Radiative decay rates of impurity states in semiconductor nanocrystals Vadim K Turkov, Alexander V Baranov, Anatoly V Fedorov, and Ivan D Rukhlenko Citation: AIP Advances 5, 107126 (2015); doi: 10.1063/1.4934595 View online: http://dx.doi.org/10.1063/1.4934595 View Table of Contents: http://aip.scitation.org/toc/adv/5/10 Published by the American Institute of Physics AIP ADVANCES 5, 107126 (2015) Radiative decay rates of impurity states in semiconductor nanocrystals Vadim K Turkov,1 Alexander V Baranov,1 Anatoly V Fedorov,1 and Ivan D Rukhlenko1,2 ITMO University, 197101 Saint Petersburg, Russia Monash University, Clayton Campus, Victoria 3800, Australia (Received 31 August 2015; accepted 12 October 2015; published online 20 October 2015) Doped semiconductor nanocrystals is a versatile material base for contemporary photonics and optoelectronics devices Here, for the first time to the best of our knowledge, we theoretically calculate the radiative decay rates of the lowest-energy states of donor impurity in spherical nanocrystals made of four widely used semiconductors: ZnS, CdSe, Ge, and GaAs The decay rates were shown to vary significantly with the nanocrystal radius, increasing by almost three orders of magnitude when the radius is reduced from 15 to nm Our results suggest that spontaneous emission may dominate the decay of impurity states at low temperatures, and should be taken into account in the design of advanced materials and devices based on doped semiconductor nanocrystals C 2015 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4934595] I INTRODUCTION Semiconductor nanocrystals is a unique class of materials which find multiple applications in modern electronics, photonics, biology, and medicine.1–4 The physical properties of such nanocrystals can be tuned over a wide range by changing their size, shape, and chemical composition Another popular method of controlling these properties is doping.5,6 The introduction of dopants allows one to tune the band gap and Fermi energy of the nanocrystal’s charge carriers,7,8 and also to alter the photoluminescence spectrum of the nanocrystal without changing its size.9,10 These and other degrees of freedom associated with the doping find semiconductor nanocrystals many applications in photonics,11–13 solotronics,14 photovoltaics,15 and biophysics.16 There are many theoretical approaches to the description of energy spectra of impurity centers in semiconductor nanocrystals.5 One of them employs the hydrogenic model of the center, which is useful for understanding the optical properties of donor impurities in low-dimensional systems.17 A donor impurity atom has more electrons than what is needed to make a chemical bond with the neighboring atoms in the semiconductor host If the atom has one extra electron, such a donor impurity is known as hydrogenic one for being analogous to a hydrogen atom Optical properties of impurity centers in semiconductor nanocrystals have been extensively studied in many papers on inter-level transitions,18,19 nonlinear optical response,20,21 and photoionization.22 Still the dominant relaxation channels of excited impurity states are under debate, and their radiative decay rates reported in literature vary greatly Here we theoretically demonstrate that the radiative decay rates of the excited impurity states depend heavily on the nanocrystal size The impurity states of a spherical nanocrystal are described using the common hydrogen model, which enables the exact solution of the Schrödinger equation and the analytical calculation of the wave functions and energy spectrum of the impurity.6 This model is based on the effective mass approximation (EMA) and widely used to calculate the energy spectra of semiconductor nanocrystals.23 Experiments showed that EMA is well justified for nanocrystals as small as nm in diameter.24 By analytically calculating the inter-level matrix elements of singlephoton optical transitions between the impurity states, we derive a general expression for the radiative decay rate of an excited impurity state We use this expression to study the radiative relaxation in 2158-3226/2015/5(10)/107126/6 5, 107126-1 © Author(s) 2015 107126-2 Turkov et al AIP Advances 5, 107126 (2015) semiconductor nanocrystals made of ZnS, CdSe, Ge, and GaAs, and show that the radiative decay rates of their impurity states can rise by a factor of 1000 when the nanocrystal radius is reduced from 15 to nm II IMPURITY ENERGY SPECTRUM Consider a hydrogen-like donor impurity inside a spherical semiconductor nanocrystal of radius R and low-frequency permittivity ε Let the impurity be an ion of electric charge Ze located in the nanocrystal center The ion is coupled through Coulomb interaction to an electron of charge −e and conduction-band effective mass mc We assume that the nanocrystal is embedded in a wide band gap dielectric matrix, so that its confining potential for electrons in the lowest-energy states can be well approximated by an infinite potential well Then the potential energy of the electron inside the nanocrystal of is given by V = −Ze2/(ε 0r) The energy spectrum of the impurity has negative-energy levels, which are located within the band gap, and positive-energy levels in the conduction band, as shown in Fig The number of negative-energy states is determined by the nanocrystal radius In particular, all the impurity states are located in the conduction band for R ≤ 1.8 r B, where r B = ε 2/(Ze2mc ) is the impurity’s Bohr radius, and there is only one state within the nanocrystal band gap if 1.8 r B ≤ R ≤ 5.1 r B.6 The solution of the Schrödinger equation with the above potential is given in spherical coordinates (r, ϑ, ϕ) by the atomic-like wave function of the form Ψnlm(r) = Fnl(r)Ylm(ϑ, ϕ), (1) where n, l, and m are the principle quantum number, orbital momentum, and its projection, respectively, Fnl is the normalized radial part of the wave function, and Ylm are the spherical harmonics Since the energy spectrum of the impurity is determined by the boundary condition Fnl(R) = 0, each energy level in the spectrum is m-fold degenerate and characterized by a pair of quantum numbers n and l For the sake of convenience, we use the standard atomic notations nQ, where Q = s, p, d, corresponds to the angular momenta l = 0, 1, 2, , to mark the impurity states In these notations, the ground impurity state is 1s To study the radiative decay rates of impurity states in semiconductor nanocrystals, we focus on four widely used semiconductors—ZnS, CdSe, Ge, and GaAs—whose material parameters are listed in Table I The size dependencies of the lowest six energies in the impurity spectrum of nanocrystals made of these semiconductors are plotted in Fig The range of the nanocrystal radii in the plots is chosen such as to put the frequencies of intraband transitions in the optical domain As we showed in our previous paper, some impurity states can become accidentally degenerate in nanocrystals of certain radii.6 This degeneracy may be completely or partially removed by the FIG Energy-level diagram of impurity located in the center of semiconductor nanocrystal of radius R The four lowest-energy states of quantum numbers (nl) = (10), (11), (12) and (20) are denoted as in atoms by 1s, 1p, 1d and 2s 107126-3 Turkov et al AIP Advances 5, 107126 (2015) TABLE I Band gaps, effective masses, low- and high-frequency permittivities, and Bohr radii of ZnS, CdSe, Ge, and GaAs;25 m is the free-electron mass ZnS CdSe Ge GaAs E g (eV) m c /m ε0 ε∞ r B (nm) 3.76 1.73 0.89 1.42 0.220 0.120 0.082 0.066 8.10 9.10 11.93 12.80 5.13 6.30 10.90 10.86 1.95 4.02 7.67 10.24 interaction of the impurity with excitations of the nanocrystal26–29 or its environment,30,31 resulting in the splitting of the degenerate levels and renormalization of the impurity energy spectrum These are manifested as an anticrossing in the size dependencies of the degenerate energy levels The accidental degeneracy in particular takes place for impurity states 1d and 2s inside nanocrystals with R = r B In accordance with Table I, the intersection of the respective energy levels is only seen in Fig 2(b) for the nm nanocrystal made of CdSe In the following analysis, we shall ignore the anticrossing of these levels for two reasons First, because the resulting energy shifts would be small compared to the gaps between these states and the adjacent states 1p and 2p And second, because transitions 1d ↔ 2s are dipole forbidden, which means that optical transitions between the renormalized states 1d and 2s would be absent III RADIATIVE RATES OF IMPURITY STATES The transition rate of electrons from the initial impurity state |i⟩ of energy Ei to the final impurity state | f ⟩ of energy E f , with the spontaneous emission of photons of energy ωfi = Ei − E f , is given FIG Size dependencies of first six impurity energy levels in (a) ZnS, (b) CdSe, (c) Ge, and (d) GaAs nanocrystals 107126-4 Turkov et al AIP Advances 5, 107126 (2015) by the Fermi’s golden rule Wfi = 2π ρω Mfi δ ω − ωfi dΩ dω, (2) where δ(x) is the Dirac delta function, ρω = 2V ε 3/2 ∞ ω (2πc)3 (3) is the density of photon states in the frequency range from ω to ω + dω and with photon polarization vectors in the solid angle dΩ, and V is the normalization volume The factor of in ρω takes into account two possible polarization states of photons The intraband matrix element in Eq (2) is given by Mfi = −e (−1) µ E µ ⟨n f l f m f |x −µ |ni l i mi ⟩, (4) µ=0,±1 where ⟨n f l f m f |x µ |ni l i mi ⟩ = Φn f l f n i l i χ µ δ m i, m i +µ , R Φn f l f n i l i = Fn f l f Fn i l i r 3dr, (5a) (5b) χ0 = δl f ,l i +1 (l i + 1)2 + m2i − δl f ,l i −1 (2l i + 1)(2l i + 3) l i2 − m2i , (2l i − 1)(2l i + 1) (5c) and χ±1 = δl f ,l i +1 (l i ± mi + 1)(l i ± mi + 2) − δl f ,l i −1 2(2l i + 1)(2l i + 3) (l i ∓ mi − 1)(l i ∓ mi ) 2(2l i − 1)(2l i + 1) (5d) The matrix element in Eq (4) is written using the covariant cyclic coordinates.32 In these coordinates, the electric field components are given by33 4π 2π ω Eµ = |E|Y1µ (ϑ, ϕ), |E| = (6) ε ∞V One can see from Eq (5) that the selection rules for intraband transitions between the impurity states in spherical semiconductor nanocrystals coincide with the Laporte selection rules,34 i.e ∆l = l f − l i = ±1 and ∆m = m f − mi = 0, ±1 It should be noted that the displacement of impurity from the nanocrystal center partially removes the degeneracy of the impurity states35 and changes the selection rules for optical transitions If the displacement is small compared to the nanocrystal radius, then the modification of the electronic subsystem can be taken into account using the perturbation theory The radiative decay rate γi of the excited impurity state |i⟩ is the sum of the rates of the allowed optical transitions from this state to all the lower-lying states | f 1⟩, | f 2⟩, , | f n ⟩ of the impurity, γi = f Wfi = ε 1/2 ∞ e Ei − E f 3 c f Φn f l f n i l i ( ) χ2+1δ m f , m i +1 + χ20δ m f , m i + χ2−1δ m f , m i −1 (7) Equations (5) and (7) constitute the main result of this paper They allow one to calculate the natural linewidths in the photoluminescence spectrum of a spherical semiconductor nanocrystal Being applicable to semiconductor nanocrystals whose wave functions are in the form of Eq (1), they can also be used to calculate radiative decay rates in nanocrystals with other kinds of spherically symmetric confining potentials It follows from Eq (7) that the radiative decay rate of an impurity state depends on the number of intraband optical transitions from this state In particular, the decay rates of states 1p, 2s, and 1d are equal to the rates of one transition each (1p → 1s, 2s → 1p, and 1d → 1p), state 3s decays through a 107126-5 Turkov et al AIP Advances 5, 107126 (2015) FIG Size dependencies of the radiative decay rates of the lowest five states of a donor impurity in nanocrystals made of (a) ZnS, (b) CdSe, (c) Ge, and (d) GaAs pair of intraband transitions (3s → 2p and 3s → 1p), and three transitions (2p → 1d, 2p → 2s, and 2p → 1s) contribute to the decay rate of state 2p Figure shows the size dependencies of the radiative decay rates of the lowest five states (but the ground state) of a donor impurity in ZnS, CdSe, Ge, and GaAs nanocrystals These rates are seen to span over a wide range from 103 to 109 s−1 In each of the four nanocrystals, the variations of the decay rates span over less than two orders of magnitude when the nanocrystal radius is varied between and 15 nm, except for state 2s in ZnS and CdSe nanocrystals In the latter case the decay rate varies by more than × 102 and × 103 times, which is due to the significant convergence of energy levels 2s and 1p with the nanocrystal radius: ∆E = E2s − E1p reduces from 80 meV at R = nm to meV at R = 15 nm in ZnS nanocrystal, and from 200 to 13 meV in CdSe nanocrystal The highest decay rate is exhibited by state 2p regardless of the nanocrystal material, whereas the slowest decays are featured by state 2s in ZnS and CdSe nanocrystals and by states 1p and 2s in Ge and GaAs nanocrystals IV CONCLUSIONS As a concluding remark, we would like to note that the spontaneous emission of photons can dominate the decay of donor-impurity states in semiconductor nanocrystals at low temperatures This happens, on one hand, since fast Auger relaxation, which is the dominant energy-loss mechanism for electrons in impurity-free nanocrystals,36 does not take place for the impurity electron in the absence of holes On the other hand, relaxation with the emission of elementary excitations of the nanocrystal itself or its environment is only possible when the energy of an elementary excitation matches the gap between a pair of impurity levels.37,38 Such a resonance can only occur for certain impurity states in the nanocrystals of particular sizes due to the size quantization effect The resonance is not required in the case of relaxation accompanied by the simultaneous emission of several excitation, but the efficiency of such processes is negligibly small 107126-6 Turkov et al AIP Advances 5, 107126 (2015) ACKNOWLEDGMENT This work was funded by Grant 14.B25.31.0002 and Government Assignment No.3.17.2014/K of the Ministry of Education and Science of the Russian Federation M V Kovalenko, L Manna, A Cabot, Z Hens, D V Talapin, C R Kagan, V I Klimov, A L Rogach, P Reiss, D J Milliron, P Guyot-Sionnnest, G Konstantatos, W J Parak, T Hyeon, B A Korgel, C B Murray, and W Heiss, ACS Nano 9, 1012 (2015) E V Ushakova, A P Litvin, P S Parfenov, A V Fedorov, M Artemyev, A V Prudnikau, I D Rukhlenko, and A V Baranov, ACS Nano 6, 8913 (2012) A S Baimuratov, I D Rukhlenko, V K Turkov, A V Baranov, and A V Fedorov, Sci Rep 3, 1727 (2013) A S Baimuratov, I D Rukhlenko, and A V Fedorov, Opt Lett 38, 2259 (2013) B I Shklovskii and A L Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, New York, 1984) A S Baimuratov, I D Rukhlenko, V K Turkov, I O Ponomareva, M Y Leonov, T S Perova, K Berwick, A V Baranov, and A V Fedorov, Sci Rep (2014) D Mocatta, G Cohen, J Schattner, O Millo, E Rabani, and U Banin, Science 332, 77 (2011) R Buonsanti and D J Milliron, Chem Mater 25, 1305 (2013) S Jana, G Manna, B B Srivastava, and N Pradhan, Small 9, 3753 (2013) 10 G Bai, M.-K Tsang, and J Hao, Adv Optical Mater 3, 431 (2014) 11 N S Karan, D D Sarma, R M Kadam, and N Pradhan, J Phys Chem Lett 1, 2863 (2010) 12 P Ginzburg and M Orenstein, J Appl Phys 103, 083105 (2008) 13 P Ginzburg and M Orenstein, J Appl Phys 104, 063513 (2008) 14 J Kobak, T Smolenski, M Goryca, M Papaj, K Gietka, A Bogucki, M Koperski, J.-G Rousset, J Suffczynski, E Janik, M Nawrocki, A Golnik, P Kossacki, and W Pacuski, Nat Commun 5, 3191 (2014) 15 D Zhitomirsky, M Furukawa, J Tang, P Stadler, S Hoogland, O Voznyy, H Liu, and E H Sargent, Adv Mater 24, 6181 (2012) 16 P Wu and X.-P Yan, Chem Soc Rev 42, 5489 (2013) 17 P Harrison, Quantum Wells, Wires and Dots (Wiley, New York, 2005) 18 H Tas and M Sahin, J Appl Phys 112, 053717 (2012) 19 A Ozmen, Y Yakar, B Cakir, and U Atav, Opt Commun 282, 3999 (2009) 20 W Xie, Phys Lett A 372, 5498 (2008) 21 J D Correa, M E Mora-Ramos, and C A Duque, J Appl Phys 115, 213105 (2014) 22 M Sahin, Phys Rev B 77, 045317 (2008) 23 D E Gomez, M Califano, and P Mulvaney, Phys Chem Chem Phys 8, 4989 (2006) 24 U Banin, C J Lee, A A Guzelian, A V Kadavanich, A P Alivisatos, W Jaskolski, G W Bryant, A L Efros, and M Rosen, J Chem Phys 109, 2306 (1998) 25 O Madelung, Semiconductors: Data Handbook (Springer, New York, 2004) 26 I D Rukhlenko, A V Fedorov, A S Baymuratov, and M Premaratne, Opt Express 19, 15459 (2011) 27 A S Baimuratov, I D Rukhlenko, M Y Leonov, A G Shalkovskiy, A V Baranov, and A V Fedorov, Opt Express 22, 19707 (2014) 28 A V Baranov and A V Fedorov, Semiconductors 38, 1065–1073 (2004) 29 A V Baranov, A V Fedorov, and Y Masumoto, Opt Spectrosc 93, 52 (2002) 30 I D Rukhlenko and A V Fedorov, Opt Spectrosc 100, 238 (2006) 31 A V Fedorov, A V Baranov, I D Rukhlenko, T S Perova, and K Berwick, Phys Rev B 76, 045332 (2007) 32 D A Varshalovich, A N Moskalev, and V K Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988) 33 V K Turkov, S Y Kruchinin, and A V Fedorov, Opt Spectrosc 110, 740 (2011) 34 B Henderson and G F Imbusch, Optical Spectroscopy of Inorganic Solids (Oxford University Press, New York, 2006) 35 J Zhu and X Chen, Phys Rev B 50, 4497 (1994) 36 P Kambhampati, J Phys Chem C 115, 22089 (2011) 37 A Fedorov, A Baranov, I Rukhlenko, and Y Masumoto, Solid State Commun 128, 219 (2003) 38 I D Rukhlenko and A V Fedorov, Opt Spectrosc 101, 253 (2006) ... calculate the radiative decay rates of the lowest-energy states of donor impurity in spherical nanocrystals made of four widely used semiconductors: ZnS, CdSe, Ge, and GaAs The decay rates were... renormalized states 1d and 2s would be absent III RADIATIVE RATES OF IMPURITY STATES The transition rate of electrons from the initial impurity state |i⟩ of energy Ei to the final impurity state | f ⟩ of. .. the radiative decay rate of an impurity state depends on the number of intraband optical transitions from this state In particular, the decay rates of states 1p, 2s, and 1d are equal to the rates