Material Parameters for Analytical and Numerical Modeling of Si and Strained SiGe Heterostructure Devices
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Material Parameters for Analytical and Numerical Modeling of Si and Strained SiGe Heterostructure Devices S. C. Jain, A. Mehra1, S. Decoutere1, W. Schoenmaker, and M. Willander2 1 IMEC, Kapeldreef 75, 3001 Leuven, Belgium 2 Chalmers University of Technology, Department of Physics, S-41296 Göteborg, Sweden ABSTRACT We present calculated values of effective masses, bandgap reduction, and Fermi energy of p-doped Si and strained p-doped SiGe layers. The calculations have been made for Ge concentrations in the range 0 to 30% and boron concentration in the range 1018 cm-3 to 3×1020 cm-3. Empirical expressions for the effective masses are given. These expressions and calculated values of the other parameters are convenient for use in computer codes for modeling device processing and performance. To validate the calculated values, we have compared them with the available experimental results. Good agreement between the calculated and the experimental values is found. INTRODUCTION SiGe heterostructures are of great topical interest [1-8]. As compared to Si BJTs, SiGe HBTs show better DC and AC characteristics. SiGe heterostructures are used in optical and optoelectronic devices [1]. Modeling the processing and performance of SiGe heterostructures requires reliable values of the effective masses, intrinsic carrier concentration ni, Fermi energy EF and bandgaps Eg of the Si and SiGe strained layers [1-5]. Since valence bands of Si and SiGe are warped and highly non-parabolic [4,5], values of the effective masses for p type SiGe depend on both dopant and Ge concentrations in a complicated manner. Bandgap, Fermi energy and ni depend on the values of the effective masses. We first calculate the effective masses. Using the calculated values of the effective masses, we calculate the other material parameters. Calculation of Carrier Concentration Effective Mass mcc and Fermi Energy EF, Manku and Nathan [5] have calculated the density of states hole mass mDOS as a function of energy E in the valence band of Si and SiGe for low values of energy. We have extrapolated the values to higher energies and fitted them numerically with the following expression, mDOS / m0 = M 0 + M 1 exp( − M 2 E ) .
(1)
Here m0 is the free electron mass and M0, M1, and M2 are the fitting parameters (see Table 1 in Appendix A). Carrier concentration effective mass mcc is defined by the following equation [5], ∞
mcc3 / 2 ( E F , T , x ) =
3/ 2 mDOS (kT )1/ 2 ( E , x ) E 1 / 2dE F1/ 2 ( E F / kT ) ∫0 1 + exp[( E − E F ) / kT ]
(2)
In Eq. (1) F1/2(EF/kT) is the Fermi-Dirac integral [4,5] and x is the Ge fraction. The carrier concentration effective mass mcc allows the calculation of the electronic properties of the layers AA4.24.1
treating the valence band as parabolic. Using effective density of states NV=2(2πmcckT/h2)3/2 we write [4], 2( 2πmcc kT NA ≅ p = h2
3/ 2
E F1/ 2 F . kT
(3)
The values of mcc and EF are calculated by solving simultaneously Eqs. (2) and (3) numerically for each doping and Ge concentration. The calculated
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