IEE5328 Nanodevice Transport Theory and Computational Tools

1. IEE5328 Nanodevice Transport Theory and Computational Tools (Advanced Device Physics with emphasis on hands-on calculations) Lecture 7: Effective Mobility in 2DEG and 2DHG of Long-Channel Thick Gate-Oxide Planar Bulk MOSFETs: Microscopic Calculation Prof. Ming-Jer Chen Dept. Electronics Engineering National Chiao-Tung University May 1, 2013 IEE5328 Prof. MJ Chen NCTU

2. Two-Dimensional Hole Gas IEE5328 Prof. MJ Chen NCTU

3. Thick Oxides Stress No Stress IEE5328 Prof. MJ Chen NCTU

4. IEE5328 Prof. MJ Chen NCTU

5. Kubo-Greenwood Formula The mobility formula in electron case is no longer valid for the hole case due to the failure of the effective mass approximation. Thus, the hole mobility formula as derived from the Boltzmann transport equation (BTE) must be used: vxμ is the group velocity of subband μ along x-direction and f0is the equilibrium Fermi distribution.

6. Kubo-Greenwood Formula 1st Subband Si (001) @ 300K FS=1MV/cm

7. ; ; • Physical Models • Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering ω Longitudinal Transverse ΔE≈61.2meV H’Hole-AC Phonon Optical ΔE≈1meV within 1/2 Brillouin zone H’Hole-OP Phonon Acoustic Si Phonon Dispersion Phonon wave vector q

8. ; ; • Physical Models • Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering The acoustic deformation potential Dac, is strongly connected to Bir-Pikus deformation potentials. According to Lawaetz, Dac can be formulated as H’Hole-AC Phonon Small vibration term Elastic scattering Isotropic approximation c11, c12, and c44 are the elastic coefficients

9. ; ; • Physical Models • Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering According to Wiley and Costato and Reggiani, the optical deformation potential Dop can have the following formalism: H’Hole-OP Phonon Small vibration term Inelastic scattering (61.2meV) Average sound velocity Isotropic approximation ωop : optical phonon frequency; nop : Bose occupation factor of optical phonons

10. ; ; • Physical Models • Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering Gate H’Hole-SR p+ p+ Small vibration term Inversion Layer n-type Substrate Elastic scattering Anisotropic scattering H’Hole-SR

11. ; ; • Physical Models • Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering Anisotropic surface roughness scattering rate

12. ; ; • Physical Models • Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering Isotropic SR scattering Anisotropic SR scattering *A. T. Pham, C. Jungemann, and B. Meinerzhagen, “Microscopic modeling of hole inversion layer mobility in unstrained and uniaxially stressed Si on arbitrarily oriented substrates,” Solid-State Electronics, vol. 52, pp. 1437-1442, May 2008.

13. ; ;

14. Two-Dimensional Electron Gas IEE5328 Prof. MJ Chen NCTU

15. Thick Oxides Under the momentum relaxation time approximation, IEE5328 Prof. MJ Chen NCTU

16. Thick Oxides Gaussian model: Exponential model: IEE5328 Prof. MJ Chen NCTU

17. Thin Oxides IEE5328 Prof. MJ Chen NCTU

18. Thin Oxides IEE5328 Prof. MJ Chen NCTU

19. IEE5328 Prof. MJ Chen NCTU

20. Inversion layer mobility in thick oxide MOSFETs can be limited to three primary scattering mechanisms:

21. Ionized Impurity Scattering Model • Coulomb-Limited Mobility Model due to Ionized Impurity Atoms in the Substrate Region - The scattering rate of ionized impurity in 3-D case can be presented by: - However, it is not the 2-D electron gas inside the MOSFET. : the ionized impurity concentration : the Debye length can be written as , where n0 is the 3-D density of the mobile carrier Nano Electronics Physics Lab @ NCTU

22. Ionized Impurity Scattering Model • Coulomb-Limited Mobility Model due to Ionized Impurity Atoms in the Substrate Region at 2-D case - The momentum conservation in the z-direction of the 3-D case at the scattering process of 2-D carriers should be replaced by the integral as: Therefore, the scattering rate of ionized impurity scattering in 2-D case from mth subband to nth subband can be expressed as: where H2D and H3D are the matrix elements for 2-D and 3-D scattering, respectively. • :The average inversion layer thickness • :the density of states for two dimensions • :the density of states for three dimensions Nano Electronics Physics Lab @ NCTU

23. Phonon Scattering Model - For intravalley phonon scattering model, the momentum-relaxation rate in the subband mth to nth: The total scattering rate in mthsubband is determined by summing up within all the subbands: Dac : deformation potential due to acoustic phonons Sl : sound velocity ρ : crystal density Wm,n: the form factor determined by the wave-functions of the mth subband and nthsubbands Nano Electronics Physics Lab @ NCTU

24. Phonon Scattering Model - For the intervalley phonon scattering model: (Incorrect Version) (1). From mth subband in twofold valleys to the nth subband in fourfold valleys: (2). From mth subband in fourfold valleys to the nth subband in twofold valleys: (3). From mth subband in fourfold valleys to the nth subband in fourfold valleys: (4). From mth subband in twofold valleys to the nth subband in twofold valleys : where Ek and Dk are deformation energy and potential at kth intervalley phonon, and Nk is the occupation number of kth intervalley phonon. Nano Electronics Physics Lab @ NCTU

25. Surface Roughness Scattering Model - The scattering rate for a Gaussian function is described as: Δ : rms height of the amplitude of surface roughness l: correlation length of surface roughness Nano Electronics Physics Lab @ NCTU

26. Electron Mobility Model • Derivation of Two-Dimensional Mobility in the Universal Mobility Region - The scattering rates of the twofold and fourfold valley: The electron mobility by using the average energy within the 2DEG in the relaxation time approximation can be given as The total universal mobility averaged over the subband occupation is described by Nano Electronics Physics Lab @ NCTU

27. Electron Mobility Model • The physical parameters for phonon and surface-roughness electron mobility used for Si in this work Nano Electronics Physics Lab @ NCTU

28. Electron Mobility Model • The Universal mobility Curve - Universal electron mobility includes phonon scatteringand surface roughness scattering, which is independent of process parameters, especially when plotted versus of high effective field (Eeff). • : total inversion layer charge density • :the surface concentration of the depletion • charge Nano Electronics Physics Lab @ NCTU

29. Electron Effective Mobility:Thick Oxides Current NEP-electron-mobility simulator was developed under the parabolic band approximation, the isotropic scattering approximation, the elastic scattering approximation (surface roughness and impurity scattering), and the momentum relaxation approximation. IEE5328 Prof. MJ Chen NCTU

30. Ionized ImpurityScattering The perturbing potential is the screened Coulomb potential: r: the distance from the scattering center LD: Debye length n: 3-D density of the mobile carriers and equal to Ninv/Zav Through the Fermi’s Golden Rule, the relaxation time due to ionized impurity scattering is: NI: 3-D impurity concentration, which is about equal to Nsub Here, q means the elementary charge.

31. ImpurityScattering The momentum relaxation time for scattering of 2-D carriers from uth subband to vth subband: P.S. Only consider intra-subband K. Hirakawa and H. Sakaki, “Mobility of the two-dimensional electron gas at selectively doped n-type AlxGa1-xAs/GaAs heterojunctions with controlled electron concentrations,” Phys. Rev. B, Condens. Matter, vol. 33, no. 12, 8291-8303, Jun. 1986. M. Lundstrom, “Fundamentals of carrier transport,” Cambridge University Press, 2000.

32. PhononScattering 1.intravalley phonon scattering model: (acoustic phonon) the momentum-relaxation time for scattering from the uth subband to the vth subband kB : Boltzmann constant , Dac : deformation potential due to acoustic phonons ρ : crystal density , Sι: sound velocity U(x): step function S. Takagi, J. L. Hoyt, J. J. Welser, and J. F. Gibbons, “Comparative study of phonon-limited mobility of two-dimensional electrons in strained and unstrained Si metal-oxide-semiconductor field-effect transistors,” J. Appl. Phys., vol. 80, no. 3, pp. 1567-1577, Aug. 1996. D.Esseni, A. Abramo, L. Selmi, and E. Sangiorgi, “Physically Based modeling of Low Field Electron Mobility in Ultrathin Single- and Double-Gate SOI n-MOSFETs”, IEEE Trans. Electron Devices, vol. 50, no.12, pp.2445-2455, 2003. K. Uchida, A. Kinoshita, and M. Saitoh, “Carrier transport in (110) nMOSFETs: Subband structures, non parabolicity, mobility characteristics, and uniaxial stress engineering,” in IEDM Tech. Dig., pp.1019-1021, 2006.

33. PhononScattering model (Correct Version) 2.intervalley phonon scattering model: (optical phonon) (a) From uth subband in twofold valleys to the vth subband in twofold valleys: (b)From uth subband in twofold valleys to the vth subband in fourfold valleys: (c)From uth subband in fourfold valleys to the vth subband in twofold valleys: (d)From uth subband in fourfold valleys to the vth subband in fourfold valleys:

34. g-type Acoustic phonon D4 D4 f-type Acoustic phonon f-type Acoustic phonon f-type f-type D2 f-type f-type f-type Acoustic phonon g-type D4 D4 g-type f-type Acousticphonon

35. Surface RoughnessScattering Yamakawa, H. Ueno, K. Taniguchi, C. Hamaguchi, K. Miyatsuji, K. Masaki, and U. Ravaioli, “Study of interface roughness dependence of electron mobility in Si inversion layers using the Monte Carlo method,” J. Appl. Phys., vol. 79, no. 2, pp. 911-916, Jan. 1996. P.S. Only consider intra-subband D.Esseni, “On the Modeling of Surface Roughness Limited Mobility in SOI MOSFETs and its Correlation to the Transistor Effective Field”, IEEE TED, Vol.51 NO.3, pp.394-401, 2004.

36. Three-Dimensional Stress Effect on Electron Mobility