Density of States in the t′-Hubbard Model: Exact Diagonalisation Results

HAL Id: jpa-00247071

https://hal.archives-ouvertes.fr/jpa-00247071

Submitted on 1 Jan 1995

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Density of States in the t’-Hubbard Model: Exact Diagonalisation Results

W. Fettes, I. Morgenstern, T. Hußlein, H.-G. Matuttis, J. Singer, C. Baur

To cite this version:

W. Fettes, I. Morgenstern, T. Hußlein, H.-G. Matuttis, J. Singer, et al.. Density of States in the

t’-Hubbard Model: Exact Diagonalisation Results. Journal de Physique I, EDP Sciences, 1995, 5 (4),

pp.455-464. �10.1051/jp1:1995139�. �jpa-00247071�

J. Phys. ^I Hauce 5

(1995)

455-464 _APRIL 1995, PAGE 455

Classification Physics Abstracts 74.20

Density of States in trie t'-Hubbard Model: Exact Diagonalisation

Results

W.

Fettes,

I.

Morgenstern,

T.

Hufllein,

H.-G.

Matuttis,

J. M.

Singer

and C. Baur

Universitàt Regensburg, Fakultàt Physik, 93040 Regensburg, Germany

(Received

18 November 1994, received in final form 5

làecember

_{1994, accepted 14 December}

1994)

Abstract. We calculated the density of states _{in a} Hubbard mortel with nearest and next

nearest neighbour hopping. Using _an exact diagonalisation technique, _we examined this mortel

in a wide _area of parameters. The density of _{states was} calculated for dilferent fillings and for

positive and negative interaction U. The additional hopping to the next nearest neighbour sites has dilferent elfects _on the gap between the particle and hole spectrum compared to the pure

Hubbard mortel.

1. Introduction

There _are several

theories,

which _are

actually favoured,

to

explain

^the

high-temperature

_su-

perconductors [1-4].

One of them is the _van Hove scenario [4]. ^{It states that the maximum} of the critical temperature Tc _is, where the Fermi _energy and _{a van} Hove

singularity

in the

density

of states _are at the _sonne

place.

From the

experiments

_we know [Si ^{that the}

doping

with maximum Tc is trot at

half-filling

for the

high

T~

superconductors.

On the other hard, the

single-baud

Hubbard model has

only

a

singularity

in the

density

of states at

half-filling

_[6].

Because of this _reasons the

single-baud

^{Hubbard model should} be extended. One

possibility

is to allow, ^{in addition} to the

hopping

to the _nearest

neighbours,

_a

hopping

to the next nearest

neighbours t').

This t'-model is described

by

the Harniltoman:

c~~cj~ ^'

~

<l,j> _~

~ ~

~ _C$C

~

+ u

£

~~

«~'J»1«

~

~

'~~i'~~" ~i~

where t is the

hopping

parameter to the nearest

neighbours (in

the

following

_we

always

set t ₌

1),

^{t' is the}

hopping

parameter to_the next nearest

neighbours,

<1,

j

> denotes the _sum

over ail nearest

neighbours

^and < 1,

j

» is the _{sum over} the next nearest

neighbours.

Il (s the interaction between the

partiales

and

c$

_{creates a}

partiale

at the lattice site with spin a

and _nw

=

c$cw

_{is the}

_partide

number operator. ^{In this} model _{a van} Hove

singularity

exists at t'

= 1

(ni,

if the interaction is _zero, U

= 0 [7].

© Les Editions de Physique 1995

We _{use an} exact

diagonalisation technique

to salve the model

(1)

_see Section 2

).

^Because of the

exponential growth

of basis states with trie system size, it is trot

possible

to deal with

large

system sizes. This _means that the

density

of _states has individual

peaks

and _{is trot a} continuous

spectrum. ^{SO the}

density

of states cannot show

singularities,

which _we would

perhaps

obtain

in infinite system sizes. Nevertheless _{we cari}

study

the eoEects of the

additionally

t'

hopping

to the

density

of states

compared

to the pure Hubbard model.

2. Calculation of trie

Spectral Density

and trie

Density

of States

As numerical

technique

_{we use} the _exact

diagonalisation

with the Lanczos method [8, 9]. One of the

advantages

of the Lanczos method is that _{one cari}

easily

calculate the Green functions

with this method

[loi.

Within the Lanczos scheme it is trot necessary to calculate ail excited states of the model. Tue Green function to tue state (flft) cari be evaluated

by

tue continued fraction

expansion [loi

(ÎPii(i H)~~

ilYi) ₌

~~ ,

(2)

~ °~

~-a2-fi

wuere _o~ and _{fl~ are} tue

diagonal

and

oOE-diagonal

elements of tue

tridiagonal

matrix Tm from tue Lanczos

algoritum.

Tue initial vector of tue Lanczos

algoritum

is tue _state

(flft).

Tue Green function

G(+)(k,

_{uJ) of tue} state

c[~J(flfo)

^{witu tue} norm 1 is defined _as:

G(+) (k,

_uJ)

= lim (flfo(ck,a(uJ + iô

H)~~c(~

_flfo)

(3)

à-o ,

G(~)(k,

_uJ)

= lim

(flfolcla

(L~ + iô

Hi ~~ck,«

_Îflfo)

(4)

wuere (flfo) ^is tue _norm

groundstate

and à is _a small real number.

The

spectral density A(+)(k,uJ)

is connected with the Green function

G(+)(k,uJ)

of the _same k-vector

through

the equation:

A(+)(k,~J)

₌

-n(G(+)(k,o~)) _(à)

Here U is the

imaginary

part of the Green function. _The

spectral density A(+)

_is the partide part ^{of the} spectrum and

A(~) _gives

_{the hole} contribution to the spectrum.

The

partide /

hole

(+ /-

part of the

density

of states

N(+)

_{is the}

sum over ail allowed momenta k of the

spectral

densities

A(+)(k,

_uJ):

Nl+)(uJ)

₌

£A(+)(k,uJ) (6)

k

and tue

complete density

of _state is tue _sum of the

partide

N~ and noie N+ part of trie spectrum.

N(uJ)

=

N~+1(uJ)

+

N~~~(uJ) (7)

In trie simulation it is of course not

possible

to determme trie limits à _- 0, therefore _we take

a small value of à

= 0.05 _or 0.10.

N°4 DENSITY OF STATES IN THE t'-HUBBARD _{MODEL 457}

ij

i j.

Il=6.0 _!',

j', j',

ii

_

U=4,0 ^{'j jj}

~

-

, J i i

z m

i

Î

_U=3.0 ⁱⁱ

à ^{j , i}

~

, ,i

~

U=2.0

"

o~

j^'

j

o

~i

u=i.o »

/

'

U=0.0

',

J

-4 0 4 8

Energy

_w

Fig. 1. Density of states N+

(w)

of the square 8 point system with the filling

(n)

= 1.00, t'

= +0.22 and the imaginary part ^à = 0.10 for several interactions U. Partiale spectrum ^{N~: solid} fine, hole

spectrum N+: _dashed fine, All _curves, except for U ₌ 0, _are shifted upwards for clarity.

3. Results

We have studied the t'-Hubbard model _{over a} wide _range of parameters. As the _van Hove

theory

_suggests the value of

t',

to be _neon the

doping

[4], we take

always positive

values for t' and

fillings

less _or

equal

half

filling.

In the square

8-point

system

(see

_{[11] we} studied the

three dioEerent

fillings (ni

= 1.00,

(ni

₌ 0.75 and

(ni

= 0.50, ^which

belong

to the number of

partides

_{ni = ni =} 4, 3 and 2. The third parameter ^that we

changed

is the interaction U.

We calculated the

density

of states for values between U

= -10 and U

= 20. But

only

in the

range U

= -6 to 6 _{we see}

changes

in the evolution of the _gap between the

partide

and hole spectrum.

In

Figure

1 the evolution of the

density

of _states is shown for t'

= +0.22 for positive values of U = 0,1,.

.,

6. The spectra ^{for U} > 0 _are shifted _up for

darity.

The spectrum for U

= 0 is

the spectrum ^{that is}

expected by solving

^the

non-interacting

model

exactly:

e(k~, ky)

=

-2t(cos(k~)

+

cos(ky))

+

4t'cos(k~) cos(ky) (8)

and

N(uJ)

=

/

_dk~

/ _{dkye(k~, ky) (9)}

no

The delta

peaks

of the energy-axis are broadened

by

the finite

imaginary

part ^à.

For the interaction U _> 0 the gap between the

partide

solid fine and hole

dash-pointed

fine spectrum ^does not open

immediately

with interaction U

#

0 in contrast to the

single-

baud Hubbard model.

Spectra

of the _pure Hubbard model

(t'

₌ 0 _cari be found in

[iii.

t'= +.38 _.

~ ii

t'= +.30

,

~ ii

/ ,

1 [ l

9 f. 1

Ql ' '

a ^a

a~

'

a , Î

c '

Q~ '

' ' o

t'"

' l'

j '~

j ^.

~'

;

[ ',

Fig. 2. - nsity of tates

U

₌

Partiale solid

fine, hole ^pectrumN+:^ashedfine, All except for t' = 0, are

hifted for _clarity.

There is _obviously ^up to U = 2 _{no or only}

a very small ^ap the partiale

hole N(+) pectrum. Only

constant

U

_and arious alues off' see

Figure 2

with

_{U =} 2.0), one sees that

for

t/

=

_o

there

is a _{gap of} about

1-fi, ^hich reduces for ^creasing t'

(

t' = +o.lo and t'

The gap _{anishes almost completely} for still _larger

t'

( t' =

+o.22

to

_{t' =} +o.38 ).

otting

the aps for _and

ioEerent values of ^{t' (see} Fig.

3)

_we

see

that the gap for

the

simple Hubbard

model

( idine

) at

fixed U is ^he

Increasing t' it ^ecomes smaller ( dashed fines ). On_the otherhand

if

_the

value

_of

gap for small interactions

U

is

about zero ^d_increases at ertain value Ug(t')

The

alue of Ug(t') mcreases ^th the value _of t'. For large interactions ( U > 6 ) here

is

a nearly ^linear dependence between U _and t' _with about the

same ^lope for ail

values ^of

We can _also

see

_this transition in the pectral ensity.

Therefore

let

us ^wonsider igure

4.

Here

we

see _the

spectral ^nsity Al+)(k,uJ) for the

momentum

k

=

(o,7r) at _ioEerent

interactions U. At _small nteractions ( U <

2 )

we see

some huge peaks

in

~(k,uJ) and _only

relatively small

peaks in the

_hole

spectrum A+(k,uJ). ^or _an

interaction

larger _than U = 3 ietuation ere th trie _partiale A~

and trie

_noie

spectrum

A+

bave

an quivalent

peak.

A similar hange in the weights

of

_the

spectral density

sonne

interaction Ug, where the gap opens

faster

( _compare Figs.

l

and 4 ). This change in

weights of

the

_spectral ensity is

seen

for ail

allowed

_omenta k.

For the

_k-vectors,

which lie

at the teraction U = o _completely in e noie _spectrum (

for

mple k

=

7r,7r) ),

N°4 DENSITY OF STATES IN THE t'-HUBBARD MODEL 459

8

~ ...U...£...

~~$

~ ""'

73 ,,'

~#

_',,'

h

^~ _'.~ÎÎ

[

~ ,':

~ ~_,

3

U _, ,'

~

"

Ç~ ~

Éd :

~ _,';

,"

Î "'1"""""

0 2 4 6 8 10

Interaction U

Fig. 3. Gap between the partiale N~ and the hale N+ _{spectrum for the}

square 8 point system

with the filling

(ni

= 1.00 for dilferent values of t': t'

= o-o

(-+),

t'

= +0.10

(-+),

t'

= +0.15

(-

-n)~ ^t' ₌ +0.22

(-

_-x ), ^t' ₌ +0.30

(- -6),

t'

= +0.38

(- -%).

ii

i

, ,

'p

₁₍

4 U=4.0

,'

_'

~

j

~ j~

= Î

~

^~ '

u md

u

~ ,

u

~

~'

u=o.o

;

3

Fig.

nteractions U the imaginary

part _à

_{= 0,10, trie} extnearest neighbour ^hopping t' =

+0.22 ^and

the _-vector k = (0, 7r). solid fine, hole spectrum: dashed fine. _Ail

curves, except

for U =

0, are^{hifted upwards} for ^clarity.

3

,

-.-"

Z

_2,5

_,.JK"'

+ 2

...1...

__.--

~ --"~"

~ j ^-'

8

j

--

j

""

__...;._...

0.5

...Î...

o 2 3 4 5 6

Interaction U

Fig. 5. Gap between the partiale N~ and the hole N+ _{spectrum for the}

square 8 point system

with the filling

in)

= 0.75 for dilferent values of t': t'

= o-o

(-+),

t'

= +0.15

(- -+),

t'

= +0.22

(-

-x ), t'= +0.25 (-

-D),

t'

= +0.30

(-

-6), ^t'= +0.50 (- -%).

the distribution _moves to the partiale spectrum.

A

possible explanation

for this behaviour is

that,

in the _pure Hubbard

model,

^t'

= o, the

Fermi surface is _square and unstable

against

the metal insulator transition which _{occurs as}

soon as the Hubbard interaction U is turned _on. As t' _is

increased,

t' > o, the Fermi surface in the free system, U ₌ o, is curved _[4] and at small interaction U _no

longer

suitable for trie metal

insulator transition.

Only

at _a

larger

value of U,

depending

_on

t',

the interaction is strong

enough

to _mix the states and the metal insulator transition _occurs.

To

summarize,

_a metal insulator transition _occurs in the t'-model _at half

filling,

but de-

layed

with respect to U. This transition at half

filling

is characteristic for models

describing high-temperature superconductors. Therefore,

tuis is anotuer indication that trie t'-mortel and tue

underlaying

_van Hove _scenario describes tue

rigut puysics

of tue

uigh-temperature

superconductors.

If _we return to

Figure

2, _{we see} for the pure Hubbard

model,

^t'

= 0,

only

_one central

peak

in the

partiale

_spectrum at _uJ m -o.8 ). ^{For t'} > o these

peaks spht

into two parts ^the

peaks

at _{uJ =} -o.8 and _{uJ m} -1.2 for t' _m +o.lo

).

The distance of these

splitted peaks

increases with

increasing

^t'. These two

peaks belong

to two dioEerent _momenta k. The lower _one results from the spectral ^densities of k ₌

(o,

_7r) and k ₌ (7r,

o)

and the _{upper one}

belongs

to the momenta k

=

(7r/2, 7r/2)

and

degenerated

momenta.

Now _we reduce the

filling

and agam

plot

the _gap between the

partiale

and the hole spectrum.

Here _we must

distinguish

between tue

filling (ni

= o.75 and

(ni

= 0.50

(Figs.

5 and

6).

In tue first _case tuere _{is a gap} between tue

partiale

and uole spectrum in tue free system, ^because tue Fermi energy lies between tue two contributions at uJ = -4t' and _{uJ =} 0 in the

density

of

states. One _{cari see} this gap of the free system U

= o _m the lowest _curve in

Figure

1 gap between the

peaks by

_{uJ =} -o.88 and _uJ

= 0

).

In the second _case the Fermi _energy lies at uJ =

4t',

and not in _a finite _{size gap} of the free system. ^{In both} cases tue _gap for 0 < t' _< 0.45 is

larger

than for tue _pure Hubbard model. At tue

filling (ni

= 0.75 this increase of the _gap witu t' is

mainly

caused

by

the

increasing

^limite size gap of tue free system

Fig.

5

).

Also _in tue _case of tue

filling (ni

= 0.50

Fig.

_GI tue gap mcreases witu

increasing

t'. But

N°4 DENSITY OF STATES IN THE t'-HUBBARD _{MODEL 461}

0.6

~

0.5

md

fi

_o4 ...j...

+

Î _~§---.-,,j---j---

0.3

',

_""

( 8

_0.2

...i.... ^'

JJ ,'

~

_{o j} ..l... ..l....__ .Îj_.___.I__..__..._____.]_,

O

0

0 2 4 6 8 10

Interaction U

Fig. 6. Gap between trie partiale N~ and the noie N+ _{spectrum for the}

square 8 point system with the filling

(ni

= 0.50 for dilferent values of t': t'

= 0.0

(-+),

t'

= +0.15

(- -+),

t'

= +0.22

(- -n),

t'

= +0.30

(-

_-x ), ^t' = +0.38

(- -6),

t'

= +0.45

(- -%),

t'

= +0.50

(- -o),

in contrast to tue

filling (n)

= 0.75 tuere is _no limite size gap ^at tue Fermi level. Moreover tue gap tends to _a constant value for

uuge

interactions U e-g- ^at t'

= +0.22 tue gap tends to

0.36,

dasued fine witu boxes in

Fig. 61.

Tue situation

changes dramatically

^{for t'} > +0.45 and

large

interactions U > 4. At t'

= +0.45 tue gap bas its maximum at U

= 4 and decreases for

larger

^{U. If} we increase t' to t'

=

+0.50,

tuen _{we see a} sudden breakdown of tue _gap between U ₌ 5.0 and U ₌ 5.3. For U _> 5.3 there is _no

longer

_{a gap} between tue

partiale

N~ and hale

spectrum N+ of the

density

of states. The _reasons for this sudden breakdown of the _{gap are} not yet ^clear.

There _{are some}

possible explanations:

the breakdown _may be _a result of the

degeneration

of the k-vectors k

=

(0,7r),

k

= (7r,

o)

and k

= (7r,7r) ^{for t'} = +0.50 _or of the special system

size. Another

possibility

is, that the accumulation of dioEerent k-vectors at the Fermi _energy

as in the _van Hove scenario _causes this breakdown.

Dagotto

et ai. found _a similar breakdown for _a

superconducting

order parameter ^{in the t J} model with the _same

filling

_[12]. Their

explanatioii

is _a

phase

transition.

In addition to the

positive

U _{case, we} have aise calculated the

density

of states and the gap between the

partiale

and the hole spectrum ^of the

density

of states for

negative

interactions U.

Trie

resulting

_gap for several

fillings

are

plotted

in

Figure

7. Here _{we see} the gaps for the three

possible fillings, in)

=1.oo 7

(a) ), (ni

₌ o.75 7

(b)

and

(ni

= o.50 7

(c) ).

^In trie cases

of

(ni

₌ 1.oo and

(ni

= o.50 tue gap decreases witu increasing ^t'. In tue case of

(ni

= 0.75

tue situation _{is a} little _more

complicated.

As above, tuere is a finite size gap between tue

partiale

and tue uole spectrum _in tue free system, ^U =

0,

witu t'

#

0. Tuerefore _{we see m}

Figure

7

(b)

tuat tue _gap increases witu

increasing

^t' for small interactions 0 _> U _> -1. But at

large

interaction U > 4 tue gap shows trie _same behaviour at trie

fillings in)

= 1.00

(a)

and

in)

= o.50

(c).

So for

uuge

interactions tuere is _a decrease of tue gap witu

increasing

^t' for ail

fillings.

Similar to tue

positive

U _case and ualf

filling

tue increase of tue _gap is almost linear witu tue interaction U for

large

interactions U _> 4, for ail

fillings

and for ail values of t'.

Given

Orly

tue

density

of states

Nl+)

_(uJ) and tue

spectral density A(+) (k,

uJ), ^{it is}

impossible

~

---j---j---j---j

+

3

,

~~T~~~~~~~~~~

Z

₂₅

, , 1--- ---J--

~

,,j

~) g

2 -~--~ fi- ^--~---~--

'

1.5

-~---~-b~jt~--

j~

^Ô,

~ l -r---~---t-

é

_°.5

_~Î ~~~Î~~~Î~~~

,

0

~ -5 -4 -3 -2 -1 0

Interaction U

Î

^~~

,

~~~T~~~T~~~~~~~~~~~

fi

35 L---L---J---~---

' _'

~~

³

,

--t---~---~

b) (

~'( ~( "~ _~~j~~~

j

1.5

~Î~~~Î~~~

,

~~~~

j -~---L---L--

Î

0.5

---1---1---1---1---~

0

-6 -5 -4 -3 -2 -1 0

Interaction U

,

5

2~

]

4.5 --p-

--+---+---+---~-

~ 35 ^L---1---J

z

,

~

3

,

-t---~---~-

~) _g

2,5 ^--~- ---J---~-

~

2

~~Î~~~~ ~~~~~Î~

fl

1.5 --~---~---+ ~-

it

^l

~~Î~~~Î~~~Î~~

o 0.5 --~---~---t--

0

-6 -5 -4 -3 -2 -1 0

Interaction U

Fig. 7. Gap between the

partiale

N~ and the hole N+ spectrum ^{for tue} _square 8 point system for negative interaction U and dilferent values of t': t'

= o-o

(-o),

t'

= +0.15

(-

-+), ^t' ₌ +0.30

(- -D).

Sllbplot

a)

filling

(n)

_=1.00,

b) in)

= 0.75 and c)

in)

= 0.50.

N°4 DENSITY OF STATES IN THE t'-HUBBARD MODEL 463

to

clarify

wuat is tue

origin

of tue gap. Tuerefore _we must calculate additional observables like correlation functions.

Projector

quantum ^{Monte Carlo} results [13] ^{indicate that the t'-model} in the

negative

U

regime

is

superconducting

_away from half

filling.

4. Conclusions

Within the

positive

U range for half

filling

t' retards the

opening

of the gap with respect to U. This retardation increases with

increasing

t'. So the metal insulator transition _{seems to} be

delayed through

_a finite value of t'. This transition supports ^{the t'-model and} the _van Hove scenario _as

possible explanations

for the

high-temperature superconductors.

At the

filling (ni

₌ 0.75 the

mcreasing

value of the _gap with

increasing

t' is dommated

by

the finite size gap in the free system, ^U ₌ ^{o. For} that _reason the _gap increases with

increasing

t'. And _at the next

possible filling, (ni

₌

o.50,

there is _a

huge magnification

with

increasing t',

until t' reaches the value t'

= +o.45. Here _{we see a} maximum of the gap ^at U ₌ 4.

Finally

at t'

= +o.50 there is _a sudden breakdown of the _{gap at} the interaction U _m 5. Tue

origm

of tuis beuaviour is not yet ^{clear. Additional} calculations _are in progress.

In the _case of

negative

interactions U tuere _{are no}

large changes

m trie evolution of the _gap for various

fillings

observed. Here the additional

hopping

t' decreases the gap with

increasing

t'. The

only exception

is the _case of the

filling (ni

₌ 0.75 with small interactions (U( < 1, there

the finite size gap for t'

#

0 dominates.

In order to decide _on the

origin

of this gap m the

negative

U

regime,

it is necessary ^to calculate other observables like correlation functions. But results obtained with the

projector

quantum ^{Monte Carlo} [13] ^{indicate that the}

negative

U t'-model is

superconducting

_away from half

filling.

Acknowledgments

The authors would like to thank D. M. Newns for

helpful

discussions.

References

iii

Monthoux P, and Pines D,, Phys. Rm. Lett. 69

(1992)

961

[2] Scalapino D.J., Loh E. and Hirsch J-E-, Phys. Rm. 835

(1987)

6694 [3] Markiewicz R-S-, Physica C217

(1993)

381.

[4] ^Newns D.M., Comments Gond. Mat. Phys. 15

(1992)

273.

[Si ^Torrance J-B-, Tokura Y., Nazzal A.I., Bezinge A., Huang ^T.C. and Parkin S-S-P-, Phys. ^Rm.

Lett. 61

(1988)

1127.

[6] Hirsch J-E-, Phys. Rm. B31

(1985)

4403.

[7] Morgenstern I., Singer J-M-, Hufllein Th. and Matuttis H.G.,"Numerical Simulation of High Temperature Superconductors", Proceedings of the 20th Int. School of Cryst., Erice, 1994, E.

Kaldis, Ed.

(Klower).

[8] Golub G-H- and Van Loan C.F., Matrix Computations

(The

Johns Hopkins University Press, Baltimore and London, 2nd edition,

1989).

[9] Cullum J. and Willoughby R-A-, Lanczos Algorithms for Large Symmetric Eigenvalue Computa-

tions (Birkhàuser, Boston, 1985) vol. I-II.

[10] ^Fulde P., Electron Correlations _in Molecules and Solids (Springer Verlag, Heidelberg,

1991).

[11] Dagotto E., Joynt R., Moreo A., Bacci S. and Gaghano E., Phys. Rm. 841

(1990)

9049.

[12] Dagotto E. and Riera J. Phys. Re~. Lett. 70

(1993)

682.

[13] Singer J. M, et ai., unpublished.