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

Time Independent Schrödinger Wave Equation - Revisited K.E. Term P.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)

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

Rectangular confinement Parabolic confinement Triangular confinement Sine + cosine Hermite PolynomialsAiry Functions Bound states calculation lab on the nanoHUB

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

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.

Strained Silicon

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.)

Other Materials Bandstructure Model GaAs Bandstructure

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

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

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

Solve Poisson equation using LU decomposition method

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

Matrix transformation to preserve symmetry Let where M is diagonal matrix with elements L i 2 Where, and Solve using the symmetric matrix H Obtain the value of φ where L is diagonal matrix with elements L i (Tan,1990)

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

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.

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

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

Subset of Simulation Results Conventional MOS Capacitors with arbitrary crystallographic orientation Silicon Subband energy Valleys 1 and 2 Confinement Direction Transport, width and confinem ent Effective mass Valleys 1 and 2 (001)mZmZ 0.19 (110)mZmZ (111)mZmZ (001)mZmZ 1.17 (110)mZmZ (111)mZmZ

Conventional MOS Capacitors with arbitrary crystallographic orientation Silicon Subband energy Valley 3 Confinement Direction Transport, width and confineme nt Effective mass Valley 3 (001)mZmZ 0.98 (110)mZmZ 0.19 (111)mZmZ (001)m xy (110)m xy (111)m xy

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

Sheet charge density Vs gate voltage Capacitance Vs gate voltage

Average Distance from Interface Vs log(Sheet charge density)

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, (2008))

Subband population (all valleys) Valley population (all valleys)

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

More Complicated Structures - 3D Confinement - Electron DensityPotential Profile

Open Systems - Single Barrier Case -

Transfer Matrix Approach

Tunneling Example and Transmission Over the Barrier

Generalized Transfer Matrix Approach Propagating domain Interface between two boundaries Transfer Matrix

Example 1: Quantum Mechanical Reflections from the Front Barrier in MOSFETs PCPBT - tool

Example 2: Double Barrier Structure - Width of the Barriers on Sharpness of Resonances Sharp resonance

Example 3: Double Barrier Structure - Asymmetric Barriers T < 1

Example 4: Multiple Identical Barrier Structure - Formation of Bands and Gaps

Example 5: Implementation of Tunneling in Particle-Based Device Simulators Tarik Khan, PhD Thesis: Modeling of SOI MESFETs, ASU Tool to be deployed

Highlights Reduced junction capacitance. Absence of latchup. Ease in scaling (buried oxide need not be scaled). Compatible with conventional Silicon processing. Sometimes requires fewer steps to fabricate. Reduced leakage. Improvement in the soft error rate. Drawbacks Drain Current Overshoot. Kink effect Thickness control (fully depleted operation). Surface states. Welcome to the world of Silicon On Insulator SOI – The Technology of the Future

Principles of Operation of a SJT The SJT is a SOI MESFET device structure. Low-frequency operation of subthreshold CMOS (Lg > 1 μm due to transistor matching) It is a current controlled current source The SJT can be thought of as an enhancement mode MESFET. T.J. Thornton, IEEE Electron Dev. Lett., 8171 (1985)

2D/3D Monte Carlo Device Simulator Description Ensemble Monte Carlo transport kernel Ensemble Monte Carlo transport kernel Generate discrete impurity distribution Generate discrete impurity distribution Molecular Dynamics routine Molecular Dynamics routine 3D Poisson equation solver V eff Routine 2D/3D Poisson equation solver V eff Routine Dopantatoms real-space position Dopantcharge assigned to the mesh nodes Device Structure Applied Bias Coulomb Force Mesh Force Particle charge assigned to the mesh points (CIC, NEC) Scattering Rates Nominal Doping Density Transmission coefficient Vasileska et al., VLSI Design 13, pp (2001).

E a i-1 a i a i+1 V i V i+1 V i-1 V(x) Gate Current Calculation 1D Schrödinger equation: Solution for piecewise linear potential: - Use linear potential approximation - Between two nodes, solutions to the Schrödinger equation are linear combination of Airy and modified Airy functions

Matrices that satisfy continuity of the wave- functions and the deri- vative of the wavefunctions

Transfer Characteristic of a Schottky Transistor

How is the tunneling current calculated? At each slice along the channel we calculate the transmission coefficient versus energy If an electron goes towards the interface and if its energy is smaller than the barrier height, then a random number is generated If the random number is such that: –r > T(E), where E is the energy of the particle, then that transition is allowed and the electron contributes to gate leakage current –r < T(E), where E is the energy of the particle, that that transition is forbidden and the electron is reflected back