^{1}, Pawel Kempisty

^{1}, Maria Ptasinska

^{2}and Stanislaw Krukowski

^{1,2,a)}

### Abstract

A critical comparison of three polarization based approaches with the fields in AlN/GaN multiple quantum wells (MQWs) systems proved that they give identical results. The direct density functional theory (DFT) results, i.e., the fields, are in qualitative agreement with data obtained within the polarization theory. The results of DFT calculations of an AlN/GaN MQW system were used in the projection method to obtain a spatial distribution of the bands in the structure with atomic resolution. In parallel, the plane averaged and c-smoothed potential profiles obtained from the solution of the Poisson equation were used to determine the electric field in the multiquantum well structures and the magnitude of dipole layers at the AlN/GaN heterostructures. The dipole layers cause potential jumps of about 2.4 V that seriously affects the band offsets. The presence of the dipole layer is in good agreement with the potential measurements by electron holography. It was shown that the wells of the width up to 4 Ga layers behave as potential minima, but the wider layers behave as standard quantum wells. The barriers up to 3 Al layers do not localize the carriers. It is shown that the Quantum Confined Stark Effect causes a huge decrease of their energies and oscillator strengths of the optical transitions, especially for wider structures. For wider wells, the strengths fall much faster for perpendicular polarization which indicates the important role of the anisotropic band offsets. A direct simulation shows that the band offset for the valence band crystal field split off hole states, i.e., pz states are different from heavy and light hole (i.e., ) states being equal to valence band offset and rough estimate of , respectively. These values are in good agreement with the recently reported measurement of AlN/GaN offsets.

The calculations reported in this paper were performed using the computing facilities of the Interdisciplinary Centre for Modelling of Warsaw University (ICM UW). The research was funded by the Polish National Science Centre on the basis of the decision DEC-2011/03/D/ST3/02071.

I. INTRODUCTION

II. CALCULATION METHOD

III. RESULTS

A. Spatial distribution of the energy bands and the electric fields

B. Electric properties of MQWs structure of the various width GaN wells and different AlN layer thick barrier

C. Optical properties of MQWs structure of various width GaN wells and different AlN layer thick barrier

D. Band offset for CH and LH and HH valence band states

E. Summary

### Key Topics

- III-V semiconductors
- 57.0
- Polarization
- 33.0
- Density functional theory
- 24.0
- Valence bands
- 24.0
- Multiple quantum wells
- 22.0

## Figures

Band structure and density of states obtained from VAPS for PBE exchange-correlation functional and LDA-1/2 method of (a) left—AlN and (b) right—GaN.

Band structure and density of states obtained from VAPS for PBE exchange-correlation functional and LDA-1/2 method of (a) left—AlN and (b) right—GaN.

DFT results of 16 Ga/16 Al layers periodic QW system: (a) total electric potential (i.e., electron energy) averaged over the surface perpendicular to c-axis (blue line); c-axis smoothed values obtained by an adjacent averaging procedure (red line); (b) averaged smoothed potential, obtained from LDA-1/2 (red line) and PBE (blue line) formulations; (d) derivation of a dipole layer localization by Gaussian fit; (c) dipole moment magnitude at the GaN/AlN heterointerface and the electric field determination in well/barrier interior in the 16 GaN/16 AlN MQWs system.

DFT results of 16 Ga/16 Al layers periodic QW system: (a) total electric potential (i.e., electron energy) averaged over the surface perpendicular to c-axis (blue line); c-axis smoothed values obtained by an adjacent averaging procedure (red line); (b) averaged smoothed potential, obtained from LDA-1/2 (red line) and PBE (blue line) formulations; (d) derivation of a dipole layer localization by Gaussian fit; (c) dipole moment magnitude at the GaN/AlN heterointerface and the electric field determination in well/barrier interior in the 16 GaN/16 AlN MQWs system.

Band energy obtained from projection of the band wavefunctions on theatom centered harmonics (shading) and the electric potential distribution surface averaged (dashed line) and c-axis smoothed (solid line) potential profiles. Left diagram—c-smoothing profiles were adjusted to AlN-33 points used and the energy band difference shift equal to 5.95 eV; right diagram—c-smoothing profiles were adjusted to GaN-34 points used and the energy band difference shift equal to 3.5 eV.

Band energy obtained from projection of the band wavefunctions on theatom centered harmonics (shading) and the electric potential distribution surface averaged (dashed line) and c-axis smoothed (solid line) potential profiles. Left diagram—c-smoothing profiles were adjusted to AlN-33 points used and the energy band difference shift equal to 5.95 eV; right diagram—c-smoothing profiles were adjusted to GaN-34 points used and the energy band difference shift equal to 3.5 eV.

Electric field in (left) AlN barrier and (right) GaN well for systems having a different thickness of an AlN barrier (top) and GaN well (bottom).

Electric field in (left) AlN barrier and (right) GaN well for systems having a different thickness of an AlN barrier (top) and GaN well (bottom).

Plane averaged and c double smoothed electric potential, obtained within LDA-1/2 approximation for systems with a different thickness of a GaN well. The barrier is 32 AlN atomic layers wide.

Plane averaged and c double smoothed electric potential, obtained within LDA-1/2 approximation for systems with a different thickness of a GaN well. The barrier is 32 AlN atomic layers wide.

DFT LDA-1/2 results: Dispersion relation (left) and the DOS projected on the atomic wavefunction, presented in function of the positions of the atoms (right) for systems of the well having 1, 2, 4, 8, 16, and 32 GaN layers and the same barrier having 16 AlN layers. The black lines are plane averaged double c-smoothed potential profiles presented as electron energy. The lines are separated by 5.96 eV in accordance to AlN conduction and valence bands energy difference.

DFT LDA-1/2 results: Dispersion relation (left) and the DOS projected on the atomic wavefunction, presented in function of the positions of the atoms (right) for systems of the well having 1, 2, 4, 8, 16, and 32 GaN layers and the same barrier having 16 AlN layers. The black lines are plane averaged double c-smoothed potential profiles presented as electron energy. The lines are separated by 5.96 eV in accordance to AlN conduction and valence bands energy difference.

Spatial overlap of electron and hole wavefunction densities obtained in LDA-1/2 approximation: (a) for different well width, i.e., 1 GaN, 4 GaN, 8 GaN, and 16 AlN layers wide barrier; (b) for different barrier width, i.e., 8 AlN, 4 AlN, 1 AlN, and 16 GaN layers wide well; (c) —short period equal width superlattices: 2 GaN/AlN, 4 GaN/AlN, and 8 GaN/AlN layers. The densities were integrated in the x-y plane; therefore, c-axis dependence is shown in the horizontal axis.

Spatial overlap of electron and hole wavefunction densities obtained in LDA-1/2 approximation: (a) for different well width, i.e., 1 GaN, 4 GaN, 8 GaN, and 16 AlN layers wide barrier; (b) for different barrier width, i.e., 8 AlN, 4 AlN, 1 AlN, and 16 GaN layers wide well; (c) —short period equal width superlattices: 2 GaN/AlN, 4 GaN/AlN, and 8 GaN/AlN layers. The densities were integrated in the x-y plane; therefore, c-axis dependence is shown in the horizontal axis.

Characteristics of the principal optical transitions in the equal width AlN/GaN MQWs system: (a) energy and (b) oscillator strength. The c parallel polarized transition, i.e., the electric field vector parallel to c-axis, is denoted by red circles; the c perpendicular polarized transitions (two degenerate modes), having electric field vector perpendicular to c-axis, are denoted by blue squares.

Characteristics of the principal optical transitions in the equal width AlN/GaN MQWs system: (a) energy and (b) oscillator strength. The c parallel polarized transition, i.e., the electric field vector parallel to c-axis, is denoted by red circles; the c perpendicular polarized transitions (two degenerate modes), having electric field vector perpendicular to c-axis, are denoted by blue squares.

Characteristics of the principal optical transitions in an AlN/GaN MQWs system having a 32 ALs thick AlN barrier: (a) energy and (b) oscillator strength. The c parallel polarized transition is denoted by red circles; the perpendicular transitions are denoted by blue squares.

Characteristics of the principal optical transitions in an AlN/GaN MQWs system having a 32 ALs thick AlN barrier: (a) energy and (b) oscillator strength. The c parallel polarized transition is denoted by red circles; the perpendicular transitions are denoted by blue squares.

Characteristics of the principal optical transitions in AlN/GaN MQWs system having 16 ALs thick AlN barrier: (a) energy and (b) oscillator strength. The c parallel and perpendicular polarized transitions are denoted by red circles and blue squares, respectively.

Characteristics of the principal optical transitions in AlN/GaN MQWs system having 16 ALs thick AlN barrier: (a) energy and (b) oscillator strength. The c parallel and perpendicular polarized transitions are denoted by red circles and blue squares, respectively.

Characteristics of the principal optical transitions in AlN/GaN MQWs system having 4 ALs thick AlN barrier: (a) energy and (b) oscillator strength. The c parallel polarized transition is denoted by red circles, the pairs of perpendicular transitions (degenerate) are denoted by blue, green, and cyan polygons.

Characteristics of the principal optical transitions in AlN/GaN MQWs system having 4 ALs thick AlN barrier: (a) energy and (b) oscillator strength. The c parallel polarized transition is denoted by red circles, the pairs of perpendicular transitions (degenerate) are denoted by blue, green, and cyan polygons.

Characteristics of the principal optical transitions in AlN/GaN MQWs system having 32 ALs thick GaN wells: (a) energy and (b) oscillator strength. The c parallel polarized transition is denoted by red circles, the perpendicular ones by blue, green, cyan, and blue-navy polygons.

Characteristics of the principal optical transitions in AlN/GaN MQWs system having 32 ALs thick GaN wells: (a) energy and (b) oscillator strength. The c parallel polarized transition is denoted by red circles, the perpendicular ones by blue, green, cyan, and blue-navy polygons.

Characteristics of the principal optical transitions in AlN/GaN MQWs system having 16 ALs thick GaN wells: (a) energy and (b) oscillator strength. The c parallel polarized transition is denoted by red circles, the perpendicular ones by blue and green polygons.

Characteristics of the principal optical transitions in AlN/GaN MQWs system having 16 ALs thick GaN wells: (a) energy and (b) oscillator strength. The c parallel polarized transition is denoted by red circles, the perpendicular ones by blue and green polygons.

Characteristics of the principal optical transition in AlN/GaN MQWs system having 4 ALs thick GaN wells: (a) energy and (b) oscillator strength. The c parallel and perpendicular polarized transitions are denoted by red circles and blue squares, respectively.

Characteristics of the principal optical transition in AlN/GaN MQWs system having 4 ALs thick GaN wells: (a) energy and (b) oscillator strength. The c parallel and perpendicular polarized transitions are denoted by red circles and blue squares, respectively.

Spatial dependence of the different symmetry band states in 32 AlN, 16 GaN ALs system obtained by their projection on the atom centered: (a) ; (b) pz; (c) s orbitals (VASP LDA-1/2 approximation). The top diagrams present three dimensional plot in which the vertical axis is the magnitude of the projection. The bottom plots contain color maps in the logarithmic color scale.

Spatial dependence of the different symmetry band states in 32 AlN, 16 GaN ALs system obtained by their projection on the atom centered: (a) ; (b) pz; (c) s orbitals (VASP LDA-1/2 approximation). The top diagrams present three dimensional plot in which the vertical axis is the magnitude of the projection. The bottom plots contain color maps in the logarithmic color scale.

## Tables

Electric fields in the wells and barriers in (×10−2 V/Å), potential jumps in V at AlN/GaN heterojunctions. For the fields calculated from Ref. 64 , we have used dielectric permittivity and . We have assumed no zero strain for , i.e., .

Electric fields in the wells and barriers in (×10−2 V/Å), potential jumps in V at AlN/GaN heterojunctions. For the fields calculated from Ref. 64 , we have used dielectric permittivity and . We have assumed no zero strain for , i.e., .

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