Combined exciton–electron excitation in quantum wells with a two-dimensional electron gas of low density

Superlattices and Microstructures, Vol. 23, No. 2, 1998

Combined exciton–electron excitation in quantum wells with a two-dimensional electron gas of low density V. P. Kochereshko, D. R. Yakovlev†, R. A. Suris A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St Petersburg, Russia

W. Ossau, A. Waag, G. Landwehr Physikalisches Institut der Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg, Germany

P. C. M. Christianen, J. C. Maan High Field Magnet Laboratory and Research Institute for Materials, University of Nijmegen, NL-6525 ED Nijmegen, the Netherlands (Received 15 July 1996)

A combined exciton–cyclotron resonance is found in the photoluminescence excitation and reflectivity spectra of semiconductor quantum wells with an electron gas of low density. In external magnetic fields an incident photon creates an exciton in the ground state and simultaneously excites an electron between Landau levels. A theoretical model is developed and suggests the dominating contribution of the exchange exciton–electron interaction. c 1998 Academic Press Limited

1. Introduction An energy spectrum of quasi-two-dimensional excitons in quantum-well (QW) semiconductor heterostructures has been studied for two extreme cases: (i) absence of free carriers in QWs, and (ii) under high concentration of the free carriers, when exciton states are screened. The second situation is realized in the doped QW structures or under high-density excitation [1]. However, detailed information about the transition between these two limiting cases has not been available until recently. Effects resulting from the exciton–electron interaction in the presence of two-dimensional electron gas (2DEG) of low density at n e a B  1, here n e is the electron concentration and a B is the exciton Bohr radius, became a subject of intensive investigation very recently [2–6]. An interest in this problem has been stimulated by the paper of Kheng et al [2] where the observation of a negatively charged exciton (X − ) in CdTe/(Cd,Zn)Te modulation-doped QWs has been reported. Immediately after negatively and positively charged excitons have been observed in GaAs/(Al,Ga)As modulation-doped QWs with the free-carrier concentration tuned by the variation of the gate voltage [4–6]. In this paper we report an observation of combined exciton–electron excitations at external magnetic fields in CdTe/(Cd,Mg)Te QWs with low concentration of additional electrons. † Also at: Physikalisches Institut der Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg, Germany

0749–6036/98/020283 + 05 $25.00/0

sm960488

c 1998 Academic Press Limited

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Superlattices and Microstructures, Vol. 23, No. 2, 1998 T = 1.6 K

Reflectivity and PL intensity (a.u.)

1s-hh

ECR

1s-lh

2s-hh

Reflectivity

PLE

1.62

1.63

1.64 1.65 Energy (eV)

1.66

1.67

Fig. 1. Reflectivity and PLE spectra of a CdTe/Cd0.74 Mg0.26 Te structure taken in a magnetic field of 7.5 T.

2. Experiment Effects determined by the exciton–electron interaction are studied in a type-I CdTe/Cd0.74 Mg0.26 Te het˚ thick QW sandwiched between 1000 A ˚ thick (16 periods) short-period supererostructure consisting of 75 A ˚ ˚ ˚ lattices (SLs) (30 A/30 A). The QW is separated from SLs by 200 A thick barriers. Tunneling probabilities for electrons and holes from the SLs into the QW are different due to the difference of their effective masses. This allows us to tune concentration of electrons in the QW by varying intensity of photoexcitation at energies exceeding a band gap of the SL. The structure has been grown by molecular beam epitaxy on the (100) oriented Cd0.97 Zn0.03 Te substrate. Photoluminescence (PL), photoluminescence excitation (PLE) and reflectivity spectra have been measured at 1.6 K and in magnetic fields up to 20 T applied perpendicular to the QW layers (Faraday geometry). A dye-laser (Pyridine) with the photon energy below the band gap of SLs has been used for a direct excitation of excitons in the QW. An Ar–ion laser with a photon energy of 2.41 eV, which exceeds the SL gaps, has been used for variation of the electron concentration in the QW. Excitation density of the indirect excitation does not exceed 10 W cm−2 and PL intensity from the QW under indirect excitation only was very small in comparison with the PL intensity taken under direct excitation. The PL spectrum of the structure consists of two lines: the X line at 1.6305 eV corresponds to the recombination of excitons localized on monolayer fluctuations of QW width, and the low-energy line (X − line) at 1.627 eV is shifted at 3.5 meV from the X line and corresponds to the bound exciton state. It has been shown recently that the binding energy of 3–3.5 meV is characteristic for the negatively charged exciton state in CdTe-based QWs and is smaller than the typical binding energy of exciton to the neutral donor which is about 4.5 meV [2, 3]. Additional indirect excitation causes strong redistribution of the PL intensity in favor of the X − line. Indirect excitation increases the probability for the X − formation by an increase in the number of electrons in the QW. In the external magnetic fields a new exciton–cyclotron resonance (ECR) line appears in the PLE and reflectivity spectra of the QW additionally to pronounced peaks of magnetoexcitons (see Fig. 1). An intensity

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1.67

T = 1.6 K

4s-hh 3s-hh 2s-hh

Energy (eV)

1.66

1s-lh 1.65 ECR 1.64

1s-hh 1.63

0

5

10 15 Magnetic field (T)

20

Fig. 2. Magnetic-field dependence of the spectral position of magnetoexciton lines and the ECR line.

of the ECR line increases strongly for higher indirect excitations (i.e. for higher n e ), meanwhile the magnetoexciton features are insensitive to the indirect illumination. It is shown in Fig. 2 that the ECR line shifts linearly in magnetic fields with a slope of about 1.14 meV T−1 , which is close to the cyclotron energy of the electron in CdTe/(Cd,Mg)Te QWs. An extrapolation of this shift to the zero field meets about the energy of the 1s of heavy-hole exciton (1s-hh). Linear shift of the ECR line shows that free carriers contribute to the observed process, while excitonic states have characteristic quadratic diamagnetic shifts in the low-field limit, which are clearly seen for 1s, 2s, 3s and 4s states of the heavy-hole exciton shown in Fig. 2. Energy for these states are fitted by the procedure suggested for the two-dimensional magnetoexcitons in [7]. The best fit with reduced exciton mass of 0.089 m 0 is shown in Fig. 2 by solid lines. Taking an electron effective mass m e = 0.11 m 0 we determine heavy-hole mass of m hh = 0.48m 0 . A strong polarization of the ECR line is observed in PLE spectra. The intensity of ECR line for σ − polarization is much stronger than for σ + polarization. At a temperature of 1.6 K and in magnetic fields larger than 2 T the free electrons in the QW are strongly polarized along the external field direction (the electron g factor in CdTe is equal to −1.644 [8]). Under these conditions the X − line appears when the spin of photoexcited exciton is opposite to the free electron orientation. But the ECR line appears when the exciton spin is parallel to the orientation of the free electron gas polarization.

3. Theory All experimental manifestations of the ECR line allow us to conclude that the effect we are dealing with is caused by the photon absorption that creates exciton and simultaneously excites an electron in 2DEG from the zero to the first Landau level. We suggest that this process be identified as a combined exciton–cyclotron resonance (ECR). The following approximations are used for the theoretical description of the process.

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1. The cyclotron energy, h¯ ωc,e = eh¯ H/m e c, is much smaller than √ the exciton binding energy and a B /L H hh1(a B is the exciton Bohr radius of an exciton, L H = h¯ c/eH is a magnetic length). In this case a modification of the exciton wavefunction by magnetic field can be neglected. 2. The two-dimensional approximation for the exciton states. 3. The low-temperature limit: kT hhµ B ge H, kT hhh¯ ωc,e , that means that the free electrons occupy the lowest energy Zeeman component of the zero Landau level. The absorption spectrum is calculated as the imaginary part of the polarizability: 1 X h80 |Pα |GihG|Pβ |80 i , χα,β = 2 G E G − E 0 − h¯ ω − iδ where |80 i and |Gi are wave functions and E 0 and E G are energies of the ground and excited states of the system, respectively, h80 |Pα |Gi is the matrix element of the αth component of the dipole moment operator, ω is the photon frequency andQδ → 0 is the imaginary addition to the frequency. We choose a + + ground state of the system as: |80 i = X an=0,X,σ =1/2 |0i, where an,X,σ is an operator of creation of a two-dimensional electron on the nth Landau level with spin σ and coordinate of the Landau oscillator X = L 2H p y /h¯ . The energy of this state is E 0 = N (h¯ ωc,e /2 − µ B ge H/2), where N is the total number of electrons of the QW, which is equal to the electron two-dimensional density n e multiplied on the QW area. Excited states of the system are first a single exciton state: Z Z Q,J,σ i=C d 2 re d 2 rh eiQR ϕ(re − rh )aσ+ (re )b+ |G J (rh )|80 i, where aσ+ (re ) and b+ J (rh ) are operators of the creation of an electron with spin σ and a heavy hole with angular momentum J . R and Q are the in-plane coordinate and wavevector of the exciton center of masses, ϕ(re − rh ) is the exciton envelope wavefunction. The energy of the state |G Q,J,σ i, measured from the ground state, is just J,σ (Q) − µ B ge σ H/2 − µ B gh J H . Secondly, there are other excited an energy of excitonic state E Q,J,σ = E ex states that are the states with one exciton with momentum h¯ Q and one electron excited from Landau level n = 0 with spin ↑ into Landau level n > 0 with spin σ 1 : Z Z Q,J,σ + d 2 re d 2 rh eiQR ϕ(re − rh )aσ+ (re )b+ |G n,σ 1 ,X ;0,↑,X 0 i = C 0 J (rh )an,X,σ 1 a0,X 0 ,↑ |80 i. The energy of this state is: Q,J,σ

J,σ (Q) + n h¯ ωc,e − µ B ge (σ 1 − 1)H/2 − µ B ge σ H/2 − µ B gh J H. E n,σ 1 ,X ;0,↑,X 0 = E ex

For the imaginary part of polarizability of the system, which describes an absorption spectrum we get equation: Z X J (dασ,J )∗ dασ,J |ϕ(0)|2 δ(h¯ ω − E ex (0) + µ B ge σ H/2 + µ B gh J H ) + n e | d 2 rϕ J (r)|2 Imχα,β = π J,σ

×

X J,σ,σ 1

(dασ,J )∗ dασ,J

Z

∞ d 2Q X 1 4π n=1 n!



Q2 L 2H 2

n

J × exp(−Q2 L 2H /2)δ(h¯ ω − E ex (Q) − n h¯ ωc,e + µ B ge (σ 1 − 1)H/2 + µ B ge σ H/2 + µ B gh J H )

where dασ,J is a band-to-band dipole matrix element. The first term of this equation corresponds to the free exciton transition and the second term corresponds to the ECR state. For the quadratic exciton dispersion J J (Q) = E ex (0) + h¯ 2 Q2 /2M a simple integration of the ECR term gives: E ex     ∞ X X M 1n,σ,J,σ 1 1 M 1n,σ,J,σ 1 n EC R 2 σ,J ∗ σ,J M (dα ) dα 2 exp − Imχα,β = 4πn e a B , ωc,e me ωc,e h¯ n=1 n! m e J,σ,σ 1

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J (0) − n h¯ ωc,e + µ B ge (σ 1 − 1)H/2 + µ B ge σ H/2 + µ B gh J H . One can see where 1n,σ,J,σ 1 = h¯ ω − E ex that the absorption spectrum of the considering system in a magnetic field consists of a free exciton line and several lines shifted from it to higher energies with an energy distance of n h¯ ωc,e (1 + m e /M), where n = 1, 2, 3 . . . and M = m e + m h . An estimation of the energy shift of the first excited state with an experimentally determined masses m e = 0.11m 0 and m h = 0.48m 0 gives a shift of 1.24 meV T−1 that is close to the experimentally observed shift of the ECR line of 1.14 meV T−1 . A calculated ratio of the integral intensity of the nth combined peak to the intensity of the exciton line, S0 , is: SnECR /S0 = 8n e L 2H = 4 f 0 /π, where f 0 is an electron occupancy of the zero Landau level. This ratio increases linearly with the electron occupancy increase and does not depend on the magnetic-field value. These theoretical results are in good agreement with the experimental findings.

Acknowledgements—This work has been supported in part by the European Commission under Contract No. CHGT-CT93-0051 (Large Facility), the Deutsche Forschungsgemeinschaft through SFB 410, the Russian Foundation for Basic Research grants No. 95-02-04061a and No. 96-02-17952 and by the program ‘Nanostructures’ of the Russian Ministry of Science.

References [1] }H. Haug and C. Klingshirn, Phys. Report 70, 315 (1981); S. Schmidt-Rink, D. S. Chemla and D. A. B. Miller, Adv. Phys. 38, 89 (1989). [2] }K. Kheng, R. T. Cox, Y. Merle d’Aubigne, F. Bassani, K. Seminadayar, and S. Tatarenko, Phys. Rev. Lett. 71, 1752 (1993). [3] }R. T. Cox, K. Kheng, K. Seminadayar, T. Baron, and S. Tatarenko, Proc. Int. Conf. High Magnetic Fields in the Physics of Semiconductors, Cambridge 1994, edited by D. Heiman World Scientific, Singapore (1995) p. 394. [4] }G. Finkelstein, H. Shtrikman, and I. Bar-Joseph, Phys. Rev. B53, R1709 (1996). [5] }R. Harel, E. Cohen, E. Linder, A. Ron, and L. N. Pfeiffer, Phys. Rev. B53, 7868 (1996). [6] }A. J. Shields, M. Pepper, D. A. Ritchie, M. Y. Simmons, and G. A. C. Jones, Phys. Rev. B51, 18049 (1995). [7] }A. H. MacDonald and D. A. Ritchie, Phys. Rev. B33, 8336 (1986). [8] }A. A. Sirenko, T. Ruf, M. Cardona, D. R. Yakovlev, W. Ossau, A. Waag, and G. Landwehr, Phys. Rev. B56, 2114 (1994).