Percolation and lasing in real 3D crystals with inhomogeneousdistributed random pores

Gennadiy Burlak n, Yessica Calderón-SeguraCIICAp, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Cuernavaca, Mor., C.P. 62209, Mexico

a r t i c l e i n f o

Keywords:PercolationSpanning clusterRandom poresIncorporated nanoemittersLasing

a b s t r a c t

We systematically study the percolation phase transition in real 3D crystals where not only the state ofpores but also their radius r and displacement s are random valued numbers. The mean values R¼ ⟨r⟩and S¼ ⟨s⟩ emerge as additional spatial scales in such an extended network. This leads to variations ofthe threshold (critical) percolation probability pC and the percolation order parameter P that become tobe the intricate functions of R and S. Our numerical simulations have shown that in such extendedsystem the incipient spanning cluster can arise even for situations where for simple periodical systemthe percolation does not exist. We analyzed the validity of the nearest neighbor's approximation andfound that such approximation is not valid for materials with large dispersivity of pores. The lasingof nanoemitters incorporated in such percolating spanning cluster is studied too. This effect can openinteresting perspectives in modern nano- and micro-information technologies.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

The radiation of disordered nanoemitters or magnetization ofspins incorporated into three-dimensional (3D) clusters in perco-lation solids is an area of active research [1–10]. At a smallconcentration of defects in such system the number of clusters isinsignificant. However, if the concentration of clusters p exceeds acertain threshold value p4pC , then spanning cluster in the systemis formed, extending over the entire volume. This cluster qualita-tively changes the properties of the medium and produces ageneralized conductivity in the system that originally does notpossess such a property [11–15].

Percolating systems are well-studied theoretically and alsopossess important applied perspectives, e.g., the creation ofvarious solid-state filters on the basis of a porous medium. Onthe other hand, percolation is a simple geometrical model thatexhibits a continuous phase transition (the second type phasetransition in the system). The simplest example of percolationmodel (site percolation) is formulated on a lattice where equalsized pores are fixed in nodes (reference lattice). It is assumed thatevery lattice site can be in one of the two states, 1 (conducting)or 0 (empty) with the occupation probability p. Two occupiedsites belong to the same cluster if they are jointed by a path of

nearest-neighbor bonds. The percolation appears, when spanningcluster manages to reach the opposite boundaries of the system.

Two pores or voids (linked to nodes) are said to communicate if thedistance between them is less than their radius (that is fixed in asimple model). Nevertheless, such a model with linked positions andfixed sizes of voids is very simple and has to be extended in order tobe adequate to the case of real crystals. In our study we normalize allthe lengths in the system to the reference lattice constant a, whichreduces the distance between nearest pores to the unity. Besides weconsider the additional factors of the randomness: (i) the radius ofa pore r is random valued function; (ii) the shift of pore s (fromthe nodes of the reference lattice) is a randomvector. In periodical cases¼0, r¼const so one can define the conducting pores as nearestneighbors if 1=2oro

ffiffiffi2

p=2 (of the nearest neighbors approximation-

NNA). However, in general r and s are random valued functions, thusthe means R¼ ⟨r⟩ and S¼ ⟨s⟩ emerge here as new spatial scales. Thisresults in considerable dependence of the threshold (critical) percola-tion probability pC and the percolation order parameter P on R,S. As itis shown in this paper, in such a random network the percolationacquires new specific features. In particular, the incipient spanningcluster can arise here even for situations where in a periodical systemthe percolation does not exist at all.

2. Basic equations

Two pores (voids) produce a cluster if they have at least oneedge in common. In such approach, the percolation occurring

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n Corresponding author. Tel.: þ52 777 3297084.E-mail address: gburlak@uaem.mx (G. Burlak).

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Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

when the size of the largest (spanning) cluster reaches the size ofthe system. It is worth noting that in 3D case, no analyticalexpression for percolation threshold is known exactly; onlynumerical data are available [11].

In order to simulate a local cluster we introduce functionΨ ij ¼ dij�ðriþrjÞ, where dij is the distance between of i and jpores, ri is the radius of i pore, and i; j¼ x; y; z. From geometricalconsiderations it is clear that Ψ ijo0 if i and j are coupled objects,otherwise, if Ψ ijZ0 such pores are uncoupled. For nearest poresin simplest geometry we have dij ¼ 1, s¼0 and ri ¼ r that yieldΨ ij ¼ 1�2r. This means that all pores are uncoupled at all ifrrrc ¼ 0:5. If

ffiffiffi2

p=24r4rc only nearest pores are coupled that

corresponds to NNA. In general case we will characterize theradius r as a random valued function distributed uniformly inthe interval ½b; a�, with b¼ ð1þαÞR, a¼ ð1�αÞR, where 0oαo1and R40 are free parameters. This allows to simulate ri asri ¼ ð1�αþ2αξiÞR and recognize the parameter R¼ ⟨r⟩ as a meanvalue. The square of standard deviation (dispersion) for r iss2r ¼ ⟨r2⟩� ⟨r⟩2 ¼ R2α2=3, so α¼

ffiffiffi3

psr=R. The random vectorial

displacement si ¼ fsx; sy; szgi is simulated by the random numbersfor every component separately. To do that it is naturally tocharacterize a component sj with j¼ x; y; z as a randomly distrib-uted number in the interval ½�β;β� that yields ⟨si⟩¼ 0 and thesquare of standard deviation s2

s ¼ β2=3 (or β¼

ffiffiffi3

pss) so sj ¼

ffiffiffi3

pss

ð�1þ2ξjÞ. Since the mean value ⟨si⟩¼ 0 , it is expected that theeffect of the random pores displacement is weaker than thecontribution of the randomness of pore sizes. (It is worth to notethat the fixed voids geometry corresponds to sr ¼ 0 and ss ¼ 0.)Now Ψij reads

Ψ ij ¼ dijþ ∑m ¼ x;y;z

ðSmi�SmjÞ2" #1=2

�2½1�αþαðξiþξjÞ�R: ð1Þ

The coupled conditions in such extended system that is Ψ ijo0allows to recognize the parameters α;β as new spatial scales. Onlyfor nearest fixed voids case (α¼ 0;β¼ 0 or sr ¼ 0;ss ¼ 0) Eq. (1)can be simplified to Ψ ij ¼ 1�2r. But in general for real crystalssr ;ssa0, so such a system acquires new properties with respectto the standard 3D percolation model. To avoid the multi inter-sections of voids we restrict our attention by case R¼ ⟨r⟩offiffiffi2

p=2� 0:707 (mean value of R is less than half of the nearest cell

diagonal) and β¼ffiffiffi3

psso

ffiffiffi2

p=2� 0:707, so sso0:408. In principle

it allows the communications not only the nearest neighbor voids(NNA), but also the voids lying at the nearest diagonal. For caseβ¼

ffiffiffi3

psso1=2 we obtain sso0:288. Thus at values ss40:3 the

nearest neighbors approximation can be inaccurate. So for largedispersivity case the validity of NNA has to be verified. The studyof such system cannot be done analytically and requires quiteintensive numerical computations. The first step deals with theidentification of the spanning cluster existence as a function of theoccupation probability p [8,9]. The percolation order parameter P(p) is defined as the ratio of the number of conducting voidsbelonging to the spanning cluster to the general number of pores.In such simulations we assumed: (i) the percolating voids to be

randomly displaced from a fixed (reference) lattice node, and(ii) the radius of a void to be a random quantity. Our simulationsshow that in this case the percolation order parameter P(p)acquires the shift from the critical dependencies well-known fora fixed geometry, see Figs. 1–3. In general the behavior of suchsystem becomes complicated function, since more parameters(average distance between voids and the average radius of voids)are involved in consideration.

3. Numerics

First we consider the case without pore displacement s¼0.However the radius r of every i (i¼ x; y; z) pore (for simplicity thespherical pore shape is considered) is a randomly valuated numberwith the mean value R¼ ⟨r⟩. The latter leads to the fact that thethreshold probability pc becomes the function of R such thatpc ¼ pcðRÞ. The following questions emerge (i) if the percolationstill exists at random r? (ii) Whether the nearest neighborapproximation remains valid for such geometry? To answer thesequestions we study the properties of the order parameter P(p) forrandom radius r that is uniformly distributed in the interval [0, R].To understand deeper how the randomness of the radius affectsthe percolation properties, it is instructive to study the case whenR¼ 0:49orc ¼ 0:5. As it was already noted, for fixed r¼ R¼ rc(case α¼0) none of the pores intersect, so that no clusters can beformed. Fig. 1 shows the dependence of order parameter P(p) forR¼ 0:49orc and various standard deviations s¼ 0:02, 0.03, 0.04,0.3, 0.4, and 0.5. In Fig. 1 panel (a) shows the case where NNA wasused, while the panel (b) shows the simulation without NNA. Thenumerical grid considered was formed by 90�90�90 sites. Forguide to eyes in both panels the curve Pth displays the dependenceP(p) for simple (periodic) 3D percolation model with fixed r¼ R¼0:64rc (such that rcoro

ffiffiffi2

p=2, α¼0), where the percolation

arises at p¼ pCC0:318 [11].We observe from Fig. 1 that for small srC0:02 the spanning

cluster does not exist (pores are not connected) and P(p)¼0.However for larger sr some (nearest) pores acquire radius r largerthan rc ¼ 0:5 that leads to creating of coupled clusters. Suchsituation is displayed in Fig. 1 for s40:03 where a spanningcluster is formed for popC ¼ 0:9. We observe that for smallso0:3 the values of the order parameter P(p) do not differsignificantly for the cases illustrated in panels (a) and (b) ofFig. 1. This means that for so0:3 the NNA is valid. Other situationtakes places for larger s40:3. We observe from Fig. 1 that fors¼0.4 (panel (a), case with NNA) the threshold value pCC0:495differs considerably from the value pCC0:4 (panel (b), case with-out NNA). This leads to conclusion that the nearest neighborsapproximation is valid only for mediums with small dispersivity ofpores s≲0:3. That is the natural result: for larger sr the radius ofa pore r can become large enough to communicate with voidslocated at longer distance.

To compare the effects of random sizes and random displace-ments now we consider different situations when size of voids are

Nomenclature

dij distance between of i and j pores, i; j¼ x; y; zri radius of i pore, i¼ x; y; zp pores occupation probabilitypC critical percolation probabilityP(p) percolation order parametersr radius a pore (random valued function)

s random vectorial displacement of a pore(from reference lattice)

R¼ ⟨r⟩ and S¼ ⟨s⟩ mean values for r and s respectivelyξi random variable distributed uniformly in ½0;1�s2r;s square of standard deviation for r and s respectively

P vector of polarization densityNi electron population density in level i

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0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

P(p

)

R=<r>=0.49. With NNAPth

σ=0.05σ=0.075σ=0.1σ=0.2σ=0.3σ=0.4

0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

P(p

)

R=<r>=0.49. Without NNA

p

Pth

σ=0.05σ=0.075σ=0.1σ=0.2σ=0.3σ=0.4

σ=0.4

σ=0.075

σ=0.1

σ=0.2σ=0.3

σ=0.3

σ=0.05

σ=0.4

σ=0.1

σ=0.2

Pth

Pth

σ=0.075

σ=0.05

Fig. 2. The order parameter P(p) of the spanning percolation cluster for different spherical pores that have fixed R¼ r ¼ 0:49orc and random vectorial displacement s withs¼ss ¼ 0:05; 0:075; 0:1; 0:2; 0:3, and 0.4. (a) With of nearest neighbors approximation (NNA). (b) Without NNA. See details in text.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

P(p

)

S=<s>=0. With NNAPth

σ=0.02σ=0.03σ=0.04σ=0.3σ=0.4σ=0.5

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

P(p

)

S=<s>=0. Without NNA

p

Pth

σ=0.02σ=0.03σ=0.04σ=0.3σ=0.4σ=0.5

Pth

Pth

σ=0.5

σ=0.4

σ=0.3

σ=0.04

σ=0.04

σ=0.03σ=0.02

σ=0.02σ=0.5σ=0.3

σ=0.03

σ=0.4

Fig. 1. The order parameter P(p) of the spanning percolation cluster for random valued radius of the spherical pores r with average value R¼ ⟨r⟩¼ 0:49orc and the standarddeviation s¼sr ¼ 0:02, 0.03, 0.04, 0.3, 0.4, and 0.5. (a) The case when the nearest neighbors approximation (NNA) is applied. (b) Without NNA. All pores have fixed (zero)displacement s¼0. See details in text.

G. Burlak, Y. Calderón-Segura / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3

equal, but all the voids have a random displacement s. Suchsituation is shown in Fig. 2 for fixed radius r¼ R¼ 0:49orc , butfor the randomly shifted pores. We observe from Fig. 2 that dueto random displacement of voids the percolation appears herealready for large enough p with ssZ0:075 though the size of voidsstill is rorc. Such behavior is new and completely different withrespect to the standard percolation model; it becomes due totaking in consideration the random (dispersive) porosity factor.For large ss40:3 the order parameter P(p) becomes quite differentfor case without NNA with respect of model where such approx-imation is applied. Besides from Figs. 1 and 2 one can see that therandomness of the void size affects P(p) more considerably thanthe randomness of displacement (with zero mean value).

In Fig. 3 we display other situation when pores have fixedradius R¼ r¼ 0:64rc, but all of them are shifted randomly fromthe reference lattice nodes. We observe from Fig. 3 that for smallss ¼ 0:1 the value pC (when the percolation appears) is quite closeto the standard model case, but for larger displacement ss40:3the value of percolation threshold pC is shifted considerably.Besides, as one can see from Fig. 3(a) and (b) for large ss ¼ 0:4the NNA model becomes inaccurate. This circumstance alreadywas predicted in Section 2.

4. Lasing of nanoemitters

In what follows we study the lasing of nanoemitters that areincorporated in a spanning percolating cluster that occupies quitelarge working space. First it is instructive to display the 3D spatialstructure of the incipient cluster for p nearly the percolationtransition. Fig. 4 exhibits such cluster (all internal uncoupledclusters are omitted) for parameters used in Fig. 1(b) (s¼0.3 atp¼ pC ¼ 0:499). (In Fig. 4 the letters i, j, and k correspond to the

numbering of sites used for our simulations numerical grid90�90�90 indicated in the paragraph 3.) From Fig. 4 one cansee the essential inhomogeneous (fractal) structure of the incipi-ent cluster [16]. (It is worth noting that in Fig. 4 the incipientpercolation cluster has a dendrite shape that depends onthe actual random sampling. Re-running simulation with another

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

P(p

)

R=0.6. With NNAPth

σ=0.1σ=0.2σ=0.3σ=0.4

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

P(p

)

R=0.6. Without NNA

p

Pth

σ=0.1σ=0.2σ=0.3σ=0.4

Pth

Pth

σ=0.1

σ=0.2

σ=0.3σ=0.4

σ=0.4

σ=0.2

σ=0.3

σ=0.1

Fig. 3. The order parameter P(p) of the spanning percolation cluster for pores with fixed R¼ r ¼ 0:64rc and different random displacement with s¼ ss ¼ 0:1; 0:2; 0:3, and0.4. (a) With nearest neighbors approximation (NNA). (b) Without NNA. For large value ssZ0:3 the difference between both cases becomes considerable. See details in text.

Fig. 4. The spatial structure of the incipient percolating cluster near the percolationthreshold corresponding to Fig. 1 (b) for s¼0.3 at p¼0.499. The letters i, j, and kcorrespond to the numbering of sites in the numerical grid 90�90�90 indicatedin the paragraph 3. Only clusters coupled to spanning cluster are shown. Theinternal clusters unconnected to the entry side are not displayed. In this config-uration considerable quantity of the nanoemitters are incorporated thought theentry side (indicated by incoming arrow) to the crystal.

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random seed value will lead to percolation cluster of differentgeometries, which will also have dendrite structure.)

In case popC the crystal consists of a lot of uncoupled internalclusters (shown in Fig. 5) that do not communicate to the input

side (such clusters are omitted in Fig. 4). In this case the workingarea of incorporated nanoemitters is small; such nanoemittersgive a scant contribution to lasing and are not considered here. Formore details about the structure of the spanning cluster in vicinityof the percolation transition, see Ref. [8]. We assume that nanoe-mitters incorporated in the spanning cluster have been inverted tothe excited state that finally leads to the lasing.

Now we briefly summarize our strategy of simulations: (i)calculating the geometry of the spanning cluster. (ii) Calculatingthe photon field E generated by spatial nanosources incorporatedin this cluster deals with the use FDTD technics [17] and analysis ofdynamic field order parameter, see [9]. (iii) Solution of nonlineardynamic coupled equations for field, polarization P and theoccupation numbers N.

Step (i) is well described in Section 3, see also Ref. [8]. Step (ii)deals with the generation of 3D photon field E by nanoemittersand it is described with details in Ref. [9].

To avoid the repetitions of [9], in what follows we willconcentrate in more details in step (iii). In this item we numeri-cally study the transitions between the energy levels that aregoverned by Maxwell equations for field E (added with polariza-tion P) and coupled to the rate equations for the occupations Ni

[17–19] for 4-level nanoemitters in the cluster. Here Ni is theelectron population density in level i and τij is the decay time cons-tant between levels i and j. The populations vary with pumpingE � ðdP=dtÞ (here P is the vector of polarization density) andspontaneous emission decay ðNi�NjÞ=τij.

The lasing effect, produced by the 4-level nanoemitters in suchconfiguration with a light-localizing random media, is quitedifficult to analyze in general. It requires the knowledge of many

Fig. 5. Spatial structure of the clusters below the percolation threshold correspond-ing to Fig. 1 (b) for s¼0.3 but p¼0.2. As opposed to Fig. 4 in this case none ofspanning clusters are formed and only multi-scaled uncoupled clusters exist. Onlysmall quantity of nanoemitters closely to the entry side (indicated by incomingarrow) can contribute to lasing.

0 5 10 15 20 25 30 350

1

2

3

4

5

Inte

nsity

, arb

itrar

y un

its

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

Pop

ulat

ion

num

bers

Time, 10−9 sec

N0

N1

N2

N3

25 26 27 28 29 300

1

2

3

4

5

N1

N2

N3

N0

Intensity of field

Fig. 6. The lasing of nanoemitters incorporated in percolating cluster shown in Fig. 4. (a) Intensity output of lasing signals, (b) the time evolution of the electron populationnumbers Ni in four-level nanoemitters. See details in text.

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parameters (decay time constants τ, frequencies ωa;b, dampingcoefficients γ, etc.). We study with details the situation whenelectrons in level 3 spontaneously decay to levels 2 and 0 withdecay time constants τ32 and τ30 respectively.

In our simulation we use τ32 ¼ τ10 ¼ 100 fs and τ21 ¼ τ30 ¼300 ps (which takes into account the dephasing time) [18]. Theinitial population density is N0 ¼ 5� 1023=m3. The governingequations for the polarization densities Pa;bðtÞ (having the resonantfrequencies ωa and ωb) read

d2Pa=dt2þγadPa=dtþω2

aPa ¼ ζaðN2�N1ÞE;ptd2Pb=dt

2þγbdPb=dtþω2bPb ¼ ζbðN3�N0ÞE; ð2Þ

where ℏωa (ℏωb) is the energy difference between levels 1 and 2(levels 0 and 3); coefficient ζa is 6πε0c3=ðω2

21τ21Þ, and the dampingcoefficients γa and γb simulate the non-irradiative energy loss.For our example γa ¼ γb ¼ 10�13 s�1 is applied. The followingdynamic rate equations for electron populations within four levelsare used:

dN3=dt ¼ �N3ð1�N2Þ=τ32�N3ð1�N0Þ=τ30þð1=ℏωbÞE � dPb=dt;dN2=dt ¼N3ð1�N2Þ=τ32�N2ð1�N1Þ=τ21þð1=ℏωaÞE � dPa=dt;dN1=dt ¼N2ð1�N1Þ=τ21�N1ð1�N0Þ=τ10�ð1=ℏωaÞE � dPa=dt;dN0=dt ¼N3ð1�N0Þ=τ30þN1ð1�N0Þ=τ10�ð1=ℏωbÞE � dPb=dt: ð3Þ

Fig. 6 displays the calculated lasing dynamics, where panel (a)shows the intensity output of the lasing signals, while panel (b)exhibits the time evolution of the electron populations Ni in fourlevels. We observed from Fig. 6 that for considered parameters inthe percolating medium the lasing effect appears at t¼25 ns.However more complete investigation of such advanced nonlinearsystem for various material parameters will be published in future.

5. Conclusions

We studied the percolation phase transition in real 3D crystalswhere not only the state of pores but also their radius r anddisplacement s are random numbers. It is shown that the thresh-old (critical) percolation probability pC and the percolation orderparameter P(p) become complicated functions of the mean values⟨r⟩ and ⟨s⟩ (or ⟨s2⟩ if ⟨s⟩¼ 0. Our numerical simulations show that insuch advanced systems the incipient spanning cluster can ariseeven for situations where for periodical fixed system (standard

model) the percolation does not exist. It is found that nearestneighbor's approximation ceases to be valid for materials withlarge dispersivity of pores. The photon field properties and laserdynamics for nanoemitters incorporated into spanning clusterclosely to the percolation phase transition were studied as well.It is shown that the raise of field intensity leads to generation of ahigh-density coherent field state. Such effect emerges due to highcontribution of the coherent nanoemitters in the spanning clusterclosely to the percolation transition. This effect can allow the useof the disordered optical nanostructures with incorporated radiat-ing nanoemitters in various applications of modern micro- andnano-information technology.

Acknowledgments

The work is partially supported by CONACyT (México) Grant no.169496.

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