Effects of hydrostatic pressure on the nonlinear optical properties of a donor impurity in a GaAs quantum ring

Physica E ] (]]]]) ]]]–]]]

Contents lists available at SciVerse ScienceDirect

Physica E journal homepage: www.elsevier.com/locate/physe

Effects of hydrostatic pressure on the nonlinear optical properties of a donor impurity in a GaAs quantum ring R.L. Restrepo a,d,n, M.G. Barseghyan b, M.E. Mora-Ramos c,d, C.A. Duque d a

Escuela de Ingenierı´a de Antioquia, AA 7516, Medellı´n, Colombia Department of Solid State Physics, Yerevan State University, Al. Manookian 1, 0025 Yerevan, Armenia c ´noma del Estado de Morelos, CP 62209, Cuernavaca, Morelos, Mexico Facultad de Ciencias, Universidad Auto d Instituto de Fı´sica, Universidad de Antioquia, AA 1226, Medellı´n, Colombia b

H I G H L I G H T S c c c c c

Binding energy and nonlinear optics in two-dimensional quantum rings. Binding energy is an increasing/decreasing function of inner/outer radius. Nonlinear optical properties have a blueshift when inner radius increase. Nonlinear optical properties show a redshift when outer radius increase. Nonlinear optical properties have a blueshift when hydrostatic pressure increases.

a r t i c l e i n f o

abstract

Article history: Received 29 August 2012 Accepted 14 September 2012

The effects of hydrostatic pressure, size quantization and impurity position on the binding energies of a hydrogenic-like donor impurity in a two-dimensional GaAs quantum ring and together with the linear and nonlinear intraband optical absorption and the relative refractive index changes are studied using the variational method and effective-mass approximation. The binding energies of 1s and 2s states are examined as functions of the structure (inner and outer radii), impurity position and hydrostatic pressure. We have also investigated the dependencies of the linear, nonlinear, and total optical absorption coefficients and relative index changes as a function of incident photon energy for different geometric configurations, hydrostatic pressure and impurity position. It is found that the variation of distinct sizes of the structure and impurity position leads to either a redshift and/or a blueshift of the resonant peaks of the intraband optical absorption and relative refractive index change spectrum. In addition we have found that the application of a hydrostatic pressure leads to a blueshift. & 2012 Elsevier B.V. All rights reserved.

1. Introduction Semiconductor nanorings or quantum rings (QRs) are attracting considerable attention due to their applications in the study of quantum mechanical phase coherence [1,2], as well as their unique physical properties, such as large excitonic permanent dipole moment [3] and persistent currents under magnetic response [4]. Currently, there is a considerable interest in the investigation of impurity states in QRs because the presence of impurities strongly affects the electronic states and optical properties of these structures [5]. Monozon et al. [6] have considered the case

n

Corresponding author. Tel.: þ574 3549090; fax: þ 574 386 11 60. E-mail address: [email protected] (R.L. Restrepo).

of an impurity center in a semiconductor QR and calculated analytically the binding energy of an electron in the presence of crossed magnetic and electric fields, taking the QR potential as an infinite barrier. Bruno-Alfonso and Latge´ [7,8] also studied the effects of an impurity in a QR subjected to an external magnetic field. They used a variational approach to obtain the 1s-like shallow-donor impurity states, with the impurity located inside the QR and also placed at the medium distance between the internal and external radii. Dias da Silva et al. [9] analyzed the effects of impurities on the Aharonov–Bohm oscillations in QRs and their subsequent effects on the photoluminescence emission. Monozon and Schmelcher [10] put forward an analytical approach to the problem of an impurity electron positioned in a QR in the presence of crossed – axially directed – homogeneous magnetic and – radially directed – electric fields. Farias et al. [11] have calculated the electronic states of a semiconductor QR in the

1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.09.030

Please cite this article as: R.L. Restrepo, et al., Physica E (2012), http://dx.doi.org/10.1016/j.physe.2012.09.030

2

R.L. Restrepo et al. / Physica E ] (]]]]) ]]]–]]]

presence of an axial uniform magnetic field and a donor impurity. The authors have shown that either the impurity and the interface roughness, which break the cylindrical ring symmetry, can strongly affect, or even destroy, the Aharonov–Bohm oscillations. Barseghyan et al. [12,13] using the variational method and the effective mass and parabolic band approximations have investigated the behavior of the binding energy and photo-ionization cross section of a hydrogenic-like donor impurity in an InAs QR ¨ with a Poschl–Teller confinement potential along the axial direction. This work has taken into account the combined effects of hydrostatic pressure and electric and magnetic fields applied in the growth direction. On other hand, the investigation of the effect of hydrogen-like impurities on the linear and nonlinear optical properties in lowdimensional semiconductor heterostructures, especially in quantum dots, also attracts pretty much interest [14–21]. To our knowledge there are only few research articles related with the effects of impurity and external influences on linear and nonlinear optical properties of QRs [22,23]. Xie [22] theoretically investigated the linear and nonlinear optical absorption coefficients of a donor impurity in a QR. The energy levels and the corresponding wave functions in a QR are obtained within the framework of the effective-mass approximation by using the matrix diagonalization method. The author has found that the linear and nonlinear optical properties of a donor impurity trapped by a QR are strongly affected by the confinement strength, the incident optical intensity and the ring radius. In Ref. [23] the same author investigated the second-order nonlinear optical rectification coefficient associated with intraband transitions in a hydrogenic QR with a two-dimensional pseudopotential in the presence of an external magnetic field. The results have shown that the secondorder nonlinear optical rectification coefficient of a hydrogenic QR are strongly influenced by the geometrical size and chemical potential of the pseudopotential, the hydrogenic impurity and the external magnetic field. In the present work we investigate the effects of hydrostatic pressure, impurity position as well as the changes of the different dimensions of the structure’s geometry on the linear and the nonlinear intraband optical transitions and the relative refractive index changes in GaAs two-dimensional QRs. The paper is organized as follows. In Section 2 we describe the theoretical framework. Section 3 is dedicated to the results and discussion, and our conclusions are given in Section 4.

2. Theoretical framework The expression for the effective mass Hamiltonian of the electron in the GaAs two-dimensional QR, in the presence of a hydrogenic donor impurity, and within the parabolic band approximation is " #   _2 1 @ @ 1 @2 e2 H¼ , ð1Þ þ r þ Vðr,PÞ n 2 2 @r 2m ðP,TÞ r @r eðP,TÞr r @f ! ! where r ¼ ½ð r  ri Þ2 1=2 is the distance from the electron to the impurity site, e is the absolute value of the electron charge. Additionally, mn ðP,TÞ and eðP,TÞ are, respectively, the hydrostatic pressure and temperature (T¼4 K in this work) dependent electron effective mass and static dielectric constant. Besides, Vðr,PÞ is the confinement potential of the QR defined as zero in the QR region and infinite elsewhere. We choose to describe the effect of the impurity by using the variational method. Our interest lies strictly in the ground (1s) and first excited (2s) states of the confined electron. Therefore, the proposed trial functions for the mentioned impurity states consist of products between the uncorrelated first confined sub-

band eigenfunctions – associated with the electron radial motion in the QR – and a 1s- and a 2s-like two-dimensional hydrogenic functions [24,25]. In consequence, the wave functions of 1s and 2s states are proposed as

Ci ðrÞ ¼ N i jðrÞeai r ,

ð2Þ

Cf ðrÞ ¼ N f jðrÞð1bf rÞeaf r ,

ð3Þ

where Ni and Nf are the normalization constants, fai , bf , af g are the variational parameters, which can be determined by also requiring Ci and Cf to form a set of orthogonal functions. In addition, the ground state wave function jðrÞ in the absence of the impurity potential has the following form [26–28]:

WðrÞ ¼ J 0 ðkrÞ þ GY 0 ðkrÞ,

ð4Þ

where ðJ0 ,Y 0 Þ are the first and second kind Bessel functions of order zero, respectively  1=2 2mn ðP,TÞ E k¼ r _2 (Er is the ground-state energy associated with the lateral confinement). On the other hand G is the constant obtained from the imposition of continuity conditions of the solutions at the interfaces. For the 1s-like state the variational procedure involves minimizing /Ci 9H9Ci S with respect to a1s in order to find the impurity ground state energy E1s . One must follow a similar procedure for the excited 2s-like state. The inclusion of hydrostatic pressure effects implies the introduction of the pressure dependence of the electron effective mass, the GaAs static dielectric constant, and on the dimensions (inner and outer radii and height) of the heterostructure. They are respectively given by [29–31] m ðP,TÞ  1 15020 meV 7510 meV þ ¼ 1þ m0 , Eg ðP,TÞ Eg ðP,TÞ þ 341 meV

ð5Þ

eðP,TÞ ¼ 12:74  expð1:67  103 kbar1 PÞ  exp½9:4  105 K1 ðT75:6 KÞ, Ri ðPÞ ¼ Ri ð0Þ½12PðS11 þ2S12 Þ1=2

ð6Þ

ði ¼ 1,2Þ,

ð7Þ

where m0 is the free electron mass and Eg ðP,TÞ is the pressureand temperature-dependent GaAs band gap, determined by the following relation: ! 0:5405 K1 T 2 1 Eg ðP,TÞ ¼ 1519þ 10:7 kbar P meV: ð8Þ T þ204 K 1

In the calculations the values S11 ¼ 1:16  103 kbar and S12 ¼ 1 3:7  104 kbar are taken. The density matrix approach allows to derive the linear and third order optical absorption coefficients and relative refractive index change. The expressions for these quantities are, respectively [32,33] 2

að1Þ ð_oÞ ¼

4poe2 ss 9M fi 9 G0 nr c ðEfi _oÞ2 þ G20

ð9Þ

and 4

32p2 oe4 Iss 9M fi 9 G0 n2r c2 ½ðEfi _oÞ2 þ G20 2 "  #  M ff M ii 2 ðEfi _oÞ2 ðG0 Þ2 þ 2Efi ðEfi _oÞ  1 , 2M fi  E2fi þ G20

að3Þ ð_o,IÞ ¼ 

Please cite this article as: R.L. Restrepo, et al., Physica E (2012), http://dx.doi.org/10.1016/j.physe.2012.09.030

ð10Þ

R.L. Restrepo et al. / Physica E ] (]]]]) ]]]–]]]

Dnð1Þ ð_oÞ nr

2

¼

Dnð3Þ ð_o,IÞ nr

2pe2 ss 9M fi 9 ðEfi _oÞ , n2r ðEfi _oÞ2 þ G20

ð11Þ

"  #  4 M ff M ii 2 16p2 e4 Iss 9M fi 9 ðEfi _oÞ 1   ¼ 1 2M fi  E2fi þ G20 n3r ½ðEfi _oÞ2 þ G20 2

 ½Efi ðEfi _oÞðG0 Þ2 ðG0 Þ2 ð2Efi _oÞ2 ,

ð12Þ

where G0 ( ¼0.4 meV) is the Lorentzian – damping-related – parameter. In expressions (9)–(12) the intensity of the incident field is labeled by I, ss is the density of the electrons in the system, nr is the refractive index, Efi ¼ Ef Ei , whereas Mfi ¼ /Cf 9r cosðfÞ9Ci S is the matrix element of the dipole operator. Our numerical calculations use ss ¼ 5  1016 cm3 , nr ¼ 3:2 and I ¼ 103 W=cm2 . The total absorption coefficient and relative refractive index change are given as follows respectively:

að_oÞ ¼ að1Þ ð_oÞ þ að3Þ ð_oÞ

ð13Þ

and

Dnð_oÞ nr

¼

Dnð1Þ ð_oÞ nr

þ

Dnð3Þ ð_oÞ nr

:

ð14Þ

3. Results and discussion In Fig. 1 we present our results for binding energy Eb, of a donor impurity of 1s and 2s states (Fig. 1(a)) and expectation value of electron-impurity distance of the same states (Fig. 1(b)) in a GaAs two-dimensional QR as a function of inner radius R1. The calculations are performed for R2 ¼ 15 nm, P¼0 and the impurity is placed at the center of QR ri ¼ ðR1 þR2 Þ=2. As we can see from Fig. 1(a) the binding energies of both states increase with the increment of the inner radius R1. The obtained behavior of Eb can be explained using the results depicted in Fig. 1(b) together with some physical reasons, presented below. From Fig. 1(b) it is clear that with the increase of the inner QR radius the expected electron-impurity distance of the 1s state increases. This happens as a result of the strengthening in the size quantization effect due to the increase of the inner radius. The electronic cloud becomes closer to the position of the impurity; that is, the center of the QR. For this reason, the intensity of the Coulomb interaction between the electron and the impurity augments, thus leading to increase of the binding energy. In the 2s state the electron density of probability is more delocalized

than the corresponding one in the 1s state. Thus, with the increment in the value of the inner radius R1 the expectation value of the electron-impurity distance increases. Then, the growth of the associated binding energy can be understood according to the following arguments: Since the corresponding binding energy is defined as the energy difference between the free electron energy E0 (without impurity) and the energy of electron in the confined impurity-related 2s state E2s , and given that the size quantization effect is weaker in this excited state, the energy difference E0 E2s – and therefore the binding energy – increases. The dependencies of the intraband absorption coefficient (Fig. 2(a)) and relative refractive index change (Fig. 2(b)) on the incident photon energy for several values of the inner radius R1 show a blueshift of the resonant peaks as long as this radial size augments. The results are for R2 ¼ 15 nm, P¼0, and the impurity is placed at the center of the QR ri ¼ ðR1 þ R2 Þ=2. As can be seen from figures the increase in R1 brings about a blueshift in the spectrum of both the intraband absorption and relative refractive index change. This can be explained by the fact that for increasing values of R1 the energy distance between 1s and 2s states (see Fig. 1(a)) enhances. Therefore, a larger photon energy is required for the electron’s transitions. Notice that the variations of the amplitudes of the – nonlinear and total – resonant peak of absorption coefficient and relative refractive not only depend on the incident light intensity but also there is a main from the magnitude of the matrix elements, Mfi. Thus it can be argued that the value of resonant peaks can augment or diminish depending on which term (matrix elements or energy difference) has the larger contribution in the expressions of absorption coefficient and relative refractive index change. The results depicted in Fig. 3 correspond to the dependencies on the outer radius, R2, of the binding energy of a donor impurity in 1s and 2s states (Fig. 3(a)), and of the expectation value of the electron-impurity distance associated with the same states (Fig. 3(b)); in a GaAs two-dimensional QR. The results presented correspond to the configuration in which R1 ¼ 5 nm, P¼0, with the impurity placed at the center of the QR, ri ¼ ðR1 þ R2 Þ=2. As we can see from Fig. 3(a) the binding energies of both 1s and 2s states are decreasing functions of the outer QR radius. This behavior of Eb in both cases can be explained by using the results shown in Fig. 3(b). With the weakening of the size quantization (with the increase of outer radius of the QR) in both cases the electron cloud moves far from impurity position and the expectation values of the electron-impurity distance in the 1s and 2s states decrease,

68

21

51

Expectation value (nm)

Binding energy (meV)

3

1s state

34 2s state

5 0 -5 0

2

4

6

8

14

2s state

7

0 10 R1 (nm)

1s state

2

4

6

8

10

Fig. 1. Binding energy of a donor impurity of a 1s and 2s states (a) and expectation values of electron-impurity distance of the same states (b) as a function of inner radius R1 in a GaAs two-dimensional quantum ring. The considering results are for R2 ¼ 15 nm, P¼ 0 and the impurity is placed at the centre of QR ri ¼ ðR1 þ R2 Þ=2. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Please cite this article as: R.L. Restrepo, et al., Physica E (2012), http://dx.doi.org/10.1016/j.physe.2012.09.030

4

R.L. Restrepo et al. / Physica E ] (]]]]) ]]]–]]]

4.5 R1 = 10 nm

R1 = 0.01 nm R1 = 5 nm

α (104 cm-1)

3.0

1.5

0.0

-1.5 24

32

40

60

64

68

2 R1 = 5 nm

R1 = 10 nm

∆n/n

1

0

-1

R1 = 0.01 nm -2 24

32

40 60 Photon energy (meV)

64

68

Fig. 2. Absorption coefficient (a) and relative refractive index change (b) as a function of photon energy: solid lines are for að1Þ and Dnð1Þ =nr , dashed lines are for að3Þ and Dnð3Þ =nr , and dotted line are for a and Dn=nr . Several values of inner radius R1 have been considered for R2 ¼ 15 nm, p¼ 0, and the impurity is placed at the centre of QR ri ¼ ðR1 þ R2 Þ=2. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

which in turn leads to a reduction in the value of the binding energy. The dependencies of the intraband absorption coefficient (Fig. 4(a)) and relative refractive index change (Fig. 4(b)) on the incident photon energy for several values of the outer radius, R2, show a redshift of the resonant peak energy positions which are a consequence of the weakening in the size quantization (the increment in electron wave function spatial localization) due to the increasing of the outer QR radius. Along the sequence of reported values of R2 the energy distance between the involved energy levels progressively decrease, and the intraband 1s-2s electron transition requires less energy. One can also see a progressive reduction in the resonant peak amplitude as long as the outer QR becomes larger. In this case the value of the resonant peak is mostly determined by the value of matrix element Mfi, which is a decreasing function of R2 in the case of the chosen configuration: R1 ¼ 10 nm, P¼0, and ri ¼ ðR1 þ R2 Þ=2. The influence of the change in the position of the donor impurity on the binding energy in a GaAs two-dimensional QR is the one presented in Fig. 5(a), together with the variation of the expected electron-impurity distance on ri (Fig. 5(b)). The calculations correspond to the configuration in which R1 ¼ 5 nm, R2 ¼ 15 nm and P ¼0. It should be again noticed that the behavior of the binding energy can be explained using the results shown in Fig. 5(b) together with the behavior of the electron energy levels with and without the presence of the impurity. Notice that in the case of a QR the density of probability of a non-correlated electron is shifted away from the center, ðR1 þR2 Þ=2. As can be seen from Fig. 5(b), in the case of ri ¼ R1 the expectation value of the electron-impurity distance of the 1s state is slightly below the one obtained when the impurity is placed at ri ¼ R2 ; but in the intermediate values this average distance is always smaller, with the minimum at ri ¼ ðR1 þ R2 Þ=2. In consequence, the binding energy shows the maximum at this value in Fig. 5(a). The results obtained for the 2s state differ significantly. This happens because in this state the electron delocalization is stronger than in the 1s state. On the other hand, the asymmetry of the binding energy and the expectation value of electron-impurity distance – as functions of the impurity position – are also bigger. The reason for the increase or decrease of the binding energy as a function of the impurity position lies only on the behavior of the 2s state energy. It is necessary to notice that in this case the 2s binding energy has a region of negative values in 5 nm r ðri Þ r 7:2 nm. It means that when the impurity moves in this region the excited

51

30 2s state

1s state

Expectation value (nm)

Binding energy (meV)

68

34 2s state

0.8 0.4 0.0 15

18

21

20

10

0 24 15 R2 (nm)

1s state

18

21

24

Fig. 3. Binding energy of a donor impurity of a 1s and 2s states (a) and expectation values of electron-impurity distance of the same states (b) as a function of outer radius R2 in a GaAs two-dimensional quantum ring. The considering results are for R1 ¼ 10 nm, P ¼0 and the impurity is placed at the centre of QR ri ¼ ðR1 þ R2 Þ=2. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Please cite this article as: R.L. Restrepo, et al., Physica E (2012), http://dx.doi.org/10.1016/j.physe.2012.09.030

R.L. Restrepo et al. / Physica E ] (]]]]) ]]]–]]]

state of the donor impurity is higher than the ground state energy of the non-correlated electron. In Fig. 6 we are presenting the dependencies of the intraband absorption coefficient (Fig. 6(a)) and relative refractive index change (Fig. 6(b)) on the incident photon energy for several values of the impurity position, ri . The results correspond to the configuration in which R1 ¼ 5 nm, R2 ¼ 15 nm and P¼0. From Fig. 6(a) and (b) it is clear that in the spectra of intraband absorption and relative refractive change there can be both blueshifts and redshifts. As we have already mentioned the

5

justification for this phenomenon lies on the energy differences between the 1s and 2s states obtained in each situation. It is readily seen that the maximum value of the photon energy necessary for an intraband transition corresponds to the case when the impurity is placed on the left edge of QR. It is worth noticing that several authors Refs. [34–38] have been investigating the dependence of the total absorption coefficient as a function of incident photon energy for different values of the light intensity. They have shown that from certain values of I the resonant peak of absorption coefficient divides into two parts. The reason for this can be found in the linear dependence of the third

6 7.0

R2 = 15 nm

2

ρi = 15 nm

R2 = 20 nm

ρi = 10 nm

3.5 α (104 cm-1)

α (104 cm-1)

4 R2 = 25 nm

0

0.0

-3.5 ρi = 5 nm

-2 32 2

36

40

44

64

68

-7.0

72

5

R2 = 15 nm

15

R2 = 20 nm

35

40

45

ρi = 5 nm ρi = 10 nm

3.5 Δn/n

Δn/n

1 R2 = 25 nm

10

7.0

0

0.0

-1

-3.5

-2 32

-7.0

ρi = 15 nm 36

40 44 64 68 Photon energy (meV)

5

72

Fig. 4. Absorption coefficient (a) and relative refractive index change (b) as a function of photon energy: solid lines are for að1Þ and Dnð1Þ =nr , dashed lines are for að3Þ and Dnð3Þ =nr , and dotted line are for a and Dn=nr . Several values of outer radius R2 have been considered for R1 ¼ 10 nm, P¼ 0, and the impurity is placed at the centre of QR ri ¼ ðR1 þ R2 Þ=2. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

15 35 40 Photon energy (meV)

45

Fig. 6. Absorption coefficient (a) and relative refractive index change (b) as a function of photon energy: solid lines are for að1Þ and Dnð1Þ =nr , dashed lines are for að3Þ and Dnð3Þ =nr , and dotted line are for a and Dn=nr . Several values of impurity position ri have been considered for R1 ¼ 5 nm, R2 ¼ 15 nm and P ¼0. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

20

40 1s state 30 20 10

2s state

0

Expectation value (nm)

50 Binding energy (meV)

10

16 2s state 12

-10

8 1s state 4 0

-20 5

10

15 ρi (nm)

5

10

15

Fig. 5. Binding energy of a donor impurity of a 1s and 2s states (a) and expectation values of electron-impurity distance of the same states (b) as a function of impurity position ri in a GaAs two-dimensional quantum ring. The considering results are for R1 ¼ 5 nm, R2 ¼ 15 nm and P¼ 0. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Please cite this article as: R.L. Restrepo, et al., Physica E (2012), http://dx.doi.org/10.1016/j.physe.2012.09.030

R.L. Restrepo et al. / Physica E ] (]]]]) ]]]–]]]

results for the intraband absorption coefficient and the relative refractive index change, respectively, as function of the incident photon energy. In this case, the parameter chosen to vary is the hydrostatic pressure P and results are for R1 ¼ 5 nm, R2 ¼ 15 nm and the impurity locates at the center of QR, ri ¼ ðR1 þ R2 Þ=2. As can be seen from figures the increase in P brings about a blueshift in the spectrum of both the intraband absorption and relative refractive index change. This is explained by the fact that for the considered values of P the energy distance between 1s and 2s states (see Fig. 7(a)) increases and therefore the electron

3.6 P=0 2.4 α (104 cm-1)

order optical absorption coefficient with respect to the incident light intensity. As can be seen from Fig. 6(a) (red dotted line), we obtain here the same kind of effect. But we must highlight that in our case the value of I has been kept fixed. So, the reason for the oscillation of the total absorption coefficient as well as the additional oscillation of the third order relative refractive index change (see Fig. Fig. 6(b), red dotted line) is the growth in the value of the dipole matrix element Mfi (see, for instance, that the amplitude of the third-order absorption peak significantly grows when ri augments). As a result of this growth it is possible to obtain even a negative total absorption coefficient. At this point, it is worth discussing the mathematical result for the total optical absorption coefficient appearing in Fig. 6(a) when ri coincides with the outer QR radius. As we can observe, the value of this coefficient at the resonant light frequency and turns out to be negative. This is a consequence of the dominant weight of the factor I (representing incident light intensity) in the magnitude of the third-order contribution. From the physical point of view, in this one-electron and stationary system, such a result can be spurious. However, it is actually possible to have a vanishing of the total absorption coefficient via the growing influence of the nonlinear response, leading to what is known as bleaching. That is, the system will become transparent for the propagation of a light signal whose frequency coincides with that of the involved energy transition. The variation of the 1s and 2s donor-impurity binding energies in a GaAs two-dimensional QR as functions of the hydrostatic pressure P are shown in Fig. 7(a), whereas the corresponding variation of the expected electron-impurity distance of the same states appears in Fig. 7(b). Our calculations take R1 ¼ 5 nm, R2 ¼ 15 nm and the impurity is placed at the center of QR, ri ¼ ðR1 þ R2 Þ=2. It is readily seen that in all cases, the influence of the hydrostatic pressure leads to an increase in the binding energy of both states (see Fig. 7(a)) with an associated reduction of the expectation vale of electron-impurity distance as long as P augments (see Fig. 8(b)). Augmenting P implies a growth in the electron effective mass and a fall in the value of the dielectric constant. With the increase of the effective mass, the first noncorrelated electron confined state goes down in energy. On the other hand, the reduction of the dielectric constant leads to the weakening of the electrostatic screening with the consequent reinforcement of the Coulombic interaction and the growth of the binding energy value. On the other hand, reducing the effective dimensions of the structure results in shortening of the effective electron-impurity distance (see Fig. 7(b)), which also leads to an increase in the binding energies. Fig. 8(a) and (b) contain our

-1.2 30 2

35

40

45

50

55

P=0 1

P = 20 kbar P = 40 kbar

0

-1

-2 30

35

40 45 50 Photon energy (meV)

55

Fig. 8. Absorption coefficient (a) and relative refractive index change (b) as a function of photon energy: solid lines are for að1Þ and Dnð1Þ =nr , dashed lines are for að3Þ and Dnð3Þ =nr , and dotted lines are for a and Dn=nr . Several values of hydrostatic pressure P have been considered for R1 ¼ 5 nm, R2 ¼ 15 nm, and the impurity is placed at the centre of QR ri ¼ ðR1 þ R2 Þ=2. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

17

55

Expectation value (nm)

Binding energy (meV)

P = 40 kbar

1.2

0.0

60 1s state

50 45 2s state

5 4 3

P = 20 kbar

∆n/n

6

0

10

20

30

2s state

16

3

2 0 40 P (kbar)

1s state

10

20

30

40

Fig. 7. Binding energy of a donor impurity of a 1s and 2s states (a) and expectation values of electron-impurity distance of the same states (b) as a function of hydrostatic pressure P in a GaAs two-dimensional quantum ring. The considering results are for R1 ¼ 5 nm, R2 ¼ 15 nm and the impurity is placed at the centre of QR ri ¼ ðR1 þ R2 Þ=2.

Please cite this article as: R.L. Restrepo, et al., Physica E (2012), http://dx.doi.org/10.1016/j.physe.2012.09.030

R.L. Restrepo et al. / Physica E ] (]]]]) ]]]–]]]

transitions will require a larger value of the photon energy. We can also observe that the amplitudes of the linear and nonlinear resonant peaks of the two optical properties considered always diminish with P. This is a consequence of the decreasing variation of the dipole matrix element Mfi as a function of P. One may notice as well that the amplitudes of the third-order resonant peaks are more strongly affected by pressure because, despite the fall of Mfi(P), the magnitude of the term 9ðM ff Mii Þ=M fi 9 is also reduced due to the decrease in the difference 9Mff M ii 9 when the hydrostatic pressure augments.

4. Conclusions In this paper we have studied the combined influence of size quantization, impurity position and hydrostatic pressure on the donor-related linear and nonlinear intraband optical absorption and relative refractive index change in GaAs two-dimensional single quantum ring. Our results show that the binding energies of first and excited states are rather sensitive to the variation of the quantum ring geometry via the modification of the size of inner and outer radii. In consequence, both 1s and 2s exciton binding energies are increasing functions of the inner radius, whereas they exhibit decreasing variations with respect to the increment in the outer quantum ring radius. On the other hand, in a configuration with a fixed value of the outer radius, augmenting the size of the inner radius leads to the blueshift of the energy position of both the optical absorption peak position and the relative refractive index change. However, by keeping fixed the value of the inner radius, the increase of the value of the outer radius causes the redshift of the optical responses. The change in the radial position of the donor impurity, from being placed at the inner radius to move to locate at the outer one, has a mixed effect on the energy position of the maxima corresponding to the optical coefficients. First, there is a blueshift and, for impurity positions above the quantum ring half-radius, these quantities are redshifted. Our results regarding the influence of the hydrostatic pressure show that both impurity states studied have augmenting energies, with a simultaneous increase of the transition energy, which reflects in the blueshift of the linear and nonlinear optical responses. The presented results can be useful in the understanding of the influences of hydrostatic pressure and hydrogenlike donor impurity on nonlinear optical properties of single quantum rings.

Acknowledgments C.A.D. is grateful to the Colombian Agencies CODI-Universidad de Antioquia (Project: E01535-Efectos de la presio´n hidrosta´tica y de los campos ele´ctrico y magne´tico sobre las propiedades o´pticas no lineales de puntos, hilos y anillos cua´nticos de GaAs-(Ga,Al)As y Si/SiO 2 ) and Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (CAD-exclusive dedication Project 2012-2013). This research was partially supported by Direccio´n de Investigacio´n de la

7

Escuela de Ingenierı´a de Antioquia (Co-supported EIA-UdeA)-Colombia and by Armenian State Committee of Science (Project no. 11B1c039). M.E.M.R. thanks Mexican CONACYT for 2012-2013 sabbatical Grant no. 180636. He is also grateful to the Universidad de Antioquia and the Escuela de Ingenierı´a de Antioquia for hospitality in his sabbatical stay. References [1] R.A. Webb, S. Washburn, C.P. Umbach, R.B. Laibowitz, Physical Review Letters 54 (1985) 2696. [2] G. Timp, A.M. Chang, J.E. Cunningham, T.Y. Chang, P. Mankiewich, R. Behringer, R.E. Howard, Physical Review Letters 58 (1987) 2814. ¨ [3] R.J. Warburton, C. Schulhauser, D. Haft, C. Schaflein, K. Karrai, J.M. Garcia, W. Schoenfeld, P.M. Petroff, Physical Review B 65 (2002) 113303. [4] D. Mailly, C. Chapelier, A. Benoit, Physical Review Letters 70 (1993) 2020. [5] A.D. Yoffe, Advances in Physics 50 (2001) 1. [6] B.S. Monozon, Mikhail V. Ivanov, P. Schmelcher, Physical Review B 70 (2004) 205336. [7] A. Bruno-Alfonso, A. Latge´, Physical Review B 61 (2000) 15887. [8] A. Bruno-Alfonso, A. Latge´, Physical Review B 71 (2005) 125312. [9] L.G.G.V. Dias da Silva, S.E. Ulloa, A.O. Govorov, Physical Review B 70 (2004) 155318. [10] B.S. Monozon, P. Schmelcher, Physical Review B 67 (2003) 045203. [11] G.A. Farias, M.H. Degani, J.A.K. Freire, J. Costa e Silva, R. Ferreira, Physical Review B 77 (2008) 085316. [12] M.G. Barseghyan, M.E. Mora-Ramos, C.A. Duque, European Physical Journal B 84 (2011) 265. [13] M.G. Barseghyan, Alireza Hakimyfard, Marwan Zuhair, C.A. Duque, A.A. Kirakosyan, Physica E 44 (2011) 419. [14] I. Karabulut, S. Baskoutas, Journal of Applied Physics 103 (2008) 073512. [15] W.F. Xie, Physics Letters A 372 (2008) 5498. [16] Bin. Li, Kang-Xian Guo, Zuo-Lian Liu, Yun-Bao Zheng, Physics Letters A 372 (2008) 1337. [17] W.F. Xie, Physica Status Solidi B 245 (2008) 101. [18] W.F. Xie, Physica B 405 (2010) 3436. ¨ zmen, M. O ¨ zgur ¨ Sezer, Superlattices and [19] Bekir C - akir, Ysuf Yakar, Ayhan O Microstructures 47 (2010) 556. [20] Lu Zhang, Zhongyuan Yu, Wenjie Yao, Yumin Liu, Han Ye, Superlattices and Microstructures 48 (2010) 434. [21] W.F. Xie, Superlattices and Microstructures 48 (2010) 239. [22] W.F. Xie, Physica Status Solidi B 246 (2009) 2257. [23] W.F. Xie, Physica Scripta 85 (2012) 055702. [24] A. Latge´, M. deDios-Leyva, L.E. Oliveira, Physical Review B 49 (1994) 10450. [25] C.A. Duque, A.L. Morales, A. Montes, N. Porras-Montenegro, Physical Review B 55 (1997) 10721. [26] Z. Barticevic, M. Pacheco, A. Latge´, Physical Review B 62 (2000) 6963. [27] J.M. Llorens, C. Trallero-Giner, A. Garcı´a-Cristo´bal, A. Cantarero, Physical Review B 64 (2001) 035309. [28] L.A. Lavene re-Wanderley, A. Bruno-Alfonso, A. Latge´, Journal of Physics: Condensed Matter 14 (2002) 259. [29] N. Raigoza, A.L. Morales, A. Montes, N. Porras-Montenegro, C.A. Duque, Physical Review B 69 (2004) 045323. [30] M.G. Barseghyan, A.A. Kirakosyan, C.A. Duque, European Physical Journal B 72 (2009) 521. [31] F.J. Culchac, N. Porras-Montenegro, J.C. Granada, A. Latge´, Journal of Applied Physics 105 (2009) 094324. [32] D. Ahn, S.-L. Chuang, IEEE Journal of Quantum Electronics 23 (1987) 2196. [33] R.W. Boyd, Nonlinear Optics, Academic Press, San Diego, 2003. [34] Wenjie Yao, Zhongyuan Yu, Yumin Liu, Boyong Jia, Physica E 41 (2009) 1382. [35] V. Prasad, P. Silotia, Physics Letters A 375 (2011) 3910. [36] M. Kirak, S. Yilmaz, M. S- ahin, M. Genc- aslan, Journal of Applied Physics 109 (2011) 094309. ¨ zmen, Journal of Luminescence 132 (2012) 2659. [37] B. C - akir, Y. Yakar, A. O [38] H. Hassanabadi, H. Rahimov, Liangliang Lu, Chao Wangc, Journal of Luminescence 132 (2012) 1095.

Please cite this article as: R.L. Restrepo, et al., Physica E (2012), http://dx.doi.org/10.1016/j.physe.2012.09.030