Table Of ContentParamagnetic-ferromagnetic transition in a double-exchange model
Eugene Kogan1 and Mark Auslender2
1 Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900,
Israel
2 Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105
3
Israel
0
(February 1, 2008)
0
2 Westudyparamagnetic -ferromagnetic transition duetoexchangeinteraction between classical
n localized magnetic moments and conduction electrons. We formulate the Dynamical Mean Field
a Approximation equations for arbitrary electron dispersion law, concentration and relation between
J exchangecouplingandtheelectronbandwidth. Solvingtheseequationswefindexplicitformulafor
8 thetransition temperatureTc. Wepresent theresults ofcalculations of the Tc for thesemi-circular
electron density of states.
]
h
PACS numbers: 75.10.Hk, 75.30.Mb, 75.30.Vn
c
e
m
I. INTRODUCTION II. DMFA EQUATIONS AND TC
-
t
a
t The double-exchange (DE) model [1–3] is one of the Spins being considered as classical vectors Sn = mn
s (with the normalization m2 =1), the problembecomes
t. basic ones in the theory of magnetism. Magnetic order- | |
a ing appears in this model due to exchange coupling be- a single electron one. The DE Hamiltonian in a single
m electron representation can be presented as
tween the core spins and the conduction electrons. The
- Hamiltonian of the model is
d Hnn′ =tn−n′ Jmn σδnn′. (2)
n − ·
o H = tn−n′c†nαcn′α−J Sn·σαβc†nαcnβ, (1) Let us introduce Green’s function
c nn′α nαβ
[ X X
Gˆ(E)=(E H)−1 (3)
2 wherecandc†aretheelectronsannihilationandcreation −
v operators, Sn is the operator of a core spin, tn−n′ is the and local Green’s function
0
electron hopping, J is the exchange coupling between a
45 core spin and n electrons, σˆ is the vector of the Pauli Gˆloc(E)= Gˆnn(E) . (4)
2 matrices, and α,β are spin indices. D E
1 The model (the core spins being treated as classical InthelastEquation,theaveragingiswithrespecttoran-
2 vectors) began to be studied several decades ago [1–3]. domconfigurationsofthecorespins. Intheframeworkof
0
Because of a recent general interest in manganites, it the DMFA approach to the problem (see [8,10] and ref-
/
t reappeared in the focus of attention (see reviews [4–7] erences therein) the local Green’s function is expressed
a
m and references therein). In particular, application of the through the the local self-energy Σˆ by the equation
Dynamical Mean Field Approximation (DMFA) [8] sub-
-
d stantially advanced our understanding of the properties Gˆ (E)=g E Σˆ(E) , (5)
loc 0
n of the DE model [9,10]. Most of the papers dealing with −
(cid:16) (cid:17)
o
themodel,startingfromclassicalpaperbyDeGennes[3], where
c
: considered the DE Hamiltonian with infinite exchange
Xiv (cahnadngwe,itwhhtihcehaidsdcirtuiocnialoffotrhethaenteixfeprlraonmataigonneotifcmsuapgenreetxic- g0(E)= N1 k (E−tk)−1 (6)
X
r properties of manganites). However lately many proper-
a ties of the model for arbitrary strength of the exchange is the bare (in the absence of the exchange interaction)
were analyzed [11]. localGreen’s function. The self-energysatisfies equation
In this paper we calculate the temperature of a
paramagnetic-ferromagnetic transition Tc in a double- Gˆ (E)= 1 , (7)
exchange model for arbitrary electron dispersion law, loc *Gˆ−1(E)+Σˆ(E)+Jm σˆ+
loc ·
concentration and relation between the exchange cou-
pling and the electron band width by formulating and where X(m) X(m)P(m),andP(m)is aprobabil-
h i≡
solving the DMFA equations. We treat the core spins as ityofagivenspinorientation(one-siteprobability). The
classical vectors. quantities Gˆ andRΣˆ are 2 2 matrices in spin space.
×
1
The difference between Eq. (7) and similar equation muchlessthechemicalpotential,sowecanconsiderelec-
appearingintheCoherentPotentialApproximation[12], tron gas as degenerate. In this case
is due to the fact that in our problem the disorder is an-
∞ µ
nealed (and not quenched). That is the probability of
f(E)...dE ...dE, (14)
a given (random) core spin configuration is determined Z−∞ ≡Z−∞
bythe energyofelectronsubsysteminteractingwiththis
where the chemical potential (in paramagnetic phase) is
configuration.
found from the equation
To understandthe DMFA assumptionallowingto find
the probability P(m) self-consistently with the solution 2 µ
of Eq. (7), notice that Eq. (7) reduces the problem of n= Img dE, (15)
−π
electronscatteringbymanyspins, toscatteringby asin- Z−∞
glespinwiththeeffectivepotentialV = Jm σ Σˆ. The
and n is the number of electrons per site.
− · −
spin is embedded in an effective medium, described by
Forthe T , afterstraightforward,thoughlengthyalge-
the Hamiltoniantk+Σˆ, and,hence,by the localGreen’s bra, we obtacin
functionGˆ . Considerthechangeinthenumberofelec-
loc
tron states in such system due to a single spin. We get 2J2 µ g
[12,13] Tc = 3π Im (Σg′−g)(1+Σg) 2J2g dE, (16)
Z−∞ " g′+g2 − 3 #
1
∆D(E,m)= Imlndet 1+ Jmσˆ+Σˆ Gˆ , (8)
−π loc where Σ and g are determined by the properties of the
h (cid:16) (cid:17) i system in the PM phase; they are the solutions of Eqs.
wherethe argumentofbothG andΣ isE+i0. Sothe
loc (5) and (11); g′ dg0(ε) . Eq. (16) is the main
change in thermodynamic potential is [14,11,15] ≡ dε ε=E−Σ
result of our paper. (cid:12)
(cid:12)
∆Ω(m)= f(E)∆D(E,m)dE, (9) Inthecaseofweakexch(cid:12)angeJ 0Eq. (16)takesthe
→
form
Z
where f(E) is the Fermi function. The DMFA approxi- 2J2 1 µ
mation for the one-site probability P(m) is: Tc = 3 N0(µ)− π Img02(E) dE. (17)
(cid:20) Z−∞ (cid:21)
P(m) exp[ β∆Ω(m)]. (10)
In fact, this equation is the MF approximation [16] for
∝ −
theRKKYHamiltonian[17],towhichtheoriginalHamil-
Eqs. (7) and (10) are the system of non-linear (inte-
tonian (1) can be reduced in the case J W.
gral) equations. However, the system is grossly simpli- ≪
In the opposite case J we have to consider only
fiedinparamagnetic(PM)phase,wherethemacroscopic →∞
one spin sub-band. (Because of the symmetry of the
magnetization M = 0. In this case P(m) = const, the
problem with respect to transformation n 2 n, ev-
averagingin Eq. (7) canbe performed explicitly, and we → −
erywhere further on we consider only the case n 1.)
obtain ≤
After we shift E, Σ and µ by J, we obtain
1 1
g(E)= 2 g−1(E)+Σ(E) J, (11) 2 µ g′+g2
T = Im dE, (18)
X(±) ± c π g′ 2g2
Z−∞ (cid:20) − (cid:21)
where Σˆ =Σˆ1, Gˆ =gˆ1=g (E Σ)ˆ1, where ˆ1 is a unity
0 where g is found from the equation
−
matrix.
In the ferromagnetic (FM) phase near the Curie tem- 1 1
g(E)= . (19)
perature, Eqs. (7) and (10) can be linearized [9,11] with 2 g−1(E)+Σ(E)
respectto M. Thus we reduce the DMFA equations to a
traditional MF equation [16]
P(m) exp( 3βTcM m). (12) III. TC FOR SEMI-CIRCULAR DOS
∝ − ·
The parameter T is formally introduced as a coefficient
c LetusapplyEq. (16)tothe caseofsemi-circular(SC)
in the expansion of ∆Ω(m) with respect to M. Non-
bare density of states (DOS)
trivial solution of the MF equation
M= m (13) 2 E 2
h i N0(ε)= 1. (20)
Ws W −
(cid:18) (cid:19)
can exist only for T < T , hence T is the ferromagnetic
c c
transition temperature. In all cases T turns out to be Then
c
2
2 0.05
N (ε)dε 2 E E
0
g (E)= = 1 . (21)
0
E ε W W −s W −
Z − (cid:18) (cid:19)
0.04
From Eq. (5) follows
0.03
Σ=E W2g/4 g−1, (22) W
− − T/c
0.02
and Eq. (11) can be written as
1 1 0.01
g = . (23)
2 E W2g/4 J
X(±) − ±
0
0 0.2 0.4 0.6 0.8 1
Eq. (16) takes the form n
FIG.1. Tc asafunctionofelectronconcentrationnfordif-
ferent relative strengthsof the exchange: J/W =.25 (dotted
J2W2 µ
T = line), J/W = 1 (dash-dotted line), J/W = 2 (dashed line),
c
6π Z−∞ and J/W =20 (solid line).
g2
Im dE. (24)
E W2g E W2g J2W2g2 IV. NONFERROMAGNETISM
− 4 − 2 − 6
(cid:16) (cid:17)(cid:16) (cid:17)
Notethatthatourmainresult(equationfortheT )in-
In the case of weak exchange J/W n1/3, we obtain c
≪ dicatesit’s ownlimitsofvalidity. InpartoftheJ/W n
−
plane,Eq. (24)givesT <0. NegativevalueofT means,
4J2 c c
T = (4y2 1) 1 y2, (25) that at any temperature, including T = 0, the para-
c 3π2W − − magnetic phase is stable with respect to appearance of
p
small spontaneous magnetic moment, and strongly sug-
where
geststhatthegroundstateinthispartofthephaseplane
2 isnonferromagnetic[18]. The mechanismsleadingtothe
n=n0(y) 1+ sin−1y+y 1 y2 . (26) destruction of ferromagnetism can be easily understood
≡ π −
(cid:16) p (cid:17) in the extreme cases of weak and strong exchange.
In the case of strong exchange J/W n1/3, Eq. (18) It is well known, that for J → 0 the DE Hamiltonian
≫ canbereducedtototheRKKYHamiltonian(seereview
takes the form [10,15]
[17] and references to the original papers therein)
Tc = W6π2 Z−µ∞Im"1−gW262g2#dE HRKKY =Xnn′ Inn′mn·mn′. (30)
Thus weak exchange between the electrons and the core
W√2 1
= 1 y2 tan−1 3(1 y2) , (27) spins generates effective long-range and sign oscillating
4π − − √3 −
(cid:20) (cid:21) effective exchange interaction between the core spins.
p p
In the opposite case J , we can reduce the DE
where n=n (y)/2. If, in addition, n 1, we obtain → ∞
0 Hamiltonian (2) to effective (classic) t J model (see
≪
−
review [19] and references therein). To do that, first di-
3
T = Wn .53Wx. (28) agonalize the exchange part of the DE Hamiltonian by
c
4√2 ≈ choosing local spin quantization axis on each site in the
direction of m, the latter being determined by polar an-
Notice, that virtual crystal approximation (for J )
gle θ and azimuthal angle φ. In this representation the
→ ∞
[3] gives
Hamiltonian is
Tc = 125Wx≈.13Wx, (29) Hnn′ =−Jσzδnn′ +tnn′(cid:18)ab∗nn′nn′ abnnn′n′ (cid:19), (31)
where
thuFsorbaeirnbgitrimarpyreecxicsheabnygea,tfahcetionrteogfr4a.linEq. (24)canbe ann′ =cosθn cosθn′ +sinθn sinθn′ei(φn′−φn)
2 2 2 2
ctiaolncualaretepdreosnelnytendumonerFicIaGll.y.1.The results of such calcula- bnn′ =sinθn cosθn′e−iφn cosθn sinθn′e−iφn′. (32)
2 2 − 2 2
3
The next step is to apply a canonical transformation describing a single spin in aneffective field, proportional
to the macroscopic magnetization, and we obtained ex-
1
H H˜ =eSHe−S =H +[S,H]+ [S[S,H]]+... (33) plicit formula for the Tc. For the semi-circular electron
→ 2
density of states we explicitly calculated the transition
whichexcludesallband-to-bandtransitions. Thiscanbe temperature as a function of the exchange interaction
achieved if we chose the operator S in the form and electron concentration.
Snn′ =−t2nJn′ (cid:18)−abn∗nn′′n abnnn′n′ (cid:19). (34) FoTunhdisatrieosneaardcmhinwiastserseudppboyrttehde IbsyraethleAcIasrdaeemliySocfieSnccie-
ences and Humanities.
We have
[S,Hex]nn′ =−tnn′(cid:18)b∗n0′n bn0n′ (cid:19)
[S,[S,Hex]]nn =2Xn′′ Inn′′(cid:18)−|bn0n′′|2 |bnn0′′|2 (cid:19), (35) [1] C. Zener, Phys.Rev. 82, 403 (1951).
whereInn′′ = tnn′′ 2/(2J). Keepingtermsuptothesec- [2] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675
| | (1955).
ond order with respect to t (and only site-diagonal part
[3] P. G. De Gennes, Phys. Rev. 118, 141 (1960).
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H˜nn′ =−Jσzδnn′ +tnn′(cid:18)an0n′ an0′n (cid:19) [5] Y1464u7,.1(A10.999I(z92y)0.u0m1)o.v and Yu. N. Skryabin, Physics-Uspekhi,
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· −
n′′ [7] D. M. Edwards, Adv.Phys. 51, 1259 (2002).
X
[8] A.Georges,G.Kotliar,W.Krauth,andM.J.Rozenberg,
InthesecondquantizationtheHamiltonianobtainedhas
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Field Approximation equations for the double-exchange
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4
