Valence BandMixing in GaAs / AlGaAs QuantumWells Adjacent to Self-Assembled InAlAs Antidots

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.

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][20][21][22].
In this work, we study optical properties of GaAs/Al-GaAs QWs in the vicinity of In 0.4 Al 0.6 As QDs and clarify the effect of the QD strain on the nearby QW.The band gap of In 0.4 Al 0.6 As is estimated to be about 1.87 eV and is larger than that of GaAs [23][24][25].For the case of In 0.4 Al 0.6 As QDs in GaAs, the energy gap difference will be larger due to the strain.Hence, when In 0.4 Al 0.6 As 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.

Composition of AlGaAs and InAlAs.
For sample preparation, we employed MBE with the use of a reflection highenergy 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 f GaAs = 1 and f AlAs = 0 43 monolayer (ML)/s.Hence, the expected Al mole fraction of Al x Ga 1-x As is x = f AlAs / f GaAs + f AlAs ~0 3 [29].To determine the In deposition rate f In , we used the specular beam intensity oscillation for the growth of InGaAs by following Ref.[30]: f In = f InGaAs − f GaAs ~0 28.

The expected In mole fraction of In
2.2.Critical Thickness of QD Formation.RHEED is a powerful tool for investigating not only the growth dynamics (Section 2.1) 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 In 0.4 Al 0.6 As 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 layerby-layer fashion, and the deposited lattice-mismatched materials form a fully strained 2D layer [3].For further deposition of 4-5 ML of In 0.4 Al 0.6 As, a spotty RHEED pattern suddenly appeared, corresponding to the growth of 3D single crystal dots [2,3,30].Therefore, the critical thickness Θ th of In 0.4 Al 0.6 As QDs on GaAs is expected to be 4-5 ML; the QD formation occurs when In 0.4 Al 0.6 As is deposited above Θ th , while a thin 2D layer is formed for the InAlAs deposition below Θ th .

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 Figure 1.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 T s 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 L w = 8 nm).After T s was lowered to 520 °C, InAlAs QDs were formed by depositing 6.1 ML of In x Al 1-x As with a nominal In composition of x = 0 4.While T s 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.
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 thickness Θ th of In 0.4 Al 0.6 As QDs on GaAs is 4-5 ML.

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.Figure 2 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.

Results and Discussion
3.1.Surface Morphology.To evaluate the sample surface profiles, we performed AFM measurements in a noncontact mode.Figures 3(a 3.2.PLE Spectra. Figure 4 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 E e-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) Δ e-hh is about 3 meV.In contrast, the higher energy peaks at about E e-lh = 1 603 and 1.607 eV for C and S (Δ e-lh = 4 meV ) 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 I e-lh of the e-lh excitation for S is larger than that for C, although I e-hh is almost the same for S and R. The ratio I e-lh /I e-hh is about 0.5 and 0.65 for C and S, respectively.The larger ratio I e-lh /I e-hh for S indicates the presence of the valence band mixing of the hh and lh states in the QW.

Excitation
Energies of e-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 the e-hh and e-lh pairs by using a simple model for a finite barrier QW.According to Ref. [31], the ground state energy E e of electrons confined in a QW (thickness L w , barrier height V e for the conduction band) is obtained by solving where ℏ and m e are Planck's constant and the electron effective mass, respectively.From equation ( 1), the ground state energies (E hh , E lh ) of heavy and light holes are also calculated by using the hh and lh effective masses (m hh , m lh ) and the barrier height V h for the valence band.The calculated excitation energies of the e-hh and e-lh pairs are E e-hh = E g + E e + E hh ~1 576 eV and E e-lh = E g + E e + E lh ~1 597 eV, 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.
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]: Here, ê is the unit polarization vector and p is the momentum operator.ψ σ e is the electron wave function at the Brillouin zone center.In the infinite barrier approximation, where s σ 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,    3 Journal of Nanomaterials and the QW is located at the origin.ψ σ h is the hole wave function at the Brillouin zone center and is written as where A σ λ 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, J z = 3/2, ±3/2) and the light holes (J, J z = 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 ψ σ h 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 ψ σ h 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 s σ | ê • 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 A σ λ ) by examining the PLE intensity.

QD Strain Potential and Perturbative Approach. When
Young's modulus E and Poisson's ratio ν can be considered a constant in the system, the strain ε ij is expressed as where χ is the Lamè displacement potential [35].By assuming that the height of the QD is written as and by remembering h d ≪ r d , the two-dimensional Fourier component of χ is approximately given as [36] Here, J 2 is the Bessel function of the second kind.ε 0 is the mismatch rate between the lattice constants of GaAs a GaAs and InAlAs a InAlAs [24]; i.e., ε 0 = a GaAs − a InAlAs a InAlAs ≅ 0 03 8 In this approximation, the piezoelectric potential φ has the simple form [36]: where e 14 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: For the potential V of the strain effect including piezoelectricity due to the QD located at (0, 0, L d ), the matrix element between the hole states is given as [37] where b and d are the deformation potential coefficients.The unperturbed wave function of the hole ground state is defined as 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 potential V in equation ( 11), the valence where n dot is the density of the randomly distributed QDs.E 0 λ and E λ q are given as

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where m 0 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 V λλ ′ 0 = 0 (see equations ( 12), ( 13), ( ).The coefficient A σ λ can be evaluated by diagonalizing equation ( 17).
3.6.Calculated Results.Some terms of the squared matrix element V λλ ′′ − q V λ ′′ λ ′ q in equation ( 17) disappear by the angular integration.The nonvanishing terms are q θ , and P ε − q R ε q θ = − R ε − q P ε q θ , where A − q B q θ is defined Because of this, the valence band mixing only occurs between J z = 3/2 and -1/2 and between J z = −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 L d is set to 5 nm.For simplicity, the ternary parameters for In 0.4 Al 0.6 As 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 06 k F (k F : Fermi wave number), indicating that only the hole states near the Brillouin zone center k ≪ k F contribute to the perturbative calculation.for the heavy and light hole states as functions of the QD density n dot .Here, the coefficient A σ λ does not depend on σ = ±.The valence band mixing occurs between J z = 3/2 and -1/2 and between J z = −3/2 and 1/2, which exhibits the same result.Note that the light hole component A σ −1/2 increases in the heavy hole state as the QD density n dot increases, while A σ 3/2 becomes large in the light hole state at large n dot .
Figure 7 shows the ratio M lh 2 / M hh 2 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 n dot .Here, we take into account the incident angle and the refraction of the excitation light.In Figure 7, we also plot the measured ratio I e-lh /I e-hh of the integrated PLE intensities for the light and heavy holes with open circles, since the ratio M lh 2 / M hh 2 is a rough estimate of the ratio I e-lh /I e-hh .Note that the ratio M lh 2 / M hh 2 increases with the increase in the QD density n dot , 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.
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.

Conclusions
We investigated the optical properties of GaAs/AlGaAs quantum wells (QWs) in the vicinity of self-assembled InA-lAs 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 In 0.4 Al 0.6 As above and below the critical thickness of the Stranski-Krastanov (SK) growth.Photoluminescence excitation (PLE) measurements showed that the InAlAs QDs increase the ratio I e-lh /I e-hh of the PLE intensities for the electron-light hole (lh) and electronheavy hole (hh) excitations; I e-lh /I e-hh is about 0.65 and 0.5 ⟨P 휀 (−q)P 휀 (q)⟩ 휃 ⟨Q 휀 (−q)Q 휀 (q)⟩ 휃 i⟨P 휀 (−q)R 휀 (q)⟩ 휃 ⟨S 휀 (−q)A 휀 (q)⟩ 휃 5 Journal of Nanomaterials 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 I e-lh and the decrease in I e-hh , which partly explains the experimental data.Ref. [24], b Ref. [34], c Ref. [25], d Ref. [35], and e Ref. [36].

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Figure 1 :
Figure 1: Schematic drawing of the cross-sectional structure of the GaAs/AlGaAs QW with embedded InAlAs QDs.

Figure 4 :
Figure 4: PLE spectra of QW samples with and without embedded InAlAs QDs.

Figures 6 (
a) and 6(b) show the calculated coefficient A σ λ

Figure 5 :
Figure 5: Squared matrix elements as functions of the wave number.

Figure 6 :Figure 7 :
Figure 6: Calculated coefficient A σ λ for heavy (a) and light hole states (b) as functions of QD density n dot .

Table 1 :
Parameters used in the ground state calculations.

Table 2 :
Material parameters used in the calculation.Luttinger parametersDielectric constant Deformation potential coefficients Poisson's ratio Piezoelectric constant γ 1