Bound States, Open Systems and Gate Leakage Calculation in Schottky Barriers Dragica Vasileska

1. Bound States, Open Systemsand Gate Leakage Calculation in Schottky BarriersDragica Vasileska

6. Time Independent Schrödinger Wave Equation - Revisited P.E. Term K.E. Term Solutions of the TISWE can be of two types, depending upon the Problem we are solving: - Closed system (eigenvalue problem) - Open system (propagating states)

7. Bound States - Eigenvalue Problem -

8. Closed Systems • Closed systems are systems in which the wavefunction is localized due to the spatial confinement. • The most simple closed systems are: • Particle in a box problem • Parabolic confinement • Triangular Confinement

21. Rectangular confinement Parabolic confinement Triangular confinement Hermite Polynomials Airy Functions Sine + cosine Bound states calculation lab on the nanoHUB

22. Summary of Quantum Effects Schred Second Generation – Gokula Kannan - • Band-Gap Widening • Increase in Effective Oxide Thickness (EOT)

23. Motivation for developing SCHRED V2.0- Alternate Transport Directions - • Conduction band valley of the material has three valley pairs • In turn they have different effective masses along the chosen • crystallographic directions • Effective masses can be computed assuming a 3 valley conduction • band model.

24. Strained Silicon

25. Arbitrary Crystallographic Orientation • The different effective masses in the • Device co-ordinate system (DCS) along • different crystallographic directions • can be computed from the ellipsoidal • Effective masses ( A Rahman et al.)

26. Other Materials Bandstructure Model GaAs Bandstructure

27. Charge Treatment Semi-classical Model Maxwell Boltzmann Fermi-Dirac statistics Quantum-Mechanical Model Constitutive Equations:

28. Self-Consistent Solution 1D Poisson Equation: LU Decomposition method (direct solver) 1D Schrodinger Equation: Matrix transformation to make the coefficients matrix symmetric Eigenvalue problem is solved using the EISPACK routines Full Self-Consistent Solution of the 1D Poisson and the 1D Schrodinger Equation is Obtained

29. 1D Poisson Equation • Discretize 1-D Poisson equation on a non-uniform generalized mesh • Obtain the coefficients and forcing function using 3-point finite • difference scheme

30. Solve Poisson equation using LU decomposition method

31. 1D Schrodinger Equation • Discretize 1-D Schrodinger equation on a non-uniform mesh • Resultant coefficients form a non-symmetric matrix

32. Matrix transformation to preserve symmetry Let Let where M is diagonal matrix with elements Li2 Where, where L is diagonal matrix with elements Li and • Solve using the symmetric matrix H • Obtain the value of φ (Tan,1990)

33. 1D Schrodinger Equation • symmetric tridiagonal matrix solvers (EISPACK) • Solves for eigenvalues and eigenvectors • Computes the electron charge density

34. Full Self-Consistent Solution of the 1D Poisson and the 1D Schrodinger Equation • The 1-D Poisson equation is solved for the potential • The resultant value of the potential is used to solve the 1-D Schrodinger equation using EISPACK routine. • The subband energy and the wavefunctions are used to solve for the electron charge density • The Poisson equation is again solved for the new value of potential using this quantum electron charge density • The process is repeated until a convergence is obtained.

35. Other Features Included in the Theoretical Model Partial ionization of the impurity atoms Arbitrary number of subbands can be taken into account The simulator automatically switches from quantum-mechanical to semi-classical calculation and vice versa when sweeping the gate voltage and changing the nature of the confinement

36. Outputs that Are Generated Conduction Band Profile Potential Profile Electron Density Average distance of the carriers from the interface Total gate capacitance and its constitutive components Wavefunctions for different gate voltages Subband energies for different gate voltages Subband population for different gate voltages

37. Subset of Simulation Results Conventional MOS Capacitors with arbitrary crystallographic orientation Silicon Subband energy Valleys 1 and 2

38. Conventional MOS Capacitors with arbitrary crystallographic orientation Silicon Subband energy Valley 3

39. Subband population – Valleys 1 and 2 Subband population – Valley 3

40. Sheet charge density Vs gate voltage Capacitance Vs gate voltage

41. Average Distance from Interface Vs log(Sheet charge density)

42. GaAs MOS capacitors Capacitance Vs gate voltage (“Inversion capacitance-voltage studies on GaAs metal-oxide-semiconductor structure using transparent conducting oxide as metal gate”, T.Yang,Y.Liu,P.D.Ye,Y.Xuan,H.Pal and M.S.Lundstrom,APPLIED PHYSICS LETTERS 92, 252105 (2008))

43. Valley population (all valleys) Subband population (all valleys)

44. Strained Si MOS capacitors Capacitance Vs gate voltage (Gilibert,2005)

45. More Complicated Structures- 3D Confinement - Electron Density Potential Profile

46. OPEN Systems