Numerical Quadrature of the Subband Distribution Functions in Strained Silicon UTB Devices Oskar Baumgartner, Markus Karner, Viktor Sverdlov, and Hans Kosina Institute for Microelectronics, TU Wien, Gußhausstraße 27–29, A–1040 Wien, Austria Email: {baumgartner|karner|sverdlov|kosina}@iue.tuwien.ac.at Abstract—In this work, the k · p method is used to calculate the electronic subband structure. To reduce the computational cost of the carrier concentration calculation and henceforth the required number of numerical solutions of the Schr¨ odinger equation, an efficient 2D k-space integration by means of the Clenshaw- Curtis method is proposed. The suitability of our approach is demonstrated by simulation results of Si UTB double gate nMOS and pMOS devices. I. I NTRODUCTION Strained silicon ultra-thin body MOSFETs are considered to be good candidates for CMOS integration in the post 22 nm technology nodes. An accurate description of such devices relies on the modeling of the subband structure. An efficient self-consistent Schr¨ odinger-Poisson model for the calculation of the electronic subband structure is presented, taking into account band nonparabolicity and arbitrary strain [1]. A two- band k · p Hamiltonian has been used for electrons and a six- band k · p Hamiltonian for holes. II. CALCULATION OF THE SUBBAND STRUCTURE The numerical modeling of the subband structure in ultra thin body SOI MOS structures relies on an accurate model of the bulk Hamiltonian. We applied a two-band k · p Hamilto- nian [2], [3] to describe the silicon conduction band around −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 10 −2 10 −3 10 −4 10 −5 kx[ 2π a 0 ] ky[ 2π a 0 ] Subband occupation density[1] Fig. 1. Occupation of the heavy hole band of Si in a 3 nm wide quantum well. The grid shows the nodes of the numerical quadrature. the X points. H =  H − H bc H bc H +  with H ∓ = E c (z)+  2 k 2 z 2m l +  2 ( k 2 x + k 2 y ) 2m t ∓  2 k 0 k z m l , H bc = D xy −  2 k x k y M . E c denotes the conduction band edge energy, m l and m t are the longitudinal and transversal electron masses, respectively, and 1 M ≈ 1 mt − 1 me . The shear strain deformation potential D = 14eV and the off-diagonal strain component  xy describe the effects of shear strain on the bandstructure. k 0 =0.15 2π a0 corresponds to the distance of the valley to the X point. To model the silicon valence band structure a 6 × 6 − k · p Hamiltonian [4] has been implemented. Following the notation of Manku it is written as H = E v I 6×6 +  S + D 0 3×3 0 3×3 S + D  + H so , where E v is the valence band edge and the perturbation matrix S and the deformation potential matrix D are given by S= ⎡ ⎣ Lk 2 x +M(k 2 y +k 2 z ) Nkxky Nkxkz Nkxky Lk 2 y +M(k 2 x +k 2 z ) Nkykz Nkxkz Nkykz Lk 2 z +M(k 2 x +k 2 y ) ⎤ ⎦ D= ⎡ ⎣ lεxx+m(εyy+εzz) nεxy nεxz nεxy lεyy+m(εxx+εzz) nεyz nεxz nεyz lεzz+m(εxx+εyy) ⎤ ⎦ As parameters for the silicon valence band structure without strain L = −6.53, M = −4.64, and N = −8.75 in units of  2 2me have been used [5]. l, m, and n are the strain deformation potentials for the valence band. The spin orbit coupling is described by the Hamiltonian H so = − E so 3 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 i 0 0 0 −1 −i 0 0 0 0 i 0 0 0 1 −i 0 0 0 1 0 −i 0 0 0 i i 0 0 −1 −i 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , with the split off energy of silicon E so = 44 meV. 978-1-4244-3927-0/09/$25.00 ©2009 IEEE