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Driven quasi-one-dimensional classical electron gas in the presence of a constriction: Pinning and depinning
ARTICLE IN PRESS
Physica E 34 (2006) 224–227 www.elsevier.com/locate/physe
Driven quasi-one-dimensional classical electron gas in the presence of a constriction: Pinning and depinning G. Piacente, F.M. Peeters Department of Physics, University of Antwerpen (Campus Middleheim), Groenenborgerlaan 171, B-2020 Antwerpen, Belgium Available online 30 June 2006
Abstract We studied the properties of a quasi-one-dimensional system of charged particles in the presence of a local Lorentzianshaped constriction. We investigated the response of the system when a time-independent external driving force is applied in the unconfined direction. Langevin molecular dynamics simulations for different values of the drive and temperature are performed. We found that the particles are pinned unless a threshold value of the driving force is reached. We investigated in detail the depinning phenomenon. The system can depin ‘‘elastically’’, with particles moving together and keeping their neighbors, or ‘‘quasi-elastically’’, with particles moving together through a complex net of conducting channels without keeping their neighbors. In the case of elastic depinning the velocity vs applied drive curves is characterized by a critical exponent b consistent with the value 23, while in the case of quasi-elastic depinning the critical exponent b has on average the value 0.94. The model is relevant e.g. for electrons on liquid helium, colloids and dusty plasma. r 2006 Published by Elsevier B.V. PACS: 71.10.w; 52.65.Yy; 45.50.Jf Keywords: Electron gas; Pinning; Depinning; Critical exponents
1. Introduction Interacting systems of charged particles, which tend to form periodic or ordered structures when the density and the temperature are low enough (i.e. Wigner crystallization), can exhibit a remarkable variety of complex phenomena when they are driven by an external force. Many of these phenomena are still poorly understood or even unexplored in mesoscopic confined systems. Transport experiments are a useful way to probe the physics and often the only way when direct methods, e.g. imaging, are not available. It is thus an interesting and challenging problem to obtain a quantitative description of the dynamics of such systems. One striking property exhibited by all these systems is pinning, i.e. at low temperature there is no macroscopic motion
Corresponding author. Tel.: +32 03 2653203.
E-mail address: [email protected] (G. Piacente). 1386-9477/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.physe.2006.03.040
unless the applied force f reaches a threshold value f c . Evidences of a quasi-one-dimensional (Q1D) gas of electrons were found recently in experiments on electrons on the surface of liquid helium where the electrons were confined by metallic gates [1]. Those electrons exhibited dynamical ordering in the form of filaments. Other Q1D systems of strongly interacting particles have been realized in complex plasmas [2] and in colloidal suspensions [3]. 2. Model and methods An infinite number of particles with coordinates r ¼ ðx; yÞ move in a plane and interact through a Yukawa-type potential. An external parabolic potential in the y-direction reduces the in-plane motion and makes the system Q1D. A constriction modeled by a Lorentzian-shaped potential centered in the origin of the axes is present and a constant external driving force f is applied in the positive x-direction.
ARTICLE IN PRESS G. Piacente, F.M. Peeters / Physica E 34 (2006) 224–227
By using dimensionless units the Langevin equations of motion can be written in the following form: 0 0 d2 x0i dx0i 1 X q expðkjri rj jÞ ¼ g jr0i r0j j dt2 dt 2 j qx0i
þ
V 00 a0 2 x0i þ xx ðT 0 Þ þ f , ð1 þ a0 2 x0i 2 Þ2
ð1aÞ
0 0 d2 y0i dy0i 1 X q expðkjri rj jÞ ¼ g dt2 dt 2 j qy0i jr0i r0j j
y0i
0
þ xy ðT Þ,
225
and v0 yi ¼ 0, which means that in the absence of thermal noise and constriction the total effect of the external driving force is a sliding of the ordered structure with a drift velocity which is directly proportional to the driving force and inversely proportional to the friction. More in general, for very large values of the external drive, one should expect that the drift velocity is v0 xi f =g, or in other words that the system behaves like a classical twodimensional Drude conductor. 3.1. Pinning
ð1bÞ
0
where r and t are the dimensionless coordinates and time, respectively, g is the friction coefficient, k the inverse screening length, V 00 is the maximum of the constriction potential which has a full width at half maximum 2=a0 . The term nðT 0 Þ is a random force, reproducing the thermal noise, with zero average and standard deviation hxiz ðtÞxjZ ðsÞi ¼ gT 0 dij dzZ dðt sÞ, where ðz; ZÞ ðx; yÞ and T 0 is the dimensionless temperature. We integrated numerically the equations of motion to study the time evolution of the system and the transport properties. In order to simulate an infinite long wire, periodic boundary conditions were introduced along the xdirection. The driving force was increased from zero by small increments. The classical model we present is suitable for the description of diverse confined driven systems, as electrons on liquid helium, colloids and complex plasmas. It should be noticed that the classical approach, which is naturally valid in the case of colloids and complex plasmas because of the microscopic size of the particles, is still valid for pure quantum objects as electrons on liquid helium when they exhibit Wigner crystallization. In this case the De Broglie thermal length lD is much smaller than the interparticle spacing, hence the quantum aspect of the original fermionic problem can be neglected.
The system is pinned until the applied driving force reaches a threshold value f c . The particles move in the direction of the driving force and accumulate in front of the constriction, that exerts a force, which is opposite to the driving force. If f of c , new static configurations are reached, in which the electrostatic repulsion and the repulsive force due to the constriction balance the driving force. The situation is depicted in Fig. 1 for N ¼ 600 particles and for a high value of the constriction barrier height. As can be seen in Fig. 1 for large V 00 the chain structure is not homogeneous and different numbers of chains can coexist. On the contrary, in the case of low constriction barrier height, the chain structures are more homogeneous, with the same number of chain present at the left and at right of the constriction, although the inter-particle distances are larger at the right than at the left of the barrier. The critical force f c is a function of the temperature T 0 : it decreases with increasing temperature, that is the thermal agitation aids the net motion of the particles. The critical force is also a function of the density, i.e. the number of particles. In our simulations we observed that for larger densities, i.e. larger N, f c becomes smaller. 3.2. Depinning
3. Dynamical properties
When the driving force f is larger than the threshold f c , the system exhibits Q1D flow in the form of channels. In
At T 0 ¼ 0 in the absence of constriction and driving force, the charged particles crystallize in a number of chains depending on the density, as shown in Ref. [4]. The presence of the constriction modifies locally the chain-like structure near the constriction barrier, but the particles are still organized in chains far away from this constriction. When a longitudinal driving force f is applied in the xdirection, the particles are then pushed along the direction of the driving force. In what follows we deal with a short range constriction, which can therefore be seen as local disturbance. An obviously important quantity to determine is the average velocity v0 of the particles as a function of the applied force f. In the absence of thermal fluctuations and in the absence of the constriction potential, the stationary solutions of Eqs. (1a) and (1b) are, respectively, v0 xi ¼ f =g
Fig. 1. Typical trajectories of the particles when the system is pinned. The plot is for a temperature T 0 ¼ 0:002, well below the melting temperature, and for k ¼ 1 and a0 ¼ 1.
ARTICLE IN PRESS 226
G. Piacente, F.M. Peeters / Physica E 34 (2006) 224–227
(a)
Fig. 2. Typical trajectories of the particles when the system is depinned for (a) elastic depinning and (b) quasi-elastic depinning. The plots are for a temperature T 0 ¼ 0:002, k ¼ 1 and a0 ¼ 1.
Fig. 2 some typical trajectories of the depinned particles are reported, for values of f just above f c . The first interesting observation is that the driven system does not break up into pinned and flowing regions or in other words, the chain-like system under investigation does not exhibit plastic flow. Once the driving force overwhelms the critical threshold f c , all the particles move together. This effect is clearly to the low dimensionality and to the specific features of the constriction barrier. The depinning can be either elastic or quasi-elastic depending on the height of the constriction barrier. In the case of a low barrier (V 00 ¼ 1 in Fig. 2(a)) the particles move such that they keep the same neighbors, thus the system depins elastically. In contrast in the case of a high barrier (V 00 ¼ 5 in Fig. 2(b)) a complex net of conducting channels is activated and the particles move without keeping their neighbors, that is the depinning is quasi-elastic. As first predicted by Fisher the elastic depinning exhibits criticality [5] and the velocity vs force curves scale as v ¼ ðf f c Þb . In Fig. 3 we report the v f curves in the case of elastic and quasi-elastic depinning. By running simulations for different number N of particles and different values of k, we noticed that the critical exponents do not depend neither on the density, i.e. on the number of chains, nor on the range of the inter-particle interaction.
(b) Fig. 3. Velocity vx vs applied drive f for the elastic depinning (a) and the quasi-elastic depinning (b) for N ¼ 600 particles. The dashed curves are the best fitted power-law behavior for T 0 ¼ 0. At zero temperature the curves exhibit a sudden jump at the depinning threshold, while at finite temperature they are smoother.
The theoretical value of the critical exponent b is 23 for twodimensional infinite systems in the case of elastic depinning, which has been confirmed e.g. in experiments and numerical simulations on colloidal systems [6,7]. A good theoretical treatment of flows which are other than elastic is still missing. For all the investigated chain configurations we obtained on average that b ’ 0:66 in the case of homogeneous channel flow, i.e. elastic depinning, and b ’ 0:94 in the case of inhomogeneous channel flow, i.e. quasi-elastic depinning. The value of the critical exponent could, therefore, be considered as a clear signature of the kind of flow occurring. It should be stressed that we found that the critical exponent in the case of elastic depinning for confined Q1D systems has the same value as for infinite systems in two dimensions.
ARTICLE IN PRESS G. Piacente, F.M. Peeters / Physica E 34 (2006) 224–227
Finally, in our simulation both in case of elastic and quasi-elastic depinning the velocity vs applied drive curves did not show hysteretic features. 4. Conclusions We studied numerically the response of a classical Q1D infinite system of particles interacting through a Yukawatype potential to an external drive in the presence of a constriction. We found that the particles are pinned up to a critical value f c of the driving force. The pinned phase is a new static phase, where the system rearranges itself in such a way to balance the external drive. For values of the driving force larger than f c , the particles can overcome the potential barrier and the system depins. We analyzed in detail the depinning phenomenon and we found that the system can depin either elastically or quasi-elastically depending on the strength of the contact point potential. The elastic depinning is characterized by a critical exponent, which is on average b23 and does not depend either on the number of chains nor on the inter-particle interaction range. This is in excellent agreement with theoretical and numerical findings on two-dimensional systems exhibiting elastic depinning. Therefore, we proof that the critical exponent for the elastic depinning has the
227
same value both for confined and infinite systems in two dimensions. The quasi-elastic depinning regime is characterized by a critical exponent b0:94. Acknowledgments This work was supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00157 ‘‘Surface Electrons’’, the Flemish Science Foundation (FWO-Vl) and Belgian Science Policy and the University of Antwerp (GOA). References [1] P. Glasson, V. Dotsenko, P. Fozooni, M.J. Lea, W. Bailey, G. Papageorgiou, S.E. Andresen, A. Kristensen, Phys. Rev. Lett. 87 (2001) 176802. [2] B. Liu, K. Avinash, J. Goree, Phys. Rev. Lett. 91 (2003) 255003. [3] K. Zahn, R. Lenke, G. Maret, Phys. Rev. Lett. 82 (1999) 2721. [4] G. Piacente, I.V. Schweigert, J.J. Betouras, F.M. Peeters, Solid State Commun. 128 (2003) 57; G. Piacente, I.V. Schweigert, J.J. Betouras, F.M. Peeters, Phys. Rev. B 69 (2004) 045324. [5] D.S. Fisher, Phys. Rev. B 31 (1985) 1396. [6] C. Reichhardt, C.J. Olson, Phys. Rev. Lett. 89 (2002) 078301. [7] Y. Cao, J. Zhengkuan, H. Ying, Phys. Rev. B 62 (2000) 4163.
Physica E 34 (2006) 224–227 www.elsevier.com/locate/physe
Driven quasi-one-dimensional classical electron gas in the presence of a constriction: Pinning and depinning G. Piacente, F.M. Peeters Department of Physics, University of Antwerpen (Campus Middleheim), Groenenborgerlaan 171, B-2020 Antwerpen, Belgium Available online 30 June 2006
Abstract We studied the properties of a quasi-one-dimensional system of charged particles in the presence of a local Lorentzianshaped constriction. We investigated the response of the system when a time-independent external driving force is applied in the unconfined direction. Langevin molecular dynamics simulations for different values of the drive and temperature are performed. We found that the particles are pinned unless a threshold value of the driving force is reached. We investigated in detail the depinning phenomenon. The system can depin ‘‘elastically’’, with particles moving together and keeping their neighbors, or ‘‘quasi-elastically’’, with particles moving together through a complex net of conducting channels without keeping their neighbors. In the case of elastic depinning the velocity vs applied drive curves is characterized by a critical exponent b consistent with the value 23, while in the case of quasi-elastic depinning the critical exponent b has on average the value 0.94. The model is relevant e.g. for electrons on liquid helium, colloids and dusty plasma. r 2006 Published by Elsevier B.V. PACS: 71.10.w; 52.65.Yy; 45.50.Jf Keywords: Electron gas; Pinning; Depinning; Critical exponents
1. Introduction Interacting systems of charged particles, which tend to form periodic or ordered structures when the density and the temperature are low enough (i.e. Wigner crystallization), can exhibit a remarkable variety of complex phenomena when they are driven by an external force. Many of these phenomena are still poorly understood or even unexplored in mesoscopic confined systems. Transport experiments are a useful way to probe the physics and often the only way when direct methods, e.g. imaging, are not available. It is thus an interesting and challenging problem to obtain a quantitative description of the dynamics of such systems. One striking property exhibited by all these systems is pinning, i.e. at low temperature there is no macroscopic motion
Corresponding author. Tel.: +32 03 2653203.
E-mail address: [email protected] (G. Piacente). 1386-9477/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.physe.2006.03.040
unless the applied force f reaches a threshold value f c . Evidences of a quasi-one-dimensional (Q1D) gas of electrons were found recently in experiments on electrons on the surface of liquid helium where the electrons were confined by metallic gates [1]. Those electrons exhibited dynamical ordering in the form of filaments. Other Q1D systems of strongly interacting particles have been realized in complex plasmas [2] and in colloidal suspensions [3]. 2. Model and methods An infinite number of particles with coordinates r ¼ ðx; yÞ move in a plane and interact through a Yukawa-type potential. An external parabolic potential in the y-direction reduces the in-plane motion and makes the system Q1D. A constriction modeled by a Lorentzian-shaped potential centered in the origin of the axes is present and a constant external driving force f is applied in the positive x-direction.
ARTICLE IN PRESS G. Piacente, F.M. Peeters / Physica E 34 (2006) 224–227
By using dimensionless units the Langevin equations of motion can be written in the following form: 0 0 d2 x0i dx0i 1 X q expðkjri rj jÞ ¼ g jr0i r0j j dt2 dt 2 j qx0i
þ
V 00 a0 2 x0i þ xx ðT 0 Þ þ f , ð1 þ a0 2 x0i 2 Þ2
ð1aÞ
0 0 d2 y0i dy0i 1 X q expðkjri rj jÞ ¼ g dt2 dt 2 j qy0i jr0i r0j j
y0i
0
þ xy ðT Þ,
225
and v0 yi ¼ 0, which means that in the absence of thermal noise and constriction the total effect of the external driving force is a sliding of the ordered structure with a drift velocity which is directly proportional to the driving force and inversely proportional to the friction. More in general, for very large values of the external drive, one should expect that the drift velocity is v0 xi f =g, or in other words that the system behaves like a classical twodimensional Drude conductor. 3.1. Pinning
ð1bÞ
0
where r and t are the dimensionless coordinates and time, respectively, g is the friction coefficient, k the inverse screening length, V 00 is the maximum of the constriction potential which has a full width at half maximum 2=a0 . The term nðT 0 Þ is a random force, reproducing the thermal noise, with zero average and standard deviation hxiz ðtÞxjZ ðsÞi ¼ gT 0 dij dzZ dðt sÞ, where ðz; ZÞ ðx; yÞ and T 0 is the dimensionless temperature. We integrated numerically the equations of motion to study the time evolution of the system and the transport properties. In order to simulate an infinite long wire, periodic boundary conditions were introduced along the xdirection. The driving force was increased from zero by small increments. The classical model we present is suitable for the description of diverse confined driven systems, as electrons on liquid helium, colloids and complex plasmas. It should be noticed that the classical approach, which is naturally valid in the case of colloids and complex plasmas because of the microscopic size of the particles, is still valid for pure quantum objects as electrons on liquid helium when they exhibit Wigner crystallization. In this case the De Broglie thermal length lD is much smaller than the interparticle spacing, hence the quantum aspect of the original fermionic problem can be neglected.
The system is pinned until the applied driving force reaches a threshold value f c . The particles move in the direction of the driving force and accumulate in front of the constriction, that exerts a force, which is opposite to the driving force. If f of c , new static configurations are reached, in which the electrostatic repulsion and the repulsive force due to the constriction balance the driving force. The situation is depicted in Fig. 1 for N ¼ 600 particles and for a high value of the constriction barrier height. As can be seen in Fig. 1 for large V 00 the chain structure is not homogeneous and different numbers of chains can coexist. On the contrary, in the case of low constriction barrier height, the chain structures are more homogeneous, with the same number of chain present at the left and at right of the constriction, although the inter-particle distances are larger at the right than at the left of the barrier. The critical force f c is a function of the temperature T 0 : it decreases with increasing temperature, that is the thermal agitation aids the net motion of the particles. The critical force is also a function of the density, i.e. the number of particles. In our simulations we observed that for larger densities, i.e. larger N, f c becomes smaller. 3.2. Depinning
3. Dynamical properties
When the driving force f is larger than the threshold f c , the system exhibits Q1D flow in the form of channels. In
At T 0 ¼ 0 in the absence of constriction and driving force, the charged particles crystallize in a number of chains depending on the density, as shown in Ref. [4]. The presence of the constriction modifies locally the chain-like structure near the constriction barrier, but the particles are still organized in chains far away from this constriction. When a longitudinal driving force f is applied in the xdirection, the particles are then pushed along the direction of the driving force. In what follows we deal with a short range constriction, which can therefore be seen as local disturbance. An obviously important quantity to determine is the average velocity v0 of the particles as a function of the applied force f. In the absence of thermal fluctuations and in the absence of the constriction potential, the stationary solutions of Eqs. (1a) and (1b) are, respectively, v0 xi ¼ f =g
Fig. 1. Typical trajectories of the particles when the system is pinned. The plot is for a temperature T 0 ¼ 0:002, well below the melting temperature, and for k ¼ 1 and a0 ¼ 1.
ARTICLE IN PRESS 226
G. Piacente, F.M. Peeters / Physica E 34 (2006) 224–227
(a)
Fig. 2. Typical trajectories of the particles when the system is depinned for (a) elastic depinning and (b) quasi-elastic depinning. The plots are for a temperature T 0 ¼ 0:002, k ¼ 1 and a0 ¼ 1.
Fig. 2 some typical trajectories of the depinned particles are reported, for values of f just above f c . The first interesting observation is that the driven system does not break up into pinned and flowing regions or in other words, the chain-like system under investigation does not exhibit plastic flow. Once the driving force overwhelms the critical threshold f c , all the particles move together. This effect is clearly to the low dimensionality and to the specific features of the constriction barrier. The depinning can be either elastic or quasi-elastic depending on the height of the constriction barrier. In the case of a low barrier (V 00 ¼ 1 in Fig. 2(a)) the particles move such that they keep the same neighbors, thus the system depins elastically. In contrast in the case of a high barrier (V 00 ¼ 5 in Fig. 2(b)) a complex net of conducting channels is activated and the particles move without keeping their neighbors, that is the depinning is quasi-elastic. As first predicted by Fisher the elastic depinning exhibits criticality [5] and the velocity vs force curves scale as v ¼ ðf f c Þb . In Fig. 3 we report the v f curves in the case of elastic and quasi-elastic depinning. By running simulations for different number N of particles and different values of k, we noticed that the critical exponents do not depend neither on the density, i.e. on the number of chains, nor on the range of the inter-particle interaction.
(b) Fig. 3. Velocity vx vs applied drive f for the elastic depinning (a) and the quasi-elastic depinning (b) for N ¼ 600 particles. The dashed curves are the best fitted power-law behavior for T 0 ¼ 0. At zero temperature the curves exhibit a sudden jump at the depinning threshold, while at finite temperature they are smoother.
The theoretical value of the critical exponent b is 23 for twodimensional infinite systems in the case of elastic depinning, which has been confirmed e.g. in experiments and numerical simulations on colloidal systems [6,7]. A good theoretical treatment of flows which are other than elastic is still missing. For all the investigated chain configurations we obtained on average that b ’ 0:66 in the case of homogeneous channel flow, i.e. elastic depinning, and b ’ 0:94 in the case of inhomogeneous channel flow, i.e. quasi-elastic depinning. The value of the critical exponent could, therefore, be considered as a clear signature of the kind of flow occurring. It should be stressed that we found that the critical exponent in the case of elastic depinning for confined Q1D systems has the same value as for infinite systems in two dimensions.
ARTICLE IN PRESS G. Piacente, F.M. Peeters / Physica E 34 (2006) 224–227
Finally, in our simulation both in case of elastic and quasi-elastic depinning the velocity vs applied drive curves did not show hysteretic features. 4. Conclusions We studied numerically the response of a classical Q1D infinite system of particles interacting through a Yukawatype potential to an external drive in the presence of a constriction. We found that the particles are pinned up to a critical value f c of the driving force. The pinned phase is a new static phase, where the system rearranges itself in such a way to balance the external drive. For values of the driving force larger than f c , the particles can overcome the potential barrier and the system depins. We analyzed in detail the depinning phenomenon and we found that the system can depin either elastically or quasi-elastically depending on the strength of the contact point potential. The elastic depinning is characterized by a critical exponent, which is on average b23 and does not depend either on the number of chains nor on the inter-particle interaction range. This is in excellent agreement with theoretical and numerical findings on two-dimensional systems exhibiting elastic depinning. Therefore, we proof that the critical exponent for the elastic depinning has the
227
same value both for confined and infinite systems in two dimensions. The quasi-elastic depinning regime is characterized by a critical exponent b0:94. Acknowledgments This work was supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00157 ‘‘Surface Electrons’’, the Flemish Science Foundation (FWO-Vl) and Belgian Science Policy and the University of Antwerp (GOA). References [1] P. Glasson, V. Dotsenko, P. Fozooni, M.J. Lea, W. Bailey, G. Papageorgiou, S.E. Andresen, A. Kristensen, Phys. Rev. Lett. 87 (2001) 176802. [2] B. Liu, K. Avinash, J. Goree, Phys. Rev. Lett. 91 (2003) 255003. [3] K. Zahn, R. Lenke, G. Maret, Phys. Rev. Lett. 82 (1999) 2721. [4] G. Piacente, I.V. Schweigert, J.J. Betouras, F.M. Peeters, Solid State Commun. 128 (2003) 57; G. Piacente, I.V. Schweigert, J.J. Betouras, F.M. Peeters, Phys. Rev. B 69 (2004) 045324. [5] D.S. Fisher, Phys. Rev. B 31 (1985) 1396. [6] C. Reichhardt, C.J. Olson, Phys. Rev. Lett. 89 (2002) 078301. [7] Y. Cao, J. Zhengkuan, H. Ying, Phys. Rev. B 62 (2000) 4163.