Charge transfer is discussed for the case when gyrotropy parameter (Hall coefficient) varies along transport л-direction and inverses its sign. This situation takes place in contacts of the serially joined materials having electron and hole types of conductivity. Spatial inhomogeneity of conductivity and inversion of Hall coefficient sign are analyzed in terms of electric potential and current density distribution. It is shown that under inhomogeneous magnetic field the steady current skinning takes place in plate sample

Mazurenko, O. N.

Sobol, V. R.

Zoli, M.

Belarusian State Pedagogical University Repository

BIANISOTROPICS 20008th International Conference on Electromagnetics of Complex MediaLisbon, Portugal, 27—29 September 2000ProceedingsEdited byAfonso M. Barbosa and Antonio L. ТораРЕПОЗИТОРИЙ БГПУ119Spatial Inversion of Gyrotropy Parameter in  Conductivity Tensor and Charge Transport PeculiaritiesV.R.Sobol1, O. N. Mazurenko1, and M. Zoli21 National Academy o f Sciences, Institute of Solid State and Semiconductor Physics Brovka Street 17, 220072 Minsk, BelarusFax: + 375-172-840888; email: sobol@ifttp.bas-net.by2 Istituto Nazionale Fisica della Materia,Dipartimento di Matematica e Fisica, Universita di CamerinoVia Madonna delle Carceri, 62032 Camerino, ItalyFax: + 39-0737-632525; email: zoli@mail.publigraphic.itAbstractCharge transfer is discussed for the case when gyrotropy parameter (Hall coefficient) varies along transport л-direction and inverses its sign. This situation takes place in contacts o f the serially joined materials having electron and hole types of conductivity.Spatial inhomogeneity o f conductivity and inversion of Hall coefficient sign are analyzed in terms of electric potential and current density distribution. It is shown that under inhomogeneous magnetic field the steady current skinning takes place in plate sample.j^ L Introduction|r Metal heterocontact between the conductors having electron and hole types of conductivity (for I example contact similar to conjunction o f A1 and Cu plate sample) is an example of bianisotropic || medium especially under the action of an external magnetic field. Bianisotropy is a result o f opposite .< signs of Hall coefficients in A1 and Cu. As a result such a contact has a transformation of conducting ; properties from the electron type to the hole that along transport direction and may be represented as |an inhomogeneous medium having gradient type o f conductivity. The inhomogeneity is not determined with only the electron structure via contact. The magnetic field itself is a reason o f ■ roagnetostimulated inhomogeneity o f conductivity and respective the potential picture rearrangement, й As a result the current density redistribution through cross section of sample takes place [1, 2]. So the excessive resistance connected with current line redistribution is a result o f conductivity |inhomogeneity stimulated with magnetic field and heterogeneous electron conducting properties in ifuch a conjunction. In this paper the double type of electric conductivity inhomogeneity stimulated by wanysotropy and by magnetic field is investigated. In other words the processes taking place in metal Bprocontacts placed in inhomogeneous magnetic field are modeled and analysed.2. Experimental and Theoretical Approachg£*e procedure o f modeling o f magnetic field inhomogeneity is based on the method of curving o f ^ent lines so that the normal local component o f magnetic field has a variation along the transportРЕПОЗИТОРИЙ БГПУ120direction in accordance with definite law. For simplicity the heterocontact under the investigation have been chosen as a symmetric one consisted of materials not Al-Cu but АГ-АГ type. Here Al+ is an usual widely used aluminum of hole type of conductivity and АГ is an farfetched electron analogue of АГ which has an electron type of conductivity in magnetic field. That is АГ has an electron type of conductivity so that its Hall coefficient R is equal -\R\ instead of R for usual aluminum. So A f  and AT type components used in experimental modeling process have the same electric resistivity tensors excluding the sign of Hall coefficient. Both components are realized on the base of usual aluminum.3. Results and AnalysisIn experiment two types of magnetic field inhomogeneity in bianisotropy contact medium are modeled. The first type of magnetic field structure is represented in Fig. 1-a) and the second type of magnetic field spatial structure is represented in Fig. 1-b).L, mm L, mmFig. 1 The topology of magnetic field: a) The inversion type of inhomogeneity; b) The symmetric type o f inhomogeneity.L, mm L, mmijmЩIFig. 2 The potential picture along sample length on opposite sides (solid and dashed lines)inhomogeneous magnetic field: a) Field topology Fig. 1-a) where the maximal field В, T: 0.Ш (diamonds), 1.4 (triangles), 4.3 (squares), 7.1 (circles); b) Field topology Fig. 1-b) where the. maximal field is 7.5 T. JЯРЕПОЗИТОРИЙ БГПУ121'fa e the potential distribution has been measured on opposite sides o f sample in accordance with Pheme of potential probes arrangement. Electric field potential picture is represented in Fig. 2 where Ц potential distribution on opposite Hall sides is shown along sample length L in contact region and §§ g t0 it. It is interesting that for inversion type of inhomogeneity o f magnetic field the potential Pcture is symmetric respectively zero field point. The strong and weak spatial dependence of potential Щ L takes place. For symmetric type of magnetic field inhomogeneity the potential dependence on I  nposite sides is different on behavior. One side has abrupt jump of potential in contact region but a^nother side shows very weak dependence on coordinate along transport direction including the Contact region. To analyze this behavior the discontinuity conditions for current density have been i used to write the equation for electric potential cp. Taking into account that the thickness o f samples Bffrected along the magnetic field is rather small in comparison with other dimensions the ^approximation had been used that the current flow picture does not influence on potential distribution lalong z-direction. The carrier motion along magnetic field is neglected and the electric potential is not function of z-coordinate. So for the two-dimensional geometry the potential equation is1 aT +1 <Pz +JXf  ± -  — Л P2 +  P£~- P 1(PxX- J^ T(Py +1 a¥ +l ) (P = 0Гуу ( 1)ere (p'x , (p"x , (--Ox and so on are the derivatives o f respective order, /3' = д(3/ck ; /3 = сот , со is armor frequency, r is a relaxation time. The next type of electric conductivity tensor is usedШ  ШШ0 = 0 ,a 11 +Тг Pl a ___ L lp------Гp ' p2l a lp T “ pPa 1------- 1(2)ere g0 is the conductivity in zero magnetic field. The component cr„ = 1//32 + a/(5 has such view ;er taking into account o f the existence o f a layer o f open electron orbits. The term responsible for s layer is cxJ/3 which must be accepted in modulo. Following such a presentation the tgnetoresistance o f aluminum рхх=1/сг0(1+аД) that is has not very strong slope in linear dependence magnetic field because the parameter alp  describing the quantity o f open electron trajectory is Йег small. Respectively the magnetoresistance for copper may be represented as the same Ijpression where the linear Kapitza law takes place because the same parameter describing the width open electron trajectories is closed to unit. The separation of variables allows to get the total asion of equation for some particular cases:У-(p =  c , It+ a/3-dx exp P'1l-i-aP+ C-> (3)ere this expression is valid when fi'/(l+ap) = const. So for the limit case a->0  one can obtain the tential distribution in approximation o f free electron gas and at nonzero a  the potential picture for иипит and copper separately and for the contact o f these materials can be obtained. Following (3) the potential dependence on sides of sample placed in inhomogeneous magnetic field has strong = b, where b is the sample width) and weak (at у  = 0) dependence on coordinate along transport fo°n. The direction of magnetic field gradient plays very important role in the potential l^ uti°n on the reason that magnetic gradient participates in the governing of transport process via exponential dependence on transverse ^-coordinate. The transverse dependence of potential onРЕПОЗИТОРИЙ БГПУу-coordinate is much higher of that along transport x-coordinate. Here under inhomogeneity the correspondence between transport electric field and Hall field is similar to homogeneous situation. So for particular case o f aluminum type conductor having very small width o f elongated trajectories the potential picture and respective current density distribution along transport direction are:1221(p =  CjJ f52d xexp ([5 'y )+  C2 ; j x =  Q 3 'exp(/3 'y ) ;  p '=  const (4)Following this expression the potential dependence on coordinate in transverse magnetic field is rather complicated than that belonging to the homogeneous magnetic field actions which is (p = C(x+fy)_ Respectively the steady current skinning takes place. Namely near one o f side the current density iJ large and near opposite side the current density is small. Magnetic field gradient inversion transforms the potential picture and the current skinning center to opposite side. This type o f dependence takes ? place in Fig.2a where the effective inversion o f magnetic field gradient sign occurs near zero point f because the conductivity is opposite on Hall effect. For Fig.2b the dependence of (p is governed with the effective magnetic field gradient which has the same sign via total sample. The opposite type of I conductivity near zero point transforms the effective magnetic actions and the abrupt increase of potential corresponds to an action o f exponent in Eq.(4) whereas the weak potential growth on opposite side is a result o f decrease o f exponential include. As to copper type conductor in heterocontact (experimental results we have no yet), the analysis o f Eq.(3) allows to conclude that the , potential picture redistribution due to double inhomogeneity also can be estimated. So for copper type conductor the width o f layer o f open trajectories is to be taken into account and the expressions for the potential dependence is:P ...J P ' f  Pcp = c A — dx exp J aУ + C2 ; —  = const (5):aft I a/3The analysis shows that for copper type conductor the potential redistribution due to inhomogeneity is not so high as in aluminum type conductor. The reason of this is the large number of elongated orbits on isoenergetic surface. The carriers o f these orbits are not so free to drift in gradient magnetic field and as a result more complicated movement o f carriers leads to more weak degree o f electric potential and current skinning.4. Conclusion:|The method of modeling of magnetic field inhomogeneity via curving of current lines allowed to create a physical picture of current flow through the aluminum based heterocontact. Aluminum based heterocontact consists o f pure aluminum sample have been bent in such manner that an effective magnetic field action is equal to the presence o f two heterocomponents having opposite Hall coefficient. Double type of inhomogeneity due to heteroconductivity in magnetic field and due to magnetic field action itself generates current density redistribution via contact region. This redistribution depends strongly on the topology of magnetic field and the excessive heat generation due to current skinning can be taken into account on the base o f data have been analyzed here.References[1] Yu. A. Dreizin and A. M. Dykhne, Zh. Eksp. Teor. Fiz., Vol. 63, p. 242, 1972 [5ov. Phys. JETP, Vol. 36, p. 127, 1972.[2] V. R. Sobol, O. N. Mazurenko, and A. A. Drozd, Fiz .Nizk. Temp., Vol. 25, p. 1211, 1999 Temp. Phys., Vol. 25, p. 907, 1999].РЕПОЗИТОРИЙ БГПУ

Spatial inversion of gyrotropy parameter in conductivity tensor and charge transport peculiarities

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