Magnetoresistance from Fermi surface topology

Extremely large nonsaturating magnetoresistance has recently been reported for a large number of both topologically trivial and nontrivial materials. Different mechanisms have been proposed to explain the observed magnetotransport properties, yet without arriving to definitive conclusions or portraying a global picture. In this work, we investigate the transverse magnetoresistance of materials by combining the Fermi surfaces calculated from first principles with the Boltzmann transport theory approach relying on the semiclassical model and the relaxation time approximation. We first consider a series of simple model Fermi surfaces to provide a didactic introduction into the charge-carrier compensation and open-orbit mechanisms leading to nonsaturating magnetoresistance. We then address in detail magnetotransport in three representative materials: (i) copper, a prototypical nearly free-electron metal characterized by the open Fermi surface that results in an intricate angular magnetoresistance, (ii) bismuth, a topologically trivial semimetal in which very large magnetoresistance is known to result from charge-carrier compensation, and (iii) tungsten diphosphide ${\mathrm{WP}}_2$, a recently discovered type-II Weyl semimetal that holds the record of magnetoresistance in compounds. In all three cases our calculations show excellent agreement with both the field dependence of magnetoresistance and its anisotropy measured at low temperatures. Furthermore, the calculations allow for a full interpretation of the observed features in terms of the Fermi surface topology. Our study thus establishes guidelines to clarifying the physical mechanisms underlying the magnetotransport properties in a broad range of materials. These results will help addressing a number of outstanding questions, such as the role of the topological phase in the pronounced large nonsaturating magnetoresistance observed in topological materials.

Figure 1

(a) Band structure of the two-band model described by the Hamiltonian of Eq. (6). (b)–(d) Fermi surfaces for Fermi energies $E_{\text{F}}=0$, 0.2 and $-0.2\mathrm{eV}$, indicated in blue, black and red in panel (a), respectively. Field dependence of (e) magnetoresistivity and (f) Hall resistivity for the three different Fermi energies.

Figure 2

(a) Field dependence of resistivity ${\rho }_{yy}$ in the anisotropic two-band model described by Hamiltonian (9). The inset shows the Fermi surface. (b) Resistivity ${\rho }_{yy}$ as a function of magnetic field orientation. The inset shows individual resistivities ${\rho }_{yy}$ of electrons (red) and holes (blue for $B\parallel z$ and purple for $B\parallel x$). The resistivity of electrons is scaled as ${\rho }_{yy}/1.62$ in order to make this curve visible.

Figure 3

(a) Field dependence of resistivities ${\rho }_{yy}$ and ${\rho }_{zz}$ for the isotropic open Fermi surface model. The magnetic field is applied along the $x$ axis. The ${\rho }_{yy}$ resistivity is multiplied by ${10}^3$ to make it visible. (b) Field dependence of resistivity ${\rho }_{zz}$ for the anisotropic open Fermi surface model with magnetic field oriented along the $x$ and $y$ axes. The insets show the corresponding Fermi surfaces.

Figure 4

(a) Fermi surface of copper. (b) Polar diagram that shows the dependence of experimentally measured $\delta {\rho }_{xx}\left(B\right)/\rho \left(0\right)$ on magnetic field direction (adopted from Ref. [62]). The measurements were performed on a single crystal of copper at $T=4.2$ K and $B=1.8\mathrm{T}$ rotated in the plane normal to the current direction. (c) Calculated anisotropy of resistivity ${\rho }_{xx}$ for magnetic field rotated in the $y-z$ plane agrees well with experimental results in panel (b) given $B\tau$ corresponds to $\omega \tau \gg 1$. (d). Resistivity ${\rho }_{xx}$ as a function of the magnitude of magnetic field $B$ for the four field directions indicated in panel (c).

Figure 5

Typical cross sections of the Fermi surface of copper for magnetic field oriented along the $z$ axis [point A in Fig. 4]. The horizontal axis is along the $k_x$ direction. Cross sections in the $k_x-k_y$ plane correspond to (a) $k_z=0$, (b) $k_z=0.3\pi /a$, (c) $k_z=0.62\pi /a$, and (d) $k_z=0.63\pi /a$. Pink and blue dashed lines highlight the closed electron and hole orbits, respectively.

Figure 6

Typical cross sections of the Fermi surface of copper that correspond to magnetic field orientations marked by points (a) B, (b) C, and (c)–(e) D in Fig. 4. The horizontal axis corresponds to the $k_x$ direction while the vertical axis is the direction parallel to ${\widehat{k}}_x\times \mathbf{B}$. In panels (a)–(c) the plane includes the $\mathrm{\Gamma }$ point, while those in panels (d) and (e) pass through points $(0,0.15\pi /a,0.15\pi /a)$ and $(0,0.52\pi /a,0.52\pi /a)$, respectively. The pink dashed lines highlight one of the open orbits in (a) and (b). Panel (c) shows open orbits extend along the $k_x$ direction. Panels (d) and (e) show closed electron and hole orbits, respectively.

Figure 7

Comparison of the experimentally measured and calculated resistivity anisotropy of bismuth. (a)–(c) Resistivity anisotropy of bismuth measured at magnetic field $B=0.5\mathrm{T}$ and temperatures $T=10$ K, $T=20$ K, and $T=15$ K, respectively, for the three indicated current directions (reproduces Fig. 5 in Ref. [25]). (d)–(f) Calculated resistivity anisotropy of bismuth for the same current directions. We assumed $B\tau =2.2\mathrm{T}\mathrm{ps}$ in our calculations, and similar results can be obtained at weaker magnetic fields.

Figure 8

(a) Schematic drawing of the Fermi surface projection of bismuth onto the $x-y$ plane normal to the trigonal axis. Hole and electron pockets are shown in blue and pink, respectively. (b) Magnetic field dependence of ${\rho }_{zz}$ for magnetic field $B$ oriented along the bisectrix and binary axes. The inset shows ${\rho }_{zz}$ for electron and hole contributions separately, with the resistivity of hole charge carriers scaled to ${\rho }_{zz}/22$ in order to fit the plot. (c),(d) Cross sections of the Fermi surface for the case of magnetic field $B$ oriented along the $x$ axis. Horizontal and vertical axes are along the $k_y$ and $k_z$ directions, respectively. (e)–(g) Cross sections of the Fermi surface for magnetic field $B$ along the $y$ axis. Horizontal and vertical axes are along the $k_x$ and $k_z$ directions, respectively. Enlarged cross sections are shown due to the very small size of the Fermi surface pockets in bismuth.

Figure 9

(a) Fermi surface of ${\mathrm{WP}}_2$ composed of a bowtie electron pocket (pink) and an open tube hole pocket (blue) extending along the $y$ direction. (b) Measured (adopted from Ref. [21]) and (c) calculated resistivity ${\rho }_{xx}$ as a function of magnetic field orientation for different magnetic field strengths, with $B\parallel b$ being the reference field orientation. (d),(e) Fermi surface cross sections for magnetic field along the $b$ and $c$ axes, respectively. (f) Individual resistivities ${\rho }_{xx}$ of electron and hole charge carriers for $B\parallel b$ and $B\parallel c$. (g) Field dependence of resistivity ${\rho }_{xx}$ for the cases of magnetic field oriented along the $b$ and $c$ directions. The inset shows experimentally measured ${\rho }_{xx}\propto B^{1.94}$ for $B\parallel b$ that can be compared with the calculated ${\rho }_{xx}\propto B^{1.91}$ scaling [red line in the main figure (g)]. (h) Individual Hall resistivities ${\rho }_H$ for electron and hole charge carriers in magnetic field oriented along the $b$ and $c$ directions. (i) Electron and hole orbits in real space for the Fermi surface cross sections shown in panels (d) and (e). The orbits self-intersect due to the presence of concave segments of the Fermi surface [64].