Dielectrophoresis model for the colossal electroresistance of phase-separated manganites
Shuai Dong,1 Han Zhu,2 and J.-M. Liu1,3,*
1Nanjing National Laboratory of Microstructures, Nanjing University, Nanjing 210093, China
2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
3International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China 共Received 24 September 2007; published 25 October 2007兲
We propose a dielectrophoresis model for phase-separated manganites. Without increase of the fraction of metallic phase, an insulator-metal transition occurs when a uniform electric field applied across the system exceeds a threshold value. Driven by the dielectrophoretic force, the metallic clusters reconfigure themselves into stripes along the direction of electric field, leading to the filamentous percolation. This process, which is time dependent, irreversible, and anisotropic, is a probable origin of the colossal electroresistance in manganites.
DOI:10.1103/PhysRevB.76.132409 PACS number共s兲: 75.47.Lx, 82.45.Un, 71.30.⫹h, 81.30.Mh
INTRODUCTION
Phase separation共PS兲can be found in various condensed matter systems. In the past half a century, the effects of elec- tric field on phase-separated liquids have been cross- fertilized by physicists, chemists, and biologists.1–5 Among them, a distinct phenomenon is dielectrophoresis:6 the mix- ing or demixing of neutral materials with different dielectric constants under a nonuniform electric field.2–5 It has also found wide application in microbiology, nanotechnology, polymer processing, etc.2,7 However, dielectrophoresis would seem unlikely in solids, which normally lack the nec- essary condition of flowability.
As for solids, PS has been demonstrated to be crucial to understanding various exotic phenomena in strongly corre- lated electronic materials,8 manganites being a typical ex- ample. In manganites, PS generally consists of ferromagnetic 共FM兲 metallic phase and insulating phase which is usually antiferromagnetic and charge ordered共CO兲.9One can under- take to understand the colossal electroresistance共CER兲, first observed in Pr0.7Ca0.3MnO3,10where an applied electric field could reduce the resistance enormously. Subsequent experi- ments found a simultaneous uprush of magnetization, imply- ing that the CER might be caused by the collapse of CO phase to FM phase.11 Although this CO-FM collapse sce- nario can lead to the CER, the estimated threshold of electric field is 1 order of magnitude larger than the experiment.12 Moreover, in a recent intriguing experiment on La0.225Pr0.4Ca0.375MnO3, no appreciable change of the mag- netization was detected when the electric field induced a re- sistance drop as large as 4 orders of magnitude.13 This anomalous CER calls for a further understanding of PS be- yond the equilibrium ground state in manganites.
In this Brief Report, we propose a dielectrophoresis model to understand the PS dynamics associated with the CER in manganites. At the beginning, it is essential to dem- onstrate the possibility of dielectrophoresis in manganites be- ing solid. Unlike liquid mixtures, the different phases in a prototype phase-separated manganite have the same chemi- cal composition. The migration of eg electrons rather than cations makes the different phases in manganites “flowable”
in a wide temperature range,14–16 satisfying the first condi-
tion of dielectrophoresis: ambulatory phases. The second condition, nonuniform electric field, can be solely self- generated by the PS, as shown in Fig. 1, without external control, which is necessary for dielectrophoresis in liquids.7 Therefore, dielectrophoresis is not impossible in manganites although they are solids.
MODEL
As shown in Fig. 1, our model is based on a two- dimensional square lattice共L⫻L,L= 64兲with a voltage drop Vapplied in theydirection. A fractionpMof the lattice sites are metallic, with the others insulating. The assumption of a fixed FM phase fraction will be checked behind. This binary mixture picture is adequate to describe the large scale PS consisting of FM and CO phases.9 A standard Metropolis algorithm is employed in our Monte Carlo simulation with the dielectrophoresis mechanism: exchange between nearest neighboring 共NN兲 sites with the probability determined by the free energy change. Considering the free energy of elec- tric fieldFEin the medium,3,4the total free energyFin our model can be written as
F=FE+FS= −1
2
冕
共r兲E2共r兲dr+冖
AdS, 共1兲where 共r兲 denotes the dielectric constant and E共r兲 is the local field at pointr.共r兲 equalsM共I兲if the phase at ris
FIG. 1.共Color online兲Left: the phase-separated lattice consisted by metallic共red or dark gray兲 and insulating 共cyan or light gray兲 sites. Right: the corresponding color image map ofV共r兲.
PHYSICAL REVIEW B76, 132409共2007兲
1098-0121/2007/76共13兲/132409共4兲 132409-1 ©2007 The American Physical Society
metallic共insulating兲, whereM共I兲is the dielectric constant of the metallic共insulating兲phase. The second item in Eq.共1兲 plays the role of obstruction during dielectrophoresis. In real manganites, many factors can be obstruction in the transition between metastable states, e.g., domain interfaces, strains, and defects,16,18 and here we simplify the obstruction by characterizing them with the interface energyFS共Ref.17兲in Eq. 共1兲. FS is proportional to the interface area S between metallic and insulating regions with a coefficientA. Within the system,E共r兲is obtained using the resistor-network共RN兲 model.19,20In the RN model, three types of resistors are dis- tributed according to the links between NN sites: RM be- tween metallic sites, RI between insulating sites, and RMI
between a metallic and an insulating site,20 set as 共RM+RI兲/ 2 here. The voltage of each siteV共r兲in the RN can be exactly solved by the Kirchhoff equations, as shown in Fig.1.
Our simulation begins with a zero-field quench preprocess from a high temperature T0 to Td. Then, at Td fixed, the voltageV changes between zero and a maximum valueVm, following an increasing-decreasing-increasing 共IDI兲 path with a fixed rate 关characterized by Monte Carlo steps 共MCSs兲兴. Given the strong contrast between the electric properties of FM and CO phases,9,21 both the ratios RI/RM
andM/I are quite large. Their values strongly depend on material, as well as doping and temperature. Experimentally, they can be determined by comparing the metal-rich limit and insulator-rich limit.9,21 To accelerate our simulation, a special case, f=RI/RM=M/I 共fⰇ1兲, will be used. More general cases without such restriction have also been tested.
It is found that the physical picture is unaffected as long as fⰇ1. The values of parameters used in our simulations are listed in TableI, if not noted explicitly.
RESULTS
The resistanceRyalong the direction of the electric field is shown in Fig.2 as a function of the external electric field E关E=V/共L− 1兲兴. Initially, the resistance is only weakly de- pendent onE. Then, the CER transition takes place onceE exceeds a threshold Ec. In the subsequent stages of increasing-decreasingE,Ryremains low and nonlinear even forE well belowEc. We find that the value ofEcincreases with the IDI rate 共not shown here兲, in agreement with the observations in La0.225Pr0.4Ca0.375MnO3,13 implying that the process is a relaxative evolution between metastable states.
We use the ratio ␦ER=Rymax/Rymin to characterize the CER,
whereRymax共Ry
min兲 is the maximum共minimum兲 of Ry before 共after兲the CER transition.␦ERappears to be proportional to f, as shown in the inset of Fig.2.
To understand the above transport behavior, four snap- shots of PS pattern taken at different stages during the IDI process are shown in Fig.3. BeforeEexceedsEc, as shown in Fig. 3共a兲, the metallic sites accumulate into clusters in favor of lower interface energy. However, once E⬎Ec, the energy gained from dielectrophoresis overcomes the inter- face energy barrier, and the PS structure is rapidly reconfig- ured into a stripelike pattern, as shown in Fig. 3共b兲. As a result of the percolation, the system has higher overall con- ductance and higher effective dielectric constant. Upon fur- ther increasing the field, the percolation path is strengthened, as shown in Fig. 3共c兲, giving rise to a slight decrease of resistance. When the field is decreasing, the percolation path is hardly destroyed except whenE is close to zero. WhenE is too small to compete against the surface tension and ther- mal energy, the percolation path will be corroded gradually and, given enough time, eventually break down, as shown in Fig.3共d兲. Nevertheless, with the parameters as listed in Table I, the main shape of percolation path can remain effective for a finite period of time, which accounts for the nonlinear transport behavior observed in Fig.2after the voltage thresh- old is first exceeded. As a result, the dielectrophoresis in our simulation is irreversible.
TABLE I. Default values of parameters in the model. Here,pM is a little larger than the experimental value共11%–14.5%兲 in Ref.
13because the percolation threshold in two-dimensional system is larger than that in three-dimensional materials. The value of f is within the experimental range for manganites共Refs.9and21兲. The other parameters are relative values.
pM I T0 Td A MCS f
20% 2 1.2 1 0.85 1.5⫻103 3⫻103
FIG. 2. 共Color online兲Resistances as a function of the applied field. Inset:␦ERas a function off.
FIG. 3. 共Color online兲Four snapshots taken from an evolution cycle of dielectrophoresis.
BRIEF REPORTS PHYSICAL REVIEW B76, 132409共2007兲
132409-2
Besides the changing rate of the electric field, the interface energy coefficient A and temperature also play important roles in modulating the dielectrophoresis. In our simulation, the dependence of 共Ec/A兲 upon 共Td/A兲 is quite nontrivial, as shown in Fig. 4. 共Ec/A兲 reaches the minimum at 共Td/A兲⬃1.1. It agrees with the experimental evidences that there is a T window for the dynamic PS 共T⬃25– 85 K兲 共Ref. 15兲 or a novel consolute critical point T⬃30 K 共Ref. 13兲 in the phase-separated La0.225Pr0.4Ca0.375MnO3. The origin could be that, at large 共Td/A兲, the thermal energy can destroy a stable percolation of the metallic phases; on the other hand, at small共Td/A兲, the relatively large surface tension can effectively prevent the metallic clusters from being reshaped into stripes. Therefore, 共Ec/A兲is larger in both cases of large and small共Td/A兲.
As stated above, the obstruction is responsible for the process-dependent conductance behavior and novel critical point. Although the parameterAcannot be directly measured in experiments, the consolute critical point does provide a way to compare our result ofEc with experimental results.
Using the parameters values taken from experiments,9,13,21 Ec in our model is estimated to be ⬃103V / m, within the same order of magnitude of the experimental data.13As we compare the physics of the present filamentous percolation picture and the CO-FM collapse picture, one notable thing is thatEc关⬃106– 107V / m共Refs.10and12兲兴required for the CO-FM collapse is indeed much larger thanEcin our model.
The reason is that the CER here is induced by merely repat- terning of the CO and FM phases, not by converting them from one to the other, which costs more energy to compen- sate for the energy difference between CO and FM phases.
Therefore, the assumption of the fixed FM phase fraction in our model is self-satisfied for the low electric field case.
Moreover, even if there does exist a possible slight increase 关⬍1%共Ref.13兲兴of FM phase fraction, the dielectrophoresis remains the most probable contribution to the CER effect, since the CER is not very sensitive to the FM fraction.13
Since the metallic filaments along the direction of electric field break the space symmetry of PS, it is natural to expect transport anisotropy. In our simulation, resistances both along共Ry兲 and across 共Rx兲 the electric field are affected, as shown in Fig.5. Before the voltage first reaches the thresh- old, the transport is isotropic, and resistances along both x
and y are highly independent of the voltage drop. At the voltage threshold, the CER transition is accompanied by the sharp emergence of transport anisotropy. WhileRyalong the direction of the electric field is drastically reduced, a simul- taneous increase ofRx across the electric field occurs irre- versibly near the threshold voltage. Furthermore, the system has the intriguing feature that it is metallic 共dRy/dT⬎0兲 along the electric field direction but insulating共dRx/dT⬍0兲 in the perpendicular direction.
DISCUSSION
Using the Bruggeman-Landauer 共BL兲 equation based on self-consistent effective medium approximation,22 the effec- tive conductance of a well-distributed and isotropic binary mixtures system can be estimated as
C= 1
2共d− 1兲关−P+
冑
P2+ 4共d− 1兲CICM兴, 共2兲 where d is the dimensionality and CI= 1 /RI 共CM= 1 /RM兲 is the conductance of insulting 共metallic兲 phases;P=CM共1 −dpM兲+CI共dpM+ 1 −d兲. Since CI/CM⬃0, Eq. 共2兲 can be simplified toC⬇CI/共1 −dpM兲 for all cases共fⰇ1兲in our model. Therefore, the effective resistance is about 共1 −dpM兲RI= 0.6RI for all values of f considered here, and the conductance is isotropic. In our simulation, the conduc- tance is isotropic and its value is indeed independent of f before the CER transition. The value ofRy⬇0.55RIis close to, but slightly smaller than, the BL estimation. The reason is that the accumulation of metallic phases into clusters will enhance the conductance. As the percolation path is formed, the Rymin can be estimated as RI/共pMf+ 1 −pM兲⬇RI/共pMf兲 sincefⰇ1. It accounts for the linear dependence of␦ERonf, as shown in the inset of Fig. 2. 共The relationship
␦ER⬀RI/RM is independent of the choice of M/I in our model.兲On the other hand, Rxmax after the CER transition is approximately RI共1 −pM兲+RMpM⬇RI共1 −pM兲. Compared with its original value, there is a relative small change 共increase兲 of Rx: Rxmax/Rxmin=共1 −pM兲/共1 − 2pM兲 by the two- dimensional BL equation.
Compared with the dielectrophoresis in liquids, there are two remarkable features in phase-separated manganites, add- ing to its appeal to the research community. Firstly, the pro- totype PS in manganites would generate inhomogeneous FIG. 4. 共Color online兲 The field threshold as a function of the
temperature. Here, the surface tension coefficientAis chosen as the unit, while the other parameters have the values in TableI.
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 0.0
0.0 0.0 0.0 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.8 0.8 0.8 0.8
R R R R////RRRRIIII
E E E E
R R R RXXXX R R R RYYYY
FIG. 5.共Color online兲Comparison of the perpendicular共Rx兲and parallel共Ry兲resistances as a function of the applied field.
BRIEF REPORTS PHYSICAL REVIEW B76, 132409共2007兲
132409-3
electric field by itself, while for liquids, great effort was made to artificially produce nonuniform electric field.4,7Yet, as in the cases of the liquids, the PS in manganites can also be controlled by applying artificially nonuniform electric field 共e.g., patterning of gates兲, offering great potential for making devices. Secondly, the diversity of electric properties between different phases in manganites, e.g., resistance and dielectric constant, is far greater than those in consolute liq- uids. In real manganites, the colossal electroresistance and/or magnetoresistance implies a colossal difference between the resistivities of metallic and insulating phases. Therefore, the main difficulties of dielectrophoresis in liquid, i.e., a large voltage required and weak effects, could be overcome in manganites.
In conclusion, we introduced the dielectrophoresis dy- namics, a phenomenon usually existing in mixed phase liq- uids, to investigate the phase separation in solid manganites.
We have shown that the colossal electroresistance could be achieved by the filamentous percolation through a dielectro-
phoresis dynamic process without increasing the metallic composition. Our results give a probable explanation of the colossal electroresistance in manganites, including its time dependence, irreversibility, and the consolute critical point.
Besides, it was also predicted that the dielectrophoresis pro- cess renders the system anisotropic. All these properties can- not be explained by the physics of single phases but arise from phase competition. The present work also highlights the analogy8between strongly correlated electronic systems and complex systems in the classical world, e.g., complex fluids.
ACKNOWLEDGMENTS
We acknowledge helpful discussions with C. Acha. This work was supported by the Natural Science Foundation of China共50332020 and 50601013兲and National Key Projects for Basic Research of China 共2006CB0L1002兲. S.D. was supported by the Scientific Research Foundation of Graduate School of Nanjing University共2006CL1兲.
1P. Debye and K. Kleboth, J. Chem. Phys. 42, 3155共1965兲; J. S.
Rowlinson and F. S. Swinton,Liquids and Liquid Mixtures共But- terworths, London, 1982兲; D. Wirtz and G. G. Fuller, Phys. Rev.
Lett. 71, 2236共1993兲.
2K. Amundson, E. Helfand, X. Quan, S. D. Hudson, and S. D.
Smith, Macromolecules 27, 6559共1994兲.
3T. Christen, Appl. Phys. Lett. 76, 230共2000兲.
4Y. Tsori, F. Tournilhac, and L. Leibler, Nature共London兲 430, 544 共2004兲.
5A. Kumar, Z. Qiu, A. Acrivos, B. Khusid, and D. Jacqmin, Phys.
Rev. E 69, 021402共2004兲.
6H. A. Pohl,Dielectrophoresis: The Behavior of Natural Matter in Nonuniform Electric Fields共Cambridge University Press, Cam- bridge, England, 1978兲.
7E. Schäffer, T. Thurn-Albrecht, T. P. Russell, and U. Steiner, Na- ture共London兲 403, 874共2000兲; R. Krupke, F. Hennrich, H. v.
Löhneysen, and M. M. Kappes, Science 301, 344共2003兲; B. H.
Lapizco-Encinas, B. A. Simmons, E. B. Cummings, and Y.
Fintschenko, Electrophoresis 25, 1695共2004兲; M. Dimaki and P.
Bøggild, Nanotechnology 15, 1095共2004兲.
8E. Dagotto, Science 309, 257共2005兲.
9M. Uehara, S. Mori, C. H. Chen, and S.-W. Cheong, Nature共Lon- don兲 399, 560共1999兲.
10A. Asamitsu, Y. Tomioka, H. Kuwahara, and Y. Tokura, Nature
共London兲 388, 50共1997兲.
11J. Stankiewicz, J. Sesé, J. García, J. Blasco, and C. Rillo, Phys.
Rev. B 61, 11236 共2000兲; A. Guha, N. Khare, A. K. Ray- chaudhuri, and C. N. R. Rao,ibid. 62, R11941共2000兲.
12S. Dong, C. Zhu, Y. Wang, F. Yuan, K. F. Wang, and J.-M. Liu, J.
Phys.: Condens. Matter 19, 266202共2007兲.
13G. Garbarino, C. Acha, P. Levy, T. Y. Koo, and S.-W. Cheong, Phys. Rev. B 74, 100401共R兲 共2006兲.
14L. Zhang, C. Israel, A. Biswas, R. L. Greene, and A. de Lozanne, Science 298, 805 共2002兲; M. Tokunaga, Y. Tokunaga, and T.
Tamegai, Phys. Rev. Lett. 93, 037203共2004兲.
15L. Ghivelder and F. Parisi, Phys. Rev. B 71, 184425共2005兲.
16W. Wu, C. Israel, N. Hur, S. Park, S.-W. Cheong, and A. de Lozanne, Nat. Mater. 5, 881共2006兲.
17D. I. Golosov, J. Appl. Phys. 91, 7508共2002兲; D. I. Golosov and D. Orgad, Phys. Rev. B 74, 104403共2006兲.
18T. Wu and J. F. Mitchell, Phys. Rev. B 74, 214423共2006兲.
19M. Mayr, A. Moreo, J. A. Vergés, J. Arispe, A. Feiguin, and E.
Dagotto, Phys. Rev. Lett. 86, 135共2001兲; S. Dong, H. Zhu, X.
Wu, and J.-M. Liu, Appl. Phys. Lett. 86, 022501共2005兲.
20S. Ju, T. Y. Cai, and Z. Y. Li, Phys. Rev. B 72, 184413共2005兲.
21B. Kundys, N. Bellido, C. Martin, and C. Simon, Eur. Phys. J. B 52, 199共2006兲.
22C. W. Nan, Prog. Mater. Sci. 37, 1共1993兲.
BRIEF REPORTS PHYSICAL REVIEW B76, 132409共2007兲
132409-4
