Intersubband and intrasubband electron scattering by acoustic-phonons in quantum wells

Journal Pre-proof Intersubband and intrasubband electron scattering by acoustic-phonons in quantum wells Nguyen Dinh Hien

PII:

S0030-4026(20)30182-0

DOI:

https://doi.org/10.1016/j.ijleo.2020.164348

Reference:

IJLEO 164348

To appear in:

Optik

Received Date:

28 September 2019

Accepted Date:

1 February 2020

Please cite this article as: Nguyen Dinh Hien, Intersubband and intrasubband electron scattering by acoustic-phonons in quantum wells, (2020), doi: https://doi.org/10.1016/j.ijleo.2020.164348

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Intersubband and intrasubband electron scattering by acoustic-phonons in quantum wells

Nguyen Dinh Hien1,2 1

Laboratory of Magnetism and Magnetic Materials, Advanced Institute of Materials

2

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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.

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Abstract

We study the intersubband and intrasubband electron scattering by acoustic-phonons in

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quantum wells by comparing the cyclotron-resonance (CR) full-width at half-maximum (FWHM) when without intersubband transitions with that when having intersubband tran-

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sitions in quantum-well structure due to electrons-acoustic phonons scattering. Using operator projection technique, the expressions for CR magneto-optical absorption powers (MOAPs) are obtained. Using graphical and numerical method, we obtain the dependence

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of the MOAPs on the photon energy, temperature, quantum-well width, and external magnetic field. We obtain the FWHM as profile of the curve based on the profile technique. The

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FWHM is calculated for both when without intersubband transitions and when having intersubband transitions. The influence of the temperature, external magnetic field, and well

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width on the FWHM have been considered and discussed. The results for the CR-peak’s FWHM in the case without intersubband transitions are compared with those in the case having inter-subband transitions. In addition, the contribution of acoustic phonons in the range of low- and high-temperatures have also been considered and compared. Key words: CR absorption FWHM, Electrons-acoustic phonons scattering, Cyclotron resonance

Preprint submitted to Elsevier

28 September 2019

1

Introduction Low dimensional semiconductor is recognized widely for technological ap-

plications in the future. Quasi-two-dimensional quantum well was expected to extremely exhibit high mobility of the electrons at low-temperature regions. It is known that at low temperature regions in carriers scattering the acoustic phonon

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plays an enhanced role and can dominate over optic-phonon [1]. In addition, the electrons transport in quantum wells in low-temperatures region (< 100K) is con-

trolled by optic-phonons scattering in high electric-field regions and by acoustic-

phonons scattering in low electric-fields. Magneto optical materials create the part

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of optical diodes or isolators where optical isolations are accomplished for the

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change in transmission of light tuned by a magnetic field. Researchers combine this two dimensional material with graphene thereby creating hybrid material to

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realize device with enhanced property and also to surmount the gap-less nature of graphene [2]. The cyclotron resonance-acoustic phonons is an important effect in study of the electrons-acoustic phonons scattering in the presence of external

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magnetic-field and it is arisen from cyclotron-resonance. The cyclotron resonance occurs in the semiconductor due to magnetic and electric fields, and the cyclotron

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frequency is equal to the photon frequency. The cyclotron resonance (CR) effect has been demonstrated to be a useful tool for obtaining direct-information about

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scattering processes, effective mass, and nonparabolicity of the valence and the conduction bands [3]. The cyclotron resonance effect has been investigated for both the theoretically [4] and experimentally [5] in bulk semiconductor, experi-

∗ Corresponding author. Email address: [email protected] (Nguyen Dinh Hien1,2 ).

2

mentally in quantum-well structures [6–8]. The study of the CR effect in quasi2D electrons systems has been of great interest to scientists. The reason for this is that for high-purity semiconductors, electrons-phonons interaction is the main type of interaction. It will help to clarify the new characteristics of 2D electron gas due to application of the external magnetic field, thereby providing new information about the crystal and magneto-optical property of quasi-2D electrons systems for manufacturing technology of optoelectronics and electronics devices. The cyclotron-resonance line-width or full-width at half-maximum is known as a

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instrument for investigation the scattering mechanism of carrier, transport behavior

in the material [5,9,10]. The FWHM due to electrons-acoustic phonons interaction was investigated, but it is still open for comparing FWHM of the CR peak when

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without intersubband transitions with that when having intersubband transitions in

GaAs quantum-well structures. The FWHM has created as the profile of the curve

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depiction the MOAPs dependence on the energy of photon [11,12]. The FWHMs have been measured in the quantum-well [13–16], quantum-wires structure [17,18],

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quantum-dots structure [19,20], monolayer materials [21,12]. However, in above investigations, the FWHM was studied for optical phonons case. In the case of acoustic phonon, only a few works have been reported [1,3,22–25]. This prop-

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erty of the acoustic phonons can be proved when we investigate the CR absorption FWHM. Thus, a further investigation for comparing FWHM of the CR peak when

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without intersubband transitions with that when having intersubband transitions in quantum-well structures is very necessary. In this investigation, the influence of the

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quantum-well width, external magnetic-field, and temperature on the FWHM of the CR peak for both when without intersubband transitions and when having intersubband transitions have been presented to compare. In addition, the contribution of acoustic phonons in the range of low- and high-temperatures have been considered and compared. The organization of study is as follows. The theoretic framework 3

for electrons-acoustic phonons scattering in quantum-well structures is introduced in Sec. 2. In Sec. 3, we present analytic results for the MOAP due to electronsacoustic phonons interaction. Sec. 4 is detailed numerical results. Finally, Sec. 5 is conclusions.

2

Theoretical framework for electrons-acoustic phonons scattering In the GaAs quantum-well, the electrons are confined in the layer GaAs leads

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to the creation of a quasi-2D electrons gas in which the movement of the electron perpendicular to the layer of the wells is quantized. The magnetic field, B, perpendicular to the layer of the wells, the in (x − y) plane motion of electron is quantized into the Landau-levels. In the case of square potential well, V (z), where V (z) = ∞

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with |z| > Lz /2, V (z) = 0 with |z| < Lz /2; |N, n, ky i and εN,n respectively are

the eigenfunction and energy eigenvalues of an electron, they are given as follows

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[26]

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1 |N, n, ky i = q ψN (x − x0 ) exp(iky y)ϕn (z), Ly 1 hωc + n2 ε0 , εn,N = (N + )¯ 2

(1)

(2)

as follows [26]

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where, in z-direction the wave-function, ϕn (z) of the an electron can be expressed s

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ϕn (z) =

2 nπz nπ sin( + ), Lz Lz 2

(3)

where n(= 1, 2, 3, ...) being index of the subband level; N (= 0, 1, 2, ...) being

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index of the Landau level; ψN (x − x0 ) is harmonic-oscillator function, where

x0 = −a2c ky , ac =

q

h ¯ c/(eB) is cyclotron radius; ky and Ly respectively is the

electron’s wave-vector in y-direction and normalization length; ωc = eB/m∗ , ε0 = h ¯ 2 π 2 /(2m∗ L2z ) being the cyclotron frequency, lowest electric-subband energy; m∗ being the electron’s effective-mass; Lz is the well width. 4

In quantum-wells the matrix element due to electrons-acoustic phonons interaction can be expressed as follows [3,9,27–30] |hi|Hep |f i|2 = |V (q)|2 |Inn0 (qz )|2 |JN N 0 (u)|2 δkf ,ki ±q⊥ , ⊥

⊥

(4)

where I

nn0

(qz ) =

Z Lz /2 −Lz /2

|JN N 0 (u)|2 =

ϕn (z) exp (iqz z)ϕn0 (z)dz,

n2 !e−u un1 −n2 n1 −n2 [Ln2 (u)]2 , n1 !

(5)

(6)

ro of

2 here u = a2c q⊥ /2; Lnn12 −n2 (u) is the Laguerre polynomial; n1 = max{N 0 , N },

n2 = min{N 0 , N }.

V (q) is a coupling factor, it describes the strength of electrons-phonons interaction and depends on the phonons mode.

h ¯ κ2 q , 2ρva V0

(7)

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|V (q)|2 =

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For the electrons-acoustic phonons interaction, V (q), is given as [3]

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where va , κ and ρ respectively are the velocity of sound, deformation potential constant and material’s mass-density; V0 = SLz being the system’s volume; q

3

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being the wave vector of acoustic phonon.

Analytic results for the MOAP due to electrons-acoustic phonons scatter-

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ing in quantum-well structures The average MOAP per unit volume due to electrons-acoustic phonons scat-

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tering in quantum-well structures can be expressed as [31] P (ω) =

E02 Re[σ+− (ω)], 2

(8)

where E0 and Re[σ+− (ω)] respectively are the electromagnetic-field amplitude and real part of the magneto-optical conductivity tensor. The term, σ+− (ω), of the quan5

tum well is given as [31] σ+− (ω) =

X fα+1 − fα i lim+ , |jα+ |2 h ¯ ω a→0 α ω ¯ − ωc − Γα (¯ ω)

(9)

with fα and fα+1 respectively are the distribution functions at the state |αi ≡ |N, n, ky i and |α + 1i ≡ |N + 1, n, ky i of the electron, and they are described by Fermi-Dirac; |jα+ |2 = |hα + 1|j + |αi|2 = 2e2 h ¯ ωc (N + 1)/m∗ ; ω is the photon frequency; ω ¯ = ω − ia(a → 0+ ). Γα (¯ ω ) is the linewidth function and given by 1

Γα (¯ ω) =

h ¯ (fα+1 − fα )

×

X

|Cαη (q)|2 [

q,η

(1 + Nq )fα+1 (1 − fη ) Nq fη (1 − fα+1 ) − h ¯ω ¯ − Eα+1,η + h ¯ ωq h ¯ω ¯ − Eα+1,η + h ¯ ωq

ro of

(

Nq fα+1 (1 − fη ) i (1 + Nq )fη (1 − fα+1 ) + h ¯ω ¯ − Eα+1,η − h ¯ ωq h ¯ω ¯ − Eα+1,η − h ¯ ωq X (1 + Nq )fη (1 − fα ) Nq fα (1 − fη ) + |Cη,α+1 (q)|2 [ − h ¯ω ¯ − Eηα + h ¯ ωq h ¯ω ¯ − Eηα + h ¯ ωq q,η

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−

)

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(1 + Nq )fα (1 − fη ) Nq fη (1 − fα ) − + ] . h ¯ω ¯ − Eηα − h ¯ ωq h ¯ω ¯ − Eηα − h ¯ ωq

(10)

Because Γα (¯ ω ) in Eq. (10) is a complex expression, it can be analysed as follows:

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Γα (¯ ω ) = Aα (ω) + iγα (ω),

(11)

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where Aα (ω) ≡ Re{Γα (¯ ω )} and γα (ω) ≡ Im{Γα (¯ ω )} respectively are the real part and imaginary part of Γα (¯ ω ). Here Aα (ω) is related to the peak shift and γα (ω) is related to the line-width of the absorption spectrum. In the quantum limit,

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h ¯ ωc  kB T . Thus, the term Aα (ω) in comparison with ωc , it can be neglected. Besides, γα (ω) is called the relaxation rate, in the absorption spectrum γα (ω) gives

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the dominant contribution for the line shape. Using the Dirac identity as follows: lim+

b→0

1 1 = P ( ) + iπδ(x), (x − ib) x

where P is the Cauchys principal value. We obtained 6

γα (ω) =

π

X

h ¯ (fα+1 − fα )

q,η

|Cα,η (q)|2

n

× [(1 + Nq )fα+1 (1 − fη ) − Nq fη (1 − fα+1 )]δ(¯ hω − Eα+1,η + h ¯ ωq ) o

+[Nq fα+1 (1 − fη ) − (1 + Nq )fη (1 − fα+1 )]δ(¯ hω − Eα+1,η − h ¯ ωq ) X π |Cα+1,η (q)|2 + h ¯ (fα+1 − fα ) q,η n

× [(1 + Nq )fη (1 − fα ) − Nq fα (1 − fη )]δ(¯ hω − Eηα + h ¯ ωq ) o

+[Nq fη (1 − fα ) − (1 + Nq )fα (1 − fη )]δ(¯ hω − Eηα − h ¯ ωq ) .

(12)

In order to calculate the term Re[σ+− (ω)] in Eq. (8), inserting Eq. (11) into

fα+1 − fα a→0 h ¯ω − h ¯ ωc − h ¯ Aα (ω) − i¯ h[γα (ω) + a] α X fα+1 − fα |jα+ |2 h ¯ω − h ¯ ωc − i¯hγα (ω) α X (fα+1 − fα )[¯ hω − h ¯ ωc + i¯hγα (ω)] |jα+ |2 2 (¯ hω − h ¯ ωc ) + [¯hγα (ω)]2 α ( ) X (fα+1 − fα )(¯ hω − h ¯ ωc ) (fα − fα+1 )¯ hγα (ω) + 2 |jα | +i . (¯ hω − h ¯ ωc )2 + [¯hγα (ω)]2 (¯ hω − h ¯ ωc )2 + [¯hγα (ω)]2 α (13) X

|jα+ |2

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lim+

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re

i ω i = ω i = ω 1 = ω

σ+− (ω) =

ro of

Eq. (9), we obtain

Finally, we obtain the real part of the magneto-optical conductivity tensor as

hγα (ω) 1 X + 2 (fα − fα+1 )¯ |jα | . 2 ω α (¯ hω − h ¯ ωc ) + [¯hγα (ω)]2

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follows:

Re[σ+− (ω)] =

(14)

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Inserting Eq. (14) into Eq. (8), we obtain the average MOAP per unit volume

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due to confined electrons-acoustic phonons scattering in the quantum-well as

P (ω) =

E02 X |jα+ |2 (fα − fα+1 )γ(ω) . 2¯ hω α (ω − ωc )2 + [γα (ω)]2

(15)

In order to calculate the term γα (ω) in Eq. (15), we use Eq. (4) to calculate the

matrix elements Cα,η (q) and Cα+1,η (q) due to confined electrons-acoustic phonons interaction into Eq. (12), and make use of the relations 7

X q

X

Z ∞ V0 Z ∞ ··· → 2 q⊥ dq⊥ dqz · · · , 4π 0 −∞

··· →

η

XX

··· ,

(16)

Nη nη

we obtain γ(ω) =

XX N 0 n0

C(2 + δn,n0 ) (fN +1,n − fN,n )

n

× [fN +1,n (1 − fN 0 ,n0 ) − fN 0 ,n0 (1 − fN +1,n )]δ(ε1 ) +[fN +1,n (1 − fN 0 ,n0 ) − fN 0 ,n0 (1 − fN +1,n )]δ(ε1 ) +[fN 0 ,n0 (1 − fN,n ) − fN,n (1 − fN 0 ,n0 )]δ(ε2 ) o

with C=

κ2 kB T , 8ρva2 h ¯ a2c Lz

(17)

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+[fN 0 ,n0 (1 − fN,n ) − fN,n (1 − fN 0 ,n0 )]δ(ε2 ) ,

(18)

(19)

re

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ε1 = (N 0 − N − 1)¯ hωc + h ¯ ω + (n02 − n2 )ε0 , ε2 = (N − N 0 )¯ hωc + h ¯ ω + (n2 − n02 )ε0 .

Where we have used the approximation for Nq + 1 ≈ Nq ≈ kB T /(¯ hωq ), with

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ωq = va q; acoustic phonon energy h ¯ ωq  h ¯ ωc . Thus, h ¯ ωq in Eq. (19) has been neglected.

δ(ε` ) (` = 1, 2) in Eq. (17) is the delta function of Dirac, it can be replaced by

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the Lorentzian width, γN,N 0 , namely as [32]

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δ(ε` ) =

1 γN,N 0 , π (ε` )2 + (γN,N 0 )2

(20)

C(2 + δnn0 )¯ h . π

(21)

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where

(γN 0 ,N )2 =

From this analytical results we obtain the influence of magnetic field, tem-

perature, and well’s width on the CR-peak’s FWHM for both the case without intersubband transitions and the case having intersubband transitions to compare by using the numerical methods described in the Abstract. 8

4

Detailed numerical results The influence of the photon energy on the MOAPs in GaAs-well structure

have been detailed numerically calculated . The material parameters used are [33– 38]: κ = 13.5 eV, ρ = 5300 kgm−3 , E0 = 5.0 × 106 V/m, va = 5370 ms−1 , m∗ = 0.067m0 with m0 is free-electron’s mass. Assuming only levels: n = 1, n0 = 1, 2,

ro of

P Harb.unitsL

and N = 0, N 0 = 1 are occupied by the electrons.

0

2

-p

1

50

100

150

re

ÑΩ HmeVL

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Fig. 1. Dependence of MOAP on the photon energy due to electrons-acoustic phonons scattering in the GaAs well structure at T = 100 K, Lz = 11 nm, and B = 12 T.

Fig. 1 shows that there are two peaks numbered from (1) to (2). The (1) peak

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at energy h ¯ ω = 20.75 meV satisfies h ¯ ω = −(N − N 0 )¯ hωc . This describes electron absorbs photon energy (¯ hω) to move from N = 0 to N 0 = 1. It is called CR peak

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without the intersubband transition (1 → 2). The (2) peak at energy h ¯ ω = 160.07 meV satisfies h ¯ ω ≡ (N 0 − N )¯ hωc + (n02 − n2 )ε0 . This describes electron absorbs

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photon energy (¯ hω) due to acoustic phonon scattering to move from N = 0 to

N 0 = 1 along with from n = 1 to n0 = 2. It is called CR peak along with the

intersubband transition (1 → 2). Fig. 2 shows the temperature increases while the CR-peaks position are constant for both the case without intersubband transitions and the case having inter9

P Harb.unitsL

º

a)

: T = 100 K

1

º

—

16

P Harb.unitsL

- - - : T = 70 K : T = 40 K

18

20

22

24

156

: T = 100 K

b) 2

- - - : T = 70 K

— : T = 40 K

158

ÑΩ HmeVL

160

162

164

ÑΩ HmeVL

Fig. 2. a) Dependence of MOAP on the photon energy at the CR peak without intersubband transitions at different values of temperatures. b) Dependence of MOAP on the photon en-

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ergy at the CR peak having intersubband transitions at different values of the temperatures. With Lz = 11 nm, and B = 12 T.

subband transitions. Thus, the above peaks position are not effected by temperature,

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but the temperature effects on the CR-peak’s value by the distribution functions Nq ,

and fN,n , fN +1,n of phonon and electron. It is because the Eq. (19) does not contain

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the temperature, but when the temperature rises as the possibility of the electronacoustic phonon scattering rises. Our calculated result is in qualitative-agreement

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with results [15,39–42]. However, the Fig. 2b) shows that the temperature effects on the FWHM of CR peak when having intersubband transitions is negligible but that is significant for the case without intersubband transitions (Fig. 2a)). This statement

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will be proved at the Fig. 3 below.

Fig. 3 shows the CR-absorption FWHM increases with the increase of the

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temperature for both the case without inter-subband transitions and the case having inter-subband transitions. It is because the system’s temperature rises as the

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possibility of the electrons-acoustic phonons scattering increases. In addition, the CR-absorption FWHM the case without inter-subband transitions varies faster than and has larger value than that for the case having inter-subband transitions. Fig. 4a) shows the position of the CR peak without intersubband transitions is not effected by the well width because the resonance condition h ¯ ω = −(N − 10

FWHM HmeVL

1.4 1.2 1.0 0.8 0.6 0.4 0.2 ç æ 0

æ

æ

æ

æ

æ æ

æ æ æ æ

ç ç ç ç ç ç æ ç ç ç æ ç ç ç æ ç ç æ

20

40

60

80

100

T HKL

ro of

Fig. 3. Dependence of the CR-absorption FWHM on the temperature for both the case without inter-subband transitions (filled circles) and the case having inter-subband transitions (empty circles) at Lz = 11 nm, B = 12 T. º

: Lz = 13 nm

a)

18

22

24

-p

P Harb.unitsL 20

ÑΩ HmeVL

Lz = 12 nm

re

16

: Lz = 11 nm

110

lP

P Harb.unitsL

—

b)

Lz = 13 nm

- - - : Lz = 12 nm

120

130

140

Lz = 11 nm

150

160

ÑΩ HmeVL

Fig. 4. a) Dependence of MOAP on the photon energy at the CR peak without intersubband

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transitions at different values of the well’s width. b) Dependence of MOAP on the photon energy for the CR peak having intersubband transitions at different values of the well’s

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width. With T = 100 K, B = 12 T.

N 0 )¯ hωc does not depend on the well width, but the well’s width effects on the

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position of the CR-peak having intersubband transitions is seen at the Fig. 4b) because the resonance condition h ¯ ω = −(N − N 0 )¯ hωc − (n2 − n02 )ε0 depends on the well width, with ε0 = h ¯ 2 π 2 /(2m∗ L2z ). However, the influence of the quantumwell width on the CR-peak’s FWHM is significant for both the case without intersubband transitions and the case having inter-subband transitions. This will be seen 11

clearly at the Fig. 5.

FWHM HmeVL

3.5 æ 3.0 æ

2.5 2.0 1.5

æ

ç

æ æ

ç ç

1.0

æ æ æ æ æ æ æ ç æ æ ç ç ç ç ç ç ç ç ç ç

5

10

15

20

ro of

Lz HnmL Fig. 5. Dependence of the CR-absorption FWHM on the well width for both the case with-

out inter-subband transitions (filled circles) and the case having inter-subband transitions

-p

(empty circles) at T = 100 K, B = 12 T.

Fig. 5 shows the CR absorption FWHM reduces with the rise of the quantum-

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well width for both the case without inter-subband transitions and the case having inter-subband transitions. It is because the increase of the quantum-well width

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as the possibility of the electrons-acoustic phonons scattering decreases. Besides, the CR absorption FWHM for the case having inter-subband transitions decreases slower and has smaller value than that for the case without inter-subband transi-

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tions, and this is seen more clearly at a narrow quantum-well’s width. Fig. 6 shows the position of CR-peak is effected by the magnetic field for both

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the case without inter-subband transitions and the case having inter-subband transitions through the Delta functions in Eq. (17) and (19), and influence of the external

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magnetic field on the CR-peak’s FWHM is significant for both the case without inter-subband transitions and the case having inter-subband transitions. This can be detailed proved at the Fig. 7 below. Fig. 7 shows the CR absorption FWHM rises with the increase of the external magnetic field for both the case without inter-subband transitions and the case 12

a)

b) B=8T

P Harb.unitsL

B = 12 T B=8T

B = 12 T

P Harb

B=4T

5

10

15

20

B=4T

25

145

150

155

ÑΩ HmeVL

160

165

ÑΩ HmeVL

Fig. 6. a) Dependence of MOAP on the photon energy at the CR peak without intersubband transitions at different values of the external magnetic field. b) Dependence of MOAP on

external magnetic field. With T = 100 K, Lz = 11 nm.

2.0

ro of

the photon energy for the CR peak having intersubband transitions at different values of the

æ

æ

æ

1.5 æ æ

1.0

æ æ

ç

0.5 æ

ç ç

ç

lP

ç

ç

-p

æ æ

ç

ç

ç

ç

ç

ç

re

FWHM HmeVL

æ æ

5

10

15

20

B HTL

na

Fig. 7. Dependence of the CR-absorption FWHM on the external magnetic field for both the case without inter-subband transitions (filled circles) and the case having inter-subband

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transitions (empty circles) at T = 100 K, Lz = 11 nm.

having inter-subband transitions. It is because the external magnetic field rises, the

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cyclotron-frequency (ωc )rises, the cyclotron radius (ac ) reduces, the confinement of electrons rise, the possibility of electrons- acoustic phonons scattering increases. Furthermore, the CR-absorption FWHM in the case without inter-subband transitions increases faster than that for the case having inter-subband transitions, and this is seen more clearly at the high-magnetic field. 13

6 FWHM HmeVL

5

ñ ó á ç

4 3 2

ð ò à

1æ

ñ ó á ç ñ ó á ñ ç ó á ñ ç ó á ç ð ò ð ð ò ð à à ò ò à à æ æ æ æ

5

ð: T = 100 K

ñ: T = 400 K

ò: T = 70 K

ó: T = 370 K

à: T = 40 K

á: T = 340 K

æ: T = 10 K

ç: T = 310 K

ñ ñ ó á ó ó ñ ñ ç á ç á á ó ó ñ ó ç ñ ó ç ç á ñ á á ç ç ð ð ð ò ò ð ð ð ð ò ò ò ò ò à à à à à à à æ æ æ æ æ æ æ

10

15

ñ ó ñ ó ó ñ á á ç á ç ç ð ò à æ

ð ò à æ

ð ò à æ

20

ro of

Lz HnmL Fig. 8. Dependence of the CR-absorption FWHM on the well width for the case without inter-subband transitions at different values of the temperatures. With B = 12 T.

-p

Fig. 8 shows the CR absorption FWHM reduces with the increase of the quantum-well width, but rises with the increase of the system’s temperature. In par-

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ticular, in the region of low temperatures (T < 100K), the CR absorption FWHM reduces with the rise of the quantum-well width faster than that in the region of high

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temperatures (T > 300K). This statement can be proved as follows: The slope of the curve caused by T = 40K has been compared with that of T = 10K is increased by about 96.4%, they are 32.7% for T = 70K and 19.2% for T = 100K

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have been compared with that of T = 40K and 70K, respectively. The CR absorption FWHM for T = 40K is increased by about 98.9%, they are 32.1% for

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T = 70K and 19.4% for T = 100K. While the slope of the curve caused by T = 340K has been compared with that of T = 310K is increased by about

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4.6%, they are 4.2% for T = 370K and 3.8% for T = 400K have been compared with that of T = 340K and 370K, respectively. The CR absorption FWHM for

T = 340K is increased by about 4.6%, they are 4.0% for T = 370K and 3.9% for T = 400K. The present results prove that in the range of low-temperature the variableness rate and the rise rate of the CR-absorption FWHM decrease with the 14

increase of temperature. Thus, in the range of the low temperatures the electronsacoustic phonons scattering are very important and should be considered in investigation the CR-absorption FWHM in the quantum-well structure. These results is in qualitative-agreement with the results of the M.P. Chaubey et al. [38].

3

ð : T = 100 K

ñ : T = 400 K

ò : T = 70 K

ó : T = 370 K

à : T = 40 K

á : T = 340 K

æ : T = 10 K

ç : T = 310 K

2

ñ ó á ç

ñ ó á ç ñ ó á ç

1á ñ ó

ç ð ò à æ

ð ò à æ

ð ò à æ

ð ò à æ

ñ ó á ç

ð ò à æ

5

ñ ó á ç

ð ò à æ

ñ ó á ç

ñ ó á ç

ñ ó á ç

ð ò à

ð ò à

ð ò à

æ

æ

æ

10

ñ ó á ç

ñ ó á ç

ð ò à

ð ò à

ð ò à

æ

æ

æ

15

20

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B HTL

ñ ó á ç

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FWHM HmeVL

4

Fig. 9. Dependence of the CR absorption FWHM on the external magnetic field for the case

re

without inter-subband transitions at different values of the temperatures. With Lz = 11 nm.

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Fig. 9 shows that the CR-absorption FWHM increases with the rise of the external magnetic field, and system’s temperature. In particular, in the range of the low temperatures (T < 100K), the CR-absorption FWHM increases with the rise

na

of the external magnetic field faster than that in the range of high temperatures (T > 300K). This statement can be proved as follows: The slope of the curve

ur

caused by T = 40K has been compared with that of T = 10K is increased by about 100%, they are 32.7% for T = 70K and 17.4% for T = 100K have been

Jo

compared with that of T = 40K and 70K, respectively. The CR absorption FWHM

for T = 40K is increased by about 100%, they are 32.3% for T = 70K and 19.0% for T = 100K. While the slope of the curve caused by T = 340K has been compared with that of T = 310K is increased by about 4.8%, they are 4.6% for T = 370K and 3.8% for T = 400K have been compared with that of T = 340K 15

and 370K, respectively. The CR absorption FWHM for T = 340K is increased by about 4.8%, they are 4.3% for T = 370K and 3.9% for T = 400K. From this result we can see that the increase rate and variableness rate of the CR-absorption FWHM decrease with the rise of temperature. Thus, in the region of low temperature the electrons-acoustic phonons scattering is dominant. This properties are in qualitative-agreement with the work’s result in Ref. [38].

Conclusions

ro of

5

In summary, intersubband and intrasubband electron scattering by acousticphonons in GaAs quantum-well structures have been theoretically investigated. The MOAP expression has been presented. From the detailed numerical results,

-p

we present main remarks: (1) The CR absorption FWHM increases with the rise of temperature, external magnetic field, but it reduces with the rise of quantum-well

re

width for both the case without inter-subband transitions and the case having intersubband transitions. (2) The CR absorption FWHMs the case without inter-subband

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transitions vary faster than and have larger value than that for the case having intersubband transitions. (3) In the range of high temperatures (T > 300K) the acoustic

na

phonons make slower and weaker contribution than that in the range of low temperatures (T < 100K). (4) In the region of low temperatures (T < 100K) the

ur

electrons-acoustic phonons scattering is dominant and should be consider for investigation the CR-absorption FWHM in quantum-well. We hope that these results

Jo

could be good information to orientate for applied and experimental investigation for nano electronic device in the future.

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