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Coordination imperfection enhanced electron–phonon interaction and band-gap expansion in Si and Ge nanocrystals
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Scripta Materialia 60 (2009) 1105–1108 www.elsevier.com/locate/scriptamat
Coordination imperfection enhanced electron–phonon interaction and band-gap expansion in Si and Ge nanocrystals Likun Pan,a,* Zhuo Suna and Changqing Sunb a
Engineering Research Center for Nanophotonics & Advanced Instrument, Ministry of Education, Department of Physics, East China Normal University, Shanghai 200062, China b School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Received 13 January 2009; revised 22 February 2009; accepted 24 February 2009 Available online 27 February 2009
Correlation between the size-enlarged Stokes shift and band-gap expansion of Si and Ge nanocrystals has been investigated in terms of the bond order–length–strength correlation. It is shown that the bond order deficiency of surface atoms dictates electron– phonon coupling and crystal binding, and enhances the size dependence of the observed blue shift in photoemission and photoabsorption of Si and Ge nanocrystals. Compared with Si, Ge nanocrystal exhibits stronger electron–phonon coupling due to larger phonon-induced deformation of the crystal lattice. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Elemental semiconductors; Nanocrystalline materials; Theory; Modeling; Optical properties
Considerable attention has been paid to the study of low-dimensional structures based on the indirect-band-gap semiconductors such as Si and Ge nanocrystals due to their fundamental physical properties and the promise they offer for applications in advanced optoelectronic and electronic devices [1,2]. The distinctive feature of Si and Ge nanocrystals compared with the corresponding bulk crystals (c-Si, c-Ge) is their efficient photoluminescence (PL) in the visible spectra region at room temperature [3,4]. Numerous theoretical models have been developed for the observed blue shift of PL, including quantum confinement theory [5], the free-exciton collision model [6], the impurity luminescent center model [7], and surface states and surface alloying mechanisms [8]. It has been suggested that the energy shift of photoemission (EPL) or band-gap (EG) follows quantum confinement that is dictated by the Coulomb interaction (/D1) between the excited electron–hole pair separated by the nanocrystal dimension D, and the kinetic energies (/D2) of the mobile carriers confined in the quantum well. However, there is fair agreement among existing theoretical calculations with experimental photoabsorption (PA) energy (EPA) and EPL data [9,10], which is thought to be due to the break* Corresponding author. Tel.: +86 21 62234132; fax: +86 21 62234321; e-mail: [email protected]
down of quantum confinement [11]. The EPL is often confused with the EG [12], which is determined by the integral of the crystal potential and the Bloch wavefunction. The PL and PA energies, however, contain contributions from electron–phonon coupling or electron– lattice interaction. The vibration of the lattice will contribute to the electron energy through the offset of the energy band structures. A complete microscopic understanding of the size dependence of the optical excitations in Si and Ge nanocrystals has not yet been achieved. The red shift of EPL with respect to EPA is known as Stokes shift [13]; this is commonly observed in semiconductor nanocrystals and is one of the most important quantities determining the optical properties of nanocrystals. As the radius increases, the Stokes shift decreases, and then disappears beyond a certain radius. A desire to understand the underlying mechanism of Stokes shift has led to various theories in which the Stokes shift was attributed to the ‘‘surface” states [14,15], the fine structure of the band edge excitons [16,17], or to the self-trapped excitons localized in the surface region due to the dangling bonds [18]. In fact, the Stokes shift is dominated mainly by electron–phonon interactions [19], which are prevalent in most nanostructures and determine a wide variety of phenomena, such as pairing of electrons in superconductors, electrical conductivity of metals and carrier scattering lengths in semiconductors [20,21]. Si and Ge nanocrystals are
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.02.046
1106
L. Pan et al. / Scripta Materialia 60 (2009) 1105–1108
indirect-band-gap semiconductors, and absorption and emission of momentum-conserving phonons are needed during their light absorption and emission processes [17,22]. The objective of this work is to report our understanding on the factors dominating electron–phonon coupling, and to establish the quantitative correlation between particle size, Stokes shift and the band-gap of nanocrystalline Si and Ge. It should then be possible provide consistent insight into the coordination number (CN)-imperfection-induced unusual behavior of photons, phonons and electrons in nanostructured semiconductors. The recently developed bond order–length– strength (BOLS) correlation [23] indicates that the CN imperfection of an atom causes the remaining bonds of the undercoordinated atoms to contract spontaneously, with a consequent increase in bond strength gain and reduction in the trapping potential well. There is then a localized densification of charge, energy and mass to the surface skin, which modifies the Hamiltonian of the nanocrystal. The perturbed Hamiltonian determines the entire band structure such as band-gap expansion [24], core-level shift [25] and Stokes shift [26]. According to the band theory [27], the band-gap EG is determined by the integration of the crystal field Vcry(r) in combination with the Bloch wave of the nearly free electron, /(r): EG ¼ 2 < /ðrÞ j V cry ðrÞ j /ðrÞ > V cry ðDÞ ¼ V cry ð1Þð1 þ dsurf Þ
ð1Þ
where Vcry(1) is the crystal field of an extended solid which sums the interatomic binding potential over the solid, and dsurf represents the surface perturbation to the crystal binding energy. Eq. (1) indicates clearly that EG is a function of the crystal potential; any perturbation to the crystal potential will vary the value of EG. The size-enhanced EG expansion follows the relation [12]: 8 EG ðDÞ ¼ EG ð1Þ½1 þ dsurf > > > > d ¼ P c ðcm 1Þ > surf > i i > < i63 ð2Þ ci ¼ VV i ffi scKi > > > > > ci ¼ d i =d ¼ 2=f1 þ exp½ð12 zi Þ=ð8zi Þg > > : where ci is the volume portion of the ith atomic layer (di thick) of the entire solid of different dimensionality (s = 1, 2 and 3 correspond to a thin plate, a rod and a spherical dot, respectively). K = D/2d is the number of atoms lined along the radius of a spherical dot, and d is the diameter of a atom in the bulk. The i is counted up to three from the outermost atomic layer to the center of the solid. cm ¼ ei =eo describes the increase in i magnitude of the bond energy upon bond relaxation. m, an adjustable parameter, varies with the nature of the bond. zi is the effective CN of an atom in the ith atomic layer and z1 = 4(1 0.75/K) [28], z2 = z1 + 2 and z3 = 12 for a spherical dot of any size. The PL or PA energies are not the band-gap but they are determined by the joint effect of crystal binding and electron–phonon coupling. Figure 1 illustrates the effect
Figure 1. Mechanisms for EPA and EPL of a semiconductor nanocrystal, involving crystal binding (EG) and electron–phonon coupling (W) (from Ref. [26]). Insert illustrates the Stokes shift from EPA to EPL.
of electron–phonon coupling and crystal binding on the EPL and EPA. The energies of the ground state (E1) and the excited state (E2) are expressed as [19]: ( E1 ðqÞ ¼ Aq2 ð3Þ 2 E2 ðqÞ ¼ Aðq q0 Þ þ EG where the constant A is the slope of the parabolas, and q has the dimensions of a wave-vector. The vertical distance between the two minima is the real EG and depends functionally on the crystal potential. The lateral displacement of E2(q0) originates from the electron–phonon coupling, which can be strengthened by enhancing the lattice vibration. Therefore, the blue shift in the EPL and EPA is a joint effect of the change in crystal binding and electron–phonon coupling. At a surface, the CN-imperfection-enhanced bond strength affects both the frequency [29] and magnitude [30,31] of lattice vibration. Hence, at the surface, the electron–phonon coupling and hence the Stokes shift will be enhanced. In the process of carrier formation and recombination, an electron is excited by a photon with energy EG + W from the ground minimum to the excited state resulting in the creation of an electron–hole pair. The excited electron then undergoes thermalization and moves to the minimum of the excited state, and eventually returns to the ground state, combining with the hole. The carrier recombination is associated with a photon emission at energy EPL = EG W. The transition processes (electron–hole production and recombination) follow the rule of momentum and energy conservation, though conservation may be relaxed for the short-range ordering found in nanocrystals. This relaxation in the conservation law is responsible for the broad peaks in the PA and PL. The insert in Figure 1 illustrates the Stokes shift, 2W ¼ 2Aq20 , from E PL to EPA. The q0 is inversely proportional to atomic distance di, and hence, Wi = A/(cid)2, in the surface region. Based on this premise, the blue shift of the EPL, the EPA and the Stokes shift can be correlated to the CN-imperfection-induced bond contraction:
L. Pan et al. / Scripta Materialia 60 (2009) 1105–1108
9 DEPL ðDÞ = EPL ð1Þ
DEPA ðDÞ ; EPA ð1Þ
B¼E
G ðDÞDW ðDÞ ¼ DE ffi EG ð1ÞW ð1Þ
A G ð1Þd
2
;
W ð1Þ EG ð1Þ
0
P
ci ½ðcm 1Þ Bðc2 i i 1Þ
i63
!
ð4Þ One can also easily calculate the size-dependent EPL, EPA and EG = (EPL + EPA)/2 with Eq. (4). Fitting the measured data gives the values of m and A for a specific semiconductor. In our previous work [26], the size dependence of both the EPL and EPA of Si nanocrystal has been studied. A combination of theoretical prediction with the measured EPL and EPA data has been used to discriminate the effect of electron–phonon coupling (B = 0.91) from the effect of crystal binding (m = 4.88) in Si nanocrystal. Figure 2a and b compares the predictions (solid line) and measurements on the size dependence of the EPL and EPA of Ge nanocrystal. Matching the prediction in Eq. (4) with the EPL and EPA data of Ge nanocrystal gives B = 4.26 and m = 7.57. The consistency of the prediction in Eq. (2) with the EG data of Ge nanocrystal further confirms the accuracy of the BOLS premise. Table 1 shows the BOLS calculation parameters and results for Si and Ge nanocrystals. Compared with Si nanocrystal (B = 0.91), Ge nanocrystal exhibits stronger electron–phonon coupling. Electron–phonon coupling is due to a phonon-induced deformation of the crystal
Figure 2. Comparison between predictions (solid line) and measurements on the size dependence of: (a) the EPL of Ge nanocrystal with Data-1 [34], Data-2 [35] and Data-3 [36]; (b) the EPA of Ge nanocrystal with Data-1 [37], Data-2 [38] and Data-3 [39]; (c) the EG of Ge nanocrystal with Data-1 [40], Data-2 [41], Data-3 [42], Data-4 [43] and Data-5 [44].
Table 1. The BOLS calculation parameters and results for Si and Ge nanocrystals. Sample
Bulk EG (eV)
d (nm)
m
B
Si Ge
1.12 0.66
0.2632 0.2732
4.88 7.57
0.91 4.26
1107
lattice acting on the charge distribution of the exciton and results in a lowering of the excitonic energy [11]. Compared to Si, Ge nanostructures exhibit more prominent quantum size effects [32,33]. The Bohr exciton radius is larger in Ge (24.3 nm) than in Si (4.9 nm): this increases the oscillator strength of the indirect transitions, which enhances the magnitude of lattice vibration and causes a larger deformation of the crystal lattice to minimize the excitonic energy. As a result, stronger electron–phonon coupling is found in Ge nanocrystal. Consistency between BOLS prediction and measurements in Ge nanocrystal shows once again the effect of atomic coordination imperfection on the unusual behavior of photons, phonons and electrons in a nanocrystal. In summary, the size-enhanced Stokes shift and the size-enlarged band-gap expansion of Si and Ge nanocrystals have been accurately described by BOLS correlation in which the bond order deficiency of surface atoms dictates electron–phonon coupling and crystal binding, and enhances the shape and size dependence of the observed blue shift in PL and PA of Si and Ge nanocrystals. This work was supported by Shanghai Natural Science Foundation (No. 07ZR14033), Shanghai Pujiang Program (No. 08PJ14043), Special Project for Nanotechology of Shanghai (No. 0752nm011), and Applied Materials Shanghai Research & Development Fund (No. 07SA12). [1] K.D. Hirschmann, L. Tsybekov, S.P. Duttagupta, P.M. Fauchet, Nature 384 (1996) 338. [2] A. Fowler, Phys. Today 50 (1997) 50. [3] M. Zacharias, P.M. Fauchet, Appl. Phys. Lett. 71 (1997) 380. [4] M.V. Wolkin, J. Jorne, P.M. Fauchet, G. Allan, C. Delerue, Phys. Rev. Lett. 82 (1999) 197. [5] P.F. Trwoga, A.J. Kenyon, C.W. Pitt, J. Appl. Phys. 83 (1998) 3789. [6] Y.D. Glinka, S.H. Lin, L.P. Hwang, Y.T. Chen, N.H. Tolk, Phys. Rev. B 64 (2001) 085421. [7] G.G. Qin, H.Z. Song, B.R. Zhang, J. Lin, J.Q. Duan, G.Q. Yao, Phys. Rev. B 54 (1996) 2548. [8] F. Koch, V. Petrova-Koch, T. Muschik, A. Nikolov, V. Gavrilenko, Microcrystalline Semiconductors: Materials Science and Devices, vol. 283, Materials Research Society, Pittsburgh, 1993, p. 197. [9] G. Ledoux, J. Gong, F. Huisken, Q. Guillois, C. Reynaud, Appl. Phys. Lett. 80 (2002) 4834. [10] B. Garrido, M. Lopez, A. Perez-Rodrigez, C. Garcia, P. Pellegrino, Nucl. Instrum. Methods Phys. Res. Sect. B 216 (2004) 213. [11] J. Marin, F. Cichos, F. Huisken, C. von Borczyskowski, Nano Lett. 8 (2008) 656. [12] C.Q. Sun, T.P. Chen, B.K. Tay, S. Li, H. Huang, Y.B. Zhang, L.K. Pan, S.P. Lau, X.W. Sun, J. Phys. D 34 (2001) 3470. [13] Q. Zhang, S.C. Bayliss, J. Appl. Phys. 79 (1996) 1351. [14] F. Koch, V. Petrova-Koch, T. Muschik, J. Lumin. 57 (1993) 271. [15] M.G. Bawendi, P.J. Carroll, W.L. Wilson, L.E. Brus, J. Chem. Phys. 96 (1997) 946. [16] A.L. Efros, M. Rosen, M. Kuno, M. Nirmal, D.J. Norris, M.G. Bawendi, Phys. Rev. B 54 (1996) 4843. [17] M. Nirmal, D.J. Norris, M. Kuno, M.G. Bawendi, A.L. Efros, M. Rosen, Phys. Rev. Lett. 75 (1995) 3728.
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[18] G. Allan, C. Delerue, M. Lannoo, Phys. Rev. Lett. 76 (1996) 2961. [19] R.A. Street, Hydrogenated Amorphous Silicon, Cambridge University Press, Cambridge, 1991, p. 279. [20] T. Fujisawa, T.H. Oosterkamp, W.G. van der Wiel, B.W. Broer, R. Aguado, S. Tarucha, L.P. Kouwenhoven, Science 282 (1998) 932. [21] S.H. Kwok, P.Y. Yu, C.H. Tung, Y.H. Zhang, M.F. Li, C.S. Peng, J.M. Zhou, Phys. Rev. B 59 (1999) 4980. [22] Y. Kanemitsu, S. Okamoto, Phys. Rev. B 58 (1998) 9652. [23] C.Q. Sun, Prog. Solid State Ch. 35 (2007) 1. [24] L.K. Pan, C.Q. Sun, B.K. Tay, T.P. Chen, S. Li, J. Phys. Chem. B 106 (2002) 11725. [25] C.Q. Sun, L.K. Pan, Y.Q. Fu, B.K. Tay, S. Li, J. Phys. Chem. B 107 (2003) 5113. [26] L.K. Pan, C.Q. Sun, J. Appl. Phys. 95 (2004) 3819. [27] M.A. Omar, Elementary Solid State Physics: Principles and Applications, Addison-Wesley, New York, 1975. [28] S. Schuppler, S.L. Friedman, M.A. Marcus, D.L. Adler, Y.H. Xie, F.M. Ross, Y.J. Chabal, T.D. Harris, L.E. Brus, W.L. Brown, E.E. Chaban, P.F. Szajowski, S.B. Christman, P.H. Citrin, Phys. Rev. B 52 (1995) 4910. [29] C.Q. Sun, B.K. Tay, X.T. Zeng, S. Li, T.P. Chen, J. Zhou, H.L. Bai, E.Y. Jiang, J. Phys. Condens. Matt. 14 (2002) 7781. [30] Q. Jiang, Z. Zhang, J.C. Li, Chem. Phys. Lett. 322 (2000) 549.
[31] F.G. Shi, J. Mater. Res. 9 (1994) 1307. [32] A.N. Kholod, V.L. Shaposhnikov, N. Sobolev, V.E. Borisenko, F.A. D’Avitaya, S. Ossicini, Phys. Rev. B 70 (2004) 035317. [33] J.R. Heath, J.J. Shiang, A.P. Alivisatos, J. Chem. Phys. 101 (1994) 1607. [34] S. Takeoka, M. Fujii, S. Hayashi, K. Yamamoto, Phys. Rev. B 58 (1998) 7921. [35] S. Takeoka, M. Fujii, S. Hayashi, K. Yamamoto, Appl. Phys. Lett. 74 (1999) 1558. [36] P. Alfaro, A. Miranda, A.E. Ramos, M. Cruz-Irisson, Braz. J. Phys. 36 (2006) 375. [37] Q. Wan, C.L. Lin, N.L. Zhang, W.L. Liu, G. Yang, T.H. Wang, Appl. Phys. Lett. 82 (2003) 3162. [38] S.Y. Ren, Solid State Commun. 89 (1994) 587. [39] C.S. Garoufalis, M.S. Skaperda, A.D. Zdetsis, J. Phys. Conference Series 10 (2005) 97. [40] E.B. Gorokhov, V.A. Volodin, D.V. Marin, D.A. Orekhov, A.G. Cherkov, A.K. Gutakovskii, V.A. Shvets, A.G. Borisov, M.D. Efremov, Semiconductor 39 (2005) 1168. [41] C. Bulutay, Phys. Rev. B 76 (2007) 205321. [42] F.A. Reboredo, A. Zunger, Phys. Rev. B 62 (2000) R2275. [43] A. Tsolakidis, R.M. Martin, Phys. Rev. B 71 (2005) 125319. [44] H.-Ch. Weissker, J. Furthmuller, F. Bechstedt, Phys. Rev. B 69 (2004) 115310.
Scripta Materialia 60 (2009) 1105–1108 www.elsevier.com/locate/scriptamat
Coordination imperfection enhanced electron–phonon interaction and band-gap expansion in Si and Ge nanocrystals Likun Pan,a,* Zhuo Suna and Changqing Sunb a
Engineering Research Center for Nanophotonics & Advanced Instrument, Ministry of Education, Department of Physics, East China Normal University, Shanghai 200062, China b School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Received 13 January 2009; revised 22 February 2009; accepted 24 February 2009 Available online 27 February 2009
Correlation between the size-enlarged Stokes shift and band-gap expansion of Si and Ge nanocrystals has been investigated in terms of the bond order–length–strength correlation. It is shown that the bond order deficiency of surface atoms dictates electron– phonon coupling and crystal binding, and enhances the size dependence of the observed blue shift in photoemission and photoabsorption of Si and Ge nanocrystals. Compared with Si, Ge nanocrystal exhibits stronger electron–phonon coupling due to larger phonon-induced deformation of the crystal lattice. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Elemental semiconductors; Nanocrystalline materials; Theory; Modeling; Optical properties
Considerable attention has been paid to the study of low-dimensional structures based on the indirect-band-gap semiconductors such as Si and Ge nanocrystals due to their fundamental physical properties and the promise they offer for applications in advanced optoelectronic and electronic devices [1,2]. The distinctive feature of Si and Ge nanocrystals compared with the corresponding bulk crystals (c-Si, c-Ge) is their efficient photoluminescence (PL) in the visible spectra region at room temperature [3,4]. Numerous theoretical models have been developed for the observed blue shift of PL, including quantum confinement theory [5], the free-exciton collision model [6], the impurity luminescent center model [7], and surface states and surface alloying mechanisms [8]. It has been suggested that the energy shift of photoemission (EPL) or band-gap (EG) follows quantum confinement that is dictated by the Coulomb interaction (/D1) between the excited electron–hole pair separated by the nanocrystal dimension D, and the kinetic energies (/D2) of the mobile carriers confined in the quantum well. However, there is fair agreement among existing theoretical calculations with experimental photoabsorption (PA) energy (EPA) and EPL data [9,10], which is thought to be due to the break* Corresponding author. Tel.: +86 21 62234132; fax: +86 21 62234321; e-mail: [email protected]
down of quantum confinement [11]. The EPL is often confused with the EG [12], which is determined by the integral of the crystal potential and the Bloch wavefunction. The PL and PA energies, however, contain contributions from electron–phonon coupling or electron– lattice interaction. The vibration of the lattice will contribute to the electron energy through the offset of the energy band structures. A complete microscopic understanding of the size dependence of the optical excitations in Si and Ge nanocrystals has not yet been achieved. The red shift of EPL with respect to EPA is known as Stokes shift [13]; this is commonly observed in semiconductor nanocrystals and is one of the most important quantities determining the optical properties of nanocrystals. As the radius increases, the Stokes shift decreases, and then disappears beyond a certain radius. A desire to understand the underlying mechanism of Stokes shift has led to various theories in which the Stokes shift was attributed to the ‘‘surface” states [14,15], the fine structure of the band edge excitons [16,17], or to the self-trapped excitons localized in the surface region due to the dangling bonds [18]. In fact, the Stokes shift is dominated mainly by electron–phonon interactions [19], which are prevalent in most nanostructures and determine a wide variety of phenomena, such as pairing of electrons in superconductors, electrical conductivity of metals and carrier scattering lengths in semiconductors [20,21]. Si and Ge nanocrystals are
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.02.046
1106
L. Pan et al. / Scripta Materialia 60 (2009) 1105–1108
indirect-band-gap semiconductors, and absorption and emission of momentum-conserving phonons are needed during their light absorption and emission processes [17,22]. The objective of this work is to report our understanding on the factors dominating electron–phonon coupling, and to establish the quantitative correlation between particle size, Stokes shift and the band-gap of nanocrystalline Si and Ge. It should then be possible provide consistent insight into the coordination number (CN)-imperfection-induced unusual behavior of photons, phonons and electrons in nanostructured semiconductors. The recently developed bond order–length– strength (BOLS) correlation [23] indicates that the CN imperfection of an atom causes the remaining bonds of the undercoordinated atoms to contract spontaneously, with a consequent increase in bond strength gain and reduction in the trapping potential well. There is then a localized densification of charge, energy and mass to the surface skin, which modifies the Hamiltonian of the nanocrystal. The perturbed Hamiltonian determines the entire band structure such as band-gap expansion [24], core-level shift [25] and Stokes shift [26]. According to the band theory [27], the band-gap EG is determined by the integration of the crystal field Vcry(r) in combination with the Bloch wave of the nearly free electron, /(r): EG ¼ 2 < /ðrÞ j V cry ðrÞ j /ðrÞ > V cry ðDÞ ¼ V cry ð1Þð1 þ dsurf Þ
ð1Þ
where Vcry(1) is the crystal field of an extended solid which sums the interatomic binding potential over the solid, and dsurf represents the surface perturbation to the crystal binding energy. Eq. (1) indicates clearly that EG is a function of the crystal potential; any perturbation to the crystal potential will vary the value of EG. The size-enhanced EG expansion follows the relation [12]: 8 EG ðDÞ ¼ EG ð1Þ½1 þ dsurf > > > > d ¼ P c ðcm 1Þ > surf > i i > < i63 ð2Þ ci ¼ VV i ffi scKi > > > > > ci ¼ d i =d ¼ 2=f1 þ exp½ð12 zi Þ=ð8zi Þg > > : where ci is the volume portion of the ith atomic layer (di thick) of the entire solid of different dimensionality (s = 1, 2 and 3 correspond to a thin plate, a rod and a spherical dot, respectively). K = D/2d is the number of atoms lined along the radius of a spherical dot, and d is the diameter of a atom in the bulk. The i is counted up to three from the outermost atomic layer to the center of the solid. cm ¼ ei =eo describes the increase in i magnitude of the bond energy upon bond relaxation. m, an adjustable parameter, varies with the nature of the bond. zi is the effective CN of an atom in the ith atomic layer and z1 = 4(1 0.75/K) [28], z2 = z1 + 2 and z3 = 12 for a spherical dot of any size. The PL or PA energies are not the band-gap but they are determined by the joint effect of crystal binding and electron–phonon coupling. Figure 1 illustrates the effect
Figure 1. Mechanisms for EPA and EPL of a semiconductor nanocrystal, involving crystal binding (EG) and electron–phonon coupling (W) (from Ref. [26]). Insert illustrates the Stokes shift from EPA to EPL.
of electron–phonon coupling and crystal binding on the EPL and EPA. The energies of the ground state (E1) and the excited state (E2) are expressed as [19]: ( E1 ðqÞ ¼ Aq2 ð3Þ 2 E2 ðqÞ ¼ Aðq q0 Þ þ EG where the constant A is the slope of the parabolas, and q has the dimensions of a wave-vector. The vertical distance between the two minima is the real EG and depends functionally on the crystal potential. The lateral displacement of E2(q0) originates from the electron–phonon coupling, which can be strengthened by enhancing the lattice vibration. Therefore, the blue shift in the EPL and EPA is a joint effect of the change in crystal binding and electron–phonon coupling. At a surface, the CN-imperfection-enhanced bond strength affects both the frequency [29] and magnitude [30,31] of lattice vibration. Hence, at the surface, the electron–phonon coupling and hence the Stokes shift will be enhanced. In the process of carrier formation and recombination, an electron is excited by a photon with energy EG + W from the ground minimum to the excited state resulting in the creation of an electron–hole pair. The excited electron then undergoes thermalization and moves to the minimum of the excited state, and eventually returns to the ground state, combining with the hole. The carrier recombination is associated with a photon emission at energy EPL = EG W. The transition processes (electron–hole production and recombination) follow the rule of momentum and energy conservation, though conservation may be relaxed for the short-range ordering found in nanocrystals. This relaxation in the conservation law is responsible for the broad peaks in the PA and PL. The insert in Figure 1 illustrates the Stokes shift, 2W ¼ 2Aq20 , from E PL to EPA. The q0 is inversely proportional to atomic distance di, and hence, Wi = A/(cid)2, in the surface region. Based on this premise, the blue shift of the EPL, the EPA and the Stokes shift can be correlated to the CN-imperfection-induced bond contraction:
L. Pan et al. / Scripta Materialia 60 (2009) 1105–1108
9 DEPL ðDÞ = EPL ð1Þ
DEPA ðDÞ ; EPA ð1Þ
B¼E
G ðDÞDW ðDÞ ¼ DE ffi EG ð1ÞW ð1Þ
A G ð1Þd
2
;
W ð1Þ EG ð1Þ
0
P
ci ½ðcm 1Þ Bðc2 i i 1Þ
i63
!
ð4Þ One can also easily calculate the size-dependent EPL, EPA and EG = (EPL + EPA)/2 with Eq. (4). Fitting the measured data gives the values of m and A for a specific semiconductor. In our previous work [26], the size dependence of both the EPL and EPA of Si nanocrystal has been studied. A combination of theoretical prediction with the measured EPL and EPA data has been used to discriminate the effect of electron–phonon coupling (B = 0.91) from the effect of crystal binding (m = 4.88) in Si nanocrystal. Figure 2a and b compares the predictions (solid line) and measurements on the size dependence of the EPL and EPA of Ge nanocrystal. Matching the prediction in Eq. (4) with the EPL and EPA data of Ge nanocrystal gives B = 4.26 and m = 7.57. The consistency of the prediction in Eq. (2) with the EG data of Ge nanocrystal further confirms the accuracy of the BOLS premise. Table 1 shows the BOLS calculation parameters and results for Si and Ge nanocrystals. Compared with Si nanocrystal (B = 0.91), Ge nanocrystal exhibits stronger electron–phonon coupling. Electron–phonon coupling is due to a phonon-induced deformation of the crystal
Figure 2. Comparison between predictions (solid line) and measurements on the size dependence of: (a) the EPL of Ge nanocrystal with Data-1 [34], Data-2 [35] and Data-3 [36]; (b) the EPA of Ge nanocrystal with Data-1 [37], Data-2 [38] and Data-3 [39]; (c) the EG of Ge nanocrystal with Data-1 [40], Data-2 [41], Data-3 [42], Data-4 [43] and Data-5 [44].
Table 1. The BOLS calculation parameters and results for Si and Ge nanocrystals. Sample
Bulk EG (eV)
d (nm)
m
B
Si Ge
1.12 0.66
0.2632 0.2732
4.88 7.57
0.91 4.26
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lattice acting on the charge distribution of the exciton and results in a lowering of the excitonic energy [11]. Compared to Si, Ge nanostructures exhibit more prominent quantum size effects [32,33]. The Bohr exciton radius is larger in Ge (24.3 nm) than in Si (4.9 nm): this increases the oscillator strength of the indirect transitions, which enhances the magnitude of lattice vibration and causes a larger deformation of the crystal lattice to minimize the excitonic energy. As a result, stronger electron–phonon coupling is found in Ge nanocrystal. Consistency between BOLS prediction and measurements in Ge nanocrystal shows once again the effect of atomic coordination imperfection on the unusual behavior of photons, phonons and electrons in a nanocrystal. In summary, the size-enhanced Stokes shift and the size-enlarged band-gap expansion of Si and Ge nanocrystals have been accurately described by BOLS correlation in which the bond order deficiency of surface atoms dictates electron–phonon coupling and crystal binding, and enhances the shape and size dependence of the observed blue shift in PL and PA of Si and Ge nanocrystals. This work was supported by Shanghai Natural Science Foundation (No. 07ZR14033), Shanghai Pujiang Program (No. 08PJ14043), Special Project for Nanotechology of Shanghai (No. 0752nm011), and Applied Materials Shanghai Research & Development Fund (No. 07SA12). [1] K.D. Hirschmann, L. Tsybekov, S.P. Duttagupta, P.M. Fauchet, Nature 384 (1996) 338. [2] A. Fowler, Phys. Today 50 (1997) 50. [3] M. Zacharias, P.M. Fauchet, Appl. Phys. Lett. 71 (1997) 380. [4] M.V. Wolkin, J. Jorne, P.M. Fauchet, G. Allan, C. Delerue, Phys. Rev. Lett. 82 (1999) 197. [5] P.F. Trwoga, A.J. Kenyon, C.W. Pitt, J. Appl. Phys. 83 (1998) 3789. [6] Y.D. Glinka, S.H. Lin, L.P. Hwang, Y.T. Chen, N.H. Tolk, Phys. Rev. B 64 (2001) 085421. [7] G.G. Qin, H.Z. Song, B.R. Zhang, J. Lin, J.Q. Duan, G.Q. Yao, Phys. Rev. B 54 (1996) 2548. [8] F. Koch, V. Petrova-Koch, T. Muschik, A. Nikolov, V. Gavrilenko, Microcrystalline Semiconductors: Materials Science and Devices, vol. 283, Materials Research Society, Pittsburgh, 1993, p. 197. [9] G. Ledoux, J. Gong, F. Huisken, Q. Guillois, C. Reynaud, Appl. Phys. Lett. 80 (2002) 4834. [10] B. Garrido, M. Lopez, A. Perez-Rodrigez, C. Garcia, P. Pellegrino, Nucl. Instrum. Methods Phys. Res. Sect. B 216 (2004) 213. [11] J. Marin, F. Cichos, F. Huisken, C. von Borczyskowski, Nano Lett. 8 (2008) 656. [12] C.Q. Sun, T.P. Chen, B.K. Tay, S. Li, H. Huang, Y.B. Zhang, L.K. Pan, S.P. Lau, X.W. Sun, J. Phys. D 34 (2001) 3470. [13] Q. Zhang, S.C. Bayliss, J. Appl. Phys. 79 (1996) 1351. [14] F. Koch, V. Petrova-Koch, T. Muschik, J. Lumin. 57 (1993) 271. [15] M.G. Bawendi, P.J. Carroll, W.L. Wilson, L.E. Brus, J. Chem. Phys. 96 (1997) 946. [16] A.L. Efros, M. Rosen, M. Kuno, M. Nirmal, D.J. Norris, M.G. Bawendi, Phys. Rev. B 54 (1996) 4843. [17] M. Nirmal, D.J. Norris, M. Kuno, M.G. Bawendi, A.L. Efros, M. Rosen, Phys. Rev. Lett. 75 (1995) 3728.
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