Generalized rate-equation analysis of excitation exchange between silicon nanoclusters and

erbium ions

Rui LiJournal Club, 02.04.08

Electrical Engineering

Boston University

A. J. Kenyon, M. Wojdak, and I. AhmadDepartment of Electronic and Electrical Engineering, University

College LondonW. H. Loh and C. J. Oton

Optoelectronics Research Centre, University of Southampton

Physics Review B 77, 035318 (2008)

Outline

• General form of coupled rate equations: two-level system

• Effective excitation cross section• Stretched exponential rise/decay• Complicated rate equation model

General form of coupled rate equations: two-level systems

Si-ncs Er

* *0A B B

Effective excitationcross section

Pacifici’s PhD thesis

Effective excitation cross section

eff

0

1A

effA

A

is misleading? γ is more robust

001

Aeff A

A

AA

(Weak pumping condition)

Stretched exponential rise

: 0 ~ 1

Stretched exponential decay

•Auger nonradiative recombination.•Variations of size, shape and environment of Si-ncs.•Energy transfer between ncs.•Different decay channels (radiative and nonradiative recombinations).•Intrinsic property of Si-ncs as indirect-gap semiconductor nanocrystals.

Delerue et al. Phys. Rev. B 73, 235318 (2006)

Two energy transfer processes?

Fujii et al. J. Appl. Phys. 95 1 (2004)

Si-ncs in SRN decay: three effective decay parameters needed to fit the data

0 10 20 30 401E-4

1E-3

0.01

0.1

1

Experimental data of SRN decay Multiexponential decay

Time (ns)

PL

Inte

nsity

(a

rb. u

nits

)

1 D1

2 D2

3 D3

0.03 4.57

0.23 1.04

0.74 0.27

ns

ns

ns

302010 )(3

)(2

)(10

DDD xxxxxx eeeyy

We need to develop a rate equation model that includes 3 exponentials

and coupling with Er ions

1 2 3

1 2 3

1 11 0

1

2 22 0

2

3 33 0

3

1

b b b b

b bSi nc P b

D

b bSi nc P b

D

b bSi nc P b

D

n n n n

n nt n n

t

n nt n n

t

n nt n n

t

Si-ncs population is divided into three parts nb1, nb2 and nb3 with different decay times and weighted absorption cross sections.

A rate equation modelwith 3 exponentials is sufficiently accurate

Fitting parameters:

D3D2D1321

Weighting factors

Population constraint

Donor emission

Origin of non-single exponential rise

Inhomogeneity may from

•Coupling coefficient γ

•Si-ncs decay time τA

•Si-ncs absorption cross section σ

•Excitation photon flux Φ

Gaussian Beam

Coupling Coefficient g

16 2 18 -2 -1 -6A

17 -3B 0i

10 cm =5 10 cm s 2 10 s

~ 10ms A 10 cm

In order to have stretched or multiexponential behavior, we need to have, for at least one class i with gi

12 3 -1~ 10 cm si

Saturation of rise rate

1. In homogeneity

2. Up-conversion

Experiments: Si-ncs lifetime shortening

Donor lifetime shorteningUnderlying assumption: Er does not

introduce significant non-radiative contributions

0 10 20 30 401E-4

1E-3

0.01

0.1

1

Experimental data of SRN decay Experimental data of Er:SRN decay Multiexponential decay Multiexponential decay with 2

Time (ns)

PL

Inte

nsi

ty (

arb

. un

its)

31

30

3

32

1 18 /

1 20 /

0.7 4 12

0.3

/

1 15 /

T

a

b

E cm

n E cm

N E cm

s

E cm s

A Gaussian beam is considered

Si 49%

Conclusion

• Coupled rate equation model

• Validity of effective excitation cross section

• Stretched (multiexponential) rise/decay (inhomogeneity τA γ σ? Φ?)

• Lifetime shortening

• How complicated a simple model should we have to fit the experimental data?

1 11 0 1 1 1

1

2 21 0 2 2 1

1

3 32 0 1 3 1

2

4 42 0 2 4 1

2

5 53 0 1

3

b ba Si nc P b b

D

b bb Si nc P b b

D

b ba Si nc P b b

D

b bb Si nc P b b

D

b ba Si nc P b

D

n nt n n n N

t

n nt n n n N

t

n nt n n n N

t

n nt n n n N

t

n nt n n

t

5 1

6 63 0 2 6 1

3

22 22 1 1 2 1 2 22

b

b bb Si nc P b b

D

Er P T a b b b up ESA PA

n N

n nt n n n N

t

N Nt N N n N n N C N t N

t

Energy coupling model with 3 “effective” decays

Si-ncs

Er

1 2 3

1 2 3 4 5 6

1

1a b

b b b b b b bn n n n n n n

• 2 coupling parameters (fast, slow)• 3 decay amplitudes

Gaussian Beam

Thank you !