Equivalence of effective medium and random resistor network models for disorder-induced unsaturating linear magnetoresistance

A linear unsaturating magnetoresistance at high perpendicular magnetic fields, together with a quadratic positive magnetoresistance at low fields, has been seen in many different experimental materials, ranging from silver chalcogenides and thin films of InSb to topological materials like graphene and Dirac semimetals. In the literature, two very different theoretical approaches have been used to explain this classical magnetoresistance as a consequence of sample disorder. The phenomenological random resistor network model constructs a grid of four terminal resistors, each with a varying random resistance. The effective medium theory model imagines a smoothly varying disorder potential that causes a continuous variation of the local conductivity. Here, we demonstrate numerically that both models belong to the same universality class and that a restricted class of the random resistor network is actually equivalent to the effective medium theory. Both models are also in good agreement with experiments on a diverse range of materials. Moreover, we show that in both cases, a single parameter, i.e., the ratio of the fluctuations in the carrier density to the average carrier density, completely determines the magnetoresistance profile.

Figure 1

Disorder-induced magnetoresistance has been historically modeled by the random resistor network (RRN) and the effective medium theory (EMT), shown on the right and the left, respectively. The former discretizes the material into a network of four terminal resistors where the resistance of each unit is random, while the latter models the material as an amalgamation of puddles where the conductivity is a continuous random variable.

Figure 2

Four-terminal resistor unit that comprises the random resistor network. Voltage differences $V_i$ between the equally spaced terminals are linearly related to the currents $I_i$ via $V_i=z_{ij}{I}_j$, where $z$ is the impedance matrix defined in Eq. (2).

Figure 3

The numerical collapse of both the RRN and EMT disorder models onto a single curve when the $x$ and $y$ axes are scaled by $B_q$ and $\mathrm{MR}\left(B_q\right)$, respectively, shows that the theories belong to the same universality class. The mathematical window shows where all possible quadratic-to-linear curves may lie. Any analytic representation of these theories would only need two parameters corresponding to $B_q$ and $\mathrm{MR}\left(B_q\right)$. Note that the disorder parameter $\eta =\frac{n_0}{n_{\mathrm{rms}}}$ covers a range spanning 2 orders of magnitude for both the EMT and the RRN.

Figure 4

Equivalence of the RRN and EMT theories: Both the EMT and RRN theories can be formulated in terms of $n_0$ and $n_{\mathrm{rms}}$. The magnetoresistance is plotted here as a function of $\mu B$ for disorder parameter $\eta =\frac{n_0}{n_{\mathrm{rms}}}=0.1,1,10$. Notice that both theories depend only on the ratio of the average carrier density to the fluctuations in carrier density and give the same value of MR in both the low-field quadratic regime and the high-field linear regime, thus establishing the equivalence of the two theories.

Figure 5

The quadratic coefficient of magnetoresistance $\mathcal{A}=\frac{\mathrm{MR}}{{\left(\mu B\right)}^2}$ in the limit $\mu B\ll 1$ and the linear coefficient of MR defined as $\mathcal{B}=\frac{\mathrm{\Delta }\mathrm{MR}}{\mu \mathrm{\Delta }B}$ in the limit $\mu B\gg 1$ for both theories. Agreement of $\mathcal{A}$ at $\eta =0.1$ is through fitting, but agreement at all other $\eta$ for $\mathcal{A}$ and $\mathcal{B}$ shows clear evidence that the two theories are equivalent. This is in contrast to the the quadratic-to-linear phenomenological formula that has been previously used to characterize MR [43], which shows quantitatively different results for the relationship between $\mathcal{A}$ and $\mathcal{B}$. Note that at large $\eta$, we have $\mathcal{A}\approx {\eta }^{-2}$ and $\mathcal{B}\approx {\eta }^{-1}$.

Figure 6

Magnetoresistance data from the graphene samples of Ref. [22] show good agreement with the EMT-RRN formalism. In these experiments, the mobility is known (one measures the mobility in the regime with low disorder by using a back gate to increase carrier density) and $\eta$ is a fit parameter. The magnetoresistance data can be used as a measure of disorder $\eta$ or vice versa.

Figure 7

Experimental magnetoresistance data using Dirac semimetal TlBiSSe (Ref. [26]), silver selenide (Ref. [21]), and indium antimonide (Ref. [44]) are compared against the EMT-RRN formalism. A back gate to increase the carrier density and measure mobility was not possible and hence there are two free parameters, $\eta$ and $\mu$, for the theory. We therefore use only the low-field data (shown in the insets) to find the best fit for both $\eta$ and $\mu$. Good agreement is found in each case in the high-field linear regime, and the fitted $\mu$ is within the range of the experimental Hall mobilities of these materials. We conclude that the disorder-induced magnetoresistance in these diverse range of materials can indeed be explained by this formalism.