Extraordinary Magnetoresistance in Hybrid Semiconductor-Metal Systems
NNovember 19, 2018 22:31 WSPC/INSTRUCTION FILE Magnetoresis-tance
Extraordinary Magnetoresistance in HybridSemiconductor-Metal Systems
T.H. Hewett and F.V. Kusmartsev
Department of Physics, Loughborough University, Loughborough, LE11 3TU, UK
We show that extraordinary magnetoresistance (EMR) arises in systems consisting oftwo components; a semiconducting ring with a metallic inclusion embedded. The im-portant aspect of this discovery is that the system must have a quasi-two-dimensionalcharacter. Using the same materials and geometries for the samples as in experiments bySolin et al. , , we show that such systems indeed exhibit a huge magnetoresistance. Themagnetoresistance arises due to the switching of electrical current paths passing throughthe metallic inclusion. Diagrams illustrating the flow of the current density within thesamples are utilised in discussion of the mechanism responsible for the magnetoresistanceeffect. Extensions are then suggested which may be applicable to the silver chalcogenides.Our theory offers an excellent description and explanation of experiments where a hugemagnetoresistance has been discovered , . Keywords : Magnetoresistance; Current paths; Strongly inhomogeneous.
1. Introduction
The magnetoresistance effect known as extraordinary magnetoresistance (EMR) wascoined by Solin et al. after the experimental discovery in 2000. This effect showedextremely large room temperature magnetoresistance values of a million percentin a 5T applied magnetic field . These values were found in composite van derPauw disks consisting of a conducting and semiconducting region of non-magneticmaterials. As a result of the observed properties of EMR it has been proposed thatimprovements could be achieved in the read heads of magnetic disk drives , .In 1997 Xu et al. discovered a large magnetoresistance in non-magnetic silverchalcogenides Ag δ Se and Ag δ T e . The addition of a small excess amount ( δ ) ofsilver atoms into the structure of the semiconductors Ag Se and Ag T e (with noappreciable magnetoresistance themselves) caused a large linear magnetoresistanceto be observed. At room temperature and in a magnetic field of 5.5T the magnetore-sistance was 200% with no sign of saturation. The magnetoresistance displayed alinear dependance on magnetic field which remained in low fields . A model for thismagnetoresistance has been proposed , , which considers the silver chalcogenidesas consisting of two components: a semiconductor and a conducting component.This is analogous to the EMR effect in systems where it may arise. Bulgadaevand Kusmartsev , have found explicit expressions for the magnetoresistance ofstrongly inhomogeneous two-phase systems by developing a method of conformalmapping transformations. This method utilises the exact dual transformation in re- a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l ovember 19, 2018 22:31 WSPC/INSTRUCTION FILE Magnetoresis-tance lating the effective conductivity of plannar inhomogeneous two-phase systems withand without an applied magnetic field. They have studied three models: the randomdroplet model (RDM); the random parquet model (RPM); and an effective mediummodel (EMM).Parish and Littlewood have modelled a strongly inhomogeneous conductor us-ing a random resistor network made from four terminal resistors, and have shownthat the properties of the magnetoresistance are similar to those of the silver chalco-genides. Abrikosov proposed an explanation of the abnormal magnetoresistanceobserved in the silver chalcogenides, namely quantum magnetoresistance, based onthe assumption that in such systems gapless dirac fermions arise. Recent experimen-tal work has verified that the magnetoresistance effect in the silver chalcogenidesis somehow related to the microstructure of the material, which has been identifiedand investigated , .In this paper van der Pauw disks consisting of a conducting and semiconductingregion of non-magnetic materials are considered. Their electrical properties, suchas conductivity and the distribution of the electric field and current, are studiedusing finite element simulations. These were based on the EMR geometry usedby Solin et al. when the effect was first discovered. The experimental results areverified and the mechanism for the effect discussed with the use of streamline plotsof total current density. The paper is structured as follows: firstly, a description ofthe approach taken in order to produce the simulations including deviations fromthe experimental geometry; secondly, the magnetoresistance results are given alongwith the streamline plots; and finally, discussion of the results in comparison withprevious experiments, with the mechanism for the magnetoresistance offered as wellas proposals for extension.
2. Simulation approach
We modelled the experimental construction of the composite van der Pauw disk,as used by Solin et al. , in the discovery of the EMR effect. The experimentalconstruction consisted of a circular disk of Indium Antimonide (InSb) of radius r b =0 . mm with a concentric inclusion of Gold (Au) of radius r a . Different geometrieswere achieved by varying the radius of the Au inclusion and quantified with the useof the filling factor α . α = r a r b = n
16 (1)Alternatively the volume fraction (f), the fraction of conducting to semiconduct-ing material, can be defined as follows. f = πr a πr b = α (2)Experimentally disks were grown on a Gallium Arsenide (GaAs) substrate, witha 1 . µm thick active layer of InSb. It was noted that the side walls between the InSbovember 19, 2018 22:31 WSPC/INSTRUCTION FILE Magnetoresis-tance r b )with a conducting inclusion (of radius r a ), including the four point contacts. Here the current isflowing through contacts A and B, while the voltage is measured across contacts C and D. and the Au regions were approximately 19 ◦ from vertical . The magnetoresistancewas determined utilising the van der Pauw method of measuring the resistivityof a thin film . This required four Au contacts to be placed equidistant aroundthe perimeter of the disk. Two adjacent contacts were used to inject the currentinto the system, with the other two contacts used to measure the resulting poten-tial difference. To measure the magnetoresistance the magnetic field was appliedperpendicular to the surface of the disk.In order to simulate this system some aspects were required to be adapted. Thefirst simplification made was to simulate the system in two dimensions. This wasjustified since the experimental disk was of a thin film construction. Consequently,experimental errors in the angle of the side walls between the two regions were ne-glected. Additionally, only the active layer of the disk was considered with all otherlayers of experimental significance ignored for the purposes of simulation. The fourcontacts on the disk were assumed to be point contacts with experimental errorsdisregarded. The simulations were carried out using the finite element analysis soft-ware package COMSOL multiphysics. With such assumptions the two-dimensionalgeometry used for simulations is given in Fig. 1.Various geometries have been considered, with the filling factor varying by 1 / r b from 0 to 15 /
16. With the geometry considered, various parameters were de-fined and quantified. The simulations were carried out with an applied field rangingfrom 0 – 5T mirroring those used experimentally . In order to measure the mag-ovember 19, 2018 22:31 WSPC/INSTRUCTION FILE Magnetoresis-tance netoresistance an electrical current was applied through contacts A and B. Usingthe van der Pauw method for measurement of the resistivity leads to the followingexpression of the magnetoresistance. M R = R ( H ) − R (0) R (0) = V CD ( H ) V CD (0) − V CD ( H ) = V C ( H ) − V D ( H ). In order to specify the required mate-rials the conductivities and charge carrier mobilities were utilised to establishthe conductivity tensor for each of the two regions. The conductivity and mo-bility values used were as close to the experimental system as possible. The val-ues of the conductivity used for InSb and Au were σ InSb = 1 . × (1 / Ω m )and σ Au = 4 . × (1 / Ω m ) respectively. With the corresponding mobilities of µ InSb = 4 . m /V s ) and µ Au = 5 × − ( m /V s ). Using these values, and theparameter β = µH , the conductivity tensor could be defined for each of the twomaterials using the following expression.ˆ σ = (cid:18) σ xx σ xy σ yx σ yy (cid:19) = (cid:32) σ β − σβ β σβ β σ β (cid:33) (4)Finally the boundary conditions were specified. On the perimeter of the entiredisk (the semiconducting region) they were given as electrical insulation, while at theinterface between the semiconducting ring and the conducting inhomogeneity theywere set to continuity. These simulations were carried out with a mesh consisting ofapproximately 60,000 triangular elements, with the mesh being more refined wherethe variation in the potential was greatest.
3. Results
Fig. 2 presents the magnetoresistance as a function of applied magnetic field. Herewe see the magnitude of the magnetoresistance reaching 3,750,000% in a geometrywith n = 15 ( f = 0 . n = 1 – 8) show saturation in the magnetoresistance at amagnetic field of approximately 1.5T. Larger inclusions require a larger magneticfield to reach their maximum magnetoresistance value and thus saturation occurs athigher magnetic fields. The geometries with n = 14 and 15 do not exhibit saturationof the magnetoresistance in a field of up to 5T. This indicates that the saturationfield increases with the size of the conducting region.The form and magnitude of the simulated results are in agreement with exper-imental data , with the values especially close at lower applied fields. The magne-toresistance being found to be 79% with n = 12 ( f = 0 . n = 13 ( f = 0 . (R-R0)/R0 (104 %) H ( T )
Fig. 2. The magnetoresistance as a function of applied magnetic field. Here the symbols corre-spond to various geometries: n = 15 ( (cid:72) ), 14 ( (cid:5) ), 13 ( • ), 12 ( ◦ ), 11 ( (cid:78) ), 10 ( (cid:77) ) and 8 ( (cid:79) ). Thelargest value of magnetoresistance seen at a field of 5T corresponds to a geometry with n = 15,with the lowest value of magnetoresistance at 5T corresponding to a geometry with n = 8. magnetic field, perpendicular to the surface of the system (z direction), only chargecarrier motion in the x-y plane is affected. Charge carriers travel with a motion un-altered by the magnetic field when their trajectory is in a parallel direction (alongthe z axis). Therefore, if the conducting inclusion does not extend the entire thick-ness of the structure alternate current paths are created (the current can avoid theconducting inclusion by travelling underneath). This causes a dramatic reductionin the magnitude of the EMR effect.The magnitude of the simulated magnetoresistance is found to be larger thanthat found experimentally for n = 14 ( f = 0 . f = 0 . α ) at 5 various applied magneticfields: H = 0.05T ( (cid:79) ), 0.1T ( (cid:77) ), 0.25T ( (cid:72) ), 1T ( (cid:78) ) and 5T ( • ) that deviate from vertical. However, very good agreement is found for smaller Auinclusions, for example a magnetoresistance of 39 × % for n = 12 ( f = 0 . × % for n = 13 ( f = 0 . α .This plot shows a good agreement to the experimental results yet some differencesare apparent. We notice that in general the magnetoresistance increases with α up to a certain value, with the value of α at which the magnetoresistance peaksincreasing with higher applied fields. At low magnetic fields (H = 0.05, 0.1 and0.25T) the magnetoresistance peaks at n = 12 and for higher fields (H = 1 and 5T)peaking at n = 13 and 15. One may notice that the data for the field of 5T doesnot fall after the peak magnetoresistance occurs. Instead the magnetoresistance isits largest for n = 15, the geometry with the largest Au inclusion. Experimentally itovember 19, 2018 22:31 WSPC/INSTRUCTION FILE Magnetoresis-tance was found that the magnetoresistance dropped after n = 13 in a 5T field and did notcontinue to increase. This difference may be a result of experimental uncertainties,especially those errors in the contacts deviating from ideal point contacts. Theerrors associated with the contacts are more pronounced in geometries where thesemiconducting ring is narrow (large values of α ).We represent the path of the total current density for a single system where n =12 (the concept generally holds for other geometries) at various applied magneticfields in Figs. 4(a) – (d). The streamlines (red) link points of the same current densitywhile the arrows (blue) show the magnitude and direction of the current flow atthat particular point in the system. These diagrams shows the current injected atpoint A and taken off at point B.In Fig. 4(a) we see the path of the current flow throughout the system with zeroapplied magnetic field. The current flow here is predominantly directed into the Auinclusion, which can be seen most clearly from the arrows (blue), with very littleof the total current flowing around the perimeter of the disk in the semiconduct-ing region. At zero magnetic field the electric field lines are perpendicular to theequipotential Au surface. With no applied magnetic field the current flow is parallelto the electric field lines and so flows in the same direction, into the Au inclusion.This flow into the conducting region constitutes a system of low resistance.In the next diagram Fig. 4(b), the applied magnetic field is 0.5T. The current isstill predominantly flowing through the Au inclusion however the magnetic field hashad the effect of distorting the current path. Now a larger proportion of the currentis being forced to flow around the semiconducting region of high resistance. Themagnetic field causes the electric field (which is still perpendicular to the surfaceof the Au inclusion) and the current flow to stray away from being parallel withthe electrical field lines. The angle between the electric field and the current (Hallangle) increases with higher applied magnetic fields.There is a continuation of this trend in Fig. 4(c) where a larger magnetic field( H = 1 T ) has caused the angle between the electric field and the current flowto increase to near 90 ◦ . This means the current flow and electric field are almostperpendicular. Thus the current is directed away from the conducting region andforced to flow through the semiconducting outer ring of much higher resistance. Wesee here that most of the current (arrow plot) is travelling from point A to point Bthrough the outer ring of the system and avoiding the lower resistance Au path.Finally, with a high applied magnetic field of H = 5 T represented in Fig. 4(d) thedirections of the current flow and the electric field are almost perpendicular. Nowthe majority of the current is flowing around the Au inhomogeneity in the region ofsemiconductor material. The application of the magnetic field causes the current toavoid the low resistance path and forces it to flow along narrow (depending on thevalue of α ) channels of semiconductor. This results in a system with a substantiallyhigher resistance than when no magnetic field is applied. This effect causes the hugegeometric magnetoresistance known as EMR that was found in experiments .The geometry of the system plays an important role in the magnetoresistanceovember 19, 2018 22:31 WSPC/INSTRUCTION FILE Magnetoresis-tance n = 12 ( f = 0 . H = 0, (b) H = 0 .
5, (c) H = 1 and (d) H = 5. effect. With a relatively small Au inclusion a large proportion of the current alreadyflows through the wide semiconducting channel at H = 0. With the application ofthe magnetic field in such geometries a small proportion of the current switches fromovember 19, 2018 22:31 WSPC/INSTRUCTION FILE Magnetoresis-tance flowing through the Au to through the semiconductor. This leads to smaller valuesof magnetoresistance with low values of α . With large Au inclusions however, moreof the current initially flows through the Au and thus gets switched to flow throughthe semiconductor when a magnetic field is applied. More of the current is affectedby the application of the magnetic field, therefore larger values of magnetoresistanceare observed. Additionally in geometries with large conducting inclusions the semi-conducting channels are narrow, this increases the resistance of current flow throughthe semiconductor material, thus further enhancing the magnetoresistance.In geometries with large conducting droplets a larger field is required to forcethe current to flow through the narrow semiconducting regions. For this reasonthe geometries with larger values of α peak in their values of magnetoresistance athigher applied fields. This explains the trend of the peak magnetoresistance valueoccurring at higher values of α at higher magnetic fields observed both here (Fig.3) and in experimental data .These current flow diagrams reinforce ideas discussed by Solin et al. concerningthe EMR effect. They regard the system as a short circuit (current flowing throughthe conducting region) when the magnetic field is small and an open circuit (currentdeflected around conducting region) when the magnetic field is large.The application of the magnetic field can be thought of as a switch; with zeroor a small applied field the majority of the current flows through the conductingAu inclusion giving a low resistance path. However, the application of a large mag-netic field forces the current to flow around the perimeter of the disk avoiding theconducting inclusion, as if there was a ring of semiconductor with a cavity in thecentre, thus creating a significantly higher resistance path, see Fig. 5.ovember 19, 2018 22:31 WSPC/INSTRUCTION FILE Magnetoresis-tance
4. Further Discussion
The EMR effect strongly depends on the geometry of the system. The physicalcomponent to the magnetoresistance (ordinary magnetoresistance) being almostnegligible for both the conducting and semiconducting materials individually. Otherlarge magnetoresistance effects originate from a large physical component that relieson the electrical properties of the materials involved. The EMR effect may thereforebe applicable to many different systems, where droplets or regions of material witha significantly higher conductivity to that of the surrounding material are observed.Due to the high economic value of Au, practical devices based on EMR wouldbe more cost effective to produce if the EMR effect was efficient and produced thehighest value of magnetoresistance with a fixed quantity of Au. In order to optimisethe amount of Au required the geometry could be modified, for example, with theimplementation of different shaped droplets (oval shaped droplets instead of circularones) .Here the effect of a single conducting droplet embedded in a semiconductor isinvestigated, by replicating the experimental geometry in which the EMR effect wasdiscovered. However, further research will be carried out investigating the effect ofmultiple conducting droplets embedded in a semiconducting material. Simulationof a random droplet model may mean the magnetoresistance follows a linear growthwith magnetic field as is desirable practically. A multiple droplet system may behard to create precisely but some systems form droplets of conducting and semicon-ducting material spontaneously. An example of this is the silver chalcogenides whichhave shown a large room temperature magnetoresistance, where a small excess ofconductor is added to a semiconducting material. The addition of this small excessamount of conducting material has been shown to form a percolating silver net-work along the grain boundaries (for already low silver excess) and even conductingdroplets in the microstructure of the material (for higher excess silver content) inanalogy with this work. It would be interesting to look at a random parquet modelas a parquet type structure has also been identified .
5. Conclusion
Since this paper has been produced a similar piece of work has come to our atten-tion, namely that of Moussa et al. , which documents finite element simulationsregarding the EMR effect. Their results show the field dependence of the currentflow, the potential on the disk periphery and compare the disk resistance with ex-periment. These simulations were carried out with a mesh consisting of 6000 nodalpoints. The results given are in excellent agreement with those presented in this pa-per and with the experimental data . Our results were calculated using a much moresophisticated mesh with the magnetoresistance explicitly calculated and comparedto the experimental data.We have investigated the EMR effect using finite element simulations. The sim-ulated geometry was considered in two-dimensions with parameters described inovember 19, 2018 22:31 WSPC/INSTRUCTION FILE Magnetoresis-tance experiments by Solin et al. so as to replicate these systems. The magnetoresistancewas calculated for different geometries ( n = 1 – 15) and in a range of magneticfields (H = 0 – 5T).We have found that a huge magnetoresistance appears, due to the geometry ofthe system, and our results show good agreement with the experimental values. Thesize of the conducting inclusion is found to affect the magnetoresistance greatly. Wehave looked at diagrams of the total current density flow throughout the samples atvarious applied magnetic fields and discussed the mechanism for the resulting hugemagnetoresistance. This model can now be utilised in the investigation of the effectof various droplet shapes on the magnetoresistance. References
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