described the carrier dynamics in QD systems using a rate equation model [13-19].

independently [19-21].

from QWs into the barrier [22].

230–300o

thermal decays.

However, it is not still clear the details of exciton capture and thermal escape in high quality QD structures. Two main ways are discussed: a) thermal escape (capture) of carriers (electron and hole) in/from QDs takes place as an exciton or correlated electron-hole pairs [17, 18]; or b) the excitons dissociate and single electrons (holes) thermally escape or capture

The first mentioned mechanism is the subsequence of the fact that the activation energy of ground state PL thermal decay measured in the regime of strong quenching matches the difference between the GaAs band gap and QD ground state (GS) energy (the sum of barrier heights for electrons and holes) [17-19]. The same effect was earlier revealed in strained InGaAs/GaAs QWs, where the activation energy of PL thermal decay corresponded to the difference between the GS energies of QW and the GaAs barrier, defined as total confinement energy, i.e. the sum of electron and hole potential depths. The main conclusion follows from this fact: excitons or electron-hole pairs are emitted

The motivation for the second mechanism was presented in [19-21]. It was shown in [19] at the InAs/GaAs QD PL investigation under low excitation and high temperature the ground state PL intensity varies quadratically with excitation density. This fact was explained as an evidence of independent occupation of QDs by electrons and holes with the probability

Other evidences for supporting the second approach were shown in [20] at the investigation of cathodoluminescence thermal quenching in InAs/GaAs self-assembled QDs at highexcitation conditions. The significant reductions in the thermal activation energies in the

suggested that excitons dissociate and the thermal reemission of single holes from QD states into the GaAs is responsible for the observed temperature dependence. The same explanation was supposed in [21] at the joint investigation of InAs QD and InGaAs QW PL

K temperature range for the ground (GS) and excited states (ES) are found. It was

which is proportional to the multiplication of their concentrations (nenp).

In DWELL structures the introduction of surrounding InxGa1-xAs/GaAs QWs changes the QD density, the elastic strain in structures, the height of potential barriers for exciton capture and thermal escape in/from QDs as well as can increase in some cases the density of nonradiative (NR) centres. In these structures the mechanism of PL intensity thermal decay could depend strongly on parameters of DWELL. Improved understanding of the operation and design of InAs/InGaAs QD-based devices could be emanated from studies involving the excitation and temperature dependences of PL in such structures.

Note the highest emission intensity of QDs was achieved in the symmetric wells with the composition of quantum well (QW) capping/buffer layers of In0.15Ga0.85As/GaAs [3]. One of the best methods of manipulating the InAs QD density and sizes inside of QWs related to controlling the QD growth temperatures [3,12-15]. But even for the optimal QD growth parameters and capping/buffer layer compositions, the InAs QD structures are characterized by photoluminescence (PL) inhomogeneity along the wafers [16-19].

The inhomogeneity of InAs QD parameters across the wafer due to the size, chemical composition and stress variations leads to a broadening of emission spectra. This type of problems in QD structures was investigated using scanning PL spectroscopy [23, 26-29], high resolution transmission electron microscopy [30], scanning tunneling microscopy [31], as well as spatially resolved scanning tunneling luminescence [32]. However, in dot-in-a well structures, where the InAs QDs coupled with InGaAs/GaAs QWs, the physical reasons of emission inhomogeneity still have to be discussed. The technology of growth of InAs QD structures has become more reliable recently enabling the systematic studies their physical properties and emission inhomogeneity of QD ensembles.

### **3. PL spectra of symmetric GaAs/In0.15Ga0.85As/GaAs QWs with the different density of InAs QDs**

The solid-source molecular beam epitaxy (MBE) in V80H reactor was used to grow the waveguide structures consisting of the layer of InAs self-organized QDs inserted into 12 nm In0.15Ga0.85As/GaAs QWs. The thickness of the buffer In0.15Ga0.85As layer was 2nm, which was grown on the 200.nm GaAs buffer layer and the 2 inch (100) GaAs SI substrate (Fig.1). Then an equivalent coverage of 2.4 monolayers of InAs QDs were confined by the capping (10nm) In0.15Ga0.85As layer and by the 100nm GaAs barrier. Investigated structures are grown under As-stabilized conditions at five different temperatures: 470C (#1), 490C (#2), 510C (#3), 525C (#4) and 535C (#5), during the deposition of the InAs active region and InGaAs wells, and at 590-610C for the rest of layers. All layers were grown with the growth rate of 0.30 ML/s, but for the QD formation the process provides deposition of 2.4 ML with the growth rate of 0.053 ML/s.

Fig. 1. The schematic design of studied QD structures

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 157

= 525°C Gs 1Es

= 510°C Gs 1Es

0

b

PL intensity (ar. un.)

50K

110K

170K

0

50K

110K

170K

230K

PL intensity (ar.un.)

230K

0.96 0.99 1.02 1.05 1.08 1.11 1.14

0.96 0.99 1.02 1.05 1.08 1.11 1.14

Emision energy (eV)

#2(a), #3(b) and #4(c) at the excitation light density of 500 W/cm2.

Fig. 2. Typical PL spectra measured at different temperatures for the DWELL structure

T gr

Emision energy (eV)

T gr

c

For AFM measurement the process of growth was stopped for satelite samples after the formation of QDs. The in-plane density of QDs changed from 1.1x1011 to 1.3x1010cm-2 (Table 1) when the temperature increases from 470 up to 535 C [4, 33]. The samples were mounted in a closed-cycle He cryostat where the temperature is varied in the range of 10 - 300 K. PL spectra were measured under the excitation of the 514.5 nm line of a cw Ar+-laser at an excitation power density form the range of 10-1000 W/cm2. The setups used for PL and PL excitation spectroscopies were presented earlier in [5] and [17], respectively.

Table 1. Average parameters of InAs QDs estimated by AFM [33]

Typical PL spectra of the DWELL structures #2, #3 and #4 measured at different temperatures at the excitation light density of 500 W/cm2 are shown in Fig.2. The structures #1 and #2 grown at low temperatures 470 and 490C are characterized by one PL band only (Fig.2a) with a higher value (Fig.3b) of the full width at half maximum (FWHM). In structures #3, #4 and #5 two PL bands appear due to the recombination of excitons localized at a ground state (GS) and at first excited state (1ES) in QDs (Fig.2b,c).

For AFM measurement the process of growth was stopped for satelite samples after the formation of QDs. The in-plane density of QDs changed from 1.1x1011 to 1.3x1010cm-2 (Table 1) when the temperature increases from 470 up to 535 C [4, 33]. The samples were mounted in a closed-cycle He cryostat where the temperature is varied in the range of 10 - 300 K. PL spectra were measured under the excitation of the 514.5 nm line of a cw Ar+-laser at an excitation power density form the range of 10-1000 W/cm2. The setups used for PL and PL

excitation spectroscopies were presented earlier in [5] and [17], respectively.

h nm

470 1.11011 12 6 0.50 113.0 0.124 490 7.01010 14 8 0.57 153.9 0.108 510 3.41010 18 13 0.72 254.3 0.086 525 1.81010 24 11 0.46 452.2 0.081 535 1.31010 28 10 0.36 615.4 0.080

Typical PL spectra of the DWELL structures #2, #3 and #4 measured at different temperatures at the excitation light density of 500 W/cm2 are shown in Fig.2. The structures #1 and #2 grown at low temperatures 470 and 490C are characterized by one PL band only (Fig.2a) with a higher value (Fig.3b) of the full width at half maximum (FWHM). In structures #3, #4 and #5 two PL bands appear due to the recombination of excitons localized

h/D single

a

QD S x 1014, cm2

0

PL intensity (ar. un.)

50K 110K 170K 230K

Surface Area of QDs, cm2

DQD nm

Table 1. Average parameters of InAs QDs estimated by AFM [33]

at a ground state (GS) and at first excited state (1ES) in QDs (Fig.2b,c).

= 490°C

0.96 0.99 1.02 1.05 1.08 1.11 1.14

T gr

Emission energy (eV)

Tgr (°C) NQD cm-2

Fig. 2. Typical PL spectra measured at different temperatures for the DWELL structure #2(a), #3(b) and #4(c) at the excitation light density of 500 W/cm2.

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 159

grown at 525 and 535oC are characterized by lower intensities of GS emission (Fig.3a), by smallest FWHM (Fig.3b), and the peak position of GS PL bands shifts into higher energy (Fig.2b,c). Note that lower PL peak energy corresponds to higher PL intensity (Fig.2b and 3a). Thus the variation non monotonous the PL intensity ans peak positions versus QD

In high excitation energy range the spectrum presents a sharp PL intensity increase due to the fundamental light absorption (at 1.51 eV) in the GaAs barriesr. In the low energy region the PL excitation spectrum can be considered as a superposition of the four absorption

E, eV 1.51 1.46-1.47 1.44 1.42 1.35-1.36 1.06-1.11 1.32-1.33

The peak A spectral position (1.46-1.47eV) is close to the GS resonant absorption related to the WL in InAs/GaAs QD structures [17, 24, 25]. Note the authors [25] registrated at 100K in PL excitation spectrum two overlapping maxima (1.45-1.47eV) which were attributed to the photon absorption in the 2D wetting layer between the heavy hole and light hole GS subbands to the GS electronic subband. In studied DWELL structures the GS resonant absorption in the 2 nm buffer In0.15Ga0.85As QW can contribute in other peaks B and C. Thus

Band gap peak A peak B peak C peak D QD-PL QW-PL

density ( QD growth temperature) has been revealed in studied QD structures.

The typical PL excitation spectrum measured at 80K is shown in Fig.4.

Fig. 4. PL excitation spectrum for the structure #3.

Table 2. Optical transitions in DWELL structures at 80K

bands: A, B, C and D (Fig.4, Table 2).

GaAs

Optical transitions

**dependences**

**4. PL excitation spectra of InAs QD structure and PL excitation power**

The density of QDs decreases monotonically with the rise of QD growth temperatures from 470 up to 535 oC (Table 1). Thus the PL intensity diminishing can be expected with reducing a QD density. The QD diameters in studied structures increase monotonically from 12 up to 24-28 nm with the rise of QD growth temperatures from 470 up to 535 oC. Thus it is possible to expect that the PL peak position in QDs has to shift monotonically to low energy.

Fig.3 presents the variation of in plane QD densities, estimated by AFM on satelite samples, as well as the average GS integrated PL intensities and the GS peak FWHM measured at 300K in studied structures.

Fig. 3. The average integrated PL intensities measured in DWELLs with QDs grown at different temperatures (a) and the FWHM and QD density in DWELLs with QDs grown at different temperatures (b).

However the PL intensity increases (Fig.3a) and the GS peak shifts to low energy in structures with QDs grown at 490 and 510oC (Fig.2a,b). On the contrary the QD structures

The density of QDs decreases monotonically with the rise of QD growth temperatures from 470 up to 535 oC (Table 1). Thus the PL intensity diminishing can be expected with reducing a QD density. The QD diameters in studied structures increase monotonically from 12 up to 24-28 nm with the rise of QD growth temperatures from 470 up to 535 oC. Thus it is possible

Fig.3 presents the variation of in plane QD densities, estimated by AFM on satelite samples, as well as the average GS integrated PL intensities and the GS peak FWHM measured at

300K

5

0.0

3.0x1010

6.0x1010

9.0x1010

QD density (cm-2

)

1.2x1011

1.5x1011

4

470 480 490 500 510 520 530 540

470 480 490 500 510 520 530 540

Temperature (ºC)

However the PL intensity increases (Fig.3a) and the GS peak shifts to low energy in structures with QDs grown at 490 and 510oC (Fig.2a,b). On the contrary the QD structures

Fig. 3. The average integrated PL intensities measured in DWELLs with QDs grown at different temperatures (a) and the FWHM and QD density in DWELLs with QDs grown at

QD Growth Temperature (°C)

3

to expect that the PL peak position in QDs has to shift monotonically to low energy.

2

300K in studied structures.

Average Integrated

PL Intensity (arb. un.)

FWHM (meV)

different temperatures (b).

25

30

35

40

45

50

0.0

0.2

1

0.4

0.6

0.8

1.0

grown at 525 and 535oC are characterized by lower intensities of GS emission (Fig.3a), by smallest FWHM (Fig.3b), and the peak position of GS PL bands shifts into higher energy (Fig.2b,c). Note that lower PL peak energy corresponds to higher PL intensity (Fig.2b and 3a). Thus the variation non monotonous the PL intensity ans peak positions versus QD density ( QD growth temperature) has been revealed in studied QD structures.

### **4. PL excitation spectra of InAs QD structure and PL excitation power dependences**

The typical PL excitation spectrum measured at 80K is shown in Fig.4.

Fig. 4. PL excitation spectrum for the structure #3.

In high excitation energy range the spectrum presents a sharp PL intensity increase due to the fundamental light absorption (at 1.51 eV) in the GaAs barriesr. In the low energy region the PL excitation spectrum can be considered as a superposition of the four absorption bands: A, B, C and D (Fig.4, Table 2).

Table 2. Optical transitions in DWELL structures at 80K

The peak A spectral position (1.46-1.47eV) is close to the GS resonant absorption related to the WL in InAs/GaAs QD structures [17, 24, 25]. Note the authors [25] registrated at 100K in PL excitation spectrum two overlapping maxima (1.45-1.47eV) which were attributed to the photon absorption in the 2D wetting layer between the heavy hole and light hole GS subbands to the GS electronic subband. In studied DWELL structures the GS resonant absorption in the 2 nm buffer In0.15Ga0.85As QW can contribute in other peaks B and C. Thus

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 161

The inhomogeneity of QD ensemble emission along the line crossed the structures #1 and #2 related mainly to the variation of PL intensity (Fig.6). The integrated PL intensity in the structure #1 is low in comparison with #2, and the variations of PL peak positions are small for both DWELLs (Fig.6): Δhvm ~0.003 eV (#1) and Δhvm ~0.010 eV (#2). Thus in QD structures with InAs QDs grown at low temperatures (470-490oC) the less integrated PL intensity (Fig.3a), the high dispersion of QD sizes and, as a result, the larger value of FWHM (Fig.3b) of GS PL bands, but the low dispersion of QD ensemble parameters along the line

0.96 0.97 0.98 0.99 1.00 1.01 1.02

300K #2

#1 300K

b

a

GS peak energy (eV)

crossed the structures #1 and #2 have been detected (Fig.6a,b).

T g

=470o

C

0

center of QD structures #1 (a) and #2 (b) to their periphery.

T g

=490o

C

Fig. 6. The PL intensity and GS PL peak positions measured along the line scan from the

As it follows from fig.7 the decrease of integrated PL intensity, measured along the line from the center to periphery in QD structures #3, #4 and #5, is accompanied by the ''blue'' energy shift of the PL maximum. The FWHM of GS PL bands in the structures #3, #4 and #5 was equal to 35-40 meV (Fig.3b) and it is smaller than those in mentioned above structures #1 and #2. The inhomogeneity of QD ensemble emission along the lines crossed the structures #3, #4 and #5 is characterized by the essential variations of GS peak positions (Fig.7): Δhvm

2000

4000

6000

0

200

400

600

800

1000

PL Intensity (arb. un.)

both layers (the In0.15Ga0.85As buffer QW and WL) are the same in all studied DWELL structures and can be responsible for the peak A, B and C in the PL excitation spectrum. The spectral position of the lowest energy absorption band (1.35 eV, peak D) is close to one of the PL band (1.31-1.32eV) caused by GS exciton recombination in the capping InGaAs QW (Table 2) [34]. Thus the peak D can be attributed to the GS resonant excitation in the capping In0.15Ga0.85As /GaAs QW (Fig.4).

Fig. 5. PL integrated intensity versus excitation power measured for all studied structures

The dependences of integrated PL intensity versus excitation light power, measured at 12K with the aim to avoid the thermal decay process, are presented in Fig.5 for all studied structures. The QD structures #2, #3 and #4 with highest emission intensity are characterized by the linear dependence of the integrated PL intensity (I) versus excitation power (I≈Pn), n=0.95-1.00 (Fig.5). In QD structures with low emission (#1, #5) the integrated PL intensity changes sublineary (n=0.72-0.93) with excitation light power (Fig.5).

### **5. Emission inhomogeneity along the QD structures**

### **5.1 Scanning PL spectroscopy at 300K**

Photoluminescence inhomogeneity of QD ensembles is studied along the scanning line crossed the wafers from the periphery of QD structures to the center. In this case PL spectra were measured at 300 K in a set of points on the QD structures under the excitation by the 804 nm line of a solid state IR laser at an excitation power density of 100 W/cm2. PL spectra were dispersed by a SPEX 500M spectrometer and recorded by a liquid-nitrogen-cooled Ge detector with a standard lock-in technique.

both layers (the In0.15Ga0.85As buffer QW and WL) are the same in all studied DWELL structures and can be responsible for the peak A, B and C in the PL excitation spectrum. The spectral position of the lowest energy absorption band (1.35 eV, peak D) is close to one of the PL band (1.31-1.32eV) caused by GS exciton recombination in the capping InGaAs QW (Table 2) [34]. Thus the peak D can be attributed to the GS resonant excitation in the capping

Fig. 5. PL integrated intensity versus excitation power measured for all studied structures

PL intensity changes sublineary (n=0.72-0.93) with excitation light power (Fig.5).

**5. Emission inhomogeneity along the QD structures**

**5.1 Scanning PL spectroscopy at 300K**

detector with a standard lock-in technique.

The dependences of integrated PL intensity versus excitation light power, measured at 12K with the aim to avoid the thermal decay process, are presented in Fig.5 for all studied structures. The QD structures #2, #3 and #4 with highest emission intensity are characterized by the linear dependence of the integrated PL intensity (I) versus excitation power (I≈Pn), n=0.95-1.00 (Fig.5). In QD structures with low emission (#1, #5) the integrated

Photoluminescence inhomogeneity of QD ensembles is studied along the scanning line crossed the wafers from the periphery of QD structures to the center. In this case PL spectra were measured at 300 K in a set of points on the QD structures under the excitation by the 804 nm line of a solid state IR laser at an excitation power density of 100 W/cm2. PL spectra were dispersed by a SPEX 500M spectrometer and recorded by a liquid-nitrogen-cooled Ge

In0.15Ga0.85As /GaAs QW (Fig.4).

The inhomogeneity of QD ensemble emission along the line crossed the structures #1 and #2 related mainly to the variation of PL intensity (Fig.6). The integrated PL intensity in the structure #1 is low in comparison with #2, and the variations of PL peak positions are small for both DWELLs (Fig.6): Δhvm ~0.003 eV (#1) and Δhvm ~0.010 eV (#2). Thus in QD structures with InAs QDs grown at low temperatures (470-490oC) the less integrated PL intensity (Fig.3a), the high dispersion of QD sizes and, as a result, the larger value of FWHM (Fig.3b) of GS PL bands, but the low dispersion of QD ensemble parameters along the line crossed the structures #1 and #2 have been detected (Fig.6a,b).

Fig. 6. The PL intensity and GS PL peak positions measured along the line scan from the center of QD structures #1 (a) and #2 (b) to their periphery.

As it follows from fig.7 the decrease of integrated PL intensity, measured along the line from the center to periphery in QD structures #3, #4 and #5, is accompanied by the ''blue'' energy shift of the PL maximum. The FWHM of GS PL bands in the structures #3, #4 and #5 was equal to 35-40 meV (Fig.3b) and it is smaller than those in mentioned above structures #1 and #2. The inhomogeneity of QD ensemble emission along the lines crossed the structures #3, #4 and #5 is characterized by the essential variations of GS peak positions (Fig.7): Δhvm

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 163

The ground state PL intensity (IPL) of InAs/InGaAs QDs is directly proportional to internal

1

*i R f f*

probabilities for electrons and holes at ground state levels given by the Fermi-Dirac

for the conduction and valence bands, respectively, measured from the QD band edges, *En*,*p* are the quantized energy levels of an electron and a hole in th conduction and valence bands of a QD, measured from the QD band edges, ND is QD density and τR is electron–hole radiative recombination time in the QD [36]. At low excitation light intensity (100 W/cm2), used during GS PL scanning, which is well below the GS saturation intensity, we can present the occupation probabilities using Maxwell -Boltzmann distribution functions *fe*, *f p*=exp(−*En*,*p*+ µ*n*,*p* /*kT*). Taking into account that excitation light power is not changed during GS PL scanning experiment, we can assume that µ*n*,*p* are constant along the scanning line

and the PL intensity variation occurs due to parameters *En*,*p*only [35]. Thus *e f* ≈ exp

1 1

temperature inhomogeneity along the wafer at the QD growth process.

*RQD RQD E E E E I*

levels in a QD. In this case for GS emission intensity at room temperature is possible to

1 1 exp( ) exp( ) *N N e p InGaAs QD*

≈ exp

where *InGaAs EGS* is energy gap between electron and hole quantized levels in narrow-gap In0.15Ga0.85As layer and *QD EGS* is energy gap between electron and hole quantized levels in

exciton binding energy *ex Ebin* in QDs. Exciton binding energies were computed as the function of QD size using 8-band **k·p** approach and are estimated as 22 meV for QDs with the base size 14 nm [37]. Thus the *ex Ebin* value is small in comparison with GS PL optical transition

the capping/buffer layers, the linear dependence of GS PL intensity versus PL peak positions in QD ensembles in semi-logarithmic plot follows from Eq.1 (Fig.7). At the same time the slope of this linear dependence is proportional to kT. The fitting procedure was applied to the analysis of PL intensity data presented in Fig.7 and the slopes of these three lines were estimated as ~25 meV that is exactly the value of kT at 300K. Thus the analysis of linear dependence in Fig.7 testifies that the variation of GS PL intensity in the QD ensemble along the line scan in the structures #3, #4 and #5 is owing to the QD size variation – its decreasing from the wafer center to the periphery [38]. The last effect can be connected with

*loc loc GS GS*

*kT kT*

*GS hv* ) equal to 1.06-1.11 eV. If the InGaAs composition is the same in any points of

*D N e p i i*

*R NR*

*i f* and *p*

. For QD emission the

*i f* are the occupation

*QD EGS kT*

*GS hv* ) is the difference between *QD EGS* and

(1)

*e Eloc kT*

, where *e*

*kT* , where *p Eloc* and *e Eloc* are the binding energy of electron and hole on GS

*, fp*=(exp[(*En*,*p* −µ*n*,*p*)/*kT*]+1]−1, where µ*n*,*p* are the quasi-Fermi-levels

which can be presented as: *R*

**5.2 Analyses of PL inhomogenouty reasons in studied QD structures at 300K**

*R*

quantum efficiency

distribution functions, *fe*

and *p f* ≈ exp

energy ( max

write:

*p Eloc*

1

QD. Finally the GS PL optical transition energy ( max

*PL R*

radiative recombination rate is [35]:

~ 0.02 eV(#3), 0.03 eV(#4) and Δhvm ~0.04 eV(#5), respectively. The integrated PL intensity of QD ensembles in the structure #3 in two- or five-fold higher than in #4 and #5, respectively. Thus in the structures #3, #4 and #5 the low dispersion of QD sizes and, as a result, less values of FWHM, but the essential PL energy shift and the dispersion of QD ensemble parameters along the line crossed the structures have been detected (Fig.7).

The emission inhomogeneity in the QD structures #1 and #2 is related mainly to the change of PL intensity owing the density variations of QDs and/or of nonradiative centers (NR) in QD structures. At the same time the PL peak positions vary negligibly (3-10 meV) testifying that the difference in QD sizes is small. In the structures #3, #4 and #5, as it follows from fig.7, the decrease of PL intensity along the line scan is accompanied by the ''blue'' energy shift of PL maximum. This effect leads at the wafer periphery to shallower QD localized states (i.e. smallest electron and hole binding energy), poorer carrier localization and a higher probability of carrier thermal escape, which reduces the integrated QD PL intensity at 300K.

Fig. 7. The PL intensity and GS PL peak positions measured along the line scan from the center of QD structures #3 (a), #4 (b) and #5 (c) to their periphery.

~ 0.02 eV(#3), 0.03 eV(#4) and Δhvm ~0.04 eV(#5), respectively. The integrated PL intensity of QD ensembles in the structure #3 in two- or five-fold higher than in #4 and #5, respectively. Thus in the structures #3, #4 and #5 the low dispersion of QD sizes and, as a result, less values of FWHM, but the essential PL energy shift and the dispersion of QD ensemble parameters along the line crossed the structures have been detected (Fig.7).

The emission inhomogeneity in the QD structures #1 and #2 is related mainly to the change of PL intensity owing the density variations of QDs and/or of nonradiative centers (NR) in QD structures. At the same time the PL peak positions vary negligibly (3-10 meV) testifying that the difference in QD sizes is small. In the structures #3, #4 and #5, as it follows from fig.7, the decrease of PL intensity along the line scan is accompanied by the ''blue'' energy shift of PL maximum. This effect leads at the wafer periphery to shallower QD localized states (i.e. smallest electron and hole binding energy), poorer carrier localization and a higher probability

0.94 0.96 0.98 1.00 1.02

GS peak energy (eV)

c #5

b #4

C

C

#3

T g =525o

a

T g =510o

C

Fig. 7. The PL intensity and GS PL peak positions measured along the line scan from the

2000

T g =535o

center of QD structures #3 (a), #4 (b) and #5 (c) to their periphery.

4000

6000

2000

4000

PL Intensity (arb. un.)

6000

8000

6000

12000

18000 24000

of carrier thermal escape, which reduces the integrated QD PL intensity at 300K.

#### **5.2 Analyses of PL inhomogenouty reasons in studied QD structures at 300K**

The ground state PL intensity (IPL) of InAs/InGaAs QDs is directly proportional to internal quantum efficiency which can be presented as: *R R NR* . For QD emission the

radiative recombination rate is [35]: 1 *D N e p i i R i R f f* , where *e i f* and *p i f* are the occupation

probabilities for electrons and holes at ground state levels given by the Fermi-Dirac distribution functions, *fe , fp*=(exp[(*En*,*p* −µ*n*,*p*)/*kT*]+1]−1, where µ*n*,*p* are the quasi-Fermi-levels for the conduction and valence bands, respectively, measured from the QD band edges, *En*,*p* are the quantized energy levels of an electron and a hole in th conduction and valence bands of a QD, measured from the QD band edges, ND is QD density and τR is electron–hole radiative recombination time in the QD [36]. At low excitation light intensity (100 W/cm2), used during GS PL scanning, which is well below the GS saturation intensity, we can present the occupation probabilities using Maxwell -Boltzmann distribution functions *fe*, *f p*=exp(−*En*,*p*+ µ*n*,*p* /*kT*). Taking into account that excitation light power is not changed during GS PL scanning experiment, we can assume that µ*n*,*p* are constant along the scanning line

and the PL intensity variation occurs due to parameters *En*,*p*only [35]. Thus *e f* ≈ exp *e Eloc kT*

and *p f* ≈ exp *p Eloc kT* , where *p Eloc* and *e Eloc* are the binding energy of electron and hole on GS levels in a QD. In this case for GS emission intensity at room temperature is possible to write:

$$I\_{PL} \approx \tau\_R^{-1} \approx \sum\_{1}^{N} \frac{1}{\tau\_{RQD}} \exp(\frac{E\_{\rm loc}^{\varepsilon} + E\_{\rm loc}^{p}}{kT}) \approx \sum\_{1}^{N} \frac{1}{\tau\_{RQD}} \exp(\frac{E\_{\rm GS}^{\rm InCaAs} - E\_{\rm GS}^{\rm QD}}{kT}) \approx \exp\frac{-E\_{\rm GS}^{\rm QD}}{kT} \tag{1}$$

where *InGaAs EGS* is energy gap between electron and hole quantized levels in narrow-gap In0.15Ga0.85As layer and *QD EGS* is energy gap between electron and hole quantized levels in QD. Finally the GS PL optical transition energy ( max *GS hv* ) is the difference between *QD EGS* and exciton binding energy *ex Ebin* in QDs. Exciton binding energies were computed as the function of QD size using 8-band **k·p** approach and are estimated as 22 meV for QDs with the base size 14 nm [37]. Thus the *ex Ebin* value is small in comparison with GS PL optical transition energy ( max *GS hv* ) equal to 1.06-1.11 eV. If the InGaAs composition is the same in any points of the capping/buffer layers, the linear dependence of GS PL intensity versus PL peak positions in QD ensembles in semi-logarithmic plot follows from Eq.1 (Fig.7). At the same time the slope of this linear dependence is proportional to kT. The fitting procedure was applied to the analysis of PL intensity data presented in Fig.7 and the slopes of these three lines were estimated as ~25 meV that is exactly the value of kT at 300K. Thus the analysis of linear dependence in Fig.7 testifies that the variation of GS PL intensity in the QD ensemble along the line scan in the structures #3, #4 and #5 is owing to the QD size variation – its decreasing from the wafer center to the periphery [38]. The last effect can be connected with temperature inhomogeneity along the wafer at the QD growth process.

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 165

**7. The analysis of Ga/In interdiffusion in QD structures with different density**

Fig. 9 presents the variation of PL peak positions versus temperature in studied structures. PL peaks shift to low energy with increasing temperature due to the optical gap shrinkage. The lines in Fig.9 are the fitting results analyzed on the base of Varshni relation that presents

> o ( )=E + *aT E T*

Fig. 9. The variation of PL peak positions versus temperature. The lines present the Varshni

The comparison of fitting parameters with the variation of energy band gap versus temperature in the bulk InAs and GaAs crystals (Table 3) has revealed that in studied QD structures the fitting parameter "a" and 'b' in the temperature range 12-250 K are very close to their values for the bulk InAs crystal only in the structures #2 and #3. But in other QD structures the fitting parameter "a" and 'b' are different a little bit from the values in the bulk InAs crystal (Table 3). The last fact testifies that the process of Ga/In inter diffusion takes place in these QD structures [33,42-44]. Note that the process of Ga/In inter diffusion in studied structures passes non monotonically versus QD growth temperatures. It means that not only temperature but some other factors discussed below

2

(2)

*T b*

**of InAs QDs**

the energy gap variation with temperature as [39]:

fitting results: 1- #1, 2- #2, 3- #3, 4-#4 and 5 - #5.

are essential as well.

### **6. Temperature dependences of PL integrated intensities and peak positions**

Two reasons can explain the variation non monotonically of PL peak positions and the PL intensity in studied QD structures (Fig.2 and 3a): (i) the change of QD composition due to the Ga/In inter diffusion between the InAs QDs and capping/buffer In0.15 Ga0.85As QW layers or (ii) the different levels of elastic strains in QD structures due to the difference in QD densities and sizes. To distinguish these two reasons PL spectra at different temperatures in the range 12-300 K have been studied. The temperature dependences of integrated PL intensities and peak positions of GS PL bands for the structures #1, #3, #5 are shown in Fig.8.

T, K

Fig. 8. Temperature dependencies of GS integrated PL intensities (a) and GS peak positions (b) measured for the structures #1, #3 and #5. Lines present the fitting results.

The process of PL thermal decay starts at 80-120 K and it is characterized by different rates in the ranges T=100-180 K, T=180-250 K and T=250-300 K. At low temperatures up to 100 K the integrated PL intensity does not change, but the energy of GS peak decreases. The process of PL intensity decay starts at 100-120K in high quility structures #2, #3 and #4, but in structures of low quality (#1, #5) this process appears at 80-90K.

164 Fingerprints in the Optical and Transport Properties of Quantum Dots

**6. Temperature dependences of PL integrated intensities and peak positions**  Two reasons can explain the variation non monotonically of PL peak positions and the PL intensity in studied QD structures (Fig.2 and 3a): (i) the change of QD composition due to the Ga/In inter diffusion between the InAs QDs and capping/buffer In0.15 Ga0.85As QW layers or (ii) the different levels of elastic strains in QD structures due to the difference in QD densities and sizes. To distinguish these two reasons PL spectra at different temperatures in the range 12-300 K have been studied. The temperature dependences of integrated PL intensities and

0 40 80 120 160 200 240 280 320

T, K

Fig. 8. Temperature dependencies of GS integrated PL intensities (a) and GS peak positions

The process of PL thermal decay starts at 80-120 K and it is characterized by different rates in the ranges T=100-180 K, T=180-250 K and T=250-300 K. At low temperatures up to 100 K the integrated PL intensity does not change, but the energy of GS peak decreases. The process of PL intensity decay starts at 100-120K in high quility structures #2, #3 and #4, but

(b) measured for the structures #1, #3 and #5. Lines present the fitting results.

in structures of low quality (#1, #5) this process appears at 80-90K.

InAs 3

a 3

Region I

b

GaAs

Region III

Region II

peak positions of GS PL bands for the structures #1, #3, #5 are shown in Fig.8.

1.00

1.04

h

max, eV

1.08

2

1

2 1

1. 470°C 2. 510°C 3. 535°C

1.12

1.16

10

100

1000

PL Intensity (arb. un.)

10000

### **7. The analysis of Ga/In interdiffusion in QD structures with different density of InAs QDs**

Fig. 9 presents the variation of PL peak positions versus temperature in studied structures. PL peaks shift to low energy with increasing temperature due to the optical gap shrinkage. The lines in Fig.9 are the fitting results analyzed on the base of Varshni relation that presents the energy gap variation with temperature as [39]:

$$E(T) = \mathbf{E}\_o - \frac{aT^2}{T+b} \tag{2}$$

Fig. 9. The variation of PL peak positions versus temperature. The lines present the Varshni fitting results: 1- #1, 2- #2, 3- #3, 4-#4 and 5 - #5.

The comparison of fitting parameters with the variation of energy band gap versus temperature in the bulk InAs and GaAs crystals (Table 3) has revealed that in studied QD structures the fitting parameter "a" and 'b' in the temperature range 12-250 K are very close to their values for the bulk InAs crystal only in the structures #2 and #3. But in other QD structures the fitting parameter "a" and 'b' are different a little bit from the values in the bulk InAs crystal (Table 3). The last fact testifies that the process of Ga/In inter diffusion takes place in these QD structures [33,42-44]. Note that the process of Ga/In inter diffusion in studied structures passes non monotonically versus QD growth temperatures. It means that not only temperature but some other factors discussed below are essential as well.

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 167

2 2

the GaAs barrier into QWs and from QWs into QDs, respectively, *NQD* , *NQW* , *NGaAs* is the

recombination rates in QWs and QDs as well as the nonradiative recombination rates in the GaAs barrier and in QWs, respectively. The values EGaAs-QW, EQW-QD, EGaAs-QD are the energy differences between: (i) the GaAs band gap and the GS energy in QWs, (ii) the GS energy in

The temperature dependence of the exciton escape rates from QWs and QDs is taking into account, but temperature dependences of other parameters (trapping and recombination coefficients, density of states, exciton thermalization rates) are neglected. This is motivated by simplifying the calculation process and by minimization of the number of parameters samulteniously with significant progress in understanding the experimental results. Several comments deal with the experimental base for such simplifications are discussed below.

In presented model the variation of the GS PL intensity (I(T)) in QDs versus temperature in

*NR InGaAs GaAs*

1

*N E E N N*

1 1 11 1 1 1 1 2 1 2 1 1 11 1 1 1 1

*GaAs QW QW QD RQW NR NR QW GaAs RQW NR QW QW NR GaAs QD QD QW NR QW RQD QD RQD QW*

*N kT N kT*

Here the radiative lifetime in QDs ( τRQD ) is assumed to be constant with T as expected theoretically for strong confinement in three dimensions [15-18]. The radiative recombination rate in InGaAS/GaAs QWs ( τRQW ), as was shown in [22], is controled mainly by nonradiative recombination processes at the T≥100K. The dependence of the nonradiative rate in InGaAs/GaAs QWs was investigated as well in [22]. Its value in QWs, as a rule, increases at low temperatures and saturates at 120-150K at a constant value depending on the quality of the structure. As result it is possible to neglect by temperature

NR1 τ and 1

to increase versus temperature due to the enlargement of the phonon number. However, it was shown in [42] that the PL rise time in a QW (dependent on the exciton capture rate into

The dominant multiphonon capture mechanism and the InGaAs/GaAs QD capture time

decreasing with temperature were confirmed experimentally in time-resolved

) for all temperatures from the range 50-300K is the same about 1-2 ps.

*G G*

, 1

multiple longitudinal optical phonon (LO) emission [41-43]. The values 1

The exciton thermalization (capture) rates from the GaAs barrier into a QW 1

1 1 1 1 1

*QW NR*

, 1

*dC E C CN CN*

*QW QD*

 *QW* ,

NR1 τ and 1

*RQD*

(5)

*QD* are the exciton capture

NR2 τ are the radiative exciton

*NR QD*

*QW*

 and 1 *QD* have

*QW* exp

*N kT*

(6)

 *GaAs QD*

*E*

and from a

 1 1

NR2 τ at the T≥120-150K.

, are the multiphonon-assisted processes deal with the scattering via

are exciton thermalization (capture) rates from

1 2 exp

, 1 τRQD

QWs and in QDs, (iii) the GaAs band gap and the GS energy in QDs, respectively.

*dt kT*

*QD*

*QD QD QD QW*

 and 1 QD QD N

density of states in QDs, QWs and in the GaAs barrier,

the stationary state can be described by the formula:

dependences of the parameters: 1 τRQW

*QD*

1

1 exp exp

Here 1

QW into QDs, 1

QWs, 1 *QW*

1 *QD*

*I T*

 QW QW N *QW*

coefficients into QWs and QDs, 1 τRQW

Table 3. The Varshni fitting parameters

### **8. Model for the dependence of PL integrated intensity versus temperature**

An analysis of the thermal behavior of QD luminescence indicates that excitons are the dominant electronic particles. Thus to modeling the dependence of the ground state PL intensity versus temperature the simple assumption is applied: the carriers are considered to behave as excitons or correlated electron-hole pairs. This assumption is a common feature of most exiting models [15-18]. The motivation of the choosing this approach in this paper will be clear from presented esperimental results as well.

For InAs/InGaAs structures, the two-stage processes of exciton capture and thermal escape in/from QDs have been considered [17, 33]. These processes in QD structures can occur not only through the wetting layer (WL) states, as was proposed earlier [16, 18], but also through the capping/buffer InGaAs QW layers [17, 33]. However in [17] the localization of nonradiative defects was considered in GaAs barrier mainly. In present model two types of nonradiative defects: in the GaAs barrier (NR1) and in the capping/buffer In0.15Ga0.85As QWs (NR2) are taken into account [33]. To simplify the rate equations, one intermediate level, referred as QW, will be considered in the model.

It is supposed that photo-generated excitons are created in GaAs barrier and in In0.15Ga0.85As QWs with generation rates GGaAs and GInGaAs, respectively. The exciton recombination takes place in the GaAs barrier, InGaAs QWs and in InAs QDs. In this model all QDs in an ensamble are assumed to be identical with the same properties and the process of carrier thermal redistribution between QDs is inessential. This is supported by the experimental fact that the FWHM of the ground state PL band is measured to be constant in the temperature range of 10-300K. The system of rate equations for exciton concentrations in the GaAs barrier (Co,), QWs (C1) and QDs (C2) can be written as:

$$\frac{d\mathbf{C}\_0}{dt} = \mathbf{G}\_{\text{CaAs}} - a\_{\text{QVV}}\mathbf{C}\_0\mathbf{N}\_{\text{QV}} + a\_{\text{QV}}\mathbf{C}\_1\mathbf{N}\_{\text{CaAs}}\exp\left(-\frac{\Delta E^{\text{GaAs}-\text{QV}}}{kT}\right) - \frac{\mathbf{C}\_0}{\tau\_{\text{NR1}}}\tag{3}$$

$$\begin{split} \frac{d\mathbf{C}\_{1}}{dt} &= \mathbf{G}\_{\text{InGaAs}} + a\_{\text{QV}} \mathbf{C}\_{0} \mathbf{N}\_{\text{QV}} - a\_{\text{QV}} \mathbf{C}\_{1} \mathbf{N}\_{\text{GaAs}} \exp\left(-\frac{\Delta E^{\text{CaAs}-\text{QV}}}{kT}\right) - a\_{\text{QD}} \mathbf{C}\_{1} \mathbf{N}\_{\text{QD}} + \\ &+ a\_{\text{QV}} \mathbf{C}\_{2} \mathbf{N}\_{\text{QV}} \exp\left(-\frac{\Delta E^{\text{QV}-\text{QV}}}{kT}\right) - \frac{\mathbf{C}\_{1}}{\tau\_{\text{RQV}}} - \frac{\mathbf{C}\_{1}}{\tau\_{\text{NR2}}} \end{split} \tag{4}$$

Structure numbers eV meV/ oK oK

 InAs [40] 0.415 0.276 93 GaAs [41] 1.519 0.540 204

**8. Model for the dependence of PL integrated intensity versus temperature**  An analysis of the thermal behavior of QD luminescence indicates that excitons are the dominant electronic particles. Thus to modeling the dependence of the ground state PL intensity versus temperature the simple assumption is applied: the carriers are considered to behave as excitons or correlated electron-hole pairs. This assumption is a common feature of most exiting models [15-18]. The motivation of the choosing this approach in this paper will

For InAs/InGaAs structures, the two-stage processes of exciton capture and thermal escape in/from QDs have been considered [17, 33]. These processes in QD structures can occur not only through the wetting layer (WL) states, as was proposed earlier [16, 18], but also through the capping/buffer InGaAs QW layers [17, 33]. However in [17] the localization of nonradiative defects was considered in GaAs barrier mainly. In present model two types of nonradiative defects: in the GaAs barrier (NR1) and in the capping/buffer In0.15Ga0.85As QWs (NR2) are taken into account [33]. To simplify the rate equations, one intermediate

It is supposed that photo-generated excitons are created in GaAs barrier and in In0.15Ga0.85As QWs with generation rates GGaAs and GInGaAs, respectively. The exciton recombination takes place in the GaAs barrier, InGaAs QWs and in InAs QDs. In this model all QDs in an ensamble are assumed to be identical with the same properties and the process of carrier thermal redistribution between QDs is inessential. This is supported by the experimental fact that the FWHM of the ground state PL band is measured to be constant in the temperature range of 10-300K. The system of rate equations for exciton concentrations in the

0 0

*dC E C G CN CN*

*dC E G CN CN C N*

exp

0 1 1

exp

*InGaAs QW QW QW GaAs QD QD QW QD*

*E CC C N kT*

*GaAs QW*

2

*GaAs QW*

1 1

*RQW NR*

1

(3)

(4)

*NR*

0 1

*dt kT*

*GaAs QW QW QW GaAs*

exp

*dt kT*

Table 3. The Varshni fitting parameters

be clear from presented esperimental results as well.

level, referred as QW, will be considered in the model.

GaAs barrier (Co,), QWs (C1) and QDs (C2) can be written as:

2

*QD QW*

1

E0 A b

#1 1.082 0.355 110 #2 1.089 0.346 98 #3 1.010 0.300 95 #4 1.049 0.330 110 #5 1.079 0.335 130

$$\frac{d\mathcal{C}\_2}{dt} = a\_{\rm QD}\mathcal{C}\_1 N\_{\rm QD} - a\_{\rm QD}\mathcal{C}\_2 N\_{\rm QW} \exp\left(-\frac{\Delta E^{\rm QV-QD}}{kT}\right) - \frac{\mathcal{C}\_2}{\tau\_{\rm RQD}}\tag{5}$$

Here 1 QW QW N *QW* and 1 QD QD N *QD* are exciton thermalization (capture) rates from the GaAs barrier into QWs and from QWs into QDs, respectively, *NQD* , *NQW* , *NGaAs* is the density of states in QDs, QWs and in the GaAs barrier, *QW* , *QD* are the exciton capture coefficients into QWs and QDs, 1 τRQW , 1 τRQD , 1 NR1 τ and 1 NR2 τ are the radiative exciton recombination rates in QWs and QDs as well as the nonradiative recombination rates in the GaAs barrier and in QWs, respectively. The values EGaAs-QW, EQW-QD, EGaAs-QD are the energy differences between: (i) the GaAs band gap and the GS energy in QWs, (ii) the GS energy in QWs and in QDs, (iii) the GaAs band gap and the GS energy in QDs, respectively.

The temperature dependence of the exciton escape rates from QWs and QDs is taking into account, but temperature dependences of other parameters (trapping and recombination coefficients, density of states, exciton thermalization rates) are neglected. This is motivated by simplifying the calculation process and by minimization of the number of parameters samulteniously with significant progress in understanding the experimental results. Several comments deal with the experimental base for such simplifications are discussed below.

In presented model the variation of the GS PL intensity (I(T)) in QDs versus temperature in the stationary state can be described by the formula:

$$\begin{aligned} \text{I} \quad \text{(I)} \quad &\frac{\text{G}\_{\text{K}\text{Ca}^{+}} + \text{G}\_{\text{CaO}^{+}}}{1 + \frac{\tau\_{\text{K}\text{PO}^{-}}^{-1} + \tau\_{\text{NH}\_{2}^{+}}^{-1}}{\tau\_{\text{-}\text{O}^{2}}^{-1} \left(\tau\_{\text{-}\text{O}^{2}}^{-1} + \tau\_{\text{NH}\_{2}^{+}}^{-1}\right) \text{N}\_{\text{QIV}}} \times \exp\left(-\frac{\text{AE}^{\text{Ca}\text{s}} - \text{G}^{\text{H}}}{kT}\right) + \frac{\text{G}^{\text{2}}\_{\text{K}\text{O}^{2}} + \tau\_{\text{NH}\_{2}^{+}}^{-1} \text{N}\_{\text{QIV}}}{\tau\_{\text{-}\text{K}^{2}}^{-1} \text{N}\_{\text{QIV}}} \exp\left(-\frac{\text{AE}^{\text{O}\text{s}} - \text{G}^{\text{H}}}{kT}\right) + \frac{\tau\_{\text{O}^{2}}^{-1} \tau\_{\text{K}\text{S}}^{-1} \text{N}\_{\text{QIV}}}{\tau\_{\text{-}\text{K}^{2}}^{-1} \left(\tau\_{\text{H}\text{O}^{2}}^{-1} + \tau\_{\text{N}\text{I}^{2}}\right) \text{N}\_{\text{QIV}}} \times \exp\left(-\frac{\text{AG}^{\text{Ca}\text{s}} - \text{G}^{\text{H}}}{kT}\right) \tag{6} \end{aligned} \tag{6}$$

Here the radiative lifetime in QDs ( τRQD ) is assumed to be constant with T as expected theoretically for strong confinement in three dimensions [15-18]. The radiative recombination rate in InGaAS/GaAs QWs ( τRQW ), as was shown in [22], is controled mainly by nonradiative recombination processes at the T≥100K. The dependence of the nonradiative rate in InGaAs/GaAs QWs was investigated as well in [22]. Its value in QWs, as a rule, increases at low temperatures and saturates at 120-150K at a constant value depending on the quality of the structure. As result it is possible to neglect by temperature dependences of the parameters: 1 τRQW , 1 NR1 τ and 1 NR2 τ at the T≥120-150K.

The exciton thermalization (capture) rates from the GaAs barrier into a QW 1 *QW* and from a QW into QDs, 1 *QD* , are the multiphonon-assisted processes deal with the scattering via multiple longitudinal optical phonon (LO) emission [41-43]. The values 1 *QW* and 1 *QD* have to increase versus temperature due to the enlargement of the phonon number. However, it was shown in [42] that the PL rise time in a QW (dependent on the exciton capture rate into QWs, 1 *QW* ) for all temperatures from the range 50-300K is the same about 1-2 ps.

The dominant multiphonon capture mechanism and the InGaAs/GaAs QD capture time 1 *QD* decreasing with temperature were confirmed experimentally in time-resolved

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 169

**In the range I** (100-180K) the activation energy (Ea (I) = 48-55 meV) of PL decay measured at low excitation power for GS PL bands (Fig.10) is very close to the energy difference between the GaAs band gap and the GS energy of the WL and/or the buffer InGaAs QW layer (peak A, B, C in Fig.4, Table 2). It can be supposed that thermal quenching of the GS PL intensity in the temperature range I is due to decreasing an exciton flow to QDs caused by thermal escape of excitons from the WL (or buffer InGaAs QW) into the GaAs barrier where they are lost through subsequent nonradiative recombination. If the above mentioned mechanism takes place in QD structures the same activation energy ( *GaAs QW E* ) has to be detected in the range I for thermal decay of PL bands connected with the GS and ES of QDs, as well as for the QW PL band. Actually the results presented in Fig.11 testify that in the range I thermal decay of PL bands deal with the GS, four ES in QDs, as well as with a QW is characterized

4 6 8 10

/T (K-1)

103

Fig. 11. Arrhenius plots for the thermal decay of the integrated PL intensity measured at

**In the temperature range II** (180-250K) the activation energy (Ea (II) = 170-173 meV) of PL decay for GS bands (Fig.10 and 11) is very close to the energy difference between the GaAs band gap and the GS energy for the capping In0.15Ga0.85As QW (peak D, Fig.4, Table 2). Thus the process of thermal decay of the QD PL intensity in the range II can be attributed to diminishing of an exciton flow to QDs caused by thermal escape of excitons from the capping In0.15Ga0.85As QW into the GaAs barrier. This explanation is confirmed as well by

50 meV

53 meV

51 meV

53 meV

GS

1ES

WL

4ES

3ES

2ES

by the same activation energy from the range 48-53meV.

241 meV

286 meV

350 meV

349 meV

160 meV 239 meV 55 meV

338 meV 55 meV

177 meV

177 meV

170 meV

168 meV

153 meV

0

the power density of 1000W/cm2 for the QD structures #3

5

10

15

PL Integrated Intensity

ln[(Imax/I(T)-1)] (arb. units)

20

25

30

35

experiments [45-47]. It was shown using the investigation of the PL rise time for the GS in QDs that its value decreases: from 6.5ps to 3.5ps upon increathing the temperature from 4 to 300K [45], or from 15ps to 7ps at the temperature rise from 50 to 300K [46]. Thus nearly twofold decreasing of the exciton capture time into QDs is revealed experimentally versus temperature. It is essential that the temperature dependence of the exciton escape rates from QWs and from QDs obviously much stronger and it is taking into account in the model.

### **9. Discussion of PL integrated intensity dependences versus temperature**

It is clear from the formula (6), that in high quality QD structures (#2, #3 and #4) with low concentrations of the NR1 and NR2 centres the PL intensity I(T) is linearly dependent on excitation power (or generation rates, GInGaAs, GGaAs), as it is demonstrated in Fig.5. In low quality QD structures (#1 and #5) photo-generated excitons recombine partially via NR1 and NR2 centres and W (T) changes Sublineary versus power (Fig.5).

The analysis of PL temperature dependences is resonable to provide for the temeparture ranges I – III separately (Fig.8). At low temperature (< 100K) the processes of exciton thermal escape from a QW into the GaAs barrier and from QDs into a QW are not essential, and the PL intensity I (T) does not change (Fig. 8).

In the temperature ranges I00-300K following the model the activation energy of PL thermal decay has correspond to the values , , *GaAs QW QW QD GaAs QD E EE* . To quantitative determination of activation energies the {Imax/I(T) – 1} dependences for the GS and ES PL bands were plotted versus temperature in Arrhenius plots (Fig.10). Three distinct linear regions (I-III) with the corresponding activation energies Ea (I), Ea (II) and Ea(III) are observed for different temperatures (Fig.10).

Fig. 10. Arrhenius plots for the thermal decay of the GS integrated PL intensity measured at the power density of 500W/cm2 for the structures #3 (1), #2 (2), #1 (3) and #5 (4).

experiments [45-47]. It was shown using the investigation of the PL rise time for the GS in QDs that its value decreases: from 6.5ps to 3.5ps upon increathing the temperature from 4 to 300K [45], or from 15ps to 7ps at the temperature rise from 50 to 300K [46]. Thus nearly twofold decreasing of the exciton capture time into QDs is revealed experimentally versus temperature. It is essential that the temperature dependence of the exciton escape rates from QWs and from QDs obviously much stronger and it is taking into account in the model.

**9. Discussion of PL integrated intensity dependences versus temperature**

and NR2 centres and W (T) changes Sublineary versus power (Fig.5).

and the PL intensity I (T) does not change (Fig. 8).

observed for different temperatures (Fig.10).

279 meV

297 meV

PL integrated intensity ln[(I0-I)/I]

(arb.un.)

It is clear from the formula (6), that in high quality QD structures (#2, #3 and #4) with low concentrations of the NR1 and NR2 centres the PL intensity I(T) is linearly dependent on excitation power (or generation rates, GInGaAs, GGaAs), as it is demonstrated in Fig.5. In low quality QD structures (#1 and #5) photo-generated excitons recombine partially via NR1

The analysis of PL temperature dependences is resonable to provide for the temeparture ranges I – III separately (Fig.8). At low temperature (< 100K) the processes of exciton thermal escape from a QW into the GaAs barrier and from QDs into a QW are not essential,

In the temperature ranges I00-300K following the model the activation energy of PL thermal decay has correspond to the values , , *GaAs QW QW QD GaAs QD E EE* . To quantitative determination of activation energies the {Imax/I(T) – 1} dependences for the GS and ES PL bands were plotted versus temperature in Arrhenius plots (Fig.10). Three distinct linear regions (I-III) with the corresponding activation energies Ea (I), Ea (II) and Ea(III) are

173 meV

173 meV

172 meV

452 meV 52 meV

170 meV

52 meV 472 meV

345678

/T (K-1 ]

103

Fig. 10. Arrhenius plots for the thermal decay of the GS integrated PL intensity measured at

the power density of 500W/cm2 for the structures #3 (1), #2 (2), #1 (3) and #5 (4).

4 3

48 meV

55 meV

2

1

**In the range I** (100-180K) the activation energy (Ea (I) = 48-55 meV) of PL decay measured at low excitation power for GS PL bands (Fig.10) is very close to the energy difference between the GaAs band gap and the GS energy of the WL and/or the buffer InGaAs QW layer (peak A, B, C in Fig.4, Table 2). It can be supposed that thermal quenching of the GS PL intensity in the temperature range I is due to decreasing an exciton flow to QDs caused by thermal escape of excitons from the WL (or buffer InGaAs QW) into the GaAs barrier where they are lost through subsequent nonradiative recombination. If the above mentioned mechanism takes place in QD structures the same activation energy ( *GaAs QW E* ) has to be detected in the range I for thermal decay of PL bands connected with the GS and ES of QDs, as well as for the QW PL band. Actually the results presented in Fig.11 testify that in the range I thermal decay of PL bands deal with the GS, four ES in QDs, as well as with a QW is characterized by the same activation energy from the range 48-53meV.

Fig. 11. Arrhenius plots for the thermal decay of the integrated PL intensity measured at the power density of 1000W/cm2 for the QD structures #3

**In the temperature range II** (180-250K) the activation energy (Ea (II) = 170-173 meV) of PL decay for GS bands (Fig.10 and 11) is very close to the energy difference between the GaAs band gap and the GS energy for the capping In0.15Ga0.85As QW (peak D, Fig.4, Table 2). Thus the process of thermal decay of the QD PL intensity in the range II can be attributed to diminishing of an exciton flow to QDs caused by thermal escape of excitons from the capping In0.15Ga0.85As QW into the GaAs barrier. This explanation is confirmed as well by

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 171

*K E kT K E kT K E kT*

The values Io and the activation energies E1, E2 and E3 for different temperature ranges (I-III) were taken from the experimental results presented in Fig.10 (Table 4). Parameters K1, K2 and K3 have been obtained from the numerical simulation procedure (Table 4) [48,49]. As it follows from Eq.6 the coefficients K1 and K3 related to the exciton nonradiative recombination rate (τNR1-1) in the GaAs barrier, but the coefficient K2 depends on the exciton nonradiative recombination rate (τNR2-1) in InGaAs QWs. Numerical simulation results presented in Table 4 have shown that the coefficients K1 and K3 related to the exciton nonradiative recombination rate (τNR1-1) in the GaAs barrier of the structure #3 are less in comparison with those in #1 and #5 [48-50]. Simultaneously the coefficient K2 related to the exciton nonradiative recombination rate (τNR2-1) in InGaAs QWs for the structure #3 is one order smaller than in #1 and #5. These facts are the reason of fast PL thermal decay and lower integrated PL intensities in structures #1 and #5 in comparison

*I I*

meV

Table 4. Fitting parameters estimated from the PL thermal decay

E2 meV

E3 meV

#1 1.1 104 53 174 451 100 2.0 105 6.7 109 #3 1.7 104 52 175 452 85 2.3 104 6.0 109 #5 8.9 103 54 173 450 100 2.3 105 109

Thus, it is shown that the excitonic nature of carriers is important for the processes of capture and thermal escape in/from QDs in DWELL structures. At low temperatures the GS and ES PL thermal decays are attributed to the reduction of exciton flow into the QDs due to the exciton thermal escape from the WL or InGaAs buffer layers (100–180 K) or from the GS of capping In0.15 Ga0.85 As layers (180–250 K) into the GaAs barrier with subsequent NR recombination. At high temperatures (250–300 K) the activation energy of PL thermal decay depends on the QD density and the quality of DWELL structures. In DWELL structures with high emission (#2,# 3, and #4) the activation energy matches the energy difference between the GaAs band gap and the GS level of QDs. In structures with weak emission (#1 and #5) the activation energy is close to the energy difference between the GS level of QDs and the GS energy level in InGaAs/GaAs QWs. The reasons of DWELL quality variation

**11. X ray diffraction study in InAs QD structures with the different densities**

It is important to discuss the reasons of the quality change in studied QD structures with different InAs QD densities. The application of the capping/buffer In*x*Ga1−*x* As layers in the QD structure has been demonstrated as an effective means to the QD density increase

with the structure #3.

Structure Io E1

versus QD density are discussed below.

**of QDs**

0 11 22 33 1 exp( ) exp( ) exp( )

(9)

K1 K2 K3

the thermal quenching of PL bands deal with the GS and 1ES-4ES in QDs, and with the capping QW. All PL bands in the range II demonstrate the activation energy from the range of 153-177 meV (Fig.11).

**At high temperatures, stage III** (250-300K), the activation energy of PL thermal decay depends on the quality of DWELL structures (Fig.10). As one follows from the formula (6) the activation energy of PL thermal decay will be close to the value *GaAs QD E* if the follows two relations are correct:

$$\frac{\left(\tau\_{RQM}^{-1} + \tau\_{NR2}^{-1}\right) \mathcal{N}\_{QW}}{\tau\_{RQD}^{-1} \mathcal{N}\_{QD}} \exp\left(-\frac{\Delta E^{QW-QD}}{kT}\right) \\
\quad \ll \frac{\tau\_{QW}^{-1} \tau\_{NR1}^{-1} \mathcal{N}\_{GaAs}}{\tau\_{RQD}^{-1} \left(\tau\_{QV}^{-1} + \tau\_{NR1}^{-1}\right) \mathcal{N}\_{QD}} \exp\left(-\frac{\Delta E^{GaAs-QD}}{kT}\right) \tag{7}$$

or after the simplification of the (7):

$$(\tau\_{RQM}^{-1} + \tau\_{NR2}^{-1})N\_{QW} \cdot \exp\left(-\frac{E\_{GS}^{QW}}{kT}\right) \ll \frac{\tau\_{QW}^{-1}\tau\_{NR1}^{-1}N\_{GaAs}}{\left(\tau\_{QW}^{-1} + \tau\_{NR1}^{-1}\right)} \exp\left(-\frac{E\_g^{GaAs}}{kT}\right) \tag{8}$$

It means that if the concentration of nonradiative defects in QWs (NR2) is low the activation energy of PL thermal decay for the GS PL band in QDs will approach to the value *GaAs QD E* . Actually for the high quality structures (#2, #3) the activation energies (452 and 472 meV) in the range III (Fig.10) are close to the *GaAs QD E* . Note this value is the sum of the barrier energy for electrons and holes in QDs. It testifies that just excitons thermally escape in the range III from QDs into the GaAs barrier with subsequent NR recombination. In the case of measurement at the high excittaion power (Fig.11) the decrease of activation energy for GS thermal decay deals with exciton re-localization from the ES to the GS of other QDs which slow down the process of GS thermal quenching.

In low efficient QD structures (#1, #5) the process of GS PL thermal decay is characterized by smaller activation energies, 279 and 297 meV (Fig.10). For these structures it is natural to supose the high concentration of the NR2 centers in InGaAs QWs. It means the conditions (7) and (8) do not satisfy. As one follows from the formula (6) in this case the activation energy of GS PL thermal decay will approach to the value *QW QD E* .

Note that obtained experimental results do not present an evidence for the exciton thermal dissociation in QWs or in QDs at high temperatures with subsiquent re-localization, thermal escape or tunneling of separated electrons or holes. If the process of exciton thermal dissociation is realized the activation energy of GS PL thermal decay should be smaller and comparative with the value of barriers for electrons or holes in the QD structures that was not observed.

### **10. Fitting of the data of PL integrated intensity thermal decay**

The fitting procedure was applied to the experimental curves for thermal decay of integrated PL intensities presented in Fig.8. As it follows from Eq.6 the integrated PL intensity thermal decay can be simulated as:

the thermal quenching of PL bands deal with the GS and 1ES-4ES in QDs, and with the capping QW. All PL bands in the range II demonstrate the activation energy from the range

**At high temperatures, stage III** (250-300K), the activation energy of PL thermal decay depends on the quality of DWELL structures (Fig.10). As one follows from the formula (6) the activation energy of PL thermal decay will be close to the value *GaAs QD E* if the follows

<<

*QW EGS kT*  *QW NR GaAs QW NR*

1 1 1 1 1

*RQD QW NR QD*

1

1

exp *GaAs QD QW NR GaAs*

*N kT*

*GaAs Eg kT*  (7)

(8)

*N E*

exp

1 1 1 11 1

<<

It means that if the concentration of nonradiative defects in QWs (NR2) is low the activation energy of PL thermal decay for the GS PL band in QDs will approach to the value *GaAs QD E* . Actually for the high quality structures (#2, #3) the activation energies (452 and 472 meV) in the range III (Fig.10) are close to the *GaAs QD E* . Note this value is the sum of the barrier energy for electrons and holes in QDs. It testifies that just excitons thermally escape in the range III from QDs into the GaAs barrier with subsequent NR recombination. In the case of measurement at the high excittaion power (Fig.11) the decrease of activation energy for GS thermal decay deals with exciton re-localization from the ES to the GS of

In low efficient QD structures (#1, #5) the process of GS PL thermal decay is characterized by smaller activation energies, 279 and 297 meV (Fig.10). For these structures it is natural to supose the high concentration of the NR2 centers in InGaAs QWs. It means the conditions (7) and (8) do not satisfy. As one follows from the formula (6) in this case the activation

Note that obtained experimental results do not present an evidence for the exciton thermal dissociation in QWs or in QDs at high temperatures with subsiquent re-localization, thermal escape or tunneling of separated electrons or holes. If the process of exciton thermal dissociation is realized the activation energy of GS PL thermal decay should be smaller and comparative with the value of barriers for electrons or holes in the QD structures that was

The fitting procedure was applied to the experimental curves for thermal decay of integrated PL intensities presented in Fig.8. As it follows from Eq.6 the integrated PL

 *N*

of 153-177 meV (Fig.11).

two relations are correct:

1 1

not observed.

2 1 exp *QW QD RQW NR QW*

or after the simplification of the (7):

1 1 2 ( )

 *RQW NR QW N* exp

*N E N kT*

other QDs which slow down the process of GS thermal quenching.

energy of GS PL thermal decay will approach to the value *QW QD E* .

**10. Fitting of the data of PL integrated intensity thermal decay**

intensity thermal decay can be simulated as:

*RQD QD*

$$I = \frac{I\_0}{1 + K\_1 \exp(-E\_1/kT) + K\_2 \exp(-E\_2/kT) + K\_3 \exp(-E\_3/kT)}\tag{9}$$

The values Io and the activation energies E1, E2 and E3 for different temperature ranges (I-III) were taken from the experimental results presented in Fig.10 (Table 4). Parameters K1, K2 and K3 have been obtained from the numerical simulation procedure (Table 4) [48,49]. As it follows from Eq.6 the coefficients K1 and K3 related to the exciton nonradiative recombination rate (τNR1-1) in the GaAs barrier, but the coefficient K2 depends on the exciton nonradiative recombination rate (τNR2-1) in InGaAs QWs. Numerical simulation results presented in Table 4 have shown that the coefficients K1 and K3 related to the exciton nonradiative recombination rate (τNR1-1) in the GaAs barrier of the structure #3 are less in comparison with those in #1 and #5 [48-50]. Simultaneously the coefficient K2 related to the exciton nonradiative recombination rate (τNR2-1) in InGaAs QWs for the structure #3 is one order smaller than in #1 and #5. These facts are the reason of fast PL thermal decay and lower integrated PL intensities in structures #1 and #5 in comparison with the structure #3.

Table 4. Fitting parameters estimated from the PL thermal decay

Thus, it is shown that the excitonic nature of carriers is important for the processes of capture and thermal escape in/from QDs in DWELL structures. At low temperatures the GS and ES PL thermal decays are attributed to the reduction of exciton flow into the QDs due to the exciton thermal escape from the WL or InGaAs buffer layers (100–180 K) or from the GS of capping In0.15 Ga0.85 As layers (180–250 K) into the GaAs barrier with subsequent NR recombination. At high temperatures (250–300 K) the activation energy of PL thermal decay depends on the QD density and the quality of DWELL structures. In DWELL structures with high emission (#2,# 3, and #4) the activation energy matches the energy difference between the GaAs band gap and the GS level of QDs. In structures with weak emission (#1 and #5) the activation energy is close to the energy difference between the GS level of QDs and the GS energy level in InGaAs/GaAs QWs. The reasons of DWELL quality variation versus QD density are discussed below.

### **11. X ray diffraction study in InAs QD structures with the different densities of QDs**

It is important to discuss the reasons of the quality change in studied QD structures with different InAs QD densities. The application of the capping/buffer In*x*Ga1−*x* As layers in the QD structure has been demonstrated as an effective means to the QD density increase

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 173

a

GaAs-layers

1 2

5

K

3 4

66.0 66.2 66.4

K

a K

59.000 59.025 59.050 59.075

GaAs bullk

66.04 66.08 66.12 66.16 66.20 66.24 66.28 66.32

GaAs-layer

460 480 500 520 540

b K

T, o C

460 480 500 520 540

K

K

b

 T, o C

2

1

3 4

2

5

Fig. 12. XRD peaks related to the diffraction of Kα1 and Kα2 lines on (400) cubic GaAs crystal plane for the structures with QDs grown at 470 (1), 490 (2), 510 (3), 525(4) and 535 oC (5).

GaAs layers

58.8 58.9 59.0 59.1 59.2 59.3

Fig. 13. XRD peaks related to the diffraction of K line on (400) cubic GaAs crystal plane in

Figures 14 presents the superposition of XRD peaks related to the diffraction of Kα1 and Kα2 lines of the X-ray Cu source from the (200) crystal planes of cubic GaAs substrate and GaAs

DWELLs with QDs grown at 470 (1), 490 (2), 510 (3), 525(4) and 535 oC (5).

0

0

2000

4000

6000

XRD intensity (arb. un.)

8000

10000

(400)

GaAs bulk

12000

80000

160000

240000

XRD intensity, arb.un.

GaAs-bulk

(400)

320000

400000

[3], to tune the GS PL transition to the 1.3 µm spectral region [3,17, 51] and to narrow a PL line width [52]. However, the structural and electronic properties of InAs QDs coupled with In*x*Ga1−*x* As/GaAs QWs are still understood partially. It is generally supposed that this type of growth provides the potential to strain engineering of structural and electronic properties due to efficient strain relaxation altering the electronic potential of capped QDs [51, 53]. Additionally the In*x*Ga1−*x* As layer reduces the inhomogeneous surface stress enhancing the ability to grow a multitude of identical uncorrelated QD layers [51]. The majority of publications considered the stress variation in the vicinity of QDs or ordered arrays of QDs [52-58].

To investigate the strain levels in QD structures the X ray diffraction (XRD) has been studied. The XRD experiments were made using the XRD equipment model of D-8 advanced (Bruker Co.) with Kα1 line from the Cu source (λ=1.5406Å). Figure 12 present the superposition of XRD peaks related to the diffraction of Kα1 and Kα2 lines of Cu source from the (400) crystal planes of the cubic GaAs substrate and GaAs layers in studied InGaAs/GaAs QWs [59.60].

Table 5. Values of 2 angles for the diffraction of Kα1, Kα2 and K X-ray beams of Cu source from the (400) and (200) cubic GaAs crystal planes [61].

As one can see the peaks (66.05o and 66.24o) related to the diffraction of Kα1 and Kα2 lines from the (400) crystal planes in GaAs QW structures #2, #3 and #4 with QDs grown at 490- 525 oC locate very close to corresponding XRD peaks (66.044o and 66.225o [61]) of the bulk cubic GaAs (Fig.12, Table 5). The last fact testifies that the level of elastic strain in InGaAs/GaAs QWs of #2, #3, #4 is minimum. In contrary in the structures with QDs grown at 470 and 535 oC the corresponding XRD peaks shift to 66.10o and 66.29o (Fig.12) testifying the higher levels of compressive strains in InGaAs/GaAs QWs of these structures.

Figure 13 presents the additional confirmation of conclusions mentioned above. It shows the superposition of XRD peaks related to the diffraction of K line from the same (400) crystal planes of cubic GaAs substrate and of GaAs layers in InGaAs/GaAs QWs. As one can see the peaks related to the K line diffraction from the (400) crystal plan of GaAs layers in DWELLs with QDs grown at 490-525 oC (59.036-59.042o) locate close to the corresponding XRD peak in the bulk GaAs (Table 5, Fig.12) for the same (400) crystal plane. Thus the level of elastic strain in these DWELLs is minimum. In DWELLs with QDs grown at 470 and 535 oC the corresponding XRD peaks shift to 59.08o testifing the higher level of compresive strain. Note that the partial relaxation of elastic strain can stimulate the creation of nonradiative recombination centers and due to this decreasing the PL intensity in the structures with QDs grown at 470 and 535 oC.

[3], to tune the GS PL transition to the 1.3 µm spectral region [3,17, 51] and to narrow a PL line width [52]. However, the structural and electronic properties of InAs QDs coupled with In*x*Ga1−*x* As/GaAs QWs are still understood partially. It is generally supposed that this type of growth provides the potential to strain engineering of structural and electronic properties due to efficient strain relaxation altering the electronic potential of capped QDs [51, 53]. Additionally the In*x*Ga1−*x* As layer reduces the inhomogeneous surface stress enhancing the ability to grow a multitude of identical uncorrelated QD layers [51]. The majority of publications considered the stress variation in the vicinity of

To investigate the strain levels in QD structures the X ray diffraction (XRD) has been studied. The XRD experiments were made using the XRD equipment model of D-8 advanced (Bruker Co.) with Kα1 line from the Cu source (λ=1.5406Å). Figure 12 present the superposition of XRD peaks related to the diffraction of Kα1 and Kα2 lines of Cu source from the (400) crystal planes of the cubic GaAs substrate and GaAs layers in studied

(400) (200)

d1, A 22

Table 5. Values of 2 angles for the diffraction of Kα1, Kα2 and K X-ray beams of Cu source

As one can see the peaks (66.05o and 66.24o) related to the diffraction of Kα1 and Kα2 lines from the (400) crystal planes in GaAs QW structures #2, #3 and #4 with QDs grown at 490- 525 oC locate very close to corresponding XRD peaks (66.044o and 66.225o [61]) of the bulk cubic GaAs (Fig.12, Table 5). The last fact testifies that the level of elastic strain in InGaAs/GaAs QWs of #2, #3, #4 is minimum. In contrary in the structures with QDs grown at 470 and 535 oC the corresponding XRD peaks shift to 66.10o and 66.29o (Fig.12) testifying

Figure 13 presents the additional confirmation of conclusions mentioned above. It shows the superposition of XRD peaks related to the diffraction of K line from the same (400) crystal planes of cubic GaAs substrate and of GaAs layers in InGaAs/GaAs QWs. As one can see the peaks related to the K line diffraction from the (400) crystal plan of GaAs layers in DWELLs with QDs grown at 490-525 oC (59.036-59.042o) locate close to the corresponding XRD peak in the bulk GaAs (Table 5, Fig.12) for the same (400) crystal plane. Thus the level of elastic strain in these DWELLs is minimum. In DWELLs with QDs grown at 470 and 535 oC the corresponding XRD peaks shift to 59.08o testifing the higher level of compresive strain. Note that the partial relaxation of elastic strain can stimulate the creation of nonradiative recombination centers and due to this decreasing the PL intensity in the

the higher levels of compressive strains in InGaAs/GaAs QWs of these structures.

(degree) K2

66.044 59.0165 1.414 66.225 31.63 2.828 31.71

21 (degree) K1

d2, A 23

(degree) K2

QDs or ordered arrays of QDs [52-58].

InGaAs/GaAs QWs [59.60].

(degree) K1

21 (degree) K

from the (400) and (200) cubic GaAs crystal planes [61].

structures with QDs grown at 470 and 535 oC.

Material 21

GaAs Bulk

Fig. 12. XRD peaks related to the diffraction of Kα1 and Kα2 lines on (400) cubic GaAs crystal plane for the structures with QDs grown at 470 (1), 490 (2), 510 (3), 525(4) and 535 oC (5).

Fig. 13. XRD peaks related to the diffraction of K line on (400) cubic GaAs crystal plane in DWELLs with QDs grown at 470 (1), 490 (2), 510 (3), 525(4) and 535 oC (5).

Figures 14 presents the superposition of XRD peaks related to the diffraction of Kα1 and Kα2 lines of the X-ray Cu source from the (200) crystal planes of cubic GaAs substrate and GaAs

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 175

temperature the surface area of QDs reduces (Table 1) and the InGaAs/GaAs interface area between the QDs enlarges. The variation non-monotonically of the integrated PL intensity versus QD density in studied structures, apparently, is connected with the competition of

460 470 480 490 500 510 520 530 540

bulk GaAs 1

QD growth temperature, o

Fig. 15. The 2Θ angles for the diffraction of Kα1 and Kα2 lines from the (400) crystal planes of

the GaAs substrate and GaAs QW layers in DWELLs with QD grown at different

temperatures (or with different QD densities).

C

2

3

4

K

K

mentioned above two effects.

66.05

66.10

66.15

66.20

XRD intensity, arb.un

66.25

bulk GaAS

66.30

66.35

66.40

layers in studied In0.15 Ga0.85As /GaAs QWs. As one can see the peaks (31.69-31.70o and 31.77-31.78o) related to the diffraction of Kα1 and Kα2 lines from the (200) crystal planes in GaAs QW layers with QDs grown at 490-525 oC locate more close to the corresponding XRD peaks (31.63o and 31.71o [22]) of the bulk cubic GaAs (Fig.14). The last fact indicates that the level of elastic strain in In0.15 Ga0.85As/GaAs QWs of #2, #3, #4 is smaller than in the structures #1 and #5.

In the QD structures with QDs grown at 470 and 535 oC the corresponding XRD peaks shift to higher angles (31.72o for Kα1 and 31.80o for Kα2) testifying the higher levels of compressive strain in the QWs of structures #1 and #5 (Fig.14). The lowest integrated PL intensities have been detected in the QD structures #1 and #5, apparently, due to the high concentration of nonradiative (NR) defects. The high level of elastic strain enhances, apparently, partial stress relaxation in the structures #1 and #5 that accompanies by the appearance of NR defects.

Fig. 14. XRD peaks related to the diffraction of Kα1 and Kα2 lines of the X-ray Cu source from the (200) crystal planes in the GaAs substrate and GaAs QW layers of studied structures: 1- #1, 2- #2, 3- #3, 4-#4 and 5 - #5.

### **12. Elastic strain in symmetric InAs QD structures with different QD densities**

Let us to discuss the reason of quality changes in studied DWELL structures. It is essential that the application of the buffer and capping InxGa1-xAs layers coupled with InAs QDs stimulates the lattice mismatch and stress decreasing in the vicinity of QDs, but simultaneously to enhance the lattice mismatch and stress increasing at the InGaAs /GaAs interface for the surface area between QDs. With reducing the QD density versus growth

layers in studied In0.15 Ga0.85As /GaAs QWs. As one can see the peaks (31.69-31.70o and 31.77-31.78o) related to the diffraction of Kα1 and Kα2 lines from the (200) crystal planes in GaAs QW layers with QDs grown at 490-525 oC locate more close to the corresponding XRD peaks (31.63o and 31.71o [22]) of the bulk cubic GaAs (Fig.14). The last fact indicates that the level of elastic strain in In0.15 Ga0.85As/GaAs QWs of #2, #3, #4 is smaller than in the

In the QD structures with QDs grown at 470 and 535 oC the corresponding XRD peaks shift to higher angles (31.72o for Kα1 and 31.80o for Kα2) testifying the higher levels of compressive strain in the QWs of structures #1 and #5 (Fig.14). The lowest integrated PL intensities have been detected in the QD structures #1 and #5, apparently, due to the high concentration of nonradiative (NR) defects. The high level of elastic strain enhances, apparently, partial stress relaxation in the structures #1 and #5 that accompanies by the

GaAs layer

2 1 3 4 5

31.50 31.75 32.00

(200)

GaAs bulk

4700 C

 4900 C

 5250 C

 5350 C

 5100 C

GaAs layer

Fig. 14. XRD peaks related to the diffraction of Kα1 and Kα2 lines of the X-ray Cu source from the (200) crystal planes in the GaAs substrate and GaAs QW layers of studied structures: 1-

**12. Elastic strain in symmetric InAs QD structures with different QD densities**  Let us to discuss the reason of quality changes in studied DWELL structures. It is essential that the application of the buffer and capping InxGa1-xAs layers coupled with InAs QDs stimulates the lattice mismatch and stress decreasing in the vicinity of QDs, but simultaneously to enhance the lattice mismatch and stress increasing at the InGaAs /GaAs interface for the surface area between QDs. With reducing the QD density versus growth

structures #1 and #5.

appearance of NR defects.

0

5000

#1, 2- #2, 3- #3, 4-#4 and 5 - #5.

10000

15000

20000

XRD Intensity , arb. un.

25000

30000

GaAs bulk

35000

temperature the surface area of QDs reduces (Table 1) and the InGaAs/GaAs interface area between the QDs enlarges. The variation non-monotonically of the integrated PL intensity versus QD density in studied structures, apparently, is connected with the competition of mentioned above two effects.

Fig. 15. The 2Θ angles for the diffraction of Kα1 and Kα2 lines from the (400) crystal planes of the GaAs substrate and GaAs QW layers in DWELLs with QD grown at different temperatures (or with different QD densities).

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 177

minimum of elastic strain ( #2, #3 and #4) are characterized by the higher PL intensity and by the shift of PL peak to 1.3µm (300K) that is important for the application in

The work was supported by CONACYT Mexico (project 130387) and by SIP-IPN, Mexico. The author thanks Dr. A. Stintz from Center of High Technology Materials at University of New Mexico, Albuquerque, USA, for growing the studied QD structures and Dr. G. Gómez

[1] D. Bimberg, M. Grundman, N. N. Ledentsov, Quantum Dot Heterostructures, Ed. Wiley

[2] V. M. Ustinov, N. A. Maleev, A. E. Shukov, A. R. Kovsh, A. Yu .Egorov, A. V. Lunev, B.

[4] Stintz, G. T. Liu, L. Gray, R. Spillers, S. M. Delgado, K. J. Malloy, J. Vac. Sci.Technol. B.

[5] T. V. Torchynska, J. L. Casas Espínola, E. Velazquez Losada, P. G. Eliseev, A. Stintz, K. J.

[6] Y. T. Dai, J. C. Fan, Y. F. Chen, R. M. Lin, S. C. Lee, H. H. Lin, J. Appl. Phys. 82, 4489

[7] M. A. Kapteyn, M. Lion, R. Heitz, and D. Bimberg, P. N. Brunkov, B. V. Volovik, S. G. Konnikov, A. R. Kovsh, and V. M. Ustinov, Appl. Phys. Lett. 76, 1573 (2000) [8] A. Duarte, E. C. F. da Silva, A. A. Quivy, M. J. da Silva, S. Martini, J. R. Leite E. A.

[9] X.Q. Meng, B. Xu, P. Jin, X.L. Ye, Z.Y. Zhang, C.M. Li, Z.G. Wang, Journal of Crystal

[10] L. Seravalli, P. Frigeri, M. Minelli, P. Allegri, V. Avanzini, S. Franchi, Appl. Phys. Lett.

[11] I. Lubyshev, P. P. Gonzalez-Borrero, E. Marega, Jr. E. Petitprez, N. La Scala, Jr. and P.

[12] C. Lobo, R. Leon, S. Marcinkevičius, W. Yang, P. C. Sercel, X. Z. Liao, J. Zou, and D. J.

[15] J. W. Tomm, T. Elsaesser, Y. I. Mazur, H. Kissel, G. G. Tarasov, Z. Y. Zhuchenko, and W.

[16] T. E. Nee, Y. F. Wu, Ch. Ch. Cheng, and H. T. Shen, J. Appl. Phys. 99, 013506

[14] W. H. Chang, T. M. Hsu, N. T. Yeh, and J. I. Chyi, Phys. Rev. B 62, 13040 (2000).

Malloy, R. Peña Sierra, Surface Science 532, 848 (2003).

Meneses and E. Lauretto, J. Appl. Phys., 93, 6279 (2003).

Alferov, N. N. Ledentsov, D. Bimberg, Appl.Phys.Lett. 74, 2815 (1999). [3] G. T. Liu, A. Stintz, H. Li, K. J. Malloy and L. F. Lester, Electron Lett, 35, 1163

V. Volovik, I. L. Krestnikov, Yu. G. Musikhin, N. A. Bert, P. S. Kopev, Zh .I.

optical fiber lasers.

**15. References**

(1999).

(1997).

(2006).

**14. Acknowledgements**

Gasga for XRD measurements.

& Sons (2001) 328.

18(3), 1496 (2000).

Growth 243, 432 (2002).

Basmaji, Appl. Phys. Lett. 68, 205 (1996).

H. Cockayne, Phys. Rev. B 60, 16647 (1999). [13] M. Grundmann and D. Bimberg, Phys. Rev. B 55, 9740 (1997).

T. Masselink, Phys. Rev. B 67, 045326 (2003).

87, 063101 (2005).

The X-ray difraction results confirm the mentioned explication. It is knom that the value of elastic deformation can be estimated by following [62]:

$$
\varepsilon = - (\theta - \theta\_o) \cot \theta\_o \tag{10}
$$

where Θ and Θo are the diffraction angles: in the strained layer (Θ) and in the reference layer without strain (Θo). The Θo value in present cases has been choosen as the diffraction angle meaning in the bulk cubic GaAs [61]. Thus in studied structures the elastic deformation (or elastic strain) is proportional to the difference between the diffraction angles measured for the GaAs QW layers and the bulk GaAs (Fig.15). Actually as it follows from Fig.15 the variation of elastic deformation in studied structures versus QD growth temperatures (QD densities) has the non monotonous behavior: decreasing in the structures #2, #3 and #4 and increasing in the structures #1 and #5 [38].

Note, that in #3, #4 and #5 the shift of the dependence of PL intensity versus PL peak position into the high energy range (Fig.7) can be explained as well by elastic strain increasing with the enlargement of QD growth temperatures. The high level of elastic strain enhances, apparently, stress relaxation partially in the low quality structures #1 and #5 with the appearance of NR defects (NR1 and NR2) in the GaAs barriers and InGaAs QWs respectively. Simultaneously the shift of GS PL peak to high energy in #5 with larger QD lateral sizes (Table 1) is the subsequence of essential compressive strain in this structure. Note that the compresive strain can stimulate the Ga/In inter-diffusion process in #1 and #5 that is accompained by the PL peak shift into high energy side as well (Fig.7) [63,64].

The low PL intensity in #1 can related to coupling and/or the coalescence of QDs (revealed by AFM for highest concentration of QDs) which can stimulate the NR defect generation, as well as the activation of exciton (or electron/hole) tunneling between the QDs [38]. Finally, the structures with less levels of elastic strain (#2, #3 and #4), have been characterized by the high PL intensity (Fig.3a) and by the shift of GS PL peak emission in low energy spectral range (Fig.2).

### **13. Conclusions**

The photoluminescence, its temperature and power dependences as well as PL inhomogeneity and X ray diffraction has been studied in the symmetric In0.15Ga1- 0.15As/GaAs quantum wells coupled with InAs quantum dots. The different QD densities in DWELLs were achieved by the variation of QD growth temperatures between 470 and 535 oC. It is shown four reasons for the variation of emission intensities, PL peak positions and PL inhomogeneity in studied QD structures: i) the high concentration of nonradiative recombination centers in the capping In0.15Ga1-0.15As layer at low QD growth temperatures (470oC), ii) the QD density and size distributions along the wafer for DWELLs with QD grown at 510-535oC, ii) the high concentration of nonradiative recombination centers in the GaAs barrier at the QD growth temperature of 535oC and iv) the non monotonous behavior of elastic strain in DWELLs versus QD density. XRD testifies that with decreasing the density of QDs from 1.1 1011 cm−2 down to 1.3 1010 cm−2 the level of compressive strain in DWELLs varies not monotonously. The DWELLs with minimum of elastic strain ( #2, #3 and #4) are characterized by the higher PL intensity and by the shift of PL peak to 1.3µm (300K) that is important for the application in optical fiber lasers.

### **14. Acknowledgements**

The work was supported by CONACYT Mexico (project 130387) and by SIP-IPN, Mexico. The author thanks Dr. A. Stintz from Center of High Technology Materials at University of New Mexico, Albuquerque, USA, for growing the studied QD structures and Dr. G. Gómez Gasga for XRD measurements.

### **15. References**

176 Fingerprints in the Optical and Transport Properties of Quantum Dots

The X-ray difraction results confirm the mentioned explication. It is knom that the value of

( )cot *o o*

where Θ and Θo are the diffraction angles: in the strained layer (Θ) and in the reference layer without strain (Θo). The Θo value in present cases has been choosen as the diffraction angle meaning in the bulk cubic GaAs [61]. Thus in studied structures the elastic deformation (or elastic strain) is proportional to the difference between the diffraction angles measured for the GaAs QW layers and the bulk GaAs (Fig.15). Actually as it follows from Fig.15 the variation of elastic deformation in studied structures versus QD growth temperatures (QD densities) has the non monotonous behavior: decreasing in the structures #2, #3 and #4 and

Note, that in #3, #4 and #5 the shift of the dependence of PL intensity versus PL peak position into the high energy range (Fig.7) can be explained as well by elastic strain increasing with the enlargement of QD growth temperatures. The high level of elastic strain enhances, apparently, stress relaxation partially in the low quality structures #1 and #5 with the appearance of NR defects (NR1 and NR2) in the GaAs barriers and InGaAs QWs respectively. Simultaneously the shift of GS PL peak to high energy in #5 with larger QD lateral sizes (Table 1) is the subsequence of essential compressive strain in this structure. Note that the compresive strain can stimulate the Ga/In inter-diffusion process in #1 and #5 that is accompained by the PL peak shift into high energy side as well (Fig.7)

The low PL intensity in #1 can related to coupling and/or the coalescence of QDs (revealed by AFM for highest concentration of QDs) which can stimulate the NR defect generation, as well as the activation of exciton (or electron/hole) tunneling between the QDs [38]. Finally, the structures with less levels of elastic strain (#2, #3 and #4), have been characterized by the high PL intensity (Fig.3a) and by the shift of GS PL peak emission in low energy spectral

The photoluminescence, its temperature and power dependences as well as PL inhomogeneity and X ray diffraction has been studied in the symmetric In0.15Ga1- 0.15As/GaAs quantum wells coupled with InAs quantum dots. The different QD densities in DWELLs were achieved by the variation of QD growth temperatures between 470 and 535 oC. It is shown four reasons for the variation of emission intensities, PL peak positions and PL inhomogeneity in studied QD structures: i) the high concentration of nonradiative recombination centers in the capping In0.15Ga1-0.15As layer at low QD growth temperatures (470oC), ii) the QD density and size distributions along the wafer for DWELLs with QD grown at 510-535oC, ii) the high concentration of nonradiative recombination centers in the GaAs barrier at the QD growth temperature of 535oC and iv) the non monotonous behavior of elastic strain in DWELLs versus QD density. XRD testifies that with decreasing the density of QDs from 1.1 1011 cm−2 down to 1.3 1010 cm−2 the level of compressive strain in DWELLs varies not monotonously. The DWELLs with

, (10)

elastic deformation can be estimated by following [62]:

increasing in the structures #1 and #5 [38].

[63,64].

range (Fig.2).

**13. Conclusions**

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 179

[40] P.G. Eliseev, H. Li, G.T. Iu, A. Stintz, T.C. Nevell, L.F. Lester, J. Mally, IEEE J. Select. 35

[41] Y.S. Huang, H. Qiang, F.H. Pollack, G.D. Pettit, P.D. Kirtchner, J.M. Woodall, H. Stagier,

[42] Torchynska, T.V., Casas Espinola, J.L., Lopez, H.M.A., Eliseev, P.G., Stintz, A., Malloy, K.J., Pena Sierra, R. *Institute of Physics Conference Series,* 174, 69-71 (2003). [43] J. L. Casas Espinola, M. Dybiec, S. Ostapenko, T. V. Torchynska and G. Polupan, J. of

[44] L. Casas Espinola, 1 T. V. Torchynska, G. Polupan, R. Pena Sierra, phys. stat. sol. (c), 4,

[45] M. De Giorgi, C. Lingk, G. von Plessen, J. Feldmann, S. De Rinaldis, A. Passaseo, M. De Vittorio, R. Cingolani, M. Lomascolo, Appl. Phys. Lett. 79, 3968 (2001).

[47] B. Ohnesorge, M. Albrecht, J. Oshinowo, A. Forchel, Y. Arakawa, Phys. Rev. B., 54,

[48] T.V. Torchynska, E. Velazquez Lozada, J.L. Casas Espinola, J. Vac. Scien. & Tech. B

[50] T. V. Torchynska, J. L. Casas Espinola, E. Velasquez Lozada, L. V. Shcherbyna, A. Stintz,

[51] E. Zhukov, A. R. Kovsh, N. A. Maleev, S. S. Mikhrin, V. M. Ustinov, A. F. Tsatsulnikov,

[53] N. T. Yeh, T. E. Nee, J. I. Chyi, T. M. Hsu, C. C. Huang, Appl. Phy. Lett. 76, 1567

[54] F. Guffarth, R. Heitz, A. Schliwa, O. Stier, N. N. Ledentsov, A. R. Kovsh, V. M. Ustinov,

[55] E. Romanov, G. E. Beltz, W. T. Fischer, P. M. Petroff, J. S. Speck, J. Appl. Phys., 89, 4523

[58] Y. Nabetani, T. Matsumoto, G. Sasikala, I. Suemune, J. Appl. Phys. 98, 063502

[59] T.V.Torchynska, J. Palacios Gomez, G. Gomez Gasga, A. Vivas Hernandez, E.

[60] J.L. Casas Espinola, T V Torchynska, J Palacios Gomez, G Gómez Gasga, A Vivas

[61] Yeh CY, Lu ZW, Froyen S and Zunger A, *Phys. Rev.* B, 46, 10086 (1992). [62] S. Ejiri, T. Sasaki, and Y. Hirose, Thin Solid Films 307, 178 (1997)

Velazquez Lozada, G. Polupan, Ye. S. Shcherbyna, J. of Physics, Conference Ser.

Hernandez and R. Cisneros Tamayo, phys.stat.solid. (c). 8, No. 4, 1391-1393

Alferov, N. N. Ledentsov, D. Bimberg, Appl. Phys. Lett. 75, 1926 (1999).

R. Pena Sierra, Localization of defects in InAs QD symmetric InGaAs-GaAs DWELL structures, Physica B: Condensed Matter, 401-402, 584-586 (2007), ISSN

M. V. Maximov, B. V. Volovik, D. A. Bedarev, Y. M. Shernyakov, P. S. Kopev, Z. I.

Topics Quant. Electron. 7 (3) (2001) 135.

L.B. Soresen, J. Appl. Pys. 70 (12) (1991) 7537

Physics: Conference Series, 61, 180-184 (2007).

[46] S. Marcinkevicius, R. Leon, Phys. Rev. B., 59, 4630 (1999).

[49] T.V. Torchynska, Superlattice and Microstructure, 45, 349-355 (2009)

[52] K. Nishi, H. Saito, S. Sugou, J. S. Lee, Appl. Phys. Lett. 74, 1111 (1999).

D. Bimberg, Phys. Rev. B, 64, 085305 (2001).

[56] Y. Y. Lin, J. Singh, J. Appl. Phys. 92, 6205 (2002).

[57] B. Yang, J. Appl. Phys. 92, 3704 (2002).

245, 012060 (2010).

n.2, 379-381 (2007).

11532 (1996).

0921-4526.

(2000).

(2001).

(2005).

(2011)

27(2), 919-922, (2009)

[17] T. V. Torchynska, J. L. Casas Espinola, L. V. Borkovska, S. Ostapenko, M. Dybic, O.

[18] S. Sanguinetti, M. Henini, M. Grassi Alessi, M. Capizzi, P. Frigeri, S. Franchi, Phys. Rev.

[20] S. Khatsevich, D. H. Rich, E.T. Kim and A. Madhukar, J. Appl. Phys. 97, 123520

[21] Yu. I. Mazur, B. L. Liang, Zh. M. Wang, D. Guzun, G. J. Salamo, Z. Ya. Zhuchenko and

[22] Bacher, C. Hartmann, H. Schweizer, T. Held, G. Mahler, H. Nickel, Phys. Rev. B. 47,

[24] Guffarth, R. Heitz, A. Schliwa, O. Stier, A. R. Kovsh, V. Ustinov, N. N. Ledentsov, D.

[25] S. Sauvage, P. Boucaud, J. M. Gerard, V. Thierry-Mieg, J. Appl. Phys. 84, 4356

[27] M. Dybiec, L. Borkovska, S. Ostapenko, T. V. Torchynska, J. L. Casas Espinola, A. Stintz

[28] M. Dybiec, S. Ostapenko, T. V. Torchynska, E.Velasquez Lozada, P. G. Eliseev, A. Stintz,

[29] M. Dybiec, S. Ostapenko, T. V. Torchynska, E.Velasquez Losada, Scanning

[30] M. Catalano, A. Taurino, M. Lomescolo, L. Vasanelli, M. De Giorgi, A. Passasco, R.

[32] S.E.J. Jacobs, M. Kemerink, P.M. Koenroad, M. Hopkinson, H.W.M. Salimink, J.H.

[34] T.V. Torchynskaa,, A. Vivas Hernandezb, G. Polupanb, E. Velazquez Lozada,

[36] T.V. Torchynska, J.L. Casas Espinola, P.G. Eliseev, A. Stintz, K.J. Malloy, R. Pena Sierra,

[38] T.V.Torchynska, A. Stintz, Some aspects of emission variation in InAs quantum

structures, Material Science and Engineering B. 176 , 331–333 (2011).

Photoluminescence study and parameter evaluation in InAs quantum dot-in-a-well

dots coupled with symmetric quantum wells, J. Applied Physics, 108, 2, 024316

and K. J. Malloy, Photoluminescence scanning on InAs/InGaAs Quantum Dot

K. J. Malloy, Photoluminescence mapping on InAs/InGaAs dot structures, phys.

Photoluminescence Spectroscopy in InAs/InGaAs Quantum Dot Structures, Appl.

[19] C. Le Ru, J. Fack and R. Murray, Rhys. Rev. B. 67, 245318 (2003).

G. G. Tarasov, Appl. Phys. Lett., 89,151914 (2006).

[26] S.-K. Eah, W. Jhe, Y. Arakawa, Appl. Phys. Lett. 80, 2779 (2002).

Structures, Applid. Surface Science, 252, 5542 (2006).

Rinaldi, R. Cingolani, J. Appl. Phys. 87, 2261 (2000). [31] Y. Temko, T. Siziki, K. Jacobi, Appl. Phys. Lett. 82, 2142 (2003).

[35] L.V. Asryan, R.A. Suris, Semicond. Sci. Technol. 11, 554 (1996).

Bimberg, phys. stat. sol. (b) 224, 61 (2001).

stat. sol. (c), 2, n.8, 2951-2954 (2005).

Walter, Appl. Phys. Lett. 83, 290 (2003). [33] T.V. Torchynska, J. Appl. Phys., 104, 074315 , n.7 (2008).

Phys. Status Solidi (a) 195 , 209 (2003).

[37] C. Pryor, Phys. Rev. B 57, 7190 (1998).

[39] Y.P. Varshni, Physica 34 , 149 (1967).

(2010).

Phys. Lett. 84, 5165-5167 (2004).

[23] T. V. Torchynska, M. Dybiec, S. Ostapenko, Phys. Rev. B. 72, 195341 (2005).

024323 (2007).

(2005).

9545 (1993).

(1998).

B 60, 8276 (1999)

Polupan, N. O. Korsunska, A. Stintz, P. G. Eliseev, K. J. Malloy, J. Appl. Phys., 101,

**0**

**7**

*Taiwan*

C. Y. Lin and Y. K. Ho

**Photoionization Cross Sections of Atomic**

With the advances of experimental techniques in fabrication and investigation of nano-scale structures, confined atomic systems become practical and useful models for the illustration of interesting phenomena arising from a system in dimensions comparable to the electronic de Broglie wavelength. The confined atomic models are widely used to study a variety of physical systems, such as impurities in quantum dots (Lin & Ho (2011)), atoms encapsulated in fullerenes (Connerade et al. (1999); Dolmatov et al. (2004)), and atoms under high pressure (de Groot & ten Seldam (1946); Michels et al. (1937)). In this article, we focus on the quantum confinement occurring in quantum dots. The emphasis is placed on the variation of electronic structures and photoionization properties of atomic impurities under the spatial confinement

As a quantum confinement system, the quantum dot has attracted considerable attention due to not only its theoretical but also practical significance. In addition to the analogies of discrete structure in their optical and electrical features between a quantum dot and an atom, the coupled quantum dots provide a model to mimic molecules with tunable bonds (Alivisatos (1996); Schedelbeck et al. (1997)). On the other hand, the quantum dots also serve as contrast agents in bioimaging for biotechnological applications (Michalet et al. (2005)). It is well known that the quantum dot with atomic impurities is a suitable model for studying the semiconductor heterostructures. Recently, the enhancement of semiconductor nano-crystal performance due to the impurities has been reported in the literature (Cao (2011)), which indicates, for instance, that magnetic impurities can be doped to tune optical and magnetic

The physical properties of confined atomic systems are greatly influenced by confinement potentials, which are unable to be determined through the direct measurement of experiment. Although *ab initio* calculations can comprehensively deal with the interaction in confined atomic systems, they may not provide a direct and simple physical interpretation. The usage of semi-empirical model potentials to mimic the interaction of confined atom and surrounding environment provides an efficient way to study the complex systems. The appropriate models, which might not treat the system comprehensively but take the important interaction into account, give a clear physical insight into complex problems. The confinement potentials associated with the structures of quantum dots are often modelled by the rectangular potential

**1. Introduction**

effect of quantum dots.

properties.

**Impurities in Spherical Quantum Dots**

*Institute of Atomic and Molecular Sciences, Academia Sinica*

## **Photoionization Cross Sections of Atomic Impurities in Spherical Quantum Dots**

C. Y. Lin and Y. K. Ho *Institute of Atomic and Molecular Sciences, Academia Sinica Taiwan*

### **1. Introduction**

180 Fingerprints in the Optical and Transport Properties of Quantum Dots

[63] J. L. Casas Espínola, T. V. Torchynska, G. Polupan, and M. Ojeda Martínez,

[64] J. Casas Espinola, V. Torchynska, G 1 . Polupan, E. Velazquez Lozada, Material Science

phys.stat.solid.(c) 8, No. 4, 1388–1390 (2011)

and Engineering, B., 165, 115- 117 (2009)

With the advances of experimental techniques in fabrication and investigation of nano-scale structures, confined atomic systems become practical and useful models for the illustration of interesting phenomena arising from a system in dimensions comparable to the electronic de Broglie wavelength. The confined atomic models are widely used to study a variety of physical systems, such as impurities in quantum dots (Lin & Ho (2011)), atoms encapsulated in fullerenes (Connerade et al. (1999); Dolmatov et al. (2004)), and atoms under high pressure (de Groot & ten Seldam (1946); Michels et al. (1937)). In this article, we focus on the quantum confinement occurring in quantum dots. The emphasis is placed on the variation of electronic structures and photoionization properties of atomic impurities under the spatial confinement effect of quantum dots.

As a quantum confinement system, the quantum dot has attracted considerable attention due to not only its theoretical but also practical significance. In addition to the analogies of discrete structure in their optical and electrical features between a quantum dot and an atom, the coupled quantum dots provide a model to mimic molecules with tunable bonds (Alivisatos (1996); Schedelbeck et al. (1997)). On the other hand, the quantum dots also serve as contrast agents in bioimaging for biotechnological applications (Michalet et al. (2005)). It is well known that the quantum dot with atomic impurities is a suitable model for studying the semiconductor heterostructures. Recently, the enhancement of semiconductor nano-crystal performance due to the impurities has been reported in the literature (Cao (2011)), which indicates, for instance, that magnetic impurities can be doped to tune optical and magnetic properties.

The physical properties of confined atomic systems are greatly influenced by confinement potentials, which are unable to be determined through the direct measurement of experiment. Although *ab initio* calculations can comprehensively deal with the interaction in confined atomic systems, they may not provide a direct and simple physical interpretation. The usage of semi-empirical model potentials to mimic the interaction of confined atom and surrounding environment provides an efficient way to study the complex systems. The appropriate models, which might not treat the system comprehensively but take the important interaction into account, give a clear physical insight into complex problems. The confinement potentials associated with the structures of quantum dots are often modelled by the rectangular potential

Impurities in Spherical Quantum Dots 3

Photoionization Cross Sections of Atomic Impurities in Spherical Quantum Dots 183

later applied to few-electron quantum dots (Kimani et al. (2008)). Both potentials have *r*2-dependence near the center of quantum dots, which is the typical character of harmonic oscillators. It should be noted that the impurity is not taken into account for the quantum

Fig. 1. Comparison of finite oscillator potential with Gaussian potential.

For the investigations of electronic structure and optical properties of atomic impurities in quantum dots, many efforts have been devoted to study hydrogenic impurity states in spherical quantum dots described by finite and infinite potential wells. The state energies of hydrogen impurity in spherical quantum dots with the infinite and finite well of rectangular potentials are explored by Chuu et al. (Chuu et al. (1992)), Yang et al. (Yang et al. (1998)), and Huang et al. (Huang et al. (1999)). Photoionization cross sections and oscillator strengths of hydrogenic impurities in spherical quantum dots are also obtained for the infinite and finite rectangular well models by Ham and Spector (Ham & Spector (n.d.)), ¸Sahin ( ¸Sahin (2008)), and Stevanovi´c (Stevanovi´c (2010)). Recently, Lin and Ho (Lin & Ho (2011)) study the photoionization of hydrogen impurities in spherical quantum dots using the finite oscillator and Gaussian potentials. Chakraborty and Ho (Chakraborty & Ho (2011)) adopt the finite oscillator potential to describe the quantum dot for exploring the autoionization resonance states of helium impurities in quantum dots. In the present work, the alkali-metal atoms as impurities in the quantum dots are studied. On the basis of the finite oscillator and Gaussian models, the energy levels and photionization cross sections subject to the quantum

The chapter is organized as follows. In Sec. 2, the FE DVR approach and complex-coordinate rotation method associated with the current work are described. The energy spectrum and photoionization cross sections varying with the different conditions of quantum dots for the lithium and sodium impurities are presented and discussed in Sec. 3. Section 4 summarizes this work and gives conclusions. Atomic units are used throughout unless otherwise noted.

dots in above-mentioned investigations.

confinement effect are illustrated.

well

$$V\_{\text{RECT}}(r) = \begin{cases} -V\_0 & r \le R; \\ 0 & r > R, \end{cases} \tag{1}$$

or the harmonic oscillator (parabolic) potential.

$$V\_{\rm HO}(r) = -V\_0 + \frac{V\_0}{R^2}r^2,\tag{2}$$

where *R* determines the size of quantum dot, and *V*0 gives the strength of confinement. The rectangular potential well has a simple but unrealistic form due to the non-parabolic shape at the center of quantum dots. Although the harmonic oscillator potential fulfils the parabolic property, the infinite depth and range of potential restrict the calculation of continuum states and fail to describe the charging of quantum dots with the finite number of electrons.

The Woods-Saxon potential given as

$$V\_{\rm WS}(r) = \frac{V\_0}{1 + \exp\left[ (R - r) / \gamma \right]},\tag{3}$$

where *γ* controls the slope of confinement potential, also has been used in the study of confined quantum system (Costa et al. (1999); Xie (2009)). It should be noted that the Woods-Saxon potential turns to be the rectangular potential well as *γ* → 0. Another confinement potential flexible to model the different type of quantum dots is the so called power-exponential potential (Ciurla et al. (2002)),

$$V\_{\rm EP}(r) = -V\_0 \exp\left[-(r/\mathcal{R})^p\right].\tag{4}$$

With the change of parameter *p*, the shape of potential is modified from the Gaussian potential *p* = 2 to the rectangular potential well *p* = ∞.

In this work, the systems of atomic impurities in spherical quantum dots characterized by finite oscillator (FO) and Gaussian potentials (Adamowski et al. (2000a;b); Kimani et al. (2008); Winkler (2004)) are investigated using the method of complex-coordinate rotation (Ho (1983); Reinhardt (1982)) in a finite-element discrete variable representation (FE DVR) (Balzer et al. (2010); Rescigno & McCurdy (2000)). The finite oscillator potential *V*FO and Gaussian potential *V*G are defined as

$$V\_{\rm FO}(r) = -A\left(1 + \frac{B}{\sqrt{A}}r\right) \exp(-\frac{B}{\sqrt{A}}r) \tag{5}$$

and

$$V\_{\mathcal{G}}(r) = -\mathcal{C} \exp(-r^2/D^2),\tag{6}$$

where A and C are the confining strength of potentials, and the radii of dots are characterized inherently by 1/*B* and *D* for FO and Gaussian potentials, respectively. Figure 1 shows the examples of both potentials. The Gaussian potential being a special case of the power-exponential potential (Ciurla et al. (2002)) has a soft boundary of the potential. The one-electron energy spectrum for a Gaussian potential has been calculated by Adamowski et al. (Adamowski et al. (2000a;b)) using the variational method with Gaussian-type basis functions. The finite oscillator potentials as weakly confining potentials of quantum dots have been used to study the two-electron quantum dots by Winkler (Winkler (2004)), and 2 Will-be-set-by-IN-TECH

−*V*0 *r* ≤ *R*;

*V*0

1 + exp [(*R* − *r*)/*γ*]

*V*EP(*r*) = −*V*0 exp [−(*r*/*R*)*p*] . (4)

exp(− *B* √*A*

*V*G(*r*) = −*C* exp(−*r*2/*D*2), (6)

where *γ* controls the slope of confinement potential, also has been used in the study of confined quantum system (Costa et al. (1999); Xie (2009)). It should be noted that the Woods-Saxon potential turns to be the rectangular potential well as *γ* → 0. Another confinement potential flexible to model the different type of quantum dots is the so called

With the change of parameter *p*, the shape of potential is modified from the Gaussian potential

In this work, the systems of atomic impurities in spherical quantum dots characterized by finite oscillator (FO) and Gaussian potentials (Adamowski et al. (2000a;b); Kimani et al. (2008); Winkler (2004)) are investigated using the method of complex-coordinate rotation (Ho (1983); Reinhardt (1982)) in a finite-element discrete variable representation (FE DVR) (Balzer et al. (2010); Rescigno & McCurdy (2000)). The finite oscillator potential *V*FO and Gaussian potential

where A and C are the confining strength of potentials, and the radii of dots are characterized inherently by 1/*B* and *D* for FO and Gaussian potentials, respectively. Figure 1 shows the examples of both potentials. The Gaussian potential being a special case of the power-exponential potential (Ciurla et al. (2002)) has a soft boundary of the potential. The one-electron energy spectrum for a Gaussian potential has been calculated by Adamowski et al. (Adamowski et al. (2000a;b)) using the variational method with Gaussian-type basis functions. The finite oscillator potentials as weakly confining potentials of quantum dots have been used to study the two-electron quantum dots by Winkler (Winkler (2004)), and

 1 + *B* √*A r*

0 *r* > *R*, (1)

*R*2 *r*2, (2)

, (3)

*r*) (5)

*V*RECT(*r*) =

*V*HO(*r*) = −*V*0 +

and fail to describe the charging of quantum dots with the finite number of electrons.

*V*WS(*r*) = *V*0

where *R* determines the size of quantum dot, and *V*0 gives the strength of confinement. The rectangular potential well has a simple but unrealistic form due to the non-parabolic shape at the center of quantum dots. Although the harmonic oscillator potential fulfils the parabolic property, the infinite depth and range of potential restrict the calculation of continuum states

or the harmonic oscillator (parabolic) potential.

The Woods-Saxon potential given as

power-exponential potential (Ciurla et al. (2002)),

*p* = 2 to the rectangular potential well *p* = ∞.

*V*FO(*r*) = −*A*

*V*G are defined as

and

well

later applied to few-electron quantum dots (Kimani et al. (2008)). Both potentials have *r*2-dependence near the center of quantum dots, which is the typical character of harmonic oscillators. It should be noted that the impurity is not taken into account for the quantum dots in above-mentioned investigations.

Fig. 1. Comparison of finite oscillator potential with Gaussian potential.

For the investigations of electronic structure and optical properties of atomic impurities in quantum dots, many efforts have been devoted to study hydrogenic impurity states in spherical quantum dots described by finite and infinite potential wells. The state energies of hydrogen impurity in spherical quantum dots with the infinite and finite well of rectangular potentials are explored by Chuu et al. (Chuu et al. (1992)), Yang et al. (Yang et al. (1998)), and Huang et al. (Huang et al. (1999)). Photoionization cross sections and oscillator strengths of hydrogenic impurities in spherical quantum dots are also obtained for the infinite and finite rectangular well models by Ham and Spector (Ham & Spector (n.d.)), ¸Sahin ( ¸Sahin (2008)), and Stevanovi´c (Stevanovi´c (2010)). Recently, Lin and Ho (Lin & Ho (2011)) study the photoionization of hydrogen impurities in spherical quantum dots using the finite oscillator and Gaussian potentials. Chakraborty and Ho (Chakraborty & Ho (2011)) adopt the finite oscillator potential to describe the quantum dot for exploring the autoionization resonance states of helium impurities in quantum dots. In the present work, the alkali-metal atoms as impurities in the quantum dots are studied. On the basis of the finite oscillator and Gaussian models, the energy levels and photionization cross sections subject to the quantum confinement effect are illustrated.

The chapter is organized as follows. In Sec. 2, the FE DVR approach and complex-coordinate rotation method associated with the current work are described. The energy spectrum and photoionization cross sections varying with the different conditions of quantum dots for the lithium and sodium impurities are presented and discussed in Sec. 3. Section 4 summarizes this work and gives conclusions. Atomic units are used throughout unless otherwise noted.

Impurities in Spherical Quantum Dots 5

Photoionization Cross Sections of Atomic Impurities in Spherical Quantum Dots 185

Based on the properties of the Lobatto shape functions and the approximation of Gauss-Lobatto quadrature for integrals, the matrix elements of kinetic-energy operator, *T* =

> � ∞ 0 d*x* d d*x χi*1 *m*1 (*x*) d d*x χi*2 *m*2 (*x*)

ensure the continuity of wave functions at end points of each finite element.

(*δi*1,*i*2 + *δi*1,*i*2±1)

1,*m*2 )

*m*1,1)

1,1 ) + *δi*1,*i*2−1*T*˜*i*2

d*f i m*1 (*xi m*) d*x*

According to Eq. (11), the first derivatives of the Lobatto shape functions at the quadrature

*m*) ∏*m*��=*m*1,*m*

The matrix of the local potential-energy operator *V*(*x*) in FE DVR has a diagonal representation with matrix element values equal to potential values at grid points, i.e.,

*ng* + *V*(*xi*+1

*ng* + *wi*+1 1

*wi*

*m*1 (*x*)*V*(*x*)*χi*2

1 )(*wi*2

*ng*,*m*2 + *δi*1,*i*2−1*T*˜*i*2

*m*1,*ng* + *δi*1,*i*2+1*T*˜*i*1

*ng*,*ng* + *T*˜*i*1+1

� (*wi*1

*m*1,*m*2 is defined as

*T*˜*i*

⎧ ⎪⎪⎨

⎪⎪⎩

� ∞ 0

⎧ ⎪⎨ *V*(*xi*

*V*(*xi ng* )*wi*

⎪⎩

d*xχi*1

*Vi*1,*i*2 *m*1,*m*2 =

*V*˜ *i m* = (*xi m*1 − *xi*

1 2*wi m*1

*m*1,*m*2 = ∑ *m*

1

*ng* + *wi*1+1 1 )

*ng* + *wi*2+1 1 )

*ng* + *wi*1+1

d*x*2 , in FE DVR are evaluated by analytic formulas,

*m*2 � = 1 2

*m*1 |*T*|*χi*2

*δi*1,*i*2*T*˜*i*1 � *m*1,*m*2 *wi*1 *m*1*wi*2 *m*2

(*δi*1,*i*2*T*˜*i*1

� *wi*2 *m*2 (*wi*1

(*δi*1,*i*2*T*˜*i*1

� *wi*1 *m*1 (*wi*2

(*δi*1,*i*2 (*T*˜*i*1

1(*x*) in Eq. (10) is in charge of connecting the adjacent elements to

1,*ng* + *δi*1,*i*2+1*T*˜*i*1

*wi*

*ng* + *wi*2+1 1 )

> d*f i m*2 (*xi m*) d*x*

> > (*xi m* − *xi m*�)

(*xi m*1 − *xi*

*m*) for *m* > 1;

1 )*wi*+1 1

(*δm*1,*ng* − *δm*1,1) for *m*1 = *m*.

*m*2 (*x*) = *δi*1,*i*2 *δm*1,*m*2*V*˜ *i*1

*ng*,1)

*m*�) for *m*1 �= *m*;

(*m*1 > 1, *m*2 > 1);

(*m*1 = 1, *m*2 > 1);

(*m*1 > 1, *m*2 = 1);

(12)

(14)

(*m*1 = *m*2 = 1),

*m*. (13)

*m*1 , (15)

for *m* = 1. (16)

The bridge basis function *χi*

*Ti*1,*i*2 *m*1,*m*2 = �*χi*1

⎧

1 2

1 2

1 2

1 2

*d f i m*1 (*xi m*) *dx* =

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

=

in which the term *T*˜*i*

points are given as

with

−1 2 d2

#### **2. Theoretical method**

#### **2.1 Finite-element discrete variable representation**

The finite-element discrete variable representation (FE DVR) is a hybrid computation scheme taking advantage of the finite-element approach and the discrete variable representation to obtain the sparse kinetic-energy matrices and the diagonal representation of potential-energy matrices. Using this hybrid approach, the kinetic-energy matrix is block diagonal with matrix elements in compact expressions, and the potential-energy matrix elements are given by the potential values at grid points. This method has been implemented to investigate a variety of interesting physical problems, such as quantum-mechanical scattering problems (Rescigno & McCurdy (2000)), bright solitons in Bose-Einstein Condensates and ultracold plasmas (Collins et al. (2004)), non-equilibrium Green ˛a˛es function calculations (Balzer et al. (2010)), and photoionization of impurities in quantum dots (Lin & Ho (2011)).

In the present work, the method of FE DVR which is detailed in references (Rescigno & McCurdy (2000)) and (Balzer et al. (2010)) is adopted to obtain the Hamiltonian matrix elements for atoms confined by quantum dots. Within the framework of FE DVR, the interval [0,*Rmax*] is divided into *ne* finite elements, in which each element between [*xi* , *xi*+1] is further subdivided by *ng* Gauss quadrature points (see Fig. 2). Taking advantage of the standard Gauss-Lobatto points *xm* and weights *wm* (Michels (1963)), we define the generalized Gauss-Lobatto points,

$$\mathbf{x}\_m^i = \frac{1}{2} [ (\mathbf{x}^{i+1} - \mathbf{x}^i) \mathbf{x}\_m + (\mathbf{x}^{i+1} + \mathbf{x}^i)] \tag{7}$$

and weights,

$$w\_m^i = \frac{w\_m}{2} (\mathbf{x}^{i+1} - \mathbf{x}^i). \tag{8}$$

It should be noted that *xi* 1 = *xi* and *xi ng* = *xi*+1 because *x*1 = −1 and *xng* = 1. In calculations, the integrals are approximated by Gauss-Lobatto quadrature,

$$\int\_{\mathbf{x}^{i}}^{\mathbf{x}^{i+1}} \psi(r) dr \simeq \psi(\mathbf{x}^{i}) w\_{1}^{i} + \sum\_{m=2}^{n\_{\S}-1} \psi(\mathbf{x}\_{m}^{i}) w\_{m}^{i} + \psi(\mathbf{x}^{i+1}) w\_{n\_{\S}}^{i}. \tag{9}$$

The wave functions are expanded in terms of local basis functions (see Fig. 2),

$$\chi\_m^i(\mathbf{x}) = \begin{cases} \frac{[f\_{n\_\mathcal{S}}^i(\mathbf{x}) + f\_1^{i+1}(\mathbf{x})]}{\sqrt{(w\_{n\_\mathcal{S}}^i + w\_1^{i+1})}} & \text{for } \mathbf{m} = 1 \text{ (bridge)};\\ \frac{f\_m^i(\mathbf{x})}{\sqrt{w\_m^i}} & \text{for else (element)}, \end{cases} \tag{10}$$

where the Lagrange interpolating polynomials or so-called Lobatto shape functions *f i m*(*x*) are given as

$$f\_m^i(\mathbf{x}) = \begin{cases} \prod\_{m' \neq m} \frac{(\mathbf{x} - \mathbf{x}\_{m'}^i)}{(\mathbf{x}\_m^i - \mathbf{x}\_{m'}^i)} \text{ for } \mathbf{x}^i \le \mathbf{x} \le \mathbf{x}^{i+1};\\ 0 & \text{for } \mathbf{x} < \mathbf{x}^i \text{ or } \mathbf{x} > \mathbf{x}^{i+1}. \end{cases} \tag{11}$$

The bridge basis function *χi* 1(*x*) in Eq. (10) is in charge of connecting the adjacent elements to ensure the continuity of wave functions at end points of each finite element.

Based on the properties of the Lobatto shape functions and the approximation of Gauss-Lobatto quadrature for integrals, the matrix elements of kinetic-energy operator, *T* = −1 2 d2 d*x*2 , in FE DVR are evaluated by analytic formulas,

*Ti*1,*i*2 *m*1,*m*2 = �*χi*1 *m*1 |*T*|*χi*2 *m*2 � = 1 2 (*δi*1,*i*2 + *δi*1,*i*2±1) � ∞ 0 d*x* d d*x χi*1 *m*1 (*x*) d d*x χi*2 *m*2 (*x*) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ 1 2 *δi*1,*i*2*T*˜*i*1 � *m*1,*m*2 *wi*1 *m*1*wi*2 *m*2 (*m*1 > 1, *m*2 > 1); 1 2 (*δi*1,*i*2*T*˜*i*1 *ng*,*m*2 + *δi*1,*i*2−1*T*˜*i*2 1,*m*2 ) � *wi*2 *m*2 (*wi*1 *ng* + *wi*1+1 1 ) (*m*1 = 1, *m*2 > 1); 1 2 (*δi*1,*i*2*T*˜*i*1 *m*1,*ng* + *δi*1,*i*2+1*T*˜*i*1 *m*1,1) � *wi*1 *m*1 (*wi*2 *ng* + *wi*2+1 1 ) (*m*1 > 1, *m*2 = 1); 1 2 (*δi*1,*i*2 (*T*˜*i*1 *ng*,*ng* + *T*˜*i*1+1 1,1 ) + *δi*1,*i*2−1*T*˜*i*2 1,*ng* + *δi*1,*i*2+1*T*˜*i*1 *ng*,1) � (*wi*1 *ng* + *wi*1+1 1 )(*wi*2 *ng* + *wi*2+1 1 ) (*m*1 = *m*2 = 1), (12)

in which the term *T*˜*i m*1,*m*2 is defined as

$$\tilde{T}\_{m\_1, m\_2}^{\dot{i}} = \sum\_m \frac{\mathbf{d} f\_{m\_1}^{\dot{i}}(\mathbf{x}\_m^{\dot{i}})}{\mathbf{d} \mathbf{x}} \frac{\mathbf{d} f\_{m\_2}^{\dot{i}}(\mathbf{x}\_m^{\dot{i}})}{\mathbf{d} \mathbf{x}} w\_m^{\dot{i}}.\tag{13}$$

According to Eq. (11), the first derivatives of the Lobatto shape functions at the quadrature points are given as

$$\frac{df\_{m\_1}^i(\mathbf{x}\_m^i)}{d\mathbf{x}} = \begin{cases} \frac{1}{\left(\mathbf{x}\_{m\_1}^i - \mathbf{x}\_m^i\right)} \prod\_{m' \neq m\_1, m} \frac{\left(\mathbf{x}\_m^i - \mathbf{x}\_{m'}^i\right)}{\left(\mathbf{x}\_{m\_1}^i - \mathbf{x}\_{m'}^i\right)} \text{ for } m\_1 \neq m;\\ \frac{1}{2\mathbf{w}\_{m\_1}^i} (\delta\_{m\_1, n\_\beta} - \delta\_{m\_1, 1}) & \text{ for } m\_1 = m. \end{cases} \tag{14}$$

The matrix of the local potential-energy operator *V*(*x*) in FE DVR has a diagonal representation with matrix element values equal to potential values at grid points, i.e.,

$$V\_{m\_1,m\_2}^{i\_1,i\_2} = \int\_0^\infty \mathrm{d}x \chi\_{m\_1}^{i\_1}(\mathbf{x}) V(\mathbf{x}) \chi\_{m\_2}^{i\_2}(\mathbf{x}) = \delta\_{i\_1,i\_2} \delta\_{m\_1,m\_2} \tilde{V}\_{m\_1 \prime}^{i\_1} \tag{15}$$

with

4 Will-be-set-by-IN-TECH

The finite-element discrete variable representation (FE DVR) is a hybrid computation scheme taking advantage of the finite-element approach and the discrete variable representation to obtain the sparse kinetic-energy matrices and the diagonal representation of potential-energy matrices. Using this hybrid approach, the kinetic-energy matrix is block diagonal with matrix elements in compact expressions, and the potential-energy matrix elements are given by the potential values at grid points. This method has been implemented to investigate a variety of interesting physical problems, such as quantum-mechanical scattering problems (Rescigno & McCurdy (2000)), bright solitons in Bose-Einstein Condensates and ultracold plasmas (Collins et al. (2004)), non-equilibrium Green ˛a˛es function calculations (Balzer et al.

In the present work, the method of FE DVR which is detailed in references (Rescigno & McCurdy (2000)) and (Balzer et al. (2010)) is adopted to obtain the Hamiltonian matrix elements for atoms confined by quantum dots. Within the framework of FE DVR, the interval [0,*Rmax*] is divided into *ne* finite elements, in which each element between [*xi*

is further subdivided by *ng* Gauss quadrature points (see Fig. 2). Taking advantage of the standard Gauss-Lobatto points *xm* and weights *wm* (Michels (1963)), we define the generalized

2 (*xi*+1 − *xi*

*ng*−1 ∑ *m*=2

1 (*x*)]

*ψ*(*xi m*)*wi*

)*xm* + (*xi*+1 + *xi*

, *xi*+1]

)], (7)

*ng* . (9)

(10)

*m*(*x*) are

(11)

). (8)

*ng* = *xi*+1 because *x*1 = −1 and *xng* = 1. In calculations,

*m* + *ψ*(*xi*+1)*wi*

for m=1 (bridge);

for else (element),

*m*�) for *xi* ≤ *x* ≤ *xi*+1;

0 for *x* < *xi* or *x* > *xi*+1.

(2010)), and photoionization of impurities in quantum dots (Lin & Ho (2011)).

[(*xi*+1 − *xi*

)*wi* 1 +

*ng* (*x*) + *f i*+1

*ng* + *wi*+1 1 )

where the Lagrange interpolating polynomials or so-called Lobatto shape functions *f i*

(*x* − *xi m*�)

(*xi m* − *xi*

The wave functions are expanded in terms of local basis functions (see Fig. 2),

*wi m* = *wm*

**2. Theoretical method**

Gauss-Lobatto points,