Melting of two dimensional solids on disordered substrate

发布于:2021-06-22 11:03:57

Melting of two dimensional solids on disordered substrate
David Carpentier and Pierre Le Doussal
CNRS-Laboratoire de Physique Th?orique de l’Ecole Normale Sup?rieure, 24 Rue Lhomond, 75231 Paris e e We study 2D solids with weak substrate disorder, using Coulomb gas renormalisation. The melting transition is found to be replaced by a sharp crossover between a high T liquid with thermally induced dislocations, and a low T glassy regime with disorder induced dislocations at scales larger than ξd which we compute (ξd ? Rc ? Ra , the Larkin and translational correlation lengths). We discuss experimental consequences, reminiscent of melting, such as size e?ects in vortex ?ow and AC response in superconducting ?lms.

arXiv:cond-mat/9712227v1 [cond-mat.dis-nn] 18 Dec 1997

Although the phase diagram of the mixed state of high Tc superconductors [1] has considerably evolved since their discovery, many questions remain. In d = 3 the low ?eld region has been proposed as the topologically ordered Bragg glass phase [2]. The role of dislocations remains unclear near melting or at higher ?elds. To study analytically the e?ect of disorder on defective solids, d = 2 appears as a natural starting point, also important for numerous experimental systems besides thin superconductor ?lms [3–6], such as Wigner crystals in heterojunctions [7] and on the surface of Helium [8], magnetic bubbles arrays [9]. In the absence of disorder, the continuous melting of a 2d crystal occurs as dislocation pairs 0 unbind at Tm and is well described by the KTNHY theory [10,11]. But no equivalent theoretical description of melting with substrate disorder exists, though it has been studied in many experiments [4,5,7,12]. Progress was made for the simpler problem of crystals with structural (i.e internal) disorder [13] where melting occurs at a temperature Tm (σ) shifted downwards by the disorder strength σ (Fig.1). As discussed in [2], [13] is equivalent to treating only the long wavelength part σ of the substrate disorder. On the other hand an analytic RG study of the 2d Bragg glass including pinning disorder g, i.e short wavelength disorder (at the cost of excluding by hand dislocations) was performed recently [14,15]. A complete treatment however should include dislocations. Indeed the general theoretical belief based on qualitative arguments and simulations [16,17] is that no true solid (neither a lattice nor a vortex glass [17]) exist in 2D in presence of disorder at T > 0, and thus there should be no true melting transition. On the other hand signatures reminiscent of melting are observed in various experiments [6,5]. This thus calls for further studies. In this Letter we derive renormalisation group (RG) equations describing a solid on a disordered substrate and allowing for dislocations. They generalize the KTNHY equations to weak pinning disorder near melting. They are obtained from the RG analysis, performed here for the ?rst time, of the generic elastic vector electromagnetic coulomb gas (VECG) with disorder [18]. Though the asymptotic RG ?ow is always towards strong coupling (disorder or/and dislocation fugacity) several studies are still possible for weak pinning disorder g but arbitrary σ 1

(see below) which strongly suggest that the pure melting transition is replaced by a sharp crossover between two very distinct regimes: (i) at higher T a weakly disordered liquid where disorder has no e?ect at large scale and dislocations are present on all length scales greater than 0 ξ+ (T ) as in the pure case (ii) at lower T a quasi Bragg glass regime, where disorder disrupts the quasi-long range order of the lattice beyond a length [19] Ra ? Rc and dislocations appear on scales larger than a new length ξd (T ). From the RG, here we estimate ξd (T ) both for T < Tm where ξd ? Rc ? a, and in the crossover region. At scales L ≤ ξd , the low T defect free glassy regime is described by the theory of [14], despite its defective nature in the thermodynamic limit. The sharp crossover predicted between the two regimes at weak disorder is analyzed and argued to account for experimental observation of 2D melting transition, and of the 2D peak e?ect.
(long wavelength)

Tm (σ)


Tm Glassy regime Liquid regime g
(short wavelength)





FIG. 1. Schematic phase diagram: σ and g are respectively the long and short (pinning) wavelength disorder. Inset: RG 0 ?ow for weak g and T ? Tm . The hatched region corresponds to the g = 0 true solid which survives at g = 0.

We emphasize that our RG equations (5) are derived assuming that dislocations are thermalized. This assumption is reasonable near Tm and allows us to analyze the crossover near melting. A fully consistent RG analysis including pinning of dislocations which become important at lower temperatures [2], is beyond the scope of this paper [20]. However, as a ?rst step, we also stud0 ied the case of no pinning disorder g = 0 at T ≤ Tm /2 by extending the analysis of [21,22] on a simpler model. A further study of melting is presented elsewhere [18].

In the absence of disorder and of dislocations a two dimensional crystal is described by a smooth 2D displacement ?eld u(r) and an elastic distortion energy: Hel = 1 2 2c66 u2 + (c11 ? 2c66 )uii ujj ij (1)


where c11 and c66 are respectively the compression and the shear elastic moduli, and uij = 1 (?i uj + ?j ui ). For 2 simplicity we study a triangular lattice of spacing a. Next, disorder is modeled by a gaussian random potential V (r) with correlator V (r)V (r′ ) = h(r ? r′ ) of range rf , and couples to the density of vortices ρ(r) = i δ(r ? ri ) according to Hp = r ρ(r)V (r). Decomposition of ρ(r) and of V (r) in Fourier components gives several couplings between u(r) and the disorder [2]. The two main contributions [23] are a random stress ?eld σij arising from the long wavelength part of the disorder, and a ‘random phase ?eld’ φν which comes from the part of the (pinning) disorder with almost the periodicity of the lattice:

each ma belonging to the reciprocal lattice, with the constraint [14] a ma = 0 as well as the usual neutrality condition. Integration over the replicated ?eld ua (r) 0 leads to an elastic VECG (instead of the purely electric studied in [14]) which in its most general form is d2 rα ?S de?ned by Z n = {b,m} C ?1 p α=1 a2 Y [bα , mα ]e where the integrals are restricted by |rα ? rβ | ≥ a, the Y [b, m] are the composite charges fugacities and C(b, m) is a combinatorial factor. The action is given by ? S[b, m] = 1 a b ? V ac (κ1 , κ2 ) ? bc 2 (3)

+ma ? V ac (κ3 , κ4 ) ? mc

+ i ma ? W ac (κ5 , κ6 ) ? bc

denoting b ? V ? b ≡ ij r,r′ bi (r)Vij (r ? r′ )bj (r′ ). The ac interactions are πVij (κ1 , κ2 ) = κac δij G ? κac Hij and 1 2 ac 2πWij (κ5 , κ6 ) = δij δ ac Φ + κac ?ij G + κac ?jk Hik . The 5 6 renormalization of this elastic VECG goes beyond the previous analysis of the ECG [27], due to the presence of new marginal operators corresponding to the elastic ing Hp = σij uij + 2 cos (Kν .u(r) + φν (r)) (2) teractions V, W . Note that usual electric/magnetic self? T a2 ν=1,2,3 r duality for (3) now reads: κ1 ? κ3 ; κ2 ? κ4 ; κ5 ? ?κ5 ; κ6 ? κ6 and exchanges direct and reciprocal latwith ei(φν (r)?φν ′ (r′ )) = δν,ν ′ δ 2 (r ? r′ ), the Kν are the tices. The full RG equations and applications to various √ ?rst reciprocal lattice vectors (of length K0 = 4π/ 3a) 2D models (depending on the de?nitions of the κ1...6 ) and the correlator σij (r)σkl (r′ ) is parametrised [24] by is presented elsewhere [18]. Here we study the model ?11 , ?66 whose bare values are ?11 = ρ2 hq=0 , ?66 = 0, (1,2) for which the κ1...6 are replica matrices of the form 0 while g = a2 ρ2 hq=K0 /T 2 where ρ0 is the mean density. κac = κi δ ac ? ?i with the matrix de?nitions : 0 i Note that if rf ≥ a, g can be greatly reduced (e.g. by 1 a factor exp(?c(rf /a)2 )) with respect to ?11,66 . For (c11 ? c66 )c66 c?1 ± γc66 (γ + c66 )?1 (4a) κ1,2 = 11 T out of plane disorder(such as in [8]), varying the relaT tive strength of the two types of disorder can be done by κ3,4 = (c66 + γ)?1 ± c?1 (4b) 11 4 changing the distance between the lattice and the disorκ5 = c66 c?1 ? γ(γ + c66 )?1 ; κ6 ? κ5 = 1 ? 2c66 c?1 (4c) 11 11 der plane. Besides this contribution, an underlying disorder potential induces local prefered orientation of the where the original parameters of the model (1,2) have lattice [25] : this new random ?eld (random torque) coubeen embedded in three replica matrices cac 11,66 = 1 1 ples to the local bond angle [11] θ = 1 (?x uy ? ?y ux ) : 2 c11,66 δ ac ? T ?11,66 and γ ac = γδ ac ? T ?γ . Without (2) Hs = r A(r).θ(r) with A(r)A(r′ ) = 4?γ δr,r′ . Even if ?γ coupling to a periodic lattice [25], one sets γ = 0. is null in the bare model, it will be renormalised to ?nite This VECG can be studied by the RG as in [27] by value (see (5)), and must be taken into account from the incrementing the hard-core cuto? a → ael in a three beginning since it is a new independent disorder strength, steps coarse-graining: reparametrisation of the interwhereas without dislocations [14] it can be absorbed in a action, which gives the dimension of the operators : 2 2 K0 rede?nition of ?66 (?66 → ?66 + ?γ ). ?l y = (2 ? G (κ1 ? ?1 ))y, ?l g = (2 ? 2π 2κ3 )g; fusion 2π To describe plastic distortions of the lattice one splits of electromagnetic charges separated from a ≤ d ≤ ael ; u = u0 + ud into a smooth phonon part u0 and a disannihilation of dipoles of diameter a ≤ d ≤ ael , which 1 placement ?eld ud,i (r) = 2π r′ Gij (r ? r′ )bj (r′ ) due to de?nes scale dependent interactions κ1...6 (l) [10]. Here edge dislocations of Burger vectors density b(r), where we restrict ourselves to charges of minimal fugacity: (i) charges b with single non zero replica component ba = G c11 ? c66 c66 r + Gij (r) = δij Φ(r) + ?ij G ?jk Hik (r) (a lattice vector of minimal length), Y (b, 0) = y (ii) c11 a c11 charges m with two (opposite) non zero replica compowith G(r) + iΦ(r) = ln(x + iy) where r = (x, y). The innent ma = ?mc = Kν , Y (0, m) = g [20]. We show after 1 i teraction Hik (r) is de?ned by [26] Hik (r) = rrrk ? 2 δik . rather tedious calculations [18] that (3) is renormalisable 2 Introducing n replicas and averaging over disorder using to lowest order within the form (4), needed for consisthe Villain form for the cosine coupling in (2) we obtain tency. We obtain the general RG equations, which, in [14,18] in each site replicated charges m = (m1 , ), the case of (1,2), generalize the KTHNY equations: 2

?l y = 2 ?

a2 K δ + σK 2 ? y + 2π I0 y 2 ? 8π T T2 2 p2 K 0 T K ?l g = 2 ? 2? g ? Bm (α)g 2 4π c66 4c66 3π ? 3π ? 2 ? ?l K ?1 = I0 y (2I0 ? I1 )y 2 , ?l c?1 = 66 4T T 2 ?l δ = 3πT 2 p2 K0 (I0 ? I1 ) g 2 2 3πp2 K0 T 2 ?l σ = (4c66 ? K)2 (I0 + I1 ) + K 2 I0 g 2 (4c66 K)2

(5a) (5b) (5c) (5d) (5e)

? where I0,1 = I0,1 (α) and I0,1 = I0,1 (? ) are modi?ed α Bessel function. We have de?ned K = 4T κ1 = 4T κ2 = 4(c11 ? c66 )c66 /c11 , σK 2 /2 = T 2 (?1 + ?2 ) = ?66 (1 ? 2c66 /c11 ) + ?11 (c66 /c11 )2 , δ = 4?γ , and [14] Bm (α) = a2 2π(2I0 (α/2) ? I0 (α)), α = 8π (K/T ? σK 2 /T 2 + δ/T 2 ) ? 2 and α = p2 K0 KT /(16πc2 ). We introduce an additional 66 parameter p analogous to the p-fold symmetry breaking ?eld in [28], setting K0 → pK0 in (2). The physical model corresponds to p = 1. We emphasize again that (5) should be valid at high enough T (near melting). We start by the simpler case of no pinning disorder g = 0, where further extensions of (5) at low T can also be given. The modi?cations induced by g > 0 are discussed later for T > Tm /2. g ? 0 is experimentally relevant when disorder (?11,66,γ ) varies very smoothly at the scale of the lattice. At high enough temperature (0) T > Tm /2, eqs. (5) show that the solid is stable at weak 1 2 disorder σ < 64π where σa2 = σ + δ/KR and at low tem√ 2 perature T < Tm (σ, δ) = KR a (1? 1 ? 64πσ) (note that 32π (0) ?11,66,γ are unrenormalized). At T = Tm (σ, δ) < Tm it undergoes a true KT like melting transition (see Fig.1) to a high temperature phase where dislocations proliferate. The correlation length at the transition is given 0 by ξ+ ? exp(const(T ? Tm )?ν ) where ν, computed in [18] depends continuously on σ and δ and vanishes for 1 σ = 64π . When the random shear and torque are null ?γ = ?66 = 0, one recovers Nelson’s results [13]. At 0 lower temperatures (T ≤ Tm /2), freezing of dislocations leads to modi?ed RG equations for T ≡ T /(2a2 σK) < 1. We showed [18] that it is possible to extend to the elastic model the approach of [22] for the simpler scalar model, by de?ning the appropriate dislocation fugacities y : ? ?l y = ? 2? 1 32πσ y , ?l δ = 0 ? (6a) (6b)

which exists for T ≤ Tg = p266 (2 ? K 0 /(4c0 ))?1 66 (much larger than Tm ). Due to the unbounded increase of ?11,66 ? l, disorder induced displacements grow as u ? ln r. The rest of the paper is devoted to studying the situation where both weak pinning disorder and dislocations are included, using eqs.(5). The RG ?ow shows a sharp crossover between two distincts regimes (see ?g.2) : a high temperature regime characterized by 0 a correlation length ξ+ (T ) una?ected by pinning disorder (i.e the same as given above), and a low temperature glassy regime where translational order decays beyond Ra ? Rc , though equilibrium dislocations are separated (apart from small dipoles of size ? a) by the larger length ? ξd (T ) (de?ned by ξd = ael , y(l? ) ? 1). We also study the crossover by parametrizing temperature and disorder 0 using the correlation lengths ξ± (T ) of melting in the absence of pinning disorder and the Larkin length in the 1 0 absence of dislocations Rc ? g ? 2τ (see (35) in [14]).

We now turn to the e?ect of pinning disorder. Let us ?rst recall that in situations where dislocations can be neglected (see below) one sets y = 0 in (5) and recovers the Bragg glass phase [14] g ? Bm (α) = 2(1 ? T /Tg ) ≡ 2τ




g c66 y
glassy regime liquid regime


ξ d (T)

0 0.0

g y
2.0 4.0 6.0 8.0


1e+02 Tm 1e+01 0.0003 0.0008 Temperature 0.0013

g=0 g=1e-7 g=1e-6 g=1e-5 g=1e-4 g=1e-3


FIG. 2. The length ξd /a as a function of temperature, for various pinning disorder strength g, and (rescaled) scale dependant parameters in the two regimes (inset).

?l (T K ?1 ) = C(1 ? q)?2 , ?l σ = Cq y 2 y ?

where q ≈ T is the ratio of frozen dislocations [29]. The 1 solid phase thus survives for σ R < 64π , as in [22], and the physics is dominated by rare favorable regions for frozen dislocations [18], leading to the phase diagram of Fig. 1. Positional correlations decay algebraically with exponent 2 ? +? ? ηG = G T (c?1 +c?1 + T c11 + 66c2 γ ), which leads at the 2 11 66 4π T
1 0 transition T = Tm /2 to 1 ≤ ηG ≤ 3 (for ?66 = ?γ = 0), 6 1 1 and at T = 0 to 24 ≤ ηG ≤ 6 (depending on c11 /c66 ).
11 66

In the low T glassy regime as l increases y(l) starts by dropping quickly to a negligible value (Fig.2) whereas g(l) 0 increases and reaches for L = ael > Rc ? Rc a plateau ? value corresponding to the Bragg glass ?xed point g ? de?ned above [14,30]. Within this range of scale K and c66 remain constant since y(l) ? 0. However the steady growth of σ and δ (g(l) being ?nite) eventually makes the eigenvalue of y positive. Since y(l) is then very small, it takes a large l? before it reaches ? 1 and explodes to a non perturbative value. Thus dislocations, which are strongly suppressed in the intermediate Bragg glass regime, appear only on very large length scales: to estimate ξd , we can neglect the y 2 terms in (5) (y(l) ? 1 up to ξd ). Integrating the corresponding RG eqs., we obtain 0 in the limit of weak disorder (large Rc ) 3

ξd ? Rc eb

ln Rc

with b = 2τ

λ(σc ? σ0 )


0 [31] where λ is a constant and Rc ? Rc (c66 (Rc )/c66 (0)) τ 0 is the true Larkin length. Note that Rc ? Rc in the low T regime, except very near the crossover region. In the high T liquid regime, y(l) increases quickly, causing g(l) to decrease to 0. Thus the correlation length is 0 the same as without pinning disorder ξ+ . However this regime cannot be considerer as an hexatic phase: since the renormalised ?γ is nonzero (computable from our RG), the corresponding random ?eld destroys [28] long0 range orientational order [18,32] beyond RH ? (Rc )2 /a 2 (since ?γ ? g ) leading to a liquid at large scales. Finally, the crossover between the two regimes is dominated at weak enough disorder by the g = 0 ?xed point (see ?g.1). Studying the RG ?ow around the KTNHY separatrix [11] we ?nd ξd in the crossover region [18] : 0 Rc 0 (ln Rc ) 4τ

ξd ≈


0 ln Rc 0 ln ξ±

1 ν

1 4τ


0 0 where the ± sign depends on T > Tm or T < Tm [33] and τ ≈ 0.8, ν as in [11] near pure melting (σ ≈ 0). When increasing pinning disorder (or decreasing temperature), ξd gradually goes from (8) to the low T behaviour (7). This sharp crossover will have consequences for ?nite size e?ects in 2D systems such as thin superconducting ?lms [5,6,4]. In experiments of narrow channel vortex ?ow in Nb3 Gea ?lms [5], one probes the visco-elastic response of the lattice for T < Tm on scales much smaller than ξd (see ?g.2) : the system responds as a solid with a ?nite shear modulus c66 , and one observes a “diverging” correlation length (or viscosity [5]) when approaching Tm (σ), as without pinning. In much larger system for T < Tm , the response of the lattice should in fact be ? 2 liquid-like with a large viscosity ν ? ξd (T ). In AC experiments [6], the 2D vortex lattice is probed on a length lω ? D/ω. By varying ω, a crossover at lωc ≡ Rωc was observed at low T in [6] between a low ω liquidlike behaviour and an activated glassy behaviour. The length Rωc strongly di?ers [6] from the corresponding Rc extracted from critical current experiments. We argue that the observed Rωc of [6] corresponds to the correlation length ξd (T ) de?ned here. Indeed the scaling of Rωc with the sample thickness d (i.e disorder strength) in [6] is consistent with (7). Finally, using (5) we compute the scale dependent c66 (l) and obtain the softening of c66 on scale Rc by dislocations. The self consistent de?nition of Rc given above then leads to a quantitative description of the increase of critical current (peak e?ect) in 2D ?lms [12] near melting, further studied in [18]. To conclude we extended the KTNHY analysis in presence of weak pinning disorder, predicted and analyzed a sharp crossover near pure melting. To go beyond our study in the low T region would necessitate a controlled RG method [18] to describe frozen topological defects.

[1] G. Blatter et al., Rev. Mod. Phys, 66, 1125 (1994) [2] T. Giamarchi and P. Le Doussal, Phys. Rev. B 52, 1242 (1995) [3] D. A. Huberman, and S. Doniach, Phys. Rev. Lett. 43, 950 (1979); D. S. Fisher, Phys. Rev. B 22, 1190 (1980) [4] P. Berghuis et al, Phys. Rev. Lett. 65, 2583 (1990) [5] M. H. Theunissen, E. van der Drift and P.H. Kes, Phys. Rev. Lett. 77, 159 (1996), and T.H. Theunissen, PhD [6] A. Yazdani, Stanford PhD thesis (1994) and ref. therein [7] E.Y. Andrei et al., Phys. Rev. Lett. 60, 2765 (1988) [8] C.G. Grimes and G. Adams, Phys. Rev. Lett. 42, 795 (1979); F.I.B. Williams, Hel. Phys. Act. 65, 297 (1992) [9] R. Seshadri and R.M. Westervelt, Phys. Rev. B 46, 5142 (1992); and ibid 5150 (1992). [10] J. Kosterlitz and D. Thouless, J. Phys C 6, 1181 (1973) [11] D.R. Nelson and B.I. Halperin, Phys. Rev. B 19, 2457 (1979); A.P. Young, Phys. Rev. B 19, 1855 (1979) [12] M.J. Higgins and S. Bhattacharya, Physica C 257, 232 (1992) and references therein [13] D.R. Nelson, Phys. Rev. B 27, 2902 (1983) [14] D. Carpentier and P. Le Doussal, Phys. Rev. B 55, 12128 (1997) [15] C. Carraro, D. R. Nelson, Phys. Rev. E 56, 797 (1997) [16] A. Shi and A. Berlinsky, Phys. Rev. Lett. 67, 1926 (1991) [17] see [1] section VIII.D.3 [18] D. Carpentier and P. Le Doussal, unpublished [19] At such high temperature, Rc ≈ Ra (see [2]) [20] Higher charges have higher naive dimensions, but composite charges should be included at low T [18]. [21] T. Nattermann et al., J. Phys. I (France) 5, 565 (1995); M. Cha and H.A. Fertig, Phys. Rev. Lett. 74, 4867 (1995) [22] S. Scheidl Phys. Rev. B 55, 457 (1997) [23] Around Tm the only other relevant harmonic induces minor di?erences [24] the correlator is given by formula (7) in Ref. [14] with the replacement ?11,66 of [14] by ?11,66 /T . [25] A periodic substrate (e.g. the atomic underlying lattice) induce a non zero γ (see text). See the study in [18] [26] The δij was absent in ?rst ref. of [11]: this only implies a di?erent de?nition for the charge fugacities y, g. [27] B. Nienhuis, in Phase transitions and critical phenomena, C. Domb and J.L. Leibovitz Eds., 11 (1987) [28] J. Cardy and S. Ostlund, Phys. Rev. B 25, 6899 (1982) [29] After completion, we learned of a recent unpublished study of the particular case g = 0 (no pinning disorder (6)) by P. Stahl, cond-mat/9710344 with similar results (though not identical because of di?erent RG schemes). [30] The low T regime can be studied perturbatively beyond Rc by choosing p ≈ pc such that Tg ? Tm (σ) = O(p ? pc ) ? 1, and extrapolate to p = 1 at the end. [31] This extends to triangular elastic lattices the estimate of [2] based on the simpler N = 1 component model. [32] J. Toner, Phys. Rev. Lett. 67, 1810 (1991) 0 0 0 [33] (8) is valid, e.g. for T > Tm , for Rc , ξ+ ? a, and ?xed 0 ln R 0 α = ln ξ0c (α < 1). For α > 1, ln ξd ? ln ξ+ . ? +