Bound States Open Systems
description
Transcript of Bound States Open Systems
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Bound States, Open Systemsand Gate Leakage Calculation in Schottky
Barriers
Dragica Vasileska
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Time Independent Schrödinger Wave Equation - Revisited
2 2
2 ( ) ( ) ( )2 *
V x x E xm 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)
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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
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-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
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Summary of Quantum Effects
• Band-Gap Widening• Increase in Effective Oxide Thickness (EOT)
Schred Second Generation – Gokula Kannan -
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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.
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Strained Silicon
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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.)
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Other Materials Bandstructure Model
GaAs Bandstructure
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Charge Treatment• Semi-classical Model
– Maxwell Boltzmann– Fermi-Dirac statistics
• Quantum-Mechanical Model Constitutive Equations:
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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
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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
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• Solve Poisson equation using LU decomposition method
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1D Schrodinger Equation
• Discretize 1-D Schrodinger equation on a non-uniform mesh
• Resultant coefficients form a non-symmetric matrix
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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)
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1D Schrodinger Equation
• symmetric tridiagonal matrix solvers (EISPACK)• Solves for eigenvalues and eigenvectors• Computes the electron charge density
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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.
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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
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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
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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
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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
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Subband population – Valley 3Subband population – Valleys 1 and 2
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Sheet charge density Vs gate voltage
Capacitance Vs gate voltage
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Average Distance from Interface Vs log(Sheet charge density)
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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))
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Subband population (all valleys)
Valley population (all valleys)
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Strained Si MOS capacitors
Capacitance Vs gate voltage(Gilibert,2005)
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More Complicated Structures- 3D Confinement -
Electron Density Potential Profile
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Open Systems- Single Barrier Case -
V(x)
x
V0E
Region 1(classically allowed)
Region 2(classically forbidden)
mk
E2
21
2
mEV
2
22
2
0
Region 3(classically allowed)
L
ikLikLLL
ikLikLLL
FeEeikDeCeLLFeEeDeCeLL
DCBAikDCBA
)()()()(
)()()0()0()0()0(
'3
'2
32
'2
'1
21
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Transfer Matrix Approach
FEMF
E
ekieki
ekieki
DC
DCMD
C
ki
ki
ki
ki
BA
LikLik
LikLik
2)()(
)()(
1
1211
21
1211
21
1211
21
1211
21
FEMF
EMMDCMB
A211
211
2 1)(mA
EET 3
1
kk
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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
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Generalized Transfer Matrix Approach
rxxik
rxxik
r
lxxik
lxxik
l
xxebeaxxebea
xrlrrr
llll
,,
)( )()(
)()(
ii
ii
lik
lik
i ee
P0
0
rrrr
Bi 1111
21
rmmlr PBPBBPM 11221
Propagating domain
Interface between two boundaries
Transfer Matrix
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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
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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
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Example 3: Double Barrier Structure - Asymmetric Barriers
T < 1
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Example 4: Multiple Identical Barrier Structure - Formation of Bands and Gaps
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Example 5: Implementation of Tunneling in Particle-Based Device Simulators
• Tarik Khan, PhD Thesis: Modeling of SOI MESFETs, ASU
Tool to bedeployed
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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
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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)
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2D/3D Monte Carlo Device Simulator Description
Ensemble Monte Carlo transport
kernel
Ensemble Monte Carlo transport
kernel
Generate discrete impurity distributionGenerate 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 DeviceStructureAppliedBias
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).
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E
ai-1 ai ai+1
Vi
Vi+1
Vi-1 V(x)
Gate Current Calculation
• 1D Schrödinger equation:
• Solution for piecewise linear potential:
ExVdxd
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
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1 2 1........T FI N BIM M M M M M
120
11
1 NT
kT
K m
' '1 1
0 0
' '1 1
0 0' '
1 1' '
1 1
1 1[ (0) (0)] [ (0) (0)]2 21 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
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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/u
m]
Gate Voltage [V]
Transfer Characteristic of a Schottky Transistor
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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