PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
Metals are often described with a free electron model where the microscopic states are those for a particle in a square well potential. The energies of the microscopic states consist just of the kinetic energy $E=\hbar^2k^2/2m$ where $m$ is the mass of the electron and $k$ corresponds to a wavelength that fits in the square well.
8.x Starting from the density of states $D(k)$ for electrons, calculate the energy density of states $D(E)$ for the free electron model in 1-, 2-, and 3-dimensions.
Since electrons are fermions, the occupation of the microscopic states is described by the Fermi-Dirac distribution,
\[ \begin{equation} f(E) = \frac{1}{\exp\left(\frac{E-\mu}{k_BT}\right)+1}. \end{equation} \]Here $\mu$ is the chemical potential, $k_B$ is Boltzmann's constant and $T$ is the temperature. The electron density is the density of states times the probability that those states are occupied integrated over all energies,
\[ \begin{equation} n = \int_{-\infty}^{\infty}D(E)f(E)dE. \end{equation} \]The Fermi energy is defined as the limit of the chemical potential at low temperature, $E_F=\mu (T=0)$. At zero temperature the Fermi-Dirac function is $f=1$ for $E < E_F$ and $f=0$ for $E > E_F$ so for low temperature the electron density is,
\[ \begin{equation} n = \int_{-\infty}^{E_F}D(E)dE. \end{equation} \]8.1 For a metal at a temperature of $T = 0$ K, the conduction electrons at the bottom of the band have an energy $E = 0$ and the conduction electrons with the highest energy have an energy $E = E_F$. Using the density of states for free electrons, what is the average energy of the conduction electrons in 2-D and in 3-D?
8.x Copper has a lattice constant of 3.615 Å. The Bravais lattice is fcc so there are 4 atoms in a cube 3.615 Å on a side. Copper is monovalent. The Debye temperature of copper is $\Theta_D = 343 \text{ K}$.
(a) Calculate the electron density of copper.
(b) Using the free electron model, calculate the electronic contribution to the specific heat capacity $c_{v,el}$ at a temperature of $T = 300 \text{ K}$.
(c) Use the Debye model to estimate the contribution of the phonons to the specific heat $c_{v,ph}$ at at a temperature of $T = 300 \text{ K}$. The phonon density of states in the Debye model is,
$$D(\omega) = \frac{9n\omega^2}{\omega_D^3}= \frac{9n\hbar^3\omega^2}{k_B^3\Theta_D^3}$$Here the Debye frequency is $\omega_D = \frac{k_B\Theta_D}{\hbar}=$ rad/s.
We can check this result by integrating over all frequencies,
$$\int\limits_0^{\omega_D}D(\omega)d\omega = \int\limits_0^{\omega_D}\frac{9n\omega^2}{\omega_D^3}d\omega =3n.$$The specific heat can be calculated from the density of states, (See: Density of states → Specific heat).
(d) At which temperature is $c_{v,el} = c_{v,ph}$?
8.2 Aluminum has a molar mass of 26.98 g/mol and a mass density of $\rho=$ 2.70 g/cm³. Calculate the electron density under the assumption that each atom contributes 3 valence electrons to the free electrons. Calculate the Fermi energy and the Fermi temperature of aluminum, $k_BT_F = E_F$.
8.3 The chemical potential of a metal is weakly temperature dependent so it can be estimated by calculating the Fermi energy. A monovalent metal has a bcc lattice and a lattice constant of $a=$ 0.2 nm. Use the free electron model to estimate the chemical potential in this case.
8.x For a two-dimensional non-interacting free electron gas it is possible to find an analytic expression for the temperature dependence of the chemical potential. Show that,
\[ \begin{equation} \mu = k_BT\ln\left[\exp\left(\frac{E_F}{k_BT}\right)-1 \right]. \end{equation} \]Many thermodynamic properties can be calculated from the density of states. Typically, the electron density is known and the chemical potential be determined. From there all other equilibrium properties can be calculated.
Chemical potential: | \begin{equation} n = \int_{-\infty}^{\infty}D(E)f(E)dE. \end{equation} | |
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Internal energy density: | \[ \begin{equation} u= \int_{-\infty}^{\infty}\frac{ED(E)}{\exp\left(\frac{E-\mu}{k_BT}\right)+1}. \end{equation} \] | |
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Specific heat: | \[ \begin{equation} c_v=\frac{du}{dT}= \int_{-\infty}^{\infty}\frac{ED(E)(E-\mu)\exp\left(\frac{E-\mu}{k_BT}\right)}{k_BT^2\left(\exp\left(\frac{E-\mu}{k_BT}\right)+1\right)^2}dE. \end{equation} \] | |
Entropy density: | \begin{equation} s=\int \frac{c_v}{T}dT=\frac{1}{T}\int\limits_{-\infty}^{\infty}D(E)\left[\frac{E-\mu}{1+\exp\left(\frac{E-\mu}{k_BT}\right)}+k_BT\ln\left(\exp\left(-\frac{E-\mu}{k_BT}\right)+1\right)\right]dE \end{equation} | |
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Helmholz free energy density: | \[ \begin{equation} f=u -Ts= \int_{-\infty}^{\infty}D(E)\left(\frac{\mu}{\exp\left(\frac{E-\mu}{k_BT}\right)+1}- k_BT\ln\left(\exp\left(-\frac{E-\mu}{k_BT}\right)+1 \right)\right)dE. \end{equation} \] | |
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These integrals are difficult to perform since the density of states often includes singularities. Sommerfeld showed that for metals, only the states near the Fermi surface contribute to most observable quantities. He used the following expansion for integrals involving the Fermi function $f(E)$,
\[ \begin{equation} \int_{-\infty}^{\infty}H(E)f(E)dE \approx K(\mu )+\frac{\pi^2}{6}k_B^2T^2\frac{dH(E)}{dE}|_{E=\mu}+\frac{7\pi^4}{360}k_B^4T^4\frac{d^3H(E)}{dE^3}|_{E=\mu}+\cdots \end{equation} \]This is known as the Sommerfeld expansion. Here the relationship between $H(E)$ and $K(E)$ is, \[ \begin{equation} K(E)=\int_{-\infty}^{E}H(E')dE', \hspace{1.5cm} H(E)=\frac{dK(E)}{dE}. \end{equation} \]
For free electrons the density of states are,
$$\text{1D: }D(E)=\frac{1}{\pi\hbar}\sqrt{\frac{2m}{E}},\qquad\text{2D: }D(E)=\frac{m}{\pi\hbar^2},\qquad\text{3D: }D(E)=\frac{(2m)^{\frac{3}{2}}}{2\pi^2\hbar^3}\sqrt{E}.$$Using these densities of states and the Sommerfeld expansion, the thermodynamic properties of free electrons can be calculated.
8.4 The density of states for a free electron gas in two dimensions is,
\[ \begin{equation} D(E)=\frac{m}{\hbar^2\pi}. \end{equation} \]Use the Sommerfeld expansion to calculate the leading order (linear) expression the specific heat for a free electron gas in two dimensions.
8.5 The specific heat of a metal is observed to be linear at low temperature, $c_v=\gamma T$. The bulk modulus of this metal has the form, $B(T) = B_0 +B_1(T)$, where $B_0$ is temperature independent and $B_1(T)$ depends on temperature. Use the free electron model to express $B_1(T)$ in terms of $\gamma$.
In the free-electron model, all of the thermodynamic properties can be expressed in terms of the electron density $n=\frac{N}{V}$. Here $N$ is the number of electrons and $V$ is the volume. Note that $\gamma$ also depends on $n$.
8.x A monovalent metal has a simple cubic Bravais lattice and a lattice constant of $a =$ 0.15 nm. Calculate the chemical potential, the specific heat, the entropy, and the Helmholtz free energy of the electrons at temperatures of 10 K and 300 K assuming that the free electron model can be used.
8.x The electron density of states for fcc gold is given below. Calculate the chemical potential, the electronic contribution to the specific heat, and the electronic contribution to the Helmholtz free energy for gold at temperatures of 10 K and 300 K.