- Email: [email protected]
intersubband transition in quantum wells
Superlattices and Microstructures 131 (2019) 86–94
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Comparison of magneto-resonance absorption FWHM for the intrasubband/intersubband transition in quantum wells Nguyen Dinh Hien a, b a b
Laboratory of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
A R T I C L E I N F O
A B S T R A C T
Keywords: Confined phonon Intrasubband transition Intersubband transition FWHM Quantum wells
We compare magneto-resonance absorption FWHM (full-width at half-maximum) for the intra subband transition (1 → 1) to that for the intersubband transition (1 → 2) in a square GaAs quantum well by using the projection operator method due to confined LO-phonon delineated by the model of Fuchs-Kliewer, Ridley, and Huang-Zhu. The numerical results show that the FWHM of the optically detected magnetophonon resonance (ODMPR) peak for the intrasubband/inter subband transition decreases with the rise of the width of well and increases with the growing temperature for all models of phonon. This is in accordance with the result has been investigated in the theory and experimental [1,2]. Furthermore, the FWHM of the above ODMPR peaks for the bulk phonon has a smaller and changes slower than it does for the confined phonon. The FWHM for the Huang-Zhu model is largest among three models of confined phonon (Fuchs-Kliewer, Ridley, and Huang-Zhu model), while it for Ridley model is smallest. This result is in accordance with the results have been investigated in the theory [3,4]. In addition, the FWHM of the ODMPR peak for the intrasubband transition has larger value than that for intersubband transition for both the bulk and confined phonon, this is in accordance with the result has been investigated in the theory and experimental [5].
1. Introduction Confined phonon effect, it is an necessary part of the depiction of electrophonon interactions and causes the rise of electrophonon scattering rates, modifies the density of states of phonon in low dimensional semiconductor structure [6–8]. There have been several theory models of the confined phonon in quantum wells such as Fuchs-Kliewer (FK) [9,10], Ridley (R) [11], and Huang-Zhu (HZ) model [12,13]. However, the Huang-Zhu model has received wide acceptance and best depicts and been in accordance with exper imental result [14,15]. MPR (magnetophonon-resonance) is the resonance scattering of electrons caused by phonon absorption or emission when the distance between two Landau levels is equal to the energy of the longitudinal optical phonon [16–18]. MPR has been of great interest to scientists because it is a powerful spectral tool to investigate properties such as effective mass measurement, restoration mechanism of carriers … Magnetophonon-resonance phenomena can be observed directly through the ODMPR (optically detected MPR) [18]. The FWHM is well known as a instrument for study the transport behavior, scattering mechanisms of carriers in the materials [19]. Hence, it may be used to probe confined electron-confined LO-phonon scattering process. The FWHM of the ODMPR peak for the intrasubband/intersubband transitions with the emission bulk phonon processes was studied, but with the emission confined phonon processes is still open. The FWHM has been measured in different kinds of semiconductors: Quantum dots E-mail address: [email protected]. https://doi.org/10.1016/j.spmi.2019.05.035 Received 18 February 2019; Accepted 22 May 2019 Available online 29 May 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
[20,21], quantum wells [22,23], and quantum wires [24,25] under the effect of the bulk phonon. Under the effect of the confined phonon, several works have been examined [3,15,26–29]. Furthermore, the FWHM of the ODMPR peak for the intra subband/intersubband transitions with the emission bulk phonon processes has examined in the theory and experimental [5]. Thus, when confined LO-phonon is considered, this is still open. We study the FWHM of the ODMPR peak for the intrasubband/intersubband transitions with the emission phonon processes for the above models of phonon in a quantum well. The dependence of the above FWHM on the width of well and the temperature of system by using the operator projection method is presented to compare. We introduce the theory models of phonon confinement in quantum wells in Sec. 2. In Sec. 3, we calculate the absorption power in quantum wells under the effect of the phonon confinement. Sec. 4 presents the numerical results in a square GaAs/AlAs quantum well and discussion. Finally, in Sec. 5 we present the conclusions. 1.1. Theory models of phonon confinement in quantum wells In this paper, we consider the GaAs/AlAs quantum well with electron is free in the x y plane; B ¼ ð0; 0; BÞ is a magnetic field, and it is applied in the z direction. For the potential VðzÞ, VðzÞ ¼ 0 for jzj < Lz =2 and VðzÞ ¼ ∞ for jzj > Lz =2, the one electron energy eigenvalues εðNnÞ and eigenfunction ϒðNnky Þ are given by Ref. [27]. � 1 ϒ Nnky ¼ pffiffiffiffi ψ N ðx Ly
� x0 Þexp iky y Φn ðzÞ
(1) (2)
εðNnÞ ¼ ðN þ 1=2Þℏωc þ n2 ε0
x0 Þ being harmonic oscillator wave function,
where n ¼ 1; 2; 3; ⋯ and N ¼ 0; 1; 2; ⋯ are the electric subband and Landau index; ψ N ðx
centered at x0 ¼
d2B ky with dB ¼ ðℏc=eBÞ1=2 is cyclotron radius; Ly , Φn ðzÞ, and ky respectively being the normalization length, electron
wave function, and wave vector of the electron in z-direction; ωc ¼ eB=m� and ε0 ¼ ℏ2 π2 =ð2m� L2z Þ being the cyclotron frequency and energy of the lowest electric subband, respectively; Lz is the width of well; m� being the effective mass of electron. Φn ðzÞ is given by Ref. [27]. sffiffiffiffi � � 2 nπz nπ sin Φn ðzÞ ¼ þ (3) Lz Lz 2 The matrix element for confined electron - LO-phonon confinement interaction with κð ¼ FK; R; HZÞ models of the confined phonon can be written as [3,27]. �� � � ��2 � �2 �2 � �2 � 2 �� � � � � � �� �� � i�Hel ph �f � ¼ e ℏωLO χ �V κ ðq? Þ� �JNN’ ðϑÞ� �Λκ;τϕ � δ f i (4) ni ;nf � k? ;k? �q? τϕ � � � � � � � � � 2ε0 V0 where � �2 � � �JNN’ ðϑÞ� ¼ n2 !e ϑ ϑn1 � � n1 ! ϑ ¼ d2B q2? =2 ; Ln1 n2 Z Λκ;niτ;nϕf ¼
Lz =2 Lz =2
n2
n2
� n1 Ln2
2
n2
� ðϑÞ
(5)
ðϑÞ is the Laguerre polynomial; n1 ¼ maxfN; N g, n2 ¼ minfN; N g. 0
0
(6)
Φ�nf ðzÞuκτϕ ðzÞΦni ðzÞdz
is the overlap integral, it has been shown in Ref. [3] with intrasubband and intersubband.Where χ � ¼ ð1=χ ∞ 1=χ 0 Þ, χ 0 and χ ∞ respectively are the static and high-frequency dielectric constants; ε0 is the vacuum dielectric constant; q? ¼ ðqx ; qy Þ is the two-dimensional LO-phonon wave vector; V0 ¼ SLz and ℏωLO ¼ 36:25 meV is the energy of the LO-phonon in the GaAs/AlAs quantum well. In the confined space, uκτϕ ðzÞ is the parallel composition of the displacement vector of τ th phonon mode, and different for the FK model [10], R model [11], and HZ model [13]; ϕ are ( ) even and the ( þ ) odd confined phonon modes. For the FK model, uκτϕ ðzÞ is written as follows: � � τπ z ; τ ¼ 1; 3; 5; ⋯ uFK ðzÞ ¼ cos (7) τþ Lz � � τπz ; τ ¼ 2; 4; 6; ⋯ uFK τ ðzÞ ¼ sin Lz
(8)
For the R model, uκτϕ ðzÞ is written as follows: � � τπz uRτþ ðzÞ ¼ sin ; τ ¼ 1; 3; 5; ⋯ Lz
(9)
87
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
� � τπz uRτ ðzÞ ¼ cos ; τ ¼ 2; 4; 6; ⋯ Lz
(10)
For the HZ model, uκτϕ ðzÞ is written as follows: � � μτ π z cτ z uHZ þ ; τ ¼ 3; 5; 7; ⋯; τþ ðzÞ ¼ sin Lz Lz � � μτ π z uHZ ðzÞ ¼ cos τ Lz
(11) (12)
ð 1Þτ=2 ; τ ¼ 2; 4; 6; ⋯
with μτ and cτ are given by �μ π � μ π tan τ ¼ τ ; τ 1 < μτ < τ 2 2 cτ ¼
2sin
(13)
�μ π � τ 2
(14)
The term V κτϕ ðq? Þ in Eq. (4) is given by Refs. [27,28]. � � V κτϕ ðq? Þ ¼ aκτϕ q2? þ bκτϕ L2z
(15)
1=2
HZ with aHZ τϕ and bτϕ are found for the HZ model as follows � � 1 2 1 2 2 aHZ c2τ ; τ ¼ 3; 5; 7; ⋯ ; bHZ τþ ¼ 1 þ cτ τ þ ¼ μτ π 6 μ2τ π2
(16) (17)
HZ 2 2 aHZ τ ¼ 3; bτ ¼ τ π ; τ ¼ 2; 4; 6; ⋯
these factors are (18)
FK;R 2 2 aFK;R τ� ¼ 1; bτ� ¼ τ π ; τ ¼ 1; 2; 3; ⋯
for the FK and R model. τϕ The term Λκ; ni ;nf in Eq. (6) can be evaluated for intrasubband transition and intersubband transition with above models of confined phonon. It is obtained for the FK model as follows: τþ ΛFK; ¼ 0; τ ¼ 1; 3; 5; ⋯ 12 τ=2 � 2 1 τ ΛFK; ¼ τð 1Þ 12 2
π
τþ ΛFK; ¼ 11 τ ΛFK; 11
2ð
τ
�
ðτþ1Þ=2
1Þ
1
τπ
� ; τ ¼ 2; 4; 6; ⋯ τ 1 � τπ ; τ ¼ 1; 3; 5; ⋯ 2 2 2 1
2
9 þ
4π
(19)
πτ
¼ 0; τ ¼ 2; 4; 6; ⋯
For the R model as follows: τþ ΛR; 12 ¼
1 ðδm;1 þ δτ;3 Þ; τ ¼ 1; 3; 5; ⋯ 2
τ ΛR; 12 ¼ 0; τ ¼ 2; 4; 6; ⋯
(20)
τþ ΛR; 11 ¼ 0; τ ¼ 1; 3; 5; ⋯
1 τ ΛR; 11 ¼ δτ;2 ; τ ¼ 2; 4; 6; ⋯ 2 For the HZ model as follows: � � 2cτ 8 1 τþ þ ΛHZ; ¼ 12 π2 9 μ2τ 1 τ ΛHZ; 12
¼ 0; τ ¼ 2; 4; 6; ⋯
τþ ΛHZ; 11
¼ 0; τ ¼ 3; 5; 7; ⋯
3 τ ΛHZ; ¼ δτ;2 11 2
1
μ2τ
�� ; τ ¼ 3; 5; 7; ⋯ 9 (21)
τ=2
ð
1Þ ð1
δτ;2 Þ; τ ¼ 2; 4; 6; ⋯
88
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
From above results, it is shown that only even modes in HZ model, odd modes in FK model, and the τ ¼ 2 mode in R model contribute to intrasubband transition. But, only odd modes in HZ model, even modes in FK model, and the τ ¼ 1; 3 modes in R model make contribution for intersubband transition. 1.2. Absorption power under the effect of the phonon confinement in quantum wells The absorption power per unit volume of quantum wells is given by N.L. Kang et al. [30] as follows: � �2 � fαþ1 ÞΘðωÞ E2 X �jþ α ðfα pðωÞ ¼ 0 2ℏω α ðω ωc Þ2 þ ½ΘðωÞ�2
(22)
where ω and E0 respectively are frequency and amplitude of the electromagnetic field; fα and fαþ1 are the Fermi-Dirac distribution � � �2 � � � � � �2 �� � ¼ �hα þ 1��jþ ��αi�� ¼ 2e2 ℏωc ðN þ 1Þ=m� . functions of electron at state �αi ¼ �N; n; ky i and �α þ 1i ¼ �N þ 1; n; ky i; �jþ α The term ΘðωÞ in Eq. (22) for bulk LO-phonon is obtained as follows: ΘðωÞ � Θb ðωÞ � � Z ∞ �J 0 ðuÞ�2 X X e ℏωLO χ Γ nn0 N;N � dq? ¼ 8πℏε0 q? fNþ1;n fN;n 0 0 0 N n � � �� � �� Nq fN 0 ;n0 1 fNþ1;n δ ε1 � 1 þ Nq fNþ1;n 1 fN 0 ;n0 � � � �� �� 1 þ Nq fN 0 ;n0 1 fNþ1;n δ εþ þ Nq fNþ1;n 1 fN 0 ;n0 1 � �2 Z ∞ �J � 0 e2 ℏωLO χ * XX Γnn0 Nþ1;N ðuÞ � þ dq? 8πℏε0 q? f f Nþ1;n N;n 0 0 0 N n � � �� � �� Nq fN;n 1 fN 0 ;n0 δ ε2 � 1 þ Nq fN 0 ;n0 1 fN;n � � � �� �� 1 þ Nq fN;n 1 fN 0 ;n0 δ εþ þ Nq fN 0 ;n0 1 fN;n 2 *
2
(23)
with Γnn’ ¼
π Lz
(24)
ð2 þ δn;n0 Þ
ε�1 ¼ ℏω þ ðN ε�2 ¼ ℏω þ ðN
0
N 0
� n2 ε0 � ℏωLO
02
1Þℏωc þ n
N Þℏωc þ n2
n
02
(25)
�
(26)
ε0 � ℏωLO
(δðεℓ Þ; ℓ ¼ 1; 2) in Eq. (23) are the Dirac’s delta functions, they are replaced by Lorentzians of width �
γ� γ� 0 0 � 1 N;N Nþ1;N � �2 ; δ ε2 ¼ �2 � � π ðε� Þ2 þ γ� π ðε� Þ2 þ γ� 0 0 1 2 N;N Nþ1;N
� γ� N;N’ ; γ Nþ1;N’ ,
namely [31].
� 1 δ ε� 1 ¼
(27)
� � � Z ∞ � �2 e2 ℏω χ * � �J 0 ðϑÞ�2 1 1 LO N;N 0 γ� � Γ ¼ þ dq N 0 q ? N ;N 2 2 nn 0 8π2 ε0 q?
(28)
� �2 � Z ∞ � �2 e2 ℏω χ * � �J � 0 1 1 LO Nþ1;N ðϑÞ 0 � Γ γ� ¼ þ dq N 0 q ? nn Nþ1;N 2 2 8π2 ε0 q ? 0
(29)
with
The term ΘðωÞ in Eq. (22) for confined LO-phonons is obtained as follows:
89
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
ΘðωÞ � Θc ðωÞ ¼
Z ∞ ;q? * X XX Λκ;n0τ;nϕ e2 ℏωτLO χ � q? dq? 8πℏε0 Lz 0 0 τ ϕ¼� fNþ1;n fN;n 0 N ;n
JN;N 0 aκτϕ q2? þ
bκτϕ
L2z � � �� � �� Nq fN 0 ;n0 1 fNþ1;n δ ϑ1 c � 1 þ Nq fNþ1;n 1 fN 0 ;n0 � � �� �o � Nq fNþ1;n 1 fN 0 ;n0 δ ϑþ þ 1 þ Nq fN 0 ;n0 1 fNþ1;n 1 c Z
Λκ;n;nτϕ0
τ;q? *
e2 ℏωLO χ XXX þ 8πℏε0 Lz 0 0 τ ϕ¼� fNþ1;n N ;n
fN;n
∞
�
(30)
JNþ1;N 0 bκτϕ aκτϕ q2? þ 2 Lz �� � fN’;n’ δ ϑ2 c �� �o fN;n δ ϑþ 2 c
q? dq? 0
� � �� Nq fN;n 1 � 1 þ Nq fN’;n’ 1 fN;n � � � Nq fN’;n’ 1 þ 1 þ Nq fN;n 1 fN’;n’ with
ε�1 ¼ ℏω þ ðN ε�2 ¼ ℏω þ ðN δ ε� 1
δ ε� 2
� c
� c
0
N
� ;q? n2 ε0 � ℏωτLO
02
1Þℏωc þ n
0
N Þℏωc þ n2
n
02
(31)
�
(32)
;q? ε0 � ℏωτLO
¼
γκ� 0 N;N � � π ðε� Þ2 þ γκ� 2 0 1 N;N
(33)
¼
γκ� 0 Nþ1;N �2 � π ðε� Þ2 þ γκ� 0 2 Nþ1;N
(34)
1
1
where � �2 � � �2 e2 ℏωτ;q? χ * � 1 1 XX�� κ;τϕ �� LO γ κ� � ¼ þ Λ N 0 0 q N ;N 2 2 τ ϕ¼�� n ;n � 8 π 2 ℏ 2 ε 0 V0 Z ∞ JN;N 0 � q? dq? bκτϕ 0 aκτϕ q2? þ 2 Lz
(35)
Fig. 1. Influence of photon energy on the optical absorption power in a quantum well at B ¼ 20:97 T, T ¼ 300 K, and Lz ¼ 12 nm for the models of confined phonon and bulk phonon. 90
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
� �2 � � �2 e2 ℏωτ;q? χ � � 1 1 XX�� κ;τϕ �� LO � Λ γ κ� ¼ þ N 0 q Nþ1;N’ 2 2 τ ϕ¼�� n;n � 8 π 2 ℏ 2 ε 0 V0 Z ∞ JNþ1;N 0 � q? dq? : bκτϕ 0 aκτϕ q2? þ 2 Lz
(36)
1.3. Numerical results in a square GaAs/AlAs quantum well and discussion The optical absorption power in a GaAs quantum well as a function of the photon energy is obtained. The material parameters used are [32,33]: χ 0 ¼ 13:18, χ ∞ ¼ 10:89, ε0 ¼ 8:85 � 10 12 C2 =Nm2 , E0 ¼ 5:0 � 106 V/m, ℏωLO ¼ 36:25 meV, m� ¼ 0:067 � m0 (m0 is the 0 free electron’s mass). Assuming that only n ¼ 1; n’ ¼ 1; 2 for confined electron and N ¼ 0; N ¼ 1 for Landau levels are occupied by the electrons and the following result is considered in the quantum limit. From Fig. 1 shows the each curve there are four peaks as follows: - The first peak at ℏω ¼ 36:25 meV, which corresponds to the condition ℏω ¼ ðN NÞℏωc . It describes that an electron in N ¼ 0 0 move to N ¼ 1 by the absorption of a photon with energy ℏω. It is called cyclotron resonance peak. - The second peak at ℏω ¼ 72:5 meV, which corresponds to the condition ℏω ¼ ðN’ NÞℏωc þ ℏωLO . It describes that an electron in 0 N ¼ 0 move to N ¼ 1 by absorption of a photon with energy ℏω along with the emission processes of a LO-phonon with energy ℏω. It is called ODMPR peak. 0 02 - The third peak at ℏω ¼ 117:09 meV, which corresponds to the condition ℏω ¼ ðN NÞℏωc ðn2 n Þε0 ℏωLO . It describes that 0 0 an electron in N ¼ 0 move to N ¼ 1 and from the n ¼ 1 can move to n ¼ 2 by absorption of a photon with energy ℏω along with the absorption processes of a LO-phonon with energy ℏωLO . It is called ODMPR peak. 0 02 - The fourth peak at ℏω ¼ 189:59 meV, which corresponds to the condition ℏω ¼ ðN NÞℏωc ðn2 n Þε0 þ ℏωLO . It describes that 0 0 an electron in N ¼ 0 move to N ¼ 1 and from the n ¼ 1 can move to n ¼ 2 by absorption of a photon with energy ℏω along with the emission processes of a LO-phonon with energy ℏωLO . It is called ODMPR peak. 0
From Fig. 2 shows the temperature doesn’t effect on the ODMPR peak’s position for the intrasubband/intersubband transition but effects on the ODMPR peak’s value through the distribution functions fNþ1;n , fN;n , and N! of electron and phonon. The temperature q
rises as the ODMPR peak’s value increases while its position is constant. This result is because the temperature of system increases as the possibility of the electrophonon scattering rises. This is in accordance with the works have been investigated in the theory [34–36]. Furthermore, effect of temperature on the ODMPR peaks for the intersubband transition (Fig. 2b)) is negligible but for the intra subband transition (Fig. 2a)) is significant. This is in accordance with the works have been investigated in the theory and experimental [5]. This is proved by the following figure (Fig. 3). From Fig. 3 proves that effect of temperature on the ODMPR peak for the intersubband transition (Fig. 3b)) is negligible but for the intrasubband transition (Fig. 3a)) is significant and particular for HZ model. This is in accordance with the works have been inves tigated in the theory and experimental [5]. In addition, from Fig. 3 shows the FWHM of the ODMPR peak for the intra subband/intersubband transition increases with the growing temperature of system for all models of phonon. This result is because the temperature of system increases as the possibility of the electrophonon scattering rises. Furthermore, the FWHM of the above peaks for the bulk phonon has a smaller and changes slower than it does for the confined phonon. It is because the possibility of the electro phonon scattering rises when phonon is confined. The FWHM for the Huang-Zhu model is largest, while it for R model is smallest among three models of confined phonon (FK, R, and HZ model). This result is in accordance with the results have been investigated in the theory by Refs. [3,4].
Fig. 2. Influence of photon energy on the ODMPR peak of optical absorption power for the intrasubband transition (Fig. a))/intersubband transition (Fig. b)) at different values of temperature. 91
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
Fig. 3. Influence of temperature on the FWHM of the ODMPR peak for the intrasubband transition (Fig. a))/intersubband transition (Fig. b)) with models of confined phonon and bulk phonon at Lz ¼ 12 nm and B ¼ 20:97 T.
From Fig. 4 shows the FWHM of the ODMPR peak for the intrasubband transition has larger value than that for intersubband transition for both confined phonon and bulk phonon, this result is in good accordance with the work has been investigated in the theory and experimental [5]. From Fig. 5a shows the well width doesn’t effect on the ODMPR peak’s position for the intrasubband transition but Fig. 5b. shows pm the well width effects on the ODMPR peak’s position for the intersubband transition through the εpm 1 and ε2 in Eqs. (25) and (26) for 0 0 intrasubband transition (n ¼ n ) and intersubband transition (n 6¼ n ). Besides, from Fig. 5. shows that the FWHM of the ODMPR peak for the intrasubband/intersubband transition decreases with the rise of the well width for both the confined phonon and bulk phonon. This is proved by the following Fig. 6 shows the FWHM of the ODMPR peak for the intrasubband/intersubband transition decreases with the rise of the well width for both the confined phonon and bulk phonon. This is in accordance with the result has been inves tigated in the theory and experimental [1,2]. Furthermore, the FWHM of the above peaks for the bulk phonon has a smaller and changes slower than it does for the confined phonon. It is because the possibility of the electrophonon scattering rises when phonon is confined. The FWHM for the Huang-Zhu model is largest, while it for R model is smallest among three models of confined phonon (FK, R, and HZ model). This result is in good accordance with the work have been investigated in the theory [3,4]. Beside, FWHM of the ODMPR peak for the intrasubband/intersubband transition decreases quickly when the value of well width increases from Lz ¼ 5 nm to Lz ¼ 30 nm for all the models of phonons. Therefore, when the well width is narrow, the confined phonon become more necessary and should be considered. The effect of confined phonon on the FWHM of the ODMPR peak for the intrasubband/intersubband transition is negligible and can be ignored when Lz > 30 nm. From Fig. 7 shows the FWHM of the ODMPR peak for the intrasubband transition has larger value than that for the intersubband transition for only the model of Huang-Zhu, this is in good accordance with the result has been investigated in the theory and experimental [5]. Furthermore, this result proves that the model of Huang-Zhu has received wide acceptance and best describes and been in agree with experimental results [14,15]. 2. Some conclusions The FWHM of the ODMPR peak for the intrasubband/intersubband transition has theoretically studied by basing on the confined
Fig. 4. Compare influence of temperature on the FWHM of the ODMPR peak for intrasubband transition to that for the intersubband transition with models of confined phonon and bulk phonon at Lz ¼ 12 nm and B ¼ 20:97 T. 92
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
Fig. 5. Influence of photon energy on the ODMPR peak of optical absorption power for the intrasubband transition (Fig. a))/intersubband transition (Fig. b)) at different values of the well width.
Fig. 6. Influence of well width on the FWHM of the ODMPR peak for the intrasubband transition (Fig. a))/intersubband transition (Fig. b)) with models of confined phonon and bulk phonon at T ¼ 300 K and B ¼ 20:97 T.
Fig. 7. Compare influence of well width on the FWHM of the ODMPR peak for intrasubband transition to that for the intersubband transition with models of confined phonon and bulk phonon at T ¼ 300 K and B ¼ 20:97 T.
phonon models of FK, R, and HZ and operator projection technique. The optical absorption power expression in quantum wells presented. Based on the obtained numerical result, we have some main remarks as follows: - The FWHM of the ODMPR peak for the intrasubband/intersubband transition increases with the growing temperature of system and decreases with the rise of the well width for both the confined phonon and bulk phonon. This is in accordance with the result has been investigated in the theory and experimental [1,2]. - The FWHM of the above peaks for the bulk phonon has a smaller value and changes slower than it does for the confined phonon models, this proclaimed clearly at the high temperature or the small well width. The FWHM for the model of Huang-Zhu is largest, 93
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
while it for R model is smallest among models of confined phonon (FK, R, HZ model). - The FWHM of the ODMPR peak for the intrasubband transition has larger value than that for intersubband transition for both the confined phonon and bulk phonon. This is in good accordance with the result has been investigated in the theory and experimental [5]. - The FWHM of the ODMPR peak for the intrasubband/intersubband transition decreases quickly when the value of well width increases from Lz ¼ 5 nm to Lz ¼ 30 nm for all the models of phonon. Thus, in this large range of the well width, the confined phonon become more necessary and should be considered in study the FWHM of the ODMPR peak for the intrasubband/intersubband tran sition. The effect of confined phonon on the FWHM of the above ODMPR peaks is negligible and can be ignored when Lz > 30 nm. - The above results prove that the model of Huang-Zhu has received wide acceptance and best describes and been in agree with experimental results [14,15]. We expect that this obtained results would be useful information to orient for experimental study and applications for nano devices in the future. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
M. Belmoubarik, K. Ohtani, H. Ohno, Appl. Phys. Lett. 92 (2008) 191906. F. Szmulowicz, M.O. Manasreh, C. Kutsche, C.E. Stutz, Mater. Res. Soc. 299 (1994) 53. S. Rudin, T.L. Reinecke, Phys. Rev. B 41 (1990) 7713. Gerald Weber, A.M. de Paula, J.F. Ryan, Semicond. Sci. Technol. 6 (1991) 397. H. Kobori, T. Ohyama, E. Otsuka, J. Phys. Soc. Jpn. 59 (1990) 2141. C.R. Bennett, K. Guven, B. Tanatar, Phys. Rev. B 57 (1998) 3994. S.G. Yu, K.W. Kim, M.A. Stroscio, G.J. Iafrate, A. Ballato, Phys. Rev. B 50 (1994) 1733. A. Svizhenko, A. Balandin, S. Bandyopadhyay, M.A. Stroscio, Phys. Rev. B 57 (1998) 4687. R. Fuchs, K.L. Kliewer, Phys. Rev. 140 (1965) A2076. J.J. Licari, R. Evrard, Phys. Rev. B 15 (1977) 2254. B.K. Ridley, Phys. Rev. B 39 (1989) 5282. K. Huang, B.-F. Zhu, Phys. Rev. B 38 (1988) 2183. K. Huang, B. Zhu, Phys. Rev. B 38 (1988) 13377. Ruf Tobias, Wald Keith, Peter Y. Yu, K.T. Tsen, H. Morkoc, K.T. Chan, Superlattice. Microst. 13 (1993) 203. J.S. Bhat, S.B. Kapatkar, S.S. Kubakaddi, B.G. Mulimani, Phys. Status Solidi B 209 (1998) 37. X. Wu, F.M. Peeters, Phys. Rev. B 55 (1997) 9333. S.Y. Liu, X.L. Lei, Phys. Rev. B 60 (1999) 10624. G.Q. Hai, F.M. Peeters, Phys. Rev. B 60 (1999) 16513. Y.J. Cho, S.D. Choi, Phys. Rev. B 49 (1994) 14301. C. Matthiesen, A.N. Vamivakas, M. Atatre, Phys. Rev. Lett. 108 (2012), 093602. C.Y. Lin, F. Grillot, N.A. Naderi, Y. Li, L.F. Lester, Appl. Phys. Lett. 96 (2010), 051118. J.M. Miloszewski, M.S. Wartak, P. Weetman, O. Hess, J. Appl. Phys. 106 (2009), 063102. F. Zhang, L. Li, X.H. Ma, Z.G. Li, Q.X. Sui, X. Gao, Y. Qu, B.X. Bo, G.J. Liu, Acta Phys. Sin. 61 (2012), 054209. H. Weman, L. Sirigu, K.F. Karlsson, K. Leifer, A. Rudra, E. Kapon, Appl. Phys. Lett. 81 (2002) 2839. H. Ham, H.N. Spector, Phys. Rev. B 62 (2000) 13599. J.S. Bhat, S.S. Kubakaddi, B.G. Mulimani, J. Appl. Phys. 72 (1992) 4966. J.S. Bhat, B.G. Mulimani, S.S. Kubakaddi, Phys. Rev. B 49 (1994) 16459. B.-H. Wei, C.S. Kim, Phys. Rev. B 58 (1998) 9623. R. Zheng, M. Matsuura, Phys. Rev. B 61 (2000) 12624. N.L. Kang, K.S. Bae, C.H. Choi, Y.J. Lee, J.Y. Sug, J.H. Kim, S.D. Choi, J. Phys. Condens. Matter 7 (1995) 8629. M.P. Chaubey, C.M. Van Vliet, Phys. Rev. B 33 (1986) 5617. J. Gong, X.X. Liang, S.L. Ban, J. Appl. Phys. 100 (2006), 023707, 1. K.S. Bae, N.L. Kang, Y.J. Cho, S.D. Choi, Phys. stat. sol. B 200 (1997) 239. K.D. Pham, L. Dinh, P.T. Vinh, C.A. Duque, H.V. Phuc, C.V. Nguyen, Superlattice. Microst. 120 (2018) 738. N.V.Q. Binh, et al., J. Phys. Chem. Solids 125 (2019) 74. L.V. Tung, P.T. Vinh, H.V. Phuc, Phys. B 539 (2018) 117.
94
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Comparison of magneto-resonance absorption FWHM for the intrasubband/intersubband transition in quantum wells Nguyen Dinh Hien a, b a b
Laboratory of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
A R T I C L E I N F O
A B S T R A C T
Keywords: Confined phonon Intrasubband transition Intersubband transition FWHM Quantum wells
We compare magneto-resonance absorption FWHM (full-width at half-maximum) for the intra subband transition (1 → 1) to that for the intersubband transition (1 → 2) in a square GaAs quantum well by using the projection operator method due to confined LO-phonon delineated by the model of Fuchs-Kliewer, Ridley, and Huang-Zhu. The numerical results show that the FWHM of the optically detected magnetophonon resonance (ODMPR) peak for the intrasubband/inter subband transition decreases with the rise of the width of well and increases with the growing temperature for all models of phonon. This is in accordance with the result has been investigated in the theory and experimental [1,2]. Furthermore, the FWHM of the above ODMPR peaks for the bulk phonon has a smaller and changes slower than it does for the confined phonon. The FWHM for the Huang-Zhu model is largest among three models of confined phonon (Fuchs-Kliewer, Ridley, and Huang-Zhu model), while it for Ridley model is smallest. This result is in accordance with the results have been investigated in the theory [3,4]. In addition, the FWHM of the ODMPR peak for the intrasubband transition has larger value than that for intersubband transition for both the bulk and confined phonon, this is in accordance with the result has been investigated in the theory and experimental [5].
1. Introduction Confined phonon effect, it is an necessary part of the depiction of electrophonon interactions and causes the rise of electrophonon scattering rates, modifies the density of states of phonon in low dimensional semiconductor structure [6–8]. There have been several theory models of the confined phonon in quantum wells such as Fuchs-Kliewer (FK) [9,10], Ridley (R) [11], and Huang-Zhu (HZ) model [12,13]. However, the Huang-Zhu model has received wide acceptance and best depicts and been in accordance with exper imental result [14,15]. MPR (magnetophonon-resonance) is the resonance scattering of electrons caused by phonon absorption or emission when the distance between two Landau levels is equal to the energy of the longitudinal optical phonon [16–18]. MPR has been of great interest to scientists because it is a powerful spectral tool to investigate properties such as effective mass measurement, restoration mechanism of carriers … Magnetophonon-resonance phenomena can be observed directly through the ODMPR (optically detected MPR) [18]. The FWHM is well known as a instrument for study the transport behavior, scattering mechanisms of carriers in the materials [19]. Hence, it may be used to probe confined electron-confined LO-phonon scattering process. The FWHM of the ODMPR peak for the intrasubband/intersubband transitions with the emission bulk phonon processes was studied, but with the emission confined phonon processes is still open. The FWHM has been measured in different kinds of semiconductors: Quantum dots E-mail address: [email protected]. https://doi.org/10.1016/j.spmi.2019.05.035 Received 18 February 2019; Accepted 22 May 2019 Available online 29 May 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
[20,21], quantum wells [22,23], and quantum wires [24,25] under the effect of the bulk phonon. Under the effect of the confined phonon, several works have been examined [3,15,26–29]. Furthermore, the FWHM of the ODMPR peak for the intra subband/intersubband transitions with the emission bulk phonon processes has examined in the theory and experimental [5]. Thus, when confined LO-phonon is considered, this is still open. We study the FWHM of the ODMPR peak for the intrasubband/intersubband transitions with the emission phonon processes for the above models of phonon in a quantum well. The dependence of the above FWHM on the width of well and the temperature of system by using the operator projection method is presented to compare. We introduce the theory models of phonon confinement in quantum wells in Sec. 2. In Sec. 3, we calculate the absorption power in quantum wells under the effect of the phonon confinement. Sec. 4 presents the numerical results in a square GaAs/AlAs quantum well and discussion. Finally, in Sec. 5 we present the conclusions. 1.1. Theory models of phonon confinement in quantum wells In this paper, we consider the GaAs/AlAs quantum well with electron is free in the x y plane; B ¼ ð0; 0; BÞ is a magnetic field, and it is applied in the z direction. For the potential VðzÞ, VðzÞ ¼ 0 for jzj < Lz =2 and VðzÞ ¼ ∞ for jzj > Lz =2, the one electron energy eigenvalues εðNnÞ and eigenfunction ϒðNnky Þ are given by Ref. [27]. � 1 ϒ Nnky ¼ pffiffiffiffi ψ N ðx Ly
� x0 Þexp iky y Φn ðzÞ
(1) (2)
εðNnÞ ¼ ðN þ 1=2Þℏωc þ n2 ε0
x0 Þ being harmonic oscillator wave function,
where n ¼ 1; 2; 3; ⋯ and N ¼ 0; 1; 2; ⋯ are the electric subband and Landau index; ψ N ðx
centered at x0 ¼
d2B ky with dB ¼ ðℏc=eBÞ1=2 is cyclotron radius; Ly , Φn ðzÞ, and ky respectively being the normalization length, electron
wave function, and wave vector of the electron in z-direction; ωc ¼ eB=m� and ε0 ¼ ℏ2 π2 =ð2m� L2z Þ being the cyclotron frequency and energy of the lowest electric subband, respectively; Lz is the width of well; m� being the effective mass of electron. Φn ðzÞ is given by Ref. [27]. sffiffiffiffi � � 2 nπz nπ sin Φn ðzÞ ¼ þ (3) Lz Lz 2 The matrix element for confined electron - LO-phonon confinement interaction with κð ¼ FK; R; HZÞ models of the confined phonon can be written as [3,27]. �� � � ��2 � �2 �2 � �2 � 2 �� � � � � � �� �� � i�Hel ph �f � ¼ e ℏωLO χ �V κ ðq? Þ� �JNN’ ðϑÞ� �Λκ;τϕ � δ f i (4) ni ;nf � k? ;k? �q? τϕ � � � � � � � � � 2ε0 V0 where � �2 � � �JNN’ ðϑÞ� ¼ n2 !e ϑ ϑn1 � � n1 ! ϑ ¼ d2B q2? =2 ; Ln1 n2 Z Λκ;niτ;nϕf ¼
Lz =2 Lz =2
n2
n2
� n1 Ln2
2
n2
� ðϑÞ
(5)
ðϑÞ is the Laguerre polynomial; n1 ¼ maxfN; N g, n2 ¼ minfN; N g. 0
0
(6)
Φ�nf ðzÞuκτϕ ðzÞΦni ðzÞdz
is the overlap integral, it has been shown in Ref. [3] with intrasubband and intersubband.Where χ � ¼ ð1=χ ∞ 1=χ 0 Þ, χ 0 and χ ∞ respectively are the static and high-frequency dielectric constants; ε0 is the vacuum dielectric constant; q? ¼ ðqx ; qy Þ is the two-dimensional LO-phonon wave vector; V0 ¼ SLz and ℏωLO ¼ 36:25 meV is the energy of the LO-phonon in the GaAs/AlAs quantum well. In the confined space, uκτϕ ðzÞ is the parallel composition of the displacement vector of τ th phonon mode, and different for the FK model [10], R model [11], and HZ model [13]; ϕ are ( ) even and the ( þ ) odd confined phonon modes. For the FK model, uκτϕ ðzÞ is written as follows: � � τπ z ; τ ¼ 1; 3; 5; ⋯ uFK ðzÞ ¼ cos (7) τþ Lz � � τπz ; τ ¼ 2; 4; 6; ⋯ uFK τ ðzÞ ¼ sin Lz
(8)
For the R model, uκτϕ ðzÞ is written as follows: � � τπz uRτþ ðzÞ ¼ sin ; τ ¼ 1; 3; 5; ⋯ Lz
(9)
87
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
� � τπz uRτ ðzÞ ¼ cos ; τ ¼ 2; 4; 6; ⋯ Lz
(10)
For the HZ model, uκτϕ ðzÞ is written as follows: � � μτ π z cτ z uHZ þ ; τ ¼ 3; 5; 7; ⋯; τþ ðzÞ ¼ sin Lz Lz � � μτ π z uHZ ðzÞ ¼ cos τ Lz
(11) (12)
ð 1Þτ=2 ; τ ¼ 2; 4; 6; ⋯
with μτ and cτ are given by �μ π � μ π tan τ ¼ τ ; τ 1 < μτ < τ 2 2 cτ ¼
2sin
(13)
�μ π � τ 2
(14)
The term V κτϕ ðq? Þ in Eq. (4) is given by Refs. [27,28]. � � V κτϕ ðq? Þ ¼ aκτϕ q2? þ bκτϕ L2z
(15)
1=2
HZ with aHZ τϕ and bτϕ are found for the HZ model as follows � � 1 2 1 2 2 aHZ c2τ ; τ ¼ 3; 5; 7; ⋯ ; bHZ τþ ¼ 1 þ cτ τ þ ¼ μτ π 6 μ2τ π2
(16) (17)
HZ 2 2 aHZ τ ¼ 3; bτ ¼ τ π ; τ ¼ 2; 4; 6; ⋯
these factors are (18)
FK;R 2 2 aFK;R τ� ¼ 1; bτ� ¼ τ π ; τ ¼ 1; 2; 3; ⋯
for the FK and R model. τϕ The term Λκ; ni ;nf in Eq. (6) can be evaluated for intrasubband transition and intersubband transition with above models of confined phonon. It is obtained for the FK model as follows: τþ ΛFK; ¼ 0; τ ¼ 1; 3; 5; ⋯ 12 τ=2 � 2 1 τ ΛFK; ¼ τð 1Þ 12 2
π
τþ ΛFK; ¼ 11 τ ΛFK; 11
2ð
τ
�
ðτþ1Þ=2
1Þ
1
τπ
� ; τ ¼ 2; 4; 6; ⋯ τ 1 � τπ ; τ ¼ 1; 3; 5; ⋯ 2 2 2 1
2
9 þ
4π
(19)
πτ
¼ 0; τ ¼ 2; 4; 6; ⋯
For the R model as follows: τþ ΛR; 12 ¼
1 ðδm;1 þ δτ;3 Þ; τ ¼ 1; 3; 5; ⋯ 2
τ ΛR; 12 ¼ 0; τ ¼ 2; 4; 6; ⋯
(20)
τþ ΛR; 11 ¼ 0; τ ¼ 1; 3; 5; ⋯
1 τ ΛR; 11 ¼ δτ;2 ; τ ¼ 2; 4; 6; ⋯ 2 For the HZ model as follows: � � 2cτ 8 1 τþ þ ΛHZ; ¼ 12 π2 9 μ2τ 1 τ ΛHZ; 12
¼ 0; τ ¼ 2; 4; 6; ⋯
τþ ΛHZ; 11
¼ 0; τ ¼ 3; 5; 7; ⋯
3 τ ΛHZ; ¼ δτ;2 11 2
1
μ2τ
�� ; τ ¼ 3; 5; 7; ⋯ 9 (21)
τ=2
ð
1Þ ð1
δτ;2 Þ; τ ¼ 2; 4; 6; ⋯
88
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
From above results, it is shown that only even modes in HZ model, odd modes in FK model, and the τ ¼ 2 mode in R model contribute to intrasubband transition. But, only odd modes in HZ model, even modes in FK model, and the τ ¼ 1; 3 modes in R model make contribution for intersubband transition. 1.2. Absorption power under the effect of the phonon confinement in quantum wells The absorption power per unit volume of quantum wells is given by N.L. Kang et al. [30] as follows: � �2 � fαþ1 ÞΘðωÞ E2 X �jþ α ðfα pðωÞ ¼ 0 2ℏω α ðω ωc Þ2 þ ½ΘðωÞ�2
(22)
where ω and E0 respectively are frequency and amplitude of the electromagnetic field; fα and fαþ1 are the Fermi-Dirac distribution � � �2 � � � � � �2 �� � ¼ �hα þ 1��jþ ��αi�� ¼ 2e2 ℏωc ðN þ 1Þ=m� . functions of electron at state �αi ¼ �N; n; ky i and �α þ 1i ¼ �N þ 1; n; ky i; �jþ α The term ΘðωÞ in Eq. (22) for bulk LO-phonon is obtained as follows: ΘðωÞ � Θb ðωÞ � � Z ∞ �J 0 ðuÞ�2 X X e ℏωLO χ Γ nn0 N;N � dq? ¼ 8πℏε0 q? fNþ1;n fN;n 0 0 0 N n � � �� � �� Nq fN 0 ;n0 1 fNþ1;n δ ε1 � 1 þ Nq fNþ1;n 1 fN 0 ;n0 � � � �� �� 1 þ Nq fN 0 ;n0 1 fNþ1;n δ εþ þ Nq fNþ1;n 1 fN 0 ;n0 1 � �2 Z ∞ �J � 0 e2 ℏωLO χ * XX Γnn0 Nþ1;N ðuÞ � þ dq? 8πℏε0 q? f f Nþ1;n N;n 0 0 0 N n � � �� � �� Nq fN;n 1 fN 0 ;n0 δ ε2 � 1 þ Nq fN 0 ;n0 1 fN;n � � � �� �� 1 þ Nq fN;n 1 fN 0 ;n0 δ εþ þ Nq fN 0 ;n0 1 fN;n 2 *
2
(23)
with Γnn’ ¼
π Lz
(24)
ð2 þ δn;n0 Þ
ε�1 ¼ ℏω þ ðN ε�2 ¼ ℏω þ ðN
0
N 0
� n2 ε0 � ℏωLO
02
1Þℏωc þ n
N Þℏωc þ n2
n
02
(25)
�
(26)
ε0 � ℏωLO
(δðεℓ Þ; ℓ ¼ 1; 2) in Eq. (23) are the Dirac’s delta functions, they are replaced by Lorentzians of width �
γ� γ� 0 0 � 1 N;N Nþ1;N � �2 ; δ ε2 ¼ �2 � � π ðε� Þ2 þ γ� π ðε� Þ2 þ γ� 0 0 1 2 N;N Nþ1;N
� γ� N;N’ ; γ Nþ1;N’ ,
namely [31].
� 1 δ ε� 1 ¼
(27)
� � � Z ∞ � �2 e2 ℏω χ * � �J 0 ðϑÞ�2 1 1 LO N;N 0 γ� � Γ ¼ þ dq N 0 q ? N ;N 2 2 nn 0 8π2 ε0 q?
(28)
� �2 � Z ∞ � �2 e2 ℏω χ * � �J � 0 1 1 LO Nþ1;N ðϑÞ 0 � Γ γ� ¼ þ dq N 0 q ? nn Nþ1;N 2 2 8π2 ε0 q ? 0
(29)
with
The term ΘðωÞ in Eq. (22) for confined LO-phonons is obtained as follows:
89
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
ΘðωÞ � Θc ðωÞ ¼
Z ∞ ;q? * X XX Λκ;n0τ;nϕ e2 ℏωτLO χ � q? dq? 8πℏε0 Lz 0 0 τ ϕ¼� fNþ1;n fN;n 0 N ;n
JN;N 0 aκτϕ q2? þ
bκτϕ
L2z � � �� � �� Nq fN 0 ;n0 1 fNþ1;n δ ϑ1 c � 1 þ Nq fNþ1;n 1 fN 0 ;n0 � � �� �o � Nq fNþ1;n 1 fN 0 ;n0 δ ϑþ þ 1 þ Nq fN 0 ;n0 1 fNþ1;n 1 c Z
Λκ;n;nτϕ0
τ;q? *
e2 ℏωLO χ XXX þ 8πℏε0 Lz 0 0 τ ϕ¼� fNþ1;n N ;n
fN;n
∞
�
(30)
JNþ1;N 0 bκτϕ aκτϕ q2? þ 2 Lz �� � fN’;n’ δ ϑ2 c �� �o fN;n δ ϑþ 2 c
q? dq? 0
� � �� Nq fN;n 1 � 1 þ Nq fN’;n’ 1 fN;n � � � Nq fN’;n’ 1 þ 1 þ Nq fN;n 1 fN’;n’ with
ε�1 ¼ ℏω þ ðN ε�2 ¼ ℏω þ ðN δ ε� 1
δ ε� 2
� c
� c
0
N
� ;q? n2 ε0 � ℏωτLO
02
1Þℏωc þ n
0
N Þℏωc þ n2
n
02
(31)
�
(32)
;q? ε0 � ℏωτLO
¼
γκ� 0 N;N � � π ðε� Þ2 þ γκ� 2 0 1 N;N
(33)
¼
γκ� 0 Nþ1;N �2 � π ðε� Þ2 þ γκ� 0 2 Nþ1;N
(34)
1
1
where � �2 � � �2 e2 ℏωτ;q? χ * � 1 1 XX�� κ;τϕ �� LO γ κ� � ¼ þ Λ N 0 0 q N ;N 2 2 τ ϕ¼�� n ;n � 8 π 2 ℏ 2 ε 0 V0 Z ∞ JN;N 0 � q? dq? bκτϕ 0 aκτϕ q2? þ 2 Lz
(35)
Fig. 1. Influence of photon energy on the optical absorption power in a quantum well at B ¼ 20:97 T, T ¼ 300 K, and Lz ¼ 12 nm for the models of confined phonon and bulk phonon. 90
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
� �2 � � �2 e2 ℏωτ;q? χ � � 1 1 XX�� κ;τϕ �� LO � Λ γ κ� ¼ þ N 0 q Nþ1;N’ 2 2 τ ϕ¼�� n;n � 8 π 2 ℏ 2 ε 0 V0 Z ∞ JNþ1;N 0 � q? dq? : bκτϕ 0 aκτϕ q2? þ 2 Lz
(36)
1.3. Numerical results in a square GaAs/AlAs quantum well and discussion The optical absorption power in a GaAs quantum well as a function of the photon energy is obtained. The material parameters used are [32,33]: χ 0 ¼ 13:18, χ ∞ ¼ 10:89, ε0 ¼ 8:85 � 10 12 C2 =Nm2 , E0 ¼ 5:0 � 106 V/m, ℏωLO ¼ 36:25 meV, m� ¼ 0:067 � m0 (m0 is the 0 free electron’s mass). Assuming that only n ¼ 1; n’ ¼ 1; 2 for confined electron and N ¼ 0; N ¼ 1 for Landau levels are occupied by the electrons and the following result is considered in the quantum limit. From Fig. 1 shows the each curve there are four peaks as follows: - The first peak at ℏω ¼ 36:25 meV, which corresponds to the condition ℏω ¼ ðN NÞℏωc . It describes that an electron in N ¼ 0 0 move to N ¼ 1 by the absorption of a photon with energy ℏω. It is called cyclotron resonance peak. - The second peak at ℏω ¼ 72:5 meV, which corresponds to the condition ℏω ¼ ðN’ NÞℏωc þ ℏωLO . It describes that an electron in 0 N ¼ 0 move to N ¼ 1 by absorption of a photon with energy ℏω along with the emission processes of a LO-phonon with energy ℏω. It is called ODMPR peak. 0 02 - The third peak at ℏω ¼ 117:09 meV, which corresponds to the condition ℏω ¼ ðN NÞℏωc ðn2 n Þε0 ℏωLO . It describes that 0 0 an electron in N ¼ 0 move to N ¼ 1 and from the n ¼ 1 can move to n ¼ 2 by absorption of a photon with energy ℏω along with the absorption processes of a LO-phonon with energy ℏωLO . It is called ODMPR peak. 0 02 - The fourth peak at ℏω ¼ 189:59 meV, which corresponds to the condition ℏω ¼ ðN NÞℏωc ðn2 n Þε0 þ ℏωLO . It describes that 0 0 an electron in N ¼ 0 move to N ¼ 1 and from the n ¼ 1 can move to n ¼ 2 by absorption of a photon with energy ℏω along with the emission processes of a LO-phonon with energy ℏωLO . It is called ODMPR peak. 0
From Fig. 2 shows the temperature doesn’t effect on the ODMPR peak’s position for the intrasubband/intersubband transition but effects on the ODMPR peak’s value through the distribution functions fNþ1;n , fN;n , and N! of electron and phonon. The temperature q
rises as the ODMPR peak’s value increases while its position is constant. This result is because the temperature of system increases as the possibility of the electrophonon scattering rises. This is in accordance with the works have been investigated in the theory [34–36]. Furthermore, effect of temperature on the ODMPR peaks for the intersubband transition (Fig. 2b)) is negligible but for the intra subband transition (Fig. 2a)) is significant. This is in accordance with the works have been investigated in the theory and experimental [5]. This is proved by the following figure (Fig. 3). From Fig. 3 proves that effect of temperature on the ODMPR peak for the intersubband transition (Fig. 3b)) is negligible but for the intrasubband transition (Fig. 3a)) is significant and particular for HZ model. This is in accordance with the works have been inves tigated in the theory and experimental [5]. In addition, from Fig. 3 shows the FWHM of the ODMPR peak for the intra subband/intersubband transition increases with the growing temperature of system for all models of phonon. This result is because the temperature of system increases as the possibility of the electrophonon scattering rises. Furthermore, the FWHM of the above peaks for the bulk phonon has a smaller and changes slower than it does for the confined phonon. It is because the possibility of the electro phonon scattering rises when phonon is confined. The FWHM for the Huang-Zhu model is largest, while it for R model is smallest among three models of confined phonon (FK, R, and HZ model). This result is in accordance with the results have been investigated in the theory by Refs. [3,4].
Fig. 2. Influence of photon energy on the ODMPR peak of optical absorption power for the intrasubband transition (Fig. a))/intersubband transition (Fig. b)) at different values of temperature. 91
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
Fig. 3. Influence of temperature on the FWHM of the ODMPR peak for the intrasubband transition (Fig. a))/intersubband transition (Fig. b)) with models of confined phonon and bulk phonon at Lz ¼ 12 nm and B ¼ 20:97 T.
From Fig. 4 shows the FWHM of the ODMPR peak for the intrasubband transition has larger value than that for intersubband transition for both confined phonon and bulk phonon, this result is in good accordance with the work has been investigated in the theory and experimental [5]. From Fig. 5a shows the well width doesn’t effect on the ODMPR peak’s position for the intrasubband transition but Fig. 5b. shows pm the well width effects on the ODMPR peak’s position for the intersubband transition through the εpm 1 and ε2 in Eqs. (25) and (26) for 0 0 intrasubband transition (n ¼ n ) and intersubband transition (n 6¼ n ). Besides, from Fig. 5. shows that the FWHM of the ODMPR peak for the intrasubband/intersubband transition decreases with the rise of the well width for both the confined phonon and bulk phonon. This is proved by the following Fig. 6 shows the FWHM of the ODMPR peak for the intrasubband/intersubband transition decreases with the rise of the well width for both the confined phonon and bulk phonon. This is in accordance with the result has been inves tigated in the theory and experimental [1,2]. Furthermore, the FWHM of the above peaks for the bulk phonon has a smaller and changes slower than it does for the confined phonon. It is because the possibility of the electrophonon scattering rises when phonon is confined. The FWHM for the Huang-Zhu model is largest, while it for R model is smallest among three models of confined phonon (FK, R, and HZ model). This result is in good accordance with the work have been investigated in the theory [3,4]. Beside, FWHM of the ODMPR peak for the intrasubband/intersubband transition decreases quickly when the value of well width increases from Lz ¼ 5 nm to Lz ¼ 30 nm for all the models of phonons. Therefore, when the well width is narrow, the confined phonon become more necessary and should be considered. The effect of confined phonon on the FWHM of the ODMPR peak for the intrasubband/intersubband transition is negligible and can be ignored when Lz > 30 nm. From Fig. 7 shows the FWHM of the ODMPR peak for the intrasubband transition has larger value than that for the intersubband transition for only the model of Huang-Zhu, this is in good accordance with the result has been investigated in the theory and experimental [5]. Furthermore, this result proves that the model of Huang-Zhu has received wide acceptance and best describes and been in agree with experimental results [14,15]. 2. Some conclusions The FWHM of the ODMPR peak for the intrasubband/intersubband transition has theoretically studied by basing on the confined
Fig. 4. Compare influence of temperature on the FWHM of the ODMPR peak for intrasubband transition to that for the intersubband transition with models of confined phonon and bulk phonon at Lz ¼ 12 nm and B ¼ 20:97 T. 92
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
Fig. 5. Influence of photon energy on the ODMPR peak of optical absorption power for the intrasubband transition (Fig. a))/intersubband transition (Fig. b)) at different values of the well width.
Fig. 6. Influence of well width on the FWHM of the ODMPR peak for the intrasubband transition (Fig. a))/intersubband transition (Fig. b)) with models of confined phonon and bulk phonon at T ¼ 300 K and B ¼ 20:97 T.
Fig. 7. Compare influence of well width on the FWHM of the ODMPR peak for intrasubband transition to that for the intersubband transition with models of confined phonon and bulk phonon at T ¼ 300 K and B ¼ 20:97 T.
phonon models of FK, R, and HZ and operator projection technique. The optical absorption power expression in quantum wells presented. Based on the obtained numerical result, we have some main remarks as follows: - The FWHM of the ODMPR peak for the intrasubband/intersubband transition increases with the growing temperature of system and decreases with the rise of the well width for both the confined phonon and bulk phonon. This is in accordance with the result has been investigated in the theory and experimental [1,2]. - The FWHM of the above peaks for the bulk phonon has a smaller value and changes slower than it does for the confined phonon models, this proclaimed clearly at the high temperature or the small well width. The FWHM for the model of Huang-Zhu is largest, 93
Superlattices and Microstructures 131 (2019) 86–94
N.D. Hien
while it for R model is smallest among models of confined phonon (FK, R, HZ model). - The FWHM of the ODMPR peak for the intrasubband transition has larger value than that for intersubband transition for both the confined phonon and bulk phonon. This is in good accordance with the result has been investigated in the theory and experimental [5]. - The FWHM of the ODMPR peak for the intrasubband/intersubband transition decreases quickly when the value of well width increases from Lz ¼ 5 nm to Lz ¼ 30 nm for all the models of phonon. Thus, in this large range of the well width, the confined phonon become more necessary and should be considered in study the FWHM of the ODMPR peak for the intrasubband/intersubband tran sition. The effect of confined phonon on the FWHM of the above ODMPR peaks is negligible and can be ignored when Lz > 30 nm. - The above results prove that the model of Huang-Zhu has received wide acceptance and best describes and been in agree with experimental results [14,15]. We expect that this obtained results would be useful information to orient for experimental study and applications for nano devices in the future. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
M. Belmoubarik, K. Ohtani, H. Ohno, Appl. Phys. Lett. 92 (2008) 191906. F. Szmulowicz, M.O. Manasreh, C. Kutsche, C.E. Stutz, Mater. Res. Soc. 299 (1994) 53. S. Rudin, T.L. Reinecke, Phys. Rev. B 41 (1990) 7713. Gerald Weber, A.M. de Paula, J.F. Ryan, Semicond. Sci. Technol. 6 (1991) 397. H. Kobori, T. Ohyama, E. Otsuka, J. Phys. Soc. Jpn. 59 (1990) 2141. C.R. Bennett, K. Guven, B. Tanatar, Phys. Rev. B 57 (1998) 3994. S.G. Yu, K.W. Kim, M.A. Stroscio, G.J. Iafrate, A. Ballato, Phys. Rev. B 50 (1994) 1733. A. Svizhenko, A. Balandin, S. Bandyopadhyay, M.A. Stroscio, Phys. Rev. B 57 (1998) 4687. R. Fuchs, K.L. Kliewer, Phys. Rev. 140 (1965) A2076. J.J. Licari, R. Evrard, Phys. Rev. B 15 (1977) 2254. B.K. Ridley, Phys. Rev. B 39 (1989) 5282. K. Huang, B.-F. Zhu, Phys. Rev. B 38 (1988) 2183. K. Huang, B. Zhu, Phys. Rev. B 38 (1988) 13377. Ruf Tobias, Wald Keith, Peter Y. Yu, K.T. Tsen, H. Morkoc, K.T. Chan, Superlattice. Microst. 13 (1993) 203. J.S. Bhat, S.B. Kapatkar, S.S. Kubakaddi, B.G. Mulimani, Phys. Status Solidi B 209 (1998) 37. X. Wu, F.M. Peeters, Phys. Rev. B 55 (1997) 9333. S.Y. Liu, X.L. Lei, Phys. Rev. B 60 (1999) 10624. G.Q. Hai, F.M. Peeters, Phys. Rev. B 60 (1999) 16513. Y.J. Cho, S.D. Choi, Phys. Rev. B 49 (1994) 14301. C. Matthiesen, A.N. Vamivakas, M. Atatre, Phys. Rev. Lett. 108 (2012), 093602. C.Y. Lin, F. Grillot, N.A. Naderi, Y. Li, L.F. Lester, Appl. Phys. Lett. 96 (2010), 051118. J.M. Miloszewski, M.S. Wartak, P. Weetman, O. Hess, J. Appl. Phys. 106 (2009), 063102. F. Zhang, L. Li, X.H. Ma, Z.G. Li, Q.X. Sui, X. Gao, Y. Qu, B.X. Bo, G.J. Liu, Acta Phys. Sin. 61 (2012), 054209. H. Weman, L. Sirigu, K.F. Karlsson, K. Leifer, A. Rudra, E. Kapon, Appl. Phys. Lett. 81 (2002) 2839. H. Ham, H.N. Spector, Phys. Rev. B 62 (2000) 13599. J.S. Bhat, S.S. Kubakaddi, B.G. Mulimani, J. Appl. Phys. 72 (1992) 4966. J.S. Bhat, B.G. Mulimani, S.S. Kubakaddi, Phys. Rev. B 49 (1994) 16459. B.-H. Wei, C.S. Kim, Phys. Rev. B 58 (1998) 9623. R. Zheng, M. Matsuura, Phys. Rev. B 61 (2000) 12624. N.L. Kang, K.S. Bae, C.H. Choi, Y.J. Lee, J.Y. Sug, J.H. Kim, S.D. Choi, J. Phys. Condens. Matter 7 (1995) 8629. M.P. Chaubey, C.M. Van Vliet, Phys. Rev. B 33 (1986) 5617. J. Gong, X.X. Liang, S.L. Ban, J. Appl. Phys. 100 (2006), 023707, 1. K.S. Bae, N.L. Kang, Y.J. Cho, S.D. Choi, Phys. stat. sol. B 200 (1997) 239. K.D. Pham, L. Dinh, P.T. Vinh, C.A. Duque, H.V. Phuc, C.V. Nguyen, Superlattice. Microst. 120 (2018) 738. N.V.Q. Binh, et al., J. Phys. Chem. Solids 125 (2019) 74. L.V. Tung, P.T. Vinh, H.V. Phuc, Phys. B 539 (2018) 117.
94