Optical intersubband transitions in quantum wires with an applied magnetic field

The intersubband optical absorption is investigated in parabolic quantum wires in the presence of a tilted magnetic fields. We show that for increasing magnetic field the intersubband absorption peak is shifted to higher energies and its amplitude is increased, too. In particular, it has been shown...

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Published in:	Semiconductor Physics Quantum Electronics & Optoelectronics
Date:	2004
Main Author:	Ibragimov, G.B.
Format:	Article
Language:	English
Published:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2004
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/119125
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Cite this:	Optical intersubband transitions in quantum wires with an applied magnetic field / G.B. Ibragimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 283-286. — Бібліогр.: 24 назв. — англ.

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spelling	Ibragimov, G.B. 2017-06-04T16:21:17Z 2017-06-04T16:21:17Z 2004 Optical intersubband transitions in quantum wires with an applied magnetic field / G.B. Ibragimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 283-286. — Бібліогр.: 24 назв. — англ. 1560-8034 PACS: 73.21.Hb, 73.21.Nm https://nasplib.isofts.kiev.ua/handle/123456789/119125 The intersubband optical absorption is investigated in parabolic quantum wires in the presence of a tilted magnetic fields. We show that for increasing magnetic field the intersubband absorption peak is shifted to higher energies and its amplitude is increased, too. In particular, it has been shown that the direction of the magnetic field plays a significant role in the intersubband optical absorption. The author would like to thank Prof. М. I. Aliev and Prof. F. М. Gashimzade for helpful discussions. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Optical intersubband transitions in quantum wires with an applied magnetic field Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	Optical intersubband transitions in quantum wires with an applied magnetic field
spellingShingle	Optical intersubband transitions in quantum wires with an applied magnetic field Ibragimov, G.B.
title_short	Optical intersubband transitions in quantum wires with an applied magnetic field
title_full	Optical intersubband transitions in quantum wires with an applied magnetic field
title_fullStr	Optical intersubband transitions in quantum wires with an applied magnetic field
title_full_unstemmed	Optical intersubband transitions in quantum wires with an applied magnetic field
title_sort	optical intersubband transitions in quantum wires with an applied magnetic field
author	Ibragimov, G.B.
author_facet	Ibragimov, G.B.
publishDate	2004
language	English
container_title	Semiconductor Physics Quantum Electronics & Optoelectronics
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
format	Article
description	The intersubband optical absorption is investigated in parabolic quantum wires in the presence of a tilted magnetic fields. We show that for increasing magnetic field the intersubband absorption peak is shifted to higher energies and its amplitude is increased, too. In particular, it has been shown that the direction of the magnetic field plays a significant role in the intersubband optical absorption.
issn	1560-8034
url	https://nasplib.isofts.kiev.ua/handle/123456789/119125
citation_txt	Optical intersubband transitions in quantum wires with an applied magnetic field / G.B. Ibragimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 283-286. — Бібліогр.: 24 назв. — англ.
work_keys_str_mv	AT ibragimovgb opticalintersubbandtransitionsinquantumwireswithanappliedmagneticfield
first_indexed	2025-11-26T15:18:47Z
last_indexed	2025-11-26T15:18:47Z
_version_	1850628150201942016
fulltext	283© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 3. P. 283-286. PACS: 73.21.Hb, 73.21.Nm Optical intersubband transitions in quantum wires with an applied magnetic field G.B. Ibragimov Institute of Physics, NAS of Azerbaijan Republic, 33, Javid av., 1143 Baku, Azerbaijan E-mail: guseyn@physics.ab.az, guseyn_gb@mail.ru Abstract. The intersubband optical absorption is investigated in parabolic quantum wires in the presence of a tilted magnetic fields. We show that for increasing magnetic field the intersubband absorption peak is shifted to higher energies and its amplitude is increased, too. In particular, it has been shown that the direction of the magnetic field plays a significant role in the intersubband optical absorption. Keywords: quantum wire, optical intersubband transitions. Paper received 23.09.03; accepted for publication 21.10.04. 1. Introduction During the past three decades, the physics of low-dimen- sional semiconductors has become a vital part of present- day research. Low-dimensional structures allow the study of a variety of new mechanical, optical and transport phenomena. In this context, one-dimensional systems have been of particular interest for past decade. There is a considerable interest in using the inter- subband transitions in low-dimensional systems. Inter- subband optical transition has attracted considerable attention. It occurs between the subbands that are cre- ated within the same band: its wavelength ranges mostly from near to far infrared. This latter transition suggests the possibility of a whole new field of applications and related physics. This follows from the unusual features which the above transition is thought to passes: large absorption coefficient [1], narrow absorption linewidth [2], large optical nonlinearity [3], fast intraband relaxa- tion [4] and reduced Auger effect [5]. These features could all be readily exploited to enhance the performance of optoelectronic devices such as lasers, detectors and opti- cal switches. The improvements of the semiconductor growth tech- niques have offered the possibility to obtain low-dimen- sional semiconductor structures with any desired well shapes. One of those structures is the so-called parabolic quantum well. Theoretically, parabolic confining potentials are very attractive, since the spectrum and wave functions of one-electron states have a simple analytical form, it is possible to derive explicit analytical expres- sions for the coefficient of absorption of high-frequency electromagnetic field. The intersubband optical absorp- tion in low-dimensional structures is well developed both in the absence [6�11] and in the presence of magnetic fields [12�19]. The magnetic field is an interesting additional pa- rameter, since it can be applied experimentally in a well- controlled way and modifies fundamentally the electronic structure. The application of a magnetic field to a crystal changes the dimensionality of electronic levels and leads to a redistribution of a density of states. The magnetic field is assumed to be tilted with respect to the normal, it serves to add an extra confining potential to the initial confinement, gives rise to two different kinds of Landau level indices, and causes a dramatic change in the energy spectrum, leading to the so-called hybrid magnetoelectric quantization . This paper reports the absorption coefficients due to intersubband optical transition in parabolic quantum wires in tilted magnetic fields.. In accordance with the generalized Kohn theorem [20], electron-electron inter- actions have no effect on the electron transitions in this case. 2. Formalism We consider the transport of an electron gas in a Q1D electron quantum wire structure as treated in [21,22], in which a Q1D electron gas is confined by two confine- 284 SQO, 7(3), 2004 G.B. Ibragimov: Optical intersubband transitions in quantum wires with ... ment frequencies ω1 and ω2 in the x and z directions, res- pectively, and the conduction electrons are free along only one direction (y direction) of the wire. Considering the magnetic field transverse tilt direction, H = (Hx, 0, Hz) with the Landau gauge, the one-electron Hamilto- nian He, eigenstates Ψnlk and eigenvalues Enlk(ky) are written as [21,22] 22 2 22 1 2 2 1 2 1 2 1 zmxm c eA p m H ωω ∗∗ ∗ ++    += (1) ( ) ( ) ( ) ( )yikzzxxL ylnynlky exp1 00 21 −−= φφψ (2) ( ) ( ) ( ) m p lnkE y ynl ~2 2121 2 21 +Ω++Ω+= hh (3) where p is the momentum operator of a conduction elec- tron, ,cosϑωω cxx cmeH == ∗ ,sinϑωω czz cmeH == ∗ 22 1 2 1 zωω +=Ω , 22 2 2 2 xωω +=Ω , yB klbx 2 110 −= , yB klbz 2 220 = and ( )( ) 1222 2 2 1 2 2 2 1 ~ −∗ −ΩΩ= zxmm ωωωω . Here lB1 = ( ) 21 1Ω= ∗mh , ( ) 21 22 Ω= ∗mlB h , 11 Ω= zb ω , b2 = 2Ω= xω and φn,l(x) represent harmonic-oscillator wave functions. For the sake of simplicity, we assume that the coupling term xzHH zx in Eq. (1) is negligible [22,23] since its contribution to the total electron energy in these systems is minor. The absorption coefficient for the case of nondegenerate electron gas in first-order perturbation theory is given by [12,13,24] ( ) ( ) , exp1 2 2 0 0      +−× ×′′′× ×             −− ∈ = ′′′ ∑∑ ωδ ωωπ α h h h yy y y y klnnlk yRy nlk nlk nlk B EE klnHnlkEf TKnc (4) where ( )ωε is the real part of the dielectric constant, n0 � the number of photons in the unit volume with frequency ω, ñ is the speed of light in vacuum, ( )[ ]TK Bωh−− exp1 gives the emission of the photons and HR is the interac- tion Hamiltonian between electrons and the radiation field. The electron distribution function f0(Enlk) for quasi- one-dimensional nondegenerate electron gas in the pres- ence of a magnetic field can be shown as ( ) ( ) ( ).exp ~ 2sinh2sinh24 21 0 TKE TKmL TKTKN f Bnlk By BB y −× × ΩΩ = hhh (5) We write the Hamiltonian HR representing the inter- action with the high�frequency field in the form ( )      + ∈ = ∗ A c e p n m e HR ε ωω π 2 0h , (6) where ε is the polarization vector of the radiation field. In the calculation of the matrix elements of HR that fol- low, the high�frequency field is assumed uniform. For this, the photon wavelength λ must be much larger than 2,1l , which imposes certain restrictions on the hybrid- oscillation frequencies 2,1Ω . If this condition is met, the electron-photon transitions are the dipole ones. Below we shall calculate the absorption for linear polarization and choose the polarization vector ε in the y direction. Then the matrix element of the electron-photon interac- tion Hamiltonian can be written as ( ) ( ) yxzyy yRy klnzmxmPnlk n m e klnHnlk ′′′−+ ∈ = =′′′ ∗∗ ∗ ωω ωω π 02 h (7) A straightforward calculation of the matrix element square in the representation (2) gives ( ) ( )[ ] ( )[ ]     ++        + +++        + +         ∈ =′′′ ′′+− ′′+′−′ ′′′ yy yy yy kknnllll bx kkllnnnn bz kkllnnzyRy ll le nn le x b en klnHnlk δδδδ ω δδδδ ω δδδω ωω π 11 2 2 11 2 1 2 0 2 2 1 2 02 1 2 1 2 2 h (8) where Kronecker symbols ),,( yykkllnn ′′′ δδδ denote the selection rules, which arise during the integration of the matrix element with respect to each direction. Plugging (5) and (8) into (4), we can further perform the sum over n(l) by writing ( )∑ =− nn αexp ) ( ) ( )∑ −∂∂−= nαα exp , summing the geometric series, it is found that the absorption coefficient for quasi-one-di- mensional electron gas is given by ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . 1exp exp 1exp 21exp exp 1exp2 exp116 2 22 1 2 2 2 1 11 1 1 2 1 2 1 2 0 2 223           −Ω Ω−Ω + −Ω Ω+ × ×        +   −Ω Ω−Ω + +   −Ω Ω+         +     × × ∈ −− = TK TK TK l TK TK TK l b x c NeTK b B b Bx B B B Bzz B h h h h h h h h ωδωδ ωωδ ωδω ωδ ω ωω ωπ α (9) To account for the smearing of the hybrid-oscillation resonance, we must replace in Eq. (9) the delta function by the Lorentzian ( ) ( ) ( )221 xx += −− τπτδτ . In this case, α(ω) has delta � function - like spikes with a halfwidth equal to τ�1, where τ is the phenomenological relaxation time. G.B. Ibragimov: Optical intersubband transitions in quantum wires with ... 285SQO, 7(3), 2004 In the limit that 2,1Ωh >>KBT and ( )TKBωh−exp << <<1 and ( )TK B2,1exp Ω−h <<1, we obtain ( ) ( ) , 1 1cos 1 1sin8 2 2 2 2 1 2 1 2221 0     Ω−+Ω + + Ω−+    Ω = ωτ ϑ ωτ ϑ ω ωπ α α ñ (10) where ( )( )∗∈= cmNå ωτα 2 0 . From Eq. (10) we see that the intersubband optical absorption shows the resonant behavior at 2,1Ω=ω . We call this resonance at points 2,1Ω=ω , a hybrid resonance. Since the hybrid-oscilla- tor frequencies 2,1Ω depend on the magnitude of the mag- netic field and the direction of the magnetic field, the position of the peaks on the α vs ω curve depends on these parameters. 3. Results and discussion Let as study the absorption coefficient more closely. The detailed description of the absorption peaks requires numerical studies of the α dependence on the radiation frequency, the magnetic field strength, and the field orientations. In Fig. 1 for B = 7.5T, we present the variation of the absorption coefficient 0αα as a function of the photon frequency. The first group of peaks corresponding to the electron transition between the Landau�level states with frequency 1Ω , while the second group peaks correspond- ing the electron transition between the Landau�level states with frequency 2Ω . By changing the direction of the magnetic field, we can tune the resonance photon energy for this transition. As seen in this figure for first group peaks, the absorption coefficient increases as the tilt angle increases. Since the confinement ( )1Ω of the subbands increases with large ϑ values, the overlap be- tween electron subbands increases, thus the magnitude of the absorption coefficient becomes larger. The reso- nance photon frequency increases with tilt angle ϑ . As seen in this figure the direction of the field is tunable parameter in intersubband optical transitions. This gives an additional degree of freedom in optical device appli- cations. For the second group absorption peaks, separation between Landau�level states is decreased with increas- ing tilt angle. Therefore, in this case intersubband opti- cal absorption is decreased in energy with increasing tilt angle. Since the confinement ( )2Ω of the subbands de- creases with large ϑ values, the overlap between electron subband decreases, thus the magnitude of the absorption coefficient becomes small. In Fig. 2, we present the absorption coefficient 0αα for the intersubband transition as a function of the incident photon frequency for H = 15T. By comparing Fig. 1 with Fig. 2, we find that the resonance photon frequency for intersubband transition increases with the increasing magnetic field. The absorption peak for the intersubband optical absorption is increased in its mag- nitude with the increasing magnetic field. It should be noted that in [18] it has been observed experimentally that the photon energy and intensity of the absorption peaks depends on magnetic fields. In [14] also found for the case of two-dimensional system was that the resonance photon frequency increases with the tilt angle ϑ . In conclusion, interssubband optical absorption in a quantum wire has been studied under an external tilted magnetic field. It is found that the absorption peak is shifted in its energy and is also increased in its magni- tude with the increasing magnetic field. We show that the intersubband optical absorption is sensitive to the tilt angle. 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 10 3 2 1 3 2 1a /a 0 w, 10 13 s �1 Fig. 1. The intersubband absorption coefficient α/α0 vs ω at vari- ous tilt angles ϑ for the quasi-one-dimensional electron gas (τ = = 5⋅1012 s, ω1= 1013 s, ω2 = 5⋅1013 s, are used) for H = 7.5Ò. The lines are, ϑ : 1 � 15°, 2 � 30°, 3 � 45°. a /a 0 w, 10 13s�1 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14 3 3 2 2 1 1 Fig. 2. The intersubband absorption coefficient α/α0 vs ω at various tilt angles ϑ for the quasi-one-dimensional electron gas (τ = 5⋅1012 s, ω1= 1013 s, ω2 = 5⋅1013 s, are used) for H = 15Ò. The lines are for ϑ : 1 � 15°, 2 � 30°, 3 � 45°. 286 SQO, 7(3), 2004 G.B. Ibragimov: Optical intersubband transitions in quantum wires with ... Acknowledgments The author would like to thank Prof. M. I. Aliev and Prof. F. M. Gashimzade for helpful discussions. References 1. L. C. West and S. J. Eglash. First observation of an extremely large-dipole infrared transition withinthe conduction band of a GaAs quantum well // Appl.Phys.Lett., 46, p.1156-1158 (1985). 2. A. Harwit and. J. S. Harris Observation of Stark shifts in quantum well intersubband transition // J. Appl. Phys. Lett., 50, p.685-687 (1987). 3. 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Optical intersubband transitions in quantum wires with an applied magnetic field

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