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Ac hopping conduction at extreme disorder takes place on the percolating cluster

Thomas B. Schr?der and Jeppe C. Dyre

DNRF Centre “Glass and Time,” IMFUFA, Department of Sciences, Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark (Dated: June 6, 2008) Simulations of the random barrier model show that ac currents at extreme disorder are carried almost entirely by the percolating cluster slightly above threshold; thus contributions from isolated low-activation-energy clusters are negligible. The e?ective medium approximation in conjunction with the Alexander-Orbach conjecture leads to an excellent analytical ?t to the universal ac conductivity with no nontrivial ?tting parameters.

arXiv:0801.0531v4 [cond-mat.dis-nn] 6 Jun 2008

Recent advances relating to ion conduction in glasses and other disordered solids include the application of multidimensional NMR techniques [1], the introduction of ac nonlinear spectroscopy [2], and elucidations of the high-frequency nearly constant loss [3]. Moreover, it was found that the old idea of ions moving by the vacancy mechanism may well be correct [4], and simulations gave new insight into the mixed-alkali e?ect [5]. Despite these and other signi?cant advances, important questions remain unanswered. For instance, it is still not understood what role is played by ion interactions for the conductivity [6], or why the random barrier model (RBM) [7, 8] represents ac conductivity data so well. The latter question is not answered below, but new simulations and arguments are presented that we believe lead to a full understanding of the physics of the RBM in the extreme disorder limit (low temperature limit). Ac conductivity is often studied also for amorphous semiconductors, electronically or ionically conducting polymers, defective crystals of various kinds, polaronic conductors, etc [7, 8]. It is a longstanding observation that all disordered solids have remarkably similar ac conductivities [9]. Universal features include [8]: At low frequencies the conductivity is constant. At higher frequencies it follows an approximate power law with an exponent less than one that increases slightly with increasing frequency. When measured in a ?xed frequency range, the exponent converges to one as temperature goes to zero. The ac conductivity is less temperature dependent than the dc conductivity and obeys time-temperature superposition (sometimes referred to as “scaling”). The frequency marking onset of ac conduction, ωm , has the same activation energy as the dc conductivity. These and other observed features are reproduced by the RBM characterized [8, 10] by ?ve assumptions: 1) All charge carrier interactions including self-exclusion are ignored; 2) Charge carrier motion takes place on a cubic lattice; 3) All lattice sites have same energy; 4) Only nearest-neighbor jumps are allowed; 5) Jump rates ∝ exp(?E/kB T ) have random activation energies with distribution p(E). In the RBM the ac conductivity σ(ω) relative to σ(0) as a function of a suitably scaled frequency becomes independent of p(E) in the extreme disorder limit, i.e., when the width of p(E) is much larger

than kB T [8]. Despite lack of non-trivial free parameters the RBM universal ac conductivity gives a good ?t to experiment [8]; more re?ned models yield results that are close to those of the RBM [11]. It is well-known that the percolation threshold determines the dc conductivity activation energy [12]. At low temperatures the particles preferably jump across the lowest barriers. The highest barriers on the percolation cluster are bottlenecks dominating the low-temperature dc conductivity. If Ec is the highest barrier on the percolating cluster, one has σ(0) ? exp(?Ec /kB T ) as T → 0 [12]. In order to have a non-zero dc conductivity of the percolation cluster, barriers slightly above the percolation threshold must be included. This de?nes the “fat percolation cluster” [8]; on length scales shorter than its correlation length the fat percolation cluster appears fractal, on longer length scales it appears homogeneous. Understanding the RBM universal ac conductivity in terms of percolation arguments is much more challenging. Traditionally [7, 13] the problem was approached “from the high-frequency side” by proceeding as follows. For (angular) frequencies ω > ωm there is a character? istic activation energy E(ω) < Ec for motion on time scales ? 1/ω; when ω decreases towards ωm one has E(ω) → Ec . Links with E ≤ E(ω) form ?nite lowactivation-energy clusters. The cluster size distribution is assumed to determine the ac conductivity. Some time ago we proposed what amounts to coming “from the lowfrequency side,” namely that all relevant motion takes place on some subset of the in?nite percolation cluster [8]. Numerical evidence for this conjecture is given below, where it is shown that contributions from low-activation energy clusters outside the fat percolating cluster are insigni?cant. Moreover, it is shown that by assuming that not just a subset, but in fact the entire percolation cluster is important, an excellent analytical approximation to the universal ac conductivity with no nontrivial ?tting parameters may be derived. The simulations of the RBM reported below refer to the Box distribution of activation energies (p(E) = 1/E0 for 0 < E < E0 , zero otherwise); ac universality in the extreme disorder limit implies that this distribution gives representative results [8]. The lowest temperature simulated is given by β = 320 where β is the inverse di-

2

-29 -30 -31 -32 log10(σ’) -33 -34 -35 -36 log10[ 1 - σ’(k)/σ’(k=12.8) ] (a)

σ’IC(ω) / σ’(ω) 0.4 k=6.4

k = 12.8 k = 3.2

0.3

Relative mass of isolated clusters (β=320): 0.33 β = 320 β = 240 β = 160 Relative mass of isolated clusters (β=160): 0.18

0.2

0.1

0

0

1

2

3 4 log10[σ’(ω) / σ(0)]

5

6

7

-1 -2 -3 -4 -5 -42 -40 -38 -36 -34 log10(ω) -32

k = 3.2 k = 4.8 k = 6.4 k = 8.0

(b) -30 -28

FIG. 2: Contribution from isolated clusters for the real part of ′ the ac conductivity, σIC (ω), relative to σ ′ (ω) as a function of the scaled real part of the conductivity, σ ′ (ω) ≡ σ ′ (ω)/σ(0) ? (cut-o?: k = 6.4). The two dashed lines mark the relative masses of isolated clusters. Their contribution, however, is much smaller than their relative mass, showing that the dominant part of the ac conduction takes place on the fat percolation cluster.

FIG. 1: Results for the ac conductivity at β ≡ E0 /kB T = 320 in rationalized units [8]; the frequency marking onset of ac conduction, ωm , is of order 10?38 . Ten independent 96 × 96 × 96 samples were simulated. (a) Real part of σ(ω) with two cut-o?’s: k = 3.2 and k = 12.8, averaged over the ten samples. The dashed line is the prediction of Eq. (2) empirically scaled to the k = 12.8 data. (b) Relative deviation from k = 12.8 as a function of frequency plotted for each of the ten independent samples.

mensionless temperature, β ≡ E0 /kB T . For β = 320 the jump rates cover more than 130 orders of magnitude, making simulations quite challenging. We used a method based on solving the Laplace transform of the master equation numerically [14]. Conductivity data for β = 320 give an excellent representation of the universal master curve for the RBM over the frequency range studied here [8]. In previously reported simulations [8] we applied an activation energy cut-o? above the percolation threshold, Ecut /E0 = Ec + k/β, where Ec = 0.2488 is the percolation energy for the cubic lattice and k a numerical constant. Jump rates for links with activation energies larger than Ecut were set to zero in order to be able to simulate large samples. Figure 1(a) presents the real part of the ac conductivity σ(ω) = σ ′ (ω) + iσ ′′ (ω) for k = 3.2 and k = 12.8 respectively at β = 320. There is little di?erence between the two data sets. The dashed line gives the prediction of the di?usion cluster approximation (DCA) combined with the Alexander-Orbach conjecture as de-

tailed below (Eq. (2)). Fig. 1(b) gives the relative errors involved for di?erent k values, taking k = 12.8 as representing the “correct” data. As expected, the errors are largest in the dc regime and decrease with increasing k. Choosing k = 6.4 gives an error of just 1-2%. We proceed to investigate the behavior with k = 6.4 in more detail. Applying this cut-o?, the links with nonzero jump rate fall into two sets, the “fat” percolating cluster and all remaining ?nite isolated clusters. The latter do not contribute to the dc conductivity. According to the traditional approaches based on cluster statistics, however, they give a signi?cant contribution to the ac conductivity as soon as ω > ωm [7, 13]. This was ? never tested numerically. Figure 2 presents the contribu′ tion from isolated clusters σIC (ω) relative to the full ac conductivity as a function of the real part of the scaled conductivity σ ≡ σ(ω)/σ(0). The dashed lines mark this ? relative mass of the isolated clusters for β = 160 and ′ β = 320, respectively. The quantity σIC (ω)/σ ′ (ω), however, is much smaller than the relative mass of isolated clusters for the range of frequencies covered in the ?gure, i.e., up to 10 billion times ωm (compare Fig. 1). Moreover, for β → ∞ the relative mass of isolated clusters goes ′ to one, whereas we ?nd that σIC (ω)/σ ′ (ω) is independent of temperature and stays insigni?cant. In summary, the dominant part of the low-temperature universal ac conductivity comes from the fat percolation cluster [8] with little contributions from isolated clusters. We now turn to the issue of analytical approximations utilizing the e?ective medium approximation (EMA) ∞ [10, 15, 16]. If G ≡ 0 P0 (t) exp(?iωt)dt where P0 (t)

3 is the probability for a particle to be at a site given it was there at t = 0 for a homogeneous system with uniform jump rate, the extreme disorder limit of the EMA self-consistency equation is ln σ = ΛβiωG where Λ is a ? numerical constant [10]. This determines a frequencydependent complex “e?ective” jump rate that is proportional to the frequency-dependent conductivity [7, 10]. Henceforth we switch to the rationalized unit system [8] where the EMA selfconsistent ac conductivity equals the complex e?ective jump rate. Because P0 (t) is a function of the e?ective jump rate times time, σt, the quantity iωG is a function of iω/σ. In the frequency range relevant for the universal ac conductivity of the extreme disorder limit corresponding to times obeying σt ? 1, one has |iωG| ? 1 [10, 17]. If d is dimension, whenever d ≥ 2 iωG as a function of iω/σ has a regular ?rst order term [7, 10]: iωG = α1 (iω/σ) + .... If Λ is absorbed into a dimensionless frequency by de?ning ω ≡ α1 Λβω/σ(0), ? the EMA universality equation [10, 17] for d ≥ 2 is σ ln σ = i? . ? ? ω (1)

10 10 10 10

1

β = 320

Slope = 1

0

~ | ln (σ) |

-1

Slope = 2/3

-2

(a) 10

-3

10

-3

10

-2

10

-1

10 ~ ~ | ω/σ |

0

10

1

10

2

1 (b) Apparent exponent 0.9 0.8 0.7 0.6 0.5 Real part Imaginary part Lines: Eq. (3) β = 320 2/3 β = 320 β = 160

~ ~ log10(σ’), log10(σ’’)

6 4 2 0 -2 -4 -4

This equation gives a qualitatively correct, but numerically inaccurate ?t to simulations [8]. In our previous works it was proposed that some unspeci?ed subset of the percolating cluster with fractal dimension df (“the di?usion cluster”) is responsible for the ac conduction [8]. If df < 2 this led to the di?usion cluster approximation (DCA): ln σ = (i? /? )df /2 [8]. If ? ω σ the di?usion cluster is the so-called backbone, one expects df = 1.7, if the di?usion cluster is the set of red bonds, one expects df = 1.1 [8]. Treating df as a ?tting parameter led to df = 1.35 [8], however, leaving the nature of the di?usion cluster as an open problem. What if not just a subset, but in fact the entire percolating cluster contributes signi?cantly to the universal ac conductivity? Random walks on a fractal structure are characterized by P0 (t) ∝ (σt)?dH /2 [18] where dH is the spectral dimension. For dH < 2 this leads to iωG ∝ (iω/σ)dH /2 . In terms of a suitably scaled frequency the EMA thus implies the DCA expression with df = dH . According to the Alexander-Orbach conjecture [19] – known to be almost correct (see, e.g., [14, 20]) – one has dH = 4/3 for the in?nite percolating cluster. If frequency is suitably scaled, this leads to the following approximation to the universal ac conductivity of the extreme disorder limit: i? ω σ ?

2/3

(c) -2 0 2 4 ~ log10(ω) 6 8 10

FIG. 3: Testing Eqs. (2) and (3). Data represent averaging over 100 independent 96 × 96 × 96 samples (β = 320, cut-o?: k = 6.4) and 100 independent 64 × 64 × 64 samples (β = 160, cut-o?: k = 6.4). For both temperatures the frequency was empirically scaled such that σ = 1 + ω in the low frequency ? ? limit, where σ ≡ σ(ω)/σ(0), ω ≡ ω???0 /σ(0) [8]. (a) shows ? ? | ln σ| as a function of |? /? | in a log-log plot. (b) shows the ap? ω σ parent exponent d ln(| ln σ |)/d ln |? /? | as a function of scaled ? ω σ frequency. A cross-over from fractal behavior (exponent 2/3) to homogeneous behavior (exponent one) is clearly visible. (c) shows the real and imaginary parts of the scaled conductivity compared to Eq.(3) (full lines).

ln σ = ?

.

(2)

As shown in Fig. 1(a) this expression provides an excellent ?t to the universal ac conductivity of the extreme disorder limit [21]. Equation (2) may be put to a more severe test, however, than just ?tting the real part

of σ (? ). Figure 3(a) tests one implication of Eq. (2), ? ω | ln σ | = |? /? |2/3 , by plotting | ln σ | as function of |? /? | ? ω σ ? ω σ in a log-log plot. A cross-over between two power-law regimes is seen, corresponding to a cross-over between Eqs. (1) and (2). In Fig. 3(b) the apparent exponent d ln(| ln σ |)/d ln |? /? | is plotted as a function of scaled ? ω σ frequency. Similar results are found by plotting the ratio of the phases of the complex numbers ln σ and i? /? ? ω σ (data not shown). The picture emerging from Figs. 3(a) and (b) is the following: Equation (2) works well whenever ω > 1; here the ?? fat percolation cluster appears fractal because over one

4 cycle the particles move less than the correlation length. At low frequencies there is a transition to the analytic behavior predicted when the dimension is larger than two (Eq. (1)); over one cycle the particles here move longer than the correlation length and consequently the fat percolation cluster appears homogeneous. The entire frequency range is accurately described by the expression i? ω σ ? i? ω σ ?

?1/3

ln σ = ?

1 + 2.66

(3)

that is plotted as the full lines in Figs. 3(a) and 3(c). The exponent ?1/3 was chosen to get agreement with Eq. (2) for |? /? | ? 1. The di?erence between Eqs. (2) and (3) ω σ is signi?cant only at such low frequencies that σ (? ) is of ? ω order unity (Fig. 3(a)). Equation (2) breaks down only for the imaginary part for ω < 1 where Eq. (2) predicts ? σ ′′ ∝ ω 2/3 instead of the observed σ ′′ ∝ ω . Numerical ? ? ? ? solutions of Eqs. (2) and (3) are provided Ref. [22]. In our opinion, the RBM must now be regarded as solved in the extreme disorder limit in the sense that a good understanding of the model’s physics is at hand, leading to an accurate description of the ac conductivity. A notable consequence of the above is that the EMA – generally believed to be inaccurate except at weak disorder – works surprisingly well in the extreme disorder limit if the “geometrical” input G is taken to re?ect the fractal geometry of the percolation cluster. It would be interesting to know whether similar results apply when the EMA is applied for the extreme disorder limit of other models. The centre for viscous liquid dynamics “Glass and Time” is sponsored by the Danish National Research Foundation (DNRF).

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5

ductivity a few decades above ωm has an approximate power-law frequency depedence with exponents in the range 0.6-0.7, thus close to 2/3 (compare Fig. 1) and consistent with experiment. It is important to note that this is not related to the exponent 2/3 appearing in Eq. (2); for instance Eq. (1) has the same mathematical property. [22] See EPAPS Document No. [number will be inserted by publisher] for [give brief description of material]. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html. See EPAPS Document No. ... .

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