Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow

Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow

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Бібліографічні деталі
Дата:2010
Автори: Dzhezherya, Yu.I., Tovstolytkin, A.I., Klymuk, E.S., Uspenskaya, L.S.
Формат: Стаття
Мова:English
Опубліковано: НТК «Інститут монокристалів» НАН України 2010
Назва видання:Functional Materials
Теми:
Modeling and simulation
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/135131
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow / Yu.I. Dzhezherya, A.I. Tovstolytkin, E.S. Klymuk, L.S. Uspenskaya // Functional Materials. — 2010. — Т. 17, № 3. — С. 363-370. — Бібліогр.: 11 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-1351312018-06-15T03:05:23Z Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow Dzhezherya, Yu.I. Tovstolytkin, A.I. Klymuk, E.S. Uspenskaya, L.S. Modeling and simulation 2010 Article Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow / Yu.I. Dzhezherya, A.I. Tovstolytkin, E.S. Klymuk, L.S. Uspenskaya // Functional Materials. — 2010. — Т. 17, № 3. — С. 363-370. — Бібліогр.: 11 назв. — англ. 1027-5495 http://dspace.nbuv.gov.ua/handle/123456789/135131 en Functional Materials НТК «Інститут монокристалів» НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Modeling and simulation
Modeling and simulation
spellingShingle Modeling and simulation
Modeling and simulation
Dzhezherya, Yu.I.
Tovstolytkin, A.I.
Klymuk, E.S.
Uspenskaya, L.S.
Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow
Functional Materials
format Article
author Dzhezherya, Yu.I.
Tovstolytkin, A.I.
Klymuk, E.S.
Uspenskaya, L.S.
author_facet Dzhezherya, Yu.I.
Tovstolytkin, A.I.
Klymuk, E.S.
Uspenskaya, L.S.
author_sort Dzhezherya, Yu.I.
title Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow
title_short Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow
title_full Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow
title_fullStr Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow
title_full_unstemmed Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow
title_sort autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow
publisher НТК «Інститут монокристалів» НАН України
publishDate 2010
topic_facet Modeling and simulation
url http://dspace.nbuv.gov.ua/handle/123456789/135131
citation_txt Autostabilization of temperature in systems with combined magnetic and resistive transition under electric current flow / Yu.I. Dzhezherya, A.I. Tovstolytkin, E.S. Klymuk, L.S. Uspenskaya // Functional Materials. — 2010. — Т. 17, № 3. — С. 363-370. — Бібліогр.: 11 назв. — англ.
series Functional Materials
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AT klymukes autostabilizationoftemperatureinsystemswithcombinedmagneticandresistivetransitionunderelectriccurrentflow
AT uspenskayals autostabilizationoftemperatureinsystemswithcombinedmagneticandresistivetransitionunderelectriccurrentflow
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fulltext Functional Materials, 17, 3, 2010 363 - Autostabilization of temperature in systems with combined magnetic and resistive transi- tion under electric current flow Yu.I. Dzhezherya, A.I. Tovstolytkin, E.S. Klymuk*, L.S. Uspenskaya** Institute of Magnetism, National Academy of Sciences of Ukraine and Ministry for Education and Science of Ukraine, 36-b Vernadsky Blvd., 03142 Kyiv, Ukraine *National Technical University “KPI”, 37 Peremogy Ave., 03056 Kyiv, Ukraine **Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russia Received February 16, 2010 It is shown that the active thermostabilization regime can be realized in systems based on materials exhibiting combined magnetic and resistive phase transition of first order, when the sample temperature under electric current is self-maintained about the threshold equal to Tc (Tc is the phase transition temperature). The dependences of the ferromagnetic phase fraction as well as the sample temperature on the electric and magnetic field have been calculated. The sys- tem working parameters necessary to realize the active thermostabilization regime have been determined. It is found that the magnetic field can effectively influence the system temperature under certain circumstances. Показано, что на основе материалов с комбинированным резистивно-магнитным фазовым переходом первого рода может быть реализован режим активной термостабилизации, когда при протекании электрического тока в образце автоматически поддерживается температура вблизи порога Tc (Tc – температура фазового перехода). Рассчитаны зависимости доли ферромагнитной фазы и температуры в образце от электрического и магнитного полей. Определены рабочие параметры системы, при которых реализуется режим активной термостабилизации. Установлено, что при определенных условиях магнитное поле эффективно влияет на температуру системы. 1. Introduction In recent years, the first order magnetic phase transitions in substituted perovskite manga- nites R1-xAxMnO3 (R – rare earth, A – alkaline or alkaline-earth element) are under a particular at- tention [1, 2]. The magnetic phase layering phenomenon observed in such materials is accompanied by the material separation into regions with different conductivities [3, 4]. So, the low-temperature ferromagnetic (FM) phase is characterized by high conductivity (as high as that of typical metals), while the conductivity of high-temperature paramagnetic (PM) phase is several orders lower [4, 5]. This unique combination of the peculiar features of different phases makes it possible to control the system conductivity by varying the external magnetic field. At the same time, there is a pos- Yu.I. Dzhezherya et al. / Autostabilization of temperature in systems ... Functional Materials, 17, 3, 2010 364 sibility to exert influence on the parameters of the phase domain structure by its heating under electric current [6, 7]. In this work, a mechanism is proposed to realize the active thermostabiliza- tion regime for the systems based on materials exhibiting combined magnetic and resistive phase transition of the first order. 2. Theoretical considerations Let the sample be assumed to be a film of the thickness L. The film can be either in FM or PM state depending on its temperature. At a certain temperature Тс, the values of thermodynamic potentials for FM and PM phases (ФFM and ФPM, respectively) become equal to one another; this means that under particular conditions, the phase coexistence is possible. This process is assumed to occur through the first-order phase transition, with both magnetization М and conductivity s exhibiting discontinuous changes. In this case s sFM PM>> . As the FM phase is characterized by a high magnetization value, its response to external magnetic field and internal magnetostatic field will be appreciable. That is why the absolute value and the direction of the external field may affect the temperature at which the transition starts, as well as the nature of the phase transition itself. Let the sample initial temperature be much lower than Тс, and the sample is in a single-do- main state. This condition can be realized easily by applying an external magnetic field Н. If the field is perpendicular to the film plane, the single-domain state is attained for H M> 4p . Consider a case when the electric field E e= E x0 is applied to the film. An electric current starts to flow through the film. As only an insignificant fraction of electrons from the conduction band near Fermi energy eF takes part in the transfer processes, the electric current exerts no di- rect influence on the s f s d- -, interband exchange which makes the system turn to the FM state below the phase transition point Tc . At the same time, there is a mechanism of indirect influence of electric field on the mag- netic subsystem, because the electric current flow is accompanied by heat release and the system heating. When the sample temperature approaches Тс, the PM phase (i.e. regions of low conductivity) starts nucleating. A further heating results in the growth of the PM phase fraction which, in turn, causes the decreasing current density and deceleration of heat release process. The latter even can become negligibly small when the FM phase volume, VFV , lowers under the percolation threshold V Vp ~ .0 3 × (V - the sample volume). In this case, the FM phase is a system of isolated domains surrounded by the PM matrix. As is shown in [8, 9], it is in the vicinity of the percolation threshold where the sharp conduc- tivity and, correspondingly, the current density changes occur in a static regime. To describe the system conductivity qualitatively, we use a series expansion of the effective conductivity seff over the volume of FM phase near to the percolation threshold. Having neglected the conductivity of PM phase, we can write: s seff FM p FM p V V V V V = < × -( ) ì í ïïï î ïïï 0 0 , , (1) where s0 - the expansion coefficient which is of the same order as sFM . As follows from Eq. (1), the effective conductivity increases sharply as the volume fraction of FM phase exceeds the percolation threshold. This effect can be used to develop the active thermo- stabilization regime for the system. The task will be solved in two steps. 3. The film phase layering in a perpendicular field When solving the problem of phase layering for the film in a perpendicular field, let us as- sume that the film thickness L exceeds considerably the domain structure period P: Yu.I. Dzhezherya et al. / Autostabilization of temperature in systems ... Functional Materials, 17, 3, 2010 365 L P>> (2) This allows us to make two important simplifications. On the one hand, such assumption will make it easier to determine the dependence of the FM phase volume on temperature and external field. On the other hand, it will allow us to neglect the nonuniformity in the temperature distribu- tion within the film plane when we solve the problem of heat exchange between the film and the surrounding medium. As follows from Eq. (1), in the vicinity of the percolation threshold, the system conductivity heavily depends on the FM phase fraction. That is why we will concentrate attention on the deter- mination of the V VFM dependence on the system temperature and external field. That dependence can be obtained from the condition of energy minimum for a magnetic sub- system, which is given by the following expression in the case under consideration [10, 11]: D SE dv Sm VFM = - × - ×( )+ ×ò e M H M H 2 , (3) where e = - -F FFM PM is the difference of the thermodynamic potentials for FM and PM phases, respectively, which can be presented in the vicinity of the phase transition point as a series ex- pansion over temperature: e = -( )k T Tc . Here, k- is the expansion coefficient (a phenomenologi- cal parameter); Tc, the temperature at which the thermodynamic potentials of the phases become equal to one another; M = M ∙ ez, the magnetization of FM phase; H = H ∙ ez, the external magnetic field directed perpendicular to the film plane; Hm, magnetostatic field of the sample; S, S, the sur- face energy density and the total area of the phase interfaces, respectively. Integration in expres- sion (3) is carried out over the volume of FM phase. The form of the expression (3) implies that the crystallographic anisotropy is neglected. For a system, a fragment of which is shown in Fig. 1, the magnetostatic field is about H M em FM zM V V»- =- ×( )4 4p p , when the condition L P>> is met. As a result, the integral (3) can be easily calculated: D SE V MH k T T V V M V V Sc FM FM»- - -( )( ) + æ è ççç ö ø ÷÷÷÷ + ×2 2 2 p . (4) The total interface area depends on the characteristic period of domain structure rather than on the FM phase volume. Therefore, the V VFM value can be easily determined from the condition of the system magnetic energy minimum (see Eq. (4)): V V HM k T T M FM c= - -( ) 4 2p . (5) Since the FM phase relative volume varies from zero to unity, the system temperature in the phase separation regime lies within the range: T T MH k T T MH kc M c- + < < + . (6) Here, to make the expression more explicit, a temperature parameter T M kM = 4 2p is introduced. This parameter determines the width of the temperature interval where the FM and PM phases coexist. The k parameter value can be determined in an indirect way, for example, using the experi- mental results from [8, 9]. According to these data, for the Ca-substituted perovskite manganites with M ~ 320 G and Tc ~ 240 K, the percolation onset temperature rises with the magnetic field increase at an average rate of 0.8 K/kOe. This corresponds to the value of k ~ 4 105× erg⋅cm–3⋅K–1. So, for the material considered in [9], the temperature interval of the phase coexistence should be TM ~ .2 5 K. However, the experimentally observed value is several times larger than that es- Yu.I. Dzhezherya et al. / Autostabilization of temperature in systems ... Functional Materials, 17, 3, 2010 366 timated in the above way. This effect, at least in part, may originate from the inhomogeneous Ca ion distribution over the sample volume. As a re- sult, Tc displays a considerable space dispersion. Such a dispersion can no doubt influence sig- nificantly the course of the phase transition. How- ever, the account for this effect is beyond the scope of this work. In what follows, we will concentrate on the development of the approach for a hypothet- ical, ideally homogeneous material. The Eq. (5) links the sample phase compo- sition with the system temperature and magnetic field. As the main task of this work is to describe the active thermostabilization regime, we should supplement the system of equations by those of a heat balance with the surrounding medium. 4. Temperature distribution and heat balance conditions When determining the equilibrium distribution of phases (see Eq. (5)), we assumed that the sample temperature is the same over its whole volume. In practice, such a condition can be satisfied only under a certain approximation. For the system shown fragmentarily in Fig. 1, the temperature distribution inhomogeneities originate from a spatial inhomogeneity of conductivity resulting from the phase separation, as well as from a permanent heat exchange with the surrounding medium across the film surface. How- ever, for some particular relationships between physical and geometrical parameters of a magnetic film and thermo-isolating layer, the temperature distribution inhomogeneity can be neglected. To derive the equation of thermal balance and determine the condition when the tempera- ture field can be considered as homogeneous, let us consider the heat conduction equation: c T t x y T x yr k s¶ ¶ -Ñ ( )×Ñ( )= ( ), , E2 , (7) where k sx y x y, , ,( ) ( )-are the film heat conduction coefficient and conductivity, modulated in the xOy plane; c, r, the material heat capacity and density being practically independent of whether the film is in a phase separated state or not; Ñ= ¶ ¶e xi i ; the right part of Eq. (7) takes into ac- count the power of heat sources in a sample. Let us solve the problem of the temperature distribution determination in a simplified way, assuming the thermostat temperature to be constant and equal to Te . Under such conditions, the system will eventually achieve a thermal balance with ¶ ¶ =T t 0 . At a stationary regime, the latent heat of a phase transition present in dynamics and accom- panying the changes in the sample phase distribution is also insignificant. This allows us to exclude the temporal part of the task and concentrate attention on the study of the spatial temperature distribution. It should be noted that for the regime settled, the heat conduction problem will include two dimensional parameters, the domain structure period P in the film plane and the film thickness L. To simplify the form of the expressions, we introduce the variables: ρρ= +( ) =x y P z Lx ye e , z . The heat conduction equation takes the form: ¶ ¶ ( )× ¶ ¶ æ è çççç ö ø ÷÷÷÷ = æ è ççç ö ø ÷÷÷÷ × - ( )¶ ¶ - ( ) ρρ ρρ ρρ ρρk r k z sT P L T  2 2 2 E22 2L æ è ççççç ö ø ÷÷÷÷÷ . (8) As follows from expression (2), the right part of Eq. (8) is a correction squared with respect to the small parameter P L . Thus, the solution of Eq. (8) can be written as a series expansion over P L( )2 : Fig. 1. A fragment of the sample with the ferro- magnetic phase domains (schematic view). Yu.I. Dzhezherya et al. / Autostabilization of temperature in systems ... Functional Materials, 17, 3, 2010 367 T T P L Tρρ ρρ, , ...z z z( ) = ( )+( ) × ( )+0 2 1 (9) The unknown function T0 x( ) should be fitted to satisfy the Fredholm alternative for a non- uniform equation: ¶ ¶ ( )× ¶ ¶ æ è çççç ö ø ÷÷÷÷ =- ( ) ¶ ( ) ¶ - ( ) ρρ ρρ ρρ ρρ ρρk k z z s T T L1 2 0 2 2 2E . (10) In our case, the necessary and sufficient condition for obtaining a non-trivial solution of this equation is that the average value of the right part of Eq. (10) is zero in the xOy-plane: ¶ ( ) ¶ =- ( ) ( ) × 2 0 2 2 2T E L z z s k ρρ ρρ __________ ________ . (11) Here, the overbars mean averaging over the film plane. The calculation of k ρρ( ) ______ is quite easy and results in k k k k kρρ( ) = = + -( )×( ) ______ eff PM FM PM FMV V . However, the determination of the average power of heat sources, s ρρ( )E2 ___________ , is troublesome, be- cause both the conductivity and local electric field intensity depend considerably on the state of the phase domain system. However, since the sharp discontinuities of the conductivity and heat producing are observed near the percolation threshold, let us use the phenomenological formula of the conductivity expansion (1) and introduce the average capacity of heat sources in the form: s s ρρ( ) = -( ) > < ì í ïïï î ïïï E E V V V V V V V FM P FM P FM P 2 0 0 2 0 ___________ , , (12) Now the solution of Eq. (8), symmetrical relative to the z = 0 plane, can be presented as T z T E z V V Veff FM P 0 0 0 0 2 2 0 2 ( ) = ( )- × × -s k , (13) where T0(0) is the integration constant which is determined from the condition of the heat flux con- tinuity in the direction perpendicular to the film surface: k keff e z L T z z T T z h ¶ ( ) ¶ = - ( ) = 0 0 0 2 . (14) So, the temperature distribution within the film is determined by the expression: T z T V V V z L L he FM P eff 0 0 2 1 2 ( ) = + × - × - × -( ) × æ è çççççç ö ø ÷÷÷÷÷÷÷ Q k k , (15) where the designation Q= × s k 0 0 2 02 E Lh is introduced. It is this quantity that is an important heat parameter of the problem. It has the temperature dimensionality and characterizes the sample heating process resulted from the electric current. Its value grows with the increase in both the thermal isolation efficiency ( h k0 ) and heat source intensity (s0 0 2E ). It follows from (15) that the increase in the thermo-isolating parameter k keff 0 gives rise to more homogeneous temperature distribution over the sample. At this stage, we can formulate the condition that, being satisfied, enables us to consider the temperature distribution as a homogeneous one over the whole film volume. It is clear that the temperature modulation over the film thickness should be considerably less than the temperature interval of a magnetic phase transition: Yu.I. Dzhezherya et al. / Autostabilization of temperature in systems ... Functional Materials, 17, 3, 2010 368 V V V L h TFM P eff M- × << k k 0 4 Q . (16) Then, as follows from the expression (15), we can neglect the temperature changes within the film thickness and consider the sample temperature T to be constant within the whole volume, so T T V V Ve FM p= + × -( )Q . (17) As was mentioned above, this approach is correct in the vicinity of the percolation threshold when V V VFM p-( ) <<1 . Therefore, taking into account that the material temperature in a phase separated state is close to Tc , it follows from (17) that: V V V T TFM p c e- - <<~ Q 1 . (18) This condition can be satisfied in various ways, for example, by increasing the system ther- mal insulation or raising the electric field intensity. Thus, expressions (5) and (17) give the solution of the problem under consideration, which makes it possible to determine the temperature Т and the fraction of the FM phase V VFM in the sample using the preset intensity values of the electric and magnetic fields: V V V H M V V T T T T T Tc TM H M Vp V Tc Te T FM p p c e M M - = - + -( ) + - = - - -( ) + 4 1 4 1 p p Q Q MM Q . (19) The condition (16) for the applicability of the theory results, i.e. the condition which deter- mines if the temperature distribution over the sample can be considered as homogeneous over the whole volume, takes the form: H M T T T V V T h L c e M p M eff4 1 4 0 p k k + -( ) - + << Q . (20) In other words, this result implies that high heat conductivity of a material and good ther- mal insulation of a system, both favor the homogeneous temperature distribution over the sample volume. It follows from Fig. 2 that for great values of Q TM , the film temperature responses weak- ly to the temperature of the surrounding medium. So, for Q TM ~ 150 , even the change of the temperature of the surrounding medium by DT Te M~ ~20 50 K results in the film temperature Fig. 2. Dependence of the sample temperature Т on the heat parameter Q= × s k 0 0 2 02 E Lh . The two lower curves are plotted for magnetic field H M= 4p , and correspond to different values of the thermo- stat temperature Te . The upper pair of curves is built for magnetic field H M= 8p for the same thermostat temperatures. Yu.I. Dzhezherya et al. / Autostabilization of temperature in systems ... Functional Materials, 17, 3, 2010 369 change by DT TM~ . ~ .0 1 0 25 K . At the same time, the film temperature can be effectively tuned by means of external magnetic field. For the case where Q TM is sufficiently great, an asymptoti- cal approximation T Tc TM H M Vp V- = × -( )4p (21) can be used. According to this approximation, the magnetic field strength increase by DH M= 4p brings about the rise of the film temperature by DT TM~ . 5. Conclusion Thus, the active thermal stabilization regime can be realized in the systems with combined magnetic and resistive phase transition of the first order placed in a perpendicular magnetic field with electric current flowing through the film. As a concluding remark, let us note that electric current influences the system magnetic susceptibility. In fact, as the system average magnetization is M M V Vz FM= × , we can find c p = = +( ) d M dH T z M 1 4 1Q , (22) with taking into account expression (19). It follows from Eq. (22) that the system susceptibility decreases with the increasing heat parameter Q TM . References 1. Y. Murakami, H. Kasai, J.J. Kim et al., Nature Nanotechnology, doi:10.1038/nnano.2009.342 (2009). 2. Yu.I. Dzhezherya , A.I. Tovstolytkin , J. Phys.: Condens. Matter., 19, 246212 (2007). 3. K. Dörr, J. Phys. D: Appl. Phys., 39, R125 (2006). 4. E. Dagotto, T. Hotta & A. Moreo, Phys. Rep., 344, 1 (2001). 5. L. Zhang, C. Israel, A. Biswas et al., Science, 298, 805 (2002). 6. N.A. Tulina, L.S. Uspenskaya, V.V. Sirotkin, et al., Physica C, 444, 19 (2006). 7. M. Tokunaga, Y. Tokunaga, T. Tamegaki, Phys. Rev. Lett., 93, 037203 (2004). 8. N. G. Bebenin, R. I. Zainullina, N. S. Chusheva et al., Fiz. Metal. Metalloved., 103, 261 (2007). 9. N. G. Bebenin, R. I. Zainullina, N. S. Bannikova et al., Phys. Rew. B 78, 064415 (2008). 10. V. G. Bar’yakhtar, I. M. Vitebskiy, D. A. Yablonskiy, Fiz. Tverd. Tela, 19, 347 (1977). 11. V. G. Bar’yakhtar, A. E. Borovik, V. A. Popov et al., Fiz. Tverd. Tela, 13, 3232 (1971). Yu.I. Dzhezherya et al. / Autostabilization of temperature in systems ... Functional Materials, 17, 3, 2010 370 Автостабілізація температури у матеріалі з резистивно-магнітним фазовим переходом першого роду при протіканні електричного струму Ю.І.Джежеря, O.І.Товстолиткін О.С.Клімук, Л.С.Успенська Показано, що на основі матеріалів з комбінованим резистивно-магнітним фазовим переходом першого роду може бути реалізований режим активної термостабілізації, коли під час протікання електричного струму у зразку автоматично підтримується температура поблизу порогу Tc (Tc – температура фазового переходу). Розраховано залежності частки феромагнітної фази та температури у зразку від електричного та магнітного полів. Визначено робочі параметри системи, за яких реалізується режим активної термостабілізації. Встановлено, що за визначених умов магнітне поле ефективно впливає на температуру системи.

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