Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures

The theory of free-carrier absorption is given for a quasi one-dimensional semiconducting structures in a quantizing magnetic field for the case when carriers are scattered by polar optical phonons and acoustic phonons and the radiation field is polarized perpendicular to the magnetic field direc...

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Datum:	2004
1. Verfasser:	Ibragimov, G.B.
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Sprache:	English
Veröffentlicht:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2004
Schriftenreihe:	Semiconductor Physics Quantum Electronics & Optoelectronics
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spelling	irk-123456789-1191242017-06-05T03:03:39Z Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures Ibragimov, G.B. The theory of free-carrier absorption is given for a quasi one-dimensional semiconducting structures in a quantizing magnetic field for the case when carriers are scattered by polar optical phonons and acoustic phonons and the radiation field is polarized perpendicular to the magnetic field direction. 2004 Article Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures / G.B. Ibragimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 279-282. — Бібліогр.: 28 назв. — англ. 1560-8034 PACS: 73.21.Hb,73.21.Nm http://dspace.nbuv.gov.ua/handle/123456789/119124 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The theory of free-carrier absorption is given for a quasi one-dimensional semiconducting structures in a quantizing magnetic field for the case when carriers are scattered by polar optical phonons and acoustic phonons and the radiation field is polarized perpendicular to the magnetic field direction.
format	Article
author	Ibragimov, G.B.
spellingShingle	Ibragimov, G.B. Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet	Ibragimov, G.B.
author_sort	Ibragimov, G.B.
title	Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures
title_short	Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures
title_full	Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures
title_fullStr	Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures
title_full_unstemmed	Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures
title_sort	theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate	2004
url	http://dspace.nbuv.gov.ua/handle/123456789/119124
citation_txt	Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures / G.B. Ibragimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 279-282. — Бібліогр.: 28 назв. — англ.
series	Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv	AT ibragimovgb theoryoffreecarrierabsorptioninthepresenceofaquantizingmagneticfieldinquasionedimensionalquantumwellstructures
first_indexed	2025-07-08T15:15:59Z
last_indexed	2025-07-08T15:15:59Z
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fulltext	279© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 3. P. 279-282. PACS: 73.21.Hb,73.21.Nm Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures G.B. Ibragimov Institute of Physics, NAS of Azerbaijan Republic, 33, Javid av., 1143 Baku, Azerbaijan E-mail: guseyn@physics.ab.az and guseyn_gb@mail.ru Abstract. The theory of free-carrier absorption is given for a quasi one-dimensional semiconducting structures in a quantizing magnetic field for the case when carriers are scat- tered by polar optical phonons and acoustic phonons and the radiation field is polarized perpendicular to the magnetic field direction. The usual resonance condition 0ωω +Ω=cP , where P is an integer and w0 and wc are the optical-phonon frequency and cyclotron fre- quency, respectively, becomes , 0ωω +Ω=P with ω~ equal to 22 ωω +ñ . The magnetic field dependence of the absorption for the transverse configuration can be explained in terms of phonon-assisted transitions between various Landau levels of carriers. Keywords: quantizing magnetic field, quantum well. Paper received 18.03.04; accepted for publication 21.10.04. 1. Introduction The application of magnetic field to crystal changes the dimensionality of electronic levels and leads to a redis- tribution of the density of states. Quantum well wires (QWW) in a magnetic field have been the subject of a few investigations [1-5]. In [1] rectangular QWWs were treated in the decoupled approximation. Concerning the theo- retical work on magnetotransport in QWWs, we are aware of the Hall resistivity treatments of Refs 2 and 3 and of magnetophonon oscillations in Ref. 4. In [5] the field- induced change in optical anisotropy was studied for a quasi - two- dimensional (Q2D) system subject to a peri- odic modulation. The effect of a magnetic field on con- ductance quantization in quasi-one �dimensional (Q1D) systems is reviewed in [6]. An essential progress in tech- niques of growth on patterned substrates and of cleaved- edge overgrowth has led to QWWs with very good opti- cal properties [7�10], thus renewing the interest for the basic properties of Q1D systems. In this work, we are interested in the effect of mag- netic field on the free carrier absorption (FCA) in semi- conductor QWWs. Over the past two decades the investi- gation of FCA in low-dimensional systems has been very intense. FCA is one of the powerful means to understand the scattering mechanisms of carriers. In bulk semicon- ductors, it accounts for the absorption of electromagnetic radiation of frequencies Ω such that Ωh < gE , where gE is the band gap [11]. In quantum well (QW) struc- tures, apart from the direct interband and intersubband optical transitions, the optical absorption can also take plase via indirect intrassubband optical transitions in which carriers absorb or emit photons with a simultane- ous scattering by phonons or other imperfections. The quantum theory for FCA in Q2D structures is well devel- oped both in the absence [12�22] and in the presence of quantizing magnetic fields [23]. In the works [23], we have extended the theory of FCA in Q2D systems in the presence of a quantizing magnetic field when phonon scattering is important, and it was found that FCA coef- ficient oscillates as a function of the magnetic field and photon frequency with resonances occuring when 0ωω ±Ω=ñP , where Ω,ñω and 0ω are the cyclotron, photon and phonon frequencies, respectively, and where P is an integer. The theory FCA has been studied theo- retically in quasi-one dimensional (Q1D) structures only in the absence of quantizing magnetic field [24�27]. In this paper, we extend the quantum theory of FCA developed previously to take into account the presence of quantizing magnetic fields. We consider FCA for the case 280 SQO, 7(3), 2004 G.B. Ibragimov: Theory of free-carrier absorption in the presence of ... when carriers are scattered by the alloy disorder, acous- tic phonons and boundary roughness. We will present a calculation of FCA coefficient for electromagnetic ra- diation polarized along the length of the wire. The mag- netic field is assumed to be perpendicular to the wire axis, so that the dispersion of one-dimensional subbands is strongly modified. 2. Formalism We consider Q1D electron gas confined in a wire of di- mensions zyx LLL ,, . We model transverse confinement via an infinite square well approximation to a heterojunction quantum well (z axis) and a parabolic potential of frequency ω (x axis). Moreover, a magnetic field B, parallel to the z axis, is applied to the wire. The electrons are free in the direction of the wire (y axis). Correspondingly, the one-electron eigenfunctions yNlkΨ and energy eigenvalues yNlkE are given by ( )     −Φ        =Ψ z yik N zy Nlk L zl exx LL y y π sin 2 0 21 (1) 0 2 22 ~2 ~ 2 1 El m k NE y cNlky ++    += ∗ h hω (2) where N = 0,1,2,�, l = 1,2,3,�, and 222 0 2 zLmE ∗= hπ , ky is the wave vector in the y direction, m* is the effective mass of the electron, cmeHñ ∗=ω is the cyclotron fre- quency, 2222 /~~ ,~ ωωωωω ∗=+= mmñ . Moreover, ( )0xxN −Φ is the well-known harmonic-oscillator wave function centered at ykRbx 2 0 ~~= with ωω ~~ cb = and ω~~2 ∗= mR h . The FCA coefficient α, which is related to the quan- tum-mechanical transition probabilities in which the car- riers absorb or emit a photon with the simultaneous scat- tering of carriers by phonons, is given by [12] ∑∈= i ii fW cn0 21 α (3) Here ∈ is the dielectric constant of material, 0n is the number of photons in the radiation field and if is the free-carrier distribution function. The sum is over all the possible initial states �i� of the system. The transition prob- abilities iW can be calculated using the standard second- order Born golden rule approximation: ( ) ( ) +Ω−−+   +−Ω−−= − +∑ qif fq qifi EEiMf EEiMfW ωδ ωδ π hh hh h 2 22 (4) Here Ei and Ef are the initial and final state energies, respectively, of electrons, Ωh is the photon energy, qωh is the phonon energy, and iMf ± are the transition matrix elements from the initial state to the final state for the interaction between electrons, photons and phonons. These transition matrix elements can be represented by the following expression: ∑         Ω−− + − =± α αα αα ω αα hhm EE iHVf EE iVHf iMf i Rs qi sR (5) where HR is the interaction Hamiltonian between the elec- trons and the radiation field, Vs is the scattering poten- tial due to the electron-phonon interaction. Using the wavefunctions given by the expression (1), the matrix elements of the electron-photon interaction Hamiltonians can be written as llNNêê yRy yy ê V n m å NlkHlNk ′′′∗         ∈Ω −= =′′′ δδδε π 2 1 02 hh (6) where V is the volume of the crystal. Here the radiation field is polarized along the wire, ε is the polarization vector of the radiation field. We shall use two different scattering processes: po- lar-optical scattering and acoustic -phonon scattering. The matrix elements nlkVlnk ysy ′′′ of electron-phonon interaction corresponding to the above two processes are equal to ( ) ( )zllyxnnqkkjysy qqqJCnlkVlnk yyy ′′±′ Λ′=′′ ,δ (7) where Jn′, n(qx,qy) is the overlap integral of the harmonic wave functions: ( ) ( ) ( ) ( )∫ ∞ ∞− ′ −Φ−−Φ= = yNyyNx yxNN kRbxqRbkRbxxiqdx qqJ 222 ,' ~~~~~~ exp (8) ( ) ( )         =Λ ∫ z L z z z zll L zl L zl ziqdz L q z ππ 0 ' sin ' sinexp 2 (9) Ñj ′ 2=Cj 2Fj(q) The function Λll′ (qz) given by Eq. (8 ) is crucial for our calculation whose suitable approximation was discussed by Ridley [28]. For the electron-polar-optic phonon interaction we have , 2 0 Vq N FPOL ± = 1 0 22 2 −∈′= ωπ heÑPOL ,       −=∈′ ∞ − 0 1 11 εε . Here, ε∞ and ε0 are the high-frequency and static dielec- tric constants of the semiconductor, respectively. As usual, we take phonon energy qωh = 0ωh ≈ const. G.B. Ibragimov: Theory of free-carrier absorption in the presence of ... 281SQO, 7(3), 2004 ,1exp 1 0 0 −         −    = TK N B ωh ,00 NN =− 100 +=+ NN . where ( )+− 00 NN describes the annihilation (creation) of the phonon. When acoustic phonon scattering is dominant, one may obtain V TKE C s Bd AC 2 2 2 2ρυ = , ( ) 1=qFAC In the case of bulk materials and at extremely strong magnetic fields, the electronic wave functions have small absolute values of momentum components parallel to the applied magnetic field. Therefore, we can neglect the qz dependence in the interaction potential given by Ñj ¢. The electron distribution function for quasi-one-di- mensional nondegenrate electron gas in the presence of magnetic field can be shown as follows: ( ) ( ) ( ) ( )         −                 Ε++ −× ×= ∗ TKm k TK lN TKm TKLLn f B y B B Bzxe Nlky ~2 exp ~21 exp ~ 2~sinh22 22 0 2 21 21 hh hh ω δ ωπ (10) where ( )∑ Ε= l BTKl /exp 0 2δ , ne is the concentration of electrons. Below, we shall use the following identities: ( )∫ ∞ ⊥⊥ = 0 2 2 ' 1 , R dqqqqJ yxnn ( ) ( )∫ ∞ ⊥⊥ ++′= 0 4 32 ' 1 2 , NN R dqqqqJ yxnn (11) ( )∫ ∞      +=Λ 0 ' 2 ' 2 1 1 2 llzzll d dqq δπ Now we make the same approximation as in [4], i.e. we take ( ) 02 ~2 2 2 =− ∗ yyy qkq m h , in δ functions. Using Eqs (4-6) and (9) in (3) and also identities (11), we obtain the following expression for the FCA coefficient for polar and acoustic phonon scattering in a Q1D semiconducting structure in the presence of a magnetic field: ( ) ( ) ( )( ){ ( ) ( ) ( )( )}00 22 0 00 22 0 0 2 23221 0 42 ~1 ~ ~ 2 11 exp 2 1 ~ 2~sinh4 )( ωωδ ωωδ ω δ δ ωωπ α hhh hhh h hh −Ω−−+−++ ++Ω−−+−× ×               +    +−        +× × Ω∈′∈ = ∑ ∑ ∗ EllNNN EllNNN ElN TK RbLmc TKne H ifif ifif ii BlN lN ll z Be POL ff ii if ( ) ( ) ( ) ( ) ( )( ){ }.~ ~ 2 11 exp 1 2 1 ~ 2~sinh2 )( 0 22 0 2 4232221 2122 Ω−−+−× ×               +     +−× ×++        + Ω∈ = ∑ ∑ ∗ hh h h EllNN ElN TK NN RbLmc TKTKnEe H ifif ii B if lN lN ll zs BBed AC ff ii if ωδ ω δ δυρ ωπ α (13) It is particularly convenient to express our results in terms of the dimensionless ratio of the FCA coefficient in presence of the magnetic field to that in the absence of the field. For scattering through acoustic phonon, we adopt the results [24] ( ) ( ) ( ) ( ) ( ) ( )ZKZZ TK Eln cLlm TKnTKEe B i ln ln ll zs BeBd AC ff ii fi 1 0 2 221321 232223 exp 21 exp 2 1 1 )( 2sinh2 0 × ×        ++ −    +× × ∈Ω = ∑ ∑ ∗ ω δ δρυ ω α ω h h h (14) where ( ) ( ) TK llnn Z B ifif 2 0 22 Ε−−−−Ω = ωhh and K1(x) is the modified Bessel function of the second kind, ωω ∗= ml h 2 . In the quantum limit, in which only the ni = nf = li = lf = 1 quantum level is occupied and ñωh >>KBT, only the lowest Landau level N=0 is ther- mally populated, the ratio ( ) ( )0/ ACAC H αα takes the par- ticularly simple form ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ Ω−+× × ΩΩ − = = fN ff Bñ BB AC AC NN TKK TKTK H hh h hh ωδ ωω ωω α α ~1 2 2~exp~2 0 1 221 21421 (15) For optical phonon scattering, the ratio takes a simi- lar form ( ) ( ) ( )Ω= ,, 0 c POL POL TF H ω α α (16) From Eqs (15)�(16), it can be seen that, the ratio de- pends only upon the magnetic field, absolute tempera- ture, and photon frequency and does not depend upon such material parameters as the values of the deforma- tion potential, sound velocity, or density of the material, although, of course, the absolute value of absorption co- efficient does depend upon the numerical values of these parameters. (12) 282 SQO, 7(3), 2004 G.B. Ibragimov: Theory of free-carrier absorption in the presence of ... 3. Discussion Thus, we have obtained general expressions for FCA co- efficients for QWWs in the presence of the quantizing magnetic field. From Eqs (12)-(13) it can be seen that, in the extreme quantum limit ( ω~h >>KBT, Ni = 0, li = lf = 1) for polar optical phonons, the FCA coefficient oscillates as a function of the magnetic field and photon frequency with resonances occuring when 0 ~ ωω ±Ω=P . Since ñω <ω~, for ω > 0, the resonances are shifted to smaller magnetic fields. For ω = 0, i.e., in the absence of con- finement, cb ωω == ~ ,1 ~2 , and we recover the usual reso- nance condition 0ωω ±Ω=ñP . For the elastic scattering by acoustic phonons, resonances are expected when 0 ~ Ω=ωP . The oscillatory dependence of the absorption on mag- netic field can be understood in terms of the Landau subband structure of the electronic energy levels in quantizing magnetic fields. As the magnetic field, and therefore ω~, increases there are fewer and fewer subbands to which the transition can place until finally. Every time that the ratio ωω ~/)( 0±Ω equals an integer value, the transition can take place with an additional subband ending as a final state. In conclusion, we predict that FCA coefficient should increase with magnetic field with an oscillatory depend- ence on the field when Ω >ω~ . The magnetic field de- pendence of the FCA coefficient is explained in terms of the field dependence of the scattering rates and the possi- bility of phonon-assisted transitions between various Landau levels when Ω >ω~. 4. Acknowledgment The author would like to thank Prof. M.I. Aliev and Prof. F.M. Gashimzade for helpful discussions. References 1. J.A. Brum and G. Bastard, Superlattices Microstruct. 4, p. 443 (1998). 2. F.M. Peeters, Quantum Hall Resistance in the Quasi-One- Dimensional Electron Gas // Phys. Rev. Lett., 61, p. 589 (1989). 3. H. Akera and T. Ando, Quantum Hall effect in quantum wires // Phys. Rev. B 39, p. 5508-5511 (1989). 4. P. Vasilopoulos, P. Warmenbol, F.M. Peeters and J.T. Dev- reese, Magnetophonon resonances in quasi-one-dimensional wires // Phys. Rev. B, 40, p. 1810-1816 (1989). 5. U. Bockelmann and G. Bastard, Interband absorption in quan- tum wires. II. Nonzero-magnetic-field case // Phys. Rev. B, 45, p. 1700-1704 (1992). 6. C.W.J. Beenakker and H. van Houten // Solid State Phys., 44, p. 1 (1991). 7. W. Wegscheider et.al. // Phys. Rev. Lett. 71, p. 4071 (1990). 8. F. Vouilloz, D.Y. Oberli, M-A. Dupertuisi, A. Gustofsson, F. Recnhardt and E. Kapon. Effect lateral confinement on valence-band mixing and polarization anisotropy in quan- tum wires // Phys. Rev. B, 57, p. 12378 (1998). 9. R. Cingolani, F. Sogana, Y. Arakawa, R. Rinaldi, M.De. Vit- torio, A. Posscase, A. Towrino, M. Gatalino, L. Vasanelli. Microphotoluminescence spectroscopy of vertically stacked in In1�xGaxAs quantum wires // Phys. Rev., 58, p. 1962-1966 (1998). 10. M. Lomascolo, P. Ciccarese, R. Cingolani and R. Rinaldi Free versus localized exciton in GaAs V-shaped quantum wires // J. Appl. Phys., 83, p. 302-305 (1998). 11. K. Seeger, Semiconductor Physics: An Introduction (Springer, New York, 1997), 6th ed. 12. H.N. Spector, Free-carrier absorption in quasi-two-dimen- sional semiconducting structures // Phys. Rev. B, 28, p. 971- 976 (1983). 13. H. Adamska and. H.N. Spector, Free-carrier absorption in quantum well structures for polar optical phonon scattering // J. Appl. Phys., 56, p. 1123-1127 (1984). 14. C. Trallero Ciner and M. Anton, Quantum Theory of Free- Carrier Absorption in Quasi-Two-Dimensional Semicon- ducting Structures // Phys. Stat. Sol (b), 133, p. 563-572 (1986). 15. V.L. Gurevich, D.A. Parshin and K.E. Stengel // Fiz.Tverd. Tela, 30, p. 1468 (1988). 16. J.S. Bhat, S.S. Kubakaddi and B.G. Mulimani, Free carrier absorption in semiconducting quantum wells for confined LO phonon scattering // J. Appl. Phys., 72, p. 4966-4968 (1992). 17. C.C. Wu and C.J. Lin Free-carrier absorption in n-type gal- lium arsenide films for polar optical phonon scattering // J. Appl. Phys., 79, p. 781-785 (1996). 18. C.C. Wu and C.J. Lin, Free-carrier absorption in n-type pi- ezoelectric semiconductor films. // J. Phys: Condens matter, 6, p. 10147-10156 (1994). 19. F.M. Gashimzade and E.V. Tahirov, Free-Carrier Absorp- tion in Quantum-Well Structures for Charged Impurity Scat- tering // Phys. Stat. Sol (b), 160, K177-K181 (1990). 20. I. Vurgaftman and J.R. Meyer TE-and TM-polarized rough- ness-assisted free-carrier absorption in quantum wells at midinfrared and terahertz wavelengths // Phys. Rev. B, 60, p. 14294-14301 (1999). 21. G.G. Zegrya and V.E. Perlin, Intraband light absorption in quantum wells at the expense of electron-electron collisions // Fiz. Tekh. Poluprovodn., 32, p. 466-471 (1998). 22. G.B. Ibragimov, Free-carrier absorption in semiconducting quantum wells for alloy-disorder scattering // J. Phys.: Condens. Matter, 14, p. 4977-4983 (2002). 23. G.B. Ibragimov, Free-carrier absorption in size-confined sys- tems in a quantizing magnetic field // Fizika, 6, p. 43-46 (2000). G.B. Ibragimov, Free-carrier magnetoabsorption in quantum well structures // Ukr. J. Phys., 48, p. 527-532 (2003). 24. S.S Kubakaddi and B.G Mulimani, Free-carrier absorption in semiconducting quantum well wires // J. Phys. C: solid State Phys. 18, p. 6647-6652 (1985). 25. H. Adamska and N. Spector, Free-carrier absorption from electrons in confined systems // J. Appl. Phys., 59, p. 619-626 (1986). 26. G.B. Ibragimov, Free-carrier absorption in semiconducting quantum well wire for alloy-disorder scattering // J. Phys.: Condens. Matter, 14, p. 8145-8152 (2002) 27. G.B. Ibragimov, Theory free-carrier absorption in quantum wires at the expense boundary roughness scattering // Semi- conductor Physics, Quantum Electronics& Optoelectronics, 6, (1), p. 9 (2003). 28. B.K. Ridley, The electron-phonon interaction in quasi-two- dimensional semiconductor quantum�well structures // J. Phys. C: Solid State Phys. 15, p. 5899-5917 (1982).

Theory of free-carrier absorption in the presence of a quantizing magnetic field in quasi-one-dimensional quantum well structures

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