- Email: [email protected]

Contents lists available at ScienceDirect

Journal of Luminescence journal homepage: www.elsevier.com/locate/jlumin

Nonlinear intersubband transitions in a parabolic and an inverse parabolic quantum well under applied magnetic ﬁeld Emine Ozturk a,n, Ismail Sokmen b a b

Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey Department of Physics, Dokuz Eylül University, 35160 Buca, Izmir, Turkey

art ic l e i nf o

a b s t r a c t

Article history: Received 29 November 2012 Received in revised form 15 July 2013 Accepted 2 August 2013 Available online 12 August 2013

In this study, both the intersubband optical absorption coefﬁcients and the refractive index changes as dependent on the magnetic ﬁeld are calculated in a parabolic (PQW) and an inverse parabolic quantum well (IPQW) for different Al concentrations at the well center. Our results show that the position and the magnitude of the linear, nonlinear and total absorption coefﬁcients and refractive index changes depend on the magnetic ﬁeld strength and the Al concentration at the well center. By decreasing the Al concentration at the well center we can obtain a red shift and a blue shift in the intersubband optical transitions for PQWs and IPQWs, respectively. Thus, the absorption coefﬁcients and the refractive index changes which can be suitable for great performance optical modulators and multiple infrared optical device applications can be easily obtained by tuning the applicable magnetic ﬁeld value and the Al concentration at the well center. & 2013 Published by Elsevier B.V.

Keywords: Parabolic and inverse parabolic quantum well Optical absorption coefﬁcient Refractive index changes Linear and nonlinear transitions Magnetic ﬁeld

1. Introduction Optical and transport properties have a strong relation with the degree localization of the electronic states in the materials. In lowdimensional semiconductor structures, such as quantum wells (QWs), quantum-well wires (QWWs), and quantum dots (QDs), the electronic localization can be changed by external ﬁelds, which are frequently used in applications to opto-electronic devices. Intersubband transition in quantum wells (QWs) has unique properties such as a large dipole moment, an ultra-fast relaxation time, and a large tunability of transition wavelength. Because of the possibility for novel devices, the optical properties of the quasi-two dimensional electron gas (2D) in a semiconductor structure has been investigated both theoretically and experimentally, and many new GaAs/Ga1 xAlxAs quantum well photodetectors based on intersubband absorption has been proposed to replace the conventional detectors [1–5]. Recent improvements in the semiconductor growth techniques have made it possible to prepare low-dimensional semiconductor structures with any desired potential shape. One of those structures is the so-called inverse parabolic quantum well (IPQW), an IPQW was fabricated experimentally by Chen et al. [6]. These structures open a new

n

Corresponding author. Tel.: +90 3462191010. E-mail addresses: [email protected], [email protected] (E. Ozturk). 0022-2313/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.jlumin.2013.08.011

ﬁeld in fundamental physics, and also offer a wide range of potential applications for semiconductor optoelectronics devices [7–11]. Some of these applications include semiconductor lasers [7], single-electron transistors [8], quantum computing [9], photodetectors [10], and high-speed electro-optical modulators [11]. Nonlinear optical properties related to the intersubband transitions within the conduction band of QWs have attracted considerable attention due to the strong quantum-conﬁnement effect, large values of dipole transition matrix elements and possibility of achieving resonance conditions. Therefore, the linear and nonlinear optical properties within the conduction band of a GaAs quantum well (QW) have been intensively studied both theoretically and experimentally [12–34]. A number of device applications based on the intersubband transition, for example, far-infrared photo-detectors [15–19], electro-optical modulators [20], all optical switch [21], and infrared lasers [22,23], have been proposed and realized. It is well known that in a semiconductor quantum well two types of optical transitions, interband and intersubband, may occur. If the quantum well structure is subjected to a magnetic ﬁeld, these transitions become very interesting. The magnetic ﬁeld is an important additional parameter, since it can be applied experimentally in a well-controlled way and modiﬁes fundamentally the electronic structure. Parallel magnetic ﬁeld has a small quantitative impact on the quantum well's energy spectrum. If a magnetic ﬁeld perpendicular to the quantum well plane is applied, however energy spectrum is changed considerably [35–37].

388

E. Ozturk, I. Sokmen / Journal of Luminescence 145 (2014) 387–392

This fact directly inﬂuences the nature of electronic and optical properties in these structures. As different from our previous studies [34,36], in this paper we investigate the linear and nonlinear optical absorptions and the refractive index change associated with intersubband transitions within the conduction band for parabolic quantum well structures (PQWs) and inverse parabolic quantum well structures (IPQWs) for different Al concentrations at the well center under the magnetic ﬁeld which is applied perpendicular to the growth direction. To our knowledge, this is the ﬁrst study comparing the nonlinear optical properties in PQWs and IPQWs for different magnetic ﬁeld values and different Al concentrations at the well center. SPQW denominates semi-parabolic quantum wells and ISPQW refers to inverse semi-parabolic quantum wells. Karabulut et al. have investigated the nonlinear optical rectiﬁcation and optical absorption in asymmetric double quantum well (ADQW) under the inﬂuence of applied electric and magnetic ﬁelds as well as hydrostatic pressure [38]. They have inferred that the nonlinear optical rectiﬁcation (NOR) would become zero for high enough magnetic ﬁeld values. In this study, our results were given mostly for B ¼0 and B¼ 20 T and these magnetic ﬁeld values are suitable for the optical properties of which are based on the electric dipole moment approximation. Nevertheless, we theoretically calculated for high value B ¼ 40 T to better illustrate the effect of the magnetic ﬁeld.

2. Theory Under the effective mass approximation, the energy levels and the wave functions for electrons in a system can be obtained by solving the Schrödinger equation with a proper Hamiltonian. The one-dimensional Schrödinger equation in the presence of magnetic ﬁeld B applied perpendicular to the growth direction [35–37] is given by ! 2 ℏ2 d e2 B2 2 þ VðzÞ þ z ψðzÞ ¼ E ψðzÞ ð1Þ 2 mn dz2 2mn c2 where mn is the electron effective mass, e is the electron charge. To solve the Schrödinger equation in the z-direction we take as base the eigenfunctions of the inﬁnite potential well with the L width. These bases are formed as rﬃﬃﬃ hn πz i 2 cos δn ð2Þ ψ n ðzÞ ¼ L L where 8 <0 δn ¼ π :2

rﬃﬃﬃﬃ μ I jM 21 j2 n βð3Þ ðω; IÞ ¼ ω εr 2ε0 nr c

e4 sv ℏ=τin 2 o2 ðE2 E1 ℏωÞ2 þ ℏ=τin n o3 2 2 jM 22 M 11 j2 ðE2 E1 ℏωÞ2 ℏ=τin þ 2ðE2 E1 ÞðE2 E1 ℏωÞ 5 44jM 21 j2 2 ðE2 E1 Þ2 þ ℏ=τin

ð6Þ and the linear and the third-order nonlinear refractive index changes can be expressed as [37,39], respectively. ð1Þ Δn ðωÞ sv e2 jM 21 j2 ðE2 E1 ℏωÞ ð7Þ ¼ nr 2n2r ε0 ðE2 E1 ℏωÞ2 þ ℏ=τin 2 ð3Þ Δn ðω; IÞ μc e4 sv I ¼ 3 jM 21 j2 2 nr 4nr ε0 ½ðE2 E1 ℏωÞ2 þ ℏ=τin 2 " ðM 22 M 11 Þ2 4ðE2 E1 ℏωÞjM 21 j2 2 ðE2 E1 Þ2 þ ℏ=τin h 2 i ðE2 E1 ℏωÞ ðE2 E1 ÞðE2 E1 ℏωÞ ℏ=τin 2 ℏ=τin ½2ðE2 E1 Þℏω

# ð8Þ

Here I is the optical intensity of incident electromagnetic wave (with the angular frequency ω) that excites the structure and leads to the intersubband optical transition, μ is the permeability, ω is the angular frequency of the incident photon, E1 and E2 denote the quantized energy levels for the initial and ﬁnal states, respectively, c is the speed of light in free space, εr is the real part of the permittivity, nr is the refractive index, sv is the carrier density,τin is the intersubband relaxation time, and the dipole moment matrix elements per electronic charge is deﬁned by Z Mf i ¼ ϕnf ðzÞ z ϕi ðzÞdz; ði; f ¼ 1; 2Þ ð9Þ The total absorption coefﬁcient is given by

if n is odd

βðω; IÞ ¼ βð1Þ ðωÞ þ βð3Þ ðω; IÞ

if n is even

ψðzÞ ¼ ∑ cn ψ n ðzÞ

ð10Þ

and the total refraction index change can be written as

and the solution in the z-direction are described by n¼1

V 0 is 228 meV for xmax ¼0.3, V 0 =g and V 0 =b are the maximum value of potential at the center of PQWs and IPQWs, respectively. After the energies and their corresponding wave functions are obtained, the ﬁrst-order linear absorption coefﬁcient βð1Þ ðωÞ and the third-order nonlinear absorption coefﬁcient βð3Þ ðω; IÞ for the intersubband transitions between two subbands can be clearly calculated as [36,37,39,40] rﬃﬃﬃﬃ μ e2 sv ℏ=τin jM 21 j2 βð1Þ ðωÞ ¼ ω ð5Þ 2 εr ðE2 E1 ℏωÞ2 þ ℏ=τin

ð3Þ

This wave function is localized in the well region, and where cn is the coefﬁcient of expansion. V(z) is the conﬁnement potential for the electron in z-direction, is taken as [35,37] 8 2 > V0 z > jzj rL=2 for PQW > g L=2 > > < 2 ð4Þ VðzÞ ¼ V 0 1 z jzj rL=2 for IPQW > L=2 b > > > > : V0 elsewhere with V 0 being the discontinuity in the conduction band edge, L the well widths. Where g and b are xmax =xC (xmax is the Al concentration at the barriers, xC is the Al concentration at the well center),

Δnðω; IÞ Δnð1Þ ðωÞ Δnð3Þ ðω; IÞ ¼ þ : nr nr nr

ð11Þ

3. Results and discussion We have theoretically investigated the linear and nonlinear intersubband optical properties for the (1–2) transition in PQWs and IPQWs under an applied magnetic ﬁeld. In this study, for numerical calculations, we have taken m* ¼0.0665 m0 (where m0 is the free electron mass), the barrier height V0 ¼228 meV (for xmax ¼ 0.3), the well widths L ¼90 Å, τin ¼ 0:14 ps; sv ¼ 3 1016 cm3 , nr ¼3.2, and I ¼0.5 MW/cm2. In Fig. 1(a, b, c and d) for PQW (g¼ 1), SPQW (g ¼2), IPQW (b¼1) and ISPQW (b¼2), respectively, we show the change of the potential proﬁles and the energies with their squared envelope

E. Ozturk, I. Sokmen / Journal of Luminescence 145 (2014) 387–392

400

400

B=0 B=20T

B=0 B=20T

g=1

g=2

300

i=2

200

100

E i (meV)

300

E i (meV)

389

200

i=2

100

i=1

i=1

0 -200

-100

0

100

0 -200

200

-100

400

400

B=0 B=20T

100

200

B=0 B=20T

b=1

b=2

300

i=2 i=1

200

100

E i (meV)

300

E i (meV)

0

z( )

z( )

200

i=2 i=1

100

0 -200

-100

0

100

200

0 -200

-100

z( )

0

100

200

z( )

Fig. 1. The change of the potential proﬁles (thick solid curves) and the energies with their squared envelope wave functions of the ground subband and the ﬁrst excited subband for different magnetic ﬁeld values for (a) PQW (g ¼ 1), (b) SPQW (g ¼2), (c) IPQW (b ¼ 1) and (d) ISPQW (b¼ 2).

Table 1 The probabilities of ﬁnding in quantum wells of the electrons, the resonance frequency values and the resonant peak values of the total absorption coefﬁcients as dependent on the magnetic ﬁeld values for different PQWs and IPQWs.

180 g=1

E2 - E1 (meV)

150 120

The probabilities of ﬁnding of the electrons

g=2

2 ψ 1

2 ψ 2

g¼1 g¼2 b ¼1 b ¼2

0.969 0.961 0.731 0.892

0.561 0.704 0.430 0.630

B ¼20 T g ¼ 1 g¼2 b ¼1 b ¼2

0.973 0.966 0.788 0.907

0.689 0.754 0.549 0.686

90 60

b=2

30

b=1

B ¼0

0

0

10

20

30

40

B (T)

Resonance Resonant peak values frequency of the total absorption ( 1014 s 1) coefﬁcients (cm 1) I ¼0

I¼0.5 MW/cm2

1.99 1.79 0.46 0.99

2917 3293 2979 3238

2328 2459 – 1778

2.14 1.89 0.59 1.09

3208 3333 2925 3237

2546 2523 – 1906

Fig. 2. Energy difference between the ground and second subbands versus the magnetic ﬁeld for different PQWs and IPQWs.

wave functions of the ground (i¼ 1) and the second subband (i¼2) under a magnetic ﬁeld (the dashed curves are for B ¼0). By increasing the magnetic ﬁeld the geometric conﬁnement of the electron increases thus it gets to be more energetic and it can penetrate into the potential barriers easily. This penetration modiﬁes the subband dispersion relations and causes a variation in the overlap function between the ground and second subband. As seen from Fig. 1, for PQW structures the energy differences between E2 E1 are greater than IPQWs. The variation of these energy differences for both PQWs and IPQWs increases with the

magnetic ﬁeld (see Fig. 2), the largest increase is for PQW. By decreasing the xC concentration, while the energy difference between E2 E1 increases for IPQWs, these differences decrease for PQWs. In the high xC concentrations, IPQWs behave like double quantum well having a parabolic potential barrier. The probabilities of ﬁnding of the electrons as dependent on the magnetic ﬁeld and the xC concentration values in the PQWs and IPQWs are given in Table 1. Thus, we suggest that the magnetic ﬁeld and the xC concentration affect the conﬁnement and localization. By considering the variation of the energy difference E2 E1 in

390

E. Ozturk, I. Sokmen / Journal of Luminescence 145 (2014) 387–392

Fig. 2, it should point out that by changing xC concentration we can obtain a blue shift or a red shift in the intersubband optical transitions. For different PQWs and IPQWs, Fig. 3 shows the linear absorption coefﬁcient as a function of the incident photon energy for two different magnetic ﬁeld values. It can be seen that as the magnetic ﬁeld increases, the linear absorption coefﬁcient changes shift toward higher energies and the magnitude of these absorption coefﬁcients increases. The main reason for this resonance shift is the increasing energy difference with increasing magnetic ﬁeld. Also, by decreasing the xC concentration, while the magnitudes of the linear absorption coefﬁcients increase for all PQWs, the photon energy values further increases and decreases for IPQW and PQW, respectively. These results are in good agreement with Refs. [37,41]. In Fig. 4, the linear refractive index changes as a function of the incident photon energy for two different magnetic ﬁeld values are shown for different PQWs and IPQWs. As seen in this ﬁgure, as the magnetic ﬁeld increases, the magnitude of the linear refractive index change decreases and also shifts toward higher energies. The main reason for this behavior is that the quantum conﬁnement varies with the increasing magnetic ﬁeld value. The quantum conﬁnement variation causes to the increment of the energy

B=0 B=40T

4000

( cm -1)

g=2

g=1

b=1

2000

1000

0 0.00

0.04

0.08

0.12

0.16

0.20

Absorption coefficient ( cm-1 )

b=2

3000

(1)

difference between the lowest two energy levels, where an optical transitions occurs. The linear refractive index change depends strongly on the Al concentration at the well center. By increasing the xC concentration, while the magnitudes of the linear refractive index change decrease for PQWs, these magnitudes increases for IPQW. Respectively, for B ¼20 T and B ¼40 T, Fig. 5a and b shows the variation of the linear, total and nonlinear absorption coefﬁcients as a function of the incident photon energy for different PQWs and IPQWs. This ﬁgure clearly shows that the peak position of the absorption coefﬁcients is sensitive to xC concentration. With decreasing xC concentration, the absorption maximum shifts towards higher (lower) energies for IPQWs (PQWs). Since the energy difference between the ground state and the ﬁrst excited state increases (decreases) for IPQWs (PQWs) as the xC concentration decreases, the peak position of absorption coefﬁcients changes. All absorption coefﬁcients have a prominent peak, but the large linear change generated by the βð1Þ ðωÞ term is opposite in sign of the nonlinear change generated by the βð3Þ ðω; IÞ term. Therefore, the total absorption coefﬁcient βðω; IÞ will be reduced. There is no shift at the resonant peak position with incident optical intensity. Therefore, the contribution of both the linear and the nonlinear terms should be considered in the calculation of βðω; IÞ near the resonance frequency ðE2 E1 ﬃ ℏωÞ, especially at higher intensity values. These resonance frequency and the

linear total nonlinear

4000 g=2 g=1

b=2

b=1

2000

0 I=0.5MW/cm2 B=20T

Photon energy ( eV ) -2000

Fig. 3. For different magnetic ﬁeld values the linear absorption coefﬁcient as a function of the photon energy for (1 2) intersubband transition for different PQWs and IPQWs.

0.00

0.04

0.08 0.12 0.16 Photon energy ( eV )

0.06 0.04

Absorption coefficient ( cm-1 )

B=0 B=40T b=1

0.02

b=2

n(1)/n r

g=2

0.00 g=1

-0.02 -0.04

0.20

linear total nonlinear

4000 b=1

g=2 g=1

b=2

2000

0 I=0.5MW/cm2 B=40T

-2000

-0.06 0.00

0.04

0.08

0.12

0.16

0.20

Photon energy ( eV ) Fig. 4. For different magnetic ﬁeld values the linear refractive index change as a function of the photon energy for (1 2) intersubband transition for different PQWs and IPQWs.

0.00

0.04

0.08

0.12

0.16

0.20

Photon energy ( eV ) Fig. 5. Linear, total and nonlinear absorption coefﬁcients as a function of the incident photon energy for different PQWs and IPQWs for (a) B¼20 T and (b) B ¼40 T.

E. Ozturk, I. Sokmen / Journal of Luminescence 145 (2014) 387–392

resonant peak values of total absorption coefﬁcient are given in Table 1 for both B ¼ 0 and B ¼20 T. The resonant peak of total absorption coefﬁcient can be bleached at sufﬁciently high incident optical intensities and this resonant peak is signiﬁcantly split up into two peaks due to the strong bleaching effect. In our study, for b¼ 1, the bleaching begins approximately at the incident optical intensity value of I ¼0.3 MW/cm2. Also it was seen that, the maximum value of linear, nonlinear and total absorption coefﬁcients depend on the xC concentration and the magnetic ﬁeld. With the decrease of the xC concentration, the nonlinear coefﬁcient decreases (increases) for IPQWs (PQWs), thus the total absorption coefﬁcients change in the magnitude. In our calculations, the bleaching intensities and the magnitude of the absorption coefﬁcient depend on the magnetic ﬁeld strength and the xC concentration in PQWs and IPQWs. Such a dependence of the exciting optical intensity on the external ﬁeld strengths and the xC concentration in PQWs and IPQWs can be very useful for several potential device applications. To show clearly the xC concentration effect on the linear, nonlinear and total refractive index changes in PQWs and IPQWs, in Fig. 6a and b we plot the variation of the refractive index changes as a function of the photon energy for (1–2) intersubband transition. As seen in Fig. 6, the refractive index changes depend strongly on the xC concentration. By increasing the xC concentration, while the magnitudes of the refractive index change decrease for PQWs, these magnitudes increases for IPQW. The total

refractive index change is signiﬁcantly reduced with increasing optical intensity by the negative nonlinear refractive index change contribution. In Fig. 7, we plot the total absorption coefﬁcient βðω; IÞ as a function of the photon energy for different PQWs, without and with applied magnetic ﬁeld. It is clear that as the magnetic ﬁeld increases, the total absorption coefﬁcients increase and also shift toward higher energies. As known, increasing the magnetic ﬁeld causes to a stronger conﬁnement of carriers and this case results in the blue shift of the absorption spectrum. By increasing the xC concentration, the magnitude of total absorption coefﬁcient change for all PQWs. By considering the variation of the energy difference, it should point out that by decreasing xC concentration we can obtain a red shift and a blue shift in the intersubband optical transitions for PQWs and IPQWs, respectively. These results are good qualitatively agreement with previous study [37]. Fig. 8 shows the variation of the total refractive index changes as a function of the incident photon energy for different xC concentrations. This ﬁgure clearly shows that the total refractive index change is related on the xC concentration. As the xC concentration decreases, the peak values of the total refractive index changes decrease (increase) for IPQWs (PQWs) and also the corresponding peak positions shift toward higher (lower) energies for IPQWs (PQWs). This shift in the peak position of the total refractive index change depends on the variation in the energy

4000 linear total nonlinear

( cm-1 )

Refractive index change

3000

0.01

g=2

2000

g=1

b=2

0.00 b=1

1000 -0.01 g=1

0

g=2

0.00

0.04

0.08

0.12

0.16

0.20

Photon energy ( eV )

-0.02 0.08

0.12

0.16

0.20

Photon energy ( eV )

Fig. 7. For different magnetic ﬁeld values the total absorption coefﬁcient as a function of the incident photon energy for different PQWs and IPQWs.

0.06

0.06 linear total nonlinear

I=0.5MW/cm 2 B=20T

0.04

B=40T

b=1

0.02

0.02 0.00

B=0

I=0.5MW/cm 2

0.04

n /nr

Refractive index change

B=0 B=40T

I=0.5MW/cm 2

0.02 I=0.5MW/cm 2 B=20T

391

b=2

g=2

0.00 g=1

-0.02

-0.02 b=2

-0.04

-0.04

b=1

-0.06

-0.06 0.00

0.04

0.08

0.12

Photon energy ( eV ) Fig. 6. Linear, total and nonlinear refractive index change as a function of the incident photon energy for (a) different PQWs and (b) different IPQWs.

0.00

0.04

0.08

0.12

0.16

0.20

Photon energy ( eV ) Fig. 8. For different magnetic ﬁeld values the total refractive index change as a function of the incident photon energy for different PQWs and IPQWs.

392

E. Ozturk, I. Sokmen / Journal of Luminescence 145 (2014) 387–392

magnetic ﬁeld and the xC concentration. The sensitivity to the quantum well shape of the intersubband transitions under the feasible magnetic ﬁeld can be used in various potential device applications. On the other hand, for intersubband transitions in different parabolic and inverse parabolic quantum wells with an applied magnetic ﬁeld we can obtain blue or red shift by tune the Al concentration at the well center.

b=1 12

2

-14

M 21 (x10 cm2 )

16

b=2

8

References

g=2 4 g=1 0 0

10

20 B (T)

30

40

Fig. 9. The squared dipole matrix elements per electronic charge as a function of the applied magnetic ﬁeld.

difference between successive energy levels which an optical transition occurs. Also, as the magnetic ﬁeld increases, the magnitudes of the total refractive index change decrease and also shift toward higher energies. The total refractive index changes of PQWs and IPQWs are changed in energy and magnitude with changing magnetic ﬁeld and the xC concentration. We show the squared dipole matrix elements per electronic charge as a function of the applied magnetic ﬁeld in Fig. 9. This ﬁgure shows that jM 21 j2 depends on the magnetic ﬁeld for IPQWs and this dependence is evident for IPQW(b¼1). The change of jM 21 j2 as dependent on the magnetic ﬁeld is almost constant for PQWs. 4. Conclusions In this study, the effect of applied magnetic ﬁeld on the linear and nonlinear intersubband absorption coefﬁcient in different parabolic and inverse parabolic quantum wells for different Al concentrations at the well center has been investigated within the framework of the effective mass approximation. The linear, nonlinear and total absorption and refractive index changes depending on the Al concentration at the well center are investigated as a function of the incident photon energy for the different magnetic ﬁelds. We have shown that by increasing xC concentration, the effective potential well shapes modify which changes the spacement of the energy levels in the quantum wells, so that this mechanism can be used as a way to control the carrier conﬁnement in these quantum well structures. We can observe that the increasing magnetic ﬁeld strength changes the separation between subbands, thus the energy differences and the linear and nonlinear absorption peaks change in magnitude and position as magnetic ﬁeld value increases. Thus, the modulation of the absorption coefﬁcients which can be suitable for good performance optical modulators and various infrared optical device applications can be easy obtained by tuning the applicable

[1] D.D. Coon, R.P.G. Karunasiri, Appl. Phys. Lett. 45 (1984) 649. [2] B.F. Levine, C.G. Bethea, K.K. Choi, J. Walker, R.J. Malik, Appl. Phys. Lett. 53 (1988) 231. [3] Y. Huang, C. Lien, Tan-Fu Lei, J. Appl. Phys. 74 (1993) 2598. [4] D. Ahn, S.L. Chuang, J. Appl. Phys. 62 (1987) 3052. [5] L.N. Pandey, T.F. George, Appl. Phys. Lett. 61 (1992) 1081. [6] W.Q. Chen, S.M. Wang, T.G. Andersson, J.T. Thordson, Phys. Rev. B 48 (1993) 14264. [7] N. Kristaedter, O.G. Schmidt, N.N. Ledentsov, D. Bimberg, V.M. Ustinov, A. Yu, A.E. Zhukov, M.V. Maximov, P.S. Kopev, Z.I. Alferov, Appl. Phys. Lett. 69 (1996) 1226. [8] E. Leobandung, L. Guo, S. Chou, Appl. Phys. Lett. 67 (1995) 2338. [9] D. Loss, D.P. Divicenzo, Phys. Rev. A 57 (1998) 120. [10] X. Jiang, S.S. Li, M.Z. Tidrow, Phys. E 5 (1999) 27. [11] S.Y. Yuen, Appl. Phys. Lett. 43 (1983) 813. [12] R. Dingle, W. Wiegman, C.H. Henry, Phys. Rev. Lett. 33 (1974) 827. [13] A. Harwitt, J.S. Harris Jr., Appl. Phys. Lett. 50 (1987) 685. [14] A. Fenigstein, A. Fraenkel, E. Finkman, G. Bahir, S.E. Schacham, Appl. Phys. Lett. 66 (1995) 2513. [15] L.C. West, J.J. Eglash, Appl. Phys. Lett. 46 (1985) 1156. [16] R.J. Turton, M. Jaros, Appl. Phys. Lett. 47 (1989) 1986. [17] F. Capasso, K. Mohammed, A.Y. Cho, Appl. Phys. Lett. 48 (1986) 478. [18] K.W. Gossen, S.A. Lyon, Appl. Phys. Lett. 47 (1985) 289. [19] K.K. Choi, B.F. Levine, C.G. Bethea, J. Walker, R.J. Malik, Appl. Phys. Lett 50 (1987) 1814. [20] R.P.G. Karunasiri, Y.J. Mii, K.L. Wang, IEEE Electron Device Lett. 11 (1990) 227. [21] S. Noda, T. Uemura, T. Yamashita, A. Sasaki, J. Appl. Phys. 68 (1990) 6529. [22] R.F. Kazarinov, R.A. Suris, Sov. Phys. Semicond. 5 (1971) 707. [23] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, A.Y. Cho, Science 264 (1994) 553. [24] A. Mathur, Y. Ohno, F. Matsukura, K. Ohtani, N. Akiba, T. Kuroiwa, H. Nakajima, H. Ohno, Appl. Surf. Sci. 113/114 (1997) 90. [25] B. Chen, K.X. Guo, R.Z. Wang, Z.H. Zhang, Z.L. Liu, Solid State Commun. 149 (2009) 310. [26] P.F. Yuh, K.L. Wang, J. Appl. Phys. 65 (1989) 4377. [27] M. Bedoya, A.S. Camacho, Phys. Rev. B 72 (2005) 155318. [28] I. Karabulut, H. Safak, M. Tomak, Solid State Commun. 135 (2005) 735; I. Karabulut, U. Atav, H. Safak, M. Tomak, Eur. Phys. J. B 55 (2007) 283. [29] H. Yıldırım, M. Tomak, Eur. Phys. J. B 50 (2006) 559. [30] N. Li, K.X. Guo, S. Shao, Superlattices Microstruct. 50 (2011) 461. 50 (2011) 461. [31] A. Keshavarz, M.J. Karimi, Phys. Lett. A 374 (2010) 2675; M.J. Karimi, A. Keshavarz, Superlattices Microstruct. 50 (2011) 572. [32] S. Baskoutas, C. Garoufalis, A.F. Terzis, Eur. Phys. J. B 84 (2011) 241. [33] U. Yesilgul, F. Ungan, E. Kasapoglu, H. Sari, I. Sokmen, Mod. Phys. Lett. B 25 (2011) 2451; F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. Sokmen, Opt.Commun. 285 (2012) 373. [34] E. Ozturk, I. Sokmen, Superlattices Microstruct. 52 (2012) 1010; E. Ozturk, I. Sokmen, Superlattices Microstruct. 50 (2011) 350; E. Ozturk, Eur. Phys. J. B 75 (2010) 197. [35] E. Kasapoglu, H. Sari, I. Sokmen, Chin. Phys. Lett. 21 (2004) 2500. [36] E. Ozturk, I. Sokmen, Superlattices Microstruct. 48 (2010) 312. [37] F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. Sokmen, J. Lumin. 132 (2012) 1627. [38] I. Karabulut, M.E. Mora-Ramos, C.A. Duque, J. Lumin. 131 (2011) 1502. [39] S. Unlu, I. Karabulut, H. Safak, Phys. E 33 (2006) 319. [40] D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. QE-23 (1987) 2196. [41] L. Zhang, H.J. Xie, Phys. Rev. B 68 (2003) 235315.

Copyright © 2023 C.COEK.INFO. All rights reserved.