Confined-acoustic-phonon-assisted cyclotron resonance viamulti-photon absorption process in GaAs quantum well structure

Huynh Vinh Phuc a,n, Nguyen Thi Thu Thao b, Le Dinh b, Tran Cong Phong b

a Department of Physics, Dong Thap University, Dong Thap 93000, Viet Namb Department of Physics and Center for Theoretical and Computational Physics, Hue University's College of Education, Hue 47000, Viet Nam

a r t i c l e i n f o

Article history:Received 22 May 2013Received in revised form22 September 2013Accepted 16 October 2013Available online 22 October 2013

Keywords:A. Quantum wellD. Phonon-assisted cyclotron resonanceD. Confined-acoustic phononD. Multi-photon absorption processD. Linewidth

a b s t r a c t

Phonon-assisted cyclotron resonance (PACR) in GaAs quantum well (QW) structure is investigated viamulti-photon absorption process when electrons interact with the confined acoustic phonon throughdeformation potential. The additional peaks in the absorption spectrum due to transitions betweenLandau levels accompanied with the emission and absorption of phonons are indicated. The dependenceof absorption power on the temperature, magnetic field and well width is presented. Using profilemethod, we obtain PACR-linewidth as profiles of the curves. The temperature, magnetic field and wellwidth dependences of the PACR-linewidth are investigated. The results are compared with those in thecase of mono-photon absorption process, as well as in the electron-bulk acoustic phonon interaction.The results show that the multi-photon absorption process is strong enough to be detected in PACR.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The phonon-assisted cyclotron resonance (PACR), which arisescommonly from the cyclotron resonance (CR), is an importanteffect in investigating the electron-phonon interaction in thepresence of a magnetic field. This is an effect indicating electrontransitions between Landau levels due to the absorption ofphotons accompanied with the absorption or emission of phonons.Recently, CR and PACR have been studied both theoretically andexperimentally in bulk semiconductors [1–8], quantum wells [9–14],quantum wires [15,16], and quantum dots [17]. In these works, themain CR peak occurs at ω¼ωc , and PACR peaks occur atω¼ pωcþωq, ðp¼ 1;2;…Þ, where ω, ωc and ωq are the frequen-cies of the electromagnetic wave, cyclotron and phonon, respec-tively. In all these studies, CR and PACR have been examined in theprocess of mono-photon absorptions. However, the study of PACRas well as PACR-linewidth via the multi-photon absorption pro-cess, which is of great importance, has not been found.

Multi-photon absorption process occurs when two or morephotons are simultaneously absorbed by the material [18]. Recently,the interest in the development of materials with multi-photonabsorption processes has increased, according to their potentialapplication in different fields of science. Multi-photon absorptionprocess has been studied on the optical power limiting [19], multi-

photon microscopy [20,21], micro-fabrication [22], up conversionlasing [23], photodynamic therapy [24], and multi-photon lumines-cence [25]. In particular, Boyd and co-workers [25] have demon-strated that the multi-photon luminescence is more sensitive tothe local fields than the single-photon luminescence. So, the study onmulti-photon process is important for understanding in detail thetransient response of semiconductors excited by electromagnetic field.

The multi-photon process is a nonlinear phenomenon. In recentyears there has been much interest in the study of the nonlinearphenomenon [26–29]. In these papers, the authors have obtainedthe first- and second-order nonlinear optical conductivity by theprojection-diagram method. Their results are very clear in thephysical interpretation and useful in the study of optical conductiv-ity in low-dimensional systems. However, analytical calculations forthese results are quite complicated, and they have not any applica-tion to investigate the nonlinear phenomenon, especially, in thenumerical calculations.

In one of our previous paper [30], based on the results ofSharma et al. [31], Cenerazio et al. [32], and Xu et al. [33,37], wesuggested a method for obtaining the explicit expression of theabsorption power in the presence of a magnetic field with themulti-photon absorption process, and applied it to study the PACRin QW when electrons interact with bulk dispersion of phononsmodes. In the mono-photon absorption process, this result reducesto that obtained by Bhat et al. [12–14], which is given by thesecond-order Born golden rule approximation. We have alsostudied the dependence of PACR-linewidth on the temperatureand magnetic field. The calculations demonstrated that PACR-

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/jpcs

Journal of Physics and Chemistry of Solids

0022-3697/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jpcs.2013.10.007

n Corresponding author. Tel.: þ84 673882919.E-mail address: huynhvinhphuc2710@gmail.com (H.V. Phuc).

Journal of Physics and Chemistry of Solids 75 (2014) 300–305

linewidth in the two-photon absorption process is smaller thanthat of the one-photon absorption process and strong enough tobe detected.

The electron-confined acoustic phonon interaction plays animportant role in narrow quantum well at low temperatures. Thisfeature of the confined acoustic phonon can be identified when westudy the PACR [14]. The purpose of the present paper is to applyour previous results [30] for studying PACR via the multi-photonabsorption process in QW when electrons interact with confinedacoustic phonons. Calculations are presented for GaAs quantumwell structure. The paper first presents the theoretical frameworkused in calculations and analytical results. It then discusses theresults in Section 3, and draws up the conclusions in the lastsection.

2. Theoretical framework and analytical results

When a uniform static magnetic field B is applied in thez-direction of a single quantum well, the electron eigenfunctionjN;n; ky⟩ and eigenvalues EN;n are given by [14]

jN;n; ky⟩¼1ffiffiffiffiffiLy

p ϕNðx�x0ÞeikyyχnðzÞ; ð1Þ

EN;n ¼ ðNþ1=2Þℏωcþn2E0; ð2Þwhere electrons are assumed to move freely in the (x; y) plane,ωc ¼ eB=mn is the cyclotron frequency, e and mn being the chargeand effective mass of electron, respectively; ϕNðx�x0Þ representsthe harmonic oscillator wave function centered at x0 ¼ �a2c ky withac ¼ ðℏ=mnωcÞ1=2 being the cyclotron radius; N¼ 0;1;2;… theLandau level index; n¼ 1;2;3;… the electric subband quantumnumber; Lz the well width, and E0 ¼ ℏ2π2=ð2mnLzÞ.

In Eq. (1), the envelope function is given as follows:

χnðzÞ ¼ffiffiffiffiffi2Lz

ssin

nπzLz

þ nπ2

� �: ð3Þ

The absorption power can be calculated by relating it to thetransition probability of the absorption of photons as follows [30]:

PðωÞ ¼ F20ffiffiffiɛ

p

8π∑iWif i; ð4Þ

where F0 is the intensity radiation; ɛ is the dielectric constant ofthe medium; fi is the electron distribution function; and Wi is thetransition probability. The sum is taken over all the initial states iof the electrons. The transition probability of absorbing photonwith simultaneously absorbing and/or emitting phonon W8

i canbe written as follows [30,33,37]:

W8i ¼ 2π

ℏ∑f∑qj⟨f jHe�pji⟩j2 ∑

þ1

ℓ ¼ �1

1ðℓ!Þ2

a0q?2

� �2ℓ

�δðEf �Ei8ℏωq�ℓℏωÞ; ð5Þwhere ⟨f jHe�pji⟩ is the transition matrix element for theelectron-phonon interaction; a0 ¼ ðeF0Þ=½mnðω2�ω2

c Þ�; Ei andEf are the initial and final state energies of electron; andq¼ ðq? ; qmÞ is the phonon wave vector. The sum is taken over allthe final state f of electrons. Through this paper, the upper sign ð�Þcorresponds to the phonon absorption, whereas the lower sign ðþÞcorresponds to the phonon emission. In Eq. (5), the index ℓcorresponds to the process of ℓ-photon.

It is known that, in the case of elastic isotropic medium, althoughthere are three different types of confined acoustic modes (shearwaves, dilatational waves, and flexural waves) [34,35], only thedilatational mode contributes to PACR in the extreme quantum limits

(n¼ n′¼ 1) [14,34–36]. The matrix elements for electron-confinedacoustic phonon interaction in QW for the dilatational mode in theextreme quantum limits can be written below:

j⟨f jHe�pji⟩j2 ¼ jVpðq? ; qt;m; ql;mÞj2ðNmþ1=281=2Þ�jJNN′ðq? Þj2jG11ðql;mÞj2δky ′;ky þqy ; ð6Þ

where Nm the distribution function of the phonon with frequencyωm, and

Vpj2 ¼�� ��Fmj2 ℏξ2

2Aρωmðq2t;m�q2x Þ2ðq2l;mþq2x Þ2 sin 2 qt;mLz

2

� �; ð7Þ

jJNN′ðq? Þj2 ¼N!N′!

e�a2c q2? =2 a2c q

2?

2

� �N′�N

LN′�NN

a2c q2?

2

� �� 2; ð8Þ

jG11ðql;mÞj2 ¼ 64 sin 2 ql;mLz2

� �1

jq2l;mL2z �4π2j; ð9Þ

where Fm is the normalization constant defined in Ref. [34], ξ thedeformation potential constant, the function LMN ðxÞ in Eq. (8) isthe associated Laguerre polynomials. The label m¼ 1;2;3;… isthe index for different branches of the confined modes, ρ is thedensity of the material, A the area of the QW, and ωm is thephonon frequency in m-branch, satisfying the condition [34]

ωm ¼ slffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2x þq2l;m

q¼ st

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2x þqt;m

q; ð10Þ

where slðstÞ denotes the velocity of longitudinal (transverse)acoustics waves, ql;m and qt;m are solutions of the system of twoequations

tan ðqt;mLz=2Þtan ðql;mLz=2Þ

¼ � 4q2xql;mqt;mðq2x �q2t;mÞ2

; ð11Þ

s2l ðq2x þq2l;mÞ ¼ s2t ðq2x þq2t;mÞ: ð12Þ

Numerical solutions of Eqs. (11) and (12) in a GaAs quantum wellare shown in Figs. 1 and 2, with st ¼ 3:35� 105 cm s�1 andsl ¼ 3:7� 105 cm s�1. These solutions are different from that in aGaN quantum well [14].

The energy dispersion of the confined dilatational acousticphonon ℏωm, evaluated for Lz¼10 nm, in GaAs quantum well isshown in Fig. 3. This can be compared with that in GaN quantumwell [14].

According to Bannov et al. [34,36], at the neighboring of theBrillouin zone center qx is much smaller and can be neglectedin comparison with ql;m and qt;m. In this case Fm becomes jFmj2 ¼

0 2 4 6 8 100

5

10

15

20

25

qxLz

q lmL z

Fig. 1. The solutions qlm shown as a function of qx for the dilatational phonon in aGaAs quantum well.

H.V. Phuc et al. / Journal of Physics and Chemistry of Solids 75 (2014) 300–305 301

4= ðLzq2l;mq4t;mÞ, and Eq. (7) reduces to

jVpj2 ¼ 2ℏξ2

ρV0ωmq2l;m sin 2 qt;mLz

2

� �: ð13Þ

The distribution function f i ¼ f N;1 in Eq. (4) for a nondegenerateelectron gas in the presence of a magnetic field can be written as

f N;1 ¼2πnea2c Lz

βe�EN;1=kBT ; ð14Þ

where ne is the electron concentration, kB the Boltzmann constant,T the temperature, EN;1 ¼ ðNþ1=2ÞℏωcþE0, and

β¼∑Ne�EN;1=kBT : ð15Þ

The transition probability in Eq. (5) contains contributions ofabsorption process of ℓ-photons. In this paper, we restrict our-selves to considering the process of absorbing two photonsðℓ¼ 1;2Þ. Using the above equations and making a straightforwardcalculation of transition probability, we obtain the followingexpression for the absorption power in the extreme quantumlimits for PACR in quantum well via the two-photon absorptionprocess caused by the confined acoustic phonon mode

PcðωÞ ¼ 2F20ffiffiffiɛ

pV20neξ

2a20π2βρLza2c

∑N;N′

e�EN;1=kBT

�∑m

Z 1

0q3?dq?

q2l;mωm

sin 2 qt;mLz2

� �

� sin 2 ql;mLz2

� � jJNN′ðq? Þj2jq2l;mL2z �4π2j

� NmδðZ�1 ÞþðNmþ1ÞδðZþ

1 Þþ a20q

2?

16½NmδðZ�

2 ÞþðNmþ1ÞδðZþ2 Þ�

�; ð16Þ

where Z8ℓ ¼ ðN′�NÞℏωc8ℏωm�ℓℏω, and the subscript c refers to

the confined-phonon mode. We can see that, in the case of one-photon absorption process ðℓ¼ 1Þ, Eq. (16) reduces to thatobtained by Bhat et al. [14], which is given by the second-orderBorn golden rule approximation. Using Eq. (16) and making astraightforward calculation of integral over q? (see Appendix A),we obtain the following expression for the absorption power:

PcðωÞ ¼ 4F20ffiffiffiɛ

pV20neξ

2a20π2βρLza6c

∑N;N′

e�EN;1=kBT∑m

q2l;mωm

� sin 2 qt;mLz2

� �sin 2 ql;mLz

2

� �1

jq2l;mL2z �4π2j� ðNþN′þ1Þ½NmδðZ�

1 ÞþðNmþ1ÞδðZþ1 Þ�

þ a208a2c

KðN;N′;2Þ½NmδðZ�2 ÞþðNmþ1ÞδðZþ

2 Þ��: ð17Þ

In Eq. (17), we have denoted (for detailed discussion of KðN;N′;2Þ,see Appendix B in Ref. [30])

KðN;N′;2Þ ¼ N!N′!

Z 1

0xN′�Nþ2e�x½LN′�N

N ðxÞ�2 dx; ð18Þ

The delta functions in Eq. (17) are replaced by Lorentzian ofinverse relaxation time Γ8

ℓ given from Eq. (A6) in Ref. [38] below

ðΓ8ℓ Þ2 ¼ 64ξ2

πℏρLza2c∑m

q2l;mωm

sin 2 qt;mLz2

� �sin 2 ql;mLz

2

� �

� 1

q2l;mL2z �4π2

ðNmþ1=281=2Þ: ð19Þ

The expression (17) can be compared with that for the absorp-tion power of the bulk acoustic phonon mode in the extremequantum limits, in a quantum well via two-photon absorptionprocess [30]

PbðωÞ ¼ 3F20ffiffiffiɛ

pV20kBTneξ

2a2064π2ℏβL2zρu2

0a6c

∑N;N′

e�EN;1=kBT

� ðNþN′þ1Þδ½ðN′�NÞℏωc�ℏω�þ a20

8a2cKðN;N′;2Þδ½ðN′�NÞℏωc�2ℏω�

�; ð20Þ

where the subscript b refers to the bulk-phonon mode.

3. Numerical results and discussions

To clarify the obtained results we numerically calculate theabsorption power for a specific GaAs quantum well. In Eq. (17),the sum over m is due to contributions by different branches of theconfined-phonon mode. Because the contribution to PACR ofthe modes of higher order is small, we shall take into accountthe three lowest phonon modes. The parameters used in ourcalculations are [12,40–42] ξ¼ 13:5 eV, ρ¼ 5:32 g cm�3, mn ¼0:067me, u0 ¼ 5:378� 105 cm s�1, ne ¼ 1023 m�3, and F0 ¼ 4:0�105 V m�1.

Fig. 4 shows the dependence of absorption power on ω=ωc dueto confined (solid curve) and bulk (dashed curve) acoustic phononmodes at Lz¼10 nm, T¼27 K and B¼10 T, corresponding tocyclotron energy 17.389 meV. In the bulk phonon modes (dashedcurve), the peaks are due to pure cyclotron resonance, whereas theadditional peaks in the confined modes (solid curve) are due toPACR. The resonance peaks caused by confined modes occur atℓω¼ pωc8ωm, and appear as satellite peaks to those caused bythe bulk modes peaks. The PACR transitions describe the fact thatan electron absorbs a photon to make a transition between Landaulevels during the absorption or emission of a confined acousticphonon of energy ℏωm. In Fig. 4, the main peak at ω¼ωc comes

0 2 4 6 8 10

10

20

30

40

qxLz

q tmL z

Fig. 2. The solutions qtm shown as a function of qx for the dilatational phononin a GaAs quantum well.

0 2 4 6 8 10

2

4

6

8

10

qxLz

Fig. 3. Energy dispersion of the confined dilatational acoustic phonon in GaAsquantum well shown as a function of qx at Lz¼10 nm.

H.V. Phuc et al. / Journal of Physics and Chemistry of Solids 75 (2014) 300–305302

from the term ðω�ωcÞ�2 in a20. The appearance of the additionalpeaks in the case of the confined-phonon is due to PACR.

In the one-photon absorption process, the condition for reso-nance transitions can be written as ω¼ pωc8ωm, ðp¼ 1;2;3;…Þ.However, the peaks corresponding to p¼1 are not obviouslyvisible, because their positions virtually coincide with the mainpeak, and their values are much smaller than those of the mainpeak. The peak at ω=ωc ¼ 1:918 is caused by the absorption of aconfined phonon with energy 1.428 meV, and the peak atω=ωc ¼ 2:074 is caused by the emission of a phonon with energy1.289 meV. These two peaks correspond to the transition withp¼2, and appear symmetrically with respect to the peak atω=ωc ¼ 2. Moreover, the first peak is higher than the secondone. This means that the phonon absorption process is moredominant than the emission one. The peak at ω=ωc ¼ 2:919 isdue to the absorption of a phonon with energy 1.417 meV, and theone at ω=ωc ¼ 3:077 is due to the confined phonon with energy1.349 meV. These peaks correspond to p¼3, and are symmetricwith respect to the peak at ω=ωc ¼ 3. This behavior of PACR in theone-photon absorption process is quite in agreement with theresults of Bhat and co-workers [14], showing that the PACR peaksare symmetric through the pure cyclotron resonance peaks.

In the two-photon absorption process, we obtained thesimilar results. In this case, the PACR peaks satisfy the condition2ω¼ pωc8ωm. The peaks at ω=ωc ¼ 0:460 result from theabsorption of a confined phonon with energy 1.374 meV, andthe one at ω=ωc ¼ 0:543 is due to the phonon with energy1.504 meV. These two peaks correspond to the transition withp¼1. The transition corresponding to p¼2 is not obviouslyvisible, because their peak positions coincide with the positionof the main peaks. The peak at ω=ωc ¼ 1:458 is due to theabsorption of a confined phonon with energy 1.459 meV, andthat at ω=ωc ¼ 1:535 is due to the emission of a phonon withenergy 1.230 meV. These two peaks correspond to p¼3. Inaddition, we can see that in all these transitions, the intensityof the peaks at p¼2 and 3 is smaller than that of the first ones,and the processes correspond to the type 1 in Ref. [13].Especially, the two-photon absorption process is strong enoughto be detected in PACR.

In Fig. 5, the absorption power is plotted versus ω=ωc for theconfined phonon modes at different values of the temperature.We can see that the PACR peaks are located at the same positionbut the peak values of absorption power increase with thetemperature. As the temperature increases, the electron-phonon scattering increases, thus, the magnitude of the absorp-tion power increases. Moreover, with the increase in thetemperature, the peaks become less sharp. At the highertemperature, contribution from higher order phonon modesincreases, making peaks indistinct. Therefore, the contributions

of the confined acoustic phonon to PACR can be only detected atlow temperatures.

Fig. 6 shows the dependence of absorption power on themagnetic field at L¼10 nm and T¼27 K. As the magnetic fieldincreases, the cyclotron frequency ωc increases, while the ratioωm=ωc decreases; thus, the distance between PACR peaksdecreases. Moreover, at the high magnetic field PACR peaksbecome sharper. Because at high magnetic field, the Landau levelgaps increase, only the several confined phonon mode, theenergies of which are comparable to the cyclotron energy con-tribute to PACR.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Parb.units

c

Fig. 4. Absorption power due to confined (solid curve) and bulk (dashed curve)acoustic phonon modes are shown as a function of ω=ωc at B¼10 T, Lz¼10 nm, andT¼27 K.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Parb.units

c

Fig. 5. Temperature dependence of absorption power due to the confined phononat B¼10 T, and L¼10 nm. The solid curve is for T¼27 K, the dashed one for 50 K,and the dotted one for 77 K.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Parb.units

c

Fig. 6. Magnetic field dependence of absorption power due to the confined phononat L¼10 nm, and T¼27 K. The dotted curve is for B¼6 T, the dashed one for 10 T,and the solid one for 14 T.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Parb.units

c

Fig. 7. Quantum well width dependence of absorption power due to the confinedphonon at B¼10 T, and T¼27 K. The solid curve is for Lz¼10 nm, the dashed one for8.5 nm, and the dotted one for 7 nm.

H.V. Phuc et al. / Journal of Physics and Chemistry of Solids 75 (2014) 300–305 303

Fig. 7 shows the quantumwell width dependence of absorptionpower on the confined phonon for B¼10 T and T¼27 K. We cansee that with a decrease in well width, the peak intensity ofabsorption power increases, and the distance between PACR peaksbecomes wider. As the well width decreases, the energy of aconfined phonon increases, leading to an increase in the distanceof PACR peaks. Moreover, the more the decrease in well width is,the more distinctly the different branches of the phonon modescontribute. Therefore, the PACR peaks are sharper when the wellwidth decreases.

We have also studied the dependence of PACR-linewidth ontemperature, magnetic field and well width. It is known that thelinewidth is often defined in terms of the full-width at half-maximum (FWHM) of the absorption power spectrum. To findout the dependence of the PACR-linewidth on temperature, wehave to specify two values of photon energy ℏωmin and ℏωmax

corresponding to a half-maximum of the absorption power. Onepair of ðT ;ℏωmax�ℏωminÞ represents one point on the curve of thegraph. Joining these points, we obtain the curve showing thedependence of linewidth on temperature. This method is calledthe profile method [43]. With this method, we obtained thetemperature dependence of the PACR-linewidth as shown inFig. 8.

From Fig. 8 we can see that in contrast to cyclotron resonance,in which the linewidth is independent of temperature at low-temperature range [9], PACR-linewidth strongly depends on thetemperature, and its variation with temperature quite agrees withffiffiffiT

p. As temperature increases, the probability of electron-phonon

scattering increases, therefore the linewidth rises. The PACR-linewidth for the confined-phonon is also found larger than thatfor the bulk-phonon. This means that phonon confinement givesrise to the probability of electron-phonon scattering. Moreover,

the PACR-linewidth in the two-photon absorption process issmaller than that in the one-photon absorption.

Fig. 9 shows the magnetic field dependence of the PACR-linewidth at T¼27 K, and Lz¼10 nm. In contrast to the bulkphonon interaction, in which the linewidth slowly increases withthe magnetic field, in the confined phonon the PACR-linewidthrapidly increases with the magnetic field. Moreover, the variationof PACR-linewidth with the magnetic field quite agrees with

ffiffiffiB

p.

This result is in good accordance with cyclotron resonance line-width [7–9].

Fig. 10 shows the dependence of PACR-linewidth on the wellwidth at B¼10 T, and T¼27 K. From the figure, we can see thatthe PACR-linewidth decreases with the well width for bothconfined and bulk phonons. The result is consistent with thatshown in some previous works [44–46]. Furthermore, the PACR-linewidth in the confined phonon varies faster and has a largervalue than it does in the bulk phonon. Thus, as the well widthdecreases, the phonon confinement becomes more importantand should not be neglected. As the well width becomes larger,the influence of phonon confinement on the PACR-linewidth isinsignificant.

4. Conclusion

In this paper, by considering the multi-photon process, we havestudied PACR in quantum well structures when electrons interactwith the confined acoustic phonon. Results are presented for aspecific GaAs quantum well. In the extreme quantum limits, onlythe dilatational mode contributes to PACR. The PACR peaks causedby the confined-modes occur at ℓω¼ pωc8ωm, and appear assatellite peaks to those caused by the bulk-mode peaks. The two-photon absorption process is strong enough to be detected inPACR. The contributions of confined acoustic phonon to PACR canbe only detected at low temperatures. In the high magnetic field,PACR peaks become sharper and only a few confined phonon modescontribute to PACR.

Using profile method, we obtained the PACR-linewidth as profile ofthe curves. The obtained results show that PACR-linewidth increaseswith temperature and magnetic field, and decreases with well width.The influence of phonon confinement on the PACR-linewidth is veryimportant at a narrow quantum well width. In addition, the values ofthe PACR-linewidth in the confined phonon is greater than andasymptotic to that in the bulk phonon when well width increases.Besides, the probability of the two-photon absorption process isalways smaller than that of the one-photon absorption. Our resultsinclude those obtained in the other works related to the one-photonabsorption process. Especially, our results are much more useful todeal with multi-photon process problems. Finally, it would be

20 40 60 80 100

0.5

1.0

1.5

2.0

T K

PACRLinewidthmeV

Fig. 8. Dependence of the PACR-linewidth on the temperature T at B¼10 T,and Lz¼10 nm. The filled (empty) squares and circles correspond to the confined-and bulk-phonons for one-photon (two-photon) absorption process, respectively.

5 10 15 200.00.20.40.60.81.01.21.4

B T

PACRLinewidthmeV

Fig. 9. Magnetic field dependence of PACR-linewidth at T¼27 K, and Lz¼10 nm.The filled (empty) squares and circles correspond to the confined- and bulk-phonons for one-photon (two-photon) absorption process, respectively.

5 10 15 20

0.5

1.0

1.5

2.0

2.5

Lz nm

PACRLinewidthmeV

Fig. 10. Well width dependence of PACR-linewidth at B¼10 T, and T¼27 K.The filled (empty) squares and circles correspond to the confined- and bulk-phononsfor one-photon (two-photon) absorption process, respectively.

H.V. Phuc et al. / Journal of Physics and Chemistry of Solids 75 (2014) 300–305304

interesting to have experimental data to check the predictions of thepresent calculations.

Acknowledgments

This work was supported by NAFOSTED-Vietnam (Grant no.103.99-2011.56).

Appendix A. Appendix

The integral over q? is calculated as follow for one-photonabsorption process

I1 ¼Z 1

0q3? dq? JNN′ðq? Þj2

��¼

Z 1

0q3? dq?

N!N′!

e�a2c q2? =2 a2c q

2?

2

� �N′�N

LN′�NN

a2c q2?

2

� �� 2

¼ 2a4c

ðNþN′þ1Þ; ðA:1Þ

and for two-photon absorption process

I2 ¼Z 1

0q3? dq? JNN′ðq? Þj2

a20q2?

16

����¼ a20

16

Z 1

0q5? dq?

N!N′!

e�a2c q2? =2 a2c q

2?

2

� �N′�N

LN′�NN

a2c q2?

2

� �� 2

¼ a204a6c

KðN;N′;2Þ: ðA:2Þ

Here we have used (see Eq. (A4) in Ref. [39])Z 1

0e� xxMþ1½LMN ðxÞ�2 dx¼

ðNþMÞ!N!

ð2NþMþ1Þ; ðA:3Þ

with M ¼N′�N.

References

[1] R. Bakanas, Y. Levinson, F. Bass, Phys. Lett. A 28 (1969) 604–605.[2] R. Lassnig, E. Gornik, Solid State Commun. 47 (1983) 959–963.[3] H. Kobori, T. Ohyama, E. Otsuka, Solid State Commun. 64 (1987) 35–39.[4] K. Nagasaka, G. Kido, S. Narita, Phys. Lett. A 45 (1973) 485–486.

[5] T.M. Rynne, H.N. Spector, J. Appl. Phys. 52 (1981) 393–396.[6] Y.J. Cho, S.D. Choi, Phys. Rev. B 49 (1994) 14301–14306.[7] J.Y. Sug, S.G. Jo, J. Kim, J.H. Lee, S.D. Choi, Phys. Rev. B 64 (2001) 235210–9.[8] H. Kobori, T. Ohyama, E. Otsuka, J. Phys. Soc. Jpn 59 (1990) 2141–2163.[9] M.P. Chaubey, C.M. Van Vliet, Phys. Rev. B 34 (1986) 3932–3938.[10] M. Singh, B. Tanatar, Phys. Rev. B 41 (1990) 12781–12786.[11] B. Tanatar, M. Singh, Phys. Rev. B 42 (1990) 3077–3081.[12] J.S. Bhat, S.S. Kubakaddi, B.G. Mulimani, J. Appl. Phys. 70 (1991) 2216–2219.[13] J.S. Bhat, B.G. Mulimani, S.S. Kubakaddi, Phys. Rev. B 49 (1994) 16459–16466.[14] J.S. Bhat, R.A. Nesargi, B.G. Mulimani, Phys. Rev. B 73 (2006) 235351.[15] T.C. Phong, L.T.T. Phuong, H.V. Phuc, Superlattices Microstruct. 52 (2012)

16–23.[16] Re. Betancourt-Riera, Ri. Betancourt-Riera, J.M. Nieto Jalil, R. Riera, Physica B

410 (2013) 126–130.[17] Ka-Di Zhu, Shi-Wei Gu, Solid State Commun. 83 (1992) 491–494.[18] FarhadH.M. Faisal, Theory of Multiphoton Processes, Plenum Press, New York,

1987.[19] G.S. He, L. Yuan, N. Cheng, J.D. Bhawalkar, P.N. Prasad, L.L. Brott, S.J. Clarson,

B.A. Reinhardt, J. Opt. Soc. Am. B 14 (1997) 1079–1087.[20] W. Denk, J.H. Strickler, W.W. Webb, Science 248 (1990) 73–76.[21] Jianyong Tang, Ronald N. Germain, Meng Cui, Proc. Natl. Acad. Sci. USA 109

(2012) 8434–8439.[22] S. Kawata, H.B. Sun, T. Tanaka, K. Takada, Nature 412 (2001) 697–698.[23] S.L. Oliveira, D.S. Correa, L. Misoguti, C.J.L. Constantino, R.F. Aroca, S.C. Zilio,

C.R. Mendonca, Adv. Mater. 17 (2005) 1890–1893.[24] J.D. Bhawalkar, G.S. He, P.N. Prasad, Rep. Prog. Phys. 59 (1996) 1041–1070.[25] G.T. Boyd, Z.H. Yu, Y.R. Shen, Phys. Rev. B 33 (1986) 7923–7936.[26] H.J. Lee, N.L. Kang, J.Y. Sug, andS.D. Choi, Phys. Rev. B 65 (2002) 195113.[27] N.L. Kang, S.D. Choi, J. Phys. A: Math. Theor. 43 (2010) 165203.[28] N.L. Kang, S.D. Choi, AIP Adv. 2 (2012) 012161.[29] N.L. Kang, S.S. Lee, S.D. Choi, AIP Adv. 3 (2013) 072104.[30] H.V. Phuc, L. Dinh, T.C. Phong, Superlattices Microstruct. 59 (2013) 77–86.[31] Parikshit Sharma, V.S. Rangra, Pankaj Sharma, S.C. Katyal, J. Phys. D: Appl.

Phys. 41 (2008) 225307.[32] E.R. Generazio, H. Spector, Phys. Rev. B 20 (1979) 5162–5167.[33] W. Xu, R.A. Lewis, P.M. Koenraad, C.J.G.M. Langerak, J. Phys.: Condens. Matter

16 (2004) 89–102.[34] N. Bannov, V. Mitin, M. Stroscio, Phys. Status Solidi 183 (1994) 131–142.[35] N. Bannov, V. Aristov, V. Mitin, M.A. Stroscio, Phys. Rev. B 51 (1995)

9930–9942.[36] N. Bannov, V. Aristov, V. Mitin, Solid State Commun. 93 (1995) 483–486.[37] W. Xu, L.B. Lin, J. Phys.: Condens. Matter 13 (2001) 10889–10900.[38] M.P. Chaubey, C.M. Van Vliet, Phys. Rev. B 33 (1986) 5617–5622.[39] M. Vasilopoulos, Phys. Rev. B 33 (1986) 8587–8594.[40] Alka Ingale, M.L. Bansal, A.P. Roy, Phys. Rev. B 40 (1989) 12353–12358.[41] D. Rönnow, N.E. Christensen, M. Cardona, Phys. Rev. B 59 (1999) 5575–5580.[42] H. Bruus, K. Flensberg, H. Smith, Phys. Rev. B 48 (1993) 11144–11155.[43] T.C. Phong, H.V. Phuc, Mod. Phys. Lett. B 25 (2011) 1003–1011.[44] H.N. Spector, J. Lee, P. Melman, Phys. Rev. B 34 (1986) 2554–2560.[45] H. Ham, H.N. Spector, Phys. Rev. B 62 (2000) 13599–13603.[46] H. Ham, H.N. Spector, J. Appl. Phys. 90 (2001) 2781–2785.

H.V. Phuc et al. / Journal of Physics and Chemistry of Solids 75 (2014) 300–305 305