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Евгений Пузырёв Sokrates T. Pantelides groupУниверситет Вандербильт,
Теннесси США
SiO2 Graphene
Collaborators
Kalman Varga, Kirill Bolotin, Physics & Astronomy Vanderbilt UniversityDan Fleetwood, Ron Schrimpf, EECS Vanderbilt UniversityUmesh Mishra, EECS University of California At Santa BarbaraXiaoguang Zhang, CNMS, G. E. Ice, MST, Oak Ridge National Lab and many more others
1. Обзор методов вычислительной физики Много-масштабное моделирование: от дефектов к ошибкам в приборах
2. Локальная структура металлических сплавов: диффузионное рассеяние и атомные смещения.
3. Дефекты в полупроводниках и поведение приборов: GaN, SiC и AlSb.
4. Проблемы функциональности материалов для мемристора TiO2 и ZnO.
5. Графен,- материал будущего или поиск ниши для применения.
Практическое применение функционала плотностиМного-масштабное моделирование
Основные методы много-масштабного моделирования
I Применение функционала плотности1. Расчеты возбужденных состояний 10-100 атомов
а) Ширина запрещенной зоныб) Положение электронного уровня дефекта
LDA+UHybrid functionalGW, absorption spectrum T(100 atoms) = 100 000 MPP
2. Расчеты из первых принципов 100-1000 атомовa) Атомные координаты и электронная
б) ПроводимостьLDA (VASP, Quantum ESPRESSO, SIESTA)
II Применение полу-эмпирических потенциаловМолекулярная динамика и расчеты методом Монте-Карло 10000-1000000 атомовКлассическая механика (LAMMPS, NAMD)
Introduction: Ion-Induced Leakage Currents
Metallization burnout after SEGR
Heavy-Ion strikes degrade or destroy dielectric layers
Lum, et al., IEEE TNS 51 3263 (2004)
Massengill, et al., IEEE TNS 48 1904 (2001)
I-V following biased irradiation of 3.3 nm SiO2 capacitors
Distinct Electrical degradation modes:
Rupture (Hard breakdown, HB)
Soft breakdown (SB)
Long-term reliability degradation (LTRD)
TRIM Calculations:
Sample geometry:
Only atomic recoils occurring IN the SiO2 layer!
High-LET ions generate O(100) eV recoils in thin oxide layers!
Отдача при низких энергиях
Methods
• Quantum Mechanical MD– DFT-LDA for energy and forces– Classical mechanics for ions– Cell sizes: 200-1000 atoms– Calculation times: 0-1000 fs
• Quantum Mechanical Transport Calculations– Complex-valued potentials at boundaries as “source” and “sink”– Non-equilibrium Green’s function method for transport properties– Orbital basis set: LaGrange functions
• Percolation Theory– Mott defect-to-defect tunneling– Node-to-node percolation model
Dynamical atomic and electronic structures
Fully QM transport calculations for underlying transport physics
Physically motivated, QM and experimentally parameterized model for
realistic device structures!
Beck, et al., IEEE TNS 55, 3025 (2008)
QM Dynamics
Arbitrary Materials System
Materials ResponseDefect Structure
QM Transport
Arbitrary Device geometry
I-V Characteristics
(~1 nm)
Ab Initio calculation of experimentally measureable device properties!
Percolation Transport
(~0.1 micron)
Много-масштабное моделирование: От дефектов к ошибкам в приборах
Time-dependent atomic and electronic structure
Вычислительный Метод Молекулярная динамика из первых пртципов
• Применение функционала плотности– DFT-LDA for energy and forces
• Классическая механика для атомных смещений• Размер ячейки 200-1000 атомов• Время 0-1000 fs
Apply KE to primary atom…
…evolve system!
Highest fidelity for bond-breaking/forming during
low-energy events
Atomic AND electronic structure!
Отдача при низких энергиях
0 29 58 femtoseconds after recoil
1
0
…Correlates with formation of electronic defect states in band gap!
Beck, et al., IEEE TNS 55, 3025 (2008)
Beck, et al., IEEE TNS 55, 3025 (2008)
Current Results: Multi-scale Model
QM Dynamics
Arbitrary Materials System
Materials ResponseDefect Structure
QM Transport
Arbitrary Device geometry
I-V Characteristics
(~1 nm)
Ab Initio calculation of experimentally measureable device properties!
Percolation Transport
(~0.1 micron)
Time-dependent atomic and electronic structure
Time-dependent atomic and electronic structure
Nikolai Sergueev
Defect: single oxygen vacancy
defect
EF
Nikolai Sergueev
0 2.0Cu
rren
t (µA
)
1.5
1.0
0.5
0 0.5 1.0 1.5Bias voltage (V)
2.0
Transport energy window
from -Vb/2 to +Vb/2
EF
Defect: single oxygen vacancy
Nikolai Sergueev
16.24 Å
16.2
4 Å
556 atoms in scattering region
Creating defects in a-SiO2
Number of oxygen to be removed: from 1 to 6
12
3
4
6
5
Amorphous SiO2 leakage currents
Nikolai Sergueev
Theoretical formalism
Tuning the model: crystalline SiO2 system
Leakage currents in thin amorphous SiO2
a-SiO2
Bias voltage Vb
Electrode Electrode
Mol. Dyn. QM Calculationsstructure
First Principles Transport QM Model
Nikolai Sergueev
Density Functional Theory + “Source and sink” method
Infinitesystem
Finitesystem
…a-SiO2…
a-SiO2
Conventional transport methods: scattering theory, open infinite system
complex potential complex potential
Our formalism:
K. Varga and S.T. Pantelides, PRL 98, 076804 (2007)
Source Sink
Nikolai Sergueev
Solve diagonalization problem:
Compute Green’s functions:
Calculate charge density:
Compute leakage current:
W=Wsource+Wsink
Flow-chart:
Nikolai Sergueev
Initial model calculations
Crystalline SiO2 – computationally fast
d
SiO2Al (100) Al (100)
How does conductance of SiO2 depend on oxide thickness d ?
Can we compute device related property ?
Nikolai Sergueev
Conductance versus thickness of SiO2
Conductance: exponential dependence as expected from tunneling
Not defected structure yet!
Nikolai Sergueev
Applying bias voltage across the device …
M. Fukuda et al, Jpn. J. Appl. Phys. (1998)
105
0 0.5 1.0 1.5 2.0Bias voltage (V)
Curr
ent (
A/cm
2 ) 108
107
106
104
103
1090.54 nm
0.8 nm1.07 nm
1.35 nm
1.61 nm
Calculations
SiO2
EF Al Al ~ 4.5 eV
We used standard Hamiltonian
Experiment
Nikolai Sergueev
Our formalism allows:--- not only to compute current and conductance
--- but also to analyze the transport mechanism
PDOS – density of states that has an amplitude on oxide atomsTransmission – describes the tunneling efficiency
Oxide statesEF
Nikolai Sergueev
Energy (eV)
Tran
smis
sion
Energy (eV)
Tran
smis
sion
1
2
3
4
5
6
Increasing number of defects …
Nikolai SergueevBias voltage (V)
Curr
ent (
µA)
1 defect
Nikolai SergueevBias voltage (V)
Curr
ent (
µA)
2 defects
Nikolai SergueevBias voltage (V)
Curr
ent (
µA)
3 defects
Nikolai SergueevBias voltage (V)
Curr
ent (
µA)
4 defects
Nikolai SergueevBias voltage (V)
Curr
ent (
µA)
5 defects
Nikolai SergueevBias voltage (V)
Curr
ent (
µA)
6 defects
Results: QM Transport Calculations
Individual defects…
QM Tunneling Probability: Convolution of
electronic DOS and spatial information
…introduce defect states with specific energy levels and localizations
Al Ala-SiO2
QM calculated I-V characteristics showing activation of discrete tunneling paths!
Nikolai Sergueev
the defects result in the step-like functions of the IV
conductance vs. oxide thickness dependence is correct
current increases with number of defects
We performed first principles quantum mechanical transport calculations and we obtained the following:
Going from atomic-scale to mesoscale description …
current-voltage dependence qualitatively agrees with experiment
First Principles Transport QM Model Percolation Model parameters
Beck, et al., IEEE TNS 55, 3025 (2008)
Current Results: Multi-scale Model
QM Dynamics
Arbitrary Materials System
Materials ResponseDefect Structure
QM Transport
Arbitrary Device geometry
I-V Characteristics
(~1 nm)
Ab Initio calculation of experimentally measureable device properties!
Percolation Transport
(~0.1 micron)
Time-dependent atomic and electronic structure
Results: Percolation Model
Parameterize defect atoms with:Position
Eigenvalue From QM MD calculation
From QM DOS calculation
Defect levels from SHI-induced defects!
Mott defect-to-defect tunneling
S. Simeonov et al. Physica Status Solidi, 13, 2004
ri – defect positionE – external fieldεi - energy level relative to EF
σi – site occupancy, [0, 1], at boundary σ=0.5ν0- Mott’s escape frequency
DOS
Defects
Iterative procedure for occupancies until Δσi < 10-7
J = ν0Σij(σiboundary-σj)
ν0 = 1.15 × 1013 s–1
Defect-to-defect tunneling
E
DOSDefects
time = 78fs22 defects
L
• L =1.4 nm • Defect energy levels• Defect atomistic map ri, εi ,σi
Defect-to-defect tunneling
ri, εi ,σi
Leakage Current Temperature Dependence
-20.0 -10.0 0.0 10.0 20.0 qE, MV/cm
Current, nA
-6 -4 -2 0 2 4
Leakage Current Time Dependence Current, nA
-2 0 2 4 6
-10.0 -5.0 0.0 5.0 10.0 15.0 20.0 qE, MV/cm
Model results in real-time defect evolution and transient currents
Defect time evolution
Energy
Space
0 200 400 500 600 time, fs
Current, nA
0.0 4.0 8.0
Num
ber o
f defects
5 10 15 20 25
Transient currentKeeps going
Results: Calculated I-V Characteristics
-20 -10 0.0 10 20 qE, MV/cm
C
urre
nt,
nA-6
-4
-2
0
2
4
Thermal smoothing
Steps showing activation of discrete tunneling paths
Asymmetric: Defect level dependence
Results: Transient I-V Characteristics6 fs 32 fs 58 fs
Thresholds in time and applied field!
Results: Transient Leakage
Transient defect-induced weakness!
E=1.5 V
E=3 V
Defects and current peaks within ~200 fs of recoil
Defects and current persists on the ns time-scale
Roughness of curve due to exponential dependence on atomic
and electronic structure!
Quantitative agreement!
Massengill, et al., IEEE TNS 48 1904 (2001)
As a result of the calculation we have direct comparison with experiment for the gate current as a function of gate voltage!
Graphene device degradation
• Graphene fabricated by mechanical exfoliation from Kish graphite
• Sweep VG with VDS=5mV
Motivation and Outline
Experiment [1]
o Graphene’s resistivity response to x-ray radiation,
ozone exposure, annealing.
o Defect related Raman D-peak appears after
x-ray irradiation in air
ozone exposure, decreases after annealing.
[1] E.-X. Zhang et al, IEEE Trans. Nucl. Sci. 58, 2961 (2011)
Theory: behavior of impurities on graphene
o Temperature and concentration dependence.
o Need to remove oxygen without vacancy formation (would H help?)
Graphene device degradation
Two-probe resistances measured on
• 10 keV irradiated graphene• pristine graphene• ozone exposed graphene (1 min) • annealed (300C for 2 hrs in 200 sccm Ar)
Graphene device degradation
Defect related D-peak
• increases x-ray exposure • decreases after temperature anneal
Ozone exposure
a)
b)
0
2000
4000
6000
8000
Anneal15 Mrad(SiO2)8 Mrad(SiO
2)
Inte
grat
ed in
tens
ity A
rea
10-keV X-ray Dose
G-Peak
D-Peak
Pre0
20
40
60
80
I D/I G
(10
0%)
Kinetic Monte-CarloKMC
Density Functional TheoryDFT
Theoretical Approach
O
O dimer
O migrationO desorption
• Defect formation energies • Migration/desorption barriers
Defect dynamics• Temperature• Initial concentration
Top
Bridge
1.3 eV
0.8 eV
0.5 eV
1.3 eV
Oxygen Removal and Vacancy Generation
CO, CO21.1 eV O2 1.1 eV
Oxygen: clustering behavior
Removal of oxygen • Pairs O2
• Triplets CO, CO2, VC
Device degradation
Residual oxygen atomVacancy
High-temperature Annealing
Concentration of vacancies exceeds concentration of residual O
T
High vs Low Temperature Anneal
T, oC
Temperature Anneal Initial Defect Concentration Dependence
Lo
Low O, High V concentration
High O concentration
vacancy
oxygen
High T: Removal of oxygen > 0.05 initial surface coverage leads to vacancy formationLow T: Oxygen stays on the surface and forms clusters
Decrease of D-peak, Increase in resistivity
surface coverage
Method to prevent defect formation during irradiation/annealing?
T initial O surface coverage
Oxygen and Hydrogen on Graphene:Binding energies, Migration and Reaction Barriers
O-H is most likely to desorb from graphene surface
Leaves carbon network intactH
O
Effect of Hydrogen On Oxygen Annealing
Oxygen/Hydrogen Concentrations
Low High
Low 2% O, 10% H
High 15% O, 1% H 15% O, 10% H
@ T = 300 C
Final defect concentrations?
Removal of residual Oxygen Causes formation of large
amount of Vacancies
t ~ 0.001 s
t ~ 1 s
t ~ 0.0001 s
t ~ 1 s
Effect of Hydrogen On Oxygen Annealing
Residual Hydrogen Forms clusters L ~ 0.5 nmNo Vacancies are formed
Higher Hydrogen concentrationHigher Oxygen concentrationHydrogen is removed Oxygen is removed
Hydrogen is removed first, Removal of residual Oxygen Causes formation of Vacancies
High O, High H concentrations
Effect of Hydrogen On Oxygen Annealing
Электронная плотность
Разложение по функциям Гаусса
Перенос заряда
pseudopsV q S q w q
Полная энергия
2
1 21
45 3 3 1 2, ln
8 4 5 2 8 2
q qw w q q and w
q q
Теория линейного отклика
Кинетическая энергия
corr corrWang Teter LDA atomT T T T
λ=1 upper limit von Weizsäcker
λ=1/9 gradient expansion second order
λ=1/5 computational Hartree-Fock
1. Phase Diagram
2. Elastic Properties
3. Defect Formation Energies
G0W0
Ширина запрещенной зоны
GaN