Magnetic Properties of Nonmagnetic Nanostructures: Dangling Bond

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Magnetic properties of nonmagnetic nanostructures: Dangling bond magnetic polaron in CdSe nanocrystals Anna Rodina, and Alexander Lev Efros Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b01566 • Publication Date (Web): 28 Apr 2015 Downloaded from http://pubs.acs.org on May 3, 2015

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Magnetic properties of nonmagnetic nanostructures: Dangling bond magnetic polaron in CdSe nanocrystals Anna Rodina† and Alexander L. Efros∗,‡ Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia, and Naval Research Laboratory, Washington DC 20375, USA E-mail: [email protected]

∗

To whom correspondence should be addressed Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia ‡ Naval Research Laboratory, Washington DC 20375, USA

†

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Abstract We predict theoretically that non-magnetic CdSe nanocrystals may possess macroscopic magnetic moments due to the formation of dangling-bond magnetic polarons (DBMPs). A DBMP is created in optically pumped nanocrystals by dynamic polarization of dangling bond spins (DBSs) at the nanocrystal surface during radiative recombination of the ground state ”dark” exciton assisted by a spin-flip of the DBS. The formation of DBMPs suppresses the radiative recombination of the dark exciton and leads to a temperature-dependent contribution to the Stokes shift of the photoluminescence. This model consistently explains the experimentally observed low-temperature photoluminescence features of non-magnetic CdSe nanocrystals as manifestations of their spin-related magnetic properties.

Keywords: dark exciton, CdSe, nanocrystal, spin flip, dynamic polarization, magnetic polaron


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The growing attention paid to semiconductor nanocrystals (NCs) is connected with their unusual and potentially useful electro-optical and transport properties, which are mainly controlled by the NC size. Since 1993, when the first reliable technology of colloidal CdSe NC growth was reported by Murray et al.,

1

practically all semiconductors have been prepared in

NC form. The electronic structure and absorption spectra of the NCs are quite well understood and can be accurately described using multiband effective mass approximations 2–4 and first principles calculations. 5,6 Surprisingly, however, after more than 20 years of research, low temperature radiative recombination in NCs is still under discussion. This is connected with ambiguity in the recombination properties of the ground exciton state, which was shown to be an optically passive (dark) exciton state in colloidal NCs. 7,8 The experimentally observed radiative recombination should be caused by the admixture of these states with optically active (bright) exciton states, e.g. via phonons or some external or internal magnetic fields. 7–9 The energy splitting between the dark exciton and the lowest bright exciton state depends strongly on the size and shape of the NC and can be as large as 20 meV in a 2.3 nm diameter CdSe NC. 8 To understand the mixing mechanism, many groups have studied the fine structure of the NC band edge photoluminescence (PL) and exciton recombination dynamics as function of temperature with and without magnetic fields. 10–20 Although activation of dark exciton recombination by an external magnetic field has been well documented, the temperature dependence of PL decay times and PL structure remain controversial and even puzzling. For example, the temperature dependence of exciton lifetimes in CdSe NCs show activation energies 10,11,13 that are significantly smaller than the bright-dark exciton splitting predicted theoretically and observed experimentally in fluorescence line narrowing (FLN) experiments. 7,21 An important hint about the mechanism of the dark exciton recombination was obtained in one of first studies of CdSe NC PL. 22 Using FLN experiments, Nirmal et. al 22 have shown that 3 ACS Paragon Plus Environment

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CdSe NCs PL consists of so called zero–phonon line (ZPL) and clearly distinguished optical phonon assisted satellites, all Stokes shifted from the photo-excitation energy. The Stokes shift of the ZPL is the dark–bright exciton splitting. Experiments show that the intensity of the ZPL dramatically increases with both temperature and magnetic field and becomes the strongest in the FLN spectra. The magnetic field induced activation of the dark exciton is obviously connected to its mixing with the bright exciton. This raises questions about the thermo-activation mechanism of dark exciton recombination. The similarity between the effects of temperature and of magnetic field on the dark exciton recombination suggests that we observe some kind of magnetic polaron which have been found in diluted magnetic semiconductors. 23,24 Indeed, the magnetic moments, which at low temperature can be aligned via their interaction with an exciton along the NC wurzite axis (c-axis), become arbitrarily oriented with a temperature rise. This creates an arbitrarily oriented internal magnetic field, which has a component oriented perpendicular to the c-axis and mixes the dark and bright exciton states similar to an external magnetic field. What is the origin of these magnetic moments? The coupling between electrons or excitons with spins of magnetic ions in Mn-doped quantum dots is well documented. 25–30 However, the commonly grown CdSe NCs do not contain magnetic impurities. It was suggested that the spin of the surface dangling bonds could be responsible for this observed phenomenon. 7,8,31 Existence of the dangling bond spins (DBSs) at the NC surfaces and their magnetization by an external magnetic field was demonstrated recently by SQUID measurements. 32,33 In this article we study theoretically the effect of the spins of the surface dangling bonds on the PL of CdSe NCs. We show that the interaction of these spins with the spin of an electron localized in the NC mixes up the dark and bright excitons allowing flip – flop spin–assisted recombination of the dark exciton. This mechanism of dark exciton recombination is shown schematically in Figure 1(a). We also show that at low temperatures, the dangling bond assisted dark exciton recombination in CdSe NCs may lead to the dynamic polarization 4 ACS Paragon Plus Environment

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Figure 1: Schematic image of the dark exciton radiative decay via the dangling bond spin flipflop assisted recombination. (a) The radiative decay of the dark exciton occurs via its spin flip-flop virtual transition to the bright exciton and the bright exciton’s consequential radiative decay. Above the critical temperature, Tc , the dangling bond spins (DBSs) are not polarized and there are always DBSs which could assist dark exciton recombination. (b) Below the critical temperature, Tc , dangling bond assisted recombination of the dark exciton results in dynamic polarization of DBSs. The polarized DBS splits the ground ±2 dark exciton into two spin sublevels and creates a dangling bond magnetic polaron (DBMP). The number of DBSs available for the flip-flop assisted recombination of the ground dark exciton state is diminished in the DBMP and its radiative decay is suppressed. of DBSs and to the formation of a dangling bond magnetic polaron (DBMP) with a magnetic moment aligned along the NC hexagonal c-axis. This process shown schematically in Figure 1(b) takes place at low temperatures if the interaction between the electron spin in the exciton and the DBS is ferromagnetic and the rate of optical pumping of the DBSs is faster than the 5 ACS Paragon Plus Environment

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DBS relaxation rate. The DBMP formation does not require a resonant or polarized excitation. The formation of the DBMP suppresses the dangling bond assisted radiative recombination and results in an additional Stokes shift of the PL connected with the polaron binding energy. A temperature rise suppresses the polaron formation. The suppression of the dangling bond magnetic polaron explains the temperature dependence of the PL decay time and the FLN structure measured in CdSe NCs. Thus, the low temperature PL of CdSe NCs is a manifestation of their magnetic properties. Dangling Bond Assisted Recombination of the Dark Exciton. In most of the commonly grown CdSe NCs, the ground exciton has momentum projection F = ±2 and is an optically passive (dark) state. The separation ∆EAF between the ground state and the lowest optically active F = ±1L state increases with decreasing NC radius a as ∆EAF ≈ 3η(a) ∝ 1/a3 and tends to saturate in very small NCs to ∆EAF ≈ 3∆/4. Here η is the electron-hole exchange constant defined in Ref. 8 and ∆ is the splitting between heavy and light hole sublevels caused by the hexagonal lattice structure and NC shape asymmetry. 8 The ∆EAF splitting depends strongly on the size and shape of NCs and varies from several meV to 20 meV in the smallest CdSe NCs. 8 As a result the low temperature PL in CdSe NCs is completely controlled by the optical properties of the ±2 dark exciton state. The spins of the surface dangling bonds generally admix the bright excitons with the dark ±2 excitons via their exchange interaction with the electron spin. This admixture makes the dark excitons optically active and results in the zero-phonon PL line. The electron-DBSs exchange interaction can be written as: ˆ ss = H

X

ˆ j = −α H ss

j

X

ˆ j, σê σ

(1)

j

ˆ j are the Pauli matrices for the electron and j-th dangling bond spins, respecwhere σê and σ


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tively. The sum is over all dangling bonds. Here we assume that the NC has a spherical shape and the exchange interaction of an electron in the ground 1Se state with all dangling bonds is the same: α = αa ˜ 30 ξ 2 (a), where α ˜ is the exchange strength constant for electrons from the conduction band and the dangling bonds at the NC surface, a0 is the lattice constant, and ξ 2 (a) is the square of the electron wave function at the NC surface. The spin-spin interaction of Eq. (1) aligns the DBSs along the spin of an electron, which due to a very strong electron–hole exchange interaction in small size CdSe NCs, is in turn aligned along the NC hexagonal axes. 8 Choosing the hexagonal axis to be along the z direction, we can rewrite the electron-DBSs exchange interaction of Eq. (1) as j ˆ ssj = H ˆ zz ˆj H +H flip−flop ,

(2)

ˆ j = −αˆ H σze σ ˆzj , zz α e j e j ˆj σˆ σ ˆ +σ ˆ− σ ˆ+ H flip−flop = − 2 + − h

(3) i

.

(4)

e,j e,j where σ ˆx,y,z are the projections of electron and dangling bond spin Pauli operators, and σ ˆ± =

σ ˆxe,j ± iˆ σye,j . ˆ j term in Eq.(2) does not change either the spin of an electron or of a dangling bond. The H zz j ˆ zz Averaging H over all DBS projections, one can write the effective contribution of the dangling

ˆ s = αˆ bonds to the energy of the exciton states as H σze Ndb ρdb . Here Ndb is the total number of ex dangling bonds in the NC and

ρdb = −

P

σzj i j hˆ

Ndb

=

− + Ndb − Ndb Ndb

(5)

is the polarization of DBSs, which is controlled by the difference between the number of dan+ − gling bond spins oriented along, Ndb , and oriented anti-parallel, Ndb , to the z axis. The statis-


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± + − tical fluctuations of Ndb from the equilibrium value, Ndb = Ndb , could lead to a spontaneous

polarization of the DBSs and result in the splitting of the ±2 dark exciton state even in zero magnetic field: ∆Eex = 2αNdb ρdb .

(6)

At the same time, the DBS polarization may cause the polarization of the dark exciton population ρex = (Nex,−2 − Nex,+2)/Nex , where Nex,±2 is the population of the ±2 exciton levels and Nex is the average exciton population of the NC. The exciton polarization ρex depends generally on the energy splitting between the ±2 exciton states ∆Eex given by Eq. (6) and temperature, as well as on the exciton spin relaxation and recombination rates. In the following we assume that the spin relaxation time τs of the dark exciton is much shorter than the exciton lifetime. In this case, the relative population of the ±2 exciton states is determined only by thermal equilibrium and the exciton polarization and can be written as:

ρex = tanh

∆Eex αNdb ρdb = tanh , 2kT kT

(7)

where T is the temperature and k is the Boltzmann constant. ˆj ˆ± = The sum of H flip−flop terms in the exchange Hamiltonian (1) can be written as Ve P j e e Aˆ∓ σˆ± = −(α/2) j σ ˆ∓ σ ˆ± , an operator which admixes the bright excitons with the dark ex-

citon. This admixture causes the radiative decay of the dark ±2 exciton via the flip of the

electron spin simultaneously with the flop of the dangling bond spin in the opposite direction. The respective radiative decay rates Γ±2 flip−flop can be found in the framework of the second order perturbation theory (See the Methods section):

Γ±2 flip−flop

=

∓ 2γex Ndb

,

γex =

α η

!2


1 . 6τ0

(8)

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Here τ0 is the radiative decay time of the 0U bright exciton state 8 and γex is the flip-flop rate. The dangling bond assisted radiative decay rate of the dark exciton, Γdb , which contributes to ZPL, is controlled by the relative population of the ±2 sublevels and their radiative decay +2 −2 rates Γ±2 flip−flop : Γdb = Γflip−flop (Nex,2 /Nex ) + Γflip−flop (Nex,−2 /Nex ). This expression can be

rewritten as: Γdb = γex Ndb [1 − ρdb ρex ] .

(9)

At high temperatures, when DBSs are not polarized (ρdb = 0) and randomly oriented, the recombination rate is proportional to the total number of dangling bonds Ndb . This recombination, however, is suppressed at low temperatures, when ρdb ρex −→ 1. We will consider the mechanism of the dynamic polarization of the DBSs at low temperatures and its consequences in the next section. For temperatures higher than 6–10 K, but low enough not to populate the first bright ±1L exciton state, ZPL dominates in bare and thin shell CdSe NCs. 16,22,34 Let us consider the size dependence of the dark exciton radiative decay rate Γdb = γex Ndb in this temperature regime. Figure 2 shows the size dependences of parameters controlling the DBS assisted recombination of the dark exciton calculated for CdSe NCs (See the Methods section for the brief description of these calculations). All curves in (a)-(d) are normalized to 1 for the NC radius a = 1.2 nm. The coupling α(a) = αa ˜ 30 ξ 2 (a) between electrons and dangling bond spins is shown in Fig. 2 (a). The insert shows the dependence of α on the potential barrier U calculated for the NC with radius a = 1.2 nm assuming a0 = 0.6 nm and α ˜ = 10 meV. The size dependence of the total exchange energy αNdb is shown in Fig. 2 (b). The size dependence of the flip-flop rate γex = [α/η]2 /(6τ0 ) is shown on Fig. 2 (c). The insert shows the dependence of γex on the potential barrier U calculated for the CdSe NC with radius a = 1.2 nm assuming a0 = 0.6 nm, α ˜ = 10 meV and τ0 = 4 ns. Finally, the size dependence of the dangling bond assisted decay


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Figure 2: Size dependent parameters of the electron–dangling bond spins exchange interaction and dangling bond assisted radiative decay of the dark exciton. (a) Exchange interaction constant α(a)/α(1.2). (b) Exchange energy α(a)Ndb (a)/α(1.2)Ndb (1.2). (c) Flip-flop rate γex (a)/γex (1.2). (d) Dangling bond assisted decay rate γex (a)Ndb (a)/γex (1.2)Ndb (1.2). The calculations were conducted for CdSe NCs with different potential barriers at the surface: U = 0.6 eV, 0.8 eV, 1.0 eV and 1.2 eV. All curves in (a )-(d) are normalized to 1 for the NC radius a = 1.2 nm. Inserts in (a) and (c) show the dependence of α(1.2) and γex (1.2), respectively, on the value of the potential barrier U calculated for the CdSe NC with radius a = 1.2 nm assuming a0 = 0.6 nm, α ˜ = 10 meV and τ0 = 4 ns. rate Γdb ∝ γex Ndb is shown in Fig. 2 (d). One can see that, although the exchange constant α and exchange energy αNdb are decreasing with increase of NC size, the dangling bond assisted recombination rate Γdb is increasing. Dynamic Polarization of DBSs and Formation of the Dangling Bond Magnetic Polaron (DBMP). The polarization of ground exciton state at low temperature, ρex , connected with fluctuations of the DBS polarization ρdb described by Eq. (7), in combination with the dangling bond spin flip assisted recombination suggest the possibility of formation of the polarized state 10 ACS Paragon Plus Environment

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with ρdb 6= 0 and ρex 6= 0. Indeed, one can envision that the dangling bond spin flip assisted recombination of the polarized dark exciton would increase ρdb , which in turn increases ρex , etc., leading to combined exciton - dangling bond magnetic polarization in the NC. Below, we will show, that such dynamic polarization and formation of the DBMP depend critically on the sign of the exchange interaction constant α. The other obvious critical condition of DBMP formation is connected with the efficiency of the DBS pumping via the exciton spin flip assisted recombination, which must be faster than the DBS relaxation. The rate of pumping is controlled by the average population of excitons in NCs, Nex . For relatively weak NC excitation, for which Nex ≪ 1, the averaged exciton population of a single NC is Nex = Gex τ = [Iex σcr (ωex )/¯hωex ]τ , where Gex is the exciton generation rate per NC, where Iex is the intensity of exciting light, ωex is the excitation frequency, σcr (ωex ) is the absorption crossection at the excitation frequency, 35 and τ is the exciton life time. This expression for Nex can be used both for continuous wave (CW) excitations and for pulsed excitations, assuming Iex is averaged over the period Tp of the pulse repetition. For pulsed excitations the number of excitons created during one pulse does not depend on the lifetime (if the pulse duration is shorter than the exciton lifetime), while averaging over repeating pulses gives the factor τ /Tp . To describe the formation of the DBMP let us consider the dynamics of the DBS polarization caused by dark exciton recombination. The rate equations for the dangling bond populations + − Ndb and Ndb can be written:

± h i h i dNdb ± ± ∓ ∓ = −Ndb 2γex Nex,∓2 + γdb + Ndb 2γex Nex,±2 + γdb , dt

(10)

where the terms 2γex Nex,±2 = γex Nex (1 ∓ ρex ) describe the rate of the DBS flip during the ± dark exciton recombination and γdb are the spin relaxation rates of the dangling bonds, which


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are not connected with exciton decay. One can see that the efficiency of the DBS pumping is proportional to the average population of excitons in NCs, Nex . According to Eq. (7) the exciton polarization ρex is determined by the energy splitting of the ±2 excitons induced by the polarized dangling bonds. It is described by the thermal equilibrium ˆ zz distribution due to fast exciton spin relaxation. In turn, as it follows from the Hamiltonian H in Eq. (3), the exciton polarization leads to the energy splitting of the DBS projections:

∆Edb = 2αρex .

(11)

− The thermal equilibrium condition, consequently, leads to the following relationship γdb = + − + γdb exp[−∆Edb /(kT )]. 36 Extracting the equation for Ndb from the equation for Ndb described

by Eqs. (10), we obtain the equation for the rate of dangling bond polarization: dρdb = −2γex Nex [ρdb − ρex ] − γdb [ρdb − Nex ρ¯db ] , dt

(12)

+ − + − where γdb = γdb + γdb , and ρ¯db = (γdb − γdb )/γdb = tanh[αρex /kT ] ≈ αρex /kT is a thermo-

equilibrium polarization of the dangling bond in the presence of the exciton. In the absence of an exciton, dangling bonds relax down to zero polarization. To take this effect into account we introduce in Eq. (12) a time averaged thermo-equilibrium polarization of the dangling bond as Nex ρ¯db . The steady state solution of Eq. (12) gives us the connection between the DBS polarization and the polarization of the exciton in the exchange field of the polarized dangling bonds

ρdb =

2ρex γex Nex + γdb Nex ρ¯db . 2γex Nex + γdb

(13)

A stability analysis shows that the trivial solution ρdb = ρ¯db = ρex = 0 of unpolarized dangling 12 ACS Paragon Plus Environment

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bonds and the exciton in NCs is always stable in the case of the antiferromagnetic exchange interaction between dangling bonds and electron spins described by α < 0. For the opposite case of ferromagnetic exchange interaction with α > 0, the stability of the trivial solution depends dramatically on the exciton pumping rate, dangling bond relaxation rate and temperature. At low pumping intensity Nex ≪ 1 and a considerable relaxation rate of the DBS polarization (i.e

db

γdb ≫ γex Nex ), only the trivial solution is possible.

Nex Ndb

ex Nex

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-4

Ndb kT

Ndb kT Figure 3: The phase diagram showing the polarized (polaron) and unpolarized (no polaron) states of the exciton and dangling bond spins. The phase separation lines calculated for Nex /Ndb = 0.0001 (blue curve), 0.001 (red curve) and 0.005 (black curve) are shown in the plane of control parameters (2γex Nex /γdb , αNdb /kT ), and in the plane of control parameters (γdb /2γex Nex , αNdb /kT ) (in the insert). The situation changes, however, with an increase of the exciton pumping rate or a decrease of the dangling bond relaxation rate. One can see that if γdb ≪ γex Nex , Eq. (13) gives ρdb =


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ρex . Substitution of this expression into Eq. (7) results in ρex = tanh [αρex Ndb /kT ], which describes the spontaneous polarization of the exciton and the dangling bonds respectively if αNdb /kT ≥ 1. Adapting the terminology used in diluted magnetic semiconductors, 23,24 we will call this spontaneously created magnetic state of the NC a dangling bond magnetic polaron (DBMP). Substituting Eq. (13) into Eq. (7) one can find a general condition for DBMP formation. In the limit case α ≪ kT it reads: 2γex Nex αNdb αNdb −1 ≥1− γdb kT kT

2

Nex . Ndb

(14)

This inequality is simplified to αNdb /kT ≥ 1 + γdb /(2γex Nex ) for the range of parameters αNdb /kT < 10 and Nex /Ndb ≤ 0.001 and gives a critical temperature of the DBMP formation:

Tc =

αNdb . k[1 + γdb /(2γex Nex )]

(15)

Figure 3 shows the phase diagram, which separates the polarized (polaron) and unpolarized (no polaron) states of the system in the plane of the control parameters (2γex Nex /γdb , αNdb /kT ). The phase separation lines are calculated for the three ratios Nex /Ndb = 0.0001, 0.001 and 0.005. The insert shows the same phase diagram plotted in other coordinates γdb /(2γex Nex ) and αNdb /kT . One can see that an increase of the exciton pumping rate and a lowering of the dangling bond relaxation rate as well as a decrease in temperature leads to polaron formation. At the same time, for a given pumping rate, the polaron is created at higher critical temperatures Tc for larger exchange energies αNdb . This shows that formation of the dangling bond magnetic polaron under the same pumping conditions is more probable in small NCs because αNdb is decreasing with increase of the NC radius (see Fig. 2). Figures 4(a) and 4(b) show the dependence of the DBS polarization ρdb and the exciton 14 ACS Paragon Plus Environment

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ex Nex

db

db

Eex 2 Ndb

Eex ex

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db

Tc Tc Tc Tc

ex Nex

Nex Ndb

Ndb kT Figure 4: Dependencies of the dangling bond magnetic polaron polarization on the control parameters. (a) Dependence of the dangling bond spin polarization ρdb on the parameter αNdb /kT . (b) Dependence of the exciton spin polarization ρex on the parameter αNdb /kT . The solid lines show ρdb and ρex dependencies calculated for the following ratios of the DBS relaxation rate to the DBS pumping rate γdb /(2γex Nex ) = 0, 1, 2, 3, which are assumed to be constant during polaron formation. The dashed curves show the same dependencies calculated self-consistently taking into account an increase of Nex due to the polaron formation assuming the relative efficiency of the DBS assisted recombination q = 0.5. The half of the dimensionless exciton energy splitting ∆Eex /αNdb is shown by an arrow for γdb /(2γex Nex ) = 3 in (a). spin polarization, ρex , on the ratio αNdb /kT . The arrows show the increase of the ratio of spin-flip to the exciton stimulated spin-flip relaxation rates γdb /(2γex Nex ) = 0, 1, 2, 3 used in calculations of different curves. For each excitation condition there is a bifurcation point on the 15 ACS Paragon Plus Environment

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axis of the control parameter αNdb /kT , above which the formation of the DBMP is possible: αNdb /kT > αNdb /kTc . The points that correspond to the critical temperatures, Tc , below which the polaron is formed, are marked for each curve. The dimensionless exciton energy splitting ∆Eex /αNdb = 2ρdb can be directly obtained from the computed DBS polarization and its half corresponds to the DBMP energy shift (shown in Fig. 4(a) by arrow for γdb /(2γexNex ) = 3). Importantly, for a given direction of the quantization axis, the signs of the dangling bond and exciton polarizations are the same. One can see from Eq. (9) that formation of the DBMP, which has the spontaneous polarization of the exciton and the DBSs, decreases the dark exciton recombination rate. At low temperatures when this polarization ρdb ρex −→ 1 the DBS assisted recombination rate of the dark exciton becomes almost completely suppressed and only phonon assisted transitions contribute to the exciton recombination. The temperature rise prevents the dynamic polarization and thus the DBMP formation. The suppression of the DBMP formation increases the DBS assisted recombination rate of the dark exciton and thus the intensity of ZPL and decreases the Stokes shift of the PL. These phenomena were observed in FLN experiments, 22 but have not until now been explained. Temperature Dependence of Dark Exciton Lifetime. As we have shown in Eq. (9) the polarization of the dangling bond and exciton spins at T < Tc leads to the decrease of the dangling bond assisted radiative recombination rate Γdb (T ) = γex Ndb [1 − ρdb (T )ρex (T )]. This increases the exciton lifetime τ and the average exciton population Nex ∝ τ . To describe this effect we introduce a new parameter q = τ (Tc )Γdb (Tc ) = τ (Tc )γex Ndb , which represents the relative efficiency of the dangling bond assisted dark exciton recombination with respect to the total decay at T > Tc . Using Eq. (9) the temperature dependence of the exciton life time can be written as τ (T < Tc ) 1 = . τ (Tc ) 1 − qρdb (T )ρex (T ) 16 ACS Paragon Plus Environment

(16)

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Figure 5: Temperature dependencies of the dark exciton life time and ZPL intensity. (a) and (b) Dependence of the dark exciton life time τ /τ (Tc ) on kT /αNdb . (c) and (d) Dependence of the ratio of intensities of the ZPL to the 1PL: IZPL /I1PL on kT /αNdb . In (a) and (c) calculations were conducted for the following ratios of the DBS relaxation rate to the DBS pumping rate γdb /2γex Nex (Tc ) = 0.1, 0.5, 1, 2, 3 and the relative efficiency of the DBS assisted recombination q = 0.5. In (b) and (d) calculations were conducted for q = 0.20, 0.33, 0.50, 0.67, 0.75 and γdb /2γex Nex (Tc ) = 1. For each excitation condition there is a critical temperature Tc , below which the temperature dependence can be observed. The temperature dependence of τ (T )/τ (Tc ) is shown in Figs. 5(a) and 5(b). One can see a significant increase of the exciton lifetime τ with temperature decrease at T < Tc . The calculations were conducted for several ratios of the DBS relaxation to the DBS pumping rates γdb /2γex Nex (Tc ). Experimentally, this parameter can be controlled by changing the excitation intensity. Indeed, the increase of the excitation intensity Iex and thus of Nex ∝ Iex decreases the ratio γdb /2γex Nex (Tc ). One can see from Fig.5(a), that decrease of γdb /2γex Nex (Tc ) increases Tc and leads to a faster increase of τ at T < Tc for a given q. Fig. 5(b) shows that for given 17 ACS Paragon Plus Environment

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excitation and relaxation conditions, the increase of τ with decreasing temperature depends strongly on the relative efficiency of the dangling bond assisted recombination q. It is clear that polaron formation may lead to a considerable increase of τ only for large q describing the relative contribution of the dangling bond assisted recombination to the total decay. The increase of Nex during polaron formation leads to a faster rise of the exciton polarization with temperature decrease and to a larger dangling bond polarization, as shown by the dashed lines in Fig. 4 calculated with q = 0.5. All further calculations take this effect into account. Temperature Dependence of Multicomponent PL Structure in FLN Experiments. The resonant band-edge excitations of CdSe NCs conducted in FLN experiments excite the lowest optically active 1L exciton, which then relaxes down to the ground ±2 dark exciton. The dark exciton decays radiatively via DBS and optical phonon assisted transitions seen in the PL as ZPL and 1PL, correspondingly. Above the critical temperature, Tc , the Stokes shift of ZPL from the excitation energy EA is equal to the distance between the 1L and the ground ±2 exciton, ∆EAF , (see Fig. 6(a)). The Stokes shift of the 1PL is equal to ∆EAF + ELO , correspondingly, where ELO is the optical phonon energy. Let us consider the temperature dependence of the relative intensities of the ZPL and 1PL in the temperature range below Tc , assuming that the thermal energy kTc is smaller than ∆EAF and the optical phonon energy ELO . At such temperatures neither the anti Stokes component of the LO phonon assisted transition nor the bright exciton contribute to PL. The optical phonon population NB (ELO , T ≤ Tc ) ≈ 0 and the rate of the optical phonon assisted radiative recom37 bination, ΓO Neglecting the contributions ph , can be considered as temperature independent.

of acoustical phonon assisted transitions to radiative and non-radiative recombination, one can write the total exciton decay rate: 1/τ = ΓO ph + Γdb . In this approximation, the ratio of intensities of ZPL, IZPL , and 1PL, I1PL , at T > Tc is


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directly related to the relative efficiency of the DBS assisted recombination, q: IZPL (T > Tc ) IZPL (Tc ) γex Ndb q = = = . O I1PL I1PL Γph 1−q

(17)

The formation of the dangling bond magnetic polaron at T < Tc leads to the following temperature dependence: IZPL (Tc ) IZPL (T < Tc ) = [1 − ρdb (T )ρex (T )]. I1PL I1PL

(18)

The temperature dependence of the ratio, IZPL /I1PL , is shown in Figs. 5(c) and 5(d). One can see that the formation of the DBMP as the temperature drops below Tc leads to a considerable decrease in the relative intensity of the ZPL line. The excitation intensity Iex also strongly affects the ratio. Indeed, the average exciton population Nex ∝ Iex increases with Iex and increases the pumping rate of the dangling bond spin polarization 2γex Nex (Tc ). Calculations performed for several values of the relative relaxation parameter γdb /2γexNex (Tc ), show that its decrease in turn increases Tc and results in suppression of ZPL at higher temperatures, see Fig. 5(c). Figure 5(d) shows that an increase in q leads to a larger variation of IZP L (T < Tc )/IZP L (Tc ) with growing temperature. Let us now consider the temperature dependence of the ZPL and 1PL Stokes shift in the temperature range below Tc when the DBMP is formed. The exchange interaction of aligned DBSs splits the ground exciton state into the two ±2 sublevels, which splits the energies ±∆Eex /2 = ±αNdb ρdb (T ), and the populations, Nex,±2 , differ from each other. This difference leads to different probabilities of phonon assisted transitions from the ±2 sublevels and the following temperature dependent Stokes shift of the 1PL:

EA − E1PL = ELO + ∆EAF + ∆E1PL = ELO + ∆EAF + αNdb ρdb ρex ,


(19)

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Figure 6: Temperature dependencies of the ZPL and 1PL Stokes shifts. (a) Schematic structure of the PL lines seen in the FLN experiments above and below the critical temperature of dangling bond polaron formation Tc . (b) Temperature dependent Stokes shift of ZPL, ∆EZPL /αNdb as a function of kT /αNdb . (c) Temperature dependent Stokes shift of 1PL, ∆E1PL /αNdb as a function of kT /αNdb . Calculations were conducted for the following ratios of the DBS relaxation rate to the DBS pumping rate γdb /(2γex Nex (Tc ) = 0.1, 0.5, 1, 2, 3 and relative efficiency of the DBS assisted recombination q = 0.5. where we assume that the splitting of the ground exciton level ∆Eex = 2αρdb Ndb is less than the line width and cannot be resolved experimentally. To describe the temperature dependence of ZPL Stokes shift one needs to take into account


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the different relaxation rates Γ±2 flip−flop of the two exciton sublevels. A straightforward calculation results in the following temperature dependent ZPL Stokes shift:

EA − EZPL = ∆EAF + ∆EZPL = ∆EAF + αNdb ρdb

ρex − ρdb . 1 − ρex ρdb

(20)

Figure 6 shows the temperature dependence of ∆EZPL /αNdb and ∆E1PL /αNdb as a function of αNdb /kT calculated for γdb /(2γex Nex (Tc ) = 0.1, 0.5, 1, 2, 3 and q = 0.5. One can see that at very low temperatures ρex ≈ 1 and ∆EZPL ≈ αNdb ρdb ≈ ∆E1PL . Discussion. Let us apply our theory to describe the experimental data on the temperature dependent Stokes shift measured in Ref. 38. Figure 7 shows the experimental data measured in CdSe NCs with radius a = 1.2 nm together with the results of theoretical calculations (blue curve) conducted for DBMP models of the dark exciton recombination. The calculations were conducted for the following parameters of the dangling-bond model: α = 0.185 meV, Ndb = 40, γdb ≈ 0.012 µs−1 , q = 0.66. We also used the average number of excitons per NC in the unpolarized state to be Nex = 0.2. For the set of parameters describing experimental data αNdb = 7.4 meV, and Γdb (T > Tc ) = γex Ndb = 0.4 µs−1 , where Tc ≈ 22 K. One can see that the DBMP model allows us to describe the experimental data concerning the temperature dependent Stokes shift in small size CdSe NCs. 38 The calculations presented in the paper are valid for the condition αNdb < ∆EAF which is fulfilled for the parameters found. For the same set of model parameters, the increase of the NC radius up to 3.5 nm decreases the characteristic energy αNdb by a factor of 5 and the critical temperature Tc for polaron formation is reduced from 22 down to 4.5 K. The average exciton population Nex = 0.2 we used corresponds to a quite strong exciton pumping, which is in agreement with experimental conditions described in Refs. 22,38. As one can see in Fig. 3, the decrease of excitation intensity Iex and consequently Nex by a factor of 10 would effectively shift the system from the


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Figure 7: Temperature dependent Stokes shift of the ZPL for the CdSe NC with radius a = 1.2 nm. Experimental data (symbols) are from Ref. 38. Results of the dangling bond magnetic polaron model are shown by the blue solid curve. Parameters used in the calculation are described in the text. The insert shows the temperature dependence of the intensity ratio IZPL /I1PL (red curve) and of the PL life time τ (green curve) calculated with the same parameters. ”polaron” to the ”no polaron” region of the phase diagram, except for at very low temperatures. This may result in intensity-dependent Stokes shifts of ZPL and 1PL in FLN measurements and could be the reason why such a large temperature dependent Stokes shift was not reported in other experiments performed under lower excitation intensities. Our theory explains also the rapid rise of ZPL and the shortening of the decay time with temperature as observed in small CdSe NCs. 22,38 In insert of Fig. 7 we show the temperature dependence of the intensity ratio IZPL /I1PL and the PL life time τ calculated for the parameters that describe the Stokes shift behavior on the temperature. Indeed, if the polaron state is formed at low temperatures T < Tc , the dangling bond assisted radiative rate is suppressed according to Eq. (9) and the intensity of the ZPL line is small. A temperature rise decreases the dangling 22 ACS Paragon Plus Environment

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bond and exciton spin polarizations down to ρex ρdb = 0 at T = Tc and causes a rapid increase of the dangling bond assisted radiative rate and ZPL intensity. The effect is more pronounced at larger excitation intensities. The relative increase of the ZPL intensity as well as the value of the PL life time τ ≈ 3µs and its decrease with the temperature increase shown in the insert of Fig. 7 are in agreement with the experimental data. 22,38 Let us discuss the parameters that control the efficiency of the dangling bond assisted recombination of the dark exciton. While the temperature and excitation power can be changed during the experiment, the number of dangling bonds at the NC surface, the exchange interaction constant and the spin relaxation rates in exciton and dangling bond system are ”intrinsic” parameters of the nanostructure. The surface related parameters, Ndb and α can be modified by varying the NC radius and, more importantly, by the capping of the NC core or by a chemical passivation of the dangling bonds. Thus, the formation of the DBMP might be not possible in core/shell NCs, while it might be observed in bare core NCs of the same radius. The mechanisms of the DBS relaxation and their dependence on the surface quality are not studied yet. The fast spin relaxation rate between the dark exciton sublevel, which is split in the averaged field of the DBSs, is critical for the dynamic polarization described in our model. Fast relaxation leads to a difference in populations of the exciton sublevels and results in different probabilities for DBS flops. The DBS polarization increases and in turn increases the exciton level splitting. Indeed, fast spin relaxation was reported in most studies of the II-VI semiconductor NCs, including bare core and core/shell structures. 31,39–41 The mechanisms responsible for this relaxation are not identified yet. The sign and magnitude of the exchange constant α play an important role in our theory. Dynamic polarization of the spin system and formation of the DBMP are possible only for the ferromagnetic type of exchange interaction between the electron spin in the exciton and DBSs. The study of the exchange constant α for the interaction between Mn ions and an electron in 23 ACS Paragon Plus Environment

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NCs show that its sign could depend on the NC size. 28,29,42 Detailed studies of the effects of the NC size and surface on the sign and magnitude of the exchange constant are beyond the scope of this paper. We would like to stress here that the DBMP is created in individual NCs by un-polarized light. In each NC the DBMP is characterized by a magnetic moment aligned along the c-axis of the NC. These magnetic moments, however, do not create macroscopic magnetization in an ensemble of randomly oriented NCs. In summary, we have developed a model of dark exciton radiative recombination assisted by its interaction with the spins of surface dangling bonds. We have shown that relatively strong excitations of small NCs make the unpolarized state of the exciton and dangling bond spins unstable at low temperatures. Optical pumping of NCs results in dynamic polarization of the dangling bond spins and formation of the dangling bond magnetic polaron. The formation of the dangling bond magnetic polaron suppresses the dark exciton radiative recombination and results in an additional Stokes shift of the dark exciton PL. This model explains the temperature dependence of the Stokes shift and the temperature activation of the dark exciton radiative recombination observed in the experiments. The model predicts NC size, temperature, excitation intensity, and dangling bond concentration dependencies of the radiative decay time, and the PL line positions, which could be verified experimentally. Methods. Calculation of Electron-DBS Flip-Flop Assisted Recombination of the Dark Exciton. The DBS assisted radiative recombination of the dark exciton is possible due to the ˆj H flip−flop terms in the exchange Hamiltonian (1). Its rate is controlled by the matrix elements e |Ve± | = |h±1/2|Aˆ∓ σ ˆ± | ∓ 1/2i| = 2|Aˆ∓ |. The radiative decay rates of the dark exciton can be

found in the framework of the second order perturbation theory, 8,43 which describes a virtual transition of the dark exciton to the bright exciton states and its consequential radiative decay.


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These calculations give the radiative decay rates Γ±2 flip−flop Γ±2 flip−flop =

h|Aˆ∓ |2 i 1 , 3η 2 τ0

(21)

where the operator h|Aˆ∓ |2 i is averaged over the DBS polarization, η is the electron-hole exchange constant, and τ0 is the radiative decay time of the 0U bright exciton state through which the radiative decay times of both 1L,U bright exciton states can be expressed. 8 Equation (21) is ˆ j term in the exchange Hamiltonian derived for the case in which corrections coming from the H zz (1) to the dark and bright exciton energies are much smaller than ∆AF and their contributions to the optical transition probabilities can be neglected. The straightforward averaging of |Aˆ∓ |2 over the dangling bond spin polarization gives: ± h|Aˆ∓ |2 i = α2 Ndb ,

(22)

± where Ndb = Ndb (1 ∓ ρdb )/2. For the F = +2 and F = −2 dark exciton states, the resulting

spin-flip assisted recombination rates

Γ±2 flip−flop =

α η

!2

Ndb (1 ∓ ρdb ) 6τ0

(23)

− + differ from each other if the dangling bond spins are polarized: ρdb = (Ndb − Ndb )/Ndb 6= 0.

Calculation of Size Dependence of the Dangling Bond Assisted Recombination Rate of the Dark Exciton. As one can see from Eq.(8), there are three independent factors contributing to this dependence: the square of the electron wave function at the NC surface ξ 2 (a), the electronhole exchange constant η(a), and the number of dangling bonds at the NC surface Ndb (a). The size dependence of the exchange constant in CdSe NCs is very well described as 8 η(a) ∝ 1/a3 . We calculate the size dependence of the square of the electron wave function at the 25 ACS Paragon Plus Environment

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NC surface for various values of the potential barrier U as 45 ξ 2 (a) ∝ sin2 (k0 a)/a3 , where k0 = (2m∗ E0 )1/2 /¯h and m∗ (E0 ) is the electron effective mass at the energy of the lowest electron level, E0 . The electron mass outside the NC is taken to be the free electron mass m0 and the continuity of the electron function at the NC surface is assumed. We have assumed a constant surface density of the dangling bonds: Ndb (a) ∝ (a/a0 )2 , to describe their dependence on the NC size, which within the above mentioned assumption can be written as Ndb (a) = Ndb (1.2)(a/1.2)2 where Ndb (1.2), is the number of dangling bonds in the a = 1.2 nm NCs. The characteristic radiative time of the bright exciton τ0 generally depends on the NC radius and the potential barrier height. For simplicity, we neglect this dependence and use τ0 ≈ 4 ns, which was calculated for CdSe NCs with a = 1.2 nm and U = 1.2 eV taking into account the depolarization factor D ≈ 0.4 . 44 Author Information Corresponding Author *E-mail: [email protected] Notes The authors declare no competing financial interests. Acknowledgements Authors thank K. Kavokin, L. Biadala, D. Yakovlev, L. Boyer, S. Crooker, and S. Erwin for helpful discussions and TU Dortmund University for the hospitality. The work of A. V. R was supported by the RFBR (Grant No. 13-02-00888) and by the Government of Russia (Project No. 14.Z50.31.0021, leading scientist M. Bayer). Al.L.E. acknowledges the financial support of the Office of Naval Research (ONR) through the Naval Research Laboratory Basic Research Program.


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References (1) Murray,C. B., Norris, D. J. & Bawendi, M. G. J. Am. Chem. Soc. 1993, 115, 8706-8715. (2) Grigoryan, G. B., Kazarayn, E. M., Efros, Al. L. and Yazeva, T. V. Sov. Phys. Solid State 1990, 32, 1031-1035. (3) Ekimov, A. I., Hache, F., Schanne-Klein, M. C., Ricard, D., Flytzanis C., Kudryavtsev, I. A., Yazeva, T. V., Rodina, A. V. & Efros, Al. L. J. Opt. Soc. Am. B 1993, 10, 100-107. (4) Kang, I. & Wise, F. W. J. Opt. Soc. Am. 1997, 14, 1632-1646. (5) Califano, M., Franceschetti, A. & Zunger, A. Phys. Rev. B 2007, 75, 115401. (6) Korkusinski, M., Voznyy, O. & Hawrylak, P. Phys. Rev. B 2010, 82, 245304. (7) Nirmal, M. D., Norris, J., Kuno, M., Bawendi, M. G., Efros, Al. L. & Rosen, M. Phys. Rev. Lett. 1995, 75, 3728-3731. (8) Efros, Al. L., Rosen, M., Kuno, M., Nirmal, M., Norris, D. J. & Bawendi, M. G. Phys. Rev B 1996, 54, 4843-4856. (9) Efros, Al. L. Phys. Rev. B 1992, 46, 7448. (10) Crooker, S. A., Barrick, T., Hollingsworth, J. A. & Klimov, V. I. Appl. Phys. Lett. 2003, 82, 2793-2795. (11) Labeau, O., Tamarat, P. & Lounis, B. Phys. Rev. Lett. 2003, 90, 257404. (12) van Driel, A. F., Allan, G., Delerue, C., Lodahl, P., Vos, W. L. & Vanmaekelbergh, D. Phys. Rev. Lett. 2005, 95, 236804. (13) Donega, C., Bode, M. & Meijerink, A. Phys. Rev. B 2006, 74, 085320. 27 ACS Paragon Plus Environment

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(14) Oron, D., Aharoni, A., de Mello Donega, C., van Rijssel, J., Meijerink, A. & Banin, U. Phys. Rev. Lett. 2009, 102, 177402. (15) Furis, M., Hollingsworth, J. A., Klimov, V. I. & Crooker, S. A. J. Phys. Chem. B 2005, 109, 15332-15338. (16) Wijnen, F. J. P., Blokland, J. H., Chin, P. T. K., Christianen, P. C. M. & Maan, J. C. Phys. Rev. B 2008, 78, 235318. (17) Biadala, L., Louyer, Y., Tamarat, Ph. & Lounis, B. Phys. Rev. Lett. 2009, 103, 037404. (18) Biadala, L., Louyer, Y., Tamarat, Ph. & Lounis, B. Phys. Rev. Lett. 2010, 105, 157402. (19) Moreels, I., Rain, G., Gomes, R., Hens, Z., Stferle, T., Thilo, M. & Rainer, F. ACS nano 2011, 5, 8033-8039. (20) Fernee, M. J., Tamarat, Ph. & Lounis, B. J. Phys. Chem. Lett. 2013, 4 609-618. (21) Bawendi, M. G., Wilson, W. L., Rothberg, L., Carroll, P. J., Jedju, T. M., Steigerwald, M. L. & Brus, L. E. Phys. Rev. Lett. 1990, 65, 1623-1626. (22) Nirmal, M., Murray, C. B. & Bawendi, M. G. Phys. Rev. B 1994, 50, 2293-2300. (23) Mackh, G., Ossau, W., Yakovlev, D. R., Waag, A., Landwehr, G., Hellmann, R. & Göbel, E. Phys. Rev. B 1994, 49, 10248-10258. (24) Yakovlev D. R. & Ossau, W. Introduction to the Physics of Diluted Magnetic Semiconductors edited by J. Kossut and J. A. Gaj, Ch. 7, p.221 (Springer-Verlag, Heidelberg 2010, Springer Series in Materials Science 144, ISBN: 978-3-642-15855-1). (25) Cheng, S.J. & Hawrylak, P. Eur.Phys.Lett. 2008, 81, 37005.


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(26) Qu F. & Hawrylak, P. Phys.Rev.Lett. 2006, 96, 157201. (27) Abolfath R. M., Hawrylak, P. & Zutic, I. Phys. Rev. Lett. 2007, 98, 207203. (28) Merkulov I. A. & Rodina, A. V. Introduction to the Physics of Diluted Magnetic Semiconductors edited by J. Kossut and J. A. Gaj, Ch. 3, p.65 (Springer-Verlag, Heidelberg 2010, Springer Series in Materials Science 144, ISBN: 978-3-642-15855-1). (29) Bussian, D. A., Crooker, S. A., Yin, M., Brynda, M., Efros, Al. L. & Klimov, V. I. Nature materials 2009, 8, 35-40. (30) Viswanatha, R., Pietryga, J. M., Klimov, V. I. & Crooker S. A. Phys. Rev. Lett. 2011, 107, 067402. (31) Johnston-Halperin, E., Awschalom, D. D., Crooker, S. A., Efros, Al. L., Rosen, M., Peng, X. & Alivisatos, A. P. Phys. Rev. B 2001, 63, 205309. (32) Seehra, M. S., Dutta, P., Neeleshwar, S., Chen, Y.-Y., Chen, C. L., Wei Chou, S., Chen, C. C., Chung-Li Dong & Adv. Mater. 2008, 20, 1656-1660. (33) Meulenberg, R. W., Lee, J. R. I., McCall, S. K., Hanif, K. M., Haskel, D., Lang, J. C., Terminello, L. J. & van Buuren, T. J. Am. Chem. Soc. 2009, 131, 6888-6889. (34) Norris, D., Efros, Al. L., Rosen, M. & Bawendi, M. Phys. Rev. B 1996, 53, 16347-16354. (35) Leatherdale, C. A., Woo, W.-K., Mikulec, F. V. & Bawendi, M. G. J. Phys. Chem. B 2002, 106, 7619-7622. (36) The dangling bond spin relaxation, generally, requires a non-conservation of the total spin of dangling bonds. The exchange Hamiltonian itself conserves the total spin of the dangling bonds which can be statistically distributed between 0 and Ndb /2. That is why the complete 29 ACS Paragon Plus Environment

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consideration of relaxation requires an introduction of the mechanism that does not conserve the total spin, for example, the magnetic dipole-dipole interaction between dangling bond spins. (37) A. V. Rodina and Al. L. Efros, unpublished. (38) Nirmal, M. Photophysics of CdSe Semiconductor Nanocrystals, p. 64 (PhD Thesis, MIT, 1996). (39) Liu, F., Biadala, L., Rodina, A. V., Yakovlev, D. R., Dunker, D., Javaux, C., Hermier, J.-P., Efros, Al. L., Dubertret, B. & Bayer, M. Phys.Rev B 2013, 88, 035302. (40) Liu, F., Rodina, A. V., Yakovlev, D. R., Greilich, A., Golovatenko, A. A., Susha, A. S., Rogach, A. L., Kusrayev, Yu. G. & Bayer, M. Phys. Rev. B 2014, 89, 115306. (41) Siebers, B., Biadala, L., Yakovlev, D. R., Rodina, A.V., Aubert, T., Hens, Z. and Bayer, M. Phys. Rev. B, 2015 91, 155304. (42) Merkulov, I. A., Yakovlev, D. R., Keller, A., Ossau, W., Geurts, J., Landwehr, G., Karczewski, G., Wojtowicz, T. & Kossut, J. Phys. Rev. Lett. 1999, 83, 1431. (43) Efros, Al. L. Semiconductor and Metal Nanocrystals: Synthesis and Electronic and Optical Properties edited by V. I. Klimov, Ch. 3, p. 103 (Marcel Dekker, New York, 2003). (44) Shabaev, A., Rodina A. V., & Efros, Al. L. Phys. Rev. B 2012, 86, 205311. (45) Shabaev, A., Efros, Al. L. & Efros, A. L. Nano Lett. 2013, 13, 5454-5461.


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Magnetic Properties of Nonmagnetic Nanostructures: Dangling Bond

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