лекция 1 обзор методов вычислительной физики

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!



Current Results: Multi-scale Model

QM Dynamics



QM Transport


I-V Characteristics

(~1 nm)



(~0.1 micron)



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


Curr

ent (

µA)

2 defects


Curr

ent (

µA)

3 defects


Curr

ent (

µA)

4 defects


Curr

ent (

µA)

5 defects


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


Current Results: Multi-scale Model

QM Dynamics



QM Transport


I-V Characteristics

(~1 nm)



(~0.1 micron)


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?)


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)


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


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


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

Technology

лекция 1 обзор методов вычислительной физики