Hydrostatic pressure and electric field effects and nonlinear optical rectification of confined excitons in spherical quantum dots

Superlattices and Microstructures 49 (2011) 264–268

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Hydrostatic pressure and electric field effects and nonlinear optical rectification of confined excitons in spherical quantum dots C.M. Duque a , M.E. Mora-Ramos b , C.A. Duque a,∗ a

Instituto de Física, Universidad de Antioquia, AA 1226, Medellín, Colombia

b

Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, CP 62209, Cuernavaca, Morelos, Mexico

article

info

Article history: Received 19 April 2010 Received in revised form 1 June 2010 Accepted 10 June 2010 Available online 10 July 2010 Keywords: Quantum dots Nonlinear optical rectification

abstract The effects of hydrostatic pressure and applied electric field on an interacting electron–hole pair in a semiparabolic quantum dot are studied under the density-matrix formalism and the effectivemass approximation. The binding energies of the first confined exciton states are calculated as a function of the confinement strength. The nonlinear optical rectification is also studied as a function of the hydrostatic pressure, the applied electric field, and the confinement strength. The results show that the resonant peak of the nonlinear optical rectification can be red or blue shifted by external probes such as hydrostatic pressure, applied electric field, and the confinement strength. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Optical nonlinearities in GaAs-based semiconducting heterostructures have been the subject of study for years. Ahn and Chuang [1] developed a calculation approach for nonlinear optical susceptibilities based on density-matrix theory which has been subsequently applied in several other works. For instance, Rosencher and Bois considered the presence of large second-order susceptibilities in asymmetric quantum wells (QWs) due to large dipolar matrix elements and the possibility of obtaining resonances at certain wavelengths thanks to an appropriate tuning of the width and shape of the well structure [2].

∗

Corresponding author. Tel.: +57 4 219 56 30. E-mail addresses: [email protected] (C.M. Duque), [email protected] (M.E. Mora-Ramos), [email protected], [email protected] (C.A. Duque). 0749-6036/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2010.06.008

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Nonlinear optical rectification (NLOR) in semiconductor heterostructures and nanostructures has become important for applications in the investigation of their electronic structures (see, for instance, [3] and references therein). This property was studied in [3] for asymmetrical semiparabolic QWs for which the optical rectification coefficient was calculated and shown to be a decreasing function of the confining potential frequency. In the particular situation of quantum dot (QD) systems, the effects of excitons, impurities and electric field on the NLOR have been considered [4], and the effects of an electric field and NLOR of excitons confined in a QD have very recently been reported [5]. On the other hand, the combined influence of an external electric field and hydrostatic pressure of the exciton states in rectangular-shaped QWs and coupled QDs has also been the subject of recent research [6]. Within this framework, the present study is concerned with the theoretical investigation of the effects of intense laser fields on the exciton binding energy and NLOR in spherical-shaped GaAs − Ga1−x Alx As QDs under the effects of applied electric field and hydrostatic pressure. Calculations are made in the effective-mass and parabolic band approximations. The paper is organized as follows. In Section 2 we describe the theoretical framework. Section 3 is dedicated to the results and discussion, and finally, our conclusions are given in Section 4. 2. Theoretical framework The present work is concerned with a single exciton in an ellipsoidal QD with semiparabolic confinement potential under the combined effect of hydrostatic pressure (P) and applied electric field (F ). In the effective-mass and parabolic band approximations the Hamiltonian of the confined exciton pair reads

− →

− →

H = H e ( re ) + H h ( rh ) −

e2

→ − → , ε|− re − rh |

(1)

with the single-particle Hamiltonians (+ for electron and − for hole) h¯ 2

− →

Hi ( ri ) = −

2 mi

− →

→ ▽2i +V (− ri ) ± |e|F zi ,

(2)

2 2 2 2 where V ( ri ) = 12 mi ω0x xi + ω0y y2i + ω0z zi2 with i = e, h and 0 ≤ x, y, z < +∞. The values x, y, z < 0 in the present model have been considered a rigid potential barrier; i.e., V (x) → ∞ for − → x < 0, V (y) → ∞ for y < 0, and V (z ) → ∞ for z < 0. Here mi and ri = (xi , yi , zi ) denote the effective mass and the carrier position, respectively, ω0k (k = x, y, z ) are the strengths of the confinement in the three directions of space, ε is the static dielectric constant of the QD material, and − → − → − → F is the strength of the applied electric field. In terms of the relative coordinate r = re − rh and





− →

− →

− →

the center-of-mass coordinate R = (me re + mh rh )/M, the Hamiltonian in Eq. (1) can be rewritten as H = Hcm + Hrel , where Hcm is the Hamiltonian of the center of mass (with well-known eigenvalues and eigenfunctions) and Hrel = −

h¯ 2

1

▽2rel + µ

2µ

2

−

1

2 ω0k (k + γk )2 − |e| F γz −

2

k=x,y,z

e2

. → ε |− r |

(3)

2 Here, γx = γy = 0 and γz = |e|F / µω0z , µ = me mh /M is the electron–hole reduced mass, and M = me + mh . The eigenfunctions, and the corresponding eigenvalues, of the Hamiltonian in Eq. (3), without the Coulomb interaction (the last term at the right-hand side), are





ψnx ny tz (x, y, z ) = φnx (x) φny (y) φtz (z + γz ),

(4)

and (0)

Enx ny tz =

−

(2 nj + 3/2)h¯ ω0j + (tz + 1/2)h¯ ω0z − 1/2 |e|F γz ,

j = x ,y

(5)

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Fig. 1. (Color online) The binding energy of the first four confined exciton states in a spherical semiparabolic QD as a function of the confinement strength.

with nx , ny = 0, 1, . . . and

φn (k) = Nn exp(−αk2 k2 /2) Hn (αk k), k = x, y, z . (6) √ αk = µ ω0k /h¯ and tz is obtained by solving the transcendental equation Htz (αz γz ) = 0. Nn is the normalization constant and Htz (k) is the Hermite functions (when tz is integer, Htz (z ) becomes the Hermite polynomials). We have defined an equivalent radius  for each direction of space in terms of the h¯ corresponding confinement strength in the form Rk = , k = x, y, z. The hydrostatic pressure µ ω0k dependences of these radii are obtained from the fractional change of the volume of the zinc blende material [7]. Based on the density-matrix approach and the perturbation expansion method, the second-order NLOR coefficient is given by [2,4]

χ0(2) =

q3 σs µ201 δ01

ε0

2 ∆E 2 ,   2 (∆E − h¯ ω) + (h¯ Γ0 )2 (∆E + h¯ ω)2 + (h¯ Γ0 )2

(7)

where σs is the electron density in the QD, ε0 is the vacuum permittivity, Γ0 = 1/τ is the relaxation rate for states 1 and 2, and h¯ ω is the photon energy. µ01 = ⟨Ψ0 |z |Ψ1 ⟩ is the electric dipole moment of the transition from the ground state, Ψ0 , to the first excited state, Ψ1 , and δ01 = |⟨Ψ1 |z |Ψ1 ⟩ − ⟨Ψ0 |z |Ψ0 ⟩|. ∆E is the absorption energy from Ψ0 to Ψ1 . In this work, the relaxation rate is τ = 0.14 ps and the electron density is taken as σs = 5.0 × 1024 m−3 . 3. Results and discussions Fig. 1 shows the calculated binding energies of the four lowest confined exciton states in a spherical semiparabolic QD as a function of the confinement strength (bottom scale) and as a function of the radius of the QD (top scale). These curves are obtained by exact diagonalization of the last term on the right-hand side in Eq. (3). The exciton binding energy is an increasing function of the confinement strength, or equivalently a decreasing function of the radius of the QD [8]. This is because of the diminution in the expectation value of the electron–hole distance and consequently the increase in Coulomb interaction. In Fig. 2, we present our results for the NLOR in a ellipsoidal GaAs QD as a function of the energy of the incident photon and for several values of the confinement strength [Fig. 2(a)], several values

C.M. Duque et al. / Superlattices and Microstructures 49 (2011) 264–268

a

b

267

c

(2)

Fig. 2. (Color online) The second-order NLOR coefficient χ0 as a function of the incident photon energy h¯ ω for a confined electron–hole pair in a ellipsoidal semiparabolic QD. Several values of (a) the confinement strength along the z-direction, (b) the applied electric field, and (c) the hydrostatic pressure have been considered.

of the applied electric field [Fig. 2(b)], and several values of the hydrostatic pressure [Fig. 2(c)]. From the analysis of Eq. (7), we can see that the function in the second factor on the right-hand side has two Lorentzian structures centered at h¯ ω = ± ∆E = ±(E2 − E1 ). Note that if ∆E increases, then there must be a blue shift of the right-hand resonant peak in the NLOR (h¯ ω = +∆E), whereas if ∆E decrease, such a resonant peak must be red shifted. When Rz increases there is a diminution in the confinement of the carriers and ∆E decreases; this condition corresponds to a red shift in the resonant peak, as can be observed in Fig. 2(a). Additionally, when Rz increases, µ201 and δ01 also increase; µ201

√

follows strictly a linear behavior and δ01 increases as R. These two combined effects are responsible for the increase in the intensity of the resonant peak of the NLOR when the strength confinement diminishes, as can be observed in Fig. 2(a). When F increases, ∆E also increases, and this is observed in the blue shift of the resonant peak of the NLOR, as can be seen in Fig. 2(b). Notice that for negative values of the applied electric field (dashed line in Fig. 2(b)) there is a red shift of the resonant peak, which is strictly associated with a decrease of ∆E. When the applied electric field increases, there is a quasi-linear diminution in µ201 . Additionally, if the electric field increases, δ01 grows with a quadratic behavior up to a maximum and then decreases. These two effects on µ201 and δ01 are combined and contribute to the decrease in the intensity of the resonant peak of the NLOR as a function of the applied electric field, such as is depicted in Fig. 2(b). The hydrostatic pressure effects have been incorporated in our model via the hydrostatic pressure dependences of the electron and hole effective masses, the dielectric constant, and the radius of the dot. With P given in kbar, we have used me = (0.067 + 5.56 × 10−4 P )m0 , mh = (0.34 − 1.0 × 10−4 P ) m0 , ε = 12.6 − 0.02 P, and Rk = Rk (0) (1 − 6.3 × 10−4 P ). Here m0 is the free electron mass and Rk (0) is the radius at P = 0. Consequently, the electron and hole effective masses in GaAs are, respectively, an increasing and a decreasing function of the hydrostatic pressure [7]. Hence, the reduced mass of the electron–hole pair is also an increasing function of the hydrostatic pressure. At the same time, both δ01 and µ201 are independent of the hydrostatic pressure and, for this reason, the intensity of the NLOR resonant peak is also independent of the hydrostatic pressure, as is shown in Fig. 2(c). However, ωj is hydrostatic pressure dependent and the pressure-induced increase/decrease in its value will result in a blue shift/red shift of the mentioned resonant peak. If P increases, then ∆E decreases and consequently a red shift in the resonant peak of the NLOR must be observed; see Fig. 2(c). 4. Conclusions The exciton binding energy and the NLOR in a semiparabolic QD is studied under the densitymatrix formalism and the effective-mass approximation. The study has considered the simultaneous effects of applied electric field and hydrostatic pressure for several values of the confinement strength. The main findings can be summarized as follows: (1) the exciton binding energy is an increasing

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function of the confinement strength, (2) the resonant peak of the NLOR is blue shifted with the increase of the confinement strength, and (3) the resonant peak of the NLOR is blue shifted/red shifted with the increase of the applied electric field/hydrostatic pressure. Finally, we would like to stress that only with an appropriate choosing of the set-up for the applied electric field (strength and direction) is it possible to tune the red shift or the blue shift of the resonant peak of the NLOR, which can provide information about the energy difference between the two first exciton states in a spherical QD. Acknowledgements The authors wish to thank Mexican CONACYT and Colombian COLCIENCIAS for support under 2008 bilateral Grant ‘‘Estudio de propiedades ópticas y electrónicas en nanoestructuras y sistemas semiconductores de baja dimensión’’. CAD is also grateful to CODI-Universidad de Antioquia (Estrategia de Sostenibilidad Grupo de Materia Condensada-UdeA, 2009–2010), Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (CAD-exclusive dedication project 2009–2010), and the Excellence Center for Novel Materials, and COLCIENCIAS, under Contract No 043-2005, for partial financial support. References [1] [2] [3] [4] [5] [6] [7] [8]

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