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Nonlinear intersubband transitions in a parabolic and an inverse parabolic quantum well under applied magnetic field
Journal of Luminescence 145 (2014) 387–392
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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
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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].
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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
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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.
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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
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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].
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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
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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.
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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
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