- Email: [email protected]
Coherent spin states and spin quantum beats in semiconductor microcavities
Available online at www.sciencedirect.com
Physica E 17 (2003) 329 – 334 www.elsevier.com/locate/physe
Coherent spin states and spin quantum beats in semiconductor microcavities Pierre Renuccia; b;∗ , Thierry Amandb , Xavier Marieb a Swiss
b Laboratoire
Federal Institute of Technology IPEQ-EPFL, CH-1015 Lausanne, Switzerland de Physique de la Mati%ere Condens&ee de Toulouse CNRS-INSA, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France
Abstract The present study evidences the in*uence of the strong light-matter coupling in semiconductor III–V microcavities on polariton spin dynamics at low excitation. Under resonant excitation, we observe a quenching of spin and alignment relaxation when the polariton is photon like. These behaviours are attributed, respectively, to the very small value of the long-range electron–hole exchange term of Coulomb interaction within the excitonic component of the quasi-particle and to the weakness of polariton–polariton Coulomb scattering via the inter-exciton short-range exchange interaction. Spin dynamics are also investigated under transverse magnetic 4eld. We observe an electron–hole spin correlation within the excitonic component of the quasi-particle under resonant excitation, and an increase of the absolute value of the electron e6ective transverse Land8e g-factor with the excitonic character of polaritons, which can be derived from a model taking into account only two classes of excitations in the lower polariton branch. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Polaritons; Coherent spin states; Spin dynamics; Quantum beats
1. Introduction Microcavity polariton dynamics is presently the subject of intense investigations, mainly in the regime of strong non-linear emission relative to polariton– polariton e=cient scattering [1]. However, much less studies have been devoted to the spin dynamics of these quasi-particles in the spontaneous emission regime. In our case, we will call polariton spin states the two excited states |Jz = ±1 with a total angular momentum Jz = ±1. We also de4ne ∗
Corresponding author. Laboratoire de Physique de la MatiBere Condens8ee de Toulouse CNRS-INSA, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France. Fax: +33-5-61-55-96-97. E-mail address: [email protected] (P. Renucci).
√ linear polariton states |X =(|JZ =+1+|J √ Z =−1)= 2 and |Y = (|JZ = +1 − |JZ = −1)=i 2 as coherent superpositions of polariton spin states. Moreover, we consider excitonic dark states |Jz = ±2 with total angular momentum Jz = ±2, that will play a key role under transverse magnetic 4eld. Spin coherence relies on the stability, on a time scale which can be longer than the optical dephasing time, of any linear superposition of two of these previously de4ned excited states. We propose here to analyze the phenomena relative to spin coherence through time and polarization-resolved photoluminescence under orientated optical pumping, with and without an applied transverse magnetic 4eld. In microcavities, in the strong coupling regime, some original spin dynamics behaviours are expected, since the lower branch dispersion curve is severely distorted
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00826-3
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P. Renucci et al. / Physica E 17 (2003) 329 – 334
and the excitonic weight of the quasi-particle can be tuned. In a 4rst part, we report experimental results on polariton spin and alignment relaxation dynamics under resonant and weak excitation [2], contrary to previous studies led in the strong non-linear emission regime [3,4]. We adapt the excitonic mechanisms [5,6] known in bare quantum wells to the polariton picture, in order to interpret the relaxation quenching observed both under circularly and linearly polarized light when the polariton is photon like. In the second part, we analyze polariton spin dynamics under transverse magnetic 4eld (Voigt con4guration), under resonant and non-resonant circularly polarized excitation. Up to now, some works have been performed in Faraday con4guration, leading to the observation of magnetopolaritons [7], Zeeman splitting [8], as well as an increase of Rabi splitting with magnetic 4eld [8]. Measurements in Voigt con4guration provide some complementary informations. It allows to evidence an electron–hole spin correlation within the excitonic component of polariton under resonant excitation. Through a model based on two di6erent classes of excitations in the polariton lower branch, we also deduce that the absolute value of e6ective transverse electron g-factor increases with the excitonic component of the polariton.
PL = (I X − I Y )=(I X + I Y ), where I X and I Y denote the two linear orthogonal components of luminescence. All the experiments are performed at a low temperature at, 12 or 1:7 K, in a super*uid helium bath. Three -microcavities have been investigated, one with four 12 nm wide Ga0:86 In0:14 As quantum wells with a Rabi splitting R ≈ 6 meV (MC1), and two others with a single 8 nm wide Ga0:95 In0:05 As quantum well with R ≈ 3:5 (MC2) and 3:7 meV (MC3). Under transverse magnetic 4eld, comparative studies have also been performed on a Ga0:94 In0:06 As bare quantum well (QW1). 3. Spin and alignment dynamics Fig. 1 displays the time-resolved circular polarization PC after a circularly polarized exciting pulse for MC2. We can distinguish two di6erent time behaviours for PC with respect to the detuning ( = EC − EX , with EX and EC relative, respectively, to the uncoupled exciton and photon cavity
2. Experimental technique and samples Spin dynamics is monitored through time and polarization resolved polariton secondary emission using the two-colours up-conversion scheme [9]. The time-resolution is limited by the laser pulse-width (∼ 1:5 ps) and the spectral resolution is about 3 meV. The excitation power is weak, in order not only to preserve the strong exciton-photon coupling, but also to ensure to stay in the non-stimulated emission regime [10]. The excitation angle is 8◦ (with respect to the sample growth axis (Oz), corresponding to an initial in-plane wave vector kp ≈ 104 cm−1 ; the detection is normal to the surface, with a small acceptance angle (about 3 × 10−3 Sr). The circular polarization degree of the emission is de4ned as PC =(I + −I − )=(I + +I − ), where I + (I − ) denote, respectively, the right (left) circularly polarized luminescence component. In the same way, we note the linear polarization degree
Fig. 1. MC2. Right axis, full square: circular polarization kinetics for = −3; 0; 4 meV at 12 K. Left axis, full line: total intensity. Vertical dashed lines indicate the temporal separation between the two kinetics regimes, de4ned by the change of the intensity slope (in log scale). These slopes are underlined by dashed lines.
P. Renucci et al. / Physica E 17 (2003) 329 – 334
0.6
M C3 0.4
δ = - 9 m eV 0.2 0
10
20
30
40
time (ps) Fig. 2. MC3. Right axis, full square: circular polarization kinetics for = −9 meV at 12 K. Left axis, full line: total intensity. The peak centred around t = 0 is due to the back-scattered laser on the surface sample.
energy modes). For ¡ 0 (Fig. 1a), there is a clear relaxation quenching. On the other hand, for ¿ 0 (Fig. 1b and c), the relaxation becomes e=cient, mainly during the second step of the emission kinetics. We measure a characteristic longitudinal spin relaxation time Ts1 ≈ 75 ps for MC2 at ¿ 0. The observation of the circular polarization relaxation quenching for ¡ 0 was con4rmed on MC3 (Fig. 2), its longer photon lifetime (8 ps) allows to make measurements on a larger time scale. We attribute the spin relaxation process to the spin *ip of the excitonic component. This theory, based on the simultaneous *ip of electron and hole spin, has been developed previously for bare QWs by Maialle et al. [5] and takes into account the electron–hole exchange, the long-range part being the dominant contribution. We generalize this result in the polariton picture [11,12]. This leads to 1 2 4 ∗ = ; (1) ex (k)|X (k)| (k): Ts1 Here ; ex (k) is proportional to the electron–hole long-range exchange matrix element for a given in-plane wave vector k; ∗ (k) represents the polariton momentum relaxation time, X (k) is the excitonic Hop4eld coe=cient, and the symbol denotes the average on the occupied polariton states within the lower branch. For ¡ 0, the polariton population is concentrated around k ≈ kp states in the k-space, as the acoustic phonon scatterings are quenched due to the distorted dispersion curve. An evaluation of each term of Eq. (1) [11,12] leads to a spin relaxation time
Linear polarization
0.8
1.0 Circular polarization
INTENSITY (arb.units)
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331
T = 12 K δ = -6 meV δ = 0 meV δ = 4 meV
0.8 0.6 0.4 0.2 0.0
0
10
20 time (ps)
30
40
Fig. 3. MC2. Linear polarization kinetics for = −6; 0; 4 meV at 12 K. Dashed lines are guides for the eyes.
Ts1 at least three orders of magnitude longer than the bare exciton one, mainly due to the weakness of the matrix element which is proportional to the small wave vector kp . This is in good agreement with the observed relaxation quenching. On the other hand, for ¿ 0, larger wave vectors are available after e=cient scatterings, which implies a stronger exchange matrix element (averaged on the whole lower branch) and a e=cient spin relaxation during the second step of the emission kinetics. This second step corresponds to the come-back of elementary excitations from large in-plane wave vectors excitonic states (with centre of mass wave vectors of the order of 105 cm−1 ) towards the centre of Brillouin zone. Actually, we thus mainly probe the exciton spin relaxation. This explains why we observe for MC2 no clear di6erence between the cases = 0 and 4 meV (Fig. 1b and c). To complete the previous study, we also performed similar experiments under linearly polarized excitation. Fig. 3 displays the time-resolved linear polarization of the MC2 microcavity following a linearly polarized exciting pulse for di6erent cavity detunings. We observe a quenching of the relaxation (characterized by the so-called spin transverse relaxation time Ts2 ) for ¡ 0. However, a fast linear polarization decay appears, during the 4rst step of kinetics, when increases. For = 0; Ts2 is about 15 ps, shorter than Ts1 (≈ 75 ps). As the linear polarization relaxation (also called alignment relaxation) becomes faster when the excitation power increases, we attribute the mechanism responsible for this behaviour to the mutual polariton spin-dependent Coulomb exchange via
(2)
Here V0 corresponds to the exciton–exciton exchange matrix element [10]. The evaluation of the term |X (kp )|8 gives thus some estimation of the strength of the process in the spontaneous regime. For = −6; 0; 4 meV, we 4nd 2 × 10−4 , 0.27, 0.70, respectively (MC2). So, at negative detuning, there is a strong reduction of the process e=ciency, due to the important photon component in the polariton, in good agreement with the alignment relaxation quenching. For ¿ 0 the Coulomb scattering process becomes more e=cient, as the exciton weight increases. 4. Spin dynamics under transverse magnetic eld Experiments under transverse magnetic 4eld usually provide some decisive informations on quantum coherences [13,14] and on electron–hole spin correlation [15]. Fig. 4a displays the polariton circular polarization oscillations under transverse magnetic 4eld of the MC2 microcavity emission for the cavity detuning = 0. Here the excitation is tuned to the InGaAs QW gap (non-resonant case). Note that the oscillations are symmetrical with respect to the time axis, and that their envelope coincides with the polariton circular polarization decay without magnetic 4eld. The beat pulsation ! depends linearly on the magnetic 4eld intensity (see inset in Fig. 4a). Fig. 4b displays the polarization oscillations under the same conditions but now the excitation is resonant with the lower polariton branch. One can note the asymmetry of the beats with respect to the time axis, since the circular polarization remains positive at early time delays. Fig. 5 shows also the polarization oscillations for = +4 meV. A
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B=0T B=2T model
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B=0T B=1T model
-1
1 ˙ V02 |X (kp )|8 : Ts2
δ =0
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ω (ps )
the exciton components of polaritons (the direct term of the Coulomb interaction is negligible in the range of wave vectors explored after a resonant excitation). This process was studied in bare QWs by Le Jeune et al. [6], and is known in microcavities as parametric scattering [10]. Due to the exchange selection rules, this scattering is a strong depolarizing process: a pair of polaritons (|X ; |X ) is scattered with an equal probability into a pair (|X ; |X ) or a pair (|Y ; |Y ). The transverse spin relaxation time Ts2 can be expressed as follows:
ω (p s )
P. Renucci et al. / Physica E 17 (2003) 329 – 334
Circular polarization
332
1.0
600
0.15 0.10 0.05 0.00
0.5
0
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2
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4
B (T) 0.0 (b)
0
100
200
300
time (ps) Fig. 4. MC2. Polariton circular polarization oscillations under transverse magnetic 4eld, and circularly polarized excitation (1:7 K). The polarization decay without magnetic 4eld is also shown. The detuning is = 0: (a) non-resonant case, B = 2 T. The parameters used in the model are: h = 2 ps, e = 200 ps, esc = 6 ps, = 200 ps, ge6 = −0:49, X = 15 eV. Inset: the pulsation ! dependence on the magnetic 4eld; dots: experimental data; solid line: linear 4t using: |!| = |ge6 |B B. (b) resonant case, B = 1 T. Same parameters as previously and h = 80 ps, X = 75 ps, ge6 = −0:45, Inset: the pulsation dependence on the magnetic 4eld; dots: experimental data; solid line: 4t using: =
(ge6 B B)2 + 2X .
careful comparison with Fig. 4b shows that the oscillations period is shorter than the one at zero detuning. The observation of polarization oscillations of the detected lower branch polaritons (k 6 5 × 103 cm−1 ) is puzzling. In bare QWs, the beats originate from the coupling between Jz =±1 states with Jz =±2 states by the transverse magnetic 4eld. Here, due to the strong coupling, the energy EJ =1 (k) lies far below the energy EJ =2 (k) (e.g. ≈ 1:75 meV at = 0 for MC2). So the mixing between |Jz = ±1 and |Jz = ±2 is extremely small, which leads to non-observable quantum beats. We interpret all the previous results as a consequence
P. Renucci et al. / Physica E 17 (2003) 329 – 334
B = 2 T B = 0 model
Circular polarization
1.0
+ 1, k
k=0 + 2, k
− 2, k
+ 1, k
− 1, k
τ esc
0.5
0.0
τ esc
τ
+ 2, k
τX τh
− 1, k
τe − 2, k
Reservoir k ≥ kp
δ = 4 m eV 0
100
200
300
time (ps)
Fig. 5. MC2. Polariton circular polarization oscillations under transverse magnetic 4eld, and resonant excitation conditions (1:7 K). The detuning is = +4 meV. The polarization decay without magnetic 4eld is also shown. The parameters used in the model are: h = 80 ps, e = 200 ps, X = 75 ps, esc = 15 ps, = 150 ps, ge6 = −0:52, X = 15 eV. Inset: scheme showing the di6erent relaxation times used in the model.
of the complex occupancy dynamics of polariton and exciton states. Large k excitations constitute a reservoir which is probed by the small k polariton emission. In the reservoir, since the |Jz =±1 and |Jz =±2 states are energetically close together, their mixing by the transverse magnetic 4eld becomes signi4cant, leading, as in bare QW [13,14], to the observed beats. We have developed a tentative model to interpret both resonant and non-resonant dynamics. Two classes of excitations are considered as suggested by the inset in Fig. 5: (i) the small k excitations (k 6 5 × 103 cm−1 ) consist of polaritons with short escape time esc , which do not relax their spin due to their very small in-plane wave vector, and uncoupled exciton states; (ii) the reservoir (R) consists of large k excitations (k ¿ kp ), in which all the spin relaxation processes between the quadruplet |Jz = ±1; ±2 are allowed: spin *ip between | + 1; k and | − 1; k exciton or polariton states, and single particle spin *ip between | ± 1; k and | ∓ 2; k or | ± 1; k and | ± 2; k states, respectively, characterized by the times X , h , e . Under transverse magnetic 4eld, the model used to describe the spin dynamics is similar to the one developed in our previous works [13,14,16]. Finally, we postulate a single transfer time between the reservoir and the detected polariton states. In non-resonant excitation conditions, electron–hole pairs photogenerated at the QW gap quickly scatter to the reservoir by
333
phonon emission, before relaxing towards the zone centre. In resonant excitation, polaritons are generated at kp ≈ 104 cm−1 ; Coulomb [1] and eventually multiple acoustic phonon scatterings allow them to reach the reservoir, from which they 4nally relax to the detected states by acoustic phonon emission. Most of the above parameters can be obtained by independent measurements performed at B = 0: esc and are deduced from intensity kinetics, while X and e are deduced from the polarization decay at long time delay under resonant and non-resonant excitation, respectively. The exchange splitting X of bare exciton is deduced from Ref. [17] and experiments performed on QW1 under transverse magnetic 4eld: we 4nd X ≈ 15 eV. Finally, the only remaining adjustable parameters are ge6 and h . Under non-resonant excitation (Fig. 4a), the hole spin *ip time is short (h ≈ 2 ps), i.e. it satis4es the criterion h ¡ ˝=X as in the case for excitons in bare quantum wells under non-resonant excitation [13,14]. The electron and the hole spins are thus uncorrelated (so that X → ∞). The polarization oscillations are the manifestation of the electron spin Larmor precession within the polariton. Their envelope decays with the electron spin *ip time e ≈ 200 ps [18]. As the hole spin is relaxed, the amplitude of the beats is maximum [13,14]. |ge6 | increases with the cavity detuning (Figs. 6a and b), which proves unambiguously that we are in the strong coupling regime; the measured value corresponds to the bare electron g-factor ge times the excitonic Hop4eld coe=cient Xk , and averaged over the reservoir. This averaging explains the rather small detuning dependence measured in Fig. 6b for MC1. In resonant excitation condition, the hole spin decay is now much longer: we 4nd h ≈ 80 ± 10 ps, in good agreement with data obtained from similar experiments performed in QW1. Note that here h ¿ ˝=X . This relative hole spin stability implies an electron– hole spin correlation at early time delays, leading to asymmetrical oscillations. The pulsation of the beats 2k + gk2 (B B)2 can be approximated by ≈ where the average is taken on the reservoir, gk = Xk ge and k = EJ =1 (k) − EJ =2 (k). The 4t parameters correspond to e6 = 2k 1=2 and |ge6 | = gk2 1=2 . In fact, we approximate e6 by X on the assumption that most of the reservoir population has a strong excitonic character. For B 6 1 T, the inequality 2e6 (ge6 B B)2
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0.56
Effective Landé I geff I factor
These studies underline the role of the population distributions at large in-plane wave vectors on spin dynamics, even for light detected near k ≈ 0.
resonant non resonant
0.52
Acknowledgements
M C2
0.48 0.44 (a)
0
2
4
6
We thank V. Thierry-Mieg at Laboratoire de Photonique et de Nanostructures in Bagneux, France for the samples growth.
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References
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[1] M. Saba, B. Deveaud, A. Mura, C. Ciuti, J.L. Staehli, D. Le Si Dang, V. Thierry-Mieg, R. Andr8e, S. Kundermann, G. Bongiovanni, J. Bloch, Nature 414 (2002) 713. [2] P. Renucci, X. Marie, T. Amand, M. Paillard, P. Senellart, J. Bloch, 25th International Conference on the Physics of Semiconductors, 2000, Osaka, Japan, Springer Proceedings in Physics Part I, 2001, p. 653. [3] M.D. Martin, L.Vi˜na, R. Ruf, E.E. Mendez, Phys. Stat. Sol. A 178 (1999) 539. [4] P. Lagoudakis, P.G. Saavidis, J.J. Baumberg, D.M. Whittaker, P.R. Eastham, M.S. Skolnick, J.S. Roberts, Phys. Rev. B 65 (2002) R161310. [5] M.Z. Maialle, E.A. de Andreada e Silva, L.J. Sham, Phys. Rev. B 47 (1993) 15776. [6] P. Le Jeune, X. Marie, T. Amand, F. Romstad, F. Perez, J. Barrau, M. Brousseau, Phys. Rev. B 58 (1998) 4853. [7] J. Tignon, P. Voisin, C. Delalande, M. Voos, R. Houdr8e, U. Oesterle, R.P. Stanley, Phys. Rev. Lett. 74 (1995) 3967. [8] T.A. Fisher, A.M. Afshar, M.S. Skolnick, D. Whittaker, J.S. Roberts, Phys. Rev. B 53 (1996) R10469. [9] X. Marie, P. Renucci, S. Dubourg, T. Amand, P. Le Jeune, J. Barrau, Phys. Rev. B. 59 (1999) R2494. [10] C. Ciuti, P. Schendimann, A. Quattopani, Phys. Rev. B. 63 (2001) R041303. [11] P. Renucci, Institut National des Sciences Appliqu8ees de Toulouse, Ph.D. Thesis, February 2002, p. 196. [12] E.A. de Andrada e Silva, G.C. La Rocca, Phys. Stat. Sol. A 190 (2002) 427. [13] T. Amand, X. Marie, P. Le Jeune, M. Brouseau, D. Robart, J. Barrau, Phys. Rev. B 78 (1997) 1355. [14] M. D’Yakonov, X. Marie, T. Amand, P. Le Jeune, D. Robart, M. Brousseau, J. Barrau, Phys. Rev. B 56 (1997) 10412. [15] M. Oestreich, D. HVagele, J. HVubner, W.W. RVuhle, Phys. Stat. Sol. A 178 (2000) 27. [16] E. Vanelle, D. Brinkmann, M. Paillard, X. Marie, T. Amand, P. Gilliot, B. HVonerlage, J. Cibert, S. Tatarenko, Phys. Rev. B 60 (2000) 15542. [17] A. Malinowski, D.J. Guerrier, N.J. Traynor, R.T. Harley, Phys. Rev. B 60 (1999) 7728. [18] A.P. Heberle, W.W. RhVule, K. Ploog, Phys. Rev. Lett. 72 (1994) 3887.
0.60
M C1
0.59 -4 (b)
-2
0
2
4
6
detuning δ (meV)
Fig. 6. Absolute value of the electron e6ective Land8e factor ge6 as a function of the cavity detuning for resonant and non-resonant excitation. (a): MC2; (b): MC1.
is satis4ed, which ensures that the beat frequency is mostly sensitive to |ge6 |. The beat frequency dependence on the magnetic 4eld is thus quasi-linear (see inset of Fig. 4b). Finally, we interpret the variations of |ge6 | reported in Fig. 6a as due to the dependence on the detuning of the population distributions in the reservoir. 5. Conclusion We have reported on the in*uence of the strong light-matter coupling in semiconductor III–V microcavities on polariton spin dynamics at low density. Under resonant excitation, we observe a quenching of spin and alignment relaxation when the polariton is photon like. Under transverse magnetic 4eld, an electron–hole spin correlation within the excitonic component of the quasi-particle is evidenced under resonant excitation, as well as an increase of the absolute value of the electron e6ective transverse Land8e g-factor with the excitonic character of polaritons.
Physica E 17 (2003) 329 – 334 www.elsevier.com/locate/physe
Coherent spin states and spin quantum beats in semiconductor microcavities Pierre Renuccia; b;∗ , Thierry Amandb , Xavier Marieb a Swiss
b Laboratoire
Federal Institute of Technology IPEQ-EPFL, CH-1015 Lausanne, Switzerland de Physique de la Mati%ere Condens&ee de Toulouse CNRS-INSA, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France
Abstract The present study evidences the in*uence of the strong light-matter coupling in semiconductor III–V microcavities on polariton spin dynamics at low excitation. Under resonant excitation, we observe a quenching of spin and alignment relaxation when the polariton is photon like. These behaviours are attributed, respectively, to the very small value of the long-range electron–hole exchange term of Coulomb interaction within the excitonic component of the quasi-particle and to the weakness of polariton–polariton Coulomb scattering via the inter-exciton short-range exchange interaction. Spin dynamics are also investigated under transverse magnetic 4eld. We observe an electron–hole spin correlation within the excitonic component of the quasi-particle under resonant excitation, and an increase of the absolute value of the electron e6ective transverse Land8e g-factor with the excitonic character of polaritons, which can be derived from a model taking into account only two classes of excitations in the lower polariton branch. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Polaritons; Coherent spin states; Spin dynamics; Quantum beats
1. Introduction Microcavity polariton dynamics is presently the subject of intense investigations, mainly in the regime of strong non-linear emission relative to polariton– polariton e=cient scattering [1]. However, much less studies have been devoted to the spin dynamics of these quasi-particles in the spontaneous emission regime. In our case, we will call polariton spin states the two excited states |Jz = ±1 with a total angular momentum Jz = ±1. We also de4ne ∗
Corresponding author. Laboratoire de Physique de la MatiBere Condens8ee de Toulouse CNRS-INSA, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France. Fax: +33-5-61-55-96-97. E-mail address: [email protected] (P. Renucci).
√ linear polariton states |X =(|JZ =+1+|J √ Z =−1)= 2 and |Y = (|JZ = +1 − |JZ = −1)=i 2 as coherent superpositions of polariton spin states. Moreover, we consider excitonic dark states |Jz = ±2 with total angular momentum Jz = ±2, that will play a key role under transverse magnetic 4eld. Spin coherence relies on the stability, on a time scale which can be longer than the optical dephasing time, of any linear superposition of two of these previously de4ned excited states. We propose here to analyze the phenomena relative to spin coherence through time and polarization-resolved photoluminescence under orientated optical pumping, with and without an applied transverse magnetic 4eld. In microcavities, in the strong coupling regime, some original spin dynamics behaviours are expected, since the lower branch dispersion curve is severely distorted
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00826-3
330
P. Renucci et al. / Physica E 17 (2003) 329 – 334
and the excitonic weight of the quasi-particle can be tuned. In a 4rst part, we report experimental results on polariton spin and alignment relaxation dynamics under resonant and weak excitation [2], contrary to previous studies led in the strong non-linear emission regime [3,4]. We adapt the excitonic mechanisms [5,6] known in bare quantum wells to the polariton picture, in order to interpret the relaxation quenching observed both under circularly and linearly polarized light when the polariton is photon like. In the second part, we analyze polariton spin dynamics under transverse magnetic 4eld (Voigt con4guration), under resonant and non-resonant circularly polarized excitation. Up to now, some works have been performed in Faraday con4guration, leading to the observation of magnetopolaritons [7], Zeeman splitting [8], as well as an increase of Rabi splitting with magnetic 4eld [8]. Measurements in Voigt con4guration provide some complementary informations. It allows to evidence an electron–hole spin correlation within the excitonic component of polariton under resonant excitation. Through a model based on two di6erent classes of excitations in the polariton lower branch, we also deduce that the absolute value of e6ective transverse electron g-factor increases with the excitonic component of the polariton.
PL = (I X − I Y )=(I X + I Y ), where I X and I Y denote the two linear orthogonal components of luminescence. All the experiments are performed at a low temperature at, 12 or 1:7 K, in a super*uid helium bath. Three -microcavities have been investigated, one with four 12 nm wide Ga0:86 In0:14 As quantum wells with a Rabi splitting R ≈ 6 meV (MC1), and two others with a single 8 nm wide Ga0:95 In0:05 As quantum well with R ≈ 3:5 (MC2) and 3:7 meV (MC3). Under transverse magnetic 4eld, comparative studies have also been performed on a Ga0:94 In0:06 As bare quantum well (QW1). 3. Spin and alignment dynamics Fig. 1 displays the time-resolved circular polarization PC after a circularly polarized exciting pulse for MC2. We can distinguish two di6erent time behaviours for PC with respect to the detuning ( = EC − EX , with EX and EC relative, respectively, to the uncoupled exciton and photon cavity
2. Experimental technique and samples Spin dynamics is monitored through time and polarization resolved polariton secondary emission using the two-colours up-conversion scheme [9]. The time-resolution is limited by the laser pulse-width (∼ 1:5 ps) and the spectral resolution is about 3 meV. The excitation power is weak, in order not only to preserve the strong exciton-photon coupling, but also to ensure to stay in the non-stimulated emission regime [10]. The excitation angle is 8◦ (with respect to the sample growth axis (Oz), corresponding to an initial in-plane wave vector kp ≈ 104 cm−1 ; the detection is normal to the surface, with a small acceptance angle (about 3 × 10−3 Sr). The circular polarization degree of the emission is de4ned as PC =(I + −I − )=(I + +I − ), where I + (I − ) denote, respectively, the right (left) circularly polarized luminescence component. In the same way, we note the linear polarization degree
Fig. 1. MC2. Right axis, full square: circular polarization kinetics for = −3; 0; 4 meV at 12 K. Left axis, full line: total intensity. Vertical dashed lines indicate the temporal separation between the two kinetics regimes, de4ned by the change of the intensity slope (in log scale). These slopes are underlined by dashed lines.
P. Renucci et al. / Physica E 17 (2003) 329 – 334
0.6
M C3 0.4
δ = - 9 m eV 0.2 0
10
20
30
40
time (ps) Fig. 2. MC3. Right axis, full square: circular polarization kinetics for = −9 meV at 12 K. Left axis, full line: total intensity. The peak centred around t = 0 is due to the back-scattered laser on the surface sample.
energy modes). For ¡ 0 (Fig. 1a), there is a clear relaxation quenching. On the other hand, for ¿ 0 (Fig. 1b and c), the relaxation becomes e=cient, mainly during the second step of the emission kinetics. We measure a characteristic longitudinal spin relaxation time Ts1 ≈ 75 ps for MC2 at ¿ 0. The observation of the circular polarization relaxation quenching for ¡ 0 was con4rmed on MC3 (Fig. 2), its longer photon lifetime (8 ps) allows to make measurements on a larger time scale. We attribute the spin relaxation process to the spin *ip of the excitonic component. This theory, based on the simultaneous *ip of electron and hole spin, has been developed previously for bare QWs by Maialle et al. [5] and takes into account the electron–hole exchange, the long-range part being the dominant contribution. We generalize this result in the polariton picture [11,12]. This leads to 1 2 4 ∗ = ; (1) ex (k)|X (k)| (k): Ts1 Here ; ex (k) is proportional to the electron–hole long-range exchange matrix element for a given in-plane wave vector k; ∗ (k) represents the polariton momentum relaxation time, X (k) is the excitonic Hop4eld coe=cient, and the symbol denotes the average on the occupied polariton states within the lower branch. For ¡ 0, the polariton population is concentrated around k ≈ kp states in the k-space, as the acoustic phonon scatterings are quenched due to the distorted dispersion curve. An evaluation of each term of Eq. (1) [11,12] leads to a spin relaxation time
Linear polarization
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1.0 Circular polarization
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T = 12 K δ = -6 meV δ = 0 meV δ = 4 meV
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0
10
20 time (ps)
30
40
Fig. 3. MC2. Linear polarization kinetics for = −6; 0; 4 meV at 12 K. Dashed lines are guides for the eyes.
Ts1 at least three orders of magnitude longer than the bare exciton one, mainly due to the weakness of the matrix element which is proportional to the small wave vector kp . This is in good agreement with the observed relaxation quenching. On the other hand, for ¿ 0, larger wave vectors are available after e=cient scatterings, which implies a stronger exchange matrix element (averaged on the whole lower branch) and a e=cient spin relaxation during the second step of the emission kinetics. This second step corresponds to the come-back of elementary excitations from large in-plane wave vectors excitonic states (with centre of mass wave vectors of the order of 105 cm−1 ) towards the centre of Brillouin zone. Actually, we thus mainly probe the exciton spin relaxation. This explains why we observe for MC2 no clear di6erence between the cases = 0 and 4 meV (Fig. 1b and c). To complete the previous study, we also performed similar experiments under linearly polarized excitation. Fig. 3 displays the time-resolved linear polarization of the MC2 microcavity following a linearly polarized exciting pulse for di6erent cavity detunings. We observe a quenching of the relaxation (characterized by the so-called spin transverse relaxation time Ts2 ) for ¡ 0. However, a fast linear polarization decay appears, during the 4rst step of kinetics, when increases. For = 0; Ts2 is about 15 ps, shorter than Ts1 (≈ 75 ps). As the linear polarization relaxation (also called alignment relaxation) becomes faster when the excitation power increases, we attribute the mechanism responsible for this behaviour to the mutual polariton spin-dependent Coulomb exchange via
(2)
Here V0 corresponds to the exciton–exciton exchange matrix element [10]. The evaluation of the term |X (kp )|8 gives thus some estimation of the strength of the process in the spontaneous regime. For = −6; 0; 4 meV, we 4nd 2 × 10−4 , 0.27, 0.70, respectively (MC2). So, at negative detuning, there is a strong reduction of the process e=ciency, due to the important photon component in the polariton, in good agreement with the alignment relaxation quenching. For ¿ 0 the Coulomb scattering process becomes more e=cient, as the exciton weight increases. 4. Spin dynamics under transverse magnetic eld Experiments under transverse magnetic 4eld usually provide some decisive informations on quantum coherences [13,14] and on electron–hole spin correlation [15]. Fig. 4a displays the polariton circular polarization oscillations under transverse magnetic 4eld of the MC2 microcavity emission for the cavity detuning = 0. Here the excitation is tuned to the InGaAs QW gap (non-resonant case). Note that the oscillations are symmetrical with respect to the time axis, and that their envelope coincides with the polariton circular polarization decay without magnetic 4eld. The beat pulsation ! depends linearly on the magnetic 4eld intensity (see inset in Fig. 4a). Fig. 4b displays the polarization oscillations under the same conditions but now the excitation is resonant with the lower polariton branch. One can note the asymmetry of the beats with respect to the time axis, since the circular polarization remains positive at early time delays. Fig. 5 shows also the polarization oscillations for = +4 meV. A
0.5
-1
0.08 0.04 0.00
0
1
2
3
B (T)
0.0
B=0T B=2T model
-0.5 -1.0 (a)
0
200
400
B=0T B=1T model
-1
1 ˙ V02 |X (kp )|8 : Ts2
δ =0
1.0
ω (ps )
the exciton components of polaritons (the direct term of the Coulomb interaction is negligible in the range of wave vectors explored after a resonant excitation). This process was studied in bare QWs by Le Jeune et al. [6], and is known in microcavities as parametric scattering [10]. Due to the exchange selection rules, this scattering is a strong depolarizing process: a pair of polaritons (|X ; |X ) is scattered with an equal probability into a pair (|X ; |X ) or a pair (|Y ; |Y ). The transverse spin relaxation time Ts2 can be expressed as follows:
ω (p s )
P. Renucci et al. / Physica E 17 (2003) 329 – 334
Circular polarization
332
1.0
600
0.15 0.10 0.05 0.00
0.5
0
1
2
3
4
B (T) 0.0 (b)
0
100
200
300
time (ps) Fig. 4. MC2. Polariton circular polarization oscillations under transverse magnetic 4eld, and circularly polarized excitation (1:7 K). The polarization decay without magnetic 4eld is also shown. The detuning is = 0: (a) non-resonant case, B = 2 T. The parameters used in the model are: h = 2 ps, e = 200 ps, esc = 6 ps, = 200 ps, ge6 = −0:49, X = 15 eV. Inset: the pulsation ! dependence on the magnetic 4eld; dots: experimental data; solid line: linear 4t using: |!| = |ge6 |B B. (b) resonant case, B = 1 T. Same parameters as previously and h = 80 ps, X = 75 ps, ge6 = −0:45, Inset: the pulsation dependence on the magnetic 4eld; dots: experimental data; solid line: 4t using: =
(ge6 B B)2 + 2X .
careful comparison with Fig. 4b shows that the oscillations period is shorter than the one at zero detuning. The observation of polarization oscillations of the detected lower branch polaritons (k 6 5 × 103 cm−1 ) is puzzling. In bare QWs, the beats originate from the coupling between Jz =±1 states with Jz =±2 states by the transverse magnetic 4eld. Here, due to the strong coupling, the energy EJ =1 (k) lies far below the energy EJ =2 (k) (e.g. ≈ 1:75 meV at = 0 for MC2). So the mixing between |Jz = ±1 and |Jz = ±2 is extremely small, which leads to non-observable quantum beats. We interpret all the previous results as a consequence
P. Renucci et al. / Physica E 17 (2003) 329 – 334
B = 2 T B = 0 model
Circular polarization
1.0
+ 1, k
k=0 + 2, k
− 2, k
+ 1, k
− 1, k
τ esc
0.5
0.0
τ esc
τ
+ 2, k
τX τh
− 1, k
τe − 2, k
Reservoir k ≥ kp
δ = 4 m eV 0
100
200
300
time (ps)
Fig. 5. MC2. Polariton circular polarization oscillations under transverse magnetic 4eld, and resonant excitation conditions (1:7 K). The detuning is = +4 meV. The polarization decay without magnetic 4eld is also shown. The parameters used in the model are: h = 80 ps, e = 200 ps, X = 75 ps, esc = 15 ps, = 150 ps, ge6 = −0:52, X = 15 eV. Inset: scheme showing the di6erent relaxation times used in the model.
of the complex occupancy dynamics of polariton and exciton states. Large k excitations constitute a reservoir which is probed by the small k polariton emission. In the reservoir, since the |Jz =±1 and |Jz =±2 states are energetically close together, their mixing by the transverse magnetic 4eld becomes signi4cant, leading, as in bare QW [13,14], to the observed beats. We have developed a tentative model to interpret both resonant and non-resonant dynamics. Two classes of excitations are considered as suggested by the inset in Fig. 5: (i) the small k excitations (k 6 5 × 103 cm−1 ) consist of polaritons with short escape time esc , which do not relax their spin due to their very small in-plane wave vector, and uncoupled exciton states; (ii) the reservoir (R) consists of large k excitations (k ¿ kp ), in which all the spin relaxation processes between the quadruplet |Jz = ±1; ±2 are allowed: spin *ip between | + 1; k and | − 1; k exciton or polariton states, and single particle spin *ip between | ± 1; k and | ∓ 2; k or | ± 1; k and | ± 2; k states, respectively, characterized by the times X , h , e . Under transverse magnetic 4eld, the model used to describe the spin dynamics is similar to the one developed in our previous works [13,14,16]. Finally, we postulate a single transfer time between the reservoir and the detected polariton states. In non-resonant excitation conditions, electron–hole pairs photogenerated at the QW gap quickly scatter to the reservoir by
333
phonon emission, before relaxing towards the zone centre. In resonant excitation, polaritons are generated at kp ≈ 104 cm−1 ; Coulomb [1] and eventually multiple acoustic phonon scatterings allow them to reach the reservoir, from which they 4nally relax to the detected states by acoustic phonon emission. Most of the above parameters can be obtained by independent measurements performed at B = 0: esc and are deduced from intensity kinetics, while X and e are deduced from the polarization decay at long time delay under resonant and non-resonant excitation, respectively. The exchange splitting X of bare exciton is deduced from Ref. [17] and experiments performed on QW1 under transverse magnetic 4eld: we 4nd X ≈ 15 eV. Finally, the only remaining adjustable parameters are ge6 and h . Under non-resonant excitation (Fig. 4a), the hole spin *ip time is short (h ≈ 2 ps), i.e. it satis4es the criterion h ¡ ˝=X as in the case for excitons in bare quantum wells under non-resonant excitation [13,14]. The electron and the hole spins are thus uncorrelated (so that X → ∞). The polarization oscillations are the manifestation of the electron spin Larmor precession within the polariton. Their envelope decays with the electron spin *ip time e ≈ 200 ps [18]. As the hole spin is relaxed, the amplitude of the beats is maximum [13,14]. |ge6 | increases with the cavity detuning (Figs. 6a and b), which proves unambiguously that we are in the strong coupling regime; the measured value corresponds to the bare electron g-factor ge times the excitonic Hop4eld coe=cient Xk , and averaged over the reservoir. This averaging explains the rather small detuning dependence measured in Fig. 6b for MC1. In resonant excitation condition, the hole spin decay is now much longer: we 4nd h ≈ 80 ± 10 ps, in good agreement with data obtained from similar experiments performed in QW1. Note that here h ¿ ˝=X . This relative hole spin stability implies an electron– hole spin correlation at early time delays, leading to asymmetrical oscillations. The pulsation of the beats 2k + gk2 (B B)2 can be approximated by ≈ where the average is taken on the reservoir, gk = Xk ge and k = EJ =1 (k) − EJ =2 (k). The 4t parameters correspond to e6 = 2k 1=2 and |ge6 | = gk2 1=2 . In fact, we approximate e6 by X on the assumption that most of the reservoir population has a strong excitonic character. For B 6 1 T, the inequality 2e6 (ge6 B B)2
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P. Renucci et al. / Physica E 17 (2003) 329 – 334
0.56
Effective Landé I geff I factor
These studies underline the role of the population distributions at large in-plane wave vectors on spin dynamics, even for light detected near k ≈ 0.
resonant non resonant
0.52
Acknowledgements
M C2
0.48 0.44 (a)
0
2
4
6
We thank V. Thierry-Mieg at Laboratoire de Photonique et de Nanostructures in Bagneux, France for the samples growth.
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References
0.61
[1] M. Saba, B. Deveaud, A. Mura, C. Ciuti, J.L. Staehli, D. Le Si Dang, V. Thierry-Mieg, R. Andr8e, S. Kundermann, G. Bongiovanni, J. Bloch, Nature 414 (2002) 713. [2] P. Renucci, X. Marie, T. Amand, M. Paillard, P. Senellart, J. Bloch, 25th International Conference on the Physics of Semiconductors, 2000, Osaka, Japan, Springer Proceedings in Physics Part I, 2001, p. 653. [3] M.D. Martin, L.Vi˜na, R. Ruf, E.E. Mendez, Phys. Stat. Sol. A 178 (1999) 539. [4] P. Lagoudakis, P.G. Saavidis, J.J. Baumberg, D.M. Whittaker, P.R. Eastham, M.S. Skolnick, J.S. Roberts, Phys. Rev. B 65 (2002) R161310. [5] M.Z. Maialle, E.A. de Andreada e Silva, L.J. Sham, Phys. Rev. B 47 (1993) 15776. [6] P. Le Jeune, X. Marie, T. Amand, F. Romstad, F. Perez, J. Barrau, M. Brousseau, Phys. Rev. B 58 (1998) 4853. [7] J. Tignon, P. Voisin, C. Delalande, M. Voos, R. Houdr8e, U. Oesterle, R.P. Stanley, Phys. Rev. Lett. 74 (1995) 3967. [8] T.A. Fisher, A.M. Afshar, M.S. Skolnick, D. Whittaker, J.S. Roberts, Phys. Rev. B 53 (1996) R10469. [9] X. Marie, P. Renucci, S. Dubourg, T. Amand, P. Le Jeune, J. Barrau, Phys. Rev. B. 59 (1999) R2494. [10] C. Ciuti, P. Schendimann, A. Quattopani, Phys. Rev. B. 63 (2001) R041303. [11] P. Renucci, Institut National des Sciences Appliqu8ees de Toulouse, Ph.D. Thesis, February 2002, p. 196. [12] E.A. de Andrada e Silva, G.C. La Rocca, Phys. Stat. Sol. A 190 (2002) 427. [13] T. Amand, X. Marie, P. Le Jeune, M. Brouseau, D. Robart, J. Barrau, Phys. Rev. B 78 (1997) 1355. [14] M. D’Yakonov, X. Marie, T. Amand, P. Le Jeune, D. Robart, M. Brousseau, J. Barrau, Phys. Rev. B 56 (1997) 10412. [15] M. Oestreich, D. HVagele, J. HVubner, W.W. RVuhle, Phys. Stat. Sol. A 178 (2000) 27. [16] E. Vanelle, D. Brinkmann, M. Paillard, X. Marie, T. Amand, P. Gilliot, B. HVonerlage, J. Cibert, S. Tatarenko, Phys. Rev. B 60 (2000) 15542. [17] A. Malinowski, D.J. Guerrier, N.J. Traynor, R.T. Harley, Phys. Rev. B 60 (1999) 7728. [18] A.P. Heberle, W.W. RhVule, K. Ploog, Phys. Rev. Lett. 72 (1994) 3887.
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M C1
0.59 -4 (b)
-2
0
2
4
6
detuning δ (meV)
Fig. 6. Absolute value of the electron e6ective Land8e factor ge6 as a function of the cavity detuning for resonant and non-resonant excitation. (a): MC2; (b): MC1.
is satis4ed, which ensures that the beat frequency is mostly sensitive to |ge6 |. The beat frequency dependence on the magnetic 4eld is thus quasi-linear (see inset of Fig. 4b). Finally, we interpret the variations of |ge6 | reported in Fig. 6a as due to the dependence on the detuning of the population distributions in the reservoir. 5. Conclusion We have reported on the in*uence of the strong light-matter coupling in semiconductor III–V microcavities on polariton spin dynamics at low density. Under resonant excitation, we observe a quenching of spin and alignment relaxation when the polariton is photon like. Under transverse magnetic 4eld, an electron–hole spin correlation within the excitonic component of the quasi-particle is evidenced under resonant excitation, as well as an increase of the absolute value of the electron e6ective transverse Land8e g-factor with the excitonic character of polaritons.