Bound States, Open Systemsand Gate Leakage Calculation in Schottky

Barriers

Dragica Vasileska

Time Independent Schrödinger Wave Equation - Revisited

2 2

2( ) ( ) ( )

2 *V x x E x

m x

K.E. Term P.E. Term

Solutions of the TISWE can be of two types, depending upon theProblem 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

-20 -10 0 10 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

distance [nm]

Ene

rgy

[eV

]

-20 -10 0 10 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

distance [nm]

ener

gy [e

V]

-100 -50 0 500

0.005

0.01

0.015

0.02

distance [nm]

ener

gy [e

V]

Rectangularconfinement

Parabolicconfinement

Triangularconfinement

Sine + cosine Hermite Polynomials Airy 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

Let where M is diagonal matrix with elements Li2

Where,

and

• Solve using the symmetric matrix H• Obtain the value of φ

where L is diagonal matrix with elements Li

(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 ResultsConventional MOS Capacitors with arbitrary crystallographic orientation

Silicon

Subband energy Valleys 1 and 2 Confinement

DirectionTransport, width and confinement Effective mass

Valleys 1 and 2

(001) mZ 0.19

(110) mZ 0.3189

(111) mZ 0.2598

(001) mZ 1.17

(110) mZ 0.2223

(111) mZ 0.1357

Conventional MOS Capacitors with arbitrary crystallographic orientationSilicon

Subband energy Valley 3

Confinement Direction

Transport, width and confinement Effective mass

Valley 3

(001) mZ 0.98(110) mZ 0.19(111) mZ 0.2598(001) mxy 0.0361(110) mxy 0.3724(111) mxy 0.1357

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-semiconductorstructure 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))

Subband population (all valleys)

Valley population (all valleys)

Strained Si MOS capacitors

Capacitance Vs gate voltage(Gilibert,2005)

More Complicated Structures- 3D Confinement -

Electron Density Potential Profile

Open Systems- Single Barrier Case -

V(x)

x

V0

E

Region 1(classically allowed)

Region 2(classically forbidden)

m

kE

2

21

2

mEV

2

22

2

0

Region 3(classically allowed)

L

ikLikLLL

ikLikLLL

FeEeikDeCeLL

FeEeDeCeLL

DCBAik

DCBA

)()(

)()(

)()()0()0(

)0()0(

'3

'2

32

'2

'1

21

Transfer Matrix Approach

FEMF

E

ek

iek

i

ek

iek

i

DC

DCMD

C

ki

ki

ki

ki

BA

LikLik

LikLik

2)()(

)()(

1

12

11

2

1

12

11

2

1

12

11

2

1

12

11

2

1

FEMF

EMMDCMB

A211

211

2 1)(

mA

EET 3

1

k

k

0

0.2

0.4

0.6

0.8

1

0.0 0.5 1.0 1.5 2.0

T(E

)

Energy [eV]

L=6 nm, V0=0.4 eV

m=6x10-32 kg

-0.2

0

0.2

0.4

0.6

0.8

1

0.0 5.0 10.0 15.0 20.0 25.0 30.0

E=0.2 eVE=0.6 eVT

(E)

Barrier thickness L [nm]

Tunneling Exampleand

Transmission Over the Barrier

Generalized Transfer Matrix Approach

rxxik

rxxik

r

lxxik

lxxik

l

xxebea

xxebeax

rlrrr

llll

,

,)(

)()(

)()(

ii

ii

lik

lik

ie

eP

0

0

rr

rrBi 11

11

2

1

rmmlr PBPBBPM 11221

Propagating domain

Interface between two boundaries

Transfer Matrix

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

VG = 0, VD > 0VG = 0, VD > 0

source

drainEC

dn/dE

dn/dE

Large potential barrier allows only few electrons to go from the source to the drain

(subthreshold conduction)

VG > VT , VD > 0VG > VT , VD > 0

source

drainEC

dn/dE

dn/dE

Smaller potential barrier allows a large number of electrons to go from the source

to the drain

PCPBT - tool

Top panel: barrier height = 0.3 eV, barrier width = 2 nm, well width = 4 nm, Middle panel: barrier height

= 0.3 eV, barrier width = well width = 4 nm; Bottom panel: barrier height = 0.3 eV, barrier width = 6

nm and well width = 4 nm.

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

Sharpresonance

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 bedeployed

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.

2/T T gf U L

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

Veff Routine

2D/3D Poisson equation solver

Veff Routine

Dopant atomsreal-space position

Dopant charge assigned to the

mesh nodes DeviceStructure

AppliedBias

Coulomb Force

MeshForce

Particle charge assigned to the mesh points (CIC, NEC)

ScatteringRates

Nominal Doping Density

Transmissioncoefficient

Vasileska et al., VLSI Design 13, pp. 75-78 (2001).

E

ai-1 ai ai+1

Vi

Vi+1

Vi-1 V(x)

Gate Current Calculation

• 1D Schrödinger equation:

• Solution for piecewise linear potential:

ExVdx

d

m

)(

2 2

22

)()( )2()1( iiiii BCAC

- Use linear potential approximation- Between two nodes, solutions to the Schrödinger equationare linear combination of Airy and modified Airy functions

1 2 1........T FI N BIM M M M M M

12

011

1 N

T

kT

Km

' '1 1

0 0

' '1 1

0 0

' '1 1

' '1 1

1 1[ (0) (0)] [ (0) (0)]

2 2

1 1[ (0) (0)] [ (0) (0)]

2 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

i i i i

FI

i i i i

N i N N i N N i N N i NBI

n N i N N i N N i N N i N

r rA A B B

ik ikM

r rA A B B

ik ik

r B ik B r B ik BM

r r A ik A r A ik A

'

' ''1 1

( ) ( )( ) ( )

( ) ( )( ) ( )

i i i ii i i i ii

i i i i i i ii i i i i

A Br B BM

r r A r Br A A

Matrices that satisfycontinuity of the wave-functions and the deri-vative of the wavefunctions

10-7

10-6

10-5

10-4

10-3

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Drain current Gate CurrentTunneling Current

Cur

rent

[A

/um

]

Gate Voltage [V]

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