Low-temperature exchange coupling between ${\mathrm{Fe}}_2{\mathrm{O}}_3$ and $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$: Insight into the mechanism of giant exchange bias in a natural nanoscale intergrowth

Exchange bias ($>1\mathrm{T}$ at $10\mathrm{K}$) has been observed in natural sample of ${\mathrm{Fe}}_2{\mathrm{O}}_3$ containing abundant nanoscale exsolution lamellae of $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$. Exchange bias is first observed below the Néel temperature of $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ $\left(55\mathrm{K}\right)$. Possible interface magnetic structures are explored within the framework of a classical Heisenberg model using Monte Carlo simulations. The simulations predict a threshold value of the ${\mathrm{Fe}}_2{\mathrm{O}}_3$ anisotropy constant, below which ${\mathrm{Fe}}^{3+}$ spins become tilted out of the basal plane in the vicinity of the interfaces. This tilting creates a $c$-axis component of magnetization in the ${\mathrm{Fe}}_2{\mathrm{O}}_3$ host that couples to the $c$-axis magnetization of the $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ lamellae. Exchange interactions across the interfaces are frustrated when the $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ lamellae contain an even number of ${\mathrm{Fe}}^{2+}$ layers, resulting in zero net exchange bias. Lamellae containing an odd number of ${\mathrm{Fe}}^{2+}$ layers, however, are negatively exchange coupled to the ${\mathrm{Fe}}_2{\mathrm{O}}_3$ host across both (001) bounding surfaces, and are the dominant source of exchange bias. Exchange bias is observed whenever there is a significant $c$-axis component to both the ${\mathrm{Fe}}_2{\mathrm{O}}_3$ magnetization and the applied field. An exchange bias of $0.9\mathrm{T}$ was obtained with an anisotropy constant of $0.1\mathrm{K}$.

Figure 1

Hysteresis loops [(a)–(c)] measured at $20\mathrm{K}$ after zero-field cooling and after (a) imparting a positive room-temperature remanence, (b) imparting a negative room-temperature remanence, and (c) completely demagnetizing the sample at room temperature. (d) The lower hysteresis branch subtracted from the upper branch in the case of negative exchange bias for six temperatures. The location of the center of each peak gives the magnitude of the exchange-bias field. (e) EB shift deduced from (d) as a function of temperature.

Figure 2

(a) Bright-field TEM micrograph of the titanohematite host, showing first generation lamellae surrounded by abundant fine-scale second generation lamellae. (b) Ti chemical map of a similar area to (a) obtained using energy-filtered imaging. Bright regions are rich in Ti and were identified as $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ by selected-area electron diffraction. (c) Higher magnification bright-field micrograph of first and second generation lamellae. A relatively large $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ lamella surrounded by a precipitate-free zone is indicated in the upper right by the white arrows. The contrast observed in the remaining host is due to the coherency strain associated with $\sim 1\mathrm{nm}$ thick precipitates of $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$. These are more clearly seen in the inset, which is an enlargement of the region indicated by the black square. The $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ precipitates appear as thin white lines, indicated by the black arrows.

Figure 3

Idealized model of an $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ precipitate within an ${\mathrm{Fe}}_2{\mathrm{O}}_3$ host. Numbered layers correspond to (001) layers of ${\mathrm{Fe}}^{3+}$ (white), ${\mathrm{Fe}}^{2+}$ (gray), and Ti (black) cations. Layers 6 and 18 are contact layers, containing a mixture of ${\mathrm{Fe}}^{2+}$ and ${\mathrm{Fe}}^{3+}$. $M_H$ is the soft lamellar moment due to imbalance between left and right pointing spins in the ${\mathrm{Fe}}_2{\mathrm{O}}_3$ host. $M_I$ is the hard lamellar moment due to imbalance between up and down pointing spins in the $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ host.

Figure 4

(Color online) Definitions of the superexchange interaction parameters $J_1–J_5$ (Table ).

Figure 5

(Color) Interface magnetic structures for odd-layered [(a) and (c)] and even-layered [(b) and (d)] $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ precipitates in an ${\mathrm{Fe}}_2{\mathrm{O}}_3$ host. The red and green cones indicate the direction of spin on ${\mathrm{Fe}}^{3+}$ and ${\mathrm{Fe}}^{2+}$ cations, respectively. Ti cation positions are indicated by the black dots. (a) Equilibrium structure for an odd-layered precipitate in zero field. The central core of the $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ precipitate (shaded light orange) carries a net downward spin $\left(M_I\right)$. The outer rim (shaded green) adopts a ferromagnetic structure with net spin $\left(M_H\right)$ at $\sim 45°$ to the basal plane. The presence of strong antiferromagnetic interactions across (100) and (001) interfaces are indicated by the solid black lines. Exchange coupling between the ferromagnetic rim and the ilmenite core is governed by weaker antiferromagnetic interactions $J_5^{2\text{−}3}$ and $J_5^{2\text{−}2}$ across the (001) interfaces (indicated by the dotted lines). (b) Equilibrium structure for an even-layered precipitate in zero field. $M_I$ is zero and $M_H$ lies much closer to the basal plane for an even-layered precipitate. Tilting $M_H$ out of plane leads to antiferromagnetic alignment of spins across one (001) interface and ferromagnetic alignment of spins across the bottom interface. (c) Exchange biased state of an odd-layered precipitate obtained in a field of $-3\mathrm{T}$ applied along [001]. The vertical component of the contact-layer spins are brought into ferromagnetic alignment with the ${\mathrm{Fe}}^{2+}$ spins of the central core. This leads to a larger net spin in the exchange biased state and a vertical shift of the hysteresis loop. (d) Exchange biased state of an even-layered precipitate. This state is degenerate with that shown in (b).

Figure 6

Average component of spin along $z$ on odd and even numbered layers of the simulation cell (closed and open circles, respectively) and the net spin along $z$ (squares) as a function of $K_{3+}$. All simulations performed at $T=10\mathrm{K}$. For $K_{3+}>0.225\mathrm{K}$, all ${\mathrm{Fe}}^{3+}$ spins (and also the ${\mathrm{Fe}}^{2+}$ spins within the contact layers) lie in the basal plane. The net spin along $z$ is negative due to the contribution from the ${\mathrm{Fe}}^{2+}$ spins of the $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ precipitate $\left(M_I\right)$. For $K_{3+}<0.05\mathrm{K}$ the ${\mathrm{Fe}}^{3+}$ spins and the contact-layer ${\mathrm{Fe}}^{2+}$ spins lie parallel and antiparallel to $z$.

Figure 7

Simulated hysteresis loops for a variety of applied field directions. Each data point represents the result of $8\times {10}^7$ Monte Carlo steps ($4\times {10}^7$ equilibration plus $4\times {10}^7$ production steps). The arrows indicate the starting point for the simulations. The net spin is defined as the component parallel to the direction of applied field. (a) An odd-layered $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ precipitate with $K_{3+}=0.1\mathrm{K}$, $T=10\mathrm{K}$, and field applied along $c$. (b) An odd-layered $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ precipitate with $K_{3+}=0.1\mathrm{K}$, $T=10\mathrm{K}$, and field applied at 45° to $c$. (c) An odd-layered $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ precipitate with $K_{3+}=0.1\mathrm{K}$, $T=10\mathrm{K}$, and field applied at 90° to $c$. (d) An even-layered $\mathrm{Fe}\mathrm{Ti}{\mathrm{O}}_3$ precipitate with $K_{3+}=0.1\mathrm{K}$, $T=10\mathrm{K}$, and field applied at 45° to $c$.

Figure 8

(Color online) Upper hemisphere stereographic projections showing the path taken by the net spin vector during the magnetization reversal process. Each figure was derived from the corresponding hysteresis loop data shown in Fig. 7. The closed and open symbols represent net magnetization vectors pointing upward and downward, respectively.