Question

Hydrogen is used in some semiconducting devices to improve their performance. Consider a material in which atomic hydrogen (H) diffuses at 1.8 × 10^-24 m²/s at 329 K.

1) When in use, the device is exposed to a large electric field. If the force on each charged hydrogen nucleus is 10 ^-11 N, use the fluctuation-dissipation relation to find out how fast they drift in the field at 329 K.

Answer 1

The question is a fairly straight forward one. The diffusion constant or the diffusivity is just the rate at which molecules diffuse from a higher concentration to lower concentration due to random motion. So, higher the diffusivity, the faster the molecules will diffuse through our device. Diffusivity for hydrogen in this material is,

The electric field exerts a force on each hydrogen nucleus that is,

The temperature at which this happens is

The fluctuation-dissipation relation says that for any dissipative process there exists a fluctuation that is enabled by the same dissipative process. This is hard to understand so lets take an example. Any small object moving in a fluid will experience drag, i.e. dissipation of kinetic energy where the energy possessed by the object is converted into heat energy (similar to your hands getting hot after sliding down a pole). This happens because of the molecules in the fluid colliding with the object continuously.

But, there is also a thing called Brownian motion. It is the random motion of a particle in a fluid. And guess how it is caused? Because of the thermal fluctuations present in the fluid. So, the drag opposing the object causes the heat but the heat also causes the object to move in brownian motion. Fascinating. We will use that same idea over here, using an equation derived by Einstein,

here is the general mobility of the particle which in our case is a hydrogen nucleus. And it is given by,

Using this in our previous equation,