# Valence Band Mixing in GaAs/AlGaAs Quantum Wells Adjacent to Self-Assembled InAlAs Antidots

Authors: Takuya Kawazu
Journal: Journal of Nanomaterials
 (2019)

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## Abstract

Optical properties of GaAs/AlGaAs quantum wells (QWs) in the vicinity of InAlAs quantum dots (QDs) were studied and compared with a theoretical model to clarify how the QD strain affects the electronic states in the nearby QW. In0.4Al0.6As QDs are embedded at the top of the QWs; the QD layer acts as a source of strain as well as an energy barrier. Photoluminescence excitation (PLE) measurements showed that the QD formation leads to the increase in the ratio Ie-lh/Ie-hh of the PLE intensities for the light hole (lh) and the heavy hole (hh), indicating the presence of the valence band mixing. We also theoretically calculated the hh-lh mixing in the QW due to the nearby QD strain and evaluated the PLE ratio Ie-lh/Ie-hh.


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## Body

## 1. Introduction

Molecular beam epitaxy (MBE), metal-organic chemical vapor deposition (MOCVD), and other deposition methods have been significantly developed over a few decades, which have made it possible to fabricate a variety of high-quality quantum nanostructures such as quantum wells (QWs), quantum wires, and quantum dots (QDs) [1]. In particular, much of the work has been done on 10 nm-scale self-assembled QDs in the Stranski–Krastanov (SK) mode due to their ease of fabrication [2–5]. Because of their potential for the advanced photonic device applications including lasers [6], charge storage devices [7–9], and interband and intersubband photodetectors [10–13], the optical and electric properties of self-assembled QDs have been actively studied by using near-infrared spectroscopy [14, 15], capacitance spectroscopy [16, 17], and resonant tunneling spectroscopy [18].In the SK growth, highly crystalline QDs are spontaneously formed just by depositing source materials on a lattice-mismatched substrate. The SK growth initially proceeds in a layer-by-layer fashion, which forms a thin wetting layer (WL). After the layer exceeds a certain critical thickness, high-density defect-free QDs are coherently formed in order to reduce the strain energy. Therefore, the strain is inevitably present inside and outside the QDs, which strongly affects the electronic states inside and outside them. Many experimental and theoretical works have been done on the QD strain and have clarified the strain effects on the electronic states [19–22].In this work, we study optical properties of GaAs/AlGaAs QWs in the vicinity of In0.4Al0.6As QDs and clarify the effect of the QD strain on the nearby QW. The band gap of In0.4Al0.6As is estimated to be about 1.87 eV and is larger than that of GaAs [23–25]. For the case of In0.4Al0.6As QDs in GaAs, the energy gap difference will be larger due to the strain. Hence, when In0.4Al0.6As QDs are embedded at the top of a GaAs/AlGaAs QW, the QD layer acts as a source of strain as well as an energy barrier, which is a suitable system for the study of the strain effects outside the QDs. By analyzing the PLE spectra and carrying out the theoretical calculation, we here show how the QD strain affects the electronic states in the nearby QW.


## 2. Materials and Methods

### 2.1. Composition of AlGaAs and InAlAs

For sample preparation, we employed MBE with the use of a reflection high-energy electron diffraction (RHEED) system. We can explore the growth dynamics of MBE by observing temporal oscillations in the intensity of the specular beam of the RHEED pattern. The intensity of the specular beam changes depending on the surface roughness. Under normal conditions, the crystal growth by MBE proceeds in a layer-by-layer fashion [26]. When a smooth complete layer is formed on the crystal surface during the MBE growth, a maximum intensity of the specular beam is observed. In contrast, the specular beam intensity becomes minimum at half-layer coverage, where 2D nuclei are grown at random positions on the crystal surface. Therefore, the oscillation period of the specular beam intensity precisely corresponds to the growth of a single molecular layer [27]. The analysis of the intensity oscillation has now been used to calibrate beam fluxes, alloy composition, and the thickness of a deposited layer [28].For MBE of III-V semiconductors, the growth rate is controlled by the group III elements. In addition, when two or more group III elements are simultaneously supplied, their growth rates are additive [29, 30]. By observing the specular beam intensity oscillations during the growths of GaAs and AlAs, we calibrated Ga and Al fluxes so that the deposition rates of Ga and Al are about fGaAs=1 and fAlAs=0.43 monolayer (ML)/s. Hence, the expected Al mole fraction of AlxGa1-xAs is x=fAlAs/fGaAs+fAlAs~0.3 [29]. To determine the In deposition rate fIn, we used the specular beam intensity oscillation for the growth of InGaAs by following Ref. [30]: fIn=fInGaAs−fGaAs~0.28. The expected In mole fraction of InxAl1-xAs is then x=fInAs/fAlAs+fInAs~0.4 [29].


### 2.2. Critical Thickness of QD Formation

RHEED is a powerful tool for investigating not only the growth dynamics (Section2.1) but also the atomic structure of crystal surfaces in situ. At the end of the buffer growth on the (100) surface of GaAs, we clearly observed a streaky 2×4 reconstruction RHEED pattern, indicating the high-quality crystalline of the buffer layer [26]. When In0.4Al0.6As is deposited on the sample surface, the streaky reconstruction changed and its intensity slowly weakened at low InAlAs coverage (<4 ML). In this initial stage, the crystal growth proceeds in a layer-by-layer fashion, and the deposited lattice-mismatched materials form a fully strained 2D layer [3]. For further deposition of 4–5 ML of In0.4Al0.6As, a spotty RHEED pattern suddenly appeared, corresponding to the growth of 3D single crystal dots [2, 3, 30]. Therefore, the critical thickness Θth of In0.4Al0.6As QDs on GaAs is expected to be 4–5 ML; the QD formation occurs when In0.4Al0.6As is deposited above Θth, while a thin 2D layer is formed for the InAlAs deposition below Θth.


### 2.3. Sample Preparation

For our study, we prepared GaAs/AlGaAs QWs with embedded self-assembled InAlAs QDs on semi-insulating GaAs (001) substrates. The sample structure is schematically shown in Figure1. First, we deposited a 300 nm thick GaAs buffer layer and a superlattice buffer (14 periods of 13 nm AlGaAs/2 nm GaAs) at the substrate temperature Ts of 570-580°C. Next, a 5 nm thick GaAs layer was grown to form a reference QW for checking the GaAs deposition rate. We then deposited a superlattice barrier layer (7 periods of 13 nm AlGaAs/2 nm GaAs) and a GaAs QW layer (the thickness Lw=8nm). After Ts was lowered to 520°C, InAlAs QDs were formed by depositing 6.1 ML of InxAl1-xAs with a nominal In composition of x=0.4. While Ts was raised to 570-580°C rapidly, we further grew a superlattice barrier layer (14 periods of 13 nm AlGaAs/2 nm GaAs) and a 10 nm thick GaAs capping layer. To investigate the QD shapes and density by atomic force microscopy (AFM) measurements, we formed InAlAs QDs again on the top of the sample under similar condition for the embedded QDs. This sample with embedded QDs is referred to as S.Figure 1
Schematic drawing of the cross-sectional structure of the GaAs/AlGaAs QW with embedded InAlAs QDs.For comparison, we also grew a sample (C) of the identical structure except that the deposited thickness of InAlAs is about 3.1 ML. In this condition, InAlAs QDs are not formed, since the critical thicknessΘth of In0.4Al0.6As QDs on GaAs is 4–5 ML.


### 2.4. PLE Measurements

PLE spectra of the samples were measured in a closed cycle helium cryostat at about 20 K by using a standard lock-in technique. The samples were excited with a Ti:sapphire laser, where the emission wavelength is variable in the range of about 710 to 830 nm. The luminescence was dispersed by a grating monochromator and detected at the lower energy tail of the QW PL peak by using an AgOCs photomultiplier tube. Figure2 shows the schematic drawing of the PLE measurement setup. The excitation laser enters the sample at an angle of about 45 degrees. Note that the direction of the excitation laser is changed due to the refraction of light, when the laser light enters the inside of the sample. The angle between the excitation laser and the vertical direction of the QW is estimated at about 10 degrees.Figure 2
Schematic drawing of the PLE measurement setup.


## 2.1. Composition of AlGaAs and InAlAs

For sample preparation, we employed MBE with the use of a reflection high-energy electron diffraction (RHEED) system. We can explore the growth dynamics of MBE by observing temporal oscillations in the intensity of the specular beam of the RHEED pattern. The intensity of the specular beam changes depending on the surface roughness. Under normal conditions, the crystal growth by MBE proceeds in a layer-by-layer fashion [26]. When a smooth complete layer is formed on the crystal surface during the MBE growth, a maximum intensity of the specular beam is observed. In contrast, the specular beam intensity becomes minimum at half-layer coverage, where 2D nuclei are grown at random positions on the crystal surface. Therefore, the oscillation period of the specular beam intensity precisely corresponds to the growth of a single molecular layer [27]. The analysis of the intensity oscillation has now been used to calibrate beam fluxes, alloy composition, and the thickness of a deposited layer [28].For MBE of III-V semiconductors, the growth rate is controlled by the group III elements. In addition, when two or more group III elements are simultaneously supplied, their growth rates are additive [29, 30]. By observing the specular beam intensity oscillations during the growths of GaAs and AlAs, we calibrated Ga and Al fluxes so that the deposition rates of Ga and Al are about fGaAs=1 and fAlAs=0.43 monolayer (ML)/s. Hence, the expected Al mole fraction of AlxGa1-xAs is x=fAlAs/fGaAs+fAlAs~0.3 [29]. To determine the In deposition rate fIn, we used the specular beam intensity oscillation for the growth of InGaAs by following Ref. [30]: fIn=fInGaAs−fGaAs~0.28. The expected In mole fraction of InxAl1-xAs is then x=fInAs/fAlAs+fInAs~0.4 [29].


## 2.2. Critical Thickness of QD Formation

RHEED is a powerful tool for investigating not only the growth dynamics (Section2.1) but also the atomic structure of crystal surfaces in situ. At the end of the buffer growth on the (100) surface of GaAs, we clearly observed a streaky 2×4 reconstruction RHEED pattern, indicating the high-quality crystalline of the buffer layer [26]. When In0.4Al0.6As is deposited on the sample surface, the streaky reconstruction changed and its intensity slowly weakened at low InAlAs coverage (<4 ML). In this initial stage, the crystal growth proceeds in a layer-by-layer fashion, and the deposited lattice-mismatched materials form a fully strained 2D layer [3]. For further deposition of 4–5 ML of In0.4Al0.6As, a spotty RHEED pattern suddenly appeared, corresponding to the growth of 3D single crystal dots [2, 3, 30]. Therefore, the critical thickness Θth of In0.4Al0.6As QDs on GaAs is expected to be 4–5 ML; the QD formation occurs when In0.4Al0.6As is deposited above Θth, while a thin 2D layer is formed for the InAlAs deposition below Θth.


## 2.3. Sample Preparation

For our study, we prepared GaAs/AlGaAs QWs with embedded self-assembled InAlAs QDs on semi-insulating GaAs (001) substrates. The sample structure is schematically shown in Figure1. First, we deposited a 300 nm thick GaAs buffer layer and a superlattice buffer (14 periods of 13 nm AlGaAs/2 nm GaAs) at the substrate temperature Ts of 570-580°C. Next, a 5 nm thick GaAs layer was grown to form a reference QW for checking the GaAs deposition rate. We then deposited a superlattice barrier layer (7 periods of 13 nm AlGaAs/2 nm GaAs) and a GaAs QW layer (the thickness Lw=8nm). After Ts was lowered to 520°C, InAlAs QDs were formed by depositing 6.1 ML of InxAl1-xAs with a nominal In composition of x=0.4. While Ts was raised to 570-580°C rapidly, we further grew a superlattice barrier layer (14 periods of 13 nm AlGaAs/2 nm GaAs) and a 10 nm thick GaAs capping layer. To investigate the QD shapes and density by atomic force microscopy (AFM) measurements, we formed InAlAs QDs again on the top of the sample under similar condition for the embedded QDs. This sample with embedded QDs is referred to as S.Figure 1
Schematic drawing of the cross-sectional structure of the GaAs/AlGaAs QW with embedded InAlAs QDs.For comparison, we also grew a sample (C) of the identical structure except that the deposited thickness of InAlAs is about 3.1 ML. In this condition, InAlAs QDs are not formed, since the critical thicknessΘth of In0.4Al0.6As QDs on GaAs is 4–5 ML.


## 2.4. PLE Measurements

PLE spectra of the samples were measured in a closed cycle helium cryostat at about 20 K by using a standard lock-in technique. The samples were excited with a Ti:sapphire laser, where the emission wavelength is variable in the range of about 710 to 830 nm. The luminescence was dispersed by a grating monochromator and detected at the lower energy tail of the QW PL peak by using an AgOCs photomultiplier tube. Figure2 shows the schematic drawing of the PLE measurement setup. The excitation laser enters the sample at an angle of about 45 degrees. Note that the direction of the excitation laser is changed due to the refraction of light, when the laser light enters the inside of the sample. The angle between the excitation laser and the vertical direction of the QW is estimated at about 10 degrees.Figure 2
Schematic drawing of the PLE measurement setup.


## 3. Results and Discussion

### 3.1. Surface Morphology

To evaluate the sample surface profiles, we performed AFM measurements in a noncontact mode. Figures3(a) and 3(b) show the AFM images of the samples C and S; 3.1 ML of InAlAs was deposited on the former sample, while the InAlAs thickness is 6.1 ML for the latter. On the sample C (Figure 3(a)), we do not observe InAlAs QDs but a low height step-terrace structure, since the deposited InAlAs amount is less than the critical thickness Θth (4–5 ML) of the QD growth. In contrast, InAlAs QDs are successfully formed on the sample S, where the QD radius rd and height hd are 13 and 2.2 nm, respectively, on average, and the dot density is about 5.3×1014m−2.Figure 3
AFM images (500×500nm2) of sample surfaces for In0.4Al0.6As depositions of 3.1 (a) and 6.1 ML (b).
(a)
(b)


### 3.2. PLE Spectra

Figure4 shows the PLE spectra of the samples C and S. While varying the energy of the exciting laser, we observed the luminescence at 1.572 and 1.576 eV for C and S (marked by arrows in Figure 4). The lower energy peaks at about Ee‐hh=1.58 and 1.584 eV for C and S are attributed to the excitation of electrons and heavy holes (e-hh), where the full width at half maximum (FWHM) Δe-hh is about 3 meV. In contrast, the higher energy peaks at about Ee‐lh=1.603 and 1.607 eV for C and S (Δe‐lh=4meV) come from the excitation of electrons and light holes (e-lh). The slight differences of the peak energies stem from a small unintentional change of the GaAs growth rate; i.e., the GaAs QW thickness of C is a little bit larger than that of S, which was noticed by the PL from the reference QW of 5 nm thick GaAs (not shown). Note that the PLE intensity Ie-lh of the e-lh excitation for S is larger than that for C, although Ie-hh is almost the same for S and R. The ratio Ie-lh/Ie-hh is about 0.5 and 0.65 for C and S, respectively. The larger ratio Ie-lh/Ie-hh for S indicates the presence of the valence band mixing of the hh and lh states in the QW.Figure 4
PLE spectra of QW samples with and without embedded InAlAs QDs.


### 3.3. Excitation Energies ofe-hh and e-lh Pairs

To confirm the assignment of the PLE peaks and also to estimate the experimental error for the QW thickness, we evaluate the excitation energies of thee-hh and e-lh pairs by using a simple model for a finite barrier QW. According to Ref. [31], the ground state energy Ee of electrons confined in a QW (thickness Lw, barrier height Ve for the conduction band) is obtained by solving
(1)tanθ=θ02θ2−1θ02=meVeLw22ℏ2,Ee=ℏ22me2θLw2,where ℏ and me are Planck’s constant and the electron effective mass, respectively. From equation (1), the ground state energies (Ehh, Elh) of heavy and light holes are also calculated by using the hh and lh effective masses (mhh, mlh) and the barrier height Vh for the valence band. The calculated excitation energies of the e-hh and e-lh pairs are Ee‐hh=Eg+Ee+Ehh~1.576eV and Ee‐lh=Eg+Ee+Elh~1.597eV, respectively. The parameters used in the calculation are listed in Table 1. Although we obtain reasonable agreement between the calculated and experimental excitation energies, the experimental values are a little bit larger than the calculated ones. Therefore, we speculate that the QW thickness is slightly smaller (<10%) than the thickness of the designed ones.Table 1
Parameters used in the ground state calculations.
mea
mhha
mlha
Veb (eV)
Vhb (eV)
Ega (eV)
0.067m0
0.35m0
0.09m0
0.259
0.159
1.519
aRef. [25] and bRef. [32]. m0: electron rest mass.


### 3.4. Optical Matrix Element

The PLE spectra are predominantly determined by the optical absorption at the energy of the excitation laser, and the intensity is approximately proportional to the squared matrix element [33, 34]:
(2)M2=∑σ,σ′ψeσe^·p^ψeσ′2.Here,e^ is the unit polarization vector and p^ is the momentum operator. ψeσ is the electron wave function at the Brillouin zone center. In the infinite barrier approximation,
(3)ψeσ=2SLwcosπzLwsσ=1Sfzsσ,where sσ is the periodic part of the s-like Bloch function with spin-up σ=+ or spin-down σ=−. S is the area of the QW. The z-axis is taken to be perpendicular to the QW surface, and the QW is located at the origin. ψhσ is the hole wave function at the Brillouin zone center and is written as
(4)∣ψhσ=∑λAλσSfz∣λ,where Aλσ is the coefficient reflecting the degree of the valence band mixing. The hole state is doubly degenerate and distinguished by σ=±. ∣λ is the Luttinger-Kohn (LK) basis, where λ indicates the heavy holes (J, Jz=3/2, ±3/2) and the light holes (J, Jz=3/2, ±1/2) for the 4×4 valence band formalism.The LK basis∣λ is isotropic in the x-y plane apart from the phase factor and diagonalizes the Hamiltonian for holes in the QW on the (001) surface. Therefore, for the QW without QDs, the hole wave function ψhσ contains only a single component of the LK basis. In contrast, when InAlAs QDs are embedded at the top of the QW, the hole wave function ψhσ consists of several components of the LK basis (see equation (4)) because of the valence band mixing due to the QD strain.From equations (2), (3), and (4), we obtain
(5)M2=∑σ,σ′∑λAλσ′sσe^·p^λ2.sσ∣e^·p^∣λ is determined only by the Bloch functions at the Brillouin zone center and does not depend on the QD strain. Therefore, we can evaluate the valence band mixing due to the strain (i.e., the magnitude of Aλσ) by examining the PLE intensity.


### 3.5. QD Strain Potential and Perturbative Approach

When Young’s modulusEand Poisson’s ratio ν can be considered a constant in the system, the strain εij is expressed as
(6)εij=∂∂xi∂xjχ,where χ is the Lamè displacement potential [35]. By assuming that the height of the QD is written as hd1−r/rd2 and by remembering hd≪rd, the two-dimensional Fourier component of χ is approximately given as [36]
(7)χq,z=1S∫e−iq→·r→χr→,zdr2=−2πS1+ν1−νε0hdq3J2rdqe−qz.Here,J2 is the Bessel function of the second kind. ε0 is the mismatch rate between the lattice constants of GaAs aGaAs and InAlAs aInAlAs [24]; i.e.,
(8)ε0=aGaAs−aInAlAsaInAlAs≅0.03.In this approximation, the piezoelectric potentialφ has the simple form [36]:
(9)φ=3e14zεxyϵ,where e14 and ϵ are the piezoelectric constant and the dielectric constant, respectively. From equations (6) and (7), we can easily prove that the hydrostatic strain disappears:
(10)εxx+εyy+εzz=0.For the potentialV of the strain effect including piezoelectricity due to the QD located at (0, 0, Ld), the matrix element between the hole states is given as [37]
(11)λ,k→+q→∣V∣λ′,k→≡Vλλ′q→=Pε+QεRεSε0Rε+Pε−Qε0−SεSε+0Pε−QεRε0−Sε+Rε+Pε+Qε,(12)Pε=e3e14ϵqxqy∫z−Ldχq,z−Ldfz2dz,(13)Qε=−32bq2∫χq,z−Ldfz2dz,(14)Rε=32bqx2−qy2−idqxqy∫χq,z−Ldfz2dz,(15)Sε=diqx+qyq∫χq,z−Ldfz2dz,where b and d are the deformation potential coefficients. The unperturbed wave function of the hole ground state is defined as
(16)∣λ,k→≡1Seik→·r→fzλ,where r→=x,y is the positional vector in the QW plain. The ordering of the valence band basis functions in equation (11) is 3/2,3/2, 3/2,−1/2, 3/2,1/2, and 3/2,−3/2.By following the second-order perturbation theory for the QD strain potentialV in equation (11), the valence band Hamiltonian matrix at the Brillouin zone center is written as [38, 39]
(17)Hλλ′=Eλ0δλλ′−ndot2∑λ′′,q→≠01Eλ′′q−Eλ0+1Eλ′′q−Eλ′0Vλλ′′−q→Vλ′′λ′q→,where ndot is the density of the randomly distributed QDs. Eλ0 and Eλq are given as
(18)Eλ0=−ℏ22m0γ1∓2γ2πLw2,Eλq=Eλ0−ℏ22m0γ1±γ2q2,where m0 and γ are the electron rest mass and the Luttinger parameter, respectively. The upper and lower signs are taken for the heavy and light holes, respectively. In our case, the first-order perturbation terms vanish because Vλλ′0=0 (see equations (12), (13), (14), and (15)). The coefficient Aλσ can be evaluated by diagonalizing equation (17).


### 3.6. Calculated Results

Some terms of the squared matrix elementVλλ′′−q→Vλ′′λ′q→ in equation (17) disappear by the angular integration. The nonvanishing terms are Pε−q→Pεq→θ, Qε−q→Qεq→θ, Rε−q→Rεq→θ, Sε−q→Sεq→θ, and Pε−q→Rεq→θ=−Rε−q→Pεq→θ, where A−q→Bq→θ is defined as
(19)A−q→Bq→θ≡∫02πS2πA−q→S2πBq→dθ.Because of this, the valence band mixing only occurs betweenJz=3/2 and -1/2 and between Jz=−3/2 and 1/2. Figure 5 shows the nonvanishing terms as functions of the wave number q. The parameters used in the calculation are listed in Table 2, and Ld is set to 5 nm. For simplicity, the ternary parameters for In0.4Al0.6As are derived from the binary parameters by linear interpolation [40]. Note that all the nonvanishing terms exponentially decrease with the increase in q at q>0.06kF (kF: Fermi wave number), indicating that only the hole states near the Brillouin zone center k≪kF contribute to the perturbative calculation.Figure 5
Squared matrix elements as functions of the wave number.Table 2
Material parameters used in the calculation.
Luttinger parameters
Dielectric constant
Deformation potential coefficients
Poisson’s ratio
Piezoelectric constant
γ1a
γ2a
ϵ/ϵ0b
bc (eV)
dc (eV)
νd
e14e (C/m2)
GaAs
6.98
2.06
12.4
-1.7
-4.5
0.31
-0.16
aRef. [24], bRef. [34], cRef. [25], dRef. [35], and eRef. [36].Figures6(a) and 6(b) show the calculated coefficient Aλσ for the heavy and light hole states as functions of the QD density ndot. Here, the coefficient Aλσ does not depend on σ=±. The valence band mixing occurs between Jz=3/2 and -1/2 and between Jz=−3/2 and 1/2, which exhibits the same result. Note that the light hole component A−1/2σ increases in the heavy hole state as the QD density ndot increases, while A3/2σ becomes large in the light hole state at large ndot.Figure 6
Calculated coefficientAλσ for heavy (a) and light hole states (b) as functions of QD density ndot.
(a)
(b)Figure7 shows the ratio Mlh2/Mhh2 of the squared optical matrix elements for the light and heavy hole states calculated by using equation (5) as a function of the QD density ndot. Here, we take into account the incident angle and the refraction of the excitation light. In Figure 7, we also plot the measured ratio Ie-lh/Ie-hh of the integrated PLE intensities for the light and heavy holes with open circles, since the ratio Mlh2/Mhh2 is a rough estimate of the ratio Ie-lh/Ie-hh. Note that the ratio Mlh2/Mhh2 increases with the increase in the QD density ndot, resulting from the heavy and light hole mixing. This trend agrees with the experimental results, which indicates at least partly the validity of our assumption; the valence band mixing in the QW is caused by the nearby QD strain.Figure 7
Calculated ratioMlh2/Mhh2 of squared optical matrix elements for light and heavy hole states as a function of the QD density ndot. The measured ratio Ie-lh/Ie-hh of the integrated PLE intensities for the light and heavy holes is also plotted with open circles.Although our model qualitatively explains the experimental results, the quantitative agreement is not satisfactory. One reason may be the simplifications in the model. We ignore the split-off valence bands [34], the excitonic effects [33], and the interface roughness of the QW [41]. In addition, the QD shape is assumed to be a thin sheet, and the strain parameters are approximated to be isotropic [35, 36]. Another reason may be the experimental errors for the thickness of the QW, the compositions of the ternary materials, and also the angle of the incident laser light. Besides, we estimated the diameter and height of embedded InAlAs QDs from the AFM images of QDs formed on the substrate surface. However, the shape of QDs grown in the SK mode often changes during overgrowth [42]. These issues will be left as the subject of a future work.


## 3.1. Surface Morphology

To evaluate the sample surface profiles, we performed AFM measurements in a noncontact mode. Figures3(a) and 3(b) show the AFM images of the samples C and S; 3.1 ML of InAlAs was deposited on the former sample, while the InAlAs thickness is 6.1 ML for the latter. On the sample C (Figure 3(a)), we do not observe InAlAs QDs but a low height step-terrace structure, since the deposited InAlAs amount is less than the critical thickness Θth (4–5 ML) of the QD growth. In contrast, InAlAs QDs are successfully formed on the sample S, where the QD radius rd and height hd are 13 and 2.2 nm, respectively, on average, and the dot density is about 5.3×1014m−2.Figure 3
AFM images (500×500nm2) of sample surfaces for In0.4Al0.6As depositions of 3.1 (a) and 6.1 ML (b).
(a)
(b)


## 3.2. PLE Spectra

Figure4 shows the PLE spectra of the samples C and S. While varying the energy of the exciting laser, we observed the luminescence at 1.572 and 1.576 eV for C and S (marked by arrows in Figure 4). The lower energy peaks at about Ee‐hh=1.58 and 1.584 eV for C and S are attributed to the excitation of electrons and heavy holes (e-hh), where the full width at half maximum (FWHM) Δe-hh is about 3 meV. In contrast, the higher energy peaks at about Ee‐lh=1.603 and 1.607 eV for C and S (Δe‐lh=4meV) come from the excitation of electrons and light holes (e-lh). The slight differences of the peak energies stem from a small unintentional change of the GaAs growth rate; i.e., the GaAs QW thickness of C is a little bit larger than that of S, which was noticed by the PL from the reference QW of 5 nm thick GaAs (not shown). Note that the PLE intensity Ie-lh of the e-lh excitation for S is larger than that for C, although Ie-hh is almost the same for S and R. The ratio Ie-lh/Ie-hh is about 0.5 and 0.65 for C and S, respectively. The larger ratio Ie-lh/Ie-hh for S indicates the presence of the valence band mixing of the hh and lh states in the QW.Figure 4
PLE spectra of QW samples with and without embedded InAlAs QDs.


## 3.3. Excitation Energies ofe-hh and e-lh Pairs

To confirm the assignment of the PLE peaks and also to estimate the experimental error for the QW thickness, we evaluate the excitation energies of thee-hh and e-lh pairs by using a simple model for a finite barrier QW. According to Ref. [31], the ground state energy Ee of electrons confined in a QW (thickness Lw, barrier height Ve for the conduction band) is obtained by solving
(1)tanθ=θ02θ2−1θ02=meVeLw22ℏ2,Ee=ℏ22me2θLw2,where ℏ and me are Planck’s constant and the electron effective mass, respectively. From equation (1), the ground state energies (Ehh, Elh) of heavy and light holes are also calculated by using the hh and lh effective masses (mhh, mlh) and the barrier height Vh for the valence band. The calculated excitation energies of the e-hh and e-lh pairs are Ee‐hh=Eg+Ee+Ehh~1.576eV and Ee‐lh=Eg+Ee+Elh~1.597eV, respectively. The parameters used in the calculation are listed in Table 1. Although we obtain reasonable agreement between the calculated and experimental excitation energies, the experimental values are a little bit larger than the calculated ones. Therefore, we speculate that the QW thickness is slightly smaller (<10%) than the thickness of the designed ones.Table 1
Parameters used in the ground state calculations.
mea
mhha
mlha
Veb (eV)
Vhb (eV)
Ega (eV)
0.067m0
0.35m0
0.09m0
0.259
0.159
1.519
aRef. [25] and bRef. [32]. m0: electron rest mass.


## 3.4. Optical Matrix Element

The PLE spectra are predominantly determined by the optical absorption at the energy of the excitation laser, and the intensity is approximately proportional to the squared matrix element [33, 34]:
(2)M2=∑σ,σ′ψeσe^·p^ψeσ′2.Here,e^ is the unit polarization vector and p^ is the momentum operator. ψeσ is the electron wave function at the Brillouin zone center. In the infinite barrier approximation,
(3)ψeσ=2SLwcosπzLwsσ=1Sfzsσ,where sσ is the periodic part of the s-like Bloch function with spin-up σ=+ or spin-down σ=−. S is the area of the QW. The z-axis is taken to be perpendicular to the QW surface, and the QW is located at the origin. ψhσ is the hole wave function at the Brillouin zone center and is written as
(4)∣ψhσ=∑λAλσSfz∣λ,where Aλσ is the coefficient reflecting the degree of the valence band mixing. The hole state is doubly degenerate and distinguished by σ=±. ∣λ is the Luttinger-Kohn (LK) basis, where λ indicates the heavy holes (J, Jz=3/2, ±3/2) and the light holes (J, Jz=3/2, ±1/2) for the 4×4 valence band formalism.The LK basis∣λ is isotropic in the x-y plane apart from the phase factor and diagonalizes the Hamiltonian for holes in the QW on the (001) surface. Therefore, for the QW without QDs, the hole wave function ψhσ contains only a single component of the LK basis. In contrast, when InAlAs QDs are embedded at the top of the QW, the hole wave function ψhσ consists of several components of the LK basis (see equation (4)) because of the valence band mixing due to the QD strain.From equations (2), (3), and (4), we obtain
(5)M2=∑σ,σ′∑λAλσ′sσe^·p^λ2.sσ∣e^·p^∣λ is determined only by the Bloch functions at the Brillouin zone center and does not depend on the QD strain. Therefore, we can evaluate the valence band mixing due to the strain (i.e., the magnitude of Aλσ) by examining the PLE intensity.


## 3.5. QD Strain Potential and Perturbative Approach

When Young’s modulusEand Poisson’s ratio ν can be considered a constant in the system, the strain εij is expressed as
(6)εij=∂∂xi∂xjχ,where χ is the Lamè displacement potential [35]. By assuming that the height of the QD is written as hd1−r/rd2 and by remembering hd≪rd, the two-dimensional Fourier component of χ is approximately given as [36]
(7)χq,z=1S∫e−iq→·r→χr→,zdr2=−2πS1+ν1−νε0hdq3J2rdqe−qz.Here,J2 is the Bessel function of the second kind. ε0 is the mismatch rate between the lattice constants of GaAs aGaAs and InAlAs aInAlAs [24]; i.e.,
(8)ε0=aGaAs−aInAlAsaInAlAs≅0.03.In this approximation, the piezoelectric potentialφ has the simple form [36]:
(9)φ=3e14zεxyϵ,where e14 and ϵ are the piezoelectric constant and the dielectric constant, respectively. From equations (6) and (7), we can easily prove that the hydrostatic strain disappears:
(10)εxx+εyy+εzz=0.For the potentialV of the strain effect including piezoelectricity due to the QD located at (0, 0, Ld), the matrix element between the hole states is given as [37]
(11)λ,k→+q→∣V∣λ′,k→≡Vλλ′q→=Pε+QεRεSε0Rε+Pε−Qε0−SεSε+0Pε−QεRε0−Sε+Rε+Pε+Qε,(12)Pε=e3e14ϵqxqy∫z−Ldχq,z−Ldfz2dz,(13)Qε=−32bq2∫χq,z−Ldfz2dz,(14)Rε=32bqx2−qy2−idqxqy∫χq,z−Ldfz2dz,(15)Sε=diqx+qyq∫χq,z−Ldfz2dz,where b and d are the deformation potential coefficients. The unperturbed wave function of the hole ground state is defined as
(16)∣λ,k→≡1Seik→·r→fzλ,where r→=x,y is the positional vector in the QW plain. The ordering of the valence band basis functions in equation (11) is 3/2,3/2, 3/2,−1/2, 3/2,1/2, and 3/2,−3/2.By following the second-order perturbation theory for the QD strain potentialV in equation (11), the valence band Hamiltonian matrix at the Brillouin zone center is written as [38, 39]
(17)Hλλ′=Eλ0δλλ′−ndot2∑λ′′,q→≠01Eλ′′q−Eλ0+1Eλ′′q−Eλ′0Vλλ′′−q→Vλ′′λ′q→,where ndot is the density of the randomly distributed QDs. Eλ0 and Eλq are given as
(18)Eλ0=−ℏ22m0γ1∓2γ2πLw2,Eλq=Eλ0−ℏ22m0γ1±γ2q2,where m0 and γ are the electron rest mass and the Luttinger parameter, respectively. The upper and lower signs are taken for the heavy and light holes, respectively. In our case, the first-order perturbation terms vanish because Vλλ′0=0 (see equations (12), (13), (14), and (15)). The coefficient Aλσ can be evaluated by diagonalizing equation (17).


## 3.6. Calculated Results

Some terms of the squared matrix elementVλλ′′−q→Vλ′′λ′q→ in equation (17) disappear by the angular integration. The nonvanishing terms are Pε−q→Pεq→θ, Qε−q→Qεq→θ, Rε−q→Rεq→θ, Sε−q→Sεq→θ, and Pε−q→Rεq→θ=−Rε−q→Pεq→θ, where A−q→Bq→θ is defined as
(19)A−q→Bq→θ≡∫02πS2πA−q→S2πBq→dθ.Because of this, the valence band mixing only occurs betweenJz=3/2 and -1/2 and between Jz=−3/2 and 1/2. Figure 5 shows the nonvanishing terms as functions of the wave number q. The parameters used in the calculation are listed in Table 2, and Ld is set to 5 nm. For simplicity, the ternary parameters for In0.4Al0.6As are derived from the binary parameters by linear interpolation [40]. Note that all the nonvanishing terms exponentially decrease with the increase in q at q>0.06kF (kF: Fermi wave number), indicating that only the hole states near the Brillouin zone center k≪kF contribute to the perturbative calculation.Figure 5
Squared matrix elements as functions of the wave number.Table 2
Material parameters used in the calculation.
Luttinger parameters
Dielectric constant
Deformation potential coefficients
Poisson’s ratio
Piezoelectric constant
γ1a
γ2a
ϵ/ϵ0b
bc (eV)
dc (eV)
νd
e14e (C/m2)
GaAs
6.98
2.06
12.4
-1.7
-4.5
0.31
-0.16
aRef. [24], bRef. [34], cRef. [25], dRef. [35], and eRef. [36].Figures6(a) and 6(b) show the calculated coefficient Aλσ for the heavy and light hole states as functions of the QD density ndot. Here, the coefficient Aλσ does not depend on σ=±. The valence band mixing occurs between Jz=3/2 and -1/2 and between Jz=−3/2 and 1/2, which exhibits the same result. Note that the light hole component A−1/2σ increases in the heavy hole state as the QD density ndot increases, while A3/2σ becomes large in the light hole state at large ndot.Figure 6
Calculated coefficientAλσ for heavy (a) and light hole states (b) as functions of QD density ndot.
(a)
(b)Figure7 shows the ratio Mlh2/Mhh2 of the squared optical matrix elements for the light and heavy hole states calculated by using equation (5) as a function of the QD density ndot. Here, we take into account the incident angle and the refraction of the excitation light. In Figure 7, we also plot the measured ratio Ie-lh/Ie-hh of the integrated PLE intensities for the light and heavy holes with open circles, since the ratio Mlh2/Mhh2 is a rough estimate of the ratio Ie-lh/Ie-hh. Note that the ratio Mlh2/Mhh2 increases with the increase in the QD density ndot, resulting from the heavy and light hole mixing. This trend agrees with the experimental results, which indicates at least partly the validity of our assumption; the valence band mixing in the QW is caused by the nearby QD strain.Figure 7
Calculated ratioMlh2/Mhh2 of squared optical matrix elements for light and heavy hole states as a function of the QD density ndot. The measured ratio Ie-lh/Ie-hh of the integrated PLE intensities for the light and heavy holes is also plotted with open circles.Although our model qualitatively explains the experimental results, the quantitative agreement is not satisfactory. One reason may be the simplifications in the model. We ignore the split-off valence bands [34], the excitonic effects [33], and the interface roughness of the QW [41]. In addition, the QD shape is assumed to be a thin sheet, and the strain parameters are approximated to be isotropic [35, 36]. Another reason may be the experimental errors for the thickness of the QW, the compositions of the ternary materials, and also the angle of the incident laser light. Besides, we estimated the diameter and height of embedded InAlAs QDs from the AFM images of QDs formed on the substrate surface. However, the shape of QDs grown in the SK mode often changes during overgrowth [42]. These issues will be left as the subject of a future work.


## 4. Conclusions

We investigated the optical properties of GaAs/AlGaAs quantum wells (QWs) in the vicinity of self-assembled InAlAs quantum dots (QDs), where the QDs act as a source of strain. We prepared two QW samples with and without the InAlAs QDs by depositing In0.4Al0.6As above and below the critical thickness of the Stranski–Krastanov (SK) growth. Photoluminescence excitation (PLE) measurements showed that the InAlAs QDs increase the ratio Ie-lh/Ie-hh of the PLE intensities for the electron-light hole (lh) and electron-heavy hole (hh) excitations; Ie-lh/Ie-hh is about 0.65 and 0.5 for the samples with and without the InAlAs QDs, respectively. We also theoretically studied how the QD strain affects the electric states of carriers in the QW. It was found that the QD strain induces the hh-lh mixing, resulting in the increase in Ie-lh and the decrease in Ie-hh, which partly explains the experimental data.



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*Source: 5349291-2019-04-28.xml*