INTRODUCTION TO MULTIFERROICS Student’s name – Umesh Chandra Thuwal Roll No. – 20109291 Department of Physics IIT Kanpur
Abstract Multiferroics are the materials in which ferroic (ferroelectric, ferromagnetic and ferroelastic) ordering coexist. These materials open up the profitable way to control magnetism by an electric field. The coexistence of an electric and magnetic field has been known for the long time, but in the last two decades, some key discoveries generated a lot of interest among researchers in this field. In this report, I provide a general introduction to multiferroics, their different mechanism which support multiferroicity, composite multiferroic and violation of inversion symmetry in multiferroic which leads to multiferroicity, ferrotoroidicity and magnetic monopole in condensed matter. I will also briefly discuss the advantages of multiferroic material and the future of multiferroicity.
Introduction
In 1994 the term multiferroic was introduced by Hans Schmid [1]. He defined the multiferroic materials as the materials in which two or more ferroic orders such as ferroelectric and ferromagnetism coexist. In a more general context, the multiferroic is defined as the coexistence of any two ferroic (ferroelectric,ferromagnetic, ferroelastic and ferrotoroidic) ordering simultaneously. At present, the researchers have extended this definition to include other long-range order such as antiferromagnetism. Although the term multiferroics was introduced around 90’s but with the introduction of magnetoelectrics, it can be said that the active research in this field was started in the ’50s If we were to talk about the technical merits, ferromagnetic and ferroelectric are different, In multiferroic, both orders may be in a single phase. These material are important for technological implication for mainly two reasons [2]. Firstly, they make it possible to control magnetic bit by electric bit and secondly, the coupling of ferroelectric and ferromagnetic materials might give some novel properties which are not present in either state alone. If in the reading and writing of magnetic bit, an electric voltage is used instead of an electric current (source of magnetic field) the loss of energy will get avoided because long build up is associated with electric current and high energy is needed. Thus multiferroic materials leads to a more smaller, faster and energy efficient data storage which is the demand of future technology. There are many material requirements which are needed to be fulfilled for a material to be called as multiferroic. For instance, ferroelectric material must be non-centrosymmetric to pass spontaneous electric polarisation and there are 31 point groups which allow this. Out of these 13 point groups allow occurrence of both the properties simultaneously. These point groups may lead to large number of possible materials exhibiting multeferroism so it is unlikely that symmetry play an important role in determining whether material will be multiferroic or not.
Important term in multiferroics 1.Ferroic Material which shows long rang order with respect to atleast one macroscopic properties and develop a dipole with conjugate field used term ferroic. There are four type of ferroic are shown in figure 1 Ferroelectric material which posseses a spontaneous polarization P that is stable and switched hysteritically by an applied electric field E. Ferromagmetic material possesses a spontaneous magnetization M that is stable and switched hysteritically by an applied magnetic field H. Ferroelastic material shows spontaneous deformation that is stable and switched hysteritically by an applied stress. Ferrotoroidic state with a domain that could be hysterically switched space-time antisymmetric external field (toroidal field).
Figure 1. four type of ferroic order, (a) ferroelasticity, (b) ferroelectricity, (c) ferromagnetism, (d) ferrotoroidicity., (a) ferroelasticity exhibit spontaneous strain, σ due to stress (b) spontaneous polarization, P, due to applied electric field , (c) spontaneous magnetization ,M, due to magnetic field (d) ferro-toroidics exhibit a vortex-like alignment of spins with a toroidization due to toroidic field.[3]
2.Magnetoelectric coupling
Landau theory describe the magnetoelectric effect in single phase material through expansion of the free energy expression as 1 1 1 F(E, H) = F − PsE − MsH − ∈ ∈ E E − μ μ H H − α E H + β E H H 0 i i i i 2 0 ij i j 2 0 ij i j ij i j 2 ijk i j k 1 + γ H E E 2 ijk i j k Where E and H are electric and magnetic field respectively. Here ∈ and 휇 are the dielectric and magnetic susceptibility.
Type 1 and type 2 Multiferroics If the magnetic and ferroelectric ordering in the material occurs independently that means the source of ferroelectric ordering and magnetic ordering are different, denoted as type 1 [2], as shown in figure 2. The coupling between magnetism and ferroelectricity in type 1 multiferroic is usually weak. In general mechanism which driven the multiferroicity in type 1 multiferroic are charge charge ordering mechanism, lone pair mechanism and geometric ferroelectrisity mechanism. If the ferroelectricity and magnetic transition emerges jointly known as the type 2 multiferroic, as shown in figure 2. Coupling between the magnetism and ferroelectricity in type 2 multiferroic is usually larger than type 1. In general spin driven mechanism is responsible for type 2 multiferroic
Figure 2 – type 1 and type 2 multiferroic, P, polarization; Si, spin at site i.[2]
Multiferroicity due to Different Mechanism As we discussed the ferroelectric and magnetic ordering coexist in these material. Four general mechanism which driven the multiferroicity [2] shown in figure 4. These four mechanism are charge ordering, lone pair, geometric effect and spin driven mechanism. In first three mechanism induced ferroelectrity is independent to magnetic ordering and the multiferroic material are type 1. In spin driven mechanism ferroelectric and magnetic transition occure jointly, type 2 multiferroic. Multiferroicity due to Charge ordering The appearance of ferroelectricity due to charge ordering [4] can be understand using Figure 3. Fig (A) consist of one dimensional chain with same (zero) charge on each site. Fig B shows same one dimensional chain with one set of sites has charge +e and other site has –e, but this arrangement not break the inversion symmetry so no net dipole moment in the arrangement. Another case of charge ordering is when a system dimerizes. Fig C shows, in this type of system strong and weak bond alternate or we can say that sites are equivalent but bonds are not. One can use the terminology, site centred charge ordering (S-CCO) for fig B and bond centred charge ordering (B-CCO) for fig C. but B-CCO arrangement is centrosymmetric thus not ferroelectric.
Figure 3 (a) Example of one dimensional neutral chain, (b) S-CCO, (c) B-CCO, (d) linear combination of S-CCO and B-CCO that is ferroelectric. Arrow indicate the polarization which is total zero in (b) and (c), but develops a microscopic moment indicated by red arrow in (d). The dashed lines indicate mirror plane of the system.[4] If combining both type of charge ordering as discussed in figure (B) and figure (C), situation changes. The situation with simultaneous S-CCO and B-CCO are shown in figure (D). In this situation inversion symmetry is broken and each molecule develop a net dipole moment, as a result whole system become ferroelectric.
In some material inequivalent bond are due to crystallographic structure and spontaneous charge ordering that can occur below certain temperature press inequivalent in site or the material can contain ion with different valence which after a structural dimerization transition induce ferroelectricity. There is some example of charge ordering multiferroic are perovskite manganites (Pr1−xCaxMnO3), Fe3O4 and LuFe2O4 (frustrated charge order).
The value of polarization in 퐿푢퐹푒2푂4 is large in comparision to these other two example. In 퐿푢퐹푒2푂4 valence electron can be distributed non-uniformaly around their host ion in the crystal lattice to form periodic superstructure. Fe atom in 퐿푢퐹푒2푂4 may form a superlattice with an alternating sequence of 퐹푒2+and 퐹푒3+ ions, shown in figure 4 (b). The ferroelectricity in LuFe2O4 is due to a combination of two factors: the bilayer character of the crystal structure, and the frustrated charge ordering leading to the formation of charged planes (e.g., negative charging of the lower layer and positive charging of the upper one). As, generally speaking, in such a situation CO may be very strong, one could in principle expect large spontaneous polarizations. The induced ferroelectricity in these material are independent of magnetic field.
Multiferroicity due to lone pair
Unbounded valence electron around the host ion. This is mainly observed in BiFeO3 [5]. This is perovskite oxide and most studied multiferroic material because of its high critical temperature In this mechanism the spatial asymmetry is created due to anisotropic distribution of and high ferroelectric polarization. In this material ferroelectricity is not due to the off centering of B site cation (퐹푒3+) rather the main reason for instability was due to the presence of steriochemically active lone pair of A site cation (Bi3+). Two 6s electron of bismuth ion not involved in the bond,thus showing the high polarizability and created a local dipole along [111] plane Figure 4 (a). The ovserved polarization in this ferroelectric marial is very large (100휇퐶/푐푚2). On the other hand the magnetization is guaranteed by B site magnetic 퐹푒3+(푑5) ions. Due to the independent origin of ferroelectric and magnetic ordering this is type 1 multiferroic. Multiferroicity due to geometric effect Geometric constraints and size effect can cause an instability in material [2]. Such type of mechanism is driven by steric effect rather than the usual change in chemical bond leads to ionic shift that result to a polar distortion and geometric ferroelectricity Figure 4(c). For example in h − RMnO3 (R = Sc, Y, In or Dy − Lu) a unit cell trimerizing lattice distortion 2 derives the force of the ferroelectric order at 푇푐 ≥ 1200퐾 with polarization 5.6 휇퐶/푐푚 . The magnetic order occure independently at very low temperature 푇푁 ≤ 120퐾. A similar behaviour is observed in h − LuFeO3 [6], hexagonal ferrites and the additional benefits of room temperature but magnetoelectric coupling remains to be demonstrated in this material.
Multiferroicity due to spin driven mechanism Magnetic order can break the inversion symmetry. In this case the multiferroic with spin ordering induced ferroelctricity [7] in which the coupling between magnetic order and ferroelectricity is obvious, or we can say that the modification should necessarily acompony a change of magnetic or ferroelectric structure and property. In very simple word electric polarization induced by magnetic ordering. Multiferroic material driven by this mechanism are type 2 multiferroic because ferroelectric ordering and magnetic ordering are correlated. In general the mechanism of spin driven can be classified into three type as shown in fig 4(d), (i) Symmetric spin exchange interaction, (ii)antisymmetric spin exchange interaction and (iii) spin ligand interaction (spin dependent p-d hybridization).
(i) Symmetric exachange interaction working between the two neighbouring spin 푆푖 and 푆푗 and induced striction along a specific crystallographic axis 휋푖푗. For example the up-up-down-down spin arrangement along the atomically alternating AB lattice can break the inversion symmetry and the inequivalent exchange striction working between the up-up (or down-down) spin pair and the up-down (or down-up) one can produce polarization (P). (ii) in antisymmetric case the polarization is due to the canted spin site 푆푖 and 푆푗. This polarization is due to spin current between site I and j. the electric polarization P is given as
P= 훼풆풊풋 × (푺풊 × 푺풋) where 훼 is coupling constant proportional to spin orbit interaction in the weak coupling case. This is often called the inverse Dzylonshikii-Moriya (DM) interaction. (iii) In spin dependent p-d hybridization locally polar bond 풆풊풍 connecting the spin site 𝑖 and ligand site 푙 can be modulated by spin direction dependent hybridization arising from spin orbit coupling. The net P can be found if sum over the crystal lattice site is not cancelled. The polarization is defined ퟐ as 푷풊풍 ∝ (푺풊. 풆풊풍) 풆풊풍. Unlike the spin exchange interaction and inverse DM interaction in the p-d hybridization mechanism only one magnetic site coupled with a ligand ion to produce polarization. We have discussed so far 4 mechanism which derives multiferroicity, the polarization value result [2] from these four mechanism are compared in Figure 5.
Figure 4 (a) lone pair ferroelectricity in 퐵𝑖퐹푒푂3, polarization along [111] plane (b) geometric ferroelectricity in hexagonal ℎ − 푅푀푛푂3 , polarization along (c) charge ordering, (d) spin driven mechanism first shows invers DM interaction, second exchange interaction and last spin dependent p-d hybridization [2]
Figure 5 Types of single-phase multiferroic materials with their maximum polarization values [2]
Composite multiferroic and thin film So far we have discussed multiferroic in which coexistence of ferroelectric and magnetic oordering in a single material. The recent identification of many new multiferroic materials limited by antiferromagnetic or weak ferromagnetic alignment by weak coupling of order parameter or by having properties that emerge well below the room temperature. The alternatives are hybrid systems that return to the historical idea of allowing ferroelectric with magnetic ion leads to the discovery of new multiferroic materials with the magnetoelectric response at room temperature. Another method to use composite multiferroic material. The first multiferroic was created from ferroelectric BaTiO3 and ferromagnetic CoFe2O4 by unidirectional solidification in the eutectic composite [8]. Composite such as to have a larger magnetoelectric effect than single-phase materials. The magnetoelectric effect in composite multiferroic material mainly occurs from mechanical analogue where ferromagnetic magnet strives and ferroelectric piezoelectric constituent are merged in granular and layer form. In first constituent, magnetostriction denotes the induction of strain by magnetic field this strain is transferred through strain coupling between constituent 1 to constituent 2, where it is converted into a voltage via the piezoelectric effect. The relative magnetoelectric coupling 10 times larger than that of single-phase multiferroic [2]. Another way to use strain, to change the size of the unit cell or one can say that to distort the unit cell in favour of the polar state that emerges independently of the magnetic order. Multiferroicity in Ba alloyed bulk SrMnO3 has obtained the stain was exerted chemically through the large size of Ba3+ ion. The precise control to strain needs a thin-film architecture. Using the thin film the polarization increased in the material, BiFeO3 is an example of this. Polarization was increased in hetroepitaxially constrained thin film of the ferromagnetic BiFeO3 [9]. Structural analysis of material shows that the structure is monoclinic in contrast to bulk which is rhombohedral. The film display a room temperature spontaneous polarization (50-60µC/cm2) an order of magnitude higher than that of the bulk (6.1µC/cm2), figure 6. Advantages of the thin film is that battery voltage applied across them can generate the electric field required to magnetoelectric phase control.
Figure 6 Ferroelectric hysteresis loop measured at a frequency of 15kHz, which shows that the film is ferroelectric with Pr ~55μC/cm2 [9] Violation of inversion symmetry The presence of elelectric dipole in ferroelectric material implies that spatial inversion symmetry is broken in the unit cell and magnetic order including ferromagnetic order implies that the time inversion symmetry is broken. In multiferroic, both symmetries present simultaneously requires the simultaneous breaking of temporal and spatial inversion symmetry. Symmetry defined the material properties and prediction of atomic structure and its charge and spin. In multiferroic due to the breaking of space and time inversion symmetry, many specific properties arise by the presence of magnetic and electric large range order simultaneously. Many aspects are related to the breaking of inversion symmetry. The multiple expansion of electromagnetic vector potential includes space-time antisymmetry term so-called an apple and toroidal moment. The spontaneous uniform alignment of the toroidal moment would establish a ferrotoroidic state with a domain that could be hysterically switched space-time antisymmetric external field (toroidal field). Consider the toroidal spin arrangement in the the unit cell where the toroidal T = ri × si, represents the magnetic whirl figure 8.
Figure 7 Magnetic toroidal moments, Hypothetical unit cell with six spins, Si, at positions ri defining a toroidal moment, T = Σi Si × ri, per unit cell. The spin arrangements of opposite toroidal moments are shown in the two sketches [2]
LuCoPO4 is the example in which the coupling of the toroidic field was established with the perpendicular electric and magnetic field. The space-time inversion also responsible for magnetic monopole because the concept of magnetic monopole closely related to the toroidal moment. A magnetic monopole is similar to an electric point charge. But according to Maxwell equation ∇.B =0, shows magnetic monopole cannot exist in free space. However, in a condensed matter system magnetic quasi monopole can exist.
Advantages of multiferroics The coexistence of both electric and magnetic ordering Leads to various advantages. In multiferroic material, we can control magnetism via an electric field, such a capability could be technologically very useful since the production of electric field need less energy than the production of magnetic field, which is used in many magnetism based devices and technology. There have been controlling the orientation of magnetism using an electric field. In the multiferroic thin film, the coupled ferroelectric and magnetoelectric are used to developing magnetoelectric devices. Much novel spintronics (spin transport electronics) devices based on it such as tunnel magnetoresistance (TMR) sensor. In advantages point of view multiferroics is a useful phenomenon for dissipationless electronics, since it enable the voltage control on spin in insulating system. For example, the energy cost per area is ∼ 7×102 Jm−2 and that per bit is around ∼7×10−10 J for MRAM of 1012 bits per m2. Compared with this value, the typical polarization in multiferroics P ∼ 2×10−3Cm−2, and the electric field to invert the magnetic polarization is E ∼ 4×106Vm−1, resulting in the energy density PE ∼ 104 Jm−3. Assuming one bit per 1μm3 = 10−18 m3, the energy cost per bit is ∼10−14 J, which is five orders of magnitude smaller than that of the present MRAM [7].
Application of multiferroics are not up to material physics it crossover to the other area of physics. Multiferroic are used to address cosmology and high energy physics. In multiferroic h-RMnO3 material distribution like a string along which ferroelectric domain. This h-RMnO3 is used to examine scaling law related to string formation in-universe after the big bang [2] upon cooling both system undergoes a topologically similar phase transition. These similarities relate the cosmological string to the distribution of line along which ferroelectric domain meet in h-RMnO3. In conclusion, The coexistence of ferroic ordering leads to a local material which is the key to the future technology as according to Moore's law, the no of transistors on a microchip almost doubles every two years. But today, nanotechnology has come under a saturation phase due to the high density of equipments in a microchip. So now multicentric materials have become important where we control one ordering by other ordering to reduce the power consumption, size and no of transistors in an electronic device.
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