MAS.020UF Introduction to Solid State Physics

## Crystal physics

If the crystal structure is known, it is possible to calcuate the energy of a crystal as a function of the temperature, applied electric field, applied magnetic field, and the lattice constant of the crystal. For experiments performed at constant temperature and constant pressure, the system will go to a minimum of the Gibbs free energy,

$$G = U - TS -pV.$$

Here $U$ is the internal energy, $T$ is the absolute temperature, $S$ is the entropy, $p$ is the pressure, and $V$ is the volume. The Gibbs free energy of crystals formed by the elements can be found at: SGTE data for pure elements.

Many observable properties can be expressed as the derivative of the Gibbs free energy. For instance if a crystal has an electric polarization $\vec{P}$, the energy will decrease if the polarization aligns with an applied electric field, $\Delta G = - \vec{P}\cdot\Delta \vec{E}$. Here $\vec{E}$ is the electric field. This means that the polarization can be written as,

$$P_i = -\frac{\partial G}{\partial E_i}\qquad i=x,\,y,\,z.$$

The electric susceptibility $\chi^E$ is the change in the polarization as a function of electric field,

$$\chi^E_{ij} = \frac{\partial P_i}{\partial E_j} = -\frac{\partial^2 G}{\partial E_i\partial E_j}$$

Further equilibrium properties related to derivatives of the free energy are:

$$\text{pyroelectric effect}\qquad \pi_i = \frac{\partial P_i}{\partial T} = -\frac{\partial^2 G}{\partial E_i \partial T}$$ $$\text{piezoelectric effect}\qquad d_{ijk} = \frac{\partial P_i}{\partial \sigma_{jk}} = -\frac{\partial^2 G}{\partial E_i \sigma_{jk}}\qquad \sigma_{ij} \text{ is the stress}$$ $$\text{magnetization}\qquad M_i = -\frac{\partial G}{\partial H_i}\qquad H_i \text{ is the magnetic intensity}$$ $$\text{magnetic susceptibility}\qquad \chi^M_{ij} = \frac{\partial M_i}{\partial H_j} = -\frac{\partial^2 G}{\partial H_i \partial H_j}$$ $$\text{entropy}\qquad S = -\frac{\partial G}{\partial T}$$ $$\text{strain}\qquad \epsilon_{ij} = -\frac{\partial G}{\partial \sigma_{ij}}$$ $$\text{thermal expansion}\qquad \alpha_{ij} = -\frac{\partial \epsilon_{ij}}{\partial T}$$

These physical properties are described by tensors. A quantity without subscripts like the temperature is a scalar = rank 0 tensor. A quantity with one subscript is a vector = rank one tensor. Quantities with two subscripts are matrices = rank 2 tensors. Quantities with $n$ subscripts are rank $n$ tensors.

The form that the tensors can take is restricted by the symmetries of the point group of the crystal.

• Be familiar with Einstein notation for tensors. In this notation, you sum over repeated indices. A scalar product $\vec{P}\cdot\vec{E}$ would be written $P_iE_i$ and the matrix equation, $\vec{P}=\chi\vec{E}$ would be written as $P_i=\chi_{ij}E_j$.