Surface Science 402–404 (1998) 880–885
Step–step interactions and correlations from 1D hard-core boson mapping Giuseppe Santoro a,b,*, Alessandro Laio a,b, Erio Tosatti a,b,c a International School for Advanced Studies (SISSA), Via Beirut 2, Trieste, Italy b Instituto Nazionale per la Fisica della Materia (INFM), Via Beirut 2, Trieste, Italy c International Center for Theoretical Physics (ICTP), Strada Costiera, Trieste, Italy Received 11 August 1997; accepted for publication 31 October 1997
Abstract The theory of the preroughening transition of an unreconstructed surface, and the ensuing disordered flat (DOF ) phase, is formulated in terms of steps. Finite terraces play a crucial role in the formulation. We start by mapping the statistical mechanics of interacting (up and down) steps onto the quantum mechanics of two species of one-dimensional hard-core bosons. The finite terraces are generated by a number-non-conserving term in the boson Hamiltonian, which forbids a mapping in terms of fermions. Once the boson problem is solved, we find the DOF phase is stabilized by short-range repulsions of like steps. On-site repulsion of up–down steps is essential in producing a DOF phase, whereas an off-site attraction between them is favorable but not required. Step–step correlations and terrace width distributions can be directly calculated with this method. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Bosons; Mapping; Step–step correlations; Step–step interactions
1. Introduction The preroughening (PR) transition and the ensuing disordered flat phase (DOF ), both predicted several years ago by Rommelse and den Nijs [1,2], are being intensively sought experimentally. Theoretically, they have been studied and characterized within certain restricted solid-on-solid (RSOS) models [1–10], as well as, more recently, by realistic simulations [11,12]. Both approaches, however, do not directly emphasize steps, terraces, and kinks, which are very crucial factors in this transition. It is therefore useful to explore this subject from a different, and perhaps more physically appealing, perspective. This need is made more urgent by the recent experimental evidences for PR on rare gas solid (111) surfaces [13–16 ], which calls * Corresponding author. Fax: (+39) 40 3878 528; e-mail:
[email protected] 0039-6028/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 97 ) 0 10 8 7 -X
for a detailed reinvestigation of the step–step interactions [10], or the reconstructive tendencies [5–8], crucial to obtain a DOF phase. Restricting our attention to low-index unreconstructed anisotropic surfaces, and following a line of thought initiated by Villain and Vilfan [17,18], we adopt a phenomenological approach based on working directly with the natural defects of such a surface, i.e. steps and terraces. We assume, as in Ref. [19], that the only relevant extended defects are monatomic steps, which can be either up or down, and run preferentially along a direction of stronger bonding. Steps of the same kind are forbidden to cross, while steps of different types can cross. Moreover, steps interact with each other, have kinks (involving the breaking of strong bonds, and as such energetically expensive), and can form finite terraces on the surface. These steps are then mapped onto world-lines of hard-core bosons in one dimension, a common mapping, for
G. Santoro et al. / Surface Science 402–404 (1998) 880–885
instance, in the theory of uniaxial commensurateto-incommensurate transitions of adsorbates [20]. Although this kind of mapping is exact only in the limit of a surface with infinitely strong anisotropy, it is known to provide very useful information about the phases and the nature of the transitions also for small, or even zero, anisotropy. Moreover, some realistic cases, such as (110) surfaces of fcc metals, are actually quite anisotropic. We show that with this method one can address the question of the presence or absence of a DOF phase, and the preroughening transition. Detailed consideration is given to the role of step–step interactions in stabilizing the DOF phase. We will also obtain quite directly the behavior of step–step correlations and terrace width distributions, not available so far, to our knowledge.
2. The model and its phase diagram We assume the steps to run preferentially in one direction, say y. The steps can only have nearest neighbor kinks. Steps running along x are thus excluded and the surface is, by construction, very highly anisotropic. Up and down steps are mathematically equivalent to the world lines of ‘‘spinup’’ and ‘‘spin-down’’ hard-core bosons in 1D, and the preferential direction in which the steps run plays the role of time in the quantum problem [21,22]. A hard-core condition is imposed to avoid crossing for steps of the same type, a physically justified restriction, in view of the large energetic cost of double-step regions. Let a be the destruction operator for a spin s i,s hard-core boson at site i (i.e. (a† )2=0). We write i,s our quantum Hamiltonian as follows [21,22]: ˆ H=−td ∑ (a† a +H.c.)−mN i,s i+1,s i,s −t) ∑ (a† a† +H.c.) 0 i,( i,3 i −t) ∑ (a† a† +H.c.) 1 i,s i+1,s: i,s −t ∑ a† a† a a ex i+1,s: i,s i+1,s i,s: i,s + ∑ Vd n n +V) ∑ n n j−i i,s j,s 0 i( i3 j>i,s i + ∑ V) n n . j−1 i,s j,s: j>i,s
(1)
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The ground state properties of the quantum Hamiltonian in Eq. (1) correspond to the temperature properties of a classical step model in the ˆ =N ˆ +N ˆ is the large anisotropy limit [21,22]. N ( 3 total number of particles. We work in the subspace ˆ ˆ N =N , for a low-index surface. ( For a vicinal ( 3 ˆ −N ˆ = surface of angle w, we would have N 3 ( L tan w.) Kinks on the steps, costing bd , correK spond to hopping terms in the quantum Hamiltonian with td3e−bdK. The chemical potential−m is proportional to the cost of a step per unit length, bd . The number-non-conserving S (BCS-like) terms proportional to t) describe pairs 0,1 of up–down steps which are created and annihilated to form finite terraces on the surface (see Fig. 1 of Ref. [21]). The term with t describes ex opposite step crossing events, and, together with the BCS-like terms, turns out to be crucial (see below). Pairwise interactions between the steps are taken into account by corresponding two-body terms in the quantum model, Vd,) , Vd terms are i−j generally repulsive. The sign of the V) terms, particularly at short range, depends on the microscopic details and is not specified. Our model allows, in principle, the description of a real surface and, in perspective, one could test it with realistic step–step interactions. We restrict here our considerations to the case of nearest neighbor interactions. Finite-size exact diagonalizations of the Hamiltonian in Eq. (1), and the bosonization techniques typical of 1D quantum problems have been used to unveil the richness of the phase diagram [21,22]. We summarize here a few results obtained with this mode [21,22]. (i) In the limit of V) 2 (t) 0 0 is relevant in such a case) and for a particular choice of parameters (the potentials are truncated to first neighbors and set to Vd =−V) =J , 1 1 z t =0, td=t)), the model maps exactly onto the ex 1 Heisenberg spin-1 chain Hamiltonian. The latter was also obtained, by den Nijs and Rommelse as the quantum mapping of RSOS models in a particular limit1 [1,2]. The DOF phase is the surface 1 The general quantum mapping of RSOS models leads to quartic terms in the spin-1 Hamiltonian (see Eqs. 3.3–3.7 in Sec. III B of Ref. [2]). The standard spin-1 Heisenberg model is obtained only if, among other things, direct opposite-step crossing events (corresponding to (S+ S− )2 terms) are explici i−1 itly forbidden. This eventually leads to T =2 (see Sec. roughening IV E of Ref. [2]).
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version of the Haldane phase [1,2]. (ii) The Heisenberg spin-1 restriction Vd =−V) is not 1 1 crucial in stabilizing the DOF phase. In particular, we observe a DOF phase even for V) =0. 1 Moreover, a DOF phase is present not only for V) =2 (spin-1 case) but also when V) is finite, 0 0 as long as positive. For general values of V) the 0 model maps onto a two spin-1/2 chain problem. (iii) The t term, describing the crossing of oppoex site steps, is important in order to stabilize a finite temperature rough phase. Thus, this term is crucial in order to obtain a model describing, at a coarse grained level, a physical surface. The reason why this mechanism is not crucial in BCSOS models is not clear to us. (iv) Terrace terms (i.e. t) ) are 0,1 important to describe correctly the universality classes of the relevant transitions. This is known in the literature [19], but never explored in detail in the present context. Moreover, in our case, the terrace terms also force us to work with hard-core bosons, as the standard Wigner–Jordan transformation to fermions does not lead to a simple local fermionic Hamiltonian. The point we want to discuss here is the role of the relative strength of the cost of a step versus interactions, in order to have a DOF phase. We illustrate this tissue with the help of the phase diagram in Fig. 1, obtained by taking V) =−Vd /10, while keeping t =td=t) =1 and 1 1 ex 1 V) =2. Temperature trajectories are (approxi0 mately) straight lines through the origin, which corresponds to the T=2 limit. As anticipated, the rough region near the origin owes its presence to a non-zero t . For t =0, as in the Heisenberg ex ex spin-1 case, roughening can take place only at T= 2. Depending on the choice of the parameters, we may have only roughening as T increases (A), or first PR and then roughening (B). PR (case B) can be found only if the non-vanishing step–step interactions are larger than the step energy cost per unit length d . This conclusion confirms results S obtained using RSOS models [10]. Given the fact that d is typically the largest ‘‘diagonal’’ energy S in the problem, this implies that a temperature trajectory will often be in a region where only roughening occurs. If, and how, long-range interactions might change this picture is an interesting and open problem.
Fig. 1. Qualitative diagram of the hard-core boson model for V) =2 with t) =td=t =1, V) =−Vd /10. The line separat0 1 ex 1 1 ing the rough phase (R) from the DOF/FLAT ones is in the Kosterlitz–Thouless class. The lines separating the FLAT and the reconstructed (REC.) phases from the DOF are, respectively, in the PR and the 2D-Ising classes. Only the main features of the phase diagram (crossing of lines with the axes, and the PR transition at a few points) have been studied quantitatively. Line (A) describes a situation where the cost of a step d is larger than the interactions, and only roughening is found S by increasing the temperature (i.e., moving towards the origin of the phase diagram). Line (B) describes a situation where d S is smaller than |V) |, and roughening is preceded by PR. 1
3. Step–step correlations and terrace distributions Correlation functions involving steps can be calculated numerically, for a given finite size, at any point in the phase diagram of our model. We will discuss here two correlation functions, i.e. step–step correlations and terrace width distributions. Let n S be the average density of steps of a single species (up or down). In general n is always different from S zero, even in the flat phase, since we do not discriminate between steps that traverse the entire sample and steps that form loops (i.e. finite terraces). Step–step correlations are defined as follows: N(s(r)=
1 n2 S
Step((0)Steps(r)=
1 n2 S
n n 0,( r,s (2)
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with s=(,3. If translational symmetry is not broken, we must have, at large distances, N(s(r2)1. The distribution of terrace sizes (along the x-direction only!) is the probability of having two steps a distance r apart without any other step in between. There are two different kinds of terraces we can look at: those delimited by two steps of the same type, and those between two different steps. Thus, we define P(s(r)=
1 n2 S
T C n 0,(
D U
r−1 a (1−n ) (1−n ) n j,( n,3 r,s j=1
(3) where, again, s=(,3. The string operator in square brackets enforces the absence of additional steps between 0 and r. Fig. 2 illustrates the behavior of N(( and N(3, at three different points in the phase diagram of the Heisenberg spin-1 chain: a rough case (J =−0.5, m=0, triangles), a DOF case z (J =1 and m=0, squares), and a flat one (J =1 z z and m=−2, pentagons). The flat case results are very simple: both N(( and N(3 converge exponentially fast (with a very short correlation length) to the large distance limit of 1. In the rough phase, instead, we have verified that the approach to 1 shows a power law tail. This is easy to prove. Rewrite first N(s in terms of density and ‘‘spin’’ correlations: N(s(r)=(1/4n2 ) ( n n ± Sz Sz ), S 0 r 0 r where the + and − signs apply, respectively, to s=( and s=3, n =n +n , and Sz =n −n . i i,( i,3 i i,( i,3 In the rough phase, density–density correlations are exponential, whereas Sz–Sz correlations have a uniform power-law tail of the form
Sz Sz =−K /(pr)2+,, which is precisely the 0 r S term responsible for the logarithm in the height– height correlation function G(r)= (h −h )2 r 0 [21,22]. The DOF case results, finally show a different behavior, with a sizeable oscillating component of the correlations. This behavior, however, reflects only a short-range effect, caused by the neighboring reconstructed (Ne´el ) phase: the oscillating part has to decrease to zero at large r, since no breaking of translational symmetry occurs in the DOF phase [1,2]. We finally discuss briefly the behavior of the distributions of terrace sizes (for simplicity, once again, in the Heisenberg spin-1 case). While in
Fig. 2. N(3 (above) and N(( (below), at three different points in the phase diagram of the Heisenberg spin-1 chain: a rough case (J =V)=−Vd =−0.6, D=−m=0, triangles), a DOF case z 1 (J =V)=−Vd =1, and D=−m=0, squares), and a flat one z 1 (J =V)=−Vd =1, and D=−m=2, pentagons). Lines are z 1 only guides to the eye.
principle it is important to know what is the probability for the surface to be flat over a distance r, this quantity has never been calculated so far. Let us consider, first, the behavior of P(3 in the rough phase. Fig. 3(a) is a plot of the logarithm of P(3 versus a scaled distance 2n r, for several S points taken inside the rough phase of the Heisenberg spin-1 phase diagram. We observe that the general behavior of P(r) is exponential in the size of the terrace r, and that a good collapse is obtained for all data if the distance r is scaled to the average separation between two steps, 1/(2n ) S . The scattering of the data for the largest r values is due to finite-size effects. The behavior of P(((r) is found to be qualitatively similar. Fig. 3(b)
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Second, in the DOF case P(((r) is one order of magnitude smaller than P(3(r), while the difference is much smaller in the rough case. These features are reasonable in view of the diluted antiferromagnetic ordering of steps, typical of the DOF phase. We will come back to a discussion of these results in a longer publication. In conclusion, we have found that: (i) a model based on steps can describe preroughening (PR), as well as roughening; (ii) the qualitative role of step–step interactions in driving PR, known already from RSOS models, is recovered in this picture; (iii) correlation functions involving steps can be calculated quite directly. One of them, which we have considered, is the terrace size distribution; (iv) in view of the additional simplicity of step models, it should be feasible, in the future, to study the role of long-ranged interactions in a more direct way than within RSOS models.
Acknowledgements
Fig. 3. (a) ln P(3 versus the scaled distance 2n r at various points S in the rough phase of the Heisenberg spin-1 chain. Full symbols: J =V)=−Vd =−0.25, D=−m=0, 0.1, 0.2, 0.3. Empty z 1 symbols: J =V)=−Vd =−0.5, D=−m=0, 0.2, 0.4. Stars: z 1 J =V)=−Vd =−0.75, D=−m=0, 0.1, 0.2, 0.3. (b) P(3 and z 1 P((, on a logarithmic scale, at two different points in the phase diagram of the Heisenberg spin-1 chain: a DOF case (J =V)=−Vd =1, and D=−m=0, squares), and a rough z 1 case (J =V)=−Vd =−0.5, D=−m=0, triangles). Full and z 1 empty symbols correspond to P(3, respectively.
illustrates the behavior of P(r) at a DOF point, corresponding to the isotropic Heisenberg point of the spin-1 chain, P(r) at a rough point is also reported for comparison. We observe, fist, that the behavior of P(s is, once again, exponential in r. Superimposed on the leading exponential, the DOF case results show a strong oscillating short range component which is again due to the neighboring reconstructed (Ne´el ) phase. Two more features are worth noticing. First, compared with the rough case, P(3(r) is larger in the DOF case for r=1, and then substantially smaller for larger values of r (and decreasing with a larger exponent).
We thank M. Fabrizio, A. Parola, and S. Sorella for many useful discussions. We acknowledge financial support from INFM, through Projects LOTUS and HTSC, and from EU, through ERBCHRXCT940438.
References [1] K. Rommelse, M. den Nijs, Phys. Rev. Lett. 59 (1987) 2578. [2] M. den Nijs, K. Rommelse, Phys. Rev. B 40 (1989) 4709. [3] J.A. Jaszczak, W.F. Saam, B. Yang, Phys. Rev. B 39 (1989) 9289. [4] J.A. Jaszczak, W.F. Saam, B. Yang, Phys. Rev. B 41 (1990) 6864. [5] G. Mazzeo et al., Surf. Sci. 273 (1992) 237. [6 ] G. Mazzeo et al., Europhys. Lett. 22 (1993) 39. [7] G. Santoro, M. Fabrizio, Phys. Rev. B 49 (1994) 13886. [8] G. Santoro, M. Vendruscolo, S. Prestipino, E. Tosatti, Phys. Rev. B 53 (1996) 13169. [9] P.B. Weichman, P. Day, D. Goodstein, Phys. Rev. Lett. 74 (1995) 418. [10] S. Prestipino, G. Santoro, E. Tosatti, Phys. Rev. Lett. 75 (1995) 4468. [11] S. Prestipino et al., Surf. Rev. Lett., in press. [12] F. Celestini, D. Passerone, F. Ercolessi, E. Tosatti, Surf. Sci. 402–404 (1998), this issue.
G. Santoro et al. / Surface Science 402–404 (1998) 880–885 [13] [14] [15] [16 ] [17] [18]
H.S. Youn, G.B. Hess, Phys. Rev. Lett. 64 (1990) 918. P. Day et al., Phys. Rev. B 47 (1993) 7501. P. Day et al., Phys. Rev. B 47 (1993) 10716. H.S. Youn et al., Phys. Rev. B 48 (1993) 14556. J. Villain, I. Vilfan, Europhys. Lett. 12 (1990) 523. J. Villain, I. Vilfan, Europhys. Lett. 13 (1990) 285.
885
[19] L. Balents, M. Kardar, Phys. Rev. B 46 (1992) 16031. [20] H.J. Schulz, B.I. Halperin, C.L. Henley, Phys. Rev. B 26 (1982) 3797. [21] G. Santoro et al., Surf. Sci. 377–379 (1997) 514. [22] A. Laio et al., in preparation.