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Nonlocal effects on intersubband optical absorption in square quantum wells Guanghui Wang ∗ Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510006, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 14 July 2014 Accepted 3 August 2015 Keywords: Optical absorption Quantum well Nonlocal effect

a b s t r a c t Based on the microscopic nonlocal optical response theory, the inﬂuence from the width and depth of the potential well on the nonlocal effects and optical absorption spectra of the square quantum well is investigated in detail. The numerical results show that the radiative shift of the spectrum line, which stems from the nonlocality of optical responses, is dependent closely on the width and depth of the potential well. It is also demonstrated that the width and depth of the quantum well have important inﬂuence on the optical absorbance from the intersubband transitions. The theoretical work may be important for the growth of the nonlocal nano-materials and the potential application in optoelectronics. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction In recent years, the spatial nonlocal linear and nonlinear optical responses from the intersubband and interband transitions in the low-dimensional quantum systems and nanostructures such as semiconductor quantum wells, quantum dots and thin ﬁlms have attracted extensive interest [1–14]. The so-called spatial nonlocal optical response is that the polarization at a spatial point is induced by the applied optical ﬁelds not only at the same point, but also at other positions within the extension of the relevant electronic (or excitonic) wavefunction. The nonlocality originates from the coupling between the microscopic current-density ﬂows associated with different spatial points, which leads to the coherent extension of the resonant exciting coupled modes of the interacting radiation–matter system [1]. One of the main reasons for the interest in the spatial nonlocal optical response of the low-dimensional quantum systems is the possibility of achieving a remarkable increase in the spatial nonlocal effects due to the quantum-size effects in the quantum systems [1–3]. In 2001, Thiele et al. [4] investigated the optical absorption of a single spherical semiconductor quantum dot in an electrical ﬁeld taking into account the nonlocal coupling between the electric ﬁeld of the light and the intensity of polarization of the semiconductor, and had shown that the nonlocal effects lead to a small size and ﬁeld dependent shift and broadening of the excitonic resonance. In 2009, McMahon et al. [5] studied nonlocal optical response of metal nanostructures with arbitrary

∗ Tel.: +86 2039310083; fax: +86 2039310083. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.08.009 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

shape, and demonstrated that nonlocal optical response can lead to the blue shift of spectrum line and nonlocal effects are particularly important in structures with apex features. Recently, the nonlocal optical responses in the AlGaAs/GaAs Pöschl–Teller quantum well and InGaN/GaN strained quantum well were studied by the present authors [6,7]. Our purpose in this article is to clarify mainly the inﬂuence from the width and depth of the potential well on optical absorption spectra from the intersubband transitions in semiconductor square quantum wells based on the theory of the microscopic nonlocal optical responses. This article is organized as follows. In Section 2, a basic theory for the spatial nonlocal optical response is presented. In Section 3, the numerical results and discussion are presented. The numerical results show that the width and depth of the potential well have important inﬂuence on optical absorbance and the radiative shift of absorption spectrum line induced by the nonlocal optical response. A brief summary is given in Section 4. 2. Theory We assume a p-polarized plane-wave electromagnetic ﬁeld is incident on a square quantum well (SQW) with the potential-well depth V0 along the z direction. One of the surfaces of the SQW is located at xy plane with z = − L/2, and the other at z = L/2. Due to the system with the translational invariance in the xy plane, one can assume the electric ﬁeld E(r, ω) and the polarizability E(z; ω) exp(ik|| · r|| ) and P(r, ω) = P(r, ω) have the forms: E(r, ω) =

P(z; ω) exp(ik|| · r|| ), where k|| = (kx , ky , 0) denotes the wave vector in the xy plane. In the case of the nonlocal linear optical response,

G. Wang / Optik 126 (2015) 4042–4045

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the electric ﬁeld E(r, ω) satisﬁes the following integral-differential equation ↔

E(z; ω) = − (ω) · ↔

ω2 c2

↔

dz (z, z ; ω) · E(z ; ω),

↔

(1)

2

2 2 2 B 2 (ω) = I [ω /c − k|| + ∂ /∂z ] − (ik|| + ez ∂/∂z)(ik|| +

with

↔

ez ∂/∂z), where I denotes the unit tensor, and ez is the unit vector ↔ in the z-direction. (z, z ; ω) is the nonlocal linear susceptibility tensor. Its nonzero elements for the two-level SQW in the lowtemperature (T → 0) and long-wavelength (k|| → 0) limits can be obtained as follows [4,13] xx (z, z ; ω) =

−e2 (εF − ε1 )2

(z) (z ), 2 2 20 ω ω + ε1 − ε2 + i 0

(2)

zz (z, z ; ω) =

ε1 − εF e2 (z)(z ), 40 m∗ ω2 ω + ε1 − ε2 + i 0

(3)

by the density-matrix method.

(z) = ϕ1 (z)ϕ2 (z) and (z) = ϕ1 (z)dϕ2 (z)/dz − ϕ2 (z)dϕ1 (z)/dz, where ϕn (z) and εn (n = 1, 2) are the envelope wave functions and the transverse energies of the nth subband in the SQW, respectively [15]. e is the electron charge. m∗ is the effective mass of an electron in conduction bands. B is the relative dielectric constant for the assumed local, isotropic background medium. 0 is the vacuum permittivity. 0 is the non-radiative relaxation rate. εF = ε1 + L 2 Nd /m∗ is the Fermi energy of the SQW system, where Nd is the donor concentration, and L is the width of the SQW [3,6]. By the Green’s function method as Refs. [1] and [3], the components of the ﬁeld E(r, ω) can be solved from Eq. (1) as follows [3,6]

Ex (z; ω) = E0x (z; ω) + ˛Nx Hxx (z; ω) + ˇNz Hxz (z; ω),

(4)

Ez (z; ω) = E0z (z; ω) + ˛Nx Hzx (z; ω) + ˇNz Hzz (z; ω),

(5)

with

+∞

Nx =

(z) Ex (z; ω)dz,

(6)

(z) Ez (z; ω)dz,

(7)

−∞ +∞

Nz = −∞

+∞

Hxx (z; ω) =

Gxx (z, z ; ω) (z )dz ,

(9)

−∞ +∞

Hzx (z; ω) =

Gzx (z, z ; ω) (z )dz ,

(10)

−∞ +∞

Hzz (z; ω) =

Gzz (z, z ; ω)(z )dz ,

(11)

−∞

=

k||

c 2

2iB

ω

+

where

1 c 2

B

Nz =

Sz (1 − ˛Mxx ) + ˛Sx Mzx , (1 − ˛Mxx )(1 − ˇMzz ) − ˛ˇMxz Mzx

(16)

+∞ Mxx = −∞ Hxx (z; ω) (z)dz, Mxz = −Mzx = where +∞ +∞ Sx = H (z; ω) (z)dz, M = H (z; ω)(z)dz, xz zz zz −∞ +∞ −∞ +∞ E (z; ω) (z)dz, Sz = −∞ E0z (z; ω)(z)dz. E0x (z; ω) and −∞ 0x E0z (z; ω) are the x- and z-component of the incident ﬁeld E0 (z; ω). Letting the observation point z→ ± ∞ in Eqs. (4) and (5), the ﬁeld outside the SQW can be obtained to have the asymptotic forms as follows [3,6]

± k|| E(z) = E0 eik⊥ z + 1, 0, ∓ Ex e±ik⊥ z , z → ±∞, k⊥

ω

(13) k⊥ /k||

sgn(z − z)

sgn(z − z)

k|| /k⊥

1/2

± Ex =

c2 2iω2 B

+∞

[˛Nx k⊥ (z) ∓ ˇNz k|| (z)]e∓ik⊥ z dz.

(17)

(18)

−∞

Then the optical absorbance can be obtained by

Ex−

Ap = 1 −

2 + 2 − 1 + Ex , E0 cos E0 cos

(19)

3. Results and discussion

eik⊥ |z−z |

(14)

ı(z − z)ez ez ,

E0 (z; ω) = E0 exp(ik⊥ z) is the incident ﬁeld, k⊥ =

(B ω2 /c 2 − k||2 )

(15)

(12)

0 e2 εF − ε1 ˇ= , 4m∗ ω + ε1 − ε2 + i 0 G (z, z ; ω)

Sx (1 − ˇMzz ) + ˇSz Mxz , (1 − ˛Mxx )(1 − ˇMzz ) − ˛ˇMxz Mzx

where is the angle of incidence.

(εF − ε1 )2

0 ˛= , 22 ω + ε1 − ε2 + i 0 e2

↔

Nx =

with

+∞

Gxz (z, z ; ω)(z )dz ,

are the nonzero elements of the tensorial Green’s function ↔ G (z, z ; ω) for the SQW in the case of the p-polarized light, and sgn(z) is sign function. Multiplying both sides of Eqs. (4) and (5) by (z) and (z), respectively, and integrating the two equations over z across the SQW, one can obtain

(8)

−∞

Hxz (z; ω) =

Fig. 1. The optical absorption spectra Ap as a function of the normalized photon energy ω/ε21 with V0 = 1.5 eV for three different QW widths: L = 5 nm (solid line), L = 15 nm (dashed line), L = 20 nm (dotted line), respectively.

the wave-vector along z direction. Gij (i, j = x, z)

In this section, we will investigate the inﬂuence from the width and depth of the potential well on optical absorption spectra from the intersubband transitions in semiconductor AlGaAs/GaAs square quantum wells. Some parameters are adopted as m∗ = 0.067m0 (m0 is the mass of a free electron), B = 13.1, Nd = 1017 /cm3 , = 700 and 0 = 4.7 meV [6]. In Fig. 1, the optical absorption spectra Ap are plotted as a function of the normalized photon energy ω/ε21 with the potential barrier V0 = 1.5 eV for three different QW widths: (a) L = 5 nm (solid

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G. Wang / Optik 126 (2015) 4042–4045

Fig. 2. The radiative shift ( ω = ω/ε21 − 1) at resonance peaks versus the quantum-well width L for three different potential barriers V0 . The inset shows the QW structure with the potential barrier V0 and the QW width L.

Fig. 3. The optical absorbance Ap at resonance peaks versus the quantum-well width L for three different potential barriers V0 .

line), (b) L = 15 nm (dashed line), (c) L = 20 nm (dotted line), respectively. From Fig. 1, it can be seen easily that there exists a radiative shift at resonance-peak position for the optical absorption spectra, in which the resonance peak occurs at ω = 1.001ε21 , 1.025ε21 and 1.07ε21 for L = 5 nm, 15 nm and 20 nm, respectively. The reason for the radiative shift is due to the nonlocality of optical responses from the intersubband transitions in the square quantum wells. In order to show how the radiative shift depends on both the width L and potential barrier V0 of the SQW, we plot the radiative shift ( ω = ω/ε21 − 1) of the absorption spectrum line at resonance in Fig. 2 versus the width of the SQW for three different potential barriers: (a) V0 = 0.3 eV, (b) V0 = 1.5 eV, (c) V0 =∞ eV, which are illustrated by the solid, dashed and dotted lines, respectively. From Fig. 2, we can see that there is certainly a dramatic radiative shift, and with the increase of the SQW width, the radiative shift will increase monotonously. The inherent physics is that the electronic coherence length increases with the width of the SQW increasing, leading to the increase of the electronic nonlocality in the SQW structure. In addition, it is worth noting that with the decrease of the potential barrier height V0 , the radiative shift will increase. This is because the tunneling effect of electrons will increase when the potential barrier height V0 decreases, which induces the nonlocal effect of electrons to be much stronger due to the expansion of the wider of electronic wave-functions. In Fig. 3, the optical absorbance Ap at resonance is depicted versus the width of the SQW for three different potential barriers: (a) V0 = 0.3 eV, (b)

Fig. 4. The optical absorbance Ap versus the potential barriers V0 for three different widths of the SQW: L = 25 nm (solid line), 20 nm (dashed line), 15 nm (dotted line), respectively.

V0 = 0.5 eV, (c) V0 =∞ eV, which are illustrated by the solid, dashed and dotted lines, respectively. It is evident in Fig. 3 that the optical absorbance Ap at resonance increases monotonously with the SQW width increasing, and for the ﬁxed width of the SQW, except that it is very narrow, the deeper the potential barrier V0 , the larger the optical absorbance Ap will be. We also plot the optical absorbance Ap at resonance in Fig. 4 versus the potential barrier V0 for three different SQW widths L: (a) L = 25 nm, (b) L = 20 nm, (c) L = 15 nm, which are illustrated by the solid, dashed and dotted lines, respectively. From Fig. 4, we can see that the optical absorbance Ap at resonance decreases with the potential barrier V0 increasing. When V0 < 0.2 eV, the decrease of the optical absorbance Ap is very rapid. When V0 > 0.2 eV, however, the decrease of the optical absorbance Ap is very slow, and ﬁnally it is to be invariable with the potential barrier V0 increasing. It is also evident in Fig. 4 that the variation of the optical absorbance Ap is very small with the variation of the SQW width when V0 is less than about 0.1 eV, and the optical absorbance Ap increases with the width of the SQW increasing when V0 is larger than about 0.1 eV. These novel optical properties should be observed in future precise experiments. The study provides also a theoretical basis for the growth of the conﬁned low-dimensional quantum systems with strong nonlocal effects. 4. Summary In this paper, the inﬂuence from the width and depth of the potential well on the nonlocal effects and optical absorption spectra in the AlGaAs/GaAs square quantum well is clariﬁed. The calculated results show that the nonlocality of optical response can lead to the radiation shift of the absorption spectrum line, which is closely dependent on the QW width and depth. In addition, the wider the SQW is, the more the radiative shift of the spectrum line and the optical absorbance will be. The theoretical work may be important for the growth of the nonlocal nano-materials, and the nonlocal optical effects of low-dimensional nanostructures may have potential application in optoelectronics. Acknowledgements This work was supported by the National Natural Science Youth Foundation of China (Grant No. 60906042) and the National Natural Science Foundation of China (Grant Nos. 10974058 and 61178003).

G. Wang / Optik 126 (2015) 4042–4045

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