On the use of a single site approximation to describe correlation in pure metals

0.11 MB·English

by  C. M. Chaves

9
0
Ontheuseofasinglesiteapproximationtodescribecorrelationinpuremetals
0
2
∗
C.M. Chavesa, , A.A.Gomesa,A.Tropera,b
n
a aCentroBrasileiro de Pesquisas F´ısicas, Rua XavierSigaud 150,Rio de Janeiro, 22290-180, RJ, Brazil
JbCentroBrasileiro dePesquisas F´ısicas, RuaXavierSigaud150,Rio de Janeiro, 22290-180, RJ, Brazil and Universidadedo Estado doRio

2 de Janeiro, Rua S˜ao Francisco Xavier524, Riode Janeiro, RJ, Brazil
2

]
l
eAbstract
-
r
tThemagneticpropertiesofpuretransition-likemetalsarediscussedwithinthesinglesiteapproximation,totakeintoaccountthe
s
electron correlation. The metal is described by two hybridized bands one of which includes the Coulomb correlation. Our results
.
tindicate thatferromagnetism follows from adequatevalues of thecorrelation and hybridization.
a
m
-Keywords: Ferromagnetic metal; Correlation;Single-siteapproximation;
d
PACS: 71.10.-w,71.10.Fd, 71.20.Be
n
o
c
[
1. Introduction approach[2] to describe the electron-electron correlation

2 throughaneffective Hamiltonian
v
The coherent potential approximation (CPA) has been
86largelyusedtoprovideasimplesinglesitedescriptionofdis- Heff =Xǫaa+iσaiσ+Xtaija+iσajσ +Xtbijb+iσbjσ (2)
2orderedalloys[?,1].AnotherapplicationofCPAprocedures i,σ i,j,σ i,j,σ
3is the description of pure metals in presence of electron- +XnaiσΣσ+Una0↑na0↓+X(Vabb+iσajσ + h.c.)
1.electron correlations[2]. In this formulation, one describes i,σ i,j,σ
90tfehceticvoerrseellaf-teenderegleycwtrhoinchbyinacosrppinoraantedsetnheeregffyedctespoenfdcoernrteleaf-- −Xna0σΣσ,
0 σ
tion.Thisself-energyisself-consistentlydeterminedbyim-
v:posingthe vanishingofthe scatteringT matrixassociated where Σσ is the self-energy associated with electron-
itoagivensitewhichexhibitsthefullCoulombinteraction. electron interaction. We recall that the main spirit of the
X
Inourmodel,themetalhastwonon-degeneratebands,a method consists in replacing a tranlationally invariant
arandb.ThefirstisaHubbard-likenarrowbandwithin-site problem, as defined by (1), by the alloy problem, defined
interactionU,hybridizedwiththesecondone,b,abroader by (2), where only the origin incorporates the Coulomb
band. The Hamiltonian we adopt to describe pure metals interaction.TheeffectiveHamiltonian(2)stillincludesthe
is then difficulty of dealing with the Coulomb intra-atomic term
attheoriginandwehavetoresorttosomeapproximation.
H=Xǫaa+iσaiσ +Xtaija+iσajσ +Xtbijb+iσbjσ (1) IntheGreen’sfunctionmethod[?,?],theequationofmo-
tionofagivenGreen’sfunctiongenerateshigherorderfunc-
i,σ i,j,σ i,j,σ
tionsandwethusneedadecouplingprocedure.Aftersome
+XUni(↑a)ni(↓a)+X(Vabb+iσajσ +Vb+aa+iσbjσ), algebra[4]we endupwiththe followingGreen’s functions:
i i,j,σ
where naiσ = a+iσaiσ ; σ denotes spin. tij denotes the tun- Gakk′,σ(w)= w−ǫ˜akδk−k′Σσ(w), (3)
neling amplitudes between neighboring sites i and j , in
eachband;ǫa isthecenterofthea-band.WefollowRoth’s the Green’sfunction ofthe barebband,
1
∗ Corresponding author. Tel: [phone] fax: [phone]- Gb0,kσ(w)= w−ǫb, (4)
k
7400
Email address: [email protected] (C.M.Chaves). andthe Green’sfunctionofthe renormalizedbband
Preprintsubmitted toElsevier 22 January 2009
Gb (w)=Gb (w)+Gb (w)VGa (w)VGb (w).(5)
kkσ 0,kσ 0,kσ kk,σ 0,kσ
Inthese equations U=4
2,0 V=0.3
V2(k)
ǫ˜a =ǫa+ ab , (6)
k k w−ǫbk 1,6
istherecursionrelationoftherenormalizedaband;ǫa and
k down
ǫb denote the bare bands. The vanishing of the T-matrix 1,2 up
k
givesaself-consistentequationfor the self-energy:
e
R
Σσ =U <na >+(U −Σσ)Fσ(w,Σσ)Σσ, (7) 0,8
0−σ
with
0,4
Fσ =XGakk,σ (8)
k 0,0
-4 -2 0 2 4
Thisisequivalenttoanalloyanalogyapproximation,where
Energy
the broadeningcorrection[3]hasbeenneglected.
Ourself-consistentnumericalresultscandescribeatran-
sition metal of the iron group (then a ≡ d and b ≡ s) or Fig.2.Realpartoftheself-energyforupanddownspinsforU =4
andV =0.3.
somef electroncompounds(a≡f andb≡s,p,d).
In fig(1) we present the density of states (DOS) of the corrections tend to reduce the range of validity of that
a-band for U = 4 and V = 0.3. In same units, the bare groundstate[6].Inthispaper,usingthesinglesiteapprox-
a band width is W = 2. The Fermi level is at ǫ = 0.45. imation, we obtain ferromagnetic solution for a set of pa-
F
This gives na = 0.475 and na = 0.390 and thus a ferro- rameters (e.g. U/W = 2 and V/W = 0.15). A more com-
↑ ↓
magnetic moment of m = 0.085 is developed. The DOS pletestudybringinguptheinterplaybetweenU andV will
a
here obtainedexhibits abimodalstructure caracterizinga be publishedelsewhere.
Hubbardstronglycorrelatedregime.
In fig (2) the realpart of Σσ is displayed for U = 4 and 3. Acknowledgement
V = 0.3. Σσ is a smooth funcion of energy and for this U
differsconsiderablyfromtheconstantHartree-Fockvalues
CMCandATaknowledgethesupportfromthebrazilian
(Σ↑ =1.56,Σ↓ =1.90).
agenciesCNP andPCI/MCT.
q
0.40
0.35 a-band σ=down
U=4 a-band σ=up
V=0.3 References
0.30
[1] H. Shiba, ProgressTheo. Phys.46 (1971)77 .
0.25
[2] L.M.Roth, in AIP Conference Proceedings 18 Magnetism and
S a 0.20 Magnetic Materials, (1973) 668.
D0 [3] J. Hubbard, Proc. Royal Soc. A281 (1964) 401 .
0.15
[4] A. N. Magalha˜es, M. A. Continentino, A Troper and A. A.
0.10 Gomes, C.B.P.F.,Notas Cient´ıficas XXIII 8(1974) 127.
[5] C. D.Batista etal, Phys.Rev. Letters 88 (2002) 187203 .
0.05 [6] D. Vollhardtetal, Adv. SolidState Phys.38 (1999) 383 .
0.00
-6 -4 -2 0 2 4 6
Energy
Fig.1.Densityofstates(DOS)ofthea-band,upanddown,forU =4
andV =0.3.TheFermilevelisatǫF =0.45.Asaresultna↑ =0.475
and na = 0.390, giving a ferromagnetic moment m = 0.085. The
↓
separation of the centers ofthe two subbands isU, as expected.
2. Summary
The traditional view of the origin of ferromagnetism in
metalshasbeenunderintensescrutinyrecently[5,6]).Con-
ventionalmean-fieldcalculationsfavorferromagnetismbut
2