Important data for Mg base Metal Hydride

MgH₂: Tetragonal (P42/mnm, 136), 4.517Å x 4.517Å x 3.02Å, Vol=61.62, 2θ(110)=27.946

Ref1: Batt, A., Polytechnic Institute of Brooklyn, Brooklyn, New York, USA. (1960)
Ref2: Ellinger et al.,J.Am.Chem.Soc.AT 77, 2647 (1955)

**************************************************************************************************

Mg: Hexagonal (P63/mmc, 194), 3.2094Å x 3.2094Å x 5.2112Å, Vol=46.48, 2θ(002)=34.398

Ref1: Jevins, A. et al. Z. Phys. Chem. (B), 40, 347 (1938)


Magnesium-Hydrogen phase diagram

















References
1987San: A. San-Martin and F.D. Manchester: Bull. Alloy Phase Diagrams, 1987, vol. 8 (5), 431-37.
1993Sha: V.I. Shapovalov, A.P. Semik, and A.G. Timchenko: Russ.Metall., 1993, vol. 3, pp. 21-24.
1999Zen: K. Zeng, T. Klassen, W. Oelerich, and R. Bormann: J.Alloys Compounds, 1999, vol. 283, pp. 213-24.

Ferroelectric Crystals

Domains:
For ferromagnetic crystals, where the domain wall consists of a transition region in which the magnetization vector turns over gradually from a given direction to the opposite one. This is because the thickness of a ferromagnetic domain wall is a result of the compromise between the exchange energy and the anisotropy energy. The former would require a domain wall as wide as possible, the latter attempts to do just the contrary, in order to avoid the directions of "hard magnetization", but the exchange term is orders of magnitude larger than the anisotropy term and the result is a rather thick domain wall.
For ferroelectric crystals, there is no real counterpart of the magnetic exchange energy. Based on a dipole interaction calculations, the interaction energies of parallel and antiparallel dipole arrays do not differ much from each other, thus allowing a narrow domain wall. Most ferroelectrics, especially those with low symmetry, has high anisotropy, which prevents the spontaneous polarization from deviating from the ferroelectric axis. Hence, the polarization vector will not rotate within the wall, but rather will decrease in magnitude, passing through zero, and increase on the other side with opposite sign. Also, the elastic energy will also favor a rather thin domain wall.

For single domain ferroelectric crystal, a hysteresis loop will be observed. When the crystal consists many domains, the loop changes. Especially when there are equal number of positive and negative domains, no piezoelectric effect can be observed and the spontaneous polarization cannot be reversed by a mechanical stress. However, an electric field can reverse the spontaneous polarization and thus the piezoelectric polarity. The relationship between strain and field is characterized by a quadratic hysteresis loop. In actual crystals, different portions of the sample alter orientation at different parts of the cycle and a smooth curve results, which is called a "butterfly loop".


Domain wall energy:
Mert modified the Bloch Wall theory in ferromanetics for the case of BaTiO₃. The total wall energy was taken to be the sum of the energy due to dipole the interactions (corresponding to the exchange energy in the ferromagnectic case) and the anisotropy energy. Calling t the wall thickness, the wall energy E_wall can be written as:
E_wall=A/t+Bt,
A and B are functions of the lattice constant, spontaneous strain and the elastic constants.
Mert estimated to be of the order of one lattice constant, the corresponding energy was estimated to of the order of 7 ergs/cm².

The process of polarization reversal
It is quite evident that when the polarization of a ferroelectric crystal is reversed by means of an external electric field, the domain configuration must undergo radical changes.
The reversal process can be interpreted in terms of two mechanisms: nucleation of domains and domain wall movements.
Under low external field, the rate of nucleation is so low that the switching is primarily governed by the nucleation process. At low field, the probability of forming new domains depends exponentially upon the applied field.
p_n=p₀e^-α/E
It follows that:
1/t_s~1/t_n=1/t_∞e^-α/E
At high field, on the other hand, the nucleation rate is so large that the swithching time is primarily determined by the domain wall motions.
Assuming that the domain wall movement can be described by:
v=d/t_d=&muE
Where d is the distance covered by the wall in motion (thickness of the sample), and μ is the mobility,we obtain:
1/t_s~1/t_d=μE/d=μV/d²
This implies that, in the region of high fields, the switching time should depend quadratically upon the thickness of the sample.
At low field, on the other hand, the sidewise expansion of the nucleated domains may bring contributions to the switching transient.
At high field, the forward motion of the domain walls is very much faster than any sidewise motion.

PbTiO3 (perovskite)

PbTiO₃ has a simple perovskite structure. Many compounds of this type undergo one or more structural phase transitions. PbTiO₃ undergoes a cubic-to-tetragonal ferroelectric phase transition at around 770K.
In terms of the strain in a thin film, equilibrium theories of epitaxy predict that, below a critical thickness, the strain coming from the lattice mismatch will be accommodated by film itself. Above the thickness, the strain will be partially relaxed by forming dislocations. If the film undergoes a strutural phase transition from a high-symmetry phase to a low-symmetry phase during cooling from the growth temperature, the epitaxial strain can be relieved by domain formation as suggested by Roitburd and Bruinsma and Zangwill.
The potential surface was initially mapped out using the Linear Augmented PlaneWave (LAPW) method, and the charge density and electronic struccure were analyzed. PbTiO₃ showed a much deeper well when tetragonal strain was included. Thus the tetragonal strain is responsible for the tetragonal ground state in PhTiO3. In BaTiO3, the Ba is quite spherical in the ferroelectric phase, whereas the Pb in PhTiO3 is not very spherical in the ferroelectric phase, and polarization of the Pb helps stabilize the large strain and the tetragonal ground state in PbTiO3.
In PbTiO3, the O 2p states strongly hybridize with the d₀ Ti⁴⁺ cation, which reduces the short-range repulsions thus allows off-center displacements.

Elastic constants of PbTiO₃:
c₁₁=1.433
c₁₂=0.322
c₁₃=0.241
c₃₃=1.316
c₄₄=0.558
c₆₆=0.556
Y=1.34
μ=0.58
ν=0.16
All terms except ν have been divided by a factor of 10¹¹N/m²

Some constants of MgO:
Y=3.105
μ=1.332
ν=0.161
Ref:B.S.Kwak, Physical Review B, Vol49, 14865, 1994

Lattice constants(Å) and TEC of various materials:
PbTiO₃:
tetragonal(RT), a=3.899(bulk); c=4.153(bulk);
cubic (823K), a=3.986,
TEC=12.6 x 10^-6.
Room temperature crystal structure of tetragonal PbTiO₃ was determined by Shirane et al.(1956) with displacements parallel to the polar axis (relative to the Pb ion at the origin):
dz_Ti=17pm
dz_OI=dz_OII=47pm

Note: Oxygen octahedra suffers no distortion in going to ferroelectric phase.
KTaO₃:
cubic(RT),a=3.989;
cubic(823K),a=4.003;
TEC=6.67 x 10^-6.
STO:
cubic(RT),a=3.905;
cubic(823K),a=3.928;
TEC=11.7 x 10^-6.
ferroelectric phase transition: 35-40K
MgO (NaCl structure):
cubic(RT),a=4.213;
cubic(823K),a=4.239;
TEC=14.8 x 10^-6.
BaTiO₃
In the tetragonal phase, the Ti and O ions move relative to Ba at the origin from their cubic position:
Ti: (1/2,1/2,1/2) to (1/2,1/2,1/2+dz_Ti)
O: (1/2,1/2,0), (1/2,0,1/2) and (0,1/2,1/2) to (1/2,1/2,dz_OI),(1/2,0,1/2+dz_OII) and (0,1/2,1/2-dz_OII
dz_Ti=5pm
dz_OI=-9pm
dz_OII=-6pm
(Harada et al.1970)
prototype cubic perovskite: >120°C;
ferro 4mm tetragonal: 120°C>T>5°C;
ferro mm orthorhombic: 5°C>T>-90°C;
ferro 3m trigonal: <-90°C
PbZrO₃:
Antiferroelectricity
Paraelectric-antiferroelectric: 230°C in zero field
but upon application of an external electric field below T_c will induce a transition to a rhombohedral ferroelectric phase.

Ref: Principles and Applications of Ferroelectrics and Related Materials by M.E. Lines and A.M. Glass

Developments of Multiferroic

Multiferroic magnetoelectrics are materials that are both ferromagnetic and ferroelectric in the same phase. For example, nickel boracite, Ni₃B₇O₁₃I. However, very few exist naturally or have been synthesized in the lab. It was proposed by N.A. Hill that the transition metal d electrons, which are essential for magnetism, reduce the tendency for off-center ferroelectric distortion.
Considering the origin of ferromagnetism, it is useful to treat the interaction as shifting the energy of the 3d band for electrons with one spin direction relative to the band for electrons with the opposite spin direction. In the stoner theory, the fundamental driving force for ferromagnetism is the exchange energy, which is minimized if all of the electrons have the same spin. However, transfering electrons from the lowest band states (occupied equally with up and down electrons) to band states of higher energy will increase the band energy. And this band energy prevents simple metals from being ferromagnetic. Depending on the 3d and 4s band structures and the Fermi energy level, some metals are ferromagnetic, such as, Fe(3d⁶4s²), Co(3d⁷4s²) and Ni(3d⁸4s²), some are not ferromagnetic, like Cu(3d¹⁰4s¹) and Zn(3d¹⁰4s²).
For ferroelectricity, it is a matter of two competing forces. One is the short-range repulsion between adjacent electron clouds, which favor the nonferroelectric symmetric structure. The opposing force is the additional bonding considerations, which might stabilize the ferroelectric phase. The short-range repulsions dominate at high temperature, resulting in the symmetric, unpolarized state. As the temperature is decreased, the stabilizing force become stronger and the polarized state becomes stable.
In 1961, G.T.Rado and VJ Folen first observed the magnetoelectric effect in Cr₂O₃

Terminologies related to Multiferroics

Ferromagnets:
A ferromagnetic material is one that undergoes a phase transition from a high-temperature phase that does not have a macroscopic magnetic moment to a low-temperature phase that has a spontaneous magnetization even in the absence of an
applied magnetic field.

ferromanetic:
--> -->
--> -->

ferrimagnetic:

-> ->
<-- <--  

antiferromagnetic:
--> -->
<-- <-- 

Ferroelectric:
Ferroelectric materials is one that undergoes phase transition from a high-temperature phase that behaves as an ordinary dielectric (so that an applied electric filed induces an electric polarization, which goes to zero when the field is removed) to a low-termperature phase with a spontaneous electric polarization whose direction can be switched by an applied field.

d₃₃, charged constant:
One can use either the direct piezoelectric effect, i.e., applying a stress and measuring the induced charge (or voltage), or use the inverse piezoelectric effect, i.e., applying a voltage (field) and measuring the induced strain.
d₃₃=(∂S₃/∂E₃)_T
Or
d₃₃=(∂D₃/∂T₃)_E
S:strain
E:electric field
D: electric displacement
T: stress
x₃ axis is the poling direction. For thin film, usually it is the normal direction to the surface.

Tensors:
piezoelectric constatnt: third-rank tensor
electrostrictive constant and elastic constant: fourth-rank tensors.
Tensor notation/matrix notation: 11/1, 22/2, 33/3, 23(32)/4, 31(13)/5, 12(21)/6

Three dimensional heteroepitaxy in self-assembled nanostructures

Take BaTiO₃-CoFe₂O₄ as an example:
1. Intrinsic similarity in crystal chemistry between perovskite and spinel. Both of them have octahedral oxygen coordination. Lattice mismatch is about 5%. CoFe₂O₄: a=0.838nm
2. Very little solid solubility in each other
3. Suitable substrate with similiar crystal structure, such as STO

The Ewald Sphere of Reflection

Incident wave vector: K_I
Diffracted wave vector:K_D
K=K_D-K_I
|K_I|=|K_D|=1/λ=|K|
Angle between K_D and K_I is 2θ, sinθ=|K|/2|K_I| --> |K|=2sinθ/λ
When θ=θ_B,
|K_B|=2sinθ_B/λ, (1)
By pure geometry, we can know, nλ=2dsinθ_B, which is Bragg's Law.
If n=1,
2sinθ_B=λ/d , (2)
Combined (1) and (2), we have: |K_B|=1/d,
Define this vector, K_B, to be g so that:
K_B=g

The reciprocal lattice is a 3D array of points, each of which we will now associate with a reciprocal-lattice rod, or relrod for short. This geometry of relrod holds even when we tilt the specimen. The reason why we have relrods is the result of the shape of the specimen. At this stage, it is purely empirical construction to explain why we can still see the DP even when the Bragg condition is not exactly satisfied.

Construction of the Ewald Sphere:
1.Pick a point along the incident beam, this is your center of the the Ewald Sphere.
2.Draw a circle with a radius of 1/λ
3.Choose the intersection of the incident beam and your Ewald Sphere as the origin point to construct the reciprocal lattice.
When the sphere cuts through the reciprocal lattice point, the Bragg condition is satisfied. When it cuts through a rod, you still see a diffraction spot, even though the Bragg condition is not satisfied.
Ewald Sphere 2D