Triplet hole propagation and high-tc superconductivity

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Ph~ica C 1 ~ - 189 ( l ~ t ) 1603-1604 North-Holland

, ,, ) •

T R I P I , E T I t ( ) I . E PROt. A ( , A T ! ( N A N D t t i ( ; t i . 7 7 S l i t ' E I ¢ ( : ( ) N D t ~ ( , T ! V l ! "~,

A.M. OLEO, "'l J. ZAANEN~ and P, HORSCH"

" M a x - P l ~ - ! n s t i t u t ffir FestkSrpeffor~chung, D-7000 Stut~gat~ 80, FederM Re~mbli¢ of G e ~ n y I Institute of Phyfies, J a # l o n i a n University, P~30-059 K ~ 6 w , Po!~nd

t AT&T Bell L a b o r a t o r i ~ , Murray HiIl, NJ 07974-2070, U.S.A.

We investigate the d ~ a m i e s of a triplet hole in the d ~ b~ekgmund by using Bora ~ppro~m~io~,0 It i~ found that the coherent propagation by the hopping t is renormalized by ~he coupling between the hMe ~nd sNn w ~ ~md that the Fermi liquid picture is lost. Arguments are gisen that triplet hdes m i ~ t # ~ tim *o g~percond~cti~gy with a high value of Te.

The main difficulty in the theoreticM description of the high Te superconductivity lies in 'dw proxinfity of these systems to the metal-insulator transition. It is commonly accepted t h a t the undoped materials are M o t t - t t u b b a r d (M-H) insulators and are well descri- bed by the isotropic spin-½ tteisenberg model. The m,'xin effort is therefore concentrated on the understanding of the dynamics of holes added by doping to the antiferromagnetic (AF) background stabilized in CuO2 planes by strong correlations on (;'u-sites. Up to now it has been assumed t h a t the added hole forms locally a singlet of Zhang-Rice type, x with the dynamics de- scribed in terms of the ~ - ,/model.-"

The recent spectral weight transfer experiments ~ gave support to the notion that the t - d model ~uffices for the description of the low energy physics. However, the results of the photoemission experiments by Bian- coni et al. ~ indicate t h a t a significant fraction of the holes have 3dz=2_,.2 (d.2) character and m a y form a lo- cal triplet. Motivated by th/s, we analyze the dynamics of a triplet hole in an A F d 9 background of C u spins, starting from the effective Hamiltonian derived in the strong correlation limit, s

H ~: E + + <i j > ,a" t: N-" (d{~f'j.,ofi,~(e)dsz _~, + b.c.) + - - ~ ~ <ij>,cr <ij>,er < i j >

Here t: stands for the hopping of the added hole of d ~ symmetry and is smaller by a factor ~ 3 than the usually considered hopping of 3d=~_¢ ( d ~ ) holes. Y is the AF exchange interaction between Cu spins. The operators + fi,m and di=,~ create a component + m = 1 , 0 , - 1 of the triplet (c~((r) = 1 , - 1 for c~ =T,~L),

and a hole of spin ~ in d,~ orbitM at site i, re~pec~ tively. There are thn'e elementary hopping proces~ for a triplet hole: (I~,Tj-,T~ I¢),(I~, Ij-*T~,Oj),(O~, T., ... T~ ,0)), with hopping amplitudes t,t/~f2 and t/2, r e ~ c ~ tively (i and j denote nn lattice sites; ~he other 5 hvps can be obtained by time- a n d / o r spin reversal). ~n the singlet t - J model, only the third type of hoppi~ag exists; the other two are a consequence of the internM degree of freedom of the triplet hole.

~¥e study the dynamics of the triplet hole by usix-~g the linear spin wave (LSW) approach of SchmStt-Rink et aL, 6 which proved to be very successful in the c,~e of the singlet t - J model with a single hole. a After rewrit- ing the HarniltoMan (1) in germs of Schwinger bosoas and making the standard mean-field approximation for a two-sublattice AF system, a one finds

;: ~=_+_*~ 12 # z~= X-"s¢ _¢~ k,(

+(~.~-d'¢ + ~¢)~-¢]

+2 ~ ~+ _{. ~ + ~ . ~ - ¢ % ( ~ 3 ¢}* - ,- ¢ + ~ , ¢ ~ _ ¢ ) + h.~.} . (21 ..2,. ..s..

Here, e~ n -z~-'yfi/t/-2, w¢ and 3 ' are , ~ the ma,gnon di- spersion and creation operator, respectively, and ~£, v~ are the LSW coherence factorsfi This equation is considerably more complicated than the equiv~ent ex- it,s.,., we note t~at the pression for the singlet model. "-"- "

n u m b e r of hole degrees of freedom is doubled. Caiii~g the up (down) spin subiatfice A (B), and adding a hole of spin 1/2 ( - 1 / 2 ) creates the triplet states of 3f, = 1 on A and of M, = 0 on B (or 0 on A and - 1 on B}, respectively. This corresponds to a physical hole added to the N6el state and is label!ed with ~_~ r., respectively. in addition, a M~ = ! state can be created on sublat;-

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1604

A.M. Ole,~ et al. I Triplet hole propagation and high-T~ superconductivity

flee B (or - 1 on A), corresponding to a total zth spin component 3/2 ( - 3 / 2 ) and labelled by ~+~,~. Second difference between the present problem and the singlet hole is the presence of a free propagation term ,~ ~ . It reflects the possibility of the triplet hole propagation without disturbing the underlaying spin background, as, for example, by 1A ~ 0B ~ 1A -~ ... processes. But like in the singlet t - J model, the coupling to magnons is present in the processes 0a --* 0B ~ ....

Using the self-consistent Born approximation, °'8 we find the 1/2 and

3/2

hole Greens functions of the form G 1 / 2 ( k , w ) = [z - el, - Z ( g , w ) - I I ( k ' , w ) ] - ' , and

Ga/2(g,w) - [ z -

R ( g , w ) ] - ' , with the selfenergies,

= 2 , - ' / , " - ' 0 ,

2 2 "*

R ( g , . , ) = 1--N - ,/,,-, -

w h e r e = W e h a v e s o l v e d t h i s

system of integral equations numerically in 2D on a

16 x 16 lattice. In the Ising limit there is no coupling of the 1/2 hole to the 3/2 defects. It suffices therefore to iterate the selfenergy E(f~, w) alone, and the problem resembles the singlet t - J model, ~ but the free propagation eg is added. It determines the main features of the spectral function

A(k,w) = ;11mG~/~(k,w),

with the sharp quasiparticle (QP) surviving at finite J only at the bottom of the band, while in the rest of the Brillouin zone, the QPs are strongly damped. If we turn on the coupling between the 1/2 and 3/2 holes, the above picture is altered in a subtle, but important way. First, one finds that the 3/2 spectral function,

B ( £ , , , ) :

_{~I Ga/~(k,w),}

m r is completely incoherent

8 7 6 , . - . 5 < 3 1 L

L

- 4 - 2 0 2 4 FIGURE 1

Spectral function

A(k,w),

as obtained for J = tz = 1 for five values of f¢ = n( '~ ~ ~-, ¥), with n = O, 1~2, 3, 4.

and no bound state develops. This can be und~-rnto,~d quMitatively by considering again the Ising limit, where the 3/2 hole splits into a delocMized 1/2 hole, and a localized spin-flip, i.e. it appears as composite particle. As a consequence of the incoherency of the 3/2 hole,

the QP character of the physical

1/2

hole is destroyed

even at low energy and wavelength due to the coupling to the zero-energy magnons with ~ =

(~r',

w). In Fig. 1 we show a typical result for the physical hale spectral function A(/~,w). Thus, in contrary to the singlet t - J model, QP behaviour is weakened by the spin background and is reminiscent of the divergence discussed in the spin-fermion model. 9

It may be shown that the ddocalization of triplet holes leads at finite hole concentration to a strong interaction with crystal field excitations which, in turn, is responsible for a qualitatively new pairing mechanism, with the attractive interaction being here of purely ki- netic origin. The hopping of triplet holes becomes ea- sier when some of the dz2 holes are excited and the d,:2 holes may hop. Therefore, one expects that the moving triplet hole is dressed by the d=2 ~ d~2 orbital excitations. Higher values of Tc are obtained for increasing energy of the charge transfer excitation, E~, as sugge- sted by the experimental data. 4a°

Summarizing, we have shown that the triplet holes exhibit qualitatively different behaviour from the singlet ones. We believe that the experimental evidence that triplet holes are present in high-To superconduc- tors may by of crucial importance for the theoretical understanding of the origin of superconductivity.

One of us (A.M.O.) acknowledges the partial support by the Polish Research Project CPBP No. 01.09.

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