One‑ and two‑photon‑induced cyclotron–phonon resonance in modified‑Pöschl–Teller quantum well

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https://doi.org/10.1007/s00339-019-2439-6

One‑ and two‑photon‑induced cyclotron–phonon resonance in modified‑Pöschl–Teller quantum well

Khang D. Pham^1,2 · Le Dinh³ · Chuong V. Nguyen⁴ · Nguyen N. Hieu⁵ · Pham T. Vinh⁶ · Le Thi Ngoc Tu⁶ · Huynh V. Phuc⁶

Received: 9 December 2018 / Accepted: 24 January 2019

Abstract

We study the two-photon-induced cyclotron–phonon resonance (CPR) effect resulting from the interaction between electrons and phonons in a quantum well with the modified-Pöschl–Teller (MPT) potential. The CPR is considered via the magneto- optical absorption coefficient (MOAC) and the full-width at half-maximum (FWHM). Numerical results are presented for a typical GaAs MPT quantum well. It is found that the quantum well parameters, such as well-depth and well-width, the magnetic field, and the temperature affect significantly the energy separation, the MOAC, and the FWHM. Obtained results agree well with previous theoretical and experimental results. Especially, we found two new rules of the well-depth and well-width-dependent FWHM, which should be tested by experimental works to examine their validities.

1 Introduction

Cyclotron–phonon resonance (CPR) is an effect describing the interaction of electrons with the longitudinal optical (LO) phonon when a quantizing magnetic field is applied to semiconductors [1, 2]. This effect is an extension of two

types of resonances connected closely to the cyclotron frequency, 𝜔_c : (i) the cyclotron resonance (CR), where the electromagnetic wave frequency, 𝛺 , is equal to 𝜔_c [3–5]

and (ii) the magneto-phonon resonance (MPR), where the LO-phonon frequency, 𝜔₀ , is equal to 𝜔_c [6, 7]. The reason for the CPR being considered to be more general in comparison with the CR and MPR is that in CPR, the transfer of electrons is the result of a combination of two actions:

the absorption of a photon of energy ℏ𝛺 occurs simultaneously with absorbing or emitting an LO-phonon. This means that the CPR arises at the photon energy ℏ𝛺=pℏ𝜔_c±ℏ𝜔₀ , where p is a positive integer. Because of its crucial role, the CPR has attracted the attention of the wide scientific com- munity from theorists [8–11] to experimentalists [12, 13].

The mentioned theoretical works indicated that it is suit- able to observe the CPR by calculating the optical absorption coefficient, which is handled based on the perturbation method. However, a drawback in this process is that it only allows to study the one-photon process but not to the two- photon one.

Because of its promising applications, especially in basic biological research as well as clinical diagnostics [14], two- photon absorption has attracted interest in recent years.

Using the 𝐤⋅𝐩 model, Padilha et al. [15] have investigated the degenerate and non-degenerate two-photon absorption for different-sized quantum dots. Their experimental results showed that when the energies of the two photons do not coincide the two-photon absorption due to intermediate-state

* Le Dinh

[email protected]

* Nguyen N. Hieu [email protected]

* Huynh V. Phuc [email protected] Khang D. Pham

[email protected]

1 Laboratory of Applied Physics, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

2 Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam

3 Physics Department, University of Education, Hue University, Hue 530000, Vietnam

4 Department of Materials Science and Engineering, Le Quy Don Technical University, Hanoi 100000, Vietnam

5 Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam

6 Division of Theoretical Physics, Dong Thap University, Cao Lanh 870000, Vietnam

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resonant enhancement increases. In addition, while experimen- tally studying the nonlinear absorption properties of Perovs- kite quantum dots, Nagamine et al. [16] showed that the first two-photon absorption peak makes a blue-shift in comparison with the first one-photon absorption one. Their results agree with earlier works reported by Nootz et al. [17] and Padilha et al. [18]. By including the two-photon process, the optical absorption coefficients have also been studied for quantum wells [19–21] and for monolayer graphene [22]. The obtained results showed that the magneto-optical properties of these nanomaterials are strongly dependent on the magnetic field, temperature, and characteristics of the materials. However, these properties of the modified-Pöschl–Teller quantum well (MPTQW) still have not received satisfactory attention.

Since being introduced by Pöschl and Teller [23] and by Rosen and Morse [24], the Pöschl–Teller potential has received the research interest of wide range of researchers. Yıldırım and Tomak investigated nonlinear optical properties including second-harmonic generation and optical coefficients [25], third- harmonic generation [26], and refractive index changes [27].

By including nonlocal effects Wang et al. [28] reported optical absorption and local-field distribution in a Pöschl–Teller quantum well. After that, because of its simplicity, the modified-Pöschl–Teller has been gradually attracting interest. This model has been successfully applied to calculate the matrix elements [29], the Ladder operations [30], the exact energy spectra [31], the influence of electric field [32], and the influence of electron–phonon scattering [33] on refractive index changes, the two particles energy spectrum [34], and the role of the interaction effective range in the bound state properties of two cold atoms in one and quasi-one-dimensional traps [35].

Inspired by the above works, here, we investigate the two- photon-induced CPR in MPTQW with the consideration of electron–phonon scattering. The purpose of the present study is to comprehend the influence of the well-shape parameters, the magnetic field, and the temperature on the magneto-optical absorption spectrum, and to find the law of variation of the full-width at half-maximum (FWHM) with these parameters.

We also propose two new expressions describing the variation of the FWHM with the well-depth and well-width.

2 Theoretical framework

2.1 Characteristics of the quantum well model Based on the effective mass approximation, in the Landau gauge 𝐀= (0,Bx, 0) with 𝐀 being the vector potential related to the magnetic field 𝐁‖Oz , the one-electron Hamiltonian in a quantum well reads

e= 1 (1)

2m^∗(𝐩+e𝐀)²+V(z),

where 𝐩 is the momentum operator, and e is the electron charge. The electron effective mass is m^∗=0.067m₀ with m₀ being the electron rest mass [36]. The corresponding eigenfunctions �𝜆⟩ and eigenenergies E_N,n are given as follows [11]:

where N=0, 1, 2,… is the Landau level (LL) index, n is the electric sub-band index, k_y and L_y are, respectively, the electron wave vector and the normalized length in the y-direction, 𝜔_c=eB∕m^∗ is the cyclotron frequency, and 𝜙_N(x−x₀) is the normalized harmonic oscillator function with x₀= −ℏk_y∕(m^∗𝜔_c) being the x-coordinate of the orbit center. The envelope eigenfunctions, 𝜉_n(z) , and eigenenergies, 𝜀_n , of the nth sub-band depend on the type of confin- ing potential, V(z). In this work, we consider the modified- Pöschl–Teller potential, which is given by [30, 32, 37]

where D=ℏ²𝛼²q(q+1)∕(2m^∗) is the well-depth. We can see that the well-shape is controlled by two parameters 𝛼 and q. However, because of the relation between D and 𝛼 as mentioned above, and for convenience, it is better to use two parameters D (related to the well-depth) and q (related to the well-width) to describe the quantum well-shape. The variation of the well-shape potential with the parameter q at D=0.34 meV is shown in Fig. 1. It is seen that the potential is perfectly symmetric and the well-width increases with the increase of parameter q.

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�𝜆⟩=�N,n,k_y⟩= e^ik^y^y

√L_y𝜙_N(x−x₀)𝜉_n(z),

(3) E_𝜆=E_N,n=

( N+1

2 )

ℏ𝜔_c+𝜀_n,

(4) V(z) = − D

cosh²(𝛼z) = −�²𝛼² 2m^∗

q(q+1)

cosh²(𝛼z), q>1,

q 2q 3

q 4 D 0.34 eV

20 10 0 10 20

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

z nm

VzeV

Fig. 1 Variation of the quantum well-shape potential with the q-parameter

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The envelope eigenfunctions of the nth sub-band in Eq. (2) are as follows [30]

where n=0, 1, 2… , 𝜂=tanh(𝛼z) , and C^q+1∕2−n_n (𝜂) are the Gegenbauer polynomials. The normalization constant A^q_n is given as follows:

The corresponding eigenenergies are

2.2 Magneto‑optical absorption coefficient

To investigate the CPR in the MPTQW, we need to have an analytic expression of the MOAC, which can be written as [38, 39]

where V₀ is the system volume, I₀ is the optical intensity, ℏ𝛺 refers to the incident photon energy, f_𝜆=f_N,n ( f_𝜆�� =f_N��,n^�� ) denotes the Fermi distribution function in the initial (final)- state. In Eq. (8), _𝜆^±��,𝜆 is the transition matrix element per unit area for the interaction of three particles (electron, photon, and phonon) in the two-dimensional system. Its expression is given as follows by Born’s second-order golden rule [40], including the 𝓁-photon absorption process [41, 42]:

(5) 𝜉_n(z) =A^q_n(1−𝜂²)^(q−n)∕2C_n^q+1∕2−n(𝜂),

(6) A^q_n =

�𝛼n!(q−n−1∕2)!(2q−2n)!

√𝜋(q−n−1)!(2q−n)!

�1∕2

.

(7) 𝜀_n = − D

q(q+1)(n−q)².

K(𝛺) = 1 (8) V₀(I₀∕ℏ𝛺)

∑

𝜆,𝜆^��

_𝜆^±��,𝜆f_𝜆(1−f_𝜆��),

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_𝜆^±��,𝜆=2𝜋 ℏ

∑

𝜆^�,𝐪

∑∞ 𝓁=1

(

||^±_𝜆��,𝜆^�||²||

|^rad_𝜆�,𝜆

||

|

2∕ℏ²𝛺² )

× (𝛼₀q)²^𝓁

(𝓁!)²2²^𝓁 𝛿(E_𝜆��−E_𝜆−𝓁ℏ𝛺±ℏ𝜔₀),

where 𝜆^′ is an intermediate/virtual state. The physical mean- ing of the transition matrix element in Eq. (9) is that the transition from 𝜆 to 𝜆^′′ occurs first by absorption 𝓁-photons of energy ℏ𝛺 to transfer from state 𝜆 to state 𝜆^′ , and then the subsequent emission or absorption of a LO-phonon of energy ℏ𝜔₀ to transfer to state 𝜆^′′ (see Fig. 2). The quantity 𝐪= (q_⟂,q_z) is the phonon wave vector of energy ℏ𝜔₀ , and 𝛼₀ is the dressing parameter.

The matrix element of the Hamiltonian for electron–phonon interaction, ^±_𝜆��,𝜆^� , is determined by [10]

where

with u=𝛼²_cq²

⟂∕2 , L^N_N^��_�^−N^�(u) is the associated Laguerre polynomials, 𝜒^∗= (1∕𝜒_∞−1∕𝜒₀) with 𝜒_∞ and 𝜒₀ being the high and static frequency dielectric constants of the materials, and 𝜖₀ is the permittivity of the vacuum.

The matrix element of the Hamiltonian for electron–photon interaction is [38]

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��

�^±_𝜆��,𝜆^�

��

�

2=�⟨𝜆^��ep�𝜆^�⟩�²=�V(q)�²�n^��,n^�(±q_z)�²

×��J_N��,N^�(q_⟂)��²N^±

0𝛿_k^��

y,k^�_y∓q_y,

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|V(q)|²=4𝜋e²𝜒^∗ℏ𝜔₀

𝜖₀V₀q² ≈ 4𝜋e²𝜒^∗ℏ𝜔₀ 𝜖₀V₀q²

⟂

,

n^��,n^�(±q_z) =� (12)

∞ 0

e^±iq^z^z𝜉^∗_n��(z)𝜉_n�(z)dz,

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||J_N��,N^�(q_⟂)||²= N^�!

N^��!e^−uu^N^��^−N^�L^N_N^��_�^−N^�(u),

� (14)

��^rad_𝜆�,𝜆

��

�

2 =�⟨𝜆^��^rad�𝜆⟩�²= 𝛺²A²₀

4 ��𝐧⋅e_𝜆�,𝜆��²,

(a) (b)

time

space

λ(k) 2γ

λ(k)

λ

(k’) q 2γ λ

(k’) λ

(k”)

q

λ

(k”)

Fig. 2 Feynman diagrams for the phonon-assisted transition due to the two-photon absorption process ( 𝓁=2 ), a phonon emission and b phonon absorption

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where 𝐧 is the polarized vector of the electromagnetic field, and e𝜆^�,𝜆=e⟨𝜆^��𝐫�𝜆⟩ is the dipole moment with 𝐫 being the position vector. Using the eigenfunctions in Eq. (2), and assuming the electromagnetic field to be linearly polarized transverse to the magnetic field, the matrix element of the position operator 𝜆^′,𝜆 is calculated as follows:

where

The summations over 𝜆 and 𝜆^′′ in Eq. (8), and over 𝜆^′ in Eq. (9) run over all quantum numbers, which are done by the transformation ∑

𝜆→∑

N,n

∑

k_y , for the 𝜆-state, for example.

For the summation over k_y , we use periodic boundary conditions [43]:

where S=L_xL_y stands for the surface area of the system. The summation over 𝐪 is transformed into the integral

To calculate the transition matrix element in Eq. (9), we need the overlap integral due to electron–phonon interaction, which is determined as follows:

where n^��,n^�(±q_z) is given in Eq. (12).

Inserting above expressions into Eq. (9), the expression of the transition matrix element will yield the following form including the two-photon absorption process ( 𝓁=1, 2)

where we have used

and denoted

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_𝜆�,𝜆=⟨𝜆^��x�𝜆⟩=⟨n^��n⟩⟨k_y^��k_y⟩⟨N^��x�N⟩

=𝛿_n�,n𝛿_k� y,k_yN^�,N,

(16)

_N�,N=x₀𝛿_N�,N+ (𝛼_c∕√ 2)

×�√

N𝛿_N�,N−1+√

N+1𝛿_N�,N+1

� .

∑ (17)

k_y

→ L_y 2𝜋 ∫

L_x∕2𝛼²_c

−L_x∕2𝛼²_c

dk_y= S 2𝜋𝛼_c²,

∑ (18)

𝐪

→ V₀ (2𝜋)² ∫

∞ 0

q_⟂dq_⟂∫

+∞

−∞

dq_z.

(19) I_n��,n^�=�

+∞

−∞

||n^��,n^�(±q_z)||²dq_z,

(20)

_𝜆^±��,𝜆= e⁴A²₀𝜒^∗ℏ𝜔₀𝛼₀² 8ℏ³𝜖₀𝛼_c²

∑

N^�

|N^�,N|²I_n��,n

× (1+2)𝛿_k_y_±q_y_,k��

y,

∑ (21)

n^�

I_n��,n^�𝛿_n�,n=I_n��,n,

where 𝛥E=E_N��,n^��−E_N,n is used to denote the transition energy difference (or energy separation), and N^±

0 =N₀+1∕2±1∕2 with N₀≡N₀(T) = [e^ℏ𝜔⁰^∕(k^B^T)−1]⁻¹ being LO-phonon distribution function.

Inserting Eqs. (20) into (8) and making the necessary cal- culations we obtain the following expression for the MOAC:

where we have denoted

with c being the speed of light, n₀ being the refractive index.

Although we cannot have an analytical result for the overlap integral in Eq. (19) and also in Eq. (21) in the general case, it could be calculated exactly in the quantum limit case, where the transition of electrons only occurs between the lowest levels. In this case, we can use the harmonic limit suggested by Dong and Lemus [30]. Note that this harmonic limit requires 𝛼→0 and D→∞ but keeping the quantity k_H=2𝛼²D still finite. However, this insistence is not always satisfied in a real system. Instead, the harmonic- limited condition will be satisfied if one can choose a big enough value of D and/or a small enough value of 𝛼 for that k= (1∕4+2m^∗D∕(ℏ𝛼)²)^1∕2≈ (2m^∗D∕(ℏ𝛼)²)^1∕2 . This approximation will work well when 2m^∗D∕(�𝛼)²≫1∕4 . Since 2m^∗D∕(ℏ𝛼)²=q(q+1) [30], the harmonic-limited condition now requires that q(q+1)≫1∕4 . In our numerical calculation, the values of the q-parameter are q=2, 3, 4 . We can see that these values of q-parameter guarantees that the harmonic-limited condition is satisfied. According to this harmonic limit, the MPT potential reduces to a harmonic oscillator potential with the frequency of 𝜔_z=2D∕(ℏ(q(q+1)^1∕2)) . Using this approximation, we obtain the following analytical result for the overlap integral for the transition between n=0 and n^��=1 states

1=N⁻ (22)

0𝛿(X⁻₁) +N⁺

0𝛿(X⁺₁),

₂= 𝛼² (23)

0

8𝛼_c²(N^�+N^��+1)[

N₀⁻𝛿(X⁻₂) +N₀⁺𝛿(X⁺₂),

(24) X^±_𝓁 =𝛥E±ℏ𝜔₀−𝓁ℏ𝛺, 𝓁=1, 2,

(25) K(𝛺) =(𝛺)∑

N,n

∑

N^��,n^��

∑

N^�,n^�

|N^�,N|²I_n��,n

×f_N,n(1−f_N��,n^��)(1+2),

(𝛺) = S²𝜒^∗e⁴𝛼²₀ℏ𝜔₀ (26) 32𝜋²cn₀𝜖²

0V₀𝛼_c⁶ℏ²𝛺,

(27) I_1,0= 1

ℏ

�

𝜋m^∗D

√q(q+1)

�1∕2

.

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Therefore, the expression of the MOAC in Eq. (25) will finally yield the following form, which includes the two- photon absorption process:

For the divergent problem-solving caused by the delta functions in Eq. (28) we will replace them by Lorentzians of width 𝛤_N^±_��

,N [44]

where |^±_𝜆��,𝜆|² is matrix element for electron–phonon interaction shown in Eq. (10) but the state 𝜆 is used instead of 𝜆^′ . The Lorentz widths presented here depend on the LL indexs.

3 Results and discussion

In this section, the numerical results for a GaAs MPTQW are presented, for which we take the following characteristics: [36, 45, 46] 𝜒_∞=10.89 , 𝜒₀=13.18 , n₀=3.2 , ℏ𝜔₀=36.25 meV, 𝛼₀=7.5 nm, and the electron density n_e=3×10¹⁶ cm⁻³ corresponding to the Fermi energy of E_F= −0.216 eV. In the following, we will only consider the transition between the two lowest states, i.e., N=0 , N^��=1 . First, we will perform the variation of the factor f_0,0(1−f_1,1) and the energy separation 𝛥E with the well-shape controlling parameters (D and q) as well as with the magnetic field and temperature. The major portion is devoted to the magneto- optical effect, in which we will present the influence of these parameters on the MOAC and FWHM.

Since the factor f_0,0(1−f_1,1) plays a vital role in the MOAC’s intensity, in Fig. 3, we show the dependence of the factor f_0,0(1−f_1,1) on the well-depth. The solid (black) curve corresponds to B=10 T, q=3 , and T=77 K. In each of the others, only one parameter is changed in comparison to that of the solid curve, while the other parameters are kept fixed. For instance, in the dashed (blue) curve, only the B-parameter is changed to 12 T compared with B=10 T in the solid one, while the factors q and T are unchanged. Similarly, in the dashed–dotted (green) and the dotted (red) curves, only the q-parameter and the T-parameter are changed, respectively. It is seen that the product f_0,0(1−f_1,1) has only a significant value in a specific range of D-parameter, labeled from D₁ to D₂ , for example.

Beyond this range, this product reduces quickly to zero.

These two specific values of D-parameter are, respectively, the solutions of the equations E_0,0=E_F and E_1,1=E_F . Our

(28) K(𝛺) =(𝛺)∑

N,N^��

∑

N^�

|_N�,N|²I_1,0f_N,0(1−f_N��,1)

× (₁+₂).

(29) (

𝛤_N^±_��

,N

)2

=∑

𝐪

|^±_𝜆��,𝜆|²,

result shows that it is better to observe the CPR in the range from D₁=0.29 eV to D₂=0.73 eV for the case described by the solid curve, for example. When the magnetic field (q-parameter) increases, this range shifts to the right (left) side corresponding to the increase (decrease) of the E_0,0 and E_1,1 with the magnetic field (q-parameter).

At higher temperature (the dotted curve), due to the thermal redistribution of the Fermi distribution functions, this range is extended leading to the reduction of the product’s value in the (D₁,D₂) interval.

Besides the factor f_0,0(1−f_1,1) , the energy separation 𝛥E is also one of the main characteristics governing the magneto-optical effect, especially the position of the MOAC’s peaks. Figure 4 depicts the dependence of 𝛥E on the well- depth for two distinguishing values of B and q. The energy separation is shown to increase linearly with rising well- depth and magnetic field, but reduces with enlarging well- width. These variation behaviors of the energy separation agree with those reported in the previous works [32, 33], except for the variation with the magnetic field, which was

Fig. 3 Dependence of the factor f_0,0(1−f_1,1) on the well-depth

Fig. 4 Dependence of the energy separation, 𝛥E , on the well-depth for two different values of B and q

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not considered in these works. In more detail, from Eqs. (3) and (7), we have

It is clear from Eq. (30) that 𝛥E is linearly proportional to the cyclotron energy, ℏ𝜔_c∼B , the well-depth, D, and q⁻¹ , as illustrated intuitively in Fig. 4.

To understand the influence of the well-depth on the magneto-phonon effect, in Fig. 5, we depict the variation of the MOAC with the photon energy for different values of the well-depth, D. The results are obtained at B=10 T, q=3 , and T =77 K. It is clear that there exist four resonant peaks in each curve. For the purpose of convenience, we label these four peaks from “(1)” to “(4)” for the solid curve, corresponding to D=0.30 eV, for example. All these peaks are the result of the cyclotron–phonon resonance (CPR) tran- sitions, which satisfying the CPR condition:

which is the direct consequence of the energy conservation law and also describes the peak positions. These peaks are generated by the resonant transition of electrons between the ground and first excited states thanks to the absorption of one (𝓁=1) or two (𝓁=2) photons and simultaneous absorption [minus sign in Eq. (31)] or emission (plus sign) of one LO-phonon of energy ℏ𝜔₀ . We can see from Fig. 5 that the phonon emission peaks [peaks “(2)” and “(4)”] are much higher in comparison to those of phonon absorption [peaks “(1)” and “(3)”]. This reveals that in the MPTQW, the phonon emission process is much more dominant than phonon absorption, similar as in other types of quantum well-shapes [19, 21, 47]. In both cases of phonon processes, the peak values caused by the two-photon process are about (30) 𝛥E=E_1,1−E_0,0=ℏ𝜔_c+ D(2q−1)

q(q+1) .

(31) 𝓁ℏ𝛺=𝛥E±ℏ𝜔₀, 𝓁=1, 2,

45% of those caused by the one-photon one. This ratio is equivalent to that reported in other types of quantum well- shapes. This means that the contribution of the two-photon process is: (i) significant and should not be ignored in studying the CPR effect and (ii) independent of the type of quantum well-shape.

Besides affecting the peak positions, the well-depth affects the intensity of the CPR peaks also. With D increasing, the peak positions make a shift toward the higher energy regime (blue-shift), and simultaneously, the peak intensities are also enhanced. The blue-shift observed here agrees with results of the previous works reported for the refractive index changes [32, 33] and can be explained directly by the linear increase of 𝛥E with the well-depth (see Fig. 4) and is shown intuitively in the left vertical axis of the inset of Fig. 5, where the peak position (caused by the one-photon process, for example) shifts linearly to the right-hand side with the increase of the well-depth.

Meanwhile, the effect of the D-parameter on the peak intensities is expressed through the overlap integral, I_1,0 , and the factor f_0,0(1−f_1,1) . This is described clearly in the variation of the factor G=(𝛺)|N^�,N|²I_1,0f_0,0(1−f_1,1) with the well-depth as shown in the inset (right vertical axis). We can see from the inset of Fig. 5 that when the well-depth increases the G-factor first increases, reaches its maximum value at D=0.64 eV, and then decreases with the further increasing D-parameter. In the specific range of D-parameter from 0.30 to 0.34 eV as shown in the main part of Fig. 5, the G-factor increases nonlinearly, leading to the enhancement of the peak intensities with the increase of well-depth. This result is opposite to that reported in the previous works [32, 33], where the refractive index changes are found to decrease with the increase of the D-parameter.

In addition to the effect on the MOAC, the well-depth also has an influence on the FWHM. Figure 6 shows the well-depth dependent FWHM at certain values of B, T, and q. Due to its proportionality to the Lorentz width, which is shown to increase with the square root of D [see Eqs. (27) and (29)], the FWHM increases with the well- depth in all four cases. This indicates that the deeper the well-depth is, the stronger electron confinement is, and as a result the strength of the electron–phonon interaction, and therefore, the FWHM becomes greater. Besides, since at the observed temperature ( T =77 K), the phonon population, N₀=4.3×10⁻³ , is much less than 1, the FWHM caused by phonon emission (which is proportional to the N₀⁺ ) is much bigger than that caused by phonon absorption (proportional to N⁻

0 ). Therefore, the phonon emission process always occurs more strongly than phonon absorption.

We now study how the FWHM depends on the controlling parameters (D, q, B, and T) through the LO-phonon- assisted broadening mechanism which is expressed in the

Fig. 5 Dependence of MOAC on photon energy for different well depths at B=10 T, q=3 , and T=77 K. The inset shows the D-dependence of the peak position of peak “(4)” (left vertical axis) and the factor G=(𝛺)|N^�,N|²I_1,0f_0,0(1−f_1,1) (right vertical axis)

(7)

Lorentz width. The well-depth dependence of the FWHM can be found as

where the FWHM and D are measured in units of meV and eV, respectively. The values of 𝛽_D listed in the first line of Table 1 are obtained from a fit to the measured FWHM data using Eq. (32), and shown clearly by the solid lines in Fig. 6.

In our understanding, they are new, and therefore, there are not any experimental data available to compare with. We hope that our studies would be a valid suggestion for future experimental investigations.

We now pay our attention to the influence of the q-parameter on the magneto-optical effect. Figure 7 shows the MOAC as functions of photon energy, ℏ𝛺 , for different values of q-parameter. Because of the complex dependence of the energy separation on the q-parameter [see Eq. (30)], the values of peak positions first increase with the q-parameter, reach maxima at q_max=1.37 , and then, the peak positions start to decrease with the further increasing q-parameter, as shown in the inset. This descending feature of the peaks positions when q>q_max leads to the red-shift behavior of the MOAC’s peaks, as shown in Fig. 7. The red-shift behavior observed here is in agreement with that reported in the

(32) FWHM(D) =𝛽_DD^1∕4,

previous works for the refractive index changes [32, 33]. We recall that the parameter q has a close relationship with the well-width [32]. For a fixed value of the well-depth D, the bigger the q-parameter is, the larger the quantum well-width will be, resulting in the decrease of quantum confinement.

This results from the downsizing of energy separation, 𝛥E , with the increase of the q-parameter, which explains the red- shift behavior of the peak positions when the parameter q increases, as illustrated in Fig. 7. Meanwhile, the effect of the q-parameter on the peak intensity is more complicated.

Figure 7 also depicts that when the q-parameter increases first the peak intensities are enhanced, reach the maximum value, and then begin to reduce with the further increasing q-parameter. This behavior of the peak intensities is illustrated clearly in the inset (right axis), in which the factor G is observed to achieve its maximum value at q=2.58 . The result obtained here is different from that reported in the previous works [32, 33], in which the peak intensities of the refractive index changes are the monotonically increasing functions of the q-parameter.

Figure 8 shows the variation of the FWHM with the q-parameter at accurate values of B, T, and D. In general, the FWHM is observed to decrease nonlinearly with the increase of the q-parameter for both phonon processes (emission and absorption) as well as for both types of photon absorption (one and two photons). As mentioned above, when the q-parameter increases, the well-width becomes wider, leading to the reduction of quantum confinement and, therefore, bringing down the electron–phonon interaction, and so does the FWHM. This result agrees qualitatively with that in the previous works [19, 21, 47], in which the FWHM is reported to decrease with increasing well-width.

To have a more accurate result, we propose a fitted expression of the FWHM as follows:

Fig. 6 Variation of the FWHM with the well-depth. The full and empty symbols stand for the one- and two-photon absorption processes, respectively

Table 1 Fitted results of parameters 𝛽_D , 𝛽_q , and 𝛽_B

Parameters Line Line Line Line

𝛽_D (meV/(eV)^1∕4) 20.13 5.03 1.31 0.33

𝛽_q (meV) 20.99 5.24 1.37 0.34

𝛽_B (meV T ^−1∕2) 4.87 1.22 0.32 0.08

Fig. 7 Dependence of MOAC on photon energy for different q-parameters at B=10 T, D=0.34 eV, and T=77 K. The inset shows the q-dependence of the peaks position (left axis) and the factor G (right axis)

(8)

where 𝛽_q is a fitting parameter, which is measured in meV and given in the second line of Table 1. The solid lines in Fig. 8 illustrate the fitted results using Eq. (33). The good agreement between the measured FWHM data (full and empty symbols) and the fitted lines confirm that Eq. (33) is a good prediction for the change of the FWHM with q-parameter. Similar to the D-dependent FWHM case [see Eq. (32)], the result in Eq. (33) is also a new prediction and its rightness needs to be tested by an experimental measure- ment in the future.

The external magnetic field is demonstrated to have a significant effect on the optical absorption properties of the low-dimensional quantum systems through the modification of energy levels. Therefore, it is necessary to study the influence of the magnetic field on the MOAC and the FWHM. In Fig. 9, the MOAC is shown as function of ℏ𝛺 for different magnetic fields B. The rise of the energy separation with B [see Eq. (30) and Fig. 4] is the cause of the blue-shift behavior of the peak positions. This blue-shift behavior is also illustrated more clearly in the left vertical axes of the inset, which shows a linear increase with the magnetic field of the peak positions. Besides, when the magnetic field increases the peak intensities heighten quickly with a nonlinear law. Mathematically, this feature can be interpreted as follows: Eq. (28) reveals that the magnetic field mainly affects the intensity of the MOAC’s peaks through the product (𝛺)|N^�,N|² which is proportional to 𝛼⁴_c or ∼B² . This relationship explains directly the increase of the factor G with the magnetic field as illustrated in the inset and so do the peak intensities. The results obtained here are in won- derful agreement with the previous works [19, 21, 47, 48].

In Fig. 10, we present the magnetic field-dependent FWHM at certain values of D, T, and q. The FWHM is found (33) FWHM(q) =𝛽_q[q(q+1)]^−1∕8,

to be an increasing nonlinear function of the magnetic field in all cases caused by the increase of the Lorentz width. This is in great accordance to what was obtained in other types of quantum well [19, 21, 47] and in graphene [22, 49, 50].

Similar to the above cases, from the sign of the √

B-dependent Lorentz width, the magnetic field-dependent FWHM (meV) is proposed as follows:

where B is measured in Tesla. The fitted values of the parameter 𝛽_B are given in the third line of Table 1 for all four cases. Solid lines in Fig. 10 illustrate the fitted results from Eq. (34). The FWHMs obtained here are a little bit smaller than those in other types of quantum well [21, 47]. This means that the electron–phonon scattering in MPTQW is weaker than in the other types.

(34) FWHM(B) =𝛽_BB^1∕2,

Fig. 8 Variation of the FWHM with the q-parameter. The full and empty symbols stand for the one- and two-photon absorption processes, respectively

Fig. 9 Dependence of MOAC on photon energy for different magnetic fields at q=3 , D=0.34 eV, and T=77 K. The inset shows the q-dependence of the peaks position (left axis) and the factor G (right axis)

Fig. 10 Variation of the FWHM with the magnetic field. The full and empty symbols stand for the one- and two-photon absorption processes, respectively

(9)

Since the temperature affects the MOAC’s peak through the factor f_0,0(1−f_1,1) and the phonon distribution function N₀ [see Eq. (28)], with the increase of temperature only the peak intensities are increased, while the peak positions remain unchanged. These features of the temperature-dependent MOAC’s peak are shown clearly in Fig. 11. The temperature- independent peak positions are the result of the independ- ence of the temperature of the CPR condition [see Eq. (31)].

The influence of the temperature on the peak intensities for the different phonon processes is different. While the peak intensities due to phonon absorption are reduced by the temperature, those of the phonon emission are strongly enhanced.

These results are different from those presented in the previous works [19, 21, 47], in which the peak intensities are observed to rise with the increase of T in both cases of phonon processes. This also implies that the CPR effect in the MPTQW is more complicated in comparison with that in the mentioned quantum wells.

The variation of the FWHM with the temperature is shown in Fig. 12, in which the FWHM is presented for all four different cases: both phonon emission and absorption cases as well as both one and two-photon processes. The FWHM is found to depend on the temperature with distinctive rules for the two cases of phonon activities. While the FWHM caused by the phonon emission increases with the temperature by the rule:

the corresponding one for the phonon absorption process is FWHM(T) =𝛾_T[N₀(T)]^1∕2 . The rule expressed by Eq. (35) is guided by the thermal broadening mechanism [51, 52], in which the stable, 𝛼_T , and the LO-phonon broadening terms, 𝛽_T , are listed in the Table 2 for both one and two-photon processes. The values of 𝛼_T and 𝛽_T obtained here are of the same order as those reported in the other models of quantum (35) FWHM(T) =𝛼_T+𝛽_TN₀(T),

wells [20, 21, 47], which are in good agreement with the experimental data measured in the multiple quantum well structure [51, 52]. The fitted values for the phonon absorption are found as 𝛾_T=15.33 meV and 3.83 meV corresponding, respectively, to one- and two-photon absorption. These values are slightly smaller than those obtained in other kinds of quantum wells [21, 47]. Similar to the mentioned works, the results obtained here are also new, and need to be veri- fied by experiments.

4 Conclusions

We have presented detailed analyses of the influence of the parameters describing the quantum well-shape of an MPTQW (D and q), the magnetic field (B) and the temperature (T) on the CPR effect. The results are discussed by investigating the role of these parameters in the change of MOAC and FWHM. Since making a significant change in the factor f_0,0(1−f_1,1) and the energy separation, all these four parameters affect sensitively not only the intensity and position of resonant peaks but also the peak profiles (FWHM) except for that the temperature does not affect peak positions. The MOAC peaks shift to higher (lower) energies when the well-depth and magnetic field (well-width) increase, while they remain unchanged with increasing

Fig. 11 Dependence of MOAC on photon energy for different tem- peratures at q=3 , D=0.34 eV, and B=10 T. The inset shows the q-dependence of the peak position (left axis) and the factor G (right axis)

Fig. 12 Variation of the FWHM with the temperature. The full and empty symbols stand for the one- and two-photon absorption processes, respectively

Table 2 Fitted results of

parameters 𝛼_T and 𝛽_T Parameters Line Line

𝛼_T (meV) 15.37 3.84 𝛽_T (meV) 7.21 1.77

(10)

temperature. The peak intensities are enhanced by the rise of well-depth and magnetic field, reduced with temperature but vary non-monotonically with the well-width.

The FWHM is observed to be a nonlinear function of all four parameters: it rises with the well-depth, magnetic field and temperature but reduces with the well-width. In all cases, the FWHM due to the phonon emission is always bigger than that caused by absorption. Besides, the FWHM caused by the two-photon absorption is always less than that caused by the one-photon one. The results obtained here for the rise of FWHM with magnetic field and temperature are in good agreement with previous theoretical and experimental works reported in other quantum well structures. Espe- cially, the expressions for the D and q-dependent FWHM described in Eqs. (32) and (33) are presented for the first time. To our understanding, they are new and their validity needs to be tested further by experiments.

Acknowledgements This work is funded by Ministry of Education and Training (Vietnam) under the project coded B2017-DHH-32.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflicts of interest.

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