Magnetic ordering versus lattice distortion in very narrow bands

Solid State Communications,

Vol. 13, pp. 1767—1770, 1973.

Pergamon Press.

Printed in Great Britain

MAGNETIC ORDERING VERSUS LATTICE DISTORTION IN VERY NARROW BANDS Anupain Madhukar IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, U.S.A. (Received 5 July 1973; in revisedform 17 August 1973 by R.H. Slisbee)

A simple criterion for antiferromagnetic ordering versus lattice distortion in very narrow band materials is obtained. The model describing these narrow band materials contains the Hubbard Hamiltonian, short ranged interatomic Coulomb and exchange interactions, and the electron—phonon interaction.

THE PHENOMENON of metal-non-metal transition

1’2 accompanied by lattice distortion (Peierls’ instability) or magnetic ordering (Mott—Hubbard transition)3 has been theoretically and experimentally known for quite some time. The occurence of such phenomenon in transition metal oxides is well documented.4 In the recent past some quasi-one dimensional organic charge transfer salts belonging to the TCNQ family have been added to this class of materials.5 For instance, NMP: TCNQ is believed to undergo a transition from a metal to anitferromagnetically ordered insulator.5

However, finite t1/s are essential to provide magnetic ordering as well as the correct charge ordered state and thus cannot be neglected.9 To the Hubbard model we add the electron phonon interaction g(q), and the short ranged (confined t~nearest neighbors only) interatomic Coulomb (ku) and exchange (Jj~)interactions. Thus our Hamiltoman is: 1C

6

— —

The recent discovery of TTF: TCNQ (Coleman et aL) showing a very high conductivity at 60°K and then

[

t,, c7

0 C10 +1 ~ fl~0fl~ ~y+ ~ ~ -

~‘a

+~~

~ c~

,

J

0+ ~

suddenly going into an insulating phase, has been suggested byphase the investigators as a transition the metallic to Peierls’ insulator. While from the situation for 11’F: TCNQ is far from clear experimentally, as well as theoretically,7’8 it does seem very appropriate to work out a simple and relatively reliable criterion determining the competition between

k~fl10 flj~,

1xo,~b~b~+

Q e~~’~i) 1”~~~ — b~t

~ g(q)n, (bq e

(1)

where c’s and b’s refer to electron and phonon operators respectively, the rest of the notation being self evident. Also t~ 1= = J~ = 0. It has been shown that in the limit ti,, -+ 0, the electron phonon interaction can be 9 written as an effective electron—electron Interaction. Though t is of order (1/10) here and not quite zero, we believe the neglect of fmite recoil effects in writing

lattice insulating grounddistorted states in and this magnetically growing class ordered of materials. In obtaining our criterion it is important to note that these materials are generally characterized by very narrow bands, of the order of 102 eV. The intramolecular Coulomb repulsion, I, is generallyof order 10’ eV, thus putting us in the regime t~~ wheret~are the single particle hopping terms. As

the e—phonon interaction as an effective e—e interaction is not quite so serious for our purposes. Thus we may write for the electron part of the Haniiltonian,

IE

~‘e

such, a Hubbard model descriptionis appropriate.

& 1767

~

cj,

+~(I—2V

]

0)~ ~ n~+

I “a

E(kjj—2V~)ninj+&E Juc70c~t’c,0’c,0 U

lice’

(2)

1768

MAGNETIC ORDERING VERSUS LATTICE DISTORTiON

where

=

V11

=~

~eiQ~rhuJ);

Vii

=

V0

(3)

is the effective e—e interaction. We use (2) to work out our required criterion employing the standard technique of Zubarev’s double time temperature dependent Green’s function’°and decoupling the two particle propagators in a modified H—F scheme. The validity of the decoupling scheme has been well established for the Hubbard model” and the solutions for one dimension compare exceedingly well with 2 The same dethe exact results of Lieb and Wu.’ coupling scheme has also been used in studying Hubbard model with interatomic interactions ~ and .J~,giving rise to the possibility of ferromagnetic insulating phase.13 Because of the well documented nature of this decoupling scheme, we omit its derivation and merely point out the salient features. (see reference 13 for details). In H—F decouplingof two particle propagators of the type ‘~c~ c0~..0c10 c~’~> we retain terms like
essential for antiferromagnetic ordering. Below we write down the equation of motion transformed into Bloch representation. In transforming from Wannier to Bloch representation we encounter correlation functions of the form,
Pg

~

Vol. 13, No.11

~I—2V0 +J~q))n~(a,a)

Pq(a,0);pq(a,a)

n~(a, — a)

=

~ (a~+~ ak_o)

(6)

and V(q), K(q) and J(q) are the Fourier transforms of the corresponding interactions. Also note that = f’OO ±

2Q.k’

We have chosen x-axis, rather than the conventionalz-axis as thetheaxis of spin quantizatlon leadthg to the following additional relations, p~(a, a) = Pg a, a) (—

~ (a, — a)

=

—

n~(— a, a)

which have also been used in obtaining (5). Finally, we also mention the obvious relation, ak0)

~

N

<4 ct’> ~iq.RI

=

(7)

k

Equation (5) contains the possibility of magnetic ordering (ferro and antiferromagnetic) as well as charge ordering. The magnetic case in the absence of electron—phonon interaction is considered in detail in reference 13. The charge ordered and antiferromagnetically ordered states are characterized by non-vanishing values of PQ and n,~respectively. For the charge ordered ground state, only Po and p~are non-vanishing in equation (5), which can then be trivially solved for Gk°k°and Gk°+°Q,k;

(akOak’O’)—(a~Oak’Q’)&kk’+(a~Oak’0)~k+Q,k’(4)

where Q is half the reciprocal lattice vector. For a half filled band, Q plays the important role that Q = 2 kF, and splits the Brilloun zone into half. It wave) wave that vector which leads (spin to Peierls’ is well of known it isQthe phonon density instability (antiferromagnetic ordering). The equation of motion thus obtained is, (~— ~

—

ao) G,~il~’ (~)+ ~3oGir (~) a~G1~Q,k’(4~,)

+ ~JQ~

—

(~) =

5kk’

b~

(5)

GJ~° (c~~) =

~1

(w+e,~) (~2

4—

1 ____________ 2

~

~

=

2ir (w

4

—

aQ) 2

(8)

Thus using the standard expression for correlation functions in terms of the spectral function of the corresponding Green’s function,10 we obtain the following determining equation for a~,the charge ordering gap parameter; ~ 2Ke~ 1e11 J 2 + a~)1t2tanh 2N

where,

0) ~ (e

+ a~)h12 /2k T } =

~ (1—2 V 0 + 2K(q) —4 V(q) — J(q)) ‘~

=

—

{K(Q) —2 V(Q) };

1~= (I — 2 V0) (9)

Vol. 13, No. 11

MAGNETIC ORDERING VERSUS LATTICE DISTORTION

For antiferromagneticstate, only ~ and nQ are nonvanishing. Equation (5) can again be solved trivially giving the following, =

,-,-Co

,

\_

—

1 ~

(c., + e )

2

—

—

=~

From (9) and (II) we obtain the following for

2

The determining equation for antiferromagnetic state gap parameter ~ is, 1

(14)

R... 2

~ ~ In the above all energies are being measured from the chemical potential ,.i, which for the case of ~f filled band is ~

1e1o) ~ (~ +

p-~> 4~

the corresponding transition temperature; 057D T,, = exp{—1IN(0)(4~—4)} (15)

(~2_4~~)

1 2

1769

~

~

+~)“1j’2kT}

~2

=

exp{~lIN(O)(21C.ff4,jJo)}

where T~and T~refer to Peierls and antiferromagnatic ordering. N(O) is the electron density of states per spin at the Fermi-energy. From (16) it is clearly seen that in the absence of electron—electron Interactions (K11 = =1 = 0), we obtain the standard expression for theelectron Peierls transition temperature of a non-interacting gas, 2

(

Thus from (9) we see that a charge ordered msulatmg state results if (2K.~— I,~— .10) > 0 and from (11) we find that an antiferromagnetically ordered insulating state results if (1~—4)> 0. To find a criterion for determining whether, for a given set of values of Kff, I,~and J~,the ground state is charge ordered or antiferromagnetically ordered we calculate the ground state energies of the two states. The ground state energy can be calculated using the standard expression in terms of the single particle Green’s function.’°For the antiferromagnetically

k

T~— 0.57 — exp {— f~Q/2N(0)Ig(Q) I 1} k~ The neglect of finite overlap effects in writing the electron—phonon interaction as an effective electron— electron interaction is not expected to be a serious short coming of the criterion (14). Inclusion of these finite overlap effects we find merely complicates the analysis without any significant change in the criterion obtainedhere. From equation (5) we could have easily obtaineda ferromagnetic insulating state characterized by finite n 0, but for the region t I the antiferromagnetic state has lower energy. As such, we considered competition between antiferromagnetic and lattice distorted states only. Comparison of (14) with experiments must unfortunately await the avallibihty of I, K and g for a given system. In the meantime however, (14) can be used to put bounds on certain parameters, knowing ‘~

ordered state, the result is, — E~ = 4ci~., —*D Coth(4ff D

and for the charge ordered state,

I

=

‘O

/~

(12)

\

a~— * D Coth ~2Keff

D ~

,

J

where D is the band width. Thus, the charge ordered state stall be the ground

others. it is believedinthat would helpful to the experimentalist the this analysis ofbedata and planning of experiments. Acknowledgements — The author wishes to acknowledge

state only if,

helpful discussions with Dr. T.D. Schultz.

‘

3\‘

REFERENCES 1. 2.

MATTISD.C. and LANGER W.D.,Phys. Rev. Lett. 25, 376 (1970). PEIERLS R.E., in Quantum Theory of Solids, p. 108, Oxford University Press, London (1955).

1770

MAGNETIC ORDERING VERSUS LATTICE DISTORTION

Vol. 13, No. 11

3. HUBBARD L,Proc. R. Soc. London, Ser. A 276, 238 (1963), and 285, 542 (1965) and references therein. 4. ADLER D. in Solid State Physics, Vol.21, (Edited by EHRENREICH H., SEITZ F. and TURNBULL D.) Academic Press, New York (1968). 5. HEEGER A.J., and GARITO A.F., in AlP Conference Proceedings-Magnetismand Magnetic Materials p. 1476 1972, (EdIted GRAHAM C.D. Jr. and RHYNE JJ.). 6.

COLEMAN L.B., et aL, Solid State Commun. (in press).

7. RICE M.J. and STRASSLER S. (to be published). 8. PATTON B. and SHAM Li. (to be published). 9. BARI R.,Fhys. Rev. B3, 2662 (1971). 10. ZUBAREV D.N.,Soy. Phys.-Usp. 3,320(1960). 11. LANGERW., PLISCHKE M. and MATHS D.,Thys. Rev. Lett. 23,1448(1969). 12. LIEB E. and WU F.,Phys. Rev. Lett. 20, 1145 (1968). 13. KISHORE R. and JOSh S.K., J. Phys. C, 4,2475 (1971). Un critére simple permettant de distinguer, dana les matériaux a bandes étroites, l’apparition d’un ordre antiferromagnétique de celle d’une distorsion du rëseau est établi. Le modéle décrivant ces matériaux a bandes étroites contient le Hamiltonien de Hubbard, les interactions interatomiques coulombiennes a courte distance ainsi que les interactions d’échange, et l’interaction électronphonon.