In-plane optical anisotropy of parabolic and half-parabolic Cd1−xMnxTe quantum wells

Solid State Communications 126 (2003) 467–471 www.elsevier.com/locate/ssc

In-plane optical anisotropy of parabolic and half-parabolic Cd12x MnxTe quantum wells K. Kowalika,*, A. Kudelskia, J.A. Gaja, T. Wojtowiczb, O. Krebsc, P. Voisinc a Institute of Experimental Physics, Warsaw University, Hoz˙a 69, 00-681 Warszawa, Poland Institute of Physics, Polish Academy of Sciences, Al. Lotniko´w 32/46, 02668 Warszawa, Poland c Laboratoire de Photonique et Nanostructures, CNRS, Route de Nozay, 91460 Marcoussis, France b

Received 4 February 2003; accepted 19 February 2003 by M. Grynberg

Abstract We report an investigation of the in-plane anisotropy of parabolic (PQW) and half-parabolic (HPQW) Cd12x Mnx Te quantum wells (QWs) grown by molecular beam epitaxy on (001) oriented GaAs substrates. For HPQW we obtain a good description using a microscopic model related to the atomic structure of interfaces. For PQW a strong anisotropy of excitonic reflectivity (not predicted by the model) is also observed. This result suggests that extrinsic contributions must be considered in realistic analysis of in-plane anisotropy of semiconductor quantum structures. q 2003 Elsevier Science Ltd. All rights reserved. PACS: 78.66; 68.35.C Keywords: A. Parabolic quantum wells; C. Oriented defects; D. Optical in-plane anisotropy

1. Introduction The optical in-plane anisotropy of (001) oriented quantum wells (QWs) associated with the breakdown of the interface symmetry, recently became a subject of many experimental and theoretical investigations (Gourdon [1], Krebs [2], Schmidt [3], Yakovlev [4], Cortez [5], Kudelski [6], Foreman [7], Magri [8], Toropov [9], Ivchenko [10]). By lowering the point group symmetry D2d of a quantum well in zincblende material to C2v -symmetry; a linear polarization anisotropy of the optical properties for light propagating along the growth direction is induced. Nonequivalence of two interfaces of a quantum well can cause the symmetry reduction. This was observed in several kinds of heterostructures made out of materials sharing no common atom [2,5], or NCA-QWs. In this case the asymmetry of left and right interfaces is predominantly due to intrinsic * Corresponding author. Tel.: þ48-22-55-32-127; fax: þ 48-22621-97-12. E-mail address: [email protected] (K. Kowalik).

difference in interfacial bonding. More generally anisotropy can appear in QWs (CA or NCA) having an asymmetric potential profile, caused either by an external (electrostatic) potential [2,5,9] or by gradients in the chemical composition [5,6]. Yet, another cause of optical anisotropy, which we shall call ‘extrinsic’ is likely to occur due to the existence of in-plane oriented defects. An example of such oriented defects is the formation of anisotropic islands when growing sub-monolayers, or the development of anisotropic roughness during the growth of thicker layers. To some extent, such extrinsic anisotropy can be controlled by using slightly miscut substrates, or vicinal surfaces [11]. There are important conceptual differences between the ‘intrinsic’ effects that respect in-plane translational invariance, and extrinsic effects that rely on its breakdown. Clearly, sorting out the intrinsic and extrinsic contributions to the observed anisotropy is an important issue if we want to use polarization resolved spectroscopy for the characterization of quantum structures. Here, we present measurements of in-plane anisotropy of parabolic and half-parabolic QWs in the CdMnTe– CdTe system. These potential profiles have been produced in molecular beam epitaxy using the ‘digital’

0038-1098/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0038-1098(03)00190-X

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technique [12]. Simulation of a gradual alloy composition is obtained by varying the ratio of well/barrier material aperture times in growth sequences having a fixed thickness of about 2 monolayers. This means that submonolayers of well or barrier materials are involved in most growth steps. Hence these structures are likely to combine anisotropies of intrinsic and extrinsic origins.

2. Samples The samples used in our experiment are Cd12x Mnx Te heterostructures grown by molecular beam epitaxy in the Institute of Physics of Polish Academy of Sciences in Warsaw. Two single QWs and one multiple quantum well (MQW) have been produced by the digital technique [13, 14]. In all cases the Mn mole fraction at the bottom of the well was x ¼ 0: The structures were grown on (001) oriented GaAs substrates with CdTe buffers, of a thickness of 3 mm for samples with HPQWs and 4 mm for the sample with PQW. The parameters of the structures are collected in Table 1. Two samples have single QWs: 09155A (half˚ ), parabolic, the well width Lz ¼ 80 ML (1 ML ¼ 3.24 A molar fraction of Mn in barrier xb ¼ 0:5) and 09205C (parabolic, Lz ¼ 82 ML; xb ¼ 0:78). The third sample 11155A contains a MQW: ten wells separated by 100 ML barriers (single well width Lz ¼ 60 ML; xb ¼ 0:25). A precise characterization of the samples was reported elsewhere for HPQW [13] structures and for PQW [14] structures. The characterization performed in Refs. [13,14] included calculation of electron and hole states in ideal parabolic or half-parabolic potentials, producing a good agreement with experimental transition energies. In this work we shall use more realistic potential profiles to account for the subtle anisotropy effects.

3. Experimental setup For the determination of in-plane anisotropy we applied an experimental setup similar to that used by Kudelski et al. [6]. The samples were mounted strain-free, immersed in liquid helium at 1.9 K and illuminated by a tungsten halogen lamp. We assume that the light collimated on the sample is randomly polarized (polarization by tungsten lamp is weak).

A linear polarizer was placed in the reflected light and was rotated during measurements. The anisotropy was measured as an amplitude variation of the reflectivity structure with polarization angle. We used also another method in which the experimental setup was modified by adding two elements. The polarization of incident light was selected by placing another linear polarizer set at 458 with respect to the (110) directions. A Babinet– Soleil compensator was also located in the light beam, its optical axis parallel to one of these directions. This configuration allowed us to obtain two spectra: that of the rotation qðvÞ of the reflected light polarization and that of the phase shift dðvÞ between two perpendicular polarizations.

4. Theoretical outline In-plane anisotropy is induced by mixing of the heavyhole and light-hole states. We follow the approach used in Ref. [6], where a variant of the model proposed by Ivchenko et al. [10] was used. The light– heavy hole mixing at G point is described in terms of a perturbation Hamiltonian, which in the base lG8 ; ^3=2l; lG8 ; ^1=2l has the form: 1 0 0 0 0 1 C B B 0 0 21 0 C C i"2 ›p B C B ð1Þ t Hl – h ¼ C 2m0 a0 l – h ›z B B 0 1 0 0C A @ 21 0 0 0 where " is Planck’s constant divided by 2p, m0 —free electron mass, a0 —lattice constant for CdTe, tl – h — dimensionless parameter characteristic for the hole mixing, ›p=›z—gradient of the QW normalized potential profile [6]. We solve Schro¨dinger equation of a hole with potential of the well and potential of interface anisotropy Hl – h : Numerical simulations of electron—as well as light—and heavy hole states were performed using effective masses 0.099, 0.512 and 0.1 for electrons, heavy holes and light holes, respectively [15,16]. CdTe – CdMnTe relative valence band offset of 0.4 was used [17]. Potential Hl – h mixes only lG8 ; 3=2l with lG8 ; 21=2l and lG8 ; 23=2l with lG8 ; 1=2l: First we solve separately one-dimensional Schro¨dinger equations for electrons as well as light and heavy holes confined in the quantum well without the anisotropy term Hl – h : We obtain C0hh ; C0lh and Ce (zero order hole and electron wavefuntions). Then we include the

Table 1 Growth parameters of investigated samples Sample name

Type of structure

QW width (ML)

Digital growth period (ML)

Barrier composition

09205C 09155A 11155A

P-QW HP-QW HP-MQW

80 80 60

2 2 1.5

0.78 0.5 0.25

K. Kowalik et al. / Solid State Communications 126 (2003) 467–471

anisotropy term in a perturbation calculation obtaining a corrected heavy-hole wavefunction: lChh l ¼ lC0hh l þ

Vl – h lC0lh l 0 0 Ehh 2 Elh

0 0 ; Elh —energies of where Vl – h ¼ kC0hh lHl – h lC0lh l and Ehh heavy-and light-hole states in the well. The degree of linear polarization A0 is defined as a ratio of difference and sum of the squared absolute values of matrix elements D ^ :

A0 ¼

lDþ l2 2 lD2 l2 lDþ l2 þ lD2 l2

D^ are dipole matrix elements of e1h1 excitonic transition for two linear polarizations 1 D^ ¼ hCe j pffiffi ðX ^ YÞjChh i 2 in our approximation we get finally: "2 Pl – h Ilh ¼ CTl – h A0 ¼ tl – h pffiffi 0 2 E0 Þ I 3m0 a0 ðEhh hh lh

ð2Þ

where D ›p E Pl – h ¼ C0hh C0lh ; ›z ð Ihh ¼ Ce ðzÞC0hh ðzÞdz

Ilh ¼

ð

Ce ðzÞC0lh ðzÞdz;

We introduce additionally normalized parameter Tl – h (tl – h divided by molar fraction of manganese in the barrier), which is expected to be composition-independent. The numerical simulation was performed, with a potential profile including the segregation effects [17]. Since spin tracing [18], routinely used for determination of intermixing length, is not applicable here, a typical value [15,18] of Lint ¼ 1 ML was used. The obtained value of dimensionless C coefficient was 0.042 and 0.038 for samples with HPQWs: 091155A and 11155A, respectively. The variation of C coefficient with intermixing length (decreased to zero and increased to 1.5 ML) does not exceed ten percent, so the choice of the precise value of the intermixing length is not critical. We have done the same calculation for the PQW. Calculated anisotropy is of course zero in the absence of interdiffusion, but becomes finite when the small asymmetry due to interface segregation is introduced. However, the obtained values for the C coefficient always remain at least an order of magnitude smaller than those obtained in the HPQWs.

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spectra obtained using the first method. The following figures show the comparison of the anisotropy to the mean reflectivity spectra for two samples: one with parabolic QW and one with half-parabolic QW. In order to eliminate parasitic polarization coming from the experimental setup (assumed to be slowly varying with photon energy) our analysis was performed on logarithmic reflectivity spectra [6]. Since the spectral response of the experimental setup was flat in the range of interest, we used directly the reflected intensity. We define dimensionless anisotropy coefficient A0 as the ratio of the line amplitudes in anisotropy and logarithm of the reflected intensity. Our analysis consisted in fitting both reflectivity and anisotropy spectra with the same Lorentzian lines, with independently chosen linear background and a scale factor A0 between the two spectra (Figs. 1 and 2). The calculated anisotropy coefficients for both used experimental methods are collected in the Table 2. We were able to describe the anisotropy values obtained for half-parabolic QWs in terms of the microscopic model discussed in Chapter 4. Values of the coefficient of hole mixing tl – h were obtained from Eq. (1). The results are presented in Fig. 3 and in Table 3. Indeed, the two values of Tl – h are very close, although the compositions differ by a factor of two. The obtained Tl – h values were compared (see Appendix) with a theoretical calculation based on HBF model [2,5]. In this approach, the parameter governing the h – l mixing is the valence band offset. The predicted value of Tl – h is 0.54, in excellent agreement with our experimental findings. Our numerical simulations yielded optical transition energies (calculated without excitonic effects). We obtained 1.652 and 1.651 eV for samples 09155A and 11155A, respectively. Experimentally determined values are lower by about 13 and 16 meV, respectively. This difference should be attributed mainly to the exciton binding energy. Theoretical estimates [16] for the binding energy performed

5. Results and analysis We observe a good agreement between the two used experimental methods. The measured anisotropy coefficients in both cases are comparable, also shapes of anisotropy spectra are similar, therefore we present only

Fig. 1. Anisotropy (dashed line, open square) and mean reflectivity (solid line, open circles) spectra for sample 09205C with PQW (symbols—data, lines—fits).

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Fig. 2. Anisotropy (dashed line, open square) and mean reflectivity (solid line, open circles) spectra for sample 11155A with HPQWs (symbols–data, lines– fits).

for rectangular CdMnTe– CdTe QWs with similar parameters produce comparable values. However, not all the features of our results can be explained in terms of the microscopic model discussed above. For example, the spectra of HPQW shown in Fig. 2 exhibit a splitting of the optical transition by about 2 meV. This effect comes probably from inhomogeneous distribution of the well width in the sample. The sharp interface may exhibit small deviations from its perfect position with respect to the crystal lattice. These deviations (by a fraction of a monolayer) modify the transition energy and introduce an enhanced interface roughness. This roughness can have a preferential orientation and contribute to the anisotropy. This explains the fact that the splitting is more pronounced in anisotropy than in the reflectivity spectrum (Fig. 2). Another feature, which cannot be explained without taking into account the realistic interface structure is a large anisotropy observed in the case of PQW (Table 2). This anisotropy, not predicted by the microscopic model, can be explained by oriented defect formation, as described in Section 1. The probability of presence of such defects in our parabolic quantum well can be much higher than in the half parabolic ones because of higher Mn mole fraction corresponding to nonequilibrium MBE growth conditions (x ¼ 0:78 cannot be obtained by equilibrium growth techniques).

Table 2 Anisotropy coefficients A0 for all studied samples obtained using two methods Sample name

A0ð1Þ (%)

A0ð2Þ (%)

09205C 09155A 11155A

4.43 2.14 1.88

4.64 2.34 2.02

Fig. 3. Calculated QW potential profile (dots), energy levels and wavefunctions (lines) for heavy and light holes (sample 09155A with single QW). Idealized half-parabolic potential shown by thin line. Note oscillating potential for consecutive monolayers, characteristic for the digital growth technique.

6. Conclusions The simulations based on a microscopic model of the interface structure describe quantitatively the in-plane optical anisotropy measured for HP-QWs. A strong anisotropy, observed for the sample with parabolic QW indicates that in a correct description of these phenomena we cannot limit ourselves to intrinsic properties of interfaces. Taking into account realistic interface structure can also explain small splittings of the reflectivity lines observed for half-parabolic QWs.

Appendix. Theoretical estimate of hole mixing coefficient We apply ‘HBF model’ of the in-plane anisotropy introduced by Krebs and Voisin [2]. In what follows we corrected some misprints of the original paper. lXl, lYl and lZl orbitals are projected onto bond directions using operators B^ and F^ (projecting on backward and forward bonds). These two operators are defined by pairing four ‘canonical’ projection operators lj1;2 l ¼ 12 l^ ðX þ YÞ þ Zl; lj3;4 l ¼ 12 l ^ ðX 2 YÞ þ Zl: With this definition B^ and F^ can be written as: B^ ¼ lj3 lkj3 l þ lj4 lkj4 l and F^ ¼ lj1 lkj1 l þ lj2 lkj2 l: Interface potential operator responsible for the Table 3 Coefficient of hole mixing tl – h and normalized coefficient of hole mixing Tl – h for samples with HPQW Sample name

xMn

tl – h

Tl – h

09155A 11155A

0.5 0.25

0.27 0.13

0.53 0.51

K. Kowalik et al. / Solid State Communications 126 (2003) 467–471

anisotropy is introduced in the form ^ ¼ VðzÞ

! a dV hðz 2 zl ÞB^ Vðzl Þ 2 8 dz l ! ! a dV þ Vðzl Þ þ hðz 2 zl ÞF^ 8 dz X

where ( hðzÞ ¼

1 lzl , a=8 0 lzl $ a=8

and the sum runs over all the anion plane positions. In the envelope function approximation it gives the interface HBF Hamiltonian: HBF ¼ H0 þ

a0 4



 Vbo ^ dðz 2 zint Þ ^ðF^ 2 BÞ 2

for an interface placed at z ¼ zint ; where Vbo is the valence potential well depth, i.e. the valence band offset between the quantum well and barrier material. This result is based on the assumption that the bond wavefunctions at the interface are the same as in the bulk. Comparing the HBF Hamiltonian with the standard empirical formulation (Eq. (1)) the dimensionless parameter tl – h can be expressed in form: tl – h ¼ Vbo

m0 a20 4"2

Using our standard parameters for CdTe– MnTe interface we obtain Vbo ¼ 0:625 eV and Tl – h ¼ 0:54; which is in excellent agreement with our experimental results Tl – h ¼ 0:51 (for sample 11155A) and Tl – h ¼ 0:53 (for sample 09155A).

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