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AlxGa1-x As quantum ring
Physica E 126 (2021) 114476
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Band alignment transition from type I to type II in GaAs / AlxGa1-x As quantum ring Rihab Sellami a, b, Afef Ben Mansour a, b, Mohamed Souhail Kehili a, b, Adnen Melliti a, b, * a b
Universit´e de Carthage, Institut Pr´eparatoire aux Etudes Scientifiques et Techniques, Laboratoire Mat´eriaux-Mol´ecules et Applications, BP51 La Marsa 2070, Tunisia Universit´e de Tunis, Ecole Nationale Sup´erieure des Ing´enieurs de Tunis, 5 Rue Taha Hussein—Montfleury, 1008, Tunis, Tunisia
A B S T R A C T
We present a study of the electronic and optical properties of carriers in GaAs/AlxGa1-xAs quantum ring (QR) as a function of the aluminum concentration (x) in the barrier and the ring height (hM). We calculated the wavefunctions and the energies of the electrons and holes using the effective mass approximation. We found that the variation of x or hM, leads to a crossover, and consequently the mixing, of the electronic states of symmetry X and Γ. Then, a change from type I to type II of the GaAs/AlGaAs quantum ring occurs for x about 0.6 and hM about 1.3 nm. To present the point of type I-type II transition, we drew a color map that shows the shift of this point with x and hM. The radiative lifetime of the carriers is calculated in both type I and type II cases. In the latter case, we have taken into account the coupling between the confined electronic states of symmetry X and Γ. A value of 7 meV for the coupling potential is deduced from a comparison between our theoretical results and experimental ones. A color map showing the variation of the lifetime with x and hM was likewise drawn. Finally, these results can be exploited in several applications like solar cells.
Author contribution Rihab Sellami: Investigation, Writing - original draft, Visualization, Afef Ben Mansour and Mohamed Souhail Kehili: Writing, Adnen Melliti: Conceptualization, Methodology, Validation, Supervision. 1. Introduction Thanks to their unique properties, like atom-like features which result in discrete density of states, there is an increased interest in the fundamental properties [1–4] and applications [5–14] of quantum dot (QD)/quantum ring (QR) systems. For these systems, we can predefine the optical and electrical properties with controllable size and geometry [15] which allows us to enlarge the fields of application of these nanostructures. The improvement of research in this area shows that one can obtain a type II band alignment such that the holes are confined in QR quantum structures (or in barriers) and the electrons are confined in the barrier (or in the quantum ring structure) [16]. We note that in these type II structures the holes and the electrons have the same symmetry (Γ). Type II band alignment allows these systems to be integrated into different applications, for example, memory storage [3–5] and solar cells [6,7]. A transition from type-I to type-II band alignment is observed for the
GaAs/AlxGa1− xAs QRs grown on GaAs (1 0 0) by droplet epitaxy via photoluminescence (PL) and time-resolved photoluminescence (TRPL) measurements with varying the Al-concentration [17]. In the case of type-I band alignment, both holes and electrons, with Γ-symmetry, are confined in the GaAs quantum structures. In the case of type-II band alignment, the holes are always confined in the GaAs quantum struc tures but the electrons, of X-symmetry, are confined in the AlGaAs barrier. In this article, we theoretically studied the electronic and optical properties of charge carriers in GaAs/AlxGa1-x As QR. First, we discussed the effect of varying Al concentration (x) and the rim height (hM) on the wavefunction and the type I to type II band alignment transition. Sec ondly, we calculated the energies and the wavefunctions of electrons near the type I-type II transition by taking into account a mixing po tential of electronic states of symmetry Γand X. We obtained an anti crossing behavior near the Γ-X conduction band crossover. Thirdly, we calculated the oscillator strength and thereafter the radiative lifetime with and without coupling. From a comparison be tween the experimental results [18] and our theoretical ones, we esti mated the coupling potential value. We found a quite strong interaction potential of 7 meV which is explained by the confinement of carriers in the QR.
* Corresponding author. Universit´e de Carthage, Institut Pr´ eparatoire aux Etudes Scientifiques et Techniques, Laboratoire Mat´eriaux-Mol´ecules et Applications, BP51 La Marsa 2070, Tunisia. E-mail address: [email protected] (A. Melliti). https://doi.org/10.1016/j.physe.2020.114476 Received 13 May 2020; Received in revised form 8 August 2020; Accepted 25 September 2020 Available online 30 September 2020 1386-9477/© 2020 Elsevier B.V. All rights reserved.
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114476
2. Theoretical model
Qc (0.65) [19] does not depend on the aluminum composition (x) ofAlx Ga1− x As. The energy gap of AlxGa1-x As is written in the following quadratic form [20]:
The height of the studied GaAs/AlGaAs QR as a function of the polar coordinates is modeled by the expression proposed by Fomin et al. [18]: } { ⎧ 2 ⎪ [hM − h0 ] 1 − [ρ/R − 1] ⎪ ⎪ ⎪ ,ρ ≤ R ⎨ h(ρ, ϕ) = h0+ {[ρ − R]/γ0 }2 + 1 (1) ⎪ ⎪ hM − h∞ ⎪ ⎪ ⎩ h(ρ, ϕ) = h∞ + , ρ>R {[ρ − R]/γ∞ }2 + 1
EgΓ(X) (Alx Ga1− x As) = (1 − x)EgΓ(X) (GaAs) + xEgΓ(X) (AlAs) − x(1 − x)CΓ(X) With CΓ(X) is the bowing parameter: CΓ = (− 0.127 + 1.310x) eV CX = 0.055 eV
Where R is the radius of the rim top, ho is the height at the center (i.e. r = 0), h∞ is the height away from the ring, and hM is the rim height; γ0 and γ ∞ determine the inner and outer slope of the rim, respectively. To have a QR structure similar to the AFM experimental results of GaAs QRs in AlxGa1− xAs [17] (inner radius: 37 nm, outer radius: 63 nm and ring height: 1.25 nm), we have chosen R = 18 nm, ho = 0, h∞ 0.6, hM = 1.2, γ0 = 5, γ∞ = 3. The wetting layer width is 2 ML. The single-particle states are calculated in the effective mass approximation without strain [19]. Indeed, the lattice mismatch be tween GaAs and AlxGa1− x As is very small. We used the following Hamiltonian for electron and hole, respectively: ⎧ − ħ2 → 1 → ⎪ ⎪ H = ∇ + VeConf (r) ∇ ⎪ ⎨ e me (r) 2
We also note that the effective mass of electrons and holes of AlxGa1was calculated with linear interpolation between GaAs and AlAs. All the values used in the computation are presented in Table 1. ¨dinger equations are solved numerically using the centered The Schro finite difference method with Dirichlet boundary conditions and the Jacobi–Davidson method. We chose a very large computation box (537.5 nm × 537.5 nm × 44.64 nm 3D) because after the transition from type I to type II, the electron wavefunction becomes spread in the AlGaAs matrix. Fig. 1 shows the energy diagram of the QR GaAs/AlxGa 1− x As, for x = 0.3 and x = 0.7. xAs
(2)
⎪ 2 ⎪ 1 → ⎪ Conf ⎩ Hh = − ħ → ∇ ∇ + Vh (r) mh (r) 2
3. Results and discussions 3.1. Electronic energy and wavefunction
The first term is the kinetic energy of the particle and is written in the Ben Daniel–Duke form. me,h(r) is the effective mass of electron and hole in the corresponding material. The second term is the confinement po tential describing the discontinuity in the band structure at the interface between the different materials. In the Γ valley, the confinement po tential is written as: { 0 in GaAs VConf = (3) e ΔEc (Γ) inAlx Ga1− x As
The composition in real GaAs/AlxGa1-xAs nanostructures grown by droplet epitaxy is not uniform. The Al composition (x) varies from the WL to the tip of the nanostructure [21,22]. However, to simplify the calculation, we supposed a uniform composition profile inside the QR and in the barrier. In Figs. 2 and 3, we present the hole and electron wavefunctions for different Al-concentration (x) and rim height (hM) respectively. We notice from Fig. 2 that the hole is confined inside the QR and the confinement increases with x. We can explain that by the increase of the hight of the AlGaAs barrier (Fig. 1). This behavior is accompanied by an increase in the hole energy (Ehh1) as a function of the Al-concentration (x) as shown in Fig. 4. For Γ valley, we notice that when x increases, the wavefunction of the ground state of the electron becomes more restricted in the GaAs region (Fig. 2 (a.1, a.2)). This behavior is accompanied by an increase in the electron energy (EΓ1 ) as shown in Fig. 5-a. This is due, as in the case of the hole, to the increase of the barrier height (Fig. 1). For the Xz valley, the wavefunction is localized in the center outside the QR (Fig. 2(b)).With the increase of x, ΔEc(X)increases. So, the confinement electron energy (EX1 ) increases (Fig. 5-b). Consequently, the wavefunction becomes less confined in the center and approaches the GaAs region (Fig. 2 (b.1, b.2)). From these analyzes, it is concluded that the localization of the wavefunction and the carriers energies are strongly affected by the
With [20]. ( ΔEc(Γ) = Qc EgΓ (Alx Ga1−
) x
(4)
As) − EgΓ (GaAs)
ΔEc(Γ) is the conduction band discontinuity in the Γ valley. EΓg (Alx Ga1− x As) is the energy gap of Alx Ga1− x As in the Γ valley The hole confinement potential is written as: { 0 in GaAs VConf = h ΔEv(Γ) inAlx Ga1− x As
(5)
With ( ΔEv (Γ) = (1 − Qc ) EgΓ (Alx Ga1−
x
) As) − EgΓ (GaAs)
(6)
ΔEv (Γ) is the valence band discontinuity. In the X valley, the electronic confinement potential is written as: { ΔEc(X) in GaAs Veconf = (7) 0 inAlx Ga1− x As
Table 1 Material parameters used throughout the paper [20]. m0 is the free electron mas.
With ( ) ΔEc(X) = EgX (Alx Ga1− x As) − EgX (GaAs) − ΔEv(Γ)
(8)
x
As) is the energy gap of Alx Ga1−
x
Parameter
GaAs
AlAs
me(m0) mh(m0) mXl (m0)
0.067 0.51 1.3
0.15 0.5 0.97
1.981 1.519 28.2 12.9
2.24 3.099 21.1 –
EgX(ev) EgΓ (ev) Ep(ev)
ΔEc(X) is the conduction band discontinuity in the X valley. EXg (Alx Ga1−
(9)
As in the X valley
εr 2
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114476
Fig. 1. Diagramme of band alignment of GaAs/AlxGa1− xAs QR for x = 0.3 (a) and x = 0.7 (b). Solid lines represent the local Γ valleys while dashed lines represent the local X valleys.
Fig. 2. Wavefunction of the ground state of the hole in the QR GaAs/AlxGa
aluminum concentration. It is noted that the energy and the wave function are also affected by the variation of the height of the ring but this effect is less pronounced.
1− x
As for hM = 1.3 nm: (a) x = 0.3 (b) x = 0.6 (z = 0 nm).
Δ = EΓ1 − EX = (EΓ1 − EX1 ) −
( ) EgX (GaAs) − EgΓ (GaAs) + ΔEc(X)
(10)
Fig. 6 shows a color map of Δ as a function of x and hM. Depending on whether delta is positive or negative, the structure will be of type I or II. The red line in this color map indicates all the points of zero delta values. On the one hand, we notice that all the values of x less than 0.56 have a negative value of Δ whatever the rim height between 1.2 nm and 1.4 nm. This is the case of a type I QR. On the other hand, the delta value is positive for all values of x superior to 0.6 for hM between 1.2 nm and 1.4 nm. So we are talking here about a type II band alignment of GaAs/ AlxGa1-xAs. For aluminum concentration between 0.57 and 0.6 the band align ment of GaAs/AlxGa1-xAs QR changes from type I to type II with the variation of the height. For example: Δ(0.6, 1.2nm) = 0.012 > 0 (type II QR) and Δ(0.6, 1.3nm) = − 0.0001 (type I QR). In the experiment, the transition from type I to type II QR is studied
3.2. Type I-type II transition In this section, we will study the transition from type I to type II of the QR GaAs/AlxGa1-xAs. In Fig. 5-a, we presented the energies of the electrons in the valleys Γ (EΓ1) and Xz (EX) as a function of x. The origin of the energy is chosen at the bottom of the Γ–symmetry conduction band of GaAs. We remark the crossover of the two curves that is a signature of the transition from type I to type II. To have the transition point of the band alignment of QR we intro duce the parameter (Δ) which is the difference between the energies of the electrons in the valleys Γ and X. It is written in the following form:
3
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Fig. 3. Wavefunction of ground state of the electron in the QR GaAs/AlxGa 1− (b.2) x = 0.6.
Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114476
x
As for hM = 1.3 nm: (a) Γ valley: (a.1) x = 0.3, (a.2) x = 0.6, (b) X valley: (b.1) x = 0.3,
by photoluminescence (PL) and photoluminescence with time resolution (TRPL) measurements. A large blue-shift with excitation intensity and a large increase in the lifetime are a signature feature for GaAs/Alx Ga1− xAs nanostructures with a type-II band alignment [17]. To identify the transition point, a variation in the aluminum concentration was made between x = 0.2 and 0.7. The results show that for about x = 0.6 we can have a change of alignment band. We conclude that our theo retical results are in good agreement with the experimental ones.
fX =
⃒∫ ⃒ ⃒ Ep ⃒⃒ ψ c,Γ (r).ψ h (r).dv⃒⃒ 2 ⃒ 2ET
(11)
With Ep is the Kane energy, ET is the type I transition energy (Fig. 1-a) and ψ c,Γ (r) and ψ h (r) are Γ -electron and hole wavefunctions respectively. The radiative lifetime is calculated using the formula [15]: 6πε0 m0 c3 ħ2 e2 nET2 fX
(12)
3.3. Calculation of X-Γ coupling
τ=
In relation to the type of the band alignment for the GaAs/AlxGa1-xAs QR structure, we will study the oscillator strength of the carriers. In the case of type I the oscillator strength is given by:
With ε0 is the dielectric vacuum permittivity, c is the speed of light,ħ is the reduced Planck constant,e is the electronic charge, n is the refractive 4
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114476
Fig. 4. Variation of the energy of the ground state of the holes (Ehh1) as a function of the concentration (x) for hM = 1.3 nm.
Fig. 6. Color map of Δ as a function of the aluminum concentration (x) and the rim height (hM). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
quantum wires and quantum dots predict a mixing of the conduction band Γ and Xz states due to the potential step at the interface [23,24]. The electron wavefunction taking into account this coupling is written in the following form.
ψ c (→ r ) = AΓ ψ c,Γ (→ r ) + AX ψ c,X (→ r) ′
(13)
Where AΓ and AX are respectively the coefficients of Γ and Xz state wavefunctions forming the mixed states. We write the electronic wavefunction in the Γ valley as following [19]:
ψ c,Γ (→ r ) = Uc,Γ (→ r )FΓ (→ r)
(14)
Uc,Γ (→ r ) is the periodic part of the Bloch function at the Γ point FΓ (→ r ): is the envelope function. In the same way, we write the electronic wavefunction in the X valley:
ψ c,X (→ r ) = Uc,X (→ r )FX (→ r)
(15)
Uc,X (→ r )is the periodic part of the Bloch function at the Xz point. The Schrodinger equation taking into account the coupling potential is written in the following form: Fig. 5. (a) Energy variation of ground states of electrons in the Γ and X valleys, EΓ1 (blue curve) and EX (red curve) respectively, as a function of the concen tration of aluminum (x)) for hM = 1.3 nm. The origin of the energy is chosen at the bottom of the Γ–symmetry conduction band of GaAs. (b) Energy variation of the ground state of electrons in X valley (EX1 ) as a function of the concentration of aluminum (x) for hM = 1.3 nm.The origin of the energy is chosen at the bottom of the X–symmetry conduction band of AlGaAs. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
′ ′ H ψ c (→ r ) = Eψ c (→ r)
(16)
With: H = H0 + V
(17)
H0: Hamiltonian of the nanostructure without coupling V: coupling potential By projecting equation (16) on ψ c,Γ and ψ c,X and using the normali zation f he wavefunctions, we get the following system (S):
index of QR material, m0 is the free electron mass. Experimentally [17], for x between 0.3 and 0.45 the lifetime grad ually increases with. From x = 0. 6, that corresponds to the type I -type II transition, a very remarkable jump is noted for the value of the lifetime but it remains finite (5.6ns) although the electron and the hole have different symmetries (X and Γ respectively) which leads, normally, to infinite lifetime. In our study and to find this behavior we consider a coupling between the Γ and X confined electronic states near x = 0.6. Theoretical models of strongly confined semiconductor systems like
⎧ ⎪ ⎨ AΓ ⎪ ⎩
v = √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ 2 v + (E − EΓ )2 √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ AX = 1 − AΓ
(18)
The energy position of both states interacting in the first-order 5
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114476
is expected to be more pronounced. For x = 0.55, 0.6, 0.65, we used the same value of v to calculate the radiative lifetime. Fig. 7 shows the variation of E- and E+ as a function of x close to x corresponding to type I-Type II transition. We remark the anticrossingtype behavior of the energies of Γ-electron and X-electron because of the Γ–X coupling. Fig. 8 summarizes the calculation of the lifetime for different Alconcentrations from 0.3 to 0.65 and for the three heights hM (1.2, 1.3, 1.4 nm). The red line indicates the transition from type I to type II. We remark that for the three heights, the radiative lifetime progressively decreases from the concentration 0.3 to the concentration correspond ing to the type I-type II transition. This is explained by the increase of the exciton energy and the overlap between the Γ-electron and hole wave functions as a function of x. The short radiative lifetime shows a high probability of electron-hole recombination in the QR structure. Above the Al-concentration corresponding to the type I-type II transition, the radiative lifetime increases very quickly. For example, for hM = 1.3 nm, it varies from 0.97ns for x = 0.55–19.47 ns for x = 0.65. So, the strong brusque increase in radiative life can be used to identify the transition from type I to type II band alignment. We also notice that the radiative lifetime decreases with the increase of the rim height. This is explained by the increase of the overlap be tween the Γ-electron and hole wavefunctions as a function of x.
Fig. 7. Variation of E− and E+ as a function of the concentration of aluminum (x).
4. Conclusion In this paper we studied theoretically the electronic and optical properties of carriers in a QR GaAs/AlxGa1-x As and the transition of band alignment for type I to type II. The investigation is done by calculating the wavefunctions and energies of electron and hole using the effective mass approximation with variation in Al-composition and the rim height. We showed that the localization of electrons and holes wavefunction is sensitive to the variation of x and hM. The increase of x allows the increase of the height of the barrier AlxGa1-xAs, therefore, more confinement of the electrons and holes wavefunction in the Γ valley. In the X valley, the increase of x and hM prevents progressively the X-electron wavefunction to penetrate inside the ring. To present the point of type I-type II transition, we drew a color map that shows the shift of this point with x and hM. This point is near x = 0.6 which is in good agreement with the experimental results. The x cor responding to the transition decreases when hM decreases. Then, we calculated the coupling between Γ-electron and X-electron states. The value of coupling potential near the x corresponding to the type I-type II transition was estimated from the comparison between the value of lifetime calculated taking into account the Γ-X coupling and the experimental one. A color map showing the variation of the lifetime with x and hM was likewise drawn. Finally, these results can be exploited in several applications like solar cells.
Fig. 8. Color map of the radiative lifetime variation as a function of the aluminum concentration(x) and the rim height (hM). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
perturbation theory can be expressed as: } { ]12 [ 1 E± = (EΓ1 + EX ) ± (EΓ1 − EX )2 + 4v2 2
(19)
/
The oscillator strength of the transition between the hole level and the lowest electron level with energy E- is obtained by replacing in ′ formula (11) ψ c,Γ (r)by the wavefunction ψ c (→ r ) corresponding to E-: fcX =
1 Ep 2 2 ⋅R ⋅AΓ ′ ⋅ ET 2
(20)
Where R is the overlap between the Γ-electron and hole wavefunctions ′ and ET is the type II transition (Fig. 1-b) energy given by: ′
ET = Ehh1 + Eg (GaAs) + E−
Declaration of competing interest
(21)
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Accordingly to equation (12), the radiative lifetime becomes:
τ=
6πε0 m0 c3 ħ2 ′ e2 nET2 fcX
(22)
References
To estimate the value of v for the concentration 0.6, we used the experimental lifetime value (5.317ns) [15] of the same concentration and equation (22). The calculation gives a value of v of the order of 7mev. This value is near to the one found in InAs/GaAs quantum dot [25]. This quite strong value of the mixing potential is explained by the confinement of electrons in quantum rings, therefore the mixing of states
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Physica E: Low-dimensional Systems and Nanostructures journal homepage: http://www.elsevier.com/locate/physe
Band alignment transition from type I to type II in GaAs / AlxGa1-x As quantum ring Rihab Sellami a, b, Afef Ben Mansour a, b, Mohamed Souhail Kehili a, b, Adnen Melliti a, b, * a b
Universit´e de Carthage, Institut Pr´eparatoire aux Etudes Scientifiques et Techniques, Laboratoire Mat´eriaux-Mol´ecules et Applications, BP51 La Marsa 2070, Tunisia Universit´e de Tunis, Ecole Nationale Sup´erieure des Ing´enieurs de Tunis, 5 Rue Taha Hussein—Montfleury, 1008, Tunis, Tunisia
A B S T R A C T
We present a study of the electronic and optical properties of carriers in GaAs/AlxGa1-xAs quantum ring (QR) as a function of the aluminum concentration (x) in the barrier and the ring height (hM). We calculated the wavefunctions and the energies of the electrons and holes using the effective mass approximation. We found that the variation of x or hM, leads to a crossover, and consequently the mixing, of the electronic states of symmetry X and Γ. Then, a change from type I to type II of the GaAs/AlGaAs quantum ring occurs for x about 0.6 and hM about 1.3 nm. To present the point of type I-type II transition, we drew a color map that shows the shift of this point with x and hM. The radiative lifetime of the carriers is calculated in both type I and type II cases. In the latter case, we have taken into account the coupling between the confined electronic states of symmetry X and Γ. A value of 7 meV for the coupling potential is deduced from a comparison between our theoretical results and experimental ones. A color map showing the variation of the lifetime with x and hM was likewise drawn. Finally, these results can be exploited in several applications like solar cells.
Author contribution Rihab Sellami: Investigation, Writing - original draft, Visualization, Afef Ben Mansour and Mohamed Souhail Kehili: Writing, Adnen Melliti: Conceptualization, Methodology, Validation, Supervision. 1. Introduction Thanks to their unique properties, like atom-like features which result in discrete density of states, there is an increased interest in the fundamental properties [1–4] and applications [5–14] of quantum dot (QD)/quantum ring (QR) systems. For these systems, we can predefine the optical and electrical properties with controllable size and geometry [15] which allows us to enlarge the fields of application of these nanostructures. The improvement of research in this area shows that one can obtain a type II band alignment such that the holes are confined in QR quantum structures (or in barriers) and the electrons are confined in the barrier (or in the quantum ring structure) [16]. We note that in these type II structures the holes and the electrons have the same symmetry (Γ). Type II band alignment allows these systems to be integrated into different applications, for example, memory storage [3–5] and solar cells [6,7]. A transition from type-I to type-II band alignment is observed for the
GaAs/AlxGa1− xAs QRs grown on GaAs (1 0 0) by droplet epitaxy via photoluminescence (PL) and time-resolved photoluminescence (TRPL) measurements with varying the Al-concentration [17]. In the case of type-I band alignment, both holes and electrons, with Γ-symmetry, are confined in the GaAs quantum structures. In the case of type-II band alignment, the holes are always confined in the GaAs quantum struc tures but the electrons, of X-symmetry, are confined in the AlGaAs barrier. In this article, we theoretically studied the electronic and optical properties of charge carriers in GaAs/AlxGa1-x As QR. First, we discussed the effect of varying Al concentration (x) and the rim height (hM) on the wavefunction and the type I to type II band alignment transition. Sec ondly, we calculated the energies and the wavefunctions of electrons near the type I-type II transition by taking into account a mixing po tential of electronic states of symmetry Γand X. We obtained an anti crossing behavior near the Γ-X conduction band crossover. Thirdly, we calculated the oscillator strength and thereafter the radiative lifetime with and without coupling. From a comparison be tween the experimental results [18] and our theoretical ones, we esti mated the coupling potential value. We found a quite strong interaction potential of 7 meV which is explained by the confinement of carriers in the QR.
* Corresponding author. Universit´e de Carthage, Institut Pr´ eparatoire aux Etudes Scientifiques et Techniques, Laboratoire Mat´eriaux-Mol´ecules et Applications, BP51 La Marsa 2070, Tunisia. E-mail address: [email protected] (A. Melliti). https://doi.org/10.1016/j.physe.2020.114476 Received 13 May 2020; Received in revised form 8 August 2020; Accepted 25 September 2020 Available online 30 September 2020 1386-9477/© 2020 Elsevier B.V. All rights reserved.
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114476
2. Theoretical model
Qc (0.65) [19] does not depend on the aluminum composition (x) ofAlx Ga1− x As. The energy gap of AlxGa1-x As is written in the following quadratic form [20]:
The height of the studied GaAs/AlGaAs QR as a function of the polar coordinates is modeled by the expression proposed by Fomin et al. [18]: } { ⎧ 2 ⎪ [hM − h0 ] 1 − [ρ/R − 1] ⎪ ⎪ ⎪ ,ρ ≤ R ⎨ h(ρ, ϕ) = h0+ {[ρ − R]/γ0 }2 + 1 (1) ⎪ ⎪ hM − h∞ ⎪ ⎪ ⎩ h(ρ, ϕ) = h∞ + , ρ>R {[ρ − R]/γ∞ }2 + 1
EgΓ(X) (Alx Ga1− x As) = (1 − x)EgΓ(X) (GaAs) + xEgΓ(X) (AlAs) − x(1 − x)CΓ(X) With CΓ(X) is the bowing parameter: CΓ = (− 0.127 + 1.310x) eV CX = 0.055 eV
Where R is the radius of the rim top, ho is the height at the center (i.e. r = 0), h∞ is the height away from the ring, and hM is the rim height; γ0 and γ ∞ determine the inner and outer slope of the rim, respectively. To have a QR structure similar to the AFM experimental results of GaAs QRs in AlxGa1− xAs [17] (inner radius: 37 nm, outer radius: 63 nm and ring height: 1.25 nm), we have chosen R = 18 nm, ho = 0, h∞ 0.6, hM = 1.2, γ0 = 5, γ∞ = 3. The wetting layer width is 2 ML. The single-particle states are calculated in the effective mass approximation without strain [19]. Indeed, the lattice mismatch be tween GaAs and AlxGa1− x As is very small. We used the following Hamiltonian for electron and hole, respectively: ⎧ − ħ2 → 1 → ⎪ ⎪ H = ∇ + VeConf (r) ∇ ⎪ ⎨ e me (r) 2
We also note that the effective mass of electrons and holes of AlxGa1was calculated with linear interpolation between GaAs and AlAs. All the values used in the computation are presented in Table 1. ¨dinger equations are solved numerically using the centered The Schro finite difference method with Dirichlet boundary conditions and the Jacobi–Davidson method. We chose a very large computation box (537.5 nm × 537.5 nm × 44.64 nm 3D) because after the transition from type I to type II, the electron wavefunction becomes spread in the AlGaAs matrix. Fig. 1 shows the energy diagram of the QR GaAs/AlxGa 1− x As, for x = 0.3 and x = 0.7. xAs
(2)
⎪ 2 ⎪ 1 → ⎪ Conf ⎩ Hh = − ħ → ∇ ∇ + Vh (r) mh (r) 2
3. Results and discussions 3.1. Electronic energy and wavefunction
The first term is the kinetic energy of the particle and is written in the Ben Daniel–Duke form. me,h(r) is the effective mass of electron and hole in the corresponding material. The second term is the confinement po tential describing the discontinuity in the band structure at the interface between the different materials. In the Γ valley, the confinement po tential is written as: { 0 in GaAs VConf = (3) e ΔEc (Γ) inAlx Ga1− x As
The composition in real GaAs/AlxGa1-xAs nanostructures grown by droplet epitaxy is not uniform. The Al composition (x) varies from the WL to the tip of the nanostructure [21,22]. However, to simplify the calculation, we supposed a uniform composition profile inside the QR and in the barrier. In Figs. 2 and 3, we present the hole and electron wavefunctions for different Al-concentration (x) and rim height (hM) respectively. We notice from Fig. 2 that the hole is confined inside the QR and the confinement increases with x. We can explain that by the increase of the hight of the AlGaAs barrier (Fig. 1). This behavior is accompanied by an increase in the hole energy (Ehh1) as a function of the Al-concentration (x) as shown in Fig. 4. For Γ valley, we notice that when x increases, the wavefunction of the ground state of the electron becomes more restricted in the GaAs region (Fig. 2 (a.1, a.2)). This behavior is accompanied by an increase in the electron energy (EΓ1 ) as shown in Fig. 5-a. This is due, as in the case of the hole, to the increase of the barrier height (Fig. 1). For the Xz valley, the wavefunction is localized in the center outside the QR (Fig. 2(b)).With the increase of x, ΔEc(X)increases. So, the confinement electron energy (EX1 ) increases (Fig. 5-b). Consequently, the wavefunction becomes less confined in the center and approaches the GaAs region (Fig. 2 (b.1, b.2)). From these analyzes, it is concluded that the localization of the wavefunction and the carriers energies are strongly affected by the
With [20]. ( ΔEc(Γ) = Qc EgΓ (Alx Ga1−
) x
(4)
As) − EgΓ (GaAs)
ΔEc(Γ) is the conduction band discontinuity in the Γ valley. EΓg (Alx Ga1− x As) is the energy gap of Alx Ga1− x As in the Γ valley The hole confinement potential is written as: { 0 in GaAs VConf = h ΔEv(Γ) inAlx Ga1− x As
(5)
With ( ΔEv (Γ) = (1 − Qc ) EgΓ (Alx Ga1−
x
) As) − EgΓ (GaAs)
(6)
ΔEv (Γ) is the valence band discontinuity. In the X valley, the electronic confinement potential is written as: { ΔEc(X) in GaAs Veconf = (7) 0 inAlx Ga1− x As
Table 1 Material parameters used throughout the paper [20]. m0 is the free electron mas.
With ( ) ΔEc(X) = EgX (Alx Ga1− x As) − EgX (GaAs) − ΔEv(Γ)
(8)
x
As) is the energy gap of Alx Ga1−
x
Parameter
GaAs
AlAs
me(m0) mh(m0) mXl (m0)
0.067 0.51 1.3
0.15 0.5 0.97
1.981 1.519 28.2 12.9
2.24 3.099 21.1 –
EgX(ev) EgΓ (ev) Ep(ev)
ΔEc(X) is the conduction band discontinuity in the X valley. EXg (Alx Ga1−
(9)
As in the X valley
εr 2
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Fig. 1. Diagramme of band alignment of GaAs/AlxGa1− xAs QR for x = 0.3 (a) and x = 0.7 (b). Solid lines represent the local Γ valleys while dashed lines represent the local X valleys.
Fig. 2. Wavefunction of the ground state of the hole in the QR GaAs/AlxGa
aluminum concentration. It is noted that the energy and the wave function are also affected by the variation of the height of the ring but this effect is less pronounced.
1− x
As for hM = 1.3 nm: (a) x = 0.3 (b) x = 0.6 (z = 0 nm).
Δ = EΓ1 − EX = (EΓ1 − EX1 ) −
( ) EgX (GaAs) − EgΓ (GaAs) + ΔEc(X)
(10)
Fig. 6 shows a color map of Δ as a function of x and hM. Depending on whether delta is positive or negative, the structure will be of type I or II. The red line in this color map indicates all the points of zero delta values. On the one hand, we notice that all the values of x less than 0.56 have a negative value of Δ whatever the rim height between 1.2 nm and 1.4 nm. This is the case of a type I QR. On the other hand, the delta value is positive for all values of x superior to 0.6 for hM between 1.2 nm and 1.4 nm. So we are talking here about a type II band alignment of GaAs/ AlxGa1-xAs. For aluminum concentration between 0.57 and 0.6 the band align ment of GaAs/AlxGa1-xAs QR changes from type I to type II with the variation of the height. For example: Δ(0.6, 1.2nm) = 0.012 > 0 (type II QR) and Δ(0.6, 1.3nm) = − 0.0001 (type I QR). In the experiment, the transition from type I to type II QR is studied
3.2. Type I-type II transition In this section, we will study the transition from type I to type II of the QR GaAs/AlxGa1-xAs. In Fig. 5-a, we presented the energies of the electrons in the valleys Γ (EΓ1) and Xz (EX) as a function of x. The origin of the energy is chosen at the bottom of the Γ–symmetry conduction band of GaAs. We remark the crossover of the two curves that is a signature of the transition from type I to type II. To have the transition point of the band alignment of QR we intro duce the parameter (Δ) which is the difference between the energies of the electrons in the valleys Γ and X. It is written in the following form:
3
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Fig. 3. Wavefunction of ground state of the electron in the QR GaAs/AlxGa 1− (b.2) x = 0.6.
Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114476
x
As for hM = 1.3 nm: (a) Γ valley: (a.1) x = 0.3, (a.2) x = 0.6, (b) X valley: (b.1) x = 0.3,
by photoluminescence (PL) and photoluminescence with time resolution (TRPL) measurements. A large blue-shift with excitation intensity and a large increase in the lifetime are a signature feature for GaAs/Alx Ga1− xAs nanostructures with a type-II band alignment [17]. To identify the transition point, a variation in the aluminum concentration was made between x = 0.2 and 0.7. The results show that for about x = 0.6 we can have a change of alignment band. We conclude that our theo retical results are in good agreement with the experimental ones.
fX =
⃒∫ ⃒ ⃒ Ep ⃒⃒ ψ c,Γ (r).ψ h (r).dv⃒⃒ 2 ⃒ 2ET
(11)
With Ep is the Kane energy, ET is the type I transition energy (Fig. 1-a) and ψ c,Γ (r) and ψ h (r) are Γ -electron and hole wavefunctions respectively. The radiative lifetime is calculated using the formula [15]: 6πε0 m0 c3 ħ2 e2 nET2 fX
(12)
3.3. Calculation of X-Γ coupling
τ=
In relation to the type of the band alignment for the GaAs/AlxGa1-xAs QR structure, we will study the oscillator strength of the carriers. In the case of type I the oscillator strength is given by:
With ε0 is the dielectric vacuum permittivity, c is the speed of light,ħ is the reduced Planck constant,e is the electronic charge, n is the refractive 4
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114476
Fig. 4. Variation of the energy of the ground state of the holes (Ehh1) as a function of the concentration (x) for hM = 1.3 nm.
Fig. 6. Color map of Δ as a function of the aluminum concentration (x) and the rim height (hM). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
quantum wires and quantum dots predict a mixing of the conduction band Γ and Xz states due to the potential step at the interface [23,24]. The electron wavefunction taking into account this coupling is written in the following form.
ψ c (→ r ) = AΓ ψ c,Γ (→ r ) + AX ψ c,X (→ r) ′
(13)
Where AΓ and AX are respectively the coefficients of Γ and Xz state wavefunctions forming the mixed states. We write the electronic wavefunction in the Γ valley as following [19]:
ψ c,Γ (→ r ) = Uc,Γ (→ r )FΓ (→ r)
(14)
Uc,Γ (→ r ) is the periodic part of the Bloch function at the Γ point FΓ (→ r ): is the envelope function. In the same way, we write the electronic wavefunction in the X valley:
ψ c,X (→ r ) = Uc,X (→ r )FX (→ r)
(15)
Uc,X (→ r )is the periodic part of the Bloch function at the Xz point. The Schrodinger equation taking into account the coupling potential is written in the following form: Fig. 5. (a) Energy variation of ground states of electrons in the Γ and X valleys, EΓ1 (blue curve) and EX (red curve) respectively, as a function of the concen tration of aluminum (x)) for hM = 1.3 nm. The origin of the energy is chosen at the bottom of the Γ–symmetry conduction band of GaAs. (b) Energy variation of the ground state of electrons in X valley (EX1 ) as a function of the concentration of aluminum (x) for hM = 1.3 nm.The origin of the energy is chosen at the bottom of the X–symmetry conduction band of AlGaAs. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
′ ′ H ψ c (→ r ) = Eψ c (→ r)
(16)
With: H = H0 + V
(17)
H0: Hamiltonian of the nanostructure without coupling V: coupling potential By projecting equation (16) on ψ c,Γ and ψ c,X and using the normali zation f he wavefunctions, we get the following system (S):
index of QR material, m0 is the free electron mass. Experimentally [17], for x between 0.3 and 0.45 the lifetime grad ually increases with. From x = 0. 6, that corresponds to the type I -type II transition, a very remarkable jump is noted for the value of the lifetime but it remains finite (5.6ns) although the electron and the hole have different symmetries (X and Γ respectively) which leads, normally, to infinite lifetime. In our study and to find this behavior we consider a coupling between the Γ and X confined electronic states near x = 0.6. Theoretical models of strongly confined semiconductor systems like
⎧ ⎪ ⎨ AΓ ⎪ ⎩
v = √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ 2 v + (E − EΓ )2 √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ AX = 1 − AΓ
(18)
The energy position of both states interacting in the first-order 5
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114476
is expected to be more pronounced. For x = 0.55, 0.6, 0.65, we used the same value of v to calculate the radiative lifetime. Fig. 7 shows the variation of E- and E+ as a function of x close to x corresponding to type I-Type II transition. We remark the anticrossingtype behavior of the energies of Γ-electron and X-electron because of the Γ–X coupling. Fig. 8 summarizes the calculation of the lifetime for different Alconcentrations from 0.3 to 0.65 and for the three heights hM (1.2, 1.3, 1.4 nm). The red line indicates the transition from type I to type II. We remark that for the three heights, the radiative lifetime progressively decreases from the concentration 0.3 to the concentration correspond ing to the type I-type II transition. This is explained by the increase of the exciton energy and the overlap between the Γ-electron and hole wave functions as a function of x. The short radiative lifetime shows a high probability of electron-hole recombination in the QR structure. Above the Al-concentration corresponding to the type I-type II transition, the radiative lifetime increases very quickly. For example, for hM = 1.3 nm, it varies from 0.97ns for x = 0.55–19.47 ns for x = 0.65. So, the strong brusque increase in radiative life can be used to identify the transition from type I to type II band alignment. We also notice that the radiative lifetime decreases with the increase of the rim height. This is explained by the increase of the overlap be tween the Γ-electron and hole wavefunctions as a function of x.
Fig. 7. Variation of E− and E+ as a function of the concentration of aluminum (x).
4. Conclusion In this paper we studied theoretically the electronic and optical properties of carriers in a QR GaAs/AlxGa1-x As and the transition of band alignment for type I to type II. The investigation is done by calculating the wavefunctions and energies of electron and hole using the effective mass approximation with variation in Al-composition and the rim height. We showed that the localization of electrons and holes wavefunction is sensitive to the variation of x and hM. The increase of x allows the increase of the height of the barrier AlxGa1-xAs, therefore, more confinement of the electrons and holes wavefunction in the Γ valley. In the X valley, the increase of x and hM prevents progressively the X-electron wavefunction to penetrate inside the ring. To present the point of type I-type II transition, we drew a color map that shows the shift of this point with x and hM. This point is near x = 0.6 which is in good agreement with the experimental results. The x cor responding to the transition decreases when hM decreases. Then, we calculated the coupling between Γ-electron and X-electron states. The value of coupling potential near the x corresponding to the type I-type II transition was estimated from the comparison between the value of lifetime calculated taking into account the Γ-X coupling and the experimental one. A color map showing the variation of the lifetime with x and hM was likewise drawn. Finally, these results can be exploited in several applications like solar cells.
Fig. 8. Color map of the radiative lifetime variation as a function of the aluminum concentration(x) and the rim height (hM). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
perturbation theory can be expressed as: } { ]12 [ 1 E± = (EΓ1 + EX ) ± (EΓ1 − EX )2 + 4v2 2
(19)
/
The oscillator strength of the transition between the hole level and the lowest electron level with energy E- is obtained by replacing in ′ formula (11) ψ c,Γ (r)by the wavefunction ψ c (→ r ) corresponding to E-: fcX =
1 Ep 2 2 ⋅R ⋅AΓ ′ ⋅ ET 2
(20)
Where R is the overlap between the Γ-electron and hole wavefunctions ′ and ET is the type II transition (Fig. 1-b) energy given by: ′
ET = Ehh1 + Eg (GaAs) + E−
Declaration of competing interest
(21)
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Accordingly to equation (12), the radiative lifetime becomes:
τ=
6πε0 m0 c3 ħ2 ′ e2 nET2 fcX
(22)
References
To estimate the value of v for the concentration 0.6, we used the experimental lifetime value (5.317ns) [15] of the same concentration and equation (22). The calculation gives a value of v of the order of 7mev. This value is near to the one found in InAs/GaAs quantum dot [25]. This quite strong value of the mixing potential is explained by the confinement of electrons in quantum rings, therefore the mixing of states
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