Optical nutation in a magnetized semiconductor quantum well structure

Superlattices and Microstructures, Vol. 26, No. 4, 1999 Article No. spmi.1999.0773 Available online at http://www.idealibrary.com on

Optical nutation in a magnetized semiconductor quantum well structure S HARMILA BANERJEE†, P RANAY K. S EN Department of Applied Physics, Shri G. S. Institute of Technology & Science, Indore 452003, India (Received 27 April 1999) Using the semiclassical coherent radiation—semiconductor interaction model, optical nutation has been analysed in a GaAs/Alx Ga1−x As quantum well structure (QWS) assumed to be immersed in a moderately strong magnetic field and irradiated by a not-too-strong near band gap resonant femtosecond pulsed Ti–sapphire laser. The finite potential well depth of the QWS and the Wannier–Mott excitonic structure of the crystal absorption edge is taken into account. The excitation intensity is assumed to be below the Mott transition where the various many-body effects have been neglected with adequate reasoning. Numerical analysis made for a GaAs quantum well of thickness ∼100 Åand the confining layers of Alx Ga1−x As with x = 0.3 at intensity I ≤ 5 × 106 W cm−2 reveals that the real and imaginary parts of the transient complex-induced polarization are enhanced with an increase in the magnetic field and their ringing behaviour confirms the occurrence of optical nutation in the QWS. c 1999 Academic Press

Key words: optical nutation, coherent transients, quantum wells, magnetized semiconductors, excitonic effect.

1. Introduction Coherent transient spectroscopy has provided considerable information on irreversible decay processes in a wide range of atomic and molecular systems [1, 2]. However, despite the presence of strong resonances in semiconductors, the rapid dephasing time made the study of coherent optical phenomena difficult in these host materials. The availability of ultrashort pulsed lasers in the subpico-to-femtosecond pulse duration regime has made it possible to realize coherent transient effects in semiconductors [3]. Ultrafast nonlinear optics including generation and applications of ultrashort pulsed lasers has been reviewed by Walmsley and Kafka [4], while Flytzanis and Hutter [5] have discussed nonlinear optics in quantum confined structures. Due to the large optical nonlinearities, semiconductor materials have established themselves as a class of attractive materials for modern optoelectronic applications. Over recent years, semiconductor nanostructures have attracted considerable attention of the researchers because of their novel optical properties. At low to moderate optical intensity, the photoinduced band-to-band transitions lead to the generation of bound electron–hole pairs known as excitons which play a nontrivial role near the semiconductor band-edge. Excitonic enhancement of optical nonlinearities is more prominent in two-dimensional (2D) quantum wells than † E-mail: [email protected]

0749–6036/99/100251 + 11 $30.00/0

c 1999 Academic Press

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in the bulk semiconductors due to stronger binding between the electron–hole pairs and shorter exciton Bohr radius. Winkler [6] has presented a theory for excitonic absorption in the semiconductor quantum well structure and claims that his numerical analysis in GaAs/AlGaAs QWS reveals a fall in the absorption coefficient due to Coulomb interaction in contrast to that in the bulk crystal. Delalande [7] has reviewed some optical properties of the semiconductor quantum wells with special emphasis on the excitonic properties. The situation becomes more interesting if the semiconductor is immersed in a magnetostatic field when it may be anticipated to yield more information about the transient behaviour of the crystal. In the presence of a magnetic field, the exciton wavefunction changes significantly if the Coulombic interaction energy is smaller than the magnetic energy. While analysing theoretically the phenomenon of nonlinear refraction in magnetised narrow direct-gap III–V bulk semiconductors using the semiclassical coherent radiation–semiconductor interaction model, Sen [8] observed the two-fold contribution of the magnetic field to the excitonic optical nonlinearities. Recently the study of magnetoexcitons in both bulk and lower-dimensional semiconductor structures has attracted keen interest [9–13]. Stark et al. [9] reported the first femtosecond measurement of nonlinear optical response of excitons in GaAs/AlGaAs quantum well structures (QWS) in the presence of a strong magnetic field when, the quasi-2D excitons become further confined to quasi-zero dimension. These authors have further shown [9] that at high magnetic fields when the cyclotron radius is much larger than the 3D exciton Bohr radius, the 2D magneto excitons behave like isolated two-level systems [10]. Glutsch, Chemla and others [11, 12] have analysed Fano resonances and four-wave mixing in bulk GaAs while Glutsch and Chemla [13] developed the semiconductor Bloch equations in a homogeneous magnetic field. Various many-body effects, while reducing the dimensionality artificially, can influence significantly the study of magnetoexcitons in a semiconductor. The possibility of enhancement in optical nonlinearity due to exciton– exciton interactions has recently drawn considerable attention [14–18]. Interestingly, the very recent analysis by Chernyak et al. [17] demonstrates the signature of two-exciton states in nonlinear optical experiments in GaAs QWS under the strong magnetic field regime. It is envisaged that the analytical investigation of the effect of external magnetic field on coherent transient optical effects in a direct-gap III–V semiconductor QWS would be of technological potentiality in the area of ultrafast optoelectronics. Accordingly, in the present paper, we have made an attempt to study analytically the phenomenon of optical nutation in a magnetized direct narrow-gap III–V QWS irradiated by ultrashort pulsed moderate intensity laser. It may be recalled that optical nutation characterized by a ringing behaviour of the transmitted signal is one of the earliest discovered coherent transient optical effects [19] and demands thorough understanding before one could propose the candidature of the material for ultrafast optoelectronic device-fabrication; especially as optical waveguides for ultrashort pulse propagation. In order to make the present analysis physically suitable for the demonstration of optical nutation in the highly magnetized QWS, we have taken some reasonable approximations which can enable us to study the transient features of the nutating signal without sacrificing much of the mathematical accuracy. At low excitation intensities, the photoinduced electronic transitions lead to the formation of weakly bound Wannier– Mott excitons in narrow gap semiconductors. Above a critical excitation density, exchange effects and the screening of the Coulomb interaction by the optically generated electron–hole paris destabilize the excitons and the electron–hole plasma is produced. We have chosen the excitation intensity slightly lower than this critical value such that the electron–hole pair density is very close to the Mott transition regime and hence, the many-body effects like exchange and correlation processes, as well as the e–h plasma effect become trivial. Moreover, the exciton–exciton interaction is also negligible in the vicinity of the Mott-density [20]. Vashishta and Kalia [21] have shown that the exchange and correlation effects are negligible in narrow-gap III–V semiconductor QWS. The band gap renormalization arising due to many-body interactions may also be neglected for moderate intensity excitation of the QWS below Mott transition [22]. Thus to make the analysis appropriate in the absence of the different many-body effects as discussed above, we have restricted the numerical analysis of optical nutation to a detuning 1 ≥ 2 T2−1 (T2 being the dephasing time constant)

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and excitation intensity I ≤ 5 × 106 W cm−2 in GaAs/Al0.30 Ga0.70 As single QWS of well thickness of the order of 3D exciton Bohr radius aex confined by relatively thick layers of Al0.30 Ga0.70 As. Apart from the various many-body effects as discussed above, one has to recognize the roles of band nonparabolicity as well as band anisotropy on the excitonic features in a quantum well. Chaudhuri and Bajaj [23] have shown that for a GaAs quantum well of thickness L ≥ aex and sandwiched between two semi-infinite layers of Alx Ga1−x As with Al concentration x ≤ 0.4, the effect of band nonparabolicity on the ground-state exciton binding-energy is almost negligible. Moreover, for the above range of Al-concentration, one may treat both static dielectric constant and the electron effective mass as identical in both GaAs and Alx Ga1−x As layers [24]. The conduction band nonparabolicity in GaAs is prominent only when the wavevector k departs from the centre of the Brillouin zone. As is evident from the band structure calculations (Ref. [22, p. 10]), for electron energy states with |k| ≤ 0.7 × 109 m−1 , this nonparabolicity can be safely neglected. Also, the electron concentration is most thickly populated in the vicinity of k ∼ 0. Hence, for near-band gap resonant optically induced band-to-band transitions with detuning 1 ≥ 2T2−1 , the effect of band nonparabolicity may be neglected for transitions between the heavy-hole valence band and the lowest conduction band in GaAs crystal. The effect of band anisotropy can be examined by including the quadratic k-terms in the k · p interaction with higher bands in Kane’s model. Ekardt [25] studied the role of a magnetic field on real excitons in bulk GaAs crystal taking into account the role of band anisotropy of zinc-blende structures and showed that the role of anisotropy could be quite small and hence may be neglected in III–V crystals.

2. Basic formulations In order to analyse the coherent transient effects in the narrow direct gap semiconductor QWS under the near-resonant not-too-strong excitation limit, we have chosen the active material to be at a low temperature. Accordingly, one may assume the band structure to be isotropic, parabolic and nondegenerate with the crystal ground state defined in terms of completely filled top-most valence band while the lowest conduction band is empty. The band off-resonance is restricted to very small values of the electron wavevector k (|k|  π/at , at being the crystal lattice constant) around the centre of the Brillouin zone such that the complications arising otherwise due to the effects of band nonparabolicity as well as anisotropies can be neglected. QWS undergoes photoinduced direct allowed band-to-band electronic transitions. The magnetostatic field B is applied along the z-axis. We have chosen GaAs/Alx Ga1−x As QWS with x = 0.3 yielding the bandgap energy difference of around 0.5 eV such that the infinite well approximation becomes too crude. Hence, we consider the potential well to be of finite depth confining the excited e–h pairs in the GaAs layer. In order to incorporate the band gap modification due to this finite well depth, we have calculated the energy eigenvalues for the lowest discrete excitonic energy state εz assuming the same to behave as an even parity state. The energy eigenvalues for the electron (hole) εz,c(v) can be obtained graphically by using the transcendental eqn [26] ! r p m c(v) εz,c(v) √ εz,c(v) tan L = V0 − εz,c(v) (1) z 2~2 where m c(v) is the effective mass of the electron in the conduction (valence) band and L z is the width of the quantum well structure. The graphical solution shows that εz,c(v) calculated by using eqn (1) is quite 2 2 different from that obtainable under infinite well approximation where one takes εz,c(v) = 2m~ π L 2 . Thus, c(v) z to be reasonable, one should calculate the band-to-band direct transition energy in the magnetoactive QWS under finite potential well depth approximation. The magnetic field modified energies at state |ki in the conduction and valence bands are given by   ~2 k 2 1 εc = εco + + εz,c + j + ~ωcc + gc β B M j (2a) 2m c 2

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and   ~2 k 2 1 0 εv = εvo − + εz,v − j + ~ωcv + gv β B M j , (2b) 2m v 2 respectively. εco (εvo ) is the energy at the centre of the lowest conduction (topmost valence) band and εco − εvo = ~ωg , the crystal band gap energy. m c(v) , gc(v) and ωcc(cv) are the effective mass, g-factor and the cyclotron frequency respectively, in the conduction (valence) band. β = eh/(2m 0 ) is the Bohr magneton while M j and j are the spin and Landau quantum numbers, respectively. Accordingly, the direct interband transition frequency in the direct-gap QWS at state |ki is defined as   ~k 2 1 eff ω Bk = ωg + + j+ c + (gc − gv )β B M j /~ (3) 2m r 2 with ~ωgeff = ~ωg + εz,c + εz,ν . c is the e–h pair cyclotron frequency, m r is the reduced mass of the e–h pair and k 2 = k x2 + k 2y . M j (= ± 21 ) is the component of angular momentum quantum number along the magnetic field and corresponds to the two possible orientations of the electron spin. Moreover, we have taken Landau quantum number j = j 0 = 0 under the near-resonant optical transitions at moderately high magnetostatic field, such that the Landau subbands only with j = 0 are considered to participate in the interband transitions. The excitonic effects in the semiconductors under the influence of an external magnetic field change with change in the field strengths. At low magnetic field strengths, the perturbation theory works quite well within a hydrogenic set of basic functions while under strong field limit, the adiabatic decoupling scheme is preferred with the exciton binding energy being given by [27] 1/2  ~eB R ∗ D. (4) εex = 3 2(2n + 1)m r R ∗ Here, R ∗ is the effective exciton Rydberg in the 3D crystal and for QWS, we take D = 1. This excitonic binding energy leads to a shift in the transition frequency. Incorporation of the excitonic effect for the lowest Landau level ( j = 0) in the discrete quasi-bosonic exciton states below the crystal band-edge modifies eqn (3) to the form ω Bkxn = ωgeff +

(gc − gν )β B M j ~k 2 c + + − 2m r 2 ~

εex 

~ n−

1 2

2

(5)

where n = 1, 2, 3, . . .. The equations of motion of the probability amplitudes of the crystal ground state a(t) and the excited e–h pair state b(k, t) can be written in the presence of the spatially uniform coherent pump field E(t) = E0 cos ωt and the screened Coulombic e–h interaction energy hk|U |k0 i as [28] X µba (k)E 0 a(t) ˙ =i e[−i(ω Bk −ω)t] b(k, t) (6a) 2~ k

and X ˙ t) + 1 b(k, hk|U |k 0 ie[i(ω Bk −ω Bk 0 )t] b(k0 , t) ~ k

iµab (k)E 0 [−i(ω Bk −ω)t] = e a(t) − 0b(k, t). 2~

(6b)

In eqns (6), we have taken 0 = γ + γinh where γ = T2−1 and γinh represents the finite inhomogeneous linewidth of the exciton transition in the QWS. Here, the interaction Hamiltonian has been taken to be of dipole type with the dipole moment operator µ(k) ˆ acting parallel to the applied electric field E(t). The

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present semiclassical formalism to represent the equations of motion of the probability amplitudes of the crystal states is considered to be valid for moderate power excitation such that the photon density is not low enough to warrant a fully quantum mechanical treatment. This assumption is in conformity with the well-accepted fact that in a wide range of optical effects including optical coherent processes, semiclassical formalism is valid and the fully quantum mechanical forms of the optical Bloch equations are similar to their semiclassical forms in case of a coherent driving light field [29]. In the absence of the radiation field (i.e. E(t) = 0), the stationary excitonic solution is obtained from (6b) as X b0 (k, t) = Bn (k) exp[i(ω Bk − ω Bn )t] (7a) n

where Bn (k) satisfies the equation (ω Bk − ω Bn )Bn (k) +

1X hk|U |k 0 iBn (k0 ) = 0. ~ 0

(7b)

n(= 1, 2, 3, . . .)

(8a)

k

Here,

εex

ω Bn = ω B −



~ n−

1 2

2 ;

representing the discrete exciton states below the crystal band edge and   c (gc − gv ) ω B = ωgeff + + βBMj. 2 ~

(8b)

In the presence of coherent radiation, we assume the solutions of (6) to be a(t) = Ae−it

(9a)

and b(k, t) =

X

Cn Bn (k)e[i(ω Bk −−ω)t]

(9b)

n

with  and Cn being two unknown parameters whose knowledge yields valuable informations about the Rabi oscillations in the QWS. Using (6) and (9), one finds µ k E 0 2 X |ψn (0)|2  = − (10) 2~ ( + ω − ω Bn + i0) n

with |µab (k)| = |µba (k)| = µk (say). Since we are interested only in the narrow direct-gap QWS, with ω−ωg ≥ 20, we have restricted ourselves to the most prominent 1s excitonic state for n = 1. Under such a situation, we find the two solutions of (10) as (ω − ω B1 + i0) + β 1 = − (11a) 2 and (ω − ω B1 + i0) − β 2 = − . (11b) 2 Here, β = 1 − 2 = [(ω − ω B1 + i0)2 + 42ex R ]1/2 is the generalized complex Rabi flopping frequency 2 for 1s excitonic state with 2ex R = µ2k ~E 0 |ψ1 (0)|2 ; ψ1 (0) being the (n = 1) 1s exciton wavefunction.

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The two solutions of  as given by (11) modify the assumed solutions (9) which on mathematical simplifications yield for n = 1   iX i X t/2 a(t) = e cos θ − sin θ (12a) β and iµk E 0 b(k, t) = |ψ1 (0)|2 ei{(X/2)+ω Bk −ω)t} sin θ (12b) ~β where X = 1 + i0, 1 = ω − ω B1 and θ = βt/2. 1 represents the detuning between the input photon energy and the magnetic field incorporated magnetoexciton transition for lowest Landau level transitions at n = 1 and can be obtained from (8). The ensemble average of the induced excitonic transition dipole moment at state |ki is defined as h i (13) hµ(k, ˆ t)i = 2µk a(t)b∗ (k, t)e(iω Bk t) . Using (12), one finds hµ(k, ˆ t)i =

2µ2k E 0 |ψ1 (0)|2 exp(iω Bk − 0)t[X | sin θ |2 + iβ cos θ sin θ ∗ ]. |β|2 ~

(14)

From (14), it is evident that the exciton wavefunction plays an important role in the estimation of the electric dipole moment and hence the induced polarization in the QWS. The magnetic field effect incorporated 1s exciton wavefunction ψ1 (r ) in a bulk semiconductor has been studied by Sen and Sen [30] using Schrödinger’s equation of motion for the exciton of reduced mass m r under the influence of an arbitrary magnetic field given by [31]   −~2 ∂ 2 1 ∂ 1 ∂2 ∂2 r2 e2 ψ i ∂ + + + − + ψ − = Eψ (15) 2m r ∂r 2 r ∂r r 2 ∂φ 2 ∂z 2 ε(r 2 + z 2 )1/2 4a 4B a 2B ∂φ in cylindrical coordinate system (r, φ, z) treating the exciton to behave like a hydrogen atom with a B being the electron cyclotron radius, and ε = ε0 εt . Since the variables in the above equation are not separable in the presence of the Coulomb interaction term, it is not possible to obtain the exact solution. Considering a relatively strong magnetic field, the detailed analysis of Sen and Sen [30] yielded the 1s exciton wavefunction for the 3D case as 2z 3/2 2 2 |ψ1 (r )| = e[−(r /a B +z/aex )] . (16a) 3 )1/2 (2πaex One may compare (16a) with the one obtained by Chernyak et al. [17] for an almost identical physical situation. Mann [32] has discussed at length how using the Schrödinger equation for a purely 2D exciton in a magnetic field one can derive the exact numerical solution. Since the QWS with a finite potential well depth may be treated as a quasi-2D system, we have retained the exponential decay factor in the above expression for the exciton wavefunction. The 1s exciton wavefunction with r ∼ 0 in a QWS can be justifiably expressed as 2z |ψ1 (0)| = e−z/aex . (16b) 2 )1/2 (2πaex For a QWS, the motion is confined in the x–y plane and the dimension along the z direction may be approximated as the electron cyclotron radius a B . Consequently, (16b) enables one to take the 1s exciton wavefunction in a magnetoactive QWS as r 2 a B −(a B /aex ) |ψ1 (0)| = e . (16c) π aex

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Equation (16c) may be compared with that obtained by Chemla and Miller [33] while studying excitonic absorption in an unmagnetized QWS. In the study of transient optical behaviour of a magnetized QWS, it is necessary that in addition to the modification in the exciton wavefunction as discussed above, one should also incorporate the change in the density-of-states in the presence of the applied magnetostatic field. For the 2D motion, the states are characterized by the cyclotron energy levels ( j + 12 )~c , located on circles with radii k x2 +k 2y . The degeneracy of the levels arises from the constraint that the centre of motion lies within the sample. Thus for a QW sample L L y m r c of dimensions L x and L y , we take the number of possible states in the range [0, m r c L x /~] as x 2π ~ i.e. the density-of-states per unit surface area is m r c eB = . 2π~ h The induced polarization in the magnetized QWS defined by the expression NS =

(17)

P(t) = Ns hµ(k, t)i

(18)

can be obtained on using eqns (14) and (17) as P(t) =

4eB E 0 2 µ |ψ1 (0)|2 [X | sin θ |2 + iβ cos θ sin θ ∗ ] exp[(iω Bk − 0)t]. |β|2 ~2

(19)

From (19), it may be noticed that the induced polarization is complex with its real part accounting for the transient dispersive behaviour and the imaginary component corresponds to the absorptive characteristics of the outcoming radiation. The role of the magnetic field on nutating signals has been examined qualitatively in the following section. The transmitted intensity can be obtained by using Maxwell’s equations under a slowly varying envelope approximation as [28] 1 IT = n 0 ε0 c|E T |2 (20a) 2 with −ik L E T (t) = P(t) (20b) ε0 for a sample of length L, n 0 and c being the crystal background refractive index and absolute velocity of light, respectively.

3. Results and discussions The numerical analysis is made for GaAs/Al0.30 Ga0.70 As QWS assumed to be irradiated by an ultrashort pulsed tunable mode-locked Ti–sapphire laser in the wavelength spectrum 790–840 nm. The possible experimental scheme to observe the optical nutation may be either the four-wave mixing geometry or the pump-probe spectroscopy with a time resolution in the range of 50–100 fs. The material parameters used for the present numerical estimations are as follows [28, 34]: m c =0.0665m 0 , m v = 0.5m 0 (m 0 = free electron mass), εt = 11.723, γ (= T2−1 ) = 5 × 1012 s−1 (T2 ∼ 200 fs), γinh = 9 × 1011 s−1 , R ∗ = 6.0 meV, L z = 130 Å, aex = 120 Å and crystal band gap energy E g = 1.52 eV. The applied magnetic field B is chosen between 10 and 15 T. Figure 1 represents the analytical behaviour of the real part of optical polarization Pr obtainable from (19) as a function of time at three fixed values of the magnetic field B. The oscillating nature of Pr at a frequency ∼ ex R establishes the occurrence of optical nutation of dispersive type in the magnetoactive GaAs/AlGaAs QWS. Moreover, with rising magnetic field, the induced polarization is found to increase and the oscillations become more prominent. We have illustrated the nature of dependence of the imaginary component of the

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8

b c

Pr (a.u.)

6

4

2

0 0.0

0.2

0.4

0.6

0.8

t (ps) Fig. 1. The real part of the induced polarization Pr as a function of time t in GaAs/Al0.30 Ga0.70 As QWS. Curve a at B = 15 T, curve b at B = 12 T and curve c at B = 10 T. The other parameters are I = 5 × 106 W cm−2 and 1 = 10 0. a b

4

c

Pi (a.u.)

2

0

–2

0.0

0.2

0.4

0.6

0.8

t (ps) Fig. 2. Time dependence of the imaginary part of the polarization Pi in GaAs/Al0.30 Ga0.70 As QWS. Curve a at B = 15 T, curve b at B = 12 T and curve c at B = 10 T. The other parameters are I = 5 × 106 W cm−2 and 1 = 10 0.

induced polarization Pi on the time of observation in Fig. 2. This figure also manifests clearly the damped ringing behaviour of the absorption process to get enhanced with rising magnetic fields. Figure 3 displays distinctly the nutating characteristics of the transmitted components of the induced transient signal as given by (20) for different magnetic fields. The curves exhibit Lorentzian profiles. It decays to half of the maximum value within a time ∼ 200 fs. This is also taken as the dephasing time T2 and thus the nutating signal helps us to estimate the relaxation time of a material. Moreover, these curves reveal a sharp increase in the transmitted signal with rising magnetic field. As can be observed from eqns (16c) and (17),

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a

30

IT (a.u.)

b

20

c

10

0 0.0

0.2

0.4

0.6

0.8

t (ps) Fig. 3. Temporal variation of transmitted intensity I T in GaAs/Al0.30 Ga0.70 As QWS. Curve a at B = 15 T, curve b at B = 12 T and curve c at B = 10 T. The other parameters are I = 5 × 106 W cm−2 and 1 = 10 0.

the exciton wavefunction as well as the density-of-states in the QWS depend upon the applied magnetic field B. This magnetic field effect on the coherent transient processes becomes significant when one achieves the condition that the cyclotron radius a B is smaller than the exciton Bohr radius aex . Calculations show that the threshold value of B for a B ∼ aex turns out to be around 5 T and one has to select the magnetic field B > 5 T in the numerical estimates of various nutating parameters in GaAs/AlGaAs QWS [32]. For magnetic fields greater than this so-called threshold value, one may observe a rise in the peak values of the induced polarization with no shift along the time axis when the magnetic field is increased. Accordingly, we have selected the range 10 ≤ B ≤ 15 T for our numerical analysis. These results are in good qualitative agreement with those of Kner et al. [16] where they found the signal peak intensity to an increase with increase of magnetic field. Experimentally, the differential transmission spectrum is usually observed to confirm the occurrence of the coherent transient effects in the active medium since the spectrum yields sensitive and reliable informations on the semiconductor transmission characteristics. However, for the sake of simplicity, we have analysed the nature of variation of the transmitted intensity IT on the detuning parameter 1 as calculated by using eqn (20) for GaAs/AlGaAs QWS in Fig. 4. The figure demonstrates the exponentially decaying nature of IT superimposed on the nutating characteristics. The figure further manifests that IT depends on the applied magnetic field and it reduces sharply as one departs even slightly away from the nearly sharp resonant photoinduced electronic transition regime. Quite interestingly, one may note that with increasing magnetic field B (from 10 T to 15 T), the transmitted intensity becomes enhanced. Such enhancement can enable one to study hot-carrier dynamics in quantum well structures with much higher precision while immersed in a moderately strong magnetostatic field.

4. Conclusions The present paper has aimed to develop a theory of coherent optical transient effects like optical nutation in a strongly magnetized semiconductor quantum well structure with finite potential well depth. The simplistic model based upon the semiclassical perturbation approach for coherent radiation–semiconductor interaction under certain realistic approximations predicts enhancement in the transient polarization and hence the dis-

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1.0 a

IT (a.u.)

0.8

0.6

0.4 b c

0.2

0.0 0

2

4 1 × (1013 s–1)

6

Fig. 4. Variation of the transmitted intensity IT with the detuning parameter 1 in GaAs/Al0.30 Ga0.70 As QWS for different magnetic fields. Curve a at B = 18 T, curve b at B = 15 T and curve c at B = 12 T.

persive as well as absorptive characteristics of the QWS with increasing magnetic field. The role of magnetic field on the quasi-2D exciton binding energy and wavefunction has been included in the analysis. Approximations such as the neglection of various many-body effects including the band gap renormalization and band characteristics in a real semiconductor in terms of band nonparabolicity and band anisotropy have been reasonably justified for nearly sharp band gap resonant band-to-band transitions. Numerical analysis is made for a single QWS made up of GaAs layer of thickness comparable to bulk exciton Bohr radius sandwiched between two thick layers of Al0.30 Ga0.70 As. The applied magnetic field is kept between 10 and 15 T. The results manifest the occurrence of optical nutation of both dispersive and absorptive types, which increase significantly with increasing magnetic field in a semiconductor quantum well structure under ultrafast moderate power coherent excitation regime. Acknowledgements—The authors are indebted to Dr Pratima Sen for many stimulating discussions. The financial supports from the Departments of Atomic Energy and Science & Technology, Government of India are gratefully acknowledged. One of the authors (SB) is grateful to the Council of Scientific and Industrial Research, New Delhi for the award of a Senior Research Fellowship.

References [1] L. Allen and J. H. Eberly, Optical Resonance and Two-level Atoms (John Wiley, New York, 1975). [2] P. L. Knight and L. Allen, Concepts of Quantum Optics (Pergamon, Oxford, 1985). [3] H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1993) 2nd edn. [4] I. A. Walmsley and J. D. Kafka, Contemporary Nonlinear Optics, edited by G. P. Agrawal and R. W. Boyd (Academic, Boston, 1992).

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