Emerging Research Landscape of Altermagnetism

Magnetism is one of the largest, most fundamental, and technologically most relevant fields of condensed-matter physics. Traditionally, two basic magnetic phases have been distinguished ferromagnetism and antiferromagnetism. The spin polarization in the electronic band structure reflecting the magnetization in ferromagnetic crystals underpins the broad range of time-reversal symmetry-breaking responses in this extensively explored and exploited type of magnets. By comparison, antiferromagnets have vanishing net magnetization. Recently, there have been observations of materials in which strong time-reversal symmetry-breaking responses and spin-polarization phenomena, typical of ferromagnets, are accompanied by antiparallel magnetic crystal order with vanishing net magnetization, typical of antiferromagnets. A classification and description based on spin-symmetry principles offers a resolution of this apparent contradiction by establishing a third distinct magnetic phase, dubbed altermagnetism. Our perspective starts with an overview of the still emerging unique phenomenology of this unconventional $d$-wave (or higher even-parity wave) magnetic phase, and of the wide array of altermagnetic material candidates. We illustrate how altermagnetism can enrich our understanding of overarching condensed-matter physics concepts and how it can have impact on prominent condensed-matter research areas.

Figure 1

Illustrative models of collinear ferromagnetism, antiferromagnetism, and altermagnetism in crystal-structure real space and nonrelativistic electronic-structure momentum space. (a) Ferromagnetic model with one spin sublattice, and corresponding distinct spin-group form, nonzero magnetization, $\mathcal{T}$-symmetry breaking momentum-independent spin splitting of bands, and isotropic $s$-wave spin-split Fermi surfaces. (b) Antiferromagnetic model with opposite-spin sublattices (red, blue) connected by inversion or translation, and corresponding distinct spin-group form, zero net magnetization, and $\mathcal{T}$-invariant spin-degenerate bands reminiscent of nonmagnetic systems. (c) Altermagnetic model with opposite-spin sublattices connected by rotation and not by translation or inversion, and corresponding distinct spin-group form, zero net magnetization, $\mathcal{T}$-symmetry breaking spin splitting with alternating sign, anisotropic sublattice spin densities, and anisotropic $d$ ($g$ or $i$)-wave spin-split Fermi surfaces. See Sec. 2b for the definitions of the distinct spin-group forms ${\mathbf{R}}_s^i$.

Figure 2

Analogy between superconductivity and magnetism. Top: conventional $s$-wave superconductivity, which has the same Cooper-pair phase around the Fermi surface and is analogous to conventional magnetism (ferromagnetism) with an excess of one spin orientation around the Fermi surface. Bottom: unconventional $d$-wave superconductivity where the Cooper-pair phase changes sign around the Fermi surface, and the anticipated analogy of unconventional $d$-wave magnetism where the excess spin orientation changes sign around the Fermi surface. The unconventional phases have the characteristic nodes highlighted by gray circles. This figure was inspired by Ref. [29].

Figure 3

(a) Schematics of the rutile $X{Y}_2$ crystal structure with antiparallel magnetic moments on $X_A$ and $X_B$ magnetic sublattices. (b) Brillouin zone of the rutile crystal and ab initio nonrelativistic calculation of a wave-vector $k_z=0$ cut of the anisotropic $d$-wave spin-polarized Fermi surface of metallic ${\mathrm{RuO}}_2$. (c) Ab initio altermagnetic spin splitting of bands in ${\mathrm{RuO}}_2$, calculated without (red and blue) and with (black) relativistic spin-orbit coupling. (d) Ab initio altermagnetic spin-split Fermi surface for selected $k_z$ values in ${\mathrm{RuO}}_2$ with correlations accounted for within the dynamical mean-field theory. (d,e) Ab initio altermagnetic spin-split bands of insulating ${\mathrm{FeF}}_2$ and ${\mathrm{MnF}}_2$, respectively. This figure is adapted from Refs. [3, 18, 20, 47].

Figure 4

(a) Schematic top view (upper panel) and 3D view (lower panel) of the ${\mathrm{RuO}}_2$ crystal with opposite spin directions on ${\mathrm{Ru}}_A$ and ${\mathrm{Ru}}_B$ sublattices depicted in red and blue, oxygen atoms shown in black, and with the depicted nonrelativistic spin-group symmetries corresponding to spin group ${}^{2}4/{}^{1}m{}^{2}m{}^{1}m$ in the notation of Ref. [16]. The curved red arrow and its label highlight the generator of opposite-spin sublattice transformations, and the generators of the halving subgroup of same-spin sublattice transformations are also highlighted (in black). Here, $C_2$ on the left of the double vertical bar is a 180° spin-space rotation transformation around an axis perpendicular to the spins, and $E$ is the spin-space identity. On the right of the double vertical bar, $C_{4z}\mathit{t}$ is a fourfold real-space rotation combined with translation, and $M_i$ are real-space mirror transformations. (b) Schematic spin arrangement on the ${\mathrm{RuO}}_2$ crystal with antiparallel (upper panel) and parallel (lower panel) spin directions and the crystallographic spin-axis orientation depicted by red and blue arrows, and with the depicted generators of the relativistic magnetic symmetry group $m^{\prime }{m}^{\prime }m$. Here, $\mathcal{T}$ is time reversal. The crossed arrow highlights that the magnetic group contains no opposite spin-sublattice transformation elements. The same magnetic group describes a fully compensated antiparallel magnetic order, a parallel magnetic order with a strong nonrelativistic ferromagnetic moment, as well as an antiparallel magnetic order with a weak uncompensated relativistic magnetization [16]. This example illustrates that the relativistic magnetic groups generally do not separate between relativistic and nonrelativistic, compensated and noncompensated, or collinear and noncollinear magnetic phases. This figure is adapted from Ref. [16].

Figure 5

(a) Spin-polarized relativistic Fermi surface highlighting the presence of the approximate spin symmetry $[C_2||C_{4z}]$, omitted by the magnetic group. (b) Relativistic momentum-resolved Berry curvature hotspots originating from the avoided crossings along the $k_{x,y}=0$ lines, whose position in the momentum space is determined by the spin symmetries in the absence of the relativistic spin-orbit coupling. (c) Fermi-surface-resolved Berry curvature of the ${\mathrm{FeSb}}_2$ altermagnet illustrates pronounced contributions to Berry curvature from the quasinodal surface $k_x=0$, $k_y=0$. (d) Brillouin zone notation. This figure is adapted from Refs. [5, 7, 16].

Figure 6

Crystal structures of selected altermagnetic candidates organized by dimensionality and conduction type. The crystal structure are discussed in detail in Refs. [7, 16].

Figure 7

(a)–(d) Ab initio spin-split band structures of depicted altermagnetic candidate materials. This figure is adapted from Refs. [6, 7, 15, 116].

Figure 8

Altermagnetic candidates identified from ab initio calculations, organized in a altermagnetic transition temperature vs altermagnetic spin-splitting strength diagram. The size of the balls scales with the largest atomic number in the crystal. This figure is adapted from Ref. [16] and references therein.

Figure 9

(a) Model relativistic Rashba spin-split bands. (b) Model of antiferromagnetic zero-magnetization crystal of ${\mathrm{BiCoO}}_3$ with magnetic symmetry $\mathcal{T}\mathit{t}$, and with broken space-inversion symmetry. (c) Model nonrelativistic altermagnetic spin splitting. (d) Model of altermagnetic crystal of ${\mathrm{RuO}}_2$ with nonrelativistic spin symmetry $[C_2\parallel C_4\mathit{t}]$. The crystal are discussed in Refs. [16, 101].

Figure 10

(a) Schematic diagram of the anisotropic ($d$-wave) exchange Fermi-liquid instability in the altermagnet. The black line corresponds to the spin-degenerate band in the nonmagnetic phase, while red and blue lines are spin-split bands in the altermagnetic phase. (b) Spin-projected (and orbital-projected) ab initio bands of ${\mathrm{RuO}}_2$ in the energy window corresponding to panel (a). (c) Schematic diagram of the isotropic ($s$-wave) exchange Fermi-liquid instability combined with anisotropic crystal potential in the altermagnet. Solid and dashed black lines correspond to the spin-degenerate bands dominated by one or the other sublattice in the nonmagnetic phase, respectively. Red and blue lines are spin-split bands in the altermagnetic phase. (d) Spin-projected (and orbital-projected) ab initio bands of ${\mathrm{RuO}}_2$ in the energy window corresponding to panel (c). Pink boxes with labels 1 and 2 correspond to the full ab initio bands of ${\mathrm{RuO}}_2$ shown in Fig. 3. This figure is adapted from Ref. [16].

Figure 11

(a,b) Model of the altermagnetic quasiparticle with quadratic dispersion around the spin-degenerate $\mathbf{\Gamma }$ point, and spin-winding number 2 around the $\mathbf{\Gamma }$ point. The model corresponds to Eq. (1). (c,d) Model of the altermagnetic spin-polarized valley quasiparticle with no spin winding in the valley. The model corresponds to Eq. (2). The center of a valley is at TRIM, and the spin polarization is opposite at TRIM $\mathbf{X}$ and $\mathbf{Y}$. (e) Schematic illustration of distinct symmetries of band structures with nonrelativistic altermagnetic and relativistic nonmagnetic valleys. (f) Real-space spin-dependent hoppings used to construct model band structures in panels (a)–(c). This figure is adapted from Refs. [6, 7, 13].

Figure 12

(a) Schematics of the longitudinal spin current in altermagnets. For an electric bias $\mathbf{E}$ applied along one of the main anisotropy axes of the spin-split Fermi surfaces, the spin-up and spin-down charge currents are parallel but of different magnitudes due to the Fermi-surface anisotropies. As a result, the longitudinal charge current is spin polarized. (b) Schematics of a GMR stack in a current-in-plane geometry. As an example, we show the antiparallel configuration of the altermagnetic-order vectors in the two electrodes ${\mathrm{AM}}_1$ and ${\mathrm{AM}}_2$. Interfaces are oriented along one of the main anisotropy axes of the spin-split bands. Energy band cuts highlight the anisotropy around the $\mathbf{\Gamma }$ point, resulting in anisotropic spin-dependent conductivities. (c) Schematics of the transverse spin current. For $\mathbf{E}$ applied in the diagonal direction between the two anisotropy axes of the spin-split Fermi surfaces, the spin-up and spin-down charge currents combine in an unpolarized longitudinal charge current and in a pure transverse spin current. (d) Spin-splitter-torque concept on a schematic of altermagnetic-altermagnetic (ferromagnetic) bilayer. A spin-current from the bottom altermagnet propagates in the out-of-plane direction and generates a spin-splitter torque on the altermagnetic (or ferromagnetic) order vector in the top layer. (e) Ab initio longitudinal spin-up and spin-down conductivities (red and blue), GMR, and the ratio of the transverse spin current relative to the longitudinal charge current (SCR) in ${\mathrm{RuO}}_2$. This figure is adapted from Refs. [8, 13].

Figure 13

(a) Schematics of a TMR stack with an insulating barrier and altermagnetic electrodes with parallel and antiparallel order vectors. Energy band cuts highlight the oppositely split valleys, resulting in valley and spin-dependent densities of states. (b) Ab initio quantum-transmission calculations of the TMR in a ${\mathrm{RuO}}_2|$ ${\mathrm{TiO}}_2|$ ${\mathrm{RuO}}_2$ tunnel junction. (c) Model spin and transport-momentum projected band structure, and relative difference between the conductances in the parallel and antiparallel configurations of the altermagnetic-order vectors in the two electrodes. The TMR is maximized for transport energies corresponding to the spin-split valleys well separated in momentum. This figure is adapted from Refs. [13, 14].

Figure 14

Summary of the emerging research landscape of altermagnetism.