- Email: [email protected]
Exciton–exciton interaction in coupled quantum wells
Solid State Communications 144 (2007) 395–398 www.elsevier.com/locate/ssc
Exciton–exciton interaction in coupled quantum wells Roland Zimmermann ∗ , Christoph Schindler Institut f¨ur Physik der Humboldt-Universit¨at zu Berlin, Newtonstr. 15, 12489 Berlin, Germany Received 15 February 2007; received in revised form 27 April 2007; accepted 1 July 2007 by the Guest Editors Available online 19 August 2007
Abstract The exciton–exciton interaction is investigated for spatially indirect excitons in coupled quantum wells. The Hartree–Fock and Heitler–London approaches are improved by a full two-exciton calculation including the van der Waals effect. Using these potentials for the singlet and triplet channel, the two-body scattering matrix is calculated and employed to derive a modified relation between exciton density and blue shift. Such a relation is of central importance for gauging exciton densities on the way toward Bose condensation. c 2007 Elsevier Ltd. All rights reserved.
PACS: 78.67.DE; 71.35.Gg; 71.35:Lk Keywords: A. Quantum wells; D. Optical properties; D. Exciton–exciton interaction
1. Introduction Coupled quantum wells (CQW) are among the promising systems for reaching excitonic Bose condensation, in particular due to the extremely long radiative lifetimes of the spatially indirect excitons therein [1]. They exhibit a strong dipolar repulsion which complicates the theoretical analysis as theories for weakly interacting Bosons are not applicable. On the other hand, the mutual repulsion leads to a distinct blue shift of the excitonic photoluminescence under high excitation, and can be used to infer the density reached in the experiment. However, the most naive approach based on a Hartree–Fock argument leads to a quite inadequate guess. In the present work, we show that the exciton density needed to reach a given blue shift would be underestimated by a factor as much as 10! In Section 2 we derive the exciton–exciton (XX) interaction from the underlying electron–hole Hamiltonian. Seminal papers by Usui and others [2] have restricted this finally to a plain Hartree–Fock (HF) level, and the well-cited result for the blue shift of ∆ = (26π/3)Ry ∗ n X a 3B for (direct) excitons in the bulk has been derived. Only recently, the failure of such a HF treatment without proper normalization of the two-exciton wave function has been pointed out by Okumura and Ogawa [3]. ∗ Corresponding author.
E-mail address: [email protected] (R. Zimmermann). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.07.044
We show that this ‘composite particle effect’ is nothing else other than the Heitler–London approach known from two-atom molecules, and improve towards a full solution of the twoexciton problem. These more accurate XX potentials (in the singlet and triplet channels) now contain the van der Waals effect. In Section 3, we use these potentials for calculating the XX scattering amplitude. The resulting quasiparticle self energy (linear in density) gives direct access to the exciton blue shift. 2. Exciton–exciton interaction revisited The single exciton wave function in a CQW is factorized into confinement functions for electron and hole and relative motion of the 1s ground state φ(r) (all vectors are understood as inplane ones). The single exciton problem is solved numerically with the potential veh (r) containing the confinement form factors (for more details see Ref. [4]). The parameter values refer to a CQW used by Snoke [5]: Two GaAs quantum wells of 10 nm width each are separated by a 4 nm wide barrier of Al0.3 Ga0.7 As. A bias field of 36 kV/cm2 places the spatially indirect exciton (binding energy B X = 3.5 meV) at a transition energy 43 meV below the direct one. The interaction between two indirect excitons contains the remaining Coulomb interactions which are not used to build the excitons, vtot (r) = vee (r) + vhh (r) − 2veh (r)
396
R. Zimmermann, C. Schindler / Solid State Communications 144 (2007) 395–398
⇒
e2 4π 0 S
2 2 . −√ r r 2 + d2
(1)
The second expression in Eq. (1) refers to the strict 2D limit with a separation d between electron and hole layers (point charge model). At large distances, the well-known dipole behavior evolves, (e2 d 2 )/(4π 0 S r 3 ). Integrated over space, the point charge model has a potential strength of Uref = de2 /(0 S ). In the most simple approach, this leads to a global repulsion between excitons producing a blue shift proportional to the exciton density, ∆ref = de2 /(0 S )n X , sometimes called ‘capacitor formula’. First, let us follow the Heitler–London approach by constructing a properly symmetrized two-exciton wave function as 1 φ(r1 )φ(r2 − R) ± φ(r1 − R)φ(r2 ) p Ψ s,t (r1 , r2 ) = √ (2) 2 1 ± O 2 (R) R with the overlap integral O(R) = drφ(r)φ(r − R). For simplicity, the hole is assumed to be much heavier than the electron, and R denotes the hole distance. Here and in the following the upper index s stands for singlet (the spatial part of the two-exciton wave function is symmetric with respect to the interchange of the two electrons, upper sign in Eq. (2)), while t denotes the triplet state (spatially antisymmetric, lower sign). Forming the expectation value of the standard electron–hole Hamiltonian with Eq. (2), we obtain an effective interaction U Hs,tL (R) =
Ud (R) ∓ Ux (R) , 1 ± O 2 (R)
(3)
where the direct and the exchange potentials are given by Z Ud (R) = + drdr0 vee (r − r0 )φ 2 (r)φ 2 (r0 − R) Z + vhh (R) − 2 drveh (r)φ 2 (r − R), (4) Z Ux (R) = − drdr0 vee (r − r0 )φ(r)φ(r − R)φ(r0 )φ(r0 − R) Z − O 2 (R)vhh (R) + 2O(R) drveh (r)φ(r)φ(r − R). (5) This level of approximation has been recently introduced into exciton physics by Okumura and Ogawa and evaluated for bulk excitons [3]. It represents an essential improvement over the standard Hartree–Fock approach [6] which neglects completely the normalization denominator in Eq. (2) and consequently in Eq. (3). As shown in Fig. 1, the HF approach even misses the proper placement of the singlet potential below the triplet one! Next, we go one step further by solving the two-exciton problem numerically (still in the heavy-hole limit, treating R as external parameter), " h¯ 2 − ∆r1 + ∆r2 − veh (r1 ) − veh (r2 − R) 2m e # − veh (r1 − R) − veh (r2 ) + vee (r1 − r2 ) + vhh (R)
Fig. 1. Exciton–exciton interaction potentials for a CQW with d = 14 nm well separation in the singlet (thick curves) and triplet channels (thin curves). The inset shows the asymptotic range on an enlarged scale.
Fig. 2. Electron density (central cut) in the two-exciton wave functions at a hole distance of R = 35 nm marked by an arrow. Dashed curves — Heitler–London, full curves — full calculation.
× Ψ s,t (r1 , r2 ) = Λs,t (R)Ψ s,t (r1 , r2 ).
(6)
The proper symmetry Ψ s,t (r1 , r2 ) = ±Ψ s,t (r2 , r1 ) is imposed as boundary condition. The eigenvalue is decomposed into the energy of two separate excitons and an effective XX potential, Λs,t (R) = −2B X + U s,t (R). This contains now the van der Waals effect which lowers the energy by a mutual deformation of the one-exciton orbitals. This energetic gain is seen (Fig. 1) but the attraction is not able to overcome the dipolar repulsion at large distances (inset of Fig. 1). This finding invalidates a possible explanation [7] of the bead formation on the outer luminescence ring of CQWs [8]. In Fig. 2 we display the electron density in the two-exciton wave function which evidences the (repulsive) correlation effect on the electrons when going from Heitler–London to the full calculation. With these XX interaction potentials we are now able to construct the many-exciton (bosonic) Hamiltonian Z X Ď − h¯ 2 Ψs (R) ∆R Ψs (R) H = dR 2M s Z 1 + dRdR0 U t (R − R0 ) 2
R. Zimmermann, C. Schindler / Solid State Communications 144 (2007) 395–398
×
X
Ď
397
Ď
Ψs (R)Ψs 0 (R0 )Ψs 0 (R0 )Ψs (R)
ss 0
Z 1 dRdR0 U t (R − R0 ) − U s (R − R0 ) 4 n Ď X Ď × Θ(s · s 0 ) Ψs (R)Ψs 0 (R0 )Ψs (R0 )Ψs 0 (R) +
ss 0 Ď
Ď
− Ψs (R)Ψs 0 (R0 )Ψs 0 (R0 )Ψs (R) o Ď Ď + 1 − 2δs,s 0 Ψs (R)Ψ−s (R0 )Ψ−s 0 (R0 )Ψs 0 (R) ,
(7)
with the exciton mass M = m e + m h in the kinetic term (first line). The exciton spin index s = se + Jh is the sum of electron spin (se = ±1/2) and heavy-hole angular momentum (Jh = ±3/2). Consequently, s runs over four values s = ±1 (bright states) and s = ±2 (dark states). The second and third lines in Eq. (7) are the standard spin-conserving XX interaction which can be written as |R, s; R0 , s 0 i → |R, s; R0 , s 0 i. The remaining lines contain exciton spin flips. Electron exchange results e.g. in |R, +1; R0 , −1i → |R, +2; R0 , −2i, while hole exchange plays no role for m h → ∞. The step function in the fifth line of Eq. (7) assures that s and s 0 have the same sign. Consequently, a term like |R, +1; R0 , −1i → |R, −1; R0 , +1i is absent. The many-exciton Hamiltonian derived by Ben-Tabou and Laikhtman [9] was restricted to Hartree–Fock potentials and contact interaction in the exchange terms, and agrees with Eq. (7) in this case. 3. Scattering amplitude and exciton blue shift The repulsive XX scattering in the low-density limit is now investigated using the two-body T matrix, s,t hk T s,t (Ω ) k00 i = Uk−k 00 −
X
s,t Uk−k 0
k0
2k0 − h¯ Ω
hk0 T s,t (Ω ) k00 i,
(8)
where k = h¯ 2 k 2 /(2M) is the single exciton dispersion. The T matrix enters the self energy as Boson-direct (first term) and Boson-exchange (second term) X Σk0 (k0 ) = 4 [hk|T D (k 2 )|ki k
+ hk|T X (k 2 )| − ki]n(k0 +2k ),
(9)
where T D = 3T t + T s and T X = (3/2)T t − (1/2)T s follow from the spin summations (see Ref. [4] for details on the T matrix approach). n() is the exciton distribution function, later on taken as low-density Maxwell–Boltzmann P expression. The total exciton density is given by n X = 4 k n(k ) since we have assumed complete spin equilibration. Only the on-shell T matrix (or complex scattering amplitude) is needed in Eq. (9) (h¯ Ω = h¯ 2 k 2 /M). According to scattering theory [10], this can be expressed via hk|T s,t (k 2 )| ± ki =
+∞ h¯ 2 X 4 (±1)m s,t M m=−∞ i − cot δm (k)
(10)
Fig. 3. Real part of the total scattering amplitude (on-shell T matrix) as a function of momentum. The Maxwell distribution (kn(2k )) is displayed for T = 6 K showing the relevance of the momentum range used. s,t exclusively through the phase shifts δm (k). They follow from the asymptotics of the scattering wave function which has to be calculated numerically using the two-exciton Schr¨odinger equation, 1 d d m2 M − r + 2 + 2 U s,t (r ) ψms,t (r ) = k 2 ψms,t (r ). (11) r dr dr r h¯
Results of the calculation for the band edge (k0 = 0) are displayed in Fig. 3 where the total scattering amplitude (square bracket in Eq. (9)) is shown as a function of momentum. As is specific for two dimensions, it starts with a logarithmic behavior −1/ log(k). A restriction to s-wave scattering (m = 0, dashed curve) as often used in the Bose gas theory would be a rather poor estimate. The overall values reached are much below the reference value of the capacitor formula (corresponding to “1” on the abscissa). This can be explained by a strong depletion of the exciton gas around a given exciton which is forced by the strong repulsion. Therefore, the resulting blue shift is much smaller than expected, which is quantified by a correction factor f (T ) in ∆ = ReΣ0 (0) = d
e2 n X f (T ), 0 S
(12)
depending on temperature T not very strongly ( f (2 K) = 0.08, f (6 K) = 0.10). An imaginary part of similar size comes out as well, which can be taken as a rough estimate of the scatteringinduced exciton line broadening. Working with a finite exciton mass M in Eq. (11) but still using an XX potential derived in the limit m h → ∞ might appear unjustified. As a numerical check we have calculated the biexciton (exciton molecule) for a single quantum well (SQW) without electric field. In this case, the singlet potential has an attractive part U s (R) < 0 which is strong enough to afford a bound state. The replacement in Eq. (11) is k 2 → −B X X M/ h¯ 2 with the biexciton binding energy B X X . For a 10 nm wide AlGaAs SQW, we found B X X = 1.1 meV while the calculated exciton binding energy was B X = 9.5 meV. The so-called Haynes factor B X X /B X = 0.12 is in reasonable agreement with variational calculations of the biexciton energy [11]. We
398
R. Zimmermann, C. Schindler / Solid State Communications 144 (2007) 395–398
have performed a further test within the strict 2D limit (zero well width) where the mass ratio σ = m e /m h is the only nontrivial parameter. Our present approach is correct for σ = 0 but worst at σ = 1 with an error of 12% for the Haynes factor [12]. For σ = 0.3 which is the relevant mass ratio for the heavyhole exciton in GaAs quantum wells, the error invoked is about 8% [13]. Therefore, we conclude that the present method of mapping the original four-particle problem to an effective twoparticle one is well justified if the holes are distinctly heavier than the electrons. References [1] L.V. Butov, A. Zrenner, G. Abstreiter, G. B¨ohm, G. Weimann, Phys. Rev. Lett. 73 (1994) 304; V. Negoita, D.W. Snoke, K. Eberl, Phys. Rev. B 60 (1999) 2661. [2] T. Usui, Progr. Theor. Phys. 23 (1959) 787; L.V. Keldysh, A.N. Kozlov, Sov. Phys. JETP 27 (1968) 521;
E. Hanamura, H. Haug, Phys. Rep. 33 (1977) 209. [3] S. Okumura, T. Ogawa, Phys. Rev. B 65 (2001) 035105. [4] R. Zimmermann, Phys. Status Solidi (b) 243 (2006) 2358. [5] D.W. Snoke, Y. Liu, Z. V¨or¨os, L. Pfeiffer, K. West, Solid State Comm. 134 (2005) 37. [6] C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, Phys. Rev. B 58 (1998) 7926; Jun-ichi Inoue, T. Brandes, Phys. Rev. B 61 (2000) 2863; G. Rochat, C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, Phys. Rev. B 61 (2000) 13856. [7] V.I. Sugakov, Solid State Comm. 134 (2005) 63; C.S. Liu, H.G. Luo, W.C. Wu, J. Phys. Cond. Mat. 18 (2006) 9659. [8] L.V. Butov, A.C. Gossard, D.S. Chemla, Nature 418 (2002) 751; D. Snoke, S. Denev, Y. Liu, L. Pfeiffer, K. West, Nature 418 (2002) 754. [9] S. Ben-Tabou de-Leon, B. Laikhtman, Phys. Rev. B 63 (2001) 125306. [10] S.A. Morgan, M.D. Lee, K. Burnett, Phys. Rev. A 65 (2002) 022706. [11] R. Denschlag, R.v. Baltz, Phys. Status Solidi (b) 215 (1999) 287. [12] M.Y.J. Tan, N.D. Drummond, R.J. Needs, Phys. Rev. B 71 (2005) 033303. [13] J. Usukura, Y. Suzuki, K. Varga, Phys. Rev. B 59 (1999) 5652.
Exciton–exciton interaction in coupled quantum wells Roland Zimmermann ∗ , Christoph Schindler Institut f¨ur Physik der Humboldt-Universit¨at zu Berlin, Newtonstr. 15, 12489 Berlin, Germany Received 15 February 2007; received in revised form 27 April 2007; accepted 1 July 2007 by the Guest Editors Available online 19 August 2007
Abstract The exciton–exciton interaction is investigated for spatially indirect excitons in coupled quantum wells. The Hartree–Fock and Heitler–London approaches are improved by a full two-exciton calculation including the van der Waals effect. Using these potentials for the singlet and triplet channel, the two-body scattering matrix is calculated and employed to derive a modified relation between exciton density and blue shift. Such a relation is of central importance for gauging exciton densities on the way toward Bose condensation. c 2007 Elsevier Ltd. All rights reserved.
PACS: 78.67.DE; 71.35.Gg; 71.35:Lk Keywords: A. Quantum wells; D. Optical properties; D. Exciton–exciton interaction
1. Introduction Coupled quantum wells (CQW) are among the promising systems for reaching excitonic Bose condensation, in particular due to the extremely long radiative lifetimes of the spatially indirect excitons therein [1]. They exhibit a strong dipolar repulsion which complicates the theoretical analysis as theories for weakly interacting Bosons are not applicable. On the other hand, the mutual repulsion leads to a distinct blue shift of the excitonic photoluminescence under high excitation, and can be used to infer the density reached in the experiment. However, the most naive approach based on a Hartree–Fock argument leads to a quite inadequate guess. In the present work, we show that the exciton density needed to reach a given blue shift would be underestimated by a factor as much as 10! In Section 2 we derive the exciton–exciton (XX) interaction from the underlying electron–hole Hamiltonian. Seminal papers by Usui and others [2] have restricted this finally to a plain Hartree–Fock (HF) level, and the well-cited result for the blue shift of ∆ = (26π/3)Ry ∗ n X a 3B for (direct) excitons in the bulk has been derived. Only recently, the failure of such a HF treatment without proper normalization of the two-exciton wave function has been pointed out by Okumura and Ogawa [3]. ∗ Corresponding author.
E-mail address: [email protected] (R. Zimmermann). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.07.044
We show that this ‘composite particle effect’ is nothing else other than the Heitler–London approach known from two-atom molecules, and improve towards a full solution of the twoexciton problem. These more accurate XX potentials (in the singlet and triplet channels) now contain the van der Waals effect. In Section 3, we use these potentials for calculating the XX scattering amplitude. The resulting quasiparticle self energy (linear in density) gives direct access to the exciton blue shift. 2. Exciton–exciton interaction revisited The single exciton wave function in a CQW is factorized into confinement functions for electron and hole and relative motion of the 1s ground state φ(r) (all vectors are understood as inplane ones). The single exciton problem is solved numerically with the potential veh (r) containing the confinement form factors (for more details see Ref. [4]). The parameter values refer to a CQW used by Snoke [5]: Two GaAs quantum wells of 10 nm width each are separated by a 4 nm wide barrier of Al0.3 Ga0.7 As. A bias field of 36 kV/cm2 places the spatially indirect exciton (binding energy B X = 3.5 meV) at a transition energy 43 meV below the direct one. The interaction between two indirect excitons contains the remaining Coulomb interactions which are not used to build the excitons, vtot (r) = vee (r) + vhh (r) − 2veh (r)
396
R. Zimmermann, C. Schindler / Solid State Communications 144 (2007) 395–398
⇒
e2 4π 0 S
2 2 . −√ r r 2 + d2
(1)
The second expression in Eq. (1) refers to the strict 2D limit with a separation d between electron and hole layers (point charge model). At large distances, the well-known dipole behavior evolves, (e2 d 2 )/(4π 0 S r 3 ). Integrated over space, the point charge model has a potential strength of Uref = de2 /(0 S ). In the most simple approach, this leads to a global repulsion between excitons producing a blue shift proportional to the exciton density, ∆ref = de2 /(0 S )n X , sometimes called ‘capacitor formula’. First, let us follow the Heitler–London approach by constructing a properly symmetrized two-exciton wave function as 1 φ(r1 )φ(r2 − R) ± φ(r1 − R)φ(r2 ) p Ψ s,t (r1 , r2 ) = √ (2) 2 1 ± O 2 (R) R with the overlap integral O(R) = drφ(r)φ(r − R). For simplicity, the hole is assumed to be much heavier than the electron, and R denotes the hole distance. Here and in the following the upper index s stands for singlet (the spatial part of the two-exciton wave function is symmetric with respect to the interchange of the two electrons, upper sign in Eq. (2)), while t denotes the triplet state (spatially antisymmetric, lower sign). Forming the expectation value of the standard electron–hole Hamiltonian with Eq. (2), we obtain an effective interaction U Hs,tL (R) =
Ud (R) ∓ Ux (R) , 1 ± O 2 (R)
(3)
where the direct and the exchange potentials are given by Z Ud (R) = + drdr0 vee (r − r0 )φ 2 (r)φ 2 (r0 − R) Z + vhh (R) − 2 drveh (r)φ 2 (r − R), (4) Z Ux (R) = − drdr0 vee (r − r0 )φ(r)φ(r − R)φ(r0 )φ(r0 − R) Z − O 2 (R)vhh (R) + 2O(R) drveh (r)φ(r)φ(r − R). (5) This level of approximation has been recently introduced into exciton physics by Okumura and Ogawa and evaluated for bulk excitons [3]. It represents an essential improvement over the standard Hartree–Fock approach [6] which neglects completely the normalization denominator in Eq. (2) and consequently in Eq. (3). As shown in Fig. 1, the HF approach even misses the proper placement of the singlet potential below the triplet one! Next, we go one step further by solving the two-exciton problem numerically (still in the heavy-hole limit, treating R as external parameter), " h¯ 2 − ∆r1 + ∆r2 − veh (r1 ) − veh (r2 − R) 2m e # − veh (r1 − R) − veh (r2 ) + vee (r1 − r2 ) + vhh (R)
Fig. 1. Exciton–exciton interaction potentials for a CQW with d = 14 nm well separation in the singlet (thick curves) and triplet channels (thin curves). The inset shows the asymptotic range on an enlarged scale.
Fig. 2. Electron density (central cut) in the two-exciton wave functions at a hole distance of R = 35 nm marked by an arrow. Dashed curves — Heitler–London, full curves — full calculation.
× Ψ s,t (r1 , r2 ) = Λs,t (R)Ψ s,t (r1 , r2 ).
(6)
The proper symmetry Ψ s,t (r1 , r2 ) = ±Ψ s,t (r2 , r1 ) is imposed as boundary condition. The eigenvalue is decomposed into the energy of two separate excitons and an effective XX potential, Λs,t (R) = −2B X + U s,t (R). This contains now the van der Waals effect which lowers the energy by a mutual deformation of the one-exciton orbitals. This energetic gain is seen (Fig. 1) but the attraction is not able to overcome the dipolar repulsion at large distances (inset of Fig. 1). This finding invalidates a possible explanation [7] of the bead formation on the outer luminescence ring of CQWs [8]. In Fig. 2 we display the electron density in the two-exciton wave function which evidences the (repulsive) correlation effect on the electrons when going from Heitler–London to the full calculation. With these XX interaction potentials we are now able to construct the many-exciton (bosonic) Hamiltonian Z X Ď − h¯ 2 Ψs (R) ∆R Ψs (R) H = dR 2M s Z 1 + dRdR0 U t (R − R0 ) 2
R. Zimmermann, C. Schindler / Solid State Communications 144 (2007) 395–398
×
X
Ď
397
Ď
Ψs (R)Ψs 0 (R0 )Ψs 0 (R0 )Ψs (R)
ss 0
Z 1 dRdR0 U t (R − R0 ) − U s (R − R0 ) 4 n Ď X Ď × Θ(s · s 0 ) Ψs (R)Ψs 0 (R0 )Ψs (R0 )Ψs 0 (R) +
ss 0 Ď
Ď
− Ψs (R)Ψs 0 (R0 )Ψs 0 (R0 )Ψs (R) o Ď Ď + 1 − 2δs,s 0 Ψs (R)Ψ−s (R0 )Ψ−s 0 (R0 )Ψs 0 (R) ,
(7)
with the exciton mass M = m e + m h in the kinetic term (first line). The exciton spin index s = se + Jh is the sum of electron spin (se = ±1/2) and heavy-hole angular momentum (Jh = ±3/2). Consequently, s runs over four values s = ±1 (bright states) and s = ±2 (dark states). The second and third lines in Eq. (7) are the standard spin-conserving XX interaction which can be written as |R, s; R0 , s 0 i → |R, s; R0 , s 0 i. The remaining lines contain exciton spin flips. Electron exchange results e.g. in |R, +1; R0 , −1i → |R, +2; R0 , −2i, while hole exchange plays no role for m h → ∞. The step function in the fifth line of Eq. (7) assures that s and s 0 have the same sign. Consequently, a term like |R, +1; R0 , −1i → |R, −1; R0 , +1i is absent. The many-exciton Hamiltonian derived by Ben-Tabou and Laikhtman [9] was restricted to Hartree–Fock potentials and contact interaction in the exchange terms, and agrees with Eq. (7) in this case. 3. Scattering amplitude and exciton blue shift The repulsive XX scattering in the low-density limit is now investigated using the two-body T matrix, s,t hk T s,t (Ω ) k00 i = Uk−k 00 −
X
s,t Uk−k 0
k0
2k0 − h¯ Ω
hk0 T s,t (Ω ) k00 i,
(8)
where k = h¯ 2 k 2 /(2M) is the single exciton dispersion. The T matrix enters the self energy as Boson-direct (first term) and Boson-exchange (second term) X Σk0 (k0 ) = 4 [hk|T D (k 2 )|ki k
+ hk|T X (k 2 )| − ki]n(k0 +2k ),
(9)
where T D = 3T t + T s and T X = (3/2)T t − (1/2)T s follow from the spin summations (see Ref. [4] for details on the T matrix approach). n() is the exciton distribution function, later on taken as low-density Maxwell–Boltzmann P expression. The total exciton density is given by n X = 4 k n(k ) since we have assumed complete spin equilibration. Only the on-shell T matrix (or complex scattering amplitude) is needed in Eq. (9) (h¯ Ω = h¯ 2 k 2 /M). According to scattering theory [10], this can be expressed via hk|T s,t (k 2 )| ± ki =
+∞ h¯ 2 X 4 (±1)m s,t M m=−∞ i − cot δm (k)
(10)
Fig. 3. Real part of the total scattering amplitude (on-shell T matrix) as a function of momentum. The Maxwell distribution (kn(2k )) is displayed for T = 6 K showing the relevance of the momentum range used. s,t exclusively through the phase shifts δm (k). They follow from the asymptotics of the scattering wave function which has to be calculated numerically using the two-exciton Schr¨odinger equation, 1 d d m2 M − r + 2 + 2 U s,t (r ) ψms,t (r ) = k 2 ψms,t (r ). (11) r dr dr r h¯
Results of the calculation for the band edge (k0 = 0) are displayed in Fig. 3 where the total scattering amplitude (square bracket in Eq. (9)) is shown as a function of momentum. As is specific for two dimensions, it starts with a logarithmic behavior −1/ log(k). A restriction to s-wave scattering (m = 0, dashed curve) as often used in the Bose gas theory would be a rather poor estimate. The overall values reached are much below the reference value of the capacitor formula (corresponding to “1” on the abscissa). This can be explained by a strong depletion of the exciton gas around a given exciton which is forced by the strong repulsion. Therefore, the resulting blue shift is much smaller than expected, which is quantified by a correction factor f (T ) in ∆ = ReΣ0 (0) = d
e2 n X f (T ), 0 S
(12)
depending on temperature T not very strongly ( f (2 K) = 0.08, f (6 K) = 0.10). An imaginary part of similar size comes out as well, which can be taken as a rough estimate of the scatteringinduced exciton line broadening. Working with a finite exciton mass M in Eq. (11) but still using an XX potential derived in the limit m h → ∞ might appear unjustified. As a numerical check we have calculated the biexciton (exciton molecule) for a single quantum well (SQW) without electric field. In this case, the singlet potential has an attractive part U s (R) < 0 which is strong enough to afford a bound state. The replacement in Eq. (11) is k 2 → −B X X M/ h¯ 2 with the biexciton binding energy B X X . For a 10 nm wide AlGaAs SQW, we found B X X = 1.1 meV while the calculated exciton binding energy was B X = 9.5 meV. The so-called Haynes factor B X X /B X = 0.12 is in reasonable agreement with variational calculations of the biexciton energy [11]. We
398
R. Zimmermann, C. Schindler / Solid State Communications 144 (2007) 395–398
have performed a further test within the strict 2D limit (zero well width) where the mass ratio σ = m e /m h is the only nontrivial parameter. Our present approach is correct for σ = 0 but worst at σ = 1 with an error of 12% for the Haynes factor [12]. For σ = 0.3 which is the relevant mass ratio for the heavyhole exciton in GaAs quantum wells, the error invoked is about 8% [13]. Therefore, we conclude that the present method of mapping the original four-particle problem to an effective twoparticle one is well justified if the holes are distinctly heavier than the electrons. References [1] L.V. Butov, A. Zrenner, G. Abstreiter, G. B¨ohm, G. Weimann, Phys. Rev. Lett. 73 (1994) 304; V. Negoita, D.W. Snoke, K. Eberl, Phys. Rev. B 60 (1999) 2661. [2] T. Usui, Progr. Theor. Phys. 23 (1959) 787; L.V. Keldysh, A.N. Kozlov, Sov. Phys. JETP 27 (1968) 521;
E. Hanamura, H. Haug, Phys. Rep. 33 (1977) 209. [3] S. Okumura, T. Ogawa, Phys. Rev. B 65 (2001) 035105. [4] R. Zimmermann, Phys. Status Solidi (b) 243 (2006) 2358. [5] D.W. Snoke, Y. Liu, Z. V¨or¨os, L. Pfeiffer, K. West, Solid State Comm. 134 (2005) 37. [6] C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, Phys. Rev. B 58 (1998) 7926; Jun-ichi Inoue, T. Brandes, Phys. Rev. B 61 (2000) 2863; G. Rochat, C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, Phys. Rev. B 61 (2000) 13856. [7] V.I. Sugakov, Solid State Comm. 134 (2005) 63; C.S. Liu, H.G. Luo, W.C. Wu, J. Phys. Cond. Mat. 18 (2006) 9659. [8] L.V. Butov, A.C. Gossard, D.S. Chemla, Nature 418 (2002) 751; D. Snoke, S. Denev, Y. Liu, L. Pfeiffer, K. West, Nature 418 (2002) 754. [9] S. Ben-Tabou de-Leon, B. Laikhtman, Phys. Rev. B 63 (2001) 125306. [10] S.A. Morgan, M.D. Lee, K. Burnett, Phys. Rev. A 65 (2002) 022706. [11] R. Denschlag, R.v. Baltz, Phys. Status Solidi (b) 215 (1999) 287. [12] M.Y.J. Tan, N.D. Drummond, R.J. Needs, Phys. Rev. B 71 (2005) 033303. [13] J. Usukura, Y. Suzuki, K. Varga, Phys. Rev. B 59 (1999) 5652.