Doped material as a virtual crystal

Besides explicit introduction of dopant species into the host material, doping can be modeled using the virtual-crystal approximation (VCA) [126–130]. In this approach, virtual atoms are constructed that interpolate between the native (A) and doping (B) atoms in the host material AC. The potential of the virtual atom modeling the doped material (A_1−xB_x)C becomes

V_VCA(r) = (1−x)V_A(r) + xV_B(r). (3.1) So the real system is mimicked by constructing its potential from fractions of the potentials of the two different compounds. Pseudoatoms can for instance also be used in (embedded) cluster calculations to provide a realistic saturation of dangling bonds [131,132].

In all-electron calculations, doping via the VCA is introduced by modifying the nuclear num-ber of the atoms in the system, which determines the numnum-ber of electrons from the condition of charge-neutrality. The dopant concentrationN_Dis controlled via the amount∆Z by which the nuclear numberZ is changed. Ap-type doped metal oxide (MO) with a hole concentration of N_holesis modeled by changing the nuclear numberZMof allnMmetal ions with a concentration ofN_M=n_M/Ω, whereΩis the supercell volume, according to

Z_M→Z_M+ ∆Z_M, (3.2)

where

∆Z_M=−N_holes/N_M. (3.3)

In analogy,n-type doping can be modeled by changing the nuclear charges of the metal atoms by ∆Z = N_electrons/N_M, whereN_electrons is the concentration of electrons. Opposed to the

3.2. Modeling charged defects with supercell methods compensating impurity method, in this approach also very small dopant concentrations can be simulated.

In this work, the concentration ofp-type dopants in the simulation cell is chosen to allow for one or two defect electrons to transfer from the defect level to the VBM. This is obviously only possible, if a defect exhibits an occupied defect level in the bandgap. The nuclear number of all magnesium atoms in the supercell is changed by

∆Z_Mg=−q/n_Mg, (3.4)

where q is the desired charge of the oxygen vacancy (+1 or +2), and n_Mg is the number of magnesium atoms in the supercell.

It is important to keep in mind the chemical and physical properties of a material, when applying the VCA for defect charge compensation. In principle, the nuclear charges can be modified in several different ways. One way is to distribute the charge of the defect among all nuclei according to their contribution to the overall nuclear charge, keeping the system neutral.

Consider again the formation of an oxygen vacancy in MgO. For an MgO unit cell withn_Mg magnesium atoms andn_Ooxygen atoms, not including the oxygen atom that is being removed, the changes in magnesium and oxygen atom charges are:

∆ZMg=−q Z_Mg

n_MgZ_Mg+n_OZ_O (3.5)

∆Z_O=−q Z_O

n_MgZ_Mg+n_OZ_O. (3.6)

However, this corresponds to simultaneous p- andn-type doping, since the reduction (in case q >0) of the cation nuclear charge (Mg) reduces the number of electrons it can give away, while the reduction of the anion nuclear charge (O) reduces its ability to accept electrons. In reality, p-type doping of MgO is achieved by substituting magnesium atoms with atoms of lower valency like lithium. This situation is modeled by distributing the defect charge only among the cations.

Thereby, delocalized states are emptied at the top of the valence band, which is composed mostly of oxygen 2p orbitals. As mentioned before, the formation of an oxygen vacancy creates a defect level occupied by two electrons in the bandgap, and therefore, electrons from this level transfer to the vacant states at the top of the valence band of the simulated doped material, which results in the electronic configuration of an F⁺or F²⁺center (Fig.3.4). The charge state of the defect can be tuned by the value by which the nuclear charge was modified, which is related to the concentration of the dopants.

The defect charge could also be distributed only among nuclei far away from the defect. This has the advantage that the immediate surrounding of the defect resembles the situation in the undoped material with the charged defect, at the expense of a more significant modification further away.

Contrary to the neutralizing background method, the reference system in the virtual crystal method should be the doped undefected system, not the perfect undoped system. Modification of the nuclear charges results in a large change in the total energy even if the modification is very small, due to the strong on-site electron-nuclear interaction. The change in the total energy

3.2. Modeling charged defects with supercell methods

Valence band Conduction

band

p-type dopant states (empty)

Valence band Conduction

band

Defect level Pristine system System with O vacancy

Figure 3.4.: F²⁺center, modeled with the VCA. Schematic band structures of the pristine doped system (left) and the system with a defect (right)

can be written as:

E_el−nuc^VC ≈X

i

Z (Z_i+ ∆Z_i)n(r)

|r−r_i| d³r (3.7)

=E_el−nuc^ref +X

i

∆Z_i

Z n(r)

|r−r_i|d³r, (3.8) where n(r) is the electron density. In fact, since the modification of the nuclear charges is inversely proportional to the number of the corresponding atoms in the supercell, the overall shift in energy is independent on the supercell size and it is quite large:

∆E =−q·X

i

Z n(r)

|r−r_i|d³r. (3.9)

As a proof of concept, the VCA method is tested for charged bulk F centers in MgO against the neutralizing background approach. The formation energies, neglecting vibrational effects, for the F⁺ and F²⁺ centers were calculated for five different supercell sizes, once with the background method and once using distributed doping (Fig.3.5), where the nuclear charge of all magnesium atoms in the system was modified. Both methods of charge compensation, constant background and VCA, yield the same total energy differences.

It becomes obvious from Fig.3.5that periodic models for charged defects can be used to study the defect concentration dependence for moderate to large concentrations. This is important, since studying isolated defects, as suitable for a cluster model, is not always justified, although it is a well-defined limiting case.