Cross effect of Coulomb correlation and hybridization in the occurrence of ferromagnetism in two shifted band transition metals

0.17 MB·English

by  C. M. Chaves

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Crosseffect ofCoulombcorrelationandhybridizationintheoccurrence of
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2
ferromagnetismintwoshiftedband transitionmetals

n
a C.M. Chavesa,∗,A.Tropera,b
J
2 aCentroBrasileiro de Pesquisas F´ısicas, Rua XavierSigaud 150,Rio de Janeiro, 22290-180, RJ, Brazil
2 bUniversidade do Estado do Riode Janeiro, Rua S˜ao Francisco Xavier524, Rio de Janeiro, RJ, Brazil

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l
e
-Abstract
r
t
sInthisworkwediscusstheoccurrenceofferromagnetism intransition-likemetals.Themetalisrepresentedbytwohybridized(V)
.
atand shifted (ǫs) bands one of which includes Hubbard correlation whereas the other is uncorrelated. The starting point is to
mtransform the original Hamiltonian into an effective one. Only one site retains the full correlation (U) while in the others the
correlationsarerepresentedbyaneffectivefield,theself-energy(single-siteapproximation).Thisfieldisself-consistentlydetermined
-
dby imposing the translational invariance of the problem. Thereby one gets an exchange split quasi-particle density of states and
nthen an electron-spin polarization for some values of the parameters (U,V,α,ǫs), α being the ratio of the effective masses of the
otwobandsand oftheoccupation numbern.
c
[

Keywords: Ferromagnetic metal; Correlation;Single-siteapproximation;
1
PACS: 71.10.-w,71.10.Fd, 71.20.Be
v
0
0
51. The model wherena =a+a ;σdenotesspin.t denotesthetunnel-
iσ iσ iσ ij
3
ing amplitudes betweenneighboring sites i and j , in each
.
1 Inrecentyearsthestudyofmagnetisminitinerantferro- band.As in Roth’s approach[4],we reduce the presence of
0magnetssuchasFe,Co,Nihasbeenthesubjectofagreat the correlationto only one site (the origin, say ), while in
9
0dealofeffortsbyseveralapproaches.Examplesarethedy- theothersactsaneffectivespinandenergydependentbut
:namical mean field theory (DMFT) [1] and the modified siteindependentfield,theself-energyΣσ.Thisfieldisself-
v
alloy analogy (MAA)[2]. In an previous work [3] we have consistently determined by imposing the vanishing of the
i
Xdevelopeda two bandmodel, consisting of a Hubbard like scatteringT matrixassociatedtotheorigin.Wethusarrive
rnarrow band( band a) with intrasite Coulomb interaction atthe effectiveHamiltonian
a
U,hybridizedwithanotherband,whichisbroadanduncor-
related (band b), through the hybridization coupling V .
ab
The two bands had the same center (symmetric regime).
Now we treat a more general situation , with a shift be- Heff =Xtaija+iσajσ +Xtbijb+iσbjσ (2)
i,j,σ i,j,σ
tweenthe centersofthe twobands.
Wereviewbrieflythemethod[3]:TheinitialHamiltonian +XnaiσΣσ+Una0↑na0↓+X(Vabb+iσajσ + h.c.)
we adoptis then i,σ i,j,σ
H=Xtaija+iσajσ +Xtbijb+iσbjσ (1) −Xσ na0σΣσ,
i,j,σ i,j,σ
+XUni(↑a)ni(↓a)+X(Vabb+iσajσ +Vb+aa+iσbjσ) ,
i i,j,σ H still includes the difficulty of dealing with the
eff
Coulombintra-atomictermattheorigin.WeusetheGreen
∗ function method [5] ; the equations of motion for the cor-
Corresponding author. Tel: [phone] fax: [phone]-
responding Green functions Gcd (w) =<< c ,d+ >> ,
7400 ijσ iσ jσ w
Email address: [email protected] (C.M.Chaves). wherec,d=a,b,are
Preprintsubmitted toElsevier 22 January 2009
wGaija,σ(w)=δij +XtailGalja,σ(w)+ΣσGaija,σ(w) ǫs is the center of the b band; as the a band is centeredat
l the origin, this parameter represents a shift in the bands.
+XVab(Ri−Rl)Gblja,σ(w)+δi0[UGa0ja,,σa(w)+ αtheiseaffepchteivneommeanssoelosgoifcathlepaaraanmdettehredbeeslcercitbrionngst.hFerormatinooowf
l
−ΣσGaa (w)]; (3) on we take kia →ki,i = x,y,z and Vab = Vba ≡ V = real
0j,σ andconstantindependent ofk .
i
wGbija,σ(w)=XtbilGblja,σ(w)+XVba(Ri−Rl)Galja,σ(w) The vanishing of the T-matrix gives further a self-
l l consistentequationfor the self-energy:
where Ga0ja,,σa(w) =<< n0−σa0σ;a+jσ >>w≡ Γa0jaσ(w) is a Σσ =U <na0−σ >+(U −Σσ)Fσ(w,Σσ)Σσ, (10)
higherorderGreenfunction,whoseequationofmotion,af-
with
tertheneglectingofthebroadeningcorrection[6,7]reduces
to Fσ(w,Σσ)=N−1XGakk,σ (11)
k
wΓaija,σ(w)=<na0−σ >δij +XtailΓalja,σ(w)+ΣσΓaija,σ(w)
l 2. NumericalResults
+XVab(Ri−Rl)<<na0−σblσ;a+jσ >> (4)
l We perform the self-consistency in both Σσ and in
+δ (U −Σσ)Gaa,a(w). <na >,for eachtotaloccupationn=<na >+<nb >.
i0 0j,σ 0,σ
The total number of electrons per site, is fixed at n = 1.6
The ressonance broadening occurs, in the terminology of
(butseebelow),alittlelessthanhalf-filling.Wewantnow
the alloy analogy (AA), when the opposite spin direction
to exhibit the combined effect of U, V, α, n, and ǫ at
s
are not kept frozen. Some remarks are in order about
T =0K.
Eq.(4): The scattering correction is already included; in
Infig(1)weplotmagnetizationversusV.Itisclearthat
the Hubbard terminology[6,7] of an AA of up and down
smallvaluesofV helpstabilizetheferromagneticorderbut
spins, this correction would correspond to disorder scat-
largeronestendtoinhibitit[2].Thisisbecausehybridiza-
tering and produces a damping of the quasi-particles.
tion, apart from changing the occupations of the a and b
Secondly, the hybridization generates a new function <<
bands, together with the ǫ increases (small V) and de-
na b ;a+ >> and its equation of motion, again after s
0−σ lσ jσ w creases(largeV)thea-densityofstatesattheFermilevel.
neglectingthe broadeningcorrection,reducesto
Infig(2)themagnetizationisplotversusǫ .Weseethat
s
the shiftthen tends to favorferromagnetism.
w <<na0−σbiσ;a+jσ >>=X(tbil <<na0−σblσ;a+jσ >> Infig(3)weplotthechargetransferencea−>borvice-
l
versaasfunctionofǫ anditisseenthatfromǫ ≈0.8on,
+V (R −R)Γaa (w)) (5) s s
ab i l ljσ this transference increases the number of a electrons thus
At this point we have an effective impurity problem; the tending tofavorferromagnetism.
directionofthe impurityspinis notfixed.
We solve explicitly the problem defined by Eq.(3),
Eq.(4)and Eq.(5), obtaining, after imposing T = 0, the
0.18
followingGreenfunction forthea band:
0.16
Gakk′,σ(w)= w−˜ǫakδk−k′Σσ(w), (6) ation0.14
Inthis equation netiz0.12
g
a0.10
M
|V |2(k)
ǫ˜a =ǫa+ ab , (7) 0.08
k k w−ǫb
k
0.06
is the recursionrelationof the a bandmodified by the hy-
0.04
bridizationV andǫa andǫb denote the barebands,with 0.1 0.2 0.3 0.4 0.5 0.6
k k
V
t (cos(k a)+cos(k a)+cos(k a))
ǫa = a x y z , (8)
k A Fig. 1. Magnetization versus hybridization V for U = 3, α = 1.5,
n = 1.6 and ǫs = 1.0. Small values of hybridization tend to favor
In this paper we use t = 1 and A = 3, in arbitrary en-
a ferromagnetism.
ergyunits. All energymagnitudes aretakeninunits oft ,
a
Infig(4)oneexhibits the dependenceofthemagnetiza-
makingthemdimensionless.Thebareabandwidthisthen
tion on the ratio of the effective masses beween the corre-
W =2.Forsimplicity we adopthomothetic bands
lated and the uncorrelated bands. We argue that the in-
ǫb =ǫ +αǫa. (9) creasingofαisproportionaltoadecreasingoftheeffective
k s k
2
of the total occupation n. We notice that small values of
nfavorsparamagnetismwhileaftersomeoccupation,here
0.18 n≈1.6,the magnetizationdropsdown.
0.16
0.14
on0.12 0.18
ati
etiz0.10 0.16
n
ag0.08 0.14
M
0.06 0.12
n
o
0.04 ati0.10
0.02 netiz0.08
g
a
0.00 M0.06
0.0 0.2 0.4 0.6 0.8 1.0
0.04
0.02
Fig.2.Magnetization versusband shiftforU =3, V =0.4, α=1.5 0.00
andn=1.6.Largervaluesofthebandshifttendtofavorferromag-
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2
netism.
n
Fig.5.Magnetization versustotal occupation nforU =3,V =0.4,
α=1.5and ǫs=1.0.
0.4
a->b Infig(6)wepresentthedensityofstates(DOS)ofthea-
0.3
bandforU =3,V =0.4andǫ =1.0.TheFermilevelisat
s
0.2 EF =0.443andamagnetizationof0.174arises.Thecom-
>b bined effect of hybridization and the band shift produces
a-
ns 0.1 a band broadening[2]. The DOS here obtained exhibits a
a
Tr bimodalstructurecaracterizingaHubbardstronglycorre-
0.0
latedregime.
b->a
-0.1 In fig (7) we show the density ofstate (DOS) of the un-
correlatedb band, for the same set of parameters,namely
-0.2 U =3, V =0.4,α =1.5 andǫ =1.0 . We verify that the
0.0 0.2 0.4 0.6 0.8 1.0 s
renormalized band remains almost unchanged when com-
paredwith the bare one.Infact, hybridizationaffects this
Fig. 3. Charge transference versus band shift for U = 3, V = 0.4 band,enlargingit,butno noticeablebmagneticmoment
α=1.5andn=1.6 arises. Moreover,it does not present a bimodal structure.
Forthesakeofcompletenesswedisplayinfig(8)asituation
massofthecorrelatedabandwithrespecttothefreeelec-
envolving the weak correlation regime, U/W << 1. Now
tron b band and hence the magnetization should also de-
crease.Infig(5)onedisplaysthemagnetizationasfunction
down
0.7 up
0.6
0.5
0.16
0.4
a
ation0.12 DOS 0.3
etiz
agn0.08 0.2
M
0.1
0.04
0.0
-4 -2 0 2 4
0.00
1.6 2.0 2.4 2.8 3.2 Energy
Fig.6.DensityofstatesofthecorrelatedabandforU =3,V =0.4
Fig. 4. Magnetization versus α, the band width relation for U =3, ,α= 1.5, n = 1.6 and ǫs = 1.0. The Fermi level is at EF = 0.443
V =0.4,n=1.6andǫs=1.0.Smallvaluesofhybridizationtendto and a magnetization of 0.174 develops. The combined effect of hy-
favor ferromagnetism. bridizationand theband shiftproduces aband broadening.
3
Fermi Level 0.070
0.6
0.5 0.068
0.4
Re 0.066
b
OS 0.3
D
0.064
0.2
HF
0.1 0.062
0.0
0.060
-4 -2 0 2 4 -2 0 2
Energy Energy
Fig. 7. Density of states of the b band for U =3, V =0.4, α=1.5 Fig.9.Realpartoftheself-energyΣσ forU =0.1,V =0.4,α=1.5,
and ǫs=1.0.The Fermilevel is atEF =0.443. n=1.6 and ǫs =1.0. In this regime Σσ shows a very weak depen-
dence on theenergy.
the abandisrenormalizedasatypicalHartree-Fock(HF)
band without exhibiting the Hubbard bimodal structure.
Moreover, from fig (9), where we plot the real part of the
2.0
self-energy,weseeatrendoftheusualHFregime,namely
analmostconstantvalueoftheself-energy.Forcomparison
we show in fig (10) the self-energy for a strong correlated 1.5
limit.
e 1.0
R
0.8 0.5
0.7
0.6
0.0
0.5 -6 -4 -2 0 2 4 6
Energy
a 0.4
S
DO 0.3 Fig.10.Realpartoftheself-energyΣσ forU =3,V =0.4,α=1.5,
0.2 n=1.6andǫs=1.0.
0.1
4. Acknowledgement
0.0
-0.1
-2 0 2 CMCandATaknowledgethesupportfromthebrazilian
Energy agenciesPCI/MCT andCNP .
q
Fig.8.Densityofstates oftheabandintheweakcouplingregime,
U = 0.1, for V = 0.4, α = 1.5, n = 1.6 and ǫs = 1.0. Notice the
absence ofbimodal structure.
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[10]C. M.Chaves andA. Troper,to bepublished.
ofthe phasediagram[10]forthe modelis inprogress.
4