We present a theoretical analysis of a spin-dependent multicomponent
condensate in two dimensions. The case of a condensate of resonantly
photoexcited excitons having two different spin orientations is studied in
detail. The energy and the chemical potentials of this system depend strongly
on the spin polarization . When electrons and holes are located in two
different planes, the condensate can be either totally spin polarized or spin
unpolarized, a property that is measurable. The phase diagram in terms of the
total density and electron-hole separation is discussed.Comment: 4 pages, 3 figures, Accepted for publication in Physical Review
  Letter

A. J. Leggett

B. I. Halperin

C. Comte

C. Tejedor

D. Jerome

G. Mahan

H. Chu

H. Haug

J. Fernández-Rossier

J. Shah

L. V. Keldysh

L. Viña

M. Z. Maialle

R. Zimmermann

T. C. Damen

T. Portengen

X. Naveh

X. Zhu

English

arXiv

Crossref

Spin Degree of Freedom in Two Dimensional Exciton Condensates

We present a theoretical analysis of a spin-dependent multicomponent condensate in two dimensions. The case of a condensate of resonantly photoexcited excitons having two different spin orientations is studied in detail. The energy and the chemical potentials of this system depend strongly on the spin polarization. When electrons and holes are located in two different planes, the condensate can be either totally spin polarized or spin unpolarized, a property that is measurable. The phase diagram in terms of the total density and electron-hole separation is discussed.This work was supported in part by CICYT of Spain under Contract No. MAT 94-0982-C02-01 and by the Comunidad Autonoma de Madrid under Contract No. AE00330y95

Fernández-Rossier, Joaquín

Tejedor de Paz, Carlos

Spin degree of freedom in two dimensional exciton condensates

Repositorio Institucional de la Universidad de Alicante

VOLUME 78, NUMBER 25 P HY S I CA L REV I EW LE T T ER S 23 JUNE 1997Spin Degree of Freedom in Two Dimensional Exciton CondensatesJ. Fernández-Rossier and C. TejedorDepartamento de Fı´sica Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain(Received 23 December 1996)We present a theoretical analysis of a spin-dependent multicomponent condensate in two dimensions.The case of a condensate of resonantly photoexcited excitons having two different spin orientations isstudied in detail. The energy and the chemical potentials of this system depend strongly on the spinpolarization. When electrons and holes are located in two different planes, the condensate can be eithertotally spin polarized or spin unpolarized, a property that is measurable. The phase diagram in termsof the total density and electron-hole separation is discussed. [S0031-9007(97)03434-0]PACS numbers: 71.35.LkIn this Letter we address the problem of a multicompo-nent quantum liquid: a condensate of excitons which holdan internal degree of freedom, namely, the third compo-nent of the angular momentum M [1]. The multicom-ponent character is twofold: on one side, the condensateis formed by pairs of distinct particles, an electron anda hole; on the other side, more than one kind of pairexists when M is taken into account. From the later pointof view, there is an evident analogy with some of thesuperfluid 3He phases in which a condensate with twocomponents appears due to triplet pairing [2]. We con-sider excitons generated in quasi-two-dimensional (2D)semiconductor heterostructures where carriers’ lifetimescan be very large. Moreover, degeneracy at the top ofthe valence band is broken usually leaving the highesthole with a third component of the angular momentumequal to 63y2. Then, four types of excitons with dif-ferent values of M exist, implying that different com-ponents of the condensate with different populations NMfor each spin are possible. The condensate properties de-pend on both the total density and relative populations.Hence, the problem offers more possibilities than thespinless exciton case considered in the literature [3–9];e.g., Josephson-type effects between the different com-ponents of the condensate [10] are now possible.We have two main reasons for paying attention to M:(i) The first is related to experiments. A promisingway to obtain a multicomponent exciton conden-sate is to excite the semiconductor with circularlypolarized light from a pulsed laser. Because of theangular momentum selection rule, two different kindsof excitons can be created, with either M ­ 11 or21. The light polarization produces a spin polarizationP ­ sN1 2 N2dysN1 1 N2d [11], where N6 are thepopulations of excitons with M ­ 61, respectively[12,13]. It is interesting to study the properties of thephotoexcited carriers with unbalanced populations NMby time-resolved photoluminescence. There are timeevolutions both on the total density due to recombinationand on P due to spin relaxation processes. Frequencyand intensity of the emitted photons with the two0031-9007y97y78(25)y4809(4)$10.00different circular polarizations allow the experimentalstudy of the condensates. It is possible to observe howthe chemical potentials of each kind of excitons in thecondensate are different; the splitting of the excitonlevels turns out to be strongly dependent on P [13].(ii) The second reason is related to the physical originof the existence of different phases and is precisely theaim of this Letter. Each phase is characterized by dif-ferent populations of each component of the condensate;e.g., in the case of a condensate with two componentswith M ­ 11 and M ­ 21, respectively, it can be ei-ther a ferromagnetic phase with N1 ­ N and N2 ­ 0 ora paramagnetic phase with N1 ­ N2 ­ Ny2 [11]. Thereason for the existence of different phases is the exclu-sion principle which plays a crucial role in the interactionbetween excitons in semiconductors. Effective massesare very light so that electron and hole wave functionsare very extended implying large overlaps which producehighly spin-dependent effective interactions between ex-citons. We will show that the main consequence of thisis that the ground state can be either spin polarized orspin unpolarized. This depends on both the carrier den-sity and the relative intensities of attractive and repulsiveinteractions that can be controlled by spatially separatingelectron and hole gases with respect to each other.Our target is to study the ground state of N electronsand N holes lying in two different planes separated by adistance d. The Hamiltonian is given byH ­Xksµ«ekayks aks 1Xk0 ,q,s0Vffq2L2ayks ayk0s 0 ak02qs0 ak1qs¶1Xk§µ«hkbyk§bk§ 1Xk0 ,q,§0Vffq2L2byk§byk0§0bk02q§0bk1q§¶1Xk,k0,q,s ,§V ehq sddL2ayks byk0§bk02q§ak1qs , (1)where k, k0, and q are 2D wave vectors, s stands for theelectron spin 61y2 s", #d and § for the hole spin 63y2s*, +d along the direction perpendicular to the 2D plane.Creation and annihilation operators are labeled as a’s forelectrons and b’s for holes with single particle energies «ek© 1997 The American Physical Society 4809VOLUME 78, NUMBER 25 P HY S I CA L REV I EW LE T T ER S 23 JUNE 1997and «hk, respectively. L2 is the area of the sample and theinteractions V ehq sdd and Vffq sf ; e, hd will be specifiedbelow.It is convenient to point out some assumptions be-hind this Hamiltonian. First of all, we suppose that theelectron-hole se-hd spatial degrees of freedom are in ther-modynamic equilibrium. This hypothesis is supported bythe fact that roughly 1 ps after the photoexciting laserpulse has finished, the e-h liquid reaches a thermal mo-mentum distribution [14]. That distribution correspondsto zero temperature when experiments are performed inthe resonant excitation regime. Since recombination andspin relaxations are processes much slower than that ther-malization [12,13], we take both the carrier density andthe spin polarization as external parameters in the theory,in the spirit of a Born-Oppenheimer framework.In order to look for exciton condensates we use a meanfield Keldysh-Kopaev formalism, i.e., a generalization ofthe BCS treatment [3,5,6]. We define two propagators Gfor each fermion, and four anomalous propagators F forthe four pairings s", *d, s#, +d, s", +d, s#, *d:Gessk, td ­ 2ikTak,s stdayk,s s0dl ,Gh§ sk, td ­ 2ikTbk,§stdbyk,§s0dl , (2)Fehs§sk, td ­ 2ikTak,s stdbk,§s0dl .Each one of the anomalous propagators is associatedwith a different type of e-h pair ss, §d and with agap (order parameter of the condensate) Ds§skd ­Pq Vehjk2qjkTaq,s s0dbq,§s0dl proportional to the densityof e-h pairs ss, §d. Following a standard mean fieldprocedure one obtains the propagators [10]Gessk, vd ­Vh+ Vh*VesVh+ Vh* 1 Vh+ D2s*skd 1 Vh* D2s+skd,(3)Fehs,§sk, vd ­2Ds§skdVh2§VesVh+ Vh* 1 Vh+ D2s*skd 1 Vh* D2s,+skd.(4)For simplicity we have not written the hole propa-gator, Gh§ sk, vd, which has a similar expression toGessk, vd simply exchanging e with h. Vh§ sk, vd ­2iv 2 f«hk 2 mh§ 2 Sh§ skdg, Vessk, vd ­ 2iv 1«ek 2 mes 2 Sesskd with mes and mh§ being electron andhole chemical potentials and S is the usual Hartree-Fockfermion self-energy.In the expressions for the propagators one observes thedifference between the condensate we are studying andthose occurring in other problems like superconductors,3He superfluid, or the spinless exciton gas. Since the pair-ing occurs among two different types of fermions, eachone with different spin, the terms VesVh§ Vh2§ appearingin the denominator produce three poles for each propa-4810gator. This causes a complicated mixing of excitationsunlike other types of condensates [15].In this paper we will focus on the case of resonantexcitation of s#, *d pairs. Under resonance conditions,holes do not relax their spin independently [12,16], aprocess which should produce s#, +d pairs with M ­22. Instead, the main effect is a relaxation of theangular momentum of the whole exciton, by means ofan intraexciton exchange spin flip, from M ­ 11 toM ­ 21 [12,16]. Therefore, we neglect the densitiesof excitons with both M ­ 62 and only two kindsof excitons are considered: those with M ­ 21 ; s", +dand those with M ­ 11 ; s#, *d having densities n2 ­N2yL2 and n1 ­ N1yL2, respectively. Such a systemwill be characterized by two quantities, the total densityn ­ NyL2 and the polarization P. In this case, eitherthe second or the third term in the denominators ofEqs. (3) and (4) disappears, one of the Vh’s can becanceled out, and each propagator only has two polesas in the BCS or superfluid 3He theory. Then onecould follow standard procedures [10] to obtain gapequations for D#,*skd and D",+skd [hereafter denoted asD1skd and D2skd, respectively] which are coupled to eachother solely through the Hartree contribution which isspin independent [15]. For an homogeneous system, theHartree term can be embedded in the chemical potentialas an energy origin so that the gap equations becomedecoupled to all the effects. Then, solving these gapequations is equivalent to looking for an extremal fromkCjH 2 m1N1 2 m2N2jCl , (5)using a wave functionjCl ­Yk,k0su1k 1 y1k ayk#byk*d3 su2k0 1 y2k0 ayk0"byk0+d j 0l . (6)m6 are the chemical potentials of 61 excitons and j 0l isthe ground state of the system without any photoexcitede-h pairs. Each set of parentheses in Eq. (6) correspondsto one of the two components of the condensate. As inthe spinless case, this wave function interpolates smoothlybetween two limits, n ! 0 and n ! ‘, where it isexact [5]. The normalization condition kC j Cl ­ 1 ofthe condensate wave function imposes the constraintssu6k d2 1 sy6k d2 ­ 1. The trial wave function (6) is validunder resonance conditions, as discussed above, and itleads to a gap equation D6skd ­Pq Vehjk2qju6q y6q . Thepractical procedure is to fix the two chemical potentialsm6 and to minimize the functional (5) in terms of theparameters u6k , y6k in a discretized k space. Since theCoulomb interaction is singular in 2D, we follow the usualprocedure [6] of adding and subtracting a term that makesthe singularity analytically integrable. From u6k , y6k weget D6skd and the density, n6 ­Pksy6k d2, for each typeof e-h pair. So we have all the ingredients to obtain theGreen functions, the energy per pair, etc. Changing m6VOLUME 78, NUMBER 25 P HY S I CA L REV I EW LE T T ER S 23 JUNE 1997we can obtain the dependence of the energy per pair foreach spin on both n and P. The results so obtained form6 as a function of n and P are directly comparablewith photoluminescence experiments which measure theenergy liberated when an exciton recombines [12,13].Let us discuss the physics involved in our meanfield theory. We will not include screening which doesnot depend on P when treated to the lowest order inperturbation theory [17], and it does not affect the splittingbetween chemical potentials d ­ m1 2 m2 which is ourmain interest in this paper. In this approach there arethree many-body contributions to the renormalization ofthe single exciton chemical potential: The first is aHartree term which gives the unique coupling betweenthe two components of the condensate. The other twocontributions are related to the Pauli principle. Oneis the exchange correction (EC), i.e., the reduction ofthe repulsive electron-electron (hole-hole) interaction dueto the fact that two identical fermions cannot occupythe same state. The last contribution is the vertexcorrection (VC), which represents the reduction of thee-h attraction due to the occupation of final states ine-h scattering processes. EC and VC contributions tothe chemical potential of a fermion with a given spin areindependent of the amount of fermions with the oppositespin. This causes a nonzero splitting, d, whenever thereis a nonzero spin polarization P, in agreement withexperiments [12,13]. All these physical properties aredescribed by a wave function (6) for the condensatewhich is separable in two components in analogy to whathappens in the A phase of superfluid 3He due to equal spinpairing (ESP) states [2]. The exciton condensate that wepresent in this paper is more flexible than ESP states of3He because P and d can be nonzero in the exciton case,while they must be zero in the ESP case.The numerical results shown in this paper are ob-tained with V ehq sdd ­ 2e2 exps2qddyeq, i.e., the Fouriertransform of V ehsr , dd ­ 2e2yepr2 1 d2, and Vqff ­2V ehq sd ­ 0d, e being the dielectric constant of the mate-rial. The single particle energy bands are supposed to beisotropic and parabolic. The spinless case has been stud-ied by Zhu et al. [8], who considered d Þ 0 because thee-h separation increases the exciton lifetime improving thecondensation conditions and avoids formation of biexci-tons. We have an additional reason to be interested in thecase d Þ 0: d is a parameter which controls the relativestrengths of EC and VC which compete with each other.Hence, by tuning d and n, one can obtain a quantumphase transition between a ferromagnetic phase (domi-nated by exchange) and a paramagnetic phase (dominatedby the vertex corrections). This is a crucial result comingout from our study of the spin degree of freedom in theexciton condensate.For all the results presented in this paper we scalethe length to the three-dimensional Bohr radius a ­h¯2eympe2, where mp ­ memhysme 1 mhd is the excitonreduced mass. Energies are taken with origin at thebottom of the conduction band and are given in units ofthe 2D Rydberg jE2D0 j ­ 2e2yea.Figure 1 shows chemical potentials m6 as a function ofthe density for P ­ 0.8 and three values of the separationd. m6 are always increasing with n because exciton-exciton interaction turns out to be globally repulsive.That means that repulsive Hartree and VC terms in theenergy dominate on the attractive EC. The essentialpoint is the behavior of d ­ m1 2 m2, which is adecreasing function of d for moderate densities, and itcan be positive or negative. This is shown more clearlyin the inset. The first conclusion is that d is in therange of meV and is consequently an experimentallyaccessible quantity [12,13]. Since the Hartree term isspin independent, it does not contribute to d which onlydepends on EC and VC. As the e-h separation increases,the VC decreases while the EC increases as a consequenceof a larger exciton size that is implied by the increase ofthe overlap between excitons. Therefore, for moderatedensities there is, for a critical value dcr . 0.12a .10 Å, a change of sign in the measurable quantity d.Within our numerical precision, this critical separationturns out to be independent on P. For d , dcr VCdominates on EC giving d . 0 while for d . dcr ECdominates and d , 0. It must be stressed that the valueof dcr is quite low and the change of sign of thesplitting effect could be observable, for example, applyingmoderately high external electric fields.Figure 2 shows the phase diagram of the sign of din terms of the physical magnitudes of interest in theproblem: na2 and dya. For high d or n the kineticenergy dominates the e-h attraction and the e-h pairsare not bound having a positive chemical potential.FIG. 1. Majority m1 (continuous lines) and minority m2(dashed lines) chemical potentials as a function of the densityn for d ­ 0.0, 0.15, and 0.30. The inset shows the splittingd ­ m1 2 m2 as a function of the e-h separation d forna2 ­ 0.15 and P ­ 0.5 and 0.8.4811VOLUME 78, NUMBER 25 P HY S I CA L REV I EW LE T T ER S 23 JUNE 1997FIG. 2. Phase diagram for both the sign of d ­ m1 2 m2and the polarization Pme of the condensate (see text). Shadedregion corresponds to a condensate of unbounded e-h pairs.Therefore, experimental determination of the splittingmust be performed in the more accessible region of lowd and n where excitons are well defined. The importantpoint is that the diagram also gives information about thepolarization of the condensate. This is easily understoodby realizing that the derivative of the total energy per pairE, in the thermodynamic limit, isµ›E›P¶N; EsN1 1 1, N21d 2 EsN1, N2d ­ d .(7)Since for fixed d and n, d does not change sign whenP varies, the derivative (7) is a monotonous function ofP. The region of the diagram with positive splitting cor-responds to an increase of the total energy E with P,i.e., the system has minimum energy when it is unpolari-zed, Pme ­ 0. However, in the region with d , 0, thederivative (7) is also negative indicating that the systemshould have minimum energy when completely polarized,Pme ­ 1. For d , dcr VC always dominates and the un-polarized state d . 0 has the minimum energy. For in-creasing separation d, the excitons are less strongly boundand EC dominates allowing the appearance of a polarizedphase with d , 0. It is difficult to predict whether, foradequate values of d and n, the spins of the condensatewould spontaneously polarize or not because this processshould occur in the presence of spin relaxation mecha-nisms that we have not considered, as mentioned above.Therefore, the best way of experimentally analyzing the4812phase diagram is to measure the sign of d instead of look-ing for magnetic properties. The sign of d would remainas given by Fig. 2 even if other physical contributionswere to be included in our analysis. Available experi-ments [12,13] are for d ­ 0 and, in the whole range ofexperimentally accessible densities, always d results to bepositive as it happens in our phase diagram.In summary, we have studied a condensate of 2D ex-citons having different spin orientations. We have shownthat the spin degree of freedom plays a very importantand nontrivial role in exciton condensation. We haveanalyzed the case of resonant excitation finding that theground state energy depends strongly on spin polariza-tion. When electron and hole components are separatedfrom each other, the relative importance of exchange andvertex correction interactions can be altered. This causesa change of sign in d. This splitting between the chemi-cal potentials of different exciton components can be mea-sured by time-resolved spectroscopy [12,13]. The sign ofd is related to the minimum energy configuration of thecondensate, i.e., spin polarized or spin unpolarized. Allthese features are shown and discussed in a phase dia-gram in n and d.We thank L. Viña and L. Brey for useful discus-sions. This work was supported in part by CICYT ofSpain under Contract No. MAT 94-0982-C02-01 and bythe Comunidad Autonoma de Madrid under ContractNo. AE00330y95.[1] The third component of the angular momentum sMd isusually denoted in the field as the spin.[2] A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975).[3] L. V. Keldysh and Yu.V. Kopaev, Fiz. Tverd. Tela 6, 2791(1964) [Sov. Phys. Solid State 6, 2219 (1965)].[4] D. Jerome et al., Phys. Rev. 158, 462 (1967); B. I.Halperin and T.M. Rice, Solid State Physics (Academic,New York, 1968), Vol. 21, p. 115; R. Zimmermann, Phys.Status Solidi B 76, 191 (1976).[5] C. Comte and P. Nozières, J. Phys. (Paris) 43, 1069(1982); P. Nozières and C. Comte, ibid. 43, 1083 (1982).[6] H. Haug et al., Prog. Quantum Electron. 9, 3 (1984).[7] C. Comte and G. Mahler, Phys. Rev. B 34, 7164 (1986).[8] X. Zhu et al., Phys. Rev. Lett. 74, 1633 (1995).[9] X. Naveh et al., Phys. Rev. Lett. 77, 900 (1996);T. Portengen et al., Phys. Rev. Lett. 76, 3384 (1996);H. Chu and Y. C. Chang, Phys. Rev. B 54, 5020 (1996).[10] G. Mahan, Many Particle Physics (Plenum, New York,1982).[11] Along this paper we take N1 $ N2.[12] T. C. Damen et al., Phys. Rev. Lett. 67, 3432 (1991).[13] L. Viña et al., Phys. Rev. B 54, R8317 (1996).[14] J. Shah, Hot Carriers in Semiconductor Nanostructures(Academic, San Diego, 1992).[15] J. Fernández-Rossier and C. Tejedor (unpublished).[16] M. Z. Maialle et al., Phys. Rev. B 47, 15 776 (1993).[17] J. Fernández-Rossier et al., Phys. Rev. B 54, 11 582(1996).

Spin degree of freedom in two dimensional exciton condensates

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