The study of the sample size on the transverse magnetoresistance of bismuth nanowires M. Zare n , R. Layeghnejad, E. Sadeghi Department of Physics, Yasouj University, Yasouj 75918-74831, Iran article info Article history: Received 9 May 2011 Received in revised form 15 November 2011 Available online 16 February 2012 Keywords: Magnetoresistance Bismuth nanowire Size effect Specular reflection abstract The effects of sample size on the galvanomagnetice properties of semimetal nanowires are theoretically investigated. Transverse magnetoresistance (TMR) ratios have been calculated within a Boltzmann Transport Equation (BTE) approach by specular reflection approximation. Temperature and radius dependence of the transverse magnetoresistance of cylindrical Bismuth nanowires are given. The obtained values are in good agreement with the experimental results, reported by Heremans et al. & 2012 Elsevier B.V. All rights reserved. 1. Introduction The galvanomagnetic transport properties of the semimetal nano- wires have been of great experimental interest in recent years [1–3]. Among the semimetals, Bismuth with a Rhombohedral structure, has received increasing attention and is distinguished from similar samples for a number of reasons: firstly, at low temperatures the carrier concentrations are much smaller by a factor of 10 000 compared to most metals, resulting in a relatively large resistivity, in spite of the long mean free path and wavelength. Secondly, the electron mean free path at low temperatures in bulk Bi can be on the order of a millimeter [4], resulting in strong ballistic effects [5]. Thirdly, the carrier effective masses are small and highly anisotropic so that they could lead to increased quantum confinement effects, with a large spatial extension of the electron wave functions. At low carrier densities, semimetal-semiconductor (SMSC) phase transitions can occur for critical wire diameter d c [6–8]. Furthermore, at this phase transitions a significant enhancement is expected in the thermoelectric figure of merit over bulk Bi [7]. There are two types of size effects observable in thin metal samples. The ordinary size effect (OSE), which is seen when the charge carrier mean free path (MFP) is comparable with or greater than the sample diameter, which results in a resistivity is higher than the bulk values, due to the additional scattering of the charge carriers on the sample surface. When the De Broglie wavelength of an electron is comparable to the dimensions of the sample, quantum size effect (QSE) becomes important and deviations from the bulk behavior in transport properties are expected. The QSE manifests itself in the oscillatory behavior of the electron density of states. Although a number of studies have been carried out on the size effect on the conductivity of thin films [9], quantum holes [10] and superlattices [11], theoretically speaking, only a small amount of progress has been achieved for the size effect on the magnetoresistance of nanowires, theoreti- cally [3,12]. Size effect in the thin metallic films (with a spherical Fermi surface) has been given by Fuchs–Sondheimer, firstly [13]. The scattering of the carriers from the surface is characterized by a parameter P, the fraction of the carriers which are reflected on the boundary surface in a speculary way. Price studied the size effect in thin metallic films with ellipsoidal Fermi surfaces and deducted that for specular reflection the conductivity approaches a finite limit for every thin film [14]. The longitude and transverse magnetoresistance have been calculated numerically [15–17]. In this paper, the galvanomagnetic properties of Bi nanowires were explained by Parrott theorem [18] by solving the Boltzmann Transport Equation. In the previous work [19] the special elec- trical resistivity of Bi thin wire was calculated per an ellipsoidal electron pocket and the results were verified by Gurvitch experi- mental results [16]. The TMR of bismuth–like cylindrical nano- wires (with three ellipsoid electron pockets and one hole pocket) was calculated using Parrott theorem extension [18]. TMR depen- dence of radius, temperature and magnetic field were studied, too. The obtained values of TMR from the framework of this method are in good agreement with the experimental results reported by Heremans et al. [17] for Bismuth nanowires. 2. Theory In order to study the size effect on the galvanomagnetic transport properties of nanowires, it is supposed that the sample Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jmmm Journal of Magnetism and Magnetic Materials 0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2012.01.034 n Corresponding author. E-mail address: Mzare@mail.yu.ac.ir (M. Zare). Journal of Magnetism and Magnetic Materials 324 (2012) 2331–2335