Magnetic Octupole Hall Effect

Abstract

d-wave altermagnets은 order parameters에 magnetic octupole이 존재한다. 우리는 이론적으로 외부에서 주입된 magnetic octupole이 d-wave altermanget에 토크를 가함을 보인다. 이 주입은 인접한 층에서의 magnetic octupole Hall efect를 통해 달성된다. 우리는 중금속 Pt에서 magnetic octupole Hall conductivity를 계산하고 spin Hall conductivity에 상응하는 거대한 값을 찾아냈다. 

 

d-파형 얼터자석(altermagnet)은 자기 옥텁극(magnetic octupole)을 질서 매개변수(order parameter)로 갖는다. 우리는 외부로부터 주입된 자기 옥텁극이 d-파형 얼터자석에 토크를 발생시킨다는 것을 이론적으로 보였다. 이러한 주입은 인접 층에서 발생하는 자기 옥텁극 홀 효과(magnetic octupole Hall effect)를 통해 실현될 수 있다. 우리는 중금속인 백금(Pt)의 자기 옥텁극 홀 전도도(magnetic octupole Hall conductivity)를 계산하였고, 그 값이 스핀 홀 전도도(spin Hall conductivity)에 필적할 만큼 상당히 크다는 것을 확인하였다. 본 연구는 기존의 스핀 홀 현상론(중금속에서 생성되고 강자성체에서 토크로 감지됨)을 일반화하여, 자기 옥텁극 홀 현상론(중금속에서 생성되고 얼터자석에서 토크로 감지됨)으로 확장한 것이다. 이는 얼터자석의 자기 배향을 전기적으로 제어하는 데 활용될 수 있다.

 

 

Introduction.– Analysis based on the spin space group has recently proposed a new class of magnetic materials dubbed altermagnet (AM). An AM has zero net magnetization, similar to antiferromagnet (AFM), but it exhibits a spin splitting of its energy bands, which does not originate from the relativistic spin-orbit coupling (SOC) and can be larger (∼ 1 eV) than the SOC-induced spin splitting. This nonrelativistic splitting of the AM can be utilized to generate spin current, spin-splitting torque, giant magnetoresistance and anomalous Hall effect.

Recently, magnetic multipoles have emerged as primary quantities in AMs. Contrary to ferromagnets (FMs), where the magnetic dipoles are the order parameter, d-wave AMs have the magnetic octupoles (MOs) as their leading-order order parameter. The MOs distinguish the d-wave from conventional AFMs and determine the nonrelativistic spin-splitting. Additionally, a Landau theory analysis on d-wave AMs pointed out that their MO 𝐎i⁢j couples linearly to their Néel vector 𝐍, where the vector direction of 𝐎i⁢j denotes the spin direction and the indices i⁢j contain the angular distribution information of the MO [Eq. (1)]. This coupling 𝐍⋅𝐎i⁢j is analogous to the coupling 𝐌⋅𝐒 between the local magnetization 𝐌 and the spin 𝐒 in FMs. This analogy motivates the possibility that MOs injected into an AM may induce torque on the AM, just as spins injected into an FM induce torque on the FM. In this Letter, we show that the injection of MO from outside indeed generates the torque on an AM. The torque contains a non-staggered component, implying that it can rotate the Néel order of the AM. This torque can survive in a configuration where the spin-Hall-induced torque vanishes identically. This torque is our first main result. As a method for injecting the MO into an AM layer, we propose the MO Hall effect (MOHE) in heavy metals (HMs). HMs such as Pt are strong sources of spin Hall current. We find that Pt also generates a large MO Hall current, which is comparable in magnitude to the large spin Hall current generated by Pt. This is our second main result.

 

Magnetic octupole.– Magnetic multipoles naturally emerge in the interaction energy between magnetization density 𝝁⁢(𝐫) and an inhomogeneous magnetic field 𝐇⁢(𝐫) [26, 27, 28, 29, 30, 31, 32]: Hint=−∫𝝁⁢(𝐫)⋅𝐇⁢(𝐫)⁢d3⁢r can be expanded into −(∫𝝁⁢(𝐫)⁢d3⁢r)⋅𝐇⁢(0)−(∫ri⁢μj⁢(𝐫)⁢d3⁢r)⁢∂iHj⁢(0)−12⁢(∫ri⁢rj⁢μk⁢(𝐫)⁢d3⁢r)⁢∂i∂jHk⁢(0). In centrosymmetric AM, both magnetic-dipolar term (∝∫𝝁⁢(𝐫)⁢d3⁢r) and magnetoelectric multipolar term (∝∫ri⁢μj⁢(𝐫)⁢d3⁢r) vanish, but the MO, (∝∫ri⁢rj⁢μk⁢(𝐫)⁢d3⁢r), can possess a finite value. Thus, the MO is the first symmetry-allowed magnetic multipole that acts as the order parameter of AM. To facilitate the evaluation of MO, we adopt the atomic-site MO operators [33, 34, 35]

Oi⁢jq≡1ℏ2⁢{Li,Lj}⁢Sq,

(1)

where Li is the orbital angular momentum operator, Sq is the spin angular momentum operator, and {A,B}=A⁢B+B⁢A. Here, {Li,Lj}/ℏ2 represents the quadrupole density of electrons constructed by the spherical tensors [36] and the generalized Stevens operator approach, {Li,Lj}/ℏ2=C⁢ri⁢rj/a02 [37, 38, 39, 40, 41] where C is constant and a0 is the Bohr radius. Note that Oi⁢jq is defined to share the same dimension as Sq, which facilitates the magnitude comparison between the spin Hall effect (SHE) and the MOHE.

Magnetic octupole Hall effect.– We present our second main finding first: MOHE in HM. Within the linear response regime, the MO current density JjOm⁢nq carrying Om⁢nq is proportional to an electric field 𝓔,

JjOm⁢nq=χj⁢iOm⁢nq⁢ℰi,

(2)

where JjOm⁢nq is calculated by the MO current operator defined by {vj,Om⁢nq}/2, χj⁢iOm⁢nq is the MO Hall conductivity (MOHC), and vj is the velocity operator. This is analogous to the SHE response, JjSq=σj⁢iSq⁢ℰi, where JjSq represents the spin current density flowing in the j direction with the spin along the q direction. To be specific, we consider an electric field along the x-direction and compare the z-flows of MO currents and spin currents. For nonmagnetic materials with the mirror reflection symmetries with respect to the z⁢x and z⁢y planes, the SHE can generate only one component JzSy, whereas the MOHE can generate multiple components, JzOx⁢yx, JzOy⁢zz, JzOx⁢xy, JzOy⁢yy, and JzOz⁢zy since MO and spin transform differently under the mirror reflections. Figure 1(a) illustrates JzOx⁢yx. Note that the spin directions in JzOx⁢yx and JzOy⁢zz are perpendicular to the spin direction carried by the spin Hall current JzSy. This will be useful in distinguishing the torque caused by MOHE from that caused by SHE [41].