Effects Limiting High-Gradient
Operation of Metallic Accelerating
Structures
G. S. Nusinovich, D. G. Kashyn,
A. C. Keser, O. V. Sinitsyn, T. M. Antonsen, Jr.,
and K. Jensen (NRL)
US High Gradient Research Collaboration Workshop,
February 9-10, 2011, SLAC, Menlo Park, CA
Focus of UMd Efforts
• UMd theoretical efforts are focused on
studying various processes limiting highgradient operation of room temperature
structures:
- multipactor in DLA structures,
- heating of micro-cracks by RF magnetic
fields,
- processes in microprotrusions.
What was done and what will be
presented here?
• Progress in studies of multipactor in DLA structures
will be presented by O. Sinitsyn.
• Studies of micro-crack heating by RF magnetic
fields were reported at two previous workshops and
described in the paper:
W. Zhu, J. Mizrahi, T. M. Antonsen, Jr. and G. S. Nusinovich,
“Temperature rise and stress induced by microcracks in
accelerating structures”, Phys. Rev. ST – A&B, 13, 121003
(2010).
• Below we focus on processes in micro-protrusions.
Possible origin of microprotrusions
• It seems possible that microprotrusions originate from
microcracks on the surface of an initially smooth structure.
• First, the surface heating in the region of high RF magnetic
field causes cracking; some experimental data indicate
that this happens when the temperature rise in an RF
pulse exceeds 100oC and this agrees with some
theoretical results (Nezhevenko - Kovalenko).
• Next, sharp rims of the cracks magnify local RF electric
field that causes the field emission of the dark current
which is a precursor of the RF breakdown.
• This scenario may take place in the regions where not
only RF magnetic field is high, but also RF electric field is
strong enough. (SLAC experiments confirm this point.)
Thermal processes in microprotrusions
•
•
•
•
Joule heating
Ion bombardment
Role of the Nottingham effect
Possible role of the Thomson effect
D. Kashyn
Joule heating of microprotrusions
• Size of microprotrusions is much smaller
than the wavelength of the RF radiation.
• The field in the vicinity of such protrusion
and inside it (final conductivity – non-zero
skin depth) can be described by the Laplace
equation instead of the Helmholtz equation.
 k  0
2
2
 0
2
Point Charge Models – monopole model,
- dipole model
Point Charge Models
• Monopole model: charges are assembled into the line such
that the n-th charge is located on the apex of n-1-th sphere.
This assembly approximates the protrusion. The electric field
can be determined analytically if the ratio an1 / an  b for all n.
All charges of the same sign – monopole model. Zero equipotential line
corresponds to the protrusion surface. Conclusion: Monopole PCM allows one to
mimic the profile of a smooth protrusion, however, such profiles are not realistic.
Point Charge Models
• Dipole PCM: to obtain more realistic protrusions the set of image
charges should be added to the original assembly.
Potential in the vicinity of protrusion


n
n
j
j
 z

Vn (  , z )  F0 a0   a0 
 a0 

2
2
2
2
a0
j 0
j 0
  (z  z j )
  ( z  z j ) 


F0 is the background field, a0 is the scaling factor, and
 j are the magnitudes of the virtual charges.
DPCM yields a sharper
profile of the asperity.
What do we solve?
We solve the Laplace’s equation
and
0  z  z s (r )
where
 2  0
zs (r )
in the region
0  r  rmax
describes the shape of the protrusion.
Laplace equation is the subject to boundary conditions:

  (r ) at z s (r )
and
xn
 0
at
z 0
We use relaxation scheme to solve the Laplace equation.
In order to specify the boundary conditions on the
the quantity:
r
zs (r )
we need to introduce
Q(r )   2r 1  z  (r)dr
'2
s
0
which is the flux through the boundary of the protrusion. The protrusion crosses
the grid at some points. Associated with each crossed grid there is a numerical
flux Qk. Values of the potential are adjusted in such a way that numerical flux
matches the real flux.
Solving the equation gives the potential distribution inside of the protrusion.
Amplification factor and field inside the protrusion
For the DPCM the amplification factor is given by
1 Vn (0, z )
 n (r  0)  
F0
z
z  z n 1
 1 a
n 1
2
0
 (z
j 1
j
2

z
)
n 1
j
a
n 1
2
0
 (z
j 1
j
2

z
)
n 1
j
b=0.9 n=40, background field 300 MV/m,
amplification factor 45
The field inside the protrusion is rather small, however, given the high conductivity
of metals this field leads to current densities on the order of 0.1-1 A/micron2
Protrusion heating
After the field and the current density inside the protrusion are determined, the
volume power density can be calculated to solve the heat equation with proper
boundary conditions:  
J E
T
2
 D T 
t
C
T  0 at z=zmin
and
T
 0 at
xn
Electron current density at the protrusion surface is equal to
a FN F 2
the Fowler-Nordheim current density:
where F  eE RF  eE0 sin t
is the periodic
RF field and aFN and bFN are constants equal to
1.37 A/eV and 6.83 /(eV1/2nm), respectively.
j FN 
t 2
 bFN  3 / 2 v 
exp 

F


The difference between RF and DC heating is
characterized by the following ratio (cf. A. Grudiev,
S. Calatroni and W. Wuench, PRST-A&B 2009):

j2
j  
0 2
FN


1
3/ 2
4
9

sin   exp  13.6618  10 v y 0 

2 0
F0

zs (r )
 1 v y 


1

 d
 sin  v y 0   
1D Model of heat propagation
In order to verify our calculation we used a simplified model. We assumed that a
uniform electric current flows through the protrusion. The current density is given by:
J  z   J FN
where
2
aapex
az 
2
aapex is the area of the last charge in the PCM assembly and az  is
the area of the corresponding layer. Under these assumption the following equation
should be solved:
A(z) is the area of the
J

E

corresponding layer
T  A( z )     D  A( z ) T   dS 
T
z 
z 

Cp
To avoid any influence of the base, we imposed the boundary condition for the zero
temperature rise deeply enough (at z=-10a0 ) inside the base.
Comparison of 1D and accurate models
The comparison shows that the temperature rise is slightly smaller for the 1D
model than for the accurate one.
20 ns pulse, b=0.9, background field is
300MV/m
Accurate model predicts a little higher
temperature rise than a 1D model.
Results of simulations
The scaling parameter a0 affects the temperature rise significantly.
copper, b=0.9, background field is 300MV/m
Increasing the height-to-base ratio
decreases the time required to melt
the protrusion
Conclusions from studying the joule heating of
microprotrusions
A theory describing the field distribution inside micro-protrusions is
developed. A simplified model is also created to test the predictions
of the theory.
The theory allows one to accurately analyze the protrusion heating
during RF pulses.
It was found that it is possible to achieve melting for the certain
geometries of the protrusions with pulse length in the range 100-600
ns.
In order to achieve melting amplification factors should be greater
than 50.
Nusinovich
Ion bombardment
• Many effects on the metallic surfaces and, in
particular, on protrusions were attributed by
some authors to ion bombardment.
• At the same time, people developing field
emitters know that operation of such emitters in
RF fields is more stable than in DC fields.
• This difference can be explained by the fact that
in the vicinity of a microprotrusion the ions
experience the ponderomotive force pushing
them away from the protrusion surface.
Ion motion
• Analytical theory (Kapitsa’s method)
• This method is based on separation of slow
motion caused by the ponderomotive force
r
and fast, but small RF oscillations: r  R  ~
• Equation for slow motion can be written as
d 2R
U
mi 2  
R
dt
• The RF field near the apex of small protrusion has
spatial distribution quite similar to the spherically
symmetrical distribution of the field of a small sphere.
This allows one to describe the potential well by
ei E0 2 1  r0  4 Such potential well pushes charges away from the center!
U
4mi


2  R 
Ion motion
• Only ions with large enough initial
velocities directed toward protrusion
2
can reach it:    *  1 ei E0  r0     r0
r0 – apex radius,
R0 – ion initial coordinate
i ,0
Critical normalized RF field
as the function of R0/r0
i ,0
2 mi c  R0 



2  R0 
2
Ion motion (PIC simulations - WARP)
t=3T
t=2T
t=T
t=0
Evolution of ion distribution (initial thermal spread in velocities).
The velocity of propagation of the ion region boundary agrees with estimates
based on Kapitsa’s method.
Phase space in r and z
after 4 RF periods
Ion motion: effect of the dark current
• Effect of the field emitted electron dark current on the
ion motion.
• This effect can be significant when ions are located
close to the dark current and far from the apex.
• To describe this effect one should add to the equation
for ion motion averaged over the RF period the radial
force due to the presence of the electron dark current
(averaged over the RF period).
• This force is equal to the ponderomotive force for ions
when their radial departure from the dark current
obeys the condition:
C , E s  M i R 5
rion,cr  930

m 2 r02
Example: C=0.01, work function – 5eV, Mi/m=2000, wavelength – 3 cm, r0=10nm
R=1, 2 and 3 micron – r^ion,cr=0.04, 1.4 and 10.8 microns, respectively.
The region where ions are attracted radially to the dark current rapidly increases
with the departure from the apex of microprotrusion.
Keser
Nottingham Effect
• Above, the thermal conductivity equation
 
J E
T
 D 2 T 
t
C
was supplemented by the boundary condition
T
0
xn
(NB: practically all parameters in the equation above are temperature dependent)
More accurate temperature distribution can be obtained when
the boundary condition takes into account the Nottingham effect
T j ( Es , Ts )

 Es , Ts 
xn
e T 
Here, we introduced an average value of the energy evolved at the surface
per emitted electron:  E , T

s
s

Nottingham Effect



N. E. is a quantum mechanical effect that
contributes to the thermal balance in field
electron emitters.
Joule heating is dominated by this effect at the
emitter tip.
N. E. was extensively studied in the realm of
field electron emitters, because it influences
the temperature rise and, hence, the emitter
operation.
Theory

The tunneling electrons are mostly from energy levels
below the Fermi level. They are replaced by more energetic
electrons from the conduction band. As a result, emitter tip
heats up. At high current densities, the energy associated
with the Nottingham effect can exceed the energy due to
Joule dissipation.
Ref. 1. G. Fursey, “Field Emission in Vacuum Microelectronics” (Kluwer Academic/
Plenum, New York, 2005).
Heating-Cooling




Similarly, if the average energy of tunneling electrons
exceed the Fermi level, the surface starts to cool down!
Since the electron supply function evolves with
temperature; there is a certain inversion temperature T*
for which
T< T* implies heating
T> T* implies cooling
Supply Function (βT = 1/kBT)
f E  ln 1 exp T EF  E 
Transmission Probability
D E 1 / 1 exp F Eo  E 
Heating - Cooling
…
Below T*⃰
above T* ⃰
A
EF
 replacement e- gives up energy (heating)
 replacement e- takes up energy (cooling)
T< T* ⃰
E
The inversion temperature depends
on the surface field gradient and on
the work function.
B
E
T>
T* ⃰
Average energy term
–
–
–
–
<ε> depends on transmission
probability & supply function but can be parameterized by T
It is often calculated using
Fowler Nordheim (FN)
equation for field emission
(heating was the concern)
BUT: at high temperatures,
FN equation gives wrong
result, and General Thermal
Field equation should be
used
Nottingham heating at high T
for dark current needs to be
reconsidered
  E  EF
E  E D E  f E dE


 D E  f E dE
F
/ F
red region: heating
blue region: cooling
Evolution of temperature profiles with time
Profiles for the
case of N. cooling
Fursey , p. 46
Significance for Us




Temperature gradients are on the order of 10^8 K/cm
Tangential stresses may exceed 2 10^9 Pa
The resulting thermo-elastic stress can destroy the
emitter tip before melting point is reached.
There is a certain controversy about realizing
such conditions: L. M. Baskin, D. V. Glazanov and G. N. Fursey
“Influence of thermoelastic stresses on the destruction of fieldemission cathode points and the transition to explosive emission”,
Sov. Phys. Tech. Phys. (1989) versus M. G. Ancona, “Thermochanical
analysis of failure of metal field emitters”, J. Vac. Sci. Technol., (1995)

This issue deserves a more detailed study
Nusinovich
Possible role of the Thomson effect
• Thomson effect is the thermoelectric effect that
occurs in conductors with a non-uniform
temperature.
• The electric field in such conductors is equal to

 j
E   T

The Thomson coefficient does not exceed

3 kB
V
 1.29  10 4
2 e
K
Then, the heat propagation equation can be written as:


(T )
1  
2
 D (T ) 
J E  T 
t
Cp
Possible role of the Thomson effect
• The Thomson effect can play an important role when
T

E
z
Our simulations of the RF field penetrating into protrusions show
that the RF electric field there (in copper) is about 3 kV/m.
Hence, this effect becomes significant when the temperature
gradient is on the order of 23 K/micron or above.
In micro-protrusions of the height on the order of 10 microns, the
non-uniform temperature rise in short pulses with high gradient of
the RF field can easily lead to the fulfillment of such conditions.
Acknowledgments: this work is supported by the Office of HEP DOE.