Ga1−xAlxAs quantum ring

Physica E 60 (2014) 95–99

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

Effects of applied lateral electric field and hydrostatic pressure on the intraband optical transitions in a GaAs=Ga1
x Alx As quantum ring A.Kh. Manaselyan a, M.G. Barseghyan a,n, A.A. Kirakosyan a, D. Laroze b,c, C.A. Duque d a

Department of Solid State Physics, Yerevan State University, Al. Manookian 1, 0025 Yerevan, Armenia Max Planck Institute for Polymer Research, D 55021 Mainz, Germany c Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile d Grupo de Materia Condensada-UdeA, Instituto de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medellín, Colombia b

H I G H L I G H T S

   

Simultaneous effects of applied lateral electric field and hydrostatic pressure on the intraband linear optical absorption coefficient. The one electron energy spectrum has been found using the effective mass approximation and an exact diagonalization technique. The hydrostatic pressure can lead to both blue- and red-shift of the intraband optical absorption spectrum. With the increase of electric field strength the absorption coefficient's peaks decrease.

art ic l e i nf o

a b s t r a c t

Article history: Received 12 November 2013 Received in revised form 4 February 2014 Accepted 14 February 2014 Available online 22 February 2014

In this paper the simultaneous effects of applied lateral electric field and hydrostatic pressure on the intraband linear optical absorption coefficient have been investigated in a two-dimensional GaAs=Ga1
x Alx As single quantum ring. The lateral confinement potential of the ring is approximated by square finite potential well model. The one electron energy spectrum has been found using the effective mass approximation and an exact diagonalization technique. We find that for fixed geometric dimensions, the hydrostatic pressure can lead to both blue- and red-shift of the intraband optical absorption spectrum, while only a blue-shift is observed as a result of an electric field. The effect of hydrostatic pressure on the absorption coefficient's peaks is negligibly small. With the increase of electric field strength the absorption coefficient's peaks decrease. & 2014 Elsevier B.V. All rights reserved.

Keywords: Quantum ring Electric field Hydrostatic pressure Intraband absorption coefficient

1. Introduction Current progress in epitaxial techniques has resulted in striving developments in the physics of quantum dots-semiconductor-based artificial atoms. Recently a lot of attention has been directed towards non-simply connected nanostructures, for example quantum rings (QR), which have been obtained in various semiconductor systems [1,2]. The fascination in QR is partially caused by a wide variety of purely quantum mechanical effects, which are observed in ring-like nanostructures [3,4]. The star amongst them is the Aharonov–Bohm effect [5], in which a charged particle is influenced by a magnetic field, resulting in magnetic-flux-dependent oscillations of the ringconfined particle energy. Incorporating an external electric field may result in significant modifications of the electron energy spectrum in

n

Corresponding author. E-mail address: [email protected] (M.G. Barseghyan).

http://dx.doi.org/10.1016/j.physe.2014.02.015 1386-9477 & 2014 Elsevier B.V. All rights reserved.

quantum heterostructures. This certainly affects a large number of their properties. For example, due to the break of axial symmetry by the lateral electric field, the energy levels of large QR exhibit a complex dispersion as a function of an electric field strength [6]. The authors of Ref. [6] also studied the field-induced polarization of the electron wave functions and found an anomalous behavior that can be correlated with the energy dispersion. In addition, the field dependence of the oscillator strength for a number of optical transitions between valence and conduction states has been calculated and analyzed. The oscillations of the single-particle energy in single QR are strongly suppressed by a lateral electric field, thus reducing the symmetry of the system [7,8]. In the presence of a lateral electric field, exceeding a particular threshold, it is possible to switch the ground state of an exciton in an Aharonov–Bohm ring from being optically active to optical inactivity [9,10]. The states of interacting electron–hole pairs in semiconductor QRs in the presence of strong lateral electric field are considered theoretically in Ref. [11]. It has been shown that along with the

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A.Kh. Manaselyan et al. / Physica E 60 (2014) 95–99

size-quantization of the charge carriers' motion, the strong external field also leads to the additional localization of particles by the angular variable. The corresponding additional energy spectrum has an equidistant character. At the same time the strong external field polarizes the electron–hole pairs and traps them at opposite sides of the QR's diameter. The resulting field dependence of the optical characteristics of the system can serve as a basis for the direct experimental observation of exciton-like complexes in semiconductor QRs. In Ref. [12], the single-electron states in a QR have been considered for the case then the energy transfer of charge carriers by a strong lateral electric field becomes comparable with the energy of their rotational motion. The corresponding energy spectrum and the envelope wave functions of the carriers in the QR have been obtained in an explicit form. It has been found that the absorption intensity increases with an increasing electric field strength. On the other hand, the linear and the nonlinear optical absorption as well as the linear and nonlinear corrections to the refractive index are calculated in a QR under the effects of an external magnetic field and parabolic and inverse square confining potentials [13]. The exact solutions for the two-dimensional motion of the conduction band electrons are used as the basis for a perturbation-theory treatment of the effect of an applied static electric field. In general terms, the variation of one of the different potential energy parameters, keeping fixed the other ones, leads to either blue-shifts or red-shifts of the resonant peaks as well as to distinct changes of their amplitudes. We note that the influence of lateral electric field on the intraband optical properties of the two-dimensional QRs, without any limit on the electric field strength, has not been systematically investigated yet. For this purpose we have studied in the present paper the simultaneous effect of lateral electric field as well as hydrostatic pressure on the intraband optical absorption coefficients of GaAs=Ga1
x Alx As single QRs. The calculations have been made in the effective mass and parabolic band approximations. The paper is organized as follows: Section 2 contains the theoretical framework, Section 3 contains the results and discussions and, finally, Section 4 is devoted to the conclusions.

(adiabatic approximation) [14,15]. Consequently, without loss of generality, our system can be considered two dimensional, with the electron confined in the plane z¼0. The structure's radii dependencies on hydrostatic pressure (P) are given by [16] h i1=2 R1ð2Þ ðPÞ ¼ R1ð2Þ ð0Þ 1
2ðS11 þ 2S12 ÞP ;

ð1Þ

where S11 and S12 are the components of the compliance tensor of GaAs (S11 ¼ 1.16  10
3 kbar
1 and S12 ¼
3.7  10
4 kbar
1). R1(0) and R2(0) are the inner and outer radii, respectively, of the QR at zero hydrostatic pressure. The one electron Hamiltonian for a single QR in the framework of the effective mass and parabolic band approximations by taking ! into account the effects of applied electric field ð F Þ and hydrostatic pressure is given by    2 ρ ∂ 1 1 ∂2 b ¼
ℏ 1 ∂ H þ þ Vðρ; x; PÞ þeF ρ cos φ; mðx; PÞ ρ2 ∂φ2 2 ρ ∂ρ mðx; PÞ ∂ρ ð2Þ where e is the absolute value of the electron charge, x is the aluminum concentration, and F is the strength of the electric field, which is directed along x-direction, mðx; PÞ is the hydrostatic pressure dependent electron effective mass, given by [17–19] " mðx; PÞ ¼ m0 1 þ

Π 2 ðxÞ 3

2 EΓ g ðx; PÞ

1

!

þ Γ þ δðxÞ Eg ðx; PÞ þ Δ0 ðxÞ

#
1 :

In this expression, m0 is the free electron mass, Π ðxÞ is the 2 interband matrix element ½Π ðxÞ ¼ ð28; 900
6290xÞ meV, Δ0 ðxÞ is the valence-band spin–orbit splitting ½Δ0 ðxÞ ¼ ð341
66xÞ meV, and the remote-band effects are taken into account via the δðxÞ parameter ½δðxÞ ¼
3:935 þ0:488 x þ 4:938 x2 . The energy gap function at the i-point ði ¼ Γ ; XÞ of the conduction band is given by Eig ðx; PÞ ¼ ai þ bi x þ ci x2 þ αi P:

2. Theoretical framework In Fig. 1 the schematic view of a single QR structure is presented. The dimensions of the heterostructure (the inner and outer radii), as well as the directions of the external applied electric field and the polarization of the incident light, are shown. The ring material is GaAs and it is embedded in a Ga1
x Alx As host matrix material. Usually, the thickness of the ring along the growth direction is smaller than the radial dimensions. Therefore, one can approximately decouple the electron motions along the growth direction and in the (x, y) plane and retain in the analysis only the first state along the z direction

ð3Þ

ð4Þ

The values of the parameters ai ; bi ; ci , and αi are shown in Table 1 [17]. For the confining potential, we have taken into account the hydrostatic pressure induced crossover of the Γ –X conduction band minima via the expression ( 0 if R1 r ρ r R2 Vðρ; x; PÞ ¼ ð5Þ Vðx; PÞ if ρ o R1 or ρ 4 R2 ; where Vðx; PÞ ¼ r

8 Γ < EΓ g ðx; PÞ
E g ð0; PÞ

if P r P 1 ðxÞ;

: EXg ðx; PÞ
EΓ g ð0; PÞ þSΓ X ðx; PÞ

if P 1 ðxÞ o P r P 2 ðxÞ:

ð6Þ

Z Here r is the fraction of the band gap discontinuity (for GaAs/ GaAlAs structures r ¼ 0.6), P 1 ðxÞ is the value of the pressure corresponding to the crossover between the Γ and X bands minima in Ga1
x Alx As, and P 2 ðxÞ is its equivalent with respect to the crossover between the Γ-band minimum at the GaAs well and

F R2 R1 X light polarization Y Fig. 1. The single quantum ring heterostructure considered in the present work. R1 and R2 stand for the inner and outer radii, respectively. The directions of the external applied electric field and polarization of the incident light are depicted.

Table 1 The parameters considered in the description of the Γ and X energy gap functions. Γ-minimum

X-minimum

aΓ ¼ 1519:36 meV bΓ ¼ 1360 meV cΓ ¼ 220 meV αΓ ¼ 10:7 meV=kbar

aX ¼ 1981 meV bX ¼ 207 meV cX ¼ 55 meV αX ¼
1:35 meV=kbar

A.Kh. Manaselyan et al. / Physica E 60 (2014) 95–99

3. Results and discussion

the X-band minimum at the Ga1
x Alx As barrier. Besides, SΓ X ðx; PÞ ¼ S0

P
P 1 ðxÞ x P

ð7Þ

is the Γ –X mixing strength coefficient, where S0 is an adjustable parameter which fits theoretical results with the experimental measurements [20,21]. In this work the value of aluminum concentration is fixed to x ¼0.3. Without an electric field (F ¼0) the eigenfunctions of the Hamiltonian (2) can be presented in the form Φðρ; φÞ ¼ Ceilφ f nl ðρÞ, where C is the normalization constant, and n ¼ 1; 2; …, and l ¼ 71; 72; … are the quantum numbers. The eigenfunctions f nl ðρÞ are obtained from a suitable linear combination of Bessel functions [22]. The corresponding eigenvalues Enl are obtained from the standard boundary conditions for the eigenfunctions. The eigenfunctions of the Hamiltonian, Eq. (2), with lateral electric field included, can be presented as a linear expansion of the eigenfunctions discussed above. The problem is solved numerically using an exact diagonalization procedure. Calculations of the optical absorption coefficient are based on Fermi's golden rule. The intraband absorbtion coefficient can be written as [23,24]

αðℏ ωÞ ¼

16π 2 β FS ℏω

εðPÞ

1=2

V

 Nif M fi j2 δðEf
Ei
ℏωÞ;

97

ð8Þ

where εðPÞ is the pressure dependent static dielectric constant for GaAs [25], V, β FS , and ℏω are the volume of the sample per QR (in this work V ¼ 6  10
18 cm3 [1]), the fine structure constant, and the photon energy, respectively, Nif ¼ N i
N f is the difference in the numbers of electrons in the initial and final states. Since we consider here only one particle problem, we have taken N i ¼ 1 for the ground state and N f ¼ 0 for all upper states. In Eq. (8) the Mfi is the matrix element of the dipole operator for the x-polarization of the incident radiation.

The numerical calculations are carried out for GaAs=Ga0:7 Al0:3 As QR with R1 ð0Þ ¼ 5 nm, and R2 ð0Þ ¼ 35 nm. In Fig. 2 the density of probability of an electron in a single QR for the ground and the first two excited states (N ¼ 1, 2, 3; where N labels the bound electronic states in increasing order of the energy) is presented for three different values of the applied electric field and zero hydrostatic pressure. As we can see from the figure, at zero applied electric field, for all states the density of probability has full axial symmetry. With the increase of electric field strength, the density of probability becomes anisotropic because of displacement of the electron towards the opposite (or the same) direction of the field. For the ground state (N ¼ 1) the electron moves, as it could be expected, opposite to the direction of the field (to the left in Fig. 1). For the excited states the density of probability can have more complicated behavior and can be shifted in the direction of the field as well as in the opposite direction. This phenomenon is associated to the orthogonality between the wave functions of the confined states of the carrier [6]. In Fig. 3 the dependencies on the applied electric field of few low laying energy levels for a confined electron in a QR for different values of the hydrostatic pressure are presented. With the increase of the electric field strength, the ground state energy always decreases due to lowering of the bottom of the confining potential, which is the analogue of the quantum confined Stark effect in quantum wells [27]. For the excited states the dependencies are non-monotonous which is the result of coupling between the states with the same parity. Also it should be mentioned that without electric field, all the states with l ¼0 are non-degenerated and all the states with l a 0 are twice degenerated. The electric field removes this degeneracy because the axial symmetry is broken. Also the field mixes the states with different quantum numbers n and l and that is why now we label the states with the new quantum number N. For the complete range of hydrostatic

Fig. 2. The probability density of an electron confined in a single GaAs=Ga0:7 Al0:3 As QR. Three different values of electric field strength are shown for P¼ 0.

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A.Kh. Manaselyan et al. / Physica E 60 (2014) 95–99

In Fig. 4 the dependencies on the electric field of the threshold energies of optical transitions from the ground state (N ¼1) to the states with N ¼3 ð1-3Þ and N ¼ 5ð1-5Þ for three values of the hydrostatic pressure are presented. It should be mentioned that transitions 1-2 and 1-4 are optically forbidden due to parity selection rule. In both cases the threshold energy increases with an increase of the electric field strength. The hydrostatic pressure, on its turn, strengthens the effect of electric field. In Fig. 5 the intraband absorption coefficient as a function of the incident photon energy is presented for various values of electric field and hydrostatic pressure. In all cases the lowest energy peak corresponds to 1-3 transition and the highest energy one to the 1-5 transition. For all values of the pressure, with the increase of electric field a blue shift in the optical spectrum is observed for both peaks. On the other hand, in the absence of electric field, the increase of pressure brings a red shift for both peaks. But, in the presence of electric field, both red and

pressure the following effects can be considered [26]: (1) the electron-effective mass is an increasing function of pressure, which itself decreases the energy; (2) from Eq. (1) it is clear that the dimensions of the QR decrease with pressure, and this forces the energy to increase; (3) the depth of the confinement potential decreases with pressure and this results in energy decrease. But the most dominant one from these effects is the change of effective mass. That is why in Fig. 3, we can see that all energy levels are decreasing with the increase of hydrostatic pressure. Also it should be mentioned, that the hydrostatic pressure strengthens the effect of the electric field on the energy levels. For example for the case P ¼0 with the increase of electric field from 0 to 50 kV/cm the ground state energy decreases by approximately 4 meV, but for the case P¼ 25 kbar the energy change is approximately 8 meV. Similar effects are visible also for excited states. In this case with the increase of the pressure the splitting between degenerated energies becoming visible for lower values of electric field. Here we can remark that these effects are mainly associated with the diminishing of the potential barrier as the hydrostatic pressure increases.

6

F=0 F = 10 kV/cm F = 20 kV/cm F = 30 kV/cm F = 40 kV/cm

3

P= 0

P = 15 kbar

P = 25 kbar

Absorption coefficient (104cm-1)

12 10

Energy (meV)

8

(1,±2)

(1,±1)

6 4 2

(1, 0)

0 0

3

6

9

12

0

3

6

9

12

0

3

6

9

12

6

3

0 6

0

3 -2

-4

0 0

10

20

30

40

50

Photon energy (meV)

Electric field (kV/cm)

Fig. 5. Dependence of the intraband optical absorption coefficient on incident photon energy in a single GaAs=Ga0:7 Al0:3 As QR. The results are presented for various values of the applied electric field strength and hydrostatic pressure.

Threshold energy (meV)

Fig. 3. Energy levels of an electron confined in a single GaAs=Ga0:7 Al0:3 As QR as a function of the applied electric field strength. Several values of the hydrostatic pressure have been considered.

15

15

15

12

12

12

E5-E1

9

E5-E1

9

6

6

6

E3-E1

3

E3-E1

E3-E1

3

3 0

0

0 0

10

20

30

40

50

E5-E1

9

0

10

20

30

40

50

0

10

20

30

40

50

Electric field (kV/cm) Fig. 4. Dependencies on electric field strength of threshold energies in a single GaAs=Ga0:7 Al0:3 As QR for the first two allowed transitions 1-3 and 1-5. Various values of hydrostatic pressure have been investigated.

A.Kh. Manaselyan et al. / Physica E 60 (2014) 95–99

blue shifts can be observed. For example for F¼10 kV/cm and P ¼25 kbar (black curve) the first peak is shifted to high energy region (blue shift) and the second one is shifted to low energy region (red shift). Also it should be mentioned that the maximum values of the peaks of absorption coefficient are observed for the case F¼0. The effect of hydrostatic pressure on absorption coefficient's peaks is negligibly small. With the increase of electric field strength both peaks are decreasing due to the decrease of overlap integrals of corresponding wave functions between the initial and final states. 4. Conclusions In this paper, we have studied the combined influence of hydrostatic pressure and lateral electric field on one-electron states in GaAs=Ga1
x Alx As single quantum rings. Also we have investigated the mentioned influences on intraband light absorption coefficient. For all values of the pressure, with the increase of electric field strength a blue shift in the optical spectrum is observed for both 1-3 and 1-5 transitions, while the hydrostatic pressure can lead both to the blue shift and to the red shift of the intraband optical absorption coefficient. Acknowledgments We thank H. Pleiner (MPI-P, Mainz) for his critical reading of the manuscript. The work of AKM, MGB and AAK was supported by NFSAT Grant nos. YSSP-13-06 and YSSP-13-03 and by the Armenian State Committee of Science (Project nos. SCS-13-1C196 and AR-13RF-093). C.A.D. is grateful to Colombian Agencies: CODIUniversidad de Antioquia (Estrategia de Sostenibilidad 2013–2014 de la Universidad de Antioquia), Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (CAD-exclusive dedication project 2013–2014), and El Patrimonio Autónomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación, Francisco José de Caldas. The work was developed with the help

99

of CENAPAD-SP, Brazil. D.L. acknowledges partial financial support from FONDECYT 1120764, Millennium Scientific Initiative, P 10-061-F, Basal Program Center for Development of Nanoscience and Nanotechnology (CEDENNA), UTA-project 8750-12.

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