Simulation 0fthe Heat Transfer
L
n the Nanocathode
V.G. Daniov
Moscowtechnicaluniversity ofcommunicationsand
informatics,
Moscowinstitute ofelectronics and mathematics National
research university Higher school of economics
MTUCI,
MIEM NRU HSE
Moscow, Russia
V.Yu. Rudnev
Moscowtechnicaluniversity ofcommunicationsand
informatics,
Moscowinstitute ofelectronics and mathematics National
research university Higher school of economics
MTUCI,
MIEM NRU HSE
Moscow, Russia
V.I. Kretov
Moscowinstitute ofelectronics and mathematics National
research university Higher school of economics
MIEM NRU HSE
Moscow, Russia
Abstract—The heat transfer processis simulated in a nano-sized cone-shaped cathode. A model of heat transfer is constructed
using the phase field system and theNottingham effect. We considerinfluence of the free boundary curvatureand the Nottingham
effect on the heat balance in the cathode.
Keywords—thermo-field emission, cathode, Nottingham effect, free boundary, phase field system, Stefan-Gibbs-Thomson
problem
1. Introduction and Statement of the
Problem
Our main goal is to simulate the heat transfer ina doped
silicon nanocathode. Thecathode has the shape of a blunted
cone and the following linear dimensions:
height of the cathode
10 15 P
diameter of the cathode base
6 P
radius of thecathode vertex rounding 15 nm
cathode vertex angle
20
D

Such a shape of the cathode is specified by the engineering
process, see Fig.1.Such cathodes are used in the electron
microscope and in other electron devices.
An obstacle for a wide use of this cathode is the instability
of electron emission. This instability is in fact caused by the
small size of the cathode.The cathode is heated due tothe Joule
effect. The current in the cathode is very large and the Joule
heat can melt the cathode.
Figure 1. REM image of the silicon
nanocathode.
The effect of the cathode
melting is confirmed
experimentally. Namely, the
produced molten(liquid) layer
does not contain a small
region near the vertex of the cathode cone. At the same time,
the cathode material remains solid near the base. So we can
observe the following sequence oflayers: solid, liquid, solid. It
is experimentally known that the liquid layer becomes solid
after some (unknown) time.
We present anexplanation of this fact in thispaper. The
motion of the free boundary (the interface between the phases)
depends on the curvature of the free boundary and the
Nottingham effect. Namely, the temperature dependence on the
free boundary curvature is determined by the Gibbs-Thomson
law [8, 9]. The Nottingham effect determines the temperature
of the cathode vertex under the thermoemission of electrons [1].
More precisely, the Nottingham effect consists in the following.
If the temperature of the cathode vertex is higher than the so-
called inverse temperature, then the vertex is cooled;if the
temperature of the cathode vertex is lower than this inverse
temperature, then the vertex is heated.
The mathematical model of the heat transferin the case of
fieldemission is known (see [6]),


()() ,
T
cTT TF
t
UO
w 
w
 div 0.j 
Here T isthetemperature, U isthedensity, c isthespecificheat, O
isthespecific heat capacity,Fis the power density of the heat
emission under the Joule and Thomson effects, and
j
is the
Open Journal of Applied Sciences
Supplement：2012 world Congress on Engineering and Technology
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current density. The function Fand the current density are
determined by formulas
2
1(,)() ,,
()
FjrtgtjT
T V


>@
()() ,jTuTT VD 

where ()T V isthe specific conductivity, ()gt is the Thomson
coefficient, u is the potential of the electric field inside of the
cathode,and ()T D is the thermoelectric coefficient.
In our modelwe use the physical parameters:
0
t 100c time scale (time of the experiment)
0
r
5
10 m

space scale (size of the cathode)
l
5
1.64 10J kg
latent heat of melting
c
678J(kgK) Specific heat
V
0.725 Nm surface tension
U
3
2330kg m
Densi ty
P
0.5m (cK) kinetic coefficient of growth
0
T 1700K Melting temperature
k
52
9.43 10mc


Thermal conductivity
O 149 W(mK)
Specific heat capacity
e
19
1.602 10C

u
absolute charge of electron
\
0.7 emittance
SB
V
824
5.6704 10J(cmK)


Stefan-Boltzmann constant
Weconsidera simplifying modificationofthismodelwhich is
adapted to the research of silicon small-size cathodes.Namely,
the Thomson effect can be neglected because of thep-n
conductivityof the silicon emitter. In this case, the
contributions of the p- and n-carriers to the thermoEMF are
mutually compensated.We assume that the current densityis
constant in the cathode sections thatareorthogonal to the
cathode axis. The value of current density was taken
approximately from experimental data.
So we reduce system (1), (2) to the one heat equation

00
2
0
.
tt TkT F
tl
r
U
w'
w

Here r
is the dimensionless coordinate and
t
is the
dimensionless time. But this is not enough. It is necessary to
add the condition at the blunted vertex of the cathode, which
corresponds to the Nottingham effect. We also need to include
the Gibbs-Thomson and Stefan conditionson the free boundary
(the interface between the phases).
We assume that the Gibbs-Thomson condition is satisfied
on the free boundary ()t* (if this free boundary is already
generated)


0
02
()
,
t
cT c
TT K
ll
V
PU
*
 v (3)
where v is the normal velocity of the free boundary and K is
the principle curvature of the free boundary.The normal n is
the outward normal to the interface betweenthe phases (from
liquid to solid). Equation (3)determinesthe linear dependence
of the temperature on the free boundary curvature and the
velocity. If we assume that the free boundary ()t* is
determines by the function ()rrt , then we have

00
()rt rt
c
v and

0
1().Krrt
Besides it is necessary to assume that the Stefan condition
is satisfied on the free boundary

()
.
t
T
k
*
w
ªº 
«»
w
¬¼ v
n
 (4)
If we get 0
0
1
r c
lt P
and
0
2
0
11
cT
r
l
V
U
 in (3), then
condition (3) becomes the usual widelyknown condition

0
() . t
TT
*

In our case,
0
0
0
r c
lt
P
o
and
0
2
0
11.
cT
r
l
V
U

So we obtain from
(2)


0
02
() . t
cT
TT K
l
V
U
*
 
At the blunted vertex of the cathode (
0
rR ) we use the
equation(see [7]

0
00
44
0
.
SB rR
rR rR
lT jl
ET
rc rec
O\V w§·

¨¸
w©¹

Here E
is the energy of the emission electrons. In the right-
hand side of equation (6),the first term determines
theNottingham effect and the second term determines the
additional radiation condition. To derive the function E we
use the approximating from [1].
On theother outer boundaries of theblunted-cone cathode,
we use Neumann-typeboundary conditions.
Finally, we obtain problem (2), (4), (5) with condition (6),
which models the thermo-field emission under our assumptions.
As was mentioned above, the liquid layer can be produced.
This fact means that the domain of our problem can changein
time.This leads to serious obstacles for the numerical
simulation.To avoid these obstacles, we use aregularization of
problem (1), (4), (5). This regularization is the phase field
model (see [2–4])

00
2
0
1,
2
tt
kF
ttl
r
TM
TU
ww
' 
ww

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
2 00 0
2
00
1()
(1 ).
rc
cgc
T
lt trl
l
VT
MM
HHMFMT
PH
U
w§·
' 
¨¸
w©¹

Here 25. F Inmodel(7), (8)
thestateofthesystemisdeterminedbythe dimensionlessparameter
(, ,,)rt MMI- (the so-called order function) in addition to the
usualphysicalparameters (temperature, density, etc.). The
function ()g M isthederivativeofthepotential ().W M This
potentialissymmetricwith respect tozeroandhastwominima at
the points 1. M r In the simple case, we have
3
() .g MMM 
The function (, ,,, )rt TI-H is a regularization of the
temperature T in (2), and H is theregularizationparameter.
Namely, if we formally let 00 H o,thenequation (7)
becomesthe heat equation (2) and equation(8) gives condition
(3) (or condition (5) as a particular case of (3)) and condition
(4).In general, the limit transition as 0 H o from the phase field
system (7), (8) to problem (2), (4), (5) is nontrivial. This
question is discussed in [5,7].
System (7), (8) is supplemented with the boundary
(Neumann-type) conditions. At the blunted vertex of the
cathode (
0
rR
) we use equation(6) for the function
. T
In our model,we take into account the fact that the
coefficients
,k
, U , O c
depend on the temperature.To obtain the
effects of melting and solidification,we also introduce the
condition of generation of aseed of the liquid phase in the solid
phase and vice versa.
2. Numerical solution and programming
Thespecialfeaturesofsystem (7), (8) are the following.First:
thecoefficient at the time derivative in (8) is
-11 0
0
10
r c
lt
P
|
.
Thisfactmeansthatthe motion of the free boundary ()t*
depends on the freeboundaryvelocity much lesser than on the
free boundary curvature, see (5).Second: thecoefficient
(thermalconductivity)
2
00
kt r isverylarge (
7
10| ) in (7).This
fact means that the temperature rapidly stabilizes in a small
volume.
Theaforesaidmeansthatit is necessary to solve system(5), (6)
toconstruct the solution on a long time interval.Thisisa
veryserious problem. Thefactis thatequation (8) is nonlinear
and its “innerinstability” generates nonlinear
waves.Thisfactleadsto
thegenerationoftheinterfacebetweenthephases(free
boundary).However, the final form(5) of the parameter (3)
shows that equation (8) has a stationary solutionat the given
temperature.
Thisfactallowsone to solvesystem (7), (8) byusing an
iterationalgorithm. We use the standard implicit difference
scheme.For every time step k , we first solve the linearized
equation (8). The sweepingisexecutedforgiven (
1
n )
times.Sowefindthestationarysolution
1k
M

ofequation (8) at the
given temperature .
k
T Next we derive the heat equation (7)
with the function
1. k
M

The sweepingisalso executedforgiven
(
2
n ) times. Sowefindthestationarytemperature
1
.
k
T

Thecomputerprogramwasproducedtoderivesystem(7), (8)
Figure 2. Dynamycs of the deviation of the dimensionless temperature
0
:T T 
0
0,tt
2
1
0, 210tt

 (At this instant of time liquid phase is
generated), 2
21, 510.tt 
(At this instant of time the left free boundary
1
()rrt
changes moving direction),
2
3
4,5 10.tt


(At this instant of time
the free boundaries merge),
0.03. H
numericallybythe abovealgorithm. Thisprogramallows one to
varythevalues of the system parametersand the computational
algorithm. Forexample, iftheStefancondition(4) and theGibbs-
Thomsoncondition
(3)donotcontainlargeorsmallparameters,thenwecanassume
12
1.nn
3. Simulation Results
In Figs. 2, 3, 4, we present the results of numerical
simulation of the liquid layer generation and the motion of the
free boundaries.
In Fig. 2,the deviation of the dimensionless temperature
0
T T is shown. At the initial time moment, the dimensionless
temperature is equal to a negative constant,see Fig.2,
0
0.t
We also assume that the cathode is solid at the initialinstant of
time 0.t This means
0
1,
t
u  see Fig.3. The boundary
conditions used here mean that the cathode vertex is cooled
because of the Nottingham effect, while the lower base is
cooled due to the Neumann-type conditions

,
R
rR
TTT
r
D
w 
w
where
R
T
is a room temperature.
Figure 3. Dynamics of the
order function .u
0
0,tt
2
10, 210tt
 (At this
instant of time liquid phase is
generated),
2
2
1, 510,tt


2
3
4,5 10.tt


(At this
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instant of time the free boundaries merge),
0.03. H
Because of the Joule heat, the temperature increases in the
middle of the cathode, while the temperaturedecreases near the
boundary points
0
rR and
.rR
because of the Nottingham
effect and the cooling of the cathode base.
Figure 4. Trajectories of the free boundaries
1()rrt
and
2().rrt
2
21, 510.tt 
(The free boundary
1
()rrt
changes moving direction),
2
3
4,5 10.tt


(The free boundaries
1
()rrt and
2
()rrt merge).
So the temperature profile has maximum inside of the
domain,where the temperature is higherthan the melting
temperature
0,T
see Fig.2,
1.tt
In the heated domain of the
cathode, the liquid layer is generated, see the profile of the
order function in Fig.3,
1.tt
Because the heat outflow due to
the Nottingham effect increases with increasing temperature,
the heating is changed by the cooling as the temperature attains
some maximum value. In Fig.4 the trajectories of the free
boundaries are plotted, the lower curve corresponds to the left
free boundary 1()rrt and the upper curve corresponds to the
right free boundary2().rrt One can see that the melting
region (distance between curves, see Fig.4) the melting region
first begins to expand (the distance between the curves along
the vertical increases) and then decreases to zero,
3.tt
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