Hubbard model with first & second neighbor hopping; the Cu3+ question in high-Tc superconductors

Nuclear Physics B (Proc. Suppl.) 5A (1988) 151-155 North-Holland, Amsterdam

151

HUBBARD MODEL WITH FIRST & SECOND NEIGHBOR HOPPING; THE Cu~+2 QUESTION IN HIGH-T

e

SUPERCONDUCTORS

T. A. KAPLAN Department of Physlcs and Astronomy, Michlgan State University, East Lanslng,

amplitudes for a l l - l e n g t h

I. INTRODUCTION I review earlier arguments I for A n d e r s o n ' s 2

then physlcal

MI 48824, U.S.A.

bonds

are allowed,

interpretation of the linear eom-

RVB (resonatlng valence bond) states being i11-

oznatlon (e.g. the zmpllcatzon of just speaking

defined,

of

how

a

reasonable

definition

was

it as

"a resonating-valence-bond

state") in

arrived at and how that motlvated consideration

terms of the p e c u l i a r i t i e s

of the Ist- and 2 n d - n e l g h b o r - h o p p l n g

that one happens to be using (VB-states in this

model I.

the H u b b a r d

Cu3+Cu 2+~

model

with

hopplng

11ttle

cludes

Cu j+.

the o x y g e n

Emery 3, and

then

which

is called see no or

Assumlng a model which insites,

first

dlscussed

case) has no justification, noted

C u 2 + C u 3+

entlrely among the Cu s i t e s ,

into questlon by e x p e r i m e n t s very

Hubbard

I then note that the usual picture for

by

by many o t h e r ~- 7 - , I make a

2rid n e i g h b o r satlsfylng

hopplng

regime.

Thus

superconductlvlty:

slb111ty

conslstent

I then d i s c u s s

of r e p r e s e n t l n g

such

with a

the pos-

a model

by a

parameters,

t I and

ground

the s l n g l e t rlse

for

thls model Anderson's

Hubbard model on the Cu sltes only.

2. DEFINITION OF "RVB'S ''9

square

RVB as

is an

ideal

ideas about

remove electrons and see

pairs

if

stick together, and or give

to s u p e r c o n d u c t i n g

model

t2,

zn the I/2-f111ed strong cou-

pected,

being

I then

the Majumdar 9 condition tl/t 2 = 72,

its ground state, piing

set

chain wlth Ist and

has precisely the nearest-nelghbor-bond

testing

result 8 recent experiment .

or meaning.

that a Hubbard-model

crude estimate of the amount of Cu 3+ to De exthe

of the basis

fluctuations.

Thls

differs

the o t h e r s

strlklngly zn thls respect from 2 studled, the l i n e a r c h a i n and

lattice,

whlch require long-range bonds

these

in the insulating case: the spln-spln correla-

so-called RVB's were introduced 2 as llnear com-

tlon function decreases so slowly for the chain

The essentlal

point

is that b e c a u s e

blnatlons of valence-bond

(VB) states (products

that the structure

shows

an

inflnlte-

peak

no prescrlpt]on

vector, and the square lattlce is most probably

for restrlctln 6 the amplltudes,

feature

factor

of palr slnglets covering all sites once), wlth

at the a n t l f e r r o m a g n e t l c

wave

the introductlon of the term RVB state amounted

long range ordered.

to merely a renamln 6 of the we11-known concept,

this model

a szn~let

I (and

However I since became aware of the fact that a

large

Hubbard model wlth hopping solely among

many

others)

amplitude when (e g.

state.

Groplng for meanlng,

assumed

that

the s l n g l e t

bonds

are all

r~earest nelghbors).

on two accounts: ance w l t h

RVB's

had

in thelr expansion in VB states only short

range

This Is reasonable

sites

might

I had inltlated a study of I regime .

in the s t r o n g - c o u p l l n g

be u n s a t i s f a c t o r y .

the Cu

Therefore

I

won't discuss that work further here, a l t h o u g h zt is an interesting model I.

(1) it seems to be in accord-

Anderson's

physlcal

descrlptlon of

RVB'~, namely they are "spln liquids", they are 2 r~ot " s p l n c r y s t a l s " ; (i]) ] f arbltrary

0920 5632/88/$03 50 ,D Elsewer Science Pubhshers B.V. (North-Holland Physics Pubhstung Division)

3.

THE Cu3+QUESTION

An e s s e n t i a l model

assumption behlnd the Hubbard

(HM) for the Cu sites

is that when e.g.

T A Kaplan / Hubbard model wtth ftrst and second netghbor hopping

152

La2Cu

04

is doped with Ba, which is formally a

2+ ion, and which replaces extra

electrons

needed

the 3+ lon La,

the

from the Cu-O system

Cu and 0 sites d

nlO

come from the Cu, 02-.

Thus

making

the H u b b a r d

t dlo and PI~t create holes in d I0 and p6 at

Here

= d td i0

i, r e s p e c t l v e l y ,

o,

i0

it Cu 3+ and leavlng model

d

for the undoped

[

n I =

material,

with sp~n

(for I : Cu site), and

which has strong

coupllng

and

d

(2>

nlo

,

0

is at

the half-filled band, predicts an antlferromagnetlc insulator (in agreement with experiment). With

thls picture,

than-half-fllled hopefully

doping

then gives a less-

band, nonzero conductlv]ty and

s u p e r c o n d u c t ivl ty.

Unfortunately,

experiment either does not see 5'10-12,

or sees

and

s~m]larly

parameters

for

explanatory.

~

p,

Ud,

YBa2Cu307

pairs. for

Cu

etc.

Sums wlth <1,J

nearest-nelghbor case

d

Ed, £p,

*

0

]'he

are

se]

f-

mean to sum over

I consider

just

the

the estLmate of the number

of Cu 3+ . very little 8 Cu 3+ (the latter in YBa2Cu3Oy , y 7).

The

argument

for

seeing

n°ne 5 , 12 is that the hlgh energy (= 10 ev) for ox]dlzlng amount

Cu 2+ would

lead

at o b s e r v a t i o n

neutrallty oxzdIz~ng

the 3-5

plausible calculat

~ons

0 2-

to

a negllglble

temperatures.

Ls obtaLned,

Essentially thls model wlth U

essentlally

Charge

~t is a r g u e d 5'8 instead.

on the basis ~3 a n d other

This of

by Ls

band

ab

used

et.al.6to

Note that Lf the Cu are assumed

to be 2+ and

the O either 2- or I-, there has

to be one 0 I- per unzt cell for y = 7.

with

analyze

The p o s s i b l e ~nteraction

: V = 0 was P a c i u ~ t e r by S h e n

thelr experlmental result~

importance V was

of the

noted earlier.

~nteratomlc 14 The single

spatlal orbltal per slte constralns the system to maxlmum ox]d~tlon states Cu 3. and neutral O. The parameters

inLt~o

calculations 6

In c o n n e c t i o n

added

are chosen

~o that

holes

to Cu I+, 02- flrst go onto Cu, unt~] all

the copper Is 2+, the next holes 3 oxygen. Thls requ]reb

golng

on the

Once the exLstence of 0 I- is accepted, one sees immediately that covalent

mlx~ng

of Cu 3+

is not only possible, but is expected.

For ex-

ample,

consider

Starting

a neighboring

Cu-O

paLr.

from Cu2*O 2- there can't be electron

transfer from Cu to O s~nce

the latter

has a

Ep

, Ed

and

For my crude that

all

Define,

Ud

' Ep - EEl + ~V

estlmate

(3)

! assume for s~mpllc~ty

the O's on the Cu-O

chains

are O

for the chains,

f~lled p shell; however Cu2+O I- wLll surely mLx 11~ : chain state wlth all Cu 2+ & O1-;

wlth Cu3+O 2- as well as wlth Cu1+O. To get one might

a very rough ~dea of how much Cu 3+

expect

consider

the quLte

natural

i.e.

the state wlth n d = n p = I. l

i

First assume strong coupling:

model 3

H = ~d ~ nd + c

p

1

Itl

~ nP + Ud ~ n d n d i i~ ]+

p + V + Up ~ n Pn I~

X "l,J~

~< AE = U d

Itl ~

n din P j

~E' =

(Ep - E d)

Up +

- V,

Ep - cd - V,

(4)

(4')

where AE (&E') is the energy chdnge on moving a

+ t

[

t

[ (dio P~o ÷ h.c.)

(I)

Cu hole (an O hole) to a nelghbor O (Cu).

o ehaln wave funetlon

is approximately

The

T A Kaplan / Hubbard model wtth first and second neighbor hoppmg

'#u ~ 11> - (t/&E) q - ( t / A E ' ) q ' .

(5)

one

hole

per site.

153

So zn thls case x : 1/12 ~

8%. tere

So

a

ballpark

reasonable.

q : ['z ( d l

represents

tl ' ° i+

hole

+dr

1-1,o l) P z o i l l "

transfer

sistent

(6)

sites and N' is the slmilar object for C u - t o - O hole

transfer. 0 is the s p i n at O - s i t e i 1 (assumed to be { or ~). The number of Cu 3+ is

Interestlngly,

with

the

to

10%

The pared

result

with

the

finding

lt'~ e x p e c t a t i o n value in ~0 is

x =

I% to

10% s h o u l d be com-

requzrement

x ~_ I/3

(]f the Cu's are assumed to be elther

not o f

\N3+~ = (t/AE) 2 "q IN3+I n, * . . -

,

conslderatzons

the

(8)

slm/lariy, ture .nd

I0

if

Cu 2. b u t of

2+ or 3+

there

is s o m e

generally causes

the 0 2- .

tend

assume). chain.

dlsordered

O f course,

of

O I- (oxldLzed 0 2- ) occurs as a m i x the

same

three

states.

]n v l e w

Here

N zs the

Since

there

numbe~

matter

is one

(whlch

of O's

in-chaln

i

in a

0 and

3

Cu's per cell, we end up with

I

x = ~ (~

whlch

(HM)

for

sight,

case holds; one can still treat a

way

the

for

(lO)

whlle

regime.

of S h e n

and C u 1 + O will

Emery'~

thls

Subst~tutlng

et.

al.5

I find

estlmates 3 give

~ 3%

the

x ~ 5%, to

18%,

other estimates bringing the lower end to ~ I% However, .2

to

even for the smaller numbers, .4,

whlch

is

not

small

Itl /AE

extreme,

(It's

one

hole

the

~ve~age

per

site

number

IS
in

of

really

the

~p = Cd,

Cu-O chains

Cu 3÷ p e r

Cu-stte

~nteractlons

of

be

this

predominantly

object

hopping

hopping

zf o n e

consider

However,

thls

had o n l y

slmpie-mlnded ol

eq.(l).

The

those

charge

states

to

certainly volds

at

object

of a

And this would

the s p i n mapplng

the

on the O-sltes

w~th

on the Cu-sltes."

be c o r r e c t

CuZ+O1; ~

least

for the

Cu2+O 1- h a s

four

spin states, whereas the other two, CuJ+o 2- and cul*o are necessarily slnglets (spln O)

Then in

the

/ which iS \nl# ,~I]1 } : I/4 be-

cause there are no

hole

tried another simple

AE = 0 with U d = Up = V = O,

and

chain

I therefore

thls

of Cu2+O I-, Cu3+O 2-

are in one-one c o r r e s p o n d e n c e

model

b a n d w l d t h / A E that's to be ~< I). Second,

At flrst

the linear chaln with strong coupling:

then the linear c o m b m a t i o n

t )2

coupling

parameters

on Cu-sltes only."

"Consider addlng a hole to the Cu2+O 2- lattlce;

F r a c t i o n of Cu's that a~e Cu 3., m

strong

hopping

model

zt might seem that one could a r g u e

states as

of

this, one might be tempted to think 'ht doesn't

(9)

JOj+ 1 splns

to

oxldatlon

hole as a hole in the I/2-fzlled H u b b a r d for

the

Cu 3+ ( o x l d l z e d Cu2.) always appears as a m l x ture o f Cu3+O 2 - w i t h Cu2+O I - and c u l + o ;

w]th

= N

of

formal valence 2- were assumed for the oxygens.

These

which

is

this result is con-

experimental

support the idea that d o p i n g

J°J-I

I%

t h e n x = I/3; as n o t e d a b o v e I+ Cu , which requires larger x.)

(7)

d d. N3+ : [ nlf n1~,

J

=

8% 8,15

f r o m O-sites to Cu-

'q IN3.I n~ = 2N - ~ ~Oln d

of

and

there

~s

To s]b11~ty

begln

idea about

to get an

o f mapplng the model as

sldered

3 - s l t e , C u 2 + O I - C u 2+ , versloll of

(I). I and

For slmpllclty, 3 be

the

Cu

of

(1) to a Cu-slte

HM (as w e l l the

some

the p o s -

the p h y s i c s ) ,

I con-

I considered U d = ~.

sites,

Let

thn O site belng 2,

TA Kaplan / Hubbard model wtth first and second netghbor hopping

154

and d r o p

the

numbers.

superscrzpts

The

mentioned

charge

the o c c u p a t z o n

and

two d o u b l e t s

other ~wo charge states, each

energies

doublets. are easily A

zn thls

to a

case.

11 2 0> and

Wlth

there

In I n 2 n3~ , just

~s 11 1 I>; the spzn gzves rzse

quartet

are

on

state,

The

I0 2 I>,

~d + 2~p

O,

is a good quantum number.

to a smaller The

mapplng

(for

the

Thus

only

the

parameter

tzbondzng

the

where A : Ep - E d + Up.

The state w ~ t h

is n o t

Here

~s

the

quartet,

the o t h e r

ones

the spllttlng

three lowest energzes of interest,

zs of

the map-

On the other hand when will

work.

Typzcal

the for

the parameters

lead to satzsfactlon

at

t

= O.

Under

reflection

so for thls very szmple case

R the mapping w111 hold.

= d3o' R P 2 0 R -I = P2o' R 2 = I) t h e s t a t e is odd, the Ist e x c i t e d s t a t e

One st111 must check to

see if the Hef f w11l also yleld all the cally

even.

two

and an-

Note that the ground state ~s 6-fold

degenerate

ground

the

are

if t zs so small that the

posslble.

of thzs condltlon, (RdloR-1

let

energy

being

values doublets.

there

to the bondzng

E3-E 2 >~ kT the m a p p l n g 0

hole).

Is r e l e v a n t ;

the order of temperatures plng

zs to the 2-szte

both doublets,

Clearly of

the

0 spzn~

of one o x y g e n

t e f f.

states,

separatlon E 3 : O,

case

correspondmg

being 2tef f.

then

and antlparallel

Heff, and wlth three holes

hoppzng

be

energles, + 4t 2

In general

I conszder

present

the s p m s

this is because ~2

but appreclable

HM, wzth Hamlltonzan

12t 2

E~ : ~ -+ ~

1 hole on each slte,

I/2,-I/2,1/2;

average Cu splns are parallel,

the

found to be

I /~

Is exactly

are not slmply

relevant

levels for dlfferent

physl-

number~ of

Wzth holes.

The 2-hole case is

corresponds

I

IF - / 7 2 + 12 ]

important

to no O - h o l e s ) .

Agam,

(~t

it turns

F = A/Itl out that the mappzng holds: ~,

the s p m

here

and charge dzstrlbut~ons

there

is a l o w - l y l n g

Stayzng

slnglet

with

and

Ud =

trzplet

~n the ground separated

from the next hlgher

leve~

state can be written as £ d + V, w h l c h

is ~ I ev.

Thus

with two holes

(i.e. 2 electrons)

by 2 E P the 2-slte HM and

the s a m e

: (I/3 + v2/4) / ( I + 2 ) hoppzng wI]1

parameter

descrlbe

tef f determzned

the

low-lylng

prevlously

states

with

the

= -[6(I + 2 ) ] - I , help of Uef f chosen approprlately. Whether or not

such

a mappzng

wi|i

con-

: : (I + u2/2) / (I ÷ 2 ) tlnue

to

occur

for

larger There

systems i~ one

zs

an

interestmg

questlon.

important

case

it seems qulte clear that a mapplng

: (I ÷ 2~ 2) / (I + v 2). where

to the Here the szte spln Szz : (I/2)

model Since 7/24, for

v

*

-I


For

as

usual

HM wzll

not

worK.

Namely

the

(nl, - nl,).

F ~ 0 we h a v e


~n1> = 3/4, and
F ~ ~, v ~ O, so ~S1z ~ = I/3,

= -I/6, and ~n > : I zn the h m l t F * ~. z Interestlngly, in the latter limit, even though

(I) on the CuO 2 planar lattice

square Cu-lattlce [tl<~Ud,Up,

planes

(as zn the

In e.g. La. CuO.)

and V>Ep-E d.

H1rsch16~pol~ted

wlth out

that for t:O two holes placed on nearest nelghbor

O's

will

more-distant

have O's

energy

when V,O

lower

than

when on

and ~p=tt] ; g e n e r a l l y

TA Kaplan / Hubbard model wtth first and second netghbor hopping

for t:O the blndlng EB:V-(Cp-[d). blndlng.)

energy for such a palr zs

(EB>O

is the

The mznlmum

condltlon

energy

for

when the holes

are nearest nelghbors occurs on moving the hole on the Cu which

when

the hole-separatlon

such rearrangement energy.

of Cu holes

the same condltlons

Is larger no will lower the

(zero hopplng) the usual HM

will not glve rlse to bzndlng of

two holes added to the ]/2-filled Cu

sublattlce;

and

(repulsion

for

]ntultlve]y,

however,

n.n.

Veff>O help.

If Veff
but certalnly

involve a palrlng mechanzsm

for the usual

of won't

it seems that the deslred

be interestlng,

from the various

band on the

addltlon holes)

mapplng mlght be posslble would

empty

A moment's reflectlon shows that under

(wlth Ueff>O)

totally

zt would

different

ones whzch purportedly occur

repulslve-lnteractlon

Hubbard

mode I.

ACKNOWLEDGMENTS I thank cusslons

manuscrzpt. cusslons

S.D.Mahantl

and

a

for extensive

careful

reading

I also acknowledge with

Phys.

Rev.

Lett.

58,

2794

(1987). 4. J.E.Hzrsch, Workshop on Mechanlsms of HzghT Supercond., Theor. Phys. Inst., U. of M~nn., Oct. 1987, proc. to be published.

is the common nelghbor of the

two O's to one of the Cu's nelghborlng O's,

3. V. J. Emery,

155

J.W

Allen,

of

dlsthe

helpful dls-

V.J.Emery

and

D. J.Scalaplno.

6. Y.Guo,J-M.Langlols,W.A.Goddard 239, 896,(1988)

I. T.A.Kap[an, Conf. on Mag. & Mag. Materlals, Nov. 1987, to appear In J.Appl. Phys. (1987); Z.Zou, (1987); T Hsu,

Ill, Sclence

7. M.Schluter,M.S.Hybertsen, N.E.Chrlstzansen, preprlnt, plus references quoted there;G.F. Chen, X.W.Wang,T. C. Leung,B.N.Harmon, Bull. Am. P h y s . S o c . 3 3 , 6 0 7 ( 1 9 8 8 ) ; R.M.Martln, S.Satpathy, ibzd. p.649. 8. P.Stelner,S.Hufner,V.K1nslnger,l.Sander,B. S1egwart,H.Schmltt,R.Schulz,S.Junk,G. Schwltzgebel,A.Gold,C.Polltls,H P.Muller, R.Hoppe,S. Kemmler-Sack,C.Kunz,Z.Phys.B69, 449 (1988). 9. C.K.Majumdar,

J.Phys. C3, 911 (1970).

10.j.M.Tranquada,S M.Heald,A.R.Moodenbaugh,M. Suenaga,Phys. Rev. B35,7187(1987). 11.E.D. Crozier, N. Alberdlng,K.R.Bauchsples~, A. J. Seary, S. Gygax, Phys. Rev. B 36, (1987). 12.P.Stelner et.al., Z.Phys. B67, 497(1987). 13.L F.Matthelss, Phys. Rev. Lett. 58, 1028 (1987), J.Yu,A J.Freeman,J.-H. Xu, ibld. p. I035. 14.C.M.Varma, S Schmltt-R1nk, St. Comm. 62, 681 (1987)

REFERENCES

2. P W.Andetson, Mat Res. Bull.8, 153 Science 235,1196(1987); G.Baskaran, P W.AnGerson, Sol. St.Comm 63, 973 P.W.Anderson, G.Baskaran, Z.Zou, Phys. Rev. Lett. 58, 2790 (1987).

W.EIIIs, 5. Z.Shen,J.W.Allen,J.J.Yeh,J.S.Kang, W.Splcer,l.Llndou,M.B.Maple,Y.D.Dallchaouch, M.S.Torzkachwh,J.Z.Sun,T H Geballe, Phys Rev.B36, 8414 (1987).

E.Abrahams,

Sol.

15.Thzs experlment is apparently controverslal (J.W.Allen, przv. comm.), and so w111 have to a w a z t zndependent exper ]mental con fl rmat ion. 16.J.E.H1rsch, przvate communzcatlon. See also J. E. H1rsch, S. Tang, E. Loh, Jr., D. J. Scalaplno, Phys. Rev. Lett. 6(3, 1668 (1988).