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Simulation of Dynamics in Two-Dimensional Vortex Systems in Random Media
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ZHANG Wei(张伟), SUN Li-Zhen(孙李真), LUO Meng-Bo(罗孟波)** Department of Physics, Zhejiang University, Hangzhou 310027
(Received 10 June 2008) Dynamics in two-dimensional vortex systems with random pinning centres is investigated using molecular dynamical simulations. The driving force and temperature dependences of vortex velocity are investigated. Below the critical depinning force 𝐹𝑐 , a creep motion of vortex is found at low temperature. At forces slightly above 𝐹𝑐 , a part of vortices flow in winding channels at zero temperature. In the vortex channel flow region, we observe the abnormal behaviour of vortex dynamics: the velocity is roughly independent of temperature or even decreases with temperature at low temperatures. A phase diagram that describes different dynamics of vortices is presented.
PACS: 74. 25. Qt, 74. 25. Fy Elastic systems interacting with random pinning potentials have attracted growing interest recently due to rich equilibrium and nonequilibrium dynamical phases. One of the important and challenging questions is how the elastic system responds to external driving force and thermal fluctuation. This problem is relevant for many systems including chargedensity waves,[1] Wigner crystals,[2] domain wall of ferromagnetic[3] and ferroelectric materials,[4] and vortex lattices in type-II superconductors.[5] The competition between elastic force and random pinning forces builds up a highly nontrivial energy landscape and thus results in a glassy response of system. At zero temperature (𝑇 = 0), driving force 𝐹 drives vortices to slide at forces larger than critical pinning force 𝐹𝑐 . While at finite temperature 𝑇 , energy barriers to motion due to pinning can always be overcome by thermal activation that results in a finite response below 𝐹𝑐 . The velocity of creep is expected to obey 𝑣(𝐹 ) ∼ exp[−𝑈eff /𝑇 ], where the effective barrier 𝑈eff is dependent on the driving force 𝐹 . Near 𝐹𝑐 , the dynamics can be described by a scaling relation and 𝑈eff decreases linearly towards 𝐹𝑐 .[6] However, for small force near 0, scaling theory of creep predicts that 𝑈eff diverges in a power law 𝑈eff ∼ 𝑈𝑐 (𝐹𝑐 /𝐹 )𝜇 with exponent 𝜇 depending on system dimensionality.[7−10] For a two-dimensional (2D) vortex system, depending on pinning strength and density, there is a special state where most of the vortices are pinned while a small portion of vortices flow in channel.[11,12] In this channel flow state, how the energy barrier depends on force at finite temperature is still unknown. The dynamics of the system at finite temperature is unclear. Elastic theory and functional renormalization group is not available for this case.[7,9] In this Letter, we study the dynamics of a 2D vortex system with random pinning centres with molecular dynamical simulations. At 𝑇 = 0, with the in-
crease of driving force, the system goes from pinned state at small force through channel moving and collective plastic moving to smectic moving state at large force. The responses of vortex dynamics in different moving states are investigated at finite temperature. For a 2D superconducting film with magnetic field perpendicular to the film surface, the magnetic field is in the form of vortex with flux Φ0 = ℎ/2𝑒. The motion of vortex is given by the overdamped equation[6,13,14] 𝑁𝑝 𝑁𝑣 ∑︁ ∑︁ 𝑑𝑟 𝑖 𝑉𝑉 𝐹 (𝑟 𝑖 − 𝑟 𝑗 ) + 𝐹 𝑉 𝑃 (𝑟 𝑖 − 𝑅𝑘 ) = 𝜂 𝑑𝑡 𝑗̸=𝑖
+ 𝐹 + 𝐹 𝑡ℎ ,
𝑘=1
(1)
where 𝜂 is the viscosity coefficient, {𝑅𝑘 } specifies the 𝑁𝑝 pinning centre positions, and 𝑟 𝑖 denotes the location of the 𝑖th vortex. The first term on the righthand side of Eq. (1) is the repulsion between vortices 𝜀0 𝐾1 (𝑟𝑖𝑗 /𝜆)ˆ 𝑟𝑖𝑗 , with 𝜆 the effecwith 𝐹 𝑉 𝑉 (𝑟 𝑖𝑗 ) = 𝜆 tive penetration depth of the film, 𝐾1 (𝑥) the firstΦ20 order modified Bessel function, 𝜀0 = the 2𝜋𝜇0 𝜆2 energy per unit of length, and 𝑟𝑖𝑗 the distance between the 𝑖th and 𝑗th vortices. The second term is the attractive pinning force of range 𝑅𝑝 , given by 2𝑟𝑖𝑝 𝐹 𝑉 𝑃 (𝑟𝑖𝑝 ) = −𝐴𝑝 𝜀0 2 Θ(𝑟𝑖𝑝 − 𝑅𝑝 )ˆ 𝑟𝑖𝑝 , with 𝑟𝑖𝑝 being 𝑅𝑝 the distance between vortex and pin, Θ the Heaviside step function, and 𝐴𝑝 the pinning strength. The third term 𝐹 is the uniform Lorentz force acting on each vortex, and the fourth one 𝐹 𝑡ℎ is the thermal noise with zero mean ⟨𝐹𝛼 (𝑡)⟩ = ⟨𝐹𝛽 (𝑡)⟩ = 0 and a correlator ⟨𝐹𝛼 (𝑡)𝐹𝛽 (𝑡′ )⟩ = 2𝜂𝑘𝐵 𝑇 𝛿𝛼𝛽 𝛿(𝑡 − 𝑡′ ) with 𝛼, 𝛽 = 𝑥, 𝑦. In our simulation, we take 𝜂 = 1 and 𝑅𝑝 = 0.2𝜆, meanwhile the units for length, energy, temperature, force, and time are taken as 𝜆, 𝜀0 , 𝜀0 /𝑘𝐵 , 𝜀0 /𝜆, and 𝜏0 = 𝜂𝜆2 /𝜀0 . When 𝑅𝑝 = 0.2𝜆, the maximum of
* Supported
by the National Natural Science Foundation of China under Grant No 20874088. whom correspondence should be addressed. Email:
[email protected]. c 2009 Chinese Physical Society and IOP Publishing Ltd ○ ** To
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2𝐴𝑝 𝜀0 = 10𝐴𝑝 . Pinning 𝑅𝑝 strength 𝐴𝑝 is supposed to be uniform, but pins are distributed randomly in the system. We mainly investigate the system with size 𝐿𝑥 = 𝐿𝑦 = 20 considering periodic boundary conditions, pinning strength 𝐴𝑝 = 0.4, and both the numbers of vortices and pins are 𝑁𝑣 = 𝑁𝑝 = 200. Equation (1) is integrated by the 2nd order Runge-Kutta algorithm with a time step 𝛿𝑡 = 0.002𝜏0 . For each simulation, 1000𝜏0 is discarded and the consequent 𝑡 = 1000𝜏0 is used for statistics. The driving force is applied along the 𝑥 direction. The average velocity of vortices in the 𝑥 direction is calculated by ⟩ ⟨ 𝑁𝑣 1 ∑︁ 𝑣𝑖𝑥 . (2) 𝑣= 𝑁𝑣 𝑖=1 pinning force is 𝐹max =
First we anneal the 2D vortex system to zero temperature from a high temperature. The vortex configuration at 𝑇 = 0 is presented in Fig. 1 for 𝐴𝑝 = 0.4 by using Delaunay triangulation method. We find that the annealed ground state is a vortex glass. This is in agreement with theory as well as experiment that the 2D system has VG at 𝑇 = 0.[15−17] Other pinning strengths give the same result.
𝐹𝑐 is about 0.4.
Fig. 2. Velocity-force curve for 2D vortices in random pinning environment at zero temperature. The maximum pinning force on vortex is 𝐹max = 4 since 𝐴𝑝 = 0.4.
Fig. 3. Dependence of normalized average velocity 𝑣/𝐹 on temperature 𝑇 for four different driving forces corresponding to four different zero temperature 𝑣 − 𝐹 regions shown in Fig. 2. The forces 𝐹 are 0.2, 1.4, 3.4 and 8 from top to bottom. The inset shows the velocity for 𝐹 = 1.4 at low temperature. Fig. 1. Delaunay triangulation of vortices at zero temperature for a subsystem with size 12 × 12. Small open circles represent pins while other symbols represent vortices with different nearest neighbours. Small solid circles represent vortices with 6 nearest neighbours, while big solid circles represent 5, big open circles represent 7, and big open circles with small squares represent other values.
From annealed vortex configurations, we then slowly increase the driving force from zero to a high value. The average vortex velocity 𝑣 at different forces is presented in Fig. 2. There are four regions in the velocity-force (𝑣 − 𝐹 ) curve: a pinned region (I) with 𝑣 = 0 and three motion regions (II, III, and IV) with different differential resistances 𝑑𝑣/𝑑𝐹 . All the vortices are still in the pinned region. Although a few vortices are not pinned directly by pins, they are pinned indirectly by other pinned vortices nearby. Because of these indirectly pinned vortices, the critical depinning force 𝐹𝑐 is then much smaller than 𝐹max . We find that
We have checked the dynamics of each vortex and recorded trajectories of vortices in the three moving states. When 𝐹 is slight larger than 𝐹𝑐 , a part of indirectly pinned vortices begins to move. These moving vortices have stable channels. This region is named as channel flow region II.[12] In region III, all vortices move intermittently and randomly. The vortex trajectories cover up all area and there is no stable channel. This region is named as plastic moving region. When 𝐹 > 𝐹max , all the vortices move in ordered channels. The system enters into smectic moving region IV as found in many simulations.[13,18] Starting from the zero-temperature vortex configurations at finite driving forces, we then slowly increase the temperature while keeping 𝐹 as a constant. The dependences of velocity on temperature are presented in Fig. 3 for four different zero temperature 𝑣 − 𝐹 regions. For 𝐹 > 𝐹max , the velocity is almost indepen-
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dent of temperature. However, for 𝐹 < 𝐹max , thermal noise assists vortex motion[19] and helps vortex depin,[20] so one expects that the velocity should increase with 𝑇 . That is true for the system in regions I and III, see Fig. 3. However, in Region II, we find that the velocity maintains unchanged or even slightly decreases with temperature as shown clearly in the inset of Fig. 3 at low temperature.
independent of temperature (at low temperature scale, 𝑇 < 0.01). Thus, it is 𝑁MV to affect the average velocity of whole system since 𝑣 = 𝑣MV 𝑁MV /𝑁𝑣 . This is the reason why 𝑣 does not increase with 𝑇 at low temperatures but maintains unchanged or even decreases.
Table 1. The number of moving vortex 𝑁MV , average velocity 𝑣, and average velocity per moving vortex 𝑣MV at different temperatures. The driving force 𝐹 = 1.4. 𝑇 𝑁MV 𝑣 𝑣MV
0 32.1 0.170 1.06
0.0025 29.9 0.161 1.08
0.0045 29.1 0.153 1.05
0.01 46.8 0.227 0.97
0.03 116.3 0.647 1.12
What causes such abnormal behaviour of vortex dynamics in the channel flow region is carefully studied. We define a moving vortex (MV) if it moves longer than 2𝑅𝑝 during one time unit. The number of MV, 𝑁MV , changes with temperature. At 𝐹 = 1.4, 𝑁MV and 𝑣 at different temperatures are listed in Table 1. At nonzero but low temperatures, such as 𝑇 = 0.0025, both 𝑁MV and 𝑣 are smaller than that at 𝑇 = 0. The reason that 𝑁MV decreases at low temperature will be explained from the channel moving of vortices. In fact, such a phenomenon can be always observed if we can obtain channel flow for different pinning strengths and pin densities. However, at slightly higher temperature, such as 𝑇 = 0.01, both 𝑁MV and 𝑣 increases fast.
Fig. 5. Semi-logarithm plot of average velocity 𝑣 versus inverse temperature 𝑇 −1 at low temperatures for 𝐹 = 0.2.
Fig. 6. Dynamic phase diagram of vortex moving states. Lines are guide for the eyes.
(a)
(b)
Fig. 4. Channel flow of vortices at different temperatures: (a) 𝑇 = 0, and (b) 𝑇 = 0.0025. Here the driving force is 𝐹 = 1.4.
At 𝑇 = 0, the channels are always winding because of random distributed pins. The winding moving trajectories of vortices at 𝑇 = 0 is presented in Fig. 4(a). At finite temperature, thermal noise force makes the moving direction of vortex fluctuate and makes the channels more winding and broaden, as shown in Fig. 4(b) at 𝑇 = 0.0025. Thus vortices have higher probability to be captured by pins, resulting in a decrease of 𝑁MV . This phenomenon is clearly observed in Fig. 4(b) where some vortices are temporarily captured by pins. In Table 1 we also list the average velocity of MV, 𝑣MV . One can see that 𝑣MV is roughly
For the case of 𝐹 = 0.2 which locates in the pinned region, we find creep motion at low temperature. The velocities at temperatures lower than 0.02 obey the creep relation 𝑣 = 𝑣0 𝑒−𝑈eff /𝑇 , (3) where 𝑈eff is the effective energy barrier on vortex, and 𝑣0 is a pre-factor. The dependence of logarithm 𝑣 on 𝑇 −1 is plotted in Fig. 5. The effective energy barrier 𝑈eff is estimated to be about 0.01, which is close to the highest temperature 0.02 used. At 𝑇 ≫ 𝑈eff , the motion will not obey the creep rule. Similar creep motions were observed in experiments.[3,21] Lastly, we present a phase diagram for vortex motion modes in Fig. 6. There is a creep region at low temperature and at low driving force (𝐹 < 𝐹𝑐 ). At moderate force and low temperature there is a region where velocity is roughly independent of temperature or slightly decreases with temperature. In this region,
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temperature has no effect on pinned vortices, but can make moving vortices fluctuate. At high temperature the velocity is roughly equal to driving force. At moderate temperature, the velocity increases nonlinearly with temperature. In summary, we have studied the dynamics of 2D vortex system with random pinning centres. At zero temperature, with an increase of applied force, the system goes from pinned state at small force through channel moving and collective plastic moving to smectic moving state at large force. Below the critical depinning force 𝐹𝑐 , we find a creep motion of the vortex at low temperature. At moderate force and low temperature, temperature does not help a pinned vortex to depin but fluctuate moving vortices, resulting in a roughly temperature independent vortex motion. A phase diagram with four different dynamic modes is presented. The results indicate that energy barrier of 2D VG systems is complicated, especially in the channel flow region.
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