- Email: [email protected]
Electrons as harmonic oscillators in degenerate semiconducting quantum dots
ARTICLE IN PRESS
Optik
Optics
Optik 119 (2008) 349–350 www.elsevier.de/ijleo
SHORT NOTE
Electrons as harmonic oscillators in degenerate semiconducting quantum dots M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/Julio Palacios 11, 9-B, 28029 Madrid, Spain
Abstract We show that for a generic sample of a semiconductor inside a parabolic quantum dot, this sample is degenerate so that a typical case relative to an n-type semiconductor is considered. In our study, by using a quantum-box approach, Fermi velocity is calculated for evaluating later the corresponding Fermi level in order to determine the aforementioned degeneracy. In addition, some considerations upon optical emission are made. r 2007 Elsevier GmbH. All rights reserved. Keywords: Parabolic quantum dot; Degenerate semiconductor; Fermi energy; Optical transitions
A quantum dot can be regarded as a quasi-zerodimensional electron system although this system is considered in three dimensions to analyze it with respect to the main possible physical processes that can take place. Within the context of nanoscience/nanotechnology, quantum dots offer a notorious interest because these systems appear as key elements for conceiving new electron devices [1]. Furthermore, quantum dots influence existing electronic devices so that quantum dots constitute remarkable elements of reference in order to study impact over, say, more conventional devices. From the point of view of optoelectronics, the investigation of the main emission processes in quantum dots represents a relevant task. In this context, quantum-dot lasers are attracting a great attention at the present day as notorious optical sources (see, for example, [1]). In the following, we will treat electrons as harmonic oscillators in an n-type semiconductor of a quantum dot (see, for instance, [2]), that is to say, we will consider parabolic quantum dots (parabolicity of the potential energy of electrons as harmonic oscillators); parabolicity with respect to kinetic energy will be also regarded. Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.02.001
In order to examine semiconductor degeneracy, we commence by calculating Fermi velocity for the electrons in a generic n-type semiconductor inside a quantum dot (as a characteristic example, let us consider nanometer-sized islands of n-type indium arsenide within a gallium arsenide matrix). We refer to the conduction electrons in the absence of external electric and magnetic fields. Given that the dot in question may be contemplated as an artificial atom, we can assume spherical symmetry for the dot so that a given electron in the dot behaves approximately as a three-dimensional harmonic oscillator in a spherical quantum box so that the corresponding parabolic potential is relatively small; electrons of the InAs islands are confined because of the GaAs matrix. At this point, let us remember that InAs is a narrow band-gap semiconductor whereas GaAs is a wide band-gap semiconductor. Then, we can write the following expression relative to energy-level spacing induced by confinement (see, for example, [3–5]): E nþ1 E n
hvF , 4r0
(1)
where En denotes electron (quantized) total energy, n ¼ 0; 1; 2; . . . is the corresponding radial quantum number, h is the Planck constant, vF is the magnitude of Fermi’s
ARTICLE IN PRESS 350
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 119 (2008) 349–350
velocity, and r0 stands for the dot radius. Since E n ¼ n þ 32 _o, where _ ¼ h=ð2pÞ and o is the angular fundamental frequency of oscillation, from relationship (1) one gets 2r0 o . (2) p Now, by using formula (2), the corresponding quasiFermi level becomes vF
1 E F ¼ mn v2F 2mn r20 o2 p2 , (3) 2 where mn stands for electron effective mass. Now we must determine o. To this end, we have 1 2 2 3 m r o ¼ _o, (4) 2 n 0 2 where the right-hand side of expression (4) is the ground-state energy of the oscillator (which coincides with the top energy of the valence band). By inspecting formulae (3) and (4), it follows that quasi-Fermi level lies on the valence band since the value of this energy level is lower than the common value of relationship (4). Then, from Eq. (4) it follows that o¼
3_ . mn r20
(5)
Now, by replacing formula (5) into relation (3), one has EF ¼
18_2 . p2 mn r20
(6)
At this point, we recall that the quasi-Fermi level lies on the valence band, that is, the semiconducting sample in the dot is degenerate (see Ref. [2]). In addition, by considering formula (6), notice that the quasi-Fermi level is inversely proportional to the square of the dot radius. The corresponding Schro¨dinger equation, which describes the electron behavior in the box, is 1 d 2 dc 2m E r (7) þ 2n c ¼ 0. 2 r dr dr _ Now from Eq. (7) it follows that d2 c 2 dc 2mn E þ 2 c ¼ 0. þ (8) dr2 r dr _ Eq. (8) together with the boundary condition cðr0 Þ ¼ 0 and the normalization condition should give the adequate solutions for the electron wavefunctions. In summary, we have treated quantum dots by means of a quantum-box model within a one-electron approach arriving at expression (6). Then, although a single electron has been assumed, a large number of electrons can be considered in a given dot since these electrons may be regarded as non-interacting particles in our context (we can think on a Fermi gas).
At this point, the consideration of the thermodynamic limit for dots with a number of electrons appreciably larger than 200 appears as an interesting problem [2]. Furthermore, let us remark the fact that the semiconductor inside the dot can be found degenerate [2] (quasi-Fermi level within the valence band). Finally, we will comment on some aspects related to optical processes in semiconductors. In this context, excitons and biexcitons play an important role in both linear and non-linear optical processes [6,7]. In particular, referring to excitons in quantum dots, an interesting example can be mentioned: the coupling and entangling of quantum states in a pair of vertically aligned and self-assembled quantum dots by regarding excitons in a single-dot molecule as a function of the separation between the dots [8]. Unfortunately, interacting electron–hole pairs (excitons) have been treated, in an appreciable part of the literature as, for instance, Ref. [9], by using non-well-grounded schemes without transparency leading to misinterpretations derived from conceptual errors. For special studies about excitons, see [10].
References [1] D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot Heterostructures, Wiley, Chichester, 1999. [2] M.A. Grado-Caffaro, M. Grado-Caffaro, A brief discussion on energy considerations for Ga As parabolic quantum dots, Act. Pass. Electron. Comp. 23 (2001) 255–257. [3] M.A. Grado-Caffaro, M. Grado-Caffaro, A theoretical analysis on the Fermi level in multiwalled carbon nanotubes, Mod. Phys. Lett. B 18 (2004) 501–503. [4] M.A. Grado-Caffaro, M. Grado-Caffaro, Theoretical evaluation of electron mobility in multi-walled carbon nanotubes, Optik 115 (2004) 45. [5] M.A. Grado-Caffaro, M. Grado-Caffaro, Fermi velocity of a single electron as harmonic oscillator in a onedimensional potential well, Mod. Phys. Lett. B, to be published. [6] J.I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1971. [7] T.V. Shahbazyan, N. Primozich, I.E. Perakis, D.S. Chemla, Femtosecond coherent dynamics of the Fermiedge singularity and exciton hybrid, Phys. Rev. Lett. 84 (2000) 2006. [8] M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z.R. Wasilewski, O. Stern, A. Forchel, Coupling and entangling of quantum states in quantum dot molecules, Science 291 (2001) 451. [9] V.M. Agranovich, V.L. Ginzburg, Crystal Optics with Considerations of Spatial Dispersion and Theory of Excitons, Nauka, Moscow, 1965 (in Russian). [10] M.A. Grado-Caffaro, M. Grado-Caffaro, A phase formalism for excitonic transitions, Optik 116 (2005) 301–302.
Optik
Optics
Optik 119 (2008) 349–350 www.elsevier.de/ijleo
SHORT NOTE
Electrons as harmonic oscillators in degenerate semiconducting quantum dots M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/Julio Palacios 11, 9-B, 28029 Madrid, Spain
Abstract We show that for a generic sample of a semiconductor inside a parabolic quantum dot, this sample is degenerate so that a typical case relative to an n-type semiconductor is considered. In our study, by using a quantum-box approach, Fermi velocity is calculated for evaluating later the corresponding Fermi level in order to determine the aforementioned degeneracy. In addition, some considerations upon optical emission are made. r 2007 Elsevier GmbH. All rights reserved. Keywords: Parabolic quantum dot; Degenerate semiconductor; Fermi energy; Optical transitions
A quantum dot can be regarded as a quasi-zerodimensional electron system although this system is considered in three dimensions to analyze it with respect to the main possible physical processes that can take place. Within the context of nanoscience/nanotechnology, quantum dots offer a notorious interest because these systems appear as key elements for conceiving new electron devices [1]. Furthermore, quantum dots influence existing electronic devices so that quantum dots constitute remarkable elements of reference in order to study impact over, say, more conventional devices. From the point of view of optoelectronics, the investigation of the main emission processes in quantum dots represents a relevant task. In this context, quantum-dot lasers are attracting a great attention at the present day as notorious optical sources (see, for example, [1]). In the following, we will treat electrons as harmonic oscillators in an n-type semiconductor of a quantum dot (see, for instance, [2]), that is to say, we will consider parabolic quantum dots (parabolicity of the potential energy of electrons as harmonic oscillators); parabolicity with respect to kinetic energy will be also regarded. Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.02.001
In order to examine semiconductor degeneracy, we commence by calculating Fermi velocity for the electrons in a generic n-type semiconductor inside a quantum dot (as a characteristic example, let us consider nanometer-sized islands of n-type indium arsenide within a gallium arsenide matrix). We refer to the conduction electrons in the absence of external electric and magnetic fields. Given that the dot in question may be contemplated as an artificial atom, we can assume spherical symmetry for the dot so that a given electron in the dot behaves approximately as a three-dimensional harmonic oscillator in a spherical quantum box so that the corresponding parabolic potential is relatively small; electrons of the InAs islands are confined because of the GaAs matrix. At this point, let us remember that InAs is a narrow band-gap semiconductor whereas GaAs is a wide band-gap semiconductor. Then, we can write the following expression relative to energy-level spacing induced by confinement (see, for example, [3–5]): E nþ1 E n
hvF , 4r0
(1)
where En denotes electron (quantized) total energy, n ¼ 0; 1; 2; . . . is the corresponding radial quantum number, h is the Planck constant, vF is the magnitude of Fermi’s
ARTICLE IN PRESS 350
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 119 (2008) 349–350
velocity, and r0 stands for the dot radius. Since E n ¼ n þ 32 _o, where _ ¼ h=ð2pÞ and o is the angular fundamental frequency of oscillation, from relationship (1) one gets 2r0 o . (2) p Now, by using formula (2), the corresponding quasiFermi level becomes vF
1 E F ¼ mn v2F 2mn r20 o2 p2 , (3) 2 where mn stands for electron effective mass. Now we must determine o. To this end, we have 1 2 2 3 m r o ¼ _o, (4) 2 n 0 2 where the right-hand side of expression (4) is the ground-state energy of the oscillator (which coincides with the top energy of the valence band). By inspecting formulae (3) and (4), it follows that quasi-Fermi level lies on the valence band since the value of this energy level is lower than the common value of relationship (4). Then, from Eq. (4) it follows that o¼
3_ . mn r20
(5)
Now, by replacing formula (5) into relation (3), one has EF ¼
18_2 . p2 mn r20
(6)
At this point, we recall that the quasi-Fermi level lies on the valence band, that is, the semiconducting sample in the dot is degenerate (see Ref. [2]). In addition, by considering formula (6), notice that the quasi-Fermi level is inversely proportional to the square of the dot radius. The corresponding Schro¨dinger equation, which describes the electron behavior in the box, is 1 d 2 dc 2m E r (7) þ 2n c ¼ 0. 2 r dr dr _ Now from Eq. (7) it follows that d2 c 2 dc 2mn E þ 2 c ¼ 0. þ (8) dr2 r dr _ Eq. (8) together with the boundary condition cðr0 Þ ¼ 0 and the normalization condition should give the adequate solutions for the electron wavefunctions. In summary, we have treated quantum dots by means of a quantum-box model within a one-electron approach arriving at expression (6). Then, although a single electron has been assumed, a large number of electrons can be considered in a given dot since these electrons may be regarded as non-interacting particles in our context (we can think on a Fermi gas).
At this point, the consideration of the thermodynamic limit for dots with a number of electrons appreciably larger than 200 appears as an interesting problem [2]. Furthermore, let us remark the fact that the semiconductor inside the dot can be found degenerate [2] (quasi-Fermi level within the valence band). Finally, we will comment on some aspects related to optical processes in semiconductors. In this context, excitons and biexcitons play an important role in both linear and non-linear optical processes [6,7]. In particular, referring to excitons in quantum dots, an interesting example can be mentioned: the coupling and entangling of quantum states in a pair of vertically aligned and self-assembled quantum dots by regarding excitons in a single-dot molecule as a function of the separation between the dots [8]. Unfortunately, interacting electron–hole pairs (excitons) have been treated, in an appreciable part of the literature as, for instance, Ref. [9], by using non-well-grounded schemes without transparency leading to misinterpretations derived from conceptual errors. For special studies about excitons, see [10].
References [1] D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot Heterostructures, Wiley, Chichester, 1999. [2] M.A. Grado-Caffaro, M. Grado-Caffaro, A brief discussion on energy considerations for Ga As parabolic quantum dots, Act. Pass. Electron. Comp. 23 (2001) 255–257. [3] M.A. Grado-Caffaro, M. Grado-Caffaro, A theoretical analysis on the Fermi level in multiwalled carbon nanotubes, Mod. Phys. Lett. B 18 (2004) 501–503. [4] M.A. Grado-Caffaro, M. Grado-Caffaro, Theoretical evaluation of electron mobility in multi-walled carbon nanotubes, Optik 115 (2004) 45. [5] M.A. Grado-Caffaro, M. Grado-Caffaro, Fermi velocity of a single electron as harmonic oscillator in a onedimensional potential well, Mod. Phys. Lett. B, to be published. [6] J.I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1971. [7] T.V. Shahbazyan, N. Primozich, I.E. Perakis, D.S. Chemla, Femtosecond coherent dynamics of the Fermiedge singularity and exciton hybrid, Phys. Rev. Lett. 84 (2000) 2006. [8] M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z.R. Wasilewski, O. Stern, A. Forchel, Coupling and entangling of quantum states in quantum dot molecules, Science 291 (2001) 451. [9] V.M. Agranovich, V.L. Ginzburg, Crystal Optics with Considerations of Spatial Dispersion and Theory of Excitons, Nauka, Moscow, 1965 (in Russian). [10] M.A. Grado-Caffaro, M. Grado-Caffaro, A phase formalism for excitonic transitions, Optik 116 (2005) 301–302.