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Linear and nonlinear phonon-assisted cyclotron resonances in parabolic quantum well under the applied electric field
Superlattices and Microstructures 71 (2014) 124–133
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Linear and nonlinear phonon-assisted cyclotron resonances in parabolic quantum well under the applied electric field Huynh Vinh Phuc ⇑, Luong Van Tung Department of Physics, Dong Thap University, Dong Thap 93000, Viet Nam
a r t i c l e
i n f o
Article history: Received 30 October 2013 Received in revised form 8 February 2014 Accepted 14 March 2014 Available online 30 March 2014 Keywords: Nonlinear optical transition Phonon-assisted cyclotron resonance Parabolic quantum well Linewidth Electric field
a b s t r a c t In this work, the linear and nonlinear phonon-assisted cyclotron resonance (PACR) in a parabolic quantum well (PQW) under the applied electric field have been theoretically studied. General analytical expressions for the absorption power are obtained by using the perturbation approach. The effect of the electric field on PACR conditions is also indicated. We investigated numerically the dependence of PACR absorption spectrum as well as PACR-linewidth on the temperature, on the external electric and magnetic fields, and on confinement frequency. The results show that the PACR absorption spectrum is affected by the Landau levels, the electric subband levels and the electric field. Furthermore, external electric field results in increasing the possibility of the electron– phonon scattering, and plays an important role in the blue shift of PACR absorption spectrum. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The study of the nonlinear optical absorption is important for understanding in detail the transient response of semiconductors stimulated by an electromagnetic field. Therefore, the linear and nonlinear optical absorption in low-dimensional semiconductor structures have been studied by many researchers in recent years [1–11]. The effects of applied electric and magnetic fields on the linear ⇑ Corresponding author. Tel.: +84 673882919. E-mail address: [email protected] (H.V. Phuc). http://dx.doi.org/10.1016/j.spmi.2014.03.022 0749-6036/Ó 2014 Elsevier Ltd. All rights reserved.
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and nonlinear optical absorption coefficients have been theoretically investigated in a large number of papers, such as: GaAs-Ga1x Alx As asymmetric double quantum wells [1,2], in V-shaped quantum well [3], in double semi-parabolic quantum wells [4,5], in a disk-shaped quantum dot with parabolic potential plus an inverse squared potential [6], in Ga1x Inx Ny As1y =GaAs double quantum wells [7], in a two-dimensional quantum dot [8], in a parabolically confinement wire in the presence of Rashba spin orbit interaction [9], in double semi-graded quantum wells [10], and in modulation-doped quantum wells [11]. The linear and nonlinear optical properties in a spherical nanolayer quantum system subjected to a uniform applied electric field directed with respect to the z-axis have been theoretically investigated within the compact-density matrix formalism and the iterative method by Chen and Xie [12]. In addition, Wu et al. theoretically investigated the polaron effects on the linear and the nonlinear optical absorption coefficients and refractive index changes in cylindrical quantum dots [13]. The linear and nonlinear absorption coefficients as well as the refractive index changes have been studied in a disc shaped quantum dot [14], in asymmetrical Gaussian potential quantum wells [15], and in a parabolic quantum well [16] under the applied electric field. It is well known that the phonon-assisted cyclotron resonance (PACR) is an effect indicating electron transitions between Landau levels due to the absorption of photons accompanied with the absorption or emission of phonons. Since the early theoretically predictions [17] and experimental observations [18], the cyclotron resonance (CR) and PACR have been studied both theoretically and experimentally in bulk semiconductors [19–22], in quantum wells [23–29], and in quantum wires [30–32]. In most of these studies, CR and PACR have been examined in the process of mono-photon absorptions (linear phenomenon). However, the study of PACR via the multi-photon absorption process (nonlinear phenomenon) in PQW, especially under the applied electric field, is still open for further investigation. Unlike the previous studies [29,32], the present work investigates the affects of the magnetic field, the temperature and the confinement frequency on the linear and nonlinear PACR absorption spectrum as well as PACR-linewidth in PQW under the applied electric field when electrons interact with LO-phonon. We also studied the effect of electric field on the PACR condition and on the shift of PACR absorption spectrum peaks as well. The paper is organized as follows: in Section 2, the theoretical framework used in calculations and the analytical results are presented, followed by numerical results and discussions of the results in Section 3. Finally, a brief conclusion is given in Section 4.
2. Model and theory We consider the transport of an electron gas in a quantum well structure, in which a one-dimensional electron gas is confined in a heterostructure by a potential UðzÞ along the z-direction, and electron motions are free in ðxyÞ plane. When a static magnetic field B is applied in the z-direction and a static electric field Ex is applied in the x-direction, in the Landau gauge for the vector potential A ¼ Bxey , the one-electron Hamiltonian, states and eigenvalues read [33,34]
ðp þ eAÞ2 þ UðzÞ þ eEx x; 2m 1 jN; n; ky i ¼ pffiffiffiffiffi expðiky yÞ/N ðx x0 Þwn ðzÞ; Ly 1 e2 E2x hxc þ en hV d ky þ ; EN;n;ky ¼ N þ 2 2m x2c
h0 ¼
ð1Þ ð2Þ N ¼ 0; 1; 2; . . . ;
ð3Þ
where p is the momentum operator of a conduction electron, m is the effective mass of a conduction electron with electron charge e ðe > 0Þ; N is the Landau level index and n denotes level quantization in z-direction, xc ¼ eB=m is the cyclotron frequency, and V d ¼ Ex =B is the drift velocity. Also, /N ðx x0 Þ represents the harmonic oscillator wave functions, centered at x0 ¼ a2c ðky þ eEx = hxc Þ. Here ky and Ly are the electron wave vector and normalization length in the y-direction, respectively. The 1=2 radius of the orbit in the ðx; yÞ plane is ac ¼ ð h=m xc Þ .
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For a parabolic quantum well (PQW) given by UðzÞ ¼ m x2z z2 =2, with the frequency xz of the confinement potential, the one-electron normalized eigenfunctions and the corresponding eigenvalues in the conduction band are, respectively, given seen in [35]
1=2 1 z2 z ; wn ðzÞ ¼ exp 2 Hn n pffiffiffiffi az 2az 2 n! paz 1 en ¼ n þ hxz ; n ¼ 0; 1; 2; . . . ; 2
ð4Þ ð5Þ 1=2
where Hn ðxÞ is the n-th Hermite polynomial and az ¼ ð h=m xz Þ . The method used in the present calculations is based on the perturbative theory and it has already been given elsewhere [29,32,36–39]; and we will not go into details here. The matrix elements for the electron-LO-phonon interaction in PQW in the presence of magnetic and electric field can be written as [27]
jhf jHep jiij2 ¼ jVðqÞj2 jJ nn0 ðqz Þj2 jJ NN0 ðq? Þj2 ðN0 þ 1=2 1=2Þdk0y ;ky qy ;
ð6Þ
where N 0 is the distribution function of LO-phonon for frequency xq ¼ x0 , with q ¼ ðqz ; q? Þ is the phonon wave vector, and
2pe2 hx0 1 1 1 ; e0 V 0 v1 v0 q2? h 0 i2 n! 2 2 n0 n Lnn n ða2z q2z =2Þ ; n 6 n0 ; jJ nn0 ðqz Þj2 ¼ 0 eaz qz =2 ða2z q2z =2Þ n! h 0 i2 N! 2 2 N 0 N jJ NN0 ðq? Þj2 ¼ 0 eac q? =2 ða2c q2? =2Þ LNN N ða2c q2? =2Þ ; N 6 N 0 ; N!
jVðqÞj2 ¼
where
ð7Þ ð8Þ ð9Þ
e0 is the permittivity of free space, v1 and v0 are the high and low frequency dielectric con-
stants, respectively, V 0 is the volume of the systems, and LM N ðxÞ is the associated Laguerre polynomials. In this paper, we examine the process of absorbing two photons. The expression of the transition probability in PQW can be written as W i ¼
Z Z þ1 V 0 a20 e2 x0 1 1 X 1 q dq dqz jJ nn0 ðqz Þj2 jJ NN0 ðq? Þj2 8pa2c e0 Lz v1 v0 N0 ;n0 0 ? ? 1 N0 dðphxc þ Dnn0 hV d qy hx0 hXÞ þ ðN0 þ 1Þdðphxc þ Dnn0 þ hV d qy þ hx0 hXÞ a 2 q2 þ 0 ? N 0 dðphxc þ Dnn0 hV d qy hx0 2hXÞ þ ðN 0 þ 1Þdðphxc þ Dnn0 þ hV d qy þ hx0 2hXÞ ; 16 ð10Þ
0
2
here p ¼ N N is an integer, a0 ¼ ðeF 0 Þ=½m ðX x2c Þ is the laser dressing parameter, Lz is the z-directional normalization lengths, and Dnn0 ¼ en0 en is the subband separation energy in quantum well. The distribution function fi ¼ fN;n;ky for a nondegenerate electron gas in the presence of an electric and magnetic field can be written below
fN;n;ky ¼
X phEx n e V 0 EN;n;ky =kB T e ; c¼ eEN;n =kB T ; hEx Lx kB TLz Cosh 2a2 Bk T c N;n B c
ð11Þ
where EN;n ¼ ðN þ 1=2Þ hxc þ en þ e2 E2x =ð2m x2c Þ; ne is the electron concentration, kB is the Boltzmann constant, and T is the temperature of the systems. For simplicity in performing the integral over q? , we replace qy in Eq. (10) by eBD x= h, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 0 Dx ¼ N þ 1=2 þ N þ 3=2 ac =2, which was presented by Vasilopoulos et al. [34]. Using Eqs. (10) and (11), and making a straightforward calculation of integral over ky ; qz and q? , we obtain the following expression of the absorption power for PACR in PQW
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XX PðXÞ ¼ AðX; xc Þ eEN;n =kB T jInn0 jf½N 0 dðphxc þ Dnn0 eEx Dx hx0 hXÞ N;n N0 ;n0
a20 ½N0 dðphxc 8a2c x0 2hXÞg: þ eEx Dx þ h
x0 hXÞ þ ðN þ N0 þ 1Þ þðN0 þ 1Þdðphxc þ Dnn0 þ eEx Dx þ h þDnn0 eEx Dx hx0 2hXÞ þ ðN0 þ 1Þdðphxc þ Dnn0
ð12Þ
In Eq. (12), the overlap integral Inn0 is defined as below
Inn0 ¼
1 n! pffiffiffi az 2 n0 !
Z 0
1
h 0 i2 0 euz uzn n1=2 Lnn n ðuz Þ duz ;
where we have set uz ¼ a2z q2z =2. The lowest orders of Inn0 are calculated as I00 ¼ pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi ¼ p=ð2az 2Þ; I02 ¼ 3 p=ð8az 2Þ; I11 ¼ 3 p=ð4az 2Þ; I12 ¼ 7 p=ð16az 2Þ, and
AðX; xc Þ ¼
F 20
ð13Þ pffiffiffi pffiffiffiffi p=ðaz 2Þ; I01
pffiffiffi ene V 20 a20 e2 x0 1 1 : 4 64p2 e0 cac Lz v1 v0
The delta functions in Eq. (12) are replaced by Lorentzians of width C, which is given as follows
C2 ¼
jInn0 je2 x0 2phe0 ðN0 NÞ
1
v1
1
v0
ðN0 þ 1=2 1=2Þ; N < N0 :
ð14Þ
The present result yields a more specific and significant interpretation of the electronic processes for emission and absorption of phonons and photons. The delta functions in Eq. (12) present the energy conservation law. This implies that when the electron undergoes a collision by absorbing energy from the electromagnetic field, its energy can only change by an amount equal to the energy of a phonon involved in the transitions. As a result, the PACR is affected by the Landau levels, the electric subband levels and electric field. The energy-conservation delta functions in Eq. (12) show resonant behavior at the PARC condition for PQW from the selection rule of transition condition as follows
‘hX ¼ phxc þ Dnn0 eEx Dx hx0 :
ð15Þ
In the absence of electric field ðEx ¼ 0Þ and without transition between the levels en ; ðDnn0 ¼ 0Þ, this condition is reduced to ‘ hX ¼ p hxc hx0 . This is the pure PACR condition, which only takes place by the magnetic field [29,32]. Furthermore, we see that the transition condition in Eq. (15) also shows the other resonant behaviors, such as: the optically detected electrophonon resonance (ODEPR) [40,41], due to the electric subband in the z-direction at the conditions hX ¼ Dnn0 hx0 ð‘ ¼ 1; p ¼ 0Þ, and also the optically detected magnetophonon resonance (ODMPR) [35], at the condition p hxc ¼ hx0 þ Dnn0 hX. 3. Results and discussion We have theoretically investigated the linear and nonlinear PACR in PQW under an applied magnetic field. In this study, for numerical calculations, we have taken 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 m3 , and F 0 ¼ 4:5 105 V=m [40,42]. In Fig. 1, the absorption power is plotted versus the photon energy at B ¼ 25 T, corresponding to the cyclotron energy hxc ¼ 43:47 meV. The main peak occurs at hX ¼ 43:47 meV describing the cyclotron resonance. The anothers peaks are due to PACR, which represents resonance transfer of electrons between Landau levels and electric subbands with absorption of photons accompanied by the absorption/emission of phonons. For a magnetic field of B ¼ 25 T, the transitions correspond to type 1 in Ref. [28], as the cyclotron energy is larger than that of the LO-phonon. By using the computational method, we easily determine that the peaks at hX ¼ 90:70 meV, 143.17 meV and 186.64 meV satisfy the condition xþ hX ¼ p hxc þ Dnn0 þ eEx D hx0 , with p = 1, 2 and 3, respectively. These three peaks correspond to the one-photon process (linear phenomenon). Moreover, the intensities of the peaks at p ¼ 2 and 3 are about 19% and 6%, respectively, of that at p ¼ 1. In the case of two-photon absorption process (nonlinear phenomenon), the condition for resonance
H.V. Phuc, L. Van Tung / Superlattices and Microstructures 71 (2014) 124–133
P arb. units
128
0
50
100
150
200
Photon Energy meV
P (arb. units)
Fig. 1. Absorption power as a function of the photon energy in a PQW with a confinement frequency xz =x0 ¼ 0:4 at B ¼ 25 T, T ¼ 77 K, and Ex ¼ 4:5 kV=cm.
0
50
100
150
200
Photon Energy (meV) Fig. 2. Absorption power is shown as a function of photon energy for different values of confinement frequency. The solid, dashed, and dotted curves correspond to xz =x0 ¼ 0:4; 0:6; and 0:8. Here B ¼ 25 T, T ¼ 77 K, and Ex ¼ 4:5 kV=cm.
x0 ; ðp ¼ 1; 2; 3Þ. However, the two peaks transitions can be written as 2 xþh hX ¼ p hxc þ Dnn0 þ eEx D corresponding to p ¼ 1 and p ¼ 3 are not obviously visible due to the overlap of peak positions. In detail, p ¼ 1 peak is coincided with the main peak and its intensity is quite smaller than that of the main one; while p ¼ 3 peak is nearly coincided with another one whose position is at 99.70 meV, as described above. The peak at hX ¼ 71:58 meV is caused by nonlinear phenomenon with p ¼ 2, which represents resonance transfer of electrons between Landau levels and electric subbands with absorption of two-photons accompanied by the emission of an LO-phonon. Two peaks at hX ¼ 8:13 meV and 16.25 meV satisfy the condition ‘ x hX ¼ p hxc þ Dnn0 eEx D hx0 with ‘ ¼ 2 (nonlinear-) and 1 (linear- phenomenon), respectively. These two peaks describe the absorption phonon process. Fig. 2 shows the dependence of PACR absorption spectrum on the confinement frequency. From the figure, it can be seen that the PACR absorption spectrum is related to the confinement frequency. When the confinement frequency rises, the PACR absorption spectrum has been reduced in magnitude and also shifted towards higher energies (blue shift). The blue shift is caused by the larger subband separation energy, Dnn0 , for a higher confinement frequency. The additional peaks in the curves for
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0
50
100
150
200
Photon Energy meV
P arb. units
Fig. 3. Absorption power is shown as a function of photon energy for different values of electric field. The solid, dashed, and dotted curves correspond to Ex ¼ 4:5; 6:5, and 8:5 kV=cm. Here B ¼ 25 T, xz =x0 ¼ 0:4, and T ¼ 77 K.
0
50
100
150
200
Photon Energy meV Fig. 4. Absorption power is shown as a function of photon energy for different values of temperature. The solid, dashed, and dotted curves correspond to T ¼ 77 K, 150 K, and 200 K. Here B ¼ 25 T, xz =x0 ¼ 0:4, and Ex ¼ 4:5 kV=cm.
xz =x0 ¼ 0:6 (dashed) and 0.8 (dotted) are separated from the main peak when confinement frequency increases. To show clearly the electric field effect on the linear and nonlinear PACR in PQW, in Fig. 3 we plot the variation of the absorption power as a function of the photon energy for different values of the electric field. From the figure, when the electric field is increased, the PACR absorption spectrum of emission phonon process shows blue shift, whereas this spectrum of absorption phonon process shows the red one. The reason for this resonance shift is the increase in the potential energy, eEx D x, for a stronger electric field. Fig. 4 shows the dependence of PACR absorption spectrum on the temperature. As in our previous studies [29,32], in PQW the PACR peaks are located at the same position but the peak values of absorption power increase with the temperature. This can be explained from the increase in the possibility of the electron–phonon scattering in higher temperature. Moreover, with the increase in the temperature, the peaks become less sharp. At the higher temperature, the linewidth of PACR peaks are more larger (see Fig. 8), making the peaks indistinct.
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P arb. units
130
0
50
100
150
200
Photon Energy meV Fig. 5. Absorption power is shown as a function of photon energy for different values of magnetic field. The solid, dashed, and dotted curves correspond to B ¼ 25 T, 27 T, and 29 T. Here T ¼ 77 K, xz =x0 ¼ 0:4, and Ex ¼ 4:5 kV=cm.
PACR Linewidth meV
12
10
8
6
4 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Fig. 6. Dependence of the PACR-linewidth on the confinement frequency at B ¼ 25 T, T ¼ 77 K, and Ex ¼ 4:5 kV=cm. The filled and empty diamonds correspond to linear and nonlinear absorption process, respectively.
Fig. 5 shows the dependence of PACR absorption spectrum on the magnetic field. As seen in Fig. 5, we notice a blue shift when the magnetic field increases. These results are in agreement with the results obtained for previous studies [29,32]. As the magnetic field increases, the Landau level separation increases, leading to the increase of the value of absorbed photon energy. The additional peaks in the curves for B ¼ 27 T (dashed) and 29 T (dotted) are separated from the main peak when magnetic field increases. For the PACR-linewidth, using profile method [43] we obtain the confinement frequency dependence of the PACR-linewidth, as shown in Fig. 6. The figure shows the dependence of PACR-linewidth on the confinement frequency. From the figure, we can see that the PACR-linewidth increases with the confinement frequency for both linear- and nonlinear absorption process. These features can be understood easily as follows: when the confinement frequency increases, the confinement strength, az , becomes smaller, so the possibility of electron–phonon scattering increases, and thus so does the linewidth. Furthermore, the PACR-linewidth in the linear phenomenon varies faster and has a larger value than it does in comparison to the nonlinear process. The linewidth in the nonlinear process is from 54% at xz =x0 ¼ 0:2 down to 33% at xz =x0 ¼ 1:6 of that in the linear absorption process. So, the
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131
PACR Linewidth meV
15
10
5
0 2
4
6
8
10
Electric Field kV cm Fig. 7. Dependence of the PACR-linewidth on the electric field at B ¼ 25 T, T ¼ 77 K, and xz =x0 ¼ 0:4. The filled and empty rectangles correspond to linear and nonlinear absorption process, respectively.
PACR Linewidth meV
14 12 10 8 6 4 50
100
150
200
250
300
350
400
Temperature K Fig. 8. Dependence of the PACR-linewidth on the temperature at B ¼ 25 T, xz =x0 ¼ 0:4, and Ex ¼ 4:5 kV=cm. The filled and empty circles correspond to the linear and nonlinear absorption process, respectively.
nonlinear absorption process cannot be neglected in studying PACR-linewidth in PQW under the applied electric field. Fig. 7 shows the dependence of PACR-linewidth on the electric field. From the figure, we can see that the PACR-linewidth increases with increasing electric field for both linear- and nonlinear absorption process. This means that electric field gives rise to the possibility of the electron–phonon scattering. Fig. 8 shows the temperature dependence of the PACR-linewidth. From the figure, we can see that the PACR-linewidth varies with the square root of temperature. This result is in good agreement with the result of the paper by Chaubey and Van Vliet [24]. Physically, this is reasonable because the possibility of electron–phonon scattering increases as the temperature rises. From Fig. 9 we can see that the PACR-linewidth increases with the magnetic field B. This result is in good agreement with the results of previous papers [21,22,24,29,30,32]. This can be explained that as B increases, the cyclotron radius ac reduces, the confinement of electron increases, the probability of electron–phonon scattering increases, so does the linewidth. Beside, we can see from the figure that PACR-linewidth varies with the square root of magnetic field, agreeing with previous results [24,29,32]. In addition, the value of linewidth in the nonlinear absorption process is about 45% of that
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PACR Linewidth meV
7 6 5 4 3 2 1 5
10
15
20
25
30
Magnetic field T Fig. 9. Dependence of the PACR-linewidth on the magnetic field at T ¼ 77 K, xz =x0 ¼ 0:4, and Ex ¼ 4:5 kV=cm. The filled and empty squares correspond to the linear and nonlinear absorption process, respectively.
in the linear absorption process. So, the nonlinear absorption process is strong enough to be detected, and cannot be neglected in studying the PACR-linewidth in PQW under the applied electric field. 4. Conclusion In the present work, we have theoretically studied the linear and nonlinear PACR in PQW under applied electric field when electrons interact with the LO-phonon. The absorption spectrum satisfies the condition ‘ x hX ¼ p hxc þ Dnn0 eEx D hx0 , (‘ ¼ 1; 2; p ¼ 1; 2; 3), and can include the other resonant behaviors such as ODEPR or ODMPR. The results show that the PACR behavior is affected by the magnetic field, the electric field, the temperature and confinement frequency. The electric field plays an important role leading to a blue shift of the PACR absorption spectrum, and it also gives rise to the possibility of the electron–phonon scattering. Using the profile method, we obtained the PACR-linewidth as profile of the curves. The values of the PACR-linewidth are found to increase with the increasing electric and magnetic fields, temperature and confinement frequency. These results are in agreement with the results obtained by previous studies. Although being smaller than the linear absorption process, the nonlinear process is strong enough to be detected in PACR, and cannot be neglected in studying the PACR-linewidth in PQW under applied electric field. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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Linear and nonlinear phonon-assisted cyclotron resonances in parabolic quantum well under the applied electric field Huynh Vinh Phuc ⇑, Luong Van Tung Department of Physics, Dong Thap University, Dong Thap 93000, Viet Nam
a r t i c l e
i n f o
Article history: Received 30 October 2013 Received in revised form 8 February 2014 Accepted 14 March 2014 Available online 30 March 2014 Keywords: Nonlinear optical transition Phonon-assisted cyclotron resonance Parabolic quantum well Linewidth Electric field
a b s t r a c t In this work, the linear and nonlinear phonon-assisted cyclotron resonance (PACR) in a parabolic quantum well (PQW) under the applied electric field have been theoretically studied. General analytical expressions for the absorption power are obtained by using the perturbation approach. The effect of the electric field on PACR conditions is also indicated. We investigated numerically the dependence of PACR absorption spectrum as well as PACR-linewidth on the temperature, on the external electric and magnetic fields, and on confinement frequency. The results show that the PACR absorption spectrum is affected by the Landau levels, the electric subband levels and the electric field. Furthermore, external electric field results in increasing the possibility of the electron– phonon scattering, and plays an important role in the blue shift of PACR absorption spectrum. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The study of the nonlinear optical absorption is important for understanding in detail the transient response of semiconductors stimulated by an electromagnetic field. Therefore, the linear and nonlinear optical absorption in low-dimensional semiconductor structures have been studied by many researchers in recent years [1–11]. The effects of applied electric and magnetic fields on the linear ⇑ Corresponding author. Tel.: +84 673882919. E-mail address: [email protected] (H.V. Phuc). http://dx.doi.org/10.1016/j.spmi.2014.03.022 0749-6036/Ó 2014 Elsevier Ltd. All rights reserved.
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and nonlinear optical absorption coefficients have been theoretically investigated in a large number of papers, such as: GaAs-Ga1x Alx As asymmetric double quantum wells [1,2], in V-shaped quantum well [3], in double semi-parabolic quantum wells [4,5], in a disk-shaped quantum dot with parabolic potential plus an inverse squared potential [6], in Ga1x Inx Ny As1y =GaAs double quantum wells [7], in a two-dimensional quantum dot [8], in a parabolically confinement wire in the presence of Rashba spin orbit interaction [9], in double semi-graded quantum wells [10], and in modulation-doped quantum wells [11]. The linear and nonlinear optical properties in a spherical nanolayer quantum system subjected to a uniform applied electric field directed with respect to the z-axis have been theoretically investigated within the compact-density matrix formalism and the iterative method by Chen and Xie [12]. In addition, Wu et al. theoretically investigated the polaron effects on the linear and the nonlinear optical absorption coefficients and refractive index changes in cylindrical quantum dots [13]. The linear and nonlinear absorption coefficients as well as the refractive index changes have been studied in a disc shaped quantum dot [14], in asymmetrical Gaussian potential quantum wells [15], and in a parabolic quantum well [16] under the applied electric field. It is well known that the phonon-assisted cyclotron resonance (PACR) is an effect indicating electron transitions between Landau levels due to the absorption of photons accompanied with the absorption or emission of phonons. Since the early theoretically predictions [17] and experimental observations [18], the cyclotron resonance (CR) and PACR have been studied both theoretically and experimentally in bulk semiconductors [19–22], in quantum wells [23–29], and in quantum wires [30–32]. In most of these studies, CR and PACR have been examined in the process of mono-photon absorptions (linear phenomenon). However, the study of PACR via the multi-photon absorption process (nonlinear phenomenon) in PQW, especially under the applied electric field, is still open for further investigation. Unlike the previous studies [29,32], the present work investigates the affects of the magnetic field, the temperature and the confinement frequency on the linear and nonlinear PACR absorption spectrum as well as PACR-linewidth in PQW under the applied electric field when electrons interact with LO-phonon. We also studied the effect of electric field on the PACR condition and on the shift of PACR absorption spectrum peaks as well. The paper is organized as follows: in Section 2, the theoretical framework used in calculations and the analytical results are presented, followed by numerical results and discussions of the results in Section 3. Finally, a brief conclusion is given in Section 4.
2. Model and theory We consider the transport of an electron gas in a quantum well structure, in which a one-dimensional electron gas is confined in a heterostructure by a potential UðzÞ along the z-direction, and electron motions are free in ðxyÞ plane. When a static magnetic field B is applied in the z-direction and a static electric field Ex is applied in the x-direction, in the Landau gauge for the vector potential A ¼ Bxey , the one-electron Hamiltonian, states and eigenvalues read [33,34]
ðp þ eAÞ2 þ UðzÞ þ eEx x; 2m 1 jN; n; ky i ¼ pffiffiffiffiffi expðiky yÞ/N ðx x0 Þwn ðzÞ; Ly 1 e2 E2x hxc þ en hV d ky þ ; EN;n;ky ¼ N þ 2 2m x2c
h0 ¼
ð1Þ ð2Þ N ¼ 0; 1; 2; . . . ;
ð3Þ
where p is the momentum operator of a conduction electron, m is the effective mass of a conduction electron with electron charge e ðe > 0Þ; N is the Landau level index and n denotes level quantization in z-direction, xc ¼ eB=m is the cyclotron frequency, and V d ¼ Ex =B is the drift velocity. Also, /N ðx x0 Þ represents the harmonic oscillator wave functions, centered at x0 ¼ a2c ðky þ eEx = hxc Þ. Here ky and Ly are the electron wave vector and normalization length in the y-direction, respectively. The 1=2 radius of the orbit in the ðx; yÞ plane is ac ¼ ð h=m xc Þ .
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For a parabolic quantum well (PQW) given by UðzÞ ¼ m x2z z2 =2, with the frequency xz of the confinement potential, the one-electron normalized eigenfunctions and the corresponding eigenvalues in the conduction band are, respectively, given seen in [35]
1=2 1 z2 z ; wn ðzÞ ¼ exp 2 Hn n pffiffiffiffi az 2az 2 n! paz 1 en ¼ n þ hxz ; n ¼ 0; 1; 2; . . . ; 2
ð4Þ ð5Þ 1=2
where Hn ðxÞ is the n-th Hermite polynomial and az ¼ ð h=m xz Þ . The method used in the present calculations is based on the perturbative theory and it has already been given elsewhere [29,32,36–39]; and we will not go into details here. The matrix elements for the electron-LO-phonon interaction in PQW in the presence of magnetic and electric field can be written as [27]
jhf jHep jiij2 ¼ jVðqÞj2 jJ nn0 ðqz Þj2 jJ NN0 ðq? Þj2 ðN0 þ 1=2 1=2Þdk0y ;ky qy ;
ð6Þ
where N 0 is the distribution function of LO-phonon for frequency xq ¼ x0 , with q ¼ ðqz ; q? Þ is the phonon wave vector, and
2pe2 hx0 1 1 1 ; e0 V 0 v1 v0 q2? h 0 i2 n! 2 2 n0 n Lnn n ða2z q2z =2Þ ; n 6 n0 ; jJ nn0 ðqz Þj2 ¼ 0 eaz qz =2 ða2z q2z =2Þ n! h 0 i2 N! 2 2 N 0 N jJ NN0 ðq? Þj2 ¼ 0 eac q? =2 ða2c q2? =2Þ LNN N ða2c q2? =2Þ ; N 6 N 0 ; N!
jVðqÞj2 ¼
where
ð7Þ ð8Þ ð9Þ
e0 is the permittivity of free space, v1 and v0 are the high and low frequency dielectric con-
stants, respectively, V 0 is the volume of the systems, and LM N ðxÞ is the associated Laguerre polynomials. In this paper, we examine the process of absorbing two photons. The expression of the transition probability in PQW can be written as W i ¼
Z Z þ1 V 0 a20 e2 x0 1 1 X 1 q dq dqz jJ nn0 ðqz Þj2 jJ NN0 ðq? Þj2 8pa2c e0 Lz v1 v0 N0 ;n0 0 ? ? 1 N0 dðphxc þ Dnn0 hV d qy hx0 hXÞ þ ðN0 þ 1Þdðphxc þ Dnn0 þ hV d qy þ hx0 hXÞ a 2 q2 þ 0 ? N 0 dðphxc þ Dnn0 hV d qy hx0 2hXÞ þ ðN 0 þ 1Þdðphxc þ Dnn0 þ hV d qy þ hx0 2hXÞ ; 16 ð10Þ
0
2
here p ¼ N N is an integer, a0 ¼ ðeF 0 Þ=½m ðX x2c Þ is the laser dressing parameter, Lz is the z-directional normalization lengths, and Dnn0 ¼ en0 en is the subband separation energy in quantum well. The distribution function fi ¼ fN;n;ky for a nondegenerate electron gas in the presence of an electric and magnetic field can be written below
fN;n;ky ¼
X phEx n e V 0 EN;n;ky =kB T e ; c¼ eEN;n =kB T ; hEx Lx kB TLz Cosh 2a2 Bk T c N;n B c
ð11Þ
where EN;n ¼ ðN þ 1=2Þ hxc þ en þ e2 E2x =ð2m x2c Þ; ne is the electron concentration, kB is the Boltzmann constant, and T is the temperature of the systems. For simplicity in performing the integral over q? , we replace qy in Eq. (10) by eBD x= h, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 0 Dx ¼ N þ 1=2 þ N þ 3=2 ac =2, which was presented by Vasilopoulos et al. [34]. Using Eqs. (10) and (11), and making a straightforward calculation of integral over ky ; qz and q? , we obtain the following expression of the absorption power for PACR in PQW
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XX PðXÞ ¼ AðX; xc Þ eEN;n =kB T jInn0 jf½N 0 dðphxc þ Dnn0 eEx Dx hx0 hXÞ N;n N0 ;n0
a20 ½N0 dðphxc 8a2c x0 2hXÞg: þ eEx Dx þ h
x0 hXÞ þ ðN þ N0 þ 1Þ þðN0 þ 1Þdðphxc þ Dnn0 þ eEx Dx þ h þDnn0 eEx Dx hx0 2hXÞ þ ðN0 þ 1Þdðphxc þ Dnn0
ð12Þ
In Eq. (12), the overlap integral Inn0 is defined as below
Inn0 ¼
1 n! pffiffiffi az 2 n0 !
Z 0
1
h 0 i2 0 euz uzn n1=2 Lnn n ðuz Þ duz ;
where we have set uz ¼ a2z q2z =2. The lowest orders of Inn0 are calculated as I00 ¼ pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi ¼ p=ð2az 2Þ; I02 ¼ 3 p=ð8az 2Þ; I11 ¼ 3 p=ð4az 2Þ; I12 ¼ 7 p=ð16az 2Þ, and
AðX; xc Þ ¼
F 20
ð13Þ pffiffiffi pffiffiffiffi p=ðaz 2Þ; I01
pffiffiffi ene V 20 a20 e2 x0 1 1 : 4 64p2 e0 cac Lz v1 v0
The delta functions in Eq. (12) are replaced by Lorentzians of width C, which is given as follows
C2 ¼
jInn0 je2 x0 2phe0 ðN0 NÞ
1
v1
1
v0
ðN0 þ 1=2 1=2Þ; N < N0 :
ð14Þ
The present result yields a more specific and significant interpretation of the electronic processes for emission and absorption of phonons and photons. The delta functions in Eq. (12) present the energy conservation law. This implies that when the electron undergoes a collision by absorbing energy from the electromagnetic field, its energy can only change by an amount equal to the energy of a phonon involved in the transitions. As a result, the PACR is affected by the Landau levels, the electric subband levels and electric field. The energy-conservation delta functions in Eq. (12) show resonant behavior at the PARC condition for PQW from the selection rule of transition condition as follows
‘hX ¼ phxc þ Dnn0 eEx Dx hx0 :
ð15Þ
In the absence of electric field ðEx ¼ 0Þ and without transition between the levels en ; ðDnn0 ¼ 0Þ, this condition is reduced to ‘ hX ¼ p hxc hx0 . This is the pure PACR condition, which only takes place by the magnetic field [29,32]. Furthermore, we see that the transition condition in Eq. (15) also shows the other resonant behaviors, such as: the optically detected electrophonon resonance (ODEPR) [40,41], due to the electric subband in the z-direction at the conditions hX ¼ Dnn0 hx0 ð‘ ¼ 1; p ¼ 0Þ, and also the optically detected magnetophonon resonance (ODMPR) [35], at the condition p hxc ¼ hx0 þ Dnn0 hX. 3. Results and discussion We have theoretically investigated the linear and nonlinear PACR in PQW under an applied magnetic field. In this study, for numerical calculations, we have taken 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 m3 , and F 0 ¼ 4:5 105 V=m [40,42]. In Fig. 1, the absorption power is plotted versus the photon energy at B ¼ 25 T, corresponding to the cyclotron energy hxc ¼ 43:47 meV. The main peak occurs at hX ¼ 43:47 meV describing the cyclotron resonance. The anothers peaks are due to PACR, which represents resonance transfer of electrons between Landau levels and electric subbands with absorption of photons accompanied by the absorption/emission of phonons. For a magnetic field of B ¼ 25 T, the transitions correspond to type 1 in Ref. [28], as the cyclotron energy is larger than that of the LO-phonon. By using the computational method, we easily determine that the peaks at hX ¼ 90:70 meV, 143.17 meV and 186.64 meV satisfy the condition xþ hX ¼ p hxc þ Dnn0 þ eEx D hx0 , with p = 1, 2 and 3, respectively. These three peaks correspond to the one-photon process (linear phenomenon). Moreover, the intensities of the peaks at p ¼ 2 and 3 are about 19% and 6%, respectively, of that at p ¼ 1. In the case of two-photon absorption process (nonlinear phenomenon), the condition for resonance
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P arb. units
128
0
50
100
150
200
Photon Energy meV
P (arb. units)
Fig. 1. Absorption power as a function of the photon energy in a PQW with a confinement frequency xz =x0 ¼ 0:4 at B ¼ 25 T, T ¼ 77 K, and Ex ¼ 4:5 kV=cm.
0
50
100
150
200
Photon Energy (meV) Fig. 2. Absorption power is shown as a function of photon energy for different values of confinement frequency. The solid, dashed, and dotted curves correspond to xz =x0 ¼ 0:4; 0:6; and 0:8. Here B ¼ 25 T, T ¼ 77 K, and Ex ¼ 4:5 kV=cm.
x0 ; ðp ¼ 1; 2; 3Þ. However, the two peaks transitions can be written as 2 xþh hX ¼ p hxc þ Dnn0 þ eEx D corresponding to p ¼ 1 and p ¼ 3 are not obviously visible due to the overlap of peak positions. In detail, p ¼ 1 peak is coincided with the main peak and its intensity is quite smaller than that of the main one; while p ¼ 3 peak is nearly coincided with another one whose position is at 99.70 meV, as described above. The peak at hX ¼ 71:58 meV is caused by nonlinear phenomenon with p ¼ 2, which represents resonance transfer of electrons between Landau levels and electric subbands with absorption of two-photons accompanied by the emission of an LO-phonon. Two peaks at hX ¼ 8:13 meV and 16.25 meV satisfy the condition ‘ x hX ¼ p hxc þ Dnn0 eEx D hx0 with ‘ ¼ 2 (nonlinear-) and 1 (linear- phenomenon), respectively. These two peaks describe the absorption phonon process. Fig. 2 shows the dependence of PACR absorption spectrum on the confinement frequency. From the figure, it can be seen that the PACR absorption spectrum is related to the confinement frequency. When the confinement frequency rises, the PACR absorption spectrum has been reduced in magnitude and also shifted towards higher energies (blue shift). The blue shift is caused by the larger subband separation energy, Dnn0 , for a higher confinement frequency. The additional peaks in the curves for
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0
50
100
150
200
Photon Energy meV
P arb. units
Fig. 3. Absorption power is shown as a function of photon energy for different values of electric field. The solid, dashed, and dotted curves correspond to Ex ¼ 4:5; 6:5, and 8:5 kV=cm. Here B ¼ 25 T, xz =x0 ¼ 0:4, and T ¼ 77 K.
0
50
100
150
200
Photon Energy meV Fig. 4. Absorption power is shown as a function of photon energy for different values of temperature. The solid, dashed, and dotted curves correspond to T ¼ 77 K, 150 K, and 200 K. Here B ¼ 25 T, xz =x0 ¼ 0:4, and Ex ¼ 4:5 kV=cm.
xz =x0 ¼ 0:6 (dashed) and 0.8 (dotted) are separated from the main peak when confinement frequency increases. To show clearly the electric field effect on the linear and nonlinear PACR in PQW, in Fig. 3 we plot the variation of the absorption power as a function of the photon energy for different values of the electric field. From the figure, when the electric field is increased, the PACR absorption spectrum of emission phonon process shows blue shift, whereas this spectrum of absorption phonon process shows the red one. The reason for this resonance shift is the increase in the potential energy, eEx D x, for a stronger electric field. Fig. 4 shows the dependence of PACR absorption spectrum on the temperature. As in our previous studies [29,32], in PQW the PACR peaks are located at the same position but the peak values of absorption power increase with the temperature. This can be explained from the increase in the possibility of the electron–phonon scattering in higher temperature. Moreover, with the increase in the temperature, the peaks become less sharp. At the higher temperature, the linewidth of PACR peaks are more larger (see Fig. 8), making the peaks indistinct.
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P arb. units
130
0
50
100
150
200
Photon Energy meV Fig. 5. Absorption power is shown as a function of photon energy for different values of magnetic field. The solid, dashed, and dotted curves correspond to B ¼ 25 T, 27 T, and 29 T. Here T ¼ 77 K, xz =x0 ¼ 0:4, and Ex ¼ 4:5 kV=cm.
PACR Linewidth meV
12
10
8
6
4 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Fig. 6. Dependence of the PACR-linewidth on the confinement frequency at B ¼ 25 T, T ¼ 77 K, and Ex ¼ 4:5 kV=cm. The filled and empty diamonds correspond to linear and nonlinear absorption process, respectively.
Fig. 5 shows the dependence of PACR absorption spectrum on the magnetic field. As seen in Fig. 5, we notice a blue shift when the magnetic field increases. These results are in agreement with the results obtained for previous studies [29,32]. As the magnetic field increases, the Landau level separation increases, leading to the increase of the value of absorbed photon energy. The additional peaks in the curves for B ¼ 27 T (dashed) and 29 T (dotted) are separated from the main peak when magnetic field increases. For the PACR-linewidth, using profile method [43] we obtain the confinement frequency dependence of the PACR-linewidth, as shown in Fig. 6. The figure shows the dependence of PACR-linewidth on the confinement frequency. From the figure, we can see that the PACR-linewidth increases with the confinement frequency for both linear- and nonlinear absorption process. These features can be understood easily as follows: when the confinement frequency increases, the confinement strength, az , becomes smaller, so the possibility of electron–phonon scattering increases, and thus so does the linewidth. Furthermore, the PACR-linewidth in the linear phenomenon varies faster and has a larger value than it does in comparison to the nonlinear process. The linewidth in the nonlinear process is from 54% at xz =x0 ¼ 0:2 down to 33% at xz =x0 ¼ 1:6 of that in the linear absorption process. So, the
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PACR Linewidth meV
15
10
5
0 2
4
6
8
10
Electric Field kV cm Fig. 7. Dependence of the PACR-linewidth on the electric field at B ¼ 25 T, T ¼ 77 K, and xz =x0 ¼ 0:4. The filled and empty rectangles correspond to linear and nonlinear absorption process, respectively.
PACR Linewidth meV
14 12 10 8 6 4 50
100
150
200
250
300
350
400
Temperature K Fig. 8. Dependence of the PACR-linewidth on the temperature at B ¼ 25 T, xz =x0 ¼ 0:4, and Ex ¼ 4:5 kV=cm. The filled and empty circles correspond to the linear and nonlinear absorption process, respectively.
nonlinear absorption process cannot be neglected in studying PACR-linewidth in PQW under the applied electric field. Fig. 7 shows the dependence of PACR-linewidth on the electric field. From the figure, we can see that the PACR-linewidth increases with increasing electric field for both linear- and nonlinear absorption process. This means that electric field gives rise to the possibility of the electron–phonon scattering. Fig. 8 shows the temperature dependence of the PACR-linewidth. From the figure, we can see that the PACR-linewidth varies with the square root of temperature. This result is in good agreement with the result of the paper by Chaubey and Van Vliet [24]. Physically, this is reasonable because the possibility of electron–phonon scattering increases as the temperature rises. From Fig. 9 we can see that the PACR-linewidth increases with the magnetic field B. This result is in good agreement with the results of previous papers [21,22,24,29,30,32]. This can be explained that as B increases, the cyclotron radius ac reduces, the confinement of electron increases, the probability of electron–phonon scattering increases, so does the linewidth. Beside, we can see from the figure that PACR-linewidth varies with the square root of magnetic field, agreeing with previous results [24,29,32]. In addition, the value of linewidth in the nonlinear absorption process is about 45% of that
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PACR Linewidth meV
7 6 5 4 3 2 1 5
10
15
20
25
30
Magnetic field T Fig. 9. Dependence of the PACR-linewidth on the magnetic field at T ¼ 77 K, xz =x0 ¼ 0:4, and Ex ¼ 4:5 kV=cm. The filled and empty squares correspond to the linear and nonlinear absorption process, respectively.
in the linear absorption process. So, the nonlinear absorption process is strong enough to be detected, and cannot be neglected in studying the PACR-linewidth in PQW under the applied electric field. 4. Conclusion In the present work, we have theoretically studied the linear and nonlinear PACR in PQW under applied electric field when electrons interact with the LO-phonon. The absorption spectrum satisfies the condition ‘ x hX ¼ p hxc þ Dnn0 eEx D hx0 , (‘ ¼ 1; 2; p ¼ 1; 2; 3), and can include the other resonant behaviors such as ODEPR or ODMPR. The results show that the PACR behavior is affected by the magnetic field, the electric field, the temperature and confinement frequency. The electric field plays an important role leading to a blue shift of the PACR absorption spectrum, and it also gives rise to the possibility of the electron–phonon scattering. Using the profile method, we obtained the PACR-linewidth as profile of the curves. The values of the PACR-linewidth are found to increase with the increasing electric and magnetic fields, temperature and confinement frequency. These results are in agreement with the results obtained by previous studies. Although being smaller than the linear absorption process, the nonlinear process is strong enough to be detected in PACR, and cannot be neglected in studying the PACR-linewidth in PQW under applied electric field. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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