- Email: [email protected]
ZnS core-shell quantum dots
Solid State Communications 151 (2011) 1743–1748
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Auger process in CdS/ZnS core-shell quantum dots Ibtissem Zgaren a,∗ , Jalloul Balti a , Sihem Jaziri a,b a
Laboratoire de Physique des Matériaux, Faculté des Sciences de Bizerte, 7021 Jarzouna, Tunisia
b
Laboratoire de Physique de la Matière Condensée, Faculté des Sciences de Tunis, Tunisia
article
info
Article history: Received 28 July 2011 Received in revised form 22 August 2011 Accepted 26 August 2011 by M. Grynberg Available online 2 September 2011 Keywords: A. II–VI semiconductors A. Core-shell D. Auger recombination D. Auger relaxation
abstract Auger processes are investigated for CdS/ZnS core-shell quantum dots. Auger recombination (AR) lifetime and electron relaxation inside the core are computed. Using the effective-mass theory and by solving a three-dimension Schrödinger equation we predict the dependence of Auger relaxation on size of coreshell nanocrystals. We considered in this work different AR processes: the excited electron (EE), excited hole (EH), multiexciton AR type. Likewise, Auger multiexciton recombination rates are predicted for biexciton. Our results show that biexciton AR type is more efficient than the other AR process (excited electron (EE) and excited hole (EH)). We also found that electron Auger relaxation P → S is very efficient in core-shell nanostructures. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The last decade has been marked by great progress in the use of colloidal semiconductor quantum dots (QDs) due to both their size-controlled electronic and optical properties [1], and large technological potential for applications in such areas as biotechnology [2], photovoltaic [3], light-emitting diodes [4], nonlinear optics [5] and lasing [6,7]. Recently, various kinds of semiconductor heterostrucutres have been fabricated. Among these, a synthesized inhomogeneous spherical quantum dot with a central semiconductor core and coated with a different semiconductor shell, the so-called coreshell QDs. To obtain a noticeable quantum efficiency of the QD luminescence, the core-shell structures can also be effectively designed in the form of colloidal particles. In these structures, the shell with the large band-gap material serves as a surface passivating layer and as a barrier assisting the electron–hole confinement in the core [8]. The stability and efficiency of the QD luminescence is a critical aspect. The pathway of recombination of electrons and holes (excitons) has, in particular, attracted a lot of attention. Experimental works [9–11] show that nonradiative AR is the dominant mechanism of electron relaxation. Auger recombination occurs when excitons recombine and instead of emitting light, transfer their energy to a nearby electron or hole, creating a ‘‘hot’’ electron or ‘‘hot’’ hole.
In semiconductor nanostructures core-shell with strong quantum confinement, Auger effects are greatly enhanced by increased carrier–carrier interactions [12]. On the one hand, the energy released from a carrier electron or hole recombination is given to another carrier in a single scattering event. On the other hand, in the bulk or a 2D quantum well, the relaxation of an excited electron to its ground state usually causes phonon emission. In strongly confined QDs electronic states are discrete and it is expected [13] that this fact prevents phonon assisted electron relaxation (phonon bottleneck). Indeed, electron relaxation rates in CdSe QDs were observed to be fast [14], and to explain this fast relaxation, an alternative mechanism has been proposed. It is supported by model calculations, namely Auger carrier–carrier scattering. For instance, it was proposed by Efros et al. [15] that in a photoexcited QD the hot electron can transfer its energy to the hole via an Auger process involving electron–hole scattering. The principal aim of our work is to investigate different Auger processes in CdS/ZnS core-shell quantum dots. Within the framework of the effective mass approximation, we calculated the electron and hole energy levels and associated wave functions. Thereafter, we focused on Auger carrier relaxation inside the core, and then we studied the biexciton AR type. 2. Model and theory 2.1. Electron and hole energy levels
∗
Corresponding author. Tel.: +216 97 532 374; fax: +216 72 590 566. E-mail address: [email protected] (I. Zgaren).
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.08.037
We consider a spherical quantum dot with a central II–VI semiconductor core (CdS) and coated with a different larger band
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I. Zgaren et al. / Solid State Communications 151 (2011) 1743–1748
Fig. 1. (a) Two-dimensional model and (b) the potential diagram of the CdS/ZnS core-shell quantum dot.
gap semiconductor material acting as a shell (ZnS). Due to the bulk energy gap difference between the two materials, a true quantum well is created within the quantum dot. In this core-shell structure the inner radius is denoted by ‘‘a’’ and outer radius ‘‘b’’ (Fig. 1). In the framework of the effective mass approximation, the Schrödinger equation of the electron in the spherical coordinate reads:
−
h¯ 2
∆ + V r Φnlm (r , θ , ϕ) = E Φnlm (r , θ , ϕ) ( ) i ∗
(1)
2mi
where m∗i is the effective mass of an electron in the i-th region − → which depends on the position vector r due to different materials in the heterostructure. Because of the variety in the band gap, the potential V (r) is different for electrons or holes in each region. The effective mass and the confinement potential are expressed as follows: m∗i =
mCdS , m∗ZnS , ∗
r ≤a a≤r ≤b
and Vi (r ) =
0, V,
0≤r ≤a a≤r ≤b ∞, r > b.
Due to the spherical symmetry in the geometry of the quantum r , we can separate dot, the potential V (r) = V (⃗ r ) where r = ⃗ the envelope function into the functions with radial and angular coordinate dependencies: Φnlm (r , θ , ϕ) = Rnl (r )Ylm (θ , ϕ) where Rnl (r ) is the radial envelope function as a function of r only and Ylm (θ , ϕ) is the so-called spherical harmonic, where n is the principal quantum number and l and m are the angular momentum quantum numbers. We denote the states by nS (l = 0, n = 1, 2, 3, . . .), nP (l = 1, n = 1, 2, 3, . . .), nD(l = 2, n = 1, 2, 3, . . .). The radial eigen function Rnl (r ) consists of three parts according to the electron position. Two cases need to be considered for the solution of Rnl (r ). In regions where electron eigen energy E satisfies the condition E > V , the solution is a linear combination of spherical Bessel function jl and Neumann function nl . It may be written as follows [16]:
Rnl (r ) =
(1) (1) (1) Rnl (r ) = Ajl knl r + Bnl knl r ,
(2) (2) (2) Rnl (r ) = Cjl knl r + Dnl knl r , (3) Rnl (r ) = 0, r > b
(2)
a≤r ≤b
(i)
=
2mi (E − Vi ) / h¯ . Given that the wave function is finite for r → 2
∗
0, we deduce B = 0. Furthermore, the wave function must satisfy the following boundary conditions [17,18] (1)
(2)
Rnl (a) = Rnl (a)
and
1 ∗
m1
(1)
dRnl dr
= r =a
1 ∗
m2
(2)
dRnl dr
Rnl (r )
(1) (1) (1) ′ ′ R ( r ) = A j k r + B n k r , r≤a l l nl nl nl ( 2 ) (+) ( 2 ) (−) (2) ′ ′ = R (r ) = C h iknl r + D hl iknl r , a ≤ r ≤ b (4) nl l (3) Rnl (r ) = 0, r > b ′ ′ ′ ′ where A , B , C and D are also normalization constants and (i)
knl =
. (3) r =a
With the requirement that the wave function has to vanish (2) sufficiently rapidly for r → ∞, we have Rnl (b) = 0 and the wave
2m∗i (Vi − E ) / h¯ 2 . By similar considerations as for E >
V , these constants can be easily fixed by solving the obtained equations. Hence, we get the charge carrier’s wave functions and eigen energies for the two considered energy cases. The electron wave function and the eigen energy will be respectively denoted Φnlm (r , θ , ϕ) and E. Similarly, we determine the energy levels and wave function of the hole. 2.2. Auger processes In order to investigate the Auger processes in CdS/ZnS semiconductor quantum dots, we consider two process types: carrier Auger relaxation and Auger recombination. First, we study the carrier Auger relaxation when the electron is inside the core (Fig. 2(a)). Once we have calculated the wave functions and energy levels of confined carriers in CdS/ZnS nanocrystals as functions of core radius and shell thickness by solving the effective mass approximation Hamiltonian, the Auger scattering rate can be computed by using Fermi’s golden rule: 1
r ≤a
where A, B, C and D are normalized constants and knl
function must be normalized. Thus, solving the obtained equations gives the electron wave function. In regions where E < V , Rnl (r ) stays the same as in the region E > V for r ≤ a and r > b. But for a < r ≤ b, the radial wave (+) (−) function is a combination of the Hankel functions hl and hl as given below:
τi,l
=
2π − (eh) 2 Ji,j,k,l δ Ei + Ek − El − Ej h¯ k
(5)
where, (Ei ; Ek ) and El ; Ej correspond to the many particle energy of the initial and final states, and the labels l, j and k, l refer respectively to the initial and final states. The electron–hole
(eh)
Coulomb scattering matrix elements Ji,j,k,l are given as follows:
(eh)
Jij;kl =
∫∫
(h)
∗ ∗ (e) Φi (⃗r ′ ) Wc (⃗r , ⃗r ′ )
d3 ⃗ r d3 ⃗ r ′ Φj (⃗ r)
× Φl(e) (⃗r ′ )Φk(h) (⃗r ) .
(6)
The Wc presents here the Coulomb interaction between the 2
electron and hole which is written as: Wc ⃗ r −⃗ r ′ = 4π ε−⃗re−⃗r ′ with | | ε is the dielectric constant of the embedding material between
I. Zgaren et al. / Solid State Communications 151 (2011) 1743–1748
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Fig. 2. Auger processes in core-shell nanocrystal quantum dots: (a) carrier relaxation within the core, (b) excited electron recombination, (c) excited hole recombination, (d) and (e) are the biexciton recombination types resulting in excited hole and excited electron excitons, respectively.
electron and hole. We introduce a phenomenological broadening Γ of the final states that allows us to replace δ (x) in Eq. (5) with a Gaussian function [19]:
√ −1 exp δ (x) = Γ 2π
−x
2
2Γ 2
(7)
where x = Ei + Ek − El − Ej . The Gaussian function presents a maximum around the exact conserved energy Ek = Ei + Ej − El . Second, we study the Auger recombination rate, considering different Auger recombination possibilities as shown in Fig. 2. We have the excited electron (Fig. 2(b)), the excited hole (Fig. 2(c)) and their biexciton variants (Fig. 2(d) and (e)). Then, the Auger recombination rate is calculated similarly to the relaxation scattering rate by using Fermi’s golden rule in Eq. (5). The calculated biexciton scattering rate is expressed in terms of excited electron (EE) and excited hole (EH) AR type as follows [20]: 1
τbiexciton
=
2
τEE
+
2
τEH
.
Table 1 Materials parameters for CdS and ZnS [21–24]. Material
CdS
ZnS
Electron effective mass m∗e Heavy hole effective mass m∗h Dielectric constant Energy gap (eV) Conduction offset (eV) Valence bond offset (eV)
0.18 0.8 5.5 2.5 0.897 0.49
0.42 0.61 5.2 3.62 – –
(8)
3. Results and discussion 3.1. Energy levels We have numerically simulated the electron and the hole energy levels, and we have investigated the effect of the core and shell size on the energy levels. The material constants used for this system are presented in Table 1. We considered first two different core radius 20 and 40 Å for a 50 Å fixed shell thickness, and then we considered two different shell thicknesses 30 and 50 Å for a fixed core radius, equal to 20 Å. In Figs. 3 and 4 we present the obtained confined states of electron–hole energy for different nanocrystal sizes. The electron and hole states are, respectively, arranged in Se , Pe , De and Sh , Ph , Dh . One the one hand, and comparing the two first QDs presented in Fig. 3(a) and (b), it turns out that the electron and hole energy
Fig. 3. Electron and hole states for CdS/ZnS core-shell quantum dot for two different core radii and the shell thickness fixed equal 50 Å.
levels depend on the CdS core radius; the electron–hole energy is relatively higher for a = 20 Å than for a = 40 Å. On the other hand, the hole mass in CdS is much heavier than the electron mass, and so, confined hole states tend to be more densely spaced than electron states.
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I. Zgaren et al. / Solid State Communications 151 (2011) 1743–1748
Fig. 4. Electron and hole states for CdS/ZnS core-shell quantum dot for two different shell thicknesses and the core radius fixed equal 20 Å.
Fig. 5. The matrix element |J (Sh , Se , Pe , kh )| for CdS/ZnS core-shell quantum dots as functions of the nanocrystal shell thickness for different core radius ‘‘a’’.
Now, when we consider the effect of shell thickness on energy states we remark that the electron and hole energies are relatively higher for greater thickness (Fig. 4(a) and (b)). We distinguish that for a fixed core radius a = 20 Å, energy levels differ from one shell thickness to another. For instance, the 1Se electron energy is about 2783 and 2680 eV, for 30 and 50 Å shell thickness respectively. It should be noted that shell thickness also influences the optical properties in CdS/ZnS core-shell QDs and it is well studied in our previous works [25,26]. 3.2. Auger relaxation Second, we consider as initial electron–hole state |1Pe 1Sh ⟩ those corresponding to the electron in the 1P state and the hole in the low-lying state 1S; and as final scattering states those that correspond to an electron occupying the 1S state and a hole in a deep state k (Fig. 2 (a)); i.e. |1Se kh ⟩. Hence, the decay rate can be computed by using the Fermi Golden rule: 1
τA
=
2 2π − (eh) JPe ,Se ,kh ,Sh δ EPe + Ekh − ESe − ESh . h¯ k
(9)
Fig. 5. shows the variation of the electron–hole Coulomb scattering matrix elements |J (Sh , Se , Pe , kh )| as a function of the shell thickness for different core radius a = 18, 20 and 22 Å. It can be seen that the Coulomb energy decreases with the increase of the shell thickness or core radius. This behavior is the same observed by Ferreira and Bastard in their work on the Auger relaxation in QDs [27]. From the Fig. 5, we would like to emphasize the significant role of the Coulomb matrix elements. As the quantum dots size increases, this Coulomb interaction decreases due to the decreasing wave function overlap. Indeed, in their work on the AR in PbSe NCs, Allan and Delerue have deduced that such Coulombic interactions are primarily governed by the state-density function [28]. Consequently, the strong size dependence indicates the nontrivial role of Coulomb scattering matrix elements. Eq. (9) gives the direct relation between time relaxation and the matrix element, and it is evident that time relaxation should also be sensitive to the nanocrystal size variations. We present in Fig. 6 the thickness dependence of P → S Auger relaxation time τA for three different core radii, a = 18 Å, a = 20 Å and a = 22 Å. Our calculations show a great dependence of time
Fig. 6. Auger relaxation lifetime in CdS/ZnS as functions of the nanocrystal shell thickness for different core radius ‘‘a’’.
relaxation on QDs size. On the one hand, it increases with shell thickness increasing; on the other hand it decreases when core radius increases. For 50 Å shell thickness for example, τA progresses from 18 ps to 215 ps corresponding to 22 Å and 18 Å respectively. However, time variation with shell thickness is clear as presented in Fig. 6, but it is more important for a smaller core radius. For 18 Å core radius, τA varies from 22 to 215 ps corresponding to a shell thickness variation from 32 to 50 Å. This variation is smaller for a greater radius; it is about 18 ps for 22 Å core radius. This behavior is observed by Wang et al. [29] for CdSe colloids. They calculated auger time relaxation τA for (P → S ) transition, using the pseudo potential-based atomistic approach and they concluded that time relaxation decreases with QDs radius increasing. They found τA = 0, 6 ps and τA = 0, 2 ps for 29 and 38 Å radius. Otherwise, the k.p-based calculation of Efros and coworkers [30] predicts Auger decay lifetimes in CdSe colloidal dots of τA ∼ 2 ps almost independently of dot size for radius between 20 and 40 Å.
I. Zgaren et al. / Solid State Communications 151 (2011) 1743–1748
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CdS/ZnS core-shell quantum dot, the carriers (electron or hole) are confined in the shell. AR lifetime is inversely proportional to Coulomb interaction and when the shell thickness increases, Coulomb interaction decreases due to the decreased wave function overlap and then the AR lifetime increases. Moreover, Nanda et al. [33] observed the same behavior for inverted core-shell. They used ZnSe/CdSe and found that biexciton Auger lifetime increases with shell thickness. They consider that this fact is due to the increasing degrees of quantum confinement. Similarly, previous experimental studies of colloidal CdS nanoparticles caped with L -glutathione (GSH) and myristic acid (MA) [10] indicate that Auger decay is extremely fast, and show that the biexciton AR rate depends on the size of QDs. The decay of multiexcitons (Biexciton, triexciton. . . ) in NCs is, however, dominated not by radiative processes but by nonradiative Auger recombination, in which the e–h recombination energy is transferred to a third carrier (an electron or a hole). 4. Conclusion Fig. 7. The Auger recombination rate results for excited electron, excited hole and the calculated biexciton type AR in CdS/ZnS heterostructures as a function of the shell thickness for fixed core radius value a = 20 Å.
Auger mechanism requires the presence of a hole to be effective; this mechanism was challenged by Guyot-Sionnest et al. [31] as an explanation of the observed P → S electron relaxation. They reported a relatively fast (10–30 ps) electron decay in CdSe dots passivated by alkylamines or dodecanethiol ligands, which act as hole traps and thus separate the photogenerated hole from the electron. Likewise, Califano [32] predict that the Auger mechanism can explain the experimentally observed intraband P → S relaxation time scale without the need to invoke any exotic relaxation.
In this paper, we have developed a theoretical treatment of the size dependent electronics and optical properties in CdS/ZnS coreshell quantum dots. We computed the electron and hole energy variation in terms of the nanocrystal size and then we focused on Auger relaxation and Auger recombination. Our results show that all these processes depend on both core radius and shell thickness. Our study revealed that the AR type biexciton is more efficient than the other AR processes (excited electron and excited hole), and for a fixed core radius value a = 20 Å, the carriers (electron or hole) are confined in the shell, which is in good agreement with recent experimental studies [10,11].
3.3. Auger recombination (AR)
References
In this part of our work we are interested in the Auger multiexciton recombination scattering rate that we will predict for biexciton. It depends on two terms, the excited electron (EE) and the excited hole (EH) AR type. We calculate it by using Eq. (8) [20], where the two terms are determined by the following expressions: 1
τEE 1
τEH
=
2 2π − (eh) JSe ,Se ,Sh ,Sk δ Eg + ESe − Eke h¯ k
(10)
=
2 2π − (eh) JSh ,Sh ,kh ,Se δ Eg + Ekh − ESh . h¯ k
(11)
Eg being the ground state exciton energy. In Fig. 7. we present the AR rate computed for excited electron (EE), excited hole (EH) and the biexciton, in terms of the shell thickness for a fixed core radius a = 20 Å. We clearly observe that for the three processes, AR rate decreases when shell thickness increases. It can be noted that AR rate variation depending on shell thickness is different for the three processes but it is considerably greater for the biexciton type. This finding results in an important increasing of biexciton AR lifetime with increasing shell thickness. Indeed our calculations implicate that τbiexciton vary from 0.5 ns to 20 ns corresponding from 36 Å to 56 Å respectively, which is almost forty times longer. Recently, a similar behavior was experimentally observed by Garcia-Santamaria et al. [11] for CdSe/CdS core-shell. They used a 15 Å core radius and observed for biexciton a great lifetime increase with increasing shell thickness. They consider that biexciton AR lifetime depends on shell thickness because the electron is delocalized in the shell region, which confirms our simulation results. Indeed, in our case the AR rate shows that in
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[27] R. Ferreira, G. Bastard, Appl. Phys. Lett. 74 (1999) 19. [28] G. Allan, C. Delerue, Phys. Rev. B 73 (2006) 205423. [29] L.-W. Wang, M. Califano, A. Zunger, A. Franceschetti, Phys. Rev. Lett. 91 (2003) 056404. [30] Al.L. Efros, V.A. Kharchenko, M. Rosen, Solid State Commun. 93 (1995) 281.
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Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Auger process in CdS/ZnS core-shell quantum dots Ibtissem Zgaren a,∗ , Jalloul Balti a , Sihem Jaziri a,b a
Laboratoire de Physique des Matériaux, Faculté des Sciences de Bizerte, 7021 Jarzouna, Tunisia
b
Laboratoire de Physique de la Matière Condensée, Faculté des Sciences de Tunis, Tunisia
article
info
Article history: Received 28 July 2011 Received in revised form 22 August 2011 Accepted 26 August 2011 by M. Grynberg Available online 2 September 2011 Keywords: A. II–VI semiconductors A. Core-shell D. Auger recombination D. Auger relaxation
abstract Auger processes are investigated for CdS/ZnS core-shell quantum dots. Auger recombination (AR) lifetime and electron relaxation inside the core are computed. Using the effective-mass theory and by solving a three-dimension Schrödinger equation we predict the dependence of Auger relaxation on size of coreshell nanocrystals. We considered in this work different AR processes: the excited electron (EE), excited hole (EH), multiexciton AR type. Likewise, Auger multiexciton recombination rates are predicted for biexciton. Our results show that biexciton AR type is more efficient than the other AR process (excited electron (EE) and excited hole (EH)). We also found that electron Auger relaxation P → S is very efficient in core-shell nanostructures. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The last decade has been marked by great progress in the use of colloidal semiconductor quantum dots (QDs) due to both their size-controlled electronic and optical properties [1], and large technological potential for applications in such areas as biotechnology [2], photovoltaic [3], light-emitting diodes [4], nonlinear optics [5] and lasing [6,7]. Recently, various kinds of semiconductor heterostrucutres have been fabricated. Among these, a synthesized inhomogeneous spherical quantum dot with a central semiconductor core and coated with a different semiconductor shell, the so-called coreshell QDs. To obtain a noticeable quantum efficiency of the QD luminescence, the core-shell structures can also be effectively designed in the form of colloidal particles. In these structures, the shell with the large band-gap material serves as a surface passivating layer and as a barrier assisting the electron–hole confinement in the core [8]. The stability and efficiency of the QD luminescence is a critical aspect. The pathway of recombination of electrons and holes (excitons) has, in particular, attracted a lot of attention. Experimental works [9–11] show that nonradiative AR is the dominant mechanism of electron relaxation. Auger recombination occurs when excitons recombine and instead of emitting light, transfer their energy to a nearby electron or hole, creating a ‘‘hot’’ electron or ‘‘hot’’ hole.
In semiconductor nanostructures core-shell with strong quantum confinement, Auger effects are greatly enhanced by increased carrier–carrier interactions [12]. On the one hand, the energy released from a carrier electron or hole recombination is given to another carrier in a single scattering event. On the other hand, in the bulk or a 2D quantum well, the relaxation of an excited electron to its ground state usually causes phonon emission. In strongly confined QDs electronic states are discrete and it is expected [13] that this fact prevents phonon assisted electron relaxation (phonon bottleneck). Indeed, electron relaxation rates in CdSe QDs were observed to be fast [14], and to explain this fast relaxation, an alternative mechanism has been proposed. It is supported by model calculations, namely Auger carrier–carrier scattering. For instance, it was proposed by Efros et al. [15] that in a photoexcited QD the hot electron can transfer its energy to the hole via an Auger process involving electron–hole scattering. The principal aim of our work is to investigate different Auger processes in CdS/ZnS core-shell quantum dots. Within the framework of the effective mass approximation, we calculated the electron and hole energy levels and associated wave functions. Thereafter, we focused on Auger carrier relaxation inside the core, and then we studied the biexciton AR type. 2. Model and theory 2.1. Electron and hole energy levels
∗
Corresponding author. Tel.: +216 97 532 374; fax: +216 72 590 566. E-mail address: [email protected] (I. Zgaren).
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.08.037
We consider a spherical quantum dot with a central II–VI semiconductor core (CdS) and coated with a different larger band
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I. Zgaren et al. / Solid State Communications 151 (2011) 1743–1748
Fig. 1. (a) Two-dimensional model and (b) the potential diagram of the CdS/ZnS core-shell quantum dot.
gap semiconductor material acting as a shell (ZnS). Due to the bulk energy gap difference between the two materials, a true quantum well is created within the quantum dot. In this core-shell structure the inner radius is denoted by ‘‘a’’ and outer radius ‘‘b’’ (Fig. 1). In the framework of the effective mass approximation, the Schrödinger equation of the electron in the spherical coordinate reads:
−
h¯ 2
∆ + V r Φnlm (r , θ , ϕ) = E Φnlm (r , θ , ϕ) ( ) i ∗
(1)
2mi
where m∗i is the effective mass of an electron in the i-th region − → which depends on the position vector r due to different materials in the heterostructure. Because of the variety in the band gap, the potential V (r) is different for electrons or holes in each region. The effective mass and the confinement potential are expressed as follows: m∗i =
mCdS , m∗ZnS , ∗
r ≤a a≤r ≤b
and Vi (r ) =
0, V,
0≤r ≤a a≤r ≤b ∞, r > b.
Due to the spherical symmetry in the geometry of the quantum r , we can separate dot, the potential V (r) = V (⃗ r ) where r = ⃗ the envelope function into the functions with radial and angular coordinate dependencies: Φnlm (r , θ , ϕ) = Rnl (r )Ylm (θ , ϕ) where Rnl (r ) is the radial envelope function as a function of r only and Ylm (θ , ϕ) is the so-called spherical harmonic, where n is the principal quantum number and l and m are the angular momentum quantum numbers. We denote the states by nS (l = 0, n = 1, 2, 3, . . .), nP (l = 1, n = 1, 2, 3, . . .), nD(l = 2, n = 1, 2, 3, . . .). The radial eigen function Rnl (r ) consists of three parts according to the electron position. Two cases need to be considered for the solution of Rnl (r ). In regions where electron eigen energy E satisfies the condition E > V , the solution is a linear combination of spherical Bessel function jl and Neumann function nl . It may be written as follows [16]:
Rnl (r ) =
(1) (1) (1) Rnl (r ) = Ajl knl r + Bnl knl r ,
(2) (2) (2) Rnl (r ) = Cjl knl r + Dnl knl r , (3) Rnl (r ) = 0, r > b
(2)
a≤r ≤b
(i)
=
2mi (E − Vi ) / h¯ . Given that the wave function is finite for r → 2
∗
0, we deduce B = 0. Furthermore, the wave function must satisfy the following boundary conditions [17,18] (1)
(2)
Rnl (a) = Rnl (a)
and
1 ∗
m1
(1)
dRnl dr
= r =a
1 ∗
m2
(2)
dRnl dr
Rnl (r )
(1) (1) (1) ′ ′ R ( r ) = A j k r + B n k r , r≤a l l nl nl nl ( 2 ) (+) ( 2 ) (−) (2) ′ ′ = R (r ) = C h iknl r + D hl iknl r , a ≤ r ≤ b (4) nl l (3) Rnl (r ) = 0, r > b ′ ′ ′ ′ where A , B , C and D are also normalization constants and (i)
knl =
. (3) r =a
With the requirement that the wave function has to vanish (2) sufficiently rapidly for r → ∞, we have Rnl (b) = 0 and the wave
2m∗i (Vi − E ) / h¯ 2 . By similar considerations as for E >
V , these constants can be easily fixed by solving the obtained equations. Hence, we get the charge carrier’s wave functions and eigen energies for the two considered energy cases. The electron wave function and the eigen energy will be respectively denoted Φnlm (r , θ , ϕ) and E. Similarly, we determine the energy levels and wave function of the hole. 2.2. Auger processes In order to investigate the Auger processes in CdS/ZnS semiconductor quantum dots, we consider two process types: carrier Auger relaxation and Auger recombination. First, we study the carrier Auger relaxation when the electron is inside the core (Fig. 2(a)). Once we have calculated the wave functions and energy levels of confined carriers in CdS/ZnS nanocrystals as functions of core radius and shell thickness by solving the effective mass approximation Hamiltonian, the Auger scattering rate can be computed by using Fermi’s golden rule: 1
r ≤a
where A, B, C and D are normalized constants and knl
function must be normalized. Thus, solving the obtained equations gives the electron wave function. In regions where E < V , Rnl (r ) stays the same as in the region E > V for r ≤ a and r > b. But for a < r ≤ b, the radial wave (+) (−) function is a combination of the Hankel functions hl and hl as given below:
τi,l
=
2π − (eh) 2 Ji,j,k,l δ Ei + Ek − El − Ej h¯ k
(5)
where, (Ei ; Ek ) and El ; Ej correspond to the many particle energy of the initial and final states, and the labels l, j and k, l refer respectively to the initial and final states. The electron–hole
(eh)
Coulomb scattering matrix elements Ji,j,k,l are given as follows:
(eh)
Jij;kl =
∫∫
(h)
∗ ∗ (e) Φi (⃗r ′ ) Wc (⃗r , ⃗r ′ )
d3 ⃗ r d3 ⃗ r ′ Φj (⃗ r)
× Φl(e) (⃗r ′ )Φk(h) (⃗r ) .
(6)
The Wc presents here the Coulomb interaction between the 2
electron and hole which is written as: Wc ⃗ r −⃗ r ′ = 4π ε−⃗re−⃗r ′ with | | ε is the dielectric constant of the embedding material between
I. Zgaren et al. / Solid State Communications 151 (2011) 1743–1748
1745
Fig. 2. Auger processes in core-shell nanocrystal quantum dots: (a) carrier relaxation within the core, (b) excited electron recombination, (c) excited hole recombination, (d) and (e) are the biexciton recombination types resulting in excited hole and excited electron excitons, respectively.
electron and hole. We introduce a phenomenological broadening Γ of the final states that allows us to replace δ (x) in Eq. (5) with a Gaussian function [19]:
√ −1 exp δ (x) = Γ 2π
−x
2
2Γ 2
(7)
where x = Ei + Ek − El − Ej . The Gaussian function presents a maximum around the exact conserved energy Ek = Ei + Ej − El . Second, we study the Auger recombination rate, considering different Auger recombination possibilities as shown in Fig. 2. We have the excited electron (Fig. 2(b)), the excited hole (Fig. 2(c)) and their biexciton variants (Fig. 2(d) and (e)). Then, the Auger recombination rate is calculated similarly to the relaxation scattering rate by using Fermi’s golden rule in Eq. (5). The calculated biexciton scattering rate is expressed in terms of excited electron (EE) and excited hole (EH) AR type as follows [20]: 1
τbiexciton
=
2
τEE
+
2
τEH
.
Table 1 Materials parameters for CdS and ZnS [21–24]. Material
CdS
ZnS
Electron effective mass m∗e Heavy hole effective mass m∗h Dielectric constant Energy gap (eV) Conduction offset (eV) Valence bond offset (eV)
0.18 0.8 5.5 2.5 0.897 0.49
0.42 0.61 5.2 3.62 – –
(8)
3. Results and discussion 3.1. Energy levels We have numerically simulated the electron and the hole energy levels, and we have investigated the effect of the core and shell size on the energy levels. The material constants used for this system are presented in Table 1. We considered first two different core radius 20 and 40 Å for a 50 Å fixed shell thickness, and then we considered two different shell thicknesses 30 and 50 Å for a fixed core radius, equal to 20 Å. In Figs. 3 and 4 we present the obtained confined states of electron–hole energy for different nanocrystal sizes. The electron and hole states are, respectively, arranged in Se , Pe , De and Sh , Ph , Dh . One the one hand, and comparing the two first QDs presented in Fig. 3(a) and (b), it turns out that the electron and hole energy
Fig. 3. Electron and hole states for CdS/ZnS core-shell quantum dot for two different core radii and the shell thickness fixed equal 50 Å.
levels depend on the CdS core radius; the electron–hole energy is relatively higher for a = 20 Å than for a = 40 Å. On the other hand, the hole mass in CdS is much heavier than the electron mass, and so, confined hole states tend to be more densely spaced than electron states.
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Fig. 4. Electron and hole states for CdS/ZnS core-shell quantum dot for two different shell thicknesses and the core radius fixed equal 20 Å.
Fig. 5. The matrix element |J (Sh , Se , Pe , kh )| for CdS/ZnS core-shell quantum dots as functions of the nanocrystal shell thickness for different core radius ‘‘a’’.
Now, when we consider the effect of shell thickness on energy states we remark that the electron and hole energies are relatively higher for greater thickness (Fig. 4(a) and (b)). We distinguish that for a fixed core radius a = 20 Å, energy levels differ from one shell thickness to another. For instance, the 1Se electron energy is about 2783 and 2680 eV, for 30 and 50 Å shell thickness respectively. It should be noted that shell thickness also influences the optical properties in CdS/ZnS core-shell QDs and it is well studied in our previous works [25,26]. 3.2. Auger relaxation Second, we consider as initial electron–hole state |1Pe 1Sh ⟩ those corresponding to the electron in the 1P state and the hole in the low-lying state 1S; and as final scattering states those that correspond to an electron occupying the 1S state and a hole in a deep state k (Fig. 2 (a)); i.e. |1Se kh ⟩. Hence, the decay rate can be computed by using the Fermi Golden rule: 1
τA
=
2 2π − (eh) JPe ,Se ,kh ,Sh δ EPe + Ekh − ESe − ESh . h¯ k
(9)
Fig. 5. shows the variation of the electron–hole Coulomb scattering matrix elements |J (Sh , Se , Pe , kh )| as a function of the shell thickness for different core radius a = 18, 20 and 22 Å. It can be seen that the Coulomb energy decreases with the increase of the shell thickness or core radius. This behavior is the same observed by Ferreira and Bastard in their work on the Auger relaxation in QDs [27]. From the Fig. 5, we would like to emphasize the significant role of the Coulomb matrix elements. As the quantum dots size increases, this Coulomb interaction decreases due to the decreasing wave function overlap. Indeed, in their work on the AR in PbSe NCs, Allan and Delerue have deduced that such Coulombic interactions are primarily governed by the state-density function [28]. Consequently, the strong size dependence indicates the nontrivial role of Coulomb scattering matrix elements. Eq. (9) gives the direct relation between time relaxation and the matrix element, and it is evident that time relaxation should also be sensitive to the nanocrystal size variations. We present in Fig. 6 the thickness dependence of P → S Auger relaxation time τA for three different core radii, a = 18 Å, a = 20 Å and a = 22 Å. Our calculations show a great dependence of time
Fig. 6. Auger relaxation lifetime in CdS/ZnS as functions of the nanocrystal shell thickness for different core radius ‘‘a’’.
relaxation on QDs size. On the one hand, it increases with shell thickness increasing; on the other hand it decreases when core radius increases. For 50 Å shell thickness for example, τA progresses from 18 ps to 215 ps corresponding to 22 Å and 18 Å respectively. However, time variation with shell thickness is clear as presented in Fig. 6, but it is more important for a smaller core radius. For 18 Å core radius, τA varies from 22 to 215 ps corresponding to a shell thickness variation from 32 to 50 Å. This variation is smaller for a greater radius; it is about 18 ps for 22 Å core radius. This behavior is observed by Wang et al. [29] for CdSe colloids. They calculated auger time relaxation τA for (P → S ) transition, using the pseudo potential-based atomistic approach and they concluded that time relaxation decreases with QDs radius increasing. They found τA = 0, 6 ps and τA = 0, 2 ps for 29 and 38 Å radius. Otherwise, the k.p-based calculation of Efros and coworkers [30] predicts Auger decay lifetimes in CdSe colloidal dots of τA ∼ 2 ps almost independently of dot size for radius between 20 and 40 Å.
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CdS/ZnS core-shell quantum dot, the carriers (electron or hole) are confined in the shell. AR lifetime is inversely proportional to Coulomb interaction and when the shell thickness increases, Coulomb interaction decreases due to the decreased wave function overlap and then the AR lifetime increases. Moreover, Nanda et al. [33] observed the same behavior for inverted core-shell. They used ZnSe/CdSe and found that biexciton Auger lifetime increases with shell thickness. They consider that this fact is due to the increasing degrees of quantum confinement. Similarly, previous experimental studies of colloidal CdS nanoparticles caped with L -glutathione (GSH) and myristic acid (MA) [10] indicate that Auger decay is extremely fast, and show that the biexciton AR rate depends on the size of QDs. The decay of multiexcitons (Biexciton, triexciton. . . ) in NCs is, however, dominated not by radiative processes but by nonradiative Auger recombination, in which the e–h recombination energy is transferred to a third carrier (an electron or a hole). 4. Conclusion Fig. 7. The Auger recombination rate results for excited electron, excited hole and the calculated biexciton type AR in CdS/ZnS heterostructures as a function of the shell thickness for fixed core radius value a = 20 Å.
Auger mechanism requires the presence of a hole to be effective; this mechanism was challenged by Guyot-Sionnest et al. [31] as an explanation of the observed P → S electron relaxation. They reported a relatively fast (10–30 ps) electron decay in CdSe dots passivated by alkylamines or dodecanethiol ligands, which act as hole traps and thus separate the photogenerated hole from the electron. Likewise, Califano [32] predict that the Auger mechanism can explain the experimentally observed intraband P → S relaxation time scale without the need to invoke any exotic relaxation.
In this paper, we have developed a theoretical treatment of the size dependent electronics and optical properties in CdS/ZnS coreshell quantum dots. We computed the electron and hole energy variation in terms of the nanocrystal size and then we focused on Auger relaxation and Auger recombination. Our results show that all these processes depend on both core radius and shell thickness. Our study revealed that the AR type biexciton is more efficient than the other AR processes (excited electron and excited hole), and for a fixed core radius value a = 20 Å, the carriers (electron or hole) are confined in the shell, which is in good agreement with recent experimental studies [10,11].
3.3. Auger recombination (AR)
References
In this part of our work we are interested in the Auger multiexciton recombination scattering rate that we will predict for biexciton. It depends on two terms, the excited electron (EE) and the excited hole (EH) AR type. We calculate it by using Eq. (8) [20], where the two terms are determined by the following expressions: 1
τEE 1
τEH
=
2 2π − (eh) JSe ,Se ,Sh ,Sk δ Eg + ESe − Eke h¯ k
(10)
=
2 2π − (eh) JSh ,Sh ,kh ,Se δ Eg + Ekh − ESh . h¯ k
(11)
Eg being the ground state exciton energy. In Fig. 7. we present the AR rate computed for excited electron (EE), excited hole (EH) and the biexciton, in terms of the shell thickness for a fixed core radius a = 20 Å. We clearly observe that for the three processes, AR rate decreases when shell thickness increases. It can be noted that AR rate variation depending on shell thickness is different for the three processes but it is considerably greater for the biexciton type. This finding results in an important increasing of biexciton AR lifetime with increasing shell thickness. Indeed our calculations implicate that τbiexciton vary from 0.5 ns to 20 ns corresponding from 36 Å to 56 Å respectively, which is almost forty times longer. Recently, a similar behavior was experimentally observed by Garcia-Santamaria et al. [11] for CdSe/CdS core-shell. They used a 15 Å core radius and observed for biexciton a great lifetime increase with increasing shell thickness. They consider that biexciton AR lifetime depends on shell thickness because the electron is delocalized in the shell region, which confirms our simulation results. Indeed, in our case the AR rate shows that in
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