Band Structure and Density of States (DOS)

As discussed previously, the horizontal axis of a band structure diagram corresponds to the wave vector k (which is proportional to the crystal momentum of the electrons), while the vertical axis represents energy E. Each point on the band structure curves corresponds to an allowed electronic state with a specific (k, E) value.

The density of states (DOS) plot, in contrast, shares the same energy axis with the band structure diagram but sums over all the allowed electronic states within a small energy interval (ΔE), regardless of k. This count is then normalized by ΔE (following the definition of a derivative), resulting in a function that describes the density of available electronic states at each energy level.

As illustrated in Figure 1, the left panel shows the band structure (black curves), while the right panel displays the DOS (red curve), both aligned on a common energy axis. In energy regions where the bands are dense, the DOS is high; where bands are sparse, the DOS is low; and in energy ranges with no bands at all, the DOS is zero.

Figure 1:Relationship between band structure and density of states.

From this relationship, we can understand the DOS as a compressed version of the band structure. It retains some of the information (such as allowed and forbidden energy regions, and the Fermi level), while omitting others. For instance, regions where the DOS is zero correspond to band gaps, while non-zero DOS values correspond to allowed bands. The Fermi level is shared by both representations.

Just like band structures, DOS plots can also be used to determine whether a solid-state system is metallic or insulating/semiconducting. If the Fermi level lies within a band (high DOS), the material is a conductor; if it lies in a band gap (zero DOS), the system is an insulator or semiconductor.

However, certain information is lost in the DOS plot. For example, the relative positions of the valence band maximum (VBM) and conduction band minimum (CBM) in k-space are not visible in the DOS, making it impossible to distinguish between direct and indirect band gaps. Similarly, for semiconductors, the effective mass of charge carriers—obtained from the curvature (second derivative with respect to k) of the band edges—is also inaccessible from the DOS alone.

Thus, if one is only interested in analyzing the electronic conductivity or band gap characteristics of a material, the DOS plot is often more concise and easier to interpret than the full band structure. It simplifies the E–k relationship into a direct focus on energy intervals that contain electronic states, and the Fermi level’s position relative to those states. This is why many papers include DOS plots without presenting the full band structure.

Projected Density of States (PDOS)

The Projected Density of States (PDOS) refers to the component of the DOS projected onto individual atomic orbitals. With PDOS, one can analyze the contribution of each atom—and even each orbital (s, p, d, f)—to the total DOS.

⚠️ Note: PDOS calculations are not always exact. The total sum of all PDOS components typically falls slightly short of the full DOS due to projection limitations. Nevertheless, PDOS provides highly valuable insights for interpreting the electronic structure and properties of materials.

Example 1: Band Gap Narrowing via Doping

Figure 2:Effect of Ns and Fs doping on the PDOS of TiO₂.

In Figure 2, the left panel shows the DOS of undoped TiO₂ with a calculated band gap of approximately 3 eV. The right panel shows the DOS of TiO₂ doped with nitrogen (N) and fluorine (F), where the band gap narrows to around 2.5 eV.

By analyzing the PDOS, where different colored curves correspond to different atomic orbitals, we gain deeper insight. In the undoped TiO₂ (left), the valence band maximum is primarily composed of O-2p orbitals. In the doped sample (right), there are new occupied states above the O-2p band, originating from N-2p orbitals. These N-derived states are responsible for the band gap narrowing.

Example 2: Bonding Analysis via PDOS

Figure 3:PDOS of an adsorption site and an adsorbed molecule.

Another application of PDOS is in analyzing chemical bonding. PDOS can be used to determine whether two spatially adjacent atoms are chemically bonded. If the PDOS of two neighboring atoms shows overlapping peaks in the same energy range, it suggests the formation of a bond.

• Important: This method only applies to atoms that are spatially close; overlapping PDOS between distant atoms does not imply bonding.

In Figure 3, red curves represent the PDOS of an adsorbed hydroxyl (OH) group, while blue curves represent the PDOS of the metal adsorption site. In all three cases, the OH group and the adsorption site are adjacent, and their PDOS plots exhibit overlapping features, indicating bonding in each scenario.

In the third case (Ta site), the bonding states appear at a much lower energy compared to those in Ni and Ir, implying weaker adsorption energy. This illustrates how PDOS can reveal bonding strength and help explain surface reactivity.

Example 3: d-Band Center Theory

Another important application of PDOS is the d-band center theory, especially for transition metal elemental catalysts (note: this applies to elemental metals, not oxides).

The theory suggests that the position of the d-band center relative to the Fermi level is a predictor of catalytic activity:

• The closer the d-band center is to the Fermi level, the higher the catalytic activity.

This framework is frequently used to rationalize trends in catalytic performance among transition metals.

Summary

This article introduced the concepts and applications of the Density of States (DOS) and Projected Density of States (PDOS):

• DOS shows how electronic states are distributed as a function of energy, enabling insights into conductivity and band gaps.

• PDOS projects the DOS onto atomic orbitals, allowing identification of contributions from specific atoms and orbitals, facilitating analysis of bonding, doping effects, and catalytic activity.

Both are essential tools in modern electronic structure analysis, and complement traditional band structure diagrams in providing a full picture of material properties.