Morphology effects of self-assembled quantum dots on the energy spectrum of magneto-excitons

Morphology effects of self-assembled quantum dots on the energy spectrum of magneto-excitons

Physica E 56 (2014) 301–305 Contents lists available at ScienceDirect Physica E journal homepage: www.elsevier.com/locate/physe Morphology effects ...

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Physica E 56 (2014) 301–305

Contents lists available at ScienceDirect

Physica E journal homepage: www.elsevier.com/locate/physe

Morphology effects of self-assembled quantum dots on the energy spectrum of magneto-excitons Pablo Villamil a,n, José Sierra Ortega b a Department of Mathematics and Physics: Theory of Condensed Matter Group, University of Sucre, Calle 32 No. 50 A – 16, Apto. 102, Edificio Murano, Barrio Venecia, Sincelejo P.O.B. 406, Sucre Colombia b Theory of Condensed Matter Research Group, University of Magdalena, Santa Marta P.O.B. 731, Colombia

H I G H L I G H T S

   

We analyzed the changes experienced by the energy spectra of a confined exciton in type II semiconductors quantum dots. We studied the lowest energy states of an exciton (X) confined in type II InP/GaInP self-assembled quantum dot. The electron is located within the point of InP and the hole is in the GaInP barrier. We study the energy levels associated with the electron and the hole, and the energy of the exciton.

art ic l e i nf o

a b s t r a c t

Article history: Received 5 April 2013 Received in revised form 31 August 2013 Accepted 27 September 2013 Available online 22 October 2013

In this paper we analyze the changes experienced by the energy spectra of a confined exciton in type II semiconductor quantum dots, considering the quantum dot as a possible functional part that, in the future devices, can be applied in spintronics, optoelectronics, and quantum information technologies. We studied the lowest energy states of an exciton (X) confined in type II InP/GaInP self-assembled quantum dot (SAQDs), with axial symmetry in the presence of a uniformly applied magnetic field in the growth direction. In our model, it is considered that the electron is located within the point of InP and the hole is in the GaInP barrier. The solution of the Schrödinger equation for this system is obtained by a variational separation process of variables in the adiabatic approximation limit and within the effective mass approximation. We study the energy levels associated with the electron and the hole, and the energy of the exciton. Due to the axial symmetry of the problem the z component of the total orbital angular momentum, Lz ¼le þlh, is preserved and the exciton states are classified according to the values of this component. Quantum dots have a finite and variable thickness, with the purpose of analyzing the effects related to the variation of the morphology and the presence of a wet layer. & 2013 Elsevier B.V. All rights reserved.

Keywords: Self-assembled Quantum dot Magneto-exciton

1. Introduction The study of systems of a few particles in semiconductors began in 1958 with M. A. Lampert [1] who established the theoretical basis of the formation of complexes with charge carriers of different types. The experimental verification of some of these complexes in silicon, doped with elements of Group III and V, was accomplished by Haynes years later [2]. The observation of these complexes was very difficult due to their formation under metastable conditions at temperatures above 5 K, and because they are weakly bound or linked, so the thermal energy exceeds the ionization energy of the system. Furthermore, they

n

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1386-9477/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2013.09.016

have very short lifetimes. This difficulty is overcome with the manufacture of semiconductor heterostructures, as was demonstrated for the first time by Shields et al. [3], where excitons were observed in QWS of GaAs–GaAlAs. The confinement leads to an increase in the exciton binding energy in an order of magnitude and a greater stability of the base state of the complex [4]. Additionally, it was found that the energy spectrum of these systems is extremely sensitive to changes in geometry, the position of the complex, in the case of localized centers, and the presence of external fields [5]. The existence of excitons in QDs was confirmed experimentally [6]; it was determined that excitons are much more stable than other heterojunctions due to increased confinement, enabling its observation even at room temperature. To calculate the binding energies in these systems, different approximational methods have been used, the variational [7–9] and the diagonalization matrix [10,11] being two of the most

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simple ones. Both methods require considerable computing volume, which grows exponentially with the number of particles, and significant limitations are present on the scope of its application. The variational method only allows an estimate of the energy of the base state, while the diagonalization method is hardly applicable to potentials different from the parabolic. Other methods are series expansion [12], fractional dimension [13], perturbation theory [14] and different numerical techniques such as elements and finite differences [15,16], trigonometric scanning [17], and other numerical solutions of differential equations [18]. All these techniques are quite complicated for the analysis of the energy spectrum when the number of particles increases. One way to avoid this difficulty is to consider the morphological characteristics of heterostructures, as in the case of SAQDs, for which the height is much smaller than the dimensions of its base, allowing the use of the adiabatic approximation (AA) [19–23]. With this approach dimensional problems can be reduced to similar problems in one or two dimensions with Hamiltonian renormalization [24]. Finally, the large volume of theoretical and experimental studies performed during the last years about semiconductor nanostructures shows the great interest from researchers to establish the factors that are more significantly influential in the energy spectrum of systems of few particles in low dimensional systems under hard confinement conditions, given that they are the ones to determine its technological applicability.

2. Theory A Wannier exciton is considered confined in a type II quantum dot of InP, with axial symmetry, which contains a wet layer and is embedded in a matrix of GaInP (see Fig. 1). The electron is confined in the quantum dot, in a thin layer of InP of varying thickness d, and the hole is located in the matrix. To account for different morphologies of the point, it is considered that the height of this depends on the distance ρ to its axis of symmetry and this dependence has the following relationship: h  n i1=n dðρÞ ¼ dw þ d0 1  ρ=R0 ϑðR0  ρÞ ð1Þ where R0 is the radius of the base of a quantum dot, dw is the wet layer thickness (WL), d0 is the maximum height of the point over the wet layer and ϑðxÞ is the Heaviside step function. The morphology of the dot is controlled by the parametern, which equals to 1, 2 or tends to infinity for dots in a pyramid conical shape, lens and disk, respectively. Experimentally, it has been found that for these SAQDs [25], the WL thickness is 2 nm approximately; the maximum height of the points above WL varies between 2 and 4 nm, while the lateral sizes are much larger and reach 70 nm. This fact is exploited to implement a similar method to the adiabatic approximation, which allows uncoupling movement in the vertical direction and in the SAQDs plane. We consider a structure consisting of a very thin layer that includes the dot and the wet layer, so that the ratio

between the height and the base point, in all geometries, is small, ðd0 þ dw Þ=R0 ⪡1. Furthermore, for mathematical simplicity it is assumed that the entire structure, which includes the matrix, is a thin layer of thickness df , significantly smaller than the base R0 (see Fig. 1). Type II quantum dots are considered as an infinite barrier, where the two particles are confined into two different thin layers, whose thickness depends on the distance ρ to its axis of symmetry. The electron is confined in the lower layer of InP and the hole in theupper potentials for  layer of GaInP. Confinement   the electron, V e ze ; ρe , and for the hole, V h zh ; ρh , are given by the following functions: ( 0; if 0 o ze odðρe Þ V e ðze ; ρe Þ ¼ ; 1; in other cases (   0; if d ρh o zh o df ð2Þ V h ðzh ; ρh Þ ¼ 1; in other cases Henceforth, the subscripts e and h refer to the electron and the hole, respectively. In the approximation of the effective mass, and using cylindrical coordinates, the dimensionless Hamiltonian e–h pair is written as H ¼ H e ðre Þ þ H h ðrh Þ þ U ðre ; rh Þ

ð3Þ

where,

  H e ðre Þ ¼ ηe ð Δe þ γ 2 ρ2e =4  iγ∂=∂φe Þ þ V e ρe ;

ð3aÞ

    H h ðrh Þ ¼  ηh  Δh þ γ 2 ρ2h =4  iγ∂=∂φh þV h ρh  2=r h

ð3bÞ

Uðre ; rh Þ ¼  2=jre  rh j þ 2=r h

ð3cÞ

The first two terms, (Eqs. (3a) and (3b)), correspond to the Hamiltonian of a particle, the electron, H e , within the lower layer (quantum dot) of InP and the hole, H h , within the upper layer of GaInP, both in the presence of a uniformly applied magnetic field in the growth direction. It is observed that in the Hamiltonian (Eq. (3b)) it includes an additional potential,  2=r h , replacing the real attraction between the hole and the electron for one attraction between the hole and a fixed electron at the origin of coordinates. The last term, (Eq. (3c)), corresponds to a disturbance equal to the difference between the electron–hole real and artificial interaction energies. In these relationships we have ηe ¼ μ=me ¼ 0:88; ηh ¼ μ=mh ¼ 0:12, where me ¼ 0:077m0 and mh ¼ 0:6m0 , the effective masses of the electron and hole, respectively, and μ ¼ me mh =ðme þ mh Þ ¼ 0:068m0 is the reduced mass. In the Hamiltonian (Eq. (3)), we have used the exciton Bohr radius, an0 ¼ ℏ2 ε=μ e2 , as an unit length, the effective Rydberg Ryn ¼ e2 =2ε an0 ¼ ℏ2 =2μ an02 , as the energy unit and γ ¼ eℏB=2μcRyn as the unit of magnetic field ! intensity B . The energies and wave functions are obtained by solving the Schrödinger equation for the electron (k ¼ e ), and the hole (k ¼ h ): H k ðrk Þψ k ðrk Þ ¼ Ek ψ k ðrk Þ;

k ¼ e; h

ð4Þ

Due to the axial symmetry of the problem, the wave functions ψ k ðrk Þ are also functions of the projection operators of orbital angular momentum L^ zk ¼  iℏ∂=∂φk . Moreover, the adiabatic approximation is used in order to separate the radial and axial variables that describe the fast and slow movements, respectively and expresses the solutions of Eq. (4) as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      ðkÞ   ψ ðnkÞ;l ðrk Þ ¼ eilk φk f nk ;lk ρk 2=d ρk sin πz=d ρk ; lk ¼ 0; 7 1; 7 2; … k k ð5Þ The exciton energy levels are determined by minimizing the ratio: Z h i Z ðkÞ ψ nðkÞ;ln ðrk ÞH k ðrk Þψ ðnkÞ;l ðrk Þdrk = ψ nðkÞ;ln ðrk Þψ ðnkÞ;l ðrk Þdrk ; E f nk ;lk ¼ k k

Fig. 1. The exciton model in type II quantum dot.

k k

k k

k k

P. Villamil, J. Sierra Ortega / Physica E 56 (2014) 301–305

ð6Þ ðkÞ

When performing the functional derivative with respect to f nk ;lk the wave equations are obtained for the radial part of the electron and the hole trial functions, of the form   2   ðeÞ   ðeÞ   l γ 2 ρ2 ηe  ρ1 ∂ρ∂ ρe ∂ρ∂ þ ρe2 þ 4 e þ γle f ne ;le ρe þ V ðefeÞf ρh f ne ;le ρe e e e e ðeÞ   ¼ Ee ðne ; le Þf ne ;le ρe ; ! 2   ðhÞ   γ 2 ρ2h 1 ∂ ∂ lh ðhÞ   þγlh f nh ;lh ρh þ V ðefhÞf ρh f nh ;lh ρh ρ þ þ ηh  ρh ∂ρh h ∂ρh ρ2h 4 ðhÞ   ¼ Eh ðnh ; lh Þf nh ;lh ρh ; π2

2

d ρe

  V ðefhÞf ρh ¼ 

;

π2 2  2    d ρh df  d ρh

Z

  2 2 sin πz=d ρh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi df ρh 2 þz2 dðρh Þ df

ð7Þ

In the approximation of a particle, the wave function of the pair e–h and the corresponding states of energy depend on four quantum numbers h h

ð8Þ

For the typical sizes of the quantum dots the electron–hole interaction energy, U ðre ; rh Þ, is much smaller than the energy of confinement. As a consequence, the functions are made (Eq. (8)) as a suitable set of basic functions. To avoid the difficulties related to the large number of basic functions required to analyze the total energy spectrum, the calculations are restricted to lower energy states. For this purpose, the different states are labeled with the different excitonic state with a unique index ordering the energies in ascending direction. EðiÞ ¼ Ee ðne ðiÞ; le ðiÞÞ þ Eh ðnh ðiÞ; lh ðiÞÞ; j Ψ i ðre ; rh Þ〉 ¼ jψ ðneeÞðiÞ;le ðiÞ ðre Þ〉jψ ðnhÞðiÞ;l h

h ðiÞ

EðiÞ o Eðiþ 1Þ; ðrh Þ〉;

i ¼ 1; 2; 3; …; N

ð9Þ

In this numerical calculation, N takes values large enough so that the energies of the lowest exciton states do not change upon increasing the value of N. The quantum numbers of a particle le y lh , considered separately for the exciton, are no longer good quantum numbers due to the presence of the term in the Hamiltonian (Eq. (3)) (Eq. (3c)), and the electrostatic interaction between the electron and the hole. However, because of the axial symmetry of the problem, the total orbital angular momentum in the direction z, Lz ¼ le þ lh , is preserved and exciton states can be classified according to the values of this projection. Its absolute value is labeled according to the notation for molecules with axial symmetry by the letter Λ; which takes the values 0; 1; 2; … In addition, the exciton states with different values of Λ, can be labeled similarly for the molecular states, with the Greek capital letter corresponding to the Latin letters for atomic states with different L. Thereby, for states with Λ ¼ 0; 1; 2, the notations used are Σ; Π y Δ, respectively. All the exciton states with non-zero values of Λ are doubly degenerate: each energy value corresponds to two states that differ in the direction of the projection of the total orbital angular momentum on the axis of symmetry. The Σ states are also doubly degenerate, a state that does not change is denoted by Σ þ and other state, whose waves functions change sign by reflection in a plane of symmetry passing through the axis is, denoted by Σ  . Since there is no coupling between the wave functions with different values of the total angular momentum, one can construct the exciton wave functions with total angular momentum as a linear combination.

h

i¼1

h ðiÞ

ðrh Þiδle ðiÞ þ lh ðiÞ;Λ

ð10Þ

To determine the unknown coefficients C i , in the framework of the Galerkin method, consider the k-vector of the discrepancy, ^  EÞjΨ Λ ðre ; rh Þi jDi ¼ ðH   N ^  E jψ ðeÞ ðr Þ〉jψ ðnhÞðiÞ;l ¼ ∑ Ci H ne ðiÞ;le ðiÞ e h

i¼1

h ðiÞ

ðrh Þ〉δle ðiÞ þ lh ðiÞ;Λ

ð11Þ

This discrepancy must be zero at least in the subspace formed by the N base functions (Eq. (9)). Therefore, the discrepancy projections (Eq. (10)) on the functions of the base (Eq. (9)) satisfy the following relationships: ^  EÞjΨ Λ ðre ; rh Þ ΨΛ′ ðre ; rh ÞjD ¼ Ψ Λ0 ðre ; rh ÞjðH N

¼ ∑ C j ðEe ðne ðiÞ; le ðiÞÞ þ Eh ðnh ðiÞ; lh ðiÞÞ  EÞδi;j þ U j;i j

¼ 0;

Ψ ne ;le ;nh ;lh ðre ; rh Þ ¼ ψ ðneeÞ;le ðre Þψ ðnhÞ;l ðrk Þ; Eðne ; le ; nh ; lh Þ ¼ Ee ðne ; le Þ þEe ðne ; le Þ

N

jΨ Λ ðre ; rh Þi ¼ ∑ C i jψ ðneeÞðiÞ;le ðiÞ ðre Þijψ ðnhÞðiÞ;l

j ¼ 1; 2; …:; N:

ð12Þ

Here, the diagonal elements of the matrix are defined as

ð13Þ U j;i ¼ Ψ j ðre ; rh Þ  2= re rh þ 2=r h Ψ i ðre ; rh Þ

R0= 10 nm, d 0= 4 nm, d w= 2 nm, d p= 1 nm 42.5 42.0 41.5

Hole Energy (R *y)

V ðefeÞf ¼

As there is no coupling between the wave functions with different values of the total angular momentum Λ, the exciton wave functions Ψ Λ can be constructed, with a total angular momentum Λ, as a linear combination:

41.0 40.5 40.0 39.5

k

j

i

39.0

h

g

f

e

d

c

b

38.5

a

DISK

38.0 37.5 0

2

4

6

8

10

R 0= 10 nm, d 0= 4 nm, d w= 2 nm, d p= 1 nm

42.0

Hole Energy (R *y)

k ¼ e; h

303

41.5

(nh, lh)

41.0

a = (1, 0) b = (1, -1) c = (1, 1) d = (1, -2) e = (1, 2) f = (1, -3) g = (1, 3) h = (1, -4) i = (1, 4) j = (1, -5) k = (1, 5)

40.5 40.0 39.5

k

39.0

j

i

h

g

f

e

d

c

38.5

b

a

38.0

PYRAMID

37.5 37.0 0

2

4

6

8

10

Fig. 2. Energies of the 20 lowest states of the hole confined in the upper layer of a type II quantum dot InP, with geometry (a) disk and (b) pyramid, immersed in a matrix GaInP based in the magnetic field. The group of curves at the top of each graphic is the same order of the bottom, but with nh ¼ 2.

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This condition (Eq. (12)) allows the reduction of the calculation of energies and wave functions of the exciton to the following eigenvalue problem: ^  E^IÞ! ðD C ¼ 0; ^ j;i ¼ ðEe ðne ðiÞ; le ðiÞÞ þEh ðnh ðiÞ; lh ðiÞÞ  EÞδi;j þ U j;i ; D

i; j ¼ 1; 2; …; N ð14Þ

3. Results and discussions In this work, we first solve the problems of a particle (Eq. (7)), using the scanning trigonometric method, finding the corresponding wave functions (Eq. (5)) and electron energies Ee ðne ; le Þand the hole Eh ðnh ; lh Þ for sufficient amount of different quantum numbersðne ; le ; nh ; lh Þ. Subsequently, the increasing energies EðiÞ, defined by the relation (Eq. (9)), are organized in the following five

arrangements EðiÞ; ne ðiÞ; le ðiÞ; nh ðiÞ; lh ðiÞ; i ¼ 1; 2; …; N . Finally, the matrix elements are calculated (Eq. (13)) and the secular equation (Eq. (14)) is solved to find the energies and the functions of the exciton itself. Fig. 2 shows the energies of the lower 22 states of the hole confined in the upper layer of GaInP in a type II quantum dot InP, with geometry of (a) disk and (b) pyramid, depending on the intensity magnetic field γ. The parameters of the quantum dots are R0 ¼10 nm, d0 ¼4 nm, dw ¼ 2 nm, and dp ¼1 nm. For γ ¼0 the R0= 10 nm, d 0= 4 nm, d w= 2 nm, d p= 1 nm

120

22

100

21

R0 = 10 nm, d0 = 4 nm, dw = 2 nm, dp = 1 nm

DISK

20

Σ STATES

64

19

*

80

10

18

60

17 16

60

8 6 20

15

14

12 11

40

13 10

9

7

5

Exciton Energy (R *y )

Electron Energy (Ry )

energies on the disk have higher values than in the pyramidshaped point. It is further noted that the energies of the hole in the disk shaped quantum dot grow faster with increasing magnetic field than the energies in the pyramid-shaped point. Also, the separation between the energies corresponding to nh ¼ 1 and nh ¼2 are greater in the disk-shaped point. In Fig. 3 are plotted energies of the lower states of the electron confined in the type II quantum dot InP, with (a) disk and (b) pyramid geometry as a function of the magnetic field. The parameters chosen for the points are R0 ¼10 nm, d0 ¼ 4 nm, dw ¼ 2 nm, dp ¼1 nm. In the graphic corresponding to the disk energy, transitions occur due to transitions in the orbital angular momentum le in γ¼ 8, approximately. All curves of the graphics (a) and (b) show that the energy in the quantum dot with pyramid geometry has higher values than the other point. Fig. 4 plots the magnetic field dependence γ with the energy of 10 Σ lowest states of the exciton confined in a type II quantum dot InP, with (a) disk and (b) pyramid geometries, embedded in a matrix of GaInP. In the two geometries, dimensions of the structure are the point radius, R0 ¼ 10 nm; wet layer thickness, dW ¼ 2 nm; point height, d0 ¼ 4 nm; and a layer above of the thickness point dp ¼ 1 nm; together forming a thin film of thickness df ¼ 7 nm. It is noted that for γ ¼0, all values of the exciton energy with the pyramid geometry point (Fig. 4b) are greater than those with the disk geometry point (Fig. 4a) [26]; due to the geometrical confinement the exciton is greater in the pyramid geometry point than the disc geometry point.

4

3

2

0

56

8 7

52

6 5

48

4

3

44

1

0

9

2

DISK 1

2

4

6

8

10

40

γ

0

2

4

6

8

10

γ 22 21

175

20 19

150

17

*

Electron Energy (Ry )

PYRAMID

125

16

15 13

100

11

75

6

18

14 12

10

9 8

7 5

4 3

50

2 1

25 0

2

4

6

8

10

1 = (1, 0) 2 = (1, 1) 3 = (1, -1) 4 = (1, 2) 5 = (1, -2) 6 = (2, 0) 7 = (1, -3) 8 = (1, 3) 9 = (2, 1) 10 = (2, -1) 11 = (1, 4) 12 = (1, -4) 13 = (2, 2) 14 = (2, -2) 15 = (1, -5) 16 = (1, 5) 17 = (2, 3) 18 = (2, -3) 19 = (2, 4) 20 = (2, -4) 21 = (2, 5) 22 = (2, -5)

γ Fig. 3. Energies of the 22 lowest states of an electron confined in a type II quantum dot InP, with geometry (a) disk and (b) pyramid, embedded in a matrix of GaInP as a function of the magnetic field.

R0=10 nm, d0=4 nm, dw=2 nm, dp=1 nm 110 10

105

Exciton Energy (R *y )

R 0= 10 nm, d0= 4 nm, d w= 2 nm, dp= 1 nm

200

9

PYRAMID

100

(ne,nh,le,lh) 1=(1,1,0,0) 2=(1,2,0,0) 3=(1,1,-1,1) 4=(1,1,1,-1) 5=(1,2,-1,1) 6=(1,2,1,-1) 7=(1,1,2,-2) 8=(1,1,-2,2) 9=(1,2,2,-2) 10=(1,2,-2,2)

8

95

7

90 6

85

Σ STATES

5

4

80 3

75 70

2

65

1

0

2

4

6

8

10

γ Fig. 4. Energies of the 10 lower states Σ of a confined exciton in a type II quantum dot InP, with geometry (a) disk and (b) pyramid, immersed in a matrix GaInP as a function of the magnetic field.

P. Villamil, J. Sierra Ortega / Physica E 56 (2014) 301–305

Fig. 4a shows that as the magnetic field increases, transitions occur in the exciton energy due to transitions in the orbital angular momentum. On a disk with R0⪢d0, the most probable position of the hole is located above the quantum disk. The magnetic field confines the wave functions in the radial direction and the hole is pressed strongly toward the center of the disk. Due to the magnetic field the electron and hole are pushed toward the center of the quantum dot which increases the Coulombic interaction between them, [27]. The electron energy increases dramatically with the augmentation of the magnetic field and thereby cancels the effect of an increased Coulomb energy (negative). Fig. 4b shows that for the range of γ there are no transitions in the exciton energy in the lower states.

4. Conclusions We studied the lowest energy states of an exciton (X) confined in a type II self-assembled quantum dot (SAQDs) InP/GaInP, with axial symmetry in the presence of a uniformly applied magnetic field in the growth direction. The electron is located inside the point of InP and the hole is located in the GaInP barrier. We studied the energy levels associated with the electron and the hole at the point, as well as the energy of the exciton. For axial symmetry of the problem the z component of the total orbital angular momentum Lz, the exciton states are preserved and classified according to the values of this component. For γ¼0, all values of the exciton energy with the pyramid-shaped geometry dot is greater than those with the diskshaped geometry dot, because the geometrical confinement exciton is greater in the pyramid-shaped geometry dot than the disk-shaped geometry dot. It is noted that as the magnetic field increases transitions occur in the exciton energy due to transitions in the orbital angular momentum.

Acknowledgments The authors gratefully acknowledge Universidad de Sucre for the total financial support in developing this work. Also, the authors gratefully acknowledge Drs. Harold Paredes and Carlos Beltrán for useful discussions and critical reading of the manuscript.

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