Band Offsets

Introduction:

The valence band offset is defined as the difference between the VBM in material B and material A at an A/B interface. Clearly, it is in general interface orientation dependent and also dependent on the strain state of the materials A and B. For a sufficiently thin layer of B on a substrate of A, we can assume a pseudomorphic interface in which B is biaxially strained to match the in-plane lattice constants of the substrate and the strain the direction perpendicular to the interface is then determined by minimizing the elastic energy. For a free-standing superlattice of thin layers of A and B the strain may be a compromise between the strains of A and B. For thick layers of B on A beyond the critical thickness, a network of dislocations forms and relaxes the strain within a few layers, so the material is then assumed to be strain free. Even if the material is strain free at the growth temperature, it can still develop a strain on cooling because of the different thermal expansion coefficients. The effects of strain on the absolute band positions relative to the average electrostatic potential in a material can be calculated from the deformation potentials. One splits the band offset in a bulk contribution

ΔE_v^b=[E_vbm(B)- E_ref(B)]- [E_vbm(A)- E_ref(A)]

and a dipole contribution

V_D=E_ref(B)-E_ref(A)

As reference we use the average electrostatic potential, where average means averaged over the plane and over a unit cell period perpendicular to the interface calculated as a running average deep in the bulk region on the A or B side where it becomes constant. The bulk contribution can be split in a unstrained contribution + a strain contribution. We can then define the "natural band offset" as the unstrained bulk band offset + a direction averaged dipole.

Alternatively, one sometimes defines the "natural band offset" by referring both materials' VBM in the unstrained state to the vacuum level by means of a surface calculation, but one then still needs to average over different surface orientations or remove the surface dipoles which are orientation dependent.

Specific Interface Band offsets

From Ref. [1,2]: The following offsets assume the 1st material is the substrate and the second is biaxially pseudomorphically strained . Values are in eV. Note, the values given originally in ref. [1] included a GW shift of each of the individual materials' VBM relative to their LDA values, which was later found to be sensitive to basis set convergence aspects and required a correction [2]. The values here are from the erratum.

Natural band offsets: From Ref. [1,2] (in eV)

Fig. 1: Natural band-offsets between ZnGeN₂, ZnSnN₂ and relevant substrates, GaN, ZnO, from Ref. [2].

Recently, the band-offset for GaN/ZnSnN₂ was reported by Wang et al. [3]

They find a much lower value of 0.39 eV. Several reasons may be responsible for this discrepancy and are currently

under further investigation.

Electron Affinity rule

Assuming no charge transfer occurs upon forming the interface, the band offsets would be given by the position of the VBM and CBM relative to the common vacuum level. These are the negatives of the ionization potential (IP) and electron affinity (EA). They can be calculated from surface calculations. The following table from [4] allows one to obtain band offsets for any pair of materials in the table

within the limitations of the electron affinity rule. They were calculated for a [100] non-polar plane of Pbn2₁ structure using the LMTO DFT method for the electrostatic potential and using GW bands for the bulk relative to the electrostatic potential. Slabs of 12 layers were used.

Table of (direct) gaps, IP and EA (in eV) [4]

Valence band offsets (eV) [4]

[1]. Atchara Punya and Walter R. L. Lambrecht, Phys. Rev. B 88, 075302 (2013) Band offsets between ZnGeN2, GaN, ZnO, and ZnSnN2 and their potential impact for solar cells

[2] Atchara P. Jaroenjittichai, Sai Lyu, and Walter R. L. Lambrecht, Phys. Rev. B 96, 079907 (2017), Erratum: Band offsets between ZnGeN2, GaN, ZnO, and ZnSnN2 and their potential impact for solar cells [Phys. Rev. B 88, 075302 (2013)]

[3] Tianshi Wang, Chaoying Ni, and Anderson Janotti, Phys. Rev. B 95, 205205 (2017), Band alignment and p-type doping of ZnSnN₂

[4] Sai Lyu and Walter R. L. Lambrecht, under review