Extreme Magnetoresistance in Magnetic Rare Earth Monopnictides

TL;DR

This paper investigates how magnetic order influences extreme magnetoresistance (XMR) in magnetic rare earth monopnictides, revealing that magnetic phases significantly modulate XMR magnitude and behavior, with CeSb showing exceptionally large and non-saturating XMR.

Contribution

It demonstrates the modulation of XMR by magnetic order in magnetic rare earth monopnictides, highlighting the role of orbital states and magnetic phases in controlling magnetoresistance.

Findings

CeSb exhibits XMR over 1.6 million percent at 9 T.

Magnetoresistance in CeSb is non-monotonic across magnetic phases.

XMR follows a non-saturating power law above 30 T.

Abstract

The acute sensitivity of the electrical resistance of certain systems to magnetic fields known as extreme magnetoresistance (XMR) has recently been explored in a new materials context with topological semimetals. Exemplified by WTe$_{2}$ and rare earth monopnictide La(Sb,Bi), these systems tend to be non-magnetic, nearly compensated semimetals and represent a platform for large magnetoresistance driven by intrinsic electronic structure. Here we explore electronic transport in magnetic members of the latter family of semimetals and find that XMR is strongly modulated by magnetic order. In particular, CeSb exhibits XMR in excess of $1.6 \times 10^{6}$ % at fields of 9 T while the magnetoresistance itself is non-monotonic across the various magnetic phases and shows a transition from negative magnetoresistance to XMR with field above magnetic ordering temperature $T_{N}$. The magnitude of the XMR is larger than in other rare earth monopnictides including the non-magnetic members and follows an non-saturating power law to fields above 30 T. We show that the overall response can be understood as the modulation of conductivity by the Ce orbital state and for intermediate temperatures can be characterized by an effective medium model. Comparison to the orbitally quenched compound GdBi supports the correlation of XMR with the onset of magnetic ordering and compensation and highlights the unique combination of orbital inversion and type-I magnetic ordering in CeSb in determining its large response. These findings suggest a paradigm for magneto-orbital control of XMR and are relevant to the understanding of rare earth-based correlated topological materials.

Equations

$${\sigma }_{xx}({\gamma }_{8^{\star }})={\sigma }_c+\sqrt{{\sigma }_c^2/4+8{\sigma }_7{\sigma }_{8^{\star }}},$$

Full text

Extreme Magnetoresistance in Magnetic

Rare Earth Monopnictides

Linda Ye

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

  

Takehito Suzuki

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

  

Christina R. Wicker

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

  

Joseph G. Checkelsky

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

**The acute sensitivity of the electrical resistance of certain systems to magnetic fields known as extreme magnetoresistance (XMR) has recently been explored in a new materials context with topological semimetals. Exemplified by WTe2 and rare earth monopnictide La(Sb,Bi), these systems tend to be non-magnetic, nearly compensated semimetals and represent a platform for large magnetoresistance driven by intrinsic electronic structure. Here we explore electronic transport in magnetic members of the latter family of semimetals and find that XMR is strongly modulated by magnetic order. In particular, CeSb exhibits XMR in excess of $1.6\times 10^{6}$% at fields of 9 T while the magnetoresistance itself is non-monotonic across the various magnetic phases and shows a transition from negative magnetoresistance to XMR with field above magnetic ordering temperature $T_{N}$. The magnitude of the XMR is larger than in other rare earth monopnictides including the non-magnetic members and follows an non-saturating power law to fields above 30 T. We show that the overall response can be understood as the modulation of conductivity by the Ce orbital state and for intermediate temperatures can be characterized by an effective medium model. Comparison to the orbitally quenched compound GdBi supports the correlation of XMR with the onset of magnetic ordering and compensation and highlights the unique combination of orbital inversion and type-I magnetic ordering in CeSb in determining its large response. These findings suggest a paradigm for magneto-orbital control of XMR and are relevant to the understanding of rare earth-based correlated topological materials. **

The rare earth monopnictides $RX$ crystallize in the NaCl structure (Fig. 1(a)) and exhibit a rich variety of magnetic ground states RareEarthHandBook . In terms of electronic structure, most $RX$ compounds are known as compensated semimetals with the conduction band deriving from rare earth 5$d$ $t_{2g}$ states and valence band from pnictogen ${}^{3/2}p$ states, located at $\Gamma$ and $X$ points in the Brillouin zone, respectively (Fig. 1(b)) CeSbARPES . Potential topological aspects of the electronic structure have recently been discussed, including Dirac semimetal nodes or topological insulating gaps along $\Gamma-X$ depending on the pnictogen LaSbTI ; LaSbCava (highlighted in blue) and an unusual four-fold degenerate Dirac surface state at $\bar{M}$ CeSbHasan (projected on to $X$ shown in green). Combining this with the $f-$electron degree of freedom suggests $RX$ may therefore be host to topological phases of correlated electrons. This is further enriched by the reports of extreme magnetoresistance (XMR) in LaSb KasuyaTransport ; LaSbCava and LaBi KasuyaTransport ; LaSbCava2 ; LaBi . While a rarity, XMR is of fundamental interest in terms of its microscopic origin and has technological relevance in magnetic sensing and related technologies XMR1 ; XMR2 . The impact of strongly correlated behavior on this remarkable transport response in semimetals has thus far remained unexplored.

The magnetism of the $RX$ compounds is distinct from that in simpler magnetic metals such as Fe, Gd, Tb and dilute magnetic semiconductors such as MnxGa1-xAs DMS owing to the combination of strongly localized $f$ electrons and low density, high mobility carriers from the $p$ and $d$ bands. The NaCl structure enforces a significant interaction between the two with the principle pnictogen wave function transfer being mediated through the rare earth wave function (and vice versa) leading to a relatively wide variety of magnetic phases. This behavior is particularly distinct for the choice of a single $f$ electron for $R=$ Ce; compounds CeP CeP , CeAs CeAs , and CeSb CeSbneutron each have rich phase diagrams characterized by the mixed $f$- orbital occupation of Ce in the lattice. We focus on strong spin-orbit CeSb here and find this magneto-orbital character has a significant impact on XMR, leading to both negative ($>$70%) and positive ($>$1,670,000%) magnetoresistance. This behavior, distinct from giant magnetoresistance (GMR) in magnetic multilayers GMR and colossal magnetoresistance (CMR) in magnetic perovskites CMR , intertwines the semimetallic structure with magnetism and represents a potential form of magneto-orbital control of XMR.

CeSb is an unusual magnetic system, exhibiting at least 14 magnetic phases in close proximity in its magnetic field $B$ and temperature $T$ phase diagram (see Fig. 1(c)) CeSbMH . The primary driving force for this complexity is the interplay between the semimetallic band electrons and Ce3+ ${}^{1}f$ states, the latter being situated near to the Fermi level $E_{F}$ Kasuya1 . In the high temperature paramagnetic phase, the preferred orbital state for the Ce ion is $\Gamma_{7}$ as expected from the cubic coordination (shown schematically in Fig. 1(d)). However, the $\Gamma_{8}$ states are at an energy only 3 meV higher, and in the magnetically ordered state the cruciform $\Gamma_{8}^{(1)}$ orbital becomes energetically favored accompanied with a shift in the electronic structure as depicted in Fig. 1(e) Kasuya1 (see supplementary information). At intermediate $T$ and $B$ a complex magnetic phase diagram arises consisting of phases built by stacking paramagnetic $\Gamma_{7}$ planes and ferromagnetic $\Gamma_{8}^{(1)}$-like planes as shown in Fig. 1(c). Neutron Boucherle and X-ray Xray scattering measurements have mapped the orbital content of these phases (the $\Gamma_{8}^{(1)}$-like planes are reported to be composed of planar orbitals that are close to a $J_{z}=|\pm 5/2\rangle$ fully polarized state we hereafter refer to as $\Gamma_{8^{\star}}$, see supplementary materials); the color scale in Fig. 1(c) reflects the $\Gamma_{8^{\star}}$ occupation $\gamma_{8^{\star}}$. Among isostructural cerium monopnictides, the phase diagram of CeSb uniquely hosts antiferromagnetic (AF), antiferro-ferromagnetic (AFFn), ferromagnetic (F), antiferro-paramagnetic (AFPn), ferro-paramagnetic (FPn), and paramagnetic (P) phases CeBi ; CeP ; Pressure ; LaCeSb .

We first examine the longitudinal resistivity $\rho_{xx}$ and transverse resistivity $\rho_{yx}$ as a function of magnetic induction $B$ at different characteristic temperatures $T$ in the phase diagram. We note that $B=\mu_{0}(H+(1-N)M)$ is corrected for demagnetization effects with the demagnetization factor $N$ calculated from the sample dimensions and magnetization $M$ measured separately (see supplementary materials). Starting with sample A1 at $T=$ 2 K in Fig. 2(a), $\rho_{xx}(B)$ shows a rapid positive magnetoresistance reaching $233,500$% of its zero field value at $B=$ 9 T. This XMR behavior is comparable to that seen in WTe2 XMR1 , LaBi LaSbCava2 ; LaBi , and LaSb LaSbCava , and significantly larger than that reported previously in CeSb R1 ; R2 ; R3 ; CanfieldCeSb ($<10,000\%$) and other magnetic $RX$ (see supplementary section S4). Kinks in $\rho_{xx}(B)$ are noticeable at intermediate $B$ corresponding to the magnetic phase boundaries between AF, AFFn, and F states in Fig. 1(c). This is also seen in $\rho_{yx}(B)$ shown in Fig. 2(b), where the vertical lines denote the phase boundaries observed on decreasing $B$. A significant hysteresis in both $\rho_{xx}(B)$ and $\rho_{yx}(B)$ is observed. The hysteresis is found to be sample dependent, similar to that reported in previous magnetization studies CeSbMH .

While at $T=2$ K four different phases are stable at different $B$, for all $B$ these phases are fully composed of $\Gamma_{8^{\star}}$ and thus have pure $\Gamma_{7}$ layer volume fraction $\gamma_{7}=0\%$. Upon increasing to $T=11$ K, the AFPn, FPn with mixed orbital character enter the phase diagram. Here, as shown in Fig. 2(c) the XMR response is weakened (538.5% at $B=9$ T) and clear non-monotonic behavior in $\rho_{xx}(B)$ is observed, with regions of both positive and negative $d\rho_{xx}/dB$. Also plotted in gray is $\gamma_{7}=100\%-\gamma_{8^{\star}}$; a correlation between intermediate regions of enhanced $\rho_{xx}$ and $\gamma_{7}$ is apparent. We expand on this below. The Hall response (Fig. 2(d)) is also sensitive to the magnetic phase boundaries, with a significant drop in magnitude in the FPn phases where $\gamma_{7}\neq 0$.

At higher $T=14$ K the same $\Gamma_{7}$ rich phases occupy a wider range of field and significant positive magnetoresistance is observed only for $B>6$ T in the F state (Fig. 2(e)) with discontinuities in $\rho_{yx}(B)$ appearing at the phase boundaries (Fig. 2(f)). At $T=19$ K, above the zero field magnetic ordering temperature $T_{N}=16$ K, positive magnetoresistance is absent up to 9 T (Fig. 2(g)), while at sufficient $B$ the system transitions from the P phase to FPn with features in $\rho_{yx}(B)$ apparent at the critical values of $B$ (Fig. 2(f)). The linear Hall effect in the P phase corresponds to a single band carrier number of $5.0\times 10^{20}$/cm3 or 0.046 e*-*/Ce. We note that the non-linear $\rho_{yx}(B)$ in the low temperature F phase resembles that observed at lowest $T$ in LaSb KasuyaTransport and LaBi LaBi ; a multi-band model must be incorporated to fully account for the behavior CeSbdHvA . More broadly, at these elevated temperatures the complex evolution of $\rho_{xx}(B)$ correlates with $\gamma_{7}$, suggestive of a connection between the orbital content and the conductivity of the system.

The evolution of the magnetoresistance is summarized in Fig. 3 where $d\rho_{xx}/dB$ is plotted. Here the large, non-saturating magnetoresistance corresponding to XMR can be seen at low $T$ with a superimposed Shubnikov-de Haas oscillation. The oscillation (frequency $=213.2\pm 0.5$ T) corresponds to the $k_{x}-k_{y}$ cross section of the $X_{Z}$ electron pocket CeSbdHvA . For higher $T>T_{N}$ there are regions of striking negative magnetoresistance, reaching magnitudes of 100 $\mu\Omega$ cm / T at the phase boundary between P and FP phases. The sharp features are suppressed in transitions between FPn phases, and an overall negative magnetoresistance is observed, reaching a magnitude of $72\%$ at $T=17$ K. It is noteworthy that this negative magnetoresistance differs from conventional field suppression of magnon scattering which follows a $B$-linear trend and is typically at the percent level at comparable $B$ Raquet .

As an aside we note that using the sharp features in $d\rho_{xx}/dB$ it is possible to construct the phase diagram of CeSb purely from transport. This is shown projected in the $B-T$ plane in Fig. 3. Closed circles are features that reproduce those seen in $M$ (see supplementary materials). Interestingly, we see an additional feature that develops in the AFF1 phase in decreasing $B$ not previously reported in magnetization studies (open circles). This may correspond to a previously unidentified phase that further enriches the phase diagram of CeSb.

A detailed comparison of transport to the orbital content across the phase diagram is motivated by recent X-ray analysis demonstrating the evolution of the localized $f$ wave function from $\Gamma_{8^{\star}}$ to $\Gamma_{7}$ with increasing $T$ across the zero field AF, AFPn, and P states Xray . As discussed above, compared to paramagnetic $\Gamma_{7}$, $\Gamma_{8^{\star}}$ enhances hopping between the neighboring Sb sites in the plane (Fig 1(d)) and therefore may be expected to lead to enhanced conductivity. This is qualitatively consistent with the correlation of $\gamma_{7}$ and enhanced $\rho_{xx}$ in Figs. 2(c), 2(e) and 2(g). To further quantify this, we model the system as a binary mixture with relatively low and high conductivity component layers composed of the $\Gamma_{7}$ and $\Gamma_{8^{\star}}$ orbitals, respectively. A common approach to conductivity in two component mixtures first developed for composites Landauer , and later applied to systems ranging from superconductors EM_SC to CMR manganites EM_CMR , is that of the effective medium. The underlying assumption is that a given region can be considered to be surrounded by a medium with uniform conductivity characteristic of the mixture MsLachlan . Denoting the conductivities of $\Gamma_{7}$ and $\Gamma_{8^{\star}}$ regions as $\sigma_{7}$ and $\sigma_{8^{\star}}$, respectively, in this model the total conductivity can be written as

[TABLE]

where $\sigma_{c}=\sigma_{7}(2-3\gamma_{8^{\star}})+\sigma_{8}(3\gamma_{8^{\star}}-1)$ (see supplementary materials). Calculating the conductivity as $\sigma_{xx}=\rho_{xx}/(\rho_{xx}^{2}+\rho_{yx}^{2})$ we estimate $\sigma_{xx}$ at each phase as its value at the central $B$ between the phase boundaries. For simplicity we focus on the region above $T_{N}$ where $\gamma_{7}$ evolves most rapidly. These values are shown as a function of $\gamma_{8^{\star}}$ in Fig. 4 as circles with the fit to Eq. (1) as the solid line. For metal-insulator mixtures this curve takes a divergent shape reflecting the percolative transition between the two end phases EM_CMR . Here the dependence is more gentle, suggesting components of comparable conductivity.

The orbital-dependent conductivity values found at each $T$ in the inset of Fig. 4, indicating a conductivity ratio of approximately 6 at $T=17$ K that diminishes on warming. While both orbital dependent conductivities are metallic, the ferromagnetic $\Gamma_{8^{\star}}$ contribution rises rapidly as $T$ is reduced, eventually leading to the high conductivity XMR state at low $T$. While this confirms the above observation of enhanced resistivity in $\Gamma_{7}$ rich states, it further suggests a manner of control for XMR by the orbital degree of freedom. If the orbital content could be manipulated at low $T$ this suggests XMR could be similarly modulated. That XMR would be absent at low $T$ if the orbital content were modified is supported by magnetotransport reports in CeP which has a $\Gamma_{7}$ rich ground state at low $T$ and shows negligible XMR at similar fields CeP_MR . Application of pressure may therefore be an effective manner to tune XMR as positive pressure is known to suppress $\Gamma_{7}$ in CeSb Pressure and negative (chemical) pressure via La doping acts in the opposite fashion LaCeSb . Alternatively, epitaxial thin films grown on appropriate substrates may realize materials with strain-controlled XMR.

The non-saturating nature of the XMR in CeSb is observed with application of larger magnetic fields. As shown in Fig. 5(a), sample C1 measured to fields above 30 T shows a similar crossover pattern from negative to positive magnetoresistance at intermediate $T$ and sharply increasing XMR at the lowest $T$. As shown in Fig. 5(b), for $T=0.44$ K the MR is in excess of $1,500,000\%$ at the highest fields and is well described by a 1.95 power law without sign of saturation (pronounced SdH oscillations are observed, see supplementary section S5). Large magnetic field also demonstrates the correlation of positive MR with field induced $\Gamma_{8^{\star}}$ planar orbitals Kasuya1 . For $T=17$ K, $\gamma_{7}$ drops to zero near 9 T after which a large, non-saturating MR emerges (see Fig. 5(c)).

To further elucidate the origin of XMR at low $T$ we compare the response of CeSb to $RX$ compound GdBi, which is expected to be similar in electronic structure but is orbitally quenched. The overall metallicity and field response is shown in Fig. 6(a) and Fig. 6(b) for CeSb sample S1 and GdBi sample T1, respectively. For CeSb, $\rho_{xx}$ drops dramatically below $T_{N}$ reaching a value of 1 n$\Omega$ m at $T=2$ K (Residual resistivity ratio RRR = 1017), while application of $B$ induces XMR. For GdBi, the behavior is similar with a drop in $\rho_{xx}$ to 1.2 n$\Omega$ m at $T=2$ K (RRR = 255) and XMR approximately one order of magnitude smaller at $B=9$ T. XMR for CeSb samples A1, B2, and B4 is shown in Fig. 6(c), the largest of which reaches $1,672,200\%$ at 9 T (RRR = 2726 and residual resistivity 77 n$\Omega$cm). This is larger than any previous report in the $RX$ family, including the non-magnetic LaBi and LaSb. For GdBi, XMR is observed as shown in Fig. 6(d), reaching values of $17,125\%$ (previous reports of GdSb have reported similar values of $2,300\%$ GdX_MR ). Whereas for CeSb multiband fitting is complicated by the various field induced transitions, GdBi remains in an antiferromagnetic state ($T_{N}=28$ K) up to $B=31$ T. In this case multiband fitting indicates a significant enhancement of mobility below $T_{N}$ and nearly compensated state at the lowest $T$ (see supplementary materials). We therefore suggest that XMR in magnetic $RX$ systems share a common origin with non-magnetic La$X$ below $T_{N}$ where magnetic scattering suppressed. We note it recently been discussed in the context of YSb that both exact compensation and moderate compensation with mobility mismatches may support this behavior YSb .

The XMR in CeSb exceeds even that reported in its non-magnetic analogs, which is unexpected from the viewpoint of the additional disorder associated with the magnetic degree of freedom. We hypothesize that this behavior is rooted in the orbitally-ordered ground state of CeSb. In particular, the planar orbital favored in the magnetic ground state provides highly mobile carriers that populate the planes formed by the type-I ordering (see inset of 6(a)). It is noteworthy that the planar orbital is favored in CeSb despite the preference for the $\Gamma_{7}$ orbital shape in the cubic crystal field of the NaCl structure and that an unusually large magnetic anisotropy pins the moments normal to the ordered planes Kasuya1 . This is not the case, for example, for NdSb R3 ; NdSb . Additionally, as type-II ordering is favored for $RX$ heavier than Eu$X$ R2 , we suggest that CeSb may realize a unique combination of orbital and magnetic ordering that gives rise to its large XMR in this configuration. This can be contrasted with GdBi, which shows moderate XMR here and has spherical orbitals supporting a type-II antiferromagnetism (see inset of 6(b)). Theoretical work may allow prediction of significant XMR in other magnetic $RX$ and related compounds along these lines.

The presence of XMR in rare earth monopnictides appears to be a ubiquitous phenomenon originating from their common semimetallic band structure. The use of rare earth elements beyond La, Y, and Lu introduces correlation effects into these systems that modulate XMR. The study here demonstrates how the anomalous ordering of crystal field states in CeSb allows this tuning with moderate $B$ and $T$. While electronic structure calculations in the various magnetic ground states of CeSb are challenging, it is noteworthy that previous calculations in the F state show bands with the character of type II Weyl points in the vicinity of the Fermi level Fcalc , indicating the possible role of topological features in these systems. Furthermore, it can be expected that magnetic order may introduce exchange effects to produce magnetically induced Weyl points for the inverted gap $\Gamma-X$ direction as have been discussed for half-Heusler systems GdPtBi1 ; GdPtBi2 . Further theoretical work is needed to confirm whether such scenarios occur and to more broadly understand the underlying electronic structure in these magnetic $RX$ systems and their potential for magneto-orbitally modified XMR.

I Methods

Single crystals of CeSb are grown using a Sn-flux method Canfield ; CanfieldCeSb from Ce (Ames Laboratory Ames , 99.99%), Sb (Alfa Aesar, 99.999%), Sn (Alfa Aesar, 99.995%) powders. They are mixed with atomic ratio Ce:Sb:Sn = 1:1:20, put in alumina crucible and sealed in quartz tube back filled with 150 torr Ar gas. The raw materials are first heated to 1050*∘*C and slowly cooled to 750 *∘C, at which point centrifuge separation of CeSb crystals from the Sn flux is performed. Single crystals of GdBi are gown using a Bi self flux method Canfield from Gd (Alfa Aesar, 99.9%) and Bi (Alfa Aesar, 99.999%). They are mixed with atomic ratio Gd:Bi = 18.5:81.5, put in alumina crucible and sealed in quartz tube. The raw materials are first heated to 1100∘*C and slowly cooled to 950 *∘*C followed by 4 days of annealing, at which point centrifuge separation of GdBi crystals from the Bi flux is performed. In both cases sub-cm size rectangular crystals are obtained and oriented with single crystal diffraction. Transport properties are measured in a commercial cryostat with a superconducting magnet. The magnetic field is applied along [001] and current flows along [100]. Field symmetrization/anti-symmetrization is performed on time-reversed field sweeps (up and down) to calculate the longitudinal/transverse resistivity and eliminate electrical pickup from contact misalignment. Magnetization is measured with a commercial SQUID magnetometer. Transport measurements at the National High Magnetic Field Laboratory are performed in a 3He cryostat in Cell-9 with a four probe method.