- Email: [email protected]
GaAs quantum dots
Physica B 479 (2015) 6–9
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Light effects in asymmetric vertically coupled InAs/GaAs quantum dots V.N. Stavrou n Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 8 July 2015 Received in revised form 27 August 2015 Accepted 13 September 2015 Available online 14 September 2015
In this paper, the dependence of circular light polarization on the size asymmetry of self-assembled coupled quantum dots (SACQDs) has been studied. The heterostructure consists of ellipsoidal shaped QDs made with InAs which are embedded in a wetting layer and are surrounded by GaAs. By considering fully spin-polarized carriers within the QD structure, the light polarization has been estimated along the plane of spin polarized electrons (or holes). Circularly polarized light strongly depends on the ratio related to the different QD volumes. In the case of elongated QDs, small interdot distance and large volume ratio, the light polarization observed along the plane (110) receives the largest value (∼90%). On the other hand, the polarization efficiency of the emitted light decreases as the QD elongation decreases and finally vanishes for axially symmetric QD caps. & 2015 Elsevier B.V. All rights reserved.
Keywords: Asymmetric SAQDs Light polarization Spin-polarized states Spin qubits vertically coupled quantum dots PACS numbers: 73.21.La 73.63.Kv
1. Introduction Semiconductor QDs are heterostructures which confine the motion of conduction band electrons and valence band holes in three dimensional (3-D) space. The carrier confinement makes the QDs significant in quantum technology due to highly tunable electronic and optical properties [1–3]. These properties make QDs suitable for building quantum bits (qubits), QD lasers, QD field effect transistors, and QD infrared photodetectors among other optoelectronic devices [4]. Several fabrication techniques have been recently used to control the growth of QDs and the quality of the samples [4]. Stranski–Krastanov (SK) random growth in molecular beam epitaxy (MBE) or metalorganic chemical vapor deposition (MOCVD) is the most commonly used growth method to create high quality QDs samples, in size and QDs shapes. It is worth mentioning that in SK growth, the first deposited layers of QD islands grow in a flat, layer-by-layer fashion which is called wetting layer. Theoretical results on the electronic structure of QDs including the wetting layer have been recently reported [5–7]. In quantum computation, the superposition of spin states of quantum dots is frequently used to create quantum bit gates [8]. Carrier spin states have been calculated using variational Monte n
Fax: þ 30 210 5531755. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.physb.2015.09.024 0921-4526/& 2015 Elsevier B.V. All rights reserved.
Carlo, spin density-functional theory and strain dependent k·p theory among other numerical schemes to perform simulations [9–11]. Measurable quantities which can be used to control the qubit performance are, among others, spin relaxation, light polarization within the QDs, quantum noise and quantum entanglement [1–3,13]. During the last decades, the polarization efficiency of the emitted light in QDs has been studied [12–14] for the case of single and coupled identical QDs. The main objective of this paper is the investigation of light polarization in a heterostructure made with two coupled QDs and different QD volumes. The results are presented in terms of parameters, e.g. energy gap, which can be experimentally measured. This paper is arranged as follows. In Section 2, the theoretical description of electron/hole states and of light polarization within the asymmetric CQDs is outlined. The results of the theoretical estimation and the conclusions are presented in Sections 3 and 4 respectively.
2. Theory The investigated heterostucture, as illustrated in Fig. 1, consists of two ellipsoidal caps (with different volumes) made with InAs. The two caps are separated by a distance D are embedded in a wetting layer of thickness 0.3 nm and are surrounded by GaAs. The strain dependent k·p theory has been employed to calculate single electron and hole wavefunctions. For the description of the k·p
V.N. Stavrou / Physica B 479 (2015) 6–9
7
all material parameters which are used in the numerical calculations are taken by Ref. [15]. The observed emitted light polarization along the plane of the polarized electron (or hole) spin can be written as
Pl =
Il(+) − Il(−) Il(+) + Il(−)
(3)
Il(±) is the intensity of emission of circularly polarized light with ± helicity for the case of spin-polarized electrons and unpolarized holes and it can be described as (±)
(±)
Il(±) = |〈ψh |ϵ^l ·p|ψe 〉|2 + |〈ψh |T ^ϵl ·p|ψe 〉|2
(4)
and the intensity of circularly polarized light for the case of spinpolarized holes and unpolarized electrons is given by (±) (±) Il(±) = |〈ψh |ϵ^l ·p|ψe 〉|2 + |〈ψh |ϵ^l ·pT |ψe 〉|2
(5) ^ϵl(±)
Fig. 1. The geometry of the self-assembled coupled quantum dot heterostructure made with InAs/GaAs with a fixed wetting layer thickness to 0.3 nm. The volume of the lower dot is V1 and for the upper dot the volume is V2. The quantum dots volume ratio is defined as Vr = V2/V1.
3. Results
theory, a basis of eight Bloch functions has been used:
{ψs, ↑, ψx, ↑, ψy, ↑, ψz, ↑, ψs, ↓, ψx, ↓, ψy, ↓, ψz, ↓ } where x, y, y are related to the three directions of the space and the arrows denote the spin. A 2 × 2 block matrix is used to describe the eight-band k·p Hamiltonian:
⎛ G (k) Γ ⎞ ^ ⎟⎟ / = ⎜⎜ ⎝ − Γ G ( − k) ⎠
where p is the momentum operator and is the circular polarization vector with helicity ± . The emitted light polarization is independent whether the injected spin-polarized carriers are electrons or holes due to the anticommutation relations between p and T.
The main research part of the current investigation is the dependence of circular light polarization on the size asymmetry of CQDs, on the interdot distance and on polarization planes for 100% spin polarized electrons (or holes). Circularly polarized light along some planes (like [110], [100] and [001] ) has been explored. The results concerning the plane [110] highlight the importance of this
(1)
where F and G are 4 × 4 matrices. The matrix F couples the spin projections ↑ and ↓ due to spin–orbit (SO) interaction and matrix G consists of the potential energy, the kinetic energy, a SO interactions and a strain dependent part. The quantum dots elongation and the width-to-height ratio are respectively given by the following equations:
e = d[110]/d[1 − 10] b = (d[110] + d[1 − 10] )/h The height h denotes the QDs height h1 or h2. The volume ratio of the quantum dots is denoted by Vr = V2/V1. Vr varies by changing the height of the QD with volume V2 and by keeping the height of the other QD fixed ( h1 = 3.3 nm , including the wetting layer). Lastly, the width-to-height ratio is fixed to b¼7.5 in all calculations of this paper. Strain and carrier confinement split the heavy hole and light hole degeneracy and the states which are denoted by |ψ 〉 and T |ψ 〉 (time reverses of each other) are doubly degenerate. Carriers spin-polarized ground and first excited states can be constructed by taking a linear combination of the states comprising the doublet and by adjusting the coefficient in order to maximize the expectation value of the pseudospin operator projected onto a direction l [13]. The adjusting complex number α maximizes the quantity
^ [〈ψ | + α ⁎〈ψ |T ] l ·S [|ψ 〉 + αT |ψ 〉] 1 + |α|2
(2)
with S the pseudospin operator in the 8-band k·p theory [13]. Furthermore, all wave functions were computed numerically on a real space grid with spacing set to the wetting layer thickness and
Fig. 2. The circular light polarization P110 dependence on the interdot distance for elongated QDs and for a variety of QD volume ratios (Vr ).
8
V.N. Stavrou / Physica B 479 (2015) 6–9
polarization plane in spintronic devices. Fig. 2 shows the circular light polarization along the plane [110] (as illustrated in Fig. 1) for 100% spin polarized carriers along the same plane. As it is shown in Ref. [6], electron/hole ground state energies increase by increasing the separation distance and the energy gap consequently increases. As the interdot distance reaches large values, the electron/hole energies of the QDs slightly change and thereafter the energy gap is almost constant. Furthermore, increasing the separation distance, the energy difference between the first excited state and ground state becomes very tiny which is the approach of single electron/hole within a single QD. On the other hand electron/hole wavefunctions strongly depend on the size of the QD. In the case of small QDs, the wavefunctions are larger than the wavefunctions corresponding to large QDs due to the strong carrier confinement. As far as the separation distance is small, the wavefunctions overlap, while they do not overlap for large interdot distance. The electron/hole wavefunctions are distributed within either the large or the small dot, which depends on the volume ratio of the QDs [6]. For spherical QD caps (e¼ 1.0), the polarization is zero because the intensities with different helicities are equal (see Eq. (3)). Circular light polarization increases as the QDs become more elongated due to the azimuthal symmetry breaking in the case of elongated QDs as it appears in (Eqs. (4) and 5). As the interdot distance takes smaller values, the polarization increases, and it decreases as the interdot distance increases. The enhancement of the light polarization is assisted by the relative size of the QDs as well. For large volume ratio, circular light polarization receives large values approaching 90% for 100% polarized carriers due to the wavefunction distribution between large and small QD, as earlier mentioned. The dependence of the energy gap as a function of the QDs volume ratio for small and large separation distances has been studied in Ref. [6]. As the ratio increases the energy gap decreases either for small or large interdot distance [6]. The dependence of circular light polarization on the energy gap is presented in Fig. 3. It is obvious that for small energy gap (large QDs) light polarization gets large values either for small or large interdot distance due to the polarization dependence on the electron/hole wavefunctions on different geometries, as above mentioned. Fig. 4 presents the polarization dependence on QD elongation for a few geometry parameters. For axially symmetric QD caps light polarization vanishes due to the fact that light intensity with different helicities have the same value. P110 increases as QDs become more elongated receiving the largest value P110 ∼ 0.9 for large QD separation distance and large QD volume ratio as a
Fig. 4. The circular light polarization P110 dependence on QDs elongation for different geometry configurations.
Fig. 5. The circular light polarization P110 dependence on the QD volume ratio, the interdot distance for two different elongated QDs.
consequence of the azimuthal symmetry breaking. Furthermore, the light polarization P110 as a function of the QDs volume ratio is presented in Fig. 5. Although the features of the results have earlier been discussed and explored, here a clear overview of influences of the QDs size on the polarization is presented Lastly, the fact that for any geometric configuration the light polarization along the plane [001] has a value close to 1 and along the [100], [010] is zero for 100% spin polarized carriers is of special importance. Therefore, the choice of the light polarization plane could be very critical for optoelectronic devices like qubits and QD lasers, among others.
4. Conclusions
Fig. 3. The circular light polarization P110 dependence on energy gap (Eg ) for different geometry configurations.
The main aim of this paper has been the development of a theory to describe the polarization efficiency of the emitted light in asymmetric self-assembled coupled quantum dots. In particular, the circular light polarization for the case in which the electron (or hole) spin is polarized along the same direction as the observed emitted light has been explored. The geometry has been chosen to
V.N. Stavrou / Physica B 479 (2015) 6–9
be comparable to the fabricated QDs [12,16]. The numerical investigation shows that the degree of polarization is very sensitive to the size asymmetry, the elongation and the separation distance of the QDs. Circular light polarization reaches the value of (∼90%) for large QD volume ratio and small separation distance. However, for small volume ratio and large interdot distance circular polarization is almost four times smaller than the highest value (Fig. 5). Consequently, for a variety of applications (e.g. spin-qubits), the samples should be fabricated by considering as a top priority the relative size of the QDs, the QDs elongation and the separation distance as well. According to the best knowledge of the author, the polarization efficiency of the emitted light in asymmetric coupled quantum dots has not been reported earlier.
Acknowledgments The work is supported by the University of Iowa under the Grant no. 52570034. The author would like to thank Prof. C.E. Pryor for useful discussions.
9
References [1] A. Tartakovskii (Ed.), Quantum Dots: Optics, Electron Transport and Future Applications, Cambridge, 2012. [2] Z.M. Wang (Ed.), Quantum Dot Devices, Springer, 2012. [3] L. Jacak, P. Hawrylak, A. Wójs, Quantum Dots, Springer, 1998. [4] M. Razeghi, Technology of Quantum Devices, Springer, 2010. [5] S. Lee, Olga L. Lazarenkova, P. Allmen, F. Oyafuso, G. Klimeck, Phys. Rev. B 70 (2004) 12. [6] V.N. Stavrou, Physica B: Condens. Matter B 407 (7) (2012) 1157. [7] C. Sun, P. Lu, Z. Yu, H. Cao, L. Zhang, Physica B: Condens. Matter B 407 (22) (2012) 4440. [8] D. Loss, D.P. DiVincenzo, Phys. Rev. B 57 (1998) 120. [9] S. Akbar, I.H. Lee, Phys. Rev. B 63 (2001) 165301; E. Rasanen, H. Saarikoski, V.N. Stavrou, A. Harju, M.J. Puska, R.M. Nieminen, Phys. Rev. B 67 (2003) 235307. [10] W.M.C. Foulkes, L. Mitas, R.J. Needs, G. Rajagopal, Rev. Mod. Phys. 73 (2001) 33. [11] C. Pryor, C. Pistol, L. Samuelson, Phys. Rev. B 56 (1997) 10404; C. Pryor, Phys. Rev. B 57 (1998) 7190. [12] Y. Chye, M.E. White, E. Johnston-Halperin, B.D. Gerardot, D.D. Awschalom, P. M. Petroff, Phys. Rev. B 66 (2002) 201301(R). [13] C. Pryor, M.E. Flatté, Phys. Rev. Lett. 91 (2003) 257901. [14] V.N. Stavrou, J. Phys.: Condens. Matter. 20 (2008) 395222. [15] I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys. 89 (2001) 5815. [16] H.Z. Song, T. Usuki, in: Zhiming M. Wang (Ed.), Self-Assembled Quantum Dots, Springer, 2008, and references therein (Chapter 9).
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Light effects in asymmetric vertically coupled InAs/GaAs quantum dots V.N. Stavrou n Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 8 July 2015 Received in revised form 27 August 2015 Accepted 13 September 2015 Available online 14 September 2015
In this paper, the dependence of circular light polarization on the size asymmetry of self-assembled coupled quantum dots (SACQDs) has been studied. The heterostructure consists of ellipsoidal shaped QDs made with InAs which are embedded in a wetting layer and are surrounded by GaAs. By considering fully spin-polarized carriers within the QD structure, the light polarization has been estimated along the plane of spin polarized electrons (or holes). Circularly polarized light strongly depends on the ratio related to the different QD volumes. In the case of elongated QDs, small interdot distance and large volume ratio, the light polarization observed along the plane (110) receives the largest value (∼90%). On the other hand, the polarization efficiency of the emitted light decreases as the QD elongation decreases and finally vanishes for axially symmetric QD caps. & 2015 Elsevier B.V. All rights reserved.
Keywords: Asymmetric SAQDs Light polarization Spin-polarized states Spin qubits vertically coupled quantum dots PACS numbers: 73.21.La 73.63.Kv
1. Introduction Semiconductor QDs are heterostructures which confine the motion of conduction band electrons and valence band holes in three dimensional (3-D) space. The carrier confinement makes the QDs significant in quantum technology due to highly tunable electronic and optical properties [1–3]. These properties make QDs suitable for building quantum bits (qubits), QD lasers, QD field effect transistors, and QD infrared photodetectors among other optoelectronic devices [4]. Several fabrication techniques have been recently used to control the growth of QDs and the quality of the samples [4]. Stranski–Krastanov (SK) random growth in molecular beam epitaxy (MBE) or metalorganic chemical vapor deposition (MOCVD) is the most commonly used growth method to create high quality QDs samples, in size and QDs shapes. It is worth mentioning that in SK growth, the first deposited layers of QD islands grow in a flat, layer-by-layer fashion which is called wetting layer. Theoretical results on the electronic structure of QDs including the wetting layer have been recently reported [5–7]. In quantum computation, the superposition of spin states of quantum dots is frequently used to create quantum bit gates [8]. Carrier spin states have been calculated using variational Monte n
Fax: þ 30 210 5531755. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.physb.2015.09.024 0921-4526/& 2015 Elsevier B.V. All rights reserved.
Carlo, spin density-functional theory and strain dependent k·p theory among other numerical schemes to perform simulations [9–11]. Measurable quantities which can be used to control the qubit performance are, among others, spin relaxation, light polarization within the QDs, quantum noise and quantum entanglement [1–3,13]. During the last decades, the polarization efficiency of the emitted light in QDs has been studied [12–14] for the case of single and coupled identical QDs. The main objective of this paper is the investigation of light polarization in a heterostructure made with two coupled QDs and different QD volumes. The results are presented in terms of parameters, e.g. energy gap, which can be experimentally measured. This paper is arranged as follows. In Section 2, the theoretical description of electron/hole states and of light polarization within the asymmetric CQDs is outlined. The results of the theoretical estimation and the conclusions are presented in Sections 3 and 4 respectively.
2. Theory The investigated heterostucture, as illustrated in Fig. 1, consists of two ellipsoidal caps (with different volumes) made with InAs. The two caps are separated by a distance D are embedded in a wetting layer of thickness 0.3 nm and are surrounded by GaAs. The strain dependent k·p theory has been employed to calculate single electron and hole wavefunctions. For the description of the k·p
V.N. Stavrou / Physica B 479 (2015) 6–9
7
all material parameters which are used in the numerical calculations are taken by Ref. [15]. The observed emitted light polarization along the plane of the polarized electron (or hole) spin can be written as
Pl =
Il(+) − Il(−) Il(+) + Il(−)
(3)
Il(±) is the intensity of emission of circularly polarized light with ± helicity for the case of spin-polarized electrons and unpolarized holes and it can be described as (±)
(±)
Il(±) = |〈ψh |ϵ^l ·p|ψe 〉|2 + |〈ψh |T ^ϵl ·p|ψe 〉|2
(4)
and the intensity of circularly polarized light for the case of spinpolarized holes and unpolarized electrons is given by (±) (±) Il(±) = |〈ψh |ϵ^l ·p|ψe 〉|2 + |〈ψh |ϵ^l ·pT |ψe 〉|2
(5) ^ϵl(±)
Fig. 1. The geometry of the self-assembled coupled quantum dot heterostructure made with InAs/GaAs with a fixed wetting layer thickness to 0.3 nm. The volume of the lower dot is V1 and for the upper dot the volume is V2. The quantum dots volume ratio is defined as Vr = V2/V1.
3. Results
theory, a basis of eight Bloch functions has been used:
{ψs, ↑, ψx, ↑, ψy, ↑, ψz, ↑, ψs, ↓, ψx, ↓, ψy, ↓, ψz, ↓ } where x, y, y are related to the three directions of the space and the arrows denote the spin. A 2 × 2 block matrix is used to describe the eight-band k·p Hamiltonian:
⎛ G (k) Γ ⎞ ^ ⎟⎟ / = ⎜⎜ ⎝ − Γ G ( − k) ⎠
where p is the momentum operator and is the circular polarization vector with helicity ± . The emitted light polarization is independent whether the injected spin-polarized carriers are electrons or holes due to the anticommutation relations between p and T.
The main research part of the current investigation is the dependence of circular light polarization on the size asymmetry of CQDs, on the interdot distance and on polarization planes for 100% spin polarized electrons (or holes). Circularly polarized light along some planes (like [110], [100] and [001] ) has been explored. The results concerning the plane [110] highlight the importance of this
(1)
where F and G are 4 × 4 matrices. The matrix F couples the spin projections ↑ and ↓ due to spin–orbit (SO) interaction and matrix G consists of the potential energy, the kinetic energy, a SO interactions and a strain dependent part. The quantum dots elongation and the width-to-height ratio are respectively given by the following equations:
e = d[110]/d[1 − 10] b = (d[110] + d[1 − 10] )/h The height h denotes the QDs height h1 or h2. The volume ratio of the quantum dots is denoted by Vr = V2/V1. Vr varies by changing the height of the QD with volume V2 and by keeping the height of the other QD fixed ( h1 = 3.3 nm , including the wetting layer). Lastly, the width-to-height ratio is fixed to b¼7.5 in all calculations of this paper. Strain and carrier confinement split the heavy hole and light hole degeneracy and the states which are denoted by |ψ 〉 and T |ψ 〉 (time reverses of each other) are doubly degenerate. Carriers spin-polarized ground and first excited states can be constructed by taking a linear combination of the states comprising the doublet and by adjusting the coefficient in order to maximize the expectation value of the pseudospin operator projected onto a direction l [13]. The adjusting complex number α maximizes the quantity
^ [〈ψ | + α ⁎〈ψ |T ] l ·S [|ψ 〉 + αT |ψ 〉] 1 + |α|2
(2)
with S the pseudospin operator in the 8-band k·p theory [13]. Furthermore, all wave functions were computed numerically on a real space grid with spacing set to the wetting layer thickness and
Fig. 2. The circular light polarization P110 dependence on the interdot distance for elongated QDs and for a variety of QD volume ratios (Vr ).
8
V.N. Stavrou / Physica B 479 (2015) 6–9
polarization plane in spintronic devices. Fig. 2 shows the circular light polarization along the plane [110] (as illustrated in Fig. 1) for 100% spin polarized carriers along the same plane. As it is shown in Ref. [6], electron/hole ground state energies increase by increasing the separation distance and the energy gap consequently increases. As the interdot distance reaches large values, the electron/hole energies of the QDs slightly change and thereafter the energy gap is almost constant. Furthermore, increasing the separation distance, the energy difference between the first excited state and ground state becomes very tiny which is the approach of single electron/hole within a single QD. On the other hand electron/hole wavefunctions strongly depend on the size of the QD. In the case of small QDs, the wavefunctions are larger than the wavefunctions corresponding to large QDs due to the strong carrier confinement. As far as the separation distance is small, the wavefunctions overlap, while they do not overlap for large interdot distance. The electron/hole wavefunctions are distributed within either the large or the small dot, which depends on the volume ratio of the QDs [6]. For spherical QD caps (e¼ 1.0), the polarization is zero because the intensities with different helicities are equal (see Eq. (3)). Circular light polarization increases as the QDs become more elongated due to the azimuthal symmetry breaking in the case of elongated QDs as it appears in (Eqs. (4) and 5). As the interdot distance takes smaller values, the polarization increases, and it decreases as the interdot distance increases. The enhancement of the light polarization is assisted by the relative size of the QDs as well. For large volume ratio, circular light polarization receives large values approaching 90% for 100% polarized carriers due to the wavefunction distribution between large and small QD, as earlier mentioned. The dependence of the energy gap as a function of the QDs volume ratio for small and large separation distances has been studied in Ref. [6]. As the ratio increases the energy gap decreases either for small or large interdot distance [6]. The dependence of circular light polarization on the energy gap is presented in Fig. 3. It is obvious that for small energy gap (large QDs) light polarization gets large values either for small or large interdot distance due to the polarization dependence on the electron/hole wavefunctions on different geometries, as above mentioned. Fig. 4 presents the polarization dependence on QD elongation for a few geometry parameters. For axially symmetric QD caps light polarization vanishes due to the fact that light intensity with different helicities have the same value. P110 increases as QDs become more elongated receiving the largest value P110 ∼ 0.9 for large QD separation distance and large QD volume ratio as a
Fig. 4. The circular light polarization P110 dependence on QDs elongation for different geometry configurations.
Fig. 5. The circular light polarization P110 dependence on the QD volume ratio, the interdot distance for two different elongated QDs.
consequence of the azimuthal symmetry breaking. Furthermore, the light polarization P110 as a function of the QDs volume ratio is presented in Fig. 5. Although the features of the results have earlier been discussed and explored, here a clear overview of influences of the QDs size on the polarization is presented Lastly, the fact that for any geometric configuration the light polarization along the plane [001] has a value close to 1 and along the [100], [010] is zero for 100% spin polarized carriers is of special importance. Therefore, the choice of the light polarization plane could be very critical for optoelectronic devices like qubits and QD lasers, among others.
4. Conclusions
Fig. 3. The circular light polarization P110 dependence on energy gap (Eg ) for different geometry configurations.
The main aim of this paper has been the development of a theory to describe the polarization efficiency of the emitted light in asymmetric self-assembled coupled quantum dots. In particular, the circular light polarization for the case in which the electron (or hole) spin is polarized along the same direction as the observed emitted light has been explored. The geometry has been chosen to
V.N. Stavrou / Physica B 479 (2015) 6–9
be comparable to the fabricated QDs [12,16]. The numerical investigation shows that the degree of polarization is very sensitive to the size asymmetry, the elongation and the separation distance of the QDs. Circular light polarization reaches the value of (∼90%) for large QD volume ratio and small separation distance. However, for small volume ratio and large interdot distance circular polarization is almost four times smaller than the highest value (Fig. 5). Consequently, for a variety of applications (e.g. spin-qubits), the samples should be fabricated by considering as a top priority the relative size of the QDs, the QDs elongation and the separation distance as well. According to the best knowledge of the author, the polarization efficiency of the emitted light in asymmetric coupled quantum dots has not been reported earlier.
Acknowledgments The work is supported by the University of Iowa under the Grant no. 52570034. The author would like to thank Prof. C.E. Pryor for useful discussions.
9
References [1] A. Tartakovskii (Ed.), Quantum Dots: Optics, Electron Transport and Future Applications, Cambridge, 2012. [2] Z.M. Wang (Ed.), Quantum Dot Devices, Springer, 2012. [3] L. Jacak, P. Hawrylak, A. Wójs, Quantum Dots, Springer, 1998. [4] M. Razeghi, Technology of Quantum Devices, Springer, 2010. [5] S. Lee, Olga L. Lazarenkova, P. Allmen, F. Oyafuso, G. Klimeck, Phys. Rev. B 70 (2004) 12. [6] V.N. Stavrou, Physica B: Condens. Matter B 407 (7) (2012) 1157. [7] C. Sun, P. Lu, Z. Yu, H. Cao, L. Zhang, Physica B: Condens. Matter B 407 (22) (2012) 4440. [8] D. Loss, D.P. DiVincenzo, Phys. Rev. B 57 (1998) 120. [9] S. Akbar, I.H. Lee, Phys. Rev. B 63 (2001) 165301; E. Rasanen, H. Saarikoski, V.N. Stavrou, A. Harju, M.J. Puska, R.M. Nieminen, Phys. Rev. B 67 (2003) 235307. [10] W.M.C. Foulkes, L. Mitas, R.J. Needs, G. Rajagopal, Rev. Mod. Phys. 73 (2001) 33. [11] C. Pryor, C. Pistol, L. Samuelson, Phys. Rev. B 56 (1997) 10404; C. Pryor, Phys. Rev. B 57 (1998) 7190. [12] Y. Chye, M.E. White, E. Johnston-Halperin, B.D. Gerardot, D.D. Awschalom, P. M. Petroff, Phys. Rev. B 66 (2002) 201301(R). [13] C. Pryor, M.E. Flatté, Phys. Rev. Lett. 91 (2003) 257901. [14] V.N. Stavrou, J. Phys.: Condens. Matter. 20 (2008) 395222. [15] I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys. 89 (2001) 5815. [16] H.Z. Song, T. Usuki, in: Zhiming M. Wang (Ed.), Self-Assembled Quantum Dots, Springer, 2008, and references therein (Chapter 9).