28 2. TUNABLE MATERIALS–CHARACTERISTICS AND CONSTITUTIVE PARAMETERS
2.3.2 HEXAGONAL FERRITES OR HEXAFERRITES–PERMANENT
MAGNETIC FERRITES
Self-Biased Ferrites e distinguishing feature of hexagonal ferrites is their high residual
magnetization, which enables the development of “self biased” microwave devices in the mw
frequency range (e.g., above 30 GHz). It is expected that hexagonal ferrites can be used for
non-reciprocal functions at frequencies up to or higher than 100 GHz. For this purpose, their
large built-in anisotropy field is exploited. However, most of them have a uniaxial permeability
tensor [18, 19]; that is, hexagonal ferrites are mostly used in the mw range in order to eliminate
the need for a very high external DC bias magnetic field.
ChemicalComposition Hexagonal ferrites are, in general, complex crystalline structures with
complicated chemical formulas, which have been given code letters. One large family based on
barium (Ba
+2
) is (Ba
+2
O)
X
(Me
+2
O)
Y
(Fe
2
O
3
)
Z
. Me
+2
is a divalent metallic ion from the first
series of transition elements or a combination of elements with valence equal to two [12]. e
most widely known representatives are BaFe
12
O
19
, with the code letter M (also known as BaM)
and Me
2
BaFe
16
O
27
, with the code letter W. Another widely used hexagonal ferrite is based on
strontium: this is SrFe
12
O
19
, known as SrM.
Saturation Magnetization e saturation magnetization and the anisotropy field of BaM and
SrM can be reduced by partial substitution of cations. In addition, for the W type Ni
2
BaFe
16
O
27
,
a partial substitution of nickel (Ni) by cobalt (Co) lowers its saturation magnetization, while
aluminum’s (Al
+3
) substitution for iron (Fe
+3
) increases both its saturation magnetization and
its anisotropy.
It is important to note that in order to achieve low microwave losses, the ferrite should be
produced with all iron ions in trivalent state so that losses related to electron transfer between
divalent and trivalent iron ions can be prevented.
2.3.3 MAGNETIC GARNETS
It was as early as 1956 that Bertaut and his coworkers [20] synthesized the first magnetic rare-
earth iron garnet in Grenoble. An avalanche of works followed and by the end of the decade,
magnetic garnets, including YIG as the most important along with spinel ferrites, had become
available to microwave engineers [19].
Crystal Structure e garnet crystal structure is very complicated. Each unit cell is composed
of formula units as A
3
B
2
C
3
O
12
, where A, B, and C are trivalent metallic cations [12]. Within
this crystal structure, metal cations are surrounded by oxygen anions in tetrahedron, octahedron,
and dodecahedron coordination, occupied correspondingly by C , B, and A ions. e YIG for-
mula is Y
3
Fe
5
O
12
or Y
+3
! A and Fe
+3
! B plus C. us, Yttrium occupies the dodecahedron
and iron occupies both the tetrahedron and octahedron sites in the crystal structure. An example
of the crystal structure of the garnet is shown in Figure 2.7.
2.3. FERRIMAGNETICS: FERRITE MATERIALS AND MAGNETIC GARNETS 29
1/8
1/8
1/8
1/8
1/4
1/4
1/21/2
A S
ites B Sites Oxygen Sites
1/2
1/8
1/8
1/8
1/8
3/8
3/8
3/8
3/8
3/8
3/8
3/8
3/8
1/8
1/8
1/8 3/8
1/83/8
3/8
3/8
Figure 2.7: Crystal structure of magnetic garnets on the plane z D 0.
e fractional numbers show the heights obove the plane. Figure 2.7 shows only the lower
half part. e upper half is rotated by 90
ı
.
e magnetic moments of the magnetic cations on different lattice sites are aligned in two
antiparallel directions. us, garnets are of a ferrimagnetic nature. In YIG, the four iron cations
are balanced (yttrium is a non-magnetic ion), resulting in one unbalanced trivalent iron cation
(five Bohr magnetons) per unit cell, which gives YIG magnetization. is phenomenon clearly
explains why magnetic garnets have a saturation magnetization lower than that of ferrites. In
garnets, only one iron ion per cell contributes to net magnetization.
Microwave Losses It is important to observe that in garnets, all metallic cations are trivalent
and there are no positions for divalent ones in the crystal lattice. us, losses due to electron
exchange between divalent and trivalent ions are clearly avoided, which ensures low losses at
microwave frequencies. Moreover, Fe ions provide weak coupling between the excitation of the
spin lattice (called “magnons”) and the excitation of the crystal lattice (phonons), constituting
30 2. TUNABLE MATERIALS–CHARACTERISTICS AND CONSTITUTIVE PARAMETERS
a negligible direct relaxation loss mechanism [11]. To minimize microwave losses, the garnet
should also be prepared as a mono-crystal so that anisotropy effects will not introduce a broad-
ening of gyromagnetic resonance (losses) [11]. Furthermore, the surface of the crystal should be
free of damage, and minimal losses are achieved with YIG single crystal spheres.
Controlling Saturation Magnetization As will be explained in the next section, saturation
magnetization (M
s
) defines the range of gyromagnetic resonance frequency. erefore, in order
to use YIG at low microwave frequencies, we need to reduce M
s
. e reduction is achieved with
the following techniques [11]:
1. Partial substitution of magnetic iron Fe
+3
cations with non-magnetic aluminum cations
Al
+3
on tetrahedral sites reduces the differences between antiparallel magnetizations. is
yields a decrease in M
s
.
2. Partial substitution of non-magnetic yttrium ions (Y
+3
) with magnetic gadolinium (Ga
+3
)
ions presents magnetization antiparallel to that of iron ions. is, again, results in M
s
re-
duction. e partial substitution of yttrium in YIG with aluminium (Al) or gadolinium
yields lower saturation magnetization. erefore, it is used for applications at lower mi-
crowave frequencies [12]. Exact practical values for M
s
can be found in commercial data
sheets, e.g., [20, 21].
In general, partial substitution of YIG cations is used with a wide range of metallic ones,
e.g., aluminum (Al
+3
), gadolinium (Ga
+3
), holmium (Ho
+3
), calcium (Ca
+3
), and vanadium
(V
+3
).
Garnets vs. Spinel Ferrites e saturation magnetization of YIG at room temperature is
about M
s
D 0:18 tesla, and its Curie temperature is T
C
D 286
ı
C. Furthermore, YIG character-
istics have smaller variations with temperature as compared to those of magnesium-manganese
ferrites. is, along with its quite lower losses, makes YIG preferable for low power and for
control-tunable applications above 3.3 GHz (stemming from its saturation magnetization).
Moreover, magnetic garnets have a rectangular hysteresis loop, making them appropriate for
latching control applications.
It should be noted at this point that ferrites still preserve the advantage of a much higher
saturation magnetization. is allows their operation in the mw frequency bands. In particular,
hexagonal ferrites with their self-biasing capability (permanent magnets) can be used at frequen-
cies of up to 100 GHz without the need for an otherwise extremely high required DC biasing
magnetic field. As will also be discussed in later sections, this is a desired operation around the
gyromagnetic resonance. at is, in turn, calculated as 28 GHz/tesla. In other words, operation
at 56 GHz requires a 2 tesla biasing field, which requires bulky electromagnets or superconduct-
ing magnets [12].
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