Optical polarization anisotropy of quantum wells induced by a cubic anisotropy of the host material

Physica E 13 (2002) 24–35

www.elsevier.com/locate/physe

Optical polarization anisotropy of quantum wells induced by a cubic anisotropy of the host material S.M. Ryabchenkoa; ∗ , Yu.G. Semenovb , A.V. Komarova , T. Wojtowiczc , G. Cywi3nskic , J. Kossutc a Institute

of Physics NAS of Ukraine, 46, Prospect Nanki Str., 03028, Kiev, Ukraine of Semiconductors Physics NAS of Ukraine, 03028, Kiev, Ukraine c Institute of Physics, Polish Academy of Sciences, 02-668, Warsaw, Poland

b Institute

Received 17 April 2001; received in revised form 9 July 2001; accepted 16 July 2001

Abstract The optical polarization anisotropy of quantum wells (QW) in structures fabricated from materials with cubic symmetry due to the anisotropic character of the Luttinger Hamiltonian for the valence band is studied. The matrix elements of interband optical transitions are shown to be di>erent for di>erent orientations of the linear polarization of light, oriented in the plane of a structure, provided that the structure growth axis does not coincide with any of the main crystallographic directions, i.e., with either [1 0 0]; [0 1 0]; [0 0 1] or [1 1 1]. The degree of this in-plane anisotropy of the polarization is calculated theoretically within an appropriate model for arbitrary structure growth orientation. The polarization anisotropy is experimentally detected for the fundamental 1hh–1e transition by studying the reBectivity from [1 2 0]-oriented Cd 0:8 Mn0:2 Te=CdTe=Cd 0:8 Mn0:2 Te QWs. At the same time the e>ect is shown to be absent in the case of QW grown along [1 0 0] direction. The measured polarization anisotropy in the former case shows a predicted
-periodicity as the polarization direction is rotated within the plane of the QW. The magnitude of the e>ect is also in qualitative agreement with the model. Quantitatively, the measured values tend to be some what larger than those predicted by the model. Surprisingly, the polarization anisotropy of the reBection by the barrier excitons is also detected. For the [1 2 0]-oriented structures this anisotropy has, approximately, the same magnitude as the one of the exciton transitions in the QWs. In the structure grown along [1 0 0], the anisotropy of the reBectivity of the barrier exciton transitions is also observed, although it is much smaller in magnitude. Some hypothetical c 2002 Elsevier Science B.V. All rights reserved. explanations of these observations are put forward.
PACS: 73.20. Dx; 71.55 Gs; 42.25.Ja Keywords: Quantum wells; Optical polarization anisotropy; Cubic semiconductors; Anisotropy of the Luttinger Hamiltonian; Light reBection spectrum

∗

Corresponding author. Tel.: +380-44-265-09-39; fax: +38044-265-15-89. E-mail address: [email protected] (S.M. Ryabchenko). c 2002 Elsevier Science B.V. All rights reserved. 1386-9477/02/$ - see front matter  PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 2 3 2 - 6

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35

1. Introduction Quantum wells (QWs) made of cubic semiconducting materials, such as AIII BV or AII BVI , often display an in-plane optical polarization anisotropy (OPA). Namely, optical properties of a QW depend on an orientation of the plane of a linear polarization of light with respect to the crystallographic axis within the QW plane. Such anisotropy was observed in [1 0 0]-oriented QWs in which there was no common anion in the barrier and the QW material [1,2]. These papers, in particular, have brought into attention the fact that, in the case of QW grown along even [0 0 1] crystallographic direction, a reduction of Td cubic symmetry of the zinc-blende-type of materials composing a quantum structure, is characterized by the point group C2v and not D2d . The one-dimensional step function interface potential (which has a full rotational symmetry) cannot describe such severe symmetry reduction. Therefore, as it was emphasized in Refs. [1,2], an additional interaction describing the observed symmetry reduction should be taken into account in the Hamiltonian itself. Even for the rectangular QWs, it is possible that the left and right interfaces are not chemically symmetric with respect to the reBection operator depending on the relative number of anion and cation layers in QW. In the case of such symmetry, a reBection followed by the rotation by
=4 is the equivalence operation for the two interfaces of a QW, and then a mutual compensation takes place of the two C2v -symmetric contributions originating in the two opposite interfaces. In the opposite case such mutual compensation of the C2v -symmetric contributions does not take place. So, we can expect the inBuence of such C2v -symmety on the hole conMnement states in a QW. The authors of Refs. [1,2] have shown that the additional C2v -symmetric terms in the Hamiltonian lead to a mixing of the heavy hole (HH) and the light hole (LH) states. Such mixing gives rise to di>erent values of the optical transition matrix elements for various in-plane directions of the light polarization. The reduction of QW symmetry down to C2v , leading to OPA, can also be induced by other reasons that lead to non-equivalence of the two interfaces. Such a situation occurs, for instance, in the case of asymmetric interface proMle for QWs with either common or non-common anions [3]. It would be interesting from

25

this point of view to examine OPA in triangular or half-parabolic QW [4]. Yet another example of OPA in the QW is observed in the polarization anisotropy of the luminescence in the structures grown on misoriented substrates having the growth plane that does not coincide with any of the main cubic axes (e.g., substrates with [3 1 1] orientation) [5 –7]. On a microscopic scale, such substrates contain atomic steps and=or corrugations. Thus, there are two directions that are not equivalent, the one along and the other perpendicular to these steps. It is natural, then, to expect the optical properties of such structures to depend on a relative orientation of the plane of the polarization of light with respect to these two particular directions. In particular, OPA due to this mechanism was considered theoretically in Ref. [7] assuming that a period of the corresponding corrugation is large. In the current paper, we examine yet another possible source of OPA in the QWs grown on planes that are not perpendicular to any of the main axes of a cubic crystal. We concentrate here on deep QWs in which the wave vector attributed to the ground state can be approximated by Kz ≈
=Lw . Here Lw is the QW width, and z is the direction normal to the QW plane. In such a case, the anisotropy components of the k · p Hamiltonian (which are wave vector dependent) should be taken into account at this value of Kz . If neither of the main cubic axes of a crystal coincides with the z-direction, these elements turn out to be non-diagonal with respect to the main part of the Hamiltonian, which causes the splitting between the heavy hole (HH) and light hole (LH) states (as opposed to the mixing of these states induced by the cubic part of the Hamiltonian). As a result, there should be a mixing of the hole states and, as a consequence, OPA will appear. This mechanism of OPA was considered for GaAs=AlAs heterostructures after the works [8,9] in a number of theoretical and experimental works (see Refs. [10 –14] and references therein). The focus of most of these works was an interplay of QW width and spin–orbital split-o> subband inBuence in the [h h l]-orientated QWs. In this work we investigate the OPA in [1 2 0]-orientated QW, which is beyond the aforementioned class of QWs orientation. So we present a theory of OPA for the most general case of [a b c]-orientated QWs. For the sake of simplicity

26

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35

the spin–orbital split-o> subband is not taken into account because CdTe-based structures with large spin–orbital splitting will be considered. The structure of the paper is as follows: in Section 2, a general relation describing the OPA of the fundamental 1hh–1e transition in the case of mixed HH and LH states are derived. Later, in Section 3, the transformation of the Luttinger Hamiltonian, originally written in a system related to the main cubic axes, to a system of coordinates related to an arbitrary orientation is carried out. Then, we give a quantitative estimate of the OPA for the particular mechanism under consideration. In Section 4, the results of our experimental studies of the OPA carried out on Cd 0:8 Mn0:2 Te=CdTe=Cd 0:8 Mn0:2 Te QW structures grown simultaneously on CdTe substrates oriented either along [0 0 1] or [1 2 0] orientations are presented. Finally, in Section 5, we discuss the obtained experimental results and compare them to the predictions of the theoretical model.

2. Parameters determining the optical anisotropy Let the HH eigenstates, 3=2 and −3=2 , obtained in the case of a lower symmetry, be represented by a linear combination of 8 valence band states of the zinc-blende semiconductor. The spin–orbit split-o> 7 states can be, in principle, taken into account as well, however, in view of a large value of spin–orbit splitting in the material of the present interests, we shall neglect them from now on. Assuming the representation with the projections of the e>ective angular momentum on z direction |M z being M = ± 3=2; ±1=2, we can write 3=2

= |3=2z F(z) + (u + iv)| − 1=2z G(z);

−3=2

= | − 3=2z F(z) + (u − iv)|1=2z G(z);

(1)

where u and v are the real and imaginary parts of coeOcients describing a small LH–HH mixing, while F(z) and G(z) are the envelope functions of the conMned ground states of the heavy and light holes, respectively. The mixing of |3=2z and |1=2z as well as that of | − 3=2z and | − 1=2z does not lead to polarization anisotropy of the interband optical transitions and is not important in the case under consideration.

In the case of a linear polarization, the operator inducing optical transitions can be generally written as (considering only its transformation properties) Vˆ = Xˆ cos ’0 + Yˆ sin ’0 ;

(2)

where ’0 is the azimuth angle of the optical polarization plane, and Xˆ ; Yˆ are the operators whose transformation properties are the same as those of the in-plane coordinates x and y, respectively. We are interested speciMcally in the interband transition matrix element M+ = 

1e

↑ |Vˆ |

3=2 ;

(3)

where 1e↑ is the wave function of the conMned conduction electron in its ground state 1e with the spin projection 1=2 (the 1e↓ is the same for electron 1e state with spin projection −1=2). With the deMnition of the basis functions |J = 3=2; M  (Ref. [15], see Appendix A) one obtains
2 |M+ |2 ˙ 12 PSF − √13 PSF PSG u2 + v2 cos(2’0 − ): (4) Here PSF = S(z)|F(z) and PSG = S(z)|G(z) are the overlap integrals of the electron and the heavy and light hole, respectively, envelope functions; S(z) is the envelope function √of conMned 1e conduction electron state, cos  = u= u2 + 2 . The maxima of the expression given by Eq. (4) as a function of the angle ’0 repeat √ Q’ =
and have an ampli√ with a period tude (1= 3)PSF PSG u2 + v2 . The optical anisotropy parameter can be derived from Eq. (4) and it reads +

=

|M+ |2max − |M+ |2min ∼ 2 PSG
2 u + v2 : =√ |M+ |2max + |M+ |2min 3 PSF

(5)

Considering the matrix element M− =  1e↓ |Vˆ | −3=2 , it leads to the same anisotropy − in the absence of a magnetic Meld.

3. HH–LH mixing in QW with arbitrary orientation The Luttinger Hamiltonian, that describes the hole states, takes the following form in the system of coordinates associated with the crystalline axes

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35

˜ ||[1 0 0]; Y ˜ ||[0 1 0]; ˜Z||[0 0 1]: X   $2 5 &1 + &2 K2 − 2&3 (JK)2 HL = 2m0 2 + 2(&3 −

&2 )(JX2 KX2

+

JY2 KY2

−√

+

:

(6)

The JX ; JY ; JZ are the usual angular momentum J = 3=2 matrices in the representation assuming the basis Ref. [15] given in Eq. (A.1). The K = − i{9=9x; 9=9y; 9=9z} is the momentum (wave vector) operator. Now we have to transform Eq. (6) in the coordinate system compatible with the geometry of the growth of the QW. To perform this task, the new oz axis will be taken to be perpendicular to the QW plane and characterized by the direction cosines {l; n; m}. The ox and oy axes are assumed to be in the QW plane. If the ox axis also lies in the plane deMned by the OZ and oz axes, then the unitary rotation transformation matrix takes the form given by Eq. (B.1). The Mrst two terms in Eq. (6) are invariant with respect to the coordinate system rotation. The rotation transformation of the remaining third, anisotropic term in Eq. (6) results in terms that are proportional to Kx2 ; Ky2 and Kz2 of which we shall retain only those which are proportional to Kz2 (the in-plane motion is beyond our present consideration; besides Kx2 ; Ky2 terms are expected to be much smaller than that proportional to Kz2 while leading to qualitatively similar mixing of the HH and LH states). The result of a lengthy but straightforward calculation is ˆ z2 ; JX2 KX2 + JY2 KY2 + JZ2 KZ2 ⇒ QK where ˆ z2 ≡ Kz2 QK



2(a4 + a2 b2 + b4 )c2 J2 + b2 )(a2 + b2 + c2 )2 x

(a2

+

2a2 b2 J2 (a2 + b2 )(a2 + b2 + c2 ) y

+

a4 + c 4 + b 4 2 J (a2 + b2 + c2 )2 z

−

2abc(a2 − b2 ) {Jx Jy } (a2 + b2 )(a2 + b2 + c2 )3=2

(7)

2ab(a2 − b2 ) {Jy Jz } a2 + b2 (a2 + b2 + c2 )3=2

 2c(a4 + b4 − c2 (a2 + b2 )) {Jz Jx } : +√ a2 + b2 (a2 + b2 + c2 )2



JZ2 KZ2 )

27

Here we have introduced the notation with the symbols a; b and c related to the directional cosines

l = a= a2 + b2 + c2 ; m = b= a2 + b2 + c2 ;
n = c= a2 + b2 + c2 and {Ji Jj } = 12 (Ji Jj + Jj Ji ): Let us assume that the last anisotropy term in Eq. (6) represents a small perturbation of the conMned QW states while all the other terms in Eq. (6) shall be considered as an unperturbed Hamiltonian. In this case, the admixture of the LH states to those of heavy holes can be calculated using the Mrst-order pertur(0) (1) (0) bation theory ±3=2 = ±3=2 + ±3=2 , where ±3=2 = |±3=2z F(z) and (1) (0) ˆ (0) (&3 − &2 )˝ ±3=2 = ∓1=2 |Q| ±3=2  m0 L2w +LH

2

TGF |

(0) ∓1=2 G(z);

(8) 

92 F(z) d z; 9z 2

(9)

where +LH is the energy splitting between the

(0) ±1=2

TGF = L2w

G(z)

(0) (LH) and ±3=2 (HH) states in a QW. The operator given by Eq. (7) leads, generally speaking, also to a small mixing of |3=2z with |1=2z and of |−3=2z with |−1=2z , not only mixing of states with the spins of the light holes reversed with respect to that of the heavy holes as in Eq. (8). The former type of mixing does not lead to the polarization anisotropy of interband optical transitions and, thus, is not important for the case under consideration. We have omitted it in Eq. (8) for simpliMcation. The matrix element of the operator Qˆ is directly calculated using Eq. (7)



(0) ˆ (0) ∓1=2 |Q| ±3=2 

=

√ c2 (a4 + b4 ) − a2 b2 (a2 + b2 ) 3 (a2 + b2 )(a2 + b2 + c2 )2 √ ±i 3

abc(a2 − b2 ) : (a2 + b2 )(a2 + b2 + c2 )3=2

(10)

28

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35

4. Optical anisotropy of the optical transitions between electron and hole QW con)ned states Eq. (10) deMnes the parameters u and v introduced already by Eq. (1). Inserting proper formulae Eq. (5), we can obtain the Mnal result for the magnitude of the optical anisotropy PSG 2(&3 − &2 )˝2 TGF += PSF m0 L2w +LH
a4 b4 + b4 c4 + c4 a4 − a2 b2 c2 (a2 + b2 + c2 ) × : (a2 + b2 + c2 )2 (11) It was observed that for high symmetry directions in the QW plane [a b c] = [1 0 0] as well as [1 1 1] the optical polarization anisotropy vanishes. Let us now estimate the value of Eq. (11) in a simple way. Assuming that the potentials due to the quantum well experienced by both the electron and the hole QW are deep enough so that approximation Kz ≈
=Lw is valid, then  2 ,z S(z) ∼ cos ; = F(z) ∼ = G(z) ∼ = Lw Lw Lw Lw ¡z¡ : (12) 2 2 In the real case an internal strain due to the lattice mismatch between the substrate, barriers and QW itself should be taken into account. The HH–LH splitting assumes the value −

+lh = ELH − EHH = +conf + +strain ; lh lh +strain lh

(13)

where the is a contribution of an internal strain; +conf represents the HH–LH splitting energy due to lh the conMnement 2,2 ˝2 &R +conf ; (14) lh = m0 L2w where the value of &R is within the range between the parameters &1 and &2 . When +strain is small and can be neglected, Eq. (11) lh simpliMes signiMcantly and becomes &3 − & 2 += &R
a4 b4 + b4 c4 + c4 a4 − a2 b2 c2 (a2 + b2 + c2 ) × : (a2 + b2 + c2 )2 (15)

Eq. (15) is independent of the QW width. This is related to the fact that in our model, the matrix element describing the wave function mixing is conf proportional to Kz2 ∼ L−2 w and ELH − EHH = +lh −2 proportional to Lw as well. This result should not be extrapolated to the case of QW with Lw → ∞ or even to the case of the bulk crystal. The anisotropic part of the Luttinger Hamiltonian leads to the OPA for electron-to-heavy-hole transitions independent from electron-to-light-hole transitions. These two kinds of OPA are of an opposite sign; therefore, they compensate each other for the HH and the LH from the zone center where these states are degenerate in the bulk crystals. Eq. (15) can be reduced to the corresponding equations of the works [12,14] in the case of [h h l]-orientated QW, i.e., a = b, by neglecting the inBuence of spin–orbital split-o> subband. Note, however, that spin–orbital split-o> subband contribution taken into account in Refs. [8,14] leads to the dependence of OPA on QW width. Let us consider now, for speciMcity, CdTe with &3 = 2:1; &2 = 1; 6; &1 = 5:3 [16], and a QW grown on a [1 2 0] plane. The numerical estimate of + based on Eq. (15) amounts to + = 4%. A more realistic model that takes into account the contribution of +stress as lh well as a realistic depth and width of the QW can result in a considerable (up to few times) modiMcation of the latter value. 5. Experiments The OPA investigations of the reBectivity spectra were carried out on structures fabricated in the Institute of Physics of the Polish Academy of Sciences in Warsaw. The structures were grown simultaneously on three CdTe substrates oriented in three di>erent crystallographic directions: [1 0 0]; [1 2 0] and [1 1 0]. Each structure contained four QWs, each having di>erent width being separated from each other by 50 nm wide Cd 1−x Mnx Te barriers. Thus, the QWs can be regarded as independent of each other. The width of the QWs was 6, 8, 12 and 20 monolayers, counting in the [1 0 0] interlayer spacing (ML100 ), which corresponds to the width of 19.4, 25.9, 38.9 T respectively, independent of the substrate and 64:8 A, orientation. The Mn molar fractions in the barriers,

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35 Optical spectra registration

±λ/4 acoustic-optical modulator (fm) Polarizer Lamp

λ/4 phase plate

29

PEM Monochromator

Lock-in amplifier (fm) Computer

Motor

> Polarization anisotropy registration

Angle meter Generator (fm) Rotatable compensating polarizer Sample in cryostat

Fig. 1. Schematic sketch of the experimental setup.

in the bu>er and in the cap layers were nominally the same, however, since the growth of a thick bu>er layer was performed at a higher temperature than that of the QW region, some small di>erence may be expected. It may be one of reasons for the complicated shape of the reBection line in the barrier exciton spectral region. The molar fraction x was determined to be x = 0:20 from the energy position of the photoluminescence line ascribed to the barrier layer exciton and with the use of the relation describing the shift of the exciton line in the bulk Cd 1−x Mnx Te with x at T = 2–4:2 K [17]: Eex CdMnTe (x) = Eex CdTe + 1:563x, where Eex CdMnTe (x) is the exciton transition energy in Cd 1−x Mnx Te while Eex CdTe = 1:596 eV—that in CdTe. Four well-resolved photoluminescence lines corresponding to the four isolated QWs were observed in the spectra from each structure. The energy positions of those PL peaks were found to be well described by calculations of the 1e–1hh exciton luminescence of rectangular QWs with the nominal values of Lw as determined by the growth rate. One of the possible methods of studying the in-plane OPA is by measuring the polarization of the luminescence detected in the direction normal to the structure plane (see, for instance Ref. [18]). However, the luminescence is usually caused by transitions between relaxed, and partially localized states, and hence the OPA of the PL can be di>erent from the anisotropy of truly interband transitions. Therefore,

we have chosen to study the OPA by measuring the reBectivity spectra collected for a near normal incidence of the light. We would like to emphasize that the OPA measurement of the reBectivity spectra is quite diOcult, especially in the case of small anisotropy. A special experimental setup was used for our measurements of the OPA of the reBected light (see Fig. 1). The light from an unpolarized broadband source (a Mlament lamp) passes through a polarizer and an acoustic-optical modulator. The polarization direction of the Mrst polarizer is set at the angle of ◦ 45 with respect to the principal direction of the modulator. The modulator introduces a di>erence in the optical path for the light polarized in two mutually orthogonal directions (principal directions of the modulator). This di>erence varies with an amplitude ±/=4 and a frequency fm ≈ 54:5 kHz. Further, the light passes through a /=4 phase plate. As a result, the light becomes linearly polarized alternatively along the two mutually perpendicular directions (we refer to them as p and s directions), whenever it encounters an extreme di>erence of ±/=4 in the optical path. This cycle of the polarization switching is repeated with the frequency fm . The p and s directions are ori◦ ented at the angles ±45 with respect to the principal optical directions of the /=4-plate. Next, the light is reBected from the sample surface and passes through a second polarizer serving as a ◦ compensator and initially set at the angle ±45 with

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35

where Rs (E) and Rp (E) are the reBectivity coeOcients of the light polarized along the axis p and s, respectively; R(E) = Ir (E)=I0 (E), where I0 (E) and Ir (E) are the intensity of the incident and reBected light, respectively, having the same energy E. The studied samples could be rotated around the axis z, normal to the plane of the structure. Therefore, it was possible to change the angle ’ between the direction p and some Mxed reference direction in the sample plane. The angle between this reference direction and the s direction is, therefore, equal to ’ + ,=2. The angle ’0 appearing in Eqs. (2), (4) acquires, then, the values equal to ’0p = ’+0 and ’0s = ’+0 +,=2 for p and s directions of the polarization, respectively. Here 0 denotes the arbitrary angle, which represents a reference direction in the plane of the sample. The measurements of both Ir (E) and 3(E) were carried out at two temperatures, namely at 4.2 and 77 K. The spectra Ir (E) were normalized to the reBection spectra from a neutral mirror in order to rule out

Ir

norm

(E )

0.3 0.2

20 ML100

8 ML100

12 ML100

0.1

6 ML100

Barrier exciton group

(a) 0.0

2

-0.5

(b)

(deg)

1 -1.0

φ 1) φ ( ϕ ),

-1.5

o

(deg)

2) φ ( ϕ +9 0 ) 1

(c)

o

respect to p and s directions. The direction of the polarization of the compensator can be changed by an angle 3, required to achieve a complete compensation, with the use of a step motor. The angle 3 is then transformed into an electrical signal fed to a computerized acquisition system. Further, the light enters into a grating monochromator. The intensity of the light is measured by a photomultiplier (PEM) as a function of the light wavelength. If the reBection of the light (of a particular wavelength) from a sample is not identical for the two polarizations, p or s, there appears a component with the frequency fm in a signal registered by the PEM. This signal, with the frequency fm , is picked up by a lock-in ampliMer whose reference signal comes from the generator, which also drives the acousticoptical modulator with the same fm . The output signal of the lock-in ampliMer is fed to the step motor that turns the polarizer-compensator. The compensating polarizer will then rotate until the component of a PEM signal with the frequency fm becomes zero. The angle 3 between the direction p and the polarization direction of compensating polarizer is given by the ratio  Rs (E) 3(E) = arctan ; (16) Rp (E)

φ (ϕ) - φ ( ϕ + 90 )

30

0

-1 1.6

1.7

1.8

1.9

2.0

E (eV) Fig. 2. Experimental data for [1 2 0]-oriented structure: (a) the normalized spectrum of the reBected light, Irnorm (E); (b) the ◦ ◦ spectra (3(E; ’) − 45 )—solid line, and (3(E; ’ + 90 ) ◦ ◦ − 45 )—dotted line, obtained for ’ = 65 ; (c) the spectrum ◦ ◦ 3(E; ’) − 3(E; ’ + 90 ) for ’ = 65 .

spectral characteristics of the light source and=or other elements of the setup. Certainly, the curves Irnorm (E) obtained in this way are not exactly equal to the spectra of R(E). This is because the spectral dependence of the reBectivity of the mirror itself is not controlled and, therefore, cannot be excluded from the data. Nevertheless, this procedure deMnitely approximates better R(E) than non-normalized Ir (E) spectra. The results of our measurements performed at 4.2 and 77 K do not di>er qualitatively, although a temperature shift of the optical transitions energy and a small broadening of the lines were observed. The temperature shift of various features observed in the reBectivity spectra in the samples grown with di>erent substrates orientations was found to be the same. The curves Irnorm (E) obtained for the unpolarized light at T = 4:2 K in [1 2 0]- and [1 0 0]-oriented structures are shown in Figs. 2a and 3a, respectively. One can see that these spectra are very similar, as far as the

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35

Ir

norm

(E )

0.3

20 ML100

0.2

8 ML100 12 ML100

6 ML100

Barrier exciton group

0.1

(a) 0.0 1.5

1

(deg)

1.0

2

0.5

φ

1) φ ( ϕ ), o

2) φ ( ϕ +9 0 )

0.0

(b)

φ(ϕ) - φ (ϕ + 90°)

(deg)

-0.5 1

(c) 0

-1 1.6

1.7

1.8

1.9

2.0

E (eV)

Fig. 3. Experimental data for [1 0 0]-oriented structure: (a) the normalized spectrum of reBected light, Irnorm (E); (b) the spectra ◦ ◦ ◦ (3(E; ’) − 45 )—solid line, and (3(E; ’ + 90 ) − 45 )—dotted ◦ line, obtained for ’ = 90 ; (c) the spectrum 3(E; ’) − ◦ ◦ 3(E; ’ + 90 ) for ’ = 90 .

energy positions of various reBectivity features and their amplitudes are concerned. The structure grown on the [1 1 0]-oriented substrate reveals features whose energy is practically equal to those of the corresponding features observed in [1 0 0]- and [1 2 0]-oriented structures, however, they are wider and are of smaller intensity. This suggests that we deal in this sample with a stronger inhomogeneous broadening of the optical transitions. The broadening results in errors of the experimental data of the OPA in this particular sample that are very large thus making them unsuitable for further analysis. The measurements of 3(E) for di>erent in-plane orientations with respect to p and s directions (for different ’) shows that 3(E) is, in fact, a function of ◦ the angle ’. The spectra (3(E) − 45 ) for [1 2 0]- and [1 0 0]-oriented structures obtained at the selected values of ’ are shown in Figs. 2(b) and 3(b), respectively. We chose the ’ orientations that corresponded to the maximum amplitude of features of 3(E) curves

31

in the region of the exciton lines. When comparing ◦ the curves for the orientations ’ and ’ + 90 , one can ◦ see that 3(E; ’) and 3(E; ’ + 90 ) coincide at the energies far from the exciton reBection lines, while they contain components that are opposite in phase in the energy region close to these lines. This means that those portions of 3(E) which are independent of ’ at the energy far from the exciton reBection lines are due to the measuring system itself, for example, the presence of the mirror or=and not exactly normal incidence of the light, etc. On the other hand, the contributions to 3(E), which does depend on ’ within the spectral region of the exciton lines area are connected to the polarization anisotropy of the probability of the transitions. In order to extract these contributions, the depen◦ dencies of the di>erence 3(E; ’)−3(E; ’+90 ) were plotted in Figs. 2(c) and 3(c). One can see that in the case of [1 2 0]-oriented sample these dependencies reveal striking features in the energy regions related to the excitonic transitions in both the barriers and in the QWs. For the [1 0 0]-oriented sample, on the other hand, no detectable peculiarities in the ◦ 3(E; ’)−3(E; ’+90 ) in the region of the QWs spectrum are observed, within the experimental errors. It is only in the region of the barrier excitons that such clear features are distinguishable. 6. Discussion The problem of formation of reBectivity spectra in the spectral region of interband optical transitions is relatively well investigated and is described in the literature both for bulk crystals [19] and for QW structures [20]. The reBection spectrum R(E) includes the regions of a background reBection Rf (E), which varies only a little in spectral range far from the optical transitions, and the regions with distinct particularities of the reBectivity 6i R(E) connected to these transitions. If the index i labels transitions in a sample, one can symbolically write R(E) = Rf (E) + 6i R(E): (17) i

In the case of the QW structures the function R(E) is determined by a spectral dependence of a complex dielectric susceptibility of the layers particularly

32

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35

in the spatial regions where the optical transitions do take place (e.g., in the barrier, in the QWs, or in the substrate). There is also an inBuence of other layers (mainly of their width) that are transparent to the incident as well as to the reBected light. The probability of each transition i can be characterized by its magnitude of the longitudinal–transversal splitting +LTi . For the bulk crystal, or for a suOciently thick barrier layer, this parameter is deMned as [20] +bulk LT =

4,d2 4˝2 e2 |P|2 = 3 2 2 ; 8b aB E0 m0 8b

(18)

where P is the interband transition matrix element. The dielectric constant of the bulk crystal 8b is assumed to be the same in the barriers and in the QWs. E0 is the energy of ith optical transition, aB stands for the Bohr radius of the exciton in the bulk crystal, d is the density of electric dipole moment of the transition. On the other hand, in the case of the quantum wells, the quantity +QW LT depends also on their width (see, e.g., Ref. [20]). The shape of the contributions 6i R(E) depends on the thickness of the layers preceding the layer, in which the optical transition i takes place, and it generally contains a maximum 6i R(E)max and a minimum 6i R(E)min . The di>erence of the energy positions of 6i R(E)max and 6i R(E)min depends on +LTi of the optical transition in question, on its damping parameter i and on inhomogeneous broadening iin hom . This di>erence can be approximated by i + iin hom under the condition (i + iin hom ) ¿ +LTi . In such a case, when (i + iin hom ) ¿ +LTi , the magnitude of the singularity connected to the ith reBectivity line Q6i R(E) = 6i R(E)max − 6i R(E)min is much smaller than Rf (E) in the vicinity of a given transition and is proportional to +LTi =(i + iin hom ). The measurements of Irnorm (E) have shown that the ratio Q6i R(E)=Rf (E) is in the range between 0.25 and 0.15 in the case of the exciton reBection both in the barrier and in the QWs, i.e., it is smaller than unity. ◦ The maximum magnitude of 3(E; ’) − 3(E; ’ + 90 ) related to the exciton transitions (shown Figs. 2c and ◦ 3c) does not exceed 1:4 (6 0:025 rad). It means that we have (Rs − Rp )(Rs + Rp ) for each value of ’.

Under these conditions    Rs (E) − Rp (E) Rs (E) ≈ 1− Rp (E) Rs (E) + Rp (E) =1+

R 6Rs (E) − 6Rp (E) 6R(E) R 6Rs (E) + 6Rp (E) Rf (E) + 6R(E) (19)

R with 6R(E) = (6Rs (E) + 6Rp (E))=2 ≈ 6Rs (E) ≈ 6Rp (E). The damping parameters of the transitions do not vary signiMcantly with the polarization of light. So, when (i + iin hom ) ¿ +LTi , the ratio 6Rs (E) − 6Rp (E) 6Rs (E) + 6Rp (E) is equal to the polarization anisotropy of the transition probability for the two light polarizations, s and p: sp (’) =

|M (’)|2s − |M (’)|2p : |M (’)|2s + |M (’)|2p

The latter quantity depends on the orientation of the axes in the plane of the structure. Taking into account the fact that the value of the di>erence 3(E) − ,=4 is small, one obtains (3(E) − ,=4)’ ≈

1 1 + (,=4)2

sp (’)

R 6R(E) : R Rf (E) + 6R(E)

(20)

Thus, the shape of the singularity in 3(E) around the optical transition energy reBects a singularity of 6i R(E) with a sign determined by the sign of sp (’). In other words, 3(E) has a maximum, 3(E)max , and a minimum, 3(E)min , near the optical transition energy with a certain di>erence Q3(E)’ = 3(E)max − 3(E)min . In the case of Rf (E) 12 Q6i R(E) one Mnds Q3i (E)’ ≈ 0:618

sp (’)

Q6i R(E) : Rf (E)

(21)

According to Eqs. (2) – (4), we can expect that + cos(2’ + 0 − ). On the basis of Eq. (10), the values  = ,, 3,, etc., are expected taking into account the values of u and v obtained for [1 2 0]-oriented structure. sp (’) =

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35 1.0

1.0

(a)

[210]

0.5

[110]

0.5 ∆φ (deg)

∆ φ (deg)

33

0.0

0.0

-0.5 -0.5

[001]

[1 1 0]

-1.0 -90

0

90

180

270

360

-1.0

φ (deg)

0

90

180

270

360

φ (deg)

1.0

Fig. 5. The dependence of Q3(E)’ on ’ for the barrier exciton reBectivity lines for the structure with [1 0 0] orientation at T = 77 K.

(b) [210]

∆ φ (deg)

0.5

0.0

-0.5 - 8 ML[100] - 6 ML[100] - 20 ML[100]

[001]

-1.0 -90

0

90

180

270

360

φ (deg)

Fig. 4. The dependence of Q3(E)’ on ’ for the structure with [1 2 0] orientation at T = 77 K: (a) in the region of the barrier exciton reBectivity lines; (b) in the region of the exciton transitions in the QWs.

Taking into account the fact that Q6i R(E)=Rf (E) ≈ 0:25– 0.15, while the estimate for the [1 2 0]-oriented structure (see end of Section 4) gives + ∼ = 0:04, one can expect the amplitude of the Q3i (E)’ dependence to vary within 0.0038–0:0062 rad, or, within ◦ 0.22– 0.36 . The dependence of Q3(E)’ on ’ for the exciton reBectivity lines in the barriers and in the QWs is shown in Figs. 4(a) and (b) for the [1 2 0]-oriented structure and in Fig. 5 for the barrier exciton lines in the [1 0 0]-oriented structure (using data obtained at T = 77 K). The positive sign of Q3(E)’ in these Mgures is chosen for the case of equivalent signs of reBec◦ tivity singularities appearing on the curves 3(E)’ –45 and 6i R(E) for the corresponding lines, and negative— in the opposite case.

One can see that the dependencies of Q3(E)’ on ’ possess a period equal to ,, as expected. The extremes of the dependence for the barrier exciton line and QW exciton lines in the [1 2 0]-oriented structure coincide. They correspond to p and s orientations along the [2 1 0] and [0 0 1] directions, respectively, both lying in the plane of this structure. The extremes of the similar ’-dependence for the barrier exciton in the [1 0 0]-oriented structure correspond to p and s orientations along [1 1 0] and [1 1R 0], respectively. The directions of the crystallographic axes in the plane of the structures were found by an observation by a microscope of the crossings pattern of micro-indentations made in the substrates. The amplitudes of the ’-dependence of Q3(E)’ ◦ in the [1 2 0]-oriented structure are equal to 0:7 for ◦ the barrier exciton reBection line, 0:7 for the exci◦ ton in the 8 ML[0 0 1] wide QW line, and 0:5 for the 6 and 20 ML[0 0 1] wide QW lines. The data for the 12 ML[0 0 1] wide QW line were not reliable, so they are left beyond the present discussion. We see that the amplitudes of the ’-dependence of Q3(E)’ in this structure exceed the above-mentioned estimates by a factor about 1.5 –2. In order to make the estimations more precise, the calculations of the line shape of the barrier and the QW exciton reBection spectra R(E) were carried out for the normal light incidence using the formulae from Ref. [20]. The model-

34

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35

ing
of 3(E) was performed by calculating arctan R1 (E)=R2 (E), where R1 (E) and R2 (E) were calculated using 8b = 9:7, (+LT1 + +LT2 )=2 = 1:25 meV and (+LT1 − +LT2 )=(+LT1 + +LT2 ) = 12 . The 1 + 1in hom = 2 + 2in hom and 12 values were Mtted to bring Q6i R(E) and Q3(E) into agreement with experiments for di>erent exciton (i.e., those in the barrier and QWs) reBection lines. The results of these calculations do not lead to changes of the previous estimations that would exceed 15 –25%. Thus, the reasons for the divergence between our theoretical estimates and the experimental data have to be ascribed to some speciMc property of the samples. At the same time, it remains unexplained as to why the OPA in the barrier is as strong as it is in the case of the QW excitons. Some possible explanations can be suggested. One may be related to step bunching on the [1 2 0] plane which leads to the OPA due to the formation of suOciently wide steps. If this is the case, the resultant OPA mechanism may be similar to that considered in Ref. [7]. Another reason can be related to the presence of in-plane strain component. Yet another reasons for the additional OPA (not mentioned here) of the reBected light are possible. As to the small OPA observed for the barrier exciton line in the [1 0 0]-oriented structure, the valence band anisotropy does not inBuence the value of the OPA in this particular case. Thus the source of this OPA may be related to small corrugations on the substrate surface as well. At the same time the absence of the OPA of the QW exciton transitions in this structure indicates that either the interface-induced anisotropy (Refs. [1,2]) can be neglected or the interfaces are nearly symmetric. 7. Conclusion The present work proves that the cubic anisotropy of the valence band dispersion law results in a mixing of the HH and the LH states which are split due to their conMnement in the QWs grown on cubic crystal substrates along various arbitrary orientations. The mixing disappears for the orientations of growth along the C4 and C3 crystallographic axes. The mixing of the HH and the LH states results in the in-plane OPA of 1hh–1e excitonic optical transitions in QWs. The

quantitative estimation of the OPA due to this mechanism is carried out in terms of the model, which assumes the form of the Luttinger Hamiltonian proper for [1 2 0]-orientation of the substrate and uses the values of various parameters suitable for CdTe. We carried out the in-plane polarization anisotropy measurements of the reBected light in the structures of Cd0:8 Mn0:2 Te=CdTe=Cd0:8 Mn0:2 Te grown by MBE at identical conditions on substrates with various orientations, [1 2 0] and [1 0 0]. We have found the existence of the OPA for 1hh–1e exciton transitions in the QWs in the [1 2 0]-oriented structure. At the same time, transitions in the [1 0 0]-oriented structure did not show any OPA for the QWs optical transitions within the limits of accuracy of our measurement. These results are in qualitative agreement with the proposed mechanism and the model developed in this paper. The angular dependence of the OPA shows a period equal to ,, as predicted by the model. The amplitude of this dependence agrees, as far as the order of magnitude is concerned, with our estimations, though, there is no quantitative agreement: the experimental values exceed the model estimation by, approximately, a factor of 1.5 –2. Some degree of the OPA is also found in the case of the barrier exciton reBection lines. The latter OPA turns out to be approximately the same in magnitude as the OPA due to the QW excitons in the structure with [1 2 0] orientation. The [1 0 0]-oriented structure reveals the anisotropy, which is smaller but nevertheless detectable beyond error. Acknowledgements The authors express their gratitude to V.I. Sugakov and A.V. Vertsimakha for discussion of some results. This work is partially supported by Grant INTAS-99-15. Appendix A. If the X , Y , Z are the basis functions (Bloch amplitudes) of the valence band states that transform as the corresponding spatial coordinates, then one can choose the basis functions of the 8 states, having the transformation properties of the angular momentum

S.M. Ryabchenko et al. / Physica E 13 (2002) 24–35

equal to 3=2, as



3 3 1

;+

2 2 = √2 (X + iY ) ↑;



3 1 1

;+

2 2 = √6 [(X + iY ) ↓ −2Z ↑ ];



3 1 1

;−

2 2 = √6 [ − (X − iY ) ↑ −2Z ↓ ];



3 3 1

;−

2 2 = − √2 (X − iY ) ↓ :

(A.1)

The basis set (A.1) di>ers from that in the original work of Luttinger and Kohn [21] by a unitary transformation. Appendix B. The unitary rotation matrix can be expressed in terms of the Euler angles , ;, and Mxing the value of the third angle & (for speciMcity we assume & = 0). Then, the sine and the cosine of these angles can be expressed in terms of the direction cosines l, m and n. The rotation matrix becomes  
nl nm √ √ − 1 − n2  1 − n2  1 − n2     l Rˆ =  √ −m  : (B.1) √ 0    1 − n2  1 − n2 l

m

n

For the sake of √ convenience, we introduce further notations, l = a= a2 + b2 + c2 , etc., which were used in Eq. (7) and below. References [1] O. Krebs, P. Voisin, Phys. Rev. Lett. 77 (1996) 1829.

35

[2] E.L. Ivchenko, A.A. Toropov, P. Voisin, Phys. Solid State 40 (1998) 1748. [3] A. Kudelski, A. Golnik, J.A. Gaj, F.V. Kyrychenko, G. Karczewski, T. Wojtowicz, Yu.G. Semenov, O. Krebs, P. Voisin, Phys. Rev. B, in print. [4] T. Wojtowicz, G. Karczewski, J. Kossut, Thin Solid Films 306 (1997) 271. [5] K. Fujiwara, N. Taukada, T. Nakayama, T. Nishino, Solid State Commun. 69 (1989) 63. [6] D.V. Korbutyak, V.G. Litovchenko, I.A. Troshchenko, S.G. Krylyuk, H.T. Grahn, K. Ploog, Semicond. Sci. Technol. 10 (1995) 422. [7] V.G. Litovchenko, D.V. Korbutyak, S.G. Kryluk, Yu.V. Kryuchenko, V.I. Sugakov, H.T. Grahn, K. Ploog, SPIE Proc. 2648 (1995) 294. [8] D. Gershoni, J.S. Wiener, S.N.G. Chu, G.A. Bra>, J.M. Vandenberg, L.N. Prei>er, K. West, R.A. Logan, T. Tanbun-Ek, Phys. Rev. Lett. 65 (1990) 1631. [9] D. Gershoni, I. Brener, G.A. Bra>, S.N.G. Chu, L.N. Prei>er, K. West, Phys. Rev. B 44 (1991) 1930. [10] S. Nojima, Phys. Rev. B 47 (1993) 13 535. [11] M.V. Belousov, V.L. Berkovitts, A.O. Gusev, E.L. Ivchenko, P.S. Kopyev, N.N. Ledentsov, A.I. Nesvizhskii, Phys. Solid State 36 (1994) 596. [12] M.V. Belousov, E.L. Ivchenko, A.I. Nesvizhskii, Phys. Solid State 37 (1995) 763. [13] E.G. Tsitsishvili, Phys. Rev. B 52 (1995) 11 172. [14] R. Winkler, A.I. Nesvizhskii, Phys. Rev. B 53 (1996) 9984. [15] Yu. G. Semenov, B.D. Shanina, Phys. Stat. Sol. B 104 (1981) 631. [16] M. Said, M.A. Kanehisa, Phys. Stat. Sol. B 157 (1990) 311. [17] J. Gaj, W. Grieshaber, C. Bodin-Deshayes, C. Cibert, G. Feuillet, Y. Merle d’Aubigne, A. Wasiela, Phys. Rev. B 50 (1994) 5512. [18] Yu.G. Kusrayev, A.V. Koudinov, I.G. Aksyanov, B.P. Zakharchenya, T. Wojtowicz, G. Karczewski, J. Kossut, Phys. Rev. Lett. 82 (1999) 3176. [19] M. Born, E. Wolf, Principles of Optics, Pergamon press, Oxford, 1964. [20] E.L. Ivchenko, A.V. Kavokin, V.P. Kochereshko, G.R. Posina, I.N. Uraltsev, D.R. Yakovlev, R.N. Bicknell-Tassius, A. Waag, G. Landwehr, Phys. Rev. B 45 (1992) 7713. [21] J.M. Luttinger, W. Kohn, Phys. Rev. 97 (1955) 869.