- Email: [email protected]
Molecular-dynamics studies of charge complexes in liquid helium
Journal of Molecular Structure (Theochem) 464 (1999) 87–93
Molecular-dynamics studies of charge complexes in liquid helium Paulo S. Branı´cio*, Jose´-Pedro Rino, Nelson Studart Departamento de Fı´sica, Universidade Federal de Sa˜o Carlos 13565-905 Sa˜o Carlos, Sa˜o Paulo, Brazil
Abstract Molecular-dynamics simulations were employed to describe the solid–liquid phase transition of charge complexes in helium films. These two-dimensional atomic-like states, called diplons, arise from the coupling between surface electrons on helium film and positive ions localized in the helium film–substrate interface. The effects of the distance between the helium film and the substrate on the structural properties of the diplon system are investigated. The melting temperature was determined by analyzing quantities like the total energy per particle, the self-diffusion coefficient, and the pair correlation function. 䉷 1999 Elsevier Science B.V. All rights reserved. Keywords: Molecular-dynamics; Phase transitions; Charge systems PACS Numbers: 64.70.Dv; 73.20.Dx; 61.20Ja
1. Introduction The study of electronic two-dimensional (2D) systems is not new, but is still of considerable interest. The possibility of experimental realization of a 2D system brings valuable evidence in order to test theories for melting in two dimensions [1]. One of the first experimental realizations of a 2D charge system was the trapping of electrons on the surface of liquid helium and other cryogenic substrates. Physically the electrons are subject to the Coulomb attraction as a result of its image charge in helium, and a short-range potential barrier coming from the Pauli exclusion principle. The effect of these two interactions over the surface electrons results in a quantum well in the direction normal to the liquid helium while the electrons are free to move in the direction parallel to the surface, hence forming a 2D electron gas. The movement of these surface electrons are affected by the scattering from helium atoms in the vapor phases * Corresponding author.
at high temperatures, and by ripplons (elementary excitations of the helium surface) at low temperatures (below 0.7 K). The electronic density over liquid helium ranges from 10 5 –10 8 cm ⫺2 which corresponds to a Fermi energy much smaller then kBT, which means that the system form a nondegenerate 2D plasma. A more interesting 2D system consists of electrons deposited over a thin film of liquid helium lying on a substrate. In this case, the interparticle interaction depends on the thickness of the film and on the characteristics of the substrate. It can be easily shown that the electron–electron interaction potential can vary from the bare Coulomb interaction (1/r) through dipolar interaction (1/r 3) depending of the film thickness and substrate. [2–4] Several interesting results can be found related to 2D electron system [5]. From the experimental evidence that ions can be injected in liquid helium [6], Monarkha and Kovdrya [7,8] proposed the existence of a system where surface electrons over film of liquid helium are trapped by the Coulomb force as a result of ions injected in the interface between the helium film and the substrate. These
0166-1280/99/$ - see front matter 䉷 1999 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(98)00538-7
88
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
simulation method is used to study the structural and thermodynamic properties of the system of interacting diplons, i.e. tightly bound electron–ion pairs moving with an effective mass m and interacting with a well-defined interparticle potential. Fig. 1 shows schematically the system of diplons in liquid helium. The effect of parameters, such as the film thickness and the dielectric constant of the substrate, in the liquid–solid phase transition is analyzed. Fig. 1. Schematic view of the geometry of the diplon system.
charge complexes, formed by ions and surface electrons, constitute an idealized 2D atomic-like state, which was labeled a diplon. For an isolated diplon, the electron is bound to a parabolic well U r ⫺e2 =1 d ⫹ 1=2kr 2 ; where 1 1 ⫹ 1=2; 1 is the substrate dielectric constant, d is the film thickness and k e2 =1 d3 . The system is stable and the minimum film thickness above which the ions cannot be detached from the substrate, in the presence of the ˚, electron field, is estimated as dmin 20–70 A depending on the substrate. The phase diagram, the ground-state energy, and the phonon spectrum have been evaluated recently within the harmonic approximation and the dislocation-mediated melting theory. [9] Experimental observation of diplons should be provided by microwave absorption of the electrons [10]. In this paper, the molecular-dynamics (MD)
2. Interaction potential The diplon–diplon interaction potential can be obtained by solving the Poisson equation with appropriate boundary conditions. If we assume the dielectric constant for the helium liquid as one, the interaction potential reads as [7,8,11] ! 4e2 1 1 V r ⫺ 1=2 1s ⫹ 1 r r2 ⫹ d2 ! ÿ e 2 1s ⫺ 1 1 1 ⫺ ⫹ 1=2 1s ⫹ 1 r r2 ⫹ 2d2 where r is the distance between diplons and d is the distance between the electron layer and the ion layer. For small interparticle distances (r Ⰶ d), we obtain essentially the Coulomb potential V(r) e* 2/r for particles confined in a plane with a renormalized
Fig. 2. Total energy per particle as a function of the temperature for the diplon system with density 1.477 × 10 8 cm ⫺2, es 17 and d 500. The vertical lines mark the hysteresis region.
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
89
Fig. 3. Comparison of the total energies for different film thicknesses, higher diplon density 1.3 × 10 10 cm ⫺2 and es 7.3.
ÿ ÿ charge e* e 1s ⫹ 3 = 1s ⫹ 1 1=2 . If the diplons are far apart, r Ⰷ d, (low-density regime) one has an asymptotic dipolar interaction V r 2e 02 d2 =r3 , ÿ 0 where e 2 1s = 1s ⫹ 1 e. Note that in contrast to the system of electrons on helium films [2] in which the dipolar-like regime is achieved only for a metallic
substrate, the diplon system is described by a dipolar interaction irrespective of the substrate. Molecular-dynamics simulations were performed for a system consisting of 2500 diplons in a rectangular box at two densities, 1.477 × 10 8 and 1.3 × 10 10 cm ⫺2. The diplon mass was taken as
90
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
Fig. 4. Difusion constant versus temperature for the same parameters of Fig. 2 used as a criterion to determine the solid–liquid transition.
m 3.819 × 10 ⫺23 g (corresponding to the Na ⫹ ion). In order to accommodate a triangular lattice the sides p of the MD box have a ratio 3=2. Periodic boundary conditions were used and the Ewald’s method was applied to handle the long-range terms of the interaction. As the simulation box is rectangular more reciprocal lattice vectors were used in the largest side of the box up to a maximum vector which depends on the value of the first peak of the correlation functions and on the convergence of the Ewald’s sum. The equations of motion were integrated using a fifth-order predictor–corrector method with a time step of
10 ⫺9s, which led to a conservation of the total energy of 1 part in 10 4 after several thousands time steps. The equilibrium was monitored through temperature fluctuations, where we have used the criterion that the equilibrium was reached if the fluctuations were less then 5%. After attain the equilibrium, time averages of the physical quantities were taken over additional 40 000 time steps. The heating and cooling procedure was done in cascade, i.e., the equilibrium configuration for a given temperature was used as input to obtain a new configuration at higher temperature. The two asymptotic limits observed in this system, the one-component electron plasma when d ! ∞, and the bare dipolar system when d Ⰶ r0, where r0 (pn) ⫺ 1/2 is the mean distance between the diplons and n is the diplon density, were used to check our results and they were reproduced for the internal energy and the value of the plasma parameter at the melting transition.
3. Results and discussion
Fig. 5. Melting temperature as a function of the film thickness for different substrate dielectric constants. The lines are guide to the eyes.
The solid–liquid phase diagram was determined from the calculation of the total energy per particle and the diffusion coefficient as a function of temperature for several film thicknesses and different substrates. The total energy as a function of temperature is displayed in Fig. 2 for a system of 2500 diplons, film thickness d 500, and substrate dielectric constant 1 s 17. The density is 1.477 ×
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
91
Fig. 6. Melting temperature as a function of the substrate dielectric constant for several film thicknesses. The lines are guide to the eyes.
10 8 cm ⫺2. The circles represent states in the solid phase while the crosses correspond to the liquid phase. Special has been paid to the region of hysteresis delimited by vertical dashed lines in Fig. 2 which indicates a first-order transition from a solid to a liquid. The points L1 and S2 correspond to the supercooled liquid and the superheated solid states. While typical times for homogeneous nucleation in the 2D electron system are about 10 000 time steps, the nucleation time for the diplon system was taken about 2000 time steps owing to the large mass of the diplon. So in order to be sure that L1 is indeed a liquid and S2 is a solid and to check the structure of other states as a function of temperature in the metastable region, the system was allowed to relax for additional 20 000 time steps before the averages were performed. So we feel confident that the circles do indeed represent the solid phase and the crosses the liquid phase. We have also investigated the effect of the density and the substrate on the total energy per particle. We show in Fig. 3 the total energy for different thicknesses, a diplon density 1.3 × 10 10 cm ⫺2, and for 1 s 7.3 (glass substrate). By increasing the density, we observe that the hysteresis region shifts to higher temperatures as an effect of the increase of the interaction energy. We still observe in Fig. 3 that the melting temperature increases with increasing the film thickness, as a consequence of the decrease of screening effects owing to the substrate. The behavior of the system in the hysteresis region can be still
inferred from the temperature dependence of the self-diffusion coefficient D for the diplon system, which is shown in Fig. 4 for the same simulation as Fig. 2. We see that D is zero for all points in the solid phase. In the liquid phase, D has nonzero increasing values increasing with increasing temperature. So, we confirm that L1 is a supercooled liquid and S2 is a superheated solid. The energy difference between the liquid and solid phases is the latent heat of melting, and taking Tm to be in the middle of the hysteresis region we estimate the entropy of melting to be approximately 0.3kB, which is the same as that found in other 2D systems. [12] From these results, we conclude that (i) L1 and S2 are supercooled liquid and superheated solid states, respectively, (ii) the system shows hysteresis, and (iii) it melts releasing latent heat. As a consequence, the system undergoes a first-order phase transition. The melting temperature Tm was determined for several kinds of substrate and film thicknesses. In Fig. 5 we display the dependence of the melting temperature with the film thickness and in Fig. 6 its dependence with the dielectric constant of the substrate. At fixed density, the melting temperature decreases as the film thickness decreases for a given substrate. However, Tm decreases when 1 s increases, for a given d. This is a direct consequence of the increase of screening effects which should come from the substrate (by increasing 1 s 10 7 (in reality the dielectric constant of a metal is infinity but in the simulations 1 s 10 7 is a very good approximation).
92
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
Fig. 7. Bond angle distribution f (u ), pair correlation function g(r) and static structure factor S(k) of the superheated solid (on the left) and of the supercooled liquid (on the right) for u s 17 and d 500.
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
We also observed that for d ⬎ ⬎ 5000, the melting temperature becomes independent of the film thickness. Note also that we recovered the melting temperature (Tm 0.28 K) for 2D electrons on a helium film deposited over a metallic substrate [12]. The structural correlations were analyzed through the angle distributions f(u ), the pair correlation function g(r) and the static structure factor S(k). The hexagonal structure is clear from all the results. The correlation functions of the state corresponding to the superheated solid and the supercooled liquid state, labeled by S2 and L1 in Fig. 1, respectively are shown in Fig. 7. Fig. 7a and b represent the angles distributions which are defined as the average angle among three nearest neighbor diplons. Bond-angle distributions are obtained from MD trajectories as follows. First, a list of all X nearest-neighbor particles around an fixed particle A is constructed. From this list, the angles ⬔X–A–X are calculated for all bonds and a histogram is made from an average over all angles involving all A particles. These distributions clearly exhibit the local structure. In the solid phase (T 0.0171 K) the bond-angle distribution is peaked at 60⬚, 120⬚ and 180⬚ with and error of ^ 0.5⬚, which correspond to the hexagonal symmetry. For lower temperatures, the peaks become sharper. In the liquid phase (T 0.0155 K) the peaks remain at the same position, but become broader owing to thermal effects. In Fig. 7c and d the pair correlation function is displayed and numbers on it indicate the number of particles in successive shells. The first peak at r (1.80 ^ 0.05)r0 with an average coordination number equals to 6, both in the solid and liquid phases, indicates clearly the hexagonal structure. The peaks are sharper in the superheated solid even though its temperature is higher than that of the supercooled liquid. Furthermore, g(r) of the superheated solid shows a small shoulder just beyond the third peak, whereas it is smooth in the supercooled liquid. Finally, in Fig. 7e and f, we show the static structure factor at those temperatures. As expected, the first peak in S(k) appears at k close to the smallest reciprocal lattice vector of the hexagonal lattice, whereas in the normal liquid phase, the peak is broader owing to thermal effects. From these results we can see that,
93
even though there is no long-range order in the liquid phase, a well-defined local order is observed with a mean distance between diplons equal to (1.80 ^ 0.05)r0 and the coordination number around 6. In conclusion, we have studied the solid–liquid transition of the system of diplons for different film thicknesses and substrates. Our MD results have indicated that the phase transition for the system is always first order, as in other 2D charge classical systems. We have obtained the dependence of the melting temperature with the dielectric constant of the substrate as well as with the He film thickness. We have also been able to observe the thermodynamical properties of systems with bare Coulomb interactions and dipolar interaction from our system by only changing external parameters. Acknowledgements This work was partially sponsored by the Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq) and the FundaC¸a˜o de Amparo a` Pesquisa do Estado de Sao Paulo (FAPESP). References [1] K.J. Strandburg, Rev. Mod. Phys. 60 (1988) 161. [2] J.-P. Rino, N. Studart, O. Hipo´lito, Phys. Rev. B 29 (1984) 2584. [3] F.M. Peeters, Phys. Rev. B 30 (1984) 2457. [4] P.M. Platzman, F.M. Peeters, Phys. Rev. Lett. 50 (1983) 2021. [5] For a detailed account of the literature see Two-Dimensional Electron Systems on Helium and Other Substrates, E.Y. Andrei (Ed.) Netherlands: Kluwer, 1997. [6] B. Maraviglia, Phys. Lett. 25A (1967) 99. [7] Yu. P. Monarkha, Yu. Z. Kovdrya, Fiz. Nizk. Temp. (8) (1982) 215 (Sov. J. Low Temp. Phys. 8 (1982) 107). [8] Yu. P. Monarkha, Fiz. Nizk. Temp. 8 (1982) 1113 (Sov. J. Low Temp. Phys. 8 (1982) 571). [9] L. Caˆndido, J.-P. Rino, N. Studart, Phys. Rev. B (1998) (to appear). [10] A.J. Dahm, Z. Phys. B 98 (1995) 333. [11] U. de Freitas, J.-P. Rino, N. Studart, in: G. Castro, M. Cardona (Eds.), Lectures on Surface Physics, Springer, Berlin, 1987 p. 177. [12] L. Caˆndido, J.-P. Rino, N. Studart, Phys. Rev. B 54 (1996) 7046.
Molecular-dynamics studies of charge complexes in liquid helium Paulo S. Branı´cio*, Jose´-Pedro Rino, Nelson Studart Departamento de Fı´sica, Universidade Federal de Sa˜o Carlos 13565-905 Sa˜o Carlos, Sa˜o Paulo, Brazil
Abstract Molecular-dynamics simulations were employed to describe the solid–liquid phase transition of charge complexes in helium films. These two-dimensional atomic-like states, called diplons, arise from the coupling between surface electrons on helium film and positive ions localized in the helium film–substrate interface. The effects of the distance between the helium film and the substrate on the structural properties of the diplon system are investigated. The melting temperature was determined by analyzing quantities like the total energy per particle, the self-diffusion coefficient, and the pair correlation function. 䉷 1999 Elsevier Science B.V. All rights reserved. Keywords: Molecular-dynamics; Phase transitions; Charge systems PACS Numbers: 64.70.Dv; 73.20.Dx; 61.20Ja
1. Introduction The study of electronic two-dimensional (2D) systems is not new, but is still of considerable interest. The possibility of experimental realization of a 2D system brings valuable evidence in order to test theories for melting in two dimensions [1]. One of the first experimental realizations of a 2D charge system was the trapping of electrons on the surface of liquid helium and other cryogenic substrates. Physically the electrons are subject to the Coulomb attraction as a result of its image charge in helium, and a short-range potential barrier coming from the Pauli exclusion principle. The effect of these two interactions over the surface electrons results in a quantum well in the direction normal to the liquid helium while the electrons are free to move in the direction parallel to the surface, hence forming a 2D electron gas. The movement of these surface electrons are affected by the scattering from helium atoms in the vapor phases * Corresponding author.
at high temperatures, and by ripplons (elementary excitations of the helium surface) at low temperatures (below 0.7 K). The electronic density over liquid helium ranges from 10 5 –10 8 cm ⫺2 which corresponds to a Fermi energy much smaller then kBT, which means that the system form a nondegenerate 2D plasma. A more interesting 2D system consists of electrons deposited over a thin film of liquid helium lying on a substrate. In this case, the interparticle interaction depends on the thickness of the film and on the characteristics of the substrate. It can be easily shown that the electron–electron interaction potential can vary from the bare Coulomb interaction (1/r) through dipolar interaction (1/r 3) depending of the film thickness and substrate. [2–4] Several interesting results can be found related to 2D electron system [5]. From the experimental evidence that ions can be injected in liquid helium [6], Monarkha and Kovdrya [7,8] proposed the existence of a system where surface electrons over film of liquid helium are trapped by the Coulomb force as a result of ions injected in the interface between the helium film and the substrate. These
0166-1280/99/$ - see front matter 䉷 1999 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(98)00538-7
88
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
simulation method is used to study the structural and thermodynamic properties of the system of interacting diplons, i.e. tightly bound electron–ion pairs moving with an effective mass m and interacting with a well-defined interparticle potential. Fig. 1 shows schematically the system of diplons in liquid helium. The effect of parameters, such as the film thickness and the dielectric constant of the substrate, in the liquid–solid phase transition is analyzed. Fig. 1. Schematic view of the geometry of the diplon system.
charge complexes, formed by ions and surface electrons, constitute an idealized 2D atomic-like state, which was labeled a diplon. For an isolated diplon, the electron is bound to a parabolic well U r ⫺e2 =1 d ⫹ 1=2kr 2 ; where 1 1 ⫹ 1=2; 1 is the substrate dielectric constant, d is the film thickness and k e2 =1 d3 . The system is stable and the minimum film thickness above which the ions cannot be detached from the substrate, in the presence of the ˚, electron field, is estimated as dmin 20–70 A depending on the substrate. The phase diagram, the ground-state energy, and the phonon spectrum have been evaluated recently within the harmonic approximation and the dislocation-mediated melting theory. [9] Experimental observation of diplons should be provided by microwave absorption of the electrons [10]. In this paper, the molecular-dynamics (MD)
2. Interaction potential The diplon–diplon interaction potential can be obtained by solving the Poisson equation with appropriate boundary conditions. If we assume the dielectric constant for the helium liquid as one, the interaction potential reads as [7,8,11] ! 4e2 1 1 V r ⫺ 1=2 1s ⫹ 1 r r2 ⫹ d2 ! ÿ e 2 1s ⫺ 1 1 1 ⫺ ⫹ 1=2 1s ⫹ 1 r r2 ⫹ 2d2 where r is the distance between diplons and d is the distance between the electron layer and the ion layer. For small interparticle distances (r Ⰶ d), we obtain essentially the Coulomb potential V(r) e* 2/r for particles confined in a plane with a renormalized
Fig. 2. Total energy per particle as a function of the temperature for the diplon system with density 1.477 × 10 8 cm ⫺2, es 17 and d 500. The vertical lines mark the hysteresis region.
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
89
Fig. 3. Comparison of the total energies for different film thicknesses, higher diplon density 1.3 × 10 10 cm ⫺2 and es 7.3.
ÿ ÿ charge e* e 1s ⫹ 3 = 1s ⫹ 1 1=2 . If the diplons are far apart, r Ⰷ d, (low-density regime) one has an asymptotic dipolar interaction V r 2e 02 d2 =r3 , ÿ 0 where e 2 1s = 1s ⫹ 1 e. Note that in contrast to the system of electrons on helium films [2] in which the dipolar-like regime is achieved only for a metallic
substrate, the diplon system is described by a dipolar interaction irrespective of the substrate. Molecular-dynamics simulations were performed for a system consisting of 2500 diplons in a rectangular box at two densities, 1.477 × 10 8 and 1.3 × 10 10 cm ⫺2. The diplon mass was taken as
90
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
Fig. 4. Difusion constant versus temperature for the same parameters of Fig. 2 used as a criterion to determine the solid–liquid transition.
m 3.819 × 10 ⫺23 g (corresponding to the Na ⫹ ion). In order to accommodate a triangular lattice the sides p of the MD box have a ratio 3=2. Periodic boundary conditions were used and the Ewald’s method was applied to handle the long-range terms of the interaction. As the simulation box is rectangular more reciprocal lattice vectors were used in the largest side of the box up to a maximum vector which depends on the value of the first peak of the correlation functions and on the convergence of the Ewald’s sum. The equations of motion were integrated using a fifth-order predictor–corrector method with a time step of
10 ⫺9s, which led to a conservation of the total energy of 1 part in 10 4 after several thousands time steps. The equilibrium was monitored through temperature fluctuations, where we have used the criterion that the equilibrium was reached if the fluctuations were less then 5%. After attain the equilibrium, time averages of the physical quantities were taken over additional 40 000 time steps. The heating and cooling procedure was done in cascade, i.e., the equilibrium configuration for a given temperature was used as input to obtain a new configuration at higher temperature. The two asymptotic limits observed in this system, the one-component electron plasma when d ! ∞, and the bare dipolar system when d Ⰶ r0, where r0 (pn) ⫺ 1/2 is the mean distance between the diplons and n is the diplon density, were used to check our results and they were reproduced for the internal energy and the value of the plasma parameter at the melting transition.
3. Results and discussion
Fig. 5. Melting temperature as a function of the film thickness for different substrate dielectric constants. The lines are guide to the eyes.
The solid–liquid phase diagram was determined from the calculation of the total energy per particle and the diffusion coefficient as a function of temperature for several film thicknesses and different substrates. The total energy as a function of temperature is displayed in Fig. 2 for a system of 2500 diplons, film thickness d 500, and substrate dielectric constant 1 s 17. The density is 1.477 ×
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
91
Fig. 6. Melting temperature as a function of the substrate dielectric constant for several film thicknesses. The lines are guide to the eyes.
10 8 cm ⫺2. The circles represent states in the solid phase while the crosses correspond to the liquid phase. Special has been paid to the region of hysteresis delimited by vertical dashed lines in Fig. 2 which indicates a first-order transition from a solid to a liquid. The points L1 and S2 correspond to the supercooled liquid and the superheated solid states. While typical times for homogeneous nucleation in the 2D electron system are about 10 000 time steps, the nucleation time for the diplon system was taken about 2000 time steps owing to the large mass of the diplon. So in order to be sure that L1 is indeed a liquid and S2 is a solid and to check the structure of other states as a function of temperature in the metastable region, the system was allowed to relax for additional 20 000 time steps before the averages were performed. So we feel confident that the circles do indeed represent the solid phase and the crosses the liquid phase. We have also investigated the effect of the density and the substrate on the total energy per particle. We show in Fig. 3 the total energy for different thicknesses, a diplon density 1.3 × 10 10 cm ⫺2, and for 1 s 7.3 (glass substrate). By increasing the density, we observe that the hysteresis region shifts to higher temperatures as an effect of the increase of the interaction energy. We still observe in Fig. 3 that the melting temperature increases with increasing the film thickness, as a consequence of the decrease of screening effects owing to the substrate. The behavior of the system in the hysteresis region can be still
inferred from the temperature dependence of the self-diffusion coefficient D for the diplon system, which is shown in Fig. 4 for the same simulation as Fig. 2. We see that D is zero for all points in the solid phase. In the liquid phase, D has nonzero increasing values increasing with increasing temperature. So, we confirm that L1 is a supercooled liquid and S2 is a superheated solid. The energy difference between the liquid and solid phases is the latent heat of melting, and taking Tm to be in the middle of the hysteresis region we estimate the entropy of melting to be approximately 0.3kB, which is the same as that found in other 2D systems. [12] From these results, we conclude that (i) L1 and S2 are supercooled liquid and superheated solid states, respectively, (ii) the system shows hysteresis, and (iii) it melts releasing latent heat. As a consequence, the system undergoes a first-order phase transition. The melting temperature Tm was determined for several kinds of substrate and film thicknesses. In Fig. 5 we display the dependence of the melting temperature with the film thickness and in Fig. 6 its dependence with the dielectric constant of the substrate. At fixed density, the melting temperature decreases as the film thickness decreases for a given substrate. However, Tm decreases when 1 s increases, for a given d. This is a direct consequence of the increase of screening effects which should come from the substrate (by increasing 1 s 10 7 (in reality the dielectric constant of a metal is infinity but in the simulations 1 s 10 7 is a very good approximation).
92
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
Fig. 7. Bond angle distribution f (u ), pair correlation function g(r) and static structure factor S(k) of the superheated solid (on the left) and of the supercooled liquid (on the right) for u s 17 and d 500.
P.S. Branı´cio et al. / Journal of Molecular Structure (Theochem) 464 (1999) 87–93
We also observed that for d ⬎ ⬎ 5000, the melting temperature becomes independent of the film thickness. Note also that we recovered the melting temperature (Tm 0.28 K) for 2D electrons on a helium film deposited over a metallic substrate [12]. The structural correlations were analyzed through the angle distributions f(u ), the pair correlation function g(r) and the static structure factor S(k). The hexagonal structure is clear from all the results. The correlation functions of the state corresponding to the superheated solid and the supercooled liquid state, labeled by S2 and L1 in Fig. 1, respectively are shown in Fig. 7. Fig. 7a and b represent the angles distributions which are defined as the average angle among three nearest neighbor diplons. Bond-angle distributions are obtained from MD trajectories as follows. First, a list of all X nearest-neighbor particles around an fixed particle A is constructed. From this list, the angles ⬔X–A–X are calculated for all bonds and a histogram is made from an average over all angles involving all A particles. These distributions clearly exhibit the local structure. In the solid phase (T 0.0171 K) the bond-angle distribution is peaked at 60⬚, 120⬚ and 180⬚ with and error of ^ 0.5⬚, which correspond to the hexagonal symmetry. For lower temperatures, the peaks become sharper. In the liquid phase (T 0.0155 K) the peaks remain at the same position, but become broader owing to thermal effects. In Fig. 7c and d the pair correlation function is displayed and numbers on it indicate the number of particles in successive shells. The first peak at r (1.80 ^ 0.05)r0 with an average coordination number equals to 6, both in the solid and liquid phases, indicates clearly the hexagonal structure. The peaks are sharper in the superheated solid even though its temperature is higher than that of the supercooled liquid. Furthermore, g(r) of the superheated solid shows a small shoulder just beyond the third peak, whereas it is smooth in the supercooled liquid. Finally, in Fig. 7e and f, we show the static structure factor at those temperatures. As expected, the first peak in S(k) appears at k close to the smallest reciprocal lattice vector of the hexagonal lattice, whereas in the normal liquid phase, the peak is broader owing to thermal effects. From these results we can see that,
93
even though there is no long-range order in the liquid phase, a well-defined local order is observed with a mean distance between diplons equal to (1.80 ^ 0.05)r0 and the coordination number around 6. In conclusion, we have studied the solid–liquid transition of the system of diplons for different film thicknesses and substrates. Our MD results have indicated that the phase transition for the system is always first order, as in other 2D charge classical systems. We have obtained the dependence of the melting temperature with the dielectric constant of the substrate as well as with the He film thickness. We have also been able to observe the thermodynamical properties of systems with bare Coulomb interactions and dipolar interaction from our system by only changing external parameters. Acknowledgements This work was partially sponsored by the Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq) and the FundaC¸a˜o de Amparo a` Pesquisa do Estado de Sao Paulo (FAPESP). References [1] K.J. Strandburg, Rev. Mod. Phys. 60 (1988) 161. [2] J.-P. Rino, N. Studart, O. Hipo´lito, Phys. Rev. B 29 (1984) 2584. [3] F.M. Peeters, Phys. Rev. B 30 (1984) 2457. [4] P.M. Platzman, F.M. Peeters, Phys. Rev. Lett. 50 (1983) 2021. [5] For a detailed account of the literature see Two-Dimensional Electron Systems on Helium and Other Substrates, E.Y. Andrei (Ed.) Netherlands: Kluwer, 1997. [6] B. Maraviglia, Phys. Lett. 25A (1967) 99. [7] Yu. P. Monarkha, Yu. Z. Kovdrya, Fiz. Nizk. Temp. (8) (1982) 215 (Sov. J. Low Temp. Phys. 8 (1982) 107). [8] Yu. P. Monarkha, Fiz. Nizk. Temp. 8 (1982) 1113 (Sov. J. Low Temp. Phys. 8 (1982) 571). [9] L. Caˆndido, J.-P. Rino, N. Studart, Phys. Rev. B (1998) (to appear). [10] A.J. Dahm, Z. Phys. B 98 (1995) 333. [11] U. de Freitas, J.-P. Rino, N. Studart, in: G. Castro, M. Cardona (Eds.), Lectures on Surface Physics, Springer, Berlin, 1987 p. 177. [12] L. Caˆndido, J.-P. Rino, N. Studart, Phys. Rev. B 54 (1996) 7046.