Bound States, Open Systems
and Gate Leakage Calculation in Schottky
Barriers
Dragica Vasileska
Time Independent Schrödinger
Wave Equation - Revisited


 V ( x) ( x)  E ( x)
2
2m * x
2
2
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
0.35
0.4
0.3
0.35
0.2
0.15
0.1
0.015
energy [eV]
energy [eV]
Energy [eV]
0.02
0.3
0.25
0.25
0.2
0.15
0.01
0.005
0.1
0.05
0
-20
Triangular
confinement
0.05
-10
0
10
20
0
-20
-10
0
10
20
distance [nm]
distance [nm]
Sine + cosine
Hermite Polynomials
0
-100
-50
0
50
distance [nm]
Airy Functions
Bound states calculation lab on the nanoHUB
Schred Second Generation
– Gokula Kannan Summary of Quantum Effects
• Band-Gap Widening
• Increase in Effective Oxide Thickness (EOT)
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 Results
Conventional MOS Capacitors with arbitrary crystallographic orientation
Silicon
Subband energy
Valleys 1 and 2
Confinement Transport, Valleys
Direction
width and 1 and 2
confinem
ent
Effective
mass
(001)
mZ
0.19
(110)
mZ
0.3189
(111)
(001)
mZ
mZ
0.2598
1.17
(110)
mZ
0.2223
(111)
mZ
0.1357
Conventional MOS Capacitors with arbitrary crystallographic orientation
Silicon
Subband energy
Valley 3
Confinement Transport,
Direction
width and
confineme
nt
Effective
mass
(001)
mZ
(110)
mZ
(111)
mZ
(001)
mxy
Valley 3
(110)
mxy
0.3724
(111)
mxy
0.1357
0.98
0.19
0.2598
0.0361
Subband
Subbandpopulation
population– –Valleys
Valley 13 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, 252105 (2008))
Valley population
(all valleys)
Subband 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)
Region 1
(classically allowed)
E
Region 2
(classically forbidden)
Region 3
(classically allowed)
V0
 2 k12
E
2m
 2  22
V0  E 
2m
x
L
 1 ( 0)   2 ( 0) 
A B  C  D
1' (0)   '2 (0)  ik ( A  B )    (C  D )
 L
L
ikL
 ikL
 2 ( L)   3 ( L)  Ce
 De  Ee  Fe
'
'

 2 ( L)   3 ( L)    Ce
 L
 De
L
  ik Ee
ikL
 Fe
 ikL

Transfer Matrix Approach
 1
 
1 
1

i
1

i




2
k
2
k
A


 C   M  C 
  
1  D
 B   1 
 1
   D 
 
1  i 
1  i 
 2 
k 2
k 
1 
k  (ik   ) L 1 
k   (ik   ) L 
1  i e
 1  i e

2

2



C    
E   M 2 E 
 D   1 
 F 
k  (ik   ) L 1 
k   (ik   ) L   F 
1  i e
 2 1  i  e

2







 A  M C   M M  E   M  E 
1  D
1 2 F 
 B 
 F 
 
 
E
T (E) 
A
2

1
m11
2
k3
k1
Tunneling Example
and
Transmission Over the Barrier
1
1
0.8
0.8
0.6
T(E)
T(E)
0.6
E=0.2 eV
E=0.6 eV
0.4
L=6 nm, V =0.4 eV
0
0.4
-32
m=6x10
kg
0.2
0.2
0
0
0.0
0.5
1.0
Energy [eV]
1.5
2.0
-0.2
0.0
5.0
10.0
15.0
20.0
Barrier thickness L [nm]
25.0
30.0
Generalized Transfer Matrix
Approach
 al e ikl ( x xl )  bl e ikl ( x xl ) , x  xl
 ( x)   ikr ( x xr )
ikr ( x  xrl )
a
e

b
e
, x  xr
r
 r
e ikili
Pi  
 0
0 

e ikili 
Propagating domain
1 1  r 1  r 
Bi  
Interface between two boundaries

2 1  r 1  r 
r  Ml  Pm Bm1  B2 P2 B1 P1 r
Transfer Matrix
Example 1: Quantum Mechanical Reflections
from the Front Barrier in MOSFETs
PCPBT - tool
dn/dE
VVGG==0,
0,VVDD>>00
dn/dE
source
Large potential barrier allows only few
electrons to go from the source to the drain
(subthreshold conduction)
VVGG>>VVTT,,VVDD>>00
drain
EC
dn/dE
dn/dE
source
Smaller potential barrier allows a large
number of electrons to go from the source
to the drain
drain
EC
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
SOI–The Technology of the Future
Welcome to the world of Silicon On Insulator
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.
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)
fT  UT / L2g
• 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
Nominal Doping Density
Generate
Generatediscrete
discrete
impurity
impuritydistribution
distribution
Dopant charge
assigned to the
mesh nodes
Dopant atoms
real-space
position
Molecular
Molecular
Dynamics
Dynamicsroutine
routine
2D/3D
Poisson
3D Poisson
equation
equationsolver
solver
VVeff Routine
eff Routine
Mesh
Force
Coulomb
Force
Applied
Bias
Particle charge
assigned to the
mesh points (CIC, NEC)
Ensemble
EnsembleMonte
Monte
Carlo
transport
Carlo transport
kernel
kernel
Vasileska et al., VLSI Design 13, pp. 75-78 (2001).
Device
Structure
Scattering
Rates
Transmission
coefficient
Gate Current Calculation
• 1D Schrödinger equation:
Vi+1
 2 d 2

 V ( x)  E
2
2m dx
• Solution for piecewise
linear potential:
 i  Ci(1) Ai ( )  Ci( 2 ) Bi ( )
Vi
E
Vi-1
V(x)
ai-1
ai
ai+1
- Use linear potential approximation
- Between two nodes, solutions to the Schrödinger equation
are linear combination of Airy and modified Airy functions
MT  M FI M1M 2 ........M N 1M BI
r1
1
 2 [ Ai (0)  ik
0
M FI  
1
r1
 [ Ai (0) 
ik0
2
r1 ' 
1
'
Ai (0)] [ Bi (0)  Bi (0)]
2
ik0

r1 ' 
1
'
Ai (0)] [ Bi (0)  Bi (0)]
2
ik0

'
'
  rN Bi ( N )  ik N 1Bi ( N ) rN Bi ( N )  ik N 1Bi ( N ) 

M BI  
rn  r A' ( )  ik A ( ) r A' ( )  ik A ( ) 
N 1 i N
N i N
N 1 i N 
 N i N
'
Bi (i ) 
  ri Bi (i )  Bi (i )   Ai (i )

Mi  

'
ri  r A ( ) A ( )   ri 1 Ai' (i ) ri 1Bi' (i ) 
i i 
 i i i
k
T  N 1
K0
1
T 2
m11
Matrices that satisfy
continuity of the wavefunctions and the derivative of the wavefunctions
Transfer Characteristic of a Schottky Transistor
Current [A/um]
10
-3
10
-4
10
-5
10
-6
10
-7
0.1
Drain current
Gate Current
Tunneling Current
0.2
0.3
0.4
0.5
Gate Voltage [V]
0.6
0.7
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