VOLUME65, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 2 JULY 1990

Spin-Wave Renormalization in Two-Dimensional Antiferromagnets with Finite Hole Density

David Yuk Kei Ko Department of Physics, University of Tokyo, Hongo 7-3-1 Bunkyo-ku, Tokyo 113, Japan

(Received 3 November 1989)

When holes are doped into a two-dimensional antiferromagnet the spin excitations are renormalized. This renormalization is calculated for finite hole densities. The spin-wave spectrum is found to soften in­stantaneously upon doping, and the spectra functions are significantly broadened. With 0.1 hole per site the spin-wave spectrum collapses. The implications on the hole mobility is discussed briefly.

PACS numbers: 75.30.Ds, 75.50.Ee

Strong electron correlation is now believed to be re­sponsible for high-7V superconductivity (SC). This has motivated a vigorous reexamination of strongly correlat­ed systems.] In addition, the parent materials are anti-ferromagnetic (AF) insulators which undergo a phase transition to superconductors when dopant holes are in­troduced. Understanding this interplay between AF or­dering and hole doping is considered important to the understanding of the SC. A fitting starting point for in­vestigation is the Hubbard model2 near half filling, which describes electrons on a lattice with nearest-neighbor transfer and on-site repulsion. At half filling with strong on-site repulsion, that is, in the large-t/ limit, it describes an AF insulator. With doping it should mimic the physics in the high-Tc materials. The Hub­bard model, however, is notoriously difficult to solve. For large t/, the simpler / - / limit which describes elec­trons moving in a Heisenberg-spin system is often taken. The problem of the interplay between AF, hole motion, and SC is now divided into the destruction of the AF or­dering by the holes. The problem is, however, still not trivial. The holes are coupled to the spin lattice and, as they move, cause the spins to misalign. Thus they be­come dressed by a cloud of spin excitations (SE), their mobility depends on the relaxation of the disrupted spins. At the same time the spin lattice is only perfect in the half-filled limit. With doping, even if the holes are im­mobile, the spin configuration is disrupted. The SE are no longer ideal AF spin waves (SW) but are renormal­ized by scattering with the vacant sites. Hole motion will further disrupt the spin ordering and contribute to dressing the SW which, in turn, dress the holes. In order to study the transition from AF to SC a solution to both the hole and the spin dynamics is needed.

The problem of hole motion has been analyzed by a number of authors.3 Nagaoka and Brinkman and Rice first considered it in the limit of no spin relaxation. Schmitt-Rink, Varma, and Ruckenstein and others have obtained results with coupling to virtual SE included. Su et al. have further included the effects of local spin dis­tortions around a hole. Their results indicate a strong dependence of the hole bandwidth on the SW stiffness. More recently, a number of authors4 have examined hole

excitations numerically in finite systems. In particular, Szczepanski et al. have shown that the hole spectral function depends significantly on the nature of the spin background. Most of the work to date, however, assume only one hole in the system; the SE correspond to that of the perfect spin lattice. When there is a finite hole densi­ty the SE spectrum will change, and a knowledge of the SW renormalization is needed to carry their results over to the real system.

The local distortions of the spin configuration around static vacancies have been considered by Nagaosa, Hatsugai, and Imada5 and Bulut et al.5 However, al­though important to the problem of high-7V SC, SW re­normalization due to a finite hole density has largely been neglected. Here results on this are presented. A Green's-function (GF) analysis is used. Although the formalism is valid for mobile holes, the localized limit has been considered. That is, a flat hole band structure has been assumed. This is a drastic approximation, but for low doping it is arguably valid. At low hole density, hole motion reduces the AF coupling and introduces frustration into the spin lattice.6 The effects on the SW will be similar to scattering off vacant sites. The results obtained in the localized limit are therefore not expected to differ significantly. With increased hole doping, as we shall report, the SE spectrum is softened and the hole mobility should increase significantly. The proper hole dispersion, itself renormalized by the changes in the SW spectrum, should then be included. For obtaining a sim­ple qualitative picture the present approximation may, however, suffice. More detailed quantitative results, in particular for the higher dopant concentrations, must wait for a more complete treatment.

The starting Hamiltonian is the spin term of the t-J model,3'4

H-JX(\ -/,7,)(1 ^ X a / a / i ^ ^ A + a/flp.

(1)

The factor (1 —f?f,)(\ —fjfj) projects out the spin coupling between two neighboring sites if one or both of them are hole occupied. This accounts for the disrup­tion of the spin lattice. The operators are defined by

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(see, e.g., Schmitt-Rink, Varma, and Ruckenstein3) f} =c/l and a, =S / + for / on an up site, and ff—cn and ai—Si~ for / on a down site. The a?a] and ataj terms correspond to the AT term in the Heisenberg Hamiltoni-an, and the remaining terms correspond to SfS], with Sf-±(S-a?aih the " + " and the " - " is for the up and the down sublattice, respectively. The four operator terms involving a and af have been neglected. Fourier

transforming and applying the Bogoliubov transforma­tion into

uDan — vDa i pup

SW t

p" - p '

i + nE

variables, where

1/2

b* 'Upa-p — VpQp and b

2 a r

p =6>p/7, gives

ssgn(/p) 2nn

1/2

n D = / / = (!

p Mpbpbp—— JL fkfk-p-p-q U -pqb -pbq + Bpq(b-pb

Apq — il —n)j{(\ + 7q-fp)(wpWq-Hrprq) — (y q +7p)(w p ^ q - l -rpWq)} ,

#pq = T « / 0 ~w){(yp+7rq)(WpWq + l1pl,q) ~ ( 1 + Yp + q)(upVq + l'pWq)} ,

, = • / ( ! - y o 2 ) 1 7 2 , and

+ &p£q (2)

/p™ i {cos(/?Jcfl)+cos(p>-fl)}. r 0),

The second term gives the scattering of the SW with the holes. The simultaneous annihilation, or creation, of ex­citations with opposite momenta ensures momentum conservation. (1 —f?fi)(l —f]fj) has been set equal to 1 —(1 —n)(f?fj+fjfj)—n2, where n is the hole densi­ty. The excitation nonconserving terms, bf-pb-q and bpbq, require two anomalous GF in addition to the normal GF. The resulting Dyson's equation, G=Go — GonG, is a 2x2 matrix equation. G, Go, and IT are the dressed and bare GF and the SW self-energy, respec­tively. Partial summation of the first-order self-energy diagrams [Fig. 1 (a)] renormalizes the SW energies by a factor of (1 — A) 2 , corresponding to averaging the holes over the lattice. The excitation lifetimes remain infinite. Quantum corrections are obtained from the higher-order terms. Summation over the second-order diagrams [Fig. 1 (b)] gives, in the localized hole limit, the self-energies

n,

n,

-pp

l-pp

/{ —qpA. • pq B - qpB pq 0) — a)D + i8 (Q + Q)p + iS g IRZAI

-qp M- pq 0) — Q)p + iS

- 2 B -qpA -pq

(0 + (Dp + i8

2k,

i p - q i

(3)

2k,

(4)

The second term in the integrals contributes to the spec-

(a) (b) / \

A ? \ /

* t «*+4 •+»>

» 4» » fr * » ^ * «fr « *4 * *fr*

++. _*» . +*+*—

FIG. 1. Irreducible self-energy diagrams included in the cal­culation. The solid and dashed lines are spin and hole propaga­tors, respectively. An arrow entering (leaving) a vertex repre­sents the annihilation (creation) of an excitation.

tral function for negative energies, and arises from the anomalous GF. The phase factor

g(x) = \-(2/x)icos-l(x)-x(\-x2)]/2}0(\-x)

1 0 -

0 8 -

0.6-

0-4-

0 2 -

0-1

(a)

"——i 1 r——i r

C

d

e

^__J

— i 1 — i — — i — i — i

\ a

b \

i 1 1 1 1 1 1 r"" i—^1

FIG. 2. Renormalized spin-wave spectrum along (a) U,t)n and (b) (/,0)/r for (curves) perfect lattice, (curve/?) 0.01, (curve c) 0.03, (curved) 0.065, (curve e) 0.09, and (curve/ ) 0.12 hole per site.

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is obtained by integrating over the initial- and final-hole Fermi circles, and corresponds to the phase space avail­able for the SW to scatter. Here x = | p — q | /2kf, 0( 1 — x) — 1 for x < 1, and 0 =0 otherwise.

The SE energies, Fig. 2, are obtained from the poles of the dressed GF. There is little change in the overall form of the dispersions. The phase space available for the SE to scatter is larger for larger wave vectors, and leads to an asymmetry for higher dopings; otherwise, the functional form remains close to that for perfect AF SW. This lack of change in the functional behavior is due to the flat hole band approximation.

More important is the softening of the SW energies upon doping. At a concentration of 0.01 hole per site the SW energies have dropped by 20%. When the doping concentration reaches 0.03 hole per site the SW energies are renormalized by half. With the functional form of the dispersions largely unchanged, the softening of the SW spectrum may be viewed as a decrease in the ex­change parameter / . Intuitively, this corresponds to a weakening of the spin-spin coupling, and signifies a reduction in the energy penalty for misaligning the spins as the holes move. The hole mobility should therefore increase sharply. In addition, the renormalized SW spectral functions, Fig. 3, are broadened with doping. This broadening is especially large for excitations near the top of the SW dispersion [Fig. 3(a)]. Since holes move via virtual SE, a broader spectral function allows this to occur over a larger energy range. The result would be a further enhancement in the hole mobility. Note that the narrowing of the spectral functions upon further doping is an artifact of a non-self-consistent cal­culation. If a dressed GF were used in the self-energy, a continual broadening would be obtained.

Further increases in the hole doping causes the SW spectrum to collapse. The critical doping for this is 0.1 hole per site. At this level, beginning with the large-wave-vector excitations, the SE energies become zero; a condensation into the zero-energy modes and a transition into some other phase is expected. The entire spectrum collapses with a further increase in doping. We note that with the softening of the SW and the high hole concen­tration the initial localized hole assumption is unlikely to be valid, and the details of the behavior in this region re­quires a more rigorous calculation. However, intuitively, improvements to the localized hole approximation should only lower the critical concentration at which this col­lapse occurs, since mobile holes will further disrupt the spin ordering. This would agree with the exact diagonal-ization results of Hasegawa and Poilblanc,4 which sug­gest that AF ordering in the t-J model disappears at a doping of about 6% in the thermodynamic limit.

There are a number of candidates for the new phase with hole doping. Nagaoka3 proved that the ground state of a Hubbard system with infinite U and a single hole is a ferromagnet. The stability of this state with more than one hole and finite U is, however, uncertain.7

\ (a) 1 1

1 1 1 1 1 I T 1 1 1 1 T'*l"

a

i"

I 1 AC

A Ab

-1.0 - 0 5 0 0 5 Energy/J

1 0

J (b)

J

T r—~i—* i "" i

le

I i

, , ,*X . IT

i

c | b

Uli

1 a

V , , , - 1 0 -0.5 0

Energy/j 0-5 10

FIG. 3. Absolute value of the spectral function of G\\ at (a) q = U/2,/r/2) and (b) 0r/4,/r/8) for (curve a) 0.005, (curved) 0.03, (curve c) 0.06, (curved) 0.09, and (curves) 0.12 hole per site.

Anderson8 proposed a resonating-valence-bond state for the doped situation. Other possibilities,1 such as para­magnetism, or a chiral spin phase, also exist and detailed calculations are needed to distinguish between them.

In conclusion, SE in a 2D antiferromagnet have been found to be very sensitive to hole doping, even in the lo­calized limit. At low doping the AF SW energies are significantly lowered; the hole mobility should increase accordingly. With further doping a collapse of the AF order soon follows. It is thus important to consider the spin and the hole dynamics on an equal footing.9

I wish to thank R. Saito and H. Kamimura for many useful discussions and for their help in Tokyo. I also thank C. Schrammel for her help. This work is support­ed by the Japan Society for the Promotion of Science.

'See, for example, Mechanisms of High Temperature Su­perconductivity, edited by H. Kamimura and A. O. Shiyama

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(Springer-Verlag, Berlin, 1989). 2J. Hubbard, Proc. Roy. Soc. London A 276, 283 (1963). 3Y. Nagaoka, Phys. Rev. 147, 392 (1966); W. F. Brinkman

and T. M. Rice, Phys. Rev. B 2, 1324 (1979); S. Schmitt-Rink, C. A. Varma, and A. E. Ruckenstein, Phys. Rev. Lett. 60, 2793 (1988); C. L. Kane, P. A. Lee, and N. Read, Phys. Rev. B 39, 6880 (1989); S. Maekawa, J. Inoue, and M. Miyazaki, in Mechanisms of High Temperature Superconductivity (Ref. 1); Z. B. Su, Y. M. Li, W. Y. Lai, and L. Yu, Phys. Rev. Lett. 63, 1318 (1989).

4K. J. von Szczepanski, P. Horsch, W. Stephan, and M. Ziegler, Phys. Rev. B 41, 2017 (1990); Y. Hasegawa and D. Poilblanc, Phys. Rev. B 40, 9035 (1989); J. Bonca, P. Prelovsek, and I. Sega, Phys. Rev. B 39, 7074 (1989); E. Dagotto, A. Moreo, and R. Joynt, Phys. Rev. B 41, 2585 (1990).

5N. Nagaosa, Y. Hatsugai, and M. Imada, J. Phys. Soc. Jpn. 59, 978 (1989); N. Bulut, D. Hone, D. J. Scalapino, and E. Y. Loh, Phys. Rev. Lett. 62, 2192 (1989).

6Th. Jolicoeur and J. C. Le Guillou, Europhys. Lett. 10, 599 (1989).

7B. Doucot and X. G. Wen, Phys. Rev. B 40, 2719 (1989); J. A. Riera and A. P. Young, Phys. Rev. B 40, 5285 (1989); E. Dagotto, A. Moreo, and T. Barnes, Phys. Rev. B 40, 6721 (1989); Y. Fang, A. E. Ruckenstein, E. Dagotto, and S. Schmitt-Rink, Phys. Rev. B 40, 7406 (1989); B. S. Shastry, H. R. Krishnamurthy, and P. W. Anderson, Phys. Rev. B 41, 2375 (1990).

8P. W. Anderson, Science 235, 1196 (1987). 9C. Gros and M. D. Johnson, Phys. Rev. B 40, 9423 (1989),

have developed a scheme to treat the hole and the spin with a single set of operators obeying Wick's theorem.