Abstract
Recently, highly anisotropic conductors, such as multilayer graphene, have been attracting much attention. The local resistivity can be determined by measuring the contact resistance; however, the theoretical expressions of contact resistance have been developed for isotropic slabs but have not been well developed for highly anisotropic film conductors. We obtain theoretical expressions of the spreading resistance below the circular contact for a highly anisotropic film on a bulk slab. The film spreading resistance of isotropic conductors deviates from the bulk spreading resistance when the film thickness is smaller than the contact radius. Nevertheless, the spreading resistance of anisotropic conducting films can be approximated by that of the bulk slabs even when the film thickness is smaller than the contact radius if the in-plane electrical conductivity is larger than the out-of-plane electrical conductivity. Owing to the high in-plane conductivity, the spreading resistance of anisotropic bulk conductors can be lowered from that predicted by the Holm’s equation obtained using the out-of-plane conductivity and the contact radius. We show that these characteristics are beneficial to use the highly anisotropic film as a cover layer when the in-plane conductivity of the film is high and the conductivity of the base slab is low.
Highlights
Highly anisotropic conductors, such as multilayer graphene, have been attracting much attention
For an infinite isotropic conductor, the theoretical ideal resistance in the conductor directly under the contact is known as the spreading resistance and is expressed by Holm’s equation[9], R = 1/(4aσ), where R represents the spreading resistance, a is the radius of the disc electrode and σ is the isotropic electrical conductivity of the conductor slab
Though the specific values relevant to multilayer graphene are used for drawing figures, the theoretical results might apply to the spreading resistance of other anisotropic materials such www.nature.com/scientificreports as conducting polymers, Bi2Te3 films, and composite materials[41,42,43,44,45]
Summary
We consider the spreading resistance for the current passing through the circular domain on an anisotropic conductor. The final result requires numerically solving an integral equation (see Fig. 3) which satisfies both boundary conditions given by Eqs. The commonly used empirical equation of the spreading resistance for isotropic materials with the electrical conductivity σ is given by R = arctan(2h/a)/(2πσa)[30]. The importance of the bottom boundary condition to the value of the spreading resistance of thin films has been pointed out by studying theoretically an isotropic conducting layer of finite width on an insulating base slab, where currents injected from the sides of the layer flow parallel to the bottom surface[31,32,33]. We study the effect of the electrical conductivity of the base slab on the spreading resistance of the anisotropic conducting layer
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