Download Imaginary Bandstructure, Si tunneling

Imaginary Band Structure and Its
Role in Calculating Transmission
Probability in Semiconductors
Jamie Teherani
Collaborators: Paul Solomon (IBM), Mathieu Luisier (Purdue)
Advisors: Judy Hoyt, Dimitri Antoniadis
Acknowledgements: Steve Laux (IBM)
1
Outline
• Introduction
– Quantum mechanics refresher
– Imaginary k-vector and WKB approximation
– Imaginary band structure
• Applications of Imaginary Band Structure
– Directional dependence of GIDL in MOSFETs
– Strain dependence of tunneling current in Si
tunnel diodes
2
Outline
• Introduction
– Quantum mechanics refresher
– Imaginary k-vector and WKB approximation
– Imaginary band structure
• Applications of Imaginary Band Structure
– Directional dependence of GIDL in MOSFETs
– Strain dependence of tunneling current in Si
tunnel diodes
3
QM Transmission Through a Square Barrier
Classical Analogy
Potential
Energy
Kinetic
Energy
Boundary Conditions
Trial Solution
4
Imaginary Momentum Vector
V(x)
V0
Plane Wave Solution
=∙
I
II
III
potential
barrier
E
k is real
k is imag.
k is real
x
Momentum in the
tunneling direction
=
2∗ − ℏ
In the basic example of a plane wave tunneling through a square potential barrier, the
momentum in the tunneling direction is imaginary inside the barrier which results in
a decaying exponential.
5
Imaginary Momentum Vector
V(x)
V0
Plane Wave Solution
=∙
I
II
III
potential
barrier
E
k is real
k is imag.
k is real
x
Momentum in the
tunneling direction
=
2∗ − ℏ
≡
= ∙ In the basic example of a plane wave tunneling through a square potential barrier, the
momentum in the tunneling direction is imaginary inside the barrier which results in
a decaying exponential.
6
Tunnel Probability
• Start with the WKB approximation for
tunneling probability
~ −2 Energy
Action Integral
a
b
x
7
Transmission Coefficient for PN Junction
E
Decaying
Exponential
conduction band
Transmitted
Wave
Initial
Wave
Energy
potential barrier
Ec
valence band
tunneling of electrons
Ev
a
b
a
x
x
b
• Start with the WKB approximation for
transmission probability
≡
~ −2 Action Integral
8
Constant Field Approximation
• For a constant field across the junction, action
integral can be rewritten as
= !
• Where F is the field, q is the charge of an
electron, and E is energy. Since:
! = 9
Constant Field Approximation
• For a constant field across the junction, action
integral can be rewritten as
= !
• Where F is the field, q is the charge of an
electron, and E is energy. Since:
! = Assumes parabolic band
=
2∗ − −ℏ
10
Constant Field Approximation
• For a constant field across the junction, action
integral can be rewritten as
Can be calculated from
= imaginary band structure
!
• Where F is the field, q is the charge of an
electron, and E is energy. Since:
! = Assumes parabolic band
=
2∗ − −ℏ
11
Outline
• Introduction
– Quantum mechanics refresher
– Imaginary k-vector and WKB approximation
– Imaginary band structure
• Applications of Imaginary Band Structure
– Directional dependence of GIDL in MOSFETs
– Strain dependence of tunneling current in Si
tunnel diodes
12
Full Band Structure of Silicon
Abhijeet Paul et al., “Band Structure Lab,” May-2006. [Online]. Available: http://nanohub.org/resources/1308.
13
Full Band Structure of Silicon
Abhijeet Paul et al., “Band Structure Lab,” May-2006. [Online]. Available: http://nanohub.org/resources/1308.
14
Full Band Structure of Silicon
The electric field provides energy for tunneling from valence band to conduction band.
Valence Bands
Conduction Bands
Abhijeet Paul et al., “Band Structure Lab,” May-2006. [Online]. Available: http://nanohub.org/resources/1308.
15
Imaginary Band Structure
[100] Tunneling in Silicon
Imaginary
k-vector, Real
k-vector
k┴≠ 0
k┴=0
k┴
i
i
k||
(X-valley)
k||
S. Laux, “Computation of Complex Band Structures in Bulk and Confined Structures,” in Computational Electronics, 2009.
IWCE '09. 13th International Workshop on, pp. 1-2, 2009.
16
Imaginary Band Structure
[100] Tunneling in Silicon
Imaginary
k-vector, Real
k-vector
k┴≠ 0
k┴=0
k┴
i
i
k||
(X-valley)
k||
S. Laux, “Computation of Complex Band Structures in Bulk and Confined Structures,” in Computational Electronics, 2009.
IWCE '09. 13th International Workshop on, pp. 1-2, 2009.
17
Imaginary Band Structure
[100] Tunneling in Silicon
Imaginary
k-vector, Real
k-vector
k┴≠ 0
k┴=0
Action Integral
k┴
i
i
k||
(X-valley)
k||
S. Laux, “Computation of Complex Band Structures in Bulk and Confined Structures,” in Computational Electronics, 2009.
IWCE '09. 13th International Workshop on, pp. 1-2, 2009.
18
Outline
• Introduction
– Quantum mechanics refresher
– Imaginary k-vector and WKB approximation
– Imaginary band structure
• Applications of Imaginary Band Structure
– Directional dependence of GIDL in MOSFETs
– Strain dependence of tunneling current in Si
tunnel diodes
19
GIDL in MOSFETs
(Gate Induced Drain Leakage)
n-MOSFET (On State)
n+ Si
p Si
n-MOSFET
Drain
n+ Si
x
thermionic emission
of electrons
Energy
Source
(1a)
Gate
Oxide
(1b)
n-MOSFET
tunneling of electrons
Energy
(1c)
n-MOSFET (Off State)
n-TFET
n-TFET (On State)
Source
(2a)
p+ Si or SiGe
Gate
Oxide
n Si
n-TFET
Drain
n+ Si
x
Energy
conduction band
tunneling of electrons
(2b)
valence band
20
Question:
What is the effect of channel
orientation on GIDL?
D
(100) Wafer
Surface
S
S
<100>
D
Notch
<110>
21
GIDL Experiment
Simplified Tunneling Equation
!
General
expression:
. 2 For a triangular
barrier:
4 1 2.4
. 2567789
3 !
[100] Tunneling
4 lobes with transverse electron mass
z
x
[100]
[110]
Tunneling Direction
Lobes with transverse
electron mass
[100]
4
y
[110] Tunneling
2 lobes with transverse electron mass
z
[110]
2
x
[111]
0
[100]
[110]
y
P. M. Solomon, S. E. Laux, L. Shi, J. Cai, and W. E. Haensch, “Experimental and theoretical explanation for the orientation
dependence gate-induced drain leakage in scaled MOSFETs,” Device Research Conference, 2009.
22
GIDL Experiment Con’t
• <110> channel MOSFETs have about 10x GIDL
current compared to <100> channel MOSFETs
P. Gilbert et al., “Technology Elements of a Common Platform Bulk Foundry Offering (Invited),” in Electron Devices
Meeting, 2007. IEDM 2007. IEEE International, pp. 259-262, 2007.
23
GIDL Experiment Con’t
Tunneling Direction
Lobes with transverse
electron mass
[100]
4
(Most Current)
[110]
2
[111]
0
(Least current)
• <110> channel MOSFETs have about 10x GIDL
current compared to <100> channel MOSFETs
P. Gilbert et al., “Technology Elements of a Common Platform Bulk Foundry Offering (Invited),” in Electron Devices
Meeting, 2007. IEDM 2007. IEEE International, pp. 259-262, 2007.
24
GIDL Experiment Con’t
10000
001-2
011-2
111-2
001-3
011-3
111-3
001-4
011-4
111-4
001-5
011-5
111-5
1000
2
Tunneling
Current
J (A/cm )
100
[011]
[111]
10
[001]
1
0.1
Vertical tunnel diodes were fabricated on
(100), (110), and (111) Si substrates to
study tunneling along these directions.
0.01
Increasing V
1E-3
3
4
5
6
7
8
9
10
11
WT (nm)
Tunneling width determined by CV
doping profile extraction and simulation.
P. M. Solomon, S. E. Laux, L. Shi, J. Cai, and W. E. Haensch, “Experimental and theoretical explanation for the orientation
dependence gate-induced drain leakage in scaled MOSFETs,” Device Research Conference, 2009.
25
GIDL Experiment Con’t
. −2 !
• Dashed lines are curves
calculated using a band
edge effective mass
– Band edge mass is
insufficient in accurately
calculating action integral
• Solid lines are calculated
from the imaginary band
structure
• Imaginary band structure is
especially needed in
analyzing tunneling in the
[110] and [111] direction
S. Laux, “Computation of Complex Band Structures in Bulk and Confined Structures,” in Computational Electronics, 2009.
IWCE '09. 13th International Workshop on, pp. 1-2, 2009.
26
Outline
• Introduction
– Quantum mechanics refresher
– Imaginary k-vector and WKB approximation
– Imaginary band structure
• Applications of Imaginary Band Structure
– Directional dependence of GIDL in MOSFETs
– Strain dependence of tunneling current in Si
tunnel diodes
27
Strain Dependence of Tunneling in
Vertical Silicon Tunnel Diodes
Objective: To determine the relative change in tunneling current due to applied uniaxial stress.
Procedure: Experimentally measure the uniaxially stressed vertical Si tunnel diodes and
compare results to theoretical calculations.
Mechanically
Applied
Uniaxial
Stress
Tunneling
Current
Vertical tunnel diodes were fabricated on
(100), (110), and (111) Si substrates to
study tunneling along these directions.
The uniaxial bending apparatus was
designed by the S. Thompson group at
University of Florida.
28
IV Plot for (001) Wafer with [100] Stress
1.E-04
1.E-05
1.E-06
z
Thermal Regime
(63mV/decade)
Uniaxial
Stress
x [100]
[110]
y
1.E-07
Current (A)
Tunneling
Current
1.E-08
Tunneling
Regime
1.E-09
1.E-10
3.E-08
Compressive
Strain
2.E-08
1.E-11
1.E-12
Trap Limited
Regime
1.E-13
Tensile
Strain
1.E-08
0.7
0.75
0.8
1.E-14
-1
Forward
Bias
-0.5
0
0.5
Voltage on N-Region (Volts)
1
1.5
Reverse
Bias
29
Relative Change in Current Compared to 0% Strain
(001) Wafer with [100] Stress
20
Percent Change in Current Compared to 0% Strain Case
15
-0.13% Strain
10
Increasing
Compressive Strain
-0.065% Strain
0% Strain
5
0% Strain (remeasured)
0% Strain (remeasured 2)
0.065% Strain
0
0.13% Strain
0.195% Strain
-5
0.195% Strain (remeasured)
0.26% Strain
0.325% Strain
-10
0.39% Strain
Increasing
Tensile Strain
-15
-20
0.2
0.4
0.6
0.8
1
Voltage on N-Region (Volts)
1.2
1.4
30
Imaginary Band Structure Calculation
for Silicon (001) Wafer with [100] stress
• Analysis of the action integral calculated using imaginary band
structure shows that the stress dependence is most sensitive
to changes in band edge energies and occupancy of the bands
3
Imaginary Band Structure of Silicon with [100] Uniaxial Strain
Imaginary k-vector (1/nm)
2.5
2
1.5
x-lobe
z-lobe
y-lobe
1
SOH VB2
b

 b dE 

T ≈ exp ∫ k x dx  = exp ∫ k x

a

 a qF 
VB1
0.5
Tunneling Action Integral
0
0
0.2
0.4
0.6
Energy (eV)
0.8
1
1.2
31
Theoretical Analysis
• Imaginary band structure calculations for (001) wafer with [100]
stress showed a large sensitivity of the transmission probability
to energy band edge movements with stress.
– Little change in tunneling mass
• Strained imaginary band structure data was not available for all
wafer/strain configurations
• Ultimately, the theoretical calculations were made without
imaginary band structure calculations, taking into account the
changes in band edge energy and occupation of the different
lobes.
– This produced a reasonable fit, since we compared changes in current
with strain for a set tunneling direction.
– Different tunneling directions were not compared to each other.
32
Relative Change in Current with Applied Strain
33
Summary of the Tunneling Analysis
34
Summary
• Imaginary k-vector, , represents a decaying
exponential wavefunction
• Imaginary band structure calculations allow
for accurate calculation of the action integral
– Band edge effective mass is insufficient to capture
tunneling in different directions
• Band structure calculations that incorporated
strain matched the trending behavior of
relative change in tunneling current vs strain
35
Acknowledgements
DARPA Steep program
IBM
Network for Computational Nanotechnology (NCN)
Jamie Teherani
Collaborators: Paul Solomon (IBM), Mathieu Luisier (Purdue)
Advisors: Judy Hoyt, Dimitri Antoniadis
Acknowledgements: Steve Laux (IBM)
36