Impact of heavy hole–light hole coupling on the exciton fine structure in quantum dots

Physica E 87 (2017) 161–165

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Impact of heavy hole–light hole coupling on the exciton fine structure in quantum dots

MARK

E. Tsitsishvili Institute for Cybernetics, Tbilisi Technical University, S. Euli 5, 0186, Tbilisi Georgian Republic

A R T I C L E I N F O

A BS T RAC T

Keywords: Quantum dots Optical–related phenomena

We present analytical results which describe the properties of the exciton ground state in a single semiconductor quantum dot (QD). Calculations are performed within the Luttinger–Kohn and Bir–Pikus Hamiltonian theory. We show in an explicit form that an interplay of the exchange interaction and the heavy hole–light hole coupling, which is due to the in-plane asymmetries of the dot shape and the strain distribution, plays an essential role. For both the bright and dark exciton, this combined effect leads to a dependence of the fine structure splitting and polarizations on the main anisotropy axis direction relative to the dot orientation. Basing on the obtained analytical expressions, we discuss some special cases in details.

1. Introduction Epitaxially grown structures of semiconductor self-assembled quantum dots (QDs), which are considered as candidate building blocks for quantum technologies, attract a great interest during the last two decades. One of the most prominent applications of QDs is as the entangled photon emitters, based on the biexciton cascade process [1,2]. While ideally polarization entanglement proposals require a degeneracy of the intermediate exciton state [1], real self-assembled semiconductor QDs (grown along the [001] direction) do not meet this condition. A splitting of the two bright exciton states in QDs, known as the fine structure splitting (FSS), is a consequence of the intrinsic atomistic asymmetry of the underlying lattice, spin–orbit interactions, and the electron–hole exchange interaction [3–6]. Numerous experimental investigations of the neutral exciton emission in single selfassembled QDs show that the growth conditions significantly influence the polarization properties and the FSS of the bright exciton [7–14]. The results are attributed in particular to the valence band mixing (VBM), which arises from the confinement asymmetry and anisotropic strains build up in a QD. Although some theoretical background has been provided in the cited experimental references, detailed analytical description of the VBM impact on the exciton fine structure in QDs is still absent, to our knowledge. Theoretical researches in this field are devoted mainly to a change of the FSS under external perturbations [15–19]. In this paper we give a consecutive analytical description of the exciton ground state in a single QD with a reduced symmetry using the Luttinger-Kohn and Bir–Pikus model. Our purpose is to understand how actually the VBM affects the exchange interaction and, respectively, changes the FSS and polarization of the bright and dark excitons in self-assembled QDs. The recieved analytical results show that here

the orientation of the main anisotropy axis with respect to the dot direction is of a paramount importance. The obtained results are applicable to QDs under external stresses and in this case directly specify a dependence of the exciton properties on the stress direction. 2. The model We consider eight exciton states confined in a self-assembled flat QD grown in the [001] (z) direction: four heavy-hole (hh) and four lighthole (lh) states associated with the projection of the total hole angular 3 1 momentum |Jz | = 2 and |Jz | = 2 , respectively. For ideal QDs, the Hamiltonian may be written as H = HQD + Hex . The first term describes the exciton energies, HQD = ∑n En |n〉〈n|, where the summation is taken over all states, En=0 for all hh exciton states, En = Elh(0) for all lh exciton states, and Elh(0) denotes the energy difference between hh and lh excitons. The second term Hex accounts for the electron–hole exchange interaction which will be discussed latter. For Stransky–Krastanov type dots, where strain is required in quantum dot formation, a major effect responsible for the heavy hole–light hole spitting is the biaxial in-plane compressive strain (known as the epitaxial strain), which lifts the heavy-hole band, so that Elh(0) > 0 . Typical valence band splittings found in quantum dot experiments are large, on the order of 10 meV, and the hole ground state is in first approximation treated as a purely heavyhole state. For real QDs, the shape elongation and anisotropic distribution of the strain built up in a dot can enhance the light-hole admixture. Although it is difficult to draw any general conclusions for quantum dots, since different growth methods can result in various shape and strain characteristics, it is usually assumed usually that the main effect is due to the in-plane anisotropy of the confinement potential and strain. Based on the Luttinger–Kohn and Bir–Pikus (LKBP) theories, where the valence band is described by a (4× 4)

http://dx.doi.org/10.1016/j.physe.2016.11.033 Received 29 March 2016; Received in revised form 26 October 2016; Accepted 30 November 2016 Available online 02 December 2016 1386-9477/ © 2016 Elsevier B.V. All rights reserved.

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slightly split by the energy δ 2 = 3b /2 . Having in mind the QDs with a nonspherical base, the Mz = ± 1 states are coupled mainly by the longrange exchange interaction [6,4], whereas the atomistic anisotropy leads to a splitting of the bright exciton states already in symmetrical QDs with C2v symmetry [3,23]. So generally, the true exciton ground state in real QDs becomes fully split. Note, however, that the Mz = ± 1 states are not mixed with the Mz = ± 2, 0 states by the exchange interaction [3,4]. The resulting exciton Hamiltonian therefore reduces to two submatrices in the {|Ψ1 〉, |Ψ2 〉} basis and the {|Ψ3 〉, |Ψ4 〉} basis, respectively.

matrix in the Bloch functions basis [20,21], the mixing in this case 1 3 occurs between the Jz = ± 2 and Jz = ∓ 2 hole states. The corresponding off-diagonal matrix element is given by (see, e.g. Ref. [12])

R = − 3 [∼ γ2(kx2 − k y2 ) − 2iγ∼3kx k y] +

3 b [εxx − εyy] − idεxy , 2

(1) → where k = {kx , k y} (x ∥[100], y ∥[010]) is the in-plane momentum operator, εij = εji are the strain tensor components with εij = εji , =2 ∼ γ = − γ with γ the Luttinger constants (in what follows we take i

i

2m 0 i

γ2 = γ3), and b and d are the deformation potential constants. Below, taking an average over the hole envelope functions 〈R〉 = ρe−2iψ , we introduce the amplitude ρ and the angle ψ, which determine the hole coupling strength and the main anisotropy axis orientation in the QD plane with respect to the [110] crystallographic axis, respectively. Consequently, for the exciton ground state, the resulting eigenfunctions can be written as (we take for simplicity the same envelope function for the heavy and light holes) |Ψ1,2 〉 ∝ C1 | ± 1h〉 − C2 e ± 2iψ | ∓ 1l〉,

(2)

|Ψ3,4 〉 ∝ C1 | ± 2h〉 − C2 e ± 2iψ |0( ± ) 〉,

(3)

3. Exciton fine structure splitting and polarization The exciton Hamiltonian in the space spanned by the |Ψ1 〉 and |Ψ2 〉 states, see Eq. (2), has a form

⎛ ⎜ E0 + ⎜ Hb = ⎜ ⎜⎜ ⎝

C1,2

tan(2Φ1) =

∑ i = x , y, z

3

⎟ ⎠

Δ1 2iΦ1 e 2

⎞ ⎟ ⎟ ⎟. C22 ⎞ Δst ⎛ 2 E0 + 2 ⎜C1 − 3 ⎟⎟⎟ ⎠⎠ ⎝ Δ1 −2iΦ1 e 2

χ1 sin 2θd + χ2 cos 2θd , χ1 cos 2θd − χ2 sin 2θd

χ1 = C12 Δh + C22 Δl cos 4α +

(6)

(7)

4 C1 C2 Δst cos 2α , 3

⎛ ⎞ 2 χ2 = 2C2 sin 2α ⎜C2 Δl cos 2α + C1 Δst ⎟ , ⎝ ⎠ 3

Elh(0)

(8)

(9)

with α = ψ − θd and Δh (Δl) denoting a splitting between the | + 1h〉 and | − 1h〉 (| + 1l〉 and | − 1l〉) bright states. As noted above, we believe that Δh (Δl) is nonzero even in a shaped-symmetric dot. The corresponding eigenstates

(4)

Note that generally the parameters ρ and ψ depend on the mutual orientation of the dot elongation axis θd and the strain direction θs, see the left panel in Fig. 1. Taking 〈R〉 = ρd e−2iθd + ρs e−2iθs , it turns out ρ = [ρd2 + ρs2 + 2ρd ρs cos 2(θd − θs )]1/2 and that generally tan(2ψ ) = [ρd sin 2θd + ρs sin 2θs][ρd cos 2θd + ρs cos 2θs]−1 . The fine structure of the exciton ground state in QDs is governed mostly by the short-range exchange interaction, which can be presented as [4,22]

Hsre = −

C22 ⎞

where

1

⎞2 ⎟ . ⎟ (0) 2 (Elh ) + 4ρ2 ⎠

−

Here the phase Φ1 is defined relatively to the [110] axis and obeys the equation

1

and the corresponding energy is E0 = 2 [Elh(0) − (Elh(0) )2 + 4ρ2 )]. The states involved in Eqs. (2)–(3) are labeled by a number denoting the projection of the total angular momentum Mz = Jz + sz and a letter h or l, denoting hh and lh exciton states, respectively. For the Mz=0, a “+” or “−” sign denoting electron spin up or down is appended to distinguish between the two lh exciton states. Remember that the |Mz | = 1 and Mz=0 states are optically active (bright excitons) with the lateral and transversal polarization, respectively. States with |Mz | = 2 are optically inactive (dark excitons). The probabilities for the exciton to be hh and lh exciton are given by

⎛ 1 ⎜ = 1± 2 ⎜⎝

Δst ⎛ 2 ⎜C 2 ⎝ 1

|X 〉 ∝

1 [|Ψ1 〉 − e 2iΦ1 |Ψ2 〉], 2

(10)

|Y 〉 ∝

1 [|Ψ1 〉 + e 2iΦ1 |Ψ2 〉] 2

(11)

are split by the energy Δ1 = |EX − EY |, which is given by

Δ1 =

[ai σi ⊗ Ji + bi σi ⊗ Ji3]. (5)

χ12 + χ22 .

(12)

We observe that both the phase Φ1 and the FSS Δ1 are affected by the hh–lh coupling and an interplay between the hole mixing and the exchange interaction as well. It should be noted that the obtained eigenstates coincide in a form with those reported in Ref. [12], but are actually very different. According to the Eq. (7), the phase Φ1 differs generally from the dot orientation θd and at fixed θd may change with ρ and α. The reason is a combined effect of the VBM and electron–hole exchange interaction, which was neglected in Ref. [12]. Further, it is easy to check that the obtained eigenstates are linearly polarized in the dot plane but are generally nonorthogonal. The corresponding polarization directions are tilted with respect to the crystallographic axes [110] and [110], respectively, at the angles (see the right panel in Fig. 1)

Here σi are the spin 1 matrices, ai and bi are the spin–spin coupling 2 constants, and the cubic (second) term in Eq. (5) is believed to be small. The linear term leads to a splitting between the bright and dark exciton states, whereas the dark states become hybridized by the cubic term. In the (considered below) isotropic approximation, where ai = a and bi = b , the bright–dark splitting Δst ≃ 3a /2 and the dark exciton is

tan φX , Y =

3 C1 sin Φ1 ± C2 sin(2ψ − Φ1) . 3 C1 cos Φ1 ± C2 cos(2ψ − Φ1)

(13)

The angle Ω between them is expressed by Fig. 1. Definition of the angles involved in the simulation. Left panel: The angle θd, θs, and ψ sets the dot orientation, the strain direction, and the main anisotropy axis, respectively. Right panel: φX and φY are the tilt angles of the |X 〉 and |Y 〉 eigenstates, and Ω is the angle between these two states.

cos Ω =

2 3 C2 C1 sin 2(ψ − Φ1) 9C14

− 6C22 C12 cos 4(ψ − Φ1) + C24

. (14)

Hence, for the bright exciton, the FSS and polarization directions, 162

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were reported in Refs. [9,11], where numerical calculations for the splitting of the emission lines and their direction of polarization in individual self-assembled CdTe/ZnTe QDs and InAlAs QDs, respectively, are presented. For the two dark exciton states, which are located energetically below the bright doublet, the exciton Hamiltonian has a form

according to the above presented results, significantly depend on the dot symmetry. For different special cases we can conclude the following. (i) For ideal QDs with D2d symmetry, there is no FSS at all. Indeed, in this case the heavy hole–light hole coupling amplitude ρ = 0 and the splitting energies Δh = Δl = 0 as well. (ii) For QDs with C2v symmetry, the dot and strain directions have to coincide with the high symmetry direction, that is θs = θd = 0 and therefore the mixing phase ψ = 0 . Consequently, the bright exciton states are linearly polarized along the [110] and [110] orthogonal directions and split generally by the energy Δ1 = |Δ + (4/ 3 ) C1 C2 Δst |, where Δ = C12 Δh + C22 Δl . For the strain-free QDs with a spherical base, there is no VBM and the FSS reduces simply to the heavy-hole exciton splitting Δ1 = |Δh |. Otherwise, the magnitude of the FSS, which varies from dot to dot, can be tuned to zero in dots with Δ < 0 , as has been observed experimentally for elongated InAs/ GaAs self-assembled QDs in Ref. [13]. In Ref. [13] a systematic correlation between the FSS and the polarization anisotropy is also reported. (Remember that the hh–lh mixing leads to the optical anisotropy which can be characterized by a linear polarization degree 7l = (Imax − Imin )/(Imax + Imin ) = 2 3 C1 C2 (3 − 2C22 )−1 (with Imax (Imin ) the emission intensity) [12]. It can happen, apparently, if a weak mixing limit is relevant and the polarization degree reduces to 7l ≃ 2C1 C2 / 3 . (iii) For real QDs with C1 symmetry, the strain and dot directions are generally different, θs ≠ θd , and the mixing parameters ρ depends on θs − θd . As a result, the FSS and the polarization directions exhibit a rather intricate dependence on θs and θd. In a particular case of weakly asymmetric QDs, in which a hole band mixing arises mainly from the anisotropic relaxation of strains in the QD plane (that is ρ ≃ ρs and ψ ≃ θs ), the polarization directions and the FSS become very sensitive to the angle θs − θd between the strain direction and the dot direction. Both the FSS Δ1 and the angle Ω (see Eqs. (12) and (14), respectively) are modulated by a sinusoidal perturbation with an amplitude defined by the exchange interaction and the light hole probability, respectively. At specific values θs − θd = nπ /2 (n is an integer), the FSS has extreme values and the angle Ω = 90°. To illustrate, we suppose that the strain direction θs is fixed, while the dot direction θd is changed. In order to evaluate numerically the FSS Δ1 and the angle Ω from Eqs. (12) and (14), respectively, we take Δst = 400 μeV , Δh = Δl = 100 μeV , ρs / Elh(0) = 0.2 , and we set θs = 15°. The evolution of Δ1 and Ω with a variation of the angle θs − θd (remember that the angle θs is fixed) is shown in Fig. 2(a) and (b), respectively. The inset shows a change of the phase Φ1 with θs − θd . We observe that the FSS is minimal when the dot and the strain directions are perpendicular and maximal when they are parallel. The angle Ω varies around the angle in 90° being acute at θs − θd < π /2 and blunt at θs − θd > π /2 . A rather sharp switching from the one mode (Ω < 90°) to the other (Ω > 90°) occurs at a critical point θs − θd = 90°, where the FSS has the lower bound. For the above chosen parameters, the splitting Δ1 varies between 72 μeV and 272 μeV. For the angles Ω and Φ1 a deviation of ± 7° and ± 16°, respectively, is observed. Note that similar features

⎛ ⎜ E0 − ⎜ Hd = ⎜ ⎜⎜ ⎝

Δst ⎛ 2 ⎜C 2 ⎝ 1

−

C22 ⎞ 3

⎟ ⎠

Δ2 2iΦ 2 e 2

⎞ ⎟ ⎟ ⎟. C22 ⎞ Δst ⎛ 2 E0 − 2 ⎜C1 − 3 ⎟⎟⎟ ⎠⎠ ⎝ Δ2 −2iΦ 2 e 2

(15)

The phase Φ2 in Eq. (15) is defined relatively to the [110] axis and obeys the equation

tan(2Φ 2 ) =

χ3 sin 4ψ , χ4 + χ3 cos 4ψ

where χ3 =

Δ2 =

χ42

+

4 2 C Δ 3 2 st

χ32

and χ4 =

(16)

δ 2 C12 .

The FS splitting is given by

+ 2χ3 χ4 cos 4ψ .

(17)

Because of the hh–lh mixing, both eigenstates

|Z1 〉 ∝

1 [|Ψ3 〉 − e 2iΦ 2 |Ψ4 〉], 2

(18)

|Z 2 〉 ∝

1 [|Ψ3 〉 + e 2iΦ 2 |Ψ4 〉] 2

(19)

contribute generally to the optical transitions polarized in the growth (z) direction. It is easy to check that the corresponding emission probability |4 1,2 |2 ∝ C22 [1 ∓ cos 2(Φ 2 − 2ψ )]. For the QDs with C2v symmetry, the main anisotropy axis direction ψ = 0 and the FSS gets the upper bound Δ2 = χ4 + χ3 at δ 2 > 0 . In this case only the upper |Z 2 〉 state makes a weak contribution in the optical transitions. 4. Impact of external stresses The obtained results can be extended to the case of QDs under the in-plane external stresses. The stress-induced hh–lh mixing is introduced through the strain tensor components 1 εxx − εyy = p (S11 − S12 )cos 2θx and εxy = 4 pS44 sin 2θx , where p is the stress magnitude, Sij are the components of the compliance tensor, and the angle θx sets the external stress direction with respect to the [100] axis. The corresponding term in the Bir–Pikus Hamiltonian R ( p ) = ρp e−2iθp (see Eq. (1)), where the angle θp determines the (stress-induced) strain direction with respect to the [110] axis. For the tensile (compressive) stress along the [110], [110], or [100] axis, π π π π θp = 2 (0), θp = 0( 2 ), or θp = 4 (− 4 ). (We take the compressive stress as negative.) Besides, the hh–lh splitting introduced in Eq. (4) is replaced by Elh ( p ) = Elh(0) + b (S11 − S12 ) p [21]. From the above presented results we immediately conclude that the

Fig. 2. (a) The FSS of the bright exciton, (b) the angle between the |X 〉 and |Y 〉 eigenstates, and (insert) the phase Φ1 as a function of the angle θs − θd between the strain direction and the dot-shape direction.

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largely from QD to QD belonging to the self-assembled QDs structure. For the stress along the [100] axis, the FSS has its lower bound Δ1 = Δh at p=0. Obviously, the polarization directions of the bright exciton states also depend on the external stress magnitude and orientation, in agreement with the previous results reported in the literature [15,18,19,24,26]. For pure QDs with C2v symmetry under the stress along the [110] ([110]) direction, the polarization direction is still set by the high symmetry axis. There are here, however, some peculiarities. Indeed, the phase Φ1, which governs a direction of the polarization, obeys the equation tan (2Φ1) = χ2 / χ1 at θd = 0 , see Eq. (7). Because χ2 = 0 and χ1 changes a sign at p = p110 , one can conclude that at p ≠ p110 the phase Φ1 differs by 90° below and after p110. When the stress is oriented, for example, along the [110] direction, from Eqs. (10)–(11) immediately follows that below p110 the |X 〉 (|Y 〉) state of the bright exciton is polarized along the [110] ([110]) axis, but the polarization jumps to the [110] ([110]) direction after p110. When the stress is along the [100]-direction, χ1 ≠ 0 at any p and χ2 = 0 at p=0. Considering the approximation C1 C2 Δst > Δ, which can be valid at a rather small p because Δst ⪢Δ, the ratio χ2 / χ1 ≈ tan (2α ) and therefore Φ1 ≈ θp . Hence the stress along the [100] direction induces a rotation of the polarization directions from their conventional directions along the [110] and [110] axes in unstrained QDs towards the [100] and [010] directions at enough large values of |p|. A behavior of the dark exciton under uniaxial stresses differs from that for the bright exciton. Indeed, we observe that Eq. (17) is invariant with respect to a replacing ψ → ψ ± π /2 , so that the FSS Δ2 is equally affected by the tensile and compressive stress and the orthogonal stresses as well. For the QDs with C2v symmetry, according to Eq. (17), 4 Δ2 = |δ 2 C12 ± 3 C22 Δst |, where a “+” or “−” sign refers to the stress along the [110] direction or to the stress along the [100] direction. So the FSS of the dark exciton also can be tuned to zero by the uniaxial stress. In contrast to the bright exciton, however, now a sign of the original splitting δ2 is crucial: at δ 2 < 0 the splitting Δ2 can be tuned to zero by the stress along the [110] direction, but at δ 2 > 0 this effect can be achieved by the stress along the [100] direction. As an example, Fig. 3(b) illustrates the case of a negative δ2, namely, we take δ 2 = −1 μeV ; other parameters are the same as in Fig. 3(a). In Fig. 3(b) is seen that while at some critical values of the tensile and compressive stress along the [110] direction the dark exciton is degenerate, the FSS increases with both the tensile and compressive stress along the [100]-direction. The observed p → −p asymmetry is due to a difference between the hh–lh splitting Elh for the tensile and compressive stress.

properties of the exciton ground state in QDs under uniaxial stress depend on the dot symmetry and the stress direction which now operates the hole mixing parameter ψ, in agreement with previous results based on the symmetry consideration [24]. Assuming that the dot parameters ( Δh , Δl , and Δst) are not changed under the external stress, a combined effect of the stress-induced hole mixing and the exchange interaction will be here of a paramount importance. Below we concentrate firstly on a behavior of the bright exciton. For symmetrical QDs with C2v symmetry, which show no built-in anisotropic strains, the angle α = ψ = θp and therefore the FSS is governed directly by the external stress direction θp. For the stress oriented along the high symmetry axis, the FSS, according to Eq. (12), is given by (20)

Δ1 = |Δ ∓ (4/ 3 ) C1 C2 Δst |,

where a “−” or “+” sign refers to the tensile (compressive) stress or compressive (tensile) stress along the [110] ([110])-direction, C1 (C2 ) is given by Eq. (4) with ρ = ρp = dS44 p /4 , and Elh(0) → Elh ( p ), as noted above. Believing that Δ > 0 , from the result (20) follows that the FSS can be tuned to zero by means of the tensile stress along the [110] direction or the compressive stress along the [110] direction. If the external stress orientation deviates from the high symmetry axis, the original symmetry is violated and the FSS cannot be tuned to zero. For example, for the stress along the [100] axis, α = ± π /4 and Eq. (12)) reduces to

Δ1 =

2

(C12 Δh − C22 Δl ) +

16 2 2 2 C1 C2 Δst , 3

(21)

Elh(0)

→ Elh ( p ) and where C1 (C2 ) are given by Eq. (4) with ρp = 3 b (S11 − S12 ) p /2 . One can proof that the elongated QDs with C2v symmetry (θd = 0 ) will show similar features. For QDs with C1 symmetry, no crossing of the bright exciton levels is possible at all, since now the orientation of the main anisotropy axis and, consequently, the stress-dependent angle α in Eqs. (8)–(12) will depart from the [110] ([110]) direction. To illustrate, the FSS versus p along the [110], [110], and [100] axes given by Eqs. (20)–(21), respectively, is presented in Fig. 3(a). Calculations are performed with a set of typical d = −5 eV , S11 = 1.17 × 10−12 cm2/dyn , b = −2 eV , parameters S12 = −0.37 × 10−12 cm2/dyn , S44 = 1.7 × 10−12 cm2/dyn [25]. For other Δh = Δl = 10 μeV , and Elh(0) = 20 meV , parameters we take Δst = 100 μeV . The obtained results are in a qualitative accordance with those based on the atomistic pseudopotential calculations for pure InAs/GaAs QDs previously reported in Refs. [24,15]. We observe in Fig. 3(a) that for the stress along the [110] and [110] directions, the FSS behaves almost symmetrically. Besides, for the considered stress region, the splitting shows a linear dependence on the stress magnitude because Δst ⪢Δh , Δl . Using the linear approximation, the zero value of Δ1 is reached at the critical stress |p110 | ≃

(0) Δh 3 Elh , Δst dS44

5. Summary In summary, the fine structure splitting of the exciton ground state in a quantum dot and the polarization of emitted photons as well are

which may be different

Fig. 3. The FSS of (a) the bright exciton and (b) the dark exciton as a function of the uniaxial stress along the [110], [100], and [110] crystallographic directions.

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E. Tsitsishvili

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significantly dependent on the orientation of the main anisotropy axis in the plane of a dot with respect to the dot orientation. Such result is a consequence of an interplay of two leading mechanisms which modify the spin states in a quantum dot. The first mechanism is given by the Luttinger–Kohn–Bir–Pikus terms which couple the hole spin states and are due to the shape asymmetry and anisotropic strains built up in a quantum dot and/or arising out of the external stresses. The second mechanism is due to the hole–electron exchange interaction which modify the overall exciton spins. The presented here analytical results, covering some special cases discussed previously in the literature, allow to predict qualitatively the excitonic properties in quantum dots with different features. Tuning of the FS splitting by external stress also falls under consideration. References [1] O. Benson, C. Santori, M. Pelton, Y. Yamamoto, Phys. Rev. Lett. 84 (2000) 2513. [2] R.M. Stevenson, R.J. Young, P. Atkinson, K. Cooper, D.A. Ritchie, A.J. Shields, Nature 439 (2006) 179. [3] Gabriel Bester, Selvakumar Nair, Alex Zunger, Phys.Rev. B 67 (2003) 161306 R. [4] M. Bayer, G. Ortner, O. Stern, A. Kuther, A.A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T.L. Reinecke, S.N. Walck, J.P. Reithmaier, F. Klopf, F. Schäfer, Phys. Rev. B 65 (2002) 195315. [5] T. Takagahara, Phys. Rev. B 62 (2000) 16840. [6] E.L. Ivchenko, G.E. Pikus, Superlattices and Other Heterostructures: Symmetry and Optical Phenomena, Springer-Verlag, Berlin, 1995; Optical spectroscopy of Semiconductor Nanostructures, Alpha Science International, Harrow, UK, 2005. [7] T. Belhadj, T. Amand, A. Kunold, C.-M. Simon, T. Kuroda, M. Abbarchi, T. Mano, K. Sakoda, S. Kunz, X. Marie, Appl. Phys. Lett. 97 (2010) 051111. [8] I. Favero, G. Cassabois, C. Voisin, C. Delalande, Ph. Roussignol, R. Ferreira,

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