Optoelectronics Devices and Applications Part 8 pot

273 for diode laser Duan et al., 2010, as one of the transistor laser parents, demonstrate considerable enhancement in optical bandwidth and gain of the device when increasing the number

ξ (damping ratio)

Simulated -3dB Bandwidth(GHz)

shows that τ TL and consequently β declines Fig 13 shows this change in optical bandwidth and current gain versus displacement of QW while bias voltage, i.e v be, forces base current

to be constant at IB=33mA for WEQW=590 Å This bias enforcement does not disturb generality of the simulation results

The opposite dependence of BW and β to WEQW, is a trade-off between TL optoelectronic characteristics Other experimental and theoretical works proved the described “trade-off”

between β and f -3db as well (Then et al.,2009),(faraji et al., 2009) They also predict analytically

the above mentioned direct dependence of f -3db on τ B,spon In (Then et al., 2008) the authors utilized an auxiliary base signal to enhance the optical bandwidth As a merge of their work and the present analysis we can find the optimum place for QW that leads to better results

for both β and f -3db of a TL It means we can use both method, i.e auxiliary base signal and

QW dislocation method, simultaneously A suggestion for finding an “optimum” QW location consists of two steps First we focus on β and make it larger by locating the QW close to emitter, e.g WEQW<300 Å, which results in BW<43 GHz Then we use auxiliary AC bias signal to trade some gain for BW It should be noted that β less than unity is not generally accepted if TL is supposed to work as an electrical amplifier

Fig 12 Calculated optical cut-off frequency (f -3db ) versus τ B,spon Bandwidth is maximum for

τ B,spon corresponding to WEQW≈730Å

Where υth is the thermal velocity of carriers, Nr is the density of possible recombination sites and σ is the cross section of carrier capture σ is a measure of the region that an electron has the possibility to capture and recombine with a hole and is proportional to well width (WQW) In the other hand, Nr depends on the hole concentration, i.e NA of the base region

So we can evaluate τB as

where G is a proportionality factor defined by other geometrical properties of the base Using this equation one can extract base recombination lifetime of base minority carriers for different base doping densities Calculations exhibit an indirect relation between τB and well

width, agreeing with a larger QW width enhancing the capture cross section for electrons Moreover, the larger NA, the greater the recombination and hence the smaller τB

The results for optical frequency response based on Statz-deMars equations of section 4 can be utilized to evaluate the optical properties of a TL for varying well width Indeed, equations (13) and (14) require τB,spon not τB, as described above, therefore threshold current should be calculated for different QW widths An expression for base threshold current of TL is as below

where n 0 is minority carrier density in steady-state (under dc base current density of J 0 ), τ cap is

the electron capture time by QW (not included in charge control model for simplicity), τ qw is the QW recombination lifetime of electron and τrb0 is the bulk lifetime (or direct recombination lifetime outside the well, also ignored in our model) The base geometry factor, ν, gives the fraction of the base charge captured in the QW and defines as (Zhang & Leburton, 2009)

where Wqw is the QW width, the factor we investigate here, Wb is the base width and xqw is the QW location, similar but not equal to previously defined parameter of WEQW By setting all the constants, one can calculate τB,spon and then small-signal optical frequency response and bandwidth of TL for a range of QW widths Optimization is also possible like what we did for QW location

6 Conclusion and future prospects

An analytical simulation was performed to predict dependence of TL optoelectronic characteristics on QW position in order to find a possible optimum place for QW Simulated base recombination lifetime of HBTL for different QW positions exhibited an increase in optical bandwidth QW moved towards the collector within the base Further investigations

of optical response prove the possibility of a maximum optical bandwidth of about 54GHz

in WEQW≈730 Å Since no resonance peak occurred in optical frequency response, the bandwidth is not limited in this method In addition, the current gain decreased when QW moved in the direction of collector The above mentioned gain-bandwidth trade-off between optoelectronic parameters of TL was utilized together with other experimental methods reported previously to find a QW position for more appropriate performance The investigated transistor laser has an electrical bandwidth of more than 100GHz Thus the structure can be modified, utilizing the displacement method reported in this paper, to equalize optical and electrical cut-off frequencies as much as possible

In previous sections we consider the analysis of a single quantum well (SQW) where there is just one QW incorporated within the base region This simplifies the modelling and math-related processes In practice, SQWTL has not sufficient optical gain and may suffer thermal heating which requires additional heat sink Modifications needed to model a multiple QW transistor laser (MQWTL) First one should rewrite the rate equations of coupled carrier and photon for separate regions between wells Solving these equations and link them by applying initial conditions, i.e continuity of current and carrier concentrations, is the next step In addition to multiple capture and escape lifetime of carriers, tunnelling of the 2-dimensional carriers to the adjacent wells should be considered For wide barriers one may use carrier transport across the barriers instead the mentioned tunnelling Simulation results

273 for diode laser (Duan et al., 2010), as one of the transistor laser parents, demonstrate considerable enhancement in optical bandwidth and gain of the device when increasing the number of quantum wells (Nagarajan et al., 1992), (Bahrami and Kaatuzian, 2010) Like the well location modelled here in this chapter, there may be an optimum number of quantum wells to be incorporated within the base region Due to its high electrical bandwidth (≥100 GHz), it is needed to increase the optical modulation bandwidth of the TL Base region plays the key role in all BJT transistors, especially in Transistor Lasers

Like Quantum-Well, base structural parameters have significant effects on optoelectronic characteristics of TL which can be modelled like what performed before during this chapter Among these parameters are base width (Zhang et al., 2009), material, doping (Chu-Kung et al., 2006), etc For example, a graded base region can cause an internal field which accelerates the carrier transport across the base thus alters both the optical bandwidth and the current gain considerably

7 References

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the differential optical gain of a quantum-well transistor laser Appl Phys Lett Vol

91, No 243508

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the effective minority carrier lifetime in the operation of a quantum-well n-p-n heterojunction bipolar light-emitting transistor of varying base quantum-well

design and doping Appl Phys Lett Vol 91, No 033505

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doi:10.1109/JQE.2009.2013215, pp (359–366)

Intersubband and Interband Absorptions

in Near-Surface Quantum Wells

Under Intense Laser Field

Nicoleta Eseanu

Physics Department, “Politehnica” University of Bucharest,

Bucharest Romania

1 Introduction

The intersubband transitions in quantum wells have attracted much interest due to their unique characteristics: a large dipole moment, an ultra-fast relaxation time, and an outstanding tunability of the transition wavelengths (Asano et al., 1998; Elsaesser, 2006; Helm, 2000) These phenomena are not only important by the fundamental physics point of view, but novel technological applications are expected to be designed

Many important devices based on intersubband transitions in quantum well heterostructures have been reported For example: far- and near-infrared photodetectors (Alves et al., 2007; Levine, 1993; Li, S.S 2002; Liu, 2000; Schneider & Liu, 2007; West & Eglash, 1985), ultrafast all-optical modulators (Ahn & Chuang, 1987; Carter et al., 2004; Li, Y

et al., 2007), all optical switches (Iizuka et al., 2006; Noda et al., 1990), and quantum cascade lasers (Belkin et al., 2008; Chakraborty & Apalkov, 2003; Faist et al., 1994)

It is well-known that the optical properties of the quantum wells mainly depend on the asymmetry of the confining potential experienced by the carriers Such an asymmetry in potential profile can be obtained either by applying an electric/laser field to a symmetric quantum well (QW) or by compositionally grading the QW In these structures the changes

in the absorption coefficients were theoretically predicted and experimentally confirmed to

be larger than those occurred in conventional square QW (Karabulut et al., 2007; Miller, D.A.B et al., 1986; Ozturk, 2010; Ozturk & Sökmen, 2010)

In recent years, with the availability of intense THz laser sources, a large number of strongly laser-driven semiconductor heterostructures were investigated (Brandi et al., 2001; Diniz Neto & Qu, 2004; Duque, 2011; Eseanu et al., 2009; Eseanu, 2010; Kasapoglu & Sokmen, 2008; Lima, F M S et al., 2009; Niculescu & Burileanu, 2010b; Ozturk et al., 2004; Ozturk et al., 2005; Sari et al., 2003; Xie, 2010)

These works have revealed important laser-induced effects:

i) the confinement potential is dramatically modified; ii) the energy levels of the electrons and, to a lesser extent, those of the holes are enhanced; iii) the linear and nonlinear absorption coefficients can be easily controlled by the confinement parameters in competition with the laser field intensity

The intersubband transitions (ISBTs) have been observed in many different material systems In recent years, apart from GaAs/AlGaAs, the InGaAs/GaAs QW structures have

attracted much interest because of their promising applications in optoelectronic and microelectronic devices: multiple-quantum-well modulators and switches (Stohr et al., 1993), broadband photodetectors (Gunapala et al., 1994; Li J et al., 2010, Passmore, et al., 2007), superluminescent diodes used for optical coherence tomography (Li Z et al., 2010) and metal-oxide-semiconductor field-effect-transistors, i e MOSFETs (Zhao et al., 2010) The optical absorption associated with the excitons in semiconductor QWs have been the subject of a considerable amount of work for the reason that the exciton binding energy and oscillator strength in QWs are considerably enhanced due to quantum confinement effect (Andreani & Pasquarello, 1990; Jho et al., 2010; Miller, R C et al., 1981; Turner et al., 2009; Zheng & Matsuura, 1998)

As a distinctive type of dielectric quantum wells, the near-surface quantum wells (n-sQWs) have involved increasing attention due to their potential to sustain electro-optic operations under a wide range of applied electric fields In these heterostructures the QW is located close to vacuum and, as a consequence, the semiconductor-vacuum interface which is parallel to the well plane introduces a remarkable discrepancy in the dielectric constant (Chang & Peeters, 2000) This dielectric mismatch leads to a significant enhancement of the exciton binding energy (Gippius et al., 1998; Kulik et al., 1996; Niculescu & Eseanu, 2010a) and, consequently, it changes the exciton absorption spectra as some experimental (Gippius

et al., 1998; Kulik et al., 1996; Li, Z et al., 2010; Yablonskii et al., 1996) and theoretical (Niculescu & Eseanu, 2011a; Yu et al., 2004) studies have demonstrated

The rapid advances in modern growth techniques and researches for InGaAs/GaAs QWs (Schowalter et al., 2006; Wu, S et al., 2009) create the possibility to fabricate such heterostructures with well-controlled dimensions and compositions Therefore, the differently shaped InGaAs/GaAs near-surface QWs become interesting and worth studying systems

We expect that the capped layer of these n-sQWs induces considerable modifications on the intersubband absorption as it did on the interband excitonic transitions (Niculescu & Eseanu, 2011) To the best of our knowledge this is the first research concerning the intense laser field effect on the ISBTs in InGaAs/GaAs differently shaped near-surface QWs

In this chapter we are concerned about the intersubband and interband optical transitions in differently shaped n-sQWs with symmetrical/asymmetrical barriers subjected to intense high-frequency laser fields We took into account an accurate form for the laser-dressing confinement potential as well as the occurrence of the image-charges Within the framework

of a simple two-band model the consequences of the laser field intensity and carriers-surface interaction on the absorption spectra have been investigated

The organization of this work is as follows In Section 2 the theoretical model for the intense laser field (ILF) effect on the intersubband absorption in differently shaped n-sQWs is described together with numerical results for the electronic energy levels and absorption coefficients (linear and nonlinear) In Section 3 we explain the ILF effect on the exciton ground energy and interband transitions in the same QWs, taking into account the repulsive interaction between carriers and their image-charges Also, numerical results for the 1S-exciton binding energy and interband linear absorption coefficient are discussed Finally, our conclusions are summarized in Section 4

2 Intersubband transitions in near-surface QWs under intense laser field

The intersubband transitions (ISBTs) are optical transitions between quasi-two-dimensional electronic states ("subbands") in semiconductors which are formed due to the confinement

277

of the electron wave function in one dimension The conceptually simplest band-structure

engineered system that can be fabricated is a quantum well (QW), which consists of a thin

semiconductor layer embedded in another semiconductor with a larger bandgap

Depending on the relative band offsets of the two semiconductor materials, both electrons

and holes can be confined in one direction in the conduction band (CB) and the valence

band (VB), respectively Thus, allowed energy levels which are quantized along the growth

direction of the heterostructure appear (Yang, 1995) These levels can be tailored by

changing the QW geometry (shape, width, barrier heights) or by applying external

perturbations (hydrostatic pressure, electric, magnetic and laser fields) Whereas, of course,

optical transitions can take place between VB and CB states, in this Section we are

concerning only with ISBTs between quantized levels within the CB

2.1 Theory

Let us consider an InGaAs n-sQW embedded between symmetrical/asymmetrical GaAs

barriers It is convention to define the QW growth as the z-axis According to the effective

mass approximation, in the absence of the laser field, the time-independent Schrödinger

where m is the electron effective mass, V z is the confinement potential in the QW  

growth direction and V self z describes the repulsive interaction in the system consisting of

an electron and its image-charge

Here  is the semiconductor dielectric constant, e20e2/ 4, and d is the distance between 0

the electron and its image-charge without laser field For the three differently shaped

n-sQWs studied in this work the potential V z reads as follows For a square n-sQW,   V z  

has the well-known form

For a semiparabolic n-sQW, V z is given by  

 

0 2

The quantities L w and L c are the well width and capped layer thicknesses, respectively; V0

is the GaAs barrier height in the QW left side (with cap layer); V r is the barrier height in the

QW right side and  is the barrier asymmetry parameter

Under the action of a non-resonant intense laser field (ILF) represented by a monochromatic

plane wave of frequency LF having the vector potential A t uA 0cosLF t, the

Schrödinger equation to be solved becomes a dependent one due to the

time-dependent nature of the radiation field Here u is the unit vector of the polarization 

direction (chosen as z-axis) By applying the translation r r  t the equation

describing the electron-field interaction dynamics was transformed by Kramers (Kramers,

describes the quiver motion of the electron under laser field action 0 is known as the

laser-dressing parameter, i e a laser-dependent quantity which contains both the laser

frequency and intensity,

0

LF

e A m



Thus, in the presence of the laser field linearly polarized along the z-axis, the confinement

potential V z  t  is a time-periodic function for a given z and it can be expanded in a

where V k is the k-th Fourier component of V z and   J v is the Bessel function of order v In

the high-frequency limit, i.e  LF 1, with τ being the transit time of the electron in the

QW region (Marinescu & Gavrila, 1995) the electron “sees” a laser-dressed potential which

is obtained by averaging the potential V z  t  over a laser field period,

Note that this approximation remains valid provided that the laser is tuned far from any

resonance so that only photon absorption was taken into account and photon emission was

disregarded

Following some basic works (Gavrila & Kaminski, 1984; Lima, C.A.S & Miranda, 1981;

Marinescu & Gavrila, 1995) the laser-dressed electronic eigenstates in the QW are the

solutions of a time-independent Schrödinger equation,

Here ~ z is the laser-dressed wave function of the electron The envelope wave functions

and subband energies in this modified potential can be obtained by using a transfer matrix

method (Ando & Ytoh, 1987; Tsu & Esaki, 1973)

In order to characterize the intersubband transitions in laser-dressed n-sQWs the E1E2

transition energy, E trE2E1, and square of the optical matrix element, 2

21

M , are worth

being calculated Our results for n-sQWs revealed that these quantities depend on the QW

width and shape as well as on laser field parameter and cap layer thickness

The dipole matrix element of the E iE transition is defined by f

using the compact density matrix method (Bedoya & Camacho, 2005; Rosencher & Bois,

1991) and a typical iterative procedure (Ahn & Chuang, 1987) The linear and nonlinear

absorption coefficients are written (Unlu et al., 2006; Ozturk, 2010) as:

  

* 2 1

eff

m k T M

Here I is the optical intensity of the incident electromagnetic wave (with the angular

frequency ex) that excites the semiconductor nanostructure and leads to the intersubband

optical transition,  is the permeability,   0n r2 with n r being the refractive index, ex is

the pump photon energy produced by a tunable laser source, E F represents the Fermi

energy, E1 and E2 denote the quantized energy levels for the initial and final states,

respectively; k B and T are the Boltzmann constant and temperature, respectively, c is the

speed of light in free space, L is the effective spatial extent of the electrons and eff in is the

intersubband relaxation time

The total absorption coefficient is given by

The absorbtion coefficients in n-sQWs under laser field depend on both laser-dressing

parameter and QW geometry (well shape, barrier asymmetry, cap layer thickness)

2.2 Electronic properties

2.2.1 Laser-dressed confinement potential and energy levels

Within the framework of effective-mass approximation the ground and the first excited

energetic levels for an electron confined in differently shaped In0.18Ga0.82As/GaAs

near-surface QWs: square (SQW), graded (GQW), and semiparabolic (sPQW) under

high-frequency laser field were calculated We used various QW widths L = 100 Å, 150 Å,

and 200 Å, different cap layer thicknesses, L , between 5 Å and 200 Å, for n-sQWs with c

symmetrical ( = 1) or asymmetrical ( = 0.6 and 0.8) barriers The small In atoms

concentration in the QW layer allowed us to take the same value of the electron effective

mass in all regions (barriers, QW), i e m0.0665m0 Also, we used E = 6.49 meV which F

corresponds to about 1.6 × 1017 electrons/cm3 in GaAs, and in= 0.14 ps (Ahn & Chuang,

1987)

Fig 1 displays the “dressed” potential profiles of the conduction band (CB) for three

differently shaped n-sQWs (SQW, GQW and sPQW) with QW width L = 150 Å, identical w

barriers ( = 1) and cap layer thickness L = 20 Å, under various laser intensities described c

by the laser parameter values 0 = 0; 40; 80 and 100 Å Only two energy levels have been

taken into account for all the n-sQWs investigated in this work: E (ground state) and1 E 2

(first excited state) They are are plotted in Fig 1, too

We see that for all studied n-sQWs the increasing of the laser parameter dramatically

modifies the potential shape which is responsible for quantum confinement of the electrons

Up to 0L w/ 2 two effects are noticeable: i) while the effective “dressed” well width (i e

the lower part of the confinement potential) decreases with the laser intensity, the width of

the upper part of this “dressed” QW increases; ii) a reduction of the effective well height at

the interface between the capped layer and the QW (z = 0)

281

Fig 1 (Color online) Laser-dressed confinement potentials of three differently shaped

In0.18Ga0.82As/GaAs near-surface QWs with identical barriers and the corresponding energy levels: ground state E (solid lines) and first excited state 1 E (dashed lines) Notations a, c, 2

e, and f stand for various laser parameter values,  = 0 (black), 40 Å (olive), 80 Å (purple), 0and 100 Å (orange), respectively QW width and cap layer thickness are L = 150 Å and w L c

= 20 Å, respectively

Therefore, under an intense laser field a distinctive blue-shift of the electronic energy levels occurs, as expected (Brandi et al., 2001; Diniz Neto & Qu, 2004; Eseanu, 2010; Kasapoglu & Sökmen, 2008; Lima, F M S et al., 2009; Niculescu & Burileanu, 2010b; Ozturk et al., 2004; Ozturk et al., 2005) This laser-induced push-up effect is more pronounced in the semiparabolic QW due to the stronger geometric confinement

For 0L w/2 a supplementary barrier having a “hill”-form appears into the well region and, as a consequence, the formation of a double well potential in the InGaAs layer is predicted Similar laser-induced phenomena were reported for a GaAs/AlGaAs square QW (Lima, F M S et al., 2009) and for coaxial quantum wires (Niculescu & Radu, 2010c) The two energy levels E (ground state) and 1 E (first excited state) depend on both laser 2

field characteristics (frequency and intensity, combined in the laser parameter) and QW confining geometry (shape, width, cap layer thickness and barrier asymmetry)

In Figs 2 A-C the energy levels E and 1 E are plotted as functions of laser parameter in 2

three differently shaped n-sQWs (SQW, GQW and sPQW) with L = 200 Å for various cap w

layer thicknesses

We observed that:

i the laser-induced increasing of the ground level energy E is more pronounced for 1



0 40 Å in all three differently shaped n-sQWs with the same L ; w

ii the first excited level energy E is almost unaltered by the laser field in the GQW and 2

sPQW for 0 40 Å; instead, in the SQW, E has a significant rising 2

The reasons are as follows: i) the ground level E is localized in the lower part of the laser-1

dressed QW and it is significantly moved up only by an intense laser field; ii) in the GQW and sPQW the stronger geometric confinement competes with the laser-induced push-up of the energy levels As a consequence, the excited level which is localized in the upper part becomes less sensitive to the laser action

10203040506070

283

405060708090

0 [Å]

406080

Also, in the range of thin cap layers, the energy levels E1 and E depend on the capped 2

layer thickness (L ) Fig 3 displays the energies c E , 1 E and transition energy, 2 E trE2E1,

vs L in a near-surface SQW with c L = 150 Å for several values of the laser parameter w

This variation is similar in GQW and sPQW

We note that, up to L  40 Å, the energies c E , 1 E and, consequently, the transition energy 2

first rapidly decrease with cap layer thickness and, for further large L values, these three c

quantities turn out to be insensitive to the dielectric effect afforded by the cap layer The reason for this behavior is that the effect of the image-charge is reduced by a thicker cap layer

2.2.2 The E1E2 transition energy

As the intense laser field induces obvious changes in the electronic levels a noticeable dependence of the E1E2 transition energy, E trE2E1, on the laser parameter, 0, is expected Also, this energy is modified by the QW confining properties Figs 4 A-C present the grouped data plot of the E tr and square of the dipole matrix element (in units of e ), 2

0 < 0M the transition energy E tr increases due to reduction of the effective “dressed” well width Instead, for 0 > 0M the subbands levels E1 and E2 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 E1 and E2 values

in SQW and GQW under laser field action Therefore, E tr may have a maximum for a certain value 0M This critical value of the laser parameter seems to have a weak dependence on the cap layer thickness (Table 1)

L [Å] 0M[Å]

(Lc = 20 Å)

0M[Å]

(Lc = 50 Å)

0M [Å]

(Lc = 100 Å)

0m[Å]

(Lc = 20 Å)

0m[Å]

(Lc = 50 Å)

0m [Å] (Lc = 100 Å)

For sPQW we note a different behavior, i.e E reduces monotonically as the laser tr

parameter increases (Fig 4C) The supplementary quantum confinement of the electron localization in sPQW comparing with SQW and GQW could be the explanation

For the three studied QW structures and for all the widths under our investigation the values of E are diminuted by increasing tr L (vertical arrows indicate the c L rising) This c

1000 1500 2000 2500 3000 3500 4000

1000 2000 3000 4000 5000

1000 2000 3000 4000 5000

Also, the transition energy depends on the QW width as Fig 5 shows for a near-surface SQW with two values of the cap layer thickness, under several laser intensities

20 30 40 50 60

100 - 200 Å (Eseanu, 2010) and in a In0.5Ga0.5As/AlAs SQW with L = 20 - 100 Å (Chui, w

1994)

By applying an intense laser field the reduction of E (as function of tr L ) becomes weaker, w

but for high laser parameter values (0 80 Å), E turn to rise with tr L , especially in the w

presence of a thick cap layer The reason for this behavior is the competition between the geometric quantum confinement and laser-induced increasing of the energy levels

In the asymmetrical n-sQWs (GQW and sPQW) another factor modifiying the transition energy appears This is the asymmetry parameter of the QW barriers, , (see Eqs 2b and 2c)

In Fig 6 the transition energy, E , as a function of the barrier asymmetry is plotted for tr

GQW and sPQW with L = 200 Å and a thin cap layer (20 Å) As seen in this figure w E is tr

enhanced by the increasing  The dependence E tr  f  is also modulated by the laser field and, to a lesser extent, by the QW shape

In the GQW the increasing of E is almost linear for all laser parameter values, but the tr

rising slope is higher under intense laser field (0 80 Å) due to the laser-induced push-up effect on the energy levels Instead, the stronger geometric confinement leads to a deviation from linearity of the curve E tr  f  in the sPQW

Now we should emphasize an important issue As a consequence of the laser-induced shift

in the subband transition energy it clearly appears the possibility that E can be tuned by tr

the joint action of the laser field and a supplementary external perturbation such as: electric

287 field (Ozturk et al., 2004; Ozturk et al., 2005) or hydrostatic pressure (Eseanu, 2010), these two cases refering to uncapped QWs The last case was named “laser- and pressure-driven optical absorption” (LPDOA) The variable cap layer thickness, especially in the range of thin films, in simultaneous action with an intense laser field could be a new method (suggested by the present study) to adjust the transition energy, E tr

20 25 30 35 40 45

2.2.3 The square dipole matrix element

The intense laser field strongly modifies the dipole matrix element of the E 1 E2 transition,

E has its maximum A similar behavior have been reported for regular (i.e uncapped)

GaAs/AlGaAs SQWs (Eseanu, 2010) The explanation for the minimum of 2

21

M can be connected with the laser intensity dependent overlap of the two wave functions 1 z

and2 z

In the presence of laser field, the SQW shape is dramatically modified: as  increases the 0lower 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:

i for 00m the second subband E2 is localized in the upper part of the “dressed” well

and the ground state E1 is still localized in the lower part As 0 increases, the ground state wave function 1 z becomes more compressed in the vicinity of z = 0 and its

overlap with the first excited wave function 2 z (which has a minimum in z = 0)

reduces

ii by further increasing0, for 00m , the E1 level is also pushed up to the wider upper part of the QW Thus the carrier confinement decreases and the two wave functions 1 z and2 z are spread out (or delocalized) in the potential barrier regions As a consequence their overlapping is enhanced

In the asymmetrical structures GQW and sPQW, 2

21

M monotonically increases with laser parameter (Figs 4 B, C) This effect is determined by the strong electron localization in the graded barriers Therefore, the two wave functions 1 z , 2 z are localized in the upper part of the laser-dressed QW

For the three studied n-sQWs and for all the widths under our investigation the values of

2

21

M are enhanced by growing L (vertical arrows indicate the c L rising in Figs 4 A, B, C) c

This effect can be explained by the broadening of the effective n-sQW width for thicker cap layers Thus, the ground state wave function 1 z becomes more extended in the heterostructure and its overlap with the first excited wave function 2 z is enhanced

2.3 Linear and nonlinear optical absorption

The intersubband linear and nonlinear absorption coefficients given by the Eqs (11)-(12) depend on the transition energy, E trE2E1, and on square of the dipole matrix element,

2

21

M In the n-sQWs under our investigation these quantities are strongly modified by the laser field parameter, but in a different manner (Fig 4) As a consequence, the absorption coefficients  1 

ex

  and  3ex,I are significantly changed by the laser intensity (included in the laser-dressing parameter 0) However, there are still four variables to take into account: the pump photon energy, ex , cap layer thickness, L , asymmetry parameter c

of the QW barriers, , and QW potential shape

In Fig 7 the dependence of the linear absorption coefficient  1 on the pump photon energy

in differently shaped n-sQWs with L = 150 Å and symmetrical barriers, under various laser w

intensities 0= 0, 40 Å, and 80 Å, is plotted The values of L are 20 Å and 200 Å c

Fig 8 displays the variation of  1 on the pump photon energy in a near-surface SQW with c

L = 50 Å under various laser intensities 0= 0, 40 Å, and 80 Å The values of L are 100 Å w

and 200 Å

One can see from Figs 7 and 8 that the increasing laser intensity generates a noticeable shift

of the absorption peak position toward higher/lower photon energies and an obvious reduction of the  1 magnitude The trend of this shift significantly depends on the QW shape and width, also on the laser parameter Several regimes occur:

1 in SQW and GQW with L w = 150 Å the absorption peaks are blue-shifted for

0 L w/ 2

  , but red-shifted for 0L w/ 2;

2 in GQW with L w = 200 Å the absorption peaks (not plotted here) are blue-shifted for all the studied 0 values, but the shift induced by a strong laser field (0= 80 Å) is smaller;

3 in sPQW with L w = 150 Å the linear absorption peaks are red-shifted for all the studied values of the laser parameter; a similar variation (not plotted here) was observed for L w

= 200 Å

4 in SQW the absorption peaks are red-shifted for L = 100 Å, but for w L = 200 Å they w

are blue-shifted (Fig 8)