Symmetry analysis and exact model for the elastic, piezoelectric, and electronic properties of inhomogeneous and strained wurtzite quantum nanostructures

A symmetry analysis and a semi analytical exact model are proposed to describe the mechanical, piezoelectric and electronic properties of strained wurtzite quantum nanostructures (QNs) with axial symmetry. An expression of the piezoelectric polarization is given as a function of inhomogeneous strains. The 3D 8x8 strained k.p Hamiltonian is reduced to 2D using the total angular momentum representation. When the spin-orbit coupling is neglected, the Hamiltonians are reduced to 1x1 and 3x3 Hamiltonians for the states in the S-shell. For all the other shells, the four-fold degeneracy is demonstrated. Simulations are performed for InN/GaN quantum dots (QDs). P.A.C.S. 71.20.Nr, 73.21.La, 77.65.Ly Electronic mail : [email protected] J. EVEN. 2 A general model for bulk strained wurtzite (WZ) semiconductors has been extended later to quantum wells (QWs) and quantum dots (QDs). A block-diagonalization of the valence band (VB) Hamiltonian is possible for a strained bulk sample or a homogeneous and unstrained cylindrical quantum nanostructure QN. The simulation of electronic and optical properties requires a precise description of elastic and piezoelectric effects. Moreover, the continuous variation of chemical composition around the nanostructure has often to be included into the theoretical description. The aim of this paper is to propose a complete symmetry analysis and a semi analytical model for mechanical, piezoelectric and electronic properties of strained and inhomogeneous WZ QNs with a cylindrical geometry by analogy to QNs in zinc-blende lattice. Symmetry properties are used as much as possible to simplify the problem. QN geometries with axial symmetry are studied in this work. The CB∞v B axis for the geometry is perpendicular to the (0001) plane (parallel to the [0001] c axis). Physical parameters for InN and GaN materials are used for the WZ lattice. If the strain field is calculated using a continuum mechanical model (elasticity), the geometry of the deformed inhomogeneous QN still corresponds to the CB∞v B symmetry. This effect is associated to the ( ) 2 / 12 11 66 C C C − = relation between elastic constants. Five independent parameters ( 11 C , 12 C , 13 C , 33 C , 44 C ) describe the mechanical properties of the WZ lattice which is related to the "transverse isotropic" materials class. The components of the strain tensor using the cylindrical representation ( r ,φ , z ) depend only on four quantities ( ) ( ) ( ) ( ) z r z r z r z r rz zz rr , , , , , , , ε ε ε ε φφ in this case. 2D calculations can then be performed in cylindrical coordinates ( r , z ) to evaluate the strains but also the piezoelectric potential. The polarization WZ P (including the pyroelectric SP P , and piezoelectric radial rad P and vertical vert P contributions) is: ( ) ( ) z zz 33 rr 31 SP r rz 15 WZ u e e P u e 2 P ε ε ε ε φφ + + + + =