CdS Core ... - ACS Publications

Subscriber access provided by MAHIDOL UNIVERSITY (UniNet)

Communication

Biexciton Auger recombination in CdSe/ CdS core/shell semiconductor nanocrystals Roman Vaxenburg, Anna Rodina, Efrat Lifshitz, and Alexander L. Efros Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b00066 • Publication Date (Web): 07 Mar 2016 Downloaded from http://pubs.acs.org on March 8, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Biexciton Auger recombination in CdSe/CdS core/shell semiconductor nanocrystals Roman Vaxenburg George Mason University, Fairfax, VA 22030, USA

Anna Rodina Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia

Efrat Lifshitz Technion - Israel Institute of Technology, Haifa 32000, Israel

Alexander L. Efros∗ Naval Research Laboratory, Washington DC 20375, USA A theoretical study of the positive and negative trion channels in the nonradiative Auger recombination of band-edge biexcitons (BXs) in CdSe/CdS core/shell nanocrystals (NCs) is presented. The theory takes into account the BX fine-structure produced by NC asymmetry and exchange interaction. The calculations show that growth of CdS shell upon CdSe core suppresses the rate of the Auger recombination via negative trion channel, while the more efficient Auger recombination via positive trion channel shows much weaker dependence on the shell thickness. The demonstrated oscillatory dependence of the BX Auger rate on the core and shell sizes is explained qualitatively in terms of overlap of the ground and excited carrier wavefunctions. The calculations show that raise of temperature accelerates the Auger recombination in CdSe/CdS NCs due to reduction of the bulk energy gaps of CdSe and CdS.

Keywords:

Auger recombination, biexciton, CdSe/CdS, core/shell, nanocrystal,

quantum dot.

∗

Author to whom correspondence should be addressed. Electronic mail: [email protected]

ACS Paragon Plus Environment

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 28

Auger recombination is a nonradiative energy dissipation channel affecting all aspects of charge carrier dynamics in multiply-excited semiconductor nanocrystals (NCs).[1] In the Auger process, the energy of a recombining electron-hole pair, instead of being emitted as a photon, is transferred nonradiatively to another carrier, which subsequently loses this energy in the form of heat. Despite being an inefficient process in wide-gap bulk semiconductors,[2] the Auger recombination is greatly enhanced in NCs made from these semiconductors.[3–7] This enhancement originates in the increased Coulomb interactions between charges confined to extremely small NC volume, reduced dielectric screening, and relaxation of the quasi-momentum conservation requirement,[1, 4, 5, 8] which otherwise limits the Auger recombination efficiency in bulk materials. In NCs, this renders the Auger recombination an efficient nonradiative de-excitation channel, unfavorable for NC luminescent applications. In particular, Auger recombination induces gain decay in NC lasers,[9, 10] limits the time available for photocarrier extraction in solar cells,[11] triggers the efficiency roll-off in NC-based LEDs,[12] and is believed to induce luminescence intermittency[13] (blinking) in single NCs used, for example, as biolabels.[14] Significant efforts have been made in developing strategies to suppress the unwanted Auger recombination. Fabrication of NCs with graded (soft) confinement potential,[15, 16] growth of quasi type-II NCs with thick shells,[17, 18] and preparation of type-II NCs[19] have been proposed. The simplest configuration in which the Auger recombination can take place in an electrically neutral NC is a doubly-excited state, or a biexciton (BX), shown schematically in Figure 1a. This band-edge BX is most relevant to the NC luminescent properties and it reflects the general physics of Auger recombination in NCs with arbitrary excitation multiplicity, both neutral and charged. Being a three-body process, the BX Auger recombination can proceed via two alternative pathways: the positive trion pathway, in which the electronhole recombination energy is transferred to another hole, and the negative trion pathway, in which the energy is transferred to another electron, as shown in Figure 1b. The total rate of the BX Auger recombination is then given by the sum of these two pathways.[20] The main factors influencing the Auger rate in NCs are the band gap energy, shape of the confining potential, carriers’ wavefunction overlap and their characteristic confinement width. The latter is determined by the NC dimensions, and, in the case of core/shell NCs, by the mutual alignment of the conduction and valence bands (i.e., band-offsets) of the constituent materials. In addition, the Auger rate can be indirectly influenced by temperature via the 2


Page 3 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

energy gap and band-offset temperature-dependence.[21]

Figure 1: (a) Ground state biexciton (BX) in CdSe/CdS core/shell nanocrystal having quasi typeII band-alignment with electrons delocalized over the entire structure and holes confined to the CdSe core. (b) Schematic of BX Auger recombination. The recombination energy of an electronhole pair can be transfered to the second electron (negative trion channel), or to the second hole (positive trion channel), ejecting them into a continuum of states above the nanocrystal potential barrier. (c) Fine-structure of the six BX band edge states with total angular momentum J = 0, 2 . The BX states are split by hole-hole exchange interaction, intrinsic hexagonal crystal field, and nonspherical nanocrystal shape, resulting in three distinct BX energy levels.[27] Possible Auger transitions between initial BX states and final exciton (X) states are shown by arrows. Due to the conservation of total angular momentum in Auger process, final X states with only J = 0, 2 are allowed. Indicated by je and jh are the possible single-particle angular momenta of the electron and hole comprising the final X states formed via the negative and positive trion channels of the BX Auger recombination.

Technologically reproducible CdSe/CdS core/thick-shell NCs demonstrated significant suppression of Auger recombination. The investigation of these ”giant” NCs, which started independently by two groups,[18, 22] finally resulted in significant reduction of lasing threshold[23] and complete suppression of blinking at room temperature.[24] The band alignment of the quasi type-II CdSe/CdS core/shell system forces the hole to be strongly confined to the CdSe core, while allowing the electron to be delocalized over the entire NC (see Figure 1a). Based on this, it was concluded that the Coulombic scattering between confined holes must be stronger than that between delocalized electrons, implying that the BX Auger process in core/shell NCs should be dominated by the positive trion pathway,[20, 25, 26] 3


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 28

which was also confirmed experimentally.[16, 20] Theoretically, however, the role of the positive and negative trion channels in BX Auger recombination in core/shell NCs has not yet been addressed. The study of the BX Auger precesses requires detailed knowledge of the band-edge BX electronic structure – a problem that was studied theoretically in CdSe NCs in the past.[27] It has been shown that the 6-fold degeneracy of the band-edge BX can be lifted by hole-hole exchange interaction, nonsphericity of the NC shape, and the intrinsic hexagonal lattice symmetry,[27] as shown schematically in Figure 1c. The resultant BX fine-structure, produced by mixing of the band-edge BX states having total angular momentum J = 0 and J = 2 , should in turn have an effect on the Auger rate. The Auger processes originating in each one of these fine-structure BX states could have different rates and different properties with respect to the positive and negative trion channels. Further, the thermal population distribution among these states would influence the total BX Auger rate measured in a NC ensemble. In this Letter we present calculations of BX Auger recombination rates in CdSe/CdS core/shell NCs. We address the role of the positive and negative trion channels in the BX Auger process, and we study their dependence on the NC dimensions, band-offsets, and temperature, taking into account the fine-structure of the band-edge BX. The calculations show that the BX Auger rate is mostly dominated by the positive trion channel when the confinement width of the ground state hole is smaller than that of the electron. We find that the Auger recombination rates of the different band-edge BXs differ for the positive trion channel, but are equal to each other for the negative trion. The observed oscillatory behavior of the Auger recombination rates with core size and shell thickness is qualitatively explained. The effect of the temperature dependence of the bulk energy gaps and thermal population of band-edge BX states on Auger recombination was considered. We use the eight-band k · p method[28, 29] to model CdSe/CdS core/shell NCs having cubic (zinc-blende) or hexagonal (wurtzite) crystal structure and being embedded in a surrounding matrix.

The wurtzite structure is modelled in spherical quasi-cubic

approximation,[30, 31] in which (i) the effect of hexagonal crystal field is expressed as small effective deformation of the unperturbed zinc-blende crystal in the [111] cubic direction, and (ii) small warping terms related to the zinc-blende valence band structure are neglected.[31, 32] This effective deformation, as well as possible non-sphericity of the NC shape, can then be treated perturbatively. The unperturbed single-particle electron and 4


Page 5 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

hole wavefunctions are characterized by the magnitude, j , and projection, m , of the total angular momentum, j = f + ` , and have a well-defined parity, π = ±1 . Here, f and ` are the Bloch and envelope wavefunction angular momenta, respectively.[29, 33, 34] The full single-particle wavefunction in the core and shell regions of the NC can be written as Φ = Ψc |uc i + Ψv · |uv i , where Ψc and Ψv are two- and six-component envelope functions, so and |uc i = (|uc+1/2 i, |uc−1/2 i) and |uv i = (|uv+3/2 i, |uv+1/2 i, |uv−1/2 i, |uv−3/2 i, |uso +1/2 i, |u−1/2 i) is

the corresponding Bloch basis.[29, 34, 35] Outside the NC, in the surrounding material, we assume that the wavefunctions have the form of a free-particle two-component spinor. The bulk material parameters used in the calculation are given in Supporting Information. The unperturbed electron and hole wavefunctions are found by a procedure similar to that in Ref. 34 with (i) modified boundary conditions to suit a core/shell heterostructure, (ii) finite potential barrier for both electrons and holes, and (iii) assumption of the ”pure” Kane model[36] to describe excited holes, which, in our case, is equivalent to setting the modified Luttinger parameters[29] γ1 and γ to zero (see also Ref. 37). The general criterion for the choice of boundary conditions is the continuity of generalized envelope probability current density normal to the heterointerfaces.[38] In CdSe/CdS NCs, there are two such interfaces – one between the CdSe core and the CdS shell, and the other between the CdS shell and the environment. At each one of these interfaces we impose the standard boundary conditions that require continuity of all components of the envelope wavefunctions and the corresponding envelope velocities,[38] as described in Supporting Information. The single-particle states involved in formation of band-edge BXs are the 1S 1/2 electron and 1S 3/2 hole levels (see Figure 1c). In the BX, the Pauli exclusion principles requires two 1S 1/2 electrons to be in the singlet configuration only. Further, the exclusion principle restricts the possible total angular momentum, J , of the two 1S 3/2 holes, and, consequently, of the BX itself, to J = 0 and J = 2 (the latter is 5-fold degenerate).[27, 39] We label these unperturbed BX states by their total angular momentum and its projection as |J, Jz i . The BX states with J = 0 and J = 2 are split by the hole-hole exchange interaction[27] and the resultant energies can be written as EJ = EBX − ∆exch (4J − 5)/8 . Here, EBX is the unperturbed BX energy and ∆exch is the exchange splitting constant calculated similarly to Refs. 27, 40 but using our single-particle hole wavefunctions described above. BX states in spherical NCs with cubic (zinc-blende) lattice are subject only to this exchange splitting. In NCs having hexagonal (wurtzite) crystal structure and/or nonspherical shape, the 5


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 28

BX states are further split by the crystal-field and shape asymmetry. The corresponding ˆ BX = −∆(ˆj 2 − 5/4)/2 − ∆(ˆj 2 − 5/4)/2 , where perturbation operator can be written as H 1z

2z

the net asymmetry splitting constant is given by ∆ = ∆hex + ∆shape , and where ˆj1z and ˆj2z is the angular momentum projections of the 1st and 2nd hole, respectively. The splitting energies, ∆hex and ∆shape , are calculated using the procedure similar to Refs. 27, 41, 42. ∆hex is expressed via the crystal field splitting constant ∆cr of the corresponding bulk material,[27, 41] and ∆shape is expressed via the NC ellipticity parameter, µ , which is negative for oblate NCs, positive for elongated ones, and is zero for spherical NCs.[27, 42] ˆ BX (with addition of the exchange splitting) constructed Diagonalization of the matrix of H between the six unperturbed BX states, |J, Jz i , produces the BX fine-structure, shown schematically in Figure 1c. The 6-fold degenerate BX is split into three distinct energy levels: the lowest E− , the highest E+ , and the middle 4-fold degenerate E2 level.[27] The E+ and E− levels are produced by mixing of the BX wavefunctions |0, 0i and |2, 0i as |±i = A± |0, 0i + B± |2, 0i , while the states belonging to the E2 energy are just the 2 1/2 ) and B± = unperturbed |2, ±1i and |2, ±2i BX states. Here, A± = 2∆/(4∆2 + D± 2 1/2 ) with D± = ∆exch ∓ (∆2exch + 4∆2 )1/2 . D± /(4∆2 + D±

The Auger recombination, which is driven by Coulomb interaction, preserves the total angular momentum of the system (as well as parity). It is therefore convenient to construct the final BX Auger states, each of which consists of one electron-hole pair, or an exciton (X), as eigenstates of the total angular momentum as well. The BX Auger process proceeding via the negative trion channel produces an excited n P 3/2 electron and a ground state 1S 3/2 hole (see Figure 1c). The resultant X states with both J = 0 and J = 2 are then constructed from these particular electron and hole states by using the standard angular momentum addition rules.[43] In contrast, the BX Auger process occurring via the positive trion channel produces a ground state 1S 1/2 electron accompanied by either one of the following excited holes: n P 1/2 , n P 3/2 , n P 5/2 (Figure 1c). The final X states with J = 0 are then constructed from the 1S 1/2 electron and n P 1/2 hole exclusively, while the X states with J = 2 can be constructed from the 1S 1/2 electron and either n P 3/2 or n P 5/2 hole. Having constructed the initial BX and final X states, we use Fermi’s golden rule to P 2 calculate the rate of the Auger recombination as R = (2π/~) X X Cˆ BX /Ef0 (kf ) ,

where X Cˆ BX ≡ M is the Coulomb matrix element,[4, 34] and the sum extends over all X states compatible with the initial state. Here, Ef0 = ∂Ef /∂k , Ef and kf are the 6


Page 7 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

energy and momentum of the excited electron or hole outside the NC, Cˆ = e2 /κ|r1 − r2 | , and κ is the dielectric constant. The Coulomb matrix elements are calculated numerically as described in Ref. 34. The total Auger rate, R , can be decomposed into a sum of positive and negative trion channels (described below), and, in general, R is different for each given initial state of the BX. We can note, however, that only a small number of distinct Coulomb matrix elements needs to be calculated because of (i) the chosen representation of the X and BX states in terms of the eigenstates of total angular momentum, which is conserved in Auger process, and (ii) the independence of the matrix elements of the Coulomb (scalar) operator on the angular momentum projection.[4, 43] Analysis shows that for the negative trion channel in spherical zinc-blende NCs (SNCs), the modulus of the matrix element is the same for all possible initial BX states. We can then write the rate of the negative trion channel as Rneg = (2π/~)|Mneg |2 /Ef0 (kf ) , where ˆ 0i| = |hXel (2, Jz )|C|2, ˆ Jz i| and Jz = 0, ±1, ±2 . Here, the notation |Mneg | = |hXel (0, 0)|C|0, |Xel (J, Jz )i indicates the final X state formed in the course of the negative trion Auger process in which an electron is excited to n P 3/2 state. On the other hand, three matrix elements are involved in the positive trion channel in SNCs. The first is for the J = 0 (1/2) 1/2 ˆ 0i , where |X1/2 (J, Jz )i is the final X state in which a hole is case, Mpos = hX (0, 0)|C|0, hl

hl

excited to the n P 1/2 state via the positive trion Auger channel. The second matrix element (3/2) 3/2 ˆ Jz i . is for J = 2 when the hole is excited to the n P 3/2 state, Mpos = hX (2, Jz )|C|2, hl

The third and last matrix element is also for the case of J = 2 , but when the hole is excited (5/2) 5/2 ˆ Jz i . As mention above, the matrix elements to the n P 5/2 state, Mpos = hX (2, Jz )|C|2, hl

are independent of Jz . Putting it together, we can write the total rate of the positive trion (0)

1/2

channel in SNCs for the BXs with J = 0 as Rpos = (2π/~)|Mpos |2 /Ef0 (kf ) , and for BX (2) (3/2) (5/2) with J = 2 as Rpos = (2π/~) |Mpos |2 + |Mpos |2 /Ef0 (kf ) . Taking into account the asymmetry-induced mixing between the BX states of SNCs, we obtain at once that the negative trion channel is unaffected by the crystal-field and/or shape asymmetry and the rate remains Rneg . However, the positive trion channel rate for (0)

(2)

± 2 the initial BX states |+i and |−i becomes Rpos = A2± Rpos + B± Rpos , and for the BX (2)

states |2, ±1i and |2, ±2i the rate remains Rpos . This analysis shows that in the presence of mixing induced in hexagonal and/or nonspherical NCs, the rates of the positive trion (0)

(2)

channel can still be represented in terms of the rates Rpos and Rpos corresponding to BXs in spherical zinc-blende NCs. Finally, the total rate of the BX Auger process can be written 7


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 28

(2)

± as a sum of the negative and positive channels as R = Rneg + Rpos or R = Rneg + Rpos ,

depending on the initial BX state.

8


Page 9 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Figure 2: Calculated dependence of the BX Auger rate and its positive and negative trion components on shell width in CdSe/CdS core/shell NCs with core radius a = 1.5 nm and conduction band-offset ∆ECB = 200 meV at room temperature, T = 300 K. (a) Cubic (zinc-blende) NCs with spherical shape. Only exchange splitting of the BX states with J = 0, 2 is present. The (0)

(2)

negative trion rate, Rneg , and the positive trion rates, Rpos and Rpos , are shown. (b) Hexagonal (wurtzite) NCs with spherical shape. Both exchange and crystal-field splittings are present. (2)

+ , R− , are shown. (c) Hexagonal The negative trion rate and the positive trion rates, Rpos , Rpos pos

(wurtzite) slightly elongated NCs with µ = 0.1 . Exchange, crystal-field, and shape asymmetry splittings are present. (d) Hexagonal slightly elongated NCs with µ = 0.1 . Shown are the negative trion rate, the net rate of the positive trion channel, and the total BX Auger rate. In panels ± , limited by their (a)-(c), the grey area shows the possible range of mixed BX Auger rates, Rpos (2)

(0)

± ≤R unperturbed components as Rpos ≤ Rpos pos .

9


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 28

Using these equations, we calculate the rates of the BX Auger process and the individual contributions of the negative and positive trion channels. Figure 2a shows the calculated dependence of the Auger rates on shell thickness in spherical zinc-blende CdSe/CdS NCs with core radius a = 1.5 nm and conduction-band offset ∆ECB = 200 meV at room temperature, (0)

T = 300 K. The blue line indicates the negative trion channel rate. The rates Rpos and (2)

(0)

Rpos of the positive trion channel are shown by dotted and solid red lines, where Rpos is (2)

larger than Rpos roughly by one order of magnitude for the shell thicknesses shown. The BX states in spherical zinc-blende NCs are split only by exchange interaction of the order ∆exch ∼ 1 meV, which is much smaller than kT at room temperature. Consequently, based on degeneracy of the BX states with J = 0, 2 , the net positive trion rate can be written as (0)

(2)

Rpos ≈ (1/6)Rpos + (5/6)Rpos . In the presence of additional asymmetry splitting, the Auger ± rates Rpos of the mixed BX states would be determined by the mixing constants, A± , B± , (2)

± ≤ and would lie within the range limited by their unperturbed components as Rpos ≤ Rpos (0)

Rpos . This range is indicated by the gray area in Figure 2. Figure 2b shows the Auger rates calculated as in Figure 2a, but with addition of crystalfield asymmetry, producing wurtzite CdSe/CdS NCs with spherical shape. As mentioned (2)

above, the negative trion rate, Rneg , as well as the positive trion rate Rpos corresponding to the E2 4-fold degenerate level of the BX fine-structure, are independent of the asymmetry perturbations. These two rates are shown in Figure 2b by solid blue and solid red curves, ± , are shown by dashed and dotted respectively. The other two positive trion rates, Rpos

red curves. Among the two perturbations present in hexagonal spherical NCs, the crystalfield energy dominates, ∆hex ∆exch . This results in |D+ | ≈ |D− | ≈ |2∆| , implying 2 ≈ 1/2 . This, in turn, leads approximate equality between the mixing constants, A2± ≈ B± + − to nearly equal positive trion rates, Rpos ≈ Rpos , as can be seen in Figure 2b.

Further, Auger rates calculated in slightly elongated (µ = 0.1) wurtzite CdSe/CdS NCs are shown in Figure 2c. The addition of the shape-asymmetry perturbation to a hexagonal NC can render the splitting constant ∆ = ∆hex + ∆shape comparable with the exchange 2 energy, ∆ ∼ ∆exch . This would result in A2± 6= B± and, consequently, in quite different + − rates Rpos and Rpos , as can be seen in Figure 2c. The net rate of the positive trion Auger

channel, as it could be measured in experiment, is given by the average of the three individual (2)

± rates, Rpos , Rpos , weighted by their corresponding Boltzmann occupation probabilities. The

resultant net rate of the positive trion channel at room temperature is shown in Figure 2d 10


Page 11 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

by red line. Finally, the total BX Auger rate is just a sum of its positive and negative trion components, and it is shown in Figure 2d by the grey line. One can see that in NCs shown in Figure 2d, the rate of the positive trion channel is generally larger than that of the negative trion channel, and that it dominates the total BX Auger rate. However, these rates can have comparable magnitude in NCs with larger core radii, as we show below. Continuing this analysis, we again consider hexagonal slightly elongated CdSe/CdS NCs at room temperature, and we calculate the dependence of the BX Auger rates on shell thickness for several fixed core radii and conduction-band offsets. Figure 3a shows the Auger rates calculated for NCs with a = 1.5 nm and ∆ECB = 50 meV. One can see that in this case the total BX Auger rate is mostly due to the positive trion channel, whose rate is generally larger than the negative trion rate. All rates show a distinct oscillatory dependence on the shell width. The amplitude of the oscillations is roughly one order of magnitude. Apart from the oscillations, the negative trion rate decreases with shell width due to a gradual increase of confinement width (and loss of high-momentum components) of the delocalized 1S 1/2 electron. In contrast, the simultaneous decrease of the positive trion rate is due to reduction of the electron-hole overlap as the electron increasingly penetrates into the CdS shell, while the confinement width of the 1S 3/2 hole is virtually insensitive to the shell thickness. Figure 3b shows the dependence of the Auger rates on shell width calculated for the same conditions as in Figure 3a, but with larger band-offset, ∆ECB = 300 meV. The reduction of the rates with shell width is slower in this case, which is especially clear for the negative trion channel. This slower reduction of the rates is due to the increased conduction band offset which limits the delocalization capability of the electron by binding it to the CdSe core more strongly. With this core radius and band-offset, however, the electron’s confinement width is still larger than that of the hole, which renders the positive trion channel rate dominant. The overall magnitude of the positive trion channel rate (and, in turn, of the total BX Auger rate) is somewhat lower in Figure 3b than in Figure 3a. This is caused by a larger energy-gap (which directly influences the Auger rate) of the NCs with larger band-offset (see inset in Figure 3c). Figures 3c and 3d show the Auger rates calculated as in panels (a) and (b) ( ∆ECB = 50 meV and ∆ECB = 300 meV, respectively), but with larger core radius, a = 2.5 nm. The overall magnitude of the rates is smaller in this case due to localization of both carriers in the 11


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 28

Figure 3: Calculated dependence of the rates of negative and positive trion channels and of the total BX Auger rate on shell thickness in hexagonal CdSe/CdS core/shell NCs having slightly elongated shape ( µ = 0.1 ) at room temperature. The negative trion rate is shown by blue line. The net positive trion rate is represented by red line. The total BX Auger rate is shown in grey. (a) NCs with core radius a = 1.5 nm and conduction band-offset ∆ECB = 50 meV. (b) a = 1.5 nm, ∆ECB = 300 meV. (c) a = 2.5 nm, ∆ECB = 50 meV. (d) a = 2.5 nm, ∆ECB = 300 meV. Inset in (c) shows calculated energy gaps of NCs in all four panels as indicated. Inset in (d) shows calculated size-dependence of the positive and negative trion Auger rates in CdSe core-only NCs.

larger volume. With the larger CdSe core, even a smaller band-offset of 50meV (Figure 3c) is capable of binding the electron to the core region, thereby producing positive and negative trion channel rates that are closer in magnitude. As a result, in this configuration, the negative trion channel contributes appreciably to the total BX Auger rate, as can be seen

12


Page 13 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

in Figure 3c. However, there is still a residual delocalization of the electron into the CdS shell which causes the reduction of the Auger rate with increasing shell width. Finally, both electron and hole are strongly confined to the core region in the case of type-I structure with a larger band-offset of 300meV and a = 2.5 nm in Figure 3d. This produces positive and negative trion rates of similar magnitudes and renders them (apart from the oscillations) independent of the shell thickness. Comparable magnitudes of the positive and negative channels is a characteristic of Auger process in core-only NCs (see inset in Figure 3d), and this ”core mode” is essentialy realized in type-I core/shell NCs in Figure 3d. The oscillatory behavior of the Auger rates demonstrated in Figures 2 and 3 is summarized in Figure 4. This Figure shows the rates of the negative and positive trion channels of the BX Auger process calculated as a function of both core radius and shell thickness in slightly elongated wurtzite CdSe/CdS NCs with µ = 0.1 and ∆ECB = 200 meV at room temperature. It can be observed that the dependence of the Auger rates on the core radius shows a qualitatively different oscillatory patterns than the dependence on the shell thickness (e.g., compare insets in Figures 4c and 4d). When viewed as a function of shell thickness, the oscillations are similar to those in Figures 2 and 3. Namely, the Auger rates oscillate with shell thickness roughly within an order of magnitude range and always remain finite (never reach zero) as the Coulomb matrix element maintains constant sign. In contrast, when viewed as a function of core radius, the oscillations are much wider and they periodically pass through zero as the matrix element changes sign. This results in appearance of ”valleys” where, at specific core sizes, the rate of either Auger channel can be low and nearly independent of shell thickness. This effect of nearly vanishing Auger rate can be clearly seen for the negative trion channel at core radii a ≈ 1.9 nm and a ≈ 3.1 nm. In the positive trion channel, this effect, while still clearly seen at a ≈ 2.0, 3.2 nm, is less pronounced since the three components of the positive trion rate do not vanish simultaneously. Similar behavior of widely oscillating and periodically vanishing Auger rate with NC size can be seen in core-only CdSe NCs shown in inset of Figure 3d. It has also been observed in Auger rate calculations in the past.[4, 15, 34, 44] The strong oscillatory dependence of the Auger rate on core size in comparison with its weaker dependence on shell thickness (see, e.g., insets in Figures 4c and 4d) can be qualitatively explained in terms of the overlap between wavefunctions of the four single-particle states participating in the Auger recombination. As shown schematically in Figure 4c and 13


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 28

Figure 4: Calculated dependence of the rate of (a) negative trion and (b) positive trion BX Auger channels on core radius and shell thickness in slightly elongated wurtzite CdSe/CdS NCs with ∆ECB = 200 meV at room temperature. The data are shown on logarithmic scale. (c),(d) Diagram of the mechanism producing two different oscillatory dependences of the Auger rate on core radius and shell thickness. Negative trion channel is used as example. The wavefunctions of the electron excited in Auger process are indicated by solid, dashed, dotted red lines. The corresponding potential profiles are shown by black solid, dashed, and dotted lines. See text for details. Insets in (c) and (d) are the negative trion Auger rates taken along the dashed lines in (a), as indicated.

4d, three of these four states (ground state electrons and holes) have nodeless envelope wavefunctions keeping constant sign. On the other hand, the fourth wavefunction, belonging to the excited carrier, oscillates and changes sign periodically. When the Coulomb matrix ele-

14


Page 15 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

ment is calculated, the volume of integration is determined mainly by the core size because the hole wavefunction is essentially zero outside the core region. As one can see in Figure 4c, the increase of the shell thickness does not change the number of nodes of the excited carrier wavefunction within the integration volume of the core. As a result, the increase of the shell thickness does not change appreciably the amount of positive and negative contributions to the integral, thereby preventing it from vanishing. The dependence of the Auger rate on shell thickness maintains only mild residual oscillations (see, e.g., in inset in Figure 4c, and in Figures 2 and 3) caused by periodic change of the overall amplitude of the excited carrier wavefunction inside the NCs with increasing core and/or shell. These amplitude changes can be seen in Figures 4c and 4d. In contrast, as shown in Figure 4d, when the core size is varied, the number of nodes of the excited wavefunction in the integration (core) region will vary. This would lead to periodic cancellation between the positive and negative contributions to the integral and consequent periodic vanishing of the Auger rate as a function of core size, as can be seen in Figure 4. The rate of nonradiative Auger recombination depends also on temperature due to its effect on the bulk energy gaps of the semiconductors comprising the NCs. In particular, in II-VI semiconductors the energy gaps decrease with temperature, leading to enhancement of the Auger rate. Figure 5a shows the Auger decay rates of BX that occur via positive and negative trion channels, calculated in spherically-shaped zinc-blende CdSe/CdS NCs having fixed core radius, a = 1.5 nm. The Auger rates are shown as a function of shell thickness at several temperatures, T = 5, 150, 300 K. The temperature dependence of the bulk energy gaps of CdSe and CdS was calculated using the phenomenological expression[45, 46] Eg (T ) = Eg (0) − 2EB /[exp(Θ/T ) − 1] , where EB and Θ are material-specific fitting parameters (see Supporting Information). The conduction band-offset was kept constant at ∆ECB = 200 meV. In the temperature range between 5 and 300 K, the energy gap of bulk CdSe decreases with temperature by ≈ 100 meV, as shown in the inset. This produces an overall enhancement of the rates of both positive and negative BX Auger channels in CdSe/CdS NCs by approximately a factor of 5, as one can see in Figure 5a. Further, an interesting behavior of the Auger recombination rate can be observed at low temperatures, where the energy gap is practically temperature independent (see the inset of Figure 5a). The rise of temperature can still influence the net Auger rate by changing the population of the three BX states, which have different rates of Auger recombination via 15


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 28

Figure 5: Biexciton Auger rates in CdSe/CdS NCs with fixed core radius, a = 1.5 nm, calculated as a function of shell thickness at temperatures T = 5, 150, 300 K. (a) Spherically-shaped zinc-blende NCs. (b) Slightly elongated wurtzite NCs. Red and blue lines represent positive and negative trion channels, respectively. Arrows indicate increase of temperature. Inset in (a) shows temperature dependence of the bulk energy gap in CdSe, calculated empirically (see Supporting Information). Grey area indicates the low-temperature regime where bulk Eg is essentially constant. Inset in (b) shows representative BX 3-level fine structure and the corresponding net positive trion rates taken at core radius and shell thickness of 1.5 nm.

positive trion channels. This effect is demonstrated in Figure 5b, which shows the Auger rates calculated as in Figure 5a, but in low temperature regime, T = 5, 40, 80 K, in slightly elongated ( µ = 0.05 ) wurtzite NCs. One can see that the net positive trion rates decrease with temperature. This decrease is due to a gradual shift of the BX population from the lowest non-degenerate BX level E− , which has higher positive trion rate, to the 4-fold degenerate level E2 , which has lower rate. Representative 3-level fine-structure of the BX and three corresponding positive trion rates in elongated wurtzite CdSe/CdS NCs with core radius and shell thickness of 1.5 nm are shown in inset of Figure 5b. At the same time, the Auger rate of the highest BX level, E+ , is higher than that of the middle level E2 . In the case of smaller splitting between the BX states, or in case of further increase of temperature, this difference in the Auger rates of positive trions could result in increasing of net Auger rate with temperature due to population changes. Our calculations confirm that growth of thick CdS shell upon small CdSe core results 16


Page 17 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

in strong suppression of the negative trion Auger recombination (Figures 2–5), while the positive trion process remains almost unaffected on average. Importantly, suppression of only the negative trion Auger process is sufficient to improve the luminescence efficiency in CdSe/CdS NCs because under constant illumination they can become negatively charged.[47, 48] In these core/shell NCs, an occasional BX formation could be an important source of the negative charging due to rapid ejection of holes out of NCs, followed by suppressed Auger recombination of the negative trions.[49] The improved luminescence in charged CdSe/CdS core/thick-shell NCs has been observed in experiment, where appearance of grey states in place of non-emitting dark states has been detected.[20, 22, 47] Similarly, full recovery of charged NC luminescence, up to intensities comparable with neutral NCs, has been observed in core/thick-shell structures.[49] Further, our results (Figure 4a,b) suggest an alternative approach of obtaining Auger-recombination-free NCs by exploiting the ”valleys” of zero Auger rate existing at several specific core sizes, but independent of shell thickness. Such a valley might have been observed recently in Ref. 50 which reported a series of nonblinking CdSe/CdS NCs with fixed core and various shell sizes. The rare residual off-periods of the samples in Ref. 50 can be attributed to the B-type blinking,[51] unrelated to the Auger quenching blinking mechanism. The theoretical work presented here does not take into account electronic correlations, which can be accounted for by using, e.g., configuration-interaction expansion of the manyelectron wavefunctions describing the initial and final states of the BX Auger process.[52] It has been shown in Ref. 53 that taking correlations into account could mildly influence the calculated Auger rate in CdSe core-only NCs. In addition, as mentioned above, in the present work we used standard boundary conditions[38] to describe the core-shell and shell-environment heterointerfaces. However, the model could be refined by accounting for the surface properties via additional parameters of general boundary conditions[34, 35, 38] describing interfaces between dissimilar semiconductor materials and/or ligands. In summary, we have presented calculations of the rates of nonradiative Auger recombination of ground state BXs in CdSe/CdS core/shell NCs, taking into account the fine-structure of the BX states induced by the NC asymmetry and hole-hole exchange interaction. We find that the rates of the Auger process originating in these fine-structure BX states differ for the positive trion channel, but are equal for the negative trion. The calculations show that the negative trion channel of the BX Auger process is suppressed in comparison with the 17


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 28

positive trion channel when the localization of the hole is stronger than that of the electron. Two different oscillatory dependencies of the Auger rates on core size and shell thickness are qualitatively explained in terms of the overlap between the three ground state single-particle states and the oscillating wavefunction of the excited carrier. Calculations show that raise of temperature enhances the rate of the BX Auger process due to reduction of the bulk energy gaps of CdSe and CdS, and that temperature modifications of the population of the BX fine-structure levels could modify the BX Auger rate at low temperatures. Supporting Information Supporting Information Available: (i) description of the boundary conditions used in our model for the single-particle electron and hole states, (ii) list of CdSe and CdS bulk material parameters, and (iii) phenomenological expressions describing the temperature-dependence of the bulk energy gaps. This material is available free of charge via the Internet at http://pubs.acs.org. Author Information The authors declare no competing financial interests. Acknowledgments R.V. acknowledges the financial support of the Office of Naval Research (ONR). A.R. acknowledges the support from the NCCR SwissMAP of the Swiss National Science Foundation. E.L. acknowledges the financial support of the Israel Council for High Education - Focal Area Technology (No. 872967), the Volkswagen Stiftung (No. 88116), the Niedersachsen-Deutsche Technion Gesellschaft E.V (No. ZN2916), and the Israel Science Foundation projects No. 1508/14 and 914/15. Al.L.E. acknowledges the financial support of the Office of Naval Research (ONR) through the Naval Research Laboratory Basic Research Program.

[1] Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Science 2000, 287, 1011-1013. [2] Abakumov, V. N.; Perel, V. I.; Yassievich, I. N. Nonradiative recombination in semiconductors; North Holland: New York, 1991. [3] de Rougemont, F.; Frey, R.; Roussignol, P.; Ricard, D.; Flytzanis, C. Appl. Phys. Lett. 1987,

18


Page 19 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

50, 1619-1621. [4] Chepic, D. I.; Efros, Al. L.; Ekimov, A. I.; Ivanov, M. G.; Kharchenko, V. A.; Kudriavstev, I. A.; Yazeva, T. V. J. Lumin. 1990, 47, 113-127. [5] Roussignol, P.; Kull, M.; Ricard, D.; de Rougemont, F.; Frey, R.; Flytzanis, C. Appl. Phys. Lett. 1987, 51, 1882-1884. [6] Achermann, M.; Bartko, A. P.; Hollingsworth, J. A.; Klimov, V. I. Nat. Phys. 2006, 2, 557-561. [7] Robel, I.; Gresback, R.; Kortshagen, U.; Schaller, R. D.; Klimov, V. I. Phys. Rev. Lett. 2009, 102, 177404-177407. [8] Efros, Al. L. in Semiconductor nanocrystals: From basic principles to applications; Efros, Al. L.; Lockwood, D. J.; Tsybeskov, L., Eds.; Springer: New York, 2003; Chapter 2. [9] Klimov, V. I.; Mikhailovsky, A. A.; Xu, S.; Malko, A.; Hollingsworth, J. A.; Leatherdale, C. A.; Eisler, H.-J.; Bawendi, M. G. Science 2000, 290, 314-317. [10] Kazes, M.; Oron, D.; Shweky, I.; Banin, U. J. Phys. Chem. C 2007, 111, 7898-7905. [11] Nozik, A. J. Physica E 2002, 14, 115-120. [12] Bae, W. K.; Park, Y.-S.; Lim, J.; Lee, D.; Padilha, L. A.; McDaniel, H.; Robel, I.; Lee, C.; Pietryga, J. M.; Klimov, V. I. Nat. Commun. 2013, 4, 2661-2668. [13] Nirmal, M.; Dabbousi, B. O.; Bawendi, M. G.; Macklin, J. J.; Trautman, J. K.; Harris, T. D.; Brus, L. E. Nature 1996, 383, 802-804. [14] Resch-Genger, U.; Grabolle, M.; Cavaliere-Jaricot, S.; Nitschke, R.; Nann, T. Nat. Methods 2008, 5, 763-775. [15] Cragg, G. E.; Efros, Al. L. Nano Lett. 2010, 10, 313-317. [16] Bae, W. K.; Padilha, L. A.; Park, Y. S; McDaniel, H.; Robel, I.; Pietryga, J. M.; Klimov, V. I. ACS Nano 2013, 7, 3411-3419. [17] Chen, Y.; Vela, J.; Htoon, H.; Casson, J. L.; Werder, D. J.; Bussian, D. A.; Klimov, V. I.; Hollingsworth, J. A. J. Am. Chem. Soc. 2008, 130, 5026-5027. [18] Garcia-Santamaria, F.; Chen, Y.; Vela, J.; Schaller, R. D.; Hollingsworth, J. A.; Klimov, V. I. Nano Lett. 2009, 9, 3482-3488. [19] Qin, W.; Liu, H.; Guyot-Sionnest, P. ACS Nano 2014, 8, 283-291. [20] Park, Y.-S.; Bae, W. K.; Pietryga, J. M.; Klimov, V. I. ACS Nano 2014, 8, 7288-7296. [21] Raino, G.; Stoferle, T.; Moreels, I.; Gomes, R.; Kamal, J. S.; Hens, Z.; Mahrt, R. F. ACS

19


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 28

Nano 2011, 5, 4031-4036. [22] Spinicelli, P.; Buil, S.; Quelin, X.; Mahler, B.; Dubertret, B.; Hermier, J.-P. Phys. Rev. Lett. 2009, 102, 136801-136804. [23] Park, Y.-S.; Bae, W. K.; Baker, T.; Lim, J.; Klimov, V. I. Nano Lett. 2015, 15, 7319-7328. [24] Nasilowski, M.; Spinicelli, P.; Patriarche, G.; Dubertret, B. Nano Lett. 2015, 15, 3953-3958. [25] Klimov, V. I. Annu. Rev. Condens. Matter Phys. 2014, 5, 285-316. [26] Park, Y.-S.; Bae, W. K.; Padilha, L. A.; Pietryga, J. M.; Klimov, V. I. Nano Lett. 2014, 14, 396-402. [27] Rodina, A. V.; Efros, Al. L. Phys. Rev. B 2010, 82, 125324-125337. [28] Pidgeon, C. R.; Brown, R. N. Phys. Rev. 1966, 146, 575-583. [29] Efros, Al. L.; Rosen, M. Phys. Rev. B 1998, 58, 7120-7135. [30] Bir, G. L.; Pikus, G. E. Symmetry and strain-induced effects in semiconductors; Wiley: New York, 1974. [31] Sirenko, Yu. M.; Jeon, J. B.; Lee, B. C.; Kim, K. W.; Littlejohn, M. A.; Stroscio, M. A.; Iafrate, G. J. Phys. Rev. B 1997, 55, 4360-4375. [32] Kapustina, A. B.; Petrov, B. V.; Rodina, A. V.; Seisyan, R. P. J. Cryst. Growth 2000, 214-215, 899-903. [33] Sercel, P. C.; Vahala, K. J. Phys. Rev. B 1990, 42, 3690-3710. [34] Vaxenburg, R.; Rodina, A.; Shabaev, A.; Lifshitz, E.; Efros, Al. L. Nano Lett. 2015, 15, 2092-2098. [35] Rodina, A. V.; Efros, Al. L.; Alekseev, A. Yu. Phys. Rev. B 2003, 67, 155312-155322. [36] Kane, E. O. J. Phys. Chem. Solids 1957, 1, 249-261. [37] Rabouw, F. T.; Vaxenburg, R.; Bakulin, A. A.; van Dijk-Moes, R. J. A.; Bakker, H. J.; Rodina, A.; Lifshitz, E.; Efros, Al. L.; Koenderink, A. F.; Vanmaekelbergh, D. ACS Nano 2015, 9, 10366-10376. [38] Rodina, A. V.; Alekseev, A. Yu.; Efros, Al. L.; Rosen, M.; Meyer, B. K. Phys. Rev. B 2002, 65, 125302-125313. [39] Efros, Al. L.; Rodina, A. V. Solid State Commun. 1989, 72, 645-649. [40] Shabaev, A.; Rodina, A. V.; Efros, Al. L.; Phys. Rev. B 2012, 86, 205311-205324. [41] Efros, Al. L. Phys. Rev. B 1992, 46, 7448-7458. [42] Efros, Al. L.; Rodina, A. V. Phys. Rev. B 1993, 47, 10005-10007.

20


Page 21 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

[43] Landau, L. D.; Lifshitz, E. M. Quantum Mechanics - Nonrelativistic Theory; Pergamon Press: Oxford, 1991. [44] Climente, J. I.; Movilla, J. L.; Planelles, J. Small 2012, 8, 754-759. [45] Logothetidis, S.; Cardona, M.; Lautenschlager, P.; Garriga, M. Phys. Rev. B 1986, 34, 24582469. [46] O’Donnel, K. P.; Chen, X. Appl. Phys. Lett. 1991, 58, 2924-2926. [47] Javaux, C.; Mahler, B.; Dubertret, B.; Shabaev, A.; Rodina, A. V.; Efros, Al. L.; Yakovlev, D. R.; Liu, F.; Bayer, M.; Camps, G.; Biadala, L.; Buil, S.; Quelin, X.; Hermier, J.-P. Nat. Nanotechnol. 2013, 8, 206-212. [48] 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, 035302-035313. [49] Galland, C.; Ghosh, Y.; Steinbruck, A.; Hollingsworth, J. A.; Htoon, H.; Klimov, V. I. Nat. Commun. 2012, 3, 908-914. [50] Qin, H.; Niu, Y.; Meng, R.; Lin, X.; Lai, R.; Fang, W.; Peng, X. J. Am. Chem. Soc. 2013, 136, 179-187. [51] Galland, C.; Ghosh, Y.; Steinbruck, A.; Sykora, M.; Hollingsworth, J. A.; Klimov, V. I.; Htoon, H. Nature 2011, 479, 203-208. [52] Wang, L.-W.; Califano, M.; Zunger, A.; Franceschetti, A. Phys. Rev. Lett 2003, 91, 056404056407. [53] Korkusinski, M.; Voznyy, O.; Hawrylak, P. Phys. Rev. B 2011, 84, 155327-155337.

21


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

For TOC only:

22


Page 22 of 28

Page 23 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Nano Letters


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


Page 24 of 28

Page 25 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

Nano Letters


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


Page 26 of 28

Page 27 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Nano Letters


Nano Letters

1 2 3 4 5 6

Page 28 of 28


CdS Core ... - ACS Publications

Recommend Documents