2-а Всеукраїнська наукова конференція молодих вчених ФІЗИКА НИЗЬКИХ ТЕМПЕРАТУР (КМВ–ФНТ–2009)
ELECTRON CORRELATIONS IN DOPED FULLERIDES:
EFFECTIVE HAMILTONIAN APPROACH
Yu. Skorenkyy1 , O.Kramar1, Yu.Drohobitskyy2
1Ternopil State Technical Unіversity, 46001 Ternopil, 56, Rus’ka St.
2Ternopil V. Hnatiuk National Pedagogical Unіversity 46027, 2, M.Kryvonis St
Introduction Despite the intensive experimental and theoretical studies, the diversity of physical properties of doped fullerides remains unexplained at a microscopic level. Doped systems A3С60 (where А=К, Rb, Cs) turn out to be metallic at low temperatures.
According to the theoretical band structure calculations, fullerides with integer band-filling parameter n should be Mott-Hubbard insulators by electric nature, because all of them possess large enough values of intra-atomic Coulomb correlation parameter U.
Doping of fullerene C60 with alkali metal leads to occurrence
of metallic
conductivity in fullerides, at low temperature a superconducting transition has been observed (Tc varying from 2.5 К for Na2KC60 to 33 К for RbCs2C60)[1].
Metallic, insulating and superconducting phases have been obtained at different band fillings n (0<
n< 6) of the lowest unoccupied molecular orbital (LUMO) in С60.
Electronic states in С60. In single- particle approximation, neglecting electron correlations, the following spectrum has been calculated [2]: 50 of 60 pz
electrons of a neutral molecule fill all orbitals up to L=4.
The lowest L=0,1,2 orbitals correspond to icosehedral states ag, t1u, hg. All states with greater L values
undergo the
icosahedral-field splitting. There are 10 electrons in partially filled L=5 state. Icosahedral splitting (L=5 → hu+t1u+ t2u) of these 11-fold degenerate orbital lead to the electronic configuration shown below.
Microscopic calculations and experimental data shows that the completely filled highest occupied molecular orbital is of hu symmetry, and
LUМО (3-fold
degenerate) has t1u
symmetry.
The Hamiltonian of doped fulleride electronic subsystem
Within the second quantisation formalism, the Hamiltonian of interacting electrons (with spin-independent interaction
) (r r Vee
) in crystal field
r Vion ,
may be written as
tn
Hi
H H 0
,
V r a r
r m ra d
H ion
2
3
0
,
r r n r n r
V r d r d
Hi tn ee
ˆ ˆ
,
3 3
. Here r
a
r ,
a
are field operators of electron creation and annihilation, respectively,
r a r a r
nˆ
. Interaction integral of zeroth- order magnitude
is on-site
Coulomb correlation (characterized by Hubbard parameter U):
r R drdr
r r R e r
U i i
2 2 2
In orbitally
degenerate system, the on-site (Hund's rule) exchange integral
r R r Rdrdr
r r R e r R r
JH i i i i
* 2 *
is of principal importance, too.
The estimations for on-site energy parameters are given in table below.
U 0.8 – 1.3 eV 1.4 – 1.6 eV 1.3 eV 2.7 eV JH 0.1-0.3 eV
0.1 eV The relevant inter- site parameters are electron hopping integral
i ion i i
i j
i V r r R
R m r r d
t
2
*
3
and inter-site exchange coupling
r R r R drdr
r r R e r R r i
j j i
J i j j i
* 2 *
Estimations of
bare half
bandwidth (w=2z|
t|, z being the number of nearest neighbors to a site, t is the
nearest neighbor hopping integral) and inter-site exchange parameters are given below.
w 0.5-0.6 eV
0.6 eV
J 0.05 eV
The resulting
Hamiltonian of doped fulleride electronic subsystem reads as
, )
2 ( 1
. . '
. . '
'
2 2
ij
j i j i
ij
i j i j i
ij ij i j i
ij
j i ij
i
i i H i
i i
i i i
i
i i
a a a a ij J
c h n a a t c
h n a a t a
a n t
n J n
n U U n
n n U a a H
w h e r e
JH
U U 2
a n d
h o p p i n g
i n t e g r a l s
tij ,
t ij t a k i n g
i n t o
a c c o u n t
t h e
c o r r e l a t e d
h o p p i n g
o f
e l e c t r o n s
a r e
i n t r o d u c e d .
Configurational representation of the Hamiltonian In a model of triply degenerate band, every site can be in one of 64 configurations depicted below
To pass from electron operator to Hubbard operators Xpl of site transition from state |l to state |pwe use relations of type
, ˆ
22 , 222 2
, 2 2
2 , 22 2
, 2 2 2
, 22 022
, 22 2
0 , 2
2 0 , 2 02
, 202 02
, 2 02
, 2 20
, 220
, 2 ,
2 ,
2 ,
2 0
,
0 , 0
, 20 0
, 0 2 0
, 0
,
0 , 20 0
, 0 2 002 , 02 020
, 20 00
, 0
00 , 0 0
0 , 0 0
0 , 0 00
, 200 000
, 00
X X
X X
X X
X
X X
X X
X
X X
X X
X
X X
X X
X
X X
X X
X
X X
X X
X a
which ensure the fulfilment of anticommutation relations
pt ikl
pt i k l j i kt j pl
i X X X
X ;
,
a nd normalizing condition,
for number
operators
lp i pl i p
i X X
X
of p
- state on site i.
In the configurational representation the model Hamiltonial takes the form Here H0 sums the
“atomic limit”
terms and the translation part may be decomposed as where n,m serve for numbering states with energies shown in figure.
Terms of Tnn
Hamiltonians form the energy subbands and terms of Tnn
describe the hybridization of these subbands.
Different hopping integrals correspond to transitions in (or between) the different subbands. The subbands of higher-energy processes appear to be narrower due to the correlated
hopping of
electrons. The relative positions and overlapping of the subbands depends on the relations between
the energy
parameters. At integer values of electron concentration (n=1, 2, 3, 4, 5) in the system the metal-insulator transition is possible.
Partial case of band integer filling n=1
Hb Hh Hex
H
H
, ,
) ( ) ( 0
i i i i
i i
i i
i i i
X X
X J U
X X
X X
X X H
0 0 0
0 0 0 00
0 0 0 00
3
2
0 , 0 0 00 , 0
0 , 00 0 0 , 0 0
, 00 00 , 0
0 , 0 0 0 0 , 0 00
, 000 000 , 00
~
~
~
~
j j i
i
j j i
j i j i
i
j j i
j i
i ij i j
b
X X
t
X X
t X
X t
X X
t X
X t H
00 , 000 00 , 0 0
, 00 000 , 00
00 , 000 0 0 , 0 0
, 0 0 000 , 00
j i
j i
j i
j j
i ij i
h
X X
X X
X X
X X
t H
ij ij
i i j
j i i
ex ijJ X X X X J Xij X
H ( 5.0) 5.0 )( , ,
Canonical transformation.
We use a variant of the method proposed in paper [10] and apply a canonical transformation
S
SHe
e
H~
To obtain
effective form of the Hamiltonian
0 0 0
2
~ 1
SH S SH
SH H
SH H
H H
H b ex h b h
Where the form of S is chosen to exclude hybridization processes
) 3 (
00 , 000 0 0 , 0 00 , 000 000 , 0
0 , 0 0 000 , 00 0
, 00 000 , 00 0 0 , 000 00 , 0
0 0 , 000 00 , 0 0 , 00 000 , 0 0 0
, 00 000 , 0 0
00 , 000 00 , 0 00 , 000 0 0 , 0 0
, 00 000 , 00
0 , 0 0 000 , 00 2
j i
j i
j i
j i
j i
j i
j i
j i
j i
j i
j i
j i
ij H
ij
X X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X J X
U S t
After the canonical transformation, effective Hamiltonian reads
as
, ,
) ( 0
~
~
ex ex
b H H
H H
H
where the last term represents “indirect exchange”
interaction
ij
i i i i j i
ex J ij X X X X X X
H~ ~( ) , , 000
with indirect exchange integral
)
~(ij
J is of
the same order of magnitude as direct exchange integral
) (ij
J and
can enhance considerably ferromagnetic tendencies.
Strong correlation limit In the limit of strong on-site correlation limit
w J
U 3 H
at partial band filling 0<n<1 one
has
ij
i i j i
ex J ij X X X X
H ( ) , ,
ij
i i j i
ex J ij X X X X
H~ ~( ) , ,
One can see that in triply degenerate model indirect exchange is of
ferromagnetic nature, in contrast to standard “non- degenerate” t-J model
Possible metal-insulator transition in the
model
T
he obtained effective Hamiltonian s allow to classify processes of subbands formation and hybridizatio
n, to
discover mechanisms of electron localization and possible magnetic ordering stabilization .
F
or an
explanation of a metallic behaviour of Mott- Hubbard system A3C60 (x=3 corresponds to the half- filled conduction band) three- fold degeneracy of energy levels and Hund’s rule coupling has
to be
properly taken into account.
T
he metal- insulation transition in the model is possible also under external pressure (which case is realized, e.g., in Rb4C60).
Peculiar concentrational
behaviour of
electrical conductivity in fullerides AnC60 may be an analog to
those of
dihalcohenides series. For example,
within DMFT
approach for doubly degenerate band the critical values of correlation parameter depend on band filling [11].
Critical values, obtained [12] within Gutzwiller approximation are Uc/2w~2,8 for twofold degenerate half-filled band, Uc/2w ~3,9 for threefold degenerate half-filled band.
Correlated hopping also modifies the metal-insulator transition conditions.
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2. N. Manini, E.
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Phys. Rev. Lett.
68, 3924, 1992.
6. P.A. Brühwiler et al. Phys. Rev.
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Rozenberg. Phys.
Rev. B. 55, R4855, 1997.
12. Jian Ping Lu.
Phys. Rev. B. 49, 5687, 1994.
t~ij tij tij
ij
J
U
0 T.
H
H
p p
Xi 1
m n
Tnm
T
,
“atomic limit”
“subbands”
H ij
J U
n ij t
J 3
~ 2 2
“direct exchange”
“hybridization”
