Phase equilibrium in two orbital model under magnetic field

0.2 MB·English

by  S. Dong

Phase equilibrium in two orbital model under magnetic field
S. Dong, X. Y. Yao, K. F. Wang, and J.-M. Liu
∗
Nanjing National Laboratory of Microstructures, Nanjing University, Nanjing 210093,China
International Center for Materials Physics, Chinese Academy of Sciences,Shenyang, China
(Dated: February 6, 2008)
The phase equilibrium in manganites under magnetic field is studied using a two orbital model,
based on the equivalent chemical potential principle for the competitive phases. We focus on the
6
magnetic field induced melting process of CE phase in half-doped manganites. It is predicted that
0
thehomogenous CE phase begins to decompose into coexisting ferromagnetic phase and CE phase
0
oncethemagneticfieldexceedsthethresholdfield. Inamorequantitativeway,thevolumefractions
2
of thetwo competitive phases in thephase separation regime are evaluated.

n
a PACSnumbers: 75.47.Lx,64.10.+h,75.47.Gk
J

9
Manganites, a typical class of strong correlated elec- variable,which,however,lackscredibletheoreticalinves-

] tron systems, have been intensively studied in the last tigation yet. In earlier studies, this important variable
l
e decade,duetotheirunusualbehaviorsofpotentialappli- was obtained from experiment or toy model [12, 13].
-
r cations such as colossal magnetoresistance (CMR). The In this letter, we attempt to study the phase equilib-
t
s double exchange (DE) mechanism can explain the mag- rium (PE) in a two orbital model. We emphasize par-
t. netic transition qualitatively, but more complex mecha- ticularly the evolvement of PS upon the magnetic field
a
nism responsible for CMR is not yet fully understood. perturbation. SowecallitPEinsteadofPSinthiswork.
m
Theidea ofphaseseparation(PS)wasrecentlyproposed Let’s begin with a simplified model Hamiltonian which
-
d to understand the physics essence underlying the amaz- has been frequently used [2]:
n ing behaviors of manganites, while more and more theo-
o retical and experimental evidences confirm the existence
[c of PS due to the intrinsic inhomogeneity [1, 2, 3]. Hami. = − tαγγ′d†iγσdiγ′σ −JH si ·Si
iαXγγ′σ Xi
Former investigations on the phase diagram of man-
v1 ganitesrevealedthe first-ordercharacterofphasetransi- +JAF Si·Sj +H· g(si+Si)(1)
6 tions between various phases, e.g. charge-ordered (CO) <Xi,j> Xi
6 insulator and ferromagnetic (FM) metal [2, 4, 5, 6]. The
1 where α is the vector connecting nearest-neighbor (NN)
insulator-metal transition in manganites can be reason-
01 ablyunderstoodastheconsequenceofpercolationofFM sites and d†iγσ(di+αγ′σ) is the generation (annihilation)
operator for e electron with spin σ in the γ(γ )-orbital
6 metal filaments embedded in the insulated matrix, and g ′
on site i(i+α); tα is the NN hopping amplitude be-
0 thereareplentyofexperimentalevidencestosupportthis rr′
/ PS framework [2, 7, 8]. Current theories on manganites tween γ and γ′-orbital (dx2 y2 as a orbital, d3z2 r2 as
at mainly stemfromthe competitionbetweenseveralinter- b orbital) along α-direction−, with tzbb = t0 > 0−(t0 is
m actions: DE, super exchange, Hund coupling, electron- taken as energy unit), txaa = tyaa = 43t0, txbb = tybb = 41t0,
d- phononinteractionandCoulombinteraction[2]. Besides, tyab = tyba = −txab = −txba = √43t0, tzaa = tzab = tzba = 0,
n the effect of quench disorder on PS dynamics is high- respectively; Si is the spin operators for t2g core on site
o lighted, especially on the large scale PS. The theoretical i, whilesi foreg itinerantelectron. The firsttermrepre-
c progress has enabled us to sketch the phase diagram in sents the kinetic energy(DE process)whichleads to FM
:
v somespecialregimesfromcalculationandidentifythePS spinarrangement. ThesecondtermistheHundcoupling
Xi regimeinparameterspacewithvariousmicroscopicmod- of eg and t2g electrons where JH > 0 is large enough to
els[9,10,11]. Nevertheless,itisstilluncleartheoretically beregardedasinfinite,sothespinofe electronisalways
g
r
a howthePSdevelops,especiallyunderanexternalpertur- parallelwith the same-site t2g spin. The super exchange
bation,e.g. magneticorelectricfield. Inotherword,it is interaction JAF >0 prefers to couple NN t2g spins anti-
ofinteresttonotonlyidentify theexistenceofPSregime, ferromagnetically;Thelasttermrepresentsthemagnetic
butalsoconcernhowthephaseseparationoccursandhow fieldcontributionwithmagneticfieldH andLandefactor
it evolves upon external perturbation, because potential g. The electron-phononcouplingandCoulombrepulsion
applications call for more sufficient theoretical interpre- are not taken into account in this Hamiltonian and their
tation. Forinstance, the CMReffect, whichis one ofthe effect on PE will be discussed below.
mostattractingtopicsinthe physicsofmanganites,may The above simplified Hamiltonian can be solved ex-
be described by the resistor network model phenomeno- actly once a prior t2g spin pattern is given. In real man-
logically, based on the percolation mechanism. In such ganites, various t2g spin patterns exist corresponding to
case the volume fraction of metal phase is the key input abundant phases, e.g. FM, antiferromagnetic (AFM),
2

1.0
(a) (b) 0.2 FM
0
DOS00..68 FCCMEAFM -1 /t0 gy/t-000..02 CCEAAFMF
r GAFM
0.4 FCME -2 ne-0.4
E
0.2 CAFM -0.6
GAFM -3 -0.8
0.0
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.0 0.2 0.4 0.6 0.8 1.0 -1.0 JAF = 0.0278t0
Energy/t0 n
-1.2
0.0 0.2 0.4 0.6 0.8 1.0
FIG. 1: (color online) (a)Density of state of FM/CE/CAFM
n
phase. HereonlyregimeE ≤0isdisplayedsincethesymmet-
rical E > 0 part is empty in ground state. In GAFM, there FIG. 2: (color online) System energy E (ground state under
is only a single energy level at 0 because of the localization zero field) as a function of eg electron concentration n, with
of eg electron. (b)Chemical potential µ as a function of eg JAF =0.0278t0. Upon increasing eg electrons n, a transition
electron concentration. The energy gap t0 for the CE phase sequenceoftheground state: GAFM→CAFM→CE→FM, is
at n=0.5 is evidentboth in (a) and (b). shown. The CE phase as ground state is possible only in the
narrow range around thehalf-filling point.
CO, orbital-ordered (OO) phases. In this work, sev-
eraltypicalt2g patterns confirmedfromexperiments: C- phases, when JAF is set as 0.0278t0 and H = 0, is plot-
type AFM (CAFM), G-type AFM (GAFM), FM and
ted in Fig.2. The preferred ground state is the energy
CE phase are chosen as the candidates to compare. minimal state. It should be GAFM as n ∼ 0 because
The CAFM phase is constructed by antiferromagnet-
the interaction is almost pure AFM super exchange. As
ically coupled one-dimension FM lines, while the CE n ∼ 0.3, the preferred state is CAFM. The CE phase
phase is constructed by antiferromagnetically coupled can appear only in a narrow regime n ∼ 0.5. When the
one-dimension zigzag FM chains and is found to be
gain from kinetic energy suppresses the loss of super ex-
CO/OO[14,15,16]. TheGAFMtakesthefamiliarAFM
changeenergyinthelargenregime,theFMbecomesthe
arrangement in all three directions. Then the Hamilto-
stable phase. With this calculation, the phase diagram
nian can be exactly solved when the Hund factor J
H over the whole concentration regime has be developed.
is simplified as infinite. The procedure of derivation is
Of course, the phase diagramof real manganites is more
straightforward and the details can be found in Ref.[2].
complex than this simple sketch for two reasons: first,
Then density of state (DOS, D(E)) of the these phases
theHamiltonianEq.(1)isoversimplifiedandsecondlythe
can be calculated from the dispersion relationship using
candidate phases chosen here are not complete but only
numericalmethod,asshowninFig.1(a). Inaddition,the
four phases. Even though, the calculated phase diagram
chemicalpotential µ of these phases is obtained simulta-
isquite similarto thatofsometypicalmanganites(zero-
neously by integrating the DOS, as shown in Fig.1(b).
temperature): e.g. Nd1 xSrxMnO3 (heren=1−x)[17].
Consequently,the groundstate energy is calculated. For −
Infact,itisshownthatthe phasediagramofotherman-
instance, the energy of FM phase can be written as:
ganites: e.g. La1 xSrxMnO3 or Pr1 xCaxMnO3, can be
− −
E(n,H) 1 E<µ 27 reproducedroughlybyadjustingthevalueofJAF,whose
= D(E)EdE+ J −3H (2) rolewillberevisitedbelow. Animportanttruthrevealed
AF
N N Z 2
here is that no matter what J , the CE phase can ei-
AF
hereN isthenumberofwholesitesandisinfiniteideally. ther appear in the narrow regime n ∼ 0.5 or simply be
Thefirstintegraltermgivestheenergyofalle electrons; unstable over the whole concentration regime.
g
term 27J /2 arises from the six NN FM correlation The abovetheoreticalapproachbasedonHamiltonian
AF
between |S| = 3/2 t2g cores in classical approximation; Eq.(1),whoseorigincanbe found fromRef.[2], allowsus
factor 3 before H is calculated by multiplying t2g spin to study the PE in PS regime. Here, a simple but repre-
3/2 with Lande factor 2. Besides, the influence of H sentative case: the PEbetween FM phase and CE phase
on D(E) has also been taken into account. The average withn=0.5,willbestudied. Itcorrespondstothemelt-
e electron concentration n ∈ [0,1]. The DOS, chemical ingprocessofatypeofCOstate(hereitisCEphase)un-
g
potential and ground energy of other phases can also be der external magnetic field. Since for FM metallic phase
calculated exactly just as the FM case. thelatticedistortionsareabsentandthee electronsare
g
However, what should be noted is that none of these delocalized [18], the electron-phonon coupling and on-
phases can be stable over the whole doping range. In site Coulomb repulsion are unimportant. On the other
order to determine which phase is the preferred one at hand, the CE phase can also be reasonably described as
a given concentration, the ground state energy of these a band insulator by this Hamiltonian at n = 0.5 case
3
[2, 16]. Therefore, Eq.(1) is suitable to deal with the PE
between FM metallic phase and CE phase, noting that -0.74
Eq.(1) can be exactly solved without scale issue. (a) H=0 FM 100 e
theTheqeuPivEaplernincecipinlecfhoermaiPcaSlspyostteemnticaalnbbeetwreepernestehnetecdomby- E/t--000..7786 CE 6800 entag
c
petitive phases, i.e. FM and CE to be considered. From 40 r
e
theabovecalculation(Fig.1(b)),itisseenthatatn=0.5 -0.80 P
20
and under zero field, the chemical potential of pure FM
phase is about −1.52t0 and that of CE phase is −t0. -0.82 0
Therefore, the chemical potential µ for the possible PS -0.80 (b) H=0.015t 100 e
systemwouldbeintherange[−1.52t0,−t0]. Thesumof 0 FM 80 ag
eg electronsinthe FMandCE phasescanbe calculated: t-00.81 CE nt
/ 60 e
E c
r
-0.82 40 e
p E<µ P
A
nA = D(E)dE (3) 20
N Z
-0.83
0
herep isthevolumefractionofA(FM/CE)phase. The
A -1.5 -1.4 -1.3 -1.2 -1.1 -1.0
upper limit of the integral should meet the equivalent
/t
0
chemical potential condition: µ = µ = µ. Then
FM CE
the following set of equations yields: FIG.3: (coloronline)VolumefractionofFM/CEphase(right
axis) and corresponding total energy system E (left axis) as
p +p =1 a function of chemical potential µ. JAF = 0.0278t0. (a)Zero
FM CE (4) fieldcase. wherethegroundstateisCEphasewithoutphase
(cid:26)nFM +nCE =0.5 separation because the energy decreases monotonously with
chemical potential; (b)under magnetic field H = 0.015t0,
By numerical method, Eq.(4) can be solved with the
where a energy minimal appears at µ = −1.13t0. The co-
prior assumed µ. Given parameters JAF =0.0278t0 and existencewiththefraction ofFMandCEphaseinthispoint
H =0,thecalculatedrelativevolumefractionofFM/CE is theground state.
phase as a function µ are shown in Fig.3(a) (right axis).
Thenbyincludingthefactorp intotheenergyequation,
A
the total system energy: E = p E +p E can as shown in Fig.4(a). Note here that the threshold of
FM FM CE CE
beencalculated,asshowninFig.3(a)(leftaxis). Itisseen percolation for FM phase can be easily surpassed under
thatbothpFM andE aremonotonouslydecreasingfunc- a field slightly higher than Hmin. For instance, only a
tions of µ. The ground state must be homogenous CE field of H = 0.009t0 is needed to obtain a FM phase
phase because the PS state goes against energy. In case of 24.7% (threshold for a three-dimensional simple cu-
of nonzero magnetic field (e.g. H = 0.015t0), the above bic bond percolation), beyond which an insulator-metal
calculation is repeated by taking the magnetic field con- transition may be expected. When H is extremely high
tribution to DOS and energy into account. The results (not shown in Fig.4(a)), the equivalent chemical poten-
are plotted in Fig.3(b). Quite interestingly, a nonzero tialconditioncannolongerbe satisfied,indicating ater-
field results in a minimal in the E − µ curve (here at mination of the PS state and the ground state will be
µ∼−1.13t0). ThePSstateassociatedwiththisminimal homogenous FM state.
energy point is more stable with respect to the homoge- The above calculationindicates that the lower thresh-
nous FM or CE phase. The value −1.13t0 is the real old of magnetic field to melt the CE phase is 0.0069t0.
chemicalpotentialofthisPSgroundstatewhichconsists Considering that t0 in low-bandwidth manganites is
of 57% CE phase and 43% FM phase. small, e.g. 0.1eV, the calculated H is about 12T,
min
By varying the value of H, one can repeat the above a value consistent with the experimental data [19, 20,
calculation and then obtain the relative volume frac- 21, 22]. Note here that H is not equal the direct en-
min
tions of the two phases as a function of H, as shown ergy gap between FM and CE phase, which is one order
in Fig.4(a). When H is low (below the lower thresh- of magnitude larger than experimental value (shown in
old Hmin ∼ 0.0069t0), the CE phase is robust against Fig.2,about0.08t0). Itmeansthattherequiredmagnetic
magnetic field perturbation, indicating the stable and field to destroy the CO insulator is strongly reduced by
homogenous CE phase as the ground state. Upon an in- PSwhichcanoccuroncetheenergyofcompetitivephases
creasingofH beyondH ,partofthe CEphasebegins is close to each other. On the other hand, as identified
min
tomeltintoFMphase,andp isdeceasingwithincreas- earlier,parameterJ playsakeyroleinPSalthoughit
CE AF
ingH underthe equivalentchemicalpotentialcondition. is the leastintrinsic interactionin manganites [2]. Other
In details, p increases rapidly once H > H , and thantheabovecaseoffield-inducedsequences,abundant
FM min
the growth becomes slower when H is further higher, phenomenaassociatedwithPSinmanganitescanbepre-
4
potential between competitive phases.
60
M 100 In summary,the principle ofchemicalpotential equiv-
e of F4500 (JaA)F=0.0278t0 (Hb=)0 80 of FM alilbernicuemhaosfbheaelnf-dinotpreoddumceadngtoaninitveesstuignadteertehxetperhnaaslemeqaugi--
ntag30 Homogenous Coexistence 60 age netic field. By employingthe twoorbitalmodel, wehave
Perce1200 CE Phase of FM & CE FM PS CE 2400 Percent pbreetwseenetnedFaMn pexhpaslieciatnsdoluCtEionphtaoset.heTphheasmeaegqnueitliicbrfiiuelmd
threshold required for melting of the CE phase has been
0 0
0 5 (10 )15 201.2 1.6 2.0( 2.4)2.8 3.2 calculated,consistentwiththeexperimentalresults. The
-3 -2
H 10 t0 JAF 10 t0 volume fractions of the two competitive phases in the
phase separation regime as a function of external mag-
netic field have been evaluated. In addition, the super
FIG. 4: (a)Melting process of the CE phase under magnetic
field. When the field is below the threshold Hmin, the ho- exchange modulated transitions between ferromagnetic,
mogenous CE phaseis robust. Then phaseseparation occurs phase separated and CE states under zero-field, is pre-
when H > Hmin. (b) Phase transitions induced by JAF un- dicted.
der zero-field condition: FM phase at weak JAF; coexistence S. Dong thanks G. X. Cao for valuable discussions.
ofFMandCEphaseatmiddleJAF;CEphaseatstrongJAF. This work was supported by the Natural Science Foun-
dation of China (50332020, 10021001, 10474039) and
National Key Projects for Basic Research of China
dicted by our model through adjusting JAF. For exam- (2002CB613303,2004CB619004).
ple, againatn=0.5,a coexistence ofFM phase and CE
phase under zero field is predicted at 0.019t0 < JAF <
0.026t0. When JAF is further reduced, a homogeneous
FMphaseasthegroundstateispossibleevenunderzero
field. The transition between three regimes: homoge-
∗
Electronic address: [email protected]
nousFMphasetoPSstatetohomogenousCEphase,are
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5
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