Pseudo diamagnetism of four component exciton condensates
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Pseudo diamagnetism of four component exciton condensates
Yuri G. Rubo and A. V. Kavokin
2, 3 Centro de Investigaci´on en Energ´ıa, Universidad Nacional Aut´onoma de M´exico, Temixco, Morelos, 62580, Mexico Groupe d’Etude des Semi-conducteurs, UMR 5650, CNRS–Universit´e Montpellier 2,Place Eug`ene Bataillon, 34095 Montpellier Cedex, France School of Physics and Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, UK (Dated: March 23, 2011)We analyze the spin structure of the ground state of four-component exciton condensates incoupled quantum wells as a function of spin-dependent interactions and applied magnetic field.The four components correspond to the degenerate exciton states characterized by ± ± ± ± PACS numbers: 78.67.-n, 71.36.+c, 42.25.Kb, 42.55.Sa
Introduction. —Cold exciton gases formed by indirectexcitons in coupled quantum wells (CQW) represent asolid state bosonic system very rich in new quantum co-herent phenomena [1]. Indirect excitons are semiconduc-tor crystal excitations with unique properties: they havelong lifetime and spin-relaxation time, can cool down tothe temperatures well below the temperature of quan-tum degeneracy, can travel over large distances beforerecombination, and can be in situ controlled by appliedvoltage [2–6]. The indirect excitons form a model systemboth for the studies of fundamental properties of lightand matter and for the development of conceptually newoptoelectronic devices [7]. Being formed by heavy holesand electrons, the indirect excitons may have four al-lowed spin projections to the axis normal to quantumwell plane: − −
1, +1, +2 [8]. The excitons with spinprojections − − Condensate at zero magnetic field —We characterizeXBEC by a four component order parameter ψ m with m = ± , ±
2. The indices m denote four allowed excitonspin projections on the structure axis. The Hamiltoniandensity H in the absence of applied magnetic field canbe written as ( ~ = 1) H = 12 M X m |∇ ψ m | − µn + 12 X m,m ′ V m,m ′ | ψ m | | ψ m ′ | + W ( ψ ∗ +2 ψ ∗− ψ +1 ψ − + ψ ∗ +1 ψ ∗− ψ +2 ψ − ) . (1)Here n = P m | ψ m | is the total polariton concentration, M is the exciton mass, and V m,m ′ = V m ′ ,m . In a generalcase, exciton-exciton interactions are described by fiveindependent constants which come from the amplitudeof spin-independent electron-hole interaction and ampli-tudes of electron-electron and hole-hole interactions inthe triplet (parallel spins) and singlet (antiparallel spins)configurations. In the following, we denote V m,m = V >
0, which describes interactions between particles havingidentical spins and accounts for the dipole-dipole and ex-change repulsion. The other parameters are V +2 , +1 = V − , − = V − V e , V +2 , − = V − , +1 = V − V h , and V +2 , − = V +1 , − = V − V x , where V e,h,x describe ex-change interactions between the particles with oppositespins: electrons, holes, and excitons, respectively. In re-alistic CQW the dipole-dipole repulsion dominates overthe exchange interaction constants, but the polarizationstate of exciton condensate is governed by small spin-dependent interaction parameters W and V x,e,h .The last term in (1) describes the mixing between darkand bright excitons. It appears due to possibility of scat-tering of two bright excitons into two dark ones and viceversa [11]. This term is switched off if one or more exci-ton spin components become empty. On the other hand,if all components are occupied, this term reduces the en-ergy of the condensate independently of the sign of W .This is assured by minimization of H over the phase fac-tor ( θ +2 + θ − − θ +1 − θ − ), where θ m is the phase of the m -th component of the order parameter: ψ m = A m e iθ m .In the following we set W > u = n ( W + V x + V e + V h ) / , (2a) u x = n ( W − V x + V e + V h ) / , (2b) u e = n ( W + V x − V e + V h ) / , (2c) u h = n ( W + V x + V e − V h ) / , (2d)as it is shown in Fig. 1, where we denoted u min =min { u x , u e , u h } . Note that it is u min that defines whichparticular two-component condensate (TCC) is realized.For example, if V x > W + V e + V h and u min = u x < V e > W + V x + V h and u min = u e <
0, the dark andbright components become circular with the same signof circular polarization. Finally, for u min = u h <
0, thedark and bright components are also circular, but of theopposite signs.In what follows, we consider the most interesting andpresumably most experimentally relevant case of a four-component condensate (FCC), where all parameters (2)
FIG. 1. Showing the state of the exciton condensate for dif-ferent values of interaction parameters defined in the text. are positive. The chemical potential in this case is µ = µ = V n − u . (3)The excitation spectrum of four-component conden-sate can be found in the usual way [12] by linearizing theGross-Pitaevskii equation generated by Hamiltonian (1)with respect to small plane-wave perturbation. The spec-trum consists in four branches. Three branches have theBogoliubov dispersion, ε = p ε ( k )[2 u + ε ( k )], where ε ( k ) = k / M and u takes the values of µ , u e , and u h .The fourth branch is gaped in k = 0 and the quasiparticleenergy is ε x ( k ) = p [ W n + ε ( k )][2 u x + ε ( k )] . (4)The gap ∆ = √ W nu x appears due to the mixing termof Hamiltonian (1). We note that its physical origin isthe same as of the gap in BCS superconductors: namely,the pare scattering . Effects of Zeeman splitting —Weak applied magneticfields affect the exciton condensate mainly due to addi-tion of the Zeeman splitting term H Z to the Hamiltonian, H Z = −
12 ( ω s + ω s ) , (5)where ω , = g , µ B B are the Zeeman splitting energiesfor single excitons and we introduced the pseudospins ofthe components s = | ψ +1 | − | ψ − | and s = | ψ +2 | −| ψ − | . The g -factors of dark and bright excitons are g = g h − g e and g = g h + g e , where g e and g h are g -factors of electrons and heavy-holes, respectively. Boththe values and the signs of g -factors can be different asthey depend substantially on the quantum well widths[13, 14]. Here and further we neglect the magnetic fieldeffect on internal orbital motion of electrons and holes inthe exciton which has been extensively studied in the past[15–18]. That orbital effect is independent on spin andresults in the exciton diamagnetic shift similar to the blueshift of atomic lines due to the Langevin diamagnetism.As we show below, the non-trivial behavior of the equi-librium state of the condensate in magnetic field is dueto the possibility of resonant scattering of dark to brightexcitons and vice versa described by the last term in theHamiltonian (1). Minimization of H + H Z over ψ ∗ m yieldsa relation between the amplitudes of the spin components( V x A − m + V e A l + V h A − l ) A m + W A − m A l A − l = (cid:18) V n − µ − ω m (cid:19) A m , (6)where l = ± , ± ω m = ± ω , for m = ± , ±
2, re-spectively. There are solutions to Eqs. (6) describingFCC, two-component (TCC) and one-component con-densate (OCC). We note that there is no solution withthree components—the equation corresponding to thesingle empty component cannot be satisfied due to the W -term in (6). Physically, it becomes clear if we re-member that each exciton is composed by an electron in+1 / − / / − / t ≡ t = A +1 A − A +2 A − , t − ≡ t = A +2 A − A +1 A − , (7)that make it possible to write the concentrations, n = A + A − and n = A + A − , and pseudospins of thebright and dark components as n , = W t , + V x − V e − V h W ( t + t ) + 2( V x − V e − V h ) n, (8) s , = ( W t , + V x ) ω , + ( V e − V h ) ω , ( W t + V x )( W t + V x ) − ( V e − V h ) n. (9)Substitution of these expressions back to definition (7)allows to link t and applied magnetic field by the equation s − t s = n − t n . (10)Since s , ∝ B this equation directly expresses magneticfield as a function of t . Finally, the change of the chemicalpotential of FCC in magnetic field is given by µ − µ = n W ( W + V e + V h − V x )( t − W ( t + 1) − t ( W + V e + V h − V x ) . (11)The system (6) can be easily solved for TCC and OCC.The signs of g -factors of bright and dark excitons de-fine which components remain occupied in high magneticfields. In what follows, we consider the case g > g > A , = 12 n + ω , − ω , V e , A − , = 0 , (12)and the chemical potential is µ − µ = n W + V x − V e + V h ) −
14 ( ω + ω ) . (13)In the region of very strong magnetic fields, the con-densate becomes one-component, A +1 = √ n , A − = A ± = 0. The chemical potential of OCC is µ − µ = n W + V x + V e + V h ) − ω . (14)The chemical potential of FCC is independent of mag-netic field in the case of equal g -factors g = g , since the (a) (b) (c) (d) FIG. 2. The magnetic-field dependencies of chemical poten-tials of four-component (green lines), two-component (bluelines), and one-component (red lines) exciton condensates.(the curves are also labeled by the number of components).Stable and metastable states of a uniform condensate are indi-cated by bold and thin parts of the curves, respectively. Theinteraction parameters are: (a) W = V x , V e = V h = 0 . V x , g /g = 0 .
5; (b) W = V x , V e = V h = 0 . V x , g /g = 0 . W = V x , V e = (3 / V x , V h = (5 / V h , g /g = 0 .
25; (d) W = 3 V x , V e = V h = 0 . V x , g /g = 0 . bright and dark excitons are polarized in the same wayand t = 1. However, the behavior of µ becomes nontrivialif g = g . In this case, if W + V x > V e + V h , the chemi-cal potential increases with magnetic field as one can seefrom (11). From the experimental point of it leads to ablue-shift of the emission line, i.e., to an apparent dia-magnetic effect. Unlike the conventional diamagnetism,this effect is only related to spin interactions and does notdepend on the orbital motion of electrons and holes. It isspecific to four-component exciton condensates and doesnot take place in two-component exciton-polariton con-densates [20, 21], where the chemical potential remainsindependent of the magnetic field up to some critical field.The effect we discuss can be referred to as pseudo-diamagnetism , since it is not related to the increase of thetotal energy of the system. In fact, the Zeeman energy(5) decreases quadratically with magnetic field, because s , ∝ B for small B [see (9)]. On the other hand, thechemical potential increases ∝ B for small B . Clearly,for weak fields this increase can be neglected comparedto the usual, orbital diamagnetism ∝ B . However, aswe show below, the additional pseudo-diamagnetic shiftbecomes dominant when the bare Zeeman energy ap-proaches the energy of exchange interactions in the exci-ton condensate and it can result in dramatic changes inthe polarization state of the system.The behavior of chemical potentials (11), (13), and(14) as functions of applied magnetic field is shown inFig. 2(a-d). For a small difference of g -factors the changesof the ground state of the condensate are continuous,as it is shown in Fig. 2(a,b). The chemical potentialof FCC slightly increases, and at some magnetic field itreaches the chemical potential of TCC. At this point theFCC transforms into TCC: the components A − and A − vanish. Subsequently, the TCC is transformed into OCCat the higher magnetic field, as it is shown in Fig. 2(b).Note also, that FCC can transform directly into OCC insome range of parameters.Very interestingly, in a wide range of parameters,namely, for a sufficiently large amplitude of the mixingterm W and strong difference of g -factors, the changeof the ground state of the condensate is discontinuous[see Fig. 2(c,d)]. In these cases, for a given magneticfield there are two solutions of Eq. (10) for parameter t corresponding to different polarization states of the con-densate. The state with a higher value of µ is metastable.The FCC disappears at B > B c , where B c is defined by dµ/dB = ∞ . If B reaches B c from below the FCC withfinite occupation of all components transforms discon-tinuously into OCC. The chemical potential jumps by afinite value in this case.The discontinuous change of the chemical potential ischaracteristic for a phase transition of the first-order. Inthis case one can expect formation of a mixed state of thesystem for B close to B c , where FCC and OCC with dif-ferent concentrations of excitons coexist, similarly to howit happens in the case of vapor-liquid phase transition.In conclusion, we have shown that applied magneticfield suppresses the mixing of dark and bight excitonsand leads to pseudo-diamagnetic increase of the chem-ical potential of exciton condensate, provided the darkand bright excitons possess different g -factors. The in-terplay between spin-dependent exciton-exciton interac-tions and Zeeman effect can lead to the first order transi-tion between four-component and one- or two-componentcondensates.We are grateful to B. L. Altshuler and M. M. Glazovfor valuable discussions. This work was supported in partby DGAPA-UNAM under Project No. IN112310 and bythe EU FP7 IRSES Project POLAPHEN. [1] L. V. Butov, A. C. Gossard, and D. S. Chemla, Nature , 751 (2002). [2] L. V. Butov, L. S. Levitov, A. V. Mintsev, B. D. 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