Magnetic polaron structures in the one-dimensional double and super-exchange model

0.39 MB·

by  E. Vallejo

4
1
Magneticpolaronstructures intheone-dimensional double and
0
2
super-exchange model

n
a
J E. Vallejoa,∗ F. Lo´pez-Ur´ıasb O. Navarroa andM. Avignonc
9 aInstitutode InvestigacionesenMateriales, Universidad Nacional Auto´noma de M´exico, Apartado Postal 70-360, 04510 M´exico D.F.,
2 M´exico.
bAdvanced MaterialsDepartment, IPICYT, Camino a laPresa San Jos´e 2055, Lomas 4a secci´on, 78216, San LuisPotos´ı, San Luis

] Potos´ı, M´exico.
elcInstitutN´eel,CentreNationaldelaRechercheScientifique(CNRS)andUniversit´eJosephFourier,BP166,38042GrenobleCedex9,France.
-
r
t
s
.
tAbstract
a
m
An analytical and numerical study of the one-dimensional double and super-exchange model is presented. A phase separation
-between ferromagnetic and anti-ferromagnetic phases occurs at low super-exchange interaction energy.When thesuper-exchange
d
interaction energygetslarger, theconductionelectronsareself-trapped withinseparatesmall magneticpolarons. Thesemagnetic
n
polarons contain a single electron inside two or three sites depending on the conduction electron density and form a Wigner
o
ccrystallization.Anewphaseseparationisfoundbetweenthesesmallpolaronsandtheanti-ferromagneticphase.Ourresultscould
[explain thespin-glass-like behaviorobservedin thenickelateone-dimensional compound Y2−xCaxBaNiO5.

1
v
Keywords: Exchange andsuper-exchange interactions, Classicalspinmodels,Phase separation
6
PACS: 75.30.Et, 75.10.Hk, 64.75.+g
5
6
7
.1. Introduction ferromagnetic. In recent literature the ferromagnetic cou-
1
0 pling caseisoften referredto as the FerromagneticKondo
4 Magnetic ordering of localized spins mediated by non- model. Independently of the sign of the coupling, the “ki-
1magnetic conduction electrons, the so-called double ex- netic”energylowering,favorsaFbackgroundoflocalspins.
:
vchange (DE) or indirect exchange, is the source of a vari- This F tendency is expected to be thwarted by AF super-
Xietyofmagneticbehaviorintransitionmetalandrare-earth exchange (SE) interactions between localized spins −→Si as
compounds [1]. Conversely, this interplay affects the mo- first discussed by de Gennes [5] who conjectured the ex-
r
ability ofthe carriersandmayleadtointerestingtransport istence of canted states. In spite of recent interesting ad-
properties such as colossal magnetoresistance in mangan- vances,ourknowledgeofmagneticorderingresultingfrom
ites.Theoriginofthe DEliesintheintra-atomiccoupling this competitionis stillincomplete.
of the spin of the itinerant electrons with localized spins Although it may look academic, the one-dimensional
−→S . In this coupling, localized and itinerant electrons be- (1D) version of this model is very illustrative and help-
i
long to the same atomic shell. According to Hund’s rule, ful in building an unifying picture. On the other hand,
thecouplingisferromagnetic(F)whenthelocalspinshave the number of pertinent real 1D systems as the nicke-
less than half-filled shells and anti-ferromagnetic (AF) for late one-dimensionalmetaloxidecarrier-dopedcompound
more than half-filled shells [2]. This mechanism has been Y2 xCaxBaNiO5[6] is increasing. In this compound, car-
−
widely used in the context of manganites [2–4]. A similar riers are essentially constrained to move parallel to NiO
couplingalsooccursinKondosystemsviatheso-calleds d chains and a spin-glass-like behavior was found at very
exchangemodel.Inthis case,localspinsarefromadsh−ell low temperature T . 3K for typical dopings x = 0.045,
(or f shellinrare-earthcompounds) while the conduction 0.095 and 0.149. Recently, it has been shown that three-
electrons are from s or p states and the coupling is anti- leg ladders in the oxyborate system Fe3BO5 may provide
evidence for the existence of spin and charge ordering re-
∗ sulting fromsuchacompetition[7].
Correspondingauthor.
Email address: [email protected] (E. Vallejo).
Preprintsubmitted toElsevier 30 January 2014
Naturally, the strength of the magnetic interactions an anti-ferromagnetic inter-atomic exchange coupling be-
depends significantly on the conduction band filling, x. tween localized spins −→S . The complete one-dimensional
i
At low conduction electron density, F polarons have DE+SEHamiltonianbecomes,
been found for localized S = 1/2 quantum spins [8].
θ
“Island” phases, periodic arrangement of F polarons cou- H = t cos i (c+c +h.c.)+J →S →S , (3)
− (cid:18)2(cid:19) i i+1 i· i+1
pled anti-ferromagnetically, have been clearly identified Xi Xi
at commensurate fillings both for quantum spins in one where θ = θ and J is the super-exchange interaction
i,i+1 i
dimension [9] and for classical spins in one [10] and two energy.
dimensions [11]. Phase separation between hole-undoped
anti-ferromagnetic and hole-rich ferromagnetic domains
3. Resultsand discussion
hasbeenobtainedintheFerromagneticKondomodel[12].
Phase separation and small ferromagnetic polarons have
In this section, we determine the complete phase dia-
been also identified for localized S = 3/2 quantum spins
graminonedimensionasafunctionofthesuper-exchange
[13].Therefore,itisofimportancetoclarifythesizeofthe
interaction energy J and the conduction electron density
polarons,andwhetheritispreferabletohaveislandphases,
x, showing that up to now the model has not revealed all
separatesmallpolaronsoreventuallylargepolarons.
itsrichness.Besidesthequantumresultsalreadypublished
[8,9,13]wefindtwotypesofphaseseparation.Inadditionto
In this paper, we present an analytical and numerical
theexpectedF-AFphaseseparationappearingforsmallJ,
study of the one-dimensional double and super-exchange
we obtain a new phase separationbetween small polarons
model. Our results provide a plausible understand-
(oneelectronwithintwoorthreesites)andAFregionsfor
ing of the ground state of the nickelate 1D compound
larger J. It is interesting to note that large polarons are
Y Ca BaNiO allowing a straightforward explanation
2−x x 5 neverfoundstableinthis limit.
of its spin-glass-like behavior [6]. The paper is organized
ThemagneticphasediagramhasbeenobtainedatT=0K
as follows. In section 2 a brief description of the model is
by using open boundary conditions on a linear chain of
given. In section 3, results and a discussionare presented.
N=60sites.Foragivenconductionelectrondensityx(0
Finally,ourresultsaresummarizedinsection4. ≤
x 0.5becauseofthehole-electronsymmetry),wehaveto
≤
optimize allthe N-1anglesθ .Forthis goal,we useanan-
i
2. The model alyticaloptimizationanda classicalMonte Carlomethod.
Theanalyticalsolutionhasbeentestedasastartingpoint
The DE Hamiltonianis originallyofthe form, inthe Monte Carlosimulation.
Ourresultsaresummarizedinfigure1,showingthewhole
H =− tij(c+iσcjσ +h.c.)−JH S→i·→σi, (1) magneticphasediagram.
iX,j;σ Xi

wherec+(c )arethefermionscreation(annihilation)op-
iσ iσ 0.30
eratorsofthe conductionelectronsatsitei, tij is the hop- 0.28 AF CP3 + CP2
ping parameter and −→σi is the electronic conduction band 0.26 CP2
spin operator. In the second term, J is the Hund’s ex- 0.24
change coupling. Here, Hund’s exchaHnge coupling is an 0.22 AF + CP3 CP3 CP3 + P2
intra-atomic exchange coupling between the spin of con- 2 S/t00..1280 Fig. 8 P2
ductionelectrons−→σiandthespinoflocalizedelectrons−→Si. J00..1146 AF + P3 FPig3. 3 T Fig. 2
This Hamiltonian simplifies in the strong coupling limit Fig. 7
0.12
JH ,alimitcommonlycalleditselftheDEmodel.We 0.10
→∞
willconsiderthelocalspinsasclassical−→Si ,areason- 0.08 AF + F F
→∞ 0.06
ableapproximationinmanycasesinviewofthesimilarity Fig. 4 Fig. 5 Fig. 6
0.04
oftheknownresults[9,12].TheDEHamiltoniantakesthe
0.02
well-knownform, 0.00
0.0 0.1 0.2 0.3 0.4 0.5
θ x
H = t cos i,j (c+c +h.c.). (2)
− i,j (cid:18) 2 (cid:19) i j
Xi,j
Fig. 1. Magnetic phase diagram as a function of the SE energy J
Theitinerantelectronsbeingnoweitherparallelorantipar- andtheconductionelectrondensityx.Adottedlineinthisdiagram
allel to the local spins are thus spinless. θ is the relative representsaguidefortheeyes.Thedifferentphasesaredescribedin
i,j
the text.
angle between the classical localized spins at sites i and
j which are specified by their polar angles φ , ϕ defined For the commensurate fillings x = 1/2 and 1/3, we
i i
withrespecttoaz-axistakenasthespinquantizationaxis recover the “island” phases with ferromagnetic polarons
of the itinerant electrons. The super-exchange coupling is (θ = 0) separated by antiferromagnetic links (θ = π),
i i
2
P2 ( ) andP3 ( ), SimilarphasesP3andCP3areobtainedforx=1/3.In
···↑↑↓↓↑↑↓↓↑↑↓↓··· ···↑↑↑↓↓↓↑↑↑↓↓↓···
figure 2 and figure 3 respectively, identified previously for CP3, two angles θ , θ are finite inside the 3-site polaron
1 2
classical [10] and S = 1/2 quantum [9] local spins. In the while, between polarons,θ =π. This phase has a general
3
quantum case, the real space spin-spin correlations illus- continuousdegeneracywithineach3-sitepolarongivenby
trate such structures. For these phases, the analytical op- the condition,
timizationimplies angles0orπ exactly. 1
cos(θ )+cos(θ )= 2. (4)
1 2 8(JS2/t)2 −
Anexample ofthis degeneracyofthe spinconfigurationis
3.0 0.5 clearly seen in fig 8 where different sets of angles (θ1,θ2)
appear within the CP3 phase. Both CP2 and CP3 evolve
2.5 towardscomplete anti-ferromagnetismasJS2/t ,see
0.4 equation (4). P3 CP3 at JS2 = 1 . The→se ∞phases
2.0 → t 4√2
0.3 result from the “spin-induced Peierls 2kF instability” due
to the modulation of the hopping with I = 1/x angles.
1.5 n
For lower commensurate fillings x < 1/3,such P polaron
0.2 I
phasesarenotfoundstable.Instead,nexttotheFphaseat
1.0
n 0.1 low J we find AF-F phase separation. Of course, an anti-
0.5 P2 phase, x=1/2 ferromagneticphasealwaysoccursatx=0.Figure1shows
thatwhentheSEinteractionenergyissmallJS2/t.0.12,
0.0
0.0 theFphaseoccursforalargeconductionelectrondensity.
00 55 1100 1155 2200 2255 3300 3355 4400 4455 5500 5555 6600 TheF-AFtransitionisgivenbytheF-AFphaseseparation
Ni (AF+F in figure 1) consisting of one large ferromagnetic
polaronwithinanAFbackgroundascanbeseeninfigures
Fig. 2. P2 phase for x = 1/2, showing N-1 angles (θ) and charge 4 and 5, for a typical value of JS2/t = 0.04. All electrons
distribution(n). Angles inthis figureare0or π exactly. are inside the polaron. The position of the polaronwithin
thelinearchainisnotimportantbecauseoftranslationde-
generacy. These figures also show charge distribution (n)
inside each polaron and a spin configuration snapshot. In
this region,the polarons’sizediminishes withthe conduc-
3.0
0.5
tionelectrondensity,(figures4and5).ForthisF-AFphase
2.5 separation,theanalyticaloptimizationimpliesangles0and
0.4
π exactlyfortheFandAFdomainsrespectively(figures4
2.0 and5).InthethermodynamiclimitN ,andforM
0.3 →∞ ≫
3 sites (M being the size of the F domain), the energy is
1.5 n obtainedas,
0.2
U JS2
1.0 = 2xcos(x π) , (5)
o
n 0.1 Nt − − t
0.5 P3 phase, x=1/3 for x xo. xo corresponds to the Maxwell construction
betwee≤ntheanti-ferromagneticenergy U = JS2 andthe
0.0 Nt − t
0.0 ferromagneticone U = 2 sin(xπ)+ JS2. x is givenby
00 55 1100 1155 2200 2255 3300 3355 4400 4455 5500 5555 6600 Nt −π t o
the followingequation,
Ni
1 JS2
x cos(x π) sin(x π)+ =0. (6)
o o o
− π t
Fig. 3.P3 phasefor x=1/3, showingthe sameas infigure2.
Thecorrespondingboundarygivenbyequation(6)isshown
The electrons are individually self-trapped in small in- bythe fulllineinfigure1.TheanalyticalresultsforN=60
dependent ferromagnetic polarons of two and three sites sitesareverycloseto this line.When the conductionelec-
respectivelyforming aWigner crystallization.Inreference trondensitygetslargerx x ,theF-AFphaseseparation
o
≥
[14], a spiral phase has been proposed instead of the P2 becomestheFphase.Thesizeoftheferromagneticpolaron
phase for x = 1/2. The ferromagnetic phase is stable for inthethermodynamiclimit(N ,M 3)isǫ= M =
weak SE interaction below P2 phase, P2 phase becomes x , for x x , and ǫ = 1 in the→fe∞rromag≫netic phaseN. An
xo ≤ o
stable for 2 1 < JS2/t < 1. For JS2/t > 1, P2 trans- important effect of lattice distortion is expected between
π − 2 4 4
forms intoa cantedpolaronphaseCP2in whichthe angle theseFandAFdomains[7].Theeffectoflatticedistortion
insidetheFislandsbecomesfinite(θ )whiletheanglebe- canbestudiedintheF-AFphaseseparationusingthefol-
1
tweenthepolarons(θ )stillkeepsthevalueπ.Acomplete lowing density matrix elements. Inside the F domain the
2
analyticalsolutioncanbe derivedinthis case. matrixelements aregivenby,
3
classical localized spins and J = 8 [14] and in the one-
H
dimensionalferromagneticKondomodel[16].Quantumre-
F AF sultsforS =3/2,showedphaseseparationwhenCoulomb
repulsion was taken into account [13]. We can see that in
3.0 0.30
thislimit,ourresultsdifferfromthoseofKoshibaeetal.[10]
n
withinthe “spin-inducedPeierlsinstability”mechanism.
2.5 2 0.25
(x=0.05; JS/t = 0.04)
2.0 0.20
0.15
1.5
n F
0.10 0.50
1.0 3.0
0.48
0.05
0.5 2.5 0.46
0.00 0.44
0.0 2.0
00 55 1100 1155 2200 2255 3300 3355 4400 4455 5500 5555 6600 0.42
Ni 1.5 0.40n
1.0 0.38
Fig. 4. AF+F phase at x = 0.05 (3 electrons) and JS2/t = 0.04, (x=0.416667; JS2/t = 0.04)
0.36
showing N-1 angles, charge distribution and a spin configuration
0.5
snapshot. n 0.34
2 xN (p)(i)π (p)(j)π 0.0 0.32
ρ = sin sin . (7) 0 5 10 15 20 25 30 35 40 45 50 55 60
i,j
M +1 (cid:18) M +1(cid:19)(cid:18) M +1(cid:19)
Xp=1 N i
BetweenF-AFdomainsandwithinAFdomainsthematrix
elements are zero. These density matrix elements suggest Fig.6.Fphaseatx= 25 (25electrons)andJS2/t=0.04,showing
60
an important lattice distortion inside F domains and null the sameas infigure4.
between F-AF domains and within AF domains. This lat-
At low concentration x < 1/3, if the SE interaction en-
tice distortion could be detectable for example using neu-
ergy increases 0.12 . JS2/t . 1 , we find a new phase
tron diffraction techniques as in La2CuO4+δ [12]. Charge 4√2
distribution is easily obtained for n = ρ figures 2-7. separationbetweenP3andAFphasesasshowninfigures1
i i,j=i
Forexampleinfigure6,chargedistributionoftheFphase and7.IttransformsintoAF+CP3forJS2/t> 1 asP3
4√2
can be observed for 25 electrons and at JS2 = 0.04. For becomesCP3.AphaselikeAF+P3(CP3)hasbeenidenti-
t
fiedusingS =3/2quantumspins[13].Figure7andfigure
8showthe AF+P3andthe AF+CP3phaseswith 12elec-
tronsamongthe 60sitesfortypicalvaluesofthe SEinter-
F AF actionenergyJS2/t=0.13andJS2/t=0.20respectively.
3.0 0.30 These phase separations consisting in P3 or CP3 phases
inanAFbackgroundandtheyaredegeneratewithphases
2.5 0.25 where the polarons can be ordered or not, while keeping
the number of F andAF bonds fixed. The phase obtained
2.0 0.20 withinthe“spin-inducedPeierlsinstability”[10]belongsto
this class. The former degeneracy unifies ideas like phase
1.5 2 0.15 separationand individualpolaronsandgivesa naturalre-
(x=0.20; JS/t=0.04)
n
sponsetothe instability atthe Fermienergyandto anin-
n 0.10
1.0 finite compressibilityas well.Inthe thermodynamic limit,
0.05 P3-AF phase separation energy is given by the following
0.5
equation
0.00
0.0 U JS2 JS2
= √2+4 x . (8)
00 55 1100 1155 2200 2255 3300 3355 4400 4455 5500 5555 6600 Nt (cid:18)− t (cid:19) − t
Ni
WefindthattheAF +F AF +P3transitionisfirstor-
→
der. In figure 1, the transition line JS2/t 0.12 between
Fig.5. Thesameas inFig. 4,but at x=0.20(12 electrons). ≃
the two phase separations AF+P3 and AF+F has been
small SE interaction,the F-AF phase separationhas been determined using the corresponding energies in the ther-
reported in two dimensions [15], in one dimension using modynamiclimit.Densitymatrixelementssuggestalarge
4
P3 AF
CP3 AF
3.0
3.0 0.5
0.5
2.5 n
2.5 n 0.4 2 0.4
2 (x=0.2; JS/t=0.2)
(x=0.2; JS/t=0.13)
2.0
2.0
0.3
0.3
1.5 n 1.5 0.n2
0.2
1.0
1.0
0.1
0.1
0.5
0.5
0.0
0.0
0.0
0.0
00 55 1100 1155 2200 2255 3300 3355 4400 4455 5500 5555 6600 00 55 1100 1155 2200 2255 3300 3355 4400 4455 5500 5555 6600
Ni Ni
Fig. 7. AF+P3 phase at x=0.20 (12 electrons) and JS2/t=0.13, Fig.8.AF+CP3phaseatx=0.20(12electrons)andJS2/t=0.20,
showing N-1 angles, charge distribution and a spin configuration showingthe sameas infigure7.
snapshot.
TheenergiesofeachCP3andCP2polaronsarerespectively
1 2JS2 and 1 JS2.
lattice distortion within each 3-site polaron in P3 phase. −8JSt2 − t −8JSt2 − t
Chargedistributionamongthe3sitesi=1,2,3insideeach Let us mention that homogeneous spiral phases (θ =
i
polaronisn =n = 1 andn = 1,see figure7. θ) could be possible groundstates. In the thermodynamic
1 3 4 2 2
limit, these phases can occur for JS2 sinπx and have
Phaseseparationalsotakesplaceforfillingsbetweenx= energy U = (sinπx)2 JS2. Howetver≥, our2πMonte Carlo
1/2 and x = 1/3 for SE interactions JS2/t ≥ 4√12. It is results sthow−th2aπt2(tJhSte2s)e−aretnever stable within the model
betweenCP3andP2orCP2due to the cantinginside the
used here. This can be proved analytically in the thermo-
P2 polaron with increasing J. The transition between the
two occurs for JS2/t = 0.25, where P2 CP2. Again, dynamic limit using the expressions we have derived for
→ the different phases, except for the T-phase for which nu-
duetothe AFlinksbetweenthe polaronsthesephasesep-
merical results are necessary. In two-dimension however,
arationsaredegeneratewithrespecttothe positionofthe
renewedinterestina spiralstate results fromexperiments
two types of polarons. Below CP3+P2,the phase labelled
indicatingaspinglassbehaviorofhigh-T La Sr CuO
T infigure 1 is a more generalcomplex phase obtainedby c 2 x x 4
−
atsmalldoping[17].
the Monte Carlomethodand canbe polaroniclike ornot.
Of course, all the phase separations involving CP3
Closeto the boundarywithCP3+P2this phase resembles
(AF+CP3, CP3+P2, CP3+CP2) present the spin con-
the P3+P2 phase separation, so as seen in figure 1, the
transitionlineJS2/t= 1 ,correspondingtoP3 CP3, figuration degeneracy (θ1,θ2) of the CP3 polarons. This
4√2 → analyticalcontinuous degeneracy is consistent with a spin
is second order. This boundary also extends in the region
glass state. Therefore, we propose that the ground state
x<1/3between AF+P3and AF+CP3.For the SE inter-
of Y Ca BaNiO for the studied hole doping x < 0.15
actionregionJS2/t> 1 (JS2/t= 1 isshownbytheshort- 2 x x 5
4 4 [6] be−longs to the AF+CP3 phase providing a plausible
dotted line in figure 1), the spin configurations (θ ,π) or
1 explanation for the observed spin-glass-like behavior. It
(π,θ ),i.eaCP2polaronplusanAFlink,belongtoallthe
2 is interesting to note that such a possibility of polarons
possible degenerateconfigurationsof CP3polarons as can
immersedintoananti-ferromagneticbackgroundhasbeen
be seen from equation (4). This means that in this region
invoked by Xu et al.[18] to fit their neutron data. Finally,
phase separation AF+CP3 may also contain a number of
we remark that the size chosen for the linear chain N=60
two-sites canted polarons CP2. Similarly, this also occurs
sites and the boundary conditions do not change the na-
withinCP3+CP2phase.Thetotalnumberofpolaronsre-
tureofthe phases involvedinthe phase diagram.
maining equal to the number of electrons; we can label it
as AF+CP3+CP2. A single energy is found in the whole
conduction electron density regime (0 x 0.5). In the
≤ ≤ 4. Conclusions
thermodynamic limit,itcorrespondsto,
U x JS2 In this work, we presented an unifying picture for the
= . (9)
Nt −8JS2 − t magnetic phase diagram of the one-dimensional DE+SE
t
5
model using large Hund’s coupling and classical local-
ized spins. The solution is in general a) phase separation
between F and AF phases for low SE interaction energy
and b) phase separation between small polaronic and AF
phases when the SE interaction is large. In a large SE
limit a Wigner crystallization and a spin-glass behavior
can be identified. A spin-glass behavior can be obtained
under the condition JS2 & 1 0.177,when CP3 phase
t 4√2 ≈
exists and could explain the spin-glass-like behavior ob-
served in the nickelate one-dimensional doped compound
Y Ca BaNiO . On the other hand, density matrix ele-
2 x x 5
−
mentssuggestanimportantlatticedistortioninthephases
involvedinthe phasediagram.
Acknowledgment
We want to acknowledge partial support from CONA-
CyT Grant-57929 and PAPIIT-IN108907 from UNAM.
E.V. would like also to thank CONACyT and DGAPA-
UNAM forfinancialsupport.
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