AlGaAs quantum well structures

Superlattices and Microstructures 40 (2006) 10–18 www.elsevier.com/locate/superlattices

Effect of polarization state of light on magnetoabsorption and optical nutation in GaAs/AlGaAs quantum well structures Saral K. Gupta, Sheetal Kapoor, Jitendra Kumar, Pranay K. Sen ∗ Department of Applied Physics, Shri G. S. Institute of Technology and Science Indore, 452003 (M.P.), India Received 23 June 2005; accepted 6 January 2006 Available online 3 March 2006

Abstract Rigorous dipole selection rules are derived for an interacting electron–hole system in a multiband magneto-active GaAs/AlGaAs semiconductor quantum well (QW). The valence band structure is modelled using an 8 × 8 Luttinger Hamiltonian under the axial approximation. For three different polarization states of input pulse radiation i.e. linearly (σ z )-, right circularly (σ + )- and left circularly (σ − )-polarized, we have studied the absorption characteristics of a GaAs/AlGaAs QW under the influence of a magnetic field applied perpendicular to the growth direction. A strong blue shift of the absorption peaks is observed with increasing magnetic field. Also, spin splitting of the absorption peaks is seen for the case of linearly polarized input radiation. Numerical estimation of optical nutation has been made for GaAs/AlGaAs under the influence of strong magnetic field and it is observed that nutating signal frequency enhances with increasing magnetic field. c 2006 Elsevier Ltd. All rights reserved.  Keywords: A. Nanostructures; Quantum well; D. Optical properties; Spin–orbit effects; Electronic band structure

1. Introduction The field of spintronics involves a whole new area of active manipulation of carrier spin dynamics. This control of spin through application of external magnetic field or by shining polarized light is envisioned to lead to novel magneto-electro-optical technology. Recently, ∗ Corresponding author. Tel.: +91 731 2434095; fax: +91 731 2432540.

E-mail address: [email protected] (P.K. Sen). c 2006 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter  doi:10.1016/j.spmi.2006.01.002

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spin effects have begun to attract considerable interest to understand the physical properties as well as in connection with several device applications such as spin transistors, spinfilters, modulators, new memory devices and quantum computers [1,2]. The interest in spin effects has been spurred by the observation of long spin-coherence times in semiconductors and their nanostructures, which make solid state implementations of quantum information processing attractive. Examples include the control of spin in quantum dots and other heterostructures for realization of quantum computing devices. To observe these processes in semiconductor structures, the ability to enhance and control the spin splitting plays a key role. The carrier spin-based technology derives importance from the fact that spin can store information due to its polarization properties. Currently used methods of polarizing electron spins include magnetic field, optical orientation and spin injection. The most widely studied systems are GaAs/AlGaAs QWs and QDs for modern spin based technology [1,3,4]. The most critical physical process needed to understand spin effects on the optical properties of nanostructures is the Zeeman splitting in a magnetic field. The splitting is determined by mixing of the spin with the electronic states. Recent advances in the understanding of spin effects on the optical properties of nanostructures include also the electron–hole (e–h) Landau level splitting [5]. This splitting in quantum wells has strong nonlinear dependence on the magnetic field. In order to understand this effect, reliable theoretical models are needed for realistic systems where the parameters such as size, shape, alloy concentration, potential of the semiconductor nanostructures are known sufficiently well. Many researchers have carried out magneto-photoluminescence studies [6,7] which has emerged as an important area of research in nanostructures where investigations are performed making use of high magnetic field that influences the electronic properties of the nanostructures. In order to investigate the optical properties of semiconductor nanostructures, we need to evaluate the interband transition dipole matrix elements between the energy levels in the  p theory, the selection rules become conduction and valence bands. In the framework of k. more involved as the hole energy levels are no longer purely heavy- or light-hole states, but an admixture of a number of components [8]. In general, valence band states show a p-type symmetry of the orbital angular momentum of the Bloch states (i.e. l = 1) with total angular momentum quantum no. j = 32 , 12 . Typically in the case of III–V semiconductors, the split-off band with j = 12 does not affect the j = 32 states significantly as the spin–orbital splitting (Δ) is quite large. Thus we consider only the transition taking place from the j = 32 states of the valence band to the s-type conduction band. In this paper, we report the results of the analytical investigation of optically induced interband electronic transitions in a single magneto-active GaAs/AlGaAs QW by using different polarizations of input radiation. For detailed understanding of the polarization dependence and  p Luttinger Hamiltonian for the valence band in III–V interband transitions, an eight-band k. semiconductor is used which includes the coupling of light-hole (lh) and heavy-hole (hh) valence subbands. It has been further modified to include the magnetic field effects. This Luttinger Hamiltonian is diagonalized and dipole matrix elements for different transitions are obtained. The inclusion of polarization dependent dipole matrix elements and e–h transition strengths allows accurate predictions of excitation spectra of quantum wells. In addition to this, we have also carried out the analyses in the time domain and theoretically studied the phenomenon of optical nutation. The valence subband mixing and magnetic field effects manifest in the form of a complicated nutation signal which gives information about the strength and frequency separation

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of the transitions from different heavy- and light-hole states. Chu et al. [9] have reported lineshape theory of magnetoabsorption for low and intermediate values of magnetic fields which includes the excitonic effects. The Coulombic effects can be easily neglected at higher magnetic fields because the magnetic confinement becomes dominant in this regime. In Section 2, we present the theoretical analysis of coherent transient process of optical nutation and absorption characteristics in a strongly magnetized GaAs/AlGaAs QW structure where the dipole matrix elements for total allowed transitions are also given. Section 3 deals with the results and discussions of numerical estimation of differential transmission spectra (DTS) and transient nutation of induced polarization in GaAs/AlGaAs semiconductor QW. The important conclusions are drawn in Section 4. 2. Theoretical formulations In this section, we develop an analytical framework to study the interband absorption characteristics and optical coherent transient processes in GaAs/AlGaAs QW under the Faraday configuration using circular and plane polarized light fields. Selection rules are derived to evaluate both allowed and forbidden transitions at wave vector k = 0. We shall first treat an idealized semiconductor with eight valence and four conduction subbands having a parabolic energy–momentum relationship. This model is subsequently used to derive simple selection rules and to obtain the expected shape of absorption spectra as a function of photon energy for direct interband transitions. 2.1. Hamiltonian formalism We begin with the well known Luttinger–Kohn [10] Hamiltonian to incorporate the multiple valence subband structure in our formalism along with the modification in the energy of the states ˚ and the aluminium concentration due to magnetic field effects. The well thickness L(= 70 A) x(= 0.05) in Al x Ga1−x As are chosen such that we have two hh and two lh states in the valence band and two electron states in the conduction band. These states being spin degenerate, split on application of magnetic field, yielding eight hole subbands and four electron states in the conduction band. For the valence band, we obtain the Luttinger Hamiltonian as an 8 × 8 matrix in our case under the influence of a moderately strong magnetic field. The electron and hole motions in the transverse x y-plane are affected by magnetic field applied along the thickness in the z-direction. The wavefunction for the valence band can be written as  f mv j (r )u vm j (r ) exp(i ky) (1) ψ v (r ) = mj

with the sum extending over the allowed values of m j , viz. ± 32 for hh and ± 12 for lh. u vm j (r ) are the degenerate Bloch functions. In our case, fmv j is comprised of two components χ v (z) and φ v (x) to account for the quantum confinement and magnetic field effects, respectively. Since the magnetic field in Landau gauge leads to a potential similar to that of a harmonic oscillator, we choose φs to be the Hermite Gaussian functions. Thus, the basis functions for the valence bands in our case can be written as  1 2 (z) φn−2 (x) u 3 (x, y, z); χhh (z) φn−2 (x) u 3 (x, y, z); χhh 2

1 χlh (z)

φn (x) u − 1 (x, y, z); 2

2

2 χlh (z)

φn (x)

u − 1 (x, y, z); 2

S.K. Gupta et al. / Superlattices and Microstructures 40 (2006) 10–18 1 χlh (z) 1 χhh (z)

φn−1 (x) u 1 (x, y, z);

2 χlh (z)

2

φn+1 (x)

u − 3 (x, y, z); 2

φn−1 (x) u 1 (x, y, z); 2

2 χhh (z)

φn+1 (x) u − 3 (x, y, z)

13

 (2)

2

where the superscripts 1 and 2 denote the two quantum well levels considered. In the conduction band, the wave functions can be written as  χe1 (z)φn−1 (x)u 1 (x, y, z); χe1 (z)φn (x)u − 1 (x, y, z); 2 2  (3) 2 2 χe (z)φn−1 (x)u 1 (x, y, z); χe (z)φn (x)u − 1 (x, y, z) . 2

2

The total 8 × 8 Hamiltonian that determines the solutions for the hole wavefunctions is given by ⎡A ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

hh1 − 3C1 B1 a Ď2

0 0 0 0

Δ12 a Ď 0

B1 a 2 Alh1 + C1 0 0 0 0 0 Δ21 a Ď

0 0 Alh1 − C1 B1 a Ď2 Δ21 a 0 0 0

0 0 B1 a 2 Ahh1 + 3C1 0 Δ12 a 0 0

0 0

Δ21 a Ď 0 Ahh2 − 3C1 B2 a Ď2 0 0

0 0 0

Δ12 a Ď B2 a 2 Alh2 + C1 0 0

Δ12 a 0 0 0 0 0 Alh2 − C1 B2 a Ď2

⎤ 0 Δ21 a ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎦ 2 B2 a Ahh2 + 3C1

(4)

Here, Aλi = E λi + h¯ ωλ (a Ď a + 1/2) with E λi being the edge energies of the different frequencies. subbands of the QW in the z-direction and

 ωλ = eB/m λ are the cyclotron √  i j i i 2 3(γ2 + γ3 )/(2lc2 m 0 ), Bi = C2 χhh |χlh , Δi j = 2γ3 C3 χhh |∂z |χlh ; C1 = κμ B B and C2 = h¯ √ C3 = h¯ 2 3/(2m 0lc ); i, j take the values 1, 2 where γ1 , γ2 , γ3 and κ are the Luttinger parameters [11]. Also a and a Ď are the √ Landau level lowering and raising operators, λ = lh, hh; m 0 is the electron rest mass, lc (= h¯ /eB) is the magnetic length and μ B (= eh¯ /(2m 0 )) is the Bohr magneton. The 8 × 8 matrix in Eq. (4) is diagonalized to obtain the energy eigenvalues and the corresponding eigenvectors W j , j ranging from 1 to 8. 2.2. Transition dipole moment matrix elements and selection rules We now calculate the transition dipole moment matrix elements according to the selection rules to find the allowed and forbidden transitions. The dipole matrix element for transitions between the valence subband states and conduction band states can be written as [12]  



μ j = ψmc s | . p|ψmv j = (5) Jm j u cm s | . p|u vm j . mj

In our case, Jm j = χ c (z)φe (x)|χ v (z)φv (x)

(6)

and the matrix element 

u c | pv |u v  = (i m 0 P/h¯ )δv,v  ;

v, v  ∈ {x, y, z}

(7)

where  is the unit vector in the direction of the electric field and p is the momentum operator. For GaAs, the value of the Kane matrix element P is calculated as 2m 0 P 2 = 22.71 eV. The dipole matrix elements must satisfy certain selection rules for a transition to be allowed [13]. These include the usual dipole selection rules which depend on the polarization

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Table 1 The dipole matrix elements for optical transitions according to selection rules mj

ms

RCP (σ + )

LCP (σ − )

Linearly polarized (σ z )

+ 32

+ 12

0

−i √0 [u c ↓ | px,y |x, y ↓]

0

+ 32 − 12 − 12 + 12 + 12 − 32 − 32

− 12 + 12 − 12 + 12 − 12 + 12 − 12

0 E i √0 [u c ↑ | px,y |x, y ↑]

0 0

0 0

0

0

i E0

0

0

i E0

0

E −i √0 [u c ↑ | px,y |x, y ↑] 6

0 E i √0 [u c ↓ | px,y |x, y ↓]

0 0

6

2

E

2



2 [u c ↓ | p |z ↓] z

3

2 [u c ↑ | p |z ↑] z 3

0 0 0

of the incident light and that of the envelope functions. Transitions are allowed only between the states with identical parity due to envelope function selection rules. Also, the dominant allowed transitions obey the selection rules Δn = 2 and −2 where n is the Landau orbital quantum number. According to these rules, if light propagating in the z-direction and polarized along the x- or y-direction is incident on a quantum well, then transitions from heavy- and light-hole states occur with the heavy-hole transitions being stronger. On the other hand, for the polarization along z, heavy holes do not participate in optical transitions and the light holes dominate. These rules can be used to predict the contributions of light and√heavy holes to absorption. For light polarized along the x-axis, the matrix element is i m 0 P/(h¯ 2) for the heavy holes and √ i m 0 P/(h¯ 6) for the light holes. The absorption, proportional to the square of the matrix element is therefore, in the ratio of 3:1. There is no contribution from holes if light is polarized along zaxis. This analysis has been further carried out for the case of left- and right-handed circularly polarized light. The dipole transition matrix elements (μj s) obtained according to these selection rules are given in Table 1. We can obtain the induced polarization in the medium using these transition dipole matrix elements and the wavefunctions obtained by diagonalization of the 8 × 8 matrix Eq. (4) [14]. The equation used to calculate the induced polarization is given by  μ j |ξe (0)ξ j (0)|2 Φˆ j + c.c. (8) P= j

where μ j is the expectation value of dipole moment calculated in terms of the hole wavefunctions. ξ j , ξe are the j th species of hole and electron wavefunctions, respectively. Φˆ j is the time dependent expansion coefficient [14] calculated using effective semiconductor Bloch Equations [15]. The induced polarization thus obtained is used to study the DTS in a semiconductor QW structure under the influence of moderately strong magnetic field defined by the relation ωL Im[P(0, ω).E 1∗ (0, ω)] . (9) DTS = 0 cηω |E 1 (0, ω)|2 Here, ηω is the background refractive index of the QW material at frequency ω while P(0, ω) and E 1 (0, ω) are obtained by taking the Fourier transforms of P(0, t) and E 1 (0, t), respectively. Eq. (9) enables one to study the absorption coefficient Δα(= −DTS/L).

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3. Results and discussions We present here the detailed numerical analyses of spectral and temporal characteristics and optical coherent transient effect of optical nutation of induced polarization in a GaAs/AlGaAs QW under the influence of a moderately strong magnetic field applied normal to the QW surface. The QW is assumed to be illuminated by a femtosecond pulsed Ti:sapphire laser. The material parameters used are: m e = 0.067m 0, Luttinger parameters γ1 = 6.85, γ2 = 2.1, γ3 = 2.9 and κ = 1.2 [11,16], dephasing time T2 = Γ −1 = 10−12 s, bulk crystal bandgap energy, ˚ E g = 1.5117 eV, QW width L = 70 A. We have studied the absorption characteristics in magnetoactive GaAs/AlGaAs QWs for different polarization states of the input radiation. A perturbative approach is developed for spin splitting in nanostructures which shows in a simple and intuitive way that confinement due to magnetic field and the polarization states of incident electromagnetic radiation are intimately connected with the blue shifting and spin splitting as shown in Fig. 1. These results provide an explanation of the dependence of optical transitions on polarization of incident radiation in quantum well structures. In Fig. 1(a), we plot the DTS for linear polarization of light for different values of magnetic field. Though our formulation is applicable for high magnetic field we have shown the spectrum at B = 0 for the sake of comparison. According to the selection rules [see Table 1], the linear polarized light excites spin-up and spin-down light holes only. Application of magnetic field results in the spin splitting of the absorption peak and splitting increases with increasing magnetic field. Similar results are reported by Barticevic et al. [17], Fr¨omer et al. [18] and Munteanu et al. [6]. The most prominent peak appearing in the Fig. 1(a), splits in two parts corresponding to light-hole spin-up and down states and the splitting increases with increase in the applied magnetic field. This enhancement in splitting can be attributed to the factor C1 defined in Eq. (4) in our theoretical model. This is a manifestation of the energy shifting of spin-up and spin-down states with increasing magnetic field. This feature is absent in absorption spectra plotted for the circularly polarized (σ + and σ − ) input radiation as shown in Fig. 1(b) and (c). The σ + and σ − polarized lights excite only one hole each in the light- and heavy-hole bands as shown in Table 1. Thus, the spin splitting due to magnetic field is not observed, as the allowed transition is either for spin-up or spin-down states. Further, it is clear from these figures that under both σ + and σ − polarized excitation, the DTS exhibits almost the same features. This has been observed experimentally by Gerlovin et al. [19]. In addition to this, the blue shift of the absorption peaks with increase in magnetic field as found in the present work agrees well with the results reported by Barticevic et al. [17]. We observe a number of small peaks in the spectra arising due to the valence band mixing effect [7]. In the absence of band mixing, the parabolic confinement leads to only two dipoletransitions for every polarization state, with the well known resonance frequencies for quantum wells. The strong coupling of light-hole and heavy-hole energy states leads to substantial shift of the resonance frequencies and introduces additional peaks in the spectra. These results obtained by us are in good agreement with those obtained by Dickmann et al, Citrin et al. and Berger et al. [20–22]. We have also plotted the imaginary part of total induced polarization as a function of time t (Fig. 2). The valence subband mixing effect manifests itself in the form of multiple oscillations in the nutating signal. The multiple oscillations in the nutation signal arise due to the interference of the radiation of different frequencies as seen from the absorption curves [23]. Moreover, the magnetic field increments result in the increase in nutating signal frequency which is due to the blue shift as seen from the absorption spectra. This may be effectively employed in

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Fig. 1. Absorption spectra for GaAs/AlGaAs quantum well with varying magnetic field for (a) plane polarized, (b) right handed circularly polarized and (c) left handed circularly polarized input radiation.

understanding various coherent transient optical processes in QW structures. Berger et al. [22] have shown experimentally that the applied magnetic field leads to faster Rabi oscillations in QW microcavity.

S.K. Gupta et al. / Superlattices and Microstructures 40 (2006) 10–18

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Fig. 2. Temporal characteristics of induced polarization in a GaAs/AlGaAs quantum well with varying magnetic field for (a) plane polarized, (b) right handed circularly polarized and (c) left handed circularly polarized input radiation.

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S.K. Gupta et al. / Superlattices and Microstructures 40 (2006) 10–18

4. Conclusions The effect of polarization state of light on magnetoabsorption and optical nutation in GaAS/ AlGaAs QW has been analyzed theoretically under Faraday configuration. We have applied the appropriate selection rules to calculate the transition dipole matrix elements for different states of polarization of the input light. The Luttinger Hamiltonian formalism has been used to include valence subband mixing in the presence of magnetic field. We find Zeeman splitting of the spinup and spin-down lines in case of light linearly polarized along the magnetic field direction. This Zeeman splitting is not observed for the circularly polarized light due to the contribution of only a single spin state following the selection rules. The spectra show the expected blue shift with the increase in magnetic field. Appearance of additional lines with magnetic field has been explained in terms of band mixing. The optical nutation signal also shows multiple oscillations due to magnetic field and band-mixing effects. The results are found to be in good agreement with experimental observations as available in the literature. Acknowledgements The authors are thankful to Dr. (Mrs.) P. Sen for fruitful discussions. The financial support from Department of Science and Technology, New Delhi is gratefully acknowledged. One of the authors (SKG) is thankful to the Council of Scientific and Industrial Research, New Delhi, for the award of Senior Research Fellowship. References [1] L.M. Woods, T.L. Reinecke, R. Kotlyar, Phys. Rev. B 69 (2004) 125330. [2] V.I. P¨uller, L.G. Mourokh, N.J.M. Horing, A.Y. Smirnov, Phys. Rev. B 67 (2003) 155309. [3] R.I. Dzhioev, V.L. Korenev, B.P. Zakharchenya, D. Gammon, A.S. Bracker, J.G. Tischler, D.S. Katzer, Phys. Rev. B 66 (2002) 153409. [4] P. Machnikowski, L. Jacak, Phys. Rev. B 69 (2004) 193302. [5] R. Kotlyar, T.L. Reinecke, M. Bayer, A. Forchel, Phys. Rev. B 63 (2001) 085310. [6] F.M. Munteanu, Y. Kim, C.H. Perry, D.G. Rickel, J.A. Simmons, J.L. Reno, Sol. State Commun. 114 (2000) 63. [7] M. Bayer, T.L. Reinecke, S.N. Walck, V.B. Timofeev, A. Forchel, Phys. Rev. B 58 (1998) 9648. [8] E.L. Ivchenko, G.E. Pikus, Superlattices and Other Heterostructures, Springer-Verlag, Berlin, 1997. [9] H. Chu, Y. Chang, Phys. Rev. B 89 (1989) 5497. [10] J.M. Luttinger, W. Kohn, Phys. Rev. 97 (1957) 869. [11] T. Darnhofer, D.A. Broido, U. R¨ossler, Phys. Rev. B 50 (1994) 15412. [12] U. Bockelmann, G. Bastard, Phys. Rev. B 45 (1992) 1688. [13] J.H. Davies, The Physics of Low Dimensional Semiconducors: An Introduction, Cambridge University Press, Cambridge, 1998. [14] S. Kapoor, J. Kumar, P.K. Sen, J. Appl. Phys. 95 (2004) 4833. [15] M. Lindberg, S.W. Koch, Phys. Rev. B 38 (1988) 3342. [16] I. Vurgaftmann, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys. 89 (2001) 5815. [17] Z. Barticevic, M. Pacheco, C.A. Duque, L.E. Oliveira, J. Appl. Phys. 92 (2002) 1227. [18] N.A. Fr¨omer, C. Sch¨uller, C.W. Lai, D.S. Chemla, I.E. Perakis, D. Driscoll, A.C. Gossard, Phys. Rev. B 66 (2002) 205314. [19] I.Ya. Gerlovin, Yu.K. Dolgikh, S.A. Eliseev, V.V. Ovsyankin, Yu.P. Efimov, I.V. Ignatiev, V.V. Petrov, S.Yu. Verbin, Y. Masumoto, Phys. Rev. B 69 (2004) 035329. [20] S. Dickmann, A.I. Tartakovskii, V.B. Timofeev, V.M. Zhilin, J. Ziman, G. Martinez, J.M. Hvam, Phys. Rev. B 62 (2000) 2743. [21] D.S. Citrin, S. Hughes, Phys. Rev. B 61 (2000) R5105. [22] J.D. Berger, O. Lyngens, H.M. Gibbs, G. Khitrova, T.R. Nelson, E.K. Lindmark, A.V. Kavokin, M.A. Kaliteevski, V.V. Zapasskii, Phys. Rev. B 54 (1996) 1975. [23] N.A. Fr¨omer, P. Kner, D.S. Chemla, R. L¨ovenich, W. Sch¨afer, Phys. Rev. B 62 (2000) 2516.