Superlattices and Microstructures 60 (2013) 508–515

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Superlattices and Microstructures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . co m / l o c a t e / s u p e r l a t t i c e s

LO-phonon-assisted cyclotron resonancelinewidth via multiphoton absorption processin cylindrical quantum wire

0749-6036/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.spmi.2013.05.038

⇑ Corresponding author. Tel.: +84 67 3882919.E-mail address: hvphuc@dthu.edu.vn (H.V. Phuc).

Huynh Vinh Phuc a,⇑, Le Thi Mai Hue 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 a b s t r a c t

Article history:Received 17 May 2013Accepted 29 May 2013Available online 6 June 2013

Keywords:LO-phonon-assisted cyclotron resonanceMultiphoton absorption processCylindrical quantum wireLinewidth

Phonon-assisted cyclotron resonance (PACR) in cylindrical quan-tum wire (CQW) is investigated via multiphoton absorption pro-cess when electrons are scattered by LO-phonon. The additionalpeaks in the absorption spectrum due to transitions between Lan-dau levels accompanied by emission and absorption of LO-phononare indicated. From curves on graphs of the absorption power, weobtain PACR-linewidth as profiles of the curves by using profilemethod. The temperature and magnetic field dependence of theabsorption power and PACR-linewidth are investigated. The resultsare compared with those in the monophoton absorption. The cal-culations demonstrate that the multiphoton absorption process isstrong enough to be detected in PACR.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Phonon-assisted magneto-optical transitions in semiconductors was first predicted theoretically byBass and Levinson in 1965 [1] and was observed experimentally in 1969 using InSb by Enck et al. [2].In a magnetic field, direct transitions can occur between Landau levels giving cyclotron-resonance(CR). Indirect transitions with the assistance of phonon interaction are called cyclotron-phonon reso-nance or phonon-assisted cyclotron resonance (PACR). The PACR is an effect indicating electron tran-sitions between Landau levels due to the absorption of photons accompanied with the absorption oremission of phonons. In recent years, there has been considerable interest in the study of CR and PACR

H.V. Phuc et al. / Superlattices and Microstructures 60 (2013) 508–515 509

both theoretically and experimentally in bulk semiconductors [2–8], in quantum wells [9–14], insuperlattices [15], and in quantum wires [16,17]. In these works, the main CR peak occurs atx ¼ xc , and PACR peaks occur at x ¼ pxc þxq; ðp ¼ 1;2; . . .Þ, where x;xc and xq are the frequenciesof the electromagnetic wave, cyclotron and phonon, respectively. In all these studies, CR and PACRhave been examined in the process of monophoton absorptions. However, the study of PACR viathe multiple photon absorption processes in CQW has not been found.

In an earlier paper [18], basing on the results of Sharma et al. [19], Generazio and Spector [20], andXu et al. [21,22], we suggested a method for obtaining the explicit expression of the absorption powerin the presence of a magnetic field with the multiphoton absorption process. In the monophotonabsorption process, this result reduces to that in previous papers [13,14], which is given by the sec-ond-order Born golden rule approximation. The purpose of the present paper is to apply our previousresult [18] for studying PACR as well as PACR-linewidth via the multiphoton absorption process inCQW when electrons interact with LO-phonon. The paper is organized as follows. In Section 2, thetheoretical framework used in calculations and the analytical results are presented. The discussionof the results are given in Section 3. The conclusion is included in the last section.

2. Theory

The wave function and energy of electron inside a cylindrical quantum wire of radius R and lengthLz, with infinite potential barrier at the surface, in the presence of a strong magnetic field parallel tothe wire axis, are given by [23]

wðr;u; zÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffi2pLzp eimueikzz/N;mðrÞ; ð1Þ

EN;m ¼�h2k2

z

2m�þ �hxc N þm

2þ 1

2þ jmj

2

� �; ð2Þ

the radial wave function in Eq. (1) is

/N;mðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðN þ jmjÞ!

N!

r1

acjmj!e�n=2njmj=2

1F1ð�N; jmj þ 1; nÞ; ð3Þ

where N ¼ 0;1;2; . . . and m ¼ 0;�1;�2; . . . are the Landau-level index and the azimuthal quantumnumber, respectively; m� is the effective mass of electron; ac ¼ ð�h=m�xcÞ1=2 is the cyclotron radius;1F1ð�N; jmj þ 1; nÞ is the general form of the confluent hypergeometric function, n ¼ r2=ð2a2

c Þ.When an electromagnetic wave characterized by a time-dependence electric field of amplitude F0

and angular frequency x is applied, the absorption power delivered to the system is given by [18]

PðxÞ ¼ F20

ffiffiffiep

8pX

i

Wifi; ð4Þ

where e is the dielectric constant of the medium, fi is the electron distribution function, and Wi is thetransition probability. The sum is taken over all the initial states i of the electrons. The transition prob-ability of absorbing photon with simultaneously absorbing and/or emitting phonon W�

i can be written[18,21,22]

W�i ¼

2p�h

Xf

Xq

jhf jHe�pjiij2Xþ1‘¼�1

1

ð‘!Þ2a0q?

2

� �2‘� dðEf � Ei � �hxq � ‘�hxÞ; ð5Þ

where the upper sign (�) corresponds to the phonon absorption, the lower sign (+) corresponds to thephonon emission. hf jHe�pjii is the transition matrix element for the electron–phonon interaction,a0 ¼ ðeF0Þ=½m�ðx2 �x2

c Þ�; Ei � EN;m and Ef � EN0 ;m0 are the initial- and final-state energies of theelectron, and q ¼ ðqz; q?Þ is the phonon wave vector. The sum is taken over all the final state f ofthe electrons. In Eq. (5), the index ‘ corresponds to the process of ‘-photon.

510 H.V. Phuc et al. / Superlattices and Microstructures 60 (2013) 508–515

The matrix elements for the electron-LO-phonon interaction in CQW in the presence of magneticfield can be written as [22–24]

jhf jHe�pjiij2 ¼Dq2?ðN0 þ 1=2� 1=2ÞjMmnm0n0 ðq?Þj

2jJNN0 ðq?Þj2dk0z ;kz�qz

; ð6Þ

where N0 is the distribution function of LO-phonon for frequency xq ¼ x0, and

D ¼ 2pe2�hx0

e0V0

1v1� 1

v0

� �; ð7Þ

Mmnm0n0 ðq?Þ ¼2R

Z R

0Jjm�m0 jðq?RÞw�m0 ;n0 ðrÞwm;nðrÞrdr; ð8Þ

wm;nðrÞ ¼1

Jmþ1ðAmnÞJm Amn

rR

� �; ð9Þ

jJNN0 ðq?Þj2 ¼ N!

N0!e�a2

c q2?=2ða2

c q2?=2ÞN

0�NLN0�N

N ða2c q2?=2Þ

h i2; N 6 N0; ð10Þ

where e0 is the permittivity of free space, v1 and v0 are the high and low frequency dielectric con-stants, respectively, V0 is the volume of the systems, Amn is the n-th zero of Bessel function of them-order, JmðAmnÞ ¼ 0. The function LM

N ðxÞ is the associated Laguerre polynomials.We consider the transitions between states in the extreme electric quantum limit

ðm ¼ m0 ¼ 0;n ¼ n0 ¼ 1Þ, and making the same approximation as in Ref. [25] (i.e. we take�h2=ð2m�Þðq2

z � 2kzqzÞ ¼ 0) in delta functions. The transition probability in Eq. (5) contains contribu-tions of absorption process of ‘-photons. In this paper, we consider the process of absorbing two pho-tons ð‘ ¼ 1;2Þ. Using Eq. (4) and making a straight forward calculation of probabilities using matrixelement for LO-phonon scattering (Eq. (6)), for nondegenerate electron gas, we obtain the expressionfor the absorption power in a CQW structure

PðxÞ¼Aðx;xcÞXN;N0

e�EN;0=kBT

�Z 1

0q?dq?jM0101ðq?Þj

2jJNN0 ðq?Þj2 N0d Z�1

þðN0þ1Þd Zþ1

þa2

0q2?

16N0d Z�2

þðN0þ1ÞdðZþ2 Þ

� � �;

ð11Þ

where

Aðx;xcÞ ¼F2

0

ffiffiffiep

neV30Da2

0

128p3�hRLzGa2c

; G ¼X

N

e�EN;0=kBT

with ne is the electron concentration, and

Z�‘ ¼ ðN0 � NÞ�hxc � �hx0 � ‘�hx; ð‘ ¼ 1;2Þ: ð12Þ

The expression for PðxÞ can now be integrated over q? (see Appendix A) to obtain

PðxÞ¼Aðx;xcÞ4a2

c

XN;N0

e�EN=kBT N0dðZ�1 ÞþðN0þ1ÞdðZþ1 ÞþðNþN0 þ1Þ a20

8a2c

N0dðZ�2 ÞþðN0þ1ÞdðZþ2 Þ� � �

:

ð13Þ

Following the collision broadening model, we replace the delta functions in Eq. (13) by Lorentzianwith the width C, which is given as follow from Eq. (A.6) of [26]

ðC�Þ2 ¼pV0D

8ð2p�hÞ2ðN0 � NÞLz

ðN0 þ 1=2� 1=2Þ; N < N0: ð14Þ

We have obtained the general expressions for the absorption power when electrons are scattered byLO-phonon. The present result yields a more specific and significant interpretation of the electronicprocess for emission and absorption of LO-phonon.

H.V. Phuc et al. / Superlattices and Microstructures 60 (2013) 508–515 511

3. Results and discussions

To clarify the obtained results we numerically calculate the expression for PACR absorption power(Eq. (13)) for a specific CQW made up GaAs/AlAs. The parameters used in our calculations are [27–29]:e ¼ 12:5;v1 ¼ 10:9;v0 ¼ 13:1;m� ¼ 0:067me;me being the mass of free electron, �hx0 ¼ 36:25 meV,ne ¼ 1023 m�3, and F0 ¼ 4:0� 105 V m�1.

Fig. 1 shows the dependence of the absorption power on x=xc at B ¼ 7 T, corresponding to thecyclotron energy �hxc ¼ 12:17 meV, which lead to the ratio x0=xc ¼ 2:978. The main cyclotron reso-nance peak occurs at x=xc ¼ 1 coming from the term ðx�xcÞ�2 in a2

0 in Eq. (13). The appearance ofthe additional peaks are due to PACR, which represents resonance transfer of electrons with absorp-tion of photons accompanied by the absorption/emission of phonons. Depending upon the Landau le-vel separation, photon, and phonon energies, the PACR transitions have many different types [14]. Fora magnetic field B ¼ 7 T, the cyclotron energy is smaller than the LO-phonon energy, and transitionscorrespond to the type 2 in Ref. [14].

The peaks at x=xc ¼ 3:978, and 4.978 satisfy the condition x ¼ pxc þx0, with p ¼ N0 � N ¼ 1 and2, respectively. Both these peaks correspond the one-photon process. Moreover, the peak value ofabsorption power at x ¼ 2xc þx0 is 38.79% of that at x ¼ xc þx0. Being at x=xc ¼ 1:989 and2.489, the peaks satisfy the condition 2x ¼ pxc þx0, with p = 1 and 2, respectively. These two peakscorrespond the two-photon process. The peak value at 2x ¼ 2xc þx0 is 29.57% of that at2x ¼ xc þx0. Beside, the peak value in the two-photon process is 81.34% of that in the one-photonprocess at p = 1. This rate is 62.00% at p = 2. So, the two-photon absorption process is strong enough tobe detected in PACR. The behavior of the PACR transitions are more explicitly described in Figs. 2 and 3.

0 1 2 3 4 5

Par

b.un

its

Fig. 1. Absorption power is shown as a function of x=xc at B = 7 T, R = 16 nm and T = 77 K.

Fig. 2. PACR transitions in the one-photon absorption process.

Fig. 3. PACR transitions in the two-photon absorption process.

0 1 2 3 4 5

Par

b.un

its

Fig. 4. Absorption power is shown as a function of x=xc for different values of temperature. The solid, dashed, and dottedcurves correspond to T = 77 K, 150 K, and 300 K. Here B = 7 T and R = 16 nm.

512 H.V. Phuc et al. / Superlattices and Microstructures 60 (2013) 508–515

Fig. 2 shows the PACR transition in the one-photon absorption process. In the figures, i, f, and j de-note the initial, final, and intermediate levels; the subscript 1 and 2 correspond to p = 1 and 2, respec-tively. The physical interpretation is clear. For example, the first transition represents the fact that oneelectron in the state i absorbs a photons x1 to jump to state j1, then emits one phonon x0 to jump tostate f1. The other transition in Fig. 2 can be explained in a similar manner. Fig. 3 shows the PACR tran-sition in the two-photon process, in which the peaks occur at 2x ¼ pxc þx0.

Fig. 4 shows the dependence of PACR absorption spectrum on the temperature. We can see that thePACR peaks are located at the same position but the peak values of absorption power increase with thetemperature. As the temperature increases, the electron–phonon scattering increases, leading to ageneral increase in the temperature of absorption power.

Fig. 5 shows the dependence of PACR absorption spectrum on the magnetic field. In the figure,when the magnetic field increases, the peak values of absorption power increase and the PACR peaksare seen to shift to the small region of x=xc . As the magnetic field increases, the cyclotron frequencyincreases, while the ratio x0=xc decreases, leading to the decrease of the value of x=xc , which satis-fying the condition ‘x=xc ¼ pþx0=xc.

We have also studied the dependence of PACR-linewidth on the temperature and magnetic field. Itis known that, linewidth is often defined in terms of the full-width at half-maximum (FWHM) of theoptical field power spectrum. Using profile method [30], we obtain the temperature dependence of thePACR-linewidth, as shown in Fig. 6.

0 1 2 3 4 5

Par

b.un

its

Fig. 5. Absorption power is shown as a function of x=xc for different values of magnetic field. The solid, dashed, and dottedcurves correspond to B = 7 T, 8 T, and 9 T. Here T = 77 K and R = 16 nm.

50 100 150 200 250 300 350 400

1

2

3

4

Temperature K

PAC

RL

inew

idth

meV

Fig. 6. Dependence of the PACR-linewidth on the temperature at B = 7 T, and R = 16 nm. The filled and empty correspond to theone-photon and two-photon absorption process, respectively.

5 10 15 20

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Magnetic field T

PAC

RL

inew

idth

meV

Fig. 7. Dependence of the PACR-linewidth on the magnetic field at T = 77 K, and R = 16 nm. The filled and empty correspond tothe one-photon and two-photon absorption process, respectively.

H.V. Phuc et al. / Superlattices and Microstructures 60 (2013) 508–515 513

The figure shows that PACR-linewidth varies with the temperature quite agrees withffiffiffiTp

. The resultis in good agreement with the result of previous paper [10]. As the temperature increases, probabilityof electron-phonon scattering increases, leading to a general increase in the temperature of PACR-linewidth. In addition, the linewidth in the two-photon absorption process is smaller than that in theone-photon absorption. So, the one-photon process makes dominant contribution to the total process.

514 H.V. Phuc et al. / Superlattices and Microstructures 60 (2013) 508–515

Fig. 7 shows the magnetic field dependence of the PACR-linewidth. From the figure, we can see thatthe PACR-linewidth increases with the magnetic field B. Because the confinement of electron increaseswhen magnetic field increases. This result is in good agreement with the results of previous papers[7,8,10,16]. Beside, we can see from the figure that PACR-linewidth varies with

ffiffiffiBp

, agreeing withprevious result [10].

4. Conclusion

We have studied the PACR in GaAs/AlAs CQW by considering the multiphoton absorption processwhen electrons interact with LO-phonon. Although being smaller than the one-photon absorption pro-cess, the two-photon process is strong enough to be detected in PACR. Our results show that the PACRis affected by the magnetic field and the temperature. Using profile method, we obtain the PACR-line-width as profile of the curves. The values of the PACR-linewidth are found to increase with increasingtemperature and magnetic field. The results are in good agreement with experimental and other the-oretical results. Present working can include the results obtained in the other works in the one-photonabsorption process.

Acknowledgment

This work was supported by the National Foundation for Science and Technology Development ofVietnam - NAFOSTED (Grant No. 103.99-2011.56), and Ministry of Education and Training of Vietnam(Grant No. B2012-20-01).

Appendix A

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

I1 ¼Z 1

0q?jM0101ðq?Þj

2jJNN0 ðq?Þj2dq? ¼

14a2

c; ðA:1Þ

and for two-photon absorption process

I2 ¼a2

0

16

Z 1

0q3?jM0101ðq?Þj

2jJNN0 ðq?Þj2dq? ¼

a20

32a4cðN þ N0 þ 1Þ: ðA:2Þ

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