EFFECTIVE HAMILTONIAN APPROACH

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 (T_c 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 a_g, t_1u, h_g. 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 h_u 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 V^ee(r−r′)⁾

in crystal field ^Vion( )^r, may be written as

t n

Hi

H H= 0+ ,

( ) ( ) ( )

∑∫ _^{− ∆+} _^

= ⁺

σ σ V r aσ r

r m ra d

H ^ion

2

3 0

h ,

(^{r r}) ( ) ( )^{n rn r}

V r d r d

H_int= ′ ^ee − ′ ′ ′

∑∫ ∫_σ′ ^{σ σ}

σ

ˆ ˆ

, 3 3

. Here ^aσ⁺( )^r, ^aσ( )^r are field operators of electron creation and annihilation, respectively, nˆ_σ( )r=a_σ⁺( ) ( )ra_σr

. 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=∫∫^ϕλ − _i² −²′^ϕλ ′− _i² ′

In orbitally degenerate system, the on-site (Hund's rule) exchange integral (^{r R}) (^{r R})_r^e_r (^{r R}) (^{r R})^drdr

J_{H i i} ′− _i ′− _i ′

−′

−

−

=∫ ∫^{ϕλ* ϕλ′ 2 ϕλ*′ ϕλ}

is of principal importance, too. The estimations for on-site energy parameters are given in table below.

U 0.8 – 1.3 eV [3,4]

1.4 – 1.6 eV [5,6]

1.3 eV [7]

2.7 eV [8]

JH 0.1-0.3 eV [9]

0.1 eV [7]

The relevant inter-site parameters are electron hopping integral

( ) ( ) (i i)

ion i i j

i V r r R

m R r r d

t  −

− ∆+

−

=∫ ^{λ λ}

λ ϕ ϕ

2

*

3 h

and inter-site exchange coupling

(^{ij ji} ) (^{r R}) (^{r R})_r^e_r (^{r R}) (^{r R})^drdr

J i j − j − i ′

′

− −

−

′=

′^λλ ∫∫^{ϕλ ϕλ ϕλ ϕλ}

λ

λ ^{* 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 [2]

0.6 eV [7]

J 0.05 eV [7]

The resulting Hamiltonian of doped fulleride electronic subsystem reads as

( )

( ) ( ) ( )

, )

2 ( 1

. . '

. . '

'

2 2

∑

∑

∑

∑

∑

∑

∑

∑

′ + ′

+ ′

+ +

+

′ ′

↓ ′

↑ +

+

+

′′

+ +

′ + +

′−

′ + + +

−

=

σ

λσ λσ λσ λσ λσ

λσ λσ λσ λσ

σ λ

λ λσ λσ λ

λσ λσ λσ

σ λ

λ λσ λσ

λσ λσ λσ

λ λ λ

λσ λσ λσ

µ

ij

j i j i

ij

i j i j i ij

j i 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

where^U′=U−2JH and hopping integrals

t′ij^, t′′_ij taking into account the correlated hopping of electrons are introduced.

Configurational representation of the Hamiltonian In a model of triply degenerate band, every site can be in one of 64 configurations depicted below

000 ↑00 ↓00 0↑0 0↓0 00↑ 00↓

0 0

↑↓0 ↓↑0 ↑0↓ ↓0↑ 0↑↓ 0↓↑ ↑↑↑ ↓↓↓

JH

U−2 3U−9JH

200 020 002

U U−3JH

↑↑0 ↓↓0 ↑0↑ ↓0↓ 0↑↑ 0↓↓

↑↑↓ ↓↓↑ ↑↓↑ ↓↑↓ ↑↓↓ ↓↑↑ 3U−7J^H

2↑0 2↓0 20↑ 20↓ ↑20 ↓20 02↑ 02↓ ↑02 ↓02 0↑2 0↓2

220 022 202 2↑↑ 2↓↓ ↑2↑ ↓2↓ ↑↑2 ↓↓2

JH

U 10

6 −

JH

U 5

3 −

↑↓

2 2↓↑ ↑2↓ ↓2↑ ↑↓2 ↓↑2

JH U 12

6 −

↑

22 22↓ 2↑2 2↓2 ↑22 ↓22 222

JH

U 17

9 − 13U−24JH

JH

U 13

6 −

To pass from electron operator to Hubbard operators X^pl of site transition from state |l〉 to state |p〉we 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_α

which ensure the fulfilment of anticommutation relations

{^Xipl;Xktj}^=δij(^{δlkXipt+δptXikl})^,

and 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.

λσ 000 0=

222 6=

λσ 4

λ λ′ 4

σ σ λ2 ′

σσ λ 2

σ λ λ 2

σ σσ σσσ

λ2 σ λ λσ

σ λ λσ

JH U 17

9 −

JH U 10

6 −

JH U 12

6 −

JH U 13

6 −

JH

U 5

3 −

JH

U 7

3 −

(U 3JH)

3 −

U JH

U−2 JH U 24 13 −

JH

U−3

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= + ∑ + +

=αβγ λ

λ λ , ,

) ( ) ( 0

( ( ))

( ⁻ )^∑( ^{+ +} )

+

+ + + +

∑ + +

−

= σ

σσ σ σ σσ

σσ σ σ σσ σ σ σ µ σ

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 00 0

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

~

~

~

~

σσ σ σ σ σ

σ σ σ σ σσ σ σ σ σ σ σ

σσ σ σ σσ σ σ α

j i j i

j i j i j i j i

j i j 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

σ σ σ σ σ σ σ σ

σ σ σσ σσ σ σ α

j i j i

j i j j

i ij i h

X X X X

X X X X t H

+ +

+

∑ ′ +

=

( )( )

∑ ^{+ + −}∑

−

=

σ λ

λ λσ

λσ σ λ σ λ λσ σ λ λσ λσ σ λ λσ λσ

ij ij

i i j

j i i

ex Jij X X X X JijX X

H ()0.5 0.5 () ^{, ,}

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

[ ]

( + )+[ ] [ ]+ + [[ ]]+^K +

+ +

= 0 0 0

2

~ H H H SH H SH SH 1SSH

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

σ σ σσ σ σ σ

σσ σ σ σ σ σ σ σ σ σσ

σ σ σσ σσ σ σ σσ σ σ

σ σ σ σ σ σ σσ σ σ σ σ

σσ σ σ σ

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

=∑

+ + +

=

γ β α λ

λ , ,

) ( 0

~

~

ex ex

b H H

H H H

where the last term represents “indirect exchange” interaction

( )

∑ ^{+ +}

−

=

σ λ λ

σ λ λσ λσ σ λ σ λ λσ σ λ λσ ij

i i i i j i

ex Jij 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 U−3JH>>w at partial band filling 0<n<1 one has

( )

∑ ⁺

−

=

σ λ λ

λσ σ λ σ λ λσ λσ λσ ij

i i j i

ex Jij X X X X

H () ^{, ,}

( )

∑ ⁺

−

=

σ λ λ

λσ σ λ σ λ λσ σ λ λσ ij

i i j i

ex Jij 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

• The obtained effective Hamiltonians allow to classify processes of subbands formation and hybridization, to discover mechanisms of electron localization and possible magnetic ordering stabilization.

• For 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.

• The 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 U_c/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.

References

1. А.В. Елецкий, Б.М. Смирнов УФН, 165, 977, 1995.

2. N. Manini, E. Tosatti. E-print cond-mat/0602134.

3. M.R. Pederson and A.A. Quong. Phys. Rev. B. 46, 13584, 1992.

4. V.P. Antropov et al. Phys. Rev. B. 46, 13647, 1992.

5. R.W. Lof et al. Phys. Rev. Lett. 68, 3924, 1992.

6. P.A. Brühwiler et al. Phys. Rev. B. 48, 18296, 1993.

7. R.L. Martin and J.P. Ritchie // Phys. Rev. B. 48, 4845, 1993.

8. R.L. Hettich et al. Phys. Rev. Lett. 67, 1242, 1991.

9. R.W.Lof et al. Phys. Rev. Lett. 68, 3924, 1992.

10. L.Didukh and O.Kramar. Condens. Matter Phys., 8, 547, 2005.

11. M.J. Rozenberg. Phys. Rev. B. 55, R4855, 1997.

12. Jian Ping Lu. Phys. Rev. B. 49, 5687, 1994.

tij

~

t′ij t_ij

( )^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”