Dislocation Mobility in a Quantum Crystal: the Case of Solid 4He
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Dislocation Mobility in a Quantum Crystal: the Case of Solid He Renato Pessoa, ∗ S. A. Vitiello, † and Maurice de Koning ‡ Instituto de Física Gleb Wataghin, Caixa Postal 6165,Universidade Estadual de Campinas - UNICAMP 13083-970, Campinas, SP, Brazil (Dated: October 11, 2018)
Abstract
We investigate the structure and mobility of dislocations in hcp He crystals. In addition to fullycharacterizing the five elastic constants of this system, we obtain direct insight into dislocationcore structures on the basal plane, which demonstrates a tendency toward dissociation into partialdislocations. Moreover, our results suggest that intrinsic lattice resistance is an essential factor inthe mobility of these dislocations. This insight sheds new light on the possible correlation betweendislocation mobility and the observed macroscopic behavior of crystalline He . He , the propotypical quantum solid,indirectly indicate that dislocations are indeed involved in macroscopic phenomenona. Theoccurrence of apparent superfluidity [4, 5] and the observation of elastic stiffening [6–8],for instance, have been linked to the mobility of dislocations. However, in contrast toclassical solids, for which an abundant body of experimental results exists, the lack of specificexperimental data concerning the behavior of dislocations prevents a direct investigation ofthis relationship. In view of these difficulties, one needs to resort to theoretical approaches.In principle, a realistic picture of dislocations and their properties is possible using path-integral or variational Monte Carlo simulations. However, given that dislocation modelingrequires a simultaneous treatment of different length scales [9], one associated with the coreproperties and the other associated with distances large compared to the atomic scale, theviability of these direct approaches remains a subject of debate [10–12].In this Letter, adopting a methodology that incorporates both of the relevant scales, weobtain fundamental insight into the properties of dislocations in the archetypal quantumcrystal: solid hcp He . We compute the intrinsic dislocation structure and mobility of fourdifferent dislocation types using the multiscale paradigm of the semi-discrete Peierls-Nabarro(PN) model [13–15]. It constitutes a hybrid continuum-atomistic approach that capturesthe long-range elastic fields as well as the lattice-discreteness effects associated with thedislocation core. All parameters in the model are determined using quantum-mechanicalexpectation values for He applying the shadow wave function (SWF) formalism [16, 17].In addition to providing key information concerning the elastic properties, our results shednew light onto the possible role of dislocations in the experimental observation of the elasticstiffening of solid and its potential connection to the apparent superfluidity in this quantumcrystal.Within the semi-discrete PN framework, a straight dislocation lying along the y -direction2s represented in terms of a set of misfit vectors ~δ i that describe the disregistry of atomicrow i relative to their counterparts on the other side of the glide plane, as depicted in Fig.1.Panel a) shows a schematic representation of two sets of atomic rows extending along the y -direction, one set on each side of the glide plane. The rows above the plane are labeledby the index i . Panel b) shows a top view, depicting a disregistry vector ~δ i for row i . Inthis manner, the total misfit associated with a given dislocation, described by the Burgersvector ~b, is thought of as distributed among the atomic rows along the x -axis, subject to theboundary conditions ~δ −∞ = 0 and ~δ ∞ = ~b . Here, we consider two-dimensional misfit vectors ~δ i = δ ei ˆ x + δ si ˆ y , with edge and screw components along the x and y directions, respectively.The equilibrium structure of the dislocation is then represented by that particular misfitdistribution ~δ i that minimizes the dislocation energy per unit length, U disl = U elastic + U misfit + U stress + C, (1)where U elastic = X i,j χ ij (cid:0) K e ρ ei ρ ej + K s ρ si ρ sj (cid:1) , (2) U misfit = X i γ ( ~δ i )∆ x, (3) U stress = X i (cid:2) τ e (cid:0) δ ei + δ ei − (cid:1) + τ s (cid:0) δ si + δ si − (cid:1)(cid:3) ∆ x, (4)and C is a constant that can be ignored [13]. In the elastic part of the dislocation energy,Eq. (2), χ ij is a discretized universal kernel [14], and K e = µ/ π (1 − ν ) and K s = µ/ π are elastic pre-factors with µ the shear modulus and ν the Poisson’s ratio. In addition, ρ ei ≡ (cid:0) δ ei − δ ei − (cid:1) / ∆ x and ρ si ≡ (cid:0) δ si − δ si − (cid:1) / ∆ x , where ∆ x is the distance between adjacentrows in the defect-free crystal. Eq.(3) represents the misfit contribution, in which γ ( ~δ ) isknown as the generalized stacking-fault (GSF) energy surface [18]. It describes the excessenergy per unit area of a crystal that is subjected to the following procedure. It is first cutinto two defect-free parts across a given plane. The two parts are then displaced relativeto each other by a vector ~δ , after which they are patched together again. An exampleconfiguration of the GSF on the basal plane of the hcp structure is the intrinsic stacking fault(ISF), in which the displacement vector ~δ describes the associated shift in the planar stacking.3n the context of the PN model, the GSF surface reflects the inter-atomic interactions in thesystem and serves to model the details of the dislocation core on the atomic scale. Finally,the stress term of Eq. (4) accounts for the work done by any external stresses, where τ e and τ s denote the magnitude of the components of the stress tensor that couple to the edge andscrew displacements, respectively [1]. The quantities that specify the model for dislocationsin a particular material are the elastic parameters µ and ν , the GSF surface γ ( ~δ ) associatedwith the glide plane of interest, and ∆ x .Here, we employ the SWF model based on the parameter set of Ref.[17] to determine thesequantities for solid hcp He (space group 194). In order to determine its elastic properties,we employ a computational cell containing 720 particles at a density of 0.0294 Å − , whichcorresponds to lattice parameters a =3.63668 Å and c=5.93866 Å, subject to standard pe-riodic boundary conditions. Sampling configurations according to the quantum-mechanicalprobability density of the SWF model using the Metropolis algorithm, we then computeexpectation values of the stress tensor [19] associated with the six independent deformationsof the periodic cell, imposing strain levels of 0.25%. Using the standard relationship betweenthe stress and strain tensors [20], we extract the five independent elastic constants of the hcp structure. The results, which, to the best of our knowledge represent the first completeestimate of the elastic constants in hcp He, are reported in Table I. The shear modulus µ = C = 17 . ± . MPa, is in good agreement with the value of 14 MPa that follows fromthe ratio of the experimental shear stress and strain values reported in [21]. Poisson’s ratio,obtained from the results in Table I, is found to be ν = 0 . . Both theoretical values arethose corresponding to shear directions in the basal plane, in which hexagonal crystals areisotropic [22].Since basal slip is known to be the dominant dislocation glide mechanism in hcp solid He [23] we focus on the properties of these particular dislocations and compute the GSF surfaceassociated with the basal plane. For this purpose, we utilize the 720-atom cell and impose aseries of 400 slip vectors ~δ in the basal plane by adjusting the periodic boundary conditionalong the c -axis. The shadow degrees of freedom of the atoms immediately adjacent to theslip plane are allowed to vary only along the c -direction to maintain the relative displacement.Using the Metropolis algorithm we then sample configurations according to the SWF andcompute the expectation value of the Hamiltonian as a function of ~δ . Subtracting theexpectation value at ~δ = 0 and dividing by the area, we then obtain the GSF surface.4n order to implement the results into the PN mode, we fit the results using a Fourierseries that reflects the lattice symmetry of the basal plane of the hcp structure: γ ( ~δ ) = P ~ G c ~ G exp( i ˜G · ~δ ) , in which we use a set of 81 two-dimensional reciprocal lattice vectors ˜G .The results are shown in Fig. 2. The perfect crystal configuration, which has GSF valueof zero, is associated with the displacement ~δ = 0 and its periodic equivalents. The ISFconfiguration, which corresponds to the displacement vector (and equivalents) ~δ = (0 , b p ) ,with b p = √ a = 2 . Å the length of a partial Burgers vector, has an excess energy of0.0063 mJ/ m .Using our estimates for the elastic properties and the GSF surface in the PN model, weinvestigate the structure and intrinsic mobility of 4 dislocation types on the basal plane: (i)screw, (ii) ◦ , (iii) ◦ , and (iv) edge. Fig. 3 shows the optimized disregistry profile for thescrew dislocation, obtained by minimizing Eq. (1) at zero external stress. As expected, giventhe low stacking-fault energy (SFE) value , it dissociates into two ◦ partial dislocations withopposite edge components, separated by an ISF area with a width of 29 atomic rows, whichcorresponds to 91.33 Å. The core width ς of the partials, defined as the distance over whichthe displacement changes from to of it total value[14], is approximately 1 atomic rowor ∼ ~δ i . At a critical stress value, the so-calledPeierls stress, an instability is reached and an equilibrium solution ceases to exist.[14, 15]In this situation the dislocation becomes free to move through the crystal. The third lineof Table II contains the Peierls stress values for the four considered dislocation types. Thelowest Peierls stress value, obtained for the 30 ◦ dislocation, is of the order of 1.5 × − MPa .This value is ∼ orders of magnitude smaller than the shear modulus, which is consistentwith the typical discrepancy between the ideal shear strength and actual yield stresses incrystals [1]. 5he above results were obtained using the SWF model, which, while providing a gooddescription of the shear modulus for hcp solid He , significantly underestimates a recentexperimental estimate for the SFE, (0 . ± .
02) mJ / m [24]. In order to explore the possibleinfluence of the SFE value on the dislocation mobility, we repeat the Peierls stress calcula-tions for the case in which the equilibrium dissociation widths of the four dislocation typesis reduced by a factor 10, which is the ratio between the experimental and theoretical SFEvalues. To this end, we apply an additional shear stress component, whose direction in theglide plane is perpendicular to the total Burgers vector of the dislocation. This stress com-ponent, known as Escaig stress [9], does not produce a force on the dislocation as a whole,but mimics a situation with a different SFE value. Using an Escaig stress of 0.4 MPa, werecompute the Peierls stress values for the four dislocations types in the basal plane. Theresults are shown in the fourth row of Table II. The effect of an increased effective SFE valuedoes not significantly affect the Peierls stress values of the model. This is consistent withearlier PN calculations in metals, in which the Peierls stress was not found to be sensitiveto dissociation width [14].Finally, we examine our results in the context of recent experiments considering themacroscopic behavior of solid He at low temperatures. In the observation[4, 5] of non-classical rotational inertia (NCRI), interpreted as a signature of superfluidity, the crucialrole of crystal defects and disorder seems firmly established. Specifically, the behavior ofdislocations has attracted a particular interest after the discovery of an unexpected increaseof the shear modulus that shows the same temperature and He impurity concentration de-pendence as the original NCRI observations [6–8]. Inspired by the continuum-elasticity basedGranato-Lücke theory [25], it has been hypothesized that this stiffening is a consequence ofa change of mobility of a network of dislocations. This network is thought to be pinned by He impurities at lowest temperatures, while it becomes mobile under warmer conditions.Analyzing the dislocation mobility results of our model, it is interesting to observe that ourlowest Peierls barrier is about 20 times larger than the shear stresses of ∼ Pa reachedin recent experiments [7]. This suggests that intrinsic lattice resistance is an essential factorwhen it comes to the mobility of dislocations on the basal plane in hcp solid He . Indeed,at the stress levels reported in these recent experiments, such dislocations would not beexpected to be mobile, not even in the absence of any pinning centers. Moreover, it is notexpected that the Peierls stress varies significantly as a function of temperature below 0.1 K,6t which the stiffening is observed, given that finite temperature path-integral Monte Carlocalculations of several properties do not show a significant temperature dependence below 1K [26]. In this context, a satisfactory explanation for the observed elastic stiffening in hcp solid He must involve intrinsic mobility issues.In summary, we employ a hybrid continuum-atomistic approach, based on the Peierls-Nabarro model and the shadow wave function formalism, to obtain direct insight into theintrinsic structural and mobility properties of dislocations in hcp solid He at zero temper-ature. In addition to providing key information concerning the elastic properties of thisprototypical quantum crystal, the results reveal a significant lattice resistance to dislocationmotion. Analyzing our results in the context of the proposed dislocation-pinning interpre-tation of the similarity between the NCRI and elastic stiffening phenomena suggests thatintrinsic lattice resistance is an essential factor when it comes to the mobility of dislocations.The proposed interpretation, which entirely ignores this element, may therefore not providea satisfactory explanation for the observed elastic stiffening in hcp solid He .The authors acknowledge financial support from the Brazilian agencies FAPESP, CNPqand CAPES. Part of the computations were performed at the CENAPAD high-performancecomputing facility at Universidade Estadual de Campinas. ∗ Electronic address: rpessoa@ifi.unicamp.br † Electronic address: [email protected] ‡ Electronic address: dekoning@ifi.unicamp.br[1] J. P. Hirth and J. Lothe,
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40 −30 −20 −10 0 10 20 30 40−0.2−0.40.00.20.40.60.81.0
Atomic row D i s p l a c e m e n t ( b ) ISF
Figure 3: Optimized displacement profile ~δ i for screw dislocation on basal plane: edge component(open circles) and screw component (filled squares) are measured in units of the Burgers vector b = a = 3 . Å. Dashed lines indicate positions of partial dislocations.Table II: Core structure and Peierls Stress for four dislocations on the basal plane for the SWFmodel for solid hcp He . Screw ◦ ◦ EdgePartial separation (Å) 91.3 109.1 119.7 127.3Peierls Stress (MPa) 0.41 ± ± ± ± . Peierls Stress corrected SFE (MPa) 0.48 ± . ± . ± . ± .01