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Spin–orbit effects on the nonlinear optical properties of a quantum dot in simultaneous electric and magnetic fields
Physica E 64 (2014) 29–32
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Spin–orbit effects on the nonlinear optical properties of a quantum dot in simultaneous electric and magnetic fields O. Aytekin a,n, S. Turgut b, M. Tomak b a b
Department of Physics, Adıyaman University, 02040 Adıyaman, Turkey Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
The nonlinear optical coefficients of a quantum dot are studied using a Woods–Saxon potential. The Rashba spin–orbit effects on the nonlinear optical properties are investigated. The competing effects of spin–orbit, dot size, electric and magnetic fields are given in detail.
art ic l e i nf o
a b s t r a c t
Article history: Received 5 February 2014 Received in revised form 30 May 2014 Accepted 16 June 2014 Available online 8 July 2014
We report on the nonlinear optical properties of a quantum dot including the Rashba spin–orbit interaction (RSOI) with external electric and magnetic fields. The effect of dot size is considered. We do not make any assumptions about the strength of the confinement. We use the numerical diagonalization of the Hamiltonian to determine the electronic structure. The confining potential is taken to be of the Woods–Saxon type. We find the effect of RSOI on nonlinear optical coefficients. & 2014 Elsevier B.V. All rights reserved.
Keywords: Quantum dot Nonlinear optics Spin–orbit
1. Introduction Increasing perfection in growth techniques allows one to design and investigate high performance, low-dimensional semiconductor structures. This enables one to design and investigate quantum dots (QD) with varying sizes, shapes and the number of electrons. The resulting physics is rich and challenging [1–3]. In recent years, the field of spin physics with the possibility of controlling and manipulating electron spins for low-dimensional semiconducting device applications have received great attention [4–11]. The theoretically interesting spin effect is the study of the
n
Corresponding author. E-mail address: [email protected] (O. Aytekin).
http://dx.doi.org/10.1016/j.physe.2014.06.018 1386-9477/& 2014 Elsevier B.V. All rights reserved.
spin–orbit interaction (SOI) in low-dimensional structures [12–17]. There are two widely used models for the SOI arising from different physical effects and having different Hamiltonian forms. One is the Dresselhaus [18] spin–orbit interaction (DSOI) caused by bulk inversion asymmetry and the other is the Rashba spin– orbit interaction (RSOI) [19] which arises from the asymmetry of the potential in reduced dimensional structures. There are a very limited number of investigations of spin–orbit effects on the nonlinear optical properties of quantum dots [20– 24]. Hassanabadi et al. have investigated the optical absorption and refractive index changes in a three-electron quantum dot in the presence of the RSOI [20]. They found that a decrease of the quantum dot radius blue shifts and amplifies the refractive index changes. Jha et al. have studied the RSOI effect on the nonlinear optical properties of parabolically confined quantum dot
30
O. Aytekin et al. / Physica E 64 (2014) 29–32
in a magnetic field [21]. It is found that the magnetic field red shifts the refractive index changes, whereas the RSOI and confinement blue shift these peaks. Vaseghi et al. have studied the influence of DSOI, electric field and dot size on the refractive index changes [22] and the optical rectification (OR) [23] in a cubic quantum dot. It is found that OR and refractive index changes decrease and occur at lower values of the incident photon energy with SOI. Vaseghi et al. have also studied the effect of RSOI and quantum confinement on the refractive index changes of a spherical quantum dot [24]. They found that the refractive index changes remain constant for different values of RSOI strength. It is also found that the spectrum moves toward higher photon energies as RSOI strength increases. In the present paper, we have focused on the nonlinear optical properties of a QD by including RSOI with electric and magnetic fields. We make no assumptions on the strength of confinement. Thus, we do not use any perturbational approach. We diagonalize the full Hamiltonian matrix to get the energy spectrum and use the density matrix approach for nonlinear optical coefficients. To the best of our knowledge, the second and the third harmonic generation in this context are not investigated before. We present the theoretical framework in Section 2. The results and discussion are presented in Section 3. A brief summary of our main findings and conclusion are given in Section 4.
2. Theoretical framework The Hamiltonian of an electron in a two-dimensional QD which is modeled by a Woods–Saxon (WS) potential and subjected to external electric and magnetic fields with RSOI is given by H¼
1 α g n eBℏ ðp þeAÞ2 þ σ ðp þ eAÞ þ V WS ðrÞ þ eFr cos φ þ σz n 2m ℏ 2mn
ð1Þ
where mn is the effective mass of the electron, p is the electron momentum operator, e is the electronic charge, σ is the Pauli spinmatrix and A is the vector potential. The electric field F is chosen parallel to the x-direction and the magnetic field B is taken to be perpendicular to the xy-plane. The second term in the Hamiltonian is the RSOI and the α parameter is the strength of the spin–orbit coupling, which is proportional to the symmetry breaking electric field in this structure. We use the symmetric gauge in which the vector potential A has the form ðB=2Þð y; x; 0Þ. The confining potential
is taken to be of the WS form V WS ðrÞ ¼
V0 ; 1 þ exp½ðR0 rÞ=γ
ð2Þ
where R0 and V0 define the dot radius and potential depth respectively and γ controls the barrier slope [25]. The last term in the Hamiltonian is the Zeeman splitting due to the magnetic field and gn is the effective Landé g-factor. The wave function is written as a superposition of a number of eigenstates of a two-dimensional isotropic harmonic oscillator. The matrix representation of the Hamiltonian is computed in this basis and diagonalized numerically. For brevity, we do not provide the reader with a full set of equations but refer the reader to our recent paper that used the same numerical technique [26].
3. Results and discussion The parameters used in this calculation are mn ¼ 0:067m0 and g n ¼ 0:44. Like our earlier work [26], the geometrical parameters are set as V 0 ¼ 310 meV, α ¼ 15 meV–nm [27], γ¼30 Å. We present our results for a narrow (R0 ¼ 100 and 200 Å) and a wider dot (R0 ¼ 500 Å). The optical rectification coefficient χ 0ð2Þ is presented as a function of the incident photon energies in Fig. 1(a) and (b), for different SOI, magnetic field values and dot sizes. Fig. 1(a) shows clearly that in a dot with radius R0 ¼ 200 Å the magnetic field and the SOI reduce the peak values of OR coefficient, and this is accompanied with a blue shift. Fig. 1(b) shows that the dot radius increases the OR peak values and there is also a blue shift. The blue shift in both cases is a result of the magnetic field which acts as an additional confining agent. The peak values depend on the dipole matrix elements which give the observed behaviour. ð2Þ Fig. 2(a) shows χ 2ω as a function of incoming photon energies. The effect of the magnetic field is similar to the case of OR coefficient because of a similar dependence on the dipole matrix elements. The effect of SOI is quite weak in this case. This may be a result of the dominance of the kinetic energy term for smaller dots. Fig. 2(b) is for a larger dot of radius R0 ¼ 500 Å. Here, the spin–orbit term is a bit more effective as the effect of the kinetic energy decreases. The third harmonic generation (THG) susceptibility χ ð3Þ 3ω is shown in Fig. 3(a) and (b) as a function of the incident photon energies for different dot sizes. The effect of the confinement and
Fig. 1. The resonant χ ð2Þ 0 values in an external electric field of F¼ 60 kV/cm for different magnetic fields, α and dot sizes.
O. Aytekin et al. / Physica E 64 (2014) 29–32
Fig. 2. The second harmonic coefficient χ ð2Þ 2ω with an electric field of F¼40 kV/cm for different dot sizes.
Fig. 3. The third harmonic coefficient χ ð3Þ 3ω for F¼ 40 kV/cm and different dot sizes.
Fig. 4. Change in the total refractive index for photon intensity of I ¼ 5 108 W=m2 , F¼ 30 kV/cm for different B and dot sizes.
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O. Aytekin et al. / Physica E 64 (2014) 29–32
the spin–orbit interaction are larger for a small dot. This can clearly be seen in Fig. 3(a). The inclusion of the magnetic field reduces the peak values. For a larger dot, the effect of the SOI is also very noticeable. Increasing magnetic field values decreases the THG coefficient for all quantum dot sizes. This may be a result of further confinement effect of the magnetic field on the wave functions. Finally, the changes in the total refractive index Δn as a function of the incoming photon energies are shown in Fig. 4 (a) and (b), for different B values and dot sizes. The refractive index change becomes smaller for a small dot with increasing magnetic field values. On the other hand, Δn becomes bigger for a larger dot with increasing magnetic field values. The effect of the SOI is rather small in this case. 4. Conclusion In this work, we first determine the electronic structure of a two-dimensional quantum dot by numerically diagonalizing the Hamiltonian matrix. The confining potential is taken to be of the WS-type. The simultaneous effects of the external electric and magnetic fields are taken into account. The main aim of the paper is to determine the effect of RSOI on the nonlinear optical properties, such as optical rectification, the second and the third harmonic generation coefficient and the change in the refractive index. We find the effect of each term in the Hamiltonian separately and simultaneously. The electric field strength is taken as constant when we compare the effects of dot size, RSOI and the magnetic field. The effect of RSOI is clearly determined in all the cases studied. The assumed WS confinement potential is not the best possible
choice since it is quite sensitive to the potential parameters it may create difficulties in the calculation of nonlinear optical coefficient.
References [1] T. Chakraborty, Quantum Dots: A Survey of the Properties of Artificial Atoms, Elsevier, Amsterdam, 1999. [2] L. Bányai, S.W. Koch, Quantum Dots, World Scientific, Singapore, 1993. [3] R.W. Boyd, Nonlinear Optics, Academic Press, New York, 2003. [4] S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. [5] S. Bandyopadhyay, M. Cahay, Appl. Phys. Lett. 85 (2004) 1814. [6] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [7] S.J. Gong, Z.Q. Yang, Phys. Lett. A 367 (2007) 369. [8] S.M. Huang, A.O. Badrutdinov, K. Kono, K. Ono, Jpn. J. Appl. Phys. 52 (2013) 04CJ02. [9] D. Stepanenko, N.E. Bonesteel, Phys. Rev. Lett. 93 (2004) 140501. [10] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molnar M.L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Science 294 (2001) 1488. [11] S.J. Pearton, C.R. Abernathy, D.P. Norton, A.F. Hebard, Y.D. Park, L.A. Boatner J.D. Budai, Mater. Sci. Eng. R 40 (2003) 137. [12] P. Pietiläinen, T. Chakraborty, Phys. Rev. B 73 (2006) 155315. [13] L. Yu, C. Fang, Chin. Phys. Lett. 28 (2011) 067303. [14] M. Valín-Rodríguez, A. Puente, L. Serra, Phys. Rev. B 69 (2004) 153308. [15] M. Florescu, P. Hawrylak, Phys. Rev. B 73 (2006) 045304. [16] R. Khordad, J. Lumin. 134 (2013) 201. [17] S. Lahon, M. Kumar, P.K. Jha, M. Mohan, J. Lumin. 144 (2013) 149. [18] G. Dresselhaus, Phys. Rev. 100 (1955) 580. [19] Y.A. Bychkov, E.I. Rashba, J. Phys. C: Solid State Phys. 17 (1984) 6039. [20] H. Hassanabadi, H. Rahimov, L. Lu, C. Wang, J. Lumin. 132 (2012) 1095. [21] P.K. Jha, M. Kumar, S. Lahon, M. Mohan, Superlatt. Microstruct. 65 (2014) 71. [22] B. Vaseghi, G. Rezaei, V. Azizi, Opt. Quantum Electron. 42 (2011) 841. [23] B. Vaseghi, G. Rezaei, V. Azizi, Physica E 44 (2012) 1241. [24] B. Vaseghi, G. Rezaei, M. Malian, Optics Commun. 287 (2013) 241. [25] S. Flügge, Practical Quantum Mechanics I, Springer-Verlag, Berlin, Heidelberg, 1971. [26] O. Aytekin, S. Turgut, V. Üstoğlu Ünal, E. Akşahin, M. Tomak, Physica E 54 (2013) 257. [27] A. Manaselyan, T. Chakraborty, Europhys. Lett. 94 (2011) 57005.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Spin–orbit effects on the nonlinear optical properties of a quantum dot in simultaneous electric and magnetic fields O. Aytekin a,n, S. Turgut b, M. Tomak b a b
Department of Physics, Adıyaman University, 02040 Adıyaman, Turkey Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
The nonlinear optical coefficients of a quantum dot are studied using a Woods–Saxon potential. The Rashba spin–orbit effects on the nonlinear optical properties are investigated. The competing effects of spin–orbit, dot size, electric and magnetic fields are given in detail.
art ic l e i nf o
a b s t r a c t
Article history: Received 5 February 2014 Received in revised form 30 May 2014 Accepted 16 June 2014 Available online 8 July 2014
We report on the nonlinear optical properties of a quantum dot including the Rashba spin–orbit interaction (RSOI) with external electric and magnetic fields. The effect of dot size is considered. We do not make any assumptions about the strength of the confinement. We use the numerical diagonalization of the Hamiltonian to determine the electronic structure. The confining potential is taken to be of the Woods–Saxon type. We find the effect of RSOI on nonlinear optical coefficients. & 2014 Elsevier B.V. All rights reserved.
Keywords: Quantum dot Nonlinear optics Spin–orbit
1. Introduction Increasing perfection in growth techniques allows one to design and investigate high performance, low-dimensional semiconductor structures. This enables one to design and investigate quantum dots (QD) with varying sizes, shapes and the number of electrons. The resulting physics is rich and challenging [1–3]. In recent years, the field of spin physics with the possibility of controlling and manipulating electron spins for low-dimensional semiconducting device applications have received great attention [4–11]. The theoretically interesting spin effect is the study of the
n
Corresponding author. E-mail address: [email protected] (O. Aytekin).
http://dx.doi.org/10.1016/j.physe.2014.06.018 1386-9477/& 2014 Elsevier B.V. All rights reserved.
spin–orbit interaction (SOI) in low-dimensional structures [12–17]. There are two widely used models for the SOI arising from different physical effects and having different Hamiltonian forms. One is the Dresselhaus [18] spin–orbit interaction (DSOI) caused by bulk inversion asymmetry and the other is the Rashba spin– orbit interaction (RSOI) [19] which arises from the asymmetry of the potential in reduced dimensional structures. There are a very limited number of investigations of spin–orbit effects on the nonlinear optical properties of quantum dots [20– 24]. Hassanabadi et al. have investigated the optical absorption and refractive index changes in a three-electron quantum dot in the presence of the RSOI [20]. They found that a decrease of the quantum dot radius blue shifts and amplifies the refractive index changes. Jha et al. have studied the RSOI effect on the nonlinear optical properties of parabolically confined quantum dot
30
O. Aytekin et al. / Physica E 64 (2014) 29–32
in a magnetic field [21]. It is found that the magnetic field red shifts the refractive index changes, whereas the RSOI and confinement blue shift these peaks. Vaseghi et al. have studied the influence of DSOI, electric field and dot size on the refractive index changes [22] and the optical rectification (OR) [23] in a cubic quantum dot. It is found that OR and refractive index changes decrease and occur at lower values of the incident photon energy with SOI. Vaseghi et al. have also studied the effect of RSOI and quantum confinement on the refractive index changes of a spherical quantum dot [24]. They found that the refractive index changes remain constant for different values of RSOI strength. It is also found that the spectrum moves toward higher photon energies as RSOI strength increases. In the present paper, we have focused on the nonlinear optical properties of a QD by including RSOI with electric and magnetic fields. We make no assumptions on the strength of confinement. Thus, we do not use any perturbational approach. We diagonalize the full Hamiltonian matrix to get the energy spectrum and use the density matrix approach for nonlinear optical coefficients. To the best of our knowledge, the second and the third harmonic generation in this context are not investigated before. We present the theoretical framework in Section 2. The results and discussion are presented in Section 3. A brief summary of our main findings and conclusion are given in Section 4.
2. Theoretical framework The Hamiltonian of an electron in a two-dimensional QD which is modeled by a Woods–Saxon (WS) potential and subjected to external electric and magnetic fields with RSOI is given by H¼
1 α g n eBℏ ðp þeAÞ2 þ σ ðp þ eAÞ þ V WS ðrÞ þ eFr cos φ þ σz n 2m ℏ 2mn
ð1Þ
where mn is the effective mass of the electron, p is the electron momentum operator, e is the electronic charge, σ is the Pauli spinmatrix and A is the vector potential. The electric field F is chosen parallel to the x-direction and the magnetic field B is taken to be perpendicular to the xy-plane. The second term in the Hamiltonian is the RSOI and the α parameter is the strength of the spin–orbit coupling, which is proportional to the symmetry breaking electric field in this structure. We use the symmetric gauge in which the vector potential A has the form ðB=2Þð y; x; 0Þ. The confining potential
is taken to be of the WS form V WS ðrÞ ¼
V0 ; 1 þ exp½ðR0 rÞ=γ
ð2Þ
where R0 and V0 define the dot radius and potential depth respectively and γ controls the barrier slope [25]. The last term in the Hamiltonian is the Zeeman splitting due to the magnetic field and gn is the effective Landé g-factor. The wave function is written as a superposition of a number of eigenstates of a two-dimensional isotropic harmonic oscillator. The matrix representation of the Hamiltonian is computed in this basis and diagonalized numerically. For brevity, we do not provide the reader with a full set of equations but refer the reader to our recent paper that used the same numerical technique [26].
3. Results and discussion The parameters used in this calculation are mn ¼ 0:067m0 and g n ¼ 0:44. Like our earlier work [26], the geometrical parameters are set as V 0 ¼ 310 meV, α ¼ 15 meV–nm [27], γ¼30 Å. We present our results for a narrow (R0 ¼ 100 and 200 Å) and a wider dot (R0 ¼ 500 Å). The optical rectification coefficient χ 0ð2Þ is presented as a function of the incident photon energies in Fig. 1(a) and (b), for different SOI, magnetic field values and dot sizes. Fig. 1(a) shows clearly that in a dot with radius R0 ¼ 200 Å the magnetic field and the SOI reduce the peak values of OR coefficient, and this is accompanied with a blue shift. Fig. 1(b) shows that the dot radius increases the OR peak values and there is also a blue shift. The blue shift in both cases is a result of the magnetic field which acts as an additional confining agent. The peak values depend on the dipole matrix elements which give the observed behaviour. ð2Þ Fig. 2(a) shows χ 2ω as a function of incoming photon energies. The effect of the magnetic field is similar to the case of OR coefficient because of a similar dependence on the dipole matrix elements. The effect of SOI is quite weak in this case. This may be a result of the dominance of the kinetic energy term for smaller dots. Fig. 2(b) is for a larger dot of radius R0 ¼ 500 Å. Here, the spin–orbit term is a bit more effective as the effect of the kinetic energy decreases. The third harmonic generation (THG) susceptibility χ ð3Þ 3ω is shown in Fig. 3(a) and (b) as a function of the incident photon energies for different dot sizes. The effect of the confinement and
Fig. 1. The resonant χ ð2Þ 0 values in an external electric field of F¼ 60 kV/cm for different magnetic fields, α and dot sizes.
O. Aytekin et al. / Physica E 64 (2014) 29–32
Fig. 2. The second harmonic coefficient χ ð2Þ 2ω with an electric field of F¼40 kV/cm for different dot sizes.
Fig. 3. The third harmonic coefficient χ ð3Þ 3ω for F¼ 40 kV/cm and different dot sizes.
Fig. 4. Change in the total refractive index for photon intensity of I ¼ 5 108 W=m2 , F¼ 30 kV/cm for different B and dot sizes.
31
32
O. Aytekin et al. / Physica E 64 (2014) 29–32
the spin–orbit interaction are larger for a small dot. This can clearly be seen in Fig. 3(a). The inclusion of the magnetic field reduces the peak values. For a larger dot, the effect of the SOI is also very noticeable. Increasing magnetic field values decreases the THG coefficient for all quantum dot sizes. This may be a result of further confinement effect of the magnetic field on the wave functions. Finally, the changes in the total refractive index Δn as a function of the incoming photon energies are shown in Fig. 4 (a) and (b), for different B values and dot sizes. The refractive index change becomes smaller for a small dot with increasing magnetic field values. On the other hand, Δn becomes bigger for a larger dot with increasing magnetic field values. The effect of the SOI is rather small in this case. 4. Conclusion In this work, we first determine the electronic structure of a two-dimensional quantum dot by numerically diagonalizing the Hamiltonian matrix. The confining potential is taken to be of the WS-type. The simultaneous effects of the external electric and magnetic fields are taken into account. The main aim of the paper is to determine the effect of RSOI on the nonlinear optical properties, such as optical rectification, the second and the third harmonic generation coefficient and the change in the refractive index. We find the effect of each term in the Hamiltonian separately and simultaneously. The electric field strength is taken as constant when we compare the effects of dot size, RSOI and the magnetic field. The effect of RSOI is clearly determined in all the cases studied. The assumed WS confinement potential is not the best possible
choice since it is quite sensitive to the potential parameters it may create difficulties in the calculation of nonlinear optical coefficient.
References [1] T. Chakraborty, Quantum Dots: A Survey of the Properties of Artificial Atoms, Elsevier, Amsterdam, 1999. [2] L. Bányai, S.W. Koch, Quantum Dots, World Scientific, Singapore, 1993. [3] R.W. Boyd, Nonlinear Optics, Academic Press, New York, 2003. [4] S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. [5] S. Bandyopadhyay, M. Cahay, Appl. Phys. Lett. 85 (2004) 1814. [6] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [7] S.J. Gong, Z.Q. Yang, Phys. Lett. A 367 (2007) 369. [8] S.M. Huang, A.O. Badrutdinov, K. Kono, K. Ono, Jpn. J. Appl. Phys. 52 (2013) 04CJ02. [9] D. Stepanenko, N.E. Bonesteel, Phys. Rev. Lett. 93 (2004) 140501. [10] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molnar M.L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Science 294 (2001) 1488. [11] S.J. Pearton, C.R. Abernathy, D.P. Norton, A.F. Hebard, Y.D. Park, L.A. Boatner J.D. Budai, Mater. Sci. Eng. R 40 (2003) 137. [12] P. Pietiläinen, T. Chakraborty, Phys. Rev. B 73 (2006) 155315. [13] L. Yu, C. Fang, Chin. Phys. Lett. 28 (2011) 067303. [14] M. Valín-Rodríguez, A. Puente, L. Serra, Phys. Rev. B 69 (2004) 153308. [15] M. Florescu, P. Hawrylak, Phys. Rev. B 73 (2006) 045304. [16] R. Khordad, J. Lumin. 134 (2013) 201. [17] S. Lahon, M. Kumar, P.K. Jha, M. Mohan, J. Lumin. 144 (2013) 149. [18] G. Dresselhaus, Phys. Rev. 100 (1955) 580. [19] Y.A. Bychkov, E.I. Rashba, J. Phys. C: Solid State Phys. 17 (1984) 6039. [20] H. Hassanabadi, H. Rahimov, L. Lu, C. Wang, J. Lumin. 132 (2012) 1095. [21] P.K. Jha, M. Kumar, S. Lahon, M. Mohan, Superlatt. Microstruct. 65 (2014) 71. [22] B. Vaseghi, G. Rezaei, V. Azizi, Opt. Quantum Electron. 42 (2011) 841. [23] B. Vaseghi, G. Rezaei, V. Azizi, Physica E 44 (2012) 1241. [24] B. Vaseghi, G. Rezaei, M. Malian, Optics Commun. 287 (2013) 241. [25] S. Flügge, Practical Quantum Mechanics I, Springer-Verlag, Berlin, Heidelberg, 1971. [26] O. Aytekin, S. Turgut, V. Üstoğlu Ünal, E. Akşahin, M. Tomak, Physica E 54 (2013) 257. [27] A. Manaselyan, T. Chakraborty, Europhys. Lett. 94 (2011) 57005.