Consider a system of N "non-interacting" electrons / cm^3, each of which can occupy either a bou

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Question

Consider a system of $N$ "non-interacting" electrons $/ \mathrm{cm}^3$, each of which can occupy either a bound state with energy $\varepsilon=-E_d$ or a free-particle continuum with $\varepsilon=\frac{p^2}{2 m}$. (This could be a semiconductor like Si with $N$ shallow donors $/ \mathrm{cm}^3$.) (a) Compute the density of states as a function of $\varepsilon$ in the continuum. (b) Find an expression for the chemical potential in the low temperature limit. (c) Compute the number of free electrons (i.e., electrons in the continuum) as a function of $T$ in the low temperature limit.

   Consider a system of $N$ "non-interacting" electrons $/ \mathrm{cm}^3$, each of which can occupy either a bound state with energy $\varepsilon=-E_d$ or a free-particle continuum with $\varepsilon=\frac{p^2}{2 m}$. (This could be a semiconductor like Si with $N$ shallow donors $/ \mathrm{cm}^3$.)
(a) Compute the density of states as a function of $\varepsilon$ in the continuum.
(b) Find an expression for the chemical potential in the low temperature limit.
(c) Compute the number of free electrons (i.e., electrons in the continuum) as a function of $T$ in the low temperature limit.

Problems and Solutions on Thermodynamics and Statistical Mechanics

U.S.T. of China… 1st Edition

Chapter 2, Problem 88

Instant Answer

Step 1

The density of states $g(\varepsilon)$ for a free particle in three dimensions is given by: \[ g(\varepsilon) = \frac{1}{(2\pi)^3} \int d^3p \, \delta\left(\varepsilon - \frac{p^2}{2m}\right) \] To convert this into an expression in terms of energy Show more…

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Consider a system of $N$ "non-interacting" electrons $/ \mathrm{cm}^3$, each of which can occupy either a bound state with energy $\varepsilon=-E_d$ or a free-particle continuum with $\varepsilon=\frac{p^2}{2 m}$. (This could be a semiconductor like Si with $N$ shallow donors $/ \mathrm{cm}^3$.) (a) Compute the density of states as a function of $\varepsilon$ in the continuum. (b) Find an expression for the chemical potential in the low temperature limit. (c) Compute the number of free electrons (i.e., electrons in the continuum) as a function of $T$ in the low temperature limit.

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