J. Semicond. > 2024, Volume 45 > Issue 2 > 022701, doi: 10.1088/1674-4926/45/2/022701
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ARTICLES

An advanced theoretical approach to study super-multiperiod superlattices: theory vs experiments

Alexander Sergeevich Dashkov1, 2, , Semyon Andreevich Khakhulin1, Dmitrii Alekseevich Shapran1, Gennadii Fedorovich Glinskii1, Nikita Andreevich Kostromin1, 3, Alexander Leonidovich Vasiliev4, 5, Sergey Nikolayevich Yakunin4, 5, Oleg Sergeevich Komkov1, Evgeniy Viktorovich Pirogov2, Maxim Sergeevich Sobolev2, Leonid Ivanovich Goray1, 2, 6, 7 and Alexei Dmitrievich Bouravleuv1, 6, 7, 8

+ Author Affiliations

 Corresponding author: Alexander Sergeevich Dashkov, dashkov.Alexander.OM@gmail.com

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Abstract: A new theoretical method to study super-multiperiod superlattices has been developed. The method combines the precision of the 8-band kp-method with the flexibility of the shooting method and the Monte Carlo approach. This method was applied to examine the finest quality samples of super-multiperiod Al0.3Ga0.7As/GaAs superlattices grown by molecular beam epitaxy. The express photoreflectance spectroscopy method was utilized to validate the proposed theoretical method. For the first time, the accurate theoretical analysis of the energy band diagram of super-multiperiod superlattices with experimental verification has been conducted. The proposed approach highly accurately determines transition peak positions and enables the calculation of the energy band diagram, transition energies, relaxation rates, and gain estimation. It has achieved a remarkably low 5% error compared to the commonly used method, which typically results in a 25% error, and allowed to recover the superlattice parameters. The retrieved intrinsic parameters of the samples aligned with XRD data and growth parameters. The proposed method also accurately predicted the escape of the second energy level for quantum well thicknesses less than 5 nm, as was observed in photoreflectance experiments. The new designs of THz light-emitting devices operating at room temperature were suggested by the developed method.

Key words: super-multiperiod superlattice, photoreflectance spectroscopy, Kane model, kp-method, energy band diagram, light amplifiers



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Fig. 1.  (Color online) The transformed PR spectrum of sample no. 1 measured at T = 300 K. Black dotted arrows indicate the experimental transition energy peaks and green solid arrows indicate the transition energy positions computed by CSM.

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Fig. 2.  (Color online) The electronic energy-band structure calculated by Kane model corresponding to sample no. 1 and 2 from Table 1. (a) Electronic energy-band structure of the SMP SL no. 1. (b) Electronic energy-band structure of the SMP SL no. 2. (c) Electronic subband energies and bandwidth of the SMP SL no. 1. (d) Electronic subband energies and bandwidth of the SMP SL no. 2.

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Fig. 3.  (Color online) Transformed PR spectra of samples no.1 (a) and no. 2 (b) from Table 1 measured at T = 300 K.

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Fig. 4.  (Color online) Electronic subband energies and bandwidth calculated by the Kane model as a function of well thickness, with a fixed barrier thickness of 2 nm. The energy zero coincides with the GaAs conduction band edge. The graph exhibits the positions for both $ \Gamma $ and $ \Pi $ point in the Brillouin zone of the SLs. The solid light green horizontal line represents the position of the Al0.3Ga0.7As (barrier layers) conduction band edge. The inset contains the energy-band diagram demonstrated in Fig. 2(c).

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Fig. 5.  (Color online) Simulated gain vs. radiation frequency for SMP SL with quantum well thicknesses between 7 and 10 nm and a fixed barrier thickness of 2 nm.

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Fig. 6.  The computed figure of merit values vs. quantum well width for SLs with a fixed barrier thickness of 2 nm.

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Fig. 7.  (Color online) Simulated scattering times of upper and lower level for two mechanisms: LO-phonon (LO), impurity scattering (imp) vs. quantum well width for SLs with a fixed barrier thickness of 2 nm.

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Table 1.   Parameters of the AlxGa1-xAs/GaAs superlattices samples under study.

No. Al(%) Doping level
(сm−3)		 Number of periods
(counts)		 Well thickness (nm) Barrier thickness (nm) Designed period
(nm)		 Period from XRD
(nm)
1 30 1016 100 10 2 12 12.0
2 30 1016 100 7 2 9 8.99
DownLoad: CSV

Table 2.   The band structure parameters used in the Kane model calculations at T = 300 K.

Parameter Material
$ \mathrm{GaAs} $ $ \mathrm{AlAs} $ $ \mathrm{Al}_x\mathrm{Ga}_{1-x}\mathrm{As} $
a (Å) 5.6532 5 5.6611 $ {a}_\mathrm{AlAs}\cdot x+{a}_\mathrm{GaAs}\cdot (1-x) $
$ {E}_\rm{g}^{\left(\Gamma \right)}\left(\rm{eV}\right) $ 1.422 3.003 $ {E}_\mathrm{AlAs}\cdot x+{E}_\mathrm{GaAs}\cdot \left(1-x\right)+ $$ \left(-0.127+1.310\cdot x\right)\cdot (1-x) $
$ {\mathrm{\Delta }}_\rm{so}\left(\rm{eV}\right) $ 0.341 0.28 $ {\mathrm{\Delta }}_\mathrm{AlAs}\cdot x+{\mathrm{\Delta }}_\mathrm{GaAs}\cdot (1-x) $
$ {m}_{\mathrm{e}}^{*}\left(\Gamma \right) $ 0.067 0.124 −
$ {\gamma }_{1}^{L} $ 6.98 3.76 −
$ {\gamma }_{2}^{L} $ 2.06 0.82 −
$ {\gamma }_{3}^{L} $ 2.93 1.42 −
$ {E}_\mathrm{P}\left(\rm{eV}\right) $ 28.8 21.1 $ {E}_\mathrm{PAlAs}\cdot x+{E}_\mathrm{PGaAs}\cdot (1-x) $
DownLoad: CSV

Table 3.   Values of the optical transition energies (eV) calculated with the Kane model.

Transition type Point in k-space
Sample no. 1 Sample no. 2
Г П Г П
e1−hh1 1.449 1.466 1.486 1.516
e1−lh1 1.454 1.485 1.497 1.549
e2−hh2 1.591 1.529 1.763 1.647
e2−lh2 1.655 1.556 1.876 1.691
e3−hh3 1.662 1.779 − −
e3−lh3 1.729 1.895 − −
DownLoad: CSV
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[3]
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[9]
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[13]
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[14]
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[16]
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[19]
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[21]
Rogowicz E, Kopaczek J, Polak M P, et al. Carrier dynamics in (Ga, In)(Sb, Bi)/GaSb quantum wells for laser applications in the mid-infrared spectral range. Sci Rep, 2022, 12(1), 12961 doi: 10.1038/s41598-022-16966-x
[22]
Cretì A, Prete P, Lovergine N, et al. Enhanced optical absorption of GaAs near-band-edge transitions in GaAs/AlGaAs core–shell nanowires: implications for nanowire solar cells. ACS Appl Nano Mater, 2022, 5(12), 18149 doi: 10.1021/acsanm.2c04044
[23]
Goryacheva V D, Mironova M S, Komkov O S. Investigation of GaAs/AlGaAs superlattice by photoreflectance method. J Phys: Conf Ser, 2018, 1038(1), 012124 doi: 10.1088/1742-6596/1038/1/012124
[24]
Kudrawiec R, Se̢k G, Ryczko K, et al. Photoreflectance investigations of oscillator strength and broadening of optical transitions for GaAsSb–GaInAs/GaAs bilayer quantum wells. Appl Phys Lett, 2004, 84(18), 3453 doi: 10.1063/1.1737065
[25]
Pikhtin A N, Komkov O S, Bugge F. Effect of electric field on the probability of optical transitions in InGaAs/GaAs quantum wells observed by photo- and electroreflectance methods. Phys Status Solidi, 2005, 202(7), 1270 doi: 10.1002/pssa.200460912
[26]
Rygała M, Ryczko K, Smołka T, et al. Investigating the physics of higher-order optical transitions in InAs/GaSb superlattices. Phys Rev B, 2021, 104(8), 085410 doi: 10.1103/PhysRevB.104.085410
[27]
Shanabrook B V, Glembocki O J, Beard W T. Photoreflectance modulation mechanisms in GaAs-AlxGa1-xAs multiple quantum wells. Phys Rev B, 1987, 35(5), 2540 doi: 10.1103/PhysRevB.35.2540
[28]
Ferizović D, Peng L, Sultana H, et al. Photoreflectance spectroscopy study of a strained-layer CdTe/ZnTe superlattice. J Appl Phys, 2011, 110(9), 093703 doi: 10.1063/1.3657785
[29]
Motyka M, Janiak F, Misiewicz J, et al. Determination of energy difference and width of minibands in GaAs/AlGaAs superlattices by using Fourier transform photoreflectance and photoluminescence. Opto−Electron Rev, 2011, 19(2), 151 doi: 10.2478/s11772-011-0021-7
[30]
Janiak F, Dyksik M, Motyka M, et al. Advanced optical characterization of AlGaAs/GaAs superlattices for active regions in quantum cascade lasers. Opt Quantum Electron, 2015, 47, 945 doi: 10.1007/s11082-014-9991-2
[31]
Hosea T J C. Estimating critical-point parameters of modulated reflectance spectra. Phys Status Solidi B, 1995, 189(2), 531 doi: 10.1002/pssb.2221890221
[32]
Jirauschek C, Kubis T. Modeling techniques for quantum cascade lasers. Appl Phys Rev, 2014, 1(1), 011307 doi: 10.1063/1.4863665
[33]
Vukmirović N, Wang L W. Quantum dots: theory. In comprehensive nanoscience and technology. London: Academic Press, 2011
[34]
Glinskii G F, Lakisov V A, Dolmatov A G, et al. Multiband coupling and electronic structure of short-period (GaAs)n/(AlAs)n (001) superlattices. Nanotechnology, 2000, 11(4), 233 doi: 10.1088/0957-4484/11/4/308
[35]
Glinskii G F, Mironova M S. Effective Hamiltonians for heterostructures based on direct-gap III–V semiconductors. The kp perturbation theory and the method of invariants. Semicond, 2014, 48, 1324 doi: 10.1134/S106378261410008X
[36]
Mironova M S, Komkov O S, Firsov D D, et al. Determination of InSb/AlInSb quantum well energy spectrum. J Phys: Conf Ser, 2014, 541(1), 012085 doi: 10.1088/1742-6596/541/1/012085
[37]
Dashkov A S, Kostromin N A, Babichev A V, et al. Simulation of the energy-band structure of superlattice of quaternary alloys of diluted nitrides. Semicond, 2023, 57(3), 207 doi: 10.21883/SC.2023.03.56237.4163
[38]
Liu P Q, Hoffman A J, Escarra M D, et al. Highly power-efficient quantum cascade lasers. Nat Photonics, 2010, 4(2), 95 doi: 10.1038/nphoton.2009.262
[39]
Callebaut H, Hu Q. Importance of coherence for electron transport in terahertz quantum cascade lasers. J Appl Phys, 2005, 98(10), 104505 doi: 10.1063/1.2136420
[40]
Matyas A, Lugli P, Jirauschek C. Role of collisional broadening in Monte Carlo simulations of terahertz quantum cascade lasers. Appl Phys Lett, 2013, 102(1), 011101 doi: 10.1063/1.4773516
[41]
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    Received: 22 July 2023 Revised: 19 October 2023 Online: Uncorrected proof: 08 December 2023Published: 10 February 2024

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      Article Navigation >  Journal of Semiconductors  > 2024  >  45(2): 022701
      Alexander Sergeevich Dashkov, Semyon Andreevich Khakhulin, Dmitrii Alekseevich Shapran, Gennadii Fedorovich Glinskii, Nikita Andreevich Kostromin, Alexander Leonidovich Vasiliev, Sergey Nikolayevich Yakunin, Oleg Sergeevich Komkov, Evgeniy Viktorovich Pirogov, Maxim Sergeevich Sobolev, Leonid Ivanovich Goray, Alexei Dmitrievich Bouravleuv. An advanced theoretical approach to study super-multiperiod superlattices: theory vs experiments[J]. Journal of Semiconductors, 2024, 45(2): 022701. doi: 10.1088/1674-4926/45/2/022701 A. S. Dashkov, S. A. Khakhulin, D. A. Shapran, G. F. Glinskii, N. A. Kostromin, A. L. Vasiliev, S. N. Yakunin, O. S. Komkov, E. V. Pirogov, M. S. Sobolev, L. I. Goray, A. D. Bouravleuv. An advanced theoretical approach to study super-multiperiod superlattices: theory vs experiments[J]. J. Semicond, 2024, 45(2): 022701. doi: 10.1088/1674-4926/45/2/022701Export: BibTex EndNote
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      Alexander Sergeevich Dashkov, Semyon Andreevich Khakhulin, Dmitrii Alekseevich Shapran, Gennadii Fedorovich Glinskii, Nikita Andreevich Kostromin, Alexander Leonidovich Vasiliev, Sergey Nikolayevich Yakunin, Oleg Sergeevich Komkov, Evgeniy Viktorovich Pirogov, Maxim Sergeevich Sobolev, Leonid Ivanovich Goray, Alexei Dmitrievich Bouravleuv. An advanced theoretical approach to study super-multiperiod superlattices: theory vs experiments[J]. Journal of Semiconductors, 2024, 45(2): 022701. doi: 10.1088/1674-4926/45/2/022701

      A. S. Dashkov, S. A. Khakhulin, D. A. Shapran, G. F. Glinskii, N. A. Kostromin, A. L. Vasiliev, S. N. Yakunin, O. S. Komkov, E. V. Pirogov, M. S. Sobolev, L. I. Goray, A. D. Bouravleuv. An advanced theoretical approach to study super-multiperiod superlattices: theory vs experiments[J]. J. Semicond, 2024, 45(2): 022701. doi: 10.1088/1674-4926/45/2/022701
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      An advanced theoretical approach to study super-multiperiod superlattices: theory vs experiments

      doi: 10.1088/1674-4926/45/2/022701
      • 1.

        Saint-Petersburg Electrotechnical University "LETI", Saint Petersburg, 197022, Russian Federation

      • 2.

        Alferov Saint-Petersburg National Research Academic University of the Russian Academy of Sciences, Saint Petersburg, 194021, Russian Federation

      • 3.

        Saint Petersburg Polytechnic University of Peter the Great, Saint Petersburg, 195251, Russian Federation

      • 4.

        Shubnikov Institute of Crystallography of Federal Scientific Research Centre “Crystallography and Photonics”, Russian Academy of Sciences, Moscow, 119333, Russian Federation

      • 5.

        National Research Center ‘Kurchatov Institute’, Moscow, 123182, Russian Federation

      • 6.

        University associated with IA EAEC, Saint Petersburg, 194044, Russian Federation

      • 7.

        Institute for Analytical Instrumentation, Saint Petersburg, 198095, Russian Federation

      • 8.

        Ioffe Institute, Saint Petersburg, 194021, Russian Federation

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      • Author Bio:

        Alexander Sergeevich Dashkov Alexander S. Dashkov got his bachelor degree in 2017 from Peter the Great St. Petersburg Polytechnic University and his master’s degree in 2019 from Alferov University of St. Petersburg. He is currently a PhD candidate at Saint Petersburg Electrotechnical University "LETI" under the supervision of Prof. Leonid Goray. His research focuses on numerical simulations of quantum cascade and super-multiperiod superlattice structures

        Dmitrii Alekseevich Shapran Dmitry A. Shapran completed his master's degree in Electronics and Nanoelectronics at St. Petersburg State Electrotechnical University "LETI" in 2021. Currently, he is a graduate student of Professor Gennadii Glinskii. The area of research is the theory of semiconductor nanoheterostructures based on compounds of groups III-V

        Gennadii Fedorovich Glinskii Gennadii F. Glinskii is a Doctor of Physical and Mathematical Sciences, Professor of the Department of Micro- and Nanoelectronics of St. Petersburg State Electrotechnical University "LETI". Area of scientific interests: theory of semiconductor nanoheterostructures and photonic crystals

        Evgeniy Viktorovich Pirogov Evgeny V. Pirogov got his bachelor's degree in 2004 and a master's degree in 2006 from the Electrotechnical University "LETI" in St. Petersburg, Russia. Currently he is a researcher at the Laboratory of Nanoelectronics at the Alferov University. His research focuses on MBE growth, A3B5 heterostructures and their properties

        Leonid Ivanovich Goray Leonid I. Goray is a Principal Researcher & Professor at Alferov University, 2008−present. He is a Principal Researcher at Saint Petersburg Electrotechnical University "LETI", 2022−present. Leonid Goray received a Ph.D. in Physics and Mathematics in Institute for Analytical Instrumentation (IAI, RAS) in 2004, and received a Dr. in Science in IAI in 2011. He has a strong background in the electromagnetic theory of diffraction and scattering by diffraction gratings, rough mirrors, and nanocrystals; Helmholtz equation, Schrodinger equation, non-linear continuum equation, neuromorphic computing, machine learning: 150+ publications, 7 patents, 2 registered software

        Alexei Dmitrievich Bouravleuv Alexei D. Bouravleuv received the Ph.D. degree from Ioffe Institute RAS in 2002 and his Dr. Sc. degree from Institute for Analytical Instrumentation RAS, St. Petersburg, Russia, in 2014. He is currently a Head of Micro- and Nano-electronics laboratory with LETI University, St. Petersburg, Russia. His research interests include the study of the growth processes of different semiconductor nanostructures and their properties

      • Corresponding author: dashkov.Alexander.OM@gmail.com
      • Received Date: 2023-07-22
      • Revised Date: 2023-10-19
        • Available Online: 2023-12-08

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