Butterfly-shaped magnetoresistance in triangular-lattice antiferromagnet Ag₂CrO₂

Abstract

Spintronic devices using antiferromagnets (AFMs) are promising candidates for future applications. Recently, many interesting physical properties have been reported with AFM-based devices. Here we report a butterfly-shaped magnetoresistance (MR) in a micrometer-sized triangular-lattice antiferromagnet Ag₂CrO₂. The material consists of two-dimensional triangular-lattice CrO₂ layers with antiferromagnetically coupled S = 3/2 spins and Ag₂ layers with high electrical conductivity. The butterfly-shaped MR appears only when the magnetic field is applied perpendicularly to the CrO₂ plane with the maximum MR ratio (≈15%) at the magnetic ordering temperature. These features are distinct from those observed in conventional magnetic materials. We propose a theoretical model where fluctuations of partially disordered spins with the Ising anisotropy play an essential role in the butterfly-shaped MR in Ag₂CrO₂.

Introduction

Compared to ferromagnets (FMs), antiferromagnets (AFMs) are much more difficult to manipulate the magnetic state. Thus, it has been believed that AFMs do not seem to have any applications¹. In the field of spintronics, since 1990s, AFMs have been mainly used as a pinned layer in spin-valve devices where parallel and antiparallel magnetic states of two FM layers induce a large resistance change and one of them has an AFM layer underneath in order to raise the magnetic coercivity^2,3. Recently, AFMs have played a more essential role in spintronics^4,5,6 because AFM-based spintronic devices are not only robust for external magnetic fields⁷ but also generate many interesting phenomena such as very fast domain wall motions^8,9, large anomalous¹⁰ and spin Hall effects¹¹, antiferromagnetic dynamics in the THz range^12,13, and so on.

When antiferromagnetically-coupled spins are arranged in a triangular shape, some interesting physical properties can be expected. For example, Mn₃X (X = Sn, Ge) has an AFM ordering phase with 120° structure above room temperature and the remnant magnetic moment in the AFM state is very small (≈0.002 μ_B per Mn atom where μ_B is the Bohr magneton)^14,15,16,17. Nevertheless, it shows an extremely large anomalous Hall effect, which seems to be inconsistent with the commonly believed picture: the anomalous Hall effect is proportional to the total magnetization¹⁸. A fictitious field due to the Berry curvature is in charge of the anomalous Hall effect^14,15,16,17. In addition, a peculiar spin Hall effect, namely magnetic spin Hall effect, has also been reported in this material^19,20. For a better understanding of the above peculiar phenomena, it is important to further perform magnetic transport measurements in triangular lattice AFM systems with electrical conductivity.

Ag₂CrO₂ is a highly conductive triangular lattice AFM with the transition temperature T_N of 24 K²¹. The crystal structure of Ag₂CrO₂ is shown in Fig. 1(a). The Cr site has an S = 3/2 moment, which is arranged in a triangular shape. It is known that the coupling between the two Cr sites in the two-dimensional (2D) plane is antiferromagnetic²¹. The conductive Ag₂ layers which are coupled to the magnetic CrO₂ layers enable us to investigate this spin system with electrical transport measurements. Such an electronic property is unique and advantageous compared to most of triangular lattice AFMs which are insulators or have very low electric conductivity.

Figure 1

(a) The crystal structure of Ag₂CrO₂. The distance between the two Cr sites in the plane is 2.93 Å. The lattice constant along the c-axis is 8.66 Å. (b) Temperature dependence of ρ_xx of a Ag₂CrO₂ thin film. The inset is a scanning electron microscope image of the Ag₂CrO₂ device. (c) Schematic of the PD state with 5 sublattices. The red, blue, and gray circles represent the up- and down- and PD-spins along the b-axis, respectively.

According to the neutron²² and muon spin resonance (μSR) experiments²³, Ag₂CrO₂ has a unique thermodynamic property, i.e., partially disordered (PD) state with 5 sublattices, at finite temperatures^24,25 [see Fig. 1(c)], which is different from the above 120° structure. In principle, the PD spin acts as a free spin since all the interactions from the nearest, the second nearest, and even the third nearest neighbors are canceled out. Thus, there should be negligibly small magnetization below T_N. Nevertheless, this compound has a finite magnetization (≈ 0.08μ_B per Cr atom) at zero field^21,24. The origin of the small magnetization is still unclear. This system is also quite interesting from the perspective of spin fluctuations in the PD state.

In this work, we performed magnetotransport measurements in order to investigate the impact of the PD spin fluctuations on the electrical transport property using a micrometer-sized Ag₂CrO₂; it is close to the single crystal. We find a butterfly-shaped magnetoresistance (MR) when the magnetic field is applied along the c-axis. Unlike the case of conventional magnetic systems, the amplitude of the butterfly-shaped MR is small at low temperatures and takes a maximum at around T_N, and disappears above T^* = 32 K. The result coincides with the MR by magnetic fluctuation, which is based on the 2D magnetic system with the uniaxial anisotropy.

Sample Fabrications and Experimental Setup

Polycrystalline Ag₂CrO₂ samples were obtained by encapsulating a mixture of Ag, Ag₂O, and Cr₂O₃ powders in a gold cell, and by baking them at 1200 °C for 1 hour under a pressure of 6 GPa²¹. The polycrystalline samples were then pounded on a glass plate in order to obtain small pieces of Ag₂CrO₂. The small grains were picked up with a scotch tape and pasted onto a thermally oxidized silicon (SiO₂/Si) substrate with several 100 nm thick gold marks. After removing the scotch tape from the substrate, another SiO₂/Si substrate without any gold marks was pushed onto the substrate with the Ag₂CrO₂ flakes and the 100 nm thick gold marks. In this process, Ag₂CrO₂ flakes thinner than ≈100 nm are left on the SiO₂/Si substrate with the gold mark, and relatively thick Ag₂CrO₂ flakes are transferred to the substrate without the gold marks²⁵. As a result, Ag₂CrO₂ flakes with a few micrometer-size remain on the substrate with the gold marks. We have fabricated 10 different devices and always observed the butterfly-shaped structure as detailed in the next section. The thicknesses of these flakes (80~120 nm) were confirmed by a commercially available atomic force microscope.

To confirm if the Ag₂CrO₂ flakes used for transport measurements are close to a single crystal, we took scanning transmission electron microscope (STEM) images shown in Fig. 2(a,b). As confirmed with the X-ray diffraction pattern for the polycrystalline samples, there is a single Ag₂CrO₂ phase²¹; no other phases such as AgCrO₂²⁶, which is a ferroelectric material. The Ag₂ and Cr layers [see Fig. 1(a)] are alternatively stacked perpendicularly to the SiO₂/Si substrate. On top and bottom of the Ag₂CrO₂ flake, some Ag clusters segregated can be seen. However, inside the Ag₂CrO₂ flake, there is no obvious cluster, indicating that the flake is close to a single crystal.

Figure 2

STEM images of a typical Ag₂CrO₂ device. (a) Bright-field STEM image of a wide area. (b) High angle annular dark-field STEM image in the area shown with the solid square in (a). The bright and dark spheres correspond to Ag and Cr atoms, respectively. (c) High angle annular dark-field STEM image in the area shown with the broken square in (a).

To perform the transport measurement, we deposited 150 nm thick Cu electrodes to the micrometer-sized Ag₂CrO₂ films, using the standard electron beam lithography and a Joule heating evaporator²⁵. A typical device is shown in the inset of Fig. 1(b). The lateral size and the thickness of the Ag₂CrO₂ flake are a few μm and 100 nm, respectively. The contact resistance between Ag₂CrO₂ and Cu is less than 1 Ω, which is comparable to a contact resistance for normal metallic junctions with almost the same junction area. This is also consistent with the STEM image in Fig. 2(c): the segregated Ag part on the left top side is continuously connected to the most top part of the Ag₂CrO₂ flake, showing that the Ag layer is exposed after the fabrication. The transport measurements have been carried out using an ac lock-in amplifier and a ⁴He flow cryostat.

Experimental Results

In Fig. 1(b), we plot the longitudinal resistivity ρ_xx of the Ag₂CrO₂ device as a function of temperature. The overall temperature dependence is metallic and there is a large resistivity drop at around T_N determined from the heat capacity measurements for polycrystalline Ag₂CrO₂ samples²¹. This resistivity change can be explained by spin fluctuation: fluctuations of paramagnetic spins at the Cr sites above T_N are strongly suppressed below T_N and the scattering rate of the conduction electrons is reduced. We also note that ρ_xx at 5 K obtained for the thin film device is about 10 times smaller than that for the polycrystalline bulk samples^21,25. From this result, we can argue that the thin film devices obtained with the mechanical exfoliation technique include much less grain boundaries compared to the polycrystalline bulk samples, resulting in a much better quality of Ag₂CrO₂. This is also supported by the STEM image in Fig. 2.

For the Ag₂CrO₂ device, we performed MR measurements with three different magnetic field (B) directions, i.e., x (in-plane along the current direction), y (in-plane perpendicular to the current direction), and z (out-of-plane along the c-axis, i.e., z || c) directions. Figure 3(a–h) shows the MRs, $\frac{{\rho }_{xx}(B)-{\rho }_{xx}(0)}{{\rho }_{xx}(0)}$, along the three directions measured at several different temperatures. At T = 5 K (<<T_N), a clear positive MR is observed at high magnetic fields when B is applied along the z-direction. With increasing temperature, the positive slope becomes flatter. At 25 K (≈T_N), the MR shows the negative sign. This trend is explained by the competition of two different mechanisms for MR, i.e., the ordinary MR and the MR related to spin fluctuation. The positive MR at T <<T_N is related to the ordinary MR by the Lorenz force because the magnetic fluctuation is suppressed in this temperature region. On the other hand, when T ~ T_N, the magnetic scattering by thermal fluctuation of spins is enhanced. This contribution to the resistivity is suppressed by the magnetic field perpendicular to the plane, producing the negative MR.

Figure 3

(a–h) MR curves at several different temperatures (T = 5, 10, 14, 22, 25, 27, 32, and 36 K). The red, blue, and green curves show the MRs when B is applied along the x, y and z-axes, respectively. The axes are defined as shown in the inset of (a). The solid and broken lines in (e) are the best fits with Eq. (2). Because of the uniaxial anisotropy, the MR has a jump at B ≈ ±0.5 T. The arrows and numbers in (f) indicate the order of the field sweep direction.

Another unique feature is the butterfly-shaped MR at B ≈ ±0.5 T. The amplitude of the butterfly-shaped MR is small when T << T_N. As we approach T_N, it becomes larger and takes a maximum at 25 K (≈T_N). The maximum value reaches more than 10% at B = 0.5 T, which is unusually large for conventional ferromagnetic materials^27,28. As we raise the temperature further, the amplitude of the MR suddenly decreases and becomes zero above T^* = 32 K.

In contrast to the MR along the z-direction, such a drastic temperature dependence of MR has not been observed when B || x and B || y, although a small negative MR can be seen below T_N. The B-angle dependence of the MR has never been studied for polycrystalline bulk Ag₂CrO₂^21,24.

To evaluate the butterfly-shaped MR observed only for B || z, we define the amplitude of the buttery-shaped MR, i.e., $\varGamma \equiv \frac{{\rho }_{xx}^{{\rm{upper}}}({B}_{{\rm{c}}})-{\rho }_{xx}^{{\rm{lower}}}({B}_{{\rm{c}}})}{{\rho }_{xx}(0)}$, and the corresponding magnetic field (B_c), as illustrated in the inset of Fig. 4(a). Γ has a small value at low temperatures and takes a maximum (15%) at around T_N. It still has a finite value even above T_N and finally disappears at T^*. B_c in Fig. 4(b) is almost constant up to T ≈ 22 K, and starts to decrease with increasing temperature and disappears at T^*.

Figure 4

(a) The amplitude of the buttery-shaped MR (Γ ) as a function of temperature. The inset shows the definitions of Γ and B_c. The red and blue curves show ρ_xx^upper and ρ_xx^lower, respectively. (b) Temperature dependence of B_c.

Similar MR effects are often observed not only in current-in-plane (CIP) giant-magnetoresistance (GMR) devices^27,28 but also in ferromagnetic^29,30 and even antiferromagnetic materials³¹. However, the present butterfly-shaped MR is essentially different from them. While commonly-used CIP-GMR devices have in-plane magnetization whose direction is the same as the current direction, Ag₂CrO₂ has a perpendicular magnetization to the basal plane and the current direction. An MR in conventional magnetic materials depends on the relative angle of magnetic domains, which is tuned by B. The amplitude of the MR is at most less than 1%³⁰ at B = 0.5 T. It decreases with increasing temperature and becomes zero above the transition temperature. In the butterfly-shaped MR, however, Γ has a maximum value of 15% near the transition temperature, which cannot be expected in conventional magnetic materials^29,30,31. These experimental facts indicate that spin fluctuations of the PD state are strongly related to the butterfly-shaped MR. To our knowledge, this is the largest MR value induced by spin fluctuations.

Discussions

What is the origin of the butterfly-shaped MR? As mentioned in the introduction, Ag₂CrO₂ is basically an antiferromagnetic metal and the PD spin behaves as a free spin. Thus there should be no spontaneous magnetization, but it has been established that the polycrystalline Ag₂CrO₂ has a small but finite spontaneous magnetization below T_N^21,22,23,24. The origin of the spontaneous magnetization is still an unsolved problem, but the PD spins should play an essential role in the butterfly-shaped MR. Here we assume the same magnetic state for the thin film device as the bulk sample, although the magnetization measurement has not been performed. We also recall that the butterfly-shaped MR in the present work appears only when B || z. These features imply that the uniform magnetic moment is along the z-axis and has a strong uniaxial anisotropy. As illustrated in Fig. 5, we assume that the PD spin is canted and has a small magnetic moment along the c-axis. In addition, we also assume that the magnetic fluctuation of the PD spin is relatively large, since the PD spin flips only at 0.5 T. In such a situation, B suppresses the spin fluctuations when it is parallel to the moment direction, while B causes a spin flip when it is antiparallel to the moment direction. Thus, a negative and butterfly-shaped MR can be explained by the suppression of spin fluctuation and the spin-flip process induced by B, respectively.

Figure 5

Schematic drawing of the spin configuration on the b-c plane (see Fig. 1(a)) expected from the experimental result.

As one of the possible models to explain the butterfly-shaped MR, we consider a 2D ferromagnetic spin system with the Ising anisotropy. Under a small magnetic field ≤1 T, we assume that the ferromagnetic magnon well approximates the low-energy magnon states of Ag₂CrO₂. The Hamiltonian is given by

$$H=-J\sum _{ < i,j > }{{\boldsymbol{S}}}_{i}\cdot {{\boldsymbol{S}}}_{j}-\Delta \sum _{i}{S}_{i}^{z}{S}_{i}^{z}-B\sum _{i}{S}_{i}^{z}$$

(1)

where J (>0) is the ferromagnetic exchange coupling between the nearest neighbor sites, Δ (>0) is the Ising anisotropy, and i and j are the site numbers. B is applied perpendicular to the 2D plane (i.e., B || z). The uniaxial anisotropy lifts the Goldstone mode, producing a spin gap proportional to the anisotropy energy 2Δ (∝ |B_c|), as illustrated in Fig. 6. For the positive magnetization, the spin gap increases (decreases) by applying the positive (negative) magnetic field and becomes zero when B = B_c (<0). The suppression of the magnetic fluctuation for B > 0 and the spin flip at B = B_c (<0) are intuitively explained by the spin gap modulated by B. The exactly same scenario is valid for the negative magnetization just by inverting the sign of B.

Figure 6

Dispersion relation of magnons with the uniaxial anisotropy for the positive magnetization. Because of the anisotropy energy 2Δ, the energy band is shifted and becomes zero when B = B_c (<0).

To see how B suppresses the MR, we compute the elastic scattering rate $\frac{1}{{\tau }_{{\rm{mag}}}}\propto {\rho }_{xx}$ due to spin fluctuations at finite T by Born approximation (See Supplementary Information for details on our theoretical model, which includes refs. ^{32,33,34,35,36}):

$$\frac{1}{{\tau }_{{\rm{mag}}}}\propto T\{{F}_{1}(1+\frac{2\Delta +{\mu }_{{\rm{eff}}}B}{2J{k}_{F}^{2}{a}_{0}^{2}})-{F}_{2}(1+\frac{2\Delta +{\mu }_{{\rm{eff}}}B}{2J{k}_{F}^{2}{a}_{0}^{2}})\}$$

(2)

where k_F is the Fermi wave number, μ_eff is the effective ferromagnetic moment, a₀ is the lattice constant between the neighboring effective ferromagnetic moments, ${F}_{1}(x)=\frac{1}{\sqrt{{x}^{2}-1}}$ and ${F}_{2}(x)=\frac{x}{\sqrt{{x}^{2}-1}}-1$. A fit to the experimental data in Eq. (2) is shown in Fig. 3(e); the butterfly-shaped MR near T_N is well-reproduced by our theoretical model. We have roughly estimated both J and Δ to be about 10 K, which is reasonable for the present case (See Supplementary Information for details on our theoretical model, which includes refs. ^{32,33,34,35,36}).

Finally, let us mention the relation between T_N and T^*. In the present experiment, the butterfly-shaped MR takes a maximum at around T_N and vanishes at T^* = 32 K. Originally, T_N is determined from the peak position of the heat capacity measurement as shown in ref. ²¹. A long range ordering emerges below T_N (~24 K), but some short range ordering with magnetic fluctuations may grow even above T_N. This was already pointed out by Sugiyama et al. from zero-field μ⁺SR measurements²³. They argued that the phase with the PD spins already grows above T_N and vanishes T^* = 28 K. Those tendencies are indeed consistent with our experimental data. In addition, it is also known that T_N is slightly shifted to the higher temperature side, by applying the magnetic field²¹. Thus, it seems to be reasonable that T^* determined from the present MR measurement is higher than T^* determined from the zero-field μ⁺SR measurement. However, we cannot make a further statement about the relation between T_N and T^* because it is difficult to perform standard measurements such as heat capacity and magnetization measurements for a tiny crystal. Further experimental and theoretical works are highly desirable to unveil the relation between the two characteristic temperatures.

Conclusions

In summary, we observed a butterfly-shaped MR in triangular-lattice antiferromagnetic Ag₂CrO₂ devices. The butterfly-shaped MR can be seen only when the magnetic field is applied along the c-axis. This fact indicates a strong uniaxial anisotropy in Ag₂CrO₂. The butterfly-shaped MR takes a maximum value of 15% at around the transition temperature, suggesting that spin fluctuations are essential. The result is well explained by the theoretical model based on the 2D magnetic system with the Ising anisotropy. Rich physics is further expected in such a magnetically frustrated system coupled to conducting electrons. In addition, the large MR at small switching fields obtained in the frustrated spin system would be useful for future device applications.