- Email: [email protected]
Laser field effect on the nonlinear optical properties of donor impurities in quantum dots with Gaussian potential
Physica B 406 (2011) 4129–4134
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Laser field effect on the nonlinear optical properties of donor impurities in quantum dots with Gaussian potential Liangliang Lu a,, Wenfang Xie a, Hassan Hassanabadi b a b
Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316 Shahrood, Iran
a r t i c l e i n f o
abstract
Article history: Received 6 July 2011 Received in revised form 26 July 2011 Accepted 27 July 2011 Available online 11 August 2011
A detailed investigation of optical properties of donor impurities in quantum dots under the influence of laser field with Gaussian potential is performed by using the matrix diagonalization method within the effective mass approximation. Based on the computed energies and wave functions, the dependence of the nonlinear optical properties on the dot size and the potential depth is investigated. The outcome of the calculation suggests that all the factors mentioned above can influence the nonlinear optical properties strongly. We also note that the increase of the laser-dressing parameter leads to important effects on the electronic and optical properties of a quantum dot. This gives a new degree of freedom in various device applications based on the intersubband transition of electrons. & 2011 Elsevier B.V. All rights reserved.
Keywords: Quantum dot Nonlinear optics Laser radiation
1. Introduction Semiconductor quantum dots (QDs) effectively confine electrons on the nanometer length scale in all spatial directions and hence may be considered as artificial atoms with discrete energy levels for electrons [1,2]. Unlike real atoms the physical properties of QDs can be easily varied, which gives theorists and experimentalists the opportunity to study quantum effects in a wellcontrolled systems. It is well-known that carrier confinement into dimensions of a few tens of nanometers provides strong blue shift of the photoluminescence features from that in the original bulk material, a clear consequence of quantum confinement in these QDs. Hence, the main features to consider in relation to QDs are geometrical shape, size, and the confining potential. The influence of spatial confinement on the energy spectra of physical systems is one of the most interesting properties to be investigated in the study of confined systems. Traditionally, the spatial confinement can be modeled by introduction of a confinement potential [3–5]. In most studies, a harmonic oscillator potential is used to describe the lateral confinement of electrons. However, the parabolic potential possesses infinite depth and range. It is inappropriate for a description of the experimentally measured charging of the QD by the finite number of excess electrons. Some experimental results suggest that the real confining potential possesses a well-like shape. When a QD is small (i.e. when its radius is comparable to the characteristic length of
Corresponding author. Tel.: þ 86 13751823828.
E-mail address: [email protected] (L. Lu). 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.07.063
the variation of the lateral potential near the edge), a good approximation offers simple smooth potentials, such as a Gaussian well, VðrÞ ¼ V0 expðr 2 =2R2 Þ [6]. Many authors have employed the Gaussian potential to study the electronic and optical properties of QDs [7–9]. On the other hand, the study of the effect of the laser fields on low-dimensional heterostructures has been considered on several occasions [10–16]. Brandi et al. [11] used the dressed atom approach to treat the influence of the laser field upon a semiconductor system. The interaction with the laser is taken into account through the renormalization of effective mass. They concluded that the binding energy is an increasing function of the laser intensity. Based on the non-perturbative theory that has been developed to describe the atomic behavior in intense highfrequency laser fields, several works have reported the effects of an intense laser field on donor impurities in low-dimensional systems. The laser dressing effects on both the Coulomb potential and the confining potential are taken into account. Niculescu et al. [12] used the transfer matrix method to investigate the effects of laser on the electronic states in GaAs/Ga1 xAlxAs V-shaped and inverse V-shaped quantum wells under a static electric field. Their conclusion is that the position of the binding energy maximum versus the impurity location in the structure can be adjusted by the intensity of the laser field. This effect can be used to tune the electronic levels in quantum wells operating under electric and laser fields without modifying the physical size of the structures. In parallel, considerable attention has been paid to the nonlinear optical properties of low-dimensional semiconductor systems such as quantum wells, QDs, and other nanostructures in recent years because they have the potential for device
4130
L. Lu et al. / Physica B 406 (2011) 4129–4134
applications in far-infrared laser amplifies, photodetectors, and high-speed electro-optical modulators [17,18]. In the nonlinear optical properties of QDs, the analysis of impurity states is inevitable and the understanding of impurities effects on the properties of QDs is crucial for evolution of the emerging nanoelectronic, an area that has attracted increasing interest due to the possibilities it opens in applied physics [19–21]. Since the operation of optoelectronic devices relies on the interaction of carriers with electromagnetic radiation, it is intriguing to investigate the effects of an intense laser field on the impurity states. Some investigations have been published about the effect of laser fields on nano-structure [22–24]. It is found that the laser field considerably affects the nonlinear optical properties. However, to the best of our knowledge, there are only a few studies on the effect of an intense laser field on the optical transitions in a QD with Gaussian potential. In this work, we will focus on studying the effect of an intense, high-frequency laser field on the nonlinear optical absorption coefficients (ACs) and refractive index (RI) changes of a hydrogenic impurity in a QD with Gaussian potential by using the matrix diagonalization method. The results are predicted to have the potential for the design of tunable optical semiconductor devices.
! ! where Vð a 0 , r Þ is the ‘‘dressed’’ potential that depends on O and I only through a0 and given by Z o 2p=o ! ! ! ! ð5Þ Vð r þ a ðtÞÞ dt: Vð a 0 , r Þ ¼ 2p 0 Hence the dressed Gaussian and Coulomb potentials are defined as follows: ! ! ! ! d !! VGS ð r , a 0 Þ ¼ 12½VGS ð9 r þ a 0 9Þ þ VGS ð9 r a 0 9Þ,
ð6Þ
! e2 1 1 !! Vcd ð r , a 0 Þ ¼ þ , ! ! ! 2e 9! r þ a 09 9 r a 09
ð7Þ
where e is the dielectric constant. The Hamiltonian has cylindrical symmetry with respect to the QD axis, i.e. z-axis, which implies that the total orbital angular momentum L (i.e. Lz) is a good quantum number. Hence, the eigenstates of the hydrogenic impurity can be classified according to the total orbital angular momentum. To obtain the eigenfunction and eigenenergy associated with the hydrogenic impurity, the Hamiltonian is diagonalized in the model space spanned by two-dimensional harmonic states: X o ! CL ¼ ci fi ð r Þ, ð8Þ i
2. Model and theory Without defining the type of a QD (i.e. deciding on the simplified but universal model), we assume that the potential of a QD can be modeled in the form of a Gaussian potential well [2], which is given by VðrÞ ¼ V0 expðr 2 =2R2 Þ,
ð1Þ
where V0 is height of the potential well and R is the range of the confinement potential, which corresponds to a radius of the quantum dot-like islands. The approach used in the present calculation is based on a non-perturbative theory that has been developed to describe the atomic behavior in intense high-frequency laser fields [10]. We consider one electron with a hydrogenic impurity confined by a two-dimensional QD with the Gaussian potential, subjected to a laser field of frequency O, whose vector potential is given by ! A ðtÞ ¼ A0 e^ cos Ot, where e^ is the unit vector. By applying the ! ! ! time-dependent translation r - r þ a ðtÞ, the semi-classical ¨ Schrodinger equation in the momentum gauge, describing the interaction dynamics in the laboratory frame of reference, was transformed by Kramers in the form ! @Cð r ,tÞ ‘2 2 ! ! ! ¼ r Cð r ,tÞ þ½Vð! r þ a ðtÞÞCð r ,tÞ: @t 2mn ! ! Here Vð r þ a ðtÞÞ is the ‘‘dressed potential’’ energy, and i‘
! a ðtÞ ¼ e^ a0 sin Ot,
ð2Þ
ð3Þ pffiffi n where a0 ¼ ð8p=OcÞ Ie=m is the amplitude of the electron oscillation in the laser field, mn being the effective mass of an electron and I is the laser intensity. By applying the Floquet approach [16], the space translated version of the Schr0¨dinger Eq. (2) can be recast in the equivalent form of a system of coupled time-independent differential equations of the Floquet components of the wavefunction c, containing the (in general complex) quasi-energy E. For the zeroth Floquet component c0 , the system reduces to the time-indepen¨ dent Schrodinger equation [16] " # ‘2 2 !! r þ Vð r , a Þ ð4Þ 0 c0 ¼ Ec0 , 2mn 2
o ! where fi ð r Þ ¼ Rni li ðrÞexpðili yÞ is the ith two-dimensional harmonic oscillator eigenstate with a frequency o and an energy ð2ni þ 9li 9 þ 1Þ‘ o. Rnl(r) is the radial wave function, given by 9l9
Rnl ðrÞ ¼ N expðr 2 =ð2a2 ÞÞr 9l9 Ln ðr 2 =a2 Þ,
ð9Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which N is the normalization constant, a ¼ ‘ =ðmn oÞ, and Lkn ðxÞ is the associated Laguerre polynomial. The radial and orbital angular momentum quantum numbers can have the following values: n ¼ 1,2 . . . ,l ¼ 0, 7 1, 7 2 . . . :
ð10Þ
This single-particle basis is used by the matrix diagonalization method in order to expand the Hamiltonian H. In practice calculation, o serves as a variational parameter to minimize the eigenvalues. Since the whole set of eigenstates of the harmonic product basis forms a complete basis in the Hilbert space, the procedure of increasing the number of linearly independent eigenstates is converging the exact result. The limits are set only by the capacity of the computer to diagonalize N N Hermitian matrices. This is achieved by extending the dimension of the model space step by step; in each step the new results are compared with previous results from a smaller space, until satisfactory convergence is achieved. In this paper, the dimension of the model space is constrained by 0 rN ¼ 2n þ 9l9 r26. If N is increased by 2, the ratio of the difference in energy is less than 0.001. We calculate the absorption coefficients and refractive index changes using the linear wð1Þ and third-order wð3Þ optical susceptibilities. The analytical forms of wð1Þ and wð3Þ are obtained from modeling a QD as a two-level system. The susceptibility w is related to the absorption coefficient aðnÞ by rffiffiffiffiffiffi m aðnÞ ¼ n 0 Im½e0 wðnÞ, ð11Þ
er
where m0 is the permeability of the system. er ¼ n2r e0 is the real part of the permittivity, nr is the medium refractive index and e0 is the permittivity of vacuum. w is the Fourier component of wðtÞ with expðiotÞ dependence. Using the compact density–matrix method, the optical absorption coefficient is given by [22]
aðnÞ ¼ að1Þ ðnÞ þ að3Þ ðn,IÞ,
ð12Þ
L. Lu et al. / Physica B 406 (2011) 4129–4134
where 4pbFS ss 2 að1Þ ðnÞ ¼ hn9Mfi 9 dðEfi hnÞ nr e2
ð13Þ
and 2
32p2 bFS ss I 4 hn9Mfi 9 dðEfi hnÞ2 n2r e4 ‘ Gff ( ) 2 9Mff Mii 9 ½ðhnEfi Þ2 ð‘ Gfi Þ2 þ 2Efi ðEfi hnÞ 1 2 E2fi þ ð‘ Gfi Þ2 49Mfi 9
are the linear and the third-order nonlinear optical absorption coefficients, respectively. e is the electronic charge of an electron, ss is the electron density in the QD, bFS ¼ e2 =4pe0 ‘ c is the fine structure constant, I is the incident optical intensity, and hn is the photon energy. The d function in Eqs. (13) and (14) is replaced by a narrow Lorentzian by means of
að3Þ ðn,IÞ ¼
dðEfi hnÞ ¼
ð14Þ
4131
‘ Gfi
p½ðhnEfi Þ2 þ ð‘ Gfi Þ2
:
ð15Þ
Here G is the phenomenological operator. Nondiagonal matrix element Gfi ðf aiÞ of operator G, which is called as the relaxation rate of fth state and ith state, is the inverse of the relaxation time Tfi for the states 9f 4 and 9i 4 , namely Gfi ¼ 1=Tfi . In addition, the susceptibility w is related to the change in the refractive index as follows: DnðoÞ wðoÞ ¼ Re : ð16Þ 2 nr 2nr Therefore, the linear and the third-order nonlinear refractive index changes are obtained by [22]
Dnð1Þ ðoÞ nr
¼
ss e2 Mfi2 ðEfi hnÞ 2er ðE21 hnÞ2 þð‘ Gfi Þ
ð17Þ
and
Dnð3Þ ðoÞ nr
¼
ss e4 Mfi2 m0 cI ½4Mfi2 ðEfi hnÞ 4er n2r ½ðEfi hnÞ2 þð‘ Gff Þ2 2
ðMff Mii Þ2
Efi ðEfi hnÞ2 ð‘ Gff Þ2 ð3Efi 2hnÞ E2fi þ ð‘ Gff Þ2
:
ð18Þ
Hence, the total refractive index change can be given as Fig. 1. The linear a ðnÞ, the third-order nonlinear a ðn,IÞ and the total aðn,IÞ optical absorption coefficients as a function of the incident photon energy hn for three different values of the laser field. ð1Þ
ð3Þ
DnðoÞ nr
¼
Dnð1Þ ðoÞ nr
þ
Dnð3Þ ðoÞ nr
:
ð19Þ
Fig. 2. The linear, the third-order nonlinear and the total changes in the refractive index as a function of the incident photon energy hn for three different values of the laser field.
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3. Results and discussions All calculations were performed using the following parameters: mn ¼ 0:067m0 (m0 is the mass of a free electron), e ¼ 12:53, nr ¼3.2, ss ¼ 5:0 1022 m3 , I ¼ 1:0 109 W=m2 , Gff ¼ 1 ps1 , and Gfi ¼ 1:4 ps1 . Figs. 1 and 2 show the linear, third-order nonlinear and total optical ACs and the total optical RI changes as a function of the photon energy hn for three different laser-field values with V0 ¼112.23 meV and R ¼10.0 nm. As seen in the figure, since the third-order term has a negative sign, according to Eqs. (14)
Fig. 3. The linear að1Þ ðnÞ, the third-order nonlinear að3Þ ðn,IÞ and the total aðn,IÞ optical absorption coefficients as a function of the incident photon energy hn for three different values of the potential depth.
and (18), the total ACs and RI changes decrease. In addition, the laser-field-induced effect is clear. It is readily seen that as the laser field value increases, the AC and RI peaks will move to the left side, which show a laser-field-induced red shift of the resonance in QDs. This is because the energy difference between the ground and excited states in QDs decreases with increasing a0 . Obviously this the result of the laser-dressed Coulomb interaction is clear and it is different from that of quantum wells [24]. On the other hand, we find that all ACs and RI changes (absolute values of third-order terms) increase with increasing the laser field values. This behavior is due to the fact that, as the value of a0 is increased, the electron wavefunction is compressed. Thus the overlapping of the wave functions of these two states, i.e. the transition matrix element which is effective on the ACs and RI changes, is enhanced. Hence one can say that in order to obtain the larger ACs and RI changes in QDs, we should induce the laser field. The height of absorption coefficient can reach the magnitude of 105/m. Compared with those existing results, the maximum value of ACs and RI changes significantly increased. Especially, it is 1–2 orders of magnitude higher than that in the case of two-dimensional parabolic QDs [22]. These results indicate that the large ACs and RI changes in a desired energy can be obtained by adjusting the laser field. Figs. 3 and 4 illustrate the linear and nonlinear ACs and RI changes in a Gaussian QD with R0 ¼10 nm and a0 ¼ 1:0 nm as a function of the incident photon energy hn for three different potential depths, i.e. V0 ¼112.23, 224.46 and 336.69 meV, respectively. The confinement effect of the potential depth seems clear. It can be seen that the absorption peak positions shift to higher energies (blueshift) with increasing V0. That is due to that the energy difference between the 1S and 1P states increases with increasing V0. We also note that the bigger the potential depth is, the bigger will be the ACs and RI changes. This is because the overlapping of the wavefunctions increases with increasing confinement, i.e. V0. In Figs. 5 and 6, the potential depth V0 is set to be 112.23 meV, the laser field a0 is set to be 1.0 nm and R are set to be 5.0, 7.5 and
Fig. 4. The linear, the third-order nonlinear and the total changes in the refractive index as a function of the incident photon energy hn for three different values of the potential depth.
L. Lu et al. / Physica B 406 (2011) 4129–4134
10.0 nm, respectively. From these figures, we can find that the size effect of the QD is obvious. The ACs and RI changes peak positions shift to the low photon energies with increasing R, which shows a quantum-size-induced redshift of the resonance in QDs. The physical origin is that, with decreasing R the increase of the energy difference between the initial and final states is increased. However, the peak values of the ACs and the RI changes are not monotonic functions of R. As seen in Figs. 5 and 6, we find that the magnitude of the ACs and the RI changes first increases with R up to a critical value, R 7:5 nm, and for further large R value they begin to decrease. Hence, we can obtain large ACs and RI changes in QDs by controlling the dot size. In order to show better the influence of the incident optical intensity I for the total ACs, in Fig. 7, we set V0 ¼112.23 meV,
Fig. 5. The linear að1Þ ðnÞ, the third-order nonlinear að3Þ ðn,IÞ and the total aðn,IÞ optical absorption coefficients as a function of the incident photon energy hn for three different values of R.
4133
R0 ¼10.0 nm and a0 ¼ 1:0 nm and plot the total ACs as a function of the incident photon energy hn for six different values of I. As the linear AC does not depend on photon intensity, whereas the third-order term increases with photon intensity as it varies quadratically with laser amplitude. Thus the total AC curve, being the sum of linear and nonlinear parts, is robust when I ¼ 0 W=m2 . If I a0 W=m2 , the total AC changes considerably with increasing optical intensity as expected, especially near the resonance frequency. When the incident optical intensity I exceeds a critical value, which indicates saturation, the nonlinear term causes a collapse at center of the total absorption peaks splitting it into two peaks. Finally the total RI changes as a function of the incident photon energy hn for four different relaxation times T¼0.14 ps, 0.25 ps, 0.35 ps and 0.5 ps is shown in Fig. 8. From this figure, we observed
Fig. 7. The total aðn,IÞ optical absorption coefficients as a function of the incident photon energy hn for six different values of the incident optical intensity.
Fig. 6. The linear, the third-order nonlinear and the total changes in the refractive index as a function of the incident photon energy hn for three different values of R.
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L. Lu et al. / Physica B 406 (2011) 4129–4134
dot radii. Moreover, the calculated results also reveal that the laser field amplitude has a significant effect on the nonlinear optical properties. Applying a laser field to QDs can make the position and height value of the optical ACs and RI change significantly. Our results are important to the experimental studies and have significant influences on improvements of optical devices such as infrared laser amplifiers, photo-detectors, and high-speed electro-optical modulators. This also gives a new degree of freedom in various device applications based on the intersubband transition of electrons.
Acknowledgments This work is financially supported by the National Natural Science Foundation of China under Grant No. 11074055.
References Fig. 8. The total changes in the refractive index as a function of the incident photon energy hn for four different values of the relaxation times.
that the relaxation time T has a great influence on the RI changes, namely, with the increase of the relaxation time, the total RI changes increase obviously. But the saturation begins at the relaxation time value of 0.5 ps. On the other hand, the relaxation rate is related not only to the materials constituting the QD, but also to some other factors, such as the boundary conditions, temperature of the system as well as electron-impurity and electron–phonon scattering interactions, etc. Hence, one should reduces the influences of these factors on the systems to achieve a large absorption coefficient.
4. Conclusion In this study, we proposed the Gaussian confining potential function to simulate the spatial confinement of QDs. The detailed nonlinear optical properties of a hydrogenic impurity in a disclike QD in the presence of an intense laser field are investigated by using the matrix diagonalization method and the compact density–matrix approach. Our results suggest that for intersubband transitions in QDs with the Gaussian potential, we can obtain a blue or red shift by tuning the potential depth or/and the
[1] A.D. Yoffe, Adv. Phys. 50 (2001) 1. [2] L. Jacak, P. Hawrylak, A. Wo´js, Quantum Dot, Springer, Berlin, Heidelberg, 1998. [3] D.S. Chuu, C.M. Hsiao, W.N. Mei, Phys. Rev. B 46 (1992) 3898. [4] E. Ley-Koo, S. Rubinstein, J. Chem. Phys. 46 (1979) 351. [5] R. Vawter, Phys. Rev. 174 (1968) 749. [6] Y. Masumoto, T. Takagahara, Semiconductor Quantum dots, Springer-Verlag, Heidelberg, New York, 2002. [7] W.F. Xie, Commun. Theor. Phys. 42 (2002) 923. [8] W.F. Xie, Commun. Theor. Phys. 47 (2007) 167. [9] Y. Kayanuma, Phys. Rev. B 44 (1991) 13085. [10] Q. Fanyao, A.L.A. Fonseca, O.A.C. Nunes, Phys. Rev. B 54 (1996) 16405. [11] H.S. Brandi, A. Latge, L.E. Oliveira, Phys. Rev. B 70 (2004) 153303. [12] E.C. Niculescu, A. Radu, M. Stafe, Superlatt. Microstruct. 46 (2009) 443. ¨ [13] U. Yesilgul, S. Sakiroglu, E. Kasapoglu, H. Sari, I. Sokmen, Physica B 406 (2011) 1441. [14] E. Ozturk, H. Sari, I. Sokmen, J. Phys. D: Appl. Phys. 38 (2005) 935. [15] A. John Peter, Phys. Lett. A 374 (2010) 2170. ¨ [16] F. Ungan, E. Kasapoglu, C.A. Duque, U. Yesilgul, S. Sakiroglu, I. Sokmen, Eur. Phys. J. B 80 (2011) 89. [17] M. S- ahin, J. Appl. Phys. 106 (2009) 063710. [18] L.E. Ivchenko, Optical Spectroscopy of Semiconductor Nanostructures, Alpha Science International Ltd., 2005. [19] S. Baskoutas, E. Paspalakis, A.F. Terzis, J. Phys.: Condens. Matter 19 (2007) 395024. [20] W.Y. Ruan, Y.-C. Chang, Phys. Rev. B 66 (2002) 115314. [21] G.H. Wang, K.X. Guo, J. Phys.: Condens. Matter 13 (2001) 8197. [22] W.F. Xie, Opt. Commun. 283 (2010) 3703. ¨ [23] C.A. Duque, E. Kasapoglu, S. Sakiroglu, H. Sari, I. Sokmen, Appl. Surf. Sci. 257 (2011) 2313. ¨ [24] H. Sari, I. Kasapoglu, I. S¨kmen, M. Gunes, Phys. Lett. A 319 (2003) 211.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Laser field effect on the nonlinear optical properties of donor impurities in quantum dots with Gaussian potential Liangliang Lu a,, Wenfang Xie a, Hassan Hassanabadi b a b
Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316 Shahrood, Iran
a r t i c l e i n f o
abstract
Article history: Received 6 July 2011 Received in revised form 26 July 2011 Accepted 27 July 2011 Available online 11 August 2011
A detailed investigation of optical properties of donor impurities in quantum dots under the influence of laser field with Gaussian potential is performed by using the matrix diagonalization method within the effective mass approximation. Based on the computed energies and wave functions, the dependence of the nonlinear optical properties on the dot size and the potential depth is investigated. The outcome of the calculation suggests that all the factors mentioned above can influence the nonlinear optical properties strongly. We also note that the increase of the laser-dressing parameter leads to important effects on the electronic and optical properties of a quantum dot. This gives a new degree of freedom in various device applications based on the intersubband transition of electrons. & 2011 Elsevier B.V. All rights reserved.
Keywords: Quantum dot Nonlinear optics Laser radiation
1. Introduction Semiconductor quantum dots (QDs) effectively confine electrons on the nanometer length scale in all spatial directions and hence may be considered as artificial atoms with discrete energy levels for electrons [1,2]. Unlike real atoms the physical properties of QDs can be easily varied, which gives theorists and experimentalists the opportunity to study quantum effects in a wellcontrolled systems. It is well-known that carrier confinement into dimensions of a few tens of nanometers provides strong blue shift of the photoluminescence features from that in the original bulk material, a clear consequence of quantum confinement in these QDs. Hence, the main features to consider in relation to QDs are geometrical shape, size, and the confining potential. The influence of spatial confinement on the energy spectra of physical systems is one of the most interesting properties to be investigated in the study of confined systems. Traditionally, the spatial confinement can be modeled by introduction of a confinement potential [3–5]. In most studies, a harmonic oscillator potential is used to describe the lateral confinement of electrons. However, the parabolic potential possesses infinite depth and range. It is inappropriate for a description of the experimentally measured charging of the QD by the finite number of excess electrons. Some experimental results suggest that the real confining potential possesses a well-like shape. When a QD is small (i.e. when its radius is comparable to the characteristic length of
Corresponding author. Tel.: þ 86 13751823828.
E-mail address: [email protected] (L. Lu). 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.07.063
the variation of the lateral potential near the edge), a good approximation offers simple smooth potentials, such as a Gaussian well, VðrÞ ¼ V0 expðr 2 =2R2 Þ [6]. Many authors have employed the Gaussian potential to study the electronic and optical properties of QDs [7–9]. On the other hand, the study of the effect of the laser fields on low-dimensional heterostructures has been considered on several occasions [10–16]. Brandi et al. [11] used the dressed atom approach to treat the influence of the laser field upon a semiconductor system. The interaction with the laser is taken into account through the renormalization of effective mass. They concluded that the binding energy is an increasing function of the laser intensity. Based on the non-perturbative theory that has been developed to describe the atomic behavior in intense highfrequency laser fields, several works have reported the effects of an intense laser field on donor impurities in low-dimensional systems. The laser dressing effects on both the Coulomb potential and the confining potential are taken into account. Niculescu et al. [12] used the transfer matrix method to investigate the effects of laser on the electronic states in GaAs/Ga1 xAlxAs V-shaped and inverse V-shaped quantum wells under a static electric field. Their conclusion is that the position of the binding energy maximum versus the impurity location in the structure can be adjusted by the intensity of the laser field. This effect can be used to tune the electronic levels in quantum wells operating under electric and laser fields without modifying the physical size of the structures. In parallel, considerable attention has been paid to the nonlinear optical properties of low-dimensional semiconductor systems such as quantum wells, QDs, and other nanostructures in recent years because they have the potential for device
4130
L. Lu et al. / Physica B 406 (2011) 4129–4134
applications in far-infrared laser amplifies, photodetectors, and high-speed electro-optical modulators [17,18]. In the nonlinear optical properties of QDs, the analysis of impurity states is inevitable and the understanding of impurities effects on the properties of QDs is crucial for evolution of the emerging nanoelectronic, an area that has attracted increasing interest due to the possibilities it opens in applied physics [19–21]. Since the operation of optoelectronic devices relies on the interaction of carriers with electromagnetic radiation, it is intriguing to investigate the effects of an intense laser field on the impurity states. Some investigations have been published about the effect of laser fields on nano-structure [22–24]. It is found that the laser field considerably affects the nonlinear optical properties. However, to the best of our knowledge, there are only a few studies on the effect of an intense laser field on the optical transitions in a QD with Gaussian potential. In this work, we will focus on studying the effect of an intense, high-frequency laser field on the nonlinear optical absorption coefficients (ACs) and refractive index (RI) changes of a hydrogenic impurity in a QD with Gaussian potential by using the matrix diagonalization method. The results are predicted to have the potential for the design of tunable optical semiconductor devices.
! ! where Vð a 0 , r Þ is the ‘‘dressed’’ potential that depends on O and I only through a0 and given by Z o 2p=o ! ! ! ! ð5Þ Vð r þ a ðtÞÞ dt: Vð a 0 , r Þ ¼ 2p 0 Hence the dressed Gaussian and Coulomb potentials are defined as follows: ! ! ! ! d !! VGS ð r , a 0 Þ ¼ 12½VGS ð9 r þ a 0 9Þ þ VGS ð9 r a 0 9Þ,
ð6Þ
! e2 1 1 !! Vcd ð r , a 0 Þ ¼ þ , ! ! ! 2e 9! r þ a 09 9 r a 09
ð7Þ
where e is the dielectric constant. The Hamiltonian has cylindrical symmetry with respect to the QD axis, i.e. z-axis, which implies that the total orbital angular momentum L (i.e. Lz) is a good quantum number. Hence, the eigenstates of the hydrogenic impurity can be classified according to the total orbital angular momentum. To obtain the eigenfunction and eigenenergy associated with the hydrogenic impurity, the Hamiltonian is diagonalized in the model space spanned by two-dimensional harmonic states: X o ! CL ¼ ci fi ð r Þ, ð8Þ i
2. Model and theory Without defining the type of a QD (i.e. deciding on the simplified but universal model), we assume that the potential of a QD can be modeled in the form of a Gaussian potential well [2], which is given by VðrÞ ¼ V0 expðr 2 =2R2 Þ,
ð1Þ
where V0 is height of the potential well and R is the range of the confinement potential, which corresponds to a radius of the quantum dot-like islands. The approach used in the present calculation is based on a non-perturbative theory that has been developed to describe the atomic behavior in intense high-frequency laser fields [10]. We consider one electron with a hydrogenic impurity confined by a two-dimensional QD with the Gaussian potential, subjected to a laser field of frequency O, whose vector potential is given by ! A ðtÞ ¼ A0 e^ cos Ot, where e^ is the unit vector. By applying the ! ! ! time-dependent translation r - r þ a ðtÞ, the semi-classical ¨ Schrodinger equation in the momentum gauge, describing the interaction dynamics in the laboratory frame of reference, was transformed by Kramers in the form ! @Cð r ,tÞ ‘2 2 ! ! ! ¼ r Cð r ,tÞ þ½Vð! r þ a ðtÞÞCð r ,tÞ: @t 2mn ! ! Here Vð r þ a ðtÞÞ is the ‘‘dressed potential’’ energy, and i‘
! a ðtÞ ¼ e^ a0 sin Ot,
ð2Þ
ð3Þ pffiffi n where a0 ¼ ð8p=OcÞ Ie=m is the amplitude of the electron oscillation in the laser field, mn being the effective mass of an electron and I is the laser intensity. By applying the Floquet approach [16], the space translated version of the Schr0¨dinger Eq. (2) can be recast in the equivalent form of a system of coupled time-independent differential equations of the Floquet components of the wavefunction c, containing the (in general complex) quasi-energy E. For the zeroth Floquet component c0 , the system reduces to the time-indepen¨ dent Schrodinger equation [16] " # ‘2 2 !! r þ Vð r , a Þ ð4Þ 0 c0 ¼ Ec0 , 2mn 2
o ! where fi ð r Þ ¼ Rni li ðrÞexpðili yÞ is the ith two-dimensional harmonic oscillator eigenstate with a frequency o and an energy ð2ni þ 9li 9 þ 1Þ‘ o. Rnl(r) is the radial wave function, given by 9l9
Rnl ðrÞ ¼ N expðr 2 =ð2a2 ÞÞr 9l9 Ln ðr 2 =a2 Þ,
ð9Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which N is the normalization constant, a ¼ ‘ =ðmn oÞ, and Lkn ðxÞ is the associated Laguerre polynomial. The radial and orbital angular momentum quantum numbers can have the following values: n ¼ 1,2 . . . ,l ¼ 0, 7 1, 7 2 . . . :
ð10Þ
This single-particle basis is used by the matrix diagonalization method in order to expand the Hamiltonian H. In practice calculation, o serves as a variational parameter to minimize the eigenvalues. Since the whole set of eigenstates of the harmonic product basis forms a complete basis in the Hilbert space, the procedure of increasing the number of linearly independent eigenstates is converging the exact result. The limits are set only by the capacity of the computer to diagonalize N N Hermitian matrices. This is achieved by extending the dimension of the model space step by step; in each step the new results are compared with previous results from a smaller space, until satisfactory convergence is achieved. In this paper, the dimension of the model space is constrained by 0 rN ¼ 2n þ 9l9 r26. If N is increased by 2, the ratio of the difference in energy is less than 0.001. We calculate the absorption coefficients and refractive index changes using the linear wð1Þ and third-order wð3Þ optical susceptibilities. The analytical forms of wð1Þ and wð3Þ are obtained from modeling a QD as a two-level system. The susceptibility w is related to the absorption coefficient aðnÞ by rffiffiffiffiffiffi m aðnÞ ¼ n 0 Im½e0 wðnÞ, ð11Þ
er
where m0 is the permeability of the system. er ¼ n2r e0 is the real part of the permittivity, nr is the medium refractive index and e0 is the permittivity of vacuum. w is the Fourier component of wðtÞ with expðiotÞ dependence. Using the compact density–matrix method, the optical absorption coefficient is given by [22]
aðnÞ ¼ að1Þ ðnÞ þ að3Þ ðn,IÞ,
ð12Þ
L. Lu et al. / Physica B 406 (2011) 4129–4134
where 4pbFS ss 2 að1Þ ðnÞ ¼ hn9Mfi 9 dðEfi hnÞ nr e2
ð13Þ
and 2
32p2 bFS ss I 4 hn9Mfi 9 dðEfi hnÞ2 n2r e4 ‘ Gff ( ) 2 9Mff Mii 9 ½ðhnEfi Þ2 ð‘ Gfi Þ2 þ 2Efi ðEfi hnÞ 1 2 E2fi þ ð‘ Gfi Þ2 49Mfi 9
are the linear and the third-order nonlinear optical absorption coefficients, respectively. e is the electronic charge of an electron, ss is the electron density in the QD, bFS ¼ e2 =4pe0 ‘ c is the fine structure constant, I is the incident optical intensity, and hn is the photon energy. The d function in Eqs. (13) and (14) is replaced by a narrow Lorentzian by means of
að3Þ ðn,IÞ ¼
dðEfi hnÞ ¼
ð14Þ
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‘ Gfi
p½ðhnEfi Þ2 þ ð‘ Gfi Þ2
:
ð15Þ
Here G is the phenomenological operator. Nondiagonal matrix element Gfi ðf aiÞ of operator G, which is called as the relaxation rate of fth state and ith state, is the inverse of the relaxation time Tfi for the states 9f 4 and 9i 4 , namely Gfi ¼ 1=Tfi . In addition, the susceptibility w is related to the change in the refractive index as follows: DnðoÞ wðoÞ ¼ Re : ð16Þ 2 nr 2nr Therefore, the linear and the third-order nonlinear refractive index changes are obtained by [22]
Dnð1Þ ðoÞ nr
¼
ss e2 Mfi2 ðEfi hnÞ 2er ðE21 hnÞ2 þð‘ Gfi Þ
ð17Þ
and
Dnð3Þ ðoÞ nr
¼
ss e4 Mfi2 m0 cI ½4Mfi2 ðEfi hnÞ 4er n2r ½ðEfi hnÞ2 þð‘ Gff Þ2 2
ðMff Mii Þ2
Efi ðEfi hnÞ2 ð‘ Gff Þ2 ð3Efi 2hnÞ E2fi þ ð‘ Gff Þ2
:
ð18Þ
Hence, the total refractive index change can be given as Fig. 1. The linear a ðnÞ, the third-order nonlinear a ðn,IÞ and the total aðn,IÞ optical absorption coefficients as a function of the incident photon energy hn for three different values of the laser field. ð1Þ
ð3Þ
DnðoÞ nr
¼
Dnð1Þ ðoÞ nr
þ
Dnð3Þ ðoÞ nr
:
ð19Þ
Fig. 2. The linear, the third-order nonlinear and the total changes in the refractive index as a function of the incident photon energy hn for three different values of the laser field.
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3. Results and discussions All calculations were performed using the following parameters: mn ¼ 0:067m0 (m0 is the mass of a free electron), e ¼ 12:53, nr ¼3.2, ss ¼ 5:0 1022 m3 , I ¼ 1:0 109 W=m2 , Gff ¼ 1 ps1 , and Gfi ¼ 1:4 ps1 . Figs. 1 and 2 show the linear, third-order nonlinear and total optical ACs and the total optical RI changes as a function of the photon energy hn for three different laser-field values with V0 ¼112.23 meV and R ¼10.0 nm. As seen in the figure, since the third-order term has a negative sign, according to Eqs. (14)
Fig. 3. The linear að1Þ ðnÞ, the third-order nonlinear að3Þ ðn,IÞ and the total aðn,IÞ optical absorption coefficients as a function of the incident photon energy hn for three different values of the potential depth.
and (18), the total ACs and RI changes decrease. In addition, the laser-field-induced effect is clear. It is readily seen that as the laser field value increases, the AC and RI peaks will move to the left side, which show a laser-field-induced red shift of the resonance in QDs. This is because the energy difference between the ground and excited states in QDs decreases with increasing a0 . Obviously this the result of the laser-dressed Coulomb interaction is clear and it is different from that of quantum wells [24]. On the other hand, we find that all ACs and RI changes (absolute values of third-order terms) increase with increasing the laser field values. This behavior is due to the fact that, as the value of a0 is increased, the electron wavefunction is compressed. Thus the overlapping of the wave functions of these two states, i.e. the transition matrix element which is effective on the ACs and RI changes, is enhanced. Hence one can say that in order to obtain the larger ACs and RI changes in QDs, we should induce the laser field. The height of absorption coefficient can reach the magnitude of 105/m. Compared with those existing results, the maximum value of ACs and RI changes significantly increased. Especially, it is 1–2 orders of magnitude higher than that in the case of two-dimensional parabolic QDs [22]. These results indicate that the large ACs and RI changes in a desired energy can be obtained by adjusting the laser field. Figs. 3 and 4 illustrate the linear and nonlinear ACs and RI changes in a Gaussian QD with R0 ¼10 nm and a0 ¼ 1:0 nm as a function of the incident photon energy hn for three different potential depths, i.e. V0 ¼112.23, 224.46 and 336.69 meV, respectively. The confinement effect of the potential depth seems clear. It can be seen that the absorption peak positions shift to higher energies (blueshift) with increasing V0. That is due to that the energy difference between the 1S and 1P states increases with increasing V0. We also note that the bigger the potential depth is, the bigger will be the ACs and RI changes. This is because the overlapping of the wavefunctions increases with increasing confinement, i.e. V0. In Figs. 5 and 6, the potential depth V0 is set to be 112.23 meV, the laser field a0 is set to be 1.0 nm and R are set to be 5.0, 7.5 and
Fig. 4. The linear, the third-order nonlinear and the total changes in the refractive index as a function of the incident photon energy hn for three different values of the potential depth.
L. Lu et al. / Physica B 406 (2011) 4129–4134
10.0 nm, respectively. From these figures, we can find that the size effect of the QD is obvious. The ACs and RI changes peak positions shift to the low photon energies with increasing R, which shows a quantum-size-induced redshift of the resonance in QDs. The physical origin is that, with decreasing R the increase of the energy difference between the initial and final states is increased. However, the peak values of the ACs and the RI changes are not monotonic functions of R. As seen in Figs. 5 and 6, we find that the magnitude of the ACs and the RI changes first increases with R up to a critical value, R 7:5 nm, and for further large R value they begin to decrease. Hence, we can obtain large ACs and RI changes in QDs by controlling the dot size. In order to show better the influence of the incident optical intensity I for the total ACs, in Fig. 7, we set V0 ¼112.23 meV,
Fig. 5. The linear að1Þ ðnÞ, the third-order nonlinear að3Þ ðn,IÞ and the total aðn,IÞ optical absorption coefficients as a function of the incident photon energy hn for three different values of R.
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R0 ¼10.0 nm and a0 ¼ 1:0 nm and plot the total ACs as a function of the incident photon energy hn for six different values of I. As the linear AC does not depend on photon intensity, whereas the third-order term increases with photon intensity as it varies quadratically with laser amplitude. Thus the total AC curve, being the sum of linear and nonlinear parts, is robust when I ¼ 0 W=m2 . If I a0 W=m2 , the total AC changes considerably with increasing optical intensity as expected, especially near the resonance frequency. When the incident optical intensity I exceeds a critical value, which indicates saturation, the nonlinear term causes a collapse at center of the total absorption peaks splitting it into two peaks. Finally the total RI changes as a function of the incident photon energy hn for four different relaxation times T¼0.14 ps, 0.25 ps, 0.35 ps and 0.5 ps is shown in Fig. 8. From this figure, we observed
Fig. 7. The total aðn,IÞ optical absorption coefficients as a function of the incident photon energy hn for six different values of the incident optical intensity.
Fig. 6. The linear, the third-order nonlinear and the total changes in the refractive index as a function of the incident photon energy hn for three different values of R.
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dot radii. Moreover, the calculated results also reveal that the laser field amplitude has a significant effect on the nonlinear optical properties. Applying a laser field to QDs can make the position and height value of the optical ACs and RI change significantly. Our results are important to the experimental studies and have significant influences on improvements of optical devices such as infrared laser amplifiers, photo-detectors, and high-speed electro-optical modulators. This also gives a new degree of freedom in various device applications based on the intersubband transition of electrons.
Acknowledgments This work is financially supported by the National Natural Science Foundation of China under Grant No. 11074055.
References Fig. 8. The total changes in the refractive index as a function of the incident photon energy hn for four different values of the relaxation times.
that the relaxation time T has a great influence on the RI changes, namely, with the increase of the relaxation time, the total RI changes increase obviously. But the saturation begins at the relaxation time value of 0.5 ps. On the other hand, the relaxation rate is related not only to the materials constituting the QD, but also to some other factors, such as the boundary conditions, temperature of the system as well as electron-impurity and electron–phonon scattering interactions, etc. Hence, one should reduces the influences of these factors on the systems to achieve a large absorption coefficient.
4. Conclusion In this study, we proposed the Gaussian confining potential function to simulate the spatial confinement of QDs. The detailed nonlinear optical properties of a hydrogenic impurity in a disclike QD in the presence of an intense laser field are investigated by using the matrix diagonalization method and the compact density–matrix approach. Our results suggest that for intersubband transitions in QDs with the Gaussian potential, we can obtain a blue or red shift by tuning the potential depth or/and the
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