Belgirate, Italy, 28-30 September 2005
THERMIONIC REFRIGERATION WITH PLANAR AND NONPLANAR ELECTRODES
- CHANCES AND LIMITS -
Y.C. Gerstenmaier* and G. Wachutka**
*Siemens AG, Corporate Technology, D-81730 Muenchen, Germany, e-mail: yge@tep.ei.tum.de
**Institute for Physics of Electrotechnology, Munich University of Technology
ABSTRACT
In this paper for the first time a precise theoretical analysis
of efficiencies and power densities for nano-scaled therm-
ionic vacuum gap devices is presented. Plane electrodes
with nano-gaps will be treated first. With nonplanar elec-
trodes (nanotips) enhanced electron field emission occurs,
giving rise to new design options. The theories of field and
thermionic emission are combined by calculating the
transmission coefficient for the surface potential barrier ex-
actly numerically for thermionic and tunnelling energies.
This then is used for current and energy transport determi-
nation and derivation of cooling and generator efficiencies.
Because of the high work function of metals, efficient
thermionic electron emission takes place only at high tem-
peratures. At low temperatures also efficiencies near Carnot
are possible but with very low power densities. The energy
exchange processes in the cathode during emission are re-
viewed and an improved model is presented.
1. INTRODUCTION
With increasing power densities in micro- and power-
electronic components thermoelectric cooling becomes
more and more interesting [1], [2]. Contrary to the tradi-
tional cooling of PCBs and modules by fans or radiators,
thermoelectric devices can cool in a controlled way in the
immediate neighbourhood of the hot spots of the power dis-
sipating devices. Especially in opto-electronic devices very
high power densities occur over a small region. However,
the efficiency (coefficient of performance = CoP) of cool-
ers with bulk thermoelectric materials is still low. The fig-
ure of merit for these materials Z T = S2 ? T /? with S the
thermo-power (Seebeck coefficient), ? the electrical con-
ductivity, ? the thermal conductivity and T the average
temperature between the high temperature side TH and cold
temperature side TC of the device does not exceed 1 very
much [3], [4]. In an inverse mode of operation thermoelec-
tric and thermionic devices can also be used as generators.
A large demand of micro-structured generators [5] and
coolers is expected in the telecommunication sector.
In recent years considerable progress has been made in
the development of thermoelectric nano-structured superlat-
tices [1, 6, 7, 8]. ZT values of 2.4 have been achieved for
phonon blocking lattices [7] (low thermal conductivity)
with temperature gradient and current flow perpendicular to
the planes of the multilayer structure. Even higher ZT val-
ues of 3.8 were obtained for quantum well confinement
multilayer structures with temperature gradient and current
transport parallel to the planes [8] (low dimensional or 2D
system).
The purpose of this work is to investigate theoretically
the potential of thermionic vacuum devices with additional
electron field emission due to non planar electrodes for
cooling applications. Because of the high work function of
metals, thermionic electron emission (i.e. by thermal excita-
tion of electrons) takes place only at very high temperatures
over 1500 K. Thermionic generators, e.g. for space mis-
sions, are known for long [9]. In [10, 11] thermionic de-
vices with plane electrodes were also proposed and calcu-
lated for cooling. Figure 1 shows the band structure over a
vacuum barrier with applied voltage V between cathode
and anode and work function W. The electron to be emitted
must have an energy higher than the electrochemical poten-
tial ? (equal to the Fermi energy EF at T = 0?K) plus W.
The line in the vacuum gap indicates the lowest possible
energy level for the electron there and gives the electric
field by its gradient. Electrons with high energy, evaporat-
ing from the cathode, reduce the average electron energy
and cool the cathode. For operation at room temperatures
work functions as low as 0.3eV are needed, which cannot
be achieved at present. For operation at 500K 0.7eV would
be sufficient. The potential barrier W is considerably lower
in solid state hetero-junctions or for barriers caused by pn-
doping [12, 13, 6]. In case of sufficiently thin barriers bal-
listic electron transport (i.e. without scattering) occurs be-
tween cathode and anode which has been analysed in [14]
by Monte Carlo simulation. For broader solid state barriers
there is no clear-cut dividing line between thermionic cool-
ing and thermoelectric Peltier cooling. On the other hand,
the efficiency of solid state devices is reduced compared to
vacuum gap devices due to the heat Fourier current over
the barrier. In [15] it has been pointed out, that it may be
important to consider the Joule losses in the cathode and its
contact resistance for an overall efficiency calculation.
When using a vacuum gap, operation at room temperature
Y.C. Gerstenmaier and G. Wachutka
Thermionic Refrigeration with Planar and Nonplanar Electrodes - Chances and Limits -
and below is conceivable by reducing the electrodes spac-
ing to a few nm so that quantum mechanical tunnelling be-
comes possible also for lower cathode temperature. A theo-
retical analysis was presented in [16]. As a result it turned
out that the electrodes spacing has to be larger than 4nm.
Otherwise excessive tunnelling of electrons from below the
Fermi level would occur, which leads to heating of the
cathode instead of cooling, because then the average elec-
tron energy increases. The CoP of such devices could be
high and their realisation is pursued since several years [17,
18]. However, the feasibility of the nano-gap concept with
planar electrodes is not proved and doubts concerning their
realisation are not cleared up.
We shall analyse vacuum nano-gap devices with plane
electrodes more precisely in section 5, both, for refrigera-
tion and current generation, following the theory developed
in section 2 and 3. In section 6 the performance of en-
hanced field emission devices with arrays of nanotips at the
emitting electrode will be investigated and compared.
2. THEORY OF ELECTRON EMISSION
In the usual theory of electron field emission [27, 28, 29]
the Sommerfeld free electron model is used, which is a
good approximation for metals. To treat semiconductors
this formalism should be extended to electrons in a periodic
effective crystal potential using Bloch wave functions. The
energy bands n(k) can in principle be obtained by band-
structure calculations. Here n denotes the band index and
a0 k the crystal momentum of the single electron, not to be
confused with its real momentum. In the interior of the ma-
terial at thermal equilibrium the number of electrons per
unit volume in states around k in a volume element d3k is
according to Fermi-Dirac statistics:
( ) 33133 4/1)/))(exp(4/))(( pi?pi kdTkkdf Bnn ?+?= kk (1)
where 1/ 8pi 3 is the density of levels for the unit k-volume.
(Each level can be occupied by two electrons.) For current
transport in the presence of electric fields and temperature
gradients the Fermi distribution would have to be replaced
by a non-equilibrium distribution function g(r, k) depend-
ing generally also on position r. Often the relaxation time
approximation with a semiclassical model of electron dy-
namics is used in calculating g [30, p. 244]:
T
fT
fefg
?
????
?
???=
))(())(()()(
))(())(()()())((),(
kkrk
a1
kkk
a1rFkkr
?
? (2)
where F is the applied external electric field, ? is the expec-
tation value of the electron?s velocity, ? () is the relaxation
time and ?T is the temperature gradient. e denotes the
magnitude of the electron charge. Expression (2) can be
justified, when the single electron is described by a wave
packet of Bloch wave functions extending over many crys-
tal grid cells. On the other hand the extension of the wave
packet has to be smaller than the length over which the ex-
ternal electric field varies appreciably. These conditions are
no longer satisfied at the surface, where strong potential
variations according to fig.1 occur on a scale of the lattice
constant.
The concept of effective electron mass and crystal mo-
mentum are attributes of a quasi-particle not subject to the
crystal potential - which has been accounted for by use of
the Bloch waves - but only to a slowly varying external po-
tential. The potential step W at the surface is caused by a
dipole layer of width not more than a lattice constant. In
calculating the transmission of the electrons through this
barrier the original crystal potential is essential and the ef-
fective mass concept cannot be used any longer. For the
electrons impinging on the surface barrier from inside the
metal therefore the real electron mass has to be used. In
[31] it is argued, that in the transmission of heterojunction
potential barriers the smaller effective mass of the two ma-
terials has to be used.
Since there is no consistent theory for current and energy
transport over the surface barrier, several heuristic ap-
proaches are used in the literature. We start on a more gen-
eral basis by considering an electron current impinging on a
surface element dS from inside the material. The normal di-
rection perpendicular to dS will be denoted in local coordi-
nates as x-direction. The probability for an electron imping-
ing with velocity component vx > 0 to tunnel through the
surface potential barrier V(r) will be denoted by D(vx),
where vx is the x-component of the expectation value of the
velocity. According to Bloch?s theory: v(k) = ?k n(k) /a0 .
Since electron density times vx gives the current density
normal to the surface, the total emission current density can
be written with the help of (1) by integration over all elec-
tron states in k-space with vx(k) > 0 :
>
??=
0)(,
3
3
4
))((/)())((
kk
kkkk
x
kdfDeJ
nxnxe
? pi
? . (3)
In this work electron current will be defined to be in the di-
rection of the particle flow, opposite to the technical direc-
tion of electrical current. In (3) the non-equilibrium distri-
bution g can be approximated by the equilibrium f near the
surface, which is a good approximation.
Restricting for the time being to the free electron model
with real electron mass m and real momentum a0 k, because
of lack of bandstructure data, n(k) reduces to
n(k) = EC + (a0 k)2/2m , (4)
EC2 EC1
?1
x
V ?2
W
T1 T2
Vacuum
Energy
Cathode
Anode
Figure 1: Thermionic em-
ission over vacuum po-
tential barrier with ap-
plied voltage V. EC is the
bottom of the conduction
band, ? the Fermi-level,
W workfunction.
Y.C. Gerstenmaier and G. Wachutka
Thermionic Refrigeration with Planar and Nonplanar Electrodes - Chances and Limits -
where EC is the energy of the bottom of the conduction
band which can be chosen to be zero.
By use of the expression (4) in (3) the electron?s veloc-
ity in x-direction becomes: vx = a0 kx /m . Ex will denote the
x-component m vx2 /2 = (a0 kx)2/2m of the total kinetic energy
E(k) = mv2/ 2 = (a0 k)2/2m. The integration in (3) is thus re-
duced to k-values with kx > 0 . With k2 = kx2 + ky2 + kz2 :
3
3
0 4))(())(( pi
kdkEEf
m
kkEDeJ
x y zk k k C
xxxe
> +=
. (5)
The 3D integral in (5) can be simplified by a suitable pa-
rameter transformation using kinetic energy E and its x-
component Ex as independent variables. Eq. (5) then reads:
?
+=
0 03 )()(
4 dEdEEDEEf
h
meJ E
xxCe
pi (6)
This result was also obtained in other context in [28] and is
used in [32]. It can be proved in a more compact way by
rewriting d3k in polar coordinates and using the definitions
of E(k) and Ex . The integrand of (6) is the differential
emission current density
+= E xxC dEEDEEf
h
meEj
03 )()(
4)( pi
and can be interpreted as the current density j(E) dE in the
energy range between E and E+dE originating from all
electron directions inside the material. The corresponding
particle flux is obtained by division by e .
In order to obtain the emitted energy current density, the
particle flux is weighted with the total energy E +EC :
? ++= 0 03 )()()(4 dEdEEDEEEEfh mJ E xxCCE pi (7)
The classical work of [29] considered one dimensional
electron emission in the free electron model from a plane
surface with constant external electric field F thus resulting
in a triangular shaped potential vacuum barrier: V(x) = (W
+?) ? e F x, where x is the distance from the surface. Nowa-
days the term Fowler-Nordheim tunnelling is often used
generally for tunnelling across triangular shaped barriers. It
is essential to include in V(x) also the Schottky image po-
tential ?e2/ 4x caused by the redistribution of charges on the
metal surface induced by a single electron in the vacuum,
which lowers the maximum W +? of V to a certain degree.
Both, for tunnelling and pure thermionic emission at high
energies (temperatures), the transmission coefficient D(Ex)
for a potential barrier V(x) depending only on x, can be ob-
tained by solution of the 3D Schr?dinger equation
),,()(),,()(
2
2
zyxEEzyxxeVm C ?? +=
+??
when the extension of the surface element dS can be con-
sidered to be infinite compared to atomic dimensions. In
that case the 3D equation is separable with ?(x, y, z) = u(x)
?(y) w(z) and for u(x) :
)()()()(
2 2
22
xuEExuxeV
dx
d
m xC +=
+? (8)
and (?a0 2/2m d 2/dy2 ) ?(y) = Ey ?(y). Similarly (?a0 2/2m
d2/dz2 ) w(z) = Ez w(z). Ex , Ey , Ez denote the components
(a0 kx)2/2m, ? of the total kinetic energy E(k) = (a0 k)2/2m.
EC defines the zero point for the potential V(x).
For nonplanar surface tip radii approaching atomic di-
mensions it is more appropriate to consider instead of (8)
the full 3D Schr?dinger eq. with the radial symmetry of the
problem and spherically symmetric V(r). Future work of us
will deal with this situation more thoroughly. Tip radii as
small as 5nm have already been reported.
3. ENERGY BALANCE IN ELECTRODES
The emitted electrons are replaced by electrons from a res-
ervoir inside the cathode or circuit of temperature TC . The
replacement electrons are scattered into unoccupied levels
near the surface. The average electron replacement energy
?r determines whether heating or cooling of the electrode
occurs. If ?r is lower than the average emitted electron en-
ergy, the cathode is cooled during emission, otherwise it is
heated by emission. There is an old debate originating from
[35] and [36] concerning the average electron replacement
energy. In [35] it was argued that ?r should be slightly be-
low the Fermi-level ?, whereas [36] assumed ?r to be equal
to ?. The theory of [35] was improved in [32] and also re-
sults in ?r < ?. In [35] and [32] the cathode reservoir tem-
perature is restricted to 0?K.
A further problem with the theory of [35], [32] is the use
of the equilibrium distribution f(E) inside the emitting cath-
ode for replacement energy current determination. The use
of f(E) inside the material must strictly lead to zero elec-
tron and energy current, since then the integration in (5)
(without the factor D(Ex)) is not restricted to kx > 0 and all
electrons contributing to the current with kx > 0 are exactly
cancelled by the same number of electrons with kx < 0. We
therefore make use of the non-equilibrium distribution g(k)
presented in (2), which leads to nonzero electron and en-
ergy current depending on the weak applied electric field
and temperature gradient inside the material. The unnatural
assumption of zero reservoir temperature then also is not
necessary. Similar to (5) the replacement electron current is
3
3
4
)(
pi
kdg
m
keJ
x y zk k k
xr = k
and now includes full integration over k. Inserting (2) with
neglected temperature gradient and E-field Fx in x-direction
and using the same variable transformation as in (6) leads
with the convention EC = 0 to
?
?
??=
0
2/3
3
2 )(
)(
3
28 dE
E
EfEEFmeJ
xr ?
pi
Again as in (7) the energy replacement current is obtained
by weighting the integrand of Jr by E and dividing by e:
Y.C. Gerstenmaier and G. Wachutka
Thermionic Refrigeration with Planar and Nonplanar Electrodes - Chances and Limits -
?
?
??=
0
2/5
3
)()(
3
28 dE
E
EfEEFmeJ
xrE ?
pi
(9)
The average electron replacement energy is given by
e JEr /Jr :
?? ??? ???= 0 2/30 2/5 )()()()( / dEEEfEEdEEEfEEr ???
The expression (??f(E)/?E) is a delta-shaped function
around ? with a peak of small width also for high tempera-
tures. Thus the integrals for ?r can be restricted to a small
interval around ? where the ? (E) are essentially constant
and therefore cancel. The integrals can then be expanded in
a series in T around T = 0 by using an expansion of E5/2 and
E3/2 in powers of (E??) of the form n H(n)(?) (E??)n /n! .
With this H(E) (??f(E)/?E) dE can be evaluated to be
?
= ?
?
??
?
+=
?
??
1 )1(2
2)2( )2()
2
12()()(
)()()(
n n
nBn nTkH
HdEEEfEH
??
?
where ? is the Riemann Zeta-function. The lower integra-
tion limit has been extended to ?? with negligible error.
With H(E) = E5/2 and E3/2 a series expansion in kB T of the
nominator and denominator of ?r is obtained and the rapidly
converging series expansion for the fraction ?r is:
...)(168019)(12011)(21 5
6
3
42
+??+=
?
pi
?
pi
?
pi?? TkTkTk BBB
r
For reasonable temperatures up to 8000 K only the first two
terms are significant. This is to be compared with [35]: ?r
= ? ? pi2 kB T /(12 ln(2)) ? O((kB T)2). Due to our result ?r
(?, T ) is a slowly increasing function of T and for T >0
slightly above the Fermi-level ?, contrary to [35], [32]. ?
itself may also depend on temperature. According to our
analysis the approximation ?r ? ?, as e.g. used in [16],
seems to be correct.
The heat current JQ due to electrons flowing from the
cathode is given by the difference in average energy be-
tween emitted and replacement electrons and therefore by
the difference of (7) and (9): JE ? JE r . Since the electron
replacement current Jr must be equal to the emission cur-
rent (6) by reason of continuity, we have JE r = ?r Je /e and:
? ?++= 0 03 )()()(4 dEdEEDEEEEfh mJ E xxrCCQ ?pi
For the complete thermionic converter it is important to
take into account also the electron current from the right
hand side electrode 2 to electrode 1 in fig.1. This is espe-
cially important for low applied voltage V, since for T2 > T1
a strong backward current can occur. The net electron cur-
rent Jen between the electrodes is obtained by using (6) for
the different values ?1, T1 and ?2, T2 of the electrodes and
performing a variable substitution E+EC1,2 ? E in the inte-
grals. The difference of both expressions leads to: (10)
? ?= C CE EE aaTTen dEdEEDEfEfh meJ )())()((4 2,21,13 ??pi
The transmission D(Ea) is here expressed as function of the
absolute electron energy Ea= EC+Ex in x-direction. EC de-
notes the maximum of both electrodes conduction band
edges: EC = Max(EC 1 , EC 2). Also use has been made of the
independence of D(Ea) on the electron?s direction, which is
true for any shape of the potential barrier provided Ea is the
same for both directions [37]. D(Ea) is zero for Ea < EC ,
since no transmission is possible in this case. When consid-
ering only Je without backward current, strictly speaking an
additional factor (1 ? f?2, T2 ) should be included in (6) to
take into account the occupied electron levels in the collec-
tor electrode. However, for the net current (10) this factor
cancels, since f1 (1 ? f2 ) ? f2 (1 ? f1 ) = f1 ? f2 . Jen depends
on V through the the Fermi-level ?2 = ?1? e V (see fig.1, V
in Volt) and the function D(Ea), since V influences the bar-
rier V(x) and the barrier in turn D.
Now it is easy to gain the net heat flow from electrode 1
to 2 by weighting the net current (10) with (E??r) and di-
viding by e :
?
??
=
C
C
E E
E aa
rTT
Qn dEdEED
TEEfEf
h
mJ
)(
)),(())()((4 112,21,1
3
??pi ?? (11)
This is the cooling power (in [W/m2]) for electrode 1 or the
heat current density leaving electrode 1. The heat current
which arrives at electrode 2 is obtained by replacing in (11)
?r(?1, T1) by ?r(?2, T2). The difference of both currents is P
= V |Jen|, the electrical power (in [W/m2]) needed for the
operation of the thermionic converter as refrigerator. In sec-
tion 5 Joule heating in the electrodes and contact resis-
tances will also be included.
4. PURE THERMIONIC EMISSION
For thermionic emission it is assumed that no tunnelling of
particles occurs. All emitted electrons have to have a en-
ergy higher than the maximum Vmax (in eV) of the potential
surface barrier. In this classical approximation the transmis-
sion coefficient D(E) is always zero for E < Vmax and D(E)
= 1 for E ? Vmax . Inserting these values for D(E) into (10),
(11) the integration can be performed exactly analytically.
The result is expressed by polylogarithm functions Lin[z] =
i =1 zi /i n with z = ?exp(?(Vmax??1,2)/kBT1,2). We derived
the following formulas, which are probably known and
valid for all real x, to obtain that result:
Li2[?exp(?x)] = ?(pi2 + 3 x2 + 6 Li2[?exp(+x)])/ 6
Li3[?exp(?x)] = +(pi2 x + x3 + 6 Li3[?exp(+x)])/ 6
For (Vmax??1,2) on the order of material workfunctions and
T below several 1000 K the magnitude of z is so small that
the Li(z) functions can be very accurately approximated by
z. Then the result for the net electron and heat current is:
( )22max2211max21
3
2
/)(exp()/)(exp(
4
TkVTTkVT
h
kmeJ
BB
Ben
??
pi
??????
=
Y.C. Gerstenmaier and G. Wachutka
Thermionic Refrigeration with Planar and Nonplanar Electrodes - Chances and Limits -
???+?
???+?
=
22max11max222
11max11max121
32
/)(exp()),(2(
)/)(exp()),(2(
)12(/4
TkVTVTkT
TkVTVTkT
hkmJ
BrB
BrB
BQn
???
???
pi
The first term of Jen (current from electrode 1 to 2) is for
Vmax = W+?1,2 the traditional thermionic Richardson cur-
rent, which is usually inferred in another way. The expres-
sion for JQn corresponds, likewise for Vmax = W+?1,2 , to the
result in [10], however, (12) is more general. Very low Vmax
can be obtained for increased voltages in nano-gap devices.
Our analysis suggests that in this case our original analyti-
cal formulas with polylogarithms should be used. On the
other hand, the tunnelling contribution in nano-gap devices
is not negligible, so that the premises D(E) = ?(E ? Vmax) is
not fulfilled.
5. COMBINED FIELD AND THERMIONIC
EMISSION - NANOMETER GAP DESIGN -
In ref. [16] a theoretical investigation of nano-gap therm-
ionic devices was performed. For nanometer distances d of
the plane electrodes multiple image forces become impor-
tant. One free electron in the vacuum gap causes an image
of opposite charge in electrode 1 and 2. The images them-
selves cause other image charges in the electrodes which
then cause new image forces. Thus an infinite series is ob-
tained for the gap potential with applied voltage Vbias:
?
?
+??+=
?
=1 2220
1
2
1
4
1
4)( nbiasd ndxdn
nd
x
e
d
xV
e
WxV
pi?
?
A factor ? has to be incorporated, because the mirror
charge position is not constant, when the electron?s position
x varies. We included the workfunction of the second elec-
trode W2 and succeeded to sum the infinite series exactly
analytical. Vd then reads:
++????
?+?+=
))1()1((412414
)/)(()(
0
2111
d
x
d
x
dd
C
x
e
eWWVdxeWxV biasd
??pi?
?
(13)
Here C = 0.5772? denotes Euler?s constant, ? is the di-
gamma function (logarithmic derivative of the gamma-
function). The potential exhibits singularities at both elec-
trodes at x =0 and x =d, which are not real. We therefore
limit the potential to EC 1 near x =0 for Vd (x)< EC 1 , and to
EC 2 near x =d for Vd (x) < EC 2 . Figure 2 shows as an ex-
ample the potential shape for d = 6 nm. As can be seen
from fig. 2 a considerable reduction of the potential maxi-
mum from originally 2 V to Vmax = 1.375 V occurs, which
is due to the image forces (the 2nd line in (13)). This effect
is for similar Vbias nearly negligible in case of large gaps
with d in the region of ?m or mm.
The electron transmission probabilities D(E) are needed
in the expressions (10), (11) for the currents, in order to
calculate the device performance. The theory of field and
thermionic emission are combined by calculating D(E) pre-
cisely numerically from (8) as interpolated function. Using
(13) for the Schr?dinger-equation (8), the wave function
u(x) can be obtained for any given electron energy E = EC
+Ex in the x-interval (0, d). In the electrodes region x < 0
and x > d , V(x) is constant (= EC 1 , EC 2 ) with plane wave
functions u(x), due to the free electron model. A transmitted
wave for an incident electron from right to left is of the
form in x < 0 : )/)(2exp( 1 CEEmxiS ?? . This gives
with S = 1 at x = 0 the start condition for u(x) in (8). For x
> d a superposition of incident and reflected wave is used:
)/)(2exp()/)(2exp( 22 CC EEmxiREEmxiY ?+?? .
Y and R are determined by the continuity condition for u(x)
and u?(x) at x = d. The transmission coefficient then is
given by D(E) = 1?|R/Y| 2 .
In fig.3 a result of this calculation is displayed. For every
value E the differential equation (8) was solved numerically
with a standard package and D(E) calculated subsequently.
The complete function D(E) in fig.3 is created as interpo-
lated function within 16 sec on a 3 GHz PC.
The Wentzel-Kramers-Brillouin method (WKB, [34]) is
often used for an approximate solution of the 1D eq. (8).
Within this approximation D(E) can be represented as:
??= dxExeVmED x
x
2
1 ))((2
2exp)(
.
x1 and x2 are the turning points of the problem, i.e.
V(x1, x2) = E . For E exceeding the potential maximum
Vmax , x1 equals x2 and D(E) is set to the classical value 1.
In that region the WKB approximation is no longer valid.
Contrary to classical mechanics there is a non vanishing
probability for the electron to be reflected, also when E >
Vmax , in the same way as there is a non vanishing probabil-
ity for transmission, when E < Vmax . Therefore D(E) < 1
also for E > Vmax. The WKB-approximation in fig.3 is not
bad. Its threshold voltage is near to Vmax = 1.375 V. Sur-
prisingly the threshold of the exact curve is higher, above
0 1 2 3 4 5 6
nm
1.5
1
0.5
0
0.5
1
1.5
2
Ve
Figure 2: Potential profile (13) in d = 6nm vacuum gap with
voltage Vbias=1.6V, W1, W2 = 1eV, EC 1=0, EC 2=EC 1?e Vbias,
?1=1eV. Top curve represents the first two terms of (13).
EC 1
EC2
?1
?2
W 1
Y.C. Gerstenmaier and G. Wachutka
Thermionic Refrigeration with Planar and Nonplanar Electrodes - Chances and Limits -
1.4V. For other barrier shapes the WKB approximation can
fail severely, as can easily be shown by comparison with
analytical solvable cases.
The standard numerical algorithm for solving the
Schr?dinger eq. (8) fails, if the gap width d increases. In the
regions with E > e V(x) the wave function u(x) is highly os-
cillatory, which necessitates an extraordinarily large num-
ber of x-steps. Therefore V(x) is approximated by a piece-
wise constant function and (8) is solved analytically in each
constant interval with continuity conditions for u(x), u?(x)
at the borders of the intervals. The solution is obtained re-
cursively very rapidly, so that the D(E) function in fig.3 can
be calculated within 1 sec with nearly equal accuracy. How-
ever, this method fails again for other barrier shapes to be
considered later. So a similar method was developed with
piecewise linear approximation to V(x). The solution in
each linear interval of V(x) is given by Airy-functions
which bears some numerical subtleties. Nevertheless, with
this the most reliable method was obtained for solving (8)
for all geometry and bias conditions. The D(E) function can
be calculated in typically less than 3 sec with this method.
In fig.4 the net heat current from electrode 1 (cooling
power) according to (11) is shown for different separations
of the electrodes and different bias voltages. The result
looks promising and is qualitatively similar to [16]. Our re-
sult includes exact D(E) functions instead of WKB ap-
proximations, the presence of the backward current from
the 2nd electrode and a precise consideration of all emission
directions because of (6), instead of effectively considering
lateral currents by a term kB T. The temperature of elecrode
1 to be cooled is 405?K, and of the collector electrode 2:
T2 = 450 ?K.
Essential for the assessment of the thermionic refrigera-
tor performance, however, is the relation of the cooling
power to the electrical power P = Vbias |Jen| needed for de-
vice operation, which is expressed by the ?coefficient of
performance? ? = JQ / P. ? is limited by the Carnot-
efficiency ?C = T2 /(T2 ?T1) ( = 9 for fig.5). Figure 5 shows
the CoP plotted against the cooling power for the different
electrode separations. Very high CoP near Carnot effi-
ciency, much higher than for thermoelectric devices, can be
obtained for extremely low cooling powers < 10?1 W/cm2 .
However, for cooling powers of technical interest > 1
W/cm2 the CoP is below or at best similar to what can be
achieved by conventional bulk-material thermoelectric de-
vices.
Additional Joule heating JJ in the cathode and its contact
resistance has also been taken into account in the CoP in
fig.5. For a contact resistance and internal electrode resis-
tance R the cooling power is reduced to JQ ? JJ = JQ ? R
Jen2 and the operation power is increased by R Jen2. Thus:
? = (JQ ? R Jen2) / (Vbias |Jen| + R Jen2) . We assumed an ef-
fective R of 10?6 ? for 1 cm2 device area. The Joule losses
have little or negligible influence except for extremely
small spacing of d = 2 nm, when very high electrical cur-
rent densities occur. The CoP is of negative sign in this
case (Joule heating larger than cooling) and therefore not
represented in fig.5.
Figure 6 shows the same as in fig.5 for largely increased
temperatures of T1 = 900 ?K and T2 = 1000 ?K. The Carnot-
efficiency is again 9. Now very large CoP at much higher
cooling powers result. The CoP are even better than for ad-
vanced nanostructured thermoelectric devices. (Again the
performance ford = 2 nm is destroyed by Joule heating.)
6 4 2 0 2 4
Log10JQ Wcm2
1
2
3
4
5
6
7
8
tne
ici
ffe
oC
fo
ecn
am
rof
re
P
6 nm
Figure 5: CoP of thermionic refrigerator for plane elect. dis-
tance: 3, 4, 6, 8, 15, 30, 60 nm. T1=405 K, T2=450 K.
60
4 nm
0 2 4 6 8 10
Vbias V
1
10
100
1000
10000
100000.
W
mc
2
8nm
Figure 4: Cooling power of thermionic refrigerator for plane
elect. distance: 2, 3, 4, 6, 8, 15, 30, 60 nm. T1=405, T2=450K
15 nm
30 nm
60 nm
0 0.5 1 1.5 2 2.5 3
Energy eV
0
0.2
0.4
0.6
0.8
1
noi
ssi
msn
arT
D
Figure 3: Transmission probability D(E) for potential profile
(13) with Vbias=1.6V (fig.2). Exact curve by numerical solu-
tion of Schr?dinger eq. (8), WKB-curve by WKB-method.
exact
WKB
Y.C. Gerstenmaier and G. Wachutka
Thermionic Refrigeration with Planar and Nonplanar Electrodes - Chances and Limits -
However, this temperature region is not suited for elec-
tronic cooling applications. Our theory allows for a calcula-
tion of the generator efficiency by the same set of data as
obtained for cooling. For the same electrode temperatures
the current reverses in the generator mode for very low
positive bias voltage, because of T2 > T1 . The device then
delivers power to the external circuit. The generator effi-
ciency ? is defined as delivered power P = Vbias |Jen| di-
vided by the heat current from the 2nd (hot) electrode. The
Carnot efficiency is ?C = (T2 ? T1) /T2 . As in the cooling
case ? is near ?C with generated powers up to 500 W/cm2.
However, for the more interesting low temperature case (T1
= 405 K, T2 = 450 K) the generated power-densities are
very small (< 10?2 W/cm2), when good efficiencies are to
be obtained.
6. EMISSION FROM NANOTIP ELECTRODES
Figure 7 shows a thermionic cooler with numerous metal
tips on the cathode electrode. Due to the small tip radii the
electric field is enhanced strongly at the tips. This leads to
a deformed electrostatic potential shape with very small
barrier width, as shown in fig. 9 along the line A-B, so that
electrons of lower energy can tunnel by field emission in-
stead of thermionic emission. Devices similar to fig.7 have
been devised in vacuum microelectronics usually with in-
clusion of a gate electrode to control the current flow [19,
20, 21]. The advantages of vacuum microelectronics in-
clude high operation temperatures, radiation hardness, and
use for very high frequencies. Also flat panel displays are
considered for this technique. Improved properties and long
term reliability are expected with carbon-nano-tubes [22] or
diamond coated tips [23, 24], however, our work is re-
stricted to metal and semiconductor field emission surfaces.
Structures like that in fig.7 have been proposed in [25],
[26] for cooling applications. In the inverse operation mode
as generator shown in fig. 8, the electric field exerts a force
in opposite direction to the electron movement.
In case of a cone with tip radius r0 which emits elec-
trons, we assume a 3D radial symmetric potential near the
surface of the spherical tip, which leads to [28, 25]:
?+++
??
+?
?+?+=
)(4
1
)2(21
1
4
)1()/)(()(
0
0
0
0
02111
xdxrx
re
rx
r
d
eWWV
e
WxV bias
r
?
?
pi?
?
(14)
Vr(x) = EC 1 for x < 0 , Vr(x) = EC 2 for x > d.
x denotes the normal distance from the spherical tip. The
image potential for the sphere as given by [25, 33] has been
added and also the image potential for the plane electrode
in the 2nd line. ? is the dielectric constant of the emitter. In
case of metals the factor (? ?1)/ (? +1) has to be omitted. d
denotes the distance from the tip to the (nearly) plane col-
lector electrode. Contrary to the last section, d now assumes
values in the ?m or mm range, to make possible a techno-
logical realisation. The calculation of the potential for the
tip array of fig.7 is obviously a 3D problem but at least in a
neighbourhood along the lines AB Vr(x) gives a useful 1D
approximation. Figure 9 shows Vr(x) for a tip radius of 10
nm. The width of the peak at ?1 = 1eV is 20nm, so the re-
duction of the peak height from 2eV to Vmax = 1.572 eV
may be more important for improved emission.
0 1 2 3 4
?m
1.5
1
0.5
0
0.5
1
1.5
2
Ve
EC 1
EC 2
?1
?2
W 1
Figure 9: Potential profile (14) for d =4?m and nanotip r0 =
10nm, bias Vbias=1.6V, W1, W2 =1eV, ?1=1eV, ?2= ?1 ?e Vbias
0 1 2 3 4 5 6 7
log10JQ Wcm2
0
2
4
6
8
tne
ici
ffe
oC
fo
ecn
am
rof
re
P
4 nm
Figure 6: CoP of thermionic refrigerator for plane elect. dis-
tance: 2, 3, 4, 6, 8, 15, 30, 60 nm. T1=900 K, T2=1000 K.
60 nm
3 nm
E-field force
opposite e?
+
e?
? +
Figure 7: Thermionic cooler
with electron field emission by
enhanced electric field at tips.
TC
E-field
TH
e?
e?
A B e?
?
TC TH
e?
A B
RL
Figure 8: Device in genera-
tor mode. E-field opposes e-
which is driven by TH > TC .
Y.C. Gerstenmaier and G. Wachutka
Thermionic Refrigeration with Planar and Nonplanar Electrodes - Chances and Limits -
We can now apply the same formalism as in section 5 to
calculate electron and heat currents at the field emitting
tips. The fraction of the tip area of the array compared to
the device area is taken into account by a reduction of JQ to
6% in fig.10. Figure 10 shows the result for high tempera-
tures (T1 = 900K, T2 = 1000K). The CoP exceeds that of
thermoelectric devices by far with good power densities.
For the low temperature case (405K, 450K), similar to fig.
5, much too small power densities below 10?3 W/cm2 arise.
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2 1 0 1 2 3 4 5
log10JQ Wcm2
0
2
4
6
8
tne
ici
ffe
oC
fo
ecn
am
rof
re
P
Figure 10: CoP of nanotip thermionic refrigerator for tip ra-
dii: 5, 10, 20, 40, 80, 160 nm. T1=900 K, T2=1000 K.
160 nm 5 nm