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Simultaneous effects of laser field and hydrostatic pressure on the intersubband transitions in square and parabolic quantum wells
Physics Letters A 374 (2010) 1278–1285
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Physics Letters A www.elsevier.com/locate/pla
Simultaneous effects of laser field and hydrostatic pressure on the intersubband transitions in square and parabolic quantum wells N. Eseanu ∗ Department of Physics I, “Politehnica” University of Bucharest, 313 Splaiul Independentei, RO-060042, Bucharest, Romania
a r t i c l e
i n f o
Article history: Received 14 April 2009 Received in revised form 28 October 2009 Accepted 27 December 2009 Available online 11 January 2010 Communicated by R. Wu Keywords: Laser-dressed quantum wells Intersubband transitions Oscillator strength Hydrostatic pressure
a b s t r a c t The intersubband transitions in square and parabolic quantum wells under simultaneous action of the hydrostatic pressure and high-frequency laser field have been investigated. We found that the laser-induced blueshift effect on the subband energy levels may be tuned by the pressure action. Our calculations revealed that the oscillator strength of the transition between the ground and the first excited levels depends on the quantum well width and shape, laser field intensity and hydrostatic pressure. This combined effect of pressure and laser field offers a new degree of freedom in the optoelectronic devices applications. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The intersubband transitions (ISBTs) in quantum wells (QWs) have attracted much interest due to their unique properties such as a large dipole moment, an ultra-fast relaxation time, and a large tunability of transition wavelength [1]. These characteristics are not only important by fundamental physics point of view but novel devices applications are also expected to be tailored on that basis. In recent years, with the availability of intense THz laser sources, strongly laser-driven semiconductor heterostructures were intensively investigated [2–7]. The optical properties of an electronic system which couples strongly to a laser field, whose frequency is of the same order of magnitude as the characteristic frequencies of the system, show a nonlinear dependence on the amplitude and frequency of the laser field. Recent THz experiments revealed interesting phenomena such as resonant absorption [8,9], THz photon assisted tunnelling [10,11], and optical bistability with tunnelling-induced interference [12]. A number of novel devices applications based on ISBTs, for example: far- and near-infrared (IR) photodetectors [13,14], electrooptical modulator [15], ultrafast all-optical modulator [16], and quantum cascade lasers (QCLs) [17–19] were proposed and investigated. In the last decade a lot of studies have been performed on QWs properties under high-intensity THz laser field. Neto et al. [20] have derived the laser-dressed quantum well potential for an
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electron in a square quantum well (SQW), in the frame of a nonperturbation theory and a variational approach. A simple scheme including the effect of laser-semiconductor interaction through the renormalization of the effective mass has been proposed by Brandi and Jalbert [21]. A systematic study on the influence of two intense, long-wavelength, non-resonant laser fields on the electron levels and density of states in GaAs/AlGaAs QWs have been performed by Enders et al. [22] within a Green’s function approach. It is shown that the laser induced blueshifts depend on the laser intensity and frequency as well as on the polarization direction. The effect of intense THz radiation on the linear optical absorption spectra of semiconductor structures was theoretically studied by Johnsen et al. [23]. Ozturk et al. [24,25] have investigated the effect of the laser field on the intersubband optical transitions for a SQW and a graded QW under external electric field. These transitions are predicted to have a narrow bandwidth which reveals a practical interest as tunable semiconductor devices. Camacho et al. [26] reported the study of the intersubband optical absorption in a superlattice made of asymmetric double QWs tailored as a three level system. By applying an external electric field they have obtained the Wannier–Stark ladder and have analysed the system as an active region of a THz laser based on electric-induced resonant tunnelling. Recently, we have reported studies about the laser field effects on the energy spectra in finite V-shaped, parabolic, square, and inverse V-shaped GaAs quantum wells under an electric field [27–29]. The absorption coefficient related to the interband transitions was discussed as a function of the laser parameter, geometric shape of the wells and the applied electric field. We also have investigated the combined effect of the hydrostatic pressure and
N. Eseanu / Physics Letters A 374 (2010) 1278–1285
high-frequency laser field on the binding energy of a hydrogenic impurity in square and parabolic GaAs/Ga1−x Alx As QWs. We established that the laser field effects on the electronic properties of the system are more pronounced than those of the pressure ones [30]. In this Letter the simultaneous effects of the hydrostatic pressure and high-frequency laser field on the intersubband transitions in square and parabolic quantum wells (SQW and PQW) are investigated. We found that the subband energy levels are blueshifted with increasing laser intensity for both structures, as expected. Also, we noted a linear dependence of the ground state energy E 1 as a function of pressure. The slopes of these curves weakly depend on the laser intensity parameter. We calculated the oscillator strength of the E 1 → E 2 (ground state to first excited level) transition and we established that it depends on the electron geometric confinement in QWs, laser intensity and hydrostatic pressure. Our results for SQW revealed a critical value of the laser intensity for which the subband energy difference E = E 2 − E 1 in SQW has a maximum, in agreement 2 with other studies [24,25]. The optical matrix element M 21 (which enters the oscillator strength f 21 ) has a minimum at approximately the same critical values. The effect of the QW width on the quantities E and f 21 is also discussed. This Letter is organized as follows. In Section 2 the method used to obtain the energy levels is described. The analysis of the calculated results is presented in Section 3 and the conclusions of our work are reported in Section 4.
For a longitudinal linear polarization of the laser radiation, in the high-frequency limit, the Hamiltonian H z (α0 , p , T ) of the electron in the growth direction of the well in the impurity absence is
H z (α0 , p , T ) = −
H=
2m∗
2 P⊥
w ,b
( p, T )
+ H z (α0 , p , T ).
(1)
2
Here
P⊥ 2m∗w ,b ( p , T )
is the kinetic energy operator in the x– y plane, p is
the hydrostatic pressure in kbar, T is the temperature in kelvin. The subscripts w and b stand for the well and barrier layer materials, respectively. The application of hydrostatic pressure modifies the barrier height and effective masses m∗w ,b ( p , T ). The expression for m∗w ( p , T ) [33,34] is
m0 m∗w ( p , T )
= 1 + E τp
2 E ( p, T )
+
g
1 E ( p, T ) + Δ g
.
(2)
ergy related to the momentum matrix element, Δ0 = 0.341 eV is the spin–orbit splitting, and E g ( p , T ) is the pressure- and temperature-dependent energy gap for the GaAs QW at the point [33]. The expression for E g ( p , T ) is
E g ( p , T ) = E g (0, T ) + bp + cp
2
(3)
where E g (0, T ) = 1.519 − (5.405 × 10−4 T 2 )/( T + 204), b = 0.0126 eV/kbar, and c = 3.77 × 10−5 eV/(kbar)2 [33]. The corresponding conduction effective mass in the barrier is obtained from a linear interpolation between the GaAs and AlAs compounds [34,35], i.e.
mb∗ ( p , T ) = m∗w ( p , T ) + 0.083xm0 . Here x is the Al concentration in the layer.
(4)
h¯ 2
d2
( p , T ) dz2
+ V sd ( z, α0 , p , T )
(5)
where V sd ( z, α0 , p , T ) is the laser “dressed” confinement potential of the quantum structure. We consider the laser radiation field represented by a monochromatic plane wave of frequency Ω . The vec (t ) = tor potential associated with the radiation field is given by A A 0 cos(Ω t ), where u is the unit vector of the polarization. The u eA quantity α0 = m∗ 0Ω is the laser-dressing parameter; e and m∗w ,b w ,b
are the charge and the effective mass of the electron, respectively. For SQW the “dressed” confinement potential is given [20,27] by the expression
V sd ( z, α0 )SQW =
⎧ 0, ⎨
| z| ∈ D 1 , b−| z|
V0
arccos α , | z| ∈ D 2 , 0 ⎩ π V 0, | z| ∈ D 3 ,
(6)
where b = L /2 is the half-width of the QW, D 1 = [0, b − α0 ), D 2 = [b − α0 , b + α0 ], and D 3 = (b + α0 , +∞). For PQW the expression of the “dressed” confinement potential [27] is
V sd ( z, α0 )PQW
⎧ V0 [2z2 + α02 ], ⎪ ⎪ 2b2 ⎪ ⎪ ⎪ ⎨ V 0 + V 0 (2z2 + α 2 − 2b2 ) arccos |z|−b 0 α0 2π b 2 = ⎪ |z|−b 2
⎪ ⎪ − α0 (3| z| + b) 1 − ( α0 ) , ⎪ ⎪ ⎩ V 0,
| z| ∈ D 1 , (7)
| z| ∈ D 2 , | z| ∈ D 3 .
Here V 0 is the barrier height for the value x of the Al concentration in the layer. Under applied pressure p, at temperature T , it is given by
V 0 ( p , T ) = Q c E g (x, p , T )
(8)
where Q c = 0.6 is the conduction band offset parameter and
E g (x, p , T ) = E g (x) + D (x) p + G (x) T .
(9)
Like in Refs. [34,36] we consider E g (x) = 1.155x + 0.37x2 , D (x) = −1.3 × 10−3 x eV/kbar, and G (x) = −1.15 × 10−4 x eV/kbar. In order to calculate the wave functions we used a variational approach in which the these functions φn ( z, p , α0 ) are written as
φn ( z, p , α0 ) =
0
Here m0 is the free electron mass, E τp = 7.51 eV is the en-
2m∗
w ,b
2. Theory The method used in the present calculation is based on a nonperturbative theory that has been developed to describe the atomic behavior in intense high-frequency laser fields [31,32]. In the effective mass approximation, the Hamiltonian for an electron confined in a QW under the combined action of hydrostatic pressure and laser field is given by
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ckn ( p , α0 )χk ( z, p ).
(10)
k
Here k is extended over all bound states in the quantum well related to the subband energies in the absence of the laser field (E k0 ) [4]. The corresponding eigenvalues associated with φn ( z) may be obtained as roots of the equation
W nk − E n − E k0 δnk ck = 0
(11)
k
where W nk = χn ( z)| V sd ( z, α0 ) − V sd ( z, α0 = 0)|χk ( z) and E n ’s are the subband energy of the electron under laser field action. After the energies and their corresponding wave functions are obtained, the optical matrix element M 21 which enters the oscillator strength f 21 (also, in the linear absorption coefficient and dipole moment) for the intersubband transition E 1 → E 2 can be calculated. This matrix element is given [24] by
M 21 =
Ψ2∗ ( z) zΨ1 ( z) dz
(12)
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where Ψ1 ( z), Ψ2 ( z) are the wave functions in the E 1 , E 2 states, respectively. The expression for the oscillator strength [37,38] is
2m0 ( E 2 − E 1 )
| M 21 |2 . (13) h¯ 2 Here m0 and h¯ are the free-electron mass and reduced Planck’s f 21 =
constant, respectively. 3. Results and discussion Within the framework of effective-mass approximation, the ground and the first excited energetic levels for an electron confined in a GaAs/Ga1−x Alx As square quantum well (SQW) and parabolic quantum well (PQW), under simultaneous action of hydrostatic pressure and high-frequency laser field, for different QW widths L = 100 Å, 150 Å, and 200 Å were calculated. We considered the electron effective mass in the QW m∗w = 0.0665m0 , the Al concentration in the layer x = 0.3 and T = 4 K. It is important to discuss the relation between the laserdressing parameter α0 , laser intensity I and strength of the laser electric field F 0 . For a given laser source the following practical equations are useful [39]:
α0 ∼ = 7.31
√
I
2 5/4 r
υ ε
(14)
,
F0 ∼ = 0.87
√
I
(15)
1/4 r
ε
where υ is the frequency in THz, I in kW/cm2 , F 0 in kV/cm2 , α0 in units of a∗B (the effective Bohr radius; in bulk GaAs a∗B = 86 Å) and εr is the high-frequency laser field dielectric constant (by following Refs. [39] and [40] at T = 4 K and p = 12 kbar we obtained εr ∼ = 10.6). The value α0 = 100 Å, at T = 4 K, p = 12 kbar and υ = 1 THz, corresponds to I ∼ = 9.25 kW/cm2 and F 0 ∼ = 1.47 kW/cm2 . In order to characterize the intersubband transitions we calculated the E 1 → E 2 transition energy E = E 2 − E 1 , the square of 2 the optical matrix element M 21 , and the corresponding oscillator strength f 21 . We found that these quantities depend on the QW width and shape, laser field intensity and hydrostatic pressure. 3.1. The ground subband energy In Fig. 1 the lowest subband energy E 1 as a function of hydrostatic pressure for SQW (A, C) and PQW (B, D) with different width L under various laser field intensities (parameter α0 ) is plotted. As seen in Fig. 1 for both SQW and PQW the E 1 values are increasing as α0 increases i.e. it occurs a “blueshift effect” on the subband energy levels, as expected [24,25,30]. Due to the additional quantum confinement in PQW structure, the subband ener-
(A)
(B)
(C)
(D)
Fig. 1. The lowest subband energy E 1 as a function of hydrostatic pressure for SQW (A, C) and PQW (B, D) with different width L under various laser field intensities. Numbers 0–10 stand for α0 = 0–100 Å, respectively.
N. Eseanu / Physics Letters A 374 (2010) 1278–1285
Fig. 2. The subband energies E 1 (ground), E 2 (first excited) and E 1 → E 2 transition energy E = E 2 − E 1 as functions of hydrostatic pressure for SQW with L = 100 Å without laser field.
gies are larger than those of the SQW and the “blueshift effect” is more pronounced. The cases L = 150 Å PQW and SQW are not represented in Fig. 1. We remark that the value α0b for which the accentuated rising of the E 1 values begins is bigger for wider QWs. For SQW, the values are: α0b = 30 Å, 40 Å, and 60 Å for L = 100 Å, 150 Å, and 200 Å, respectively. For PQW, the values α0b are smaller and well-size dependent too: α0b = 20 Å, 30 Å, and 40 Å for the above mentioned L values, respectively. We note a slowly decreasing of the E 1 values with hydrostatic pressure for both SQW and PQW. This effect is determined by the pressure induced augmentation of the electron effective mass (Eqs. (2), (3)) and by the reduction of the barrier height with pressure (Eqs. (8), (9)). Generally, for SQWs the slopes of the dependencies E 1 = f ( p ), in absolute value, are enhanced by the laser intensity as for PQW they are diminuted. Also, in PQWs the energy levels are more sensitive to the hydrostatic pressure, the E 1 = f ( p ) slopes absolute values being slightly larger than in SQW case. As an example, for L = 100 Å, these slopes are: 0.21 meV/kbar (PQW) versus 0.18 meV/kbar (SQW) in the presence of laser field (α0 = 30 Å), and 0.22 meV/kbar versus 0.15 meV/kbar without laser radiation. Again, the stronger geometric confinement in PQW could be the explanation. In Fig. 2 the subband energies E 1 (ground), E 2 (first excited) and E 1 → E 2 transition energy E = E 2 − E 1 as functions of hydrostatic pressure for SQW with L = 100 Å, without laser field, are plotted. These energies linearly decrease with p; the slopes, in absolute value, are: 0.145 meV/kbar for E 1 , 0.576 meV/kbar for E 2 and 0.431 meV/kbar for E. The reason of this reduction is the same as presented for E 1 . 3.2. The E 1 → E 2 transition energy Fig. 3 presents the grouped data plot of the E 1 → E 2 transition 2 energy ( E = E 2 − E 1 ) and the square of the matrix element (M 21 ) as a function of the laser field intensity in case of SQW (A, C) and PQW (B, D), for different widths and various pressure values. As seen in Figs. 3A, C for SQWs the subband energy differences increase up to a certain value of the laser parameter, α0M , and then they begin to decrease. The critical laser parameter α0M increases for larger QWs (Table 1). The α0M values for L = 100 Å and 150 Å, in the absence of the hydrostatic pressure, are in good agreement with those reported in Refs. [24,25]. As suggested in Refs. [24,25], for α0 < α0M the energy difference E increases due to reduc-
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tion of the effective “dressed” well width. Instead, for α0 > α0M the subbands levels E 1 and E 2 tends to localize in the upper part of the laser-dressed well (which has a larger width) and thus they become closer to each other. Our calculations show a difference between the increasing rates of E 1 and E 2 values in SQWs under laser action. Therefore, E may have a maximum for a certain value α0M . This critical value of the laser intensity seems to slowly vary with the hydrostatic pressure (Table 1). For PQWs we note a different behavior i.e. E reduces monotonically when the laser parameter increases (Figs. 3B, D). The supplementary quantum confinement of the electron localization in PQW comparing with SQW could be the explanation. For both QW structures and for all the widths under investigation the values of E are reduced as the pressure increases (the arrow indicates the pressure rising). This effect can be explained by the stronger reduction of the excited level energy values E 2 with increasing pressure. By taking into account the variation of the subband energy difference E with the laser parameter α0 it clearly appears the possibility to increase or decrease the quantity E by applying an intense laser field in simultaneous action with an electric field [24,25] or with hydrostatic pressure (present work). In the last case we may refer this phenomenon as “laser- and pressure-driven optical absorption” (LPDOA). 3.3. The square optical matrix element Now we will discuss the values of the square optical matrix 2 element M 21 as a function of the intensity laser field for QW structures under study. Again, the behavior is different for SQWs and PQWs. 2 For SQWs, M 21 decreases down to a certain value of the laser parameter (α0m ) and then it begins to increase (Figs. 3A, C). These 2 critical values α0m corresponding to the minima of M 21 are very close to those for which the transition energies E = E 2 − E 1 have 2 its maxima (see Table 1). The explanation for the minimum of M 21 can be connected with the laser intensity dependent overlap of the two wave functions Ψ1 ( z) and Ψ2 ( z). In the presence of laser field the SQW shape is modified: as α0 increases the lower part of the confinement potential becomes more and more narrow while the upper part becomes wider. Therefore, the energy subbands will be pushed up to the top of the well. We may identify two regimes: (a) for α0 < α0m the second subband E 2 is localized in the upper part of the “dressed” well and the ground state E 1 is still localized in the lower part. As α0 increases, the ground state wave function Ψ1 ( z) becomes more compressed in the vicinity of z = 0 and its overlap with the first excited wave function Ψ2 (z) (which has a minimum in z = 0) reduces; (b) by further increasing α0 , for α0 > α0m , the ground energy E 1 is also pushed up to the wider upper part of the QW. Thus the carrier confinement decreases and the two wave functions Ψ1 (z) and Ψ2 (z) are spread out (or delocalized) in the potential barrier regions. As a consequence their overlapping is enhanced. The critical laser parameter α0m increases for larger QWs and seems to have a weak dependence on the hydrostatic pressure. 2 For PQWs the square matrix element M 21 increases monotonically with laser parameter (Figs. 3B, D). This effect is determined by the strong electron localization in the graded barriers. Therefore, the two wave functions Ψ1 ( z), Ψ2 ( z) are localized in the upper part of the laser-dressed QW. 2 As seen in Fig. 3 the values of M 21 decrease with hydrostatic pressure for all the situations under our study. This effect may be explained as follows: the barrier height V 0 ( p , T ) is reduced under
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N. Eseanu / Physics Letters A 374 (2010) 1278–1285
(A)
(B)
(C)
(D)
2 ) as a function of the laser field intensity in case Fig. 3. The grouped data plot of the E 1 → E 2 transition energy E = E 2 − E 1 and the square of the matrix element (M 21 of SQW (A, C) and PQW (B, D) for different widths and various pressure values. The arrow indicates the pressure rising: p = 0 (dotted lines), p = 6 kbar (dashed lines), and p = 12 kbar (solid lines).
Table 1 The laser parameter critical values L [Å] 100 150 200
2 α0M (for the maximum of E) and α0m (for the minimum of M 21 ).
( p = 0)
α0M [Å]
( p = 6 kbar)
α0M [Å]
α0M [Å]
(p = 12 kbar)
α0m [Å] (p = 0)
α0m [Å]
α0m [Å]
26.1 51.1 76
26.5 51.6 76.2
27.2 52.0 77.2
28.1 54.0 80.0
28.2 54.6 80.3
28.3 55.4 80.3
(p = 6 kbar)
(p = 12 kbar)
Table 2 The oscillator strength extreme values for the cases investigated in this work. L [Å]
f 21 min
Conditions for f 21 min
100 150 200
11.9 12.5 12.6
PQW, PQW, PQW,
α0 = 40 Å, p = 12 kbar α0 = 70 Å, p = 12 kbar α0 = 100 Å, p = 12 kbar
hydrostatic pressure action (Eqs. (8) and (9)) and consequently the two wave functions are “spread” into the barriers. 3.4. The transition oscillator strength We calculated the E 1 → E 2 transition oscillator strength f 21 which is an useful quantity in the experimental works [37]. Our results revealed that the f 21 values depends on the quantum well width and shape, laser intensity and hydrostatic pressure. In
f 21 max
Conditions for f 21 max
14.8 14.6 14.7
SQW, SQW, SQW,
α0 = 0, p = 0 α0 = 30 Å, p = 0 α0 = 50 Å, p = 0
Table 2 there are listed the f 21 extreme values for the cases under our investigation. The f 21 value for SQW, without pressure and without laser field, is in reasonable agreement with those reported in Refs. [37,38]. In Fig. 4 the variation of the E 1 → E 2 oscillator strength as a function of hydrostatic pressure for SQW (A) and PQW (B) with L = 200 Å under various intensity laser field α0 is plotted. For both QW structures and for all the width values under our investigation the oscillator strength f 21 linearly decreases with pressure. The explanation should be connected with the linear reduction of
N. Eseanu / Physics Letters A 374 (2010) 1278–1285
(A)
1283
(B)
Fig. 4. The oscillator strength f 21 as a function of hydrostatic pressure for SQW (A) and PQW (B) with L = 200 Å under various laser field intensities. Numbers 0–10 are the same as in Fig. 1.
(A)
(B)
Fig. 5. The dependence of the transition energy E = E 2 − E 1 on the QW width for different pressure values in the presence of laser field (α0 = 40 Å) and without laser (α0 = 0): (A) SQW; (B) PQW. The arrow indicates the pressure rising (see the caption of Fig. 3). 2 both the transition energy E and square matrix element M 12 as pressure increasing. Concerning the slopes of the f 21 = g ( p ) curves we may distinguish two cases. For small size QWs (not represented here) these slopes are not significantly sensitive to the laser field. Instead, for large QWs we observe a different behavior of SQWs and PQWs. For PQWs the fact that the slopes for various α0 have approximately the same value results in a linear dependence of f 21 on the laser parameter (Fig. 4B). For SQWs we observe that in the range α0 40 Å the slope is bigger than that for 70 Å α0 100 Å (Fig. 4A).
3.5. The influence of QW width on the intersubband transition E 1 → E 2 It is useful to discuss the influence of the QW width on the transition energy E and on corresponding oscillator strength. In Fig. 5 the dependence of the transition energy E = E 2 − E 1 on the QW width, for different pressure values, in the presence of laser field (α0 = 40 Å) and without laser (α0 = 0), for SQW (A) and PQW (B), respectively, is plotted. We observe that in the absence of laser radiation E reduces as L increases for both quantum struc-
tures, as expected, because the levels E 1 and E 2 become closer to each other. Under laser field action the transition energy E shows different behaviour for SQWs and PQWs. For SQWs, E decreases as L increases because under intense laser field α0 > α0M the subbands levels E 1 and E 2 tends to localize in the upper part of the laser-dressed well (as mentioned above). Instead, for PQWs our calculations reveal that the decreasing rates of E 1 and E 2 values with L are approximately the same. This fact results in a weak dependence of E on the PQW width showing a “flat” maximum around L ∼ = 145 Å. In Fig. 6 the dependence of the oscillator strength f 21 on the QW width, for different pressure values, in the presence of laser field (α0 = 40 Å) and without laser (α0 = 0), for SQW and PQW, respectively, is plotted. We observe that for QWs, in the absence of laser radiation, f 21 slowly decreases with L. For p = 0 this result is in agreement with that reported in Ref. [37]. Instead, under laser field action f 21 weakly increases for larger SQWs. For PQWs the oscillator strength is slightly enhanced by the QW size, with and without laser field. We can explain this effect as fol-
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(A)
(B)
Fig. 6. The dependence of the oscillator strength f 21 on the QW width for different pressure values in the presence of laser field (α0 = 40 Å) and without laser (α0 = 0): (A) SQW, (B) PQW. Notations a, b, c are associated with hydrostatic pressure values p = 0, 6 kbar, 12 kbar, respectively, for α0 = 40 Å. Notations a0 , b0 , c0 are the corresponding ones for α0 = 0.
lows: E decreases with L or remains almost unchanged (Fig. 5B) 2 but M 21 strongly increases (Figs. 3B, D) with L. This is because in larger QWs the energy levels E 1 and E 2 are localized in the QW lower part and the wave functions overlapping is enhanced. So, 2 f 21 ∼ E · M 21 may be enhanced by the QW width. Interesting aspects may be appear in pressure dependence of the subband populations with and without laser field action. For similar nanostructures, in the absence of laser radiation, the carrier concentrations (ntotal , n1 and n2 ) decrease with hydrostatic pressure [41]. The laser field strongly modifies the effective QW width and the subband populations depends on QW width [42]; so, important effects are expected. We intend to investigate them in a future work. 4. Conclusions Summing up, the study of the intersubband transitions in square and parabolic quantum wells (SQW and PQW) under simultaneous action of the hydrostatic pressure and high-frequency laser field, for different well widths, reveals interesting properties. We found that the laser induced “blueshift” effect on the subband energies E 1 and E 2 (ground and first excited levels) with increasing laser parameter α0 may be tuned by the pressure action. We calculated the transition energy E = E 2 − E 1 , the square 2 of the optical matrix element M 21 , and the corresponding oscillator strength f 21 . The dependence of these three quantities on the QW width and shape, laser field intensity and hydrostatic pressure is discussed. For SQW case we established a critical value of the laser intensity for which the energy difference E has a maximum and the 2 square matrix element M 12 which enters the optical absorption coefficient has a minimum. Our calculations show that the oscil2 lator strength f 21 ∼ E · M 21 linearly depends on the hydrostatic pressure and may be enhanced by the QW width. To the best of our knowledge this is the first study on the simultaneous effects of intense high-frequency laser field and hydrostatic pressure on the intersubband transitions in SQW and PQW. These effects create the possibility of novel optoelectronic devices applications. Acknowledgements The author would like to express her special thanks to Professor E.C. Niculescu for helpful discussions and constant support.
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Physics Letters A www.elsevier.com/locate/pla
Simultaneous effects of laser field and hydrostatic pressure on the intersubband transitions in square and parabolic quantum wells N. Eseanu ∗ Department of Physics I, “Politehnica” University of Bucharest, 313 Splaiul Independentei, RO-060042, Bucharest, Romania
a r t i c l e
i n f o
Article history: Received 14 April 2009 Received in revised form 28 October 2009 Accepted 27 December 2009 Available online 11 January 2010 Communicated by R. Wu Keywords: Laser-dressed quantum wells Intersubband transitions Oscillator strength Hydrostatic pressure
a b s t r a c t The intersubband transitions in square and parabolic quantum wells under simultaneous action of the hydrostatic pressure and high-frequency laser field have been investigated. We found that the laser-induced blueshift effect on the subband energy levels may be tuned by the pressure action. Our calculations revealed that the oscillator strength of the transition between the ground and the first excited levels depends on the quantum well width and shape, laser field intensity and hydrostatic pressure. This combined effect of pressure and laser field offers a new degree of freedom in the optoelectronic devices applications. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The intersubband transitions (ISBTs) in quantum wells (QWs) have attracted much interest due to their unique properties such as a large dipole moment, an ultra-fast relaxation time, and a large tunability of transition wavelength [1]. These characteristics are not only important by fundamental physics point of view but novel devices applications are also expected to be tailored on that basis. In recent years, with the availability of intense THz laser sources, strongly laser-driven semiconductor heterostructures were intensively investigated [2–7]. The optical properties of an electronic system which couples strongly to a laser field, whose frequency is of the same order of magnitude as the characteristic frequencies of the system, show a nonlinear dependence on the amplitude and frequency of the laser field. Recent THz experiments revealed interesting phenomena such as resonant absorption [8,9], THz photon assisted tunnelling [10,11], and optical bistability with tunnelling-induced interference [12]. A number of novel devices applications based on ISBTs, for example: far- and near-infrared (IR) photodetectors [13,14], electrooptical modulator [15], ultrafast all-optical modulator [16], and quantum cascade lasers (QCLs) [17–19] were proposed and investigated. In the last decade a lot of studies have been performed on QWs properties under high-intensity THz laser field. Neto et al. [20] have derived the laser-dressed quantum well potential for an
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electron in a square quantum well (SQW), in the frame of a nonperturbation theory and a variational approach. A simple scheme including the effect of laser-semiconductor interaction through the renormalization of the effective mass has been proposed by Brandi and Jalbert [21]. A systematic study on the influence of two intense, long-wavelength, non-resonant laser fields on the electron levels and density of states in GaAs/AlGaAs QWs have been performed by Enders et al. [22] within a Green’s function approach. It is shown that the laser induced blueshifts depend on the laser intensity and frequency as well as on the polarization direction. The effect of intense THz radiation on the linear optical absorption spectra of semiconductor structures was theoretically studied by Johnsen et al. [23]. Ozturk et al. [24,25] have investigated the effect of the laser field on the intersubband optical transitions for a SQW and a graded QW under external electric field. These transitions are predicted to have a narrow bandwidth which reveals a practical interest as tunable semiconductor devices. Camacho et al. [26] reported the study of the intersubband optical absorption in a superlattice made of asymmetric double QWs tailored as a three level system. By applying an external electric field they have obtained the Wannier–Stark ladder and have analysed the system as an active region of a THz laser based on electric-induced resonant tunnelling. Recently, we have reported studies about the laser field effects on the energy spectra in finite V-shaped, parabolic, square, and inverse V-shaped GaAs quantum wells under an electric field [27–29]. The absorption coefficient related to the interband transitions was discussed as a function of the laser parameter, geometric shape of the wells and the applied electric field. We also have investigated the combined effect of the hydrostatic pressure and
N. Eseanu / Physics Letters A 374 (2010) 1278–1285
high-frequency laser field on the binding energy of a hydrogenic impurity in square and parabolic GaAs/Ga1−x Alx As QWs. We established that the laser field effects on the electronic properties of the system are more pronounced than those of the pressure ones [30]. In this Letter the simultaneous effects of the hydrostatic pressure and high-frequency laser field on the intersubband transitions in square and parabolic quantum wells (SQW and PQW) are investigated. We found that the subband energy levels are blueshifted with increasing laser intensity for both structures, as expected. Also, we noted a linear dependence of the ground state energy E 1 as a function of pressure. The slopes of these curves weakly depend on the laser intensity parameter. We calculated the oscillator strength of the E 1 → E 2 (ground state to first excited level) transition and we established that it depends on the electron geometric confinement in QWs, laser intensity and hydrostatic pressure. Our results for SQW revealed a critical value of the laser intensity for which the subband energy difference E = E 2 − E 1 in SQW has a maximum, in agreement 2 with other studies [24,25]. The optical matrix element M 21 (which enters the oscillator strength f 21 ) has a minimum at approximately the same critical values. The effect of the QW width on the quantities E and f 21 is also discussed. This Letter is organized as follows. In Section 2 the method used to obtain the energy levels is described. The analysis of the calculated results is presented in Section 3 and the conclusions of our work are reported in Section 4.
For a longitudinal linear polarization of the laser radiation, in the high-frequency limit, the Hamiltonian H z (α0 , p , T ) of the electron in the growth direction of the well in the impurity absence is
H z (α0 , p , T ) = −
H=
2m∗
2 P⊥
w ,b
( p, T )
+ H z (α0 , p , T ).
(1)
2
Here
P⊥ 2m∗w ,b ( p , T )
is the kinetic energy operator in the x– y plane, p is
the hydrostatic pressure in kbar, T is the temperature in kelvin. The subscripts w and b stand for the well and barrier layer materials, respectively. The application of hydrostatic pressure modifies the barrier height and effective masses m∗w ,b ( p , T ). The expression for m∗w ( p , T ) [33,34] is
m0 m∗w ( p , T )
= 1 + E τp
2 E ( p, T )
+
g
1 E ( p, T ) + Δ g
.
(2)
ergy related to the momentum matrix element, Δ0 = 0.341 eV is the spin–orbit splitting, and E g ( p , T ) is the pressure- and temperature-dependent energy gap for the GaAs QW at the point [33]. The expression for E g ( p , T ) is
E g ( p , T ) = E g (0, T ) + bp + cp
2
(3)
where E g (0, T ) = 1.519 − (5.405 × 10−4 T 2 )/( T + 204), b = 0.0126 eV/kbar, and c = 3.77 × 10−5 eV/(kbar)2 [33]. The corresponding conduction effective mass in the barrier is obtained from a linear interpolation between the GaAs and AlAs compounds [34,35], i.e.
mb∗ ( p , T ) = m∗w ( p , T ) + 0.083xm0 . Here x is the Al concentration in the layer.
(4)
h¯ 2
d2
( p , T ) dz2
+ V sd ( z, α0 , p , T )
(5)
where V sd ( z, α0 , p , T ) is the laser “dressed” confinement potential of the quantum structure. We consider the laser radiation field represented by a monochromatic plane wave of frequency Ω . The vec (t ) = tor potential associated with the radiation field is given by A A 0 cos(Ω t ), where u is the unit vector of the polarization. The u eA quantity α0 = m∗ 0Ω is the laser-dressing parameter; e and m∗w ,b w ,b
are the charge and the effective mass of the electron, respectively. For SQW the “dressed” confinement potential is given [20,27] by the expression
V sd ( z, α0 )SQW =
⎧ 0, ⎨
| z| ∈ D 1 , b−| z|
V0
arccos α , | z| ∈ D 2 , 0 ⎩ π V 0, | z| ∈ D 3 ,
(6)
where b = L /2 is the half-width of the QW, D 1 = [0, b − α0 ), D 2 = [b − α0 , b + α0 ], and D 3 = (b + α0 , +∞). For PQW the expression of the “dressed” confinement potential [27] is
V sd ( z, α0 )PQW
⎧ V0 [2z2 + α02 ], ⎪ ⎪ 2b2 ⎪ ⎪ ⎪ ⎨ V 0 + V 0 (2z2 + α 2 − 2b2 ) arccos |z|−b 0 α0 2π b 2 = ⎪ |z|−b 2
⎪ ⎪ − α0 (3| z| + b) 1 − ( α0 ) , ⎪ ⎪ ⎩ V 0,
| z| ∈ D 1 , (7)
| z| ∈ D 2 , | z| ∈ D 3 .
Here V 0 is the barrier height for the value x of the Al concentration in the layer. Under applied pressure p, at temperature T , it is given by
V 0 ( p , T ) = Q c E g (x, p , T )
(8)
where Q c = 0.6 is the conduction band offset parameter and
E g (x, p , T ) = E g (x) + D (x) p + G (x) T .
(9)
Like in Refs. [34,36] we consider E g (x) = 1.155x + 0.37x2 , D (x) = −1.3 × 10−3 x eV/kbar, and G (x) = −1.15 × 10−4 x eV/kbar. In order to calculate the wave functions we used a variational approach in which the these functions φn ( z, p , α0 ) are written as
φn ( z, p , α0 ) =
0
Here m0 is the free electron mass, E τp = 7.51 eV is the en-
2m∗
w ,b
2. Theory The method used in the present calculation is based on a nonperturbative theory that has been developed to describe the atomic behavior in intense high-frequency laser fields [31,32]. In the effective mass approximation, the Hamiltonian for an electron confined in a QW under the combined action of hydrostatic pressure and laser field is given by
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ckn ( p , α0 )χk ( z, p ).
(10)
k
Here k is extended over all bound states in the quantum well related to the subband energies in the absence of the laser field (E k0 ) [4]. The corresponding eigenvalues associated with φn ( z) may be obtained as roots of the equation
W nk − E n − E k0 δnk ck = 0
(11)
k
where W nk = χn ( z)| V sd ( z, α0 ) − V sd ( z, α0 = 0)|χk ( z) and E n ’s are the subband energy of the electron under laser field action. After the energies and their corresponding wave functions are obtained, the optical matrix element M 21 which enters the oscillator strength f 21 (also, in the linear absorption coefficient and dipole moment) for the intersubband transition E 1 → E 2 can be calculated. This matrix element is given [24] by
M 21 =
Ψ2∗ ( z) zΨ1 ( z) dz
(12)
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N. Eseanu / Physics Letters A 374 (2010) 1278–1285
where Ψ1 ( z), Ψ2 ( z) are the wave functions in the E 1 , E 2 states, respectively. The expression for the oscillator strength [37,38] is
2m0 ( E 2 − E 1 )
| M 21 |2 . (13) h¯ 2 Here m0 and h¯ are the free-electron mass and reduced Planck’s f 21 =
constant, respectively. 3. Results and discussion Within the framework of effective-mass approximation, the ground and the first excited energetic levels for an electron confined in a GaAs/Ga1−x Alx As square quantum well (SQW) and parabolic quantum well (PQW), under simultaneous action of hydrostatic pressure and high-frequency laser field, for different QW widths L = 100 Å, 150 Å, and 200 Å were calculated. We considered the electron effective mass in the QW m∗w = 0.0665m0 , the Al concentration in the layer x = 0.3 and T = 4 K. It is important to discuss the relation between the laserdressing parameter α0 , laser intensity I and strength of the laser electric field F 0 . For a given laser source the following practical equations are useful [39]:
α0 ∼ = 7.31
√
I
2 5/4 r
υ ε
(14)
,
F0 ∼ = 0.87
√
I
(15)
1/4 r
ε
where υ is the frequency in THz, I in kW/cm2 , F 0 in kV/cm2 , α0 in units of a∗B (the effective Bohr radius; in bulk GaAs a∗B = 86 Å) and εr is the high-frequency laser field dielectric constant (by following Refs. [39] and [40] at T = 4 K and p = 12 kbar we obtained εr ∼ = 10.6). The value α0 = 100 Å, at T = 4 K, p = 12 kbar and υ = 1 THz, corresponds to I ∼ = 9.25 kW/cm2 and F 0 ∼ = 1.47 kW/cm2 . In order to characterize the intersubband transitions we calculated the E 1 → E 2 transition energy E = E 2 − E 1 , the square of 2 the optical matrix element M 21 , and the corresponding oscillator strength f 21 . We found that these quantities depend on the QW width and shape, laser field intensity and hydrostatic pressure. 3.1. The ground subband energy In Fig. 1 the lowest subband energy E 1 as a function of hydrostatic pressure for SQW (A, C) and PQW (B, D) with different width L under various laser field intensities (parameter α0 ) is plotted. As seen in Fig. 1 for both SQW and PQW the E 1 values are increasing as α0 increases i.e. it occurs a “blueshift effect” on the subband energy levels, as expected [24,25,30]. Due to the additional quantum confinement in PQW structure, the subband ener-
(A)
(B)
(C)
(D)
Fig. 1. The lowest subband energy E 1 as a function of hydrostatic pressure for SQW (A, C) and PQW (B, D) with different width L under various laser field intensities. Numbers 0–10 stand for α0 = 0–100 Å, respectively.
N. Eseanu / Physics Letters A 374 (2010) 1278–1285
Fig. 2. The subband energies E 1 (ground), E 2 (first excited) and E 1 → E 2 transition energy E = E 2 − E 1 as functions of hydrostatic pressure for SQW with L = 100 Å without laser field.
gies are larger than those of the SQW and the “blueshift effect” is more pronounced. The cases L = 150 Å PQW and SQW are not represented in Fig. 1. We remark that the value α0b for which the accentuated rising of the E 1 values begins is bigger for wider QWs. For SQW, the values are: α0b = 30 Å, 40 Å, and 60 Å for L = 100 Å, 150 Å, and 200 Å, respectively. For PQW, the values α0b are smaller and well-size dependent too: α0b = 20 Å, 30 Å, and 40 Å for the above mentioned L values, respectively. We note a slowly decreasing of the E 1 values with hydrostatic pressure for both SQW and PQW. This effect is determined by the pressure induced augmentation of the electron effective mass (Eqs. (2), (3)) and by the reduction of the barrier height with pressure (Eqs. (8), (9)). Generally, for SQWs the slopes of the dependencies E 1 = f ( p ), in absolute value, are enhanced by the laser intensity as for PQW they are diminuted. Also, in PQWs the energy levels are more sensitive to the hydrostatic pressure, the E 1 = f ( p ) slopes absolute values being slightly larger than in SQW case. As an example, for L = 100 Å, these slopes are: 0.21 meV/kbar (PQW) versus 0.18 meV/kbar (SQW) in the presence of laser field (α0 = 30 Å), and 0.22 meV/kbar versus 0.15 meV/kbar without laser radiation. Again, the stronger geometric confinement in PQW could be the explanation. In Fig. 2 the subband energies E 1 (ground), E 2 (first excited) and E 1 → E 2 transition energy E = E 2 − E 1 as functions of hydrostatic pressure for SQW with L = 100 Å, without laser field, are plotted. These energies linearly decrease with p; the slopes, in absolute value, are: 0.145 meV/kbar for E 1 , 0.576 meV/kbar for E 2 and 0.431 meV/kbar for E. The reason of this reduction is the same as presented for E 1 . 3.2. The E 1 → E 2 transition energy Fig. 3 presents the grouped data plot of the E 1 → E 2 transition 2 energy ( E = E 2 − E 1 ) and the square of the matrix element (M 21 ) as a function of the laser field intensity in case of SQW (A, C) and PQW (B, D), for different widths and various pressure values. As seen in Figs. 3A, C for SQWs the subband energy differences increase up to a certain value of the laser parameter, α0M , and then they begin to decrease. The critical laser parameter α0M increases for larger QWs (Table 1). The α0M values for L = 100 Å and 150 Å, in the absence of the hydrostatic pressure, are in good agreement with those reported in Refs. [24,25]. As suggested in Refs. [24,25], for α0 < α0M the energy difference E increases due to reduc-
1281
tion of the effective “dressed” well width. Instead, for α0 > α0M the subbands levels E 1 and E 2 tends to localize in the upper part of the laser-dressed well (which has a larger width) and thus they become closer to each other. Our calculations show a difference between the increasing rates of E 1 and E 2 values in SQWs under laser action. Therefore, E may have a maximum for a certain value α0M . This critical value of the laser intensity seems to slowly vary with the hydrostatic pressure (Table 1). For PQWs we note a different behavior i.e. E reduces monotonically when the laser parameter increases (Figs. 3B, D). The supplementary quantum confinement of the electron localization in PQW comparing with SQW could be the explanation. For both QW structures and for all the widths under investigation the values of E are reduced as the pressure increases (the arrow indicates the pressure rising). This effect can be explained by the stronger reduction of the excited level energy values E 2 with increasing pressure. By taking into account the variation of the subband energy difference E with the laser parameter α0 it clearly appears the possibility to increase or decrease the quantity E by applying an intense laser field in simultaneous action with an electric field [24,25] or with hydrostatic pressure (present work). In the last case we may refer this phenomenon as “laser- and pressure-driven optical absorption” (LPDOA). 3.3. The square optical matrix element Now we will discuss the values of the square optical matrix 2 element M 21 as a function of the intensity laser field for QW structures under study. Again, the behavior is different for SQWs and PQWs. 2 For SQWs, M 21 decreases down to a certain value of the laser parameter (α0m ) and then it begins to increase (Figs. 3A, C). These 2 critical values α0m corresponding to the minima of M 21 are very close to those for which the transition energies E = E 2 − E 1 have 2 its maxima (see Table 1). The explanation for the minimum of M 21 can be connected with the laser intensity dependent overlap of the two wave functions Ψ1 ( z) and Ψ2 ( z). In the presence of laser field the SQW shape is modified: as α0 increases the lower part of the confinement potential becomes more and more narrow while the upper part becomes wider. Therefore, the energy subbands will be pushed up to the top of the well. We may identify two regimes: (a) for α0 < α0m the second subband E 2 is localized in the upper part of the “dressed” well and the ground state E 1 is still localized in the lower part. As α0 increases, the ground state wave function Ψ1 ( z) becomes more compressed in the vicinity of z = 0 and its overlap with the first excited wave function Ψ2 (z) (which has a minimum in z = 0) reduces; (b) by further increasing α0 , for α0 > α0m , the ground energy E 1 is also pushed up to the wider upper part of the QW. Thus the carrier confinement decreases and the two wave functions Ψ1 (z) and Ψ2 (z) are spread out (or delocalized) in the potential barrier regions. As a consequence their overlapping is enhanced. The critical laser parameter α0m increases for larger QWs and seems to have a weak dependence on the hydrostatic pressure. 2 For PQWs the square matrix element M 21 increases monotonically with laser parameter (Figs. 3B, D). This effect is determined by the strong electron localization in the graded barriers. Therefore, the two wave functions Ψ1 ( z), Ψ2 ( z) are localized in the upper part of the laser-dressed QW. 2 As seen in Fig. 3 the values of M 21 decrease with hydrostatic pressure for all the situations under our study. This effect may be explained as follows: the barrier height V 0 ( p , T ) is reduced under
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N. Eseanu / Physics Letters A 374 (2010) 1278–1285
(A)
(B)
(C)
(D)
2 ) as a function of the laser field intensity in case Fig. 3. The grouped data plot of the E 1 → E 2 transition energy E = E 2 − E 1 and the square of the matrix element (M 21 of SQW (A, C) and PQW (B, D) for different widths and various pressure values. The arrow indicates the pressure rising: p = 0 (dotted lines), p = 6 kbar (dashed lines), and p = 12 kbar (solid lines).
Table 1 The laser parameter critical values L [Å] 100 150 200
2 α0M (for the maximum of E) and α0m (for the minimum of M 21 ).
( p = 0)
α0M [Å]
( p = 6 kbar)
α0M [Å]
α0M [Å]
(p = 12 kbar)
α0m [Å] (p = 0)
α0m [Å]
α0m [Å]
26.1 51.1 76
26.5 51.6 76.2
27.2 52.0 77.2
28.1 54.0 80.0
28.2 54.6 80.3
28.3 55.4 80.3
(p = 6 kbar)
(p = 12 kbar)
Table 2 The oscillator strength extreme values for the cases investigated in this work. L [Å]
f 21 min
Conditions for f 21 min
100 150 200
11.9 12.5 12.6
PQW, PQW, PQW,
α0 = 40 Å, p = 12 kbar α0 = 70 Å, p = 12 kbar α0 = 100 Å, p = 12 kbar
hydrostatic pressure action (Eqs. (8) and (9)) and consequently the two wave functions are “spread” into the barriers. 3.4. The transition oscillator strength We calculated the E 1 → E 2 transition oscillator strength f 21 which is an useful quantity in the experimental works [37]. Our results revealed that the f 21 values depends on the quantum well width and shape, laser intensity and hydrostatic pressure. In
f 21 max
Conditions for f 21 max
14.8 14.6 14.7
SQW, SQW, SQW,
α0 = 0, p = 0 α0 = 30 Å, p = 0 α0 = 50 Å, p = 0
Table 2 there are listed the f 21 extreme values for the cases under our investigation. The f 21 value for SQW, without pressure and without laser field, is in reasonable agreement with those reported in Refs. [37,38]. In Fig. 4 the variation of the E 1 → E 2 oscillator strength as a function of hydrostatic pressure for SQW (A) and PQW (B) with L = 200 Å under various intensity laser field α0 is plotted. For both QW structures and for all the width values under our investigation the oscillator strength f 21 linearly decreases with pressure. The explanation should be connected with the linear reduction of
N. Eseanu / Physics Letters A 374 (2010) 1278–1285
(A)
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(B)
Fig. 4. The oscillator strength f 21 as a function of hydrostatic pressure for SQW (A) and PQW (B) with L = 200 Å under various laser field intensities. Numbers 0–10 are the same as in Fig. 1.
(A)
(B)
Fig. 5. The dependence of the transition energy E = E 2 − E 1 on the QW width for different pressure values in the presence of laser field (α0 = 40 Å) and without laser (α0 = 0): (A) SQW; (B) PQW. The arrow indicates the pressure rising (see the caption of Fig. 3). 2 both the transition energy E and square matrix element M 12 as pressure increasing. Concerning the slopes of the f 21 = g ( p ) curves we may distinguish two cases. For small size QWs (not represented here) these slopes are not significantly sensitive to the laser field. Instead, for large QWs we observe a different behavior of SQWs and PQWs. For PQWs the fact that the slopes for various α0 have approximately the same value results in a linear dependence of f 21 on the laser parameter (Fig. 4B). For SQWs we observe that in the range α0 40 Å the slope is bigger than that for 70 Å α0 100 Å (Fig. 4A).
3.5. The influence of QW width on the intersubband transition E 1 → E 2 It is useful to discuss the influence of the QW width on the transition energy E and on corresponding oscillator strength. In Fig. 5 the dependence of the transition energy E = E 2 − E 1 on the QW width, for different pressure values, in the presence of laser field (α0 = 40 Å) and without laser (α0 = 0), for SQW (A) and PQW (B), respectively, is plotted. We observe that in the absence of laser radiation E reduces as L increases for both quantum struc-
tures, as expected, because the levels E 1 and E 2 become closer to each other. Under laser field action the transition energy E shows different behaviour for SQWs and PQWs. For SQWs, E decreases as L increases because under intense laser field α0 > α0M the subbands levels E 1 and E 2 tends to localize in the upper part of the laser-dressed well (as mentioned above). Instead, for PQWs our calculations reveal that the decreasing rates of E 1 and E 2 values with L are approximately the same. This fact results in a weak dependence of E on the PQW width showing a “flat” maximum around L ∼ = 145 Å. In Fig. 6 the dependence of the oscillator strength f 21 on the QW width, for different pressure values, in the presence of laser field (α0 = 40 Å) and without laser (α0 = 0), for SQW and PQW, respectively, is plotted. We observe that for QWs, in the absence of laser radiation, f 21 slowly decreases with L. For p = 0 this result is in agreement with that reported in Ref. [37]. Instead, under laser field action f 21 weakly increases for larger SQWs. For PQWs the oscillator strength is slightly enhanced by the QW size, with and without laser field. We can explain this effect as fol-
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(A)
(B)
Fig. 6. The dependence of the oscillator strength f 21 on the QW width for different pressure values in the presence of laser field (α0 = 40 Å) and without laser (α0 = 0): (A) SQW, (B) PQW. Notations a, b, c are associated with hydrostatic pressure values p = 0, 6 kbar, 12 kbar, respectively, for α0 = 40 Å. Notations a0 , b0 , c0 are the corresponding ones for α0 = 0.
lows: E decreases with L or remains almost unchanged (Fig. 5B) 2 but M 21 strongly increases (Figs. 3B, D) with L. This is because in larger QWs the energy levels E 1 and E 2 are localized in the QW lower part and the wave functions overlapping is enhanced. So, 2 f 21 ∼ E · M 21 may be enhanced by the QW width. Interesting aspects may be appear in pressure dependence of the subband populations with and without laser field action. For similar nanostructures, in the absence of laser radiation, the carrier concentrations (ntotal , n1 and n2 ) decrease with hydrostatic pressure [41]. The laser field strongly modifies the effective QW width and the subband populations depends on QW width [42]; so, important effects are expected. We intend to investigate them in a future work. 4. Conclusions Summing up, the study of the intersubband transitions in square and parabolic quantum wells (SQW and PQW) under simultaneous action of the hydrostatic pressure and high-frequency laser field, for different well widths, reveals interesting properties. We found that the laser induced “blueshift” effect on the subband energies E 1 and E 2 (ground and first excited levels) with increasing laser parameter α0 may be tuned by the pressure action. We calculated the transition energy E = E 2 − E 1 , the square 2 of the optical matrix element M 21 , and the corresponding oscillator strength f 21 . The dependence of these three quantities on the QW width and shape, laser field intensity and hydrostatic pressure is discussed. For SQW case we established a critical value of the laser intensity for which the energy difference E has a maximum and the 2 square matrix element M 12 which enters the optical absorption coefficient has a minimum. Our calculations show that the oscil2 lator strength f 21 ∼ E · M 21 linearly depends on the hydrostatic pressure and may be enhanced by the QW width. To the best of our knowledge this is the first study on the simultaneous effects of intense high-frequency laser field and hydrostatic pressure on the intersubband transitions in SQW and PQW. These effects create the possibility of novel optoelectronic devices applications. Acknowledgements The author would like to express her special thanks to Professor E.C. Niculescu for helpful discussions and constant support.
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