Energy Levels and Density of States in Solids

In solid-state physics, understanding how electrons behave within a material is crucial. Electrons in isolated atoms occupy discrete energy levels. However, when atoms come together to form a solid, these discrete levels broaden into energy bands due to the interactions between neighboring atoms. The way these energy levels are distributed and the number of available states within these bands significantly influence the material's electrical and thermal properties. We will explore this concept across different dimensions: one, two, and three dimensions.

The Concept of Density of States (DOS)

The density of states, often abbreviated as DOS, is a fundamental concept that describes the number of available electronic states per unit energy interval per unit volume. It tells us how densely packed the energy levels are at a particular energy. Mathematically, it is represented as $D(E)$, where $E$ is the energy. A higher DOS at a certain energy means there are more available states for electrons to occupy at that energy.

Density of States in One Dimension (1D)

Consider electrons confined to move along a single line, like electrons in a quantum wire or along a chain of atoms. In a 1D system, the energy of an electron is related to its wave vector $k$. For a free electron, the energy is given by $E = \frac{{\hbar^2 k^2}}{{2m}}$, where $\hbar$ is the reduced Planck constant, $k$ is the wave vector, and $m$ is the electron mass.

In one dimension, the wave vector $k$ can range from $-\infty$ to $+\infty$. The relationship between energy $E$ and wave vector $k$ is parabolic. The density of states $D(E)$ in 1D is proportional to $E^{-1/2}$.

To derive this, we consider the number of states within a range of $k$. In 1D, the number of states in a range $dk$ is $\frac{L}{2\pi} dk$, where $L$ is the length of the 1D system. Since $E = \frac{{\hbar^2 k^2}}{{2m}}$, we have $k = \sqrt{\frac{{2mE}}}{{\hbar}}$. Differentiating with respect to $E$, we get $dk = \sqrt{\frac{{2m}}{{\hbar^2}}} \frac{1}{2\sqrt{E}} dE$.

The number of states in $dE$ is then $\frac{L}{2\pi} dk = \frac{L}{2\pi} \sqrt{\frac{{2m}}{{\hbar^2}}} \frac{1}{2\sqrt{E}} dE$.

The density of states per unit length is $D(E) = \frac{1}{L} \frac{dN}{dE} = \frac{1}{2\pi} \sqrt{\frac{{2m}}{{\hbar^2}}} \frac{1}{2\sqrt{E}} = \frac{1}{2\pi\hbar} \sqrt{\frac{{2m}}{E}}$.

Therefore, $D(E) \propto \frac{1}{\sqrt{E}}$. This means that as energy increases, the density of states decreases. At low energies, there are many available states, but they become sparser at higher energies.

Density of States in Two Dimensions (2D)

In a 2D system, like electrons in a thin film or a quantum well, electrons can move in a plane. The energy is given by $E = \frac{{\hbar^2 (k_x^2 + k_y^2)}}{{2m}} = \frac{{\hbar^2 k^2}}{{2m}}$, where $k^2 = k_x^2 + k_y^2$.

The wave vector $k$ now has a magnitude and direction in 2D. The number of states within a wave vector magnitude $k$ is proportional to the area in k-space, which is $\pi k^2$. The number of states in a range $dk$ is $\frac{A}{4\pi^2} (2\pi k) dk$, where $A$ is the area of the 2D system.

Since $E = \frac{{\hbar^2 k^2}}{{2m}}$, we have $k = \sqrt{\frac{{2mE}}}{{\hbar}}$ and $dk = \sqrt{\frac{{2m}}{{\hbar^2}}} \frac{1}{2\sqrt{E}} dE$.

Substituting these into the expression for the number of states in $dE$: $\frac{A}{4\pi^2} (2\pi \sqrt{\frac{{2mE}}}{{\hbar}}) (\sqrt{\frac{{2m}}{{\hbar^2}}} \frac{1}{2\sqrt{E}} dE) = \frac{A}{4\pi^2} 2\pi \sqrt{\frac{{2m}}{{\hbar^2}}} \frac{\sqrt{E}}{1} \frac{1}{2\sqrt{E}} dE = \frac{A}{2\pi} \frac{\sqrt{2m}}{\hbar} dE$.

The density of states per unit area is $D(E) = \frac{1}{A} \frac{dN}{dE} = \frac{1}{2\pi} \frac{\sqrt{2m}}{\hbar}$.

This result is remarkable: the density of states in a 2D system is constant and independent of energy. This constancy has significant implications for the electronic properties of 2D materials like graphene.

Density of States in Three Dimensions (3D)

In a typical solid, electrons can move in all three spatial dimensions. The energy is given by $E = \frac{{\hbar^2 (k_x^2 + k_y^2 + k_z^2)}}{{2m}} = \frac{{\hbar^2 k^2}}{{2m}}$.

The number of states within a wave vector magnitude $k$ is proportional to the volume in k-space, which is $\frac{4}{3}\pi k^3$. The number of states in a range $dk$ is $\frac{V}{8\pi^3} (4\pi k^2) dk$, where $V$ is the volume of the 3D system.

Using $k = \sqrt{\frac{{2mE}}}{{\hbar}}$ and $dk = \sqrt{\frac{{2m}}{{\hbar^2}}} \frac{1}{2\sqrt{E}} dE$ as before:

Number of states in $dE = \frac{V}{8\pi^3} (4\pi (\frac{{2mE}}}{{\hbar^2}})) (\sqrt{\frac{{2m}}{{\hbar^2}}} \frac{1}{2\sqrt{E}} dE) = \frac{V}{8\pi^3} 4\pi \frac{2m}{\hbar^2} \sqrt{E} \sqrt{\frac{{2m}}{{\hbar^2}}} \frac{1}{2\sqrt{E}} dE = \frac{V}{8\pi^3} \frac{4\pi}{\hbar^2} \sqrt{\frac{{2m^3}}{{\hbar^2}}} dE = \frac{V}{2\pi^2} \frac{(2m)^{3/2}}{\hbar^3} \sqrt{E} dE$.

The density of states per unit volume is $D(E) = \frac{1}{V} \frac{dN}{dE} = \frac{1}{2\pi^2} \frac{(2m)^{3/2}}{\hbar^3} \sqrt{E}$.

Therefore, $D(E) \propto \sqrt{E}$. This indicates that in a 3D system, the density of states increases with the square root of energy. At low energies, there are fewer states, and they become more abundant as energy increases.

Electrical and Thermal Conductivities

The electrical and thermal properties of solids are largely governed by the behavior of electrons and phonons (lattice vibrations). In metals, free electrons are primarily responsible for both electrical and thermal conduction.

Electrical Conductivity ($\sigma$)

Electrical conductivity measures how easily an electric current can flow through a material. It is the reciprocal of resistivity ($\rho$). When an electric field is applied across a conductor, free electrons experience a force and drift in a particular direction, creating an electric current.

The classical Drude model provides a simple picture. It assumes that electrons in a metal behave like a gas of free particles that collide with the fixed ions in the lattice. These collisions impede the flow of electrons.

The drift velocity ($v_d$) acquired by an electron under an electric field ($E_{field}$) is given by $v_d = -\frac{{eE_{field}\tau}}{{m}}$, where $e$ is the electron charge, $m$ is its mass, and $\tau$ is the average time between collisions (relaxation time).

The current density ($J$) is given by $J = n e v_d$, where $n$ is the number of free electrons per unit volume. Substituting the expression for $v_d$:

$J = n e (-\frac{{eE_{field}\tau}}{{m}}) = -\frac{{ne^2\tau}}{{m}} E_{field}$.

Ohm's law in differential form is $J = \sigma E_{field}$. Comparing this with the Drude model result, we get the electrical conductivity:

$\sigma = \frac{{ne^2\tau}}{{m}}$.

This formula highlights that conductivity increases with the number of free charge carriers ($n$) and the relaxation time ($\tau$), and decreases with increasing electron mass ($m$). In metals, $\tau$ is primarily determined by scattering from lattice vibrations (phonons) and impurities.

Thermal Conductivity ($\kappa$)

Thermal conductivity measures a material's ability to conduct heat. Heat can be transported in solids by two main mechanisms: electrons and phonons. In metals, the electronic contribution is usually dominant.

Similar to electrical conductivity, the Drude model can be extended to describe electronic thermal conductivity. When a temperature gradient exists, electrons with higher kinetic energy in the hotter region tend to diffuse towards the colder region, carrying energy with them.

The electronic thermal conductivity ($\kappa_e$) can be approximated using the Wiedemann-Franz law, which we will discuss shortly. Qualitatively, it depends on the number of free electrons, their average speed, and the relaxation time. A higher density of free carriers and longer mean free path (related to $\tau$) lead to better thermal conduction.

Phonons, which are quantized lattice vibrations, also contribute to thermal conductivity ($\kappa_{ph}$). In insulators, this is the primary mechanism. Phonons scatter off each other, impurities, and boundaries, limiting thermal transport.

The total thermal conductivity is the sum of the electronic and phononic contributions: $\kappa = \kappa_e + \kappa_{ph}$.

The Wiedemann–Franz Law

The Wiedemann–Franz law is an empirical relationship observed in metals that connects their electrical and thermal conductivities. It states that the ratio of the thermal conductivity ($\kappa$) to the electrical conductivity ($\sigma$) is proportional to the temperature ($T$).

Mathematically, the law is expressed as:

$\frac{{\kappa}}{{\sigma}} = LT$.

Here, $L$ is the Lorentz number, which is theoretically predicted to be a universal constant for many metals, especially at high temperatures.

Theoretical Basis of the Wiedemann–Franz Law

The Wiedemann–Franz law arises naturally from the Drude model when applied to both electrical and thermal conduction by electrons. We already have $\sigma = \frac{{ne^2\tau}}{{m}}$.

For thermal conductivity, considering electrons as a classical gas with average kinetic energy $\frac{3}{2} k_B T$ (where $k_B$ is the Boltzmann constant), the electronic thermal conductivity is approximately given by:

$\kappa_e \approx \frac{1}{3} n C_v v_e \lambda$, where $C_v$ is the specific heat capacity per electron, $v_e$ is the average electron speed, and $\lambda = v_e \tau$ is the mean free path.

For a classical electron gas, $C_v = \frac{3}{2} k_B$. The average electron speed is related to temperature. A more rigorous derivation using the Boltzmann transport equation and considering the energy distribution of electrons leads to:

$\kappa_e = \frac{{\pi^2}}{3} \frac{{n k_B^2 T}}{{m}} \tau$. (This is the result from a more advanced treatment).

Now, let's calculate the ratio $\frac{{\kappa_e}}{{\sigma T}}$:

$\frac{{\kappa_e}}{{\sigma T}} = \frac{{\frac{{\pi^2}}{3} \frac{{n k_B^2 T}}{{m}} \tau}}{{(\frac{{ne^2\tau}}{{m}}) T}} = \frac{{\pi^2 k_B^2}}{{3 e^2}}$.

This ratio is indeed the Lorentz number, $L = \frac{{\pi^2 k_B^2}}{{3 e^2}}$.

Let's calculate the value of $L$: $k_B \approx 1.38 \times 10^{-23} \, \text{J/K}$ $e \approx 1.60 \times 10^{-19} \, \text{C}$ $L \approx \frac{{\pi^2 (1.38 \times 10^{-23})^2}}{{3 (1.60 \times 10^{-19})^2}} \approx \frac{{9.87 \times 1.90 \times 10^{-46}}}{{3 \times 2.56 \times 10^{-38}}} \approx \frac{{18.75 \times 10^{-46}}}{{7.68 \times 10^{-38}}} \approx 2.44 \times 10^{-8} \, \text{J}^2/\text{K}^2\Omega$.

The theoretical value of the Lorentz number is approximately $2.44 \times 10^{-8} \, \text{J}^2/\text{K}^2\Omega$.

Validity and Deviations

The Wiedemann–Franz law holds reasonably well for many metals, particularly at high temperatures (where $T \gg \Theta_D$, the Debye temperature) and for pure metals where electron scattering is the dominant mechanism for both electrical and thermal resistance.

Deviations occur due to several factors:

• Phonon Contribution: In some materials, especially at low temperatures, the lattice contribution ($\kappa_{ph}$) to thermal conductivity can be significant, causing $\kappa/\sigma T$ to be larger than the theoretical $L$.
• Electron-Phonon Scattering: At low temperatures, electron-phonon scattering becomes less frequent, increasing $\tau$ and thus $\sigma$. However, the thermal conductivity might not increase proportionally, leading to deviations.
• Quantum Effects: At very low temperatures and in low-dimensional systems, quantum mechanical effects and the nature of electron-phonon interactions can lead to significant deviations.
• Impurities and Defects: Impurities and defects scatter both electrons and phonons, affecting both conductivities. While they reduce electrical conductivity, their effect on thermal conductivity can be complex.

Example: Comparing Metals

Consider copper and aluminum, both excellent conductors. Copper has a lower electrical resistivity (higher $\sigma$) and lower thermal resistivity (higher $\kappa$) than aluminum. The Wiedemann–Franz law predicts that their $\kappa/\sigma T$ ratios should be similar. Indeed, experimental values for $L$ for copper and aluminum are close to the theoretical value, confirming the law's applicability.

However, consider a material like bismuth, which is a poor electrical conductor but a relatively good thermal conductor. This indicates that phonon contribution might be significant, or the electronic structure leads to different scattering mechanisms.

Summary of Key Concepts

The energy levels in solids form bands, and the density of states ($D(E)$) describes how these states are distributed. The dimensional dependence of $D(E)$ ($E^{-1/2}$ for 1D, constant for 2D, $E^{1/2}$ for 3D) is fundamental. Electrical and thermal conductivities in metals are primarily due to free electrons, described by the Drude model. The relaxation time ($\tau$) is a critical parameter. The Wiedemann–Franz law elegantly links these two transport properties through the Lorentz number $L$, highlighting the common role of electrons in both electrical and thermal conduction. While the law is a powerful approximation, deviations arise from phonon contributions and quantum effects, especially at low temperatures.