The disordered electron-hole recombination in multi-quantum well was investigated using analytical method based on the rate equations. The results show extreme broad distribution of the recombination time which depends exponentially on the distances between the recombining excitons. The energies at each localised state shows an energy splitting between the electronic ground state and the first excited state of 0.0038eV

Emetere, Moses

Isah, K. U.

Umaru, Ahmadu

Uno, U. E.

Covenant University Repository

IJFPS, Vol. 2, No.4, pp. 52-57 , Dec, 2012 M.E.Emetere  52   On The Effect of Electron-Hole Recombination in Disordered GaAs-Aa1-xALAs Multi-quantum Well Structure  Uno E. Uno, Moses E. Emetere*, Isah K.U and Umaru Ahmadu   Federal University of Technology, Minna, Nigeria P.M.B 65, Minna, Nigeria   Email: emetere@yahoo.com   (Received Sep 2012; Published Dec 2012)  ABSTRACT The disordered electron-hole recombination in multi-quantum well was investigated using analytical method based on the rate equations. The results show extreme broad distribution of the recombination time which depends exponentially on the distances between the recombining excitons. The energies at each localised state shows an energy splitting between the electronic ground state and the ﬁrst excited state of 0.0038eV.  Key words: Disordered structure, semiconductor, energy transition and hopping length     INTRODUCTION All semiconductors possess a certain degree of disorder due to their alloy structure and imperfect interfaces. This order gives rise to localized state and in turn affects the optical properties of the system. Recently, recombination of electron-hole in a disordered structure (e.g. organic semiconductor (Shushin, 2011), graphene bilayer (Efimkin, Kulbachinskii, & Lozovik, 2011) e.t.c.) has been the focus of discussion because of its importance in constructing electronics device. Small disordered one-dimensional structures was the first to be investigated (Kramer & MacKinnon, 1999),it was discovered that when there is randomness, complete localization of non-interacting electrons and holes occur which initiated a further research on the interaction of electron - hole i.e. conducted at strong coulomb interaction (Leadbeater, Römer, & Schreiber, 1999) and weakly electron-hole interaction in a random potential (Kalyanasundaram & Grätzel, 1998) to produce excitons with long and short lifetime respectively. The recombination of electron-hole rate in disordered has proven to be important in photocatalytic reactions with semiconductor particles, rectification, photoconductivity and transistor behavior (Kalyanasundaram & Grätzel, 1998). The choice of 𝐺𝑎𝐴𝑠 −𝐴𝑎1 − 𝑥 𝐴𝐿𝐴𝑠 in this paper is to due to two facts. First, reawakening a new dimension into research work of the past decade and secondly is because of its potential for applications in optoelectronic devices. The effects of type II alignment in 𝐺𝑎𝐴𝑠/𝐴𝑙𝐴𝑠 double layer structures was achieved by the appropriate choice of the 𝐺𝑎𝐴𝑠 and 𝐴𝑙𝐴𝑠 layers’ width (Ashoori et al., 1993; Reed, 1993) and was initially grown by molecular beam epitaxy (Isu, Jiang, & Ploog, 1987; Ohya, Ishida, & Mori, 1984) and metal organic chemical vapour deposition (Ishibashi, Kidoguchi, Sugahara, & Ban, 2000). In the existing theoretical studies of disordered structures, it was assumed that electrons and holes are strongly spatially correlated in the form of excitons (Dal Don et al., 2004). This assumption is not always valid because of variation in the application of the disorder potential. This paper is an attempt to present the analytical method based on set of rate equations to determine the uncorrelated electrons and holes in quantum wells. The simulations in section three were done using source codes from Matlab and polynomial derivations. THEORY To enable a better environment for our work, the following theoretical assumptions were made. 1) The disorder in quantum well of 𝐺𝑎𝐴𝑠 − 𝐴𝑎1−𝑥𝐴𝑙𝐴𝑠 is caused by their alloy composition and imperfect interfaces which makes it create a set of localized states than trap photo-generated charge IJFPS, Vol. 2, No.4, pp. 52-57 , Dec, 2012 M.E.Emetere  53  carriers. The carriers are captured after photo-generation. 2) The carriers can either recombine or perform a photo– assisted hopping transition to other localized states. The probabilities depend exponentially on the distance involved. The rate of recombination with localized hole at a rate is  Γ𝜎(𝑅) ≈ 𝜏𝑜−1 exp  −2𝑅𝛼            (1) Where 𝜏𝑜  depends on the particular recombination mechanism. In the case of radioactive recombination, 𝜏𝑜  is of the order of the exciton radioactive life time. 3) The rate for a charge carrier to perform a non radioactive hopping  transition from an occupied state 𝑖 to an empty localized  state 𝑗 over a distance 𝑟𝑖𝑗  is determined by the Miller, Blum, Glennon, and Burton (1960). Γ𝑖𝑗 = 𝑉𝑜exp⁡[−2𝑟𝑖𝑗𝛼−𝜉𝑗−𝜉𝐼+ 𝜉𝑗−𝜉𝐼  2𝐾𝑇        (2) Where 𝜉𝑗  and 𝜉𝐼 are the energies of states 𝑗 and 𝑖 respectively and 𝑉0 is the attempt-to-escape-frequency of the 1012𝑠−1.  4)  Applying the approach of (Marshall, 2000), one divides the energy range where the localized states are distributed into a set of m energy slices with a given width and formulates the rate equation for carrier densities in those energy slices. The time of the carrier concentration 𝑛𝑘  in slice number 𝑘 is determined by the equation 𝑑𝑛𝑘𝑑𝑡=   𝑛𝑗Γ𝑗→𝑘 − 𝑛𝑘Γ𝑘→ − 𝑛𝑘Γ𝜎𝑚𝑗=𝑖𝑗≠𝑘      (3) Where Γ𝑗→𝑘  denotes the rate of a charge carrier transition from a state in slice 𝑖 to a state in slice  𝑗, and Γ𝜎  is the recombination rate. 5) Considering the transition rate 𝑇𝑘↓ from the slice 𝑘 downward in energy, since the transition occurs through energy loss hopping; only the tunnelling term remains in Eq 2. Therefore, one can write the downward transition rate as;  𝑇𝑘↓ = 𝑉𝑜exp⁡(−2𝑅𝑘𝛼)           (4) Where 𝑅𝑘  is the hopping distance determined by energy below 𝐸𝑘  . In two dimensional cases it can be estimated as  𝑅𝑘 =  𝜋  (𝑑𝑗 − 𝑛𝑖 𝑡 )𝑚𝑖=𝑘  −12         (5) Where 𝑑𝑗  denotes the concentration of localized states in the energy 𝑗. Using equation 4, we can derive the downward hopping rate Γ𝑗→𝑘  between two energy slices as a fraction of 𝑇𝑘↓.Therefore, the downward in energy transition rates between states in two energy slices 𝑘 and  , (𝐸𝑘 > 𝐸𝑗 ) can be written in the form  𝑇𝑘→𝑗 = 𝑉𝑜 exp  −2𝑅𝑘𝛼 ∗𝑑𝑗−𝑛𝑖 𝑡   𝑑𝑗−𝑛𝑖 𝑡  𝑚𝑖=𝑘        (6) We can also calculate the transition rate for carriers between two slices 𝑗 and 𝑘 upward in energy can be derived from downward transition rate 𝑇𝑘→𝑗 = 𝑇𝑘→𝑗𝑑𝑘−𝑛𝑘  𝑡 𝑑𝑗−𝑛𝑖 𝑡 exp  𝜉𝑘−𝜉𝑗𝐾𝑇           (7) Hence, upward in energy transition, the expression becomes 𝑇𝑘→𝑗 = 𝑉𝑜 exp  −2𝑅𝑘𝛼−𝜉𝑘−𝜉𝑗𝐾𝑇 ∗𝑑𝑗−𝑛𝑖 𝑡   𝑑𝑗−𝑛𝑖 𝑡  𝑚𝑖=𝑘    (8) 6) If the concentration of charge carriers at time 𝑡 is  𝑛, the recombination rate would be of the order Γ𝜎(𝑛−1/2). Therecombination rate in equation (3) can be expressed as the product of the density of filled electron state n and the probability 𝑛2  from the filled hole state of a distance α from filled electron state n;  Γ𝜎 ≈ 𝜏𝑜−1𝑛(𝑡)𝛼2           (9) Applying Eq3-9 we obtain the luminescence spectrum as; 𝐼 ℏ𝜔, 𝑡 = 𝛼  𝑛 𝑜 ∞−∞ ℏ𝜔 + 𝜉, 𝑡 𝑛 𝑘  𝜉, 𝑡 𝑑𝜉    (10)  Where 𝑛 𝑜 (𝜉, 𝑡) and 𝑛 𝑘  𝜉, 𝑡  denote the densities of electron and holes respectively. SIMULATIONS OF DERIVATIONS   Figure.1 Effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and 𝑅 > 𝛼 in the second Taylors expansion   Figure.2 Effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and 𝑅 > 𝛼 in the second Taylors expansion   Figure.3 Effects on the downward energy transition when  𝑉𝑜 = 1𝐻𝑧  and 𝑅 > 𝛼 in the third Taylors expansion IJFPS, Vol. 2, No.4, pp. 52-57 , Dec, 2012 M.E.Emetere  54   Figure.4 effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧  and R>α in the third Taylors expansion  Figure.5 Effects on the downward energy transition when𝑉𝑜 = 1𝐻𝑧  and R>α in the fourth Taylors expansion  Figure.6 Effects on the downward energy transition when  𝑉𝑜 = 1012𝐻𝑧  and 𝑅 > 𝛼 in the fourth Taylors expansion  Figure.7 Effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and 𝑅 > 𝛼 in the general Taylors expansion  Figure.8 Effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and 𝑅 > 𝛼 in the general Taylors expansion  Figure.9 Downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and 𝑅 < 𝛼 in the second Taylors expansion  Figure.10 Effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and 𝑅 < 𝛼 in the second Taylors expansion  Figure.11 Downward energy transition when 𝑉𝑜 =1𝐻𝑧 and 𝑅 < 𝛼 in the third Taylors expansion IJFPS, Vol. 2, No.4, pp. 52-57 , Dec, 2012 M.E.Emetere  55   Figure.12 The downward energy transition when 𝑉𝑜 =1𝐻𝑧 and 𝑅 < 𝛼 in the general Taylors expansion.   Figure.13 Downward energy transition when 𝑉𝑜 =1𝐻𝑧 and 𝑅 < 𝛼 in the general Taylors expansion  Figure.14 Downward energy transition when 𝑉𝑜 = 1012𝐻𝑧 and 𝑅 < 𝛼 in the general Taylors expansion.   The simulations on equations (4) were carried as shown in the figure above. The Taylors expansion was applied to analyze the features of the various changes in the downward energy transitions. Figure (1) describes the effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and 𝑅 > 𝛼 in the second Taylors expansion. Figure (2) describes the effects on the downward energy transition when 𝑉𝑜 = 1012𝐻𝑧  and R>α in the second Taylors expansion. Figure (3) describes the effects on the downward energy transition when 𝑉𝑜 =1𝐻𝑧  and 𝑅 > 𝛼 in the third Taylors expansion. Figure (4) describes the effects on the downward energy transition when 𝑉𝑜 = 1012𝐻𝑧  and 𝑅 > 𝛼 in the third Taylors expansion. Figure (5) describes the effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧  and 𝑅 > 𝛼 in the fourth Taylors expansion. Figure (6) describes the effects on the downward energy transition when 𝑉𝑜 = 1012𝐻𝑧  and R>α in the fourth Taylors expansion. Figure (7) describes the effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and R>α in the general Taylors expansion. Figure (8) describes the effects on the downward energy transition whenVo=1012Hz and R>α in the general Taylors expansion. Figure (9) describes the effects on the downward energy transition when 𝑉𝑜 =1𝐻𝑧 and R<α in the second Taylors expansion. Figure (10) describes the effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and 𝑅 < 𝛼 in the second Taylors expansion. Figure (11) describes the effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and 𝑅 < 𝛼 in the third Taylors expansion. Figure (12) describes the effects on the downward energy transition when𝑉𝑜 = 1𝐻𝑧 and 𝑅 < 𝛼 in the fourth Taylors expansion. Figure (13) describes the effects on the downward energy transition when 𝑉𝑜 = 1𝐻𝑧 and R<α in the general Taylors expansion. Figure (14) describes the effects on the downward energy transition when 𝑉𝑜 = 1012𝐻𝑧 and R<α in the general Taylors expansion. The values of the downward transition energies gotten from the figures above were assumed to be equal to the upward transition energies.    Figure.15 Diagrammatic representation of the term  (𝜉𝑘 − 𝜉𝑗 )/𝑘𝑡 when  𝑉𝑜 = 1𝐻𝑧  and 𝑅 > 𝛼 in the second Taylors expansion.  Figure.16 Diagrammatic representation of the term  (𝜉𝑘 − 𝜉𝑗 )/𝑘𝑡 when 𝑉𝑜 = 1𝐻𝑧  and 𝑅 > 𝛼 in the general Taylors expansion.  The values for the term (𝜉𝑘 − 𝜉𝑗 )/𝑘𝑡 for almost all the simulations were the same. Two figures (15, 16) were highlighted below to explain its features. Figure (15) IJFPS, Vol. 2, No.4, pp. 52-57 , Dec, 2012 M.E.Emetere  56  describes the effects on the term (𝜉𝑘 − 𝜉𝑗 )/𝑘𝑡  when 𝑉𝑜 = 1𝐻𝑧  and 𝑅 > 𝛼 in the second Taylors expansion. Figure (16) describes the effects on the term (𝜉𝑘 − 𝜉𝑗 )/𝑘𝑡 when 𝑉𝑜 = 1𝐻𝑧  and 𝑅 > 𝛼 in the general Taylors expansion. RESULTS AND DISCUSSION Figures (1-4) have the same shape when 𝑅 > 𝛼, which shows that beyond the linearity between the localisation length (α) and hopping length (R), the possibility of electron-holes localizing within a site becomes lesser as the attempt- to-escape-frequency (𝑉0) increases. At this point, the concentration (R) remains constant i.e. 0.2 while the localisation length ranges between 0.0012 and 0.0014. Most importantly, it shows the time dependence of localisation length in a disordered semiconductor. Figures (5, 6) show a different feature. It expresses the effects of the disordered 𝐺𝑎𝐴𝑠 − 𝐴𝑎1 − 𝑥𝐴𝐿𝐴𝑠 on the localisation length, hopping length and attempt-to-escape-frequency (𝑉0). The three quantities have the same relations (positive parabolic) to one another with higher magnitudes. This feature in figures (5, 6) simply means that the effect of the localization of the particles on various defects caused by either fluctuations in the well width or by remote charged impurities at increases the calculated lifetime in dependence on the localization length, well width, and other input parameters (Citrin, 1993). Also, increasing the attempt-to-escape frequency, the luminescence lines from the quantum wells are strongly shifted towards higher downward transition energies. Figures (7, 8) are the resultant effect of figures (1, 6). This shows that the indirect recombination time in single quantum wells do not only depend on the applied electric field but on other parameters as hopping distance. The features (shown in figures 9, 14) when 𝑅 < 𝛼 revealed strange progression. Figures (9,11,13) whose attempt-to-escape frequency is unity, has a gradual overturn of the shape showing the slow relaxation of quantum particles through a system of localized states which are signatures of transient photoconductivity and optical properties of disordered systems (Monroe, 1985; Takagahara, 1999). When 𝑉𝑜 = 1012𝐻𝑧 (figures 10, 12, 14), 𝐺𝑎𝐴𝑠 − 𝐴𝑎1 − 𝑥𝐴𝐿𝐴𝑠 in particular, experiences an energy relaxation which ultimately depends on the hopping distance. According Baranovskii, Zvyagin, Cordes, Yamasaki, and Thomas (2002) in an exponential density of state, the fastest upwards transitions occur towards a preferred energy level known as the transport energy. Since the hopping transport is governed by upwards transitions from filled states near the Fermi level to empty states (Godet, 2003), we analyzed a balance equilibrium condition (as can be seen in figures (15,16)) where (𝜉𝑘 − 𝜉𝑗 )/𝑘𝑡 = 200. Secondly, a decrease or increase in the attempt-to-escape frequency had no influence on the (𝜉𝑘 − 𝜉𝑗 )/𝑘𝑡 . Adopting the experiments of (Golub et al., 1999) where the  𝐺𝑎𝐴𝑠 − 𝐴𝑎1 − 𝑥𝐴𝐿𝐴𝑠  sample was held in an optical cryostat at temperature 𝑇 = 1.4 𝐾, 𝜉𝑘  and  𝜉𝑗  which is the energies at each localized state shows an energy splitting between the electronic ground state and the first excited state to be 0.0038𝑒𝑉. CONCLUSION The filling factors of all the energy slices considered are time-dependent and are hence, on equilibrium values. It could be concluded that the distribution of recombining carriers is essentially non equilibrium at low temperatures. We would like to mention that  in the system with less disorder, the light emission can be treated using semiconductor Photoluminescence (PL) equation in Eq (10) (Kira, Jahnke, Hoyer, & Koch, 1999). The energy splitting between the electronic ground state and the first excited state (0.0038𝑒𝑉) enables the peculiarity of the 𝐺𝑎𝐴𝑠 − 𝐴𝑎1 − 𝑥𝐴𝐿𝐴𝑠 semiconductor to technological advancement in optoelectronics.      REFERENCE Ashoori, R., Stormer, H., Weiner, J., Pfeiffer, L., Baldwin, K., & West, K. (1993). N-electron ground state energies of a quantum dot in magnetic field. Physical Review Letters, 71(4), 613-616.  Baranovskii, S., Zvyagin, I., Cordes, H., Yamasaki, S., & Thomas, P. (2002). Electronic transport in disordered organic and inorganic semiconductors. Journal of non-crystalline solids, 299, 416-419.  Citrin, D. (1993). Radiative lifetimes of excitons in quantum wells: Localization and phase-coherence effects. Physical Review B, 47(7), 3832.  Dal Don, B., Kohary, K., Tsitsishvili, E., Kalt, H., Baranovskii, S., & Thomas, P. (2004). Quantitative interpretation of the phonon-assisted redistribution processes of excitons in Zn_ {1-x} Cd_ {x} Se quantum islands. 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On The Effect of Electron-Hole Recombination in Disordered GaAs-Aa1-xALAs Multi-quantum Well Structure

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On The Effect of Electron-Hole Recombination in Disordered GaAs-Aa1-xALAs Multi-quantum Well Structure

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