Question

(a) At \(670 \mathrm{K}, \mathrm{CaF}_{2}(\mathrm{mp}=1691 \mathrm{K})\) doped with \(1 \% \mathrm{NaF}\) has an electrical conductivity of \(0.1 \Omega^{-1} \mathrm{m}^{-1} .\) Suggest how this conductivity arises. (b) The value of \(T_{\mathrm{c}}\) for \(\mathrm{YBa}_{2} \mathrm{Cu}_{3} \mathrm{O}_{7}\) is \(93 \mathrm{K}\). Sketch the change in electrical resistivity as a function of temperature as \(Y B a_{2} C u_{3} O_{7}\) is cooled from 300 to \(80 \mathrm{K} .\) How does the shape of this graph differ from those that describe the change in resistivity with temperature for a typical metal and a typical semiconductor?

Short answer

(a) Conductivity in doped CaF2 arises from Na+ ions creating vacancies, allowing ionic movement. (b) The resistivity of YBa2Cu3O7 drops to zero below 93 K, unlike metals (gradual decrease) and semiconductors (increase).

Step-by-step solution

Understand the Problem

In part (a), we are tasked with explaining how the electrical conductivity in doped CaF2 arises. For part (b), we need to sketch the change in electrical resistivity of a superconductor YBa2Cu3O7 as it is cooled below its critical temperature and compare this with typical metals and semiconductors.

Analyze Conductivity Mechanism in Doped CaF2 (Part a)

The doped CaF2 has a conductivity likely due to the introduction of Na+ into the Ca2+ lattice. This substitution leaves a vacancy in the structure, facilitating the movement of fluoride ions, which increases ionic conductivity. Such ionic movement is responsible for the conductivity in this case.

Describe the Behavior of Superconductors (Part b)

Start by noting that YBa2Cu3O7 is a type-II superconductor with a critical temperature of 93 K. At temperatures above 93 K, it behaves like a regular metal, but below 93 K, it transitions into a superconducting state, showing zero resistivity. This signifies an abrupt drop in resistivity to zero as temperature decreases past 93 K.

Sketching and Analyzing Resistivity Graph

For a superconductor, plot resistivity on the y-axis and temperature on the x-axis from 300 K to 80 K. The graph shows high resistivity above 93 K, sharply dropping to zero below 93 K. By contrast, typical metals show a continuous decrease in resistivity with decreasing temperature. Semiconductors, on the other hand, show increasing resistivity as temperature decreases.

Key concepts

YBa2Cu3O7

YBa2Cu3O7, often referred to as YBCO, is a compound known for its remarkable superconducting properties. Superconductivity is a state where a material can conduct electricity without resistance, meaning no energy is lost as heat. YBa2Cu3O7 belongs to a group of materials known as high-temperature superconductors. These materials show superconducting properties at temperatures much higher than traditional superconductors, although still far below room temperature.

In the case of YBa2Cu3O7, the critical temperature, denoted as \(T_c\), is 93 K. It's labeled as a type-II superconductor, which means it can sustain higher magnetic fields compared to type-I superconductors before losing its superconducting capabilities. This property makes YBa2Cu3O7 and other high-temperature superconductors particularly useful in creating strong electromagnets used in MRI machines, maglev trains, and other advanced technologies.

Understanding the structure and composition of YBa2Cu3O7 helps in developing materials with better superconducting properties. This compound's structure includes layers of copper oxide, which are believed to play a crucial role in its superconductivity. Studying these layers in detail continues to be an active area of research.

Electrical conductivity

Electrical conductivity defines how easily a material allows the flow of electric current. It's a crucial characteristic in materials science, heavily influencing their technological applications. Conductivity in YBa2Cu3O7 showcases a significant change as it transitions from a normal state to a superconducting state.

In a material like YBCO, which we consider a ceramic, the electrical conductivity at higher temperatures is similar to that of metals, allowing current to flow with some resistance. As the temperature drops below its critical temperature (93 K for YBCO), the material enters its superconducting phase. This causes the resistivity to drop abruptly to zero.

In contrast, metals generally show a linear decrease in resistivity with decreasing temperature. However, they never achieve zero resistivity. Semiconductors, on the other hand, behave differently; their resistivity increases as the temperature falls, making them poor conductors at lower temperatures.

For YBa2Cu3O7, this unique behavior of complete loss of resistivity makes it invaluable for applications that require efficient energy transmission.

Doping

Doping is a method used to alter the electrical properties of a material by adding impurities. In the context of superconductivity, doping can significantly influence the critical temperature and the overall conductivity of a compound.

For example, in the exercise with CaF2 doped with 1% NaF, the Na+ ions substitute some of the Ca2+ ions in the lattice. This substitution leads to vacancies or "holes" in the lattice structure which allows the fluoride ions to move more freely. This movement facilitates ionic conductivity.

Similarly, doping in superconductors like YBa2Cu3O7 is used to optimize their superconducting transition temperature and enhance their electrical properties. By adjusting the amount of oxygen in the compound through doping, researchers can fine-tune the properties of YBCO to suit specific needs. The intention of doping is always to maximize the material's electrical efficiency and stability, making it tailor-fit for advanced technological applications.

Phase transition

A phase transition refers to a change in the state of matter when external conditions like temperature or pressure are altered. The transition from a normal conducting state to a superconducting state in materials like YBa2Cu3O7 is a fascinating example of a phase transition.

As YBa2Cu3O7 is cooled from room temperature to below 93 K, it undergoes this dramatic phase transition. When it reaches its critical temperature, \(T_c\), the resistivity drops to zero, marking the onset of superconductivity.

Understanding this phase transition helps scientists develop better materials for power systems, transportation, and sophisticated devices. In the case of metals, the change is more gradual and less dramatic, as they don't transition into a superconducting state simply by cooling. Instead, they simply exhibit lowered resistivity until they reach low enough temperatures.

In semiconductors, unlike superconductors, the resistivity increases as they are cooled down. Each type of material reacts differently to temperature changes, making the understanding of phase transitions pivotal in materials science.