Electronic Structures of Shallow Acceptors Confined in Si/SiGe Quantum Well Structures
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the quantization axis z), the acceptor Hamiltonian expressed in electron energy is given by the following operator,
(1)
H = -[Hkin +Hc++HW]
Hqw contains the confinement potential due to the valence-band discontinuity and the influence of a deformation potential, and HC and H"' are the potential of the acceptor impurity center and of the image-charges due to the mismatch of the dielectric constant and kinetic energy. The details of each term in Hamiltonian are referred to [8,9]. The modification of the QW potential, H', must include the effects of the deformation potential. That is, Hqw will contain a square-well potential, A-hIh, for the heavy hole (hh) and the light hole (lh), and includes a potential difference (Vp) between the hh and lh band edges in the well due to built-in strain, thus: F
--- AEhhj`
+
VP.
(2)
The band alignment of pseudomorphic Si/SiGe was clearly identified by low excitation photoluminescence as type II [10] with the main band offset in the valence band and only a very small offset in the conduction band. In our calculations, we have assumed that the valence-band offset AE'w for heavy holes is equal to the band-gap energy difference between the strained Si -,Ge, and bulk Si material. D.J. Robbins et al. [11] have reported the band-gap energy of a strained SiGe layer, which is given by Eg(Si,_,Ge ) = Eg(Si) - 0.896x + 0.396x2 (eV).
(3)
Therefore the band offset for heavy holes is given by AEq, = 0.896x - 0.396x2 (eV).
(4)
In this expression of the heavy-hole band offset, the hydrostatic and uniaxial stress corrections due to the lattice mismatch between SiGe alloy and Si are all included. Because the hydrostatic stress correction is the same for heavy and light holes, the band offset for light holes is given [ 12] by
AEq,hw = AEqw - Es.
(5)
Here the shear stress energy E, is given by
E,= -(•' -€, where 4' = -,
= b[1+ 2C12 ]e C11 and ASO- C+Aa (I + Eo+ 9 ( 2
2 V(
A3"
)2 ) . Aso
274
The biaxial deformation is given by E = a° -a(x). Parameter a0 is the lattice constant of Si a(x) material, and a(x) is the lattice constant of the corresponding well layer Si1 5_ Ge5 . A,0 is a parameter of the spin-orbit splitting. Quantity b is the deformation potential constant, and C,2 and CH are the stiffness constants. The energy separation between the hh and lh bands is given by VP = (ý] - '2). The parameters for the Si,-_Ge, alloy are obtained by a linear interpolation between the Si and Ge parameters, which can be found in Ref.7. Once the strain effects are properly included in the acceptor Hamiltonian, the energy levels of the shallow acceptor states and corresponding wave functions can be calculated. The oscillator strength of the transitions from the ground state to the excited states can be obtained according the dipole transition rule for transitions between acceptor states [6-9]. We would like to point out that even the modification of the band offset due to coupling with the split-off band has been included in the calculation, as described in Equations (4-7), in the above 4 x 4
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