Geometry-Dependent Auger Recombination Process in

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Geometry-Dependent Auger Recombination Process in Semiconductor Nanostructures Yan He, and Gang Ouyang J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 09 Oct 2017 Downloaded from http://pubs.acs.org on October 9, 2017

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The Journal of Physical Chemistry

Geometry-Dependent Auger Recombination Process in Semiconductor Nanostructures Yan He, and Gang Ouyang* Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China

Abstract The geometry-dependent Auger recombination (AR) rate of semiconductor nanostructures has been investigated based on atomic-bond-relaxation correlation mechanism. We found that the increasing of dimension is of great benefit to suppress the AR process due to reduction of the Coulomb interaction between electron and hole. The AR lifetime increases as the size decreases with a Eg7 / 2 D 7 dependence. Moreover, the AR rate of nanostructures can be achieved though modulating the related geometry parameters. Our results are consistent with the available evidence, implying that the proposed model could be expected to be a general approach to deal with AR process in semiconductor nanostructures.

*

Corresponding author: [email protected]

1

ACS Paragon Plus Environment


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1. Introduction Auger recombination (AR), the energy of a recombining electron-hole pair is transferred nonradiatively to another carrier, has received comprehensive attention in fundamental scientific issues, especially in a confined nanostructure which can enhance the AR process dramatically due to the increase of Coulomb electron-electron coupling and kinematic restrictions imposed by energy and momentum conservation.1-3 Being a three-body process, the biexciton AR can proceed via two alternative pathways: the negative trion pathway (two electrons and a hole) and the positive trion pathway (an electron and two holes), as shown in Figure 1. In nanostructures, the AR process has been identified as the main reason for the decay dynamics of multiexcitons,4 induces fluorescence intermittency5 and decrease of photoluminescence quantum efficiency in light-emitting diodes,6 etc. Therefore, understanding the effect of the geometry parameters on the AR rate for nanostructures is one of the significant challenging problems in optoelectronic and photovoltaic devices. To date, significant efforts have been made in developing strategies to study the AR in nanostructures both from theoretically7-13 and experimentally.14-20 In particular, Efros et al.7,9,12,13 and Klimov et al.14-18, respectively, investigated the size and shape dependence of AR rate in nanocrystals (NCs), which open up a new field of research in AR rate for low-dimensional materials. Moreover, they indicated that the AR rate, 1 / τ A , of NCs satisfies: 1 / τ A ∝ D p , where p is a constant and D denotes the diameter of NCs. However, the p was questioned as the experimental results demonstrate p is changed from 2 to 4.9,1,21-23 which is lower than that of theoretical calculations, 5 and 7,7,24 and simulations, ~6.25,26 Even though some works showed that the reason for this discrepancy originates from the effect of surface and interface,

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Kobayashi et al.27-29 reported that the AR does not depend on surface defects and capping reagents in any size regions and they found the p of NCs in the diameter range from 2.3 to 3.5 nm, p=4.3, is smaller than that of the diameter change from 3.3 to 4.5 nm, p=6.3, which make p in nanostructures become more mysterious. Therefore, it is necessary to understand how the size influences p and establish a systematic study to pursue the mechanism regarding the difference of p between experimental measurements and theoretical predictions. In addition, the shape effect plays the crucial role for the AR process as well. Compared to the NCs, the AR process in nanowires (NWs) can be treated as a two-particle collision (exciton-exciton) rather than a three-particle collision (electron-hole-electron or electron-hole-hole) as the transformation effect from zeroto one-dimensional structures, which leads to the AR rate in NWs different from that in NCs.14,30 Even more than that, Aerts et al. 31 and Taguchi et al.

32

reported that the

AR rate can be, respectively, reduced and enhanced in NWs as the low charge density along the axial direction and the large surface area. Importantly, Zegrya et al.33-35 indicated that the AR process can be affected by three fundamentally different mechanisms of nonthreshold, quasithreshold and threshold types, which makes the AR phenomenon change with the geometric parameters. In addition, the band structure, exciton binding energy, surface states and phonon-assisted have also been considered

on

AR

process in

semiconductor

nanostructures.36-39 Although

considerable efforts have been put forth to suppress AR process by modulating the size and shape in nanostructures both from experimentally and theoretically, the discrepancy of AR rate between experiments and theories under varies size, length and shape is still unsolved. In order to investigate the influence of the geometric effect on the AR rate and

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provide a full understanding of the physical origin that evaluates the AR process in semiconductor nanostructures, in this contribution we carried out a quantitative method to explore the influence of size and shape on the AR rate from the pespective of atomistic origin. In our case, we focus on the confinement role induced by the size and shape on the energy bandgap and AR. Our results show that the AR lifetime in nanostructures is determined by the geometry parameters. Furthermore, the underlying mechanism on geometry-dependent AR lifetime is clarified based on the atomic-bond-relaxation (ABR) correlation mechanism and Fermi’s golden rule.

2. Theoretical methodologies In general, surface effect induced by the surface defects has a signification impact on the physical-chemical properties of specimen, especially for the nanostructures with high surface-to-volume (SVR) ratio. According to ABR consideration, the coordination deficiency and dangling bonds at the end parts would make the edge atoms more active and induce the remaining bonds of the less-coordinated atom become shorter and stronger spontaneously, resulting in the system at a self-equilibrium state.40-42 Therefore, some physical quantities such as cohesive energy will be different from that of the bulk counterparts. Theoretically, the cohesive energy of a specimen is defined as the energy required breaking the atoms of the solid into isolated atomic species, i.e., Ecoh = zEb , z

and Eb are the

coordination numbers (CNs) and the single bond energy in bulk, respectively. Taking into account the discrepancy between surface and core interior, the cohesive energy of a nanostructure can be expressed as: Ecoh = ∑ N i zi Ei + ( N − ∑ Ni )zb Eb , where zi i ≤ ns

i ≤ ns

and N i are the CNs and atomic numbers of surface atoms, N is the number of total

4


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atoms, Ei denotes the single bond energy of surface atoms, ns represents the number of surface shells. Besides, in terms of band theory, the width of bandgap ( Eg ) can be calculated by the first Fourier coefficient of the crystal potential energy, which is in proportional to the mean cohesive energy per bond, Eg ∝ Ecoh z N ,43 where

z = ∑ γ i ( zi − zb ) + z b is the mean CNs, γ i = σci d0 / D is the SVR ratio, σ is the i ≤ ns

shape

factor

of

a

spherical

( σ =6 )

NC

and

a

NW

( σ = 4 ),

ci = 2 /(1 + exp((12 − zi ) / 8zi )) is the bond contraction coefficient, d0 and D denote the bond length of the bulk and the diameter of the nanostructures. Therefore, we have

Eg =

b b  Ecoh Eg zb Eg  − md =  ∑ γ i zib ci − 1 + 1 z N Eb z i < n s 

(

)

(1)

Egb is the bandgap of the bulk case, md denotes the bond nature indicator.40 According to the method reported by Chepic et al.,7 the negative trion AR rate from the excited state ψ i to the final state ψ

f

can be calculated by using Fermi’s

golden rule, 1

τA

−

=

2π h

∑M

2 A

δ (E final − Einitial )

(2)

k ,l , m

with r r M A = ψ kf,l ,m V ( r1 , r2 )ψ i

r r r r where τ A− is the lifetime of negative trion AR, V (r1 − r ) = e 2 / (κ r1 − r2 ) is the Coulomb potential for the electron, h = h / 2π , h is Planck’s constant, Einitial and

E final are the energies of the initial and final states, respectively, κ is the dielectric constant. In the case of that the angle integration gives the nonzero matrix elements, M A , only for final state with l = 1 and m = M , M is the projection of 5



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the total momentum in the spherical potential. We consider that the matrix element, M A , is independent of M because of the spherical symmetry, i.e., M = 0 . Therefore, the calculation of the matrix element is carried out by the standard technique of the multipole expansion of the Coulomb potential. Introducing dimensionless variables of integration x ' = r1 / a and x = r2 / a , we obtain

  E g  2  Em   1 + g  1 − 2   2 E0   2 E f 8 C0 1 = χ 2 (κ s , κ g )  2 τ A− 27 ∆ + (1, k g ) + ∆2− (1, k g ) where

C0

2

is

a

  2  2  (κ s a ) 4  e 2  E f MA 4   (κ g a )  h V κ  h

constant,

(3)

κ g2 = [( Egm / 2) 2 − E 2 ]/( h 2 V ) 2

,

κ s2 = [E 2 − ( E g / 2) 2 ]/(h 2 V ) , E gm are the bandgap in the matrix material, V the 2

Kane matrix element, E f the energy of a free electron, a the radius of a sphere, E 0 the energy of the size-quantized electron ground state, E the energy of the state

measured from the middle of the energy gap, k g the quasi-momentum of the ejected electron,

χ (κ s , κ g ) −1 = 1 − sin(2κ s a) / 2κ s a + sin 2 (κ s a)( E g + 2 E ) / κ g a( E gm + 2 E )

,

∆ + (1, k g ) and ∆ − (1, k g ) are as a function of spherical Bessel functions of the second kind and the matrix element. In the case that the ejected particle have large final 2

momenta, then ∆ + (1, k g ) ⇒ 1 /( k g a) 2 , ∆ − (1, k g ) ⇒ 0 , h 2 V /( aE g ) 2 ( k g , κ g ) , where

f (κ s a, κ g a ) is as a function of ( κ s a, κ g a ), the integration J < ( J > ) is carried out over the volume of the structure (the glass matrix surrounding of the structure). Therefore, the integrated M A can be written as a product of a rapidly oscillating

r function and a smooth function f (r ) . Thus, it can be rewritten as below with the momentum

that

Auger

electron

has

in

the

6


final

state

kf

,

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rr r M A = f (r ) exp(ik r ) ≈ f ' a ( ak f ) −2 + f '' a ( ak f ) −3 + f ''' a (ak f ) −4

,

where

the

coefficients of (ak f ) −2 and (ak f ) −3 can be vanished due to the continuity of the wave function at the surface of nanosystem. Therefore, M A is proportional −4

to f ''' ( ak f ) and the AR rate is

ε 1 = A'  q E τ A−  g

7

2 m  Eg − Eg  

(

)

(4)

where A' is a constant, E gm is the

energy gap of the matrix material, ε q is the

quantum size energy that can be written as ： ε q =

isotropic NC or quantum dot (QD), ε q =

h 2π 2 6me

 1   + 1 + 1  . For an  a2 a2 a2  y y   x

h 2π 2  2 x 2 + 1  h 2π 2   for the , while = ε q 6me D 2  x 2  2me D 2

NW, where x = L / D , me is the effective mass of electron, D is the diameter of NCs (QDs) or NWs, L denotes the length of NWs. Thus, the lifetime of negative trion AR is 7

τA

−

  x 2  2 m  E g − E g = A E g D 2  2  2 x + 1  

(

)

−1

(5)

Based on “statistical” arguments,1,44 the biexciton AR can be described in terms of a superposition of two independent positive-trion- and negative-trion-like pathways, implying that its Auger lifetime can be presented as: τ A XX = (2 / τ A− + 2 / τ A+ ) −1 , where

τ A is the lifetime of positive trion AR. Also, in the case of idealized structures with +

identical conduction and valence bands, we have τ A− = τ A+ , which leads to

τ A = τ A / 4 .45 Note that we assume that the energy gap in the nanostructures is much XX

−

smaller than that in the matrix material so that the Egm − Eg is close to a constant in 7



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our calculation.

3. Results and discussion In order to illustrate the multiple exciton recombination phenomenon of the semiconductor nanostructures with different geometry effects (including size and length), we take the typical CdSe nanostructures (shown in the inset of Fig. 2a), including NCs and NWs, as an example to elucidate the effect of dimension and shape on the trion and biexciton AR lifetime. Notably, in our case the nanostructure is capped by a perfect coating layer. Therefore, the influence of the surface such as carrier traps which can assist AR on the recombination process can be ignored. Using the relationships mentioned above, we calculate the negative trion, τ A − , biexciton, τ A XX , and AR lifetime of CdSe nanostructures (including NCs and NWs) at first. As plotted in Figure 2a, we can see that the τ A − and τ A XX of CdSe NCs decrease monotonically with reducing size. The physical origin can be attributed to the enhancement of the Coulomb interaction which can increase the recombination rate of electron-hole pairs.4,9 Moreover, the log-log dependence of the AR lifetime versus diameter is approximately linear, which is well consistent with the experimental measurements and simulations.4,13,15,21,23,26,27,32,46-49 Figure 2b shows the size-dependent τ A − and τ A XX of CdSe NWs. Clearly, the AR process of CdSe NWs is similar with that of CdSe NCs, and the increase trend of AR lifetime of CdSe NWs is slightly less than that of CdSe NCs with increasing size. For instance, the slope of the AR lifetime versus size in CdSe NWs is nearly equal to that of CdSe NCs in large size ( D > 6 nm ), while it appears obvious difference when the size is less than 6 nm. In fact, it can be ascribed to the effect of energy bandgap as it increases with

8


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decreasing size (as shown in the inset of Fig. 2 (b)), which can make the AR lifetime enhancement and result in the variant trend of AR process change. However, the difference between the bandgap of NWs and NC becomes gradually increase due to influence of the quantum confinement in those of two types of nanostructures ( EgNC > E gNW ), resulting in the slope of the AR lifetime versus size of NWs is less than that of NCs. Furthermore, we found that the biexciton AR lifetime of CdSe NWs with NC a large length is about 3 times large than that of CdSe NCs, i.e., τ ANW XX ≈ 3τ XX . This can A

be simply interpreted as the difference of quantum confinement, which leads to the quantum size energy and Hamiltonian of those two nanostructures different from each other.50,51 Htoon et al.

14

and Taguchi et al.32, respectively, claimed that the biexciton

AR lifetime of CdSe NWs is a factor of 2 to 8 and 2 to 5 larger than that of CdSe NCs with a fixed diameter of 4.6 and 4.1 nm. Moreover, Yang et al.52 reported that the biexciton lifetimes of the nanorods are 3 to 4 times longer than those of the QDs with similar energy gap. Noticeably, in our case, the AR lifetime of NCs with diameter D is slightly larger than that of NWs at the same size and D / L = 1 . For example, the AR lifetime of NCs is 34.6 ps, while it is about 28 ps for the NWs with D =L =4 nm , which are, respectively, 3 and 4-5 times small than that of NWs with an infinity length. In fact, the discrepancy of AR lifetime between NWs and NCs can be attributed to the change of bandgap. As the transformation effect from zero- to one-dimensional structures, the confinement effect and band structure would differ from each other. Therefore, it should be noted that the AR lifetime of NWs with D =L , may not satisfy NW the formula: τ ANW XX ( L = ∞ ) =3τ XX ( L = D ) , while it can be suitable for the NCs, i.e., A

τ ANW ≈ 3τ ANC . XX

XX

To investigate the influence of the geometry parameters on the CdSe NWs AR

9



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process, ABR method and Fermi’s golden rule are carried out to calculate the biexciton AR lifetime, as shown in Fig. 3. Evidently, the τ A − and τ A XX show a first-rapid increase with increasing size and then keep a constant as the size beyond 30 nm or the aspect ratio, x = L / D , beyond 8 (shown in the inset of Fig. 3). The difference of AR lifetime between short-NWs and long-NWs is mainly due to the quantum size energy and band structure change with the length of the NWs. Taguchi

et al.32 indicated that the nanorod (NR) biexciton lifetime depends weakly on or is almost independent of the NR volume with a fixed diameter. However, some of the studies found that the NR biexciton lifetime increases linearly with increasing NWs length.14 The reason of this difference may be due to the effect of surface carrier traps-assisted.22,32 Also, by studying the available works, we found that the biexciton AR lifetime in some specimen such as PbSe NWs increases with the volume even if coated by a well-passivated surface.31,53 To pursue the deeper physical mechanism, we notice that the Bohr radius, aB , of PbSe is 46 nm31 (5.4 nm for CdSe

31

), which

suggests that the Coulomb interaction for a larger length of PbSe NWs is still larger than that of bulk case, resulting in the AR lifetime depends on the length. In order to clarify the relationship between band structure and AR process, we calculate the biexciton AR lifetime of CdTe and PbSe NCs compared with that of CdSe NCs, as shown in Figure 4, and the bandgap of those three types of nanostructures are also shown in the inset of Figure 4. It should be noted that the biexciton AR lifetime and bandgap are, respectively, increasing and diminishing as the size of NCs increases, suggests that the competition between this two aspects determines the AR process of the nanostructures. Furthermore, the slope of the AR lifetime versus size changes with not only the size, but also the specimen. For instance, the p of the CdSe NCs in thin (thick) structure is 4.9 (6.3), while it is 4.6 (6.2) and 3 10


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(5) for CdTe and PbSe NCs, respectively. It can be ascribed to the size-dependent energy bandgap. On the one hand, when the size of the nanostructure is less than the Bohr radius, the Coulomb interaction and bandgap would enhance dramatically, which drop the slope of the AR lifetime versus size and the value of p steadily due to τ A − ∝ Eg7 / 2 D 7 . Kobayashi et al.27,28 reported that the biexciton AR lifetime of CdTe QDs is proportional to D 4.6 in the diameter range from 2.3 to 3.5 nm, while it is proportional to D 6.3 in the diameter from 3.3 to 4.5 nm. Jain et al. 26 and Vaxenburg

et al.25 claimed that the values of p in CdSe NCs are, respectively, 6.24 and 6.5 by using atomistic tight-binding wave functions and eight-band k·p model. Importantly, our predictions are well consistent with the related calculations. On the other hand, it is well known that each specimen has its unique Bohr radius which determines the band offset and the Coulomb interaction. Therefore, the larger Bohr radius would provide more remarkable effect of Coulomb interaction and band offset on the AR, thereby allowing for low p. For example, as plotted in Figure 4, we can see that the p of CdSe NCs ( aB = 5.4 nm ) in thin (thick) structure is 4.9 (6.3), which is larger than that of CdTe ( aB = 7.3 nm 28) and PbSe NCs ( aB = 46 nm ). Therefore, it is concluded that an effective way to suppress AR process in low-dimensional nanostructures through decreasing the quantum confinment effect.

Also, we can prospect that

coating an epitaxial layer is of great benefit to suppress the AR process because the layer can not only reduce the quantum confinement, but also increase the size of the nanostructures. Note that in our case the system is an idealized structure and the influence of surface trap effects are ignored so that our predictions may not be good fitted with some experimental measurements, but it is exceedingly well agreement with the simulations.

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4. Conclusions In summary, we have proposed a theoretical model to address the geometry-dependent AR rate of NCs and NWs based on ABR correlation mechanism and Fermi’s golden rule. Our results show that increasing the dimension of nanostructures is of great benefit to suppress the AR process due to reduction of Coulomb interaction between electron and hole. We find that the AR lifetime of the nanostructures increases as the diameter decreases with a Eg7 / 2 D 7 dependence, which leads to the log-log dependence of the AR lifetime versus diameter is approximately linear with

D p and the

p

changes with the size of the

nanostructures. Moreover, the result of AR process that is determined by the coupling effect of band structure and size suggests that the AR lifetime is inversely proportional to the Bohr radius. These results indicate that suppression of the AR process can be achieved by not only decreasing the quantum confinement effect that can improve the p, but also changing the shape of the nanostructures that can modulate the quantum size energy and confinement condition. Therefore, our approach provides an effective method to explore the AR process of the semiconductor nanostructures.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11574080).

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I.;

Htoon,

H.

Two

Types

of

Luminescence

Blinking

Revealed

by

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Figure captions Figure 1. A cartoon of the AR process is shown for a negative trion, τ A − , a positive trion, τ A + , and a biexciton. Eke and Ekh denote the conduction and valence band states.

Figure 2. Size-dependent τ A − and τ A XX of CdSe NCs (a), and CdSe NWs (b) at fixed L = 1000 nm . The inset of (b) shows size-dependent bandgap of CdSe NCs and NWs at fixed L = 1000 nm . The data of bandgap are obtained from Refs. 13, 21, 54, 55, and 56. The necessary parameters for bulk CdSe are obtained from Refs. 59 and

(

60. EgCdSe = 1.7 eV , h0CdSe = 0.26 nm and A Egm − Eg

)

−1

= 6 × 10− 4 .

Figure 3. Length-dependent τ A − and τ A XX of CdSe NWs at fixed D = 4 nm . The inset shows aspect ratio, x , dependence of τ A − and τ A XX of CdSe NWs at fixed

D = 4 nm .

Figure 4. Size-dependent τ A XX of CdSe NCs (a), CdTe NCs (b), and PbSe NCs (c). The inset of (a), (b), and (c) show size-dependent bandgap of CdSe NCs, CdTe NCs, and PbSe NCs, respectively. The data of bandgap of CdTe NCs and PbSe NCs are obtained from Refs. 27, 57, 58. The necessary parameters for bulk CdTe and PbSe are obtained from Refs. 58, 59, and 60, EgCdTe = 1.6 eV , EgPbSe = 0.28 eV , h0CdTe = 0.28 nm ,

(

and h0PbSe = 0.31 nm . A Egm − Eg

)

−1

are, respectively, 7.2 × 10 −4 and 4.8 × 10 −3 for

CdTe and PbSe.

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Figure 1

21



Figure 2

5

10

τΑ

τΑ

Ref. (26) Ref. (21) Ref. (4) Ref. (47) Ref. (48) Ref. (46) Ref. (15) Ref. (49) Ref. (32)

Ref. (23) Ref. (46) Ref. (13)

2

4

XX

4

Auger lifetime [ ps ]

10

3

10

2

10

−

1

10

0

10

6

8

10 12 14

D [ nm ]

(a)

6

10

3.5

L = 1000 nm

Eg [ eV ]

3.0 2.5


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4

10

NW Ref. (55) Ref. (56)

NC Ref. (54) Ref. (21) Ref. (13)

8

16

2.0 1.5 0

4

12

τA NC

20

-

D [ nm ]

τA NC XX

2

10

τA NW

NC

Ref. (14) D =4.0 nm NC Ref. (14) D =4.6 nm NW Ref. (14) D =4.6 nm NC Ref. (32) D =4.2 nm NW Ref. (32) D =4.1 nm

-

τA NW XX

2

4

6

8

10 12 14

D [ nm ]

(b)

22


Page 23 of 25

Figure 3

D = 4 nm 100 500

10 τΑ

−



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


400

300

τΑ

200

−

τΑ

XX

100

τΑ

XX

0 1

x

Ref. (32)

10

100

1 1

10

100

L [ nm ]

23


1000


Figure 4 4

10

CdSe NC p = 4.9 p = 6.3 Ref. (26) Ref. (49)

2

τΑ [ ps ]

10

Eg [ eV ]

XX

3 0

10

(a)

CdSe NC Ref. (54) Ref. (21)

2

0

6

12

18

D [ nm ]

-2

10

2

4

6

8

10 12 14

D [ nm ] 4

10

CdTe NC p = 4.6 p = 6.2 Ref. (27)

2

τΑ [ ps ]

10

(b)

XX

3

CdTe NC Ref. (27)

Eg [ eV ]

0

10

2

0

6

12

18

D [ nm ] -2

10

2

4

6

8

10 12 14

D [ nm ] 4

10

XX

CdSe NC CdTe NC PbSe NC p=3 p=5 Ref. (53) Ref. (44) Ref. (36)

3

Eg [ eV ]

2

10

τΑ [ ps ]

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 24 of 25

0

10

(c)

PbSe NC Ref. (57) Ref. (58)

2

1

0 0

5

10

15

20

D [ nm ]

-2

10

2

4

6

8

10 12 14

D [ nm ] 24


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TOC graphic


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


4

10

τΑXX τΑ −

D

2

10

D 0

10

3

6

9

12 15

D [ nm ]

25


Geometry-Dependent Auger Recombination Process in

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