (b). In the case of pressure, we theeffectoftheDMisgreatlyreducedastheCratomsforma
performedstructurerelaxationandGGA+Ucalculations. tightoctahedralcluster.Thisisconsistentwithourcalculation
resultsthatmagneticmomentstendtobeinthexyplane.
Ourresultssuggestthatsimilartocupratesandiron-based
where α,β label sublattices and (cid:104)ij(cid:105) and (cid:104)(cid:104)ij(cid:105)(cid:105) denote the superconductors,themagneticfluctuationsareresponsiblefor
NN and next NN pairs. With J ,J(cid:48) being AFM and J be- the superconductivity in A Cr As . If this is the case, it is
1 1 2 2 3 3
ingFM,H hastheIOPmagneticgroundstateasaclassical interesting to ask what type of pairing is favored from the
M
spin model or in the large S-limit. The magnetic excitations specific magnetic fluctuations predicted here. We argue that
canbecalculatedbyemployingtheHolstein-Primakofftrans- thefavoritepairingislikelytobespin-tripletpairingfromthe
formationwithinthelinearspin-waveapproximation. Asthe magneticfluctuations. AsshowninAppendixA,theCr1and
1D chain described by H has a C symmetry with six Cr Cr2 atoms make major contributions to different bands near
M 3v
atoms in the unit cell, there must be three acoustic and three the Fermi level. Thus, as the pairing is expected to be dom-
optical spin excitations. Each of these three modes must be inatedbytheintra-bandpairing, intherealspacepicturethe
composedofoneA andtwodegeneratedE modes. Thede- majority pairing takes place within each sublattice that only
1
tailedanalyticresultsofthespinwavesaregiveninAppendix containsonetypeofCratoms. Theeffectivemagneticfluctu-
B. A typical spin wave spectrum is plotted in Fig.4(b). The ationswithineachsublatticealongthethechaindirectioncon-
acousticspinwavedispersionshowsalineardependencewith nectedbyJ areclearlyFMfromtheaboveresults.Therefore,
2
smallk ,whichindicatestheAFMnatureofthemagneticex- ifweignoretheparitybreakingcausedbyalkalineatoms,the
z
changecouplings. pairing from the magnetic fluctuations is very likely to be a
Although there is no direct experimental measurement on p-wavespintripletasthecorrelationeffectforbidsanyonsite
magneticpropertiesinthesematerials,themagneticexchange pairing. If we consider that the pairing is determined by the
coupling parameters obtained from our calculation are con- local FM exchange coupling[33], the induced gap function ,
sistent with recent experimental measurements in CrAs[31], ∆ ∝ sinkz, whichshouldbecharacterizedbylinenodeson
a MnP-type orthorhombic crystal structure and also a super- theFermisurfaceinthekz =0planeinreciprocalspace.
conductorunderpressure[32]. ThemagneticorderinCrAsis In summary, we predict the Q1D superconductors,
a double helimagnetic structure. Along the one-dimensional A Cr As ,areclosetoanovelin-outco-planarmagneticor-
2 3 3
helimagnetic structure, the measurements have shown that it deredstateandshareatypicalphasediagramsimilartothose
5
ofQ2Dhightemperaturesuperconductors,cupratesandiron- Acknowledgments We thank S. M. Nie and Q. Xie for the
basedsuperconductors. Thepredictionqualitativelyexplains helponcalculations.Theworkissupportedby”973”program
thenonFermi-liquidbehaviorsobservedinthesesystemsand (Grant No. 2010CB922904, No. 2012CV821400 and No.
suggeststhatthesuperconductivityinthesesystemsaredriven 2015CB921300), the National Science Foundation of China
by electron-electron correlation effects. We also predict that (Grant No. NSFC-1190024, 11175248 and 11104339) and
T canbemaximizedinthesesystemsbyapplyingcertainex- the Strategic Priority Research Program of CAS (Grant No.
c
ternalpressure. ThenewmaterialscanbeanidealQ1Dsys- XDB07000000). Wealsowanttonotethattheauthorsin[29]
tem to understand the intimate relation between magnetism have checked our results and have acknowledged to us that
andsuperconductivity. theymissedtheIOPgroundstate.
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6
AppendixA:ElectronicstructuresofA Cr As (a) (b)
2 3 3
The band structure and density of states(DOS) for
K Cr As withexperimentalstructuralparametersareshown
2 3 3
in Fig.5 (a) and (b),(c), respectively. From the band struc-
ture and DOS, we find that Cr1 ions have more d electrons
thanCr2ions,whichisconsistentwithanalysisfromthecrys-
tal structure. It will lead to different magnetic moments at
Cr1andCr2sitesinmagneticstates. TheFermisurfacesare
showninFig.7(a), whicharesimilartothoseinRef.29. The
band structure and DOS for Rb Cr As , shown in Fig.6, are
2 3 3
similartothoseofK Cr As . Themaindifferenceisthatthe
2 3 3
3DFermisurfaceisquitedifferentfromthatofK Cr As and
2 3 3
there is an additional electron Fermi surface near A(0,0,π) FIG. 7: Fermi surfaces of K Cr As (a) and Rb Cr As (b) with
2 3 3 2 3 3
point,asshowninFig.7(b). experimentalparametersintheparamagneticstate.
AppendixB:Spinexcitations
ThemodelofCroctahedralclustersisshowninFig.8. The
magnetic excitations in the effective magnetic Hamiltonian,
Eq.2,forA Cr As canbecalculatedbyusingtheHolstein-
2 3 3
1
K s
0.5 CrC1 rd1x dy,xdtzox,2dt-ayy2zl AAss ppyx Primakofftransformation,
Cr1 dz2 As pz
V) Cr1 dxy,dx2-y2 Cr2 dxy,dx2-y2
gy (e 0 Cr1 Cdrx1z, ddyzz2 Cr2C drx2z, ddyz2z
er Cr2 dxy,dx2-y2
En Cr2 dxz,dyz
Cr2 dz2
-0.5
-1
Γ K M Γ A H L A 0 10 20 30 40 50 0 1 2 3 4 5
(cid:113)
FIG.5: BandstructureandDOSofK2Cr3As3 withexperimental Sα+i = 2S−a†αiaαiaαi,
parametersintheparamagneticstate.
(cid:113)
S− =a† 2S−a† a ,
αi αi αi αi
1
Sz =S−a† a , (B1)
αi αi αi
total K s
0.5 Cr1 dxy,dx2-y2 CrC1 rd1xC yd,rdx1zx ,2dd-zyy22z AAAsss pppyxz
V) Cr1 dxz,dyz Cr2 dxy,dx2-y2
gy (e 0 Cr2 dxCy,rd1x 2d-yz22 Cr2C drx2z ,ddzy2z
Ener Cr2C drx2z ,ddyzz2
-0.5
-1
Γ K M Γ A H L A 0 10 20 30 40 50 60 70 0 1 2 3 4 5 whereαlabelssublatticeandα = (1,2,3,4,5,6)andaαi is
a bosonic operator. Then, keeping only the linear terms and
FIG.6: BandstructureandDOSofRb2Cr3As3 withexperimental using Fourier transformation, the magnetic Hamiltonian can
parametersintheparamagneticstate.
beexpressedas,
7
(cid:88)
H = H (1,2)+H (2,3)+H (3,1)+H (4,5)+H (5,6)+H (6,4)
M 0 0 0 0 0 0
k
(cid:88)
+ H (1,4)+H (1,6)+H (2,4)+H (2,5)+H (3,5)+H (3,6)
1 1 1 1 1 1
k
(cid:88)
+ H (1,1)+H (2,2)+H (3,3)+H (4,4)+H (5,5)+H (6,6), (B2)
2 2 2 2 2 2
k
SJ SJ 3 1
H (1,2) = 1(a† a +a a† )+ 1(a† a +a† a )− SJ (a† a† +a a )− NJ S2, (B3)
0 4 1k 2k 1k 2k 2 1k 1k 2k 2k 4 1 1k 2−k 1k 2−k 2 1
SJ(cid:48) k 3 k
H (1,4) = 1cos( z)(a† a +a† a )− SJ(cid:48)cos( z)(a a +a† a† )
1 2 2 1k 4k 4k 1k 2 1 2 1k 4−k 1k 4−k
+ SJ(cid:48)(a† a +a† a )−J(cid:48)NS2, (B4)
1 1k 1k 4k 4k 1
H (1,1) = 2SJ cos(k )(a† a )−2SJ cos(k )(a† a +a† a )+J NS2. (B5)
2 2 z 1k 1k 2 z 1k 1k 1k 1k 2
InthebasisΦ†(k)=(a† ,a† ,a† ,a† ,a† ,a† ,a ,a ,a ,a ,a ,a ),themagneticHamiltonianis,
1k 2k 3k 4k 5k 6k 1−k 2−k 3−k 4−k 5−k 6−k
1(cid:88)
H = Φ†(k)h(k)Φ(k)−S(S+1)N(3J +6J(cid:48) −6J ), (B6)
M 2 1 1 2
k
 
A B B D 0 D 0 C C E 0 E
 B A B D D 0 C 0 C E E 0 
 B B A 0 D D C C 0 0 E E 
D D 0 A B B E E 0 0 C C 
 
 0 D D B A B 0 E E C 0 C 
 
D 0 D B B A E 0 E C C 0 
h(k) =  , (B7)
 0 C C E 0 E A B B D 0 D 
 
 C 0 C E E 0 B A B D D 0 
 
 C C 0 0 E E B B A 0 D D 
 
 E E 0 0 C C D D 0 A B B 
 
 0 E E C 0 C 0 D D B A B 
E 0 E C C 0 D 0 D B B A
A = J S+2J(cid:48)S−2J S+2J Scosk , (B8)
1 1 2 2 z
1
B = J S, (B9)
4 1
3
C = − J S, (B10)
4 1
1 k
D = J(cid:48)Scos( z), (B11)
2 1 2
3 k
E = − J(cid:48)Scos( x). (B12)
2 1 2
The spectrum of the spin wave is given in Fig.9, with J(cid:48) = weplotthespectrumofspinwave(Fig.4(b))inthemaintext
1
0.7J ,J = −0.15J . In A Cr As , the bond lengths in with J = J ,J = 0.9J and J(cid:48) = 0.7J ,J =
1 2 1 2 3 3 1Cr1 1 1Cr2 1 1 1 2
Cr1As1 and Cr2As2 planes are different, which leads to dif- −0.15J . Compared with Fig.9, there are gaps on the Bril-
1
ferentJ couplingsinCr1As1andCr2As2planes. Therefore, louinZoneboundary,indicatingtheintrinsictwosublattices.
1
8
3
6
1
5
2
4
FIG.8: ModeloftheCroctahedralclusterinA Cr As .
2 3 3
FIG. 9: The calculated spin-wave dispersion of K Cr As along
2 3 3
k direction, with J = J = J , J(cid:48) = 0.7J and J =
z 1Cr1 1Cr2 1 1 1 2
−0.15J .
1