# Ways to experimentally control the chemical potential of a solid state system

When working in the grand canonical ensemble we write the grand potential as $\Omega = \Omega (T,V,\mu)$. In this case we are taking the chemical potential $\mu$ to be an independent variable. This would imply that this ensemble is best suited to situations in which the chemical potential is an experimentally controlled parameter.

If this is the case, then what are some ways in which the chemical potential can be controlled experimentally?

Specifically I am interested in a solid state system which is being modeled as electrons on a lattice. In theory, by adjusting the chemical potential we could control the filling fraction of electrons for this system, taking it from band insulator to conductor or from Mott insulator to conductor. Is this type of tuning possible with real systems? If so, what are some realistic applications and/or specific limitations of these techniques.

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## 1 Answer

The use of the chemical potential $\mu$ as state variable is useful in situations where composition $N$ is variable and/or cannot be easily controlled. From an experimental point of view the chemical potential is fixed when the system is in contact with energy and particle reservoirs. At equilibrium, the chemical potential of the system equals that of the reservoir $\mu = \mu_\mathrm{res}$. Thus by modifying the parameters of the reservoir you can control the chemical potential of the system.

A typical example is when the system is a layer of molecules adsorbed in a surface. This is an open system and composition is variable --and generally unknown--. By using a gas of the same molecules as reservoir you can fix the value of the chemical potential of the open system. The chemical potential $\mu_\mathrm{res}$ of the gas can be obtained from evaluating the fundamental equation of the gas or, if this is not available, from integrating the Gibbs Duhem relation if the equation of state of the gas is known.

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Thanks @juanrga. I have actually done the problem of a surface with absorption sites in contact with a gas.  I recall at the end putting the chemical potential of the gas in terms of its number density.  Then by controlling the density of the gas we control the chemical potential of the surface.  This is what you said, by modifying parameters of the reservoir you can control the $\mu$.  Do you by chance know any of the corresponding parameters for solid state systems? – Todd R Dec 13 '12 at 1:31
@ToddR The chemical potential of electrons in solids is defined in exactly the same way as the chemical potential of a chemical species. There is no 'special' parameters for solids but the usual ones: Temperature $T$, composition $N$, external electric potential $\phi$ (if any)... Notice that, in presence of electric fields, some authors rename the chemical potential to "electrochemical potential", whereas solid-state physicists like to use a radically different terminology: "Fermi energy". – juanrga Dec 16 '12 at 12:29
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