**Perovskites** - A family of structures named after the mineral perovskite, which has a chemical formula of $\ce{CaTiO3}$. - **General Formula** - The general formula $\ce{ABX_3}$ describes the entire family. The aristotype structure is cubic. - **A Site** - A large metal ion, often a group I, group II or lanthanide ion. It is 12 coordinate. - **B Site** - A smaller metal ion site that is inside an oxygen octahedra. Often contains transition metal ions or $\ce{Al^{3+}}$ or $\ce{Ga^3+}$. - **X Site** - Most commonly oxygen, however can also be a halogen. **Hybrid Metal Organic Perovskites** - Perovskites can also be synthesised with small organic molecules in the A site and larger ions in the B site (*e.g.* $\ce{Pb^2+}$). **Applications** - The diverse selection of features found in perovskites lead to a large number of applications. - **Ferroelectrics** - The most common ferroelectrics are members of the perovskite family. They have widespread use in electronic devices. - **Ionic Conductors** - Solid state electrolytes such as $\ce{La_xLi_yTiO3}$ are being developed for use in lithium-ion batteries. - **Photovoltaics** - Photoferroic devices such as $\ce{(CH3NH3)PbI3}$ are being investigated for use in solar cells. - **Superconductors** - Many of the most commonly used high-temperature superconductors are defect perovskite structures, such as $\ce{YBa_2Cu_3O_7}$ - **Others** - Include magnetoresistance and catalysis. ## Crystal Chemistry **Control** - There are two major factors - **ionic size** and **ionic charge**. **Structure Field Map** - For the composition $\ce{ABO3}$, it can be seen that the perovskite structure is very accommodating to various ionic size ratios. **Cell Constraints** - There are two ways the size of the unit cell can be constrained. - **Along [110] Direction** - The size of the A site ion can constrain the unit cell along the face-diagonal. ($a=\sqrt 2(r_{A}+r_{O})$) - **Along [100] Direction** - The size of the B site ion can constrain the unit cell along a line through the middle of the cell. ($a= 2(r_{A}+r_{O})$) - **Ionic Radii** - The ionic radii of the elements depend on both their coordination and spin state - consult Shannon. - **Perfect Perovskite** - In a perfect perovskite, the desired unit cell size from each of these factors is equal and no distortion is necessary. **Distortions** - The perovskite structure can tolerate when these two factors do not equal and will distort to stabilise the structure. - **Tolerance Factor, $t$** - The level of mismatch is quantified by the tolerance factor. $t=\frac{r_A+r_{O}}{\sqrt 2 (r_B+R_O)}$ - **Limits** - Generally, tolerance factors between 0.8 and 1.05 can be tolerated. - **$1.0<t<1.05$** - When the A site ion is too large, the perovskite has a tetragonal distortion. - **B Site Ion** - The B site ion is now too small to bond efficiently to its octahedra and therefore moves off the centre of the octahedron. ![[asg-notes-2.png#invert|cm]] - ***e.g. $\ce{BaTiO3}$*** - In the room temperature phase the titanium ion is displaced along the $c$-axis to create a tetragonal unit cell. - **Second Order Jahn-Teller Distortion** - Can also be used to explain the effect. ^4cd0e1 - **Definition** - A second-order or pseudo Jahn-Teller effect is where a distortion in symmetry facilitates the mixing of ground and low-lying excited states in a way that reduces total energy. - In this case, by moving off centre the ions empty orbital can interact better with the oxygen 2p orbitals and therefore further reduce their energy. ![[St Andrews/02 Public Notes/02.02 University Notes/CH5518 Blockbuster Solids/attachments/ipad-8.png]] - $t<1$ - When the A-site ion is too small, the A-site ion may displace and the octahedra can tilt to reduce the size of the A-site cavity. This may be referred to as the $\ce{GdFeO3}$ type structure. - **Unit Cell Expansion** - The addition of tilts reduces the symmetry of the structure and hence result in larger unit cells. - ***e.g. $\ce{CaTiO3}$*** - The $\ce{Ca^2+}$ ion is a bit too small and hence it is displaced and results in an octahedral tilts. The resulting structure is orthorhombic. ![[asg-notes-3.png#invert]] **Glazer Tilt Notation** - Due to their ubiquity, a common notation has been developed to describe tilts. - **System to Consider** - Glazer tilt considers a 2 x 2 x 2 array of linked $\ce{BO6}$ octahedra. Should view the octahedra head-on (ignore assigned axes) from three directions. - **Symbols** - A rotation is described by a symbol $a$, $b$ or $c$. If the magnitude of the tilt is the same, they share the same symbol. - **Superscript** - The superscript indicates the type of tilt. 0 indicates no tilt, + indicates an in-phase tilt and - indicates an antiphase tilt. - If the tilt is antiphase, viewing it down it axis will show both the top octahedra and the one below it. - Only antiphase tilts remove a mirror plane perpendicular to the tilt axis. - **Ordering** - Although there are conventions, the ordering does not matter if there are three unique axes. As with tetragonal cells, tend to have the last tilt as the unique one. ![[tilley-1.png#invert|cl]] - ***e.g. $\ce{CaTiO3}$*** - In the above example in blue, calcium titanate is assigned $a^-a^-c^+$. Here we see the same in phase tilts in two axes and then an antiphase tilt on another. **Pauling’s Rules** - Pauling established five rules that help to describe how stable a structure is. 1. _A coordinated polyhedra of anions is formed about each cation; the cation-anion distance being determined by the radius sum and the coordination number of the cation determined by the radius ratio._ 2. _In a stable coordination structure the electric charge of each anion tends to compensate the strength of electrostatic valence bonds reaching it from the cations at the centre of the polyhedra of which it forms a corner._ - **Electrostatic Valence Rules -** Arise from this rule, essentially the charge on a particular ion must be balanced by an equal and opposite charge from the surrounding ions. - **Expression -** This can be expressed for a $\ce{M^{m+}}$ cation surrounded by $\ce{ nX^{x-} }$ anions: $ s=\frac mn \qquad \sum s = x $ 3. _The presence of shared edges, and particularly shared faces, in a coordinated structure tends to decrease its stability; this effect is large for cations with large valence and small coordination, and especially so when the radius ratio approaches the lower limit of stability for the polyhedron._ 4. _In a crystal containing different cations those with large valence and small coordination number tend not to share polyhedron elements with each other._ 5. _**Rule of Parsimony -** The number of essential different kinds of constituents in a crystal tends to be small._ - _i.e._ the repeating units will tend to be identical because each atom in the structure is most stable in a specific environment. ### Symmetry **Importance** - A materials properties arise directly from its symmetry. **Lattice -** An arrangement of **points** in space having the same symmetry such that the lattice looks the same from whichever point you view it. - **Lattice Types** - There are four possible lattice types; primitive, body-centred, face-centred and side-centred **Motif/Basis -** A group of atoms positioned in relation to each lattice point. **Unit Cell -** A high symmetry volume that is a repeat unit of allows a 3D structure to be defined along with a lattice. - **Definition -** A unit cell is defined by six parameters - the dimensions; _a, b & c_ and the angles; _α, β & γ._ - **Crystal Systems -** There are seven possible shapes of unit cell that have different relationships between the parameters. | System | Minimum Symmetry | Unique Parameters | Example | |:------------:|:----------------:|:--------------------------------:|:-----------:| | Triclinic | None | $a, b, c, \alpha, \beta, \gamma$ | $P\,\bar 1$ | | Monoclinic | $2$ or $m$ | $a, b, c, \beta$ | $P\, 2_1/c$ | | Orthrohombic | 3x ($2$ or $m$) | $a, b, c$ | $P\, mm2$ | | Trigonal | $3$ | $a, \alpha$ | $R\, 3$ | | Tetragonal | $4$ | $a, c$ | $P\,4/mmm$ | | Hexagonal | $6$ | $a,c,\gamma=120\degree$ | $P\, 6mm$ | | Cubic | 4x ($3$) | $a$ | $F\,m\bar 3m$ | **Primitive Unit Cell -** The smallest possible unit cell choice that only contains one lattice point. **Conventional Unit Cell -** Sometimes a different unit cell is chosen to construct a larger unit cell with simpler geometry. **Bravais Lattice -** The combination of a crystal system and a lattice. - **Number -** There are 14 lattices, less than the expected 28 (4 lattice × 7 crystal systems) - **Discrepancy -** Some lattices are incompatible with some crystal systems and some combinations are equivalent, _e.g. I Tetragonal = B Tetragonal._ ![[St Andrews/02 Public Notes/02.02 University Notes/CH5518 Blockbuster Solids/attachments/notion-1.webp|cl]] **Space Group Symbols** - There are 230 space groups that are all given symbols as shown in the example column of the table above. - **Viewing Directions** - After the lattice type, up to three symmetry elements are given. The viewing directions they refer to vary depending on the crystal system. ![[hoffmann-1.png#invert|cl]] **Point Groups** - Each space group has a corresponding point group for which the translational symmetry has been stripped away. - *e.g.* The point group of $C\, mc2_1$ is $mm2$. **Enantiomorphic Space Groups** - Groups are chiral when they have no improper rotations ($\bar 1$ or $m$ etc.) **Unique Polar Axis** - A unique polar axis is one that is not perpendicular to a $n$-fold axis or mirror plane. - **Polar Space Groups** - Space groups that have a unique polar axis a referred to as polar. - If equivalent positions are generated, there must be an axis that contains no bar terms, *e.g.* $\bar z$. - **Identifying Polar Groups** - Tend to have either one symmetry element or two mirror planes and one rotational axis. >[!aside] Symmetry Requirements >**Equivalent Positions** - Examination of the equivalent positions for 3 orthorhombic space groups shows how the centrosymmetric nature and non-polar nature are independent. > >| $P\, mmm$ | $P\, mm2$ | $P\, 222$ | |:----------------------:|:------------:|:------------------:| | $x,y,z$ | $x,y,z$ | $x,y,z$ | | $\bar x,y, z$ | $\bar x,y,z$ | - | | $x,\bar y,z$ | $x,\bar y,z$ | - | | $\bar x,\bar y, z$ | $\bar x,y,z$ | $\bar x, \bar y,z$ | | $x,y,\bar z$ | - | - | | $\bar x,y,\bar z$ | - | $\bar x,y,\bar z$ | | $x,\bar y, \bar z$ | - | $x,\bar y,\bar z$ | | $\bar x,\bar y,\bar z$ | - | - | > >$P\, 222$ - Although it lacks inversion (non-centrosymmetric), all axes are perpendicular to $2$-fold rotation axis. >$P\, mm2$ - Again lacks an inversion centre and this time does not have any equivalent positions that contain $\bar z$. Therefore is polar. ### Phase Transitions **Gibbs Phase Rule -** Describes the number of parameters that can be varied independently, the degrees of freedom, while the number of phases in the equilibrium is preserved. - **System -** A specification of the range of compositions of interest. $ F=C-P+2 $ - **Degrees of Freedom, $F$ -** The number of independent variables, which could be temperature, pressure or composition. - **Composition -** Composition is only a degree of freedom for when it refers to the composition of a phase, _i.e. for a liquid or solid solution._ It is **not** a degree of freedom for mixtures. - **Components, $C$** - A chemically independent constituent of the system that can undergo independent variation in the different phases. - _e.g. H₂O has one component as it must be the same 2:1 ratio of H:O, FeO has two components as a non-stoichometric Fe₁₋ₓO phase exists._ - **Number of Phases, $P$ -** The number of phases in the equilibrium. **Condensed Gibbs Phase Rule** - When considering solids, we can often assume a constant pressure. $F=C-P+1$ **Phase Transition** - Phase transitions can be characterised on both a structural or thermodynamic basis. - **Thermodynamic** - Defined based on the derivatives of $G$. The free energies of each phase are equal at the phase transition. - **First Order** - $dG/dT$ and $dG/dP$ are discontinuous, they show an abrupt change. - **Second Order** - The first derivates are continuous but the second derivates are discontinuous. - **Structural:** - **Reconstructive** - A major structural change that involves making and breaking bonds. Can be kinetically prohibited. - Must be a first order transition. - **Displacive** - A minor structural change that is not kinetically prohibited. - May be a second order transition, but may be first order too. - **Order-Disorder** - A transition where there is a change to the level of order in the system. *e.g. a mixed ionic site segregates* - This can be a major, more reconstructive transition or can be a minor, more displacive transition. **Thermodynamic/Crystallographic Variables** - The two types of thermodynamic transitions give different behaviours for a variety of the thermodynamic variables and crystallographic parameters. - **First Order** - Clear discontinuities are observed in the volume, enthalpy and entropy for a first order transition. This is also seen in the crystallographic parameters. - **Second Order** - In second order transitions, the phase change is more likely to be indicated in a continuous variation with a change in the gradient. - **Latent Heat** - A first order transition has latent heat and hence a large spike is observed in the heat capacity. ![[asg-notes-6.png#invert|cl]] **Phase Coexistence** - The two transition types differ on the coexistence of phases. - **First Order** - As there may be a kinetic barrier, there may be some phase coexistence around the transition temperature. Additionally this may give rise to hysteresis in transition temperature. - **Second Order** - The lack of kinetic barrier means it should be single phase above and below the transition temperature. ### Variants **Cation Ordering** - Perovskites containing multiple ions for the same sites may be ordered if they have different sizes or charges. - **Disordered Structures** - If the cations have similar sizes and charges, they tend to form a disordered solid solution, *e.g. $\ce{K_{1-x}Na_{x}NbO3}$*. - **Checkerboard** - If there is a 50:50 mix of the two ions, they can segregate so that the B1 ion is fully surrounded by B2 ions and vice versa. Also referred to as rock salt. - **Layered** - In a similar manner, a layered structure can be form where the ions alternate along one axis but stay the same on the other axes. ![[asg-notes-4.png#invert|cl]] > [!aside] Motivation for Ordering > The electrostatic valence rules can be used to understand the reason for the ordering. > > ![[St Andrews/02 Public Notes/02.02 University Notes/CH5518 Blockbuster Solids/attachments/ipad-9.png]] > > Only when an oxygen is bonded to both W and Mg is the valence rule fulfilled. Otherwise there is an unfavourable concentration of charge. **Hexagonal Perovskites** - Classic perovskites are based on cubic close packing of the $\ce{[AO]3}$ layers but this can instead be hexagonal close packed. - **Face Sharing** - Unlike the corner sharing ccp packing, hcp necessitates the $\ce{BO6}$ octahedra to face-share. - **Requirements** - These variants normally occur when $t \gg 1$, often with $\ce{Ba+}$ on the A-site. - **Intergrowths** - Can get more complex phases that are result of intergrowths of cubic and hexagonal layers. - **Notation** - They are name liked 9R, where the number indicates the number of structural layers in the cell while the letter indicates the type of stacking. - ***Example*** - $\ce{BaMnO3}$ forms a variety of intergrowth hexagonal perovskites. ![[cr0c00622_0001.webp#invert|cl]] **Layered Perovskites** - Perovskites with a quasi-2D nature that lack the full 3D connectivity of a standard perovskite. - $n$ - In these structures, $n$ determines the number of perovskite layers before the disruption. - **Oxygen-Rich** - These stoichiometries tend to be more rich in oxygen. ![[custom-tilley-2.png#invert]] - **Aurivillius $\ce{(Bi2O2)(A_{n-1}B_{n}O_{3n+1})}$** - Varying thickness of perovskite slabs punctuated with corrugated layers of $\ce{B2O2}$. - *e.g.* $\ce{Bi2WO6}$ an example of the $n=1$ structure. - **Ruddlesden-Popper $\ce{A_{n+1}B_{n}O_{3n+1}}$** - Create perovskite slabs with cuts along [100] and then shift the second layer by $(a/2, b/2, 0)$. - **Rock Salt** - Can also be interpreted as a succession of perovskite and rocks salt layers. - *e.g.* $\ce{K2NiF4}$ is a $n=1$ example, $\ce{Ca3Ru2O7}$ is a $n=2$ example. - **Carpy-Galy, $\ce{A_{n}B_{n}O_{3n+2}}$** - The perovskites slabs are now cut along the [110] direction. - *e.g.* $\ce{Ba2Mn2F8}$ is a $n=2$ example. ![[asg-notes-5.png#invert|cm]] ## Diffraction **Diffraction** - The scattering of radiation in the crystal structure can result in constructive and destructive interference that allows information to be gleaned about the structure. - **X-Ray Scattering** - In the case of X-rays, radiation is scattered by the electron density surrounding the atom. - **Neutron Scattering** - In the case of neutrons, radiation is spherically scattered by the nuclei, which acts as a point scatterer. - **Miller Planes** - In either case, the scattering centres will create Miller planes throughout the structures. - **Interference** - Radiation that is scattered by the second Miller plane will travel an extra distance $x$ and hence interfere with radiation scattered by the first Miller plane. - **Constructive Interference** - If this radiation travels an extra distance $x$ equal to a whole number of wavelengths, constructive interference occurs and the radiation’s intensity is strengthened. - **Destructive Interference** - Otherwise, the radiation destructively interferes and the intensity is diminished. - **Bragg’s Law** - This requirement can be expressed as Bragg’s law. $n\lambda =2d_{hkl}\sin \theta$ **Diffraction Experiments** - Diffraction experiments tend to fall into two classes: - **Single Crystal** - A single crystal is measured in a goniometer and the resulting 2D diffraction pattern is measured at numerous different angles. - **Information Content** - The intensities of each spot can be directly determined, which allows (with a bit of work) a full structure solution. - **Powder XRD** - In a powder XRD experiment, hundreds or thousands of crystallites are measured at once, each with a random orientation. - **Overlapping** - This results in a 2D pattern of concentric circles which can be represented as a 1D pattern. - **Information Content** - As peaks can be coincident and overlapping, structure solutions are much harder to determine (tends not to be done). **Scanning Variable** - There are two ways to get a spectrum of $d$-spacings. - **Angle-Dispersive** - A constant wavelength source is used with a moving detector. This therefore measures intensity variation with $2\theta$. - **Energy-Dispersive** - A variable wavelength source is used with a fixed detector. In the case of neutrons, referred to as time-of-flight as the energy of the neutron can be determined when it hits the detector. **Information Content** - Even powder XRD patterns hold a wealth of content about the material. - **Peak Positions** - The peak positions (through Bragg’s law) hold information about the size, shape and symmetry of the unit cell. - **Peak Intensities** - The peak intensities (through the structure factor) hold information about the location and types of atom in the cell. - **Peak Width/Shape** - The peak widths and shape hold information about the particle size, strain and homogeneity. - **Peak Shape** - Normally described by a mix of Lorentzian broadening (more for strain) and Gaussian broadening (more for size). **$d$-Spacing** - The $d$-spacing for a particular *hkl* peak can be found with the following materials. $\begin{align} \text{Cubic:} && \frac{1}{d^{2}}&=\frac{h^{2}+k^{2}+l^{2}}{a^{2}} \\ \text{Tetragonal:} && \frac{1}{d^{2}}&=\frac{h^{2}+k^{2}}{a^{2}} + \frac{l^{2}}{c^{2}} \\ \text{Orthorhombic:} && \frac{1}{d^{2}}&=\frac{h^{2}}{a^{2}} +\frac{k^{2}}{b^{2}} +\frac{l^{2}}{c^{2}} \\ \end{align}$ **Systematic Absences** - All translational symmetry elements give rise to systematic absences in the diffraction pattern. - **Finding the Space Group** - These systematic absences are key in determining the space group of the structure, but this sometimes cannot just be determined by powder diffraction. - **Lattice Centring** - The most commonly encountered systematic absences. - **Primitive Cell -** No systematic absences. - **Body-Centred Lattice -** Absent if $h+k+l=2n+1$, *i.e.* the sum of the *hkl* indices must be even. - **Face-Centred Lattice -** Absent if $h=2n+1;\, k=2n+1;\, l=2n+1$, *i.e.* the hkl integers must be all odd or even to be present. - Also stated as $h+k,\ k+l ,\ h+l$ must all equal $2n$. - **Reciprocal Lattice** - A face-centred real lattice creates a body-centred reciprocal lattice and vice versa. **Pattern Indexing** - The unit cell can be found from powder XRD pattern through the process of indexing. - **Cubic Indexing -** With a singular parameter $x$, quite simple to index the cell. - **Tetragonal Indexing -** Convert the peaks into $1/d^2$ and then examine how they relate to each other. - *e.g.* If the first two peaks are not multiples to each other, then likely one is a 001 peak and the other is a 100 peak. Proceed to try different combinations for the subsequent peaks. - **Uniqueness** - A certain solution is not unique for the the set of data and large cells can be constructed. - **In Practice -** Generally completed computationally where different solutions are compared and evaluated. >[!example] Indexing Example >The following $2\theta$ data was indexed to give a primitive tetragonal cell with $a=5.199$ Å and $c=8.402$ Å. > ![[St Andrews/02 Public Notes/02.02 University Notes/CH5518 Blockbuster Solids/attachments/ipad-7.png|cl]] >**Tips**: >1. Remember to divide $2\theta$ by 2 in the formula. >2. Try cubic indexing first by dividing all $d^{-2}$ by the $d^{-2}$ of the first peak. > - **Primitive/Body-Centred** - If primitive, you should be missing the 7th peak. If that occurs, try doubling the index to get a body-centred cell. > - **Face-Centred** - If all peaks are $n$/3, then try a face-centred solution. >3. If not, try tetragonal indexing. > - Take the first two peaks $d^{-2}$ (assuming they aren’t an integer number of each other) and multiply by 4 and 9. Check for matches and assign them. > - Assign the unique axis by multiplying $d^{-2}$ by 2. One should have a match, this will be $a$ and $b$ parameter. > - Should be able to assign the rest of peaks from this. **Multiplicities** - In 1D powder data, a peak may be made up of a number of symmetry-equivalent peaks. - *e.g.* For a {100} peak in a cubic system, there are 6 reflections that contribute - $(100),\ (\bar 100),\ (010),\ (0\bar 1 0),\ (001),\ (00\bar 1)$. - **Cubic Cells** - In a cubic cell, peaks can reach very high multiplicities. - **Reduction in Symmetry** - Moving to tetragonal and orthorhombic systems, the multiplicities of each peak begins to reduce substantially. **Reduction in Symmetry** - Two primary effects: - **Increasing Unit Cell Size** - Comparing an ordered double perovskite and a disordered standard one, new peaks appear at completely different positions. - **Lowering Temperature -** With lower temperatures, lower symmetry cells become more stable and hence new peaks may appear in VT data. - **Lower Symmetry** - Peak splitting may occur upon the reduction in symmetry. - *e.g.* - Moving from a cubic cell to tetragonal the 100 peak ($M=6$) splits into two distinct peaks - 100 peak ($M=4$) and 001 peak ($M=2$). - **Separation** - A cell may only become slightly tetragonal and hence identifying this peak splitting may be challenging. **Vegard’s Law** - The unit cell size is proportional to the composition of a solid solution. For $\ce{A_{1-x}B_x}$, $a=(1-x)a_{A}+xa_B$ ### Neutrons vs X-Rays **X-Ray Scattering** - In the context of diffraction, X-rays are scattered by the electron density surrounding the atom through Thomson scattering. - **Atomic Scattering Factor, $f_a$** - The atomic scattering factor at 0° scales with atomic number squared, $Z^2$. - **Angle** - At a higher angle (or scattering vector $Q$) the intensity of scattering falls off. ![[notion-2.webp]] ![[sydney-1.png#invert]] **Neutron Scattering** - Neutrons are scattered by the nuclei, which are modelled as point scatterers. - **Spherical Scattering** - Neutron scattering is spherically symmetric, it is of the same intensity in all directions. - **Scattering Lengths** - The intensity of scattering is determined by its scattering length, which basically randomly varies between atoms and isotopes. - **Negative Scattering Lengths** - Some isotopes have negative scattering lengths which result in a change in phase. - **Isotopic Substitution** - As different isotopes have different scattering lengths, this can be exploited to get better scattering, *e.g. deuteration*. - **Zero Scattering Alloys** - Atoms can be combined to create a zero scattering alloy for sample environment, otherwise V is used due to almost 0 scattering length. **Synchrotron X-Rays** - High intensity X-rays are produced in synchrotrons that accelerate electrons to close to the speed of light. - **Examples** - Diamond is the UK national synchrotron, ESRF, DESY and MAX IV in Europe. **Neutron Sources** - There are two primary types of neutron sources. - **Spallation Sources** - Protons are accelerated inside a synchrotron and then fired at a target (commonly tungsten) which spallates neutrons.*e.g. ISIS Neutron and Muon Source, Oxfordshire or PSI, Switzerland* - **Time-Of-Flight** - Polychromatic neutrons are generated in pulses and hence the energy of a neutron can be determined from the time after the pulse. - **Reactor Sources** - Neutrons are generated in a nuclear reactor. Greater intensities allow the neutrons to be monochromated. *e.g. ILL, Grenoble* - ***e.g HRPD Instrument at ISIS*** - The high resolution powder diffractometer at ISIS was the first instrument at ISIS to receive neutrons in 1984. - **Design** - It is the furthest instrument from the target so that it has the highest resolution. - The resolution improves at longer distances as the energy-dispersed neutrons travel at different speeds and hence separate out more. $ \frac{\Delta d}{d}= \left[\left(\frac{\Delta L}{L_{1}+L_{2}}\right)^{2}+\left(\frac{\Delta t}{t}\right)^{2}+(\Delta \theta \cot \theta)^{2}\right]^\frac{1}{2}$ - **Detectors** - HRPD had four banks of detectors, one at high angles, two at 90° and one at low angles. The backscattering detectors have the highest resolution. **Neutron Advantages** - There are many reasons to justify the expense and inconvenience of neutron diffraction. - **Light Element Sensitivity** - Determination of light element locations and occupancies are now possible as they have a large contribution to the structure factor for neutrons. ^18e8fa - **Discriminate Similar $Z$ Atoms** - As the neutron scattering length varies randomly with $Z$, can easily discriminate between atoms of similar atomic mass. *e.g. Mn and Fe* - **Form Factor** - As neutrons are spherical scatters, can collect high quality data at much higher angles than X-rays. - This allows better determination of parameters that are dependent on $2\theta$ such as displacement parameters. - **Sample Environment** - Utilisation of zero scattering alloys and vanadium allows a wide range of sample environment to be used. **X-Ray Advantages** - There are still many reasons why neutron diffraction might not be the best choice, synchrotron X-rays have a number of advantages. - **Signal-to-Noise Ratio** - Synchrotron X-rays are orders of magnitude brighter than neutron sources, allowing better quality data to be collected. - Much faster collection times, much less sample required. - **Resonant Scattering** - Scattering at wavelengths close to an absorption edge results in a strong enhancement of scattering. - **Focusing** - X-ray optics are much more developed than neutron optics, therefore allowing tighter focusing etc. - **Bad Scatters** - Some elements are bad at neutron scattering (incoherent scatterers etc.) and hence X-rays are the only way to get good data. ### Magnetic Structure **Magnetic Structures** - Magnetic structures can be determined by both X-rays and neutrons. - **Neutrons** - It is very easy to determine magnetic structures using neutrons. - **X-Rays** - It quite difficult to determine magnetic structures using X-rays as it requires a specialised beamline and only probes the surface. **Magnetic Unit Cells** - The neutrons are scattered differently depending on the spin state of the unpaired electron. - **Expansion** - Therefore, magnetically ordered materials have expansion of their unit cells in a magnetically ordered state. - **Magnetic Peaks** - This results in additional peaks being present in the diffraction pattern. ![[asg-notes-1.png#invert]] ## Dielectric Properties **Insulating Perovskites** - Insulating materials can be classed into a number of groups that describe the response of the material to some stimulus. - **Crystal Chemistry** - Only certain crystal symmetries allow these response to occur. **Dielectric Material** - A dielectric material is an electrical insulator that can be polarised by an applied electric field. - **Constraints** - All point groups can be dielectric. - **Applications** - Materials with a high dielectric constant are used for capacitors. **Piezoelectric Material** - A subset of dielectrics in which polarisation occurs as a result of applied mechanical stress. - **Constraints** - All non-centrosymmetric point groups (with one exception) exhibit piezoelectricity. - **Applications** - The ability to precisely convert electrical energy to mechanical energy and vice versa is very useful. Used in microphones, watches, loudspeakers, mechanical actuators etc. **Pyroelectrics** - A subset of piezoelectric that show an electrical polarisation in the absence of applied field. This is referred to as a **spontaneous polarisation.** - **Temperature Dependence** - The extent of this polarisation is dependent on the temperature. - **Constraints** - Only 10 point groups with a unique polar axis can be pyroelectrics. - **Applications** - The variation of electrical repones with temperature can be used for IR detection and thermal imaging. **Ferroelectrics** - A subset of pyroelectrics that exhibit a spontaneous polarisation that can be switched by application of electric field. - **Applications** - Widely used for memory applications. Also tend to be the best dielectric materials. ![[tilley-2.png#invert|cl]] **Common Examples** - $\ce{BaTiO3}$ and $\ce{Pb(Zr,Ti)O3}$ (PZT) are some of the most ubiquitous examples. PZT especially has very good properties, with a stronger polarisation. - **Lead-Free** - There is a drive to replace PZT with a lead-free replacement. Although in normal use it is completely inert, possibility of ‘escape’ in manufacturing and disposal. - **Alternatives** - $\ce{(K,Na)NbO3}$, $\ce{(Bi,Na)TiO3}$ and $\ce{BiFeO3}$ are being developed as alternatives. **Ferroelectric Mechanism** - Application of a charge to an insulator can result to the local polarisation of the ion. - **Perovskite** - In the case of perovskites this can arise from the local displacement of A and/or B-site cations. - **Ferroelectric Difference** - Unlike normal dielectrics, in a ferroelectric this polarisation is energetically favoured even after the removal of field, *i.e.* polarisation is maintained when field is switched off. - **Hysteresis Loop** - Polarisation-field measurement show a hysteresis loop, where a non-zero coercive field has to be applied to switch the polarisation. - **Curie Temperature, $T_{C}$** - As with magnetism, there is some temperature at which the ferroelectric becomes paraelectric, where it is centrosymmetric and its spontaneous polarisation is removed. ![[tilley-3.png#invert|cs]] **Compositional Requirements** - Ferroelectrics require cations that are prone to displacements (see tolerance factor), but this is not sufficient. **Proper Ferroelectricity** - In these structures, the transition from paraelectric to ferroelectric is directly prompted by the electronic nature of the cation. - The electric polarisation is caused directly by correlated polar displacement of charges, it is the primary order parameter. - **$d^0$ Mechanism** - The $d^0$ mechanism is widespread and occurs when the B-site cation is prone to the [[#^4cd0e1|second order Jahn-Teller effect]]. - ***Examples*** - $\ce{Ti^4+, Nb^5+, W^6+}$ all can use this mechanism. - **$s^2$ Mechanism** - The $s^{2}$ mechanism is the other widespread mechanism. It occurs when the A-site contains a stereochemically active $s^2$ lone pair that results in very high polarisabilities. - ***Examples*** - $\ce{Pb^2+, Bi^3+}$ use this mechanism. $\ce{BaTiO3}$ - A classic ferroelectric example that uses the $d^0$ mechanism. It has 3 phase transitions. - **Polarisation Direction** - As temperature is reduced, the polarisation axis changes direction. ![[batio3_transitions.jpg#invert]] $\ce{Pb(Zr,Ti)O3}$ **(PZT)** - The ability to vary the Zr:Ti ratio makes this material very tunable. - $\ce{Pb(Zr_{0.48}Ti_{0.52})O3}$ - This composition has the best properties - **Morphotropic Phase Boundary (MPB)** - A phase boundary where the properties of material are enhanced. - **Reasoning** - It is unknown why the properties are enhanced here, may be due to a phase mixture of lower symmetry present. ![[tilley-4.png#invert|cm]] **Improper Ferroelectricity** - A less common type of ferroelectricity, where the ferroelectric displacement is a secondary effect of a different underlying structural instability. - The polarisation is a secondary effect of a non-polar distortion, it is not the primary order parameter. - *i.e.* the ferroelectricity is a by-product of some geometric effect. - **Strength** - Improper ferroelectricity tends to be a weaker effect that proper ferroelectricity. - ***Example*** - $\ce{YMnO3}$ is a triangular based lattice that shows polarisation due to a two down, one up arrangement of the $\ce{Y^3+}$ ions. This is the result of $\ce{MnO5}$ polyhedral tilting. ![[Atomic-rearrangements-associated-to-improper-ferroelectricity-in-YMnO-3-in-which-the.png#invert|c]] **Hybrid Improper Ferroelectricity** - A more modern type of ferroelectricity that is the result of trilinear coupling. - Two non-polar distortions couple together and allow a knock-on polarisation to occur. - *e.g.* $\ce{Ca3Mn2O7}$ - A Ruddlesden-Popper $n=2$ structure where a rotation and tilt mode couple to enable ferroelectricty. ## Electrical and Magnetic Properties ### Metallic Properties > [!warning] Note > I don’t actually know how much of the metallic properties is in the course, but it has come up in past papers. Most of it is in CH3712 however. **Metal vs. Insulator** - Whether a metal is an insulator, semiconductor or metal depends on its crystal chemistry and the temperature. - **Band Structure** - The metallic properties of a material can be explained by its band structure. Below is a representative band structure of a perovskite. ![[woodward-1.png#invert|cm]] **Hubbard Parameter, $U$** - Quantifies the energy to move an electron from one metal atom to another that already has an electron. $\ce{M^{3+} + M^{3+} -> M^{4+} + M^{2+}}$ - **Effect** - This has the effect of splitting a partially filled band in two, separated by an amount $U$. **Band Width, $W$** - Depending on the overlap between the metal orbitals of different atoms, the bands will have a finite width. ![[notion-4.webp|cl]] **ZSA Classification** - Zaanen, Sawatzky and Allen classified materials into a number of types. - **Band Insulator** - Insulators where the bands are either fully filled or unfilled and hence there is no possibility of conduction. - **Mott Insulator** - If $U>W$, the Hubbard sub-bands remain separated and hence the electrons cannot move freely. - **Charge-Transfer Insulator** - The lower Hubbard sub-band is pushed below a $p$ orbital of the oxygen. Therefore, any semiconduction is completed by promotion of an electron out the $p$ orbital into the metal sub-band. - **Metallic Materials** - If $W>U$, the band width is greater than the Hubbard splitting and hence the material acts as a metal. This can also occur if the top Hubbard sub-band overlaps with the $p$ orbital. ![[notion-3.webp|c]] **Metal-Insulator Transition** - Changes to the crystal with changing temperature can result in a transition between a metallic state and an insulating one. ### Magnetic Properties > [!recap] Magnetism Recap > > **Magnetism** - A property arising from unpaired electrons in the structure interacting and aligning. > - **Magnetic Moment, $\mu$** - The spin-only magnetic moment for an electron is given by $\mu_{eff}=2\sqrt{(S(S+1))}\cdot \mu_B$ where each unpaired electron contributes $S=\frac{1}{2}$. There may also be an orbital contribution, but this can be ignored for the $3d$ transition metals. > - **Magnetic Susceptibility, $\chi$** - Quantifies how easily a material is magnetised, $\chi=\frac{M}{H}$ where $M$ is the magnetisation and $H$ is the magnetic field density. > - **Curie-Weiss Law** - Describes the variation of magnetic susceptibility with temperature $\chi = \frac{C}{T-\theta}$ where $C$ is the Curie constant describing the magnetic moments and $\theta$ is the Weiss constant describing the interaction between the dipoles. > >**Magnetic Ordering -** There are four ways of arranging magnetic dipoles in a solid: >- **Paramagnetic ($\theta = 0$) -** Weak interaction between the dipoles and therefore the dipoles are randomly aligned. > - **Antiferromagnetic ($\theta < 0$) -** Strong antiparallel interaction between the dipoles and therefore there is no net magnetic moment. > - **Ferrimagnetic ($\theta > 0$) -** There is strong interaction between dipoles. Although they are aligned anti-parallel, one direction is weaker and so there is a net magnetic moment. > - **Ferromagnetic ($\theta > 0$)-** There is a strong parallel interaction between dipoles, giving a strong net magnetic moment. > > **Curie Temperature, $T_C$** - The temperature at which the ferromagnetic behaviour ‘turns off’ and the material becomes paramagnetic. Corresponds to $\theta$ for ferromagnetic and ferrimagnetic materials. > **Néel Temperature** - The corresponding temperature for antiferromagnets. >**Curie-Weiss Plot** - A Curie-Weiss plot of $\chi^{-1}$ against $T$ can be used to determine the Curie temperature. > > ![[St Andrews/02 Public Notes/02.02 University Notes/CH5518 Blockbuster Solids/attachments/ipad-10.png]] > > **Exchange** - There are three primary types for dipoles to interact to give magnetic ordering. > - **Direct (Cation-Cation)** - Simple interactions between neighbouring atoms. Intervening atoms decreases the strength and hence only significant for metals. > - **Double Exchange** - Exchange mechanism that is mediated by electrons moving between anions and cations (*i.e.* for metals). > - **Superexchange* - Magnetic exchange is mediated by both anions and cations for insulators. > - **Heisenberg Exchange Interaction** - This interaction can be described by the exchange Hamiltonian, $\hat H = -2 J(\mathbf{S_{1}\cdot S_{2}})$ where $J$ is exchange consonant and $\mathbf{S}_{i}$ is the spin at site $i$. **Perovskite Magnetism** - Magnetism in insulating transition metal oxides is usually mediated by superexchange . Generally the superexchange is antiferromagnetic. - **Mechanism** - The covalency of the anion-cation $\sigma$ bonds results in preferred spins for the particular orbitals. In particular the metal $e_g$ orbitals can interact with the oxygen $p_z$ orbitals. - **Control** - The magnetism can be carefully controlled by the design of the crystal. **Goodenough-Kanamori Rules** - A set of rules that predict the magnetic outcome of linking two paramagnetic cations via superexchange. - Predicts this based on the geometry and $d$ orbital occupancy. - **$T_C/T_{N}$ Prediction** - Can also give very rough indication of the magnetic transition temperature, work quite well for simple perovskites. - $d^{3}-d^3$ **180° Exchange** - Exchange between two unoccupied $e_g$ orbitals at 180° will be antiferromagnetic. - **Imparted Spin** - The covalency results in the oxygen electrons spin being imparted to the empty $e_g$ orbital. As parallel spins are preferred, this enforces antiferromagnetic exchange. - $d^5-d^5$ **180° Exchange** - Exchange between two occupied $e_g$ orbitals at 180° will be antiferromagnetic. Strongest interaction. - **Enforced Spin** - The spin of the $e_g$ electron is enforced by the Pauli exclusion principle to be opposite the oxygen electrons spin due the covalency. - $d^3-d^5$ **180° Exchange** - Exchange between one occupied and one unoccupied $e_g$ orbital at 180° will be ferromagnetic. - One spin is enforced, one spin is imparted; net result is the cations having the same spin. - $d^5-d^5$ **90° Exchange** - Exchange between two occupied $e_g$ orbitals at 90° will be weakly ferromagnetic. - **Enforced Spins** - The spins are again enforced to be opposite the oxygen orbitals. However, Hund's rule means the oxygen orbitals are more stable if they are parallel. ![[St Andrews/02 Public Notes/02.02 University Notes/CH5518 Blockbuster Solids/attachments/ipad-11.png|cl]] **Zener Double Exchange** - A ferromagnetic exchange mechanism that relies on the electrons hopping between oxygens and variable oxidation state metal cations. - *e.g.* An electron hops between the $\ce{Mn^{3+}}$ ion to the $\ce{Mn^4+}$ ion via the oxygen *p* orbital. - **Filled** - As the oxygen orbital is already filled, the electron from the $\ce{Mn^3+}$ pushes an electron of the same spin onto the neighbouring $\ce{Mn^4+}$ ion. - **Ferromagnetic Ordering** - This only favourable if the spins on the manganese ions are aligned. ![[tilley-9.png#invert|cl]] ### Examples **$\ce{LnNiO3}$ Series** - Substitution of the lanthanide can have a substantial impact on the properties of the material. ![[asg-notes-7.png#invert]] - **Crystal Structure** - $\ce{LnNiO3}$ adopts a $P\, bnm$ structure with a $a^-a^-b^+$ tilt system, which is the same as $\ce{CaTiO3}$. - **Decreasing Ionic Radii** - The ionic radius decreases from Pr to Lu, hence resulting in an increasing in tilting and a decrease in M-O-M angle. - **Overlap** - This results in decreasing $d-p$ overlap which decreases the strength of the superexchange and lowering the Néel temperature. ![[asg-notes-8.png#invert|cm]] $\ce{La_{1-x}Sr_xVO3}$ - Increasing doping of $\ce{Sr^{2+}}$ for $\ce{La^3+}$ results in an increase in the amount of $\ce{V^4+}$ compared to $\ce{V^3+}$. - **Smaller U** - This results in a decrease in the Hubbard U parameter as it easier to transfer an electron to $\ce{V^4+}$. - **Metallic Behaviour** - Therefore doping moves it from semiconducting state at $x=0$ to a metallic state at $x=0.2$. **Colossal Magnetoresistance (CMR)** - Where a materials electrical resistance drops dramatically when a magnetic field is applied. - The application of magnetic field stabilises the metallic ferromagnetic state and hence drops the resistance substantially. - **Magnetoresistance, $MR$** - The resistance that arises from the application of magnetic field, $H$. $MR = \frac{\rho_H-\rho_0}{\rho_{0}}$ - **Applications** - Not much magnetic field is required to result in a considerable change in conductivity, making these very useful. $\ce{La_{1-x}Sr_{x}MnO3}$ - The solid solution shows very different properties depending on $x$, very industrially relevant. ![[asg-notes-9.png#invert|cm]] - **End Members** - Both $\ce{LaMnO3}$ and $\ce{SrMnO3}$ are antiferromagnetic insulators at low temperatures. - **A-Type AFM** - $\ce{LaMnO3}$ is an A-type AFM, where there are ferromagnetic basal planes that interact with other planes antiferromagnetically. - **Paramagnetic Phase**- At high temperature the structure loses its AFM properties. As $\ce{Mn^3+}$ ($t_{2g}^3e_{g}^{1}$) is Jahn-Teller active, the octahedra elongate along one axis. - **Orbitally Ordered** - This effect is cooperative and hence the direction of elongation has long-range order. - **$x=0.33$** - This stoichiometry is a ferromagnetic metal. There is a decrease in the Jahn-Teller effect and an increase in $t$, reducing structural distortion. - **Metal** - The addition of $\ce{Mn^4+}$ lowers U and creates a metal. - **Double Exchange** - The metallic nature results in a double exchange mechanism that gives ferromagnetism. - **Metal-Insulator Transition** - The metal-insulator is coincident with the change in ferromagnetic ordering. - **CMR** - The series $\ce{La_{1-x}A_{x}MnO3}$ show colossal magnetoresistance. The choice of A ion can alter the level of CMR shown. - **Controlling Jahn-Teller Distortion** - The distortion due to the $\ce{Mn^3+}$ can be controlled by the amount of $\ce{Mn^4+}$ but also by the radius of the other ion. ### Multiferroics **Ferroic** - A general term for crystals that have an internal structure that can be switch from stable orientation to another that is equally stable via application of a suitable driving force. - *e.g. ferroelectricity driven by electric field, ferromagnetism driven by magnetic field, ferroelasticity driving by strain* **Multiferroics** - A material the exhibits multiple ferroic effects. - **Type I** - The simultaneous presence of more than one ferroic order that occur at different temperatures and via different mechanisms. This does not have to be cooperative. *e.g.* $\ce{BiFeO3}$ - **Type II** - The ferroic orders are directly coupled and hence both orders occur at the same temperature. - **Magnetoelectric Coupling** - The coupling of ferroelectricity and ferromagnetism where the magnetic ordering breaks the inversion symmetry and directly induces the ferroelectricity. - **Memory Applications** - This has potential applications in data storage, where we could write with an electric field and read with a magnetic field. - **Magnetostriction** - A material changing shape or dimensions when it magnetisation changes as result of the magnetic alignment inducing strain. **Geometry-Based Mixed FM-AFM Order** - In $\ce{HoMnO3}$, the geometry of the Mn-O-Mn chains give rise to ferromagnetism and ferroelectricity. [^1] - **Alternating Bond Angles** - Along the chain the bond angles alternate with opposite oxygen ion displacements due to geometric tilting. - **Alternating Magnetism** - This changes the exchange paths between the ions and hence results in antiferromagnetic exchange being preferred for some and ferromagnetic exchange being preferred for some. - **Displacement** - All the oxygen ions will displace in the same direction to optimise the superexchange and hence give rise to a polarisation and ferroelectricity. ![[asg-notes-10.png|cm]] **Charge-Ordered Mixed FM-AFM Order** - In $\ce{Ca3Co^{II}Mn^{IV}O6}$, there are chains of oxygen octahedra with alternating $\ce{Co^{II}}$ and $\ce{Mn^{IV}}$ centres. [^1] - **Alternating Magnetism** - The material alternates between antiferromagnetic and ferromagnetic exchange. - **Optimisation** - To optimise this interaction, the ions move closer to the ion that they have ferromagnetic exchange with. This breaks the centrosymmetry of the structure. - **Polarisation** - These displacements naturally generate an electrical polarisation and ferroelectricity. ![[asg-notes-11.png|cm]] **Frustrated Magnetic Structures** - Competing magnetic interactions become frustrated can result in interesting structures, such as spiral spin arrangements in $\ce{TbMnO3}$. - **Dzyaloshinskii-Moriya (DM) Interaction** - Ligand displacement results in a weaker antisymmetric exchange and the canting of spins, *i.e.* dipoles move from perfect antiferromagnetic arrangement. - **Maths** - The Heisenberg interaction is a dot product, the DM interaction is a cross product. Cross product results in perpendicular displacement. - **Ligand Displacement** - A $\ce{GdFeO3}$ type structure may for structural reasons have the ligand displacements that result in the DM interaction. Without it, the material would be AFM (dipoles in plane in diagram). - **Not Ferroelectric** - In this case, the material is not ferroelectric as the ligand displacements are non-polar. - **Inverse DM Interaction** - In a spin spiral arrangement, an inverse DM interaction induces ligand displacements and polarisation. - **Periodicity** - The spin spiral arrangement has a large periodicity (may be incommensurate). - **Magnetostriction** - Through the inverse DM interaction, the magnetism induces a ligand displacement to stabilise the arrangement. ![[asg-notes-12.png#invert|cm]] ### Superconductivity **Superconductors** - Materials in which the resistivity drops to zero below a critical temperature $T_C$. Other properties include: - **Meissner Effect** - Magnetic flux is excluded from the sample. - **Superconductivity Breakdown** - Above a critical magnetic field, $H_C$, the superconductivity breaks down. ![[St Andrews/02 Public Notes/02.02 University Notes/CH5518 Blockbuster Solids/attachments/ipad-12.png|cl]] **History**: - **1911** - The superconductivity of Hg below 4.2 K was discovered by Kammerlingh Onnes after the liquefaction of He. - **1950s** - The Bardeen-Cooper-Schrieffer (BCS) model was developed that explained superconductivity of materials where $T_C < 10 \rm\ K$. - **1986** - High temperature superconductivity was discovered in copper oxides by Bednorz and Muller. **BCS Theory** - BCS theory is based on coupling between electrons and phonons to create pairs of electrons called Cooper pairs. - **Cooper Pair** - The electrons in the Cooper pair have equal and opposite spins and equal and opposite momenta to form a system with zero spin and zero momentum. - **Isotope Effects** - BCS theory predicts that $T_{C}\propto M^{-1/2}$, which is experimentally reproduced. - **0 K State** - At absolute zero, all electrons are in Cooper pairs, they all share the same energy state and are correlated. - **Breaking Cooper Pairs** - For the material to absorb or emit energy, the Cooper pair must be broken. The energy required to do this is the superconducting energy gap $E_g$. - **Critical Temperature** - At the critical temperature, the energy gap $E_g$ goes to 0 and hence all the Cooper pairs are broken. - **Conductivity** - Normal metallic resistance arises from electron-phonon scattering that changes the electrons momentum. As Cooper pairs have no momentum, they are scattered with no momentum change and hence have no resistance. - **Above $T_C$** - BCS superconductors are poor metals above $T_C$ due to strong electron-phonon interactions. **High $T_C$ Superconductors** - A number of superconductors have been found with a $T_{C}$ above 140 K. - **BCS Theory** - The highest $T_C$ BCS theory predicts is 40 K, there must be an alternate undiscovered mechanism out there. **Cuprate Superconductors** - The most famous examples of these high $T_C$ superconductors contain copper. - **Common Features** - These superconductors share a number of common features: - **Electronics** - Mixed valent metal centres with strong $3d-2p$ hybridisation, *e.g.* $\ce{Cu^{2\pm\delta}}$ - **Coordination** - Copper in square planar, square pyramidal and JT-distorted octahedral coordination. - **Structure** - 2D layered structures with superconducting $\ce{CuO2}$ sheets and insulating M-O blocks that act as charge reservoirs along the $c$ axis. ![[woodward-2.png#invert]] - **$\ce{CuO2}$ Layers** - The copper oxide layers are the superconducting part of the structure. - $\ce{CaCuO2}$ - A modified perovskite structure where on oxygen is removed to leave sheets of copper oxide in square planar coordination. - **Charge Reservoirs** - These layers are more insulating. They allow for modification of the $\ce{CuO2}$ layers. - **Doping** - The doping of this layer can be charge balanced by copper ions. - ***p*-Type** - Doping $\ce{Ba^2+}$ for $\ce{La^3+}$ in $\ce{La_{2-x}Ba_{x}CuO4}$ results in an oxidation state of $\ce{Cu^{2+x}}$ and *p*-type conduction. - **Overbonded** - Bond valence analysis indicates $\ce{Cu^2+}$ is overbonded (O pushed too close) and hence doping alleviates this compressive stress. - ***n*-Type** - Doping $\ce{Ce^{4+}}$ for $\ce{Nd^3+}$ in $\ce{Nd_{2-x}Ce_{x}CuO4}$ results in an oxidation state of $\ce{Cu^{2-x}}$ and *n*-type conduction. - **Underbonded** - Bond valence analysis indicates $\ce{Cu^2+}$ is underbonded (O too far away) and hence doping alleviates this tensile stress. - **Optimal $x$** - The optimal amount of doping tends to be *approx*. 0.15. - **Standard Phase Diagram** - For most perovskite superconductors, the phase diagram shows a superconducting dome. ![[tilley-8.png#invert|cm]] - **Overall Conductivity** - Even though the 2D $\ce{CuO2}$ layers are the nominally superconducting part, the entire structure is superconducting. Above $T_C$ the resistivity is much greater along the $c$ axis. - $\ce{La_{2-x}Ba_{x}CuO4}$ - The first cuprate superconductor had a $T_C$ of 32 K. It has a Ruddlesden-Popper $n=1$ structure. The parent structure is not a superconductor. - **Dopant** - Sr was later found to slight increase $T_{C}$ if used instead of Ba. ![[tilley-5.png#invert|cm]] - **Electron-Doped Superconductors** - Most cuprates are hole-doped but $\ce{Nd_{2-x}Ce_{x}CuO4}$ is electron-doped due to the aforementioned underbonding. - $\ce{YBa2Cu3O_{7-\delta}}$ **(YBCO)** - The first superconductor discovered with a $T_C$ above the boiling point of liquid nitrogen. $\ce{YBa2Cu3O_{6.95}}$ has the highest $T_C$ of 93 K. - **Target Compound** - The target stoichiometry did not form and instead formed a triple perovskite. - **Oxygen Stoichiometry** - The superconductivity is best when $\delta$ is small and the square planar sheets can chain together. - **Charge Reservoir** - Here the square planar $\ce{CuO2}$ and the $\ce{BaO}$ acts as the charge reservoir. ![[tilley-6.png#invert]] - **Intergrowths** - The $T_C$ can be pushed higher in intergrowth structures with more complicated structures. - *e.g.* $\ce{Bi2Sr2Ca_{n-1}Cu_{n}O_{4+2n}}$ (BiSCCO) - Has a $T_C$ of 110 K when $n=3$. ![[tilley-7.png#invert|cm]] --- ## References + Footnotes Tilley R., *Perovskites: Structure-Property Relationships*, 2016 Woodward P. M., Karen P., Evans J. S. O., Vogt T., *Solid State Materials Chemistry*, 2021 [^1]: See [van den Brink and Khomskii, *J. Phys.: Condens. Matter*](https://doi.org/10.1088/0953-8984/20/43/434217) for more details.