Sizable nuclear spin polarization is pumped in individual InP/GaInP dots in a
wide range of external magnetic fields B_ext=0-5T by circularly polarized
optical excitation. We observe nuclear polarization of up to ~40% at Bext=1.5T
and corresponding to an Overhauser field of ~1.2T. We find a strong feedback of
the nuclear spin on the spin pumping efficiency. This feedback, produced by the
Overhauser field, leads to nuclear spin bi-stability at low magnetic fields of
Bext=0.5-1.5T. We find that the exciton Zeeman energy increases markedly, when
the Overhauser field cancels the external field. This counter-intuitive result
is shown to arise from the opposite contribution of the electron and hole
Zeeman splittings to the total exciton Zeeman energy

Skiba-Szymanska, J.

Nikolaenko, A. V.

Chekhovich, E. A.

Tartakovskii, A. I.

Makhonin, M. N.

Drouzas, I.

Skolnick, M. S.

Krysa, A. B.

English

arXiv

J. Skiba-Szymanska

E. A. Chekhovich

A. E. Nikolaenko

A. I. Tartakovskii

M. N. Makhonin

I. Drouzas

M. S. Skolnick

A. B. Krysa

Crossref

Overhauser effect in individual
                    
                      
                        In

The coupled cluster method (CCM) is applied to a spin-half model at zero temperature which interpolates between a triangular lattice antiferromagnet (TAF) and a kagomé lattice antiferromagnet (KAF). The strength of the bonds which connect kagomé lattice sites is J, and the strength of the bonds which link the non-kagomé lattice sites to the kagomé lattice sites on an underlying triangular lattice is J′. Our results are found to be highly converged, and our best estimate for the ground-state energy per spin for the spin-half KAF (J′ = 0) is -0.4252J. The amount of classical ordering on the kagomé lattice sites is also considered, and it is seen that this parameter goes to zero for values of J′ very close to the KAF point. Further evidence is also presented for CCM critical points which reinforce the conjecture that there is a phase near to the KAF point which is very different to that near to the TAF point (J = J′)

Farnell, D. J J

Bishop, R. F.

Gernoth, K. A.

The University of Manchester - Institutional Repository

RAPID COMMUNICATIONSPHYSICAL REVIEW B, VOLUME 63, 220402~R!Coupled cluster treatment of an interpolating triangle-kagoméantiferromagnetD. J. J. Farnell, R. F. Bishop, and K. A. GernothDepartment of Physics, University of Manchester Institute of Science and Technology (UMIST), P.O. Box 88,Manchester M60 1QD, United Kingdom~Received 9 March 2001; published 3 May 2001!The coupled cluster method~CCM! is applied to a spin-half model at zero temperature which interpolatesbetween a triangular lattice antiferromagnet~TAF! and akagome´ lattice antiferromagnet~KAF!. The strengthof the bonds which connectkagome´ lattice sites isJ, and the strength of the bonds which link the non-kagome´lattice sites to thekagome´ lattice sites on an underlying triangular lattice isJ8. Our results are found to behighly converged, and our best estimate for the ground-state energy per spin for the spin-half KAF (J850) is20.4252J. The amount of classical ordering on thekagome´ lattice sites is also considered, and it is seen thatthis parameter goes to zero for values ofJ8 very close to the KAF point. Further evidence is also presented forCCM critical points which reinforce the conjecture that there is a phase near to the KAF point which is verydifferent to that near to the TAF point (J5J8).DOI: 10.1103/PhysRevB.63.220402 PACS number~s!: 75.10.Jm, 03.65.Ca, 75.50.Eelaexlaaronntiis-mteenoticdcoheednta-deFfoits,atulesedsFtheodalf-er-cen’’esl,Our knowledge of the zero-temperature properties oftice quantum spin systems has been enhanced by thetence of exact solutions, mostly fors51/2 one-dimensionasystems, and by approximate calculations for higher qutum spin number and higher spatial dimensionality. Of pticular note have been the density matrix renormalizatigroup ~DMRG! calculations1 for one-dimensional~1D! andquasi-1D spin systems, although the DMRG has, as yet,been so conclusively applied to systems of higher spadimensionality. Similarly, quantum Monte Carlo~QMC!calculations2,3 at zero temperature are limited by the extence of the infamous sign problem, which in turn is oftenconsequence of frustration for lattice quantum spin systeWe note that for nonfrustrated systems one can often demine a ‘‘sign rule’’4 which completely circumvents thminus-sign problem.A good example of a spin system for which, as yet,sign rule has been proven is the spin-half triangular latHeisenberg antiferromagnet~TAF!. The fixed-node quantumMonte Carlo~FNQMC! method5 has, however, been applieto this system with some success, although the resultsstitute only a variational upper bound for the energy. Otapproximate methods6–9 have also been successfully applito the spin-half TAF, and most, but not all, such treatmepredict that about 50% of the classical Ne´el-like ordering onthe three equivalent sublattices remains in the quantum cIn particular, series expansion results6 give a value for theground-state energy ofEg /N520.551, although the corresponding value for the amount of remaining classical orof about 20% is almost certainly too low. This spin-half TAmodel therefore constitutes a very challenging problemsuch approximate methods. However, the spin-halfk gome´lattice Heisenberg antiferromagnet~KAF! poses an evenmore difficult problem, because, like the TAF, not only ishighly frustrated and no exactly provable ‘‘sign rule’’ existbut also the classical ground state is infinitely degenerCareful finite-sized calculations10–12 have, however, beenperformed for the quantum spin-half KAF, and these resindicate that none of the classical Ne´el-like ordering seen inthe TAF remains for the quantum KAF model. The best0163-1829/2001/63~22!/220402~4!/$20.00 63 2204t-is-n---otalas.r-en-rsse.rre.ts-timate for the ground-state energy of the KAF via finite-sizcalculations12 stands atEg /N520.43. Furthermore, serieexpansion results6 indicate that the ground state of the KAis disordered. Indeed, a variational calculation13 which uti-lized a dimerized basis also found that the ground state ofKAF is some sort of spin liquid.In this article we wish to apply the coupled cluster meth~CCM! to a model which interpolates between the spin-hTAF and spin-half KAF models, henceforth termed theJ-J8model ~illustrated in Fig. 1!. The Hamiltonian is given byH5J(^ i , j &si•sj1J8($ i ,k%si•sk , ~1!where^ i , j & runs over all nearest-neighbor~NN! bonds on thekagome´ lattice, and$ i ,k% runs over all NN bonds which connect thekagome´ lattice sites to those other sites on an undlying triangular lattice. Note that each bond is counted onFIG. 1. TheJ-J8 model is illustrated in diagram~a!, where thebonds of strengthJ betweenkagome´ lattice sites are indicated bythe thick solid lines and the non-kagome´ bonds of strengthJ8 on theunderlying triangular lattice sites are indicated by the ‘‘brokelines. The triangular lattice Heisenberg antiferromagnet~TAF! isillustrated in diagram~b!, and it is noted that the two models arequivalent whenJ5J8. The quadrilateral unit cells for both caseare also illustrated. TheJ-J8 model contains four sites per unit celwhereas the TAF has only one site per unit cell.©2001 The American Physical Society02-1,aldefoo-.otofein-tags-toeentthhesns:thecter,eo-red-lactrsy tozedserthelct-RAPID COMMUNICATIONSD. J. J. FARNELL, R. F. BISHOP, AND K. A. GERNOTH PHYSICAL REVIEW B63 220402~R!and once only. We explicitly setJ51 throughout this paperand we note that atJ851 we thus have the TAF and atJ850 we have the KAF.We now briefly describe the general CCM formalism,though for further details the interested reader is referreRefs. 9, 14, and 15. The exact ket and bra ground-stateergy eigenvectors,uC& and ^C̃u, of a general many-bodysystem described by a HamiltonianH,HuC&5EguC&; ^C̃uH5Eg^C̃u, ~2!are parametrized within the single-reference CCM aslows:uC&5eSuF&; S5(IÞ0SICI1 ,^C̃u5^FuS̃e2S; S̃511(IÞ0S̃ICI2 . ~3!The single model or reference stateuF& is required to havethe property of being a cyclic vector with respect to twwell-defined Abelian subalgebras ofmulti-configurationalcreation operators$CI1% and their Hermitian-adjoint destruction counterparts$CI2[(CI1)†%. Thus,uF& plays the role ofa vacuum state with respect to a suitable set of~mutuallycommuting! many-body creation operators$CI1%. Note thatCI2uF&50, ; IÞ0, and thatC02[1, the identity operatorThese operators are furthermore complete in the many-bHilbert ~or Fock! space. Also, thecorrelation operator Sisdecomposed entirely in terms of these creation opera$CI1%, which, when acting on the model state, ($CI1uF&%),create excitations from it. We note that although the maniHermiticity (^C̃u†5uC&/^CuC&) is lost, the normalizationconditions ^C̃uC&5^FuC&5^FuF&[1 are explicitly im-posed. Thecorrelation coefficients$SI ,S̃I% are regarded asbeing independent variables, and the full set$SI ,S̃I% thusprovides a complete description of the ground state. Forstance, an arbitrary operatorA will have a ground-state expectation value given asĀ[^C̃uAuC&5^FuS̃e2SAeSuF&5Ā~$SI ,S̃I%!. ~4!We note that the exponentiated form of the ground-sCCM parametrization of Eq.~3! ensures the correct countinof the independentand excited correlated many-body cluters with respect touF& which are present in the exacground stateuC&. It also ensures the exact incorporationthe Goldstone linked-cluster theorem, which itself guarantthe size extensivity of all relevant extensive physical quaties.The determination of the correlation coefficients$SI ,S̃I%is achieved by taking appropriate projections ontoground-state Schro¨dinger equations of Eq.~2!. Equivalently,they may be determined variationally by requiring tground-state energy expectation functionalH̄($SI ,S̃I%), de-fined as in Eq.~4!, to be stationary with respect to variation22040-ton-l-dyrsst-tefsi-ein each of the~independent! variables of the full set. Wethereby easily derive the following coupled set of equatiodH̄/dS̃I50⇒^FuCI2e2SHeSuF&50, IÞ0; ~5!dH̄/dSI50⇒^FuS̃e2S@H,CI1#eSuF&50, IÞ0. ~6!Equation~5! also shows that the ground-state energy atstationary point has the simple formEg5Eg~$SI%!5^Fue2SHeSuF&. ~7!It is important to realize that this~bi-!variational formulationdoesnot lead to an upper bound forEg when the summationsfor S andS̃ in Eq. ~3! are truncated, due to the lack of exaHermiticity when such approximations are made. Howevone can prove that the important Hellmann-Feynman threm is preserved in all such approximations.In the case of spin-lattice problems of the type considehere, the operatorsCI1 become products of spin-raising operatorssk1 over a set of sites$k%, with respect to a modestate uF& in which all spins point ‘‘downward’’ in somesuitably chosen local spin axes. The CCM formalism is exin the limit of inclusion of all possible such multispin clustecorrelations forSandS̃, although in any real application thiis usually impossible to achieve. It is therefore necessarutilize various approximation schemes withinS and S̃. Thethree most commonly employed schemes previously utilihave been:~1! the SUBn scheme, in which all correlationinvolving only n or fewer spins are retained, but no furthrestriction is made concerning their spatial separation onlattice; ~2! the SUBn-m subapproximation, in which alSUBn correlations spanning a range of no more thanm ad-jacent lattice sites are retained; and~3! the localized LSUBmscheme, in which all multispin correlations over all distinlocales on the lattice defined bym or fewer contiguous sitesare retained.For the interpolatingJ-J8 model described by Eq.~1!, wechoose a model stateuF& in which the lattice is divided intothree sublattices, denoted$A,B,C%. The spins on sublatticeAare oriented along the negativez axis, and spins on sublattices B and C are oriented at1120° and2120°, respec-tively, with respect to the spins on sublatticeA. Our localaxes are chosen by rotating about they axis the spin axes onsublatticesB andC by 2120° and1120°, respectively, andby leaving the spin axes on sublatticeA unchanged. Underthese canonical transformations,sBx →2 12sBx 2A32sBz ; sCx →2 12sCx 1A32sCz ,sBy→sBy ; sCy →sCy , ~8!sBz →A32sBx 212sBz ; sCz →2A32sCx 212sCz .2-2istdrafodeme-ar-trynofthe-his,Aicalstoig.on-re-thendthsomnRAPID COMMUNICATIONSCOUPLED CLUSTER TREATMENT OF AN . . . PHYSICAL REVIEW B63 220402~R!The model stateuF& now appears mathematically to conspurely of spins pointing downwards along thez axis, and theHamiltonian ~for J51) is given in terms of these rotatelocal spin axes as,H5 (^ i→ j &H 2 12 sizsjz1 A3l4 ~sizsj11sizsj22si1sjz2si2sjz!1l8~si1sj21si2sj1!23l8~si1sj11si2sj2!J1J8 ($ i→k%H 2 12 sizskz1 A3l4 ~sizsk11sizsk22si1skz2si2skz!1l8~si1sk21si2sk1!23l8~si1sk11si2sk2!J . ~9!Note thati and j run only over theNK sites on thekagome´lattice, whereask runs over those non-kagome´ sites on the~underlying! triangular lattice.N indicates the total numbeof triangular-lattice sites, and each bond is counted onceonce only. The symbol→ indicates an explicitbond direc-tionality in the Hamiltonian given by Eq.~9!, namely, thethree directed nearest-neighbor bonds included in Eq.~9!point from sublattice sitesA to B, B to C, andC to A forboth types of bond. We now perform high-order LSUBmcalculations for this model via a computational procedurethe Hamiltonian of Eq.~9!. The interested reader is referreto Refs. 9 and 15 for a full account of how such high-ordCCM techniques are applied to lattice quantum spin systeWe note that for the CCM treatment of theJ-J8 modelpresented here the unit cell contains four lattice sites~seeFig. 1!. By contrast, previous calculations9 for the TAF useda unit cell containing only a single site per unit cell. Hencthe J-J8 model has many more ‘‘fundamental’’ configurations than the TAF model at equivalent levels of approximtion. However, we find that those configurations which anot equivalent for theJ-J8 model butare equivalent for theTABLE I. CCM results for the ground-state energy per spin asublattice magnetization of the TAF and KAF models usingLSUBm approximation withm5$2,3,4,5,6%. CCM critical values,Jc8 , of theJ-J8 model~with J51), which are themselves indicatorof a phase transition point in the true system, are also given. Cparison is made in the last row with the results of other calculatioKAF TAF J-J8m Eg /NK MK Eg /N MK Jc82 20.37796 0.8065 20.50290 0.85783 20.39470 0.7338 20.51911 0.8045 20.6834 20.40871 0.6415 20.53427 0.7273 20.2175 20.41392 0.5860 20.53869 0.6958 20.2446 20.41767 0.5504 20.54290 0.6561 20.088` 20.4252 0.366 20.5505 0.516 0.060.1cf. 20.43a 0.0a 20.551b 0.5caSee Refs. 10 and 11.bSee Ref. 6.cSee Refs. 7 and 8.22040ndrrs.,-eTAF have CCM correlation coefficients$SI ,S̃I% which be-come equal at the TAF point,J851. Hence, the CCM naturally and without bias reflects the extra amount of symmeof the J-J8 model at this one particular point. This is aexcellent indicator of the validity of the CCM treatmentthis model. The results for theJ-J8 model atJ851 thus alsoexactly agree with those of a CCM previous treatment ofTAF. Our approach is now to ‘‘track’’ this solution for decreasing values ofJ8 until we reach acritical value ofJc8 atwhich the solution to the CCM equations breaks down. Tis associated with a phase transition in the real system9,15and results forJc8 for this model are presented in Table I.simple ‘‘heuristic’’ extrapolation~e.g., see Ref. 15! of theseresults gives a value ofJc850.060.1 for the position of thisphase transition point. This result indicates that the classthree-sublattice Ne´el-like order, of which about 50% remainfor the TAF, completely disappears at a point very nearthe KAF point (J850).The results for the ground-state energy are shown in F2 and in Table I. These results are seen to be highly cverged with respect to each other over the whole of theFIG. 2. CCM results for the ground-state energy per spin ofJ-J8 model ~with J51) using the LSUBm approximation withm5$2,3,4,5,6%. The boxes indicate the CCM critical points,Jc8 , and asimple extrapolation in the limitm→` implies thatJc850.060.1.FIG. 3. CCM results for the sublattice magnetization of theJ-J8model ~with J51) using the LSUBm approximation with m5$2,3,4,5,6%.e-s.2-3oninesdsumi-hiI.nglyaso,nt.sisotbetatetedher--d.hewnRAPID COMMUNICATIONSD. J. J. FARNELL, R. F. BISHOP, AND K. A. GERNOTH PHYSICAL REVIEW B63 220402~R!gion 0<J8<1. A simple heuristic extrapolation9,15 may beattempted for these results for varyingJ8 by plottingLSUBm results form5$3,4,5,6% against 1/m2 and perform-ing a linear extrapolation of these data as was dpreviously9 for the TAF only. These results are givenTable I for the KAF and TAF models. We believe that thextrapolated results are among the most accurate resultthe ground-state energies of the TAF and KAF ever founWe now wish to describe how much of the original clasical ordering of the model state remains for the quantsystem. If one considers non-kagome´ lattice sites then thespins on these sites are effectively ‘‘frozen’’ into their orignal directions~of the model state! at J850. Hence, we be-lieve that the relevant quantity to be considered for tmodel is the average value ofskz ~again after rotation of thelocal spin axes! wherek runs only over theNK kagome´ lat-tice sites, given byMK522NK(k51NKskz . ~10!The results forMK are presented in Fig. 3 and in TableThe puzzling ‘‘upturn’’ of MK for the LSUB5 data is anartifact, and such behavior only ever occurs9 when one entersv.cr22040efor.-sa phase in which the model state becomes an increasibad starting point. Again, we extrapolate9,15 these results forthe KAF by plotting LSUBm results form5$3,4,5,6% against1/m and performing a linear extrapolation of these data,was done previously9 for the TAF. Although the extrapolatedvalue forMK specifically at the KAF point remains nonzerthe LSUB6 result goes to zero very close to the KAF poiCCM results are thus fully consistent with the hypothethat, unlike the TAF, the ground state of the KAF does ncontain any Ne´ l ordering.It has been shown in this article that the CCM mayused to provide highly accurate results for the ground-senergy of theJ-J8 model ~with J51) which interpolatesbetween the TAF and KAF models. Indeed, the extrapolaresults for the ground-state energy for the KAF ofEg /NK520.4252 and for the TAF ofEg /N520.5505 are amongthe most accurate yet determined for these models. Furtmore, the amount of classical ordering~evaluated on thekagome´ lattice sites only! yields results which are fully consistent with the hypothesis that the KAF is fully disordereCCM critical points also reinforce the conjecture that tclassically ordered phase evident for the TAF breaks dovery near to the KAF point.d-C..1S. R. White and R. M. Noack, Phys. Rev. Lett.68, 3487~1992!;S. R. White, ibid. 69, 2863 ~1992!; Phys. Rev. B48, 10 345~1993!.2K. J. Runge, Phys. Rev. B45, 12 292~1992!; 45, 7229~1992!.3A. W. Sandvik, Phys. Rev. B56, 11 678~1997!.4R. F. Bishop, D. J. J. Farnell, and J. B. Parkinson, Phys. Re61, 6775~2000!.5M. Boninsegni, Phys. Rev. B52, 15 304~1995!.6R. R. P. Singh and D. A. Huse, Phys. Rev. Lett.68, 1766~1992!.7B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre, Phys. ST49, 192 ~1993!; Phys. Rev. B50, 10 048~1994!.8T. Jolicoeur and J. C. LeGuillou, Phys. Rev. B40, 2727~1989!.9C. Zeng, D. J. J. Farnell, and R. F. Bishop, J. Stat. Phys.90, 327B.~1998!.10B. Bernu, P. Lechimant, and C. Lhuillier, Phys. Scr.T49, 192~1993!.11P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, and P. Sinzingre, Phys. Rev. B56, 2521~1997!.12C. Waldtmann, H.-U. Everts, B. Bernu, P. Sindzingre,Lhuillier, P. Lecheminant, and L. Pierre, Eur. Phys. J. B2, 501~1998!.13C. Zeng and V. Elser, Phys. Rev. B51, 8318~1995!.14R. F. Bishop, Theor. Chim. Acta80, 95 ~1991!.15R. F. Bishop, D. J. J. Farnell, S. E. Kru¨ger, J. B. Parkinson, JRichter, and C. Zeng, J. Phys.: Condens. Matter12, 6887~2000!.2-4

Overhauser effect in individual InP/GaInP dots

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