1. High field enhancement due to the surface changes
V. Zadin, S. Parviainen, K. Kuppart, K. Eimre, S. Vigonski, A. Aabloo, F. Djurabekova
CLIC workshop 2015

2. Mechanisms behind field emitting tips?
Field emitters with aspect ratio ~100
Voids or precipitates as possible mechanisms responsible for generating the emitters
Multiplication of betas
Influence of surface roughness
Emitter on top of emitter ….
Dynamic surface changes, thermal effects?
Can low aspect ratio features lead to high field enhancement?
Possible mechanism also in changing work function
V. Zadin, University of Tartu

3. Simulated systems Coupled electric, mechanical, thermal interactions d
Electric field deforms sample and causes emission currents
Emission currents lead to current density distribution in the sample
Material heating due to the electric currents
Electric and thermal conductivity temperature and size dependent
(Deformed) sample causes local field enhancement h d
Dc El. field ramped up to MV/m
Comsol Multiphysics 4.4 (and 5)
Nonlinear Structural Materials Module
AC/DC module
HELMOD (Combined Electrodynamics, Molecular dynamics)
Simulated materials: Copper
V. Zadin, University of Tartu

4. MD vs. FEM in nanoscale MD – exaggerated el. fields are needed
MD simulations are accurate, but time consuming
FEM is computationally fast, but limited at atomistic scale
Very similar protrusion shape to MD
Material deformation starts in same region
Maximum field enhancement is 2 times
E0~ 2000 MV/m
V. Zadin, University of Tartu

5. Deformation at realistic electric field strength
Void formation starts at fields > 400 MV/m
Material is plastic only in the vicinity of the defect
Thin slit may be formed by combination of voids or by a layer of fragile impurities
Field enhancement factor ~2.4
Thin material layer over the void acts like a lever, decreasing the pressure needed for protrusion formation
V. Zadin, University of Tartu

6. Polycrystalline Cu in high electric fields
Initial sample
O.Saresoja, Advanced Engineering Materials 14, 265 (2012)
Cu sample obtained from an explosive welding simulation
Severe plastic deformations due to the applied stress and temperature
Similar treatment and conditions as during breakdown
Defect reduction methods:
Conjugate-Gradient minimization scheme to relax the lattice
simulated annealing to grow the grains and remove stacking faults
Final sample contains several defect free grains and a number of surface intersecting grain boundaries
Opportunities to study grain boundary effects and influence of surface roughness
Final optimized and relaxed sample
V. Zadin, University of Tartu

7. The influence of surface roughness
1. Atomistic surface detection using common neighbor analysis:
Imperfect surface leads to nonuniform stress distribution
MD simu. must be coupled to el. field calculations
Coordination analysis to find the surface atoms
The surface is imported into COMSOL Multiphysics for Finite Element Analysis
the electric field distribution
mechanical stresses in the sample
Deformation of the polycrystalline copper under el. field:
Using already existing ED-MD (HELMOD) code or
Coupling the FEM simulations to LAMMPS
Simulations with uniform pressure over surface already demonstrated mass transport starting from surface roughness
2. Surface reconstruction in FEM using splines:
3. Calculating the surface roughness enhanced el. field:
V. Zadin, University of Tartu

8. Multiplication of betas
r_1
We can see different surface modifications leading to small β
Large β is needed
Multiplication of field enhancement factors
Can explain observed high beta values
Incorporates surface roughness
r_1/r_2<0.1 is needed to observe significant influence
r_2
Max. enhancement
Reference sim.
V. Zadin, University of Tartu

9. Schottky conjecture – reducing the aspect ratio of emitter
How to identify the shape of the surface defect causing field enhancement?
h/r= 23
β~80
FN plot is characterized by beta and emission surface area
htotal=3nm
d=10nm
1/E
ln(I/E2)
h/r=10
Compared geometries:
High aspect ratio emitter
Low aspect ratio emitter standing on top of a protrusion
Field enhancement of protrusion β~3-4
h/r= 17
β~17
Both emitters have similar height but different „thickness“
Shape of the top part is the same - equal emission area
Beta is fitted by adjusting the geometry
h/r=6
d=10nm
htotal=2.6nm
V. Zadin, University of Tartu

10. Rising tip in el. field
σMises, max<< 1Pa
σMises, max=68 MPa
Field emitting tip, rising from the surface is assumed
Simulation starts, when the emitter is ~40o angle
Simulation ends when fast increase of field enhancement factor starts
σMises, max=165 MPa
Dynamic behavior of field enhancement factor
Elastic deformation up to ~90MV/m
Corresponding field enhancement factor ~20
Rising tip can cause significant increase of the field enhancement
Elastic limit
V. Zadin, University of Tartu

11. Field enhancement by „dynamic tip“
Comparison of static (reference) and dynamic emitters
Static emitter does not change the shape during simulation
Dynamic emitter deforms elastoplastically
100 MV/m
γ - slope
Direct calculation from simulation
From FN plot
Beta from static tip 18 22
Beta from dynamic tip
18-33
11.5
The slope in dynamic case is larger, as the applied field is the same, compared to the reference case, but now, the current is higher due to locally more enhanced field.
New concept – how hard it is to remove electrons from metal…
If the surface loses protrusions, it becomes harder to extract the electrons to hold the emission current. Thus, we need to increase the local field.
100 MV/m
ln(I/E2)
Beta decreases 2-3 times during dynamic deformation of emitter
Instead of growing emitters, we have decreasing emitters?
Evaporation of surface protrusions?
V. Zadin, University of Tartu

12. General Thermal Field model
Simulations of emission currents over large surfaces
Thermionic emission: high temperature, low field
Field emission: low temperature, high field
Combined effects : general thermal field equation:
Special interest:
Intermediate region where thermal contribution is significant
V. Zadin, University of Tartu
K. L. Jensen, J. Appl. Phys. (2007)

13. Heating and emission currents
Local emission currents – connection to the experiment
Field emitters as nanowires
Emission current density
F(Kn)
Size dependence of electric and thermal conductivity
Conductivity in nanoscale emitters is significantly decreased (more than 10x for sub-nanometer tip)
Knudsen number to characterizes nanoscale size effects
Wiedemann-Franz law for thermal conductivity
Heat equation in steady state
Fully coupled currents and temperature
Emission currents concentrated to the top of the tip
Fast, exponential temperature rise in the emitter
V. Zadin, University of Tartu

14. Current density in ED-MD and FEM models
Electric field
HELMOD/FEM
Local current density and el. field
Different solutions methods for el. field using FEM and HELMOD
Discretization in HELMOD tied to atomic structure
FEM geometry represented by perfectly cylindrical and hemispherical structures
Good comparison between obtained electric fields
The current density dependence on local electric field for FEM and HELMOD.
Apex el. fields are compared
FEM and HELMOD implementations agree, validating the results
V. Zadin, University of Tartu

15. Emission currents & temperature using FEM and HELMOD
Sensitivity to numerical effects:
Electric field calculations
Emission current integration algorithms
Difficulties at estimating material heating
Both FEM and HELMOD represent surface incorrectly (smooth, continuous for FEM and discrete for HELMOD)
Significant difference due to integration algorithms from fundamentally different surfaces
Both approaches capture the same general behavior!
V. Zadin, University of Tartu

16. Influence of temperature – FN plot
Simulation of single emitter
Fully coupled currents, temperature and external field
Emission current is integrated over whole surface
Taller emitters demonstrate smaller thermal effects
high local E is reached faster
Thermal effects influence lower applied fields
FN equation assumes static system
Thermal effects introduce a dynamic component
Problem – effect remains in low current region
Possible use – allows us to estimate the actual size of the emitter?
V. Zadin, University of Tartu

17. Some conclusions
Field enhancement due to single protrusion is not sufficient
Additional mechanisms are needed
Multiplication of betas
Thermal effects or dynamic surface changes?
Emission currents are now calculated using general thermal field model
Thermal effects can have significant influence over the field enhancement
Dynamic surface changes can lead to modification of measured β
V. Zadin, University of Tartu

18. Thank you for your attention!