Extended Kohler$^,$s Rule of Magnetoresistance

A notable phenomenon in topological semimetals is the violation of Kohler$^,$s rule, which dictates that the magnetoresistance $MR$ obeys a scaling behavior of $MR = f(H/\rho_0$), where $MR = [\rho_H-\rho_0]/\rho_0$ and $H$ is the magnetic field, with $\rho_H$ and $\rho_0$ being the resistivity at $H$ and zero field, respectively. Here we report a violation originating from thermally-induced change in the carrier density. We find that the magnetoresistance of the Weyl semimetal, TaP, follows an extended Kohler$^,$s rule $MR = f[H/(n_T\rho_0)]$, with $n_T$ describing the temperature dependence of the carrier density. We show that $n_T$ is associated with the Fermi level and the dispersion relation of the semimetal, providing a new way to reveal information on the electronic bandstructure. We offer a fundamental understanding of the violation and validity of Kohler$^,$s rule in terms of different temperature-responses of $n_T$. We apply our extended Kohler$^,$s rule to BaFe$_2$(As$_{1-x}$P$_x$)$_2$ to settle a long-standing debate on the scaling behavior of the normal-state magnetoresistance of a superconductor, namely, $MR$ ~ $tan^2\theta_H$, where $\theta_H$ is the Hall angle. We further validate the extended Kohler$^,$s rule and demonstrate its generality in a semiconductor, InSb, where the temperature-dependent carrier density can be reliably determined both theoretically and experimentally.


I. Introduction
The magnetic-field-induced resistance change is conventionally termed as magnetoresistance (MR) 1 . In 1938, Max Kohler 2 developed a rule to account for the magnetoresistances in metals. Kohler's rule states that the magnetoresistance MR should be a function of the ratio H/r0 of the magnetic field H to the zerofield resistivity r0. That is, the field dependence of the magnetoresistances should exhibit a scaling behavior of MR = f(H/r0), where MR = [r(H)-r0]/r0 with r(H) and r0 being the resistivity at H and zero field at a fixed temperature, respectively. Kohler's rule of the magnetoresistances has been observed in materials [3][4][5][6][7][8][9][10][11][12][13][14][15] beyond metals and recently has been extensively used to understand novel magnetoresistance behavior such as the 'turn-on' temperature behavior of the magnetoresistance in topological materials [5][6][7][8][9][10][11][12][13][14][15] . Violations of Kohler's rule have been often reported  and used as evidence for phase transitions 16,17 or for emergent new physics [18][19][20][21][22] . Here we explore Kohler's rule of magnetoresistances in Weyl semimetals, where both its validity 5,6,8 and violation [23][24][25]27,28 have been reported, with the aim to uncover the role played by the inevitable thermally induced change in the density of carriers on the scaling behavior. We find that the magnetoresistance of the Weyl semimetal, tantalum phosphide (TaP), follows an extended Kohler's rule MR = f[H/(nTr0)], where nT describes the relative change induced by thermal excitation in the carrier density, with nT = 1 denoting the original Kohler's rule. We outline an innovative approach to obtain nT without knowing the values of the carrier density, providing a new way to probe the temperature dependence of carrier density. We show that nT is associated with the Fermi level and the dispersion relation, thereby revealing information on the electronic bandstructure. We offer a fundamental description of the violation and validity of Kohler's rule in terms of different temperature responses of nT. In particular, Kohler's rule is expected to be violated in materials with low carrier density where a noticeable density change due to thermal excitation occurs, while Kohler's rule should hold in materials where the carrier density is high enough such that its thermally-induced change is experimentally indiscernible. Furthermore, by investigating the scaling behavior of the normal-state magnetoresistance in superconducting BaFe2(As1-xPx)2 crystals, we demonstrate that our extended Kohler's rule can account for the scaling MR ~ tan 2 qH where qH is the Hall angle, which was first observed in cuprate superconductors YBa2Cu3O7-d and La2SrxCuO4 [18] and has led to a long-standing debate in describing the normal-state magnetoresistance of a superconductor [19][20][21][33][34][35][36][37][38][39][40][41] .
We also demonstrate the generality of our extended Kohler's rule in an undoped narrow-gap semiconductor, indium antimonide (InSb). At high temperatures, InSb is a compensated two-band system with high-mobility electrons and low-mobility holes, enabling us to determine the carrier density via Hall measurements and use its temperature dependence to test the validity of the nT term in our extended Kohler's rule.

II. MATERIALS AND METHODS
Data reported here were obtained from two TaP crystals (sample TP1 and sample TP2), two BaFe2(As1-xPx)2 crystals with x = 0.25 (sample PL) and x = 0.5 (sample PH), and one undoped InSb crystal (sample IS).

Preparation of samples.
(1) TaP: Centimeter-sized single crystals of TaP were grown using the vapor transport method in two steps. In the first step, 3 grams of Ta (Beantown Chemical, 99.95%) and P (Beantown Chemical, 99.999%) powders were weighed, mixed, and ground in a glovebox. The mixed powders were sealed in an evacuated quartz tube which was subsequently heated to 700°C and sintered for 20 hours for a pre-reaction. In the second step, the obtained TaP powder along with 0.4 grams of I2 (Sigma Aldrich, ³99.8%) were sealed in a new evacuated quartz tube, and was subsequently placed in a two-zone furnace with zone temperatures of 900°C and 950°C, respectively. The crystal growth time was 14 days. Due to the very high electrical conductivity of TaP, it is difficult to carry out high-precision electrical measurements on the as-grown crystals. To increase the electrical resistance of the samples, we polished the crystals down to a thin slab along the c-axis (thickness of tens of micrometers). Electrical leads were gold wires glued to the crystals using silver epoxy H20E. (2) BaFe2(As1-xPx)2: crystals of BaFe2(As1-xPx)2 with doping levels of x = 0.25 and x = 0.50 were grown using the self-flux method 42 . Highpurity flakes of Ba (99.99%, Aldrich) and powders of FeAs and FeP (homemade from Fe, As, and P, 99.99%, Aldrich) were thoroughly mixed and placed in an Al2O3 crucible, which was then sealed in an evacuated quartz tube under vacuum and placed in a Lindberg box furnace. Crystals of plate shapes with lateral dimensions up to 2 mm and thicknesses up to 200 µm were obtained by heating up to 1180 • C and then cooling down to 900 • C at a rate 2 • C/min. Electrical leads of gold wires with a diameter of 50 µm were attached to the crystals using silver epoxy H20E. (3) InSb: a crystal of 5 mm ´ 5 mm ´ 0.5 mm was purchased from the MTI corporation. It was cut into pieces with desired lateral dimensions. Gold pads of ~100 nm thick were deposited on locations pre-defined using photolithography. Electrical leads were fabricated by attaching 50 µm diameter gold wires to the pads with silver epoxy H20E.

Resistance measurements. We conducted resistance measurements to obtain both Rxx(H) and Rxy(H)
curves at various fixed temperatures, enabling the calculation of the resistivities rxx(H) = Rxxwd/l and rxy(H) = Rxyd, where w, d, and l are the width, thickness of the sample and the separation between the voltage contacts, respectively. The magnetic field is applied along the c-axis of the crystals. The magnetoresistance is defined as MR = [rxx(H) -r0)]/r0, where rxx(H) and r0 are the resistivities at a fixed temperature with and without the presence of a magnetic field, respectively. In some cases, we obtained rxx(T) curves from the measured rxx(H) curves at fixed temperatures to avoid nonequilibrium temperature effects. Data for TaP and InSb were obtained using the conventional four-probe DC electrical transport measurement technique while those for BaFe2(As1-xPx)2 were obtained using a low-frequency lock-in method.
The key results of this work on TaP are displayed in  Fig.2b are nearly in parallel with each other, suggesting that a single temperature-dependent multiplier to MR (y-axis) or to H/r0 (x-axis) could cause them to overlap or collapse onto the same curve. Here we tackle the latter case and uncover the underlying physics.
For the convenience of the forthcoming discussions we designate 1/nT as the temperature dependent multiplier to H/r0 (x-axis) in the MR ~ H/r0 curves in Fig.2b. In practice, we scale all the MR curves to the In the analysis above we purposely used nT in the denominator to couple with r0 in Eq We reveal in the discussion below that H and the mobility µi = et/mi* (i ³ 1) are interconnected as Hµi in the MR expression. Also, it is the temperature-induced change in the carrier density is responsible for the violation of Kohler's rule. Hence, our extended Kohler's rule expression in Eq.1 provides a unique way to reveal the temperature dependence of the carrier density.
In the normal state of a cuprate superconductor the temperature-induced change in the carrier density can be attributed to the existence of a pseudogap or Mott -Wannier excitons of weakly bound electrons and holes 41 . Differing from the linear temperature dependence of the carrier density used to explain the violation of Kohler's rule in cuprate superconductors 41 , the nT obtained in our samples can be approximately described by nT ~ T 2 (Fig.S4), which can be attributed to the thermally-induced change in the carrier density, as elaborated below.
Since equal number of holes are created, Eq.2 can also be used to calculate the thermally-induced change in the hole density. In TaP, the density of electron and holes are close to each other (Fig.S3c). We focus only on electron density in the following discussions.
The energies E1 and E2 of the Weyl nodes W1 and W2 in TaP have been determined by bandstructure calculations 44 and by ARPES measurements 45 . The theoretical DoS roughly follows ~ at up to ~0.3 eV and becomes nearly constant at higher energies 44,47 . As an estimate we use i.e., a slight increase in the theoretical E1 value towards the experimental one while keeping the value of E2 unchanged.
The above discussion indicates that the relative position of the Fermi level to the bottom of the conduction band and the top of the valence band, i.e., the density of electrons and holes at T = 0 K affects the temperature dependence of nT. As presented in Fig.S5b, the temperature-induced change in nT mostly comes from the electron band. In Fig.S6a we present nT versus T curves calculated using different values of EF in the electron band. It is clear that the sensitivity of nT to T depends strongly on the Fermi level. In our TaP samples with EF » 50 meV to the bottom of the conduction band, nT nearly doubles when the temperature is increased from T = 2 K to 300 K. However, it becomes challenging to experimentally resolve the change in nT in the same temperature range when the Fermi level is increased to 200 meV. In this case, nT » 1 and Kohler's rule should hold within experimental errors. Thus, the Fermi energy, i.e., electron density at T = 0 K, plays a key role in Kohler's rule. It directly explains why Kohler's rule is violated in type-I Weyl semimetals while it is upheld in their type-II counterparts since the latter typically have much higher electron densities 5,6,8 . The validity of Kohler's rule in conventional metals is also not a surprise. They have much higher Fermi energies of a few eVs and electron densities of 10 28~29 m -3 [48], which is 3~4 orders of magnitude higher than that of TaP, making thermally induced changes in the carrier density irrelevant. In

III.2. Extended Kohler's rule versus other alternative scaling forms of the magnetoresistance
In addition to topological materials, violations of Kohler's rule were often reported in cuprates and iron-based superconductors as well as other topologically trivial materials and various alternative MR scaling forms have been introduced 18-20,33-41 . Among them, the most common one is where qH = arctan(rxy/rxx) is the Hall angle. This MR scaling behavior was first reported in cuprate superconductors, with gH = 1.7 and gH = 1.5-1.7 for underdoped and optimally doped YBa2Cu3O7-d, respectively, and gH = 13.6 for La2SrxCuO4 [18]. It has numerous explanations [18][19][20][21][37][38][39][40][41] including the spincharge separation scenario of the Luttinger liquid 21 , current vertex corrections and spin density wave 20 .
Below we show that the scaling Eq.3 is a natural outcome of our extended Kohler's rule Eq.1 in a compensated two-band material when the carrier mobility is very low, with gH being an indicator of the ratio of the hole and electron mobility. We also obtain similar scaling forms to Eq.1 and Eq.3 for a lowmobility single-band system with an anisotropic Fermi surface as well as non-compensated two-band and multi-band systems.
For a compensated two-band system where ne = nh = n, the second term in the denominator of Eq.S1 At very low mobilities, MR® 0 and rxx » r0, resulting in Eq.3 and Eq.4 to be equivalent. That is, in a compensated two-band system with very low mobilities, our extended Kohler's rule in Eq.1 will lead to the scaling behavior of Eq.3.
The same conclusion can be reached for a nearly compensated two-band system, if the mobilities and/or the magnetic fields are low, such that the second term in the denominator of both Eq.S1 and Eq.S2 as well as the second term in the numerator of Eq.S2 become negligible. As revealed in TaP below, aµ is indeed temperature insensitive (Fig.5b). Thus, our extended Kohler's rule Eq.1 can be expressed as Eq.3 for a compensated system as well as for a nearly compensated system with very low mobilities and/or at very low fields if the mobilities are high.
Underdoped YBa2Cu3O7-d [49,50] are indeed two-band materials. On the other hand, the optimaldoped YBa2Cu3O7-d (with Tc = 90 K) which also shows the scaling behavior akin to Eq.3 is believed to be single-band system with an anisotropic Fermi surface 41  along the x-direction and is the ratio ( Y / S ) of the carrier mobilities S and Y along the x and y directions. In the meantime, we can also obtain TaP is a nearly compensated two-band system with high mobilities, as manifested by the large MRs ( Fig.2a) and non-linear rxy curves (Fig.S7). Thus, the scaling behavior in Eq.3 is expected to fail when the MRs become significantly large, as confirmed by the plot in Fig.4a, which shows that Eq.3 is roughly valid for MR < 10% and yields a value of gH = 9 at T = 300 K. On the other hand, Eq.4 is an approximate expression of Eq.1 at low magnetic fields and should be valid over a wider field range by avoiding the influence of rxx(H) in Eq.3. As plotted in Fig.4b, MR ~ (rxy/r0) 2 indeed allows us to more reliably derive the gH values (Fig.5a). The corresponding µh/µe changes from 1.39 at T = 300 K to 1.24 at T = 2 K (Fig.5b), leading to a nearly temperature-independent aµ with a very small change (< 0.35%) from T = 300 K to T = 2 K (Fig.5b).
In both Kohler's theory 2 and the derivations by Luo et al. 41 , Ht appears as a product that is inseparable in the expression for MR = f(Ht) and hence has been proposed as a modified Kohler's rule 33,36,41 . The magnetoresistances in the normal state of La2-xSrxCuO4 and KxFe2−ySe2 single crystals were indeed found to follow the scaling behavior of MR = f(Ht) if t ~ T -1 [33] and t ~ T -2 [36] are respectively assumed. However, t is not a parameter that can be conveniently obtained from resistivity measurements. As discussed in section III.1, MR = f(Ht) does not consider the possible role of the carrier's effective mass and t should be replaced with µ, which can have more than one value, as demonstrated in Eq.S10 and Eq.S11 as well as Eq.S15 and Eq.S16. Thus, its applications are limited, particularly for single band systems with temperature independent effective mass. It will usually fail in two-band and multiband systems. For example, by assuming single relaxation time for all carriers we can re-write Eq.S15 as The discussions in the preceding section indicate that Eq. 1, our extended Kohler's rule applied to a semimetal also provides a sensible explanation for the alternative scaling rule presented in Eq.3, which has been used routinely to account for the normal-state magnetoresistances in several classes of superconductors [19][20][21][33][34][35][36][37][38][39][40][41] . Here, we experimentally confirm the applicability of Eq.1 on two superconducting BaFe2(As1-xPx)2 crystals with x = 0.25 and 0.5, respectively. Their corresponding superconducting transition temperatures are 31 K and 22 K, as revealed by the temperature dependence of the zero-field resistivity presented in Fig.S8.
In the over-doped crystal with x = 0.5, we found that the magnetoresistance obeys Kohler's rule is temperature dependent (inset of Fig.S9b), similar to that found in TaP (Fig.5a). In the under-doped crystal with x = 0.25, we did observe both the violation of Kohler's rule (Fig.6b) and the validity of the scaling Eq.3 (Fig.6c). As indicated in Fig.6c and the gH values in its inset, MR versus tan 2 qH curves at T ³ 50 K overlap each other while those at T < 45 K show a slight parallel shift, with an increase of gH from ~7 at T = 45 K to ~9 at T = 32 K. In contrast, our extended Kohler's rule, Eq.1, works well over the entire temperature range as shown in Fig.6d. Similar to gH, the derived nT (inset of Fig.6d) also shows a significant change in its temperature dependence at T ≈ 50 K. At T ³ 50 K, the temperature dependence of nT is roughly linear. Interestingly, it can also be described by nT = n0 + aTe -D/k B T (dashed line in the inset of Fig.6d with n0 = 0.7, a = 5.4´10 -3 , and D = 5.18 meV), analogous to the temperature dependence of the carrier density arising from carriers thermally excited over a pseudogap D in cuprates 51,52 . At T £ 45 K, nT changes with temperature at a much higher rate. The temperature (~50 K) at which nT changes its temperature sensitivity is coincident with that of a transition into an antiferromagnetic orthorhombic phase 35 (inset of Fig.S8 and discussion in its caption). While further investigations are needed to account for the temperature behavior of nT at temperatures above and below ~50 K, the results in Fig.6  ni is practically same as that (nH) obtained from Hall measurements ( Fig.S2 and caption). This allows a further verification of nT using the experimentally determined carrier density.
We present typical rxx(H) curves of InSb around room temperature in Fig.7a. We focused on data obtained at T ³ 240 K to avoid interference of quantum magnetoresistance that can occur at lower temperatures 54 and the contribution to the magnetoresistance by the residual impurity in the nominally undoped crystal (Fig.S10 and caption). As expected for a semiconductor, rxx increases with decreasing temperature (also Fig.S10a). Figure 7c shows that Kohler's rule is violated in InSb. In fact, the curves in the Kohler's rule plot are separated from each other even further, compared to those prior to the scaling (Fig.7b). This is in contrast to those shown in Fig.2a&2b and in Fig.6a&b for a semimetal (TaP) and for a superconductor [BaFe2(As2-xP)2] in the normal state, because their zero-field resistivity r0 has opposite temperature dependence to that of the semiconducting InSb. On the other hand, our extended Kohler's rule, Eq.1, can collapse all the data into a single curve (Fig.7d). The temperature dependence of the derived nT can be well described theoretically (Fig.S11b), unveiling band gaps comparable to those determined from other methods in the literature (inset of Fig.S11b). It is nearly indistinguishable to that of the experimental Hall carrier density nH obtained from the rxy(H) curves (Fig.S11a) as well as carrier density ni (Fig.S12c) obtained by simultaneous fittings of rxx(H) and rxy(H) curves using the two-band model (Fig.S12a). These results evidently prove the validity of our extended Kohler's rule Eq.1 in semiconducting InSb, further demonstrating its generality. As presented in Fig.7e, a plot of MR versus tan 2 qH can also collapse all data into one common curve, which becomes however nonlinear with increasing magnetic field. This indicates that MR is not proportional to tan 2 qH, i.e., the scaling provided by Eq.3 is not valid. The reason is that Eq.3 is deducible from Eq.4 only when the carrier mobility is very low so that the magnetoresistance MR is negligible, i.e., rxx(H) » r0. On the other hand, Fig.7f shows that the general form Eq.4 does work well in InSb over the entire field range, confirming that in a compensated two-band system, Eq.4, i.e., MR = gH(rxy/r0) 2 , can be derived from the extended Kohler's rule Eq.1, regardless of the value of the carrier mobility.

IV. CONCUDING REMARKS
Since it was proposed more than 80 years ago for orbital magnetoresistances in non-magnetic metals,  TaP)               Supplementary Information:

Extended Kohler's Rule of Magnetoresistance
Jing Xu et al.
Equations for calculating the magnetoresistivities of a two-band system 1,9 Scaling behavior of magnetoresistance in an anisotropic single-band system with low carrier mobility (S1) For clarity in discussions below we re-write them as However, carriers at energy levels near EF also contribute to the conductance 1 with R * = /( ? > ). Since SS ≈ ? at low carrier mobility, Eq.S8 can be expressed as: Eq.S8 and Eq.S9 indicate that a low-mobility anisotropic single-band system can have similar scaling forms as those (Eq.5 and Eq.6) of a two-band system discussed in the text.
We can also re-write Eq.S6 as: where I * = > ? . That is, the magnetoresistance of an anisotropic single-band system will follow the extended Kohler's rule Eq.1 if I * is temperature independent. It may seem to be contradictory to claim that I * (= > ? ) has no temperature dependence while can be temperature dependent. As demonstrated below, all in I * get cancelled out because ~ and ? ~ 1/ .
As an example, we consider a case that is equally distributed among R ≤ S ≤ Q . In this case, = /( R − Q ); ? = 2/[ ( R + Q )]; = ( R " − Q " )/[4( R − Q )]; resulting in That is, I * is indeed temperature independent because R and Q have the same temperature behavior (if the temperature does not induce a change in the anisotropy).
Likewise, we obtain a temperature independent R * : The scalings of Eq.S15 and Eq.16 can be valid at high fields in materials with low mobilities or at low fields in those with high mobilities, as long as ( ≪ 1. Eq.S16 indicates that 7 represents temperature dependence of the carrier density, i.e., 7 ~ g ( ), if the densities/mobilities from different bands have the same or similar temperature dependence, i.e.,   can result in unexpected outcomes. For example, in materials where Kohler's rule holds, i.e., nT in Eq.1 should be close to 1 and insensitive to temperature, the derived carrier densities can double or triple when the temperature is increased from 2 K to 300 K 14,26 . Here we use the measured zero-field resistivity r0 to reduce the number of fitting parameters to three, and simultaneously fit both rxx(H) and rxy(H). The fitted curves, particularly those to the rxx(H) data, deviate significantly from the experimental data. One possible reason is that the two-band model is valid only for an isotropic system and is not suitable to describe the magnetoresistivities of TaP that has anisotropic Fermi pockets. Fittings with the two-band model yields values of ne and nh that are close to each other and increase with temperature. However, they are nearly a factor of 6 larger than the theoretical values. Furthermore, the fittings generate larger ne than nh, in contrast to those obtained from bandstructure calculations and Eq.2.  used g(e) ~ e a (a = 1/2 and 1) at e from 0 to ¥ in the calculations, as typically done for free electrons. As discussed in the text for TaP, DoS can deviate from g(e) ~ e a at high energies (e.g. e > 300 meV). However, the contribution to the total electron density from the high energy levels (e.g. e > 300 meV in TaP) is negligible (<0.1%).   The sample is a single crystal that is undoped but N-type due to residual impurities. The peaks in the rxx(T) curves in (a) and the bending in the rxy(T) curve in the inset of (b) at temperatures between 150 K and 175 K are due to a transition from the classical intrinsic state to the impurity dominated state, where quantum effects were reported 54   and rxy(H) curves at T = 300 K simultaneously using Eq.S1 and Eq.S2. We neglected the contributions of the impurities at T ³ 240 K and used ne = nh = ni. We also used the measured r0 value and the relationship of r0 = e(neµe+nhµh) = nie(µe+µh) to reduce the number of fitting parameters. This led us to only two free parameters of ni and µe while µh can be calculated from µh = r0/nie-µe. (c) and (d) Temperature dependence of the derived parameters of ni, µe and µh. (c) indicates that the carrier density ni obtained from the two-band model analysis is indistinguishable from the Hall carrier density nH (Fig.S10b). This is consistent with the results in Fig.S2, which showed that the Hall carrier density, nH, is equal to the carrier density, ni, at very large ratios of µe/µh (it changes slightly from 190 at 360 K to 199 at 240 K for the µe and µh in(d)).