Interpreting Defect and Energy Level Diagrams

A beginner's guide to understanding defect and energy level diagrams obtained from first-principles calculations

Written by Prashun Gorai

The formation energy of point defects is calculated using the supercell approach described here. The results of the defect calculations are typically presented in the form of "defect diagrams" and "energy level diagrams" – both are discussed below. These diagrams are useful for qualitative assessment of defect and doping properties (e.g., dopability) as well as quantitative determination of defect and electronic charge carrier (electrons, holes) concentrations. The discussion here will focus on semiconductors and insulators i.e., materials with a band gap.

Defect Diagrams

In a defect diagram, the defect formation energy $(\Delta E_{D,q})$ is plotted against the Fermi energy $(E_\mathrm{F})$, which is typically referenced to the valence band maximum i.e., $E_\mathrm{F(VBM)} = 0$. A schematic defect diagram is shown below, where $E_g$ is the band gap.

Schematic defect diagram

The defect diagrams are plotted at a chosen elemental chemical potential ($\mu_i$) condition. Here, $\mu_i$ for every element i should be within the range where the material is thermodynamically stable against decomposition into competing phases. These ranges are determined from a grand potential phase diagram.

How are these defect diagrams plotted?

Each set of connected lines (red solid, red dotted, blue solid, blue dotted) represents a defect D, which could be, for example, a cation vacancy or substitution dopant. The slope of the line is the charge state q of the defect D. Recall that $\Delta E_{D,q}$ has a linear dependence on $E_\mathrm{F}$. Therefore, the equation for calculating $\Delta E_{D,q}$ represents an equation of a straight line with slope q.

$$\Delta E_{D,q} = E_{D,q} - E_{H} + \sum_i{n_i \mu_i} + E_{corr} + ~ qE_\mathrm{F}$$

For a defect D, conventionally only the charge state q with the lowest formation energy at a given $E_\mathrm{F}$ is plotted as shown below. This way of plotting the data is motivated by two reasons: (1) to improve readability of the defect diagrams, and (2) the charge state q with lowest $\Delta E_{D,q}$ has the highest concentration at that $E_\mathrm{F}$. This is shown schematically for a defect.

Charge states with lowest formation energy at given Fermi energy is plotted

Even though only the charge state q with lowest $\Delta E_{D,q}$ is plotted at a given $E_\mathrm{F}$, a non-zero Boltzmann population of other charge states exists, in reality.

What does the charge state/slope of a defect signify and how is related to doping?

Charged defects create electronic carriers – electrons and holes. A neutral defect (e.g. isoelectronic doping, alloying) does not create electronic carriers. A positively-charged defect has a positive slope in a defect diagram and represent a donor-like defect. In other words, the defect ionizes to a positively-charged state by giving up (donating, hence donor) electron(s). Donors tend to make the material n-type. Similarly, negatively-charged defects are acceptor-like and they tend to make the p-type. Whether a material is actually doped n- or p-type depends on several other factors. We will return to the topic of doping once we have understood a few additional concepts.

Deep donor and acceptor defects

What is the significance of a charge transition level in defect diagram?

The charge transition level (q/q') of a defect is the Fermi energy at which the formation energy of charge states q and q' are equal. On a defect diagram, charge transition levels (CTLs) are identified by points at which the slope of the defect lines charge. A defect can have multiple CTLs or they can be absent within the band gap i.e. $0 \leq E_\mathrm{F} \leq E_g$. In the schematic below, defect D1 has no CTLs, D2 has one CTL, and D3 has multiple CTLs.

Charge transition levels of shallow and deep defects

The charge transition levels represent the approximate energetic location of defect states (a.k.a. defect levels) inside the band gap. These defect states give rise to many interesting phenomena that have implications for the electronic, optical, magnetic properties and many more. It is beyond the scope of this tutorial to delve into these effects. At this point, the mere acknowledgement of the existence of these defect states is sufficient.

What are shallow and deep defects and how to find them in defect diagrams?

Charged defects need to be ionized to make the associated electronic carriers available for conduction. If this ionization energy is large, the electronic carriers remain "bound" to the defect site, but if it is small (typically within a few $k_\mathrm{B}T$), the bound electronic carriers are excited into conduction states. The energetic location of a CTL relative to a band edge is essentially the thermal energy needed to ionize the defect. For a donor defect (donates electrons), the CTL relative to the conduction band edge is relevant. Similarly, for an acceptor defect (accepts electron/donates holes), the CTL location from the valence band edge is relevant.

Ionization energy of donor and acceptor defects

How far the CTL is from the relevant band edge determines if a defect is "shallow" or "deep" (and how deep). When a defect has no CTLs inside the band gap, it implies that the defect state(s) lies inside the bands. Such defects are "shallow" defects – they are readily ionized and the associated electronic carriers are available as conduction carriers (also, free carriers). Another case of a shallow defect is when the CTL is close to the corresponding band edge (donor CTL relative to CBM, acceptor CTL relative to VBM), typically within a few $k_\mathrm{B}T$. The electrons(holes) in the defect states can be easily thermalized (remember thermal energy ~ $k_\mathrm{B}T$) into the conduction(valence) band. The schematic below illustrates these two cases for shallow donor defects – one with no CTL inside the band gap and another with CTL close to the conduction band edge.

Shallow donor defect has defect states inside or close to the conduction band

By extension, it should be now clear that "deep" defects has CTLs far from the corresponding band edges, typically more than few $k_\mathrm{B}T$. Electronic carriers are trapped at these deep defect sites; such carriers cannot contribute to the electrical conductivity, which is only due to free carriers. Deep defects states are also sometimes referred to as mid-gap states. Next time you see a defect diagram with CTLs far from the band edges, you can quickly conclude that there are deep defects present. However, it is also important to consider the concentration of such defects, which brings us to the next important topic – calculation of defect concentrations.

Deep donor and acceptor defects

In rare cases, an extremely deep donor defect will have CTLs lie inside the valence band. Analogously, an extremely deep acceptor will have CTLs inside the conduction band.

What is the effect of elemental chemical potentials on the defect diagram and how is related to the synthesis/growth conditions?

The defect formation energy depends on the elemental chemical potentials ($\mu_i$).

$$\Delta E_{D,q} = E_{D,q} - E_{H} + \sum_i{n_i \mu_i} + E_{corr} + ~ qE_\mathrm{F}$$

As discussed above, defect diagrams are typically plotted at a fixed set of $\mu_i$s, where i are the elements in the chemical phase space. For example, the defect formation energy of ZnO will depend on the chemical potentials of Zn and O. Similarly, for KGaSb4, the defect energetics depend on the chemical potentials of K, Ga, and Sb. Mathematically, changing $\mu_i$ shifts the defect lines vertically in a defect diagram (changes y-axis intercept), but the dominant charge q at a given $E_\mathrm{F}$ and the CTLs remain unchanged. This is schematically shown in the figure below.

Effect of changing chemical potential on defect energetics

The elemental chemical potentials intimately depend on the chemical environment prevalent during growth/synthesis. Formation of defects involve exchange of atoms between the material and external elemental reservoirs. This exchange requires a specific amount of energy, which is determined by the chemical potentials of the elemental reservoirs ($\mu_i$). Here, the chemical conditions during growth dictate $\mu_i$.

Let us consider the example of ZnO grown under oxygen-rich (O-rich) and oxygen-deficient (O-poor) conditions. Under what conditions would oxygen vacancy formation in ZnO be more favorable? We can arrive at the answer by simply thinking about the process. Oxygen vacancy formation involves the removal of O atoms from ZnO and placing it in the external O reservoir. Under O-rich conditions, this external reservoir already has a high concentration of O, so placing additional O atoms would require a lot more energy. In contrast, under O-poor conditions, where the reservoir has a lower concentration of oxygen, removal of O from ZnO and placing it in this reservoir should be energetically cheaper. Therefore, O vacancy formation is more favorable under O-poor conditions. A detailed mathematical treatment on the effect of elemental chemical potentials and growth conditions on defect energetics will be added later.

How do we calculate the defect concentrations and associated electron and hole concentrations?

The defect concentration at a given temperature is calculated from the Boltzmann probability:

$$[D_q] = N_s.exp\left(- \frac{\Delta E_{D,q}}{k_\mathrm{b}T} \right)$$

where $[D_q]$ is the concentration of defect D in charge state q, $N_s$ is the number of atomic sites where D can form, $k_\mathrm{B}$is the Boltzmann constant, and T is the temperature. The exponential term is the essentially the probability of forming the D, which when multiplied with the volumetric (or areal) site concentration, given the volumetric defect concentration. Typically, defect concentrations are expressed in numbers per unit volume ($\mathrm{cm}^{-3}$).

The equation for calculating defect concentration using Boltzmann probability implicitly includes the configurational entropy contribution. However, it does not include the vibrational entropy contribution, which is generally negligible in most cases. The latter is an important consideration when defect formation involves elements that are gases under standard conditions. In such cases, chemical potential of the elemental (gaseous) reservoir has a strong T dependence.

Looking at the defect diagram, the defect formation energy depends on the Fermi energy ($E_\mathrm{F}$), but it is not clear at which $E_\mathrm{F}$ should the defect concentration be calculated. For this, we need to consider charge neutrality.

The defect and electronic carrier (electron, hole) concentrations are calculated by imposing the condition of overall charge neutrality. This means, for the material to be charge neutral, the total number of positive and negative charges must be balanced. The positive charges include donor defects (+ve slope in defect diagrams) and holes, and negative charges are acceptors (-ve slope in defect diagrams) and electrons.

First, we need to choose a relevant temperature, typically, the synthesis or annealing temperature at which defects form and equilibrate. Next, we calculate the acceptor and donor defect concentrations as a function of $E_\mathrm{F}$. Mathematically, $\Delta E_{D,q} = c + q.E_\mathrm{F}$, where c is the y-axis intercept in a defect diagram. Neutral defects do not participate in charge balance. The electron and hole concentrations are given by the density of states g(E) and the Fermi-Dirac distribution f(E). Mathematically, $[e] = \int_{E_C}^{\infty}g(E)f_c(E)dE$ and $[h] = \int^{E_V}_{-\infty}g(E)f_v(E)dE$, where $E_C$ and $E_V$ are the conduction and valence band extrema, respectively, and $f_c(E)$ and $f_v(E)$are the Fermi-Dirac distribution functions for the conduction and valence bands, respectively. Here, $f_c(E)$ depends on $E_\mathrm{F}$, $E_C$, and $T$. Similarly, $f_v(E)$depends on $E_\mathrm{F}$, $E_V$, and $T$. Please refer to standard semiconductor textbooks for equations, including the simplified formula to calculate electron and hole concentrations in the non-degenerate limit using band effective mass (instead of performing the integrals discussed above).

Note how the only unknown in the equations to calculate defect and electron and hole concentrations is $E_\mathrm{F}$! By imposing the condition of charge neutrality, we determine the equilibrium Fermi energy ($E_\mathrm{F,eq}$).

$$\sum_{d}\left( q_d[D_{d}] \right) + [h] = \sum_{a}\left( q_a[D_{a}] \right) +[e]$$

where a and d denote acceptors and donors, respectively. $q_a$and $q_d$ are the charge states of the corresponding acceptor and donor defects, respectively. Terms in [] denote concentrations. The left-hand side is the total number of positive charges and the right-hand side the negative charges. The charge neutrality equation is solved self-consistently (often, numerically) to obtain $E_\mathrm{F,eq}$. Once $E_\mathrm{F,eq}$ is determined, we can calculate the defect and electron and hole concentrations.

There are several open-source Python scripts that are available for numerically solving $E_\mathrm{F,eq}$. I recommend py-sc-fermi developed by the groups of David Scanlon and Benjamin Morgan. A paper describing the software is available here.

The equilibrium Fermi energy is also often referred to as pinned Fermi level. Strict practitioners of the field may argue that Fermi level pinning is not an appropriate term in this context.

This tutorial focuses on the equilibrium thermodynamics of defect formation. Defects can and do form under non-equilibrium condition, but determination of their concentrations is not straightforward without detailed experimental inputs. It is a specialized topic that is beyond the scope of this guide designed for beginners.

Can we guess the approximate location of the equilibrium Fermi energy? What can we learn?

We can guess, to a first order, the approximate location of the equilibrium $E_\mathrm{F}$. The exact location must be determined mathematically by self consistently solving the charge neutrality condition, as described above. To locate the approximate equilibrium $E_\mathrm{F}$, one must identify the "defect lines" (in a defect diagram) corresponding to the lowest-formation energy donor and acceptor defects. Remember that neutral defects do not participate in charge balance. The equilibrium $E_\mathrm{F}$is, in most cases, in the vicinity of where the lowest-energy donor and acceptor defect lines intersect.

How do we qualitatively interpret the defect diagrams in the context of doping?

We learned above that donor-like defects tend to donate electrons while acceptor-like defects tend to accept electrons (create holes). Rarely does a material contain only donor (or acceptor) defects; often, both donor-like and acceptor-like defects are present. Whether a material is overall n- or p-type, depends on the relative concentrations of donor and acceptor defects. Intuitively, if the concentration of donors (+ve charged defects) is much higher than acceptors (-ve charged defects), one can imagine that the material will be n-type. Mathematically, this can be understood by charge balance discussed above. If the +ve charged defects are in higher concentration, the overall charge balance is restored by creating more electrons (-ve charges) than holes. As such, the material will have a high concentration of electrons, making it n-type. Here, we have discussed the scenario of "self-doping" by the native defects. However, in practical applications, materials are often doped with extrinsic dopants e.g., n-type Si doped with P, n-type ZnO doped with Al.

Materials cannot be extrinsically doped n- or p-type at will (otherwise life would have been easy!). A lot depends on the native defect chemistry. In other words, whether a material can be doped n- or p-type depends critically on the donors and acceptors natively present in material e.g., Si vacancy in Si or O vacancy in ZnO. Native defects, as the name suggests, are inherently present in the material and are not introduced through extrinsic doping. Therefore, one cannot bypass the existence of such native defects. The formation energetics of these native defects determine if a material is n- or p-type (or both, or neither) dopable. Here, dopable means the possibility of extrinsically doping a material. The figure below showcases four different scenarios for materials that are (1) p-type dopable, (2) n-type dopable, (3) both p- and n-type dopable ("ambipolar"), and (4) neither p- nor n-type dopable (typical insulators).

Different doping scenarios. Defects labelled "donor" and "acceptor" are native defects, while "dopant" refers to extrinsic dopant.

Scenario 1: